Cold-Formed Steel Design
Cold-Formed Steel Design
Fifth Edition
Wei-Wen Yu
Missouri University of Science and Technology
Rolla, Missouri
Roger A. LaBoube
Missouri University of Science and Technology
Rolla, Missouri
Helen Chen
American Iron and Steel Institute
Washington, DC
This edition first published 2020
©2020 John Wiley & Sons, Inc.
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Library of Congress Cataloging-in-Publication Data
Names: Yu, Wei-Wen, author. | LaBoube, Roger A., author. | Chen,
Helen, author.
Title: Cold-formed steel design / Wei-Wen Yu and Roger A. LaBoube, Missour University of Science
and Tech; Helen Chen, American Iron and Steel Institute.
Description: Fifth edition. | Hoboken : Wiley, 2020. | Includes
bibliographical references and index.
Identifiers: LCCN 2019023220 (print) | LCCN 2019023221 (ebook) | ISBN
9781119487395 (cloth) | ISBN 9781119487388 (adobe pdf) | ISBN
9781119487418 (epub)
Subjects: LCSH: Building, Iron and steel. | Sheet-steel. | Thin-walled
structures. | Steel—Cold working.
Classification: LCC TA684 .Y787 2020 (print) | LCC TA684 (ebook) | DDC
624.1/821—dc23
LC record available at https://lccn.loc.gov/2019023220
LC ebook record available at https://lccn.loc.gov/2019023221
Cover Design: Wiley
Cover Image: Courtesy of Steel Framing Alliance
Set in 10/12pt TimesLTStd by SPi Global, Chennai, India
10 9 8 7 6 5 4 3 2 1
CONTENTS
CHAPTER 1
Preface
ix
INTRODUCTION
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
CHAPTER 2
General Remarks
Types of Cold-Formed Steel Sections and Their Applications
Metal Buildings and Industrialized Housing
Methods of Forming
Research and Design Specifications
General Design Considerations of Cold-Formed Steel Construction
Economic Design and Optimum Properties
Design Basis
Serviceability
1
2
7
13
15
21
26
27
35
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
37
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
37
44
45
45
47
48
48
51
53
54
55
57
General Remarks
Yield Stress, Tensile Strength, and Stress–Strain Curve
Modulus of Elasticity, Tangent Modulus, and Shear Modulus
Ductility
Weldability
Fatigue Strength and Toughness
Influence of Cold Work on Mechanical Properties of Steel
Utilization of Cold Work of Forming
Effect of Temperature on Mechanical Properties of Steel
Testing of Full Sections and Flat Elements
Residual Stresses Due to Cold Forming
Effect of Strain Rate on Mechanical Properties
v
vi
CONTENTS
CHAPTER 3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
59
3.1
3.2
3.3
59
59
3.4
3.5
3.6
3.7
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
General Remarks
Definitions of Terms
Structural Behavior of Compression Elements and Effective Width Design
Criteria
Perforated Elements and Members
Direct Strength Method and Consideration of Local and Distortional
Buckling
Plate Buckling of Structural Shapes
Additional Information
61
97
100
117
117
FLEXURAL MEMBERS
119
4.1
4.2
4.3
4.4
4.5
4.6
119
119
183
209
216
216
General Remarks
Bending Strength and Deflection
Design of Beam Webs
Bracing Requirements of Beams
Torsional Analysis of Beams and Combined Bending and Torsional Loading
Additional Information on Beams
COMPRESSION MEMBERS
217
5.1
5.2
5.3
5.4
5.5
5.6
217
218
226
228
228
General Remarks
Column Buckling
Local Buckling Interacting with Yielding and Global Buckling
Distortional Buckling Strength of Compression Members
Effect of Cold Work on Column Buckling
North American Design Formulas for Concentrically Loaded Compression
Members
5.7 Effective Length Factor K
5.8 Built-Up Compression Members
5.9 Bracing of Axially Loaded Compression Members
5.10 Design Examples
5.11 Compression Members in Metal Roof and Wall Systems
5.12 Additional Information on Compression Members
230
234
236
237
238
247
250
COMBINED AXIAL LOAD AND BENDING
251
6.1
6.2
6.3
6.4
6.5
6.6
6.7
251
251
253
261
265
266
283
General Remarks
Combined Tensile Axial Load and Bending
Combined Compressive Axial Load and Bending (Beam–Columns)
Member Forces Considering Structural Stability
North American Design Criteria for Beam–Column Check
Design Examples
Additional Information on Beam–Columns
CLOSED CYLINDRICAL TUBULAR MEMBERS
285
7.1
7.2
7.3
285
285
285
General Remarks
Types of Closed Cylindrical Tubes
Flexural Column Buckling
CONTENTS
7.4
7.5
7.6
CHAPTER 8
CHAPTER 9
CHAPTER 10
CHAPTER 11
CHAPTER 12
Local Buckling
North American Design Criteria
Design Examples
vii
286
289
293
CONNECTIONS
297
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
General Remarks
Types of Connectors
Welded Connections
Bolted Connections
Screw Connections
Power-Actuated Fasteners
Other Fasteners
Rupture Failure of Connections
I- or Box-Shaped Compression Members Made by Connecting Two
C-Sections
8.10 I-Beams Made by Connecting Two C-Sections
8.11 Spacing of Connections in Compression Elements
297
297
297
316
327
331
334
336
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
345
9.1
9.2
9.3
9.4
9.5
9.6
345
345
358
367
378
380
General Remarks
Steel Shear Diaphragms
Structural Members Braced by Diaphragms
Shell Roof Structures
Metal Roof Systems
Shear Walls
337
340
342
CORRUGATED SHEETS
381
10.1 General Remarks
10.2 Applications
10.3 Sectional Properties and Design of Arc- and Tangent-Type
Corrugated Sheets
10.4 Sectional Properties and Design of Trapezoidal-Type Corrugated
Sheets
381
381
COMPOSITE DESIGN
389
11.1 General Remarks
11.2 Steel-Deck-Reinforced Composite Slabs
11.3 Composite Beams or Girders With Cold-Formed Steel Deck
389
389
390
LIGHT-FRAME CONSTRUCTION
393
12.1 General Remarks
12.2 Framing Standards
12.3 Design Guides
393
393
406
381
386
viii
CONTENTS
APPENDIX A
THICKNESS OF BASE METAL
407
APPENDIX B
TORSION
409
APPENDIX C
FORMULAS FOR COMPUTING CROSS-SECTIONAL PROPERTY 𝛽 y
421
APPENDIX D
DEFINITIONS OF TERMS
423
NOMENCLATURE
429
ACRONYMS AND ABBREVIATIONS
443
CONVERSION TABLE
445
REFERENCES
447
INDEX
513
PREFACE
This fifth edition of the book has been prepared to provide
readers with a better understanding of the analysis and design
of the thin-walled, cold-formed steel structures that have
been so widely used in building construction and other areas
in recent years. It is a revised version of the first author’s
book, Cold-Formed Steel Design, fourth edition, published
by John Wiley & Sons, Inc. in 2010. All the revisions are
based on the 2016 edition of the North American Specification, which incorporated the Direct Strength Method
into the main body of the Specification, and reorganized
the chapters to be consistent with hot-rolled steel design
specification,1.411 published by American Institute of Steel
Construction.
The material was originally developed for graduate
courses and short courses in the analysis and design of
cold-formed steel structures and is based on experience in
design, research, and development of the American Iron and
Steel Institute (AISI) and North American design criteria.
Throughout the book, descriptions of the structural
behavior of cold-formed steel members and connections
are given from both theoretical and experimental points of
view. The reasons and justification for the various design
provisions of the North American specification are discussed
at length. Consequently the text not only will be instructive
for students but also can serve as a major source of reference
for structural engineers and researchers.
To reflect the change in format and the inclusion of the
Direct Strength Method into the main body of the Specification, all chapters have been completely revised according to
the reorganized layout of the North American Specification
and framing standards.
Chapter 1 includes a general discussion of the application
of cold-formed steel structures and a review of previous and
recent research. It also discusses the development of design
specifications and the major differences between the design
of cold-formed and hot-rolled steel structural members.
Because of the many research projects in the field that
have been conducted worldwide during the past 43 years,
numerous papers have been presented at various conferences
and published in a number of conference proceedings and
engineering journals. At the same time, new design criteria
have been developed in various countries. These new developments are reviewed in this chapter. New Sections 1.8 and
1.9 discuss the AISI Specification’s design basis for strength
and serviceability.
Since material properties play an important role in the
performance of structural members, the types of steel and
their most important mechanical properties are described in
Chapter 2. The mechanical properties of ASTM A1063 steel
sheets are added in Table 2.1.
In Chapter 3, the strength of thin elements and design
criteria are discussed to acquaint the reader with the fundamentals of buckling modes to be considered in cold-formed
steel design, such as local and distortional buckling and postbuckling strength of thin plates, and with the basic concepts
used in design. The analytical and numerical approaches
for determining local and distortional buckling strengths are
discussed in this chapter. This chapter also introduces the
definitions of commonly used terms in cold-formed steel
design. The concepts of the Effective Width Method and the
Direct Strength Method are discussed with the limits of applicability of these methods.
Chapter 4 deals with the design of flexural members.
The contents have been reorganized to be consistent with
the 2016 edition of the North American Specification. This
chapter discusses the flexural member strengths due to global
buckling, local buckling interacting with global buckling,
and distortional buckling. It also includes new and revised
design provisions on inelastic reserve capacity of beams,
members with holes, shear strength of webs, web crippling
ix
x
PREFACE
strength and combination with bending, bearing stiffeners
in C-section beams, bracing requirements, combination of
bending and torsion, and beams having one flange attached
to a metal roof system.
The design procedures for compression members are
discussed in Chapter 5. The contents have been reorganized
to be consistent with the 2016 edition of the North American Specification. This chapter discusses the compressive
member strengths due to global buckling, local buckling
interacting with global buckling, and distortional buckling.
It also includes provisions about the design of built-up
members, bracing requirements, and compression members
having one flange attached to a metal roof system.
In the 2016 edition of the North American specification,
the Direct Analysis Method was introduced to consider the
second-order effect in structural analysis. This Direct Analysis Method is discussed in Chapter 6. In addition, revisions
have been made on the design of beam–columns using ASD,
LRFD, and LSD methods.
Chapter 7 covers the design of closed cylindrical tubes.
This revised chapter reflects the rearrangement of design
provisions in the North American specification.
Like the member design, the design of connections has
been updated in Chapter 8 using the ASD, LRFD, and
LSD methods with additional and revised design provisions for bearing strength between bolts and connected
parts, combined shear and tension in fasteners, block shear
strength, revised design information on screw connections,
and power-actuated fasteners.
Because various types of structural systems, such as
shear diaphragms and shell roof structures, have become
increasingly popular in building construction, Chapter 9
contains design information on these types of structural
systems. Revisions are made reflecting the new North American standard of AISI S310 for profiled steel diaphragm
panels.
The sectional properties of standard corrugated sheets are
discussed in Chapter 10 because they have long been used in
buildings for roofing, siding, and other applications. Minor
revisions have been made in the chapter.
Steel decks are widely used in building construction.
Consequently the updated information in Chapter 11 on
their use in steel-deck-reinforced composite slabs and
composite beams is timely.
In 2015, the AISI design standards for cold-formed steel
framing were consolidated. These standards are specifically
applicable for residential and commercial construction. As
a result, Chapter 12 has been completely rewritten based on
new and consolidated AISI standards.
It is obvious that a book of this nature would not have
been possible without the cooperation and assistance of
many individuals, organizations, and institutions. It is based
primarily on the results of continuing research programs on
cold-formed steel structures that have been sponsored by
the American Iron and Steel Institute (AISI), the ASCE,
the Canadian Sheet Steel Building Institute (CSSBI), the
Cold-Formed Steel Engineers Institute (CFSEI) of the Steel
Framing Alliance (SFA), the Metal Building Manufacturers
Association (MBMA), the Metal Construction Association
(MCA), the National Science Foundation (NSF), the Rack
Manufacturers Institute (RMI), the Steel Deck Institute
(SDI), the Steel Framing Industry Association (SFIA), the
Steel Stud Manufacturers Association (SSMA), and other
organizations located in the United States and abroad. The
publications related to cold-formed steel structures issued
by AISI and other institutions have been very helpful for the
preparation of this book.
The first author is especially indebted to his teacher, the
late Dr. George Winter of Cornell University, who made
contributions of pronounced significance to the building
profession in his outstanding research on cold-formed steel
structures and in the development of AISI design criteria. A
considerable amount of material used in this book is based
on Dr. Winter’s publications.
Our sincere thanks go to Mr. Robert J. Wills, Vice President, Construction Market Development, Steel Market
Development Institute (a business unit of the American Iron
and Steel Institute), for permission to quote freely from
the North American Specification, Commentary, Design
Manual, Framing Standards, Design Guides, and other AISI
publications. An expression of appreciation is also due to
the many organizations and individuals that granted permission for the reproduction of quotations, graphs, tables, and
photographs. Credits for the use of such materials are given
in the text.
We are very grateful to Mrs. Christina Stratman for her
kind assistance in the preparation of this book. The financial
assistance provided by the Missouri University of Science
and Technology through the first author’s Curators’ Professorship and the sponsors for the Wei-Wen Yu Center for
Cold-Formed Steel Structures is appreciated.
This book could not have been completed without the help
and encouragement of the authors’ wives, Yueh-Hsin Yu and
Karen LaBoube, and husband, Chunwei Huang, as well as
for their patience, understanding, and assistance.
Wei-Wen Yu
Roger A. LaBoube
Hong (Helen) Chen
Rolla, Missouri
March 2019
CHAPTER 1
Introduction
1.1 GENERAL REMARKS
In steel construction, there are two main families of structural members. One is the familiar group of hot-rolled
shapes and members built up of plates. The other, less
familiar but of growing importance, is composed of sections
cold formed from steel sheet, strip, plate, or flat bar in
roll-forming machines or by press brake or bending brake
operations.1.1,1.2,1.3∗ These are cold-formed steel structural
members. The thickness of steel sheet or strip generally
used in cold-formed steel structural members ranges from
0.0149 in. (0.378 mm) to about 14 in. (6.35 mm). Steel plates
and bars as thick as 1 in. (25.4 mm) can be cold formed
successfully into structural shapes.1.1,1.4,1.314,1.336,1.345
Although cold-formed steel sections are used in car
bodies, railway coaches, various types of equipment, storage
racks, grain bins, highway products, transmission towers,
transmission poles, drainage facilities, and bridge construction, the discussions included herein are primarily limited
to applications in building construction. For structures other
than buildings, allowances for dynamic effects, fatigue, and
corrosion may be necessary.1.314,1.336,1.345,1.417
The use of cold-formed steel members in building
construction began in about the 1850s in both the United
States and Great Britain. However, such steel members
were not widely used in buildings until around 1940. The
early development of steel buildings has been reviewed by
Winter.1.5–1.7
∗ The
references are listed at the back of the book.
Since 1946 the use and the development of thin-walled
cold-formed steel construction in the United States have
been accelerated by the issuance of various editions of
the “Specification for the Design of Cold-Formed Steel
Structural Members” of the American Iron and Steel Institute
(AISI).1.267,1.345 The earlier editions of the specification were
based largely on the research sponsored by AISI at Cornell
University under the direction of George Winter. It has been
revised subsequently to reflect the technical developments
and the results of continuing research.1.267,1.336,1.346,1.416,1.417
In general, cold-formed steel structural members provide
the following advantages in building construction:
1. As compared with thicker hot-rolled shapes,
cold-formed light members can be manufactured
for relatively light loads and/or short spans.
2. Unusual sectional configurations can be produced
economically by cold-forming operations (Fig. 1.1),
and consequently favorable strength-to-weight ratios
can be obtained.
3. Nestable sections can be produced, allowing for
compact packaging and shipping, as well as for
developing efficient structural applications.
Figure 1.1 Various shapes of cold-formed sections.1.1
1
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
2
1
INTRODUCTION
4. Load-carrying panels and decks can provide useful
surfaces for floor, roof, and wall construction, and in
other cases they can also provide enclosed cells for
electrical and other conduits.
5. Load-carrying panels and decks not only withstand
loads normal to their surfaces, but they can also act as
shear diaphragms to resist force in their own planes if
they are adequately interconnected to each other and to
supporting members.
Compared with other materials such as timber and
concrete, the following qualities can be realized for
cold-formed steel structural members1.8,1.9 :
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Lightness
High strength and stiffness
Ease of prefabrication and mass production
Fast and easy erection and installation
Substantial elimination of delays due to weather
More accurate detailing
Nonshrinking and noncreeping at ambient temperatures
Formwork unneeded
Termite proof and rot proof
Uniform quality
Economy in transportation and handling
Noncombustibility
Recyclable material
The combination of the above-mentioned advantages can
result in cost savings in construction (www.buildsteel.org).
1.2 TYPES OF COLD-FORMED STEEL SECTIONS
AND THEIR APPLICATIONS
Cold-formed steel structural members can be classified into
two major types:
1. Individual structural framing members
2. Panels and decks
The design and the usage of each type of structural
member have been reviewed and discussed in a number of
publications.1.5–1.75,1.267–1.285,1.349,1.358,1.418
1.2.1
Individual Structural Framing Members
Figure 1.2 shows some of the cold-formed sections generally
used in structural framing. The usual shapes are channels
Figure 1.2
Cold-formed sections used in structural framing.1.6
(C-sections), Z-sections, angles, hat sections, I-sections,
T-sections, and tubular members. Previous studies have
indicated that the sigma section (Fig. 1.2d) possesses several
advantages, such as high load-carrying capacity, smaller
blank size, less weight, and larger torsional rigidity as
compared with standard channels.1.76
In general, the depth of cold-formed individual framing
structural members ranges from 2 to 16 in. (50.8 to 406 mm),
and the thickness of material ranges from 0.0329 to 0.1180
in. (0.836 to 2.997 mm). In some cases, the depth of individual members may be up to 18 in. (457 mm), and the thickness of the member may be 12 in. (12.7 mm) or thicker in
transportation and building construction. Cold-formed steel
plate sections in thicknesses of up to about 34 or 1 in. (19.1 or
25.4 mm) have been used in steel plate structures, transmission poles, and highway sign support structures.
In view of the fact that the major function of this type
of individual framing member is to carry load, structural strength and stiffness are the main considerations
in design. Such sections have commonly been used as
primary framing members in buildings having multiple
stories in height.1.278 In 2000, the 165-unit Holiday Inn in
Federal Way, Washington, utilized eight stories of axial load
bearing cold-formed steel studs as the primary load-bearing
system.1.357 Figure 1.3 shows a mid-rise construction
building. Cold-formed steel for mid-rise construction has
become popular for these buildings that typically may
range from 4 to 12 stories high. Chapter 12 provides additional discussion of cold-formed steel applications for lowand mid-rise construction. Additional information may
also be obtained at www.buildsteel.org. In tall multistory
buildings the main framing is typically of heavy hot-rolled
TYPES OF COLD-FORMED STEEL SECTIONS AND THEIR APPLICATIONS
3
Figure 1.3 Building composed entirely of cold-formed steel sections. Courtesy of Don Allen.
shapes and the secondary elements may be of cold-formed
steel members such as steel joists, studs, decks, or panels
(Figs. 1.4 and 1.5). In this case the heavy hot-rolled steel
shapes and the cold-formed steel sections supplement each
other.1.264
As shown in Figs. 1.2 and 1.6–1.10, cold-formed sections
are also used as chord and web members of open web steel
joists, space frames, arches, and storage racks.
1.2.2
Panels and Decks
Another category of cold-formed sections is shown in
Fig. 1.11. Historically, these sections are generally used for
roof decks, floor decks, wall panels, siding material, and
bridge forms. Recently, profiled deck has been used for
shear wall. Some deeper panels and decks are cold formed
with web stiffeners.
4
1
INTRODUCTION
Figure 1.4
Composite truss–panel system prefabricated by Laclede Steel Company.
Figure 1.5 Cold-formed steel joists used together with hot-rolled shapes. Courtesy of Stran-Steel
Corporation.
The depth of panels generally ranges from 9/16 to 7 12 in.
(14.2 to 191 mm), and the thickness of materials ranges
from 0.018 to 0.075 in. (0.457 to 1.91 mm). This is not to
suggest that in some cases the use of 0.012-in. (0.305-mm)
steel-ribbed sections as load-carrying elements in roof and
wall construction would be inappropriate.
Steel panels and decks not only provide structural
strength to carry loads, but they also provide a surface on
which flooring, roofing, or concrete fill can be applied,
as shown in Fig. 1.12. They can also provide space for
electrical conduits, or they can be perforated and combined
with sound absorption material to form an acoustically
conditioned ceiling. The cells of cellular panels are also
used as ducts for heating and air conditioning.
In the past, steel roof decks were successfully used
in folded-plate and hyperbolic paraboloid roof construction,1.13,1.22,1.26,1.30,1.34,1.35,1.72,1.77–1.84 as shown in
Figs. 1.13 and 1.14. One of the world’s largest cold-formed
steel primary structures using steel decking for hyperbolic
paraboloids, designed by Lev Zetlin Associates, is shown in
TYPES OF COLD-FORMED STEEL SECTIONS AND THEIR APPLICATIONS
Figure 1.6
Cold-formed steel sections used in space frames. Courtesy of Unistrut Corporation.
Figure 1.7 Cold-formed steel members used in space grid system. Courtesy of Butler Manufacturing Company.
5
6
1
INTRODUCTION
Figure 1.8 Cold-formed steel members used in a 100 × 220 × 30-ft (30.5 × 67.1 × 9.2-m)triodetic
arch. Courtesy of Butler Manufacturing Company.
(a)
(b)
Figure 1.9 Hangar-type arch structures using cold-formed steel sections. Courtesy of Armco Steel
Corporation.1.6
Fig. 1.15.1.82 Roof decks may be curved to fit the shape of an
arched roof without difficulty. Some roof decks are shipped
to the field in straight sections and curved to the radius of an
arched roof at the job site (Fig. 1.16). In other buildings, roof
decks have been designed as the top chord of prefabricated
open web steel joists or roof trusses (Fig. 1.17).1.85,1.86 In
Europe, TRP 200 decking (206 mm deep by 750 mm pitch)
has been used widely. In the United States, the standing seam
metal roof has an established track record in new construction and replacement for built-up and single-ply systems in
many low-rise buildings.
Figure 1.11 also shows corrugated sheets, which are often
used as roof or wall panels and in drainage structures. The use
of corrugated sheets as exterior curtain wall panels is illustrated in Fig. 1.18a. It has been demonstrated that corrugated
sheets can be used effectively in the arched roofs of underground shelters and drainage structures.1.87–1.89
The pitch of corrugations usually ranges from 1 14 to 3 in.
(31.8 to 76.2 mm), and the corrugation depth varies from
1
to 1 in. (6.35 to 25.4 mm). The thickness of corrugated
4
steel sheets usually ranges from 0.0135 to 0.164 in. (0.343 to
4.17 mm). However, corrugations with a pitch of up to 6 in.
(152 mm) and a depth of up to 2 in. (50.8 mm) are also available. See Chapter 10 for the design of corrugated steel sheets
based on the AISI publications.1.87,1.88 Unusually deep
corrugated panels have been used in frameless stressed-skin
construction, as shown in Fig. 1.18b. The self-framing
corrugated steel panel building proved to be an effective
blast-resistant structure in the Nevada tests conducted in
1955.1.90
METAL BUILDINGS AND INDUSTRIALIZED HOUSING
7
Figure 1.10 Rack structures. Courtesy of Unarco Materials Storage.
Figure 1.11 Decks, panels, and corrugated sheets.
Figure 1.19 shows the application of standing seam roof
systems. The design of beams having one flange fastened to a
standing seam roof system and the strength of standing seam
roof panel systems are discussed in Chapter 4.
In the past four decades, cold-formed steel deck
has been successfully used not only as formwork but
also as reinforcement of composite concrete floor and
roof slabs.1.55,1.91,1.103 The floor systems of this type of
composite steel deck-reinforced concrete slab are discussed
in Chapter 11.
1.3 METAL BUILDINGS AND INDUSTRIALIZED
HOUSING
Single-story metal buildings have been widely used in
industrial, commercial, and agricultural applications. Metal
8
1
INTRODUCTION
Figure 1.12 Cellular floor panels. Courtesy of H. H. Robertson Company.
Figure 1.13
Company.
Cold-formed steel panels used in folded-plate roof. Courtesy of H. H. Robertson
Figure 1.14 Hyperbolic paraboloid roof of welded laminated steel deck. Reprinted from Architectural Record, March 1962. Copyright by McGraw-Hill Book Co., Inc.1.79
METAL BUILDINGS AND INDUSTRIALIZED HOUSING
Figure 1.15 Super bayhangar for American Airlines Boeing 747s in Los Angeles.1.82 Courtesy of
Lev Zetlin Associates, Inc.
Figure 1.16 Arched roof curved at job site. Courtesy of Donn Products Company.
Figure 1.17 Steel deck is designed as the top chord of prefabricated open web steel joists. Courtesy
of Inland-Ryerson Construction Products Company.
9
10
1
INTRODUCTION
Figure 1.18 (a) Exterior curtain wall panels employing corrugated steel sheets.1.87 (b) Frameless
stressed-skin construction. Courtesy of Behlen Manufacturing Company.
building systems have also been used for community
facilities such as recreation buildings, schools, and
churches.1.104,1.105 Metal buildings provide the following
major advantages:
1.
2.
3.
4.
5.
Attractive appearance
Fast construction
Low maintenance
Easy extension
Lower long-term cost
In general, smaller buildings can be made entirely
of cold-formed sections (Fig. 1.20), and relatively large
buildings are often made of welded steel plate rigid frames
with cold-formed sections used for girts, purlins, roofs, and
walls (Fig. 1.21).
The design of pre-engineered metal buildings is often
based on the Metal Building Systems Manual issued by the
Metal Building Manufacturers Association (MBMA).1.360
The 2012 edition of the MBMA manual is a revised version
of the previous manual. The new manual includes (a) load
METAL BUILDINGS AND INDUSTRIALIZED HOUSING
Figure 1.19 Application of standing seam roof systems. Courtesy
of Butler Manufacturing Company.
application data [International Building Code (IBC) 2006
loads], (b) crane loads, (c) serviceability, (d) common
industry practices, (e) guide specifications, (f) AISC-MB
certification, (g) wind load commentary, (h) fire protection,
(i) wind, snow, and rain data by U.S. county, (j) a glossary, (k) an appendix, and (l) a bibliography. In addition,
MBMA also has published the Metal Roof Systems Design
Manual.1.361 It includes systems components, substrates,
specifications and standards, retrofit, common industry
practices, design, installation, energy, and fire protection.Additional information may be located at www.mbma
.com.
The design of single-story rigid frames is treated extensively by Lee et al.1.107 In Canada the design, fabrication, and
11
erection of steel building systems are based on a standard of
the Canadian Sheet Steel Building Institute (CSSBI).1.108
Industrialized housing can be subdivided conveniently
into (1) panelized systems and (2) modular systems.1.109,1.278
In panelized systems, flat wall, floor, and roof sections
are prefabricated in a production system, transported to
the site, and assembled in place. In modular systems,
three-dimensional housing unit segments are factory built,
transported to the site, lifted into place, and fastened together.
In the 1960s, under the School Construction Systems
Development Project of California, four modular systems of
school construction were developed by Inland Steel Products Company (modular system as shown in Fig. 1.17),
Macomber Incorporated (V-Lok modular component system
as shown in Fig. 1.22), and Rheem/Dudley Buildings (flexible space system).1.110
In 1970 Republic Steel Corporation was selected by the
Department of Housing and Urban Development under the
Operation Breakthrough Program to develop a modular
system for housing.1.111 Panels consisting of steel facings
with an insulated core were used in this system.
Building innovation also includes the construction of
unitized boxes. These boxes are planned to be prefabricated
of room size, fully furnished, and stacked in some manner
to be a hotel, hospital, apartment, or office building.1.25,1.112
For multistory buildings these boxes can be supported by a
main framing made of heavy steel shapes.
In the past, cold-formed steel structural components
have been used increasingly in low-rise buildings and
residential steel framing. Considerable research and
Figure 1.20 Small building made entirely of cold-formed sections. Courtesy of Stran-Steel
Corporation.1.6
12
1
INTRODUCTION
Figure 1.21 Standardized building made of fabricated rigid frame with cold-formed sections for
girts, purlins, roofs, and walls. Courtesy of Armco Steel Corporation.
Figure 1.22
V-Lok modular component system. Courtesy of Macomber Incorporated.
METHODS OF FORMING
development activities have been conducted continuously by numerous organizations and steel companies.1.21,1.25,1.27,1.28,1.113–1.116,1.280–1.301 In addition to the study
of the load-carrying capacity of various structural components, recent research work has concentrated on (1) joining
methods, (2) thermal and acoustical performance of wall
panels and floor and roof systems, (3) vibrational response
of steel decks and floor joists, (4) foundation wall panels, (5)
trusses, and (6) energy considerations. Chapter 12 provides
some information on recent developments, design standards, and design guide for cold-formed steel light-frame
construction.
In Europe and other countries many design concepts
and building systems have been developed. For details, see
Refs. 1.25, 1.140–1.143, 1.117, 1.118, 1.268, 1.270, 1.271,
1.273, 1.275, 1.290, 1.293, and 1.297.
1.4 METHODS OF FORMING
Three methods are generally used in the manufacture of
cold-formed sections such as illustrated in Fig. 1.1:
1. Cold roll forming
2. Press brake operation
3. Bending brake operation
1.4.1 Cold Roll Forming1.1,1.119
The method of cold roll forming has been widely used for
the production of building components such as individual
structural members, as shown in Fig. 1.2, and some roof,
floor, and wall panels and corrugated sheets, as shown in
Fig. 1.11. It is also employed in the fabrication of partitions,
frames of windows and doors, gutters, downspouts, pipes,
agricultural equipment, trucks, trailers, containers, railway
passenger and freight cars, household appliances, and other
products. Sections made from strips up to 36 in. (915 mm)
wide and from coils more than 3000 ft (915 m) long can be
produced most economically by cold roll forming.
The machine used in cold roll forming consists of pairs
of rolls (Fig. 1.23) which progressively form strips into the
final required shape. A simple section may be produced by
as few as six pairs of rolls. However, a complex section may
require as many as 15 sets of rolls. Roll setup time may be
several days.
The speed of the rolling process typically ranges from
20 to 300 ft/min (6 to 92 m/min). The usual speed is in the
range of 75–150 ft/min (23–46 m/min). At the finish end, the
completed section may be cut to required lengths by an automatic cutoff tool without stopping the machine. Maximum
13
cut lengths are usually between 20 and 40 ft (6 and 12 m).The
flat sheet may be cut to length prior to the rolling process.
As far as the limitations for thickness of material are
concerned, carbon steel plate as thick as 34 in. (19 mm) can be
roll formed successfully, and stainless steels have been roll
formed in thicknesses of 0.006–0.30 in. (0.2–7.6 mm). The
size ranges of structural shapes that can be roll formed on
standard mill-type cold-roll-forming machines are shown in
Fig. 1.24.
The tolerances in roll forming are usually affected by the
section size, the product type, and the material thickness.
The following limits were given by Kirkland1.1 as representative of commercial practice, but do not necessarily represent
current industry tolerances:
Piece length, using
automatic cutoff
Straightness and twist
1
−
± 64
1
8
in. (0.4–3.2 mm)
1
−
± 64
1
4
in. (0.4–6.4 mm) in
10 ft (3 m)
Cross-sectional dimensions
Fractional
Decimal
Angles
1
−
± 64
1
16
in. (0.4–1.6 mm)
±0.005–0.015 in.
(0.1–0.4 mm)
±1∘ –2∘
Table 1.1 gives the fabrication tolerances as specified by
the MBMA for cold-formed steel channels and Z-sections
to be used in metal building systems.1.360 All symbols
used in the table are defined in Fig. 1.25. The same tolerances are specified in the standard of the CSSBI.1.108 For
light steel framing members, the AISI framing standard
S240-151.400,1.432 includes manufacturing tolerances for
structural members. These tolerances for studs and tracks
are based on the American Society for Testing and Materials
(ASTM) standard C955-11. See Table 1.2 and Fig. 1.26. For
additional information on roll forming, see Ref. 1.119.
1.4.2
Press Brake
The press brake operation may be used under the following
conditions:
1. The section is of simple configuration.
2. There is a small required quantity.
The equipment used in the press brake operation consists
essentially of a moving top beam and a stationary bottom
bed on which the dies applicable to the particular required
product are mounted, as shown in Fig. 1.27.
14
1
INTRODUCTION
Figure 1.23 Cold-roll-forming machine.
RESEARCH AND DESIGN SPECIFICATIONS
Figure 1.24
Table 1.1
Size ranges of typical roll-formed structural shapes.1.1
MBMA Table on Fabrication Tolerances1.360
Dimension
Geometry
D
B
d
𝜃1
𝜃2
Hole location
E1
E2
E3
S1
S2
F
P
L
Chamber, C
Minimum thickness t
Note: 1 in. = 25.4 mm.
+
Tolerances, in.
−
3
16
3
16
3
8
3
16
3
16
1
8
3∘
5∘
3∘
5∘
1
8
1
8
1
8
1
16
1
16
1
8
1
8
1
8
1
8
1
8
1
8
1
16
1
16
1
8
1
8
1
8
1
4
(
15
L.ft
10
)
, in.
0.95 × design t
Simple sections such as angles, channels, and Z-sections
are formed by press brake operation from sheet, strip, plate,
or bar in not more than two operations. More complicated
sections may take several operations.
It should be noted that the cost of products is often dependent upon the type of the manufacturing process used in
production. Reference 1.120 indicates that in addition to the
strength and dimensional requirements a designer should
also consider other influencing factors, such as formability,
cost and availability of material, capacity and cost of manufacturing equipment, flexibility in tooling, material handling,
transportation, assembly, and erection.
1.5
1.5.1
RESEARCH AND DESIGN SPECIFICATIONS
United States
1.5.1.1 Research During the 1930s, the acceptance and
development of cold-formed steel members for the construction industry in the United States faced difficulties due to the
lack of an appropriate design specification. Various building
codes made no provision for cold-formed steel construction
at that time.
Since cold-formed steel structural members are usually
made of relatively thin steel sheet and come in many
different geometric shapes in comparison with typical
hot-rolled sections, the structural behavior and performance
of such thin-walled, cold-formed structural members under
loads differ in several significant respects from that of
16
1
INTRODUCTION
Figure 1.25
Symbols used in MBMA table.1.360
D
E
G
A
B
H
F
K
C
I
Stiffening Lip Length
OVERBEND
MEASUREMENT
J–Flange Width
C
Figure 1.26
FLARE
MEASUREMENT
Manufacturing tolerances.1.400,1.432
RESEARCH AND DESIGN SPECIFICATIONS
17
ASTM C 955-11 Manufacturing Tolerances for Structural Members1.400,1.432
Table 1.2
Dimension𝑎
Item Checked
Studs, in. (mm)
Tracks, in. (mm)
A
Length
+3/32 (2.38)
–3/32 (2.38)
+1/2 (12.7)
–1/4 (6.35)
B𝑏
Web Depth
+1/32 (0.79)
–1/32 (0.79)
+1/32 (0.79)
+1/8 (3.18)
C
Flare
Overbend
+1/16 (1.59)
–1/16 (1.59)
+0 (0)
–3/32 (2.38)
D
Hole Center Width
+1/16 (1.59)
–1/16 (1.59)
NA
NA
E
Hole Center Length
+1/4 (6.35)
–1/4 (6.35)
NA
NA
F
Crown
+1/16 (1.59)
–1/16 (1.59)
+1/16 (1.59)
–1/16 (1.59)
G𝑐
Camber
1/8 per 10 ft (3.13 per 3 m)
1/32 per ft (2.60 per m) 1/2 max (12.7)
H𝑐
Bow
1/8 per 10 ft (3.13 per 3 m)
1/32 per ft (2.60 per m)
1/2 max (12.7)
I
Twist
1/32 per ft (2.60 per m)
1/2 max (12.7)
1/32 per ft (2.60 per m)
1/2 max (12.7)
J
Flange Width
+1/8 (3.18)
–1/16 (1.59)
+1/4 (6.35)
–1/16 (1.59)
K
Stiffening Lip Length
+1/8 (3.18)
–1/32 (0.79)
NA
𝑎
All measurements are taken not less than 1 ft (305 mm) from the end. See Fig. 1.26 for symbol definitions.
Outside dimension for stud; inside for track.
𝑐
1/8 inch per 10 feet represents L/960 maximum for overall camber and bow. Thus, a 20-foot-long member has 1/4-inch permissible
maximum; a 5-foot-long member has 1/16-inch permissible maximum.
𝑏
Figure 1.27
Press braking.1.2,1.16
heavy hot-rolled steel sections. In addition, the connections
and fabrication practices which have been developed for
cold-formed steel construction differ in many ways from
those of heavy steel structures. As a result, design specifications for heavy hot-rolled steel construction cannot possibly
cover the design features of cold-formed steel construction
completely. It soon became evident that the development of
a new design specification for cold-formed steel construction
was highly desirable.
Realizing the need for a special design specification and
the absence of factual background and research information,
the AISI Committee on Building Research and Technology
(then named the Committee on Building Codes) sponsored a
research project at Cornell University in 1939 for the purpose
of studying the performance of light-gage cold-formed steel
structural members and of obtaining factual information for
18
1
INTRODUCTION
the formulation of a design specification. Research projects
have been carried out continuously at Cornell University and
other universities since 1939.
The investigations on structural behavior of cold-formed
steel structures conducted at Cornell University by Professor
George Winter and his collaborators resulted in the development of methods of design concerning the effective width
for stiffened compression elements, the reduced working
stresses for unstiffened compression elements, web crippling of thin-walled cold-formed sections, lateral buckling of
beams, structural behavior of wall studs, buckling of trusses
and frames, unsymmetrical bending of beams, welded
and bolted connections, flexural buckling of thin-walled
steel columns, torsional–flexural buckling of concentrically
and eccentrically loaded columns in the elastic and inelastic
ranges, effects of cold forming on material properties, performance of stainless steel structural members, shear strength
of light-gage steel diaphragms, performance of beams and
columns continuously braced with diaphragms, hyperbolic
paraboloid and folded-plate roof structures, influence of
ductility, bracing requirements for channels and Z-sections
loaded in the plane of the web, mechanical fasteners for
cold-formed steel, interaction of local and overall buckling, ultimate strength of diaphragm-braced channels and
Z-sections, inelastic reserve capacity of beams, strength of
perforated compression elements, edge and intermediate
stiffeners, rack structures, probability analysis, and C- and
Z-purlins under wind uplift.1.5–1.7,1.31,1.121,1.122,1.133–1.136
The Cornell research under the direction of Professor
Teoman Pekoz included the effect of residual stress on
column strength, maximum strength of columns, unified
design approach, screw connections, distortional buckling of beams and columns, perforated wall studs, storage
racks, load eccentricity effects on lipped-channel columns,
bending strength of standing seam roof panels, behavior of
longitudinally stiffened compression elements, probabilistic
examination of element strength, direct-strength prediction
of members using numerical elastic buckling solutions,
laterally braced beams with edge-stiffened flanges, steel
members with multiple longitudinal intermediate stiffeners,
design approach for complex stiffeners, unlipped channel
in bending and compression, beam–columns, cold-formed
steel frame design, and second-order analysis of structural
systems and others.1.220,1.273,1.302–1.308,1.346,1.362,1.363
In addition to the Cornell work, numerous research
projects on cold-formed steel members, connections, and
structural systems have been conducted at many individual
companies and universities in the United States.1.121.1.143,
1.267,1.302–1.305,1.309,1.311,1.346,1.362–1.366,1.419–1.423 Forty-three
universities were listed in the first edition of this book
published in 1985.1.352 Research findings obtained from
these projects have been presented at various national and
international conferences and are published in the conference proceedings and the journals of different engineering
societies.1.43,1.117,1.118,1.124–1.132,1.144–1.147,1.272–1.276,1.302–1.308,
1.367–1.377
Previously, the ASCE Committee on Cold-Formed
Members conducted surveys of current research on coldformed structures and literature surveys.1.133–1.134,1.135,1.136,
1.139–1.141 Thirty-eight research projects were reported in
Ref. 1.136. In Ref. 1.141, about 1300 publications were
classified into 18 categories. These reports provide a useful
reference for researchers and engineers in the field of
cold-formed steel structures.
In 1990, the Center for Cold-Formed Steel Structures was
established at the University of Missouri–Rolla to provide
an integrated approach for handling research, teaching, technical services, and professional activity.1.312 In 1996, the
Center for Cold-Formed Steel Structures conducted a survey
of recent research. Reference 1.309 lists 48 projects carried
out in seven countries. In October 2000, the center was
renamed the Wei-Wen Yu Center for Cold-Formed Steel
Structures (CCFSS) at the Fifteenth International Specialty
Conference on Cold-Formed Steel Structures.1.378
1.5.1.2 AISI Design Specifications As far as the design
criteria are concerned, the first edition of “Specification
for the Design of Light Gage Steel Structural Members”
prepared by the AISI Technical Subcommittee under the
chairmanship of Milton Male was issued by the AISI in
1946.1.5 This allowable stress design (ASD) specification
was based on the findings of the research conducted at
Cornell University up to that time and the accumulated practical experience obtained in this field. It was revised by the
AISI committee under the chairmanships of W. D. Moorehead, Tappan Collins, D. S. Wolford, J. B. Scalzi, K. H.
Klippstein, S. J. Errera, and R. L. Brockenbrough in 1956,
1960, 1962, 1968, 1980, 1986, 1996, 2001, 2007, 2012, and
2016 to reflect the technical developments and results of
continuing research.
In 1991, the first edition of the load and resistance factor
design (LRFD) specification1.313 was issued by AISI under
the chairmanship of R. L. Brockenbrough and the vice chairmanship of J. M. Fisher. This specification was based on the
research work discussed in Ref. 1.248. In 1996, the AISI
ASD Specification1.4 and the LRFD Specification1.313 were
combined into a single specification1.314 under the chairmanship of R. L. Brockenbrough and the vice chairmanship
of J. W. Larson. The revisions of various editions of the
AISI Specification are discussed in Ref. 1.267. In Ref. 1.315,
Brockenbrough summarized the major changes made in the
1996 AISI Specification. See also Ref. 1.316 for an outline of
RESEARCH AND DESIGN SPECIFICATIONS
the revised and new provisions. In 1999, a supplement to the
1996 edition of the AISI Specification was issued.1.333,1.335
The AISI Specification has gained both national and
international recognition since its publication. It has been
accepted as the design standard for cold-formed steel
structural members in major national building codes. This
design standard has also been used wholly or partly by most
of the cities and other jurisdictions in the United States
having building codes. The design of cold-formed steel
structural members based on the AISI Specification has been
included in a large number of textbooks and engineering
handbooks.1.13,1.149–1.158,1.269,1.277,1.318–1.320,1.350–1.358,1.412
1.5.1.3 North American Specifications The above
discussions dealt with the AISI Specification used in the
United States. In Canada, the Canadian Standards Association (CSA) published its first edition of the Canadian
Standard for Cold-Formed Steel Structural Members in
1963 on the basis of the 1962 edition of the AISI Specification with minor changes. Subsequent editions of the
Canadian Standard were published in 1974, 1984, 1989, and
1994.1.177,1.327 The 1994 Canadian Standard was based on
the limit states design (LSD) method, similar to the LRFD
method used in the AISI specification except for some
differences discussed in Section 1.8.3.1
In Mexico, cold-formed steel structural members have
always been designed according to the AISI specification.
The 1962 edition of the AISI design manual was translated
into Spanish in 1965.1.201
In 1994, Canada, Mexico, and the United States implemented the North American Free Trade Agreement
(NAFTA). Consequently, the first edition of North American
Specification for the Design of Cold-Formed Steel Structural
Members (NAS) was developed in 2001 by a joint effort
of the AISI Committee on Specifications, CSA Technical
Committee on Cold-Formed Steel Structural Members, and
Camara Nacional de la Industria del Hierro y del Acero
(CANACERO) in Mexico.1.336 It was coordinated through
the AISI North American Specification Committee chaired
by R. M. Schuster. This 2001 edition of the North American Specification was accredited by the American National
Standard Institute (ANSI) as an American National Standard
(ANS) to supersede the AISI 1996 Specification and the
CSA 1994 Standard with the approval by CSA in Canada
and CANACERO in Mexico.
The North American Specification provides an integrated
treatment of ASD, LRFD, and LSD. The ASD and LRFD
methods are for use in the United States and Mexico, while
the LSD method is used in Canada. This first edition of the
North American Specification contained a main document in
Chapters A through G applicable for all three countries and
19
three separate country-specific Appendices A, B, and C for
use in the United States, Canada, and Mexico, respectively.
The major differences between the 1996 AISI Specification and the 2001 edition of the North American Specification were discussed by Brockenbrough and Chen in Refs
1.339 and 1.341 and were summarized in the CCFSS Technical Bulletin.1.338
In 2004, AISI issued a Supplement to the 2001 Edition of
the North American Specification that provides the revisions
and additions for the Specification.1.343,1.344 This supplement
included a new Appendix for the design of cold-formed steel
structural members using the direct-strength method (DSM).
This new method provides alternative design provisions for
determining the nominal axial strengths of columns and flexural strengths of beams without using the effective widths of
individual elements. The background information on DSM
can be found in the Commentary of Ref. 1.343 and Chapters
3 through 6.
The first edition of the North American Specification
was revised in 2007.1.345 It was prepared on the basis of
the 2001 Specification,1.336 the 2004 supplement,1.343
and the continued developments of new and revised
provisions. The major changes in the 2007 edition of
the North American specification were summarized in Refs.
1.346–1.348. In this revised Specification, some design
provisions were rearranged with editorial revisions for
consistency. The common terms used in the Specification
were based on the Standard Definitions developed by a
joint AISC–AISI Committee on Terminology.1.380 In addition to Appendix 1 on the DSM, Appendix 2 was added
for the second-order analysis of structural systems. For
the country-specific design requirements, Appendix A is
now applicable to the United States and Mexico, while
Appendix B is for Canada. Subsequent editions of the North
American specification have been issued in 20121.416 and
2016.1.417 The major changes to these specification editions
are summarized by Refs. 1.424–1.426.
The North American specification has been approved by
the ANSI and is referred to in the United States as AISI S100.
It has also been approved by the CSA and is referred to in
Canada as S136.
1.5.1.4 AISI Design Manuals In addition to the issuance
of the design specification, AISI published the first edition of
the Light Gauge Steel Design Manual1.5 in 1949, prepared by
the Manual Subcommittee under the chairmanship of Tappan
Collins. It was subsequently revised in 1956, 1961, 1962,
1968–1972, 1977, 1983, 1986, 1996, 2002, 2008, 2013, and
2018.1.349,1.427,1.428
The 2002 AISI Design Manual was based on the 2001
edition of the North American Specification.1.336,1.340
20
1
INTRODUCTION
It included the following six parts: I, Dimensions and
Properties; II, Beam Design; III, Column Design; IV,
Connections; V, Supplementary Information; and VI, Test
Procedures. Design aids (tables and charts) and illustrative examples were given in Parts I, II, III, and IV for
calculating sectional properties and designing members
and connections. Part I also included information on the
availability and properties of steels that are referenced in the
Specification. It contains tables of sectional properties of
channels (C-sections), Z-sections, angles, and hat sections
with useful equations for computing sectional properties.
The development of this 2002 AISI Design Manual was
discussed by Kaehler and Chen in Ref. 1.342.
Following the issuance of the 2007 edition of the Specification, AISI revised its Design Manual in 20081.349 on
the basis of the second edition of the North American
Specification.1.345 As for previous editions of the Design
Manual, the data contained in the AISI design manual
are applicable to carbon and low-alloy steels only. They
do not apply to stainless steels or to nonferrous metals
whose stress–strain curves and some other characteristics of
structural behavior are substantially different from those of
carbon and low-alloy steels. For the design of stainless steel
structural members, see Ref. 1.429.
It should also be noted that at the present time there
are standardized sizes for studs, joists, channels, and tracks
produced by the light-steel framing manufacturing companies as defined by the AISI North American Standard for
Cold-Formed Steel Framing—Product Data.1.379 The design
aids for those frequently used members are included in
the AISI Design Manual. Except for the AISI designated
sections, the sections listed in the tables of Part I of the
AISI design manual are not necessarily stock sections with
optimum dimensions. They are included primarily as a guide
for design.
In some other countries, the cold-formed steel shapes may
be standardized. The standardization of shapes would be
convenient for the designer, but it may be limiting for particular applications and new developments.
1.5.1.5 AISI Commentaries Commentaries on several
earlier editions of the AISI design specification were
prepared by Professor Winter of Cornell University and
published by AISI in 1958, 1961, 1962, and 1970.1.161 In the
1983 and 1986 editions of the Design Manual, the format
used for the simplified commentary was changed in that
the same section numbers were used in the Commentary
as in the Specification. For the 1996 edition of the Specification, the AISI Commentary, prepared by Wei-Wen Yu,
contained a brief presentation of the characteristics and the
performance of cold-formed steel members, connections,
and systems.1.310 In addition, it provided a record of the
reasoning behind and the justification for various provisions
of the AISI Specification. A cross reference was provided
between various provisions and the published research data.
The Commentary on the 2001 edition of the North American Specification1.337 was prepared on the basis of the
AISI Commentary on the 1996 Specification with additional
discussions on the revised and new design provisions. In the
Commentary on the 2007 and subsequent editions of the
North American Specification, comprehensive discussions
with extensive references are included for the new provisions, particularly for Appendices 1 and 2. For details, see
Refs. 1.346, 1.430, and 1.431.
In Refs. 1.62, 1.73, and 1.174, Johnson has reviewed some
previous research work together with the development of
design techniques for cold-formed steel structural members.
1.5.1.6 Other Design Standards and Design Guides In
addition to the AISI Design Specifications discussed in
Sections 1.5.1.2 and 1.5.1.3, AISI also published “Overview
of the Standard for Seismic Design of Cold-Formed Steel
Structures—Special Bolted Moment Frames”1.381,12.47
and the ANSI-accredited North American standards for
cold-formed steel framing, including (a) general provisions, (b) product data, (c) floor and roof system design,
(d) wall stud design, (e) header design, (f) lateral design,
(g) truss design, and (h) a prescriptive method.1.387 These
standards have been developed by the AISI Committee
on Framing Standards since 1998. In 2015, the AISI
Committee on Framing Standards merged the framing
standards into a single document, North American Standard for Cold-Formed Steel Structural Framing, AISI
S240.1.432 A companion specification, the North American
Standard for Cold-Formed Steel Framing—Nonstructural
Members, AISI S2201.433 was introduced. The uses of
these standards for residential and commercial construction are discussed in Chapter 12. Furthermore, AISI also
published numerous design guides: Direct Strength Method
(DSM) Design Guide,1.383 Cold-Formed Steel Framing
Design Guide,1.384,1.434 Steel Stud Brick Veneer Design
Guide,1.385,1.435 A Design Guide for Standing Seam Roof
Panels,1.386 and others. In addition, the Cold-Formed Steel
Engineers Institute (CFSEI) has developed and published
numerous technical notes and design guides on a broad
range of design issues (www.cfsei.org).
In the past, many trade associations and professional organizations had special design requirements for using coldformed steel members as floor decks, roof decks, and wall
panels,1.103,1.162,1.330–1.332 open web steel joists,1.163 transmission poles,1.45,1.48,1.164,1.321,1.322,1.323 storage racks,1.165,
1.166,1.407–1.410
shear
diaphragms,1.167–1.169,1.388,1.389
GENERAL DESIGN CONSIDERATIONS OF COLD-FORMED STEEL CONSTRUCTION
composite slabs,1.103,1.170,1.324,1.325,1.390 metal buildings,1.106,
light framing systems,1.171 guardrails, structural supports for highway signs, luminaries, and traffic
signals,1.88 and automotive structural components.1.172,1.173
The locations of various organizations are listed at the end
of the book under Acronyms and Abbreviations.
1.360,1.361
1.5.2 Other Countries
In other countries, research and development for cold-formed
steel members, connections, and structural systems have
been actively conducted at many institutions and individual companies in the past. Design specifications and
recommendations are now available in Australia and New
Zealand,1.69,1.175,1.326,1.391 Austria,1.176 Brazil,1.392 Canada,
1.177–1.180,1.327,1.393 the Czech Republic,1.181 Finland,1.182
France,1.183,1.184 Germany,1.196–1.198,1.396 India,1.185 Italy,
1.394 Japan,1.186 Mexico,1.397 the Netherlands,1.187,1.395 the
People’s Republic of China,1.188 the Republic of South
Africa,1.189 Sweden,1.191.1.193 Romania,1.190 the United
Kingdom,1.49,1.72,1.194,1.195 Russia,1.199 and elsewhere (see
http://coldformedsteel.pbworks.com/w/page/16065164/Front
Page). Some of the recommendations are based on LSD.
The AISI Design Manual has previously been translated into
several other languages.1.200–1.204
In the past, the European Convention for Constructional
Steelwork (ECCS), through its Committee TC7 (formerly
17), prepared several documents for the design and testing of
cold-formed sheet steel used in buildings.1.205–1.214 In 1993,
the European Committee for Standardization published Part
1.3 of Eurocode 3 for cold-formed, thin-gage members and
sheeting.1.328 This work was initiated by the Commission of
the European Communities and was carried out in collaboration with a working group of the ECCS. The design of
cold-formed steel sections is also covered in Refs. 1.66, 1.69,
1.215, 1.216, 1.217, and 1.268.
With regard to research work, many other institutions
have conducted numerous extensive investigations in the
past. References 1.40–1.43, 1.71, 1.117, 1.118, 1.124–1.147,
1.158, 1.218, 1.237, 1.268–1.276, 1.302–1.309, and 1.362–
1.377 contain a number of papers on various subjects related
to thin-walled structures from different countries. Comparisons between various design rules are presented in Refs.
1.239 and 1.240.
The following is a brief discussion of some considerations
usually encountered in design.
1.6.1 Local Buckling, Distortional Buckling,
and Postbuckling Strength of Thin Compression
Elements
Since the individual components of cold-formed steel
members are usually thin with respect to their widths, these
thin elements may buckle at stress levels less than the yield
stress if they are subject to compression, shear, bending, or
bearing. Local buckling of such elements is therefore one of
the major design considerations.
It is well known that such elements will not necessarily fail
when their buckling stress is reached and that they often will
continue to carry increasing loads in excess of that at which
local buckling first appears.
Figure 1.28 shows the buckling behavior and postbuckling
strength of the compression flange of a hat-section beam with
1.6 GENERAL DESIGN CONSIDERATIONS
OF COLD-FORMED STEEL CONSTRUCTION
The use of thin material and cold-forming processes results
in several design features for cold-formed steel construction
different from those of heavy hot-rolled steel construction.
21
Figure 1.28
Consecutive load stages on hat-shaped beam.1.7
22
1
INTRODUCTION
with different types of compression elements. The current
design methods for beams, columns, and beam–columns are
discussed in Chapters 4, 5, and 6, respectively.
During recent years, distortional buckling has been
considered as one of the important limit states for the
design of cold-formed steel beams and columns having
edge-stiffened compression flanges. Design provisions have
been added in the current North American specification. For
details, see Chapters 4 and 5.
1.6.2
Figure 1.29
Consecutive load stages on I-beam.1.7
a compression flange having a width-to-thickness ratio of
184 tested by Winter. For this beam the theoretical buckling
load is 500 lb (2.2 kN), while failure occurred at 3460 lb
(15.4 kN).1.7
Figure 1.29 shows the buckling behavior of an I-beam
having an unstiffened flange with a width-to-thickness ratio
of 46.1.7 The beam failed at a load about 3.5 times that at
which the top flange stress was equal to the theoretical critical
buckling value. These pictures illustrate why the postbuckling strength of compression elements is utilized in design.
Prior to 1986, different procedures were used in the
AISI Specification for the design of beams and columns
Torsional Rigidity
Because the torsional rigidity of open sections is proportional
to t3 , cold-formed steel sections consisting of thin elements
are relatively weak against torsion. Figure 1.30 shows the
twist of a channel-shaped unbraced beam when it is loaded
in the plane of its web. In this case, the shear center is outside
the web and the applied load initiates rotation.
Since cold-formed steel sections are relatively thin and
in some sections the centroid and shear center do not coincide, torsional–flexural buckling may be a critical factor
for compression members. In addition, distortional buckling
may govern the design for certain members used as beams or
columns.
1.6.3
Stiffeners in Compression Elements
The load-carrying capacity and the buckling behavior of
compression components of beams and columns can be
Figure 1.30 Twist of unbraced channel loaded in plane of its web1.6 : (a) before loading; (b)
near-maximum load.
GENERAL DESIGN CONSIDERATIONS OF COLD-FORMED STEEL CONSTRUCTION
improved considerably by the use of edge stiffeners or intermediate stiffeners. Provisions for the design of such stiffeners have been developed from previous research. However,
this type of stiffener generally is not practical in hot-rolled
shapes and built-up members.
1.6.4 Variable Properties of Sections Having Stiffened
or Unstiffened Compression Elements
For a section having a stiffened, partially stiffened, or
unstiffened compression element, the entire width of the
element is fully effective when the width-to-thickness ratio
of the element is small or when it is subjected to low
compressive stress. However, as stress increases in the
element having a relatively large width-to-thickness ratio,
the portions adjacent to the supported edges are more structurally effective after the element buckles. As a result, the
stress distribution is nonuniform in the compression element.
When using the Effective Width Method for design of such
members the sectional properties are based on a reduced
effective area.
The effective width of a compression element not only
varies with the unit stress applied but also depends on its
width-to-thickness ratio. For a given beam having a compression flange with a relatively large width-to-thickness ratio,
the effective section modulus Se decreases with an increase
in the yield stress of steel used because the effective width of
the compression flange becomes smaller when it is subjected
to a higher unit stress. The strength of such a beam is therefore not directly proportional to the yield stress of the steel.
The same is true for the compression members.
When using the Direct Strength Method for design, full,
unreduced section properties are used.
In addition to bolted and welded connections, screws and
power-actuated fasteners are often used for cold-formed steel
construction. Design provisions for determining the shear
and tensile strengths of screw and power-actuated fastener
connections are included in the current North American
specification.
1.6.6
Web Crippling Strength of Beams
Web crippling is often a critical problem for cold-formed
steel structural members for two reasons. First, the use of
stamped or rolled-in bearing stiffeners (or stiffeners under
concentrated loads) is frequently not practical in cold-formed
steel construction. Second, the depth-to-thickness ratio of
the webs of cold-formed steel members is usually large and
generally exceeds that of hot-rolled shapes. Figure 1.31 illustrates the pattern of web crippling of an I-section.
Special design criteria for web crippling of cold-formed
steel sections included in the North American Specification
have been developed on the basis of extensive research.
1.6.7
Thickness Limitations and Corrosion Protection
The ranges of thickness generally used in various types
of cold-formed steel structural members are described in
Section 1.2. However, they should not be considered as thickness limitations.
For the design of cold-formed steel structural members
the important factors are the width-to-thickness ratio of
compression elements and the unit stress used; the thickness
1.6.5 Connections
For bolted connections the thickness of connected parts is
usually much thinner in cold-formed steel construction than
in heavy construction. The steel sheet or strip may have
a small spread between yield stress and tensile strength.
These are major influences that make the behavior of the
cold-formed steel bolted connection differ from that of
heavy construction, particularly for bearing and tension
stress. Modified design provisions have been developed in
the Specification for cold-formed steel bolted connections.
In welded connections, arc welds (groove welds, arc spot
welds, arc seam welds, fillet welds, and flare groove welds)
are often used for connecting cold-formed steel members to
each other as well as for connecting cold-formed sections to
hot-rolled shapes. Arc spot welds without prepunched holes
and arc seam welds are often used for connecting panels or
decks to supporting beams or to each other.
23
Figure 1.31
Test for web crippling strength of thin webs.1.6
24
1
INTRODUCTION
of the steel itself is not a critical factor. Members formed
of relatively thin steel sheet will function satisfactorily if
designed in accordance with the North American Specification.
The durability of lightweight steel construction has been
studied by Cissel and Quinsey.1.241,1.242 It was found that
the durability of cold-formed steel sections is primarily
dependent upon the protective treatment applied to the sheet
and not necessarily upon the thickness of the sheet itself.1.243
For galvanized cold-formed steel there is high corrosion
resistance. Available data indicate that the corrosion rate of
galvanized sheets in the atmosphere is practically linear; that
is, for the same base-metal thickness a sheet having double
the weight of coating of another sheet can be expected
to last twice as long before rusting of the base metal sets
in.1.244–1.246 References 1.398, 1.399, and 1.436 present a
better understanding of how galvanizing provides long-term
corrosion protection to steel members. It is therefore unnecessary to limit the minimum thickness for cold-formed
steel sections merely for the purpose of protecting the steel
from corrosion. The accepted methods of protection were
discussed in Section 5 of Part III of the 1977 AISI design
manual1.159 and the minimum metallic coating requirements
for framing members are specified in the AISI general provisions for cold-formed steel framing.1.400,1.432 In addition, the
CFSEI technical note outlines available corrosion-resistant
materials for cold-formed steel framing members and makes
recommendations for buildings at various distances from
the ocean and for different exposure conditions within an
individual building.1.401,1.436 Tests of coil-coated steel panels
are reported in Ref. 1.329.
material of the section is considered to be concentrated
along the centerline or midline of the steel sheet and the area
elements are replaced by straight or curved “line elements.”
The thickness dimension t is introduced after the linear
computations have been completed. Thus the total area
A = L × t and the moment of inertia of the section I = I′ × t,
where L is the total length of all line elements and I′ is the
moment of inertia of the centerline of the steel sheet. The
properties of typical line elements are shown in Fig. 1.32.
Example 1.1 illustrates the application of the linear method.
Example 1.1 Determine the full section modulus Sx of the
channel section shown in Fig. 1.33a. Use the linear method.
SOLUTION
The midline of the cross section is shown in Fig. 1.33b.
1. Flat width of flanges (element 1):
𝐿f = 1.5 − 0.292 = 1.208 in.
2. Distance from x–x axis to centerline of flange:
0.105
3.0 −
= 2.948 in.
2
3. Computation of properties of 90∘ corner (element 2)
(Fig. 1.33c):
0.105
= 0.240 in.
𝑅′ = 0.1875 +
2
𝐿c = 1.57(0.240) = 0.377 in. (Fig. 1.32)
𝑐 = 0.637(0.240) = 0.153 in.
(Fig. 1.32)
4. Flat width of web (element 3):
1.6.8
Plastic Design
A complete plastic design method is not included in the
North American specification because most cold-formed
steel shapes have width-to-thickness ratios considerably in
excess of the limits required by plastic design.1.148 Such
members with large width-to-thickness ratios are usually
incapable of developing plastic hinges without local buckling or distortional buckling. However, since 1980 the AISI
specification has included design provisions to utilize the
inelastic reserve capacity of flexural members. The requirements are retained in the North American specification. For
details, see Sections 4.2.2.6 and 4.2.3.1.2.
𝐿w = 6.0 − 2(0.292) = 5.416 in.
5. Distance from x–x axis to center of gravity (c.g.) of
corner:
5.416
𝑦=
+ 0.153 = 2.861 in.
2
6. Linear 𝐼𝑥′ , moment of inertia of midlines of steel sheets:
Flanges: 2(1.208)(2.948)2 = 21.00
Corners: 2(0.377)(2.861)2 = 6.17
Web: 12 (5.416)3 = 13.24
Total: 40.41 in.3
7. Actual Ix:
1.6.9 Linear Method for Computing Properties
of Formed Sections
Because the thickness of the formed section is uniform, the
computation of properties of such sections can be simplified
by using a linear or “midline” method. In this method the
𝐼𝑥 = 𝐼𝑥′ 𝑡 = 40.41(0.105) = 4.24 in.4
8. Section modulus:
𝐼
4.24
𝑆𝑥 = /𝑥 =
= 1.41 in.3
3.0
𝑑 2
GENERAL DESIGN CONSIDERATIONS OF COLD-FORMED STEEL CONSTRUCTION
Figure 1.32
Figure 1.33
25
Properties of line elements.1.159
Example 1.1.
The accuracy of the linear method for computing the
properties of a given section depends on the thickness
of the steel sheet to be used and the configuration of the
section. For the thicknesses of steel sheets generally used
in cold-formed steel construction, the error in the moment
of inertia determined by the linear method is usually
negligible, particularly for relatively deep sections made
of thin material. For example, as indicated in Table 1.3,
the expected errors in the computed moment of inertia
of the two arbitrarily chosen channel sections as shown
in Fig. 1.34 are less than 1% if the material is 14 in. or
thinner.
For cylindrical tubes, the error in the computed moment
of inertia about the axis passing through the center of the
tube determined by the linear method varies with the ratio of
mean diameter to wall thickness, D/t; the smaller the ratio,
the larger the error. The expected errors in the moment of
inertia are approximately 2.7 and 0.2% for D/t ratios of 6 and
20, respectively, if the wall thickness is 14 in. Errors smaller
26
1
INTRODUCTION
Figure 1.34
Sections used for studying the accuracy of the linear method.
Table 1.3 Expected Error in Ix
Channel
Section
A
B
Thickness of
Material (in.)
Expected
in Ix (%)
0.50
0.25
0.10
0.50
0.25
0.10
3.3
0.7
0.1
0.6
0.15
0.02
conducted by an independent testing laboratory or by a
manufacturer’s laboratory. It is not the intent of the North
American provision, however, to substitute load tests for
design calculations.
A detailed discussion on the method of testing is beyond
the scope of this book. However, when tests are found
necessary to determine structural strength or stiffness of
cold-formed sections and assemblies, Chapter K of the North
American specification1.417 and Part VI of the AISI Design
Manual1.428 should be used for the evaluation of test results
and the determination of allowable load-carrying capacities.
Note: 1 in. = 25.4 mm.
1.6.11
than the above values are expected for materials thinner than
1
in.
4
The Direct Strength Method Design Guide1.383 indicates
that the use of midline dimensions ignoring the corner is
adequate for analysis unless the corner radius is larger than
10 times the thickness.
1.6.10
Cold Work of Forming
It is well known that the mechanical properties of steel are
affected by cold work of forming. The North American specification permits utilizing the increase in yield stress from
a cold-forming operation subjected to certain limitations.
Sections 2.7 and 2.8 discuss the influence of cold work on
the mechanical properties of steel and the utilization of the
cold work of forming, respectively.
Tests for Special Cases
In Section 1.1 it was indicated that in cold-formed steel
construction unusual sectional configurations can be
economically produced by cold-forming operations.
However, from the point of view of structural design,
the analysis and design of such unusual members may
be very complex and difficult. In many cases it may be
found that their safe load-carrying capacity or deflection
cannot be calculated on the basis of the design criteria
presently included in the North American specification.1.417
For this case the North American Specification permits
their structural performance to be determined by load tests
1.7 ECONOMIC DESIGN AND OPTIMUM
PROPERTIES
The basic objective of economic design is to achieve the
least expensive construction that satisfies the design requirements. One of the conditions required for the low cost of the
erected structure is that the weight of the material be kept to a
minimum, which is associated with the maximum structural
efficiency.
It has been shown by numerous investigators that for
a given loading system the maximum efficiency can be
DESIGN BASIS
obtained when the member strengths for all the possible
modes of failure are the same.
In practice, such ideal conditions may not be obtained
easily because of unavoidable limitations, such as preselected shapes and specific dimensional limitations. However,
it can be shown that in some cases there may be a possible
mode of failure or limit state that will result in a maximum
efficiency within the practical limitations.
The efficiency of the use of high-strength steel depends on
the type of mode of failure. Under certain conditions, such
as long columns having large slenderness ratios, the failure
is usually limited by overall elastic buckling. For this case
the use of high-strength steel may not result in an economic
design because the performance of structural members under
the above-mentioned conditions will be the same for different
grades of steel. For this reason the use of high-strength steel
for these cases may not be justified as far as the overall cost
is concerned.
In any event the general aim should always be to utilize
the full potential strength of the steel that can be used in
fabrication by designing the detail outline of the section
for maximum structural efficiency. Flexibility of the
cold-forming process to produce an endless variety of
shapes is ideal for this purpose.1.225,1.247,1.402–1.406
1.8 DESIGN BASIS
Prior to 1996, the AISI issued two separate specifications for
the design of cold-formed steel structural members, connections, and structural assemblies. One was for the Allowable
Strength Design (ASD) method1.4 and the other was for the
Load and Resistance Factor Design (LRFD) method.1.313
These two design specifications were combined into a single
specification in 1996.1.314 Both methods have been used
for the design of cold-formed steel structures, even though
they may or may not produce identical designs. When the
North American specification was developed in 2001, 2007,
2012, and 2016, the Limit States Design (LSD) method
was included in the Specification for use in Canada. The
ASD and LRFD methods are only used in the United States
and Mexico. Because the design provisions are based on
strengths (moment, force, etc.) instead of stresses, the ASD
method has been redefined as allowable strength design.
In 2016 the Effective Width Method and Direct Strength
Method are presented within the main body of the Specification. Both methods may be used for the design of
cold-formed steel structures, even though they may or may
not produce identical designs.
The North American Specification has been approved by
the ANSI and is referred to in the United States as AISI S100.
It has also been approved by the CSA and is referred to in
Canada as S136.
27
According to Section A1.2 of the North American Specification, the nominal strength and stiffness of cold-formed
steel elements, members, assemblies, connections, and
details shall be determined in accordance with the provisions provided in Chapters A through M, Appendices 1
and 2, and Appendices A and B of the North American
Specification. When the composition or configuration of
such components is such that calculation of the strength
and/or stiffness cannot be made in accordance with those
provisions (excluding Chapter K), structural performance
should be established from one of the following methods:
1. Determine the available strength (allowable strength
for ASD or design strength for LRFD and LSD) or stiffness by tests undertaken and evaluated in accordance
with Section K2.1.1(a) of the Specification.
2. Determine the available strength or stiffness by rational
analysis with confirmatory tests. Specifically, the available strength is determined from the calculated nominal
strength by applying the safety factor or the resistance factor in accordance with Section K2.1.1(b) of
the Specification.
3. Determine the available strength or stiffness based
on appropriate theory and engineering judgment. The
available strength is determined from the calculated
nominal strength by applying the safety and resistance
factors given in Section A1.2(c) of the Specification.
It should be noted that for a limit state already provided
in the main Specification the safety factor should not be less
than the applicable Ω and the resistance factor should not
exceed the applicable 𝜙 for the prescribed limit state.
1.8.1
Allowable Strength Design
Since the issuance of the first AISI Specification in 1946,
the design of cold-formed steel structural members and
connections in the United States and some other countries
has been based on the ASD method. In this method, the
required strengths (axial forces, bending moments, shear
forces, etc.) for structural members and connections are
computed from structural analysis by using the nominal
loads or specified working loads for all applicable load
combinations, as discussed in Section 1.8.1.2. The allowable strength permitted by the specification is determined
by the nominal strength and the specified safety factor.
1.8.1.1 Design Format for the ASD Method For the ASD
method, the required strength R should not exceed the allowable strength Ra as follows:
𝑅 ≤ 𝑅a
(1.1)
28
1
INTRODUCTION
Based on Section A4.1.1 of the North American Specification, the allowable strength is determined by Eq. (1.2):
𝑅
𝑅a = n
(1.2)
Ω
where Rn = nominal strength
Ω = safety factor corresponding to Rn (see
Table 1.4)
In Eq. (1.2), the nominal strength is the strength or
capacity of the element or member for a given limit state
or failure mode. It is computed by the design equations
provided in Chapters B through M, Appendices 1 and 2, and
Appendices A and B of the North American Specification.
The safety factors provided in Chapters C through M and
Appendices of the North American Specification are summarized in Table 1.4. These safety factors are used to compensate for uncertainties inherent in the design, fabrication, and
erection of structural components and connections as well
as uncertainties in the estimation of applied loads. It should
also be noted that for the ASD method only a single safety
factor is used to compensate for the uncertainties of the
combined load.
1.8.1.2 Load Combinations for the ASD Method The
design provisions for nominal loads and combinations are in
accordance with Appendix A of the North American Specification. The following discussion is applicable only to the
ASD method.
a. Nominal Loads. The North American Specification does
not provide any specific dead load, live load, snow, wind,
earthquake, or other loading requirements for the design
of cold-formed steel structures. Section B2 of the AISI
Specification merely states that the nominal loads shall be
as stipulated by the applicable building code under which
the structure is designed or as dictated by the conditions
involved. In the absence of an applicable building code,
the nominal loads shall be those stipulated in ASCE/SEI
7-10, Minimum Design Loads for Buildings and Other
Structures.1.416
For the impact loads on a structure, reference may be
made to the AISC publication1.411 for building design
and the MBMA publication for the design of metal
buildings.1.360
In addition to the above-mentioned loads, due consideration should also be given to the loads due to (1)
fluids with well-defined pressures and maximum heights,
(2) weight and lateral pressure of soil and water in
soil, (3) ponding, and (4) self-straining forces and
effects arising from construction or expansion resulting
from temperature, shrinkage, moisture changes, creep
in component materials, movement due to different
settlement, and combinations thereof.
b. Load Combinations for ASD. In Section B3.2 of the North
American specification, it is specified that the structure
and its components shall be designed so that the allowable
strengths equal or exceed the effects of the nominal loads
and load combinations as stipulated by the applicable
building code under which the structure is designed or, in
the absence of an applicable building code, as stipulated
in the ASCE Standard ASCE/SEI 7.1.416
When the ASCE Standard is used for allowable strength
design, the following load combinations should be considered:
1. 𝐷
2. 𝐷 + 𝐿
(1.3a)
(1.3b)
3. 𝐷 + (𝐿r or 𝑆 or 𝑅)
(1.3c)
4. 𝐷 + 0.75𝐿 + 0.75(𝐿r or 𝑆 or 𝑅)
(1.3d)
5. 𝐷 + (0.6𝑊 or 0.7𝐸)
(1.3e)
6. 𝐷 + 0.75(0.6𝑊 ) + 0.75(𝐿r or 𝑆 or 𝑅)
(1.3f)
7. 0.6𝐷 + 0.6𝑊
(1.3g)
8. 0.6𝐷 + 0.7𝐸
(1.3h)
where D
E
L
Lr
R
S
W
1.8.2
= dead load
= earthquake load
= live load
= roof live dead
= rain load
= snow load
= wind load
Load and Resistance Factor Design
During recent years, the LRFD method has been used in
the United States and other countries for the design of steel
structures.1.313,1.345,1.411,1.417 The advantages of the LRFD
method are (1) the uncertainties and the variabilities of
different types of loads and resistances are accounted for by
use of multiple factors, and (2) by using probability theory,
all designs can ideally achieve a consistent reliability. Thus,
the LRFD approach provides the basis for a more rational
and refined design method than is possible with the allowable
strength design method.
In order to develop the load and resistance factor design
criteria for cold-formed, carbon, and low-alloy steel structural members, a research project was conducted at the
University of Missouri–Rolla under the direction of
Wei-Wen Yu with consultation of T. V. Galambos and
M. K. Ravindra. This project, which was initiated in 1976,
DESIGN BASIS
was sponsored by the AISI and supervised by the AISI
Subcommittee on Load and Resistance Factor Design.1.248
Based on the studies made by Rang, Supornsilaphachai,
Snyder, Pan, and Hsiao, the AISI Load and Resistance
Factor Design Specification for Cold-Formed Steel Structural Members with Commentary was published in August
1991 on the basis of the 1986 edition of the AISI ASD Specification with the 1989 Addendum.1.313,3.152 The background
information and research findings for developing the AISI
LRFD criteria were documented in 14 progress reports of the
University of Missouri–Rolla and are summarized in Refs.
1.248 and 3.153–3.159. As previously discussed, the 1996
edition of the AISI Specification included both the ASD and
LRFD methods in a single standard for the first time.
1.8.2.1 Design Format for the LRFD Method As
discussed in Section 1.8.1.1, the allowable strength design
method employs only one safety factor for the combined
load under a given limit state. A limit state is the condition
in which a structure or component becomes unfit for service
and is judged either to be no longer useful for its intended
function (serviceability limit state) or to have reached its
ultimate load-carrying capacity (strength limit state). For
cold-formed steel members, typical limit states are yielding,
buckling, postbuckling strength, shear lag, web crippling,
excessive deflection, and others. These limits have been
established through experience in practice or in the laboratory, and they have been thoroughly investigated through
analytical and experimental research.
Unlike allowable strength design, the LRFD approach uses
multiple load factors and a corresponding resistance factor
for a given limit state to provide a refinement in the design
that can account for the different degrees of the uncertainties
and variabilities of analysis, design, loading, material properties, and fabrication. The design format for satisfying the
structural safety requirement is expressed in Eq.(1.4)1.417 :
𝑅u ≤ 𝜙𝑅n
(1.4)
where Ru = required strength or required resistance for
factored loads
∑
= γi QI
γi = load factor corresponding to Qi
QI = load effect
Rn = nominal strength
ϕ = resistance factor corresponding to Rn
ϕRn = design strength
The nominal strength Rn is the total strength of the element
or member for a given limit state, computed according to the
applicable design criteria. The resistance factor ϕ accounts
29
for the uncertainties and variabilities inherent in Rn , and it
is usually less than unity, as listed in Table 1.4. The load
effects Qi are the forces (axial force, bending moment, shear
force, etc.) on the cross section determined from the structural analysis and γi are the corresponding load factors that
account for the uncertainties and variabilities of the applied
loads. The load factors are usually greater than unity, as given
in Section 1.8.2.2.
For the design of cold-formed members using carbon and
low-alloy steels, the values of 𝜙 and Rn are given in the North
American Specification.1.417
1.8.2.2 Nominal Loads, Load Factors, and Load Combinations for the LRFD Method The design provisions for
nominal loads and load combinations are provided in Section
B3 of the North American Specification for use in the United
States and Mexico. The following discussion is applicable
only to the LRFD method:
a. Nominal Loads. The design requirements for nominal
loads to be used for the LRFD method are the same as
that used for the ASD method.
b. Load Factors and Load Combinations for LRFD. Section
B3.2 of the North American Specification specifies that
the structure and its components shall be designed so that
design strengths equal or exceed the effects of the factored
loads and load combinations stipulated by the applicable
building code under which the structure is designed or,
in the absence of an applicable building code, as stipulated in the ASCE Standard, Minimum Design Loads for
Buildings and Other Structures, ASCE/SEI 7.
When the ASCE Standard is used for the LRFD method,
the following load factors and load combinations should be
considered for the strength limit state1.416 :
1.
2.
3.
4.
5.
6.
7.
1.4𝐷
1.2𝐷 + 1.6𝐿 + 0.5(𝐿r or 𝑆 or 𝑅)
1.2𝐷 + 1.6(𝐿r or 𝑆 or 𝑅) + (𝐿 or 0.5𝑊 )
1.2𝐷 + 1.0𝑊 + 𝐿 + 0.5(𝐿r or 𝑆 or 𝑅)
1.2𝐷 + 1.0𝐸 + 𝐿 + 0.2𝑆
0.9𝐷 + 1.0𝑊
0.9𝐷 + 1.0𝐸
(1.5a)
(1.5b)
(1.5c)
(1.5d)
(1.5e)
(1.5f)
(1.5g)
All the symbols are defined in item (b) of Section 1.8.1.2. For
the above load combinations, exceptions are as follows:
1. The load factor on L in combinations (3), (4), and (5) is
permitted to equal 0.5 for all occupancies in which the
minimum uniformly distributed live load L in Table 4-1
30
1
INTRODUCTION
of ASCE/SEI1.415 is less than or equal to 100 psf, with
the exception of garages or areas occupied as places of
public assembly.
2. In combinations (2), (4), and (5), the load S shall be
taken as either the flat-roof snow load or the sloped-roof
snow load.
Each relevant strength limit state shall be investigated.
Effects of one or more loads not acting shall be investigated.
The most unfavorable effects from both wind and earthquake
loads shall be investigated, where appropriate, but they need
not be considered to act simultaneously.
The ASCE Standard does not provide load factors and
load combinations for roof and floor composite construction
using cold-formed steel deck. For this construction,the Steel
Deck Institute NSI/SDI C-2017 Standard for Composite Steel
Floor Deck—Slabs1.437 load combinations should be used.
1.8.2.3 Design Strength 𝝓Rn The design strength is the
available strength of a structural component or connection to
be used for design purposes. As shown in Eq. (1.4), design
strength is obtained by multiplying the nominal strength or
resistance Rn by a reduction factor 𝛟 to account for the
uncertainties and variabilities of the nominal strength.
1.8.2.3.1 Nominal Strength or Resistance Rn The
nominal strength or resistance Rn is the capacity of a structural component or connection to resist load effects (axial
force, bending moment, shear force, etc.). It is usually determined by computations using specified material properties
and dimensions in the design criteria derived from accepted
principles of structural mechanics and/or by tests, taking
account of the effects of manufacturing and fabrication
processes. For the design of cold-formed members using
carbon and low-alloy steels, Chapters C through M of the
North American Specification1.417 provide the equations
needed for determining the nominal strengths of tension
members, flexural members, compression members, closed
cylindrical tubular members, wall studs, connections, and
joints. It should be noted that for the purpose of consistency
the same nominal strength equations are used in the North
American specification for the ASD and LRFD methods.1.417
In 2004, the Direct Strength Method was added in the
North American Specification. This method may be used
to determine the axial strengths of columns and beams
subjected to bending and to shear.1.417
1.8.2.3.2 Resistance Factor 𝝓 The resistance factor 𝜙 is
a reduction factor to account for unavoidable deviations of
the actual strength from the nominal value prescribed in
the design specification. These deviations may result from
the uncertainties and variabilities in (1) the material properties (i.e., yield stress, tensile strength, modulus of elasticity,
etc.), (2) the geometry of the cross section (i.e., depth, width,
thickness, etc., to be used for computing area, moment of
inertia, section modulus, radius of gyration, etc.), and (3) the
design methods (i.e., assumptions, approximations of theoretical formulas, etc.).
In the development of the AISI LRFD criteria,3.159 the
resistance factors were derived from a combination of
(1) probabilistic modeling,3.160,3.161 (2) calibration of the
new criteria to the ASD approach,3.162 and (3) evaluations
of the new LRFD criteria by judgment and past experience.
The development was aided by a comparative study of the
ASD and LRFD methods.3.157,3.163 The procedures used
for developing the resistance factors for cold-formed steel
design consisted of the following five steps:
1. Analyze the available information and test data to
obtain the statistical value (mean values and coefficients of variation) of resistance and load effects.
2. Assume the mean values and coefficients of variation
of the variable for which no statistical information is
available.
3. Compute the reliability index implied in the applicable
ASD specification.
4. Select the target reliability index.
5. Develop the resistance factors according to the selected
target reliability index for different types of members
with the limit state being considered.
Details of steps 1, 2, and 3 are presented in several progress
reports of the University of Missouri–Rolla and are summarized in Ref. 3.162. Based on the probability distribution
shown in Fig. 1.35 and the first-order probabilistic theory, the
reliability index 𝛽 can be computed by Eq. (1.6)1.248,3.152 :
ln(𝑅 ∕𝑄 )
𝛽= √ m m
𝑉R2 + 𝑉Q2
(1.6)
where Rm
= mean value of resistance
= Rn (Pm Mm Fm )
Qm = mean value of load effect
of variation of resistance, = 𝜎 R /Rm
VR = coefficient
√
VQ
𝜎R
𝜎Q
Pm
=
=
=
=
𝑉P2 + 𝑉M2 + 𝑉F2
coefficient of variation of load effect, = 𝜎 Q /Qm
standard deviation of resistance
standard deviation of load effect
mean ratio of experimentally determined
ultimate load to predicted ultimate load of test
specimens
DESIGN BASIS
Figure 1.35
Mm
=
Fm
=
VP
VM
VF
=
=
=
31
Definition of reliability index.
mean ratio of tested material properties to
specified minimum values
mean ratio of actual sectional properties to
nominal values
coefficient of variation of ratio P
coefficient of variation of ratio M
coefficient of variation of ratio F
The reliability index 𝛽 is a measure of the safety of the
design. As shown in Fig. 1.35, a limit state is reached when
ln(R/Q) = 0. The shaded area represents the probability of
exceeding the limit states. The smaller the shaded area is, the
more reliable the structure is. Because the distance between
the mean value of [ln(R/Q)]m and the failure region is defined
as 𝛽[𝜎 ln(R/Q) ], when two designs are compared, the one with
the larger 𝛽 is more reliable.
As far as the target reliability index 𝛽 0 is concerned,
research findings indicated that for cold-formed steel
members the target reliability index may be taken as 2.5 for
gravity loads. In order to ensure that failure of a structure
is not initiated in connections, a higher value of 3.5 was
selected for connections using cold-formed carbon steels.
Even though these two target values are somewhat lower
than those recommended by the ASCE code (i.e., 3.0 and
4.5 for members and connections, respectively),3.149 they
are essentially the same targets used for the AISC LRFD
Specification.3.150 For wind loads, the same ASCE target
value of β0 = 2.5 is used for connections in the LRFD
method. For flexural members such as individual purlins,
girts, panels, and roof decks subjected to the combination of
dead and wind loads, the target β0 value used in the United
States is reduced to 1.5. With this reduced target reliability
index, the design based on the LRFD method is comparable
to the allowable strength design method.
On the basis of the load combination of 1.2D + 1.6L with
an assumed D/L ratio of 15 and the available statistical data, it
can be shown that the resistance factor 𝜙 can be determined
as follows1.310 :
1.521𝑀m 𝐹m 𝑃m
(1.7)
𝜙=
)
( √
exp 𝛽0 𝑉R2 + 𝑉Q2
where β0 is the target reliability index. For practical reasons,
it is desirable to have relatively few different resistance
factors. Table 1.4 lists the resistance factors for the design
of various types of members and connections using carbon
and low-alloy steels.
If the 𝜙 factor is known, the corresponding safety factor Ω
for allowable strength design can be computed for the load
combination 1.2D + 1.6L as follows1.310 :
/
1.2𝐷 𝐿 + 1.6
Ω=
/
𝜙(𝐷 𝐿 + 1)
where D/L is the dead-to-live load ratio for the given condition.
1.8.3
Limit States Design
The LSD method is currently used in Canada for the design
of cold-formed steel members, connections, and structural
assemblies. The methodology for the LSD and LRFD
methods is the same, except that load factors, load combinations, target reliability indexes, and the assumed dead-to-live
ratio used for the development of the design criteria are
different. As a result, the resistance factors used for the
LSD method are usually slightly smaller than that for the
LRFD method in the main document of the North American
Specification.
The design provisions for the LSD method are provided in
Section B3 of the North American Specification. For the LSD
32
1
INTRODUCTION
Table 1.4 Safety Factors 𝛀 and Resistance Factors 𝝓 Used in North American Specification 1.417
Type of Strength
(a) Stiffeners
Bearing stiffeners
Bearing stiffeners in C-section beams
(b) Tension members
For yielding of gross section
For rupture of net section away from connection
For rupture of net section at connection (see connections)
(c) Flexural members
Yielding and global buckling
Local buckling interacting with yielding and global buckling
Distortional buckling
Beams having one flange through fastened to deck or
sheathing
i. For general cross-sections and system connectivity
ii. For C- or Z-sections with specifically defined systems
Beams having one flange fastened to a standing seam roof
system
Web design
Shear strength
Web crippling
Built-up sections
Single web channel and C-sections
Single web Z-sections
Single hat sections
Multi-web deck sections
Combined bending and web crippling
(d) Concentrically loaded compression members
Yielding and global buckling
Local buckling interacting with global buckling
Distortional buckling
(e) Closed cylindrical tubular members
Bending strength
Axial compression
(f) Compression members having one flange through fastened to
deck or sheathing
i. For general cross-sections and system connectivity
ii. For C- or Z-sections with specifically defined systems
(g) All-steel design of wall stud assemblies
Wall studs in compression
Wall studs in bending
(h) Diaphragm construction
(i) Rational engineering analysis for members
ASD
Safety
Factor, Ω
LRFD
Resistance
Factor, 𝜙
LSD
Resistance
Factor, 𝜙
2.00
1.70
0.85
0.90
0.80
0.80
1.67
2.00
0.90
0.75
0.90
0.75
1.67
1.67
1.67
0.90
0.90
0.90
0.90
0.90
0.90
1.67
1.67
1.67
0.90
0.90
0.90
0.85
0.90
—a
1.60
0.95
0.80
1.75–2.00
1.65–2.00
1.65–2.00
1.75–2.00
1.65–2.45
1.70
0.75–0.85
0.75–0.90
0.75–0.90
0.75–0.85
0.60–0.90
0.90
0.60–0.75
0.65–0.80
0.65–0.80
0.65–0.75
0.50–0.80
0.75–0.80
1.80
1.80
1.80
0.85
0.85
0.85
0.80
0.80
0.80
1.67
1.80
0.95
0.85
0.90
0.80
1.80
1.80
0.85
0.85
0.80
0.80
1.80
1.67
2.15–3.00
2.00
0.85
0.90–0.95
0.55–0.80
0.80
0.80
0.90
0.40–0.75
0.75
DESIGN BASIS
Table 1.4
33
(Continued)
Type of Strength
(j) Welded connections
Groove welds
Tension or compression
Shear (welds)
Shear (base metal)
Arc spot welds
Welds in shear
Connected part in shear
Welds in Tension
Arc seam welds
Welds in shear
Connected part in shear
Top arc seam welds in shear
Fillet welds
Longitudinal loading (connected part)
Transverse loading (connected part)
Welds
Flare groove welds
Transverse loading (connected part)
Longitudinal loading (connected part)
Welds
Resistance welds
Rupture
(k) Bolted connections
Bearing strength
Shear strength of bolts
Tensile strength of bolts
Rupture
(l) Screw connections
Combined shear and pul-over
Combined shear and pul-out
Rupture
(m) Power-actuated fastener (PAF) connections
PAF in tension
pull-out in tension
pull over
PAF in shear
Bearing and tilting
Pull-out in shear
Rupture
(n) Rational engineering analysis for connections
ASD
Safety
Factor, Ω
LRFD
Resistance
Factor, 𝜙
LSD
Resistance
Factor, 𝜙
1.70
1.90
1.70
0.90
0.80
0.90
0.80
0.70
0.80
2.55
2.20–3.05
2.50–3.00
0.60
0.50–0.70
0.50–0.60
0.50
0.40–0.60
0.40–0.50
2.55
2.20
2.60
0.60
0.70
0.60
0.50
0.60
0.55
2.55–3.05
2.35
2.55
0.50–0.60
0.65
0.60
0.40–0.50
0.60
0.50
2.55
2.80
2.55
2.35
2.50
0.60
0.55
0.60
0.65
0.60
0.50
0.45
0.50
0.55
0.75
2.22–2.50
2.00
2.00
2.22
3.00
2.35
2.55
3.00
0.60–0.65
0.75
0.75
0.65
0.50
0.65
0.60
0.50
0.50–0.55
0.55
0.65
0.75
0.40
0.55
0.50
0.75
2.65
4.00
3.00
2.65
2.05
2.55
3.00
3.00
0.60
0.40
0.50
0.60
0.80
0.60
0.50
0.55
0.50
0.30
0.40
0.55
0.65
0.50
0.75
0.50
Note: This table is based on Chapters D through J and Appendices A and B of the 2016 edition of the North American Specification for the
Design of Cold-Formed Steel Structural Members.1.345
𝑎
See Appendix of the North American specification for the provisions applicable to Canada.
34
1
INTRODUCTION
Table 1.5 Equivalent Terms for the LRFD and LSD
Methods
LRFD (U.S. and Mexico)
LSD (Canada)
Design strength, 𝜙Rn
Flexural strength
Nominal load
Nominal strength, Rn
Required strength, Ru
Specification
Strength
Factored resistance, 𝜙Rn
Moment resistance
Specified load
Nominal resistance, Rn
Effect of factored loads, Rf
Standard
Resistance
method, a few different terms are defined in the North American Specification. Table 1.5 lists some equivalent terms used
for the LRFD and LSD methods.
1.8.3.1 Design Format for the LSD Method According
to Section B3.2.3 of the North American Specification for
LSD requirements, structural members and connections
shall be designed such that the factored resistance equals
or exceeds the effect of factored loads. The design shall be
performed in accordance with the following equation1.345 :
𝜙𝑅n ≥ 𝑅f
where Rf
Rn
ϕ
ϕRn
=
=
=
=
effect of factored loads
nominal resistance
resistance factor (see Table 1.4)
factored resistance
The above equation for the LSD method is similar to
Eq. (1.4) for the LRFD method.
1.8.3.2 Specified Loads, Load Factors, and Load Combinations for LSD Method The design requirements for specified loads, load factors, and load combinations are provided
in Appendix B of the North American specification. The
following discussions are applicable only to the LSD method
for use in Canada. All design tables referred in items (b)
and (c) are based on Appendix B of the 2007 edition of the
Specification1.345 :
a. Specified Loads. The load provisions contained in
the North American Specification are consistent with
the 2015 edition of the National Building Code of
Canada.1.438
b. Load Factors and Load Combinations for LSD.
According to Section B2, the effect of factored loads
for cold-formed steel design should be determined on
the basis of the applicable load combinations. Appropriate load factors shall be used for principal and
companion loads.
1.8.3.3 Factored Resistance 𝝓Rn In the development
of the LSD criteria, the resistance factors were determined
from a live-load factor of 1.50 and a dead-load factor of
1.25 according to the National Building Code of Canada.
The target safety indexes are 3.0 for members and 4.0 for
connections. In order to determine the loading for calibration, it was assumed that 80% of cold-formed steel is used
in panel form and 20% for structured members. An effective
load factor was arrived at by assuming the live-to-dead
load ratio and their relative frequencies of occurrence. For
nominal resistance Rn see Section 1.8.2.3. The same nominal
resistance equations are used in the main document of the
North American specification for the ASD, LRFD, and
LSD methods with specific Canadian design requirements
provided in Appendix B of the North American Specification
for some cases.
1.8.3.4 Design Using the LSD Method Even through the
load factors, load combinations, and resistance factors for
the LSD method are somewhat different as compared with
LRFD, the methodology and design procedures for these
two methods are the same. For the purpose of simplicity, the
discussions and design examples presented in this book are
based on the ASD and LRFD methods. The design procedures for the LRFD method can be used for the LSD method
with appropriate terms and 𝜙 factors.
1.8.4
Units of Symbols and Terms
The North American Specification was written so that any
compatible system of units may be used except where explicitly stated in the document. The unit systems adopted in the
North American Specification are:
1. U.S. customary units (force in kilopounds or kips and
length in inches)
2. SI units (force in newtons and length in millimeters)
3. MKS units (force in kilograms and length in centimeters)
The conversions of the U.S. customary units into SI units
and MKS units are given in parentheses throughout the text
of the Specification. Table 1.6 is a conversion table for
these units.
For the purpose of simplicity, only U.S. customary units
are used in this book for design examples.
SERVICEABILITY
Table 1.6
Length
Area
Force
Stress
Conversion Table1.417
To Convert
To
Multiply By
in.
mm
ft
m
in.2
mm2
ft2
m2
kip
kip
lb
lb
kN
kN
kg
kg
ksi
ksi
MPa
MPa
kg/cm2
kg/cm2
mm
in.
mm
ft
mm2
in.2
m2
ft2
kN
kg
N
kg
kip
kg
kip
N
MPa
kg/cm2
ksi
kg/cm2
ksi
MPa
25.4
0.03937
0.30480
3.28084
645.160
0.00155
0.09290
10.7639
4.448
453.5
4.448
0.4535
0.2248
101.96
0.0022
9.808
6.895
70.30
0.145
10.196
0.0142
0.0981
1.9 SERVICEABILITY
Section 1.8 dealt only with the strength limit state. A structure should also be designed for the serviceability limit state
as required by the AISI Specification.1.417
Serviceability limit states are conditions under which
a structure can no longer perform its intended functions. Strength considerations are usually not affected by
35
serviceability limit states. However, serviceability criteria
are essential to ensure functional performance and economy
of design.
The conditions that may require serviceability limits are
listed in the AISI Commentary as follows1.310,1.417 :
1. Excessive deflections or rotations that may affect
the appearance or functional use of the structure and
deflections which may cause damage to nonstructural
elements.
2. Excessive vibrations that may cause occupant discomfort of equipment malfunctions.
3. Deterioration over time, which may include corrosion
or appearance considerations.
When checking serviceability, the designer should
consider appropriate service loads, the response of the
structure, and the reaction of building occupants.
Service loads that may require consideration include
static loads, snow or rain loads, temperature fluctuations,
and dynamic loads from human activities, wind-induced
effects, or the operation of equipment. The service loads are
actual loads that act on the structure at an arbitrary point in
time. Appropriate service loads for checking serviceability
limit states may only be a fraction of the nominal loads.
The response of the structure to service loads can normally
be analyzed assuming linear elastic behavior. Serviceability
limits depend on the function of the structure and on the
perceptions of the observer. Unlike the strength limit states,
general serviceability limits cannot be specified that are
applicable to all structures. The North American Specification does not contain explicit requirements; however, guidance is generally provided by the applicable building code.
In the absence of specific criteria, guidelines may be found
in Refs.1.439, 3.164–3.167, and 3.202–3.206.
CHAPTER 2
Materials Used in Cold-Formed
Steel Construction
2.1 GENERAL REMARKS
Because material properties play an important role in the
performance of structural members, it is important to be
familiar with the mechanical properties of the steel sheets,
strip, plates, or flat bars generally used in cold-formed steel
construction before designing this type of steel structural
member. In addition, since mechanical properties are greatly
affected by temperature, special attention must be given by
the designer for extreme conditions below −30∘ F (−34∘ C)
and above 200∘ F (93∘ C).
Seventeen steels are specified in the current edition of
the North American Specification1.417 for structural applications. These steels are identified in ASTM standards for
sheet material as SS or, in the case of high-strength, low-alloy
steels, as HSLAS or HSLAS-F steels:
ASTM A36, Carbon Structural Steel
ASTM A242, High-Strength Low-Alloy Structural Steel
ASTM A283, Low and Intermediate Tensile Strength Carbon
and Steel Plates
ASTM A500, Cold-Formed Welded and Seamless Carbon
Steel Structural Tubing in Round and Shapes
ASTM A529, High-Strength Carbon-Manganese Steel of
Structural Quality
ASTM A572, High-Strength Low-Alloy Columbium-Vanadium Structural Steel
ASTM A588, High-Strength Low-Alloy Structural Steel
with 50 ksi (345 MPa) Minimum Yield Point to 4 in.
(100 mm) Thick
ASTM A606, Steel, Sheet and Strip, High-Strength,
Low-Alloy, Hot-Rolled and Cold-Rolled, with Improved
Atmospheric Corrosion Resistance
ASTM A653, Steel Sheet, Zinc-Coated (Galvanized) or
Zinc-Iron Alloy-Coated (Galvannealed) by the Hot-Dip
Process
ASTM A792, Steel Sheet, 55% Aluminum-Zinc AlloyCoated by the Hot-Dip Process
ASTM A847, Cold-Formed Welded and Seamless HighStrength, Low-Alloy Structural Tubing with Improved
Atmospheric Corrosion Resistance
ASTM A875, Steel Sheet, Zinc-5% Aluminum Alloy-Coated
by the Hot-Dip Process
ASTM A1003, Steel Sheet, Carbon, Metallic- and
Nonmetallic-Coated for Cold-Formed Framing Members
ASTM A1008, Steel, Sheet; Cold-Rolled, Carbon, Structural, High-Strength Low-Alloy, High-Strength Low-Alloy
with Improved Formability, Solution Hardened, and Bake
Hardenable
ASTM A1011, Steel, Sheet and Strip, Hot-Rolled, Carbon,
Structural, High-Strength Low-Alloy and High-Strength
Low-Alloy with Improved Formability
ASTM A1039, Steel, Sheet, Hot-Rolled, Carbon, Commercial and Structural, Produced by the Twin-Roll Casting
Process
ASTM A1063, Steel, Sheet, Twin-Roll Cast, Zinc-Coated,
(Galvanized) by the Hot-Dip Process
See Table 2.1 for the mechanical properties of these 17
steels.
In addition to the above-listed steels, other steel sheet,
strip, or plate may also be used for structural purposes
provided such material conforms to the chemical and
mechanical requirements of one of the listed specifications
or other published specification that establishes its properties
and suitability1.417 for the type of application. Specification
Section A3.2 includes additional specific requirements for
using other steels.
From a structural standpoint, the most important properties of steel are as follows:
1.
2.
3.
4.
5.
6.
7.
8.
Yield stress
Tensile strength
Stress–strain characteristics
Modulus of elasticity, tangent modulus, and shear
modulus
Ductility
Weldability
Fatigue strength
Toughness
In addition, formability and durability are also important
properties for thin-walled cold-formed steel structural
members.
37
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
Table 2.1 Mechanical Properties of Steels Referred to in Section A3.1 of the AISI North American Specification1.417,1.428
Carbon structural steel
High-strength, low-alloy
structural steel
A36
A242
Low- and
intermediate-tensile-strength
carbon steel plates
A283
A
B
C
D
A500
Round tubing
A
B
C
D
Shaped tubing
A
B
C
D
A529 Gr. 50
Cold-formed welded and
seamless carbon steel
structural tubing in rounds
and shapes
High-strength
carbon–manganese steel
High-strength, low-alloy
columbium–vanadium
steels of structural quality
High-strength, low-alloy
structural steel with 50 ksi
minimum yield point
38
Fu /Fy
58–80
70
1.61
1.40
23
21
46
67
1.46
21
—
—
—
—
24
27
30
33
45–60
50–65
55–75
60–80
1.88
1.85
1.83
1.82
30
28
25
23
—
—
—
—
33
42
46
36
45
58
62
58
1.36
1.38
1.35
1.61
25
23
21
23
—
—
—
—
—
39
46
50
36
50
45
58
62
58
70–100
1.15
1.26
1.24
1.61
1.40
25
23
21
23
21
—
—
—
—
—
—
4 in. and under
55
42
50
55
60
65
50
70–100
60
65
70
75
80
70
1.27
1.43
1.30
1.27
1.25
1.23
1.40
20
24
21
20
18
17
21
ASTM
Designation
Steel Designation
55
A572 Gr. 42
50
55
60
65
A588
Minimum
Tensile
Strength Fu
(ksi)
Minimum
Elongation (%)
in 2 in. Gage
Length
3
4
3
4
Thickness
(in.)
Minimum Yield
Stress Fy
(ksi)
—
and under
36
50
to 1 12
Hot-rolled and cold-rolled
high-strength, low-alloy
steel sheet and strip with
improved corrosion
resistance
Zinc-coated or zinc–iron
alloy-coated steel sheet
55% aluminum–zinc
alloy–coated steel sheet by
the hot-dip process
Cold-formed welded and
seamless high-strength,
low-alloy structural tubing
with improved atmospheric
corrosion resistance
A606
Hot rolled as
rolled
Hot rolled
annealed or
normalized
Cold rolled
A653 SS
33
37
40
50 Class 1
50 Class 3
50 Class 4
55
HSLAS
40
50
55 Class 1
55 Class 2
60
70
80
HSLAS-F
40
50
55 Class 1
55 Class 2
60
70
80
A792 SS
Gr. 33
37
40
50 Class 1
50 Class 4
A847
—
50
70
1.40
22
—
45
65
1.44
22
—
45
65
1.44
22
—
—
—
—
—
—
—
33
37
40
50
50
50
55
45
52
55
65
70
60
70
1.36
1.41
1.38
1.30
1.40
1.20
1.27
20
18
16
12
12
12
11
—
—
—
—
—
—
—
40
50
55
55
60
70
80
50
60
70
65
70
80
90
1.25
1.20
1.27
1.18
1.17
1.14
1.13
22
20
16
18
16
12
10
—
—
—
—
—
—
—
40
50
55
55
60
70
80
50
60
70
65
70
80
90
1.25
1.20
1.27
1.18
1.17
1.14
1.13
24
22
18
20
18
14
12
—
—
—
—
—
—
33
37
40
50
50
50
45
52
55
65
60
70
1.36
1.41
1.38
1.30
1.20
1.40
20
18
16
12
12
19
(continuous)
39
Table 2.1 (Continued)
Steel Designation
Zinc–5% aluminum
alloy–coated steel sheet by
the hot-dip process
Metallic- and
nonmetallic-coated carbon
steel sheet
Cold-rolled steel sheet,
carbon structural,
high-strength, low-alloy
and high-strength,
low-alloy with improved
formability
40
ASTM
Designation
A875 SS
Gr. 33
37
40
50 Class 1
50 Class 3
HSLAS
Gr. 50
60
70
80
HSLAS-F
Gr. 50
60
70
80
A1003 ST
Gr. 33 H
37 H
40 H
50 H
A1008
SS:
Gr. 25
30
33 Types 1 & 2
40 Types 1 & 2
HSLAS:
Gr. 45 Class 1
45 Class 2
50 Class 1
50 Class 2
Minimum
Tensile
Strength Fu
(ksi)
Fu /Fy
Minimum
Elongation (%)
in 2 in. Gage
Length
45
52
55
65
70
1.36
1.41
1.38
1.30
1.40
20
18
16
12
12
50
60
70
80
60
70
80
90
1.20
1.17
1.14
1.13
20
16
12
10
—
—
—
—
50
60
70
80
60
70
80
90
1.20
1.17
1.14
1.13
22
18
14
12
—
—
—
—
33
37
40
50
See note
See note
See note
See note
1.08
1.08
1.08
1.08
10
10
10
10
—
—
—
—
25
30
33
40
42
45
48
52
1.68
1.50
1.45
1.30
26
24
22
20
—
—
—
—
45
45
50
50
60
55
65
60
1.33
1.22
1.30
1.20
22
22
20
20
Thickness
(in.)
Minimum Yield
Stress Fy
(ksi)
—
—
—
—
—
33
37
40
50
50
—
—
—
—
Hot-rolled steel sheet and
strip, carbon, structural,
high-strength, low-alloy
with improved corrosion
resistance
55 Class 1
55 Class 2
60 Class 1
60 Class 2
65 Class 1
65 Class 2
70 Class 1
70 Class 2
HSLAS-F:
Gr. 50
60
70
80
A1011
SS:
Gr. 30
33
36 Type 1
36 Type 2
40
45
50
55
HSLAS:
Gr. 45 Class 1
45 Class 2
50 Class 1
50 Class 2
55 Class 1
55 Class 2
60 Class 1
60 Class 2
65 Class 1
65 Class 2
70 Class 1
70 Class 2
—
—
—
—
—
—
—
—
55
55
60
60
65
65
70
70
70
65
75
70
80
75
85
80
1.27
1.18
1.25
1.17
1.23
1.15
1.21
1.14
18
18
16
16
15
15
14
14
—
—
—
—
50
60
70
80
60
70
80
90
1.20
1.17
1.14
1.13
22
18
16
14
—
—
—
—
—
—
—
—
30
33
36
36
40
45
50
55
49
52
53
58–80
55
60
65
70
1.63
1.58
1.47
1.61
1.38
1.33
1.30
1.27
21–25
18–23
17–22
16–21
15–21
13–19
11–17
9–15
—
—
—
—
—
—
—
—
—
—
—
—
45
45
50
50
55
55
60
60
65
65
70
70
60
55
65
60
70
65
75
70
80
75
85
80
1.33
1.22
1.30
1.20
1.27
1.18
1.25
1.17
1.23
1.15
1.21
1.14
23–25
23–25
20–22
20–22
18–20
18–20
16–18
16–18
14–16
14–16
12–14
12–14
(continuous)
41
Table 2.1 (Continued)
Steel Designation
Hot-rolled, carbon,
commercial and structural
steel sheet
Steel sheet in coils and cut
lengths produced by the
twin-roll casting process.
42
ASTM
Designation
HSLAS-F:
Gr. 50
60
70
80
A1039
Gr. 40
50
55
60
70
80
A1063/A1063M-11 SS
Gr. 33
37
40
45
50
55
60
70
80 Class 1
80 Class 2
HSLAS
Class1
45
50
55
60
65
70
80
Minimum
Tensile
Strength Fu
(ksi)
Fu /Fy
Minimum
Elongation (%)
in 2 in. Gage
Length
60
70
80
90
1.20
1.17
1.14
1.13
22–24
20–22
18–20
16–18
40
50
55
60
70
80
55
65
70
70
80
90
1.38
1.30
1.27
1.17
1.14
1.13
15–20
11–16
9–14
8–13
7–12
6–11
—
—
—
—
—
—
—
—
—
—
33
37
40
45
50
55
60
70
80
80
45
52
55
60
65
70
70
80
90
83
1.36
1.41
1.38
1.33
1.30
1.27
1.17
1.14
1.13
1.04
20
18
15
13
11
9
8
7
6
—
—
—
—
—
—
—
—
45
50
55
60
65
70
80
60
65
70
75
80
85
95
1.33
1.30
1.27
1.25
1.23
1.21
1.19
18
15
13
11
11
8
7
Thickness
(in.)
Minimum Yield
Stress Fy
(ksi)
—
—
—
—
50
60
70
80
—
—
—
—
—
—
HSLAS
Class 2
45
50
55
60
65
70
80
—
—
—
—
—
—
—
45
50
55
60
65
70
80
55
60
65
70
75
80
90
1.22
1.20
1.18
1.17
1.15
1.14
1.13
18
15
13
11
11
8
7
Notes:
1. The tabulated values are based on ASTM standards.2.1
2. 1 in. = 25.4 mm; 1 ksi = 6.9 MPa = 70.3 kg/cm2 .
3. Structural Grade 80 of A653, A875, and A1008 steel and Grade 80 of A792 are allowed in the North American Specification under special conditions. For these
grades, Fy = 80 ksi, Fu = 82 ksi, and elongations are unspecified. See North American Specification for the reduction of yield stress and tensile strength.
4. For Type L of A1003 steel, the minimum tensile strength is not specified. The ratio of tensile strength to yield stress shall not be less than 1.08. Type L of A1003 steel
is allowed in the North American Specification under special conditions.
5. For A1011 steel, the specified minimum elongation in 2 in. of gage length varies with the thickness of steel sheet and strip.
6. For A1039 steel, the larger specified minimum elongation is for the thickness under 0.078–0.064 in. The smaller specified minimum elongation is for the thickness
under 0.064–0.027 in. For Grades 55 and higher that do not meet the requirement of 10% elongation, Sections A3.1.2 and A3.1.3 of the North American Specification
shall be used.
43
44
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
2.2 YIELD STRESS, TENSILE STRENGTH,
AND STRESS–STRAIN CURVE
2.2.1
Yield Stress Fy and Stress–Strain Curve
The strength of cold-formed steel structural members
depends on the yield point or yield strength of steel, except
in connections and in those cases where elastic local buckling or overall buckling is critical. In the 2016 edition of the
North American Specification and in this book, the generic
term yield stress is used to denote either yield point or yield
strength. As indicated in Table 2.1, the yield stresses of
steels listed in the North American specification range from
24 to 80 ksi (165 to 552 MPa or 1687 to 5624 kg/cm2 ).
There are two general types of stress–strain curves,
as shown in Fig. 2.1. One is of the sharp-yielding type
(Fig. 2.1a) and the other is of the gradual-yielding type
(Fig. 2.1b). Steels produced by hot rolling are usually
sharp yielding. For this type of steel, the yield stress is
defined by the level at which the stress–strain curve becomes
horizontal. Steels that are cold reduced or otherwise cold
worked show gradual yielding. For gradual-yielding steel,
the stress–strain curve is rounded out at the “knee” and the
yield stress is determined by either the offset method or
the strain-underload method.2.2,2.3
Figure 2.2 Determination of yield stress for gradual-yielding
steel: (a) offset method; (b) strain-underload method.
In the offset method, the yield stress is the stress corresponding to the intersection of the stress–strain curve and
a line parallel to the initial straight-line portion offset by a
specified strain. The offset is usually specified as 0.2%, as
shown in Fig. 2.2a. This method is often used for research
work and for mill tests of stainless and alloy steels. In the
strain-underload method, the yield stress is the stress corresponding to a specified elongation or extension under load.
The specified total elongation is usually 0.5%, as shown in
Fig. 2.2b. This method is often used for mill tests of sheet
or strip carbon and low-alloy steels. In many cases, the yield
stresses determined by these two methods are similar.
2.2.2
Figure 2.1 Stress–strain curves of carbon steel sheet or strip:
(a) sharp yielding; (b) gradual yielding.
Tensile Strength
The tensile strength of steel sheets or strip used for
cold-formed steel sections has little direct relationship
to the design of such members. The load-carrying capacities
of cold-formed steel flexural and compression members
are usually limited by yield stress or buckling stresses that
are less than the yield stress of steel, particularly for those
compression elements having relatively large flat-width
ratios and for compression members having relatively large
DUCTILITY
slenderness ratios. The exceptions are tension members and
connections, the strength of which depends not only on the
yield stress but also on the tensile strength of the material.
For this reason, in the design of tension members and
connections where stress concentration may occur and the
consideration of ultimate strength in the design is essential,
the North American Specification includes special design
provisions to ensure that adequate safety is provided for the
ultimate strengths of tension members and connections. As
indicated in Table 2.1, the minimum tensile strengths of the
steels listed in the North American Specification range from
42 to 100 ksi (290 to 690 MPa or 2953 to 7030 kg/cm2 ). The
ratios of tensile strength to yield stress, Fu /Fy , range from
1.08 to 1.88. Previous studies indicated that the effects of
cold work on cold-formed steel members depend largely
upon the spread between tensile strength and yield stress of
the virgin material.
2.3.3
2.3.1 Modulus of Elasticity E
The strength of members that fail by buckling depends
not only on the yield stress but also on the modulus of
elasticity E and the tangent modulus Et . The modulus of
elasticity is defined by the slope of the initial straight portion
of the stress–strain curve. The measured values of E on
the basis of the standard methods2.4,2.5 usually range from
29,000 to 30,000 ksi (200 to 207 GPa or 2.0 × 106 to 2.1 ×
106 kg/cm2 ).2.76,2.77 A value of 29,500 ksi (203 GPa or
2.07 × 106 kg/cm2 ) has been used by AISI in its specifications for design purposes since 19462.78 and is retained in
the North American Specification.1.345 This value is slightly
higher than 29,000 ksi (200 GPa or 2.0 × 106 kg/cm2 )
currently used in the AISC specification.1.411
2.3.2 Tangent Modulus Et
The tangent modulus is defined by the slope of the stress–
strain curve at any point, as shown in Fig. 2.1b. For
sharp yielding, Et = E up to the yield stress, but with gradual
yielding, Et = E only up to the proportional limit. Once the
stress exceeds the proportional limit, the tangent modulus Et
becomes progressively smaller than the initial modulus of
elasticity. For this reason, for moderate slenderness the
sharp-yielding steels have larger buckling strengths than
gradual-yielding steels. Various buckling provisions of
the North American Specification have been written for
gradual-yielding steels whose proportional limit is usually
not lower than about 70% of the specified minimum yield
stress.
Shear Modulus G
By definition, shear modulus G is the ratio between the shear
stress and the shear strain. It is the slope of the straight-line
portion of the shear stress–strain curve. Based on the theory
of elasticity, the shear modulus can be computed by the
following equation2.52 :
𝐺=
𝐸
2(1 + 𝜇)
(2.1)
where E is the tensile modulus of elasticity and 𝜇 is
Poisson’s ratio. By using E = 29,500 ksi (203 GPa or 2.07 ×
106 kg/cm2 ) and μ = 0.3 for steel in the elastic range, the
value of shear modulus G is taken as 11,300 ksi (78 GPa
or 794 × 103 kg/cm2 ) in the North American Specification.
This G value is used for computing the torsional buckling
stress for the design of beams, columns, and wall studs.
2.4
2.3 MODULUS OF ELASTICITY, TANGENT
MODULUS, AND SHEAR MODULUS
45
DUCTILITY
Ductility is defined as the extent to which a material can
sustain plastic deformation without rupture. It is not only
required in the forming process but also needed for plastic
redistribution of stress in members and connections, where
stress concentration would occur.
Ductility can be measured by (1) a tension test, (2) a bend
test, or (3) a notch test. The permanent elongation of a tensile
test specimen is widely used as the indication of ductility. As
shown in Table 2.1, for the customary range in thickness of
steel sheet, strip, or plate used for cold-formed steel structural
members, the minimum elongation in 2 in. (50.8 mm) of gage
length varies from 10 to 30%.
The ductility criteria and performance of low-ductility
steels for cold-formed steel members and connections
have been studied by Dhalla, Winter, and Errera at Cornell
University.2.6–2.9 It was found that the ductility measurement in a standard tension test includes (1) local ductility
and (2) uniform ductility. Local ductility is designated
as the localized elongation at the eventual fracture zone.
Uniform ductility is the ability of a tension coupon to
undergo sizable plastic deformations along its entire length
prior to necking. This study also revealed that for the
different ductility steels investigated the elongation in 2 in.
(50.8 mm) of gage length did not correlate satisfactorily
with either the local or the uniform ductility of the material.
In order to be able to redistribute the stresses in the plastic
range to avoid premature brittle fracture and to achieve
full net-section strength in a tension member with stress
concentrations, it was suggested that (1) the minimum local
elongation in 12 in. (12.7 mm) of gage length of a standard
tension coupon including the neck be at least 20%; (2) the
minimum uniform elongation in 3 in. (76.2 mm) of gage
46
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
length minus the elongation in 1 in. (25.4 mm) of gage
length containing neck and fracture be at least 3%; and
(3) the tensile strength–yield stress ratio Fu /Fy be at least
1.05.2.9 In this method, the local and uniform elongations are
established in accordance with the AISI Standard Method
for Determining the Uniform and Local Ductility of Carbon
and Low-Alloy Steels, AISI S9032.106 or the following
procedure:
1. Tensile coupons are prepared according to ASTM
designation A370,2.2 except that the length of the
central reduced section [ 12 in. (12.7 mm) uniform
width] of the coupon is at least 3 12 in. (89 mm). Gage
lines are scribed at 12 -in. (12.7-mm) intervals along the
entire length of the coupon.
2. Upon completion of the coupon test, the following two
permanent plastic deformations are measured:
a. Linear elongation in 3 in. (76.2 mm) of gage length
e3 , in inches, including the fractured portion, preferably having occurred near the middle third of the
gage length
b. Linear elongation in 1 in. (25.4 mm) of gage length
e1 , in inches, including the fractured portion
3. The local elongation 𝜀l and the uniform elongation 𝜀u
(both in percents) are computed as follows:
𝜀l = 50(5𝑒1 − 𝑒3 )
(2.2)
𝜀u = 50(𝑒3 − 𝑒1 )
(2.3)
For a rough differentiation of low-ductility from higher
ductility steels, Ref. 2.9 suggests that (1) the minimum
elongation in 2 in. (50.8 mm) of gage length be at least 7%
and (2) the minimum tensile strength–yield stress ratio be at
least 1.05.
These research findings and suggestions have received
careful review and consideration during the revision of
the AISI specification in 1980. Section A3.3.1 of the 1986
edition of the AISI specification requires that the tensile
strength–yield stress ratio Fu /Fy be not less than 1.08 and
the elongation in 2 in. (50.8 mm) of gage length be not less
than 10% for steels not listed in the specification and used
for structural members and connections. These requirements
are slightly higher than the corresponding values suggested
in Ref. 2.9 because the AISI Specification refers to the
conventional tensile tests.2.2 These minimum requirements
would ensure adequate ductility. When 8 in. (203 mm) gage
length is used, the minimum elongation is 7%. This specified
elongation was derived from the conversion formula given
in Section S30 of Ref. 2.2. The above-discussed ductility
requirements were contained in the 1996 edition of the AISI
Specification and the North American Specification.
In the 1980s, a low-strain-hardening ductile (LSHD) steel
that has reasonable elongation but very low Fu /Fy ratio
was developed for building purlins and girts. The results of
coupon tests reported in Ref. 2.10 show that, even though
the Fu /Fy ratio of this type of steel is less than the specified
minimum ratio of 1.08 and the elongation in 2 in. (50.8 mm)
of gage length may be less than 10%, the local elongation
in 12 in. (12.7 mm) of gage length across the fracture exceeds
20%, and the uniform elongation outside the fracture exceeds
3%. On the basis of a limited number of tests, the same
publication indicates that the AISI design provisions for flexural members with regard to effective width, lateral buckling,
and nominal bending strength based on initiation of yielding
are applicable to beams fabricated from LSHD steel. Consequently, the AISI Specification permitted the use of such
material in the 1989 Addendum and the 1996 edition for the
design of purlins and girts which support roof deck or wall
panels provided that the steel can satisfy the requirements of
Section A3.3.1.1.4,1.314 The same requirements are retained
in Section A3.2.1 of the North American Specification.1.417
When purlins and girts are subject to combined axial load and
bending, the applied axial load P should not exceed 15% of
the allowable axial load, Pn /Ωc , for the ASD method because
the use of such a LSHD steel for columns and beam–columns
would require additional study. For the LRFD and LSD
methods, P/𝜙c Pn ≤ 0.15. In the above expressions, Pn is the
nominal column strength, Ωc is the safety factor for column
design, P is the column load based on the factored loads, and
𝜙c is the resistance factor.
The required ductility for cold-formed steel structural
members depends mainly on the type of application and the
suitability of the material. The same amount of elongation
that is considered necessary for individual framing members
may not be needed for such applications as multiple-web
roof panels and siding, which are formed with large radii
and are not used in service with highly stressed connections
or other stress raisers. For this reason, the performance of
structural steel (SS) Grade 80 of A653, Grade 80 of A792,
SS Grade 80 of A875, and SS Grade 80 of A1008 steels
used for roofing, siding, and similar applications has been
found satisfactory, even though for these grades of steels the
Fu /Fy ratios are less than the North American requirements
(1.03 vs. 1.08) and elongations are unspecified. The use of
such steels, which do not meet the North American ductility
requirements of Sections A3.1.1 and A3.1.2, is permitted
by Section A2.3.3 of the Specification for multiple-web
configurations provided that (1) the yield stress Fy used
for beam design is taken as 75% of the specified minimum
yield stress or 60 ksi (414 MPa or 4218 kg/cm2 ), whichever
is less, and (2) the tensile strength Fu used for connection
design is taken as 75% of the specified minimum tensile
WELDABILITY
strength or 62 ksi (427 MPa or 4359 kg/cm2 ), whichever
is less.
In the past, studies have been made to determine the
ductility of SS Grade 80 of A653 steel and the performance of flexural members and connections using such a
low-ductility steel.2.53–2.61 Based on the research findings
reported by Wu, Yu, LaBoube, and Pan in Refs. 2.53, 2.54,
2.79, 2.80, and 3.124, the following exception clause was
added in Section A3.3.2 of the supplement to the 1996
edition of the AISI specification for determining the flexural
strength of multiple-web decks using SS Grade 80 of A653
steel and similar low-ductility steels.1.333 The same requirements are retained in Section A3.1.3 of the North American
Specification1.417 :
A reduced specified minimum yield stress, Rb Fsy , is used
for determining the nominal flexural strength [resistance] in
Chapter F, for which the reduction factor, Rb , is determined in
accordance with (a) or (b):
(a) Stiffened and Partially Stiffened Compression Flanges
For w/t ≤ 0.067E/Fsy
𝑅b = 1.0
(2.4)
For 0.067E/Fsy < w/t < 0.974E/Fsy
[
]0.4
wF sy
𝑅b = 1 − 0.26
− 0.067
tE
(2.6)
(b) Unstiffened Compression Flanges
For w/t ≤ 0.0173E/Fsy
For a concentrically loaded compression member with a closedbox section that is made of steel with a specified minimum
elongation between three to ten percent, inclusive, a reduced
radius of gyration (Rr )(r) shall be used when the value of the
effective length KL is less than 1.1L0 , where L0 is given by
Eq. 2.9, and Rr is given by Eq. 2.10.
√
𝑅b = 1.0
For 0.0173E/Fsy < w/t ≤ 60
the provisions of Section K2.1 of the North American
Specification. Design strengths based on these tests should
not exceed the strengths calculated according to Chapters
C through J, Appendices A and B and Appendices 1 and
2 of the North American Specification using the specified
minimum yield stress Fy and the specified minimum tensile
strength Fu .
In the 2000s, research was conducted by Yang, Hancock,
and Rasmussen at the University of Sydney2.81,2.82 to study
the use of low-ductility steel (Grade 80 of A792 steel) for
concentrically loaded closed-box compression members. It
was found that for the design of such compression members
90% of the specified minimum yield stress can be used as Fy
for determining the critical column buckling stress Fn and
the slenderness factor 𝜆c in Chapter 5. In addition, the elastic
column flexural buckling stress Fe should be determined by
using a reduced radius of gyration (Rr )(r) to allow the interaction of local and flexural buckling of closed-box sections.
The reduction factor Rr varies from 0.65 at KL = 0 to 1.0 at
KL = 1.1L0 , in which L0 is a specific length at which the local
buckling stress equals the column flexural buckling stress.
Consequently, in 2004, the following design provisions
were added in Section A2.3.2 of the Specification as Exception 2 and retained in Section E2.1.11.417 in 2016 to permit
the use of low-ductility steels for concentrically loaded
closed-box compression members:
(2.5)
For 0.974E/Fsy ≤ w/t ≤ 500
𝑅b = 0.75
(2.7)
𝐿0 = 𝜋r
√
𝑅b = 1.079 − 0.6
wF sy
tE
47
𝐸
𝐹crl
𝑅r = 0.65 +
(2.8)
where E = modulus of elasticity
Fsy = specified minimum yield stress as specified
in Section A3.3.1 ≤ 80 ksi (550 MPa, or
5620 kg/cm2 )
t = thickness of section
w = flat width of compression flange
For steels used in composite slabs, the requirements of
ANSI/SDI C shall be followed exclusively.1.437
Alternatively, the suitability of such steels for any configuration shall be demonstrated by load tests according to
0.35(KL)
≤ 1.0
1.1𝐿0
(2.9)
(2.10)
where Fcrl = minimum critical buckling stress for section
determined by Eq. (3.17)
r = radius of gyration of full unreduced cross section
KL = effective length
L0 = length at which local buckling stress equals
flexural buckling stress
Rr = reduction factor
2.5
WELDABILITY
Weldability refers to the capacity of a steel to be welded
into a satisfactory, crack-free, sound joint under fabrication
conditions without difficulty. It is basically determined by
48
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
the chemical composition of the steel and varies with types
of steel and the welding processes used.
“Structural Welding Code—Sheet Steel” (ANSI/AWS
D1.3) provides welding processes for shielded metal arc
welding (SMAW), gas metal arc welding (GMAW), flux
cored arc welding (FCAW), and submerged arc welding
(SAW).
The design of welded connections is discussed in
Section 8.3.
2.6
FATIGUE STRENGTH AND TOUGHNESS
Fatigue strength is important for cold-formed steel structural
members subjected to vibratory, cyclical, or repeated loads.
The basic fatigue property is the fatigue limit obtained from
the S–N diagram (S being the maximum stress and N being
the number of cycles to failure), which is established by tests.
In general, the fatigue–tensile strength ratios for steels range
from 0.35 to 0.60. This is for plain specimens; the fatigue
strength of actual members is often governed by details or
connections. For cold-formed steel members, the influence of
repeated and cyclic loading on steel sections and connections
has been studied at the University of New Mexico, the United
States Steel Research Laboratory,2.11–2.13 the University of
Manitoba,2.62 and elsewhere.2.87–2.91
In 2001, the AISI Committee on Specifications developed
the fatigue design provisions on the basis of Klippstein’s
research work (Refs. 2.11, 2.12, 2.83–2.85) as summarized
by LaBoube and Yu in Ref. 2.72 and the AISC Specification. These design criteria for cold-formed steel members
and connections subjected to cyclic loading are included in
Chapter M of the North American Specification.1.417
In general, the occurrence of full wind or earthquake loads
is too infrequent to warrant consideration in fatigue design.
Therefore, Section M1 of the North American Specification
states that evaluation of fatigue resistance is not required for
wind load applications in buildings. In addition, evaluation
of fatigue resistance is not required if the number of cycles
of applications of live load is less than 20,000.
When fatigue design is essential, cold-formed steel
members and connections should be checked in accordance
with Chapter M of the North American Specification with
due consideration given to (1) member of cycles of loading,
(2) type of member and connection detail, and (3) stress
range of the connection detail.2.86
Toughness is the extent to which a steel absorbs energy
without fracture. It is usually expressed as energy absorbed
by a notched specimen in an impact test. Additionally, the
toughness of a smooth specimen under static loads can be
measured by the area under the stress–strain diagram. In
general, there is not a direct relation between the two types
of toughness.
2.7 INFLUENCE OF COLD WORK ON
MECHANICAL PROPERTIES OF STEEL
The mechanical properties of cold-formed steel sections are
sometimes substantially different from those of the steel
sheet, strip, plate, or bar before forming. This is because the
cold-forming operation increases the yield stress and tensile
strength and at the same time decreases the ductility. The
percentage increase in tensile strength is much smaller than
the increase in yield stress, with a consequent marked reduction in the spread between yield stress and tensile strength.
Since the material in the corners of a section is cold worked
to a considerably higher degree than the material in the flat
elements, the mechanical properties are different in various
parts of the cross section. Figure 2.3 illustrates the variations
of mechanical properties from those of the parent material
at the specific locations in a channel section and a joist
chord after forming tested by Karren and Winter.2.14 For this
reason, buckling or yielding always begins in the flat portion
due to the lower yield stress of the material. Any additional
load applied to the section will spread to the corners.
Results of investigations conducted by Winter, Karren,
Chajes, Britvec, and Uribe2.14–2.17 on the influence of cold
work indicate that the changes of mechanical properties due
to cold work are caused mainly by strain hardening and
strain aging, as illustrated in Fig. 2.4,2.15 in which curve
A represents the stress–strain curve of the virgin material.
Curve B is due to unloading in the strain-hardening range,
curve C represents immediate reloading, and curve D is the
stress–strain curve of reloading after strain aging. It is interesting to note that the yield stresses of both curves C and
D are higher than the yield stress of the virgin material and
that the ductilities decrease after strain hardening and strain
aging. In addition to strain hardening and strain aging, the
changes in mechanical properties produced by cold work are
also caused by the direct and inverse Bauschinger effect.
The Bauschinger effect refers to the fact that the longitudinal compression yield stress of the stretched steels is
smaller than the longitudinal tension yield stress, as shown
in Fig. 2.5a.2.17 The inverse Bauschinger effect produces
the reverse situation in the transverse direction, as shown in
Fig. 2.5b.2.17
The effects of cold work on the mechanical properties
of corners usually depend on (1) the type of steel, (2) the
type of stress (compression or tension), (3) the direction
of stress with respect to the direction of cold work (transverse or longitudinal), (4) the Fu /Fy ratio, (5) the inside
radius–thickness ratio (R/t), and (6) the amount of cold work.
In general, the increase of the yield stress is more pronounced
for hot-rolled steel sheets than for cold-reduced sheets.
Among the above items, the Fu /Fy and R/t ratios are the
most important factors to affect the change in mechanical
INFLUENCE OF COLD WORK ON MECHANICAL PROPERTIES OF STEEL
49
Figure 2.3 Effect of cold work on mechanical properties in cold-formed steel sections2.14 :
(a) channel section; (b) joist chord.
properties of formed sections. Virgin material with a large
Fu /Fy ratio possesses a large potential for strain hardening.
Consequently, as the Fu /Fy ratio increases, the effect of cold
work on the increase in the yield stress of steel increases.
Small inside radius–thickness ratios R/t correspond to a large
degree of cold work in a corner, and therefore, for a given
material, the smaller the R/t ratio, the larger the increase in
yield stress (Fig. 2.6).
Investigating the influence of cold work, Karren derived
the following equations for the ratio of corner yield stress to
virgin yield stress2.16 :
𝐹yc
𝐹yv
=
𝐵c
(𝑅∕𝑡)𝑚
where
𝐹
𝐵c = 3.69 uv − 0.819
𝐹yv
𝑚 = 0.192
(2.11)
(
𝐹uv
− 0.068
𝐹yv
𝐹uv
𝐹yv
)
(2.12)
(2.13)
50
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
properties, the tensile yield stress of the full section may be
approximated by using a weighted average as follows:
𝐹ya = CF yc + (1 − 𝐶)𝐹yf ≤ 𝐹uv
(2.14)
where Fya = average full-section tensile yield stress
Fyc = average tensile yield stress of corners,
= Bc Fyv /(R/t)m
Fyf = average tensile yield stress of flats
C = ratio of corner area to total cross-sectional area
Figure 2.4 Effects of strain hardening and strain aging on stress–
strain characteristics.2.15
Good agreement between the computed and the tested
stress–strain characteristics for a channel section and a joist
chord section is shown in Figs. 2.7 and 2.8.
Over the past four decades, additional studies have been
conducted by numerous investigators. These investigations
dealt with the cold-formed sections having large R/t ratios
and with thick materials. They also considered residual stress
distribution, simplification of design methods, and other
related subjects. For details, see Refs. 2.18–2.29, 2.73, 2.74,
and 2.93. References 2.63–2.65 present research findings
related to stainless steels.
Figure 2.5 (a) Bauschinger effect. (b) Inverse Bauschinger
effect.2.17
Figure 2.7 Tensile stress–strain relationship of roll-formed
channel section.2.14
Figure 2.6 Relationship between Fyc /Fyv and R/t ratios based on
various values of Fuv /Fyv .2.16
and Fyc
Fyv
Fuv
R
t
=
=
=
=
=
corner yield stress
virgin yield stress
virgin tensile strength
inside bend radius
sheet thickness
Figure 2.6 shows the relationship of Fyc /Fyv and R/t for
various ratios of Fuv /Fyv . With regard to the full-section
Figure 2.8 Tensile stress–strain relationship of roll-formed joist
chord.2.14
UTILIZATION OF COLD WORK OF FORMING
2.8 UTILIZATION OF COLD WORK
OF FORMING
Section A3.3 of the North American Specification permits
utilization of the increase in material properties that results
from a cold-forming operation. For axially loaded compression members and flexural members whose components are
such that the section is not subject to a strength reduction from local buckling, that is the reduction factor 𝜌 for
strength determination is unity (1.0) as calculated according
to Section 3.3 for each of the section, the design yield stress
Fya of the steel should be determined on the basis of either
(1) full-section tensile tests or (2) stub column tests or (3)
should be computed by Eq. (2.14).∗
In the application of Eq. (2.14), Fyf is the weightedaverage tensile yield stress of the flat portions determined in
accordance with Section 2.10 or the virgin yield stress Fyc
is the tensile yield stress of corners, which may be either
computed by Eq. (2.11) or obtained from Fig. 2.6 on the
basis of the material used and the R/t ratio of the corner.
The formula does not apply where Fuv /Fyv is less than 1.2,
R/t exceeds 7, and/or the maximum included angle exceeds
120∘ . The increase in yield stress of corners having R/t ratios
exceeding 7 was discussed in Refs. 2.18, 2.24, and 2.73.
When the increased strength of axially loaded tension
members due to cold work is used in design, the yield stress
should either be determined by full-section tensile tests or be
computed by Eq. (2.14).
It should be noted that when Eq. (2.14) is used, Fya
should not exceed Fuv . This upper bound is to limit stresses
in flat elements that may not have significant increases in
yield stress and tensile strength as compared with virgin
properties.
The North American Specification limits the provisions for the utilization of the cold work of forming to
those sections of the specification concerning tension
members (Chapter D), bending strength of flexural members
[Chapter F excluding Section F2.4], concentrically loaded
compression members (Chapter E), combined axial load
and bending (Section H1), cold-formed steel light-frame
construction (Section I4), and purlins, girts, and other
members (Section I6.2). For other provisions of the specification the design of the structural member should be based
on the mechanical properties of the plain material prior to the
forming operation.
For the effective width method, when the strength increase
from the cold work of forming is used in design, the value of
Fya obtained from tests or calculations may be used for Fy
when checking each of the component elements for 𝜌 = 1.0.
If 𝜌 is found to be less than unity by substituting Fya for Fy , a
portion of the strength increase may be used as long as 𝜌 = 1.0
for all component elements.
The following examples illustrate the use of the North
American provisions on the utilization of cold work for determining the average yield stress of steel.
Example 2.1 Determine the average tensile yield stress of
steel Fya for the flange of a given channel section to be used as
a beam (Fig. 2.9). Consider the increase in strength resulting
from the cold work of forming. Use the North American
specification and A653 SS Grade 40 steel (Fy = 40 ksi and
Fu = 55 ksi).
SOLUTION
1. Check North American Requirements.
a. In order to use Eq. (2.14) for computing the average
tensile yield stress for the beam flange, the channel
must have a compact compression flange, that is,
𝜌 = 1.0. Assume that on the basis of Sections 3.3.1,
3.3.2, and 3.3.3 the reduction factor 𝜌 is found to be
unity; then Eq. (2.14) can be used to determine Fya .
b. When Eq. (2.11) is used to determine the tensile
yield stress of corners, Fyc , the following three
requirements must be satisfied:
𝐹uv
𝑅
≥ 1.2
≤7
𝜃 ≤ 120∘
𝐹yv
𝑡
Since the actual values are Fuv /Fyv = 55/40 = 1.375,
∘
R/t = 0.1875/0.135 = 1.389, and 𝜃 = 90 , which all
satisfy the North American requirements, Eq. (2.11)
can be used to determine Fyc .
∗ Section
A3.3.2 of the Commentary on the North American Specification
permits the use of cold work of forming for beam webs whose reduction
factor 𝜌 is less than unity but the sum of b1 and b2 is such that the web is
fully effective. This situation only arises when the web width–flange width
ratio, ho /bo , is less than or equal to 4.
51
Figure 2.9
Example 2.1
52
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
2. Calculation of Fyc . According to Eq. (2.11),
[
]
𝐵c
𝐹yc =
𝐹
(𝑅∕𝑡)𝑚 yv
where
(
)
(
)
𝐹uv
𝐹uv 2
− 1.79 = 1.991
− 0.819
𝐵c = 3.69
𝐹yv
𝐹yv
(
)
𝐹uv
𝑚 = 0.192
− 0.068 = 0.235
𝐹yv
Therefore,
[
]
1.735
𝐹yc =
(40) = 1.627(40) = 65.08 ksi
(1.389)0.196
3. Calculation of Fya . By using Fyc = 65.08 ksi, virgin
yield stress Fyf = 40 ksi, and
𝐶=
total cross − sectional area of
two corners
full cross − sectional area of flange
2 × 0.054
= 0.254
=
(2 × 0.054) + (2.355 × 0.135)
the average tensile yield stress of the beam flange can
be computed from Eq. (2.14) as follows:
𝐹ya = CF yc + (1 − 𝐶)𝐹yf = 46.37 ksi < 𝐹uv
(OK)
The above value of Fya can be used for tension and
compression flanges. It represents a 16% increase in
yield stress as compared with the virgin yield stress of
steel.
Example 2.2 Determine the average yield stress of steel
Fya for the axially loaded compression member, as shown in
Fig. 2.10. Consider the increase in strength resulting from the
cold work of forming. Use A1011 SS Grade 33 steel (Fy = 33
ksi and Fu = 52 ksi).
Figure 2.10
Example 2.2.
2. Calculation of Fyc . From Eq. (2.11),
(
)
(
)
𝐹uv
𝐹uv 2
𝐵c = 3.69
− 1.79 = 1.991
− 0.819
𝐹yv
𝐹yv
(
)
𝐹uv
𝑚 = 0.192
− 0.068 = 0.235
𝐹yv
and
[
𝐹yc =
[
]
]
𝐵c
1.991
=
𝐹
(33)
(𝑅∕𝑡)𝑚 yv
(1.389)0.235
= 1.843(33) = 60.82 ksi
SOLUTION
1. Check North American requirements.
a. Determination of reduction factor. Assume that on
the basis of Sections 3.3.1, 3.3.2, and 3.3.3 the
reduction factor 𝜌 is found to be unity; then Eq.
(2.14) can be used to determine Fya .
b. Review of Fuv /Fyv , R/t, and 𝜃:
𝐹uv
52
=
= 1.576 > 1.2
𝐹yv
33
𝑅 0.1875
=
= 1.389 < 7
𝑡
0.135
𝜃 = 90∘ < 120∘
(OK)
3. Calculation of Fya . By using
𝐹yc = 60.82 ksi
𝐶=
=
(OK)
𝐹yf = 33 ksi
total corner area
total area of full section
8(0.054)
= 0.17
2.54
and
(OK)
Therefore, Eq. (2.11) can be used to compute Fyc .
𝐹ya = CF yc + (1 − 𝐶)𝐹yf = 37.73 ksi < 𝐹uv
(OK)
The above computed Fya represents a 14% increase in yield
stress when the strength increase from the cold work of
forming is considered only for the corners.
EFFECT OF TEMPERATURE ON MECHANICAL PROPERTIES OF STEEL
In the 1994 Canadian Specification1.177 Eq. (2.15) was
used to compute the average tensile yield stress 𝐹y′ of the
full section of tension or compression members. This simpler
equation is also applicable for the full flange of flexural
members:
5𝐷
𝐹y′ = 𝐹y + ∗ (𝐹u − 𝐹y )
(2.15)
𝑊
where D = number of 90∘ corners. If other angles are
used, D is the sum of the bend angles divided
by 90∘ .
W* = ratio of length of centerline of full flange of
flexural members or of entire section of tension
or compression members to the thickness t.
This value provides a good agreement with the value of
Fya = 46.37 ksi computed in Example 2.1 on the basis of the
AISI Specification.
Example 2.4 For the I-section used in Example 2.2, determine the average yield stress of steel 𝐹y′ for the compression
member. Use Eq. (2.15).
SOLUTION
By using the data given in Example 2.2, the following values
can be obtained:
𝐹y = 33 ksi
𝑊∗ =
Equation (2.15) was based on a study conducted by
Lind and Schroff.2.19,2.25 By using a linear strain-hardening
model and Karren’s experimental data,2.16 Lind and Schroff
concluded that the increase in yield stress depends only on
the R/t ratio and the hardening margin (Fu − Fy ). In order to
take the cold-work strengthening into account, it is merely
necessary to replace the virgin yield stress by the virgin
tensile strength over a length of 5t in each 90∘ corner. Reference 2.30 indicates that the R/t ratio has little or no effect
on the average tensile yield stress of the full section because
when R/t is small, the volume undergoing strain hardening
is also small, whereas the increase in yield stress is large.
Conversely, when R/t is large, the volume is proportionately
large, but the increase in yield stress is small.
Example 2.3 For the channel section used in Example 2.1,
determine the average yield stress of steel 𝐹y′ for the beam
flange by using Eq. (2.15).
SOLUTION
Based on the type of steel and the dimensions used in
Example 2.1, the following values can be obtained:
𝐹y = 40 ksi
𝑊∗ =
𝐷=2
∘
(arc × length of two 90 Corners)
+ (flat width of flange)
=
𝐹u = 52 ksi
𝐷 = 2(4) = 8
2 × length of midline of each channel section
𝑡
2[2(0.3775 × 1.355) + 4.355
+ 4 (1.57)(0.1875 + 0.0675)]
0.135
= 139.6
From Eq. (2.15), the average yield stress of steel for the
compression member is
5(8)
(52 − 33) = 38.44 ksi
139.6
The above value is about 2% greater than the value of
Fya = 37.73 ksi computed in Example 2.2 on the basis of the
AISI Specification.
In the late 1990s, additional cold-formed steel members
were tested at the University of Waterloo. Based on the
available test results and the analysis, Sloof and Schuster
concluded that the AISI and Canadian design approaches
produce nearly identical results when only the virgin
mechanical properties are used.2.73,2.74 Consequently, the
AISI approach was adopted in the North American specification for the use in Canada, Mexico, and the United
States.
𝐹y′ = 33 +
𝐹u = 55 ksi
𝑡
2(1.57)(𝑅 + 𝑡∕2) + 2.355
=
0.135
2(1.57)(0.1875 + 0.0675) + 2.355
=
= 23.38
0.135
By using Eq. (2.15), the average yield stress of steel for
the beam flange is
𝐹y′ = 40 +
53
5(2)
(55 − 40) = 46.42 ksi
23.38
2.9 EFFECT OF TEMPERATURE ON
MECHANICAL PROPERTIES OF STEEL
The mechanical properties of steel discussed in Section 2.2
are based on the data obtained from tests conducted at room
temperature. These mechanical properties will be different if
the tests are performed at elevated temperatures.
The effect of elevated temperatures on the mechanical
properties of steels and the structural strength of steel
numbers has been the subject of extensive investigations
for many years.2.31–2.37,2.94–2.100,2.105,5.169,5.181 In Ref. 2.34,
Uddin and Culver presented the state of the art accompanied
by an extensive list of references. In addition, Klippstein
54
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
Figure 2.11 Effect of temperature on mechanical properties of low-carbon steel: (a) steel plates
(A36); (b) steel sheets.2.35,2.36
has reported detailed studies of the strength of cold-formed
steel studs exposed to fire.2.35,2.36 Recent studies regarding
the performance of cold-formed steel members exposed to
fire and elevated temperatures are contained in Refs. 1.419,
1.421, and 1.422. The effect of elevated temperatures on the
yield stress, tensile strength, and modulus of elasticity of
steel plates and sheet steels is shown graphically in Fig. 2.11.
For additional information on steel plates, see Ref. 2.32.
It should be noted that when temperatures are below zero
the yield stress, tensile strength, and modulus of elasticity
of steel are generally increased. However, the ductility and
toughness are reduced. Therefore, great care must be exercised in designing cold-formed steel structures for extreme
low-temperature environments, particularly when subjected
to dynamic loads.5.128 Reference 1.229 discusses structural
sandwich panels at low temperature.
The load-carrying capacities of structural members are
affected by the elevated temperature not only because the
mechanical properties of steel vary with temperature but
also because the thermal stresses may be induced due to the
restraint of expansion and secondary stresses may be developed due to the additional deformation caused by thermal
gradients.
2.10 TESTING OF FULL SECTIONS
AND FLAT ELEMENTS
When testing of full sections and flat elements is required to
determine the yield stress, the procedures and methods used
for testing and the evaluation of test results should be based
on Chapter K of the North American Specification.1.417
Figure 2.12 illustrates the typical full-section tension
specimen and the compression tests conducted at Cornell
University for the investigation of the influence of cold
work.2.14
Because welding influences the material properties due
to cold work,2.14 the effect of any welding on mechanical
properties of a member should be determined on the basis
of full-section specimens containing within the gage length
such welding as the manufacturer intends to use. Any
necessary allowance for such effect should be made in the
structural use of the member.1.314
In addition to the tests for determining material properties, Chapter K of the North American Specification1.417
also includes the test requirements for determining structural performance. These provisions can be used to obtain
design values when the composition or configuration of
elements, assemblies, connections, or details of cold-formed
steel structural members is such that calculation of their
strength cannot be made in accordance with Chapters D
through M of the specification. Test Standards for use with
the 2016 edition of the North American Specification are
summarized in Section K1.1.417 The test standards are available at www.aisistandards.org. These test methods are
dealing with (a) a rotational–lateral stiffness test method
for beam-to-panel assemblies, (b) a stub-column test
method for determining the effective area of cold-formed
steel columns, (c) standard methods for determination
of uniform and local ductility, (d) standard test methods
for determining the tensile and shear strength of screws,
(e) test methods for mechanically fastened cold-formed
steel connections, (f) standard procedures for panel and
anchor structural tests, (g) a test standard for the cantilever
test method for cold-formed steel diaphragms, (h) a base
test method for purlins supporting a standing seam roof
system, (i) a standard test method for determining the
web crippling strength of cold-formed steel beams, (j)
a test method for distortional buckling of cold-formed
steel hat-shaped compression members, (k) a method for
flexural testing cold-formed steel hat-shaped beams, (l) a
test procedure for determining a strength value for a roof
panel-to-purlin-to-anchorage device connection, (m) a test
standard for hold-downs attached to cold-formed steel
structural framing, (n) a test standard for joist connectors
RESIDUAL STRESSES DUE TO COLD FORMING
Figure 2.12
55
(a) Typical full-section tension specimen. (b) Full-section compression test.2.14
attached to cold-formed steel structural framing, (o) a test
method for through-the-punchout bridging connectors, (p)
a test method for nonstructural interior partition walls with
gypsum board, (q) a test method for fastener-sheathing local
translational stiffness, (r) a test method for fastener-sheathing
rotational stiffness, and (s) a test method for flexural
strength and stiffness of nonstructural members. Detailed
discussion of test methods is beyond the scope of this
book.
2.11 RESIDUAL STRESSES DUE TO COLD
FORMING
Residual stresses are stresses that exist in members as a
result of manufacturing and fabricating processes. In the
past, the distribution of residual stresses and the effect
of such stresses on the load-carrying capacity of steel
members have been studied extensively for hot-rolled
wide-flange shapes and welded members.2.42–2.44 For these
structural shapes, the residual stresses are caused by uneven
cooling after hot rolling or welding. These stresses are
often assumed to be uniform across the thickness of the
member.
Based on a selected residual stress pattern in W-shapes,
Galambos derived a general formula for the stress–strain
relationship of hot-rolled wide-flange cross sections.2.45
He concluded that residual stresses cause yielding earlier
than is expected if they are neglected, and they cause a
reduction in the stiffness of the member. As shown in
Fig. 2.13, even though the effect of the residual stress may
Figure 2.13 Effect of residual stress on stress–strain relationship
of hot-rolled W-shapes.
not be very great as far as the ultimate stress is concerned,
the residual stress will nevertheless lower the proportional
limit, and the inelastic behavior of these members cannot
be predicted correctly without consideration of the residual
stress.
In the past, the residual stress distribution in cold-formed
steel members has been studied analytically and experimentally by a number of investigators.2.26,2.27,2.46–2.49,2.104
Figure 2.14, adapted from Ref. 2.26, shows Ingvarsson’s
measured residual stresses in the outer and inner surfaces of
56
2
MATERIALS USED IN COLD-FORMED STEEL CONSTRUCTION
Figure 2.14 Measured longitudinal residual stress distribution in (a) outer and (b) inner surfaces of
cold-formed steel channel.2.26
a channel section. The average measured residual stresses
for the same channel section are shown in Fig. 2.15. It is
expected that the effect of such stresses on the stress–strain
relationship of cold-formed members is similar to that for
hot-rolled shapes, even though for the former the residual
stress results from cold rolling or cold bending.
In the design of cold-formed steel members, the AISI
Specification buckling provisions have been written for a
proportional limit that is considerably lower than the yield
stress of virgin steel. The assumed proportional limit seems
justified for the effect of residual stresses and the influence
of cold work discussed in Section 2.7.
EFFECT OF STRAIN RATE ON MECHANICAL PROPERTIES
Figure 2.15
57
Average measured longitudinal residual stresses in cold-formed steel channel.2.26
2.12 EFFECT OF STRAIN RATE ON
MECHANICAL PROPERTIES
The mechanical properties of sheet steels are affected by
strain rate. References 2.50, 2.51, 2.66–2.71, 2.101, and
2.102 present a review of the literature and discuss the
results of the studies for the effect of strain rate on material properties of a selected group of sheet steels and the
structural strength of cold-formed steel members conducted
by Kassar, Pan, Wu, and Yu. This information is useful for
the design of automotive structural components and other
members subjected to dynamic loads. In Ref. 2.103, Rhodes
and Macdonald discuss the behavior of plain channel section
beams under impact loading.
CHAPTER 3
Strength of Thin Elements and
Design Criteria
Figure 3.1 Local buckling of compression elements: (a) beams,
(b) columns.
3.1 GENERAL REMARKS
In cold-formed steel design, individual elements of coldformed steel structural members are usually thin and the
width-to-thickness ratios are large. These thin elements may
buckle locally at a stress level lower than the yield stress
of steel when they are subject to compression in flexural
bending, axial compression, shear, or bearing. Figure 3.1
illustrates local buckling patterns of certain beams and
columns, where the line junctions between elements remain
straight and angles between elements do not change.
Since local buckling of individual elements of coldformed steel sections has often been one of the major
design criteria, the design load should be so determined
that adequate safety is provided against failure by local
instability with due consideration given to the postbuckling
strength.
It is well known that a two-dimensional compressed
plate under different edge conditions will not fail like
one-dimensional members such as columns when the theoretical critical local buckling stress is reached. The plate
will continue to carry additional load by means of the redistribution of stress in the compression elements after local
buckling occurs. This is a well-known phenomenon called
postbuckling strength of plates. The postbuckling strength
may be several times larger than the strength determined by
critical local buckling stress, as discussed in Chapter 1.
In view of the fact that the postbuckling strength of a
flat plate is available for structural members to carry additional load, it would be proper to design such elements of
cold-formed steel sections on the basis of the postbuckling
strength of the plate rather than based on the critical local
Figure 3.2 Distortional buckling: (a) channel flange, (b) deck
with intermediate stiffeners.
buckling stress. This is true in particular for elements having
relatively large width-to-thickness ratios. The use of postbuckling strength has long been incorporated in the design
of ship structures, aircraft structures, and cold-formed steel
structures.
In addition to local buckling, the cold-formed steel
cross-section under the compression may also be subject
to distortional buckling, where two or more elements start
to rotate about fold line(s), as illustrated in Figures 3.2(a)
and (b).
Before discussing any specific design problems, it is
essential to become familiar with the terms generally used
in the design of cold-formed steel structural members and to
review the structural behavior of thin elements.
3.2
DEFINITIONS OF TERMS
The following definitions of some general terms are often
used in cold-formed steel design.1.417 For other general terms
and the terms used for ASD, LRFD, and LSD methods, see
Definitions of Terms in Appendix D.
59
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
60
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
1. Unstiffened Compression Element (u.c.e.). An unstiffened compression element is a flat compression element that
is stiffened at only one edge parallel to the direction of stress.
As shown in Fig. 3.3(a), the vertical leg of an angle section
and the compression flange of a channel section and an
inverted-hat section are unstiffened compression elements.
In addition, the portion of the cover plate in the built-up
section beyond the center of connection is also considered
as an unstiffened compression element if the spacing of the
connections are close enough.
2. Stiffened or Partially Stiffened Compression Element
(s.c.e.). A stiffened or partially stiffened compression
element is a flat compression element of which both
edges parallel to the direction of stress are stiffened by a
web, flange, stiffening lip, intermediate stiffener, or the
like (Fig. 3.3(b)). For the built-up section illustrated in
Fig. 3.3(b), the portion of the compression flange between
two centerlines of connections can be considered as a stiffened compression element if the spacing of the connections
meets the requirement of Section 8.11 on the spacing of
connections in compression elements.
3. Multiple-Stiffened Element. A multiple-stiffened
element is an element that is stiffened between webs, or
Figure 3.3(a)
Figure 3.3(c)
elements.
Sections with multiple-stiffened compression
between a web and a stiffened edge, by means of intermediate stiffeners that are parallel to the direction of stress
(Fig. 3.3(c)). The portion between adjacent stiffeners or
between a web and an intermediate stiffener or between an
edge and an intermediate stiffener is called a “subelement.”
See Section 3.3.3.3 for other limitations.
4. Flat Width w. The flat width w used in the design of
cold-formed steel structural members is the width of the
straight portion of the element and does not include the
bent portion of the section. For unstiffened flanges, the flat
width w is the width of the flat projection of the flange
measured from the end of the bend adjacent to the web
to the free edge of the flange, as shown in Fig. 3.4a. As
shown in Fig. 3.4b, for a built-up section the flat width of
Sections with unstiffened compression elements.
Figure 3.3(b) Sections with stiffened or partially stiffened compression elements.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Figure 3.4
Figure 3.5
Flat width of unstiffened compression elements.
Flat width of stiffened compression elements.
the unstiffened compression element is the portion between
the center of the connection and the free edge. The flat width
w of a stiffened element is the width between the adjacent
stiffening means exclusive of bends, as shown in Fig. 3.5a.
For the composite section shown in Fig. 3.5b, the flat width
of the stiffened compression flange is the distance between
the centers of connections.
5. Flat Width–Thickness Ratio. The flat width–thickness
ratio is the ratio of the flat width of an element measured
along its plane to its thickness. In the North American
Specification, design provisions for tension, compression, and flexural members are provided in Chapters D
through H. These design provisions are limited to certain
flat width–thickness ratios, as outlined in Section B4.1 of
the North American Specification.1.417 These limitations
are excerpted as shown in Table 3.1, which also includes
limitations for other components for completeness, such as
stiffeners and nominal yield stress.
6. Effective Design Width b. The effective design width
b is a reduced design width for computing sectional properties of flexural and compression members considering local
buckling. Figure 3.7 shows effective design widths of flexural
and compression members.
7. Thickness t. The thickness t used in the calculation
of sectional properties and the design of cold-formed
sections should be the thickness of base steel. Any thickness of coating material should be deducted from the
overall thickness of steel. See Appendix A for the thickness
of base metal. In Section B7.1 of the North American
Specification, it is specified that the uncoated minimum
61
thickness of the cold-formed product as delivered to the
job site shall not at any location be less than 95% of the
thickness used in the design. An exception is at bends,
such as corners, where the thickness may be less due to
cold-forming effects. However, the thinning is usually on the
order of 1–3% and can be ignored in calculating sectional
properties.
8. Effective Width Method. A design method that
considers local buckling of cold-formed steel members
by reducing the element width under nonlinear stress
distribution to an effective width under a simplified linear
stress distribution. The effective cross-section properties,
which are calculated based on the effective portions of the
cross-section, are then used in strength determination. See
Section 3.3 for details.
9. Direct Strength Method. A design method that provides
prediction of member strengths without use of effective
widths. See Section 3.5 for details.
10. Safety Factor Ω and Resistance Factor 𝜙. For the
design of cold-formed steel structural members, different
safety factors and resistance factors are used in the design
provisions of the North American Specification in accordance with the type of structural behavior.
3.3 STRUCTURAL BEHAVIOR OF COMPRESSION
ELEMENTS AND EFFECTIVE WIDTH DESIGN
CRITERIA
3.3.1
3.3.1.1
Stiffened Compression Elements
Stiffened Elements under Uniform Compression
3.3.1.1.1 Yielding The strength of a stiffened compression element, such as the compression flange of a hat section,
is governed by yielding if its w/t ratio is relatively small.
It may be governed by local buckling as shown in Fig. 3.8
at a stress level less than the yield stress if its w/t ratio is
relatively large.
3.3.1.1.2 Elastic Local Buckling Stress of Plates
Considering a simply supported square plate subjected
to a uniform compression stress in one direction, it will
buckle in a single curvature in both directions, as shown in
Fig. 3.9. However, for individual elements of a section, the
length of the element is usually much larger than the width,
as illustrated in Fig. 3.8.
The critical buckling stress of a plate as shown in Fig. 3.10
can be determined by solving Bryan’s differential equation
based on small-deflection theory (i.e., the significant
62
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Table 3.1 Limits of Applicability for Member Design in Chapters E Through H of the North American
Specification1.347 by the Effective Width Method and the Direct Strength Method
Criteria
Limiting
Variables𝑎
Stiffened element in compression
Edge-stiffened element in compression
𝑤∕𝑡𝑏
b/t
Unstiffened element in compression
Stiffened element in bending (e.g., a
web)
𝑑∕𝑡𝑏
h/t
Inside bend radius
Simple edge stiffener length/width
ratio
Edge stiffener type
Maximum number of intermediate
stiffeners in w
Maximum number of intermediate
stiffeners in b
Number of intermediate stiffeners in h
Nominal yield stress
Effective Width Method
Direct Strength Method
≤ 500
≤ 160
R/t
𝑑𝑜 ∕𝑏𝑜
≤ 500
≤ 90 for ls ≥ la
≤ 60 for ls < la
≤ 60
< 200 for unstiffened web
≤ 260 for bearing stiffenerc
≤ 300 for bearing and
intermediate stiffenerc
≤ 10d
≤ 0.7
𝑛𝑓
Simple only
4
Simple and complex
4
𝑛fg
2
2
𝑛w
𝐹𝑦
0
<80 ksi (552 MPa)e
4
<95 ksi (655 MPa)e
≤ 60
≤ 300
≤ 20
≤ 0.7
Note:
𝑎
Variable definitions (see Figure 3.6 for illustration):
w
t
b
bo
d
do
h
R
nf
nfe
nw
Fy
=
=
=
=
=
=
=
=
=
=
=
=
Flat width of stiffened compression element (disregard intermediate stiffeners)
Thickness of element
Flat width of element with edge stiffeners (disregard intermediate stiffeners)
Out-to-out width of element with edge stiffeners (disregard intermediate stiffeners)
Flat width of unstiffened element (disregard intermediate stiffeners)
Out-to-out width of unstiffened element (disregard intermediate stiffeners)
Depth of flat portion of web measured along plane of web (disregard intermediate stiffeners)
Inside bend radius
Number of intermediate stiffeners in stiffened compression element
Number of intermediate stiffeners in edge stiffener
Number of intermediate stiffeners in stiffened element under stress gradient (e.g. web)
Nominal yield stress
𝑏
Stiffened compression elements with w/t > 250 and unstiffened compression elements with d/t > 30 are likely to have noticeable deformations
prior to developing their full strength.
𝑐
Bearing and intermediate stiffener requirements in accordance with Specification Section F5.1.
𝑑
For inside bend R/t ratios larger than 10, rational engineering analysis in accordance with Specification Section A3 is permitted.
𝑒
See Specification Section A3 for additional limitations.
bo
d do
R
nfe
h
t
w
Simple lip
nw
nf
Complex lip
b
Figure 3.6
Illustration of Variables in Table 3.1.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Figure 3.7 Effective design width of flexural and compression members.
63
64
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.8 Local buckling of stiffened compression flange of
hat-shaped beam.1.161
Figure 3.9
Square plate subjected to compression stress.
Figure 3.10 Rectangular plate subjected to compression stress.
deflection at buckling is of the order of the thickness of the
plate or less) as follows:
𝜕4𝜔
𝜕4𝜔
𝜕4𝜔 𝑓 𝑡 𝜕2𝜔
+2 2 2 + 4 + 𝑥
=0
4
𝐷 𝜕𝑥2
𝜕𝑥
𝜕𝑥 𝜕𝑦
𝜕𝑦
where
𝐷=
(3.1)
Et3
12(1 − 𝜇2 )
and 𝐸 = modulus of elasticity of steel, =
29.5 × 103 ksi (203 GPa or 2.07 × 106 kg∕cm2 )
𝑡 = thickness of plate
𝜇 = Poisson’s ratio, = 0.3 for steel in the elastic range
𝜔 = deflection of plate perpendicular to surface
𝑓𝑥 = compression stress in 𝑥 direction
If m and n are the numbers of half sine waves in the x and y
directions, respectively, the deflected shape of the rectangular
plate as shown in Fig. 3.10 may be represented by a double
series:
∞ ∞
∑
∑
m𝜋y
m𝜋x
𝜔=
𝐴mn sin
sin
(3.2)
𝑎
𝑤
𝑚=1 𝑛=1
The above equation is satisfied with boundary conditions
because for 𝑥 = 0, a and 𝑦 = 0, w (a and w being the length
and width of the plate, respectively) the computed deflection
equals zero. Since 𝜕 2 ω∕𝜕𝑥2 = 0 and 𝜕 2 ω∕𝜕𝑦2 = 0 at four
edges, Eq. (3.2) also satisfies the condition that the edge
moments equal zero because
( 2
)
𝜕 𝜔
𝜕2𝜔
𝑀𝑥 = −𝐷
+𝜇 2
𝜕𝑥2
𝜕𝑦
( 2
)
𝜕 𝜔
𝜕2𝜔
𝑀𝑦 = −𝐷
+
𝜇
𝜕𝑦2
𝜕𝑥2
Solving Eq. (3.1) by using Eq. (3.2), one can then obtain the
equation
[ (
]
)
∞ ∞
2
2 2
2𝜋2
∑
∑
𝑓
𝑡
𝑛
𝑚
𝑚
𝐴mn 𝜋 4
+ 2
− 𝑥
2
𝐷 𝑎2
𝑎
𝑤
𝑚=1 𝑛=1
n𝜋y
m𝜋x
sin
=0
(3.3)
𝑎
𝑤
It is obvious that the solution can be obtained if either
𝐴mn = 0 or the quantity in square brackets equals zero. The
former condition means that no buckling will occur, which is
not applicable to this particular case.
By solving
( 2
)2
𝑓 𝑡 𝑚2 𝜋 2
𝑛2
4 𝑚
𝜋
+ 2
− 𝑥
=0
2
𝐷 𝑎2
𝑎
𝑤
× sin
one can obtain an equation for critical local buckling stress
as follows:
[ ( )
( )]2
𝑛2 𝑎
𝐷𝜋 2
𝑤
+
𝑓cr = 𝑓𝑥 =
(3.4)
𝑚
𝑎
𝑚 𝑤
tw2
In Eq. (3.4) the minimum value in square brackets is 𝑛 = 1,
that is, only one half sine wave occurs in the y direction.
Therefore
kD𝜋 2
𝑓cr =
(3.5)
tw2
where
( )]
[ ( )
1 𝑎 2
𝑤
+
𝑘= 𝑚
(3.6)
𝑎
𝑚 𝑤
Substituting the value of D in Eq. (3.5), Eq. (3.7) represents a general equation for critical local buckling stress for
a rectangular plate subjected to compression stress in one
direction:
𝑘𝜋 2 𝐸
𝑓cr =
(3.7)
12(1 − 𝜇2 )(𝑤∕𝑡)2
The value of k used in Eq. (3.7) is shown in Fig. 3.11 for
different a/w ratios.
It should be noted that when the a/w ratio is an integer,
the value of k equals 4. This value of k is also applicable for
relatively large a/w ratios.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
65
Table 3.2 Values of k for Determining Critical
Buckling Stress3.2
𝑓cr = 𝑘
Figure 3.11
Buckling coefficient for flat rectangular plates.3.1
From Fig. 3.11 and Eq. (3.6) it can be seen that the transition from m to 𝑚 + 1 half sine waves occurs at the condition
when the two corresponding curves have equal ordinates,
that is,
( )
( )
( )
( )
1 𝑎
1
𝑤
𝑎
𝑤
+
= (𝑚 + 1)
+
𝑚
𝑎
𝑚 𝑤
𝑎
𝑚+1 𝑤
or
√
𝑎
= 𝑚(𝑚 + 1)
𝑤
For a long plate,
𝑎
≅𝑚
𝑤
or
𝑎
𝜆=
≅𝑚
(3.8)
𝑤
where 𝜆 is the length of the half sine wave.
Equation (3.8) indicates that the number of half sine waves
increases with the increase of a/w ratios. For a long plate, the
length of the half sine waves equals approximately the width
of the plate, and therefore square waves are formed, as shown
in Fig. 3.10.
In structural engineering, the long plate having a relatively
large a/w ratio is of particular interest because such a long
plate often represents the case of individual elements of the
sections generally used in structures. As shown in Fig. 3.11,
whenever the aspect ratio a/w exceeds about 4, a value of
𝑘 = 4 can be used for determining the critical buckling stress
for a plate simply supported along four edges and subjected
to compression stress in one direction, that is,
𝑓cr =
𝜋2𝐸
3(1 − 𝜇2 )(𝑤∕𝑡)2
(3.9)
Eq. (3.9) is also applicable to a square plate.
The values of k for a long rectangular plate subjected to
different types of stress (compression, shear, or bending)
𝜋2𝐸
12(1 − 𝜇2 )(𝑤∕𝑡)2
Case Boundary Condition
Type of
Stress
Value of k for
Long Plate
(a)
Compression 4.0
(b)
Compression 6.97
(c)
Compression 0.425
(d)
Compression 1.277
(e)
Compression 5.42
(f)
Shear
5.34
(g)
Shear
8.98
(h)
Bending
23.9
(i)
Bending
41.8
and under different boundary conditions (simply supported,
fixed, or free edge) are tabulated in Table 3.2.
3.3.1.1.3 Buckling of Plates in the Inelastic Range
When the compression stress in a plate in only one direction exceeds the proportional limit of the steel, the plate
becomes an anisotropic plate which has different properties
in different directions of the plate.
In 1924 Bleich proposed the following differential
equation for inelastic buckling3.3 :
)
( 4
√ 𝜕4𝜔
𝑓 𝑡 𝜕2𝜔
𝜕4𝜔
𝜕 𝜔
= 0 (3.10)
𝜏 4 +2 𝜏 2 2 + 4 + 𝑥
𝐷 𝜕𝑥2
𝜕𝑥
𝜕𝑥 𝜕𝑦
𝜕𝑦
where 𝜏 = 𝐸𝑡 ∕𝐸, and 𝐸𝑡 is the tangent modulus of steel.
66
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Applying the modified boundary conditions, one can then
obtain the following critical buckling stress for plastic buckling of the plate:
√
√
𝑘𝜋 2 EE𝑡
𝑘𝜋 2 𝐸 𝜏
=
(3.11)
𝑓cr =
12(1 − 𝜇2 )(𝑤∕𝑡)2
12(1 − 𝜇2 )(𝑤∕𝑡)2
The wavelength for a long plate is
√
(3.12)
𝜆 = 𝜏𝑤
√
√
In Eqs. (3.10) and (3.11), 𝜏 = 𝐸𝑡 ∕𝐸 is the plasticity
reduction factor for a simply supported plate subjected to
a uniform compression stress in one direction [case (a)
of Table 3.2]. This factor varies with the type of loading
and the edge support conditions. For example, a value
of 𝐸𝑠 ∕𝐸 has been found to be an appropriate plasticity
reduction factor for case (c) of Table 3.2. The value 𝐸𝑠 is
the secant modulus. It has been used in the “Specification
for the Design of Cold-Formed Stainless Steel Structural
Members.”1.160,3.4,3.5,3.11,3.249 Additional information on
local buckling coefficients and plasticity reduction factors
can be found in Refs. 3.1 and 3.6–3.11.
3.3.1.1.4 Postbuckling Strength and Effective Design
Width Unlike one-dimensional structural members such as
columns, stiffened compression elements will not collapse
when the buckling stress is reached. An additional load can
be carried by the element after buckling by means of a redistribution of stress. This phenomenon is known as postbuckling strength and is most pronounced for elements with large
w/t ratios.
The mechanism of the postbuckling action can easily be
visualized from a square-plate model as shown in Fig. 3.12.
It represents the portion abcd of the compression flange of
the hat section illustrated in Fig. 3.8. As soon as the plate
starts to buckle, the horizontal bars in the grid of the model
Figure 3.13 Consecutive stages of stress distribution in stiffened
compression elements.
will act as tie rods to counteract the increasing deflection of
the longitudinal struts.
In the plate, the stress distribution is uniform prior to its
buckling, as shown in Fig. 3.13a. After buckling, a portion
of the prebuckling load of the center strip transfers to the
edge portion of the plate. As a result, a nonuniform stress
distribution is developed, as shown in Fig. 3.13b. The redistribution of stress continues until the stress at the edge
reaches the yield stress of the steel and then the plate begins
to fail (Fig. 3.13c).
The postbuckling behavior of a plate can be analyzed
by using large-deflection theory. The following differential
equation for large-deflection buckling of a plate was introduced by von Karman in 1910:
𝜕4𝜔
𝜕4𝜔
𝜕4𝜔
+
2
+
𝜕𝑥4
𝜕𝑥2 𝜕𝑦2
𝜕𝑦4
)
( 2
𝑡 𝜕 𝐹 𝜕2 𝜔
𝜕2𝐹 𝜕2𝜔
𝜕2𝐹 𝜕2𝜔
=
2
−
2
+
𝐷 𝜕𝑦2 𝜕𝑥2
𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑦 𝜕𝑥2 𝜕𝑦2
(3.13)
where F is a stress function defining the median fiber stress
of the plate, and
𝜕2𝐹
𝜕2𝐹
𝜕2𝐹
𝑓
=
𝜏
=
−
𝑦
xy
𝜕𝑥𝜕𝑦
𝜕𝑦2
𝜕𝑥2
It has been found that the solution of the differential
equation for large-deflection theory has little application in
practical design because of its complexity. For this reason, a
concept of “effective width” was introduced by von Karman
et al. in 1932.3.12 In this approach, instead of considering
the nonuniform distribution of stress over the entire width
of the plate w, it is assumed that the total load is carried by a
fictitious effective width b subject to a uniformly distributed
stress equal to the edge stress 𝑓max , as shown in Fig. 3.14.
The width b is selected so that the area under the curve of the
actual nonuniform stress distribution is equal to the sum of
the two parts of the equivalent rectangular shaded area with
a total width b and an intensity of stress equal to the edge
stress 𝑓max , that is,
𝑓𝑥 =
𝑤
Figure 3.12 Square-plate model for postbuckling action.1.161
∫0
fdx = bf max
(3.14)
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
67
Figure 3.14 Effective width of stiffened compression element.
Figure 3.15
It may also be considered that the effective width b represents a particular width of the plate that just buckles when the
compressive stress reaches the yield stress of steel. Therefore, for a long plate the theoretical value of b may be determined as follows:
𝑓cr = 𝐹𝑦 =
or
√
𝑏 = Ct
𝜋2𝐸
3(1 − 𝜇2 )(𝑏∕𝑡)2
√
𝐸
𝐸
= 1.9𝑡
𝐹y
𝐹y
(3.15)
(3.16)
where
𝜋
𝐶=√
= 1.9
3(1 − 𝜇2 )
μ = 0.3
(3.17)
Equation (3.16) is the von Karman formula for the design
of stiffened elements derived in 1932.
Whenever 𝑤 > 𝑏,
𝑓cr =
𝜋2𝐸
3(1 − 𝜇2 )(𝑤∕𝑡)2
or
√
𝑤 = Ct
𝐸
𝑓cr
[Eq. (3.9)]
(3.18)
From Eqs. (3.16) and (3.18), the following relationship of b
and w can be obtained:
√
𝑓cr
𝑏
(3.19)
=
𝑤
𝐹y
Based on his extensive investigation on light-gage coldformed steel sections, Winter indicated that Eq. (3.16) is
equally applicable to the element in which the stress is
below the yield stress.3.13 Therefore Eq. (3.16) can then be
rewritten as
√
𝐸
(3.20)
𝑏 = Ct
𝑓max
where 𝑓max is the maximum edge stress of the plate. It may
be less than the yield stress of steel.
Experimental determination of effective width.3.13
In addition, results of tests previously conducted by
Sechler and Winter indicate that the term C used in Eq. (3.20)
depends primarily on the nondimensional parameter
√
( )
𝐸
𝑡
(3.21)
𝑓max 𝑤
It has been
√ found that a straight-line relationship exists
between 𝐸∕𝑓max (t/w) and the term C, as shown in
Fig. 3.15. The following equation for the term C has been
developed by Winter on the basis of his experimental
investigation3.13,3.14 :
]
[
( )√
𝐸
𝑡
(3.22)
𝐶 = 1.9 1 − 0.475
𝑤
𝑓max
It should be noted√that the straight line in Fig. 3.15 starts at
a value of 1.9 for 𝐸∕𝑓max (𝑡∕𝑤) = 0, which represents the
case of an extremely large w/t ratio with relatively high stress.
For this particular case, the experimental determinations are
in substantial agreement with von Karman’s original formula
[Eq. (3.16)].
Consequently, in 1946 Winter presented the following
modified formula for computing the effective width b for
plates simply supported along both longitudinal edges:
√
[
]
( )√
𝐸
𝐸
𝑡
1 − 0.475
(3.23)
𝑏 = 1.9𝑡
𝑓max
𝑤
𝑓max
It should be noted from Eq. (3.23) that the effective width
depends not only on the edge stress 𝑓max but also on the
w/t ratio.
Eq. (3.23) may be written in terms of the ratio of 𝑓cr ∕
√
√
[
]
𝑓max as
𝑓cr
𝑓cr
𝑏
1 − 0.25
(3.24)
=
𝑤
𝑓max
𝑓max
From the above equation it can be shown that a compressed
plate is fully effective, 𝑏 = 𝑤, when the ratio of w/t is less
√
( )
than
𝐸
𝑤
= 0.95
(3.25)
𝑡 lim
𝑓max
and that the first wave occurs at a stress equal to 𝑓cr ∕4.
68
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.16 Correlation between test data on stiffened compression elements and design criteria.3.15
In summary, it may be considered that Eqs. (3.23) and
(3.24) are generalizations of Eqs. (3.16) and (3.19) in two
respects: (1) by introducing 𝑓max for 𝐹y , the equations can
be applied to service loads as well as to failure loads, and (2)
by introducing empirical correction factors, the cumulative
effects of various imperfections, including initial deviations
from planeness, are accounted for.
During the period from 1946 to 1968, the AISI design
provision for the determination of the effective design width
was based on Eq. (3.23). A longtime accumulated experience
has indicated that a more realistic equation, as shown in Eq.
(3.26), may be used in the determination of the effective
width b1.161 :
√
[
]
( )√
𝐸
𝐸
𝑡
1 − 0.415
(3.26)
𝑏 = 1.9𝑡
𝑓max
𝑤
𝑓max
Figure 3.16 illustrates the correlation between Eq. (3.26)
and the results of tests conducted by Sechler and Winter.
It should be noted that Sechler’s tests were carried out on
disjointed single sheets, not on structural shapes. Hence the
imperfect edge conditions account for many low values in his
tests.
In view of the fact that Eq. (3.26) correlates well with the
stiffened compression elements with little or no rotational
restraints along both longitudinal edges (i.e., 𝑘 = 4), this
equation can be generalized as shown below for determining
the effective width of stiffened elements having different
rotational edge restraints:
√
[
]
( )√
kE
kE
𝑡
𝑏 = 0.95𝑡
1 − 0.208
(3.27)
𝑓max
𝑤
𝑓max
where k is the local buckling coefficient. The above equation
has been used in the Canadian standard.1.177
In Ref. 3.16, Johnson pointed out that Eq. (3.27) can be
modified for the effects of inelastic buckling by replacing E
by 𝜂E, where 𝜂 is a plasticity reduction factor.
It should be noted that Eq. (3.26) may be rewritten in terms
of the 𝑓cr ∕𝑓max ratio as follows:
√
√
[
]
𝑓
𝑓cr
𝑏
cr
1 − 0.22
(3.28)
=
𝑤
𝑓max
𝑓max
Therefore, the effective width b can be determined as
𝑏 = 𝜌w
(3.29)
where the reduction factor 𝜌 is given as
√
1 − 0.22∕ 𝑓max ∕𝑓cr
𝜌=
√
𝑓max ∕𝑓cr
1 − 0.22∕𝜆
≤1
(3.30)
𝜆
In Eq. (3.30), 𝜆 is a slenderness factor determined as
√
√
𝑓max
𝑓max [12(1 − 𝜇2 )(𝑤∕𝑡)2 ]
𝜆=
=
𝑓cr
𝑘𝜋 2 𝐸
√
)
(
( ) 𝑓
𝑤
1.052
max
(3.31)
=
√
𝑡
𝑓
cr
k
=
in which k, w/t, 𝑓max , and E were previously defined. The
value of 𝜇 was taken as 0.3.
Figure 3.17 shows the relationship between 𝜌 and 𝜆. It can
be seen that, when 𝜆 ≤ 0.673, 𝜌 = 1.0.
Based on Eqs. (3.29)–(3.31), the 1986 edition of the
AISI Specification adopted the nondimensional format in
Section B2.1 for determining the effective design width b for
uniformly compressed stiffened elements.3.17,3.18 The same
design equations are retained in Section 1.1 of Appendix
1 in the 2016 edition of the North American Specification
with some format changes and symbol redesignation of local
buckling stress 𝑓cr to 𝑓cr𝓁 as follows:
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Figure 3.17
69
Reduction factor 𝜌 vs. slenderness factor 𝜆.
a. Strength Determination
𝑏 = 𝜌w
(3.32)
where 𝑏 = effective design width of uniformly
compressed element for strength
determination
𝑤 = flat width of compression element
𝜌 = local buckling reduction factor and is
determined as follows:
𝜌 = 1 when 𝜆 ≤ 0.673
(3.33)
= (1 − 022∕𝜆)∕𝜆 when 𝜆 > 0.673
(3.34)
𝐸 = modulus of elasticity
𝑓 = maximum compressive edge stress with
which the effective width is computed
Since Eq. (3.35) is a simpler equation than the design procedure prescribed in the North American Specification, this
equation is used in this book for computing the slenderness
factor 𝜆.
Figure 3.19 is a graphic presentation of Eq. (3.34).
It can be used for determination of the effective design
width of stiffened elements with a given w/t ratio and
𝜆 = plate slenderness factor and is given as
√
( )2
𝑓
𝑘𝜋 2 𝐸
𝑡
fcr𝓁 =
𝜆=
2
𝑓cr𝓁
12(1 − 𝜇 ) 𝑤
w
Substituting 𝑓cr𝓁 into the equation for 𝜆 and using 𝜇 = 0.3
result in the following simplified equation:
( )√𝑓
1.052 𝑤
𝜆= √
(3.35)
𝑡
𝐸
𝑘
Actual Element
f
where 𝑘 = plate buckling coefficient
= 4.0 for stiffened elements supported by a
web on each longitudinal edge as shown in
Fig. 3.18. Values for different types of
elements are given in the applicable sections
of the North American Specification.
𝑤 = width of stiffened compression element
𝑡 = thickness of compression element
b/2
b/2
Effective Element, b, and Stress, f,
on Effective Elements
Figure 3.18
Uniformly compressed stiffened elements.1.417
70
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
compressive
√stress. Note that the limiting w/t ratio, (𝑤∕𝑡)lim ,
is 219.76∕ 𝑓 . When 𝑤∕𝑡 ≤ (𝑤∕𝑡)lim , no reduction of the
flat width is required for stiffened elements supported by a
web on each longitudinal edge.
b. Serviceability Determination
𝑏𝑑 = 𝜌𝑤
[Eq. (3.32)]
where 𝑏𝑑 = effective design width of compression
element for serviceability
determination
𝑤 = flat width of compression element
𝜌 = reduction factor determined from
a. Strength Determination
The plate slenderness factor 𝜆 is determined as
√
( )
fd
1.052 𝑤
(3.36)
𝜆= √
t
E
k
where 𝑓𝑑 is the computed compressive stress in the
element being considered and k, t, and E are the same
as that defined above for strength determination.
Equation (3.36) provides a conservative estimate of effective width 𝑏𝑑 for serviceability. It is included in Section 1.1 of
the North American Specification as Procedure I. Figure 3.19
can also be used for the determination of the effective width
for serviceability using this procedure.
Figure 3.19
For stiffened compression elements supported by a web
on each longitudinal edge, a study conducted by Weng and
Pekoz indicated that the following method can yield a more
accurate estimate of the effective width 𝑏d for serviceability,
and it is included in the North American Specification as
Procedure II:
𝜌 = 1 when 𝜆 ≤ 0.673
(3.37)
𝜌 = (1.358 − 0.461∕λ)∕λ when 0.673 < 𝜆 < 𝜆c (3.38)
√
𝜌 = (0.41 + 0.59 𝐹𝑦 ∕𝑓𝑑 − 0.22∕λ)∕λ when 𝜆 ≥ 𝜆c
(3.39)
where
𝜆𝑐 = 0256 + 0.328(𝑤∕𝑡)
√
𝐹𝑦 ∕𝐸
(3.40)
and 𝜆 is defined by Eq. (3.36).
Example 3.1 For the given thin plate supported along both
longitudinal edges as shown in Fig. 3.20, determine the
following items using the U.S. customary unit:
1. Critical buckling stress
2. Critical buckling load
3. Ultimate load
Given:
𝑡 = 0.06 in. 𝐸 = 29.5 × 103 ksi 𝐹y = 50 ksi 𝜇 = 0.3
Reduction factor 𝜌 for stiffened compression elements.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
71
Example 3.2 Compute the effective design width of the
compression flange of the beam shown in Fig. 3.21 using the
U.S. customary unit.
a. For strength determination—assume that the compressive
stress in the flange without considering the safety factor
is 25 ksi.
b. For serviceability determination—assume that the
compressive stress in the flange under the service load is
15 ksi.
Figure 3.20
Example 3.1
SOLUTION
1. Critical Buckling Stress [Eq. (3.7)]. Since the aspect
ratio is 4, use 𝑘 = 4.0 as follows:
𝑓cr𝓁 =
𝑘𝜋 2 𝐸
12(1 − 𝜇2 )(𝑤∕𝑡)2
3
=
4(3.1416)2 (29.5 × 10 )
12(1 − 0.32 )(6∕0.06)2
= 10.665 ksi
2. Critical Buckling Load. 𝑃cr𝓁 = Af cr𝓁 = 6(0.06) ×
(10.665) = 3.839 kips.
3. Ultimate Load. The ultimate load can be computed
from the effective width b determined by Eq. (3.32).
From Eq. (3.35),
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
√
(
)
1.052 6.0
50
= √
0.06
29,500
4
= 2.166
Since 𝜆 > 0.673, 𝜌 is calculated using Eq. (3.34):
1 − 0.22∕𝜆
𝜌=
𝜆
1 − 0.22∕2.166
=
2.166
= 0.415
SOLUTION
1. Strength Determination. As the first step, compute 𝜆
using Eq. (3.35) with the following values:
𝑘 = 4.0
𝑤 = 15.00 − 2(𝑅 + 𝑡)
= 15.00 − 2(0.1875 + 0.105) = 14.415 in.
𝑤 14.415
=
= 137.286
𝑡
0.105
𝑓 = 25 ksi
𝐸 = 29,500 ksi
1.052
𝜆 = √ (137.286)
4
√
25
= 2.102
29,500
Since λ > 0.673, compute the reduction factor 𝜌
according to Eq. (3.34):
1 − 0.22∕2.102
= 0.426
2.102
Therefore, the effective design width for strength
determination is
𝜌=
b = 𝜌𝑤 = (0.426)(14.415) = 6.14 in.
2. Serviceability Determination. By using Eq. (3.36) and
fd = 15 ksi,
√
1.052
15
𝜆 = √ (137.286)
= 1.628
29,500
4
Therefore, the effective design width and the ultimate load
are computed as follows:
𝑏 = 𝜌w = 0.415(6.0) = 2.49 in.
𝑃ult = 𝐴eff 𝐹y = (2.49)(0.06)(50) = 747 kips
It is seen that here the ultimate load of 7.47 kips is almost
twice the critical buckling load of 3.839 kips.
Figure 3.21 Example 3.2.
72
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Since 𝜆 > 0.673,
1 − 0.22∕1.628
𝜌=
= 0.531
1.628
Therefore, the effective design width for serviceability determination is
𝑏d = 𝜌w = (0.531)(14.415) = 7.654 in.
Example 3.3 Calculate the effective width of the compression flange of the box section (Fig. 3.22) to be used as a beam
bending about the x axis. Use 𝐹y = 33 ksi. Assume that the
beam webs are fully effective and that the bending moment is
based on initiation of yielding. Use the U.S. customary unit.
SOLUTION
Because the compression flange of the given section is a
uniformly compressed stiffened element, which is supported
by a web on each longitudinal edge, the effective width of the
flange for strength determination can be computed by using
Eqs. (3.32)–(3.35) with 𝑘 = 4.0.
Since the bending strength of the section is based on initiation of yielding, 𝑓 = 𝐹𝑦 , 𝑦 ≥ 2.50 in. Therefore, the slenderness factor 𝜆 can be computed from Eq. (3.35), that is:
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
)√
(
1.052 6.1924
33
=√
0.06
29,500
4.0
= 1.816
Figure 3.22
Since λ > 0.673, use Eqs. (3.32) and (3.34) to compute the
effective width b as follows:
)
(
1 − 0.22∕𝜆
𝑤
𝑏 = 𝜌w =
𝜆
(
)
1 − 0.22∕1.816
=
(6.1924) = 3.00 in.
1.816
The above discussion on the structural design of stiffened compression elements is based on the Effective Width
Method described in the North American Specification.1.417
In other countries the design equations for determining the
effective design width may be different. For example, in the
Japanese Standard1.186 the effective design width is independent of the flat-width ratio w/t. This approach is similar to
Eq. (3.41), which was derived by Lind et al.3.20,3.21 on the
basis of their statistical analysis of the available experimental
results:
√
𝐸
(3.41)
𝑏 = 1.64𝑡
𝑓max
Equation (3.41) was used in the Canadian Standard during
the period from 1974 through 1984.
In the British Standard the effective design width is determined by using the w/t ratio and the design equations given
in Ref. 1.194. For the design of steel decks and panels,
European recommendations1.209,1.328 have adopted Winter’s
formula as given in Eqs. (3.32)–(3.35). In Refs. 1.183, 1.184,
and 3.22 the French recommendations give an effective
design width similar to that permitted by the North American
Specification.
Example 3.3
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
73
Figure 3.23 Comparison of effective design widths for load determination by using various design
specifications.
The effective design widths of the stiffened compression element as determined by several design specifications have been compared in Fig. 3.23. Reference 1.147
included comparisons of different specifications being used
in Australia, China, Eastern Europe, Japan, North America,
and Western Europe. Additional information on effective
design width can be found in Refs. 3.23–3.37.
3.3.1.1.5 Influence of Initial Imperfection on Effective
Design Width The load-carrying capacity of stiffened
compression elements is affected by the initial imperfection
of the plate. The larger the initial imperfection, the smaller
the capacity.
The influence of initial imperfection on the effective
design width has previously been studied by Hu et al.3.38
and by Abdel-Sayed.3.39 Figure 3.24 shows the theoretical
ratio of effective width to actual width, b/w, affected by
various values of initial imperfections. Imperfect plates have
also been studied extensively by Dawson and Walker,3.40
Sherbourne and Korol,3.41 Hancock,3.42 and Maquoi and
Rondal.3.43
3.3.1.1.6 Influence of Impact Loading on Effective
Design Width Previous discussion on effective width
was concerned with the compression elements subjected to
static loading. This type of loading condition is primarily
applicable to the design of cold-formed steel members used
in building construction.
As indicated in Section 1.1, cold-formed steel members
are also used in car bodies, railway coaches, various types
of equipment, storage racks, highway products, and bridge
construction, all of which are subjected to dynamic loads.
Since members subjected to dynamic loads behave differently than those subjected to static loads, the question
arises as to whether direct application of the AISI design
criteria based on static loading is appropriate. In order to
develop the necessary information on this topic, research
work has been conducted by Culver and his collaborators
at Carnegie Mellon University to study analytically and
experimentally the behavior of thin compression elements,
beams, and columns subjected to dynamic or time-dependent
loading.3.44–3.49 It was found that the effective design width
formula [Eq. (3.26)] satisfies both the static and the dynamic
results to the same degree of accuracy.3.44,3.46 Figure 3.25
shows the correlation between the test data and Eq. (3.26) .
In this figure β′ is the ratio of the time duration of the stress
pulse to the fundamental period of the compression flange
treated as a simply supported plate. This subject has also
been studied at the University of Missouri–Rolla under a
project on automotive structural components.3.50,3.67–2.71 In
Ref. 2.103, Rhodes and Macdonald reported channel section
beams under static and impact loading.
3.3.1.1.7 Influence of Corner Radius on Effective
Design Width For an element with ends connected to other
elements through corners (see Fig. 3.26), the calculation
74
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.24 Effect of initial imperfection on effective design width.
Figure 3.25
Correlation between effective design width formula and test data.3.46
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Figure 3.26
Influence of Corner Radius1.431
of the element effective design width usually assumes that
the corners are as simply supported conditions. However,
as the corners are getting larger, the restraints from other
elements are weakened. Research studies6.39,3.253 showed
that when 𝑅∕𝑡 > 10, the corner influence needs to be considered. One approach is to use numerical analysis method to
determine the local buckling stress of the element, such as
Direct Strength Method. For 10 < 𝑅∕𝑡 ≤ 20, the following
rational reduction through the buckling coefficient, k, was
proposed3.253 and was included in the Commentary1.431 on
the 2016 edition of the North American Specification:
𝑘𝑅 = 𝑘 𝑅𝑅1 𝑅𝑅2
(3.42)
𝑅𝑅1 = 1.08 − (𝑅1 ∕𝑡)∕50
(3.43)
𝑅𝑅2 = 1.08 − (𝑅2 ∕𝑡)∕50
(3.44)
where
k is the plate buckling coefficient determined from
Section 3.3.
Figure 3.27
75
3.3.1.1.8 Influence of Intermittent Connection Spacing
on Effective Design Width For built-up composite
sections, the spacing of intermittent connections is limited
by Section I1.3 of the North American Specification. When
the spacing exceeds the limits, the effective design width
of the uniformly compressed elements restrained by intermittent connections can be determined in accordance with
Section 1.1.4 of Appendix 1 of the North American Specification. For detailed design provisions, see Subsection
3.3.3.2 of this chapter.
3.3.1.2 Beam Webs and Stiffened Elements with Stress
Gradient When a flexural member is subjected to bending
moment, the compression portion of the web may buckle
due to the compressive stress caused by bending. Figure 3.27
shows a typical pattern of bending failure of beam webs.
Prior to 1986, the design of beam webs in the United
States was based on the full web depth and the allowable
Typical bending failure pattern for channel sections.3.60
76
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
bending stress specified in the 1980 and earlier editions of
the AISI Specification. In order to unify the design methods
for webs and compression flanges, the “effective web depth
approach” was adopted in the 1986 edition of the AISI
Specification.1.4 The same design approach was used in the
1996 AISI Specification1.314 and is retained in the North
American Specification.1.336,1.345,1.416,1.417
3.3.1.2.1 Web Buckling due to Bending Stress The
buckling of disjointed flat rectangular plates under bending
with or without longitudinal loads has been investigated by
Timoshenko, Schuette, McCulloch, Johnson, and Noel.3.1,3.2
The theoretical critical buckling stress for a flat rectangular plate under pure bending can be determined by
Eq. (3.45):
𝑘𝜋 2 𝐸
𝑓cr𝓁 =
(3.45)
12(1 − 𝜇2 )(ℎ∕𝑡)2
where h is the depth of the web and k is the buckling coefficient. For long plates the value of k was found to be 23.9
for simple supports and 41.8 for fixed supports as listed in
Table 3.2. The relationships between the buckling coefficient and the aspect ratio a/h are shown in Fig. 3.28. When a
simply supported plate is subjected to a compressive bending
stress higher than the tensile bending stress, the buckling
coefficient k is reduced according to the bending stress ratio
fc /ft as shown in Fig. 3.29.3.7
In practice the bending strength of a beam web not
only is affected by the web slenderness ratio h/t, the
Figure 3.28
ratio, a/h.3.1
Bending buckling coefficients of plates vs. aspect
aspect ratio a/h, and the bending stress ratio 𝑓c ∕𝑓t , but
also depends on the mechanical properties of material (E,
𝐹y , and 𝜇) and the interaction between flange and web
components. In addition, the buckling coefficient k for the
web is influenced by the actual edge restraint provided
by the beam flange. Because the derivation of an exact
analytical solution for the stability and the postbuckling
strength of plate assemblies is extremely cumbersome,
the AISI design criteria have been based on the results of
tests.
3.3.1.2.2 Postbuckling Strength and Effective Depth
of Webs In the past, several design formulas for computing
the effective web depth have been developed by Bergfelt,
Thomasson, Kallsner, Hoglund, DeWolf, Gladding,
LaBoube, and Yu3.51–3.61 to account for the actual buckling
strength and the postbuckling behavior of beam webs. The
effective web depth approach has been used in several
specifications.3.62,3.63 In 1986, Cohen and Pekoz3.64 evaluated the test results reported by LaBoube and Yu,3.59–3.61
Cohen and Pekoz,3.64 Kallsner,3.54 Johnson,3.65 He,3.66 and
van Neste3.67 and developed the needed design formulas
for webs connected to stiffened, unstiffened, and partially
stiffened compression flanges. Some statistical data on the
correlation are given in Ref. 3.17.
Consequently, design equations were included in Section
B2.3 of the 1986 edition of the AISI Specification for
computing the effective width of webs and stiffened
elements with a stress gradient as shown in Fig. 3.30.
The same equations were used in the 1996 edition of the
AISI Specification.
Because the AISI design equations for computing the
effective width of webs implicitly assumed that the beam
flange provided beneficial restraint to the web, the test
data on flexural tests of C- and Z-sections summarized by
Schafer and Pekoz3.168 indicated that the AISI equations can
be unconservative if the overall web depth (ℎ0 ) to overall
flange width (𝑏0 ) ratio exceeds 4. In 2001, due to the lack
of a comprehensive method for handling web and flange
interaction, the North American Specification adopted the
following two-part approach in Section B2.3 for computing
the effective width of beam webs and other stiffened
elements under a stress gradient. The same design equations
are retained in Section 1.1.2 of the 2016 edition of the
Specification:
a. Strength Determination
i. For webs under a stress gradient (𝑓1 in compression
and 𝑓2 in tension as shown in Fig. 3.30a):
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓)
(3.46)
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
77
Figure 3.29 Buckling coefficient k for simply supported plates subjected to nonuniform longitudinal bending stress.3.7 Reproduced with permission from Chatto & Windus, London.
For ℎ0 ∕𝑏0 ≤ 4
𝑏e
𝑏1 =
3+𝜓
1
𝑏2 = 𝑏e when 𝜓 > 0.236
2
𝑏2 = 𝑏𝑒 − 𝑏1 when 𝜓 ≤ 0.236
𝑏1 =
(3.47a)
(3.47b)
(3.47c)
For ℎ0 ∕𝑏0 > 4
𝑏e
3+𝜓
𝑏𝑒
𝑏2 =
− 𝑏1
1+𝜓
𝑏1 =
(3.48a)
(3.48b)
In addition, 𝑏1 + 𝑏2 shall not exceed the compression
portion of the web calculated on the basis of the effective section.
ii. For other stiffened elements under a stress gradient (𝑓1
and 𝑓2 in compression as shown in Fig. 3.30b)
𝑘 = 4 + 2(1 − 𝜓)3 + 2(1 − 𝜓)
(3.49)
𝑏e
3−𝜓
𝑏2 = 𝑏e − 𝑏1
(3.50a)
(3.50b)
In the above expressions,
𝑏1 = effective width as shown in Fig. 3.30
𝑏2 = effective width as shown in Fig. 3.30
𝑏e = effective width b determined in accordance
with Eq. (3.32) through Eq. (3.35) with f1
substituted for f and k determined from Eq.
(3.46) or (3.49)
𝑏0 = out-to-out width of compression flange as
shown in Fig. 3.30c
𝑓1 , 𝑓2 = stresses shown in Fig. 3.30 calculated on basis
of effective section, where f1 and f2 are both
compression, f1 ≥ f2
ℎ0 = out-to-out depth of web as shown in Fig. 3.30c
𝑘 = plate buckling coefficient
𝜓 = |𝑓2 ∕𝑓1 | (absolute value)
78
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
(a)
(b)
(c)
Figure 3.30 (a, b) Stiffened elements with stress gradient and webs.1.417 (c) Out-to-out dimensions
of webs and stiffened elements under stress gradient.1.417
b. Serviceability Determination. The effective widths used
in determining serviceability shall be calculated in accordance with Eqs. (3.46)–(3.50) except that 𝑓d1 and 𝑓d2
are substituted for 𝑓1 and 𝑓2 , where 𝑓d1 and 𝑓d2 are the
computed stresses 𝑓1 and 𝑓2 based on the effective section
at the load for which serviceability is determined.
In the foregoing design provisions, Eqs. (3.47a), (3.47b),
and (3.47c) for ℎ0 ∕𝑏0 ≤ 4 were adopted from the 1996
edition of the AISI Specification, except that the stress ratio
𝜓 is defined as an absolute value. Equations (3.48a) and
(3.48b), originally developed by Cohen and Pekoz,3.64 were
selected for ℎ0 ∕𝑏 > 4. As compared with the 1996 AISI
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Specification, the change of Eq. (3.48b) for 𝑏2 would result in
somewhat lower strengths when ℎ0 > 4𝑏0 . It should be noted
that due to the use of an absolute value for 𝜓, some signs
were changed in the design equations of the North American
Specification.
Example 3.4 For the box section used in Example 3.3,
it can be shown that the distance from the top compression fiber to the neutral axis is 2.908 in. if the beam webs
are fully effective. Check these two beam webs and determine whether they are fully effective according to Eqs.
(3.46)–(3.48) for strength determination. Use 𝐹y = 33 ksi
and the U.S. customary unit.
SOLUTION
From Fig. 3.31, stresses 𝑓1 and 𝑓2 are computed as follows:
)
(
2.754
= 31.25 ksi (Compression)
𝑓1 = 33
2.908
)
(
1.938
𝑓2 = 33
= 22.00 ksi (Tension)
2.908
According to Eqs. (3.46) for webs under a stress gradient,
|𝑓 |
𝜓 = || 2 || = 0.704
| 𝑓1 |
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓)
= 4 + 2(1 + 0.704)3 + 2(1 + 0.704) = 17.304
ℎ = 4.693 in.
h 4.693
=
= 78.22 < 200
t
0.06
OK
(see Section 3.2 for the maximum h/t ratio). The effective
depth be of the web can be computed in accordance with Eqs.
(3.32)–(3.35) for uniformly compressed stiffened elements
with 𝑓1 substituted for f, h/t substituted for w/t, and the k
79
value computed above. From Eq. (3.35),
√
( ) 𝑓
1.052 ℎ
1
𝜆= √
𝑡
𝐸
𝑘
√
1.052
31.25
(78.22)
= 0.644
=√
29,500
17.304
Since 𝜆 < 0.673, 𝑏e = ℎ = 4.693 in. Because ℎ0 ∕𝑏0 =
5.00∕6.50 < 4 and 𝜓 > 0.236, Eqs. (3.47a) and (3.47b) are
used to compute 𝑏1 and 𝑏2 as follows:
𝑏𝑒
4.693
=
= 1.267 in.
3+𝜓
3 + 0.704
1
𝑏2 = 𝑏𝑒 = 2.347 in.
2
𝑏1 + 𝑏2 = 1.267 + 2.347 = 3.614 in.
𝑏1 =
Since 𝑏1 + 𝑏2 > 2.754 in., the compression portion of the
web is fully effective.
3.3.2
Unstiffened Compression Elements
3.3.2.1 Unstiffened
Elements
under
Uniform
Compression
3.3.2.1.1 Yielding An unstiffened compression element,
such as the flange of the I-shaped column shown in
Fig. 3.32a, may fail in yielding if the column is short
and its w/t ratio is less than a certain value. It may buckle as
shown in Fig. 3.32b at a predictable unit stress, which may
be less than the yield stress, when its w/t ratio exceeds that
limit.
3.3.2.1.2 Local Buckling The elastic critical local buckling stress for a uniformly compressed plate can also be determined by Eq. (3.7), which gives
Figure 3.31 Example 3.4.
𝑓cr𝓁 =
𝑘𝜋 2 𝐸
12(1 − 𝜇2 )(𝑤∕𝑡)2
[Eq. (3.7)]
80
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.32 Local buckling of unstiffened compression elements.1.6
where 𝐸
μ
𝑤∕𝑡
𝑘
=
=
=
=
modulus of elasticity
Poisson’s ratio
flat width–thickness ratio
constant depending upon conditions of
edge support and aspect ratio a/w
For a long rectangular plate simply supported along three
sides, with one unloaded free edge as shown in Fig. 3.33,
𝑘 = 0.425. However, when the restraining effect of the web
is considered, k may be taken as 0.5 for the design of an
unstiffened compression flange.
If the steel exhibits sharp yielding and an unstiffened
compression element is ideally plane, the element will buckle
at the critical stress determined by Eq. (3.7) with the upper
limit of 𝐹y (Fig. 3.34). However, such ideal conditions may
not exist, and an element with a moderate w/t ratio may
buckle below the theoretical elastic buckling stress.
On the basis of experimental evidence a straight line B
is drawn in Fig. 3.34 representing those stresses at which
sudden and pronounced buckling occurred in the tests. The
1980 edition of the AISI Specification considered
√ that the
upper limit of such buckling is at 𝑤∕𝑡 = 63.3∕ 𝐹y and the
Figure 3.33 Buckling coefficient for rectangular plates simply
supported along three sides with one unloaded edge free.3.7
√
endpoint of the line is at 𝑤∕𝑡 = 144∕ 𝐹 y .3.68 ∗ In this region
√
the element will buckle inelastically. If 𝑤∕𝑡 ≤ 63.3∕ 𝐹𝑦 ,
the element will fail by yielding, represented by horizontal
line A.
∗ When the yield stress of steel is less than 33 ksi (228 Mpa or 2320 kg/cm2 ),
the endpoint of line B is 𝑤∕𝑡 = 25.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
81
Figure 3.34 Maximum stress for unstiffened compression
elements.1.161
Additional experimental and analytical investigations on
the local buckling of unstiffened compression elements in
the elastic range have been conducted by Kalyanaraman,
Pekoz, and Winter.3.8,3.69–3.71 These studies considered the
effects of initial imperfection and rotational edge restraint
on the local buckling of compression elements. By using the
procedure outlined in Ref. 3.8, a more realistic value of the
local buckling coefficient can be calculated for compression
elements of cold-formed steel members.
Figure 3.35 shows the correlation between some test data
and the predicted maximum stresses.
3.3.2.1.3 Postbuckling Strength When the w/t ratio of an
unstiffened element exceeds about 25, the element distorts
more gradually at a stress about equal to the theoretical
local buckling stress (curve D in Fig. 3.34) and returns to its
original shape upon unloading because the buckling stress
is considerably below the yield stress. Sizable waving can
occur without permanent set being caused by the additional
stress due to distortion. Such compression elements show a
considerable postbuckling strength.
Based upon the tests made on cold-formed steel sections
having unstiffened compression flanges, the following
equation has been derived by Winter for the effective
width of unstiffened compression elements, for which the
postbuckling strength has been considered3.13 :
√
[
]
( )√
𝐸
𝐸
𝑡
𝑏 = 0.8𝑡
1 − 0.202
(3.51)
𝑓max
𝑤
𝑓max
where 𝑓max is the stress in the unstiffened compression
element at the supported edge (Fig. 3.36). Curve E in
Fig. 3.34 is based on Eq. (3.51) and represents the ultimate
Figure 3.35 Correlation between test data on unstiffened
compression elements and predicted maximum stress.1.161,3.13
Figure 3.36
Effective width of unstiffened compression element.
strength of the element, which is considerably larger than
the elastic buckling stress.
Based on a selected local buckling coefficient of 𝑘 = 0.5,
Eq. (3.51) can be generalized as follows:
√
[
]
( )√
kE
kE
𝑡
1 − 0.286
(3.52)
𝑏 = 1.13𝑡
𝑓max
𝑤
𝑓max
where k is the local buckling coefficient for unstiffened
compression elements. Figure 3.37 shows a comparison
between Eq. (3.27) for stiffened elements and Eq. (3.52) for
unstiffened elements.
Equation (3.51) can also be written in terms of 𝑓cr ∕𝑓max as
√
√
(
)
𝑓cr
𝑓cr
𝑏
1 − 0.3
(3.53)
= 1.19
𝑤
𝑓max
𝑓max
where 𝑓cr is the elastic local buckling stress determined
by Eq. (3.7) with a value of 𝑘 = 0.5. The above equation
82
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.37
elements.
Comparison of generalized equations for stiffened and unstiffened compression
is practically identical to the empirical formula derived by
Kalyanaraman et al. on the basis of some additional results
of tests.3.70
Based on Eq.(3.53), the reduction factor 𝜌 for the effective
width design of unstiffened elements can be determined as
follows:
1.19(1 − 0.3∕𝜆)
𝜌=
(3.54)
𝜆
where 𝜆 is defined in Eq. (3.35).
Prior to 1986, it had been a general practice to design
cold-formed steel members with unstiffened flanges by using
the allowable stress design approach. The effective width
equation was not used in the AISI Specification due to lack
of extensive experimental verification and the concern for
excessive out-of-plane distortions at service loads.
In the 1970s, the applicability of the effective width
concept to unstiffened elements under uniform compression was studied in detail by Kalyanaraman, Pekoz, and
Winter.3.69–3.71 The evaluation of the test data using 𝑘 = 0.43
is presented and summarized by Pekoz in Ref. 3.17, which
shows that Eq. (3.34) gives a conservative lower bound to the
test results of unstiffened compression elements. In addition
to the strength determination, the same study also investigated the out-of-plane deformations in unstiffened elements.
The results of theoretical calculations and test results on
sections having unstiffened elements with 𝑤∕𝑡 = 60 are
presented in Ref. 3.17. It was found that the maximum
amplitude of the out-of-plane deformations at failure can be
twice the thickness as the w/t ratio approaches 60. However,
the deformations are significantly less at service loads.
Based on the above reasons and justifications, the
following provisions were included for the first time in
Section B3.1 of the 1986 AISI Specification for the design
of uniformly compressed unstiffened elements. The same
approach is retained in Section 1.2.1 of the 2016 edition of
the North American Specification:
a. Strength Determination. The effective widths b of unstiffened compression elements with uniform compression are
determined in accordance with Eqs. (3.32)–(3.35) with the
exception that k is taken as 0.43 and w is as defined in
Section 3.2. See Fig. 3.4.
b. Serviceability Determination. The effective widths 𝑏d
used in determining serviceability are calculated in accordance with Eqs. (3.32)–(3.35) except that f is replaced
with 𝑓d and 𝑘 = 0.43.
Example 3.5 Determine the critical buckling stress and
critical buckling load for the thin sheet simply supported at
three edges and one edge free, as shown in Fig. 3.38. Use
U.S. customary unit.
Figure 3.38
Example 3.5.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
SOLUTION
1. The critical buckling stress of the unstiffened compression element based on Eq. (3.7) is
𝑓cr𝓁 =
=
𝑘𝜋 2 𝐸
12(1 − 𝜇2 )(𝑤∕𝑡)2
0.425𝜋 2 (29.5 × 103 )
= 7.808 ksi
12(1 − 0.32 )(4∕0.105)2
In the above calculation, the value of 𝑘 = 0.425 is
slightly conservative (see Fig. 3.33).
2. The critical buckling load is
𝑃 cr𝓁 = Af cr𝓁 = 4(0.105)(7.808) = 3.279 kips
Example 3.6 Calculate the effective width of the compression flange of the channel section (Fig. 3.39) to be used as a
beam. Use 𝐹y = 33 ksi. Assume that the beam web is fully
effective and that lateral bracing is adequately provided. Use
U.S. customary unit.
SOLUTION
Because the compression flange of the given channel is
a uniformly compressed unstiffened element which is
supported at only one edge parallel to the direction of the
stress, the effective width of the flange for strength determination can be computed by using Eqs. (3.32)–(3.35) with
𝑘 = 0.43.
According to Eq. (3.35), the slenderness factor 𝜆 for 𝑓 =
𝐹y is
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
)
(
)√
(
33
1.8463
1.052
= √
= 1.651
0.06
29,500
0.43
Figure 3.39
Example 3.6.
83
Since 𝜆 > 0.673, use Eqs. (3.32) and (3.34) to calculate
the effective width b as follows:
)
(
1 − 0.22∕𝜆
𝑤
𝑏 = 𝜌w =
𝜆
(
)
1 − 0.22∕1.651
=
(1.8463) = 0.97 in.
1.651
3.3.2.2 Unstiffened Elements with Stress Gradient In
concentrically loaded compression members and in flexural
members where the unstiffened compression element is
parallel to the neutral axis, the stress distribution is uniform
before buckling. However, in some cases, such as the lips
of the beam section shown in Fig. 3.40, which are turned
in or out and are perpendicular to the neutral axis, the
compression stress is not uniform but varies in proportion to
the distance from the neutral axis.
An exact determination of the buckling condition of such
elements is complex. When the stress distribution in the lip
varies from zero to the maximum, the buckling coefficient
k may be obtained from Fig. 3.41.3.7 The local buckling
of unstiffened elements under nonuniform compression was
discussed by Kalyanaraman and Jayabalan in Ref. 3.169.
In Section B3.2 of earlier editions of the AISI
Specification,1.314,1.336 the effective widths of unstiffened compression elements and edge stiffeners with stress
gradient were treated as uniformly compressed elements
with the stress f to be the maximum compressive stress in
the element. This conservative design approach was found
to be adequate by Rogers and Schuster on the basis of the
comparisons made with the available test data.3.170
In the early 2000s, additional investigations on the unstiffened elements with stress gradient were carried out by Yiu
and Pekoz at Cornell University3.207,3.208 and by Bambach
and Rasmussen at the University of Sydney.3.209–3.216 These
studies included plain channels bending about the minor axis,
so that the unstiffened elements are under a stress gradient
with one longitudinal edge in compression and the other
longitudinal edge in tension. According to the studies of the
University of Sydney, the effects of the stress distribution
in unstiffened elements on the effective width are shown in
Fig. 3.42.1.346 It can be seen that the effective width of an
unstiffened element increases as the stress at the supported
edge changes from compression to tension.
Subsequently, in 2004, new design provisions were added
in Section B3.2 of the North American Specification for
determining the buckling coefficient k, the reduction factor
𝜌, and the effective width b of the unstiffened elements and
edge stiffeners with stress gradient.1.343 These provisions can
be used not only for unstiffened elements under a stress
84
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.40
Unstiffened lip subjected to stress gradient.
k = plate buckling coefficient defined in this section or,
otherwise, as defined in Section 3.3.1.1
t = thickness of element
w = flat width of unstiffened element, where w/t ≤ 60
𝜓 = |𝑓2 ∕𝑓1 | (absolute value)
(3.55)
𝜆 = slenderness factor defined in Section 3.3.1.1 with
f equal to the maximum compressive stress on the
effective element
𝜌 = reduction factor defined in this section or, otherwise,
as defined in Section 3.3.1.1
Figure 3.41 Buckling coefficient for unstiffened compression
elements subjected to nonuniform stress.3.7
gradient with both longitudinal edges in compression but
also for unstiffened elements under a stress gradient with one
longitudinal edge in compression and the other longitudinal
edge in tension. These design provisions were kept in Section
1.2.2 in the 2016 edition of the North American Specification
with some revisions to the definitions of stresses f1 and f2 so
that the effective width of unstiffened elements is determined
iteratively due to a shift of the neutral axis location. The
following excerpts are adapted from Section 1.2.2 of the
2016 edition of the North American Specification:
(a) Strength Determination. The effective width, b, of an
unstiffened element under stress gradient shall be determined in
accordance with Section 3.3.1.1 with f equal to the maximum
compressive stress on the effective element and the plate buckling coefficient, k, determined in accordance with Section 1.2.2
of the Specification, unless otherwise noted. For the cases where
𝑓1 is in compression and 𝑓2 is in tension, ρ in Section 3.3.1.1
shall be determined in accordance with this section.
1. When both 𝑓1 and 𝑓2 are in compression (Fig. 3.43), the plate
buckling coefficient shall be calculated in accordance with
either Eq. (3.56) or Eq. (3.57) as follows:
If the stress decreases toward the unsupported edge
(Fig. 3.43a),
𝑘=
1.2.2 Unstiffened Elements and Edge Stiffeners
with Stress Gradient
The following notation shall apply in this section of the Specification:
b = effective width measured from the supported edge,
determined in accordance with Eqs. (3.32)–(3.35)
with f equal to the maximum compressive stress on
the effective element and with k and 𝜌 being
determined in accordance with this section
𝑏0 = overall width of unstiffened element of unstiffened
C-section member as defined in Fig. 3.45
𝑓1 , 𝑓2 = stresses shown in Figs. 3.43, 3.44, and 3.45
calculated on the basis of the gross section, where 𝑓1
and 𝑓2 are both compression, 𝑓1 ≥ 𝑓2
h0 = overall depth of unstiffened C-section member as
defined in Fig. 3.45
0.578
𝜓 + 0.34
(3.56)
If the stress increases toward the unsupported edge
(Fig. 3.43b),
𝑘 = 0.57 − 0.21𝜓 + 0.07𝜓 2
(3.57)
2. When 𝑓1 is in compression and 𝑓2 in tension (Fig. 3.44),
the reduction factor and plate buckling coefficient shall be
calculated as follows:
i. If the unsupported edge is in compression (Fig. 3.44a):
⎧1 when 𝜆 ≤ 0.673(1 + 𝜓)
⎪
⎪
1 − [0.22(1 + 𝜓)]∕𝜆
𝜌 = ⎨(1 + 𝜓)
𝜆
⎪
⎪
when 𝜆 > 0.673(1 + 𝜓)
⎩
(3.58)
𝑘 = 0.57 + 0.21𝜓 + 0.07𝜓 2
(3.59)
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Figure 3.42 Reduction factor ρ vs. slenderness factor λ for unstiffened elements with stress
gradient.1.431
Figure 3.43
Unstiffened elements under stress gradient, both longitudinal edge in compression.1.417
Figure 3.44 Unstiffened elements under stress gradient, one longitudinal edge in compression and
the other longitudinal edge in tension.1.417
85
86
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.45
Unstiffened elements of C-section under stress gradient for alternative methods.1.417
ii. If the supported edge is in compression (Fig. 3.44b), for
𝜓 <1
⎧1 when 𝜆 ≤ 0.673
⎪
⎪
(1 − 0.22𝜆)
𝜌 = ⎨(1 − 𝜓)
(3.60)
𝜓
𝜆+
⎪
⎪
when 𝜆 > 0.673
⎩
𝑘 = 1.70 + 5𝜓 + 17.1𝜓 2
and for 𝜓 ≥ 1
(3.61)
𝜌=1
The effective width, b, of the unstiffened elements of an
unstiffened C-section member is permitted to be determined
using the following alternative methods, as applicable:
1. Alternative 1 for unstiffened C-sections: When the unsupported edge is in compression and the supported edge is in
tension (Fig. 3.45a),
𝑏=𝑤
where
when 𝜆 ≤ 0.856
(3.62)
𝑏 = 𝜌w when 𝜆 > 0.856
(3.63)
√
𝜌 = 0.925∕ 𝜆
(3.64)
𝑘 = 0.145(bo ∕ho ) + 1.256
(3.65)
(b) Serviceability Determination. The effective width 𝑏d used
in determining serviceability shall be calculated in accordance
with Specification Section 1.2.2a, except that 𝑓d1 and 𝑓d2 are
substituted for 𝑓1 and 𝑓2 as shown in Figs. 3.43, 3.44, and 3.45,
respectively, at the load for which serviceability is determined.
The applications of the above design provisions are illustrated in Examples 3.7 and 4.2.
3.3.3
Uniformly Compressed Elements with Stiffeners
3.3.3.1 Uniformly Compressed Elements with a Simple
Lip Edge Stiffener An edge stiffener is used to provide a
continuous support along a longitudinal edge of the compression flange to improve the buckling stress. Even though in
most cases the edge stiffener takes the form of a simple lip
(Fig. 3.46), other types of stiffeners, as shown in Fig. 3.47,
can also be used for cold-formed steel members.3.78,3.173
In order to provide the necessary support for the compression element, the edge stiffener must possess sufficient
rigidity. Otherwise it may buckle or displace perpendicular
to the plane of the element to be stiffened.
0.1 ≤ 𝑏o ∕ℎo ≤ 1.0
2. Alternative 2 for unstiffened C-sections: When the supported
edge is in compression and the unsupported edge is in tension
(Fig. 3.45b), the effective width is determined in accordance
with Section 3.3.1.2.
Where stress, 𝑓1 , occurs at the unsupported edge as in
Figs 3.43(b), 3.44(a), and 3.45(a), the design stress, f, shall
be taken at the extreme fiber of the effective section, and
𝑓1 is the calculated stress, based on the effective section, at
the edge of the gross section. If the only elements not fully
effective are unstiffened elements with stress gradient, as in
Figure 3.45(a), the stresses 𝑓1 and 𝑓2 are permitted to be
based on the gross section, f taken equal to 𝑓1 , and iteration
is not required.
The extreme tension fiber in Figs. 3.44(b) and 3.45(b) shall
be taken as the edge of the effective section closer to the
unsupported edge.
Figure 3.46
Figure 3.47
Edge stiffener.
Edge stiffeners other than simple lip.3.78
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Both theoretical and experimental investigations on the
local stability of flanges stiffened by lips and bulbs have been
conducted in the past.3.75–3.82,3.217–3.220 The design requirements included in the 1986 and 1996 editions of the AISI
Specification for uniformly compressed elements with an
edge stiffener are based on the analytical and experimental
investigations on adequately stiffened elements, partially
stiffened elements, and unstiffened elements conducted by
Desmond, Pekoz, and Winter3.75,3.76 with additional studies
carried out by Pekoz and Cohen.3.17 Those design provisions were developed on the basis of the critical buckling
criterion and the ultimate-strength criterion. In this design
approach, the design requirements recognize that the needed
stiffener rigidity depends on the width-to-thickness ratio of
the plate element being stiffened. The interaction between
the plate element and the edge stiffener is compensated for
in the design equations for the plate buckling coefficient,
the reduced effective width of a simple lip edge stiffener,
and the reduced area of other stiffened shapes. Because a
discontinuity exists in the 1996 AISI design provisions, in
2001, Dinovitzer’s expressions were adopted in the 2001
edition of the North American Specification for determining
the constant “n” to eliminate the discontinuity.3.221
In 2007, the design provisions were revised to limit the
design equations for applying only to simple lip edge stiffeners due to the fact that previous equations for complex
lip stiffeners were found to be unconservative, in comparison with the nonlinear finite-element analysis conducted by
Schafer, Sarawit, and Pekoz.3.222 These revised provisions
are retained in the 2016 edition of the north American Specification.
According to Section 1.3 of the 2016 edition of the North
American Specification, the effective width of the uniformly
compressed elements with a simple lip edge stiffener can
be calculated by the following equations. For other stiffener
shapes, the design of member strength may be handled by
the direct strength method provided in the Specification.
(a) Strength Determination.
For w/t ≤ 0.328S
𝐼a = 0 (no edge stiffener needed)
𝑏=𝑤
1
𝑏1 = 𝑏2 = 𝑤 (see Fig. 3.48)
2
′
𝑑s = 𝑑𝑠
(3.66)
(3.67)
𝑆 = 1.28
𝐸
𝑓
(3.72)
w is the flat dimension defined in Fig. 3.48; t is the thickness
of the section; 𝑙a , the adequate moment of inertia of the
stiffener so that each component element will behave as a
stiffened element, is defined as
(
(
)3
)
𝑤∕𝑡
4 𝑤∕𝑡
4
𝐼𝑎 = 399𝑡
− 0.328 ≤ 𝑡 115
+ 5 (3.73)
𝑆
𝑆
b is the effective design width; 𝑏1 , 𝑏2 are the portions of effective design width as defined in Fig. 3.48; 𝑑s is the reduced
effective width of the stiffener as defined in Fig. 3.48 and
used in computing overall effective sectional properties; 𝑑𝑠′ is
the effective width of the stiffener calculated in accordance
with Section 3.3.2.1 or 3.3.2.2 (see Fig. 3.48); and
(𝑅𝐼 ) =
𝐼𝑠
≤1
𝐼𝑎
(3.74)
where 𝑙s is the moment of inertia of the full section of stiffener about its own centroidal axis parallel to the element to
be stiffened. For edge stiffeners, the round corner between
the stiffener and element to be stiffened is not considered as
a part of the stiffener:
1 3
(3.75)
(𝑑 𝑡sin2 𝜃)
12
See Fig. 3.48 for definitions of other dimensional variables.
The effective width b in Eqs. (3.69) and (3.70) shall be
calculated in accordance with Section 3.3.1.1 with the plate
buckling coefficient k as given in Table 3.3, where
)
(
𝑤∕𝑡
1
≥
(3.76)
𝑛 = 0.582 −
4𝑆
3
𝐼𝑠 =
(b) Serviceability Determination. The effective width 𝑏d
used in determining serviceability shall be calculated as in
item (a), except that 𝑓d is substituted for f, where 𝑓d is the
computed compressive stress in the effective section at the
load for which serviceability is determined.
According to Ref. 3.17, the distribution of longitudinal
stresses in a compression flange with an edge stiffener is
shown in Fig. 3.49 for three cases.
(3.68)
Table 3.3 Determination of Plate Buckling
Coefficient k
For 𝑤∕𝑡 > 0.328𝑆
( )
1
(𝑏)(𝑅𝐼 ) (see Fig. 3.48)
2
𝑏2 = 𝑏 − 𝑏1 (see Fig. 3.48)
(3.69)
ds = 𝑑𝑠′ (𝑅I )
(3.71)
𝑏1 =
√
where
87
(3.70)
Simple Lip Edge Stiffener (140∘ ≥ θ ≥ 40∘ )
𝐷∕𝑤 ≤ 0.25
3.57 (𝑅I )𝑛 + 0.43 ≤ 4
0.25 < 𝐷∕𝑤 ≤ 0.8
(4.82 − 5𝐷∕𝑤) (𝑅I )𝑛 + 0.43 ≤ 4
88
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
w
D
θ
d
D, d = Actual stiffener dimensions
Stress f for Compression Flange
b1
b2
d's = Effective width of stiffener
calculated according to
Section 1.2.1 or 1.2.2 of
North American Specification
ds = Reduced effective width of
stiffener
Stress f3 for Lip
ds
d's
d
Centroidal Axis
Figure 3.48
Elements with simple lip edge stiffener.1.417
The design criteria are intended to account for the inability
of the edge stiffener to prevent distortional buckling by
reducing the local buckling coefficient k for calculating the
effective design width of the compression element. Because
the empirical equations were derived on the basis of the tests
of back-to-back sections with strong restraint against web
buckling, past research demonstrated that these AISI design
equations may provide unconservative strength predictions
for laterally braced beams and columns with edge-stiffened
flanges when the distortional buckling mode of the compression flange is critical.3.168 Additional discussions of distortional buckling are given in Section 4.2.4 for beams and
Section 5.4 for columns.
Research conducted by Young and Hancock on channels
with inclined simple edge stiffeners using a yield stress of
450 MPa (65.3 ksi or 4588 kg/cm2 ) indicated that the design
strength predicted by the North American Specification are
conservative for all channels with outward and inward edge
stiffeners, when the flange w/t ratios are between 20 and
30, but are slightly unconservative for channels with the
flange w/t ratios between 40 and 50, except for channels
with inward edge stiffeners.3.219 For channels having flange
w/t ratio of 65, the North American Specification predicts
unconservative results.
Example 3.7 Compute the effective width of the compression flange of the channel section with an edge stiffener as
shown in Fig. 3.50. Assume that the channel is used as a
beam and that lateral bracing is adequately provided. Use
𝐹y = 33 ksi. Also compute the reduced effective width of the
edge stiffener.
SOLUTION
1. Effective Width of Compression Flange. Because the
compression flange is a uniformly compressed element
with an edge stiffener, its effective width should be
determined according to Eqs. (3.66)–(3.76).
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
Figure 3.49 Stress distribution in edge-stiffened flange.3.17
Figure 3.50
Example 3.7
89
90
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
As the √
first step, the flat width w, w/t ratio, and
𝑆 = 1.28 𝐸∕𝑓 are computed as follows:
𝑤 = 3.5 − 2(𝑅 + 𝑡) = 3.163 in.
𝑤 3.163
=
= 42.17
𝑡
0.075
√
√
29,500
𝐸
𝑆 = 1.28
= 1.28
= 38.27
𝑓
33
0.328𝑆 = 0.328(38.27) = 12.55
Since 𝑤∕𝑡 > 0.328S, 𝑏 < 𝑤, the effective width of the
compression flange can be determined by using the
following k value: For the given simple lip edge stiffener with θ = 90∘ and 𝐷∕𝑤 = 0.72∕3.163 = 0.228,
which is less than 0.25, according to Table 3.3,
3.57(𝑅𝐼 )𝑛 + 0.43 ≤ 4
where
Use 𝑛 = 1∕3. The local buckling coefficient is
𝑘 = 3.57(0.251)1∕3 + 0.43
= 2.68 < 4
OK
Use 𝑘 = 2.68 to calculate the effective width of the
compression flange by using Eqs. (3.32)–(3.35) as
follows:
√
( )√𝑓
1.052
33
1.052 𝑤
(42.17)
=√
𝜆= √
𝑡
𝐸
29500
𝑘
2.68
= 0.906 > 0.673
The effective width of the compression flange is
(
)
1 − 0.22∕𝜆
𝑏 = 𝜌w =
𝑤
𝜆
(
)
1 − 0.22∕0.906
=
(3.163)
0.906
= 2.643 in.
(𝑅I ) = 𝐼s ∕𝐼a ≤ 1
[Eq. (3.74)]
]
[
𝑤∕𝑡
1
≥
𝑛 = 0.582 −
4𝑆
3
[Eq. (3.76)]
For the simple lip edge stiffener,
𝑑 = 𝐷 − (𝑅 + 𝑡) = 0.551 in.
and
1 3
[Eq. (3.75)]
𝑑 𝑡 = 1.047 × 10−3 in.4
12
Based on Eq.(3.73),
(
)3
𝑤∕𝑡
𝐼𝑎 = 399𝑡4
− 0.328
𝑆
(
)3
42.17
= 399(0.075)4
− 0.328
38.27
𝐼s =
Based on Eqs. (3.69) and (3.70), the effective flange
widths 𝑏1 and 𝑏2 (Fig. 3.48) are determined as follows:
( )
𝑏
(𝑅𝐼 )
𝑏1 =
2
)
(
2.643
(0.251) = 0.332 in.
=
2
𝑏2 = 𝑏 − 𝑏1 = 2.643 − 0.332 = 2.311 in.
2. Reduced Effective Width of Edge Stiffener. The
effective width of the edge stiffener under a gradient
can be determined by using Section 3.3.2.2. According
to Eq. (3.56),
|f |
0.578
where 𝜓 = || 2 ||
𝑘=
𝜓 + 0.34
| 𝑓1 |
In the above equation, the compressive stresses 𝑓1
and 𝑓2 as shown in Fig. 3.51 are calculated on the basis
= 5.852 × 10-3 in.4
The above computed value should not exceed the
following value:
(
)
(
)
𝑤∕𝑡
42.17
𝑡4 115
+ 5 = (0.075)4 115
+5
𝑆
38.27
= 4.168 × 10−3 in.4
Use Ia = 4.168 × 10−3 in.4 Therefore
𝐼𝑠
1.047 × 10−3
=
= 0.251 < 1 OK
𝐼𝑎
4.168 × 10−3
(
)
42.17
= 0.582 −
= 0.307 < 13
4(38.27)
𝑅𝐼 =
𝑛
Figure 3.51
Stress distribution in edge stiffener.
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
of the gross section as follows:
)
(
4.8312
= 31.886 ksi
𝑓1 = 33
5.0
)
(
4.28
𝑓2 = 33
= 28.248 ksi
5.0
Therefore,
| 28.248 |
| = 0.886
𝜓 = ||
|
| 31.886 |
and
0.578
= 0.471
𝑘=
0.886 + 0.34
The k value of 0.471 calculated above for the edge
stiffener under a stress gradient is slightly larger than
the k value of 0.43 for unstiffened elements under
uniform compression.
The effective width of the edge stiffener can be determined as follows:
𝐷 − (𝑅 + 𝑡)
𝑑
=
= 7.35
𝑡
𝑡
𝑓 = 𝑓1 = 31.886
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
(
)
√
31.886
1.052
= √
(7.35)
29,500
0.471
= 0.370 < 0.673
𝜌 = 1.0
91
The effective width of the edge stiffener as shown in
Fig. 3.50 is
𝑑𝑠′ = 𝑑 = 0.551 in.
The reduced effective width of the edge stiffener is
𝑑𝑠 = 𝑑𝑠′ (𝑅I ) = 0.551(0.251) = 0.138 in.
3.3.3.2 Uniformly Compressed Elements Restrained
by Intermittent Connections This section is used to
determine the effective width of a plate fastened to a
deck or other structures, where the intermittent fastener
spacing may not be close enough to ensure that the plate
is fully effective. Figure 3.52(a) shows a deck plus plate
composite cross-section subjected to bending. The connected
plate is virtually in uniform compression as illustrated in
Figure 3.52(b).
Luttrell and Balaji8.92 and Snow and Easterling8.99 developed a method to determine the effective width of the
compression plate. The method recognizes the postbuckling
strength of the compression plate after local buckling waves
are formed between connections. Two possible stages are
considered: when the compressive stress in the plate f is less
than the critical compressive stress, 𝐹𝑐 , calculated based
on “column-like” buckling of the plate, the effective width
is determined based on uniformly compressed stiffened
element; when compressive stress, f, exceeds 𝐹𝑐 , an equivalent width is determined to provide the approximate force
contribution of the buckled plate in resisting the buckled
shape (bending of the plate).
The design provisions were incorporeatd in to the 2012
edition of the North American Specification and is retained in
Section 1.1.4 of the 2016 edition of the Specification. These
provisions are adapted in the following:
1.1.4 Uniformly Compressed Elements Restrained
by Intermittent Connections
Figure 3.52(a)
Built-up deck.1.431
Figure 3.52(b)
The provisions of this section shall apply to compressed
elements of flexural members only. The provisions shall be
limited to multiple flute built-up members having edge-stiffened
cover plates. When the spacing of fasteners, s, of a uniformly
Plate in compression.1.431
92
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.52(c)
Dimension illustration of cellular deck.1.417
where Fy = Design yield stress of the compression
element restrained by intermittent
connections
d = Overall depth of the built-up member
f = Stress in compression element restrained by
intermittent connections when the
controlling extreme fiber stress is Fy
compressed element restrained by intermittent connections is
not greater than the limits specified in Specification Section
I1.3, the effective width shall be calculated in accordance with
Specification Section 1.1. When the spacing of fasteners is
greater than the limits specified in Specification Section I1.3,
the effective width shall be determined in accordance with (a)
and (b) below.
(a) Strength Determination. The effective width of the uniformly
compressed element restrained by intermittent connections
shall be determined as follows:
1. When 𝑓 < 𝐹𝑐 , the effective width of the compression
element between connection lines shall be calculated in
accordance with Specification Section 1.1.
2. When 𝑓 ≥ 𝐹𝑐 , the effective width of the compression
element between connection lines shall be calculated in
accordance with Specification Section 1.1, except that
the reduction factor, 𝜌, shall be the lesser of the value
determined in accordance with Eq. (3.34) and the value
determined by Eq. (3.77):
𝜌 = 𝜌𝑡 𝜌𝑚
where
𝜌t = 1.0
(3.77)
for 𝜆𝑡 ≤ 0.673
𝜌t = (1.0 − 0.22∕λt )∕λt for 𝜆𝑡 > 0.673
√
where
𝜆𝑡 =
𝐹𝑐
𝐹cr𝓁
where Fc = Critical column buckling stress of
compression element
= 3.29 E∕(s∕t)2
(3.78)
The provisions of this section shall apply to shapes that meet
the following limits:
1.5 in. (38.1 mm) ≤ d ≤ 7.5 in. (191 mm),
0.035 in. (0.889 mm) ≤ t ≤ 0.060 in. (1.52 mm),
2.0 in. (50.8 mm) ≤ s ≤ 8.0 in. (203 mm),
33 ksi (228 MPa or 2320 kg∕cm2 ) ≤ Fy ≤ 60 ksi (414
MPa or 4220 kg∕cm2 ), and
(5) 100 ≤ 𝑤∕𝑡 ≤ 350.
(1)
(2)
(3)
(4)
The effective width of the edge stiffener and the flat portion,
e, shall be determined in accordance with Specification Section
1.3(a) (or Section 3.3.3.1 of this book) with modifications as
follows:
For 𝑓 < 𝐹𝑐
𝑤=𝑒
(3.82)
For 𝑓 ≥ 𝐹𝑐
(3.79)
For the flat portion, e, the effective width, b, in Eqs. (3.69)
and (3.70) shall be calculated in accordance with Specification
Section 1.1(a) (or Eqs. (3.32)– (3.35) in this book) with
(3.80)
where s = Center-to-center spacing of connectors in
line of compression stress
E = Modulus of elasticity of steel
t = Thickness of cover plate in compression
Fcr𝓁 = Critical buckling stress defined in Eq.
(3.45), where w is the transverse spacing of
connectors
( )√
𝐹𝑦
tF 𝑐
≤ 1.0
(3.81)
𝜌𝑚 = 8
𝑓
df
(i) w taken as e,
(ii) if 𝐷∕𝑒 ≤ 0.8
k is determined in accordance with Table 3.3
if 𝐷∕e > 0.8
k = 1.25, and
(iii) 𝜌 calculated using Eq. (3.77) in lieu of Eq. (3.34)
where w = Flat width of element measured between
longitudinal connection lines and exclusive of
radii at stiffeners
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
93
e = Flat width between the first line of connector
and the edge stiffener. See Fig. 3.52(c).
D = Overall length of stiffener as defined in
Specification Section 1.3 (or Section 3.3.3.1 of
this book)
For the edge stiffener, ds and Ia shall be determined using w′
and f ′ in lieu of w and f, respectively.
𝑤′ = 2𝑒 + minimum of (0.75𝑠 and 𝑤1 )
(3.83)
𝑓 = Maximum of (𝜌𝑚 𝑓 and 𝐹𝑐 )
(3.84)
′
where f ′ = Stress used in Specification Section 1.3(a)
for determining effective width of edge
stiffener
Fc = Buckling stress of cover plate determined
in accordance with Eq. 3.80
w′ = Equivalent flat width for determining the
effective width of edge stiffener
w1 = Transverse spacing between the first and
the second line of connectors in the
compression element. See Fig. 3.52(c).
The provisions of this section shall not apply to single flute
members having compression plates with edge stiffeners.
(b) Serviceability Determination. The effective width of the
uniformly compressed element restrained by intermittent
connections used for computing deflection shall be determined in accordance with Section (a) except that:
(1) 𝑓𝑑 shall be substituted for f, where 𝑓𝑑 is the computed
compression stress in the element being considered at
service load, and
(2) The maximum extreme fiber stress in the built-up
member shall be substituted for 𝐹𝑦 .
3.3.3.3 Uniformly Compressed Elements with Intermediate Stiffeners
3.3.3.3.1 Uniformly Compressed Elements with Single
Intermediate Stiffener In the design of cold-formed steel
beams, when the width-to-thickness ratio of the stiffened
compression flange is relatively large, the structural efficiency of the section can be improved by adding an intermediate stiffener as shown in Fig. 3.53.
The buckling behavior of rectangular plates with central
stiffeners is discussed in Ref. 3.7. The load-carrying
capacity of an element with a longitudinal intermediate
stiffener has been studied by Höglund,3.72 König,3.73 König
and Thomasson,3.74 Desmond, Pekoz, and Winter,3.75–3.77
Pekoz,3.17 and Yang and Schafer.3.223
In the study of Bernard, Bridge, and Hancock,3.171,3.172
both local buckling and distortional buckling in the
Figure 3.53
Section with single intermediate stiffener.
compression flange of profiled steel decks were discussed
by the researchers.
As far as the design provisions are concerned, the 1980
and earlier editions of the AISI Specification included the
requirements for the minimum moment of inertia of the
intermediate stiffener for multiple-stiffened compression
elements. When the size of the actual intermediate stiffener
did not satisfy the required minimum moment of inertia,
the load-carrying capacity of the member had to be determined either on the basis of a flat element disregarding the
intermediate stiffener or through tests. For some cases, this
approach could be unduly conservative.3.17
The AISI design provisions were revised in 1986 on the
basis of the research findings reported in Refs. 3.75–3.77.
In that method, the buckling coefficient k for determining
the effective width of subelements and the reduced area
of the stiffener was calculated by using the ratio 𝐼s ∕𝐼a ,
where 𝐼s is the actual stiffener moment of inertia and 𝐼a is
the adequate moment of inertia of the stiffener determined
from the applicable equations. The same design requirements
were retained in the 1996 edition of the AISI Specification. Because a discontinuity could occur in those equations,
the design provisions were revised in the 2001 edition of
the North American Specification by adopting Dinovitzer’s
expressions to eliminate the discontinuity.3.221
In the 2007 edition of the North American Specification, the design of uniformly compressed stiffened elements
with a single intermediate stiffener was merged with the
stiffened elements having multiple intermediate stiffeners.
Section 3.3.3.3.2 provides the AISI design requirements for
this particular case by using the number of stiffeners equal to
unity (i.e., 𝑛 = 1) in Eqs. (3.92) and (3.93). See Example 4.7
for the application of these equations.
3.3.3.3.2 Uniformly Compressed Elements with Multiple Intermediate Stiffeners In beam sections, the normal
stresses in the flanges result from shear stresses between
the web and flange. The web generates the normal stresses
by means of the shear stress which transfers to the flange.
The more remote portions of the flange obtain their normal
stress through shear from those close to the web. For this
reason there is a difference between webs and intermediate
94
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
stiffeners. The latter is not a shear-resisting element and
does not generate normal stresses through shear. Any normal
stress in the intermediate stiffener must be transferred to it
from the web or webs through the flange portions. As long
as the subelement between web and stiffener is flat or is only
very slightly buckled, this stress transfer proceeds in an unaffected manner. In this case the stress in the stiffener equals
that at the web, and the subelement is as effective as a regular
single-stiffened element with the same w/t ratio. However,
for subelements having larger w/t ratios, the slight waves of
the subelement interfere with complete shear transfer and
create a “shear lag” problem which results in a stress distribution as shown in Fig. 3.54.
In the 1996 edition of the AISI Specification, the design
requirements for uniformly compressed elements with
multiple intermediate stiffeners and edge-stiffened elements
with intermediate stiffeners included (a) the minimum
moment of inertia of the full stiffener about its own
centroidal axis parallel to the element to be stiffened,
(b) the number of stiffeners considered to be effective, (c)
the “equivalent element” of the entire multiple-stiffened
element for closely spaced stiffeners with an “equivalent
thickness,” (d) the reduced effective width of subelement
having 𝑤∕𝑡 > 60, and (e) the reduced effective stiffener
area when the w/t ratio of the subelement exceeds 60. The
reasons for using the above requirements are discussed by
Yu in Ref. 1.354.
In the past, the structural behavior and strength of
cold-formed steel members with multiple longitudinal
intermediate stiffeners have been investigated by Papazian,
Schuster, and Sommerstein,3.174 Schafer and Pekoz,3.175 3.176
Acharya and Schuster,3.177,3.178 Teter and Kolakowski,3.224
and Schafer.3.225 Some of these studies considered the
distortional buckling of the entire stiffened elements as a
unit (Fig. 3.55a) and local buckling of the subelements
between stiffeners (Fig. 3.55b). The AISI Specification and
the Canadian Standard have been compared with analytical
Figure 3.55 Buckling modes of multiple-stiffened elements with
longitudinal intermediate stiffeners3.176 : (a) distortional buckling
mode; (b) local buckling mode.
and experimental results. It has been found that the 1996
AISI design requirements were nearly 20% unconservative
for the 94 members studied.3.175,3.176 Based on the experimental and numerical studies, a method for calculating
the ultimate strength of stiffened elements with multiple
intermediate stiffeners was proposed by Schafer and Pekoz
in Ref. 3.176. This method involves the calculation of the
critical local buckling stress for the subelement and the
distortional buckling stress for the entire multiple-stiffened
element. Because the experimental and numerical data
revealed that the overall (distortional) buckling mode
usually dominated the behavior, a modified effective width
equation was proposed for the entire multiple-stiffened
element by using the proposed plate buckling coefficient to
determine the reduction factor.
Consequently, in 2001, the design provisions were revised
to reflect those additional research findings.1.336,3.176 The
same requirements are retained in Section 1.4.1 of the 2016
edition of the North American Specification for determining
the effective width of uniformly compressed stiffened
elements with single or multiple intermediate stiffeners as
given below:
1.4.1 Effective Widths of Uniformly Compressed Stiffened Elements with Single or Multiple Intermediate
Stiffeners
Figure 3.54 Stress distribution in compression flange with intermediate stiffeners.1.161
The following notation shall apply as used in this section.
Ag = gross area of element including stiffeners
As = gross area of stiffener
be = effective width of element, located at centroid of
element including stiffeners; see Fig. 3.57
STRUCTURAL BEHAVIOR OF COMPRESSION ELEMENTS AND EFFECTIVE WIDTH DESIGN CRITERIA
95
Figure 3.56 Plate widths and stiffener locations.1.347
bo = total flat width of stiffened element; see Fig. 3.56
bp = largest subelement flat width; see Fig. 3.56
ci = horizontal distance from edge of element to
centerline(s) of stiffener(s); see Fig. 3.56
Fcr𝓁 = plate elastic buckling stress
F = uniform compressive stress acting on flat element
h = width of elements adjoining stiffened element (e.g.,
depth of web in hat section with multiple
intermediate stiffeners in compression flange is equal
to h; if adjoining elements have different widths, use
smallest one)
Isp = moment of inertia of stiffener about centerline of flat
portion of element; radii that connect the stiffener to
the flat can be included
k = plate buckling coefficient of element
kd = plate buckling coefficient for distortional buckling
kloc = plate buckling coefficient for local subelement
buckling
Lbr = unsupported length between brace points or other
restraints which restrict distortional buckling of
element
R = modification factor for distortional plate buckling
coefficient
n = number of stiffeners in element
t = element thickness
i = Index for stiffener “i”
𝜆 = slenderness factor
𝜌 = reduction factor
The plate buckling coefficient, k, shall be determined from
the minimum of Rkd and 𝑘loc , as determined in accordance with
Specification Section 1.4.1.1 or 1.4.1.2, as applicable:
The effective width shall be calculated in accordance with
Eq. (3.85) as follows:
( )
𝐴𝑔
(3.85)
𝑏e = ρ
𝑡
If 𝐿br < 𝛽𝑏o , 𝐿br ∕𝑏o is permitted to be substituted for 𝛽
to account for increased capacity due to bracing.
b. Serviceability Determination. The effective width, 𝑏d , used
in determining serviceability shall be calculated as in Specification Section 1.4.1.1(a), except that 𝑓d is substituted
for f, where 𝑓d is the computed compressive stress in the
element being considered based on the effective section at
the load for which serviceability is determined.
where
{
𝜌=
1 when 𝜆 ≤ 0.673
(1 − 0.22∕𝜆)∕𝜆 when 𝜆 > 0.673
√
where
𝜆=
where
𝐹cr𝓁
𝑓
𝐹cr𝓁
𝜋2𝐸
=𝑘
12(1 − 𝜇 2 )
(3.86)
(3.87)
(
𝑡
𝑏𝑜
)2
(3.88)
𝑘 = minimum of Rkd and 𝑘loc
(3.89)
R = 2 when 𝑏o ∕ℎ < 1
𝑅=
11 − bo ∕h
1
≥
5
2
when 𝑏o ∕ℎ ≥ 1
(3.90)
1.4.1.1 Specific Case: Single or n Identical Stiffeners,
Equally Spaced
For uniformly compressed elements with single, or multiple
identical and equally spaced, stiffeners, the plate buckling coefficients and effective widths shall be calculated as follows:
a. Strength Determination3.240
𝑘loc = 4(bo ∕bp )2
𝑘d =
where
(3.91)
2 2
(1 + β ) + γ(1 + n)
β2 [1 + δ(n + 1)]
1
𝛽 = [1 + γ(n + 1)] ∕4
(3.92)
(3.93)
where
𝛾=
𝛿=
10.921sp
𝑏𝑜 𝑡3
𝐴𝑠
𝑏𝑜 𝑡
(3.94)
(3.95)
1.4.1.2 General Case: Arbitrary Stiffener Size, Location, and Number
For uniformly compressed stiffened elements with stiffeners of
arbitrary size, location, and number, the plate buckling coefficients and effective widths shall be calculated as follows:
96
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
a. Strength Determination
𝑘loc = 4(bo ∕bp )2
(3.96)
∑
(1 + β2 )2 + 2 γi ωi
i=1
𝑘d =
(
)
n
∑
β2 1 + 2 δi ωi
n
(3.97)
i=1
where
(
𝛽=
2
n
∑
) 1∕4
γi ωi + 1
(3.98)
i=1
Figure 3.57 Effective width locations.1.347
where
𝛾i =
10.92(Isp )i
bo t3
(
)
c
𝜔i = sin2 π i
bo
(A )
𝛿i = s i
bo t
(3.99)
(3.100)
(3.101)
If 𝐿br < β𝑏o , 𝐿br ∕𝑏o is permitted to be substituted for 𝛽
to account for increased capacity due to bracing.
b. Serviceability Determination. The effective width, 𝑏d , used
in determining serviceability shall be calculated as in Specification Section 1.4.1.2(a), except that 𝑓d is substituted
for f, where 𝑓d is the computed compressive stress in the
element being considered based on the effective section at
the load for which serviceability is determined.
It should be noted that according to Eq. (3.85), the effective width of the uniformly compressed stiffened elements with
multiple intermediate stiffeners is determined from an overall
equivalent flat width (𝐴g ∕𝑡), in which 𝐴g is the gross area
of the stiffened element including intermediate stiffeners. The
equation used for computing the reduction factor, 𝜌, is the same
as Eq. (3.34) , except that in the calculation of slenderness factor
𝜆, the plate buckling coefficient, k, is the lesser of Rkd and
𝑘loc , and the width-to-thickness ratio is based on 𝑏o ∕𝑡, in which
𝑏o is the total overall flat width of the stiffened element. See
Fig. 3.56. As shown in Fig. 3.57, the effective width is placed at
the centroidal line of the entire element including the stiffeners
for the calculation of the effective sectional properties.
3.3.3.3.3 Edge-Stiffened Elements with Intermediate
Stiffeners For the design of edge-stiffened elements with
intermediate stiffeners, if the overall flat width-to-thickness
√
ratio (𝑏o ∕𝑡) is small (i.e., 𝑏o ∕𝑡 ≤ (0.328𝑆 = 0.42 𝐸∕𝑓 ),
the flat subelements and intermediate stiffeners can be fully
effective. However, if the 𝑏o ∕𝑡 ratio is large, three buckling
modes are possible, as shown in Fig. 3.58.1.346,3.226
In order to provide new requirements for computing the
effective width of edge-stiffened elements with intermediate
stiffeners, Section 1.4.2 of the North American Specification
includes the following design provisions1.417 :
1.4.2 Edge-Stiffened Elements with Intermediate
Stiffener(s)
a. Strength Determination. For edge-stiffened elements with
intermediate stiffener(s), the effective width, 𝑏e , shall be
determined as follows:
• If 𝑏o ∕𝑡 ≤ 0.328𝑆, the element is fully effective and no
local buckling reduction is required.
• If 𝑏o ∕𝑡 > 0.328S, the plate buckling coefficient, k, is
determined in accordance with Section 3.3.3.1, but with
𝑏o replacing w in all expressions.
If k calculated from Section 3.3.3.1 is less than 4.0
(𝑘 < 4), the intermediate stiffener(s) is ignored and the
provisions of Section 3.3.3.1 are followed for calculation of the effective width.
If k calculated from Section 3.3.3.1 is equal to 4.0
(𝑘 = 4), the effective width of the edge-stiffened
element is calculated from the provisions of Section
3.3.3.3.2, with the following exception: R calculated in
accordance with Section 3.3.3.3.2 is less than or equal
to 1, where
bo = total flat width of edge-stiffened element
See Sections 3.3.3.1 and 3.3.3.3.2 for definitions of
other variables.
b. Serviceability Determination. The effective width, 𝑏d , used
in determining serviceability shall be calculated as in (a)
above, except that 𝑓d is substituted for f, where 𝑓d , is the
computed compressive stress in the element being considered based on the effective section at the load for which
serviceability is determined.
In the above criteria, the modification factor (R) for the distortional plate buckling coefficient is limited to less than or equal to
1.0 due to the fact that the edge-stiffened element does not have
the same web rotational restraint along the side-supported edge
stiffener.
For the calculation of effective sectional properties, the effective width (𝑏e ) of the edge-stiffened element with intermediate
stiffeners is placed at the centroidal line as shown in Fig. 3.57.
The centroidal line is located on the basis of the gross areas of
subelements and intermediate stiffeners without using the edge
stiffener.
PERFORATED ELEMENTS AND MEMBERS
97
Figure 3.58 Buckling modes in an edge-stiffened element with intermediate stiffeners.1.417
The adequacy of this approach was demonstrated by the
stub compression tests performed by Yang and Hancock in
2003.3.219
3.4 PERFORATED ELEMENTS AND MEMBERS
In cold-formed steel structural members, holes are sometimes provided in webs and/or flanges of beams and columns
for duct work, piping, bracing, and other construction
purposes. For steel storage racks (Fig. 1.10), various types
of holes are often used for the purpose of easy assembly.
The presence of such holes may result in a reduction of
the strength of individual component elements and of the
overall strength of the member depending on the size, shape,
and arrangement of holes, the geometric configuration of
the cross section, and the mechanical properties of the
material used.
The exact analysis and the design of steel sections having
perforated elements are complex, in particular when the
shapes and the arrangement of the holes are unusual. Even
though limited information is available for relatively thick
steel sections,1.148,1.165,3.84–3.86 on the basis of previous
investigations,3.87–3.90 these design criteria may not be
completely applicable to perforated cold-formed steel
sections due to the fact that local buckling is usually a major
concern for thin-walled structural members.
For perforated cold-formed steel structural members the
load-carrying capacity of the member is usually governed
by the buckling behavior and the postbuckling strength of
the component elements. The critical buckling loads for
perforated plates and members have been previously studied
by numerous investigators.3.91–3.111,3.227–3.233 The effect of
Figure 3.59 Effect of circular hole on buckling coefficient in
compression.3.99
circular holes on the buckling coefficients in compression
is shown in Fig. 3.59. Figure 3.60 shows the effect of a
central square hole on the buckling coefficient for a simply
supported square plate, in which the top curve was computed
by the finite-element method developed by Yang.3.112 The
test data obtained from the testing of beams and columns are
also shown in these two figures.3.99
In Figs. 3.59 and 3.60, k is the buckling coefficient for
square plates without holes, 𝑘c is the buckling coefficient
for perforated square plates having a circular hole, 𝑘s is the
buckling coefficient for perforated square plates having a
square hole, d is the diameter of circular holes, h is the width
of square holes, and w is the width of the plate.
The postbuckling strength of perforated compression
elements has also been studied by Davis and Yu in Ref.
3.99. It was found that Winter’s effective width equation for
98
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
3.4.1 Uniformly Compressed Stiffened Elements
with Circular Holes
Based on the Cornell study presented in Ref. 3.100, limited
design provisions have been included in the AISI Specification since 1986. Section 1.1.1 of the North American
Specification1.417 includes the following provisions for determining the effective width of uniformly compressed stiffened
elements with circular holes (Fig. 3.62):
a. Strength Determination. The effective width, b, shall be
calculated by either Eq. (3.102) or Eq. (3.103) as follows:
For 0.50 ≥ 𝑑h ∕𝑤 ≥ 0, 𝑤∕𝑡 ≤ 70, and the distance between
centers of holes ≥ 0.50𝑤 and ≥ 3𝑑h ,
Figure 3.60 Effect of square hole on buckling coefficient in
compression.3.99
a solid plate [Eq. (3.26)] can be modified for the determination of the effective width of perforated stiffened elements.
Even though the buckling load for the perforated stiffened
element is affected more by square holes than by circular
holes, the postbuckling strength of the elements with square
and circular holes was found to be nearly the same if the
diameter of a circular hole was the same as the width of a
square hole.
The effect of perforations on the design of industrial steel
storage racks has been accounted for by using net section
properties determined by stub column tests.1.165
Considering the effect of holes on the shear buckling
of a square plate, the reduction of the buckling coefficients has been studied by Kroll,3.113 Rockey, Anderson,
and Cheung,3.114,3.115 and Narayanan and Avanessian.3.101
Figure 3.61 shows the buckling coefficients in shear affected
by holes.
⎧𝑤 − 𝑑 when 𝜆 ≤ 0.673
(3.102)
h
⎪
⎪ 𝑤[1 − 0.22∕𝜆 − 0.8𝑑h ∕𝑤 + 0.085𝑑h ∕(𝜆w)]
𝑏=⎨
𝜆
⎪
⎪ when 𝜆 > 0.673
(3.103)
⎩
In all cases,
𝑏 ≤ 𝑤 − 𝑑ℎ
where w
t
dh
𝜆
=
=
=
=
flat width
thickness of element
diameter of holes
as defined in Section 3.3.1.1 with k = 4.0
b. Serviceability Determination. The effective width, 𝑏d , used
in determining serviceability shall be equal to b calculated
in accordance with Eqs. (3.32)–(3.35) except that 𝑓d is
substituted for f, where 𝑓d is the computed compressive
stress in the element being considered.
3.4.2 Uniformly Compressed Stiffened Elements
with Noncircular Holes
For uniformly compressed stiffened elements with noncircular holes such as the perforated web element of steel studs
Figure 3.61
shear.3.114
Effect of circular hole on buckling coefficient in
Figure 3.62 Uniformly compressed stiffened elements with
circular holes.
PERFORATED ELEMENTS AND MEMBERS
Figure 3.63
99
Uniformly compressed stiffened elements with noncircular holes.1.417
shown in Fig. 3.63, the effective width of the perforated
web can be determined by assuming the web to consist of
two uniformly compressed unstiffened elements with the flat
width one on each side of the hole. The effective design width
of these unsiffened compression elements can be calculated
in accordance with Section 3.3.2.1 or the effective area of the
perforated web can be determined from stub-column tests.
The unstiffened strip approach was studied by Miller and
Pekoz at Cornell University in the 1990s.3.186 Test results
indicated that this method is generally conservative for the
wall studs tested in the Cornell program. This approach has
long been used in the Rack Manufacturers Institute (RMI)
Specification for the design of perforated rack columns.1.156
Since 1996, similar requirements were used in the AISI Specification for the design of wall studs under specific limitations. The same requirements were moved from the previous
Section D4 to Section B2.2 of the 2007 edition of the Specification , and was retained in Section 1.1.1 of the 2016 edition
of the Specification:
a. Strength Determination. A uniformly compressed stiffened element with noncircular holes shall be assumed to
consist of two unstiffened strips of flat width, c, adjacent
to the holes (see Fig. 3.63). The effective width, b, of each
unstiffened strip adjacent to the hole shall be determined
in accordance with Eqs. (3.32)–(3.35), except that the plate
buckling coefficient, k, shall be taken as 0.43 and w as c.
These provisions shall be applicable within the following
limits:
1. Center-to-center hole spacing, 𝑠 ≥ 24 in. (610 mm),
2. Clear distance from the hole at ends, 𝑠end ≥ 10 in.
(254 mm),
3. Depth of hole, 𝑑h ≤ 2.5 in. (63.5 mm),
4. Length of hole, 𝐿h ≤ 4.5 in. (114 mm), and
5. Ratio of the depth of hole, 𝑑h , to the out-to-out width,
𝑤𝑜 , 𝑑h ∕𝑤o ≤ 0.5.
Alternatively, the effective width, b, is permitted to
be determined by stub-column tests in accordance with
the test procedure, AISI S902.
b. Serviceability Determination. The effective width, 𝑏d , used
in determining serviceability shall be calculated in accordance with Eqs. (3.32)–(3.35).
It should be noted that the effective area should be based
on the lesser of the total effective design width of two unstiffened elements and the effective design width determined for
the stiffened element with the flat width, w. The calculation
of the effective area for the steel stud having noncircular web
perforations is illustrated in Example III-2 of the 2017 edition
of the AISI Design Manual.1.428
3.4.3 C-Section Webs with Holes under Stress
Gradient
In the past, numerous studies have been conducted to investigate the structural behavior and strength of perforated
elements and members subjected to tension, compression,
bending, shear, and web crippling.3.179–3.193,3.197–3.200 Based
on the research work conducted by Shan et al. at the University of Missouri–Rolla,3.184,3.197 the following requirements
have been included in Section 1.1.3 of the Specification for
determining the effective depth of C-section webs with holes
under stress gradient1.417 :
a. Strength Determination. When 𝑑h ∕ℎ < 0.38, the effective
widths, 𝑏1 and 𝑏2 , shall be determined by Section 3.3.1.2
by assuming no hole exists in the web. When 𝑑h ∕ℎ > 0.38,
the effective width shall be determined by Section 3.3.2.1,
assuming the compression portion of the web consists of
an unstiffened element adjacent to the hole with 𝑓 = 𝑓1 ,
as shown in Fig. 3.64.
b. Serviceability Determination. The effective widths shall be
determined by Section 3.3.1.2 by assuming no hole exists
in the web.
Because the above requirements are based on the experimental study, these provisions are applicable only within the
following limits:
1. 𝑑h ∕ℎ < 0.7
2. ℎ∕𝑡 ≤ 200
100
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.64 C-section webs with holes under stress gradient.
Figure 3.65
Virtual hole method for multiple openings.1.431
3.
4.
5.
6.
Holes centered at mid-depth of the web
Clear distance between holes ≥ 18 in. (457 mm)
Noncircular holes, corner radii ≥ 2t
Noncircular holes, 𝑑h ≤ 2.5 in. (64 mm) and Lh ≤
4.5 in. (114 mm)
7. Circular hole diameter ≤ 6 in. (152 mm)
8. 𝑑h ≥ 9∕16 in. (14 mm), where
d = depth of web hole
h = depth of flat portion of web measured along
the plane of the web
t = thickness of web
Lh = length of web hole
b1 , b2 = effective widths defined by Fig. 3.30
Although these provisions are based on the tests of
C-sections having the web hole centered at mid-depth of
the section, the provisions may be conservatively applied to
sections for which the full unreduced compression region of
the web is less than the tension region. Otherwise, the web
strength must be determined by tests.1.333
The design provisions apply to any hole pattern that fits
within equivalent virtual holes, as shown in Figs. 3.65 and
3.66. Figure 3.65 shows the dimensions 𝐿h and 𝑑h for a
multiple-hole pattern that fits within a noncircular virtual
hole, while Fig. 3.66 illustrates the dimension 𝑑h for a
rectangular hole that exceeds the limits of 2.5 in. (64 mm) ×
4.5 in. (114 mm) but still fits within an allowable circular
virtual hole. For each case, the provisions apply to the
geometry of the virtual hole, not the actual hole or
holes.1.333
For the effect of web holes on the shear strength and web
crippling strength of C-sections, see Section 4.3 on the design
of beam webs. Extensive studies of perforated elements
and members have been conducted by numerous investigators. See Refs. 3.228–3.232, 3.242–3.248, 3.250–3.253.
Design provisions using the Direct Strength Method have
been developed and will be discussed in Chapters 4 and 5.
3.5 DIRECT STRENGTH METHOD AND
CONSIDERATION OF LOCAL AND
DISTORTIONAL BUCKLING
The Direct Strength Method provides a consistent design
procedure for determining cold-formed steel member
strengths under different buckling failure modes. This
method was developed by Schafer and Peoz3.254,3.255
in 1990s and was continued developing by other
researchers.3.281–3.293 In 2004, this method was adopted
into the North American Specification as Appendix 1.1.343
In 2016, this method was incorporated into the main body
of the Specification1.417 and is considered as an equivalent
design method to the Effective Width Method.
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
Figure 3.66
Virtual hole method for opening exceeding limit.1.431
The Direct Strength Method is based on the same assumption as the Effective Width Method3.255 : the member strength
is the function of elastic buckling and the yielding of the
material. Therefore a good estimate of the elastic buckling
will result in a better prediction of member strength. To
realistically predict the member buckling strengths, the
Direct Strength Method analyzes the buckling of the whole
cross-section instead of individual elements, which ensures
that the compatibility and equilibrium are maintained at the
element junctures. To capture the postbuckling behaviors, the
method calibrated the strength expressions with numerous
test data.3.254,3.255 Figures 3.67a and 3.67b show that the
strength prediction expressions have a good agreement
with the test data. Through these test data, the geometric
limitations are established as provided in Table 3.1. For
members outside the limitations, the Specification permits
to use the rational engineering analysis to determine the
member strengths and apply the safety and resistance
factors provided in Specification Section A1.2(c)1.417 :
Ω = 2.00 (ASD) and 𝜙 = 0.80 (LRFD) and 0.75 (LSD).
If test data available, the Specification also permits to use
the safety and resistance factors provided in the relevant
sections in Specification Chapters E through H provided
those tests which are performed per Specification Section
K21.417 , and the calculated resistance factor, 𝜙, is greater
than that in Chapters E through F. Detailed provisions are
provided in Section B4.2 of the Specification.1.417
Figure 3.68a and 3.68b plotted the compression and
bending strengths of a member that is laterally braced
against global (lateral torsional) buckling. The curves
1.5
Local: Eq. (5.65)
Distortional: Eq. (5.70)
Local
1
Distortional
0.5
0
0
1
101
2
3
λ =
4
5
6
7
λ =
Figure 3.67a Comparison between the test data and the nominal axial strengths calculated by DSM
for concentrically loaded pin-ended columns.1.431
8
102
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
1.5
Local: Eq. (4.92)
Distortional: Eq. (4.102)
Local
1
Distortional
0.5
0
0
1
2
λ =
3
4
5
λ =
Figure 3.67b Comparison between the test data and the nominal flexural strengths calculated by
DSM for laterally braced beams.1.431
Local: Eq. (5.65)
Distortional: Eq. (5.70)
Figure 3.68a
Local and distortional direct strength curves for a braced column.1.431
show the local and distortional post-buckling strengths as
compared to the elastic buckling, where the local buckling
strength possesses a higher post-buckling strength than the
distortional buckling. The figures also indicate that similar
to the local buckling, the distortional buckling should be
considered even if the member is braced against the global
buckling.
The application of the Direct Strength Method to determine the member strengths due to different buckling modes
will be discussed in details in Chapters 4 and 5. This section
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
103
Inelastic Bending Reserve
Considered in Specification Sections
F2.4.2 and F3.2.3
Inelastic Bending
Reserve Ignored
in Sections
F2.1 and F3.2.1
Figure 3.68b Local and distortional direct strength curves for a laterally braced beam1.431 where
the referenced section numbers are those in the Specification.1.417
is focused on the determination of the elastic local and
distortional buckling, which will be used in the Direct
Strength Method in later chapters. The global buckling will
be discussed in Chapters 4 and 5.
3.5.1 Local Buckling
The element local buckling has been fully discussed in
Sections 3.1 to 3.4. Expressions that are used for the Effective
Width Method are provided.
To consider the local buckling of the whole cross-section,
the following numerical and analytical solutions may be
employed.3.256,1.417,1.431
1. Numerical Solutions. Numerical methods such as
the shell finite element method,3.262–3.264 the finite
strip method,3.257–3.260 and generalized beam theory
(GBT)3.265,3.266,3.294 can be used to determine the
member local, distortional, and global buckling. Even
though the shell finite element method provides a
flexible way to model members with different shapes
or support conditions, the method generally requires
the user to visually determine buckling modes and the
buckling modes are often coupled. The general beam
theory was originally developed by Schardt3.265 and
extended by Davies et al.3.266 The method is capable of
generating the buckling signature curve, as shown in
Figs. 3.69 and 3.70, which can be used to identify the
buckling modes and determine the buckling loads to be
used in design. Open software that determines distortional buckling of C- and Z-Section members3.261 can
be downloaded from (www.civil.ist.utl.pt/gbt/). The
finite strip method for cold-formed steel was pioneered
at the University of Sydney and the long-used program
(THIN-WALLED) is commercially available at (www
.civil.usyd.edu.au/case/thinwall.php).3.260 Through partial research support of AISI, an open-source and
free finite-strip method program (CUFSM) was
developed by Schafer et al.3.267,3.268 The software
can be downloaded from (www.ce.jhu.edu/bchafer/
cufsm).3.259 The finite strip method can be used to
determine the buckling loads and moments of prismatic members with arbitrary cross-section. The
method has also been extended to determine the shear
buckling,3.269–3.271,5.109,3.272 generalized end boundary
conditions,1.432 members with holes,3.273,3.274,3.275 and
so forth.
Shown in Figs. 3.69 and 3.70 are the buckling
analysis signature curves for C-Section (9CS2.5×059)
obtained from CUFSM. The figures show that the
C-Section member subjects a local buckling at short
buckling wavelength. The local minimum buckling
wavelength is at or near the outer dimensions of
the member cross-section, the distortional buckling
104
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
Figure 3.69
method.1.431
Compression elastic buckling analysis of C-section (9CS2.5 × 059) with finite-strip
typically occurs between three and nine times the out
dimensions, and the global buckling occurs at much
longer wavelength.1.383 The global buckling load or
moment can be selected from the signature curve based
on actual unbraced length.1.383
2. Analytical Solutions. The local buckling of an element
can be determined by Eq. (3.45)
𝑘𝜋 2 𝐸
(3.45)
12(1 − 𝜇2 )(𝑤∕𝑡)2
where k is the buckling coefficient which can be determined from Table 3.2 for typical boundary conditions,
w is the flat width of the element, and t is the thickness
of the element. For an interconnected element in a
cross-section, it is difficult to determine actual fixities.
Therefore, the above equation only can provide an estimated critical buckling stress. In addition, depending
on the dimensions and the fixities, each element on
a cross-section may predict different critical buckling stresses. The North American Specification1.417
requires: for a compression member, the minimum
𝑓cr𝓁 among all the elements on the cross-section be
used to determine the member local buckling force;
and for flexural member, the 𝑓cr𝓁 , which results in the
smallest stress level when linearly extrapolated to the
fcr𝓁 =
extreme compression fiber, will be used to determine
the local buckling moment. Since the restraints at the
element junctures are not accurately modeled, this
analytical approach could be very conservative.
3.5.2
Distortional Buckling
Since 1962, the distortional buckling problem of coldformed steel members has been studied by Douty,4.19
Haussler,4.20 Desmond, Pekoz and Winter,3.76,3.77
Hancock,1.69,4.163,4.164,1.358,4.223 Lim and Rhodes,4.293 Kwon
and Hancock,4.196 Hancock, Rogers, and Schuster,4.165
Lau and Hancock,5.109–5.111 Serrette and Pekoz,4.158–4.162
Buhagiar, Chapman, and Dowling,4.166 Davies and
Jiang,4.167,4.1.68,4.197 Schafer and Pekoz,3.168,3.175,3.176,3.195
Bambach,
Merrick,
and
Hancock,3.173
Bernard,
3.171,3.172
Bridge, and Hancock,
Ellifritt, Sputo, and
Haynes,4.186 Kavanagh and Ellifritt,4.188 Ellifritt,
Glover, and Hren,4.169 Jonson,4.198,4.199 Bradford,4.200
Sarawit and Pekoz,4.201 Camotim, Silvestre, and
Dinis,3.286,3.294,4.202,4.203,4.207,4.208,4.214,4.225
Nuttayasakul
and Esterling,4.204 Cortese and Murray,4.205 Yu,4.209 Yu
and Schafer,4.206,4.210,4.217 Chodraui, Malite, Goncalves,
and Neto,4.211,4.213 Schafer, Sarawit, and Pekoz,4.212
Schafer and Adany,4.215 Yap and Hancock,4.216 Yu and
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
Figure 3.70
method.1.431
105
Bending elastic buckling analysis of C-section (9CS2.5 × 059) with finite-strip
Lokie,4.218 Javaroni and Goncalves,4.219 Mahaarachchi and
Mahendran,4.220 Georgescu,4.221 Pham and Hancock,4.222
Schafer, Sangree, and Guan,4.223 Yap and Hancock,4.222
Bambach,4.227 and others. Some of the past research findings and the development of the AISI design criteria for
distortional buckling strength of cold-formed steel members
are well summarized in the AISI commentary1.346,1.431 and
direct-strength method design guide.1.383 Section 13.4.1 of
the SSRC guide1.412 presents detailed discussions of the
available research work on distortional buckling.
According to Section 13.2.3 of the SSRC Guide, Lau and
Hancock’s analytical model5.109 is in widest use and is based
primarily on the assumption that the flange acts as an isolated
column undergoing flexural–torsional buckling, while the
web provides elastic restraint to the flange. This model was
subsequently improved to include more consistent treatment
of the web. Their model is used in the Standards of Australia
and New Zealand.1.391 In 1999, Schafer and Pekoz further
developed the model to allow for the impact of applied
stresses on the web’s rotational stiffness, thus allowing for
the case when distortional buckling is triggered by instability
of the web as opposed to the flange.3.168 Schafer and Pekoz’s
model is used in the North American Specification,1.345,1.417
and is enclosed item (2) Analytical Solutions, below.
Similar to the determination of local buckling, distortional
buckling can be analyzed numerically and analytically.
1. Numerical Solutions. Same as discussed in the numerical solutions for local buckling, all the methods
mentioned for local buckling analysis in Section
3.5.1(1) can be used for determining the distortional
buckling force and moments. See Section 3.5.1(1) for
details.
2. Analytical Solutions. The following analytical expressions for C- or Z-Section members with simple or
complex stiffeners, are derived by Schafer3.280 and
verified for complex stiffeners by Schafer et al.3.222
The following distortional buckling force and moment
expressions are excerpted from the 2016 edition of the
North American Specification.1.417
2.3.1.3 Distortional Buckling (Fcrd , Pcrd )
The provisions of this section shall apply to any open
cross-section with stiffened flanges of equal dimension where
the stiffener is either a simple lip or a complex edge stiffener.
The elastic distortional buckling load, Pcrd , shall be calculated
as follows:
(3.104)
𝑃crd = 𝐴g 𝐹crd
106
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
where 𝐴g = Gross cross-sectional area
𝐹crd =
k𝜙fe + k𝜙we + k𝜙
(3.105)
̃
k𝜙wg
k𝜙fg + ̃
where
𝑘𝜙fe = Elastic rotational stiffness provided by the
flange to the flange∕web juncture
[
]
( )4
I2xyf
π
2
2
EIxf (xof − hxf ) + ECwf − E
(x − hxf )
=
L
Iyf of
( )2
π
GJf
+
L
(3.106)
𝑘𝜙we = Elastic rotational stiffness provided by the web to
flange∕web juncture
=
Et3
6ho (1 − μ2 )
Lm = Distance between discrete restraints that restrict distortional buckling (for continuously restrained members Lm = Lcrd )
Variables 𝐴f , 𝐽f , 𝐼xf , 𝐼yf , 𝐼xyf , 𝐶wf , 𝑥of , 𝑦of , and ℎxf are defined
in Table 3.4, and variables 𝐿x , 𝐿y are unbraced length for
bending about the x and y axis, respectively; 𝐿t is the unbraced
length for torsion; E, G, 𝜇 are modulus of elasticity, shear
mdulys and Poisson’s ratio, respectively; and 𝐴g is the gross area
of the cross-section.
2.3.3.3 Distortional Buckling (Fcrd , Mcrd )
The provisions of this section are permitted to apply to any
open cross-section with a single web and single edge-stiffened
compression flange extending to one side of the web where the
stiffener is either a simple lip or a complex edge stiffener. The
elastic distortional buckling moment, 𝑀crd , shall be calculated
as follows:
𝑀crd = 𝑆𝑓 𝐹crd
(3.107)
where
where
𝐹crd = 𝛽
ho = Out-to-out web depth (See Fig.3.30(c))
t = Base steel thickness
k𝜙 = Rotational stiffness provided by restraining
elements (brace, panel, sheathing) to
flange/web juncture of member (zero if the
flange is unrestrained).If rotational stiffness
provided to the two flanges is dissimilar, the
smaller rotational stiffness is used.
̃
𝑘𝜙fg = Geometric rotational stiffness demanded by
flange from flange/web juncture
(
⎧
( )2 ⎪
𝜋
𝐴𝑓
=
𝐿 ⎨
⎪
⎩
⎡
𝐼
⎢(𝑥 − ℎ )2 xyf
xf
⎢ of
𝐼yf
⎣
)
(
𝐼xyf
−2𝑦of (𝑥of − ℎxf )
𝐼yf
+
ℎ2xf
⎫
⎪
+ 𝐼xf + 𝐼yf ⎬
⎥
⎪
⎦
⎭
where
𝐿 = Minimum of 𝐿crd and 𝐿m
where
𝐿crd =
Lm
(3.108)
web from flange∕web juncture
( )2 th3
π
o
L 60
where 𝐿 = Minimum of 𝐿crd and 𝐿m where
[
{
6𝜋 4 ℎo (1 − 𝜇 2 )
𝐼xf (𝑥of − ℎxf )2 + 𝐶wf
𝐿crd =
𝑡3
−
2
𝐼xyf
𝐼yf
(3.109)
]} 1∕4
(𝑥of − ℎxf )
2
{
[
4𝜋 4 ℎo (1 − 𝜇 2 )
𝐼xf (𝑥of − ℎxf )2 + 𝐶wf
𝑡3
−
̃
𝑘𝜙wg = Geometric rotational stiffness demanded by
=
(3.112)
̃
𝑘𝜙fg + ̃
𝑘𝜙wg
= 1.0 ≤ 1 + 0.4(L∕Lm )0.7 (1 + M1 ∕M2 )0.7 ≤ 1.3
(3.113)
)2
𝑦2of ⎥
𝑘𝜙fe + 𝑘𝜙we + 𝑘𝜙
where 𝛽 = A value accounting for moment gradient, which is
permitted to be conservatively taken as 1.0
⎤
+
(3.111)
(3.110)
2
𝐼xyf
𝐼yf
]
(𝑥of − ℎxf )
2
𝜋 4 ℎo 4
+
720
} 1∕4
(3.114)
= Distance between discrete restraints that
restrict distortional buckling (for continuously restrained members Lm = Lcrd )
M1 and M2 = Smaller and larger end moments,
respectively, in the unbraced segment
(Lm ) of the beam; M1 /M2 is positive
when the moments cause reverse
curvature and negative when bent in
single curvature
= Elastic rotational stiffness provided by
k𝜙fe
the flange to the flange/web juncture,
given in Eq. (3.106)
= Elastic rotational stiffness provided by
k𝜙we
the web to the flange/web juncture
]
[
( )2 19ℎ
( )4 ℎ 3
3
Et3
𝜋
𝜋
o
o
(3.115)
+
+
=
𝐿
60
𝐿 240
12(1 − 𝜇 2 ) ℎo
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
Geometric Flange Plus Lip Properties for C- and Z-Sections1,2,3
Table 3.4
b
b
d
θ
h
θ
h
d
Af = (b + d)t
Af = (b + d)t
Jf = 1∕3bt3 + 1∕3dt3
Jf = 1∕3bt3 + 1∕3dt3
Ixf =
t(t2 b2 + 4bd3 + t2 bd + d4 )
12(b + d)
Ixf =
t(t2 b2 + 4bd3 − 4bd3 cos2 (θ) + t2 bd + d4 − d4 cos2 (θ))
12(b + d)
Iyf =
t(b4 + 4db3 )
12(b + d)
Iyf =
t(b4 + 4db3 + 6d2 b2 cos(θ) + 4d3 bcos2 (θ) + d4 cos2 (θ))
12(b + d)
Ixyf =
tb2 d2
4(b + d)
Cwf = 0
Ixyf =
tbd2 sin(θ)(b + d cos(θ))
4(b + d)
Cwf = 0
xof =
b2
2(b + d)
xof =
b2 − d2 cos(θ)
2(b + d)
hxf =
−(b2 + 2db)
2(b + d)
hxf =
−(b2 + 2db + d2 cos(θ))
2(b + d)
hyf = yof =
−d2
2(b + d)
hyf = yof =
−d2 sin(θ)
2(b + d)
Notes:
1. b, d, and h are mid-line dimensions of cross-section.
2. x–y axis system is located at the centroid of the flange with x positive to the right from the centroid, and y positive
down from the centroid. Table 3.4 does not include the effect of corner radius. More refined values are permitted.
3. Variables are defined as follows:
Af
t
Jf
Ixf
Iyf
Ixyf
Cwf
xof
yof
hxf
hyf
=
=
=
=
=
=
=
=
=
=
=
Cross-sectional area of flange
Thickness of cross-section
St. Venant torsion constant of flange
x-axis moment of inertia of flange
y-axis moment of inertia of flange
Product of the moment of inertia of flange
Warping torsion constant of flange
x distance from centroid of flange to shear center of flange
y distance from centroid of flange to shear center of flange
x distance from centroid of flange to flange/web junction
y distance from centroid of flange to flange/web junction
107
108
3
k𝜙
̃
𝑘𝜙fg
̃
𝑘𝜙wg
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
= Rotational stiffness provided by a restraining
element (brace, panel, sheathing) to the
flange/web juncture of a member (zero if the
compression flange is unrestrained)
= Geometric rotational stiffness demanded by
the flange from the flange/web juncture, given
in Eq. (3.108)
= Geometric rotational stiffness demanded by
the web from the flange/web juncture
( )2 ⎫
⎧
⎪ [45360(1 − 𝜉web ) + 62160] 𝐿
⎪
ℎo
⎪
⎪
( ℎ )2
⎪ + 448𝜋 2 + o [53 + 3(1 − 𝜉 )]𝜋 4 ⎪
2
web
ℎ 𝑡𝜋 ⎪
⎪
𝐿
= o
⎬ (3.116)
( )2
( )4
13440 ⎨
𝐿
𝐿
⎪
⎪
𝜋 4 + 28𝜋 2 ℎ
+ 420 ℎ
⎪
⎪
𝑜
o
⎪
⎪
⎪
⎪
⎩
⎭
where 𝜉 web = (f1 – f2 )/f1 , stress gradient in the web, where
f1 and f2 are the stresses at the opposite ends
of the web, f1 > f2 , compression is positive,
tension is negative, and the stresses are
calculated on the basis of the gross section
(e.g., pure symmetrical bending, f1 = –f2 ,
𝜉 web = 2 )
(5) 2 ≤ ho /bo ≤ 8, and
(6) 0.04 ≤ D sin𝜃/bo ≤ 0.5
where
For compression members:
Fcrd = 𝛼kd
where
For C- and Z-sections that have no rotational restraint of the
flange and that are within the dimensional limits provided in
this section, Eq. (3.117) can be used to calculate a conservative prediction of distortional buckling stress, 𝐹crd , provided the
following dimensional limits are met:
(1)
(2)
(3)
(4)
50 ≤ ho /t ≤ 200,
25 ≤ bo /t ≤ 100,
6.25 < D/t ≤ 50,
45∘ ≤ 𝜃 ≤ 90∘ ,
(
t
bo
)2
(3.117)
𝛼 = A value that accounts for the benefit of an
unbraced length, Lm , shorter than Lcr , but
can be conservatively taken as 1.0
= (𝐿𝑚 ∕𝐿cr )𝓁𝑛(𝐿𝑚 ∕𝐿cr )
for 𝐿𝑚 < 𝐿cr
(3.118)
Lm = Distance between discrete restraints that
restrict distortional buckling
(
)
bo Dsin𝜃 0.6
≤ 10ho
ho t
(
)
b Dsin𝜃 1.4
≤ 8.0
𝑘𝑑 = 0.05 ≤ 0.1 o
ho t
𝐿cr = 1.2ho
The AISI Cold-Formed Steel Design Manual1.428 has
provided 𝐿crd , 𝑘𝜙fe , 𝑘𝜙we , ̃
𝑘𝜙fg , ̃
𝑘𝜙wg , 𝐹crd , or 𝐹crd ∕𝛽 in
tables for stud/joist stock sections. These tables can be used
in design.
Simplified Method for Unrestrained C- and Z-Sections
with Simple Lip Stiffeners
π2 E
12(1 − 𝜇 2 )
= 1.0 for 𝐿𝑚 ≥ 𝐿cr
All other variables are defined in Specification Section
2.3.1.3.
3. Simplified Analytical Solutions. For C- or Z-Section
members with simple lip stiffeners, the following
simplified expressions can be used to predict the
distortional buckling stress. The following provisions
are adapted from Sections 2.3.1.2 and 2.3.3.3 of the
Commentary1.431 on the 2016 edition of the North
American Specification.
ho = Out-to-out web depth as defined in Figure
3.30(c)
bo = Out-to-out flange width as defined in
Figure 3.30(c)
D = Out-to-out lip dimension as defined in
Figure 3.48
t = Base steel thickness
𝜃 = Lip angle as defined in Figure 3.48
(3.119)
(3.120)
𝐸 = Modulus of elasticity of steel
𝜇 = Poisson’s ratio of steel
For flexural members:
𝐹crd = 𝛽𝑘𝑑
where
𝜋2E
12(1 − 𝜇 2 )
(
𝑡
𝑏o
)2
𝛽 = A value accounting for moment gradient,
which is permitted to be conservatively taken
as 1.0
= 1.0 ≤ 1 + 0.4(L∕L𝑚 )0.7 (1 + M1 ∕M2 )0.7 ≤ 1.3
where
(3.121)
(3.122)
L = Minimum of Lcr (per Eq. 3.119) and Lm
Lm = Distance between discrete restraints that
restrict distortional buckling (for continuously
restrained members Lm = Lcr )
M1 and M2 = Smaller and larger end moment, respectively,
in the unbraced segment (Lm ) of the beam;
M1 /M2 is positive when the moments cause
reverse curvature and negative when bent in
single curvature
(
)
bo Dsin𝜃 0.7
≤ 8.0
(3.123)
𝑘d = 0.5 ≤ 0.6
ho t
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
4. Determination of Rotational Restraints, k𝜙
In Eqs. (3.105) and (3.112) the rotational stiffness k𝜙
accounts for the rotational restraints to the compression
flange(s). For example, in cold formed steel framing
systems, structural sheathings that attached to studs or
joist members will provide rotational restraints. Such
restraints can be calculated using the following expressions, which was developed based on the research by
Schafer et al.3.276, 4.223 and was adopted in Appendix 1
of the 2015 edition of the North American Standard for
Cold-Formed Steel Structural Framing.1.432
The rotational stiffness, k𝜙 , shall be determined in accordance
with the following:
𝑘𝜙 = (1∕𝑘𝜙w + 1∕𝑘𝜙c )−1
(3.124)
where
𝑘𝜙w = Sheathing rotational restraint
= EI w ∕𝐿1 + EI w ∕𝐿2 for interior members (joists or rafters)
with structural sheathing fastened on both sides (3.125)
= EI w ∕𝐿1 for exterior members (joists or rafters) with
structural sheathing fastened on one side
(3.126)
where EIw = Sheathing bending rigidity
= Values as specified in Table 3.5(a) for
plywood and OSB
= Values as specified in Table 3.5(b) for
gypsum board permitted only for
serviceability calculations
Table 3.5(a)
Span Rating
24/0
24/16
32/16
40/20
48/24
16oc
20oc
24oc
32oc
48oc
1,2 Plywood
3-ply
66,000
86,000
125,000
250,000
440,000
165,000
230,000
330,000
715,000
1,265,000
L1 , L2 = One-half joist spacing to the first and second
sides respectively, as illustrated in Fig. 3.71
k𝜙c = Connection rotational restraint
= Values as specified in Table 3.6 for fasteners
spaced 12 in. o.c. (305 mm) or closer
Table 3.5(b) 1 Gypsum Board Bending Rigidity
Effective Stiffness (Typical Range), EIw
Board Thickness
(in.) (mm)
EI (Lbf-in.2 /in.) of
width (N-mm2 /mm)
0.5
(12.7)
0.625
(15.9)
1500 to 4000
(220,000 to 580,000)
3000 to 8000
(440,000 to 1,160,000)
Note:
1. Gypsum board bending rigidity is obtained from the Gypsum
Association.
5. Distortional Buckling of C- or Hat Sections Subject to
Bending with Lips in Compression
The following analytical solution was developed by
Glauz3.277 for C-Section or Hat sections subject to
bending where both lips are in compression and the
flanges are under the stress gradient.
and OSB Sheathing Bending Rigidity, EIw (lbf-in2 /ft)
Strength Parallel to
Strength Axis
Plywood
4-ply
5-ply
66,000
86,000
125,000
250,000
440,000
165,000
230,000
330,000
715,000
1,265,000
66,000
86,000
125,000
250,000
440,000
165,000
230,000
330,000
715,000
1,265,000
109
OSB
60,000
86,000
125,000
250,000
440,000
165,000
230,000
330,000
715,000
1,265,000
Notes:
1. To convert to lbf-in.2 /in., divide table values by 12.
To convert to N-mm2 /m, multiply the table values by 9.415.
To convert to N-mm2 /mm, multiply the table values by 9.415.
2. Plywood and OSB bending rigidity are obtained from APA.
Stress Perpendicular to
Strength Axis
Plywood
3-ply
4-ply
5-ply
3,600
5,200
8,100
18,000
29,500
11,000
13,000
26,000
75,000
160,000
7,900
11,500
18,000
39,500
65,000
24,000
28,500
57,000
615,000
350,000
11,000
16,000
25,000
56,000
91,500
34,000
40,500
80,500
235,000
495,000
OSB
11,000
16,000
25,000
56,000
91,500
34,000
40,500
80,500
235,000
495,000
110
3
Table 3.6
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
1 Connection
Rotational Restraint
T
(mils)
t
(in.)
k𝜙c
(lbf-in./in./rad)
k𝜙c
(N-mm/mm/rad)
18
27
30
0.018
0.027
0.03
78
83
84
348
367
375
33
43
54
68
97
0.033
0.043
0.054
0.068
0.097
86
94
105
123
172
384
419
468
546
766
k𝜙we = Elastic rotational stiffness provided by
the web to the flange/web juncture
[
(
)2
(
)4 ]
1 𝜋ℎ𝑒
1 𝜋ℎ𝑒
Et3
+
1+
=
6 𝐿
120 𝐿
6ℎ𝑒 (1 − 𝜇2 )
(3.128)
Note:
1. Fasteners spaced 12 in. (25.4 mm) o.c. or less.
The elastic distortional buckling moment, Mcrd , is calculated as follows:
𝑀crd = 𝑆f 𝐹crd
[Eq. (3.111)]
𝑘𝜙fe + 𝑘𝜙we + 𝑘𝜙
[Eq. (3.112)]
̃
𝑘𝜙fg + ̃
𝑘𝜙wg
̃
𝑘𝜙fg = Geometric rotational stiffness demanded by
the flange from the flange/web juncture
=
where
𝐹crd = 𝛽
𝛽, L and Lm , k𝜙fe are defined in Specification Section 2.3.3.3
included in Section 3.5.2(2).
{
6(1 − 𝜇2 ) [
𝐿crd = 𝜋ℎ𝑒
𝐶wf + 𝐼xf (𝑥of − ℎxf )2
𝑡3 ℎ3𝑒
} 1∕4
)]
(
2
𝐼xyf
1
+
× 1−
(3.127)
𝐼xf 𝐼yf
120
[
( )2 {
𝜋
𝐼xf + 𝐼yf + 𝐴𝑓 ℎ2xf + 𝑦2of − 2𝑦of (𝑥of − ℎxf )
𝐿
)]}
(
( )2
𝐼xyf
𝜋
𝐼yf 𝜉𝑓
(3.129)
𝜓𝑓 +
×
𝐼yf
𝐿
̃
𝑘𝜙wg = 0
Sf = Gross elastic cross-sectional modulus referenced
to the compression fiber of the flange/web
juncture, the point at which ho is measured
√
ℎe = 3.5
𝐼yf
𝐴f
+ ℎ2xf
y
x
z
L
Sheathing
df
Lf
wtf
Joist or Wall
Spacing
Joist or Wall
Framing
Interior Joist
or Wall Example
Figure 3.71
Exterior Joist or
Wall Stud
Illustration of L1 and L2 for sheathing rotational restraint1.432
(3.130)
111
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
fcg
f1 (compression)
f2 (tension)
Flange/web
Juncture
Shear center of
stiffened flange
Centroid of
stiffened flange
hxf
xof
Flange stresses for bending about axis parallel to web.
Figure 3.72
𝜉 f = (f1 – f2 )/f1 , stress gradient in the flange,
where f1 is the stress at the extreme
compression fiber of the flange, f2 is the
stress at the flange/web juncture,
compression is positive, tension is negative,
and the stresses are calculated on the basis of
the gross section (see Fig. 3.72)
𝜓 f = fcg /f1 , stress ratio in the flange, where f1
is the stress at the extreme compression
fiber of the flange, fcg is the stress at the
centroid of the flange, compression is
positive, tension is negative, and the stresses
are calculated on the basis of the gross
section (see Figure 3.72)
Example 3.8 For the C-section shown in Fig. 3.73, determine the distortional buckling moment when bending about
x axis using the simplified analytical solution based on
Eq. (3.121) and the expressions given in Specification
Section 2.3.3.3.
SOLUTION
A. Distortional Buckling Moment Determined by the Simplified Method
In order to use Eq. (3.121), the following geometric limits
should be checked as the first step:
50 ≤ (ℎo ∕𝑡 = 133.33) < 200
25 ≤ (𝑏o ∕𝑡 = 46.67) < 100
OK
OK
6.25 < (𝐷∕𝑡 = 9.6) < 50 OK
45∘ ≤ (𝜃 = 90∘ ) = 90∘ OK
2 ≤ (ℎo ∕𝑏o = 2.86) < 8 OK
0.04 ≤ (𝐷 sin 𝜃∕𝑏o = 0.21) < 0.5
OK
From Eq. (3.121), the elastic distortional buckling stress is
calculated as follows:
( )2
t
𝜋2E
Fcrd = kd
[Eq. (3.121)]
12(1 − 𝜇2 ) bo
Figure 3.73
Example 3.8
Based on Eq. (3.120), the plate buckling coefficient for
distortional buckling is
(
[
)
]0.7
𝑏 Dsinθ 0.7
3.5(0.72) sin 90∘
= 0.6
= 1.4
𝑘crd = 0.6 o
ℎo t
10(0.075)
Since 0.5 < 𝑘d < 8, use 𝑘d = 1.40. Because 𝑀1 and 𝑀2 are
not given in the problem, use 𝛽 = 1.0 as a conservative value.
Therefore,
𝜋 2 (29,500) ( 0.075 )2
= 17.14 ksi
𝐹d = (1)(1.40)
12(1 − 0.32 ) 3.5
The elastic modulus relative to the extreme compression
fiber,which can be calculated using using linear method (Fig.
1.32), is
𝑆f = 4.11 in.3
The critical elastic
[(Eq.(3.104)] is
distortional
buckling
moment
𝑀crd = 𝑆f 𝐹crd = (4.11)(17.14) = 70.45 in-kips
112
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
B. Distortional Buckling Moment Based on Specification
Section 2.3.3.3
Based on the equations listed in Table 3.4, the geometric
flange properties for the C-section can be computed as
follows. The reason for these calculations is that the
mechanical model for prediction of distortional buckling
strength considers the flange itself as a “column” which
may undergo restrained flexural–torsional buckling, and the
restraint comes from the web and any additional attachments
ℎ = ℎ0 − 𝑡 = 10.000 − 0.075 = 9.925 in.
𝑏 = 𝑏0 − 𝑡 = 3.500 − 0.075 = 3.425 in.
𝑑 = 𝐷 − 𝑡∕2 = 0.720 − 0.075∕2 = 0.6825 in.
𝐴f = (𝑏 + 𝑑)𝑡 = (3.425 + 0.6825)(0.075) = 0.308 in.2
𝐼𝑥f = 𝑡[𝑡2 𝑏2 + 4bd3 + 𝑡2 bd + 𝑑 4 ]∕12(𝑏 + 𝑑)
= (0.075)[(0.075)2 (3.425)2 + 4(3.425)(0.6825)3
+ (0.075)2 (3.425)(0.6825)
+ (0.6825)4 ]∕12(3.425 + 0.6825) = 0.00708 in.4
𝐼𝑦f = 𝑡[𝑏4 + 4bd3 ]∕12(𝑏 + 𝑑)
= (0.075)[(3.425)4 + 4(0.6825)(3.425)3 ]∕12(3.425
+ 0.6825)
= 0.376 in4 .
𝐼xyf = tb2 𝑑 2 ∕4(𝑏 + 𝑑)
= (0.075)(3.425)2 (0.6825)2 ∕4(3.425 + 0.6825)
= 0.0249 in4 .
𝑥of = 𝑏2 ∕2(𝑏 + 𝑑) = (3.425)2 ∕2(3.425 + 0.6825)
= 1.428 in.
𝑦of = −𝑑 2 ∕2(𝑏 + 𝑑) = −(0.6825)2 ∕2(3.425 + 0.6825)
= −0.0567 in.
ℎ𝑥f = −[𝑏2 + 2db]∕2(𝑏 + 𝑑)
= −[(3.425)2 + 2(0.6825)(3.425)]∕2(3.425 + 0.6825)
= −1.997 in.
𝐽f = [bt3 + dt3 ]∕3
= [(3.425)(0.075)3 + (0.6825)(0.075)3 ]∕3
= 0.000578 in4 .
𝐶wf = 0.0 in.6
According to Eq. (3.114), the critical unbraced length of
distortional buckling, Lcrd , can be computed as follows:
[
{
4𝜋 4 ℎo (1 − 𝜇2 )
𝐼xf (𝑥of − ℎxf )2 + 𝐶wf
𝐿crd =
𝑡3
−
{
=
2
𝐼xyf
𝐼yf
]
(𝑥of − ℎxf )2
𝜋 4 ℎ𝑜 4
+
720
} 1∕4
4𝜋 4 (10.000)(1 − 0.32 )
(0.075)3
]
[
(0.00708)[1.428 − (−1.997)]2 + 0−
2
×
0.0249
[1.428 − (−1.997)]2
0.376
}1∕4
𝜋 4 (10.000)4
+
= 27.07 in.
720
Assume that the distortional buckling length L equals 𝐿crd ,
𝐿 = 𝐿crd = 27.07 in.
From Eq. (3.106), the elastic rotational stiffness provided
by the flange to the flange/web juncture, 𝑘𝜙fe , is
[
]
( )4
I2xyf
π
2
2
EIxf (xof − hxf ) + ECwf − E
k𝜙fe =
(x − hxf )
L
Iyf of
( )2
π
+
GJf
L
⎡(29,500)(0.00708)
⎤
⎥
)4 ⎢[1.428 − (−1.997)]2
(
π
⎢
=
(0.0249)2 ⎥
⎥
27.07 ⎢+(29,500)(0) − (29,500)
(0.376) ⎥
⎢
⎣[1.428 − (−1.997)]2
⎦
)2
(
π
(11,300)(0.000578)
+
27.07
= 0.429 in.-kips∕in.
From Eq. (3.115), the elastic rotational stiffness provided
by the web to the flange/web juncture, 𝑘𝜙we , is
[
]
( )2 19h
( )4 h 3
Et3
3
π
π
o
o
+
+
k𝜙we =
L
60
L 240
12(1 − μ2 ) ho
)2
(
⎡ 3
(19)(10.000) ⎤
π
+⎥
(29,500)(0.075)3 ⎢ 10.000 + 27.07
60
=
)4
(
⎢
⎥
3
2
(10.000)
π
12(1 − 0.3 ) ⎢
⎥
27.07
240
⎣
⎦
= 0.391 in.-kips∕in.
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
From Eq. (3.108), the geometric rotational stiffness
demanded by the flange from the flange/web juncture, ̃
𝑘𝜙fg ,
is
( )2
π
̃
k𝜙fg =
L
⎫
⎧ ⎡
( )2
⎤
⎪
⎪ ⎢(x − h )2 Ixyf
− 2yof (xof − hxf )⎥
xf
⎪
⎪ ⎢ of
Iyf
⎥
(
)
× ⎨Af
+ Ixf + Iyf ⎬
Ixyf
⎢
⎥
⎪
⎪ ⎢
× I
+ h2xf + y2of
⎥
⎪
⎪ ⎣
yf
⎦
⎭
⎩
)2 ⎫
(
⎧
⎤
⎡
2 0.0249
[1.428
−
(−1.997)]
⎪
⎪
⎢
0.376 ⎥⎪
⎪
⎥
⎢
)2 ⎪
(
) ⎥⎪
(
(0.308) ⎢−2(−0.0567)
π
=
⎨
⎥⎬
⎢[1.428 − (−1.997)] 0.0249
27.07 ⎪
0.376
⎥⎪
⎢
⎪
⎦⎪
⎣+(−1.997)2 + (−0.0567)2
⎪+0.00708 + 0.376
⎪
⎩
⎭
= 0.0220 (in.-kips∕in)∕ksi
From Eq. (3.116), the geometric rotational stiffness
demanded by the web from the flange/web juncture, ̃
𝑘𝜙wg ,
where ξweb = 2 for pure bending.
( )2
⎧
⎫
⎪ [45360(1 − ξweb ) + 62160] L + 448π2 + ⎪
h
o
⎪
⎪
( h )2
⎪
⎪
o
4
[53
+
3(1
−
ξ
)]π
2
web
ho tπ ⎪
⎪
L
̃
k𝜙wg =
⎬
( )2
( )4
13,440 ⎨
⎪
⎪
π4 + 28π2 hL + 420 hL
⎪
⎪
o
o
⎪
⎪
⎪
⎪
⎩
⎭
⎧
⎫
[45360(1 − 2) + 62160]
⎪
⎪
)2
(
27.07
2
⎪
⎪
+
448π
+
10.000
⎪
⎪
(
)2
10.000
⎪
⎪
4
[53
+
3(1
−
2)]π
2 ⎪
⎪
27.07
(10.000)(0.075)π
=
⎨
(
)2
)4 ⎬
(
13440
27.07
⎪ π4 + 28π2 27.07
⎪
+ 420 10.000
10.000
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
= 0.00286 (in.-kips∕in.)∕ksi
From Eq. (3.112), the elastic distortional buckling stress
𝐹crd is
𝑘𝜙fe + 𝑘𝜙we + 𝑘𝜙
𝐹crd = 𝛽
̃
𝑘𝜙fg + ̃
𝑘𝜙wg
In the above equation, use 𝛽 = 1.0 as a conservative
value (i.e., ignoring moment gradient). Since no rotational
113
restraints are provided to the compression flange of the
beam, 𝑘𝜙 = 0. Therefore,
0.429 + 0.391 + 0.0
= 32.98 ksi
0.0220 + 0.00286
which is 92% higher than the elastic distortional buckling
stress computed in item A above using the simplified provision on the basis of Section 2.3.3.3 of the Commentary1.431
on the Specification. From item A, Sf = 4.11 in.3 Based on
Eq. (3.111), the critical elastic distortional buckling moment,
𝑀crd , is
𝐹crd = 1.0
𝑀crd = 𝑆f 𝐹crd = (4.11)(32.98) = 135.55 in.-kips
Example 3.9 Determine the distortional buckling force
𝑃crd and 𝑀crd of C-Section 9CS2.5×059 (designation per
AISI Cold-Formed Steel Design Manual1.428 ) using the
numerical and analytical solutions. The member yield stress,
𝐹y = 55 ksi.
SOLUTION
Numerical Solution. The cross-section geometry and
gross section properties of 9CS2.5×059 can be obtained
from the 2017 edition of the AISI Cold-Formed Steel
Design Manual1.428 : 𝐴g = 0.881 in.2 ; Ix = 10.3 in.4 , and
Sx = 2.29 in.3 . From CUFSM analysis, the signature
curves under the compression and bending are as shown
in Figs. 3.70 and 3.69, respectively. From the ploted
deformed shapes, the distortional buckling occurred at
Pcrd ∕Py = 0.26, and 𝑀crd ∕𝑀y = 0.85. The distortional
buckling force and moment can be calculated as follows:
𝑃y = 𝐴g 𝐹y = (0.881 in.2 )(55 ksi) = 48.455 kips
𝑃crd = 0.26 𝑃y = (0.26)(48.455 kips) = 12.598 kips
𝑀y = 𝑆x 𝐹y = (2.29 in.3 )(55 ksi) = 125.95 kip-in.
𝑀crd = 0.85 My = (0.85)(125.95) = 107.058 kip-in.
Analytical Solution. Instead of using the expressions
provided in Specification Sections 2.3.1.3 and 2.3.3.3 to
determine the distortional buckling stresses, AISI Design
Manual Tables III-4 and II-7 can be used. This example
will illustrate how to use the Design Manual to determine
the distortional buckling stresses.
Since there is no indication of rotational restraints to
members’ flanges in this example, 𝑘𝜙 = 0. From AISI Design
Manual Table III-4, the following values are obtained:
114
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
𝐿crd = 28.6 in.; 𝑘𝜙fe = 0.153 kips; ̃
𝑘𝜙fg = 0.00705 in.2 ;
2
̃
𝑘𝜙we = 0.123 kips; 𝑘𝜙wg = 0.00867 in. ; 𝐹crd = 17.6 ksi.
The distortional buckling force: 𝑃crd = 𝐴g 𝐹crd =
(0.881 in.2)(17.6 ksi) = 15.506 kips
From AISI Design Manual Table II-7,
𝑘𝜙fg = 0.00863 in.2 ;
𝐿crd = 25.8 in.; 𝑘𝜙fe = 0.221 kips; ̃
2
𝑘𝜙wg = 0.00181 in. ; 𝐹crd ∕β = 41.2 ksi.
𝑘𝜙we = 0.209 kips; ̃
Assume moment distribution influence is not considered in
this example, 𝛽 = 1
𝑀crd = 𝑆f 𝐹crd = (2.29 in.3 )(41.2 ksi) = 94.348 kip-in
For cases where rotational restraints are provided to the
compressive flange, for example, structural sheathings
or metal sheets are attached to the flange, the rotational stiffness k𝜙 can be determined analytically using
Eq. (3.124) or experimentally (refer to AISI test standard,
AISI S9183.279 .)
Example 3.10 Determine the local and distortional buckling moment 𝑀cr𝓁 and 𝑀crd of U-shaped section as shown
in Fig. 3.74 when the lips are in compression. Assume the
member is laterally braced and does not subject to global
buckling. The yield stress 𝐹y = 33 ksi.
SOLUTION
The numerical method of finite strip method is employed to
determine the local and distortional buckling moment. By
using the CUFSM software, the following buckling signature
curve is obtained:
In Fig. 3.75, two numbers are shown below the local
minima of the signature curve: the first number is the
buckling half-wavelength, and the second number is the load
factor, i.e. 𝑀cr ∕𝑀y , where 𝑀y is the yield moment. The
buckled cross-sections illustrate the buckling mode taking
place at the corresponding minimum. As shown in the figure,
the local buckling occurs at the half-wavelength of 6.4 in.
and the corresponding load factor is 2.39; the distortional
buckling occurs at the half-wavelength of 67 in. and load
factor is 0.86. From the numerical analysis output, the yield
moment 𝑀y = 161 in.-kips. Therefore, the local buckling
and distortional buckling moments can be calculated as
follows:
𝑀cr𝓁 = 6.4𝑀y = 384.8 in.kips
𝑀crd = 0.86𝑀y = 138.5 in.-kips
The results indicate that the distortional buckling moment
is much lower than the local buckling, and likely controls
member strength.
The distortional buckling moment can also be determined
analytically using the equations provided in Section 3.5.2(5).
The following calculations are based on those equations
given in Section 3.5.2(5):
Calculate the centerline dimensions:
Note, the equations in Table 3.4 and the symbols in Section
3.5.2(5) are based on a coordinate system with the x axis as
the symmetric axis.
ℎ = 8 − 0.105 = 7.895 in.;
𝑏 = 8 − 0.105 = 7895 in.;
𝑑 = 1.34 − 0.105∕2 = 1.2875 in.
Af = (b + d)t = 0.9642 in.2
Jf = 1∕3bt3 + 1∕3dt3 = 0.003543 in.4
Ixf =
t(t2 b2 + 4bd3 + t2 bd + d4 )
= 0.0676 in.4
12(b + d)
Iyf =
t(b4 + 4db3 )
= 6.117 in.4
12(b + d)
tb2 d2
= 0.2954 in.4
4(b + d)
=0
Ixyf =
Cwf
xof =
b2
= 3.394 in.
2(b + d)
hxf =
−(b2 + 2db)
= −4.501 in.
2(b + d)
hyf = yof =
Figure 3.74
Example 3.10
−d2
= −0.0903 in.
2(b + d)
From numerical analysis, the centroidal axis is 3.188 in.
from the bottom flange and 4.812 in. from the top of the lips:
𝑐c = 4.812 in.; 𝑐t = 3.188 in.
DIRECT STRENGTH METHOD AND CONSIDERATION OF LOCAL AND DISTORTIONAL BUCKLING
115
6
5
Load Factor
4
3
6.4, 2.39
2
1
67.0, 0.86
0
1
10
100
1000
Length
Figure 3.75
U-shaped cross-section buckling signature curve
Since the centroid axis is closer to the bottom flange, the
compression stress controls:
𝑓1 = 𝐹y = 33 ksi; 𝑓2 = −𝑓1 × 𝑐t ∕𝑐c = −21.863 ksi
The stress at flange centroid:
𝑓cg = 𝑓1 (|ℎxf | + 𝑡∕2 − 𝑐t )∕𝑐c = 8.647 ksi (Note: 𝑓cg is
positive since it is under compression.)
ξf = (𝑓1 − 𝑓2 )∕𝑓1 = 1.662
ψf = fcg ∕f1 = 0.262
√
Iyf
ℎe = 3.5
+ h2xf = 18.05 in.
Af
{
[
(
)]
I2xyf
6(1 − μ2 )
2
Cwf + Ixf (xof − hxf ) 1 −
𝐿crd = πhe
Ixf Iyf
t3 h3e
+
1
120
} 1∕4
= 72.52 in.
Assume no bracing for distortional buckling is provided,
therefore, 𝐿m > 𝐿crd ,
𝐿 = minimum (𝐿crd , 𝐿m ) = 𝐿crd = 72.52 in.
[
(
)2
(
)4 ]
1 πhe
Et3
1 πhe
1+
+
𝑘𝜙we =
6 L
120 L
6he (1 − μ2 )
= 0.383 kips
[
]
( )4
I2xyf
π
EIxf (xof − hxf )2 + ECwf − E
𝑘𝜙fe =
(x − hxf )2
L
Iyf of
( )2
π
GJf = 0.421 kips
+
L
[
( )2 {
π
̃
k𝜙fg =
Ixf + Iyf + Af x2hf + y2of − 2yof (xof − hxf )
L
)]}
(
( )2
Ixyf
π
I
= 0.0318 in2
ψf +
×
Iyf
L yfξf
̃
k𝜙wg = 0
Assume β, which accounts for moment gradient, is
ignored, i.e. β = 1
The distortional buckling stress is calculated:
𝐹crd = 𝛽
𝑘𝜙fe + 𝑘𝜙we + 𝑘𝜙
̃
𝑘𝜙fg + ̃
𝑘𝜙wg
= 25.294 ksi
From numerical analysis,
Moment inertia 𝐼 = 23.245 in.4
Section modulus 𝑆f = 𝐼∕𝑐c = 4.83 in.3
𝑀crd = 𝐹crd 𝑆f = (9.957)(4.83) = 122.2 in.-kips
Compared with the numerical analysis result 𝑀crd =
138.5 in-kips. the analytical solution is reasonable close.
3.5.3 Influence of Perforation on Local
and Distortional Buckling
Researchers have observed the influence of holes on
the local buckling.3.273–3.275 Both numerical and analytical approaches are provided in the North American
Specification1.417 to consider the hole effects.
116
3
STRENGTH OF THIN ELEMENTS AND DESIGN CRITERIA
3.5.3.1 Local Buckling
(a) Numerical Solutions. The numerical shell finite
element method can be used to consider the hole
effect. However, since shell finite element software
may not be readily available for regular users, the
following approximate method was developed by
Moen and Schafer,3.278 which can be used in conjunction with the finite strip method to predict the local
buckling stress. The method assumes that the local
buckling occurs as either buckling of the unstiffened
strip adjacent to the hole at the net cross-section
(𝑃cr𝓁h ) or as local buckling of the gross cross-section
between the hole (𝑃cr𝓁nh ), and:
𝑃cr𝓁 = min(𝑃cr𝓁nh , 𝑃cr𝓁h )
(3.131a)
Both 𝑃cr𝓁nh and 𝑃cr𝓁h can be determined using the
finite strip method such as CUFSM. To ensure a
consistent comparison of 𝑃cr𝓁nh and 𝑃cr𝓁h , the reference load applied in the analysis needs to be the
same. More detailed discussion can be found in the
Commentary1.431 of the 2016 edition of the North
American Specification. The same approach can
also be used to determine the beam local buckling
moment:
𝑀cr𝓁 = min(𝑀cr𝓁nh , 𝑀cr𝓁h )
(3.131b)
where 𝑀cr𝓁nh and 𝑀cr𝓁h are the buckling moments
of the gross cross-section (no hole) and the net
cross-section (with hole), respectively.
(b) Analytical Solutions. For compression members, refer
to Section 3.5.1(2). For the element with hole: the
local buckling stress, 𝐹cr𝓁 , should be calculated as
both unstiffened elements at the hole location (for
determining Pcr𝓁h ) and as an element where the hole
is not located (for determining 𝑃cr𝓁nh ). For the unstiffened element, the buckling stress 𝐹cr𝓁 should be modified by multiplying a ratio 𝐴net ∕𝐴g to account for the
net section. The smallerest 𝐹cr𝓁 from the element with
hole and the other element is then used to determine
𝑃cr𝓁 of the cross-section.
For flexural members, refer to Section 3.5.1(2). For
the element with hole: the local buckling stress, 𝐹cr𝓁 ,
should be calculated as both unstiffened elements at
the hole location (for determining 𝑀cr𝓁nh ) and as an
element where the hole is not located (for determining
𝑀cr𝓁nh ). For the unstiffened element, the buckling
stress 𝐹cr𝓁 should be modified by multiplying a ratio
𝑆fnet ∕𝑆fg to account for the net section. The smallest
𝐹cr𝓁 extrapolated to the extreme compression fiber is
then used to determine 𝑀cr𝓁 of the cross-section.
3.5.3.2 Distortional Buckling
(a) Numerical Solutions. Similar to the consideration
of local buckling, an approximate method that can
be used in conjunction with the finite strip method
was developed.3.278 To implement the method, a
finite strip analysis is performed with the gross
cross-section to identify the critical distortional buckling half-wavelength, 𝐿crd . Then the thickness of the
element with hole (not just the portion of the hole)
is modified from t to tr using the expression below.
This revised thickness is to account for the reduced
restraints of the element with hole to other elements
when distortional buckling occurs.
(
)
𝐿h 1∕3
𝑡𝑟 = 𝑡 1 −
(3.132)
𝐿crd
where 𝐿h is the length of the hole. This simplification
is only appropriate for cases of flat-punched discrete
holes in the web or flange (or both).
For patterned holes along web3.275 , the following
reduction expression should be used:
(
)
𝐴web,net 1∕3
𝑡𝑟 = 𝑡
(3.133)
𝐴web,gross
where t is the thickness of the web, 𝐴web,net is net area
of the web along the full member length; 𝐴web,gross is
the gross area of the web along the full member length.
The finite strip analysis is then performed using the
modified cross section with the reference load consistent with the gross area. The determined critical distortional buckling half-wavelength, 𝐿crd (based on the
gross cross-section), and the corresponding distortional buckling load, Pcrd , can then be obtained. This
method has been validated for compressive members
and is recommended for use for flexural members.
(b) Analytical Solutions. For compression members
meeting the requirements of Specification Section
2.3.1.3, the analytical expressions given in Section
2.3.1.3 can be used except the thickness, t, in Eqs.
(3.107) and (3.109) is replaced by tr determined per
Eq. (3.121) for flat-punched holes, or thickness, t, in
Eqs. (3.107), (3.109), and (3.110) is replace by 𝑡r per
Eq. (3.122) for patterned holes.
For flexural members meeting the requirements of
Specification Section 2.3.3.3, the analytical expressions given in Section 2.3.3.3 can be used except the
thickness t in Eqs. (3.115) and (3.116) is replaced by
tr determined per Eq. (3.121) for flat-punched holes or
t in Eqs. (3.114), (3.115) and (3.116) is replaced by 𝑡r
determined per Eq. (3.122) for patterned holes.
ADDITIONAL INFORMATION
117
AISI Cold-Formed Steel Design Manual 2017 edition1.431
contains design examples for members with holes.
3.6 PLATE BUCKLING OF STRUCTURAL SHAPES
Section 3.3 discussed the local buckling of stiffened and
unstiffened compression elements, for which the edges were
assumed to be simply supported. If the actual restraining
effects of adjoining cross-sectional elements are taken into
account, the plate buckling coefficient k for box sections,
channels, and Z-sections used as columns can be found from
Fig. 3.76. These curves are based on the charts developed
by Kroll, Fisher, and Hei-merl.3.116 Additional information
can be found from Refs. 1.94, 1.158, 3.8, 3.80, 3.117–3.123,
3.195, and 3.196 and in Chapter 5 on compression members.
The advantages of using a numerical solution for the
design of cold-formed steel members are discussed by
Schafer and Pekoz in Refs. 3.195 and 3.196. The element
interaction can be handled properly by the numerical
solution.
3.7 ADDITIONAL INFORMATION
The strength of thin compression elements and the current
design criteria were discussed in this chapter on the basis of
the publications referred to in the text. Additional information on the strength of compression elements and members
Figure 3.76
Plate buckling coefficient k for side h′ of columns.
can also be found from the publications included in the list
of references for this chapter.
The structural behavior of webs of beams subjected to
shear or bearing is discussed in Chapter 4 on the design of
flexural members. The buckling behavior of closed cylindrical tubular members is discussed in Chapter 7.
3. Bracing requirements
4. Shear lag
5. Flange curling
CHAPTER 4
Flexural Members
In general, long-span, shallow beams are governed by
deflection and medium-length beams are controlled by
bending strength. For short-span beams, shear strength may
be critical.
For design tables and charts, reference should be made to
Part II of the AISI Design Manual.1.428
4.2
4.2.1
BENDING STRENGTH AND DEFLECTION
Introduction
In the design of flexural members, sufficient bending strength
must be provided, and at the same time the deflection of
the member under service loads should not exceed specific
limitations.
4.1 GENERAL REMARKS
Beams are used to support transverse loads and/or applied
moment. Cold-formed steel sections such as I-sections,
C-sections (channels), Z-shapes, angles, T-sections, hat
sections, and tubular members (Fig. 1.2) and decks and
panels (Fig. 1.11) can be used as flexural members.
In the design of cold-formed steel flexural members,
consideration should first be given to the moment-resisting
capacity and the stiffness of the member, which may not be a
constant value due to the noncompactness of the thin-walled
section and the variation of the moment diagram. Second,
the webs of beams should be checked for shear, combined
bending and shear, web crippling, and combined bending
and web crippling. In addition, flexural members must be
braced adequately to ensure their sufficient lateral–torsional
buckling strength. Bracing to compression flange of a
flexural member can also increase member’s distortional
buckling strength.
Unlike hot-rolled heavy steel sections, in the design of
thin-walled cold-formed steel beams, special problems such
as shear lag and flange curling are also considered to be
important matters due to the use of thin material. Furthermore, the design of flexural members can be even more
involved if the increase of steel mechanical properties due
to cold work is to be utilized.
Based on the above general discussion, the following
design features are considered in this chapter with some
design examples for the purpose of illustration:
1. Bending strength and deflection
2. Design of webs for shear, combined bending and
shear, web crippling, and combined bending and web
crippling
A. ASD Method. According to Section 1.8.1.1, the
ASD method requires the member flexural or bending
strength to satisfy the following requirement:
𝑀 ≤ 𝑀a
(4.1)
where M is the required flexural strength or bending
moment for ASD computed from the load combinations discussed in Section 1.8.1.2 and 𝑀a is the allowable flexural strength or bending moment determined
by Eq.(4.2):
𝑀
𝑀a = n
(4.2)
Ωb
where Ωb = 1.67 is the safety factor for flexural or
bending strength where the bending strength is determined in Section 4.2. For those members that their
geometry or material property is outside the limits
provided in Table 3.1, rational engineering analysis
may be applied to determine the member strength. In
this case, Ωb = 2.00 should be used according to the
North American Specification Section A1.2(c). 𝑀n in
Eq. (4.2) is the smallest nominal flexural strength or
moment determined from the following four design
considerations:
1. Strength of initiation of yielding and global
lateral–torsional buckling in accordance with
Section 4.2.2
2. Strength of global buckling interacting with local
buckling in accordance with Section 4.2.3
3. Strength of distortional buckling in accordance
with Section 4.2.4
4. Strength of beams having one flange attached to
deck or sheathing determined in accordance with
Section 4.2.8
119
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
120
4
FLEXURAL MEMBERS
In addition to the above-listed four cases, consideration should also be given to shear lag problems for
unusually short span beams (see Section 4.2.10). Interaction of bending with web shear, web crippling, and
torsions should be considered as well.
B. LRFD Method. According to Section 1.8.2.1, the
LRFD method requires the member flexural or
bending strength to satisfy the following requirement:
𝑀u ≤ 𝜙b 𝑀n
(4.3)
where 𝑀u is the required flexural strength or bending
moment for LRFD computed from load combinations
(see Section 1.8.2.2);
𝜙b = 0.90 is the resistance factor where the bending
strength is determined in Section 4.2. For member
strength determined in accordance with rational engineering analysis, 𝜙b = 0.80 should be used according
to the North American Specification Section A1.2(c).
C. LSD Method. According to Section 1.8.3.1, the LSD
method requires that the member flexural or bending
strength to satisfy the following requirement
𝑀f ≤ 𝜙b 𝑀n
(4.4)
where 𝑀f is the bending moment for LSD computed
from load combinations (see Section 1.8.3.2); 𝑀n is
the nominal flexural resistance; and 𝜙b = 0.90 is the
resistance factor where the bending strength is determined in Section 4.2. For member strength determined
in accordance with rational engineering analysis, 𝜙b =
0.75 should be used according to the North American
Specification.
methods for determining the lateral–torsional buckling
strength of singly-, doubly-, and point-symmetric sections
according to the actual number and location of braces. The
design of braces is discussed in Section 4.4.
4.2.2.1 Doubly and Singly Symmetric Sections When a
simply supported, locally stable I-beam is subject to a pure
moment M as shown in Fig. 4.2, the following differential
equations for the lateral–torsional buckling of such a beam
are given by Galambos in Ref. 2.45:
where 𝑀
𝐸
𝐺
𝐼y
𝐶w
EI 𝑦 𝑢iv + 𝑀𝜙′′ = 0
(4.6)
EC𝑤 𝜙iv − GJ𝜙 + 𝑀𝑢′′ = 0
(4.7)
=
=
=
=
=
pure bending moment
modulus of elasticity
shear modulus, = 𝐸∕2(1 + μ)
moment of inertia about the 𝑦 axis
warping constant of torsion of the cross
section (see Appendix B)
𝐽 = St. Venant torsion constant of cross section
∑
approximately determined 13 𝑏i 𝑡3𝑖
𝑢 = deflection of shear center in x direction
𝜙 = angle of twist
The primes indicate differentiation with respect to z.
Considering the simply supported condition, the end
sections cannot deflect or twist; they are free to warp, and no
end moment exists about the y axis. The boundary conditions
are
𝑢(0) = 𝑢(𝐿) = 𝜙(0) = 𝜙(L) = 0
𝑢′′ (0) = 𝑢′′ (𝐿) = ϕ′′ (0) = ϕ′′ (𝐿) = 0
4.2.2 Yielding and Lateral–Torsional Buckling
Strength
A cold-formed cross-section with small w/t ratio may fail by
yielding. The yielding moment is defined by Eq. (4.5):
𝑀y = 𝑆fy 𝐹y
(4.5)
where 𝑀y is the yield moment and 𝑆fy is the elastic section
modulus of cross-section relative to extreme fiber in first
yielding. As shown in Fig. 4.1, on a balanced section (Fig. 4.1
(a)) or a section with neutral axis closer to the tension flange
(Fig. 4.1(c)), 𝑆fy is the section modulus about compression
flange; while for the section with neutral axis closer to the
compression flange (Fig. 4.1(b)), tension flange will yield
first, 𝑆fy should be the section modulus about the tension
flange.
In general, cold-formed steel flexural members with open
cross-sections tend to twist and deflect laterally due to small
lateral–torsional resistance. This section contains the design
(4.8)
(4.9)
The solution of Eqs. (4.6) and (4.7) gives the following
equation for the critical lateral buckling moment:
√
(
)
𝑛2 𝜋 2 ECw
n𝜋
EI 𝑦 GJ 1 +
(4.10)
𝑀cre =
𝐿
GJL2
where L is the span length and 𝑛 = 1, 2, 3, ⋯
The deflected shape of the beam is
)
(
n𝜋z
(4.11)
𝜙 = 𝐶 sin
𝐿
and the lateral deflection u can be determined by
𝑢=
CML2 sin(n𝜋z∕𝐿)
𝑛2 𝜋 2 EI 𝑦
(4.12)
The deflection history of the I-beam is shown in Fig. 4.3.
When 𝑀 ≤ 𝑀cre prior to lateral–torsional buckling taking
place, the beam deflects in the y direction. The vertical deflection 𝑣 can be obtained from Eq. (4.13) for in-plane bending,
EI 𝑥 𝑣′′ = −𝑀
(4.13)
BENDING STRENGTH AND DEFLECTION
121
Figure 4.1 Stress distribution for yield moment: (a) balanced sections; (b) neutral axis close to
compression flange (initial yielding in tension flange); (c) neutral axis close to tension flange (initial
yielding in compression flange).
From Eq. (4.10), for 𝑛 = 1, the lowest critical moment for
lateral–torsional buckling of an I-beam is equal to
𝑀cre
𝜋
=
𝐿
√
(
)
𝜋 2 EC𝑤
EI 𝑦 GJ 1 +
GJL2
(4.16)
Figure 4.2 Simply supported beam subjected to end moments.
For I-beams (Fig. 4.4)
Solving Eq. (4.13) and using the boundary conditions
𝑣(0) = 𝑣(𝐿) = 0, the deflection equation is2.45
[( ) ( ) ]
ML2
𝑧
𝑧 2
−
𝑣=
(4.14)
2EI 𝑥
𝐿
𝐿
When the beam buckles laterally, the section rotates about
the center of rotation 𝐶LB . This point is located at a distance
of 𝑦LB below the shear center of the section as determined by
Eq. (4.15),
𝑢
ML2
(4.15)
𝑦LB ≈ = 2 2
𝜙 𝑛 𝜋 EI 𝑦
𝑏3 td2
24
3
𝑏 𝑡
𝐼𝑥 ≈
6
𝐶w ≈
(4.17)
(4.18)
Equation (4.16) can then be rewritten as follows:
√
𝑀cre =
𝜋
𝐿
EI 𝑦 GJ +
𝐸 2 𝐼𝑦2 𝑑 2 ( 𝜋 )2
4
𝐿
(4.19)
122
4
FLEXURAL MEMBERS
lateral–torsional buckling stress1.161,3.84,4.15 :
(
)
√
𝜋 2 Ed
4GJ𝐿2
𝐼yc − 𝐼yt + 𝐼y 1 + 2
(4.21)
𝜎cre =
2𝐿2 𝑆xc
𝜋 𝐼y 𝐸𝑑 2
Figure 4.3 Positions of I-beam after lateral–torsional buckling.
Figure 4.4 Dimensions of I-beam.
where 𝑆xc is the section modulus relative to the compression fiber and 𝐼yc and 𝐼yt are the moments of inertia of the
compression and tension portions of the full section, respectively, about the centroidal axis parallel to the web. Other
symbols were defined previously. For equal-flange sections,
𝐼yc = 𝐼yt = 𝐼y ∕2 Eqs. (4.20) and (4.21) are identical.
For other than simply supported end conditions, Eq. (4.21)
can be generalized as given in Eq. (4.21a) as follows1.337 :
√
(
)
4GJ(𝐾t 𝐿t )2
𝜋 2 Ed
𝐼yc − 𝐼yt + 𝐼y 1 + 2
𝜎cre =
2(𝐾y 𝐿y )2 𝑆xc
𝜋 𝐼y 𝐸𝑑 2
(4.21a)
In the above equation, 𝐾𝑦 and 𝐾t are effective length
factors and 𝐿𝑦 and 𝐿t are unbraced lengths for bending about
the y axis and for twisting, respectively.
As previously discussed, in Eq. (4.21a) the second term
under the square root represents the St. Venant torsional
rigidity, which can be neglected without much loss in
economy. Therefore Eq. (4.21a) can be simplified as shown
in Eq.(4.22) by considering 𝐼y = 𝐼yc + 𝐼yt and neglecting
the term of 4GJ(𝐾t 𝐿t )2 ∕π2 𝐼y Ed2 :
𝜎cre =
Consequently the critical stress for lateral–torsional buckling of an I-beam subjected to pure bending is given by
𝑀cre
𝑀 𝑑
= cre
𝑆x
2𝐼x
√
√(
)2 (
) ( )2
√
JI y
π2 𝐸 √ 𝐼y
𝐿
=
+
2
2
2𝐼
𝜋d
2(𝐿∕𝑑)
2(1 + 𝜇)𝐼x
x
𝜎cre =
(4.20)
where 𝑆x is the section modulus and 𝐼𝑥 is the moment
of inertia of the full section about the x axis. The unpublished data of 74 tests on lateral–torsional buckling of
cold-formed steel I-sections of various shapes, spans,
and loading conditions have demonstrated that Eq. (4.20)
applies to cold-formed steel sections with reasonable
accuracy.1.161
In Eq. (4.20) the first term under the square root represents
the strength due to lateral bending rigidity of the beam, and
the second term represents the St. Venant torsional rigidity.
For thin-walled cold-formed steel sections, the first term
usually exceeds the second term considerably.
For simply supported I-beams with unequal flanges, the
following equation has been derived by Winter for the elastic
𝜋 2 Ed𝐼yc
(𝐾y 𝐿y )2 𝑆xc
(4.22)
Equation (4.22) was derived on the basis of a uniform
bending moment. It is rather conservative for the case of
unequal end moments. For this reason it may be modified
by multiplying the right-hand side by a bending coefficient
𝐶b 1.161,3.84 as given in Eq. (4.23):
𝜎cre =
𝐶𝑏 𝜋 2 𝐸
(𝐾𝑦 𝐿𝑦 )2 𝑆xc ∕(𝑑𝐼yc )
(4.23)
where 𝐶b is the bending coefficient, which can conservatively be taken as unity. During the period from 1968 to
1996, the bending coefficient was calculated from 𝐶b =
1.75 + 1.05(𝑀1 ∕𝑀2 ) + 0.3(𝑀1 ∕𝑀2 )2 but must not exceed
2.3. Here 𝑀1 is the smaller and 𝑀2 the larger bending
moment at the ends of the unbraced length, taken about the
strong axis of the member. The ratio of end moments 𝑀1 ∕𝑀2
is positive when 𝑀1 and 𝑀2 have the same sign (reverse
curvature bending) and negative when they are of opposite
signs (single curvature bending).
The above equation for 𝐶b was replaced by the following
equation in the 1996 edition of the AISI specification and is
retained in the North American Specification:
12.5𝑀max
𝐶b =
2.5𝑀max + 3𝑀A + 4𝑀B + 3𝑀C
BENDING STRENGTH AND DEFLECTION
where 𝑀max = absolute value of maximum moment in
unbraced segment
𝑀A = absolute value of moment at quarter point
of unbraced segment
𝑀B = absolute value of moment at centerline of
unbraced segment
𝑀C = absolute value of moment at three-quarter
point of unbraced segment
The above equation for bending coefficient 𝐶b was derived
from Ref. 4.156. It can be used for various shapes of moment
diagrams within the unbraced segment and gives more accurate results for fixed-end beams and moment diagrams which
are not straight lines.
Consequently, the simplified, elastic critical moment for
lateral–torsional buckling of doubly symmetric I-beams can
be calculated from the elastic critical buckling stress given in
Eq. (4.23) and the section modulus relative to the compression fiber as follows:
𝐶b 𝜋 2 EdI yc
𝑀cre = 𝜎cre 𝑆xc =
(4.24)
(𝐾y 𝐿y )2
The above design formula was used in Section C3.1.2
(b) of the 1996 edition of the AISI Specification for doubly
symmetric I-sections except that 𝐾y = 1 and 𝐿y = 𝐿. In
the 2016 edition of the North American Specification,
Eq. (4.24) (or Eq. (4.23) for buckling stress) was limited
to doubly-symmetric I-Sections since the equation was
derived from I-Section members and was found that it may
be unconservative for singly-symmetric sections.1.431
It should be noted that Eq. (4.23) applies to elastic buckling of cold-formed steel beams when the computed theoretical buckling stress is less than or equal to the proportional
limit σpr . However, when the computed stress exceeds the
proportional limit, the beam behavior will be governed by
inelastic buckling. For extremely short beams, the maximum
moment capacity may reach the full plastic moment 𝑀p for
compact sections. A previous study4.16 has indicated that for
wide-flange beams having an average shape factor of 10/9,
10
10
𝑀p =
(4.25)
𝑀y =
𝐹 𝑆
9
9 y x
where 𝑀p = full plastic moment
𝑀y = yield moment
This means that the stress in extreme fibers may reach
a hypothetical value of 10
𝐹 when 𝐿2 𝑆xc ∕dI yc ≈ 0 if the
9 y
elastic section modulus is used to compute the moment.
As in the previous design approach for compression
members (Ref. 1.4), the effective proportional limit (or the
upper limit of the elastic buckling) may be assumed to be
equal to one-half the maximum stress, that is,
( )
𝐹𝑦 = 0.56𝐹𝑦
(4.26)
𝜎pr = 12 10
9
123
As shown in Fig. 4.5, assuming 𝐾y = 1 and 𝐿y = 𝐿, the
corresponding 𝐿2 𝑆xc ∕dI yc ratio for 𝜎cr = 𝜎pr is 1.8π2 ECb ∕𝐹y .
When the theoretical critical stress exceeds 𝜎pr , the critical stress for inelastic buckling may be represented by the
following parabolic equation:
[
( )]
1 𝐹𝑦
(4.27)
(𝜎cr )𝐼 = 𝐹𝑦 𝐴 −
𝐵 𝜎cr
where A and B are constants that can be determined by the
following conditions:
1. When 𝐿 = 0,
(𝜎cr )𝐼 =
10
𝐹
9 y
(4.28)
2. When 𝐿2 𝑆xc ∕(dI yc ) = 1.8π2 EC𝑏 ∕𝐹𝑦 ,
(𝜎cr )I = 0.56𝐹y
(4.29)
By solving Eq. (4.27), A and B are found as follows:
𝐴=
10
9
(4.30)
𝐵 = 3.24
(4.31)
Therefore Eq. (4.27) can be rewritten as
[
)]
(
𝐹y
10
1
𝜎crI = Fy
−
9
3.24 σcre
[
)]
(
10 10 𝐹y
= Fy
(4.32)
−
9
36 σcre
where 𝜎cre is the elastic buckling stress for lateral–
torsional buckling and (𝜎cr )I is the theoretical equation
for lateral–torsional buckling in the inelastic range.
4.32
4.23
Figure 4.5 Maximum lateral–torsional buckling stress for
I-beams (𝐾𝑦 = 1 and 𝐿𝑦 = 𝐿).
124
4
FLEXURAL MEMBERS
Even though the maximum stress computed by Eq. (4.32)
as shown in Fig. 4.5 is larger than 𝐹y , a conservative approach
has been used by AISI to limit the maximum stress to 𝐹y .
By using the inelastic critical buckling stress given in Eq.
(4.32) and the section modulus relative to the compression
fiber, the inelastic critical moment for lateral–torsional buckling of I-beams can be computed as follows:
(𝑀cr )I = (𝜎cr )I 𝑆xc ≤ 𝑀y
]
[
10
10 𝑀yc
=
≤ 𝑀y
1−
9
36 𝑀cre
(4.33)
where 𝑀y and 𝑀yc are the yield moment and yield moment
about compression fiber, respectively; and 𝑀cre is the elastic
critical moment defined in Eq. (4.24). Equation (4.33) was
used in Section C3.1.2 of the 1996 edition of the AISI Specification for (𝑀cr )e > 0.56𝑀y as shown in Fig. 4.6. Hill has
demonstrated that the equations derived for I-sections can
also be used for channels with satisfactory accuracy.4.17
For cold-formed steel design, Eqs. (4.23) and (4.32) were
used in the 1968 and 1980 editions of the AISI Specification
to develop the design equations for lateral–torsional buckling of I-beams and channels. In the 1986 and 1996 editions
of the AISI Specification, in addition to the use of Eqs. (4.24)
and (4.33) for determining the critical moment, new design
formulas for lateral–torsional buckling of singly and doubly
symmetric sections bending about the symmetry axis perpendicular to the web3.17,6.11 were added:
√
(4.34)
𝑀cre = 𝐶𝑏 𝑟0 𝐴 𝜎ey 𝜎𝑡
where A is the full cross-sectional area and
𝜋2𝐸
𝜎e𝑦 =
(𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 )2
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
(4.35)
(4.36)
Eq. (4.33)
Eq. (4.24)
Figure 4.6 Maximum lateral–torsional buckling moment for
I-beams (𝐾y = 1 and 𝐿y = 𝐿).
where 𝐾y , 𝐾t = effective length factors for bending about
the y axis and for twisting
𝐿y , 𝐿t = unbraced length for bending about the
y√axis and for twisting
𝑟0 = 𝑟2𝑥 + 𝑟2𝑦 + 𝑥20
𝑟𝑥 , 𝑟𝑦 = radii of gyration of the cross section about
the centroidal principal axes
𝑥0 = distance from the shear center to the
centroid along the principal x axis, taken as
negative
Other terms were defined previously. For singly symmetric
sections, the x axis is the axis of symmetry oriented such that
the shear center has a negative x coordinate. The basis for
Eq. (4.34) is discussed by Pekoz in Ref. 3.17. A comparison
of Eqs. (4.24) and (4.34) shows that these two equations give
similar results for channels having 𝐼𝑥 > 𝐼𝑦 .3.17 However, for
channel sections having 𝐼𝑥 < 𝐼𝑦 with large 𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 ratios,
the simplified Eq. (4.24) provides very conservative results
as compared with Eq. (4.34).
For singly symmetric sections bending about the
centroidal axis perpendicular to the symmetry axis, the
elastic critical moment based on the flexural–torsional
buckling theory can be computed by using Eq. (4.37):
√
C𝑠 𝐴𝜎ex [𝑗 + 𝐶𝑠 𝑗 2 + 𝑟20 (𝜎𝑡 ∕𝜎ex )]
(4.37)
𝑀er =
𝐶TF
where 𝐶s = +1 for moment causing compression on shear
center side of centroid
𝐶s = −1 for moment causing tension on shear center
side of centroid
𝜎ex =
𝜋2𝐸
(𝐾𝑥 𝐿𝑥 ∕𝑟𝑥 )2
(4.38)
𝐶TF = 0.6 − 0.4(𝑀1 ∕𝑀2 ), where 𝑀1 is the smaller
and 𝑀2 the larger bending moment at the ends
of the unbraced length and 𝑀1 ∕𝑀2 , the ratio
of end moments, is positive when 𝑀1 and 𝑀2
have the same sign (reverse curvature bending)
and negative when they are of opposite sign
(single curvature bending). When the bending
moment at any point within an unbraced length
is larger than that at both ends of this length
and for members subject to combined axial
load and bending moment, 𝐶TF shall be taken
as unity.
𝐾𝑥 = effective length factor for bending about the
x axis
𝐿𝑥 = unbraced length for bending about the 𝑥 axis
BENDING STRENGTH AND DEFLECTION
and
𝑗=
1
2𝐼𝑦
∫𝐴
Table 4.1 Coefficients K in Eq. (4.41)3.3
)
(
𝑥3 dA +
∫𝐴
xy2 dA − 𝑥0
125
(4.39)
= 𝛽y ∕2 (see Appendix C for computation of 𝛽y ) (4.40)
Other terms were defined previously.
The derivation of Eq. (4.37) is presented in Chapter 6 for
beam–columns.
It should be noted that Eqs. (4.34) and (4.37) can be used
only when the computed value of 𝑀cre does not exceed
0.56𝑀y , which is considered to be the upper limit for the
elastic buckling range. When the computed 𝑀cre exceeds
0.56𝑀y , the inelastic critical moment can be computed from
Eq. (4.33). The elastic and inelastic critical moments are
shown in Fig. 4.7.
The equations developed above for the uniform bending
moment can also be used for other loading conditions with
reasonable accuracy.1.161,4.18 If more accurate results are
desired, the theoretical critical value for a concentrated load
at the center of a simply supported beam can be computed
as3.3
√
EI 𝑦 GJ
𝑃cre = 𝐾
(4.41)
𝐿2
where K is a coefficient to be taken from Table 4.1 based
on the parameter GJL2 ∕ECw . For symmetrical I-sections,
𝐶w ≈ 𝐼y 𝑑 2 ∕4, where d is the depth of the section.
For a uniformly distributed load, the critical load is
√
EI 𝑦 GJ
𝑤cre = 𝐾
(4.42)
𝐿3
where K is to be taken from Table 4.2.
Eq. (4.33)
Loads Act At
GJ
𝐿2
ECw
0.4
4
8
16
24
32
48
64
80
96
160
240
320
400
Centroid
Top Flange
Bottom Flange
86.4
31.9
25.6
21.8
20.3
19.6
19.0
18.3
18.1
17.9
17.5
17.4
17.2
17.2
51.3
20.2
17.0
15.4
15.0
14.8
14.8
14.9
14.9
15.1
15.3
15.6
15.7
15.8
145.6
50.0
38.2
30.4
27.2
26.3
23.5
22.4
21.7
21.1
20.0
19.3
18.9
18.7
Table 4.2 Coefficients K in Eq. (4.42)3.3
Loads Act At
GJ
𝐿2
ECw
0.4
4
8
16
24
32
48
64
80
128
200
280
360
400
Centroid
Top Flange
Bottom Flange
143.0
53.0
42.6
36.3
33.8
32.6
31.5
30.5
30.1
29.0
29.0
28.8
28.7
28.6
92.9
36.3
30.4
27.4
26.6
26.1
25.8
25.7
25.7
26.0
26.4
26.5
26.6
26.6
222.0
77.3
59.4
48.0
43.4
40.4
37.6
36.2
35.1
33.3
32.1
31.4
31.0
30.7
Eq. (4.24)
Eq. (4.34)
Unbraced length
Figure 4.7 Elastic and inelastic critical lateral buckling moments
for members bending about centroidal axis perpendicular to the
web.
4.2.2.2 Point-Symmetric Sections Point-symmetric sections such as Z-sections with equal flanges will buckle laterally at lower strengths than doubly and singly symmetric
sections. A conservative design approach has been used in
the previous AISI Specification and is also used in the North
American Specification, in which the elastic critical moment
is taken to be one-half of those permitted for I-beams or channels. Therefore instead of using Eq. (4.34), the following
126
4
FLEXURAL MEMBERS
equation is used for determining the elastic critical moment
for point-symmetric Z-sections bending about the centroidal
axis perpendicular to the web:
√
1
(4.43)
𝑀cre = 𝐶𝑏 𝑟0 𝐴 𝜎ey 𝜎𝑡
2
In lieu of Eq. (4.43), the following simplified equation
can be used to calculate the elastic critical moment for
Z-sections:
𝐶𝑏 𝜋 2 EdI yc
𝑀cre =
(4.44)
2(𝐾𝑦 𝐿𝑦 )2
All symbols used in Eqs. (4.43) and (4.44) are defined in
Section 4.2.2.1.
4.2.2.3 Closed-Box Sections Closed sections such as
box shapes have relatively larger torsional stiffness as
compared with open sections such as I-beams, C-sections,
and Z-sections discussed in Sections 4.2.2.1 and 4.2.2.2. As
far as lateral–torsional buckling is concerned, these closed,
double-web sections are more stable than single-web open
sections, and therefore any use of closed-box sections will
result in an economical design if lateral–torsional stability
of the beam is essential.
In Ref. 4.18, Winter indicated that for closed-box beams
the bending strength is unaffected by lateral–torsional
buckling even when the length-to-width ratio is as high as
100 for a steel having a yield stress of 33 ksi (228 MPa or
2320 kg/cm2 ).
Previous editions of the AISI Specification contained
in Section D3.3 a conservative design provision for
lateral–torsional buckling of closed-box beams, in which
laterally unbraced box sections can be designed without any
strength reduction for lateral–torsional buckling consideration if the ratio of the unsupported length to the distance
between the webs of the section does not exceed 0.086
𝐸∕𝐹y . In 1999, this design requirement was replaced by
Section C3.1.2.2 in the Supplement to the 1996 edition of
the AISI Specification.1.333 The same design provisions are
retained in the North American Specification, except that
the laterally unbraced length, L, was clarified to be 𝐾𝑦 𝐿𝑦 .
For a closed-box section subjected to a uniform bending
moment as shown in Fig. 4.2, the elastic critical moment for
lateral–torsional buckling is3.84
√
𝜋
𝑀cre =
EI 𝑦 GJ
𝐿
All terms are defined in Section 4.2.2.1, except that the
torsional constant J may be determined by the following
equation for a closed-box section having a uniform
thickness4.157 :
2𝑏2 𝑑 2 𝑡
𝐽=
(4.45)
𝑏+𝑑
where 𝑏 = midline or centerline dimension of flange
𝑑 = midline or centerline dimension of web
𝑡 = wall thickness
When a closed-box section is subject to a nonuniform
bending moment, the above equation for the elastic critical
moment can be modified by a bending coefficient 𝐶b as
follows:
𝐶 𝜋√
𝑀cre = 𝑏
EI 𝑦 GJ
(4.46)
𝐿
Consequently, the elastic critical lateral–torsional buckling stress 𝐹cre can be determined by Eq. (4.47):
𝐶𝑏 𝜋 √
EI 𝑦 GJ
(4.47)
𝐹cre =
𝐾 𝑦 𝐿𝑦 𝑆 𝑓
in which 𝑆f is the elastic section modulus of the full unreduced section relative to the extreme compression flange, 𝐾𝑦
is the effective length factor, and 𝐿𝑦 is the unbraced length
for bending about the y axis.
4.2.2.4 Lateral–Torsional Buckling with Hole Influence
The lateral–torsional buckling stress, 𝜎cre or moment, 𝑀cre ,
discussed previously is based on beams without holes. The
existence of holes along the member length will reduce
the member bending rigidity, EI, and consequently reduce
the buckling strength.3.252,3.278,4.294,4.295,6.39 The numerical
shell finite element method can be used to determine the
global buckling moment. However, the complication in identifying the different buckling modes and software not readily
available to regular design engineers make it not feasible in
design. The finite strip method is not applicable since the
section properties affected by holes cannot be considered
simply by revising the thickness. Therefore, the analytical method developed by Moen and Schafer3.278,4.294,4.295
is recommended. Based on the research and experiment
verifications,4.295 the lateral–torsional buckling moment
of doubly- and singly-symmetric section members with
patterned holes can be approximated by using the average
section properties in the buckling expression as developed
in the previous sections:
√
(
)
𝜋
𝜋2
𝑀cre =
EI y,avg GJ avg + ECwnet
𝐾 𝑦 𝐿𝑦
(𝐾𝑡 𝐿𝑡 )2
(4.48)
where 𝐼𝑦 ,avg and 𝐽avg are the average moment of inertia about
the y axis and average torsional constant, respectively; 𝐶wnet
is the warping constant of the net section. The existence of the
hole creates a discontinuity that interrupts warping torsion
resistance along the member. It is therefore a net warping
constant is used. 𝐶wnet can be determined by assuming thickness at the hole equals zero, it can also be determined using
software such as CUFSM by setting the thickness of the
element at the hole equals zero.
BENDING STRENGTH AND DEFLECTION
A complete list of the average properties of cross-section
is provided in Table 4.3, which can also be used to determine
the global buckling loads for columns with holes later in
Chapter 5.
The above analytical method was first adopted into the
2012 edition of the North American Specification1.416 and
is retained in 2016 edition of the Specification.1.417
The hole influence on the member strength could be
counterintuitive. The Commentary1.431 on the 2016 North
American Specification provided the following guidance:
“Rules of thumb on the influence of holes in both compression and flexural members are: (1) rectangular or elongated
holes typically reduce local buckling strength more than
square and circular holes; (2) web holes always decrease
distortional buckling strength; (3) holes always reduce
global (Euler) buckling strength; (4) the more holes along a
member, the more the strength decreases; (5) hole patterns,
such as those typically present in storage rack columns,
can reduce strength as much as discrete holes; and (6)
adding edge stiffeners to holes increases local buckling
strength more than distortional buckling and global buckling
strength.”
4.2.2.5 North
American
Design
Criteria
for
Lateral–Torsional Buckling Strength of Singly-, Doubly-,
and Point-Symmetric Sections Sections 4.2.2.1–4.2.2.4
discussed how to determine the member lateral–torsional
buckling moment and how to consider the inelastic moment
strength when the buckling moment exceeds 0.56𝑀y .
The North American Specification developed the design
provisions based on those expressions. The following
excerpts are adapted from Sections F2.1 and F2.2 of the
2016 edition of the North American Specification, which
provides the needed design equations for computing the
critical lateral–torsional buckling stress.1.417 The applications of the North American design criteria are illustrated in
Examples 4.1–4.3.
F2.1 Initiation of Yielding Strength
The nominal flexural strength [resistance], Mne , for yielding and
global (lateral–torsional) buckling considering capacity up to
first yield shall be calculated in accordance with Eq. (4.49).
𝑀ne = 𝑆f 𝐹n ≤ 𝑀y
(4.49)
where 𝑀ne = Nominal flexural strength [resistance] for
yielding and global buckling
𝑆f = Elastic section modulus of full unreduced
section relative to extreme compression fiber
𝑀y = 𝑆fy 𝐹y
where 𝑆fy = Elastic section modulus of full unreduced crosssection relative to extreme fiber in first yielding
𝐹y = Yield stress
Fn shall be determined as follows:
For 𝐹cre ≥ 2.78Fy
𝐹n = 𝐹y
(4.51)
For 2.78 𝐹y > 𝐹cre > 0.56𝐹y
(
)
10𝐹y
10
𝐹n =
𝐹 1−
9 y
36𝐹cre
(4.52)
For Fcre ≤ 0.56Fy
𝐹𝑛 = 𝐹cre
(4.53)
where 𝐹cre = Critical elastic lateral–torsional buckling stress,
determined in accordance with Specification
Sections F2.1.1 to F2.1.5, as applicable, or
Specification Appendix 2.
F2.1.1 Singly or Doubly Symmetric Sections Bending
About Symmetric Axis
The elastic buckling stress for singly or doubly symmetric
sections bending about the symmetric axis shall be calculated
as follows:
C r A√
𝜎ey 𝜎t
(4.54)
𝐹cre = b 0
Sf
where
𝐶b =
12.5Mmax
2.5Mmax + 3MA + 4MB + 3MC
(4.55)
where 𝑀max = Absolute value of maximum moment
in unbraced segment
𝑀𝐴 = Absolute value of moment at quarter point
of unbraced segment
𝑀B = Absolute value of moment at centerline
of unbraced segment
𝑀C = Absolute value of moment at three-quarter
point of unbraced segment
𝐶b = is permitted to be conservatively taken as unity
for all cases. For cantilevers or overhangs
where the free end is unbraced, 𝐶b shall be
taken as unity.
𝑟0 = Polar radius of gyration of cross-section about
shear center
√
(4.50)
127
=
𝑟2𝑥 + 𝑟2𝑦 + 𝑥20
(4.56)
128
4
FLEXURAL MEMBERS
Table 4.3
Average Cross-Sectional Properties𝒂
Average Properties
Formulas
𝐴g 𝐿g + 𝐴net 𝐿net
Cross-sectional area
𝐴avg =
Moment of inertia about axis of buckling
𝐼avg =
Saint-Venant Torsion constant
𝐽avg =
Shear center x-coordinate relative to centroid
𝑥0,𝑎v𝑔 =
Shear center y-coordinate relative to centroid
𝑦0,𝑎v𝑔 =
L
𝐽g 𝐿g + 𝐽net 𝐿net
𝑟0,𝑎v𝑔 =
Polar radius gyration about shear center
𝑎
L
𝐼g 𝐿g + 𝐼net 𝐿net
𝐿
𝑥0,𝑔 𝐿𝑔 + 𝑥0,net 𝐿net
𝐿
𝑦0,𝑔 𝐿𝑔 + 𝑦0,net 𝐿net
√
𝐿
𝑥20,𝑎v𝑔 + 𝑦20,𝑎v𝑔 +
𝐼𝑥,𝑎v𝑔 +𝐼𝑦,𝑎v𝑔
𝐴𝑎v𝑔
Definition of variables:
𝐴g , 𝐴net
𝐿g
𝐿net
𝐿
𝐼g , 𝐼net
=
=
=
=
=
=
𝐽g , 𝐽net
𝑥0,g 𝑥0,net
𝑦0 ,g , 𝑦0 ,net
𝑟0,g , 𝑟0 ,net
𝐼xy,g , 𝐼xy,net
=
=
=
=
=
Gross and net area, respectively
Segment length without holes
Length of holes or net section regions
Unbraced length about the axis of buckling
𝐿g + 𝐿net
Moment of inertia of gross or net cross-section about the axis of buckling,
respectively
Saint-Venant torsion constant of gross or net cross-section, respectively
Shear center x-coordinate relative to centroid for gross or net section, respectively
Shear center y-coordinate relative to centroid for gross or net section, respectively
Polar radius gyration about shear center of gross or net cross-section, respectively
Product of inertia of gross or net cross-section, respectively
where 𝑟x , 𝑟y = Radii of gyration of cross-section about
centroidal principal axes
𝑥0 = Distance from centroid to shear center in
principal x-axis direction
𝐴 = Full unreduced cross-sectional area
𝑆f = Elastic section modulus of full unreduced
cross-section relative to extreme compression
fiber
𝜎ey =
𝜋2E
(Ky Ly ∕ry )2
(4.57)
where 𝐸 = Modulus of elasticity of steel
𝐾𝑦 = Effective length factor for bending about y axis
𝐿y = Unbraced length of member for bending about
y axis
𝜎t =
[
]
𝜋 2 𝐸𝐶w
1
𝐺𝐽
+
(Kt Lt )2
Ar20
(4.58)
where 𝐺 = Shear modulus of steel
𝐽 = Saint-Venant torsion constant
of cross-section
𝐶w = Torsional warping constant of cross-section
𝐾t = Effective length factor for twisting
𝐿t = Unbraced length of member for twisting
For singly-symmetric sections, the x axis shall be the axis
of symmetry.
Alternatively, for doubly-symmetric I-sections, 𝐹cre is
permitted to be calculated using the equation given
𝐹cre =
Cb π2 EdI yc
Sf (Ky Ly )2
(4.59)
where 𝑑 = Depth of section
𝐼yc = Moment of inertia of compression portion
of section about centroidal axis of entire section
parallel to web, using full unreduced
section
BENDING STRENGTH AND DEFLECTION
F2.1.2 Singly-Symmetric Sections Bending About
Centroidal Axis Perpendicular to Axis of Symmetry
The elastic buckling stress, 𝐹cre , for singly-symmetric sections
bending about the centroidal axis perpendicular to the axis of
symmetry shall be calculated as follows, where the x axis is the
symmetric axis of the cross-section oriented such that the shear
center has a negative x-coordinate:
]
[
√
C 𝐴𝜎
𝐹cre = s ex j + Cs j2 + r02 (σt ∕σex )
(4.60)
CTF Sf
where 𝐶s = +1 for moment causing compression on shear
center side of centroid
= −1 for moment causing tension on shear center
side of centroid
𝜎ex =
𝜋2E
(Kx Lx ∕rx )2
(4.61)
where 𝐾x = Effective length factor for bending about xaxis
𝐿x = Unbraced length of member for bending about
x axis
𝐶TF = 0.6 − 0.4 (M1 ∕M2 )
(4.62)
where 𝑀1 and 𝑀2 = The smaller and the larger bending
moment, respectively, at the ends of the
unbraced length in the plane of bending;
𝑀1 ∕𝑀2 , the ratio of end moments, is
positive when 𝑀1 and 𝑀2 have the
same sign (reverse curvature bending)
and negative when they are of opposite
sign (single curvature bending). When
the bending moment at any point within
an unbraced length is larger than that at
both ends of this length, 𝐶TF shall be
taken as unity
]
[
1
𝑗=
x3 dA +
xy2 dA − x0
(4.63)
∫A
2I𝑦 ∫A
where 𝑥0 = Distance from centroid to shear center in principal
x-axis direction, taken as negative
Other variables are defined in Specification Section F2.1.1.
F2.1.3 Point-Symmetric Sections
The elastic buckling stress, 𝐹cre , for point-symmetric Z-sections
bending about x axis that is perpendicular web and through the
centroid is permitted to be calculated as follows:
C r A√
𝐹cre = b 0
(4.64)
σey σt
2Sf
Alternatively, 𝐹cre is permitted to be calculated using
Eq. (4.65):
Cb π2 EdI 𝑦𝑐
𝐹cre =
(4.65)
2Sf (K𝑦 L𝑦 )2
Variables are defined in Specification Section F2.1.1.
129
F2.1.4 Closed-Box Sections
For closed-box section members, if the laterally unbraced length
of the member is less than or equal to 𝐿u , as calculated in Eq.
(4.66), the global buckling does not need to be considered, and
the nominal stress, 𝐹n = 𝐹y .
0.36𝐶b 𝜋 √
EGJIy
(4.66)
𝐿u =
Fy Sf
where 𝐽 = Torsional constant of closed-box section
𝐼y = Moment of inertia of full unreduced section about
centroidal axis parallel to web
𝐹y = Yield stress
Other variables are defined in Specification Section F2.1.1.
If the laterally unbraced length of a member is larger than
Lu , as calculated in Eq. (4.66), the elastic buckling stress, 𝐹cre ,
for bending about the symmetric axis shall be calculated as
follows:
Cb 𝜋 √
EGJIy
(4.67)
𝐹cre =
Ky Ly Sf
F2.1.5 Other Cross-Sections
For cross-sections other than those defined in Specification
Sections F2.1.1 through F2.1.4, the elastic buckling stress is
permitted to be determined in accordance with Specification
Section 2.2 of Appendix 2.
F2.2 Beams With Holes
For shapes whose cross-sections have holes, Fcre shall consider
the influence of holes in accordance with Specification
Appendix 2.
Exception: For the Effective Width Method, where hole sizes
meet the limitations of Specification Appendix 1.1.3, the provisions of this section shall not be required.
Even though the existence of holes does affect the
global buckling moment, North American Specification1.417
provides some relief when the Effective Width Method is
used and the hole dimensions are within the limitations of
Specification Appendix 1.1.3.
From Fig. 4.7, it can be seen that when a member’s
unbraced length is less than length, 𝐿u , the global buckling
does not occur. 𝐿u can be derived from the stress changes
between Eqs. (4.51) and (4.52) by setting 𝐹cre = 2.78𝐹y . The
following equations are given in Section 1.4 of Part II of the
AISI Cold-Formed Steel Design Manual1.428 for computing
𝐿u :
(a) For singly, doubly, and point-symmetric sections,
[
⎧
(
)2 ]0.5 ⎫
𝐶2
⎪
⎪ GJ
GJ
+
+
𝐿𝑢 = ⎨
⎬
2𝐶
𝐶
2𝐶
1
1
1
⎪
⎪
⎭
⎩
0.5
(4.68)
130
4
FLEXURAL MEMBERS
SOLUTION
As shown in Fig. 4.7, when 𝐾y 𝐿y ≤ 𝐿u , 𝑀ne = 𝑀y (or 𝐹n =
𝐹y ). The elastic critical lateral–torsional buckling stress of
the I-section is determined according to Section F2.1 of the
North American Specification. For 𝐹n = 𝐹y ,
where
1. For singly- and doubly-symmetric sections,
(
)2
𝜋 2 ECw
7.72 𝐾𝑦 𝐹𝑦 𝑆f
𝐶1 =
C2 =
AE
𝐶b 𝜋𝑟𝑦
(𝐾t )2
2. For point-symmetric sections,
(
)2
30.9 𝐾𝑦 𝐹𝑦 𝑆𝑓
𝐶1 =
AE
𝐶𝑏 𝜋𝑟𝑦
𝐹cre ≥ 2.78𝐹y
𝜋 2 EC𝑤
C2 =
(𝐾𝑡 )2
(b) For I-, C-, or Z-sections bent about the centroidal axis
perpendicular to the web (x axis), in lieu of (a), the following
equations may be used:
1. For doubly-symmetric I-sections:
)0.5
(
0.36𝐶𝑏 𝜋 2 EdI yc
(4.69a)
𝐿𝑢 =
𝐹𝑦 𝑆𝑓 (𝐾𝑦 )2
2. For point-symmetric Z-sections:
)0.5
(
36𝐶𝑏 𝜋 2 EdI yc
𝐿𝑢 =
𝐹𝑦 𝑆𝑓 (𝐾𝑦 )2
3. For closed-box sections:
0.36𝐶b 𝜋 √
𝐿u =
EGJI𝑦
𝐹y 𝑆 f
(4.69b)
(4.70)
In addition, Part II of the Design Manual provides beam
design charts for determining the nominal flexural strengths
of C-sections and Z-sections with lips. These charts were
prepared for 𝐹y = 33, 50 and 55 ksi (228, 345, and 379 MPa;
2.32, 3.52, and 3.87 × 103 kg∕cm2 ) with Cb = 1.0. The
torsional unbraced length (𝐾t 𝐿t ) is assumed to be equal to
the unbraced length about the y axis (𝐾y 𝐿y ).
Example 4.1 An I-beam with cross-section shown
in Fig. 4.8 is used as a simply supported beam with
a span length of 10 ft to support a uniform load (see
Fig. 4.8). Determine what is the maximum unbraced length
such that the beam will not subject to lateral torsional
buckling.
Substituting Eq. (4.59) for 𝐹cre into the above expression
yields
𝐶𝑏 𝜋 2 EdI yc
≥ 2.78𝐹𝑦
𝑆𝑓 (𝐾𝑦 𝐿𝑦 )2
Therefore,
𝐾 y 𝐿y ≤
√
𝐶b 𝜋 2 EdI yc ∕(2.78𝐹y 𝑆f )
where 𝐸 = 29.5 × 103 ksi
𝐶b = 1.0 (assumed value)
𝐷 = 8.0 in.
𝐹𝑦 = 50 ksi
𝐾y = 1.0
𝐼yc = 0.724 in.4 (see the following calculation)
The section properties are calculated using the center-line
dimensions: corner radius r = 3/16 in.+ t/2 = 0.255 in.,
flange flat width b = 2 in. − (𝑟 + 𝑡∕2) = 1.6775 in.; flat
web depth 𝑎 = 8 in –2(𝑟 + 𝑡∕2) = 7.355 in. Corner length =
1.57𝑟 = 0.400 in.
Calculation of A, Iyc , Sf :
Element
Flanges
Corners
Webs
Total
Distance from y
Axis, xci (in.)
Area Ai (in.2 )
Ai xci 𝟐 (in.4 )
4(1.6775)(0.135) = 0.9059 2–b∕2 = 1.1613
4(0.400)(0.135) = 0.2162 0.363r + t∕2
= 0.160
2(7.355)(0.135) = 1.9859 𝑡∕2 = 0.0675
3.108
Iflanges = 4 ×
1
0.135(1.6775)3
12
1.2215
0.0055
0.0090
1.2360
= 0.2124
I𝑦 = 1.4484 in.4
𝐼yc = 12 𝐼y = 0.724 in4
𝑆𝑓 = 6.54 in.3 (see the following calculation)
Element
Area Ai (in.𝟐 )
Flanges
Corners
Webs
Total
4(1.6775)(0.135) = 0.9059
4(0.05407) = 0.2163
2(7.355)(0.135) = 1.9859
3.1081
2𝐼web = 2 ×
Figure 4.8
Example 4.1.
1
12
Distance from
Mid depth yi (in.)
3.9325
3.8436
0
× (0.135)(7.355)3
𝐼𝑥 =
𝑆f =
26.1570 in.4
= 6.54 in.3
𝐼𝑥
8∕2
Ai y𝟐i (in.𝟒 )
14.0093
3.1955
0
17.2048
8.9522
BENDING STRENGTH AND DEFLECTION
131
The maximum unbraced length between lateral supports is
√
(1)𝜋 2 (29,500)(8)(0.724)
𝐿y =
= 43.1 in.
2.78(50)(6.54)
Alternatively, the above unbraced length can be calculated
directly from Eq. (4.69a). Actually, the beam may be braced
laterally at one-third span length with an unbraced length
of 40 in., as shown in Fig. 4.9. For segment CD, 𝐶b = 1.01,
which is practically the same as the assumed value of 1.0.
Figure 4.10
𝐶b =
Example 4.2.
12.5(wL2 ∕8)
2.5(wL2 ∕8) + 3(7wL2 ∕128)
+ 4(12wL2 ∕128) + 3(15wL2 ∕128)
= 1.30
Figure 4.9 Lateral supports.
Example 4.2 Determine the allowable uniform load
considering the lateral–torsional buckling only if the I-beam
used in Example 4.1 is braced laterally at both ends and
midspan. See Fig. 4.10. Use the value of 𝐶b determined
by the formula included in the North American Specification and 𝐹𝑦 = 50 ksi. Use the ASD method and the LRFD
method with an assumed dead load–live load ratio 𝐷∕𝐿 = 15 .
SOLUTION
A. ASD Method
1. Nominal Moment for Lateral–Torsional Buckling
Strength. From Example 4.1, 𝑆𝑓 = 6.54 in.3 and
𝐼yc = 0.724 in.4 Considering the lateral supports at
both ends and midspan and the moment diagram shown
in Fig 4.10, the bending coefficient 𝐶b for segment AB
can be calculated by using Eq. (4.55) as follows:
𝐶b =
12.5𝑀max
2.5𝑀max + 3𝑀1 + 4𝑀2 + 3𝑀3
where 𝑀max = wL2 ∕8 at midspan
𝑀1 = 7wL2 ∕128 at 1∕4 point of unbraced
segment
𝑀2 = 12wL2 ∕128 at midpoint of unbraced
segment
𝑀3 = 15wL2 ∕128 at 3∕4 point of unbraced
segment
Using Eq. (4.59), with 𝐾𝑦 = 1.0,
𝐹cre =
=
𝐶𝑏 𝜋 2 EdI yc
𝑆𝑓 (𝐾𝑦 𝐿𝑦 )2
(1.30)𝜋 2 (29, 500)(8)(0.724)
= 93.11 ksi
(6.54)(5 × 12)2
0.56𝐹y = 28.00 ksi
2.78𝐹y = 139.00 ksi
Since 2.78 𝐹y > 𝐹cre > 0.56𝐹y , from Eq. (4.52)
(
)
10𝐹𝑦
10
𝐹𝑛 =
𝐹𝑦 1 −
9
36𝐹cre
(
)
10(50)
10
=
(50) 1 −
= 47.27 ksi
9
36(93.11)
Based on Eq. (4.49), the nominal moment for lateral–
torsional buckling strength is
𝑀ne = Sf 𝐹n = (6.54)(47.27) = 309.15 in.-kips
3. Allowable Uniform Load. The allowable moment based
on lateral–torsional buckling strength is calculated as
follows:
(𝑀𝑎 ) =
𝑀ne
309.15
=
= 185.12 in.-kips
Ω𝑏
1.67
The maximum moment at midspan is wL2 /8 ft-kips:
1 2
wL (12) = 185.12 in.-kips
8
132
4
FLEXURAL MEMBERS
Then the allowable uniform load is
The allowable moment is
𝑤 = 1.234 kips∕ft
𝑀a = 𝑀D + 𝑀L = 30.24 + 151.22 = 181.46 in.-kips
It should be noted that the allowable load computed above
is based on the consideration of lateral–torsional buckling. It
should also be checked for bending due to local and distortional buckling. In addition, shear, web crippling, deflection,
and other requirements should be checked, as applicable.
B. LRFD Method
Using the same method employed above for the ASD
method, the governing nominal moment for lateral–torsional
buckling strength is
The allowable uniform load can be calculated as
follows:
1 2
wL (12) = 181.46 in.-kips w = 1.21 kips∕ft
8
It can be seen that the allowable uniform load computed
on the basis of the LRFD method is similar to that computed
from the ASD method. The difference is only about 2%.
𝑀ne = 309.15 in.-kips
Example 4.3 For the singly symmetric channel section
(8 × 2 × 0.06 in.) shown in Fig. 4.11, determine the nominal
moment, 𝑀ne , for lateral–torsional buckling strength
according to Section F2.1 of the 2016 edition of the North
American Specification. Assume that the channel is used as
a simply supported beam to support a concentrated load at
midspan and lateral supports are located at one-fourth of the
span length. Use 𝐹y = 33 ksi, 𝐾𝑦 𝐿𝑦 = 𝐾t 𝐿t = 2.5 ft.
The design moment is
𝜙𝑏 𝑀ne = 0.90(309.15) = 278.24 in.-kips
Based on the load combination of Eq. (1.5a), the required
moment is
𝑀u = 1.4𝑀D
where MD is the bending moment due to dead load. Similarly,
based on the load combination of Eq. (1.5b), the required
moment is
𝑀u = 1.2𝑀D + 1.6𝑀L = 1.2𝑀D + 1.6(5𝑀D ) = 9.2𝑀D
where 𝑀L is the bending moment due to live load. A comparison of the above computations indicates that for a given
member the load combination of Eq. (1.5b) allows a smaller
moment 𝑀D than the load combination of Eq. (1.5a). Therefore, the bending moment 𝑀D can be computed from 𝑀u =
𝜙b 𝑀ne as follows:
SOLUTION
1. Sectional Properties. By using the design formulas
given in Part I of the AISI Design Manual,1.428 the
following full-section properties can be calculated:
9.2𝑀D = 278.24 in.-kips
Therefore,
𝐴 = 0.706 in.2
𝑥0 = −0.929 in.
𝑆f = 1.532 in.3
𝑟0 = 3.14 in.
𝑟𝑥 = 2.945 in.
𝐽 = 0.000848 in.4
𝑟𝑦 = 0.569 in.
𝐶w = 2.66 in.6
2. Elastic Critical Lateral–Torsional Buckling
Stress, Fcre . Because the given singly symmetric
channel section is subject to a moment bending
about the symmetry axis (x axis), the elastic critical
278.24
𝑀𝐷 =
= 30.24 in.-kips
9.2
ML = 5𝑀𝐷 = 151.22 in.-kips
Figure 4.11
Example 4.3.
BENDING STRENGTH AND DEFLECTION
lateral–torsional buckling stress can be determined
according to Eq. (4.54) as follows:
𝐶 𝑟 𝐴√
𝜎e𝑦 𝜎t
𝐹cre = b 0
𝑆f
For two central segments, the value of 𝐶b can be
computed from Eq. (4.55) as follows:
12.5𝑀max
𝐶b =
2.5𝑀max + 3𝑀1 + 4𝑀2 + 3𝑀3
where 𝑀max = PL∕4 at midspan
𝑀1 = 5PL∕32 at 1/4 point of unbraced
segment
𝑀2 = 6PL∕32 at midpoint of unbraced
segment
𝑀3 = 7PL∕32 at 34 point of unbraced
segment
Then
𝐶b =
𝜎ey =
12.5(PL∕4)
2.5(PL∕4) + 3(5PL∕32)
+ 4(6PL∕32) + 3(7PL∕32)
= 1.25
𝜋 2 (29, 500)
𝜋2E
=
2
(Ky Ly ∕ry )
(2.5 × 12∕0.569)2
= 104.73 ksi
[
]
𝜋 2 ECw
1
𝜎t =
GJ +
(Kt Lt )2
A𝑟20
[
1
=
(11, 300)(0.000848)
(0.706)(3.14)2
]
𝜋 2 (29, 500)(2.66)
+
(2.5 × 12)2
= 125.0 ksi
Therefore, the elastic critical lateral–torsional buckling stress is
(1.25)(3.14)(0.706) √
Fcre =
(104.74)(125.0)
(1.532)
= 206.96 ksi
3. Critical Lateral–Torsional Buckling Stress, Fn .
0.56𝐹y = 18.48 ksi
2.78𝐹y = 91.74 ksi
Since 𝐹cre > 2.78𝐹y , the member segment is not
subject to lateral–torsional buckling. 𝐹n = Fy
4. Nominal Moment 𝑀ne . According to Eq. (4.49) the
nominal moment based on lateral–torsional buckling
is:
𝑀ne = 𝑆f 𝐹n = (1.532)(33) = 50.56 in.-kips
133
It should be noted that 𝑀ne is the nominal moment
based on lateral–torsional buckling. The member
design strength should also consider the local buckling
and distortional buckling, as applicable.
4.2.2.6 Inelastic Reserve Strength The inelastic reserve
strength in considering yielding and global buckling is shown
in Fig. 3.68b in the region with 𝑀n ∕𝑀y surpassing 1, where a
linear function between 𝑀p and 𝑀y is used. To determine the
inelastic reserve strength, the following design provisions,
developed based on the research work by Shifferaw and
Schafer,4.296 were adopted in the 2012 edition of the North
American Specification, and is retained in the 2016 edition
of the Specification:
F2.4.2 Direct Strength Method
The nominal strength (resistance), Mne , considering inelastic
flexural reserve capacity is permitted to be considered in accordance with the provisions of this section:
For 𝑀cre > 2.78𝑀y
√
My ∕Mcre − 0.23
Mne = Mp − (Mp − My )
0.37
≤ Mp
(4.71)
where
𝑀cre = Critical elastic lateral–torsional buckling moment
= 𝑆𝑓 𝐹cre
(4.72)
where 𝑆𝑓 = Elastic section modulus of full unreduced
cross-section relative to extreme
compression fiber
𝐹cre = Critical elastic lateral–torsional buckling
stress, determined in accordance with
Specification Appendix 2 or
Section F2.1
𝑀𝑦 = Member yield moment in accordance with
Specification Section F2.1
𝑀p = Member plastic moment
= Zf Fy
(4.73)
where 𝑍f = Plastic section modulus
𝐹y = Yield stress
4.2.3 Local Buckling Interacting With Yielding
and Global Buckling
For members with larger w/t ratio, the local buckling interacting with the lateral–torsional buckling will result in
134
4
FLEXURAL MEMBERS
reduced member strength. Both the Effective Width Method
and the Direct Strength Method can be used to determine
the reduced strength due to local buckling.
For members with small w/t ratio, like hot-rolled steel,
it is possible for member cross-section to reach yielding in
compression and/or tension sides and the inelastic reserve
capacity may be considered.
Both of above two situations will be considered by the
Effective Width Method and the Direct Strength Method in
the following subsections.
4.2.3.1 Effective Width Method The Effective Width
Method considers the local buckling interacting with the
beam lateral–torsional buckling by reducing the section
modulus, Sf , in global buckling Eq. (4.49) to the effective
section modulus, Se , which is calculated based on stress
level Fn as determined in Specification Section F2.1. The
effective width of the compression flange and the effective
depth of the web can be computed from the design equations
presented in Chapter 3.
Therefore, the nominal strength for local buckling interacting with the global buckling is calculated by using
Eq. (4.74):
𝑀n𝓁 = 𝑆e 𝐹n ≤ 𝑆et 𝐹y
(4.74)
where 𝐹n = nominal stress considering lateral–torsional
buckling per Eqs. (4.51) to (4.53)
𝑆e = elastic section modulus of effective section
calculated with compression fiber at 𝐹n
𝑆e = elastic section modulus of effective section
with respect to the extreme compression
fiber
𝑆et = elastic section modulus of effective section
calculated with tension fiber at 𝐹y
As discussed in Section 4.2.2, when the neutral axis
is close to the compression flange, it is possible that the
yielding initiating from the tension flange controls the
design. Equation (4.74) therefore requires that the nominal
strength 𝑀n𝓁 be less than 𝑆et 𝐹y .
In cold-formed steel design, the effective section modulus,
𝑆e or 𝑆et , is usually computed by using one of the following
two cases:
1. If the neutral axis is closer to the tension than to
the compression flange, as shown in Fig. 4.1c, the
maximum stress occurs in the compression flange, and
therefore the plate slenderness factor 𝜆 and the effective width of the compression flange are determined by
the w/t ratio and 𝑓 = 𝐹n in Eq. (3.35). Of course, this
procedure is also applicable to those beams for which
the neutral axis is located at the middepth of the section,
as shown in Fig. 4.1a.
2. If the neutral axis is closer to the compression than
to the tension flange, as shown in Fig. 4.1(b), the
maximum stress of 𝐹y occurs in the tension flange.
The stress in the compression flange depends on the
location of the neutral axis, which is determined by
the effective area of the section. The latter cannot be
determined unless the compressive stress is known.
The closed-form solution of this type of design is
possible but would be a very tedious and complex
procedure. It is therefore customary to determine
the sectional properties of the section by successive
approximation.
The calculation of the nominal moment on the basis of
initiation of yielding and the determination of the design
moment are illustrated in Examples 4.4–4.7.
4.2.3.1.1 Members with Holes For members with a hole,
the effective widths of the flat elements adjacent to the hole
are treated as unstiffened elements. Other elements are determined as described in Specification Appendix 1 or Chapter
3 of this book. Note, if the holes in a member are within the
limitations of Specification Section 1.1.3, the Specification
permits that the lateral–torsional buckling stress, 𝐹n , being
calculated based on gross cross-section, i.e., the hole influence can be ignored in considering global buckling.
Example 4.4 Determine the local buckling strength
interacting with the lateral–torsional buckling using the
Effective Width Method. Use the ASD and LRFD methods
to check the adequacy of the I-section with an unstiffened compression flange as shown in Fig. 4.12. Assume
the beam is simply supported with ends and midspan
lateral bracing as shown in Fig. 4.10. The dead-load
moment 𝑀D = 30 in.-kips and the live-load moment
𝑀L = 140 in.-kips.
Figure 4.12
Example 4.4.
BENDING STRENGTH AND DEFLECTION
SOLUTION
A. ASD Method
1. Lateral–torsional buckling strength. From Example
4.2, the nominal stress and nominal moment due to
lateral–torsional buckling are 𝐹n = 47.27 ksi, and
𝑀ne = 309.15 in.-kips, respectively.
2. Local buckling strength interacting with the lateral–
torsional buckling. The nominal strength considering
the local buckling can be determined per Section
4.2.3.1. From Eq. (4.74),
𝑀𝑛𝓁 = 𝑆𝑒 𝐹𝑛 ≤ 𝑆et 𝐹𝑦
The effective section modulus is calculated based on
the stress Fn , as shown below:
(a) Calculation of Sectional Properties. The
sectional properties of the corner element can
be obtained from Table 4.4 or approximated using
3
the linear method (see Fig. 1.32). For 𝑅 = 16
in.
and 𝑡 = 0.135 in.,
𝐼𝑥 = 𝐼𝑦 = 0.0003889 in.4
𝐴 = 0.05407 in.2
OK according to section 3.2.
Since the compression flange is an unstiffened
element and the neutral axis is either at middepth or
closer to the tension flange, use Eqs. (3.32)–(3.35)
with 𝑘 = 0.43 and 𝑓 = 𝐹n = 47.27 ksi. Therefore
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
√
1.052
47.27
(12.426)
=√
29,500
0.43
𝑤=
4.000
− (𝑅 + 𝑡) = 1.6775 in.
2
𝑤
= 1.6775∕0.135 = 12.426
𝑡
1 − 0.22∕𝜆
𝜆
1–0.22∕0.798
=
= 0.907
0.798
𝑏 = 𝜌w = (0.907)(1.6775)
𝜌=
[Eq. (3.34)]
= 1.523 in.
By using the effective width of the compression
flange and assuming the web is fully effective, the
location of the neutral axis, the moment of inertia
𝐼𝑥 and the elastic section modulus of the effective
section 𝑆e can be computed as shown in table on the
following page.
One 90∘ Corner, Dimensions and Properties
Table 4.4
Dimensions
Thickness t
(in.)
0.135
0.105
0.075
0.060
0.048
0.036
[eq. (3.35)]
= 0.798 > 0.673
𝑥 = 𝑦 = 0.1564 in.
For the unstiffened flange,
135
Properties
Inside Radius
R (in.)
Moment of Inertia
𝐼𝑥 = 𝐼𝑦 (in.4 )
Centroid Coordinates
𝑥 = 𝑦(in.)
Area A
(in.2 )
Blank Width
(in.)
0.1875
0.1875
0.0938
0.0938
0.0938
0.0625
0.0003889
0.0002408
0.0000301
0.0000193
0.0000128
0.00000313
0.1564
0.1373
0.0829
0.0734
0.0658
0.0464
0.05407
0.03958
0.01546
0.01166
0.00888
0.00452
0.3652
0.3495
0.1865
0.1787
0.1724
0.1170
Notes:
(1) Stock width of blank taken at t/3 distance from inner surface.
(2) 1 in. = 25.4 mm.
136
Element
4
FLEXURAL MEMBERS
Area 𝐴(in.2 )
Top flange
2(1.523)(0.135) = 0.4111
Top
2(0.05407) = 0.1081
corners
Webs
2(7.355)(0.135) = 1.9859
Bottom
2(0.05407) = 0.1081
corners
Bottom
2(1.6775)(0.135) = 0.4529
flange
Total
3.0661
∑
(Ay)
12.4288
𝑦cg = ∑
=
= 4.054 in.
3.0661
𝐴
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓)
Distance
from Top
Fiber
y (in.)
Ay(in.3 )
0.0675
0.1564
0.0278
0.0169
0.0019
0.0026
4.0000
7.8436
7.9434
0.8479
31.7746
6.6506
7.9325
3.5928
28.500
12.4288
66.930
= 4 + 2(1 + 0.971)3 + 2(1 + 0.971) = 23.26
From Fig. 4.12,
Ay2 (in.4 )
ℎo = out-to-out depth of web
= 8.00 in.
𝑏o = out-to-out width of the compression
flange of each channel = 2.00 in.
Since ℎo ∕𝑏o = 4, then use Eq. (3.47a),
𝑏𝑒
𝑏1 =
3+𝜓
where 𝑏e is the effective width of the web determined in accordance with Eqs. (3.32)–(3.35) with
f1 substitued for f and 𝑘 = 23.26 as follows:
Since 𝑦cg > 𝑑∕2 = 4.00 in., initial yield occurs
in the compression flange. Prior to computing the
moment of inertia, check the web for full effectiveness by using Fig. 4.13 and Section 3.3.1.2 as
follows:
ℎ = 7.355 in.
ℎ 7.355
=
= 54.48 < 200
𝑡
0.135
OK according to Section 3.2
[Eq. (3.44)]
√
1.052
43.51
(54.48)
𝜆= √
= 0.456 < 0.673
29,500
23.26
)
3.7315
= 43.51 ksi (compression)
4.054
)
(
3.6235
𝑓2 = 47.27
= 42.25 ksi (tension)
4.054
| 𝑓 | | 42.25 |
| = 0.971
𝜓 = || 2 || = ||
|
| 𝑓1 | | 43.51 |
𝑓1 = 47.27
[Eq. (3.46)]
(
𝜌=1
𝑏𝑒 = ℎ = 7.355 in.
47.27
3.7315”
4.054”
3.6235”
3.946”
Figure 4.13 Stress distribution in webs.
[Eq. (3.33)]
BENDING STRENGTH AND DEFLECTION
𝑏𝑒
[Eq. (3.47a)]
3+𝜓
7.355
=
= 1.852 in.
3 + 0.971
Since 𝜓 > 0.236,
𝑏1 =
1
𝑏2 = 𝑏𝑒 = 3.6775 in.
2
b1 = 𝑏2 = 1.852 + 3.6775 = 5.5295 in. [Eq. (3.47b)]
Since 𝑏1 + 𝑏2 is greater than the compression
portion of the web of 3.7315 in., the web is fully
effective as assumed. Since the neutral axis (ycg =
4.054 in.) is very close to the assumed location
(𝑦cg = 4.00 in), no iteration is necessary. The total
𝐼𝑥 is determined as
∑
(Ay2 ) = 66.930
( )
1
(0.135)(7.355)3 = 8.9522
2Iweb = 1
12
(∑ )
−
A (y2cg ) = −(3.0661)(4.054)2 = −50.391
Ix = 25.491 in.4
The elastic section modulus relative to the top
fiber is
𝐼
25.4912
𝑆e = x =
= 6.288 in.3
𝑦cg
4.054
B. LRFD Method
1. Nominal and Design Moments. The nominal moment
for the LRFD method is the same as that used for the
ASD method, that is,
𝑀𝑛𝓁 = 297.23 in.-kips
The design moment is
𝜙b 𝑀𝑛𝓁 = 0.9(297.23) = 267.51 in.-kips
2. Required Moment. According to the load factors and
the load combinations discussed in Section 1.3.3.3, the
required moment for the given dead-load moment and
live-load moment can be computed as follows:
𝑀u1 = 1.4𝐷
= 1.4(30) = 42.00 in. − kips
𝑀u2 = 1.2𝐷 + 1.6𝐿
= 1.2(30) + 1.6(150)
= 276.00 in. − kips ⇐ controls
Since 𝑀u < 𝜙b 𝑀n , the I-section is also adequate for the
LRFD method.
Example 4.5 For the C-section with an edge stiffener as
shown in Fig. 4.14, determine the allowable moment (𝑀a )
about the x axis for the ASD method and the design moment
(𝜙b 𝑀n ) for the LRFD method. Assume that the yield stress
3. Nominal and Allowable Moments. The nominal
moment considering the local buckling interacting with
the lateral–torsional buckling is
𝑀𝑛𝓁 = 𝑆e 𝐹n = (6.288)(47.27) = 297.23 in.-kips
Compared with the nominal moment due to yielding
and lateral–torsional buckling (𝑀ne = 309.15 in.kips)
calculated in Example 4.2, the nominal moment due
to local buckling interacting with the global buckling
controls.
Therefore, the allowable moment is
𝑀
297.23
𝑀𝑎 = 𝑛𝓁 =
= 178.0 in.-kips
Ω𝑏
1.67
4. Required Moment. Based on the ASD load combination discussed in Section 1.3.1.2, the required moment
for the given dead-load moment and live-load moment
is computed as follows:
𝑀 = 𝑀D + 𝑀L = 30 + 140 = 170 in.-kips
Since 𝑀 < 𝑀a , the I-section is adequate for the ASD
method.
137
Figure 4.14
Example 4.5 (same as Fig 3.50).
138
4
FLEXURAL MEMBERS
of steel is 50 ksi and that lateral bracing is adequately
provided. Determine the nominal moment due to local buckling with initiation of yielding. The linear method can be used
to determine the sectional properties.
SOLUTION
A. ASD Method
1. Calculation of Sectional Properties. In order to
simplify the calculation, line elements, as shown in
Fig. 4.15a, are used for the linear method.
i. Corner Element. (Figs. 1.32 and 4.15a)
1
𝑅 = 𝑅 + 𝑡 = 0.131 in.
2
′
Arc length:
𝐿 = 1.57𝑅′ = 1.57(0.1313) = 0.2063 in.
𝑐 = 0.637𝑅′ = 0.637(0.1313) = 0.0836 in.
ii. Effective Width of the Compression Flange. For
the given C-section with equal flanges, the neutral
axis is located either at the middepth or closer to
the tension flange. Therefore, use 𝑓 = 𝐹y = 50 ksi
to compute the effective width of the compression flange according to Section 3.3.3.1a. For the
compression flange,
𝑤 = 3.50 − 2(𝑅 + 𝑡) = 3.1624 in.
𝑤 3.1624
=
= 42.17
𝑡
0.075
From Eq. (3.72)
√
√
29,500
𝐸
= 1.28
= 31.09
𝑆 = 1.28
𝑓
50
0.328𝑆 = 10.20
Since w/t > 0.328 S, use Eq. (3.73) to compute
the required moment of inertia of the edge stiffener
𝐼a as follows:
(
)3
4 𝑤∕𝑡
− 0.328
𝐼a = 399𝑡
𝑆
(
)3
42.17
= 399(0.075)4
− 0.328
31.09
= 13.72 × 10−3 in.4
(a)
The above computed value should not exceed the
following value:
)
(
𝑤∕𝑡
4
𝐼a ≤ 𝑡 115
+5
𝑆
(
)
42.17
= (0.075)4 115
+5
31.09
= 5.093 × 10−3 in.4
Therefore, use 𝐼a = 5.093 × 10−3 in.4 For the
simple lip edge stiffener used for the given channel
section,
𝐷 = 0.720 in.
𝑑 = 𝐷 − (𝑅 + 𝑡) = 0.5512 in.
𝑑 0.5512
=
= 7.35
𝑡
0.075
(b)
Figure 4.15
(a) Line elements. (b) Compression stresses f1 and f2 .
By using Eq. (3.75), the moment of inertia of the
full edge stiffener is
𝐼s =
𝑑3𝑡
= 1.047 × 10−3 in.4
12
BENDING STRENGTH AND DEFLECTION
From Eq. (3.74),
𝑅I =
𝐼s
= 0.206 < 1.0
𝐼a
OK
The effective width b of the compression flange
can be calculated as follows:
0.72
𝐷
=
= 0.228
𝑤 3.1624
From Eq. (3.76),
𝑤∕𝑡
42.17
= 0.582 −
4𝑆
4(31.09)
1
= 0.243 <
3
𝑛 = 0.582 −
Use 𝑛 = 13
Since 𝐷∕𝑤 < 0.25 and θ = 90∘ for the simple
lip edge stiffened, from Table 3.3
𝑘 = 3.57(𝑅I )𝑛 + 0.43
= 3.57(0.206)1∕3 + 0.43 = 2.54 < 4.0
OK
Use 𝑘 = 2.54 to calculate the plate slenderness factor for the compression flange as
follows:
(
)
( )√𝑓
1.052
𝑤
𝜆=
√
𝑡
𝐸
𝑘
(
)
√
50
1.052
= √
(41.17)
= 1.146 > 0.673
29,500
2.54
The effective width of the compression
flange is
(
)
1 − 0.22∕𝜆
𝑏 = 𝜌w =
𝑤
𝜆
(
)
1 − 0.22∕1.146
=
(3.1624) = 2.230 in.
1.146
From Eqs. (3.69) and (3.70),
( )
1
1
bR𝐼 = (2.230)(0.206) = 0.230 in.
𝑏1 =
2
2
𝑏2 = 𝑏 − 𝑏1 = 1.146 − 0.230 = 2.00 in.
iii. Reduced Effective Width of the Edge Stiffener.
The effective width of the edge stiffener under
a stress gradient can be determined according to
Section 3.3.2.2. From Eq. (3.56),
𝑘=
0.578
𝜓 + 0.34
|f |
where 𝜓 = || 2 ||
| f1 |
139
In the above equations, the compression stresses
𝑓1 and 𝑓2 (Fig 4.15b) are calculated on the basis of
the gross section as follows:
)
(
4.8312
= 48.312 ksi
𝑓1 = 50
5.0
(
)
4.28
𝑓2 = 50
= 42.80 ksi
5.0
Therefore
| 42.80 |
| = 0.886
𝜓 = ||
|
| 48.312 |
and
0.578
= 0.471
𝑘=
0.886 + 0.34
The k value of 0.471 calculated above for the
edge stiffener under the stress gradient is slightly
larger than the k value of 0.43 for unstiffened
elements under uniform compression.
The effective width of the edge stiffener can be
determined as follows:
𝑑 𝐷 − (𝑅 + 𝑡)
=
= 7.35
𝑡
𝑡
𝑓 = 𝑓1 = 48.312 ksi
)
(
( )√𝑓
𝑑
1.052
𝜆=
√
𝑡
𝐸
𝑘
(
)
√
48.312
1.052
= √
(7.35)
29,500
0.471
= 0.456 < 0.673
𝜌 = 1.0
The effective width of the edge stiffener is
𝑑s ′ = 𝑑 = 0.551 in.
The reduced effective width of the edge
stiffener is
𝑑s = 𝑑s ′ (𝑅I ) = 0.551(0.206) = 0.113 in.
The above calculation
the compression stiffener
effective.
indicates
is not
that
fully
iv. Location of Neutral Axis and Computation of 𝐼x
and Sx .
a. Location of Neutral Axis. Assuming that the
web element (element 7 in Fig. 4.16) is fully
effective, the neutral axis can be located by
using the following table. See Fig. 4.16 for
dimensions of elements.
140
4
FLEXURAL MEMBERS
| f | | 41.05 |
| = 0.847
𝜓 = || 2 || = ||
|
| f1 | | 48.44 |
k = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓) = 20.30
From Fig 4.14, the out-to-out web depth h0 =
10.00 in. and the out-to-out compression flange
width b0 = 3.50 in. Since h0 ∕b0 = 10.00∕3.50
= 2.86 < 4, use Eq. (3.47a),
𝑏1 =
𝑏𝑒
3+𝜓
where 𝑏e is the effective width of the web determined in accordance with Eqs. (3.32)–(3.35)
with 𝑓1 substitued for f and 𝑘 = 20.30 as
follows:
Figure 4.16 Effective lengths and stress distribution using fully
effective web.
Element
1
2
3
4
5
6
7
Total
𝑦cg
Effective
Length L (in.)
0.5512
2(0.206) = 0.4120
3.1624
2(0.206) = 0.4120
2.2300
0.1130
9.6624
16.5430
Distance
from Top
Fiber y (in.)
∑
(Ly) 89.3135
=
= ∑
= 5.399 in.
16.5430
𝐿
9.5556
9.9148
9.9625
0.0852
0.0375
0.2254
5.0000
ℎ 9.6624
=
= 128.83 < 200 OK
𝑡
0.075
√
48.44
1.052
(128.83)
𝜆= √
29,500
20.30
= 1.219 > 0.673
1 − 0.22∕𝜆
𝜌=
= 0.672
𝜆
𝑏𝑒 = 𝜌h = (0.672)(9.6624) = 6.4931 in.
𝑏1 =
Ly(in.2 )
5.2670
4.0849
31.5054
0.0351
0.0836
0.0255
48.3120
89.3135
Use Section 3.3.1.2 in this volume or Section
1.1.2 of the North American Specification to
check the effectiveness of the web element.
From Fig. 4.16,
)
(
5.2302
= 48.44 ksi (Compression)
f1 = 50
5.399
)
(
4.4322
f2 = 50
= 41.05 ksi (Tension)
5.399
𝑏𝑒
6.4931
=
= 1.6878 in.
3+𝜓
3 + 0.847
Because 𝜓 > 0.236, Eq. (3.47b) is used to
compute b2 :
1
𝑏2 = 𝑏e = 3.2465 in.
2
𝑏1 + 𝑏2 = 4.9343 in.
Since the value of 𝑏1 + 𝑏2 is less than
5.2302 in. shown in Fig. 4.16, the web element
is not fully effective as assumed. The neutral
axis should be relocated by using the partially
effective web. The procedure is iterative as
illustrated below.
b. Location of Neutral Axis Based on Ineffective
Web Elements. As the first iteration, the ineffective portion of the web can be assumed as
follows:
5.2302 − (𝑏1 + 𝑏2 ) = 5.2302 − 4.9343
= 0.2959 in.
Therefore, the effective lengths of all
elements are shown in Fig. 4.17 using partially
effective web.
BENDING STRENGTH AND DEFLECTION
141
1–0.22∕1.233
= 0.666
1.233
𝑏e = 𝜌h = 6.4352 in.
𝜌=
𝑏e
= 1.6820 in.
3+𝜓
1
𝑏2 = 𝑏𝑒 = 3.2176 in
2
𝑏1 = 𝑏2 = 4.8996 in.
b1 =
Because the above computed value of 𝑏1 + 𝑏2
is less than the previous value of 4.9343 in. by
0.7%, additional iterations are required.
For the second iteration, the ineffective
portion of the web is
5.2922 − (𝑏1 + 𝑏2 ) = 5.2922 − 4.8966
= 0.3926 in.
By using the same procedure shown above,
the neutral axis can be relocated as follows:
Figure 4.17 Effective lengths and stress distribution using
partially effective web (first iteration).
Element
Element
1
2
3
4
5
6
7
8
𝑦cg =
Effective
Length L (in.)
Distance from
Top Fiber y (in.)
0.5512
0.4120
3.1624
0.4120
2.2300
0.1130
7.6787
1.6878
16.2471
9.5556
9.9148
9.9625
0.0852
0.0375
0.2254
5.9919
1.0127
Ly(in.2 )
5.2670
4.0849
31.5054
0.0351
0.0836
0.0255
46.0100
1.7092
88.7207
88.7207
= 5.461 in.
16.2471
From Fig. 4.17,
𝑓1 = 48.45 ksi (compression)
𝑓2 = 40.01 ksi (tension)
ℎ
k = 19.83
= 128.83
𝑡
√
48.45
1.052
(128.83)
𝜆= √
29,500
19.83
= 1.233 > 0.673
𝜓 = 0.826
1
2
3
4
5
6
7
8
Effective
Length
L (in.)
Distance from
Top Fiber
y (in.)
0.5512
0.4120
3.1624
0.4120
2.2300
0.1130
7.5878
1.6820
16.1504
9.5556
9.9148
9.9625
0.0852
0.0375
0.2254
6.0373
1.0098
𝑦cg =
Ly(in.2 )
Ly2 (in.3 )
5.2670
4.0849
31.5054
0.0351
0.0836
0.0057
45.8098
1.6985
88.5098
50.3298
40.5009
313.8727
0.0030
0.0031
0.0057
276.5675
1.7151
682.9977
88.5098
= 5.481 in.
16.1504
From Fig. 4.18,
𝑓1 = 48.46 ksi (compression)
𝑓2 = 39.68 ksi (tension)
ℎ
= 128.83
𝑡
√
1.052
48.46
𝜆= √
(128.83)
29,500
19.68
𝜓 = 0.819
k = 19.68
= 1.238 > 0.673
𝜌=
1–0.22∕1.238
= 0.664
1.238
142
4
FLEXURAL MEMBERS
𝐼𝑥′ = 234.6337 in.3
𝐼𝑥 = 𝐼𝑥′ 𝑡 = (234.6337)(0.075) = 17.598 in.4
17.598
= 3.211 in.3
5.481
2. Nominal and Allowable Moments. The nominal
moment for considering local buckling is
𝑆𝑥 =
𝑀n𝓁 = 𝑆e 𝐹y = 𝑆x 𝐹y = 3.211(50)
= 160.55 in.-kips
The allowable moment is
𝑀
160.55
𝑀𝑎 = 𝑛𝓁 =
= 96.14 in.-kips
Ω𝑏
1.67
B. LRFD Method
The nominal moment for the LRFD method is the same as
that computed for the ASD method. From item A above, the
nominal moment about the x axis of the C-section is
𝑀𝑛𝓁 = 160.55 in-kips
Figure 4.18 Effective lengths and stress distribution using
partially effective web (second interaction).
𝑏e = 𝜌h = 6.4158 in.
𝑏e
= 1.6800 in.
3+𝜓
1
𝑏2 = 𝑏e = 3.2079 in.
2
𝑏1 = 𝑏2 = 4.8879 in.
𝑏1 =
Because the above computed value of 𝑏1 + 𝑏2
is approximately equal to the value of 𝑏1 + 𝑏2
computed from the first iteration, it is acceptable. Better accuracy can be obtained by using
additional iterations.
c. Moment of Inertia and Section Modulus. The
moment of inertia based on line elements is
𝐼1′ =
1
(0.5512)3
12
=
0.0140
1
(0.113)3
12
=
0.0001
𝐼6′ =
=
1
(7.5878)3
12
𝐼8′ =
1
(1.6820)3
12
𝐼7′
= 36.4054
= 0.3965
∑
(Ly2 ) = 682.9977
𝐼𝑧′ = 719.8137 in.3
(∑ )
−
𝐿 (𝑦2cg ) = −(16.1504)(5.481)2 = −485.1800
The corresponding available moment:
𝜙b 𝑀𝑛𝓁 = 0.9(160.55) = 144.50 in.-kips
Example 4.6 For the hat section with a stiffened compression flange as shown in Fig. 4.19, determine the allowable
moment (𝑀a ) about the x axis for the ASD method and the
design moment (𝜙b 𝑀𝑛𝓁 ) for the LRFD method. Assume that
the yield stress of steel is 50 ksi, and the member does not
subject to lateral–torsional buckling. Determine the nominal
moment due to local buckling with initiation of yielding. The
linear method can be used to determine the member sectional
properties.
A. ASD Method
1. Calculation of Sectional Properties. In order to use
the linear method, midline dimensions are shown in
Fig. 4.20.
i. Corner Element. (Figs. 1.32 and 4.20)
1
𝑅′ = 𝑅 + 𝑡 = 0.240 in.
2
Arc length:
𝐿 = 1.57𝑅′ = 0.3768 in.
c = 0.637R′ = 0.1529 in.
ii. Location of Neutral Axis.
a. First Approximation. For the compression
flange,
𝑤 = 15 − 2(𝑅 + 𝑡) = 14.415 in.
𝑤
= 137.29
𝑡
BENDING STRENGTH AND DEFLECTION
Figure 4.19
Example 4.6
Figure 4.20
Line elements.
Using Eqs. (3.32)–(3.35) and assuming 𝑓 =
𝐹y = 50 ksi,
√
1.052
50
𝜆 = √ (137.29)
= 2.973 > 0.673
29,500
4
1 − 0.22∕2.973
𝜌=
= 0.311
2.973
𝑏 = 𝜌w = 0.311(14.415) = 4.483 in.
By using the effective width of the compression flange and assuming that the web is fully
effective, the neutral axis can be located as
follows:
Element
1
2
3
4
5
Total
Effective
Length L (in.)
2 × 1.0475
2 × 0.3768
2 × 9.4150
2 × 0.3768
=
=
=
=
2.0950
0.7536
18.8300
0.7536
4.4830
26.9152
Distance
from Top
Fiber
y (in.)
∑
(Ly) 122.7614
=
ycg = ∑
= 4.561 in.
26.9152
L
9.9475
9.8604
5.0000
0.1396
0.0525
143
Ly(in.2 )
20.8400
7.4308
94.1500
0.1052
0.2354
122.7614
144
4
FLEXURAL MEMBERS
Because the distance 𝑦cg is less than the
half depth of 5.0 in., the neutral axis is closer
to the compression flange and, therefore, the
maximum stress occurs in the tension flange.
The maximum compressive stress can be
computed as follows:
)
(
4.561
= 41.93 ksi
𝑓 = 50
10 − 4.561
Since the above computed stress is less than
the assumed value, another trial is required.
b. Second Approximation. After several trials,
assume that
𝑓 = 40.70 ksi
𝜆 = 2.682 > 0.673
𝑏 = 4.934 in.
Distance
Effective
from Top
Element Length L (in.) Fiber y (in.)
1
2
3
4
5
Total
2.0950
0.7536
18.8300
0.7536
4.9340
27.3662
𝑦cg =
122.7850
= 4.487 in.
27.3662
that is,
9.9475
9.8604
5.0000
0.1396
0.0525
Ly(in.2 )
Ly2 (in.3 )
20.8400 207.3059
7.4308 73.2707
94.1500 470.7500
0.1052
0.0147
0.2590
0.0136
122.7850 751.3549
(
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓)
= 4 + 2(2.245)3 + 2(2.245) = 31.12
From Fig 4.19,
ℎ0 = out-to-out depth of web
= 10.00 in.
= 15.00 in.
Since ℎ0 ∕𝑏0 = 10∕15 = 0.667 < 4, then use Eq.
(3.47a),
𝑏𝑒
𝑏1 =
3+𝜓
where be is the effective width of the web determined in accordance with Eqs. (3.32)–(3.35) with
𝑓1 substitued for f and 𝑘 = 31.12 as follows:
ℎ 9.415
=
= 89.67 < 200
OK
𝑡 0.105
√
1.052
38.04
(89.67)
= 0.607 < 0.673
𝜆= √
29,500
31.12
𝑏e = ℎ = 9.415 in.
𝑏e
= 2.218 in.
3+𝜓
Since 𝜓 > 0.236,
1
𝑏2 = 𝑏e = 4.7075 in.
2
𝑏1 + 𝑏2 = 6.9255 in.
𝑏1 =
)
4.487
= 40.69 ksi
10 − 4.487
Since the above computed stress is close to
the assumed value, it is OK.
iii. Check the Effectiveness of the Web. Use Section
3.3.1.2 in this volume or Section 1.1.2 of the North
American Specification to check the effectiveness
of the web element. From Fig. 4.21,
)
(
4.1945
𝑓1 = 50
= 38.04 ksi (compression)
5.513
)
(
5.2205
𝑓2 = 50
= 47.35 ksi (tension)
5.513
|𝑓 |
𝜓 = || 2 || = 1.245
| 𝑓1 |
𝑓 = 50
𝑏0 = out-to-out width of compression flange
Because the computed value of 𝑏1 + 𝑏2 is
greater than the compression portion of the web
(4.1945 in.), the web element is fully effective.
iv. Moment of Inertia and Section Modulus. The
moment of inertia based on line elements is
( )
1
(9.415)3 = 139.0944
2𝐼3′ = 2
12
∑
(Ly)2 = 751.3549
𝐼z′ = 890.4493 in.3
(∑ )
−
𝐿 (𝑦2cg ) = −27.3663(4.487)2 = −550.9683 in.3
𝐼x′ = 339.4810 in.3
The actual moment of inertia is
𝐼𝑥 = 𝐼𝑥′ 𝑡 = (339.4810)(0.105) = 35.646 in.4
The section modulus relative to the extreme
tension fiber is
35.646
𝑆𝑥 =
= 6.466 in.3
5.513
2. Nominal and Allowable Moments. The nominal
moment for section strength is
𝑀𝑛𝓁 = 𝑆e 𝐹y = 𝑆x 𝐹y = (6.466)(50) = 323.30 in.-kips
The allowable moment is
𝑀
323.30
𝑀a == 𝑛𝓁 =
= 193.59 in.-kips
Ωb
1.67
BENDING STRENGTH AND DEFLECTION
Figure 4.21
145
Effective lengths and stress distribution using fully effective web.
B. LRFD Method
The nominal moment for the LRFD method is the same as
that computed for the ASD method. From item A above, the
nominal moment about the x axis of the hat section is
𝑀𝑛𝓁 = 323.30 in.-kips
The corresponding design moment is:
𝜙b 𝑀𝑛𝓁 = 0.9(323.30) = 290.97 in.-kips
Example 4.7 For the section with an intermediate stiffener
as shown in Fig. 4.22, determine the allowable moment (𝑀a )
about the x axis for the ASD method and the design moment
(𝜙b 𝑀𝑛𝓁 ) for the LRFD method. Use the linear method with
𝐹y = 33 ksi. The nominal moment is determined by initiation
of yielding.
2.2662
2.3962
Figure 4.23
2.435
2.565
Example 4.7; line elements.
SOLUTION
A. ASD Method
1. Calculation of Sectional Properties. Using the linear
method as shown in Fig. 4.23.
i. Corner Element. (Figs. 1.32 and 4.23)
1
𝑅′ = 𝑅 + 𝑡 = 0.1313 in.
2
Arc length:
𝐿 = 1.57𝑅′ = 1.57(0.1313) = 0.2063 in.
𝑐 = 0.637R′ = 0.637(0.1313) = 0.0836 in.
Figure 4.22
Example 4.7.
ii. Location of Neutral Axis Based on Section
3.3.3.3.1 in this volume or Section 1.4.1 of the
North American Specification.
146
4
FLEXURAL MEMBERS
a. First Approximation. For the top compression
flange,
𝑏o = 12 − 2(𝑅 + 𝑡) = 11.6624 in.
𝑏p = 𝑤 = 5.5686 in.
= 24.027
The modification factor for the distortional
plate buckling coefficient can be computed as
follows:
ℎ = 5 − 2(𝑅 + 𝑡) = 4.6625
𝑏o 11.6624
=
= 2.501 > 1
ℎ
4.6625
11 − 𝑏o ∕ℎ 11 − 2.501
𝑅=
=
5
5
1
= 1.700 >
OK
2
𝐴s = [2(0.7) + 4(0.2063)](0.075)
= 0.1669 in.2
𝐴g = 2(5.5686)(0.075) + 0.1669
= 1.0022 in.2
The moment of inertia of the full intermediate
stiffener (elements 7, 8, and 9 in Fig. 4.23) about
its own centroidal axis is given as
]
[ ( )
1
(0.7)3 + 4(0.2063)×
2 12
𝐼s =
(0.075)
(0.35 + 0.0836)2
the moment of inertia of the stiffener about
the centerline of the flat portion of the element
as
2
= 0.015923 + 0.1669(0.4813)2
and the plate local buckling coefficient for the
subelement [Eq. 3.91)] as
𝑘loc = 4(𝑏0 ∕𝑏p ) = 4(11.6624∕5.5686)
2
= 17.545
From Eqs. (3.92)–(3.95),
10.92𝐼sp
10.92(54.585 × 10−3 )
𝛾=
=
𝑏0 𝑡3
(11.6624)(0.075)3
0.596068
= 121.150
0.00492
𝐴
0.1669
= 0.1908
𝛿= s =
𝑏0 𝑡 11.6624 × 0.075
=
𝛽 = [1 + 𝛾(𝑛 + 1)]1∕4
= [1 + 121.150(1 + 1)]0.25 = 3.949
𝑘d = plate buckling coefficient
for distortional buckling
=
=
(1 + 𝛽 2 )2 + 𝛾(1 + 𝑛)
𝛽 2 [1 + 𝛿(𝑛 + 1)]
(1 + 3.9492 )2 + 121.150(1 + 1)
3.9492 [1 + 0.1908(1 + 1)]
= 40.846 > (𝑘loc = 17.545)
𝑘 = minimum of (Rkd and 𝑘loc ) = 17.545
Then
𝑓cr𝓁 =
𝑘𝜋 2 𝐸
12(1 − 𝜇2 )(𝑏0 ∕𝑡)2
=
= 54.585 × 10−3 in.4
2
Rkd = (1.700)(24.027)
use
= 15.923 × 10−3 in.4
𝐼sp = 𝐼s + 𝐴s (0.35 + 0.0836 + 0.0477)
Since
(17.545)𝜋 2 (29, 500)
12(1 − 0.32 )(11.6624∕0.075)2
= 19.346 ksi
Assuming that 𝑓 = 𝐹y = 33 ksi yields
√
√
𝑓
33
𝜆=
=
𝐹cr𝓁
19.346
= 1.306 > 0.673
1 − 0.22∕1.306
𝜌=
= 0.637
1.306
From Eq. (3.85), the effective width of
elements 7, 8, 9, and 10 located at the centroid
of the top flange, including the intermediate
stiffener, is given as
( )
)
(
𝐴g
1.0022
= 0.637
𝑏e = 𝜌
𝑡
0.075
= 8.512 in.
The location of the centroid from the top fiber
of the flange is
𝑦7−10 =
2(5.5686)(0.057)(0.075∕2)
+𝐴s (0.7∕2 + 3∕32 + 0.075)
𝐴g
= 0.1176 in. from top fiber
BENDING STRENGTH AND DEFLECTION
Element
Effective
Length L (in.)
Distance from
Top Fiber
y (in.)
1
2 × 0.5812 = 1.1624
4.5406
2
2 × 0.2063 = 0.4126
4.9148
3
2 × 3.1624 = 6.3248
4.9625
4
2 × 0.2063 = 0.4126
4.9148
5
2 × 4.6624 = 9.3248
2.5000
6
2 × 0.2063 = 0.4126
0.0852
8.5120
0.1176
7–10
Total
26.5618
∑
Ly 65.0686
𝑦cg = ∑ =
= 2.4497 in. < 2.5 in.
26.5618
𝐿
)
(
2.4497
(33) = 31.70 ksi
𝑓=
5 − 2.4497
Ly(in.2 )
5.2780
2.0278
31.3868
2.0278
23.3120
0.0352
1.001
65.0686
𝑏0 = out-to-out width of
1 − 0.22∕𝜆
= 0.650
𝜆
( )
𝐴g
𝑏e = 𝜌
= 8.6857 in.
𝑡
𝑦7−10 = 0.1176 in.from top fiber
1
2
3
4
5
6
7–10
Total
1.1624
0.4126
6.3248
0.4126
9.3248
0.4126
8.6857
26.7355
4.5406
4.9148
4.9625
4.9148
2.5000
0.0852
0.1176
65.089
= 2.435 in.
26.7355
)
(
2.435
(33) = 31.33 ksi
𝑓=
5 − 2.435
𝑦cg =
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓) = 27.47
= 5.00 in.
𝜌=
Element
2.2662
= 29.156 ksi
2.565
2.3962
𝑓2 = 33
= 30.828 ksi
2.565
|f |
𝜓 = || 2 || = 1.126
| f1 |
𝑓1 = 33
ℎ0 = out-to-out depth of web
𝑓 = 31.3 ksi
√
𝑓
𝜆=
= 1.272 > 0.673
𝑓cr𝓁
Distance from
Top Fiber
y (in.)
Since the computed stress is close to the
assumed value of 31.3 ksi, it is OK.
To check if the web is fully effective, refer to
Fig. 4.23:
From Fig 4.22,
Since the computed compression stress, f, is
considerably less than 33 ksi, additional trials
are required. After several trials, it was found
that the stress should be about 31.3 ksi.
b. Additional Approximation. Assume
Effective
Length
L (in.)
147
Ly(in.2 )
Ly2 (in.3 )
5.2780
2.0278
31.3868
2.0278
23.3120
0.0352
1.0214
65.089
23.9653
9.9664
155.7570
9.9664
58.2800
0.0030
0.1201
258.058
compression flange
= 12.00 in.
Since ℎ0 ∕𝑏0 = 5∕12 = 0.417 < 4, then use
Eq. (3.47a)
𝑏e
𝑏1 =
3+𝜓
where be is the effective width of the web determined in accordance with Eqs. (3.32)–(3.35)
with 𝑓1 substitued for f and 𝑘 = 27.47 as
follows:
ℎ 4.6624
=
= 62.17 < 200 OK
𝑡
0.075
√
29.156
1.052
(62.17)
= 0.392 < 0.673
𝜆= √
29,500
27.47
𝑏e = ℎ = 4.6624 in.
𝑏e
4.6624
=
= 1.13 in.
3+𝜓
3 + 1.126
Since 𝜓 > 0.236,
1
𝑏2 = 𝑏e = 2.3312 in.
2
b1 = 𝑏2 = 3.462 in. > 2.2662 in.
b1 =
The web is fully effective as assumed.
iii. Total Ix and Sx .
1
(4.6624)3 = 16.8918
12
1
2𝐼1′ = 2 × (0.5812)3 = 0.0327
12
2(𝐼2′ + 𝐼4′ + 𝐼6′ ) = 0.0020
2𝐼5′ = 2 ×
148
4
FLEXURAL MEMBERS
∑
(Ly2 ) = 258.0345
274.9610
(∑ )
2
−
𝐿 (𝑦2cg ) = −(26.7355)(2.435 ) = −158.521
𝐼x′ = 116.440 in.3
𝐼x = 𝐼x′ 𝑡 = 116.440(0.075) = 8.7330 in.4
8.7330
= 3.4047 in.3
5 − 2.435
2. Nominal and Allowable Moments. The nominal
moment for due to initiation of yielding is
𝑆x =
𝑀𝑛𝓁 = 𝑆e 𝐹y = 𝑆x 𝐹y
= (3.4047)(33) = 112.355 in.-kips
The allowable moment is
𝑀
112.355
= 67.28 in.-kips
𝑀a = 𝑛𝓁 =
Ωb
1.67
B. LRFD Method
The nominal moment for the LRFD method is the same as
that computed for the ASD method. From the above calculations, the nominal moment about the x axis of the section
is
𝑀𝑛𝓁 = 112.355 in.-kips
The corresponding design moment is:
𝜙b 𝑀𝑛𝓁 = 0.90(112.355) = 101.12 in.-kips
4.2.3.1.2 Inelastic Reserve Capacity of Beams Prior
to 1980, the inelastic reserve capacity of beams was not
included in the AISI Specification because most cold-formed
steel shapes have width-to-thickness ratios considerably in
excess of the limits required by plastic design. Because of
the use of large width-to-thickness ratios for the beam flange
and web, such members are usually incapable of developing
plastic hinges without local buckling.
In the 1970s research work on the inelastic strength of
cold-formed steel beams was carried out by Reck, Pekoz,
Winter, and Yener at Cornell University.4.1–4.4 These studies
showed that the inelastic reserve strength of cold-formed
steel beams due to partial plastification of the cross section
and the moment redistribution of statically indeterminate
beams can be significant for certain practical shapes. With
proper care, this reserve strength can be utilized to achieve
more economical design of such members. In Europe, a study
has been made by von Unger on the load-carrying capacity
of transversely loaded continuous beams with thin-walled
sections, in particular of roof and floor decks with trapezoidal profiles.4.5 In addition, the buckling strength and
load-carrying capacity of continuous beams and steel decks
have also been studied by some other investigators.4.6–4.9
In order to utilize the available inelastic reserve strength
of certain cold-formed steel beams, design provisions based
on the partial plastification of the cross section were added in
the 1980 edition of the AISI Specification. The same provisions were retained in the 1986 and the 1996 editions of the
AISI specification and the North American Specification.
According to F2.4.1 of the 2016 edition of the Specification,
the nominal strengths 𝑀ne of those beams satisfying certain
specific limitations can be determined on the basis of the
inelastic reserve capacity with a limit of 1.25𝑆e 𝐹y . The
nominal moment 𝑀𝑛𝓁 is the maximum bending capacity
of the beam by considering the inelastic reserve strength
through partial plastification of the cross section. The
inelastic stress distribution in the cross section depends on
the maximum strain in the compression flange, 𝜀cu . Based on
the Cornell research work on hat sections having stiffened
compression flanges,4.1 the design provision included in
Section F2.4.1 of the North American Specification limits
the maximum compression strain to be 𝐶y 𝜀y , that is,
𝜀cu = 𝐶y 𝜀y
(4.75)
where 𝜀y = yield strain, = 𝐹y ∕E, in.∕in.
𝐸 = modulus of elasticity of steel,
29.5 × 103 ksi (203 GPa or
2.07 × 106 kg∕cm2 )
𝐹y = yield stress of steel, ksi
and 𝐶y is the compression strain factor determined as
follows:
1. Stiffened compression elements without intermediate
stiffeners:
a. When 𝑤∕𝑡 ≤ 𝜆1 ,
𝐶y = 3.0
b. When 𝜆1 < 𝑤∕𝑡 < 𝜆2 ,
𝐶y = 3 − 2
(
𝑤∕𝑡 − 𝜆1
𝜆2 − 𝜆1
(4.76)
)
(4.77)
c. When 𝑤∕𝑡 ≥ 𝜆2 ,
𝐶y = 1.0
(4.78)
where
1.11
𝜆1 = √
𝐹y ∕𝐸
1.28
𝜆2 = √
𝐹y ∕𝐸
The relationship between 𝐶y and the w/t ratio of the
compression flange is shown in Fig. 4.24.
BENDING STRENGTH AND DEFLECTION
149
Eq. (4.76)
Eq. (4.77)
Eq. (4.78)
Figure 4.24 Factor Cy for stiffened compression elements without intermediate stiffeners.
2. Unstiffened compression elements:
i. Unstiffened compression elements under a stress
gradient causing compression at one longitudinal
edge and tension at the other longitudinal edge:
⎧3.0
(4.79𝑎)
𝜆 ≤ 𝜆3
⎪
⎪3 − 2[(𝜆 − 𝜆3 )∕(𝜆4 − 𝜆3 )]
𝐶y = ⎨
(4.79𝑏)
⎪𝜆3 ≤ 𝜆 ≤ 𝜆4
⎪1 𝜆 ≥ 𝜆4
(4.79𝑐)
⎩
where 𝜆 is the slenderness factor defined in Section
3.3.1.1,
𝜆3 = 0.43
(4.79d)
𝜆4 = 0.673(1 + 𝜓)
(4.79e)
and 𝜓 is defined in Section 1.2.2 of the Specification.
ii. Unstiffened compression elements under a stress
gradient causing compression at both longitudinal
edges:
(4.79f)
𝐶y = 1
iii. Unstiffened compression elements under uniform
compression:
(4.79g)
𝐶y = 1
3. Multiple-stiffened compression elements and compression elements with edge stiffeners
𝐶y = 1.0
(4.80)
No limit is placed on the maximum tensile strain in the
North American Specification.
For the above requirements, Eqs. (4.79a)–(4.79g) were
added in 2004 for sections having unstiffened elements under
a stress gradient.1.343 These added design equations are based
on the research work conducted by Bamback and Rasmussen
at the University of Sydney on I- and plain channel sections
in minor-axis bending.4.194,4.195 The 𝐶y values are dependent
on the stress ratio 𝜓 and slenderness factor 𝜆 of the unstiffened element with the stress gradient determined in Section
3.3.2.2.
On the basis of the maximum compression strain εcu
allowed in Eq. (4.75), the neutral axis can be located by using
Eq. (4.81), and the nominal moment 𝑀ne can be determined
by using Eq. (4.82) as follows:
𝜎dA = 0
(4.81)
𝜎ydA = 𝑀𝑛𝓁
(4.82)
∫
∫
where 𝜎 is the stress in the cross section.
For hat sections Reck, Pekoz, and Winter gave the
following equations for the nominal moments of sections
with yielded tension flange and sections with tension flange
not yielded:
a. Sections with Yielded Tension Flange at Nominal
Moment.4.1 For the stress distributions shown in
Fig. 4.25, Eqs. (4.83)–(4.88) are used for computing
the values of 𝑦c , 𝑦t , 𝑦p , 𝑦cp , 𝑦tp , and Mn . For the purpose of
simplicity, midline dimensions are used in the calculation:
𝑦c =
𝑏t − 𝑏c + 2𝑑
4
(4.83)
150
4
FLEXURAL MEMBERS
Figure 4.25 Stress distribution in sections with yielded tension flange at nominal moment.4.1
𝑦t = 𝑑 − 𝑦c
(4.84)
𝑦c
𝜀cu ∕𝜀y
(4.85)
𝑦cp = 𝑦c − 𝑦p
(4.86)
𝑦p =
𝑦tp = 𝑦t − 𝑦p
𝑀𝑛𝓁
(4.87)
)
4
1
= 𝐹y 𝑡 𝑏c 𝑦c + 2𝑦cp 𝑦p + 𝑦cp + (𝑦p )2
2
3
(
)
]
1
(4.88)
+2𝑦tp 𝑦p + 𝑦tp + 𝑏y 𝑦t
2
[
Figure 4.26
(
b. Sections with Tension Flange Not Yielded at Nominal
Moment.4.1 For the stress distribution shown in
Fig. 4.26, 𝑦c is computed from the following quadratic
equation:
𝑦2c
(
)
1
− 𝐶y + 𝑦c (𝑏c + 2𝐶y 𝑑 + 𝐶y 𝑏t )
2−
𝐶y
− (𝐶y 𝑑 2 + 𝐶y 𝑏t 𝑑) = 0
(4.89)
Subsequently, the values of 𝑦t , 𝑦p , and 𝑦cp can be computed
from Eqs. (4.84),(4.85), and (4.86), respectively.
Stress distribution in sections with tension flange not yielded at nominal moment.4.1
BENDING STRENGTH AND DEFLECTION
If 𝑦p > 𝑦t , then the case in part (b) above applies and the
nominal moment Mn𝓁 is computed as follows:
[
(
)
𝑦cp
2
𝑀𝑛𝓁 = 𝐹y 𝑡 𝑏c 𝑦c + 2𝑦cp 𝑦p +
+ (𝑦p )2
2
3
( )
( )]
𝜎t
𝜎t
2
+ (𝑦t )2
+ 𝑏y 𝑦t
(4.90)
3
𝐹y
𝐹y
In Eq. (4.90), 𝜎t = 𝐹y 𝐶y 𝑦t ∕𝑦c .
It should be noted that according to Section F2.4.1
of the North American Specification, Eqs. (4.88) and
(4.90) can be used only when the following conditions
are met:
151
SOLUTION
A. ASD Method
1. Dimensions of Section. By using the midline dimensions and square corners, the widths of compression
and tension flanges and the depth of webs are computed
as follows:
a. Width of compression flange:
𝑏c = 3 − 0.105 = 2.895 in.
b. Width of tension flange:
𝑏t = 2(1.34 − 0.105∕2) = 2.576 in.
c. Depth of webs:
1. The member is not subject to twisting or to lateral,
torsional, or flexural–torsional buckling.
2. The effect of cold work of forming is not included in
determining the yield stress 𝐹y .
3. The ratio of the depth of the compression portion of the
web to its thickness does not exceed 𝜆1 .
4. The shear force does not exceed 0.35𝐹y for ASD
and 0.6𝐹y for LRFD and LSD times the web area
(ht for stiffened elements or wt for unstiffened
elements).
5. The angle between any web and the vertical does not
exceed 30∘ .
𝑑 = 3 − 0.105 = 2.895 in.
All dimensions are shown in Fig. 4.28a. Check the
effective width of the compression flange:
)
(
3
+ 0.105 = 2.415 in.
w=3−2
16
w 2.415
=
= 23
t
0.105
k = 4.0
f = Fy = 33 ksi
√
1.052
33
λ = √ (23)
29,500
4
= 0.405 < 0.673
It should also be noted that, when applicable, effective
design widths should be used in the calculation of sectional
properties.
Example 4.8 For the hat section (3 × 3 × 0.105 in.) shown
in Fig. 4.27, determine the allowable moment (𝑀a ) about the
x axis for the ASD method and the design moment (𝜙b 𝑀𝑛𝓁 )
for the LRFD method. Consider the inelastic reserve capacity
according to Section F2.4.1 of the 2016 edition of the North
American Specification. Use 𝐹y = 33 ksi and assume that
lateral support is adequately provided.
b = w = 2.415 in.
Therefore, the compression flange is fully effective.
2. Strain Diagram. The w/t ratio of the stiffened
compression flange is given as
𝑤
= 23
𝑡
1.11
1.11
𝜆1 = √
= 33.2
=√
33∕29,500
𝐹y ∕𝐸
Since 𝑤∕𝑡 < (𝜆1 = 33.2), according to Eq. (4.76),
𝐶y = 3.0. Therefore, 𝜀cu = 3 𝜀y , as shown in
Fig. 4.28b.
3. Stress Diagram. The values of 𝑦c , 𝑦t , 𝑦p , 𝑦cp , and 𝑦tp
are computed by using Eqs. (4.83)–(4.87) as follows:
𝑏t − 𝑏c + 2𝑑
2.576 − 2.895 + 2 × 2.895
=
4
4
= 1.368 in.
𝑦c =
Figure 4.27
Example 4.8.
𝑦t = 𝑑 − 𝑦c = 2.895 − 1.368 = 1.527 in.
𝑦c
1.368
𝑦p =
=
= 0.456 in.
𝜀cu ∕𝜀y
3
152
4
FLEXURAL MEMBERS
Figure 4.28
Stress distribution: (a) midline dimensions; (b) strain; (c) stress.
𝑦cp = 𝑦c − 𝑦p = 1.368 − 0.456 = 0.912 in.
𝑦tp = 𝑦t − 𝑦p = 1.527 − 0.456 = 1.071 in.
All dimensions are shown in Fig. 4.28c.
4. Nominal Moment Mn𝓁 . In order to utilize the inelastic
reserve capacity, the North American Specification
requirements must be checked:
𝑦c
1.368
=
= 13.03 < (𝜆1 = 33.2) OK
𝑡
0.105
Therefore, the nominal moment is
(
)
[
4
1
M𝑛𝓁 = Fy t bc yc + 2ycp yp + ycp + (yp )2
2
3
(
)
]
1
+2ytp yp + ytp + bt yt
2
[
= (33)(0.105) (2.895)(1.368) + 2(0.912)
(
)
1
4
× 0.456 + × 0.912 + (0.456)2
2
3
(
)
1
+ 2(1.071) 0.456 + × 1.071
2
]
+ (2 × 1.288)(1.527)
B. LRFD Method
The nominal moment for the LRFD method is the same as
that computed for the ASD method. From item A above, the
nominal moment about the x axis of the hat section is
𝑀𝑛𝓁 = 40.93 in.-kips
The corresponding design moment is:
𝜙𝑏 𝑀𝑛𝓁 = (0.9)(40.93) = 36.84 in.-kips
Example 4.9 For the I-section with unequal flanges as
shown in Fig. 4.29, determine the allowable moment (𝑀a )
about the x axis for the ASD method and the design moment
(𝜙b 𝑀n ) for the LRFD method. Consider the inelastic reserve
= 41.43 in.-kips
5. Based on the method illustrated in Example 4.6, 𝑆e for
the given hat section is 0.992 in.3 The nominal moment,
𝑀𝑛𝓁 should be limited to 1.25𝑆e 𝐹y according to the
Speccfication. Therefore
1.25𝑆e 𝐹y = 1.25(0.992)(33) = 40.93 in.-kips < 𝑀𝑛𝓁
6. Allowable Moment Ma . Because 𝑀𝑛𝓁 exceeds
1.25𝑆𝑒 𝐹𝑦 , use
M𝑛𝓁 = 1.25Se Fy = 40.93 in.-kips
Ma =
M𝑛𝓁
40.93
=
= 24.51 in.-kips
Ωb
1.67
Figure 4.29
Example 4.9.
BENDING STRENGTH AND DEFLECTION
capacity and use 𝐹y = 50 ksi. Assume that the lateral support
is adequately provided to prevent lateral buckling.
SOLUTION
A. ASD Method
1. Dimensions of Section. By using the midline dimensions and square corners, the widths of compression
and tension flanges and the depth of webs are computed
as follows: The flat width of the unstiffened compression flange according to Section 3.2 is
(
)
3
𝑤 = 2.5 − (𝑅 + 𝑡) = 2.5 −
+ 0.135
16
= 2.1775 in.
𝑤 2.1775
=
= 16.13
𝑡
0.135
For 𝑓 = 𝐹y = 50 ksi in the top fiber and k = 0.43 for
the unstiffened flange,
√
1.052
50
𝜆= √
(16.13)
= 1.065 > 0.673
29,500
0.43
(
)
1 − 0.22∕𝜆
𝑏 = 𝜌w =
𝑤 = 1.622 in.
𝜆
)
(
)
(
𝑏c
1
0.135
3
= 𝑏 + 𝑅 + 𝑡 = 1.622 +
+
2
2
16
2
= 1.877 in.
𝑏c = 3.754 in.
The width of the tension flange is determined as
𝑏t
𝑡
0.135
=1− =1−
= 0.9325 in.
2
2
2
𝑏t = 1.865 in.
153
The depth of the webs is given as
𝑑 = 8.0 − 𝑡 = 8.0 − 0.135 = 7.865 in.
All midline dimensions are shown in Fig. 4.30a.
2. Strain Diagram. For an unstiffened compression
flange under uniform compression, 𝐶y = 1.0. Therefore, 𝜀cu = 𝜀y , as shown in Fig. 4.30b.
3. Stress Diagram. The values of 𝑦c , 𝑦t , 𝑦p , and
𝑦tp are computed by using Eqs. (4.83)–(4.87) as
follows:
𝑏t − 𝑏c + 2𝑑
1.865 − 3.754 + 2(7.865)
=
4
4
= 3.46 in.
𝑦c =
𝑦t = 𝑑 − 𝑦c = 7.865 − 3.46 = 4.405 in.
𝑦c
𝑦p =
= 𝑦c = 3.46 in.
𝜀cu ∕𝜀y
𝑦cp = 0
𝑦tp = 𝑦t − 𝑦p = 4.405 − 3.46 = 0.945 in.
All dimensions are shown in Fig. 4.30c.
4. Nominal Moment. In order to satisfy the North American Specification requirements for using the inelastic
reserve capacity, check the 𝑦c ∕𝑡 ratio against the limit
of λ1 :
𝑦c
3.46
=
= 25.63
𝑡
0.135
1.11
1.11
𝜆1 = √
= 26.96
=√
50∕29,500
𝐹y ∕𝐸
Figure 4.30 Stress distribution: (a) midline dimensions; (b) strain; (c) stress.
154
4
FLEXURAL MEMBERS
Since 𝑦c ∕𝑡 < λl , OK. Therefore, the nominal
moment is determined as
(
)
]
[
4
1
𝑀𝑛𝓁 = 𝐹y 𝑡 𝑏c 𝑦c + (𝑦p )2 + 2𝑦tp 𝑦p + 𝑦tp + 𝑏t 𝑦t
3
2
[
4
= 50(0.135) (3.754 × 3.46) + (3.46)2 + 2(0.945)
3
(
)
]
1
× 3.46 + × 0.945 + 1.865(4.405)
2
= 301.05 in. − kips
5. Following the same method illustrated in Example 4.4,
the effective section modulus based on the yielding,
𝑆e = 6.247 in.3 The nominal moment, 𝑀𝑛𝓁 should be
limited to 1.25𝑆e 𝐹y according to the Specification:
(a) For 𝜆𝓁 ≤ 0.776
𝑀𝑛𝓁 = 𝑀ne
(4.91)
(b) For λ𝓁 > 0.776
[
𝑀𝑛𝓁
where λ𝓁 =
(
) ](
)
𝑀cr𝓁 0.4
𝑀cr𝓁 0.4
= 1-0.15
𝑀ne
𝑀ne
𝑀ne
√
𝑀ne ∕𝑀cr𝓁
(4.92)
(4.93)
𝑀ne = Nominal flexural strength [resistance] for
lateral–torsional buckling as defined in
Specification Section F2
𝑀cr𝓁 = Critical elastic local buckling moment,
determined in accordance with Specification
Appendix 2
1.25 𝑆e 𝐹y = (1.25)(6.247)(50)
= 390.4 in. − kips > 𝑀𝑛𝓁
5. Nominal Moment and Allowable Moment. Because
𝑀𝑛𝓁 is less than 1.25𝑆e 𝐹y , use 𝑀𝑛𝓁 for the nominal
moment, that is,
𝑀𝑛 = 𝑀𝑛𝓁 = 301.05 in. − kips
The allowable design moment is
𝑀
301.05
𝑀a = 𝑛 =
= 180.27 in.-kips
Ω𝑏
1.67
B. LRFD Method
The nominal moment for the LRFD method is the same as
that computed for the ASD method. From item A above,
the nominal moment about the x axis of the I-section with
unequal flanges is
𝑀n = 301.05 in. − kips
The corresponding design comment is:
𝜙𝑏 𝑀𝑛 = (0.9)(301.05) = 270.95 in.-kips
4.2.3.2 Direct Strength Method The Direct Strength
Method considers the strength due to local buckling as the
function of the local-buckling moment, 𝑀cr𝓁 , and the yield
stress, 𝐹y . How to obtain the local buckling moment using
readily available software or through analytical approach is
discussed in Section 3.5. The following design provisions,
developed by Schafer and Pekoz3.254,3.255 , were first adopted
into the North American Specification in 2004 as Appendix
1. In the 2016 edition, they are incorporated into the main
body of the Specification:1.417
F3.2.1 Members Without Holes
The nominal flexural strength [resistance], 𝑀𝑛𝓁 , for considering
interaction of local buckling and global buckling shall be determined as follows:
4.2.3.2.1 Members with Holes. As discussed in
Section 3.5, the Direct Strength Method for considering
members with holes was developed by Moen and Schafer
et al.3.273–3.275 The design provisions was introduced into
Appendix 1 of the 2012 edition of the North American Specification. In 2016, these design provisions were incorporated
into the main body of the Specification1.417 as Section
F3.2.2, and are excerpted as follows:
F3.2.2 Members With Holes
The nominal flexural strength [resistance], 𝑀𝑛𝓁 , for local buckling of beams with holes shall be calculated in accordance with
Specification Section F3.2.1, except 𝑀cr𝓁 shall be determined
including the influence of holes:
𝑀𝑛𝓁 ≤ 𝑀ynet
(4.94)
where 𝑀ynet = Member yield moment of net cross-section
= Sfnet Fy
(4.95)
where 𝑆fnet = Net section modulus referenced to the extreme
fiber at first yield
𝐹y = Yield stress
Design examples will be provided to illustrate the provisions.
4.2.3.2.3 Members Considering Local Inelastic Reserve
Strength The local inelastic reserve strength is plotted
in Fig. 3.68b in the region where 𝑀n exceeds 𝑀y . Based
on the research by Shifferaw and Schafer,4.296 the inelastic
reserve strength can be considered when the predicted
lateral–torsional buckling strength, 𝑀ne is greater than yield
moment, i.e., 𝑀ne > 𝑀y . Similar to the Effective Width
Method to predict the inelastic reserve strength, the ratio
of the maximum compressive strain to the yield strain is
limited to 3. For cross-sections with the first yielding in
BENDING STRENGTH AND DEFLECTION
tension, it is recommended that the ratio of the maximum
tensile strain to the yield strain be limited to 3 as well. The
following design provisions are excerpted from the North
American Specification:1.417
F3.2.3 Members Considering Local Inelastic Reserve
Strength
Inelastic reserve capacity is permitted to be considered as
follows, provided λ𝓁 ≤ 0.776 and 𝑀ne ≥ 𝑀y :
(a) Sections symmetric about the axis of bending or sections
with first yield in compression:
2
)(𝑀𝑝 − 𝑀𝑦 )
𝑀𝑛𝓁 = 𝑀𝑦 + (1 − 1∕𝐶𝑦𝓁
(4.96)
(b) Sections with first yield in tension:
2
𝑀𝑛𝓁 = 𝑀yc + (1 − 1∕𝐶𝑦𝓁
)(𝑀𝑝 − 𝑀yc ) ≤ 𝑀yt3 (4.97)
where λ𝓁 =
√
My ∕Mcrl
(4.98)
𝑀ne = Nominal flexural strength [resistance] as
defined in Specification Section F2
√
(4.99)
C𝑦𝓁 = 0.776∕𝜆𝓁 ≤ 3
𝑀cr𝓁 = Critical elastic local buckling moment,
determined in accordance with Specification
Appendix 2
𝑀p = Member plastic moment as given in
Eq. (4.73)
𝑀y = Member yield moment in accordance with
Specification Section F2.1
𝑀yc = Moment at which yielding initiates in
compression (after yielding in tension).
Myc = My may be used as a conservative
approximation
Myt3 = My + (1 − 1∕C2yt )(Mp − My )
(4.100)
𝐶yt = Ratio of maximum tension strain to yield
strain
= 3
4.2.4 Distortional Buckling Strength
The flexural strength of cold-formed steel beams bending
about the major axis may be limited by local buckling, or
lateral–torsional buckling. For members with edge-stiffened
flanges, the flexural strength may also be limited by distortional buckling. As shown in Fig. 4.31, the local buckling
mode of a C-section for major-axis bending consists of buckling of the compression portion of the web, the compression flange, and edge stiffener without movement of the
line junction between the flange and edge stiffener. For this
type of limit state, the section strength of the member is
155
determined according to Section 4.2.3. For the flange distortional buckling mode, the flange and edge stiffener rotate
about the flange–web junction with some rotational resistance provided by the web. This mode of failure occurs
at considerably longer wavelengths than local buckling but
generally shorter wavelength than lateral–torsional buckling.
The distortional buckling may also take place in sections
as shown in Fig. 3.2(b) and 3.2(c), where the portion of
the flange (with intermediate stiffeners as shown in Fig.
3.2(b)) or the portion of the lip and the flange (as shown in
Fig. 3.2(c)) starts to rotate about the junction(s) between the
flange and web.
Distortional buckling may occur simultaneously with
local buckling.1.358,1.358,4.208,4.223,4.224 Research work indicated that the local–distortional interaction is generally
weak and that if this limit state is included in the design
requirements the resulting capacities are not consistent
with observations.1.412 Therefore, no design provisions are
currently included in the North American Specification for
this limit state. For detailed discussion of modal interactions,
see Section 13.4.4 of the SSRC guide.1.412
In earlier years, distortional buckling has not been specifically considered for the design of cold-formed steel members
having edge-stiffened compression flanges. The AISI design
provisions provided by Desmond et al.3.76 for uniformly
compressed element with a simple lip edge stiffener account
for the inability of the stiffener to prevent flange buckling by reducing the local buckling coefficient k to less
than 4.0 for the partially stiffened compression flange. The
reduced buckling coefficient is then used to compute the
effective width of the flange element. However, in 1992
Kwon and Hancock found that the AISI approach is unconservative for distortional buckling of C-sections composed of
high-strength steel using a yield stress of 80 ksi (550 MPa or
5624 kg/cm2 ).4.196 In 1999, Schafer and Pekoz indicated that
the AISI-reduced local buckling coefficient is only intended
to be used in conjunction with the specific effective width
expressions and is not actually the elastic buckling coefficient for distortional buckling.1.412,3.226 In addition, the tests
conducted by Yu and Schafer showed that the AISI effective width method is inadequate to account for distortional
buckling.4.206,4.210
The introduction of the Direct Strength Method3.254,3.255
provided a means to determine the member strength due to
distortional buckling. As described in Section 3.5, the key to
determine the member strength due to distortional buckling is
to obtain the distortional buckling moment, 𝑀crd . Numerical
and analytical solutions on how to obtain 𝑀crd are discussed
in detail in Section 3.5.2.
In 2004, the distortional buckling strength design provisions developed by Shafer and Pekoz3.254,3.255 were adopted
156
4
FLEXURAL MEMBERS
Figure 4.31
C-section purlin buckling stress versus half wavelength for major-axis bending.1.69
in Appendix 1 of the Norther American Specification. In
2012, the distortional buckling strength for members with
holes3.273–3.275 and inelastic reserve strength4.296 were added.
These provisions are then incorporated into the main body
of the North American Specification in 2016. These design
provisions are excerpted below.
F4.2 Members With Holes
The nominal flexural strength [resistance], 𝑀nd , for distortional
buckling shall be calculated in accordance with Specification
Section F4.1, except 𝑀crd shall be determined including the
influence of holes, and when λd ≤ λd2 then:
For λd ≤ λd1
𝑀nd = 𝑀ynet
F4.1 Members Without Holes
The nominal flexural strength [resistance], 𝑀nd , shall be calculated in accordance with Eq. (4.101) or Eq. (4.102).
For λd ≤ 0.673
𝑀nd = 𝑀y
(4.101)
For λd1 > λd ≤ λd2
𝑀nd = Mynet −
√
λd = 𝑀y ∕𝑀crd
where
𝑀y = 𝑆fy 𝐹y
(4.102)
where
λd =
)
√
My ∕Mcrd
(4.107)
(4.108)
(4.103)
(4.104)
where 𝑆fy = Elastic section modulus of full unreduced
cross-section relative to extreme fiber in first
yielding
𝐹y = Yield stress
𝑀crd = Sf Fcrd
Mynet − Md2
(λd − λd1 )
λd2 − λd1
)0.5 (
)0.5
(
⎤ M
⎡
Mcrd
crd
⎥
My
≤ ⎢1 − 0.22
⎥ My
⎢
My
⎦
⎣
For λd > 0.673
)0.5 (
)0.5
(
⎤ 𝑀
⎡
𝑀crd
crd
⎥
𝑀y
𝑀nd = ⎢1 − 0.22
⎥ 𝑀y
⎢
𝑀y
⎦
⎣
(
(4.106)
(4.105)
where 𝑆f = Elastic section modulus of full unreduced
cross-section relative to extreme compression
fiber
𝐹crd = Elastic distortional buckling stress calculated in
accordance with Specification Appendix 2
where 𝑀crd = Distortional buckling moment including
influence of holes
λd1 = 0.673 (Mynet ∕My )3
(4.109)
λd2 = Limit of distortional slenderness transition
= 0.673 [1.7(My ∕Mynet )2.7 − 0.7]
𝑀d2 = [1-0.22(1∕λd2 )](1∕λd2 )My
(4.110)
(4.111)
𝑀y = Member yield moment as given in
Eq. (4.104)
𝑀ynet = Member yield moment of net cross-section
as given in Eq. (4.95)
157
BENDING STRENGTH AND DEFLECTION
F4.3 Members Considering Distortional Inelastic
Reserve Strength
Inelastic reserve capacity is permitted to be considered as
follows, provided λd ≤ 0.673:
(a) Sections symmetric about the axis of bending or sections
with first yield in compression:
2
)(𝑀𝑝 − 𝑀𝑦 )
𝑀nd = 𝑀𝑦 + (1 − 1∕𝐶yd
(4.112)
(b) Sections with first yield in tension:
2
)(𝑀𝑝 − 𝑀yc ) ≤ 𝑀yt3
𝑀nd = 𝑀yc + (1 − 1∕𝐶yd
where
√
My ∕Mcrd
√
𝐶yd = 0.673∕𝜆𝑑 ≤ 3
λd =
(4.113)
(4.114)
(4.115)
𝑀crd = Critical elastic distortional buckling moment,
determined in accordance with Specification
Appendix 2 or Section 2.3.3.3 in this volume
𝑀p = Member plastic moment as given in Eq. (4.73)
𝑀y = Member yield moment in accordance with
Specification Section F2.1
𝑀yc = Moment for yield in compression as defined in
Specification Section F3.2.3
𝑀yt3 = Maximum moment for yielding in tension as
given in Eq. (4.100)
The yield moment [(Eq. (4.104)] is
𝑀y = 𝑆f𝑦 𝐹y = 𝑆f 𝐹y = (4.11)(50) = 205.50 in.-kips
The slenderness factor for distortional
[(Eq. (4.103))] is
√
√
𝑀y
205.50
𝜆d =
=
= 1.71
𝑀crd
70.45
buckling
Since λd > 0.673, the nominal moment for distortional
buckling can be computed according to Eq. (4.102) as
follows:
[
(
) ](
)
𝑀crd 0.5
𝑀crd 0.5
𝑀nd = 1-0.22
𝑀y
𝑀y
𝑀y
[
) ](
)
(
70.45 0.5
70.45 0.5
= 1-0.22
(205.50)
205.50
205.50
= 104.82 in.-kips
Based on ASD, the allowable moment for distortional
buckling is
𝑀a =
𝑀nd
104.82
=
= 62.77 in.-kips
Ωb
1.67
Based on LRFD, the design moment for distortional buckling is
𝜙𝑏 𝑀nd = 0.9(104.82) = 94.34 in.-kips
Example 4.10 For the C-section used in Example 4.5
(See Fig. 4.14 for cross-section dimensions), determine
the available moment for distortional buckling according
to ASD and LRFD. Use the elastic distortional buckling
stresses based on the simplified provision of Section 3.5.2(3)
and the more precise provision of Section 3.5.2(2). Determine the member strength assuming the member is fully
braced against lateral–torsional buckling.
SOLUTION
A. Distortional Buckling Strength Based on Section 3.5.2(3)
The distortional buckling stress of the C-section has been
calculated in Example 3.8. From the example, the distortional buckling stress based on the simplified method is:
𝐹crd = 17.14 ksi
The elastic section modulus relative to the compression
fiber is
𝑆f = 4.11 in.3
and the distortional buckling moment
𝑀crd = 70.45 in.-kips
B. Distortional Buckling Strength Based on Section 3.5.2(2)
From Example 3.8, the distortional buckling stress based on
more accuracy method presented in Section 3.5.2(2) is
𝐹crd = 32.98 ksi
and
𝑀crd = 135.55 in.-kips
Also from item A, 𝑀y = 205.50 in.-kips. The slenderness
factor for distortional buckling [Eq. (4.103)] is
√
√
𝑀y
205.50
𝜆d =
=
= 1.231 > 0.673
𝑀crd
135.55
The nominal moment for distortional buckling according
to Eq. (4.102) is
[
(
) ](
)
𝑀crd 0.5
𝑀crd 0.5
𝑀nd = 1-0.22
𝑀y
𝑀y
𝑀y
[
) ](
)
(
135.55 0.5 135.55 0.5
= 1-0.22
(205.50)
205.50
205.50
= 137.08 in.-kips
158
4
FLEXURAL MEMBERS
Based on ASD, the allowable moment for distortional
buckling is
𝑀a =
137.08
= 82.08 in.-kips
1.67
Based on ASD, the design moment for distortional buckling is
𝜙𝑏 𝑀nd = 0.9(137.08) = 123.37 in.-kips
It is noted that the distortional buckling strength based on
the more accurate buckling analysis provides 31% higher
moment than the strength predicted based on simplified
buckling stress prediction.
C. Member Strength
Since the member does not subject the lateral–torsional
buckling, the member strength is controlled by local
and distortional buckling. From the calculation in
Example 4.5, the allowable and design moments due to local
buckling are:
ASD method: 𝑀a = 96.14 in.-kips
LRFD method: 𝜙b 𝑀𝑛𝓁 = 144.50 in.-kips
Comparing with the member strengths due to distortional
buckling calculated under Item B, the member strength is
controlled by distortional buckling. Therefore the member
strength is:
ASD method: 𝑀a = 82.08 in.-kips
LRFD method: 𝜙b 𝑀n = 123.37 in.-kips
In Tables II-7, II-8, and II-9 of the AISI Design Manual,
the computed distortional buckling properties are provided
for the representative C-shapes, stock studs/joist, and
Z-shapes with lips, respectively. The values in these tables
have been calculated for use with Section 3.5.2(2). Examples
of using finite strip method to determine the local, distortional buckling moments are also provided in Examples
II-2B, II-4B, II-6B, II-7B and II-15 of the 2017 edition of
the AISI Design Manual.1.428
Example 4.11 A beam with geometry as shown in
Fig. 4.32 with web hole at mid-depth of the web. The
hole depth 𝑑h = 1.5 in., and hole length 𝐿h = 4.5 in. Hole
spacing 𝑠 = 24.0 in. o.c. Assume that the beam is laterally
braced against lateral–torsional buckling. The member yield
stress 𝐹y = 33 ksi. Determine member strength using the
Direct Strength Method.
SOLUTION
1. Determine elastic local buckling moment:
Since the local buckling wavelength is short, the local
buckling could occur within the hole length or between
the holes. Therefore, the local buckling of both the gross
and net cross-sections should be considered. The software
CUFSM is used to determine the buckling moment. For
the net cross-section, the section properties can be calculated using CUFSM by setting element thickness at the hole
equals 0. The signature curves for gross cross-section and
cross-section with hole are shown in Figs. 4.33(a) and (b).
The yield moment based on gross and net cross-section
can be obtained from the program output:
𝑀y = 25.648 in.-kips., and 𝑀ynet = 25.507 in.-kips
From Fig. 4.33(a), the local buckling between the holes,
𝑀cr𝓁 = 1.45𝑀y with buckling wavelength 𝐿cr𝓁 = 3.2 in.,
and within the hole (Fig. 4.33(b)), 𝑀cr𝓁 = 0.78𝑀ynet and
𝐿cr𝓁 = 4.4 in., which is less than the hole length (4.5 in.).
Therefore, it is possible that local buckling takes place within
the hole length. Comparing both local buckling moments, the
local buckling within the hole controls: 𝑀cr𝓁 = 0.78𝑀ynet =
19.90 in.-kips.
The member strength based on local buckling can be determined using Section 4.2.3.2:
Since the member is fully braced, 𝑀ne = 𝑀y .
√
√
λ𝓁 = 𝑀ne ∕𝑀crl = 25.648∕19.90 = 1.135 > 0.776
) ](
)
𝑀crl 0.4
𝑀crl 0.4
= 1-0.15
𝑀ne
𝑀ne
𝑀ne
[
) ](
)
(
19.90 0.4
19.90 0.4
= 1-0.15
(25.648)
25.648
25.648
[
𝑀𝑛𝓁
(
= 20.02 in.-kips < 𝑀ynet OK
The allowable moment for ASD method:
𝑀a = 𝑀𝑛𝓁 ∕Ωb = (20.02)∕(1.67) = 11.99 in.-kips
The design moment for LRFD method:
ϕb 𝑀𝑛𝓁 = (0.9)(20.02) = 18.02 in.-kips
2. Determine distortional buckling moment, 𝑀nd :
For a member with a hole, the distortional buckling
moment can be estimated using the method described in
Section 3.5.3.2 by modifying the web thickness to take into
BENDING STRENGTH AND DEFLECTION
y
159
y
1.625 in.
0.500 in.
0.0712 in.
0.0451 in.
6.000 in.
dh = 1.5 in.
x
x
Section A-A
Section B-B
Figure 4.32 Example 4.11.
3
Load factor
2.5
2
1.5
Lcrd = 15.3 in.
L.F. = 1.65
Lcrl = 3.2 in.
L.F. = 1.45
1
0.5
0
100
Figure 4.33
101
Length
(a) Local and distortional buckling of gross cross-section.
consideration of the hole influence and then calculating the
distortional buckling at the wavelength where distortional
buckling takes place at its gross cross-section.
From Fig. 4.33(a), the distortional buckling occurs at
wavelength 𝐿crd = 15.3 in.
102
The modified thickness of the web is calculated per Eq.
(3.132):
(
)1∕3
)
(
𝐿
4.5 1∕3
𝑡𝑟 = 𝑡 1 − ℎ
= (0.0451) 1 −
𝐿crd
15.3
= 0.04016 in.
160
4
FLEXURAL MEMBERS
3. Member Strength:
Comparing the available moments calculated under
items 1 and 2, the local buckling controls the design. The
member strength is therefore,
Load factor
2
1.5
ASD method: 𝑀a = 11.99 in.-kips
LRFD method: ϕb Mn = 18.02 in.-kips
1
0
Example 4.12 A joist (800S200-97) with its cross-section
and section properties shown in Fig. 4.34. The beam is braced
at 24 in. o.c. Determine the member strength using the Direct
Strength Method.
Lcrl = 4.4 in.
L.F. = 0.78
0.5
100
101
Length
Figure 4.33 (b) Local buckling of cross-section with hole.
Modify the thickness of the flat width of the web and calculate the distortional buckling moment at the 𝐿crd = 15.3 in.,
𝑀crd = 1.46𝑀y . Comparing with the distortional buckling
moment (𝑀crd = 1.65𝑀y ) of the gross cross-section, the
distortional buckling moment is reduced about 12%.
The nominal moment due to distortional buckling can be
determined per Section 4.2.4 (Specification Section F4.2):
√
√
𝑀𝑦
1
=
= 0.828
𝜆𝑑 =
𝑀crd
1.46
√
√
𝑀𝑦
25.5
=
= 0.7785
𝜆𝑑 =
𝑀crd
42.07
SOLUTION
A. Global buckling strength:
The global buckling strength can be determined in accordance with Section 4.2.2. For C-section members, the global
buckling moment is determined by Eq. (4.54):
𝐹cre =
Cb ro A √
𝜎ey 𝜎t
Sf
y
2.000 in.
0.625 in.
R = 0.1525 in.
Ix = 11.2 in.4
Iy = 0.577 in.4
λd2 = 0.673 [1.7(My ∕Mynet )2.7 − 0.7]
= (0.673)[1.7(25.648∕25.507)2.7 − 0.7] = 0.212
Since λd > λd2 , the nominal moment due to distortional
buckling is determined using Specification Section F4.1
except 𝑀crd includes the hole influence:
Since λd > 0.673,
[
(
) ](
)
𝑀crd 0.5
𝑀crd 0.5
𝑀nd = 1 − 0.22
𝑀𝑦
𝑀𝑦
𝑀𝑦
Sf = 2.8 in.3
8.000 in.
J = 0.00438 in.4
Cw = 7.68 in.6
x
rx = 2.97 in.
ry = 0.674 in.
t = 0.1017 in.
ro = 3.28 in.
= [1 − 0.22(1.46)0.5 ](1.46)0.5 (25.648) = 23.64 in.-kips
The allowable moment for ASD method:
𝑀a = 𝑀nd ∕Ωb = (23.64)∕(1.67) = 14.16 in.-kips
The design moment for LRFD method:
𝜙b 𝑀nd = (0.9)(23.64) = 21.28 in.-kips
A = 1.27 in.2
Figure 4.34
Example 4.12.
161
√
140∕1071 − 0.23
≤ Mp = 169.85 − (169.85 − 140)
0.37
= 159.2 in.-kips < 169.8 in.-kips OK
BENDING STRENGTH AND DEFLECTION
where 𝜎ey and 𝜎t are determined by Eqs. (4.57) and (4.58),
respectively:
Assume the member unbraced length 𝐾y 𝐿y = 24 in. and
𝐾t 𝐿t = 24 in. and 𝐶b = 1.0
π2 (29,500)
π2 E
=
= 229.6 ksi
(Ky Ly ∕ry )2
(24∕0.674)2
[
]
𝜋 2 EC𝑤
1
1
𝜎𝑡 =
GJ +
=
Ar0
(1.27)(3.28)
(𝐾𝑡 𝐿𝑡 )2
[
]
𝜋 2 (29,500)(7.68)
× (11,300)(0.00438) +
(24)2
𝜎ey =
= 287.7 ksi
(1.0)(3.28)(1.27) √
(229.6)(287.7) = 382.4 ksi
2.80
= 𝐹cre 𝑆f = (382.4)(2.8) = 1071 in.-kips
𝐹cre =
𝑀cre
Since 𝐹cre > 2.78𝐹y = 139 ksi, inelastic reserve strength
can be considered using Specification Section F2.4.2:
The plastic section modulus is determined as follows:
Since the section is symmetric, the centroid x axis is at the
mid-height. Determine the centroid of the section above the
centroid x axis using linear method (Fig. 1.32):
Elements
Length, 𝑙i
Dist. To top fiber, 𝑦i
𝑙i 𝑦i
lip
Flange
Web
Corners
Total
0.3708
1.4916
3.7458
2x0.3193
6.2467
0.4396
0.0508
2.1271
0.1247
0.1630
0.0758
7.9677
0.0796
8.2861
Centroid of the portion above the cross-section neutral
axis, 𝑦′top = 4 − 8.2861∕6.2467 = 2.6735 in.
The plastic section modulus:
𝑍f = (A∕2)(2y′c )
= (6.2467)(0.1017)(2 × 2.6735)
= 3.3969 in3
𝑀p = 𝑍f 𝐹y = (3.3969)(50) = 169.85 in.-kips
𝑀y = 𝑆f 𝐹y = (2.8)(50) = 140 in.-kips
Since 𝑀cre > 2.78𝑀y , nominal moment considering
inelastic reserve is determined by Eq. (4.71):
√
My ∕Mcre − 0.23
Mne = Mp − (Mp − My )
0.37
B. Local buckling strength interacting with global buckling:
Using CUFSM program, the following results are obtained
(Fig. 4.35):
𝑀y = 141.66 in.-kips (note the numerical analysis result
may differ from the value calculated using the linear
method.)
Load factor for local buckling (i.e., 𝑀cr𝓁 ∕𝑀y ), LF local =
2.76
Load factor for distortional buckling (i.e., 𝑀crd ∕𝑀y ),
LF dist = 2.13
Therefore, the local and distortional buckling moments are
𝑀cr𝓁 = 2.76𝑀y = 2.76(141.66) = 390.98 in.-kips
𝑀crd = 2.13𝑀y = 2.13(141.66) = 301.74 in.-kips
The local buckling slenderness factor is calculated:
√
√
𝜆𝓁 = 𝑀𝑦 ∕𝑀cr𝓁 = 1∕2.76 = 0.602
Since 𝑀ne > 𝑀y and λ𝓁 < 0.776, inelastic reserve
strength can be considered by using Specification Section
F3.2.3:
√
√
C𝑦𝓁 = 0.776∕λ𝓁 ≤ 3 = 0.776∕0.602 = 1.135 < 3 OK
For section symmetric about the axis of bending, the
nominal moment is determined by Eq. (4.96)
2
M𝑛𝓁 = My + (1 − 1∕C𝑦𝓁
)(Mp − My ) = (141.66)
+ (1–1∕1.1352 )(169.85 − 141.66) = 148.0 in.-kips
C. Distortional buckling strength:
As described in Item B, the distortional buckling moment
from CUFSM program:
𝑀crd = 301.74 in.-kips
The distortional buckling slenderness factor is calculated
using Eq. (4.103):
√
√
λd = My ∕Mcrd = 1∕2.13 = 0.685
Since λd > 0.673, inelastic reserve strength cannot be
considered, the distortional buckling strength is determined
by Specification Section F4.1:
[
(
) ](
)
𝑀crd 0.5
𝑀crd 0.5
𝑀nd = 1 − 0.22
𝑀𝑦
𝑀𝑦
𝑀𝑦 = [1 − 0.22(2.13)0.5 ](2.13)0.5 (141.66)
= 140.4 in.-kips
162
4
FLEXURAL MEMBERS
8
7
Load factor
6
5
4
3
Lcrl = 4.6,
L.F. = 2.76
2
1
0
100
Lcrl = 13.5,
L.F. = 3.13
101
Length
Figure 4.35
102
Signature Curve of 800S200-97.
D. Member strength:
By checking the cross-section dimensions (not shown), the
cross-section satisfies the limits given in Table 3.1. Therefore
the safety and resistance factors provided in Chapter F are
applicable.
Since the same safety factors for ASD method and the
same resistance factors for LRFD method are applied
to member strengths due to different buckling failures,
the member strength can be determined by the minimum
nominal strength as calculated in Items A to C.
𝑀n = minimum (𝑀ne , 𝑀𝑛𝓁 , 𝑀nd )
= minimum(159.2, 148.0, 140.4) = 140.4 in.-kips
The available strengths are calculated:
ASD method: 𝑀a = 𝑀n ∕Ωb = 140.4∕1.67 = 84.1 in.-kips
LRFD method: 𝜙b 𝑀n = 0.9(140.4) = 126.3 in.-kips
In this example, the distortional buckling controls the
design. It is possible to increase the member distortional
buckling strength by restraining the compression flange from
rotating about the flange and web juncture, thus increase
the member distortional buckling moment. For example, by
attaching the structural sheathing to the compression flange,
the structural sheathing is capable to provide the rotational
stiffness, 𝑘ϕ , which can be determined in accordance with
Section 3.5.2(4). By including the term 𝑘ϕ in Eq. (3.105), or
in the numerical analysis, the distortional buckling moment
is expected to be increased.
4.2.4.1 Laterally Unbraced Compression Flanges The
problems discussed in Sections 4.2.2 and 4.2.3 dealt with the
type of lateral–torsional buckling of I-beams, C-sections, and
Figure 4.36 Three possible types of supporting elastic frame for
equivalent column.4.19
Z-shaped sections for which the entire cross section rotates
and deflects in the lateral direction as a unit. But this is not
the case for U-shaped beams and the combined sheet stiffener
sections as shown in Fig. 4.36. For the latter, when it is loaded
in such a manner that the brims and the flanges of stiffeners
are in compression, the tension flange of the beams remains
straight and does not displace laterally; only the compression flange tends to buckle separately in the lateral direction,
accompanied by out-of-plane bending of the web, as shown
in Fig. 4.37, unless adequate bracing is provided.
Prior to 2004, this buckling phenomenon was considered
as lateral–torsional buckling. With the introduction of distortional buckling design provisions in the Supplement 1 to
the 2001 North American Specification1.343 , the flange along
with web buckling is considered as a distortional buckling,
and the member strength can be determined using the Direct
Strength Method. In the following section, both approaches
are discussed:
163
BENDING STRENGTH AND DEFLECTION
Figure 4.38
Figure 4.37
where 𝐴web is the area of the web and 𝐶c and 𝐶t are the
distance from the neutral axis to the extreme compression
fiber and the extreme tension fiber, respectively (Fig. 4.38).
Consequently, the equation of equilibrium of the compression flange is
[
] 2
𝐴web
𝑑 𝑥a
𝑑4𝑥
EI f 4a + 𝜎cr 𝐴f +
=0
12𝐶c ∕(3𝐶c − 𝐶t ) dz2
dz
(4.119)
and the corresponding nontrivial eigenvalue leads to
Force normal to buckled flange.4.19
4.2.4.1.1 Considering the Unstable Flange as Lateral
Torsional Buckling The precise analysis of the lateral–
torsional buckling of U-shaped beams is rather complex. Not
only to the compression flange and the compression portion
of the web act like a column on an elastic foundation, but
also the problem is complicated by the weakening influence
of the torsional action of the flange. For this reason, the
design procedure for determining the allowable design stress
for laterally unbraced compression flanges has been based
on the considerable simplification of an analysis presented
by Douty in Ref. 4.19. See Section 2 of Part V of the 2002
edition of the AISI Cold-Formed Steel Design Manual.1.340
When the compression flange of a U-shaped beam is
subject to the critical bending forces 𝜎cr 𝐴f (𝜎cr being the critical stress and 𝐴f the area of the flange), the component of
these forces normal to the buckling flange is
𝑞f = 𝜎cr 𝐴f
𝑑 2 𝑥a
dz2
(4.116)
See Fig. 4.37. In the same manner, the component on a unit
strip of the buckled web as shown in Fig. 4.38 is
𝑞w = 𝜎𝑡w
𝑑2𝑥
dz2
𝐴web
𝑑 2 𝑥a
12𝐶c ∕(3𝐶c − 𝐶t ) dz2
𝜎cr =
where
√
𝑟=
(4.117)
(4.118)
𝜋2𝐸
(𝐿∕𝑟)2
𝐼f
𝐴f + 𝐴web ∕[12𝐶c ∕(3𝐶c − 𝐶t )]
(4.120)
(4.121)
which is the radius of gyration of the effective column
consisting of the compression flange and a part of
the compression portion of the web having a depth of
[(3𝐶c − 𝐶t )∕12𝐶c ]𝑑, where d is the depth of the beam.
The above analysis is for the type of column supported on
an elastic foundation where the elastic support is provided
by the remaining portion of the web and the tension flange
acting together as an elastic frame. The effect of torsional
weakening in the combined flexural–torsional stability of
the effective column can be determined by the theorem of
minimum potential energy4.19 :
𝑈 = 𝑉1 + 𝑉2 + 𝑈w
L
=
As a result, the total lateral force 𝑅a transmitted to the
compression flange by the buckled web is
𝜎cr
Force normal to buckled web.4.19
1
[EI 𝑦 (𝑢′′ )2 + ECw (𝜙′′ )2 + GJ(𝜙′ )2 ]dz
2 ∫0
𝐿
1
(𝐶1 𝑢2 − 2𝐶2 u𝜙 + 𝐶3 𝜙2 )dz
(4.122)
2 ∫0
( )
]
𝐿 [
𝐼p
𝑃
−
(𝜙′ )2 dz
(𝑢′ )2 + 2𝑦0 𝑢′ 𝜙′ +
2 ∫0
𝐴
+
164
4
FLEXURAL MEMBERS
where 𝑈 = change in entire potential energy of system
consisting of effective column and its
supporting elastic frame
𝑉1 = strain energy accumulated in bent and
twisted column
𝑉2 = strain energy of deflected supporting frame
𝑈w = change in potential energy of external forces
acting on system
𝐼 = moment of inertia of column about its
vertical y axis
𝑈 = horizontal displacement of shear center
Φ = rotation of column
𝐽 = torsional constant of column
𝑦0 = vertical distance between shear center and
centroid of column
𝐼p = polar moment of inertia of column about its
shear center
𝐶w = warping constant
2 )
𝐶1 = 𝛿𝜙 ∕(𝛿u 𝛿𝜙 − 𝛿u𝜙
2 )
𝐶2 = 𝛿u𝜙 ∕(𝛿u 𝛿𝜙 − 𝛿u𝜙
2
𝐶3 = 𝛿u ∕(𝛿u 𝛿𝜙 − 𝛿u𝜙 )
δu = horizontal displacement of shear center due
to unit load
δu𝜙 = horizontal displacement of shear center due
to unit moment
𝛿ϕ = rotation of column due to unit moment
By solving Eq. (4.122) and applying considerable simplifications, the following expressions can be obtained for
the stability of the effective column on an elastic foundation taking the torsional weakening of the flange into
consideration4.19 :
)
(
⎧𝑇 1 + 𝛽𝐿2 𝑃 when 𝛽𝐿2 ∕𝑃 ≤ 30 (4.123)
e
e
2𝑃
𝜋
e
⎪
√
)
⎪ (
𝛽𝐿2
𝑃cr = ⎨𝑇 0.6 + 2
𝑃e
2𝑃
𝜋
𝜋
e
⎪
⎪
when 𝛽𝐿2 ∕𝑃e > 30
(4.124)
⎩
where 𝑃cr
𝑃e
𝛽
𝐷
=
=
=
=
critical load of equivalent column
Euler critical load, π2 EI∕𝐿2
spring constant, 1∕𝐷
lateral deflection of column centroid due to a
unit force applied to web at level of column
centroid
𝐿 = unbraced length of equivalnt column
and T, the torsional reduction factor, is determined as
follows:
{
ℎ
if 𝐿 ≥ 𝐿′
(4.125)
𝑇 = 𝑇0 =
ℎ + 3.4𝑦0
)( )
( ) (
ℎ
𝐿
𝐿
=
if 𝐿 < 𝐿′
𝑇0
(4.126)
𝐿′
ℎ + 3.4𝑦0
𝐿′
√
√
where 𝐿′ = 𝜋 4 2𝐼(ℎ∕𝑡)3 = 3.7 4 𝐼(ℎ∕𝑡)3
𝑦0 = distance from centroid of equivalent column
to its shear center
ℎ = distance from tension flange to centroid of
equivalent column
For beams with a large distance between bracing, the
following expression for 𝑃cr may be used:
√
𝑃cr = 𝑇0 4𝛽EI
(4.127)
From the value of 𝑃cr given above, the equivalent slenderness ratio (𝐿∕𝑟)eq can then be determined as follows:
√
( )
490
𝐿
𝜋2𝐸
=𝑘
=√
(4.128)
𝑟 eq
𝑃cr ∕𝐴c
𝑃 ∕𝐴
cr
c
where k is an experimental correction factor for the postbuckling strength and equals 1/1.1 and 𝐴c is the cross-sectional
area of the equivalent column.
The allowable compression stress 𝐹a for the ASD method
can be computed from the column formula (Chapter 5) on the
basis of this equivalent slenderness ratio. To obtain the allowable compression bending stress in the extreme compression
fiber 𝐹b′ , the axial stress 𝐹a may be extrapolated linearly from
the centroid level and adjusted for the different factors of
safety used for beam yielding and column buckling, that is,
( )
Ω
𝐶c
𝐹b′ = c
(4.129)
𝐹a
Ωb 𝑦c
where Ωc = safety factor used for column buckling
Ωb = safety factor used for beam yielding
𝑦c = distance from neutral axis of beam to centroid
of equivalent column
The design method developed in Ref. 4.19 has been
compared with the results of more than 100 tests (Fig. 4.39).
It has been found that discrepancies are within about
30% on the conservative side and about 20% on the
nonconservative side.
Based on the analysis and simplifications, the following
10-step design procedure has been included in the AISI
design manual since 1962.1.159 1.349
1. Determine the location of the neutral axis and define
as the “equivalent column” the portion of the beam
BENDING STRENGTH AND DEFLECTION
√
𝐿′ = 3.7
4
𝐼
( )3
ℎ
𝑡
165
(4.134)
where 𝐼 = moment of inertia of equivalent
column about its gravity axis
parallel to web, in.4
𝐿 = unbraced length of equivalent
column, in.
If 𝐶 ≤ 30, compute
Figure 4.39
(
)
𝛽𝐿2
𝑃cr = TPe 1 + 2
𝜋 𝑃e
4.19
Comparison between analysis and tests.
If 𝐶 > 30, compute
from the extreme compression fiber to a level that is
a distance of [(3𝐶c − 𝐶t )∕12𝐶c ]𝑑 from the extreme
compression fiber. In this expression, 𝐶c and 𝐶t are
the distances from the neutral axis to the extreme
compression and tension fibers, respectively, and d is
the depth of the section.
2. Determine the distance 𝑦0 measured parallel to the
web from the centroid of the equivalent column to
its shear center. (If the cross section of the equivalent
column is of angle or T shape, its shear center is at
the intersection of the web and flange; if of channel
shape, the location of the shear center is obtained from
Section 4.4. If the flanges of the channel are of unequal
width, for an approximation take w as the mean of the
two flange widths, or compute the location of the shear
center by rigorous methods. See Appendix B.)
3. To determine the spring constant 𝛽, isolate a portion
of the member 1 in. (25.4 mm) long, apply a force of
0.001 kip (4.45 N) perpendicular to the web at the
level of the column centroid, and compute the corresponding lateral deflection D of the centroid. Then the
spring constant is
0.001
𝐷
(4.130)
ℎ
ℎ + 3.4𝑦0
(4.131)
𝛽=
4. Calculate
𝑇0 =
where h is the distance from the tension flange to the
centroid of the equivalent column in inches.
5a. If the flange is laterally braced at two or more points,
calculate
290,000𝐼
𝑃e =
(4.132)
𝐿2
𝛽𝐿2
(4.133)
𝐶=
𝑃e
√
𝑃cr = TPe (0.60 + 0.635)
𝛽𝐿2
𝑃e
(4.135)
(4.136)
In both cases, if 𝐿 ≥ 𝐿′′ ,
𝑇 = 𝑇0
and if 𝐿 < 𝐿′ ,
𝑇0 𝐿
(4.137)
𝐿′
5b. If the flange is braced at less than two points, compute
√
(4.138)
𝑃cr = 𝑇0 4𝛽EI
𝑇 =
6. Determine the slenderness ratio of the equivalent
column,
( )
490
KL
=√
(4.139)
𝑟 eq
𝑃cr ∕𝐴c
where 𝐴c is the cross-sectional area of the equivalent
column.
7. From Eqs. (5.51), and (5.52) compute the stress 𝐹n
corresponding to (KL∕𝑟)eq .
8. The design compression bending stress based on
previous factors of safety is
( )
𝐶c
(4.140)
≤ 𝐹y
𝐹b2 = 1.15𝐹n
𝑦c
where 𝐶c = distance from neutral axis of beam to
extreme compression fiber
𝑦c = distance from neutral axis of beam to
centroid of equivalent column
The critical moment is 𝑀c = 𝐹b2 𝑆f .
Use Eq. (4.74) to compute 𝑀n .
Example 4.13 Determine the design compression bending
stress in the compression flanges (top flanges of the U-shaped
section shown in Fig. 4.40. Assume that the compression
166
4
FLEXURAL MEMBERS
Figure 4.41
Figure 4.40
Example 4.13.
Equivalent column.
and 4.44). The centroid of the equivalent column can
be located as follows:
flanges are laterally braced at the third points with unbraced
lengths of 48 in. The yield point of steel is 33 ksi.
SOLUTION
1. Location of Neutral Axis and Determination of
Equivalent Column (Fig. 4.40)
a. Location of Neutral Axis.
Distance
from
Top
Ay
Fiber
y (in.) (in.3 )
Element Area A (in.2 )
1
2
3
4
5
Total
2(1.0475)(0.105)
2(0.0396) (Table 4.4)
2(7.415)(0.105)
2(0.0396)
7.415(0.105)
=
=
=
=
=
0.2200
0.0792
1.5572
0.7920
0.7786
2.7142
0.0525
0.1373
4.0000
7.8627
7.9475
0.0116
0.0109
6.2288
0.6227
6.1879
13.0619
13.0619
= 4.812 in.
2.7142
𝐶t = 8.0 − 4.812 = 3.188 in.
𝐶c =
b. Equivalent Column. Based on step 1 of the procedure, the equivalent column used in the design is an
angle section as shown in Fig. 4.41. The depth of the
equivalent column can be determined as follows:
]
)
[
(
3𝐶c − 𝐶t
3(4.812) − 3.188
× 8.00
𝑑=
12𝐶c
12(4.812)
= 1.558 in
2. Determination of y0 (Distance from Centroid of
Equivalent Column to Its Shear Center). (Figs. 4.41
Distance
from
Top
Fiber
y (in.)
Ay (in.3 )
Element Area 𝐴(in.2 )
1
2
6
Total
𝑦cg =
1.0475(0.105)
= 0.1100 0.0525
0.0396 (Table 4.4) = 0.0396 0.1373
1.2655(0.105)
= 0.1329 0.9253
0.2825
0.0058
0.0055
0.1230
0.1343
0.1343
= 0.475 in.
0.2825
From Appendix B it can be seen that the shear center
of an angle section is located at the intersection of two
legs. Therefore, the distance y0 between the centroid
and the shear center of the equivalent column is
1
𝑦0 = 𝑦cg − 𝑡 = 0.475 − 0.0525 = 0.4225 in
2
3. Calculation of Spring Constant 𝜷. The spring
constant 𝛽 can be computed from Eq. (4.130) as
0.001
𝐷
for a portion of the member 1 in. in length. Here D is
the lateral deflection of the column centroid due to a
force of 0.001 kip applied to the web at the level of the
column centroid (Fig. 4.42). Using the moment–area
method (Fig. 4.43), the deflection D can be computed:
𝛽=
𝐷=
(7.4725)3
3EI × 103
+
(7.4725)2 (7.895)
2EI × 103
where E = 29.5 × 103 ksi and
1
𝐼=
(0.105)3 = 96.5 × 10−6 in.4
12
BENDING STRENGTH AND DEFLECTION
Figure 4.42
Force applied to web for computing spring constant.
Figure 4.44
167
Dimensions of equivalent column.
The I values of the individual elements about their
own centroidal axes parallel to the web are
1
𝐼1′ =
(0.105)(1.0475)3 = 0.0101
12
𝐼2′ = 0.0002
𝐼6′ = 0.0000
Figure 4.43
𝐼1′ + 𝐼2′ + 𝐼6′ = 0.0103
∑
(Ax2 ) = 0.0745
Lateral deflection of equivalent column.
𝐼𝑧 = 0.0848 in.4
Therefore
359.50
(29.5 × 10 )(96.5 × 10−6 )103
= 0.1263 in.
1
1
=
𝛽=
= 7.918 × 10−3
3
126.3
𝐷 × 10
𝐷=
3
4. Computation of T0 [Eq. (4.131)]
𝑇0 =
ℎ
7.525
=
= 0.840
ℎ + 3.4𝑦0
7.525 + 3.4(0.4225)
5. Determination of Pcr . In order to determine 𝑃cr , we
should first compute the moment of inertia of the
equivalent column about its y axis parallel to the web
(Fig. 4.44) as follows:
Element
1
2
6
Total
𝑥cg
Area
𝐴 (in.2 )
0.1100
0.0396
0.1329
0.2825
0.1023
=
= 0.362
0.2825
Distance
from
z Axis,
x (in.)
0.8163
0.1373
0.0525
in.
Ax (in.3 )
Ax2 (in.4 )
0.0898
0.0055
0.0070
0.1023
0.0733
0.0008
0.0004
0.0745
−
(∑ )
𝐴 (𝑥2cg ) = −0.2825(0.362)2 = −0.0370
𝐼𝑦 = 0.0478 in.4
and
𝐴 = 0.2825 in.2
Since the compression flange is braced at the third
points, the values of 𝑃e , C, and L′ can be computed
from Eqs. (4.132)–(4.134):
290,000(0.0478)
𝐼
𝑃e = 290,000 2 =
𝐿
482
= 6.016 kips
𝛽𝐿2
7.918 × 10−3 (48)2
=
= 3.032
𝑃e
6.016
√
√
)
( )3
(
4
4
ℎ
7.525 3
′
𝐿 = 3.7 𝐼
= 3.7 0.0478
𝑡
0.105
= 42.61 in.
𝐶=
Since 𝐶 < 30 and 𝐿 > 𝐿′ , from Eq. (4.135),
(
)
𝛽𝐿2
𝑃cr = 𝑇0 𝑃e 1 + 2
𝜋 𝑃e
[
]
7.918 × 10−3 (48)2
= 0.840(6.016) 1 +
𝜋 2 (6.016)
= 0.840(6.016)(1 + 0.307)
= 6.605 kips
168
4
FLEXURAL MEMBERS
6. Determination of (KL∕r)eq . For the equivalent column
[Eq. (4.139)]
( )
490
KL
=√
𝑟 eq
𝑃 ∕𝐴
cr
c
490
=√
= 101.3
6.60∕0.2825
7. Determination of Compression Stress Fn . From
Eq. (5.54),
𝐹cre =
𝜋 2 (29, 500)
𝜋2𝐸
=
(101.3)2
(KL∕𝑟)2eq
= 28.37 ksi
√
√
𝐹y
33
=
𝜆c =
𝐹cre
28.37
= 1.08 < 1.5
𝐹n = (0.658𝜆c )𝐹y = (0.6581.08 )(33)
2
2
= 20.25 ksi
8. Design Compression Bending Stress [Eq. (4.140)].
( )
𝐶𝑐
𝐹𝑏2 = 1.15𝐹𝑛
𝑦𝑐
4.812
= 25.84 ksi
4.337
< (𝐹𝑦 = 33 ksi) OK
= 1.15 (20.25)
Once the design compression bending stress is computed,
the critical or nominal moment can be calculated as 𝑀n =
𝐹b2 𝑆e .
In 1964, Haussler presented rigorous methods for determining the strength of elastically stabilized beams.4.20 In his
methods, Haussler also treated the unbraced compression
flange as a column on an elastic foundation and maintained
more rigor in his development. A comparison of Haussler’s
method with Douty’s simplified method indicates that the
latter may provide a smaller critical stress.
In the early 1990s, the flexural behavior of standing seam
roof panels with laterally unsupported compression flanges
was restudied by Serrette and Pekoz.4.158–4.162 Based on the
available test data and the analytical results from elastic
finite-strip buckling analysis, the authors introduced two
design methods in Ref. 4.161 to estimate the maximum
moment capacity of sections subjected to an interaction
between local and distortional buckling. It was assumed that
distortional buckling may be taken as local overall buckling
behavior. Both methods used the design philosophy currently
used in the North American Specification for local–lateral
buckling interaction. Method A used a derived analytical
expression for distortional buckling and method B used a
modified version of Douty’s formulation discussed in this
section. It was concluded that method A gives somewhat
better results than method B and is consistent with the present
formulation for flexural, torsional, and torsional–flexural
buckling.
According to the 2008 edition of the AISI Cold-Formed
Steel Design Manual, this type of buckling problem can be
solved by using the direct-strength method. For this reason,
the above 10-step design procedure has been removed from
the 2008 edition of the Design Manual.
4.2.4.1.2 Considering the Unstable Flange as Distortional Buckling When the compressed flange buckles laterally, the flange along with the web rotates about the juncture
of the web and the tension (stable) flange as illustrated in
Fig. 4.37. The member strength can be determined using the
Direct Strength Method, while the buckling moment can be
determined numerically or analytically in accordance with
Chapter 3. The following example illustrates how to use the
Direct Strength Method to determine the member strength.
Example 4.14 Determine the member strength of the
U-shaped section with its dimension shown in Fig. 4.40.
Assume that the compression flanges are laterally unbraced.
The yield point of steel is 33 ksi.
SOLUTION
1. Buckling moments.
From Example 3.10, the numerical analysis output
shows that the U-shaped section is susceptible to local
and distortional buckling. The local buckling occurs at
the buckling wavelength Lcr𝓁 = 6.4 in. and the corresponding load factor = 2.39. The distortional buckling
occurs at a buckling wavelength Lcrd = 67 in. and the
corresponding load factor = 0.86. From the numerical
analysis, the yield moment My = 161 in.-kips. Therefore, the local and distortional buckling moments are
obrained as:
𝑀cr𝓁 = 6.4My = 384.8 in.-kips
𝑀crd = 0.86My = 138.5 in.-kips
2. Strength due to local buckling.
Per Section 4.2.3.2, the nominal moment due to local
buckling can be determined using Eqs. (4.91) and
(4.92).
Since the member is lateral stable,
𝑀ne = 𝑀y = 161 in.-kips
BENDING STRENGTH AND DEFLECTION
Local buckling slenderness factor is calculated per Eq.
(4.93):
√
√
λ𝓁 = Mne ∕Mcr𝓁 = 161∕384.8 = 0.647 < 0.776
Therefore,
𝑀𝑛𝓁 = 𝑀ne = 161 in.-kips
From cross-section dimension check not shown, the
section meets the limits given in Table 3.1. Therefore,
the safety factor 1.67 for ASD and the resistance factor
of 0.90 for LRFD can be used:
ASD method: 𝑀a = 𝑀𝑛𝓁 ∕Ωb = 161∕1.67 = 96.4 in.
-kips
LRFD method: 𝜙b 𝑀𝑛𝓁 = (0.9)(161) = 144.9 in.-kips
3. Strength due to distortional buckling.
The distortional buckling slenderness factor is calculated per Eq. (4.103):
√
√
λd = My ∕Mcrd = 161∕138.5 = 1.078 > 0.673
Therefore,
) ](
)
Mcrd 0.5
𝑀crd 0.5
𝑀nd = 1 − 0.22
My
My
[
]
(
)
)
(
138.5 0.5 138.5 0.5
My = 1 − 0.22
(161)
161
161
[
(
= 118.86 in.-kips
The member strengths due to distortional buckling are
calculated
ASD method: 𝑀a = 𝑀nd ∕Ωb = (118.86)∕(1.67) =
71.17 in.-kips
169
virgin material. The effects of cold work were completely
neglected.
When the effects of cold work are utilized in the determination of bending strength, the computation can be
performed by one of the following two design approaches.
1. Consider the increase in yield stress at corners due to
cold work and neglect the effects of cold work in all flat
portions of the section. As discussed in Chapter 2, the
increase in yield stress can be found either by the use of
Eq. (2.11) or by tests.
2. Consider the effects of cold work for corners and all flat
elements. Equation (2.14) can be used to compute the
average yield stress of the entire section.
In either design approach, the following procedures may be
used2.17 :
1. Subdivide the section into a number of elements.
Assume a position of the neutral axis and the strain in
the top fiber. Compute the strains in various elements
based on the assumed neutral axis and the top fiber
strain.
2. Determine the stresses from the stress–strain relationship of the material in various elements for the
computed strains.
3. Locate the neutral axis by iteration until
∑
𝜎Δ𝐴 = 0
is satisfied. Then the bending moment can be approximated by
∑
𝑀=
𝜎yΔ𝐴
LRFD method: 𝜙b Mn = (0.9)(118.86) = 106.97 in.-kip
4. Member strength:
Since the member does not subject to lateral–
torsional buckling the member strength is controlled
by local and distortional buckling. By comparing the
member strengths due to local buckling (item 2) and
the strengths due to distortional buckling (item 3), the
distortional buckling controls the design. Therefore
the member design strengths are:
ASD method: 𝑀a = 71.17 in.-kips
LRFD method: 𝜙b 𝑀n = 106.97 in.-kip
4.2.5 Effects of Cold Work on Bending Strength
The bending strength of cold-formed steel sections discussed
above was based on the mechanical properties of the
where 𝜎 = stress
ΔA = area for element
𝑦 = distance between center of gravity of each
element and neutral axis
Results of the study by Winter and Uribe indicate that for
the steels commonly used in thin-walled cold-formed steel
construction, considering the effects of cold work only in the
corners of the formed sections, the moment capacities can
be increased by 4–22% compared with those obtained when
neglecting cold work.2.17
If the effects of cold work are considered in both the flats
and the corners, the increase in bending strength ranges from
17 to 41% above the virgin value.
It can be seen that a substantial advantage can be obtained
by using the increase in strength of the material. Figure 4.45,
170
4
FLEXURAL MEMBERS
1. For unstiffened compression flanges,
√
𝐸
𝑤 = 0.43𝑡
𝑓
(4.141)
2. For stiffened compression flanges supported by a web
on each longitudinal edge,
√
𝐸
𝑤 = 1.28𝑡
𝑓
where 𝑤
𝑡
𝐸
𝑓
=
=
=
=
(4.142)
flat width for compression flange
thickness of steel
modulus of elasticity
maximum compressive edge stress in the
element without considering the safety factor
The economic design of continuous beams and long-span
purlins is discussed in Refs. 4.11 and 4.12.
Figure 4.45 Comparison of ultimate moments computed for three
different conditions.2.17
reproduced from Ref. 2.17, shows a comparison of the
ultimate moments computed for three different conditions.
It should be noted that the effects of cold work as shown in
Fig. 4.45 may not be directly applied to other configurations
because the relative influence of corners or flats on the
increase in bending strength depends mainly on the configuration of the section and the spread between the tensile
strength and yield stress of the virgin material. Attention
should be given to the limitations of Section A3.3.2 of the
North American Specification when the effects of cold work
are used in design.
4.2.6
Economic Design for Bending Strength
The above discussion and design examples are based on
the fact that the allowable design moment is determined
for a given section for which the dimensions are known.
In the design of a new section, the dimensions are usually
unknown factors. The selection of the most favorable
dimensions can be achieved by using the optimum design
technique. This is a very complex nonlinear problem
which can only be solved by computer analysis.1.247
However, if the depth and the thickness of the section
are known, previous study has shown that the maximum
moment-to-weight ratio usually occurs in the neighborhood
of the flange width determined by Eq. (4.141) or (4.142) as
applicable:
4.2.7
Deflection of Flexural Members
For a given loading condition, the deflection of flexural
members depends on the magnitude, location, and type of
the applied load, the span length, and the bending stiffness
EI, in which the modulus of elasticity in the elastic range is
29.5 × 103 ksi (203 GPa or 2.07 × 106 kg∕cm2 ) and I is the
moment of inertia of the beam section.
Similar to the bending strength calculation, the determination of the moment of inertia I for calculating the deflection
of steel beams can be calculated based on the either the Effective Width Method or the Direct Strength Method:
(a) The Effective Width Method is used: I is determined
using the effective areas of the compression flange, edge
stiffer, and beam web, for which the effective widths
are computed for the compressive stress developed from
the bending moment. If the compression flange, edge
stiffer, and beam web are fully effective, the moment
of inertia is obviously based on the full section. In this
case, the moment of inertia is a constant value along the
entire beam length. Otherwise, if the moment of inertia
is on the basis of the effective areas of the compression
flange, edge stiffener, and/or beam web, the moment
of inertia may vary along the beam span because the
bending moment usually varies along the beam length,
as shown in Fig. 4.46.
(b) The Direct Strength Method is used: I is considered
linearly proportional to the strength of the section that
is determined at the service stress of the interest:
𝐼 = 𝐼g (𝑀d ∕𝑀) ≤ 𝐼g
(4.143)
BENDING STRENGTH AND DEFLECTION
171
SOLUTION
From Example 4.4, the allowable moment for the given
I-section is 178.0 in.-kips. The estimated compressive stress
in the top fiber under the allowable moment is
Mycg
178.0(4.063)
=
= 28.31 ksi
𝑓=
𝐼𝑥
25.382
The same stress of 𝑓 = 28.31 ksi will be assumed in the
calculation of the effective design width for deflection calculation.
By using Eqs. (3.32)–(3.34), and (3.36) and the same
procedure employed in Example 4.4, the effective width bd
of the unstiffened flange can be computed as follows:
Figure 4.46 Bending moment and variable moments of inertia for
two-span continuous beam under uniform load.4.14
where 𝐼g is the moment of gross cross-section; 𝑀d is the
minimum of the member strengths determined according
to Sections 4.2.2, 4.2.3.2, and 4.2.4 but with 𝑀y replaced
with M, and M is the moment at the service loads to be
considered. Detailed design provisions can also be found
in Section L2 of the 2016 edition of the North American
Specification.
In the design of thin-walled cold-formed steel sections,
the method to be used for deflection calculation is based on
the accuracy desired in the analysis. If a more exact deflection is required, a computer program or a numerical method
may be used in which the beam should be divided into a
relatively large number of elements according to variable
moments of inertia. The deflection calculation for such a
beam is too complicated for hand calculation. On the other
hand, if an approximate analysis is used, the deflection of
a simply supported beam may be computed on the basis of
a constant moment of inertia determined for the maximum
bending moment. The error so introduced is usually small
and on the conservative side.3.13 For continuous spans, the
deflection of the beam may be computed either by a rational
analysis4.13 or by a method using a conventional formula
in which the average value of the positive and negative
moments of inertia 𝐼1 and 𝐼2 will be used as the moment
of inertia I.4.14 This simplified method and other approaches
have been used in Refs. 4.6 and 4.7 for a nonlinear analysis
of continuous beams.
Example 4.15 Determine the moment of inertia of the
I-Section (Fig. 4.12) to be used for deflection calculation
when the I-section is loaded to the allowable moment as
determined in Example 4.4 for the ASD method. The Effective Width Method is used.
𝑤 = 1.6775 in.
𝑤
= 12.426
𝑡
𝑘 = 0.43
𝑓d = 28.31 ksi
1.052
(12.426)
𝜆= √
0.43
𝜌 = 1.0
√
28.31
= 0.618 < 0.673
29,500
𝑏d = 𝑤 = 1.6775 in.
Using the full width of the compression flange and
assuming the web is fully effective, the neutral axis is located
at the middepth (i.e., ycg = 4.0 in.). Prior to computing the
moment of inertia, check the web for effectiveness as
follows:
)
(
4 − 0.3225
= 26.03 ksi (compression)
𝑓1 = 28.31
4
)
(
4 − 0.3225
= 26.03 ksi (tension)
𝑓2 = 28.31
4
|𝑓 |
𝜓 = || 2 || = 1.0
| 𝑓1 |
𝑘 = 4 + 2(1 − 𝜓)3 + 2(1 − 𝜓) = 24.0
As in Example 4.4, ℎ0 ∕b0 = 4. Use Eq. (3.55a),
𝑏e
3+𝜓
where 𝑏e is the effective width of the web determined
in accordance with Eqs. (3.32) through (3.36) with 𝑓1
substitued for f and 𝑘 = 24.0 as follows:
ℎ
= 54.48
𝑡
√
26.03
1.052
= 0.348 < 0.673
𝜆 = √ (54.48)
29,500
24
𝜌 = 1.0
𝑏1 =
172
4
FLEXURAL MEMBERS
𝑏e = ℎ = 7.355 in.
𝑏e
7.355
=
= 1.839 in.
3+𝜓
3+1
Since 𝜓 > 0.236,
1
𝑏2 = 𝑏e = 3.6775 in.
2
𝑏1 + 𝑏2 = 1.839 + 3.6775 = 5.5165 in.
𝑏1 =
The same stress of 𝑓 = 24.37 ksi will be assumed in the
calculation of the effective design width for deflection determination.
Using Eqs. (3.32)–(3.36) and the same procedure
employed in Example 4.6, the effective width 𝑏d of the
stiffened compression flange is computed as follows:
𝑤 = 14.415 in.
𝑤
= 137.29
𝑡
𝑘 = 4.0
Since 𝑏1 + 𝑏2 is greater than the compression portion of
the web of 3.6775 in., the web is fully effective as assumed.
Because both the compression flange and the web are fully
effective, the moment of inertia 𝐼𝑥 of the full section can be
computed as follows:
Area
A (in.2 )
Element
𝑀𝑥 𝑦cg
𝐼𝑥
=
178.0(4.0)
= 27.22 ksi
26.1570
In view of the fact that the computed stress of 27.22 ksi
is less than the assumed value of 28.31 ksi, the moment of
inertia 𝐼𝑥 computed on the basis of the full section can be
used for deflection calculation without additional iteration.
Example 4.16 Compute the moment of inertia of the hat
section (Fig. 4.19) to be used for deflection calculation when
the hat section is loaded to the allowable moment as determined in Example 4.6 for the ASD method. The Effective
Width Method is used.
SOLUTION
1. First Approximation. From Example 4.6, the
allowable moment is 193.59 in.-kips. The estimated
compressive stress in the top flange under the allowable
moment is
𝑓=
𝑀𝑥 𝑦cg
𝐼𝑥
√
1.052
24.37
𝜆 = √ (137.29)
29,500
4
= 2.076 > 0.673
1 − 0.22∕2.076
𝜌=
= 0.431
2.076
𝑏d = 𝜌w = 0.431(14.415) = 6.213 in.
Distance
from
Middepth
y (in.) Ay2 (in.4 )
Flanges 4(1.6775)(0.135) = 0.9059 3.9325
14.0093
Corners
4(0.05407) = 0.2163 3.8436
3.1955
0
Webs
2(7.355)(0.135) = 1.9859 0
Total
3.1081
17.2048
1
(0.135)
2𝐼web = 2 × 12
× (7.355)3 =
8.9522
𝐼𝑥 = 26.1570 in.4
𝑓=
𝑓d = 24.37 ksi
193.59(4.487)
=
= 24.37 ksi
35.646
By using the effective width of the compression flange and
assuming the web is fully effective, the moment of inertia
can be computed from the line elements shown in Fig. 4.20
as follows:
Element
Distance
from
Top
Fiber
y (in.)
Ly (in.2 ) Ly2 (in.3 )
Effective
Length
L (in.)
1
2(1.0475) =
2.0950 9.9476
20.8400 207.3059
2
3
4
5
Total
2(0.3768) = 0.7536 9.8604
2(9.415) = 18.8300 5.0000
2(0.3768) = 0.7536 0.1396
6.2130 0.0525
28.6452
7.4308 73.2707
94.1500 470.7500
0.1052
0.0147
0.3262
0.0171
122.8522 751.3584
𝑦cg =
122.8522
= 4.289 in.
28.6452
The total 𝐼𝑥 is determined as follows:
( )
1
(9.415)3 = −139.0944
2𝐼3′ = 2
12
∑
(Ly2 ) = −751.3584 − 890.4528
(∑ )
−
𝐿 (𝑦2cg ) = −28.6452(4.289)2
= −526.9434
𝐼𝑥′
= −363.5094 in.3
𝐼𝑥 = 𝐼𝑥′ 𝑡 = 363.5094(0.105) = 0 − 38.168 in.4
BENDING STRENGTH AND DEFLECTION
The compressive stress in the top fiber is
𝑀𝑥 𝑦cg
193.59(4.289)
𝑓=
=
𝐼𝑥
38.168
= 21.75 ksi < the assumed value (no good)
2. Second Approximation. Assuming 𝑓d = 21.00 ksi and
using the same values of w/t and k,
√
1.052
21.00
𝜆 = √ (137.29)
= 1.927 > 0.673
29,500
4
𝜌 = 0.460
𝑏d = 𝜌w = 6.631 in.
Effective
Length
L (in.)
Element
1 to 4
5
Total
22.4322
6.6310
29.0632
Distance
from
Top
Fiber
y (in.)
0.0525
Ly (in.2 )
Ly2 (in.3 )
122.5260
0.3481
122.8741
751.3413
0.0183
751.3596
ycg = 4.289 in.
The total 𝐼𝑥 is
2𝐼3′ = −139.0944
∑
(Ly2 ) = −751.3596 − 890.4540
(∑ )
−
𝐿 (𝑦2cg ) = −29.0632(4.228)2
= −519.5333
𝐼𝑥′ = −370.9207 in.3
𝐼𝑥 = 𝐼𝑥′ 𝑡 = −38.947 in.4
𝑓=
𝑀𝑥 𝑦cg
𝐼𝑥
=
193.59(4.228)
= 21.01 ksi
38.947
Since the computed value of f is close to the assumed value
of 21.00 ksi, the moment of inertia for deflection calculation
under the allowable moment is 38.947 in.4 It is of interest to
note that the difference between the I values computed from
the first and second approximations is only about 2%.
173
roofs or panels provide more or less lateral support and
rotation restraints to the connected beam and to the flange,
and increase the beam bending capacity. Two approaches are
used to account such restraints in determining the member
strength:
1. Use the empirical equations developed based on experimental study. This approach simplifies the design but
is limited to certain beam and panel sizes, and beam
span lengths that are tested, and
2. Use the Direct Strength Method in which the buckling
moments are determined considering the effects of roof
or panel covering and span continuity. This method is
essentially applicable to any cross-section types and
span lengths. The complexity of this approach is how
to numerically model the system so that the roof or
panel restraints and the structural system connectivity
are realistically modeled.
Both approaches are discussed in the subsections.
4.2.8.1 Strengths for Members with General CrossSections and Connectivity The Direct Strength Method
provides a means to directly determine the strength of the
members that are connected to sheathings through-fastened
or with standing seam clips. The method requires that the
member buckling moment considers the restraints from
the connected sheathing or panels. Research work4.297–4.303
has been conducted to model these restraints with springs
which posessess translational and rotational stiffnesses.
Methods on how to determine the stiffness based on the
deck or sheathing configuration and connectivities are
provided in the study. 4.305 A finite element method such
as MASTAN24.312 was then used to determine the member
(global) buckling moment. Using this numerical analysis
in metal building wall and roof systems were documented
for bare deck through-fastened to members,4.398,4.303 and
for through-fastened and standing seam insulated metal
panels.4.301
In 2016, the following design provisions were introduced
into the North American Specification:
I6.1.2 Flexural Member Design
4.2.8 Beams in Metal Roof and Wall Systems
The nominal flexural strength [resistance], Mn , shall be the
minimum of 𝑀ne , 𝑀𝑛𝓁 , and 𝑀nd as given in Specification Sections I6.1.2.1 to I6.1.2.3. For members meeting the
geometric and material limits of Specification Section B4, the
safety and resistance factors shall be as follows:
Beams in metal roof and wall systems usually have one of
the flanges attached to metal roofs or wall panels through
through-fastened connections or sliding clips. These metal
Ωb = 1.67
𝜙b = 0.90
= 0.85
(ASD)
(LRFD)
(LSD)
174
4
FLEXURAL MEMBERS
For all other members, the safety and resistance factors in
Specification Section A1.2(c) shall apply. The available strength
[factored resistance] shall be determined in accordance with the
applicable method in Specification Section B3.2.1, B3.2.2, or
B3.2.3.
I6.1.2.1 Lateral–Torsional Buckling
The nominal flexural strength [resistance], Mne , for
lateral–torsional buckling shall be calculated in accordance
with Specification Section F2, except 𝐹cre or 𝑀cre shall be
determined including lateral, rotational, and composite stiffness
provided by the deck or sheathing, bridging and bracing, and
span continuity.
I6.1.2.2 Local Buckling
The nominal flexural strength [resistance], 𝑀𝑛𝓁 , for local buckling shall be calculated in accordance with Specification Section
F3, except 𝐹cr𝓁 or 𝑀cr𝓁 shall be determined including lateral,
rotational, and composite stiffness provided by the deck or
sheathing.
I6.1.2.3 Distortional Buckling
The nominal flexural strength [resistance], 𝑀nd , for distortional
buckling of girts and purlins shall be calculated in accordance
with Specification Section F4, except 𝑀crd shall be determined
including lateral, rotational, and composite stiffness provided by
the deck or sheathing.
I6.1.3 Member Design for Combined Flexure and Torsion
The nominal flexural strength [resistance], 𝑀n , for members in
combined flexure and torsion shall be reduced by applying the
reduction factor, R, determined in accordance with Specification
Eq. H4-1.
A design example that illustrates Specification Section I6.1
is provided in the 2017 edition of the AISI Cold-Formed Steel
Design Manual.1.428
4.2.8.2 Strengths for Members with Specific CrossSections and Connectivity The design methods provided
in the following subsections are applicable within the given
limitations.
4.2.8.2.1 Beams Having One Flange Through Fastened
to Deck or Sheathing When roof purlins or wall girts are
subject to the suction force due to wind load, the compression
flange of the member is laterally unbraced, but the tension
flange is supported by the deck or sheathing. The bending
capacity of this type of flexural member is less than the fully
braced member but is greater than the laterally unbraced
condition because of the rotational restraint provided by the
panel-to-purlin (or girt) connection. The rotational stiffness
has been found to be a function of the member thickness,
sheet thickness, fastener type, and fastener location.
In the past, the bending capacity of flexural members
having the tension flange through-fastened to deck or
sheathing has been studied by a large number of investigators in various countries.4.30–4.40 Based on the results of these
studies, reduction factors for the effective yield moment have
been developed for simple- and continuous-span conditions.
These factors are given in Section I6.2.1 of the 2016 edition
of the North American Specification. For the convenience of
readers, the following excerpts are adapted from the North
American Specification:
I6.2.1 Flexural Members Having One Flange
Through-Fastened to Deck or Sheathing
This section shall not apply to a continuous beam for the
region between inflection points adjacent to a support or to a
cantilever beam.
The nominal flexural strength [resistance], 𝑀n , of a C- or
Z-section loaded in a plane parallel to the web, with the tension
flange attached to deck or sheathing and with the compression flange laterally unbraced, shall be calculated in accordance
with Eq. 4.143. Consideration of distortional buckling in accordance with Specification Section F4 shall be excluded. The
safety factor and resistance factors given in this section shall be
used to determine the allowable flexural strength or design flexural strength [factored resistance] in accordance with the applicable design method in Specification Section B3.2.1, B3.2.2, or
B3.2.3.
𝑀n = R M𝑛𝓁𝑜
(4.143)
Ωb = 1.67
(ASD)
𝜙b = 0.90
(LRFD)
= 0.90
(LSD)
where R = A value obtained from Specification Table I6.2.1-1
for C- or Z-sections
𝑀𝑛𝓁𝑜 = Nominal flexural strength with consideration of
local buckling only, as determined from
Specification Section F3 with 𝐹n = 𝐹y or
𝑀ne = 𝑀y
Specification Table I6.2.1-1C- or Z-Section R Values
Simple Span
Member Depth Range, in. (mm)
d ≤ 6.5 (165)
6.5 (165) < d ≤ 8.5 (216)
8.5 (216) < d ≤ 12 (305)
8.5 (216) < d ≤ 12 (305)
Continuous Span
Profile
C or Z
C or Z
Z
C
Profile
C
Z
R
0.60
0.70
R
0.70
0.65
0.50
0.40
BENDING STRENGTH AND DEFLECTION
The reduction factor, R, shall be limited to roof and wall
systems meeting the following conditions:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
Member depth ≤ 12 in. (305 mm).
Member flanges with edge stiffeners.
60 ≤ depth/thickness ≤ 170.
2.8 ≤ depth/flange width ≤ 5.5.
Flange width ≥ 2.125 in. (54.0 mm).
16 ≤ flat width/thickness of flange ≤ 43.
For continuous span systems, the lap length at each interior
support in each direction (distance from center of support to
end of lap) is not less than 1.5d.
Member span length is not greater than 33 feet (10 m).
Both flanges are prevented from moving laterally at the
supports.
Roof or wall panels are steel sheets with 50 ksi (340 MPa
or 3520 kg/cm2 ) minimum yield stress, and a minimum
of 0.018 in. (0.46 mm) base metal thickness, having a
minimum rib depth of 1-1/8 in. (29 mm), spaced at
a maximum of 12 in. (305 mm) on centers and attached in
a manner to effectively inhibit relative movement between
the panel and member flange.
Insulation is glass fiber blanket 0 to 6 in. (152 mm) thick,
compressed between the member and panel in a manner
consistent with the fastener being used.
Fastener type is, at minimum, No. 12 self-drilling or
self-tapping sheet metal screws or 3/16 in. (4.76 mm) rivets,
having washers with 1/2 in. (12.7 mm) diameter.
Fasteners are not standoff type screws.
Fasteners are spaced not greater than 12 in. (305 mm) on
centers and placed near the center of the member flange, and
adjacent to the panel high rib.
The ratio of tensile strength to design yield stress shall not
be less than 1.08.
If variables fall outside any of the above-stated limits, the
user shall perform full-scale tests in accordance with Section
K2.1 of the Specification or apply a rational engineering analysis procedure. For continuous purlin and girt systems in which
adjacent bay span lengths vary by more than 20%, the R values
for the adjacent bays shall be taken from the simple-span values
in Specification Table I6.2.1-1. The user is permitted to perform
tests in accordance with Specification Section K2.1 as an alternative to the procedure described in this section.
For simple-span members, R shall be reduced for the effects
of compressed insulation between the sheeting and the member.
The reduction shall be calculated by multiplying R from Specification Table I6.2.1-1 by the following correction factor, r:
𝑟 = 1.00 − 0.01𝑡i when 𝑡i is in inches
(4.144)
𝑟 = 1.00 − 0.0004𝑡i when ti is in millimeters
(4.145)
where 𝑡i = Thickness of uncompressed glass fiber blanket
insulation.
175
4.2.8.2.2 Flexural Members Having One Flange
Fastened to a Standing Seam Roof System Standing
seam roofs were first introduced in the 1930s.4.171 Because
standing seam roof panels are attached to supporting purlins
with a clip that is concealed in the seam, this type of roof
system has proved to be a cost-effective roof membrane
due to its superior weather tightness, its ability to provide
consistent thermal performance, its low maintenance
requirements, as well as its ability to adjust to thermal
expansion and contraction.4.172
For C- or Z-purlins supporting a standing seam roof
system, the bending capacity is greater than the bending
strength of an unbraced member and may be equal to the
bending strength of a fully braced member. The bending
capacity is governed by the nature of the loading, gravity or
uplift, and the nature of the particular standing seam roof
system. Due to the availability of numerous types of standing
seam roof systems, the method to determine the member
strength through experiments was developed, and was first
added in the 1996 edition of the AISI Specification and is
retained in Appendix A of the 2016 edition of the North
American Specification as Section I6.2.2 for beams having
one flange fastened to a standing seam roof system. In this
section, it is specified that the available flexural strength of a
C- or Z-section loaded in a plane parallel to the web with the
top flange supporting a standing seam roof system shall be
determined using discrete point bracing and the provisions
of Chapter F of the Specification or shall be calculated as
follows:
𝑀n = 𝑅𝑀𝑛𝓁𝑜
(4.145)
Ωb = 1.67
(ASD)
𝜙b = 0.90
(LRFD)
where R is the reduction factor determined in accordance
with AISI S908 and Mn𝓁o is the nominal flexural strength
with consideration of local buckling only, as determined
from Section 4.2.3 with 𝐹n = 𝐹y or 𝑀ne = 𝑀y . For additional design information, see Ref. 4.172, which includes
detailed discussion and design examples using standing seam
roof systems. The major advantage of the base test is that
a simple span test may be used to predict the performance
of continuous-span systems for reducing experimental costs.
The concepts for the base test was developed by T. M. Murray
and his associates at Virginia Polytechnic Institute & State
University.
In Canada, this type of member is designed in accordance with Specification Chapter F based on discrete bracing
provided.
176
4.2.9
4
FLEXURAL MEMBERS
Strength of Standing Seam Roof Panel System
Under gravity loading, the nominal strength of standing seam
roof panel systems can be determined according to Chapter
F of the Specification because the load-carrying capacity of
usual panels can be calculated accurately. The strength of this
type of panel system can also be determined by the AISI
S906, Test Standard for Determining the Load-Carrying
Strength of Panels and Anchor-to-Panel Attachments for
Roof or Siding Systems Tested in Accordance with ASTM
E1592, in accordance with Section I6.3.1 of the North American Specification.1.345
For uplift loading, the nominal strength of standing seam
roof panels and their attachments or anchors cannot be calculated with accuracy; therefore, it can only be determined by
tests using the AISI S906 with the requirements and exceptions prescribed in Section I6.3.1 of the Specification on the
use of Factory Mutual FM4471, Corps of Engineers CEGS
07416, and ASTM E1592. The load combinations including
wind uplift are provided in Section I6.3.1a of Appendix A of
the Specification.
The evaluation of test results should follow the AISI S906.
When three or more assemblies are tested, safety factors
(not less than 1.67) and resistance factors (not greater than
0.9) shall be determined in accordance with the procedure
of Specification Section K2.1.1 (c) with the target reliability index and statistical data provided in Section I6.3.1 of
the Specification. The justifications for these variables are
discussed in the AISI Commentary.1.431 When the number
of physical test assemblies is less than 3, a safety factor of
2.0 and resistance factors of 0.8 (LRFD) and 0.7 (LSD) shall
be used.
4.2.10 Unusually Wide Beam Flanges and Unusually
Short Span Beams
distance from the web, as shown in Fig. 4.47a for a box-type
beam and an I-section. This phenomenon is known as
shear lag.
Analytical and experimental investigations of the problem
on shear lag have previously been conducted by Hildebrand
and Reissner,4.41 Winter,4.42 Miller,4.43 and Tate.4.44,4.45 This
subject has been investigated by Malcolm and Redwood,4.46
Parr and Maggard,4.47 Van Dalen and Narasimham,4.48 and
Lamas and Dowling4.49 and in Refs. 4.50–4.55.
In their paper, Hildebrand and Reissner concluded that
the amount of shear lag depends not only on the method of
loading and support and the ratio of span to flange width
but also on the ratio of G/E and the ratio 𝑚 = (3𝐼w + 𝐼s )∕
(𝐼w + 𝐼s ), where 𝐼w and 𝐼s are the moments of inertia of webs
and of cover plates, respectively, about the neutral axis of the
beam.
Based on the theory of plane stress, Winter analyzed the
shear lag problem and developed tabular and graphic data,4.42
from which the effective width of any given beam section
can be obtained directly for use in design. The ratios of the
maximum and minimum bending stresses in beam flanges
were computed and verified by the results of 11 I-beam
tests. It was indicated that shear lag is important for beams
Figure 4.47 (a) Stress distribution in both compression and
tension flanges of beams due to shear lag.
When beam flanges are unusually wide, special consideration should be given to the possible effects of shear lag
and flange curling, even if the beam flanges, such as tension
flanges, do not buckle. Shear lag depends on the type of
loading and the span-to-width ratio and is independent of the
thickness. Flange curling is independent of span length but
depends on the thickness and width of the flange, the depth
of the section, and the bending stresses in both flanges.
4.2.10.1 Shear Lag For conventional structural members
with ordinary dimensions, the effect of shear deformation on
flange stress distribution is negligible. However, if the flange
of a beam is unusually wide relative to its span length, the
effect of shear deformation on bending stress is pronounced.
As a result, the bending stresses in both compression and
tension flanges are nonuniform and decrease with increasing
Figure 4.47 (b) Analytical curve for determining effective width
of flange of short-span beams.4.42
177
BENDING STRENGTH AND DEFLECTION
Table 4.5
Ratio of Effective Design Width to Actual Width for Wide Flanges
L/wf
Loading Condition
Investigator
6
8
10
12
16
20
Hildebrand and Reissner
0.830
0.870
0.895
0.913
0.934
0.946
Hildebrand and Reissner
0.724
0.780
0.815
0.842
0.876
0.899
Hildebrand and Reissner
0.650
0.710
0.751
0.784
0.826
0.858
Hildebrand and Reissner
Winter
Miller
Hildebrand and Reisser
0.686
0.550
—
0.610
0.757
0.670
—
0.686
0.801
0.732
0.750
0.740
0.830
0.779
0.870
0.850
0.895
0.894
0.936
0.945
0.778
0.826
0.855
0.910
Hiildebrand and Reisser
Winter
Miller
0.830
0.850
—
0.897
0.896
—
0.936
0.928
0.875
0.957
0.950
0.977
0.974
0.985
0.984
0.991
0.995
with wide flanges subjected to concentrated loads on fairly
short spans; the smaller the span-to-width ratio, the larger
the effect. For beams supporting uniform loads, shear lag is
usually negligible except that the L/wf ratio is less than about
10 as shown in Fig. 4.47b. Winter also concluded that for a
given span-to-width ratio the effect of shear lag is practically
the same for box beams, I-beams, T-beams, and U-shaped
beams.
Table 4.5 is a summary of the ratios of effective design
width to actual width based on the results obtained by several
investigators.4.45
In Table 4.5, 𝑤f is the width of the flange projection
beyond the web for I-beams and half the distance between
webs for multiple-web sections (Fig. 4.47a); L is the span
length. It should be noted that the values obtained by Hildebrand and Reissner were for 𝐺∕𝐸 = 0.375 and 𝑚 = 2.
As far as the design criteria are concerned, the “effective
width” concept used in the design of compression elements
(Section 3.3) can also be applied to the design of beams
whenever the shear lag problem is critical.
Based on the results of Winter’s investigation,4.42 design
provisions for shear lag have been developed as included
in Section B4.3 of the North American Specification.1.417 It
is specified that when the effective span L of the beam is
less than 30𝑤f and when it carries one concentrated load
or several loads spaced farther apart than 2𝑤f , the ratio of
effective design width to actual width of the tension and
30
compression flanges shall be limited to the value given in
Table 4.6 in accordance with the 𝐿∕𝑤f ratio.
In the application of Table 4.6 the effective span length
of the beam is the full span for simple-span beams, the
distance between inflection points for continuous beams, or
twice the length of cantilever beams. The symbol 𝑤f indicates the width of the flange projection beyond the web for
I-beams and similar sections or half the distance between
webs for multiple-web sections, including box or U-type
sections (Fig. 4.47a). When I-beams and similar sections are
Table 4.6 Maximum Allowable Ratio of Effective
Design Width to Actual Width for Short-Span, Wide
Flanges
L/wf
30
25
20
18
16
14
12
10
8
6
Effective Design Width (b) /Actual Width (w)
1.00
0.96
0.91
0.89
0.86
0.82
0.78
0.73
0.67
0.55
178
4
FLEXURAL MEMBERS
Example 4.17 Compute the nominal moment for the beam
section shown in Fig. 4.48a if it is used to support a concentrated load on a simple span of 2 ft. Assume that the minimum
yield stress of steel is 40 ksi.
stiffened by lips at outer edges, wf shall be taken as the sum of
the flange projection beyond the web plus the depth of the lip.
The tabulated ratios in Table 4.6 are also plotted in
Fig. 4.47b for comparison with the analytical values. The
AISI design values are slightly larger than the analytical
results when 𝐿∕𝑤f ratios exceed about 16.
Although the above-discussed provision relative to shear
lag is applicable to tension and compression flanges, local
buckling in compression as discussed in Section 3.3 may
be a critical factor and should also be investigated separately. The shear lag problem is of particular importance in
the analysis and design of aircraft and naval structures. In
cold-formed steel building construction, however, it is infrequent that beams are so wide that they would require considerable reduction of flange widths.
For members designed by the Direct Strength Method,
Commentary on the Specification1.431 recommended that the
ratio of effective width (b) to the actual width (w) be replaced
by the corresponding ratio of 𝑀n ∕𝑀y . In building construction, in the cases of short spans under concentrated loads,
web crippling is typically controlling the limit state. Therefore, web crippling (Section 4.3.6) must be checked.
SOLUTION
From Fig. 4.48a,
1
𝑤f = (3.25) − 0.135 = 1.490 in.
2
30𝑤f = 44.70 in.
𝐿 = 2 ft = 24 in.
Since 𝐿 < 30𝑤f and the beam is subject to a concentrated
load, shear lag is an important factor.
Using Table 4.6 for 𝐿∕𝑤f = 16.1, the ratio of effective
design width to actual width is 0.86. The effective design
widths for both compression and tension flanges are
𝑏′ = 0.86 × 1.49 = 1.28 in.
See Fig. 4.48b.
(a)
(b)
Figure 4.48
Example 4.17.
BENDING STRENGTH AND DEFLECTION
To check if the web is fully effective according to
Section 3.3.1.2,
(
)
1
𝑓1 = 40
× 2.6775 = 35.7 ksi (compression)
3
𝑓2 = 35.7 ksi (tension)
|𝑓 |
Ψ = || 2 || = 1.0
| 𝑓1 |
𝑘 = 4 + 2[1 + 1]3 + 2[1 + 1] = 24
179
𝑤 = 12 (3.25) − (0.1875 + 0.135) = 1.3025 in.,
𝑤 1.3025
=
= 9.62
𝑡
0.135
)
(
√
40
1.052
(9.62)
λ= √
= 0.568 < 0.673
29,500
0.43
𝑏 = 𝑤 = 1.3025 in.
The nominal moment is
From Fig 4.48a,
𝑀n = 𝑆𝑥 (full section)𝐹𝑦
ℎ0 = out-to-out depth of web
= 6.00 in.
𝑏0 = out-to-out width of the compression flange of
In view of the fact that the nominal moment determined
above for local buckling consideration is larger than that
determined for shear lag, the nominal moment of 136 in.-kips
will govern the design.
each channel
= 3.25∕2
= 1.625 in.
Since ℎ0 ∕𝑏0 = 3.69 < 4, then use Eq. (3.47a),
𝑏e
3+𝜓
where 𝑏e is the effective width of the web determined in
accordance with Eqs. (3.32–3.35) with 𝑓1 substistuted for f
and 𝑘 = 24 as follows:
√
1.052(5.355∕0.135) 35.7∕29,500
𝜆=
= 0.296
√
24
𝑏1 =
Since λ < 0.673, 𝜌 = 1.0,
𝑏e = 5.355 in.
𝑏e
5.355
=
= 1.339 in.
3+𝜓
3+1
For 𝜓 = 1, which is larger than 0.236,
𝑏1 =
1
1
𝑏2 = 𝑏e = × 5.355 = 2.678 in.
2
2
𝑏1 + 𝑏2 = 1.339 + 2.678 = 4.017 in.
Example 4.18 For the tubular section shown in Fig. 4.49,
determine the nominal moment if the member is to be used
as a simply supported beam to carry a concentrated load at
midspan. Assume that the span length is 5 ft and Fy = 50 ksi.
SOLUTION
1. Nominal Moment Based on Effective Width of
Compression Flange. For the compression flange,
𝑤 8 − 2(3∕32 + 0.06) 7.693
=
=
𝑡
0.06
0.06
= 128.2
Based on Eqs. (3.32)–(3.35),
(
)
( )√𝑓
1.052
𝑤
𝜆=
√
𝑡
𝐸
𝑘
)
(
√
50
1.052
(128.2)
=
= 2.776 > 0.673
√
29,500
4
Since 𝑏1 + 𝑏2 is larger than the compression portion of the
web of 2.6775 in., the web is fully effective.
Based on the method discussed previously, the effective
section modulus is
𝑆e = 3.4 in.3
The nominal moment is
𝑀n = 𝑆e 𝐹y = 3.4 × 40 = 136 in. − kips
The nominal moment determined above for shear
lag should be checked for local buckling. Since
Figure 4.49 Example 4.18.
180
4
FLEXURAL MEMBERS
1 − 0.22∕𝜆 1 − 0.22∕2.776
=
= 0.332
𝜆
2.776
𝑏 = 𝜌w = 0.332(7.693) = 2.554 in.
𝑏e = 0.871(4.692) = 4.087 in.
𝜌=
See Fig. 4.50a.
Assume that the web is fully effective. The distance ycg
can be determined as follows:
Area 𝐴 (in.2 )
Element
1
2
3
4
5
Total
Distance
from
Top
Fiber
y (in.)
2.554 × 0.06 = 0.1532
2 × 0.01166 = 0.0233
2 × 4.6925 × 0.06 = 0.5631
2 × 0.01166 = 0.0233
7.6925 × 0.06 = 0.4616
1.2245
0.030
0.073
2.500
4.927
4.970
𝑏e
4.087
=
= 1.141 in.
3+𝜓
3 + 0.581
Since 𝜓 > 0.236, b2 = be ∕2 = 4.087∕2 = 2.0435 in., and
b1 + b2 = 1.141 + 2.0435 = 3.1845 > 2.968 in. (compression portion of the web). The web is fully effective. The total
𝐼𝑥 is determined as
∑
(Ay2 ) = 15.4872
𝑏1 =
Ay (in.3 )
Ay2 (in.4 )
0.0046
0.0017
1.4078
0.1148
2.2942
3.8231
0.00014
0.0013
3.5194
0.5656
11.4019
15.48717
∑
(Ay)
𝑦cg = ∑
= 3.122 in.
𝐴
1
𝐼webs = 2 × (0.06)(4.6925)3 = 1.0333
2
(∑ )
−
𝐴 (𝑦2cg ) = −1.2245(3.122)2 = −11.9351
𝐼𝑥 = 4.5850 in.4
The section modulus is
𝐼
4.5850
𝑆𝑥 = 𝑥 =
= 1.469 in.3
𝑦cg
3.122
The nominal moment about the x axis is
𝑀n = 1.469(50) = 73.45 in.-kips
1. To check if the web is fully effective (see Fig. 4.50b),
(
)
2.968
= 47.53 ksi (compression)
3.122
)
(
1.724
𝑓2 = 50
= 27.61 ksi (tension)
3.122
| 𝑓 | 27.61
𝜓 = || 2 || =
= 0.581
| 𝑓1 | 47.53
𝑓1 = 50
𝑘 = 4 + 2[1 + 0.581]3 + 2[1 + 0.581] = 15.066
From Fig. 4.49,
ℎ0 = out-to-out depth of web = 5.00 in.
𝑏0 = out-to-out width of the compression flange
2. Nominal Moment Based on Shear Lag Consideration. According to Figs. 4.47 and 4.49,
8 − 2(0.06)
= 3.94 in.
2
5 × 12
𝐿
=
= 15.23 < 30
𝑤f
3.94
Because the 𝐿∕wf ratio is less than 30 and the member
carries a concentrated load, additional consideration for
shear lag is needed. Using Table 4.6,
Effective design width
= 0.845
Actual width
Therefore the effective design widths of compression and
tension flanges between webs are (Fig. 4.51)
𝑤f =
0.845[8 − 2(0.06)] = 6.6586 in.
= 8.00 in.
Since ℎ0 ∕b0 = 0.625 < 4, use Eq. (3.47a),
𝑏1 =
𝑏e
3+𝜓
where be is the effective width of the web determined in
accordance with Eqs. (3.32)–(3.35) with 𝑓1 substituted for f
and 𝑘 = 15.066 as follows:
(
)
)√
(
47.53
1.052
4.692
𝜆= √
0.06
29,500
15.066
= 0.851 > 0.673
1 − 0.22∕𝜆 1 − 0.22∕0.851
𝜌=
=
= 0.871
𝜆
0.851
Using the full areas of webs, the moment of inertia about
the x axis is
𝐼𝑥 = 4[3.22356(0.06)(2.5 − 0.03)2
+ 0.01166(2.5 − 0.073)2 ] + 2
1
(0.06)(4.6925)3
12
= 6.046 in.4
and the section modulus is
𝐼
𝑆𝑥 = 𝑥 = 2.418 in.3
2.5
The nominal moment is
𝑀n = 2.418(50) = 120.9 in.-kips
BENDING STRENGTH AND DEFLECTION
181
(a)
(b)
Figure 4.50 (a) Effective width of compression flange for postbuckling strength. (b) Webs order
stress gradient.
4.2.10.2 Flange Curling When a beam with unusually
wide and thin flanges is subject to bending, the portion of the
flange most remote from the web tends to deflect toward the
neutral axis. This is due to the effect of longitudinal curvature
of the beam and bending stresses in both flanges. This subject
was studied by Winter in 1940.4.42
Let us consider an I-beam which is subject to pure bending
as shown in Fig. 4.52. The transverse component q of the
flange force 𝑓av 𝑡 per unit width can be determined by
𝑞=
Figure 4.51 Effective design widths of compression and tension
flanges for shear lag.
3. Nominal Moment for Design. From the above calculation, the nominal moment for design is 73.45 in.-kips.
Shear lag does not govern the design.
where 𝑓av
𝑡
d𝜙, dl, 𝑟b
𝐸
𝐼
𝑑
𝑓 𝑡
𝑓av 𝑡 dϕ 𝑓av 𝑡
2𝑓 2 𝑡
= av = av
=
dl
𝑟b
EI∕𝑀
Ed
=
=
=
=
=
=
average bending stress in flanges
flange thickness
as shown in Fig.4.52
modulus of elasticity
moment of inertia of beam
depth of beam
(4.146)
182
4
FLEXURAL MEMBERS
Figure 4.52
Flange curling of I-beam subject to bending.4.42
If the value of q is considered to be a uniformly distributed
load applied on the flange, the deflection or curling at the
outer edge of the flange can be computed by the conventional
method for a cantilever plate, namely,
)
(
) (
qw4f
𝑓av 2 𝑤4f
(4.147)
(1 − 𝜇2 )
=3
𝑐f =
8𝐷
𝐸
𝑡2 𝑑
where 𝑐f = deflection at outer edge
𝑤f = projection of flange beyond web
𝐷 = flexural rigidity of plate, = Et3 ∕12(1 − μ2 )
By substituting 𝐸 = 29.5 × 103 ksi and 𝜇 = 0.3 in
Eq. (4.147), the following formula for the maximum width
of an unusually wide stiffened or unstiffened flange in
tension and compression can be obtained:
√
√
4 100𝑐f
1800td
𝑤f =
×
𝑓av
𝑑
√
√
= 0.061𝑡 dE∕𝑓av 4 100𝑐f ∕𝑑
(4.148)
where 𝑐f = permissible amount of curling, in.
𝑓av = average stress in full unreduced flange width,
ksi (𝑤f , t, and 𝑑 were defined previously)
When members are designed by the effective design width
procedure, the average stress equals the maximum stress
times the ratio of the effective design width to the actual
width. Equation (4.148) is included in Section L3 of the
North American Specification to limit the width of unusually
wide flanges.
The above formula for determining wf is derived on the
basis of a constant transverse component q. As soon as flange
curling develops, the distance from the flange to the neutral
axis becomes smaller at the outer edge of the flange. This
results in the reduction of bending stresses. Therefore the
values of q vary along the flange as shown in Fig. 4.52. Since
the amount of cf is usually limited to a small percentage
of the depth, the error in the determination of wf by using
Eq. (4.148) is negligible and on the conservative side.
The above approximate treatment for I-beams can also be
applied to the design of box and U-type beams, except that
for the latter the flanges of the closed box beams may be
regarded as simple plates freely supported at webs and that wf
is to be measured as half of the distance between webs. Using
the same analogy, one can determine the flange curling cf for
closed box sections as follows:
)
(
) (
𝑓av 2 𝑤4f
𝑞(2𝑤f )4
5
(1 − 𝜇2 )
×
=5
𝑐f =
384
𝐷
𝐸
𝑡2 𝑑
(4.149)
A comparison of Eqs. (4.147) and (4.149) indicates that
the use of Eq. (4.148), which is derived on the basis of
I-beams, to determine wf for box beams, may
√ result in
a possible error of 13%. This is because 4 5∕3 = 1.13.
However, this discrepancy can be reduced if the restraint of
webs and the variable values of the transverse component q
are taken into consideration.
No specific values are given by the North American Specification for the maximum permissible amount of curling.
However, it is stated in the AISI Commentary that the amount
of curling that can be tolerated will vary with different kinds
of sections and must be established by the designer. An
amount of curling on the order of 5% of the depth of the
section is usually not considered excessive.
Assuming 𝑐f ∕𝑑 = 0.05, Eq. (4.148) can be simplified as
√
tdE
𝑤f = 0.37
𝑓av
In general, the problem of flange curling is not a critical
factor to limit the flange width. However, when the appearance of the section is important, the out-of-plane distortion
should be closely controlled in practice.
In general, the problem of flange curling is not a critical
factor to limit the flange width. However, when the appearance of the section is important, the out-of-plane distortion
should be closely controlled in practice.
Example 4.19 Determine the amount of curling for the
compression flange of the hat section used in Example 4.6
when the section is subjected to the allowable moment. Use
the ASD method.
SOLUTION
The curling of the compression flange of the hat section can
be computed by Eq. (4.148). In the calculation,
1
𝑤f = (15.0 − 2 × 0.105) = 7.395 in.
2
𝑡 = 0.105 in.
𝑑 = 10.0 in.
40.69
4.934
×
= 8.34 ksi
𝑓av =
1.67
14.415
183
DESIGN OF BEAM WEBS
Using Eq. (4.148),
√
√
0.061 × 0.105 × 10 × 29,500 4 100𝑐f
7.395 =
×
8.34
10
√
4
= 26.77 𝑐f
𝑐f = 0.0058 in.
4.3 DESIGN OF BEAM WEBS
under concentrated loads can be designed as compression
members. The nominal strength, 𝑃n , for bearing stiffeners
is the smaller of the values determined by 1 and 2 as
follows:
(4.150)
1. 𝑃n = 𝐹wy 𝐴c
2. 𝑃n = Nominal axial load evaluated according to Specification Section E3.1 with 𝐴e replaced by 𝐴b . Specification Section E3.1 is provided in Section 5.6 of this
book.
4.3.1 Introduction
Not only should thin-walled cold-formed steel flexural
members be designed for bending strength and deflection as
discussed in Section 4.2 but also the webs of beams should
be designed for shear, bending, combined bending and shear,
web crippling, and combined bending and web crippling.
In addition, the depth of the web should not exceed the
maximum value permitted by Section B4.1 of the North
American Specification.
When the Effective Width Method is used, the maximum
allowable depth-to-thickness ratio h/t for unreinforced webs
is limited to 200, in which h is the depth of the flat portion
of the web measured along the plane of the web and t
is the thickness of the web. When bearing stiffeners are
provided only at supports and under concentrated loads,
the maximum depth-to-thickness ratio may be increased to
260. When bearing stiffeners and intermediate shear stiffeners are used simultaneously, the maximum h/t ratio is 300.
These limitations for h/t ratios are established on the basis of
the studies reported in Refs. 3.60 and 4.56–4.60. When the
Direct Strength Method is used, the h/t is limited to 300 for
any of the above conditions. If a web consists of two or more
sheets, the h/t ratios of the individual sheets shall not exceed
the maximum allowable ratios mentioned above.
The following discussions deal with the minimum requirements for bearing and shear stiffeners, the design strength
for shear and bending in beam webs, the load or reaction to
prevent web crippling, and combinations of various types of
strengths.
4.3.2 Stiffener Requirements
Section F5 of the 2016 edition of the North American
Specification, provides the following design requirements
for attached bearing stiffeners and shear stiffeners. When
the bearing stiffeners do not meet these requirements, the
load-carrying capacity for the design of such members can
be determined by tests.
a. Bearing Stiffeners. For beams having large h/t ratios,
bearing stiffeners attached to beam webs at supports or
Ωc = 2.00 (ASD)
{
0.85 (LRFD)
𝜙c =
0.80 (LSD)
where
2
⎧18𝑡 + 𝐴s for bearing stiffeners at interior
⎪ support or under concentrated load (4.151)
𝐴c = ⎨ 2
⎪10𝑡 + 𝐴s for bearing stiffeners
⎩ at end support
(4.152)
𝐴s = cross-sectional area of bearing stiffeners
𝐹wy = lower value of 𝐹y for beam web,
or 𝐹ys for stiffener section
⎧
⎪𝑏1 𝑡 + 𝐴s for bearing stiffeners at
⎪ interior support or under
⎪
𝐴𝑏 = ⎨ concentrated load
⎪𝑏 𝑡 + 𝐴 for bearing stiffeners
s
⎪ 2
⎪ at end support
⎩
[
( )
]
𝐿st
𝑏1 = 25𝑡 0.0024
+ 0.72 ≤ 25𝑡
𝑡
[
( )
]
𝐿st
+ 0.83 ≤ 12𝑡
𝑏2 = 12𝑡 0.0044
𝑡
(4.153)
(4.154)
(4.155)
(4.156)
𝐿st = length of bearing stiffener
𝑡 = base steel thickness of beam web
In addition, the specification stipulates that w/ts ratios
for the stiffened and unstiffened elements of cold-formed
√
steel bearing stiffeners should not exceed 1.28 𝐸∕𝐹ys
√
and 0.42 𝐸∕𝐹ys , respectively. In the above expressions,
𝐹ys is the yield stress and 𝑡s is the thickness of the stiffener
steel.
It should be noted that Eq. (4.150) is used to prevent end
crushing of bearing stiffeners, while the second 𝑃n is used
to prevent column buckling of the combined web stiffener section. The equations for computing the effective
184
4
FLEXURAL MEMBERS
areas 𝐴b and 𝐴c and the effective widths 𝑏1 and 𝑏2
are adopted from Nguyen and Yu.4.59 Figures 4.53 and
4.54 show the effective areas 𝐴c and 𝐴b of the bearing
stiffeners.
b. Bearing Stiffeners in C-Section Flexural Members. For
two-flange loading (Figs. 4.68c and 4.68d) of C-section
flexural members with bearing stiffeners that do not meet
the above requirements of Section 4.3.2a, the nominal
strength 𝑃n should be determined as follows:
𝑃n = 0.7(𝑃wc + 𝐴e 𝐹y ) ≥ 𝑃wc
Ωc = 1.70
{
0.90
𝜙c =
0.80
(4.157)
(ASD)
(LRFD)
(LSD)
where 𝑃wc is the nominal web crippling strength for
C-section flexural members calculated in accordance with
Eq. (4.197) for single web members, at end or interior locations; 𝐴e is the effective area of the bearing
stiffener subjected to uniform compressive stress, calculated at yield stress; and 𝐹y is the yield stress of the
bearing stiffener steel. Equation (4.157) is based on the
research conducted by Fox and Schuster at the University of Waterloo.1.299,4.229–4.231 The program investigated
the behavior of 263 stud- and truck-type bearing stiffeners in cold-formed steel C-section flexural members
subjected to two-flange loading at both interior and end
locations. This equation is applicable within the following
limits:
1. Full bearing of the stiffener is required. If the bearing
width is narrower than the stiffener such that one of the
stiffener flanges is unsupported, 𝑃n shall be reduced by
50%.
2. Stiffeners are C-section stud or track members with a
minimum web depth of 3 12 in. (89 mm) and minimum
base steel thickness of 0.0329 in. (0.84 mm).
3. The stiffener is attached to the flexural member web
with at least three fasteners (screws or bolts).
4. The distance from the flexural member flanges to the
first fastener(s) is not less than d/8, where d is the
overall depth of the flexural member.
5. The length of the stiffener is not less than the depth of
the flexural member minus 38 in. (9.53 mm).
6. The bearing width is not less than 1 12 in. (38.1 mm).
c. Shear Stiffeners. All shear stiffeners shall be designed to
satisfy the following requirements for spacing, moment of
inertia, and gross area:
1. Spacing a between Stiffeners. When shear stiffeners
are required, the spacing shall be based on the nominal
shear strength 𝑉n permitted by Section 4.3.3.2f and the
following limits:
(
)2
260
𝑎≤
ℎ
(4.158)
ℎ∕𝑡
𝑎 ≤ 3ℎ
(4.159)
2. Moment of Inertia Is of Shear Stiffeners. With reference to an axis in the plane of the web, the moment
Figure 4.53 Effective area Ac of bearing stiffener: (a) at end support; (b) at interior support and
under concentrated load.
Figure 4.54 Effective area Ab of bearing stiffener: (a) at end support; (b) at interior support and
under concentrated load.
185
DESIGN OF BEAM WEBS
of inertia of a pair of attached stiffeners or of a single
stiffener shall satisfy the following requirements:
)
(
ℎ 0.7𝑎
(4.160)
𝐼s ≥ 5ht3
−
𝑎
ℎ
( )4
ℎ
𝐼s ≥
(4.161)
50
3. Gross area As of Shear Stiffeners. The area 𝐴s shall
satisfy the requirement of Eq. (4.162):
]
[
1 − 𝐶𝑣 𝑎
(𝑎∕ℎ)2
𝐴s ≥
YDht
−
√
2
ℎ (𝑎∕ℎ) + 1 + (𝑎∕ℎ)2
(4.162)
where
⎧ 1.53Ekv
when 𝐶v ≤ 0.8
⎪
2
⎪ 𝐹y (ℎ∕𝑡)
√
𝐶v = ⎨
Ekv
⎪ 1.11
when 𝐶v > 0.8
⎪ ℎ∕𝑡
𝐹y
⎩
⎧4.00 + 5.34
when 𝑎∕ℎ ≤ 1.0
⎪
(𝑎∕ℎ)2
𝑘v = ⎨
4.00
⎪5.34 +
when 𝑎∕ℎ > 1.0
⎩
(𝑎∕ℎ)2
where 𝑓v = actual shear stress
𝑉 = total external shear force at a section
𝑄 = static moment of area between the extreme
fiber and the particular location at which the
shear stress is desired, taken about neutral axis
𝐼 = moment of inertia of entire cross-sectional
area about neutral axis
𝑡 = width of section where shear stress is desired
Even though Eq. (4.167) gives the exact value at any location, it has been a general practice to use the average value
in the gross area of the web as the shear stress for design
purposes. This average shear stress can be computed by using
the following equation:
𝑓v =
(4.163)
𝑉
htw
(4.164)
where ℎ = depth of the flat portion of the web measured
along the plane of the web
𝑡w = thickness of the web
(4.165)
The use of Eqs. (4.167) and (4.168) is illustrated in
Example 4.20.
(4.166)
and 𝑎 = distance between shear stiffeners
𝑌 = yield stress of web steel∕yield stress of
stiffener steel
𝐷 = 1.0 for stiffeners furnished in pairs
= 1.8 for single-angle stiffeners
= 2.4 for single-plate stiffeners
Most of the above requirements for shear stiffeners
are based on the AISC Specification1.148 and the study
reported in Ref. 4.59.
d. Nonconforming Stiffeners. According to Section F5.3
of the North American specification, the available
strength of members with stiffeners that do not meet the
requirements of Section 4.3.2a, 4.3.2b, or 4.3.2c, such
as stamped or rolled-in stiffeners, shall be determined
by tests in accordance with Section K2 of the Specification or rational engineering analysis in accordance with
Section A1.2(c) of the Specification.
Example 4.20 Determine the shear stress distribution at
the end supports of the uniformly loaded channel shown in
Fig. 4.55. Assume that the load is applied through the shear
center of the cross section so that torsion is not involved.∗
See Appendix B for a discussion of the shear center.
SOLUTION
1. Exact Shear Stress Distribution Using Eq. (4.167).
For simplicity, use square corners and midline dimensions as shown in Fig. 4.56 for computing the exact
shear stresses at various locations of the section.
a. Shear stress at points 1 and 4:
𝑉 = 𝑅A = 1.5 kips
𝑄1 = 𝑄4 = 0
VQ1
=0
It
b. Shear stress at points 2 and 3:
(𝑓v )1,4 =
𝑄2 = 𝑄3 = 1.4325(0.135)
4.3.3 Shear
VQ
It
(4.167)
(
1
× 6.865
2
)
= 0.664 in.3
VQ2
1.5(0.664)
(𝑓v )2,3 =
=
= 0.941 ksi
It
7.84(0.135)
4.3.3.1 Shear Stress In the design of beams, the actual
shear stress developed in the cross section of the beam can be
calculated by using the following well-known equation4.61 :
𝑓v =
(4.168)
∗ When
the load does not pass through the shear center, see Appendix B.
186
4
FLEXURAL MEMBERS
Figure 4.55
Example 4.20.
the beam is probably governed by shear yielding. The
maximum shear stress at the neutral axis can be computed
by Eq. (4.169):
𝐹y
(4.169)
𝜏y = √
3
in which τy is the yield stress in shear and 𝐹y is the yield
stress in tension.The nominal shear strength for yielding
can be determined by the shear stress given in Eq. (4.169)
and the web area, ht, as follows:
(
)
𝐹y
𝑉n = √
(4.170)
(ht) ≅ 0.60𝐹y ht
3
Figure 4.56
Shear stress distribution.
c. Shear stress at point 5:
(
)(
)
1
1
× 6.865
× 6.865
𝑄5 = 𝑄2 + 0.135
2
4
3
= 1.46 in.
VQ5
1.5(1.46)
(𝑓v )5 =
=
= 2.07 ksi
It
7.84(0.135)
2. Average Shear Stress on Beam Web by Using
Eq. (4.168)
𝑉
1.5
=
𝑓v =
= 1.65 ksi
htw
6.73(0.135)
From the above calculation it can be seen that for the
channel section used in this example the average shear stress
of 1.65 ksi is 25% below the maximum value of 2.07 ksi.
4.3.3.2 Shear Strength of Beam Webs without Holes
a. Shear Yielding. When a beam web with a relatively small
h/t ratio is subject to shear stress, the shear capacity of
where 𝑉n is the nominal shear strength, h is the depth of
the flat portion of the web, and t is the thickness of the web.
b. Elastic Shear Buckling. For webs with large h/t ratios,
the shear capacity of the web is governed by shear
buckling. Figure 4.57 shows a typical pattern of shear
failure.4.56 Based on studies by Southwell and Skan
on shear buckling of an infinitely long plate, the
plate develops a series of inclined waves, as shown
in Fig. 4.58.4.62,4.63 The elastic critical shear buckling
stress can be computed by Eq. (4.171)∗ :
𝜏cr =
where 𝑘v
𝐸
𝜇
ℎ
𝑡
=
=
=
=
=
𝑘v 𝜋 2 𝐸
12(1 − 𝜇2 )(ℎ∕𝑡)2
(4.171)
shear buckling coefficient
modulus of elasticity of steel
Poisson’s ratio
depth of plate
thickness of plate
∗ The problem of shear buckling of plane plates has also been studied by
Timoshenko and other investigators. For additional information see Refs.
3.1 and 4.63.
DESIGN OF BEAM WEBS
Figure 4.57
Figure 4.58
187
Typical shear failure pattern (h/t = 125).4.56
Shear buckling of infinitely long plate4.63 : (a.) simply supported edges; (b.) fixed edges.
In Eq. (4.171), the value of 𝑘v varies with the
supporting conditions and the aspect ratio a/h (Fig. 4.59),
in which a is the length of the plate. For a long plate the
value of 𝑘v was found to be 5.34 for simple supports and
8.98 for fixed supports, as listed in Table 3.2.
Substituting 𝜇 = 0.3 in Eq. (4.171),
0.904𝑘v 𝐸
(4.172)
𝜏cr =
(ℎ∕𝑡)2
Thus if the computed theoretical value of τcr is less than
the proportional limit in shear, the nominal shear strength
188
4
FLEXURAL MEMBERS
By substituting the values of τpr and τcri into Eq. (4.174),
one can obtain Eq. (4.176) for the shear buckling stress in
the inelastic range, that is,
√
0.64 𝑘v 𝐹y 𝐸
(4.176)
𝜏cr =
ℎ∕𝑡
Consequently, the nominal shear strength in the
inelastic range can be obtained from Eq. (4.177):
√
√
0.64 𝑘v 𝐹y 𝐸
(ht) = 0.64𝑡2 𝐾v 𝐹y 𝐸 (4.177)
𝑉n =
ℎ∕𝑡
Figure 4.59 Shear buckling stress coefficient of plates versus
aspect ratio a/h.3.1
for elastic buckling can be obtained from Eq. (4.173):
0.904𝑘v 𝐸
0.904𝑘v Et3
(ℎ∕𝑡)
=
(4.173)
ℎ
(ℎ∕𝑡)2
The above elastic critical buckling stress (τcr ) and the
resultant (𝑉cr ) are based on the web alone ignoring the
interaction from the flanges. Aswegan and Moen4.306
developed analytical expressions which enable to take
into consideration of the interactions of the connected
elements. These equations are provided in Section 2.3.5
of the North American Specification.1.417 The shear
buckling can also be determined numerically using the
semi-analytical finite strip method4.307 or spline finite
strip method4.308 developed by Hancock and Pham.
c. Shear Buckling in Inelastic Range. For webs having
moderate h/t ratios, the computed theoretical value of τcr
may exceed the proportional limit in shear. The theoretical value of the critical shear buckling stress should
be reduced according to the change in the modulus of
elasticity. Considering the influence of strain hardening
observed in the investigation of the strength of plate
girders in shear,4.64 Basler indicated that Eq. (4.174) can
be used as the reduction formula:
√
𝜏cr = 𝜏pr 𝜏cri
(4.174)
√
where τpr = 0.8τy = 0.8(𝐹y ∕ 3) is the proportional limit
in shear and the initial critical shear buckling stress is
given as
𝑘v 𝜋 2 𝐸
(4.175)
𝜏cri =
12(1 − 𝜇2 )(ℎ∕𝑡)2
𝑉cr =
d. Direct Strength Method. The research by Pham and
Hancock4.309,4.310 showed that considerable tension field
action is available for local buckling if a web is fully
restrained at the loading and support points over its full
depth by bolted connections. This post-buckling strength
can be considered using the direct strength method
that Pham and Hancock has developed. In addition, for
members with transvers stiffeners, the stiffness contribution can also be included in web strength through the
Direct Strength Method as long as the critical shear buckling force, 𝑉cr , used in the method includes the transverse
stiffener effects. Since it is difficult to develop analytical
solutions for shear buckling of webs with transverse stiffeners, numerical methods are employed. The methods
such as: semi-analytical finite strip method4.307 or spline
finite strip method4.308 can be used to determine the shear
buckling force with stiffness contribution.
The Direct Strength Method design for determining
shear strength was first adopted into Appendix 1 of the
2012 edition of the North American Specification, and
was incorporated into the main body of the 2016 edition
of the Specification as Section G2.2.
e. Safety Factors. Prior to 1996, the AISI ASD Specification employed three different safety factors (i.e., 1.44 for
yielding, 1.67 for inelastic buckling, and 1.71 for elastic
buckling) for determining the allowable shear stresses
in order to use the same allowable values for the AISI
and AISC Specifications. To simplify the design of shear
elements for using the allowable stress design method,
the safety factor for both elastic and inelastic shear buckling was taken as 1.67 in the 1996 edition of the AISI
Specification. The safety factor of 1.50 was used for
shear yielding to provide the allowable shear stress of
0.40Fy , which has been used in steel design for a long
time. The use of such a smaller safety factor of 1.50 for
shear yielding was justified by long-standing use and by
the minor consequences of incipient yielding in shear as
compared with those associated with yielding in tension
and compression.
189
DESIGN OF BEAM WEBS
In the 2001 and 2007 editions of the North American
Specification, the constant used in Eq. (4.176) for determining the inelastic shear buckling stress was reduced
slightly from 0.64 to 0.60 on the basis of Craig’s calibration of the test data of LaBoube and Yu.1.346,4.56,4.232,4.233
For the purpose of simplicity, a single safety factor of 1.60
was used in the 2007 edition the Specification for shear
yielding, elastic and inelastic shear buckling for the ASD
method with a corresponding resistance factor of 0.95
for LRFD and 0.80 for LSD. These safety and resistance
factors are retained in Section G2 of the 2016 edition of
the Specification.1.417 With these minor revisions of safety
factor and design equations, for ASD, the North American Specification reduces 6% of the allowable inelastic
shear buckling strength as compared with the AISI 1996
ASD Specification. For beam webs with large h/t ratios,
the North American Specification allows 4% increase of
the allowable strength for the elastic shear buckling.
f. North American Design Criteria for Shear Strength of
Webs without Holes. Based on the foregoing discussion
of the shear strength of beam webs, the 2016 edition of
the North American Specification includes the following
design provisions in Section G2 for the ASD, LRFD, and
LSD methods.1.417 In the 2016 edition of the Specification, the equation format has been revised from shear
stress to shear resultants by multiplying the web shear
area.
where
𝐴w = Area of web element
= ht
where h = Depth of flat portion of web measured along plane
of web
t = Web thickness
𝐹y = Design yield stress as determined in accordance
with Specification Section A3.3.1
𝑉cr = Elastic shear buckling force as defined in
Specification Section G2.3 for flat web alone, or
determined in accordance with Specification
Appendix 2 for full cross-section of prequalified
(Table 3.1) members
E = Modulus of elasticity of steel
𝑘v = Shear buckling coefficient, determined in
accordance with Specification Section G2.3
G2.2 Flexural Members With Transverse Web
Stiffeners
For a reinforced web with transverse web stiffeners meeting the
criteria of Specification Section G4, and spacing not exceeding
twice the web depth, this section is permitted to be used to
determine the nominal shear strength [resistance], 𝑉n , in lieu of
Specification Section G2.1.
For λv ≤ 0.776,
(4.183)
𝑉n = 𝑉y
For λv > 0.776,
[
G2.1 Flexural Members Without Transverse Web
Stiffeners
The nominal shear strength [resistance], 𝑉n , of flexural members
without transverse web stiffeners shall be calculated as follows:
For 𝜆𝑣 ≤ 0.815,
(4.178)
𝑉n = 𝑉 y
For 0.815 < λv ≤ 1.227
√
𝑉cr 𝑉y
√
= 0.60𝑡2 𝐸𝑘𝑣 𝐹y
𝑉n = 0.815
= 0.904Ekv 𝑡 ∕h
√
𝜆𝑣 =
𝑉𝑦
𝑉cr
𝑉cr
𝑉𝑦
)0.4 ] (
𝑉cr
𝑉𝑦
)0.4
𝑉𝑦
(4.184)
Other variables are defined in Specification Section G2.1.
(4.179b)
G2.3 Web Elastic Critical Shear Buckling Force, V cr
(4.180b)
(4.181)
𝑉y = Yield shear force of cross-section
= 0.6 𝐴w 𝐹y
(
where 𝑉cr = Elastic shear buckling force as defined in
Specification Section G2.3 for flat web alone, or
determined in accordance with Specification
Appendix 2 for full cross-section of prequalified
(Table 3.1) members
The shear buckling force, 𝑉cr , of a web is permitted to be determined in accordance with this section:
𝑉cr = 𝐴w 𝐹cr
(4.180a)
3
where
𝑉𝑛 = 1 − 0.15
(4.179a)
For λv > 1.227
𝑉n = 𝑉cr
(4.182b)
(4.182a)
(4.185)
where 𝐴w = Web area as given in Eq. (4.182b)
𝐹cr = Elastic shear buckling stress
=
𝜋 2 Ek𝑣
12(1 − 𝜇 2 )(ℎ∕𝑡)2
where 𝐸 = Modulus of elasticity of steel
𝑘v = Shear buckling coefficient calculated in
accordance with (a) or (b) as follows:
(4.186)
190
4
FLEXURAL MEMBERS
(a) For unreinforced webs, 𝑘v = 5.34
(b) For webs with transverse stiffeners satisfying the requirements of Specification Section G4
when
𝑎∕ℎ ≤ 1.0
𝑘𝑣 = 4.00 +
5.34
(𝑎∕ℎ)2
(4.187a)
when
𝑎∕ℎ > 1.0
𝑘𝑣 = 5.34 +
4.00
(𝑎∕ℎ)2
(4.187b)
where 𝑎 = Shear panel length of unreinforced web element
= Clear distance between transverse stiffeners of
reinforced web elements
Other variables are defined in Specification Section G2.1.
For the ASD method, the allowable shear stresses in webs
are shown in Fig. 4.60 by using 𝐸 = 29,500 ksi (203 GPa,
or 2.07 × 106 kg∕cm2 ). Table 4.7 gives the allowable shear
stresses for 𝐹y = 33 and 50 ksi (228 and 345 MPa, or 2320
and 3515 × 106 kg∕cm2 ). Examples 4.21–4.23 illustrate the
applications of the shear design provisions.
Example 4.21 Use the ASD and LRFD methods to determine the available shear strength for the I-section used in
Example 4.4. Use 𝐹y = 50 ksi.
SOLUTION
A. ASD Method
The depth-to-thickness ratio of each individual web element
is
ℎ 8 − 2(0.135 + 0.1875) 7.355
=
=
= 54.48
𝑡
0.135
0.135
Based on the North American design criteria, the value
of 𝑘v for unreinforced webs is 5.34. Therefore, according to
Section G2.3 of the North American Specification,
𝐹cr =
𝜋 2 Ek𝑣
𝜋 2 (29,500)(5.34)
=
2
2
12(1 − 𝜇 )(ℎ∕𝑡)
12(1 − 0.32 )(54.48)2
Figure 4.60 Allowable shear stress in webs for the ASD method.
The allowable shear strength for the I-section having two
webs is
2𝑉
2(29.79)
= 37.24 kips
𝑉a = n =
Ωv
1.60
B. LRFD Method
Using the same nominal shear strength computed in item
A, the design shear strength for the I-section having double
webs is
2𝜙v 𝑉n = 2(0.95)(29.79) = 56.60 kips
= 47.97 ksi
𝑉cr = 𝐴w 𝐹cr = (7.355)(0.135)(47.97) = 47.63 kips
From Specification Section G2.1:
𝑉y = 0.6Aw Fy = 0.6(7.355)(0.135)(50) = 29.79 kips
√
√
𝑉𝑦
29.79
=
𝜆=
= 0.791
𝑉cr
47.63
Since λ < 0.815
𝑉n = 𝑉y = 29.97 kips
Example 4.22 Use the ASD and LRFD methods to determine the available shear strength for the channel section used
in Example 4.5. Use 𝐹y = 50 ksi.
SOLUTION
A. ASD Method
For the given channel section, the depth-to-thickness ratio of
the web is
ℎ 10 − 2(0.075 + 0.09375) 9.6625
=
=
= 128.83
𝑡
0.075
0.075
DESIGN OF BEAM WEBS
Table 4.7
191
Allowable Shear Stresses for ASD Method, ksi
𝐹y = 33 ksi
𝐹y = 50 ksi
a/h
a/h
h/t
0.5
1.0
2.0
3.0
>3
0.5
1.0
2.0
3.0
>3
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
220
240
260
280
300
12.4
12.4
12.4
12.4
12.4
12.4
12.4
12.4
12.4
12.4
12.4
11.6
11.0
10.4
9.8
9.3
8.5
7.3
6.3
5.4
4.7
12.4
12.4
12.4
12.4
12.4
11.3
10.3
9.4
8.7
7.9
6.9
6.1
5.4
4.8
4.3
3.9
3.2
2.7
2.3
2.0
1.7
12.4
12.4
12.4
11.6
10.4
9.3
8.5
7.3
6.3
5.4
4.7
4.1
3.7
3.3
2.9
2.6
2.2
1.8
1.6
1.3
1.2
12.4
12.4
12.4
11.1
9.9
8.9
8.0
6.7
5.7
4.9
4.3
3.8
3.3
3.0
2.7
2.4
2.0
1.7
1.4
1.2
1.1
12.4
12.4
12.2
10.7
9.5
8.6
7.4
6.2
5.3
4.5
4.0
3.5
3.1
2.7
2.5
2.2
1.8
1.5
1.3
1.1
1.0
18.8
18.8
18.8
18.8
18.8
18.8
18.8
18.8
17.6
16.4
15.3
14.3
13.5
12.7
11.7
10.6
8.7
7.3
6.3
5.4
4.7
18.8
18.8
18.8
17.4
15.5
13.9
12.7
10.8
9.2
7.9
6.9
6.1
5.4
4.8
4.3
3.9
3.2
2.7
2.3
2.0
1.7
18.8
18.8
16.4
14.3
12.7
10.6
8.7
7.3
6.3
5.4
4.7
4.1
3.7
3.3
2.9
2.6
2.2
1.8
1.6
1.3
1.2
18.8
18.3
15.6
13.7
11.9
9.6
8.0
6.7
5.7
4.9
4.3
3.8
3.3
3.0
2.7
2.4
2.0
1.7
1.4
1.2
1.1
18.8
17.5
15.0
13.2
11.0
8.9
7.4
6.2
5.3
4.5
4.0
3.5
3.1
2.7
2.5
2.2
1.8
1.5
1.3
1.1
1.0
Notes:
𝑎
Values above the single underlines are based on Eq. (4.178); values between the single underlines and double underlines are based on
Eq. (4.179); and values below the double underlines are based on Eq. (4.180). For the case of a/h > 3, 𝑘v = 5.34.
𝑏
1 ksi = 6.9 MPa or 70.3 kg/cm2 .
Based on 𝑘v = 5.34 and 𝐹y = 50 ksi,
𝐹cr =
𝜋 2 Ek𝑣
𝜋 2 (29,500)(5.34)
=
12(1 − 𝜇2 )(ℎ∕𝑡)2
12(1 − 0.32 )(128.8)2
= 8.578 ksi
𝑉cr = 𝐴w 𝐹cr = (9.6625)(0.075)(8.578) = 6.217 kips
From Specification Section G2.1:
𝑉y = 0.6𝐴w 𝐹y = 0.6(9.6625)(0.075)(50) = 36.23 kips
√
√
𝑉y
26.23
𝜆=
=
= 2.4 > 1.227
𝑉cr
6.217
Therefore,
𝑉n = 𝑉cr = 6.217 kips
The allowable shear strength using the ASD method for
the design of the given channel section is
𝑉a =
𝑉n
6.22
=
= 3.89 kips
Ωv
1.60
B. LRFD Method
Based on the same nominal shear strength computed in item
A, the design shear strength using the LRFD method for the
design of the channel section is
𝜙v 𝑉n = (0.95)(6.22) = 5.91 kips
Example 4.23 Use the ASD and LRFD methods to determine the available shear strength for the hat section used in
Example 4.6. Use 𝐹y = 50 ksi.
192
4
FLEXURAL MEMBERS
SOLUTION
A. ASD Method
The depth-to-thickness ratio of the web is
ℎ 10 − 2(0.105 + 0.1875) 9.415
=
=
= 89.67
𝑡
0.105
0.105
Based on 𝑘v = 5.34 and 𝐹y = 50 ksi,
𝜋 2 Ek𝑣
𝜋 2 (29, 500)(5.34)
𝐹cr =
=
12(1 − 𝜇2 )(ℎ∕𝑡)2
12(1 − 0.32 )(89.67)2
= 17.71 ksi
𝑉cr = 𝐴w 𝐹cr = (9.415)(0.105)(17.71) = 17.505 kips
From Specification Section G2.1:
𝑉y = 0.6𝐴w 𝐹y = 0.6(9.415)(0.105)(50) = 49.429 kips
√
√
𝑉y
49.429
λ=
=
= 1.680 > 1.227
𝑉cr
17.505
Therefore, 𝑉n_single = 𝑉cr . For two webs, the total nominal
shear strength is
𝑉n = 2𝑉cr = 2(17.505) = 35.01 kips
𝑉𝑎 =
𝑉n
35.01
=
= 21.88
Ωv
1.60
B. LRFD Method
Based on the same nominal shear strength computed in item
A, the design shear strength using the LRFD method for the
design of the hat section is
𝜙v 𝑉n = (0.95)(35.01) = 33.26 kips
4.3.3.3 Shear Strength of C-Section Webs with Holes
When holes are present in beam webs, the effect of web
perforation on the reduction of shear strength of C-sections
was investigated in the 1990s by Shan et al.,3.184,3.197
Schuster et al.,3.187 and Eiler.3.192 In these studies, three
hole geometries (rectangular hole with corner fillets,
circular hole, and diamond-shaped hole) were considered
in the test programs. Based on the results of research
findings, the design provisions were added in the supplement to the 1996 edition of the AISI Specification in
19991.333 as Section C3.2.31.336,1345 and is retained in
the 2016 edition of the North American Specification as
Section G3. 1.417
Based on the Specification provisions, for C-section webs
with holes, the shear strength shall be calculated in accordance with Specification Section G2 for section without
holes and multiplied by the reduction factor 𝑞s , as defined
in Eq. (4.188a) or (4.188b), under the same limits given in
Section 3.4.3:
⎧
𝑐
⎪1.0 when 𝑡 ≥ 54
𝑞s = ⎨ 𝑐
𝑐
⎪ 54𝑡 when 5 ≤ 𝑡 < 54
⎩
⎧ ℎ − 𝑑h
for circular holes
⎪
𝑐 = ⎨ 2 2.83
⎪ ℎ − 𝑑h for noncircular holes
⎩2
2
(4.188𝑎)
(4.188𝑏)
(4.189𝑎)
(4.189𝑏)
where 𝑑h = depth of web hole
ℎ = depth of flat portion of web measured along
plane of the web
𝑡 = web thickness
Similar to Section 1.1.1 of the Specification (Section 3.4.3
in this volume), the above design provisions for circular and
noncircular holes apply to any hole pattern that fits within an
equivalent virtual hole, as shown in Figs. 3.65 and 3.66.
4.3.4
Bending
Webs of beams can buckle not only in shear but also due to
the compressive stress caused by bending, for example, at the
location of a maximum moment. Figure 3.27 shows a typical
pattern of bending failure of beam webs.
The web buckling stress due to bending and the postbuckling strength of flat beam webs are discussed in
Section 3.3.1.2. The same section also discusses the AISI
design equations for computing the effective design depth
of beam webs.
For beam webs having relatively large depth-to-thickness
ratios, the buckling of web elements becomes more important. The structural efficiency of such beam webs can be
improved by adding longitudinal stiffeners in the compression portion of the web, as shown in Fig. 4.61. References
4.60, 4.65, and 4.66 present the studies made by Nguyen
and Yu on the structural behavior of longitudinally reinforced
beam webs.
Figure 4.61 Typical sections for longitudinally reinforced beam
specimens.
193
DESIGN OF BEAM WEBS
In Europe, the design methods for profiled sheeting and
sections with stiffeners in the flanges and webs are provided
in Refs. 1.209 and 3.56.
The Direct Strength Method provided in Section G2.2 of
the North American Specification can handle the available
bending strength of beams having longitudinal web stiffeners.
4.3.5 Combined Bending and Shear
When high bending and high shear act simultaneously, as
in cantilever beams and at interior supports of continuous
beams, the webs of beams may buckle at a lower stress than
if only one stress were present without the other. Such webs
must be safeguarded against buckling due to this combined
bending and shear.
The critical combination of bending and shear stresses
in disjointed flat rectangular plates has been studied by
Timoshenko.3.2 Figure 4.62 shows the interaction between
𝑓b ∕𝑓cr and τ∕τcr , in which 𝑓b is the actual computed bending
stress, 𝑓cr is the theoretical buckling stress in pure bending,
𝜏 is the actual computed shear stress, and τcr is the theoretical buckling stress in pure shear. It can be seen from
Fig. 4.62 that for a/h ratios ranging from 0.5 to 1.0 the relationship between 𝑓b ∕𝑓cr and τ∕τcr can be approximated by
Eq. (4.190), which is a part of the unit circle:
( )2 ( )2
𝑓b
𝜏
+
=1
(4.190a)
𝑓cr
𝜏cr
or
√
( )2 ( )2
𝑓b
𝜏
+
≤1
(4.190b)
𝑓cr
𝜏cr
beam webs subjected to the combined bending and shear
stresses:
√
( )2
(
)
𝑓v
𝑓bw 2
+
≤1
(4.191)
𝐹bw
𝐹v
where 𝑓bw = actual compression stress at junction of flange
and web
𝐹bw = allowable compressive stress
𝑓v = actual average shear stress
𝐹v = allowable shear stress,
For additional information on simply supported plates
under combined shear and uniformly distributed longitudinal
stresses, see Ref. 3.3.
In the past, the structural strength of beam webs subjected
to a combination of bending and shear has been studied by
LaBoube and Yu.4.57,4.67 The results of 25 beam tests indicated that Eq. (4.190a), originally developed for a disjointed
individual sheet, would be conservative for beam webs with
adequate transverse stiffeners, for which a diagonal tension
field action can be developed. Based on the test results shown
in Fig. 4.63, the following interaction equation was developed for beam webs with transverse stiffeners satisfying the
requirements of Section 4.3.2:
0.6
𝑓b
𝑓b,max
+
By using proper safety factors, the following interaction formula can be used for the allowable stress design of
𝜏
𝜏max
= 1.3
(4.192)
Eq. (4.192)
Eq. (4.190a)
Figure 4.62
T∕Tcr .4.63
Interaction
relationship
between
𝑓𝑏 ∕𝑓cr
and
Figure 4.63
Interaction diagram for T∕Tmax and fb ∕fb,max .
194
4
FLEXURAL MEMBERS
where 𝑓b,max = maximum computed stress governing
bending
τmax = maximum computed stress governing shear
for reinforced web
Accordingly, the allowable stress equation for webs reinforced with adequate transverse stiffeners can be expressed
as follows:
)
(
𝑓
𝑓bw
(4.193)
+ v ≤ 1.3
0.6
𝐹bw
𝐹v
Equation (4.193) is applicable only when 𝑓bw ∕𝐹bw > 0.5 and
𝑓𝑣 ∕𝐹v > 0.7.
For other conditions, the design of beam webs is governed
by either the allowable bending stress or the allowable shear
stress.
Instead of using stress ratios in Eqs. (4.191) and (4.193),
the 1986 and 1996 editions of the AISI Specification and
the North American Specification use strength ratios (i.e.,
moment ratio for bending and force ratio for shear) for
the interaction equations. The following design criteria are
adapted from Section H2 of the North American Specification for combined bending and shear. In addition, in the
2016 edition of the North American Specification, consistent
format of ASD, LRFD and LSD are used, i.e., strengths with
subscript “a” are denoted as allowable strength when ASD
is used, and design strengths when LRFD or LSD is used.
The shear forces or moments with a top bar are denoted as
the demands calculated based on ASD, LRFD or LSD load
combinations. All the variables are defined in the following
excerpted provisions:
where: 𝑀 = Required flexural strength [moment due to
factored loads] in accordance with ASD, LRFD,
or LSD load combinations
𝑉 = Required shear strength [shear force due to
factored loads] in accordance with ASD, LRFD
or LSD load combinations
𝑀a = Available flexural strength [factored resistance]
when bending alone is considered, determined in
accordance with Specification Chapter F
𝑉a = Available shear strength [factored resistance]
when shear alone is considered, determined in
accordance with Specification Sections G2 to G4
𝑀𝑎𝓁𝑜 = Available flexural strength [factored resistance]
for globally braced member determined as
follows:
(a) For members without transverse web stiffeners, 𝑀𝑎𝓁𝑜 is determined in accordance
with Specification Section F3 with Fn = Fy
or Mne = My , and
(b) For members with transverse web stiffeners,
𝑀𝑎𝓁𝑜 is the lesser of
(1) Available strength [factored resistance]
determined in accordance with Specification Section F3 with Fn = Fy or Mne =
My , and
(2) Available strength [factored resistance]
determined in accordance with Specification Section F4.
𝐹n = Global flexural buckling stress as defined in
Specification Section F2
𝐹y = Yield stress
𝑀ne = Nominal flexural strength [resistance]
considering yielding and global buckling,
determined in accordance with Specification
Section F2
𝑀y = Member yield moment in accordance with
Specification Section F2.1
H2 Combined Bending and Shear
For beams subjected to combined bending and shear, the
required flexural strength [moment due to factored loads], 𝑀,
and the required shear strength [shear force due to factored
loads], 𝑉 , shall not exceed 𝑀a and 𝑉a , respectively.
For beams without shear stiffeners as defined in Specification Section G4, the required flexural strength [moment due to
factored loads], 𝑀, and the required shear strength [shear force
due to factored loads], 𝑉 , shall also satisfy the following interaction equation:
√
(
)2 ( )2
𝑀
𝑉
+
≤ 1.0
(4.194a)
𝑀𝑎𝓁𝑜
𝑉𝑎
For beams with shear stiffeners as defined in Specification
Section G4, when M∕M𝑎𝓁𝑜 > 0.5 and V∕𝑉a > 0.7, 𝑀 and 𝑉
shall also satisfy the following interaction equation:
(
0.6
𝑀
𝑀𝑎𝓁𝑜
)
(
+
𝑉
𝑉𝑎
)
≤ 1.3
(4.194b)
Figure 4.64 shows the interaction formulas using the ASD
method for the design of beam webs subjected to the
combination of bending and shear. These design criteria are
based on Eqs. (4.194a) and (4.194b). A study conducted
by Almoney and Murray indicated that combined bending
and shear is a possible limit state for a continuous lapped
Z-purlin system and that the current design provisions
accurately predict the failure load.4.173
4.3.6
Web Crippling
4.3.6.1 Web Cripping Strength of Beam Webs without
Holes When the end and load stiffeners are not used in
thin-walled cold-formed steel construction, the webs of
beams may cripple due to the high local intensity of the load
or reaction. Figure 4.65 shows the types of failure caused by
DESIGN OF BEAM WEBS
Eq. (4.194b)
Eq. (4.194a)
Figure 4.64 Interaction formulas for combined bending and shear
using ASD method.
Figure 4.65
Web crippling of unfastened beams.1.161
web crippling for I-beams and hat sections unfastened to the
support. The web crippling of I-beam webs is also shown in
Fig. 1.31.
The buckling problem of separate flat rectangular plates
under locally distributed edge forces has been studied
by numerous investigators, including Sommerfeld,4.68
Timoshenko,4.69 Leggett,4.70 Hopkins,4.71 Yamaki,4.72
Zetlin,4.73 White and Cottingham,4.74 Khan and Walker,4.75
Khan, Johns, and Hayman,4.76 and others.3.7 Based on Refs.
4.75 and 4.76, the buckling load for the plates subjected to
locally distributed edge forces as shown in Figs. 4.66 and
4.67 can be computed as
𝑘𝜋 2 Et3
(4.195)
12(1 − 𝜇2 )ℎ
where k is the buckling coefficient depending on the ratios of
N/h and a/h as given in these two figures.
For steel beams having webs connected to flanges, theoretical and experimental investigations on web crippling
under partial edge loading have been conducted by Lyse
and Godfrey,4.77 Rocky, Bagchi, and El-gaaly,4.78–4.83,4.94
Roberts and Neware,4.181 Bergfelt,4.84 Edlund,4.102 and
others. However, the theoretical analysis of web crippling
for cold-formed steel flexural members is rather complicated
because it involves the following factors:
𝑃cr =
195
1. Nonuniform stress distribution under the applied load
and adjacent portions of the web
2. Elastic and inelastic stability of the web element
3. Local yielding in the immediate region of load
application
4. Bending produced by eccentric load (or reaction) when
it is applied on the bearing flange at a distance beyond
the curved transition of the web
5. Initial out-of-plane imperfection of plate elements
6. Edge restraints provided by beam flanges based on
the fastened condition to the support and interaction
between flange and web elements
7. Inclined webs for decks and panels
For these reasons, in the United States, the AISI
1946–1968 design provisions for web crippling were based
on the extensive experimental investigations conducted at
Cornell University by Winter and Pian4.85 and by Zetlin4.73
in the 1940s and 1950s. The AISI design previsions were
revised in 1980, 1986, and 1996 based on the additional
research work conducted at the University of Missouri-Rolla
by Hetrakul and Yu,4.58,4.94 Yu,4.86 Bhakta, LaBoube, and
Yu,4.174 and Cain, LaBoube, and Yu.4.175 These modified
provisions have been used for high-strength steels, high h/t
ratios, and the increase of end-one-flange web crippling
strength for Z-sections fastened to the support member. In
these experimental investigations, the web crippling tests
have been carried out under the following four loading
conditions for beams having single unreinforced webs and
I-beams:
1.
2.
3.
4.
End one-flange (EOF) loading
Interior one-flange (IOF) loading
End two-flange (ETF) loading
Interior two-flange (ITF) loading
All loading conditions are illustrated in Fig. 4.68. In
Figs. 4.68a and 4.68b the distances between bearing plates
were kept to no less than 1.5 times the web depth in order to
avoid the two-flange action.
The developments of the 1996 and earlier AISI design
requirements for web crippling and the research background
information were summarized by Yu in the third edition of
the book.1.354
In Canada, the study of web crippling of cold-formed steel
members was initiated at the University of Waterloo by Wing
and Schuster in 1981.4.88,4.89,4.98 In the 1990s, Prabakaran
and Schuster developed the following unified web crippling equation for the CSA standard S136-94 with variable
coefficients1.177,4.177,4.238 :
√
𝑃n = Ct2 𝐹y (sin 𝜃)(1 − 𝐶𝑅 𝑅∗ )
√
√
× (1 + 𝐶𝑁 𝑁 ∗ )(1 − 𝐶𝐻 𝐻 ∗ )
(4.196)
196
4
FLEXURAL MEMBERS
Figure 4.66 Buckling coefficient k for simply supported plates subjected to two opposite locally
distributed edge forces.1.216,4.75 Reproduced with permission from Walker, A. C. (Ed.), Design and
Analysis of Cold-Formed Sections, International Textbook Co., Glasgow and London, 1975.
Figure 4.67 Buckling coefficient k for simply supported plates subjected to one locally distributed
edge force.4.76
in which C is the web crippling coefficient, 𝐶𝐻 is the web
slenderness coefficient, 𝐶𝑁 is the bearing length coefficient,
𝐶𝑅 is inside bend radius coefficient, 𝐻* = ℎ∕𝑡, 𝑁* = 𝑛∕𝑡,
and 𝑅* = 𝑟∕𝑡. All coefficients were listed in three separate
tables for built-up sections (I-beams), shapes having single
webs, and deck sections (multiple webs). Subsequently,
Beshara and Schuster developed additional web crippling
coefficients based on the above-mentioned research data and
the later investigations conducted at (a) the University of
Missouri-Rolla by Santaputra,4.233 Santaputra, Parks, and
Yu,4.97,4.104 Langan, LaBoube, and Yu,3.185,3.198 and Wu,
Yu, and Laboube4.183,4.192 ; (b) the University of Waterloo
by Gerges,4.239 Gerges and Schuster,4.178 Beshara,4.240 and
Beshara and Schuster4.241–4.243 ; and (c) the University of
Sydney by Young and Hancock.4.182
In 2001, the North American specification adopted the
unified web crippling equation in for determining the
nominal web crippling strengths of the following five
different section types:
1.
2.
3.
4.
5.
Built-up sections
Single-web channel and C-sections
Single-web Z-sections
Single hat sections
Multiweb deck sections
The limitations of h/t, N/t, N/h, R/t, and 𝜃 for using
these tabulated coefficients are given in all five tables. Also
included in these tables are safety factors for ASD and resistance factors for LRFD and LSD.
197
DESIGN OF BEAM WEBS
Figure 4.68 Loading conditions for web crippling tests: (a) EOF loading; (b) IOF loading; (c) ETF
loading; (d) ITF loading.
The 2001 design provisions were retained in the 2007
edition of the North American Specification, in which an
alternative method was added for end-one-flange loading
condition on a C- or Z-section with an overhang on one side
of the bearing plate. This method was based on the research
findings of Holesapple and Laboube.4.235 Based on extensive testing, more web crippling coefficients were developed by Wallace for both the unfastened and fastened cases
of end-one-flange loading for multiweb deck sections.4.246
These design provisions for web crippling strengths of webs
without holes are included in Section G5 of the 2016 edition
of the North American Specification.
4.3.6.2 Web Cripping Strength of C-Section Webs
with Holes Since 1990, the structural behavior of perforated web elements of flexural members subjected to web
crippling and a combination of web crippling and bending
has been investigated at the University on Missouri-Rolla
by Langan et al.,3.185,3.198 Uphoff,3.199 Deshmukh,3.200 and
Laboube, Yu, Deshmukh, and Uphoff.3.193 It was found that
the reduction in web crippling strength is affected primarily
by the size of the hole as reflected in the dh /h ratio and the
location of the hole x/h ratio. New reduction equations for
the web crippling strength of C-section webs with holes
were developed for inclusion in Supplement No. 1 to the
1996 edition of the AISI Specification.1.333–1.335 The same
design equations were retained in Section G6 of the North
American Specification. It should be noted that the design
equations for determining the reduction factor can only be
applicable for the C-sections with web holes subjected to
the limitations listed in Section G6 of the Specification.
4.3.6.3 North American Design Criteria for Web Cripping
The following excerpts are adapted from Sections G5 and
G6 of the 2016 edition of the North American Specification, which provide the design equations for computing
the available web crippling strengths of cold-formed steel
members, and the treatments of C-Section members with
holes. The loading cases (EOF, IOF, ETF, and ITF) are shown
in Fig. 4.69. Examples 4.24–4.26 illustrate the applications
of design equations.
G5 Web Crippling Strength of Webs Without Holes
The nominal web crippling strength [resistance], Pn , shall be
determined in accordance with Eq. (4.197) or Eq. (4.198),
as applicable. The safety factors and resistance factors in
Tables 4.8 to 4.12 shall be used to determine the allowable
strength or design strength [factored resistance] in accordance
with the applicable design method in Specification Section
B3.2.1, B3.2.2, or B3.2.3.
(
𝑃𝑛 = Ct 𝐹𝑦 sin 𝜃
2
√
1 − 𝐶𝑅
√ )
ℎ
1 − 𝐶ℎ
𝑡
𝑅
𝑡
)(
1 + 𝐶𝑁
√
𝑁
𝑡
)
(
×
(4.197)
198
4
FLEXURAL MEMBERS
Figure 4.69
Application of AISI loading cases.
where 𝑃n = Nominal web crippling strength [resistance]
𝐶 = Coefficient from Table 4.8, 4.9, 4.10, 4.11,
or 4.12
𝑡 = Web thickness
𝐹y = Design yield stress as determined in accordance
with Specification Section A3.3.1
𝜃 = Angle between plane of web and plane of bearing
surface, 45∘ ≤ θ ≤ 90∘
𝐶𝑅 = Inside bend radius coefficient from Table 4.8, 4.9,
4.10, 4.11, or 4.12
𝑅 = Inside bend radius
𝐶𝑁 = Bearing length coefficient from Table 4.8, 4.9,
4.10, 4.11, or 4.12
𝑁 = Bearing length (3/4 in. (19 mm) minimum)
𝐶ℎ = Web slenderness coefficient from Table 4.8, 4.9,
4.10, 4.11, or 4.12
ℎ = Flat dimension of web measured in plane of web
Alternatively, for an end one-flange loading condition on a
C- or Z-section, the nominal web crippling strength [resistance],
𝑃𝑛c , with an overhang on one side, is permitted to be calculated
as follows, except that 𝑃nc shall not be larger than the interior
one-flange loading condition:
𝑃nc = 𝛼𝑃n
(4.198)
where 𝑃nc = Nominal web crippling strength [resistance] of
C- and Z- sections with overhang(s)
𝛼=
1.34(𝐿o ∕ℎ)0.26
≥ 1.0
0.009(ℎ∕𝑡) + 0.3
(4.199)
where 𝐿o = Overhang length measured from edge of bearing
to the end of the member
𝑃n = Nominal web crippling strength [resistance] with
end one-flange loading as calculated by
Eq. (4.197) and Tables 4.8 to 4.12
199
DESIGN OF BEAM WEBS
θ = 90∘ . In Table 4.9, for interior two-flange loading or reaction
of members having flanges fastened to the support, the distance
from the edge of the bearing to the end of the member shall
be extended at least 2.5h. For unfastened cases, the distance
from the edge of the bearing to the end of the member shall be
extended at least 1.5h.
Table 4.10 shall apply to single web Z-section members
where ℎ∕𝑡 ≤ 200, N∕t ≤ 210, N∕h ≤ 2.0, and θ = 90∘ . In
Table 4.10, for interior two-flange loading or reaction of
members having flanges fastened to the support, the distance
from the edge of the bearing to the end of the member shall
be extended at least 2.5h; for unfastened cases, the distance
from the edge of the bearing to the end of the member shall be
extended at least 1.5h.
Table 4.11 shall apply to single hat section members where
ℎ∕𝑡 ≤ 200, 𝑁∕𝑡 ≤ 200, 𝑁∕ℎ ≤ 2, and θ = 90∘ .
Table 4.12 shall apply to multi-web section members where
ℎ∕𝑡 ≤ 200, 𝑁∕𝑡 ≤ 210, 𝑁∕ℎ ≤ 3, and 45∘ ≤ θ ≤ 90∘ .
Eq. (4.198) shall be limited to 0.5 ≤ 𝐿o ∕ℎ ≤ 1.5 and ℎ∕𝑡 ≤
154. For 𝐿o ∕ℎ or h/t outside these limits, 𝛼 = 1.
Webs of members in bending for which h/t is greater than 200
shall be provided with means of transmitting concentrated loads
or reactions directly into the web(s).
𝑃n and 𝑃nc shall represent the nominal strengths [resistances]
for load or reaction for one solid web connecting top and bottom
flanges. For hat, multi-web sections and C- or Z-sections, 𝑃n or
𝑃nc shall be the nominal strength [resistance] for a single web,
and the total nominal strength [resistance] shall be computed by
multiplying 𝑃n or 𝑃nc by the number of webs at the considered
cross-section.
One-flange loading or reaction shall be defined as the condition where the clear distance between the bearing edges of
adjacent opposite concentrated loads or reactions is equal to or
greater than 1.5h.
Two-flange loading or reaction shall be defined as the
condition where the clear distance between the bearing edges
of adjacent opposite concentrated loads or reactions is less
than 1.5h.
End loading or reaction shall be defined as the condition
where the distance from the edge of the bearing to the end of
the member is equal to or less than 1.5h.
Interior loading or reaction shall be defined as the condition
where the distance from the edge of the bearing to the end of the
member is greater than 1.5h, except as otherwise noted herein.
Table 4.8 shall apply to I-beams made from two channels
connected back-to-back where ℎ∕𝑡 ≤ 200, 𝑁∕𝑡 ≤ 210, N∕h ≤
1.0, and θ = 90∘ . See Section G5 of the 2016 edition of the
Commentary1.431 for further explanation.
Table 4.9 shall apply to single web channel and C-section
members where ℎ∕𝑡 ≤ 200, 𝑁∕𝑡 ≤ 210, 𝑁∕ℎ ≤ 2.0, and
Table 4.8
G6 Web Crippling Strength of C-Section Webs With
Holes
Where a web hole is within the bearing length, a bearing stiffener
shall be used.
For beam webs with holes, the available web crippling
strength [factored resistance] shall be calculated in accordance
with Specification Section G5, multiplied by the reduction
factor, Rc , given in this section.
The provisions of this section shall apply within the following
limits:
(a) 𝑑h ∕ℎ ≤ 0.7,
(b) ℎ∕𝑡 ≤ 200,
Safety Factors, Resistance Factors, and Coefficients for Built-Up Sections per Web
USA and
Mexico
Support and Flange Conditions
Load Cases
Fastened to Stiffened or Partially One-Flange
Support
Stiffened Flanges
Loading or
Reaction
Stiffened or Partially One-Flange
Stiffened Flanges
Loading or
Reaction
Unfastened
Two-Flange
Loading or
Reaction
Unstiffened Flanges One-Flange
Loading or
Reaction
C
𝐶𝑅
𝐶𝑁
𝐶ℎ
ASD LRFD Canada
Ωw
𝜙w
LSD 𝜙w
Limits
End
10
0.14 0.28 0.001
Interior 20.5 0.17 0.11 0.001
2.00
1.75
0.75
0.85
0.60
0.75
𝑅∕t ≤ 5
𝑅∕t ≤ 5
End
10
0.14 0.28 0.001
Interior 20.5 0.17 0.11 0.001
2.00
1.75
0.75
0.85
0.60
0.75
𝑅∕t ≤ 5
𝑅∕t ≤ 3
End
15.5 0.09 0.08
Interior 36
0.14 0.08
0.04
0.04
2.00
2.00
0.75
0.75
0.65
0.65
𝑅∕t ≤ 3
End
10
0.14 0.28 0.001
Interior 20.5 0.17 0.11 0.001
2.00
1.75
0.75
0.85
0.60
0.75
𝑅∕t ≤ 5
𝑅∕t ≤ 3
200
4
FLEXURAL MEMBERS
Table 4.9 Safety Factors, Resistance Factors, and Coefficients for Single Web Channel and C-Sections
USA and
Mexico
Support and Flange Conditions
Load Cases
C
𝐶𝑅
𝐶𝑁
𝐶ℎ
ASD LRFD Canada
Ωw
𝜙w LSD 𝜙w
Limits
1.75
1.65
0.85
0.90
0.75
0.80
𝑅∕𝑡 ≤ 9
𝑅∕𝑡 ≤ 5
End
7.5 0.08 0.12 0.048 1.75
Interior 20 0.10 0.08 0.031 1.75
0.85
0.85
0.75
0.75
End
4 0.14 0.35 0.02
Interior 13 0.23 0.14 0.01
1.85
1.65
0.80
0.90
0.70
0.80
𝑅∕𝑡 ≤ 12
𝑅∕𝑡 ≤ 12𝑑 1 ≥
4.5 in.
(110 mm)
𝑅∕𝑡 ≤ 5
End
13 0.32 0.05 0.04 1.65
Interior 24 0.52 0.15 0.001 1.90
0.90
0.80
0.80
0.65
𝑅∕𝑡 ≤ 3
End
4 0.40 0.60 0.03
Interior 13 0.32 0.10 0.01
1.80
1.80
0.85
0.85
0.70
0.70
𝑅∕𝑡 ≤ 2
𝑅∕𝑡 ≤ 1
End
2 0.11 0.37 0.01
Interior 13 0.47 0.25 0.04
2.00
1.90
0.75
0.80
0.65
0.65
𝑅∕𝑡 ≤ 1
Fastened to Stiffened or Partially One-Flange
Support
Stiffened Flanges
Loading or
Reaction
Two-Flange
Loading or
Reaction
End
4 0.14 0.35 0.02
Interior 13 0.23 0.14 0.01
Unfastened Stiffened or Partially One-Flange
Stiffened Flanges
Loading or
Reaction
Two-Flange
Loading or
Reaction
Unstiffened Flanges One-Flange
Loading or
Reaction
Two-Flange
Loading or
Reaction
Notes:1 d = Out-to-out depth of section in the plane of the web.
(c)
(d)
(e)
(f)
(g)
Hole centered at mid-depth of web,
Clear distance between holes ≥ 18 in. (457 mm),
Distance between end of member and edge of hole ≥ d,
Noncircular holes, corner radii ≥ 2t,
Noncircular holes, 𝑑h ≤ 2.5 in. (63.5 mm) and Lh ≤ 4.5 in.
(114 mm),
(h) Circular holes, diameters ≤ 6 in. (152 mm), and
(i) 𝑑h > 9∕16 in. (14.3 mm).
where 𝑑h = Depth of web hole
ℎ = Depth of flat portion of web measured along plane
of web
𝑡 = Web thickness
𝑑 = Depth of cross-section
𝐿h = Length of web hole
For end one-flange reaction (Eq. (4.197) with Table 4.9)
where a web hole is not within the bearing length, the reduction
factor, Rc , shall be calculated as follows:
𝑅c = 1.01 − 0.325dh ∕h + 0.083x∕h ≤ 1.0
𝑁 ≥ 1 in. (25.4 mm)
(4.200)
For interior one-flange reaction (Eq. (4.197) with Table 4.9)
where any portion of a web hole is not within the bearing length,
the reduction factor, Rc , shall be calculated as follows:
𝑅c = 0.90 − 0.047𝑑h ∕h + 0.053x∕h ≤ 1.0
(4.201)
𝑁 ≥ 3 in. (76.2 mm)
where 𝑥 = Nearest distance between web hole and edge of
bearing
𝑁 = Bearing length
In addition to the research work mentioned in Sections
4.3.6.1 and 4.3.6.2, the web crippling strengths of
cold-formed steel beams have been studied by numerous
researchers over the past decade. For more information on
web crippling, the reader is referred to Refs. 4.105–4.107,
4.179–4.181, 4.184, and 4.248–4.273.
4.3.7
Combined Web Crippling and Bending
In Section 4.3.6, the web crippling limit state of cold-formed
steel beams was discussed. The design formulas were used
to prevent any localized failure of webs resulting from the
bearing pressure due to reactions or concentrated loads
without consideration of the effect of other stresses.
DESIGN OF BEAM WEBS
Table 4.10
201
Safety Factors, Resistance Factors, and Coefficients for Single Web Z-Sections
USA and
Mexico
Support and Flange Conditions
Load Cases
Fastened to Stiffened or Partially One-Flange
Support
Stiffened Flanges
Loading or
Reaction
Two-Flange
Loading or
Reaction
Unfastened Stiffened or Partially One-Flange
Stiffened Flanges
Loading or
Reaction
Two-Flange
Loading or
Reaction
Unstiffened Flanges One-Flange
Loading or
Reaction
Two-Flange
Loading or
Reaction
Table 4.11
C
𝐶𝑅
𝐶𝑁
ASD LRFD Canada
Ωw
𝜙w
LSD 𝜙w
𝐶ℎ
Limits
0.02
0.01
1.75
1.65
0.85
0.90
0.75
0.80
𝑅∕𝑡 ≤ 9
𝑅∕𝑡 ≤ 5.5
End
9 0.05 0.16 0.052
Interior 24 0.07 0.07 0.04
1.75
1.85
0.85
0.80
0.75
0.70
𝑅∕𝑡 ≤ 12
𝑅∕𝑡 ≤ 12
End
5 0.09 0.02 0.001
Interior 13 0.23 0.14 0.01
1.80
1.65
0.85
0.90
0.75
0.80
𝑅∕𝑡 ≤ 5
End
13 0.32 0.05 0.04
Interior 24 0.52 0.15 0.001
1.65
1.90
0.90
0.80
0.80
0.65
𝑅∕𝑡 ≤ 3
End
4 0.40 0.60
Interior 13 0.32 0.10
0.03
0.01
1.80
1.80
0.85
0.85
0.70
0.70
𝑅∕𝑡 ≤ 2
𝑅∕𝑡 ≤ 1
End
2 0.11 0.37
Interior 13 0.47 0.25
0.01
0.04
2.00
1.90
0.75
0.80
0.65
0.65
𝑅∕𝑡 ≤ 1
End
4 0.14 0.35
Interior 13 0.23 0.14
Safety Factors, Resistance Factors, and Coefficients for Single Hat Sections per Web
USA and
Mexico
Support
Conditions
Fastened to
Support
Unfastened
Load Cases
One-Flange Loading
or Reaction
Two-Flange Loading
or Reaction
One-Flange Loading
or Reaction
End
Interior
End
Interior
End
Interior
C
𝐶𝑅
𝐶𝑁
𝐶ℎ
ASD
Ωw
LRFD
𝜙w
Canada
LSD 𝜙w
4
17
9
10
4
17
0.25
0.13
0.10
0.14
0.25
0.13
0.68
0.13
0.07
0.22
0.68
0.13
0.04
0.04
0.03
0.02
0.04
0.04
2.00
1.80
1.75
1.80
2.00
1.80
0.75
0.85
0.85
0.85
0.75
0.85
0.65
0.70
0.75
0.75
0.65
0.70
In practical applications, a high bending moment may
occur at the location of the applied concentrated load in
simple span beams. For continuous beams, the reactions
at interior supports may be combined with high bending
moments and/or high shear. Under these conditions, the web
crippling strength as determined in Section 4.3.6 may be
reduced significantly due to the effect of bending moment.
The interaction relationship for the combination of bearing
pressure and bending stress has been studied by numerous
Limits
𝑅∕𝑡 ≤ 5
𝑅∕𝑡 ≤ 10
𝑅∕𝑡 ≤ 10
𝑅∕𝑡 ≤ 5
𝑅∕𝑡 ≤ 10
researchers.4.58,4.80,4.81,4.86–4.88,4.90–4.92,4.237,4.244,4.245,4.249,
4.252,4.265,4.266,4.271 Based on the results of beam tests with
combined web crippling and bending, interaction formulas
have been developed for use in several design specifications.
4.3.7.1 Shapes Having Single and Multiple Unreinforced Webs For the AISI Specification, the interaction of
combined bending and web crippling has been recognized
since 1980. The interaction equation used in the 1980,
202
4
Table 4.12
FLEXURAL MEMBERS
Safety Factors, Resistance Factors, and Coefficients for Multi-Web Deck Sections per Web
USA and
Mexico
Support
Conditions
Fastened to
Support
Unfastened
Load Cases
One-Flange Loading
or Reaction
Two-Flange Loading
or Reaction
One-Flange Loading
or Reaction
Two-Flange Loading
or Reaction
End
Interior
End
Interior
End
Interior
End
Interior
C
𝐶𝑅
𝐶𝑁
𝐶ℎ
ASD
Ωw
LRFD
𝜙w
Canada
LSD 𝜙w
4
8
9
10
3
8
6
17
0.04
0.10
0.12
0.11
0.04
0.10
0.16
0.10
0.25
0.17
0.14
0.21
0.29
0.17
0.15
0.10
0.025
0.004
0.040
0.020
0.028
0.004
0.050
0.046
1.70
1.75
1.80
1.75
2.45
1.75
1.65
1.65
0.90
0.85
0.85
0.85
0.60
0.85
0.90
0.90
0.80
0.75
0.70
0.75
0.50
0.75
0.80
0.80
Limits
𝑅∕𝑡 ≤ 20
𝑅∕𝑡 ≤ 10
𝑅∕𝑡 ≤ 20
𝑅∕𝑡 ≤ 5
Note: Multi-web deck sections are considered unfastened for any support fastener spacing greater than 18 in. (460 mm).
1986, and 1996 editions of the Specification was developed by Hetrarul and Yu in 1978 on the basis of the test
results for combined web crippling and bending achieved at
Cornell University,4.85 the University of Missouri-Rolla,4.58
and United States Steel Research Laboratory.4.92 See
Figs. 4.70, 4.71, and 4.72. The research findings and the
development of design equations were summarized by Yu in
the third edition of the book.1.354
Due to the adoption of the unified web crippling equation
by the North American Specification in 2001, the interaction equations for the combination of bending and web crippling were reevaluated by LaBoube, Schuster, and Wallace
in 2002 using Section 4.3.6.3 for determining the web crippling strength.4.244,4.245 The experimental data were based on
research studies conducted by Winter and Pian,4.85 Hetrarul
and Yu,4.58,4.94 Yu,1.354,4.86 and Young and Hancock.4.249 The
safety factors for ASD and the resistance factors for LRFD
and LSD were selected from the calibration of test data using
the procedures given in Section K2 of the North American
Specification.
Consequently, the following design equations were
developed for the combined bending and web crippling of
cold-formed steel beams in the Specification.
a. For shapes having single unreinforced webs,
( ) (
)
1.33
𝑃
𝑀
ASD method∶ 0.91
+
≤
𝑃n
𝑀𝑛𝓁𝑜
Ω
(4.202a)
( ) (
)
𝑃
𝑀
LRFD and LSD methods∶0.91
+
𝑃n
𝑀𝑛𝓁𝑜
≤ 1.33𝜙
(4.202b)
b. For shapes having multiple unreinforced webs such as
I-sections made of two webs or similar sections that
provide a high degree of restraint against rotation of the
web,
)
( ) (
1.46
𝑃
𝑀
+
≤
(4.203a)
0.88
𝑃n
𝑀𝑛𝓁𝑜
Ω
)
( ) (
𝑃
𝑀
+
≤ 1.46𝜙
(4.203b)
0.88
𝑃n
𝑀𝑛𝓁𝑜
where 𝑃 = required strength [factored resistance] for
concentrated load or reaction in the presence
of bending moment determined in
accordance with ASD, LRFD, or LSD load
combinations
𝑃n = nominal strength for concentrated or
reaction in absence of bending moment
determined in accordance with
Section 4.3.6
𝑀 = required flexural strength at, or immediately
adjacent to, the point of application of the
concentrated load or reaction, 𝑃
𝑀𝑛𝓁𝑜 = nominal flexural strength [resistance] about
the x axis in the absence of axial load,
determined in accordance with Section 4.2.3
with 𝐹n = 𝐹y or 𝑀ne = 𝑀y
Ω = safety factor for combined bending and web
crippling = 1.70
𝜙 = resistance factor for combined bending and
web crippling
= 0.90 (LRFD)
= 0.80 (LSD)
DESIGN OF BEAM WEBS
Figure 4.70 Graphic presentation for web crippling (IOF loading) and combined web crippling and
bending for specimens having single unreinforced webs.4.58
Figure 4.71
Interaction between
√ web crippling and bending for I-beam specimens having unrein-
forced webs [when ℎ∕𝑡 ≤ 2.33∕
𝐸y ∕𝐸 and 𝜌 = 1].4.58
203
204
4
FLEXURAL MEMBERS
Interaction between
having unre√ web crippling and bending for I-beam specimens
√
inforced webs [when ℎ∕𝑡 ≤ 2. 𝐹y ∕𝐸 combined with 𝜌 < 1, and 2.33∕ 𝐹y ∕𝐸 < ℎ∕𝑡 ≤ 200
Figure 4.72
combined with any value of w/t] (IOF loading).4.58
Equations (4.202) and (4.203) are shown graphically in
Fig. 4.73.
Since the safety factor for bending (Section 4.2.3), Ω𝑏 , and
the safety factor for web crippling (Section 4.3.6), Ω𝑤 , are
Eq. (4.203a)
Eq. (4.202a)
Figure 4.73 ASD interaction equations for combined web crippling and bending (shapes having single and multiple unreinforced
webs).
not the same as the safety factor for combined bending and
web crippling, the required strength should not exceed the
available strength, i.e., 𝑀 ≤ 𝑀𝑎𝓁𝑜 and 𝑃 ≤ 𝑃a .
Example 4.24 below illustrates the applications of the
design equations.
In the North American Specification, an exception clause
is included for the interior supports of continuous spans
using the decks and beams as shown in Fig. 4.74. This is
because the results of continuous beam tests of steel decks4.86
and several independent studies of individual manufacturers
indicate that for these types of members the postbuckling
behavior of beam webs at interior supports differs from the
type of failure mode occurring under concentrated loads on
single-span beams. This postbuckling strength enables the
member to redistribute the moments in continuous beams.
For this reason, Eq. (4.202) may be found to be conservative for determining the load-carrying capacity of continuous spans on the basis of the conventional elastic design.
If localized distortions of webs over interior supports as
shown in Fig. 4.75 are permitted, the inelastic flexural reserve
capacity due to partial plastification of beam cross section
and moment redistribution may be used, as discussed in
Section 4.3.8.
4.3.7.2 Nested Z-Shapes In the 1996 edition of the AISI
Specification, design provisions were added for nested
DESIGN OF BEAM WEBS
Figure 4.74
205
Sections used for exception clauses1.346 : (a) decks; (b) beams.
Figure 4.75 Two-span continuous beam tests using uniform loading.
Z-shapes. The same design equations are retained in the
North American Specification. When two nested Z-shapes
are subject to a combination of bending and concentrated
load or reaction, these members shall be designed to meet
the following requirements of Section H3 of the 2016 edition
of the North American Specification.
)
( ) (
1.65
𝑃
𝑀
+
≤
(4.204a)
ASD method∶ 0.86
𝑃n
𝑀𝑛𝓁𝑜
Ω
)
( ) (
𝑃
𝑀
+
LRFD and LSD methods∶ 0.86
𝑃n
𝑀𝑛𝓁𝑜
≤ 1.65𝜙
(4.204b)
where 𝑀𝑛𝓁𝑜 is the nominal flexural strength for two nested
Z-sections determined in accordance with Section 4.2.3 with
𝐹n = 𝐹y or 𝑀ne = 𝑀y and Ω = 1.70, and 𝜙 = 0.90 (LRFD)
and 0.75 (LSD).
The above two equations were originally derived by
LaBoube, Nunnery, and Hodges from the experimental
work summarized in Ref.4.176 and reevaluated in 2003 using
revised web crippling equations. These equations are valid
only for the shapes that meet the following limits: ℎ∕𝑡 ≤ 150,
𝑁∕𝑡 ≤ 140, 𝐹y ≤ 70 ksi (483 MPa, or 4920 kg/cm2 ), and
𝑅∕𝑡 ≤ 5.5. In addition, the following conditions shall also
be satisfied: (1) the ends of each section shall be connected
to another section by a minimum of two 1∕2-in.- (12.7-mm-)
diameter A307 bolts through the web, (2) the combined
section is connected to the support by a minimum of
two 1∕2-in.- (12.7-mm-) diameter A307 bolts through the
flanges, (3) the webs of the two sections are in contact,
206
4
FLEXURAL MEMBERS
where ℎ∕𝑡 = [10 − 2(0.105 + 0.1875)]∕0.105
= 89.67 < 200
𝑁∕𝑡 = 3.5∕0.105 = 33.33 < 210
𝑁∕ℎ = 3.5∕[10 − 2(0.105 + 0.1875)]
= 0.37 < 2.0
𝑅∕𝑡 = 0.1875∕0.105 = 1.786 < 9
θ = 90∘
Figure 4.76
Example 4.24.
and (4) the ratio of the thicker to thinner part does not
exceed 1.3.
Example 4.24 For the channel section shown in Fig. 4.76
to be used as a simply supported beam:
1. Determine the allowable end reaction 𝑃max to prevent
web crippling by considering it as a one-flange loading
condition with 𝑁 = 3.5 in.
2. Determine the allowable interior load to prevent web
crippling by considering the load as a one-flange
loading condition with 𝑁 = 5 in. and assuming that
the applied bending moment M at the location of the
interior load is less than 42% of the allowable bending
moment 𝑀𝑛𝓁𝑜 ∕Ω permitted if bending stress only
exists.
3. Same as item 2, except that the applied bending
moment M at the location of the interior load is equal
to the allowable bending moment 𝑀𝑛𝓁𝑜 ∕Ω.
Use 𝐹y = 50 ksi and the ASD method. Assume that the
bottom flange of the beam is fastened to the support.
SOLUTION
From Table 4.9 for a simple web channel, 𝐶 = 4,
𝐶𝑅 = 0.14, 𝐶𝑁 = 0.35, 𝐶ℎ = 0.02, and Ω𝑤 = 1.75.
Therefore,
√
𝑃n = 4(0.105)2 (50)(sin 90∘ )[1 − 0.14 1.786]
√
√
× [1 + 0.35 33.33][1 − 0.02 89.67]
= 4.39 kips∕web
The allowable end reaction is then
𝑃
4.39
𝑃a = n =
= 2.51 kips
Ωw
1.75
2. Allowable Interior Load for Web Crippling (M <
𝟎.𝟒𝟐Mn𝓁o ∕Ω). According to Eq. (4.197), the nominal
web crippling strength for interior load without considering the effect of the bending moment is
[
√ ][
√ ]
𝑅
𝑁
1 + 𝐶𝑁
𝑃n = Ct 𝐹y sin 𝜃 1 − 𝐶𝑅
𝑡
𝑡
]
[
√
ℎ
× 1 − 𝐶ℎ
𝑡
2
From Table 4.9 for interior one-flange loading, 𝐶 = 13,
𝐶𝑅 = 0.23, 𝐶𝑁 = 0.14, 𝐶ℎ = 0.01, and Ωw = 1.65. Therefore,
√
𝑃n = 13(0.105)2 (50)(sin 90∘ )[1 − 0.23 1.786]
√
√
× [1 + 0.14 33.33][1 − 0.01 89.67]
= 8.13 kips
1. Allowable End Reaction for Web Crippling
(One-Flange Loading). Based on Eq.(4.197), the
nominal web crippling strength can be computed as
follows:
[
√ ][
√ ]
𝑅
𝑁
1 + 𝐶𝑁
𝑃n = Ct 𝐹y sin 𝜃 1 − 𝐶𝑅
𝑡
𝑡
[
√ ]
ℎ
× 1 − 𝐶ℎ
𝑡
2
The allowable interior load for web crippling is
𝑃a = 𝑃n ∕Ωw = 8.13∕1.65 = 4.93 kips. Because the applied
moment M at the location of the interior load is less than
42% of the allowable bending moment 𝑀𝑛𝓁𝑜 ∕Ω permitted
if bending stress only exists, Eq. (4.202a) is not applicable.
See Fig. 4.73. For this case, the computed allowable interior
load of 8.13 kips should be used without any reduction due
to combined web crippling and bending.
3. Allowable Interior Load for Web Crippling
(M = Mn𝓁o ∕𝛀 = Mn𝓁o ∕1.70). From item 2, the
DESIGN OF BEAM WEBS
computed nominal interior load for web crippling is
8.13 kips. Because 𝑀 = 𝑀𝑛𝓁𝑜 ∕1.70, which is less
than 𝑀 = 𝑀𝑛𝓁𝑜 ∕1.67 for bending, the beam is to
be designed for the allowable bending moment for
combined bending and web crippling. The applied
concentrated load should be reduced according to Eq.
(4.202a), as shown in Fig. 4.73, in order to account for
the effect of the bending moment. Accordingly,
)
(
Ω𝑃
Ω𝑃
= 0.363
+ 1.0 = 1.33 or
0.91
𝑃n
𝑃n
For this case, the allowable interior load for web crippling is
0.363𝑃n
𝑃a =
1.70
= 1.74 kips <
(
𝑃n
= 4.93kips
Ωw
)
OK
Example 4.25 Use the ASD method to determine the
allowable end reaction for the single hat section used in
Example 4.6 to prevent web crippling. Assume that the
length of the bearing is 3.5 in. and 𝐹y = 50 ksi. Also use
the LRFD method to determine the design web crippling
strength. Assume that the hat section is fastened to the
support.
SOLUTION
A. ASD Method
ℎ 10 − 2(0.105 + 0.1875)
=
= 89.67 < 200
𝑡
0.105
𝑁
3.5
=
= 33.33 < 200
𝑡
0.105
𝑁
3.5
=
= 0.37 < 2.0
ℎ
10 − 2(0.105 + 0.1875)
𝑅 0.1875
=
= 1.786 < 5
𝑡
0.105
𝜃 = 90∘
Equation (4.197) can be used for the design of this hat
section having single unreinforced webs.
[
√ ][
√ ]
𝑅
𝑁
2
1 + 𝐶𝑁
𝑃n = Ct 𝐹y sin 𝜃 1 − 𝐶𝑅
𝑡
𝑡
[
√ ]
ℎ
× 1 − 𝐶ℎ
𝑡
From Table 4.11 for one-flange loading, 𝐶 = 4, 𝐶𝑅 =
0.25, 𝐶𝑁 = 0.68, 𝐶ℎ = 0.04, Ωw = 2.00, and 𝜙𝑤 = 0.75.
207
Therefore,
√
𝑃n = 4(0.105)2 (50)(sin 90∘ )[1 − 0.25 1.786]
√
√
× [1 + 0.68 33.33][1 − 0.04 89.67]
= 4.49 kips∕web
The allowable end reaction, per web, is
𝑃
4.49
𝑃a = n =
= 2.25 kips∕web
Ωw
2.00
For two webs, the total allowable end reaction is
2𝑃a = 2(2.25) = 4.50 kips
B. LRFD Method
The nominal web crippling strength for the LRFD method
is the same as that computed for the ASD method. From
item A above, the nominal web crippling strength for the end
reaction is
𝑃n = 4.49 kips∕web
The design strength to prevent web crippling of the hat
section having two webs is
2𝜙w 𝑃n = 2(0.75)(4.49) = 6.74 kips
Example 4.26 Use the ASD and LRFD methods to determine the allowable end reaction for the I-section used in
Example 4.4 to prevent web crippling. Assume that the
length of the bearing is 3.5 in. and 𝐹y = 50 ksi. The dead
load–live load ratio is assumed to be 15 . Assume that the
I-section is not fastened to the support.
SOLUTION
A. ASD Method
As the first step, check the AISI limits on h/t, N/t, N/h, and
R/t for using Eq. (4.197):
ℎ 8 − 2(0.135 + 0.1875)
=
= 54.48 < 200
𝑡
0.135
𝑁
3.5
=
= 25.93 < 210
𝑡
0.135
𝑁
3.5
=
= 0.48 < 1.0
ℎ
7.355
𝑅 0.1875
=
= 1.39 < 5
𝑡
0.135
Because the above ratios are within the North American
Specification provisions’ limits, use Eq. (4.197) to determine
the allowable end reaction for web crippling:
[
√ ][
√ ]
𝑅
𝑁
1 + 𝐶𝑁
𝑃n = Ct2 𝐹y sin 𝜃 1 − 𝐶𝑅
𝑡
𝑡
[
]
√
ℎ
× 1 − 𝐶ℎ
𝑡
208
4
FLEXURAL MEMBERS
From Table 4.8 for unfastened end one-flange loading,
𝐶 = 10, 𝐶𝑅 = 0.14, 𝐶𝑁 = 0.28, 𝐶ℎ = 0.001, Ωw = 2.00,
and 𝜙w = 0.75. Therefore,
√
𝑃n = 10(0.105)2 (50)(sin 90∘ )[1 − 0.14 1.39]
√
√
× [1 + 0.28 25.93][1 − 0.001 54.48]
= 18.32 kips
For the I-section having double webs, the total allowable
end reaction is
2𝑃
2(18.32)
𝑃a = n =
= 18.32 kips
Ωw
2.00
B. LRFD Method
Use the same equation employed in item A for the ASD
method, the nominal web crippling strength for the I-section
having double webs is
𝑃n = 2(18.32) = 36.64 kips
The design web crippling strength is
𝜙w 𝑃n = 0.75(36.64) = 27.48 kips
Based on the load combination of Eq. (1.5a), the required
strength is
𝑃u = 1.4𝑃D
Similarly, based on the load combination of Eq. (1.5b), the
required strength is
𝑃u = 1.2𝑃D + 1.6𝑃L = 1.2𝑃D + 1.6(5𝑃D )
= 9.2𝑃D
Controls
where 𝑃D = end reaction due to dead load
𝑃L = end reaction due to live load
Using 𝑃u = 𝜙w 𝑃n , the end reaction due to dead load can
be computed as follows:
9.2𝑃D = 27.48 kips
𝑃D = 2.99 kips
𝑃L = 5𝑃D = 14.95 kips
The allowable end reaction to prevent web crippling on the
basis of the LRFD method is
𝑃 = 𝑃D + 𝑃L = 2.99 + 14.95 = 17.94 kips
It can be seen that for the given I-section the LRFD method
permits a slightly smaller end reaction than the ASD method.
The difference is about 2%.
4.3.8
Moment Redistribution of Continuous Beams
In Section 4.2, the increase of bending moment capacity due
to the plastification of the cross section are discussed. Studies
of continuous beams and steel decks conducted by Yener
and Pekoz,4.2 Unger,4.5 Ryan,4.9 and Yu4.86 indicate that
the inelastic flexural reserve capacity of continuous beams
due to moment redistribution may be used in the design of
cold-formed steel sections provided that the following conditions are met:
1. The member is not subject to twisting, lateral, torsional,
or torsional–flexural buckling.
2. The effect of cold forming is not included in determining the yield stress 𝐹y .
3. Localized distortions caused by web crippling over
supports are permitted.
4. Reduction of the negative moment capacity over interior support due to inelastic rotation is considered.
5. Unreinforced flat webs of shapes subjected to a combination of bending and reaction are designed to meet
the requirements of Section 4.3.7.1, in which 𝑃n is
nominal web crippling load computed from Section
4.3.6 and 𝑀𝑛𝓁𝑜 is the nominal bending moment defined
in Section 4.2.3.1.2. The values of P and M should not
exceed the available strength 𝑃a and 𝑀a , respectively.
4.3.9
Additional Information on Web Crippling
During the past 25 years, the web crippling strengths of
various sections have been studied by numerous investigators. Additional information on web crippling can be found
in Ref. 4.95–4.107, 4.177–4.184, and 4.234.4.273.
4.3.10
Effect of Holes on Web Strength
Cold-formed steel members have been widely used in
residential and commercial construction. Holes are usually
punched in the webs of joists and wall studs for the installation of utilities. Additional research has been conducted to
study the effect of holes on bending strength, shear strength,
web crippling strength, and the combination thereof. Design
Guide for Cold-Formed Steel Beams with Web Penetration
was published by the AISI in 1997.4.185 For further information, see Refs. 1.284, 3.179, 3.181, 3.184, 3.185, 3.187,
3.189, 3.190, 3.192, and 3.193.
In 1999, the AISI supplement to the 1996 edition of the
Specification included additional design provisions for (a)
C-section webs with holes under stress gradient (see Section
3.4.3 in this volume), (b) shear strength of C-section webs
with holes (Section 4.3.3.3), and (c) web crippling strength
of C-section webs with holes (Section 4.3.6.2). These design
provisions are retained in the North American specification.
BRACING REQUIREMENTS OF BEAMS
4.4 BRACING REQUIREMENTS OF BEAMS
4.4.1 Single-Channel (C-Section) Beams
4.4.1.1 Neither Flange Connected to Sheathing When
single-channel sections (C-sections) are used as beams,
adequate bracing must be provided to prevent rotation about
the shear center, as shown in Fig. 4.77, if the load is applied
in the plane of the web.
The shear center is the point through which the external
load must be applied in order to produce bending without
twisting. It is located on the axis of symmetry at a distance
m from the midplane of the web of the channel section. The
value of m can be determined approximately by Eq. (4.205)
or Eq. (4.206) for different types of flanges∗ :
1. For channels without stiffening lips at the outer edges,
𝑚=
𝑤2f
2𝑤f + 𝑑∕3
(4.205)
2. For channels with stiffening lips at the outer edges,
(
[
)]
𝑤f dt
4𝐷2
𝑚=
𝑤f 𝑑 + 2𝐷 𝑑 −
4𝐼𝑥
3𝑑
(4.206)
where m = distance from shear center to midplane of
web of a channel section
𝑤f = projection of flanges from inside face of
web (for channels with flanges of unequal
widths, 𝑤f shall be taken as the width of
the wider flange)
d = depth of channel
D = overall depth of simple lip (edge
stiffener)
𝐼𝑥 = moment of inertia of one channel about
its centroidal axis normal to web
(a) Bracing of Channel Beams. Since the spacing
between braces is usually larger than the spacing of connections required for connecting two channels to form an
I-beam, each channel section may rotate slightly between
braces and results in additional stress. For this reason,
braces must be arranged and designed so that the rotation
of the beam is small and the additional stresses will not
significantly affect the load-carrying capacity of the channel
section.
The spacing and strength of bracing required to counteract
the twisting tendency of channel beams have been investigated theoretically and experimentally by Winter, Lansing,
and McCalley.4.108 A simplified design method has been
developed on the basis of the studies of braced and unbraced
channels and verified by test data.
It has been found that even for impractically small spans
the unbraced channel section [depth = 12 in. (305 mm),
flange width = 3 12 t has been found that even for impractically small spans t = 0.135 in. (3.4 mm)] made of steels
having a yield stress of 33 ksi (228 MPa, or 2320 kg/cm2 )
can only carry less than half of the load that each continuously braced channel would carry before yielding.4.108 See
Fig. 4.78 for the bracing ratio 𝑎∕𝑙 = 1.0. In addition, the
angle of rotation at the midspan exceeds 2∘ if the span length
is larger than 40 in. (1016 mm) (Fig. 4.79).
However, as shown in Fig. 4.78, for braced channels, even
for the spacing of braces equal to 0.478 times the span length,
the ultimate loads for all practical purposes are the same as
for continuous bracing. This fact indicates that the localized overstresses at corners do not affect the strength of
∗ See Appendix B for the location of the shear center for other open sections.
Figure 4.77 Rotation of channel section about its shear center.
209
Figure 4.78
Effect of use of bracing.4.108
210
4
Figure 4.79
FLEXURAL MEMBERS
Results of analysis for channel beam indicated.4.108
the channel sections, since plastic redistribution allows the
initially understressed portions of the section to carry additional load. Furthermore, analyses for a great variety of practical loading conditions show that it is unnecessary to provide
more than three braces between supports in order to limit the
overstresses to 15% of the simple bending stress 𝑓 ′ = Mc∕𝐼,
which equals the yield stress at a load equal to the design load
times the safety factor, except that additional bracing should
be provided at the location of a concentrated load.4.108 This
criterion has been used in the past as a basis in the development of the earlier AISI Specifications for a minimum
number of braces. The same study also showed that for the
15% limitation of overstress the rotation of the section at
1
of the span length under
midspan for a deflection of 360
design load would not exceed 1.5∘ , as shown in Fig. 4.80.
The above discussion dealt with the number of braces
required to limit the additional stress induced by the twisting
of channels between braces. The lateral force to be resisted
by bracing can be determined by calculating the reaction of
a continuous beam consisting of half a channel loaded by a
horizontal force 𝐹 = Pm∕𝑑 since the load P applied in the
plane of the web is equivalent to the same load P applied at
the shear center plus two forces applied at both flanges, as
shown in Fig. 4.81.
Based on the research work and practical considerations,
the following AISI design criteria have been developed and
included in the AISI Specifications during the period from
1956 through 1996 for bracing single-channel beams when
they are loaded in the plane of the web and neither flange is
braced by deck or other means:
Figure 4.80 Rotations for load resulting in central vertical deflections equal to span/360.4.108
Figure 4.81
Lateral force for design of brace for channels.
Braces are to be attached to both the top and bottom flanges
of the section at the ends and at intervals not greater than
one-quarter of the span length in such a manner as to prevent
tipping at the ends and lateral deflection of either flange in
either direction at intermediate braces. Additional bracing is to
be placed at or near the center of the loaded length if one-third
or more of the total load is concentrated over a length of
one-twelfth or less of the span of the beam. However, when all
loads and reactions on a beam are transmitted through members
which frame into the section in such a manner as to effectively
restrain the section against the rotation and lateral displacement,
no additional braces will be required.
In the early 1990s, beam tests conducted by Ellifritt,
Sputo, and Haynes4.186 showed that for typical sections a
BRACING REQUIREMENTS OF BEAMS
midspan brace may reduce service load horizontal deflections and rotations by as much as 80% when compared to
a completely unbraced beam. However, the restraining effect
of braces may change the failure mode from lateral–torsional
buckling to distortional buckling of the flange and lip at
a brace point. The natural tendency of the member under
vertical load is to twist and translate in such a manner as
to relieve the compression on the lip. When such movement is restrained by intermediate braces, the compression
on the stiffening lip is not relieved and may increase. In this
case, distortional buckling may occur at loads lower than
that predicted by the lateral–torsional buckling equations
(Sections F2 and F3 of the AISI Specification).
The same research4.186 has also shown that the AISI
lateral–torsional buckling equations predict loads which
are conservative for cases where only one midspan brace
is used but may be unconservative where more than one
intermediate brace is used. Based on such research findings,
Section D3.2.2 of the AISI specification was revised in
1996 to eliminate the requirement of quarterpoint bracing.
Consequently, Section D3.2.2 of the 1996 edition of the AISI
Specification included the following three requirements for
spacing of braces:
1. Braces shall be designed to avoid local crippling at the
points of attachment to the member.
2. When braces are provided, they shall be attached in
such a manner as to effectively restrain the section
against lateral deflection of both flanges at the ends and
at any intermediate brace points.
3. When all loads and reactions on a beam are transmitted through members which frame into the section
in such a manner as to effectively restrain the section
against torsional rotation and lateral displacement,
no additional braces will be required except those
required for strength according to Section C3.1.2 of
the Specification (Section F3 of the 2016 edition of the
Specification).
Figure 4.82
211
C-purlin on sloped roof.
this case, both components 𝑃𝑥 and 𝑃𝑦 should be considered
in design.
(c) North American Design Criteria. In order to
provide a general method for determining the lateral force
for the design of braces, equations were provided in Section
C2.2.1 of the North American Specification for the design of
C-section or channel beams.1.343,1.345,1.417 It is specified that
each intermediate brace, at the top and bottom flanges of
C-section members, shall be designed with resistance forces
of 𝑃L1 and 𝑃L2 , where 𝑃L1 is the brace force required on
the flange which is located in the quadrant with both x and
y axes positive, 𝑃L2 is the brace force on the other flange.
The x axis is the centroidal axis perpendicular to the web,
and the y axis is the centroidal axis parallel to the web. The
x and y coordinates shall be oriented such that one of the
flanges is located in the quadrant with both positive x and y
axes as shown in Fig. 4.83 for uniform load and Fig. 4.84 for
concentrated load. Consequently, for C-section members,
the brace forces 𝑃L1 and 𝑃L2 can be determined as follows:
1. For uniform loads (Fig. 4.83),
(
)
𝑊 𝑥 𝑀z
𝑃 L1 = 1.5 −
+
(4.207)
2
𝑑
(
)
𝑊
𝑀
𝑃 L2 = 1.5 − 𝑥 − z
(4.208)
2
𝑑
The above requirements 2 and 3 are retained in Section C2
of the 2016 edition of the North American Specification with
minor editorial revisions.
(b) Effect of Slope and Eccentricity. The foregoing
discussion dealt only with a simple case, for which the
gravity load is applied in the plane of the web of a horizontal
beam as shown Fig. 4.81. For general design practices,
the applied load can be either a gravity load or an uplift
load acting at any location along the beam flange. When
C-purlins are used for sloped roofs, the top flange of the
purlin is subject to an inclined load as shown in Fig. 4.82. For
Figure 4.83
C-section member subjected to the uniform load.
212
4
FLEXURAL MEMBERS
2. For concentrated loads (Fig. 4.84),
𝑃 L1 = −
𝑃 𝑥 𝑀z
+
2
𝑑
(4.210)
𝑃 𝑥 𝑀z
−
(4.211)
2
𝑑
When a design load (factored load) acts through the
midplane of the web, that is, 𝑃 𝑦 = 𝑃 , 𝑃 𝑥 = 0, and 𝑒sx =
𝑚, then
( )
𝑚
𝑃 L1 = −𝑃 L2 =
𝑃
(4.212)
𝑑
𝑃 L2 = −
Figure 4.84
load.1.346
C-section member subjected to a concentrated
When the uniform load 𝑊 acts through the midplane
of the web, that is, 𝑊 𝑦 = 𝑊 , 𝑊 𝑥 = 0, and 𝑒sx = 𝑚, then
( )
𝑚
𝑃 L1 = −𝑃 L2 = 1.5
𝑊
(4.209)
𝑑
where 𝑊 𝑥 , 𝑊 𝑦 = components of design load (factored
load) 𝑊 parallel to the x and y axis,
respectively; 𝑊 𝑥 and 𝑊 𝑦 are
positive if pointing to the positive x
and y direction; respectively
𝑊 = design load (factored load) (applied
load determined in accordance with
the most critical load combinations
for ASD, LRFD, or LSD, whichever
is applicable) within a distance of
0.5a on each side of the brace
a = longitudinal distance between
centerline of braces
d = depth of section
m = distance from shear center to
midplane of web of C-section
M 𝑧 = −𝑊 𝑥 𝑒sy + 𝑊 𝑦 𝑒sx , torsional moment
of 𝑊 about shear center using
right-hand rule
𝑒sx , esy = eccentricities of load components
measured from the shear center and
in the x and y directions, respectively
It should be noted that in Eqs. (4.207)–(4.209) the
constant of 1.5 is used to account for the larger reaction
at the first interior support of a continuous beam, which
consists of half a channel loaded by a uniform horizontal
force. In addition, it considers the fact that the assumed
uniform load may not be really uniform. The Specification adopts a conservative approach for uneven uniform
loading with the shift of loading location along the beam
flange.
where 𝑃 𝑥 , P𝑦 = components of design load (factored
load) 𝑃 paralle to the x and y axis,
respectively; 𝑃 𝑥 and 𝑃 𝑦 are positive if
pointing to the positive x and y
directions, respectively
𝑃 = design concentrated load (factored load)
within a distance of 0.3a on each side of
the brace, plus 1.4(1 - l/a) times each
design concentrated load located farther
than 0.3a but not farther than 1.0a from
the brace; the design concentrated load
(factored load) is the applied load
determined in accordance with the most
critical load combinations for ASD,
LRFD, or LSD, whichever is applicable
l = distance from concentrated load to the
brace
𝑀 𝑧 = −𝑃 𝑥 𝑒sy + 𝑃 𝑦 𝑒sx , torsional moment of 𝑃
about shear center using right-hand rule
General Provisions. In both items 1 and 2 above, the
bracing forces 𝑃 𝐿1 and 𝑃 𝐿2 are positive where restraint
is required to prevent the movement of the corresponding
flange in the negative x direction.
Where braces are provided, they shall be attached in such
a manner as to effectively restrain the section against lateral
deflection of both flanges at the ends and at any intermediate
brace points.
When all loads and reactions on a beam are transmitted
through members which frame into the section in such a
manner as to effectively restrain the section against torsional
rotation and lateral displacement, no additional braces shall
be required except those required for strength (resistance)
according to Chapter F of the North American Specification.
Example 4.27 Use the ASD method to determine the
allowable uniform load and design lateral braces for the
channel section used as a simple beam shown in Fig. 4.85.
BRACING REQUIREMENTS OF BEAMS
213
where 𝑀max = 93.50 in.-kips, maximum moment at
po int C
𝑀1 = 69.11 in.-kips, at 14 point of unbraced
length
𝑀2 = 78.66 in.-kips, at midpoint of
unbraced length
𝑀3 = 86.79 in.-kips, at 34 point of unbraced
length
Figure 4.85
Example 4.27.
Therefore
12.5(93.50)
2.5(93.50) + 3(69.11) + 4(78.66) + 3(86.79)
= 1.15
𝐶b =
Assume that braces are attached to both top and bottom
flanges of the channel section at both ends and at intervals
equal to one-quarter of the span length. Use A1011 SS Grade
50 steel (𝐹y = 50 ksi).
SOLUTION
1. Determination of Allowable Uniform Load Based
on Section Strength. By using the design procedure
illustrated in Example 4.4, the allowable moment based
on local buckling is
𝑀𝑎𝓁𝑜 =
𝑆e 𝐹y
Ωb
=
(3.123)(50)
1.67
= 93.50 in.-kips
For the concentrated load of 2 kips, the moment at
point C is
1
1
PL(12) = (2)(8)(12) = 48 in.-kips
4
4
The moment permissible for the uniform load is
𝑀C =
𝑀 = 𝑀𝑎𝓁𝑜 − 𝑀C = 93.50 − 48
= 45.50 in.-kips
1 2
wL (12) = 45.50 in.-kips
8
𝑤 = 0.474 kip∕ft including weight of beam
2. Determination of Allowable Uniform Load Based
on Lateral–Torsional Buckling Strength. Using an
interval of 24 in., the allowable moment for portion BC
(Fig. 4.85) on the basis of lateral–torsional buckling
strength (Section 4.2.2.5) can be determined as follows:
From Eq. (4.54),
𝐹cre =
𝐶b =
𝐶b 𝑟 0 𝐴 √
𝜎e𝑦 𝜎t
𝑆f
12.5𝑀max
2.5𝑀max + 3𝑀1 + 4𝑀2 + 3𝑀3
Based on Section F2.1.1 of the North American
specification as given in Eq. (4.56),
√
𝑟0 = 𝑟2𝑥 + 𝑟2𝑦 + 𝑥20 = 3.09 in.
Using the full cross-sectional area,
𝐴 = 1.554 in.2
From Eq. (4.57),
𝜎e𝑦 =
𝜋2𝐸
(𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 )2
𝜋 2 (29,500)
= 158.0 ksi
(1 × 2 × 12∕0.559)2
From Eq. (4.58),
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
[
1
(11,300)(0.00944)
=
(1.554)(3.09)2
]
𝜋 2 (29,500)(5.56)
+
(1 × 2 × 12)2
=
= 196.60 ksi
In the above equation, the values of J and 𝐶w
are computed from the Design Manual1.428 or from
Appendix B in this volume. Therefore, the elastic
critical lateral–torsional buckling stress is
(1.15)(3.09)(1.554) √
(158.0)(196.60)
𝐹cre =
3.269
= 297.72 ksi
For the yield stress of steel, 𝐹y = 50 ksi,
0.56𝐹y = 28.00 ksi
2.78𝐹y = 139.00 ksi
Since 𝐹cre > 2.78 𝐹y , the member segment is not
subject to lateral–torsional buckling at a bending
214
4
FLEXURAL MEMBERS
moment equal to 𝑀y . The nominal moment should
be determined in accordance with Section 4.2.3.1 in
this volume or Section F3 of the North American
specification as follows:
𝑀ne = 𝑆e 𝐹y = (3.123)(50) = 156.15 in.-kips
The allowable moment is
𝑀
156.15
= 93.50 in.-kips
𝑀a = ne =
Ωb
1.67
Because the above allowable moment for
lateral–torsional buckling is the same as that allowed
for section strength in item 1, the allowable uniform
load is
𝑤 = 0.474 kip∕ft including weight of beam
3. Design of Braces. Assuming that the gravity loads are
applied through the midplane of the web, based on
Eqs. (4.209) and (4.212), the braces used at midspan
should be designed to resist the following forces
𝑃 L1 and 𝑃 L2 :
( )
( )
𝑚
𝑚
𝑃 L1 = −𝑃 L2 = 1.5
𝑊 +
𝑃
𝑑
𝑑
where 𝑑 = 8.0 in.
𝑤2f
(1.865)2
𝑚 =
=
2𝑤f + 𝑑∕3 2(1.865) + 8∕3
= 0.544 in.
𝑚
0.544
=
= 0.068
𝑑
8.0
𝑊 = −2(0.474) = −0.948 kips
𝑃 = −2 kips
Therefore,
𝑃 L1 = −𝑃 L2 = 1.5(0.068)(−0.948) + 0.068(−2)
= −0.232 kips
1
4
The braces used at points (points B and D) should be
designed to resist the following forces 𝑃 L1 and 𝑃 L2 :
( )
𝑚
𝑃 L2 = −𝑃 L2 = 1.5
𝑊
𝑑
= 1.5(0.068)(−0.948) = −0.097 kips
4.4.1.2 Top Flange Connected to Sheathing Section
4.4.1.1 dealt only with C-sections or single-channel beams
when neither flange is connected to deck or sheathing material. For C-sections having through-fastened or standing
seam sheathing attached to the top flange, each anchorage
device shall be designed according to Sections I6.4.1 and
I6.4.2 of the North American Specification to restrain the
flanges so that the maximum top-flange lateral displacements
with respect to the purlin reaction points do not exceed the
span length divided by 360. For details, see Section 9.5.
4.4.2
Z-Beams
4.4.2.1 Neither Flange Connected to Sheathing For
Z-beams, when a load is applied in the midplane of the
web, the section does not tend to rotate in the same manner
as channels because the shear center coincides with its
centroid, as shown in Fig. 4.86a. However, in view of the
fact that in Z-sections the principal axes are oblique, even
though such a section is loaded vertically, it will deflect
vertically and horizontally, as shown in Fig. 4.86b. When
the section deflects in the horizontal direction, the applied
load will also move with the beam and is no longer in the
same plane with the reactions at both ends. As a result, the
section also twists in addition to vertical and horizontal
deflections. The additional stress caused by the twist reduces
the load-carrying capacity of the member.
a. Bracing of Z-Beams. The investigation carried out
by Zetlin and Winter to study the bracing requirement for
Z-beams consisted of testing 19 beams of three different
shapes.4.109 An approximate method of analysis indicated
that the braced Z-beams can be analyzed in the same way
as braced channels, except that the total fictitious horizontal
load caused by the actual vertical load should be determined by P(𝐼xy ∕𝐼𝑥 ). In order to prevent the movement of the
Z-beam along the axis perpendicular to the web, the resistance brace forces P(𝐼xy ∕2𝐼𝑥 ) should be applied to the top
and bottom flanges (Fig. 4.86b). A term K′ is used for the
value of 𝐼xy ∕(2𝐼𝑥 ) in Section C2.2 of the North American
Specification for Z-beams.
For simplification of design, the vertical and horizontal
deflections and the corresponding stresses can be determined
by the summations of the values computed for the actual and
fictitious loads by using the following modified moments of
inertia (𝐼mx and 𝐼my )4.109 :
𝐼m𝑥 =
𝐼m𝑦 =
2
𝐼𝑥 𝐼𝑦 − 𝐼xy
𝐼𝑦
2
𝐼𝑥 𝐼𝑦 − 𝐼xy
𝐼𝑥
(4.213)
(4.214)
Figure 4.86 Z-section subjected to a vertical load through plane
of the web.
215
BRACING REQUIREMENTS OF BEAMS
b. Effect of Slope and Eccentricity. When the Z-section
member is used for sloped roofs, the top flange is subject to
an inclined load as shown in Fig. 4.87. The applied load can
be either a gravity load or an uplift load acting at any location
on the top flange. For this case, both components 𝑃𝑥 and 𝑃𝑦
should be considered for design.
c. North American Design Criteria. For Z-beams, a
general method is also provided in Section C2.2.1 of the
North American specification for determining the resistance
brace forces 𝑃 𝐿1 and 𝑃 𝐿2 as follows:
𝑃 L2
)
𝑊 𝑥 𝑀z
= 1.5 𝑊 𝑦 𝐾 −
+
2
𝑑
(
)
𝑊
𝑀
𝑥
z
= 1.5 𝑊 𝑦 𝐾 ′ −
−
2
𝑑
′
𝑃 L1 = 𝑃 L2 = 1.5
Figure 4.87
roof.
Figure 4.88
𝐼xy
2𝐼𝑥
(4.216)
)
𝑊
𝑃 L1 = 𝑃 𝑦 𝐾 ′ −
(4.215)
When the uniform load W acts through the midplane of
the web, that is, 𝑊 𝑦 = 𝑊 and W𝑥 = 0,
(
and 𝑊 , 𝑊 𝑥 , 𝑊 𝑦 , 𝑑, 𝑀 𝑧 , esx and esy were defined in
Section 4.4.1.1(c). The justification for using a factor
of 1.5 in Eqs. (4.215)–(4.217) was discussed in Section
4.4.1.1.
2. For concentrated loads (Fig. 4.89),
1. For uniform loads (Fig. 4.88)
(
𝑃 L1
where 𝐾 ′ = 𝐼xy ∕(2𝐼𝑥 )
𝐼xy = product of inertia of full unreduced
section
𝐼𝑥 = moment of inertia of full unreduced
section aboutxaxis
(4.217)
Z-section subjected to an inclined load on sloped
Z- section member subjected to a uniform load.
𝑃 𝑥 𝑀z
+
2
𝑑
(4.218)
𝑃 𝑥 𝑀z
+
(4.219)
2
𝑑
When a design load (factored load) acts through the plane of
the web, that is, 𝑃 𝑦 = 𝑃 and P𝑥 = 0,
(
)
𝐼xy
𝑃 L1 = 𝑃 L2 =
𝑃
(4.220)
2𝐼𝑥
𝑃 L2 = 𝑃 𝑦 𝐾 ′ −
In Eqs. (4.218), (4.219) and (4.220), K′ , 𝐼𝑥 , and 𝐼xy are
defined under item 1 above; 𝑃 𝑥, 𝑃 𝑦, 𝑃 , 𝑑, and M𝑧 were
defined in Section 4.4.1.1.
The general provisions of Section 4.4.1.1 for C-section
members are equally applicable to Z-section members. The
commentary on the 2016 edition of the North American
specification also provides the design equations for Z-section
member rests on a sloped roof.1.431
4.4.2.2 Top Flange Connected to Sheathing When
Z-sections are used for roof construction to support the
attached sheathing directly, the 1986 AISI Specification
through the 2016 North American Specification included
brace force equations that were based on the work by
Murray and Elhouar,4.110 Lee and Murray,4.274 Seek and
Figure 4.89
load.1.431
Z-section member dubjected to a concentrated
216
4
FLEXURAL MEMBERS
Murray,4.275,4.276,4.277 and Sears and Murray.4.278 For the
current design requirements, see Section 9.5.
4.4.3
I-Beams
For I-beams, braces should be attached to top and bottom
flanges at both ends. According to Section 4.2.2.5, if 𝐹n is
greater than or equal to 2.78𝐹y and 𝑆𝑒 = 𝑆f , no intermediate
braces are required, except that additional braces should be
placed at the locations of concentrated loads.
In case the value of 𝐹n is less than 2.78𝐹y but greater than
0.56𝐹y , the intervals of braces should not exceed the required
unbraced length determined from Eqs. (4.49) and (4.52). If
𝐹cre is less than or equal to 0.56𝐹y , the required unbraced
length should be determined from Eqs. (4.49) and (4.53).
The design of braces is not specified in the 2016 edition
of the Specification. However, the bracing members may be
designed for a capacity of 2% of the force resisted by the
compression portion of the beam. This is a frequently used
rule of thumb but is a conservative approach, as proven by a
rigorous analysis.
4.4.4
Continuous Lateral Bracing for Beams
When the compression flange of the cold-formed steel
beam is closely connected to decking or sheathing material
as to effectively restrain lateral deflection of the flange
and twisting of the member, previous studies made by
Winter1.157,4.111 indicated that the required resistance to be
provided by decking may be approximated as follows:
(
)
𝛽id
(4.221)
𝐹req = 𝑑i
𝛽id ∕𝛽act
where 𝐹reg = required lateral force provided by decking
𝑑i = initial crookedness
𝛽act = extensional stiffness of decking material,
AE/L′ , in which A is area of decking, E is
modulus of elasticity, and L′ is length
and 𝛽 id , the spring constant of elastic support, is computed
as follows:
2
⎧ 𝜋 (𝑃 − 𝑃 ) when 𝛽 𝐿2 ≤ 30𝑃
(4.222)
e
id
e
⎪ 𝐿2 cr
)2
𝛽id = ⎨ 𝜋 2 𝑃 ( 𝑃
e
cr
⎪
− 0.6 when 𝛽ib 𝐿2 > 30𝑃e (4.223)
⎩ 4𝐿2 𝑃e
where 𝐿 = length of beam
𝑃e = Euler critical load, = π2 EI yc ∕𝐿2
𝑃cr = critical load for compressed half of beam
buckling out of its plane as a column
𝐼yc = moment of inertia of compressed portion of
beam about its weak axis
During the past four decades, the strength and behavior of
diaphragm-braced beams loaded in the plane of the web have
been studied by numerous investigators at Cornell University
and several other institutions. The published research reports
and technical papers provide a better understanding of such
a complicated problem.4.110–4.136,4.274–4.278 These documents
contain valuable background information for developing new
design recommendations for channels and Z-sections when
one flange is connected to deck or sheathing material.
4.5 TORSIONAL ANALYSIS OF BEAMS
AND COMBINED BENDING AND TORSIONAL
LOADING
4.5.1
Torsional Analysis of Beams
In the design of beams, if the transverse load does not pass
through the shear center of the cross section, the beam is
subject to a combination of plane bending and torsional
moment.2.45,4.104 The types of stress caused by plane bending
and torsion are discussed in Appendix B.
4.5.2
Combined Bending and Torsional Loading
When a beam is subject to a combination of bending and
torsion, the longitudinal and shear stresses caused by plane
bending and torsion are discussed in Appendix B. The calculations of these types of stress are illustrated in Example B1
of Appendix B.
For the design of such a beam subjected to combined
bending and torsion, the nominal flexural strength, 𝑀x ,
calculated from Section 4.2.3 with 𝐹n = 𝐹y or 𝑀ne = 𝑀y
for bending alone should be reduced to take into account the
effect of torsion. Section H4 of the North American Specification can be used to determine the reduction to nominal
flexural strength under combined bending and torsional
loading.1.417,1.431 For detailed discussion, see Section B4 of
Appendix B.
In addition to the reduction of the nominal flexural strength
for combined bending and torsion, the reduction of nominal
shear strength can be handled in a similar manner on the basis
of the shear stresses caused by plane bending and torsion.
The calculations of different shear stresses are also discussed
in Appendix B.
4.6
ADDITIONAL INFORMATION ON BEAMS
The structural strength of cold-formed steel purlins has been
investigated by a large number of researchers and engineers.
For further information on this subject, the reader is referred
to Refs. 4.137–4.139, 4.141–4.155, 4.187–4.191,1.414,
2.103, and 4.279–4.292.
Numerical analysis approaches for determining member
flexural strength with consideration of sheathing rotational
and lateral resistances have been investigated.4.297–4.306
Design examples are provided in this research and also in
the AISI Cold-Formed Steel Design Manual.1.428
CHAPTER 5
Compression Members
5.1 GENERAL REMARKS
Similar to the heavy hot-rolled steel sections, thin-walled
cold-formed steel compression members can be used to carry
a compressive load applied through the centroid of the cross
section. The cross section of steel columns can be of any
shape that may be composed entirely of stiffened elements
(Fig. 5.1a), unstiffened elements (Fig. 5.1b), or a combination of stiffened and unstiffened elements (Fig. 5.1c).
Unusual shapes and closed cylindrical tubular sections are
also often found in use.
Cold-formed sections are made of thin material, and in
many shapes the shear center does not coincide with the
centroid of the cross-section. Therefore in the design of such
compression members, consideration should be given to the
following limit states depending on the configuration of the
section, thickness of material, and column length used:
1. Yielding
2. Overall column buckling (or global buckling)
a. Flexural buckling: bending about a principal axis
b. Torsional buckling: twisting about shear center
c. Flexural–torsional buckling: bending and twisting
simultaneously
3. Local buckling of individual compression elements
interacting with the yielding or global buckling
4. Distortional buckling of open cross-sections with
edge-stiffened flanges
Design provisions for the global flexural buckling
and the effect of local buckling on column strength
have long been included in the AISI Specification. The
Figure 5.1 Types of compression members: (a) members
composed entirely of stiffened elements; (b) members composed
entirely of unstiffened elements; (c) members composed of both
stiffened and unstiffened elements.
provisions for flexural–torsional buckling were added
to the specification in 1968 following a comprehensive
investigation carried out by Winter, Chajes, Fang, and Pekoz
at Cornell University.1.161,5.1,5.2
The design provisions have been based on the unified
approach developed in 1986 and discussed by Pekoz in Ref.
3.17. This approach consists of the following steps for the
design of axially loaded compression members:
1. Calculate the elastic column buckling stress (flexural,
torsional, or flexural–torsional) for the full unreduced
section.
2. Determine the nominal failure stress (elastic buckling,
inelastic buckling, or yielding).
3. Calculate the nominal column load based on the
governing failure stress and the effective area.
4. Determine the design column load from the nominal
column load using the specified safety factor or the
resistance factor.
In 2007, the design provisions for determining the distortional buckling strength of I-, Z-, C-, hat, and other open
sections having edge-stiffened flanges were added in the
North American Specification.1.345
In 2016, the Direct Strength Method has been moved
from Appendix 1 into the main body of the North American
Specification. The Effective Width Method and the Direct
Strength Method are considered as equivalent methods in the
Specification.
For column design tables and example problems, reference
should be made to Part III of the 2017 edition of the AISI
Design Manual.1.428
The column strengths for different failure modes are
discussed in subsequent sections of this chapter. References 5.3–5.8, 5.100, 5.110, 5.114, 5.126, 5.133, and
5.141–5.155 deal with some previous and recent studies on
columns.
217
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
218
5.2
5.2.1
5
COMPRESSION MEMBERS
COLUMN BUCKLING
Yielding
It is well known that a very short, compact column under
axial load may fail by yielding. For this case, the yield load
is simply
(5.1)
𝑃y = AF y
where A = full cross-sectional area of column
Fy = yield stress of steel
5.2.2
Flexural Column Buckling
Figure 5.2
5.2.2.1 Elastic Buckling A slender axially loaded column
may fail by overall flexural buckling (or global buckling)
if the cross-section of the column is a doubly symmetric
shape (I-section), closed shape (square or rectangular tube),
closed cylindrical shape, or point-symmetric shape (Z-shape
or cruciform). For singly symmetric shapes, flexural buckling is one of the possible failure modes, as discussed in
Section 5.2.3.2. If a column has a cross-section other than
the above-discussed shapes but is connected to other parts
of the structure such as wall sheathing material, the member
can also fail by flexural buckling. For other possible buckling
modes, see Section 5.2.3.
The elastic critical buckling load for a long column can be
determined by the Euler formula:
𝑃𝑐𝑟e =
where Pcre
E
I
L
K
=
=
=
=
=
𝜋 2 EI
(KL)2
limit as discussed in Section 2.7, Eq. (5.3) would not be
suitable for columns made of gradual-yielding steel having
small and moderate slenderness ratios. This is because when
the stress is above the proportional limit, the column will
buckle in the inelastic range.
5.2.2.2 Inelastic Buckling For the flexural column buckling in the inelastic range, two concepts have been used
in the past. They are the tangent modulus method and the
reduced-modulus method.2.45,3.3
The tangent modulus method was proposed by Engesser
in 1889. Based on this method, the tangent modulus load is
(5.2)
Euler buckling load
modulus of elasticity
moment of inertia
column length
effective length factor
Substituting 𝐼 = Ar2 in Eq. (5.2) or dividing Eq. (5.2) by
the full area A, the following Euler stress for elastic column
buckling can be obtained:
𝜋2𝐸
(5.3)
(KL∕𝑟)2
where KL/r is the effective slenderness ratio and r is the least
radius of gyration.
Equation (5.3) is graphically shown as curve A in
Fig. 5.2, which is applicable to the ideal columns made of
sharp-yielding-type steel having stress–strain characteristics
as shown in Fig. 2.1a without consideration of residual stress
or effects of cold working. In view of the fact that many steel
sheets and strips used for cold-formed structural members
are of the gradual-yielding type as shown in Fig. 2.1b and
the cold-forming process tends to lower the proportional
𝜎𝑐𝑟e =
Flexural column buckling stress.
𝑃T =
𝜋 2 𝐸t 𝐼
(KL)2
(5.4)
and the critical buckling stress is
𝜎T =
𝜋 2 𝐸t
(KL∕𝑟)2
(5.5)
where Et is the tangent modulus.
In 1895 Jasinky pointed out that the tangent modulus
concept did not include the effect of elastic unloading.
Engesser then corrected his theory and developed the
reduced- or double-modulus concept, in which
𝑃r =
𝜋 2 𝐸r 𝐼
(KL)2
or
𝜎R =
𝜋 2 𝐸r
(KL∕𝑟)2
(5.6)
where Er = reduced modulus, E(I1 /I) + Et (I2 /I)
I1 = moment of inertia about neutral axis of the
area on unloading side after buckling
12 = moment of inertia about neutral axis of the
area on loading side after buckling
Shanley5.9 carefully did the experiments and analytical
investigation, concluding:
COLUMN BUCKLING
1. The tangent-modulus concept gives the maximum
load up to which an initially straight column remains
straight.
2. The actual maximum load exceeds the tangent
modulus load, but it cannot reach the reduced-modulus
load.
Many other investigators have proved Shanley’s findings
and have indicated that for the case studied the maximum
load is usually higher than the tangent modulus load by 5%
or less.2.45
In view of the fact that the tangent modulus strength
provides an excellent prediction of the actual column
strength, the Column Research Council∗ has suggested that
design formulas for steel columns should be on the basis of
the tangent modulus concept.3.84 For this reason, whenever
the computed Euler stress is above the proportional limit,
the tangent modulus should be used to compute the buckling
stress.
The tangent modulus can be determined by the techniques described in Technical Memorandum 2 of the Structural Stability Research Council, “Notes on the Compression
Testing of Metals,”3.84,1.158,1.412 However, it is impossible to
provide stress–strain curves and values of tangent moduli for
all types of sheets and strip, in particular when the cold work
of forming is utilized. In the design of hot-rolled shapes, the
Structural Stability Research Council has indicated that Eq.
(5.5) can be conservatively approximated by the following
formula if the effect of residual stress is considered and the
effective proportional limit is assumed to be equal to one-half
the yield stress1.161,3.84 :
( 2 )
(
)
( )2
𝐹y
𝐹y
KL
𝜎T = 𝐹y 1 −
(5.7)
= 𝐹y −
4𝜎𝑐𝑟e
𝑟
4𝜋 2 𝐸
in which 𝐹y is the minimum yield stress. The above formula
can also be used for cold-formed sections if the residual stress
induced by cold forming of the section and the stress–strain
characteristics of the gradual-yielding steel sheets and strip
are considered.
√
2𝜋 2 𝐸∕𝐹y is the
As shown in Fig. 5.2, the value of
limiting KL/r ratio corresponding to a stress equal to 𝐹y ∕2.
When the KL/r ratio is greater than this limiting ratio,
the column is assumed to be governed by elastic buckling, and when the KL/r ratio is smaller than this limiting
ratio, the column is to be governed by inelastic buckling.
Equation (5.7) has been used for the design of cold-formed
steel columns up to 1996.
∗ The
Column Research Council has been renamed Structural Stability
Research Council.
219
In the 1996 edition of the AISI Specification, the design
equations for calculating the nominal inelastic and elastic
flexural buckling stresses were changed to those used in the
AISC LRFD Specification as follows3.150 :
(𝐹n )I = (0.658𝜆𝑐 )𝐹y
[
]
0.877
(𝐹n )𝑐𝑟e =
𝐹y
𝜆2c
2
when
𝜆c ≤ 1.5
(5.7a)
when
𝜆𝑐 > 1.5
(5.3a)
where (𝐹𝑛 )I is the nominal inelastic buckling
√ stress, (𝐹n )𝑐𝑟e
is the nominal elastic buckling stress, 𝜆c = 𝐹y ∕𝜎cre is the
column slenderness parameter, in which 𝜎𝑐𝑟e is the theoretical elastic flexural buckling stress of the column determined
by Eq. (5.3).
The reasons for changing the design equations from Eq.
(5.7) to Eq. (5.7a) for the nominal inelastic buckling stress
and from Eq. (5.3) to Eq. (5.3a) for the nominal elastic
buckling stress are as follows:1.159
1. The revised column design equations [Eqs. (5.7a) and
(5.3a)] are based on a different basic strength model
and were shown to be more accurate by Pekoz and
Sumer.5.103 In this study, 299 test results on columns
and beam–columns were evaluated. The test specimens included members with component elements
in the post–local buckling range as well as those
that were locally stable. The test specimens included
members subjected to flexural buckling as well as
flexural–torsional buckling, to be discussed in Section
5.2.3.
2. Because the revised column design equations represent the maximum strength with due consideration
given to initial crookedness and can provide the better
fit to test results, the required safety factor for the
ASD method can be reduced. In addition, the revised
equations enable the use of a single safety factor for
all λc values even though the nominal axial strength of
columns decreases as the slenderness increases due to
initial out-of-straightness. With the use of the selected
safety factor and resistance factor given in the Specification (Section 5.6), the results obtained from the
ASD and LRFD approaches would be approximately
the same for a live load–dead load ratio of 5.0.
Figure 5.3 shows a comparison of the nominal critical flexural buckling stresses used in the 1986 edition of the ASD
specification, the 1991 edition of the LRFD Specification,
and the 1996 edition of the combined ASD/LRFD Specification. The North American specification uses the same
equations as the 1996 AISI Specification.
220
5
COMPRESSION MEMBERS
Figure 5.3
Comparison between the critical buckling stress equations.
It should be noted that by using Eqs. (5.7a
√ and (5.3a)
2𝜋 2 𝐸∕𝐹y to
the limiting KL/r ratio is changed from
√
4.71
𝐸∕𝐹y corresponding to an assumed proportional
limit of 0.44 𝐹y . This revised limiting KL/r ratio is being
used in the 2005 edition of the AISC Specification for the
design of structural steel members for compression.1.411 For
cold-formed steel design, Eqs. (5.7a) and (5.3a) are retained
in the 2016 edition of the North American Specification for
the design of concentrically loaded compression members.
For details, see Section 5.6.
Figure 5.4 Displacement of a nonysmmetric section during
torsional–flexural buckling.5.2
5.2.3 Torsional Buckling and Flexural–Torsional
Buckling
Usually, closed cross-sections will not buckle torsionally
because of their large torsional rigidity. For open thin-walled
cross-sections, however, three modes of failure are considered in the analysis of overall instability (flexural buckling,
torsional buckling, and flexural–torsional buckling) as previously mentioned.
When an open cross-section column buckles in the
flexural–torsional mode, bending and twisting of the section
occur simultaneously. As shown in Fig. 5.4, the section
translates u and v in the x and y directions and rotates an
angle ϕ about the shear center. This problem was previously
investigated by Goodier, Timoshenko, and others.5.10,5.11,3.3
It has been further studied by Winter, Chajes, and Fang for
development of the AISI design criteria.5.1,5.2
The equilibrium of a column subjected to an axial load P
leads to the following differential equations5.2,5.11 :
EI 𝑥 𝑣iv + Pv′′ − Px0 𝜙′′ = 0
(5.8)
EI 𝑦 𝑢iv + Pu′′ + Py0 𝜙′′ = 0
(5.9)
ECW 𝜙iv − (GJ − Pr 20 )𝜙′′
+ Py0 𝑢′′ − Px0 𝑣′′ = 0
where Ix
Iy
u
v
𝜙
x0
y0
E
=
=
=
=
=
=
=
=
(5.10)
moment of inertia about the x axis
moment of inertia about the y axis
lateral displacement in x direction,
lateral displacement in y direction
angle of rotation
x coordinate of shear center
y coordinate of shear center
modulus of elasticity, = 29.5 × 103 ksi
(203 GPa, or 2.07 × 106 kg/cm2 )
G = shear modulus, = 11.3 × 103 ksi (78 GPa,
or 794 × 103 kg/cm2 )
COLUMN BUCKLING
221
J = St. Venant torsion constant of cross
∑
section, 13 𝑙i 𝑡3i
Cw = warping constant of torsion of cross
section (Appendix B)
ECw = warping rigidity
GJ = torsional rigidity
r0 = polar radius of gyration
√of cross section
about shear center, = 𝑟2𝑥 + 𝑟2𝑦 + 𝑥20 + 𝑦20
ry , ry = radius of gyration of cross section about x
and y axes
All derivatives are with respect to z, the direction along the
axis of the member.
Considering the boundary conditions for a member with
completely fixed ends, that is, at 𝑧 = 0, 𝐿,
𝑢=𝑣=𝜙=0
𝑢 =𝑣 =𝜙 =0
′
′
′
(5.11)
and for a member with hinged ends, that is, at z = 0, 𝐿,
𝑢=𝑣=𝜙=0
𝑢 =𝑣 =𝜙 =0
′′
′′
′′
(5.12)
Equations (5.8)–(5.10) result in the following characteristic
equation:
r02 (𝑃𝑐𝑟e − 𝑃𝑥 )(𝑃𝑐𝑟e − 𝑃𝑦 )(𝑃𝑐𝑟e − 𝑃𝑧 )
− (𝑃𝑐𝑟e )2 (𝑦0 )2 (𝑃𝑐𝑟e − 𝑃𝑥 )
− (𝑃𝑐𝑟e )2 (𝑥0 )2 (𝑃𝑐𝑟e − 𝑃𝑦 ) = 0
(5.13)
where the Euler flexural buckling load about the x axis is
given as
𝜋 2 EI 𝑥
𝑃𝑥 =
(5.14)
(𝐾𝑥 𝐿𝑥 )2
the Euler flexural buckling load about the y axis as
𝑃𝑦 =
𝜋 2 EI 𝑦
(𝐾𝑦 𝐿𝑦 )2
the torsional buckling load about the z axis as
[ 2
]( )
𝜋 ECw
1
𝑃𝑧 =
+ GJ
2
(𝐾t 𝐿t )
𝑟20
(5.15)
Figure 5.5
Doubly symmetric shapes.
center coincides with the centroid of the section (Fig. 5.5),
that is,
𝑥0 = 𝑦0 = 0
(5.17)
For this case, the characteristic equation becomes
(𝑃𝑐𝑟e − 𝑃𝑥 )(𝑃𝑐𝑟e − 𝑃𝑦 )(𝑃𝑐𝑟e − 𝑃𝑧 ) = 0
(5.18)
The critical buckling load is the lowest value of the following
three solutions:
(5.19)
(𝑃𝑐𝑟e )1 = 𝑃𝑥
(𝑃𝑐𝑟e )2 = 𝑃𝑦
(5.20)
(𝑃𝑐𝑟e )3 = 𝑃𝑧
(5.21)
An inspection of the above possible buckling loads indicates that for doubly-symmetric sections the column fails
either in pure bending or in pure torsion, depending on the
column length and the shape of the section. Usually compression members are so proportioned that they are not subject to
torsional buckling. However, if the designer wishes to evaluate the torsional buckling stress σt , the following formula
based on Eq. (5.16) can be used:
[
]
𝜋 2 ECw
1
𝜎𝑡 = 2 GJ +
(5.22)
(𝐾t 𝐿t )2
Ar0
The critical stress for flexural buckling was discussed in
Section 5.2.2.
(5.16)
and KL is the effective length of the column; theoretically,
for hinged ends 𝐾 = 1, and for fixed ends 𝐾 = 0.5.
The buckling mode of the column can be determined by
Eq. (5.13). The critical buckling load is the smallest value of
the three roots of 𝑃𝑐𝑟e . The following discussion is intended
to indicate the possible buckling mode for various types of
cross-section.
5.2.3.1 Doubly-Symmetric Sections For a doubly-symmetric sections, such as an I-section or a cruciform, the shear
5.2.3.2 Singly-Symmetric Sections Angles, channels
(C-sections), hat sections, T-sections, and I-sections with
unequal flanges (Fig. 5.6) are singly-symmetric shapes. If
the x axis is the axis of symmetry, the distance y0 between
the shear center and the centroid in the direction of the y axis
is equal to zero. Equation (5.13) then reduces to
(𝑃𝑐𝑟e − 𝑃𝑦 )[𝑟20 (𝑃𝑐𝑟e − 𝑃𝑥 )(𝑃𝑐𝑟e − 𝑃𝑧 ) − (𝑃𝑐𝑟e 𝑥0 )2 ] = 0
(5.23)
For this case, one of the solutions is
(𝑃𝑐𝑟e )1 = 𝑃𝑦 =
𝜋 2 EI 𝑦
(𝐾𝑦 𝐿𝑦 )2
(5.24)
222
5
COMPRESSION MEMBERS
Figure 5.6
Singly symmetric shapes.
(Pcre)
Figure 5.7 Comparison of P𝑐𝑟e with Px , Py , and Pz for hat section
(Kx Lx = Ky Ly = Kt Lt = L).
which is the critical flexural buckling load about the y axis.
The other two solutions for the flexural–torsional buckling
load can be obtained by solving the following quadratic
equation:
𝑟20 (𝑃𝑐𝑟e − 𝑃𝑥 )(𝑃𝑐𝑟e − 𝑃𝑧 ) − (𝑃𝑐𝑟e 𝑥0 )2 = 0
(5.25)
Letting 𝛽 = 1 − (𝑥0 ∕𝑟0 )2 ,
]
[
√
1
2
(𝑃𝑐𝑟e )2 =
(𝑃𝑥 − 𝑃𝑧 ) + (𝑃𝑥 − 𝑃𝑧 ) − 4𝛽𝑃𝑥 𝑃𝑧
2𝛽
(5.26)
]
[
√
1
(𝑃𝑐𝑟e )3 =
(𝑃𝑥 + 𝑃𝑧 ) − (𝑃𝑥 + 𝑃𝑧 )2 − 4𝛽𝑃𝑥 𝑃𝑧
2𝛽
(5.27)
Because (𝑃cr )3 is smaller than (𝑃cr )2 , Eq. (5.27) can be
used as the critical flexural–torsional buckling load, which is
always smaller than 𝑃𝑥 and 𝑃𝑧 , but it may be either smaller
or larger than 𝑃𝑦 [Eq.(5.24)] (Fig. 5.7).
Dividing Eq. (5.27) by the total cross-sectional area A,
the following equation can be obtained for the elastic
flexural–torsional buckling stress:
]
[
√
1
𝜎TFO =
(𝜎e𝑥 + 𝜎t ) − (𝜎e𝑥 + 𝜎t )2 − 4𝛽𝜎e𝑥 𝜎t (5.28)
2𝛽
where σTFO is the elastic flexural–torsional buckling stress
and
𝑃
(5.29)
𝜎ex = 𝑥
𝐴
𝑃
𝜎t = 𝑧
(5.30)
𝐴
In summary, it can be seen that a singly symmetric
section may buckle either in bending about the y axis∗ or
in flexural–torsional buckling (i.e., bending about the x
axis and twisting about the shear center), depending on the
dimensions of the cross-section and the effective column
length. For the selected hat section used in Fig. 5.7, the
critical length 𝐿cr , which divides the flexural buckling
mode and the flexural–torsional buckling mode, can be
determined by solving 𝑃𝑦 = (𝑃𝑐𝑟e )3 . This means that if
the effective length is shorter than its critical length, the
flexural–torsional buckling load (𝑃𝑐𝑟e )3 , represented by
curve AB, will govern the design. Otherwise, if the effective
length is longer than the critical length, the load-carrying
capacity of the given member is limited by the flexural
buckling load 𝑃𝑦 , represented by curve BC. The same is true
for other types of singly symmetric shapes, such as angles,
channels, T-sections, and I-sections having unequal flanges.
In view of the fact that the evaluation of the critical
flexural–torsional buckling load is more complex as
compared with the calculation of the Euler load, design
charts, based on analytical and experimental investigations, have been developed for several commonly used
sections,5.1,1.159 from which we can determine whether a
given section will buckle in the flexural–torsional mode.
Such a typical curve is shown in Fig. 5.8 for a channel
section. If a column section is so proportioned that
flexural–torsional buckling will not occur for the given
length, the design of such a compression member can then
be limited to considering only flexural, local, or distortional
buckling. Otherwise, flexural–torsional buckling must also
be considered.
As indicated in Fig. 5.8, the possibility of overall column
buckling of a singly symmetric section about the x axis
may be considered for three different cases. Case 1 is for
flexural–torsional buckling only. This particular case is
characterized by sections for which 𝐼𝑦 > 𝐼𝑥 . When 𝐼𝑦 < 𝐼𝑥 ,
the section will fail either in case 2 or in case 3. In case
2, the channel will buckle in either the flexural or the
flexural–torsional mode, depending on the specific ratio of
b/a and the parameter tL∕𝑎2 , where b is the flange width, a
is the depth of the web element, t is the thickness, and L is
the effective length. For a given channel section and column
length, if the value of tL∕𝑎2 is above the (tL∕𝑎2 )lim curve,
∗ It
is assumed that the section is symmetrical about the x axis.
223
COLUMN BUCKLING
Figure 5.8
Figure 5.9
Figure 5.10
Buckling mode for channel section.5.2
Figure 5.11 Buckling mode curves for hat sections.5.2
provided for determining the critical length for angles,
channels, and hat sections. From this type of graphic design
aid, the critical length can be obtained directly according to
the dimensions and shapes of the member.
The preceding discussion deals with flexural–torsional
buckling in the elastic range for which the compression stress
is less than the proportional limit. Members of small or
moderate slenderness will buckle at a stress lower than the
value given by the elastic theory if the computed critical
buckling stress exceeds the proportional limit.
Similar to the case of flexural buckling, the inelastic
flexural–torsional buckling load may be obtained from
the elastic equations by replacing E with 𝐸t and G with
𝐺(𝐸t ∕𝐸), where 𝐸t is the tangent modulus, which depends
on the effective stress–strain relationship of the entire cross
section, that is, for inelastic flexural–torsional buckling:
( )
𝐸t
(5.31)
𝑃𝑥
(𝑃𝑥 )T =
𝐸
( )
𝐸t
(𝑃𝑧 )T =
(5.32)
𝑃𝑧
𝐸
( )
𝐸t
(𝑃𝑐𝑟e )T =
(5.33)
𝑃𝑐𝑟e
𝐸
Buckling mode curve for angles.5.2
5.2
Buckling mode curves for channels.
the section will fail in the flexural buckling mode. Otherwise
it will fail in the flexural–torsional buckling mode. In case 3,
the section will always fail in the flexural mode, regardless
of the value of tL∕𝑎2 . The buckling mode curves for angles,
channels, and hat sections are shown in Figs. 5–9 to 5–11.
These curves apply only to compatible end conditions,
that is,
𝐾 𝑥 𝐿𝑥 = 𝐾 𝑦 𝐿𝑦 = 𝐾 t 𝐿t = 𝐿
In Part VII of the 1996 edition of the AISI Design
Manual,1.159 a set of design charts such as Fig. 5.12 were
With regard to the determination of 𝐸t , Bleich3.3 indicates
that
)]
[ (
𝜎
𝜎
1−
(5.34)
𝐸t = CE
𝐹y
𝐹y
where
𝐶=
1
(𝜎pr ∕𝐹y )(1 − 𝜎pr ∕𝐹y )
(5.35)
𝐹y and 𝜎pr being the yield stress and proportional limit of
the steel, respectively. The values of C obtained from an
experimental study5.2 ranged from 3.7 to 5.1. Based on Eq.
(5.34) and using 𝐶 = 4 (assuming 𝜎pr = 12 𝐹𝑦 ), the tangent
modulus Et for the inelastic buckling stress is given by
(
)
𝜎TFT
𝜎TFT
𝐸t = 4𝐸
1−
(5.36)
𝐹y
𝐹y
224
5
COMPRESSION MEMBERS
Figure 5.12
AISI chart for determining critical length of channels.1.159
where 𝜎 TFT is the inelastic flexural–torsional buckling
stress. Substituting the above relationship into Eq. (5.33),
the following equation for inelastic flexural–torsional
buckling stress can be obtained:
(
)
𝐹y
𝜎TFT = 𝐹y 1 −
(5.37)
4𝜎TFO
in which 𝜎TFo is the elastic flexural–torsional buckling stress
determined by Eq. (5.28). Equation (5.37) is shown graphically in Fig. 5.13.
To verify the design procedure described above, a total of
eight columns were tested for elastic flexural–torsional
buckling and 30 columns were tested for inelastic
flexural–torsional buckling at Cornell University.5.2 The
results of the inelastic column tests are compared with Eq.
(5.37) in Fig. 5.14.
Similar to the case for flexural column buckling, Eq.
(5.37) has been used in the AISI specification up to 1996. In
the 1996 edition of the specification, the nominal inelastic
flexural–torsional buckling
stress was computed by Eq.
√
(5.7a), in which 𝜆𝑐 = 𝐹y ∕𝜎TFO .
Following an evaluation of the test results of angles
reported by Madugula, Prabhu, Temple, Wilhoit, Zandonini,
and Zavellani,5.12–5.14 Pekoz indicated in Ref. 3.17 that
angle sections are more sensitive to initial sweep than lipped
channels. It was also found that the magnitude of the initial
sweep equal to L/1000 would give reasonable results for
the specimens considered in his study. On the basis of the
findings summarized in Ref. 3.17, an out-of-straightness
of L/1000 was used in Section C5.2 of the 1996 edition of
the AISI Specification for computing additional bending
moment 𝑀y (for ASD) or 𝑀uy (for LRFD), and is retained
in Section H1.2 of the 2016 edition of the North American
Specification. Based on the research findings of Popvic,
Hancock, and Rasmussen, reported in Ref. 5.100, the North
American Specification uses the additional bending moment
due to an out-of-straightness of L/1000 only for singly
symmetric unstiffened angles for which the effective area
at stress 𝐹y is less than the full unreduced cross-sectional
area, or the member local buckling strength (𝑃n𝓁 ) is less
than the member global buckling strength (Pne ). For the
structural strengths of angles and channels, additional information can be found in Refs. 5.15, 5.19, 5.101, 5.102, 5.110,
5.117, 5.119, 5.123, 5.125, 5.128, 5.131–5.134, 5.137, and
5.156–5.173.
5.2.3.3 Point-Symmetric Sections A point-symmetric
section is defined as a section symmetrical about a point
225
COLUMN BUCKLING
Figure 5.13
Maximum stress for flexural–torsional buckling.
been carried out at Cornell.5.2 It was found that plain and
lipped Z-sections will always fail as simple Euler columns
regardless of their size and shape, provided that the effective length for bending about the minor principal axis is
equal to or greater than the effective length for twisting.
The structural strengths of point-symmetric Z-sections have
been discussed by Rasmussen5.153,5.154 and Thottunkal and
Ramseyer.5.174
Figure 5.14 Correlation
investigations.5.2
of
analytical
and
experimental
(centroid), such as a Z-section having equal flanges or a
cruciform section.5.17 For this case the shear center coincides
with the centroid and 𝑥0 = 𝑦0 = 0.
Similar to doubly symmetric sections, Eq. (5.13) leads to
(𝑃𝑐𝑟e − 𝑃𝑥 )(𝑃𝑐𝑟e − 𝑃𝑦 )(𝑃𝑐𝑟e − 𝑃𝑧 ) = 0
(5.38)
Therefore, the section may fail in either bending (𝑃𝑥 or 𝑃𝑦 )
or twisting (𝑃𝑧 ), depending on the shape of the section and
the column length. It should be noted that the x and y axes
are principal axes.
Although the curve for determining the buckling mode
is not available for Z-sections, a limited investigation has
5.2.3.4 Nonsymmetric Sections If the open section has
no symmetry either about an axis or about a point, all three
possible buckling loads 𝑃𝑐𝑟e are of the flexural–torsional
type. The lowest value of 𝑃𝑐𝑟e is always less than the lowest
of the three values 𝑃𝑥 , 𝑃𝑦 , and 𝑃𝑧 .
In the design of compact nonsymmetric sections,
the elastic flexural–torsional buckling stress 𝜎TFO may
be computed from the following equation by trial and
error1.159,1.349 :
(
3
𝜎TFO
𝜎e𝑥 𝜎e𝑦 𝜎t
+
)
(
𝛼−
2
𝜎TFO
𝜎e𝑦 𝜎t
)
(
𝛾−
𝜎TFO 𝜎TFO 𝜎TFO
+
+
=1
𝜎e𝑥
𝜎e𝑦
𝜎t
2
𝜎TFO
𝜎e𝑥 𝜎t
)
(
𝛽−
2
𝜎TFO
)
𝜎e𝑥 𝜎e𝑦
(5.39)
226
5
COMPRESSION MEMBERS
In the calculation, the following equation may be used for
the first approximation:
𝜎TFO = [(𝜎e𝑥 𝜎e𝑦 + 𝜎e𝑥 𝜎t + 𝜎e𝑦 𝜎t )
√
(𝜎e𝑥 𝜎e𝑦 + 𝜎e𝑥 𝜎t + 𝜎e𝑦 𝜎t )2
−
−4(𝜎e𝑥 𝜎e𝑦 𝜎t )(𝛾𝜎e𝑥 + 𝛽𝜎e𝑦 + 𝜎t )]
[
]
1
×
(5.40)
2(𝛾𝜎e𝑥 + 𝛽𝜎e𝑦 + 𝜎t )
where
𝜎e𝑥 =
𝜋2𝐸
(𝐾𝑥 𝐿𝑥 ∕𝑟𝑥 )2
𝜋2𝐸
(𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 )2
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
𝜎e𝑦 =
and
(5.41)
(5.42)
(5.43)
E = modulus of elasticity, =29.5 × 103 ksi
(203 GPa, or 2.07 × 106 kg/cm2 )
KL = effective length of compression member
rx = radius of gyration of cross section about the
x axis
ry = radius of gyration of cross section about the
y axis
A=√
cross-sectional area
r0 = 𝑟2𝑥 + 𝑟2𝑦 + 𝑥20 + 𝑦20
G = shear modulus, =11.3 × 103 ksi (78 GPa, or
794 × 103 kg/cm2 )
J = St. Venant torsion constant of cross section
𝛼 = 1 − (x0 /r0 )2 – (y0 /r0 )2
𝛽 = 1 − (x0 /r0 )2
𝛾 = 1 − (y0 /r0 )2
x0 = distance from shear center to centroid along
principal x axis
y0 = distance from shear center to centroid along
principal y axis
Cw = warping constant of torsion of cross section
(Appendix B)
5.2.4
the global buckling equations derived in Section 5.2.1–5.2.3,
except the corresponding average section properties provided
in Table 4.3 are used. The following expressions can be used
for determining the column global buckling load:
Global buckling load 𝑃𝑐𝑟e is calculated:
𝑃𝑐𝑟e = 𝐴g 𝐹𝑐𝑟e
(5.44)
Where 𝐴g is the gross cross-section area, and 𝐹𝑐𝑟e is the
smallest buckling stress of considering applicable flexural,
torsional or flexural-torsional buckling with consideration of
hole effect:
𝜋 2 EI avg
(5.45)
For flexural buckling∶𝐹𝑐𝑟e =
𝐴g (KL)2
where 𝐼avg = average moment inertia about the axis of buckling as defined in Table 4.3; KL = effective unbraced length
about the axis of buckling; 𝐴g = gross cross-section area.
Note: the gross cross-section area used in the buckling stress
equation [Eq. (5.45)] is for converting the buckling load to
the uniform compressive stress at the ends of the columns,
and it should be not be confused with 𝐴avg .
[
]
𝜋 2 ECwnet
1
For torsional buckling ∶ 𝜎𝑡 =
+
GJ
avg
(𝐾t 𝐿t )2
𝐴g 𝑟20,avg
(5.46)
In Eq. (5.46), 𝑟0,avg and 𝐽avg are average section properties
defined in Table 4.3, and 𝐶wnet is the warping constant of the
net cross-section.
For flexural–torsional buckling:
√
1
𝜎TFO =
[(𝜎e𝑥 + 𝜎t ) − (𝜎e𝑥 + 𝜎t )2 − 4𝛽avg 𝜎e𝑥 𝜎t ]
2𝛽avg
(5.47)
where 𝜎ex is determined using Eq. (5.45) with the x axis
as the axis of buckling, 𝜎t is given in Eq. (5.46), and 𝛽avg
is calculated using the average section properties, 𝑥0,avg and
𝑟0,avg (defined in Table 4.3), as follows:
(
)
𝑥0,avg 2
𝛽avg = 1 −
(5.48)
𝑟0,avg
Refer to Section 4.2.2.4 for further discussions on some
basic rules of thumb on the influence of holes.
Effect of Holes on Global Column Buckling
As discussed in Section 4.2.2.4, the existence of holes along
the member length will reduce the member bending rigidity,
EI, and consequently reduce the member global buckling
load.3.278,4.294,4.295,6.39 Even though numerical methods such
as the shell finite element method can be used to determine
the member global buckling loads, the method developed by
Moen and Shafer3.278 is recommended: for members with
holes, the global buckling load can be approximated by using
5.3 LOCAL BUCKLING INTERACTING
WITH YIELDING AND GLOBAL BUCKLING
Cold-formed steel compression members may be so proportioned that local buckling of individual component plates
occurs before the applied load reaches the overall collapse
load of the column. The interaction effect of the local and
yielding or overall column buckling may result in a reduction
of the overall column strength.
LOCAL BUCKLING INTERACTING WITH YIELDING AND GLOBAL BUCKLING
In general, the influence of local buckling on column
strength depends on the following factors:
1. Shape of cross section
2. Slenderness ratio of column
3. Type of governing overall column buckling (flexural buckling, torsional buckling, or flexural–torsional
buckling)
4. Type of steel used and its mechanical properties
5. Influence of cold work
6. Effect of imperfection
7. Effect of welding
8. Effect of residual stress
9. Interaction between plane components
10. Effect of perforations
During the past 60 years, investigations on the
interaction of local and yielding or overall buckling
in metal columns have been conducted by numerous
researchers.1.11,3.70,5.20–5.63 Different approaches have been
suggested for analysis and design of columns.
5.3.1 Effective Width Method
The effect of local buckling on column strength was considered in the AISI Specification during the period from 1946
through 1986 by using a form factor Q in the determination of allowable stress for the design of axially loaded
compression members.1.418 Accumulated experience has
proved that the use of such a form factor is a convenient
and simple method for the design of cold-formed steel
columns.1.161
Even though the Q-factor method has been used
successfully in the past for the design of cold-formed
steel compression members, additional investigations at
Cornell University and other institutions have shown that
this method is capable of improvement.3.17,5.26–5.28,5.49 On
the basis of test results and analytical studies of DeWolf,
Pekoz, Winter, Kalyanaraman, and Loh, Pekoz show in Ref.
3.17 that the Q-factor approach can be unconservative for
compression members having stiffened elements with large
width-to-thickness ratios, particularly for those members
having slenderness ratios in the neighborhood of 100. On
the other hand, the Q-factor method gives very conservative
results for I-sections having unstiffened flanges, especially
for columns with small slenderness ratios. Consequently,
the Q factor was eliminated in the 1986 edition of the
AISI Specification. In order to reflect the effect of local
buckling on column strength, the nominal column load is
determined by the governing critical buckling stress and the
effective area 𝐴e instead of the full sectional area. When
227
𝐴e cannot be calculated, such as when the compression
member has dimensions or geometry outside the range of
applicability of the generalized effective width equations of
the AISI Specification, the effective area 𝐴e can be determined experimentally by stub column tests as described in
Ref. 3.17. For C- and Z-shapes and single-angle sections
with unstiffened flanges, the nominal column load has been
limited by the column buckling load, which is calculated by
the local buckling stress of the unstiffened flange and the
area of the full, unreduced cross section. This requirement
was included in Section C4(b) of the 1986 edition of the
AISI Specification. It was deleted in 1996 on the basis of the
study conducted by Rasmussen and Hancock (Refs. 5.101
and 5.102).
The current North American design provisions are
presented in Section 5.6 followed by design examples.
5.3.1.1 Effect of Holes on Local Buckling For members
with holes, the flat elements beside the hole is considered
unstiffened element with 𝑘 = 0.43. If the hole dimensions
are within the limits of AISI Specification Section 1.1.1, it
is permitted that the hole influence on the global buckling
can be ignored if the Effective Width Method is used to
determine the strength of local buckling interacting with the
global buckling.
5.3.2
Direct Strength Method
Instead of using the Effective Width Method, the member
strength due to local buckling interacting with yielding or
global buckling can be determined by the Direct Strength
Method.3.196 The method predicts the member strength
based on the member local buckling load and the material yield strength. A comprehensive discussion on how
to obtain member local buckling load is provided in
Section 3.5.
The Direct Strength Method was introduced in the 2007
edition of the North American Specification as Appendix 1.
In 2016, the Direct Strength Method was incorporated into
the main body of the Specification and is considered as an
equivalent method with the Effective Width Method. The
excerpted design provisions from the 2016 North American
Specification were included in Section 5.6.
5.3.2.1 Effect of Holes on Local Buckling The Direct
Strength Method considers hole effects through the elastic
buckling load (𝑃cr𝓁 ) in predicting the member strength.
The existence of holes will result in reduced buckling load
and consequently reduced member strength. The simplified
method for predicting Pcr𝓁 is provided in Section 3.5.3,
and the Specification design provisions are provided in
Section 5.6.
228
5
COMPRESSION MEMBERS
5.4 DISTORTIONAL BUCKLING STRENGTH
OF COMPRESSION MEMBERS
5.4.2 Design Criteria for Distortional Buckling
Strength of Open-Section Compression Members
5.4.1
In 2007, the North American Specification included
Appendix 1 for the design of cold-formed steel structural members using the Direct Strength Method. These
design provisions were incorporated into the main body of
the Specification in the 2016 edition and were excerpted in
Section 5.6.
Research Work
The distortional buckling mode for flexural members was
discussed in Section 4.2.4. For column design, flange
distortional buckling is also one of the important failure
modes for open cross-sectional compression members
having edge-stiffened flanges as shown in Fig. 5.15. This
type of buckling mode involves the rotation of each flange
and lip about the flange–web junction.
During the past three decades, the distortional buckling
modes of compression members have been studied by
Hancock,5.108,1.358 Lau and Hancock,5.109–5.111 Charnvarnichborikarn and Polyzois,5.112 Kwon and Hancock,5.113,5.114
Hancock, Kwon, and Bernard5.115 Schafer,3.195 Davies,
Jiang, and Ungureanu,4.168 Bambach, Merrick, and Hancock,
3.173 and others. Earlier research findings were well summarized by Hancock in Ref. 1.69. In the same publication,
Hancock also discussed the background information on the
1996 Australian design provisions for distortional buckling
of flexural members and compression members.
Since 1999, additional studies of distortional buckling of compression members have been conducted by
Kesti and Davies,5.175 Schafer,5.176 Young and Yan,5.177
Camotim, Dinis, and Silvestre,5.178–5.180,5.183,5.184, 5.186,5.187
Ranawaka and Mahendran,5.181 Tovar and Sputo,5.182 Rao
and Kalyanaraman,5.185 and others. Some of the research
findings are summarized in Refs. 1.346, 1.383, and 1.412.
For the analytical model used for development of design
provisions, see Section 4.2.4.
5.4.3
Members with Holes
Similar to predicting the local buckling strength for members
with holes, the hole influence on the distortional buckling
strength is considered in the distortional buckling load,
which is discussed in Section 3.5.3.
The member’s ultimate strength is limited by the capacity
of the net cross-section 𝑃ynet = 𝐴net 𝐹y . The member strength
is then transmitted from 𝑃ynet ,3.252,5.212 through inelastic
region, to the elastic buckling strength as shown in Fig. 5.16.
The complete design provisions are provided in Section 5.6.
5.5 EFFECT OF COLD WORK ON COLUMN
BUCKLING
The discussions in Sections 5.1 to 5.4 were based on the
assumption that the compression members have uniform
mechanical properties over the entire cross section. However,
as shown in Fig. 2.3, the yield stress and tensile strength of
the material vary from place to place in the cross section
due to the cold work of forming. The column strength of
the axially loaded compression member with nonuniform
Figure 5.15 Rack section column buckling stress versus half wavelength for concentric
compression.1.69
EFFECT OF COLD WORK ON COLUMN BUCKLING
Figure 5.16
Direct Strength Method (DSM) distortional buckling strength curve.
Eq. (5.49)
Figure 5.17
sections.2.17
Comparison
of
column
curves
for
channel
In order to investigate the strength of cold-formed
compression members subjected to an axial load, six specimens made of channels back to back have been tested by
Karren and Winter at Cornell University.2.14,2.17 The test data
are compared graphically with Eqs. (5.5), (5.7) and (5.49)
in Fig. 5.17. In addition, four pairs of joist sections have
also been tested at Cornell. Results of tests are compared in
Fig. 5.18.2.17
Based on the test data shown in Figs. 5.17 and 5.18,2.17
it may be concluded that with the exception of two channel
tests Eq. (5.49) seems to produce a somewhat better correlation because it considers the variable material properties
mechanical properties throughout the cross section may
be predicted by Eq. (5.49) on the basis of the tangent
modulus theory if we subdivide the cross section into j
subareas, for which each subarea has a constant material
property2.14,2.17,5.64,5.65 :
𝜎T =
where
229
𝑗
𝜋2 ∑
𝐸 𝐼
𝐴(KL)2 𝑖=1 t𝑖 𝑖
Eq. (5.49)
(5.49)
𝐸t𝑖 = tangent modulus of ith subarea at a
particular value of strain
I𝑖 = moment of inertia of ith subarea about
neutral axis of total cross section
Figure 5.18
sections.2.17
Comparison of column curves for joist chord
230
5
COMPRESSION MEMBERS
over the cross section. Equations (5.5) and (5.7), based on the
average of compressive and tensile yield stresses, also predict
satisfactory column buckling stress in the inelastic range with
reasonable accuracy, particularly for columns with a slenderness ratio around 60. Equation (5.7) could provide a lower
boundary for column buckling stress if the tensile yield stress
is to be used.2.14
E2.1 Sections Not Subject
Flexural-Torsional Buckling
5.6 NORTH AMERICAN DESIGN FORMULAS
FOR CONCENTRICALLY LOADED
COMPRESSION MEMBERS
where E = Modulus of elasticity of steel
K = Effective length factor determined in
accordance with Specification Chapter C
L = Laterally unbraced length of member
r = Radius of gyration of full unreduced crosssection about axis of buckling
Based on the discussions of Sections 5.1 to 5.4, appropriate
design provisions are included in the North American specification for the design of axially loaded compression members.
The following excerpts are adapted from Chapter E of the
2016 edition of the North American Specification for the
ASD, LRFD, and LSD methods:
E2 Yielding and Global (Flexural, Flexural-Torsional,
and Torsional) Buckling
The nominal axial strength [resistance], 𝑃ne , for yielding, and
global (flexural, torsional, or flexural-torsional) buckling shall
be calculated in accordance with this section. The applicable
safety factor and resistance factors given in this section shall
be used to determine the available axial strength [factored
resistance] (𝜙C 𝑃ne or 𝑃ne ∕ΩC ) in accordance with the applicable
design method in Specification Section B3.2.1, B3.2.2, or
B3.2.3.
(ASD)
𝜙C = 0.85
(LRFD)
= 0.80
(LSD)
F𝑐𝑟e =
where
√
λC =
𝐹y
𝐹𝑐𝑟e
(5.51)
(5.52)
(5.53)
where Fcre = Least of the applicable elastic global (flexural,
torsional, and flexural–torsional) buckling
stresses determined in accordance with
Specification Sections E2.1 through E2.5 or
Appendix 2
Fy = Yield stress
π2 E
(KL∕r)2
𝑅r = 0.65 +
where
where Ag = Gross area
Fn = Compressive stress and shall be calculated as
follows:
2
or
(5.54)
E2.1.1 Closed-Box Sections
For a concentrically loaded compression member with a
closed-box section that is made of steel with a specified
minimum elongation between three to ten percent, inclusive, a
reduced radius of gyration (𝑅r )(r) shall be used in Eq. (5.50)
when the value of the effective length KL is less than 1.1 L0 ,
where L0 is given by Eq. (5.55), and 𝑅r is given by Eq. (5.56).
√
𝐸
𝐿0 = 𝜋𝑟
(5.55)
𝐹cr𝓁
(5.50)
For λC ≤ 1.5 𝐹n = (0.658λc )𝐹y
(
)
0.877
For λC > 1.5 𝐹n =
𝐹y
λ2c
Torsional
For doubly-symmetric sections, closed cross-sections, and any
other sections that can be shown not to be subjected to torsional
or flexural-torsional buckling, the elastic flexural buckling
stress, 𝐹𝑐𝑟e , shall be calculated as follows:
𝑃ne = 𝐴g 𝐹n
ΩC = 1.80
to
0.35(𝐾𝐿)
1.1𝐿0
(5.56)
L0 = Length at which local buckling stress equals
flexural buckling stress
r = Radius of gyration of full unreduced crosssection about axis of buckling
E = Modulus of elasticity of steel
Fcr𝓁 = Minimum critical buckling stress for crosssection calculated by Eq. (3.7)
Rr = Reduction factor
KL = Effective length determined in accordance
with Specification Chapter C
E2.2 Doubly- or Singly-Symmetric Sections Subject
to Torsional or Flexural-Torsional Buckling
For singly-symmetric sections subject to flexural–torsional
buckling, 𝐹𝑐𝑟e shall be taken as the smaller of 𝐹𝑐𝑟e calculated in
accordance with Specification Section E2.1 and F𝑐𝑟e calculated
as follows:
√
1
F𝑐𝑟e =
(5.57)
[(σex + σt ) − (σex + σt )2 − 4𝛽𝜎ex σt ]
2β
Alternatively, a conservative estimate of 𝐹𝑐𝑟e is permitted to
be calculated as follows:
σt σex
(5.58)
F𝑐𝑟e =
σt + σex
where
𝛽 = 1 − (𝑥0 ∕𝑟0 )2
(5.59)
NORTH AMERICAN DESIGN FORMULAS FOR CONCENTRICALLY LOADED COMPRESSION MEMBERS
where r0 = Polar radius of gyration of cross-section about
shear center
√
= rx2 + ry2 + x20
(5.60)
where rx , ry = Radii of gyration of cross-section about
centroidal principal axes
x0 = Distance from centroid to shear center in
principal x-axis direction, taken as negative
𝜎t =
where
A
G
J
E
Cw
Kt
[
]
π2 ECw
1
GJ
+
(Kt Lt )2
Ar20
(5.61)
=
=
=
=
=
=
Full unreduced cross-sectional area of member
Shear modulus of steel
Saint-Venant torsion constant of cross-section
Modulus of elasticity of steel
Torsional warping constant of cross-section
Effective length factor for twisting determined
in accordance with Specification Chapter C
Lt = Unbraced length of member for twisting
𝜎ex =
π2 E
(Kx Lx ∕rx )2
(5.62)
where Kx = Effective length factor for bending about the
x axis determined in accordance with Chapter
C
Lx = Unbraced length of member for bending about
x-axis
For singly-symmetric sections, the x-axis shall be selected as
the axis of symmetry.
For doubly-symmetric sections subject to torsional buckling,
𝐹𝑐𝑟e shall be taken as the smaller of 𝐹𝑐𝑟e calculated in accordance with Specification Section E2.1 and 𝐹𝑐𝑟e = 𝜎t , where 𝜎t is
defined in accordance with Eq. (5.61).
For singly-symmetric unstiffened angle sections for which
the effective area (𝐴e ) at stress 𝐹y is equal to the full unreduced
cross-sectional area (A) for effective width method, or 𝑃n𝓁 = 𝑃ne
from Specification Section E3 for Direct Strength Method, 𝐹𝑐𝑟e
shall be computed using Eq. (5.54) where r is the least radius of
gyration.
E2.3 Point-Symmetric Sections
For point-symmetric sections, 𝐹𝑐𝑟e shall be taken as the lesser of
𝜎t as defined in Specification Section E2.2 and 𝐹𝑐𝑟e as calculated
in Specification Section E2.1 using the minor principal axis of
the section.
231
E2.5 Sections With Holes
For shapes whose cross-sections have holes, 𝐹𝑐𝑟e shall
consider the influence of holes in accordance with Specification Appendix 2. Alternatively, compression members with
holes are permitted to be tested in accordance with Specification
Section K2.
Exception: For the Effective Width Method, where hole sizes
meet the limitations of Specification Appendix 1.1.1, the provisions of this section shall not be required.
E3 Local Buckling Interacting With Yielding and
Global Buckling
The nominal axial strength [resistance], 𝑃n𝓁 , for local buckling
interacting with yielding and global buckling shall be calculated
in accordance with this section. All members shall be checked
for potential reduction in available strength [factored resistance]
due to interaction of the yielding or global buckling with local
buckling. This reduction shall be considered through either the
Effective Width Method of Specification Section E3.1 or the
Direct Strength Method of Specification Section E3.2.
The applicable safety factors and resistance factors given
in this section shall be used to determine the available axial
strength [factored resistance] (𝜙C 𝑃n𝓁 or 𝑃n𝓁 ∕ΩC ) in accordance
with the applicable design method in Specification Section
B3.2.1, B3.2.2, or B3.2.3.
ΩC = 1.80 (ASD)
𝜙C = 0.85 (LRFD)
= 0.80 (LSD)
E3.1 Effective Width Method
For the Effective Width Method, the nominal axial strength
[resistance], 𝑃n𝓁 , for local buckling shall be calculated in accordance with the following:
𝑃n𝓁 = Ae Fn ≤ Pne
(5.63)
where Fn = Global column stress as defined in Specification
Section E2
Ae = Effective area calculated at stress Fn ,
determined in accordance with Specification
Sections E3.1.1 and E3.1.2
Pne = Nominal strength [resistance] considering
yielding and global buckling, determined in
accordance with Specification Section E2
E2.4 Nonsymmetric Sections
Concentrically loaded angle sections shall be designed for an
additional bending moment as specified in the definitions of 𝑀 x
and 𝑀 y in Specification Section H1.2.
For shapes whose cross-sections do not have any symmetry
either about an axis or about a point, 𝐹𝑐𝑟e shall be determined
by Specification Appendix 2 or rational engineering analysis.
Alternatively, compression members composed of such shapes
are permitted to be tested in accordance with Specification
Section K2.
E3.1.1 Members Without Holes
For members without holes, except closed cylindrical tubular
members, 𝐴e shall be determined from the summation of the
thickness times the effective width of each element comprising
the cross-section. The effective width of all elements shall be
232
5
COMPRESSION MEMBERS
determined in accordance with Specification Appendix 1 at
stress 𝐹n .
E3.1.2 Members With Circular Holes
For members with circular holes, 𝐴e shall be determined from
the effective width in accordance with Specification Appendix
1.1.1(a), subject to the limitations of that section. If the number
of holes in the effective length region times the hole diameter
divided by the effective length does not exceed 0.015, 𝐴e is
permitted to be determined by ignoring the holes, i.e., in accordance with Specification Section E3.1.1.
E3.2 Direct Strength Method
For the Direct Strength Method, the nominal axial strength
[resistance], 𝑃n𝓁 , for local buckling shall be calculated in accordance with Specification Sections E3.2.1 and E3.2.2.
E3.2.1 Members Without Holes
For λ𝓁 ≤ 0.776; 𝑃n𝓁 = 𝑃ne
(5.64)
]
[
(
)0.4 (
)
Pcr𝓁 0.4
P
Pne
For λ𝓁 > 0.776; 𝑃n𝓁 = 1 − 0.15 cr𝓁
Pne
Pne
(5.65)
where
√
(5.66)
λ𝓁 = Pne ∕Pcr𝓁
Pne = Global column strength as defined in
Specification Section E2
Pcr𝓁 = Critical elastic local column buckling load,
determined in accordance with Specification
Appendix 2
The applicable safety factor and resistance factors given
in this section shall be used to determine the available axial
strength [factored resistance] (𝜙C 𝑃nd or𝑃nd ∕ΩC ) in accordance
with the applicable design method in Specification Section
B3.2.1, B3.2.2, or B3.2.3.
ΩC = 1.80 (ASD)
𝜙C = 0.85 (LRFD)
= 0.80 (LSD)
E4.1 Members Without Holes
The nominal axial strength [resistance], 𝑃nd , for distortional
buckling shall be calculated in accordance with the following:
For λd ≤ 0.561; 𝑃nd = 𝑃y
(5.69)
)0.6 (
)0.6
(
⎤ P
⎡
P
⎥ crd
For λd > 0.561; 𝑃nd = ⎢1 − 0.25 crd
Py
⎥ Py
⎢
Py
⎦
⎣
(5.70)
where
√
λd = 𝑃y ∕𝑃crd
(5.71)
where
𝑃y = 𝐴g 𝐹y
where
(5.72)
𝐴g = Gross area of cross-section
𝐹y = Yield stress
𝑃crd = Critical elastic distortional column buckling
load, determined in accordance with
Specification Appendix 2
E4.2 Members With Holes
E3.2.2 Members With Holes
The nominal axial strength [resistance], 𝑃n𝓁 , for local buckling
of columns with holes shall be calculated in accordance with
Specification Section E3.2.1, except 𝑃cr𝓁 shall be determined
including the influence of holes and:
𝑃n𝓁 ≤ 𝑃ynet
(5.67)
𝑃ynet = 𝐴net 𝐹y
(5.68)
where
where Anet = Net area of cross-section at the location of
a hole
Fy = Yield stress
The nominal axial strength [resistance], 𝑃nd , for distortional
buckling of columns with holes shall be calculated in accordance
with Specification Section E4.1, except 𝑃crd shall be determined
including the influence of holes, and if λd ≤ λd2 then:
(5.73)
For λd ≤ λd1 ; 𝑃nd = 𝑃ynet
(
)
𝑃ynet − 𝑃d2
For λd1 < λd ≤ λd2 ; 𝑃nd = 𝑃ynet −
(λd − λd1 )
λd2 − λd1
(5.74)
where
√
λd = 𝑃y ∕𝑃crd
(5.75)
)
(
𝑃ynet
(5.76)
λd1 = 0.561
𝑃y
E4 Distortional Buckling
The nominal axial strength [resistance], 𝑃nd , for distortional
buckling shall be calculated in accordance with this section.
The provisions of this section shall apply to I-, Z-, C-, hat, and
other open cross-section members that employ flanges with edge
stiffeners.
λd2
𝑃d2
)0.4
(
⎡
⎤
𝑃y
⎢
= 0.561 14.0
− 13.0⎥
⎢
⎥
𝑃ynet
⎣
⎦
[
( )1.2 ] ( )1.2
1
1
= 1 − 0.25
𝑃y
λd2
λd2
(5.77)
(5.78)
NORTH AMERICAN DESIGN FORMULAS FOR CONCENTRICALLY LOADED COMPRESSION MEMBERS
where
𝑃y = 𝐴g 𝐹y
(5.79)
𝑃ynet = 𝐴net 𝐹y
(5.80)
233
𝐴𝑔 = Gross area
𝐴net = Net area of cross-section at the location of a
hole
𝐹y = Yield stress
5.6.1 Additional Comments on North American
Design Formulas
In addition to the discussions of Section 5.6, the following
comments are related to some of the North American design
provisions for concentrically loaded compression members:
1. Safety Factor. In the 1986 and earlier editions of the
AISI Specification, the allowable axial load for the ASD
method was determined by either a uniform safety factor
of 1.92 or a variable safety factor ranging from 1.67 to
1.92 for wall thickness not less than 0.09 in. (2.3 mm) and
𝐹e > 𝐹y ∕2. The use of the smaller safety factors for the type
of relatively stocky columns was occasioned by their lesser
sensitivity to accidental eccentricities and the difference
in structural behavior between the compression members
having different compactness. The latter fact is illustrated
by the stress–strain curves of the more compact and less
compact stub-column specimens, as shown in Fig. 5.19. In
the experimental and analytical investigations conducted by
Karren, Uribe, and Winter,2.14,2.17 both types of specimens
shown in Fig. 5.19 were so dimensioned that local buckling
would not occur at stresses below the yield stress of the
material. Test data did indicate that the less compact stub
column (curve A for cold-reduced killed press braked hat
section) reached the ultimate compressive load at strains
in the range of 3 × 10−3 –5 × 10–3 in./in., after which the
load dropped off suddenly because yielding was followed
by local crippling. However, the more compact stub column
(curve B for hot-rolled semi killed roll-formed channel
section) showed a long stable plateau and, after some strain
hardening, reached the ultimate load at much higher values
of strain in the range of 16 × 10−3 –27 × 10–3 in./in. For this
reason, the use of a smaller safety factor for more compact
sections is justified and appropriate as far as the overall
safety of the compression member is concerned.
As discussed in Section 5.2.2.2 on inelastic buckling, the
AISI design equations were changed in 1996 on the basis of
a different strength model. These equations enable the use of
a single safety factor of 1.80 for all 𝜆c values. Figure 5.20
shows a comparison of design axial strengths permitted
by the 1986 and 1996 AISI Specifications and the 2007
North American Specification. It can be seen that the design
Figure 5.19 Stress–strain curves for more compact and less
compact stub columns.2.17
Figure 5.20 Comparison between the design axial strengths, 𝑃d ,
for the ASD method.
capacity is increased by using the 1996 and 2007 Specifications for thin columns with low slenderness parameters
and decreased for high slenderness parameters. However, the
difference would be less than 10%. The design provisions in
the 2016 edition is consistent with the 2007 edition of the
North American Specification.
For the LRFD method, the differences between the
nominal axial strengths used for the 1991, 1996, and 2007
LRFD design provisions are shown in Fig. 5.21. The resistance factor of 𝜙c = 0.85 is the same as the 1999 AISC
Specification3.150 and the 1991 edition of the AISI LRFD
Specification.3.152
In the 2007 edition, similarly in the 2016 edition of the
North American Specification, the safety factor for the ASD
method and the resistance factor for the LRFD method are
the same as that used in the 1996 AISI Specification.
234
5
COMPRESSION MEMBERS
Figure 5.21 Comparison between the nominal axial strengths, 𝑃n ,
for the LRFD method.
2. Maximum Slenderness Ratio. The maximum allowable slenderness ratio KL/r of compression members has
been preferably limited to 200, except that during construction the maximum KL/r ratio should not exceed 300.1.346 This
limitation on the slenderness ratio is the same as that used by
the AISC for the design of hot-rolled heavy-steel compression members. Even though the design formulas are equally
applicable to columns having slenderness ratios larger than
200, any use of the unusually long columns will result in an
uneconomical design due to the small capacity to resist buckling. In 1999, the AISI Committee on Specifications moved
the KL/r limit from the Specification to the Commentary.1.333
3. Flexural–Torsional Buckling. The simplified equation
for flexural–torsional buckling [Eq. (5.58)] is based on the
following formula given by Pekoz and Winter in Ref. 5.66:
1
1
1
+
𝑃𝑥 𝑃z
(5.81)
1
1
1
=
+
𝜎TFO
𝜎e𝑥 𝜎t
(5.82)
𝑃TFO
=
or
For singly symmetric angle sections with unstiffened legs,
a study conducted at the University of Sydney by Popovic,
Hancock, and Rasmussen5.100 indicated that the relatively
compact angle section will not fail in a flexural–torsional
buckling mode if the effective area (𝐴e ) under yield stress
𝐹y is equal to the full, unreduced cross-sectional area (A).
For this case, the concentrically loaded angle section can
be designed for flexural buckling alone in accordance with
Sections E2.1 and E3 of the Specification.
4. Point-Symmetric Sections. As discussed in Section
5.2.2.3, point-symmetric sections may fail either in flexural
buckling about the minor principal axis or twisting. The
provision was added in the North American Specification in
2001 for the design of concentrically loaded point-symmetric
reactions and was retained in Section E2.3 of the 2016 edition
of the Specification.
5. Design Tables. Part III of the AISI Design Manual1.428
contains a number of design tables for column properties
and nominal axial strengths of C-sections with and without
lips including stock studs/joists and tracks. The tables are
prepared for 𝐹y = 33, 50, and 55 ksi (228, 345, and 379 MPa,
or 2320, 3515, and 3867 kg/cm2 ). In addition, Tables III-4
to III-6 of the Design Manual provide computed distortional buckling properties and strengths under axial load for
the representative C-shapes, stock studs/joists, and Z-shapes
with lips, respectively. The values in these tables have been
calculated in accordance with equations given in Specification Section 2.3.1.3.
5.7
EFFECTIVE LENGTH FACTOR K
In steel design, lateral bracing is generally used to resist
lateral loads, such as wind or earthquake loads, or to increase
the strength of members by preventing them from deforming
in their weakest direction.4.111 The use of such bracing may
affect the design of compression members.
In Sections 5.2–5.6, the effective length KL of the column
was used to determine buckling stresses. The factor K (a ratio
of the effective column length to the actual unbraced length)
represents the influence of restraint against rotation and/or
translation at both ends of the column.
The theoretical K values and the design values recommended by the Structural Stability Research Council are
tabulated in Table 5.1. In practice, the value of 𝐾 = 1 can
be used for columns or studs with X-bracing, diaphragm
bracing, shear-wall construction, or any other means that
will prevent relative horizontal displacements between both
ends.1.161 If translation is prevented and restraint against rotation is provided at one or both ends of the member, a value
of less than 1 may be used for the effective length factor. The
effect length factor, K, can also be taken as 1 if the structural
analysis takes into consideration of moment magnifications
due to joint translation and the member deformation (detailed
discussion is provided in Chapter 6.)
In the design of trusses, it is realized that considerable
rotational restraint could be provided by continuity of the
compression chord as long as the tension members do not
yield. In view of the fact that for the ASD method tension
members are designed with a safety factor of 1.67 and
compression members are designed with a relatively large
EFFECTIVE LENGTH FACTOR K
235
Table 5.1 Effective Length Factor K for Axially Loaded Columns with Various End
Conditions
a
Source: Reproduced from Guide to Stability Design Criteria for Metal Structures, 4th ed., 1988.
(Courtesy of John Wiley & Sons, Inc.)
safety factor of 1.80, it is likely that the tension members will
begin to yield before the buckling of compression members.
Therefore, the rotational restraint provided by tension
members may not be utilized in design as discussed by
Bleich.3.3 For this reason, compression members in trusses
can be designed for 𝐾 = 1.1.161 However, when sheathing
is attached directly to the top flange of a continuous chord,
additional research has shown that the K value may be taken
as 0.75,5.107 as discussed in Section 12.2.1.6.
For unbraced frames, the structure depends on its own
bending stiffness for lateral stability. If a portal frame is not
externally braced in its own plane to prevent sidesway, the
effective length KL is larger than the actual unbraced length,
as shown in Fig. 5.22, that is, 𝐾 > 1. This will result in a
reduction of the load-carrying capacity of columns when the
sidesway is not prevented.
For unbraced portal frames, the effective column length
can be determined from Fig. 5.23 for the specific ratio of
(𝐼∕𝐿)beam ∕(𝐼∕𝐿)col and the condition of the foundation. If
the actual footing provides a rotational restraint between
hinged and fixed bases, the K value can be obtained by
interpolation.
Figure 5.22
Laterally unbraced portal frame.1.161
The K values to be used for the design of unbraced multistory or multibay frames can be obtained from the alignment
chart in Fig. 5.24.1.158 In the chart, G is defined as
∑
(𝐼 ∕𝐿 )
𝐺= ∑ c c
(5.83)
(𝐼b ∕𝐿b )
236
5
COMPRESSION MEMBERS
between individual shapes, the effective slenderness ratio
KL/r is replaced by the modified effective slenderness ratio
(KL∕𝑟)m calculated by Eq. (5.84):
√
( )
( )2 ( )2
𝑎
KL
KL
=
+
(5.84)
𝑟 m
𝑟 0
𝑟i
Figure 5.23
frames.1.161
Effective length factor K in laterally unbraced portal
in which Ic is the moment of inertia and 𝐿c the unbraced
length of the column and Ib is the moment of inertia and 𝐿b
the unbraced length of the beam.
In practical design, when a column base is supported by
but not rigidly connected to a footing or foundation, G is
theoretically infinity, but unless actually designed as a true
friction-free pin, it may be taken as 10. If the column end
is rigidly attached to a properly designed footing, G may be
taken as 1.0.1.158,5.67
In the use of the chart, the beam stiffness 𝐼𝑏 ∕𝐿𝑏 should be
multiplied by a factor as follows when the conditions at the
far end of the beam are known:
1. Sidesway is prevented:
1.5 for far end of beam hinged
2.0 for far end of beam fixed
2. Sidesway is not prevented:
0.5 for far end of beam hinged
0.67 for far end of beam fixed
After determining 𝐺A and 𝐺B for joints A and B at two
ends of the column section, the K value can be obtained from
the alignment chart of Fig. 5.24 by constructing a straight line
between the appropriate points on the scales for 𝐺A and 𝐺B .
5.8
BUILT-UP COMPRESSION MEMBERS
For built-up compression members composed of two
sections in contact, the available axial strength (factored
axial resistance) shall be determined in accordance with
Chapter E of the North American Specification (Section
5.6 in this volume) subjected to modification as necessary.
Based on Section I1.2 of the 2016 edition of the North American specification, if the buckling mode involves relative
deformations that produce shear forces in the connections
where (KL∕𝑟) = overall slenderness ratio of entire
0
section about built-up member axis
𝑎 = intermediate fastener or spot weld
spacing
𝑟i = minimum radius of gyration of full
unreduced cross-sectional area of an
individual shape in built-up member
See Section 5.6 for definition of other symbols. In addition, the fastener strength (resistance) and spacing shall
satisfy the following:
a. The intermediate fastener or spot weld spacing a is limited
such that 𝑎∕𝑟i does not exceed one-half the governing
slenderness ratio of the built-up member.
b. The ends of a built-up compression member are connected
by a weld having a length not less than the maximum
width of the member or by connectors spaced longitudinally not more than four diameters apart for a distance
equal to 1.5 times the maximum width of the member.
c. The intermediate fastener(s) or weld(s) at any longitudinal
member tie location are capable of transmitting a force in
any direction of 2.5% of the available strength (compressive resistance) of the built-up member.
In the above design criteria, Eq. (5.84) was added to the
North American Specification since 2001 on the basis of the
1999 AISC Specification and the 1994 CSA Standard.1.117
The overall slenderness ratio, (KL∕𝑟)0 , is computed about
the same axis as the modified slenderness ratio, (KL∕𝑟)m .
The (KL∕𝑟)m ratio replaces KL/r for both flexural and
flexural–torsional buckling.
Section I1.2 of the North American Specification
includes the above three requirements concerning intermediate fastener spacing a, end connection of the built-up
member, and the applied force for the design of intermediate
fastener(s). The intermediate fastener spacing requirement
(a) for [𝑎∕𝑟i ≤ 0.5 (KL∕𝑟)] is to prevent flexural buckling
of individual shapes between intermediate connectors to
account for any one of the connectors becoming loose or
ineffective. Requirements (b) and (c) are related to connection design as discussed in Section 8.9 and illustrated in
Example 8.6.
BRACING OF AXIALLY LOADED COMPRESSION MEMBERS
237
Figure 5.24 Alignment charts developed by L.S. Lawrence for effective length of column in continuous frames.1.158 Courtesy of Jackson & Moreland Division of United Engineers & Constructors, Inc.
For the research work on built-up compression members,
more studies have been made by Yang and Hancock,5.188,5.191
Brueggen and Ramseyer,5.189 Stone and LaBoube,5.190
Young and Chen,5.192 and others.
2[4 − (2∕𝑛)]
For LRFD and LSD∶ 𝛽br =
𝐿b
In the 2007 edition of the North American Specification, new
design provisions were included for bracing of axially loaded
compression members, and the provisions are retained in
the 2016 edition of the Specification with modifications.
According to Section C2.3 of the Specification, the required
brace strength (brace force) and required brace stiffness to
restrain lateral translation at a brace point for an individual
compression member shall be calculated by Eqs. (5.85) and
(5.86), respectively, as follows:
The required brace strength is determined:
𝑃rb = 0.01𝑃 ra
(5.85)
𝑃 ra
𝜙
)
(5.86b)
𝜙 = 0.75 for LRFD and 0.70 for LSD.
where
5.9 BRACING OF AXIALLY LOADED
COMPRESSION MEMBERS
(
𝑃rb = required brace strength (brace force) for a
single compression member with an axial
load 𝑃 ra
𝑃 ra = required compressive axial strength
(compressive force) of individual
concentrically loaded compression
member to be braced, which is calculated
in accordance with ASD, LRFD or LSD
load combinations
𝛽 rb = minimum required brace stiffness to brace
a single compression member
n = number of equally spaced intermediate
brace locations
Lb = distance between braces on individual
concentrically loaded compression member
to be braced
The minimum required brace stiffness is determined:
For ASD∶ 𝛽br =
2[4 − (2∕𝑛)]
(Ω𝑃 ra )
𝐿b
Ω = 2.00
(5.86a)
In the 2007 edition of the North American Specification,
the required brace strength (brace force) and the required
brace stiffness are determined based on the nominal strength
of the column to be braced. The design provisions have been
238
5
COMPRESSION MEMBERS
revised in the 2016 edition to determine the required brace
strength and the brace stiffness based on the axial force in the
compression member to be braced. This change will result
in a more economical design. However, if the design load in
the compression member may change during the design, the
required brace strength and the required brace stiffness may
be determined based on the compression member nominal
strength, i.e., replace 𝑃 ra in Eq. (5.85) with nominal strength
of the compression member to be braced, Pn ; and (Ω𝑃 ra ) in
Eq. (5.86a) and (𝑃 ra ∕𝜙) in Eq. (5.86b) with 𝑃n .
The above design requirements were developed from a
study conducted by Green, Sputo, and Urala in 2004.5.193
These provisions for lateral translation assume that the braces
are perpendicular to the compression member being braced
and located in the plane of buckling. The stiffness requirements include the contributions of the bracing members,
connections, and anchorage details.1.346 It should be noted
that these requirements are only for lateral translation of the
compression member. They do not account for torsion or
flexural–torsional buckling, which may be designed through
rational analysis or other methods. Additional design information on bracing can be found from the publications of
Helwig and Yura,5.194 Green, Sputo, and Urala,5.195 Sputo
and Turner,5.196 Sputo and Beery,5.197 Tovar, Helwig, and
Sputo,5.207 and others.
5.10
DESIGN EXAMPLES
Figure 5.25
Example 5.1.
2. Nominal Strength due to Global Buckling Pne . Since
the square tube is a doubly symmetric closed section,
it will not be subject to torsional or flexural–torsional
buckling. It can be designed for flexural buckling.
For closed-box sections, the radius of gyration may
need to be revised per Specification Section E2.1.1:
𝐹cr𝓁 =
=
𝑘𝜋 2 E
12(1 − μ2 )(𝑤∕𝑡)2
(4.0)𝜋 2 (29,500)
12(1 − 0.32 )[(7.415)∕(0.105)]2
√
𝐿0 = πr
𝐸
= π(3.212)
𝐹cr𝓁
√
= 21.358 ksi
29,500
= 374.78 in.
21.385
0.35(KL)
0.35(10 × 12)
= 0.65 +
= 0.752
1.1𝐿0
1.1(374.78)
Since KL = 120 in < 1.1𝐿0 = 412.26 in., the radius of
gyration needs to be modified:
𝑅𝑟 = 0.65 +
Example 5.1 Determine the allowable axial load based
on ASD and LRFD methods for the square tubular
column shown in Fig. 5.25. Assume that 𝐹y = 40 ksi,
𝐾x 𝐿x = 𝐾y 𝐿y = 10 ft, and the dead load–live load ratio is
1
. Both the Effective Width Method and the Direct Strength
5
Method are employed in determining the member local
buckling strength.
SOLUTION
1. Sectional Properties of Full Section
𝐴 = 4[7.415 × 0.105 + 0.0396] = 3.273 in.2
[
(
)2 ]
1
1
3
= 2(0.105)
(7.415) + 7.415 4 − × 0.105
12
2
2
+ 4(0.0396)(4 − 0.1373) = 33.763 in.
√
√
𝐼𝑥
33.763
𝑟𝑥 = 𝑟𝑦 =
=
= 3.212 in.
𝐴
3.273
According to Eq. (5.54), the elastic flexural buckling
stress 𝐹cre is computed as follows (with the modified
rm is used):
𝐾𝐿 10 × 12
=
= 49.69 < 200
OK
𝑟𝑚
2.415
𝜋 2 (29,500)
𝜋2𝐸
=
= 117.92 ksi
2
(KL∕𝑟𝑚 )
(49.69)2
√
√
𝐹y
40
=
𝜆c =
= 0.582 < 1.5
𝐹𝑐𝑟e
117.92
𝐹𝑐𝑟e =
w = 8.00 − 2(R + t) = 7.415 in.
𝐼𝑥 = 𝐼𝑦
𝑟𝑚 = (𝑅𝑟 )(𝑟𝑥 ) = (0.752)(3.212) = 2.415 in.
4
𝐹n = (0.658𝜆c )𝐹y = (0.6580.582 )40
2
2
= 34.71 ksi
Pne = 𝐹𝑛 𝐴 = (34.71)(3.273) = 113.6 kips
3. Nominal Strength Due to Local Buckling Interacting with Global Buckling, Pn𝓁 .
DESIGN EXAMPLES
a. Effective Width Method. Because the given square
tube is composed of four stiffened elements, the
effective width of stiffened elements subjected to
uniform compression can be computed from Eqs.
(3.32)–(3.35) where 𝑓 = 𝐹𝑛 determined in Item 2
and 𝑘 = 4.0:
𝑤 7.415
=
= 70.619 < 500 OK
𝑡
0.105
√
( ) 𝐹
1.052 𝑤
n
𝜆𝓁 = √
𝑡
𝐸
𝑘
(
)
√
34.71
1.052
𝜆𝓁 =
(70.619)
= 1.274
√
29,500
4
Since 𝜆𝓁 > 0.673, from Eq. (3.32),
𝑏 = 𝜌w
where
1 − 0.22∕𝜆 1 − 0.22∕1.274
=
= 0.649
𝜆
1.274
Therefore,
𝜌=
𝑏 = (0.649)(7.415) = 4.812 in.
The effective area is
𝐴e = 3.273 − 4(7.415 − 4.812)(0.105)
= 2.180 in.2
𝑃n𝓁 = 𝐹n 𝐴e = (34.71)(2.180) = 75.67 kips
b. Direct Strength Method. The local buckling of the
tubular section can be determined using Eq. (3.45):
𝐹cr𝓁 =
=
𝑘𝜋 2 𝐸
12(1 − 𝜇2 )(𝑤∕𝑡)2
(4.0)𝜋 2 (29,500)
12(1−0.32 )[(7.415)∕(0.105)]2
= 21.385 ksi
Pcr𝓁 = 𝐹cr𝓁 𝐴 = 69.985 kips
A numerical analysis is also performed using the
CUFSM program. The following results are obtained
from the program output:
𝐴 = 3.2785 in.2
𝑃y = 131.14 kips
From the analysis the local buckling is the only mode
in addition to global buckling and the buckling load
factor (𝑃cr𝓁 ∕𝑃y ) = 0.4692.
The local buckling load 𝑃cr𝓁 = 0.4692 𝑃y =
61.53 kips.
239
The above numerical analysis result is close to result
from Eq. (3.45). In this example, the result from the
numerical analysis, i.e. Pcr𝓁 = 61.53 kips, is used.
√
√
𝑃ne
113.6
𝜆𝓁 =
=
= 1.36 > 0.776
𝑃cr𝓁
61.53
Therefore, the local buckling strength is calculated:
[
(
) ](
)
𝑃cr𝓁 0.4
𝑃cr𝓁 0.4
𝑃ne
𝑃n𝓁 = 1 − 0.15
𝑃ne
𝑃ne
[
) ](
)
(
61.53 0.4 61.53 0.4
= 1 − 0.15
(113.6)
113.6
113.6
= 78.46 kips
4. Nominal Strength Due to Distortional Buckling,
Pnd . The given closed section does not subject to
distortional buckling. Therefore, this buckling mode
does not need to be considered.
5. Member Strength. Since the section satisfied the limitations given in Table 3.1, the safety and resistance
factors provided in Specification Chapter E can be
used. Since the safety and the resistance factors are the
same for all the failure modes: Ωc = 1.80 for ASD and
𝜙c = 0.85 for LRFD, the member strength is controlled
by the minimum nominal strength calculated in Items
2 and 3, i.e., the nominal strength due to local buckling interacting with the global buckling controls in this
example.
Based on the Effective Width Method prediction:
The nominal strength: 𝑃n = 75.67 kips
Allowable strength 𝑃a = 𝑃n ∕Ωc = 75.67∕1.80 =
42.04 kips
Design strength 𝜙c 𝑃n = (0.85)(75.67) = 64.32 kips
Based on the Direct Strength Method prediction:
The nominal strength: 𝑃n = 78.46 kips
Allowable strength 𝑃a = 𝑃n ∕Ωc = 78.46∕1.80 =
43.59 kips
Design strength 𝜙c 𝑃n = (0.85)(78.46) = 66.69 kips
The calculated values from the Effective Width
Method and the Direct Strength Method are very close
in this example.
6. Allowable Loads.
Based on the load combination of dead and live
loads, the required load is
𝑃u = 1.2𝑃D + 1.6𝑃L = 1.2𝑃D + 1.6(5PD ) = 9.2PD
where PD = axial load due to dead load
PL = axial load due to live load
240
5
COMPRESSION MEMBERS
By using 𝑃u = ϕc 𝑃n , the values of 𝑃D and 𝑃L are
computed as follows:
Based on the Effective Width Method:
64.32
𝑃D =
= 6.99 kips
9.2
𝑃L = 5𝑃D = 34.96 kip
Therefore, the allowable axial load is
𝑃a = 𝑃D + 𝑃L = 41.95 kips
Based on the Direct Strength Method:
66.69
= 7.25 kips
9.2
𝑃L = 5𝑃D = 36.2 kip
𝑃D =
be computed:
A = 2.24 in.2
𝐼𝑥 = 22.1 in.4
𝑟𝑥 = 3.15 in.
𝐾𝑥 𝐿𝑥 (1)(12 × 12)
=
= 45.714
𝑟𝑥
3.15
𝐾 𝑦 𝐿𝑦
𝑃a = 𝑃D + 𝑃L = 43.45 kips
Example 5.2 Use the ASD and LRFD methods to determine the available strengths of the I-section (Fig. 5.26) to
be used as a compression member. Assume that the effective length factor K is 1.0 for the x and y axes and that the
unbraced lengths for the x and y axes are 12 and 6 ft, respectively. Also assume that 𝐾t 𝐿t = 6 ft. Use 𝐹y = 33 ksi . The
intermediate fastener spacing is assumed to be 12 in.
SOLUTION
1. Properties of Full Section. Based on the method used
in Chapter 4, the following full section properties can
𝑟𝑦 = 1.37 in.
2. Nominal Strength due to Global Buckling, Pne .
Since the given I-section is a doubly symmetric
section, the nominal buckling stress will be governed
by either flexural buckling or torsional buckling as
discussed in Section 5.2.3.1.
a. Elastic Flexural Buckling. By using Eq. (5.54), the
elastic flexural buckling stress can be computed as
follows:
Therefore, the allowable axial load is
It can be seen that the allowable axial loads determined by the ASD and LRFD methods are practically
the same.
𝐼𝑦 = 4.20 in.4
𝑟𝑦
=
(1)(6 × 12)
= 52.555
1.37
Use
𝐾𝐿
= 52.555
𝑟
Since the slenderness ratio KL/r is governed by
the column buckling about the y axis of the I-section,
which involves relative deformations that produce
shear forces in the connections between individual
channels, the modified slenderness ratio (KL∕𝑟)m
should be used to compute the elastic flexural buckling stress 𝐹cre . Based on Eq. (5.84),
√
( )2 ( )2
( )
𝑎
KL
KL
=
+
𝑟 m
𝑟 0
𝑟i
where (KL/r)0 = 52.555
a = intermediate fastener
spacing, = 12 in.
ri = radius of gyration of a
channel section about its y
axis, = 1.08 in.
Therefore,
(
KL
𝑟
√
)
=
m
(52.555)2 +
(
12
1.08
)2
= 53.717
Since 𝑎∕𝑟𝑖 < 0.5(KL∕𝑟)m , requirement 1 of
Section 5.8 is satisfied,
Figure 5.26 Example 5.2.
𝐹𝑐𝑟e =
𝜋 2 (29,500)
𝜋2𝐸
=
= 100.902 ksi
2
(53.717)2
(KL∕𝑟)m
241
DESIGN EXAMPLES
b. Elastic Torsional Buckling. Using Eq. (5.22) of
Section 5.2.3.1, the torsional buckling stress is
[
]
𝜋 2 ECw
1
𝐹𝑐𝑟e = 𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
where A = √
2.24 in.2
𝑟2𝑥 + 𝑟2𝑦
√
= (3.15)2 + (1.37)2 = 3.435 in.
G = 11,300 ksi
J = 0.00418 in.4
Cw = 70.70 in.6
E = 29,500 ksi
Kt Lt = 6 ft
r0 =
Therefore
𝐹𝑐𝑟e =
1
(2.24)(3.435)2
[
]
𝜋 2 (29,500)(70.70)
(11,300)(0.00418) +
(6 × 12)2
= 152.02 ksi
The nominal buckling stress 𝐹n is determined by
using the smaller value of the elastic flexural buckling stress and torsional buckling stress, that is,
𝐹𝑐𝑟e = 100.902 ksi
√
√
𝐹y
33
=
𝜆c =
= 0.572 < 1.5
𝐹𝑐𝑟e
100.902
From Eq. (5.51),
𝜆2c
𝐹n = (0.658 )𝐹y = (0.658
0.5722
)(33) = 28.777 ksi
From Eq. (5.50),
𝑃ne = 𝐴𝑔 𝐹𝑛 = (2.24)(28.777) = 64.460 kips
a. Effective Width of the Compression Flanges
(Section 3.3.3.1). From Eq. (3.72) with f = Fn ,
√
√
29,500
𝐸
= 1.28
= 40.982
𝑆 = 1.28
𝐹𝑛
28.777
0.328𝑆 = 13.442
𝑤2 2.6625
=
= 35.50
𝑡
0.075
Since 𝑤2 ∕𝑡 > 0.328𝑆, use Eq. (3.73) to compute
the required moment of inertia of the edge stiffener
𝑙a as follows:
(
)3
𝑤2 ∕𝑡
𝐼a = 399𝑡4
− 0.328
𝑆
(
)3
35.50
= 399(0.075)4
− 0.328 = 0.002 in.4
40.982
The above computed value should not exceed the
following value:
[(
)
]
115(𝑤2 ∕𝑡)
4
𝐼𝑎 = 𝑡
+5
𝑆
[(
)
]
115(35.50)
= (0.075)4
+ 5 = 0.0033 in.4
40.982
Therefore, use 𝑙𝑎 = 0.002 in.4 For the simple lip
edge stiffener,
𝐷 = 0.7in.𝑑 = 0.5313 in.
𝑑
0.5313
=
= 7.08
𝑡
0.075
By using Eq. (3.75), the moment of inertia of the
full edge stiffener is
3
𝑑 3 𝑡 (𝑤1 ) 𝑡
=
12
12
3
(0.5313) (0.075)
=
= 0.000937 in.4
12
From Eq. (3.74),
𝐼𝑠 =
𝑅𝐼 =
3. Nominal Strength due to Local Buckling Interacting with Global Buckling, Pn𝓁 . Both the Effective
Width Method and the Direct Strength Method can be
used to determine the nominal strength. Both methods
are illustrated.
Effective width method:
Lip ∶ 𝑤1 = 0.7 − (𝑅 + 𝑡) = 0.5313 in.
Flange ∶ 𝑤2 = 3.0 − 2(𝑅 + 𝑡) = 2.6625 in
Web ∶ 𝑤3 = 8.0 − 2(𝑅 + 𝑡) = 7.6625 in.
𝐼𝑠
0.000937
=
= 0.469 < 1.0
𝐼𝑎
0.0020
OK
The effective width b of the compression flange
can be calculated as follows:
𝐷
0.7
=
= 0.263
𝑤2
2.6625
From Eq. (3.76),
𝑤2 ∕𝑡
35.50
= 0.582 −
4𝑆
4 × 40.982
1
= 0.365 >
3
𝑛 = 0.582 −
242
5
COMPRESSION MEMBERS
Use 𝑛 = 0.365. Since 0.25 < 𝐷∕𝑤2 < 0.8 and
θ = 90∘ ,
(
)
5𝐷
𝑘 = 4.82 −
(𝑅I )𝑛 + 0.43
𝑤2
= [4.82 − 5(0.263)](0.469)0.365 + 0.43
Use 𝑘 = 3.09 to compute the effective width of
the compression flange.
From Eqs. (3.32)–(3.35),
( ) 𝐹
1.052 𝑤2
1.052
𝑛
(35.50)
𝜆= √
=√
𝑡
𝐸
𝑘
3.09
= 0.664 < 0.673
𝐴e = 2.24 − [4(0.5313 − 0.249)
+ 2(7.6625 − 3.969)](0.075)
= 2.24 − 0.639 = 1.601 in.2
e. Nominal Axial Strength due to Local Buckling. The
nominal load is
= 3.09 < 4.0
√
d. Effective Area 𝐴e .
√
28.777
29,500
𝜌 = 1.0
𝑏 = 𝑤2 = 2.6625 in.
𝑃n𝓁 = 𝐴e 𝐹n = (1.601)(28.777) = 46.07 kips
Direct Strength Method:
The nominal strength, 𝑃n𝓁 , of the I-Section can be
considered as the sum of the local buckling of the
individual C-Sections. Using the numerical analysis
software CUFSM, the following analysis results are
obtained for the single C-Section:
Yield strength, 𝑃y = 36.802 kips
Local buckling load factor (i.e., 𝑃cr𝓁 ∕𝑃y ): LF local =
0.41
b. Effective Width of Edge Stiffeners
w1 0.5313
=
= 7.084
𝑡
0.075
√
( ) 𝐹
1.052 𝑤1
𝑛
𝜆= √
𝑡
𝐸
𝑘
√
1.052
28.777
(7.084)
=√
29,500
0.43
= 0.355 < 0.673
𝑑𝑠′ = 𝑤1 = 0.5313 in.
Based on Eq. (3.71), the reduced effective width of
the edge stiffener is 𝑑𝑠 = 𝑅𝐼 𝑑𝑠′ = (0.469)(0.5313) =
0.249 in. < 𝑑𝑠′
OK
c. Effective Width of Web Elements
𝑤3 7.6625
=
= 102.167 < 500
OK
𝑡
0.075
√
( ) 𝐹
1.052 𝑤3
𝑛
𝜆= √
𝑡
𝐸
𝑘
√
1.052
28.777
= √ (102.167)
29,500
4
= 1.678 > 0.673
1 − 0.22∕𝜆 1 − 0.22∕1.678
𝜌=
=
= 0.518
𝜆
1.678
𝑏 = 𝜌𝑤3 = (0.518)(7.6625) = 3.969 in.
Therefore, the local buckling load is:
𝑃cr𝓁 = LF local 𝑃y = (0.41)(36.802) = 15.09 kips
The I-Section local buckling load is
2𝑃cr𝓁 = 2(15.09) = 30.18 kips
Using Eqs. (5.64) to (5.66):
√
√
𝑃ne
64.460
=
= 1.46 > 0.776
𝜆𝓁 =
𝑃cr𝓁
30.18
[
(
) ](
)
𝑃cr𝓁 0.4
𝑃cr𝓁 0.4
𝑃ne
𝑃n𝓁 = 1 − 0.15
𝑃ne
𝑃ne
[
) ](
)
(
30.18 0.4
30.18 0.4
= 1 − 0.15
(64.460)
64.460
64.460
= 42.32 kips
4. Nominal Strength due to Distortional Buckling,
Pnd . Because edge-stiffened flanges are used for the
I-section, the nominal axial load for distortional buckling should be checked according to Section E4 of
the North American Specification as provided here in
Section 5.6.
The distortional buckling load, 𝑃crd , can be determined analytically using the equations provided Specification Section 2.3.1.3 which is included in Section
3.5.2. The distortional buckling load is assumed to be
the sum of the individual C-Sections. The geometric
DESIGN EXAMPLES
flange plus lip properties of C-section are provided in
Section 3.5.2.
ℎ = ℎo − 𝑡 = 8.000 − 0.075 = 7.925 in.
𝑏 = 𝑏 − 𝑡 = 3.000 − 0.075 = 2.925 in.
1
1
𝑑 = 𝐷 − 𝑡 = 0.700 − × 0.075 = 0.6625 in.
2
2
𝐴𝑓 = (𝑏 + 𝑑)𝑡 = (2.925 + 0.6625)(0.075)
= 0.269 in.2
𝐼𝑥f = 𝑡[𝑡2 𝑏2 + 4bd3 + 𝑡2 bd + 𝑑 4 ]∕12(𝑏 + 𝑑)
= (0.075)[(0.075)2 (2.925)2 + 4(2.925)(0.6625)3
+ (0.075)2 (2.925)(0.6625)
+ (0.6625)4 ]∕12(2.925 + 0.6625)
= 0.00637 in.4
𝐼𝑦f = 𝑡[𝑏4 + 4db3 ]∕12(𝑏 + 𝑑)
= (0.075)[(2.925)4 + 4(0.6625)(2.925)3 ]∕12(2.925
+ 0.6625)
= 0.243 in.4
𝐼xyf = tb2 𝑑 2 ∕4(𝑏 + 𝑑)
= (0.075)(2.925)2 (0.6625)2 ∕4(2.925 + 0.6625)
= 0.0196 in.4
𝑥of = 𝑏2 ∕2(𝑏 + 𝑑) = (2.925)2 ∕2(2.925 + 0.6625)
= 1.192 in.
𝑦of = −𝑑 2 ∕2(𝑏 + 𝑑)
= −(0.6625)2 ∕2(2.925 + 0.6625)
= −0.0612 in.
ℎxf = −[𝑏 + 2db]∕2(𝑏 + 𝑑)
2
2
= −[(2.925) + 2(0.6625)(2.925)]∕2(2.925 + 0.6625)
= −1.733 in.
𝐽f = [bt3 + dt3 ]∕3
= [(2.925)(0.075)3 + (0.6625)(0.075)3 ]∕3
= 0.000504 in.4
𝐶wf = 0.0 in.6
According to Eq. (3.110), the critical unbraced
length of distortional buckling, 𝐿crd , can be computed
as follows:
{
𝐿𝑐𝑟d =
6𝜋 4 ℎ0 (1 − 𝜇2 )
𝑡3
243
[
𝐼𝑥f (𝑥0f − ℎxf )2 + 𝐶wf
]}1∕4
2
𝐼xyf
(𝑥 − ℎxf )
𝐼𝑦f 0f
{ 4
[
6𝜋 (8.000)(1 − 0.32 )
=
0.00637
(0.075)3
−
2
× (1.192 − (−1.733))2 + 0 −
}]1∕4
(0.0196)2
0.243
× (1.192 − (−1.733))2
= 25.35 in.
Since (𝐿m = 𝐿y = 72 in.) > 𝐿crd , use L = 25.35 in.
The elastic rotational stiffness provided by the flange
to the flange/web juncture, 𝑘𝜙fe , can be computed from
Eq. (3.106) as follows:
[
( )4
𝜋
𝑘𝜙fe =
EI 𝑥f (𝑥0f − ℎxf )2 + ECwf
𝐿
]
2
( )2
𝐼xyf
𝜋
2
−𝐸
(𝑥0f − ℎxf ) +
GJ 𝑓
𝐼𝑦f
𝐿
)4 [
(
𝜋
=
(29,500)(0.00637)[1.192
25.35
−(−1.733)]2 + (29,500)(0.0) − (29,500)
] (
)2
(0.0196)2
𝜋
2
[1.192 − (−1.733)] +
0.243
25.35
(11,300)(0.000504)
= 0.373 in. − kips∕in.
From Eq. (3.107), the elastic rotational stiffness
provided by the web to the flange/web juncture,
𝑘ϕwe , is
𝐾𝜙we =
Et3
6ℎ0 (1 − 𝜇2 )
(29,500)(0.075)3
6(8.00)(1 − (0.3)2 )
= 0.285 in. − kips∕in.
=
Since no sheathing is attached to the I-section, 𝑘𝜙 =
0. From Eq. (3.108) the geometric rotational stiffness
demanded by the flange from the flange/web juncture,
244
5
COMPRESSION MEMBERS
̃
𝑘𝜙fg , is
̃
𝑘𝜙fg =
( )2
𝜋
𝐿
{
[
(
𝐴f (𝑥0f − ℎxf )
(
− 2𝑦0f (𝑥0f − ℎxf ) ×
}
2
𝐼xyf
𝐼xyf
)2
𝐼𝑦f
)
𝐼𝑦f
Therefore the distortional buckling load for
I-Section is twice the the distortional buckling load of
a single C-Section:
]
+ ℎ2xf + 𝑦20f
+ 𝐼𝑥f + 𝐼𝑦f
)2 {
[
𝜋
(0.269) [1.192 − (−1.733)]2
25.35
)
(
0.0196 2
×
− 2(−0.0612)[1.192 − (−1.733)]
0.243
)
]
(
0.0196
+ (−1.733)2 + (−0.0612)2
×
0.243
}
+ 0.00637 + 0.243
(
=
= 0.0166 (in. − kips∕in.)∕ksi
From Eq. (3.109), the geometric rotational stiffness
demanded by the web from the flange/web juncture,
̃
𝑘𝜙wg , is
( )
( )2 th3
𝜋
0
̃
𝑘𝜙wg =
𝐿
60
)2 ( (0.075)(8.00)3 )
(
𝜋
=
25.35
60
= 0.00983 (in. − kips∕in.)∕ksi
From Eq. (3.105), the elastic distortional buckling
stress 𝐹crd is
𝐹𝑐𝑟d =
𝑘𝜙fe + 𝑘𝜙we + 𝑘𝜙
̃
𝑘𝜙fg + ̃
𝑘𝜙wg
0.373 + 0.285 + 0.0
= 24.90 ksi
0.0166 + 0.00983
The distortional buckling load is
=
𝑃𝑐𝑟d = 2LF dist 𝑃y = 2(0.73)(36.81) = 53.74 kips
The above numerical solution is very close to the
analytical solution. In this example, the analytical solution 𝑃crd = 55.78 kips is used to determine the distortional buckling strength in this example.
Based on Eq. (5.71),
𝑃𝑦 = 𝐴𝑔 𝐹𝑦 = (2.24)(33) = 73.92 kips
√
√
𝑃y
73.92
𝜆d =
=
= 1.151 > 0.561
𝑃𝑐𝑟d
55.78
From Eq. (5.70), the nominal axial load for distortional buckling based on Section C4.2(b) of the specification is
[
(
) ](
)
𝑃𝑐𝑟d 0.6
𝑃𝑐𝑟d 0.6
𝑃nd = 1 − 0.25
𝑃y
𝑃y
𝑃y
[
) ](
)
(
55.78 0.6 55.78 0.6
= 1 − 0.25
(73.92)
73.92
73.92
= 49.25 kips
5. Member Strength. By inspection, the given I-section
satisfies the limits of Table 3.1. Therefore the safety and
resistance factors given in Specification Chapter E can
be used to determine the member strength. Since safety
and resistances factors for all the buckling modes are
the same: ΩC = 1.80 for ASD and 𝜙c = 0.85 for LRFD,
the member strength is governed by the minimum
nominal strength due to global buckling (𝑃ne ), local
buckling (𝑃n𝓁 ) and distortional buckling (𝑃nd ). From
the calculations above, the local buckling interacting
with global buckling governs the design:
Using the Effective Width Method:
𝑃𝑐𝑟d = 𝐴g 𝐹d = (2.24)(24.90) = 55.78 kips
𝑃n = 𝑃n𝓁 = 46.07 kips
The numerical analysis method can also be used
to determine the distortional buckling load. Using
CUFSM program, the following results are obrained
for a single channel section:
Therefore, the available strengths are calculated:
ASD method: 𝑃a = 𝑃n ∕Ωc = 46.07∕1.80 = 25.59 kips
LRFD method: 𝜙c 𝑃n = (0.85)(46.07) = 39.16 kips
Yield strength, 𝑃y = 36.81 kips
Using the Direct Strength Method: 𝑃n = 𝑃n𝓁 =
42.32 kips
Load factor for distortional buckling,
ASD method: 𝑃𝑎 = 𝑃n ∕Ωc = 42.32∕1.80 = 23.51 kips
LF dist (i.e., 𝑃𝑐𝑟d ∕𝑃y ) = 0.73
LRFD method: 𝜙c 𝑃n = (0.85)(42.32) = 35.97 kips
DESIGN EXAMPLES
245
𝑎 = 8 − 0.135 = 7.865 in.
𝑏 = 3 − 12 × 0.135 = 2.9325 in.
𝑐=0
𝑏 2.9325
=
= 0.373
7.865
𝑎
𝑐
=0
𝑎
0.135
𝑡
=
= 0.0022
2
(7.862)2
𝑎
Figure 5.27
From Fig. 5.12, it can be seen that because the
value of 𝑡∕𝑎2 is so small, it is difficult to obtain
the accurate value of the critical length 𝐿cr by using
the graphic method.
b. Theoretical Solution. As shown in Fig. 5.7 and
discussed in Section 5.2.3.2, the critical length can
be determined by solving the following equation:
Example 5.3.
Example 5.3 For the channel section shown in Fig. 5.27:
1. Determine the critical length 𝐿cr below which the
flexural–torsional buckling mode is critical.
2. Use the ASD and LRFD methods to determine the
available strengths.
Assume that 𝐾𝑥 𝐿𝑥 = 𝐾𝑦 𝐿𝑦 = 𝐾t 𝐿t = 6ft. Use 𝐹y =
50 ksi.
SOLUTION
1. Sectional Properties of Full Section. By using the
equations given in Part I of the AISI Design Manual
or the methods discussed previously in this book, the
following sectional properties can be computed:
𝐴 = 1.824 in.
2
𝑚 = 1.040 in.
𝐼𝑥 = 17.26 in.
4
𝐽 = 0.01108 in.
𝐼𝑦 = 1.529 in.4
𝐶w = 16.907 in.6
𝑟𝑥 = 3.076 in.
𝑥0 = 1.677 in.
𝑟𝑦 = 0.916 in.
𝑟0 = 3.622 in.
𝛽 = 0.7855
4
2. Critical Unbraced Column Length Lcr . The discussion of Section 5.2.3.2 indicates that the critical
unbraced column length that divides the flexural and
flexural–torsional buckling modes can be determined
by either a graphic method or a theoretical solution, as
illustrated below.
a. Graphic Method. For the given channel section, the
2
values of 𝑏∕𝑎, 𝑐∕𝑎, and 𝑡∕𝑎 according to Fig. 5.12
are as follows:
𝑃𝑦 = (𝑃𝑐𝑟e )3
]
[
√
1
=
(𝑃𝑥 + 𝑃𝑧 ) − (𝑃𝑥 + 𝑃𝑧 )2 − 4𝛽𝑃𝑥 𝑃𝑧
2𝛽
Since the same full area is to be used for computing
𝑃𝑦 , 𝑃𝑥 , and 𝑃𝑧 , the following equation may also be used
to determine 𝐿cr :
]
[
√
1
2
𝜎e𝑦 =
(𝜎e𝑥 + 𝜎t ) − (𝜎e𝑥 + 𝜎t ) − 4𝛽𝜎e𝑥 𝜎t
2𝛽
where
𝜎e𝑦 =
𝜋 2 (29,500)
𝜋2𝐸
=
(𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 )2
(𝐿∕0.916)2
𝜋 2 (29,500)
𝜋2𝐸
=
(𝐾𝑥 𝐿𝑥 ∕𝑟𝑥 )2
(𝐿∕3.076)2
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
[
1
=
(11300)(0.01108)
(1.824)(3.622)2
]
𝜋 2 (29,500)(16.907)
+
𝐿2
𝜎e𝑥 =
It should be noted that, in the equations of 𝜎ey ,
𝜎ex , and 𝜎𝑡 , 𝐾𝑥 𝐿𝑥 = 𝐾𝑦 𝐿𝑦 = 𝐾t 𝐿t = 𝐿. By solving the
above equations, the critical length is 91.0 in.
3. Nominal and Allowable Loads.
a. Nominal Strength due to Global Buckling, 𝑃ne . In
view of the facts that the channel section is a singly
symmetric section and that the given effective
246
5
COMPRESSION MEMBERS
length of 72 in. is less than the computed critical
length of 91 in., the nominal axial load for the
given compression member should be governed by
flexural–torsional buckling.
In case the critical length is not known, both
flexural buckling and flexural–torsional buckling
should be considered. The smaller value of the
elastic flexural buckling stress and the elastic
flexural–torsional buckling stress should be used to
compute the nominal buckling stress 𝐹n .
i. Elastic Flexural Buckling Stress: By using Eq.
(5.54) of Section E2.1 of the North American
Specification, the elastic flexural buckling stress
about the y axis can be computed as follows:
𝐾 𝑦 𝐿𝑦
6 × 12
=
= 78.60 < 200
OK
𝑟𝑦
0.916
(𝐹𝑐𝑟e )𝑦 =
𝜋 2 (29,500)
𝜋2𝐸
=
(𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 )2
(78.600)2
= 47.13 ksi
ii. By using Eq. (5.57) of Section E2.2 of the AISI
Specification, the elastic flexural–torsional buckling stress can be determined:
1
[(𝜎 + 𝜎t )
(𝐹𝑐𝑟e )TF =
2𝛽 e𝑥
√
− (𝜎e𝑥 + 𝜎t )2 − 4𝛽𝜎e𝑥 𝜎t ]
where
𝜋 2 (29,500)
𝜋2𝐸
=
𝜎e𝑥 =
(𝐾𝑥 𝐿𝑥 ∕𝑟𝑥 )2
(6 × 12∕3.076)2
= 531.41 ksi
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
=
1
(1.824)(3.622)2
[
]
𝜋 2 (29,500)(16.907)
(11,300)(0.01108) +
(6 × 12)2
= 44.92 ksi
Substituting the values of 𝛽, 𝜎e𝑥 , and 𝜎t into the
equation of (𝐹cre )TF , the elastic flexural–torsional
buckling stress is
From Eq. (5.51),
𝐹n = (0.658𝜆c )𝐹y = (0.6581.065 )(50)
2
= 31.10 ksi
𝑃ne = 𝐴g 𝐹n = (1.824)(31.10) = 56.73 kips
b. Nominal Strength due to Local Buckling Interacting
with Lateral Torsional Buckling, 𝑃𝑛𝓁
Effective Width Method
i. Flange Elements
𝑤 = 3 − (𝑅 + 𝑡) = 3 − (0.1875 + 0.135)
= 2.6775 in.
𝑤 2.6775
=
= 19.83 < 60 OK
𝑡
0.135
𝑘 = 0.43
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
√
1.052
31.10
(19.83)
=√
29,500
0.43
= 1.033 > 0.673
0.22∕𝜆
0.22∕1.033
𝜌=1−
=1−
𝜆
1.033
= 0.762
𝑏 = 𝜌w = (0.762)(2.6775) = 2.040 in.
ii. Web Elements
𝑤 = 8 − 2(𝑅 + 𝑡) = 8 − 2(0.1875 + 0.135)
= 7.355 in.
𝑤 7.355
=
= 54.48 < 500 OK
𝑡
0.135
𝑘 = 4.0
√
1.052
31.10
𝜆 = √ (54.48)
29,500
4.0
= 0.930 > 6.673
0.22∕0.930
𝜌=1−
= 0.821
0.930
𝑏 = 𝜌w = (0.821)(7.355) = 6.038 in.
(𝐹𝑐𝑟e )TF = 44.07 ksi < (𝐹𝑐𝑟e )y = 47.13 ksi
Use 𝐹𝑐𝑟e = 44.07 ksi
√
√
𝐹y
50
=
𝜆c =
= 1.065 < 1.5
𝐹𝑐𝑟e
44.07
2
The effective area is
𝐴e = 𝐴 − [2(2.6775 − 2.040)
+ (7.355 − 6.038)](0.135)
COMPRESSION MEMBERS IN METAL ROOF AND WALL SYSTEMS
= 1.824 − 0.350 = 1.474 in.2
𝑃n𝓁 = 𝐴e 𝐹n = (1.474)(31.10) = 45.84 kips
Direct Strength Method
i. Local buckling load:
By using the finite strip method software,
CUFSM, the local buckling load is obtained as
follows:
𝑃y = 𝐴g 𝐹y = (1.8233)(50) = 91.165 kips
The local buckling load factor from the numerical
analysis is
LF (𝑖.𝑒., 𝑃cr𝓁 ∕𝑃y ) = 0.62
Therefore, the local buckling load:
𝑃cr𝓁 = (0.62)𝑃y = 56.52 kips
ii. Local buckling slenderness factor is obtained
from Eq. (5.66):
√
√
𝜆𝓁 = 𝑃ne ∕𝑃cr𝓁 = 56.73∕56.52 = 1.002 > 0.776
Therefore the nominal strength is determined by
Eq. (5.65):
[
(
) ](
)
𝑃cr𝓁 0.4
𝑃cr𝓁 0.4
𝑃n𝓁 = 1 − 0.15
𝑃ne
𝑃ne
𝑃ne
[
) ](
)
(
56.52 0.4 56.52 0.4
= 1 − 0.15
(56.73)
56.73
56.73
= 48.16 kips
c. Member Strength. Since the cross-section, does
not subject to distortional buckling, the member
strength is governed by global buckling and the local
buckling interacting with the global buckling. In this
example, the nominal strength due to local buckling
controls.
From the Effective Width Method:
𝑃𝑛 = 45.84 kips
Available strengths are:
ASD method: 𝑃a = 45.84∕1.80 = 25.48 kips
LRFD method: 𝜙c 𝑃n = (0.85)(45.84)
= 38.96 kips
From the Direct Strength Method:
𝑃𝑛 = 48.16 kips
ASD method: 𝑃a = 48.16∕1.80 = 26.76 kips
LRFD method: 𝜙𝑐 𝑃n = (0.85)(48.16)
= 40.94 kips
247
5.11 COMPRESSION MEMBERS IN METAL ROOF
AND WALL SYSTEMS
This section considers compression members with one flange
attached to metal roof or wall panels. These metal roofs
or panels provide more or less lateral support and rotational restraints to the connected compression member, and
increase member axial capacity. Two approaches are used to
account such restraints in determining member strength:
1. Use the empirical equations developed based on experimental study. This approach simplifies the design but is
limited to certain member and panel sizes, and member
span lengths that are tested.
2. Use the Direct Strength Method, in which the buckling
loads are determined considering the effects of roof
or panel covering and span continuity. This method is
essentially applicable to any cross-section types and
span lengths. The complexity of this approach is how
to numerically model the system so that the roof or
panel restraints and the structural system connectivity
are realistically modeled.
Both approaches are discussed in the subsections.
5.11.1 Strengths for Members with General
Cross-Sections and Connectivity
The Direct Strength Method provides a means to directly
determine the strength of the members that are connected to
sheathings through-fastened or with standing seam clips. The
method requires that the member buckling force considers
the restraints from the connected sheathing or panels.
Research work4.297–4.303 has been conducted to model these
restraints with springs which possess translational and
rotational stiffnesses. Methods on how to determine the
stiffness based on the deck or sheathing configuration and
connectivities are provided in the study.4.305 Finite element
method such as MASTAN24.312 was then used to determine
the member buckling load. Using this numerical analysis
in metal building wall and roof systems were documented
for bare deck through-fastened to members,4.398,4.303 and
for through-fastened and standing seam insulated metal
panels.4.301
In 2016, the following design provisions were introduced
into the North American Specification:
I6.1.1 Compression Member Design
The nominal axial strength [resistance], 𝑃n , shall be the
minimum of 𝑃ne , 𝑃n𝓁 , and Pnd as given in Specification Sections
I6.1.1.1 to I6.1.1.3. For members meeting the geometric and
material limits of Specification Section B4, the safety and
248
5
COMPRESSION MEMBERS
resistance factors shall be as follows:
ΩC = 1.80 (ASD)
𝜙C = 0.85 (LRFD)
= 0.80 (LSD)
For all other members, the safety and resistance factors in
Specification Section A1.2(c) shall apply. The available strength
[factored resistance] shall be determined in accordance with the
applicable method in Specification Section B3.2.1, B3.2.2, or
B3.2.3.
I6.1.1.1 Flexural, Torsional, or Flexural–Torsional
Buckling
The nominal compressive strength [resistance], 𝑃ne , for flexural, torsional, or flexural–torsional buckling shall be calculated in accordance with Specification Section E2, except 𝐹𝑐𝑟e
or 𝑃𝑐𝑟e shall be determined including lateral, rotational, and
composite stiffness provided by the deck or sheathing, bridging
and bracing, and span continuity.
I6.1.1.2 Local Buckling
The nominal compressive strength [resistance], 𝑃n𝓁 , for local
buckling shall be calculated in accordance with Specification
Section E3, except 𝐹n or 𝑃cr𝓁 shall be determined including
lateral, rotational, and composite stiffness provided by the deck
or sheathing.
unbraced member. The partial restraint for weak-axis buckling is a function of the rotational stiffness provided by the
panel-to-purlin connection. It should be noted that Eq. (5.87)
is applicable only for the roof and wall systems meeting the
conditions listed in Section D6.2.3 of the North American
Specification. This equation is not valid for sections attached
to standing seam roofs.
The following excerpt is adapted from Section I6.2.3 of
the 2016 edition of the North American Specification.
I6.2.3 Compression Members Having One Flange
Through-Fastened to Deck or Sheathing
These provisions shall apply to C- or Z-sections concentrically loaded along their longitudinal axis, with only one flange
attached to deck or sheathing with through fasteners.
The nominal axial strength [resistance] of simple span or
continuous C- or Z-sections shall be calculated in accordance
with (a) and (b). Consideration of distortional buckling in accordance with Specification Section E4 shall be excluded.
(a) The weak axis nominal strength [resistance], 𝑃n , shall be
calculated in accordance with Eq. (5.87). The safety factor and
resistance factors given in this section shall be used to determine
the allowable axial strength or design axial strength [factored
resistance] in accordance with the applicable design method in
Specification Section B3.2.1, B3.2.2, or B3.2.3.
𝑃n = C1 C2 C3 AE∕29500
Ω = 1.80 (ASD)
I6.1.1.3 Distortional Buckling
The nominal compressive strength [resistance], 𝑃nd , for distortional buckling shall be calculated in accordance with Specification Section E4, except 𝑃crd shall be determined including
lateral, rotational, and composite stiffness provided by the deck
or sheathing.
5.11.2 Compressive Strengths for Members
with Specific Cross-Sections and Connectivity
5.11.2.1 Compression Members Having One Flange
Trough Fastened to Deck or Sheathing In 1996, new
design provisions were added in the AISI Specification for
calculating the weak-axis capacity of axially loaded C- or
Z-sections having one flange attached to deck or sheathing
while the other flange unbraced. The same design equations
with minor modifications are included in the 2007 edition
of the North American Specification and retained in Section
I6.2.3 of the North American Specification. Equation (5.87)
was developed by Glaser, Kaehler, and Fisher,5.104 and is
also based on the work contained in the reports of Hatch,
Easterling, and Murray5.105 and Simaan.5.106
When a roof purlin or wall girt is subject to windor seismic-generated compression forces, the axial load
capacity of such a compression member is less than that
of a fully braced member but greater than that of an
(5.87)
𝜙 = 0.85 (LRFD)
= 0.80 (LSD)
where
𝐶1 = (0.79x + 0.54)
(5.88)
𝐶2 = (1.17αt + 0.93)
(5.89)
𝐶3 = α(2.5b–1.63d) + 22.8
(5.90)
where x = For Z-sections, fastener distance from outside
web edge divided by flange width, as shown in
Figure 5.28
= For C-sections, flange width minus fastener
distance from outside web edge divided by flange
width, as shown in Figure 5.28
𝛼 = Coefficient for conversion of units
= 1 when t, b, and d are in inches
= 0.0394 when t, b, and d are in mm
= 0.394 when t, b, and d are in cm
t = C- or Z-section thickness
b = C- or Z-section flange width
d = C- or Z-section depth
A = Full unreduced cross-sectional area of C- or
Z-section
E = Modulus of elasticity of steel
= 29,500 ksi for U.S. customary units
249
COMPRESSION MEMBERS IN METAL ROOF AND WALL SYSTEMS
Figure 5.28
Definition of x.
= 203,000 MPa for SI units
= 2,070,000 kg/cm2 for MKS units
a
(5.91)
b
b−a
(5.92)
For C-section, x =
b
Eq. (5.87) shall be limited to roof and wall systems meeting
the following conditions:
For Z-section, x =
𝑡 ≤ 0.125 in. (3.22 mm),
6 in. (152 mm) ≤ 𝑑 ≤ 12 in. (305 mm),
Flanges are edge-stiffened compression elements,
70 ≤ 𝑑∕𝑡 ≤ 170,
2.8 ≤ 𝑑∕𝑏 ≤ 5,
16 ≤ flange flat width∕t ≤ 50,
Both flanges are prevented from moving laterally at the
supports,
8. Steel roof or steel wall panels with fasteners spaced 12 in.
(305 mm) on center or less and having a minimum rotational lateral stiffness of 0.0015 k/in./in. (10,300 N/m/m
or 0.105 kg/cm/cm) (fastener at mid-flange width for
stiffness determination) determined in accordance with
AISI S901,
9. C- and Z-sections having a minimum yield stress of 33
ksi (228 MPa or 2320 kg∕cm2 ), and
10. Span length not exceeding 33 feet (10.1 m).
1.
2.
3.
4.
5.
6.
7.
(b) The strong axis available strength [factored resistance] shall
be determined in accordance with Specification Sections E2
and E3.
5.11.2.2 Compression of Z-Section Members Having
One Flange Fastened to a Standing Seam Roof In
2002, design provisions were prepared for calculating
the weak-axis nominal strengths of concentrically loaded
Z-section members having one flange attached to a standing
seam roof. Equation (5.93) was developed by Stolarczyk,
Fisher, and Ghorbanpoor5.198 to predict the lateral buckling
strength on the member using a buckling stress (𝑘af RF y ),
for which the flexural stress (RF y ) is determined from
uplift tests according to the AISI S908, Test Standard for
Determining the Flexural Strength Reduction Factor of
Purlins Supporting a Standing Seam Roof System. It should
be noted that in Eq. (5.93), the gross area A is used instead
of the effective area 𝐴e because the compressive stress in
the Z-section is generally not large enough to result in a
significant reduction in the effective area. This equation can
only be used for a roof system meeting the conditions listed
in Section I6.2.4 of Appendix A in the North American
specification.
The following excerpt is adapted from Section I6.2.4
(Appendix A) of the 2016 edition of the North American
specification:1.345
I6.2.4 Z-Section Compression Members Having One
Flange Fastened to a Standing Seam Roof
These provisions shall apply to Z-sections concentrically loaded
along their longitudinal axis, with only one flange attached to
standing seam roof panels. Alternatively, design values for a
particular system are permitted to be based on discrete point
bracing locations, or on tests in accordance with Specification
Section K2.
The nominal axial strength, Pn , of simple span or continuous
Z-sections shall be calculated in accordance with (a) and (b).
Consideration of distortional buckling in accordance with Specification Section E4 is permitted to be excluded.
Unless otherwise specified, the safety factor and the resistance factor provided in this section shall be used to determine
the available strengths in accordance with the applicable design
method in Specification Section B3.2.1 or B3.2.2.
(a) For weak-axis available strength
𝑃n = 𝑘af RF y 𝐴
(5.93)
Ω = 1.80 (ASD)
𝜙 = 0.85 (LRFD)
where For d/t ≤ 90
kaf = 0.36
For 90 < d/t ≤ 130
𝑘af = 0.72 −
d
250t
(5.94)
For d/t > 130
kaf = 0.20
R = Reduction factor determined from uplift
tests performed using AISI S908
A = Full unreduced cross-sectional area of
Z-section
d = Z-section depth
t = Z-section thickness
Fy = Design yield stress determined in
accordance with Specification Section
A3.3.1
Eq. (5.93) shall be limited to roof systems meeting the
following conditions:
250
5
COMPRESSION MEMBERS
1. Purlin thickness, 0.054 in. (1.37 mm) ≤ t ≤ 0.125 in.
(3.22 mm),
2. 6 in. (152 mm) ≤ d ≤ 12 in. (305 mm),
3. Flanges are edge-stiffened compression elements,
4. 70 ≤ 𝑑∕𝑡 ≤ 170,
5. 2.8 ≤ 𝑑∕b < 5, where 𝑏 = Z-section flange width,
flange flat width
< 50,
6. 16 ≤
t
7. Both flanges are prevented from moving laterally at the
supports, and
8. Yield stress, 𝐹y ≤ 70 ksi (483 MPa or 4920 kg∕cm2 ).
(b) The available strength about the strong axis shall be determined in accordance with Specification Sections E2 and E3.
5.12 ADDITIONAL INFORMATION
ON COMPRESSION MEMBERS
Additional analytical and experimental studies have been
conducted by many investigators. References 5.69–5.92
report on the research findings on doubly symmetric
sections, box sections, channels, Z-sections, and multicell plate columns. The strength evaluation and design
of cold-formed steel columns are discussed in Refs.
5.116–5.140, 5.199–5.206, 5.208, and 5.209 report on
more studies on compression members. Additional publications can be found from other conference proceedings and
engineering journals.
CHAPTER 6
Combined Axial Load and Bending
6.1 GENERAL REMARKS
Structural members are often subject to combined bending
and axial load either in tension or in compression. In the
1996 edition of the AISI Specification, the design provisions for combined axial load and bending were expanded
to include specific requirements in Section C5.1 for the
design of cold-formed steel structural members subjected to
combined tensile axial load and bending. The same requirements are retained in the North American Specification.
When structural members are subject to combined
compressive axial load and bending, the design provisions
are given in Section H1.2 of the 2016 edition of the North
American Specification. This type of member is usually
referred to as a beam–column. The bending may result from
eccentric loading (Fig. 6.1a), transverse loads (Fig. 6.1b), or
applied moments (Fig. 6.1c). Such members are often found
in framed structures, trusses, and exterior wall studs. In steel
structures, beams are usually supported by columns through
framing angles or brackets on the sides of the columns. The
reactions of beams can be considered as eccentric loading,
which produces bending moments.
The structural behavior of beam–columns depends on the
shape and dimensions of the cross section, the location of
the applied eccentric load, the column length, the condition
of bracing, and so on. For this reason, previous editions of the
AISI Specification have subdivided design provisions into
the following four cases according to the configuration of the
cross section and the type of buckling mode1.4 :
1. Doubly symmetric shapes and shapes not subjected to
torsional or flexural–torsional buckling
2. Locally stable singly symmetric shapes or intermittently fastened components of built-up shapes, which
may be subject to flexural–torsional buckling, loaded
in the plan of symmetry
3. Locally unstable symmetric shapes or intermittently
fastened components of built-up shapes, which may
be subject to flexural–torsional buckling, loaded in the
plan of symmetry
4. Singly symmetric shapes which are unsymmetrically
loaded
The early AISI design provisions for singly symmetric
sections subjected to combined compressive load and
bending were based on an extensive investigation of
flexural–torsional buckling of thin-walled sections under
eccentric load conducted by Winter, Pekoz, and Celibi at
Cornell University.5.66,6.1 The behavior of channel columns
subjected to eccentric loading has also been studied by
Rhodes, Harvey, and Loughlan.5.34,6.2–6.5
In 1986, as a result of the unified approach, Pekoz indicated that both locally stable and unstable beam–columns can
be designed by the simple, well-known interaction equations.
The justification of the AISI design criteria was given in
Ref. 3.17. The 1996 design criteria were verified by Pekoz
and Sumer using the available test results.5.103 In the 2007
edition of the North American Specification, in addition to
the use of the first-order elastic analysis to compute the
required compressive axial strength (P) and flexural strengths
(𝑀𝑥 and 𝑀𝑦 ), reference is also made for the use of the
second-order analysis in accordance with Appendix 2 of the
Specification. Furthermore, the Specification also requires
that each individual ratio in the interaction equations shall
not exceed unity.
In the 2016 edition of the North American Specification, the direct analysis method was introduced. This method
requires that the structural stability and individual member
stability shall be considered in structural analysis and design.
The consideration of stability results in additional axial loads
and moments, which should be considered in member design.
Detailed discussion is provided in Section 6.4.
6.2 COMBINED TENSILE AXIAL LOAD
AND BENDING
6.2.1
Tension Members
For the design of tension members using hot-rolled
steel shapes and built-up members, the AISC
Specifications1.148,3.150,1.411 provide design provisions
for the following three limit states: (1) tensile yielding in the
full cross-section between connections, (2) tensile rupture
in the effective net cross-section at the connection, and (3)
block shear rupture at the connection.
251
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
252
6
COMBINED AXIAL LOAD AND BENDING
Figure 6.1 Beam–columns: (a) subject to eccentric loads;
(b) subject to axial and transverse loads; (c) subject to axial loads
and end moments.
For cold-formed steel design, Section C2 of the 1996 AISI
Specification provided Eq. (6.1) for calculating the nominal
tensile strength of axially loaded tension members, with a
safety factor for the ASD method and a resistance factor for
the LRFD method as follows:
𝑇n = 𝐴n 𝐹y
(6.1)
Ωt = 1.67
(ASD)
𝜙t = 0.95
(LRFD)
where Tn = nominal tensile strength
An = net area of the cross section
Fy = design yield stress
In addition, the nominal tensile strength was also limited
by Section E3.2 of the 1996 Specification for tension in
connected parts.
When a tension member has holes, stress concentration
may result in a higher tensile stress adjacent to a hole to be
about three times the average stress on the net area.6.36 With
increasing load and plastic stress redistribution, the stress in
all fibers on the net area will reach the yield stress, as shown
in Fig. 6.2. Consequently, the AISI specification has used
Eq. (6.1) for determining the maximum tensile capacity of
axially loaded tension members since 1946. This AISI design
approach differs significantly from the AISC design provisions, which consider yielding of the gross cross-sectional
area, rupture of the effective net area, and block shear. The
reason for not considering the rupture criterion in the 1996
AISI Specification was mainly due to the lack of research
data relative to the shear lag effect on tensile strength of
cold-formed steel members.
In 1995, the influence of shear lag on the tensile capacity
of bolted connections in cold-formed steel angles and channels was investigated by Carril, Holcomb, LaBoube, and Yu
at the University of Missouri–Rolla. Design equations were
recommended in Refs. 6.23–6.25 for computing the effective
net area. This design information enables the consideration
of rupture strength at connections for angles and channels.
The same study also investigated the tensile strength of staggered bolt patterns in flat-sheet connections.
On the basis of the results of past research, provisions
were revised in the 1999 Supplement to the 1996 edition
of the Specification for the design of axially loaded tension
members.1.333 The same design provisions were included in
Appendix A of the 2001 and 2007 editions of the North
American Specifications1.336,1.346 for the United States and
Mexico. For Canada, the design of tension members has been
based on Appendix B of the North American Specification.
In 2012, tensile member design provisions for the United
States and Mexico (Appendix A) and for Canada (Appendix
B) were unified, and these design provisions were retained
in the 2016 edition of the North American Specification as
excerpted below.
D1 General Requirements
For axially loaded tension members, the available tensile
strength [factored resistance] (𝜙t 𝑇n or 𝑇n ∕Ωt ) shall be the lesser
of the values obtained in accordance with Sections D2 and D3,
where the nominal strengths [resistance] and the corresponding
safety and resistance factors are provided. The available
strengths [factored resistance] shall be determined in accordance with the applicable design method in Specification
Section B3.2.1, B3.2.2, or B3.2.3.
The nominal tensile strength [resistance] shall also be limited
by the connection strength of the tension members, which is
determined in accordance with the provisions of Chapter J of
the Specification.
D2 Yielding of Gross Section
The nominal tensile strength [resistance], 𝑇n , due to yielding of
the gross section shall be determined as follows:
𝑇n = Ag Fy
Ωt = 1.67 (ASD)
𝜙t = 0.90 (LRFD)
Figure 6.2
Stress distribution for nominal tensile strength.
= 0.90 (LSD)
(6.2)
253
COMBINED COMPRESSIVE AXIAL LOAD AND BENDING (BEAM–COLUMNS)
where Ag = Gross area of cross-section
Fy = Design yield stress as determined in accordance
with Specification Section A3.3.1
D3 Rupture of Net Section
The nominal tensile strength [resistance], Tn , due to rupture of
the net section shall be determined as follows:
𝑇n = An Fu
(6.3)
Ωt = 2.00
(ASD)
𝜙t = 0.75
(LRFD)
= 0.75
(LSD)
6.3 COMBINED COMPRESSIVE AXIAL LOAD
AND BENDING (BEAM–COLUMNS)
where An = Net area of cross-section
Fu = Tensile strength as specified in Specification
Section A3.1
6.3.1 Shapes Not Subjected to Torsional or
Flexural–Torsional Buckling1.161
6.2.2 Members Subjected to Combined Tensile Axial
Load and Bending
When cold-formed steel members are subject to concurrent
bending and tensile axial load, the member shall satisfy the
interaction equations given below which are prescribed in
Section H1.1 of the 2016 edition of the North American
Specification.
H1.1 Combined Tensile Axial Load and Bending
The required strengths [effects of factored loads] 𝑇 , 𝑀 x , and
𝑀 y shall satisfy the following interaction equations:
𝑀y
𝑀x
𝑇
+
+
≤ 1.0
𝑀axt 𝑀ayt 𝑇a
(6.4a)
𝑀y
𝑀x
𝑇
+
−
≤ 1.0
𝑀ax 𝑀ay 𝑇a
(6.4b)
where 𝑀 x , 𝑀 y = Required flexural strengths [moment due to
factored loads] with respect to centroidal axes
in accordance with ASD, LRFD, or LSD load
combinations
𝑇 = Required tensile axial strength [tensile axial
force due to factored loads] in accordance
with ASD, LRFD, or LSD load combinations
𝑀axt , 𝑀ayt = Available flexural strengths [factored
resistances] with respect to centroidal axes in
considering tension yielding
(6.5a)
= Sft Fy ∕Ωb (ASD)
= 𝜙b Sft Fy (LRFD, LSD)
Ωb = 1.67
𝜙b = 0.90 (LRFD and LSD)
Max , May = Available flexural strengths [factored
resistances] about centroidal axes in
considering compression buckling, as
determined in accordance with Chapter F
Ta = Available tensile axial strength [factored
resistance], determined in accordance with
Chapter D
(6.5b)
where Sft = Section modulus of full unreduced section
relative to extreme tension fiber about
appropriate axis
Fy = Design yield stress determined in accordance
with Section A3.3.1
When a doubly symmetric open section is subject to axial
compression and bending about its minor axis, the member
may fail flexurally at the location of the maximum moment
by yielding, local or distortional buckling. However, when
the section is subject to an eccentric load that produces a
bending moment about its major axis, the member may fail
flexurally or in a flexural–torsional mode because the eccentric load does not pass through the shear center.
For torsionally stable shapes, such as closed rectangular
tubes, when the bending moment is applied about the
minor axis, the member may fail flexurally in the region of
maximum moment, but when the member is bent about its
major axis, it can fail flexurally about the major or minor
axis, depending on the amount of eccentricities.
If a doubly symmetric I-section is subject to an axial load
𝑃 and end moments 𝑀, as shown in Fig. 6.3a, the combined
axial and bending stress in compression is given in Eq. (6.6)
as long as the member remains straight:
𝑓=
𝑃
𝑀𝑐
𝑃
𝑀
+
= +
𝐴
𝐼
𝐴
𝑆
= 𝑓a + 𝑓b
where 𝑓
𝑓𝑎
𝑓𝑏
𝑃
A
𝑀
c
I
S
=
=
=
=
=
=
=
=
=
(6.6)
combined stress in compression
axial compressive stress
bending stress in compression
applied axial load
cross-sectional area
bending moment
distance from neutral axis to extreme fiber
moment of inertia
section modulus
It should be noted that in the design of such a beam–
column using the ASD method, the combined stress should
254
6
COMBINED AXIAL LOAD AND BENDING
where 𝑀 max = maximum bending moment at mid-length
𝑀 = applied end moments
Φ = amplification factor
Figure 6.3 Beam–column subjected to axial loads and end
moments.
be limited by certain allowable stress F, that is,
𝑓a + 𝑓b ≤ 𝐹
(6.7)
or
𝑓𝑎 𝑓𝑏
+
≤ 1.0
𝐹
𝐹
As discussed in Chapters 3, 4, and 5, the safety factor
for the design of compression members is different from the
safety factor for beam design. Therefore, Eq. (6.7) may be
modified as follows:
𝑓a 𝑓b
+
≤ 1.0
𝐹a 𝐹b
(6.8)
where Fa = allowable stress for design of compression
members
Fb = allowable stress for design of beams
If the strength ratio is used instead of the stress ratio,
Eq. (6.8) can be rewritten as follows:
𝑃
𝑀
+
≤ 1.0
𝑃a 𝑀a
where 𝑃
Pa
𝑀
Ma
=
=
=
=
(6.9)
where 𝑃e = π2 EI(KLb )2 is the elastic column buckling load
(Euler load).
For ASD method, applying a safety factor Ωc to 𝑃e ,
Eq. (6.11) may be rewritten as
ASD method∶Φ =
1
1 − Ωc 𝑃 ∕𝑃e
(6.12)
If the maximum bending moment 𝑀 max is used to replace
𝑀, the following interaction formula can be obtained from
Eqs.(6.9) and (6.11) or (6.12):
Φ𝑀
𝑃
+
≤ 1.0
(6.13)
𝑃a
𝑀a
where amplification factor Φ is determined by Eq. (6.11) for
LRFD and LSD methods, and Eq. (6.12) for ASD method.
It has been found that Eq. (6.13), developed for a member
subjected to an axial compressive load and equal end
moments, can be used with reasonable accuracy for braced
members with unrestrained ends subjected to an axial load
and a uniformly distributed transverse load. However, it
could be conservative for compression members in unbraced
frames (with sidesway) and for members bent in reverse
curvature. For this reason, the interaction formula given in
Eq. (6.13) should be further modified by a coefficient 𝐶m , as
shown in Eq. (6.14), to account for the effect of end moments:
𝐶 Φ𝑀
𝑃
+ m
≤ 1.0
(6.14)
𝑃a
𝑀a
In Eq. (6.14) 𝐶m can be computed by Eq. (6.15) for
restrained compression members braced against joint translation and not subjected to transverse loading:
applied axial load, = A𝑓 𝑎
allowable axial load, = AFa
applied moment, = S𝑓 𝑏
allowable moment, = SFb
Equation (6.9) is a well-known interaction formula which
has been adopted in some specifications for the design of
beam–columns. It can be used with reasonable accuracy for
short members and members subjected to a relatively small
axial load. It should be realized that in practical application,
when end moments are applied to the member, it will be bent,
as shown in Fig. 6.3b, due to the applied moment 𝑀 and
the secondary moment resulting from the applied axial load
𝑃 and the deflection of the member. The maximum bending
moment at mid-length (point C) can be represented by
𝑀 max = Φ𝑀
It can be shown that the amplification factor Φ may be
computed by1.161,2.45
1
Φ=
(6.11)
1 − 𝑃 ∕𝑃e
(6.10)
𝐶m = 0.6 − 0.4
𝑀1
𝑀2
(6.15)
where 𝑀 1 ∕𝑀 2 is the ratio of the smaller to the larger end
moment.
When the maximum moment occurs at braced points,
Eq. (6.16) should be used to check the member at the braced
ends:
𝑃
𝑀
+
≤ 1.0
(6.16)
𝑃a0 𝑀a
where 𝑃a0 is the available axial strength for KL∕𝑟 = 0.
COMBINED COMPRESSIVE AXIAL LOAD AND BENDING (BEAM–COLUMNS)
255
Figure 6.4 Interaction relations for the ASD method.
Furthermore, for the condition of small axial load, the
influence of 𝐶m Φ is usually small and may be neglected.
Therefore when 𝑃 ≤ 0.15𝑃𝑎 , Eq. (6.9) may be used for the
design of beam–columns.
The interaction relations between Eqs. (6.9), (6.14) and
(6.16) are shown in Fig. 6.4. If 𝐶m is unity, Eq. (6.14) controls
over the entire range.
In Table 6.1 below, the values of 𝐶m are summarized,
which is similar to Refs. 1.148 and 6.6. The sign convention
for the end moments is the same as that used for the moment
distribution method (i.e., the clockwise moment is positive
and counterclockwise moment negative).
In categories A and B of Table 6.1, the effective length of
the member is used in computing 𝑃n . The effective length in
the direction of bending is to be used for computing 𝑃Ex or
𝑃Ey whichever is applicable.
In category C, the actual length of the member (𝐾 = 1.0)
is to be used for all calculations. For this case, the value of
𝐶m can be computed by using the following equations5.67,6.6 :
⎧
Ω𝑃
⎪1 + 𝜓 c
𝑃E
⎪
𝐶m = ⎨
⎪
𝑃
⎪1 + 𝜓 𝑃
E
⎩
𝛿 = maximum deflection due to transverse
loading based on ASD, LRFD or LSD
load combinations
𝑀 0 = maximum moment between supports due
to transverse loading based on ASD,
LRFD, or LSD load combinations
𝑃E = 𝑃Ex or 𝑃Ey , whichever is applicable
Values of 𝜓 are given in Table 6.2 for various loading
conditions and end restraints.5.67,1.148,3.150
The interaction equations (6.9), (6.14), and (6.16) have
been adapted by the North American Specification prior to
the 2016 edition. With the consideration of the second-order
effects in determining the member forces, the moment
magnification as well as the end moments effects (𝐶m )
are taken into consideration, the interaction equation then
simplified to the format of Eq. (6.9). Discussion on design
of system stability and beam-column check are provided in
Sections 6.4 and 6.5.
(ASD)
(6.17𝑎)
6.3.2 Open Sections That May Be Subject
to Flexural–Torsional Buckling5.66,6.1
(LRED and LSD)
(6.17𝑏)
When singly symmetric and nonsymmetric open sections
are used as beam–columns, these members may be subject
to flexural–torsional buckling. The following discussion is
based primarily on Ref. 6.1.
where 𝜓 = 𝜋 2 𝛿EI∕(𝑀 0 𝐿2 ) − 1
256
6
COMBINED AXIAL LOAD AND BENDING
Table 6.1 Values of Cm 6.6,1.148
Category
A
Loading
Conditions 𝑃 ∕𝑃𝑎 > 0.15
Computed moments
maximum at end; no
transverse loading; joint
translation not prevented
M
𝐶m
𝑀2
0.85
Remarks
𝑀 1 < 𝑀 2, 𝑀 1 ∕𝑀 2 negative
as shown. Check both
Eqs. (6.9) and (6.14)
B
𝑀2
Computed moments
maximum at end; no
transverse loading; joint
translation prevented
0.6 − 0.4(±𝑀 1 ∕𝑀 2 )
Check both Eqs. (6.9) and
(6.14)
C
Transverse loading; joint
translation prevented
𝑀 2 using Eq. (6.9)
𝐶m per Eq. (6.17)𝑎
𝑀 2 or 𝑀 3 (whichever is
larger) using Eq. (6.14)
𝐶m per Eq. (6.17)𝑎
Check both Eqs. (6.9) and
(6.14)
𝑎
In lieu of using Eq. (6.17), the following values of 𝐶m may be used: For members whose ends are restrained, 𝐶m = 0.85. For members whose
ends are unrestrained, 𝐶m = 1.0
Table 6.2 Values of 𝝍 and Cm 6.6,1.148
The differential equations of equilibrium governing
the elastic behavior of such members are given in
Eqs. (6.18)–(6.20)3.2 :
Cm
𝜓
Case
0
ASD
EI 𝑥 𝑣iv + Pv′′ − Px0 𝜙′′ + 𝑀𝑦 𝜙′′ = 0
(6.18)
EI 𝑦 𝑢 + Pu + Py0 𝜙 − 𝑀𝑥 𝜙 = 0
(6.19)
iv
1.0
1.0
Ω 𝑃
0.4 𝑃c
E
−0.4
1−
−0.4
1 − 0.4
1−
0.4 𝑃𝑃
E
Ωc 𝑃
𝑃E
1 − 0.4 𝑃𝑃
Ω 𝑃
0.2 𝑃c
E
0.2 𝑃𝑃
E
−0.2
1−
−0.3
1 − 0.3
−0.2
LRFD
and LSD
1 − 0.2
E
1−
Ωc 𝑃
𝑃E
1 − 0.3 𝑃𝑃
Ωc 𝑃
𝑃E
1 − 0.2 𝑃𝑃
E
E
′′
′′
′′
ECw 𝜙iv − GJ𝜙′′ + (Pr20 + 𝛽𝑥 𝑀𝑥 + 𝛽𝑦 𝑀𝑦 )𝜙′′
+ Py0 𝑢′′ − Px0 𝑣′′ − 𝑀𝑥 𝑢′′ + 𝑀𝑦 𝑣′′ = 0 (6.20)
where Ix
Iy
u
v
𝜙
x0
y0
=
=
=
=
=
=
=
moment of inertia about x axis
moment of inertia about y axis
lateral displacement in x direction
lateral displacement in y direction
angle of rotation
x coordinate of shear center
y coordinate of shear center
COMBINED COMPRESSIVE AXIAL LOAD AND BENDING (BEAM–COLUMNS)
E = modulus of elasticity, = 29.5 × 103 ksi
(203 GPa, or 2.07 × 106 kg/cm2 )
G = shear modulus, = 11.3 × 103 ksi (78 GPa, or
794 × 103 kg/cm2 )
J = St. Venant torsion constant of cross section, =
1∑ 3
𝑙𝑖 𝑡𝑖
3
Cw = warping constant of torsion of cross section
(Appendix B)
cross section about
r0 = polar radius of gyration of√
√
shear center, = 𝐼0 ∕𝐴 = 𝑟2𝑥 + 𝑟2𝑦 + 𝑥20 + 𝑦20
P = applied concentric load
Mx , My = bending moments about x and y axes,
respectively
𝛽𝑥 =
1
𝑦(𝑥2 + 𝑦2 )dA − 2𝑦0
𝐼𝑥 ∫𝐴
(6.21a)
(see Appendix C)
𝑀𝑥 = Pe𝑦
(6.22)
𝑀𝑦 = Pe𝑥
(6.23)
Consequently, Eqs. (6.18)–(6.20) can be rewritten as
EI 𝑥 𝑣iv + Pv′′ − Pa𝑥 𝜙′′ = 0
(6.24)
EI 𝑦 𝑢iv + Pu′′ + Pa𝑦 𝜙′′ = 0
(6.25)
ECw 𝜙iv +
2
(𝑃 𝑟0
− GJ)𝜙′′ + Pa𝑦 𝑢′′ − Pa𝑥 𝑣′′ = 0
(6.26)
𝑎 𝑥 = 𝑥0 − 𝑒 𝑥
(6.27)
𝑎𝑦 = 𝑦0 − 𝑒𝑦
(6.28)
where
𝐼0
(6.29)
𝐴
The solution of Eqs. (6.24)–(6.26) is shown in Eq. (6.30) by
using Galerkin’s method:
2
(6.21b)
All primes are differentiations with respect to the z axis.
Assume that the end moments 𝑀𝑥 and 𝑀𝑦 are due to the
eccentric loads applied at both ends of the column with equal
(a)
biaxial eccentricities 𝑒𝑦 and 𝑒𝑥 (Fig. 6.5). Then the moments
𝑀𝑥 and 𝑀𝑦 can be replaced by
𝑟0 = 𝛽𝑥 𝑒𝑦 + 𝛽𝑦 𝑒𝑥 +
1
𝑥(𝑥2 + 𝑦2 ) dA − 2𝑥0
𝛽𝑦 =
𝐼𝑦 ∫𝐴
257
′ ⎤ ⎧𝑢 ⎫
⎡ 𝑃e𝑦 − 𝑃
0
−Pa𝑦 𝐾13
0
⎢
⎥⎪ ⎪
′
0
𝑃
−
𝑃
Pa
𝐾
𝑣
⎢
e𝑥
𝑥 23 ⎥ ⎨ 0 ⎬
⎢
⎥⎪ ⎪
2
′
′
⎣−Pa𝑦 𝐾31 Pa𝑥 𝐾32 𝑟0 (𝑃ez − 𝑃 )⎦ ⎩𝜙0 ⎭
(b)
Figure 6.5 (a) Unsymmetrically loaded hat section. (b.) Hat section subjected to an eccentric load
in the plane of symmetry.
258
6
COMBINED AXIAL LOAD AND BENDING
⎫
⎧
⎪
⎪
𝑃2
𝑒 𝐾
−
⎪
⎪
𝑃e𝑦 𝑥 1
⎪
⎪
⎪
⎪
2
𝑃
=⎨
⎬
−
𝑒𝑦 𝐾2
𝑃e𝑥
⎪
⎪
(
) ⎪
⎪
𝑎𝑦 𝑒𝑥 𝑎𝑥 𝑒𝑦
⎪−𝑃 2
−
𝐾3 ⎪
⎪
⎪
𝑃e𝑦
𝑃e𝑥
⎭
⎩
dimensions of the cross section, the column length, and
the eccentricity of the applied load.
The structural behavior discussed above can be explained
by the solution of differential equations [Eqs. (6.24)–(6.26)].
When the eccentric load is applied in the plane of symmetry
of the section, as shown in Fig. 6.5b, 𝑒𝑦 = 𝑦0 = 0.
Equation (6.30) can be changed to the following
two formulas:
𝑃2
𝑒 𝐾
(6.34)
(𝑃e𝑦 − 𝑃 )𝑢0 = −
𝑃e𝑦 𝑥 1
]{ }
[
′
Pa𝑥 𝐾23
𝑃e𝑥 − 𝑃
𝑣0
=0
(6.35)
2
𝜙0
Pa 𝐾 ′
𝑟 (𝑃 − 𝑃 )
(6.30)
where
𝑃e𝑦 = 𝐾11
𝜋 2 EI 𝑦
(6.31)
𝐿2
𝜋 2 EI 𝑥
𝑃e𝑥 = 𝐾22
(6.32)
𝐿2
)
(
1
𝜋2
(6.33)
𝑃e𝑧 = 2 𝐾33 ECw 2 + GJ
𝐿
𝑟0
and 𝑢0 , 𝑣0 , and 𝜙0 are coefficients for deflection components.
The coefficients K for various boundary conditions are listed
in Table 6.3.
6.3.3
𝑥
32
0
e𝑥
in which Eq. (6.34) represents the behavior of a beam–
column deforming flexurally without twist and Eq. (6.35) is
related to flexural–torsional buckling.
If flexural failure governs the maximum strength of the
beam–column, the design of singly symmetric shapes is to
be based on the interaction formulas similar to those used in
Section 6.3.1 for doubly symmetric shapes.
However, if the singly symmetric section fails in flexural–
torsional buckling, the following critical buckling load can
be determined by the equation derived from Eq. (6.35) by
setting the determinant of the coefficient equal to zero:
√
(𝑃e𝑥 + 𝑃e𝑧 ) ± (𝑃e𝑥 + 𝑃e𝑧 )2 − 4𝛽𝑃e𝑥 𝑃e𝑧
𝑃TF =
(6.36)
2𝛽
where
(𝑥 − 𝑒 )2 2
𝛽 = 1 − 0 2 𝑥 𝐾23
(6.37)
𝑟0
For members having simply supported ends and
subjected to concentric loading (that is, 𝑒𝑥 = 0, 𝐾23 = 1.0),
Eq. (6.36) can be changed to Eq. (6.38), which was used in
Section 5.2.3.2 for axially loaded compression members:
√
1
𝑃TFO =
[(𝑃𝑥 + 𝑃𝑧 ) − (𝑃𝑥 + 𝑃𝑧 )2 − 4𝛽𝑃𝑥 𝑃𝑧 (6.38)
2𝛽
Singly Symmetric Open Shapes
Channels, angles, and hat sections are some of the singly
symmetric open shapes. If these members are subject to
bending moments in the plane of symmetry (x axis, as shown
in Fig. 5.6), they may fail in one of the following two ways1 :
1. The member deflects gradually in the plane of
symmetry without twisting and finally fails by
yielding or local buckling at the location of maximum
moment.
2. The member starts with a gradual flexural bending in
the plane of symmetry, but when the load reaches a
critical value, the member will suddenly buckle by
flexural–torsional buckling.
in which 𝛽 = 1 − (𝑥0 ∕𝑟0 )2 as previously defined in
Chapter 5.
The type of failure mode, which will govern the maximum
strength of the member, depends on the shape and
Table 6.3 Coefficients K6.1
Boundary
Conditions
at z = 0, L
𝑢′′ = 𝑣′′ = 𝜙′′ = 0
𝑢′′ = 𝑣′ = 𝜙′′ = 0
𝑢′ = 𝑣′ = 𝜙′′ = 0
𝑢′′ = 𝑣′ = 𝜙′ = 0
𝑢′ = 𝑣′ = 𝜙′ = 0
𝑢′′ = 𝑣′′ = 𝜙′ = 0
K11
K22
K33
K1
K2
K3
′
𝐾13
′
𝐾31
′
𝐾23
′
𝐾32
K23
1.0000
1.0000
4.1223
1.0000
4.1223
1.0000
1.0000
4.1223
4.1223
4.1223
4.1223
1.0000
1.0000
1.0000
1.0000
4.1223
4.1223
4.1223
1.2732
1.2732
⋯
1.2732
⋯
1.2732
1.2732
⋯
⋯
⋯
⋯
1.2732
1.2732
1.2732
1.2732
0.6597
0.6597
0.6597
1.0000
1.0000
0.5507
1.4171
1.0000
1.4171
1.0000
1.0000
1.4171
0.5507
1.0000
0.5507
1.0000
0.5507
0.5507
1.0000
1.0000
1.4171
1.0000
1.4171
1.4171
0.8834
1.0000
0.5507
1.0000
0.8834
0.8834
1.0000
1.0000
0.8834
COMBINED COMPRESSIVE AXIAL LOAD AND BENDING (BEAM–COLUMNS)
259
From Eq. (6.36) it can be seen that the computation of
the flexural–torsional buckling load is time consuming for
design use. A previous study made by Peköz, Celebi, and
Winter indicated that the flexural–torsional buckling load
may be computed by the following interaction formula if the
load is applied on the side of the centroid opposite from that
of the shear center6.1 :
𝑃 𝑒
𝑃TF
+ TF 𝑥 = 1.0
(6.39)
𝑃TFO
𝑀T
where
where PTF = flexural–torsional buckling load for eccentric
load having an eccentricity of ex
PTFO = flexural–torsional buckling load for
concentric load [Eq. (6.38)]
MT = critical moment causing tension on shear
center side of centroid
If the eccentric load is applied on the side of the shear
center opposite from that of the centroid, the critical moment
causing compression on the shear center side of the centroid,
𝑀c , can be computed as follows:
In Eq. (6.39), if we apply the modification factor given in
the equation
𝐶TF
(6.40)
1 − 𝑃TF ∕𝑃e
to the moment 𝑃TF 𝑒x , as done previously in Section 6.3.1, the
interaction formula can be written as
𝐶TF (𝑃TF 𝑒𝑥 )
𝑃TF
+
= 1.0
(6.41)
𝑃TFO (1 − 𝑃TF ∕𝑃e )𝑀T
In the above equation, the factor 𝐶TF is the same as 𝐶m used
in Section 6.3.1.
Equation (6.41) can be used to determine the theoretical elastic flexural–torsional buckling load 𝑃TF for singly
symmetric sections under eccentric loads applied on the side
of the centroid opposite from that of the shear center.
The critical moment 𝑀T used in Eq. (6.41) can be obtained
from the following equation:
[
]
√
′ 𝐼
𝑃
𝑃
1
e𝑧
e𝑥
0
2
𝑀T = − 2 𝛽𝑦 𝑃e𝑥 − (𝛽𝑦 𝑃e𝑥 )2 + 4𝐾23
𝐴
2𝐾
23
(6.42)
where
′
𝑃e𝑧
𝐾23
(
)
𝑒 𝑥 𝛽𝑦 𝐴
= 𝑃e𝑧 1 +
!
𝐼0
√
′ 𝐾′
= 𝐾23
(see Table 6.3)
23
(6.43)
(6.44)
For simply supported end conditions, Eq. (6.42) can be
simplified and rearranged as
√
(
)
⎡
𝑃e𝑧 ⎤⎥
2
2
⎢
𝑀T = −𝑃e𝑥 𝑗 − 𝑗 + 𝑟0
(6.45)
⎢
𝑃e𝑥 ⎥
⎣
⎦
or
√
(
)
⎛
𝜎t ⎤⎥
𝑀T = −𝐴𝜎e𝑥 ⎜𝑗 − 𝑗 2 + 𝑟20
(6.46)
⎜
𝜎e𝑥 ⎥
⎦
⎝
𝑗=
𝛽𝑦
2
=
1
2𝐼𝑦
(
∫𝐴
𝑥3 dA +
∫𝐴
)
xy2 dA − 𝑥0
𝜋2𝐸
(𝐾𝑥 𝐿𝑥 ∕𝑟𝑥 )2
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾𝑡 𝐿𝑡 )2
Ar0
𝜎e𝑥 =
⎡
𝑀c = 𝐴𝜎e𝑥 ⎢𝑗 +
⎢
⎣
(6.48)
(6.49)
√
𝑗2
(
+
(6.47)
𝑟20
)
𝜎t ⎤⎥
𝜎e𝑥 ⎥
⎦
(6.50)
Both Eqs. (6.46) and (6.50) were used in Eq. (4.37) for
determining the elastic critical moment for latera1–torsional
buckling strength.
For the ASD method, the allowable load for flexural–
torsional buckling in the elastic range can be derived from
𝑃TF by using a safety factor of 1.80. The inelastic buckling
stress can be computed by the equation that was used for
flexural–torsional buckling of axially loaded compression
members (Chapter 5).
So far we have discussed the possible failure modes for a
singly symmetric section under eccentric load. However, the
type of failure that will govern the maximum strength of the
beam–column depends on which type of failure falls below
the other for the given eccentricity. This fact can be shown
in Fig. 6.6a. For the given hat section having 𝐿∕𝑟𝑥 = 90,
the section will fail in flexural yielding if the load is applied
in region I. Previous study has indicated that for channels,
angles, and hat sections the section always fails in flexural yielding when the eccentricity is in region I (that is,
𝑒𝑥 < −𝑥0 ). When the eccentricity is in region III (that is,
0 < 𝑒𝑥 ), the section can fail either in flexural yielding or
in flexural–torsional buckling. Therefore, both conditions
(flexural yielding and flexural–torsional buckling) should be
checked. For the given hat section shown in Fig. 6.6a, when
the load is applied at the center of gravity, the section will
buckle flexural–torsionally at a load PTFO that is smaller
than the flexural buckling load 𝑃ey . At a certain eccentricity
in region II (that is, −𝑥0 < 𝑒𝑥 < 0), the failure mode changes
from flexural–torsional buckling to simple flexural failure.
It can also be seen that in this region small changes in eccentricity result in large changes in failure load. Thus any small
inaccuracy in eccentricity could result in nonconservative
designs.
260
6
COMBINED AXIAL LOAD AND BENDING
Figure 6.6 Strength of eccentrically loaded hat section6.1 : (a) load vs. ex ; (b) stress vs. ex .
For bending about the axis of symmetry (i.e., when the
eccentric load is applied along the y axis as shown in
Fig. 6.7), the following equation for determining the elastic
critical moment can be derived from Eq. (6.30) on the basis
of 𝑒𝑥 = 0, 𝑒𝑦 ≠ 0, 𝑃 = 0, and Pe𝑦 = 𝑀𝑥6.1 :
√
𝑀𝑥 = 𝑟0 𝑃e𝑦 𝑃e𝑧
√
= 𝑟0 𝐴 𝜎e𝑦 𝜎t
(6.51)
MEMBER FORCES CONSIDERING STRUCTURAL STABILITY
261
moment magnification due to member deformation as well as
joint translation, effective length factor, K, moment magnification factor, Φ, and coefficient, 𝐶m , can be taken as 1 in a
beam–column interaction check.
In the 2016 edition of the North American Specification,
the requirement of design for system stability is added based
on the AISC Specification and the research of Sarawit and
Pekoz.1.346,6.39,6.43 A structure is considered stable only when
it can maintain in equilibrium under deformed shape when
subjected to applied loads. The effects which shall be considered in structural analysis and design are specified in Section
C1 of the North American Specification.
Three methods are described in the Specification that can
be used for structural stability analysis:
Figure 6.7 Hat section subjected to an eccentric load applied
along the y axis.
For the case of unequal end moments, Eq. (6.51) may
be modified by multiplying by a bending coefficient 𝐶𝑏 as
follows:
√
𝑀𝑥 = 𝐶b 𝑟0 𝐴 𝜎e𝑦 𝜎t
(6.52)
The above equation was used in Eq. (4.34) for lateral–
torsional buckling strength consideration.
6.4 MEMBER FORCES CONSIDERING
STRUCTURAL STABILITY
Section 6.3 discussed the traditional effective length method
for the design of cold-formed steel beam–columns. The
required strengths (forces and moments) are from the firstorder elastic analysis of the undeflected structural geometry. In the interaction equation [Eq. (6.14)], the moment
magnification factor is used to account for the effect of loads
acting on the deflected shape of a member between joints
(P–𝛿 effect). This method also accounts for the effects on
the sidesway stability (P-Δ effect) of unbraced frames using
the effective length factor K.
In the early 2000s, Sarawit and Pekoz conducted an extensive study on industrial steel storage racks at Cornell University sponsored by the Rack Manufacturers Institute and the
American Iron and Steel Institute.1.346, 6.39, 6.43 It was shown
that the second-order analysis gives more accurate results
than the effective length approach. Subsequently, on the basis
of the research findings of Sarawit and Pekoz and the AISC
Specification,1.411 the 2007 edition of the North American
Specification added Appendix 2 to cover the second-order
analysis approach as an alternative method to determine the
required strengths of members for beam–column design.
Since the second-order analysis enables us to consider the
1. Direct Analysis Method Using Rigorous Second-Order
Elastic Analysis. This method considers structural
initial imperfection, stiffness modifications, and
second-order effects in the elastic analysis. The initial
imperfection can be considered by adding the imperfections to the initial structural geometry, or by adding
the notional loads (see Specification Section C1.1.1.2
on determination of notional loads). Since the member
forces and moments include the second-order effects,
the effective length factors 𝐾x and 𝐾y can be taken as 1
in member available strength calculation. This method
can be used for any type of structures and under any
type of loading conditions.
2. Direct Analysis Method Using Amplified First-Order
Elastic Analysis. This method still takes into consideration of initial imperfection and the stiffness reduction as the first method in structural analysis. However,
the method only uses the first-order elastic analysis
of undeflected structural geometry. The axial forces
and moments are then modified by factors 𝐵1 and 𝐵2
(See Specification Section C1.2.1.1 for calculation) to
account of P-𝛿 and P-Δ effects. This method is limited
to structures that support the gravity loads primarily
through nominally vertical columns, walls, or frames.
3. Effective Length Method. This method performs the
structural analysis as described in Item 2, except
no stiffness reduction is needed. The member forces
obtained should be modified by 𝐵1 and 𝐵2 as described
in Item 2. This method is only applicable to structures
support gravity loads primarily through nominally
vertical columns, walls or frames and the ratio of
maximum second-order drift to maximum first-order
drift is equal to or less than 1.5.
The North America Specification also permits other
methods to be used for design as long as the method takes
262
6
COMBINED AXIAL LOAD AND BENDING
into consideration of the factors that affect the structural
stability.
The following provisions for considering structural
stability are excerpted from the 2016 edition of the North
American Specification.1.431 :
C1 Design for System Stability
This section addresses requirements for the elastic design of
structures for stability. System stability shall be provided for the
structure as a whole and for each of its elements. The effects
of all of the following on the stability of the structure and its
elements shall be considered:
(a) Flexural, shear, and axial member deformations, and
all other component and connection deformations that
contribute to displacements of the structure;
(b) Second-order effects (including P-Δ and P-𝛿 effects);
(c) Geometric imperfections;
(d) Stiffness reductions due to inelasticity, including the effect
of residual stresses and partial yielding of the cross-section;
(e) Stiffness reductions due to cross-section deformations or
local and distortional buckling;
(f) Uncertainty in system, member, and connection stiffness
and strength.
All load-dependent effects shall be calculated at a level of
loading corresponding to LRFD load combinations, LSD load
combinations, or 1.6 times ASD load combinations.
Any rational method of design for stability that considers all
of the listed effects is permitted, including the methods identified
in Specification Section C1.1, C1.2, or C1.3 within the limitations stated therein.
C1.1 Direct Analysis Method Using Rigorous
Second-Order Elastic Analysis
The direct analysis method of design, which consists of the
calculation of required strengths [effects due to factored loads]
in accordance with Specification Section C1.1.1 and the calculation of available strengths [factored resistance] in accordance
with Specification Section C1.1.2, is permitted for all systems.
C1.1.1 Determination of Required Strengths
For the direct analysis method of design, the required strengths
[effects due to factored loads] of components of the structure
shall be determined from an analysis conforming to Specification Section C1.1.1.1. The analysis shall include consideration
of initial imperfections in accordance with Specification Section
C1.1.1.2 and adjustments to stiffness in accordance with Specification Section C1.1.1.3.
C1.1.1.1 Analysis
It is permitted to use any elastic analysis method capable of
explicit consideration of the P-Δ and P-𝛿 effects by capturing
the effects of system and member displacements, respectively,
on member forces.
Alternatively, it is permitted to use any elastic analysis
method capable of explicit consideration of the P-Δ effects
by capturing the effects of system displacements on member
forces. The required flexural strength [effect due to factored
loads], 𝑀, shall then be taken as the moment resulting from
such an analysis amplified by B1, where B1 is determined in
accordance with Specification Section C1.2.1.1.
C1.1.1.2 Consideration of Initial Imperfections
Initial imperfections at the points of member intersection shall
be considered as provided by either (A) or (B) below. Additionally, it is permitted, but not required, to consider imperfections
in the initial position of points along members.
(A) Direct Geometric Consideration of Initial Imperfections
In all cases, it is permitted to account for the effect of initial
imperfections by including the imperfections directly in the
analysis. The structure shall be analyzed with points of intersection of members displaced from their nominal locations. The
magnitude of the initial displacements shall be the maximum
amount considered in the design; the pattern of initial displacements shall be such that it provides the greatest destabilizing
effect.
In the analysis of structures that support gravity loads
primarily through nominally vertical columns, walls, or frames,
where the ratio of maximum second-order elastic analysis
story drift to maximum first-order elastic analysis story drift
(both determined for LRFD or LSD load combinations or 1.6
times ASD load combinations, with stiffnesses as specified in
Specification Section C1.1.1.3) in all stories is equal to or less
than 1.7, it is permissible to include initial imperfections only
in the analysis for gravity-only load combinations and not in the
analysis for load combinations that include applied lateral loads.
(B) Consideration of Initial Imperfections Through Application
of Notional Loads
For structures that support gravity loads primarily through nominally vertical columns, walls, or frames, it is permitted to use
notional loads to represent the effects of initial imperfections in
accordance with the requirements of this section. The notional
load shall be applied to a model of the structure based on its
nominal geometry.
(1) Notional loads shall be applied as lateral loads at all
levels. The notional loads shall be additive to other lateral
loads and shall be applied in all load combinations, except
as indicated in (3), below. The magnitude of the notional
loads shall be:
(6.53)
𝑁i = (1∕240)α𝑌𝑖
where α =
=
𝑁𝑖 =
Yi =
1.0 (LRFD or LSD)
1.6 (ASD)
Notional load applied at level i
Gravity load applied at level i from LRFD,
LSD, or ASD load combinations, as
applicable
MEMBER FORCES CONSIDERING STRUCTURAL STABILITY
Where the applicable project or other quality assurance
criteria stipulate a more stringent imperfection criteria,
(1/240) in the above equation is permitted to be replaced
by a lesser value.
(2) The notional load at any level, 𝑁𝑖 , shall be distributed
over that level in the same manner as the gravity load
at the level. The notional loads shall be applied in the
direction that provides the greatest destabilizing effect.
(3) For structures in which the ratio of maximum secondorder elastic analysis story drift to maximum first-order
elastic analysis story drift (both determined for LRFD
load combinations or LSD load combinations, or 1.6
times ASD load combinations, with stiffnesses adjusted
as specified in Specification Section C1.1.1.3) in all
stories is equal to or less than 1.7, it is permitted to
apply the notional load, 𝑁𝑖 , only in gravity-only load
combinations and not in combinations that include other
lateral loads.
C1.1.1.3 Modification of Section Stiffness
The analysis of the structure to determine the required strengths
[effects due to factored loads] of components shall use reduced
stiffnesses, as follows:
(a) A factor of 0.90 shall be applied to all stiffnesses considered
to contribute to the stability of the structure. Additionally,
it is permitted, but not required, to also apply the stiffness
reduction to those members that are not part of the lateral
force resisting system.
(b) An additional factor, Tb , shall be applied to the flexural
stiffnesses of all members whose flexural stiffnesses are
considered to contribute to the stability of the structure.
For α P∕Py ≤ 0.5,
Tb = 1.0
(6.54)
For 𝛼 P∕𝑃y > 0.5,
Tb = 4(αP∕𝑃y )[1 − (αP∕𝑃y )]
(6.55)
where 𝛼 = 1.0 (LRFD or LSD)
= 1.6 (ASD)
P = Required axial compressive strength
[compressive force due to factored loads]
using LRFD, LSD, or ASD load combinations
𝑃y = Axial yield strength
= 𝐹y 𝐴g
(6.56)
where Fy = Yield stress
Ag = Gross area of cross-section
(c) In lieu of using Tb < 1.0 where 𝛼P∕𝑃y > 0.5, it is permitted
to use Tb = 1.0 for all members if a notional load of
(1∕1000)αYi is applied at all levels, in the direction
specified in Specification Section C1.1.1.2, in all load
263
combinations. These notional loads shall be added to those
stipulated in Specification Section C1.1.1.2, except that
Specification C1.1.1.2(3) shall not apply.
(d) Where components comprised of materials other than coldformed steel are considered to contribute to the stability
of the structure, stiffness reductions shall be applied to
those components as required by the codes and specifications governing their design.
C1.1.2 Determination of Available Strengths [Factored
Resistances]
The available strengths [factored resistances] of members and
connections shall be calculated in accordance with the provisions of Specification Chapters D, E, F, G, H, I, J, and K, as applicable, with no further consideration of overall structure stability.
The flexural buckling effective length factors, 𝐾y and 𝐾x , of all
members shall be taken as unity unless a smaller value can be
justified by rational engineering analysis.
Bracing intended to define the unbraced lengths of members
shall have enough stiffness and strength to control member
movement at the braced points, and shall be designed in accordance with Specification Section C2.
When initial imperfections in the position of points along a
member are considered in the analysis in addition to imperfections at the points of intersection as stipulated in Specification
Section C1.1.1.2, it is permissible to take the flexural buckling
strength of the member in the plane of the initial imperfection
as the cross-section strength. The available strengths [factored
resistances] due to torsional, flexural–torsional, local, and distortional buckling of compression members shall be as specified in
Specification Chapter E.
C1.2 Direct Analysis Method Using Amplified FirstOrder Elastic Analysis
The direct analysis method of design, which consists of the
calculation of required strengths [effects due to factored loads]
in accordance with Specification Section C1.2.1 and the calculation of available strengths [factored resistance] in accordance
with Specification Section C1.2.2, shall be limited to structures
that support gravity loads primarily through nominally vertical
columns, walls, or frames.
C1.2.1 Determination of Required Strengths [Effects due
to Factored Loads]
For the direct analysis method of design, the required strengths
[effects due to factored loads] of components of the structure
shall be determined from an analysis conforming to Specification Section C1.2.1.1. The analysis shall include consideration
of initial imperfections in accordance with Specification Section
C1.2.1.2 and adjustments to stiffness in accordance with Specification Section C1.2.1.3.
C1.2.1.1 Analysis
The required flexural strength [moment due to factored
loads], M , and required axial strength [axial force due to
264
6
COMBINED AXIAL LOAD AND BENDING
factored loads], P, of all members shall be determined as
follows:
M = B1 Mnt + B2 M𝓁𝑡
P = Pnt + B2 P𝓁𝑡
(6.57)
(6.58)
where 𝐵1 = Multiplier to account for P-𝛿 effects, determined
for each member subject to compression and
flexure, and each direction of bending of the
member in accordance with Specification
Eq. (6.59), with B1 taken as 1.0 for members not
subject to compression
𝐵2 = Multiplier to account for P-Δ effects, determined
for each story of the structure and each direction
of lateral translation of the story using Eq. (6.62)
M𝓁𝑡 = Moment from first-order elastic analysis using
LRFD, LSD, or ASD load combinations, as
applicable, due to lateral translation of the
structure only
Mnt = Moment from first-order elastic analysis using
LRFD, LSD, or ASD load combinations, as
applicable, with the structure restrained against
lateral translation
M = Required second-order flexural strength [moment
due to factored loads] using LRFD, LSD or ASD
load combinations, as applicable
P𝓁𝑡 = Axial force from first-order elastic analysis using
LRFD, LSD or ASD load combinations, as
applicable, due to lateral translation of the
structure only
Pnt = Axial force from first-order elastic analysis using
LRFD, LSD or ASD load combinations, as
applicable, with the structure restrained against
lateral translation
P = Required second-order axial strength
[compressive force due to factored loads] using
LRFD, LSD or ASD load combinations, as
applicable
The P-𝛿 effect amplifier 𝐵1 shall be determined in accordance
with Eq. (6.59), in which 𝑃 shall be determined by iteration or
is permitted to be taken as Pnt + P𝓁𝑡 .
𝐵1 = 𝐶m ∕(1 − αP∕Pe1 ) ≥ 1.0
(b) For beam–columns subject to transverse loading between
supports, 𝐶m shall be determined either by analysis or
conservatively taken as 1.0 for all cases.
𝑃e1 = Elastic critical buckling strength of the member in
the plane of bending, calculated based on the
assumption of no lateral translation at member
ends
(6.61)
= π2 kf ∕(𝐾1 𝐿)2
where 𝑘f = Flexural stiffness in the plane of bending as
modified in Specification Section C1.2.1.3
L = Unbraced length of member
𝐾1 = Effective length factor for flexural buckling in
the plane of bending, Ky or Kx , as applicable,
calculated based on the assumption of no
lateral translation at member ends
= 1.0 unless analysis justifies a smaller
value
The P-Δ effect amplifier B2 for each story and each direction of lateral translation shall be calculated as follows:
B2 = 1∕[1 − (αPstory )∕Pe,story )] ≥ 1.0
(a) For beam–columns not subject to transverse loading
between supports in the plane of bending,
(6.60)
(6.62)
where 𝑃 story = Total vertical load supported by the story
using LRFD, LSD, or ASD load
combinations, as applicable, including
loads in columns that are not part of the
lateral force-resisting system
Pe,story = Elastic critical buckling strength for the
story in the direction of translation being
considered, determined by sidesway
buckling analysis or taken as:
(6.59)
where α = 1.00 (LRFD or LSD)
= 1.60 (ASD)
Cm = Coefficient assuming no lateral translation of the
frame determined as follows:
𝐶m = 0.6 − 0.4(𝑀1 ∕𝑀2 )
where 𝑀1 and 𝑀2 = Smaller and larger moments,
respectively, at the ends of that
portion of the member unbraced
in the plane of bending under
consideration. 𝑀1 and 𝑀2 are
calculated from a first-order
elastic analysis. 𝑀1 ∕𝑀2 is
positive when the member is bent
in reverse curvature, negative
when bent in single curvature.
Pe,story = RM HF∕ΔF
(6.63)
RM = 1.0 − 0.15(Pmf ∕Pstory )
(6.64)
where
where Pmf = Total vertical load in columns in the story
that are part of moment frames, if any, in the
direction of translation being considered
= 0 for braced frame systems
H = Height of story
NORTH AMERICAN DESIGN CRITERIA FOR BEAM–COLUMN CHECK
ΔF = Interstory drift from first-order elastic
analysis in the direction of translation being
considered, due to story shear, F,computed
using the stiffness as required by
Specification Section C1.2.1.3
F = Story shear, in the direction of translation
being considered, produced by the lateral
forces using LRFD, LSD, or 1.6 times ASD
load combinations
Where ΔF varies over the plan area of the structure in
a three-dimensional system with rigid diaphragms, it shall
be the average drift weighted in proportion to vertical
load or, alternatively, the maximum drift in the story. In
two-dimensional systems with flexible and semi-rigid
diaphragms, ΔF shall be evaluated at each independent
frame (i.e., line of resistance), or alternatively taken as the
maximum drift in the story.
C1.2.1.2 Consideration of Initial Imperfections
Initial imperfections shall be considered as provided by Specification Section C1.1.1.2(a) or C1.1.1.2(b).
C1.2.1.3 Modification of Section Stiffness
Section stiffness modifications shall be made as required by
Specification Section C1.1.1.3.
C1.2.2 Determination of Available Strengths [Factored
Resistances]
The available strengths [factored resistances] of members and
connections shall be calculated as provided by Specification
Section C1.1.2.
C1.3 Effective Length Method
The use of the effective length method shall be limited to the
following conditions:
(a) The structure supports gravity loads primarily through
nominally vertical columns, walls, or frames.
(b) The ratio of maximum second-order drift to maximum firstorder drift (both determined for LRFD load combinations,
LSD load combinations, or 1.6 times ASD load combinations) in all stories is equal to or less than 1.5, as determined
based on nominal unreduced stiffness.
C1.3.1 Determination of Required Strengths [Effects
of Factored Loads]
For the design, the required strengths [effects due to factored
loads] of components of the structure shall be determined from
an analysis conforming to Specification Section C1.3.1.1. The
analysis shall include consideration of initial imperfections in
accordance with Specification Section C1.3.1.2.
265
C1.3.1.1 Analysis
The analysis shall be performed in accordance with the requirements of Specification Section C1.2.1.1, except that nominal
stiffnesses shall be used in the analysis and Specification Section
C1.2.1.3 shall not apply.
C1.3.1.2 Consideration of Initial Imperfections
Notional loads shall be applied in the analysis as required by
Specification Section C1.1.1.2(b).
C1.3.2 Determination of Available Strengths [Factored
Resistances]
The available strengths [factored resistances] of members and
connections shall be calculated in accordance with the provisions of Specification Chapters D, E, F, G, H, I, J, and K, as
applicable.
The flexural buckling effective length factors, 𝐾𝑥 and 𝐾𝑦 , of
members subject to compression shall be taken as specified in
(a) or (b), below, as applicable:
(a) In braced frame systems, shear wall systems, and other
structural systems where lateral stability and resistance to
lateral loads do not rely on the flexural stiffness of columns,
𝐾𝑥 and 𝐾𝑦 of members subject to compression shall be taken
as 1.0, unless rational engineering analysis indicates that a
lower value is appropriate.
(b) In moment frame systems and other structural systems in
which the flexural stiffnesses of columns are considered
to contribute to lateral stability and resistance to lateral
loads, 𝐾𝑥 and 𝐾𝑦 , or elastic critical buckling stress, 𝐹cre , of
those columns whose flexural stiffnesses are considered to
contribute to lateral stability and resistance to lateral loads
shall be determined from a sidesway buckling analysis of
the structure; 𝐾𝑥 and 𝐾𝑦 shall be taken as 1.0 for columns
whose flexural stiffnesses are not considered to contribute to
lateral stability and resistance to lateral loads.
Exception: It is permitted to take 𝐾𝑥 or 𝐾𝑦 , as applicable,
as 1.0 in the design of all columns if the ratio of maximum
second-order drift to maximum first-order drift (both determined for LRFD or LSD load combinations or 1.6 times
ASD load combinations) in all stories is equal to or less
than 1.1.
Bracing intended to define the unbraced lengths of members
shall have enough stiffness and strength to control member
movement at the braced points, and shall be designed in accordance with Specification ection C2.
6.5 NORTH AMERICAN DESIGN CRITERIA
FOR BEAM–COLUMN CHECK
Since the 2016 edition of the North American Specification requires that the P-δ and P-Δ effects are considered
in member force determination, moment magnification and
the sidesway effects then do not need to be considered in
266
6
COMBINED AXIAL LOAD AND BENDING
the beam–column interaction check. The following are the
design provisions adapted from Section H1.2 of the 2016
edition of the North American specification for the design
of beam–columns.
H1.2 Combined Compressive Axial Load and Bending
The required strengths [effects due to factored loads] P, M𝑥 , and
My shall be determined in accordance with Specification Section
C1. Each individual ratio in Eq. (6.65) shall not exceed unity.
For singly-symmetric unstiffened angle sections with unreduced effective area or 𝑃𝑛𝓁 = 𝑃ne , My is permitted to be taken
as the required flexural strength [moment due to factored loads]
only. For other angle sections or singly-symmetric unstiffened
angles for which the effective area (𝐴e ) at stress 𝐹𝑦 is less than
the full unreduced cross-sectional area (A), or 𝑃n𝓁 < 𝑃ne , My
shall be taken either as the required flexural strength [moment
due to factored loads] or the required flexural strength [moment
due to factored loads] plus (P)𝐿∕1000, whichever results in a
lower permissible value of P.
My
M
P
+ x +
≤ 1.0
Pa Max May
(6.65)
where P = Required compressive axial strength
[compressive axial force due to factored loads]
determined as required in Specification
Section C1, in accordance with ASD, LRFD, or
LSD load combinations
𝑃a = Available axial strength [factored resistance],
determined in accordance with Specification
Chapter E
Mx , My = Required flexural strengths [moment due to
factored loads], determined as required in
Specification Section C1, in accordance with
ASD, LRFD, or LSD load combinations
𝑀ax , 𝑀ay = Available flexural strengths [factored resistances]
about centroidal axes, determined in accordance
with Specification Chapter F
Pn𝓁 = Nominal axial strength [resistance] for local
buckling defined in Specification Section E3.2
𝑃ne = Nominal axial strength [resistance] for yielding
and global buckling defined in Specification
Section E2
6.6
DESIGN EXAMPLES
Example 6.1 Check the adequacy of the tubular section
described in Example 5.1 if it is used as a beam–column
to carry axial loads: 𝑃D = 4 kips, 𝑃L = 20 kips, and end
moments: 𝑀D = 8 in.-kips and 𝑀L = 40 in.-kips (Fig. 6.8).
The yield stress of steel is 40 ksi. The member is simply
supported at both ends and the member length is 10 ft. The
Figure 6.8
Example .6.1
member is assumed to be bent in single curvature. Use
the ASD and LRFD methods.
SOLUTION
For the purpose of illustrating various design procedures
permitted by the North American Specification, four
different design cases are demonstrated in this design
example: ASD using EWM, LRFD using EWM, ASD using
DSM and LRFD using DSM. The reader can use any one
case for his/her design.
Sectional Properties of Full Section. From Example 5.1,
the sectional properties of the full section are as follows:
A = 3.273 in2
𝐼𝑥 = 𝐼𝑦 = 33.763 in.4
𝑟𝑥 = 𝑟𝑦 = 3.212 in.
PART I: ASD USING EWM:
1. Member Strength.
i. Axial strength:
From the design procedure discussed in
Chapter 5, the nominal axial strength of the column
with the 10-ft unbraced length was computed in
Example 5.1 using the Effective Width Method:
Pn = 75.67 kips
At the braced point, the nominal axial strength
𝑃no is computed for KL∕r = 0 (i.e., 𝐹n = 𝐹y
= 40 ksi).
For stiffened compression elements,
√
1.052
40
𝜆 = √ (70.619)
= 1.368 > 0.673
29,500
4.0
0.22∕1.368
= 0.613
1.368
𝑏 = 𝜌w = (0.613)(7.415) = 4.545 in.
𝜌=1−
DESIGN EXAMPLES
267
𝑤 7.415
=
= 70.619 < 500
OK
𝑡
0.105
( )√𝑓
1.052 𝑤
𝜆= √
𝑡
𝐸
𝑘
√
1.052
40
= √ (70.619)
= 1.368 > 0.673
29,500
4.0
𝐴e = 3.273 − 4(7.415 − 4.545)(0.105) = 2.068 in.2
𝑃no = 𝐴e 𝐹n = (2.068)(40) = 82.720 kips
ii. Bending Strength:
a. Lateral–Torsional Buckling Strength. Because
the tubular member is a closed box section,
the lateral–torsional buckling strength of the
member can be determined by using Section
4.2.2.3 in this volume or Section F2.1.4 of the
2016 edition of the North American specification. According to Eq. (4.66),
0.36𝐶b 𝜋 √
EI 𝑦 GJ
Lu =
𝐹𝑦 𝑆f
1 − 0.22∕𝜆 1 − 0.22∕1.368
=
= 0.613
𝜆
1.368
𝑏 = 𝜌w = (0.613)(7.415) = 4.545 in.
𝜌=
By using the effective width of the compression
flange and assuming the web is fully effective, the
neutral axis can be located as follows:
in which
Cb = 1.0 for combined axial load and bending
2𝑏2 𝑑 2 𝑡 2(8 − 0.105)4 (0.105)
=
= 51.67 in.4
𝑏+𝑑
2(8 − 0.105)
1
𝑆f = × 33.763 = 8.44 in.3
4
Therefore
0.36(1)𝜋
Lu =
40 × 8.44
√
× (29,500)(33.763)(11,300)(51.67)
𝐽=
= 2, 554.7 in.
Since the unbraced length of 120 in. is less
than 𝐿u , lateral–torsional buckling will not
govern the design.
𝑀ne = 𝑆𝑓 𝐹𝑦 = (8.44)(40) = 337.6 in.-kips
b. Local buckling strength.
The nominal flexural strength about the x-axis
should be determined according to Section F3.1
of the North American Specification. Since the
member does not subject to lateral–torsional
buckling, the local buckling is determined based
on yield stress 𝐹y .
i. Determine the effective cross-section.
𝑡
R′ = 𝑅 + = 0.240 in. (corner element)
2
𝐿 = 1.57𝑅′ = 0.377 in. (arc length)
𝑐 = 0.637𝑅′ = 0.153 in.
For the stiffened compression flange,
w = 8-2(𝑅 + 𝑡) = 8 − 2(0.1875 + 0.105)
= 7.415 in.
Element
Effective
Length L
(in.)
Distance
from Top
Fiber y (in.)
Ly (in.2 ) Ly (in.3 )
Compression 4.545
0.0525
0.239
0.013
flange
Compression 2 × 0.377
0.1395
0.105
0.013
corners
= 0.754
Webs
2 × 7.415
4.0000
59.320 237.280
= 14.830
Tension
2 × 0.377
7.8605
5.927 46.588
corners
= 0.754
Tension
7.415
7.9475
58.931 468.352
flange
28.298
124.522 752.248
124.522
𝑦cg =
= 4.400 in.
28.298
> 𝑑∕2 = 8∕2 = 4.000 in.
The maximum stress of 40 ksi occurs in the
compression flange as summed in the calculation.
Check the effectiveness of the web. Use Section
3.3.1.2 to check the effectiveness of the web
element. From Fig. 6.9,
)
(
4.1075
= 37.341 ksi (compression)
f1 = 40
4.400
)
(
3.3075
𝑓2 = 40
= 30.068 ksi (tension)
4.400
| 𝑓 | 30.068
𝜓 = || 2 || =
= 0.805
| 𝑓1 | 37.341
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓)
= 4 + 2(1 + 0.805)3 + 2(1 + 0.805)
= 19.371
268
6
COMBINED AXIAL LOAD AND BENDING
Figure 6.9
Stress distribution in webs using fully effective webs.
ℎ 7.415
=
= 70.619 < 200
OK
𝑡 0.105
√
37.341
1.052
(70.619)
= 0.601 < 0.673
𝜆= √
29,500
19.371
𝑏e = ℎ = 7.415 in.
Because ℎ0 ∕𝑏0 = 8.00∕8.00 = 1 < 4 and 𝜓 >
0.236, Eqs. (3.47a) and (3.47b) are used to compute
𝑏1 and 𝑏2 :
𝑏e
= 1.949 in.
3+𝜓
1
𝑏2 = 𝑏e = 3.708 in.
2
𝑏1 + 𝑏2 = 5.657 in.
b1 =
Because the computed value of 𝑏1 + 𝑏2 is
greater than the compression portion of the web
(4.1075 in.), the web is fully effective. The moment
of inertia based on line element is
( )
1
(7.415)3 = 67.949
2I′web = 2
12
∑
(Ly2 ) = 752.248
𝐼𝑧 = 820.197 in.3
−(Σ𝐿)(𝑦cg )2 = −(28.298)(4.40)2
= −547.849 in.3
𝐼𝑥′ = 272.348 in.3
The actual moment of inertia is
I𝑥 = 𝐼𝑥′ (𝑡) = (272.348)(0.105) = 28.597 in.4
The section modulus relative to the extreme
compression fiber is
𝑆e𝑥
28.597
=
= 6.499 in.3
4.40
ii. The nominal moment due to local buckling interacting with yielding is
𝑀n𝓁 = 𝑆e𝑥 𝐹𝑦 = (6.499)(40) = 259.960 in. − kips
Since the above local buckling strength is calculated at the yield stress, the nomnal local buckling strength at the braced point is also equal this
value, i.e.,
𝑀n𝓁o = 259.960 in.-kips
c. Member bending strength. Since the member
does not subject to distortional buckling, the
member strength is the smaller of the strengths
due to global and local buckling, and the
strength due to local buckling governs. Therefore, from the calculation above, member
bending strength is:
𝑀n = 𝑀n𝓁 = 259.96 in.-kips
At the braced point, only the local buckling
needs to be considered, and the the nominal
strength is:
𝑀no = 𝑀n𝓁o = 259.96 in.-kips
2. Applied Axial Load and Moments.
Since there is no lateral translation, the effective
length method can be employed. This method needs
to consider the second-order effect, but the stiffness
reduction does not need to be considered.
Since the column has no lateral translation at both
ends,
𝑃 𝓁𝑡 = 0, 𝑀 𝓁𝑡 = 0
i. Forces and moments from the ASD load combination:
𝑃 nt = 𝑃D + 𝑃L = 4 + 20 = 24 kips
𝑀 nt = 𝑀D + 𝑀L = 8 + 40 = 48 in.-kips
DESIGN EXAMPLES
ii. Multipliers 𝐵1 and 𝐵2 :
Since there is no translation,
𝑃 = 𝑃 nt = 24 kips
𝛼 = 1.6 (For ASD Method)
𝐵1 is determined by Eq. (6.59)
𝐵1 = 𝐶m ∕(1 − αP∕Pe1 ) ≥ 1.0
where
𝐶m = 0.6 − 0.4(𝑀1 ∕𝑀2 )
For column with single curvature bending and
end moments are equal, 𝑀1 = −𝑀2 , therefore
From Part I Item 1: Pno = 82.72 kips, Mno =
259.96 in. − kips
Since the axial strength Pno is larger than Pn , and
also since no force modifications are needed at the
braced point (i.e., both B1 and B2 equal 1), based on
the results of the interaction checks under item 3 i,
the interaction check at the braced point should be
satisfied.
PART II: LRFD USING EWM:
1. Member Strengths.
From Part I Item 1, the member strengths using the
Effective Width Method are obtained as follows:
𝐶m = 0.6 − 0.4(−1) = 1.0
𝑃n = 75.67 kips
𝑃e1 is calculated without stiffness reduction for
using the effective length method:
𝑃𝑒1 = 𝜋 2 𝑘𝑓 ∕(𝐾1 𝐿)2 = 𝜋 2 (EI 𝑥 )∕(𝐾1 𝐿)2
= 𝜋 2 (29,500)(33.763)∕[(1.0)(120)]2
= 682.65 kips
𝐵1 =
𝐶𝑚
1 − 𝛼 𝑃𝑃
𝑒1
=
1.0
24
1 − 1.6 682.65
= 1.06
iii. Member forces: The member forces considering
the structural stability are determined by Eqs.
(6.57) and (6.58)
M = B1 Mnt + B2 M𝓁𝑡 = (1.06)(48) = 50.88 in.-kips
P = Pnt + B2 P𝓁𝑡 = 24 kips
3. Beam-column interaction check.
i. Interaction check between the supports:
From the calculation of Item 1, the following
member strengths are obtained:
𝑃n = 75.67 kips
𝑀n = 259.96 in.-kips
The corresponding available strengths are
𝑃a = 75.67∕1.80 = 42.04 kips
𝑀a = 259.96∕1.67 = 155.67 in.-kips
Using the member forces calculated in Item 2, the
following interaction is performed using Eq. (6.65):
24
50.88
𝑃
𝑀
+
=
+
= 0.898 < 1.0 OK
𝑃𝑎 𝑀𝑎
42.04 155.67
ii. Interaction check at the braced point. Equation
(6.16) can be used to check the beam–column for
the yielding requirement at braced points.
269
𝑀n = 259.96 in.-kips
At braced point:
𝑃no = 82.72 kips
𝑀no = 259.96 in.-kips
The corresponding available strengths using the
LRFD are
𝑃a = (0.85)(75.67) = 64.32 kips
𝑀a = (0.9)(259.96) = 233.96 in.-kips
𝑃no = (0.85)(82.72) = 70.31 kips
𝑀ao = 𝑀a = 233.96 in.-kips
2. Applied Axial Loads and Moments.
As discussed in Part I Item 2, since the column has
no lateral translation at both ends,
𝑃 𝓁𝑡 = 0, 𝑀 𝓁𝑡 = 0
i. Forces and moments from the LRFD load combination:
𝑃 nt = 1.2PD + 1.6PL = 1.2(4) + 1.6(20)
= 36.8 kips
𝑀 nt = 1.2MD + 1.6ML = 1.2(8) + 1.6(40)
= 73.6 in.-kips
ii. Multipliers 𝐵1 and 𝐵2 :
Since there is no translations,
𝑃 = 𝑃 nt = 36.8 kips
𝛼 = 1.0 (For LRFD Method)
B1 is determined by Eq. (6.59)
𝐵1 = 𝐶m ∕(1 − 𝛼P∕𝑃e1 ) ≥ 1.0
270
6
COMBINED AXIAL LOAD AND BENDING
[
(
) ](
)
61.53 0.4
61.53 0.4
= 1 − 0.15
(131.14)
131.14
131.14
From Part I Item 2 discussions:
𝐶m = 1.0
= 86.15 kips
𝑃e1 = 682.65 kips
𝐵1 =
𝐶𝑚
1 − 𝛼 𝑃𝑃
𝑒1
=
1.0
36.8
1 − 1.0 682.65
= 1.06
iii. Member forces: the member forces considering the
structural stability are determined by Eqs. (6.57)
and (6.58)
M = 𝐵1 𝑀 nt + 𝐵2 𝑀 𝓁𝑡 = (1.06)(73.6) = 78.02 in-kips
P = 𝑃 nt + 𝐵2 𝑃 𝓁𝑡 = 36.8 kips
3. Beam-column interaction check.
i. Interaction check between the supports:
From the results of Part II Item 1, and member
forces obtained above:
36.8
78.02
𝑃
𝑀
+
=
+
= 0.906 < 1.0 OK
𝑃𝑎 𝑀𝑎
64.32 233.96
ii. Interaction check at the braced point. Equation
(6.16) can be used to check the beam–column for
the yielding requirement at braced points.
Since the axial strength 𝑃ao is larger than 𝑃a , and
also since no force modifications are needed at the
braced point (i.e., both 𝐵1 and 𝐵2 equal 1), based on
the results of the interaction checks under item 3(i),
the interaction check at the braced point should be
satisfied.
PART III. ASD USING DSM:
1. Member Strengths Using DSM.
i. Axial strength:
From the design procedure discussed in
Chapter 5, the nominal strength was computed
in Example 5.1 as
𝑃n = 78.46 kips.
The nominal strength, 𝑃no , at the braced point is
governed by the local buckling. From Example 5.1,
𝑃crl = 61.53 kips, 𝑃no is calculated at 𝑃ne = 𝑃y =
131.14 kips, obtained from Example 5.1:
√
√
Py
131.14
=
𝜆𝓁 =
= 1.46 > 0.776
Pcr𝓁
61.53
Therefore, the local buckling strength is calculated
[
) ](
)
(
𝑃cr𝓁 0.4
𝑃cr𝓁 0.4
Pno = Pn𝓁𝑜 = 1 − 0.15
𝑃𝑦
𝑃𝑦
𝑃𝑦
ii. Bending Strength:
a. Lateral–Torsional Buckling Strength. The result
of the lateral–torsional buckling strength from
the Direct Strength Method is the same as the one
using the Effective Width Method. From Part I
Item iia,
𝑀ne = 337.6 in.-kips
b. Local buckling strength.
By using the CUFSM program analysis, the
following results are obtained:
The
yield
moment
𝑀𝑦 = 343.05 in.
− kips(Note, this numerical solution of the
yield moment is not exactly the same as from
hand calculated yield moment.)
The load factor due to local buckling
LF local (i.e., 𝑀crl ∕𝑀𝑦 ) = 0.64
Therefore, the local buckling moment (based
on 𝑀𝑦 from the numerical analysis):
𝑀cr𝓁 = (0.64)(343.05) = 219.55 in.-kips
Since there are no global buckling from Part I,
item 1 analysis, 𝑀ne = 𝑀𝑦 . To obtain the local
buckling strength, 𝑀𝑦 value from the numerical analysis is used in the following calculation.
From Section 4.2.3.2,
√
√
√
𝑀𝑦
𝑀ne
343.05
=
=
𝜆𝓁 =
𝑀cr𝓁
𝑀cr𝓁
219.55
= 1.25 > 0.776
[
(
) ](
)
𝑀cr𝓁 0.4
𝑀cr𝓁 0.4
M𝑛𝓁 = 1 − 0.15
𝑀ne
𝑀ne
𝑀ne
[
) ]
(
219.55 0.4
= 1 − 0.15
343.05
)0.4
(
219.55
×
(343.05) = 250.96 in.-kips
343.05
At the braced point, the local buckling
strength is the same as the above, i.e.,
𝑀n𝓁𝑜 = 250.96 in-kips
c. Member bending strength. Since the member
does not subject to distortional buckling, the
member strength is the smaller of the strengths
due to global and local buckling. From the calculation above, the local buckling governs:
𝑀𝑛 = 𝑀n𝓁 = 250.96 in.-kips
DESIGN EXAMPLES
This is also the strength at the braced point,
i.e., 𝑀no = 250.96 in.-kips.
2. Applied Axial Loads and Moments.
The applied load and moment based on the ASD
method are calculated in Part I Item 2, from which the
following member forces are obtained:
M = 𝐵1 𝑀 nt + 𝐵2 𝑀 𝓁t = (1.06)(48) = 50.88 in.-kips
P = 𝑃 nt + 𝐵2 𝑃 𝓁t = 24 kips
3. Beam-Column Interaction Check.
From the calculation of Part II Item 1, the following
member strengths are obtained:
𝑃n = 78.45 kips
𝑀n = 250.96 in.-kips
At the braced point:
𝑃no = 86.15 kips
𝑀no = 250.96 in.-kips
The corresponding available strengths using ASD
are
𝑃a = 78.45∕1.80 = 43.58 kips
𝑀a = 250.96∕1.67 = 150.28 in.-kips
𝑃ao = 86.15∕1.80 = 47.86 kips
𝑀ao = Ma = 150.28 in.-kips
i. Interaction check between the supports:
The interaction check can be performed using
Eq. (6.65):
24
50.88
𝑃
𝑀
+
=
+
= 0.889 < 1.0 OK
𝑃𝑎 𝑀𝑎
43.58 150.28
ii. Interaction check at the braced point. Based on
the member strengths and the applied loads at the
braced point, the interaction check at the braced
point should be satisfactory.
PART IV. LRFD USING DSM:
1. Member Strengths.
From Part III Item 1, the member strengths using the
Direct Strength Method are obtained as follows:
𝑃n = 78.45 kips
The corresponding available strengths using LRFD are
𝑃a = (0.85)(78.45) = 66.68 kips
𝑀a = (0.9)(250.96) = 225.86 in.-kips
𝑃ao = (0.85)(86.15) = 73.23 kips
𝑀ao = 𝑀a = 225.86 in.-kips
2. Applied Axial Loads and Moments.
The applied load and moment based on the LRFD
method are calculated in Part II Item 2, from which the
following member forces are obtained:
M = B1 Mnt + B2 M𝓁t = (1.06)(73.6) = 78.02 in-kips
P = Pnt + B2 P𝓁t = 36.8 kips
3. Beam-Column Interaction Check.
i. Interaction check between the supports:
From the member strengths from Part IV Item 1
and the member forces from Item 2, the following
interaction check is performed:
36.8
78.02
𝑃
𝑀
+
=
+
= 0.897 < 1.0 OK
𝑃𝑎 𝑀𝑎
66.68 225.86
ii. Interaction check at the braced point. Based on
the member strengths and the applied loads at the
braced point, the interaction check at the braced
point should be satisfactory.
From the interaction checks in Parts I–IV, both ASD and
LRFD with strengths determined using either EWM or DSM
indicate that the column is adequate in supporting the given
loads.
Example 6.2 If the I-section used in Example 5.2 is to
be used as a beam–column as shown in Fig. 6.10, what is
′
the maximum allowable transverse load P applied at the
mid-span length? Assume that the axial load is 20 kips and
the beam is laterally supported at A, B, C, D, and E. Use
𝐹y = 33 ksiand the ASD method. The intermediate fastener
spacing is assumed to be 12 in.
SOLUTION
For the purpose of illustration, both the Effective Width
Method (EWM) and the Direct Strength Method (DSM) are
used to determine the member strengths in Parts I and II,
𝑀n = 250.96 in.-kips
At the braced point:
𝑃no = 86.15 kips
𝑀no = 250.96 in.-kips
271
Figure 6.10
Example 6.2
272
6
COMBINED AXIAL LOAD AND BENDING
Figure 6.11 Moment diagram for the continuous beam.
respectively. The reader can use either of the methods in
his/her design. The interaction checks are then performed in
Part III.
Sectional Properties of Full Section. From Example 5.2,
the sectional properties of the I-section are as follows:
A = 2.24 in.2
𝐽 = 0.00418 in.4
𝐼𝑥 = 22.1 in.4
𝐶w = 70.70 in.6
𝑆𝑥 = 5.53 in.3
𝑟0 = 3.435 in
𝐼𝑦 = 4.20 in.4
𝑟𝑥 = 3.15 in.
𝑆𝑦 = 1.40 in.3
𝑟𝑦 = 1.37 in.
Applied Axial Load and Moments. Since the continuous
′
beam is subject to symmetric loads P in two equal spans, the
moment diagram can be drawn as shown in Fig. 6.11. The
positive and negative moments are
5
5 ′
𝑃 (10)(12) = 18.75𝑃 ′ in. − kips
+MB = 𝑃 ′ 𝐿 =
32
32
3
3 ′
−𝑀C = 𝑃 ′ 𝐿 =
𝑃 (10)(12) = 22.5𝑃 ′ in. − kips
16
16
As given in the problem, the applied axial load is
𝑃 = 20 kips
PART I: MEMBER STRENGTH DETERMINED USING
EWM.
1. Computation of Pn .
a. Nominal Strength Due to Global Buckling, Pne .
i. By using Eq. (5.54), the elastic flexural buckling
stress can be computed as follows:
𝐾𝑥 𝐿𝑥 1 × 10 × 12
=
= 38.10
𝑟𝑥
3.15
𝐾 𝑦 𝐿𝑦
𝑟𝑦
1 × 5 × 12
=
= 43.80 < 200
1.37
OK
Since the slenderness ratio (KL∕𝑟 = 𝐾𝑦 𝐿𝑦 ∕𝑟𝑦 )
is governed by the column buckling about the
y axis of the I-section, which involves relative
deformations that produce shear forces in the
connections between individual channels, the
modified slender ratio (KL∕𝑟)m should be used
to compute the elastic flexural stress 𝐹cre . Based
on Eq. (5.84),
√
( )
( )2 ( )2
𝑎
KL
KL
=
+
𝑟 m
𝑟 0
𝑟i
Where (KL∕𝑟)0 = 43.80
a = intermediate fastener
spacing, = 12 in.
ri = radius of gyration of a
channel section about its y
axis, = 1.08 in.
Therefore,
√
)
(
( )
12 2
KL
= (43.80)2 +
= 45.19
𝑟 m
1.08
𝜋 2 (29,500)
𝜋2𝐸
𝐹cre =
=
= 142.57 ksi
(45.19)2
(KL∕𝑟)2m
ii. Elastic Torsional Buckling. From Eq. (5.61), the
torsional buckling stress is
(
)
𝜋 2 ECw
1
Fcre = 𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
=
1
(2.24)(3.435)2
(
)
𝜋 2 (29,500)(70.70)
× (11,300)(0.00418) +
(5 × 12)2
= 218.13 ksi
The nominal buckling stress 𝐹n is determined
by using the smaller value of the elastic flexural buckling stress and torsional buckling stress,
that is,
Fcre = 142.57 ksi
√
√
𝐹y
33
=
= 0.481 < 1.5
𝜆c =
𝐹cre
142.57
From Eq. (5.51),
Fn = (0.658𝜆c )𝐹y = (0.6580.481 )(33) = 29.95 ksi
2
2
𝑃ne = 𝐹n 𝐴 = (29.95)(2.24) = 67.09 kips
b. Nominal Strength Due to Local Buckling Interacting with Global Buckling, Pn𝓁 .
From Example 5.2, the flat widths of the edge
stiffener, flange, and web are
𝑤
w1 = 0.5313 in. 1 = 7.084
𝑡
DESIGN EXAMPLES
𝑤3
= 102.167500
OK
𝑡
𝑤
𝑤3 = 7.6625 in. 3 = 102.167500
OK
𝑡
i. Effective Width of Compression flange. The
effective widths are determined at stress
𝐹n = 29.95 ksi. From Eq. (3.72)
√
√
29,500
𝐸
S = 1.28
= 1.28
= 40.17
𝑓
29.95
𝑤2 = 7.6625 in.
0.328𝑆 = 13.18
𝑤2
= 35.50
𝑡
Since 𝑤2 ∕𝑡 > 0.328𝑆, use Eq. (3.73) to
compute the adequate moment of inertia of
the edge stiffener 𝑙a as follows:
Ia = 399𝑡4 [(𝑤2 ∕𝑡)∕𝑆 − 0.328]3
= 399(0.075)4 [35.50∕40.17 − 0.328]3
= 0.0022 in.4
The above computed value should not exceed
the following value:
Ia = 𝑡4 [115(𝑤2 ∕𝑡)∕𝑆 + 5]
= (0.075)4 [115(35.50)∕40.17 + 5] = 0.0034 in.4
Therefore, use 𝐼a = 0.0022 in.4 For the simple
lip edge stiffener,
1
,
3
use 𝑛 = 0.361
Since 0.25 < 𝐷∕𝑤2 < 0.8, and θ = 90∘ ,
k = [4.82 − 5D∕w2 ](𝑅𝐼 )𝑛 + 0.43
= [4.82 − 5(0.263)](0.426)0.361 + 0.43
= 3.006 < 4.0
Use 𝑘 = 3.006 to compute the effective
width of the compression flange. From
Eqs. (3.32)–(3.35),
√
1.052
29.95
𝜆= √
(35.50)
= 0.686 > 0.673
29,500
3.006
1 − 0.22∕𝜆 1 − 0.22∕0.686
=
= 0.99
𝜆
0.686
𝑏 = 𝜌𝑤2 = (0.99)(2.6625) = 2.636 in.
𝜌=
ii. Effective Width of Edge Stiffeners
w
= 7.084
𝑡
√
29.95
1.052
(7.084)
= 0.362 < 0.673
𝜆= √
29,500
0.43
𝑑s′ = 𝑤1 = 0.5313 in.
Based on Eq.(3.71) , the reduced effective width
of the edge stiffener is
𝐷 = 0.7 in.
𝑑 = 0.5313 in.
d 0.5313
=
= 7.084
𝑡
0.075
By using Eq. (3.75), the moment of inertia of the
full edge stiffener is
( )
1 3
1
(0.5313)3 (0.075)
Is =
𝑑 𝑡=
12
12
= 0.000937 in.4
From Eq. (3.74),
𝐼
0.000937
= 0.426 < 1.0
RI = s =
𝐼a
0.0022
= 0.361 >
273
OK
The effective width b of the compression flange
can be computed as follows:
0.7
D
=
= 0.263
𝑤2
2.6625
From Eq. (3.76),
n = [0.582 − (𝑤2 ∕𝑡)∕(4𝑆)]
= [0.582 − (35.50)∕(4 × 40.17)]
ds = 𝑅𝐼 𝑑s′ = (0.426)(0.5313)
= 0.226 < 𝑑s′
OK
iii. Effective Width of Web Elements
𝑤
= 102.167
𝑡
√
1.052
29.95
𝜆 = √ (102.167)
= 1.712 > 0.673
29,500
4.0
1 − 0.22∕1.712
= 0.509
1.712
𝑏 = 𝜌𝑤3 = (0.509)(7.6625) = 3.900 in.
𝜌=
iv. Effective Area Ae
Ae = 2.24 − [4(0.5313 − 0.226)
+ 4(2.6625 − 2.636)
+2(7.6625 − 3.900)](0.075)
= 1.576 in.2
274
6
COMBINED AXIAL LOAD AND BENDING
Therefore, the nominal strength using the
effective width method is
𝐶𝑏 =
Pn𝓁 = 𝐴e 𝐹n = (1.576)(29.95) = 47.20 kips
At location C, assume that the global buckling
is restrained at the support. The local buckling
is calculated at stress level, 𝐹n = 𝐹y = 33 ksi.
Using the same procedure above, the following
results are obtained:
+ 3(0.25𝑀𝐵 )
= 1.67
Therefore, 𝐹cre =
c. Nominal Strength Based on Distortional Buckling, 𝑃nd . According to Section E3 of the North
American specification, Example 5.2 shows that
the nominal axial load for distortional buckling
based on 𝐿m = 𝐿y = 72 in. is 𝑃nd = 49.25 kips.
For this example, 𝐿m = 𝐿y = 60 in., which is
also greater than 𝐿crd = 25.35 in. computed from
Example 5.2. The same nominal axial load for
distortional buckling can also be used for this case,
i.e. 𝑃nd = 49.25 kips.
d. Nominal Axial Strength of the Member, 𝑃n . The
nominal strength is the minimum of 𝑃ne , 𝑃nl and 𝑃nd .
Based on the calculations in items 1(a) to 1(c), the
local buckling governs at locations B and D. At location C, assume global and distortional buckling are
retrained, and only the local buckling is considered.
Therefore,
At location B and D: 𝑃n = 𝑃𝑛𝓁 = 47.20 kips
At Location C: 𝑃nC = 𝑃𝑛𝓁𝐶 = 49.73 kips
2. Computation of 𝑀nx
a. Lateral–Torsional Buckling Strength, 𝑀ne . For
segment AB, 𝐾y 𝐿y = 5 ft. According to Eq. (4.59),
Fcre =
𝐶𝑏 𝜋 2 EdI yc
𝑆𝑓 (𝐾𝑦 𝐿𝑦 )2
In the above equation,
12.5𝑀max
C𝑏 =
2.5𝑀max + 3𝑀1 + 4𝑀2 + 3𝑀3
where Mmax = MB at point B
M1 = 0.25MB at 1/4 pint of unbraced
segment
M2 = 0.50MB at midspan of unbraced
segment
M3 = 0.75MB at 34 point of unbraced
segment
𝜋 2 (29,500)(1.67)(8)(4.20∕2)
(5.53)(5 × 12)2
= 410.32 ksi
0.56 𝐹y = 18.48 ksi
𝐴e (for 𝐹n = 33 ksi) = 1.507 in2
𝑃n𝓁𝐶 = 𝐴e 𝐹n = 49.73 kips
12.5(𝑀𝐵 )
2.5(𝑀𝐵 )
2.5(𝑀𝐵 ) + 3(0.25𝑀𝐵 )
2.78𝐹y = 91.74 ksi
Since 𝐹cre > 2.78𝐹y , 𝐹n = 𝐹y = 33 ksi.
𝑀ne = 𝑆x 𝐹n = (5.53)(33) = 182.49 in.-kips
For segment BC, 𝐾y 𝐿y = 5 ft, 𝑀B = 18.75 𝑃 ′ and
𝑀c = 22.5 𝑃 ′ in.-kips. The value of Cb is
C𝑏 =
=
12.5𝑀max
2.5𝑀max + 3𝑀1 + 4𝑀2 + 3𝑀3
12.5(22.5𝑃 ′ )
2.5(22.5𝑃 ′ ) + 3(8.4375𝑃 ′ )
+ 4(1.875𝑃 ′ ) + 3(12.1875𝑃 ′ )
= 2.24
𝜋 2 (29,500)(2.24)(8)(4.20∕2)
𝐹cre =
= 550.37 ksi
(5.53)(5 × 12)2
Since 𝐹cre > 2.78 𝐹y , 𝐹n = 𝐹y = 33 ksi,
𝑀n𝑒 = 𝑆𝑥 𝐹n = (5.53)(33) = 182.49 in. − kips
b. Local Buckling Strength, Mn𝓵 .
The effective widths are determined based on the
maximum compressive stress 𝐹n = 33 ksi. For the
corner element,
1
1
R′ = 𝑅 + 𝑡 = 0.09375 + × 0.075 = 0.1313 in.
2
2
The arc length is
𝐿 = 1.57 𝑅′ = 0.206 in.
𝑐 = 0.637𝑅′ = 0.0836 in.
For the compression flange, the effective width
for 𝑓 = 𝐹y = 33 ksi is 𝑏 = 2.430 in. For the
compression edge stiffener, the compression stress
is conservatively assumed to be 𝑓 = 𝐹y = 33 ksi.
Following the same procedure used in item 3.b, the
effective width of the edge stiffener at a stress of
33 ksi is 0.184 in. See item A.1.iii of Example 4.5
for using Sections 3.3.2.2 and 3.3.3.1 to determine
the reduced effective width of the edge stiffener. By
using the effective widths of the compression flange
DESIGN EXAMPLES
and edge stiffener and assuming the web is fully
effective, the neutral axis can be located as follows:
Effective
Length L (in.)
Element
Compression
flange
Compression
corners
Compression
stiffeners
Webs
Tension
stiffeners
Tension
corners
Tension
flange
2 × 2.430
= 4.860
4 × 0.206
= 0.824
2 × 0.184
= 0.368
2 × 7.6625
= 15.325
2 × 0.5313
= 1.063
4 × 0.206
= 0.824
2 × 2.6625
= 5.325
28.589
118.6122
𝑦cg =
8.589
= 4.149in
Distance from
Top Fiber
y(in.)
Ly(in.2 ) Ly2 (in.3 )
0.0375
0.182
0.007
0.0852
0.070
0.006
0.2608
0.096
0.025
4.0000
61.300 245.200
7.5656
8.042
𝑘 = 4 + 2(1 + 0.925)3 + 2(1 + 0.925) = 22.116
ℎ 𝑤3
=
= 102.167
𝑡
𝑡
√
31.66
1.052
(102.167)
= 0.749 > 0.673
𝜆= √
29,500
22.116
1 − 0.22∕0.749
= 0.943
0.749
𝑏e = 0.943(7.6625) = 7.229 in.
𝜌=
Because ℎ0 ∕𝑏0 = 8.00∕3.00 = 2.67 < 4 and 𝜓 >
0.236, Eqs. (3.47a) and (3.47b) are used to compute
𝑏1 and 𝑏2 :
60.843
7.229
= 1.842 in.
3 + 0.925
7.229
𝑏2 =
= 1.842 in.
3 + 0.925
𝑏1 + 𝑏2 = 1.842 + 3.614 = 5.456 in.
b1 =
7.9148
6.522
51.620
7.9625
42.400 337.610
118.612 695.311
Since 𝑦cg > 𝑑∕2 = 4.000 in., the maximum
stress of 33 ksi occurs in the compression flange as
assumed in the above calculation.
The effectiveness of the web is checked according
to Section 3.3.1.2. From Fig. 6.12,
)
3.9802
= 31.66 ksi
4.149
)
(
3.6822
𝑓2 = 33
= 29.29 ksi
4.149
| 𝑓 | 29.29
𝜓 = || 2 || =
= 0.925
| 𝑓1 | 31.66
𝑓1 = 33
275
(
(compression)
(tension)
Because the computed value of 𝑏1 + 𝑏2 is
greater than the compression portion of the web
(3.9802 in.), the web is fully effective.
The moment of inertia based on line elements is
( )
1
2I′web = 2
(7.6625)3 = 74.983
12
( )
1
′
(0.184)3 = 0.001
=2
2𝐼comp.stiffener
12
( )
1
′
(0.531)3 = 0.025
=
2
2𝐼tension
stiffener
12
∑
(Ly2 ) = 695.311
𝐼𝑧 = 770.320 in.3
−(Σ𝐿)(𝑦cg )2 = −(28.589)(4.149)2 = −492.137
𝐼𝑥′ = 278.183 in.3
Figure 6.12 Stress distribution in webs.
276
6
COMBINED AXIAL LOAD AND BENDING
The actual moment of inertia is
𝐼𝑥 =
𝐼𝑥′ (𝑡)
= (278.183)(0.075) = 20.864 in.
4
The section modulus relative to the extreme
compression fiber is
20.864
= 5.029 in.3
Se𝑥 =
4.149
The nominal moment due to local buckling is
M𝑛𝓁 = 𝑆e 𝐹𝑛 = (5.029)(33) = 165.96 in. − kips
Since the above effective section modulus is determined at 𝐹n = 𝐹y , the local buckling strength at
Location C will be the same, i.e.,
𝑀𝑛𝓁𝐶 = 165.96 in. − kips
c. Distortional Buckling Strength, Mnd . The nominal
moment for distortional buckling can be computed
according to Section F4 of the North American
Specification. Following the procedure illustrated
in item B of Example 3.8 and using Eqs. (3.111)–
(3.116), the computed rotational stiffnesses are as
follows:
𝑘𝜙fe = 0.534 in.-kips∕in.
𝑘𝜙we = 0.482 in.-kips∕in.
𝑘𝜙 = 0
̃
𝑘𝜙fg = 0.0203(in. − kips∕in.)∕ksi
̃
𝑘𝜙wg = 0.00205(in. − kips∕in.)∕ksi
Use a conservative value of 𝛽 = 1.0 in.
Eq. (3.112),
0.534 + 0.482 + 0
𝐹crd = (1.0)
= 45.46 ksi
0.0203 + 0.00205
𝑀crd = 𝑆f 𝐹crd = (5.53)(45.46) = 251.39 in. − kips
From Eqs. (4.101)–(4.105),
𝑀y = 𝑆f 𝐹y = (5.53)(33) = 182.49 in. − kips
√
√
𝑀𝑦
182.49
𝜆d =
=
= 0.852 > 0.673
𝑀crd
251.39
𝑀nd = [1 − 0.22(𝑀crd ∕𝑀𝑦 )0.5 ](𝑀crd ∕𝑀𝑦 )0.5 𝑀𝑦
[
) ](
)
(
251.39 0.5 251.39 0.5
= 1 − 0.22
182.49
182.49
× (182.49)
= 158.88 in. − kips
d. Nominal strength, Mnx . At locations B and D,
the nominal flexural strength, 𝑀nx , is the minimum
of 𝑀ne , 𝑀n𝓁 , and 𝑀nd . Based on the computed
results under Items 2a to 2c, the distortional buckling strength governs. Therefore,
𝑀n𝑥 = 𝑀nd = 158.88 in. − kips
At Location C, it is assumed that the global buckling and distortional buckling are restrained, and
only the local buckling needs to be considered.
Therefore,
𝑀nxC = 𝑀𝑛𝓁𝐶 = 165.96 in-kips
PART II: MEMBER STRENGTH DETERMINED
USING DSM.
1. Computation of Pn .
a. Nominal Strength due to Global Buckling, Pne .
The result of the nominal strength due to global
buckling using the direct strength method is the
same as that obtained from Part I Item 1a:
𝑃ne = 67.09 kips
b. Nominal Strength due to Local Buckling Interacting with Global Buckling, Pn𝓁 .
The local buckling of the I-Section member
can be analyzed by considering the local buckling of the two individual C-Section members.
The axial buckling load of the I-Section member
is the sum of two individual C-Section buckling loads. From Example 5.2, the local buckling
strength using the Direct Strength Method is
obtained,
𝑃𝑛𝓁 = 42.32 kips.
The local buckling strength at support C is calculated based on the yielding 𝑃y . From the numerical
analysis results given in Example 5.2, the I-Section
yield strength and the buckling load are:
𝑃y = 2(36.802) = 73.604 kips
𝑃cr𝓁 = 2(15.09) = 30.18 kips
Therefore, the local buckling strength, 𝑃n𝓁 , based on
𝑃y is
√
√
𝑃𝑦
73.604
=
= 1.56 > 0.776
𝜆𝓁 =
𝑃cr𝓁
30.18
[
) ](
)
(
𝑃cr𝓁 0.4
𝑃cr𝓁 0.4
P𝑛𝓁𝐶 = 1 − 0.15
𝑃𝑦
𝑃𝑦
𝑃𝑦
DESIGN EXAMPLES
[
(
)0.4 ]
30.18
73.604
(
)0.4
30.18
×
(73.604) = 46.12 kips
73.604
c. Nominal Strength Based on Distortional Buckling, Pnd .
From Part I, item c, the distortional buckling
strength is obtained as 𝑃nd = 49.25 kips.
d. Nominal Axial Strength of the Member, Pn .
The nominal strength is the minimum of 𝑃ne , 𝑃nl
and 𝑃nd . Based on the calculations in Part II items
1(a) to 1(c), the local buckling governs.
At location B and D:
= 1 − 0.15
𝑃n = 𝑃𝑛𝓁 = 42.32 kips
At location C:
𝑃𝑛C = 𝑃𝑛𝓁𝐶 = 46.12 kips
2. Computation of Mnx .
a. Lateral–Torsional Buckling Strength, Mne . From
analysis under Part I item a, the same lateral
torsional buckling strength 𝑀ne is obtained for the
Direct Strength Method:
𝑀ne = 182.49 in.-kips
b. Local Buckling Strength, 𝑀nl . The Direct Strength
Method requires determination of the member local
buckling moment. The I-section member local
buckling moment can be calculated as the sum
of the local buckling moments of two individual
C-Section members. Using the CUFSM software
analysis, the following results are obtained for the
single C-Section member:
Yield moment∶𝑀y_C-sect = 91.767 in.-kips
Local buckling load factor∶
𝐿𝐹𝑙𝑜𝑐𝑎𝑙 (𝑖.𝑒., 𝑀𝑐𝑟𝓁∕𝑀𝑦 ) = 2.04
Therefore, the local buckling moment of a single
C-Section is
𝑀𝑐𝑟𝓁_𝐶-𝑆𝑒𝑐𝑡 = (𝐿𝐹𝑙𝑜𝑐𝑎𝑙 )𝑀𝑦 = (2.04)(91.767)
= 187.2047 in, −kips
I-Section buckling moment, Mcr𝓁 = 2Mcr𝓁 -Sect =
2(187.2047) = 374.41 in.-kips
From Section 4.2.3.2,
√
√
𝑀ne
182.49
=
= 0.698 < 0.776
𝜆𝓁 =
𝑀cr𝓁
374.41
M𝑛𝓁 = 𝑀ne = 182.49 in.-kips
277
The result above indicates that the member does not
subject local buckling if the interaction of elements
are taken into consideration.
The local buckling strength at Location C can
be obtained by replacing 𝑀ne with 𝑀y . From the
CUFSM analysis results for the single C-Section
member, the I-Section member yield moment is
obtained as,
𝑀y = (2)(91.767) = 183.534 in.-kips
√
√
𝑀𝑦
183.534
=
𝜆𝓁 =
= 0.70 < 0.776
𝑀cr𝓁
374.41
𝑀𝑛𝓁𝐶 = 𝑀𝑦 = 183.534 in.-kips
The above result indicates that member does not
subject to local buckling at Support C as well.
c. Distortional Buckling Strength, Mnd . From the
calculation in Part I Item 2c, the distortional buckling strength is 𝑀nd = 158.88 in.-kips
d. Nominal strength, Mnx . At locations B and D, the
nominal flexural strength, 𝑀nx , is the minimum of
𝑀ne , 𝑀n𝑙 and 𝑀nd . Based on the computed results
under Part II items a to c, the distortional buckling
strength governs. Therefore,
Mn𝑥 = 158.88 in. − kips
At Location C, it is assumed that the global and
distortional buckling are restrained, and only the
local buckling need to be considered. Since no local
buckling taken place, the nominal strength is equal
to the yield moment: 𝑀nxC = 183.534 in. − kips
PART III: MEMBER FORCES CONSIDERING
STRUCTURAL STABILITY
Since the beam-column does not subject to joint translation,
the effective length method is applicable.
a. Computation of Cmx . Based on Specification Section
C1.2.1.1, for beam-columns subject to transverse loading
between supports, 𝐶m = 1.0
b. Computation of B𝟏 and B𝟐 , Member Forces and
Moments. Since there is no lateral translations at
member ends, 𝐵2 = 1.
At locations B and D:
𝑃 nt = 20 kips
𝑀 nt = 18.75 P′
Since there is no translation, 𝑃 𝓁t = 0 and 𝑀 𝓁t = 0
278
6
COMBINED AXIAL LOAD AND BENDING
Therefore,
The available strengths are:
𝑃 = 𝑃 nt = 20 kips
𝑃𝑒1 = 𝜋 2 𝑘𝑓 ∕(𝐾1 𝐿)2
where 𝑘f is the stiffness in the plane of bending, and 𝐾1 L
is the effective length in the plane of bending. 𝐾1 = 1.0.
In this example,
𝑘f = EI x and 𝐾1 𝐿 = 10 ft.
𝑃𝑒1 = 𝜋 2 (29500)(22.1)∕(120)2 = 446.84 kips
𝛼 = 1.6 for ASD method
𝐶𝑚
1.0
=
= 1.077
𝐵1 =
20
𝑃
1 − 1.6 446.84
1−𝛼
𝑃a = 𝑃n ∕Ωc = 42.32∕1.80 = 23.51 kips
𝑀a = 𝑀n ∕Ωb = 158.88∕1.67 = 95.14 in-kips
From Eq. (6.65),
𝑀𝑥
20
20.19𝑃 ′
𝑃
+
=
+
≤ 1.0
𝑃𝑎 𝑀ax 23.51
95.14
)(
)
(
20
95.14
= 0.704 kips
𝑃′ = 1 −
23.51
20.19
At location C:
𝑃no = 46.12 kips
𝑀no = 158.88 in. − kips
𝑃𝑒1
𝑀 = B1 𝑀 nt = 1.077(18.75𝑃 ′ ) = 20.19P′
At locations C, B2 = 1.0:
𝑀 = 𝑀 nt = 22.5P′
𝑃 = 𝑃 nt = 20 kips
c. Interaction Check Using EWM.
At location B and D:
𝑃a = 𝑃n ∕Ωc = 47.20∕1.80 = 26.22 kips
𝑀a = 𝑀n ∕Ωb = 158.88∕1.67 = 95.14 in-kips
From Eq. (6.65),
𝑀𝑥
20
20.19𝑃 ′
𝑃
+
=
+
≤ 1.0
𝑃𝑎 𝑀ax 26.22
95.14
)(
)
(
20
95.14
= 1.12 kips
𝑃′ = 1 −
26.22
20.19
At location C:
𝑃a = 𝑃n ∕Ωc = 49.73∕1.80 = 27.63 kips
𝑀a = 𝑀n ∕Ωb = 165.96∕1.67 = 99.38 in-kips
From Eq. (6.16):
20
22.5𝑃 ′
𝑃
𝑀
+
=
+
≤ 1.0
𝑃𝑎0 𝑀𝑎 27.63
99.38
)(
)
(
20
99.38
= 1.22 kips
𝑃′ = 1 −
27.63
22.5
d. Interaction Check Using DSM.
At location B and D:
From the member strengths calculated in Part II, the
available strengths are obtained:
At location B and D:
𝑃n = 42.32 kips; 𝑀n = 158.88 in. − kips
The corresponding available strengths are:
𝑃ao = 𝑃no ∕Ωc = 46.12∕1.80 = 25.62 kips
𝑀ao = 𝑀a = 95.14 in-kips
From Eq. (6.16):
𝑀𝑥
20
20.19𝑃 ′
𝑃
+
=
+
≤ 1.0
𝑃𝑎 𝑀ax 25.62
95.14
)(
)
(
20
95.14
= 1.03 kips
𝑃′ = 1 −
25.62
20.19
′
e. Allowable Load P . Based on calculations in Part III
items c and d, the allowable load for the ASD method is
1.12 kips when the EWM is used. If the DSM is used,
𝑃 ′ = 0.704 kips. For the LRFD method, the load factors
and combinations given in Section 1.8.2.2 should be used
and 𝛼 = 1.0 is used in structural stability consideration.
Example 6.3 For the braced channel column shown
in Fig. 6.13, determine the allowable load if the load at
both ends are eccentrically applied at point A (that is,
𝑒𝑥 = +2.124 in.) along the x axis (Fig. 6.13a). Assume
𝐾x 𝐿x = 𝐾y 𝐿y = 𝐾z 𝐿z = 14 ft. Use 𝐹y = 50 ksi and the
ASD method.
SOLUTION
1. Properties of Full Section. From the equations given
in Part I of the AISI Design Manual,1.428 the following
full section properties can be computed:
𝐴 = 1.553 in.2
𝑥 = 0.876 in.
𝐼𝑥 = 15.125 in.4
J = 0.00571 in.4
𝑆𝑥 = 3.781 in.3
𝐶𝑤 = 24.1 in.6
DESIGN EXAMPLES
Figure 6.13
𝑟𝑥 = 3.12 in.
j = 𝛽y ∕2 = 4.56 in.
𝐼𝑦 = 1.794 in.4
𝑟0 = 3.97 in
𝑆𝑦 = 0.844 in.3
𝑥0 = 2.20 in.
2. Applied Axial Load and End Moments
𝑃 = axial load to be determined
𝑀𝑥 = 0
𝑀𝑦 = 2.124𝑃 in.-kips
3. Computation of 𝑃n
a. Nominal Buckling Strength due to Global Buckling, Pne .
i. Elastic Flexural Buckling Stress. Since 𝐾𝑥 𝐿𝑥 =
𝐾𝑦 𝐿𝑦 and 𝑟𝑥 > 𝑟𝑦 ,
(1)(14 × 12)
𝐾𝐿 𝐾𝑦 𝐿𝑦
=
=
𝑟
𝑟𝑦
1.075
= 156.28200
OK
𝜋2𝐸
𝜋 2 (29,500)
𝐹cre =
(KL∕𝑟)2
Example 6.3
where
(
𝛽=1−
𝜎e𝑥 =
𝑟𝑦 = 1.075 in.
=
(156.28)2
279
𝑥0
𝑟0
)2
(
=1−
2.20
3.97
)2
= 0.693
𝜋2𝐸
(𝐾𝑥 𝐿𝑥 ∕𝑟𝑥 )2
𝜋 2 (29,500)
= 100.418 ksi
(1 × 14 × 12∕3.12)2
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐾t 𝐿t )2
Ar0
=
=
⎡(11,300)(0.00571) +⎤
1
⎢ 𝜋 2 (29,500)(24.1) ⎥
⎥
(1.553)(3.97)2 ⎢
⎣ (1 × 14 × 12)2 ⎦
= 12.793ksi.
Therefore
⎤
⎡
(100.418 + 12.793) −
⎥
⎢√
1
𝐹cre =
(100.418 + 12.793)2 − 4(0.693)⎥
2(0.693) ⎢⎢
⎥
×(100.418)(12.793)
⎦
⎣
= 11.921ksi
ii. Elastic Flexural–Torsional Buckling Stress.
According to Eq. (5.57),
√
1
𝐹cre =
[(𝜎 + 𝜎t ) − (𝜎e𝑥 + 𝜎t )2 − 4𝛽𝜎e𝑥 𝜎t
2𝛽 e𝑥
= 12.269 > 11.921 ksi
Use 𝐹cre = 11.921 ksi to compute 𝐹n :
√
√
𝐹y
50
𝜆c =
=
= 2.048 > 1.5
𝐹cre
11.921
280
6
COMBINED AXIAL LOAD AND BENDING
From Eq. (5.52),
[
[
]
]
0.877
0.877
𝐹n =
=
𝐹
(50)
y
(2.048)2
𝜆2c
= 10.455ksi
𝑃ne = AF n = (1.553)(10.455)
= 16.24 in.-kips
b. Nominal Buckling Strength due to Local Buckling, Pn𝓁 .
i. Effective Width of Compression Flange
√
√
29,500
𝐸
𝑆 = 1.28
= 1.28
= 67.992
𝑓
10.455
0.382𝑆 = 25.973
𝑤
= 2.415∕0.105 = 23 < 60
𝑡
Since w/t < 0.328S,
OK
𝐼a = 0
𝑏 = 𝑤 = 2.415 in. The flange is fully effective.
𝑅I = 1
𝑑s = 𝑑s′
ii. Effective Width of Edge Stiffeners
w 0.5175
=
= 4.929
𝑡
0.105
√
10.455
10.52
(4.929)
= 0.149 < 0.673
𝜆= √
29,500
0.43
The local buckling strength can also be
obtained using the Direct Strength Method:
Using the software CUFSM, the following
numerical analysis results are obtained:
Yield load: 𝑃y = 77.586 kips
Load factor for local buckling, LF local (i.e.,
𝑃cr𝓁 ∕𝑃y ) = 0.54
Therefore, 𝑃cr𝓁 = LF local 𝑃y = (0.54)(77.586)
= 41.894 kips
Following Eqs. (5.64) to (5.66):
√
√
𝜆𝓁 = 𝑃ne ∕𝑃cr𝓁 = 16.24∕41.894
= 0.623 < 0.776
P𝑛𝓁 = 𝑃ne = 16.24 kips
The above result from the Direct Strength
Method indicates that with the consideration of
the flange and web interactions, the section does
not subject to local buckling under the given
global buckling load.
c. Nominal Buckling Strength due to Distortional
Buckling, Pnd .
The distortional buckling load 𝑃crd can be
computed according to Section 2.3.1.3 of the
North American Specification, which is enclosed in
Section 3.5.2 of this book. Following the procedure
illustrated in item 4 of Example 5.2, the computed
rotational stiffnesses are as follows:
𝑘𝜙fe = 1.069 in. − kips∕in.
𝑘𝜙we = 0.782 in. − kips∕in.
𝑑𝑠′ = 𝑑 = 0.5175 in.
𝑘𝜙 = 0
Therefore,
̃
𝑘𝜙fg = 0.0286(in. − kips∕in.)∕ksi
d𝑠 = 0.5175 in.
The edge stiffener is fully effective.
iii. Effective Width of Web Element
w 7.415
=
= 70.619 < 500
𝑡
0.105
√
10.455
10.52
𝜆 = √ (70.619)
29,500
4.0
̃
𝑘𝜙wg = 0.0162(in. − kips∕in.)∕ksi
OK
= 0.699 > 0.673
1 − 0.22∕0.699
= 0.980
0.699
𝑏 = 𝜌w = (0.980)(7.415) = 7.267 in.
𝜌=
iv. Effective Area 𝐴e
Ae = 1.553 − (7.415 − 7.267)(0.105) = 1.537 in.2
𝑃𝑛𝓁 = 𝐴e 𝐹n = (1.537)(10.455) = 16.069 kips
From Eq. (3.105), the elastic distortional buckling
stress is
𝑘𝜙fe + 𝑘𝜙we + 𝑘𝜙
1.069 + 0.782 + 0
𝐹crd =
=
0.0286 + 0.0162
̃
̃
𝑘𝜙fg + 𝑘𝜙wg
= 41.32 ksi
The distortional buckling load is
𝑃crd = 𝐴g 𝐹crd = (1.553)(41.32) = 64.17 kips
Note, the distortional buckling load can also be
obtained using the numerical analysis. Using software, CUFSM, the following results are obtained:
Yield load: 𝑃y = 77.586 kips
DESIGN EXAMPLES
Load factor for distortional buckling, LF dist (i.e.,
𝑃crd ∕𝑃y ) = 0.80 Therefore,
𝑃crd = LF dist 𝑃y = (0.80)(77.586) = 62.07 kips
The numerical analysis result is very close
to the analytical result. The analytical result
𝑃crd = 64.17 kips will be used to determine the
distortional buckling strength as shown below.
The yield load based on the calculated area from
item 1 is
𝑃y = AF y = (1.553)(50) = 77.65 kips
Based on Eq. (5.71),
√
√
𝑃y
77.65
=
= 1.100 > 0.561
𝜆d =
𝑃crd
64.17
From Eq. (5.70), the nominal axial load for distortional buckling based on Section E4 of the North
American Specification is
[
(
) ](
)
𝑃crd 0.6
𝑃crd 0.6
𝑃y
𝑃nd = 1 − 0.25
𝑃y
𝑃y
[
) ](
)
(
64.17 0.6 64.17 0.6
= 1 − 0.25
(77.65)
77.65
77.65
= 53.81 kips
d. Nominal Strength of Member, Pn . The nominal
strength of the column is the minimum of 𝑃ne , 𝑃n𝓁
and 𝑃nd . Comparing the results calculated in Items
a to c, the following results are obtained from the
EWM and the DSM:
From the EWM, the strength is governed by the
local bucking, 𝑃n = 16.069 kips.
From the DSM, the strength is governed by the
flexural buckling, 𝑃n = 16.24 kips.
In this example, since the nominal strength from
both methods are very close, 𝑃𝑛 = 16.069 kips is
used for evaluating the combined compressive axial
load and bending.
4. Application of Mn .
a. Lateral–Torsional Buckling Strength, 𝑀ne .
According to Eq. (4.60), the elastic critical
lateral–torsional buckling stress for bending
about the centroidal axis perpendicular to the
symmetry axis for a singly symmetric channel
section is
𝐹cre
𝐶 𝐴𝜎
= s e𝑥
CTF 𝑆f
⎡
⎢𝑗 + 𝐶s
⎢
⎣
√
(
𝑗2
+
𝑟20
)
𝜎t ⎤⎥
𝜎e𝑥 ⎥
⎦
where Cs
A
𝜎 ex
𝜎t
j
r0
CTF
Sf
=
=
=
=
=
=
=
=
281
–1
1.553 in.2 (see item 1)
100.418 ksi (see item 3.a.ii)
12.793 ksi (see item 3.a.ii)
4.56 in. (see item 1)
3.97 in. (see item 1)
1.0 [see Eq. (4.62)]
Sy = 0.844 in.3 (see item 1)
Substituting all values into the equation for 𝐹cre ,
the elastic critical buckling stress is
(−1.553)(100.418)
𝐹cre =
(1)(0.844)
4.56 −
⎡√
⎤
(
)⎥
⎢
12.793
2
2
⎢ 4.56 + 3.97
⎥
100.418 ⎦
⎣
= 39.744 ksi
0.56 𝐹y = 0.56(50) = 28 ksi
2.78 𝐹y = 2.78(50) = 139 ksi
Since 2. 𝐹y > 𝐹cre > 0.56𝐹y , use Eq. (4.52) to
compute 𝐹n , that is,
(
)
10𝐹y
10
𝐹n = 𝐹y 1 −
9
36𝐹cre
)
(
10
10 × 50
= 36.141 ksi
= (50) 1 −
9
36 × 39.744
Following the same procedure used in item 3b,
the elastic section modulus of the effective section
calculated at a stress of 𝑓 = 𝐹n = 36.141 ksi in the
extreme compression fiber is
𝑆f = 𝑆𝑦 = 0.844 in.3
𝑀ne = 𝑆f 𝐹𝑛 = 30.503 in. − kips
for lateral − torsional buckling strength
b. Nominal moment due to local buckling interacting
with lateral-torsional global buckling, 𝑀n𝓁 .
The maximum compressive stress 𝑓 = 𝐹n =
36.141 ksi occurs in the extreme fiber of edge
stiffeners and that both flanges are fully effective,
as shown in Fig. 6.14. For edge stiffeners,
√
36.141
1.052
(4.929)
= 0.276 < 0.673
𝜆= √
29,500
0.43
𝑏 = 𝑤 = 0.5175 in.
Check if flange is fully effective. From Fig. 6.14,
)
(
1.8315
= 31.164 ksi (compression)
𝑓1 = 36.141
2.124
282
6
COMBINED AXIAL LOAD AND BENDING
36.141 ksi
Figure 6.14 Stress distribution in flanges.
)
0.5835
= 9.929 ksi
2.124
|𝑓 |
9.929
𝜓 = || 2 || =
= 0.319
| 𝑓1 | 31.164
𝑓2 = 36.141
(
(tension)
𝑘 = 4 + 2(1 + 𝜓)3 + 2(1 + 𝜓)
= 4 + 2(1.319)3 + 2(1.319) = 11.222
√
1.052
31.164
(23)
𝜆= √
= 0.235 < 0.673
29,500
11.222
𝑏e = 𝑤 = 2.415 in.
Since ℎ0 ∕𝑏0 = 3.0∕0.81 = 3.70 < 4, use
Eq. (3.47a),
𝑏1 =
𝑏e
2.415
=
= 0.728 in.
3+𝜓
3.319
Since 𝜓 > 0.236, 𝑏2 = 𝑏𝑒 ∕2 = 2.415∕2 =
1.2075 in.,
𝑏1 + 𝑏2 = 0.728 + 1.2075 = 1.9355 in.
Because the computed value of 𝑏1 + 𝑏2 is
greater than the compression portion of the flange
(1.8315 in.), the flange is fully effective.
In view of the fact that all elements are fully
effective, the section modulus relative to the
extreme compression fiber is 𝑆e = 𝑆y (for full
section) = 0.844 in.3 :
𝑀𝑛𝓁 = 𝑆e 𝐹n = 0.844(36.141) = 30.503 in. − kips
c. Nominal strength due to distortional buckling. The
distortional buckling moment can be determined
analytically using the equations provided in Section
3.5.2(5) or using the numerical analysis method.
In this example, the software CUFSM is used to
determine the distortional buckling moment. The
following results are obtained from the software:
𝑀y = 43.2 in.-kips
The distortional buckling load factor, LF dist (i.e.,
𝑀crd ∕𝑀y ) = 2.15
Therefore the distortional buckling moment,
𝑀crd = LF dist 𝑀y = (2.15)(43.2) = 92.88 in.-kips
Using Eqs. (4.101) to (4.103), the nominal
strength is determined as follows:
√
√
𝜆𝑑 = 𝑀𝑦 ∕𝑀crd = 43.2∕92.88
𝑀nd
= 0.682 > 0.673
[
) ](
)
(
𝑀crd 0.5
𝑀crd 0.5
= 1 − 0.22
𝑀𝑦
𝑀𝑦
𝑀𝑦
ADDITIONAL INFORMATION ON BEAM–COLUMNS
[
(
) ](
)
92.88 0.5 92.88 0.5
(43.2)
= 1 − 0.22
43.2
43.2
= 42.91 in.-kips
d. Nominal Moment of Member 𝑀n . The nominal
moment is the minimum of 𝑀ne , 𝑀n𝓁 and 𝑀nd .
From the results calculated in Items 4a to 4c, the
lateral–torsional buckling controls, Therefore,
𝑀n = 30.503 in. − kips
5. Determine Member Forces in Considering Structural Stability.
Since the column has no differential lateral translation between its supports, 𝐵2 = 1. In addition, 𝑃 𝓁𝑡 = 0
and 𝑀 𝓁𝑡 = 0
𝑃 = 𝑃 nt = 𝑃
𝑀 nt = 𝑃ex = 2.124P
a. Computation of Cmy . Based on Eq. (6.60),
(
)
𝑀1
𝐶m𝑦 = 0.6 − 0.4
= 0.6 − 0.4(−1.0) = 1.0
𝑀2
b. Computation of B1 .
From Eq. (6.61)
𝑃𝑒1 = 𝜋 2 𝑘𝑓 ∕(𝐾1 𝐿)2
where 𝑘f = EI y , 𝐾1 = 1.0, and 𝐿 = 14 ft.
𝑃𝑒1 = 𝜋 2 (29500)(0.844)∕(14 × 12)2 = 8.707 kips
From Eq. (6.59), in which α = 1.6 for ASD
method,
𝐶𝑚
1.0
1.0
𝐵1 =
=
=
𝑃
1 − 0.184𝑃
𝑃
1 − (1.6)
1−𝛼
8.707
𝑃𝑒1
Based on Equation (6.57),
2.124𝑃
1 − 0.184𝑃
6. Interaction Check. Using Eq. (6.65),
𝑀 = 𝐵1 𝑀nt =
𝑃
𝑀
+
≤ 1.0
𝑃𝑎 𝑀𝑎
where
𝑃a = 𝑃n ∕Ωc = 16.069∕1.80 = 8.927 kips
𝑀a = 𝑀n ∕∕Ωb = 30.503∕1.67 = 18.365 in.-kips
2.124𝑃
𝑃
1
−
0.184𝑃 ≤ 1.0
+
8.927
18.365
283
Try 𝑃 = 2.83 kips,
2.124(2.83)
1 − 0.184(2.83)
3.4
+
= 1.0
8.927
18.365
Check the beam-column at the braced point using
𝑃 = 2.83 kips
At the braced point:
𝑃 = 2.83 kips, 𝑀 = 𝑀 nt = 2.83 × 2.124 = 6.01
in.-kips
Using the same procedure illustrated in item 3, the
effective area at stress 𝐹y = 50 ksi is
𝐴e = 1.141 in.2
𝑃𝑛0 = 𝐴e 𝐹n = 1.141(50) = 57.05 kips
Using Eq. (6.16) to check perform the interaction
check at the braced point:
2.83
6.01
𝑃
𝑀
+
=
+
= 0.377 < 1.0 OK
𝑃𝑎0 𝑀𝑎
57.05 18.365
7. Allowable Load P. Based on Eqs. (6.65) and (6.16),
the allowable load P is 2.83 kips, which is governed by
the stability requirement.
6.7 ADDITIONAL INFORMATION
ON BEAM–COLUMNS
The readers may also refer to Refs. 6.7–6.35, 5.103, 5.135,
and 6.37–6.43 for other beam–column design information.
In the examples provided in this section, the effective
length method has been used in considering structural
stability. If the member has joint translations, the member
stability may be required to be considered by other methods
outlined in Section 6.4. Design examples using different
methods are provided in the AISI Cold-Formed Steel Design
Manual.1.428
Some new research work has been accomplished in
beam-column design. The research work by S. Torabian,
et al.6.44 proposed a direct strength prediction which is based
on the local, distortional, and global buckling under the
compression and bending combined action. This approach
is expected to provide a more mechanically sound solution
to the strength of cold-formed steel beam-columns, and the
predicted strength is in average 20% higher than conventional method. The method is applicable for any types of
cross-section and combinations of 𝑃 -𝑀x -𝑀y loading, and is
a powerful tool for generating optimized shapes.
CHAPTER 7
Closed Cylindrical Tubular
Members
7.1 GENERAL REMARKS
The design of square and rectangular tubular sections as
flexural and compression members is discussed in Chapters 3
to 6. This chapter deals with the strength of closed cylindrical
tubular members and the design practice for such members
used as either flexural or compression members.
Closed cylindrical tubular members are economical
sections for compression and torsional members because of
their large ratio of radius of gyration to area, the same radius
of gyration in all directions, and the large torsional rigidity.
In the past, the structural efficiency of such tubular members
has been recognized in building construction. A comparison made by Wolford on the design loads for round and
square tubing and hot-rolled steel angles used as columns
indicates that for the same size and weight round tubing will
carry approximately 2 12 ; and 1 12 times the column load of
hot-rolled angles when the column length is equal to 36 and
24 times the size of the section, respectively.7.1
7.2 TYPES OF CLOSED CYLINDRICAL TUBES
The buckling behavior of closed cylindrical tubes, which will
be discussed later, is significantly affected by the shape of
the stress–strain curve of the material, the geometric imperfections such as out of roundness, and the residual stress. It
would therefore be convenient to classify tubular members
on the basis of their buckling behavior.
In general, closed cylindrical tubes may be grouped as
(1) manufactured tubes and (2) fabricated tubes.7.2 Manufactured tubes are produced by piercing, forming and welding,
cupping, extruding, or other methods in a plant. Fabricated
tubes are produced from plates by riveting, bolting, or
welding in an ordinary structural fabrication shop. Since
fabricated tubes usually have more severe geometric imperfections, the local buckling strength of such tubes may be
considerably below that of manufactured tubes.
Manufactured structural steel tubes include the following
three types:
1. Seamless tubes
2. Welded tubes
3. Cold-expanded or cold-worked tubes
For the seamless tubes, the stress–strain curve is affected
by the residual stress resulting from cooling of the tubes. The
proportional limit of a full-sized tube is usually about 75%
of the yield stress. This type of tube has a uniform property
across the cross section.
Welded tubes produced by cold forming and welding steel
sheets or plates have gradual-yielding stress–strain curves,
as shown in Fig. 2.2 due to the Bauschinger effect and the
residual stresses resulting from the manufacturing process.
The proportional limit of electric resistance welded tubes
may be as low as 50% of the yield stress.
Cold-worked tubes also have this type of gradual yielding
because of the Bauschinger effect and the cold work of
forming.
7.3
FLEXURAL COLUMN BUCKLING
The basic column formulas for elastic and inelastic buckling discussed in Chapter 5 [Eqs. (5.3a) and (5.7a)] are
usually applicable to tubular compression members having
a proportional limit of no less than 70% of the yield stress.
For electric resistance welded tubes having a relatively low
proportional limit, Wolford and Rebholz recommended the
following formulas on the basis of their tests of carbon steel
tubes with yield stresses of 45 and 55 ksi (310 and 379 MPa
or 3164 and 3867 kg/cm2 )7.3 :
√
√
⎡
𝐹y ( KL )⎤
KL
3𝜋 2 𝐸
2
⎥
⎢
𝜎T = 𝐹y 1 − √
for
≤
2
⎥
⎢
𝑟
𝐹y
3 3 𝜋 𝐸 𝑟 ⎦
⎣
(7.1)
√
𝜋2𝐸
KL
3𝜋 2 𝐸
for
(7.2)
𝜎e =
>
2
𝑟
𝐹y
(KL∕𝑟)
where Fy , E, K, and L are as defined in Chapter 5. The radius
of gyration r of closed cylindrical tubes can be computed as
√
𝐷o2 + 𝐷i2
𝑅
𝑟=
≃√
(7.3)
4
2
285
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
286
7
CLOSED CYLINDRICAL TUBULAR MEMBERS
Figure 7.1
Test data for column buckling of axially loaded cylndrical tubes.3.84
where Do = outside diameter
Di = inside diameter
R = mean radius of tube
The correlation between the test results and Eqs. (5.3),
(5.7), (7.1), and (7.2) is shown in Fig. 7.1.3.84,7.4,7.5 Also
shown in this figure are the test data reported by Zaric.7.6
Because closed cylindrical tubes are commonly used in
offshore structures, extensive analytical and experimental
studies of the strength of tubular members have been made
by numerous investigators throughout the world.7.7–7.15
7.4
LOCAL BUCKLING
Local buckling of closed cylindrical tubular members can
occur when members are subject to
1.
2.
3.
4.
5.
Axial compression
Bending
Torsion
Transverse shear
Combined loading
Each item will be discussed separately as follows.
7.4.1
Local Buckling under Axial Compression
When a closed cylindrical tube is subject to an axial compressive load, the elastic stability of the tube is more complicated
than is the case for a flat plate. Based on the small-deflection
theory, the structural behavior of a cylindrical shell can be
expressed by the following eighth-order partial differential
equation7.16 :
(
)
1 4
Et 𝜕 4 𝜔
𝜕2𝜔
8
∇ 𝜔 + ∇ 𝑁𝑥 2 +
=0
(7.4)
𝐷
𝜕𝑥
DR2 𝜕𝑥4
where
∇8 𝜔 = ∇4 (∇4 𝜔)4 𝜔 =
and x
y
𝜔
Nx
t
R
E
=
=
=
=
=
=
=
𝜕4𝜔
𝜕4𝜔
𝜕4𝜔
+2 2 2 + 4
4
𝜕𝑥
𝜕𝑥 𝜕𝑦
𝜕𝑦
(7.5)
coordinate in x direction
coordinate in tangential direction
displacement in radial direction
axial load applied to cylinder
thickness of tube
radius of tube
modulus of elasticity of steel, = 29.5 × 103 ksi
(203 GPa or 2.07 × 106 kg/cm2 )
3
Et
D = 12(1−𝜇
2)
𝜇 = Poisson’s ratio, = 0.3
See Fig. 7.2 for dimensions of a closed cylindrical tube
subjected to axial compression.
For a given closed cylindrical tube the buckling behavior
varies with the length of the member. For this reason, from
the structural stability point of view, it has been divided into
the following three categories by Gerard and Becker7.16 :
1. Short tubes, Z < 2.85
2. Moderate-length tubes, 2.85 < Z < 50
3. Long tubes, Z > 50
287
LOCAL BUCKLING
Figure 7.2 Cylindrical tube subjected to axial compression.
Figure 7.3 Local buckling of moderate-length tubes.
Here
𝐿2 √
𝐿2
1 − 𝜇2 = 0.954
(7.6)
Rt
Rt
For very short tubes (i.e., the radius of the tube is extremely
large compared with its length), the critical local buckling
stress is
𝜋 2 𝐸(𝑡2 ∕12)
𝑓cr =
(7.7)
(1 − 𝜇2 )𝐿2
𝑍=
which is identical with the Euler stress for a plate strip of unit
width.
For extremely long tubes, the tube will buckle as a column.
The critical buckling load is
𝜋 2 EI
(7.8)
𝐿2
where I is the moment of inertia of the cross section of the
tube,
(7.9)
𝐼 = 𝜋𝑅3 𝑡
𝑃cr =
Therefore, for long tubes the critical buckling stress is
( )
𝜋2𝐸 𝑅 2
(7.10)
𝑓cr =
2 𝐿
Moderate-length tubes may buckle locally in a diamond
pattern, as shown in Fig. 7.3. The critical local buckling
stress is
𝑡
𝑓cr = CE
(7.11)
𝑅
According to the classic theory (small-deflection theory)
on local buckling, the value of C can be computed as
1
𝐶=√
= 0.605
3(1 − 𝜇2 )
(7.12)
Therefore
( )
( )
𝐸
𝑡
𝑡
= 0.605𝐸
𝑓cr = √
𝑅
3(1 − 𝜇2 ) 𝑅
(7.13)
Whenever the buckling stress exceeds the proportional
limit, the theoretical local buckling stress is in the inelastic
range, which can be determined by
( )
𝑡
𝑓cr = aCE
𝑅
Here a is the plasticity reduction factor,7.2
)1∕2 ( ) ( )
(
𝐸s
𝐸t 1∕2
1 − 𝜇2
𝑎=
𝐸
𝐸
1 − 𝜇p2
where 𝜇
𝜇p
Es
El
E
=
=
=
=
=
(7.14)
(7.15)
Poisson’s ratio in the elastic range, = 0.3
Poisson’s ratio in the plastic range, = 0.5
secant modulus
tangent modulus
modulus of elasticity
Results of numerous tests indicate that the actual value of
C may be much lower than the theoretical value of 0.605 due
to the postbuckling behavior of the closed cylindrical tubes,
which is strongly affected by initial imperfections.
The postbuckling behavior of the three-dimensional
closed cylindrical tubes is quite different from that of
two-dimensional flat plates and one-dimensional columns.
As shown in Fig. 7.4a, the flat plate develops significant transverse-tension membrane stresses after buckling
because of the restraint provided by the two vertical edges.
These membrane stresses act to restrain lateral motion, and
therefore the plate can carry additional load after buckling.
For columns, after flexural buckling occurs, no significant
transverse-tension membrane stresses can be developed to
restrain the lateral motion, and therefore, the column is free
to deflect laterally under critical load.
For closed cylindrical tubes, the inward buckling causes
superimposed transverse compression membrane stresses,
and the buckling form itself is unstable. As a consequence
of the compression membrane stresses, buckling of an
axially loaded cylinder is coincident with failure and occurs
288
7
Figure 7.4
CLOSED CYLINDRICAL TUBULAR MEMBERS
Buckling patterns of various structural components.
suddenly, accompanied by a considerable drop in load
(snap-through buckling).
Since the postbuckling stress of a closed cylindrical tube
decreases suddenly from the classic buckling stress, the
stress in an imperfect tube reaches its maximum well below
the classic buckling stress (Fig. 7.5).
On the basis of the postbuckling behavior discussed above,
Donnell and Wan developed a large-deflection theory which
indicates that the value of C varies with the R/t ratio as shown
in Fig. 7.6, which is based on average imperfections.7.17
In the past, the local buckling strength of closed cylindrical tubes subjected to axial compression has been
studied at Lehigh University,7.18–7.20 the University of
Alberta,7.21 the University of Tokyo,7.22 the University of
Figure 7.6
Figure 7.5 Postbuckling behavior of flat plates, columns, and
cylindrical tubes.7.16
Toronto,6.30 and others.7.11–7.15,7.31 The test data obtained
from previous and recent research projects have been used
in the development and improvement of various design
recommendations.1.4,7.23–7.25
7.4.2
Local Buckling under Bending
The local buckling behavior in the compression portion of
a flexural tubular member is somewhat different compared
with that of the axially loaded compression member.
On the basis of their tests and theoretical investigation,
Local buckling imperfection parameter versus R/t ratio.7.16
NORTH AMERICAN DESIGN CRITERIA
Gerard and Becker7.16 suggested that the elastic local
buckling stress for bending be taken as 1.3 times the
local buckling stress for axial compression. This higher
elastic buckling stress for bending results from the beneficial
effect of the stress gradient that exists in bending. However,
some investigators have indicated that there is not much
difference between the critical stress in bending and that in
axial compression.3.84
The bending strength of closed cylindrical tubes has
been studied by Sherman7.26 and Stephens, Kulak, and
Montgomery.7.21
7.4.3 Local Buckling under Torsion
The theoretical local buckling stress of moderate-length
tubes subjected to torsion can be computed by7.2
( )5∕4 ( )1∕2
0.596𝑎
𝑅
𝑡
(𝜏cr )torsion =
𝐸
𝑅
𝐿
(1 − 𝜇2 )5∕8
( )5∕4 ( )1∕2
𝑅
𝑡
= 0.632aE
(7.16)
𝑅
𝐿
where 𝜏 cr is the critical shear buckling stress due to torsion
and
)3∕4 ( )
(
𝐸s
𝐸
1 − 𝜇2
𝑎=
= 1.16 s
(7.17)
2
𝐸
𝐸
1 − 𝜇p
Previous studies indicate that the effect of imperfection on
torsional postbuckling is much less than the effect on axial
compression. Test data indicate that due to the effect of initial
imperfection the actual strength of the member is smaller
than the analytical result.
7.4.4 Local Buckling under Transverse Shear
In Ref. 7.2, Schilling suggests that in the elastic range the
critical shear buckling stress in transverse shear be taken as
1.25 times the critical shear buckling stress due to torsion,
that is,
( )5∕4 ( )1∕2
𝑅
𝑡
(𝜏cr )transverse shear = 1.25 × 0.632aE
𝑅
𝐿
( )5∕4 ( )1∕2
𝑅
𝑡
= 0.79aE
(7.18)
𝑅
𝐿
7.4.5 Local Buckling under Combined Loading
The following interaction formula may be used for any
combined loading7.2 :
( ) ( )2
𝑓
𝜏
=1
(7.19)
+
𝑓cr
𝜏cr
where
f
fcr
𝜏
𝜏 cr
=
=
=
=
289
actual computed normal stress
critical buckling stress for normal stress alone
actual computed shear stress
critical buckling stress for shear stress alone
7.5 NORTH AMERICAN DESIGN
CRITERIA1.314,1.345,1.417
The AISI design criteria for closed cylindrical tabular
members were revised in the 1986 and 1996 editions of
the Specification on the basis of Refs. 1.158, 7.5, and 7.30.
In 1999, the design equation for determining the effective
area was simplified.1.333 The same design criteria were
retained in the 2007 edition of the North American Specification with rearrangement of section numbers and editorial
revisions.1.345 For additional information, the reader is
referred to Refs. 8.25–8.32.
7.5.1
Local Buckling Stress
Considering the postbuckling behavior of the axially
compressed cylinder and the important effect of the initial
imperfection, the design provisions included in the AISI
Specification were originally based on Plantema’s graphic
representation7.27 and the additional results of cylindrical
shell tests made by Wilson and Newmark at the University
of Illinois.7.28,7.29
From the tests of compressed tubes, Plantema found that
the ratio Fult /Fy depends on the parameter (E/Fy )(t/D), in
which t is the wall thickness, D is the mean diameter of
the closed tubes, and Fult is the ultimate stress or collapse
stress. As shown in Fig. 7.7, line 1 corresponds to the collapse
stress below the proportional limit, line 2 corresponds to the
collapse stress between the proportional limit and the yield
stress (the approximate proportional limit is 83% of Fy at
point B), and line 3 represents the collapse stress occurring
at yield stress. In the range of line 3, local buckling will not
occur before yielding, and no stress reduction is necessary.
In ranges 1 and 2, local buckling occurs before the yield
stress is reached. In these cases the stress should be reduced
to safeguard against local buckling.
As shown in Fig. 7.7, point A represents a specific value
of (E/Fy )(t/D) = 8, which divides yielding and local buckling.
Using E = 29.5 × 103 ksi (203 GPa or 2.07 × 106 kg/cm2 ), it
can be seen that closed tubes with D/t ratios of no more than
0.125 E/Fy are safe from failure caused by local buckling.
Specifically, Plantema’s equations are as follows7.4 :
1. For D/t < 0.125 E/Fy (yielding failure criterion represented by line 3),
𝐹ult
=1
(7.20)
𝐹y
290
7
CLOSED CYLINDRICAL TUBULAR MEMBERS
Figure 7.7
Ultimate strength of cylindrical tubes for local buckling.
2. For 0.125 E/Fy < D/t < 0.4 E/Fy (inelastic buckling
criterion represented by line 2),
( )( )
𝐹ult
𝐸
𝑡
+ 0.75
(7.21)
= 0.031
𝐹y
𝐹y
𝐷
The correlation between the available test data and
Eq. (7.24) is shown in Fig. 7.8.
Let A be the area of the unreduced cross section and A0 be
the reduced area due to local buckling; then
3. For D/t > 0.4E/Fy (elastic buckling criterion represented by line 1),
( )( )
𝐹ult
𝐸
𝑡
(7.22)
= 0.33
Fy
𝐹y
𝐷
(7.25)
Based on a conservative approach, AISI specifies that
when the D/t ratio is smaller than or equal to 0.112E/Fy , the
tubular member shall be designed for yielding. This provision is based on point A1 , for which (E/Fy ) (t/D) = 8.93.
When 0.112E/Fy <D/t < 0.441E/Fy , the design of closed
cylindrical tubular members is based on the local buckling
criteria. For the purpose of developing a design formula for
inelastic buckling, point B1 was selected by AISI to represent
the proportional limit. For point B1 ,
( )( )
𝐹ult
𝐸
𝑡
= 2.27 and
= 0.75
(7.23)
𝐹y
𝐷
𝐹y
Using line A1 B1 , the maximum stress of tubes can be
represented by
( )( )
𝐹ult
𝐸
𝑡
+ 0.667
(7.24)
= 0.037
𝐹y
𝐹y
𝐷
AF ult = 𝐴0 𝐹y
(
or
𝐴0 =
𝐹ult
𝐹y
)
𝐴
(7.26)
Substituting Eq. (7.24) into Eq. (7.26), the following
equation can be obtained for D/t < 0.441E/Fy :
[
]
0.037
+ 0.667 𝐴 ≤ 𝐴
(7.27)
𝐴0 =
(𝐷∕𝑡)(𝐹y ∕𝐸)
where D is the outside diameter of the closed cylindrical
tubular member.
7.5.2
Compressive Strength
When a closed cylindrical tubular member is subject to
a compressive load in the direction of the member axis
passing through the centroid of the section, the AISI design
provision was changed in 1996 and 1999 to reflect the
results of additional studies of closed cylindrical tubular
members to be consistent with Section C4 of the 1996
specification.1.314 The following equations are now included
NORTH AMERICAN DESIGN CRITERIA
291
Figure 7.8 Correlation between test data and AISI criteria for local buckling of closed cylindrical
tubes under axial compression.
in Section E2 of the 2016 edition of the North American
Specification for determining the nominal axial strength
Pne of closed cylindrical tubular members having a ratio
of outside diameter to wall thickness, D/t, not greater than
0.441E/Fy .1.314,1.333,1.345,1.417
𝑃ne = 𝐹n 𝐴e
𝐹n = (0.658 )𝐹y
For 𝜆c >1.5,
(7.29)
[
]
0.877
𝐹y
𝐹n =
𝜆2c
where
√
𝜆c =
𝐹y
𝐹cre
]
0.037
𝐷 0.441𝐸
+ 0.667 𝐴 ≤ 𝐴 for
≤
𝐴o =
(DF y )∕(tE)
𝑡
𝐹y
(7.34)
where
(7.28)
where Pne is the nominal axial strength of the member and
Fn is the flexural buckling stress determined as follows: For
𝜆c ≤ 1.5,
𝜆2c
[
(7.30)
(7.31)
In the above equations Fcre is the elastic flexural buckling stress determined according to Section E2.1 of the
specification and Ae is the effective area of the cylindrical
tubular member under axial compression determined as
follows1.333 1.343, 1.345 :
(7.32)
𝑅 = 𝐹y ∕(2𝐹e ) ≤ 1.0
(7.33)
=
=
=
=
=
area of full unreduced cross section
outside diameter of closed cylindrical tube
modulus of elasticity of steel
yield stress
thickness
Equations (7.28)–(7.34) can be summarized in Fig. 7.9. It
can be seen that Eq.
√ (7.32) gives Ae = Ao when 𝜆c = 0 and
Ae = A when 𝜆c = 2. The latter is due to the fact that for
long columns the stresses at which the column buckles are
so low that they will not cause local buckling before primary
buckling has taken place.
Consequently, for the design of axially loaded closed
cylindrical tubular members, the allowable axial load Pa for
the ASD method is determined by Eq. (7.35):
𝑃
𝑃a = ne
(7.35)
Ωc
where Ωc = 1.80 is the safety factor for axial compression.
For the LRFD and LSD methods, the design axial strength is
𝜙c Pn , in which 𝜙c equals 0.85 for LRFD and 0.80 for LSD.
7.5.3
𝐴e = 𝐴o + 𝑅(𝐴 − 𝐴o )
A
D
E
Fy
t
Bending Strength
In Section 7.4.2, it was pointed out that for closed cylindrical tubular members the elastic local buckling stress for
292
7
CLOSED CYLINDRICAL TUBULAR MEMBERS
Figure 7.9
Nominal compressive load of cylindrical tubular members.
bending is higher than the elastic local buckling stress for
axial compression. In addition, it has been recognized that
for thick closed cylindrical members subjected to bending
the initiation of yielding does not represent the failure condition, as is generally assumed for axial loading. For relatively compact members with D/t ≤ 0.0714E/Fy , the flexural strength can reach the plastic moment capacity, which
is at least 1.25 times the moment at first yielding. As far as
the local buckling is concerned, the conditions for inelastic
buckling are not as severe as the case of axial compression due to the effect of the stress gradient. Based on the
results of previous studies, the following design provisions
for determining the nominal flexural strength are included
in Section F2.3 of the 2016 edition of the North American
Specification for closed cylindrical tubular members having
D/t ≤ 0.441E/Fy 1.314,1.333,1.345,1.417 :
𝑀ne = 𝐹n 𝑆f
(7.36)
1. For D/t ≤ 0.0714E/Fy ,
𝐹n = 1.25𝐹y
2. For 0.0714E/Fy < D/t ≤ = 0.318E/Fy ,
(
)
0.020(𝐸∕𝐹y )
𝐹n = 0.970 +
𝐹y
𝐷∕𝑡
(7.37)
(7.38)
3. For 0.318E/Fy < D/t ≤ 0.441E/Fy :
𝐹n =
0.328𝐸
𝐷∕𝑡
(7.39)
where Mne = nominal flexural strength for yielding
and local buckling
D = outside diameter of cylindrical tube
t = wall thickness
Fn = nominal flexural bending stress
Sf = elastic section modulus of full,
unreduced cross section relative to
extreme compression fiber
The allowable flexural strength Ma for the ASD method is
determined by using Eq. (7.40):
𝑀a =
𝑀ne
Ωb
(7.40)
where Ωb = 1.67 is the safety factor for bending. For the
LRFD and LSD methods, the design flexural strength is
𝜙b Mn , in which 𝜙b equals 0.95 for LRFD and 0.90 for LSD.
The nominal flexural strengths based on the critical flexural buckling stresses from Eqs. (7.37)–(7.39) are shown
graphically in Fig. 7.10. As compared with the 1980 edition
of the AISI specification, it can be shown that the increases of
the nominal moment range from about 13 to 25% according
to the value of (E/Fy )(t/D).
7.5.4
Combined Bending and Compression
The interaction formulas presented in Chapter 6 can also
be used for the design of beam–columns using closed
cylindrical tubular members. The nominal axial strength
DESIGN EXAMPLES
Figure 7.10
Nominal flexural strength of cylindrical tubular members.
and nominal flexural strength can be obtained from Sections
7.5.2 and 7.5.3, respectively.
1. Sectional Properties of Full Section
𝐴=
7.6 DESIGN EXAMPLES
Example 7.1 Use the ASD and LRFD methods to determine the available axial strength for the closed cylindrical
tube having 10 in. outside diameter to be used as an axially
loaded simply supported column. Assume that the effective
column length is 15 ft, the yield stress of steel is 33 ksi,
and the thickness of the tube is 0.105 in.
SOLUTION
A. ASD Method
Using the North American design criteria, the limiting D/t
ratio is
( )
29,500
𝐸
𝐷
= 0.441
= 0.441
= 394.23
𝑡 lim
𝐹y
33
The actual D/t ratio is
10
𝐷
=
= 95.24 < 394.23
𝑡
0.105
293
OK
𝜋(𝐷o2 − 𝐷i2 )
4
𝜋[(10.0)2 − (10.0 − 2 × 0.105)2 ]
= 3.264 in.2
=
4
√
𝐷o2 + 𝐷i2
𝑟=
4
√
(10.0)2 + (10.0 − 2 × 0.105)2
=
= 3.500 in.
4
2. Nominal Axial Strength Pn
a.
KL 15 × 12
=
= 51.43
𝑟
3.50
According to Eq. (5.56), the elastic flexural buckling
stress is
𝐹cre =
𝜋 2 (29,500)
𝜋2𝐸
=
= 110.08 ksi
(KL∕𝑟)2
(51.43)2
294
7
CLOSED CYLINDRICAL TUBULAR MEMBERS
b. Based on Eq. (7.31),
√
√
𝐹y
33
𝜆c =
=
= 0.548 < 1.5
𝐹cre
110.08
𝐹n = (0.658𝜆c )𝐹y = (0.6580.548 )(33) = 29.10 ksi
2
2
c. Based on Eqs. (7.32), and (7.33), (7.34)
𝐴e = 𝐴o + 𝑅(𝐴 − 𝐴o )
where
𝑅=
𝐹y
2𝐹e
33
= 0.150 < 1.0
OK
2 × 110.08
[
]
0.037
+ 0.667 𝐴
𝐴o =
(𝐷∕𝑡)(𝐹y ∕𝐸)
[
]
0.037
=
+ 0.667 (3.264)
(95.24)(33∕29, 500)
=
= 3.311 in.2
Because 3.311 > A = 3.264 in.2 , use Ao =
3.264 in.2 Therefore,
The actual D/t ratio is
10
𝐷
=
= 166.67 < 394.23
𝑡
0.06
1. Sectional Properties of Full Section
𝜋[(10.0)2 − (10.0 − 2 × 0.06)2 ]
= 1.874 in.2
4
√
(10.0)2 + (10.0 − 2 × 0.06)2
𝑟=
= 3.51 in.
4
2. Nominal Axial Strength Pn
a.
KL 15 × 12
=
= 51.28
𝑟
3.51
𝜋 2 (29,500)
𝜋2𝐸
𝐹cre =
=
(KL∕𝑟)2
(51.28)2
𝐴=
= 110.72 ksi
b. Based on Eq. (7.31),
√
√
𝐹y
33
𝜆c =
=
= 0.546 < 1.5
𝐹e
110.72
𝐹n = (0.658𝜆c )𝐹y = (0.6580.546 )(33)
2
c. Based on Eqs. (7.32), (7.33), and (7.34),
𝐴e = 𝐴o + 𝑅(𝐴 − 𝐴o )
From Eq. (7.28), the nominal axial load is
𝑃ne = 𝐹n 𝐴e = (29.10)(3.264) = 94.98 kips
2
= 29.13 ksi
Ae = 3.264 + (0.150)(3.264 − 3.264)
= 3.264 in.2
OK
where
𝑅 = 𝐹y ∕(2𝐹e ) = 33∕(2 × 110.72)
3. Allowable Axial Load Pa . From Eq. (7.35), the allowable axial load is
𝑃
94.98
𝑃a = ne =
= 52.77 kips
Ωc
1.80
B. LRFD Method
For the LRFD method, the design axial strength is
𝜙c 𝑃n = 0.85(94.98) = 80.73 kips
= 0.149 < 1.0 OK
[
]
0.037
𝐴o =
+ 0.667 (1.874)
(166.67)(33∕29, 500)
= 1.622 in.2
Since Ao <A = 1.874 in.2 , use Ao = 1.622 in.2
𝐴e = 1.622 + (0.149)(1.874 − 1.622)
= 1.660 in.2
The nominal axial strength is
Example 7.2 All data are the same as those of Example 7.1,
except that the thickness of the tube is 0.06 in.
𝑃ne = 𝐹n 𝐴e = (29.13)(1.660)
= 48.36 kips
SOLUTION
A. ASD Method
Use the same procedure employed in Example 7.1:
( )
𝐸
𝐷
= 0.441
= 394.23
𝑡 lim
𝐹y
3. Allowable Axial Load Pa . From Eq. (7.35), the allowable axial load is
𝑃
48.36
= 26.87 kips
𝑃a = ne =
Ωc
1.80
DESIGN EXAMPLES
B. LRFD Method
For the LRFD method, the design axial strength is
𝜙𝑐 𝑃n = 0.85(48.36) = 41.11 kips
Example 7.3 Use the ASD and LRFD methods to determine the available flexural strength of the closed cylindrical
tubes used in Examples 7.1 and 7.2 if these tubes are to be
used as flexural members.
2. Use the data given in Example 7.2,
𝐹y = 33 ksi
𝐷o = 10 in.
𝑡 = 0.06 in.
𝐷
0.441𝐸
= 166.67 <
𝑡
𝐹y
OK
a. Section modulus of Full Section. The section
modulus of the 10 -in. tube having a wall thickness
of 0.06 in. is
(10.0)4 − (9.88)4
= 4.628 in.3
𝑆f = 0.098175
10.0
b. Nominal Flexural Strength Mne . From Eq. (7.36),
SOLUTION
A. ASD Method
1. Use the data given in Example 7.1,
𝐹y = 33 ksi
𝐷o = 10 in.
𝑀ne = 𝐹n 𝑆f
𝑡 = 0.105 in.
𝐷
0.441𝐸
= 95.24 <
𝑡
𝐹y
295
OK
a. Section Modulus of Full Section. The section
modulus of the 10-in. tube having a wall thickness
of 0.105 in. is
𝜋(𝐷o4 − 𝐷i4 )
𝐷o4 − 𝐷i4
𝑆f =
= 0.098175
32𝐷o
𝐷o
(10.0)4 − (9.79)4
= 7.99 in.3
10.0
b. Nominal Flexural Strength Mn . From Eq. (7.36),
= 0.098175
Since 0.0714E/Fy < D/t < 0.318E/Fy , the nominal
flexural strength is
[
]
(29, 500∕33)
(33)(4.628)
𝑀ne = 0.970 + 0.020
166.67
= 164.53 in. -kips
c. Allowable Flexural Strength Ma . The allowable flexural strength is
𝑀
164.53
= 98.52 in. -kips.
𝑀a = ne =
Ωb
1.67
0.0714(29,500)
0.0714𝐸
=
= 63.83
𝐹y
33
B. LRFD Method
Using the LRFD method, the design flexural strengths for the
closed cylindrical tubes used in Examples 7.1 and 7.2 can be
computed as follows:
0.318(29,500)
0.318𝐸
=
= 284.27
𝐹y
33
1. For the closed cylindrical tube used in Example 7.1, the
nominal flexural strength computed in item A above is
𝑀ne = 𝐹n 𝑆f
Since 0.0714E/Fy < (D/t = 95.24) < 0.318E/Fy ,
according to Eq. (7.38), the nominal flexural
strength is
[
]
(𝐸∕𝐹y )
𝐹y 𝑆f
𝑀ne = 0.970 + 0.020
𝐷∕𝑡
[
]
(29,500∕33)
= 0.970 + 0.020
(33)(7.99)
95.24
= 305.26 in. -kips
c. Allowable Flexural Strength Ma . Based on Eq.
(7.40), the allowable flexural strength is
𝑀
305.26
= 182.79 in. -kips
𝑀a = ne =
Ωb
1.67
𝑀ne = 305.26 in. -kips
The design flexural strength is
𝜙b 𝑀ne = 0.95(305.26) = 290.00 in. -kips
2. For the closed cylindrical tube used in Example 7.2, the
nominal flexural strength computed in Item A above is
𝑀ne = 164.53 in. -kips
The design flexural strength is
𝜙b 𝑀ne = 0.95(164.53) = 156.30 in. -kips
CHAPTER 8
guide, except that the shear strength of the fastener may
be quite different from that of bolts and different failure
modes such as pullout and inclination of fasteners should
also be considered. Additional information on the strength of
connections should be obtained from manufacturers or from
tests. Section 8.7 gives a brief discussion on the application
of cold rivets and press-joints.
8.3
Connections
8.1 GENERAL REMARKS
In Chapters 4–7 the design of individual structural members,
such as beams, columns, tension members, and cylindrical
tubular members, to be used in cold-formed steel construction has been discussed. It is often found that such structural members are fabricated from steel sheets or structural
components by using various types of connections. In addition, connections are required for joining individual members
in overall structures.
In this chapter the types of connections generally used in
cold-formed steel structures, the design criteria for various
types of connections, the requirements to fabricate I- or
box-shaped beams and columns by connecting two channels,
and the spacing of connections in compression elements are
discussed.
For connection design tables and example problems, reference should be made to Part IV of the Design Manual.
As a general rule of the AISI North American Specification, all connections should be designed to transmit the
maximum design force in the connected member with proper
regard for eccentricity.
8.2 TYPES OF CONNECTORS
Welds, bolts, cold rivets, screws, power-actuated fasteners,
and other special devices such as metal stitching and
adhesives are generally used in cold-formed steel
connections.1.428,1.161,8.1–8.10,8.63–8.65,8.95,8.102,8.103 The AISI
North American Specification contains provisions in
Chapter J for welded, bolted, screw and power-actuated
fastener connections, which are most commonly used.
In the design of connections using cold rivets, the AISI
provisions for bolted connections may be used as a general
WELDED CONNECTIONS
Welds used for building construction may be classified as
either arc welds or resistance welds.
Arc welding is a group of processes in which metals were
welded together by using weld metal at the surfaces to be
joined without the application of mechanical pressure or
blows.
Resistance welding is a group of welding processes where
coalescence is produced by the heat obtained from resistance
to an electric current through the work parts held together
under pressure by electrodes.
8.3.1
Arc Welds
Arc welds are often used for erection, connecting
cold-formed steel members to each other, or connecting
cold-formed steel members to hot-rolled framing members.
Several types of arc welds generally used in cold-formed
steel construction are:
1.
2.
3.
4.
5.
6.
Groove welds
Arc spot welds (puddle welds)
Arc seam welds
Fillet welds
Flare groove welds
Top arc seam welds
Figure 8.1 shows different types of arc welds.
Arc spot welds used for thin sheets are similar to plug
welds used for relatively thicker plates. The difference
between plug welds and arc spot welds is that the former are
made with prepunched holes, but for the latter no prepunched
holes are required. A hole is burned in the top sheet and
then filled with weld metal to fuse it to the bottom sheet or
structural members. Similarly, arc seam welds are the same
as slot welds, except that no prepunched holes are required
for the former.
The American Welding Society (AWS) has established
certain welding symbols. Figure 8.2 shows the basic symbols
and the standard locations of the elements of a welding
symbol used in cold-formed steel structures.8.11
With regard to the research work on arc welds, the earlier
AISI design provisions for fillet welds and arc spot welds
297
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
298
8
CONNECTIONS
Figure 8.1 Types of arc welds: (a) groove welds in butt joints; (b) arc spot welds; (c) arc seam
welds; (d) fillet welds; (e) flare bevel groove weld; (f) flare V-groove weld.
were based on the results of 151 tests conducted in the 1950s
at Cornell University.1.161 In the 1970s a total of 342 additional tests on fillet, flare bevel, arc spot, and arc seam welded
connections were carried out at Cornell University under the
sponsorship of the AISI.8.12,8.13 The structural behavior of the
most common types of arc welds used for sheet steel has been
studied in detail. Based on the research findings at Cornell
University summarized by Pekoz and McGuire8.12,8.13 and
a study by Blodgett of the Lincoln Electric Company,8.14
the first edition of the “Specification for Welding Sheet
Steel in Structures” was developed by the Subcommittee
on Sheet Steel of the AWS Structural Welding Committee
in 1978.8.15 The second edition of this document, entitled
“Structural Welding Code—Sheet Steel,” was issued by the
AWS in 1989.8.16 The current edition of Structural Welding
Code—Sheet Steel was published in 2008.8.96 Based on the
same data, in 1980, the AISI design provisions for arc welds
were revised extensively to reflect the research results. The
same design provisions were used in the 1986 AISI specification with additional design formulas included in the 1989
Addendum for tensile load of arc spot welds. Minor revisions
were made in 1996 with new figures added for the design of
flare bevel groove welds. In the supplement to the 1996 AISI
Specification, design equations are used to replace tabular
values for determining the nominal shear strength of resistance welds. In 2007 new design equations were provided
in the AISI North American Specification for tension on arc
spot welds and for the shear strength of sheet-to-sheet arc
spot welds. The 2012 edition of the Specification introduced
design provisions for combined shear and tension on arc spot
welds as well as design provisions for top arc seam welds.
The following sections summarize the research findings
on the structural strengths of various types of arc welds.
As discussed in Refs. 8.12 and 8.13, the thickness of steel
sheets used in the Cornell test program ranged from 0.019
to 0.138 in. (0.48 to 3.5 mm). The yield points of materials
varied from 33 to 82 ksi (228 to 565 MPa). All specimens
were welded with E6010 electrodes.
8.3.1.1 Arc Spot Welds Based on the results of 126 tests
on arc spot welds, it was found that the limit states of arc
spot welds include shear failure of welds in the fused area,
WELDED CONNECTIONS
Figure 8.2
299
Standard symbols for welded joints.8.11
Pus = ultimate shear capacity per weld, kips
As = fused area of arc spot weld, in.2
𝜏 u = ultimate shear strength of weld metal,
which was taken as 0.75 Fxx in Refs. 8.12
and 8.13, ksi
Fxx = tensile strength of weld metal according
to strength-level designation in AWS
electrode classification, ksi
de = effective diameter of fused area, in.
tearing of the sheet along the contour of the weld with the tear
spreading across the sheet at the leading edge of the weld,
sheet tearing combined with buckling near the trailing edge
of the weld, and shearing of the sheet behind the weld.8.12,8.13
In addition, some welds failed in part by peeling of the weld
as the sheet material tore and deformed out of its own plane.
An evaluation of the test results indicates that the
following equations can be used to predict the ultimate
strength of connections joined by arc spot welds.
where
8.3.1.1.1 Shear Strength of Arc Spot Welds The
ultimate shear capacity per arc spot weld can be determined
by8.12
)(
)
(
3𝜋 2
3
𝜋 2
𝑃us = 𝐴s 𝜏u =
(8.1)
𝑑e
𝐹xx =
𝑑 𝐹
4
4
16 e xx
Based on the test data on 31 shear failures of arc spot
welds, it was found that the effective diameter of the fused
area can be computed as8.12,8.13
𝑑e = 0.7𝑑 − 1.5𝑡 ≤ 0.55𝑑
(8.2)
300
8
where
d = visible diameter of outer surface of arc
spot weld
t = base thickness (exclusive of coatings) of
steel sheets involved in shear transfer
CONNECTIONS
The correlation between the computed ratios of de /d and
the test data is demonstrated in Fig. 8.3. Figure 8.4 shows
the definitions of the visible diameter d and the effective
diameter de .
8.3.1.1.2 Strength of Sheets Connected to a Thicker
Supporting Member by Using Arc Spot Welds On the
basis of his analysis of stress conditions in the connected
sheets around the circumference of the arc spot weld, Blodgett pointed out that the stress in the material is a tensile
stress at the leading edge, becoming a shear stress along the
sides, and eventually becoming a compressive stress at the
trailing edge of the weld, as shown in Fig. 8.5.8.14,8.16, 8.96 If
the strength of welded connections is governed by transverse
tearing of the connected sheet rather than by shear failure of
the weld, the ultimate load, in kips, per weld was found to be
𝑃ul = 2.2tda 𝐹u
where
(8.3)
da = average diameter of arc spot weld at
mid-thickness of t, in.; = d – t for single
sheet and = d – 2t for multiple sheets
(Fig. 8.4)
t = total combined base steel thickness of
sheets involved in shear transfer, in.
Fu = specified minimum tensile strength of
connected sheets, ksi
The same study also √
indicated that Eq. (8.3) is applicable
only when 𝑑a ∕𝑡 ≤ 140∕ 𝐹u .
Figure 8.4 Definitions of d, da , and de in arc spot welds1.314:
(a) single thickness of sheet; (b) double thickness of sheet.
For thin sheets, failure will occur initially by tension at the
leading edge, tearing out in shear along the sides, and then
buckling near the trailing edge of the arc spot weld. By using
the stress condition shown in Fig. 8.6, Blodgett developed
the following equation for determining the ultimate load, in
kips per weld8.14,8.16 :
𝑃u2 = 1.4tda 𝐹u
(8.4)
√
Equation (8.4) is applicable only when 𝑑a ∕𝑡 ≥ 240∕ 𝐹u .
Figure 8.3 Correlation between de /d ratios and test data according to plate thickness.8.13
WELDED CONNECTIONS
301
Figure 8.5 Tension, compression, and shear stresses in arc spot
weld.8.14,8.16,8.96
Figure 8.7 Comparison of observed and predicted ultimate loads
for arc spot welds.8.13
design equation was added in Section E2.2 of the 1989
Addendum to the 1986 edition of the AISI Specification:
Figure 8.6
Tension and shear stresses in arc spot weld.8.14,8.16,8.96
√
√
For 140∕ 𝐹u < 𝑑a ∕𝑡 < 240∕ 𝐹u , the ultimate load per
weld can be determined by the following transition equation:
)
(
960𝑡
tda 𝐹u
𝑃u3 = 0.28 1 + √
(8.5)
𝑑a 𝐹u
Figure 8.7 provides a graphic comparison of the observed
ultimate load Puo and the predicted ultimate load Pup
according to Eq. (8.1), (8.3), (8.4), or (8.5), whichever
is applicable.8.12 Figure 8.8 summarizes Eqs. (8.3)–(8.5),
which govern the failure of connected sheets.
8.3.1.1.3 Tensile Strength of Arc Spot Welds In
building construction, arc spot welds have often been
used for connecting roof decks to support members such as
hot-rolled steel beams and open web steel joists. This type of
welded connection is subject to tension when a wind uplift
force is applied to the roof system.
Prior to 1989, no design information was included in the
AISI Specification to predict the tensile strength of arc spot
welds. Based on Fung’s test results8.17 and the evaluation of
test data by Albrecht8.18 and Yu and Hsiao,8.19 the following
𝑃ut = 0.7tda 𝐹u
in which Put is the ultimate tensile capacity per weld in kips.
The symbols t, da , and Fu were defined previously.
The above design criterion was revised in the 1996 edition
of the AISI Specification because the UMR tests8.66,8.67 have
shown that two possible limit states may occur. The most
common failure mode is sheet tearing around the perimeter
of the weld. This failure condition was affected by the
sheet thickness, the average weld diameter, and the material tensile strength. The nominal tensile strength of concentrically loaded arc spot welds can be determined by the
following equations depending on the Fu /E ratio:
1. For Fu /E < 0.00187,
[
( )]
𝐹u
𝑃n = 6.59 − 3150
tda 𝐹u ≤ 1.46tda 𝐹u
𝐸
(8.6a)
2. For Fu /E ≥ 0.00187:
𝑃n = 0.70tda 𝐹u
(8.6b)
In some cases, the tensile failure of the weld may occur.
The tensile strength of the arc spot weld is based on the cross
section of the fusion area and the tensile strength of the weld
302
8
CONNECTIONS
Figure 8.8 Failure load for arc spot welds.
metal. Therefore, for this type of failure mode, the nominal
tensile strength can be computed by Eq. (8.7):
𝜋𝑑e2
(8.7)
𝐹
4 xx
where de is the effective diameter of fused area and Fxx is the
tensile strength of weld metal.
It should be noted that Eqs. (8.6) and (8.7) are subject to
the following limitations:
𝑃n =
𝑒min ≥ 𝑑
𝐹xx ≥ 60 ksi(414 MPa, 4220 kg∕cm2 )
𝐹u ≤ 82 ksi(565 MPa, 5770 kg∕cm2 )
𝐹xx > 𝐹u
where emin is the minimum distance measured in the line of
force from the center line of a weld to the nearest edge of
Figure 8.9
an adjacent weld or to the end of the connected part toward
which the force is directed. Other symbols were previously
defined. When the spot weld is reinforced by a weld washer,
the tensile strength given by Eqs. (8.6a) and (8.6b) can be
achieved by using the thickness of the thinner sheet.
Equations (8.6) and (8.7) were derived from tests for
which the applied tensile load imposed a concentric load on
the spot weld, such as the interior welds on a roof system
subjected to wind uplift, as shown in Fig. 8.9. For exterior
welds which are subject to eccentric load due to wind uplift,
tests have shown that only 50% of the nominal strength can
be used for design. At a lap connection between two deck
sections (Fig. 8.9), a 30% reduction of the nominal strength
was found from the tests.8.66,8.67
An analysis of the UMR data by LaBoube8.97 indicated
that the nominal tensile strength could be determined based
on the ductility of the sheet, the sheet thickness, the average
Interior weld, exterior weld, and lap connection.
WELDED CONNECTIONS
weld diameter, and the material tensile strength as follows:
( )2
𝐹
(8.8)
𝑃n = 0.8 u tda 𝐹u
𝐹y
303
tion was experimentally studied by Luttrell, and based on a
review of the data by LaBoube,8.98 the nominal strength is
given by
𝑃n = 1.65tda 𝐹u
(8.11)
The following limits apply to the use of Eq. (8.11):
8.3.1.1.4 Combined Shear and Tensile Strength of Arc
Spot Welds Based on an experimental study performed by
Stirnemann and LaBoube,8.100 the behavior of an arc spot
weld subjected to combined shear and tension forces can be
evaluated by either Eq. (8.9) or Eq. (8.10):
[
]
]
[
𝑃uv
𝑃ut 0.6
+
≤ 1.0
(8.9)
L Pnt
L Pnv
where L = 1.0 for Fu /Fy ≥ 1.23, L = 0.75 for Fu /Fy < 1.04,
Pnt is defined by Eq. (8.8), Pnv is defined by Eq. (8.4), or
(8.5), and Put and Puv are the applied tension and shear force,
] [
]
[
𝑃uv
𝑃ut
+
≤ 1.0
(8.10)
L Pnt
L Pnv
where L = 1.0 for Fu /Fy ≥ 1.23, L = 0.60 for Fu /Fy < 1.04,
Pnt is defined by Eq. (8.8), Pnv is defined by Eq. (8.3), (8.4) or
(8.5), and Put and Puv are the applied tension and shear force.
The experimental study focused on six variables that were
deemed to be the key parameters that could influence the
strength of the arc spot weld connection. These variables
were the sheet thickness; sheet material properties including
yield stress, tensile strength and ductility of the sheet; visible
diameter of the arc spot weld; and the relationship between
the magnitude of the shear force and tension force. Based
on an analysis of the test results, the Steel Deck Institute
Diaphragm Design Manual9.111 interaction equation was
found to provide an acceptable estimate of the strength
of the arc spot weld connection and was adopted by the
Specification.
8.3.1.1.5 Strength of Sheet-to-Sheet Connections Using
Arc Spot Welds The sheet-to-sheet arc spot weld connec-
Figure 8.10
𝐹u ≤ 59 ksi(407 MPa, 4150 kg∕cm2 )
𝐹xx > 𝐹u
0.028 in. (0.71 mm) ≤ 0.0635 in. (1.61 mm)
8.3.1.2 Arc Seam Welds As shown in Fig. 8.10, an arc
seam weld consists of two half-circular ends and a longitudinal weld. The ultimate load of a welded connection is
determined by the shear strength of the arc seam weld and
the strength of the connected sheets.
8.3.1.2.1 Shear Strength of Arc Seam Welds The ultimate shear capacity per weld is a combined shear resistance
of two half-circular ends and a longitudinal weld, as given by
(
)
3𝜋 2 3Lde
(8.12)
𝑑e +
𝐹xx
𝑃us =
16
4
in which L is the length of the seam weld, not including
the circular ends. For the purpose of computation, L should
not exceed 3d. Other symbols were defined in the preceding
discussion.
8.3.1.2.2 Strength of Connected Sheets by Using Arc
Seam Welds In the Cornell research project a total of 23
welded connections were tested for arc seam welds. Based on
the study made by Blodgett8.14 and the linear regression analysis performed by Pekoz and McGuire,8.12,8.13 the following
equation has been developed for determining the strength of
connected sheets:
𝑃ul = (0.625𝐿 + 2.4𝑑a )tF u
(8.13)
Equation (8.13) is applicable for all values of da /t.
Figure 8.11 shows a comparison of the observed loads
and the ultimate loads predicted by using Eq. (8.13).
Arc seam weld joining sheet to supporting member.1.314,1.345
304
8
CONNECTIONS
with Fy of 65 ksi (448 MPa, 4570 kg/cm2 ) or higher the weld
throat failure does not occur in materials less than 0.10 in.
(25.4 mm) thick and the AISI North American specification provisions based on sheet strength are satisfactory for
materials less than 0.10 in. (25.4 mm) thick.
Strength of Connected Sheets by Using Fillet Welds
1. Longitudinal Welds. A total of 64 longitudinal
fillet welds were tested in the Cornell study.8.12,8.13
An evaluation of the test data indicated that the
following equations can predict the ultimate loads of
the connected sheets for the failure involving tearing
along the weld contour, weld shear, and combinations
of the two types of failure:
)
(
𝐿
tLF u for 𝐿∕𝑡 < 25 (8.15a)
𝑃ul = 1 − 0.01
𝑡
𝑃u2 = 0.75tLF u for 𝐿∕𝑡 > 25
(8.15b)
Figure 8.11 Comparison of observed and predicted ultimate
loads for arc seam welds.8.13
8.3.1.3 Fillet Welds Fillet welds are often used for lap and
T-joints. Depending on the arrangement of the welds, they
can be classified as either longitudinal or transverse fillet
welds. (“Longitudinal” means that the load is applied parallel
to the length of the weld; “transverse” means that the load is
applied perpendicular to the length of the weld.)
From the structural efficiency point of view, longitudinal
fillet welds are stressed unevenly along the length of weld
due to varying deformations. Transverse fillet welds are more
uniformly stressed for the entire length. As a result, transverse welds are stronger than longitudinal welds of an equal
length. The following discussion deals with the strength of
welded connections using both types of fillet welds.
in which Pu1 and Pu2 are the predicted ultimate loads
per fillet weld. Other symbols were defined previously.
2. Transverse Welds. Based on the results of 55 tests on
transverse fillet welds, it was found that the primary
failure was by tearing of connected sheets along, or
close to, the contour of the welds. The secondary failure
was by weld shear. The ultimate failure load per fillet
weld can be computed as
𝑃u3 = tLF u
(8.16)
8.3.1.3.1 Shear Strength of Fillet Welds If the strength
of welded connections is governed by the shear capacity of
fillet welds, the ultimate load per weld can be determined as
3
(8.14)
𝑃us = 𝑡w LF xx
4
where tw = effective throat dimension
L = length of fillet weld
and Fxx was defined previously. As used in Eqs. (8.1) and
(8.12), the shear strength of the weld metal is assumed to be
75% of its tensile strength.
Research at the University of Sydney by Teh and
Hancock8.101 has determined that for high-strength steels
Figure 8.12 Comparison of observed and predicted ultimate
loads for longitudinal fillet welds.8.13
WELDED CONNECTIONS
305
Figure 8.15 Longitudinal flare bevel weld.1.314,1.345
8.3.1.4.2 Strength of Connected Sheets by Using Flare
Groove Welds If the strength of weld connections is
governed by the connected sheets, the ultimate load per weld
can be determined as follows:
1. Transverse Welds.
Figure 8.13 Comparison of observed and predicted ultimate
loads for transverse fillet welds.8.13
𝑃ul = 0.833tLF u
(8.18)
2. Longitudinal Welds. If t ≤ tw < 2t or if the lip height
is less than the weld length L,
𝑃u2 = 0.75tLF u
(8.19)
If tw ≥ 2t and the lip height is equal to or greater
than L,
(8.20)
𝑃u3 = 1.5tLF u
Figure 8.14
Transverse flare bevel weld.1.314,1.345
Figures 8.12 and 8.13 show comparisons of the observed
and predicted ultimate loads for longitudinal and transverse
fillet welds, respectively.
8.3.1.4 Flare Groove Welds In the Cornell research, 42
transverse flare bevel welds (Fig. 8.14) and 32 longitudinal
flare bevel welds (Fig. 8.15) were tested. It was found that the
following formulas can be used to determine the predicted
ultimate loads.
8.3.1.4.1 Shear Strength of Flare Groove Welds The
ultimate shear strength of a flare groove weld is
3
(8.17)
𝑃us = 𝑡w LF xx
4
The above equation is similar to Eq. (8.14) for fillet welds.
Figure 8.16 Comparison of observed and predicted ultimate
loads for transverse flare bevel welds.8.13
306
8
CONNECTIONS
groove welds used in butt joints, arc spot welds, arc seam
welds, fillet welds, and flare groove welds.
8.3.2.3 Groove Welds in Butt Joints For the design of
groove welds in butt joints (Fig. 8.1a), the nominal strength
Pn and the applicable safety factor and resistance factor are
given in Section J2.1 of the AISI Specification as follows:
a. Tension or Compression Normal to Effective Area or
Parallel to Axis of Weld.
𝑃n = Lte 𝐹y
(8.21)
Ω = 1.70 (ASD)
{
0.90 (LRFD)
𝜙=
0.80 (LSD)
b. Shear on Effective Area. Use the smaller value of either
Eq. (8.22) or (8.23):
𝑃n = Lte (0.6𝐹xx )
Figure 8.17 Comparison of observed and predicted ultimate
loads for longitudinal flare bevel welds.8.13
Ω = 1.90 (ASD)
{
0.80 (LRFD)
𝜙=
0.70 (LSD)
Figures 8.16 and 8.17 show comparisons of the observed
and predicted ultimate loads for transverse and longitudinal
flare bevel welds, respectively.
Lte 𝐹y
𝑃n = √
3
8.3.2
Ω = 1.70 (ASD)
{
0.90 (LRFD)
𝜙=
0.80 (LSD)
AISI Design Criteria for Arc Welds
8.3.2.1 Thickness Limitations In previous editions of the
AISI Specifications, the design provisions have been used for
cold-formed members and thin elements with a maximum
thickness of 12 in. (12.7 mm). Because the maximum material
thickness for using the AISI Specification was increased to
1 in. (25.4 mm) in 19778.21 and the structural behavior of
weld connections for joining relatively thick cold-formed
members is similar to that of hot-rolled shapes, Section
J2 of the AISI North American Specification is intended
only for the design of arc welds for cold-formed steel
3
members with a thickness of 16
in. (4.76 mm) or less.1.417
3
When the connected part is over 16
in. (4.76 mm) in thickness, arc welds can be designed according to the AISC
specification.1.148,3.150,1.411
8.3.2.2 Criteria for Various Weld Types Section 8.3.1
discussed ultimate strengths of various weld types. The ultimate load Pu determined in Section 8.3.1 for a given type
of weld is considered to be the nominal strength of welds,
Pn , used in Section J2 of the AISI North American Specification. The following are the AISI design provisions for
(8.22)
(8.23)
where Pn = nominal strength (resistance) of groove
weld
Fxx = tensile strength electrode classification
Fy = yield stress of the lowest strength base
steel
L = length of weld
te = effective throat dimension of groove weld
Equations (8.21) (8.22), and (8.23) are the same as the
AISC Specification.1.411 The effective throat dimensions
for groove welds are shown in Fig. 8.18.
8.3.2.4 Arc Spot Welds (Puddle Welds) Section J2.2
of the AISI North American specification includes the
following requirements for using arc spot welds:
1. Arc spot welds should not be made on steel where the
thinnest connected part is over 0.15 in. (3.81 mm) thick
or through a combination of steel sheets having a total
thickness of over 0.15 in. (3.81 mm).
WELDED CONNECTIONS
307
Ω = 2.20 (ASD)
{
0.70 (LRFD)
𝜙=
0.60 (LSD)
√
√
b. For 0.815 𝐸∕𝐹u < (𝑑a ∕𝑡) < 1.397 𝐸∕𝐹u ,
[
]
√
𝐸∕𝐹u
𝑃n = 0.280 1 + 5.59
tda 𝐹u (8.25b)
𝑑a ∕𝑡
Ω = 2.80 (ASD)
{
0.55 (LRFD)
𝜙=
0.45 (LSD)
√
c. For 𝑑a ∕𝑡 ≥ 1.397 𝐸∕𝐹u ,
𝑃n = 1.40tda 𝐹u
Figure 8.18 Effective dimensions for groove welds.
2. Weld washers should be used when the thickness of the
sheet is less than 0.028 in. (0.711 mm). Weld washers
should have a thickness of between 0.05 (1.27 mm) and
0.08 in. (2.03 mm) with a minimum prepunched hole of
3/8 in. (9.53 mm) diameter.
3. The minimum allowable effective diameter de is 3/8 in.
(9.5 mm).
4. The clear distance measured between the end of the
material and edge of the weld, enet , as discussed in
Section 8.8:
5. The distance from the centerline of any weld to the end
or boundary of the connected member should not be less
than 1.5d. In no case should the clear distance between
welds and end of member be less than 1.0d.
6. The nominal shear strength Pn of each arc spot weld
between sheet or sheets and supporting member should
not exceed the smaller value of the loads computed by
Eqs. (8.24) and (8.25):
i. Nominal Shear Strength Based on Shear
Capacity of Weld
𝜋𝑑e2
(0.75𝐹xx )
4
Ω = 2.55 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
𝑃n =
(8.24)
Ω = 3.05 (ASD)
{
0.50 (LRFD)
𝜙=
0.40 (LSD)
In the above requirements and design formulas
for arc spot welds,
Pn = nominal shear strength of an arc spot weld
d = visible diameter of outer surface of arc
spot weld (Fig. 8.4)
da = average diameter of arc spot weld at mid
thickness of t (Fig. 8.4), = d – t for a
single sheet and = d – 2 t for multiple
sheets (not more than four lapped sheets
over a supporting member)
de = effective diameter of fused area (Fig. 8.4),
= 0.7d – 1.5t but ≤ 0.55d
t = total combined base steel thickness
(exclusive of coating) of sheets involved
in shear transfer above the plane of
maximum shear transfer
Fxx = filler metal strength designation in AWS
electrode classification
Fu = specified minimum tensile strength of
steel
iii. Nominal Shear Strength for Sheet-to-Sheet
Connections.
ii. Nominal Shear Strength for Sheets of Connected
to Thicker Member
√
a. For da /t ≤ 0. 𝐸∕𝐹u ,
𝑃n = 2.20tda 𝐹u
(8.25c)
(8.25a)
𝑃n = 1.65tda 𝐹u
Ω = 2.20 (ASD)
{
0.70 (LRFD)
𝜙=
0.60 (LSD)
(8.26)
308
8
CONNECTIONS
7. The nominal tensile strength Pn of each concentrically
loaded arc spot weld connecting sheets and supporting
members should be the smaller value of the loads
computed by Eqs. (8.27) and (8.28):
i. Nominal Tensile Strength Based on Capacity of
Weld
1
(8.27)
𝑃n = 𝜋𝑑e2 𝐹xx
4
ii. Nominal Tensile Strength Based on Strength of
Connected Sheets
( )2
F
(8.28)
𝑃n = 0.80 u tda 𝐹u
Fy
For panel and deck applications for both Eqs. (8.27)
and (8.28),
Ω = 2.50 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
These equations are also shown in Fig. 8.8. The
background information on tensile strength was
discussed previously.
iii. Combined Shear and Tension on an Arc Spot Weld
For arc spot weld connections subjected to a combination of
shear and tension, the following interaction check shall be
applied:
( )1.5
𝑇
≤ 0.15, no interaction check is required.
If
𝑃at
( )1.5
𝑇
If
> 0.15,
𝑃at
(
)1.5 ( )1.5
𝑉
𝑇
+
≤1
(8.29)
𝑃av
𝑃at
where
For other applications for both Eqs. (8.27) and
(8.28),
Ω = 3.00 (ASD)
{
0.50 (LRFD)
𝜙=
0.40 (LSD)
Both Eqs. (8.27) and (8.28) are limited to the
following conditions: tda Fu ≤ 3 kips (13.34 kN),
emin ≥ d, Fxx ≥ 60 ksi (410 MPa, 4220 kg/cm2),
Fu ≤ 82 ksi (565 MPa, 5770 kg/cm2 ) (of connecting
sheets), and Fxx > Fu . All symbols were defined
previously.
It should be noted that Eq. (8.24) is derived from
Eq. (8.1). Equations (8.25a) (8.25b), and (8.25c) are
based on Eqs. (8.3), (8.5) and (8.4), respectively.
Figure 8.19
𝑇 = Required tensile strength per connection
fastener determined in accordance with
ASD, LRFD, or LSD load combinations
𝑉 = Required shear strength per connection
fastener, determined in accordance with
ASD, LRFD, or LSD load combinations
Pat = Available tension strength in accordance
with Eqs. (8.27) and (8.28)
Pav = Available shear strength as given by
Section Eqs. (8.25) and 8.26
In addition, the following limitations shall be satisfied:
1.
2.
3.
4.
5.
Fu ≤ 105 ksi (724 MPa or 7380 kg/cm2 ),
Fxx ≥ 60 ksi (414 MPa or 4220 kg/cm2 ),
tda Fu ≤ 3 kips (13.3 kN or 1360 kg),
Fu /Fy ≥ 1.02, and
0.47 in. (11.9 mm) ≤ d ≤ 1.02 in. (25.9 mm).
Example 8.1.
WELDED CONNECTIONS
Example 8.1 Use the ASD and LRFD methods to determine the allowable load for the arc spot welded connection
shown in Fig. 8.19. Use A1011 Grade 45 steel (Fy = 45 ksi,
Fu = 60 ksi). Assume that the visible diameter of the arc spot
weld is 34 in. and the dead load–live load ratio is 15 .
SOLUTION
A. ASD Method
Prior to determination of the allowable load, the AISI
requirements for using arc spot welds are checked as
follows:
1. Since the thickness of the connected sheets is less than
0.15 in., arc spot welds can be made.
2. Because the thickness of the connected sheets is over
0.028 in., weld washers are not required.
3. The visible diameter d is 34 in., and
𝑑a = 𝑑 − 𝑡 = 0.75 − 0.075 = 0.675 in.
𝑑e = 0.7𝑑 − 1.5𝑡 = 0.70(0.75) − 1.5(0.075)
Use
𝑑e = 0.4125 in. >
3
in.
8
(minimum size) OK
4. The distance from the centerline of any weld to the end
of the sheet is
1.25 in. > (1.5𝑑 = 1.125 in.)
OK
5. The clear distance between welds is
2 − 𝑑 = 1.25 in. > d OK
The clear distance between welds and end of
member is
1
1.25 − = 0.875 in. > 𝑑
2
OK
6. The allowable load for the ASD method is based on the
following considerations:
a. Tensile Load for Steel Sheets. Based on Sections D2
and D3 of the AISI specification:
i. For yielding [Eq. (6.2)],
𝑃al =
ii. For fracture away from the connection [Eq. (6.3)],
𝑃a1 =
𝐴 𝐹
𝑇n
= n u
Ωt
2.00
(4.5 × 0.075)(60)
= 10.125 kips
2.00
Use Pa1 = 9.09 kips.
b. Tensile Load Based on End Distance (e = 1.25 in.)
By using Eq. (8.68),
=
𝐴nv = 2ntenet
= 2[2(1.25 − 0.375) + 2(2.00 − .75)](0.075)
= 0.638 in2
𝑉n = 0.6𝐹u 𝐴nv = 0.6(60)(0.638) = 22.95 kips
𝑃𝑎2 = 22.95∕2.50 = 9.18 kips
c. Shear Capacity of Welds. By using Eq. (8.24) and
E60 electrodes,
𝑃a3 =
(4𝜋𝑑e2 ∕4)(0.75𝐹xx )
Ω
= 9.43 kips
= 0.4125 in. = 0.55𝑑
𝐴g 𝐹y
𝑇n
=
Ωt
1.67
(4.5 × 0.075)(45)
1.67
= 9.09 kips
309
=
𝜋(0.4125)2
(0.75 × 60)
2.55
d. Strength of Connected Sheets Around Welds.
(
)
√
𝑑𝑎
29,500
0.675
=
= 9 < 0.815
= 18.07
𝑡
0.075
60
By using Eq. (8.25a)
4(2.20tda 𝐹u )
Ω
4(2.20)(0.075)(0.675)(60)
=
2.20
= 12.15 kips
𝑃a4 =
On the basis of the above considerations, the allowable load for the ASD method is the smallest value,
that is, 9.09 kips, which is governed by the tensile
load for yielding of steel sheets.
B. LRFD Method
As the first step of the LRFD method, the AISI requirements
for using arc spot welds should be checked as the ASD
method. From items 1–5 for the ASD method, the layout of
the spot welds are satisfied with the AISI requirements.
The design strength 𝜙Pn for the LRFD method is based on
the following considerations:
=
a. Tensile Load for Steel Sheets. Based on Sections D2 and
D3 of the AISI specification:
310
8
CONNECTIONS
i. For yielding [Eq. (6.2)],
𝜙t 𝑃n = 𝜙t 𝑇n = 𝜙t (𝐴g 𝐹y )
= (0.90)(4.5 × 0.075)(45) = 13.67 kips
ii. For fracture away from the connections [Eq. (6.3)],
𝜙t 𝑃n = 𝜙t 𝑇n = 𝜙t (𝐴n 𝐹u )
= (0.75)(4.5 × 0.075)(60) = 15.19 kips
Use 𝜙t 𝑃n = 13.67 kips.
b. Tensile Load Based on End Distance (e = 1.25 in.). Using
Eq. (8.68) for four spot welds,
𝐴nv = 2ntenet
= 2[2(1.25 − 0.375) + 2(2.00 − .75)](0.075)
= 0.638 in2
𝑉n = 0.6𝐹u 𝐴nv = 0.6(60)(0.638) = 22.95 kips
𝜙𝑃n = 0.6(22.95) = 13.77 kips
c. Shear Capacity of Welds. By using Eq. (8.26) and E60
electrodes,
)
(
1
𝜙𝑃n = 𝜙(4) 𝜋𝑑e2 (0.75𝐹xx )
4
= (0.60)(𝜋)(0.4125)2 (0.75 × 60) = 14.43 kips
d. Strength of √
Connected Sheets Around Welds. Since
𝑑a ∕𝑡 < 0.815 𝐸∕𝐹u , use Eq. (8.25a),
𝜙𝑃n = 𝜙(4)(2.20tda 𝐹u )
The allowable load based on the LRFD method is
𝑃D + 𝑃L = 1.49 + 7.43 = 8.92 kips
It can be seen that the LRFD method permits a slightly
smaller allowable load than the ASD method. The difference
between these two design approaches for this particular case
is less than 2%.
8.3.2.5 Arc Seam Welds For arc seam welds (Fig. 8.10),
Section J2.3 of the AISI North American Specification specifies that the nominal shear strength (resistance) Pn of an arc
seam weld is the smaller of the values computed by Eqs.
(8.30) and (8.31):
i. Nominal Shear Strength Based on Shear Capacity of
Weld.
( 2
)
𝜋𝑑𝑒
𝑃n =
(8.30)
+ Ld𝑒 0.75𝐹xx
4
ii. Nominal Shear Strength Based on Strength of
Connected Sheets.
𝑃n = 2.5tF u (0.25𝐿 + 0.96𝑑𝑎 )
(8.31)
For Eqs. (8.30) and (8.31),
Ω = 2.55 (ASD)
𝜙 = 0.60 (LRFD)
= 0.50 (LSD)
d = width of arc seam weld
L = length of seam weld not including
circular ends (for computation purposes,
L should not exceed 3d)
= (0.70)(4)(2.20 × 0.075 × 0.675 × 60)
= 18.71 kips
Based on the above four considerations, the controlling
design strength is 13.67 kips, which is governed by the tensile
load for yielding of steel sheets.
According to the load factors and load combinations
discussed in Section 1.8.2.2, the required strength for the
given dead load–live load ratio of 15 is computed as follows:
𝑃u2 = 1.2𝑃D + 1.6𝑃L = 1.2𝑃D + 1.6(5𝑃D ) = 9.2𝑃D
(Eq. 1.5b)
where
PD = applied load due to dead load
PL = applied load due to live load
The definitions of da , de , Fu , and Fxx and the requirements for minimum edge distance are the same as those
for arc spot welds.
Equation (8.30) is derived from Eq. (8.12) and Eq.
(8.31) is based on Eq. (8.13).
iii. Shear Strength of Top Arc Seam Sidelap Welds
The nominal shear strength [resistance], Pnv , for longitudinal loading of top arc seam sidelap welds shall be
determined in accordance with Eq. (8.32). The following
limits shall apply: (a) hst ≤ 1.25 in. (31.8 mm), (b) Fxx
≥ 60 ksi (414 MPa), (c) 0.028 in. (0.711 mm) ≤ t ≤
0.064 in. (1.63 mm), and (d) 1.0 in. (25.4 mm) ≤ Lw ≤
2.5 in. (63.5 mm).
𝑃nv = [4.0(F𝑢 ∕F𝑠𝑦 )–1.52](𝑡∕L𝑤 )0.33𝐿𝑤𝑡 𝐹𝑢
Use Pu = 9.2PD . By using 9.2PD = 13.67 kips,
Ω = 2.60
(ASD)
𝑃D = 1.49 kips
𝜙 = 0.60
(LRFD)
𝑃L = 5𝑃D = 7.43 kips
= 0.55
(LSD)
(8.32)
WELDED CONNECTIONS
where
hst = nominal seam height.
Fxx = Tensile strength of electrode
classification
Lw = Length of top arc seam sidelap weld
t = Base steel thickness (exclusive of
coatings) of thinner connected sheet
Pnv = Nominal shear strength [resistance] of
top arc seam sidelap weld
Fu = Specified minimum tensile strength of
connected sheets
Fsy = Specified minimum yield stress of
connected sheets
8.3.2.6 Fillet Welds According to Section J2.5 of the
AISI North American Specification, the design strength of a
fillet weld in lap and T-joints should not exceed the values
computed by Eq. (8.33) for the shear strength of the fillet
weld and by Eq. (8.34) or Eq. (8.35) for the strength of the
connected sheets as follows:
i. Nominal Strength Based on Shear Capacity of Weld.
For t > 0.10 in. (3.8 mm),
𝑃n = 0.75𝑡w LF xx
(8.33)
Ω = 2.55 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
ii. Nominal Strength Based on Strength of Connected
Sheets
a. Longitudinal Loading. When L/t < 25,
)
(
𝐿
tLF u
𝑃n = 1 − 0.01
(8.34a)
𝑡
Ω = 2.55 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
Figure 8.20
311
When L/t > 25,
𝑃n = tLF u
(8.34b)
Ω = 2.35 (ASD)
{
0.65 (LRFD)
𝜙=
0.60 (LSD)
b. Transverse Loading
𝑃n = tLF u
(8.35)
Ω = 2.35 (ASD)
{
0.65 (LRFD)
𝜙=
0.60 (LSD)
where Pn = nominal strength of a fillet weld
L = length of fillet weld
tw = effective throat, =0.707w1 or 0.707w2 ,
whichever is smaller
w1 , w2 = leg size of fillet weld (Fig. 8.20)
The definitions of t, Fu , and Fxx are the same as those
used for arc spot welds. It should be noted that Eqs. (8.32),
(8.33), and (8.34) are based on Eqs. (8.14), (8.15), and (8.16),
respectively.
Example 8.2 Use the ASD method to determine the allowable load for the welded connection using fillet welds, as
shown in Fig. 8.21. Assume that A570 Grade 33 steel sheets
and E60 electrodes are to be used.
SOLUTION
From Table 2.1, the yield point and the tensile strength of
A1011 Grade 33 steel are 33 and 52 ksi, respectively. The
allowable load P can be determined as follows:
1. Allowable Tensile Load for Steel Sheet. Based on
Sections D2 and D3 of the AISI Specification:
Leg sizes of fillet welds1.4: (a) lap joint; (b) t-joint.
312
8
CONNECTIONS
i. Nominal Strength Based on Shear Capacity of Weld.
For t > 0.10 in. (3.8 mm),
𝑃n = 0.75𝑡w LF xx
(8.36)
Ω = 2.55 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
Figure 8.21 Example 8.2.
i. For yielding [Eq. (6.2)],
𝐴g 𝐹y
𝑇
𝑃al = n =
Ωt
1.67
(2.0 × 0.105)(33)
1.67
= 4.15 kips
=
ii. For fracture away from the connections [Eq. (6.3)],
𝑇
𝐴 𝐹
𝑃al = n = n n
Ωt
2.00
(2.0 × 0.105)(52)
= 5.46 kips
=
2.00
Use Pa1 = 4.15 kips.
2. Allowable Load for Longitudinal Fillet Welds. Since
L/t = 2/0.105 = 19.05 < 25, use Eq. (8.34a),
[1 − 0.01(𝐿∕𝑡)]tLF u
𝑃L =
Ω
[1 − 0.01(19.05)](0.105)(2)(52)
=
2.55
= 3.46 kips per weld
Using two longitudinal welds
𝑃a2 = 2𝑃L = 2 × 3.46 = 6.92 kips
Because the thickness of steel sheet is less than
0.10 in., it is not necessary to use Eq. (8.30).
Since Pa1 <Pa2 , the allowable tensile load is governed by
the tensile capacity of steel sheet, that is,
𝑃a = 4.15 kips
The use of the LRFD method can be handled in the same
way as Example 8.1.
8.3.2.7 Flare Groove Welds On the basis of Section
J2.6 of the AISI North American specification, the nominal
strength of each flare groove weld should be determined as
follows:
ii. Nominal Strength Based on Strength of Connected
Sheet
a. Transverse Loading (Fig. 8.14)
𝑃n = 0.833tLF u
(8.37)
Ω = 2.55(ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
b. Longitudinal Loading (Figs. (8.22a–8.22f)) If
t ≤ tw < 2t or if the lip height is less than the
weld length L,
(8.38a)
𝑃n = 0.75tLF u
If tw > 2t and the lip height is equal to or greater
than L,
(8.38b)
𝑃n = 1.50tLF u
Using Eqs. (8.37) and (8.38),
Ω = 2.80 (ASD)
{
0.55 (LRFD)
𝜙=
0.45 (LSD)
In Eqs. (8.36) through (8.38),
Pn = nominal strength (resistance) of flare groove
weld
h = height of lip
L = length of the weld
tw = effective throat of flare groove weld filled
flush to surface (Figs. 8.22c and 8.22d)
For flare bevel groove weld tw = 5/16R
For flare V-groove weld tw = 1/2R [3/8R when
R > 1/2 in. (12.7 mm)]
tw = effective throat of flare groove weld not filled
flush to surface = 0.707 w1 or 0.707 w2 , whichever
is smaller. (Figs. 8.22e and 8.22f)
tw = larger effective throat than those above shall
be permitted if measurement shows that the
welding procedure to be used consistently yields a
large value of tw
R = radius of outside bend surface
w1 , w2 = leg of weld (see Figs. 8.22e and 8.22f)
Fu , Fxx were defined previously
WELDED CONNECTIONS
313
(d)
(c)
Figure 8.22 (a) Shear in longitudinal flare bevel groove
weld.1.314,1.345,1.417 (b) Shear in longitudinal flare V-groove
weld.1.314,1.345,1.417 (c) Flare bevel groove weld (filled flush to
surface, w1 = R).1.314,1.345,1.417
It should be noted that Eqs. (8.36), (8.37), and (8.38) are
derived from Eqs. (8.17), (8.18), (8.19), and (8.20).
Example 8.3 Use the ASD method to design a welded
connection as shown in Fig. 8.23 for the applied load of
15 kips. Consider the eccentricity of the applied load. Use
A606 Grade 50 steel (Fy = 50 ksi and Fu = 70 ksi) and E70
electrodes.
SOLUTION
Considering the eccentricity of the applied load, it is desirable to place the welds so that their centroids coincide with
the centroid of the angle section. It should be noted that weld
Figure 8.22 (d) Flare bevel groove weld (filled flush to surface,
w1 = R).1.314 (e) Flare bevel groove weld (not filled flush to surface,
w1 > R).1.314 (f) Flare bevel groove weld (not filled flush to surface,
w1 < R).1.134
L1 is a flare groove weld, weld L2 is a transverse fillet weld,
and weld L3 is a longitudinal fillet weld.
Let P2 be the allowable load of end weld L2 . By using Eq.
(8.35) for transverse fillet welds
tLF u
Ω
(0.135)(2.0)(70)
=
= 8.04 kips
2.35
Taking moments about point A,
𝑃2 =
P(1.502) − P1 (2.0) − 𝑃2 (1.0) = 0
15(1.502) − 𝑃1 (2.0) − 8.04(1.0) = 0
𝑃1 = 7.25 kips
𝑃3 = 𝑃 − (𝑃1 + 𝑃2 ) ≃ 0
314
8
CONNECTIONS
Figure 8.23
For the flare groove weld subjected to longitudinal
loading, with the assumption that t ≤ tw < 2t, the allowable
load according to Eq. 8.38a is
0.75tLF u
Ω
0.75(0.135)(1)(70)
=
= 2.53 kips∕in.
2.80
The required length L1 is
𝑃a =
𝐿1 =
𝑃1
7.25
=
= 2.87 in.
𝑃a
2.53
Use L1 = 3 in.
For weld length L3 , use the minimum length of 34 in.
specified in Section 2.3.3.1 of the AWS code, even though
P3 is approximately equal to zero.
Example 8.3.
8.3.4
Resistance Welds
Resistance welds (including spot welding and projection
welding) are mostly used for shop welding in cold-formed
steel fabrication (Fig. 8.24).
The nominal shear strengths for spot welding (Table 8.1)
are based on Section E2.6 of the 1996 AISI Specification,
which is based on Ref. 8.33 for outside sheets of 0.125 in. or
less in thickness and Ref. 8.34 for outside sheets thicker than
0.125 in. The safety factor used to determine the allowable
shear strength is 2.5 and the resistance factor used for the
LRFD method is 𝜙 = 0.65. Values for intermediate thicknesses may be obtained by straight-line interpolation.
8.3.3 Additional Design Information on Welded
Connections
The preceding discussion and design examples were based
on the AISI North American Specification. Blackburn and
Sputo8.120 performed a comprehensive study of the available arc spot weld data and concluded that the Specification
equations were generally conservative and offered recommendations for improvements for the design provisions. For
additional information concerning details of welded connections, workmanship, technique, qualification, and inspection,
the reader is referred to the AWS code.8.96
In addition to the research work conducted at Cornell
and the design criteria being used in the United States,
other research projects on welded connections have
been conducted by Baehre and Berggren,8.4 Stark
and Soe-tens,8.22 Kato and Nishiyama,8.23 Snow and
Easterling,8.99 and others1.419,1.420 . These references also
discuss design considerations and testing of welded connections. An economic study of the connection safety factor
has been reported by Lind, Knab, and Hall in Ref. 8.24.
Design information on tubular joints can be found in Refs.
8.25–8.32 and 8.68–8.70.
Figure 8.24
Resistance welds.
Table 8.1 Nominal Shear Strength for Spot
Welding1.314
Thickness of
Thinnest
Outside
Sheet (in.)
0.010
0.020
0.030
0.040
0.050
0.060
0.070
Nominal
Shear
Strength per
Spot (kips)
Thickness of
Thinnest
Outside
Sheet (in.)
Nominal
Shear
Strength per
Spot (kips)
0.13
0.48
1.00
1.42
1.65
2.28
2.83
0.080
0.090
0.100
0.110
0.125
0.190
0.250
3.33
4.00
4.99
6.07
7.29
10.16
15.00
WELDED CONNECTIONS
The above tabulated values may also be applied to pulsation welding and spot welding medium-carbon and low-alloy
steels with possibly higher shear strengths. It is interesting
to note that if the shear strength specified in the AISI Specification is used for spot welding galvanized steel sheets, a
relatively larger safety factor may be obtained for the ASD
method.8.35
It should be noted that special welding procedures may
be required for the welding of low-alloy steels. In all cases,
welding should be performed in accordance with the AWS
recommended practices.8.33,8.34,8.104
In 1999, the following equations for the nominal shear
strength of spot welds were developed to replace the tabulated values given in Table 8.11.333 :
1. For 0.01 in. (0.25 mm) ≤ t < 0.14 in. (3.56 mm),
⎧144𝑡1.47 (for 𝑡 in inches and
⎪
Pn in kip)
⎪
𝑃n = ⎨
1.47
(for 𝑡 in mm and
⎪5.51𝑡
⎪
Pn in kN)
⎩
(8.39a)
(8.39b)
2. For 0.14 in. (3.56 mm) ≤ t < 0.18 in. (4.57 mm),
⎧43.4𝑡 + 1.93 (for 𝑡 in inches and
⎪
Pn in kips)
⎪
𝑃n = ⎨
⎪7.6𝑡 + 8.57 (for 𝑡 in mm and
⎪
Pn )
⎩
(8.40𝑎)
(8.40b)
The upper limit of Eq. (8.39) was selected to best fit the
data provided in Table 2 of Ref. 8.33 and Table 1 of Ref.
8.34. Equation 8.40 is limited to t ≤ 0.18 in. (4.57 mm) due
to the thickness limit set forth in the AISI North American
Specification.
Table 8.1 and Eqs. (8.39) and (8.40) provide only the
nominal shear strength for spot welding. If the tensile
strength of spot welding is required, it can be obtained either
from tests or from the following empirical formulas for
tensile and shear strengths proposed by Henschkel8.36 :
1. Tensile Strength
[
2. Shear Strength
𝑎
+ 𝑐 − (fC + 𝑔Mn )
𝐹u − 𝑏
[
(
)]
Mn
S = t𝐹u 𝐷 𝛼 − 𝛽 𝐶 +
20
N = tensile strength of spot
welding
S = shear strength of spot welding
t = sheet thickness
Fu = tensile strength of steel sheet
C = carbon content
Mn = manganese content
D = weld nugget diameter
a, b, c, f, g, 𝛼, 𝛽 = coefficients determined from
test results (see Ref. 8.36 for
detailed information)
where
It should be noted that Henschkel’s study was based on the
following ranges of material:
1. Thickness of steel sheet: 0.008–0.500 in. (0.2–
12.7 mm)
2. Tensile strength of material: 37,500–163,800 psi
(258–1129 MPa)
3. Carbon content: 0.01–1.09%
4. Manganese content: 0.03–1.37%
From the above two equations, the relationship between
tensile and shear strengths of spot welding can be expressed
as follows:
𝑎
𝑁
=
𝑆
(𝐹u − 𝑏)(𝛼 − 𝛽C − 0.05𝛽Mn )
𝑐 − fC − 𝑔Mn
𝛼 − 𝛽C − 0.05𝛽Mn
Using the constants given in Ref. 8.36, it can be seen
that for the steels specified in the AISI North American
Specification the tensile strength of spot welding is higher
than 25% of the shear strength.
See Example 8.8 for the design of welded connections
using resistance welds.
+
where t is the thickness of the thinnest outside sheet.
N = t𝐹u 𝐷
315
]
8.3.5 Shear Lag Effect in Welded Connections
of Members
When a tension member is not connected through all
elements, such as when an angle is connected through
only one leg, the stress distribution in the cross section is
nonuniform. This phenomenon is referred to as “shear lag,”
which has a weakening effect on the tensile capacity of the
member.
For the design of hot-rolled steel shapes, the AISC Specification uses the effective net area Ae for determining the
nominal strength. The effective net area is computed as
𝐴e = 𝑈sl 𝐴n
in which U is the reduction factor and An is the net area.
316
8
CONNECTIONS
For cold-formed steel design, the following Specification Section E2.7 was added in the Supplement in
19991.333 and was retained for subsequent editions of the
Specification1.345,1.417
but Usl shall not be less than 0.5
where
J6.2 Shear Lag Effect in Welded Connections
of Members Other Than Flat Sheets
The nominal strength of a welded member shall be determined in
accordance with Section D3. For fracture and/or yielding in the
effective net section of the connected part, the nominal tensile
strength, Pn , shall be determined as follows:
𝑃nt = 𝐴e Fu
(8.41)
Ω = 2.50 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
1. When the load is transmitted only by transverse welds:
A = area of directly connected elements
Usl = 1.0
2. When the load is transmitted only by longitudinal welds
or by longitudinal welds in combination with transverse
welds:
A = gross area of member, Ag
Usl = 1.0 for members when the load is transmitted directly
to all of the cross-sectional elements
Otherwise, the reduction coefficient Usl is determined as
follows:
a. For angle members:
Usl = 1.0 −
1.20𝑥
< 0.9
𝐿
(8.42)
but U shall not be less than 0.4
b. For channel members:
Usl = 1.0 −
0.36𝑥
< 0.9
𝐿
The above design provisions were adapted from the AISC
design approach. Equations (8.42) and (8.43) are based on the
research work conducted by Holcomb, LaBoube, and Yu at the
University of Missouri–Rolla on bolted connections.6.24,6.25
8.4
where Fu is the tensile strength of the connected part as specified
in Section A3.1 or A3.1.2 and Ae = AUsl is the effective net area
with Usl defined as follows:
(8.43)
x = distance from shear plane to
centroid of the cross section
(Fig. 8.25)
L = length of longitudinal welds
(Fig. 8.25)
BOLTED CONNECTIONS
The structural behavior of bolted connections in cold-formed
steel construction is somewhat different from that in
hot-rolled heavy construction, mainly because of the thinness of the connected parts. Prior to 1980, the provisions
included in the AISI Specification for the design ofdbolted
connections were developed on the basis of the Cornell tests
conducted under the direction of George Winter.8.37–8.40
These provisions were updated in 19801.4 to reflect the
results of additional research performed in the United
States4.30,8.41–8.46 and to provide a better coordination with
the specifications of the Research Council on Structural
Connections8.47 and the AISC.1.148 In 1986, design provisions for the maximum size of bolt holes and the allowable
tension stress for bolts were added in the AISI Specification.
The 1996 edition of the Specification combined the ASD
and LRFD design provisions with minor revisions. The
shear lag effect on bolted connections were considered in
the Supplement to the 1996 Specification. New bearing
equations were adopted in the 2001 edition1.336 and retained
in the 2007 edition of the Specification. The 2012 and 2016
Specifications included provisions for bearing connections
having short-slotted holes.
Figure 8.25 Determination of x for sections using fillet welds.1.333,1.346,1.414
BOLTED CONNECTIONS
317
8.4.1 Research Work and Types of Failure Mode
Since 1950, numerous bolted connections using thin sheets
with A307 bolts and A325 high-strength bolts have been
tested at Cornell University and other institutions. The
purposes of these research projects were to study the structural performance of bolted connections and to provide
necessary information for the development of reliable
design methods. A summary of this research is provided
by Yu.1.354
Results of tests indicate that the following four basic
types of failure usually occur in the cold-formed steel bolted
connections:
1. Longitudinal shearing of the sheet along two parallel
lines (Fig. 8.26a)
2. Bearing or piling up of material in front of the bolt
(Fig. 8.26b)
3. Tearing of the sheet in the net section (Fig. 8.26c)
4. Shearing of the bolt (Fig. 8.26d)
These four failure modes are also illustrated in Fig. 8.27.
In many cases, a joint is subject to a combination of different
types of failure modes. For example, the tearing of the sheet
is often caused by the excessive bolt rotation and dishing of
the sheet material.8.45,6.23
8.4.1.1 Longitudinal Shearing of Steel Sheets (Type I
Failure) When the edge distance e as shown in Figs. 8.27a
and 8-28 is relatively small, connections usually fail in longitudinal shearing of the sheet along two parallel lines.
Figure 8.27 Types of failure of bolted connections: (a) longitudinal shear failure of sheet (type I); (b) bearing failure of sheet (type
II); (c) tensile failure of sheet (type III); (d) shear failure of bolt
(type IV).
Figure 8.28
Figure 8.26
Types of failure of bolted connections.8.37
Dimensions s and e used in bolted connections.
Research1.354 has shown that for bolted connections
having small e/d ratios the bearing stress at failure can be
predicted by
𝜎b
𝑒
=
(8.44)
𝐹𝑢
𝑑
318
8
CONNECTIONS
Figure 8.29 Determination of x for sections using bolted connections.1.346,1.417
where
𝜎 b = ultimate bearing stress between bolt and
connected part, ksi
Fu = tensile strength of connected part, ksi
e = edge distance, in.
d = bolt diameter, in.
Equation (8.44) is based on the results of bolted connection tests with the following parameters8.46 :
3
− 1 in. (4.8–25.4 mm)
Diameter of bolt d: 16
Thickness of connected part t: 0.036–0.261 in. (0.9–6.6 mm)
Edge distance e: 0.375–2.5 in. (9.5–63.5 mm)
Yield point of steel Fy : 25.60–87.60 ksi (177–604 MPa,
1800–6150 kg/cm2 )
Tensile strength of steel Fu : 41.15–91.30 ksi (284–630 MPa,
2890–6420 kg/cm2 )
e/d ratio: 0.833–3.37
d/t ratio: 2.61–20.83
Fu /Fy ratio: 1.00–1.63
The dimension of the specimens and the test results are given
in Ref. 8.45.
By substituting 𝜎 b = Pu /dt into Eq. (8.44), Eq. (8.45) can
be obtained for the required edge distance e,
𝑃
𝑒= u
(8.45)
𝐹u 𝑡
This equation was also used for the specifications of the
Research Council on Structural Connections8.47,8.48 and the
AISC.1.148 In 2016, this equation was replaced by the shear
rupture equation discussed in Section 8.8.
8.4.1.2 Bearing or Piling Up of Steel Sheet (Type II
Failure) When the edge distance is sufficiently large (i.e.,
for large e/d ratios), the connection may fail by bearing
or piling up of steel sheet in front of the bolt, as shown
in Fig. 8.27b. Additional studies indicate that the bearing
strength of bolted connections depends on several parameters, including the tensile strength of the connected part, the
thickness of the connected part, the types of joints (lap joints
or butt joint), single-shear or double-shear conditions, the
diameter of the bolt, the Fu /Fy ratio of the connected part,
the use of washers, the “catenary action” of steel sheets, and
the rotation of fasteners. In 2010, Yu and Xu8.122 studied
bolted connections having oversized and short-slotted holes.
Based on their study, new bearing factors, C, and modification factors, mf , were adopted by the specification.
Equation (8.46) was developed for determining the ultimate
bearing capacity on the basis of the applicable parameters. This equation was developed from the available test
data8.105,8.106 :
(8.46)
𝑃n = 𝐶 𝑚f 𝑑 𝑡 𝐹u
where
C = bearing factor, determined in accordance
with Table 8.2
mf = modification factor for type of bearing
connection determined in accordance
with Table 8.3
d = nominal bolt diameter
t = uncoated sheet thickness
Fu = tensile strength of sheet.
It should be noted that Eq. (8.46) is applicable only when
the deformation around the bolt holes is not a design consideration. If the deformation around the bolt holes is a design
consideration, research has determined that the nominal
bearing strength is given by the following equation6.25 :
𝑃n = (4.64𝑡 + 1.53) 𝑑 𝑡 𝐹u (with 𝑡 in inches)
(8.47a)
For SI units:
𝑃n = (0.183𝑡 + 1.53) 𝑑 𝑡 𝐹u
(with 𝑡 in mm)
(8.47b)
All symbols were defined previously.
The above design equations were developed from the
research conducted at the University of Missouri–Rolla to
recognize the hole elongation prior to reaching the limited
bearing strength of a bolted connection.6.24,6.25 The movement of the connection was limited to 0.25 in. (6.4 mm),
which is consistent with the permitted elongation prescribed
BOLTED CONNECTIONS
Table 8.2
319
Bearing Factor C1.417
Connections With Standard
Holes
Connections With Oversized
or Short-Slotted Holes
Thickness of
Ratio of Fastener
Connected Part, t,
Diameter to Member
C
in. (mm)
Thickness, d/t
0.024 ≤ t < 0.1875
(0.61 ≤ t < 4.76)
d/t < 10
10 ≤ d/t ≤ 22
d/t > 22
Ratio of Fastener
Diameter to Member
C
Thickness, d/t
3.0
4 – 0.1(d/t)
1.8
d/t < 7
7 ≤ d/t ≤ 18
d/t > 18
3.0
1 + 14/(d/t)
1.8
Note:
𝑎
Oversized or short-slotted holes within the lap of lapped or nested Z-members as defined in Section J3 are permitted to be
considered as standard holes.
Table 8.3
Modification Factor mf for Type of Bearing Connection1.417
Type of Bearing Connection
mf
Single shear and outside sheets of double shear connection using standard holes with washers
under both bolt head and nut
Single shear and outside sheets of double shear connection using standard holes without washers
under both bolt head and nut, or with only one washer
Single shear and outside sheets of double shear connection using oversized or short-slotted holes
parallel to the applied load without washers under both bolt head and nut, or with only one
washer
Single shear and outside sheets of double shear connection using short-slotted holes perpendicular
to the applied load without washers under both bolt head and nut, or with only one washer
Inside sheet of double shear connection using standard holes with or without washers
Inside sheet of double shear connection using oversized or short-slotted holes parallel to the
applied load with or without washers
Inside Sheet of Double Shear Connection Using Short-Slotted Holes Perpendicular to the Applied
Load With or Without Washers
1.00
0.75
0.70
0.55
1.33
1.10
0.90
Note:
Oversized or short-slotted holes within the lap of lapped or nested Z-members as defined in Section J3 are permitted to be
considered as standard holes.
in the AISC specification for hot-rolled steel shapes and
built-up members.
8.4.1.3 Tearing of Sheet in Net Section (Type III Failure)
In bolted connections, the type of failure by tearing of the
sheet in the net section is related to the stress concentration
caused by
1. The presence of holes
2. The concentrated localized force transmitted by the bolt
to the sheets
Previous tests conducted at Cornell University for
connections using washers under the bolt head and nut have
indicated that plastic redistribution is capable of eliminating
the stress concentration caused by the presence of holes even
for low-ductility steel.8.39 However, if the stress concentration caused by the localized force transmitted by the bolt
to the sheet is pronounced, the strength of the sheet in the
net section was found to be reduced for connections having
relatively wide bolt spacing in the direction perpendicular
to the transmitted force. The effects of the d/s ratio on
the tensile strength of bolted connections with washers is
discussed in Ref. 1.354.
An additional study conducted at Cornell on connections
using multiple bolts has shown that the sharp stress concentration is much relieved when more than one bolt in line is
used and the failure in the net section in two-bolt (r = 12 ) and
three-bolt (r = 13 ) tests occurred at a much higher stress than
320
8
CONNECTIONS
in a single-bolt (r = 1) connection. The following formulas
have been developed to predict the failure stress in the net
section:
[
( )]
𝑑
𝑑
𝐹u ≤ 𝐹u when
≤ 0.3
𝜎net = 1 − 0.9𝑟 + 3𝑟
𝑠
𝑠
(8.48)
𝑑
> 0.3
(8.49)
𝑠
failure stress in net section, ksi
force transmitted by bolt or bolts at the
section considered divided by the force in
the member at that section
bolt diameter, in.
spacing of bolts perpendicular to line of
stress, in.
ultimate tensile strength of steel sheets,
ksi
𝜎net = 𝐹u
where
𝜎 net =
r =
d =
s =
Fu =
when
The correlations between Eq. (8.48) and the test data is
discussed by Yu in Ref. 1.354. The test data reflects the
following parameters8.46 :
Diameter of bolt d: 14 – 1 − 18 in. (6.4–28.6 mm)
Thickness of steel sheet t: 0.0335–0.191 in. (0.9–4.9 mm)
Width of steel sheet s: 0.872–4.230 in. (22–107 mm)
Yield point of steel Fy : 26.00–99.40 ksi (179–685 MPa,
1830–6990 kg/cm2 )
Tensile strength of steel Fu : 41.15–99.80 ksi (284–688 MPa,
2890–7020 kg/cm2 )
d/s ratio: 0.063–0.50
d/t ratio: 3.40–21.13
When washers are not used and when only one washer is
used in bolted connections, the failure stress in the net section
𝜎 net can be determined by
[
( )]
𝑑
𝐹u ≤ 𝐹u
(8.50)
𝜎net = 1.0 − 𝑟 + 2.5𝑟
𝑠
The correlation between Eq. (8.50) and the test data is
presented by Yu in Ref. 1.354.
Research conducted at the University of Sydney revealed
that for flat sheet connections having multiple rows of bolts
in the line of force the strength reduction represented by Eqs.
(8.48) and (8.50) is not required.8.107
Fox and Schuster8.123 performed additional studies and
developed shear lag reduction factors for flat sheet connections subject to tension rupture. Based on studies for sections
other than flat sheets, shear lag reduction factors were developed by Teh and Gilbert8.124 that apply to both single and
multiple bolts in the line of the force, and single and double
shear connections and are in the 2016 edition of the Specification. See Section 8.8.
8.4.1.4 Shearing of Bolt (Type IV Failure) A number
of double-shear and single-shear tests were performed at
Cornell University in the 1950s to study the type of failure
caused by shearing of the bolt.8.37,8.38 It was found that
the shear–tension strength ratio is independent of the bolt
diameter, and the ratios are equal to about 0.62 and 0.72
for double-shear and single-shear tests, respectively. In view
of the fact that the failure by shearing of the bolt is more
sudden than that in the sheets being connected, a conservative shear-to-tension ratio of 0.6 has been used in the past
for both double- and single-shear conditions in the development of design provisions, even though the extremes of test
values ranged from 0.52 to 1.0; that is, the type of failure by
shearing of the bolt occurs at a strength equal to 0.6 times the
tensile strength of the bolt.
8.4.2
AISI Design Criteria for Bolted Connections
Based upon the results of tests summarized in Section 8.4.1
and past design experience, Section J3 of the 2016 edition of
the AISI North American Specification includes a number
of requirements for the design of bolted connections are
summarized herein.
8.4.2.1 Thickness Limitations On the basis of the same
reasons discussed in Section 8.3.2.1 for the design of welded
connections, Section J3 of the Specification is applicable
only to the design of bolted connections for cold-formed
3
in. (4.8 mm) in thicksteel members that are less than 16
3
ness. For materials not less than 16 in. (4.8 mm), the AISC
specification 1.411 should be used for the design of bolted
connections in cold-formed steel structures.
8.4.2.2 Materials Prior to 1980, the AISI design provisions concerning the allowable shear stresses for mechanical
fasteners were limited to A307 and A325 bolts. Because
the maximum thickness for cold-formed steel members was
increased in 1977 from 12 in. (12.7 mm) to 1 in. (25.4 mm),
other high-strength bolts, such as A354, A449, and A490
bolts, were added to the 1980 specification for bolted
connections.
In view of the fact that A325 and A490 bolts are available
only for a diameter of 12 in. (12.7 mm) and larger, whenever
smaller bolts [less than 12 in. (12.7 mm) in diameter] are
required in a design, A449 and A354 Grade BD bolts should
be used as equivalents of A325 and A490 bolts, respectively.
For other types of fasteners, which are not listed in
Section J3 of the AISI North American Specification, drawings should indicate clearly the type and size of fasteners to
be employed and the design force.
BOLTED CONNECTIONS
8.4.2.3 Bolt Installation The requirement for bolt installation was added to the AISI Specification since 1980
to ensure that bolts are properly tightened according to
acceptable practice. Because the required pretension in bolts
usually varies with the types of connected part, fasteners,
applied loads, and applications, no specific provisions are
provided in the AISI North American Specification for
installation.
The effect of torques on the strength of bolted connections
has been studied in the past and was reported in Ref. 8.45.
8.4.2.4 Maximum Sizes of Bolt Holes The 1986 and the
1996 editions of the AISI Specification include the maximum
sizes of standard holes, oversized holes, short-slotted holes,
and long-slotted holes, as shown in Table 8.4. Standard holes
should be used in bolted connections, except that oversized
and slotted holes may be used as approved by the designer.
Additional requirements are given in the AISI North American Specification for the use of oversized and slotted holes.
Shear rupture provisions are found in Section J6 of the
2016 specification, see Section 8.8 for a discussion of this
limit state. In addition to the shear rupture requirements,
Section J3.1 and J3.2 of the AISI North American Specification also includes the following requirements concerning
minimum spacing and edge distance in the line of stress:
1. The minimum distance between centers of bolt
holes should not be less than 3d. In addition, the
minimum distance between centers of bolt holes shall
provide clearance for bolt heads, nuts, washers and the
wrench.
2. The distance from the center of any standard hole to the
end or other boundary of the connecting member should
not be less than 1 12 d.
Table 8.4
321
3. For oversized and slotted holes, the clear distance
between edges of two adjacent holes should not be less
than 2d and the distance between the edge of the hole
and the end of the member should not be less than d.
8.4.2.5 Tensile
Strength
of
Connected
Parts
at Connection Prior to 1999, the tensile strength on
the net section of connected parts was determined in accordance with Specification Section E3.2 in addition to the
requirements of Specification Section C2. In Section E3.2,
the nominal tensile strength on the net section of the bolt
connected parts was determined by the tensile strength of
steel Fu and the ratios r and d/s. These design equations
represent the shear lag effect on the tensile capacity of flat
sheets with due consideration given to the use of washers
and the type of joints, either a single-shear lap joint or a
double-shear butt joint.
During recent years, research work has been conducted
by Holcomb, LaBoube, and Yu at the University of
Missouri–Rolla to study the effect of shear lag on the
tensile capacity of angles and channels as well as flat steel
sheets.6.24,6.25 The same project included a limited study of
the behavior of bolted connections having staggered hole
patterns. It was found that when a staggered hole pattern is
involved the net area can be determined by a design equation
′
using the well-known parameter s 2 /4g.
Based on the research findings, the 2016 edition of
the AISI North American Specification includes design
guidance in Section J6 to deal with the determination of
the nominal tensile strength for (a) flat sheet connections
not having staggered hole patterns, (b) flat sheet connections having staggered hole patterns, and (c) structural
shapes including angles and channels. See Section 8.8 for a
discussion of these provisions.
Maximum Sizes of Bolt Holes1.417
Nominal
Bolt
Diameter,
d (in.)
Standard
Hole
Diameter,
dh (in.)
Oversized
Hole
Diameter,
dh (in.)
Short-Slotted
Hole
Dimensions
(in.)
Long-Slotted
Hole
Dimensions
(in.)
d < 1/2
1/2 ≤ d < 1
d=1
d≥1
d + 1/32
d + 1/16
11/8
d + 1/8
d + 1/16
d + 1/8
11/4
d + 5/16
(d + 1/32) by (d + 1/4)
(d + 1/16) by (d + 1/4)
(11/8) by (15/16)
(d + 1/8) by (d + 3/8)
(d + 1/32) by (21/2 d)
(d + 1/16) by (21/2 d)
(11/8) by (21/2)
(d + 1/8) by (21/2 d)
Note:
1. The alternative short-slotted hole is only applicable for d = 1/2 in.
Alternative
Short-Slotted
Holea Dimensions
(in.)
9/16 by 7/8
322
8
CONNECTIONS
8.4.2.6 Bearing Strength between Bolts and Connected
Parts
a. Deformation Around the Bolt Holes Is Not a Design
Consideration. The nominal bearing strength (resistance)
of the connected sheet for each loaded bolt is given in
Section J3.3.1 of the Specification as
𝑃n = 𝐶𝑚f 𝑑𝑡𝐹u
(8.51)
Ω = 2.50 (ASD)
{
0.60 (LRFD)
𝜙=
0.50 (LSD)
where C = bearing factor, determined in accordance
with Table 8.2
mf = modification factor for type of bearing
connection determined in accordance
with Table 8.3
d = nominal bolt diameter
t = uncoated sheet thickness
Fu = tensile strength of sheet
b. Deformation Around the Bolt Holes Is a Design
Consideration. When the movement of the connection
is critical and the deformation around bolt holes is a
design consideration, nominal bearing strength should
also be limited by Eq. (8.47), according to Section
J3.3.2 of the 2016 edition of the Specification. For Eqs.
(8.47a) and (8.47b), Ω = 2.22 (for ASD), 𝜙 = 0.65 (for
LRFD), and 𝜙 = 0.55 (for LRFD). See Section 8.4.1.2
for additional discussion.
8.4.2.7 Shear and Tension in Bolts Section J3.4 of the
AISI North American specification specifies that the nominal
bolt strength Pn resulting from shear, tension, or a combination of shear and tension shall be calculated as follows:
𝑃 n = 𝐴b 𝐹
Fnt = nominal tensile stress from Table 8.5
Fnt = nominal shear stress from Table 8.5
fv = required shear stress
and Ω and 𝜙 are also from Table 8.5. In addition, the required
shear stress shall not exceed the allowable shear stress Fnv /Ω
(ASD) or the design shear stress 𝜙Fnv (LRFD) of the fastener.
In Table 8.5, the allowable shear and tension stresses
specified for A307, A325, and A490 bolts are approximately the same as those permitted by the AISC1.411 and the
Research Council on Structural Connections for bearing-type
connections.8.108 Slightly smaller allowable shear stresses
are used for A449 and A354 Grade BD bolts with threads
in the shear planes as compared with A325 and A490 bolts,
respectively. Such smaller shear stresses are used because
the average ratio of the root area to the gross area of the
1
-in. (6.4-mm) and 38 -in. (9.5-mm) diameter bolts is 0.585,
4
which is smaller than the average ratio of 0.670 for the 12 -in.
(12.7-mm) and 1-in. (25.4-mm) diameter bolts. According to
Ref. 1.159, these design values provide safety factors ranging
from 2.25 to 2.52 against the shear failure of bolts.
Example 8.4 Determine the allowable load for the bolted
connection shown in Fig. 8.30. Use four 12 -in.-diameter
A307 bolts with washers under the bolt head and nut. The
steel sheets are A570 Grade 33 steel (Fy = 33 ksi and
Fu = 52 ksi). Use ASD and LRFD methods. Assume that
the dead load–live load ratio is 15 and that the deformation
around bolt holes is not a design consideration.
SOLUTION
A. ASD Method
In the determination of the allowable load, consideration
should be given to the following items:
(8.52)
where Ab is the gross cross-sectional area of the bolt.
When bolts are subject to shear or tension, the nominal
stress Fn is given in Table 8.5 by Fnv for shear or Fnt for
tension. The applicable values of Ω and 𝜙 are also given in
the same table. When bolt tension is involved, the pull-over
strength of the connected sheet at the bolt head, nut, or
washer shall be considered.
When bolts are subject to a combination of shear and
tension,
⎧
Ω𝐹nt
𝑓 ≤ 𝐹nt
⎪1.3𝐹nt −
𝐹nv v
⎪
⎪
(for ASD method)
𝐹nt′ = ⎨
𝐹
⎪1.3𝐹nt − nt 𝑓v ≤ 𝐹nt
𝜙𝐹nv
⎪
⎪
(for
LRFD method)
⎩
where
• Shear, spacing, and edge distance in line of stress
(Section 8.4.2.5)
• Tensile strength of connected parts at connection
(Section 8.4.2.6)
(8.53𝑎)
(8.53𝑏)
Figure 8.30
Example 8.4.
BOLTED CONNECTIONS
Table 8.5
323
Nominal Tensile and Shear Strength for Bolts 1.417
Nominal Tensile Strength
Fnt , ksi (MPa)
Bolt Type
ASTM A307 Grade A Bolts
Nominal Shear Strength
Fnv , ksi (MPa)a
1/4 in. ≤ d <1/2 in.
(6.4 mm ≤ d
< 12 mm)
d ≥ 1/2 in.
(12 mm)
1/4 in. ≤ d <1/2 in.
(6.4 mm ≤ d
< 12 mm)
d ≥ 1/2 in.
(12 mm)
40 (280)
45 (310)
24 (169)b
27 (188)b
NA
90 (620)
NA
ASTM F3125 Grade A325/A325M Bolts:
• When threads are not excluded from
shear planes
• When threads are excluded from shear
planes
54 (372)
68 (457)
ASTM A354 Grade BD Bolts:
• When threads are not excluded from
shear planes
• When threads are excluded from shear
planes
101 (700)
61 (411)
68 (457)
84 (579)
84 (579)
48 (334)
54 (372)
68 (457)
68 (457)
113 (780)
ASTM A449 Bolts:
• When threads are not excluded from
shear planes
• When threads are excluded from shear
planes
81 (560)
90 (620)
ASTM F3125 Grade A490/A490M Bolts:
• When threads are not excluded from
shear planes
• When threads are excluded from shear
planes
68 (457)
NA
113 (780)
NA
84 (579)
Threaded Parts:
• When threads are not excluded from
shear planes
• When threads are excluded from shear
planes
0.675 Fu c
0.400 Fu
0.450 Fu
0.563 Fu
0.563 Fu
0.75 Fu
Notes:
𝑎
For end-loaded connections with a fastener pattern length greater than 38 in. (965 mm), Fnv should be reduced to 83.3 percent of the tabulated
values. Fastener pattern length is the maximum distance parallel to the line of force between the centerline of the bolts connecting two parts
with one faying surface.
𝑏
Threads permitted in shear planes.
𝑐
Tensile strength of bolt.
324
8
CONNECTIONS
• Bearing strength between bolts and connected parts
(Section 8.4.2.7)
• Shear strength in bolts (Section 8.4.2.8)
1. Shear, Spacing, and Edge Distance in Line of Stress.
The distance from the center of a standard hole to the
nearest edge of an adjacent hole is
1
1
𝑒1 = 2 − (𝑑 + 1∕16) = 2 − (1∕2 + 1∕16)
2
2
= 1.72 in.
The distance from the center of a standard hole to the
end of the plate in the line of stress is
Since e2 < e1 , the allowable load should be determined by e2 .
Because Fu /Fy = 52/33 = 1.58 > 1.08, according
to Eq. (8.73), the allowable shear strength of the
connected sheet using four bolts can be computed as
Anv = 2net = 2(4)(0.105)(1.0 − 1∕2(1∕2 + 1∕16))
= 0.604 in2
Vn = 0.6Fu 𝐴nv = 0.6(52)(0.604) = 18.84 kips
P𝑎2 = 18.84∕2.22 = 8.48 kips
In addition, some other AISI requirements should
be checked on the basis of Section J3.1 of the AISI
North American Specification or Section 8.4.2.5 in this
volume as follows:
a. Distance between centers of bolt holes:
OK
b. Distance from center of any standard hole to end of
plate:
(
)
1
1 in. > 1 𝑑 = 0.75 in.
OK
2
2. Tensile Strength of Steel Sheets. Based on the AISI
design criteria, the allowable tensile strength of the
steel sheet can be determined under the following
considerations:
Based on Chapter D of the 2016 edition of the AISI
North American specification, the allowable tensile
strength can be computed as follows:
i. For yielding [Eq. (6.2)],
𝐴g 𝐹y
𝑇
𝑇a = n =
Ωt
1.67
=
(4.0 × 0.105)(52)
= 10.92 kips
2.00
Use Ta = 8.30 kips for the requirement of
Chapter D of the specification.
=
According to Section J6.2 of the 2016 specification
for flat sheet connections not having staggered holes,
the tension rupture is determined by the following:
Ant = Ag−nbdht = 4𝑥0.105 − 2𝑥(1∕2 + 1∕16)𝑥0.105
= 0.302 in2
𝑒2 = 1 in.
2 in. > (3𝑑 = 1.5 in.)
ii. For fracture away from the connection [Eq. (6.3)],
𝐴 𝐹
𝑇
𝑇a = n = n u
Ωt
2.00
(4.0 × 0.105)(33)
= 8.30 kips
1.67
Ae = Usl 𝐴e = (0.9 + 0.1𝑑∕𝑠)Ant
(
(
))
1
= 0.9 + 0.1𝑥 ∕2 𝑥0.302 = 0.279 in2
2
Pnt = Fu 𝐴𝑒 = 52𝑥0.279 = 14.53 kips
Pa = Pnt ∕Ω = 14.53∕2.22 = 6.54 kips
Because Pa < Ta , use Pa = 6.54 kips
3. Bearing Strength between Bolts and Steel Sheets.
According to Section J3.3.1 of the 2016 edition of the
Specification, the allowable bearing strength per bolt is
Cmf 𝐹u dt
𝑃
𝑃a = n =
Ω
2.50
𝑑
= 4.76
𝑡
Therefore C = 3.0 (Table 8.2) and mf = 1.0 (Table 8.3).
( )
1
(0.105) = 8.19 kips
= (3)(1.0)(52)
2
8.19
𝑃a =
= 3.27 kips
2.50
The allowable bearing strength for four bolts is
𝑃3 = 4 × 3.27 = 13.10 kips
4. Shear Strength in Bolts. From Table 8.5, the nominal
shear stress for the 12 -in.-diameter A307 bolts is 27 ksi
and the gross area of the bolt is 0.196 in.2 Therefore,
the allowable shear strength for four bolts is
4(𝐴b 𝐹nv )
𝑃4 =
Ω
4(0.196)(27)
=
= 8.82 kips
2.4
Comparing P1 , P2 , P3 , and P4 , the allowable load
for the given bolted connection is 6.54 kips, which is
governed by the tension rupture strength of sheet A at
section a–a.
BOLTED CONNECTIONS
325
B. LRFD Method
For the LRFD method, the design considerations are the
same as for the ASD method. The design strength can be
calculated by applying some of the values used for the ASD
method.
4. Shear Strength in Bolts. Based on Table 9, the design
shear strength of four bolts is
1. Shear, Spacing, and Edge Distance in Line of Stress.
Using Eq. (8.73) for Fu /Fy > 1.08, the design shear
strength of a connected sheet using four bolts is
Comparing the values of (𝜙Pn )1 , (𝜙Pn )2 , (𝜙Pn )3 ,
and (𝜙Pn )4 , the controlling design strength is 9.44 kips,
which is governed by the tension rupture strength of
sheet A at section a–a.
𝐴nv = 2net = 2(4)(0.105)(1.0 − 1∕2(1∕2 + 1∕16))
= 0.604 in2
𝑉𝑛 = 0.6Fu Anv = 0.6(52)(0.604) = 18.84 kips
𝜙𝑃𝑎2 = 0.65𝑥18.44 = 12.25 kips
2. Tensile Strength of Steel Sheets.
Based on Chapter D of the 2016 edition of the specification, the design tensile strength of the connected
sheet is:
i. For yielding [Eq. (6.2)],
𝜙(𝑃n )4 = 4(0.65)(𝐴b 𝐹nv ) = 4(0.65)(0.196)(27)
= 13.759 kips
The required strength can be computed from Eqs. (1.5a)
and (1.5b) for the dead load–live load ratio of 15 .
From Eq. (1.5a),
(𝑃u )1 = 1.4𝑃D + 𝑃L = 1.4𝑃D + 5𝑃D = 6.4𝑃D
From Eq. (1.5b),
(𝑃u )2 = 𝑙.2𝑃D + 1.6𝑃L = 1.2𝑃D + 1.6(5𝑃D ) = 9.2𝑃D
and 9.2PD controls.
The dead load PD can be computed as follows:
9.2𝑃D = 9.44 kips
𝜙t 𝑇n = (𝜙t )(𝐴g 𝐹y ) = (0.90)(4 × 0.105)(33)
𝑃D = 1.026 kip
= 12.474 kips
𝑃L = 5𝑃D = 5.130 kips
ii. For fracture away from the connection [Eq. (6.3)],
𝜙t 𝑇n = 𝜙t (𝐴n 𝐹n ) = (0.75)(4 × 0.105)(52)
= 16.38 kips
Use 𝜙tTn = 12.474 kips for the requirement of
Chapter D of the specification.
According to Section J6.2 of the 2016 specification
for flat sheet connections not having staggered holes,
the tension rupture is determined by the following:
A𝑛𝑡 = A𝑔−nbdht = 4𝑥0.105 − 2𝑥(1∕2 + 1∕16)𝑥0.105
= 0.302 in2
A𝑒 = U𝑠𝑙 𝐴𝑒 = (0.9 + 0.1𝑑∕𝑠)A𝑛𝑡
(
(
))
1
= 0.9 + 0.1𝑥 ∕2 𝑥0.302 = 0.279 in2
2
P𝑛𝑡 = F𝑢 𝐴𝑒 = 52𝑥0.279 = 14.53 kips
𝜙Pa = 0.65𝑥14.53 = 9.44 kips
Because 𝜙Pa < Ta , use Pa = 9.44 kips
3. Bearing Strength between Bolts and Steel Sheets.
The nominal bearing strength per bolt is the same as
calculated for the ASD method, that is, Pn = 8.19 kips.
The design bearing strength between four bolts and the
steel sheet is
(𝜙𝑃n )3 = 4(0.60)(8.19) = 19.656 kips
The total allowable load based on the LRFD method is
Pa = PD + PL = 6.156 kips. It can be seen that the LRFD
method and the ASD method provide essentially the same
available strength.
Example 8.5 Check the adequacy of the bearing-type
connection as shown in Fig. 8.31. Use four 12 -in.-diameter
A325 bolts and A606 Grade 50 steel sheets (Fy = 50 ksi
and Fu = 70 ksi). Assume that washers are used under the
bolt head and nut and that threads are not excluded from
shear planes. Use standard holes and the ASD method. The
deformation around bolt holes is not a design consideration.
SOLUTION
1. Shear, Spacing, and Edge Distance in Line of Stress.
Since the inside sheet is thicker than the sum of the
thickness of both outside sheets and the distance from
the center of the hole to the end of the plate is the same
for inside and outside sheets, the outside sheets will
govern the design.
Using Eq. (8.73), the allowable design shear strength
of the outside sheet can be computed as follows:
Anv = 2(4)(0.105)(1.0 − 1∕2(1∕2 + 1∕16))
= 0.604 in2
326
8
CONNECTIONS
Figure 8.31
Vn = 0.6Fu 𝐴nv = 0.6(70)(0.604) = 25.36 kips
In addition, the following requirements should also
be checked:
a. Distance from center of hole to edge of adjacent hole:
OK
b. Distance between centers of bolt holes:
2 in. > (3𝑑 = 1.5 in.) OK
c. Distance from center of the hole to end of plate:
(
)
1
1 in. > 1 𝑑 = 0.75 in.
OK
2
2. Tensile Strength of Steel Sheets. Based on Chapter D
of the 2016 edition of the specification, the allowable
tensile strength of the outside sheet can be computed as
follows:
i. For yielding [Eq. (6.2)],
𝐴g 𝐹y
(4.0 × 0.105)(50)
=
1.67
1.67
= 12.57 kips > 9 kips OK
𝑇a =
ii. For fracture away from the connection [Eq. (6.3)],
𝑇a =
𝐴n 𝐹u
2.00
As in Example 8.4, An = 0.30 in.2 , and
(0.30)(70)
= 10.5 kips > 9 kips OK
2.00
According to Section J6.2 of the 2016 specification
for flat sheet connections not having staggered holes,
𝑇a =
the tension rupture is determined by the following:
Ant = Ag−nbdht = 4𝑥0.105 − 2𝑥(1∕2 + 1∕16)𝑥0.105
(P𝑎 )2 = 25.36∕2.22 = 11.43 kips
1
2 − (1∕2 + 1∕16) = 1.72 in.
2
> (𝑒 = 1.0 in.)
Example 8.5
= 0.302 in2
Ae = Usl 𝐴e = (0.9 + 0.1𝑑∕𝑠)Ant
(
(
))
1
= 0.9 + 0.1𝑥 ∕2 𝑥0.302 = 0.279 in2
2
Pnt = Fu 𝐴e = 70𝑥0.279 = 19.53 kips
Pa = Pnt ∕Ω = 19.53∕2.22 = 8.80 kips
Because Pa < Ta , use Pa = 8.80 kips < 9 kips, NG
By inspection, it is not necessary to check the inside
sheet for tensile strength because it is thicker than the
sum of the thicknesses of both outside sheets.
3. Bearing Strength between Bolts and Steel Sheets.
The allowable bearing strength can be obtained using
Section 8.4.2.7 for the inside sheet and the outside
sheets which are used in a double-shear connection:
a. For the inside sheet, the allowable bearing strength is
C𝑚f 𝐹u dt (3.0)(1.33)(70)(0.5)(0.25)
=
Ω
2.50
(
)
18
= 13.96 kips∕bolt >
= 4.5 kips∕bolt
OK
4
b. For the outside sheets, the allowable bearing strength
is
C 𝐹 dt (3)(1.0)(70)(0.5)(0.105)
𝑃a = 𝑚f u =
Ω
2.50
(
)
18
= 4.41 kips∕bolt <
= 4.5 kips∕bolt
NG
4
4. Shear Strength in Bolts. When threads are not
excluded from shear planes, the nominal shear stress
for A325 bolts can be obtained from Table 8.5 that is,
𝑃a =
𝐹nv = 54 ksi
SCREW CONNECTIONS
327
The allowable shear strength for the double shear
condition is
(2)(𝐴b 𝐹nv ) (2)(0.196)(54)
=
𝑃a =
Ω
2.4
(
)
18
= 8.82 kips∕bolt >
= 4.5 kips∕bolt
OK
4
On the basis of the above calculations for the ASD
method, it can be concluded that the given connection
is not adequate for the applied load of 18 kips. The same
design considerations should be used for the LRFD
method.
8.4.3 Additional Design Information on Bolted
Connections
The research work reviewed at the beginning of Section
8.4 dealt mainly with the previous studies conducted in the
United States. The design criteria discussed in Section 8.4.2
were based on the 2016 edition of the AISI North American
specification.1.417
Additional research work on bolted connections has
been conducted by Baehre and Berggren,1.25,8.4 Stark and
Toma,8.5,8.49,8.50 Marsh,8.51 LaBoube,8.52 Zadanfarrokh
and Bryan,8.71 Carril, Holcomb, LaBoube, and Yu,6.23–6.25
Seleim and LaBoube,8.72 Kulak and Wu,8.73 Wheeler, Clarke,
Hancock, and Murray,8.74 Rogers and Hancock,2.55–2.61 and
other researchers.1.362–1.366,1.419-1.422 The criteria for the
bolted connections and the additional information on
mechanical fasteners have been published in Refs. 8.4, 8.7,
8.8, and 8.53. See also other design specifications mentioned
in Chapter 1.
Additional numerical design solutions may be found in
Ref. 1.428.
8.5 SCREW CONNECTIONS
Screws can provide a rapid and effective means to fasten
sheet metal siding and roofing to framing members and to
make joints in siding and roofing, as shown in Fig. 8.32. They
can also be used in steel framing systems and roof trusses and
to fasten gypsum sheathing to metal studs and tracks.
Figure 8.33 shows some types of self-tapping screws
generally used in building construction.8.2 Self-drilling
tapping screws are to be in compliance with ASTM
C1513.8.109 Guidance for selection of screws can be found
in the publication of the Cold-Formed Steel Engineers
Institute.8.110
8.5.1 AISI Design Criteria
The AISI design provisions for screw connections were
initially developed in 1993.8.83 The background information
Figure 8.32 Application of self-tapping screws.8.1
on these AISI design criteria is summarized by Pekoz in
Ref. 8.54. Based on the ECCS Recommendations and the
British Standard with the results of over 3500 tests from
the United States, Canada, Sweden, United Kingdom, and
the Netherlands, Pekoz developed the requirements were
developed as given in Section E4 of the 1996 edition of the
AISI Specification for the design of screw connections and
were essentially unchanged in the 2016 edition of Section
J4 the Specification.
Additional design provisions were included in the 2016
specification for the limit states of combined shear and
pull-out, combined shear and pull-over, and combined shear
and tension in screws. The combined shear and pull-out
provisions are based on studies by Luttrell8.125 and LaBoube
and Zwick.8.126 Research pertaining to the behavior of a
screw connection has been conducted at the Missouri University of Science and Technology by Francka and LaBoube8.127
and based on the findings of this research, equations were
derived that enable the evaluation of the strength of a screw
connection when subjected to combined shear and tension.
The provisions to account for shear and tension interaction
in screws are based on the rational engineering analysis and
are the same strength interaction as that used for bolts,
These provisions do not preclude evaluation of any limit
state on any through manufacturer or independent laboratory testing. The safety and resistance factors for any
nominal strength [resistance] established through testing
should be determined using provisions of Section K2 of the
specification.
328
8
CONNECTIONS
Figure 8.33
Types of self-tapping screws.8.3 Courtesy of Parker-Kalon Corporation.
J4 Screw Connections
The following notation applies to this section of the Specification:
d = nominal screw diameter
Ω = 3.00 (ASD)
𝜙 = 0.50 (LRFD)
= 0.40 (LSD)
Pnv = nominal shear strength [resistance] per screw
Pnvs = nominal shear strength [resistance] of the
screw
Pnts = nominal shear strength [resistance] of the
screw
Pnot = nominal pull-out strength [resistance] per
screw
Pnov = nominal pull-over strength [resistance] per
screw
t1 = thickness of member in contact with screw
head or washer (Figs. 8–35 and 8–36)
t2 = thickness of member not in contact with screw
head or washer (Figs. 8-35 and 8-36)
Fu1 = tensile strength of member in contact with
screw head or washer
Fu2 = tensile strength of member not in contact with
screw head or washer
All J4 requirements shall apply to screws with 0.08 in.
(2.03 mm) ≤ d ≤ 0.25 in. (6.35 mm). The screws shall be thread
forming or thread cutting, with or without a self-drilling point.
Alternatively, design values for a particular application shall
be permitted to be based on tests according to Chapter K. For
diaphragm applications, Section I2 shall be used.
Screws shall be installed and tightened in accordance with the
manufacturer’s recommendations.
The nominal screw connection strengths [resistances] shall
also be limited by Chapter D.
J4.1 Minimum Spacing
The distance between the centers of fasteners shall not be less
than 3d.
J4.2 Minimum Edge and End Distance
The distance from the center of a fastener to the edge of any part
shall not be less than 1.5d. If the end distance is parallel to the
force on the fastener, the nominal shear strength [resistance] per
screw, Pns , shall be limited by Section J6 of the specification.
J4.3 Shear
J4.3.1 Connection Shear Limited by Tilting and Bearing
The nominal shear strength [resistance] per screw, Pns , shall be
determined as follows: For t2 /t1 < 1.0, Pns shall be taken as the
smallest of
𝑃nv = 4.2(𝑡32 𝑑)1∕2 𝐹u2
(8.54)
𝑃nv = 2.7𝑡1 dF u1
(8.55)
𝑃nv = 2.7𝑡2 dF u2
(8.56)
For t2 /t1 ≥ 2.5, Pns shall be taken as the smaller of
𝑃nv = 2.7𝑡1 dF u1
(8.57)
𝑃nv = 2.7𝑡2 dF u2
(8.58)
For 1.0 < t2 /t1 < 2.5, Pns shall be determined by linear interpolation between the above two cases.
J4.3.2 Shear in Screws
The nominal strength [resistance] of the screw shall be Pnvs
as reported by the manufacturer or determined by independent
laboratory testing. In lieu of the value provided in Section J4, the
safety factor or resistance factor is permitted to be determined
in accordance with Section K2.1 and shall be taken as 1.25Ω ≤
3.0 (ASD), 𝜙/1.25 ≥ 0.5 (LRFD),or 𝜙/1.25 ≥ 0.4 (LSD).
J4.4 Tension
For screws that carry tension, the head of the screw or washer,
if a washer is provided, shall have a diameter dh or dw not
less than 5/16 in. (7.94 mm). The nominal washer thickness
shall be at least 0.050 in. (1.27 mm) for t1 greater than 0.027 in.
(0.686 mm) and at least 0.024 in. (0.610 mm) for t1 equal to
SCREW CONNECTIONS
or less than 0.027 in. (0.686 mm). The washer shall be at least
0.063 in. (1.60 mm) thick when 5/8 in. (15.9 mm) < dw ≤ 3/4 in.
(19.1 mm).
J4.4.1 Pull-Out
The nominal pull-out strength [resistance], Pnot , shall be calculated as follows:
𝑃not = 0.85𝑡c dF u2
(8.59)
where tc is the lesser of the depth of the penetration and the
thickness, t2 .
J4.4.2 Pull-Over
The nominal pull-over strength [resistance], Pnov , shall be calculated as follows:
𝑃nov = 1.5𝑡t 𝑑w 𝐹u1
(8.60)
where 𝑑w′ is the pull-over diameter determined in accordance
with (a), (b), or (c) as follows:
(a) For a round head, a hex head (Fig. 8.37), or hex washer
head (Figure 8.37) screw with an independent and solid steel
washer beneath the screw head,
𝑑w′ = 𝑑h + 2𝑡w + 𝑡1 ≤ 𝑑w
where dh = screw head diameter or hex washer head
integral washer diameter
tw = steel washer thickness
dw = steel washer diameter
(b) For a round head, a hex head, or a hex washer head screw
without an independent washer beneath the screw head:
𝑑w′ = d𝑢 but not larger than 3∕4 in.(19.1 mm)
(c) For a domed (nonsolid and independent) washer beneath the
screw head (Fig. 8.37, it is permissible to use 𝑑w′ as calculated
in Eq. (8.67) with dh , tw , and t1 , as defined in Fig. 8.37. In the
equation, 𝑑w′ cannot exceed 34 in. (19.1 mm).
J4.4.3 Tension in Screws
The nominal strength [resistance] of the screw shall be Pnts
as reported by the manufacturer or determined by independent
laboratory testing. In lieu of the value provided in Section J4,
the safety factor or resistance factor shall be permitted to be
determined in accordance with Section K2.1 and shall be taken
as 1.25Ω ≤ 3.0 (ASD), 𝜙/1.25 ≥ 0.5 (LRFD),or 𝜙/1.25 ≥ 0.4
(LSD).
J4.5.1 Combined Shear and Pull-Over
J4.5.1 ASD Method
For screw connections subjected to a combination of shear and
tension forces, the following requirements shall be met:
𝑄
𝑇
1.10
+ 0.71
≤
𝑃nv
𝑃nov
Ω
(8.61)
329
In addition, Q and T shall not exceed the corresponding allowable strength determined by Sections J4.3 and J4.4, respectively,
where Q = required allowable shear strength of
connection
T = required allowable tension strength of
connection
Pnv = nominal shear strength of connection
= 2.7t1 dFu1
Pnov = nominal pull-out strength of connection
= 2.7t1 dw Fu1
dw = larger of screw head diameter or washer
diameter
Ω = 2.35
J4.5.2 LRFD or LSD Methods
For screw connections subjected to a combination of shear and
tension forces, the following requirements shall be met:
𝑄
𝑇
+ 0.71
≤ 1.10𝜙
𝑃nv
𝑃nov
(8.62)
In addition, Q and T shall not exceed the corresponding allowable strength determined by Sections E4.3 and E4.4, respectively,
where Q = required shear strength [factored shear force]
of connection
T = required tension strength [factored tensile
force] of connection
Pnv = nominal shear strength [resistance] of
connection
= 2.7t1 dFu1
Pnov = nominal pull-out strength [resistance] of
connection
= 2.7t1 dw Fu1
dw = larger of screw head diameter or washer
diameter
𝜙 = 0.65 (LRFD)
= 0.55 (LSD)
Equations (8.61) and (8.62) shall be valid for connections that
meet the following limits:
1. 0.0285 in. (0.724 mm) ≤ t1 ≤ 0.0445 in. (1.13 mm)
2. No. 12 and No. 14 self-drilling screws with or without
washers
3. dw ≤ 0.75 in. (19.1 mm)
4. Fu1 ≤ 70 ksi (483 MPa, or 4920 kg/cm2)
5. t2 /t1 ≥ 2.5
For eccentrically loaded connections that produce a nonuniform pull-over force on the fastener, the nominal pull-over
strength [resistance] shall be taken as 50% of Pnov .
J4.5.2 Combined Shear and Pull-Out
For a screw connection subjected to combined shear and
pull-over, the required shear strength,V, and required tension
strength, T, shall not exceed the corresponding available
strength.
330
8
CONNECTIONS
In addition, the following requirement shall be met:
1.15
V
T
+
≤
Pnv Pnot
Ω
(ASD)
(8.63a)
V
T
+
≤ 1.15𝜙
Pnv Pnot
(LRFD)
(8.63b)
where
𝑃nv = Nominal shear strength [resistance] of sheet per screw
= 4.2(𝑡32 𝑑)1∕2 Fu2
(8.64)
𝑃not = Nominal pull-out strength [resistance] of sheet per screw
= 0.85𝑡c dFu2
(8.65)
Ω = 2.55
(ASD)
𝜙 = 0.60
(LRFD)
= 0.50
Figure 8.34
Comparison of tilting and bearing.1.310,1.346,1.431
(LSD)
Eq. 8.58 shall be valid for connections that meet the following
limits:
1. 0.0297 in. (0.754 mm) ≤ t2 ≤ 0.0724 in. (1.84 mm),
2. No. 8, 10, 12, or 14 self-drilling screws with or without
washers,
3. Fu2 ≤ 121 ksi (834MPa or 8510 kg/cm2), and
4. 1.0 ≤ Fu /Fy ≤ 1.62.
Figure 8.35 Design equations for t2 /t1 ≥ 2.5.
J4.5.3 Combined Shear and Tension in Screws
For screws subjected to a combination of shear and tension
forces, the required shear strength, 𝑉 , and required tension
strength, 𝑇 , shall not exceed the corresponding available
strength.
In addition, the following requirement shall be met:
1.3
V
T
+
≤
Pnvs Pnts
Ω
(ASD)
(8.66a)
V
T
+
≤ 1.3𝜙
Pnvs Pnts
(LRFD)
(8.66b)
where 𝑉 = Required shear strength, determined in
accordance with ASD, LRFD, or LSD load
combinations
𝑇 = Required tension strength, determined in
accordance with ASD, LRFD, or LSD load
combinations
Pnvs = Nominal shear strength of screw as reported
by manufacturer or determined by
independent laboratory testing
Pnts = Nominal tension strength of screw as reported
by manufacturer or determined by
independent laboratory testing
Ω = 3.0
𝜙 = 0.5
When using the above design provisions, the AISI
Commentary recommends that at least two screws should
Figure 8.36 Design equations for t2 /t1 ≤ 1.0.
be used to connect individual elements.1.431 This provides
redundancy against undertorquing, overtorquing, and so on
and limits lap shear connection distortion of flat unformed
members such as straps. Table 8.6 lists the nominal diameters
for the common number designations for screws.
Screw connections loaded in shear can fail either in one
mode or in a combination of several modes. The failure
modes include shearing of the screw, edge tearing, tilting
and subsequent pull-out of the screw, and bearing failure
of the joined materials. Tilting of the screw followed by
thread tearing out of the lower sheet reduces the connection
shear capacity from that of the typical bearing strength of the
connection as shown in Fig. 8.34.1.431
With regard to the tilting and bearing failure modes, two
cases are considered in the specification, depending on
the ratio of thicknesses of the connected members. If the
head of the screw is in contact with the thinner material
as shown in Fig. 8.35, tilting is not a design consideration
when t2 /t1 ≥ 2.5. However, when both members are the
same thickness, or when the thicker member is in contact
with the screw head as shown in Fig. 8.36, tilting must also
be considered when t2 /t1 ≤ 1.0. Use linear interpolation for
1.0 < t2 /t1 < 2.5.
POWER-ACTUATED FASTENERS
Table 8.6 Nominal Body Diameter for
Screws1.431
Nominal Diameter
for Screws
Designation
in.
mm
0
1
2
3
4
5
6
7
8
10
12
1/4
0.060
0.073
0.086
0.099
0.112
0.125
0.138
0.151
0.164
0.190
0.216
0.250
1.52
1.85
2.18
2.51
2.84
3.18
3.51
3.84
4.17
4.83
5.49
6.35
Screw connections subjected to tension can fail by either
pulling out of the screw from the plate (pull-out) or pulling of
material over the screw head and the washer (pull-over) or by
tension fracture of the screw. For the failure mode of pull-out,
Eq. (8.59) was derived on the basis of the modified European
Recommendations and the results of a large number of tests.
For the limit state of pull-over, Eq. (8.60) was derived on the
basis of the modified British Standard and the results of a
series of tests. The statistic data on these tests are presented
by Pekoz in Ref. 8.54.
8.5.2 Additional Information on Screw Connections
The North American Standard for Cold-Formed Steel
Structural Framing1.423 stipulates that a properly installed
screw shall extend through the connection a minimum of
three exposed threads. Also, the guidance is provided for
remediation if a screw is stripped during installation or if
the screws are spaced closer than the AISI North American
specification requirements.
Lease and Easterling8.113 determined that the design provisions in Section J4 of the Specification are valid for applications that incorporate 6 38 in. (162 mm) or less of compressible
insulation.
During recent years, research work on screw connections
has been conducted by Xu,8.76 Daudet and LaBoube,8.77
Serrette and Lopez,8.78 Rogers and Hancock,2.57,2.59 Kreiner
and Ellifritt,8.80 Anderson and Kelley,8.81 Sokol, LaBoube,
and Yu,8.82 Zwick and LaBoube,8.114 Carr, Mansour,
and Mills,8.115 Fulop and Dubina,8.116 Mahendran and
Maharachchi,8.117,8.118 and other researchers1.416,1.421,1.422 .
331
For design examples see Refs. 1.428 and 1.434.
8.6
POWER-ACTUATED FASTENERS
Power-actuated fastening is used by virtually all building
trades. It is used to attach electrical boxes and conduit,
to attach lath, to hang sprinklers and ductwork, to build
concrete formwork, and to attach floor and roof decking. The
most common application in the cold-formed steel framing
industry is the attachment of a bottom wall track to concrete
or structural steel supports. Power-actuated fastening was
first developed in the 1920s and has been in widespread use
in the United States for decades.
Power-actuated fastening systems are produced by a
number of manufacturers. Although some systems offer
special features, all consist of three components: fastener,
powder load, and power-actuated tool. The CFSEI Tech
Note8.119 provides more information.
Specification Section J5, provisions for determining the
available strengths were developed based on the study by
Mujagic et al.8.127 .
8.6.1
AISI Design Criteria
The steel thickness of the substrate not in contact with
the Power-Actuated Fastener (PAF) head are limited to
a maximum of 0.75 in. (19.1 mm). The steel thickness of
the substrate in contact with the PAF head is limited to a
maximum of 0.06 in. (1.52 mm). The washer diameter is to
not exceed 0.6 in. (15.2 mm) in computations, although the
actual diameter may be larger. PAF diameter is limited to a
range of 0.106 in. (2.69 mm) to 0.206 in. (5.23 mm).
For diaphragm applications, the provisions of Section I2
of the specification are to be used.
Alternatively, the available strengths for any particular application are permitted to be determined through
independent laboratory testing, with the resistance factors,
𝜙, and safety factors, Ω, determined in accordance with
Section K2 of the specification. The values of Pntp and Pnvp
are permitted to be reported by the manufacturer.
The following notation applies to Section J5 of the specification:
a = Major diameter of tapered PAF head
d = Fastener diameter measured at near side
of embedment
= ds for PAF installed such that entire
point is located behind far side of
embedment material
dae = Average embedded diameter, computed
as average of installed fastener
diameters measured at near side and far
side of embedment material
332
8
CONNECTIONS
ds
d’w
E
Fbs
Fu1
Fu2
Fuh
Fut
Fy2
HRCp
𝓁dp
= ds for PAF installed such that entire point
is located behind far side of embedment
material
= Nominal shank diameter
= Actual diameter of washer or fastener
head in contact with retained substrate
≤ 0.60 in. (15.2 mm) in computation
= Modulus of elasticity of steel
= Base stress parameter
= 66,000 psi (455 MPa or 4640 kg/cm2)
= Tensile strength of member in contact
with PAF head or washer
= Tensile strength of member not in contact
with PAF head or washer
= Tensile strength of hardened PAF steel
= Tensile strength of non-hardened PAF
steel
= Yield stress of member not in contact
with PAF head or washer
= Rockwell C hardness of PAF steel
= PAF point length. See Figure 8.37
Figure 8.37
Pnb = Nominal bearing and tilting strength per
PAF
Pnos = Nominal pull-out strength in shear per
PAF
Pnot = Nominal pull-out strength in tension per
PAF
Pnov = Nominal pull-over strength per PAF
Pnt = Nominal tensile strength per PAF
Pntp = Nominal tensile strength of PAF
Pnv = Nominal shear strength per PAF
Pnvp = Nominal shear strength of PAF
t1 = Thickness of member in contact with PAF
head or washer
t2 = Thickness of member not in contact with
PAF head or washer
tw = Steel washer thickness
J5.1 Minimum Spacing, Edge and End Distances
The minimum center-to-center spacing of the power-actuated
fasteners (PAFs) and the minimum distance from the center of
the fastener to any edge of the connected part, regardless of the
direction of the force, are provided by Table 8.7.
Geometric Variables in Power-Actuated Fasteners (PAFs)
POWER-ACTUATED FASTENERS
Table 8.7 Minimum Required Edge and Spacing
Distances in Steel1.417
PAF Shank
Diameter, ds , in.
(mm)
Minimum PAF
Spacing in.
(mm)
Minimum Edge
Distance in.
(mm)
0.106 (2.69) ≤ ds <
0.200 (5.08)
0.200 (5.08) ≤ ds <
0.206 (5.23)
1.00 (25.4)
0.50 (12.7)
1.60 (40.6)
1.00 (25.4)
J5.2.3 Pull-Over Strength
The nominal pull-over strength, nov, is permitted to be computed
in accordance with Eq. (8.69), and the following safety factor
or resistance factors shall be applied to determine the available
strength:
𝑃nov = 𝛼w 𝑡1 d′w 𝐹u1
(8.69)
Ω = 3.00 (ASD)
𝜙 = 0.50 (LRFD)
= 0.40 (LSD)
where
J5.2 Fasteners in Tension
The available tensile strength per PAF is the minimum of the
available strengths determined by the applicable J5.2.1 through
J5.2.3. The washer thickness, tw , limitations discussed in J4
shall apply, except that for tapered head fasteners, the minimum
thickness, tw , is to be not less than 0.039 in. (0.991 mm). The
thickness of collapsible premounted top-hat washers is not to
exceed 0.020 in. (0.508 mm).
J5.2.1 Tension Strength of Power-Actuated Fasteners
The nominal tension strength of PAFs, Pntp , is permitted to be
calculated in accordance with Eq. (8.67), and the following
safety factor or resistance factors are be applied to determine
the available strength:
𝑃ntp = (𝑑∕2)2 π𝐹uh
333
𝛼 w = 1.5 for screw-, bolt-, nail-like flat heads or
simple PAF, with or without head washers
(see Figures 8.37(a) and 8.37(b))
= 1.5 for threaded stud PAFs and for PAFs
with tapered standoff heads that achieve
pull-over by friction and locking of the
pre-mounted washer (see Figure 8.37(c)),
with a/ds ratio of no less than 1.6 and
(a – ds ) of no less than 0.12 in. (3.1 mm)
= 1.25 for threaded stud PAFs and for PAFs
with tapered standoff heads that achieve
pull-over by friction and locking of
pre-mounted washer (see Figure J5-1(c)),
with a/ds ratio of no less than 1.4 and
(a – ds ) of no less than 0.08 in. (2.0 mm)
= 2.0 for PAFs with collapsible spring washer
(see Figure 8.37(d))
(8.67)
J5.3 Power-Actuated Fasteners (PAFs) in Shear
Ω = 2.65 (ASD)
The available shear strength shall be the minimum of the available strengths determined by the applicable sections 8.6.1.3.1
through 8.6.1.3.5.
𝜙 = 0.60 (LRFD)
= 0.50 (LSD)
Fuh in Eq. 8.67 is to be calculated with Eq. (8.68). Alternatively, for fasteners with HRCp of 52 or more, Fuh is permitted
to be taken as 260,000 psi (1790 MPa).
𝐹uh = 𝐹bs 𝑒(HRC𝑝 ∕40)
(8.68)
where
J5.3.1 Shear Strength of Power-Actuated Fasteners
The nominal shear strength of PAFs, Pnvp , is permitted to be
computed in accordance with Eq. (8.70), and the safety factor
and resistance factors shall be applied to determine the available
strength:
𝑃nvp = 0.6(𝑑∕2)2 𝜋𝐹uh
(8.70)
𝑒 = 2.718
Ω = 2.65 (ASD)
𝜙 = 0.60 (LRFD)
J5.2.2 Pull-Out Strength
The nominal pull-out strength, Pnot, shall be determined
through independent laboratory testing with the safety factor
or the resistance factor determined in accordance with Section
K2 of the specification. Alternatively, for connections with the
entire PAF point length, 𝓁dp , below t2, the following safety
factor or resistance factors are permitted to determine the
available strength:
Ω = 4.00 (ASD)
𝜙 = 0.40 (LRFD)
= 0.30 (LSD)
= 0.55 (LSD)
where
Fuh is determined in accordance with Section J5.2.1.
J5.3.2 Bearing and Tilting Strength
For PAFs embedded such that the entire length of PAF point
length, 𝓁dp , is below t2 , the nominal bearing and tilting strength,
Pnb, is permitted to be computed in accordance with Eq. (8.71),
and the following safety factor or resistance factors shall be
applied to determine the available strength:
𝑃nb = 𝛼b 𝑑s 𝑡1 𝐹u1
(8.71)
334
8
CONNECTIONS
Ω = 2.05 (ASD)
𝜙 = 0.80 (LRFD)
= 0.65 (LSD)
where
𝛼 b = 3.7 for connections with PAF types, as shown
in Figs. 8.37(c) and 8.37(d)
= 3.2 for other types of PAFs
Eq. J5.3.2-1 shall apply for connections within the following
limits:
1. 𝑡2 ∕𝑡1 ≥ 2,
2. t2 ≥ 1/8 in. (3.18 mm), and
3. 0.146 in. (3.71 mm) ≤ ds ≤ 0.177 in. (4.50 mm).
J5.3.3 Pull-Out Strength in Shear
For PAFs driven in steel through a depth of at least 0.6t2 ,
the nominal pull-out strength, Pnot , in shear is permitted to
be computed in accordance with Eq. (8.72), and the following
safety factor and the resistance factors are to be applied to determine the available strength:
𝑃nos =
1.8 0.2
𝑡2 (𝐹𝑦2 𝐸 2 )1∕3
𝑑ae
30
𝜙 = 0.60 (LRFD)
= 0.50 (LSD)
Eq. J5.3.3-1 shall apply for connections within the following
limits:
1. 0.113 in. (2.87 mm) ≤ t2 ≤ 3/4 in. (19.1 mm), and
2. 0.106 in. (2.69 mm) ≤ ds ≤ 0.206 in. (5.23 mm).
For PAF design examples see Refs. 1.428 and 1.434.
OTHER FASTENERS
The 2019 edition of the AISI North American Specification
provides design provisions only for welded connections
(Section 8.3), bolted connections (Section 8.4), screw
connections (Section 8.5) and power-actuated connections
(Section 8.6). There are a number of other types of fasteners
which are used in cold-formed steel construction. The
following provides a brief discussion on other fasteners.
8.7.1
In the design of a joint using blind rivets, the following
general recommendations may be used:
(8.72)
Ω = 2.55 (ASD)
8.7
1. Pull-Stem Rivets. As shown in Fig. 8.38, pull-stem
rivets can be subdivided into three types:
a. Self-Plugging Rivets. The stem is pulled into but
not through the rivet body and the projecting end is
removed in a separate operation.
b. Pull-Through Rivets. A mandrel or stem is pulled
completely out, leaving a hollow rivet.
c. Crimped-Mandrel Rivets. A part of the mandrel
remains as a plug in the rivet body.
2. Explosive Rivets. Explosive rivets have a chemical
charge in the body. The blind end is expanded by
applying heat to the rivet head.
3. Drive-Pin Rivets. Drive-pin rivets are two-piece rivets
consisting of a rivet body and a separate pin installed
from the head side of the rivet. The pin, which can be
driven into the rivet body by a hammer, flares out the
slotted ends on the blind side.
Rivets
Blind rivets and tubular rivets are often used in cold-formed
steel construction. They are used to simplify assembly,
improve appearance, and reduce the cost of connection.
8.7.1.1 Blind Rivets8.3 Based on the method of setting,
blind rivets can be classified into pull-stem rivets, explosive
rivets, and drive-pin rivets:
1. Edge Distance. The average edge distance is two times
the diameter of the rivet. For lightly loaded joints, the
distance can be decreased to one and a half diameters; for heavily loaded joints, an edge distance of three
diameters may be needed.
2. Spacing. The spacing of rivets should be three times the
diameter of the rivet. It may be desirable to decrease or
increase the spacing depending upon the nature of the
load.
3. Tension and Bearing Stresses. The tension stress on
the net section and the bearing stress may be determined
by the method used for bolted connections.
4. Shear Stress. The shear stress on rivets should be
obtained from the manufacturer.
8.7.1.2 Tubular Rivets8.3 Tubular rivets are also often
used to fasten sheet metal. The strength in shear or
compression is comparable to that of solid rivets. Nominal
body diameters range from 0.032 to 0.310 in. (0.8 to 7.9 mm).
1
The corresponding minimum lengths range from 32
to 14 in.
(0.8 to 6.4 mm). When tubular rivets are used to join heavyand thin-gage stock, the rivet head should be on the side of
the thin sheet.
8.7.2
Press Joints and Rosette Joints
8.7.2.1 Press Joints Press joining is a relatively new technique for joining cold-formed steel sections. It has many
advantages over conventional connection techniques.8.64,8.65
OTHER FASTENERS
335
Figure 8.38 Types of blind rivets and methods of setting8.3: (a) pull-stem rivets; (b) explosive
rivets; (c) drive-pin rivets.
steel sheets to be
connected
punch
die
shearing of metal
1.
2.
Figure 8.39
lateral deformation of
steel as die spreads
3.
finished press join
4.
Sequence of forming press joint.8.65
The joint is formed using the parent metal of the sections to
be connected. The tools used to form a press joint consist
of a male and female punch and die. Figure 8.39 shows the
sequence of forming a press joint.
Press joining can be used for fabrication of beams, studs,
trusses, and other structural systems. The structural strength
and behavior of press joints and fabricated components and
systems have been studied recently by Pedreschi, Sinha,
336
8
CONNECTIONS
Table 8.8
Rupture
Figure 8.40
Safety Factors and Resistance Factors for
Connection Type
Ω (ASD)
𝜙 (LRFD)
𝜙 (LSD)
Welds
Bolts
Screws and
Power-Actuated
Fasteners
2.50
2.22
3.00
0.60
0.65
0.50
0.75
0.75
0.75
Rosette joint.8.87
J6 Rupture
The provisions of this section shall apply to steel-to-steel
welded, bolted, screw, and power-actuated fastener (PAF)
connections within specified limitations. The design criteria
of this section shall apply where the thickness of the thinnest
connected part is 3/16 in. (4.76 mm) or less. For connections
where the thickness of the thinnest connected part is greater than
3/16 in. (4.76 mm), the following specifications and standards
shall apply:
1. ANSI/AISC 360 for the United States and Mexico, and
2. CSA S16 for Canada
Figure 8.41
Rosette-joining process.8.87
Davies, and Lennon at Edinburgh University.8.64,8.65,8.84–8.86
8.7.2.2 Rosette Joints Rosette joining (Fig. 8.40) is also
a new automated approach for fabricating cold-formed steel
components such as stud wall panels and roof trusses.8.87,8.88
It is formed in pairs between prefabricated holes in one
jointed part and collared holes in the other part. The joining
process is shown in Fig. 8.41.
During recent years, the strength and behavior of the
Rosette joints and the fabricated thin-walled sections
have been investigated by Makelainen, Kesti, Kaitila, and
Sahramaa at the Helsinki University of Technology.8.87 The
tests were compared with the values calculated according
to the 1996 edition of the AISI Specification supported
by a distortional buckling analysis on the basis of the
Australian/New Zealand Standard.
8.8
RUPTURE FAILURE OF CONNECTIONS
In the design of connections, consideration should also be
given to the rupture strength of the connection along a plane
through the fasteners. In 2016, the AISI Specification was
revised to include t provisions for rupture strength.1.417
For connection types utilizing welds or bolts, the nominal
rupture strength, Rn, shall be the smallest of the values obtained
in accordance with Sections J6.1, J6.2, and J6.3, as applicable.
For connection types utilizing screws and PAFs, the nominal
rupture strength, Rn, shall be the lesser of the values obtained
in accordance with Sections J6.1 and J6.2, as applicable.
The corresponding safety factor and resistance factors given
in Table 8.8 shall be applied to determine the allowable strength
or design strength in accordance with the applicable design
method in Section B3.2.1, B3.2.2, or B3.2.3.
J6.1 Shear Rupture
The nominal shear rupture strength, Pnv , shall be calculated in
accordance with Eq. (8.73).
𝑅nv = 0.6𝐹u 𝐴nv
(8.73)
where Fu = Tensile strength of connected part as specified
in Section A3.1 or A3.2
Anv = Net area subject to shear (parallel to force):
For a connection where each individual fastener pulls through
the material towards the limiting edge individually:
𝐴nv = 2n 𝑡enet
(8.74)
where n = Number of fasteners on critical cross-section
t = Base steel thickness of section
enet = Clear distance between end of material and
edge of fastener hole or weld
I- OR BOX-SHAPED COMPRESSION MEMBERS MADE BY CONNECTING TWO C-SECTIONS
337
For a beam-end connection where one or more of the flanges
are coped:
(8.75)
𝐴nv = (ℎwc –𝑛b 𝑑h )𝑡
where
hwc = Coped flat web depth
nb = Number of fasteners along failure path being
analyzed
dh = Diameter of hole
t = Thickness of coped webJ6.2 Tension Rupture
The nominal tensile rupture strength, Pnt , shall be calculated in
accordance with Eq. (8.76) as follows:
𝑃nt = 𝐹u 𝐴e
where
where
where
(8.77)
Usl = Shear lag factor determined in Table 8.9
Ant = Net area subject to tension (perpendicular to
force), except as noted in Table 8.2
= Ag − nb dh t + 𝑡Σ[s′2 ∕(4g + 2dh )]
Shear rupture of beam-end connection.1.310,1.346,1.431
equations are based on the assumption that one of the failure
paths fractures and the other yields. The shear yield stress is
taken as 0.6Fy and the shear strength is taken as 0.6Fu .
(8.76)
Ae = Effective net area subject to tension
= 𝑈sl 𝐴nt
Figure 8.42
(8.78)
Ag = Gross area of member
s’ = Longitudinal center-to-center spacing of any
two consecutive holes
g = Transverse center-to-center spacing between
fastener gage lines
nb = Number of fasteners along failure path being
analyzed
dh = Diameter of a standard hole
t = Base steel thickness of section
Fu = Tensile strength of connected part.
Table 8.9 indicates that the shear lag factor for weld connections is unchanged in the 2016 Specification.
8.9 I- OR BOX-SHAPED COMPRESSION
MEMBERS MADE BY CONNECTING TWO
C-SECTIONS
I-sections fabricated by connecting two C-sections back to
back are often used as compression members in cold-formed
steel construction. In order to function as a single compression member, the C-sections should be connected at a close
enough spacing to prevent buckling of individual C-sections
about their own axes parallel to the web at a load equal to
or smaller than the buckling load of the entire section. For
this reason, Section I1.2 of the AISI Specification limits the
maximum longitudinal spacing of connections to
𝑆max =
where
smax =
L =
rI =
J6.3 Block Shear Rupture
The nominal block shear rupture strength, Pnr , shall be determined as the lesser of the following:
where
𝑃nr = 0.6𝐹y Agv + 𝑈bs 𝐹u Ant
(8.79)
𝑃nr = 0.6𝐹u 𝐴nv + 𝑈bs 𝐹u 𝐴nt
(8.80)
Agv = Gross area subject to shear (parallel to force)
Anv = Net area subject to shear (parallel to force)
Ant = Net area subject to tension (perpendicular to
force), except as noted in Table J6.2-1
Ubs = Nonuniform block shear factor
= 0.5 for coped beam shear conditions with
more than one vertical row of connectors
= 1.0 for all other cases
Fy = Yield stress of connected part
Fu = Tensile strength of connected part
In some connections, a block of material at the end of the
member may tear out, as shown in Fig. 8.43. The design
rcy =
Lrcy
(8.81)
2𝑟1
maximum permissible longitudinal
spacing of connectors
unbraced length of compression member
radius of gyration of I-section about the
axis perpendicular to the direction in
which buckling would occur for the given
conditions of end support and
intermediate bracing
radius of gyration of one C-section about
its centroidal axis parallel to the web
This requirement ensured that the slenderness ratio of the
individual C-section between connectors is less than or equal
to one-half of the slenderness ratio of the entire compression
member in the case that any one of the connectors may be
loosened or ineffective.
Box-shaped sections made by connecting two C-sections
tip to tip are also often found in use in cold-formed steel
structures due to the relatively large torsional rigidities and
their favorable radius of gyration about both principal axes.
The foregoing requirement for maximum spacing of connectors for I-shaped members was also applicable to box-type
compression members made by C-sections tip to tip, even
though it is not specified in the AISI Specification.1.310
338
8
CONNECTIONS
Table 8.9 Shear Lag Factors for Connections to Tension Members1.417
Description of Element
Shear Lag Factor, Usl
(1) For flat sheet connections not having staggered
hole patterns
(2) For flat sheet connections having staggered
hole patterns
(3) For other than flat sheet connections
(a) When load is transmitted only by transverse
welds
(b) When load is transmitted directly to all the
cross- sectional elements
(c) For connections of angle members not meeting
(a) or (b) above
𝑈sl = 0.9 + 0.1 d∕s
𝑈sl = 1.0
𝑈sl = 1.0 and
Ant = Area of the directly connected elements
𝑈sl = 1.0
For a welded angle:
𝑈sl = 1.0 − 1.20 x∕L ≤ 0.9
but Usl shall not be less than 0.4.
For a bolted angle: 𝑈sl =
(d) For connections of channel members not
meeting (a) or (b) above
1
0.5𝑏1
2𝑥
1.1 +
+
𝑏2 + 𝑏1
𝐿
For a welded channel:
𝑈sl = 1.0 − 0.36 x∕L ≤ 0.9
but Usl shall not be less than 0.5.
For a bolted channel: 𝑈sl =
1.1 +
Notes:
1
𝑏𝑓
𝑏𝑤 + 2𝑏𝑓
+
𝑥
𝐿
𝑥 = Distance from shear plane to centroid of
cross-section (Fig. 8.29)
L = Length of longitudinal weld or length of
connection (Fig. 8.29)
s = Sheet width divided by number of bolt holes
in cross-section being analyzed
d = Nominal bolt diameter
b1 = Out-to-out width of angle leg not connected
b2 = Out-to-out width of angle leg connected
bf = Out-to-out width of flange not connected
bw = Out-to-out width of web connected
In 2016, for two cross sections in contact, Section I1.2 of
the Specification stipulates the use of a modified slenderness
ratio as defined in Section 5.8 in this volume when computing
the available axial strength of a compression member. When
using Section I1.2, the following fastener requirements are
to be satisfied:
1. The intermediate fastener or spot weld spacing, a, is
limited such that a/ri does not exceed one-half the
governing slenderness ratio of the built-up member.
2. The ends of a built-up compression member are
connected by a weld having a length not less than
the maximum width of the member or by connectors
spaced longitudinally not more than four diameters
apart from a distance equal to 1.5 times the maximum
width of the member.
3. The intermediate fastener(s) or weld(s) at any longitudinal member tie location are capable of transmitting
a force in any direction of 2.5% of the nominal axial
strength (compressive resistance) of the built-up
member.
Ref. 1.432 provides the following exception to the above
requirement 2. Where a built-up axial load bearing section
I- OR BOX-SHAPED COMPRESSION MEMBERS MADE BY CONNECTING TWO C-SECTIONS
Figure 8.43
339
Block shear rupture in tension.1.310,1.346,1.431
comprised of two studs oriented back-to-back forming an
I-shaped cross-section is seated in a track in accordance with
the requirements of Section C3.4.3 and the top and bottom
end bearing detail of the studs consists of support by steel or
concrete components with adequate strength and stiffness to
preclude relative end slip of the two built-up stud sections,
the compliance with the end connection provisions is not
required.
Example 8.6 In Example 5.2 fasteners were spaced 12 in.
on center to achieve the built-up member shown in Figure
5. Fastener requirement 1 above was evaluated in Example
5. Evaluate fastener requirements 2 and 3 if the fastener
is a 1-in.- long flare V-groove weld at each flange to web
junction. Assume Fxx = 60 ksi.
SOLUTION
1. The maximum width of the member is the web depth
8 in. Therefore at each end of the member 8 in. of weld
is required.
2. The intermediate 1-in. groove welds must be capable
of transmitting a force of 2.5% of the nominal axial
strength, Pn . From Example 5.2, Pn = 46.07 kips.
Thus, the required transmitting force is 0.025 × 46.07
kips = 1.15 kips. The welds must be capable of
transmitting this force in any direction or this implies
that the welds must be evaluated for both the transverse
loading and longitudinal loading in accordance with
Section J2.5 of the AISI North American Specification:
a. Transverse loading (for 1 in. of weld length):
𝑃n = 0.833tLF u = 0.833 × 0.075 × 1 × 45
= 2.81 kips
For the ASD method,
𝑃
2.81
𝑃a = n =
= 1.10 kips per weld × 2
Ω
2.55
= 2.20 kips > 1.15 kips
For the LRFD method,
𝜙𝑃n = 0.60 × 2.81 = 1.69 kips per weld × 2
= 3.37 kips > 1.15 kips
b. Longitudinal loading (for 1 in. of weld length):
3
1
R=
+ 0.075 = 0.1688 in. 𝑅 <
32
2
Therefore 𝑡w = 12 𝑅 = 0.0844 in Because t ≤ tw
< 2t,
𝑃n = 0.75tLF u = 0.75 × 0.075 × 1 × 45
= 2.53 kips
340
8
CONNECTIONS
For the ASD method,
𝑃
2.53
𝑃a = n =
= 0.90 kip per weld × 2
Ω
2.80
= 1.80 kips > 1.15 kips
For the LRFD method,
𝜙𝑃n = 0.55 × 2.53 = 1.39 kips per weld × 2
= 2.78 kips > 1.15 kips
8.10 I-BEAMS MADE BY CONNECTING TWO
C-SECTIONS
In cold-formed steel construction, I-beams are often fabricated from two C-sections back to back by means of two
rows of connectors located close to both flanges. For this type
of I-beam, Section I1.1 of the AISI North American Specification includes the following limitations on the maximum
longitudinal spacing of connectors:
𝐿 2gT s
𝑠max =
≤
(8.82)
6
mq
where L = span of beam
g = vertical distance between rows of
connectors nearest top and bottom flanges
Ts = available strength of connectors in tension
q = design load (factored load) on the beam
for spacing of connectors (use nominal
loads for ASD, factored loads for LRFD)
m = distance from shear center of one
C-section to midplane of its web
Figure 8.44 Tensile force developed in the top connector for
C-section.
The maximum spacing of connectors required by Eq.
(8.82) is based on the fact that the shear center is neither
coincident with nor located in the plane of the web and that
when a load Q is applied in the plane of the web, it produces
a twisting moment Qm about its shear center, as shown in
Fig. 8.44. The tensile force of the top connector Ts can then
be computed from the equality of the twisting moment Qm
and the resisting moment Ts g, that is,
Qm = 𝑇s 𝑔
(8.85)
or
Qm
(8.86)
𝑔
Considering that q is the intensity of the load and that s is
the spacing of connectors, then the applied load is Q = qs/2.
The maximum spacing smax in Eq. (8.82) can easily be
obtained by substituting the above value of Q into Eq. (8.86).
The determination of the load intensity q is based upon the
type of loading applied to the beam:
𝑇s =
1. For a uniformly distributed load,
For simple C-sections without stiffening lips at the outer
edges,1.431∗
𝑤2f
𝑚=
(8.83)
2𝑤f + 𝑑∕3
For C-sections with stiffening lips at the outer edges,
(
[
)]
𝑤f dt
4𝐷2
𝑚=
𝑤f 𝑑 + 2𝐷 𝑑 −
(8.84)
4𝐼𝑥
3𝑑
where wf = projection of flanges from inside face of
web (for C-sections with flanges of
unequal widths, wf shall be taken as the
width of the wider flange)
d = depth of C-section or beam
t = thickness of C-section
D = overall depth of lip
Ix = moment of inertia of one C-section about
its centroid axis normal to the web
∗ See
Appendix B for the location of the shear center.
𝑞 = 3𝑤′
considering the fact of possible uneven loads.
2. For concentrated load or reaction,
𝑃
𝑞=
𝑁
′
where w = uniformly distributed load based on
nominal loads for ASD, factored loads
for LRFD
= concentrated load or reaction based on
nominal loads for ASD, factored loads
for LRFD
N = length of bearing
If the length of bearing of a concentrated load or reaction
is smaller than the spacing of the connectors (N < s), the
required design strength of the connectors closest to the load
or reaction is
𝑃𝑚
(8.87)
𝑇r = s
2𝑔
I-BEAMS MADE BY CONNECTING TWO C-SECTIONS
341
Figure 8.46 Example 8.7.
The radii of gyration of the box-shaped section (6 × 5 ×
0.105 in.) about the x and y axes are
Figure 8.45
Example 8.6.
where Ps is a concentrated load (factored load) or reaction
based on nominal loads for ASD, factored loads for LRFD.
It should be noted that the required maximum spacing
of connectors, smax , depends upon the intensity of the load
applied at the connection. If a uniform spacing of connectors
is used over the entire length of the beam, it should be determined at the point of maximum load intensity. If this procedure results in uneconomically close spacing, either one of
the following methods may be adopted,
1. The connector spacing may be varied along the beam
length according to the variation of the load intensity.
2. Reinforcing cover plates may be welded to the flanges
at points where concentrated loads occur. The strength
in shear of the connectors joining these plates to the
flanges shall then be used for Tr , and the depth of the
beam can be used as g.
In addition to the above considerations on the required
strength of connectors, the spacing of connectors should not
be so great as to cause excessive distortion between connectors by separation along the top of flange.
Example 8.7 Determine the maximum longitudinal
spacing of welds for joining two 6 × 2 12 × 0.105-in. channels tip to tip to make a box-shaped section (Fig. 8.45) for
use as a simply supported column member. Assume that the
column length is 10 ft.
SOLUTION
Using the method described in Chapters( 3–5, the radius
)
of gyration of the single-channel section 6 × 2 12 × 0.105
about the y axis is
𝑟cy = 0.900 in.
𝑟𝑥 = 2.35 in.
𝑟𝑦 = 1.95 in.
Since ry < rx , the governing radius of gyration for the
box-shaped section is r = 1.95 in. Based on Eq. (8.82), the
maximum longitudinal spacing of welds is
Lrc𝑦
(10 × 12)(0.900)
𝑠max =
=
= 27.7 in.
2𝑟
2 × 1.95
Use 27 in. as the maximum spacing of welds.
Example 8.8 Use the ASD and LRFD methods to determine the maximum longitudinal spacing of 14 -in. A307
bolts joining two 6 × 1 12 × 0.105-in. C-sections to form an
I-section used as a beam. Assume that the span length of the
beam is 12 ft, the applied uniform load is 0.4 kip/ft, and the
length of the bearing is 3.5 in. (Fig. 8.46). Assume that the
dead load–live load ratio is 13 .
SOLUTION
A. ASD Method
1. Spacing of Bolts between End Supports. The
maximum permissible longitudinal spacing of 14 -in.
bolts can be determined by Eq. (8.82) as follows:
𝐿 1
= (12 × 12) = 24 in.
𝑠max =
6
6
and
2gT s
𝑠max ≤
mq
Use
( )
1
𝑔 = d − 2(𝑡 + 𝑅) − 2
4
= 6.0 − 2(0.105 + 0.1875) − 0.5 = 4.915 in.
From Table 8.5,
gross area × nominal tensile stress of bolts
𝑇s =
Ω
0.049 × 40.5
=
= 0.88 kip
2.25
342
8
CONNECTIONS
From Eq. (8.83),
From Eq. (8.85),
(1.49 −
m=
0.105)2
2(1.49 − 0.105) +
6
3
= 0.402 in.
From Eq. (8.85),
1
𝑞=
(3 × 0.40) = 0.10 kip∕in.
12
Then based on Eq. (8.82),
2(4.915)(0.88)
= 215 in.
0.402(0.1)
Since the maximum longitudinal spacing determined by
L/6 will govern the design, use 24 in. as the maximum
spacing of bolts between end supports.
2. Spacing of Bolts at End Supports. The maximum
spacing of bolts at the end supports can be computed
as follows:
2gT s
𝑠max ≤
mq
in which
P
6 × 0.4
q=
=
= 0.686 kip∕in.
𝑁
3.5
and g, Ts , and m are the same as those used in item 1
above. Then
2(4.915)(0.88)
𝑠max ≤
= 31.4 in.
0.402(0.686)
Use Smax = 24 in. Since N < Smax , from Eq. (8.87) the
required design strength of bolts closest to the reaction
is
𝑃 𝑚 0.4(6)(0.402)
𝑇r = s =
2𝑔
2(4.915)
smax ≤
= 0.098 kip < 0.88 kip
(furnished design strength of bolts) OK
B. LRFD Method
1. Spacing of Bolts between End Supports. From item
A.1, the maximum spacing of bolts is
1
𝑠max = 𝐿 = 24 in.
6
and
2gT s
𝑠max ≤
mq
where g = 4.915 in. From Table 8.5,
𝑇s = 𝜙𝑇n = (0.75)(𝐴g 𝐹nt )
= (0.75)(0.049 × 40.5) = 1.49 kips
From Eq. (8.82),
𝑚 = 0.402 in.
q=
3(1.2w′𝐷 + 1.6′𝐿 )
12
3(1.2 × 0.1 + 1.6 × 0.3)
=
= 0.15 kip∕in.
12
Based on Eq. (8.82),
𝑠max ≤
2(4.915)(1.49)
= 242.9 in.
(0.402)(0.15)
Use Smax = 24 in.
2. Spacing of Bolts at End Supports. The maximum
spacing of bolts at end supports can be computed as
follows:
2gT s
𝑠max ≤
mq
in which
𝑃
q=
𝑁
𝑃 = 6(1.2 × 0.1 + 1.6 × 0.3) = 3.6 kips
3.6
𝑞=
= 1.029 kips∕in.
3.5
2(4.915)(1.49)
= 35.4 in.
𝑠max ≤
(0.402)(1.029)
Use smax = 24 in. Since N < smax , from Eq. (8.87) the
required design strength of bolts closest to the reaction
is
𝑃 𝑚 3.6(0.402)
𝑇r = s =
2𝑔
2(4.915)
= 0.147 kip
< 1.49 kip (furnished design strength) OK
8.11 SPACING OF CONNECTIONS
IN COMPRESSION ELEMENTS
When compression elements are joined to other sections
by connections such as shown in Fig. 8.47, the connectors
must be spaced close enough to provide structural integrity
of the composite section. If the connectors are properly
spaced, the portion of the compression elements between
rows of connections can be designed as stiffened compression elements.
In the design of connections in compression elements,
consideration should be given to:
1. Required shear strength
2. Column buckling behavior of compression elements
between connections
3. Possible buckling of unstiffened elements between the
center of the connection lines and the free edge
SPACING OF CONNECTIONS IN COMPRESSION ELEMENTS
343
Figure 8.48 Example 8.8.
Figure 8.47
Spacing of connectors in composite section.1.431
The requirement of item 2 is based on the following Euler
formula for column buckling:
For this reason, Section I1.3 of the AISI North American Specification contains the following design criteria: The
spacing s in the line of stress of welds, rivets, or bolts
connecting a cover plate, sheet, or a nonintegral stiffener in
compression to another element shall not exceed:
1. that which is required to transmit the shear between the
connected parts on the basis of the design strength per
connection
specified elsewhere herein; nor
√
2. 1.16𝑡 (𝐸∕𝑓𝑐 ), where t is the thickness of the cover
plate or sheet, and fc is the stress at design load in the
cover plate or sheet; nor
3. three times the flat width, w, of the narrowest
unstiffened compression element tributary
√ to the
connections, but need not be less than 1.11𝑡 (𝐸∕𝐹y ),
√
√
if 𝑤∕𝑡 < 0.50 𝐸∕𝐹y , or 1.33𝑡 (𝐸∕𝐹y ),if 𝑤∕𝑡 ≥
√
0.50 𝐸∕𝐹y , unless closer spacing is required by 1 or
2 above.
In the case of intermittent fillet welds parallel to the direction of stress, the spacing shall be taken as the clear distance
between welds plus 12 in. (12.7 mm). In all other cases,
the spacing shall be taken as the center-to-center distance
between connections.
Exception: The requirements of this section of the Specification do not apply to cover sheets which act only as
sheathing material and are not considered as load-carrying
elements.
According to item 1, the spacing of connectors for the
required shear strength is
total shear strengths of connectors × I
s=
VQ
where s = spacing of connectors, in.
I = moment of inertia of section, in.4
V = total shear force, kips
Q = static moment of compression element
being connected about neutral axis, in.3
𝜎cr =
𝜋2𝐸
(KL∕𝑟)2
(8.88)
√
by substituting 𝜎 cr = 1.67fc , K = 0.6, L = s, and 𝑟 = 𝑡∕ 12.
This provision is conservative because the length is taken
as the center distance instead of the clear distance between
connections, and the coefficient K is taken as 0.6 instead of
0.5, which is the theoretical value for a column with fixed
end supports.
The requirement of item 3 is to ensure the spacing of
connections close enough to prevent the possible buckling
of unstiffened elements.
Additional information can be found in Refs. 8.60–8.62
and 8.92–9.94.
Example 8.9 Use the ASD method to determine the
required spacing of resistance spot welds for the compression member made of two channels and two sheets (0.105 in.
in thickness), as shown in Fig. 8.48. Assume that the member
will carry an axial load of 45 kips based on the yield point
of 33 ksi and an unbraced length of 14 ft.
SOLUTION
Using a general rule, the following sectional properties for
the combined section can be computed:
A = 3.686 in.2
𝐼𝑥 = 26.04 in.4 𝐼𝑦 = 32.30 in.4
𝑟𝑥 = 2.65 in. 𝑟𝑦 = 2.96 in.
The spacing of spot welds connecting the steel sheets to
channel sections should be determined on the basis of the
following considerations:
1. Required Spacing Based on Shear Strength. Even
though the primary function of a compression member
is to carry an axial load, as a general practice, built-up
compression members should be capable of resisting a
shear force of 2% of the applied axial load, that is,
𝑉 = 0.02(45) = 0.9 kip
344
8
CONNECTIONS
If the shear force is applied in the y direction, then the
longitudinal shear stress in line a–a is
𝑉 𝑄x
vt =
𝐼𝑥
Since
(
𝑠(vt) = 𝑠
VQ𝑥
𝐼𝑥
)
= 2 × shear strength per spot
2 × shear strength per spot × 𝐼𝑥
VQ𝑥
2(2.10)(26.04)
= 42.1 in.
0.9(9 × 0.105)(3.0 + 0.105∕2)
In the above calculation, the shear strength of resistance spot welds is obtained from Eq. (8.39) using a
safety factor of 2.50.
If the shear force is applied in the x direction, then the
shear stress is
VQ
vt =
𝐼𝑦
=
and
s=
=
𝑓c =
𝑃
45.0
=
= 12.2 ksi
𝐴 3.686
Then
√
s = 1.16(0.105)
then
s=
in which
2 × shear strength per spot × 𝐼𝑦
VQ𝑦
2(2.10)(32.30)
0.9[(6 × 0.105 × 3.0625)+
(2 × 1.385 × 0.105 × 3.808)]
= 49.6 in.
2. Required Spacing Based on Column Buckling of
Individual Steel Sheets Subjected to Compression.
Based on the AISI requirements, the maximum spacing
of welds is
√
𝐸
s = 1.16t
𝑓c
29,500
= 6.0 in.
12.2
3. Required Spacing Based on Possible Buckling of
Unstiffened Elements.
s = 3𝑤 = 3 × 0.75 = 2.25 in.
However, based on item 3 of the AISI requirements,
𝑤
0.75
=
= 7.14
𝑡
0.105
√
√
29,500
𝐸
= 0.50
0.50
= 14.95
𝐹y
33
√
Since 𝑤∕𝑡 < 0.50 𝐸∕𝐹y , the required spacing determined above need not be less than the following
value:
√
√
29,500
𝐸
1.11𝑡
= 1.11(0.105)
= 3.48 in.
𝐹y
33
Comparing the required spacings computed in items
1, 2, and 3, a spacing of 3.5 in. may be used for the
built-up section.
If the LRFD method is used in design, the shear force
applied to the member should be computed by using the
factored loads and the design shear strength should be
determined by 𝜙Pn .
CHAPTER 9
Shear Diaphragms and Roof
Structures
Figure 9.1 Shear diaphragms.
9.1 GENERAL REMARKS
A large number of research projects conducted throughout
the world have concentrated on the investigation of the
structural behavior not only of individual cold-formed
steel components but also of various structural systems.
Shear diaphragms and roof structures (including purlin roof
systems, folded-plate, and hyperbolic paraboloid roofs) are
some examples of the structural roof systems that have been
studied.
As a result of the successful studies of shear diaphragms
and roof structures accompanied by the development of new
steel products and fabrication techniques, the application
of steel structural assemblies in building construction has
increased rapidly.
In this chapter the research work and the design methods
for the use of shear diaphragms and roof structures are
briefly discussed. For details, the reader is referred to the
related references.
9.2 STEEL SHEAR DIAPHRAGMS
9.2.1 Introduction
In building construction it has been a common practice to
provide a separate bracing system to resist horizontal loads
due to wind load, blast force, or earthquake. However, steel
floor and roof panels (Fig. 1.11), with or without concrete
fill, are capable of resisting horizontal loads in addition to the
beam strength for gravity loads if they are adequately interconnected to each other and to the supporting frame.1.6,9.1,9.2
The effective use of steel floor and roof panels can therefore
eliminate separate bracing systems and result in a reduction
of building costs (Fig. 9.1).
For the same reason, wall panels can provide not only
enclosure surfaces and support normal loads but also
diaphragm action in their own planes.
In addition to the utilization of diaphragm action, steel
panels used in floor, roof, and wall construction can be
used to prevent the lateral buckling of beams and the
overall buckling of columns.4.115–4.120 Previous studies
made by Winter have shown that even relatively flexible
diaphragm systems can provide sufficient horizontal support
to prevent the lateral buckling of beams in floor and roof
construction.4.111 The load-carrying capacities of columns
can also be increased considerably if they are continuously
braced with steel diaphragms.4.116
9.2.2
Research on Shear Diaphragms
Because the structural performance of steel diaphragms
usually depends on the sectional configuration of panels,
the type and arrangement of connections, the strength
and thickness of the material, span length, loading function, and concrete fill, the mathematical analysis of shear
diaphragms is complex. At the present time, the shear
strength and the stiffness of diaphragm panels can be
determined either by tests or by analytical procedures.
Since 1947 numerous diaphragm tests of cold-formed steel
panels have been conducted and evaluated by a number of
researchers and engineers. The diaphragm tests conducted in
the United States during the period 1947–1960 were summarized by Nilson in Ref. 9.2. Those tests were primarily
sponsored by individual companies for the purpose of developing design data for the diaphragm action of their specific
panel products. The total thickness of the panels tested generally range from 0.04 to 0.108 in. (1 to 2.8 mm). Design
information based on those tests has been made available
from individual companies producing such panels.
345
Cold-Formed Steel Design, Fifth Edition. Wei-Wen Yu, Roger A. LaBoube and Helen Chen.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
346
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
In 1962 a research project was initiated at Cornell
University under the sponsorship of the AISI to study the
performance of shear diaphragms constructed of corrugated
and ribbed deck sections of thinner materials, from 0.017 to
0.034 in. (0.4 to 0.9 mm) in total thickness. The results of
diaphragm tests conducted by Luttrell and Apparao under
the direction of George Winter were summarized in Refs.9.3,
9.4, and 9.5. Recommendations on the design and testing of
shear diaphragms were presented in the AISI publication,
“Design of Light Gage Steel Diaphragms,” which was issued
by the institute in 1967.9.6
Since 1967 additional experimental and analytical studies
of steel shear diaphragms have been conducted throughout
the world. In the United States, research projects on
this subject have been performed by Nilson, Ammar,
and Atrek,9.7–9.9 Luttrell, Ellifritt, and Huang,9.10–9.13
Easley and McFarland,9.14–9.16 Miller,9.17 Libove, Wu,
and Hussain,9.18–9.21 Chern and Jorgenson,9.22 Liedtke
and Sherman,9.23 Fisher, Johnson, and LaBoube,9.24–9.26
Jankowski and Sherman,9.90 Heagler,9.91 Luttrell,9.92 and
others. The research programs that have been carried out in
Canada include the work of Ha, Chockalingam, Fazio, and
El-Hakim9.27–9.30 and Abdel-Sayed.9.31
In Europe, the primary research projects on steel shear
diaphragms have been conducted by Bryan, Davies, and
Lawson.9.32–9.3 The utilization of the shear diaphragm
action of steel panels in framed buildings has been well
illustrated in Davies and Bryan’s book on stressed skin
diaphragm design.9.39 In addition, studies of tall buildings
using diaphragms were reported by El-Dakhakhni in Refs.
9.40 and 9.41.
In the past, shear diaphragms have been studied by
Caccese, Elgaaly, and Chen,9.96 Kian and Pekoz,9.97
Miller and Pekoz,9.98 Easterling and Porter,9.99 Serrette
and Ogunfunmi,9.100 Smith and Vance,9.101 Elgaaly and
Liu,9.102 Lucas, Al-Bermani, and Kitipornchai,9.103,9.104
Elgaaly,9.105 Lease and Easterling,9.110 and others1.419-1.422 .
References 1.269, 9.106, 9.111, 9.123, and 9.124 provide
additional design information on the design and use of shear
diaphragms.
In addition to the shear diaphragm tests mentioned above,
lateral shear tests of steel buildings and tests of gabled
frames with covering sheathing have been performed by
Bryan and El-Dakhakhni.4.113,4.114 Recent studies and design
criteria for cold-formed steel framed shear walls will be
discussed in Chapter 12. The structural behavior of columns
and beams continuously braced by diaphragms has also
been studied by Pincus, Fisher, Errera, Apparao, Celebi,
Pekoz, Winter, Rockey, Nethercot, Trahair, Wikstrom, and
others.4.115–4.120,4.135,4.136 More recently, experimental work
by Wang et al.9.112 and analytical research by Schafer and
Hiriyur 9.113 have extended the state of the art. This subject
is discussed further in Section 9.3.
In order to understand the structural behavior of shear
diaphragms, the shear strength and the stiffness of steel
diaphragms are briefly discussed in subsequent sections.
9.2.2.1 Shear Strength of Steel Diaphragms Results of
previous tests indicate that the shear strength per foot of steel
diaphragm is usually affected by the panel configuration, the
panel span and purlin or girt spacing, the material thickness
and strength, acoustic perforations, types and arrangements
of fasteners, and concrete fill, if any.
9.2.2.1.1 Panel Configuration The height of panels has
considerable effect on the shear strength of the diaphragm if
a continuous flat-plate element is not provided. The deeper
profile is more flexible than are shallower sections. Therefore
the distortion of the panel, in particular near the ends, is more
pronounced for deeper profiles. On the other hand, for panels
with a continuous flat plate connected to the supporting
frame, the panel height has little or no effect on the shear
strength of the diaphragm.
With regard to the effect of the sheet width within a panel,
wider sheets are generally stronger and stiffer because there
are fewer side laps.
9.2.2.1.2 Panel Span and Purlin Spacing Shorter span
panels could provide a somewhat larger shear strength than
longer span panels, but the results of tests indicate that the
failure load is not particularly sensitive to changes in span.
The shear strength of panels is increased by a reduction of
purlin spacing; the effect is more pronounced in the thinner
panels.
9.2.2.1.3 Material Thickness and Strength If a continuous flat plate is welded directly to the supporting frame,
the failure load is nearly proportional to the thickness of
the material. However, for systems with a formed panel, the
shear is transmitted from the support beams to the plane of
the shear-resisting element by the vertical ribs of the panels.
The shear strength of such a diaphragm may be increased by
an increase in material thickness, but not linearly.
When steels with different material properties are used, the
influence of the material properties on diaphragm strength
should be determined by tests or analytical procedures.
9.2.2.1.4 Acoustic Perforations The presence of
acoustic perforations may slightly increase the deflection of
the system and decrease the shear strength.
9.2.2.1.5 Types and Arrangement of Fasteners The
shear strength of steel diaphragms is affected not only by
STEEL SHEAR DIAPHRAGMS
347
the types of fasteners (welds, bolts, sheet metal screws, and
others) but also by their arrangement and spacing. The shear
strength of the connection depends to a considerable degree
on the configuration of the surrounding metal.
Previous studies indicate that if the fasteners are small
in size or few in number, failure may result from shearing
or separation of the fasteners or by localized bearing or
tearing of the surrounding material. If a sufficient number
of fasteners are closely spaced, the panel may fail by elastic
buckling, which produces diagonal waves across the entire
diaphragm.
The shear strength will be increased considerably by
the addition of intermediate side lap fasteners and end
connections.
due to perimeter beams and neglecting the influence of the
diaphragm acting as a web of a plate girder.
For simplicity the combined deflections due to shear
stress, seam slip, and local distortion can be determined
from the results of diaphragm tests. If shear transfer devices
are provided, the deflection due to relative movement
between marginal beams and shear web will be negligible.
The above discussion is based on the test results of shear
diaphragms without concrete fill. The use of concrete fill
will increase the stiffness of shear diaphragms considerably,
as discussed in the preceding section. When the advantage
of concrete fill is utilized in design, the designer should
consult individual companies or local building codes for
design recommendations.
9.2.2.1.6 Concrete Fill Steel panels with a concrete fill
provide a much more rigid and effective diaphragm. The
stiffening effect of the fill depends on the thickness, strength,
and density of the fill and the bond between the fill and the
panels.
The effect of lightweight concrete fill on shear diaphragms
has been studied by Luttrell.9.11 It was found that even though
this type of concrete may have a very low compressive
strength of 100–200 psi (0.7–1.4 MPa), it can significantly
improve the diaphragm performance. The most noticeable
influence is the increase in shear stiffness. The shear strength
can also be increased, but to a lesser extent.
9.2.3
9.2.2.1.7 Insulation Based on the results of full-scale
diaphragm tests, Lease and Easterling9.110 concluded that the
presence of insulation does not reduce the shear strength of
the diaphragm. Insulation thicknesses of up to 6 38 in. were
studied.
9.2.2.2 Stiffness of Shear Diaphragms In the use of shear
diaphragms, deflection depending upon the stiffness of the
shear diaphragms is often a major design criterion. Methods
for predicting the deflection of cold-formed steel panels
used as diaphragms have been developed on the basis of
the specific panels tested. In general, the total deflection of
the diaphragm system without concrete fill is found to be a
combination of the following factors9.2,9.39 :
1.
2.
3.
4.
Deflection due to flexural stress
Deflection due to shear stress
Deflection due to seam slip
Deflection due to local distortion of panels and relative
movement between perimeter beams and panels at end
connections
The deflection due to flexural stress can be determined
by the conventional formula using the moment of inertia
Tests of Steel Shear Diaphragms
In general, shear diaphragms are tested for each profile or
pattern on a reasonable maximum span which is normally
used to support vertical loads. The test frame and connections should be selected properly to simulate actual building
construction if possible. Usually the mechanical properties
of the steel used for the fabrication of the test panels should
be similar to the specified values. If a substantially different
steel is used, the test ultimate shear strength may be corrected
on the basis of Ref.9.114
During the past, cantilever, two-bay, and three-bay steel
test frames have been generally used. Another possible test
method is to apply compression forces at corners along a
diagonal. Nilson has shown that the single-panel cantilever
test will yield the same shear strength per foot as the threebay frame and that the deflection of an equivalent three-bay
frame can be computed accurately on the basis of the
single-panel test. It is obvious that the use of a cantilever test
is economical, particularly for long-span panels. References
9.42–9.44 and 9.114 contain the test procedure and the
method of evaluation of the test results.
The test frame used for the cantilever test is shown
in Fig. 9.2a and Fig. 9.2b shows the cantilever beam
diaphragm test.
The three-bay simple beam test frame is shown in
Fig. 9.3a, and Fig. 9.3b shows the test setup for a simple
beam diaphragm test.
The test results can be evaluated on the basis of the average
values obtained from the testing of two identical specimens
if the deviation from the average value does not exceed 10%.
Otherwise the testing of a third identical specimen is required
by Refs. 9.42–9.44. The average of the two lower values
obtained from the tests is regarded as the result of this series
of tests. According to Ref. 9.43, if the frame has a stiffness equal to or less than 2% of that of the total diaphragm
assembly, no adjustment of test results for frame resistance
348
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
Figure 9.2
(a) Plan of cantilever test frame.9.6 (b) Cantilever beam diaphragm test.9.1
need be made. Otherwise, the test results should be adjusted
to compensate for frame resistance.
The ultimate shear strength Su in pounds per foot can be
determined from
𝑝
𝑆𝑛 = 𝑛
(9.1)
𝑏
where (Pult )avg = average value of maximum jack loads
from either cantilever or simple beam
tests, lb
b = depth of beam indicated in Figs. 9.2a
and 9.3a, ft
The computed ultimate shear strength divided by the
proper load factor gives the allowable design shear Sdes in
pounds per linear foot. (See Fig. 9.4 for the tested ultimate
shear strength of standard corrugated steel diaphragms.)
According to Ref. 9.42, the shear stiffness G′ is to be
determined on the basis of an applied load of 0.4(Pult )avg
for use in deflection determination.∗ For the evaluation of
∗ Reference
9.44 suggests that the shear stiffness G′ is to be determined on
the basis of a reference level of 0.33(Pult )avg . If the selected load level is
beyond the proportional limit, use a reduced value less than the proportional
limit.
STEEL SHEAR DIAPHRAGMS
349
Figure 9.3 (a) Plan of simple beam test frame.9.6 (b) Simple beam diaphragm test.9.1
shear stiffness, the measured deflections at the free end of the
cantilever beam or at one-third the span length of the simple
beam for each loading increment can be corrected by the
following equations if the support movements are to be taken
into account:
[
]
{
𝑎
Δ = 𝐷3 − 𝐷1 + (𝐷2 + 𝐷4 ) for cantilever tests (9.2)
𝑏
= 12 (𝐷2 + 𝐷3 − 𝐷1 − 𝐷4 )
for simple beam tests
(9.3)
where D1 , D2 , D3 , and D4 are the measured deflections at
locations indicated in Figs. 9.2a and 9.3a and a/b is the ratio
of the diaphragm dimensions. The load-deflection curve can
then be plotted on the basis of the corrected test results.
The shear deflection for the load of 0.4(Pult )avg can be
computed from
Δ′s = Δ′ − Δ′b
(9.4)
where Δ′s = shear deflection for load of 0.4(Pult )avg
Δ′ = average value of deflections obtained from
load–deflection curves for load of 0.4(Pult )avg
Δ′b = computed bending deflection
350
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
Figure 9.5 Cross section of corrugated sheets.9.42
profile:
𝐺′ =
Figure 9.4 Tested ultimate shear strength of standard 2 12 × 12 -in.
corrugated galvanized steel diaphragms.9.42
In the computation of Δ′b the following equations may be
used for cantilever beams. The bending deflection at the free
end is
Pa3 (12)2
(9.5)
Δ′b =
3EI
For simple beam tests, the bending deflection at one-third
the span length is
Δ′b =
5Pa3 (12)2
6EI
(9.6)
In Eqs. (9.5) and (9.6),
P = 0.4(Pult )avg , lb
E = modulus of elasticity of steel, 29.5 × 106 psi
(203 GPa, 2070 GPa)
I = moment of inertia considering only perimeter
members of test frame,
= Ab2 (12)2 /2, in.4
A = sectional area of perimeter members CD and GE in
Figs. 9.2a and 9.3a, in.2
a, b = dimensions of test frame shown in Figs. 9.2a and
9.3a, ft
Finally, the shear stiffness G′ of the diaphragm can be
computed as
( )
𝑃 ∕𝑏
0.4𝑃 max 𝑎
(9.7)
=
𝐺′ = ′
𝑏
Δs ∕𝑎
Δ′s
The shear stiffness varies with the panel configuration
and the length of the diaphragm. For standard corrugated
sheets, the shear stiffness for any length may be computed
by Eq. (9.8) as developed by Luttrell9.5 if the constant K2
can be established from the available test data on the same
Et
[2(1 + 𝜇)𝑔]∕𝑝 + 𝐾2 ∕(Lt)2
(9.8)
where G′ = shear stiffness, lb/in.
E = modulus of elasticity of steel, = 29.5 × 106 psi
(203 GPa, 2070 GPa)
t = uncoated thickness of corrugated panel, in.
𝜇 = Poisson’s ratio, = 0.3
p = corrugation pitch, in. (Fig. 9.5)
g = girth of one complete corrugation, in.
(Fig. 9.5)
L = length of panels from center to center of end
fasteners, measured parallel to corrugations,
in.
K2 = constant depending on diaphragm cross
section and end-fastener spacing, in.4
Knowing G′ from the tested sheets, the constant K2 can be
computed as
[
]
Et 2(1 + 𝜇)𝑔
−
(9.9)
(Lt)2
𝐾2 =
𝐺′
𝑝
Figure 9.6 shows graphically the tested shear stiffness for
0.0198-in. (0.5-mm-) thick standard corrugated diaphragms.
Based on a review of diaphragm test methods, AISI
developed the Test Standard for Cantilever Test Method
for Cold-Formed Steel Diaphragms.9.114 In 2013 this test
standard was broadened to address both static and cyclic
loading conditions.9.114 According to Ref. 9.114, the test
frame stiffness may influence the test results and therefore
adjustments to compensate for frame resistance and stiffness
are provided.
The nominal diaphragm web shear strength, Sn , which is
the load per unit length across the full frame test, is calculated by
𝑃
(9.10)
𝑆n = n
𝑏
where
(
)
𝑃fn
⎧𝑃
−
𝑃
−
0.02
max 𝑃
⎪ max
max
⎪ 𝑃fn
(9.10𝑎)
𝑃n = ⎨if 𝑃 > 0.02
)
⎪ max ( 𝑃
fn
⎪𝑃max if
≤ 0.02
(9.10𝑏)
𝑃max
⎩
STEEL SHEAR DIAPHRAGMS
351
Figure 9.6 Tested shear stiffness for 2 12 × 12 -in.. standard corrugated steel diaphragms. Thickness
of panels = 0.0198 in.9.42
P
Pmax
Pd a
G' = Δ b
d
Pd = 0.4 Pmax
Δd
Figure 9.7
Δn
Typical load–net deflection curve.9.114
where Pmax = maximum applied load P to test frame
Pfn = load P from testing of bare frame at deflection
equal to deflection for load of Pmax for
strength
b = depth of diaphragm test frame and dimension
parallel with load, P
Equation (9.10) adjusts the test load for the contribution
of the frame stiffness.
For design the nominal diaphragm web strength is reduced
by either a safety factor or resistance factor as described
in Chapter B of AISI S310.9.124 See Section 9.2.5 in this
volume.
According to Ref. 9.114, the shear stiffness G’ is determined at the test load 𝑃d = 0.4𝑃max as illustrated by Fig. 9.7.
The shear stiffness is computed by Eq. (9.11),
(
)
𝑃d 𝑎
𝐺′ =
(9.11)
Δd 𝑏
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
(
⎧0.4𝑃
− 0.02
max − 0.4𝑃max
⎪
⎪
𝑃
if 0.4𝑃fd > 0.02
𝑃d = ⎨
max
⎪
𝑃
⎪0.4𝑃max
if 0.4𝑃fd ≤ 0.02
⎩
max
𝑃fd
0.4𝑃max
Depth, b
)
4
(9.11𝑎)
2
Load, P
Lv
Lv
CL Beam
1
The shear net deflection illustrated by Fig. 9.8 may be
determined from either diagonal measurements or rectangular measurements. Both measurement methods are
acceptable. However, the rectangular method is the historical method. Common existing analytical methods9.11,9.115
were verified using the rectangular method.
When using diagonal measurements the net shear deflection, Δn , is computed as follows:
√
𝑎2 + 𝑏2
Δn = (|Δ1 | + |Δ2 |)
(9.12)
2𝑏
Depth, b
A
CL Beam
C
D
Steel deck
panel
Tension
reaction
Bearing
reaction
Legend:
Displacement device mounted on the ground at frame
corners measuring movement of the test frame.
Arrow indicates direction of positive reading.
Figure 9.9
Rectangular displacement measurements.9.114
where Δ1, 2 = displacement measured at points 1 or 2
(see Fig. 9.8)
a = length of diaphragm test frame
b = depth of diaphragm test frame
When using the rectangular displacements (Fig. 9.9), the
net shear deflection is computed by Eq. (9.13),
(
)
(Δ2 + Δ4 )𝑎
(9.13)
Δn = Δ3 − Δ1 +
𝑏
where Δ1,2,3,4 is the displacement measured at points 1, 2,
3, or 4 (see Fig. 9.9).
The number of tests and the conditions for acceptance of
test results are described in Ref. 9.114.
Lv
2
Length, a
Lv
Lv
B
Wire
B
(9.11𝑏)
where Pd = load P at which diaphragm stiffness is
determined
Pfd = load P from testing of bare test frame at
deflection equal to deflection for load of
0.4 Pmax for stiffness
Δd = net shear deflection of diaphragm test at load
0.4 Pmax
Load, P
3
A
Length, a
where
Lv
352
1
D
C
Steel deck
panel
Tension
reaction
Bearing
reaction
Legend:
Displacement device mounted on top
flange of test frame. Arrow indicates
direction of positive reading.
Figure 9.8
Diagonal displacement measurements.9.114
9.2.4 Analytical Methods for Determining Shear
Strength and Stiffness of Shear Diaphragms
In Section 9.2.3 the test method to be used for establishing
the shear strength and stiffness of shear diaphragms was
discussed. During the past two decades, several analytical
methods have been developed for computing the shear
strength and the stiffness of diaphragms. The following five
methods are commonly used:
1. Steel Deck Institute (SDI) method9.45,9.106, 9.111,9.123
STEEL SHEAR DIAPHRAGMS
2.
3.
4.
5.
6.
Tri-Service method9.46
European recommendations9.47
Nonlinear finite-element analysis9.9
Simplified diaphragm analysis9.36
Metal Construction Association (MCA) method9.115
references AISI S3109.124 the design shear strength can be
determined as follows:
⎧ 𝑆n
for ASD method
⎪
𝑆d = ⎨ Ωd
⎪ 𝑆 d = 𝜙d 𝑆 n
for LRFD method
⎩
For details, the reader is referred to the referenced documents and publications. The North American Standard for
the Design of Profiled Steel Diaphragm Panels9.124 incorporates the analytical methods of both the SDI and MCA
methods.
With regard to the use of the European recommendations,
a design guide was prepared by Bryan and Davies in 1981.9.48
This publication contains design tables and worked examples
which are useful for calculating the strength and stiffness of
steel roof decks when acting as diaphragms.
In addition to the above publications, general design rules
on steel shear diaphragms can also be found in Ref. 9.116.
where Sd = available shear strength for diaphragm,
lb/ft.
Sn = in-plane diaphragm nominal shear strength
established by calculation or test, lb/ft
Ωd = safety factor for steel diaphragm shear as
specified in Table 9.1
𝜙d = resistance factor for steel diaphragm shear as
specified in Table 9.1
In Table 9.1, the safety factors and resistance factors are
based on the statistical studies of the nominal and mean
resistances from full-scale tests.1.346 The values in Table 9.1
reflect the fact that the quality of mechanical connectors
is easier to control than welded connections. As a result,
the variation in the strength of mechanical connectors is
smaller than that for welded connections, and their performance is more predictable. Therefore, a smaller factor of
safety, or larger resistance factor, is justified for mechanical connections.9.124 For other diaphragm system conditions
such as wood supports, structural concrete supports, structural concrete fill and insulating concrete fill, see Ref. 9.124.
For mechanical fasteners other than screws Ωd shall not
be less than Table 9.1 values for screws and 𝜙d shall not be
greater than Table 9.1 values for screws.
The safety factors for earthquake loading are slightly
larger than those for wind load due to the ductility demands
required by seismic loading.
The stress in the perimeter framing members should
be checked for the combined axial stresses due to the
9.2.5 Design of Shear Diaphragms
Steel shear diaphragms can be designed as web elements of
horizontal analogous plate girders with the perimeter framing
members acting as the flanges. The primary stress in the web
is shear stress, and in the flanges the primary stresses are axial
stresses due to bending applied to the plate girder.
It has been a general design practice to determine the
required sections of the panels and supporting beams based
on vertical loads. The diaphragm system, including connection details, needed to resist horizontal loads can then be
designed on the basis of (1) the shear strength and the stiffness of the panels recommended by individual companies for
the specific products and (2) the design provisions of local
building codes.
For the design of shear diaphragms according to Section
I2 of the AISI North American Specification,1.345 which
Table 9.1 Safety Factors and Resistance Factors for Diaphragms with Steel Supports and No
Concrete Fill9.124
Load
Type or
Combinations
Including
Wind
Earthquake
and All Others
𝑎
Limit State
Panel Buckling𝑎
Connection-Related
Connection
Type
Welds
Screws
Welds
Screws
353
Ωd
(ASD)
𝜙d
(LRFD)
𝜙d
(LSD)
Ωd
(ASD)
𝜙d
(LRFD)
𝜙d
(LSD)
2.15
2.00
3.00
2.30
0.75
0.80
0.55
0.70
0.60
0.75
0.40
0.55
2.00
0.80
0.75
Panel buckling is out-of-plane buckling and not local buckling at fasteners.
354
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
A = area of perimeter framing member, in.2
b = distance between centroids of perimeter
members, measured perpendicular to span
length of girder, ft
Figure 9.10
ragm.9.42
Portal frame building with wall and roof diaph-
gravity load and the wind load or earthquake applied to the
structure.
As shown in Fig. 9.10, the axial stress in the perimeter
framing members due to horizontal load (wind load or earthquake) can be determined by
𝐹
𝑀
𝑓=
=
𝐴
Ab
(9.15)
where f = stress in tension or compression, psi
F = force in tension or compression, lb, = M/b
M = bending moment at particular point investigated,
ft-lb
Table 9.2
Usually the shear stress in panels and the axial stress
in perimeter members of such a diaphragm assembly are
small, and it is often found that the framing members and
panels which have been correctly designed for gravity loads
will function satisfactorily in diaphragm action with no
increase in size. However, special attention may be required
at connections of perimeter framing members. Ordinary
connections may deform in the crimping mode if subjected
to heavy axial forces along connected members.
If the panels are supported by a masonry wall rather than
by a steel frame, tensile and compressive reinforcements
should be provided for flange action within the walls adjacent
to the diaphragm connected thereto.
In addition to designing shear diaphragm and perimeter
members for their strengths, the deflection of shear
diaphragms must also be considered. The total deflection of
shear diaphragms may be computed as
Δtotal = Δb + Δs
(9.16)
where Δtotal = total deflections of shear diaphragm, in.
Δb = bending deflection, in.
Δs = shear deflection, including deflection due
to seam slip and local distortion, in.
The bending deflection and shear deflection can be
computed by the formulas given in Table 9.2 for various
types of beams subjected to various loading conditions.
Deflection of Shear Diaphragms 9.42
Type of Diaphragm
Loading Condition
Simple beam (at center)
Uniform load
Load P applied at center
Load P applied at each
Cantilever beam (at free end)
1
point of span
3
Uniform load
Load P applied at free end
Δb
Δs 𝑎
5wL4 (12)2
384EI
PL3 (12)3
48EI
23PL3 (12)3
648EI
wa4 (12)3
8EI
3
Pa (12)3
3EI
wL2
8𝐺′ 𝑏
PL
4𝐺′ 𝑏
PL
3𝐺′ 𝑏
wa2
2𝐺′ 𝑏
Pa
𝐺′ 𝑏
𝑎
When the diaphragm
is constructed with two or more panels of different lengths, the term G′ b should be
∑
replaced by 𝐺𝑖′ 𝑏𝑖 ,where 𝐺𝑖′ and bi are the shear stiffness and the length of a specific panel, respectively.
STEEL SHEAR DIAPHRAGMS
In Table 9.2 the formula for determining the shear deflection of a diaphragm is similar to the method of computing
the shear deflection of a beam having relatively great depth.
It can be derived from the following equation4.45 :
Δs =
Vv
dx
∫ 𝐺′ 𝑏
(9.17)
where V = shear due to actual loads
v = shear due to a load of 1 lb acting at section
where deflection is derived
′
G = shear stiffness of diaphragm
b = width of shear diaphragm or depth of
analogous beam
In practical design, the total horizontal deflection of
a shear diaphragm must be within the allowable limits
permitted by the applicable building code or other design
provisions. The following formula for masonry walls has
been proposed by the Structural Engineers Association of
California:
Allowable deflection =
where h
t
E
f
=
=
=
=
ℎ2 𝑓
0.01Et
(9.18)
unsupported height of wall, ft
thickness of wall, in.
modulus of elasticity of wall material, psi
allowable compressive stress of wall
material, psi
9.2.6 Special Considerations
The following are several considerations which are essential
in the use of steel panels as shear diaphragms.
1. If purlins and girts are framed over the top of perimeter
beams, trusses, or columns, the shear plane of the panels
may cause tipping of purlin and girt members by eccentric
loading. For this case, rake channels or other members should
be provided to transmit the shear from the plane of the panels
to the flanges of the framing member or to the chords of the
truss.
2. Consideration should be given to the interruption
of panels by openings or nonstructural panels. It may be
assumed that the effective depth of the diaphragm is equal to
the total depth less the sum of the dimensions of all openings
or nonstructural panels measured parallel to the depth of
the diaphragm. The type of panel-to-frame fasteners used
around the openings should be the same as, and their spacing
355
equal to or less than, that used in the tests to establish the
diaphragm value.
3. When panels are designed as shear diaphragms, a note
shall be made on the drawings to the effect that the panels
function as braces for the building and any removal of the
panels is prohibited unless other special separate bracing is
provided.
4. The performance of shear diaphragms depends
strongly on the type, spacing, strength, and integrity of
the fasteners. The type of fasteners used in the building
should be the same as, and their spacing not larger than, that
used in the test to establish the diaphragm value.
5. Diaphragms are not effective until all components are
in place and fully interconnected. Temporary shoring should
therefore be provided to hold the diaphragms in the desired
alignment until all panels are placed, or other construction
techniques should be used to make the resulting diaphragm
effective. Temporary bracing should be introduced when
replacing panels.
6. Methods of erection and maintenance used for the
construction of shear diaphragms should be evaluated
carefully to ensure proper diaphragm action. Proper
inspection and quality control procedures would be
established to ensure the soundness and spacing of the
connections.
For other guidelines on practical considerations, erection,
inspection, and other design information, see Refs. 9.45,
9.111, 9.123 and 9.124. Design examples are also provided in
Design Examples for the Design of Profiled Steel Diaphragm
Panels Based on AISI 310-16, AISI D310-17.9.126
Example 9.1 Use the ASD method to design a longitudinal
bracing system by using steel shear diaphragms for a mill
building as shown in Figs. 9.11a and 9.11b. Assume that
a wind load of 20 psf is applied to the end wall of the
building and that 9/16” x 2 1/2” Form Deck steel sheets having
a design thickness of 0.0179 in. are used as roof and wall
panels.
SOLUTION
A. Alternate I. Longitudinal X-bracing is usually provided
for a mill building in the planes of the roof, side walls, and
lower chord of the truss,∗ as shown in Figs. 9.8c and d.
The intent of this example is to illustrate the use of shear
diaphragms in the planes of the roof and side walls instead of
∗ See
Alternate II, in which the X-bracing is eliminated in the plane of the
lower chord of the truss.
356
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
Figure 9.11 (a) Mill building. (b) End elevation. (c) Wind bracing in planes of roof and side walls,
side elevation. (d) Bracing in plane of lower chord. (e) Assumed area of wind load to be carried at A,
B, C, and D. (f) Shear diaphragms in planes of roof and side walls, side elevation. (g) Assumed area
of wind load to be carried in planes of roof and bottom chord of truss. (h) End wall columns run all
the way to roof plane.
the X-bracing system. The design of the roof truss and other
structural members is beyond the scope of this example.
1. Wind Panel Loads. The wind panel loads at A, B, C, and
D can be computed as follows based on the assumed area
in Fig. 9.11e:
(
)
1
𝑊A = 20 9.375 × 12.5 + × 8.385 × 4.193 (3)
2
= 2700 lb(4)
STEEL SHEAR DIAPHRAGMS
𝑊B = 20(16.77 × 4.193) = 1410 lb(5)
(
)
1
𝑊C = 20 8.385 × 4.193 + × 11.25 × 7.5 (6)
2
= 1550 lb(7)
(
1
𝑊D = 20 20.635 × 12.5 + 5.625 × 7.5 + × 7.5(8)
2
)
× 15.0 = 7130 lb
2. Shear Diaphragm in Plane of Roof. Considering that the
planes of the side walls and the lower chord of the truss
are adequately braced, the wind loads to be resisted by the
roof panels used as a shear diaphragm are 12 WC and WB as
indicated in Fig. 9.11f. Consequently, the shear developed
along the eave struts is
𝑣=
775 + 1410 2185
=
= 15.6 lb∕ft
140
140
From the Diaphragm Tables, Section 7 of Steel Deck
on Cold-Formed Steel Framing,9.127 the nominal shear
strength for the corrugated sheets having a base metal
thickness of 0.0119 in. is given as 295 lb/ft. Using a
safety factor of 2.00 as recommended by AISI for
screw connectors, the allowable shear strength for the
design is
295
= 147 lb∕ft
𝑆des =
2.00
Since the allowable shear is much larger than the actual
shear value of 15.6 lb/ft developed in the roof due to the
wind load, the roof panels are adequate to resist the wind
load applied to the end wall, even though no intermediate
fasteners are provided. Usually intermediate fasteners are
used for roof panels, and as a result, additional strength
will be provided by such fasteners.
3. Shear Diaphragm in Plane of Side Wall. As far as the
shear diaphragms in the planes of the side walls are
concerned, the total load to be resisted by one side wall as
shown in Fig. 9.11f is
1
𝑃 = 𝑊A + 𝑊B + 𝑊C + 𝑊D
2
= 2700 + 1410 + 775 + 7130 = 12,015 lb
The effective diaphragm width 𝑏eff is the length of
the building with the widths of doors and windows
subtracted. This is based on the consideration that the
wall panels are adequately fastened to the perimeter
members around openings, that is,
𝑏eff = 140 − (16.0 + 3 × 7.5 + 7.0 + 5.0)
= 89.5 ft
357
Therefore the shear to be resisted by the diaphragm is
𝑣=
12,015
𝑃
=
= 134 lb∕ft
𝑏eff
89.5
or the required static ultimate shear resistance should be
𝑆u = 𝑆d × SF = 134 × 2.0 = 268 lb∕ft
Since the average nominal shear strength for the
0.0179-in.-thick form deck spanning at 3 ft is 295 lb/ft,
which is larger than the computed value of 268 lb/ft,
the corrugated sheets are adequate for shear diaphragm
action. In determining the wind load to be resisted in the
planes of roof and side walls, assumptions may be made
as shown in Fig. 9.11g. Based on this figure, the wind
load to be resisted by the shear diaphragm in the plane of
the roof is 2250 lb, which is slightly larger than the load
of 2185 lb used previously. The total load to be used for
the design of the shear diaphragm in the planes of side
walls is the same as the load computed from Fig. 9.11f.
4. Purlin Members. It should be noted that the shear force in
the plane of roof panels can cause the tipping of purlins
due to eccentricity. Rake channels or other means may be
required to transmit the shear force from the plane of roof
panels to chord members. This can become important in
short, wide buildings if purlins are framed over the top of
trusses.
B. Alternate II. When end-wall columns run all the way to
the roof plane, as shown in Fig. 9.8h, there is no need for
X-bracing in the plane of the bottom chord. For this case, the
force to be resisted by one side of the roof diaphragm is
1
𝑃 = 20 × [(12.5 + 20) × 30] = 9750 lb
2
and
9750
= 69.7 lb∕ft.
140
The above shear developed along the eave struts is smaller
than the allowable shear of 147 lb/ft for 0.0179-in.-thick form
deck. Therefore, the roof panels are adequate to act as a
diaphragm.
For side walls, the shear to be resisted by the diaphragm is
𝑣=
𝑣=
9750
𝑃
=
= 109 lb∕ft
𝑏eff
89.5
or the required static ultimate shear resistance is
𝑆u = 109 × 2.0 = 218 lb∕ft
Since the above computed Su is less than the
nominal shear strength of 295 lb/ft for the 0.0179-in.
-thick form deck, the wall panels are also adequate to act as
a shear diaphragm. The X-bracing in the plane of the side
walls can therefore be eliminated.
358
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
9.3 STRUCTURAL MEMBERS BRACED
BY DIAPHRAGMS
9.3.1 Beams and Columns Braced by Steel
Diaphragms
In Section 9.2 the application of steel diaphragms in building
construction was discussed. It has been pointed out that in
addition to utilizing their bending strength and diaphragm
action the steel panels and decking used in walls, roofs,
and floors can be very effective in bracing members of
steel framing against overall buckling of columns and lateral
buckling of beams in the plane of panels. Both theoretical and experimental results indicate that the failure load
of diaphragm-braced members can be much higher than
the critical load for the same member without diaphragm
bracing.
In the past, investigations of thin-walled steel open
sections with and without bracing have been conducted
by numerous investigators. Since 1961 the structural
behavior of diaphragm-braced columns and beams has
been studied at Cornell by Winter, Fisher, Pincus, Errera,
Apparao, Celebi, Pekoz, Simaan, Soroushian, Zhang, and
others.14.115–4.118,4.123,4.125–4.128,4.133,4.136,9.50–9.57 In these
studies, the equilibrium and energy methods have been
used for diaphragm-braced beams and columns. In addition to the Cornell work, numerous studies have been
conducted at other institutions and several individual steel
companies.4.119–4.122,4.124,4.129–4.132,4.134,4.135,9.58–9.66,9.125
9.3.2
Figure 9.12
Buckling of studs between fully effective
fasteners.
Diaphragm-Braced Wall Studs
Because the shear diaphragm action of wall material can
increase the load-carrying capacity of wall studs significantly, the effect of sheathing material on the design load of
wall studs was previously considered in Sections D4(b) and
D4.1–D4.3 of the AISI Specification.1.314 However, it should
be noted that the AISI design requirements are now given
in the North American Standard for Cold-Formed Steel
Structural Framing1.432 (AISI S240-15), and are limited only
to those studs that have identical wall material attached to
both flanges. When unidentical wall materials are attached
to two flanges, the reader is referred to Refs. 9.52–9.55
or the rational design method of AISI S240 may be used
based on the weaker wall material. Refer to Chapter 12
for information regarding sheathing braced wall stud
design.
Prior to the 2004 Supplement to the AISI Specification1.343
consideration was given to the structural strength and stiffness of the wall assembly. As far as the structural strength is
concerned, the maximum load that can be carried by wall
studs is governed by either (1) column buckling of studs
between fasteners in the plane of the wall (Fig. 9.12) or (2)
Figure 9.13
Overall column buckling of studs.
overall column buckling of studs (Fig. 9.13). The following
discussion deals with the critical loads for these types of
buckling. A discussion of the current AISI S240, a rational
design method is presented in Section 12.2.2.4.
9.3.2.1 Column Buckling of Wall Studs between
Fasteners When the stud buckles between fasteners,
as shown in Fig. 9.12, the failure mode may be (1) flexural
buckling, (2) torsional buckling, or (3) torsional–flexural
buckling, depending on the geometric configuration of the
cross section and the spacing of fasteners. For these types
of column buckling, the critical loads are based on the
stud itself, without any interaction with the wall material.
359
STRUCTURAL MEMBERS BRACED BY DIAPHRAGMS
Therefore the design formulas given in Sections 5.2.23 and
5.2.3 are equally applicable to these cases.
and
9.3.2.2 Overall Column Buckling of Wall Studs Braced
by Shear Diaphragms on Both Flanges The overall
column buckling of wall studs braced by sheathing material
has been studied extensively at Cornell University and
other institutions. The earlier AISI provisions were developed primarily on the basis of the Cornell work.9.52–9.55
Even though the original research has considered the shear
rigidity and the rotational restraint of the wall material that
is attached either on one flange or on both flanges of the
wall studs, for the purpose of simplicity, the AISI design
requirements are provided only for the studs braced by
shear diaphragms on both flanges. In addition, the rotational
restraint provided by the wall material is neglected in the
AISI provisions.
Based on their comprehensive studies of wall assemblies,
Simaan and Pekoz have shown several stability equations
for determining the critical loads for different types of
overall column buckling of wall studs.9.55 The following
buckling load equations are used for channels or C-sections,
Z-sections, and I-sections having wall materials on both
flanges:
where Q = shear rigidity for two wallboards,
kips
d = depth of channel or C-section, in.
(9.19)
b. for torsional–flexural buckling
[
)
1 (
𝑃𝑥 + 𝑃zQ
𝑃cr =
2𝛽
]
√
2
− (𝑃𝑥 + 𝑃zQ ) − 4𝛽𝑃𝑥 𝑃zQ
𝑄𝑑 2
4𝑟20
(9.24)
Other symbols are as defined in Sections 5.4
and 5.7.
2. Z-Sections
a. for torsional buckling about z axis
𝑃cr = 𝑃zQ = 𝑃𝑧 +
𝑄𝑑 2
4𝑟20
(9.25)
b. for combined flexural buckling about x and y axes
)√
[(
1
𝑃cr =
12
(9.26)
𝑃𝑥 + 𝑃𝑦 + 𝑄
2
]
√
(
)
)2
(
2
𝑃𝑥 + 𝑃𝑦 + 𝑄 − 4 𝑃𝑥 𝑃𝑦 + 𝑃𝑥 𝑄 − 𝑃xy
−
where
𝑃xy =
𝜋 2 EI xy
(𝐾𝑥 𝐾𝑦 𝐿2 )
(9.27)
and Ixy is the product of the inertia of wall studs, in.4
3. Doubly Symmetric I-Sections
a. for flexural buckling about y axis
1. Singly Symmetric Channels or C-sections
a. for flexural buckling about y axis
𝑃cr = 𝑃𝑦 + 𝑄
𝑃zQ = 𝑃𝑧 +
𝑃cr = 𝑃𝑦 + 𝑄
(9.28)
b. for flexural buckling about x axis
𝑃cr = 𝑃𝑥
(9.29)
c. for torsional buckling about z axis
(9.20)
where Pcr is the critical buckling load in kips, the
euler flexural buckling load about the x axis of wall
studs in kips is
𝜋 2 EI x
(9.21)
𝑃𝑥 =
(𝐾𝑥 𝐿𝑥 )2
the Euler flexural buckling load about the y axis of
wall studs in kips is
𝜋 2 EI 𝑦
(9.22)
𝑃𝑦 =
(𝐾𝑦 𝐿𝑦 )2
the torsional buckling load about the z axis of wall
studs in kips is
[ 2
]( )
𝜋 ECw
1
(9.23)
+ GJ
𝑃𝑧 =
(𝐾𝑡 𝐿𝑡 )2
𝑟20
𝑃cr = 𝑃zQ = 𝑃𝑧 +
𝑄𝑑 2
4𝑟20
(9.30)
By using the above equations for critical loads, the
critical elastic buckling stress 𝜎 cr can be computed as
𝜎cr =
𝑃cr
𝐴
(9.31)
9.3.2.3 AISI Design Criteria for Wall Studs The AISI
North American Framing Standard1.432 permits sheathing
braced design in accordance with an appropriate theory, tests,
or rational engineering analysis. The following excerpts are
adapted from Section D4 of the 1996 edition of the AISI
Specification for the design of wall studs.1.314 These excepts
may be considered an appropriate theory for design. A
discussion of the current AISI North American Standard
360
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
for Cold-Formed Steel Structural Framing is presented in
Chapter 12.
Table D4
D4 Wall Studs and Wall Stud Assemblies1.314
Sheathing2
k
kN
3/8 in. (9.5 mm) to
5/8 in. (15.9 mm)
thick gypsum
Lignocellulosic board
Fiberboard (regular or
impregnated)
Fiberboard (heavy
impregnated)
24.0
107.0
0.008
12.0
7.2
53.4
32.0
0.009
0.007
14.4
64.1
0.010
Wall studs shall be designed either on the basis of an all-steel
system in accordance with Section C or on the basis of sheathing
in accordance with Section D4.1–D4.3. Both solid and perforated webs shall be permitted. Both ends of the stud shall
be connected to restrain rotation about the longitudinal stud
axis and horizontal displacement perpendicular to the stud
axis.
(a) All-Steel Design. Wall stud assemblies using an all-steel
design shall be designed neglecting the structural contribution of the attached sheathings and shall comply with
the requirements of Section C. In the case of circular web
perforations, see Section B2.2, and for noncircular web
perforations, the effective area shall be determined as
follows:
The effective area, Ac , at a stress Fn shall be determined
in accordance with Section B, assuming the web to consist
of two unstiffened elements, one on each side of the perforation, or the effective area, Ae , shall be determined from
stub-column tests.
When Ae is determined in accordance with Section
B, the following limitations related to the size and
spacing of perforations and the depth of the stud shall
apply:
1. The center-to-center spacing of web perforations shall
not be less than 24 inches (610 mm).
2. The maximum width of web perforations shall be the
lesser of 0.5 times the depth, d, of the section or 2 12 inches
(63.5 mm).
3. The length of web perforations shall not exceed 4 12 inches
(114 mm).
4. The section depth-to-thickness ratio, d/t, shall not be less
than 20.
5. The distance between the end of the stud and the near
edge of a perforation shall not be less than 10 inches
(254 mm).
(b) Sheathing Braced Design. Wall stud assemblies using a
sheathing braced design shall be designed in accordance
with Sections D4.1–D4.3 and in addition shall comply with
the following requirements:
In the case of perforated webs, the effective area, Ae , shall
be determined as in (a) above.
Sheathing shall be attached to both sides of the stud and
connected to the bottom and top horizontal members of the
wall to provide lateral and torsional support to the stud in
the plane of the wall.
Sheathing shall conform to the limitations specified
under Table D4. Additional bracing shall be provided
during construction, if required.
Sheathing Parameters1
𝛾
Qo
length/length
1. Notes:
2. The values given are subject to the following limitations: All
values are for sheathing on both sides of the wall assembly.
3. All fasteners are No. 6, type S-12, self-drilling drywall
screws with pan or bugle head, or equivalent.
4. All sheathing is 12 in. (12.7 mm) thick except as noted.
5. For other types of sheathing, Qo and 𝛾 shall be permitted to
be determined conservatively from representative
small-specimen tests as described by published documented
methods (see Commentary).
The equations given are applicable within the following
limits:
Yield strength, 𝐹y ≤ 50 ksi (345 MPa)
Section depth, 𝑑 ≤ 6.0 in. (152 mm)
Section thickness, 𝑡 ≤ 0.075 in. (1.91 mm)
Overall length, 𝐿 ≤ 16 ft (4.88 mm)
Stud spacing, 12 in. (305 mm) minimum; 24 in. (610 mm)
maximum
D4.1 Wall Studs in Compression
For studs having identical sheathing attached to both flanges and
neglecting any rotational restraint provided by the sheathing, the
nominal axial strength, Pn , shall be calculated as follows:
𝑃n = 𝐴c 𝐹n
(9.32)
Ωc = 1.80
(ASD)
𝜙c = 0.85
(LRFD)
where Ae is the effective area determined at Fn and Fn is the
lowest value determined by the following three conditions:
(a) To prevent column buckling between fasteners in the plane of
the wall, Fn shall be calculated according to Section C4 with
KL equal to two times the distance between fasteners.
(b) To prevent flexural and/or torsional overall column buckling,
Fn shall be calculated in accordance with Section C4 with Fc
taken as the smaller of the two 𝜎 CR values specified for the
STRUCTURAL MEMBERS BRACED BY DIAPHRAGMS
following section types, where 𝜎 CR is the theoretical elastic
buckling stress under concentric loading:
1. Singly symmetric C-sections
(9.33)
𝜎CR = 𝜎e𝑦 + 𝑄a
]
[
√
)2
(
1
𝜎CR =
(𝜎e𝑥 + 𝜎t𝑄 ) −
𝜎e𝑥 + 𝜎t𝑄 − (4𝛽𝜎e𝑥 𝜎t𝑄 )
2𝛽
(9.34)
2. Z-Sections
𝜎CR = 𝜎t + 𝑄t
(9.35)
{
𝜎CR =
1
2
(
)√
12
𝜎e𝑥 + 𝜎e𝑦 + 𝑄a
√
√ (
)2
√⎡
⎤⎫
√
𝜎e𝑥 + 𝜎e𝑦 + 𝑄a − ⎥⎪
−√⎢
⎢4(𝜎 𝜎 + 𝜎 𝑄 − 𝜎 2 )⎥⎬
exy ⎦⎪
e𝑥 a
⎣ e𝑥 e𝑦
⎭
𝜎CR = 𝜎e𝑦 + 𝑄a
(9.37)
𝜎CR = 𝜎e𝑥
(9.38)
𝜎exy =
𝜋 2 EI xy
2
AL
𝜋2𝐸
𝜎e𝑦 =
(𝐿∕𝑟𝑦 )2
[
]
𝜋 2 ECw
1
𝜎t = 2 GJ +
(𝐿)2
Ar0
𝜎t𝑄 = 𝜎t + 𝑄t
(
)
𝑠
𝑄 = 𝑄o 2 − ′
𝑠
(9.39)
(9.40)
(9.41)
(9.48)
𝐹n [(𝜎e𝑥 − 𝐹n )(𝑟20 𝐸0 − 𝑥0 𝐷0 ) − 𝐹n 𝑥0 (𝐷0 − 𝑥0 𝐸0 )]
(𝜎e𝑥 − 𝐹n )𝑟20 (𝜎t𝑄 − 𝐹n )! − (𝐹n 𝑥0 )2
(9.49)
2. Z-sections
𝐸1 =
𝐹n [𝐶0 (𝜎e𝑥 − 𝐹n ) − D0 𝜎exy ]
2
(𝜎e𝑦 − 𝐹n + 𝑄a )(𝜎e𝑥 − 𝐹n ) − 𝜎exy
𝐹n 𝐸0
𝜎t𝑄 − 𝐹n
(9.50)
(9.51)
𝐶1 =
𝐹n 𝐶0
𝜎e𝑦 − 𝐹n + 𝑄a
(9.52)
𝐸1 = 0
where x0 is the distance from shear center to the centroid
along the principal x axis in inches (absolute value) and C0 ,
E0 , and D0 are initial column imperfections which shall be
assumed to be at least
(9.42)
𝐶0 = 𝐿∕350 in a direction parallel to the wall
(9.43)
𝐷0 = 𝐿∕700 in a direction perpendicular to the wall
(9.54)
(9.44)
𝐸0 = 𝐿∕(𝑑 × 10, 000), rad, a measure of the initial
(9.53)
twist of the stud from the initial ideal,
unbuckled shape
(9.45)
and
𝐴 = area of full unreduced cross section
𝐿 = length of stud
𝑄t = (𝑄d2 )∕(4Ar20 )
𝑑 = depth of section
𝐼xy = product of inertia
1. Singly symmetric C-sections
𝐹n 𝐶0
𝐶1 =
𝜎e𝑦 − 𝐹n + 𝑄a
3. I-sections
where s = fastener spacing, in. (mm); 6 in.
(152 mm) ≤ s ≤ 12 in. (305 mm)
s′ = 12 in. (305 mm)
𝑄o = see Table D4
𝑄a = 𝑄∕𝐴
where C1 and E1 are the absolute values of C1 and E1 specified
below for each section type:
𝐶1 =
3. I-Sections (doubly symmetric)
In the above formulas:
𝜋2𝐸
𝜎e𝑥 =
(𝐿∕𝑟x )2
(c) To prevent shear failure of the sheathing, a value of Fn shall
be used in the following equations so that the shear strain of
the sheathing, 𝛾, does not exceed the permissible shear strain,
𝛾. The shear strain, 𝛾, shall be determined as follows:
(
)]
( )[
𝐸1 𝑑
𝜋
𝐶1 +
(9.47)
𝛾=
𝐿
2
𝐸1 =
(9.36)
361
(9.46)
(9.55)
If 𝐹n > 0.5 𝐹y , then in the definitions for 𝜎cy , 𝜎cx , 𝜎cxy , and
𝜎tQ , the parameters E and G shall be replaced by E′ and G′ ,
respectively, as defined below:
4EF n (𝐹y − 𝐹n )
(9.56)
𝐸′ =
𝐹y2
( ′)
𝐸
𝐺′ = 𝐺
(9.57)
𝐸
Sheathing parameters Qo and 𝛾 shall be permitted to be
determined from representative full-scale tests, conducted and
evaluated as described by published documented methods (see
Commentary), or from the small-scale-test values given in
Table D4.
362
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
D4.2 Wall Studs in Bending
For studs having identical sheathing attached to both figures, and
neglecting any rotational restraint provided by the sheathing, the
nominal flexural strengths are 𝑀nxo , and 𝑀nyo , where
For sections with stiffened or partially stiffened compression
flanges:
Ωb = 1.67
(ASD)
𝜙b = 0.95
(LRFD)
For sections with unstiffened compression flanges:
Ωb = 1.67
(ASD)
𝜙b = 0.90
completely inoperative, the allowable design load will still
be sufficient.
In Section D4.1(b) of the specification, Eqs.(9.33)–(9.38)
were derived from Eqs. (9.19)–(9.29) with 𝐾𝑥 = 𝐾𝑦 = 𝐾t =
1.0. The type of torsional buckling of doubly symmetric
I-sections [Eq. (9.30)] is not considered in the AISI requirements because it is not usually a failure mode.
The design shear rigidity Q for two wallboards was determined in the 1980 and 1986 editions of the Specification as
(LRFD)
The nominal flexural strengths Mnxo and Mnyo about the
centroidal axes are determined in accordance with Section C3.1,
excluding the provisions of Section C3.1.2 (lateral buckling)
D4.3 Wall Studs with Combined Axial Load and
Bending
The required axial strength and flexural strength shall satisfy the
interaction equations of Section C5 with the following redefined
terms:
Pn = Nominal axial strength determined according to Section
D4.1
Mnx and Mny in Equations C5.2.1-1, C5.2.1-2, and C5.2.1-3 for
ASD or C5.2.2-1, C5.2.2-2, and C5.2.2-3 shall be replaced by
nominal flexural strengths, Mnxo and Mnyo , respectively.
In the foregoing AISI design provisions, Section D4.1(a)
is based on the discussion given in this volume (Section
9.3.2.1), except that the effective length KL is taken as
two times the distance between fasteners. Thus, even if an
occasional attachment is defective to a degree that it is
Figure 9.14
𝑄 = 𝑞𝐵
(9.58)
in which the value 𝑞 was defined as the design shear rigidity
for two wallboards per inch of stud spacing. Based on the
discussions presented in Ref. 9.55, 𝑞 can be determined by
2𝐺′
(9.59)
SF
where the diaphragm shear stiffness of a single wallboard for
a load of 0.8Pult is
0.8𝑃ult ∕𝑏 0.8𝑃ult ( 𝑎 )
kips∕in.
(9.60)
=
𝐺′ =
Δd ∕𝑎
Δd
𝑏
𝑞=
where Pult = ultimate load reached in shear diaphragm
test of a given wallboard, kips (Fig. 9.14)
Δd = shear deflection corresponding to a load of
0.8Pult , in. (Fig. 9.14)
a, b = geometric dimensions of shear diaphragm
test frame, ft (Fig. 9.14)
SF = safety factor, =1.5
The reason for using 0.8Pult for G′ is that the shear deflection and thus the shear rigidity at the ultimate load Pult are
Determination of shear rigidity, Q .9.55
STRUCTURAL MEMBERS BRACED BY DIAPHRAGMS
not well defined and reproducible. A safety factor of 1.5 was
used to avoid premature failure of the wallboard.
By substituting the equation of G′ and the safety factor
into Eq. (9.59), the design shear rigidity for wallboards on
both sides of the stud can be evaluated as
0.53𝑃ult ( 𝑎 )
𝑞=
(9.61)
Δd
𝑏
Based on the results of a series of shear diaphragm tests
using different wallboards with No. 6, type S-12, self-drilling
dry-wall screws at 6- to 12-in. (152- to 305-mm) spacing,
some typical values of q0 have been developed and were
given in Table D4 of the 1980 and 1986 editions of the
AISI Specification. In this table, the value of 𝑞 0 was
computed by
𝑞
𝑞0 =
(9.62)
2 − 𝑠∕12
where s is the fastener spacing, in.
In the 1996 edition of the AISI Specification, the equation
for the design shear rigidity 𝑄 for sheathing on both sides
of the wall was rewritten on the basis of a recent study
of gypsum-sheathed cold-formed steel wall studs. In Ref.
9.108, Miller and Pekoz indicated that the strength of gypsum
wallboard-braced studs was observed to be rather intensive to
stud spacing. Moreover, the deformations of gypsum wallboard panel (in tension) were observed to be localized at
the fasteners, and not distributed throughout the panel as in
a shear diaphragm. The 𝑄o values listed in Table D4 were
determined from 𝑄o = 12 q0 , in which the q0 values were
363
obtained from the 1986 edition of the AISI Specification. The
values given in Table D4 for gypsum are based on dry service
conditions.
In addition to the requirements discussed above, the AISI
specification considers the shear strain requirements as well.
In this regard, Section D4.1(c) specifies that the computed
shear strain 𝛾 according to Eq. (9.47)–(9.52) and for a value
of Fn should not exceed the permissible shear strain of the
wallboard 𝛾 given in Table D4 of the specification. It can be
seen that the shear strain in the wallboard is affected by the
initial imperfections of wall studs, for which some minimum
values for sweep, camber, and possible twist of studs are
recommended in Eqs. (9,53)–(9.55) to represent the general
practice.
Example 9.2 Use the ASD and LRFD methods to compute
the allowable axial load for the C-section shown in Fig. 9.15
if it is to be used as wall studs having a length of 12 ft.
Assume that the studs are spaced at every 12 in. and that
1
-in.-thick gypsum boards are attached to both flanges of the
2
stud. All fasteners are No. 6, type S-12, self-drilling drywall
screws at 12-in. spacing. Use 𝐹y = 33 ksi. Assume that the
dead load–live load ratio is 15 .
SOLUTION
A. ASD Method
1. Sectional Properties. Using the methods discussed in
Chapters 4 and 5 and the AISI Design Manual, the
Figure 9.15 Example 9.2.
364
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
following sectional properties can be computed on the
basis of the full area of the given C-section:
𝐴
𝐼𝑥
𝑟𝑥
𝐽
𝑥0
=
=
=
=
=
0.651 in.2
2.823 in.4
2.08 in.
0.00109 in.4
1.071 in.
𝐼𝑦
𝑟𝑦
𝐶w
𝑟0
=
=
=
=
0.209 in.4
0.566 in.
1.34 in.6
2.41 in.
2. Allowable Axial Load. According to Section D4.1 of the
AISI Specification, the allowable axial load for the given
stud having identical sheathing attached to both flanges
and neglecting any rotational restraint provided by the
sheathing can be determined by Eq. (9.32) as follows:
𝐴𝐹
𝑃a = e n
Ωc
where the nominal buckling stress Fn is the lowest value
determined by the following three conditions:
(Fn )1 = nominal buckling stress for column buckling
of stud between fasteners in the plane of
the wall
(Fn )2 = nominal buckling stress for flexural and/or
torsional–flexural overall column buckling
(Fn )3 = nominal buckling stress to limit shear strain
of wallboard to no more than the permissible
value
a. Calculation of (Fn )1 . In order to prevent column buckling of the stud between fasteners in the plane of
the wall, consideration should be given to flexural
buckling and torsional–flexural buckling of the singly
symmetric C-section. In the calculation of the elastic
buckling stress, the effective length KL is taken to be
two times the distance between fasteners,
i. Nominal Buckling Stress for Flexural Buckling
KL = 2 × spacing of screws
KL 24
= 42.40
=
𝑟
𝑟𝑦
Using Eq. (5.56)
𝜋 2 (29,500)
𝜋2𝐸
=
= 161.95 ksi
(KL∕𝑟)2
(42.40)2
√
√
𝐹y
33
𝜆c =
=
= 0.451 < 1.5
𝐹e
161.95
𝐹e =
From Eq. (5.54),
𝐹n = (0.658𝜆c )𝐹y = (0.6580.451 )(33)
= 30.31 ksi
where
𝜋 2 (29,500)
𝜋2𝐸
=
= 2187 ksi
(KL∕𝑟𝑥 )2
(24∕2.08)2
[
]
𝜋 2 𝐸𝐶W
1
𝜎t =
GJ
+
(KL)2
𝐴𝑟20
[
1
1
=
11,300 × 0.00109
2
0.651(2.41)2
]
𝜋 2 (29,500)(1.34)
+
(24)2
𝜎e𝑥 =
= 182.4 ksi
Therefore
𝐹e = 179.2 ksi
√
√
𝐹y
33
=
𝜆c =
= 0.429 < 1.5
𝐹e
179.2
From Eq. (5.54),
𝐹n = (0.658𝜆c )𝐹y = (0.6580.429 )(33)
2
2
= 30.55 ksi
The governing nominal buckling stress is the
smaller of the values Fn computed in items (i) and
(ii) above, that is,
(𝐹n )1 = 30.31 ksi
= 2 × 12 = 24 in.
2
ii. Nominal Buckling Stress for Torsional–Flexural
Buckling. From Eq. (5.57),
]
[
√
1
2
(𝜎e𝑥 + 𝜎t ) − (𝜎e𝑥 + 𝜎t ) − 4𝛽𝜎e𝑥 𝜎t
𝐹e =
2𝛽
( )2
𝑥0
= 0.803
𝛽 =1−
𝑟0
2
b. Calculation of (Fn )2 . In order to prevent flexural
and/or torsional–flexural overall column buckling, the
theoretical elastic buckling stress Fe should be the
smaller of the two 𝜎 CR values computed for the singly
symmetric C-section as follows: For flexural overall
column buckling,
𝜎CR = 𝜎e𝑦 + 𝑄a
[Eq. (9.33)]
For torsional–flexural overall column buckling,
[
1
(𝜎e𝑥 + 𝜎t𝑄 )
𝜎CR =
2𝛽
]
√
− (𝜎e𝑥 + 𝜎tQ )2 − 4𝛽𝜎e𝑥 𝜎tQ
Eq. (9.34)
STRUCTURAL MEMBERS BRACED BY DIAPHRAGMS
In Eq. (9.33)
𝜎e𝑦 =
𝜋2𝐸
(KL∕𝑟𝑦 )2
=
Use the smaller value given in Eqs. (9.33)a and
(9.34)a:
𝜋 2 (29,500)
(12 × 12∕0.566)2
= 4.50 ksi
𝑄
𝐴
From Eq. (9.44) and Table D4 of the AISI Specification,
)
(
) (
𝑠
12
𝑄 = 𝑄o 2 − ′ = (24.0)2 −
𝑠
12
= 24 kips
𝐹e = 40.41 ksi
√
√
𝐹y
33
=
= 0.904 < 1.5
𝜆c =
𝐹e
40.41
𝑄a =
24
𝑄a =
= 36.87 ksi
0.651
According to Eq. (9.33), the theoretical elastic critical buckling stress is
𝜎CR = 4.50 + 36.87 = 41.37 ksi
[Eq. (9.33𝑎)]
In Eq. (9.34)
𝛽 = 0.803
𝜎e𝑥 =
𝜋 2 (29,500)
𝜋2𝐸
=
= 60.75 ksi
(KL∕𝑟𝑥 )2
(12 × 12∕2.08)2
𝜎t𝑄 = 𝜎t + 𝑄t
(
)
𝜋 2 ECW
1
𝜎t = 2 GJ +
𝐿2
Ar0
[
1
1
11, 300 × 0.00109
𝜎t =
2
2
0.651(2.41)
𝜋 2 (29,500)(1.34)
(12 × 12)2
= 8.23 ksi
𝑄t =
=
(𝐹n )2 = (0.658𝜆c )𝐹y = (0.6580.904 )(33)
2
c. Calculation of (Fn )3 . In items (a) and (b) it was found
that in order to prevent column buckling the nominal
buckling stress should not exceed 23.44 ksi. According
to Section D4.1(c) of the AISI Specification, in order
to prevent shear failure of the sheathing, a value of Fn
should be used in the given equations so that the shear
strain of the sheathing, 𝛾, computed by Eq. (9.47) does
not exceed the permissible value of 𝛾 = 0.008 in./in.,
which is given in Table D4 of the AISI Specification
for 12 -in.-thick gypsum board.
Based on Eq. (9.47), the shear strain of the sheathing
can be computed as follows:
(
)
𝑑
𝜋
𝐶1 + 𝐸1
𝛾=
𝐿
2
]
𝐶1 =
𝐸1 =
𝐹n 𝐶0
𝜎e𝑦 − 𝐹n + 𝑄a
𝜎t𝑄 = 8.23 + 48.00 = 56.23 ksi
From Eq. (9.34),
[
1
1
=
(60.75 + 56.23)
2(0.803)
2
√
⎤
(60.75 + 56.23)2 − 4(0.803)⎥
−
× (60.75)(56.23)
⎥
⎦
= 40.41 ksi (9.34𝑎)(1)
[Eq. (9.48)]
𝐹n [(𝜎e𝑥 − 𝐹n )(𝑟20 𝐸0 − 𝑥0 𝐷0 )
−𝐹n 𝑥0 (𝐷0 − 𝑥0 𝐸0 )]
(𝜎e𝑥 − 𝐹n )𝑟20 (𝜎t𝑄 − 𝐹n ) − (𝐹n 𝑥0 )2
[Eq. (9.49)]
4Ar20
(24)(5.45)2
= 48.00 ksi
4(0.651)(2.41)2
2
= 23.44 ksi
𝑄𝑑 2
and
𝜎CR
From Eq. (5.54),
where
where
+
365
As the first approximation, let
𝐹n = (𝐹n )2 of item 𝑏 = 23.44 ksi
𝐿
12 × 12
=
= 0.411 in.
350
350
𝐿
12 × 12
𝐸0 =
=
= 0.0026 rad.
𝑑 × 10,000 5.50 × 10,000
𝐿
12 × 12
𝐷0 =
=
= 0.206 in.
700
700
𝐶0 =
Therefore, from Eqs. (9.48) and (9.49),
𝐶1 =
23.44 × 0.411
= 0.537
4.50 − 23.44 + 36.87
366
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
23.44[(60.75 − 23.44)(2.412 × 0.0026
−1.071 × 0.206) − 23.44 × 1.071
𝐸1 =
computed for different design considerations:
(𝐹n )1 = 30.31 ksi
(0.206 − 1.071 × 0.0026)]
(𝐹n )2 = 23.44 ksi
(60.75 − 23.44)(2.41)2 (56.23 − 23.44)
(𝐹n )3 = 17.50 ksi
− (23.44 × 1.071)2
= −0.0462
Use an absolute value, E1 = 0.0462. Substituting the
values of C1 , E1 , L, and d into Eq. (9.47), the shear
strain is
)[
]
(
0.0462 × 5.5
𝜋
0.537 +
𝛾=
12 × 12
2
= 0.0145 in.∕in. > (𝛾 = 0.008 in.∕in.)
Since the computed 𝛾 value for 𝐹n = 23.44 ksi is
larger than the permissible 𝛾 value, a smaller Fn value
should be used. After several trials, it was found that
a value of 𝐹n = 17.50 ksi would give the permissible
shear strain of 0.008 in./in. as shown below.
Try 𝐹n = 17.50 ksi. Since 𝐹n > (𝐹y ∕2 = 16.5 ksi),
use E′ and G′ to compute the values of 𝜎 e x, 𝜎 e y, 𝜎 t ,
and 𝜎 t Q.
𝐸′ =
4𝐸𝐹n (𝐹y − 𝐹n )
𝐹y2
4(𝐸)(17.50)(33 − 17.50)
= 0.996E, ksi
(33)2
( ′)
𝐸
𝐺′ = 𝐺
= 0.996𝐺 ksi
𝐸
( ′)
𝐸
𝜎e𝑥 = 60.75
= 60.51 ksi
𝐸
( ′)
𝐸
𝜎e𝑦 = 4.50
= 4.48 ksi
𝐸
( ′)
𝐸
𝜎t = 8.23
= 8.20 ksi
𝐸
=
𝜎t𝑄 = 𝜎t + 𝑄t = 8.20 + 48.00 = 56.20
The smallest value of the above three stresses should
be used for computing the effective area and the allowable axial load for the given C-section stud, that is,
𝐹n = 17.50 ksi
e. Calculation of Effective area. The effective area should
be computed for the governing nominal buckling stress
of 17.50 ksi.
i. Effective Width of Compression Flanges (Section
3.5.3.2)
√
√
29,500
𝐸
𝑆 = 1.28
= 1.28
= 53.55
𝑓
17.50
1
= 17.52
3
𝑤1
1.625 − 2(0.136 + 0.071)
=
𝑡
0.071
1.211
=
= 17.06
0.071
Since 𝑤1 ∕𝑡 < 𝑆∕3, 𝑏1 = 𝑤1 = 1.211 in. The
flanges are fully effective.
ii. Effective Width of Edge Stiffeners (Section
3.5.3.2)
𝑤2
0.500 − (0.136 + 0.071)
=
𝑡
0.071
0.293
=
= 4.13 < 14
OK
0.071
( )√𝑓
1.052 𝑤2
𝜆= √
𝑡
𝐸
𝑘
√
1.052
17.50
(4.13)
=√
29,500
0.43
𝑆
= 0.161 < 0.673
𝐶1 = 0.302 in
𝑑s′ = 𝑤2 = 0.293 in.
𝐸1 = 0.0238 rad
𝑑s = 𝑑s′ = 0.293 in.
and
𝛾 = 0.008 in.∕in.
(𝛾 = 0.008 in.∕in.) OK
Therefore (𝐹n )3 = 17.50 ksi.
d. Determination of Fn . From items (a), (b), and (c), the
following three values of nominal buckling stress were
The edge stiffeners are fully effective.
iii. Effective Width of Webs (Section 3.5.1.1)
𝑤3
5.50 − 2(0.136 + 0.071)
=
𝑡
0.071
5.086
=
= 71.63
0.071
SHELL ROOF STRUCTURES
1.052
𝜆 = √ (71.63)
4
√
17.50
29,500
= 0.918 > 0.673
1 − 0.22∕𝜆 1 − 0.22∕0.918
=
𝜆
0.918
= 0.828
𝜌=
𝑏3 = 𝜌𝑤3 = 0.828(5.086) = 4.211 in.
iv. Effective Area Ae
𝐴e = 𝐴 − (𝑤3 − 𝑏3 )(𝑡)
= 0.651 − (5.086 − 4.211)(0.071)
= 0.589 in.2
f. Nominal Axial Load and Allowable Axial Load. Based
on 𝐹n = 17.50 ksi and 𝐴e = 0.589 in.2 , the nominal
axial load is
𝑃n = 𝐴e 𝐹n = (0.589)(17.50) = 10.31 kips
The allowable axial load is
𝑃
10.31
𝑃a = n =
= 5.73 kips
Ωc
1.80
B. LRFD Method. For the LRFD method, the design
strength is
𝜙c 𝑃n = (0.85)(10.31) = 8.76 kips
Based on the load combinations given in Section 1.82, the
governing required axial load is
𝑃u = 1.2𝑃D + 1.6𝑃L = 1.2𝑃D + 1.6(5𝑃D ) = 9.2𝑃D
Using 𝑃𝑢 = 𝜙𝑐 𝑃𝑛 ,
8.76
𝑃D =
= 0.95 kips
9.2
𝑃L = 5𝑃D = 4.75 kips
The allowable axial load is
𝑃a = 𝑃D + 𝑃L = 5.70 kips
It can be seen that both ASD and LRFD methods give
approximately the same result for this particular example.
It should also be noted that the C-section stud used in this
example is selected from page I-12 of the 1996 edition
of the AISI Cold-Formed Steel Design Manual.1.159 This
C-section with lips is designated as 5.5CS 1.625 × 071.
The AISI North American standard for Cold-formed
Steel Framing—General Provisions13.1 prescribes an
industry-adopted designator system for cold-formed steel
studs, joists, and track sections. For details, see Ref. 13.1.
367
The preceding discussion and Example 9.2 dealt with wall
studs under concentric loading. For studs subjected to axial
load and bending moment, the design strength of the studs
should be determined according to Sec. C5.2 of the 2012
edition of AISI North American Specification.1.345 A study
of wall studs with combined compression and lateral loads
was reported in Ref. 9.66. Additional studies on the behavior
of steel wall stud assemblies and developments of a structural system using cold-formed steel wall studs have been
conducted and reported in Refs. 9.93–9.95.
For fire resistance ratings of load-bearing steel stud walls
with gypsum wallboard protection, the reader is referred to
AISI publications.9.67,1.277
It should be noted that the AISI design provisions1.314,1.345
for wall studs permit (a) all-steel design and (b) sheathing
braced design of wall studs with either solid or perforated
webs. For sheathing braced design, in order to be effective,
sheathing must retain its design strength and integrity for the
expected service life of the wall.
For the case of all-steel design, the approach of determining effective areas in accordance with 1996 edition of
AISI Specification1.345 Section D4.1 is being used in the RMI
Specification1.165 for the design of perforated rack columns.
The validity of this approach for wall studs was verified
in a Cornell project on wall studs reported by Miller and
Pekoz.9.98 The limitations for the size and spacing of perforations and the depth of studs are based on the parameters used
in the test program. For sections with perforations which do
not meet these limits, the effective area can be determined by
stub column tests.
9.4
9.4.1
SHELL ROOF STRUCTURES
Introduction
Steel folded-plate and hyperbolic paraboloid roof structures
have been used increasingly in building construction for
churches, auditoriums, gymnasiums, classrooms, restaurants, office buildings, and airplane hangars.1.77–1.84,9.68–9.76
This is because such steel structures offer a number of advantages as compared with some other types of folded-plate
and shell roof structures to be discussed. Since the effective use of steel panels in roof construction is not only
to provide an economical structure but also to make the
building architecturally attractive and flexible for future
change, structural engineers and architects have paid more
attention to steel folded-plate and hyperbolic paraboloid
roof structures during recent years.
The purpose of this discussion is mainly to describe the
methods of analysis and design of folded-plate and hyperbolic paraboloid roof structures which are currently used in
engineering practice. In addition, it is intended to review
briefly the research work relative to steel folded-plate and
368
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
shell roof structures and to compare the test results with those
predicted by analysis.
In this discussion, design examples will be used for illustration. The shear strength of steel panels, the empirical
formulas to determine deflection, and the load factors used
in various examples are for illustrative purposes only. Actual
design values and details of connections should be based
on individual manufacturers’ recommendations on specific
products.
9.4.2
Folded-Plate Roofs
9.4.2.1 General Remarks A folded-plate structure is a
three-dimensional assembly of plates. The use of steel panels
in folded-plate construction started in this country about
1960. Application in building construction has increased
rapidly during recent years. The design method used in engineering practice is mainly based on the successful investigation of steel shear diaphragms and cold-formed steel
folded-plate roof structures.1.77–1.81,1.84,9.1,9.2,9.70
9.4.2.2 Advantages of Steel Folded-Plate Roofs Steel
folded-plate roofs are being used increasingly because they
offer several advantages in addition to the versatility of
design:
1. Steel roof panels
2. Fold line members at ridges and valleys
3. End frames or end walls
In general, the plate width (or the span length of roof
panels) ranges from about 7 to 12 ft (2.1 to 3.7 m), the slope
of the plate varies from about 20∘ to 45∘ , and the span length
of the folded plate may be up to 100 ft (30.5 m). Since unusually low roof slopes will result in excessive vertical deflections and high diaphragm forces, it is not economical to
design a roof structure with low slopes.
In the analysis and design of folded-plate roofs, two
methods are available to engineers. They are the simplified method and the finite-element method. The former
provides a direct technique that will suffice for use in the
final design for many structures. The latter permits a more
detailed analysis for various types of loading, support, and
material.1.81
In the simplified method, steel roof panels as shown in
Fig. 9.18 are designed as simply supported slabs in the transverse direction between fold lines. The reaction of the panels
is then applied to fold lines as a line loading, which resolves
1. Reduced Dead Load. A typical steel folded plate
generally weighs about 11 lb∕ft2 (527 N∕m2 ), which
is substantially less than some other types of folded
plates.
2. Simplified Design. The present design method for steel
plate roofs is simpler than the design of some other
types of folded plates, as discussed later.
3. Easy Erection. Steel folded-plate construction requires
relatively little scaffolding and shoring. Shoring can be
removed as soon as the roof is welded in place.
9.4.2.3 Types of Folded-Plate Roofs Folded-plate roofs
can be classified into three categories: single-bay, multiplebay, and radial folded plate, as shown in Fig. 9.16. The folded
plates can be either prismatic or nonprismatic.
The sawtooth folded-plate roof shown in Fig. 9.16b has
been found to be the most efficient multiple-bay structure
and is commonly used in building construction. Figure 9.17
shows a folded-plate structure of the sawtooth type used for
schools.
9.4.2.4 Analysis and Design of Folded Plates A
folded-plate roof structure consists mainly of three components, as shown in Fig. 9.18:
Figure 9.16 Types of folded-plate roofs: (a) singlebay; (b) multiplebay; (c) radial.
SHELL ROOF STRUCTURES
Figure 9.17
Company.
369
Cold-formed steel panels used in folded-plate roof. Courtesy of H. H. Robertson
Figure 9.18 Folded-plate structure.
itself into two components parallel to the two adjacent plates,
as shown in Fig. 9.19. These load components are then
carried by an inclined deep girder spanned between end
frames or end walls (Fig. 9.18). These deep girders consist
of fold line members as flanges and steel panels as a web
element. The longitudinal flange force in fold line members
can be obtained by dividing the bending moment of the
deep girder by its depth. The shear force is resisted by the
diaphragm action of the steel roof panels. Therefore the
shear diaphragm discussed in Section 9.2 is directly related
to the design of the folded-plate structure discussed in
this section.
In the design of fold line members, it is usually found
that the longitudinal flange force is small because of the
considerable width of the plate. A bent plate or an angle
section is often used as the fold line member.
Referring to Fig. 9.18, an end frame or end wall must
be provided at the ends of the folded plates to support the
Figure 9.19
Force components along fold lines.
end reaction of the inclined deep girder. In the design of
the end frame or end wall, the end reaction of the plate
may be considered to be uniformly distributed through the
entire depth of the girder. Tie rods between valleys must
be provided to resist the horizontal thrust. If a rigid frame
is used, consideration should be given to such a horizontal
thrust.
370
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
When a masonry bearing wall is used, a steel welding plate
should be provided at the top of the wall for the attachment
of panels. It should be capable of resisting the force due to
the folded-plate action.
Along the longitudinal exterior edge, it is general practice
to provide a vertical edge plate or longitudinal light framing
with intermittent columns to carry vertical loads. If an exterior inclined plate is to cantilever out from the fold line, a
vertical edge plate will not be necessary.
In addition to the consideration of beam strength, the
deflection characteristics of the folded-plate roof should also
be investigated, particularly for long-span structures. It has
been found that a method similar to the Williot diagram for
determining truss deflections can also be used for the prediction of the deflection of a steel folded-plate roof. In this
method the in-plane deflection of each plate should first be
computed as a sum of the deflections due to flexure, shear,
and seam slip, considering the plate temporarily separated
from the adjacent plates. The true displacement of the fold
line can then be determined analytically or graphically by
a Williot diagram. When determining flexural deflection, the
moment of inertia of the deep girder may be based on the area
of the fold line members only. The shear deflection and the
deflection due to seam slip should be computed by the empirical formulas recommended by manufacturers for the specific
panels and the system of connection used in the construction.
In some cases it may be found that the deflection due to seam
slip is negligible.
Example 9.3 Discuss the procedures to be used for the
design of an interior plate of a multiple-bay folded-plate
roof (Fig. 9.20) by using the simplified ASD method.
Given:
Uniform dead load wD , psf (along roof surface)
Uniform live load wL , psf (on horizontal projection)
Span L, ft
Unit width B, ft
Slope distance b between fold lines, ft (or depth of analogous
plate girder)
SOLUTION
1. Design of Steel Panels—Slab Action in Transverse
Direction
1
1
ft-lb
𝑀1 = × 𝑤L 𝐵 2 + 𝑤D bB
8
8
Select a panel section to meet the requirements of
beam design and deflection criteria.
2. Design of Fold Line Members. The vertical line
loading is
𝑤 = 𝑤 L 𝐵 + 𝑤D 𝑏
lb∕ft
Figure 9.20
Example 9.3
The load component in the direction of the inclined
girder is w′ . The total load applied to the inclined deep
girder EF is 2w′ :
1
ft-lb
× 2𝑤′ 𝐿2
8
Select a fold line member to satisfy the required
moment.
3. Design for Plate Shear
1
lb
𝑉 = 2𝑤′ × 𝐿 = 𝑤′ 𝐿
2
𝑉
𝑣=
lb∕ft
𝑏
Select an adequate welding system on the basis of
the manufacturer’s recommendations.
4. Deflection
a. In-Plane Flexural Deflection (considering that the
inclined plate is temporarily separated)
𝑀2 =
Δb =
where
5
2𝑤′ 𝐿4
×
(12)
384
EI
in.
(
)
1 2
𝑏
2
𝐸 = 29.5 × 106 psi
𝐼 = 2𝐴f
b. In-Plane Shear Deflection (including the deflection
due to seam slip)
2𝑤′ 𝐿2
in.
2𝐺′ 𝑏
where G′ is the shear stiffness of steel panels
obtained from diaphragm tests, lb/in. See
Section 9.2.
c. Total In-Plane Deflection
Δs =
Δ = Δb + Δs
in.
d. Total Vertical Deflection. After the in-plane deflection is computed, the maximum vertical deflection
of fold line members can be determined by a Williot
diaphragm, as shown in Fig. 9.21.
9.4.2.5 Research on Folded-Plate Roofs Full-size
folded-plate assemblies have been tested by Nilson at
371
SHELL ROOF STRUCTURES
Figure 9.21
deflection.
Williot diagram used for determining total vertical
Cornell University1.77 and by Davies, Bryan, and Lawson at
the University of Salford.9.77 The following results of tests
were discussed in Ref. 1.77 for the Cornell work.
The test assembly used by Nilson was trapezoidal in
cross section and was fabricated from 1 12 -in.- (38-mm-) deep
cold-formed steel panels as plates (five plates) and 3 12 × 14 ×
3 12 -in.(89 × 6.4 × 89-mm) bent plates as fold line members.
The span length of the test structure was 42 ft, 6 in. (13 m),
and the width of the assembly was 14 ft (4.3 m), as shown in
Fig. 9.22. The test setup is shown in Fig. 9.23. It should be
noted that the jack loads were applied upward because of the
convenience of testing.
In Ref. 1.77 Nilson indicated that the experimental structure performed in good agreement with predictions based
on the simplified method of analysis, which was used in
Example 9.3. It was reported that the tested ultimate load
was 11% higher than that predicted by analysis and that
the observed stresses in fold line members were about 20%
lower than indicated by analysis due to neglect of the flexural contribution of the steel panel flat-plate elements. In
view of the fact that this difference is on the safe side and
the size of the fold line members is often controlled by
practical considerations and clearance requirements rather
Figure 9.22
than by stress, Nilson concluded that no modification of the
design method would be necessary. As far as the deflection of
the structure is concerned, the measured values were almost
exactly as predicted.
In the 1960s, AISI sponsored a research project on coldformed steel folded plates at Arizona State University
to study further the methods of analysis and design of
various types of folded-plate roofs, including rectangular
and circular planforms. In this project, both the simplified
analysis and the finite-element approach were studied in
detail by Schoeller, Pian, and Lundgren.1.81
For a multiple sawtooth folded-plate structure with a
span of 40 ft (12.2 m), the analytical results obtained from
the simplified method and the finite-element method are
compared as follows1.81 :
Maximum fold
line force, lb
Maximum plate
shear, lb/ft
Simplified
Method
Finite-Element
Method
22,500
23,400
1,024
958
9.4.2.6 Truss-Type Folded-Plate Roofs The above
discussion is related to the analysis and design of
membrane-type folded-plate roofs in which the steel roof
panels not only support normal loads but also resist shear
forces in their own planes. This type of structure is generally
used for spans of up to about 100 ft (30.5 m).
For long-span structures, a folded-plate roof may be
constructed by utilizing inclined simple trusses as basic
units, covering them with steel panels. In this case steel
panels will resist normal loads only. The design of basic
Steel folded-plate assembly.1.77
372
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
Figure 9.23
Setup for test of folded-plate assembly.1.77
trusses is based on the conventional method. Additional
information on the design and use of folded-plate roofs can
be found in Refs. 1.84 and 9.78.
9.4.3
Hyperbolic Paraboloid Roofs
9.4.3.1 General Remarks The hyperbolic paraboloid roof
has also gained increasing popularity during recent years
due to the economical use of materials and its attractive
appearance. The hyperbolic paraboloid shell is a doubly
curved surface which seems difficult to construct from steel
but in fact can be built easily with either single-layer or
double-layer standard steel roof deck panels. This is so
because the doubly curved surface of a hyperbolic paraboloid
has the practical advantage of straight-line generators as
shown in Fig. 9.24.
Figure 9.24
Surface of hyperbolic paraboloid.9.79
Figure 9.25 shows the Frisch restaurant building in Cincinnati, Ohio, which consists of four paraboloids, each 33.5 ft
(10.2 m) square, having a common column in the center and
four exterior corner columns, giving a basic building of 67 ft
(20.4 m) square. The roof of the building is constructed of
laminated 1 12 -in. (38-mm) steel deck of 26-in.- (660-mm-)
wide panels. The lower layer is 0.0516-in.- (1.3-mm-) thick
steel panels, and the upper layer is 0.0396-in.- (1-mm-) thick
steel panels placed at right angles and welded together. The
reason for using a two-layer laminated hyperbolic paraboloid
roof was to achieve additional stiffness and resistance to
point loading. The use of 0.0516-in.- (1.3-mm-) thick steel
panels in the lower layer was for ease of welding. The
roof plan and the structure details of the Frisch restaurant
building, designed by H. T. Graham, are shown in Fig. 9.25.
In 1970 Zetlin, Thornton, and Tomasetti used hyperbolic
paraboloids to construct the world’s largest cold-formed steel
superbay hangars for the American Airlines Boeing 747s in
Los Angeles and San Francisco, California (Fig. 9.26).1.82
The overall dimensions of the building shown in Fig. 9.26 are
450 ft (137 m) along the door sides and 560 ft (171 m) at the
end wall. The central core of the building is 100 ft (30.5 m)
wide and 450 ft (137 m) long. The hangar area is covered by
a 230-ft (70-m) cantilever on each side of the core.
As shown in Figs. 9.26 and 9.27, the roof system is
composed of 16 basic structural modules. Each roof module
consists of a ridge member, two valley members, edge
members, and the warped hyperbolic paraboloids, as indicated in Fig. 9.27. The ridge and valley members are
SHELL ROOF STRUCTURES
373
Figure 9.25 Roof plan and structural details of Frisch Restaurant, Cincinnati, Ohio, of hyperbolic paraboloid construction.1.79 Reprinted from Architectural Record, March 1962. Copyright by
McGraw-Hill Book Co., Inc.
In order to be able to use this type of structural system
in any area of the world, prestressed cables are incorporated
into the shell structures (Fig. 9.26). Since the structural strand
cables induce a prestress in the shell, the system is readily
adaptable to any geographic site.
A comparison of various types of designs indicate
that this type of building with its hyperbolic paraboloids
weighs approximately 40% less than a conventional steel
construction.
Figure 9.26 Roof plan of superbay hangar. Courtesy of Lev Zetlin
Associates, Inc.1.82
hot-rolled steel shapes. The hyperbolic paraboloids were
made of cold-formed steel decking consisting of a flat
0.0934-in.- (2.4-mm-) thick sheet, 26 in. (660 mm) wide,
with two 9-in.- (229-mm-) wide by 7 12 -in.- (191-mm-) deep
0.0516-in.- (1.3-mm-) thick steel hat sections welded to the
flat sheets. Figure 9-28 shows a typical cross section of the
steel deck used.
9.4.3.2 Types of Hyperbolic Paraboloid Roofs The
surface of a hyperbolic paraboloid may be defined by two
methods.1.80 As shown in Fig. 9.29, with the x, y, and z axes
mutually perpendicular in space, the surface is formulated
by two straight lines called generators. One line, parallel to
the xz plane, rotates about and moves along the y axis; the
other, parallel to the yz plane, rotates about and moves along
the x axis. The intersection of the generators is contained in
the surface of the hyperbolic paraboloid.
Figure 9.30 shows several types of hyperbolic paraboloid
roofs which may be modified or varied in other ways to
achieve a striking appearance.
374
9
SHEAR DIAPHRAGMS AND ROOF STRUCTURES
Figure 9.27 Construction of superbay hangar. Courtesy of Lev Zetlin Associates, Inc.1.82
Figure 9.28 Cross section of steel deck used in superbay hangar.
Courtesy of Lev. Zetlin Associates, Inc.1.82
In general, type I is the most pleasing of the shapes
available. The edge beams are in compression and are
usually tubular members. For this type of roof, the most
serious problem is the horizontal thrust at the supporting
columns. Usually the columns are kept short in order to
transfer the thrust down to the floor where tie rods can be
hidden. Four units of this type with a common center column
probably provide the most rigid roof structure, as shown
in Fig. 1.
Type II is an inverted umbrella, which is the easiest and the
least expensive to build. The edge members of this type are in
tension, and engineers usually use angles as edge members.
Type III is the most useful type for canopy entrance structures. The edge members connected to the columns are in
compression and are usually tubes, while the outside edge
members are in tension and could be angles or channels. In
some cases, one-half of the roof may be kept horizontal and
the other half tilts up.
Generally speaking, type IV is the most useful of all
the available shapes. The entire building can be covered
with a completely clear span. The horizontal ties between
columns on four sides can be incorporated in the wall
construction.
9.4.3.3 Analysis and Design of Hyperbolic Paraboloid
Roofs The selection of the method of analysis of hyperbolic
paraboloid roofs depends on the curvature of the shell used. If
the uniformly loaded shell is deep (i.e., when the span–corner
depression ratio a/h shown in Fig. 9.24 is less than or equal to
approximately 5.0), the membrane theory may be used. For
the cases of a deep shell subjected to unsymmetrical loading
and a shallow shell, the finite-element method will provide
accurate results.9.80
In the membrane theory, the equation of the surface of a
hyperbolic paraboloid can be defined from Fig. 9.31. Assume
𝑏
𝑐
𝑎
ℎ
=
and
=
(9.63)
𝑐
𝑥
𝑧 𝑦
cy hxy
𝑧=
=
= kxy
(9.64)
𝑎
ab
where 𝑘 = ℎ∕ab, in which h is the amount of corner depression of the surface having the horizontal projections a and b.
If we rotate coordinate axes x and y by 45∘ , as shown
in Fig. 9.32, the equations for two sets of parabolas can be
obtained in terms of the new coordinate system using x′ and
y′ .9.81 Substituting
(9.65)
𝑥 = 𝑥′ cos 45∘ − 𝑦′ sin 45∘
𝑦 = 𝑦′ cos 45∘ + 𝑥′ sin 45∘
(9.66)
into Eq. (9.61), one can obtain a new equation for z in terms
of x′ and y′ ,
1
(9.67)
𝑧 = 𝑘[𝑥′2 − 𝑦′2 ]
2
In Eq. (9.61), if the value of x′ remains constant, as
represented by line a′ –a′ in Fig. 9.33, the equation for the
parabolic curve can be written as follows, where the negative
sign indicates concave downward:
1
𝑧′ = − ky′2
2
(9.68)