Design of Piping Systems
Design of Piping Systems
Pullman Power Products
A Wheelabrator-Frye Company
Revised Second Edition
AWILEY·INTERSCIENCE PUBLICATION
JOHN WILEY & SONS
New York' Chichester' Brisbane' Toronto
Copyright @ 19U, 1956
hy
The 1\1. W. Kellogg Company
All Rights Reserved
Reproduction or translation of any part of this work beyond
that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner
is unlawful. Requests for permission or further information
should be addressed to the Permissions Department, John
Wiley & Sons, Jne.
Revised Second Edition
20
19
18
17
16
15
14
Nothing contained in Design of Piping Systems is to be construed
B8 granting any right of manufacture, sale or use in connection
with any method, apparatus or product covered by Letters
Patent} nor as insuring anyone against liability for infringement
of Letters Patent.
ISBN 0 471 46795 2
Library of Congress Catalog Card Number: 56-5573
Printed in the United States of America
Preface
A volume bearing tbe title Design of Piping Systems, devoted solely to the study
of expansion stresses and reactions in piping systems, was privately published by
The M. W. Kellogg Company early in 1941. It made available for the first time
an adequately organized, eomprehcnsive analytieal method for evaluating the
stresses, reactions, and deflections in an irregular piping system in space, unlimited
as to the character, location, or number of concentrated loadings or restraints.
It was the culmination of an intensive, widespread effort to meet the recognized
need for refined analysis capable of general application to the increasing number
of critical piping services required by technological progress, and to the increasingly
severe problems which they posed. The timely availability of this reliahle and
versatile approach, now widely known as the Kellogg General Analytical Method,
made it possible to provide satisfactory design for the avalanche of critical and
pioneering piping requirements associated with World War II plant design, and
proved to be a major step in accelerating acquaintance with accurate thermal
expansion analysis and appreciation of its potentialities for more extensive application.
Since the war, technological progress and the trend to larger scale, more complex
units has continued unabated, while the attendant increased pressures, temperatures, and structural complexities have resulted in larger pipe sizes, heavier wall
thicknesses, and a marked increase in alloy construction. Concurrently, the
wartime-fostered universal acceptance of adequate piping flexibility analysis for
critical service has paved the way for more searching examination of the over-all
economics of erected piping by relating potential fabrication, materials, and
operating savings to increased engineering costs. Earlier concepts, which regarded
piping as trivial and expendable, are fast disappearing in view of the rising costs
of field corrections and loss of plant operation - and also with the recognition
that piping represents an increasing percentage of initial plant expenditure.
The importance of sound piping design is now well recognized not only by
designers and users, but also by authorities concerned with public safety. The
Code for Pressure Piping Committee (ASA B31.1) has increased its membership
and activity over the past several years and a Conference Committee has been
organized, composed of the chief enforcement authorities of each State or Province
that has adopted a portion or all of the Code. Significant improvements in the
rules have already resulted in the revised minimum (and now mandatory) require-
ments for piping flexibility. With this trend, the ASA Code is now rapidly assuming
the status of a mandatory Safety Code, whereas previously it had served designers
and users primarily as a recommended design practice guide.
The critical shortage of engineering personnel during World War II prevented
the completion of sections on other aspects of piping design that had been planned
for inclusion in the original edition of Design of Piping Systems. As the shortage
persisted, considerable time elapsed before resumption of work could be considered.
Meanwhile, many requests for extension and suggestions for improvement were
,.
vi
PREFACE
received
., from readers of the text already published. Review of these and other
developments in light of extended experience led to the conclusion that a new
edition was warranted': As the work got under way, it was soon evident that
broadening of the subject matter would have to be limited to treatment of the
structural phase of piping design; coverage of the entire field, including fluid flow,
system design and layout, valve design, piping fabrication and erection, etc., would
require much more than the desired single volume.
It is the objective of this Second Edition to supplement Code rules and other
readily available information with specific mechanical design approaches for entire
piping systems as well as their individual eomponents and to provide background
information which will engender ~nderstanding, eompetent application of analytical
results, and the exercise of good judgment in handling the many special situations
which must be faced on critical piping. In line with this objective, the opening
chapter presents a eondensed treatise on the physics of materials. It is followed
by a comprehensive study of the eapacity of piping to carry various prescribed
loadings. The utilization of materials is then considered, not only in relation to
fundamental knowledge but also on the basis of eonventionally accepted practices.
The present edition also includes a greatly augmented treatment of local flexibility
and stress intensification, and a chapter on simplified methods of flexibility analysis
eontains several newly developed approaches which should prove helpful for general
assessment of average piping, or in the planning stage of the design of critical piping.
The Kellogg General Analytieal Method, now extended to include all forms of
loading, has been improved in presentation by the use of numerous sample calculations to illustrate applieation procedures, and by placing the derivations of the
formulas in an appendix. Included in this edition are chapters on expansion joints
and on pipe supports that offer, it is believed, the first broad treatment of these
items with regard to critical piping. The rising significance of vibration, both
structural and fluid, is recognized in the final chapter, which was also prepared
especially for this edition. For ready accessibility of information, the charts and
tables most frequently needed for reference have been grouped at the end of the
text, and a detailed subject index has been provided.
THE M. W. KELLOGG COMPANY
The M.W. Kellogg Company became a subsidiary of Pullman Incorporated in 1944,
and in 1975 was re-named Pullman Kellogg. In 1977, the Power Piping, Chimney
and Mechanical Construction Operations of Pullman Kellogg became the Pullman
Power Products division of Pullman Incorporated.
1
Acknowledgments
This volume is based on the broad experience, background, and mechanical
engineering accomplishment of The M. W. Kellogg Company in the field of piping
design. It reflects the numerous achievements and contributions of the Company
to effective piping design for high temperature and pressure service. As with the
First Edition, the preparation of this book has been sponsored by the Fabricated
Products Division of which Waldo McC. McKee is Sales Manager. This work
eould be brought to realization only through the cooperation of the entire engineering staff of the Company and, in particular, of the Piping Division.
Certain individual contributions deserve specific acknowledgment. H. Wallstrom
provided the major original contributions to the Kellogg General Analytical Method
and its extensions (Chapter 5 and Appendix A). He was ahly assisted in this work
by Mrs. Catherine R. Gardiner.
Professor E. Orowan of the Massachusetts Institute of Technology, retained
consultant of The M. W. Kellogg Company, is responsible for the contents of
Chapter 1.
J. J. Murphy and N. A. Weil collaborated in composing Chapters 2 and 3 and
assisted in the preparation of Chapters 1 and 7. Chapter 4 is the result of a cooperativeeffort between H. Wallstrom and N. A. Weil; L. C. Andrews is credited with the
writing of Chapter 6.
Credit for the most significant contributions to Chapters 7 and 8 is due to
E. F. Sheaffer. M. Yachter, assisted by S. Meerbaum, prepared Chapter 9 and
Appendix B.
In addition to eredits for Chapters, the following special contributions are
acknowledged. J. J. Rush and M. Hartstein developed The Guided Cantilever
Method of Chapter 4. L. Morrison contributed to the general phases of piping
design. Valuable suggestions were supplied by M. G. Schar on Chapter 8 and by
S. Chesler on Chapter 9. Credit is due to J. T. McKeon for his notable comments
and assistance in reviewing and proof-reading this volume. L. Mylander is to be
commended for co-ordinating portions of this work.
The task of assembling and editing the Second Edition was earried out by
E. F. Sheaffer. N. A. Weil performed the review and inserted corrections for the
second printing of this Edition. The entire project has been under the direction of
D. B. Rossheim, who has guided the design principles and philosophies embodied
in this work.
As is the case with most advances in the engineering art, the First Edition and
this significantly extended Second Edition of Design of Piping Systems have greatly
benefited from the research and contributions of other investigators. Their many
valuable contributions are covered in the lists of referenees at the ends of the various
ehapters and in the "Historical Review of Bihliography" of Appendix A.
R. B. SMITH
Vicc-President, Engineering
The M. W. Kellogg Company
vii
In :Memory of
DAVID B. ROSSHEIM
In all of his career, Mr. Hossheim's ability,
dedication and friendlincss wcre an inspiration
to his associates and won for him cyeryone's
affcction and respect.
.'
Contents
.~.
Nomenclature
Chapter 1
Strength and Failure of Materials
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Stable and Unstable Deformations
Plasticity
A. Plastic Deformation under Uniaxial Stress,
2; B. Triaxial Stress: Yield Conditions, 3;
C. Plastic Stress-strain Relationships for Triaxial Stress, 4.
Failure by Plastic Instability
A. Instability of Plastic Extension: the Ultimate Tensile Strength, 5j B. Instability of the
Plastic E,.;pansion of Tubes, Vessels, and
Plates, 6; C. Ultimate Stress and Working
Stress, 7.
Creep
A. The Andrade Analysis of the Creep Curve,
8; B. Transient Creep, 9; C. Viscous Creep,
10j D. Creep under Triaxial Stress, 11; E. The
Mechanism of Creep, 11; F. Evaluation and
Engineering Usc of Creep Testa, 12; G. Creep
Fracture, 13.
Types of Fracture; Molecular Cohesion; the
Griffith Theory
Ductile Fractures
The Brittle Fracture of Steel (HNotch Brittleness")
Fatigue
A. General Features, 20; B. The Mechanism
of Fatigue, 22; C. Influence of a Superposed
Steady Stress, 23; D. Influence of a Compound State of Stress, 25; E. Influence of
Notches and of Surface Flaws, 25; F. Fatigue
Testa on Specimens VB. Fatigue Tests on Structural Parts, 26; G. Periodically Varying
Thermal Stresses, 26; H. Thermal Fatigue, 27;
J. Damage by Overstress, 27; K. Corrosion
Fatigue, 28.
xiii
2.4
I
2.5
I
2
2.6
2.7
5
8
3.1
3.2
3.3
3.4
3.5
3.8
16
20
3.9
3.10
3.11
3.12
3.13
3.14
2.1
2.2
2.3
Codes and Standards
Design Considerations: Loadings
Design Limits, Allowable Stresses, and Allowable
Stress RangCB
30
30
4.4
4.5
4.6
32
4.7
34
47
48
50
Local Components
52
Pipe Bends: Structural Loading (Static and Cyclic)
Pipe Bends: Internal Pressure
Miter Bends
Bends and Miters: Summary
Branch Connections: Static Pressure Loading
Branch Connections: Repeated Loading
Branch Connections: Comparison with Code Requirements
Branch Connections: Practical Considerations and
Summary
Corrugated Pipe
Bolted Flanged Connections: General Background
Bolted Flanged Connections: Practical Considerations
Joints Between Dissimilar Materials
Other Components
Piping and Equipment Intereffects
52
Chapter 4
Simplified Method for Flexibility Analysis
4.1
4.2
4.3
Chapter 2
Design Assumptions, Stress Evaluation,
ond Design Limits
43
Chapter 3
3.6
3.7
13
15
Stress Evaluation
a. Internal Pressure up to 3000 psi Maximum,
43; b. Internal Pressure over 3000 psi, 44;
c. External Pressures, 46; d. Expansion, 47;
e. Other Loading, 47.
Combination of Stress: Stress Intensification and
Flexibility Factors
Evaluation of Deflections and Reactions
Design Significance of Inspection and Testa
60
60
61
62
66
67
69
70
74
77
79
81
83
90
90
Scope und Merits of Approximate Methods
91
Thermal Expansion
Preliminary Segregation of Lines with Adequate
92
Flexibility: Code Rules
94
Selected Chart-form Solutions
97
Approximate Solutions
The Simplified General Method for Squarc-<:orner
102
Systems
Approximating the Effeot of Curved Pipe and
107
Other Components
CONTENTS
x
Chapter 5
Flexibility Analysis by the General
Analytical Method
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.H
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
Scope and Field of Application of the General
Analytical Method
Calculating Aids
General Outline of Operations
The Solution of Simultaneous Equations
Single Plane Cnlculati~ns
Inclined Members and Changes in Stiffness
Circular Members
General Shape Coefficients
The Secondary Term
Effects of Direct and SheRr Forces
Working Planes and Cyclic Permutation
i\Iultiplnne Pipe Lines with Two Fixed Ends
Hinged Joints and Partially Constrained Ends
Skewed I\fembers
Branched Systems
Intermediate Restraints
Calculation of Deformations at any Point
Symmetrical Pipe Lines
Inversion Procedures
Cold Springing
Weight Loading
Wind Londing
Chapter 6
Flexibility Analysis by 1\lodcl Test
6.1
6.2
6.3
6.4
6.5
Chapter 8
Supporting, Restraining, and Bracing
the Piping System
115
115
116
117
117
119
120
123
125
125
127
127
128
129
13·1
145
146
153
157
157
166
170
185
8.1
8.2
8.3
8.4
8.5
8.6
8.7
Chapter 9
Vibration: Pre\'ention and Control
9.1
9.2
9.3
9.4
9.5
198
The Experimental Approach
198
The Routinized ~lodel Test
198
The Kellogg Model Test
200
The Kellogg Model Test Laboratory and Equipment
201
Typical Model Tests
202
9.6
9.7
9.8
Chapter 7
Approaches for Reducing Expansion Effccts:
Expansion Joints
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Introduction
Sources of Excessive Expansion Effects
Approaches for Reducing Expansion Effects
Packed Type Expansion Joints
Bellows Type Expansion Joints
a. Discussion, 214; b. Bellows Details, 214;
c. Support and Protection of Bellows, 216;
d. Fabrication of Bellows Joints, 217; c. Establishing Purchasing Hequirements for Bellows
Joints, 219; f. Materials and Deterioration,
220; (1. Fatigue Basis for Prediding Bellows
Life, 220; h. Testing and Quality Control of
Bellows Joints, 222.
Expansion Joints with Built-In Constraints
Establishing Expansion Joint Movement Demands
210
210
210
210
212
214
223
226
9.9
231
Terminology and Basic Functions
231
Layout Considerations to Facilitate Support
233
The Elements of the Supporting System: Their
Selection and Location
236
Fixtures
243
Pipe Attachments
248
Structures and Structural Connections
251
Erection and Maintenance of the Supporting, Re254
straining, and Bracing System
257
Introduction
257
Fundamental Considerat,ions in Piping Vibration 258
a. Definitions, 258; b. Types of Vibration, 258;
c. Sources of Periodic Excitation, 259; d. Vibration Prevention and Control, 259.
Structural Natural Frequency Calculations
260
a. The Spring-Mass Model, 260; b. Frequency
and Mnss Effectiveness Factors for Different
End Constraints, 261; c. Variable Stiffness and
Variable Mnss, 263; d. Combined BendingTorsion, 264; e. Approximate Natural Frequencies of Pipe Bends \yith Two Members
(Vibration Perpendicular to Plane of Bend),
265; j. Plates and Hadial Mode in Pipe, 266.
Structural Resonance and Magnification Factors 267
DarnplOg of St.ructural Vibrations
270
Q. Hydraulic Snubbers, 270; b. Elastic Foundations for Rotating I\Tachinery, 271.
Acoustic Natural Frequency Calculations
273
a. The Organ Pipe and Resonators, 273;
b. Special Cases of Multiple Resonator FormuIns,274; c. Piping Systems with Branches and
Enlargements, 276.
Acoustic Resonance and Magnification Factors
277
Flow Pulsation Smoothing
279
a. Tuned Resonators, 279; b. Surge Tanks,
279; c. Gas Pulsation Dampener Principles,
280; d. Acoustic Expansion Tank, 281; e. Comparison of Gas Pulsation Smoothing Devices,
282; f. Hydraulic Hammer, 283; g. Magni':'
tude and Direction of Forccs on Piping Bends,
285.
Illustration of Vibration Analysis of a Simple
285
Piping System
a. General Data and Estimates, 285; b. Estimates of Structural Natural Frequencies of
Piping System, 285; c. Estimate of Lower
Bounds of Structural Natural Frequencies, 286;
d. Effect of Elasticity of Machine Foundation,
286; c. Estimate of Hydraulic Snubber Force
and Damping Requirement for Reduction of
Amplitude of Vibration, 287; j. Resonance
Effect due to Wind Velocity, 287; g. Estimate
of Acoustic Nntural Frequencies, 287; h. Estimate of Acoustic Frequency of the System
Corresponding to its First Harmonic (2nd
Mode), 288; i. Estimates of Some Possible
Resonator Frequencies, 288; j. Estimate of
CONTENTS
n.lO
Volume and Pressure Drop Hequirement of
Hydraulic FiI\ers (Bottles) in the Compressor
Di~charge Lines, 290- k. Tuned Resonator
....
Geometry,290.
Piping Vibration "Trouble Shooting"
291
a. Background, 291; b. Vibration Measurement, 292; c. "Trouble- Shooting" Procedure,
293.
Appendix A
History and Dcdvation of Piping
Flexibility Analysis
A.l
A.2
A.3
Appendix B
Derivation of Acoustic Vibration Formulas
B.l
B.2
B.3
B.4
295
History of Piping Flexibility and Stress Analysis 295
Bibliography on Piping Flexibility and Stress
Analysis
297
Dp.rivation of the General Analytical Method
299
Multiple Resonator of nth Order
General Characteristic Equation for a Branched
Piping System
Tuned Resonator Relations
Simplified Surge Filter Analysis
Appendix C
Charts and Tables
C- 1 Properties and Weights of Pipe
C- 2 Thermal Expansion, Carbon and Alloy Steels
C- 3 Modulus of Elasticity, Carbon and Alloy Steel&C. 4 Chart for Criterion in Par. 620 (a) in Code for Pressure Piping ASA B31.1
C- 5 Length of Leg Required, Two-Member System,
Both Ends Fixed, Thermal Expansion in Plane of
Members
328
328
329
331
333
xi
C- 6 Moments and Forces, Two-Member System, Both
Ends Fixed, Thermal Expansion in Plane of
Members
345
C. 7 Length of Leg H..equired, Two-Member System,
Both Ends Fixed, One Support Displaced in the
Direction of Adjoining Member
346
C- 8 Moments and Forces, Two-Member System, Both
Ends Fixed, One Support Displaced in the Direction of the Adjoining Member
C- 9 Length of Leg RequiIled, Two-member System,
Both Ends Fixed, One Support Displaced Normal
to Plane of Members
C-I0 Moments and Forces, Two-Member System, Both
Ends Fixed, One Support Displaced Normal to
Plane of Members
C-11 Required Height, Symmetrical Expansion Loop
C-12 Moments and Forces, Symmetrical Expansion
Loop
C-13 Guided Cantilever Chart
C-14 Correction Factor I, Guided Cantilever Method
C-15 Design Data: Tri~onometric Constants for Circular Members
C-16 Span va. Stress, Horizontal Pipe Lines, Uniform
Load
C-17 Span VB. Natural Frequency and VB. Deflection,
Horizontal Pipe Lines, Uniform Load
C-18 Correction Factors for Use with Charts C-16 and
CC17
347
348
349
350
351
352
353
354
356
357
358
Appendix D
336
336
341
342
343
344
A Mntrix Method of Piping Analysis
nod The Usc of Iligital Computers
359
5A-l Introduction
5A-2 Derivation of the Shape Coefficient Matrix
5A~3 A Matrix Method of Piping Analysis
5A-4 An Example
SA-5 Selected Bibliography
Index
361
362
369
372
378
379
Nomenclature:
Definitions of Principal Symbols
Meaning
Symbol
Horizontal coordinate to midpoint of member
in working plane.
b .. " " .... Vertical coordinate to midpoint of member in
working plane.
c
. Distance of the '.. . orking plane from the origin;
viscous damping coefficient.
Co,C aa , ew... _ Trigonometric constants.
Critical damping coefficient.
Cc •••••••••
d
, Diameter; inside diameter.
e
. Unit linear thermal expansion for a temperature difference a.T; base of Napierian
logarithms.
j
.. Frequency; factor.
in
. Natural frequency.
g
, Gravitational constant.
h
, Bend characteristic (=lR/Tm'l)j pitch of half
corrugation of an expansion bellowsj gradient of pipe supports.
.
Offset range of an expansion joint.
h r •••.
Imaginary unit (= -v=J:).
i
.
k
..
Flexibility factor of pipe in bendingj spring
constant.
1.."
, Length, span of pipe betwecn supports.
m
. Mass.
n
. Material constant, exponent in fatigue equation.
p, , , , , , , ", Pressure (load per unit area).
q" ...... ... Plastic constraint factorj shape coefficient
known as the secondary term.
r
.
Radius.
rio •.•....
Inside radius.
Mean radius.
Tm ••······ .
To • •••••.
Outside radius.
8
.
Shape coefficient; steady stress component.
8 a , Saa, Sl a, etc. Shape coefficients.
t" ........ , Time, thickness.
U, U 01 U' 0, e~. Shape cocfficien~.
V, V o , Vloo, etc. Shape cocfficien~.
w ...
Width, unit weight load.
Unit loads in the X-, y~, and z-directiona
respectively.
X,III z
. Coordinate axes, coordinates of a point.
a
.
Symbol
Meaning
A"""
A.F...
Area; activation energyj free end.
Attenuation factor.
Material constant.
Cold spring factor; ~locity of sound; constant.
Diameter.
Young's modulus of ehsticityj joint efficiency.
Young's modulus of elasticity at ambient
temperature.
Young's modulus of ehsticity at operating
tempcrature.
Forcc.
Force component in the direction of axis
indicated by subscript. Second subscript,
if used, refers to the source of the force.
Shear modulus, diameter of thc effective
gasket reaction on a flange.
Moment of inertia.
Polar moment of inertia.
Constant.
Length,
Moment.
Magnification factor.
Bcnding moment in the plane of the member.
Bending moment transverse to the plane of
the member.
Torsional moment.
Moment component referred to ongm and
about axis indicated by subscript. Second
subscript, if used, refers to the source of
thc moment.
Moment component about axis indicated by
subscript. Second subscript, if used, refers
to the source of the moment.
Any bending moment.
Number of cycles, rpm.
Origin.
Fixed end.
Point, concentrated load.
Quotient, stiffness ratio, flow rate.
B""".
C"""" "
D".""",
E""" " "
E c ••••••••.
F"""".
F:r, FII , FII
<
••
G""" "
j",
J"""",
K",
L"",,,,,,
M"""""
M.F.
M,,,,,,,,, ,
M'b
.
M,
.
MII,MII,MII .
Mo",.,·"
N" "
0"""".
0' .
p"""",
Q"""" "
xiii
NOMENCLATURE
xiv
Meaning
Symbol
Centerline radius of torus or curved member
(pipe bend or elbow)~atio.
Universa.l gas constant.
ii
.
S
. Fatigue strengthj stress, amplitude of alternating tensile stress component; shape
coefficientj Strouhal number.
Bending stress in the plane of the member.
S,
.
Bending stress transverse to the plane of the
S'b
.
member.
St .. ········ Allowable stress for a material at ambient
temperature.
S,
. Allowable stress for a material at operating
temperature.
.
Torsional stress.
S,
Allowable stress range.
SA
..
Resultant bending stress.
SB .
Computed ma.ximum stress range.
SB
.
Ultimate tensile strength (conventional
Su
.
stress).
Temperature, amplitude of alternating shear
T ......
stressj period of vibration.
Velocity, energy; shape coefficient,
u ..
Volume; shape coefficient.
V
..
Total uniform load.
W .
Yield stress in uniaxial tension; resultant
y
..
expansion.
Section modulus.
R .....
z
..
Meaning
Symbol
Surface energy (work for creating new surface
per unit area); angle; coefficient of linear
expansion.
Longitudinal stress intensification factor;
angle.
Shear strain, transverse stress intensification
factor, ratio of specific heats.
Transhtory displacement; deflection.
Normal (tensile or compressive) strain.
Logarithmic strain.
Principal strains.
Viscous damping coefficient (damping ratio).
Coefficient of viscosity.
Angle.
Wave length.
Acoustic conductivity.
Poisson's ratio.
Density.
Normal (tensile or compressive) true stress.
Principal stresses (true).
Shear stress.
Angle.
Angle.
Angular frequency.
Restrained linear thermal expansion.
a ..
p.
')'
.... , .....
0..
E •••.
E· . . . . . .
.
r· .
..
O• ..
x...
"..
P . . . . ..
p
.
tI'
..
ttl, u:!. tl'3 • ••
T
..
4>
..
..
",
w •••..•
A
<1>
..
.
.
Angle.
CHAPTER
I
Strength and Failure of Materials*
ally, though inappropriately, entitled "Strength of
Materials"). In the present chapter, only the conditions of failure hy non-elastic deformation or fracture
will be considered in detail. Failure by excessive deformation will be discussed in the first four sections, and
failure by fracture in subsequent parts of the chapter.
N the simplest cases, the failure of a structural
part occurs when a certain function of the stress
or strain components reaches a critical value.
The designer must know, then: (a) how the stresses
and strains can be calculated from the applied load;
(b) what are the critical combinations of stress and
strain at which failure occurs.
The first question belongs to the field of applied
mechanics (elasticity, mathematical theory of the
plastic field, and mathematical rheology). In relation to piping systems, it will be treated in detail in
subsequent chapters of this book.
The second question is concerned with the mechanical properties of solids, which is a chapter of
the physics of solids. It is a relatively new field of
science; until about 30 years ago, the mechanisms
of fracture and of plastic deformation were almost
unknown. Since 1920, however, the progress in this
field has been rapid; at the same time, the demands
on the designer's understanding of the mechanical
behavior of materials have gone far beyond what is
generally available in the traditional textbooks.
Hence, it is appropriate to introduce the treatment
of piping system design in this book with a brief but
up-to-date sketch of the mechanical properties of
solids.
Failure of a structural part can occur by
(a) excessive elastic deformation,
(b) excessive non-elastic (plastic or viscous) deformation, or
(c) fracture.
The calculation of elastic deformations and of the
conditions of elastic instability is the main subject
of books dealing with applied elasticity (tradition-
I
1.1
Stable and Unstable Deformations
A structure ceases to be serviceable if it suffers
excessive deformation. The deformations leading
to its failure may be elastic (i.e., deformations that
disappear when the stress is removed), or nonelastic; the latter may be plastic (i.e., depending only
on the deforming stress but not on the duration of
its action), or they may represent a creep (Le., they
may increase or decrease with time at constant
stress).
Moderate deformations (elastic or non-elastic)
may be beneficial in that they can redistribute the
stress in a structural part or between several structural parts and so prevent its rise to levels at which
fracture can occur.
In many cases, the deformation leads to changes
of the shape of the body that cause an increase of
the stresses produced by a given load. The simplest
examples of this are elastic buckling, and the plastic
extension of a rod in the course of which its cross
section diminishes and the stress for a given load increases; if this increase is not counterbalanced by
strain hardening, it leads to accelerated disruption.
Such phenomena represent an elastie instability if
the deformation is elastic, and a plastic instability
if it is essentially plastic. Plastic instabilities are of
great importance in the design of tubes and pressure
vessels.
In what follows, failure by plastic instability will
be treated separately, after the section dealing with
"Prepared by Dr. Egan Orowan, George Westinghouse
ProCessor of M~banical Engineering, Mnssachusetta Institute of Technology.
1
DESIGN OF PIPING SYSTEMS
2
q
q
,
o
FIG, 1.1
n
Yield stress-strnin curve of copper in compression.
After Cook nnd Larke [I),
FIG. 1.2
Stress-strain curve of the Uideally plastic" material.
A familiar type, the stress-strain curve of copper, is
plastic failure without instability.
failure by creep will be considered.
Subsequently,
1.2 Plasticity
A. Plastic
Deformation
under
Uniaxial
Stress. As mentioned above, pure plasticity is defined as a non-elastic type of deformation without
time influence. In uniaxial deformation, the plastic
strain, is determined by the value of the stress <r at
which the deformation takes place
<r =
f(')
(Ll )
Elastic deformations also obey a law of this form;
however, they are reversible, while in plastic deformation the relationship (eq. 1.1) is valid only for
increasing stress. When the stress is reduced, the
plastic strain remains approximately unaltered.
By its definition, pure plasticity means the absence of creep. No material satisfying this requirement is known; however, the behavior of ductile
metals and other crystalline materials at not too
high temperatures (compared with their melting
point) can be described approximately as plastic.
The stress required for plastic deformation (often
denoted by Y) is the yield stress. 1 Its dependence
(eq. 1.1) upon the preceding plastic strain is represented graphically by the "stress-strain curve" (more
accurately, it would be called the yield stress-strain
curve). The stress-strain curves of metals cannot be
represented by a simple mathematical expression.
For strains that are neither too small nor too large,
they can often be approximated by a parabola
(j
=
shown in Fig. 1.1.
For the calculation of the distribution of stress
and strain in plastieally deformed bodies, drastically
simplified types of stress-strain curve must be used.
Except in a few of the simplest cases, it is usually
assumed for this purpose that yielding starts suddenly when a critical stress value is reached, and
that it progresses thereafter at a constant stress-in
other words, that there is no strain hardening.
Figure 1.2 shows the corresponding stress-strain
eurve of the "ideally plastic" material. It must be
kept in mind that such a curve represents a sensible,
though rough, approximation only if the plasti("
strain is large compared with the elastic strain. In
the initial part of the stress-strain curve of a typical
metal (compare Fig. 1.1), the deviation from the
elastic line increases gradually and the idealized
curve (Fig. 1.2) does not represent an approximation.
A few materials (notably, low-carbon steels) show
the so-called "yield phenomenon": plastic deformation starts suddenly when the stress reaches the
value of the "yield point." After its start, the stress
required for further deformation may remain constant for a time, or drop immediately to a lower value
(the "lower yield point"), as shown in Fig. 1.3. If
such a stress drop occurs, the initial yield point is
called the "upper yield point."
Of particular interest to the designer is the stress
at which the plastic strain (or the total strain)
constant X En
At small plastic strains, as well as at very large ones,
however, the stress-strain curve is usually quite dif-
ferent from the parabola representing it for moderate
strains. In addition, the stress-strain curves of
different metals are, as a rule, different in character.
lIn the treatment of plasticity, the term llyicld stress"
means the stress required for (initiating CT continuing) plastic
deformation; owing to the presence of strain hardening, it
changes ,vith the plastic strain.
o
FIG. 1.3
•
Yield strcss--strain curve of an annealed
100v~carbon steel.
«
STRENGTH AND FAILURE OF MATERIALS
reaches the maximum permissible value. If the
stress-strain curve is of the character shown in
Fig. 1.1, the value of the yield....tress at which the
strain reaches some specified permissible amount
(e.g., 0.2% or 0.02%) is called the 0.2% (or 0.02%)
Ilyield strength" or "proof stress." Since the word
"strength" is reserved in scientific usage for the
fracture stress, the term "proof stress" will be used
in the present chapter. If the yielding is discontinuous, as in Fig. 1.3, the entire range of commonly
permissible strains, up to 1% or even 3%, lies on the
horizontal part of the curve; in this case, the lower
yield point takes the place of the proof stress. The
upper yield point is a capricious quantity which can
be obliterated by relatively small stress concentrations or small plastic deformations, so that the
designer cannot rely on it.
Naturally, the proof stress is altered by preceding
plastic deformation ("cold work"). Let OBD be
the stress-strain curve of an annealed metal and OE
the elastic line (Fig. 1.4); A is the point at which a
critical strain of, say, 0.2% is reached. After straining in tension to B and removing the load (point C),
a material is obtained of which the stress-strain
curve in tension is CFD. The point F at which the
permissible strain of 0.2% is reached is now higher
than A, owing to the preceding strain hardening.
On the other hand, if the same material, prestrained
in tension to B, is subjected to compression, the
microscopic residual stresses remaining in it give
rise to perceptible plastic deformation even at very
low compressive stresses, and the stresswstrain curve
in compression CG deviates from the elastic line
strongly from the beginning. This softening of the
material to reverse deformation is called the "Bauschinger effect." The hysteresis loop BCF observed
when the stress is removed and then applied again is
essentially the same phenomenon, due to directional
microscopic residual stresses in a plastically strained
material.
A mild heating (stress-relieving) after the deformation removes the residual stresses responsible for
the Bauschinger effect and restores the proof stress
for reverse deformation more or less to the increased
level of the proof stress for deformation continuing
in the initial direction.
B. Triaxial Stress: Yield Conditions. So far,
only uniaxial stressing has been considered. If a
general (triaxial) state of stress is present, with
principal stresses 0"1 ~ 0"2 ~ U3, yielding in a material without a sharp yield point occurs when a certain
mathematical expression containing the principal
stresses reaches a critical value. Of several "yield
3
o
f
o
,
G
FIG. 1.4
Increase of the proof stress by cold work; the
Bauschingcr effect.
conditions" suggested, only two have been found
compatible with observations and at the same time
simple enough for practical use: the Tresca (maximum shear stress) condition, and the von Mises
(maximum octahedral stress) condiHon.
The Tresca yield condition [2] a.ssumes that
yielding occurs when the maximum shear stress,
equal to one-half of the difference between the
algebraically greatest and smallest principal stresses,
reaches a critical value. It is expressed by
", -
"3 =
Y
(1.2)
where Y is the yield stress in uniaxial tension or com...
pression. With the Tresca condition, the inter
mediate principal stress ha.s no effect on yielding.
The Mises yield condition [3] assumes that yield.
ing occurs when the "effective" shear stress2
T,U=
I
~/
2
2
;:;V ("'-"2) +("2-"3) +("3-"')
2v2
2
(1.3)
reaches the critical value of the yield stress in pure
shear, i.e., one-half of the yield stress Y in tension
Expressed in terms of the uniaxial yield stress Y, it
can be written as
1 ~/
2
2
Y= V'2V ("1-"2) +("2-"3) +("3-"')
2
(J .4)
2The Hoctahedral" shear atreas differs from the right-hand
aide of eq. 1.3 by having the fnetor !, instead of 1/2 V2, before
the square root. The factor 1/2V2haa the convenien~e that it
makes the right-hand side of cq. 1.3 equal to the maximum
shear stress in the case of a uniaxial stress, i.e., for 0"2 = 0"3 "'" O.
DESIGN OF PIPING SYSTEMS
4
The Mises condition is often called the "shear strain
energy condition," since, in an isotropic material,
the right-hand side of eq. 1.3 or ·>leA is proportional
to that portion of the total energy which corresponds
to the shear deformations. For anisotropic materials,
however, the shear strain energy depends in general
upon the hydrostatic componcnt (pressure or tcnsion) of the state of stress [4]. The attainment of a
critical value of the shear strain energy, therefore,
cannot be a eondition of plastic yielding, which,
except at extreme pressures, is not influenced by the
hydrostatic eomponent of the stress.
A eharacteristic feature of the Mises condition is
that the intermediate princip,l stress· has an influence
on the occurrence of yielding. Only if is equal to
the highest or lowest principal stress does eq. 1.4
coineide with the Tresca condition (eq. 1.2). The
greatest divergence between the two conditions is
present when the intermediate prineipal stress
is
the mean value of the extreme ones
u,
u,
u,
= ~(UI
+ U3)
In this ease, eq. 1.4 beeomes
2
V3
Y = Ul - U3 ~ 1.15Y
(1.5)
That is to say, the maximum principal stress difference at yielding is about 15% higher according to
thc Mises condition than that given by the Tresca
condition.
Experimcnts indicate that the behavior of metals
with no sharp yield point, as a rule, is intermediate
between the Tresca and the Mises yield conditions,
usually somewhat closer to the latter. For mathematical investigations of stress and strain distribution in plastically deformed bodies, the Mises condition is often simpler to handle.
For materials with an upper and a lower yield point
there is no reliable criterion for the onset of yielding
at the upper yield point, sinee this quantity is
f'xtremely sensitive to slight non-uniformities of
stress distribution and to the size of the specimen {5].
As mentioned, however, the upper yield point is of
little importance to the designer, since the allowable
stress must be based on the lower yield point, which
is the stress required for the first Ltiders' bands to
widen. From this it follows at once that the yield
eondition in this case eannot be the Mises condition.
Since the Ltiders' bands are sheared layers embedded
between still rigid blocks of the material, only the
shear stress aeting in their plane can cause them to
become thicker, and the intermediate prineipal
stress which is parallel to the Ltiders' layer and
perpendieular to the direetion of shear in the layer
must be ineffective. Consequently, the appropriate
yield condition in this case must be closer to the
Tresca condition.
C. Plastic Stress-strain Relationships for
Triaxial Stress. In the preeeding section, the eonditions of plastic yielding were considered. If they
are satisfied and yielding occurs, the question of
importance to the designer is how the resulting
strains are determined by the applied state of stress.
The difficulties of this problem become evident if one
considers the faets that the resulting deformation
depends on the sequence in which the stress components are applied, and that, owing to the Bauschinger effect, the slightest deformation destroys the
initial isotropy of the material and makes reverse
deformation easier than eontinued deformation. A
plausible solution has been given only for the simple
case of an idcally plastic isotropic material (strain
hardcning and the Bauschinger effeet being ignored).
According to this solution, a given triad of principal
stresses 0"1, 0"2, U3 is related to the increment of
plastic strain arising during its application; this
increment is to be added to the plastic strains
created by preceding actions of stresses. Aecording
to Levy and Mises,
0'1 = OX{UI - ~(U2 + U3)]
+ u,)]
0'3 = OX{U3 - ~h + u,)]
0" = oX{u, -
!<U3
(1.6)
where OEt, OE2, OE3 are simultaneous increments of the
principal strains, and 0).. is a parameter determining
the extent of the deformation. The Levy-Mises
equations determine only the ratios of the principal
strain increments; the absolute amounts depend on
how long the straining is continucd at the constant
principal stresses 0"1, 0"2, 0"3.
In the literature, occasionally the stress-strain
relationship
+ U3)]
" = X[u, - ~(U3 + Ul)]
'3 = X[U3 - ~(UI + U2)]
'I = X[UI - ~ (U2
(1.7)
If the principal stresses remain invariant
during the deformation, these equations represent
simply the integrated form of the Levy-Mises equations; if not, they are incorrect. These equations are
sometimes referred to as the "deformation theory,"
as contrasted with the Levy-Mises "incremental
theory."
For strain-hardening materials, several authors
is used.
•
STRENGTH AND FAILURE OF MATERIALS
5
have suggested the generalized stress-strain relationship
Torr = f(-Yorr)'~
(1.8)
where Toff is the effective shear stress defined by
eq. 1.3, and -yorr the effective shear strain defined by
the analogous equation
-Yorr= ~v (El -E,)'+ (E,-E3)2+ (E3 -El)'
V2
(1.9)
Equation 1.8 has not yet received sufficient experimental verification; it can be a satisfactory approximation only if the anisotropy due to preceding
plastic deformation can be neglected.
o
+1
-I
l'tt"
FIG. 1.5 Considcrc's geometrical construction of the maximum load point and of the ultimate tensile stress.
Differentiation of eq. loll gives
dl/l = -dA/A
1.3
Failure by Plastic Instability
A. Instability of Plastic Extension: the Ultimate Tensile Strength. Like elastic, so plastic or
viscous deformation may also lead to buckling, e.g.,
of a compressed column, or of a thin-walled tube
under external pressure. The treatment of such
cases is analogous to that of elastic buckling, but the
literature of plastic and viscous buckling is relatively
small. For details, reference should be made to the
published literature [6].
A ease of plastic instability of great historical and
practical importance is that occurring in the tensile
test. Initially, the extension is uniform; unless
fracture intervenes, however, the tensile load reaches
a maximum in the course of the test, and at the same
time a neck begins to develop. Further extension is
then concentrated in the neck and ceases everywhere
else in the specimen. The maximum load, divided
by the initial cross-sectional area, is called the
Hultimate tensile strength ll or "ultimate tensile
stress"; its significance for engineering design will
be discussed in detail in Part C of the present section.
Let u = U(E) be the equation of the (true) yield
stress-strain curve of a purely plastic material in
uniaxial tension; the strain used is the linear strain
defined as
E =
(I - 10 )/10
(1.10)
where I is the current length of the tensile specimen
and 10 its initial length. Since the volume]! does not
change significantly during plastic deformation, the
product of length I and cross-sectional area A in the
range of uniform extension remains constant:
lA = IoA o = ]!
(1.11)
The load F = uA reaches a maximum when
Combination of this equation with eq. 1.13 leads to
(1.14)
du/u = dl/l
Equation 1.10 can be written as
I = 100 + E)
from which
dl = lodE
From the last two equations
dl/l = dE/ (I + E)
Introduced into eq. 1.14, this results in
du/dE = u/(I + E)
(1.12)
du/u = -dA/A
(1.13)
or
(1.15)
Equation 1.15, representing the condition for the
load to reach a maximum during the tensile test,
has a simple geometrical meaning. Let the stressstrain curve dE) be plotted in Fig. 1.5, and let the
point P on the negative strain axis have the distance
1 from the origin; i.e., the same distance as the
point Q on the positive strain axis representing
E = 1 = 100% extension.
For any point of the
stress-strain curve, du/dE is the gradient of the
tangent line, and u/(1 + E) the gradient of the line
connecting the point (u, E) with the point P. The
condition for the load maximum is equality of these
gradients; i.e., the maximum occurs at the point M
in which a line drawn from P is tangent to the stressstrain curve. The ordinate AM of the point of
contact is the (true) stress at maximum load; OA is
the tensile strain Eu at maximum load. This theory
of the maximum load point was given by Considere
in 1885 [7].
The ultimate stress," defined as the maximum load
divided by the initial cross-sectional area,
Su = Fmox/A o
dF=udA+Adu=O
(1. lOa)
(1.16)
3Since in the scientific treatment of this field the word
"strength" ought to be reserved to a. fmcture stress, the ultima.te strength will he-nceforth be called Uultimnte stress."
DESIGN OF PIPING SYSTEMS
6
q
p
,.
_-,O_,.jA
FIG. 1.6 Determination of the instability stress on the true
stress logarithmic strain curve in tension.
is not identical with the true stress at maximum load
O'm
= Fmnx/A
(1.17)
due to the decrease of thc load-carrying cross
section occurs also when a tube or a hollow sphere is
subjected to internal pressure [8, 9J. It is remarkable that the instability condition in these cases is
not identical with that for the rod under tension, and
the maximum prcssure withstood by the tube or the
spherical shell cannot be derived from the knowledge of the ultimate tcnsilc stress. In view of the
practical importance of these cases, their characteristic features should be pointcd out.
For a hollow sphere of radius r and (small) wall
thickness t, under an internal pressure p, the tensile
stress q is given by
pr z... = 27rftu
(1.21)
The volume of the shell is
The relation between them is
S./um = A/A o
vV /4...t
(1.23)
r =
S./um = 10 /1
Substituting eq. 1.23 into eq. 1.21 and observing that
the volume remains constant during plastic deformation,
in view of eq. 1.ll. According to eq. 1. lOa,
10/1 = 1/(1 + ,)
p = 4v,,·/V tl'u = C,tHu
Consequently,
1
Btl = O"m--1
+ '.
2rp = 2tu
(1.19)
Hence,
d,'=~
1 +,
Substitution of this in eq. 1.15 'gives
du/d,' = u
(1.20)
Figure 1.6 shows the corresponding graphical
determination of the maximum load point from the
logarithmic stress-strain curve: the subtangent PA
at the maximum load point is unity.
B. Instability of the Plastic Expansion of
Tubes, Vessels, and P1atcs. Plastic instability
(1.25)
V = 2...rt per unit of length
and
+',
log, (1 + ,)
(1.24)
For a thin-walled closed tube,
(1.18)
where Eu is the "uniform strain" at the moment of
the load maximum. In Fig. 1.5, PO = 1;
and AM = u m ; from the similarity
PA = 1
of the triangles PMA and PUO it follows, therefore, that the intercept 0 U of the ordinate axis
between the origin and the tangent PM drawn from
P to the stress-strain curve is the ultimate strcss.
A similar graphical construction can be obtained
if the logarithmic strain is used instead of the linear
strain. The relationship between logarithmic strain
E* and linear strain E is
=
(1.22)
hence
which can be written as
"
V = 47rf Zt
p = (2.../V)t Zu = CztZu
(1.26)
For a square plate of edge length I and thickness t,
extended uniformly in all directions in its plane by
tensile forces F acting upon its edges,
F = Itu
(1.27)
and
V = IZt
hence,
(1.28)
For the tensile specimen under uniaxial tension,
alrcady considercd, the corresponding relationship
would be
(1.29)
where t is the thickness of the (round) rod.
It is seen that the pressure p or the force F as a
function of the thickness of thc specimen is given in
all cases by an expression of the type
p (or F) = Ctnu
(1.30)
where n = 2 for the tensile rod and the thin-walled
tube, -~- for the thin-walled hollow sphere, and! for
the uniform-biaxially extended plate.
STRENGTH AND FAILURE OF MATERIALS
The maximum load or maximum pressure at whieh
the extension heeomes unstable is obtained from
7
Yield SIr01$ (f
dp (or dF) =0
In view of eq. 1.30, this means
nln-l".
dt + tn d". = 0
o
or
V,
n(dt/t) = -d"./".
E-
Sirain
( hin-walled tube)
f--_.'';'---l
For the hollow sphere, the tube, and the plate,
(Thin-walled hollow sphere
dt/t = -d,', where " is the logarithmic strain per-
f----;:--:-,- 1 .-:-:--,.-;--1
pendicular to the wall or the plate. Thus, the eondition of instability is
d"./d,' = n".
logorilhmic
(1.31)
For the sphere, this is
(Rod under uniQ:dal tension)
--,,--,----,--'.,,---::-;---:----1
Plolo under two equal mutually perpendicular tensions
FIG. 1.7 Graphical construction of maximum load or maximum pressure in various cases of tensile loading.
d"./d,' = (3/2)".
for the tube
d"./d,' = 2".
and for the pWie
d"./d,' = (1/2)".
For the t=ile rod, dt/t is the inerement of the
transverse logarithmic strain; since the volume is
constant, this is - (1/2)d,', where d,' is the increment of the longitudinal logarithmic strain. Thus,
d"./d,' = ".
as before (cf. eq. 1.20).
Figure 1.7 shows the corresponding graphical
construetion, quite analogous to that in Fig. 1.6,
carried out for the four cases. It shows that the
instability point on the stress-strain curve (true
maximum stress vs. greatest logarithmic strain)
is different for eaeh.
Particularly interesting is the practically important case of the thick-walled cylinder under
internal pressure. The solution of this problem has
first been published by Manning [1OJ; see also
MacGregor, Coffin, and Fisher [11]. The relatively
simple calculation shows that here, too, the pressure
reaehes a maximum as the tube expands plastically,
and then drops. The maximum pressure (often
called "bursting pressure") can be calculated
successfully from thc stress-strain curve of the
material. It is remarkable, however, that it cannot
be derived from a single point of the curve and the
corresponding tangent. In the thick-walled tube,
the strain depends on the distance from the axis;
at any moment during plastic deformation, states
of stress and strain extending over a morc or less
wide region of the stress-strain curve are present.
As a consequence, the maximum pressure cannot be
calculated without the knowledge of the entire stressstrain curve, or at least a substantial part of it.
In other words, the maximum pressure withstood
by the thick-walled tube cannot be derived from
any single Il working stress."
C. IDtimate Stress and Working Stress. The
ultimate tensile stress has served in the past generally, and still serves in many cases, as a basis for
deriving design (working) stresses; for this purpose,
it is divided by a so-called safety factor. Has this
conventional procedure a realistic basis? From the
preceding considerations, the anSwer can be easily
recognized.
There are two types of failure by plastic deformation. In the first, the structure becomes
unserviceable by suffering an inadmissible amount of
distortion; in the second, it is destroyed by plastic
disruption. In many practical cases, the second
possibility either cannot occur (e.g., if the loading is
flexural or compressive), or is of minor importance
beeause the consequences of failure by excessive
distortion are not significantly aggravated by subsequent disruption. In the design of pipes and pressure vessels, on the other hand, a moderate plastic
deformation may be no more than a nuisance;
the danger that must be excluded is disruption
(bursting).
If the practieally important type of failure is due
to distortion, the design must be based on the stress
at which plastic deformation reaches the maximum
permissible value, i.e., on the "yield strength" or
uproof stress." As is seen from the Considere
construction of the maximum load and of the
ultimate strength (Figs. 1.5 and 1.6), there is no
general relationship between the ultimate strength
and the proof stress (or, in the case of the annealed
DESIGN OF PIPING SYSTEMS
8
'-
(')
-1
FIG. 1.8
o
linear Slrain E
Uniform extension (strain outside region of neck)
for different types of materials.
low-carbon steels, the lower yield point); the old
practice of deriving the working stress from the
ultimate strength by means of a fictitious safety
factor has then no justification. A ccrtain exception
to this is the case in which different batches of the
same type of material are compared (e.g., different
deliveries of a low-carbon steel) ; the proof stress, or
the lower yield point, may (but need not) be thcn
approximately proportional to the ultimate strength.
If the only practically important type of failure
is plastic disruption (bur&ting), the working stress
should be derived, as a rule, from the load or pressure
at which plastic instability leading to rupture sets in
(the possibility of brittle or fatigue fracture should
be disregarded in this section; it will be treatcd
further below). The structure is then dimensioned
so that the design load or design pressure is a
certain fraction of the rupture load or bursting
pressure. For a rod under uniaxial tension, the
corresponding working stress is the ultimate tensile
stren~h divided by an appropriate safety factor
(which, in this case, is not a fictitious one).
It is to be kept in mind that the maximum load is
given by the ultimate tensile stress only in the case
of a structural part under uniaxial tension. For a
tube, or a pressure vessel, the maximum pressure
occurs at a (conventional or true) stress that may be
very different from the ultimate stress, as will be
discussed in morc detail in Chapter 2. In exacting
cases, therefore, the maximum load or maximum
pressure cannot be derived from the ultimatc
tensile strcss but must be obtained by accurate
calculation based on -the stress-strain curve, or from
a model experiment. Often, however, this is not
nccessary. If the ultimate stresses for tension and
for the plastic expansion of a tube differ by only
10% to 20%, and the safety factor may be anything
between 3 and 6 according to tradition or codc
regulations, it may not be worth .carrying out an
accurate design stress determination for a structural
part of subordinate importance.
The Considere construction "hows that the
ultimate stress is fundamentally unrelated not only
to the behavior of the material at small, but also to
that at large, strains. In particular, the knowlcdgc
of the ratio between the ultimate and the proof
stress gives no indication of the fracture strain:
fracture may occur immediately after the maximum
load point, or at strains 10 or 50 times higher than
the maximum load strain. The simple tensile test.
in which only the maximum load but not the stressstrain curve is measured, however, may give a
quantity that is extremely useful for judging the
ductility of the material for certain uses. This
q~antity is the uniform extension, i.e., the strain
at which thc load. maximum is reached and necking
starts (GA in Fig. 1.5). Since practically no further
extension takcs place outside the ncck after this
has been initiated, the uniform extension can easily
be measured on the fractured tensile spccimen if
this is long enough to contain parts sufficiently
removed both from the ncck and from the heads'
of the specimen. A material with small uniform
extension (a few per cent) is disrupted easily in
tension and is therefore unsuitable for drawing
operations (wire or deep drawing). At the same
time, however, it may show a high ductility (i.e.,
reduction of area at fracture), so that it may be
eminently suitable to operations involving large
plastic strains without tension. Thus, pure nickel,
tin, or lead are very unsuitable for drawing, but
extremely good for operations like bending or cold
extrusion; austenitic chromium-nickel steels, on the
other hand, have much less ductility but they are,
owing to their large uniform extension, very suitable
for drawing. Figure 1.8 shows how the shape of the
stress-strain curve is related to the uniform strain.
Materials with a fairly sudden yield and littlc strain
hardening afterwards, like pure nickel, lead, or tin.
have sharply hent stress-strain curves of the typc
A; the tangent construction gives for them a small
uniform strain. On the other hand, materials that
strain-harden slowly but steadily in the initial part
of the stress-strain curve, like copper, brass, or
18/8 Cr-Ni steel (type B in Fig. 1.8), havc a large
uniform strain, independent of whether fracture
occurs soon after necking or is preceded by a large
reduction of area
1.4 Crecp
A. The Andrade Analysis of the Creep Curve.
If a matcrial can undergo progressivc deformation
4The U.S.A. standard specimen is not long enough for this
purpose; a useful specimen can be obtained, however, by
increasing its gage length from 2" to 4".
.
STRENGTH AND FAILURE OF MATERIALS
at constant stress, it is said to show creep. The
sunplest type of deformation that eorresponds to
t~is defi?ition is viscosity: a material is ealled purely
VISCOUS If the rate of straining, d·yjdt is a function of
the stress,j(r) and does not depend on the deformation already undergone
d'yjdt = f(r)
~;:~L~t=+b+L
Timo
FIG. 1.10
Purely PIa1tie
SIro;n
d'Y
dt
(1.33)
= 11-
the material is said to show Newtonian viscosity'
the constant ~ is the <XJejficient of .viscosity. Most of
the common liquids are of the Newtonian type.
Th~ creep behavior of metals, particularly at not
too hIgh temperatures, is markedly different from
pure viscosity. If a constant load is applied to a
te~ile specimen (as is usual in technological creep
testmg) and the strain plotted as a function of time
usually curves of type A in Fig. 1.9 are obtained:
S?lid sol~tions with a tendency to develop a sharp
yIeld ~Olnt (a-brass, Monel metal, Nickel silver)
~ay g~ve curv:es of the type C; other alloys show an
mductIOn perlOd, as seen in curve D. However,
~urve A can be regarded as the pure type observed
If no structural changes occur during creep. It
shows that the rapid, almost sudden, extension that
foll?ws the application of the load is followed by a
penod of deceleration; before fracture occurs there
is a period of acceleration, and between the ~eriods
of deceleration and acceleration there is an interval
of constant creep rate which may be quite long, or
may be merely a point of inflexion.
In his analysis of creep, Andrade [12J found that
the final acceleration is usually a trivial consequence
of the increase of stress due to the decrease of crosssectional area in the course of the constant-lord
tension test. If the experiment is carried out at
constant tensile stress, the acceleration disappears in
Frodllro
Slrain
A
ViKOUI
Creep
Croop
Andrade's analysis of the creep curve.
many cases and eurves of type B are obtained. A
period of final aeceleration is frequently observed
even at constant stress; however, it is always due to
structural changes taking place during creep, and so
curve B can be regarded as representing the pure
and simple type of creep curve.
In his pioneering experiments, Andrade has observed that the slope of the straight parts towards
which the creep curve tends asymptotically depends
strongly on the temperature. At sufficiently low
temperature, the asymptote becomes horizontal and
the creep rate vanishes in the course of time. The
period of deceleration, on the other hand, is always
present, even in the neighborhood of absolute zero.
From this, Andrade concluded that the creep curve
(B in Fig. 1.9) represents the superposition of two
essentially different creep processes, which follow
the sudden straining after the application of the load.
The first component is the dccelerating one, the rate
of which disappears with time; this is at present
called transient creep. Superposed to this, at least if
the temperature is not too low, is a constant-rate
creep process, usually called viscous creep because
its rate depends, roughly speaking, only on the
applied stress and not on the preceding amount of
strain. Figure 1.10 shows Andrade's analysis of the
creep process: the observed creep strain is the sum
of the purely plastic (plus elastic) strain which follows immediately the application of the stress, the
transient creep strain, and the viscous creep strain.
B. Transient Creep. At low temperatures
(below, say, one-third of the absolute melting point)
viscous creep is insignificant and transient creep
dominates; hence its alternative name "cold creep."
At high temperatures (in the hot-creep range), the
tranSIent component is often negligible beside the viscous one; hfmce the name "hot creep" for the latter.
____- - - c
In Andrade's original experiments, which were of
relatively short duration, the transient creep curve
could be represented by the expression
'Y = 'Yo + C..;yt
Time
FIG. 1.9
Tron~jllnt
(1.32)
If. the functional relationship is ,imple proportionahty (Newton's law of viscosity),
T
9
Types of creep curves for various mnterialB.
(t = time).
(1.34)
At lower temperatures, however, the logarithmic
expression [13]
'Y = 'Yo + Clog t
(1.35)
DESIGN OF PIPING SYSTEMS
10
ViKOUS
Croep
Rale
SIren
FIG. 1.11
Stress dependence of the viscous creep rate of
lead wires at 17 C. After Andrade.
fits the curve better. All transitional types between
the Andrade formula and the logarithmic formula
can be observed, as well as curves which represent a
more-than-Iogarithmic decrcase of the creep rate.
C. Viscous Creep. The viscous component is
often represented by a reasonably straight curve, as
shown schematically in Fig. 1.10, if the duration of
the test is not very long. Otherwise structural
changes (recrystallization, precipitation, etc.) are
almost invariably present, and then the rate of
viscous creep may increase, decrease, or irregularly
fluctuate in the course of time. This is the basic
factor that makes the extrapolation and practical
use of ereep tests difficult.
The experiments of Andrade [12] have shown that
viscous creep in metals is far from being Newtonian
(eq. 1.33); it is vanishingly small up to a certain
stress region and then increases vcry rapidly with
the stress. Figure 1.11 shows the curve given by
Andrade for the viscous creep rate of lead wires at
17 C as a function of the applied stress. The character of the curve resembles that of the "Bingham
material," an idealized material often referred to in
rheology (Fig. 1.12, in which the stress is plotted as
ordinate according to convention). The Bingham
material is assumed to have a sharp yield point, and
to show linear increase of the strain rate with the
stress above thc yield point. The behavior of
metals at high temperatures differs from that of the
Bingham material in that the increase of the viscous
creep rate with the stress, as shown in Fig. 1.11, is
much more rapid than a linear increase. Expressions suggested for its dependence are, e.g., the
following ones:
dy/dt = AT"
Norton [14J
(1.36)
dy/dt = A(e"' - 1)
Soderberg [15]
(1.37)
It seems certain that no such simple expression
can represent generally a process depending strongly
on complicated structural features of the material.
However, one of the above expressions, or perhaps
another simple relationship, may well be found
accurate enough for practical purposes in the case of
an individual material.
The temperature dependence of viscous crcep
showl:5 a similar picture. Like all thermal reactions,
it is ultimately governed by the Boltzmann expression for the frequency of thermal activations; without further structural complications, this would lead
approximately to an exponential dependence of the
creep rate upon the reciprocal absolute temperature:
dy/dt = Ce- AlkT
(1.391
where A is the "activation energy" for the creep
process, k is Boltzmann's constant = 1.37 X
10-16 erg!" C, and T is the absolute temperature.
It can be shown [17] that,. in Newtonian viscou,
flow, A is practically independent of the applied
stress whereas C is proportional to the stress; on the
other hand, in plastic deformation based on crystalline slip, the increase of the strain rate dy/dt with
the increase of the applied stress is due mainly to the
decrease of the activation energy A with increasing
stress [18, 19]. In the case of crystalline plasticity,
C may be regarded as a constant because its dependence upon the stress is small relative to that of
the exponential. That this is true for the ereep of
metals can be seen in the following way: dy/dt is the
strain per unit of time; its reciprocal is the time
required for unit increase of the creep strain. Now
creep fracture (see subsection G below) takes place
after a strain of 1% = 1/100; the time t elapsing
between the application of the load and fracture i,
related to the mean creep rate rly / dt by
I/I00t = dy/dt
Introduction of this into eq. 1.39 gives
!/I00t = Ce- AlkT
(1.39a)
Shear
Strcss
Yield SIron
or
dy/dt = A sinh (ar)
Nadai [16J
where A, n, and a are constants.
(1.38)
flow Role
FIG. 1.12 Definition of the Bingham material.
11
STRENGTH AND FAILURE OF MATERIALS
and the latter
or, if the logarithm of base 10 is taken,
log(100t/f)
+ log C = O..4 34A/kT
(1.3gb)
According to Larson and Miller [20j, the dependence
of the fracture time upon the temperature for various stresses is often satisfactorily represented by
eq. 1.3gb with values of log C that vary, for different
materials and experimental conditions, between 15
and 23 if t is counted in hours. Thus, log C is in
fact almost constant. Its order of magnitude can
be derived theoretically in a simple way. It is well
known that, for processes of this kind, the activation
energy is always around 1 electron volt (ev) at room
temperature. If it were significantly higher (say,
2 ev), thermal activation would be so sluggish that
the creep rate would become too small to be observable; if it were somewhat lower (say, 0.5 ev), the
creep rate would be too high to be followed experimentally. At room temperature, kT is -lo ev, so
that A/kT = 40. As a representative example, let
it be assumed that the fracture strain! is 4% and
that fracture occurs after 1000 hours. With these
values, eq. 1.3gb gives
log C = 0.434 X 40 - log (25,000) = 13
For A = 1.5 ev, A/kT = 60, log C would be 22.6.
The observed values of C, therefore, correspond to a
range of activation energies between about 1 and
1.5 ev.
It should be remarked that, however narrow the
range of the observed values of log C is, it would be
dangerous to use eq. 1.3gb for extrapolating creep
test results to times exceeding the duration of the
test by a factor of 10 or more, because during the
extrapolated time interval structural changes (e.g.,
precipitation, grain boundary oxidation) may occur
and the permissible stress for a given service time
may be reduced far below the extrapolated value
(see Subsection G, "Creep Fracture").
D. Creep under Triaxial Stress. The problem
of how to obtain the principal creep rates for general
triaxial states of stress has been trcated by Soderberg [15J. His solution is a rational extension of the
treatment of three-dimensional cases in the theory
of plasticity, and is in fair accord with the available
experience. According to Soderberg, the basic viscous stress-creep rate relationship is a functional
relationship between the effective shear-stress and
the effective shear-strain rate, where the former is
Telf =
1 . 1)2
. J;:: V ("I - "2
2v2
+ ("2 - "3) 2 + ("3 - "I) 2
(1.40)
'.....
'Yelf =
1
v'2 y' ('1 - '2)2 + ('2 - '3)2 + ('3 - '1)2
(1.41)
i, <:2, and t3 being the principal strain rates; volume
constancy demands that
'1 + '2 + '3
=
0
(1.42)
Thus, the general viscous creep law would be
Telf = !('Yolf)
(1.43)
analogous to the three-dimensional stress-strain relationship suggested for purely plastic materials (cf.
eq. 1.8). The relative magnitudes of the principal
crecp rates are assumed to be given by the LevyMises equations
'1 = Clu, - !("2 + "3)J
'2 = C["2 - !("3 + "1)1
'3 = C["3 - !('" + "2)]
(1.44)
The common factor C on the right-hand side is no
longer indeterminate as in the case of ideal plasticity:
it is determined by the condition that, if the principal strain rates are substituted on the right-hand
side of eq. 1.43, the correct value of Telf must result.
Details of practical calculations are found in Soderberg's paper.
E. The Mechanism of Creep. Although the
details of the mechanism of transient creep are far
from being clear, there is no doubt that it is a consequence of thermal vibrations enforcing slip when
superposed to a sufficiently high applied stress. In
the course of the creep process, the material hardens
and thermal vibrations are then less and less frequently able to produce local slip; this is the cause
of the gradual disappearance of transient creep.
The fact that transient creep can be observed down
to the lowest temperatures is due to the circumstance
that the applied stress must always be high enough
to cause at least a small amount of sudden plastic
strain before transient creep can be observed. If
it is sufficient to cause slip without any thermal help,
very slight thermal fluctuation should be capable of
producing local slip at the points where the applied
stress is nearly high enough to induce slip without
thermal help.
It has been found that viscous creep itself is a
compound process. At least two different mechanisms can produce it, and often thc two act simultaneously. The first type of viscous creep is called
recovery creep. After thc application of the load,
12
DESIGN OF PIPING SYSTEMS
the rapid plastic defonnotion produces strain hardening which raises the yield stress to the level at
which it equals the applied stress "nd thus can resist
the load. If the temperature is high enough, however, thermal recovery or even recrystallization
gradually reduce the strain hardening. In order to
carry the applied load, therefore, the material must
strain-harden further until the amount of strain
hardening lost by recovery is replaced. This means
that, in every unit of time, additional plastic strain
arises, the amount of which is just sufficient to make
up for the strain hardening removed hy recovery.
The second important type of viscous creep is due
to sliding between the grains of a polycrystalline
metal when a stress acts at a sufficiently high temperature. At low temperatures, the grain boundary
is a strong part of the structure: it resists the slip in
the grains. At a high temperature, however, the
boundary becomes soft and viscous and is an element
of weakness. The tungsten filaments of incandescent
lamps, which work at the highest temperature used
in engineering, can be preserved from gradual deformation by their own weight only by being made of
single crystals, without grain boundaries present.
F. Evaluation and Engineering Use of Creep
Tests. Transient (cold) creep is of great practical
importance, e.o;., in prestressed reinforced concrete
design. However, since its evaluation does not in-
volve complex problems to the engineer, and since
the problems in which it plays a role are somewhat
specialized, it will not be treated here.
In many high-temperature applications of metals,
the viscous creep strain during the lifetime of the
equipment is so much greater than the initial transient creep strain that the latter is frequently neglected (sometimes with no sufficient justification).
In such cases, the usual practical rule is to assume
that the long-time creep rate on which the design
should be based is equal to the "minimum creep
common rule, therefore, has to be supplemented by
the condition that the constant-rate part of the
creep curve must extend over a long time, sufficient
for the disappearance of the transient component,
in order that the minimum creep rate can be identified with that of the viscous creep.
Since structural parts must often have a service
life of 10 or 20 years, whereas crecp tests cannot be
extended in engineering practice beyond about one
year (often they must be obtained within a few
weeks), the extrapolation of creep test results to the
service life is the central problem of creep testing.
Some of the extreme short-time testing methods suggested between the two wars failed because their
authors were unaware of the compound nature of
creep. Unless the test is extended long enough for
the transient component to become relatively small,
it cannot give even an approximate idea of the magnitude of the viscous component. The present conventional methods of creep testing usually avoid this
pitfall; they can be subdivided into the following
three classes:
1. Abridged tests. The creep strain is measured
as a function of time.for a few stresses around the
probable service stress, at the service temperature,
and extrapolated to the service life.
2. Mechanically acceleruled tests. The maximum
pennissible creep strain is enforced within the time
availablc for the test by a suitably increased stress.
From several such tests at different stresses, the
stress is plotted as a function of the time after which
the pennissible strain is reached, and the curve extrapolated to the service life to give the permissible
service stress.
3. Thermally accelerated tests. The maximum per-
viscous component alone, but the sum of the viscous
missible creep strain is enforced within the time
available for the test by a suitably Increased temperature. From such tests at a few different stresses
and temperatures, the stress is plotted as a function
of the tcst temperature and of the time required for
reaching the permissible strain, and extrapolated to
the service life and service temperature.
The abridged test would give a correct extrapolation if structural changes taking place in the material
during its service life could be discounted. Thermally and mechanically accelerated tests are in
principle more likely to lead to errors because they
take place under stress and temperature conditions
and the residual transient creep rates.
different from those in service.
rate" observed in a constant-load tension creep test l
i.e., to the creep rate in the straight part of curve A
in Fig. 1.9. Although in the hands of the experienced
creep practitioner this prescription usually worh
fairly wcll, strictly spcaking it is fundamcntally
wrong. When the minimum creep rate occurs, transient creep mayor may not have disappeared. If it
has not, the minimum creep rate is not that of the
In extreme
cases, solely the acceleration of transient creep, due
to the decrease of the cross-scctionalarea, may give
rise to curves of type A, Fig. 1.9, at low temperatures
where no trace of viscous creep can be present. The
However, occasion-
ally certain structural changes that would occur
during the service life but do not take placc during
the abridged test may be observed in the mechanically or thermally accelerated test. Then these
STRENGTH AND FAILURE OF MATERIALS
tests, although less correct in principle, may lead to
a better extrapolation. No general extrapolation
method can take into account the highly individual
reactions of materials to stress and temperature, and
the likelihood of grossly erroneous results can only
be reduced by an intimate knowledge of the metallurgical, structural, and plastic properties of the
material.
G. Creep Fracturc. The grain boundaries of
polycrystalline metals, being places of atomic disorder, behave like a two-dimensional glass. They
have a softening range of temperature (roughly
identical with the "equicohesive temperature") in
which they change from being.a hard structural
component to being the softest. At very high temperatures their effective viscosity is so low that, at
low stresses, most of the deformation is localized in
them: the grains slide almost as rigid units on their
neighbors. This leads to the opening up of gaps
between the grains, and finally to the type of fracture peculiar to high temperature creep: at first
sight, it appears almost brittle.
The strain at which creep fracture occurs depends
on the stress and the temperature. At low stress and
high temperature the deformation within the grains
is insignificant compared with the effect of sliding of
the grains upon their neighbors, and thus the fracture
strain is small. However, the variation of the fracture strain in a given range of stress and temperature
is always very small compared with the simultaneous
variation of the creep rate. The latter may change
in the ratio 10,000,000 to 1 while the fracture strain
increases, for instance, from 2% or 3% to 10% or
15%. Consequently, the fracture time is usually
inversely proportional to the mean creep rate, to a
fair approximation.
The creep fracture test5 consists ill applying to
the specimen a constant tensile load and recording
the time elapsing to fracture. This test is simpler
and easier to perform than the standard ereep test
because strain measurements are omitted. It is
required for design whenever the material has such
poor ductility under ereep conditions that fracture
may occur before the maximum permissible creep
strain is reached. Since creep strains exceeding 1%
are not often permitted (pressure vessels and pipes
are an exception), and fracture occurring after less
than 1% strain is infrequent, the creep fracture
test is usually unnecessary. It is nevertheless
widely used because it can be interpreted as a crude
creep test. As mentioned above, the fracture strain
varies within relatively narrow limits, so that the
bIn the creep !ester's vernacular, "stress rupture" teat.
13
creep fracture test represents a creep test in which
the time required for a certain strain (the fracture
strain) is measured for various stresses and temperatures. The great shortcoming of the test is not so
much the variation of the fracture strain as the fact
that it is always performed at high stress levels in
order to obtain fracture within 1000 or, at most,
10,000 hours. It has been shown by many experimenters, particularly by Grant and his collaborators [21], that the creep rate may change abruptly
even after 10,000 hours owing to some structural
change (e.g., coarsening of a precipitate, or oxidation). For this reason, extrapolation from highstress short-time tests to the long-time service
behavior is impossible, unless it is known (from a
thorough investigation of the material extending
Dver years) that no structural changes may be
expected in the time interval between the duration of
the routine creep test and the service life.
1.5 Types of Fraeturc; Molccular Cohesion;
thc Griffith Theory
Fracture is the disintegration of a body into fragments under mechanical stresses. If a certain type
of fracture occurs in a given material when a stress
component reaches a eritical value, this is called the
strength or fracture stress. Many types of fracture,
however, do not take place at a characteristic value
of a stress component.
Until about 20 years ago it was not realized that
there are many fundamentally different types of
fracture obeying quite different laws. They can be
classified into two main groups: brittle fractures and
ductile fractures. The former occur with little or no
plastic (or other non-elastic) deformation; the
mechanism of the latter essentially involves plastic
deformation. The mechanism of brittle fracture was
elucidated long before that of ductile fractures,
mainly by the work of A. A. Griffith in 1920 [22J.
Griffith's effort was directed to the e~planation of the
extraordinary discrepancy between the very high
values of strength inferred from the magnitude of
the intermolecular and interatomic forces, and the
observed values of the tensile strength, which are
usually hundreds or thousands of times lower.
The way in which the tensile cohesion of a material
is determined by the attractive and repulsive forces
between its molecules is illustrated in Fig. 1.13.
Suppose that a crystal contains atomic planes with
the spacing b perpendicular to the direction of tension. As the tension is raised, the spacing b increases.
The net interatomic force acting between two parts
of the crystal across the gap between two atomic
DESIGN OF PIPING SYSTEMS
14
At1roclivll force
01---+--;......-==--Intllrmolowlor
Spcu::ing
Ropulsivo Force
FIG. 1.13 The dependence of the intermolecular forces upon
the molecular spacing.
planes vanishes if no tension is applied; in this case,
the attractive and repulsive forces cancel. If a
tension is .applied and the atomic spacing increases,
the repulslVe forces diminish more rapidly than the
attractive ones; the excess of the attractive forces
over the repulsive ones balances the applied tension.
As the atomic spacing in the direction of tension
increases, the repulsive forces become insignificant,
and the tensile force transmitted through the crystal
lattice must then start to diminish with increasing
strain owing to the decrease of the attractive forces
with increasing separation of the atoms. Consequently, the net atomic force transmitted through a
cross section must have a maximum, equal to the
highest external force the material can withstand
i.e., its strength. From the general knowledge
the atomic forces it can be estimated that the maximum must occur when the spacing of the atomic
planes has increased by a large fraction of its initial
value; for an order-oi-magnitude estimate, it may
be assumed to occur when the atomic spacing has
increased by some 25% or 50%. If Hooke's law
were applicable for such large strains, the tensile
strain would be between 0.25 and 0.5 and the corresponding tensile stress, i.e., the molecular strength
of the material,
(1.45)
"m = 0.25E to 0.5E
01
where E is Young's modulus. Instead of approaching the order of magnitude indicated by eq. 1.45, the
measured tensile strengths are extremely low. The
strength of ordinary sheet glass is about 1/1000 of its
Young's modulus; that of rock salt crystals, less
than 1/10,000. It was known to physicists before
Griffith that the most likely cause of the discrepancy
was the presence of invisibly small cracks or other
flaws which produce stress concentrations and thus
raise the applied stress to high local values. It was
Griffith, however, who calculated the critical value
of the applied tensile stress " at which a crack of
atomic sharpness and of length c, starts to propag~te.
He used the following approach. When the crack
extends, the surface area of its walls increases and
this requires energy for overcoming the attr~ctive
forces between the atoms separated by the crack.
If the grips between which the specimen is pulled do
not move during the crack propagation process, the
only source from which the necessary surface energy
can be obtained is the elastic energy released as the
crack extends. Let dS be the surface energy needed
for enlarging the crack by an infinitesimal amount,
and dW the elastic energy released simultaneously.
The crack can propagate only if dW is at least as
large as dS; thus,
(1.46)
dW = dS
is the condition for the crack being ju.st able to
propagate under the tensile stress. It will be seen
that the stress needed for propagating a crack
decreases as the length of the crack increases; once
condition 1.46 is satisfied, therefore, the crack will
extend rapidly, and fracture will occur.
Griffith carried out this idea in the simple case of a
plate containing an internal crack of length 2c
(Fig. 1.14). It can be shown that the effect of such
a crack upon the fracture stress of the plate is equal
to that of an external crack of length (depth) c in
one of the side edges of the plate. A sharp and flat
internal crack of length 2c can be regarded as an
elliptical hole of major axis 2c and an extremely
short minor axis; the stress distribution around it
when the plate is put under a tensile stress " was
calculated by Inglis in 1913 [23]. From this the
excess energy in the plate, due to the presence of the
FIG. 1.14 Plaw with a nat elliptical hole (=crack).
STRENGTH AND FAILURE OF MATERIALS
crack, is obtaincd as
TV = 1r,?c IE
2
.~
per unit thickness of tbe plate, where E is Young's
modulus; if c inereases by dc, the released elastic
energy is
On the other hand, the increase of the length of the
crack is 2dc, and the increase of its wall surface area
is 4dc per unit thickness of the plate; consequently,
if" is the work required for ereating a new surface of
unit area, the inerease of the total surface energy is
dS = 4adc
(1.47)
Equating dTV and dS gives
u = ~2aE
(1.48)
1rC
This is the famous Griffith equation for the tensile
Strength of a brittle material containing an internal
crack of length 20, or a surfaee crack of depth c. In
the calculation, it has been assumed that the problem
is two-dimensional, and that the plate is very large
in both directions, but at the same time thin compared with the length of the crack; if it is thiek, the
factor (1 - v2 ) has to be applied to the denominator
under the square root, " being Poisson's ratio.
For glasses of the ordinary types, the crack length
c necessary to explain the observed tensile strength
is of the order of 1 micron. In glasses, the dangerous
eracks are almost always at the surface; tensilc
stresses confined to the interior are relatively harmless. This is the explanation of the high strength of
"tempered glass," obtained by quenching glass from
the softening temperature by an air blast. By the
time the interior has become rigid, the surface has
cooled down considerably; when subsequently the
rigid interior cools, it puts the surface layers under a
tangential compressive stress. Any tensile stress
produced by external forces is diminished at the
surface by the residual compression. In the interior,
the residual stress is tensile, but this is of no con-
15
stresses. The discussion of the complete answer is
beyond the scope of this chapter [24]; the result is
that, so long as the highest compressive principal
stress is less than three times the highest tensile
principal stress, fracture should occur when the
greatest tensile principal stress reaches the value of
the tensile strength deduced for uniaxial tension
(eq. 1.48); the algebraically smaller principal
stresses have no influence. According to the theory,
the compressive strength should be eight times the
tensile strength if the material is isotropic and contains cracks randomly distributed in all directions.
Thus, the theory confirms partially a well-known
statement found in textbooks on the strength of
materials concerning the condition of brittle failure:
in the essentially tensile region of principal stresses,
failure does obey the maximum tensile stress criterion. However, the maximum tensile stress condi-
tion cannot be valid for any state of stress. If it
were, the compressive strength of brittle materials
would be infinitely high. This shortcoming of the
textbook rule has been corrected by the Griffith
theory, in the way just mentioned.
One of the most important results of the work of
Griffith is the realization that the strength of a
brittle material is determined by the flaws it contains. This is strikingly illustrated by glass, the
strength of which can be made a hundred times
higher than normal, if by a special design (fibre glass)
the worst cracks are made ineffective.
1.6 Duetile Fractures
The Griffith theory and the fracture condition
(eq. 1.48) are applicable only to fracture of the
cleavage type ("brittle fracture"). In addition to
this, there is a large group of fractures in which separation into fragments occurs as a consequence of
certain plastic deformation processes; these are the
"ductile" fractures. The simplest ductile fractures
are straightforward geometrical consequences of
plastic deformation; a wire of gold, e.g., breaks in
tension by the formation of a neck which becomes
thinner and thinner until it is drawn out to two
needle points in contact. Similarly, single crystals
sequence because there arc no sharp cracks present
of zinc or cadmium may break, after slow extension
from which fracture may start. Thus, the strength
of the glass is strongly increased.
The Griffith theory explains very satisfactorily
the strength properties of completely brittle materials sueh as glass; for detailed treatment, reference
should be made to the literature [24].
An interesting feature of the theory is the answer
it gives to the question of strength under triaxial
at a high temperature, when one part of the crystal
slips off the other along a slip plane in which the
deformation has become concentrated.
The nature of the fracture process is less obvious
in the common fibrous fracture of ductile metals,
which produces the bottom of the cup in the cupand-cone fracture. However, it seems to be fundamentally the same type of geometrical attcnuation
DESIGN OF PIPING SYSTEMS
16
Constroincd yield Ilren "Y
Brittle (decvogo) fracture
Ten~ilo
Stresl (J
F _
~ _
~
. - Brit1lo strength B
D Ductile (fibrotn)
froch,lTe
certain precipitation hardened alloys, can be sheared
off during tightening after a small amount of plastic
twist. Another instance is that of extremely creepresistant alloys which may fail by creep fracture at
high temperatures after a relatively small creep
strain.
1.7 The Brittle Fracture of Steel ("Notch
Brittleness")
Plastic Tensilo
Strain E
FIG. l.Ip Scheme of the classical triaxial tension theory of
notch brittleness, after Mesnager [25]. Ludwik [261, and
Orowan [27].
as in the preceding examples, repeated many time"
on a microscopic scale in the surface of fracture.
Shear fracture, which forms the sides of the cup and
the cone, is a somewhat different phenomenon. The
plastic deformation leads here to the propagation of
a crack at the tip of which there is a high concentration of strain, destroying locally the cohesion of the
material.
A ductile fracture cannot obey the Griffith condition (eq. 1.48). This can be realized in the following
simple way: The plastic deformation mechanism
which leads to ductile fracture is not essentially dependent on the elastic moduli of the material; it
could take place even if Young's modulus were infinitely high. On the other hand, cleavage fracture
of the Griffith type would be impossible in a perfectly rigid material; an infinitely high value of E
in eq." 1.48 would give an infinitely high tensile
strength.
One of the conditions governing ductile fracture
can be easily recognized: it coincides with the condition of the particular type of plastic deformation
which is responsible for the fracture. Thus, in the
tensile fracture by neck attenuation the only fracture condition is that the tensile load must reach
the value of the yield stress in the neck, multiplied
by the cross-sectional area of the neck and by the
plastic constraint factor. In shear fracture, too,
this is a necessary condition for the propagation of
the shear crack. Another condition, however, must
also be satisfied: the shear strain at the tip of the
crack must reach the critical value at which the cohesion disappears.
Ductile fractures usually occur after the structure has become unserviceable by excessive plastic
deformation. However, if the material has a low
ductility, shear fracture or other types of ductile
fracture may occur after very little deformation. A
threaded bolt of a low-ductility material, such as
Low-carbon and medium-carbon steels behave ill
a maimer that is not a mere intermediate case between glassy brittleness and high ductility. A common structural steel can be very ductile in the
ordinary tensile test, with no sign of a potential
brittleness, but it can break with little or no visibJ"
plastic deformation if it contains a crack or a notch.
The classical triaxial-tension theory of noteh
brittleness was put forward by Mesnager [25] and,
independently, by Ludwik [26J. In a form modified
according to the present state of knowledge [27J, its
prineiple is illustrated by Fig. 1.15. The abscissa in
this figure is the tensile strain and the ordinate th"
tensile stress; Y represents the ordinary tensile yield
stress-strain curve. The theory assumes that a material suffers brittle (cleavage) fracture when the
tensile stress reaches a critical value B ("brittle
strength") which, in its dependence upon the plastic
strain, is given schematically by the curve B. In
the ordinary tensile test, ductile fracture occurs at
the point D on the curve Y, before the tensile stress
reaches the value of the brittle strength. However,
if the specimen contains a notch or a crack, plastic
constraint raises the value of the tensile stress
reached during plastic yielding to q Y, where q, the
Uconstraint factor," is greater than 1. The curve qY
may intersect the curve of the brittle strength B
Compreuive
Force
I
IT
Fridionat constraint
upon specimen
--I
--
I
I_Tend elley 10
spread
m
FIG. 1.16 The ongm of plastic constraint in a notched
tensile specimen illustrated by the frictional constraint acting
upon a flat compression specimen.
STRENGTH AND FAILURE OF MATERIALS
before the plastic strain is high enough to produce
ductile fracture, and so brittle fracture may occur
at F.
.....
The way in which plastic constraint arises is iilustrated in Fig. 1.16. Suppose that I is a coin compressed plastically between two hard cylinders, I I
and III. The necessary mean compressive stress is
higher than the yield stress Y in uniaxial compression: it has to overcome, not only the resistance Y
of the material to plastic deformation, but also the
frictional resistance of the compression blocks (indicated by the arrows) to the lateral spread of the coin.
The radial frictional forces, together with the axial
pressure, create a state of triaxial compression (n
hydrostatic pressure superposed to an axial pressure).
The mean axial stress reqnired for plastic compression is then not Y bnt qY > Y; of this, Y is required
for the plastic deformation itself, and (q - 1) Y
for overcoming the friction.
Figure 1.16 can also be regarded as representing a
circumferentially notched cylindrical specimen, I
being the notch core and II, III the full sections of
the specimen. If the specimen is plastically extended, the conditions are similar to the case of the
compressed coin, with the shear cohesion between
the core and the adjacent parts of the specimen
replacing the friction. As before, the axial stress
required for producing plastic deformation in the
core must be higher than the yield stress Y.
Plastie constraint is fundamentally different from
elastic stress concentration. It cannot arise without
some preceding plastic deformation; moreover, its
magnitude depends on the depth and sharpness of
the noteh in a very different way. In pure elasticity,
the stress eoncentration at the tip of a notch becomes
infinitely high as the radius of curvature of the tip
converges towards zero. In contrast to this, the
plastic constraint factor of a circumferential notch
such as is illustrated in Fig. 1.16 inereases only to a
value of the order of 3, instead of rising towards
infinity, as the tip radius is redueed to zero [27].
This is the reason why so many ductile metals
cannot be made to fracture in a brittle manner by
the application of a sharp crack: if, for any value of
Work
01
fracture
Tomporature
FIG. 1.17 Extreme types of transition curvcs.
17
TOnlilo
Slreuos
Strongly tonllroinod yield ilro""
opproximolely 3Y
Brittlo Itrongth 8 \
~
~
IT.
~
Yield siren in tensIon Y
I
T
Tomporoture of
comploto ombrittlemenl
.-1
Temperaturo
Tranlition temperaturo
botwoon notch briHI.n~
and full dUdility
FIG. 1.18 Davidcnkov-Wittrnan Theory of the transition
between brittle and ductile fracture, as modified by the author.
the plastic strain, the brittle strength is more than
about 3 times higher than the yield stress, plastic
constraint alone cannot raise the tensile stress to
the fracture level.
An important feature of notch brittleness is the
existence of a transition temperature between notchbrittle, and purely ductile, behavior. Figure 1.17
shows the dependence of the work of fracture, as
measured with a Charpy or Izod pendulum hammer,
on the temperature in low-carbon steels. Above a
certain temperature region it has a high value, and
the fracture of the notched specimen is entirely of
the fibrous type. At low temperatures, the fracture
work is extremely small, and the fracture is entirely
of the eleavage ("crystalline") type. Between these
two temperature regions, there is a transition zone
in which the fracture work drops rapidly with decreasing temperature; at the same time, the area of
cleavage in the surface of fracture increases towards
100 per cent. With some materials, the transition
zone is so narrow that one can speak of a "transition
temperature"; in other cases, e.g., of many lo\\"alloy ferritic steels, it is spread over hundreds of
degrees F.
Figure 1.18 shows schematically how the classical
theory interpreted the transition phenomenon [28].
Y is the curve giving the temperature dependence of
the yield stress; the curve q Y (= 2 or 3 times Y),
therefore, represents the highest tensile stress that
an atOlnically sharp crack can produce during plastic
yielding. Experiments and theory show that the
temperature dependence of the bri ttle strength B
must be less strong than that of Y or q Y; this is
schematically indicated in the figure. It is seen that
the tensile fraeture is entirely brittle below the temperature T., even in the absence of any notch. If 11
notch or crack of ma.ximum sharpness is present,
brittle fracture is possible below the temperature 7'"
but not above it.
18
DESIGN OF PIPING SYSTEMS
Recent investigations [29J have shown that the
fundamental cause of brittle fracture in normally
ductile steels is not plastic~onstraint but the abnormally high velocity-dependence of the yield strcss
of ferritic steels. Thc experiment from which this
can be recognized is as follows: The edge of a lowcarbon steel plate is provided with a brittle crack by
forcing a chisel into a notch at a low tempcrature.
If the plate is subjected to tension at room temperature, it is found that the brittle crack is unable to
propagate as a brittle crack. Instead, large plastic
deformations arise around its tip, accompanied by
some fibrous crack propagation; after this, the frac-
ture suddenly reverts from the ductile to the cleavage
type and the newly created brittle crack runs across
the plate. This shows that, at low rates of straining,
plastic deformation in microscopically small regions
around the tip of a brittle crack cannot create the
degree of triaxiality of tension necessary for brittle
fracture; quite large deformations, such as can be
seen with the naked eye and felt with the fingers,
arc required.
However, once cleavage cracking
starts again, it runs at high speed and without large
plastic deformations.
The simplest interpretation of these/,bservations
is that in the brittle fracture of steel the stress is
raised to the level of the brittle strength by the high
rate of plastic deformation around the tip of a running crack rather tban by plastic constraint. Without a sufficiently high velocity of the crack, the
production of the plastic constraint necessary for
cleavage fracture requires such extensive plastic
deformations that the fracture, though of the cleavage type, is far from being brittle, i.e., of low energy
consumption. Triaxiality of tension, then, is probably no more than one of several ways of initialing
cleavage fracture; the cleavage fracture is then trans-
formed into brittle cleavage fracture by the velocity
effect upon the yield stress as the crack gathers speed.
The rather exceptional combination of ductility
witb potential brittleness in steel may be understood
now as being a consequence of another exceptional
property of low-carbon steels, the unusually strong
dependence of their yield stress upon the rate of
straining [30, 31]. The yield stress of copper or
aluminum increases only some 10 to 20 per cent between "static" and ballistic testing speeds; for lowcarbon stecls, however, increases of 100 and 200
per cent have been recorded.
Why such large deformations are needed for starting cleavage fracture at the tip of a crack under slow
tension is a question not yet answered. It has been
suggcsted that plastic constraint alonc cannot raise
the tensile stress to the fracture level in typical cases
of notch brittleness under static loading; it must be
aided by strain hardening, and this requires considerable plastic deformation. Howcver, brittle
fracture can start in a welded structure with very
little plastic deformation. The plastic strains produced by thermal expansion and contraction during
welding and the corresponding strain hardening can
hardly be made responsible for this, because the
thermal strains seem too small to take the material
beyond the region of yield into that of strain
hardening.
The final question is this: What is the condition
under which thc cleavage crack arising from the intermediate ductile crack in static loading becomes a
rapidly running crack, in which the velocity-increase
of the yield stress can replace the heavy plastic de-
i
i'
!
formation necessary around a slowly extending crack
to produce cleavage? A crack can run rapidly under
static load only if the work rcquired for its propagation is obtained from the elastic energy stored in
the specimen. It was seen in Section 1.5 that the
Griffith equation (1.48), by virtue of its derivation,
is the condition for the crack propagation work to
be supplied from the released elastic energy; however, it cannot be applied directly to brittle fracture
in steel. It has been found [27J that cleavage fracture in low-carbon steel around room temperature
is not quite brittle; there is a thin cold-worked layer
at the surface of fracture, representing an energy of
cold work of about 2 X 106 ergs/cm2 • This is around
1000 times greater than the surface energy of steel;
the work of crack propagation per unit area of the
crack walls, therefore, is given by the plastic surface
work p, beside which the surface energy is negligible.
If the plastic surface work per unit area of the cleavage fracture can be treated on the same footing as
the surface energy, the condition for the work of
propagation of a brittle crack in steel to be supplied
by the simultaneously released elastic energy is
[24, 32J
(1.49)
f
I
instead of the Griffith equation (1.48). In eq. 1.49
the factor ;/2/11' has been omitted to indicate that
the equation does not pretend to be accurate enough
for this factor to matter.
Brittlc cleavage fracture in steel, therefore, requires the fulfilment of two conditions:
1. The temperature must be below the transition
range;
2. The applied stress must satisfy the crack propagation equation (1.49).
j
STRENGTH AND FAILURE OF MATERIALS
The first condition is satisfied by most structural
steels, at least at low winter temperatures. The designer, therefore, can avoid the possibility of brittle
fracture only by taking care that the crack propagation condition should not be satisfied. The simplest,
although practical1y not always easy or even feasible, way to do this is to avoid the presence of cracks
exceeding in length a certain limit. The smal1er the
crack length c, the higher is the (mean) tensile
stress <T in the plate at which crack propagation is
possible. Since the stress cannot rise above the
yield stress Y, the length of the smal1est crack that
can start brittle fracture is obtained from eq. 1.49 as
Co =
Ep/Y'
(1.50)
Cracks below this length are harmless (unless, of
course, they can grow by a non-brittle mechanism
which does not require the fulfilment of the crack
propagation condition, eq. 1.49). If, therefore, the
possibility of cracks exceeding in length the critical
value Co can be eliminated by careful fabrication or
inspection, brittle fracture cannot occur even below
the transition temperature. With E=3X107 psi=
2 X 10" dyne/em', p = 2 X 10· erg/em', and
Y = 6 X 104 psi = 4.1 X 109 dyne/em' for the
strain-hardened steel, the critical minimum crack
length Co is obtained from eq. 1.50 as
Co = 0.25 em = 0.1 in.
To avoid any crack exceeding this length is difficult
and costly, but not impossible, as is shown by the
occasional use of non-aging low-carbon steels for
pressure vessels at liquid-air temperatures.
Alternatively, the designer may attempt to keep
the stress level so low that eq. 1.49 is not satisfied
even though the longest cracks unavoidably present
might exceed the critical length Co. If, e.g., the
presence of cracks of 0.4 in. length cannot be excluded, the tensile stress must be kept below 30,000
psi; cracks of 1 in. length would set an upper safe
limit of about 19,000 psi to the strcss, and so on.
Natural1y, the propagation condition (1.49) may
not be the only condition that must be satisfied
before brittle fracture can occur. If eq. 1.49 is correct, brittle fracture cannot occur below the stresses
derived from it; however, some other, more exacting
condition may in some cases set a higher limit, so
that fracture in fact may not occur at stresses as low
as correspond to eq. 1.49. A simple example of this
is the case of a steel plate containing a brittle crack
and subjected to slowly applied tensile stress, as in
the experiments described above. Although the
stress given by eq. 1.49 may be quite low, the crack
19
cannot start to propagate before considcrable plastic
deformation takes place around its tip, and the
stress required for this may be quite elose to the
yield stress of the plate. In other words, in this
case an initiation condition must be satisfied besides
the propagation condition, and the former is more
exacting.
In recent experiments 133] in which the difficulty
of crack initiation was overcome by a wedge hammered into the crack by the impact of a bullet, fracture could not be provoked below a fairly clearly
recognizable stress level which depended on the conditions of the experiment (notch angle, plate size,
etc.). Since the mechanics of the crack initiation by
wedge impaet is very eomplex, it is diffieult to reeognize the significance of this result. The observed
stress threshold is probably due to the neeessity to
satisfy some crack initiation condition; whether this
condition is of more general significance, or a particular characteristic of the wedge impact experiment,
is an open question.
The practical importance of brittle fractures in
steel structures has rapidly increased in recent times,
owing to the widespread use of welding and of hightensile steels. Welding results in high residual tensile
stresses adjaeent to the seam, and it may also eause
structural damage (e.g., grain boundary oxidation).
This may lead to the formation of eracks whieh ean
run across the weld seam and wreck the entire structurc in a fraction of a second. The high yield stress
of many modcrn steels, obtained by al10ying additions, cold work, or heat treatment, may lure the
designcr to the use of working strcsses under which
spontaneous crack propagation becomes possible
(cf. eq. 1.49). Clearly, an uncritical raising of the
design stresses on the ground of the increased yield
stress is entirely unjustified, unless the transition
range is also considerably lowered. If the latter
condition is not satisfied, higher yield stress may
merely mean that the working stress is no longer
determined by the yield stress but by the necessity
of avoiding brittle fracture.
Good ductility (high fracture strain, reduction of
area) in the ordinary tensile test ending with ductile
fracture does not mean increased immunity to brittle
fracture in the case of ferritic steels. The possibility
of brittle fracture can be assessed only by determining the transition curve of the steel and estimating
the size of the most dangerous crack that may be
present. For low-carbon steels, it appears that a
fracture work of 15 ft-Ib in the V-notch Charpy
test at the lowest service temperaturc gives a high
degree of protection against brittle fracture even if
DESIGN OF PIPING SYSTEMS
20
nomena can be classified according to their physical
Table 1.1
Static
.~
cause as mechanical or chemical.
Cyclic
:Mechanical
Creep fracture
Ordinary cyclic
fatigue
Chemical
Delayed fracture
of glass; stress
Corrosion fatigue
corrosion
cracks cannot be avoided. This figure, however,
does not apply to harder steels. If a heat-treated
high-tensile steel of 160,000 psi yield stress gives a
V-notch Charpy value of 15 ft-Ib; the deformation
of the notch-bend specimen is only about one-quarter
of that of a plain low-carbon steel with the same
Charpy value but a yield stress of only 40,000 psi.
The 15 ft-Ib high-tensile steel, therefore, has a
tendency to brittle fracture comparable to that of
a hot-rolled low-carbon steel with a Charpy value
of 4 ft-lb.
If a steel is to be used in the brittle-fracture
danger zone of temperature and stress, careful
design and workmanship are of the greatest importance. Sharp stress concentrations, such as abrupt
cross-sectional changes, sharp thread profiles, or
blind root welds, must be avoided, and the formation
of cracks during fabrication and heat treatment
prevented. On important equipment, or where
failure may endanger lives, particular attention
must be given to careful inspection and to the
removal of internal stresses.
1.8 Fatigue
A. General Features. The term "fatigue" is
used if a specimen breaks under a load which it has
previously withstood for a length of time, or during
a load cycle which it has previously withstood a
number of times. There is a remarkably sharp
distinction between those cases of fatigue in which
only the total duration of loading matters while it is
of secondary importance whether the load is steady
or interrupted, and those where only the number of
load cycles matters and the duration of the cycles is
of a subordinate importance. The first type of
fatigue is called static, the second cyclic.
Purely elastic deformation cannot cause fatigue;
all it does is to strain atomic bonds, and these
cannot wear out. Fatigue can be the consequence
either of non-elastic deformations (I.e., of lattice
injuries or intergranular displacements it produces),
or of chemical or physicochemical processes accelerated by the applied load. Thus, fatigue phe-
In this way, a twofold subdivision of fatigue
phenomena is obtained, as illustrated in Table I. L
An example of static mechanical fatigue is creep
fracture, already discussed in Section 1.4. A littleknown case of static fatigue is that observed in the
brittle fracture of steels which may occur suddenly
after prolonged steady loading. The time delay
between the application of the stress and the occurrence of fracture must be due to a slowly progressing
deformation process; the rate of this process may be
determined by the rate at which carbon atoms
diffuse in the iron lattice. Thus, the delayed brittle
fracture of steel may possibly represent a case of
physicochemical static fatigue.
The cause of the static fatigue of glass is undoubtedly physicoehemical [34J. It is known that
air (probably mainly its moisture content) reduces
the surface energy of mica by a factor of 10 or 12.
It must also reduce the surface energy of glass;
consequently, the Griffith crack propagation condition (eq. 1.48) rimy be fulfilled for a given stress
u and crack length c in the presence of air (I.e., when
a has a lowered value), but not in vacuum. In this
case, the crack can only propagate at the rate at
which air or moisture can diffuse to its tip. After a
period of slow propagation with the help of absorbed
moisture, the crack length may increase to the value
at which the applied stress can propagate the crack
even without the reduction of the surface energy by
moisture; fracture then occurs suddenly. The
physicochemical nature of the delayed fracture in
glasses is verified by the observation that static
fatigue is absent in vacuum.
The best known type of static fatigue due to
chemical action is stress corrosion, of which the
"season cracking" of cold-worked brass and the
"caustic embrittlement" of steel are familiar examples. In some cases, its cause is the precipitation
of a phase in the grain boundary which deprives the
adjacent parts of the grains of an element that
increases the resistance to ehemical attack [35]. III
the case of some austenitic Cr-Ni steels, for instance.
ehromium carbide may segregate in the boundar)'
during heating in a certain temperature region, and
the boundary regions of the grains are then depleted
in chromium. Crack propagation by solution of the
more easily attaeked (more anodic) boundary laye"
cannot progress, however, without the presence of l:l
tensile stress which opens up the crack and provides
space for the corrosion products. Under the applied
stress plastie deformation occurs at the tip of the
STRENGTH AND FAILURE OF MATERIALS
crack; this may disrupt protective layers, and the
increased frec energy of the deformed region makes it
more susceptible to attack (more anodic). Whether
these two effects represent important causes of stress
log S
(Siren
omplitude)
Ill.g., Fllrritic moterials
corrosion is not certain.
Stress-corrosion cracking can progress not only
along the grain boundaries but also across the grains;
brass single crystals crack under tension in the
presence of ammonia much like polycrystalline
brass [36, 37J. This suggests the possibility of a
stress-corrosion mechanism similar to that of the
static fatigue of glass [38J. The effective surface
energy of the crack walls which enters into the
Griffith equation (1.48) can be lowered not only by
adsorption but even more radically by chemical
combination between the corrosive agent and the
metal atoms; .consequently, a crack may propagate
in the presence of a corrosive medium by cleavage
under a relatively low tensile stress while, in the
absence of corrosion, the propagating stress demanded by eq. 1.48 may be higher than the yield
stress so that crack propagation by cleavage is
impossible. Obviously, the effect of the adsorptive
or corrosive is to cut the cohesive bonds between the
atoms of the crack walls at an early stage of the
cleavage proeess, by converting them into chemical
or van del' Waals bonds between the atoms of the
crack walls and the atoms, molecules, or ions of the
adsorptive or corrosive agent.
In accordance with its chemical origin, the susceptibility of metals to stress corrosion is extremely
specific. Thus, for instance, the caustic embrittlement of Cr-Ni-Mo low-alloy steels apparently can be
avoided by omitting anyone of the three alloying
elements.
Corrosion fatigue differs from stress corrosion in
that it occurs only if the stress varies cyclically. It
is fairly insensitive to the duration of the cycles (i.e.,
to the total duration of stress application). Corrosion fatigue starts with the appearance of surface
pits which then spread and join up to form surface
grooves not unlike the cracks on the bark of a birch
tree. These pits and blunt cracks apparently de"clop because they give rise to stress concentrations
SIren
Time
FIG. 1.19
Typical stress cycle.
21
Genctrol type, e.g., light 0110)'$
Log N (Number of cydes
10 fradure)
FIG. l.?O
Representative fatigue fracture stress
curves for metals.
where the inereased elastic energy or plastic deformation locally raises the free energy; at these spots the
material is electrolytically more soluble in the eorrosive solution (more anodic) than its surroundings.
Another possible reason for the local attack is that
the plastic deformation at the pits or cracks may
prevent the formation of protective (passive) layers.
Those features of corrosion fatigue which are of
quantitative interest to the designer will be mentioned briefly after the treatment of ordinary
meehanical cyclic fatigue. The chemical mechanisms
of corrosion fatigue, like those of stress corrosion, are
too specific to allow any general treatment. In
what follows. therefore, the main emphasis will be
laid on common mechanical fatigue, which is the
most important fatigue phenomcnon from the point
of view of the engineer
The existence of mechanical fatigue of materials
under cyclic stressing was established by Rankine
in 1843, and the basic laws of the phenomenon were
investigated experimentally by L. Wohler between
1852 and 1869. To describe it in clear terms, a
simple terminology should first be introduced.
Generally, a cyclic stress is the superposition of a
steady stress s and an alternating stress of amplitude
8 and range 28 (Fig. 1.19). The stress amplitude
that causes fracture after N cycles will be called the
fatigue strength for N cycles; if it tends towards a
finite value for infinitely increasing N, this will be
called the limiting fatigue strength or, briefly, the
fatigue limit. In the literature, the fatigue strength
is usually called fatigue endurance; however,
there is no reason why the correct technical term
"strength" for a fracture stress should not be used in
this case also. The fatigue strength depends, in
general, on the steady stress superposed upon the
purely alternating stress.
If the logarithm of 8 (the stress amplitude) is
plotted as a function of the logarithm of N (the
number of cycles to fracture), curves of the type
shown in Fig. 1.20 are obtained. Plain carbon steels
DESIGN OF PIPING SYSTEMS
22
FIG. 1.21
Effect of high-amplitude fatigue on silver
chloride sheet.
usually have a clearly defined fatigue limit; recent
experiments indieate that this may be a consequence
of the phenomenon of strain aging shown by sueh
steels. Nonferrous materials may also give curves
showing, .more or less clearly, two straight parts
connected by a curved transition region; however,
the second straight part is usually not quite horizontal but slightly descending. In such cases, there
is no clear fatigue limit within the experimentally
accessible values of N. The fatigue strength on
which the design must be based is then that for the
number of cycles which the structure must withstand during its intended life.
n. The Mechanism of Fatigue. A revealing
observation about the mechanism of fatigue is that
the fatigue crack, in general, seems to run along slip
planes, not cleavage planes [39, 40]. This cRll be
recognized without ambiguity in iron where slip
planes and cleavage planes never coincide.
It seems that alternating slip can lead to the de-
velopment of high tensile stresses in the slip planes
due to a progressive warping of the slip "packets" in
the course of cyclic straining. Figure 1.21 shows the
waviness developed during a high-amplitude fatigue
test in some of the large grains in a polycrystalline
silver chloride sheet. The development of tensile
stresses during the warping of slip planes may be
understood by means of the dislocation theory of
plastic deformation [41, 42]; if the stress is high
enough, it can cause local fracture.
However, the local tensile stresses which arise in
the course of prolonged alternating slip do not provide a sufficient explanation of mechanical fatigue.
If the material strain-hardens with plastic deformation, the first stress cycle ought to harden it so that
no further slip can occur unless the stress amplitude
of the following eycles is progressively increased;
how, then, can alternating slip continue in tests.. at
constant stress amplitude? On the other hand,
observations show that alternating slip continues,
with gradually decreasing amplitude, even in safe
ranges of stress; how can it then be explained that,
in such cases, even hundreds of millions of nonelastic strain cycles are insufficient for accumulating
the amount of internal stress and lattice damage
necessary for fracture? Thus, the basic questions
of fatigue are (1) How is progressive slip and structural damage possible under cycles of constant stress
amplitude; and (2) How are safe ranges of stress
possible?
The answer to these questions is given by the
general theory of fatigue [42, 24], whieh is concerned
with those typical features of the fatigue phenomenon
which are largely indepcndent of the individual
molecular mechanism of the fatigue damage. A
quantitative description of the theory would require
too much space to be presented in this chapter; however, a qualitative outline of the main points can be
gh:en briefly.
The salient point is that in cyclic stressing
progressive plastic deformation soon becomes confined to relatively small regions (e.g., at the tips of
small cracks, or in particularly unfavorably situated
grains) which are then surrounded by more or less
purely elastic material. Now it is easily seen that.
if a largely elastic specimen is subjected to cycles of
constant stress amplitude, a small plastic region
embedded in it will experience stress cycles of
increasing and strain cycles of decreasing amplitude.
This is a consequence of progressive strain hardening: as the yield stress of the plastic region rises, its
elastic surroundings have to exert upon it increasing
stress amplitudes to enforce further plastic deformation. By Hooke's law the elastic surroundings must
then snffer increasing strain amplitudes, and so the
strain amplitude in the plastic region decreases
beeause the sum of the two strain amplitudes must
remain constant for a given amplitude of stress
applied to the specimens as a whole.
The gradual decrease of the plastic strain amplitude explains why safe ranges of stress are possible.
It can be shown [42, 43] that the total (integrated
absolute) amount of plastic strain in an elastically
23
STRENGTH AND FAILURE OF MATERIALS
embedded strain-hardening plastic region always converges towards a finite value as the number of cycles
increases toward infinity. This liIPit value of the
total plastic strain decreases with the decrease of
the stress amplitude applied to the specimen.
Below a certain stress amplitude the total plastie
strain can never reach the critical value necessary
for producing that combinat\on of strain hardening
(i.e., of the local stress amplitude) and structural
damage at which fracture occurs. On the other
hand, if the local plastic region fractures, a small
crack arises and gives rise to a region of stress
concentrations in which plastic deformations may
now begin. A repetition of the above process may
lead to the extension of the crack and finally to the
fracture of the specimen.
An interesting point emerging from the theory is
that a fatigue fracture can arise without any reduction of the cohesion (strength) by structural
damage. Strain hardening alone may raise the
stress in plastic regions gradually to the fracture
level even if the initial strength of the material is
not redueed in the course of the alternating plastic
straining. In most real cases, however, increase of
the local stress by strain hardening and reduction
of the strength by structural injuries probably ~o
hand in hand.
Observations indicate that, in reality, the last
traees of alternating slip never disappear; there is
apparently a minimum value of the plastic strain
amplitude below which no strain hardening is
produced. This can be recognized most directly
from the fact that the width of the hysteresis loop
decreases but does not vanish during cyclic stressing.
It may be mentioned that the general theory of
fatigue leads to a semiquantitative derivation of the
typical shape of the log S-log N curve, and it also
explains the remarkable fact (see below) that the
influence of the steady stress upon the fatigue
strength is, as a rule, very small and sometimes
imperceptible up to the value of the static yield
stress.
To sum up, it can be said that the typical features
of fatigue under cycles of constant stress amplitude
follow directly from the fact that plastic deformation
is not uniformly distributed but, after an initial
deformation that may possibly extend over most of
the specimen, becomes confined to a few local
regions. Once plastic flow becomes locally concentrated, the conditions governing the development
of fatigue cracks can be investigated by a general
consideration of the change of stress and strain
amplitudes in plastic regions embedded in elastic
surroundings subjected to cycles of constant stress
amplitude. As far as the general. features of the
fatigue phenomenon are concerned, the molecular
nature of the fatigue process is of secondary importance; in particular, fatigue fracture might coneeivably occur without any decrease of the cohesion,
solely by the rise of the local stress by strain hardening to the fracture level.
C. Influence of a Superposed Steady Stress.
Figure 1.22 shows the dependence of the fatigue
strength (limiting stress amplitude) of three plaincarbon steels on the steady stress (mean stress of the
cycle) according to the experiments of Pomp and
Hempel [44, 45, 46]; the dash-dotted lines at 45°
to the coordinate axes are the loci of the points at
which the maximum stress of the cycle (including
the steady stress) reaches the conventional elastic
limit (in the present case, the 0.2% proof stress).
The curves reflect, first of all, a general feature of the
dependence of the fatigue strength upon the steady
stress: up to the elastic limit, they represent straight
100
CUNO 1; 0.1%
&.
CUNO 2; O.21C
M
~
..,"
""
~
v
"~
'0
;;;
m
~
!
:;;
'"
UJ
0
0
Y1
50
Y2
$,
Y3
100
Mean SIren of Cycle. 10 3 p1i
150
200
FlO. 1.22 Dependence of fatigue strength on steady stress in plain carbon steels.
DESIGN OF PIPING SYSTEMS
24
.~
elOO
{
------:--~-~
u
o
f
~ so
'j
\
\
\
" o±o----:':---_.,.---~·---_--.::.u
.,;
50
100
150
200
s. M~(ln SIren 01 Cycle. 10) Pi;
FIG. 1.23
Dependence of fatigue strength on steady stres£'
in patented (0.62% C) steel wire.
lines which slopc downwards only slightly with
incrcasing stcady stress. Occasionally this line is
horizontal; in all cases, the influence of the mean
stress on the fatigue strength is small.
Another feature of Fig. 1.22 is the rapid change
of the character of the curve at the elastic limit.
The slope changes abruptly with the onset of significant plastic deformations; the curves show a
distinct increase of the fatigue strcngth (limiting
safe stress amplitude supcrposed to the steady stress)
at the end of the elastic region. This "step" at the
elastic limit is followed by a second abrupt change
of slope, during which the fatigue strength declines
with further increase of the steady stress.
With strongly cold-worked metals, proof stress
and ultimate stress nearly coincide. In such cases
only the first part of the "step" seen in Fig. 1.22
can be observed. An example is shown in Fig. 1.23
("patented" steel ,,~re, 0.62% 0) [45J.
If a rod is eircumferentially notched (e.g., if it
is threaded), its static yield and ultimate stresses
referred to the smallest cross section are higher than
S = So ( 1 -
Elc~lic limit
of eyelc. S
."
p
u
\0.,
Elallic limil
~ 'cr:::------""" (initial yield '!fem)
vi 25
~'t'
" end If Whitworth threoded
bers. overogo vol"e:
So
Y pla'n
-:':":---:;:::O;:'--'>L_,,~-L:.~-~,,
Mean SI'en. I, of eytle. 10 3 Pi;
FIG. 1.24
(1.51)
S"" S (I_"!")
of (yel(\'
"
0:':0
S:)
where S is the fatigue strength at the steady stress s,
So its value for s = 0, and Sp = OQ, a stress parameter that determines the position of the line PQ.
For many decades in the past, the dependence of
the fatigue strength upon the steady stress was
usually represented by the Goodman diagram in
which the assumption was made that the strcss
parameter Sp = OQ can be identified with the ultimate stress; the Goodman diagram is indicated
in Figure 1.25 by the dashed line PU where OU is
the ultimate stress. Goodman's idea was that the
line would have to go through the point at which
f1failure" would occur in purely static tension. As
can bc seen from the discussion in Section 1.3, this
argumcnt is invalid: the ultimate stress is not a
stress at which fracture occurs but merely the
Sl,cU
Ampliludc
Ploin bers
"
those for the smooth rod (owing to "plastic constraint" exertcd by the adjacent larger sections);
its fatigue strength, however, is rcduccd by the
stress concentration present. Figure 1.24 shows fatigue strength curves for 1 in. and Ii in. Whitworth
threaded rods of the carbon steel which, in the form
of smooth cylindrical specimens, gives curve 1 in
Fig. 1.22 (this curve is repeated in Fig. 1.24) 1461.
The designer is mainly interested in stresses
within the elastie limit; for this reason, the present
considerations will be confined to the first part of
the curves in Fig. 1.22. This can be represented
schematically as a straight line connccting the point
P of the fatigue strength in purely alternating
stressing with a point Q on the abscissa nxis (cf.
Fig. 1.25); as before, the dash-dotted 45° line
represents the elastic limit beyond which curve
deviates from the line PQ. In the str~ss range of
interest to the designer, the effect of the steady
stress is therefore given by the equation
Comparison of fatigue strengths of plain and
threaded bars of 0.1 % C steels.
1
:
"'~:
............'i
Goodman liM
~. 0 . . . . . ,
:
Obl(\'N(\'d
-.....:::,j ~
..........................
u........ Q
O;'------"I-~
Mean (,!cody) Sire". ,
FlO. 1.25 The influence of steady (mean) stress upon
the fatigue limit.
STRENGTH AND FAILURE OF MATERIALS
(conventional) stress at which the maximum load is
reached and necking begins in the static tensile
test. For this reason, the ultimate stress point U
has no place on any curvc showing the dependence
of the fatigue strength upon the steady stress, and
much less on the straight line that forms the initial
elastic part of such curves. In the experimental
curves shown in Fig. 1.22, for instance, the extension
of the initial straight part may intersect the abscissa
axis quite far from the point U of the ultimate
stress; the stress parameter Sp in eq. 1.51 and the
position of thc point Q can only be derived from
fatigue tests. The only point that can be made in
defense of the Goodman line is that its errors, however large, usually lie in the safe direetion.
D. Influence of a Compound State of Stress.
Relatively little is known about the eondition
of fatigue fracture for cyclieally varying triaxial
states of stress. However, a practically important
case, that of a shaft subjected to cyclic torsion and
bending simultaneously, has been investigated in
detail by Gough and Pollard [47]. They found that
for a given (large) number of cycles, those corresponding values Sand T, respectively, of the tensilestress amplitude due to bending and of the shearstress amplitude due to torsion at which fraeture
oceurs are determined approximately by the relationship
(1.52)
where So is the fatigue strength for the same number
of cycles in pure bending, and To the fatigue (shear)
strength in pure torsion.
E. Influence of Notches and of Surface Flaws.
Stress raisers are relatively unimportant in ductile
metals under static stress, because plastic flow levels
down the stress at thc stress concentrations. In
cyclic stressing, the situation is different: local
cyclic straining produces progressive strain hardening with consequent rise of the local stress. If the
strain hardening could continue with cyclic plastic
deformation at a finite rate, no matter how small
the plastic strain amplitude, it would finally raise
the yield stress until no plastic deformation could
occur. The local alternating stress amplitude and
the effective stress concentration factor would then
be the same as in a purely elastic body of the same
geometry.
Experience shows that this is not the case in
fatigue. The effect of notches, cracks, and surface
flaws is usually much greater than in static stressing,
but it is still far below what it would be for a purely
25
elastic material. The simplest explanation of this
remarkable fact is to assume that cyclic straining
ceases to produce strain hardening when the strain
amplitude becomes too small (sec above); if this is
the case, the material at the tip of thc crack never
becomes quite elastic and thc stress can never reach
the level of the elastic stress concentration. Different materials have different "notch sensitivities" O
(not to be confused with the notch sensitivity for
static brittle fracture). Some of them, like grey
lamellar cast iron or certain bronzes, are almost
insensitive to the presence of small sharp cracks or
notches; their q value will therefore be close to zero.
Others, like hard steels, arc very sensitive, with q
in the neighborhood of 1.
Similarly, the surface quality has an influence
upon the fatigue strength of ductile metals that is
between those for a completely brittle material
likc glass and for a ductile metal under static stress.
Occasionally, the fatigue strength of extruded lightalloy rods with the extrusion skin has been found
to be as low as one-half of the fatigue strength of a
machined specimen of the same rod. In some cases,
the fatigue strength can be raiseO considcrably by
surface rolling or shot blasting (e.g., for heat-treated
spring steels); in others, such a treatment has no
significant beneficial influence (e.g., with many
light alloys). Excessive surface rolling or shot
blasting in materials of limited ductility may even
reduce the fatigue strength by producing surface
cracks.
There is a difference of great importance between
the fatigue strength of a ductile metal and the
(static) strength of a brittle material like glass. In
the latter case, the strength can be raised sometimes
by a factor of 10 or even 100 if surface cracks arc
very carefully avoided. In ductile metals, it is
relatively easy to improve the quality of the surface
so that any remaining flaws have no influence on the
fatigue strength. However, this does not raise the
fatigue strength spectacularly because plastic deformations set in as soon as the elastic limit is exceeded,
and they produce cracks after sufficiently prolonged
cyclic stressing in a way that is now more or less
understood. For this reason, there is no hope that
the fatigue strength may be raised much above the
elastic limit. If, on the other hand, the elastic limit
is raised by strain hardening, precipitation hardcn6A conventional quantitative definition of the relative notch
sensitivity q. in cyclic stress is q = (kl - l)/(k e - 1) where
k e is the elastic stress concentration factor for a given notch
llnd kl is the factor by which the fatigue strength is reduced
by the prcsence of the notch. Of course, q depends in gencral
on the size and shape of the notch.
26
DESIGN OF PIPING SYSTEMS
ing, or in any other way, a decrease of ductility is
unavoidably associated with the increase of the
fatigue strength.
.~
multiply bent, relatively thin tube the stresses are
much lower than in a straight bar fixed at two cross
F. Fatigue Tests on Specimens VB. Fatigue
Obviously, the action of a stress upon a material
is quite independent of how it is produced; consequently, the fatigue effect of a thermal stress cycle
is identical with that of a mechanical load cycle
involving the same stresses at the same temperatures.
Compared with the ordinary fatigue test, the only
new factor introduced by the thermal eycling of a
rigidly supported specimen is that, together with the
stress, the temperature also varies during the cycle.
If the temperature amplitude is relatively small, the
fatigue effects of a thermal cycle will be the same as
those of an isothermal load cycle involving the same
stresses at a constant temperature equal to a suitably
chosen mean temperature of the thermal cycle. That
this can be so even for cycles of considerable temperature amplitude is indicated by recent experiments of
Coffin [49J. The equivalent mean temperature of
Tests on Structural Parts. The strength of
structural parts under static load can usually· be
calculated with reasonable accuracy on the basis of
tests performed on specimens. Stress concentrations
either do not matter (in very ductile materials), or
they can be ealculated by methods given in the
theory of elasticity. The situation, however, is very
different in cyclic stressing. The effective stress concentration factors depend here not only on the geometry and On the elastic constants, but in the first line
on the Hnotch sensitivity" of the material, which
depends On the size of the notch. Whenever a
structural part has a strongly non-uniform stress
distribution, therefore, its fatigue properties cannot
be calculated from tests on specimens with any
reasonable accuracy. If the structural part cannot
be overdimensioned so as to exclude any danger, it is
necessary to carry out full-scale fatigue tests on it
[48]. This is particularly important, of course, in
the case of aircraft structures. As already mentioned,
attention must be given in any case to possible differences between the fatigue behavior of specimens
with carefully machined surfaces and specimens or
structural parts with surfaces as they will be present
in the structure.
There is a morc trivial reason why so often conclusions drawn from experiments with specimens are not
fulfilled by structures. Fatigue tests are usually constant stress tests, occasionally constant strain tests.
On the other hand, if a structure is subjected to cycles
of constant load or deformation amplitude, some of
its elements (for instance, regions of stress concentra-
tions) will be under cycles of increasing stress
amplitnde and decreasing strain amplitude, for the
reason explained above in connection with the general
theory of fatigue. It follows, then, that the results
of constant amplitude tests cannot be applied directly
to the calculation of the fatigue strength of structurcs
with non-uniform stress distribution.
A general
method of calculation iu such cases has been given
[43]; for the present, however, lack of experimental
data prevents the practieal use of this method except
in the simplcst cascs.
G. Periodically Varyiug Thermal Stresses.
If a body is rigidly clampcd at two points, incrcase
or decrease of its temperaturc gives rise to thcrmal
stresses in it. The magnitude of these stresses depends not only on the temperature change and on the
material, but also on the shape of the body; in a
sections.
the cycle, however, is not necessarily the mean value
of its highest and lowest temperatures. If the magnitude of fatigue damage is determined mainly by the
amount of plastic deformation, the temperature of
the equivalent isothermal cycle will lie nearer to the
maximum than to the mean temperature of the
thermal cycle because the material is softest, and
plastic deformation greatest, in the high temperature
part of the cycle. The opposite behavior (the lowtemperature part of the cycle being of dominating
importance) may conceivably also occur. If the
specimen is a straight rod or tube with fixed ends, it
is always in tension during the low-temperature part
of the eycle. If now the tensile part of the cycle is
more likely to produce fatigue damage than the compressive part, the effect. of the thermal cycle may be
closer to that of an isothermal cycle with the same
stress range taking place near the lowest temperature
of the stress cycle.
A new factor appears (both in thermal and in
purely mechanical cycling) if the temperature is so
high that the strain hardening and the structural
damage due to plastic deformation are currently removed during the cyclic straining.
In this case, the
progressive changes which represent cyclic fatigue
cannot develop.
Nevertheless, fracture may occur
owing to a different phenomenon which has been
treated already under the heading of creep fracture.
At very high temperatures (in the hot creep range),
the grain boundaries become soft, and the consequent
relative displacements between the neighboring
grains open up cracks which finally can lead to
fracture ("static fatigue"). At first sight, it might
27
STRENGTH AND FAILURE OF MATERIALS
seem that this eannot occur under purely cyclic stress
because the displacements produced by the tensile
part of the eycle are reversed ''by the compressive
part. However, the compressive part cannot undo
all damage done by the tensile part, and so fatigue
fracture can also occur under purely cyclic stress,
although much more slowly than under a steady
fatigue fraduro SIren Curvo
tensile stress.
Lazan and Westberg [501 have carried out experiments in the interesting transition region just below
the hot-creep range; they applied both purely cyclic
and purely static stresses and intermediate types of
loading with a static stress superposed upon a steady
,;tress. Figure 1.26 illustrates some of their results.
~\s in room temperature experiments, a relatively
low mean stress has only a slight influenee upon the
fatigue strength if the duration of the test is not too
long. If the time to fracture is 150 hours or longer,
creep predominates over cyclic fatigue, and the
steady component of the cycle becomes important
from the beginning. The vertical parts of the curves
show that the static fatigue strength is almost uninfluenced by the cyclic component until the cyclic
stress amplitude becomes higher than about one-half
of the static stress. The observed curves, therefore,
consist essentially of a nearly horizontal part representing cyclie fatigue (except in very prolonged tests,
as mentioned above), and of a vertical part representing creep fracture. The transition between the
horizontal and the vertical part is the region in which
cyclic and static fatigue are of comparable importance.
If a material has been cold worked and then subjected to plastic deformation at a higher temperature, it may soften more than if it had been subjected
to the effect of the higher temperature alone without
. 30
~
~
Speed of Cycling
214,000 reversals/hr.
Time 10 frodufe
J::::::",=:::",,::,=f:::::':'"}(5. 50, 150, 500. 1500 hrs
.;
'0
~
u 20
'0
E
;;;
.~
g 10
"•
Domage Line
Log N (N = Number of slreu cydM)
FIG. 1.27 The damage area in fatigue.
further deformation [19b, 51]. In the course of the
deformation, its relatively highly hardened structure
changes to the less hardened structure characteristic
of deformation at the higher temperature. A similar
strain-softening effect can also be observed in fatigue
tests with previously strongly cold-worked materials
[52]. This, however, does not mean that strain
hardening is not an important factor in fatigue.
Local regions of stress concentration, e.g., at the tip
of a fatigue crack, may well harden under cyclic
stressing, while the static yield stress of the prestrained bulk material decreases by thermal recovery
with or without strain softening.
H. Thermal Fatigue. The most severe case of
cyclic thermal stressing takes place when the surface
of a metal is rapidly heated to a high temperature
and then cooled again. This occurs in hot rolls, gun
barrels, etc.; if the temperature amplitude is high,
the usual effect is the formation of surface cracks
("crazing") which gradually spread inwards. Frequently this cannot be prevented; the life of the body
can be prolonged, however, if the surface is machined
off before the cracks become too deep. In other cases,
thermal cracking would occur with most materials
but can be avoided by the use of special metals, such
as, e.g., the 12% Cr steel used for rolls in continuous
sheet glass manufacture.
Anisotropic metals such as zinc, or metals that
suffer phase transformations in the temperature
range to which they are subjected, can suffer plastic
deformations on a microscopic scale within the grains
.,;
o OL--~-:-,0:'--L,-L--'c20:-"--.'--'3~O­
.. Meon Siren of Cycle, 10 3 psi
FIG, 1.26 Fatigue-creep rupture interaction curves for
N-155 at 1500 F. After Lazan and Westberg.
which are confined and distorted by their neighbors,
even if there is no significant temperature gradient
present. This may result in progressive structural
damage during thermal cycling.
J. Damage by Overs tress. If a material is subjected to stress amplitudes above the fatigue limit,
DESIGN OF PIPING SYSTEMS
28
it may suffer permanent damage which reduces its
fatigue strength for suhsequently applied cycles of
lower stress amplitude. It seems that those combinations of stress amplitude and number of cycles above
which permanent damage occurs lie in the area D
(Fig. 1.27) between the high-stress part of the
log S-log N curve and a line below it which joins the
curve at the bend [53]. This line, shown dashed in
Fig. 1.27, is the "damage line." The permanent
damage suffered in the damage area D eonsists probably in the formation of small cracks.
K. Corrosion Fatigue. If the cyclically stressed
material is in a chemically active solution, its fatigue
strength may be substantially lowered. Whether in
this case an approximate fatigue limit exists is not
certain; as in stress corrosion, the phenomenon is so
strongly influenced by the individuality of the metal
and of the surrounding solution that the only general
statement that ean be made about it is a warning
against premature extrapolations to even slightly
different metals and solutions.
References
1. M. Cook and E. C. Larke, "Resistance of Copper and
Copper Alloys to Homogeneous Deformation in Com-
pression/' J. Inst. Afetals, Vol. 71, p. 371 (1945).
2. M. Tresca, /IM6moire SUf Ie poin~nnage et la thCoric
mccanique de la d6£ormll.tion des metaux," Campt. rend.,
Vol. 68, pp. 1197-1201 (1869).
3. R. von Mises, "Mcchanik der festen Korper im plastischdeformablen Zustand," Nachr. kgl. Ges. Wiss. Afath.-Phys.
Klasse, 1913, pp. 582-592.
4. R. Hill, The Mathematical Theory of Plasticity, p. 20.
The Clarendon Press, Oxford, 1950.
5. J. L. M. Morrison, liThe Yield of Mild Steel with Particular
Reference to Effect of Size of Specimen," Proc., Insf.
Mech. Engr,. (Landon), Vol. 142. pp. 193-223 (1940).
6. O. Hoffman and G. Sachs, Inlroductwn to the Theory of
Plasticity for Engineers, McGrnw~Hill Book Co., New
York, 1953.
7. M. Considcre. "L'emploi du fer et de l'acier dans les constructions," Ann. Ponts et Chaussees, 6th Series, Vol. 9,
pp. 574-775 (1885).
S. G. Sachs and J. D. Lubahn, IIFailure of Ductilc Metals in
Tension," Trans. ASME, Vol. 68, pp. 277-279 (1946).
9. H. W. Swift, HPlastic Instability Under Plane Stress,"
Journal of Mech. & Phys. of Solids, Vol. I, No.1, pp. 1-18
(October. 1952).
10. W. R. D. Manning, uThe Overstrain of Tubea by Internal
Pressure/' Engineering, Vol. 159, pp. 101-102, 183-184
(1945); also uThe Design of Compound Cylinders for
High Pressure Scrvice/' Engineering, Vol. 161, pp. 349352 (1947).
11. C. W. MacGregor, L. F. Coffin, Jr., and J. C. Fisher,
uThe Plastic Flow of Thick-Walled Tubea with Large
Strains," J. Appl. Phy,;;., Vol. 19, pp. 291-297 (1948);
also uPartially Plastic Thick-Walled Tubes," J. Franklin
IMt., Vol. 245, pp. 135-158 (1948).
12. E. N. do. C. Andrade, liThe Flow in Metals Under Large
Constant Stresses," Proc. Roy. Soc., Series A, Vol. 90,
pp. 329-342 (1914).
13. P. Phillips, "The Slow Stretch in Indiarubber, Glass, and
Metal Wires when Subjected to a Constant Pull," Phil.
Mag., 6th Series, Vol. 9, pp. 513-531 (1905).
14. F. H. Norton, Creep of Steel at High Temperatures,
McGraw-Hill Book Co., New York, 1929.
15. C. R. Soderberg, "The Interpretation of Creep Tests
for Machine Design," Trans. ASME, Vol. 58, pp. 733-743
(1936 ).
16. A. Nadai, "The Influence of Time upon Creep. The
Hyperbolic Sine Creep Law," S. Timoshenko 60th Anniversary Vol., Macmillan Co., New York, 1938.
17. E. Orowan, uDiscussion on Plastic Flow in Metals,"
Proc. Roy Soc., Series A, Vol. 168, p. 307 (1938)j also
Proc. First Nat. Congr. Appl. Mecl!., June, 1951, p. 453.
J. \V. Edwards, Ann Arbor, Mich., 1952.
18. R. Becker, uUber die Plastizitiit amorpher und kristalliner fester Korper," Physik. Z., Vol. 26, p. 919 (1925);
also Z. Tech. Physik., Vol. 7, p. 547 (1926).
19. E. Growan, "The Creep of Metals," Z. Physik., Vol. 98,
p. 382 (1935); also "The Creep of Metals'" Trans. West of
Scotland Iron Steel IMt., pp. 45-96 (1947).
20. F. R. Larson and J. Miller, "A Time-Temperature Relationship for Rupture and Creep Stresses," Trans. ASME,
Vol. 74, pp. 765-771 (1952).
21. N. J. Grant and A. G. Bucklin, On the Extrapolation of
Short-Time Stress-Rupture Data, ASM Preprint No. 18,
1949.
22. A. A. Griffith, uThe Phenomena of Rupture and Flow in
Solids," Trans. Roy. Soc., Seriea A, Vol. 221, pp. 163-198
(1920-21); also First Internal. Congr. Appl. Mech., p. 55,
Delft, 1924.
23. C. E. Inglis, flStresses in a Plate due to the Presence of
Cracks and Sharp Corners," Trans. Inst. Naval Archil.,
Vol. 55, Part I, pp. 219-230 (1913).
24. E. Orowan, uFracture and Strength of Solids," Reports
on Progress in Physics, Vol. 12,. pp. 185-232 (1949).
25. A. Mesnager, Rtunum des Membres Franrais et Belges de
l'Association Intemationale des Methode d'Essais, pp. 395405, December, 1902.
26. P. Ludwik and R. Scheu, no-ber Kerbwirkungen bei
Flusseiscn," Stahl und Eisen, Vol. 43, pp. 999-1001 (1923).
27. E. Orowan, "Notch Brittleness and the Strength of
Metals," Trans. Inst. Engrs. Shipbuilders Scot., Paper
No. 1063, pp. 165-215, December, 1945.
28. N. N. Davidenkov and F. \'littman, "Mechanical Analysis of Impact Brittleness," Phys.-Techn. Ins!. (U.S.S.R),
Vol. 4, p. 308 (1937).
2D. D. K. Felbeck and E. Orowan, "Experiments on Brittle
Fracture of Steel Plates," Welding J. (N.Y.), Res. Suppl.,
Vol. 20, No.7 (1955).
30. M. J. Manjoine, IIInOuence of Rate of Strain and Tem~
perature on Yield Stresses of Mild Steel," J. Appl.
Jfechanics, Vol. 11, pp. A211-218 (1944).
31. G. 1. Taylor, "Testing of Materials at High Rates of
Loading." J. Inst. Civ. Engrs., Vol. 26, pp. 486-519.
(1946).
32. E. Growan, "Fundamentals of Brittle Behavior in
Metals," in William M. Murray, cd., Fatigue and Fracture
of Metals: A Symposium, pp. 139-167, John Wiley & Sons,
New York, 1952.
STRENGTH AND FAILURE OF MATERIALS
33. F. J. Feely, Jr., and M. S. Northup, HStudy of Brittle
Failure in Tank Steels," presented at the Midyear Mtg.,
Am. Petro lost., in Houston, Texas, May, 1954.
34. E. Orowan, flA Type of plastiC Deformation New in
Metals/' Nature, London, Vol. 149, p. 643 (1942).
35. G. Akimow, HEine neue Theorie der StrukturkarrOs1Qn/'
Karrosion u. MetaUschutz, Vol. 8, p. 197 (1932).
36. G. Wassermann, HUntersuchungen tiber den Vorgang cler
Spannungekorrosion/' Z. Metalll..:unde, Vol. 34, p. 297
(1942).
37. G. Edmunds, /lSeasan Cracking of Brass," ASTM Symp.
on Stress-Cfm'. Crad.:ing in Metals, p. 67 (1944).
38. E. Orowan, in a paper presented before The Electrochem.
Soc., BostoD, Oct. 4, 1954.
39. J. A. Ewing and J. C. \V. Humfrey, "The Fracture of
Metals Under Repeated Alternating Stress," Trans.
Roy Soc., Series A, Vol. 200, pp. 241-250 (1903).
40. F. A. McClintock, "On Direction of Fatigue Cracks in
Polycrystalline Ingot Iron/' J. Appl. }.{echaniC8, Vol.
19, pp. 54-56 (1952).
41. E. Orowan, HDislocations and Mechanical Properties" in
Dislocaticns in Metals, AIMEI New York, 1954.
42. E. Orowan, ICTheory of the Fatigue of Metals," Proc.
Roy. Soc., Series A, Vol. 171, pp. 79-106 (1939).
43. E. Orowan, IlStress Concentrations in Steel under Cyclic
Load," Welding J., (N.Y.), Res. Suppl., Vol. 17, pp. 273,282s, June, 1952.
44. A. Pomp and M. Hempel, flDauerfestigkeitsscbaubilder
von SUihlen bei verschiedenen Zugmittelapannugen unter
BerUcksichtigung der PrUfstabform/ ' Mitt. Kai.serWilhelm-Inst. Eis<njorsch. Dusseldorf., Vol. 18, pp. 1-14
(1936).
29
45. A. Pomp and M. Hempel, "Dauerprtifung von Stahldrehten unter wechselnder Zugbeansprunchung,l' }.[iU. KaiserWilhelm-Inst. Ei.senjorsch. DUsseldorf, Vol. 19, pp. 237246 (1937).
46. A. Pomp and M. Hempel, HDauerfestigkeitsschaubilder
von Gekerbten und Kaltverformten Stahlen Bowie von
1"- und 11 11- Schrauben bei Verschiedenen Zugmittelspannungen," MiU. Kai.ser-Wilhelm-Inst. Eisenforsch.
Dusseldorf, Vol. 18, pp. 205-215 (1936).
47. H. J. Gough and H. V. Pollard, "The Strength of Metals
under Combined Alternating Stresses," Proc. Inst. Mech.
Engrs. (London), Vol. 131, pp. 3-04 (1935).
48. R. L. Templin, IIDesigning for Fatigue" in William M.
Murray, ed., Fatigue and Fracture of Metals; A Symposium, pp. 131-138, John Wiley & Sonsl New York,
1950.
49. L. F. Coffin, Jr., IIA Study of the Effecta of Cyclic Thermal
Stresses on a Ductile Metal," Trans. ASME, Vol. 76,
No.6, pp. 931-950 (19M).
50. B. J. Lazan and E. Westberg, HEffect of Tensile and Compressive Fatigue Stress on Creep, Rupture and Ductility
Properties of Temperature-Resistant Materialft," Proc.
ASTM, Vol. 52, pp. 837-855 (1953).
51. J. E. Darn, A. Goldberg, and T. E. Teitz, liThe Effect of
Thermal~mechanicalHistory on the Strain Hardening of
Metals." AIME Tech. Pub. No. 2445, 1948.
52. N. H. Polakowski, HSoftening of Certain Cold-worked
Metals Under the Action of Fatigue Loads," ASTM
Preprint No. 74, 1954.
53. H. J. French! "Fatigue and Hardening of Steels," Trans.
Am. Soc. Steel Treating, Vol. 21, pp. 899-946 (1933).
CHAPTER
2
Design Assumptions, Stress Evaluation,
and Design Limits
personnel and the interests of the general public
dictate that all feasible precautions be exercised.
Maximum assurance of safety, however, would
require complete examination of all materials and
fabrication by the best available means and with
duplicate independent inspection. Even so, absolute assurance of safety could not be attained due to
personnel fallibility and the limitations in sensitivity
of available methods of nondestructive examination.
With this realization, in the practical approach of
achieving adequate safety economically, lower levels
of quality are aceepted on the basis of including compensating safety factors in design, which are the
combined result of experience and reasoning. Many
inconsistencies still exist in current practice relative
to quality requirements of materials and fabrication,
and in the value placed on various degrees of inspection, tests, and nondestructive examination.
It should be appreciated that Codes and Standards
can establish only a level of minimum requirements
for average service, based on knowledge, experience,
and the consensus of qualified individuals. Many
circumstances relating to service operation, materials
and fabrication, inspection limitations, or to unusual
design deserve special consideration if the resulting
piping systems are to be reasonably free from maintenance, and provide satisfactory length of life with
safe operation. To assist the piping engineer in the
exercise of good judgment on these special problems,
this chapter offers approaches which largely depend
on well-established practical experience.
HE previous chapter passed over the problem
of ealculating stresscs and strains from the
applied load in order to concentrate on certain fundamental knowledge from the physics of
solids which, it was pointed out, is relatively new
and as yet largely unformulated for use in routine
design engineering. The present chapter offers a
general examination of the factors which enter into
the evaluation of stresscs in piping systcms due to
various external and internal loadings, their association with design limits and Code rules, and finally,
their significance and application to practical design.
T
\Vith the increasing complexity, size, and economic
significance' of piping installations, it is necessary
to look beyond the limits of ordinary piping design
practice and to give attention to the expericnces of
designers in related fields, particularly that of pressure vessel design. Indeed, there is often no logical
distinction between pressure vessels and piping.
Therefore, appropriate comments relative to comparative piping and pressure vessel design approaches
are given frequently in the discussions which follow.
In further consequence of the economic importance
of present-day piping installations it is necessary,
just as in the design of structures and pressure cquipment, to effect a careful and realistic compromise
betwcen design features (not overlooking materials,
fabrication, and inspection requirements) and the
overall plant economics (first cost plus maintenance
and contingency for damages to property and personnel in event of failure). Safety of operating
2.1
lPiping is n. major item in process plants, running from 50
to 75 per cent of the total plant cost.. Similar significant
expenditures are incurred in power generation and marine
Codes and Standards
The objective of Code rules and Standards (apart
from fixing dimensional values) is to achieve mini-
propulsion installations.
30
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
mum requirements for safe construction; in other
words, to provide public protection by defining those
material, design, fabrication, and inspection requirements whose omission may radically increase operating hazards. Absolute assurance of safety would
require perfect design, materials, and fabrication;
this is seldom, if ever, achieved. On the other hand,
experience with Code rules has demonstrated that
the probability of disastrous failure can be reduced
to the extremely low level necessary to protect life
and property by suitable minimum requirements and
safety factors. Obviously, it is impossible for general
rules to anticipate other than conventional service,
and it would be uneconomic for them to provide for
corrosion, erosion, fatigue, shock, or potential brittle
fracture, except to the degree that such conditions are
known to be present. Suitable precautions are, therefore, entirely the responsibility of the design engineer
guided by the needs and speeifications of the user.
A listing of all Standards and Specifications coneerning piping design, together with their mandatory
effective edition references, appears in an appendix
of the Code for Pressure Piping (ASA B31.1).
Those which affect the mechanical design of piping
are briefly commented on in the following paragraphs, relative to their basic approach and significant details.
One of the difficulties which often confronts designers of vessels and piping, as related to Code
requirements and particularly local governmental
regulations, is the proper classifieation of borderline
pressure equipment. Currently (1955), neither the
ASME nor the ASA Code contains definitions for
vessels or piping which are helpful in this respect.
While the Code Committees have considered this
matter, no common agreement has been reached.
Some items in piping systems often considered and
fabricated as part of the piping, e.g. pulsation dampeners, are classed as pressure vessels in some States.
In doubtful eases it is advisable for the user to cheek
with the local authorities, especially in localities
having regional pressure vessel laws.
ASI\1E Boiler and Pressure Vessel Code. Section I,
Pmver Boilers, cont.'\ins rules for the pressure design of boiler
piping within the specified boiler limits which are associa.ted
with appropriate steam and feed-water stop valves. The
design, fabrication, and inspection requirements of the ASME
Unfired Pressure Vessel Code, Section VIII, are often used by
reference in company specifications to supplement the Piping
Code. Section IX of the ASME Code is the universal basis
for qualification of welding procedures and operators of all
pressure equipment.
ASA B31.1: Code for Pressure Piping. This is the
standard "Piping Code" which includes sections on Power,
31
Gas & Air, Oil, District Heating, Refrigeration, Oil Transmission, Gas Transmission and Distribution Systems (ASA
B3 1.1.8-1955), and Chemical Piping. Its basic or general
supporting sections deal with requirements for internal
pressure, flexibility, materials, fabrication, and testing.
At the present writing (1955), this Code is in the process of
evolution from a Design Practice to a Safety Code. The
Gas Transmission and Distribution Section has been adopted
by several States and ie under consideration by others; the
entire Code is used as a basis of enforcement in several U. S.
cities and in the Provinces of Canada. In recognition of this
trend, a Conference Committee similar to that of the ASME
Boiler Code and composed of the Chief Inspection Authority
of each State and each Canadian Province which has adopted
the (Piping) Code, has been appointed. At the same time a
procedure was established to provide interpretations in the
form of Cases, which again parallels the ASME Boiler Code
procedure.
This transition is largely due to recognition by publie
authorities that pipe line failures associated with a sudden
release of stored energy arc potentially as dangerous as pressure vessel failures. Experience with piping systems also
demanded a change in the former attitude that thermal expansion strains could not be responsible for a major failure.
Although this type of failure is due to fatigue rather than to n
single application of strain loading it can be a definite hazard
in most services.
ASA B9: Safety Code for Mechanical Refrigeration.
This Code contains, in Section 9, brief rules for pressure and
general design of p1ping for this specific service.
Piping for Ships. Such piping requires spedal consideration because of added strains from the motions of the ship.
Naval vcssels are subject to added shock due to Budden
maneuvering, gunfirc, explosions, etc. Requirements for
merchant and naval vessels are contained in the following:
Standards:
U. S. Navy, BUfCSU of Ships: General Machinery Specifications; Gcneral Specifications for Building Naval Vessels.
American Bureau of Shipping: Rules for Building and elMsing Vcssels.
United States Coast Guard: 1Inrine Engineering Regulations and Material Specifications.
Lloyd's Register of Shipping Rules
Flange and Fitting Standards. The B1G group of ASA
Standards apply to pipe-fitting details. Although their significance is primarily dimensional, they involve design factors
which should be appreciated. These are summarized in the
following sections:
Sleel Flanges. The proportions of scparate flanges and those
integrated with fittings were established many years ago, bascd
on simplified cantilever analysis. In 1953 the stecl flangcs
were reinvestigated according to prcsent ASME Boiler Code
formulas. New ratings were established for two general
classes of gaskets and facing details. These appear in ASA
Standard, B1G.5-1953, and also in the AS ME Codes. The
basis of the neW ratings is rC'~orded in Appendix D of the
B1G.5 Standard. The calculated stress in the flanges shows
appreciable variation with size, series, and facings. A stress
of 8750 psi at the primary pressure rating was selected for the
purpose of establishing Class A ratings. Class B ratings are
approximately 83% of Class A ratings. In the creep range at
or above the primary rating temperature, ASME Power
Boiler Code strcsscs are adhercd to. For temperaturcs up to
32
DESIGN OF PIPING SYSTEMS
650 F the ratings :ire based on allowable stresses, which nrc
approximately 60% of the yield strength. This is similar to
the allowable stress basis of the Piping Code, Section 3, Oil
Piping. Ratings between 650 F a'nd the primary service
temperature are established by a straight line transition.
In general, the bolting, particularly when alloy steel, is of
substantially greater strength than the flanges, which cnn be
distorted by overtightening. This excess bolt strength is
significant in the ability of ABA flanges to transmit line moments, as discussed later in Chapter 3.
Steel Flanged Fitling 1'hickness. Fitting thicknesses were
originally established for cast-carbon steel by npplicntion of
the Barlow (outside diameter) formula with nn allowable
stress of 7000 psi at the primary pressure rating, and applying
a 50% increase in thickness as a "shape factor. J' This approach
was later extended to other cast and forged materials. The
fitting thicknesses in the 1953 issue of BIG.5 are based on this
same allowable stressJ which is 80% of the value used for
rating Class A flanges, using the primary service pre.ssure
and the modified Lam6 formula now common to the Codes.
An excess thickness of 50% is provided for all flanged fittings
in recognition of the reinforcment required at the side outlets
of tees, bonnet necks of valves and similar branch connections J
as well as for elbows, whether or not they have side branch
connections.
Steel Butt lVelding Filling Thickness. For cast- or wroughtbutt welding fittings the thickness required by ASA Standar r1
B16.9 at the welding ends is the same as that of the pipe size
and schedule with which they are intended w be used. Ingtead of establishing minimum wall thicknesses or "shape
factors" as is done for flanged fittings J this Standard requires
only that the bursting strength be not less than that of a pipe
of the corresponding material, size, and schedule number; the
pressure-temperature rating then becomes identical with that
of the intact pipe.
API-ASME Code for Unfired Pressure Vessels. This
pressure vessel Code is sometimes used as a reference in
company specifications. Except (or the absence of mandawry
random examination requirements, its provisions are essentially the same as Section VIII of the ASl\,fE Boiler and Pressure Vessel Code.
API Standards. In addition to material specifications for
line pipe, threads J etc., the American Petroleum Institute has
standards for certain types of iron or steel valves for refinery
or drilling and production service (API Standards 600, fiC
and 6D) and (or ring-joint flanges (API Standard 6B). The
flanges and ratings utilized in Standard 600 are based on ASA
standards. Standards 6C and 6D assign separate pressure
ratings for "pipe line service" and IIdrilling and production
service" at 100 F. In addition w utilizing ASA St.andard
flanges, API Standard 6B includes a special "2900 lb" series.
This is similar to the original assignment of a 4000 lb rating
to 1500 lb series flanges, drilled one size smaller, which was
advanced and used by The M. W. Kellogg CompanYJ except
that the design was refined J in accordance with calculations
.using ASME Code formulas, by Messrs. Petrie and Watts
of the Crane Company and Standard Oil Company (Indiana),
respectively. The API Standard assigns a 100 F rating of
7500 psi for pipe line service, and 10,000 psi for drilling and
production service. For the latter service, materials with
higher tensile and .yield strengths are required. The API
ratings for ASA flanges are the same as ASA ratings for pipe
line service; for drilling and production service the 600 Ib and
higher series are required w have increased physical properties
and accordingly nrc assigned ratings about 33% higher.
Other Standards: Other Standards which contribute to
piping design are those of the Manufacturers Standardization
Society of the Valve and Fittings Industry (1\1SS Standard
Practices)J American Water Works Association (AWWA),
American Gas Association (AGA), Federal Specifications
Board (FSB), and Association of American Railroads (AAR).
These are (or the most part dimensional standards and rating
hlbles (or specific piping and fittings.
2.2
Design Considerations: Loadings
A piping system constitutes an irregular space
frame into which strain and attendant stress may
be introduced by the initial fabrication and erection,
and also may exist due to various circumstances
during operation, standby, or shutdown. In its
erected position, a piping system is subject to loads
due to dead weights (pipe, fittings, insulation), snow
or ice J contents of the line J wind load for exposed
piping, and earthquake or other shock loading in
special situations. Internal (or external) pressure
loads may be imposed in service or off stream. The
restraint of thermal expansion provided by terminal
and intermediate anchors, guides, and stops introduces thermal stresses in piping due to temperature
changes. Further stress may be introduced by the
movement2 of terminal equipment, foundations or
buildings nnder temperature changes or other loadingJ or from any influence affecting the relative position of the line, anchors J or intermediate restraints.
The dead load effects, except contents, are usually
maintained at all times, while wind or earthquake
effects will be variable and reach maximnm design
values infrequently, if ever. Pressure and temperature changes usually occnr simultaneously, bnt may
be independent or have a variably dependent relationship. They may be relatively uniform for entire
service periods, or involve swings of variable
duration.
Dead load and wind or earthquake effects 011 piping are no different than for conventional structures
while pressure effects are essentially the same as
those encountered on pressure vessels or boilers.
Overall expansion effects differ from those on structures exposed to ambient temperature changes, in
that the range of temperature variation on piping is
much greater.
For many problems, the designer must consider
more than one service condition, as well as start-up,
shutdown, and emergency conditions; for example,
a specific plant may involve more than one feed
J
J
2Frequently termed Hextrancous" movement by piping
designers.
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
stock or several alternate products which may require different processing pressure and temperatures. Many plants involve~highly inflammable,
toxic, or otherwise unusual fluids, or specialized
machinery and equipment which must be carefully
isolated from air or contaminants. Start-up and
shutdown may require protracted periods of warming up, cooling ofT, or operations such as purging,
washing down, piekling or passivating, solvent cleaning, air-steam decoking, ete., each of which may
introduce entirely different combinations of temperature and pressure over given portions of piping sys-
tems. Temperature differences, or other loading
more· severe than normal service conditions, may
result where circumstances dictate that parts of a
system be heated successively. A proper appreciation of these various possibilities requires an adequate knowledge of the process design, operation,
instrumentation, and control of the connected
equipment or entire plant. It is not unusual for
start-up and shutdown procedure to be governed
by mechanical design limitations rather than to suit
process only.
For exhaust steam vacuum service, opinion differs
as to whether the design temperature for thermal
expansion effects should be based on the normal
operating temperature under vacuum conditions
plus an occasional rise to 212 F, which temperature
would be approached with loss of vacuum, or on
212 F, as though it were the normal operating temperature. The first approach is consistent with the
handling of other operating upsets, it being recognized that at reduced capacity or after lengthy
periods of operation or with abnormally high cooling
water temperatures, higher absolute pressures and
corresponding temperatures may occur. It is therefore concluded that design considering the 212 F
case as an abnormal short duration (not an operating) temperature is reasonably logical.
The Piping Code (ASA B31.1-1955) is deficient
in adequate rules for protection against overpressure.
The requirements of the ASME Boiler Code, Section VIII, for safety valves, etc., are a useful guide
but require modification to suit common piping
practice. Pipe wall thickness is generally established
for a design pressure equal to the maximum (nonshock) service pressure, without provision for a
margin between service and design pressure, and
safety valves are generally set to relieve at about
10% above the design pressure. This is in contrast
with pressure vessel practice, where at least onc valve
must be set to open at or below the design pressure.
Differenoes,;also exist in the maximum overpressure
33
allowed while a safety valve is blowing; for oil piping
a 33}% increase is often used, compared to 10% on
pressure vessels, except under exposure to external
fire when 20% is allowed. This situation will probably be rectified when adequate rules for protection
against overpressure are provided in the Piping Code.
The static effect of individual loadings forms only
one phase of the broad subject of the design of piping systems. It is equally important to consider the
duration, frequency, and manner of application of
each loading, and their mutual occurrence. Both
pressure and temperature stress, if applied in a sufficient number of repetitive applications, may result
in fracture by fatigue. Failure may be accelerated
by the dynamie influence of very sudden changes of
pressure or temperature. Dynamic effeets may also
introduce the possibility of direct shock failure,
apart from the brittle fractures associated with metallurgical considerations or ferritic steels at temperatures below the transition range. While failure due
corrosion or metallurgical changes is not a subject
for this book, it should be mentioned that the level
of stress in the piping or the occurrence of plastic
flow may be a contributing factor in some cases.
Failure by stress corrosion is an important example.
The loading; which have been diseussed can be
segregated for design purposes into two categories:
1. Those representing the application of external
forces which, if excessive, would cause failure independent of strain.
2. Those representing the application of a finite
external or internal strain. These are generally
introduced through temperature change.
The design consideration of individual loadings
may be approached on the basis of the duration,
frequency, nature, and probability of their occurrence. Individual loadings may be:
a. Present during extended normal operation but
not during off-stream condition.
b. Maintained throughout the service life.
c. Occasional and of short duration as well as low
cumulative duration (including start-up and shutdown conditions).
d. Emergency or abnormal conditions of short
duration.
For proper establishment of design assumptions,
it is necessary to have an adequate appreciation of
all direct and contingent requirements to which the
piping system will be subjected, and also to understand the interrelations between the behavior of
structures and materials, according to our present
state of knowledge. It is the aim of this chapter to
provide useful assistance toward the first objective.
'0
34
DESIGN OF PIPING SYSTEMS
Chapter I, together with the references cited, should
prove valuablc in establishing a reasonablc and
broad fundamental understanding~of the flow and
fracture of metallic materials.
2.3 Design Limits, Allowable Stresses, and
Allowable Stress Ranges
In the preceding section of this chapter, piping
system loadings have been grouped into two categories: external effects which, if excessive, might
oause direct failure, and strain effects attcndant to
temperature change. Categories for individual
loadings were also suggested, depending on the duration, frequency, and nature of the loading. This
section is devoted to the discussion of the nature of
stresses for the various forms of loading common to
piping, as well as to a consideration of allowable
stresses and an examination of the design limits
which are not directly provided for by conventional
allowable stresses and nominal safety factors.
When considering basic allowable stress values, it
is appropriate to distinguish between primary,
secondary, and loealized stresses. Although there
is probably no aecepted definition of primary and
seeondary stresses in piping systems, the following
eriteria will be advanced for purposes of this discussion:
Primary stresses are the direet, shear, or bending
stresses generated by the imposed loading which are
necessary to satisfy the simple laws of equilibrium
of internal and external forces and moments. Among
the primary stresses due to external effects are the
direct longitudinal and circumferential stresses due
to internal pressure and the bending and torsional
stresses due to dead load, snow and icc, wind, or
earthquake. In addition there are the direct, bending, and torsional stresses due to restrained thermal
loading, the external forces being supplied in this
case by the line anchors or other restraints. In
general the level of primary strcsscs directly measures
the ability of a piping system to withstand the
imposcd loadings safely. Accordingly, those stresses
due to sustained external loading (categories (a) and
(b) of Section 2.2) are controlled to the Code allowable stress value for the operating temperature.
Some overstress is allowed for temporary external
loadings (categories (c) and (d)).
Secondary stresses are usually of a bending nature,
varying from positive to negative across the pipewall thickness and arising generally because of differential radial deflection of the pipe wall. A most
important example of secondary stresses is that of
the eircumfcrential bending stresses in a curved
pipe subject to bending, discussed in Chapter 3.
Secondary stresses are not a source of direct failure
in ductile materials upon single load application.
If above the yield strength they merely effect local
deformation which results in a redistribution of the
loading and a reduction of the stress in the operating
condition. If the applied loading is cyclic, however,
they establish a local strain range corresponding
essentially to their full original magnitude. They
thus constitute a potential source of fatigue failure.
Localized stresses are those which die away rapidly
\\~thin a short distance from their origin. Examples
are the bending stresses in the hub of a flange, at a
sharp eone-to-eylinder junction, or at the inside
diameter of a branch connection. Localized bending
stresses can be considered equivalent in significance
to secondary stresses. It is possible in some eases for
the plastic flow which may result from an initial overstress to alter the contour of the pipe to a stronger
shape. This would lower the local strain range during subsequent applications of the loading and the
fatigue resistance would be raised accordingly.
Allowing large initial amounts of localized deformation carries the risk, however, of propagating flaws
in the base material, particularly in welds, and of
initiating cracks in less ductile heat-affected zones
adjacent to welds.
The Pressure Vessel and Piping Codes contain
tables of albwable stresses at various temperatures
which are related only to the primary static-loading
stresses (categories (a) and (b) of Section 2.2). The
level of localized stresses at nozzles, branch connections, in heads, etc., is only loosely and indirectly
controlled by formula and shape requirements and
may easily be 100% or more above that of the
primary circumferential pressure membrane stress.
Due to the lack of adequate analyses or to the difficulty attendant to their evaluation, many secondary
and localized stresses are neglected by the Codes,
such as the bending stresses in vessel or pipe shells
due to piping reactions, although the Code may warn
the designer to consider such loadingo.
Two criteria are associated with piping stresses.
One is the so-called "Code allowable stress" at the
operating temperature, familiar to all designers of
pressure equipment; the other one is the somewhat
less known Hallowable stress range," which is derived
from Code allowable stresses and which has appeared
in the Piping Code since 1942 as the basis for expansion and flexibility design. The application of each
of these criteria is covered later in this section in
connection with specific loadings.
The allowable stress is a function of the matcrial
....
_~
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
properties and safety factors as associated with specific design, fabrication, and inspection requirements.
Experience with the pressure ves""l Codes as presently constituted has shown that pressure and other
maintained loading can be sustained by average
equipment within this allowable stress limit for an
indefinite period. Also, it is not uncommon to allow
moderate short durations of overload or overtemperature due to abnormal or emergency circumstances. In a more precise approach, however, such
overloads should properly be assessed on an integrated basis with respect to duration and frequency.
In the following pages, the various considerations
influencing the serviceability or safety of piping systems are summarized and augmented by current
opinion as to advisable limits of stress, or other design criteria.
For Pressure Loadiug: In the 1952 ASME
Boiler and Pressure Vessel Code, the basis for the
allowable stresses for ferrous materials in both Section I, Power Boilers, and Section VIII, Unfired
Pressure Vessels is given in Appendix P of Section VIII. This appendix is important as a general
reference not only for its explanation of the basis of
allowable stresses given in the Code but also for its
guidance in setting stress values for similar materials.
For nonferrous materials Appendix Q (Section VIII,
Unfired Pressure Vessels) similarly establishes the
basis of allowable stresses.
The allowable stresses for Section 1, Power Piping,
of the ASA B31.1-1955 Code for Pressure Piping
are identical with those of the ASME Power Boiler
Code; those of Section 3, Oil Piping, within refinery
limits, are in agreement in the creep range with
Section VIII of the ASME Code. At lower temperatures, the safety factor on tensile strength is lower
than that of the Unfired Pressure Vessel Code, allowable stresses being limited to one-third of the minimum tensile strength or 60% of the minimum yield
strength. The other sections of the Code for Pressure
Piping are intended for either ambient or relatively
moderate temperature service, with allowable stresses
in varying percentages of the yield strength Sv or
tensile strength Su as indicated below.
Section 2. Ga~ and air piping: 0.6 to 0.72 Sv
Section 3. Oil transmission lines outside refinery
limits: 0.85 Sv
Section 4. District heating systems: 0.25 Su
Section 5. Refrigeration piping systems: 0.25 Su
Section 8. Gas transmission and distribution piping systems: 0.72 Sv max.
The assignment of higher allowable stresses for high
L
35
yield-strength materials operating at temperatures
below the creep range, and recognition of yield
strength enhanced by cold work and/or heat treatment, reduces the margin of safety provided by the
Piping Code for unassessed stresses and for fatigue
life under cyclic conditions. In addition, Sections 2
and 8 use nominal rather than minimum pipe-wall
thickness, which further diminishes safety margins.
The dependence of fracture (and bursting) stress
upon the shape of the part is quite properly recognized in Chapter 1. This effect, however, is one that
is commonly ignored in ordinary design praeticc and
in the Codes whieh represent such practice. Hence,
the Code safety factors against bursting, related
only to fracture of conventional tensile test specimens, must be regarded as nominal values which are
not necessarily the actual safety factors for the bursting of a cylindrical vessel under pressure, or for any
other general shape. While an exact evaluation of
the disparity between safety factors for a tensile test
specimen and those for a tube requires a complete
knowledge of the plastic stress-strain properties of
the material, a general evaluation for a wide range
of materials is made possible by certain reasonable
assumptions.
At first, the material under consideration is considered to obey the effeetive stress-strain relationship
of eq. 1.8, stresses being dependent upon strains in
accordance with the deformation theory of HenckyMises (eq. 1.7). Further, it is assumed that a function of the type
(2.1)
where
true stress in uniaxial tension
strain in uniaxial
tension
Band n = assumed material constants,
0'1 =
E*l = logarithmic
can adequately describe the stress-strain curve in
uniaxial tension. The types of stress-strain curves
obtainable from eq. 2.1 through a variation of the
constant n (sometimes referred to as the strainhardening exponent) are shown in Fig. 2.1. 3
From the foregoing assumptions, it can be shown
that the engineering (conventional) stress in a tensile bar, at the instant of attaining the maximum
load, is given by
(2.2)
Su = B(n/e)"
where Su = ultimate (conventional) tensile stress
e = 2.71828 = base of natural logarithms
Band n are as previously defined.
aB, also called the Hhardncss factor/' is simply the true
stress value at a logarithmic axial strain of 1.0.
I
n = O. Ideol PIOllitity
= 1.0 l---::......::.::......::.=~::::=:;;::::=:::::::;?""~
;;.
I
•
.8
•
.6
..5
.4
•
;;;
~
C
~
I
DESIGN OF PIPING SYSTEMS
36
.2
~
o~--:------:------o
.2
..4
.6
.8
1.0
logarithmic Strain, E~
}i'IG.
2.1
Analytical representation of the tensile stress-strain
curve for various values of n.
This instability stress value is identical with the
conventional "ultimate tensile stress."
In a thin-walled cylindrical pressure vessel the
conventional circumferential stress at instability (at
the maximum sustainable pressure) can be expressed
as
(2.3)
For a structure in uniaxial tension a design based
on 11k of the ultimate tensile stress (as given by
eq. 2.2) represents a true safety factor of k. For
pressure vessels the safety factor should appropriately be applied to eq. 2.3. If, instead, safety factors are related to the ultimate tensile stress for
pressure vessel design, then the quotient
Q = S,ISu = 1.155(0.577)n
(2.4)
will indicate whether the real safety factor against
bursting, on a single application of overpressure, is
larger (Q > 1) or smaller (Q < 1) than the nominal
or presumed value of k, Le.,
(S.F.",,,,,) = Q X (S.F.t,n,'on)
(2.5)
A plot of eq. 2.4 in Fig. 2.2 gives values of Q for
values of n ranging from 0 to 0.5 and shows that
(for materials behaving as assumed) the safety factor for bursting of thin cylindrical vessels will be
larger th~n the tensile safety factor when n is less
than 0.263 and smaller when n exceeds this value.
In commonly encountered materials the strainhardening cxponent n varies from about 0.05 to 0.15
for greatly eold-worked or tempered materials and
is within the range of 0.2 to 0.45 for soft annealed
metals. Carbon and low-alloy stecls generally have
n values from 0.15 to 0.25. Within this range Q has
a value barely exceeding 1.0. Thus, if t of the ultimate tensile stress is used as a basis for design, an
actual safety factor equal to or somewhat higher
than 4.0 on bursting will apply to cylindrical pressure vessels (of carbon or low-alloy steel), as proved
by numerous static destruction tests. Similar comments apply to Codes using a different fraetion of
the ultimate tensile stress as a design basis. Thus,
for the ASA B31.1 Code, Section 3, which limits design stresses to t of the ultimate tensile stress, a
safety faetor of around 3.0 will be available against
bursting of thin-walled cylinders. With other materials or with departures from the simple eylindrical tube, however, it would appear that the shape
effect may bear investigation for more accurate
assessment of bursting conditions.
In the creep range a similar safety factor does not
exist. That is, if ereep continues while the pressure
is maintained, fracture will inevitably take place
after a suffieiently long time. Hence, the design
stress is selected to avoid failure within the service
life period.
For the case where 100% of the extrapolated 10'
hour creep fracture stress is allowed by the Code,
and if this value governs the design stress (i.e. it is
lower than the stress causing 1% creep extension
in 10' hours), it would appear that the "life factor"
(actual vs. desired life) may be no more than 1.0.
In other words, if the desired design life is also
10' hours (about 11.4 years), fracture should follow
when the design life is exhausted. Admittedly, there
are only a few ferrous metals whose extrapolated
stress value for creep fraeture is less than the stress
producing 1% creep in 10' hours. However, even
for these metals, no case of fracture following intended life is known in the annals of the industry,
although many pressure vessels have operated in the
creep range for periods eonsiderably exceeding
11.4 years.
One reason for this lies in the fact that the allowable long-time design stress values (for both creep
and creep rupture) are obtained by extrapolation
1.2
------=:--.2
..
Vclue of n
ol-_~--~-+~--~-~-
o
.1
~
,
~.3
.S
FIG. 2.2 The "safety factor ratio" Q as 1\ function of n.
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
from short-time tests. Although not strietly admissible, this extrapolation generally leads to acceptable
results for the ereep values as shO\V!l in Fig. 2.3. On
the other hand, in the very short-time creep rupture
tests comparatively high stresses are used. As mentioned in Chapter 1, this tends to promote intracrystalline deformation, with an ensuing high
ductility. At the longest commercial testing periods
(generally 10 4 hours) the stresses are much lower;
intracrystalline deformation is largely absent, and
the ductility is considerably lower, although the
stresses are still higher than those producing 1%
elongation in the same time. The respective position of these stress values does not change even
when the loading period is increased to 10' hours.
However, the conventional log-log extrapolated value
based upon test results up to 10 4 hours in duration
may in some cases yield a fictitious rupture strength
at 105 hours which is below the 1% creep stress
value, as shown in Fig. 2.3. The unrealistic aspect
of this extrapolation pal,tially explains why pressure
vessels do not fracture after 11-12 years even if
extrapolated test data would prediet this in eases
where the creep fracture value governs design.
Structural Effects. The Piping Code rules
ASA B31.1-1955 require that primary stresses due to
weight of pipe, fittings and valves, eontained fluid
and insulation, and other sustained external loadings
be maintained within the hot allowable stress Sh.
Occasional effects sueh as wind and earthquake
should have little influence on the fatigue life of the
piping system or ereep at high temperature. Therefore, they ean be treated more liberally, similar to
AISC (American Institute of Steel Construction)
practiees, where 33!% higher stress is allowed for
the separate effects of wind or earthquake superimposed on the basic lo~ding.
In average piping systems, structural loading is
not investigated in an overall fashion; instead it is
eontrolled by standardized practices and details. In
extreme cases of large or stiff piping it is advisable to
evaluate the complete loading. Attention should be
directed to those loadings which can occur simultaneously, so as to obtain an integrated equivalent
eyelic strain as discussed in Section 2.6 and under
"Temporary Loadings" in this section.
Structural instability or collapse of piping under
longitudinal loading, such as is encountered in
columns, is possible only under unusual circumstances. Collapse by circumferential buckling is
more likely to occur, although tbe thickness-to-radius
ratios ordinarily used in piping applications are
usually high enough to prevent this. As a design
L
~
37
o Creep Fracture Tel! Data
)( Creep Role
"'-cu~......
Creep Siren
for 1% Creep Rolc
ElClropololcd
~Cfccp""frQclUrC CUNC
~
----J'--;,
rTruc Crcllp-Fracture
........... ~_
' ,____ Curve
-
Siren lor 1% Creep Rale
_...
in 10 5 hours
E:drapolotcd Siren for Frodurll in 10' hour1
J
...
(d1l1ign slreu)
10
10 2
10 3
10'
10'
Time, hau'" (log. scolc)
FIG. 2.3 Comparison of extrapolated and actual creep-fmcture curves for a typical material at constant temperature.
criterion to guard against circumferential buckling,
it is suggested that primary longitudinal compressive
stresses should not be permitted to exceed 0.07 Etlr,
where E is Young's modulus of elasticity, t is the
wall thickness, and r is the radius.
The allowable stress range was suggested initially
by Rossheim and Markl [I] as a measure of the
permissible strain range in a cycle of load application
to guard against the possibility of a fatigue failure
after a given number of cycles. It is selected so that
it will be applicable to ductile materials and to
average commercial pipe surface conditi'ons at the
location of highest stress (strain) range. The principal cyclic loadings are restrained thermal expansion and pressure, although weight of contents and
occasional effects such as wind and earthquake are
also repetitive in nature. A cycle of external loading
usually varies from the full presence of the loading
during operation to its complete removal under offstream conditions; the distribution of the associated
internal strain between the cold and hot ends of the
cycle may on the other hand vary due to the dependence of strains on the material properties at each
temperature and the presence of initial fabrication
stresses or residual stresses set up as the result of
plastic flow.
With the erection and completion of the final
joint of each leg of a piping system, internal stress
may be introduced by cold pull, weld shrinkage, or
flange makeup. This establishes an initial state of
stress, limited only by the yicld point of the material.
With temperature change on the first period of
operation, expansion strain is superimposed on the
residual fabrication strains. If the total exceeds the
elastic limit at any point, yielding occurs, leading to
relaxation of the initial fabrication stresses and a
redistribution of the thermal strain. Prolonged
elevated temperature will serve to further reduce the
DESIGN OF PIPING SYSTEMS
38
~AI_
A,_
I
l
,
t=l
FIG. 2.4
~,
~
I
Representation of bar for ca.lculation of plastic
strain concentration factor.
hot stresses by creep, at a rate proportional to the
combined stress (expansion, pressure, weight, etc.).
The reduction of the stress due to thermal strain
loading by plastic flow or creep at the operating
temperature is termed "relaxation." The relaxed
strain reappears at the cold end of the temperature
cycle with reversed sign.
For moderate-temperature piping, the division of
thermal strain between the hot and cold condition is
adjusted during the initial cycle in an amount
dictated by the initial residual fabrication stress and
the thermal-stress magnitude. The imposition of a
temporary overload during operation can effect a
further strain shift from the hot to the cold condition.
For higher temperatures, where creep occurs, strain
adjustment continues until the combincd stress at
the operating temperature is reduced to the relaxation limit. For convenience in design this is generally
assumed to be the Code allowable stress level. Although such adjustment takes place, it is important
to grasp the fact that the strain range per cycle does
not change and that the ability of the pipe material
to sustain t·he range is a function of both its hot and
cold properties. The process wherein the pipe line
seeks an equilibrium condition, and the resulting
self-adjustment accomplished by yielding and creep,
is termed "self-springing."
Self-adjustment may he minimized by prespringing
(cold springing) which consists of incorporating prestress during erection. Since this practice is particularly useful in controlling initial reactions so as
to protect connccted equipment it will bc discussed
in that regard under the heading of Piping and
Equipment Intereffects in Section 3.14.
As to whether prespringing offers advantages
beyond controlling thc initial hot reaction, a general
answer cannot readily bc given. In the 1942 edition
of thc Piping Code, the allowable stress range could
in effect be increased when 50% or more prespring
was provided by the permissible reduction in the
expansion loading to two-thirds. The 1955 edition
provides a uniform stress range regardless of the
initial strain condition. This is based on the reasoning that fatigue life is primarily dependent on the
range of strain which is unaffected by prestress, and
that the piping system seeks an equilibrium condition by self-springing. Credit for prespring is, however, still permitted when estimating maximum hot
and cold reactions on terminal equipment. By
prespringing, the plastic flow which the line may
have to undergo on the first, or first few cycles, in
order to effectively sclf-spring itsclf, can be avoided
entirely or appreciably reduced. This is sometimes
considered advantageous in minimizing the risk of
an early failure due to "follow-up elasticity" effects
should there be a highly localized weak link in the
system. Howevcr, from a fatigue standpoint, no
benefits are attributed to cold springing once selfspring has been effected. The advantage of prespring in this respect is more important for piping
which is to operate at temperatures in the creep
range. The proposition has also been advanced that
the hot plastic flow associated with self-springing
will detract from the final available ductility under
high temperature ucreep" conditions; in reality, the
mechanism of self-springing is probably more nearly
akin to fabrication hot forming operations. In this
light, the only clearcut conclusion that can be drawn
is that prespringing can have only advantageous and
no deleterious effects, especially as concerns initial
terminal reactions. Therefore, it is a desirable
practice when economically justified and effectively
carried out.
The 1955 Piping Code rules call attention to the
possibility of an undesirable amount of creep in areas
of reduced strength, such as short runs of reduced
size in highly stressed zones under certain conditions.
The possibility of the unit strain in local highly
stressed areas being magnified under conditions of
plastic flow by reason of the follow-up elasticity of
the more lowly stressed areas is not generally appreciated. In order to gain a better understanding, it
is of interest to study a simple analogue consisting
of a bar having a section of reduced area, as ShO\Vfi in
Fig. 2.4, restrained at the ends and subjected to
cyclic heating and cooling. The bar will be assnmed
to be made of an ideally elastic-plastic material
(non-strainhardening).
Let this bnr now be subjected to cyclic henting
nnd cooling of constant amplitude, to a level which
causes plastic flow in member 1 on each cycle. It
can be shown then that during any thermal halfeyele
(from heating to cooling or vice versa), other than
the first heating operation, the total (elastic plus
plastic) unit strain in member 1 is given by
(2.6)
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
where
" =
elastic strain range limit
= Su'
+ SUh . _
E,
Eh
39
All values eokuleled
for A,/A 2=0.5
7
(2.7)
e = unit linear thermal expansion for a
temperature rise of !J.T.
L = total length.
L" L 2 = lengths of members 1 and 2.
AI, A 2 = cross-sectional areas of members
1 and 2.
Suo, Suh = yield strength at the cold and hot
temperatures, respectively.
Eo, Eh = Young's modulus of elasticity at
the cold and hot temperatures
respeetively.
Had this bar been analyzed on the assumption that
all strains remained elastic, the ealculated unit strain
range in member 1 would be given by:
L
eLl
£c = _.....:~1
(2.8)
+ AlL,
A,L I
The strain given by eq. 2.6 is higher than that
indieated by eq. 2.8, and the ratio of the two can be
termed the strain magnification factor f3, which is
given by the following equation, valid for " ;:0: "
f3, =
1+
AlL,
AzL I
[1 - ~J
(2.9)
Ec
This is an extremely interesting result, since Ec is the
maximum unit strain calculated by applieation of
elastic theory and Ec is the maximum unit-strain
range which the material can accept without allowing
plastic flow on each cycle. So long as " does not
exceed " there is no magnification factor. The
magnification factor for Ec greater than £c is given by
eq. 2.9. Figure 2.5 is a plot of this equation for a
speeific ratio of AliA, = 0.5 and shows the magnification factor as a function of .,1., and L2 /L,; high
values can be reached which would materially reduce
the fatigue life of such a bar. The magnification
factors increase as area Al approaches area A z . At
first thought this might be unexpected; the explanation is that, as A,IA 2 approaches unity, the portion
of the calculated strain in member 2 which is never
developed, but instead causes plastic flow in member
1, increnses ns a dircct function of At!A,.
From this simple analogue it can be generalized
that, in any system which is stressed so that plastic
flow occurs over a portion of the total length only,
the unit strain is magnified in the portion undergoing
L
o ~,
-,:--73--':'--:'--:.:--:7:--,:--79-':':O-:'l1'--:12::"'-J~.
FIG. 2.5
',/t.
Relio of CelClJlahtd Elastic Strein Ronge 10
Available Eleslie Siroin per HoI! Cydo
Strnin magnification in a locally weakened bar.
such flow by the follow-up elasticity of the more
lowly stressed portion. It is not necessary that the
area of the critical portion be less than the remainder.
All that is necessary is that plastic flow occurs prefer. entially in the critical portion rather than over the
rest of the system. Lower mechanical properties can
have the same effect as reduced area. Systems
stressed in bending are subject to this effect even
when of uniform properties and size due to the nonuniform stress distribution which prevails. Strain
magnification will occur whether the plastic flow is
due to exceeding the elastic limit or is due to operation at high temperature where the plastic flow and
strain magnification factor would be a function of
time per cycle.
Similar conclusions were obtained in a receht
paper by Robinson [2J. Analyzing a few selected
piping systems operating at elevated temperatures
(in the creep range), he found that severe strain
concentrations can exist in layouts where the. maximum stress is limited to a very short length of the
piping, and where the follow-up elasticity of the
remainder of the system is great. These findings
are in agreement with those of the previously presented analysis for strain concentrations under
plastic flow conditions.
The allowable stress range limits established by
the Piping Code are such that plastic flow duc to
expansion effects is not permitted to occur with each
cycle. Both yielding and creep effects have been
considered in bnsing the hot portion of the allowable
range on the hot yield or creep strength, whichever
governs. Repetitive strain magnification over substantial lengths of the piping should, therefore, not
occur. For lines \vhich are not presprung, it is, however, possible for some such strain magnification to
occur during the initial operating period, while the
40
DESIGN OF PIPING SYSTEMS
line is undergoing self-springing. Since this occurs
only once it must be considered in an entirely different light and would have no influence on fatigue
life.
The bar analogue presented above was used to
derive magnification factors aSSUming that the weak
area was initially known and that an elastic analysis
of stress conditions was made. The analogue could
be readily modified to show the extremely high local
magnification factor which would exist at a defect
in a bar of uniform area, which is sufficiently serious
to cause local plastic flow. It is well known that
fatigue failure follows rapidly in the presence of such
a defect.
The allowable stress range, as associated \\~th the
various types of repeated loading, is discussed in detail in the following treatment of specific loadings.
minimum of 7000 cycles of operation without failure.
Local and secondary stresses are kept within this
limit by the stress-intensification factors. For a
number of cycles greater than 7000 the stress range
is reduced by a factor relating the allowable stress
range to the number of cycles as determined by
ambient temperature fatigue tests on carbon-steel
pipe. The reduction factor has a lower limit of 0.5.
Some adjustment of these factors, particularly for
materials other than carbon steel, will undoubtedly
be necessary as further fatigue information is
obtained.
The possibility of fatigue failure under the cyclic
straining conditions present in piping systems has
been questioned by many individuals. The propositions were variously advanced that the internal
strain loading associated with thermal cycling can-
Since thermal expansion
not initiate fatigue cracks, or that the stress-relieving
occurs as a finite strain load associated predominantly with bending effects, fracture on initial application is unlikely to occur in ductile materials.
Fractures resulting from repeated applications of
thermal strain loading are similar to fatigue failure
under mechanical loading. Therefore, the allowable
stress or strain range must be related to the number
of cycles anticipated during the life of the piping
system. Failure will occur in the zone of highest
and annealing effects at elevated temperatures would
prevent the propagation of such cracks. As indicated in Chapter 1, reasoning should lead to the
Expansion Stresses.
cyclic strain, whether primary, localized, or second-
ary. For this reason it is necessary to apply stresE
intensification factors for any individual piping component wherever stresses above the level of the primary stresses are introduced. Due to the importance
of such stresses from a fatigue standpoint, Chapter 3
is entirely devoted to recording present knowledge
of stress intensification in various components of
piping systems as well as their influence on flexibility.
Overall design is based on the stress range for the
critical component, as established by its intensification factor and the nominal primary stress at its
location.'
The basic allowable stress range established for
thermal expansion stresses in the 1955 Piping Code
1.25S,
+ 0.25Sh
where S, = allowable stress at ambient temperature
Sh = allowable stress at operating temperature,
has been selected with the objective of providing a
~Since the pressure vessel codes do not provide rules for
thermal expansion loading, it is desirable to check the effect
of comparatively stiff piping on vessel shells of low thickness/
radius ratio. This is accomplished in the manner outlined in
Chapter 3 for terminal connections.
opposite conclusion; furthermore, experimental veri-
fication that fatigue under constrained thermal
loading does occur. is provided by the work of
L. F. Coffin, Jr. [3, 4,], who demonstrated that
fatigue failure is primarily associated with the range
of cyclic plastic strain, while stress or strain relief is
of a secondary order of influence.
The Code allowable stress range cited above assumes that longitudinal stresses due to pressure and
other sustained external loadings are not over the
basic hot allowable stress, Sh. For hot lines the
expansion stresses at operating temperatures are
assumed to be gradually lowered by yielding and
creep, so as to be carried essentially as an off-stream
or cold stress. If the longitudinal stress due to sustained loadings is less than Sh, the Code permits the
unused portion to be applied to extend the stress
range available for expansion effects. Therefore the
Code, in effect, permits a total maximum allowable
stress range equal to 1.25(S, + Sh), for thermal
expansion stress combined with stresses from other
sustained loadings. For service temperatures below
the occurrence of significant creep, the total per-
missible longitudinal stress (both bending and direct)
is equivalent to approximatcly 1.25 times the yield
strength for power piping and 1.38 to 1.5 times the
yield strength for oil piping.
In general, Code design is simplified for general
use; at best it considers only average static conditions and establishes minimum design r~quirements,
placing dependence on the safety factor to take care
of unassessed stress conditions. The cyclic nature
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
of loading and the possibility of fatigue failure are
not specifically considered, except in thc Piping
Code's treatment of piping llexibility for thermal
expansion. It might be asked why the fatigue design
approaeh is currently limited to piping expansion
analysis. This is due to the fact that the Unfired
Pressure Vessel Code rules limit primary pressure
stresses in ferritic materials to 62~% of the yield
stress and 25% of the tensile strength. This provides a reasonable margin against the possibility of
fatigue due to loealized and secondary stresses, which
may be 100% or more above this allowable stress,
for the type of cyclic conditions normally encountered in most pressurc vessel services. By comparison, thermal strains playa greater role in the design
of piping, which would be seriously affected economically (and would be virtually impractical in the
case of large stiff systems) if total stress including
expansion effects were to be held within the Code
allowable stress at the operating temperature.
Spurred by this necessity, experience and analytical
work have led to the Piping Code's more advanced
treatment of thermal strains, and to rules which
recognize the influence of number of cycles, hot and
cold material properties, and local stress intensifications.
It remains for the piping engineer and designer to
reeognize any unusual demands imposed by the design or service on piping systems. The following
topics, in particular, are not at present adequately
covered by the minimum Code design.
Shock or Dynamic Loading. Shock or dynamic
loading conditions warrant special consideration because of the added stress which can be introduced
by the rate of application of the motivating influence
and the fact that the yield point of steel can be appreciably raised by very rapid loading. Localized
yielding at points of stress concentration may be
inhibited under such conditions and fracture more
readily initiated. The general subjcct of vibrations
which are a source of concern from a fatigue stand-
point is treated in Chapter 9. The more significant
dynamic loadings which enter into piping design can
be listed as follows:
Earthquake. The accelerations associated with
earth tremors are generally of the order of 1 to
S ft/sec 2 • These values represent about 3% to 25%
of the 32.2 ft/sec 2 acceleration of gravity. For this
reason, earthquake design is commonly approached
by applying a horizontal force acting at the center
of gravity of the structures; this force is 10% to 20%
of the structure weight, depending on the maximum
accelerations recorded for the locality considered.
L
41
Earthquake loading is not normally assumed in design unless it is specifically required for the locality
concerned. Some consideration has been given to
requiring that all structures be checked for some
minimum lateral thrust of this type, lower than in
recognized earthquake zoncs, but this is not the
practice at present.
Gun Fire. Piping on warships is somctimes
checked for the dynamic effect due to the firing of
guns.
Waler Hammer or Flow Surge EjJec/',. The Piping
Code contains water hammer allowances for cast
iron pipe, in the form of a required increase in design
pressure.
On steel pipes no standard allowance is
made for flow surge or hammer, and allowances arc
usually made only on high-head water flow lines,
such as penstocks. The shock pressure due to sudden stopping of a liquid is a function of its velocity,
stoppage time, and the elasticity of the pipe. Pressure surge effects are present wherever reciprocating
pumps or compressors are used. The accompanying
mechanical vibrations may in certain cases be sufR
ficient to result in fatigue failure, if not promptly
corrected. This subject is treated in more detail in
Chapter 9.
Brittle Fracture in Ferritic Steel. The potential dangers of the brittle fracture of steel structures
were made clear during World War II and after by
the numerous failures of merchant ships, and by
occasional partial or complete failures of bridges,
pressure spheres, gas-transmission piping, and storage tanks. The phenomenon and conditions under
which fracture may occur were discussed in Chapter 1. From the practical design standpoint it has
becn realized for a long time that, as ambient temperatures are reduced, the hazard of brittle fracture
in ferritic stcels is increased. As a result, the Pressure Vessel Codes have required for many years that
for services below -20 F (excluding applications
for service at prevailing ambient temperature, such
as outdoor pressure storage tanks), ferritic materials
have an impaet value of at least 15 it-Ib, at the lowest intended service temperature as determined by
keyhole or U-notch Charpy specimens.
The numerous fraetures of ships and other struetures have resulted in extensive investigations for
the causes underlying brittle behavior. While no
complete praetical remedy for avoidance of brittle
fracture has resulted, several factors have been reeognized to have important influence. Although individual impact or equivalent testing of each plate,
bar, or tube at the lowest service temperature still
provides the best assurance as to its transition tem-
42
DESIGN OF PIPING SYSTEMS
perature, there is definite evidence that average
transition temperatures are lowered and the incidence of failures significantly reduced, within the
range of ambient temperatures, by using openhearth or electric-furnace steels, controlling the
manganese-carbon ratio of plates over ! in. thick,
and by employing killed steels made to fine-grain
practice, particularly for 'thicknesses over 1 in. (see
ASTM Spec. A131-53T for example). Normalizing
is also desirable for important plate applications over
1 in. although none of the ASTM Specifications for
structural steel at present requires this in any thickness; however, ASTM Specification A131-53T in
paragraph 4 (b) mentions that plates over 1i in. may
be required to be produced to special specifications.
The ASTM Specification A373-54T covers structural steel for welding and is similar to AI31 except
that it makes no reference to fine-grain practice for
plates over 1 in. or to special requirements over
1i in. The development of these specifications and
their gradually more widespread use in the construction of ships, tankage, and other structures at insignificant increase in cost is an encouraging trend.
Though it represents only a modest start it indicates
that much more could be accomplished by economic
steel specification control and that its extension to
all pressure services is a necessary undertaking.
The experimental work also showed that a significant improvement in performance can be achieved
through careful design by the avoidance of high
stress concentration or areas of high local restraint
(e.g., ship hatch corner design). Significantly, all
such failures have been triggered off by a relatively
minor flaw or notch, the majority of which were
associated with welds. Apparently, in addition to
the possibility of welds containing small cracks, the
local residual stress pattern associated with them is
a factor. The latter plays a significant role, not only
in initiating crack propagation, but in accelerating
the crack propagation speed to a level where it can
continue as a spontaneous process through a much
lower stress field. This is in keeping with the theory
given in Chapter 1.
Non-ductile Materials. Cast iron and other
non-ductile materials are usually confined to relatively low temperature service when used for pressure parts. Bending stresses for these materials
must be kept within well-defined allowable values
(for cast iron, usually I! times the allowable stress
for tension). Bell-and-spigot or packed joints of a
design incapable of taking longitudinal stress are
provided with anchors at the end of each run, with
expansion absorbed by movement at the packed
joints.
For low-temperature underground lines
expansion provision is usually not necessary.
Temporary Loadings. An allowance of 33!%
above the basic allowable hot stresses established
for oil piping in the Piping Code has been suggested
for temporary loadings due to wind or earthquake.
Stresses due to occasional brief overloads in operation can be similarly treated; such might be occasioned by minor upsets in operating conditions or by
starting-up or shutdown conditions. For power
piping applications the ASME and ASA Codes
specifically recognize occasional operating variations
in pressure and temperature, allowing the following
increase in the calculated stress due to internal
pressure:
1. 15% during 10% of the operating period.
2. 20% during 1% of the operating period.
This permissible overstress is intended to cover the
surges expeeted to occur due to the heat lag of large
boilers when the output is suddenly decreased. It is
not recommended as a general design practice for
normal operation variations in pressure or tempera-
ture as it is better to design for the maximum pressure and temperature conditions expected to occur
in regular operation.
However, brief temperature
or pressure upsets may be treated on this basis.
provided they are such as to require quick remedial adjustments in operation to restore normal
conditions.
Severe upset or emergency loadings sometimes call
for immediate drastic corrective measures and may
require shutting down the unit. Wherever practicable the same limit as proposed for temporary
loadings should be observed, but the nature and
probability of the emergency often requires special
consideration. In the case of piping where design io
controlled by creep and stress-rupture properties.
analysis of the ability of the system to sustain an
occasional short duration emergency can be based on
the short-time properties of the materi&1 or, if more
frequent, on the permissible creep stresses for the
shorter time period involved, by evaluation of the
cumulative creep for service and unusual conditions.
No standard guide can be given. More study and
tests are desirable to assess the cumulative effect of
short-duration high overloads and long-duration
normal loads. It is known that, for a given total
period of overloading, the number of times the
loading is applied has a significant effect, being more
damaging as the frequency of application increases
for a constant total duration of the overload.
Where basic allowable stresses are set higher or are
established by cold-worked properties (e.g. gas
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
transmission line piping), overstress due to temporary loading should be avoided.
Abnormal temperature differences may occur due
to upsets or during start-up operations, which can
cause thermal expansion stresses higher than assumed
for the normal design eondition. When infrequent
compared to the normal design condition, some
increase in the permissible stress range can be justified. For example, when working to the rules of the
1951 ASA Piping Code, The M. W. Kellogg Company designed for emergency thermal expansion conditions using a 50% increase over the basic allowable
stress range. A more appropriate design approach
would be one which would determine the number of
cycles at the Code allowable strESS range which
would be equivalent to the number of cycles under
the diverse conditions aetually anticipated. Assuming a basie relation between number of cycles Nand
stress range SR of the form
N = (K/S R )"
the equivalent number of cycles N, at a stress SA
ean be established roughly as
S,)" Nt + (S,)"
N, = ( SA
SA N, + ... (SD)"
SA N D
(2.10)
where K and n are constants for the material.
Nt is the number of eycles producing an overload stress St.
N 2 is the number of cycles producing an Overload stress S2, etc.
N D is the number of expected operating cycles
on the normal design basis.
S D is the corresponding calculated stress.
SA is the Code allowable stress range for
7000 cycles.
Since the Code stress range is intended to provide for
a minimum of 7000 cycles at a stress SA, if N, does
not exceed 7000, the design may be considered
equivalent to a Code design. Tests on carbon-steel
pipes [5] indicated that n can be taken equal to 5.
Without similar test data, the use of n = 5 for other
materials is open to some question.
2.4 Stress Evaluation
Stress evaluation is commonly limited to primary
direct, bending, and torsional stresses which, in piping, result from the effect of pressure, weight, and
thermal expansion. Localized and secondary stresses
which do not affect the overall system are not ordinarily evaluated directly although their influence on
L
4:1
the cyclic or fatigue life under thermal expansion is
taken into account through so-called stress intensification factors. The following discussion presents
background information and comments to aid understanding of the current approach in treating various
loadings.
2.4a Internal Pressure up to 3000 psi Maximum. In their present status, the Pressure Vessel
Codes already mentioned are stated to be applicable
when the pressure does not exceed 3000 psi. Pressures above this may require special attention to
design and fabrication details, closures, branch connections, etc., in view of the heavier wall and thickness/diameter ratio involved. Actually, any such
limit is strictly arbitrary and should more properly
be established as a pressure/stress limit so that the
influence of different materials and the effect of
temperature would be included.
For the most common surface of revolution,. the
cylinder, the so-called inside diameter (or membrane)
and outside diameter (or Barlow) formulas were
first used for thickness/diameter below and above
0.1 respectively. These were later supplanted by
the mean diameter formula and, more recently, by
the universally adopted formula approximating the
results of the Lame formula. All these formulas
may be expressed in a common manner as follows:
S = (pr;/t) + Kp
(2.11 )
where p = internal pressure.
r i = inside radius.
t = wall thickness.
K = constant having values between a and 1.
If K is given the value of 0, the inside diameter formula is obtained; for K = 0.5, the mean diameter;
for K = 1.0, the outside diameter. When the value
of 0.6 is used, stresses are obtained which correlate
reasonably well for values of t up to about 0.5r, with
the recognized inside circumferential stress formula
of Lame. This approximation, discovered by H. C.
Boardman, was rapidly adopted for moderatetemperature piping by both Pressure Vessel and
Piping Codes, while for piping in the creep range it
is considered applicable if a further adjustment of
K is made as covered later in this section. Similar
relationships, which approximate the direct cireumferential pressure stress at the inner-wall surface for
other shapes of revolution, are presented in Table 2.1.
For dished heads it may be noted that the Code also
relates the design of torispherical and ellipsoidal
heads to the sphere formula, which is suitably modified by a correction factor to correspond with the
44
DESIGN OF PIPING SYSTEMS
Table 2.1 Internal Pressure-Circumferential
Stress Formulas for Elastic Conditions
Shape
S
P
SEt
Cylinder
-P [r,
EA
r, + 0.6t
SE - 0.6p
+ 0.6tl
Use (~) in place of Tj in the cylinder formulas.
Cone*"
CaSa
2SEt
Sphere
W'
Torus (pressure inside)f
BE
General shape of revolutiont
[RR-- 0.5r'J
pK
pr,
SB - IJK
2~t 11', + 0.2tl
r,-+-0.2t
2SE - 0.2p
fi
[I - 2R,
..>:!...J
SEt
R-
( R -
E..
[(R R-- 0.5r,) r, + KtJ
0.5,,) r,+ Kt
Et
S
;t[(1-2~},+KtJ
f'i
Tj
(r2R ) r, + Kt
1 -
1
where Ti = inside radius (usc meridional radius in general formula, i.e., radius from axis of revolution and normal to
surface, see Fig. 2.6).
E = weld joint efficiency.
R = torus center line bend radius.
R 1 = actual radius of curvature in meridional plane at the point in question (positive if concave to pressure)
(Fig. 2.6).
a = ! cone included (apex) angle.
K = 0.6
C+ L,/R)
(use absolute value).
S = circumferential stress.
p = internal pressure.
"'Not covered by Piping Code at present.
tNot given in nny code at present.
membrane stresses associated with their contour.
The pressure design of shell openings for nozzles,
manholes, and branch connections is based on the
simple maintenance of the original cross-sectional
area, by replacement of the removed metal by reinforcement Immediately adjacent to the weakened
area. Flanges and cover plates involve primarily
bending stresses; the direct stresses in these components are commonly neglected due to their lesser
magnitude. Speeifie formulas are given in the Codes
for their pressure design.
2.4b Internal Pressure over 3000 psi. The
Codes at present (1955) do not cover the design of
high-pressure vessels, although this subject has received considerable attention in the last two decades.
Many problems arise at high pressure for which conventional code details are either totally unsuited or
present an undesirable choice. Examples are: nozzle
reinforcements which, within Code limits for rein~
forcement, entail extremely abrupt changes in section, cones, etc., involving inside corner radii which
are small in comparison with the wall thickness. As
the pressure is increased, practical limits are reached
for design as covered by Code rules. In the following it is attempted to summarize the practices which
FIG. 2.6 The meridional radius of curvature for
l'hells of revolution.
are followed in the design of shells, heads, closures,
and connections of high-pressure piping.
The Lame formula and the Rankine (Maximum
Principal Stress) criterion, on which the ASME
Boiler Code and ASA Code for Pressure Piping are
based, no longer predict general yielding or rupture
within reasonable limits when the thickness/diameter
ratio exceeds approximately 0.20. Although the
error is on the safe side, the deviation becomes
greater the more the thickness/diameter ratio is increased. For initiation of yielding the Maximum
Shear or Maximum Shear-8train Energy Theories
are in good agreement with experimental evidence,
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
as mentioned in Chapter 1. Either of these theories
may be used to practical advantage as general yielding or bursting criteria when appliiJtl ineonjunction
with plastic stress analysis.
For thick cylinders, yielding of the inside fibers
leads to eompressive residual stresses in the plastically deformed portion of the wall when pressure is
removed, increased stress in the outer fibers under
pressure loading, and greater uniformity of shear
stresses throughout the wall thickness. This redistribution of stresses due to plastic flow is termed
Hauto-frettage"; it was first employed for casting
guns in the early nineteenth century. Later, greater
eontrol and uniformity of stress distribution was attained by shrinking suceessive closely maehined shell
layers on to each other, thus producing a thickwalled cylinder, whose inner layers are in a state of
45
Circumfcrlmliot
Slro"
Axial l-++-iH-i
SlrMS
I
Ton~ilc
I
Zora
I
Compreuivo
I
Radial
Siren
precompression.
The fact that initial yielding of the inner fibers
occurs at only a fraction of the pressure corresponding to general yielding distinguishes thick-walled
vessels from thin-walled shells. Since the pressure to
produce failure in thick-walled vessels is more
properly associated with plastic rather than elastic
criteria, a valid design of these structures can be
based on plastic analyses, and related to the general
yielding and bursting conditions. The various
approaches which have been suggested are discussed
in the following paragraphs.
Modifl£d Elastidty. This approximate solution
assumes that a safety factor of 4 on bursting is maintained so long as yielding of the inside fibers is
avoided at the design pressure. This approach also
requires that the stress at the mean wall thickness,
as calculated by the Lame formula, does not exceed
the usual allowable (0.25S.) value. The safety
factor assumed by this analysis is likely to be in
error on the unsafe side.
A ulo-freUage. The wall is assumed to be in two
layers with the inner layer taken to be in a state of
preeompression, attained by applying a suitable
overpressure and yielding the inner fibers. The
stress is then calculated by the Lame formula considering the initial prestrcssed condition. Thc results
will be similar to the preceding approximate approach for the same safety factor.
Partial and Complete Plasticity. Stress analyses
of eylinders having an inner plastie-elastic zone and
an outer elastic zone are available in many text
books dealing with plasticity. These solutions are
generally based on the assumption of an idealized
material which is elastic up to the yield stress and
plastic (non-work-hardening) at the yield value.
Elastic CO$e
FIG. 2.7 Typical stress variation in a pipe under elastic
or creep conditions.
For a severely cold-working material the assumption
that the strain is the sum of an clastic strain obeying
Hooke's law and a plastic strain can be considerably
in error. Special analyses have also been worked out
for strain-hardening materials.
Plasticity analyses are generally based on the
assumptions that (1) elastic strains are negligible in
eomparison with plastic strains; (2) the volume of
the matcrial remains constant during deformation;
and (3) the length of the pipc is unchanged under
the application of pressure. The distribution of circumferential stresses changes completely from the
elastic results, the maximum in the plastic range
occurring at the outside fiber. The shear strcss also
tends to be constant through the wall thickness, but
remains a maximum at the inner fiber.
Figure 2.7
illustratcs the difference in stress distribution. For a
thick-walled cylinder of an ideally plastic (non-workhardening) material, Ni,dai [6J gives the following
formulas at the onset of general yielding:
S
log, (r,/r)J
log, (r,/ri)
= p[1 <x
-p[log, (r,/r)J
log, (ro/ri)
2S, = S'X - S'X = 1
age
P
(/)
To Ti
(2.12)
(2.13)
(independent of r)
(2.14)
46
DESIGN OF PIPING SYSTEMS
where Sc% = circumferential stress at any radius r.
S" = radial stress at any point T.
8,
=
r0
r~'
= outside radius.
= insidel radius.
shear stress.
r
=
,""'
radius at point in question.
The value of 2S, is equal to S, at the outside radius.
If this is accepted as a suitable criterion of general
yielding or bursting, it is interesting to know that
eq. 2.14 can be closely approximated by the simple
mean diameter formula.
Spnrred on by an interesting paper by Burrows
and Buxton [7J on available formulas for cylinders
under internal pressure, the ASA B31 Committee
appointed a special task group to study the subject
and recommend a simple appropriate formula for the
design of heavy-walled piping in the creep range.
This task group recommended that the value of K in
the simple formula of eq. 2.11 be gradually modified
from 0.6 to 0.3 at temperatures over 900 F for
ferritic steels and over 1050 F for austenitic steel.
This recommendation was approved and the formulas
for piping in the ASA Piping Code, Sections 1 and 3,
and the ASME Power Boiler Code now include this
provision.
The formulas given in eqs. 2.12 to 2.14 will provide
a reasonably good answer for the behavior of thickwalled cylinders made of materials with only a mild
strain-hardening tendency. Where a more e~act
evaluation of probable performance is desired; the
stress distribution should be evaluated from t.he
actual stress-strain characteristics of the material [8, 9, lOJ. An analysis of thick-walled cylinders
under internal pressure in the creep range has also
been advanced by Bailey [11J.
Concerning the practical design details of thick
shells, an effort should be made to avoid stress raisers
in the form of abrupt changes of section at the location of openings, nozzles, and intersections. The
observance of these rules, coupled with careful control of materials and fabrication, and with adequate
testing, may permit a reduction in the overall
nominal safety factor without diminishing (and
possibly improving) the real safety factor. With the
trend to higher pressures and temperatures, more
adequate use of material is imperative. Lower safety
factors for simple surfaces of revolution or for construction of controlled low stress intensification is
also necessary [12J.
2.4c External Pressures. External pressure
loading involves, in addition to control of direct
stresses, the consideration of stability. Direct
stresses for external pressures are governed by the
same formulas as for internal pressure, except that
the signs of all of the equations containing the pressure p have to be reversed, indicating compression
stress.
Stability of cylinders -against collapse is well
covered by the rules of the ASME Boiler Code,
Section VIII, which provide for the design of both
unstiffened and stiffened cylinders of all Code
materials. For an explanation of the Code charts,
reference should be made to a paper by E. O. Bergman [13J. This paper also contains an extensive
bibliography on this subject. Similar to columns,
the limiting compressive load which a cylinder will
sustain is related to its equivalent slenderness, end
conditions, and deviations from true contour. In
the case of long unstiffened cylinders (length/
diameter over about 10), the collapsed contour
approximately follows a figure 8 outline, consisting
of two complete lobes. Consequently, an unstiffened
cylinder may be compared with a fixed-end column
whose length eqnals one-half of its circumference.
For stiffened cylinders, the nnmber of lobes increases
as the length-between-stiffeners/diameter is decreased, with a corresponding increase in collapse
pressure. The Code design of a stiffened shell
establishes a shell thickness and combined moment
of inertia for the stiffener and shell to assure the
stability of the entire shell section. This results in
heavier stiffeners than wonld be obtained by a
design approach wherein the stiffener loading is
based on division of load between the connected
shell and stiffener under pressure, and the elastic
conditions up to the point of collapse. The collapsing pressure of heads (which in early Code
editions involved a flat reduction in allowable
external pressure to 60% of that allowed for internal
pressure) is now predicated on the collapse pressure
of a complete sphere having a radius equal to that of
the spherical part of the head.
The ASME rules attempt to maintain the same
nominal safety factor of 4 against collapse under
external pressure as is used against bursting under
internal pressure. There is some reason to question
whether this is entirely logical, since the effect of
localized stresses or stress concentrations, such as at
branch connections, may be entirely different. Also,
the degree of hazard in the event of failnre will
generally be appreciably less for external pressure,
although hazard mnst still be judged independently
for individual applications. In addition, the Code
rules maintain the same safety factor for failure by
elastic instability as for failure by plastic yielding,
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
except for small tubes where a variable lower safety
faetor is reeognized. The practice of the Structural
Steel Codes in reducing the saf"t'y factor on columns
as the length/radius-of-gyration is reduced appears
logical. For vessels or pipes a similar practice could
be followed by lowering the safety factor to 2 on the
yield point as a suitable function of diameter/thickness, but this practice is not yet recognized.
2.4<1 Expansion. The evaluation of external
reactions at terminal points and intermediate restraints of piping systems is given in detail in
Chapters 4 and 5. The expansion forces in space
systems will generally result in 3 force and 3 bcndingmoment components at each terminal point. The
number of such components is reduced with partial
end fixation.
The evaluation of the terminal reactions permits
the ealculation of the three moments (2 bending and
1 torsional) at any point in the pipe line by the
application of statics. These moments, in turn,
permit the designer to calculate the stresses by
utilizing the section moduli of the pipe. The contribution of direct forces for the expansion stresses
in piping systems is generally insignificant, unless
the piping layout is extremely stiff.
For simplicity the Piping Code provides that
expansion stresses be calculated with the cold
(ambient temperature) modulus of elasticity. The
design values of Poisson's Ratio and the torsional
modulus for expansion stresses likewise refer to this
temperature. The Code also provides thermal
expansion data for evaluating the change in length
over any temperature range. This usc of roomtemperature data avoids the necessity of using
elevated-temperature properties, which may be less
accurately determined. With the principal strain
generally present at atmospheric temperature due to
pre- or self-springing, the Code practice of using the
lI cold" values of mechanical properties is entirely
sound.
2.4e Other Loading. Other loading which may
act on piping systems includes: the weight loads of
the piping, including structural members; the weight
of the insulation, and contents; snow and ice loading;
wind loading if exposed; loading due to aeceleration
imparted by earth tremors; speeial shock loading,
sueh as gun fire or moving vehicles; and unbalaneed
statie pressure or flow effects.
It is possible to include any or all of these loads in
a complete solution, following the methods of
Chapter 5. Ordinarily, these effects are not sufficiently critical to warrant the extra engineering cost
of this more precise approach. Instead, they are
47
indirectly controlled in a standardized way (e.g.,
support standards) or individually estimated and
controlled so that the sum of all effects will approximately meet the same combined stress criterion.
For large-diameter or otherwise stiff piping systems,
particularly where expensive materials are involved
or where the inereased spaee for additional flexibility
would require enlarged buildings or other eonsiderable expense, every contribution to the overall strain
should be evaluated by a simultaneous solution.
Weight effects are eonveniently minimized by the
provision of adequate supports. Where sueh supports
just balanee the weight reaction, they ean be validly
ignored in the expansion analysis. This condition is
seldom achieved even with elaborate compensating
spring hangers. However, average piping is sufficiently stiff so that the local restriction due to some
support friction or unbalance is not a serious factor.
For separate estimation, conventional column and
beam analysis of individual critical. members, or
frame analysis of combinations of members is recom-
mended. Wind and dynamic effects can be similarly
treated. Unbalanced pressure effects are resisted
wherever possible by rigid stops or ties which are
taken into account in the flexibility analysis, unless
such provisions would adversely affect the behavior
of the line. In the latter case a careful analysis may
be made to determine whether the pipe itself ean be
designed to earry the loads. If not, the unbalanced
pressure effect must be handled by special design
arrangements.
2.5
Combination of Stress: Stress Intensifica-
tion and Flexibility Factors
The 1955 Piping Code rules for flexibility eontain
the following equation for the combination of stresses
due to thermal expansion:
SE = VSb'
+ 4St'
(2.15)
where SE = equivalent stress to be compared with
the allowable thermal expansion stress
range, psi.
Sb = resultant longitudinal bending stress
psi = {3M b/Z.
S, = resultant torsional shear stress psi =
M,j2Z.
M b = resultant bending moment, lb-in.
M, = resultant torsional moment, lb-in.
Z = section modulus of pipe, in. 3
{3 '" stress intensification faetor.
This equation is based on the Maximum Shear
Theory and for convenient comparison with Code
48
DESIGN OF PIPING SYSTEMS
allowable stress range, eq. 2.15 represents two times
the maximum shear stress due to expansion loading.
As stated in Section 2.3, the Pipillg Code establishes
a separate limit of Sh for the maximum longitudinal
stress due to pressure, weight, and other external
sustained loadings, with the provision that., if sueh
loadings do not add up to Sh, the differenee may be
used to increase the allowable stress range for expansion effeets. This approach has been adopted for
convenience in praetical design ealeulations. It is
obvious that, when using combined-stress formulas
and a specific yield criterion, stresses from all loadings should be included to determine the principal
stresses before combining them. On the other hand,
from a fatigue failure standpoint, the loadings which
cause cyclie stresses are the most significant. There
is, therefore, reasonable logie in eombining these
separately for comparison with an allowable stress
range. Actually, so long as the allowable stress range
is adjusted to suit the methods of calculation and
stress combination which will be used, designs arrived
at by various approaches ean be made substantially
the same. Simplicity of applieation has been the
objeetive of the Code.
The Code's use of the maximum shear-stress
criterion for expansion stresses represents a departure
from the evaluation of stresses elsewhere in the Code,
where only principal stresses are considered. While
a uniform eriterion would be preferable to avoid confusion and permit better assessment of safety factors,
there is greater need for closer evaluation of eyclie
strain loadings whieh may lead to a fatigue failure.
The approach laid down above is reeommended for
ordinary practice, in view of the mandatory require-
ments of the Code and the relative simplicity of
handling expansion stresses separately. For critical
applications, or where loadings are simultaneously
analyzed, it is more appropriate to evaluate all
stresses prior to eombining them and compare them
to the total allowable stress range 1.25 (S, + Sh).
The additional provision that the principal stress
due to long-time sustained loadings other than expansion should not exceed Sh, must also be observed.
For convenient reference the following formulas
are given:
Then the resultant principal stresses at the outside
fiber ean be written as
+ Sp + V4S," + (SL - Sp)2J
S2 = 0.5[SL + Sp - V4S," + (SL - Sp)21
S, = 0.5[SL
S3 = 0
and the eombined "equivalent" stress for the respeetive yield eondition beeomes
Maximum Shear Theory (Tresca)
The greater of S, as given above or
(2.17)
Distortion-Energy Theory (Mises)
V3S,"
+ Sr} + S? - SLSp
(2.18)
Use of the maximum shear theory is favored for
eonsisteney with the Piping Code.
In the sample ealeulations in Chapters 4 and 5 the
Piping Code rules are followed. The examples in
Chapter 4 involve expansion alone; in Chapter 5,
Sample Calculations 5.14, 5.15, and 5.16 include
weight or wind effects.
In the General Analytieal Method, the influence
of localized effeets on deflections and rotations is
provided for by the inclusion of flexibility factors
with t.he shape constant.s. In effect, this eompensat.es
for the additional displacements by providing an
increase of the length of the member to a so-called
virtual length, producing the desired relative deflection. The net influence of t.his increased flexibility
is to decrease reactions and nominal primary stresses.
This greater flexibility of local components, such
as bends, is the result of localized stresses whose
magnit.ude above the nominal primary stress level is
expressed by a stress-intensification factor, whose use
is mandat.ory in the new Piping Code rules. These
rules cont.ain suggest.ed flexibility and stress fact.ors
for usual piping component.s, wit.h the provision of
allowing the alternate use of experimentally determined factors.
2.6
Let S L = maximum longitudinal stress due to
pressure, weight, and other sustained
loading plus expansion stress Sh as defined above.
Sp = cireumferential pressure stress.
S, = shear stress due to torsion as previously
defined.
(2.16)
Evaluation of Deflections and Reactions
Line movements or deflections are of interest in the
design of yielding supports, sueh as spring hangers,
and in establishing clearances for the free expansion
movement of a large-diameter or eomplex line.
Sample Caleulation 5.10 in Chapter 5 illustrates that
the evaluation of deflections by the Kellogg General
Analytical Method requires little extra effort after
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
the reactions havc been determined. Line movements at any point are also readily determined hy
Model Test for any condition of lil1iding, as discussed
in Chapter 6.
It must be appreciated that calculated deflections
establish only a range of movement; the absolute
position of any point at a given time is, in addition,
dependent upon the combined effects of initial fabrication stress, relaxation and creep, changes in dead
load, adjustment of hangers, and local temperature
differences at the cross section. Except for temporary
overload of terminal equipment, etc., a line may be
adjusted to any desired initial position so that the
movement range occurs over the desired location.
Equipment may be protected against erection overload by thermal unloading (controlled local stress
relief) as discussed in Chapter 3.
Since maintained loads, such as piping weight and
insulation, are essentially constant, deflection calculations are ordinarily confined to expansion effects.
In general, the effect of maintained loads (such as
piping weight and insulation) and transient loads
(such as contents, snow, and wind loads) are effectively limited by properly placed and designed supports, guides, or tics. The significant movements
will then be associated only with thermal expansion,
and deflection calculations can be confined to this
effect.
The calculations in Chapters 4 and 5 and the model
tests in Chapter 6 give, as their first reSUlt, the
reactions of the supports on the piping system.
These forces and moments arc determined on the
basis of a strain equivalent to the total expansion
and using the modulus of elasticity and Poisson's
ratio at atmospheric temperature. They do not
include the influence of initial stresses due to fabrication. The resulting reaction range will be immediately realized in its full magnitude only for piping
systems subjected to 100% cold spring. Beyond this
consideration, it is important to know the maximum
reactions to be expected in the hot and cold conditions for the purposc of examining their effect on
terminal equipment. The Piping Codc provides the
following rules on this subject:
(2.19)
R, = CR, or
(2.20)
(2.21)
The value of R, is taken from equations 2.20 or 2.21,
·i9
whichever is greater, and with the further condition
that
Bh
E, .
h
- X - IS less t an 1
BE
Eh
where C = cold spring factor varying from 0 for no
cold spring to 1 for 100% cold spring.
BE = maximum computed cquivalent expansion stress (per eq. 2.15).
E, = modulus of elasticity in the cold condition.
E h = modulus of elasticity in the hot condition.
R, = range of reactions corresponding to the
full cxpansion range based on E,.
Rc and Rh represent the maximum reactions
estimated to occur in the cold and hot
conditions, respectively.
Obviously, the Codc formulas for reactions, based
upon a division of strains between the ambient and
service temperatures, are somewhat arbitrary.
Equation 2.19 attempts only to establish the initial
magnitude of the hot reaction for purposes of checking the capacity of equipment to take such effects.
Equations 2.20 and 2.21, in turn, are aimed at establishing the maximum value of cold reactions, either
as obtained through initial cold springing, or due to
subsequent self-springing under service conditions.
The signs (directions) of the hot and cold reactions
are always opposed to each other. For temperaturcs
in the creep range, the hot reaction will eventually
be lowered to a value roughly corresponding to the
design allowable creep stress Bh • This value approxi-
Bh
mately corresponds to Rh = S R" whereas the cold
E
reaction increases to the value given by cq. 2.2l.
Equations 2.19 and 2.20 arc applicable to a multiplane system only when the prespring is applied as a
uniform percentage in each direction. In practice
therc may be instances where prespringing in a preferred direction only may be sufficient and be utilized
because it is simpler to carry out. For such a case
the reactions for the actual prespring to be applied
should be calculated by an appropriate analytical
mcthod in place of eq. 2.20. For the most complctc
control of prespring, an analysis of the type shown
in Samplc Calculation 5.13 is recommended.
When prespring is not specified, or is not adequately
controlled, the reactions due to fabrication may in
exceptional cases correspond to yield-point stress in
the system, unless thermal unloading has been employed. Fabrication residual strains will be reduced
when the piping system is first heated if the combined
50
DESIGN OF PIPING SYSTEMS
expansIOn and residual stresses exceed the yield
strength. The fabrication strain so relieved is not
reestablished during subsequent'"$ervice, nor will it
affect the fatigue life of the piping system. Its
significance lies largely in the load which it introduces
on equipmcnt or foundations as long as it lasts.
The individual hot and cold reaction values are of
interest mainly for judging their effect on sensitive
equipment, such as pumps and turbines which involve
maintaining close clearances and alignment; they are
also of interest in connection with foundation. design.
In regard to localized stresses in the shell of terminal
equipment, however, the reaction range rather than
the magnitude of the individual hot and cold reaetions
is the signifieant faetor. This aspect is dieusssed in
more detail in Chapter 3.
2.7
Design Significance of Inspection
and
Tests
Wall thiekness caleulations, when dealing with
pipe or a eylindrical shell, have always included a
scale tests and the use of small specimens have
proven valuable in investigations of certain aspects
of this problem, partieularly for establishing general
trends or for the quality eontrol of procedures and
actual fabrieation.
The level of quality of design, materials, and fabrieation attained, as assured by adequate inspection
and tests, is at maximum economic effectiveness when
the individual factors are eontrolled to the same degree [12J. Overemphasis on any aspect does not
ordinarily lessen the hazards attendant to the
negleeted factors, so that the probability of failure
is not proportionately redueed.
There is some opinion that pipe girth joints are less
eritieal than longitudinal welds. This view stems
from the fact that longitudinal pressure stresses are
approximately only half the circumferential stress.
It ignores the fact that expansion and struetural
effects usually make longitudinal stresses the eriterion
of design, and that weakness in a longitudinal direction causes a local weakness circumferentially.
In
so-called "joint efficiency" for welded seams which
addition, initial flaws, in propagating, tend to change
has usually been applied only to eircumferential
pressurc stresses. For structural loading the joint
efficiency is sometimes neglected. It is also generally
disregarded in flexibility calculations or compressive
loading and most situations where only bending
orientation for maximum influence from the maximum stresses present.
stress is involved.
The term "joint efficiency" is a holdover from
riveted construction, where a definite breaking
strength could be associated with a speeific design.
On welded joints, where weakening effects sueh as
rivet holes are absent, it is not difficult to provide
design strength equal to the base material as
evidenced by procedure tests of sample welds, even
for lapped joints. Better criteria of the reliability
and performance of a welded joint are its eapaeity to
take deformation as a measure of its safety against
cracking, the absence of weakening defects as assured
by examination, pressure tests, and mechanical tests
of occasional complete joints or specimens; for high-
temperature service the tests should be carried out
both at room and serviee temperature, but this is not
eurrent praetice except for special applications.
Assessment of performanee tests and degree of
examination would lead to establishment of a
Hquality factor/' rather than a "joint efficiency."
Where repetitive loading is involved, the potential
influence of the design details, fabrication quality,
and basic strueture of the weld- and heat-affected
zone of the parent metal ean apparently be aecurately
evaluated only by full-scale fatigue tests under combined loadings and temperature eycling. However,
Adequate pressure testing, as praetieed on pressure
vessels, often presents economic problems in piping.
Shop tests of irregular or large-diameter runs require
special fittihg~ or else extra welds for closures.
Adequate field. pressure tests require the installation
of blinds and often extra flanged joints, in order to
protect lower pressure vessels and terminal equipment; they sometimes require special rigging for
inspection access, and temporary supports. Such
eases require individual treatment. When the field
test is adequate the shop test ean be waived by
mutual agreement. In judging the adequaey of the
field test, the degree of inspection and level of test
stress should be jointly considered.
A water- or liquid-pressure test fulfills dual funetions.
The design, materials, and fabrication are
checked to a reasonable minimum extent by a pressure test based on 1.5 times the design. pressure in-
creased by the ratio of cold to hot allowable stresses
(S,/S,,). During t.esting, there is an opportunity t.o
detect leaks due to cracks, poro;;ty, or other flaws
whieh extend through the wall. These objeetives
are accomplished at minimum hazard when the
testing fluid is essentially incompressible, thereby
limiting the stored energy. During such a test,
should a break initiate, there is immediate loss
of pressure, usually before extensive damage is
done or fragments detaehed and propelled through
space.
DESIGN ASSUMPTIONS, STRESS EVALUATION, AND DESIGN LIMITS
The detection of leaks can be accomplished with
equal or greater effectiveness at lower pressures by
using liquids of lower surface tension properties, or
by reducing the surface tension by additives, or by
the use of air or other gas. Air pressures of 5 to 10
psi usually suffice for the detection of leaks with
equal or better effectiveness than water at full test
pressure. The Vessel Codes permit air tcsts as a
snbstitute for water tests at a rednced stress level
(83.3% of hydro test for ASME Code and 73% for
.\.PI-ASME Code), while Section 3 of the Piping
Code limits air tests to 50 psi. The Vessel Codcs
require that pressure be applied in successive stages
to minimize high-energy rnpture hazards. The precautions exercised should be in step with the size,
volume, stored energy, test stress level, and quality
of inspection.
The effectiveness of a test in proving the sonndness
of a strncture decreases rapidly as the pressure recedes
from I! timp," the equivalent cold working pressnre.
It is donbtful that air tests, at the level prescribed
in the Codes, accomplish much in this direction other
than detection of gross omissions or deficiencies.
These, for the most part, shonld have been revealed
by carefnl visual inspection. In addition, one application of pressure at or near normal design stress will
often not reveal poor welds or even lengthy cracks,
unless already extending through the wall. Higher
pressure tests are more effective as a result of greater
overstress in weak areas, and the initiation of plastic
flow in distorted or poorly fit-up areas. However,
there is certainly no complete assurance that a strncture is safe as a result of successfully passing a single
pressure test.
Equal or greater assurance of soundness can be
obtained by radiographic examination of all welds
coupled with a pressure test at the design pressure.
For magnetic materials where thicknesses do not
exceed! in., a magnetic powder examination inside
and outside in lieu of radiographic examination can
also be considered acceptable. This is not intended
to imply that weld inspection and tests are interchangeable. Instead, they mnst be considered
51
simultaneously in evalnating safety. For heavywalled or critical-service piping, all practicable
inspection procedures and tests are desirable and
necessary for adequate safety.
References
1. D. B. Rossheim and A. R. C. 1Iarkl, liThe Significance of,
and Suggested Limits for, the Stress in Pipe Lines due
to the Combined Effects of Pressure and Expansion,"
Trans. ASME, Vol. 62, No 5 (1940) .
2. E. L. Robinson, IlSteum Piping Design to :Minimize Creep
Concentrations," presented at Annual Mtg. of A81-IE,
New York, 1954.
3. L. F, Coffin, Jr., u1\ Study of the Effects of Cyclic Thermal
Stresses on a Ductile Metal," ASME Paper No. 53-A-76,
presented in December, 1953.
4. L. F. Coffin, Jr., "The Problem of Thermal Stress Fatigue
in Austenitic Steels at High Temperature," presented at
AST!vf meeting, Chicago, June, 1954.
5. A. R. C. Markl, "Fatigue Tests of Piping Components,"
Trans. ASME, Vol. 74, No.3, pp. 287-303 (1951).
6. A. Nadai, Plasticity, McGraw-Hill Book Co., New York,
1931.
7. W. J. Buxton and \V. P. Burrows, flFormula for Pipe
Thickness," Trans. ASME, Vol. 73, pp. 575-587 (July,
1951).
8. \V. R. D. Manning, "The Overstrain of Tubes by Internal
Pressure/' Engineering, Vol. 159, pp. 101-102, 183-181
(1945).
9. C. W. MacGregor, L. F. Coffin, Jr., and J. C. Fisher,
"Partially Plastic Thick-Walled Tubes," J. Franklin Insl.,
Vol. 245, pp. 135-158 (1948).
10. C. W. MacGregor, L. F. Coffin, Jr., and J. C. Fisher,
"The Plastic Flow of Thick-Walled Tubes with Large
Strains," J. Appl. Phys., Vol. 19, pp. 291-297 (19·18).
11. R. W. Bailey, "Creep Relationships and Their Application to Pipes, Tubes, and Cylindrical Parts Under Internal
Pressurc," Proc. Inst. Mech. Engrs. (London), Vol. 164,
pp. 425-431 (1951).
12. J. J. Murphy, C. R. Soderberg, Jr., and D. B. Rossheim,
IIConsidcmtions Affecting More Economic but Equally
Safe Pressure Vessel Construction Utilizing; Either Present-Day Ductile or New High-Strength Less-Ductile
Materials," API Paper presented at St. Louis, l\by 10,
1955.
13. E. O. Bergman, liThe New-Type Code Chart fOf the
Design of Vessels Undcr External Pressure," ASME Paper
No. 51-A-137, presented at Atlantic City, Novembef,
1951.
CHAPTER
3
Local Components
T
leads to greater flexibility than could be accounted
for by bar theorics. A year later, the first theoretical
treatment of the subject was published by von
Karman [2], who investigated the stress distribution
in curved tubes subjected to in-plane bending. l
At about the samc time Lorenz [3] and Marbec [41
independently furnished a solution of this problem,
using Castigliano's t[worem in their work instead of
the principle of minimum potential energy as used by
Karman. Hovgaard continued Karman's work and
arrived at an identical solution through a different
approach [5] while Karl [6] refined the solution by
HIS chapter will consider important component~ of a piping system other than straight
pipe, including flanges, bends, miters, corrugated pipe, branch connections, and terminal connections, all of which are designated herein as "local
components" since individually they usually occupy
a limited length of the total pipe run. The localized
stress pattern which they introduce often significantly increases the flexibility of the entire piping
system at the expense of stress intensification or
strain concentration at their location. It is the
intent of this chapter to offer a digest of current
knowledge about each local component, and discuss
practical application to the design of piping. Aceurate evaluation of stress and deflection for localized
effects is often complex, or even impossible with
present knowledge; as a result simplifying assumptions and shortcut solutions are resorted to, some
of which will be discussed herein.
considering morc terms in the series expansion for the
basic variables. In 1943 Vigness [7] extended the
theory to include the case of out-of-plane bending of
curved pipes. These theoretical investigations readily establish the following points:
1. The elementary bending theory for bars, which
assumes a linear variation of longitudinal stresses~
cannot account for the actual stress distribution in
curved tubes under external bending loads. In
reality, the longitudinal bending stresses in the
extreme fibers are greatly relieved by the ovalization
(flattening) of the cross section, which, undcr different loading eonditions, takes the forms shown in
Fig. 3.1. At the same time the maximum stresses
are shifted nearer the neutral axis, as shown in Fig.
3.1 Pipe Bends: Structural Loading (Static
and Cyclic)
Pipe bends are curved bars with an annular cross
section, whose reaction to external loading is complex. Visual observation, as well as scattered tests,
established quite early that the elementary theory of
elasticity is inadequate to account for the peculiar
properties of tubular bends. Despite this fact, considerable time passed before a satisfactory analysis
was undertaken. While theories are sufficiently
advanced today to account for the major aspects of
the behavior of pipe bends, many refinements of this
problem still demand clarification and a further extension of theoretical inquiry.
Systematic investigation of pipe bends began in
1910, when Bantlin [1] observed and reported on the
phenomenon of ovalization, and on the fact that it
3.2.
2. This altered bending-stress distribution, in
turn, decreases the bending-moment resistance of
the section.
The ratio of the resulting increased
lIn-plane bending refers to the case in which the pipe is
subject to bending by forces or moments applied in the plane
of the bend. Out..of-plane bending designates the case in
which the forces or moments act perpendicularly to the plane
of the bend. Obviously, these two cases enn be combined to
give a solution for forces at moments acting in any arbitrary
plane.
52
53
LOCAL COMPONENTS
deflection to that predicted by conventional beam
theory is termed the "flexibility factor" for that
member.
.~
3. The maximum longitudinal stresses in pipe
bends will differ from those generated in straight
tubing of equal dimensions. High circumferential
bending stresses are set up as well. For pure (inplane) bending, theory indicates that the peak
stresses will be the circumferential stresses near the
neutral axis (a = 0) of the pipe. The ratio of the
maximum stress in the curved pipe bend to that
which would exist in straight pipe subjected to the
same moment is termed Hstress intensification fac-
Bending
(b) Theory of
Curved
Th~'Y
Pi~
(0) Elemenlory
MOl(. longitudinal "reu OCCUI1
at angle at (leCl Fig_ 3.8)
FIG. 3.2
Distribution of longitudinal stresses in curved pipes.
tor. JJ
These findings were subsequently reexamined by
Beskin [8], who found that the previously established
results were applicable only when the bend characteristic2 was comparatively large; as the characteristic diminished, the results became increasingly diyergent. Instead of a maximum flexibility factor
for in-plane bending of 10, and a maximum stress
intensification of about 3.5, as implied by earlier
analyses for the mathematical limit of h = 0, Beskin
found that both flexibility and stress intensification
factors become infinite at this extreme value.
Further investigation showed that Karman's solution would have yielded rcsults identical with those
of Beskin, had the Fourier expansion been carried
to more terms than one.
Treating the problem of in-plane bending of curved
lubes by means of the theory of thin shells, Clark
and Reissner [9, \OJ found that the Lorenz, Karman,
Karl, and Beskin solutions merely represented
higher order approximations (in the order mentioned), and confirmed Karl's findings [6J that alter-
nate solutions by the principle of minimum potential
energy (used by Karmim) and the principle of least
work (adopted by Lorenz, Karl, and Beskin), establish upper and lower limits for the true rigidity of
the tube. The Clark-Reissner solution is obtained in
terms of a trigonometric series expansion for the
stress function and meridional angle change. By
retaining only two terms of each series expansion,
and limiting the range to h > 0.5, the Clark-Reissner
approach becomes equivalent to Karl's solution.
For h < 0.5, the number of terms needed for satisfactory accuracy increases rapidly; therefore, an
asymptotic solution was investigated. Making
assumptions which hold true when h is much smaller
than I, closed-form solutions were obtained which
are startlingly simple. All analyses dealing with the
problem of bending of curved tubes predict equal
flexibility factors for in-plane or out-of-plane bendmg.
!{arman's original solution (first approximation)
for the flexibility factor, k, is
2The bend characteristic is h = tR/rm2 , where t = wall
thickness of pipe, R = radius of bend, and rm = mean radius
of pipe.
9
k= 1+ 12h2+1
Second, third, and nth approximations [11] have the
form
SECTION A-A
(a) In-plane
Bending
Al,
'longcmh
forced
together}
(b) In-plane
Bending
(3.1 )3
(el Ov'-of-ploM
Bending
(I(lngenh
forced
2
k
9+0.255/h
= 1+ 12h2+ 1.3400+0.00750/h2
k
9+0.3003/h2 +O.OO\o587/h·'
= 1+ 12h2+ 1.4004+0.0l3946/h2+O.00001276/h 4
(3.3)3
(3.2)3
opClf1)
I
I
J
A
k=l+
9
12h 2 +I-j
(3.4)3
In eq. 3.4, j is a function of h; for known values of h
FIG. 3.1
Ovalization (flattening) of pipe bends under external
bending moments:. Exaggerated.
3In eqs. 3.1 to 3.4 the rigorous mathematical analysis would
demand that h(l - v2 )-Ji be used instead of h.
DESIGN OF PIPING SYSTEMS
the magnitude of j can be obtained by interpolation
from the following table:
Out-?f-plane {LOngitudinal
bendmg
~.
h
j
0 0.05
1 0.7625
0.1
0.5684
0.2
0~074
0.3
0.1764
0.5
0.07488
0.75
0.03526
1.0
0.02026
(3.6)
bending Circumferential 'Y' = L80/h%
(3.7)5
a, = 0.82h li
a,
5Rigorously, the correct value in eq. 3.7 should be:
L80 (1 - v2 )-n
"
h"
60
(1) -
-Asymplolic Solulion 1.:5 (Clark _ Reislnor)
{2)----BelOkin's (lorge.radius bends)
(3)---Symoflds and Parduo's (Small-rodius bends)
(') --Von Karman's nih Approximation
20
{S)----Approximolo, l.~O (Vinat _ DelBuono)
(For small·radius thkk-woUed bends)
2
FIG. 3.3
(3.10)
Equations 3.6, 3.7, and 3.10 are obtained from the
asymptotic analysis of Clark and Rcissner. Equations 3.8 and 3.9 represent empirical proposals made
by The M. W. ICellogg Company, and Markl [12J.
respectively. The various stress intensification faetors are charted in Figs. 3.4 to 3.7, whereas
io
plotted in Fig. 3.8.
These results convey that for either in-plane or
out-of-plane loading, the circumferentialstrcss in the
neighborhood of IX = 0' will first exceed the yield
point. Tllis stress is a pure bending strcss (excluding
internal pressure effects), varying from a positive
maximum at the outer surface to a negative maximum
at the inner walL A slight amount of yielding, leaying the elasticity of the pipe wholly unimpaircd, will
materially relieve this-stress, as has been observed in
experiments [13J. Similar deductions can be made
concerning the maximum longitudinal stress at the
outer surface. Pronounced yielding will ensue only
when the maximum longitudinal stress at the middle
surface also exceeds the yield point. Therefore, in
.tv = 0.3 is assumed in eqs. 3.5, 3.6, 3.7, and 3.10.
'Yi = -
(3.91
a,
Stress intcnsification factors, unfortunately, differ
for in-plane and out-of-plane bending. In general,
in-plane bending leads to higher circumferential
stress maxima than out-of-plane bending for identical
pipe bends subjected to equal bending moments.
For longitudinal stresses exactly the opposite of this
statement holds true, as witnessed by eqs. 3.6 to 3.9.
The stress intensification factors at the outer surfaces,
valid only for small values of the bend characteristic
(h < 0.5), have the following expressions:
13, = 0.84/h%
Circumferential 'Y, = L50/h%
In this same range the variation of angle
(pertaining to the largest longitudinal stress, as shown in
Fig. 3.2) with the characteristic h can be given !to
Beskin's solution for the flexibility factor cannot
be expressed in closed form; his numerical results,
which merge with IG\.rmfm's nth approximation, are
plotted in Fig. 3.3. Clark and Reissner's asymptotic
solution yields the following expression, valid for
small values of h:
(3.5)'
k = L65/h
In-plane {LOngitudinal
13, = L08/h % (3.8)
Flexibility factorB for in~plane or out--of-plane bending.
55
LOCAL COMPONENTS
described above reveals that, in addition to dealing
only with pipes having a eonstant eurvature of the
center line, constant cross-sectional properties, and
being made of an isotropie and homogeneous material obeying Hooke's law, the analyses are based on
the following assumptions:
1. Plane seetions remain plane and the neutral
axis retains its original length after loading.
2. Longitudinal and eireumferential stresses are
the opinion of many investigators the stress intensification factor of greatest practical significance is the
one pertaining to maximum longiwdinal stress existing at the middle surface of the pipe wall thickness."
Closer scrutiny of the theoretical developments
6Fatiguc tests do not support this. In fatigue, cracks opcn
up perpendicular to the actual maximum stress which is the
circwnferential stress at the inner pipe wall. This is to be
expected, since under reversed strain loading beyond the
elastic fange, as applied in a fatigue test, initial plastic flow is
of little help in alleviating the range of strain at each point.
principal stresses.
0.8.
(l) _ _ koymplotic Solulion h2/J (Clark - Roinner)
10
B
6
(3)_ •. _Symond$ and Porduo', (Smoll-rodiu, ~nch)
1.2
Approllimalo. h27i (Viuol- 001 Buono)
(4)
(For unoll-rodiu$ lhick-walled bend,)
--•06
.02
••
.2
.08 .1
-,
h -~
.6
.8
•
2
1.0
'.
FIG.3.4
In-plane bending: outer surface longitudinal stress intensificg,tion factor.
1.80
(l)-A5ymplotk Solution h2/J (Clork - Rlliunllr)
3)
(2) -
-
Be$kin', (large.radiul bllndd
10
B
{:n----Symond5 and Pardue', (Short-radiul bllnd$)
6
(4)
ApproximoI1l, ~~B (Vinal - Del Buono)
(for smoll·radius thick-walled bend$)
~
't1
.E
•
•
0
~
~
.~
1)
2
:;;
E
;;;
.6
.6
.4
.3
':-J,-L.-:':-.LJ,.u.__-!---IL-LJi.....L...L.!--.LL-_-..L--l--l
.03
.04
.06
.08 .1
.2
.4.6.6
1.0
2
4
h=~
,~
FIo. 3.5
In~plane bending: outer surface circumferential stress intensification factor.
56
DESIGN OF PIPING SYSTEMS
the curved pipe is acted upon by pure bending
moments. According to St. Venant's prineiple, local
disturbances imposed at the boundaries will cancel
a short distance therefrom. In this light, assumptions
2 and 3 can also be adopted as having reasonable
validity.
3. The bending moment has a constant value for
the entire length of the bend.
4. Radial and longitudinal strains are uniform
through the wall thickness.
5. Circumferential strains produce pure bending,
and thus vanish at the middle surface of the pipe
wall.
6. The radius of the bend is much greater and the
wall thickness is small compared with the diameter
of the pipe.
Assumption 1 is fundamental to the theory of
elastieity and can be accepted as being true. The
second and third conditions will be satisfied only if
10
•
....
(1)- -
-"""-"",/-'1.)
.... ....
The remaining assumptions deserve closer scrutiny.
Assumptions 4 and 5, dealing with the strains developed under loading, are idealized simplifications of
the actual strain distribution, and will be in accord
with the actual strains only when R/r.. > 10 (i.e.,
larger than a "five diameter" bend). For shortradius bends, characterized by 1 < R/r.. < 10, it
8eskin', {large.radilli bondd
(2)----Syrnonds and Parduo', (Small-radius ben<h)
""
(3)--Appra:dmafo.
~.~~ (Weil)
lL-.L.l....JL.L.LL.L-_ _L--l.----'-----'---'-...L.I--'-:"~__===.t;=_"'=='
.03
FIG.3.6
Out-of-plnnc bending: outer surface longitudinal stress intensification factor.
.
1..50
(l)--ApproximQlo, h'2/3 (Mark!)
10
(2)- -Beskin', (large rgdiV$ bends)
•
(3)----Symonds and Parduo's \,)mall radius bends)
6
!
1.0
••
•6
••
•3 '----.J~_'_-'-.L.JLL.LL_ _---.J_
.02
.0-4
.06 .08 .1
:2'
____'__-'--'--l.-1__LLL_ _____'__
. . . . 6 . 8 1.0
2
____'____l
h =.!!.
.'•
FICi.3.7
Out-of-plane bending: outer surface circumferential stress intensification factor.
LOCAL COMPONENTS
has been shown [7, 13] that under in-plane bending
(reducing the radius of curvature) the circumferential strcsses do not vanish at the middle layer (see
assumption 5).
The last assumption plainly limits the accuracy of
the foregoing theories to thin-walled, large-radius
tube bends; the generally accepted view is that these
analyses are proper only if both conditions, namely
that Rlrm and rmlt be greater than 10, are simultaneously satisfied. Sinee Beskin's derivation indicates
that at h > 1.0 the flexibility and stress intensification factors become generally negligible, it is of
interest to note that this development is not conditioned upon the above-stated limitation on wall
thickness.
To extend the validity of previous analyses,
Symonds and Pardue [14] undcrtook to investigate
the effect of Rlrm ratios considerably less than 10,
(2'::; Rlrm ::; 3). It may be pointed out that under
these conditions the wall thickness ratio assumes a
much greater importance; the fact that the "shortradius" development is based on thin-shell theory
plainly limits the range of accuracy to about
h = 0.2 for Rlrm = 2, or h = 0.3 for Rlrm = 3.
The Symonds-Pardue theory represents a first-order
approximation to the influence of Rlrm, and shows
that for short-radius bends (long- and short-radius
welding elbows), the flexibility factor suffers little
change, but stress intensification factors are generally
higher than for large-radius pipe bends, as seen in
Figs. 3.3 to 3.7. As might be expected, the results
of this work merge with Beskin's solution, as Rlrm
and displacements (flattening or ovalization). Jo.:nd
restraints tending to oppose ovalization, (straight
pipe tangents to a minor degree, flanges or terminal
connections to a severe dcgrcc) will lower flexibility
and stress intensification factors; in these cases the
theory will give higher values than those actually
operative. Thus deviations between theory and
actual behavior will be greater the more severe the
end restraint, or for a given end restraint, the lesser
the subtended arc of the pipe bend.
Having elaborated on the underlying assumptions
and results obtained from an analytical approach,
it is enlightening to examine how the theories compare with results obtained from experimental work.
Investigations must be separated into tests performed under static r.onditions and those relating to
fatigue conditions, since they represent fundamentally different types of loading.
The first significant static tests were made by
Hovgaard [5, 13, 15, 161. who proved that experiments were in close agreement with theoretical pre-
dictions for the flexibility, distortion and stresses of a
given system, although calculated stresses showed
smaller extremes than those actually observed. It
must be added that Hovgaard's tests were performed
mostly within the limitations of his theory: stro.ss
distribution measurements were confined to sections
remote from the disturbing effects of type-of-loading
or end-fixity conditions, and most of the experiments
were limited to large-radius bends (Rlrm > 10).
Similar observations were made by other investi-
gators [17, 18, 19,20,21,22], who again found the
longitudinal stresses to be slightly in excess of theoretical values. It was also observed [21] that the
flexibility of pipe bends for in-plane bending was
greater than that predicted by theory. This deviation was small, but consistent, and was ascribed to
increases to 10.
Lastly, all of the theories described apply rigorously
only to endless toroidal sections. If the curved tube
is not endless, the theory is accurate only if the end
conditions allow the development of idealized strains
90·
80·
70·
60·
so·
li0,,(0
57
(I) --Clark - Reinner
(I)
(2)----Asymplotic Solution O.82h l/1
(Clark and Relnnerl
0
~
.5:.30°
<i
20·
10·.';-_ _-=:_-'--!:-l-!:,.-'-:':-'--:------:--..L-J'-.LLJ~.Ll:_--_!:_--'--,J
.
.01
.02
.04
.06
.08
.1
.2
....6.8
1.0
2.0
".0
h= !!
,~
FJO. 3.8 In-plane bending: angle 'l:l corresponding to location of m!\ximum longitudinal fiher stress.
58
DESIGN OF PIPING SYSTEMS
the fact that the theory of curved tubes did not take
into account secondary influences predicted by the
theory of curved bars. Tests condticted under outof-plane bending [7J, in turn, showed that the rigidity
of pipes was greater (Le., the flexibility less) than
indicated by analysis. This was attributed to the
restraining effect of straight tangents applied to the
ends of the quarter pipe bend. Stresses meanwhile
were smaller than was anticipated from theoretieal
research.
A thorough investigation on the effeet of end conditions was carricd out by Pardue and Vigncss [23J.
Dealing first with flexibility factors, they found that
even the most detailed theory [14J was capable of
predicting flexibility factors for out-of.-plane bending
only if the pipe bcnd merged with a straight tangent
of sufficient length. Substituting a flange for the
tangent at either end resulted in a drastic drop of
flexibility; when both ends were flanged, flexibility
dropped even further. Right-angle bends were
subject to these reductions in a greater degree than
U-bends, confirming the logical expectation that the
smaller the subtended angle of a pipe bend the
greater will be its sensitivity to disturbanees caused
by end restraints.
Almost identieal statements apply to the stress
intensification factors. The theory is in agreement
with actual behavior only insofar as the bend is
furnished with sufficiently long straight tangents,
the experimental values being generally a shade on
the high side. With an increased degree of end
fixity, this correlation breaks down. Applying
flanges to both ends of the bend initiates a much
greater reduction of the stress intensification factor
than using one flange and one straight tangent; and
again, right-angle bends were subject to these modifying effects to a greater degree than U-bends.
Additional confirmation of theoretical results
was provided by a series of tests carried out by
Gross and Ford [24, 25, 26J. These tests proved
that, in line with theoretical predictions, the cir-
cumferential stress in the vicinity of " = 0 was the
largest absolute stress; in the tests carried to failure,
the cracks always ran along the side of the bend at
about the location of the neutral axis. Contrary to
assumption 5 of the theory, however, the circum-
ferential stress at the middle layer did not vanish.
Application of strain gages to the external and internal faces proved that the maximum stresses were
always situated on the inner surface of the bend,
which may explain the observation [25, 27J that
ciency in the theory, Gross [25J suggested that the
transverse compression, ignored in the analytical
work, be taken into account. He also presented an
approximate derivation for this quantity, and proved
that adding this stress component to the others
included in the theory actually brings experiments
and analysis into good accord.
The examination of test results also eonfirmed
that the theory of maximum distortion energy predicted quite accurately the load and location at
which incipient yielding occurs.
No such criterion
could be advanced for the ultimate load-carrying
eapacity of the bends, except for noting that failure
(which took place by collapse under a moment shortening the chord of the bend) occurred at a load whieh
was generally twice as large as that required to
initiate yielding.
Vissat and DelBuono [28J describe the results of
tests on welding elbows with a ratio of R/rm equal
to 2 or 3. Restricting the tests to in-plane bending
and fitting experimental points by analytical expressions, the following relations were proposed for
flexibility and stress intensification factors:
Flexibility factor
k = 1.40/h
(3.11)
Stress intensification factors (in-plane bending)
f3, = 1.2/h"
(3.12)
'Yi = 1.07/ho. 78
(3.13)
As seen in Figs. 3.3 to 3.5, these proposals result in
lower flexibility and circumferential stress intensification factors, but higher longitudinal stress intensification factors as compared with the data of all
other theories. While these results can be uscd on
welding elbows having the characteristics investigated, some reservation should be exercised, since
on the thick-walled short-radius bends used in this
work the restraining influence of straight tangents
or flanges has not received sufficient evaluation.
The next point of interest is to examine whether
the conclusions drawn above remain valid if the
bend is acted upon by repeated cyclic loading rather
than a single static load. Obviously, fatigue conditions will hardly modify the flexibility of a sound
local component. What is sought through a fatigue
test, therefore, is the practical effect of stress intensification on the number of cycles to failure.
In addition to manner of loading, fatigue tests
cracks in pipe bends are initiated on the inner face
differ in two aspects from static tests: in the manner
of measuring stress intensification, and in the weight
and penetrate outwards. To account for this defi-
given to plastic flow.
In static tests, the stress
59
LOCAL COMPONENTS
intensification factor denotes the ratio of actual peak
stresses to those developed in a straight member of
identical dimensions (for pure beftding, the reference
stress is MIZ).' In fatigue, the effective stress
intensification factor relates the stresses causing
failure over a given number of cycles in a straight
pipe tangent (or polished bar) to those initiating
fracture in the test piece subjected to an equal
amount of stress cycles.
As regards the significance of plastic strains,
static~stress measurements
are strongly dependent
upon the presence of plastic flow with its attendant
,·edistribution of loading and stress-mitigating effect.
In the fatigue test the local strain range per cycle is
the
significant value determining performance;
therefore, a redistribution of stresses due to plastic
strains has only a minor significance. While it is
common practice to use and establish design practices in terms of stresses based on elastic theory,
it should be appreciated when dealing with fatigue
that in reality these stresses are being used as a
suitable index of the strains involved.
Fatigue tests on piping components were initiated
by Rossheim and Markl [29], followed by a detailed
research program carried out by Markl [12, 30].
Since it was felt that the stress peaks developed in
local components as compared with straight runs of
pipe constituted the desired fundamental information, stress intensification factors were based on a
comparison with S-N diagrams obtained for straight
commercial finish pipe, containing butt welds, a
clamped edge or similar stress raisers. A stress
intensification factor of unity was assigned to the
latter for practical reasons.
The first finding of interest was that, while the
S-N curves for both straight pipes and welding
elbows of carbon steel seemed to reach no endurance
limit within the number of cyclcs cmploycd (2 X 10"
cycles max), both curvcs wcrc approximately straight
and parallcl to cach other on a log-log plot. This
indicated that the stress intensification factor could
be given as a constant, rcgardlcss of the number of
stress cycles involved.
In comparing test results with theory, it was
found that Beskin's or the Symonds-Pardue theory
predicted quite accurately the flexibility of the bend
or elbow, as well as the type and location of failure.
The agreement between tests and theory for stress
intensification factors was less satisfactory; however,
a reasonably good correlation was obtained if the
test results were drawn into comparison with only
one-half of the maximum theoretical stress intensi7Where Z = I ITo is the section modulus of the cross section.
fication factor (referring to circumferential stresses
for both in-plane and out-of-plane bending).
When considering the signifieanee of this finding,
it is important to note that Markl's reference point
of unity for the test results is not a theoretical but
a practical, one. In the first place, Markl [30] found
that the clamped edge used in the earlier tests involved a stress intensification factor of about 1.5, as
compared to pipes with a tapered end. The remaining factor of about !.4 needed to bring experiments
and theory into agreement may perhaps be attributed to the stress raisers inherent in commercial
finish pipe as compared to the theoretically considered smooth homogeneous tube. Markl could
have changed his reference point of unity and assigned different test factors; however, he found that
butt welds involved the same stress intensification
as the clamped edge. Therefore, he reasoned that a
base line which would include the effect of such
normally encountered stress raisers would be much
more satisfactory for practical design. This reasoning has been supported by practical designers and
by the ABA Code for Pressure Piping Committee.
It is, nonetheless, an important point which should be
kept in mind, particularly in connection with the
practical application of any theoretically derived
factors for other piping components.
Recourse to eqs. 3.7 and 3.9 indicates that the
experimentally found reduction factor of 2.0 leads
to design stress intensification factors (when referred
to the aforementioned base line) of the following
magnitude:
'Ii =
0.90/"% for in-plane bending,
'10 =
0.75/"% for out-of-plane bending
It happens that these stress intensification factors
are very close to the theoretical value of the longitudinal factor for in-plane bending, which has long
been in customary use instead of the more proper
circumferential factor. The seeming justification
of this latter practice stemmed from Hovgaard's
findings that permanent overall deformation of the
pipe bend occurred only when the longitudinal stress
at the middle surface exceeded the yield stress.
The recommendations of the revised ASA B3l.!
Code for Pressure Piping are derived principally
from these experimental observations. For both
in-plane and out-of-plane bending, the Code recommends that the stress intensification factor
fJ = 'I = 0.90/"% :::: 1.0
(3.!4)
he used, the choice of a single factor having been
60
DESIGN OF PIPING SYSTEMS
2R+t..
__,Ir'-"_+_'"-'.II:
:~O
_--t--'~"-'-\- a = 0
u~~o
•
FIG. 3.9
2R-r.. u:
2(R-r",1 0
Distribution of circumferential stresses in pipe bend
subjected to internal prCS811.re.
accepted for practical rcasons only. The flexibility
factor, as proven by theory and experiments, is
given as
k = 1.65/h
The resulting stress dbtribution is shown in Fig. 3.9.
A more elaborate investigation [32J and tests carried
out on curved pipes under internal pressure [251
confirmed the general validity of eqs. 3.15 and 3.16,
and showed that maximum stresses will be reached
at the line of the bend having the least radius of
curvature (crotch), as predicted by eq. 3.16. Yielding will first occur at this point. Despite this, it is
not normal practice to apply these formulas to thc
design of pipe bends.
When external loading and intcrnal pressure arc
imposed simultaneously on a pipe bend, experimental
results [26J show (as should be expected) that maximum circumferential stresses occurring for external
moment loading alone will be reduced by the prescnce of intcrnal pressure. While the prcsence of
internal pressure will slightly reduce tbe flexibility
of the bend [13, 24J, the stress, whether referring to
principal stresses or combined stress, will also be
mitigated [26].
(3.5)
3.3 Miter Bcnds
Additional fatigue tests again proved the restraint
of straight tangents upon the full developmcnt of
the flexibility and stress magnification factors; as in
static tests, this influence was increasingly accentuated with reduction of the subtended angle of the
pipe bend. As a rough measure, it could be stated
that both flexibilty and stress intensification factors
were reduced from their full value for a quarter-bend
elbow to unity, as the subtended angle of the bend
approached zero. This rule was upset only at thc
very small arc bends, where the disturbing effect of
closcly spaccd wclds obliterated the restraining
influence of the straight tangents, causing a concomitant risc in the stress intensification factor.
Particularly for the less severe services, changes in
direction are not infrequently made by mitering
straight pipe (Fig. 3.10). Yet miter "bends" have
received much less attention in the literature than
curved pipes. Nevertheless, it was shown by Zeno
[33J, who investigated the flexibility of a five-section
right-angle miter bend (h = 0.0158), that the theoretical flexibility values of curved pipes were ap-
3.2 Pipe Bends: Internal Pressure
The foregoing theories and experiments dealt
solely with pipe bends subj ected to external loadings.
In addition to this effect the pipe wall will be stressed
by the pressure of the fluid in the system. For a
curved pipe subjected to a pressure p, the longitudinal and circumferential membrane stresses are
givcn approximatcly by [31J
,
-~2""
...
R __
-
(3.15)s
2R+rm siua TmP
qc=
2(R + rmsina) t
'.
(3.16)s
8Notice that these formulas are identical with the equa~
tiona for straight pipes, except for the first fraction ap!>caring
in eq. 3.16.
5
2r,p,
2«J
...
-
= miter lp<long
cp=mitor anglo
R=equivalont ~nd rodilJ5 =
t cot r,p
'm=moon radiln of pipe
FIG.3.10 Geometry of miter bends.
I
~
LOCAL COMPONENTS
proached as the tangents were made sufficiently
long" Similar indications were obtained by Gross
and Ford [26], who measured str&ses and flexibility
on a miter bend of h = 0.0483, and found them
reasonably well predicted by the theory of curved
tubes.
Little additional information is available concerning the properties of welded miter joints under static
loads, since in addition to difficulties encountered
with plain pipe bends, miter joints arc subject to
yariations introduced by differences in fit-up and
welding as well as in the arrangement of segments.
..\.vailable evidence, however, seems to indicate that
the flexibility is less and stress intensification is
greater for miters than for plain bends of the same
major dimensions.
Miter bends have also been subjected to intensive
cyclic testing [12J with the finding that their behavior could be predicted with reasonable accuracy
through analogy with curved pipes when the propel'
characteristic variables were included. From geometry (see Fig. 3.10) the radius of the tangent arc of
a bend can be expressed as R = ts cot </>, where
8 = miter spacing at center line, and ¢ = miter
angle. If there is but a single miter 01' if the miter
spacing becomes large, however, this radius loses its
significance and an effective radius was suggested,
empirically expressed as R = rm (1 + cot </»/2.
Thus the bend characteristic assumes the following
form:
tR
cot </> Is
..
Ii = - 2 = - - - 2 for small miter spacmg,
Tm
2 Tm
s
- - tan </> < 1
rm
(3.17)
l+cot</>t
..
h = -'-'----=- - for large miter spacmg,
2
rm
s
-tan</»1
(3.18)
By using this bend characteristic with the expression
derived for curved pipe, eq. 3.14, values of stress
intensification arc obtained which show a good correlation with the tests. The flexibility factor was
somewhat smaller than for plain curved pipes, and
resembled that for welding elbows with one flange
and one plane tangent. The flexibility factor for
9Without tanp;ents, i.e., with flat plates welded directly to
the end of the Jast miter segments, the flexibility for in-plane
bending was {oJ..lnd to be reduced to only 3% of the thcorcti('nlly predicted value for a bend.
61
miter bends can henee be given as
k = 1.52/h" :::: 1.0
(3.19)
with h taken as the lesser of the values obtained from
eqs. 3.17 or 3.18. These results are incorporated
in the recommendations of the ASA 1331.1 Code
for Pressure Piping.
3.4 Bends and Miters: Summary
Pipe bends depart from conventional beam theory
chiefly as a result of distortion (ovalization) of the
cross section under bending. Under static loading,
theories predict the flexibility, maximum stresses,
and occurrence of incipient yielding with good
accuracy for bends with plain tangents whose subtended angle is larger than 90'. At present theories do
not consider the restraining influence of straight
tangents (particularly significant for curved pipes
whose bend angle is less than 90'), nor can they
effectively deal with the inhibiting tendency of severe
end restraints, such as flanges or terminal connections. To evaluate the characteristics of components falling into this category, reference must be
made to such test data as are available.
In actual service, idealized static loading conditions are seldom encountered. A certain amount of
plastic flow will always take place, enabling the bend
to carry loads in excess of those predicted by the
classical elasticity theory. The significance of theoretically calculated stress values is further reduced
by the fact that evell straight piping with a commercial finish carries an inherent stress-raising factor,
and that the performance of bends is, for practical
reasons, referred to teat of butt-welded or clampedend pipes rather than polished test specimens. Not
only experimental evidence but also a long history of
successful design practice support these facts.
These considerations will hardly affect the flexibility factor. Therefore, it is sound practice to use
the theoretically derived value of this factor, as
given by eq. 3.5. When considering stress intensification factors, however, it is sufficient to base
calculations on only one-half of the theoretically
predicted value; as supported by Markl's fatigue
tests, the appropriate equation for this factor is given
by eq. 3.14.
The increase of membrane stresses in pipe bends
subjected to internal pressure loading (as compared
to straight pipes), is generally not significant. In
fact, tests demonstrated that a static load alone will
lead to higher localized stresses than a combination
of this static load and a moderate internal pressure.
The effect of internal pressure on bends can, there-
DESIGN OF PIPING SYSTEMS
62
3.5
Branch Connections:
Loading
Static
Pressure
The junction of a braneh with a header, usually
referred to as a branch connection, is inherently a
(b) RcinlQrdng Saddlo
C,ola;
point of structural weakness in piping. Not onl)'
the absenee of metal in the header opening but also
the abrupt directional changes and oftentimes sharp
variations in cross section give rise to severe stress
intensification.
While this handicap of a branch
connection can be overcome to a large extent by
(0) M (c). wilh Shooldof_
fX'd~ Added
__
1'o.~~
-Bn;ln~h
Pipo
..J
(I) ReinfQr(ing Collo!
reinforcement and by the use of favorable contours.
it is difficult to achieve the ideal of developing a
strength equal to that of the intact pipe.
Branch connections must be designed first in re-
gard to their ability to resist static loads. This will
be the concern of the present section, while the effect.
of repeated loads will be considered in Section 3.6.
S~ction 3.7 will present a short review of various
pertinent Code rules, and Section 3.8 will give a
summary together with. practical design recommendations.
An involved geometrical shape and the strong
influence of certain secondary effects lO make the
analytical investigation of branch connections sub-
FIG. 3.11
Types of reinforcement for branch connections.
jected to pressure or structural loading prohibitively
difficult. Consequently, investigations dealing with
this subject are largely confined to experimental
research. The salient information on the action of
branch connections subjected to internal pressure
fore, be ignored in normal applications. An investi-
gation of this effect, in line with the principles laid
down in the text, is warranted only for very critical
serVIce.
The stress intensification and flexibility factors of
short bends (subtended angle less than 90°) are
known to be less than those indicated above. Despite this fact, it is recommended that no reduction
for either faetor be used on short bends, since the
experimental evidence on this subject is not conclusive.
Miter bends, as a rule, have lower flexibility and
higher maximum stresses than those pertaining to
curved pipes of similar dimensions. By this token,
the appropriate design value for the flexibility factor
of miter bends can be obtained from eq. 3.19. The
stress-raising factor will be givcn by eq. 3.14, with
the bend characteristic given by the smaller value
obtained from eqs. 3.17 or 3.18. These design criteria for miter bends originate from tests conducted
on 4 in. miters only. For large-diameter miter bends,
fit-up and fabrieation difficulties are likely to lead to
more severe eonditions than would be indieated by
the design rules stated above.
is summarized in Table 3.1 although mention shonld
also be made of a few tests reported in references
[34, 35, 36, 37, 38, 39]. Figure 3.11 shows various
types of reinforcements which have been proposed.
From Table 3.1 and its underlying tests, the folio\\'ing conclusions can be drawn: Unreinforced fullsized connections are deficient in both yielding and
bursting strength. This deficiency decreases as the
branch becomes smaller in comparison to the header.
The limited number of tests seems to indicatc that
an unreinforeed 90° intersection dcvelops the full
bursting strength when the ratio of branch to header
diameter does not exceed !. Unpublished tests on
30 in. diameter pipe and the general experience with.
pressure vessels, however, show that this rule canllot
be extended beyond the commonly available sizes of
commercial pipes.
The addition of a pad reinforcement is beneficial
in that it permits the fabricated connection to develop almost the full bursting pressure of the header.
lOSuch as the existence of longitudinal bending in the hender
due to removal of part of its wall, /lnd the interplay of radial
displacements of both header and branch under internal.
pressure.
LOCAL COMPONENTS
Pad reinforcements, however, afford little restraint
against plastic flow and are, therefore, ineffective in
raising the yielding pressure of""'he intersection to
the desired value [42, 43].
Several alternatives have been advanced to eliminate the shortcomings of unreinforced or pad-reinforced intersections. The reinforcing saddle [44],
shown in Fig. 3.11b, adds reinforcement around the
highly stressed areas of the crotch and shoulder.
The complete encirclement pad is pietured in Fig.
3.11e. This proposal [45] extends the reinforcement
Table 3.1
No.
Angle
35
of
Inrer~
Authors
section,
Header Branch dep'ccs
in.
in.
---
beyond the region where most failures of the padreinforced branch connections originated. An extension of this eoncept [46] supplements the encircling band with shoulder pads, as shown in Fig. 3.11d.
While no test results are presented, the authors of
these proposals have stated that the performance of
full-sized branch connections reinforced in accordance with these alternate details was entirely adequate under pressure loading. On the other hand,
the horseshoe-and-gusset reinforcement (Fig. 3.11e),
due to its extreme rigidity, led to stress concentra-
Summation of Internal Pressure Test Results on Piping Branch Connections
Size of
fuJI.
63
Type
of
Reinforcemont·
Pressure as per cent
Pressure as per cent of
of that supported by
that supporWd by
right-angle
intact header
intersection
at Proportionallimit
at
Bursting
sin €X
Remarks
atProporat
tionallimit Bursting
Everett
&
40
McCutchan
8
8
90
None
38.5
69.6
Crane
8
12
8
12
4
6
8
12
90
90
90
90
None
76.9
61.5
101.1
-
98.9
93.0
2'
2,
90
HorstlShoe
and gusset
-
38.5
-
-
90
None
-70.0
-70.0
Averaged
test values
11.9
10
6.2
'.2
90
90
None
)81.0
)82.5
ra1U'" averaged for
two tests
7.5
11.9
11.9
90
80
80
80
85.0
96.0
90.0
90.0
110.0
>100.0
6
6
90
Pad
Collar
Gusset & pad
Unbalanced
triform
Balanced
triform
74.5
79.0
11.9
7.5
7.5
7.5
7.5
121.0
>100.0
-8
-8
90
Welding
-
_96.0
70.0
79.0
83.0
60.0
70.0
79.0
86.0
60.0
66.0
66.0
50.0
4'.0
61.0
65.0
'0.0
Co.
41
48
49
Se.abloom
---
None
Pad
Pad
None
91.4
-
Blair
N.Gross
Averaged
values
Averaged
value
l<.~
11.9
6
48
Blair
10
·1
-
-
11.9
6
11.9
6
-
10
4
10
·1
-
-
90
90
90
60
60
60
'5
45
45
60
30
None
None
None
None
None
None
None
None
None
None
None
56.0
50.0
'3.0
51.0
63.0
3·1.0
·800 Fig. 3.1I for identification of types of reinforcement.'!.
tLast column denotes l/(C03CC a - 0.5 cot a) instead of sin a for equal-sized Y~inWrscctions.
85.6
70.9
71A
54.5
61.5
90.0
48.6
85.6
83.5
76.7
7IA
55.7
70.9
92$
57.1
}1.0
Averaged
value
}0$66
Averaged
value
}0.707
Averaged
value
0.880
0.502
Y-i::onnectiont
DESIGN OF PIPING SYSTEMS
tions of such magnitude that the intersection sustained only 38.5% of the bursting pressure of the
intact header.
.~.
Collar reinforcements of the type shown in Fig.
3.lIf wcre pioneered by the Swiss firm of Sulzcr
Brothers, Ltd. [47J, as early as 1928. Experiments
[48J indicated that this method bad characteristics
similar to the pad-type reinforcement. As a furthcr
improvement, Blair [48J suggested that the stiffening
collar be supplemented by a third horseshoe encircling the bottom of the header. He gave the name
Htriform" to the resulting arrangement, shown in
Fig. 3.lIg. lIB Table 3.1 shows, triforms performed
very satisfactorily, considering both yield and bursting pressures.
While tests confirm the effectiveness of triforms,
this type of reinforcement requires intricate fitting
and welding which does not lend itself to radiographic
examination. In high-temperature service the ribbed
construction leads to thermal gradients. Furthermore, the sharp re-entrant corners suggest high
stress concentrations which may not be revealed in
static-pressure tests but would become critical under
repeated loaning. American experience with the tri-
form is quite limited, hence in the United States it
is regarded as" novel approach until its performance
is more adequately assessed.
Welding tees, Fig. 3.lIh, are preferred structurally
to fabricated welded intersections, especially where
the size of the branch i. equal to or approximates the
size of the run. Recently, cast tees proven to be
sound by radiographic and magnetic particle examination and by hydrostatic test, arc finding increased
acceptance. Only a few articlcs [49J are publishcd
concerning the design and strength properties of
drawn tees subjected to internal pressure. The reason for this lies in the requirements of ASA Standard
B16.9, which prescribes that welding tees must be
able to withstand the full bursting pressure of
straight pipe in sizes for which they are intended.
On the other hand, the Standard makes no demands
regarding the pressure to be supported by drawn
tees at their yield strength.
Despite the presence of high stresses at the internal
;urface of the crotch [49J, welding tees, in general,
involve lesser fabrication difficulties and stress concentrations than those associated with welded intcrsections. Their performance with regard to bursting,
based on the Standard and thc meager test data that
are available, is also satisfactoryll Wclding tccs
llThc cylindrical we tested by Gross f49L which failed at
96% of the pipe bursting pressure, would not comply with
American Standard rcquircmcn ts.
having a fl spheroidal J1 intersection zone are claimed
to develop an increased resistance to yielding under
internal pressure. Branch connections subject. to
extremely high internal pressures are usually forged
and bored [50J.
Test results dealing with acute-angle (inclined)
branch connections are even more scarce than those
for right-angle intersections, being confined to those
reported by Blair [48], and assembled here in Table
3.1. This evidence indicates that within the limits of
the experiments (i.e., branch angles from 30' to 90'),
the strength of such intersections is roughly proportional to the sine of the branch angle for both full and
reducing sizes, which is equivalent (as Blair proposed) to basing reinforcement simply on the area
removed from the sidewall of the header. The
quantity of test data, however, is hardly sufficient to
support any such conclusion. Intuitively it would
seem that acute anglc branch connections are further
weakened by theincreased stress conccntration at the
crotch due to the elliptical shape of the cut-out, and
by the pressure load transfer through the reentrant
crotch corner. Recognition of these effects has led
to the current ASA Code requircments which will be
discussed in a subsequent section.
The foregoing material deals with branch connections in pipes subjected to internal pressure loading.
Closely related to this field is the snbjcct of nozzle,
and opcnings in pressure vessels. While no welldrawn division exists between these two fields, two
criteria may be mentioned, which help to separate
these problems. First, in pressure vessels the diameter of thc branch (nozzle) is usually small as compared to that of the header (vessel). This fact
diminishcs some of the secondary effects to lower
levels. Secondly, the wall thickncss to diameter ratio
is also exceedingly small in large-sized pressurc
vessels. This permits the investigator to study thc
effects of openings in pressure vessels by means of
flat-plate analogics, which would otherwise be of little
use or validity to the designer of piping branch connections. It should be remembered that even on
pressure vessels, the theorctical flat-plate analogie,
cannot asscss the effect of the hydrostatic end pull
exerted by the branch on the vessel. This cffect can
be evaluatcd on the basis of recent contributions by
Bijlaard [51J and Hoff [52J.
The theoretical approach to the problem of
stresses around nozzles in pressure vessels has
traditionally consisted of the investigation of flat
plates with a circular opening reinforced in the
manner shown in Fig. 3.12. Among thcse analytical
studies [53, 54, 55, 56J Beskin's work [56J is thc most
65
LOCAL COMPONENTS
Rim (pipo collar)
complete. In this study, combinations of "rim-type"
and "flat ring-type" reinforcements, applied sym-
metrically on both sides of the pillte, are investigated.
Based on the principle that the distortion energy
governs yielding, stress intensification factors are
gi\·cn in terms of the HetTective stress" rather than
anyone of the principal stresses.
The results of Beskin's investigation arc shown in
condensed form in Table 3.2. Since the idealized
stress condition in shells under internal pressure can
be decomposed into a hydrostatic and uniaxial circumferential stress of equal mangitudes, the last
column of stress intensification factors in Table 3.2,
headed by "Average" reflects upon the conditions
prevailing around nozzles in pressure vessels. As
can be seen, both the rim-type reinforcement and
the doubler-rim combination (with at least 50% of
the reinforcing area supplied to thc rim) are quite
effective in diminishing the peak stresses existing
around unreinforccd openings. In both cases, best
results are obtained when the ratio of total reinforcement to hole area (reinforcement ratio) is in the
neighhorhood of 0.8-1.0; at this ratio the average
stress intensification factor is reduced to a level of
about 1.35. By contrast, the pad-type reinforcement
Table 3.2 "Effecthe Stress" Concentrations around
Circular Holes in Flat Plates, Reinforced by
Various Methods
Stress Intensification Facror
HI"
Rt
t'
t
R'
R
--
-
I
Area of Reinforcement
Area of Cutr-Out
I
Biaxial Uniaxial A
Tension Tension verage Rim IDOUblerl Total
Rim Reinforcement only
0
0.4
0.8
1.2
1.0
1.0
1.0
1.0
I
1.0
1.0
1.0
1.0
2.0
1.32
1.08
1.01
I
I
3.0
1.52
1.63
1.76
2.50
1042
1.35
1.38
004
0.8
1.2
-
-
0
004
0.8
1.2
004
0.4
0.8
0.8
0.8
1.2
004
0.4
0.8
0.8
0.8
1.2
0.2
004
0.8
0.8
0.8
1.2
Doubler Plate Reinforcement only
-
-
-
2.0
3.0
2.0
3.0
5.0
5.0
104
1.2
1.8
1.4
1.2
1.3
1.52
1.43
1.37
1.22
1.03
1.00
2.30
I, 2.04
, 2.17
I
1.90
1.75
I 1.91
1.74
1.77
1.56
1.39
I 1.85 I 1.42
-
-
Doubler and Rim Combined
0.2
004
004
004
0.6
2.0
2.0
3.0
5.0
5.0
1.2
104
1.2
J.l
J.l5
1.36
1.01
1.02
1.04
1.00
1.78
1.61
1.73
1.72
1.85
1.57
1.31
1.38
1.38
1.43
0.2
004
0.4
004
0.6
004
004
0.4
0.6
I
I
,
H-t_
FIG.3.12
LR'
F10t Plato
R.:Ii~ I,
Edgc reinforcement of cirr.ular cut-outs
in flat plates.
(doubler plate only) is less efficient in reducing stress
peaks. The dimensions incorporated in Table 3.2
are characterized by Fig. 3.12.
The plastic behavior of flat plates having a circular
cut-out reinforced by a pipe collar was also investigated [57J. The analysis was restricted to ideally
plastic materials (no strain hardening) which obey
the maximum shear-stress flow condition. It was
found that for a "full-strength" reinforcement (load
at fully plastic condition in reinforced plate equal to
or greater than that referring to intact plate), the
dimensions of the pipe collar, as shown in Fig. 3.12,
must satisfy the equations:
H
1 + tn/R
tn/R
for R/t n :0; ~
1 + tn/R
- =
for R/l n 2: ~
I
VI + 2(tn/R)2 - 1
H
(3.20)
These equations bear resemblance to the results of
the elastic analysis, indicating that, for a given
amount of reinforcement (inH = constant), maximum effectiveness is achieved when the reinforcement is concentrated near the opening (small In and
large H). Naturally these results can be considered
to retain their validity only within consistent limits. 12
Furthermore, for strain-hardening materials or large
plastic strains eqs. 3.20 can be used, at best, only as
a rough guide. This plastic analysis was recently
extended [58J to reinforcements of various cross
sections.
Experimental work has corroborated the basic
findings and, in some respects, the numerical results
of theoretical work. It was found [59J that unreinforced circular openings in either heads or cylindrical
vessels led to stress concentrations in excess of those
12For instance, the assumption of a very high and very
slender rim would violate the fundamental condition that the
stress distribution be constant over the height of the rim.
66
DESIGN OF PIPING SYSTEMS
predicted by flat-plate theory. 13
These peak
stresses diminished, in general, with a reduction of
the ratio of the diameter of the opening to that of the
vessel.
Full-scale tests also indicated that excessive stiffening led to greatly increased bending moments just
beyond the toe of the weld attaching the nozzle.
Optimum conditions were obtained [55, 59, 60J by
concentrating the reinforcing metal near the opening. With appropriately reinforced openings the
stress concentration factors wer(; successfully limited
to the theoretically predicted value of about 1.35.
For small openings these optimum results were
achieved by a pipe collar whose height-to-thickness
ratio varied between 3 and 4. Flat doubler-type
reinforcements were found to be ineffective, as pre~
conventional manner.
Strain readings, howeveI\
were taken on both faces of the vessel. For reinforcement ratios" of roughly 0.23, 0.615, and 1.0,
the maximum stress concentration factors on the
nozzle side were about 2.8, 2.3, and 1.8, respectively.
These results are in reasonable agreement with
previous experimental and theoretical work. Significantly different values were, however, found on the
internal face, the maximum stress concentration
values here being equal to 3.7, 2.8, and 2.4. This
showed that an increase in thickness alone cannot
bring about the desired reduction of peak stresses.
and that only a moderate improvement in the stress
distribution on the unreinforced face can accrue from
reinforcement applied to the opposite side.
Actually, the circular opening is not the ideal
dieted by theory; the stress concentration in these
cases was in the vicinity of 3.0 at the longitudinal
axis of the opening.
Further proof of these results was offered by
Schoessow and Brooks [6IJ. Heavy rim-type reinforcements (reinforcement ratio 1.06-1.15) were only
shape for a cut-out in pressure vessels. Stress concentrations in a plate are minimized if the shape of
moderately effective, reducing stress concentrations
from 2.50 for an unreinforced hole to about 2.05.
Heavy doubler and thin rim combinations, however,
would call for an elliptical cut-out with a major-
effectively reduced stress intensifications to between
1.26-1.51, roughly in line with the predictions of
the flat-plate analogies.
The tests described above were all conducted with
the reinforcement being applied only to one side of
the vessel.
Stress measurements were likewise
limited to the external surface. In contrast, theoretical predictions are based on the assumption that
the reinforcement is applied in equal proportion to
both faces of the plate. The question, therefore,
arose: if reinforcement is applied to one face only,
what conditions will prevail on the unreinforced side?
An experimental answer to this question was soon
forthcoming. It was shown [62J that the analogue
prediction indicated the correct trend only as long
as the reinforcement was applied symmetrically,
as assumed by theory. Reinforcement applied to
one face benefited only the surface onto which it
was attached [55J; the stress pattern in the other
face, however, remained essentially the same as it
had been in the unreinforced opening. Further proof
came from tests conducted more recently by Gross
[49J. The reinforcement was applied to one side
only, by welding nozzles to the vessel opening in the
13Strcss concentration maxima for the hoop stresses occurred
at the ends of the hole diamewrs parallel to the axis of the
vessel, with some values as high as 5.5 in contrast to the
theoretical prediction of 2.5.
the opening is an ellipse with an axis ratio equal to
the Uratio of biaxiality/' the major axis of the
ellipse being aligned with the direction of the
greatest principal stress.
In pressure vessels, thi~
to-minar-axis ratio .of 2 with the minor axis being
in the longitudinal direction; the stress concentration
associated with the unreinforced opening then becomes 1.5 (as contrasted to 2.5 for the circular hole).
Both an analytical investigation [63J and experimental work [61J verified the desirable qualities of
reinforced elliptical openings. In the tests, an increase of the reinforcement ratio from 0.16 to 1.13
lowered the maximum stress intensification factor
from 1.40 to 1.19. While these results establish
the sound concept of elliptical nozzles, it must be
added that the fabrication nnd preparation of nozzles
of this type would be beset by severe difficulties.
These may overshadow the desirable aspects by
increasing the cost of elliptical pipe attachments to a
prohibitively high level.
3.6
Branch Connections: Repeated Loading
Having considered the performance of branch connections under internal pressure, attention will now
be focused on their behavior under repeated external
loads, such as imposed by thermal expansion of the
pipe line. Although this subject received some consideration in Blair's paper, the most detailed information is found in Markl's work [12, 64].
These tests produced the following results: Failures of, full-size unreinforced intersections occurred
at locations similar to those of curved pipes. The
14Effective height of reinforcement taken equal to radius of
finished opening.
67
LOCAL COMPONENTS
stress intensification factor could be correlated
reasonably well with that for a single miter bend
(see eq. 3.14) if the characteristic'variable was taken
to be
(3.21)'5
h = tlr",
Reinforced fabricated intersections cannot be
categorized with equal facility, since the amount of
metal incorporated in the reinforcement and the
manner of its distribution will affect the stress intensification and flcxibility factors. In an attempt
to formulate a rule which would correlate reasonably
well with limited tests on 4 in. size pipe and be applicable to most reinforced branch connections,
Mark! [12J proposed that the avcrage thickness of
the headcr and branch at the crotch, t" be assumed
as the governing factor. Assuming that reinforced
intersections otherwisc behave like unreinforced
ones, the characteristic variable would then become
h = (~)2.5 -.£
t
r",
(3.22)16
whilc the stress intcnsification factor is again obtained from eq. 3.14."
These results rcfer to tests where the assemblies
were loaded through the branch; loading straight
through the header proved to be less severe in all
cases. Furthermore, it was shown that the direction
of bending (in- or out-of-plane) did not seriously
influence these results, so that onc factor can be used
in practical design. While Markl's work represents a
marked advance in practical design approach, it
must be conceded that the experimental data are
rather limited. More work would certainly be desirable to check its applicability to large-diamctcr
piping and to reducing-size branch connections.
Data regarding the performance of full-size ASA
standard welding tees under repeated extcrnalloading can again be found in Markl's papers. Assuming
that the metal thickness available in the crotch zonc
and the crotch radius arc thc controlling variables,
the characteristic variable was expressed in the form:
h = (~)2.5
t
-.£
rm
(1 + rrm
o
)
(3.23)18
16This equation is obtained by simply substituting ¢ = 45 0
in eq. 3.18 for single miter bends.
16The design formula of the Code is given in a modified
form of eq. 3.22.
17The qua.ntity tin eq. 3.22 is the thickness of the pipe used
in the stress calculation. The intensification factor of eq.
3.14 is again applied to this pipe. These results refer to fullsized intersections and should be used with discretion for
other cases.
ISIn the Code, the recommended formula for the charac-
where
t, = effective thickness = average of crotch
and side wall thicknesses.
r c = crotch radius.
Experimental results for three different 4 in. commercial welding tees were in reasonable agreement
with stress intensification factors obtained from eq.
3.14, if eq. 3.23 was adopted for determining the
characteristic variable.
The flexibility factors associated with unreinforced
and reinforced fabricated intersections or welding
tees havc not reccived sufficient attention. Rough
tests secm to indicate that the added flexibility of
full-size branch connections is small; that is to say,
the branch will act as if it were fixed at the hcader,
whereas the header will retain the flexibility of an
intact pipe. These results, however, are open to
question since full-sized intcrsections should approach single miter bends in flexibility. In addition,
flexibility of the branch would be expected to increase
for reducing-size intersections (sec, e.g., eq. 3.27 in
Section 3.14). Lacking specific thcoretical or experimental results, and in order to rcmain on the safe
side, it is suggested that a value of 1.0 be assumed
for the flexibility of all types of braneh eonnections.
3.7
Branch Connections: Comparison with
Code Requirements
It is of interest to compare now the experimental
data with established design practice as expressed
by Code requirements for 90° (perpendicular) branch
connections. The Code for Pressure Piping, ASA
B31.1, Section 6, utilizes the area replaccment
method, requiring that the area removed from the
wall of header (referring to the required minimum
wall thickness times the diameter of the finished
opening) be replaced by thc cxcess thickness available in the header or nozzle wall plus any mctal
applied to the interscction in the form of reinforcement. This reinforcement is considered to have
value only within the rectangular Hreinforcement
zone," the length and height of which is limited as
shown in Fig. 3.13. In the subsequent derivation,
the following nomenclature is used:
I
tIl = minimum thickness of header less corrosion
tE = minimum thickness of branch allowanoc.
Rll = radius of headcrl
t'd
R B = rad ·lUS 0 fb rane housle.
w = leg of fillct weld.
teristic variable of welding tees has been simplified to h = 4.4
tlr by making assumptions for t. and T c which conservatively
reflect customary proportions.
68
DESIGN OF PIPING SYSTEMS
Area to be
reploced
Jlll;n!Otcllmeol
Zone
I"IG.3.13
U
+-+1"" '.-
Area of reinforcement," as specified by the Piping
Code, ASA B31.1, Seetion 6.
PH = maximum service pressure permissible for
intact header.
S = allowable stress at operating temperature.
II = thickness of header required by Code for
given size, service pressure, and operating
temperature.
12 = thickness of branch required by Code for
given size, service pressure, and operating
temperature.
p = allowable pressure permitted by Code for
the completed manifold.
Ip = thickness of reinforcement pad (if used).
According to the Code, the required thicknesses
ean be expressed as
pR/l
II = S
pR B
RB
+ OAp ; (,= S + OAp =-11'
R/l '
_
I /l-
p/lR Il
S
+ OAPIl
The "area to he rcplaced" is A = 21 1 (R B - tB)
For an unreinforced intersection with a branch
not heavier than the header, the available excess
metal within the "zone of reinforcement" can be
given by
A R = 2(1/l - 1r)(RB - tB)
+ 5tB(tB - (2) + w 2
Equating A R to A, and using the wall-thicklIess
expressions, yields the following result:
p
PIl
R/l _ 0.4
III
+
0.8R/l(RB - IB)
RBIB
0
2
2 A
OAIIl(RB - IB)
tB
0.2w
+
(3.24)
+
The IIpressure reduction ratio," pip/I, expresses
the decrease in allowable pressure for the completed
unreinforced branch connection as compared to the
intact header of the same size. As an example
assume standard pipe sizes, w = 'i- in. for 4 in.
branches or smaller, w = i in. for larger branch sizes,
and a corrosion allowance of 0.1 in. The Hpressure
reduction ratios" may then be obtained for various
header and branch sizes from eq. 3.24. The results
of these calculations for the speeifie ease are tabulated in Table 3.3 except that braneh eonnections
not exceeding 2 in. or 25% of the header size are
shown with a 100% rating sinee the Code permits
this arbitrarily.
An examination of this table indicates the following trends:
L For equal-size interseetions (proceeding along
the diagonal of Table 3.3), the pressure reduction
ratio decreases with increasing pipe sizes to a limiting
value of 50%. This is in reasonable accord with
experimental evidence, although tests earried out
on full-size intersections up to 12 in. did not show
bursting-pressure reduction ratios below 65%.
2. Increasing the size of the branch connection
for a given header (moving from left to right in a
given row of Table 3.3) deereases the pressure reduction ratio. Although this trend is borne out by
tests, the Code reduction ratio appears conservative
for small-size headers, sinee it permits only 56% of
the "intaet header pressure" to be applied to halfsize intersections with an 8 in. or 12 in. header, as
contrasted to the 90-100% obtained in experiments.
More complete and more searehing experimental
data would, however, be necessary to justify closer
evaluation of certain sizes and proportions.
3. An inerease of header size for a given braneh
size over 2 in. (traversing Table 3.3 from top to
bottom of a speeific eolumn) results in a reduced
allowable pressure. This is eontrary to the limited
experimental evidenee, whieh shows that the bursting pressure developed by a braneh connection
increases as the diameter ratio between the branch
and header pipes becomes smaller. Again more
searching experimental data are desirable.
4. The arbitrary Code provision assigning 100%
for welded braneh pipes not exceeding 2 in. or 25%
of the header size, while reasonable from test results,
introduees abrupt breaks in allowable ratings. A
smoother transition is desirable.
The ASA B3L1 Code rules applicable to oblique
branch eonnections are mandatory for branch angles
not less than 45° and when the braneh/header
diameter ratio is not less than 1/4. These rules, which
reeognize the higher stress intensification in the
acute eroteh, require that the replacement area be
LOCAL COMPONENTS
equal to the area removed from the header multiplied by a factor of (2 - sin a) where a is the branch
angle. The rules (for branch/header ratios of 1/4
and larger) make no distinction between full-sized
and reducing branches, a practice which appears
1. The design stress used provides a considerable
margin for local overstress.
2. Highly localized stress can be relieved by local
yielding. Such yielding induces local residual stresses
of the opposite sign in the off-stream condition, so
that the area can operate on a "stress range" basis
in the same manner that thermal expansion strains
may be absorbed in piping systems.
3. Most applications do not involve a very large
number of cycles. Therefore, the design need not
insure that stresses be kept at all times below the
endurancc limit of the material.
4. Experience to date is largely confined to steel,
which normally acts in a ductile manner.
Thus, although this experience has been generally
satisfactory, those serviee failures (and all of the
laboratory fractures) that occurred in pressure
vessels and pipe lines to date have, almost without
exception, been shown to originate at branch connections 01 local attachments. Therefore, good
engineering demands that careful judgment be
exercised when selecting designs and fabrication
details, and that fabrication quality be adequately
controlled. Poor fit-up, welding, and lack of root
penetration on welded branches, can easily furnish
added stress-raising effeets whieh ean lead to
failure.
somewhat contrary to experience.
3.8
Branch Connections: Practical Considerations and Summary
In the foregoing sections giving the highlights of
available test and analytical data, it has been noted
that stress concentrations can be expected to be
present around all circular openings and branch
connections, and that even for the most carefully
designed reinforcement, the factor is· not likely to
subside below 1.3. The question naturally arises
as to the practical significance of such effects. The
answer at present must be sought primarily in experience. Service experience using nominal design
allowable stress values (as established by Section 3,
Oil Piping, of the ASA B31.l Code for Pressure
Piping, and the concept of the simple replacementof-area method) has been reasonably good despite
the fact that design and attachment details and
fabrication quality used have not always been as
good as they should be. This fact may be accounted
for by the following considerations:
Table 3.3
~
Size
Header
69
Pressure Reduction Ratios in Per Cent for Unreinforced Intersections·
1"
l~"
2"
2 "
,
3"
89
67
65
62
59
100
100
100
100
100
100
100
100
66
64
62
60
58
56
56
56
56
56
56
56
63
61
58
57
55
55
55
55
55
55
55
1
4"
6"
8"
10"
57
55
55
56
56
56
56
56
55
55
55
55
56
56
56
12"
14"
16/1
18"
20"
24"
53
53
53
53
53
52
Size
1"
I!"
2"
2 J2·"
3'~
4"
6"
8"
10"
12 f1
14"
16"
18"
20"
24"
100
100
100
80
77
100
100
100
100
100
100
100
100
100
100
100
100
71
69
65
100
100
100
100
100
100
100
100
100
59
57
56
54
54
55
54
55
55
55
60
58
56
56
57
57
57
57
57
54
55
54
55
55
55
54
54
54
55
55
53
54
54
54
*Based on the Code for Pressure Piping, ASA B31.l for: standard weight pipe with 0.1 corrosion allowance; leg of fillet weld:= ill
{or branches 4/f or smaller, and.g ll for larger branch sizes.
70
DESIGN OF PII'ING SYSTEMS
As design stress levels and temperatures inerease,
greater attention must be given to reinforcement
details. The same is true when d~sign stresses are
raised in proportion to enhaneed physieal properties
of material obtained by eold work, since the effect of
localized stresses becomes much more serious.
Full-size 90° branch connections are difficult to
fabricate by welding without appreciable distortion,
particularly when a pad-type reinforeement is used.
This diffieulty inereases with the size of the header.
It is best to avoid such connections wherever it is
economically justified. In critical service, welding
tees, when available, should always be used in
preference to fabricated welded intersections. Integral reinforcement obtained by nsing a heavier pipe
for the header (or for both header and branch) is
generally preferred to built-up construetion and is
satisfactory for most applications. Sharp earners
at the interseetion should be avoided by the use of
coneave weld fillets. Fabrication presents eonsiderably greater problems as the size of the braneh
relative to the header is increased and must receive
special care when this ratio exceeds 50%, particularly
for header sizes above 12 in. On headers of large
size with small openings (branch to header diameter
ratio less than 50%), the method of reinforcement
should be guided by the principles established for
the reinforcement of nozzles on pressure vessels.
The greatest benefit from a given amount of reinforcing metal will be obtained by concentrating the
reinforcement nea~ the finished opening. Flow
considerations permitting, the effectiveness of the
reinforcement can be increased by application of
the reinforcing metal to the inside, as well as outside,
surface of the header. The use of elliptical nozzles
may be considered for extremely severe service
conditions, since they extend the possibility of reducing stress concentrations to the limiting value
of 1.
For the design of special heavy-walled fittings in
critical service The M. W. Kellogg Company has
found the rather simple design appI'oach given in
Fig. 3.14 satisfactory. This is, in effect, an analysis
designed to control the average membrane stress
within the chosen limits, and includes a correction
for non-uniform stress through the wall thickness
equivalent to using the mean diameter cylindrical
hoop stress formula instcad of the inside diameter
formula. Therefore, it assures a fitting strength
roughly equal to the connecting pipe. The regions
over which the pressure area and metal areas are
averaged are arbitrarily selected as being in reasonable accord with experience.
The use of gusset or rib stiffeners is not recommended, due to the high stress concentrations likely
to exist at their ends or adjacent to the attachment
welds. They are even more objectionable on hot
piping, since the ribs act as cooling fins and local
thermal stresses are imposed; if such stiffeners are
used on hot piping the thermal effects should be
minimized by the application of heavy insulation.
The effect of structural loadings other than pressure and cyclic loadings must be given due consideration. Markl's work in establishing suggested stress
intensification factors for piping flexibility analyses
is a good start, but more work is necessary on other
sizes and reducing branches. Where an individual
flexibility analysis is not warranted, yet expansion
stresses are expectcd to be at or near Code levels at
the branch location (with the moment loading being
carried through the branch), it is recommended that
branch connections be reinforced to develop the fnll
strength of the header, even if the operating pressure
may not require it.
The selection of design and fabrication details as
well as the methods and extent of inspection must
be in line with the expe'cted severity of service. Weld
details which minimize distortion and promote best
root-welding conditions are to be favored. As an
example, setting a branch on a pipe and welding it
before the hole in the header is cut will reduce distortion when the branch pipe is large compared to the
header; set-on construction also permits the use of a
backing ring.
Regarding inspection methods, a magnetic particle
examination should be favored for magnetic materials; for non-magnetic materials, a penetrant oil
examination is quite practical and is recommended
for important services. Radiographic examinations
of branch attachments are being increasingly used as
a quality control check; although they are useful in
controlling the general quality level of an individual
operator's work, such radiographs cannot be interpreted to assure the absence of cracks unless many
angled shots are taken. An indiscriminate appraisal
of radiographic examination may create an unwarranted degree of assurance regarding absence of
harmful defects.
3.9
Corrugated Pipe
As pointed out in the introduction and in Chapter
7, straight corrugated pipe provides intermediate
flexibility' between a rigid piping system and an
p.xpansion joint system. Hs use may be advantageous
where acute space limitations exist, or where reactions on equipment attendant to stiff or large-size
LOCAL COMPONENTS
71
--
o
D,
I"
p(E+t A )
p(E+i A)
A
A
90· ELBOW
TEE
G
t>( +.8
Z
+t2 coS-2-
G
"D,
"2
+ tlCOS 2'
p(E + !Al
S... ~
A
USE ALSO FOR
.. ~" ELBOW
WYE OR 45' ELBOW
LATERAL
NOMENCLATURE
A, B
-
NETAL
... RE .... ISQ.IN.)
E,F
-
INDICATED PRESSURE AREA, (SQ.INI
Os. ~,- lNSlDE 01AWETER OF FITTINGS.IIN.I
G,h,k -
P
13
-
Sa -
S....
\.1
2
-
0<./3 -
INDICATED LEHGTHS,IlH.I
DESIGN PRESSURE. AT DESIGN TENPERATUR'. (P$IC)
ALLOWA8LE STRESS AT DESIGN TENPERATURE, (PSI)
INDICATED NETAt. THICKNESS, IIN.I
AVERAGE .. [TAL, THICKNESS OF fLAT
INDICATED ANOL.ES.
SURF"I;E, (INJ
FIG.3.14 Special hc:\vy wall fittings: check of reinforcement for internal prc5surc.
72
DESIGN OF PIPING SYSTEMS
pipe must be reduced witbout further addition to
pressure drop, process problems or similar factors.
The fact tbat corrugations greatly increase the
flexibility of a straight cylindrical tube has long been
appreciated. However, it is less well known that this
reduction in stiffness is obtained by the introduction
of bending stresses, the level of which must be controlled for satisfactory service; also, it is not always
appreciated that a corrugated bend may be less
flexible than a plain pipe bend due to the fact that
the corrugations resist ovalization.
Corrugations were initially obtained by hot rollforming processes sueh as are used for shaping the
flues of Scotch Marine boilers. This imposed limitations on the depth and pitch so that resort was made
to uniform localized heating and controlled collapsing.
This process resulted in some upsetting and a
sharper radius of curvature at the crown. More
reccntly, equipment has been designed which provides rolled corrugations of greater depth. The
mean diameter of rolled corrugations is usually that
of the original pipe, while those formed by controlled
collapsing have a mean diameter roughly equal to
the initial outside diameter of the pipe. Corrugating
can only be accomplished on straight pipe, so that
bends must be formed afterward. Creased bend
construction is usually limited to sharp radii bends,
commonly 2 to 3 diameters in radius; these are formed
by heating plain pipe on one side and bending so as
to bulge out the corrugations on the inside of the
bend.
Early tests [21, 65] showed that, for bends having
a five-diameter radius, corrugated construction provided no greater flexibility than plain bends, and
that for smaller radii corrugated or creased curved
pipes are usually less flexible than plain bends; also,
that the torsional stiffness of corrugated pipe was
slightly greater than that of straight pipe of the same
nominal diameter.
In the foregoing tests, as well as later ones [66], it
was established that a corrugated bend derives its
flexibility in bending or direct loading mainly from
a change in axial length (through an increase on the
tension and a decrease on the compression side), as
compared to a plain bend, which derives its increased
deflection from ovalization of the cross section, and
the attendant modified stress distribution. A
creased pipe bend takes an intermediate position
between these extremes, deriving its flexibility on
the plain portion by ovalization, and on the creased
portion by change in length.
Consistent flexibility values for corrugated and
creased 6 in. diameter pipes wcre obtained from static
tests by Dennison [67], who related test results to
the calculated values for plain pipes of the same
dimensions,19 as given by elementary beam theory.
Corrugated bends were found to have higher flexibility factors than creased bends, although a good
approximation for both configurations was 6.0.
Nominal stress intensification factors (denoting the
ratio between the endurance limits'· of small
polished specimens to that of the actual piping component) were obtained from fatigue tests, indicating
an average factor of 8 for both corrugated and creased
pipes of the type tested. With one exception, incipient cracks in corrugated pipes originated on the
inside surface and penetrated outward, while 011
creased bends the cracks always initiated on the
external surface.
The foregoing tests have led to the acceptance of
an assumption of uniform flexibility factors for commercial corrugated or creased pipes. For the stl'es~
intensification factors, however, it was felt that Ullduly conservative values resulted by basing the
comparison on the endurance limit of polished hars.
Therefore, further tests were carried out by Rossheim and Markl [29], in which the fatigue strength"
of plain tangents was taken as a basis of comparison
for stress intensification factors. Based on these
cxperiments a stress intensification factor of 2.5 was
suggested as reasonable for finan-cyclic" service (i.e.
less than 20,000 stress cycles), whereas for "cyclic"
service (up to 500,000 reversals) a stress intensification factor of 5.0 was proposed. A flexibility
factor of 5.0 was suggested as a conservative value
for average commercial creased or corrugated pipe.
These values form the basis of the current suggested
values in the ASA B31.1 Code, viz: a flexibility factor of 5.0 and a stress intensification factor of 2.5
for usual commercial corrugated or creased components under bending or direct axial loading. The
Code also suggests a flexibility factor of 0.9 and
strcss factor of unity for torsional loading.
In contrast to the uniform values recommended
by the Code for all sizes and shapes of corrugated
pipes, the actual flexibility and stress factors are a
function of the size and wall thickness of the pipc
19The term "plain pipe of the same dimensions" refers to a
pipe having the same diameter and wall thickness as that used
for making the corrugated (creased) pipe in consideration.
2011Endurance limit" denotes the alternating stress which a
specimen can infinitely sust.ain in a fatigue test. In actu:d
tests 2 X 10 6 c)'c1es arc taken to be cquivalent to "an infinih'"
number of cycles.
21HFntigue strength," as opposed to "endurance limit,"
denotes the average maximum alternating stress which :L
specimen can sustain for n given number of stress cycles.
LOCAL COMPONENTS
and particnlarly the depth and pitch of the corrugations. Test results show that an increase in the
depth of the corrugation will improve its flexibility,
but increase the stress intensification factor.
The greatly simplified ca.se of a curved beam
shown on Fig. 3.15, which is obtained after segmenting a corrugated pipe into strips of unit width, can
be analyzed readily. This analogue will indicate
higher flexibility and lower stresses than those existing in the actual structure. Nonetheless, it is useful
in roughly predicting the influence of dimensional
changes, and for comparison with established service.
The results obtained from this analogue, assuming
that v = 0.3 and the corrugation pitch is 4r, are:
Flexibility factor
= 1I"[(3r/t)
+ .09]
Theoretical stress intensification
factor
= [(6r/t) + 1]
Stress intensification factor compared to plain pipe (Code basis) = 0.5[(6r/t) + 1]
These relations are plotted in Fig. 3.16, which shows
the strong. dependence of both of these design factors on the ratio of r It?' An increase in the pitch
of the corrugations with other items unchanged
would decrease the flexibility faetor; likewise, a
change in the shape of the corrugations to a more
rigid shape would decrease the flexibility, but also
decrease the stress intensification factor.
A detailed analytical evaluation of stresses in corrugated components is extremely difficult. Moreover, if the manufacturing tolerances and variations
in shape or between successive corrugations are considered, the theoretical treatise becomes impractical.
_.\n approximate solution was developed by Donnell
22Duc to the simplifications assumed here, the ratio of pipe
radius to pipe wall thickness Rlt docs not enter the solution.
I
"
M
_' ( I
)~
M
FIG. 3.15
Analogue representation for analysis of
corrugated pipes.
100
1000
70
700
.
73
400
'0
200
10
100
7.0
!70
2
'.0
~
€.w
~
'.0
'0
1.0
10
0.7
7.0
Piping eo&. F1uibli1y Fodor
0.'
'.0
0.2
2.0 O~--"--~8
---:"',---:':,,---','=,---:,,
2 r/I, Ratio of Corrvgmion [Hpth to p;~ Walt Thidneu (Pitch'" Twic. Depth)
FIG. 3.16
II
Stress intensification' "and flexibility factors in
analogue" solution for corrugated pipes.
[68] for V-shaped and semicircular corrugations under
concentric axial loads, and the results were extended
by inference to corrugations of elliptical and sinusoidal cross sections. Tests on thin-walled corrugated
pipes, having corrugations in reasonable accord \\~th
the specific shapcs analyzcd, were in good agreement
with theory. For corrugation shapes a.s normally
produced in heavier-walled pipe, the analysis can
be accepted only as a rough approximation.
A more rccent theoretical approach [9] treats the
effects of both axial loads and internal pressure, but
restricts the analysis to thin-walled cylindrical bodies
of relatively large diameter, so that thc results are
applicable to corrugated light-gage expansion joints
rather than pipe. This analysis is not readily
adapted to design, since the final rcsults are given
in terms of unfamiliar functions, whose numerical
values are not tabulated in standard mathematical
tables. For a more detailed treatment of the expansion joint bellows, reference should be made to
Chapter 7.
In summation, corrugated pipes have practical
application when added flexibility for stress or endreaction reduction must be obtained in extremely
limited space, thereby permitting the retention of
74
DESIGN OF PIPING SYSTEMS
a rigid piping system and avoiding the use of expansion joints. Its use is best confined to straight
lengths, since it will have little, if· any, advantage
over plain pipe when used for bends. The same may
be said for ereased bends, which offer no significant
advantage over plain bends in flexibility and may
involve higher stress intensification. Corrugated
pipe, properly designed, is capable of carrying the
axial internal pressure thrust in common with
straight pipe, but it is important to note that the
stress intensification factor applies to the longitudinal
pressure load, as well as other loadings; yielding, or
creep, will result if the combined static or dynamic
loadings exceed established limits. For applications
in the creep range an accurate evaluation of the
stress intensification factor for the particular corrugation used is desirable. Occasionally, corrugated
pipe is used to localize plastic deformation which
might occur during extreme upset conditions; for
such service the range of local unit plastic strain
determines the number of cycles which can be sustained (see Chapter 7). It is advisable that limit
stops or equivalent means be provided to limit overall yielding.
3.10 Bolted Flanged Conneetions: General
Baekgronnd
Flanged connections provide for the ready joining
or separating of portions of a piping system to facilitate inspection or cleaning, or to avoid in-position
welding or heat treatment. Their influence on the
performance of a piping system involves evaluation
of (1) the effect of the flange as a local component,
and (2) the effect of the forces and moments transmitted through a flange on its ability to maintain a
tight seal.
Analysis of flanged joints was limited to cantilever
approximations until the advance made in 1927 by
Waters and Taylor [69J. Combining the elastic
behavior of a flat plate with a cylinder treated as a
beam on an elastic foundation, they obtained expressions for the circumferential, radial, and axial
stresses in flanges with short cylindrical hubs of
constant thickness. These theoretical results were
reasonably well substantiated by tests, and offered a
reliable basis for the evaluation of loose-hubbed
flanges (Vain Stone, threaded, lapped) within the
ASA range of dimensions [70, 71, 72J. In subsequent
years this derivation was extended by Holmberg
and Axelson [73J to flanges integral with the pipe
wall.
These analyses were limited to hubs of uniform
thickness, although the desirability of increased hub
thickness at the flange-hub intersection was generally recognized. An exhaustive theoretical analysis of this problem [74, 75J was undertaken by
Waters, Rossheim, Wesstrom, and Williams. It
included both bolt-moment and pressure effects, and
could be applied to straight, single, or double taper
combinations of hub contour, with fillets simulated
by tangent tapers. A complete analysis, including
direct pressure and pressure discontinuity stress, is
complex for other than a straight hub; the considerable effort involved in this approach is justifiable
only on high-pressure, large-diameter flanges in
critical service. For usual services and flange
proportions it has been found that the direct-pressure
and pressure-discontinuity effects can be neglected,
and the flange subjected to a uniformly distributed
external moment equal to the product of the bolt
load, gasket load, end pressure load, and their respective lever arms. The junction of the ring and
hub is assumed to undergo zero radial displacement,
the bolt load to be unaffected by changes in pressure,
and ideal elasticity to be maintained without yielding
or creep. Despite these simplifying assumptions, the
analysis has proven adequate for most problems
when coupled with' a suitable choice of design
stresses, gasket factors, etc. to provide ample
margin for these effects. Originally introduced into
the ASME Unfired Pressure Vessel Code on a 'permissive basis, its general acceptance soon led to its
adoption as a mandatory requirement. One widespread usage is on exchanger flanges, for which it.
has been approved by the Tubular Exchanger Manufacturers Association (TEMA) and applied to the
standard flanges in their rules; it has also been widely
used in eonnedion with rerating ASA standard
flanges.
Experimental work [76, 77, 78J has shown that the
theoretical formulas closely predicted the stresse,;
developed under various loading conditions. It hao
also been shown that a reasonably uniform gasketload distribution can be expected only when the
maximum bolt pitch is a function of the bolt diameter
and the flange thickness, and that, within the normal
range of flange dimensions, the width of the flange
has no appreciable effect upon the load distribution.
Taylor Forge & Pipe Works' Modern Flange Design
[79J presents the formula in terms of bolt diameter,
flange thickness, and gasket factor. This formuh'
originated in the M. W. Kellogg Company and has
been widely and successfully used in practical design.
Present ASME flange-stress formulas are stated
for flange and hub dimensions assumed in advance,
leading to time-consuming trial-anci-error solution.
75
LOCAL COMPONENTS
Simplified methods have been developed [80, 81] to
aid the designer in quickly arriving at well-proportioned economical flanges. Othi!!" practical suggestions are contained in Madern Fkmge Design [79].
For certain relatively mild low-pressure services,
such as water works, thin and relativcly wide flanges
with soft gaskets located inside the bolts have been
successfully used. Such flanges usually cannot be
justified by the Code design approach. The explanation of their satisfactory use must be sought in
recognition of higher stresses, use of soft gaskets, and
the possibility of the flanges contacting each other
at the OD, establishing thereby a limiting countermoment before excessive strains are developed. For
considerations involved in such special service,
reference can be made to the paper by Waters and
Williams-[82] and thc discussion thereof.
The design assumption that the initial bolt load
remains constant for any magnitude of the internal
pressure hai also been explored. It has been shown
[74, 76, 83, and others] that a hydrostatic end force
may either increase or decrease the initial bolt load,
depending upon the relative position of the gasket
reaction and the elastic properties of tbe assembly.
With customary flanges, exemplified by ASA Standard B16.5, the bolt load decreases slightly with application of internal pressure, since the net moment on
the flange ring increases, which in turn causes increased flange rotation and a decrease in the distance
between flanges at the bolt circle. The bolt stress
is at all times a function of the summation of the
strains of the entire assembly, and their individual
moduli of elasticity. Since the modulus decreases
with temperature rise, the bolt load will likewise
decrease as the temperaturc of the assembly is uniformly raised. If the temperature of the components
is not uniform, the differential strains will alter the
bolt stress in proportion. Except in unusual cases
these effects are not of practical significance, since
the flange design permits pretightening to a level
sufficient to compensate for bolt-load reduction,
local yielding, or creep over the period of time
established by the material properties and tem-
superimposed hydrostatic load would have the simple
effect of subjecting both springs to equal amounts of
added tensile strain. In this simplified representation; leakage would occur when the tensile strain
imparted to the flange due to the hydrostatic load
offsets the compressive strain set up by tightening
the bolts (i.e., the spring representing the flange is
under no force and returns to its original length).
This concept was presented by Dolan [84] who gave
pictorial representation to this interpretation by
means of a simple force-extension diagram. Needless to say, the elastic coupling concept is a considerable oversimplification of actual conditions in a
flange which must include all components, changing
moment and rotational effects, temperature, creep,
etc. as treated, for example, in [82J.
Code rules establish two criteria which must be
satisfied to maintain a gasketed. joint free from
leakage. The one establishes a minimum initial unit
gasket seating load, and the other a ratio of gasket
load to internal pressure for operating conditions;
both are related to the gasket material and construction. As to the effective width of gasket, arbitrary
assumptions are made which are related to the
flange-facing details arid relative concentration of
loading; double this width is used for application of
the gasket operating pressure factor.
Actually, the performance of a gasketed surface
depends not only upon the elastic properties of the
gasket material as influenced by its design details,
but also upon the bolt load and deflection of the
flanges (initially and under pressure); the maximum gasket load (at the inner and outer edge or at
projections); the gasket thickness and physical
properties; and the surface finishes of gasket and
flange, which determine the elastic and plastic
deformation attendant to an initial seal. In actual
installations, the varieties of gasket stiffness, surface
finish, and imperfections of gasket and flanges, as
well as the properties of the contained fluid or gas,
can cause wide variation in the minimum load and
gasket pressure to maintain tightness. The Code
rules for minimum load and gasket factor are, there-
pp.rature.
fore, approximations of average conditions for flanges
An understanding of flange leakage may be obtained by idealizing the assembly as two elastically
coupled bodies, the bolts on one hand, and the flanges,
including the gasket, on the other. The complete
joint is then represented schematically as two springs
with different initial lengths and stiffnesses. When
the joint is tightened, this initial length difference is
eliminated by submitting the bolt spring to tension
and the stiffer flange spring to compression. A
of usual proportions. For this reason, these rules
are not mandatory but merely tentative.
Where very stiff f1ftnges or minimum-width gaskets
are involved, the entire width of the gasket may be
at essentially the same unit load; conversely, for thin
flexible flanges or wide gaskets, the gasket load may
be much lower than predicted by Code rules. Extremely soft gaskets, such as gum rubber, deflect
directly undcr the internal pressure, and flow later-
.. _.
__..._ - -
76
DESIGN OF PIPING SYSTEMS
ally to equalize and distribute tbe gasket load. Due
to this behavior, they ean often be made tight under
very low unit gasket loading or, at times, even in the
absenee of a net gasket load from the bolts. Speeial
facing details or gasket designs to promote selfsealing tendencies are successfully used where the
service permits the use of soft gasket material, or
where extremely high working pressure can be
utilized to provide sufficient load to seal harder
gaskets. Some facing details are based on mechani-
creep due to temperature effects, it is necessary to
cal concepts, such as ring-type joints; others, as for
instance, lens rings, benefit from reduced or line
contact.
"There elastic conditions arc maintained, leakage
sient temperatures. As a continuation of this work,
of a properly fabricated and assembled flanged joint
should not occur if the initial bolt load is sufficient
to maintain the required gasket load above the
longitudinal loads developed by pressure and structural effects, and to compensate for the expected reductions in bolt load due to flange deflection and
change in elastic modulus. The influence of structural loading is treated further in the next section.
Flange bolts are ordinarily made up at ambient
temperature; as the temperature is raised in service,
the temperature of the bolts, flanges, and pipe may
no longer be the same, either during the transient
heating period or in the equilibrium service condition.
Since the bolts receive heat through limited contact
with the flanges they will respond more slowly to
changes; similarly, non-integral flanges, such as the
Van Stone type, will lag behind the pipe under temperature change. In the absence of insulation, these
temperature differences will be much greater. Where
flow temperatures fluctuate rapidly or where external
influences (such as rain on exposed flanges) upset the
equilibrium between bolts and flanges, joints may
leak due to loss of gasket sealing load. This reduction in sealing pressure is traceable either to expan-
sion differences, or to yielding resulting from temporary overload. If serious yielding does not occur
the joint will eventually re-establish the same gasket
load, although it is possible that the temporary
leakage will have caused wire-drawing and prevent
re-establishment of a seal.
In aetual installations, completely elastic conditiOilS are almost never realized j aside from localized
yielding, creep will be present in high-temperature
service and to some extent at all temperatures,
particularly for non-metallic, non-ferrous, or highly
stressed gaskets. Under repeated load applications
the degree of yielding or creep with respect to time
is increased.
In order to deal with the effects of plastic flow or
consider both transient and constant thermal conditions. Of these two, the transient thermal state is
often the more important, since it will generally lead
to higher stresses and a greater amount of yielding.
This case was investigated by Bailey [85] for loosering and integrally welded flanges. Making a few
,implifying assumptions, he showed that stresses in
welded integral flanges were only 60% of those in the
less intimately connected loose flanges under tranBailey l'Ildertook to investigate the effcct of crecp
upon thc elastic stress distribution. Again both the
loose and welded integral flanges were considered for
various creep strength ratios of bolt material to
flange material. The effect of bolt holes was taken
into consideration through an analogy with the
tensile creep-relaxation properties of a solid strip of
metal versus a strip of metal with a series of holes of
varying pitch and diameter. The effect of thermal
bending moment acting on a joint due to pipe expansion was omitted, on the assumption that under creep
conditions external forces would in time be reduced
to negligible magnitude. The analysis showed that
the tightness duration of the flanged joint (as defined
by the time at which the stresses would fall below a
permissible level or permit leakage) is a function of
flange thickness for given material properties of
flange and bolts. It was also found that an optimum
flange thickness exists for each joint, which increases
as the ratio between the creep resistance of the bolts
and flanges becomes greater. These optima were
generally greater than required by the Code. As
would be expected the analysis indicated the desirability of having high elastic strains in bolts and
flanges combined with high creep resistance. Thesc
properties are in opposition, since high stresses cause
accelerated creep. For given materials certain
changes in design will, however, provide increased
elasticity without increased stresses, e.g. increasing
the effectivc bolt length; similarly, changing the
material of any component to one of equal elasticity,
but greater creep resistance, will improve high temperature performance.
Bailey's analytical work was not followed by experiments of sufficient extent to prove or disqualify
the conclusions reached. However, an interesting
report, including some test data on various aspects
of flange dcsign, was published by Gough [86J on
this subject.
Thc fatigue characteristies of various types of
stecl flanges subject to repeated bending strains received attention in an investigation carried out at
LOCAL COMPONENTS
atmospheric temperature by Markl and George [87].
Using a constant-displacement type fatigue-testing
machine, on 4 in. standard weight and 0.080 in. wall
pipe with 300 lb ASA standard RF flanges, fatigUe
failure occurred almost invariably in the pipe proper
adjacent to the flange attachment, where there is a
marked change in contour, and not in the flange or
bolts. A few tests were made with 600 psi internal
pressure; in these tests leakage well in advance of
failure was noted only on threaded joints. Gasket
leakage was not experienced when bolts were pretightened to 40,000 psi, although it was encountered
when they were tightened to only 20,000 psi. The
S-N diagrams of all types of flanges investigated
were represented by straight lines on a log-log plot,
which were parallel among themselves and with the
lines obtained for straight tangents or butt-welded
pipes. This made it possible to assign single stress
intensification factors to each of the various types
of flanges investigated; these results are listed in
Table 3.4. The superiority of the welding neck
flange is in line with service experience with regard
to suitability for critical service.
The relatively poor performance of the lap joint
flanges was rather surprising since such flanges have
a fairly good service record; the lap thickness used
was the same as the pipe wall and the poor results
were attrihuted to inadequate strength of the lap
to carry the high bending moments imposed, the
lap apparently rocking back and forth on the gasket.
In general, stress intensification factors increase with
increasing abruptness of cross-sectional changes in
the flange at the pipe connection. The welding neck
flange with its smooth transition exhibits no perceptible stress-raising tendency, whereas threaded
flanges, due to stress concentrations present in the
threads, carry an intensification factor of about 2.30.
The effect of a seal weld covering all exposed threads,
as used in some services, was not investigated. It
should also be kept in mind that in elevated-temperature service the load distribution on flange
details involving double welds wonld be less favorable, and that additional thermal stresses would
result from temperature differences between the
pipe and flange.
For services where creep or severe cyclic effects
are present, greater attention must be paid to the
reduction or elimination of stress raisers. Fillet
radii should be generous, and sharp corners should
be avoided. Stud bolts with continuous threads or
with unthreaded portions machined to the root diameter should be used in preference to headed bolts,
which involve sharp fillets under the heads and the
77
Table 3.4 Stress Intensification Factors for
Various Flanges
Welding neck flange
Socket welding flange (double welded)
Slip-on or forged ring flanges (double welded)
Slip-on or socket welding flanges (single welded)
Lap joint flanges
Threaded flanges
1.00
1.15
1.25
1.30
1.60
2.30
thread runout. For satisfactory performance, bending in studs should also be held to a minimum.
3.n Bolted .Flangcd Connections: Practical
Considerations
Experience indicates that the design rules of the
ASME Unfired Pressure Vessel Code are generally
entirely adequate for the design of special flanges,
with gaskets located inside the bolts, for service
under internal pressure. For flanges having full
face gaskets, or for any design which permits the
development of a counter-moment reaction outside
the bolt circle, there is no recognized standard design approach; a special ASME Code Committee is
currently (1955) working on tbis problem.
For piping applications it is necessary to consider
the effect of other loadings in combination with internal pressure. These are usually longitudinal
forces and bending or torsional moments due to
weight, wind, or thermal expansion of the pipe line.
By far the majority of piping flange applications in
the United States utilize ASA Standard B16.5 flanges.
The ASA Standard gives allowable pressure-temperature ratings but offers no guidance as to permissible
bending loadings. These flanges have been customarily -used up to the allowable ratings without
any check of their capacity to carry additional loadings. While occasional difficulties due to such loads
have been encountered, their service record must he
considered very good. Unsatisfactory performance
occurred generally only with pipes having an appreciable excess strength or corrosion allowance and a
high value of thermal expansion. An example is the
use of 150 lb standard flanges with low-pressure,
high-temperature heavy-walled pipe. A discussion
of the influence of bending and torsional moments
on ASA flanges is included in the paper by Rossheim
and Markl [29]; results of tests on 4 in. 300 lb standard flanges are given in the paper by Markl and
George [87]. ASA flange strengths, when judged by
ASME Code analytical methods, are by no means
uniform, and the piping designer shonld be aware
that there is greater reserve strength in the smaller
sizes and lower-pressure classes than in the larger
sizes and higher-pressure classes. Good design prac-
78
DESIGN OF PIPING SYSTEMS
tice indicates the desirability of keeping flanged
joints at a minimum for operating and maintenance
conditions and, insofar as possible, locating needed
flanges in the most advantageous location with respect to applied moments.
For routine design investigations of the effects of
loading other than internal pressure on flanges
covered by ASME Code rules, The M. W. Kellogg
Company has found it" is satisfactory to calculate
first the maximum load per inch of gasket circumference due to the applied longitudinal bending
moment and force. Then the internal pressure equivalent to this loading is determined. Finally the
Code design approach is applied, assuming an
internal pressure equal to the design internal prese
sure plus the calculated pressure equivalent to the
other loads. The equivalent pressure, expressed
in formula form, is: P. = (16MhrG3 ) + 4F/"G2 ,
where M = 10llgltudiilal nending moment in.-Ib; G=
diameter of effective gasket reaction as dcfincd by
Code, in.; and F = longitudinal force, lb. The
flange is checked for a pressure of P = P. + Pd,
where Pd = design operating pressure. Taking
the moment on the gasket center line is consistent
with analysis and experience which indicates that.
with a properly pretightened flange, the bolt load
changes very little when a moment is applied,
whereas the gasket loading changes appreciably.
The most important point for practical design is
to establish a proper allowable stress·for such checks.
For steady loading other than thermal expansion the
same allowable stress as for internal pressure alone
should be used; for temporary (short duration)
loading, an increase of 33.3% in the basic design
stresses is suggested.. Loading due to thermal expansion can be treated on a "stress range" basis
similar to the treatment of thermal stresses in the
pipe itself; allowable stresses for both bolts and
flange can be established accordingly. From a stress
standpoint there should be no question about this
procedure. From a flange leakage standpoint the
validity of this approach is somewhat questionable,
particularly under creep conditions. Nonetheless,
it is only by such an approach that thc dcmonstratcd
capacity' of flanges to take rather sizeable moment
effects ean be reasonably justificd. Thc M. W. Kcllogg Company's satisfactory experience in checking
thermal moment effects has becn bascd on an allow..
able stress for both flange and bolts of ~ (S, + Sh)
as prescribed by the 1951 ASA Standard B31.1 Codc
for piping
With adequately pretightened flange bolts the
thermal moment appears during the first heating
cycle. If the operating temperature is sufficiently
high, both the bolt stresses and thermal flange
moments will gradually relax due to creep in the
pipe line; leakage while hot would then depend on
the relative relaxation ratcs. Assuming substa';tial
hot relaxation, a sizeable thermal moment of opposite sign would develop when the line returns to
atmospheric temperature, and leakage may occur if
pressure is applied in the cold condition. Thus there
are probably factors in the problem not adequately
assessed, and whether the increased stress range now
permitted by the Code can be applied to flange design without affecting tightness is not established.
The influence of torsional moments may be investigated as indicated in the Rossheim-Markl
paper [29]; as shown therein the capacity of ASA
flanges to take torsional moment is less than for
10!1gitudinal moment. This is generally true of all
flanges, unless special mechanical means such as
dowels or keys for transmitting torsion are provided;
caution is therefore in order when high torsional
moments may be imposed.
Generally, however,
flange leakage is not as much of a problem under
torsion as it is under bending.
Occasionally, special applications may warrant a
more extensive study of stress-strain relations in the
flanged joint. This may be done by adapting the
approach presented by Wesstrom and Bergh [83],
and Blick [88, 89].
When selecting gasket dimensions for hot flanges
it is well to make the gasket as wide as can be satisfactorily seated by the initial ASME Codc dcsign
bolt load, rather than use a narrow width which will
just avoid extensive initial yielding or "crushing./l
This will insure maximum resistance to creep under
operating conditions.
In petrolcum service applications the tightncss
performance of high-temperature flanges is usually
improved by leaving the flanges uninsulated and
providing a weather shield only. The flange and
holts then operate at a lowcr metal temperature and
relaxation is slowed. Where heat loss requires it,
the shicld can be lightly insulatcd. In powcr piping,
however, heat loss is much more a matter of concern
j
making full insulation gcncrally a ncccssity.
By ·fa.· the greatest leakage troubles with flanged
joints arise due to rapid temperature changes or
quenches which create sizeable tcmperature differcnces within thc flangc componcnts. Whcrc thcse
conditions can be anticipated, flanges are preferably
avoided; if used, great care is warranted in their
selcction and location. The mating of dissimilar
typcs of flangcs, such as Van Stone and integral
LOCAL COMPONENTS
types, generally tends to exaggerate diffieulties
arising from temperature differences.
The problem of dissimilar flanged joints is briefly
discussed in Section 3.12.
3.12 Joints Between Dissimilar Materials
Individual piping systems may involve more than
one material, or may be eonneeted to equipment
of different metal analysis, so that the influenee
of intermediate or terminal joints between materials
of different physieal properties must be considered.
A principal faetor influeneing these dissimilar joints
is the difference in expansion characteristics; others
are variations in hardness, electrolytic potential,
structure, duetility, and stiffness.
The transition in material may oceur at a bolted
flange, a threaded eoupling or union, a rolled eonnection, a weld, or at a special transition piece.
Potential diffieulties in the form of leakage or failure
due to thermal eycling are related to the range and
frequeney of temperature ehange, the differenees in
material properties, and the details of the dissimilar
joint.
For low-temperature service, all of these types of
eonnections are successfully used. As the temperature is raised, threaded and rolled joints require
that the higher expansion material be inside. With
further increasing temperatures, the elastic stress
interaetion is no longer maintained; this leads to
leakage and finally structural failure. Seal welding
does not permit appreciably higher temperatures, as
struetural strength is not usually improved. All of
these forms of connections involve significant stress
raisers and localized areas of high deformations.
Hence, where repetitive cycles of wide temperature
range are involved, the probability of fatigue failure
dictates against their use.
For flanged joints consisting of integral buttwelded flanges of dissimilar material, (but each of
like analysis with the pipe to which it is attached),
leakage is dictated by the initial bolt stress and
gasket sealing properties, as compared with the
differential expansion between the bolts on the one
hand, and the flange and gasket on the other.
With certain types of gaskets, leakage may also
depend on the differential radial expansion at the
gasket center line. By using lapped or Van Stone
flanges, the flange and bolt temperatures are reduced, and the flanges and bolts can be made of the
same type of material, with differential expansion
limited to the Van Stone lips and gasket. The gasket
load must be sufficient to restrain the relative radial
movement of the lips, or the gasket must be of a
79
design which will maintain a seal under differential
expansion movement. While severe local stresses
proportional to the restraint are eaused in the lips
(which must also carry the bending due to pressure
and structural loading) , these stresses can and should
be confined to wrought, earefully contoured material,
so that reasonable fatigue life and satisfactory tightness can be obtained for high temperatures. Dissimilar flanged joints of special design using bellowstype gaskets or pressure sealing have been utilized
in power plant service [90J. For joints between
ferritic and austenitic steel, the design selected should
be such as to permit the use of ferritic alloy bolts;
austenitic bolts are unsatisfactory because of their
low yield strength and high coefficient of expansion.
Bolted joints generally are avoided for extremetemperature service, a construction not involving
gasket sealing being preferred. With welded eonstruction, leakage is no longer a factor. Satisfactory
service in this case depends upon the local strainrange differential, on the number of applications of
this strain range, and finally, upon the metallurgical
factors assoeiated with presence of a weld.
Using the most common material combination
as an example, the austenitic steels have a coefficient
of expansion of about 10.4 X 10-6 in./in./F (within
the temperature range 70-1200 F), whereas the
low-chrome-alloy steels have a coeffieient of about
8. X 10-» in./in./F. When joined with a sharp
interface, this difference will induce stresses when the
joint is heated or cooled. For a butt joint between
pipes of equal thickness the maximum thermal
stress will be the circumferential (hoop) stress
developing at the junction, whieh has a magnitude of
u = !E flT flo:
(3.25)
Here E represents the modulus of elasticity (assumed
identical for both metals), flT the temperature
change, and flo: designates the differenee between the
eoefficients of expansion for the two metals. Applying a stress-relief at 1200 F to the junction and cooling the pipe then to 100 F, flT becomes 1100 F,
flo: is equal to 2.4 X 10-»IF. Taking now E =
29 X 10· psi, a eircumferential stress of 38,300 psi is
calculated (tensile in austenitic material and
compressive in ferritie material).
The foregoing analysis assumes a thin eylinder
and evaluates only the differential radial deflection
at the mean radius. In addition to this effect, there
is also a discontinuity at the interface due to the
differential ehange in thickness of the two materials
whieh will introduce further loeal radial stresses of
equal magnitude, tensile in the one material and
80
DESIGN OF PIPING SYSTEMS
Al1denitic
Austenilic
Stelll
Weld
changing the weld bevel so that the heat-affected
zone on the ferritic side was more inclined (Fig. 3.17).
These tests did not, however, include the effect of
differential expansion stresses, so that improved
performance resulting from inclining the ferritil.:
heat-affected zone may in this case be attributed
largely to the lower axial stress component in this
Inside and oullide 9.ou"d smooth gnd
wbstanliolly lhllh cher welding
FIG.3.17
Dissimilar weld joint, 15° interface.
compressive in thc other. The resulting state of
equal biaxial stress will raise the calculated stresses
by a faetor of 1/(1 - v). A more detailed analysis
would disclose some edge-bending stresses across the
junction.
A simple butt weld approaches the above assumptions and appears at least the equal of other possible
details from a standpoint of stress magnitude. Stress
is, however, only one factor determining performance; its influence must be weighed along with that
of other factors. One of these factors is the particular
detail of the joint. For example, eonnecting an
austenitic pipe braneh to a ferritic header would
force almost all of the differential strain into the
austenitic part. This would increase the maximum
stress by a factor of about 3.6, uS compared to the
simple butt weld.
Ai, in the ease of overall thermal expansion strains
in piping systems, the performance of dissimilar
joints would be dependent on the number of cycles
and the strain range per cycle. While such joints
generally give satisfactory service in constant
temperature operation with relatively rew temperature cycles and an absence of sudden quenching
conditions, many joints subjected to more severe
eonditions have failed. Investigations conducted
on this subject [27, 91, 92, 93], and experience
indicate that metallurgical factors and flaws seriously
affect performauce when associated with plastie
deformation due to yielding or creep. The heataffected zone on the ferritie side of the dissimilar
weld has been shown to be the most eritical zone,
due possibly to reduced duetility of the mixed analysis in the fusion zone and metallurgical ehanges
during the course of the test or serVICe whieh result
in strain concentration at this location. From a
stress standpoint the superposition of internal
pressure loading and external longitudinal loading
reduced the number of eycles which could be withstood in a hot fatigue test at constant tcmperature.
While all welds withstood a large number of cycles,
a somewhat improved performance was obtained by
critical zone.
When Carpenter et al. [92J changed from a hot
fatigue test to a thermal quenching test, the number
of cycles which could be withstood dropped drastically, failures being experienced after76 to 318 cyclcs,
as compared to 89,000 or more for the hot test.
Weisberg and Soldan [93J, in a separate series of
tests, obtained no failures after 100 quench cycles.
Under such drastic quenching appreciable additional
thermal stresses are introduced because of transient
temperature differences across the wall thickness.
Thc possibility of improving the performance of
dissimilar joints by inclining the bond line originated
with Thc M. W. Kellogg Company and was aimed
at a largely longitudinal interface rather than one
transverse to the pipe axis. This method has becn
incorporated in the design of special joints made by
the Kelcaloy process (Fig. 3.18). Tests [91, 93] and
service experience over a period of six years show
excellent performance. Whereas differential expansion strcsscs are likcly to bc somewhat higher
than in a simple butt joint, the primary advantages
over a conventional weld or a wcld simulating thc
construction lie in the essentially longitudinal
interface and in the unique manufacturing process23
Thc physical properties of metal deposited by this
process are consistently superior to those obtained
23Described by Blumberg and Bunn in their discussion of
Weisberg's paper·IOI].
-;
-'-- ~~ - - -. ; -._'-..,..I
In'erface
Av"enitic Slool
\
f. rrilic 110el
I
I
I
I
- - - - I- -,-.-:!>-L
Ferrai, Sleel
Ir'orfaco
,
AV"(Ini~ic Slool
I
P= f-f-L-.-::_...:--.: A r:L 1-FIG. 3.18
i
-1------ -i-
.
-
--- '---
Kelealoy transition pieces.
LOCAL COMPONENTS
by ordinary casting or welding mcthods. Austcnitic
sections, for example, are significantly free of micro-
fissuring. In addition, progressive"'and rapid solidification around the entire circumference of the bond
zone occurs simultaneously, resulting in greater uniformity, minimum residual stresses, and less acute
material transition and heat-affccted zones.
The Kelcaloy process is also bcing used to produce
joints with a simple butt bond substantially transverse to the pipe axis. Their principal advantage
over welded joints again lies in the metallurgical
superiority and relative soundness inherent in the
process of manufacture. Additional advantages
are: only one heat-affected zone compared to two in
a conventional weld, and adaptability of the process
to produce and closely control special chemical
analyses. AB an example of the latter advantage,
carbon migration at the interface (which has been
experienced at ferritic-austenitic junctions and
which hastened some failurcs) can be combated by
introducing a carbide stabilizer, such as columbium,
into the chrome-molybdenum steel, leading to an
analysis which is not generally available.
While this discussion has emphasized that metallurgical aspects greatly influence dissimilar weld
performance, detailed discussion of this subject is
not within the scope of this book. Principal factors,
however, can be listed as follows:
1. Carbon migration resulting in a carbon-depleted zone in the ferritic steel near the austenitic
weld interface.
2. Formation of sigma phase in the austenitic
matcrial near the interface.
3. Abrupt change in structure and physical properties of weld metal and heat-affected zonc resulting
in a lImetallurgical notch."
4. Tendency of austenitic weld deposits toward
81
type of design and fabrication technique is decided
upon, every effort should be made to eliminate
mechanical stress raisers.
\Veld reinforcement
should be built up to effeet an anneal of the preeeding
layer and then should be removed without notches
or other surface stress raisers near the weld; machining is preferred where praetieable. Baeking rings
should be avoided, and controlled inside contour
welds (K-Weld) should be used where back welding
eannot be aeeomplished. The joint should be examined for soundness by the best nondestruetive
methods applieable.
3.13
Other Components
Various components other than those speeifically
mentioned in individual sections of this chapter
may be encountered, but insofar as their influence on
the flexibility and fatigue performance of the system
is eoneerned, the prineiples outlined in this chapter
usually ean be applied as neeessary. Some deserve
at least brief additional eomment.
A valve should be able to earry loading from
attaehed piping similar to a standard tee of eomparable pressure-temperature rating, but as a further
consideration should be suffieiently free from warpage or distortion to pernait operation and tight
shutoff. These problems are within the provinee of
the individual manufacturer; their engineering has
advaneed eonsiderably in recent years, particularly
for high-pressure, high-temperature serviee. As
in the ease of flanges, eare should be exercised in using
low pressure rating valves with relatively heavy
walled pipe, since the imposed moment as erected
or due to expansion may be beyond their structural
capacity. For large-diameter piping, valves are
sometimes used with venturi ports (partieularly
when motor operated) or standard valves one or
micfofissuring.
two sizes smaller are used with reducers in conven-
5. Oxidation or other corrosive notching at the
ferritic material junction accelerated by local strain.
The discussion in reference [92] will be found interesting and instructive. There is still a great deal to
be learned about austenitic and dissimilar-joint
tional lines. The bodies of venturi valves are usually designed with consideration for the moment
which may be applied by the larger piping. Similarly, when using standard smaller valves, sueh
welding and the service performance of austenitic
Flanged jittings of either cast or wrought steel
(ASA Standard BIG.5) are eapable of developing the
full struetural strength of their flanges; for further
eomments on the moment eapaeity of flanges see
Section 3.11 of this ehapter. This applies to tees,
crosses and elbows. The pressure rating of such
fittings is given in the ASA Standard and commented
on in Chapter 2.
Screwed and socket welding jittings (ells, tees, crosses,
unions, and couplings), whether cast or wrought
welds. The same can bc said in general about the
high-temperature performance of the heat-affected
zones of all types of welds under plastic dcformation
and creep conditions. Where weld difficulties have
been encountered in service, the preponderance of
cracking has been associated with heat-affected
zones.
In important praetical applieations of dissimilar
joints for high-temperature service, no matter what
piping moments must be given consideration.
DESIGN OF PIPING SYSTEMS
82
M = Applied Moment (in lin.)
...---------..
j'
"
t,
(')
t,
p
inlerflOl
preuure
psi
a
(I)
•
"--- M -----f = Ex1cmol axial end force (lbs.)
(potitive in diredion shown)
(Totol end force = F plus hydrostatic
end load due 10 pr~sure p)
FIG.3.19
Conical transition.
steel, are similarly furnished to nominal pressure
ratings and are limited to small sizes and generally
to moderate service conditions. Screwed joints are
obviously limited in their capacity for transmittng
torsional moment. Actually, in plumbing practice
they are often relied upon to relieve thermal expansion by permitting a small angular rotation of
one tbread upon the other, a practice, however,
which not infrequently results in leakage. In tension or bending, screwed fittings ean be depended
upon to be equal in strength to unthreaded pipe of
the same rating, but due allowance for metal removed
in threading must be made in determining the wall
thickness of the pipe. For cyclic effects (mechanical
or thermal), screwed joints involve the stress-raiser
effect of the threads, an effect not entirely eliminated
even with heavy seal welds. At the higher temperatures, seal welding is usually necessary to prevent leakage. Socket welding fittings also involve
the stress intensification effect of fillet welds, but are
snperior to threaded joints if the welds are adequate.
Threaded and socket-welded fitting joints would be
expected to involve stress intensifications in line
with the fatigue test resnlts obtained for flanges of
threaded and single-welded socket types.
The thicknesses of blind flanges in ASA Standard
B16.5 are the same as those of functional flanges.
For nonstandard cover plates, the rules of the
ASME Unfired Pressure Vessel Code may be used;
it should be mentioned, however, that the ASA blind
flanges are of lesser thickness than those resulting
from the application of these rules. EUipsoidal
welding caps are commonly used, as presently covered by ASA Standard BI6.9. Individual applications for larger sizes or special shapes, including flat
heads, may be checked using the rules of the ASME
Unfired Pressure Vessel Code.
Markl [12J has fatigue-tested smoothly contoured
commercial red1.l.cers and finds a stress intensification
factor of unity justified. For the design of special
conical reducers reference can be made to the interpretive report of the work of the Design Division
of the Pressure Vessel Research Committee [94] on
pressure vessel heads. For the particular case of a
sharp cone-to-cylinder junction the local stresses
at the intersection can be closely approximated by
using the familiar beam on an elastic foundation
analysis and treating the cone as though it were a
cylinder having a radius equal to the meridional
radius of the cone at the junction. The resulting
stresses due to an internal pressure p and external
loads F and M are given by the following formulas
(with Poisson's ratio >, taken as 0.3):
Ontside intersection (Point (1) in Fig. 3.19)
'F1.816C3 + (pR/2tn cos a)
I
+
l +
l c, +
I +
Sll
Cone
=
s" = -C, + (pR/tn cos a) 'I' 0.546C3
SI =
Cyl.
'F1.816n'C3
s, = - C2
(pR/2t)
(pR/t) 'I' 0.546n'C3
Re-entrant intersection (Point (2) in Fig. 3.19\
SI
= ±1.816C3 + (pR/2t,n cos a)
+ (pR/t 2n cos a) ± 0.546C3
Cone So: =
SIZ
Cyl.
=
S,2 =
± 1.816n'C3
C2
(pR 2 /2t,)
2
(pR 2/12) ± 0.546n C3
In these equations upper signs refer to the stresses in
the outer fibers and lower signs to those in the inner
fibers. The subscripts I and c refer to longitudinal
and circumferential stresses, respectively; a positive
sign denotes tension. The constants appearing above
are given by the following expressions:
C, = ~'l [C5( vncosa+ ~,)
- C. (2vn cos a + 1 +
1~') J
c, = ~4 [C 5 (vn cos a + ~2)
+ C. (n2 + 1 + V n 2cos a
)J
LOCAL COMPONENTS
1
C3 = -z- [C.(Vn cos a + 1) + C.(n Z - 1)]
n C.
.~
C. = n Z + lz + 2
n
(vn cos a + + Vncosa
1
1
)
VR[PR
F ..Rz
MJ tan a, for mCS = 2.57 F
2 + 2..R+
tersection (1)
=
J
.
v'R.[PRz
F + -M tana form257 - - - + -. t2 1.5
2
2..R z ..R?
'
tersection (2)
C.= 0.85 pR (1
t
= -0.85 pR z
t2
1_) for intersection (1)
n cos a
(1 __1_)
for intersection (2)
neosa
n = t,/t for intersection (1)
=
tdt-, for intersection (2)
It has been assumed in the above that the intersections arc far enough apart (about 2
~cos
RI, min) so
a
that their local effects do not influence each other
,ignificantly. The maximum fiber load due to the
external moment is taken as though it were uniform
around the circumference; this approximation is
considered to be on the safe side.
The special case when I, cos a = t is of interest
for intersection (1). When n = llcos a the stress
formulas for this intersection reduce to:
z
Sll = 'f'3.63C7 cos a + (pRI2!)
Cone (
s" = - (1 + cosz a ± 1.089 cosz a)C, + (pRlt)
'f'3.63C, + (pRI2/)
s, = - (1 + cosz a ± 1.089)C, + (pRI!)
SI
Cyl. (
=
where C 7
2.57 sin a cos a
= 1 + 6 cosz a + cos" a
[VRJ[PR F M2J
F 2 + 2..R + ..R
For consistent treatment with other stress intensifications in the ASA Code for Pressure Piping, rolledpipe data should be chosen as a basis of comparison.
Therefore, the calculated maximum stress as given
by the above formulas should be divided by two when
comparing with the usual expansion stress limits.
3.14
Piping and Equipment IntcrctTccts
In the over-all picture a piping system and the
mutually eonnected equipment, structures, founda-
83
tions, and soil constitute an integrated structural
system with equilibrium of interloading effects.
Each part of this structural system is influeneed by
its individual environment, e.g. pressure, temperature, weight, etc. ,. as well as by the effects transmitted
from attached parts of the system.
Ordinarily, supporting structures, foundations, ami
soil are subject only to ambient temperatures, and
are sufficiently rigid so that deflections under pipe
expansion, etc., are small enough to be neglected.
Sometimes, however, temperature rise is unavoid~
able in .steel structures; slender or high structures
may also, in combination with their foundations,
involve significant deflections under even moderate
reactions. Connected equipment will undergo dimensional changes which may augment or decrease
the thermal expansion loading. The fabrication and
assembly of such an integrated structural system
necessarily involves deviations from nominal dimensions. Hence the fitting of piping, in combination
with weld shrinkage, sets up initial internal stresses
which at the weakest location of the system may
equal the yield strength. All such conditions must
be recognized and provided for by the piping
designer.
For simplification in analysis, the ends of a piping
system are usually considered fixed at the equipment
connections. Obviously, this is a limiting condition
for the maximum reaction; localized bending or direct
loading of the equipment, by causing deflection or
rotation, serve to reduce the piping reaction. The
result will be an intermediate fixity between fixedand hinged-end conditions. While the conventional
assumption of complete fixation may seem unnecessarily severe, it must not be inferred that excessive
additional safety results. It is possible to deviate
from fixed-end assumptions without increased risk
of fatigue damage to connection equipment only
when analysis is made of the bending stresses in the
equipment whose localized deflections are being
utilized. When dealing with rotating or other equipment where alignment is sensitive to distortion,
piping can seldom be permitted to exceed the stiffness obtained with fixed-end assumptions.
Considerable misunderstanding on the part of
equipment designers relative to piping reactions has
existed in the past. Manufacturers sometimes have
made it a condition of their warranty that 1W piping
reactions be transmitted to their equipment. In
other cases, forces have been limited to unreasonably
low values, while completely ignoring the more important effects of bending moments. Such impractical
criteria, however, are detrimental to all concerned,
84
DESIGN OF PIPING SYSTEMS
including the equipment manufaeturer, as the piping
designer is left without a usable guide.
reaetions on sueh equipment have been suggested.
These are no more than rules of thumb which repre-
At present, the more progressive manufacturers
~ent experience alone without supporting analysis.
Various factors are used as indices, either individually or in combinations, such as weight, cross section,
suction or discharge nozzle sizes (and rating), cubic
volume, and pressure shell thiekness-to-diameter
ratio. For many years piping reaction limits of
1000 to 3000 pounds were speeified regardless of
size or details of the equipment involved. Sueh
limits are now generally considered meaningless.
Other approaches relate metal cross section to
rlifferent unit values for resultant forces and moments. Pressure shell thickness/radius ratios are
also employed sometimes to establish the potential
maximum pressure whieh that portion of the shell
can withstand. A morc accurate evaluation of
local moment capacity of surfaees of revolution is
given elsewhere in this section.
Cubic volume is not a significant parameter. It
has been used largely in the absence of equipment
weight, by assuming an overall density 2 to 5 times
that of water. Weight, where obtainable, is a more
suitable parameter and is usually inereased by the
estimated weight of the contents. In either approaeh, the weight is eonsidered as the maximum
value whieh the resultant foree may attain.
Suction or discharge nozzle sizes provide a more
eomprehensive index of rotating equipment design,
sinee they reflect equipment size. Pressure may be
assumed to maintain a rough balance between pipe
and equipment stiffness. Based on a survey of
acceptable piping designs for equipment piping,
Rossheim and Markl [29] proposed the eube of (pipe
aD plus 3 in.) as a eriterion to whieh eonstants were
applied to establish the maximum ax-ial and lateral
forces in pounds or bending moments in foot pounds.
attempt to provide more realistie load carrying capacities or offer to check their equipment, on large
critical units, for the reactions of the proposed piping.
The problems which they face should he appreciated
by the piping designer. With moving parts and the
need for close clearances, all strain must be carefully
controlled to avoid misalignment, rubbing, binding,
excessive wear, or other maloperation. At the same
time, the iuvolved and discontinuous contours usually
required are not amenable to a reliable evaluation of
either deflections or stresses. Thus, the manufacturer is often forced to rely on the judicious use of
experience or the projection of occasional test data
to individual equipment.
The magnitude of effects at sensitive equipment
should he kept low particularly when cost is not
significantly increased hy so doing. This objective
may be approached by providing local pipe flexibility in complex systems to favor the equipment,
using local restraints to take reactions directly or
to force deflection into other portions of the system
(discussed further in Chapter 8); by an over-all
increase in the flexibility of the system; or by a
favorable relative positioning of the equipment.
The potential influence of effects on equipment may
often be moderated by the selection of types whieh
are relatively insensitive to distortion and misalignment. Also, the location of equipment should he
such as to keep pipe sizes to a minimum insofar as
praetieable, a point of partieular concern in regard
to the larger lines (e.g. pump suction and driver
exhaust lines).
In the absenee of suitable manufacturer's data or
applicable experienee, approximations for limiting
Table 3.5
Allowable End Reaction Exerted b:y Connected Piping on Pumps, Turbine Casings, and Pressure
Vessels
\Volosewick for Maximum Temperature 650 Ft
Type of End
Reaction
Forces,lb
Rossheim-
Markl'
Actual
Value
l\Iaximum
1.50D'
250D
100D
60D'
2700D
Moments, in.-Ib
Radial reaction, including weight of pump riser,
etc.
Tangential reactions, any direction .
Longitudinal bending moment .
Circumferential bending moment
Twisting moment
4-Point Support
3.25D'
2-Point Supports
Actual
Value
Maximum
4,000
1,500
300D
40,000
1700D
2,700
900
10,000
22,000
18,000
Allowable
85D
Allowable
·In this column D denotes the OD of the pipe increased by 3 in.
tD is equal to the sum of the nominal diameters of suction and discharge decreased by 15% for every 50F increase over 650 F.
85
LOCAL COMPONENTS
Wolosewick [95J additionally varied allowable reactions to suit the type support (2 to 4 point), and
service temperature. The limits advanced in these
two papers are tabulated in Table 3.5.
The Rossheim-Markl study also brought out the
fact that expansion stresses in the piping studied
ranged from 1000 to 6000 psi. This and subsequent
experience led to the following practice used with
success by The M. W. Kellogg Company for the
past five years. The combined stress due to bending
and torsion is calculated for an assumed pipe having
a size and wall thickness equal to that of the nozzle
and connecting pipe, respectively. This stress is
limited to 6000 psi.
In establishing limits for pipe loads, consideration
must also be given to the capacity of equipment
supports: that is, anchor bolts, bed plate, steel structure, and foundation each in turn must be able to
accommodate the pipe loads.
The effect of localized concentrated forces and
moments on shells is of widespread importance in the
design of piping. The resulting bending and direct
stresses and their effect on fatigue life are important
factors in establishing satisfactory structural design
not only for tees, branch connections on pipes, and
nozzles on pressure vessels, but also for supporting
saddles, lugs, trunions, legs, hangers, and similar
attachments. Due to a lack of symmetry and
variation in cross section, the theoretical analysis
of these local effects is not only laborious but, up to
the present time, has been accomplished only for
special limited cases.
An intensive investigation of the problem of local
loadings on cylindrical shells was begun in 1952
by a spccial subcommittec of the Prcssure Vessel
Research Committee Design Division. The first
results of this program were presented in P. P.
Bijlaard's papers [51, 96J dealing with the effect of
radial loads and local moments, and evaluating the
case of a localized uniformly distributed radial load
acting over a finite area of a cylindrical surface.
.\. comparison of analytical rcsults [51} with values
extrapolated from experimental results available in
the literature [97, 98] shows reasonable agreement.
An additional theoretical treatment covering the
application of local circumferential and longitudinal
bending moments has reccntly bccn published [52J.
These investigations, as well as experience on the
behavior of surfaces of revolution under localized
effects, provide a general understanding of the moment distribution and stress patterns attendant to
such loading. Individual analyses, however, arc
exceedingly lengthy and involved, so that the aim is
L
to provide simplified approaches with a reasonable
understanding of the extent of their deviation from
more accurate solutions.
Along this line, The M. W. Kellogg Company has
made use of an approximate solution based on the
bending of a beam on an elastic foundation, to evaluate the local shell stresses resulting from a nozzle
bending moment, or radial thrust on a cylindrical
or spherical shell. The piping moment is simulated
by a uniform circumferential radial line load equal
to the maximum reaction (lb per linear in.) at the
edge of the nozzle neck. With this unit load, the
shell bending and resulting stress is established
similar to the effect of a narrow shrink ring loading,
the unit moment being applied to the shell thickness
or combined shell and pad thickness where reinforced. The formula used in this approach is:
S =
1.l7VR
t1.5
[F, + 1.5F,]
(3.26)
where S = local longitudinal bending stress in
shell, psi.
R = meridional radius of shell, in.
t = effective local thickness of shell, in.
(shell thickness plus reinforcing pad
thickness).
F 1 = unit loading due to applied longitudinal2 ' bending moment (Ib per linear
in.).
= M (1rR n 2 where M = moment, in.-lb,
and Rn = mean radius of nozzle connection.
F 2 = unit loading due to a radial thrust, lb
per m.
=' P(21rR n where P =
total thrust, lb.
The combined local stress due to thermal reactions
and internal pressure is held to the same total allowable stress range as for the piping itself [1.25
(S, + SI/ )J. As noted previously, the thermal reactions must be based on the full expansion and the
cold modulus of elasticity. The individual hot and
cold reactions cannot be used for this purpose,
While this check is not precise, it has resulted in
safe designs over a considerable period of years. At
least, it provides a simple method for consistent
design and, when more precise methods are developed, it will afford a basis by which the results of
the new proposals may be assessed in terms of past
experience. There is one further interesting observa24For a. circumferentio.l moment it is believed thnt the
bending stress (circumferential) may 'be in some cases up to
several times that indicated for a. longitudinal moment.
DESIGN OF PIPING SYSTEMS
86
tion which can be made. If a moment loading only is
1.11VR M
assumed, eq. 3.26 becomes S =
;.5
X --2' If
t
"lrR n
the thickness of the nozzle is t n this may be written as
S = (1.17Vlr~~5)
t
X-rrR
~t
n
moment, making no correction necessary for such
n
The portion of this expression enclosed in brackets
represents the stress intensification factor as used
in the Piping Code. For identical shell and nozzle
thicknesses the factor reduces to
(R/t)l'. The
similarity of this to the factor suggested by the Piping Code for full-size tee intersections, 0.9(R/t)%, is
noteworthy.
The same shrink ring loading approach has been
used by The M. W. Kellogg Company to determine
the rotations of nozzle connections due to the local
deflection of the vessel shell under the influence of
1.17
piping moments, viz:
</> =
2.46M
E
[!!:...-J)'
r t
m
2
Thus, as shown in Section 5.13, the rotation of the
nozzle becomes expressible in the form of shape coefficients which can be added to the usual summation"
to obtain the equations. No deformation of the shell
plate is assumed to occur as a result of torsional
(3.27)
where
cP = angular rotation, radians.
M = moment acting at nozzle, It-lb.
r m = mean radius of nozzle, in.
R = radius of vessel, in.
t = thickness of vessel shell, including reinforcing pad, in.
E = modulus of elasticity of shell material,
Ib/in 2
loading.
An influence which is sometimes neglected, save
for its effects on the piping proper, is end pressure reaction. Sometimes a piping system is referred
to as an open end cylinder" with the implication
that longitudinal pressure stresses are absent. Thi,
would be true only in straight runs between infinitely
stiff vessels which undergo no deflection under the
H
pressure reaction of the pipe cross section, or where
frictionless expansion joints are provided.
With the introduction of expansion joints, or
other provisions incapable of transmitting the full
longitudinal pressure stress, the end pressure reac-
tion at each fitting, or opposite eaeh nozzle on a
vessel, must be considered in the overall structural
design. Provisions for carrying this unbalanced
reaction must not.adversely affect the freedom of the
joint to absorb intended movements. Further discussion on this point will be found in Chapter 7.
Piping Reaetions. When a piping system is designed to the Piping Code, the maximum expected
hot and cold reactions, and the reaction range due
to thermal expansion, are established by the Code
rules (sec Chapter 2) for the design of anchors and
checking of terminal equipment.
Erection stresses
and hence the initial cold reactions are not included,
As is the case with the preceding treatment of
however, since these are related to fabrication and
stress, this equation for rotation is connected with a
erection details and cannot be predicted in magnitude or sign unless adequate means are provided for
their control. Upon heating, the initial effects com-
longitudinal moment. Where this moment acts in a
circumferential direction, tests indicate that the
flexibility may be several times greater. However,
the loealized stress is not likely to be lowered, so that,
for the sake of simplicity the same approach is used
for moments in either direction. This approximation
enables the piping designer to deal with partial end
fixation by introduction of a virtual length as follows:
The end rotation of a cantilever subjected to a
moment applied at the end is
<I> =
144ML/EI
(3.28)
where M and L have dimensions of ft-lb and ft
respectively. Equating this expression to eq. 3.27
yields
L = 0.017 I (R/r m 2 t)"
(3.29)
where L represents the virtual length of a fictitious
extension having the same rigidity as the pipe line.
bine with those due to expansion, their magnitude
being limitcd by the yield point at the service
temperature. Subsequently, relaxation lowers the
maintained service stress as dictated by the material
creep properties and the relieved strain reappear.
as a cold stress at ambient temperature, the adjustment of strain between service and ambient conditions being termed "self-springing." The Piping
Code cold reaction reflects this self-sprung state and
does not consider the initial (as erected) condition;
thc Code hot reaction on the other hand reflects
the maximum expected reaction (without erection
stresses) rather than the final reaction after adjustment.
Careful erection and, in particular, controlled prespringing before service can be used to limit the
maximum reactions, by assuring their occurrence
LOCAL COMPONENTS
predominately or entirely at the ambient temperature. The Piping Code rules for reactions allow for
the effect of prespring, which gaina practical significance in this respect only when it is 50% or greater.
Emphasis on the provision of maximum prespring,
approaching 100%, is usually limited to large critical
equipment where maximum assurance against possible distortion at high service temperatures is
essential.
Where a piping system is designed to meet limiting reactions, or for other reasons is to operate at
stresses of a low order, (as is the case with large
turbines, compressors, etc.), the magnitude of the
initial cold reactions as erected may be many times
that of the reactions corresponding to expansion, if
special procedures are not followed. For such systems prespring is a necessity, and should be accomplished in an effective manner. Temporary supports
should involve no restriction which will not exist in
service, making it usually desirable that the final
joint be at a low elevation, and located where the
permanent supports alone will suffice. Formerly,
prespring was accomplished by the accurate fabrication of a final section to offset the free (and presumably unrestrained) line by the desired amount
of prespring on each axis. Subsequent forcing of the
ends together was then presumed to provide the required amount of prespring. This approach ordinarily ignored the rotations. Recently, therefore,
The M. W. Kellogg Company has followed the practice of establishing, by precalculation (see Chapter 5),
the desirable locations of forces and moments to be
applied to introduce the moments required for the
desired magnitude of prestress, and simultaneously
bring the ends for the final joint into alignment.
This carefully measured loading is maintained while
the final joint is welded or bolted up, and post heattreated. To avoid possible additional plastic deformation in this final weld, an adjacent location in
the pipe can be stress relieved before this operation
is accomplished on the weld. Prespring can be
further controlled by the use of strain gages to check
the degree of accuracy to which the desired result
is being achieved.
In the conventional assembly of piping to pumps,
turbines, etc., damage by distortion or misalignment
due to fabrication effects can be avoided by thermally unloading the completed piping near the
terminal equipment. This is accomplished by controlled local heating similar to stress relief, or, in
less critical instances, by merely locally heating a
circumferential area with one or more torches to
reduce the fabrication effects at that location to
87
the yield stress of the material for the applied
temperature.
References
1. A. Bantlin, "Formanderung und Beanapruchung federnder
Ausgleichrohre/' Z., V.D.l., Vol. 54, pp. 43-49 (1910).
2. Th. von Karman, flUber die Formiinderung dfumwandiger
Rohre, insbesonders federnder Ausgleichrohre," Z.,
V.D.I., Vol. 55, pp. 1889-1895 (1911).
3. H. Lorenz, IlDie Biegung Krummer Rohre." Z. Physik,
Vol. 13, pp. 768-774 (1912).
4. M. Marbec, flFlexibilite des tubes," Bulletin de l'Associalion Technique Maritime, Vol. 22, pp. 441-457 (1911).
5. 'V. Hovgaard, "The Elastic Deformation of Pipe Bends,"
J. Moth. and Phys., M.I.T., Vol. 6, No.2, pp. 69-118
(1926).
6. H. Karl, HBiegung gekrummtcr, dunnwanwger Rohre,"
Z. Angew, Math. Mech., Vol. 23, pp. 331-345 (1943).
7. I. Vigness, "Elastic Properties of Curved Tubes,"
Trans. ASME, Vol. 65, pp. 105-120 (1943).
8. L. Deskin, uBending of Curved Thin Tubes/' J. Applied
Mechanics, Vol. 12, pp. 1-7 (1945).
9. E. Reissner, "On Bending of Curved Thin-Walled Tubes,"
Proc. Nal. Acod. Sci. US, Vol. 35, pp. 204-209 (1949).
10. R. A. Clark and E. Reissner, HBending of Curved Tubes,"
Advances in Applied Mechanics, Vol. 2, pp. 93-122,
Academic Press, 10(;., 1951.
11. H. E. Jenks, HDiscussion to W. H. Shipman's paper:
'Design of Steam Piping to Care for Expansion,' " Trans.
ASME, Vol. 51, pp. 443-444 (1929).
12. A. R. C. Markl, UFatigue Tests of Piping Components,"
Trans. ASME, Vol. 74, pp. 287-303 (1952).
13. 'V. Hovgaard, "Tests on High-Pressure Pipe Bends,"
J. Math. and Phys., M.I.T., Vol. 8, No.4, pp. 293-344
(1929).
14. P. S. Symonds and T. E. Pardue, uCharactcrigtics of
Short-Radius Tube Bends," Naval Res. Lab. Report No.
0-2761, Feb. 18, 1946.
15. W. Hovgaard, HDeformation of Plane Pipes; Further
Research on Pipe Bends," J. Math. and Phys., M.I.T.,
Vol. 7, Nos. 3 and 4, pp. 198-238 and 239-297 (1928).
16. W. Hovgaard, IIStresses in Three-Dimensional Pipe
Bends," Trans. ASME, Vol. 57, pp. 401-415 (1935).
17. A. M. ~ahl, ffStresscs and Reactions in Expansion Pipe
Bends," Trans. ASME, Vol. 50, pp. 241-255 (1923).
18. J. R. Finniecome, "The Flexibility of Plain Pipes," Engineer, Vol. 146, pp. 162-165, 199-200, 218-219, 248-248
(1928).
19. A. M. Wahl, J. 'V. Bowley, and G. Back, flStresses in
Turbine Pipe Bends," Mech. Eng., Vol. 51, pp. 823-828
(1929).
20. S. Crocker and A. McCutchan, "Frictional Resistance
and Flexibility of Seamless-Tube Fittings Used in Pipe
Welding," Trans. ASME, Vol. 53, pp. 215-245 (1931).
21. E. T. Cope and E. A. Wert, uLaad-Deflection RelatioDB
for Large Plain, Corrugated, and Creased Pipe Bends,"
Trans. ASME, Vol. 54, pp. 115-159 (1932).
22. F. M. Hill, "Solving Pipe Problems," },{ed!. Eng., Vol. 63,
pp. 19-22 (1941).
23. T. E. Pardue and I. Vigness, "Properties of Thin-Walled
Curved Tubes of Short-Bend Radius," Trans. AHME,
Vol. 73, pp. 77-84 (1951).
88
DESIGN OF PIPING SYSTEMS
24. N. Gross, HDiscussion to J. R. Finniccome's paper,"
Imt. Mem. Eng., Proc., Vol. 158, p. 377 (1948).
25. N. Gross, "Experiments on Short-Radius Pipe Bends,"
British 'Veld. Res. Assn. Pubt. FE~6/9, 195!.
26. N. Gross and H. Ford, "The Flexibility of Short-Radius
Pipe Bends," Heating and Air Treatm. Engrs'j Vol. 16,
pp. 152-155, 197-200, 210-216 (1953).
27. W. C. Stewart and W. G. Schrcitz, IIThermal Shock and
Other Comparison Tests of Austenitic and Ferritic Steels
for Main Steam Piping-A Summary Heport," 'i'rans.
ASME, Vol. 75, pp. 1051-1072 (1953).
28. P. L. Vissat and A. J. DelBuono, HIn_Plane Bending
Properties of Welding Elbows," Trans. ASME, Vol. 77,
pp. 161-175 (1955).
29. D. B. Rosshcim and A. It C. Markl, liThe Significance of,
and Suggested Limits for, the Stress in Pipe Lines Due to
the Combined Effects of Pressure and Expansion," Trans.
ASME, Vol. 62, pp. 443-460 (1940).
30. A. R. C. Markl, "Fatigue Tests of Welding Elbows and
Comparable Double-Mitre Bends," Tran.s. ASME, Vol.
69, pp. 869-879 (1947).
31. H. Lorenz, IITheorie der Rohrenfedermanometer," Z.,
V.D.I., Vol. 54, pp. 1865-1867 (1910).
32. \V. H. Dean, liThe Distortion of a Curved Tube due to
Internal Pressure," Phil. Mag., Vol. 28, Ser. 7, pp. 452464 (1939).
33. D. n. Zeno, Discussion to reference 23, Trans. ASME,
Vol. 73, pp. 84-87 (1951).
34. D. S. Jacobus, HOpenings in Cylindrical Drums," ftfecll.
Eng., Vol. 54, pp. 368-370 (1932).
35. F. L. Everett and A. McCutchan, HInvestigation of Stress
Conditions in a FuU-8ize Welded Branch Connection,"
Tram;. ASME, Vol. 60, pp. 399-410 (1938).
36. T. \V. Greene, "Importance of Design Control for Welded
Piping Systems," Welding J. (N.Y.), Vol. 15, pp. 7-11
(1936).
37. S. Crocker, "Welded Fabrication of Manifolds nnd Special
Fittings for Piping," Heating, Piping and Air Cond., Vol.
9, pp. 15-20 (1937).
38. A. 1IcCutehan, uDripping High Pressure, High Temperature Steam Piping," Heating, Piping and A£r Cond., Vol.
10, pp. 501-503 (1938).
39. E. W. P. Smith, IITnilored Pipe Joint.s," Power, Vol. 82,
pp. 142-145, 200 (1938).
40. Crane Co. Reports, "8" Pipe Header with S" Reinforced
Side Outlet/' It L. No. 4455, Aug. 26, 1937j "Welded Steel
PipeHeaders,"RL--1455,EP-D-62, May 25, 1939 i "Welded
Nozzles on Piping," RL-4857, EP-D-82, Aug. 9, 1939.
41. E. H. Seabloom, "Welded Pipe Headers and Their Reinforcement," Welding J. (N.Y.), Vol. 20, pp. 577-586
(1941 ).
42. L. W. Tuttle, "Import.'l.nce of Controlling Design and
Fabrication of Welded Joints for Gathering Lines, ~...Ianifolds nnd Drips," American Gas Association, Distribution
Conference Paper, April, 1\)36.
43. F. C. Fantz nnd W. G. Hooper, uWelded Nozzles and
Their Reinforcement," WeldingJ. (N.Y.), Vol. 19, pp. 119125 (1940).
44. F. C, .Fantz, HDcsign and Fabrication of High Pressure,
High Temperature Welded Piping," Heating, Piping and
Air Cond., Vol. 10, pp. 329-332 (1938).
45. E. R. Scabloom, Discussion to reference 35, Trans.
ASMB, Vol. 61, pp. 170-175 (1939).
46. A. McCutchan, HFabrication Details for High Pressure
and Temperature Piping," Heating, Piping and Air Cond.,
Vol. 11, pp. 141-144, 215-218 (1939).
47. Sulzer Technical Reviews No.3, 1934, ct. seq., to No.2,
1941, Sulzer Brothers Limited, Winterthur, Switzerland.
48. J. S. Blair, "Reinforcement of Branch Pieces," Engineering, Vol. 162, pp. 1-4, 217-221, 508-511, 529-533,
553-556, 577-581, 588, 605-606 (1946).
49. N. Gross, HResearches on Welded Pressure Vessels and
Pipelines," Briti.sh Welding J., Vol. I, pp. 149-159 (1954).
50. J. H. Sandaker, J. A. Markovits, and J{. B. Bredtschneider, "High Pressure (10,300 psi) Piping, Flanged
Joints, Fittings and Valves for Coal-Hydrogenation
Service," 7'rans. ASME, Vol. 72, pp. 365-372 (Hl50).
51. P. i). Bijlnard, HStresses from Radial Loads in Cylindrical
Pressure Vessels," Welding J. (N.Y.), Vol. 33, pp. 615s623, (1954).
52. N. J. HafT, f1Line Load Applied Along Generators of Thin~
Walled Circular Cylindrical Shells of Finite Length,"
Quart. Appl. Math., Vol. 11, pp. 411-425 (1954).
53. G. W. Watts and \V. R. Burrows, Discussion to reference
56, Trom. ASME, Vol. 56, pp. 139-140 (1934).
54. S. Timoshenko, "On Stresses in a Plate with a Circular
Hole," J. Fr,anklin Imt., Vol. 197, pp. 505-516 (1924).
55. E. Siebel and S. Schwaigerer, l/Neuere Untersuchungen
an DampfkesSelteilen und Behiiltern,"Z., V.D.l., Forachungsheft 400, Vol. 11 (1940). Also abstracted in
f1Reinforcement of Openings in Pressure Vessels,"
Welding J. (N.Y., Res. Suppl., Vol. 19, pp. 238s-240s
(1940 ).
56. L. Deskin, f1Strengthening of Circular Holes in Plates Under Edge Loads," J. Appl. Mechanics, Vol. 11, pp~ 140148 (1944).
57. H. J. Weiss, W. Prager and P. G. Hodge, Jr., <lLimit
Design of a Full Reinforcement for a Circular Cuwut in a
Uniform Slab," J. Appl. .Mechanics, Vol. 19, pp. 397-401
(1952).
58. E. Levin, "On. Heinforced Circular Cutouts," J. Appl.
Meehoni"" Vol. 20, pp. 546-552 (1953).
59. J. H. Taylor and E. O. Waters, HThe Effect of Openings
in Pressure Vessels," Trans. ASME, Vol. 56, pp. 119-132
(1934).
60. E. Mikocki, "Verstiirkung von Ausschnitten in Zylindrischen Miinteln von Druckbehiiltern," Die WamIC,
Vol. 61, pp. 660-664 (1938).
61. G. J. Schoessow and E. A. Brooks, "Analysis of Experimental Data Regarding Certain Design Features of
Pressure Ves..'iels," Trans. ASAfE, Vol. 72, pp. 567-577
(1950).
62. W. G. 1-1arskell, W. B. Carlson, A. A. Wells, A. N.Kinkead and A. L. Tannahill, HExperimental and Analytical
Determinations of the Stress Systems in a Welded Pressure Ves.<;el," British Welding Res. Assn., Heport F.E.
12/20, London, 1952.
63. A. A. Wells, "On the Plane Stress Distribution in an
Infinite Plate with a Rim Stiffened Elliptical Opening:,"
Quart. J. Meeh. Appl. Moth., Vol. 3, pp. 23-31 (1950).
64. A. R. C. Markl, flPiping Flexibility Analysis," ASME
1'rans., Vol. 77, pp. 127-149 (February, 1955).
65. A. I\'I. Houser and S. Hirschberg, "Flexibility of Plain
and Creased Pipe Bends," Power, Vol. 74, No. 16, pp.
568-571 (1931).
66. E. T. Cope, and E. A. Wert, "Some Changes of Shape
LOCAL COMPONENTS
Chc.racteriatics of a Smooth, a Corrugated, and n Creased
Bend Under Load," abstracted in Mech. Eng., Vol. 54,
pp. 875-876 (1932).
67. R. L. Dennison, "The Strength an~'Flexibility of Corrugated and Creased Bend Piping," J. Am. Soc. Naval
Engrs., Vol. 47, pp. 340-432 (1935); sec also Engineering,
Vol. 14, pp. 103-105, 215-217, 297--300 (1936).
68. L. H. Donnell, HThe Flexibility of Corrugated Pipes Undcr
Longitudinal Forces and Bending," Trans. ASME, Vol.
54, pp. 69-75 (1932).
69. E. O. \Vaters and J. H. Taylor, "The Strength of Pipe
Flanges," Mech. Eng., Vol. 49, pp. 531-542 (1927).
70. C. O. Sandstrom, "Bolta and Flanges for Tanka and Heat
Exchangers," Chern. & Metallurg. Eng., Vol. 40, pp. 67-71
(1933).
71. R. W. Bailey, HFlanged Pipe Joints for High Temperatures and Pressures," Engineering, Vol. 144, pp. 364365, 419--421, 490-492, 538-539, 61iHi17 and 674-676
(1947).
72. T. M. Jasper, H. Gregersen, and A. M. Zoellner, IIStrength
and Design of Covers and Flanges for Pressure Vessels
and Piping," Heating, Piping and Air Cond., Vol. 8,
pp. 605--608, 672--674 (1936); Vol. 9, pp. 43-47,109-110,
112, 174-176, 178,243-244,246,311--312 (1937).
73. E. O. Holmberg and K. Axelson, "Analysis of Stresses in
Circular Plates and Rings," Trans. ASltfE, Vol. 54,
pp. 13-28 (1932).
74. E. O. Waters, D. B. Rossheim, D. B. Wesstrom, and F. S.
G. Williams, Development of General Formulas for Bolted
Flanges, Taylor Forge & Pipe Works, Chicago, Ill., 1937.
75. E. O. Waters, D. B. \Vesstroffi, D. B. Rossheim, and
F. S. G. Williams, IfFormulas for Stresses in Bolted
Flange Connections/' Trans. ASME, Vol. 59, pp. 161169 (1937).
76. D. B. Rossheim, E. H. Gebhardt, and H. G. Oliver,
lITests of Heat Exchanger Flanges," Trans. ASltfE,
Vol. 60, pp. 305-314 (1938).
77. J. D. Mattimore, N. O. Smith-Petersen, and H. C. Bell,
IfDesign of Flanged Joints for Valve Bonnets," Trans.
ASME, Vol. 60, pp. 297-303 (1938).
78. G. W. \VattsandE. C. Petrie, "The Design of Flanges and
Flanged Fittings," Valve World, Vol. 36, pp. 121-129
(1939).
79. A{ adem Flange Design, Bull. 502, 3rd Ed., Taylor Forge
& Pipe Works, Chicago, Ill., 1950.
80. W. F. Jacp, "A Design Procedure for Integral Flanges
with Tapered Hubs," Trans. ASME, Vol. 73, pp. 569571 (1951).
81. J. J. Murphy, "Discussion to W. F. Jaep's paper 178J,"
Trans. ASME, Vol. 73, pp. 572-573 (1951).
tt
89
82. E. O. Waters nnd F. S. G. Williams, HStress Conditions in
Flnnged Joints for Low Pressure Service," Trans. ASME,
Vol. 74, pp. 135-148 (1951).
83. D. B. Wesstrom and S. E. Bergh, "Effect of Internal
Pressure on Stresses and Strains in Bolted-Flanged Connections," Trans. ASME, Vol. 73, pp. 553-558 (1951).
84. T. J. Dolan, uLoad Relations in Bolted Joints," Mech.
Eng., Vol. 64, pp. 607--611 (1942).
85. R. W. Bailey, "Thermal Stresses in Piping Joints for High
Pressures and Temperatures," Engineering, Vol. 137,
pp. 445-447, 506-507 (1934).
86. H. J. Gough, IlPipe Flanges Research-Firat Report of
the Pipe Flanges Research Committee,/I Engineering,
Vo\. 141, pp. 243-245, 271-273 (1936).
87. A. R. C. Markl and H. H. George, UFatigue Tests on
Flanged Assemblies," Trans. ASME, Vol. 72, pp. 77-87
(1950).
88. R. G. Blick, "Bending Moments and Leakage at Flanged
Joints," Petroleum Refiner, Vol. 29, pp. 129-133 (H)50).
89. R. G. Blick, "Interaction of Pressure and Bending at
Pipe Flangcs," ASME Paper No. 51-Pet-9, (1951).
90. E. C. Bailey, H. C. Schroeder, and I. H. Carlson, "Mechanical Joint Experience in High Pressure-Temperature
Steam Piping," Valve World, Vol. 49, pp. 34--39 (1952).
91. H. Weisberg, "Cyclic Heating Test of Main Steam Piping
Joints Between Ferritic and Austenitic Steels-SCwaren
Generating Station," Trans. ASME, Vol. 71, pp. 643649 (1949).
92. O. R. Carpenter, N. C. Jessen, J. L. Oberg, and R. D.
\Vylie, "Some Considerations in the Joining of Dissimilar
Metals for High-Temperature High-Pressure Service,"
Proc. ASTM, Vo\. 50, pp. 869-860 (1950).
93. H. Weisberg and H. M. Soldan, "Cyclic Heating Test,
Main Steam Piping Materials and Welds, Sewaren
Generating Station," ASME Paper No. 53-A-I51, December, 1953.
94. Pressure Vessel Researth Committee, Design Division,
"Report on the Design of Pressure Vessel Heads,"
Welding J. (N.Y.), Vol. 32, pp. 31a-51. (1953).
95. F. E. Wolosewick, UEquipment Stresses Imposed by
Piping," Petroleum Refiner, Vol. 29, pp. 89-91 (1950).
90. P. P. Bijlaard, llStresses from Local Loadings in Cylindrical Pressure Vessels," ASME Paper No. 54-Pet-7
(1954).
97. G. J. Schocssow and E. A. Brooks, "Stresses in a Cylindrical Shell Due to Nozzle or Pipe Connection," J.
Appl. Me<:hanics, Vol. 12, pp. 107-112 (1945).
98. R. J. Roark, IIStrength and Stiffness of Cylindrical Shells
under Concentrated Loading," l'ram;. ASME, Vol. 57,
pp. A147-A152 (1935).
CHAPTER
4
Simplified Method for Flexibility Analysis
simple piping configurations of two-, three-, or fourmember systems having two terminals with complete
fixity and the piping layout usually restricted to
square corners. Solutions are usually obtained from
charts or tables. The approximate methods falling
into this category are limited in scope of direet application, but they are sometimes usable as a rough
guide on more complex problems by assuming subdivision into anchored sections fitting the eontours
of the presolved cases. However, the inexperienced
analyst is eautioned not to extend these solutions
beyond the restricted proportions of their geometry.
2. Methods restricted to square-corner, singleplane systems with two fixed ends, but without limit
as to the number of members.
3. Methods adaptable to space configurations with
ITH piping, as with other structures, the
analysis of stresses may be carried to varying degrees of refinement. At one extreme
lie mere comparisons with layouts which have met
the test of service; at the other extreme are comprehensive methods involving long and tedious computations and commensurate engineering expense.!
The many approaches lying in between are compromises which have a scope and value not readily
definable since their accuracy and general reliability
are so heavily dependent on the skill and experience
of the user. These so-called "simplified methods,"
nevertheless, fulfill an important need. In capable
hands, and with ample allowance for their limitations, they serve to provide the quick rough check
demanded in establishing an initial layout while
avoiding the use of the more refined calculations
unnecessarily for false starts. rv[oreover, their use
allows the final confirmation by comprehensive
methods to be more safely postponed, when necessary, in order to even out the work load of specialists
usually employed for the purpose. In cases of noncritical service, moderate expansion requirements,
or small pipe diameters, the availability of generous
safety margins may make certain simplified methods
acceptable for final analysis.
W
4.1
square corners and two fixed ends.
4. Extensions of the previous methods to provide
for the special properties of curved pipe by indirect
means, usually a virtual length eorrection factor.
This chapter covers a number of approximate solutions including a recently developed simplified version of the General Analytical Method presented in
Chapter 5. It discusses the fundamental assumptions and range of applicability of each of these
methods and is complemented by illustrative examples. In the interest of a clear and concise
presentation, detailed derivations and procedures
are omitted but references are given to published
technical literature. The shortcut solutions presented here were selected either for their ease of
application or for their relative accuracy; numerous
other approaches proposed in the literature involve
various combinations of simplicity and accuracy but
those given are generally deemed to be most representative. The methods described all involve the
usual assumptions for analyses to the Theory of
Scope and Merits of Approximate Methods
Approximate approaches are built upon a variety
of simplifying assumptions which range from minor
to drastic significance. All such approaches may be
classified in four groups, as follows:
J. Approximate methods dealing only with special
IThc economic disadvantage of the comprehensive methods
has been offset considerably by developments in model testing
(8CC Chapter 6) and, more recently, by the rapid progress in
programmed automatic computers (see Chapter 5).
90
91
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
Elasticity which have been incorporated in the
General Method itself, and which are discussed in
Appendix A3.
.~
The principal weakness of all approximate methods
is that, with current limitations of mathematical
analysis in treating the unbounded geometrical complexity of piping layouts, there exists no means of
assessing the maximum error involved. With any
given approximate method, a layout can be devised
for which the analysis will lead to vastly misleading
results. Hence, the "accuracy" of an approximate
method is largely hypothetical, while considerations
of "degree" or "probability" of accuracy are also
not realistic. In this vein, there is no intent, in the
use of examples common to the methods, to convey
any true comparison of their accuracy, but rather
to give an appreciation of the manner of application
and, only in a. general way, to indicate limitations in
their use.
Piping flexibility calculations provide accuracy in
proportion to their completeness. Once simplifying
assumptions of unassessable accuracy are incorporated, it serves no purpose to employ excessive
refinements in the remainder of the work. When
close results are essential on important or intricate
piping systems, the use of approximate methods is
questionable. It is usually more effective and less
time consuming for organizations equipped to
handle comprehensive solutions to proceed directly
with the General Method, particularly if programmed automatic computation and model testing
are available.
4.2
Thermal Expansion
Most engineering materials respond to a temperature rise by a nearly proportionate increase in linear
dimensions. If the temperature change is uniform
throughout a homogeneous part, the dimensional
End, Unrestrained (froe to expand)
An""'
Point ,
FlO. 4,1
~J
"
Expansion at various points on a header.
ly = Projected length for
computing .1. 'I
U = Anchor Dl51onc:e
x
llt= Projected length for computing ill'
FIG. 4.2
Computing components of restrained
thermal expansion.
increase will likewise be uniform along all direetions.
The change!!. of any dimension L is calculated from
the relationship
!l = Le
(4.1)
where e = unit linear thermal expansion 2
(dimensionless if both !!. and L
are in the same units)
The application of this equation to the determination of unrestrained expansion is illustrated in
Fig. 4.1 for a header with a single point of anchorage
and with the two ends free to expand.
In the usual case, however, the piping will have
more than one point of anchorage or connection to
equipment and consequently will be subject to restraint whenever expansion differentials arise. If
the piping is not uniform in temperature throughout
or if it is made up of several materials having different coefficients of expansion, the differences in temperature or material must be taken into account in
the expansion calculations. The expansions of the
equipment to which the piping attaches must be
similarly treated.
The first basie operation in the determination of
the thermal expansion stresses (set up as a result of
restraint) requires the ealculation of the unrestrained
expansion of the piping, whieh is the expansion that
would take place with regard to an assumed single
referenee point, all movement proceeding from that
2Values of e in the Code, ABA B 31.1, cover most of the
metals commonly used in piping. These tables are based
on a datum temperature. of 70 F, which is considered as most
nearly representing the condition under which the average
installation" is made. A more careful selection of this datum
may be warranted if its effect on the temperature difference
is significant. Chart C-2 of Appendix C gives the slightly
different values of e used in the sample calculations of this
book, which Were in preparation before the Code values were
adopted.
92
DESIGN OF PIPING SYSTEMS
point without interference of any kind. If there is
no displacement of the anchors, the calculated resultant expansion between them i~, called the resultant restrained thermal expansion of the piping system.
The components of this expansion are conveniently
computed directly from the projection of the anchor
distances on the respective axes. This procedure is
illustrated in Fig. 4.2, where, by eq. 4.1,
Az = eL;x = net restrained expansion in the x direction.
Ay = eLtJ = net restrained expansion in the y direction.
A = cU = resultant restrained expansion, i.e.,
More complex cases involving more than one tem-
perature range for parts of the 'system, or terminal
displacement due to equipment expansion, will be
treated in the illustrative problems in this and in
the following chapter. Only cases where the temperature is constant over a measurable length of
piping are shown; however, thermal gradients along
piping runs can usually be readily approximated as
to their contribution to the expansion of the leg in
which they occur.
The second basic operation in calculating strtoSes
due to thermal expansion is the determination of the
forces and moments which must be applied to the
ends of the system (which are imagined to have
temporary initial freedom for expansipn) in order
to return them to their actual fixed positions. This
operation of structural analysis is di"tinguished by
its involvement with irregular configurations' and
the necessity for conversion of deflection (expansion)
into reactions and stress. It occupies the principal
role in the General AnalytIcal Method of the next
chapter and is equally involved ill this chapter, although it is obscured in certam of the approximate
approaches.
4.3 Preliminary Segregation of Lincs with
Adequate Flexibility; Code Rules
A large amount of piping in conventional layouts
possesses satisfactory inherent flexibility for the intended service. Thus the piping engineer, faced
with the problem of effectively apportioning the
time to be spent on a project, is immediately confronted with the need for recognition of such piping
with a minimum of attention to each line. Approxi-
mate solutions or simple rules of thumb are therefore essential. The results obtained cannot be ex-
pected to compare with those established by the use
of acceptable analytical methods, but in the hands
of a competent designer they serve to assist in the
recognition of' totally inadequate flexibility, and
serve as a base line for sbarpening judgment by association of occasional results with accurate analysis.
Piping flexibility, in providing for tbe changes in
length which result from thermal expansion of piping and connecting equipment, must be adequate
to serve two purposes:
1. To control within acceptable limits the piping
reactions on connected equipment located between
or at the terminals of the line.
2. To maintain stresses in the pipe itself within
a range so that direct or fatigue failure and joint
leakage are avoided.
Where sensitive equipment (due to close clearance
on moving parts, high speed, etc.) is involved, an
accurate flexibility analysis is usually advisable,
since approximate approaches are apt to be particularly unreliable for reaction evaluation. Accurate
calculations are also advisable for hazardous contents in relation to an installation location where
strength is seriously reduced, as at high temperatures; for unusually stiff piping due to size, thickness,
configuration, etc.; for economic use of expensive
materials; for definitely cyclic service; or when approximate analyses indicate overstress. Most of
these criteria are quite general and subject to opinion in their significance and manner of application.
They may be grouped under four headings: strength
requirements; reaction hazards; service hazards;
economics.
Positive assurance that the minimum required
strength for satisfactory service is attained is possible only by complete analysis; however, for not
too complex piping systems which consist predominantly of straight runs not concentrated too near
the line of thrust through the anchor points, approximations of reasonable but varying accuracy are
attainable. A designer can develop, for a given
shortcut approach, an idea of its limitations and
range of accuracy for average problems provided
he has a reasonably adequate knowledge and experience with both approximate and accurate analyses.
Hazards attendant to excessive reactions are covered in Chapters 2, 3, and 8. It should be noted
that approximate methods generally do not give the
reactions. In many of those which do, the indications are unreliable. In particular, neglect of the
flexibility of curved members will result in abnormally high values which provide little guidance in
93
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
assessing the capacity of sensitive equipment to
absorb such cffccts.
Service hazards are related in 'Part to the character of the line contents and the energy contained;
and in part to the typc of plant, its location, and
the operating conditions (pressurc, tcmperature,
etc.). For example, a line containing light hydrocarbons at moderate pressure and at a temperature
approaching their flash point would be considered
to require complete design; a line containing the
same material at the same pressure but at a much
lower temperature would be considered average
service in a rcfinery, but might properly be considered a critical service in a gas generating system
located in a populated area. Definitely cyclic service, by increasing the hazard of fatiguc failure, makes
it necessary that lines be analyzed. A limit of 7000
complete cycles during the full life of the system is
considered consistent with present design criteria in
defining nancyclic service.
Regarding considerations of strength, reaction
hazards, and service hazards, some opinion has
favored the establishment of arbitrary limits of
pipe size, pressure, and temperature above which
piping would be considered critical with detailed
analysis required. It would be typical of such an
approach to require analysis wben, simultaneously,
1. Maximum nominal operating metal temperature exceeds 800 F.
2. Service pressure exceeds 15 psi.
3. Nominal pipe size exceeds 6 in.
Other. have favored a single criterion based on the
energy stored which would be a function of compressibility, volume, and pressure; as an alternative it
has been proposed tbat a maximum temperature be
also applied.
Such a criterion is more logically established for
a partiCUlar industry or type of plant; for this reason such provisions have not been incorporated into
the Piping Code. To provide a substitute simple
criterion for the recognition of those systems requiring detailed analysis, efforts have been made to establish a rule of thumb capable of giving a rough
idea of relative flexibility. Various attempts to
devise a parameter expressing the dominant effects
of configuration geometry have led to the selection
of the ratio of developed length to distance between
anchors as the simplest useful stiffness index for the
purpose. This is the basis of the formula in the 1955
edition of the Piping Code (ASA B31.1), which cont.ains requirements for mandatory examination of
the flexibility of piping systems to avoid requiring
complete analyses on all piping if
..
DY
U'(H _ 1)2:0; 0.03
(4.2)
where D = nominal pipe size, in.
Y = resultant of restrained thermal expansion and net linear terminal displacements, in.
U = anchor distance (length of straight line
joining terminal or anchor points), ft.
Ii = ratio of developed pipe length to anchor
distance, dimensionless.
This formula is given graphically in Chart C-4 of
Appendix C. *
Equation 4.2 does not directly evaluate stresses;
however, its formulation provides that when the
left side reaches the value of 0.03, the inherent
flexibility of the piping is at the acceptable limit.
Thus, the actual maximum stress range BE contained
in eq. 4.2 can be found from:
33.3DY
SE = U 2 (H _ 1)2 SA
(4.3)
where SA = allowable stress range.
It has been stated that the Code equation represents no more than a rule of thumb, and in cases of
unfavorable configuration it can doubtless be
grossly misleading. Nevertheless, it is interesting
to note that in the few· examples of average configuration presented in Sample Calculations 4.1 to
4.4, inclusive, it comes very close to the results calculated by the General Analytical Method. This
comparison is shown in Table 4.1.
Sample Calculation 4.1
Material: ASTM A-lOG, Gr. A
Design temperature: T = 900F
Unit expansion from 70 F:
0.078 in./ft
Type of service: Oil piping
Code allowable stress range:
SA = 21,625 psi
Nominal pipe size: D = 10 in.
Developed length: L = 100 ft
Anchor distance: U = 56.6 ft
U/D = 5.66
H = L/U = 1.77
L
1'51
---,
~
From Chart C-4
H' = 1.68
H' < H; formal calculations are not mandatory.
·The Piping Code formula (Section 621) is given as
DY/(L - U)2 ::; 0.03. This can be rearranged into the form
of Eq. (4.2) by the substitution of R ~ LjU.
_-_._--------------
DESIGN OF PIPING SYSTEMS
94
Sample Calculation 4.2
U/D = 5.85
Material: ASTM A-106,
Gr. A
Design temperature: T =
900F
Unit expansion from 70 F:
0.078 in./ft
Type of service: Oil piping
Code allowable stress range:
SA = 21,625 psi
Nominal pipe size: D = 10 in.
Developed length: L ~ 115 ft
Anchor distance: U = 58.5 ft
U/D = 5.85
R = L/U = 1.97
Y/U = 0.065
R = L/U = 1.97
ASA B3Ll Code Criterion Chart G-4
R t = 1.61
R t < R; formal calculations are not mandatory.
I--Z5'
Method
Maximum Longi-
tudinal Thermal
Stress for:
Sample Calc. 4.1
Sample Calc. 4.2
Sample Calc. 4.3
ASA B31.1 Code Criterion Chart G-4
t
R = 1.67
R
t
< R; formal calculations are not mandatory.
ASA B3I.I
Code
General Analytical
Method, Square-
Criterion
16,800
10,250
10,250
Corner Solution
16,750
11,650
8,900
Resultant Y = '\12.082 + 3.082 + .078 2 = 3.8 in.
While a rule of this nature fills a very definite
need, good judgment must still be exerciscd in the
case of certain lines, exempt by this rule, as to
whether detailed analysis should be made in con-
Sample Calculation 4.3
Material: ASTM A-106,
Gr. A
Design temperature: T =
900 F
Unit expansion from 70 F:
0.078 in./ft
Type of service: Oil piping
Code allowablc stress range:
SA = 21,625 psi
Nominal pipe size: D = 10 in.
Table 4.1
--...
4.4 Selected Chart-form Solutions
Results identical with those of Sample Calculation 4.2.
Sample Calculation 4.4
Material: ASTM A-106,
Gr. A
Design temperature: T =
650F
Unit expansion from 70 F:
0.052 in./ft
Type of service: Oil piping
Code allowable stress range:
SA = 23,000 psi
Nominal pipe size: D ~ 10 in.
Developed length: L = 115 ft
Anchor distance: U ~ 58.5 ft
sideration of combinations of size, temperature, pressure, nature of contents, etc. previously discussed.
1
Expansion and terminal displacements:
= 2.08 in.
x-direction: 0.052 X 40
y-direction: 0.052 X 40 + 2 .- 1 = 3.08 in.
z-direction: 0.052 X 15
= 0.78 in.
Special solutions have been frequently presented
in the literature by means of a variety of formulas,
charts, or tables [1, 2, 3], which are both time saving
and convenient for simple configurations. Each
solution applies only to a particular configuration,
although the proportions of the legs are permitted
to vary. Since the number of variables which may
be conveniently handled is limited, these solutions
are restricted with regard to the number of legs ill
the configuration. With judgment and experience,
segmenting of more intricate systems permits wider
use, although generally at the expense of considerable hazard of error and with little saving in time
over a complete solution by a more versatile approach. The selected cases included herein are limited to four which are believed unavailable elsewhere
in the form given. These cases, shown in Fig. 4.3,
are set up primarily for convenient use in establish-
ing preliminary layouts, and provide directly the
dimensions required rather than the stress for a sel
of assumed dimensions.
An assumption common to all of the chart solutions presented is that the modulus of elasticity is
taken to be 29 X 106 psi. The charts are based on
accurate analysis so that for the square corner eases
17t
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
I
A~
ca-
~
----1-,
l~}
95
-L
l~}
c
C
(b) Two-member System. with One
(a) Two-memb4lr System Sl,Ibjeded
to Thermal Expaniion
Support Displaced in rile
Diredion of the Adioining Member
r
I
A
"
(c) Two-member System,. with
One Support Displaced
,
-- ...I
Gv!d.
;;..-----
A'1
-----,
, ,, K,
K,l
Normal 10 Inilial Plone
.t
Guid ~
--
-B
"B'
"
1
(d) Symmetrical bponsion loop Subjoded
to Thermol ExpaMion
FIG. 4.3
Representative cases for chart-form solutions.
given the results obtained will be as accurate as the
charts can be read. These chart solutions (see
Charts G-5, G-7, C-9, and G-U in Appendix C)
may be used for the determination of the length of
leg required for a given allowable stress range. For
cases where terminal reactions on connected equipment are important, such reactions may be obtained
from Charts C-6, G-8, C-10, and C-12, also in
Appendix C.
The charts are constructed so that stress is given
in terms of the SA, which may be selected to suit
the material, etc. involved. For partial solutions,
the designer may vary the value of SA to suit his
judgment as to the eontribution of the remainder
of the system to the overall flexibility, or, where
sueh is not involved, to use a fixed value (such as
SA = 18,000 psi) in applying these eharts for design purposes.
The first ease, Fig. 4.3(a), deals with a twomember right-angle system under thermal expansion. The required data are the nominal diameter
of the pipe, the length L of the longer leg AB, the
allowable stress range, SA, and the unit linear thermal expansion e. The length KL of the shorter leg
BC which will hold the stress to the allowable limit
is then found with the aid of Chart C-5. From the
supplementary Chart C-6, the moments and forees
acting on the end points are easily eomputed.
The procedure is readily apparent in Sample Caleulation 4.5.
Sample Calculation 4.5. Given a two-member
right-angle system made of 4 in. Schedule 40 ASTM
A-53, Grade A earbon-steel pipe. The leg AB is
10 ft and the operating temperature in oil piping
service is 530 F. Find
a. The required length of BC and
b. The moments and forees at A and C.
a.
The unit expansion e from a 70 F datum of carbon
steel at 530 F = .040 in. 1ft, and SA = 23,220 psi.
Enter Chart C-5 with LSAI107 e = 10 X 23,2201
10,000,000 X 0.040 = .581
Read over to the curve representing 4 in. pipe
and then down to the value of K which is 0.59. The
required length of leg BC is therefore K X L =
0.59 X 10 ft = 5.9 ft.
b.
Enter Chart G-6 with K = 0.59
Read
Al = 0.6
A 2 = 0.245
As = 0.102
A. = 0.212
The moment of inertia I for 4 in. Sehedule 40
pipe = 7.23 in'
IelL 2 = 7.23 X .040/100 = .00289
IelL = 7.23 X .040/10 = .0289
~~-----
96
DESIGN OF PIPING SYSTEMS
Therefore:
The reactions therefore become:
- F rc = - 600,000 X .00289 = -17301b
F rA ~ -Frc = -1750 Ib
- F"c = +245,000 X .00289 ~ +710 lb
FilA ~ -Fllc =
7851b
102,000 X .0289 = 2940 ft-Ib
M ,A ~
.5780 ft-Ib
w,
ill zC = -212,000 X .0289 = -6120 ft-Ib
The second ehart-form solution was developed for
a two-member system subjeeted to a terminal displacement in its own plane. Figure 4.3(b) shows
end A displaced in the direction of the adjacent leg
(in this case, leg A B). Structurally, this is equivalent to a horizontal movement of support C to the
left. This displacement, however, is now perpendiculal' to the supported leg BC. With proper discretion, therefore, this solution is adaptable to
support movements both parallel and perpendicular
to the supported leg.
The required data are identieal to those of the
previons ease. The length of leg at which the stress
equals the allowable value is found from Chart C-7.
The reaction forceR and moments are then secured
from Chart 0-8.
Sample Calculation 4.6. Support A of the system shown in Fig. 4.3 (b) is transposed in the
direction of leg AB through a distance of 2 in.
Leg AB is 22 ft long; the system is made of 6 in.
Schedule 80 ASTM A-I06, Grade A carbon-steel
pipe to be used in power piping service at 580 F.
Find
a. The required length of leg BC
b. The reaction forces and moments.
a.
Under the conditions given above, SA = 18,000 psi
and L 2 S A /10 7 fi = .435. If Chart 0-7 is entered
with this ordinate, one can read over to the line for
6 in. pipcs and down to an abscissa value of K = 0.8.
The required length of leg BC is therefore 17.0 ft.
b.
The moment of inertia for a
is I = 40.49 in'
10"(1 fi/L 3 ) = 701
and
-18,800 ft-Ib
The third case is shown in Fig. 4.3(c). It is concerned with a two-member right-angle system which
is subjected to a displacement normal to the plane
of the members. Given the nominal diameter of the
pipe, the length L of the longer leg, the allowable
stress range SA., and the displacement .6., the required
length KL of the length BC is found by the use of
Chart C-9. From Chart C-IO, the moments and
the forees acting on the end points are found. This
procedure is illustrated in Sample Calculation 4.7.
Sample Calculation 4.7. End C of the twomember system shown in Fig. 4.3(c) is displaeed
upwards by I in. The members consist of 14 in.
OD X i in. thick ASTM A-100, Grade B pipes.
The length of leg AB is 15 ft, and the design temperature is 950 F. Find
a. The required length of BC and
b. The moments and forces at A and C.
a.
SA = 20,125 psi for oil piping.
Enter Chart C-9 with
L 2 S.. /IO' fi = .588
Read over to the curve representing 14 in.
pipe and then down to thc valuc of K, which is
0.24. The required length of leg BC is thercfore
K X L = 0.24 X 15 = 3.60 ft.
b.
The moment of inertia, I, of 14 in. OD X i in.
pipe is cqual to 372.8 i "
Ifi/L 3 = 372.8 X 1/3375 ~ .]]05
°
in. Schedule 80 pipe
10'(1b./L') ~ 16,740
Enter Chart C-8 with K = 0.8 and read
I fi/L' = 372.8 X 1/225 ~ 1.057
Enter Chart C-IO with K ~ 0.24
Read
.1,= ]]5
A, = 2.30
A, =
A, = 1.03
.1 3 = 70.0
.1 3 =
A, = 24.5
.345
A, ~ 1.12
2.1
A, = 43
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
Read
Therefore:
A,
-F yc = 12,100 Ib
.55
M XA =
-3480 ft-lb
A 2 = .90
M,.
116,000 ft-Ib
F'A
M,c =
-40,600 ft-lb
111,., = -1I1'B = +273,000 X .86 = 235,000ft-lb
M,c =
+71,300 ft-lb
4.5
-F xB = -68,300 X .55 = -37,600Ib
Approximate Solutions
The fourth case is a graphical solution for the
The methods covered in this section arc approxi-
familiar and important symmetrical expansion loop!
mations, all of which are limited to square-corner
shown in Fig. 4.3(d). Chart G-11 is entered with
the outside diameter D, the effective distance L
between the anchors or guides, the allowable stress
range SA, and the expansion A between the anchors.
The required height K,L is found for any value of
K,L. From Chart C-12, the forces acting on the
anchor points and the moments acting on the guides
are computed.
Sample Calculation 4.8. Given a loop of
20 in. OD X! in. thick ASTM A-135, Grade A
pipe. K,L is 20 ft. Guides are located 10 ft on
either side of the loop, so that L = 40 ft. The distance between anchors A' and B ' is 100 ft. The
line temperature is 425 F and is used for oil piping.
Find
configurations. Although several solutions which
fall in this category have been advanced, the two
presented are selected because they appear to
achieve fair reliability with the greatest simplicity.
These arc the Guided Cantilever and the MitchellBridge Methods, both of which are applicable to
three-dimensional piping systems. The fundamental
assumptions and guides for application will be given,
followed by illustrative examples. For a more detailed description of these methods, the reader is
referred to the literature [4, 5, 6]. For important
piping these methods should not be relied upon as
the final check; their use by personnel other than
those with adequate background and experience is
apt to lead to serious errors. They can be used to
advantage, however, for the following purposes:
a. For approximate assessment of the lIexibility
of average piping, and to check lines not meeting
the criteria of Section 4.3.
b. On critical piping, for layout assistance in
arriving at a suitable system for detailed analysis.
c. On noncritical piping, to establish the location
of restraints without unduly impairing the lIexibility
of the system.
The Guided Cantilever Method. This method
is intuitively familiar to many piping designers. Its
fundamental concepts are partially used in the sidesway analysis of frames. The assumptions underlying this method can be listed as follows:
1. The system has only two terminal points; it is
composed of straight legs of pipe of uniform size
a. The required height of K,L and
b. The forces acting at points A I and B ' and the
moments acting at points A and B.
a.
The unit linear thermal expansion for carbon steel
at 425 F = 0.030 in. 1ft. t., therefore, = 100 X
0.03 = 3 in. SA = 19,890 psi (ignoring Code permission to exclude longitudinal joint efficiency)
L'SA
--- =
10' Dt.
.0531
Enter Chart C-11 with .0531
Read over to the curve representing K, = 0.5
and down to the value of K, which is 0.32. K 2 L is
therefore 40 X 0.32 = 12.80 ft.
b.
The moment of inertia for 20 in. OD X ! in. thick
pipe = 1457 in.'
It.
L3
1457 X 3
64,000 = .0683
1457 X 3
1600 = 2.73
Enter Chart C-12 with K, = 0.5 and K 2 = 0.32
1
97
and thickness with square-corner intersections.
2. All legs arc parallel to the cuordinate axes.
3. The thermal expansion in a given direction is
absorbed only by legs oriented perpendicular to this
direction.
4. The amount of thermal expansion a given leg
can absorb is inversely proportional to its stiffness.
Since the legs arc of identical cross section, their
stiffnesses will vary according to the inverse v,.Iue
of the cube of their lengths.
5. In accommodating thermal expansion, the legs
DESIGN OF PIPING SYSTEMS
98
_ ------~6y.
c2
j
6ltb
----=- h
y
J;
0
-+.:.1
II
C-.
L = length of leg, ft.
E = modulus of elasticity, psi.
D = external diameter of pipe, in.
"I!!!mpliM'
no'rotation of
tongontl at COrMr
Ora
Ii",,= Expo"".' of . . '" lI.
Oyg=Expanslon of I.g b, l!J,"y
act as guided cantilevers; that is, they are subjected
to bending under end displacements, but no end rotation is permitted. This condition is pictured in
Fig. 4.4 for the simplc two-member system. 3
According to assumptions 3 and 4 the individual
legs absorb the following portion of the thermal
expansion in the x-direction:
where
L3
T-L3 _ T-L 3 t!.x
x
(4.4)
lateral defleetion in the x-direction
for the leg under consideration, in.
L = length of the leg in question, ft.
t!.x = overall thermal expansion of system
Ox =
in x-direction, in.
T-L3 - T-L} = sum of cubed length of all legs perpendicular to the direction considered (in this case meaning the legs
parallel to the y- and z-directions).
Similar equations can be written for the lateral deflections in the y- and z-directions. The schematic
distribution of thermal expansions to the various
members of a space-bend is shown in Fig. 4.5.
The deflection capacity of a cantilever of the type
stipulated by assumption 5 (and shown in Fig. 4.4)
can be given as:
48L2 SA
0=--
ED
(4.5)
"A refinement taking cnd rotation into account is explained
later.
=
SA = allowable stress range, psi.
FIG. 4.4 Deflections assumed to occur in a single-plane system under the guided cantilever approximation.
OX =
° permissible deflection of leg, in.
where
For convenience, this equation has been plotted in
Chart C-13, Appendix C, based on the value
E = 29.0 X 106 psi.
A first (preliminary) evaluation merely requires
now the calculation of 0" 0., and 0, from eq. 4.4 and
from eq. 4.5 (or Chart C-13) for each leg. If ox.
0., and 0, are all less than 0, every leg possesses a
sufficient deflection capacity, and the system can
be regarded to be adequately flexible.
°
This comparison is most conveniently carried out
on Form R as shown in Sample Calculations 4.9.
4.10, and 4.11. The stepwise process of the analysis
is clearly indicated in these forms, which are selfexplanatory in conjunction with the foregoing discussion.
In Sample Calculations 4.9, 4.10, and 4.11, the
condition that > Om (where Om denotes the largest
of 0" 0., and 0, in any leg) is satisfied for all legs
save the one next to the far terminal. In this case
a further refinement is warranted in recognition of
the actual rotation which takes place at intersections. This refinement is accomplished through the
use of a correction factor f, which allows for the
reduction of bending moment, due to the rotation of
the leg adjacent to the one considered. The value
of f for the appropriate case is obtained from
Chart C-14, depending on the position of the leg
and its length relative to adjacent members. If the
corrected deflection capacity of the leg, fo, is larger
than 0." the leg is considered to be sufficiently
flexible. The use of the correction factor is shown
in Steps 9 and 10 of the Sample Calculations.
The ratio of om/fo indicates the proportion of the
allowable stress range that has been used up by the
leg in accommodating thermal expansion. This
°
o
d
(0) x-diroction
j
All..l1y,A'I.:::ThermalexponslolU
In the 1'-, y-, zdirodionl, ...tpectivoly
6...
FIG. 4.5
Deflections assumed to occur in a multiplane system under the guided cantilever approximation.
99
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
,
I, 'S 1,
.~
J
11
<
~
.,
'"
1-2Sc...f
PIPE DATA
CEG
NO
!"":"
U,
L
A
NOK $IZE
/0'
0-1
y
10'
l'I"lL Tt«.
~.5· 1-2
X
2!/.9
~~
<I
15' .5'375
EO' /25'()t)() .l.o!!
Z
l-V.TrS1lAL
TD1P.
J-3
3-J
"
900F
-,"'.. .078%
L .....
Se~VICl:
S.l~
1I,
25'
I~ ",
.s.
2.21
.7.
7.37
3.;;;8
J'.,f
2./
6, -
L'
L'
,-~
-
.0'
l<7tJO
-
IS,;[5
L ,rs;-
(fit""
c.......,.
::£ L -L
,,-,~
/.473
-
:2 S(;
..
• "'".."..
i-
>c-
6.-
L'
L
L
..'
I
So-
c. ..... y
c- ,...
f'
_&7
/
.545
.50
1.70 3.57 /55S0
.335
r
I~
I ISO
0,;:;./.. . '."
2U.z5
./..t) '.078 '" 3./2
40'.078"'3.12
......,"" 6 _"ClIeo'" 6 ... _£t..U:r.ts.. 0 ..
L (LA-L".') 12(;000
~ (L'-L/I 19000
5.. ,5.:1 .... 5.. ) NO ""1'/."~
MOMOIT RANGf U
I
".A,
!THE MW KELLCX?G col PIPING FLEX~".ILlTY
AND STRESS ANALYSIS
(L~-L':J
•
POINT
MAXIMUM 6fNOIIUi
0.1,.( ..... ".
"
TIO ..3 ....n: NI'CU$"I'l'l' Ul'<less s~ COI'lPOllfHT AT
.... 0 101"... 0 ..... ~.. 05U ....T r"/lM'"M-S TE~l'lltlAlS (fT III
4
.. .,II! OC'$ll:lI!b.
ROUGH
"'Le.
<
An
--
wee'up
APPROXIMATION
S,Z
2.9GO
.38800
FO M
CALC NO
,
1: 1,
'5
\'J~,
I
<
a
~
s.l
1--25'--.1'
,
PIPE DATA
LEG
NO
"':"
L
6.-
L'
A
L'
tl°KS1ZE
10·
~'I
'"z
15'
JJ75
.08/
WAll THK
.le5
I-Z
y
1<1 •
/000
.0N
z
Z9.9
'::
Z'3
x
3-
K'lTl'I:!LAI..
uT£/'lP.
... _
c ....... ~., ..
".
.100 ;:: 4'S
.078 ~
"
,-~
~~~L'
>c.
15'
3)7S
£0'
1?50tX 3.01
25'
:n;l
2./{J
"'"
PIP,,",,6
SA'I, 21GZ5
•
Lis. (Fit"..
2.£1
,
1.41)
.335
.J7
.47
-
-
5, -
~~
n
·
'A'
(~~J
.008
.027
2.2/
/.00
7.)7
g,
.13
'.C-'i
'"
.h.-
"'''
• "0"
So-
L
I
<;t........
c.-I'"
f'
SA,f"
I
/.5
1.4.1
I.M/I
9-.1'0
r
.SCl
1. 7 0
!J.~
I!J 220
MOM("IT RAHGt fI
HArlMlJM 6fHllJ/Ui
POINT
~
23350
.7.
SfI.V/C-t:
.0, .1.0 x ."7.5 .. 5.12'
".A,
L (L·-L,o') /2J1.!J75
./"'.o7$~3.1?
I (L'-l'
/5"'.07d .. 1.17
~(L·-L/ /45000
22)7S
......- II ."nooo 6 .. _~""«A;,"" 0,.
6 ... 15~ ••"151) NO '-VI".r,.r.ot (,oL("'A'
T, .. ,,~ IIR" J.lIrC.,.:\.Alty ..... L~:li' S~ COMPOH!tlr AT
.... 0 .............. ov,,,s .., AT T""N'''Alo$ TErMlIML~ {fr 15
...."'.. o;>e~I""'l>.
Tl-£ MW. KELLCX?G col PieiNG FLEX~~.ILlrr
AND STRESS ANALYSIS
ROUGH
PPROXIMA'"ION
1
AC
•
.. tCNID
An
0
S
.130dO
FO M
CALC NO 4.kJ
100
DESIGN OF PIPING SYSTEMS
r1
y
l5
,J=J 1
.~
D
<
~
)7
.L/?s'
PIPE. DATA
NO~StZf
l'lAlL T~
z
~Ttl1lAL
_.-
TeMP.
CEG
NQ
0-'
.JCO' /·2
1!9.9 Z·j
'0'
A!,O(;
9001'
LI"'''''''
..078 ;;.
Sfltv/U
Oll/"11'Pt16
~
U.
-.,
L
L'
/5'
3j '5
10'
/5'
,<700
6,-
o c'
--~
,0'
.004
'375
"
"' rd 25 M
--;;
4-5
.>-,4
J.()o
25' 15(;2
.15
.--
~, ~
L'
fill:~'
I~
.,1.7
- .0M
77 .081
- 3.0/
2.18
S"l" 2/G25
~,
/5;,071) -/'17'
A, 40 -.076 ..:3./2
~,
4f7 • • (;'7d-=3./2
::f (Ls_L".J) /45000
1: (L'-l
f!2 fJ75
I(L~ L z 121375
_ _ G li'l<CU_
,.-
S
L {S;;" (Flto.,
.h-
50-
'''0...
L
c. ........,.
f
c.-,...
fS
(:-.,)
.7C;
.j!5
T
/.5
/.4.1
/.C88
/.4'"
Z.21
,7/;'
r
0.5
/.7tJ
J.57 /.$Z;'O
ro'
z.el
CAse
7.'7 8../
3.G8
2./
6 ..{~S'" 0"
6.. ,5.:1,"'15,,) ,.... ,.v..::,.~Go,l..C:<II. ...·
AND STRESS ANALYSIS
!THE MW KEll0G3 CQfPIPING ~ttuXd~'U YPPROXIHATIDN
'"eel<'.
'T<
23350
0
53000
5
F
--
,roo
~
MOMENT R,AtHi~ 01 POINT
H.AJ:IHUM. fH:Hllm<i
TlOO'S J\RIO: "'ECII~RY Ut<Less Sir COMPONENT AT
.....0 "'_ _ .. I<J\MlfU AT T .. _ , ..,u.S
TfC~'IlI(ALS (''" Lit
...... ~,l"l!!t>.
I~
M
CALC NO
.II
permits an estimate of the actual stress range in the
leg by the formula
Om
(4.6)
SE = fo SA
cal Method in Table 4.2. While these results are
indicative of the accuracy for average configurations,
for extreme conditions the results may be much less
favorable
where BE = estimated stress range in leg, psi.
SA = allowable stress range, psi.
Om = largest of component deflections Ox, all'
or 0, by eq. 4.4, in.
o = deflection capacity of leg by Chart
C-13, in.
f ~ correction factor by Chart C-14.
Table 4,2
The estimated moment range, which is of interest
at the terminal points, is then found from the relationship
M = SEZ
(4.7)
b
12
where M b = moment range of maximum bending
component, ft-Ib.
Z = section modulus of pipe, in., from
Table C-l, Appendix C.
In Sample Calculations 4.9, 4.10, and 4.11 the
evaluation of the stress and moment ranges are
shown in Steps 11 (last column) and 12, respectively.
For a partial indication of the reliability of the
Guided Cantilever Method, the stresses so obtained
are compared with the results of the General Analyti-
l\'fethod
Layout according
to:
Sample Calc. 4.9
Sample Calc. 4.10
Sample Calc. 4.11
Guided Cantilever
General
Analytical
Maximum Stress
r-.Iaximum Stress
Value
15,550
13,220
13,220
16 ,750
Location
4
5
5
Value
11,650
8,900
Location
4
0
S
The greatest asset of the Guided Cantilever
Method is extreme simplicity, and applicability to
any space configuration with two points of fixity.
In general, least accuracy is realized when the system consists of legs of greatly disproportionate
length, or when terminal displacements are present
in addition to thermal expansion. From the limited evidence available to date, it appears that the
error will normally be on the safe side.
The terminal moment range given by the Guided
Cantilever Method is no more than a crude indication of the moment reactions actually present at
the supports. In cases where the moment reactions
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
101
govern design, this method is of use only as a preliminary evaluation.
The Mitchell-Bridge Method. Consider an
arbitrary piping system limited to two terminal
points, and subjected to thermal expansion; if all
Centroid of Pipe from 0 to 1
restraint is removed at one terminal, this end of the
Centroid of Entire
piping would move out through a distance determined by the linear expansion. To restore the
actual fixed condition, end forces and moments are
applied to the "freed" end, of such magnitude as to
return it exactly to its initial position. These forces
and moments can be expressed as a single force the
line of action of which is called the thrust axis. This
approximate method is based on the proposition
that on most piping systems the thrust axis can be
located empirically with reasonable accuracy. Once
the thrust axis is establisbed, the problem is rendered
statically determinate.
Mitchell [4J originally assumed the thrust axis,
which passes through the center of gravity of the
piping system, to be parallel to the line connecting
the anchor points. This assumption, valid only for
limited configurations, was subject to variable and
extensive error for the many shapes of piping layouts encountered in practice. Figure 4.6 shows a
configuration subjected to thermal expansion, in
which terminal moments are obviously present, yet
the original Mitchell Method would predict them to
Configuration
be zero.
Improvement of the original Mitchell Method
required a more reliable location of the thrust axis.
Bridge [5J proposed that the layout be divided into
two halves of identical developed lengths. The
centers of gravity of these half portions are then
connected to form the "gravity axis," the center of
gravity for the whole system lying half way between those of the two halves. The Mitchell axis
is now drawn (line through e.g. of system parallel
True Thruil Axis
Midpoint of tho
Developed length
A
Centroid of Pipe "
from 1 to A
Thnnl Axis Auvmed by Bridge
FIG. 4.7 The modification to Miwhell's thrust axis location
suggested by Bridge.
to the anchor line) and the thrust axis is taken as
the line bisecting the angle between the "gravity"
and Mitchell axes. This sequence of operations for
locating the thrust axis is shown in the representative example of Fig. 4.7 for a single-plane layout.
To preserve simplicity, the centers of gravity for
the halves of the layout are located by eye. For
single-plane systems of not too great complexity,
usable accuracy can be achieved; for multi plane or
involved one-plane systems, considerable error is
likely to arise. For these latter applications Randolph [6J proposed that the centers of gravity be
located by calculation. The labor involved, however, then approaches that of the General Analytical
Method and since the resnlts are still of unassessable accuracy, the choice over the General Method
is decidedly questionable.
Once the thrust axis is located, the problem becomes statically determinate. There remains the
conversion of the evaluation of linear expansion to
the reaction resultant force along the thrust line
Centroid
~u1f Mia Assumed
\ ~
by Mitchell
. "'"
FIG. 4.6
_
Illustration of the error involved in Mitchell's
assumption regarding the thrust axis.
and subsequently the stress at any locatioll) which
involves conventional struetural approaches, with
the conversion of thrust to moment usually accom-
plished graphically.
The basic limitations of the Mitchell-Bridge
Method are: (1) Two terminal points; (2) Proper
orientation of the thrust axis is dependent on experience and ability of the user.
Variation of the cross section of various runs of a
system can be taken care of by assigning specific
102
DESIGN OF PIPING SYSTEMS
weights (inversely proportional to the moments of
inertia) to the individual legs when locating the
centers of gravity of the half-sY8tems. It is not
necessary that all legs be in an orthogonal arrangement. The flexibility of bends can also be incorporated by means of the approximations of Section 4.7,
one of which was advanced by Randolph [6J.
The layouts investigated by the Guided Cantilever
Method in Sample Calculations 4.9, 4.10, and 4.11
were re-evaluated by an experienced designer using
the Mitchell-Bridge Method. While detailed calculations are not reproduced) 4 a comparison of the
magnitude and location of maximum stresses with
those obtained by the General Analytical Method
is given in Table 4.3, which shows that the maximum stresses are consistently underestimated;
furthermore, the most highly stressed lo~ation is in
sharp contrast to the results obtained by comprehensive calculations.
It has been claimed [6J that the Mitchell-Bridge
Method gives results within ±20% of an accurate
method, which is quite optimistic in view of the
limitations discussed above; also, the comparison
in Table 4.3 indicates an error of 26.1% for Sample
Calculation 4.9.
Table 4.3
Method
Modified Mitchell
General Analytical
Maximum Stress
Maximum Stress
Layout according
to:
Value
Location
Value
Location
Sample Calr. 4.9
Sample Calc. 4.10
Sample Calc. 4.11
12,350
8 13S0
6,580
2
4
4
16,750
1l,650
8,900
4
0
5
4.6 The Simplified General Method for Squarecorner Systems
The solution presented in this section is a limited
form of the General Analytical Method, which can
be followed step by step by anyone aeenstomed to
routine arithmetical computations. This Simplified
General Method is applicable to single- and multiplane eonfigurations subjected to thermal expansion
and external movement, which satisfy the following
conditions:
1. Two completely fixed ends.
2. No intermediate restraints.
3. No branches. 5
4. Straight runs only.
·For n detailed description of a systematic step-wise solution, the reader is referred to the available literature (4, 5, 6\.
5J'he effect of branches of the main system may generally be
5. All legs orthogonal (at right angles to each
otherJ with square-corner intersections).
For systems accurately represcnted within these
limitations, the accuracy is the same as that of the
General Method. Because of this accuracy and the
methodical attack, this method is highly recommended over any approximate method involving a
comparable degree of effort.
Bcfore describing the actual procedure in making
the calculations, a brief discussion of fundamentals,
conventions, and terminology is in order.
First, the unrestrained expansion of the system is
determined. One end is then arbitrarily choscn as
the fixed end to reduce the analysis to a cantilever
systcm. The other end, the so-called free end, is
imagined to be loaded with forces and moments of
unknown value. These unknowns can then be
found from the condition that they must prodncc
deflections of the free end which nullify the displacements of that end due to thermal expansion.
The coordinate system is standardized as:
X-axis-horizontal and positive to the right.
Y-axis-vertical and positive upward.
Z-axis-horizontal and positive towards the observer.
The dircctional signs are consistently applied to distances, displacements (expansions or deflections),
and forces. Signs of angular displacements (rotations) and moments will be positive in the counterclockwise dircction facing the related positive axis.
The origin of the coordinate system may be at any
point on or outside of the pipe line. A location
toward the center of the system promotes accuracy
by a more nearly uniform level of dimensional coefficients. Where members coincide with the axes
certain coefficients will be zero, and where symmetrical adjacent runs are present, extension of this
symmetry to their placement with respect to the
origin will advantageously duplicate the coefficients.
The pipe line is subdivided into individual straight
legs, which are called members. If a straight run
contains a change in stiffness such as a size reduction, it must be treated as two members.
To assist the reader with thc actual application
of the method, a stcp-by-step procedure will be
given, followed by sample calculations presented on
form sheets which arc largely self-explanatory. The
use of a calculating machine is convenient although
not essential.
J
neglected if they are less than 50% of the size of the main rUll.
Of course, the necessary flexibility of the branches lhemselvef'
must not be overlooked.
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
The solution involves three more or less distinct
stages:
1. Setting up the problem.
2. Making the computations.
3. Interpreting the results.
The first and third stages require familiarity respectively with the method and the general requirements of the Piping Code; the second is purely
routine computation.
The set-up procedure, following Form Sheet A,
consists of the following steps:
Step 1. Gather prcrcquisite data. Includc pipe
material, nominal size, design temperature, as well
as the dimensions of the layout.
Step 2. Draw a working sketch. This should be
to scale, or at least to reasonablc proportions, as an
aid in interpreting results. If simplifications are
made, show the piping as it is to be calculated.
Where there is significant expansion of the equipment to which the line connccts, indicate the distances to the anchor point as infinitely stiff members
by the use of broken lines. Designate one end as
the fixed end, denoting it 0', the other end as the
free end, denoting it A. Locate the origin 0 so as to
minimize computation. Number the remaining
points 1, 2, 3, etc. proceeding from the fixed to the
free end.
Step 8. Enter the following:
Outside diameter D, in., from Table C-l, Appendix C.
Wall thickness t, in., from Table C-l, Appendix C.
Moment of inertia l, in 4 , from Table C-l,
Appendix C.
Section modulus Z, in 3 , from Table C-l, Appendix C.
Bend radius R, and bend characteristic h, not
used unless thc approach of Section 4.7 is used.
Flexibility factor k, and stress intensification
factor, f3, not used for curved mcmbers unless the
approach of Section 4.7 is used. In such a case and
for other components obtain k and f3 from Piping
Code (see also Chapter 3 hcrein).
Hot modulus of elasticity, Eh(lb/in?), and cold
modnlus of elasticity, E,(lb/in. 2 ), from Piping Code,
ASA B 31.1."
Stiffness ratio Q = ElIENl N, which expresses
the relative stiffness of any membcr N. The product
El of a group of members is considered as unity,
Geode values for modulus of elasticity and linear expansion
are not used in the sample calculations of this book il.S these
calculations were in preparation before the Code data were
adopted. The data used nre given in Appendix CJ Charts C-3
!lod C-2 respectively.
103
and the shape coefficients for members of different
moment of inertia are corrected by Q. Thc stiffness
ratio should be selected so as to keep the magnitude
of the coefficients within reasonable limits.
Material and temperature as given.
Unit thermal expansion, e, ft/ft, from Piping Code
ASA B 31.1.
Cold spring factor, C.
Hot allowable stress, Sh, and cold allowable stress,
S" from Piping Code ASA B 31.1.
Step 4. Calculate the component free expansion
movements .6 X1 .6. tlJ and {).z' Include the expansion
of the equipment and of other members assumed to
be rigid. Prefix the sign in accordance with the
direction of the imaginary movement of the free end
with respect to the fixcd end.
Step 5.
Compute the products Ehl A,/144,
Ehl A./144, Ehl A,/144, being careful to use the
value of E h l/144 for that size pipe for which Q = 1
was selected.
The work now proceeds to the second stage which
consists entirely of computations. The form sheets
involved depend on the problem as follows:
a. For a single-plane system with expansion in the
plane, Sheet B is used.
b. For a single-plane system with expansion normal to the plane, Sheet C is used.
c. For a single-plane system with expansion both
in and normal to the plane, the forces and moments
at the coordinate origin, 0, are computed on Sheets
Band C. The remaindcr of the calcnlation, including the transfer of moments to the various
points and the combining stresses, is done on Sheet F.
d. For a multiplane system Sheets D, E, and F
are used.
The steps of the compntation stage are:
Step 6. Select Sheet B, C, or D depending on
the problem. Identify the members as 0'-1, 1-2,
etc. at the top of the page. List k, Q, and £ from
Sheet A. Compnte £2/12. Indicate the position
(I, II, or III) of each member and determine the
distances a, b, e for each in the position involved.
Enter a, b, e with the proper sign. Compute the
shape coefficients A, A a1 A UI etc. for each member
in accordance with the formulas listed in the columns at the left. Sum A, A" A b, etc. across and
enter the totals in the last colnmn at the right.
Step 7. Determine the forces and moments at
the origin.
a. On Sheet B: This computation is made immediately below the calculation of the shape coefficients, and the symbols A, A a , A b, etc., refer to the
sum of these coefficients shown in the last column.
104
DESIGN OF PIPING SYSTEMS
First, the coordinates of the elastic center, x, and
y" are calculated. Second, the constants in relation
to the elastic center, m12, m22, and mIl, are determined. Third, the constant n33 is calculated after
which N rand Ny can bc determined. The forces
F;r; and Fu can now be computed in accordance with
the formulas given, and finally, the moment at the
origin, A1~, is calculated.
b. On Sheet C: This computation is also made
immediately below the calculation of the shape coefficients, and the symbols B, Bbl Ebb, C, Ca , Caa
refer to the sum of these coefficients shown in the
last column. The order of computation is the same
as above: Xli and yz are calculated, B' bb and Of aa
arc computed, ma3 is determined, after which F Zl
M r, and My are obtained.
c. On Sheet E: The symbols A, A a • A b, ctc., E,
B bl Bel etc., and 0, Gel Ca , etc. refer to the sum of
the shape coefficients in the last column on Shect D.
After the elastic center coordinates have been calculated the constants mu,. m12, m22, m23, m33 and
m'3 can be computed. From these a second set of
constants, nU, nlZ, n22, nI3, na3, and n23 are calculated, and Zt, 1: 21 N;r;, Nil, Nt. are computed. The
forces are now determined followed by the moments.
Step 8. Transfer the moments obtained in Step 7
to the various points in the line. For a single-plane
system with expansion in the plane only, the computation is made in the space provided at the bottom
of Sheet B. For a single-plane system with expansion
normal to the plane, the computation is made at
the bottom of Sheet C. For a single-plane system
with expansion both in and normal to the plane,
Sheet F is used as an aid in properly combining the
stresses. For a multiplane system, Sheet F is used.
Indicate consecutively from the free end A the
points at which the moments are required. Enter
the coordinates x, y, z of point A with their proper
signs in relation to the origin. Enter x, y, z for each
succeeding point in relation to the one preceding it.
Fill in at point A the moment M x, My, M" obtained
from Step 7. Perform the operations indicated on
the next three lines using the F x , F y , F z from Step 7.
For example, on Sheet B, enter l)f; (the moment at
the origin) under point A, multiply Fr by the coordinate y previously listed, and - F y by x. Add
ft1 z + yF x - xFv to obtain 111' zA, the moment at
point A. Enter the M"A in the following column
on the M, line and perform the multiplications, yF r
and - xF II' using the x and y coordinates in that
column. Add to obtain M, for the 2nd point.
Proceed in this manner to point 0'. A check at
point 0' may be obtained bv setting in the coordi-
nates X, V, of 0' in rtaation to the origin, and transferring the moments directly from the origin to
point 0', using the same proeedure as at point A.
Step 9. Choose the point of maximum stress.
On Sheet B where only the bending stress in the
plane is involved, this presents no difficulty, but in
Sheets C and F, it may be necessary to evaluate
the stress at several points in order to find the maximum. Once the position of the point is selected,
the strcsses are combined as indicated.
In all of these steps (6 through 9) in the computation stagc it cannot be emphasized too strongly
that careful attention must be given to the algebraic
signs involved. Checking at the completion of each
step by a person other than the calculator himself
is strongly recommended.
Step 10. Complete the analysis by entering on
Sheet A the results for the cold and the hot condition in accordance with the formulas given in Section 2.6 of Chapter 2. The signs given are those of
acting forces and moments at the terminal points
and are determined from the calculation as follows:
Cold Condition
Fixed End 0'
Free End ,1
Opposite sign
Same sign
Hot Condition
Same sign
Opposite sign
Also included in the summary of results is the
maximum expansion stress and the point at which
it occurs, and, for purposes of comparison, the allowable stress range.
The final stage, the interpretation of results, will
be discussed in Chapter 5. Thc calculator is cautioned to examine the results:
a. In order to determine whether they are in general agreement with what is expected from the configuration and the displacements.
Calculations
giving unlikcly results should be inspected for sign
and arithmetical errors.
b. In the light of the assumptions made, to decide
whether the results are on the liberal or conservative
side and to make a judgment concerning the practicality of the layout.
The following examples are given to illustrate the
procedure:
Sample Calculation 4.12. .-\ single-plane system
of uniform size wjth expansion in the plane only.
The infinitely stiff member 0'-1, normal to the
plane, is cold. It is shown only in order that the
numbering of the points may be consistent with the
examples which follow.
Sample Calculation 4.13. The single-plane system
of 4.12 above with expansion both in and normal to
the planc. The forces and the momcnts at the
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
D.
I-A
/0.75
t
I
Z
/60.7
t?9.90
MEMBERS
/50
.365
"
~
o Ehtfl
i:!
_
I
UJ
/
-.L...
~.
W
a.
6,
ii:
70 F
0
'"
900 F
V)
6,
1-~
ti
MOMENTS (I=r -tel
COLD CONDITION
ST£P 10
AND FORCES (Le)
5", ~ q,500 p:li
5<:-/6.000 psi
5£ .~;•.';~,,:,;~rED
ACTING ON RESTRAINTS
HOT CONDITION
A
I
I
POINT
M.
M
::J
II)
>-
A
3
'"
238,280
"'6.238,280
fACTORC-O
25.00'
"C\.
~ i:!
·00<.50
(01.0 ~PRlfo:G
~rRt5$
..I
M.
F.
Ul
W
a:
I
+
"1',
,50
-/ .775
580.
580.
/390 +1390.
• /6,750 psi
+30,00
+/9550.
+
900.
-
+ 2/70.
"T POINT A
..
90a
170.
SA' ~i~~:~te>Xl.
HE MW KELLDGG COl. PIPING_,L~~I,~ILlI
~ AND STRESS ANALYSIS
ORIGINAL DATA AND RESULTS
,
2-0
,
0-3
I
I
Q
.
I
I
I
I
10.00
-15.00
I f;. 00
50.00
Zfi.OO
-7.50
0
12.50
All,A\o,El
b
-5.00
0
-25.00
-so 00
IN TlIlf CCII.
8.33
f8.75
208 ..33
5Z.08
L
VEo~~'T~~!hR LXZ
:
u
~~ 1 I
kQL
4·A
"A
b'A
d'
b'A
a·A..
b
A - TIOO.OO
:z:-
= +.50
..
UHIC llB.OW
I
t50.00
-2550.00
'"/04.875.00
t 8583.25
+ 10.00 + 15 .00
-A'Jt,'
•
+.50.00
_1O':s:oz~ _A·":.o~x..c:fI275·00
1000"'11
:z_13(:,00.00
mit
• -13.40
• + 39. 4 7
m"
N1'·-1+4000
~ ~ i1ii
I ., -40.80
tl1J1.Mn
~+337.B.o
._/84.9.0
"",
POIHT
=+152.90
A
3
X
+Z5.00 -25.00
+N.·f'1u .-349.25 tN.·"'lt
+Nyollt lla- 554 .aB +Noj.mll
__90·p3
0
I
--
loOOm... +3S424'.a
-50.00
Clr"TS~s
-1,"0.00 -112.S0
0
+3/2.00
-.250.00 -1250.00
-50.00
0
-15G25.00
0
+750.00
0
A,
0
+SZ08.Z5
A.. +2250.00 +1125.00
0
+41,""'G.SO tGl.SOUoo
Abb + .33.3.33
_/4875·00 +A..
tA.... "+'04,44~a +A..
+1‫סס‬:oo
F,
Fx
2
0
0
-15.00
+50.00
0
73 -3q898 +23,234
-2/
COI::"I-
L:
-(MIrY
~
5HAPt'
+100·00
:; -25.50
101
5\1" THE
+-Z5.00
~I='+~.-ZSSO.oo
A
II
'-A
--
A
A
Ab
cbAa. tA :/12 d'Aa
A,,+ALy'z
b'A
X _+ Aa _ :t 50.00
_
I
b
QL
x'
+45 07
0
-45,207
-F~.x
- 54J32
.54132
0
I
-A4
m"
= +1(;10.38
=' +1.I<DS.26
w.•
.
• + 0·'6
P
Z
+",,0.80
B
•
_+/082.(;,.' $ •• F"QMb
1-1.
I
I(CI-u:.c/<)
0
-/5.00
-10.00
+9041
+~o41
0
+32479
a-2f':'71.<t>
PLANE SYSTEM- EXPANSION IN PLANE
A
308S8
2.9.90
I
.401 4
-~
-21973
I SINGLE
-Mb
-Fx·~%
_10.00
Z 479
POINT
+BS5aZS
+FlI,Xx
2/973 +10500
0
STRE::S" CALCUL.ATION
.~
!OOOrT)z;r •
I
N:Ja- IEJ.6:t.
000 nu
c+554.6B
,1.'04,,4'"-83
• +8553.2
+19547
1M". -30800 +23,234 -2/~73 +lo.50 r;,. +19547
THE M
KELLOG~ CO
PIPINq FLEXIBILITV AND STRESS ANALYSIS
.
.J
K£
'-2
LtNGnt 1)F Me,..,e.IIIR
AeSCIIl.5,., 01' CU.lTI!R
OF(lR"'V1T'fOF 1"\:e"'~fR
I ~OIN"'Tf OF ClNnROF
RAV'.TY OF MfM"ER
POSITION t
HI!
O....TE
k
ST'FFN.S~ FAf;TOR
HOA.lZ. "l1!."U
o.
~
-2/,625 psi
F
A
CAL NO.4·IE
~""
H PC <:0
IC'~"T~
FOR HORI%.C.VlRT.~ct'1f1l
FLI!XII511.I'TY FACTOIIt
•
IF)
STEP 2
<i
~ -.26000'
EI'IAX/IH
EhlbyhH
Eh IlI x/14.
C>
"
-7".26000'
-~
~
-
0'
0'D BRANCH
Z
...woo{~
<:
X
,/ I
0
AlO66R.A
.00650
x
1 0//
$JUMID
900 r
TEMP.
e
1'./ ~
"'X~y
~
V)
~ MATERIAL A-I06GR.
~
(
en
'~UIJ
Q.
.1,933,400
Q
U1
y
0
g g
'"
I
I
i'.I5
29.0
EI;" 10·
Z Ec" 10-(0
/
1
2
h'
k
I.
0'-1
105
12400
C ONVERS10ti TO CODE R\JU.S:
•
,. '.35
SlE=Ecl'Eh, SI!-'6:. 740
11.Rc RO.E'<: ~ · c Oll.
E' S\iSh
(WttrCH!"'[R.l~GR£A,Tt:R. • • ~4 • ~
R R' = 1- !Ie c I
=
R I - CALo:;.REACTIONS BA~o oNE
cALc-wo,.4.12
106
DESIGN OF PIPING SYSTEMS
I-A·
MEMBE.RS
D.
/60.7
I
Z
29.'
R
h
k
'~"
I
-106GR.A
MATERIAL
900 F
TEMP.
.00650
w
e
a BRANCH O'-A
Z
f4ao
6x
W
<l.
ii:
~-.Z6000
6,
.00<=)
-"
,,- 9750
0'
:J
M
0:
F •-
M,
Fx
'"w
".
1
"
-
>~
• 650
+
•+
-•
675
- I 775
1390
580
- /390
30
30
580
+
.......
•.
<ltl:,:'f'"VI"~r.'~~£I~1U
PO_
:ITI"iI'~II.5S F'A.CTOft
LfNGTl-lO""1CM8lrll
~:6Rl~rr:'~~R
0'
•
+
00
900 +
170 - 2110
40
40
KQL
'Bb+B loYl!!
L3qL.
b,,,,
.·c
.·c
~;:-+c&\
r
5.00
C -I- (/8.00
Clft,lTS
•
25·00
/5.00
-7.$0
0
/2.50
lNTliC COL·
-5.00
0
8.33
16.75
5;>258.25
_C·x';I'._
./9
C~ ... + 9258.0<&
Mx-+F:t·Yll.--/089
""'~
x
~
I
I
--
0
B
-1-/0.00
r/9.50
SHAPe
cot'iI'l"
UIo(~OIlI.O""
-25.00 -50.00
$2.08
208.33
11
I
--
I
e,
Bb,Bbt"erC.
L:
50.00
0
+50.00
+32.50
-1250. 00 -IG25.OO
·2~25.oo
0
T4/~6<&>.50 +812'$0.00
flZ324a80
0#:13.00
1'15.00
·b
"bb +333.30
C
SU"CTHe
I
I
I
of GiS.oo
0
0
=f.04 ~x = + ~ "'" -2925.00=_2(;,/
!t-CCl,Q,.....
3-'"
0-3
I
I
I
c. -/95.00 -1/2.50
c•• f2<f325.CXJ 7-(/25.00
•.c"
<l·C12.+C L~le
CAL NO. 4./3
50.00
T
xqc
b·e
".n • - -
i5.00
~~ ,
b·e
-21.625 p.si
10.00
0
J.3"QL
...
SA~ ~ilfe~t~l.et.l<i
C
,, a
b
:!lT~I!:~:lo
"!$lif?r lsi
2.0
p
X
5f·~=6~rED
31800 - {OOO
13000 - 10SO
19550 .f30 00
1-2
I
I
b
~gl~';;'i$f'o~e~l~~I1:~
HO , 10( 151
~~~~"~ii L/'lZ
P'OlSI T'ON
•
S"'"Q500 p:li
Sc -t<iooo psi
HOT CONDITION
A
HE MW. KELLDGG• col. PIPING"~~~~\~llITY
AND STRESS ANALYSIS
ORIGINAL DATA AND RESULTS
F1.rXUllL1T"( FAC,"OIt
.1
ZS.OO'
STEP /0
"'" FORCES Lel ACTING
ON RESTRAINTS
- La) AND
-12500
F,
A
9001"
3
'""
MOME.NTS (~T
COLD CONDITION
-20675
- 8450
STEP 2
'"
- 39,360
Mx
0
t:;
-6,238,280
POINT
..J
'0'
<:>
l"6.Z3~Z80
FACTORC·O
lJ)
---
00
~
6y
COLD 5PRING
....
00 A1$vNlf
x
Mx d..--{"
~
11=)
" Z
, ~I
o~//
qgr
If,·
e;
"'"
"f.26000
.00{
Er,IllX/1 4
e.,I11I'IrH
E Illzhou
F.I i\
X
/0
0
!i!
0
~
<{
t
,
'""
">
I
I
215
29.0'
23993400
D En ~ 10·
2 Ec." 10·
o fhII
Q
,
21 15:00' Y
.365
t
g
0-1
10.7S
f
112·00
ebb_+ 123,249.8 0
+112.00
-/25.00
+118.00
-13/2.'50
-1-5208-25
9258.2.5
+~Q.'*+ 9.258·06
+8bb_+o 4G,B37. Bo
)000,""_+ 5G~09S'.Br;;
+ 5.00
n
M'x
M'.
M';
%::1'29.90
If='~.
1'=/
1:.40/34
M',
M'
·8·~llI_
7(;;,4 1 2.00
e'~lJ"'+
"'15-+ 5&·/0
Mt
46,837.80
f:t=-~k~'~u-+4/. 70
1"'01"'''
M';I_~"':a:·x';l. -2
,
:5TR-Il:.:l:5 CAL.CUL.ATION
POSITION
A
I!StafMt -
psi.
Psi.
Sf-vSiit45?..
P:5I.
baf,BMb'"
Mx
CONVERSION TO COOl!: RUL£:3:
r-f":t'!:I
S£-{t.c:k ) SfRc:/R.'''(Ec:,k .. }C Olf I!c:k.. } ~
M'
M,
fFz'}(
M'
.... "(lI;t1eVE~ 13 GRE.... Tflt,} ..
5c.-
R""R'~(l'i"3C)=
R'. GAl.e, REACTIONS 801;$£0 Ot.l
THE M.W. KELLO(i~ co. I tllt.~"t.~ ~t7t.~I!!! IS~5Te~~~x~":'~fslg~:~~~~T~I~LANE :tB
=.
-
~
OlU4lolO.e
e.o.... c;... ca.
th
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
,
POilU
CONVERSION
TO CODE RULES
5,
,
-,904.13
5',
"E'
./6760
5,
~ ·.£.L·C
R'
Ell
"
*
*.
'll'ttICH£VER IS GRC,t,UR
".64
.'
(1-
~ C)
". ./
-frJ
of z08S"
,
,.'
0
M',
I
996
M, . - 2• I'll;
1/043
-FIt,z
0
M',
t /04/
M,
-21973
,
M'
R I • CALCULATED
REACTIONS
BASED ON Eh
"4$"207
-so 132
-30898
POSITION
f
,
w',
w~
w.
+ 104/
w~
w',
wy
w,
-3a898
w',
w'
,
w;
w,
It- '1'16
'lrE
Z
29.90
vSb +
....
25,., NI
S ~s +4
+45, • 5'
TMt N.W. KtLLOGG
..
0
906
+
996
/089
0
0
1089
-!OR9
Z
Z
+23234
-4.5"20
/-"-4/32.
0
+23234
-29 3
- /089
0
417
,
-
- .72
2.
0
.,. /5.00
"" 1.5:00
-$04.13
t216 26
f
.,.3247
n
f
-1-3180
...
+
+/0S"06
I
4/7·
+3/R07
- 6262
I
+/95"62
..,.t2QQIll
-2/
0
904
+32
N9S.
0
2
4/'70
-/089
"3247.
672
- n - 280
- 626
1-1. 54 2
t;zR /-129-'f4
- 1:28
--f-/OSOo
-Z/973
0
/$:'00
- 0.00
0.00
+15"4-
CO"'4111T
I
4/8
/2401
400
1$4 20 QOO
124!S'
PlrlHG
FUXI8IL1TY. AND STRESS ,lIULYSlS
MOMENTS AND STR(SSES
origin used on Sheet F to transfer the moments to
every point in the system are those which havc previously been computed on Sheet B for 4.12.
Sample Cakulation ,;.1 4. A multiplane system
similar to the single-plane one used in 4.12 and 4.13
above, but here the infinitely stiff member 0'-1 is
replaced by an equivalent lcngth of 10 in. pipe.
Samplc Cakulalion 4.15. A second multiplane
system.
4.7
0
0
Z
-
O'...(cH£C1<
0
0
-- -- -- ---
ZORS
0
0
0'
I
- '5.00
of34
SoO' M.
S-b"f
996
- 3089_
0
m
/0'
T
---
- 1043
2 -
0
,.
n
50.00
0
+
w'
I
--
t5
M,
of x.J
-F r'x
2
0
+25:00
"2165,2.(;
I "1.70
/089
P.S.1
.B.L
.E.L ~
R' • Ell'
5 I:
3
- 25':00
-50.00
,
*'1..35
A
,
107
Approximating the Effeet of Curved Pipe
and Other Components.
A principal factor contributing to the complcxity
of accurate piping structural analysis is the differing rigidity and attendant secondary stress distribution of local components (discusscd in Chapter
3). As pointed out, the increased flexibility (i.e.
deflection and rotation) which they introduce results in reduced reactions (forces and moments),
while the concurrent localized stresses serve to limit
the fatigue life in proportion to the strain range at
their location. The Piping Code Rules providc adequate coverage of flexibility and stress intensification factors for curvcd pipe and approximations for
miters and corrugated pipe, but for the present
(1955) are confined to rough stress intensification
CALC. - . .6.
~~W:[~~"':"-C""r'
I ~11Io1
NO.
C,ll.C-HD.
r
4.13
factors alone for other components whose flexibility
factors are either not satisfactorily established or
else not reduced to usable criteria. The significance
of flexibility factors diminishes as the relative run
of curved to straight pipe or number of other components is rednced; on the other hand a single stress
intensification at a location of maximum primary
stress constitutes the weak link on which the fatigue
life of the entire system is based.
Repeated reference has been made to the "square
corner" defined as a direct intersection between connecting straight runs, which allows no angular displacement of one tangent with rcspect to the other.
It is commonly portrayed as a single miter in which
uniform stiffness exists without local effect. Actually this condition is never attained. A miter involves sizable intersection stress with proportionate
influence on flexibility and local stress as explained
in Chapter 3. Relatively heavy pipc fittings probably constitute a closer approach although at the
expense of some local increase in stiffness. This
subject is mentioned here in clarification of terminology and not as a contributing factor for evaluation
in approximate solutions.
The assumption of "square corners contributes
materially to the simplification of piping structural
Jl
DESIGN OF PIPING SYSTEMS
108
MEMBERS ,(LL
10.15
O.
.-!JG5
C
1($0·7
I
<
R
~l
h
<{
~ Eh" 10-
•
~400
3
~ MATERIAL
~
TEMP.
soo ~
e
.00G$0
a BRANCH
Z
«
W
0.
0-
0'-04
~
I /
Z
STEP t-
0
0
0
-
6,
-+.l:"Gooo
6,
-4d
"-.<?Gooo
6,
-/5.
·-.09750
.=
STEP 4~
3
I
STGP 5
G,e3tJ.2lXJ
-e:!J39UO
MOMENTS (FT -LB)
'OLD 5PRlIJ
FACTORC~O
COLD CONDITION
o'
11) POINT
A
IM,
.5875
3Eo
.J
M
- Z77,!;)
590
::J
M,
-(POZ£>
U)
STep 10
psi
Sf ~~....';..o;,-~:::T£t1
f'lZ 775
- 107.5
- 1775
18.Zoo
.1t,G50
G60
- /3.50
SA·~i~~:~L.eloj(l,
'So
"30
660
- 1350
30
90
f-
0
p~i
5c~/Gooo
A
1-
- 4Eo - 460
Sh~ <:;500
o'
,. 83
- eeoo
1:30
ACTING ON RESTRAINTS
HOT CONDITION
AND FORCES L6)
,.
0:
I
<::.6.00'
Et,lloX/,,,,4 1f-GiZ".3aeso
F,
F
F,
.1 F)
\J)
f"IloY!r4-.f
E~]loX/I ••
W
x
~,
M,
r
x
0'
I
.laP 61<.11
w
0
,~//
ZI.S
e9.0
Z Ec"IO-
O Eht
Ul
Q
STep"
I
I
k
F./ ;X
Il
Z9.90
Z
,M,
r
."",,
~TAt:SS
..
f-
-
PSt
AY POINT 0'
• z!.tDeS p5i
",0
HE MW. KELLOGG COl. PIPING)::~EXI8111TY
AND SIRESS ANALYSIS
ORIGINAL DATA AND RESULTS
A
"....TlO:K'7 ,
H
CALC NO. 4/
STEP G
ll·U:1S
I
Y·AXIS
..,..,
,-SOL
o· ... b
0· ... 0
0· ... 0
b. "'b
1
b. Ab· A.L 1l2
b.Ab
'"
'.0
'"
"b.8e
"b.a
'.0
b.e e
....,
O· ... b
..
b.8b· II I. I/12
b.Sb
,
eBc
1.30L
•.
",
"
'0
e.cc
c.CC·CI.~2
(L'ClI
lI'ClI
'"
'"
....... K(LLOOG CON'A"Y
15.00
50LlO
15.00 -
/'.i.oo -
7.50
0
,.
l!'J.00
le.50
THE COll/h7H
10.00
5.00
0
25.00
-
50·00
8E.lOW
208,33 +-
5Z.08
7.50
0
0
8.:33
18.75
:ar
II:
/
J
-
1.00
I
ere m
0
0
I
II
-
I
-
A.A.,.Ab
,.
2:
0
+5,2 0 8.25
+41,(;(,6.50 tt;,z, '500.00
"" /2,.910.751
1-106,4-"19.80
."
, ,.
~
-
10,00 {-
1,950.00 + 33'3.30
15.00 i
/50.00 -
IIZ.5O
" + 1,125.00
Ilbe -
r 1,500.00 -I-
Bee T
e.Ce
1.00
10.00
15.00
'00 + 4,!387.~ + 2,25aOC 1-
.., ,
1I,ClI·C Ll/ll
,."
c.Co
1.00
- 2..745,00
- 11,950.00
• S L~12
"
SUM TH£
SHAPt:
COEFFICIEHn
1.00
/'00
• + Z92.!iO 150.00 1/2.50 + 0
312.50
f
" - 195.00
5aoo
0
l,e50,00 - ,,250,00
"'00 + Z,9Z5,OO + 750.00
-1'5,G25.00
0
0
8bb
"
3-A
0'3
1.00
1.00
18.75
b,Sb
<'
2-0
1.00
19.50 {-
o•
e
1-2
1.00
1.00
+
ro smQ
o. "'b
'.0
-
LrllZ
t
0 .... •• L /12
1.301.
,
......
...,.
'"
•••
•, -
•,
III
"
I
,
•,
Z· ... llIS
.}j
~f IJ
0'-1
II(NaE
rORNUL"'S 1'01'1 WENDERS P...,t"'U.[L TO
f
/,11:5.00
15.00 T
1,125.00
0
50.00
19.50
0
0
0
0
0
0
ID.OO -f
333.30
0
13.00 +
" - ZZ5,OO -
195.00
112.50
- /,687.50
0
0
"
,~
0
0
'" 1-,'Z5.00
I- 3. 375.0 +Z.925.0 -1-1/2.-5.00
'oo
"Pl"G rLUlIIUTY, a STI'IU5. ANALYSIS
IlULTI- PLAME SYSTUIS
50.00 +
,.
32.50
+ lIZ/50
- 1.t2 !i,(XJ
124,749,80
t 1,12.5.00
0
G5·00 + 25.00
0
0
0
+ 31Z.50
0
0
+
~
-
0
t 5.208-25
CALC. -"6
CHEcKED
OAT[ 7. 'ezI5"3
.;i/I'
133.00
/12.50
2Z0.00
- 1,687.50
f /,125.00
0
0
127,00
- ,a075-00
0
0
+41,666.50 fBIJ!50.00
0
119,50
242·50
0
0
0
0
IIZ.50
+
25·00
- /.2.50.00 - I,GZ6.00
15.0ej +
0
f-
50.00 +-
15.00
rOil II
,
CAl.c. "0.
11~"33-Z
4/4
109
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
.
f,' .~=-2.745.00
r
A
Ji9.:!>O
.
• • ~","2,.50
,
C
S7£P
.
'I
_22.'17
...
'C ec
••
43 .
oA'bb
2/
B,
B
..
-C.I/ .-
-220.00
..3075.00
12(.33. ZS
/024·67
·c'
12,970.75
492.34
-ebb
/24,749.60
_S.y.l
74437.24
50,3/2.56
....
• aleC
,Coo
..
..
.-ebb ..
/2.478.4/
"
• AQa
1024."7
'C'OCI
/2.478.4/
'B'bb
....
1Z,27/.49
503/2.5("
62,584.05
/3.503.08
IOOOrn,: •
·cca "
-C,.),.l),
1000m'I' -17. 522..29
1000mu _+I{,II.4'
1000m,.
.. -r
m n .. +1.6/
mu '
+"2..5.0
.m".m n "
51'1.81
m" ,+ 44. 43
.m u
m~
••
.m".m n -.
84Z.23
"n
~ {mull • ~
..
'"
2.<;Z
-IOY,"."'1-0
.mlZ·m u •
.mtJ,m zz •
-1098.82
n I••
'"
""6, .!..-::;- -.34473
N ll "-144000'
N•. n"
Ny.n'l
Nl·nn
~,
N •• ~ • ..!.=.,..3"1-473
y
1: 1
144000
~
·
"·
i
'"ft
.,../2'12.7
N: --144000
• f IJ.
~ f ·:,
r
0
0
", ·
POIHT
,
Eh - 1.35
5,
0
"
5, .11,566 ps.,
..B.c.... • .E.L, C
£h
OR
." y.l
.'
fu- • .E.L.~
-fl')'
"*
·f I'X
-f x·I
R'
Eh
,
",
SE
WHicHEVER IS GREATER
-.33
Rh
R' ' j
{;,.2(;,
-<75,40
- 2/.3 7
• F•• l y
~ FI·ll
r
-12.00
"
--
- 428
... 2207
0
r/0 77
+ /77q
- 00
+ /
-18252
'F x')' +·N 77tJ
- /8, 21/
0
-f Y'x
r3.J.7ZQ
-18211
42
-I- 1.("'51!?
-m,.,n,••
r; ..
nALC.
~
CKEcKEO ......
DATE
7· Z
J;';.
O'rCHEGt'"
0
+50.00
0
-10.00
0
0
... 15.00
-15.00
- 10.00
+ /5.00
0
~
~
~
+1077
- "37
0
0
- ,,"
-428
0
42
.,..15.s IJ?
-" 77"
"
-fA 252
3:t37
-4'"
-- ~
".
,./34 J'"
----+-
-""
"
+
_2drA
-/70
-
+07"Q
+
-33'"
-2454
",?o.~~7
.,.20237
+
_nr?
-1-/77. ~
-428
+
0
,/324
7'-10/.3/
..,./0/3/
+
+8.
0
-/8; 252- +IOAi:;'
-1-(..754
0
0
'202
"f"/Q8S
-(,75.40
~
+
-/82.52
0
+ 6754
0
+202
+873Q
1-873'1
#- B7,::tQ-
.401.3
~
/
",
s'b"'
/37
2St -/ MI
S ~S~+4S,
,3507
74550000
"',
63
+4S1 • 5'
TKE ... w ~(I,.LOGG
4/4
0'
"
St d
,
I
",.,
"Z,
"'
"'
0
".8:17'1
".
CALC.
2
",
,',
~ I8,Z 52
1;0"'"
'53
",
",
37.<5"
-/5,5/4
", .
+/77113
",
"f"
-f.·)'l"
"
",
.... 33.72-
• f y'.1 •
..,.14('.
,',
III
1t3,09G.-Z7
, /G".70
" . 'fo88.28
- 273<5
Jff
II
72. ""
1'8.1'-8.95
N. nl • 0
Ny'"u •
N z nn'
POSITIOH
I
19, 251. 33
18,1t;8.95
. Ii .. -
··
- 4 ZS
-4414
+/077
..
·m'l·n'l • -
lO,
-1.50
37, 420.28
-15.00
~
0
J mil.mwn" "
0
0
'I- 1077
20
0
--
0
1'~6.
---=7.$'0/ . .3 0
30'.9S
292.86
.... Z'i..28
+ 7/.53
+ 97.81
", ·
-.3337
- 25. 00
- IfOd7.50
·
.,. 378.80
N. n'l ·
Ny.n u
, 957.72,.
·
Nz·"u
· , 12./;"1" . ... 1349./(;
- 574
.mu·m"
n
_
u
-48,45
-
·...
.m,z·m'l·
-290.34
-378 . .so
0
.', -
R' "CALCULATED
REACTIONS
BA.SED ON Eh
'n
- 28.20
- 20.25
+
"',
",
~"CI. ; Cl
PIPE
~
-676.40
..,./.]49.16
.,. 88,Z
£c. S'E
R'
A
25.00
- 50.00
,
,
",
"." - "37
E,
~(m'lll
PI PING fUXUllLlTY a STRESS "NALYSIS
MULTI' PLAK[ SYSHNS
'" N.If. KELLOGO cOMP"NY
CON .... ERSION
TO CODE RULES
2780.43
Z.2S
--.. 2778J8
_Cm,.ll
-
.m'l·mu _
'"\,'"'n •
m n • .,./j.50
-/7.52.
/m,z'
844.d.3
2.60
7.
3{,/ .
/2 27/.49
,SbC' -IIZS.OO
• B.l••)'. ' of273co. 4<;,
IDOOm.".+
-
~ 24.21
+ 127.00
'A ob .. -11.950.00
-A.f"I, •
5 572. 29
1000
- I. <.5
+/3.3.00
"
"
,A'oa
I 028.91
.c'ce
Jl • •
.
.
100.33
-A.l l ._
l
1028.9/
_ 2.03
C
1125,.00
• A ao
9(;,.09
If ••
1.
.B'
.. .3'1'1.
..
m:;' 44,428.12
.c'
l
.2
r.89
lo •
.6,,:
-a.l
IIZ5.00
-c.:/ ..
·
" ' "A"t-119,50
43,399.21
'
'~hh
·
-242.50
All
~,
r.d5
.... /.33.00
106,449.80
'
.A..
-A.'1,1 • _ ('.3,050. 59
J
7
.~~ r112. 50
B +/27.00
w
CO",PAIfY
PIPING
fLE1(1611,.ITY. UID STRESS ANALYSIS
MOWENTS AIiD STRESSES
"",""
C41,.C
.-7. ..r. 8
';.~~~I(E~_1;:,,": %4.
I
~ORN HO
CALc.. Ho.
,
4.14
no
DESIGN OF PIPING SYSTEMS
,
All
M£MBERS
D.
Fj~x
10.75
. <5
1"0. 7
Zt1, 90
t
I
Z
::: 6A.,
R
z:.
h
k
~
«
IFy
Q E -/0Z Ec:; .. rO~
9,0
9300
o Ell I
0'
I
~ MATERIAL .(·IOt; 6R.A
III
Q
~
W
TEMP.
900 F
e
.oo~so
Z
l:I
.,.t,£dtJ(qor,5Q.
a BRANCH
6'1
",+.097. 0'
-40t»~
"-.2&-<J00'
6r.
a-.Zt.OOO'
<;
)(
W
[
0'- A
_ a
e.lAx/, 4
EnlOY!I,"
a:
F
Fr.
-.
STeP /0
AND FORCES (LaJ ACTIN(l ON RESTRAINTS
HOT CONDITION
0'
.tl
.,./3.(,75 +15 700
.,.
+ 625
.,.
00
- /800
Zt;O
+ 2(;,,0
+ /090
-/090
.,.. ,350
- 3. "0
MOMENTS (F=T ·L5)
COl.D CONDITION
0'
A
275
-325
- /(}O
:SO.,.
0
.,. /0
_ /0
_ 20
.,. ()
0
.,. 10
:5
Flt'
f1:
280
z.J8. '80
fl,I.ozl H
COLO 5PRIU
F,ACTORC·O
lJ) POINT
~
Mx
M
U)
Mz
Ul
~
2.33'1..%0
-
TI-£ MW KELLOGG COl PIPINGO~'m~~~tll 'bf:rNf JbR~tM~~LYSIS
fORJ,lUI."S
X·AXIS
I·AX!1
Y·AXIS
~~ ~
,
,
"
."
,
I
....
."
U
b.•
,
b
'"
I.301.
,.,
b.'
0-3
3-A
SUM TilE
,
,
1.00
1.00
SHAPE
COEFFICIENT'S
1.00
/.00
1.00
1.00
I. 00
A,Aa., Ab,
,
15.00
15.00
- 7. 50
50.00
0
25.00
ETC, IN TH£
COLUMN
BELOW.
b
-IS.OO
-10.00
10.00
- 15;00
- 5.00
0
- 25. 00
-5'0.00
-12.S0
5"2.08
,
r 7. 50
0
0
0
l.,/,2
IR.7S
111:
8.33
18.7S
208.33
I
It
.."""
/
I
It
--
0
=
, + 19. SO +10.00 +15.00 +50.00 + 3Z.S0
-/5'0,00 - 1/2.S0
0
0
"'b --292.50
-/250.00 -1(;.25,00
195.00
50.00
0
+ 2925:00
'" f4387.50
-+ '150.00
0
0
1.3QL
'0<
",
b'
b'
bo
'b
- ISO. 00
- 50.00
0
0
0
0
0
"bllc
"bo,
"bo,
b·llb
b·llb o ll LljlZ
b.llb
'bb ""500,00 +333.30
e.lI c 0 II 1.~12
C.Co
,."
c,Cc
o.Co·c L1ltz
'"
"
,"
'"
... W. l((LLOGU
"
'"
C.Cc·C Ll"z
a.co
COlol .... KT
-555.00
-3120.00
+3'75.00
177'2S0
f'/2S,1'l'l.80
~8/1S0.oo
" + Ill. 50
"
r/2Zoo
0
't 2250.00 tIl Z 5.00
0
0
'"
0
'bb + 1950.00 1333.30
"4"".5'0
, + 15.00 + 10. 00 +19,$0 .; 50. 00 + 25'.00
"
L
/
I
b. lob
"
0.4./5
/.00
b. Ab· ... 1.1/12
:.c
p.5i
2-0
b. "'b
",
.2Ii · (,25
/.00
u,
UQL
5A·~~~:~1J(ij
/.00
a-loa
"
psi
AT POINTA
k
0.1. .... 1.11t2
to,
:lTRl:~~
.. 8'100
STEP G
1- 2-
Ub
c Il~
p:ti
p5i
Sr:.~~~TEO
CAL
a.Ab
O· ... b
Sc" It;oOO
0'- I
""
roll IolEWBERS 'ARALLEL TO
Sh" 6500
'b, - 1125.00
+IIZ!>.oO
0
0
0
'"
, + Is.ao t13.00 .,.. /5.00
SO
0
0
" -1'//2.
22 s.oo -/9s.00 - 1/2. so
"
,~
-1'87,50
+-1/25'.00
0
0
.I.,KG fLUlIlLlTT, e Sfll(SS, AN"'USIS
MULTI-PL.ANt STSTtilS
- 2700.00
- 200.00
+ 14,5'00.00
+/5.'25.00
1'105'99'1.$0
r4J"'.50 t-'Z Soo.OO
+5208.2$
0
+(;.333.2S"
""5'..00
+25,00
1-133.00
0
-3/2.S0
-200.00
0
-S3?SO
0
0
0
0
+ SZOB,2S
0
0
0
0
0
'" 13375.00 't29?S'.oo
+-1/2S.00
'"
+-1/9.50
-1250.00 -121>0.00
-312. so
0
~~~O~J$
DAft
7· 22 ·53
'011111
,
CAL.C. Polo.
-/'87.5"0
+'333.2S
.,. 7425.00
4.15
SIMPLIFIED METHOD FOR FLEXIBILITY ANALYSIS
.
, ~ .~" - 3120. 00
I
/27.00
to
A
'f· .~
C
.
'.
..
I • • • ~.h -
- 24.57
, -/.50
- 200. 00
+-/33.00
A,
IZ • + A
- SS5,O()
T- I Z 7. 00 • -
.B~
1;333.2$
"
7','&7.3"
·8.,/ o.
·Jl bll
48032.44
..
-A.I/ •.
....
'C ec
•• '034.00
48,532.44
oA'bb
-A.)',.1(,. -13,1;35.99
• B.I I ,'• •
99,"0.99
IOOOrn n ,
1000 mil s _
+ 54.57
mIla
'ilia Ill» • •
S70. S2
.mll'm n "
~(m,,}1
99.80
..
470.72' ..
- Z4.88
'~,.mu '
{mull ••
p
."U.mntill
s -SZS. 97
144000
Ny"
N,'
.r::~~
.i , .
274S.42
C.eD
.m".rn u "
-(m"l
~fr;0~8 • i ".,..30989
·f
l
tl u
N! ti'l
Ny.tl u
N,.tl u
F
,:t. · - 79c3
· r 18 / $'
", · -(;148
a
tHE
E,
4
3
,
,
o
o
-
F,
-257.2.q
1'5.'
F,
",
"'.352.49
OIl;
,
1.35
Eh
Ec s't
Eh
"8848
St
SE
~ s..E..t...
RI
Eh
c
,.'
*-.*".~
"*
M',
WHICHEVER IS GR[ATER
./
=~
.5.00
-I-
·F x'z
i C)
R I • CALCULATED
24.80
- JS (,88
-1-27165
- (,14
-271&.5
o
"'17. (..25
IS 108
-I- IJ
-1-179<4
-4(,3("
_
+
o
M'y
-
Mz
4(,
477
o
e-
·F~·y
-II oc,q
-1-12 8~4
-F y.x
0
1.1.'2
... /795
III
o
.,." 477
o
-
t- 17&. CZ
N, tl u •
tlt:/.03
+352.4-9
'fy.I,.
-474"
-fl·y,·
-.32/
- IIOC'I
", .
'A~
/fALC.
CH[CI([O
;t1. J B
OATE 7~ ZZ S3
p
I
o
o
O'(CIIECK)
- /5.00
- 10.00
..,. 15.00
257. Z8
-tlOsc..c,o
- 2(;,23
..,.. 1~29q
1324'1
- C,148
-I-1t,z99
~
-G148
CALC, NO, 4./S
o
o
fl5,OO
~
1;011.. NO E
0'
-
o
~
-(.148
14,84
N•.
Ny.tl u •
F, •
1410
-10.00
-IZ 025
- C,14
o
+ .35Z5
-I-13G7~
'" I7q~
3491
-I- 136 7(;,
..,..179&
-528
+/
,
o
<,
-3491
/10(,9
o
-llo(,q
+
-34 I
"520
.,..2573
+3
r3G8
,
+7/10
-IIOc"q
.,f(;'Z9Q
o
o
o
"'5230
... '803
",7803
-12.Rt.4
-1-1 95
", .. ZO, /30. 4.
tI,. " +
t- 3BG
o
o
o
o
-mw'llJ •
el,12
o
o
"
" - 318.07
", .. 20,448.53
.
· , 848. 8~
· , /7•.
· tI08r;. ,"2
CO
+
2
o
+1 qe.
rI
- 2.49
2S,C87./9
5238 .••
-m".tlll _ •
.. 20448.53
rn u '
.mll·tl il
,S'9.9'
•
- 15.00
-150.00
25.00
.
+ 50.31
'/79C
o
o
o
- 50.00
Z'~
~.02
*~ll·
",
·F
-248S: 50
mn ,
54!". IS
· +
", .
, +/OBc..C,O
- f ,y
2
1000rn"
STilUS ANALYSIS
.IiIUlTI - PLANE SYSHIiiS
'O'NT
CONVERSION
TO COOE RULE S
50.3/4-.7,
- 1(;87. SO
f
~
' fl ·,)'
-f, .•, •
1'11'1 N6 fLfXUlIlin
"'.'1'. !CElLOGG CO",pAHY
s297.00
45,017. 75'
• Ceo"
-C.I)'.I,
.. ,
• mu·m'J'
-F y '"
..
GO, 982. 0 S-
'"18.82
- 99.20
S19. CE
-m,t m" •
Ny.lh
N,.tln
+.30989
l
'n
N,.tl u
•
4 S; 0/7.75
m n "f.?9?
54.70
· -_1'Z.91
- 39.59
F, · -257. 28
• ..!..",-1I~21
:£,
+ 999/.2'
'n .. 273'1.22
-mll.m u s - 99.'0
- 28.24.mlJ·mu •
tlu • -/27. 74
-SOJ.09
Nt.s_Et,ID.!
..
.a'bb
. - 7?a.oo
- 4S08.74
mn • + 1I.~4
- 9.9'
IlI U "
7425.00
lOS; f'9'1. BO
.B'bb
IOoamu " •
T /4 Sao. co
oB be ,
- 4.00
/33.00
-2700.00
-ebb
oo
S33G.80
• AQQ
rOOOm,"t II 33'.' 4
+3'75.00
+Aob.
-332.30
2/28.00
·..- 5297.00
'C'
5"9'19.84
• alee
CO.34.0o
S&C. 44
,c'cc
..
..
-8.,/ ··..,
.c·
'C eo
-c.•, e
333.41
77t. 2.50
2425. 70
.A'ae " S33C .80
..
..
rCOom"' 54,
.c'
.
III
'. " . ad , 1/ 9. so , - 22.59
4.37
.A~
-C.,/ • _ 299.2,
C
B,
,.5999.84
.B'
,333.2S
.!'.<
.y ••
- 1.,7
+-1/9.50
-A.'1 J. • _
z
200.00
7
B
~
125, 199.80
'
.A..
STEP
"-
+207'
'('I Zqq
17803
REACTIONS
BASED ON Eh
11
III
POSITION
v'
,
v~
.',
.,
- 15C,88
v',
v',
v'
,
v"
- 4&3&
,
v'
,
v,
1-17q5
1
v'
,
v'
PIPE
Z
/0'
ZQ.9
!.
a
Sb'lj3 Mb
s'bc f "',
2S1 I Ml
~
Sb.... S·~+4S,
Sb. tl: +4S, • S'
THE U W. I([ll-OCO
COU'ANl
.401J4
/
'-2 '1(,
18(;1
720
43.6'Z4000
(;OS
'1'''.0
~I-H'8IllTY.
lotto STRESS ANALYSIS
uOllons ."0 STFl.HSES
CALC
"'!J G
~~~~~E°4'.1(,.":"~
Ira".. NO r
CALC. 1<0
4. /5
112
DESIGN OF PIPING SYSTEMS
analysis and permits the relatively simple application of the General Analytical Mcthod to problems
involving two points of fixation ae' covered in Section 4.6 of this chapter. It is continued in the further
approximations, i.c. guided cantilever and assumed
thrust axis approaches, of Sections 4.4 and 4.5. For
Schedule 40 or heavicr pipe curved to a radius of
five diameters or over, flexibility factors are neglected
with little error; however, the developcd lcngth and
disposition relative to the neutral axis differs from
that of tangent straight elements with a resulting
effect on the system stiffness.
There is fairly extensive successful experience
with the design of piping systems to the stress
range of hs, + Sh) (which was in effect until the
1955 revision of the Code) on the basis of squarecorner analysis neglecting both flexibility and stress
intensification factors. These eompanion assumptions tend to offset eaeh other insofar as stress
evaluation is concerned, provided details involving
high stress intensifieation are avoided. The reactions obtained, on the other hand, are always on the
high side. Sueeessful past experience might be taken
to indicate that piping has operated safely at somewhat higher peak stresses than nominally ealeulated.
This experience, however, has been predominantly
with steel pipe of sehedule 40 or heavier thiekness
which did not involve exeeptionally high iJ factors.
With a greater trend toward use of thin walled
pipe, coupled with the recently inereased allowable
stress range, there is increased need to take stress
intensification into consideration.
The Piping Code rules, as revised in 1955, require
that, when using approximate methods, the effect
of stress intensifications be taken into account.
This requiremcnt would be satisfied if a correction
factor applied to the stress calculatcd by the squarecorner approach would always assure that the adjusted stress is not lcss than would result from
application of thc Gencral Analytical Mcthod. An
obviously safe means would be to apply thc full
stress intensification factor to the stress at the
square corners. However, comparative calculations
show that this seriously ovcrcstimates the effect because of thc ncglected flexibility, and results in uneconomical and unnecessary provision of excess pipe
length. Much effort has been devoted toward
developing a simple guide for a safe, yct not unduly
conservative correction. Despite this effort no simple
rule has evolved, since the configuration of the line
and thc location, as well as amount of curved pipe,
add up to a eomplex influence. In the majority of
eascs no eorrection for stress is needed because of
the compensating influence of the flexibility factor,
and because the maximum stress, before application
of a stress intensification factor, is sometimes not
located at a bend. Naturally, this is not a generally
safe assumption, since there will be many instances
where a correction is necessary. In such, an empirical correction factor of the order of v'~ instead of iJ
will usually be ample. It is best, however, for the
designer to explore typical configurations by comparative ealculations in order to reinforce his judgment for specific applications. As mentioned previously, it is not necessary to apply the correction
factor to the reactions since they will be on the safe
side. Where the reactions so estimated prove too
high and a more accurate evaluation is desired, there
is no substitute for a solution by the General
Analytical Method.
Efforts have been made to reduce the inaccuracy
of the square-eomer assumption by factors which
eorrect the deflcction contribution of the individual
component and apply its influence on the overall
piping system at the correct relative location. The
additional flexibility which curved pipe exhibits
may be conveniently expressed as an increase in
length or "virtual length" whieh would be required
to produee the same deflection on the basis of unit
flexibility, and may be extended to the squareeorner equivalent of a bend as follows:
L, = virtual length of bend, ft
R = radius of bend, ft
k = flexibility factor of bend
For a 90° bend the developed length of the squareeorner equivalent is 2R and
L, = l.57kR
The additional virtual length to be applied to the
square-corner equivalent in simulation of bend flexibility is given by
L, - 2R = R(1.57k - 2)
The simplest locations for the application of this
additional virtual length are at the intersection
forming the square corner, or else at the center of
the bend to which the square corner is equivalent.
The first of these locations overvalues, and the second undervalues, the stiffness eontribution to thc
piping system. A more aeeurate location would bc
to apply this excess at the center of gravity of the
pipe bend; however, the considerable added effort
is unwarranted in view of the still approximate results obtained. A further altcrnative would be to
1
r
Table 4.4
Comparison of Various Corrections Applied to a Square Corner Solution in Order to Approximate the Effect of Curved Members
r.-P
Ll
L,
Value and location of maximum stress range, as calculated by
12-ln., Schoo. 20 (.25")
AST1I A-lOG Grade B
pipe
Tm u-630F, Tmin70F E - 28.8 X 10'
psi
General
Analytical
Method
Per Cent
Devia!'n'
I 1, I R I h I k i P I Valua ILoc'n I
1----------- ---12
12
1.5
.12
12.7
3.35
I
1--1-1--1-1--1--11--1-1
a,
12
12
5
.38
4.3
1.52
0.0
40,100
1 - - - 1 - 1 - - 1 - - 1 - - 1 - '11--1--1
a,
b,
I
12
12
1.85
.92
1.12
0.0
64,800
~ - - - - - - - - - - --11- -I --I
18
18
-----II-I-I
12
12
1.5
5
.12
12.7
.38
4.3
3.35
1.52
r-;;---------~_12
29,800
36,600
a
a
0.0
0.0
31,900
I
37,800
a
'37,800
1
1_2_~~~ 50,900,
0.0
37,800
I
a
I
I
38,100
a
0.0
45,200 I a
24
12
5
.38
4.3
1.52
42,600
a
0.0
45,200
24
12
12
.92
1.85
1.12
40,300
a
0.0
45,200
a
,60
12
1.5
.12
12.7
3.35
89,000
a
0.0
93,500
n
0.0
-1--1
93,500
a
F
--------- --
r;;;----------
I 60
60
12
12
5
12
.38
. .92
Layout and service conditions as
shown in Sample Calculation Nos.:
(All adjacentl'gJ! connected by ,bor!radius welding elbows).
4.3
1.52
85,700
-a-I
I 21,600
+66.7
- 3.3
+25.8
I
a
I
I
I 30,500 1
27,200
1-
a
+25.7 137'300
- 6.1
1--1--1
1--1--1
. 35,500
n
!-I
-12.2
37,900
- 5.1
I 85,200
-
+]6.7
a
+ 5.9
a
+ 4.3
I-I
185,600 1--1
- 9.1
a
I
32,800
I
I
I
I 21,900
I
1 19,900
I +45.4
- 0.8
25,900
a
!
a
I +36.2
1-- 6.3
27,200
I
!--I
1 89,800
1
a
- 0.9 168'100
-11.2
70,300
1
1--1
+ 0.1 195'300 l_a_
I
88,300
a
-51.5
81,500
(
4.12
17,400
a
0.0
16,700
A
+ 4.6
12,900
0
+26.3
13,600
4.14
15,500
0
0.0
11,600
0'
+26.6
11,600
0
+26.6
4.15
12,100
0
0.0
8,900
A
+26.5
I 9,300
0
+23.2
,
I
12,100
+33.2
a
- 0.7
-60.4
!
27,800
n
a
a
+35.6
1
93,500
I
I
a
I-I
0.0
I +66.2
I
I +42.4
1 40,600
a
a,
b,
a
I
58,300
,b,
a,
21,100
1--
I a I -39.8
1 34,800
I
I
I
~
1
I
s::
I +67.5
~
,
I
::l
! +33.9
- 1.9
a
1.12
,
I I:
1,--1
I 45,300 I
I
I
- 0.7
I
a,
b,
[fJ
Deviat'n*
I
a
a
I
I
I
I
+52.9
b
1 38,400 1
1.85
-
I
-------
15,000
+35.7 . 1 16,000
b
1-1-1
I
I-I
a
~130,ooo 1 a
-2~1...:'::900
+ 7.6
3.35
I
I
a
I
12.7
-29.1
I
....
~
Per Cent
Loc'n
Value
I
a,
I -18.6 I 35,200
.12
I
b
1 29,300
+61.0 I 31,600 1
+41.9
1.5
1
19,150
a
12
---------------
1
I
I
1 29,600
a
24
!
+14.1
1--1--1
+50.8
,
I
b
a,
1--1
n,
b,
1 19,500
Distributed proportionally to members
Per Cent
Loc'n I Devial'n*
Value
1
1--1-1
+35.0
,
a,
12
-40.5
I-I
31,900
Per Cent
Deviat'n-
1--1-1
a,
,b,
Loc'o
Value
Deviat'n·
I-I
31,900
Cenrer of gravity
of elbow
Corner
Per Cent
Loc'n
Value
I
0.0
22,700
Modified Square-corner Solution. Stress intensification factor
considered. "Excess virtual length" of bend concentrated at
Squarc-corner Solution
(Stress inrens. and flexibility factors ignored.)
a
a
_
a
81,100
a
"'l
o
I ~
~....
....t:::I
+32.0
I
§
§
I
+13.6
I >
I
+18.0
....
Z
I >
I +23.5
t"'
><
[fJ
I
[fJ
-39.1
0 I +22.3
12,100
A ~
--I----I- - -I----=.:::.:....j
I +22.4
9,700
0'
i +20.6
7,500
A
0
--9,600
0
• Percentage deviation ... 100 (He - Ra)/Re. wbero He - exact result obtained from General Analytical Method. and Ra ... result obtained from approximate method.
deviat.ion denotes erron; on the unsafo side.
+37.8
.--, +38.0
A positive value of the percentage
<0
114
DESIGN OF PIPING SYSTEMS
distribute the additional virtual lengths between the
straight members representing the square-corner
equivalent; this will generally ...,Iso overvalue the
stiffness contribution to the system. Table 4.4
shows for a simple system the effect on the actual
stress range of these alternate location assumptions
including the stress intensification factor of the bend,
as compared with the General Analytical Method
results and those of a square-corner solution with
both the stress intensification and flexibility factors
ignored. It also demonstrates the futility of attempting to get good correlation with the General Analytical Method by approximate methods, however
refined.
It can be seen, therefore, that although approximate analyses have a place in piping system analysis,
the extent of their utility depends strongly on the
experience and judgment of the designers. When
used for final designs, by far the best results will
be obtained through the simplified square-corner
method of Section 4.6, thus avoiding unassessable
errors in analysis. Added refinements, such as discussed in the preceding paragraph, are usually not
warranted, since where greater accuracy is needed
the General Analytical solution will prove no more,
burdensome in the long run and will provide the
only reliable results.
References
1. E. A. 'Vert and S. Smith, Design of Piping for Flexibility
with Flex-Anal Charls, Blaw-Knox Co" Power Piping Div.,
Pittsburgh, Pa., 1940.
2. S. W. Spielvogel, Piping Stress Cakulations Simplified,
Lake Success, New York, 4th printing, 195!.
3. lIZ_, Jr., U-, and Expansion U-Bends," Paper No. 4.02 of
Piping Engineering, Tube Turns, Inc., Louisville, Ky.,
1951.
4. C. T. ~·,'Iitchell, "Graphic Method for Determining Expansion Stresses in Pipelines," Tram. ASME, Vol. 52, pp. 16776 (1930).
5. T. E. Bridge, Hliow to Design Piping with Required
Flexibility," Heating, Piping and Air Cond., Vol. 22, No. 10,
p. 94; No. 11, p. 94; No. 12, p. 92 (1950); Vol. 23, No.1,
p. 136; No.2, p. 107 (1951).
6. L. F. Randolph, IIEnd Reactions and Stresses in 2 nnd 3Dimensional Pipe Lines: A Simplified Method of Calculation," Imp. Chem. Industries, Ltd., Billingham Div.,
March 17, 1953.
7. \V. A. Wilbur, "Thermal Stresses in Piping Systems,"
Petroleum Refiner, Vol. 32, pp. 143-148, 163-168, 174-181
(1953).
CHAPTER
Flexibility Analysis by the General
Analytical Method
I
consequence, may be included for very stiff lines
where they assume significant proportions. In
addition to the effects attendant to restrained thermal expansion, concentrated and uniformly distributed loads such as those due to gravity, static
pressure, and effect of wind may be included;
N the previous chapter, simplified and approximate methods were presented for the calculation
of stresses and reactions in piping systems sub-
jected to thermal expansion. The brevity and ease
of application of these methods was achieved by
the omission or approximation of certain influences
dynamic or impact conditions reducible to an equiva-
on over-all elastic behavior. Such solutions have
their place in preliminary and rough analyses, but
for final checking of piping systems whose dimen-
lent static loading can also be handled by this
approach.
Naturally, a method of such scope is not as readily
mastered as the simplified or approximate methods;
furthermore, the analysis of an elaborate system is
bound to be time consuming because of the large
number of variables involved in its geometry. It
has been demonstrated that, for equal accuracy, the
required effort cannot be reduced beyond that obtained by the advantageous selection of the coordinate system origin. On the other hand, accuracy and
speed are greatly improved by a universal systematized approach with a high degree of organization and
carefully planned form sheets. Calculating time is
further reduced when the work is performed by a
group assigned more or less exclusively to piping
lIexibility problems; for such use, the method is
ideally suited and has been widely adopted. The
organized approach which it provides is directly
adaptable to programmed automatic computing
machines and has been universally employed for
this purpose [3J. The economic attraction of automatic machine computations has been greatly enhanced by achievements on programming developed
by The M. W. Kellogg Company, which makes this
approach practical for problems of virtually unlimited complexity even with machines of limited
storage capacity.
sions or service performance are critical, a method
is needed which combines accuracy, versatility, and
comprehensiveness. These requirements arc met
by The Kellogg General Analytical Method.
5.1
Scope and Field of Application of the General Analytical Method
Originally presented in the first edition of the
Design of Piping Systems in 1941 [I], and subsequently by Wallstrom [2], The General Analytical
Method appears herein in its most recently extended
form. By this method, stresses, reactions, and
deformations of any piping system can be evaluated,
confined only by the conditions of elastic behavior
and static loading. The number of straight legs
and local components, such as circular arcs, miter
bends, corrugated tangents and bends, connections,
flanges, valves, is unlimited; individual elements
may be oriented in any direction, arranged in any
order, and may vary in stiffness, size, thickness, or
elastic constants. There is no restriction to the
number of points of complete or partial fixation
either at the terminals or at intermediate locations.
The method is not confined to a consideration of
bending and torsion alone; the effects of axial or
shear forces on the dellection, while usually of minor
115
116
DESIGN OF PIPING SYSTEMS
The inherent accuracy of the method itself is
more than adequate for engineering design purposes,
and in some eases might be consideted unnecessarily
refined in view of variations in piping dimensions
and tolerances. The significance of substituting
rigid square corners for elbows or tees is discussed in
Chapter 4. Omission of the so-called "secondary
term" and direct and shear effects is discussed in
this chapter along with the treatment of those
subjects.
Apparent errors may arise in the interpretation
of the results of a flexibility analysis. The assumption of linear elasticity would appear to introduce a
gross error in systems which acquire self-spring
through operation under creep conditions. However, if the expansion stress limits are maintained as
proposed in Chapter 2, plastic action will practically
cease as soon as the full self-spring has been realized.
The piping will then operate elastically with respect
to thermal effects, and the range of stress which the
piping undergoes in a single cycle of temperature
change will be dependably predicted. This is likewise true of the ranges of reactions and deflections.
Absolute values of reactions or deflections are not
predictable since the redistribution of stresses which
occurs in the process of acquiring self-spring is not
taken into account with present methods. However,
maximum expected reactions, satisfactory for all
design purposes, can be established by the method
given in Chapter 2 whieh also makes clear that the
range of stress and the range of reaetions are the
important fatigue performanee indices.
Computational errors afC a serious problem on
oecasional ealculations and ean be equally so on
routine calculations in the absenee of proper organization and eare. While it is possible to eheck the
final results obtained when using any method, the
General Analytical Method is advantageous in that
it has been set up to permit the checking of calculations at progressive intervals in the progress of the
work, reducing the time lost to a minimum.
The economics of accurate piping flexibility analysis is greatly improved by the use of automatic
data processing machines. If manual computation
is employed it is advantageous for the work to be
performed by specialists who are able to save time by
memorizing many of the operations. The material
in this chapter is devoted exclusively to application
of the method so as to provide a convenient text
for the training of such specialists and can be readily
mastered by routine calculators without advanced
mathematical training. The supporting derivation,
which is presented in Appendix A together with a
brief history and a fairly complete bibliography on
the subject, does not have to be mastered to perform
the routine calculations described in the present
chapter.
Calculating Aids
In all but the simplest cases, computations according to the General Analytical Method invite the use
of some kind of calculating aid. In some eases
slide rule results are not entirely dependable; hence,
for routine work, the ten place digital calculating
machine has become more or less standard equipment
and can be depended upon to maintain sufficient
accuracy. Actually, as will be apparent in the examples to follow, it is rarely necessary to use such
a machine to its full capacity throughout the computation. Experience has shown that carrying two
decimal places in the shape coefficients, five decimal
places for multipliers in the equations, and two
decimal places for the resulting forces and moments
will usually assure a satisfactory cheek of the equations. Naturally, the accuracy of the results will
only equal that inherent in the data entering the
calculation; hence, in the final tabulations the results
are rounded off.
Automatic programmed computing machines drastically reduce computation time and thereby make
it practical and economical to analyze piping systems
of any degree of complexity. The various computers
available differ primarily in operating speed, and
in storage capacity or (fmemory," with the larger
installations minimizing the need for intermediate
manual operations. Even machines of relatively
limited capacity may be used effeetively for eomplex
analyses by resorting to inversion procedures described in Section 5.19.
At the present time most automatic computers
represent expensive installations the economic
utilization of which requires broad application to
many accounting and engineering calculations rather
than exclusive use for piping problems. Experience
with piping calculations at The 1\1. W. Kellogg Company Electronic Computer Laboratory indicates
that with proper scheduling of the work the over-all
economies as well as delivery time are significantly
better than those of the most effieient manual piping
computations. It is worthy of note, too, that these
savings are aecomplished, in all but the simplest
piping configurations, in spite of the greater time
and care needed for preparing and ehecking information fed to or processed by the machine. The outlook
is for increasing application of computers in piping
flexibility analysis. It would be a mistake, however,
5.2
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
to underestimate the burden and responsibility
resting upon the engineer who must prepare and
interpret all significant inform!ltion and who originates the program of instructions which prescribes
the steps the machine must follow.
5.3
General Outline of Operations
Before entering into the details of the General
Analytical Method, it may be advisable to outline
briefly the complete process, and in so doing, to note
the relationship with the Simplified General Method
covered in the previous chapter.
The first stage of the work consists of recording
the given data and setting up the problem so that
physical and geometrical properties of the piping
are expressed in numerical form, which operations
are parallel to the set-up procedure of Section 4.6,
but are expanded to include elbows and bends and
the flexibility and stress intensification factors of
these components. This is readily apparent in the
examples which follow. It can hardly be overemphasized that great care must be exercised to
avoid sign errors. This is a particular source of
difficulty to the beginner, who is advised to master
thoroughly the sign convention described in Chapter
4 before attempting any calculations.
Numerical calculations are performed in the
second stage and although the General Analytical
Method and its simplified counterpart involve the
same general principles, they differ to a marked
degree in execution. While in the Simplified Method
the approach is completely formulated, in the
General Method the work is performed in a number
of distinct steps, the basic procedures of which are
the same regardless of variations in complexity of
geometry, loading, constraints, etc. These steps
include:
a. Computation of the shape coefficients for each
member.
b. Summation of the shape coefficients (this operation may include a number of intermediate summations before the final coefficients are obtained).
c. Solution of a system of simultaneous linear
equations in which the summed shape coefficients
become the coefficients of the unknown forces and
moments, while the known terminal displacements,
elastic constants, and moments of inertia are represented in the constant tenns.
d. Transfer of moments to various points (the
moments computed are referred to the origin; hence
it is necessary to apply suitable transformations to
obtain the effects at the ends or in other desired
locations).
117
e. Calculation of stresses at significant points.
f. Adjustment of forces and moments to obtain
the anticipated initial and ultimate effects on equipment, etc., taking into account cold spring if it is
employed.
g. Calculation of deflections at significant points.
These are the principal steps, but special operations, discussed later in the chapter, are required for
the more complex problems.
The third stage of the flexibility analysis involves
the evaluation of the results. Calculated stresses
are compared with allowable stress ranges at significant locations as discussed in Chapter 2, while
terminal and other loeal effects are considered in
accordance with Chapter 3. Rough comparison
with calculations of similar piping is advisable whenever possible to confinn generally the assumptions
and the reliability of the results. For highly critical
piping, comparison by model testing is usually
desirable.
5.4 The Solution of Simultaneous Equations
One of the important steps of the General Analytical Method is the solution of the system of simultaneous equations which appear in every problem.
Although such systems of equations can be solved
by several different methods, the one discussed in
this section has been found to be highly efficient.
Since experience has shown that the solution of
equations is one of the most difficult steps for a
beginner to master, it will be described here in
considerable detail.
The equations are always first degree or linear.
The variables are unknown moments and forces, Of,
in special cases, unknown rotations and deflections.
Each equation is related either to a certain rotation!
in which case it is called a moment or rotation equation, or to a certain displacement, when it is called a
force or displacement. equation. In the case of a
single-plane line with two end points and with
expansion in the plane only, three unknowns, one
moment and two forces, must be determined by the
solution of three equations. If another branch in the
same plane is added, three more unknowns must be
computed and so on, the number of unknowns or of
simultaneous equations being 3(n - 1) where n is
the number of end points. For a line in space with
two fixed ends six unknowns, three moments! and
three forces must be evaluated. For branched
systems in space! the number of equations is
6(n - 1). Stops or guides providing partial fixation
require one additional equation for each component
of fixation
us
DESIGN OF PIPING SYSTEMS
To simplify the explanation which follows, three
equations arc used. However, the procedure is
general and applies to any numbe~' of equations of
the type arising in piping analysis. If the unknown
moments and forces are written in a given order in
each equation horizontally, and the equations corresponding to the moments and forees are written
in the same order vertically, the matrix formed by
the coeffieients of the unknowns will be symmetrical about the principal diagonal. In the procedure
described below, the top equation is always eliminated in such a way as to maintain symmetry in the
remaining coefficients.
process with these equations reduces the number
of equations to one and permits the determination
of Ft..
A complete solution with a description of every
step including the eheck is given in the following
numerical problem. The equations
lOF, +
20F, -
30F, =
100
20F, + 100F, -
90F, =
500
-30F, -
90F,+ 120F, = -1200
are written:
Given a set of three such simultaneous equations:
Equation No.
F,
Fy
F,
AF,+BF,+DF,= -Elt>.,
1
+10
+20
-30
+ 20
2
3
- 30
- 90
+120
BF,+CF,+GF,= -Elt>.y
DF, + G Fy+H F, = -Elt>.,
The constants are transposed to the left-hand side
and the whole is written:
F,
F,
F,
Constant
+A
+B
+D
+ El t>.,
+B
+C
+G
+EI~v
+D
+G
+H
+ El t>.,
Table 5.1 illustrates the solution. The three
given symmetrical equations 1, 2, and 3 are reduced
to two which are likewise symmetrical, as shown on
lines 5 and 8 of the table. A repetition of the same
+100
- 90
Constant
- 100
- 500
+1200
The complete solution is given in Table 5.2.
It will be noted that the coefficients to the left
of the principal diagonal fall out as the equations
are reduced. The form sheets presented later for use
in solving equations take advantage of this faet by
omitting entirely that part of the solution.
The calculator is. cautioned that all coefficients
along the principal diagonal must be positive in
sign. A negative sign means that an error has been
made either in the solution of the equations up to
that point, or in the calculation of the shape coefficients.
A complete check by substitution of the unknowns
into each of the equations is essential for the method
Table 5.1 Method of Solving Simultaneous Equations
Equation No.
and Operation
Line
F,
F,
F,
+A
+B
+D
+Elt>.,
-1
B
A
D
A
- Elf:1%
-A-
+B
+C
+G
+Elt>.,
4. Multiplylinel by (-BIA) from line 2..
B
--·A
A
B
--·B
B
-_.f)
B
- A
(Elt>.,)
5. Add line 3 and 4. .................
0
B'
+C--
Bf)
+G-A
+EI (t>., - ~ t>.,)
No.
1.
Equation 1.. .
...........
2. Divide eq. 1 by (-A) . ..............
3. Equation £ ..
. .................
/l
/l
/l
Constant
6. Equation S.. . . .......
+D
+G
+H
+Elt>..
7. Multiply line 1by (- f) I ,1) from line 2..
D
--·A
A
D
- -·B
A
f)
-_.f)
A
- A (Elt>.,)
8. Add line 6 and line 7.................
0
D'
+II - A
+El(t>.. - ~t>.,)
+G _ BD
/l
f)
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
the distribution of error in succeeding equations
may form a clue as to the source.
of solution presented. The sum of the products of
the unknowns times the cocfficicnts in each equation
must equal the related constant with reversed sign.
It is recommended that the multiplications for the
check be performed as each unknown is found. In
this manner, the products for the check are completed as the final unknown is determined. When a
summation indicates that a particular equation does
not prove, the error may be sought within that
equation. However, it is advisable first to complete
the eheck summations for all the equations, since
Table 5.2
Line
No.*
EquatiOD
1
1
2
119
5.5
Single Plane Caleulations
When the General Analytical Method is applied
to a simple flexibility problem, the calculation is
brief and contains only the essentials to the case at
hand. A single-plane square-corner piping system
with two fixed ends is a problem of this caliber.!
IThis type of problem can also be solved by the Simplified
General Method presented in Chnp~r 4, Section 4.6.
Complete Solution of Three Equations Using Simple Numerical Coefficients
F,
+
10
1.00
+
20
2.00
+
30
3.00
+
Con-
Operation Going
Operation Going
stant
Down
Up
100
10.00
Divide line 1 by -10
Multiply F, coefficient of line 2 by
-50.00
Multiply F. coeffi3
4
IF. =
+
100.00
+
40.00
20
+
100
5
20
6
7
o
o
+
30
9
+
+
60
1.00
+
30
0.50
+
+
o
o
o
+
30
+
120.00
+
o
o
15.00
+
15.00
1.00
Multiply
Multiply
F 1/ coeffi-
cients of
lines 1, 4 ,
cients of
lines 1, 4,
9 by
-20.00
-400
-2000
+1800
F z coefficients of
lines I, 4,
9 by
-50.00
+1500
+4500
-6000
9 by
5.00
1200
Multiply line 1 by
+3.00, the F, coeffi-
+
150
Multiply line 6 by
+0.50, the F, coeffi-
750
50.00
Add lines 9, 10, and II
Divide line 12 by -15
cient in line 7
Constant
- 100
- 500
+1200
"Line numbers correspond to lines on standard three equation form sheet.
.....
Multiply F, coefficient of line 7 by
-50.00
F II = -25.00
+ 5.00 ~ -20.00
50.00
Multiply
-100.00
-1000
-2000
+3000
{cient in line 2
300
Add lines 4 and 5
5.00 Divide line 6 by - 60
300
90.00
F coeffi%
14
15
16
+
25.00
- 2.00, the F II coeffi-
200
cient in line 2
F, =
Check
Multiply line 1 by
60
90
cient of line 2 by
-20.00
F. ~ +40.00
-150.00 + 10.00
~ -100.00
10.00
500
+
60
12
13
+
40
30
11
150.00
90
20.001
8
10
-
o
o
o
120
DESIGN OF PIPING SYSTEMS
y
'-------,
FIG. 5.1
The z-plane.
Single-plane systems are usually drawn and calculated in the z-plane (Fig. 5.1). The sketch is
made and the given data recorded on Form A in
accordance with Steps 1 through 5 described for the
Simplified General Method. If there is expansion
in the plane only, the following steps arc taken.
Step 6. On Form B-1 enter the following as
indicated:
Member number.
Shape (horizontal, vertical, or inclined).
Length of member L, ft.
Distances a and b, ft, i.e. the x- and y-coordinates
respectively, of the midpoint of the straight member.
Value of L 2 /12.
Step 7. Have Steps 1 to 6 checked.
Step 8. Compute the shape coefficients for each
member in accordance with these formulas:
Shape
Coefficient
Horizontal
Member
Vertical
Member
The foregoing procedure is exemplified in Sample
Calculation 5.1. The same system is calculated in
Chapter 4 as Sample Calculation 4.10, and a comparison between the two shows that identical results
are obtained. While the Simplified General Method
has the advantage of ease of computation for the
uninitiated, the method in the present chapter is
more fundamental and hence, more versatile. Both
methods involve the same amount of work.
A single-plane system with expansion perpendicular to the plane only, requires the solution of two
rotation and one displacement equations. These
equations are entirely independent of those for
expansion in the plane. For a line such as that
shown in Sample Calculation 5.1, if there is expansion in the z-direction. M r, M., and F, must be
found. This case can be solved by following the
procedure and using the form sheets described for
multiplane systems in Section 5.12.
5.6 Inclined Members and Changes in Stiffness
The procedure for calculating a line having straight
members which are inclined to the coordinate axes
departs bnt little from that described in Section 5.5.
The only difference lies in the use of the general
formulas in the calculation of the shape coefficients.
S = kQL
B
kQL
kQL
Sa = a X S
Ba
Bb
Bab
Baa
Bbb
aXB
bX B
b X Ba(Or a X Bb)
2
a X Ba
BL /12
b X Bb
aXB
bX B
b X Ba(Or a X Sb)
a X Sa
b X Bb
BL2/12
Bb = b X S
+
+
Sum the shape coefficients, s, sa, Sb etc., across to
obtain the final coefficients.
Step 9. Enter the final coefficients and the constants into the equation as indicated on the form
sheet. For complete fixation at the ends, end rotation 0: is zero and the constant in the moment
equation is therefore zero. The constants E/* tl z
and E/*D." arc transferred from Form A, the term
EI* = E hI/144 denoting stiffness expressed in Ib-ft2 •
Step 10. Have Steps 8 and 9 checked.
Step 11. Solve the simultaneous equation and
substitute the values obtained back into the equation to check the answer, as explained in Section 5.4.
Step 12. Compute the moments at the various
points, find the point of maximum stress, and tabulate results.
Slep IS. Have Step 12 checked.
+ s(L2/12) cos ex sin ex
2
2
Baa = (a X Sa) + s(L /12) cos ex
2
Sbb = (b X Sb) + B(L 2/12) sin ex
Bab = (b X sa)
where ex is the angle of inclination measured from
the positive horizontal axis to the member as shown
in Fig. 5.2, positive in counterclockwise and negative in clockwise direction.
Sample Calculation 5.1 is calculated for 10 in.
standard pipe, A-lOG, Grade A material. It is
apparent that for the same expansion the magnitude
of the moment and the forces is in direct proportion
to the stiffness EI. Hence, for any pipe with a
stiffness of ENI N the moment and forces may be
attained by dividing those shown in the example by
Q = El/ENI N, or multiplying the summation coefficients by Q and solving the equations.
y
y
a
-ex
b
L
-L__ ,
FIG. 5.2
L-L
,
Angle of inclination of straight members.
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
O'A
MEMBERS
0
10.75
=
t
I
h
k
~
o £,,11
Z
Q
f---o'
C.$. -A'I(X,-GRA
TEMP.
~
9OO~
e
W
Q BRANCH
.00650
Z
+40.00 (.0 0(50)=
8.
+.2/.;000
8,
-Gzse280
~J6%/144
0
MOMENTS (I=T -LB)
COLD CONDITiON
COLD 5PRIN'G
ACTORC0'
tJ) POINT
:5
Mx
M
0:
F.
F
F
AND FORCES LO)
ACTING ON RESTRAINTS
A
- 12S00
.,.. S80
- /390
0'-/
0'
k
Q
R
L
- 19 77S
A
a
_
-
L ti?
S
5000-
Sb
-
~.P_lOO
S
Sbb
o
5OToo
()
/25 00
333 30
o
z 250'00
EQt/lfT/dNS -
1
1"~
t.,.l~
I
25 00 208 33
SO 00
O....TE'Z.-/" "'D
s
~
F M A
CAL NO. 5"-,
Sf)
"0 00
52 08
100 dO
SO 00
-z 550
-15 625 ()O
0
0
-
a
5
'1
000()
/0
>=.
+ /CO 00 1"~b -2. SSoo
Fy
-~ ...
815. ()
,;3: :l:
-I
0
CONSTANT
()
0
-l.eaOCQ
"25500
"'50000
w2l11734'
·23 f)S 3
"'/08Z8S
0
2
+Sbb '111 49
0 ·l> ... b'" 87$ 00 .EI411. +6 Z e
psi
.2/1"ZS" psi
• c.
00
25 00
0
/ SO
-I ZS 00 -125000
4/ 6
M,
po,..,. A,
25 40
o
-
liZ SO
-
o 00
00
1S
IS 00
I
OD
1 SO
o
10 0
!>
5 ..b
I
,t,T
5A·~i~~t~1.. .~<:o
2·A
0-2
1M
10 00
IS
00 5 00
8
3
.((" 7Z5
9(){)
r 1.!J90
I 00
b
p~i
SC a '6oo0 psi
Sf ·~:,,~~6~rtP
.,. 19S50 + 030
.,. 900
900
.,.- Z 170
2170
- sea
1-0
I
SHApe:
Sh a 6500
HOT CONDITION
THE MW KELLOGG COl PIPINGORIGINAL
FLEXI,I!ILlTY AND STRESS ANALYSIS
DATA AND RESULTS
MEI.l&
II
25.00'
~TRf:~:'
M,
'"w
A
'I
rb Z38280
EIl16X/144
EIoI6ylr.f.4
I-
~
"" ',"
-40.00 (.0 0&50)·
-.2"000
0
8,
ii:
IF)
•
/,00
~ MATERIAL
W
>
23W,J400
'"
ll.
0
;Y\ •
"x d..-{"
S;;
Z/.50
Z9.CJO
2 Ec"IO·
«
,
~
Eh" JO·
~
Fj
15.00'
'bO.7
19,90
"R
,M,
,
,~,
121
STRESS CALCULATION
POINT
M'~
z
so 0
]22
DESIGN OF PIPING SYSTEMS
Y
MEMBERS
O~z,
'-A
0
8.4>25
hlo2!i...
C
I
.$22
.,so
72.49
28.14
'" .81
8.49(.,
Z
M,
M,
,
R
h
~
Ul
Q
~
TEMP.
a.
a.
\i
+'072QO
~y
Q
. ,----~
.OOZ!i4 co
'/"10.00
'"
+,OZS40
0
~,
X
~J
~
:5
8.M':
Z
0
r28.6tdO IoZ6"4).
~,
,
15.00'
2.58
C.S.-A'IOb-G/?A
4,JO F
.00254
e
BRANCH
W
5.00'
1.00
~ MA ERIAL
-<
~y
IF,
M.b//
k
0 Eh" 10·
2'l.4lJ
Z Ec" 10-'-'
29.00
o EnII
137'13240
'"aZ
7\ ,
F,
Y
/'
O'
+1004150
'/".;1.'10..350
0
E"t6X//44
E"IlIYIrH
E"IlI:lJ411
MOMENTS (FT -LB)
COLD CONDITION
OLD SPRIIJG
FACTOR C.tI) POINT
I-'
M,
M
Ul
M,
W
IX:
Sh. 1.3 600 p:li
5<;. 1(., 000 psi
ACTING ON RESTRAINTS
HOT CONDITION
AND FORCES (LB)
O'
A
+ I08ZS
/410
_ 61150
'/" 141{)
-$50
+$50
5E ·~;..~~S~1I:0
~TR~:5:5
S
F,
F
F,
• 10 ZZ!' psi
I<T POINTA
SA' ~f~~t~L.. .l!lo.Ia.
.Z,3400 psi
THE MW. KELLOGG 001. PIPINGORIGINAL
FLEXI.BllITY AND SIRESS ANALYSIS
DATA AND RESULTS
0'2
• c.
C"
"<0
O ....TE I~-
A
F
.~
CALC NO. $.Z
-~
o·~,q
2'"
'Oi· CD$ " ...£1
- - CDs~N·15
' lli t
k
Q
R
/ ee
0'o
L
•b
L
12 50
eo
e
1000
1500
5 eo
[)
o
-
/
"
*-
1000
/93
o ee
-
.2 50
8
seo
o
1/"
o
-
()
-JJJf/9
8
33J 30
1 12500
()
o eo
o
2c:i
S89
I 1/9 49
2
18 1!'
ISDd
3
0'
S
+ S
IA~O
,
-8
11
1"5bb
1-1 ' "
~
;0
I
2.
77 -Gab
~Z
7" 2
+ 9S,"
-'.00000
M't
F
C\-IE.CK
I -4'5 '6' +-2861$6 p209510
0
'2 +-IDDJ!~ ·z..t~~!k' ~JB9 0
~oo.,. 1 ~
?;-7'S08.3{.05 -1S"S"6 8
387
-3S0350
POINT
O'
I
0
2----;.-.y
-5,"_0
-/S:t70
M%
_ S6S
..,F.·y
t- ZIIS
~F·"
M't
F"
.,.
l.2..~'''''' flO 985 I
-b 08
,U·h'"
.00
{.2.
~
+-! 8
/
-!oS
+8.<::
0
--;;:: 00
-
0
_
Q
'5
0
f"
S4<J
- /JS<i>5
tS
7"JO.5
""8.3
'1-05'29
-.3 70
o
0'
M',
1082.7
Z
8
s' "f/H.ft
o
~
F
... ,
.
5TRESS CALCULATION
POINT
f ~ +/ 00 15 fo 1'2/Z.
36
+.tJ
0
08
-.,
+ 102./ .,
.,.5.00
.,.5.00
'1-/082
0
5. .. f.I..~,'i'/
1-1 f03
F~/~/O
{Q
202 '8
~~'N
.2/4 98
66777
CONSTANT
F.
F
202.
w~,,_
o
+3 50 79
1-1 2 ~"
o
-4 94 81
-~ ';;20
+~b
- L 00000
U 1
S, 8
8
M,
1"~
I
2:::
'I·ts
',>.9
/"' .
8.
.71.38
/. 'ffl.';',j
~
9',"0
I~ CO~VER510N TO CODE RUl.ES:
30mS:::(l.
o
lSr:/022S
Rc R. ( t./~,,) C oAt
1:/.;
+-J 0260/, WHICHEveR l~Gll.rAT~p. _
RII
- r- 'lC)=
1-1 36
R ..C/U.C REACTIONS 8ASED ON
h
.,.Z3.
.00
t-
+- 3 2.;
THE MWKELLOGG 00 I PIPING F~~"drtLl~UN'jp ?~~Ht ANALYSIS
3
M
.N
$ .•
a
\
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
Sample Calculation 5.2 solves a system having
two sloping members one of which is of smaller size
than the remainder of the pipe.~· The shape coefficients are computed from the formulas given above
using an inclination angle, a = +45 for member
0'-1, and a = -30° for member 2-A. It will be
noted that the members for 0' to 2 are of 8 in. pipe
while member 2-A is of 6 in. pipe with a Q value of
2.58. In evaluating the stresses, the reduced section
modulus of the 6 in. pipe accounts for the fact that,
while the maximum moment is at point 0', the
0
maximum stress is at point A.
5.7
Circular Members
Members sueh as bends and elbows which are in
the form of circular arcs require more labor in the
calculation of the shape coefficients than do straight
members. Hence, the substitution of square corners
is common practice in flexibility analysis. This
matter was discussed at SOme length in Section 4.7
wherein it was shown that, in many cases, squarecorner solutions are not advisable.
The calculation of circular members is introdueed
by means of another single-plane line. The procedure of Section 5.5 again applies with a few exceptions. In the Pipe and Expansion Data on Form A,
it is necessary to enter the values of R, the radius
of the are, in feet. The flexibility characteristic h,
the flexibility factor k, and the stress intensifieation
factor (j are determined in accordance with the
Piping Code (see also Chapter 3 herein). The shape
coefficients for the circular members are calculated
from the following relationships:
s=kQRip
sab=(sXab)+(s'aXb)
+ (s'bXa)+S'ab
sa= (sXa)+s'a
Saa = (sXa')+ (s'aX2a)+s'aa
Sbb= (sXb 2)+ (s'bX2b )+s'bb
Sb= (sXb)+S'b
where
s'o = kQR 2ca
s'b=kQR 2Cb
and
s'ab=kQR 3cab
s' aa = kQR 3caa
Sf bb = kQR'J Cbb
b = vertical distance from origin to center of arc,
ft.
ip = arc of member, radians (when uscd directly).
a = angle measured counterclockwise from positive horizontal axis to the initial tangent
(more easily visualizcd as the angle between
a negative vertical axis and the normal at
the initial point of tangency).
The calculations of the shape coefficients for circular membcrs are facilitated by the use of Form D.
This form has space for two different mcmbers and
also provides for thc calculation of the additional
coefficients needed for expansion out of the plane
or for multiplane lines' The arrangement of the
form provides for a convenient sequence of computa-
tion. The procedure is as follows:
The given constants, k, Q, R, a, ~, a, and bare
listed in the respective spaces and 2a, a 2, 2b, b2, and
ab are calculated. The trigonometric constants ip,
in radians, Cal Cb, Cab, Caat ebb are entered. For the
most commonly occurring shapes (ip = 90° and
a = 0°, 90°, 180°, or 270°), numerical values of the
trigonometric constants are given on the form sheet.
A more complete tabulation of these constants will
be found in Table e-15 in Appendix C, which
includes additional values for both ip and a. The
functions kQR, kQR 2, kQR 3 are then calcuhted.
The coefficients s, s' 0' 8' b, s' ob, Sf ao are computed,
each succeeding cocfficient being the cross product
of column 1 by the adjacent trigonometric constant.
The multiplications s X a, S X b, etc., s' a X b,
s' a X 2a, etc., are performed and the summations
made vertically to obtain Sa, Sb, etc.
To illustrate the solution of a system with curved
members, the line calculated previously with square
corners as Sample Calculation 5.1 is presented as
Sample Calculation 5.3. It is assumed that the
bends ani made with long-radius welding elbows,
since they are the most commonly used fittings.
2Multiplanc shape coefficients arc given in Section 5.8.
y
a
cos a - cos (a + ip)
sina - sin (a + ip)
Cab = 0.25[cos 2(a + ip) - COS 2a]
Caa = 0.5ip - 0.25[sin 2(a + q:,) - sin 2a]
Cbb = 0.5ip + 0.25[sin 2(a + ip) - sin 2a]
=
Cb =
Ca
b
As indicated in Fig. 5.3
a =
horizontal distance from origin to centcr of
are, ft.
123
.L-.L--====_ _ x
FIG. 5.3
Angles for circular members.
124
DESIGN OF PIPING SYSTEMS
O'-A
10.7S
MEMBERS
....,
0
~
15.00'
11;0.7
29.90
I.ZS
I
Z
R
.zo
h
k
~o Eh" 10M
e
"'
8.00
~
Z.~O
Ax
Ijj
.OO~50
"~
sW
-.ZI;,OOO
FACTORCII) POINT
•
Mx
"
AND FORCES (Le)
COLD CONDITION
A
0
ACTING ON RESTRAINTS
HOT CONDITION
A
0
- I aZ5
-330
+-12$75
-t- iJ:?O
- '1 10
.,. 910
-t-/j70
840
Fx
F
F,
0'· I
,
,A.T PC1NT4
5A·~it&;':~~l!o./"
"Zt (QZ5' psi
Q
s-b
-/1
LJ
--
D
I
\J
--
0
I 00
/ 2$
1.3 00 -
" 5
800
b
.5
0
f
-
~b
-
t
25-
5/
i.e. -
'.<.. ~:r,4f'
(((;8 7
::.
2
I
.:5 50
f
~
F",
M t
/80728
I
2
f
:3
F
t/~'/2G(70 ofllf? (,8<f
216
()
d
.,5 bb ~
~f.,1:
-6 ... b
f{,
"
~
"'8.7,
.,. /. 25
M"
Iz9t1.3
.F.·y
-1='"
~20c./.j
-'1-38(0
0
- 627
I 18
.,.a'83
"tJ2
.,..,5
~
SOI.3
F
-1,00000
_
al
.3
6Z
+18
-/ /
.~.....
/I
F
0
I
- 122G.
.,.. /. 25
/,25
+- ,
_ I
-
18
231
o
"
'2
9
POINT
/1
M'~
1251
,
8
Z
0
.3
22
o
Z·6.0
tJl.34
()/
5 .fllM".
IZ8S.3
8~~
COWVERS10N To coo~ RvLe:S;
, .;.(Ec lSi: := I 350
f· 12 Z
6S
4al
I
0/
5TRE"SS CALCULATION
a
'9 .u·"" .,.
~
-
-I.
~¥a
CONSTANT
J4-1
0/ 61
2
~
,
+12,50
0
-
,
'"
6. .=...L
f/5 101
3
_15.00
-/0.00
&.t
I
0
W ~2
.,J '2
I 13
0
y
M'1.
:i
Z
0'
F
-I-
-
CHE.CK
x
/.:S
-
-
I
tJ
4-
I ~
r;.:;;S:U,,~
Gf845 38 5G
59:~ 00
-~.:o.
0
16.:55
_ §12881. 1020HZ
~238.3
8 3 6 M S 032 +/5(;';2385 "<:;2 .3 0
POINT
/3 1.3
I
1/3
IQ ..•
25
'5 ~
a
4 ~8
S
_
q ,
2
00
liEn
F"
+~b
/ 40
_
o -
- 1. 00000
-/2
0
o
, ., M,r I
l.,A ~
25
,71
4
•o
3 J
333 It;,
5bb
- I~ 7
L
4
_
0'
"
4
-
4
0
0'-11
800
0
/00
F
A
CAL NO. 5·3
".
4·5
/E
•
CALC.
Hr:CKE
3-4
8 00
~
ps>i
Sf .~~;:~rEO
rSOD
-/370
2-
R
L
,
p~j
5G·~()OO
-t-ZZ 2ZS
500
I· 2
I
5HAP(
L
51'1" &.:rOO
• 17 ~S"O psi
Y AND STRESS ANALYSIS
THE MW KELLDGG col PIPING_~U;~I,!I.lIl
ORIGINAL DATA AND RESULTS
MIMlif.
(
ZS".OO'
~TRr:55
M
M,
Vl
•
~.}
0
EIl ItroM44 +(qZ18280
fhltroy/rH
-6 Z38ZSO
0
Eh 1A IIr"
MOMENTS (FT MLe)
COLO 5?RIN
W
X
r.~OOO
A,
0:
~
Mx 6.--{,y
IF,
X
+4
-40.00 .0 {)(,S"O)'"
Ay
I.J
0
8b
-GR.A
a BRANCH
:z
r 40.00 (.ODuSO =
UJ
ll.
ll.
•
F./
,
If-!-
~
ZI.SO
29.00
Z EG"IO~
Of II
G899J4IJO
Vl
1.00
Q
~ MATERIAL C.s. AIO
900 F
~ TEMP.
w
Y
Y
'. 2
q 3
8
t
o
-/ z '-
•5
o
062
+ /. 2.5
o
+ 'Z38/1
0
1"11"19
-
z:
l¥rn
RcR'·{~/i.lcOII(
..... lCHE\/~I'lI'GllJ.\,TEIl =
•
70
Rh R :::(1- C)c:
RooCllL.C fUACTIOHS8ASEO OtHh
'" .r
.,. 23 .
o
·25
+ €2
a
- / /
rIc
- 32. (;,3
- ZZ" "2.3
THE MWKELLOGG CO I PIPING F~I7.'jj~tLHl;HP mH~ ANALYSIS
c~~-.
out
'"
.,..2$.000
-50·00
-12. 4-3
.,.2.S065
- 3 355
_ zz 23..3
.N
s.
..
,
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
c = perpendicular distance of the projected plane
from the parallel coordinate plane, ft. (See
Scction 5.11.)
The sketch on Form A shows the location of the
curved members. The data are taken from Sample
Calculation 5.1, with the addit'ion of the R, h, k,
and {J. On Form B-1, the a and b distances of the
members are set up in accordance with the instructions given in Section 5.5 and the straight members
are computed. The circular members are computed
on Form D-1, Sample Calculation 5.5, and the
results are transferred to Form B- 1. The rest of
the calculation is made as in previous cases.
To expedite the calculations, it is recommended that
the formulas for horizontal (a = 0) and vertical
(a = 90°) members be committed to memory.
For circular arcs in space problems, the shape
coefficients to be calculated are as given in Table 5.4.
The symbols k, Q, R, the trigonometric constants
Co, Cb, etc., and the distances a and b are defined
in Sections 5.6 and 5.7. The distance C is defined as
for straight members.
5.8 General Shape Coeffieients
When either the expansions or the members lie
outside a single plane, it is necessary to calculate
additional shape coefficients beyond those already
given in the preceding sections. For straight members the complete set of coefficients to be computed
are given, together with their formulas, in Table 5.3.
The quantity r m denotes the mean radius of the
cross section in feet and the distances a, b, and c
are defined as follows:
5.9 The Secondary Term
The shape coefficient q, known as the secondary
term, represents the effect of a transverse bending
moment on torsional rotation and a torsional moment
on transverse bending rotation. Its magnitude is
relatively small when compared with the other
shape coefficients and varies with the transverse
flexibility factor, becoming zero when this factor is
equal to 1.3. It is also zero for vertical or horizontal
straight members and for circular members if the
a = horizontal distance of the midpoint of the
straight member concerned from the coordinate origin, ft.
b = vertical distance of the midpoint of the
straight member concerned from the coordinate origin, ft.
Table 5.3
Coefficient
s
s.
s,
Sab
cu,
cv,
a=a
kQL
aXs
bXs
kQL
a Xs
b X s
0
0
s
s.
cXu
1.3QL
bX V
C X v
1.3QL
aXu
cxu
s
C X v
(1.3 - k)QL cos a sin a
cXq
(k cos' a + 1.3 sin' a)QL
(a X u) - (b X q)
cXu
(k sin' a + 1.3 cos' a)QL
(b X v) - (0 X q)
C X V
b X Sa
b X Sa
b X Sa + 12 cos ex sm a + c X cq
eX U o
c X Vo
eX U o
C X Vo
eX U o
C X Va
aXsa+cXcv
a X Sa + 12 cos 2 a + c X CV
0
+ c'lq
The effect of omitting q in the calculations of a
particular pipe line is illustrated in Fig. 5.4. A
quarter-circular bend with equal tangents is fixed
ex = 90°
kQL
aX.
b X.
0
q
cq
u
u,
cu
v
v,
cv
trigonometric coefficient Cab is zero.
Shape Coefficients for Straight Members
a=O
s,
sL'
SL2
.
SD2
SOli
+ cZv
ax so+
8bb
+ c2 u
bXs,+cXcu
bX8'+12+ CXcu
sL'
b X 8b +12sin'la + eX cu
U oa
+ Voo
sL'
a X u, + b X v, + 12
.L'
o X u, + b X v, + 12
sL'
o X u, + b X v, + 12
U
V
2.6Qrm 'L
2.6Qrm 'L
0.5Qrm 'L
IV
2.6Qrm 'L
0.5Qr m 'L
2.6Qrm 'L
0
0
2.6Qrm 'L
Qr m 'L(2.6 cos'a + 0.5 sin'a)
Qr m 'L(2.6 sin' a + 0.5 cos' a)
Qrm 'L(2.! sin a cos a)
S
12 +cXct
i
125
sL'
126
DESIGN OF PIPING SYSTEMS
Table 5.4 Shape Coefficients for Circular :Members
8.
8,
q
eq
u
+
u.
u'.
(u X a) (c X u)
eu
v
v.
e'
Sail
at both ends against rotation but one end is displaced
relative to the other in a direction perpendicular to
the plane of the line. The moments and forces
resulting from the displacement of end A are plotted
for various values of the bend radius and for the
flexibility factors k = 1, 1.3, and 5. The continuous
lines include the term q while the dashed lines
neglect q. Inspection of these curves shows that if
the circular member forms a small part of the total
line, the effect of omitting q in the calculations is
negligible. However, if the circular member represents the entire pipe line, errors of considerable
magnitude will arise. For most pipe lines, circular
members form but a small part of the whole; therefore, the omission of q is justified. By so doing, a
kQRiP
8'. + (a X 8)
8', + (b X 8)
QR(k - 1.3)e.,
eX q
QR(kc" + 1.3c•• )
8
(q X b)
QR(kc•• + 1.3c,,)
v'. + (v X b) - (q X a)
(c X v)
8'., + (s'. X b) + (s', X a) + (s X ab)
+ (cq X c)
eX U o
+ C'1.q
cu.
ev.
c X Va
+ (2a X s'.) + (s X a') + (cv X c)
[U'•• + V'•• J + (2a XU'.) + (2b X v'.)
+ (U X a') + (v X b')
+ C:lV
8bb + C'lU
U oo + V
s'••
Saa
8'" + 2b X 8', + (s X b') + (cu XC)
Oll
substantial amount of time is saved in manual calcu-
lations because the solution of equations and the
calculation of shape coefficients U OJ Va and (u oo + Vao)
- 2ab X q
2.6Qrm 'RiP
Qr m 'R(2.6c" + 0.5c•• )
Qrm 'R(2.6c•• + 0.5c,,)
IV
-2.1Qrm 'Rc.,
where
s'a = kQR'!c a
S'b = kQR2 Cb
s' ab = kQR3 cab
S'ca = kQR3caa
S' bll = kQR'JCbb
U' 0 = 1.3QR zca
V' 0 = 1.3QR2cb
u'oo + v ' oo = 1.3QWhIl
S
U
V
O':.A
MEMBER5
0
r
7.2&9
.24/
~
~
«
I
Z
3J./4
R
h
k
~.oz
9.02
.0
A
Z Eo:;'" 10-
?9.o
.5. 52.3 .333
<Il
Q
~
TEMP.
1.0
~ MATERIAL
•
Q BRANCH
w
Z
6,
W
ii:
~II_/---I.
.00
24.~
o EIlI/
6y
77
_~ 0 __!\Z,-_-i'3~_-+I~_-x
.105
2.29
a Eh'"ro·
'a."
is simplified.
To show the effect of omitting q in case of a practical pipe line, the well-known Hovgaard configuration [4,5] has been calculated both with and without
the q-term. The summary sheet of this calculation is
shown as Sample Calculation 5.4. It is seen that
in this case the effect of the secondary term is entirely negligible.
It should be noted that the secondary term is
neglected in all examples given in subsequent seetions
A
50F
.OO(;O~:I
T .M4~7
r.O
(;0
L. ..
b,z
£" ll.X/'44
Ef,UY/IH
E~I6J/IH
.03 I
r 3d.1 07d
406 520
+216!JSO
MOMENTS (FT -LB)
OLD5PRING
q
ACTDRC.
:5
Mx
M
l!)
M:r.
l1J
Fx
F
a::
:t
AND FORCES Le)
ACTING ON RESTRAINTS
'1 INCL(HUlJ
(IIOT" CONb.)
{Nor <01'10.
O'
A
O'
A
- 41010
rflSO
+ 7970
-4{,70
"t-IZOO
'1'/1090
-/750
-/7/0
""7920
-4>710
-S280
-1'1750
/710
-
,.. f&,40
l/) POINT
t-
HS;LCC'r~D
-"''''so
-1'11080
-S450
-/1501"/1S0
-/no
-1'/720
-
1.40
t
4.40
(,
1-£ MW KELLOGG COl p INGOfR~~M~II:ITbA~ A~J,Rmut~~LYSIS
p.:Ii
psi
5hSe.
Sf·~:..=~TED
~TRr:!>S
psi
~T
POINT
SA' ~~~~';"t6Rl....JLa;,
_
CAL
pf,j
O. 5:4
127
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
:>1
of this chapter. In most configurations this modification reduces calculating time without introducing
any appreciable error.
.....
~'
. ~/z/~
5.10 Effects of Direct and Shear Forces
Direct tension or compression as well as shear
coefficients, dcnoted S, U, V, lV, will, like the secondary term, also be neglected in the subsequent
portions of this chapter. These effccts ordinarily
have little significance in practical piping systems
and it would appear that their neglect would always
yield results on the safe side. Neverthelcss, they
may be included readily when they are considered
of interest in the calculation of abnormal layouts.
The expressions necessary to determine the shape
coefficients were given in Tables 5.3 and 5.4 in
Section 5.8. These shape coefficients are added to
the shape coefficients previously described as follows:
uoo+voo+S; Sbb+C 2U+V; saa+ c2v + U ;
Sab
5.11
i;r "
t~
e"
""JAy - - -_.-
fl.'
£I A'l' _1000 Las
I~
CtJRVf.:S SHOWN DOTTEO 00
HOT IHCUJOE SECONDARY TERM _/
[,/ ,.,
M"
,
"" ,
01.3
I
,
.~
'I
/
,
"'""M
•
I
•
• I2
"'1 1o
>- 1/(0' •
--- •
; 7~~.,
.~ •
-
- --
iF"
~f:;';
- -1,7, ::-F, , , • , •
•
I
7
•
2
10
RADlUS R (FT.)
FIG. 5.4 Effect of the secondary term on a symmetrical 900
bend with various lengths of tangcnta.
jeeted on the coordinate planes which are oriented
as shown in Fig. 5.5.
The counterclockwise sequence of the axes should
be noted. The transfer from one projected plane
to the next in suecession is achicved by changing the
designations XI YI and 2, in the order shown in the
following triangle:
,
Z
/\
I
Y
,
,
,
}-----y
)-----,
J-----,
FIG. 5.b
I
/1 •
~
['7; '\
3This is in contrast to the Simplified General !v1cthod of
Section 4.6 where each member was considered in three
planes.
,
S'
'/
V
When the piping lies in more than one plane, the
solution of the flexibility problem increases in complexity. To introduce the third dimension, each
member is assigned to a "working plane" for the
calculation of the shape coefficients.'
A working plane is designated by the coordinate
axis to which it is perpendicular. That is, the x-plane
is any vertical plane at right angles to the x-axis;
the y-plane is any horizontal plane at right angles
to the y-axis, and the z-plane is any vertical plane
at right angles to the z-axis. The planes which pass
through the coordinate origin are called coordinate
planes. All other planes are identified by their
perpendicular distances, designated c, from their
respective coordinate planes. For calculation of
the shape coefficients, the working planes are pro-
y_Plone
~
z
~
iA
Working Planes and Cyclic Permutation
,
~
hi-I-~ ,•
CURVES SHOWN IN FUlL
INCLUDE SECONDARY TERN
+ c"g + lV
x_Plano
t-_" 2•
/
,
z-Plone
Projections of the working planes.
- - - --
---,-,
---,------
128
DESIGN OF PIPING SYSTEMS
This is the principle of cyelic permutation which
permits the formulas developed for one plane to be
converted into formulas for the two remaining
planes by a simple change of subscripts. Thus, formulas for the x-plane are developed from the z-plane
by substituting y for x as the horizontal axis and
z for y as the vertical axis.
Correspondingly, F XI
FVI and 1I1 z become F'y, FrJ and !lIz. respectively.
Figure 5.6 shows a pipe line sketched in the standard
10
coordinate system with the working planes indicated.
Figures 5.6a to 5.6e show the projections of the
members on the coordinate planes. The c-values
give the distances from the coordinate planes, with
the subscript denoting the respective plane.
In the breakdown of a piping system, each member, whether straight or curved, is assigned to a
plane compatible with its loeation in the line. No
member may be assigned to more than one plane,
nor changed from one plane to another while the
solution is in progress.
This is important in con-
nection with straight members running parallel to a
coordinate axis since there is always a choice of two
c,
possible working planes to whieh anyone of these
members ean be assigned at the option of (,he caleulator. This does not hold for sloping or curved
members which define their own planes and leave
the calculator no choice.
5.12
In order that the reader may readily follow the
somewhat more complex operations of the multiplane calculations, a step by step procedure is again
•
c,
•
/
•
/
Multiplanc Pipe Lines with Two Fixed
Ends
:8-.
i-y
It-Plane
CIt = - '1
(a)
given.
Steps 1 to 5. Steps 1 to 5 duplicate those given in
Chapter 4. Multiplane systems are sketched on
Form A unless more space is required for the drawing
in which case it is made on Form A-I.
Form A is
then used only for data and results. The vertical
legs of space systems are customarily represented
as parallel to the y coordinate axis.
Step 6. The calculation of the shape coefficients
requires the assignment of members into working
planes. For the beginner, it is recommended that
sketches be made of the projections of the members
in their respective planes; with practice they may
be mentally visualized. It is a practieal rule to
y-Plano
c)'=o
c,
arrange the subdivisions so that a minimum number
(b)
of working planes is created, and to favor the plane
including the coordinate axes so as to reduce the
number of operations required in setting up the
simultaneous equations.
Step 7. On Form D-2 enter the quantities k, Q,
R, a, b, C, L, L 2 /12 for each member. Since it is
necessary to sum the coefficients for each plane
7
•
,-
separately, the data should be arranged so that this
can be done smoothly.
Step 8. Have Steps 1 to 7 checked.
Step 9. Compute the shape coefficients for each
/
c,
x-Plano
C.=O
FlO. 5.6
9
(d)
A
(e)
Sketch of line showing working planes.
member in accordance with the formulas given in
Section 5.8.
When curved members are involved,
auxiliary Form D-l is used to compute s, Sa, Sb, Sab,
L
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
Saa, Sbb, U, V, q, Uo, Va and (u oo + voo). These values
are then entered on Form D-2 where the coefficients
cq, CU, CV, CUo, CV a, and (sat'+ c2q), (saa + c2v),
2
(SOb + c u) are calculated.
Step 10. Sum all the coefficients for the x, y,
and z plancs scparately.
Step 11. Have Stcps 9 and 10 checked.
Step 12. Transfer the summations from Step 10
to Form D-3, entering them as indicated according
to their planes. Sum the contributions from each
plane to obtain thc final cocfficients Ax%> A xu' etc.
for the equations. Enter these on the equation form,
Form E-l, together with the constants from Form A.
Step 13. Have Step 12 checked.
Step 1J,. Solve and check the simultaneous cquations.
Step 15. Transfer the values obtained in Step 14
to Form F-1 and calculate the moments at the significant points using the point-to-point transfer
described in Step 8 for the Simplified General
Method in Chapter 4.
Step 16. Determine the maximum stress. This is
done in a manner similar to that for the Simplified
General Method, but since it is now possible to
handle inclined members, a provision is made to find
the bending moment transverse to the plane, and
the torque.
The plane in which the point lies defines ]lib, but
111' band 111, are found by an application of the
formulas in Table 5.5.
The operations are performed on Form F-1 by
following the guide shown for the respective planes.
Points on members which are parallel to the coordinate axes (i.e. where a = 0 0 or 900 ) may be said to
lie in two of the three possible planes. When no {3
factor is involved, both alternatives will give the
same stress. When {3 is involved, however, the point
in question must be placed in the plane which will
give the higher stress because {3 is applied to bending
stresses only.
Step 17. Enter results on Form A.
Step 18. Have Steps 15, 16, and 17 checked.
Sample Calculation 5.5 illustrates the computation
of a multiplane system with two points of fixation.
129
Table 5.5
Members in the x-plane;
Mb = AI':
M'b = li{'scosa -AI'vsjna
!tit =ltf'lIcosa+M'ssina
Members in the y-plane:
J,f, ~
M',
= },['zcosa - :tf',sina
/tft = Af' cos a + M' sin a
M1b
J:
:I:
Members in the z-plane:
M, ~ M'.
ltf'b = .M'l/cosa - .M' z sin 0:
Aft = Af'z cos ex + !if'" sin a
The reader will note that this line was computed
with square corners in Chapter 4 as Sample Calculation 4.13.
5.13 Hinged Joints .and Partially Constrained
Ends
In the systems considered thus far in this chapter,
the fully anchored connection was the only type of
terminal constraint discussed since it closely represents the fixity of most piping. In some cases, e.g.
jointed systems as described in Chapter 7, open
ended lines, etc., different end conditions may exist.
The end may be fixed against translation but be
free to rotate; it may be free to move in one direction
but not in another; in fact, there may be freedom of
any combination of the six components of deformation (three rotary, three translatory). These cases
can be handled with the aid of the equations in
Table 5.6, which lists the general equations of a
pipe line with two ends, A and a', subject to any
deformation. The shape coefficients are summed
from A to 0' to obtain the summation coefficients.
One of the simplest cases is that of the end which
is free to pivot, a practical cxample of which is the
system with the hinged expansion joint shown in
Fig. 5.7. Whcn thcrc is only one hinged end, the
simplest solntion is to locate the origin of the coordinate system at the center of thc hingc.
Sincc the example is shown in the z-plane, the first,
second, and sixth columns and rows in Table 5.6
Table 5.6 General Equations of a Pipe Line Subject to Deformation at Either End
j.lf:z:
+A;:;:z:
+A:z:u
+A;u
+B;:;:z:
+B:z: v
+B;u
Mu
M.
Fx
+A.7;u
+A zz
+A uz
+A u
+Bu
+Bzv
+Bu
+B u
+Buz
+B u
+Czz
+C:l: V
+Cn
+A uu
+A uz
+B vz
+B uu
+B v1
F,
+B:l: u +B:u
+Buz
+B zu +Bu
+C:z:v +C:u
+Cuu +Cvz
+Cvz +C"
+B uu
Constant
F.
+El'(8 xA - 8.0')
+ El' (8,A - 8,0')
+EI'(8,A - 8.0')
+EI'[o.. - oxo' - A.A
+EI'[ouA - 0uo' - AuA
+EI'[o'A - 0,0' - A.A
+ (YA 8. A - ZA 8,A) - (Yo' 8.0' - Zo' 8uo')]
+ (ZA 8xA - XA 8. A) - (zo' 8xo' - Xo' 8.0') I
+ (x A 8uA -YA8.. ) - (xo,8 uo' -Yo·8.0')]
--~-----------------
130
DESIGN OF PIPING SYSTEMS
.365
Z
/GO.7
29.90
R
1.2~
h
k
.20
~ Eh'S
~
a
Q
Z9.00
Z3993400
/.00
MATERIAL
C.S.1l10{ ~"!/1
.
TEMP.
QOD F
e
BRANCH
.()(){;!>O
Z
-_2.(;000
-IS.OO .00(;50)"
-.09750
.,...(; Z.1S Z$o
E,,16lh~4
-23393(;0
MOMENTS (FT -lB)
COLD CONDITiON
FA(TORC,O
O'
POINT
- 6'800
+ 7/75
-r15"'00
T
UJ
Z 925
417S
Fy
F
F,
230
- Z30
+ 550
.30
+
"'-- SSo
+ /2 87S
80
30
~
180·
MEMBER N!1/-t
I 90"
SHAPELJ
I ao TRICiONOMETRIC.
I z5 CONSTANTS
1000
I .57oe
c.
IZ 0
00
kOR: .. 12 0
c,
I
kOR)t
15 J C'b
c~
15
'OR
785
c bb
kQRJ
~
78S<
R
'OR
'OR
•
I Z5 ~cl>b+l
QR
R
QR
R
QR
QR'
•
·
25 1'''u:> UC bb 7
ZS (K·l~)CoI.b
0
-/ 00
56 I.~C..
56
I.~Cb
/95
j·:Sf
kQR ..
k R'
kQRl
kQR' ~
k R' •
k
R
QR
QR
QR
QR'
·
,
·
QRZ.
QR·+
b
ZA
Zb
CALCULATION
15 71
-
1575
27 50
.s-a - rz
+~',tL
• ZI taZ$" P5 i
..,
s·,
15 71 'a
~',
V"
913
0
-q.t1
tU'Q
"2 ()4U
A
F
CAL NO.5.S
10<£
~J
A.'r£ f:r·;1·
+s .. a 1
5:._b
+G"~Z.ll
+~'b-a
• IZ
-II' s,
• ,. +u_a -
JD()()
c
ALe.
><1:
+!!l ...4b t
+~ab
q.
psi
...
ab , /7 19 .3.tl+~$ b' ,
/ 6
-- ZI 501S ZOb
OF SHAPE COEFF"ICIENTS
+ Z 51
/96f.
Z9 IJ
, "15 3
75
01 +$_b -
- 50
0
-2
14 sAb
+u·a.
1/ ~I +V_b l
_q.a
-q _lab
OJ
- , 57
• - ,
-
tv:,
0
,
_
+lO .. b l
.
.. 5'....,.
+,j;!r"U' -
12 28
.J
5d •
n IZ GENERAL
.j.u...·za
OJ tV' .Z.b _
00
",,'011
-
I
-
,,,
+v.b
Sf
K'bl>
,
It. s"
I
ORMUL
/ 2 c a' [C05 a: -GOS (o<.+f>J
Cb' (SINO< ~ SIN (O(+~)J
0
5585 Cdb~ .z.s[co:!> z.(<>lTI)~C04 ze>()
~~
G,U' .S.·.Z5[s'o,l z.(oo;+t)~.s'1J ~lKJ
Cbb: .sf+.zs@I'" ~(01.+ iV~511'l zoo;]
38 ........ ·v"" ./
CooRDINATES
a'
800
25 b
I 25 ab
1 6 b'
0
SHAPED
0
eb
I 00 TRlaONOMETRI
za
• 50 zob
COt-J:!lTANT5
CALCULATiON 01=" SHAPE: COeFFIC1E:NTS
~
.~_b 19 (A .s_aD +
ItJ (J(J
Z45/ +S"b~ ... 2451
5 71
5748
mr~_A,
+
+s;".b _ 15
+s',a
ca
12 j()
~:."u(JtJtJo
12
JI 25
+:'·b
c,
1250 t!l'b"il. _ 1.2.
/250
/"
000
' .... Z.b~
'0
~
k
Q
R
s .,
4
COORDINATES
a.
-
,
AT POINT
SA' ~iJfJ~t~L~
~
f
Q
!>TRf$$
• 15"4S0
- 400 + 400
-- .GO
"' .GO
GO
GO
THE MW KELLOGG col PIPINGORIGINAL
FU;:XI,I!'pTY AND STRESS ANALYSIS
DATA AND RESULTS
k
Sf 'r::;"C:~b~T(D
A
-I- 0000
M
M,
0:
HOT CONDITION
BSO
-- /-300
T
5h" 6$'00 p~j
5<:" /6 000 psi
ACTING ON RE'5TRAINTS
O'
- 7500
(
I
500
7S0
My
W
AND FORCES Le)
•
25.00'
A
I-
'5
S
7
0
r
'X-PLANE ex .. -/5:00'
-~ Z38 ZSO
COLD 5PlWJG
g
g
z
Eh1t.X/j44
Ehlto)'!tH
II)
O·
'f". z~ooo
6,
,
IF,
r
I"
-40,00 (.(}~50)..
Ay
S1
Z/4
~y
My
00
+40.00 r.oO~So) ..
Ay
W
0.
0.
b
{bL
.
2/. SO
,
,0
2
Z.tiO
IO~
2 E" 11
W
/5.00'
".00
Z E G ,. 10-
'~"
F.I 0- •
y
/0.75
"I
<:
y
O~A
ME.MBERs
0
•
" • ·•
•
,
..
C.b
C aa
150
ebb
MeMBER N~.J-4
190
5-'
25 kCt>b"~)C..."
•
4+~"
0
000
0
0420
T5'",b
/'/
t
s· ...
v·,
q.
-
15 7
13 +<.\.01. 1.3
~q.b
0
"
tU'o
1 ,
II 41
.v. -
I
50
-q ••
0
t 2113
/I 4-(
0
,
.
."_b l ,
Sab
I
1424
","'" , • 0, .u:".,1V""• •
+V~·Z\
Uo
-
9 3
Vll
~
q 38 u..oo'tVllO
,/. 12
+"'bb
tLLwll,t
-q.Zo1b
<1... ·101 _
,
•
.. 5'"...
u"
.,
.. ..
,
,
-
-•
• "
000
• 7454• 7854-
25
25 (It-J»)C 4 b
/<6 l..!loC
6 t.~Cb t
5 I.:>f
Vll
•
~
"
8
Sbb
0
'00
<'0
'0
.,.5JO/lo "0,57
> •. 5JOO +'.570/10
••. OllOO + ',0000 -',0000 _I,OllOll
~HIOOO .,.00llO "'.0000 ~H'OllO
-.5ll"0 t.5"00 _.50ll0 +.s"o"
".1I1S" ".7.'" ...1"/10. ".7.05"
+.1tl!> .. ... JIS!>" ... '1'05.:; •. 741>4
t ••• oo + •. 5ll00 -1,lllOO _'.1IllOO
-j,IOOo +'.lIll00 +1. 0
-'.llllOll
+t.","Zll +l.04Z0 U,04Z
1:.04Z0
•
07 S ... "
.,
~
IfU C
C
0
e ..
,
C •
l,:'C
I.:l.e
2~'.J
PIPING FLEXIBI~TY AND STRESS Ati{'LYSI&
<+~
.~
[THE MW KELLOGG CO. SHAPE
COEFFI IENTS FOR CIRCULA MEM R5 P"'T~"'''O
.
_
'FQ"'M
0-'.5
C .....LC H9
FLEXIBILITY ANALYSIS BY THE GENEUAL ANALYTICAL METHOD
«
•
8 00
k
a
R
kaR
kaR
kaR
SHAPE
CONSTANTS
/ 25
kQIP
/
kaR
IS (,,3
/
kQRJ
OR
R
OR
• -
c.
c.
C.b
Cu
/
785
25 ~cl>bit..x..u
R
aR
2
5.
1",:::.-
1,"Ct>
'.:!lot
"
- 000
000
•
c• ·
TRIGONOMETRI
R
COt.J~TANT5
kQRlt
c.
c ••
c ..
R
Cb'
kQRI.
kQR~
.
GlR
lCI;bbH~
GlR
OR
4+L~'
SHAPE
+v~
u. • '138 v"
,
•
-7
-
tU;oo+v""
.-
C btl:
H
12 l,I."o+vo~ ."
~
S.
v·.
q.
tu·.,.
+$.lltl
+s:".b
+:>'b"a
+i!>.tl
-+-15""
..:.._u
"''''tl
tu.~.,.~
w·
+V"b~
-<i"a
-'q .. U1b
-tV'"
1.4.... za
"V;"Zb
.
+
z
/-2
2-3
b'
•
t$_b~
+
-J
S·D"lob
., ·
tEO'bb
5 ••
~
,.,
'·:!IC:b
~t>
f
•
"0
.,78 ..... •. 185
".las.
+'."000 -',SOOO -1.1~000
'1.&000 .'.&000 -+'.)0
·'.3000
.r.O'Iro 1'z.o.zo .C·O-+lO .E,41tlO
+I,~OOO
~.
0 ,","'II!.
'0
".7"5'" •. ra ..... ....7"".. •.7.......
I"·'.!l!>'"
C •
c
~,c..
0
", "0
'1.~10jl
"".:>'0 "·"'Oll "",5706
+1.0000 "',0000 -'·0000 -1.0000
".0000 .1.0000 "'.0000 -1.0000
-, ..000 ..... 000 -."000 ....5000
C
C
C..
',~C:
&RS
·
•
-+-s'......
""b
...
....
I< '8
.s 1+', ZSfrolN Z (":'t t)-SrN z«J
za.
Sb
.q ~b
+U'o
..
0'
COORDINATES
ab
a' +
,b
to
OF
COEFFICIENTS
CALCULATION
SHAPE
5"
-/
~'bb
-tV_b l 21
q c." [cos <:( ~cos (OI."!"{»]
'Q_zab
Cb"(SIN"" - SIN (e'\ + ill)]
0
.j.u'o"la
5 .. Cab'C .Z5@Ol!lo z(O(+f)-co~ Z~]
OJ
tV'
"Zb
'i3
G,u.' .51)". ZS[sIO..j z((X.. t)-srlJ Zoot]
2
.u:.,.1V""
lL",,1'VoO
I
PIPING FLEXIBI~TY AND STRESS AtiNifSI
. SHAPE COEFFI IENTS FOR CIRCULA
M
x
..~ ~~
0
0
+
KELLOGG CO
THE MW
.. .
12 28
0
0
+s'tl
u·,
.~
l
+$'.;1.
·•
0
OJ
+~'""b
,
+ 7 ,
• '"/5 ,
SOb
- 11-'558024 s.,GENERAL FORMULAS
•
-~S'b
+u "a.
-q'4
0
-2
1. 1loC b
PLANE
MEMBER
/250 +~·b·.a
+Y_l> -
.q.t>
+u'o
..
So.
1.1loC
I.~.
...., - """
/j
0
MEMBER N~
(k·I,~)c...b
QRl +
QRZ t
QR& ..
- 48 75
/ >,
+ 7/4 s,
/3 +uoa + If 41
0420
a
kaR
k R'
v·,.
0
-
/5
~.+
02
J
«
k
•-
s-+
.., •
COORDINATES
,b
-GO
b' .. Z J'J, SO
/ 5<
2>0 'b - 97 SO .ab
CALCULATION OF SHAPE COEFFICIENTS
l
+~"Ab
/5 7/ 1>S.-a
/ fA +5_b ~
2 5/ Hid.' ..J7
"J +s_a
t6'•• Z.a _
.~:'.b
+~'4
12 so
/ 25
•
Cb'
(1C·1.~)c,l.b
s .,
d8
(lOGO
- '00
00'
" 5 !I.Cu+UCllb
QR- ...
k
MEMBER N~5'(;
VO
/ 00 TRIGONOMETRIC.
1000
/2 0
..
\J ... •• "
270
131
•
-
-5
FORM
D-'
CALC H9 55
A
7-A
D
SHAPE
800
/00
800
/00
/25
k
Q
R
L
1500
a
/000
750
b
-
/500
/8 75
/500
-
ISO 00
liZ 50
o
cq
0
v
/9.50
/9500
Z9250
/500
Vo
liZ 0
u
uo
-
cu
-
cv
- ZZ500
S"b .. cZq -/IZ5aD
CU o
Z 9Z5 ao
-168750
487.5 00
5bb+C U
.5 51Z 50
u.oo+V. o
..3 07[; 00
CVD
5u.+C l V
THE MW KELLOGG CO
/lO
800
I 00
/25
47 0
23 75
0
U5
/3/3
563
o 25 0 18 75 5000
o
o
o
0
0
0
0
.38
/302
/88 02
47 0/
875
/57/
/250
/5 7/
4750
/57/
2375
/3963
/3/25 - Z285/ 9375 7 t4
0
7 t4
3/1 84 - 14/ 67
49
7 t4
0
7/4 - 1/87 5/) - 778
- !l87 50 - 32/6 90
o
000
0
000
o
0000000
1/ 38
9/3
/250
.9/3
6175
9/3
2375/3677
/70 3- /2757 9375 938
0
938
3/1 84 - r.§f!.~
o
0000000
87.5
9/3
IOZ5
9/3
4750
9/3
3088
/3077
492 938
0
938 - 1/8750 - 447/2 - / 43 75 - 3 '4639
o
0000000
73894
/0598
0
/07
0
- 35580 -15 59/ 88 -/5/0/69
o
o
0
0
0
0
0
0
000
000
o
/90875 3 26/6
86589
5 ¢
0
.554 52/0 3 //3827/
33/.
.5 4
0
5 4 381,/8 0 38 (,6
59 7.5 00/3690453
t7 509
8(,589
2230 38 618 50 2/ 9 0 82398 33 48 0/ 7/
No rkli~I!'JLI~htF"j, S ";S ANALYSIS
~
--: <
-.5.
I'Z 0
/500
z.-
-/3 ~
- / Z
750-
/Z5
IZ -
~
2 .,
x
y
M,
, /5
.u
o.
M,
, .y
f
[ .. Au
" /45 77 "A.
M,
~
0
.q
130 71 +q
0
O.
~
0
0
+A."t.1
.. Bllx
++
/5·
y
.,
r
0
-v,
[
+A"
~/5/
, .u
.. /,3G 77
771"A n
.6 •
, .u
+/9 50 .u,
+<,
.5
-1-/39 6J +Sb
.A. '1-/59 /3 +B ZK
Y ++
,
[
• 8.
-
//2
-
//2 50 .6
-cu
0
0
0
0
.
1/2 50 -Sa
,
+
_v,
0
//2 50 .B u .
+3 24< J9
+3 l,g. N
'",
+C,
0
0
-Sa
+
'Sb
0
-<q
0
0
1
x
1
+Co
1
h,
Fy
F,
1
1
I
-<q
0
+Cy
,Sb
'<q
0
0
.u,
.6 •
/95 00
-<U
2/6 90
-Sa
-
-
-
/50 00
225 00
80 1/
30 //
1292 50 +<q
+
-v,
0
_cq
,
,
141/ 90 +6
-3
-<u
/4/ 67
0
4J /7 +Bt.'l'.
0
+ /
075 00 -cu."
- 2 925 00 -r::.Vr/
-cYo
-e..ab-ctq
1
Z +Sbll+C'U. f /3" 904 '3 'S41,-c q + /5 /0/ 69 .cu o
+12 1714(, i"C I('t:
+/
I +c""
d39 479 53 .c,
0
(,8 50
/250 ,cab~c%'1 + /
/25 00
X
41.1.00+"'00
+ 1
(,8 50
y "s.u.+ clv
x
+$t111+C'
Y
I1~Vco
Z
+iu+c:v
I. ... e)'."
, 5
-cu."
-e..-o
, // J82 7/
0
+ /6 895 2/ +Cyt.
+ /
/25 00
)( +J..... +c 1
+4-
8l 00
+5bt"+c'~
y
z: +1.1....+'1"" t148
_;L~NBIL):rY A~9. STRESS ANALYS.IS
THE MW KELLOGG COl. :::IPING
PLANE SUMMATION OF SHAPE COEFFICIENTS
,
,
2
,
wJO'149'-
•
•
, + />f.
IfS
-
"'J8nU1f -S{.5n5JI
- ~J/q6J'Z -<1"901'J
•
-1%1114'1
'''Le::,
~
CI-l~CI(
O"'TIl ,
-
•
-I-S" 98f
- '8/ '99
•
.,.1I'~lf711 .. 10008"
0
0
-'2J1'"
r "724- .. lJ8JJ'
.,.1480563 .,. 09(,77.3 'UJ9f71
Fx
a
z
,/
+/~U
, h liZ 5, Fz5'. J'
7
•a
•
29 6S
•
• -- m77/ 1<78 -13 58/
/8
0
I"
•
•
. lIZ
50S 1/
• /5/ 71
•
•
• - ;.a //
, 'Si• 77 •• -, UZa
I"
iZ5
• +Z
/80 U
H, , + /5• /3 - - .-, 44 0.-In , //4 ZS
'",
•
•
M.
_ON5TANTS
f3
•
-I
a
I. 000 00
Hx~2.ll3
2
•
So
.
0
50
1.000 00
0
,. f
0
I"
, lIS.• It' -3
- 1.00 00
H. -/I ZBZ 13
4
.., ••..
0
."
M ,
J
/
./.
0
0
0
0
0
"0
0
0
0
9' ~ 2 728
'"- "/ SD -2 20.
0
0
.,.,Jt
+I .... 7 0+1. m zS<
'7 5J +I /7•
•
•
•0
•
. 15' ,4 •• Of - ~
a
, ,21 185 7/ rI +<1 138 78
.. " '4/ -
,.
- ."
f)
-
F. -
7
1.000 00
~.,
'7
~-
,I.
,.
" ••
'?!-
-. . .", .,
0
, • 8 .,,.
-/
..
'S -Z Doli ,::12
68_
-8
, <¥~
9+ /0
,HJ
1l...4 •
=1J. '"
'"• '"
IZ -I
-
1,000 00 f
F. +
So +
G
0
0
0
0
18' S4
22 8<
/3 4'
11 -1
" - ~/J U
0
=" ,
~ •• -
'7J 157
-
,
THE MW KELLOGG CO
0
W 9Z
JO 70
9J 4
/ 3. 231l t§g
/:!9 21 +1
~
0
8"- IlZ -2 <21 20
-
/8 - I
-4
.000 00 V//.
•
0
/.16 "
~
IF.
5. 3/
a STRESS ANALYSIS (. .. ~ •
FOkM ['1
6-EQUATlONS
ft,:~~;~~_~'ff' CAL~·S.5
PIPW<; .L£,J<lBI\..ITY
132
0/7/
-
767/
CALC. NO. 55
-
.,,201.'l-f.o
• +<5
• - 1809'
-/ ,r .115 "'/1 'lH05 ¥-41?lJ20
-2.,.350
•
0
•
4
"'202''''
•
5 -2388S8
•
rl08 coSt:.
n
_._c_
1: +Cz:r. tlS3
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
TO POINT
CCNVERSIO"l
,
COO" RULES
~=
y
/.3~4M
z
F,
F
F
St. !a.. S'l:
E,
S· ,
/5450
PSI
M,
...F "Z
&. ~,c .R
R"
Rc
~
Ec::
R' ""En
-F'Z·v
S"
~f:; ..fJ' Zf.J
15~O
f44()8
0
+
1/ 282
+
716(,
4040
14408
M,
HOT MOOULUS. I Eh
-
0
1530 -
0
GO.
I 070
SIN '"
lat; ...
0
0
7 fur; ...
J S~5 -
-
0
z
t.y ...
M'
+M'"
M'
.M'
~M'cz
~M'"
"SINO!;
.M'
M'
''::'It.I 0<-
PLAtJE
M.
>ccoso<.
,e
E. _
Y
r,;--
SI!_~'S'1!.
&
F
F
F
PSI
Ee,C
Re_8.. ~
R' - Eh
S"
..
•..F","x •
-F,,"z
M
~.
.. F."y
-
,
0
0
'" •
'" -
50S
/9Z5 0
I 20/ /9Z -
12. 006 fO aSI -
•
MODULlJ~, E..
7' •
7'
0
0
0
/8/ !OMI 505 t
12.01
/0777 ...
,
0
7~8
Z 8/7 ...
•
-
/8/
07
19189 010
-
8 4fZ ...
-
7'
8"
7'
0
/07
S
{Z
SO,
1201
77/G
/
•
2.uO
IfZ
•-
•
,
2/38
3103000
11447
CAe
H • .:r,
•"
aRM
A.
4
C
-I
05.5
Z:. 00
soao
0
59.3
64[
0
84<
0
140'
0
1 JOl
+
/G
., , -
0
Z2 81Z
R'· CAU:ULATl!O
REACTIONS eA~fD oN SIN (II.
,
Z.~.75
0
M
~= (J-JC)
HoT
A
+- 360.51
+
+
-F
WHtCHEV'E~ IS GRCA"TER
~
•
- 403.97
M,
..F .z
.R
R' -lh
0
,IPING, :'LE~~~ILlTY AND STRESS ANALYSIS
MOMENTS AND STRESSES
z
So'
7'8 ...
/8/
-5,
THE MWKELLOGG COl,
,
0
2SS -
85Z
111450000
.45
TO POINT
2049 ~
0
2049 +
•
z""
2St.-fMt
cooe ·RiJLES
1.25
1.25
0
99' -
/0 519
5' -f,BM b
5 .5
47.5(J 0
211'.1
0
0
2 }23 -
-
~
~
-- ---- - -- -•
Z.UO
5 ·f M
"5
-
•
0
Z IZ!J -
f-12 l
CONVERSiON
99~
G,
J
/.25
1.25 0
%
M,
5 *"5'
0
0
0
•
10 081
M'.
.. M'>c
" -.
"
Z. 1Z3
4
12..50
- " ZS5 - Ie0 777
0
- Z 04!J
- 0 255 - Z 049
- 0I I
- Z IH
/81
- 0134 .dO/!,4
"CO.!> 0(
:L
-
1.25 ...
1.25
0
J:.-od-:J 0
519 Z 04'3070
0
0
J 070
COSo<.
y
Z
1Z"78
-
R'· CALCULATED
REACTIONS e"~E"D OH
PIPE
-
"0
-
.. M'cz
SO>
IZ 876 -
•
•
0
8.75 ...
0
~
~o'o
~= I
..M':r
.M'
+
403.97
9'0.50
59.3/
Z 123 +
/4- ADS -
-F."z.
M
-M'
•
/0.00
IS.OO -
J
Z
/
0
0
/5.00
+F'Z·x
+F.·y
M',
•
-•
7
IS.ao
/80
~= (1-" C)
,
•-
0'
·-4''''' ··
, -• 4''0 - - •
., •
·•
M
M
WHICHEva IS GREATf
-
133
{sox
•
·••
•-
-
--
Z 123
0
Z.~"
843
/80
/483
0
I 303
1/ 2Sl
20/99
24 O{J
{!l0!'
coso<.
PLAtJE
M•
y
z
M'
M'
.M,
M'
• M'
-M
-M'~
-M'.
"SINO!;
..M'z
M'
:L
... M'"
+M'z
M'
M'
1(1)1t.l0<-
IP
Z
)((,050(
M',
"cosec:
M,
f-12 Z
,e
5 ·f M
S' -f,BM b
25 =fMt
5 .S .45
5
'5'
"5
-S',
THE MWKELLOGG CO.l,
"IPING. r_L~~I.",ILlTY AND STRESS ANALYSIS
MOMENTS AND STRESSES
CALC.
•"'
~,<:.
4_
-$
FORM - I
oS
I
a
134
DESIGN OF PIPING SYSTEMS
y
I
r------~_l::o'
FIG. 5.8
A1- - - - - - - - ,
Pipe line with OUel end hinged.
Flo. 5.7
are omitted, and since !l1:;:1 GXA, 0%0 ' , 01/.1, OyO" 0:0'1
all are equal to zero the equations may be
written thus:
XA, Y,l
F,
Fy
Constant
Sb
-Sa
-
8bb
-Sab
+EI*ll:cA
-Sab
+Saa
+EI*.1 I1 A
EI*8 z A
F, and F yare obtained by solving the deflection
equations; the rotation at A may now be obtained
from the first equation,
A more general case is shown in Fig. 5.8. The
pipe line has two hinge points which are not located
at the terminals. For convenience, the origin is
placed at a hinge point. Noting that the moments
are zero at both hinge points, the following equations
are written:
System with two hinge joints.
of the moment equation for line AB by the known
forces F;e and Fy'
Sample Calculation 5.6 covers a typical hinged
expansion joint system. In this simple case XB = 0,
Fx = 0, and FlI = EI*6. l1 /s aa . The case of three
rotation joints where thc flexibility of the piping is
not involved is covered in Chapter 7.
It is also possible to handle problems where the
clast'idly of the terminal connection has a significant
effect. Of such cases, the most important is that
of the rotation of a nozzle on a cylindrical shell. 4
In Section 3.14 it was shown that the rotation of
the nozzle due to shell deflection could bc cxpressed
in tcrms of a virtu'al length L, representing the
length of a fictitious extension with the same rigidity
as that of thc pipe line. This value of L is used in
the formulas of Table 5.7 to calculate a set of shapc
coefficients expressing the cffect of nozzle rotation.
The shape coefficients obtained are simply added
Equation
M,(=O)
F,
Fy
Constants
1
+s
+Sb
-Sa
+Sb
+Sbb
-Sa
-Sab
+Saa
EI*[(0'8 - 0',8)
(0" - 0',,)]
EI*[ - c"
Y8(0'8 - 0',8)]
EI*[-c,y - X8(0,8 - 0',8)1
2
3
-Sab
where the shape coefficients are summed from 0' to A.
Eliminating the rotations in the deflection equations:
F z(XB8bb -
YBSab)
+ F y(YBSaa - XnSab)
= EI*(x8c"
+ YBC,y)
AB F:r;YB = FyXn since ft.f z = 0,
F =
EI*(x8'c" + X8YUC,,,)
,
Xn , Sbb - 2 XnYnSab
Vn Saa
+'
F =
11
+ XIIY8C,,)
2XllYBSab + VI/Saa
EI* (Y8' c,,,
Xn 2S bb -
The rotation at B, (0,8 - o',B), ca.l be obtained
from either of the eqs. (2) or (3), after which the
rotation at 0, (0" - 0',,), can be obtained from
eq. (1). The rotation 0' ,B is obtained by multiplying the respective shape coefficients, (Sb and -sa),
+
+
to those ordinarily calculated; hence a detailed example is not included. However, the results of a
calculation made both with and without the effect
of nozzle rotation are shown in Section 3.14.
5.14 Skewed Members
When a member docs not lie in a coordinate plane
nor in a plane which is parallel to a coordinate plane
it is said to be skewed. To calculate such a member
or group of members) it is necessary to introduce an
auxiliary coordinate system, one coordinate plane of
which is parallel to a plane which includes the skewed
member, and the origin and one axis of which coin41f the piping is relatively stilT and the shell is relatively
flexible) the conventional assumption of rigid fixation is
likely to lead to high indicated reactions and consequent
heavy nozzle reinforcement whereas recognition of the elasticity of the shell would show that reactions are low and reinforcement is not required.
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
'-3
MEMBERS
0
t
I
Z
004/
44./X)
R
<:
,-.
/S3Z.~!iO
.5"0
2 '$14.8
/051.8
Z-3
7~0
3·5
0
'-3
1S"33410
.M
2.2.70
29.00
22.70
29.00
S<15'.:'f
.1.00
IZ.lde
Z Ec"l0-
o Eht/
Ul
Q
~
TEMP.
1000 F
e
.0074S
-
Z
h.
UJ
ll.
ll.
hy
0
~
• 1.1.17,1
'"
W
M,
MOMENTS (FT -LB)
AND FORCES L8)
ACTINq ON RESTRAINTS
HOT CONDITiON
A
COL.D CONDITION
0'
Sh- 76"00
5c· /8750
F.
F
-(;8 SSG
0
r 1200
-/200
h
THE MW. KELLOGG COl PIPING_':.L~~I.~I~IIY
AND S,!.RESS ANALYSIS
ORIGINAL DATA AND RESULTS
Table 5.7
psi
Sf ·~A~~'tg~rED
.
SUo
AT POINT I
. "' ., -
o"....TE
A
CAL NO.s.f.:,
Shape Coefficients Expressing the Effect of Nozzle Rotation
R
8
Sa
s,
q
cq
u
u.
eu
v
v.
cv
8" + c'q
euo
L
a X L
b X L
-L sin a cos a
eX q
L X cos 2 0:
(0 X u) - (b X q)
eX u
L X sin:! a
(b X v) - (0 X q)
c X V
(a X 8,)
c X ~to
cV o
C X Vo
8••
c'v
(a X 8.)
8"
c'u
(b X 8,)
ltD"
Voc
(a XU.,)
where the virtual length L is given by
a
a
a
L
L
aXL
bXL
aXL
bXL
o
o
o
o
o
L
8.
cXL
o
o
o
8,
+ (c X cq)
+ (c Xcv)
+ (c X cu)
+ (b X v
o)
cXL
a X Sb
a X Sb
o
C X Sa
C X Sb
(a X 8.)
b X 8,
+ (c Xcv)
o
a X Sa
(b X 8,) + (C X CU)
8"
(R)ll
t
I
L ~ 0.017-,
Tm
b
o
o
L
1 = moment of inertial in. '\ of the pipe in the system corresponding to Q = 1.
r m = mean radius of nozzle, in.
R = mean radius of vessel, in.
t = thickness of shell \vith pad included, in.
psi
5A·~f~~~t~l'lo.XO
• as 300 psi
Position of
Nozzle
+
+
+
p~i
:>TI'Il:S5
S M
a:
-44.00'
·'S~8721,J(,O
lJ) POINT
M.
~
2
...
8
57.n'
COLO 5PRlIJ
Fw:TORC.O
I-
~I
,
h.
Ekl.6X/J 4f"IOy! H
Ej,IOr!r44
IF)
"1,- -
x
~
-28.25" 00 I!N -1.00 (.O074{.;
-1-18,.13'(.00151 '" -.03~4!J'
~
~I
~I
...
.0074G
D BRANCH
-«
~*
vi"
x
M.~Y
"I
fi!l~ •
Elg ~
~I~ :Ii
~,
• ~
loaO f:
.. / 7\
o·
T-
f---"
~ MATERIAL '%CJ!. f'ZNO. I%CR.'Z/1(}.
w
y
,. ~I~
1:;
,
1/.8
DE .. 10-
Y
M,
S ••
MEMBER
62.m
_14
h
k
~
3-4
IX.50
.75
34.8.52S.0
135
136
DESIGN OF PIPING SYSTEMS
Table 5.8
+A'u
Formulas for Plane Rotating about x-Axis
M,
F,
F,
F,
+.-1'zvcos a
-A':u sina
+A'%I COSa
+A'zli sina
+B';%%
+B'zlIcos a
-E'n sin a
+B';u cosa
+B'zl/sina
+Jl'lIl1cos2a
+A'1I1 cos 2a
+B'U%C050:
+B'lIl1cos2a
+B'lIl COs 2 a
-B'usina
+B'usin2a
-B"/lsin 2 a
+A'us in 2 a
f
sin 2a
+ (11'
1111 A u) - 2 -
_ (B'
-A'", sin 2a
+A'ucos 2 a
+A'uusin2a
+ B') sin22a + (B '
III
+B'lIzsin ex
-E'I/I sin:! a
+ (B ' 1111 _ B':: ) sin22a
+ (B ' zv + B') :;in2 20:
+C'ZII coso:
-C'z: sin a
+C' cosO!
+C sin a::
+C'1I11 cos 2a
+C'u: cos 2a
(C ' _ C' ) sin 20:'
+ F z sin a
+B'" cos 2 a
+B'1I11 sin:! ex
II'
Zl
1
%11
+
+C'usin2a
F': = Fz
U
2
">;r:z
) ::-ill 2a
2
III/
-G'1I1 sin 2a
F', = F,casa - FI/sina
+C'u cos 2 a
+C'1I11 sin 2 a
+C'lIzsin 2a
M'.,. = Af;f:
:A['" = .AlII COSa + .M,sina
111', = ltl,cosa -Afl/sina
Table 5.9
Formulas for Plane Rotating about y-Axis
+.t1';r:llcosa
+A.';r:z cos 2a
+:1'II: sina
( ,1 ' IZ -
+
, s i n 2a
A. u) - 2 -
+B';r:;r:cos 2 a
+B';r:1I cos a
+B'usin2a
+B'zll sin a
-B':;r:sin:la
+ (8 '
'
B') sin 2a
+ (B ;r:: +
u
2
+A'u sin2a
F,
F,
M,
+A'usin:la
_ B' ) sin 2a
u
2
+B"II cos 2 a
G'n
+A'ucos:!a
1111
+B'u; coso:
+A'uzsin 2a
F' JI = F 1/ cos a
v'
u
_ E'"
+B'II: cos a
+A'II:cos a
+B'II;r: COsa
-A';r:lI sina
+B'u: sin 0:
+B'1I11
-B'II;r:sin a
+A'"cas:!o:
+A';r:;r:sin 2 a
+B':;r: cos 2 a
+B':lIcos a
+B'u cos:! 0:
+B';r:;r: sin:! a
_ (B'
B') ~!n 2a
- . -t';r:: sin 2a
-B';r:: sin 2 0:
' _ B' ;r:;r: ) sin2 2a
+ (B u
-B';r:1I sin 0:
u+
n
2
+C';r:;r: cos 2 a
+C';r:1/ cos a
+C'II cos 2a
+C'z: sin:! 0:
+C'II: sin a
+(C':: - C',rr) sil1 2a
2
+C';r:: sin 2a
----------~-------
F';r: = F;r: cosa - F, sin a
F'II = F II
F': = F,caso: + F;r:sina
Jf';r: = Af;r: cos 0: - .M: sin a
;1['11 = Af ll
..11': = ltfzcosa + .M;r:sina
+C'''II
+C/I/: cos a
-C';r:lI sina
+C/", em;:! a
+C';r:;r:sin:!o:
-C'rz sin 2a
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
cide with those of the principal coordinate system.
The shape coefficients are computed the regular way
in this auxiliary system and tMh transformed to the
principal coordinate system by means of the formulas
of Tables 5.8, 5.9, and 5.10.
Figure 5.9 shows a pipe line in which members 2-5
are shown in a plane rotated
off the standard
y-axis, and members 8-A arc shown in a plane rotated "2° off the standard x-axis. The entire system
is shown broken up into working planes. Since two
different skews are involved, they must be handled
separately. It is possible to set up skewed members
in each of two skewed planes. To illustrate this,
members 2 to 5 are shown in both x'- and v'-planes,
depending upon the selection of the prime axes and
of the corresponding direction of the angle ",.0
Members 8-A are shown only in a z' -plane, but
the reader will see they could have been set up in
an x' -plane also. However, the shape coefficients
are computed only once in one or other of the two
possible planes.
The c-value for members 0'-2 is readily apparent
by inspection as K,. For members 8-A, remember-
",°
liThe rotation of the planes follows the general rule: counterclocl..-wise positive; clockwise negative; 80 that a in Fig. 5.ge
is a positive, and in Fig. 5.9/, a negative, angle.
137
ing that c is the perpendicular distance of the
projected plane from the parallel plane at the coordinate origin, c is equal to K 2 sin 0::2 shown on
Fig. 5.9g. The signs of K, and K, sin "2 indicate
the direction of these distances from the coordinate
origin in their auxiliary planes. It will be noted
that member 2-3 could have been placed in the
z-plane with members 0'-2 as an inclined member.
The choice is arbitrary. Likewise member 8-9
could have been placed with members 6-8. Also
member 12-A could have been placed in a z- or
x-plane, but here, in the interest of keeping down
the number of working planes, it is most advisedly
calculated as part of the z'-plane.
Sample Calculation 5.7 shows the main steam system for a power station, calculated from the boiler
header to the throttle valve (at 0). The expansions
of the turbine leads and connections and the superheater header (usually given by the manufacturers)
are included. The turbine leads as indicatcd by
dotted lines are assumed to be infinitely stiff. The
members 0' to 2 are calculated in the z'-plane for
" = - 45°. Once the coefficients are converted on
Form D-5 according to the formulas given in
Table 5.9, the procedure is the same as for the
previous multi plane systems, except that the co-
Table 5.10 Formulas for Plane Rotating about z-Axis
M,
M.
+A.':;r:rcos 2 a
+A'zli cos 2a
+..1'1111 sin 2 a
+ (1'
I
%;I;
-A' 1111 )8in2a
2
+A' cos a
%E
-A'I/ z sino:
-A'zlI sin 2a
2
+A ' l/l/ cos a
+A'n sin 2 a
+A' Zli sin 2a
F.
F,
+B' cos 2 a
+B';rll cos:! a
M.
+A'l/' cosa
+A' == sin a
%%
1
F,
+B';rz cos a:
+B'lIl1 si n :l a
-B lIZ sin a
_ (B' Zli + B'l/z) Sin 2a
2
+ (B' n _ B' ) sin22a
+B ' l/ Z cos 2 a
-B'zvsin2a
+ (B' n _ B' vv ) sin22a
+B'l/l/cos 2 a
+B' zz sin 2 a
,
,sin 2a
+(B '" + B ,.) - 2 -
+B' zz cos a
-B'J:v sin a
+B';rvcos a
+B'J:rsin a
2
+C'zz cos a
+C'vv sin 2 a
-B'll: sin a
2
l/l/
+B ' v: cos a
+B'z: sin a
+B' ..
+C' Zli cos 2a
+ (C ' n _ C' lIli ) sin22a
+C'=: cos a
+C'lIV cos 2 a
+C' %I' sin 2 a
+C'Zll sin 2a
+C'u: cosa'
+C' z: sin a
-C'lI:
sin a
-C' zv sin 2a
Jlf'. = "'II:
F'r = Fzcosa + Fvsina
F'v = Fvcosa - Fzsina
F'I: = F:
1.'/'Z = Mzcosa + Musina
Jlf'lI = Ml/cosa - },fzsina
+C' ..
DESIGN OF PIPING SYSTEMS
138
0'
I ,
3
,
2
9
,
5
0
5
12
, K,
,
6
0
,
7
0/,
K,
,
,
y-Plono
Cz = 0
I
(b)
0
D'
2
(c)
5
x'-Plono
C;=O
•
(e)
3
2
(d)
x
x
o
7
6
,
Jr:=t"
"
0'
z-Pklllo
C)'=O
5
K2 (0$ ((2
,
•
6
6
A
,
10
"
, sl,
3r'=-{90~O/1)O
2
FIG. 5.9
y'-Plono
C';=O
11
9
12
,
A
z'-Plone
cz'= - K2 $in ((2
(f)
'-------------x'
Multiplane configuration with skewed members.
,
~
4L- o'
M.
730-
<
Y\
M,
IF,
$tJpe~HeAreR HE/iWR.
9
MW KELLOG:J CO PIPING FLE I I~)&~~~ STRE S ANALYSIS
CotINl:CT/ON
"TC
•
os.
F M k
CALC NO 5.
(9)
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
2-0
0'-/
13. U5
1.3.626>
MEMBERS
o
~
1149. 'i
If.e.tJ
1/49. '?
/Ca.d
R
6.00
/.50
h
.3.4/
.•s
1.07
I. 'IS
k
Eh -10·"
< ,
Z
~
2 Ec.-ro·
Q
I
fOO/'
~,
e
z
Cy,O
29.00
I~ 266 9e.O
Yo C.€'.
I
/.00
% CR.
/000 F
~ TEMP.
W
e
D BRANCH
.00726
Z
<:
-92. 75 (.0072(,)-. OfIt;7 - .011"7·
1000 F
.0072~
0':'0
6,
a.
-: 54900
-/7.25
6z
(.007Z'
P:AN<
•
EhIltoM ..4
·/~7K~ 030
Eh1bylt+
-"'If SIt. '100
S YS To!'N(
EttIllllj 44
-22 702 Ita
R07A7Ef)
MOMENTS (FT -La)
COLD CONDITION
COLD SPRIIJCi
FACTORC.o
o'
~/74!iO
"", 'fOO
M.
-/"',600
-,?:"SC
F,
F
.,. 91>0
-9{;0
.,. 2.020
Fz
~370
-2020
-370
Q
R
/
MEMBER N~3'2
4$"
SHAPE ~
00 TRIGONOMETRIC.
6 00
64Z
CONSTANTS
-
kQRl+
385
3852
k;OR l
kOJ;,) +
t
c..
Cb
Cab
-
C".!.
+
t
78
707
Z J
25{)
6/-J
14
S .,
."
I<OR J
311l
3 /Z
kORl
23/ 12
OR
OR •
OR
b 00 kCbb"·~'.u· J9.8.2.f!
"00 "Cu ""\·'Ct>t>-:lli7J9I
6 00 (K.'~lCoIb
"
36 ()O '_)C~.
1
QR
3' 00
I,;) CD
_~ Of,
"00
1.3.
1 VOl
R
QR~
..
cbb
a.
;
'.
m
•" ""u'''' ,
q.
0
_Q.b
kORl
kClR' ~
kQR' ..
k R •
"
•
"I..q~
t
c...
s·,
+
000
c ...
c....
. 785000
b
..
Cbl>
""
Z.:m
•.,.,...,
(KI)}' .. b
0
q.
aRt ~
I!>C
1.3 Cb
... I 3000
I 3000
QRa t
1.3f
.. tOilO
QR
QR •
QR
QRl.
I
785
/<
.,
kC tob,,", )Col. ~_tJ~~.~
~<l.. I.~'b
2
4
t l DOOO
Cb
+ 1/ 2'1 +~'b·a. ~
Z8 S.. b
0
"'\J..a. l
/
" "u'" - o
o
3 0
R
29
,,20700
psi
FO M A
CALC NO. 5.7
fe',,"u -
.::-.
+v~
+ 5/ 4/ v"
a.~"J5b
0'
/.
tj /022
/1 I
tV'b l 13f70 Z
"Q.tatl
10
,,"u·o·l..l _-~o
.. u",,+~~
/5 ......0 .. v~~
() !> ..
Z70
0
J 83 ;./
-qo!)
tU'"
...
eb
I 73
51i78
/
+
III 6/
+
JJ',o
5...
+
f '17 S~b
+li !5t129
GENERAL FORMULAS
c:... ,(cD5a:.eOS(lX.-+~)J
Cb·(SIl,j<".-SIN(Q(~i})]
c
b" "2:S[C05 ~(<>It~)-CD~ Zll(]
+ JJ 11 tvo·Zb tl-l!.ffi21 c.
,
/
/ H ..... t~ I- /
st·"ZS[s"'1 Z(>l,+~)-SIIJ
z:.,;J
122054. ebb' .s f+, Z Sfr.IN ;: (>l.-t ~)-SIN z.,;]
OJ/ /8
.,.
COoRDINATES
0 75 ab
5
7' a"
.,.
61
+4
3
b~
+6
0 z.ab
/
/
CALCULATION OF SHAPE COEFFrClENTS
&0 "':!i~~ to
t/ .S·b ~ 31 ~5 +s·a.b 4'(; 10 +:!> .... <-20 5 67 +S.b~ "'Z
t5'... .,. ., 40 /,
/
+s;...b
5 JO ~:""l.l
.If 'k +:!i'b'a..,. 9 94
/
... "·... D .,.
,3 () /
+.s"&.>.
't
5 18
Zd
3 B3 ,u,.:s 0
0
71
'(,d88
+!;'l>1>
5
;
1 ~
{>
90
5HAPE ' " '
/ 00 TRIGONOMETRIC
U
kOR'
~.""'C
V>lfCO<£O M
O'-'T£
t5£.,.
I 50 . CON:5TANT5
leQR ..
psi
AT POINT
... 6:,..b _/ Jr,
212J
... ~'ab
Q
k
-
MEMBER N2 _0
90
:.TAfSS
.6.325
"25
+S'b
".
01
5 ... : 7.800
5c ·'S;OOO
Z4
"'!>"4
,.,
Y
C1<' ... 6<;.75'
COORDINATES
b
t
(,7 26 ~b
~ 9S$
a. t O O ' b l
39/
2850 2.b ~ 'JI Z loab
CALCULATION O~ SHAPE COEFFICIENTS
of,'ts·a ~ 718Z~S.t>'1- J '9"'5,,~t>"4
{; ... s.ac 1 lJ42 +S; .. b'+Z280431
of
'OR
'!?~ __
5E .~:..~;,'~~rro
PIPING _,LEXI.~llIl Y AND STRESS ANALYSIS
ORIGINAL DATA AND RESULTS
ZZS'
oc
/01
'"
X PO,NI:.
0
-12 725
"'o,aoo
I
~-;?,,_?§~
l'I
ACTING ON RE5TRAINTS
HOT CONDITION
M
,
;A"'~
~--;;'-=--~
_
y< AXIS
M,
HE MWKELLOGG COl
'----...-,'R
('<
0
ABOVT
AND FORcl:S Le)
tI) POINT
UI
IX:
.~
Z'
C'Z·-"3.4S'
-./~5Z4
0(_-45AUXII.IARY COOIU)/,vA7£
:5
lI1
rtf
-.75t>71
-7Z.75(0072C -.020S3.
UI
I-
~y
IF,
Y PLANE
22.70
/.00
~ MATERIAL
-:J\ ,
F,
=
< /
22.70
29.00
18/2G81(;
o f II
lI1
y
x
1.~(J7S
/.'''7$
I
139
140
DESIGN OF PIPING SYSTEMS
,
Q
R
to oa
'OR
'OR
4
5 ••
•
000
/,
, I'
+/ (J{)OO
85Z
..
'OR
kQRJ
o
5_.
2J IZ
OR
~ 00 ~'bb'tl~e.u
R
00 k<:.a"'''bI> J
t, 00 (1':., ~)C.. b
o
36 00 '3C..
i I JOoo
OR •
OR
OR
00
I ()
QR) ..
k
R
64_
¢
8 2
c...
,
kaRl
leQR' ..
1 /
l,~f
q.
C ... b
,
54
.. 157
I 0000
-10000
..
000
5'.
78.54
s-.
s....
~V_bl
I
-'l....
0
.v;, .,.
4" 80 'V' wZb Z
·q~za.b
".
• ~I Yo
\I
a.
..
25
b
'-.J
Zd
..
Jjj 50
eb
~
.,.
<::4.[CO$CC-C;OS{""' ... ;.)]
Cb'(SIN",·srN(c< .. ;:'J]
0
+u'o-l.l.7 /3 00 C"'b~ .z.s[eol!o z(.. ~t)-CO$ ZOI:)
+ 4(,80
;.uoo+v;"
===:.'(2 lL.oo.v~.
MEMBER N9: 5·
SHAPE
t~'t.1;>
t
/8. '2
11 t 72
+(A
4 Silt>
t 7 151 19
'4305 GENERAL FORMULAS
;:r84 S... b
?t; .u-a..
-tV_b i
o
o
Jooo
210
90
Cb
(
/0 08
II
II
l 0420
TRIC;ONOMETRI
CONSTANTS
6
k R'
1.~Cb
ex
•
/ 0
Q
leQR ..
b'
CONSTANTS
'Ii
kQRl ~
I<O,,~
MEMBER N~7·(.
qo'
0<
tJ7
of
0"
SHAPE r"'\.
I 00 TRIC40NOMETRIC.
U
40 c,u.' .5t·.lS(5U< Ze"'~tJ-5IoJ <:o<J
fO ct>t> •• st-t.tS@IIIol.,(>i.",IJ-srNz.",J
3 (,
COoRDINATES
..
0 75
ab
121 50
;:ab
G9
r
+50'01. -
38
Z +5'b _
38
2
"
QR
GR ..
QR
/
"
..
KC~I,HX...
OJ
KCoJ.~+U'b
(
~r:QRa..
0
00
00
r.~c
q.
Q
R
'OR
o
kQR ..
kORJt
(, 42
.,2
J 52
,.:eo,
/5 17
'0
o
(]
0
-q"ll.
MEMBER N'!H-IO
'/0
SHAPE
so.
f
e.
MOO
0000
000
Cb
C.t>
kQRJ
23 /l
ebb
-
1
7 5
(} ~Cbbi"l~'.u
R
(, (){) k<uH'CbI:>
OR •
(,00 (k·I.))CoLb
o
, 00
,.)C:;'"
00
OR
00
1.~Cl;>
000
OR' .. 2/(, 00
I.~.
(}420
a
I
..
PO
b
R
~
kaR'.
f
fO
54
f
1/ t7 ..u·a. t-
q.
o
... ,,~l
"1.~100
+I.oCoo ".0000 ·'.0000 ·,.ocoo
c
-1.000Q .. '.0000 "".0000 _',0000
_.$oeo t.I\OOO -,SOOO ".5000
".7.11 S"
",
.,,,, ..... 7.6 ..
".l"!>'"
7<!1S
HI:> ;',785..'
"".).0 ~'.5Q
pl.'ooo ·'.1>000
-' .• 000 +' •• 000+'. 0
- •• ~OOO
::':
210
~,.$IO"
~
".57t8
_
5Z
Ct>
.,.
MOO
00
1 /2
CAb
-
50017
..
1.3/12
ebb
•
'85-
.'l....
o
ooo.;;·n)C"b
00
l.!IC
{)O
1.lloCb
/ 00
r.~.
-I
, I
..z
-
II
"
..
q.
..
/8/52
i
LJ
+
181 .5l
0 22 "'0
,.
""
+\1.' -za _ x-; U C",b" ,ZS[GOllo ~( .. ~t)~CO:l ZIX)
4(;80 ......
SI- •. lS[Slt-/ze"' .. t)-srl-JzO<]
;'IJ.oo"v;"
- () Cbb··sfot.::S@-IIol.,(o<.",'V·srNz'"'J
?
.............0. +JI
WZb.4ItJOc.......
.a.
1
too
b
,
COORDINATES
22 75 ab
i
13 5tJ at.
td
.,.
12 170
t::b
.,.
4550
3(,00
+
til
~
tab
CALCULATiON OF SHAPE COEFFICIENTS
/008 .llo . . t48 .S-b 1 Z 32 .. S'"~ab t-I
92 ~5"" ..
88 -tS.b" +.5 '/7 PO
~!o'.a. 2
~5A"b _
-' 6:""2.4, t4
"~'b t- 38 2 ·,'b·a.,
alIi
•• '" /m ,.
"'''·.. D 1/5
,-~
f"
/.1..52
o () .s.
?
/I 11
702
q(. Sob
,
5 S ....
o
c
c
c..
c •
.,
~
ttJ.wlll •
1/ t7
o
~~_'
..
'8
G 00 ~Cl>bH)(.U /8(,1if.
(, 00 ~IH~Cb / 8(,/
.7
• $i~ ~ 5..b
f / I.
II S...;.
1& Silt>
•
H.I."a..1
/2 GENERAL FORMULAS
l
tV'1> t"t-V'b ?
I. C:a.[co:;cc-GOS(<X ... ;')]
(J
·'l.zab
0
Cb"[Su·.. ",,·srN(C'I-t;:'J]
.v;, ,
5'.
C ....LC N'I?
_
2/ q(;, sb
(,702
0
- 46,0
-Q.I>
ttJ.'o
SHAPE
CA
c.......
QR
QR •
QR
QR~ ..
QRl ..
QRa ..
'8/!i2
'::1
leo
+'.!io'OO
-t$'l>l>
/0 0
1/ /7
COt-J5Tp.,NT5
kQRI. 7-31 IZ
k R •
=
C
~5:u.
5_.
()iJ TRIGONOMETRl
kQR ..
k R'
kQRI
t
$bb
o
.,.
MEMBER N9:9'·8
/
~
+
CoORDINATES
48 1j ",b.,.
a... J(, 00 b
.. Z J7/5f.,
Z~
{ 00 2.b , 9750 z.ab
U
CALCULATION OF SHAPE COEFF.CIENTS
+S;"b" ~
10 O/) i's.-a. (, 48 ';'5~1> r 4'14<) +'''lI.b ~2 9 H +s"al
72
+,.", 8 2
+~:""b
/
t.·•• u
8
~~'b " JB 52 +'·tl"lI. 1
J l
+5'~~U' ,,3 75 70
/1
Uo
"00
0
-q~Z.11>
tU,,"za -I
.t!o',..b
040
R
Q
/8/52
...
-'-ISHAPE COEFFICIENTS FOR CIRCULAR MEMBERS ;"'~~I<~0l!_ -
/80
c ol4
/d7
3
"'''0 -
-(
f
/ /2
k
5".t>
1
'15 12 Sal>
HJ.~11.~·2
18
.V"b~"'<1/
4 80
+7z,
1151-:6~ ~~
"Z 0410
0(
2 {1
(,
fO!! 12 Sb
£,
46 80 .v~.lt> • ~
C
.u:...,fVO' ~ 7 :.:~..
u..
+ II J7 Yo
~
J
ll. oo "Voo
9/ I ,.,)
+1.0-41'0 -tr.o.. ro .".0-41'0 C.O"~O
FORM 0-1
' 'OGG CO"PIPINC FLEXIBILITY AND STRESS ANALYSIS
~~' ~
1.~Cb
kOR
OR
-
00 TRl40NOMETRIC
(, 00
CONSTANTS
'QR
o
000
HE MW. KEL L:
,
,
/J /7 +u.a. t
8
?<
0
•
"6'bb
008 5 ...
/
o
(K'13)C a b
~r:- ~
+$.b~
+-3
::::~ ~Z ~~~ ~'7: Z "b Z.O -4
1'"& ... 1:>
..s'u.
3112
bl
OJ 0(,
'!"
CALCULATiON OF"" SHAPE COEFFICIENTS
/008 ... =- ..... /4 (,4 ,j.s·b. (,.IZ3(,+Soab ..87Z''' ... S ..... d
kQR'. 211
k R •
a1
- 2
o
a
12 .V"b~ ~5
-q w l:.11>
680
1700
000
(}4Z0
.. V'o
'
-1(, 80 ;''''~'ll;>
u:".tv""
u.,
, l i J 2t v"
+
C
0
'it 1l.",,1'VOO 8 /9 /
,I::~
'.J
PIPING FLEXIBILITY AND STRESS ANALYSIS
~;;~'P
THE MW
, , KELLOGG CO SHAPE COEFFICIENTS FOR CIRCULAR "'EMBE RSPAn -
&
+
·,,'bb.
1(, $bb
18152
15. ~
O"O""Z
"1,$>00 "".~l
"."'0e ~,.$lO"
"1.0000 "'.0000 -1.00 0 _1.0000
·'-0000 "',0000 .'.0000 -',0000
-.'000 +,6000 ·,5000 •. 5000
",1.'"
".1.11'"
... 7.115"
~.7"!>"
.. . .,,,~ ..... 7e .... +,76~" .... TB-!>oo+
.1.)000.'.50
_1,.000 · '.• 000
·"'000 "".'0"" ..... 0
-'.""00
u.0-4ro "".0.'0 ,r,o_"o 1:,0.1"
,~
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
8-?
9-5
9-8
SHAPE
LJ-D
k
107
Q
100
600
R
L
/2
""-7-6
I
I (){)
100
850
82 25
/850
874
15
~ a;
22175
54 75
54 7$
11 2'
7625
2 75
54 75
-
54 7S
(" 02-
"98
822S
6",35
80 75
43 5
4345
63'
S
m
~U
mM
8~
WW
8 U
Sa.
2/ %
Z 8a<l38
807 12
&,99 /3
44/1 59
5"4/ 7/
~b
~4
lm9~
~M
7/8 87
0
0
a
~~2mU
q
cq
0
0
u
II 17
li o
eu
v
0
0
0
0
a
'70 25
II 17
/I
f03 &4
mU2
~
~~
W~4Rn
3
4e, /9
ft.1I 56
/I 17
30e 92
loll 5(,
W499
8 SO
1!i7 2
44> 38
o
CUt>
Iro70~/~~49/~~~uomu~m
c~
~~n~~M~4~378Wq4~~96
"iI:3
934,)(,
312 20 I 24:. 56
49359 - 1(;,'" 41 - "''''000
44 555 Sf., 25 294 1,4 69 85/
-2353730 -II 168 82 -34. 706 12
-4()
97 94 -I] 565 09 -54 163 03
55 077.35 28070 24 83 147 59
75 '27 20 37 940871/3 5(,8 07
flO 481 97 42 70.3 54 153 185 51
so
su."t"clv 3J 5('; 07 42/498 73 98/44
82 983 00 3. /9/ 2S
sbtlTC·U 40 (,34 09 Z(,S (,44 99 4 &'34 09 3(, 083 50 385 9~ (,7
u.."o+v..o
8 /92 14 221 22/ 24 80 43/ '7 77 7/4 OJ 381 $9 08
THE MW KELLOGG CO elf No
S
y
PLANE
MEMBER
~~Wll):i,lF')
0-/1
10-9
5-4
C-If
/334
85/ 32
I 094 72
0
t!
0
874383/ZS7
54/ 7/
251'OS
798 7(,
.3 75 - 1(;,'" 41
5Jb Ib
113[,383
IS If
Sab ... c~q
Vo
II 17
300 92
/I 5'
4 (d)
30Y M
37S 85
0
o
5674 30
/2Z 17
2. &,2 74
J 384 83
5
32
&, b888l
"/5 IS 83 Of#985 21 4/4 7Z 12 9$39/ //8 033 63
cv
foIl Sf,
I .5
100
6
2S
4/ /3
2875
2-0'
D
107
I 00
I
L
a
b
c
Z'(o; .. -45"
2-1
1-0'
l(
PLAf'JE
MEMBER
l·n
c:;'
:SS ANALYSIS
,,"
y
4·3
~
~
5.7
y
c·o
y
C-9 5·2
0-9 5-2
SHAPE"
k
Q
R
L
48 75
a
o
b
107
I
lOt!
100
107
100
10
60
107
100
100
~(){)
10
600
16 75
150
{,o
/43
0
2438
()
198 05
487S
5475
4125
0
{J
0
Z3
48 75
16 75
24087
18 75
IS
0
(, 51
..33 75
145825/4250
b 75
b4 01
{,1 Zb
110 88
82
8225
82 ZS
0
.3 53
94 92
10 08
b 51
S 04
33 75
10512
53 71
44 S9
0
W7~
bffl~
~u
s.
o
10 08
21 96
So
I~Q
~n
~v
~u
t,]i4i.o
4M.,.2S
97 21
4t"" 25
~9W3~%39%M
0 0 0 0 0 0 0 0 0 0
cq
Uo
0
63 38
0
cu
0
u
0
0
1/ 17
I&:' 7S
UO 87
20 ZZ
0
0
v
Vo
4875
118853
CV
0
0
0
5 11 b +clq
0
118ft> /I
0
0
821("
13 187 32
0
0
J 85S 33
cu o
0
cVo
0
0
s.u.-t'c~\'
1/17
S'91"
2187
b
1192
0
19 SO
0
0
0
1/ t1
liZ 37
9187.3
11/7
0
8 4;;'
0
5 93
0
0
0
d.J 88
0
7Z-98
0(; 8(;
6980
5141
f}4ir..7
49 (,7
69584
8774
0
2102'312102.31
IS(}O
651
5243375
7087104;;'2
biB 75
h3118
41i:> 11
3bb IS
St,970 3817 19 41{~J:f..0
918 73
53545
43099
0
I !88sl (7
1 1885/7
0
S 9.52 72 3 437 98 .3 IO?' 25
0
l.b 810>"138 2h Sf:*, 38
0
9?A235741(642Z847
0192119$(921195
tJ
51 % 91 34 27440 ';0 1/5 8~
0
lIto .354 IS 11(:, 354 IS
0
7&:, ,,% 08 4448387 -3 84 90
0
'''0 %Z J4 1,,0 9hZ 34
5~+CU~'~~27~U~~M~~m~~nn~~M~Wa~mJh05n~.38ft9~
u.(l<)+V.O
38 631 30 31 39
77 69 142 SI 25 {j,
THE MW KELLOGG CO "I~ No
9 3{' wt 81
Z7 272 44 l{, 031 18
1i~~oJL1g".AF"jP. Sr.~ SS ANALYSIS
IZ 820 09 Zit;, 554
,.
".
I · " .•
0 229 3&1 01
....
~
M.
+V +A;. X
-? ~
M,
My
x
o I A'
,..
15
I>
-rq
...
01
A)'
I~ +lJ(.xxcOS a: ,.. 7 60 +A'xt cot c(
~ +AU St""t« .,.. 6 67 +A'yr. Slr-Ja.::
<
,
it. +AJU: 51""2«
0
+A1I.Y
+AJ(.K
.,.. '4 27
~
r
0
o I
0
+A~: CO~Za:
0
+IA'n· IiU )""'
r
l'
0
+Axt
l'
~l
~
+A'yy
~
+A y "
Q
Xl:
0
'1'1
'2 57
~
~
+q I
oJ
t.A:
-I- e~o: COS '2tt.
0
93 +B'lt SI!oJZa:
+(6'lit +8'zx )Sft.ltc( .,..
-o~
0
+e~y COSec:
0
+ B'tY 5lJJ a:.
7S
,.
ItBy,l( .,..
546 '6
.,..
386 20
'2 57 -A'xy 'S'~d.
0
+B'r:: SINd:
-
S64 81
+A Yl
-
0
J4
tAu co,~ a:
,.. 6 67
. . e:
t"\:"'"5 11/1ce.
,.. 7 60
~ e~l SIIol'lc:(
k
~
-A~% SU-l-Zd:.
~
tAn
1i(l3'rr·e~,.) ~
+B r /(
>'-14 Z7
~
,
,
~'3
+BtK
+"-.
+8':x
+5_
~
1
x cos ce.
~
,..
0
178 61
I
.,..
,..
1 +
c~" cos%a:
it.
1ee
+C~t 5'''' tee
~
+CJliI(
"
~
.. C'rt 51"!
+e~y
72
-
.,..
~."ir F>\. C05c;( -Fr 51""«
~y: F
SIt..! a;
Fr.> F''I: cOSd:+ FA 51t..!a:
co,1a: +.50
r
THE M.W. KELLO<:i<:i GO.
-e'tK 5tl.l t c:e
.,..
+8~y
-cq
+0
+cv
... BYy
0
+f/yy
0
+6yt
-0"-
0
- S"
1+6'Y1
85"1
798 76
564 &,
- e1yX. ~l"" IX.
.,..
"86 EO
tB yt
32
r
0
623 28
.. B~y .:5INc:e
"'B ty
-
I 068 67
-
69 86' eo
-Cl.l. o
~ C"n c.os?ee
$b8 07 .S<1.b oC q
+C'x y
951 01
0
GOI 98
+ B'lt c.os'" a:.
0
466 69
tS',l(J( 51""fa:.
0
-(e;:I,,+e~I)~ -
7592
-
7S 92
-!'Br:
-evo I+ C'
xt ""3" "'6 I l
1
,. 06 7M 04
+C~1 Co8"
-
49 39l 9t
,.. 76
+Cyt 6''-1 «
-
38 299 Z2 +(C~%.C~J()~ - 19 808 7Z
-
"7 692 13
S9l 76
0
+C~y
~1
it.
+C
~
+e)'1
+CM%
-cvo
19
808 72
1+ yt
.,.. 54 163 03
a::
,.. 38 2' Z2
-C'K,/ !51""((
-49 39Z 9t
.,. C'Y% CO 5
yy
-
-'54.b· c~q,. C-
&3 /47 59
,. &3
147 S9
+C y:
,.. 83 147 59
SI"-l'let
:51"1"
+.50
I~
-70
t' + llootVo o I
-I
~
~
it
L
-.50
PIPIIJ<:i FLEX/BILl""" AIJD ST'RE:SS AIoJAI..:r'SIS
SKEW MEMBERS I N A COORDINATE SYSTEM ROTATlN<; ABouT THE-y'AXIS
+Ct'tcosta::
"
0
+exx tSlt.J'Ze(
-C·I(z.51~'Zc:(.
+C u
Me
CA.1.c.
-
1/ 093 69
/ ~ ;>
·53
,../$.1 18S 5'1
,.. '" 5'92 76
,.. S6 784 04
"'34 706 12
r/68 082 92
FOI't,I.{NO.
D-5'
CA.1.C.No.
5.7
Cl-tf,CKlI:O
Me
....
W
~
o>,j
::::"':l
....Z
o
w
..:
w
..,
-34- 706 12
,. 98 670 b8
I:l
i:'j
+~'z.t
-
• C14 o
0
r
+6'lY cos¢.
64
547 36
l'
S47 36
170
623 28
+6'y: c..osa::
+~
+cq
+B~y
-
Z46!S6
- I 170 64
+6 J1C t:
135 29
I
x' .f.Su.+c'lv +c·~:;
-45'
1"
'" -.707//
co." -1'.707/1 cos?"
Mr'IM z co~cz:.+Mx 51""1:(,
ge
;{e; t· e~x) ¥"
t' +S4<1+C"V 1+ y)' r
My ~ My
-V.
+e~r co~"a::
601
1
Ilot' B'1Cl
466 69
X' t5bt:l+C.~U.t C'
M~:JA"coS"-M:r s,,,,,a:
-'d
6GO 00
0
I
X' +l.loo+Yoo +e'
Z' +5bb+C tL 1 Jli" r/l..3
Z
094
F,
92
70 0'
+B'rIC co~ d.
tS ... A'z:z.
'z;- ~
yl~
0
0
,.. 12 57
~,
0
+'+CV
t'&;yt C05 «.
X
~ ~
+B~j(
93
-V
yl
Fy
I+e;,
z-
)(' +V I +A'
l' +tJ.. J
F,
0
-cq
tA'
i:'j
:::
w
~
2
..
x
y .u
z .v
[ +-A u
M,
, 14
v 98
M,
I
91
Z
.
'1116 86
.q
.,2,30 04 vA,
M,
I,
I
0
0
0
t=
0
0
vq I
~
, IZ0 67 ..
A,,~I f
I
,, ""
v
[
.,.at 95 +A .a .
w
vA
.r ZII .
0
0
vu,
HV
75 92 +8 ..,..
/78 hi
_84 83 -c.
3
4
.oq
69 vB
4·
v /
170 64
, 4 7/G
-cu
v /
17 -v,
Hq
-oq
93 +8 x )(
,
-
-v,
, .s
/35 Z9
v2
v
712 6Z
494 G
v3
J4Z 58 .. Bu.
0
vov
0
-/
. .7 -6.
".
+104 6Z
-5
-4
v 7 772 78 +6r. ...
, 98 670 (,8
x "Uoo+Y""
559 08 ·cu. ..
y "'~,u."'cr.v fl&:'O %2 34 -ev"
-ou
-v,
f
v3
.U,
0
.B •
068 67
<24 30
+/1
-
.0.
_cq
0
-cuo
231 994 55
-4"2 300 4(1 ... c .o;l.
, 83 /47 59
/I 09J 69
x +~bL>+Cr "385 996 67 ·"~b-cr.q - 1/8 oJJ 63
y Uoa+Voo 1'22'1 J8/ 01 -cu." - 19 ZIf 95
·cvtI
z ,u.+ctv
[ .c,
"'''98 525 Z7 +CVl. -/48 339 27
f/(,8 082 qz
x +s.u,+c1. v +(;,3& /9/ 25
b49 /9Z 10 +C ...
MEMBERS 0'·0
Y 1' S bb+d-
z
, -401$ .307
•
3
GAl-G.
c:wrCKIi.
a"'TI!.
".
•
,
..
+
101<
230
-
L 000 00
"'2122
0
M.
MY
.•
~
+ 1.1 1'14
..
H,
v
-
ZZ25
0
•
0
0
-4 0' '9
0
+/1 0" 17
0
'17 7110
0
0
-18 SJ 8
+1' 7#5 58
77 8 -1/
94
13 50 + 4 / .
41 -
0
0
0
'7 17
• .,.so
-so OlUU9
+iI/',
0
-1/'143
- 1.00 00 -.14- 927 71
-
0
0
". -
"7 m ()() -33
4
0
•
-
0
~ONSTANTS
"
." 53107
0
I~
Fx
9' v
1.000 00
3
..,..,. ."- ..S.
Fz
F'I
se
75 19' +.
'9'
00
+.J 31 12
3J ,03 -I' '0
J/8
-21
52
/I
<7
'72
0
0
+/1 61 17
-4 02 69
•
• --I 45' "'2
, + 731• 95
•
•
+ W 93
•
•0
111' - I 908
+
0
4
... ..
85
>01 55
75 •
v
" -231
"'/'" "4 08
"m 15 _<1''2
99<
.,.
01'
/0
0
0
,- 4.
25 0&- /I(JJ 15 .,.:.
0
5.5 -/3'J 1<8 I""
0
'18 "
0
0
201 08
~
dl +M9215 /8 .,. I
0
-271
51
I. rJ47 68 02 -74 /8$ 11 -24 08 OS -IJ i1~ 0
1.000 00 V
141 /2 v
IFIt .,.
961 88'
H+
~81
31~ +
5
,,~,
2 z -/ 3
-, .
" '"
'08 21
-
-".
..,
0 12
f!l!l. ~. Ii§.
0
0
os. 020 24' -2, z< 9/
-
,
0
0
0
•
-'3 01
, •-/1 888':30 -,,' 1288
12' 9 -/32 S9~
..
- .. 000 M00 -+
f<
609 4J
m 00 + / 790 61
'/7 61 •
/<; "Im 013
I -22 7'2 I'
-2$ 104 0
0
.
•
-.
-- 9
-z
~
-
21 9-11 ~ 1.1<
CI of -Bo
O~
5 -II. 13
/
COrlPlNG FI.(XlBl\.ITY B STRESS A"lALY$I$ <: .. ~.
6-EQUATIONS
143
.0<
1.000 00
Fo
THE MW. KEllOGG C
•
.
37248
.'
~;f.ii.~
0117 673 5/
-
CALC. NO. 5. 7
"-H
+44
+/'7""23 -29U572
0
0
- 6 <II
""32'191
-272
- /6 233
0
+ISOO8a91 ,.'UV10Z? -UISf87R .. 28%19
4 -/~2SI10
""6"O2~ #S24219Go .,.'251I-J19. -9JtH/vIO -6~'113 ,130 rl./7/127JJ
0
5 -58.1'19/12
-75()'J'5~ -</IS&oZI!O <l-1IafJSISJ, -SSZ53411 ... 9'1$IS.JN
,.1.1<1136991 -:2"8299 -51Zo 27.1 -22J'-14907 -Z99tVo19S "'4J60982 .,.Z271%51'
0
-44Z(.S
f.]7..3 )'1'1 J1
L1",,+V.. o
1: +Czr.
-leiNG Xl~?<.'81L1TY_A~. STRESS ANAlYS),
THE MW KELLOGG COi,. PLANE
SUMMATION OF SHAPE COEFFICIENTS
2
hi. 17
75 J2
0
-
-s.u;.c~'1
+sw+Cfu.
10Z 13/
G8I 8Z
," 976
34
,so
"'6 .7
170 &I
4/7 159
- 7 ••0 51
v
.5/ 01
-/I
do.
:r ... c ,u'
-4
- 2
IZ9 .4 +Bt:t.
75 92
-87 69Z 13
- /9 80 72
_185 3/9 45
-258 Z54 /2 ~GV"
-/110 354 /5 -S.tb-cZq - 2G 8e,< 3B
[ TAn -I22Z 53 +61.1<
Z
-sa
0
oz.
-
.B •
0
+ /4 77
F. z
- I
,
-cu
-Sa
.u,
•y .u 1'/03 M .c,
+V
F
.s.
75 9Z
I·
0
0
0
+/2Z
X
F,
I
931
FORM E-I
CALC. N-&-'l
DESIGN OF PIPING SYSTEMS
144
CONVERSION TO
00I~T
COOl! RULES
J<
~"-h ..
Y
SE"'~hC.S'1!.
I:h
P.5'
80.75
/7.2.5
--
-
-
.Fz·x
R'
-F x ' z
+z933'
-leGZ?
rl0798
&,..(f_?C)
R'.3
M
Ml:
R.h _
R'
..F.· ....
- OJ(
/
R' .. (ALCULATEO
M l'
ReACTIONS eA~ED ON S"·J '"
HOT MOOU'LUS, Eh
<::050'.
Y
z:
PLAI-JE
M'
..M'"
M'
... M'
~c;o:':>'"
-M'
~M'",
~M'.
~5H'I'"
.. M'",
.. M~.
..N!'
J(:':>'>.lC'-
.. M'
.. M'l:
.. N!'
><CO.6",
/90[}
-I-/071tJ
.. //425
..,.. /5080
..
..
..
..
,.
7
0
0
.00
-t:.OO
--
a
r 8. Sa
-3/447
-I:} .541
..,.. 12/06
f-Il/So
0
0
.,.
5/447
0
a
--
+/5/'13
l'
2425
0
+ 127G
+ II OIB
-
,.. ~.oo
.
-
Z/ I
.,. /2/oG
z235
- 2/9'1 -r/Z/S()
/27G8 + 4750 - 344.3
2235
0
47 - /934/
-.3{
+ "425 + ,50g0 +/5/93
43 -13Gff
/0026
2443
78.3.3
0
1446
0
0
-/58 B87 + Z 139 r/? Gf
8575 r /3135
1.3 Gil -IOOZG
2443 -I- I/O/g +24153
+
~
0
-
0
8~G
2302 - /583
.395
,.- /022
.,. 5957
/696
5
(;.00
A
- 6.5/
a
0
R£ ... 0
~
0
- /. G
- 6./8
- IZ 7Z5 - /542 - ?.; ss
.. 1= ·z
34 Jo4 - Z 139
- /246' -F
-)007"
S59
0
- 1542
M
- f2 72'1
-27tJ
- :1/4
17455
WHICHEvEF: IS GREATER M
M',
.3
- 4.?5
Z
- , a./.1
of
~
S'c
)l
,
- I./. tl"
sa.
-• 1.06
..£.x.....- +_!."oJ.7. (;/
~t
+ 372.4/3
~ -
oR
-e...
R'
f
~ + 96J.88
51!"
&::E'G.G
R' th
&_Ec
7.1. 5
r
Z
l:.
(J'
l'
0
57t!J3
- 5783 - 8/93
2?G
+ 4750
3443
259 +3fDZS9
+-24153 n
a
- 57.83
0
0
,.+- ,2G lOG
2S'l +-3G 259 +3047G
0
Mt>
L
M'b
>:
M,
f .. 12 l
I~. 625 18.8
/3
5 ·f M
PIPE
Z
5'b"'f,8M b
25t,"'fMt
5 +5 t45
S '''5'
+45 ''''SIi!
THE MWKELLOGG COl
CONVERSION TO
COC!! RULES
E, •
r,;-
POINT
,
y
t.:?77.$3
z
Se>= ~.S'e
F
F
F
S, &;'ff'C
OR
R'
h
~ Ec ~
C'ZS 1'5'
M.
-F •
S"
M
WHICHEVER IS GREATER
tFl--X
-F x ' z
0
M
""-. (1-- c)
R'
3
tF.·y
,
RFACTIONS BA5EO oti 5t'" '"
HoT MOOULUS, E.,
•
·M·
+M'l(
.M·
··-
- 'x -
~. I
tM'"
_M'"
:E
tM'"
...M':
-
M'
PLAt.JE
M,
.M·
l(c.o~«:
-M'.
)($INOl
M',
.M·
.M·
K!ll"'~
"c.os,rx
M,
PIPE
13.(;Z5
Z
f .. 12 Z
I- .8
jj
5 of M
5' -ft3M b
25t-fMt
5 +5 +45
S't5'
-
•
/0
0
O.()fI
G,OO -
-
1'2 'SO +0
2GIG7 +
3B 3/7 +
9ZZG -
,
0
0
9ZZG 047'
t;7 713 0
37 237 -
38 JI7 +12. lOG Z. 235
2844
9<'
0
-
-
5783 ..
3 4d3 +
37237
.5 183
0
4-3 OlO -
-
0
0
1&,75 -
-
2~44
33 1~5
-
0
534.9 -
3
J,.
-
0
IG 145 +
12 702. t
43020 0
a
4:?> 020
,
p
II
~,OO
a
-
FORM F-I
C l
0.5.
a
0
48,75
a
0
0
0
0
-- - -- 4"
~,Co
5.34
12 lOr,
0
/7455 IZ' 70'2 -I
Z 235 .5 783
IG 250 43 020 0
/2 10&. ..
3091d +
+" -5
THE MWKELLOGG CO.l.
-
0
0
17 4S5
-
-
(;250
18 158
0
I 908
30914
,-
a
9,8 358
•
&>7444
0
f7 455
0
0
11 455
/ ~O8
0
0
1908
&>7443
0
0
G1443
I,M
CO!;«
z
y
M'
70,25
0
,.
, .3 2.IM
, -•
J\' - C.... LCULATEO
M',
-
0
+ 9133,88
iF ·z
*=
8
+ eOl7,GI
-£h
R
YIPING:5~~'~NTXN~N~T~lRl~fsANALYSIS
,
2
.7
4
1908
-
-
0
1908
0
17455
/
455
07109
1,00
4195
"0
1 Z41
2455 000
4 955
ylelNG. !,LE~~~ILlTY AND STRESS ANALYSIS
MOMENTS AND STRESSES
CAL
.~.
H
-
•
ORM • I
C I
0
I
I
1
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
effieients from the skewed planes are added on
Form D-3 above the spaces for the X-, y-, and
z-planes.
.\00
5.15 Branchcd Systcms
The procedure for calculating a plpll1g system
with branches follows that for a line with only two
end points except for ccrtain steps in obtaining the
equations, in particular, the calculation of the
movements, and the summation of the shape coefficients. Only those operations which are unlike
those for two point systems, previously described
in Section 5.12, are diseussed in this section.
In calculating the movements, expansion is considered to be directed from the fixed end, designated
0', to the end points of each of the branches, called
free ends, which for the four-braneh system shown
in Fig. 5.10 are designated P, Q, and 8. Therefore,
the movement in the x- and y-directions from 0' to
each free end and the products EI*D. x and EI*D..
must be detennined separately.
In caleulating the shape coefficients, it is good
practice to set them up and sum them separately
for eaeh branch involved. The braneh from the
first intersection between two free end branehes to
the fixed end 0' is called the eommon braneh with
respect to the said free end branehes. Henee, the
common branch is O'R for branches P and Q, O'T
for 8 and P, and O'R for 8 and Q. In setting up
the equations for the free end branches, the shape
coefficients are summed from the free end under
consideration to the fixed end 0'. The coefficients
thus obtained are multiplied by the respective reactions at the free ends, designated as AI: p , F:x.p, Fyp;
M: QJ F:x.Q, FyQ ; Jil zs , F:x.sI FI/s, The effect of one
free-end branch on another is obtained by multiplying the reactions of the former by the coefficients of
their common braneh. Designating the shape coeffieients as A, B, C, D, G, H and indicating their
,--------1s
'1--------1 p
-I---------j Q
o,If-
--'
I
FIG. 5.10 Four-branch system.
145
Table 5.11
Con-
}.l/zp FZ1' FliP
M:Q 1"%Q 1"IIQ
M:s F;r;s Fus
AI' B1' Dl'
Bl' C1' G1'
D1' G1' Hl'
AR BR DR
BR CR GR
DR GR HR
,1T BT DT
BT CT GT
DT GT HT
0
1<:/* .1;rJ>
E/*J.yp
AR BR DR
liR CR GR
DR GR HR
AQ BQ DQ
BQ CQ GQ
DQ GQ HQ
DR
All BR
BR CR GR
DR GR IIR
0
HI* ..\rQ
HI* .iuQ
AT B,· DT
ET CT GT
DT GT liT
An ER Dn
En CR GR
Dn GR liR
As B s Ds
Tis Cs Gs
Ds Gs lIs
::-tttut
--0
HI'" ..\zs
HI'" J.!/s
---
Table 5.12
M:l' M:Q 1\[:8 1"Z1'
Ap An AT Bp
AQ AR En
,1s BT
Cp
F.Q F:s
BR BT
TiQ TiR
ER Es
CR CT
CQ Cn
Cs
1"111'
FIiO FilS
Dp
DR
DT
Gp
Gil
GT
DT
DR
Ds
GT
GR
Gs
lIT
lIn
lIs
DR
DQ
DR
GR
GQ
GR
Ill' IIR
IIQ
Consttlnt
0
0
0
El* !1::p
EI*!1:rQ
El* ..lrS
EI*!1IJJ'
EI*!:J./lQ
HI'" .ius
sum from a point under consideration to the fixed
end 0' with the corresponding subscript letter, the
system of equations given in Table 5.11 ean be set up.
It is usually advantageous to arrange the equations so that the rotation equations first appear, and
then the deflection equations, since the constants
for the former are O. This is shown in Table 5.12.
In this rearrangement the symmetry of the coefficients is preserved. As in previous problems,
only the quantities on and to the right of the principal diagonal, need be set up on the equation form
sheet. Once obtained, the equations are solved in
the usual manner.
The moments are calculated on Sheet F-1. Here
again, each branch is grouped separately. The
forces and moments obtained from the equations
are the reactions at the coordinate origin of the
branches P, Q, and S. Accordingly, these values
are entered on the form sheet for the appropriate
branch; thus the moments and forces with the subseript 8 are for the braneh 8-1', the moments and
forces with the subscript P are for the braneh P-T,
and those with subseript Q are for branch Q-R.
For the branch 1'-R, the sum of P and 8 is used;
for the branch 0' -R, the sum of 8, P, and Q.
In setting up the coordinates for the calculation
of the moments at various points, a point-by-point
DESIGN OF PIPING SYSTEMS
146
.-------1'
.I-------IQ
upon by the restraint of branch R-Q. If the members of the branch R-Q are now reduced to zero
length so that point Q coincides with point R, all
the coefficients with the subscript Q will become
zero.
The moment, forces, and expansions with
carried out in a manner similar to that shown for
that subscript may be given the subscript R. The
six equations which could be written for this singleplane system would still be valid although, since
the point R is now an anchor, it would be simpler to
solve the system as two separate lines, O'-R and
R-P. Consider, however, that the line is allowed
to pivot at point R, so that the moment at point R
caused by the restraint is zero. As shown in Fig. 5.llb,
the system now becomes a single line with fixed
terminals of points 0' and P and with no translatory
displacement permitted at point R.
The first six equations of Table 5.13 are written
in accordance with Table 5.11, based on summed
shape coefficients A, B, ·..H for which the subseripts
P and R indicate summations from those points respectively to the fixed end 0'. The seventh equation expresses the fact that there is no moment
restraint at point R. The unknown rotation at
point R, 8,R, is eliminated by multiplying eq. 4
by -YR and +XR and by adding these equations
to eqs. 5 and 6 respectively as shown in Table 5.14.
Finally, to satisfy the relation expressed by eq. 7,
the coefficients in the M'R column are multiplied
by (-YR) and (+XR), and the products are added
to the coefficients in the F zR- and FuR-columns
respeetively. The equations obtained are given in
Table 5.15.
It should be noted that if M'R = -yRFzR +
xRFuR is written out as illustrated in Table 5.14,
the correct signs are obtained for the constants by
which the rotation equation is multiplied for addition to the two deflection equations.
For a multi plane line the following moment ex-
six equations in Sample Calculation 5.5, Form E-l.
pressions must be written:
o'f------'
FIG. 5.11a
Three-branch system,
method is used, proceeding from S to T, from P
to T, T to R, from Q to R, and from 0' to R, For
the junction point R the sum of the moments must
equal zero.
This condition serves as a check for
t,he calculations.
When the results are entered on Sheet A, it must
be remembered that 0' is the end assumed "fixed,"
and P, Q, and S are the ends assumed "free." The
guide for the signs given in Chapter 4 also applies
here.
As a suitable example, the three-branch system
given as Sample Calculation 5.7 has been enlarged
to include the flexibility of the leads from the
throttle valve to the turbine. This calculation is
labeled 5.8.
The working planes for these leads are shown on
Form A-I. The computation and the summation
of the new coefficients are shown on Forms ])-2 and
t.he summations with previous coefficients from
Sample Calculation 5.8 are shown on Forms ])-3.
Since this is a multiplane system with 3 points of
fixation, 12 simultaneous equations are required for
t he solution. These are set up and solved with the
standard procedure on Forms B-2 and B-3 and the
moments and stresses are determined OIl Forms F-l.
The check of these equations is not shown but is
5.16
Intermediate Restraints
Discussion of the details of the various stops and
guides used as intermediate restraints will be re~erved for Chapter 8. The present section will give
A/xR = -ZnFIJR + YnFzR
forFllandFzstops
and AJ yll = -xRF zR + ZnFxR
for F: and F.r: stops
the procedure for including their effects in the flexibility analysis. Although applicable to any type of
restraint the treatment will be confined for simplicity of presentation to restraints which prevent
~-------lP
translatory but not rotary movement.
The problem of the intermediate restraint may
bc approached from the branched system discussed
in Section 5.15. Referring to Fig. 5.lla, the branches
O'-R-P may be considered a pipe line which is acted
o'f-------'
FIG.
5.11b
Intermediate restraints.
FLEXIBILITY ANALYSIS BY THE GENEHAL ANALYTICAL METHOD
147
7
21. (J()'
z
MW KELLOCI3 C
MEMBERS A-O· B'O
/0.50
D
1.432
t
I
42'1.8
Z
81.8
1.25
R
h
1.04
k
'.(,5
~
</
Q Eh>lIO·
22.70
Z Ec"l029.00
'"
e
w
a~RANCH
z
</
22.70
2'1.00
t8/Xd9(,O
/.00
<I
22.70
2'9.00
~CR.
#'1. 0:.
I-Yo CK'.
1000 F
/000 F
/000
.00726
.0072e.
.00726
0',8
EhI !lz/144
- 22 702/20
Ay
ii:
A,
Ehf~J(,Ij44
COLO 5PRIIJG
FACTORC·O
lJ) POINT
S
'"
W
0:
.aone
qj
F
F
TI-£ MW KELLOGG col
0
IF,
Z
Y
1.25'1 I-0
/8
X
~y
M.
j
/3
N
/7
X
~X
z
/4.00
1.00
x~
o
r
2
I.ZS'D
'"
/.25' Z
.
Y PLANt
C'j- 0
'YPLANE
C,:/"O
Sh. 7800 p:li
5c' 15,000 psi
AND FORCeSlL6) ACTING ON RESTRAINTS
HOT CONDITION
O·
A
-q 75
+ 2075
+6000
of 2400
+ 870
., 11/0
+ 280
., 400
-22225
-
,
Se: '~;..o;.o~.'~;~r[D
S
1" 3175
+ 425
- 20{,50
- .30
- 880
230
- 100
ISO
PIPING_FL~) ISILITY AND SIRESS ANALYSIS
=:.rRt5S
• 4/75
DATA AND RESULTS
'"'
O ....TE
psi
J.,T POII'/TA
5A·~i~~~tBRl.:>.l(;;
CAl.C.
ORIGINAL
0
17
"
M.
M
F.
~r \1
,,51 I-
15\1
F.
B
14.00
MOMENTS (FT -L8)
COLD CONDITION
M.
c;z:' -5.50
Y
_A
8
~•
Z PLAN!
I. SO
.0<
EntA)'/I~4
A.
"'
U"S7S
1/4'1.9
1.95
O~A
CLCN05-
Z PLANE
Cr .t5.50
',,",.6
C:.OO
-9l.75(j)07Z6 -.041(,7
-.0'1/67- .75671
-72.75
:02083 ..·.$4900
17.25 (.oone. ...
.12524
-/37/GS 030
- 99 516 G(,O
W
0-
...
f.(.~7S
1.07
2.""
I
0:'
/3.<'25
1/49."'1
I"d.d
Mk
S ANALYSIS
y
/·0
13.<'25
3.41
'"
o fhth
Z
Q
« MATERIAL
n:
TEMP.
x
PIPING FLE I ILITY AND STR
SKETCH
• • E'.
"Z~
r:s
.20.700 psi
A
F
CAL NO. 58
liS
DESIGN OF PIPING SYSTEMS
PLANE
y
Z
A
MEMS.ER
IS
/S-14
.., /4-/.3
1.,3-/2
A-/3
16
26
12
k
Q
R
L
- /65
2 "8
I
7S
.o
II 0
700
.. 5
a
1400 -
127.
b
~
I 5
o
5
.55
1102-
869
/I 71.1
.3082
I 74-
c
-
.3 0
12
•
1809
'.
5. 2
'b
8. 76
cq
0
U
2.352
76
32 8 /2 36
180
10 394268
776
.. 5
U c
-
cu
v
v.
Z
5 a b·C: q
25
{,
68
o
560
8 9
1139
"8
8 71
438
39
2426
o
o
-3 8
1574 /6 5/
-
o
62
64 41
62
7 6
48
6H2
4 S
4268
20~
-
5 74
0
27
-
210
o
o
o
o
o
222
.. 7
o
06
-1092 9
5,u:,.C v
246
o
71 6
o
19 IS
4 S.
0
o
o
o
/I
88
62 7
23 8
2008
0
,J 9S
~
776
.1 4
400
- I
I
o
o
I 93 39 .3 O~_L 96 8
5bb"'CTu 1/6 I))
378/
1Jl t/~J 2 38 IS
lLoo ... V....
S 06647 I 09 J I 819 2 8 J2S 92
TI1E MW KELLOGG CO elf NO '!~I,<>!L1~~,r?~, ."SANAL'fS!S
l
o
o
I
-/8/104- s.B./. -1/865
cVo
21.1
I2S
0
8376
9950
1172fi:9 -
cv
cu. ..
/.3-0
D
SHAPE
L
12-0
-
22
42
-
2M'
o
.107
6£
I~
88
o
o
o
91 4.307
261".3
"Nt ):4'.
'B~ 8 RANCH
Z
PLANE
MEMBER
8-/9
SHAPE
I
Y
/9-/8
/tN7
B·n
17-110
Itc -0
/7-0
\]
--
L:
LJ - -
2:
/M
k
2 oS
Q
208
L
-
b
c
12
/400
463
550-
.5
/809
ZSJ 2(,
5b
83 7("
a
cq
u
-
cu
8 lo"i
3082
117 7/
US 74 -
$11.0
5& 7/
{)
87 7/
0
0
0
0
0
2352
77b
104 31 -
.3fJ83
/,ZIO
215 74 - (,4141
43 ~8 - 1(,,951 - 341 55
7 76
40 07
b5 92
4250
8801
.dZM) - ZZOJ9 - 3"<:57
S474
0
-I Z2733
574 15 I IS/, 57 .3 571 7(,
2338
0
- 4840b
I 83039 ,) 0(,1 'If:, 898524
z.n 81
931. 31 Z 33815
I 4a9 63
I 84982 8 3Z592
32128 -
46()('8
4 092 (N
5t>t>.C: Lt
I 14>803
SO"" 47
1,1.""."'."
-
()
S,u.c·v
425-
2/3
-
125
{)
0
/ 5/
o
3 95
18 09
8.J7b
cv
9950
5 tl b .c~q - / 172 59
C Lt"
I 8/1 04
-
/I OZ
0
/29 3(,
-
425
70()
{)
SsG
{)
v
Vo
cv"
/2.75 125
550-
,j 80
54
UO
/ 25
1150
(,75
a
268
2(,8
/25
R
L
/65
268
THE MW KELLOGG CO elf NO
,gf!!~IU'):tA11f\r'~; r~S ANALYSIS
8 b9
4384
-
-
-
.395
o
o
7 7b
3843
o
0
2008
68 10
395
0
0
o
0
425
14 13/
0
o
0
21 09
a
o
0
o
0
tZl47
.3 07
b887
0
19S ""
1,887
".
o
/I 39
- f_,*-42" -
1 76
-
1139
Z426
'.-
19 IS
~2 f:,9
o
2257
-
4 25
o
21 09
o
o
2'11 34
.3 07
I
FLEXillILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
Mx
~
2 ., +230
x
,y
.~
+V
E +-A u
C.
M,
19 /5
_65 9Z +q
+
~315
M,
0
.q
0
/I +A. v
+A.::lt-
0
+231 95
I
•
,x ., __ 6220 08
+V
, w·
/0
E .A
+3/4 13 t-Ayt.
I.
9J
.cq
-cq
a
9.1 .. Bxx
+V
a
H'
-v,
-'.
0
-c~
0
a
. 4 75 6992 .8,
-- 34/ 55
.. B v "
, .u '-~ n .u,
y
0
.u,
75 9z
'Sb
0
0
=
c
Fv
Fx
I
•
.~
_22 57 .cv
, +s .57 6. YO.
02
-cq
~,
J 134t 58
•+
'2 '9 -c~
.v,
56 5
3 7' 81 +B n .
- a
.cv
a
+G.
.cq
4 439 34 .6
7 772 78
•
-Y,
-s,
0
0
.u,
0
.8 •
12 '4
-/I
-cu
-
'9. 54
-
88 0/
-Sa
b8 /0
-,
H9
778 5
-/I
616 17
- 3 9
4/
-_ /0
96 81
759
6~
.cq
•
a
41t5 _cq
58L 7/
_ /0 538 98 +5:.:-
0
-
0
87 71
75 92
r +A n +3()2 70 .. 6:)(
7 86 '9 .6,
-4&.2 0010
-23/ ~ 55
/92 /0
-cu. ..
-c::v"
x ~Uoo+Voo
y +:>.u.... c1v .,.
a -5...b·d q +_ 3 21 09
29/ 34 ~CV"
·c:u o
571 76
Z +Sbb+C'l,l + 2 338 15 oSu,-C'q
227
1: ",c"'>C ?'5J 82/ 9 +C"y -«,/ 07,5 07 ... C 1(:1: -228 01 0
-/48 J3 27
+-G98 525 2
•-
-
"
)(
+5t>t>+C'
.. ,
."..t>-cZq
53 -cu. ..
z. -K.u.+czv
~
985 24 *CV.
r. ",cv v +7()7 775. +Cyl:.
y
"A" .BRANCH - MEMBIiRS O!.A
U~V..O
0
t
-... /48 8233
I~ 06
+//17 .7
)( ...'....+c
'
y "'Sb't..+~ +
z: +Uoet+V"" .8
THE MW KELLOGG
;
~
x
y
,
Mx
o.
I
.,
+230
.u
.v
• /9 /5
+65 91 +q
r +A"", .,.315 1/
A"
col.PLANE
PIPING XLEXIBILITY Af.'!'? STRE2.':>.. ANALYSIS
SUMMATION OF SHAPE COEFFICIENTS
M,
M,
I
=
C
,x,
+V
YO
<U
L .A
9J +6 x "
•
0
I
+ 20
+ 62 /0
+314- 1.1 -t-A t.
x
-v,
--
0
+u,
0
.c,
75 9' "t'Bl<v
t12' ,
-'a
0
0
-cu
•.,.
ot"B v ,.
- 3 '2d
.,.2t25
,
"
x
YO.
.,.
I ... C......
c~ .6,v
7 772 78
-cu
-Y,
0
87 7/
8" 49
'1'649 192 /0
-5,
.B,
-cu."
29/ N -cv"
3J8 /5 -S41>~clq
• 2 82/
'1',5/
.e x
x
".B* ,]jRANCH ~ MEMBERS - 0 1• .,8
-cu
"<Z " -v, Z 9/7 J2 .s"t.
-
ifl 55 ·+cq
-loUoo+\loo
Z -loSbO+<'u.
•
-sa
'8/0
+
y ... ~......""c~v 'I'
-
~,
0
• II
.cv
0
+G.
.,.•
•-
0
+6,..""
/I IZ 9
.cq
12< _cq
."
5
/7
.75
"
~
c -
.L
58' 7/
J 9Z
- /0 )8 98 +Bt.z _Z31
-4(.2 J.
9 55
,
-ev"
.,. / 227 3J -eua - 3 5121 ,
-5"'b- cZ q _
C
~4'1 0730
,
,.C 'et.
52
-~.b-clg
",Sbt>+cZ
y U.-a+lloo .,.
2'4 5J -CUo
z. t+s,u +CZY .. 8
832f -ell"
1:. +Cyy
po 707 775 ~ ...c vz.
)( n .....+c z
+Sbb+c"-~
y
% +Uo<>+Y"..
1:
THE MW KELLOGG COl ,!;;'~~ sG~~WI~~',)Ai\'A~E '&~M~~h~1!5
0
880
.J 95
.U, 4- ,410 '6Z 8/
0
-
- 769 54
534 58
-cq
a
.u
.u,
y +'
22 1<. .cv
+5 + 57
I: ..Au .,.J42 70 +61;)(
,
..
75 92
"b
Hq
-cq
0
I.Al<tl'f
0
r1.l1 951
Fv
Fx
93
a
.q
0
..
I
•
0
-
M
CALC.NO.5.B
,- -
CICIO
o. •
a
3
325 92
.,. 118'- 002 50
1: +en.
<: .... L.c.
Glol
/
"'C,z.
CAl.C.
CH
ox,
<;1(1;:
I
-
~-J.3
- 2 5 58 40
98 J392
-
-
0
4r·
"'
-/"'7 55 21
rl/77 ~73
to
3 '7
.,. 8 325 92
+//86 002 50
-
M
CALC. NO. 5.8
~---------------------
.....
<:>
'"
~
H x
I
f'
I'
-
/'f,,;1\
_
.J1611 r
I. 0
I.
0 0+
2 +
-
I
r +
2.' -
IM;o.a
HYA
HYB
Z.JtJtJ4
()
0
7,M &3
31Gb
(J
0
0
0
-
31.:5 I
0
0
+
93 +
.,.
l/.t;~
0
0
0
0'"
(,8 -
-
0
0
0
0
25 .,.
co 7tJ /3 Co2 -
25 +
otJ/7tJ 20 '1 -
0
0
0
0
/4 I
1.00 00
;,';"23
"f-
3/
/3.,.
3' -
IMYA +
()
3/ / fr.oo 00 _
2
J.,
+
PXA
r
-
2.3/ 9.
0
tJ
[To
fr, rA
+,
13 f
tH}l 95 -
ddt 9.
23
3.
-
-
'5
/7M
1 -
.,ft.
-
5
55
e ~
Eo
0"
Z
-
/.J t 12 85 4
-
3
0
I f1
0
0
'3.!J
OJ, 2,
0
0 . , . . 14 132 I
0
0.,.
/
0
0
I
-
0
9.
()
:131
f-
0
43
-
3
1-5
- 4- 02 6
;:y~
P'YA
1c.
~
2 1'.3
240 '13 -II
+3 34
3 ZZ -Ia 'a
lsi"\"<
201(.;S
-10
_?!M.
FZA
"+
0
0
1-"
0
0
0
0
0
+ /,
/0
0
et>m,l't. B£LO.....
"41~7'
0
C.?"T.G -
0
""
011 9
c:, I 06 + I
~ 2
0
0
'"
05,(,5
t.
+ II 6
81
0
'S
0
/
0
0-
tJ
0
0
0
0
+- II 61
8/
8
ZG; (,1
0
"
b
0
0
-
0
0
-.3
1$ -
-..3 62 0
0
0
+-1/ U
I
C
+ 10
Z
-
0
0
11/27
14 t%
0
0
0
0 . . , . I 124< 8.
I ! f4' -
I. 000,00
0
0
IMY'S -
I 3'J. 3/
0
5 +
02
0
tJ +
0
2 9(, + 2 91:. 48
_
75/::3 h
r.5' 2(, 00 + 4
I(
f-.3
o
2ZtS3.,.
0
-
o
0
;o!
IMU'
~ 8
- /0
2.2 -
-+
22. -
In
0
0
0
B 1n9
0
0
0 . , .3
0
-2.
0
0
"3 9.
II 12 I 01 -
la'" -1-
- B
T Z
-It. I'.
:.'l
"2/
0
~
0
0
"2
(.6 - 2.
- 2.
_ 1/ 12, 4- -
0
0
:.:' 22 'l ~ +
(J
75 qZ
22
0
0
9
l./3
03 -
f" 8
0
0
0
0
....
ee,.
0/4
<0
T
~ l56 - G 123 " - Ie. 1.5 18 ~30 I t.89
"
+
o
-
o
-
G
IMz.e
of' B
of'
+-
5
/
0
7.5
22 '1,5 of'
.3 42 +
22. (,.9
.3
~
/ 01 -
0
(]
0
0
0
0
0
I"
(]
0
0
0
3~ 43 f/.3 12 -
0
0
.% 43
/3 /2
0
+
0'1.:I1
, q 43.,
11(" '10
Z
/,
03 -
_/0
Z
n- d
- 5
~ z
PIPING FLEXIBILITY AND STRESS ANALYSIS
09.31
~~;~~.o
o"n:
'1.
n;.,";:'" b~
/I.
z,3 • .!5.3
~
:s"'0
Z
~
(JJ
0<
~
( JJ
Ie.. .x - + - 3
3898
q ~ +
I ...
/I
'13"18 ~
7 (, 'f' 8 /9 0
~
1#.2t. - 3139.M2 - Z 3.5
1.000100 -/4 .3 48 -IS',U 51 .... 2413342 +1(; '633 ++ /2 33 'oz - 3 38.3 - '/ a '6 'f'2/
2
+.3 'lJ, 1.5""
-
&J\5~ 13 II ..,.
22 -
0.3 -
o
1:
+
22. -
7
In
THE M.W KELLOGG CO
..,. 2
0
03 / +
.3
.,.
.:3
0 0 0 0 0
0 0 0 0 0 0 0 0
3{) 7 +
222
... 7
Z4 "" 717,;
-/0 -S/" _1/1/4< 61 ~:s r:t.i 3"
1.100< 00 ::J,S1/5 - 25 96 10 - 25
'f'34 8 " ' +-3
/3 +"
of
1'-3
-
0(;
Il: +5'
0
(JJ
~
I
0 0 0 0 0 0 0 0 0
o
t 'l
-.36
0
0
r8
ONSTANTS
69a
., Zd 4a
'7<
r /0 'i
0
0
.,. 12
-
b +24 Ir.....d/Q
tJ
~ ~;r;;
'5 '2 +
J 5. .,.. 2: 91 2. - 7
5$ 41 - 2 '5 (A - Z 4- 18 +5
20 50 .,..
4
Z
I~ 2 - 4 02 1 2 0 +1;
8
- : I ";4', 'J '(,/
.,. /
02
0
FrlJ
- 7
0
0
0
0
Fo",,, "-2-
CALC.
No. :S.8
FLEXIBILITY ANALYSIS BY THE GENEHAL ANALYTICAL METHOD
-I
I
fT]
$;
~
A
fT]
r
5
(j)
(j)
0
.Q
FxA
Fx8
FYA
hA
F'B
7 +"5/ 8t1 'I TG4 192/0 - 461 073 07 -462 040 -ZZ8401 70 - e.31 99<:. 55
/8 Z'1 90779 /829 33 + I 87J 09 + 1.5 B8
JOI 4~
Z 86 8245 6646 +
280 ZS +
lB' - ' l 737 5/ - 5(, 48 Zr,
+/54 l Z9 +IG4 W
0
0
+
/l
0
+
/B
3 'ISS 60 -339/ 67
55
7G
0
5" Z9
-t04- /07 6-f. - 201 830 0' +273 9B/ 73 +289 lB' 33 + I Z8~ 67 + 128(, G
- ZB 583 9' - .30 7<5 57 + 4B 52B U+3.3 '40 +
lEB 07 +
/880
G5/ 07
:>: +352 4/5 73 +3X ZJ5 4< -/3 .<5B 34 -NO 05B 79 - 51 J/5 04
1.000 00 I 0/0 8' +
/4-5 6/ +
/4' 56
35 9 +
397 47 +
ZG zr,
No 55 or
735 B7 07 +
48 42 ~
9.3 39 +
i
-
IF,t'~ or
'"
- /829
- 7 8G27 -.,
-
~
;:;:
~
,;go
r
~-
g-<
»
-<»
~~
-
Z
:l>
'(f)
=<
(f)
-
- /4 855 ZI
- 35
9.5 + 82 5/Z Z/ + 8/578 G3
1 .. 94+ , l3 0/ +
0 ' h5
0
0
0
0
0
0
- JIO 0'0 ' h /623 M- /823 h4
7ZZ 55
- 40 OU B' 'Z7 35 ' l - E052 /5
:S~2 1(, - t
950{,
504
3/
05 +
'98 93+
fj() (,/ + (,(.070 48
- / 5 '50 3/ +
~ 258 <B
+ 88/4 47 ZO 1. 00 00 +
7/0 0'
70 B8 +
00 +
o 5 +
IZ7 U
+
l
,
IFYB
-
'"
"
-
2 o +1177 m 5/
- /92 0/ 98
/B' B4Z 44
- Z7 OS 7(. - Z9587 3Z
- 382 90M - 405 3'7 75
-Bh 797 /2 - 58 80/
8 II 8/1
/ 24 /24
747/ 98- 7 '0 /
0444 '5
- 90 4'/ /B - 7' 04/ 69 - 28 07 B'
4
42 77
444/
:>: +J,<, "'7 / +3(,3 5277
1.000 00 991 50
+
177
99 5 If,A
12 +118& OOl 50
- /87 M'i
BZG
- 4i1 552 7Z
- 39 B35 /5
B/I
/74
75&9 97
2827
- 90
- 78 liB 14
4 443 <5
-
• X>
He
,.
"00
"0
~~;t>
.'"s."
"»."
,."
z'"
om
"'0
~~
0
0
0
0
0
0
-/37
+
-
c€-0 f0
+ 158 ~ ~I
+ 1 BG 0/
-
713
- '195/(,
0
0
0
0
~
0 f- 54 31716
-- 15i 3141(,
/47
+
963
-
qq~~
-
5J 5133J
"
- " .,
-
-
-3(,0 4{,Z5
:>. +
{, O~I 80
+
119 lO
0>
IF'8
0
0
0
0
0
0
-+ /54 707 651
4/31
-
U
+
68/
ZZ 10Z IZO
0
0
0
0
0
0
"
.
'"
389 ZZ
-/37 '8 3
0
0
0
II filS(,
-
000
..
- '"
-l
»
"
-"
- ,
•
- ,.
1M
, + 07 76 04 - /483
::0
(f)
(f)
-" 'G8
-
(f)
fTl
,g~!J
-
-
~
.
..
- 46/ 073 a -Z31
455 5B7 40
907 79 0533 + /874 0' + /85 BB
4 +
B7 45
lBO Z5 +
30/
+,.0
0
~9.3
02 .;./49
0
~
0
+ /5 8B 69
0
- Z 90B /5
7" 70
- N'j 57 90 +270 9U 44 +28:'
/ Z; +
/ Z7Z 31 + / "i7Z 3/
-33 /13 5' + 5Z 23/ 99 + 3'- a5 58 +
lOZ 47 +
tal 4Z
-"0 097 0' + 14/ 66/0 + /4/ 17 03 + 5/ B7I 30 + 57 ZlO 9Z
3/ B + Z B 8 - /9 7447 -/9 " B
>: + 4. 410 50 +
1.00 00 47 9/+ 4 l8 84 + 45l 07
ZII Z
8//4/85 9/II/ 84 +
Z 66 +
6Z9 70 IFx8 +
27
-148
23,33
148
9 + 107 775 04+'982
339 Z7
- 45 05 89- 399(,
+ 9300 B3 + ,/ '§l48
Z38 B
I 9f9 / H 8092 45 + 8704 B6
0
0
0
0
0
0
0
0
/ 727 /5
77 46 - 388 Z4 4~ /727/
zq53
3B6
5
3/9
7<6
319
Z9
- B'
41 B8 - Z " 7< - 20·453 zq
m J /
om 49+ 4ZZo 41 + 4219 63
(,732 - 65 957 03
b7 -r 154 983 74 ~ + IS 7
o 00 - /001 7 +
475 3+
m 3/
540
+
4255
+
87
+
'139
B +651 Btl 5' - 4'2 3004
CONSTANTS
"
- .,
-
151
-+ 19¥.l
(, 733 ~
-
- /0/
+
".,"
'" "
M (,[5
730
77 {,3
- 77 70l 10
0
0
0
-
0
0
0
,
-+ ,
714
- <5 ~
-+00 l30
93/ 7i3
- 108(" 755
20/03
DESIGN OF PIPING SYSTEMS
152
SI!.~.S'1!
-1'1.110
Y
.00
r ,.5.,s'C
F,
F
F
+ 2.35 .8Z
8
+
,. 35.0
.2
-
0
+F
WHICHEVl:R IS GR£AlER
*"0
+Fz.-x
-F,1l"z
~ .. (1.§ C)
M,
~d
- ·x
M
0,,"
-- 0/<1'0
/2
+Fx'y
M,
5110J '"
HoT MODULUS. E\,
coso/;
y
z
PLANE
M'
+M'"
-M'"
M'
.M·
-M'"
.. c.o~o<;
0
XSIN ot
207
M'.
,
-M
..M'z
M'
"
.. M'I(
+M''Z
IPE
Z
Z
22ZIQ
l(~INot
9
2"106
3/
/
I
I
Z
",casto;
" M,
f-12 l
~
~
0
0
·00
M
0
2
5' -f,8M b
2St."'fMt,.
.30.3/
dO
5 .5 •• 5
••5
-
Z 0
Z
IJ
5 '+5'
92t
9/
2d35/
8,,0
I
M'b
5b.. f
:flO
f 21993
0
M.
.M'
.M·
./.5
~
2o"
/
~
+2
-,. "
-
0
+
2
222/
+
+/
23
-22541
3
---2/ 0 <
,. <t
-1-/23
,.
0
R' .. CALCULATED
RE"ACTIONS eA~eD OH
x
-
00
- /
DO
-Z .200
•
,.
4/75 PS,
M.
&-if·C
DR
Rl - h
·z
& Ec ~ -F • .R' "Eh
S',
M
S, •
- / .00
2
~~ /. 277
,. 0'
,.+ O..25
A
,
POINT
CONVERSIOt-l TO
COD!! RuLES
.s
/0.
<, 0
Uo
327.5
.30,"'CO.l PIPING FLEXI.".ILlTY AND STRESS ANALYSIS
.
MOMENTS AND STRESSES
=5,
THE MWKELLOGG
CA_
"'
o.
FORM
~. C
-I
o .:5".11
Table 5.13
Equation
J.-frJ>
I
+Al'
+Bl'
+D p
2
3
4
5
6
7
F,p
+Bp
+C p
+G p
+Bn
+CR
+GR
+AR
+B R
+DR
Af'zR =
}'f rR
F,p
+D p
+Gp
+H p
+Dn
+Gn
+HR
1I-f r R
+An
+BR
+D1/
+An
+Bn
+DR
+ YnFzR - XnFyR = 0
F'R
+BlI
+ClI
+Gn
+Bn
+Cn
+Gn
F,R
+D 1I
+GlI
+H R
+Dn
+Gn
+HR
Constant
0
+EI*il,p
+EI*il,l'
-EI*O:R
+EI*(+il,R - YRe'lI)
+EI*(+il'R + XRe,R)
Table 5.14
Af zit
EquatiOiI
M: p
F,p
F,p
= (-ynF zn + XllFI/R)
5
-YRxEq.4
+B"
+C"
+G"
-YnBll
-YllD ll
+B"
I> 5'
+BR
+CR
-YnAu
-ynEn
+G"
-ynA n
-YnDn
Ij
+D R
+GR
+H R
+x" X Eq. 4
+xnA R
+xnB n
+xnD R
L ~ 6'
+D"
+GR
+xRAn
+xnB ll
+IlI,
+xRD R
Fzll
+C"
Constant
F'R
+GR
EI*(+d.:rn - Y1l8: R )
-YnAn
-yRB n
-YnDn
EI' (+Yne,R)
+B"
+Cu
-ynBn
+GR
-YnAn
EI*(+il,,,)
+D"
-YnDu
+GR
+IlR
+xRAll
+xnBn
+xnD n
+DR
+G"
+xnA n
+xRB R
+Il R
+xRD R
EI*( +il,R + Xne,R)
EI*(-xne.R)
EI* (+il,R)
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
elimination of the rotation equation at C is shown
on Form J. The equation solution is omitted.
at a point R with coordinates XR, Yn, zn in reference
to the origin.
If the stops are located at the' coordinate origin,
the eliminations are obviously unnecessary and the
equations may be written directly.
Sample Calculation 5.9 uses the piping of Sample
Calculation 5.8 'vith a z-stop at the origin and v-stop
at point C. The summation of the shape coefficients
for O'-C is shown on Form D-3; the other summations are shown in previous examples. The
5.17
Calculation of Dcformations at any Point
After the simultaneous equations have been solved.
and consequently the moments and forces are known)
it is a simple matter to calculate the deflections and
rotations at any given point in the piping. By
summing the shape coefficients from the terminal
to the point in question and arranging them in
Table 5.15
Equation
M,p
F,p
F,p
1
+Ap
+B p
+Dp
2
+B p
+C p
+G p
3
+D p
+Gp
+Hp
5'
[+BR
[+CR ]
-YRB.
[+GR ]
-yRD R
[+GR ]
+xRBR
HR
i+
_+xRDRJ
]
-YRAn
6'
[rD. ]
+x.AR
F,R
[+D R
F'R
[ +BR
-YRAR
[ +C R
Constant
] +xnA n ]
[+GR
l
-YRBR ]
+xnB n J
[+H.
[+G R
-YnDn ]
+xRD. ]
[+CR
-2y nB R
+y.'A R
l
]
+xRB R ]
[+GR
-ynD n
-ZnYnAR
0
+EZ*t;xp
+ EZ*t;yp
-ynD R ]
[+GR
+xRB.
-znynA R
+ EZ*t;x.
+2x RDR
[+HR
+ EZ*t;YR
1
+Xn 2A R _
J. .00'
.co'
4
~
'9,00'
/.$0'
7
os
.00'
z/.oo·
~
A
b
d
"
~
153
~$Q'
"
:£.so'
" ~@.'ol,
••
MW KELL083 CO PIING
LE I I'§KEf<!~ S RESS ANALYSIS
~
...TII<: • • .,; _
fAt<C LC NO 5.9
DESIGN OF PIPING SYSTEMS
154
lJ·a
R-O
O. 0
.43z.
MEMBERS
o
t
/o.so
/'.. ()
I.
/.432
t.
r
O'-t
.
/
/.
Z .
B'l
/149.9
Z
/6
.25
R
h
;. ()4
k
/.(,5
xl
a7
I. "'.,.
x/
o
2
2
1.$0
""
I.
.0
.e;.,.
o
29.00
.00
. oa
.00
'812
~ MATERIAL
~
w
TEMP.
e
.d
o oCR.
~ "" cR.
coo r
f.
/COOr
.0072G
.00 lG
UJ
- 72. 7S (.M 72G
n.
n.
6,
-17.25 .1)072
E"IbX/l44
~ /3 1{;84J
-99.:1'IG"'~
- 22 702/2
l:h 10 Y!lH
E,.IllI/IH
(OLD 5PRIl-iG
F,ACTORC,O
cR.
10(10':
Z~
. ()()72~
d
a BRANCH d_1?
7.$ ·~nZG -,il?167
~ Ax - 9Z. .04/"7.
-.7,5671
~.
ct'.
"
/aoof
0'- lJ
-17.25(.IJ072f-
..
_ '/Z52?
- /de; 2(;.8 '130
1
MOMENTS (FT-LB)
COLP CONDITION
AND FORCES (L6) ACTING ON RESTRAINTS
HOT CONDITION
(J'
II
0
+.52..5
+600
V)
POINT
l-
M"
U)
M'l
_
.,...3
W
OC
Fx
,J.
00
1=
of
0
-I-- 3/ 0
-
0
F:r.
+$(;6
of--ZO
S M
-
~
00
i
x
M,
M,
I
z~.:
., I~
+ 98 ?/~
, .u +7Z 98 .q
y
.v
E .. Au + 18G /. .. A"v
..,
.
,
M,
+
0
0
0
~ ,~
.q
... A IiI .,.
- I
,
0
1-/22
y
+ ~3 46
0
.
-'0
'E
-'
-c~
'5,
+u o
7.5 92 ... 8 "V
, 8 83
I7EJ 61
4~
2
30
.
r /03 4 • Uo
~
, +5
L +-A H
• 70 i87 H'
'f'
• 4 7/~ '11
+
8A< 17
-Yo
",
/88 78 +Btl< + 7 772 78 +B
,
,
-5.
+ ?
+UQO+Voo
(.
F
M A
CALC NO.5.
f
t
.,.389 55'1.08 -cu."
'S,,-b-c'q
+Sbt>+C'u
-I~49 19Z
R,
- 4 /759
IJ5 '9
+
Z
/ a
7/2 6'
-5.
494 ~
+
~
- Z tb2 3/
-c~
-'0
54
- 7 690
95/ 0/
• 3 1>42 8 +8"",
+
0
0
0
.CY
~,~
!...J
+ ~-
,SD
+~o
- 068'67 - - " •- - -.12. ~
0
- 5 ~i1~~ --!:.~
_cg
0
1/ 04G3Q
+6"1
~ f}jJ. {~ ~ - - ...!{1-
- /0
G70:(;8 ~ -
Iy f;;-':.. ,zv ~/60 96,134 .eVa
I +C""
-,
>(ED
OATE
.cq
.6 •
0
- 4 02 :~q .B,
:!:..._.I.,iL tl. ---- It-L i l l ~1
.u
-cu
H
-4<;90
c... c.
COl
psi
S
SA' ~i~';;"tBRL...e"'G;
• 247tJO psi
AT POI"'T
-cq
+/90 20 f'A Yl
-
:'>TR~:;S
• 63.r0
o
.c,
+
psi
Sf ·~i..~~6~T[D
Fv
-cq
-
p~i
5c· /5; 000
0
+
92
0
0
.oq
9J ... 8"x
+ /2 I~L
.u
E .A
F,
+
0
~
0
I
9JI
5hS 73"0
+ 2G25
THE MW KELLOGG COl PIPINGO~1~~~lt'T6~~ A~mut~~LYSIS
0:
.IJtJ726·~
- %0.7
.oZo83'
-:5' 00
"'-/Z5
-
(,0 Ie, +B1:'I;
75 92
- /? ONI'fl2
I2L ~92 /~
-258 25 IZ..~~ =-L<li
35 l-?. .Solb-c:q =-g~
-II~
.cu o
~
-462 3all40 i'C Ill;
-Z31 9945
-/I
• 83 14 5
093
x +!>t>b+C' +J.R5 99- , "'"ab- c1 q I~ 03.'
y u.,.,.. ...oo HIG 55 40 -cu." - /9 2JL ~
-evo
z. fK..u+ClV
1:. .C YV
+C VI:. -I4R 33 2
""85 69'
1-/{,8 082 92
/0 +c.)lV
~~
-STOP" SUMMAr/ON - H~HLJE~S - O'-C
••
x +s"..I :+c 1 ... +~3G
/9/ 25
,2:. 1" S bb+d- +360 7 73
:z:: +U..,+Voo
THE MW KELLOGG
COI~JP'NG,.rLEXIB'llTY AI~':J. ~,..!RESS ANALYS.b
PLANE SUMMATION OF SHAPE COEFFICIENTS
r
... C J:l.
t2~~';K'O "'...... C'.
O~
It.
II_ 2.
_
1104 8¥> 90
-
CA C.NO.
I
I
1..
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
equation form, the multiplication with the respective
moments and forces and the proper summation of
these products will yield EI*It;, EI*8 u, EI*8"
EI*(/j*% - ll.), EI*(/j*u - llu), EI*(/j*, - ll,) for
the point in question where O*XI 0*111 0*:: are the
deflections with reference to the origin. Any
extraneous movement at the terminal point must,
of course, be included in the summation. In other
words, the pipe line from the terminal to the point
in question is treated as an independent line whose
free-end deformations can be determined since the
moments and forces are known.
For the next point for which the deflections are
required the same procedure is repeated with the
shape coefficients summed from the first to the
155
second point. Tins part of the pipe line is now considered as an independent line for which the terminal deformations are those calculated for the
previous point.
+ yO:
0* = 011 - xO:: + zOx
Since
0* x = Ox - 20"
tI
+ xOJl
0* z = 0:: - yO,;
the actual deflections, Ox, all' 0:, can be computed.
A deflection calculation is conveniently made on
Form G. The summations of shape coefficients
from the terminal to the point in question are entered
under the columns for the respective moments and
forces, skipping a line between each row. The equa• .,'.1S Fw:.
.93
0
Tot 8.78
.,.7772.18
...3342.58
.. 3/.39
0
0
-10560./6
2300.40 .,'85698..66 I H .2 -/oSU) If,
~2331.33 -35" 05:40 • t542.30 .. "7/.33
--.3373.9
0
~
'37/.33
-
.8
I
·lt9
9.
-/0560./6
.. -
32
.F.
~7S.92
1.57
.,,18.7$
-4188.
,
COEFr/CJENT
POINT
t;-= 1.27753
ps,
$<,
&=it·
R·
~
R'
C
h
OR
Ec:
~
:rlh
f":"qG2.
.3 0,
M.
-2
(;.3
-..3502,,"
REACTIONS eA~eD oN
SI"" 0<
HoT MODULUS. Ek
z
M'
M,
.M·
-M'"
.CO~o<
.M'
M·
"
IPE
0
:L
- <;../.
0
~,
-
.5
-
-- o
+ I
-
8
0
.
I
024
, .3
- -'02
-
-
55
,,'~Z93.Z4
.-181 S'2O.25
- -"0
-
0
66/
- 0
-
~,
L •
>It Ino
AT" ""-~
-
0
0
- .5 7Z
080 +
- 239
- 0
- ZG
• 285.,
:55 of 1.34 7.5
of 20~4-
-0 2
- 0
r
Sl
-"2~ 103
I
+U
0
Z
+
-- ---
.00
. .50
.. --
•
0
-6.co
0
--
4512
CALC NO 5.9
"
0
+ 6,
--
J
-
5
-
I
OGZ4
.5/
0 0
02/
0
7
90l
~ 14/373.01
.
-/6544 -20752 -.40347
- .33Gt
ZQ5.Z6 ·1
3
OJ
/
'02
-.33
7
.553
-
~3
-"
234'12
0
of /90:Z
/
, .-
~
+r
+-z.z
Z51(;
""
0
-.5
Z~
-
-
-.5
0
/
'3~
PLAt-l£
.!>INOl
M',
.~INO'o
'0::05-«
M,
( .. 12 l
Z
1A&2 "88
/3
5b.. f M
S'b"f,8M b
2St,-fMl
..,
5 .5 .4S
""5'
.,;:
FLEXIBILITY AND STRESS ANALYSIS '"
THE MW.KELLOGG COr'PIPING
,
MOMENTS AND STRESSES
•
L
-~
413
oU
-' Be;
CO!>co;
y
M'
.M'"
.. M'l"
6
413
- 'x, -2
.0.
- --"
-F" 'z -/6 C>/o
M
+16 9
, +1.3 /2
~F.·y ,..
77So
R' .. CALCULATf.D
.M'"
-- /I
-I
- ,oG -
'.5'
5.57
.F", .;1(
~= /
-M':" -M'l:
:>:
'5
1'.
I'
I'
M
~: (l-:~C)
R'
3
.M·
M'
+ Sf).
-F •
S"
i¥- 0
M',
Y
+~·Z
WHICHEveR IS GRfA1ER
,
+
z
Sf: ~ .5'£
0
,
·E B
33'.27 U
-,S<S69 5.. "0 •-/Z5'2.30
"''''-1'5 (-4/88.83 ..
rn::£ MW. KELL()G:; col PIPING FLEXIBILITY AND STRESS ANALYSIS
CONVERSIO/'i TO
COD!!! RULES
-75.9Z
- 10560./{,
aRM • I
I
0
7
DESIGN OF PIPING SYSTEMS
156
CONVERSION TO
10
8
POINT
y
- o. z
,
0
Fit
F
+'i'C,·8
i;-:= /'277S3
Sf:~' S'E
E,
51!:
'347p.51
k.c
Re
R' :1<£.,
R~.,EC
R'
Eh
-
f-3
f-
F
,F... ·,
-F·
~
S't"
-
6<!- "
2
~F~.z
~=(1-jC)
M
/
~F.:~
z
~. - CALCULAT~D
M z
RE"....CTIONS eA~EO ON 5' ..... 0<
HoT MOOULUS,Eh
~OS~
)(
M'.
6
62/0
30
-M'
-M'",
-M'.
XS'N DC
.M'
.. M'"
'OM'
l(:!>I"'C>-
M'
... M':z:
.. M'
",coso-;
5
-
,;,
-.;L.5 '" 6G
.3
f-
-,u
,. '"
3
-3C;68/
...,
0
08
"'57
... 5
B
-
"
,.;
.560
f-
03
+45
+
- /f63
+Z9'f03
-.10
- 0'
/
"so
R
/5
.00
+F -z
M.
-F •
M
WHICHEVEIt 15 GREATeR
&R' -
+Fzox
-FJ('
M
~dl-!<>
,
0
R
R'· CA1.C.tJLAT~O
REACTIONS BA~I!:D otl
Slt.l '"
1
7
/
0
-+-I""
B54-
-
-7150
-340
-+ /323
0
- 3(
- 2. 8.3
- 3(
0
0
0
0
-{DOl
-.,3""0.;1
S4
,5
0
- 3(
- 10 Ie.
0
.3
0
C<'
0
,. / 0
-
+ 2
+1 0
+ {ool
-
/(J/6
+/
,.
-
-
- 0 92
+540
-
~
L/.
0
4QZ.
'4S-
0
-
"0
0
0
0
- /016
.5
-
- / 0
0
-/0(;0
:>:
·CO~I'(
(1
lI!>lNoo:
+38
M'b
.M'
M'
- 0 0
- 00
1(~1""D<.
I(O:;:O:&CII.
M,
IPE
Z
8.
f-I
5b"
l
13
0
M
IS.sG
S' -f.8M b
25 =fMt
+5 +45
5
S '.5"
·.s • s',
THE MWKELLOGG col.
FORM-I
C I
0
- --
x
10..$0
o
,
M.
-tM',,-
-2
- .2
coso<;
'tM''t
5
0
PLAIJE
.M'
M'
-3
.~
z
MM'"
04-
+11.38
K
HOT MOOULUS, E...
-M'z
o
o
0
0
/2
(;;1.34-
0
"'
OU
+
-
-
0
,~
-
-
--
~
-
/3
1 0
I-'
+ .25
- 5937 -(0103
0
- 34-
- "', -
.F.-y
~.
+
-
CoO,LC.
i'IPING J55~',\l1-LnNtN~T~l~Ei€sANALYSIS
-z .s.
F
s.
.M.
30
~9'8
Y
-6. '5
,
F
,.- 2. ,Z F
&,E'£,.G OR
R' f"h
Rc Ec: ~
~'Eh
-M'
f-
z. ,ado
,
M'
.M
0
- 1,32
2
To POINT
SE- ~.St
y
0
0
8.00
M'
.
2
5.,50
,
--
3
Z
6'
-/
S
·00
.. 5l;
£1,--
M'
3
0
0
0
(1
-.3GG
{
0
-2
+
~~
4
+ /3 lIZ +.5
'6
()
0
M
E, _
5<.
..
-
--
-
0
00
B l
THE MWKELLOGG col
CONVERSION
CODe RULES
-
-
+
0
o
25t-fMt
+45
2
--
Z5GGG
0
.SO
E
-/4.
o
o 0
o
M'b
S'b"f.8M b
5 " .. 5'
81
6
/2
~
+
0
o
XGO:''''
5b.. r
-1'/
/I
·0
- .00
.00
PLANE
Mb
M,
7
o
-
o
- 2
-.5
tJ~
-..30
GG4Z
02
-
0
o
-1"..5
--
-~. ao
:f'
- ZG4G
-8£3-2
C
-f.3C> '..S
- .30 '04
Z
M'
tM'
IPE
f .. 12 l
/3·'25 /68.8
Ii
o
0
-33. .5
0
+4(;4(;6
0
o
-,.O()
--
-
-
Y
M'
.M'1t
L
-/?t?Z
.... .3 8
G
o
tFz';rc
R' ;;: 0
-
6
-;'''... .3 "9-~
M
Rc
Z
.00
~~Z~+GZI04
OR
WHICHEVER IS GREAlfR
~;::
-~.oo
c
- i .00
//
,
o
Coce: RuLES
'" KI,
"'"'"'
AND'i.TRE.S.?ANALYSIS
PIPING F~!;~I.~.ILlTY
MOMENTS AND STRESSES
CAe
"
-
ORM
C l
•I
m
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
157
y
tions are completely written out. The moments and
forees are entered at the heads of the columns, and
the products of these and the shape coefficients are
placed in the blank lines. Extraneous movements
or earry-overs from previous points are listed at the
extreme left and the produets at the right of the
sheet performed. The thermal expansion constants
are computed in accordance with formulas given on
the form. A summation horizontally produces the
indicated sum (EI*Ox, EI*o% for example) from
whieh rotations in radians and defleetions in feet are
found.
t
AI-----i
r.o:;----l--·
o'
.I------J
FIG. 5.12
y
Single-plane symmetrical system:
three points of fixation.
y
o'
0'
Consideration must be given to intervening actions
(branches or stops) between the end point and the
point in question. In Sample Calculation 5.10 the
deflections have been calculated for the piping shown
in Sample Calculation 5.9. The rotations and deflections at point 0 are calculated twice; by summation
from end A, and again from point 0'. This provides
a useful check when points from different branches
are computed. From point A to 0 there are no
intervening stops so that the calculation proceeds
as deseribed in the preceding paragraph. Between
0' and 0 there is a stop at point C; the moment
and forces due to this must be dropped out at C.
It is a useful check to note that the 0v at point C and
the 0, at point 0 are zero because of the stops.
5.18
A
FIG. 5.13
A
'b
•
A
Multiplane symmetrical system:
three points of fixntion.
0'
~ymmetrieal Pipe Lines
One way of reducing the labor involved in flexibility calculations is to take advantage of symmetry.
The proeedure varies somewhat with the system;
nevertheless, the following examples will illustrate
the general approach. It should be noted that symmetry must be complete, that is, the pipe size, temperature, and material must be the same for corresponding branches. Also, the coordinate origin must
be located on the center line of symmetry.
Consider first a single plane system such as that
shown in Fig. 5.12. Because of symmetry, no bending or rotation occurs in the common branch 0'-0.
Thus thc momcnts and forces for the A branch can
be obtaincd by considering line OA only, assuming
member 0'-0 infinitely stiff.
Next consider a multiplane system as shown in
Fig. 5.13. Due to symmetry, the only deformations
of the common branch 0'-0 occur in the x-plane.
Thus, My, M" and F% are zero for line 0'-0. If
the shape coefficients for this branch are computed
in the x-plane, only s, Sa, Sb, 8 a bt SaaJ and 8bb apply
and their values are doubled in order to account for
the loading from the two branches. In other words,
the common branch may be considcred split in two,
•
•
FIG. 5.14
A
Multiplnne symmetrical system:
four points of fixation.
each having half the moment of inertia of the pipe,
and consequently a Q-value of 2. Aeeordingly, the
problem can be solved with six equations for line
0'-o-A, using the aforementioned six coefficicnts
for branch 0'-0 and all the coefficients for branch
O-A.
In Fig. 5.14, the rotations 0v and 0, and the translation 0% are zero at point O. The system ean be
solved by setting up the following nine equations:
5.19
Inversion Procedures
As previously stated, the General Analytical
Method can be applied to the flexibility analysis
of any type of piping configuration. However, the
number of simultaneous equations necessary to
solve the problem increases as the degree of complexity of the system increases, i.e. as thc number
of points of fixation of the system increases. Furthermore, the time required to solve a set of simul-
...
~
flCTRAN£O"S
I
O~OCV:-R~E:6sv~R _
EI" B"o'
+
o
-
Mx
I
I
My
25 !dc,{" 18[,J("
1
13~ 9/Z 1.93 +
0
+
198.Z0
IZ73Z8 +
0
-
I
Fx
962.89 +
75.92 +
Fv
I
F z
3170.4,5 +
334Z.58 -
rE1Z.."," -":E:i;;,:r-7eTz68 9601
5("4.3(,, '/"///////////71090.54 - EI- 6 1l
J<
Xt;
33.7.5
+ 78.75
-I
/ ,31(,013
0
z
0
80.75 ZE 1- /7.Z-.5
80.75 z' - I7.Z5
EI-c
Y
1-
y~ +
PO IN-f C
ex' O"-3Z'
73103 + 10598151 - 4340 Z33 + 1.803.1, x' - 45.00 y' +.OMZ7
40Z9.69
"" /'/.
/// /'/ - .......
c . . tt
" v
•
.0"
v"
IT". "/I"
U,
.... ,
Ii:I • .:I7V////////////J
V//////////.i/ 'l/.
r\
1'1
1'1
:2QQn,,,Q
v
U
l.. .:.JUfO U.:.JU
U
VUV 'TV
0
+ f4 Z34 /4/ ~ +
47143V//////////// 2:'.::::::
:/'///+.000£6
EI'ew'
+
.93
0
+
188.78 + 777Z.78 - 10560.16 75.92 %'i"//////"/V////////,,@ ·El" e.
:%%~ e. 0°-4'
Z3869
0
+ Z584. Z47 + 7484 33Z - 33 48Z 571 4Z 84. "/////
// ZI8 707 '0 0'~ -.OOIZI
EI"oxo'
+
75.9Z,- 4029.b9r
777Z.78+G49/9Z.lO-46Z300.40-Z31994.55+ E I cx'
-t-EI-Svz
-EI-ery
=E "51<
0)(1+.,((,,8"
- 75534781- I 1485631+ 46901 5.ZI+I 064 186 8551+ 6Z5100 58/1-14.57'1[7631-130 9Z8 4441 C 59 ZZO 5851
0
0
1+ 70 7451651 +.390'
EI',yo' 1+ 334Z.58T-' 0
T'105.0.161-4.Z300.401+685698.661-148339.Z71-El" e., I+El"e,'
I+EI"e.x j-EI"'5;
I'vl 0
1- 857906581
0
1-14¢58IZGz.I-44514443ZI+ZI741104561-837/~7501
0
1-IOGZ.80501- 73813M34z11
0
EI',zo' 1-7690.541+ II04G.39F
75.9ZI-Z31994.551-148339.Z71+11t:>484G.901-EI" e., r~EI·e;x'-I·El"ez' I-EI".,
1.,h.l5"
o
1+ 197 385 4001-12851,89331- 103943591- 223 385 2321-470331 90"1+1,573'1299"1
o T ' 1591 07" 1- ZZ 701 Z24I- Z/943341 -.012'
/ ' J25"-;:"'\'11!G39r=
10121+
9"2.891915.981+
564.3.V/.//.///?'/.!%1 x I 0
IY I 0
I, I 0
I POINT 0
EI'e«
1+
43.881
0
)
0
I
0
I
0
I
0
l=a"e.lx,I+78.75IY,I+80.75I'.1+17.25Io.10'-II'
+ 11,80366EIi' eye
+
471431
0
39Z 8/~' 0 .
0
0
3Z1 51{,~k Z4 15tl(2///,wr~~-=-.OOOI3
EI' eze
0
0
+
33.t5
0
569.70
0
;;;;
:0"$/;:00 - E1' e,
;; /'£ e, 10'-5'
- '1.187071
0
I
0
1341551
0
1+ 5Z18341
0
V#/V/ff#/J////V...0?/A1+ 26897Z1W/#.//////I +.00148
E'I'M:c
I
0
1
0
0
I
0
1
0
L._.o
I·El·ex'
I·E1'O,< I-EI'O,Y I=EI"&.'-[1;;T+I.74"
+ /ZQ--9(,5 750\
0
I
0
I
0
~
0
I
0
EI'O~e
0
0
569.70
0 + IZ 820.09
+1136459901
0
r--'O
~57"-53Gr'- 0
1-1174Z94.1
'0'1
0
1-IOHG80501
0
)- 37884701 -.OZ!'
EI'Ue
I
0
1+
51,9.701
0
r---O--r'O
1+ IZ820.091+El"O., I-El"Oy. I-EI'n'
I-E!"',
1&,1 0
+- 220 796&
EI' e,o'
4 777983
M2
1/ 639 +
0
+
0
+
~?"IQr"
o
-,J
-
r'
o
1-
I
I
;01+
'r
. //,:/."/
EI
ElxA
90Z 1"7
0
+
f1
11
0
0
82.18
0
0
409.65
'J,OO?'1
n
:2.'7'7(J:2:2
-
Z 11,3 +
1,53.31,
55430"
0
+.00306
8, 0°-.5
o
~
o"'l
...
...Z
"'0
"'0
"
(fJ
..,~
;::
(fJ
+.00148
'll+.145
+/.74"
EI ¥ OXA
+ 7 553478
1+
425.Z61
3 77583ZT+450988ZI
EI OZA
EI'oYA
...
t"j
+
o
EI'S A
't
ot"j
(fJ
.1+
590.961+
--1+ 5001,6131+
0
0
/ZZ7.33!" .9Z49.7tr=:::=484.0.I-El·ex<
98z50ZI- 751923/171'8981
0
I+El~ey'
I+El"8,x
1- 105Z87041 -0-
o
X.y.Z. COOR'OIN .....TE& OF THE POINT
x•• V •• 2. ... COORCI"'AT~~ O~ T>1e! END POINT FftO"",, WI-HCH
THE COI>J~T""'NTS ARe SU ..... lo.1CO UP.
THE MW KELLOGG CO
I
.Il·.aX~)(t.
)'."Y-Y.
Z -Z-l.e
rrr-e.z'
l'El'"
16 v l-.25
1- 37844041 -.OZI'
I=EI-o%
~2.1 0
2504
0
I
NOTE: ~~~,~t;'~iT~~i"e.~~~~~Tj5~~H~O::f~~R.~~J~a.o~g~I!.~T~H~~OoN~6AR~~6A~eT~~~~~D~~:NT.
ARE 5UMME.O UP F"J:tOM THE FREE £"lO.
PIPING FLEXIBILITY AND STRESS ANALYSIS
DEFLECTIONS
C.... I-c.
A.M.e.
CHEC:Io(I!:O
O ....T.
12·23 53
p
FORM G
CALC. NO.
5.10
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
159
Table 5.16
M.A
Au
MUA
A.u
A uu
M. A
An
Au.
An
F. A
Bu
Bu.
Bn
Cu
.~
FuA
B. u
B uu
B.u
C. u
Cuu
F. A
Bn
Bu.
EI*8;r;
-I
Bn
Cn
0
0
0
0
-I
0
0
0
-I
0
0
-1
0
.-I
0
0
0
0
0
Cn
Cu.
EI*ou
0
0
0
0
EI*Ou
0
EI*O.
0
0
EI*o.
0
0
0
EI*o"
0
0
0
0
0
-1
COllstant
0
0
0
EI*~x
EI*6 u
EI*tl,
Table 5.17
].f~·A
MUA
M.A
F. A
FUA
F. A
EI*O:t
EI*Ou
EI*O.
EI*oz
EI*ou
EI*o"
Constant
-I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
an
a.u
a..
a. u
an
b.u
b..
a..
au.
bu.
buu
au.
a..
bn
bu
bu.
b..
bu.
b..
c. u
b..
b. u
C. u
Cuu
K,
K,
K,
c..
Cu.
c..
-I
-1
-1
-1
-1
taneous equations increases greatly as the number of
equations increases. Roughly, if the solution of six
equations takes a given amount of time, it would
take four times longer to solve twelve equations,
nine times longer to solve eighteen equations, and so
on. Therefore, eighteen equations representing
four points of fixation for a multiplane system are
considered as an economic limit in manual calculations.
If equations numbering more than eighteen are encountered in setting up the calculations of a complex
piping system, it is generally advantageous to
use the inversion and re-inversion procedures discussed below. In these solutions a complex piping
system is divided into several simpler parts and
several sets of six or twelve equations are solved
instead of one set of a great many equations. The
time saved by this method increases as the number
of equations increases. In addition, the ealculations
can be performed by several people at the same time
and checking is also simplified.
The inversion methods are very well suited for
automatic electronic computing machines which
have a limited number of storage units. The time
element, which makes manual computations of exceedingly complicated systems prohibitive, is no
longer an important factor, because of the speed of
the machines. By programming the calculations in
accordance with the inversion procedures, there is
no limit to the complexity of the piping systems
which may be handled.
To illustrate the inversion procedure, the branched
system of Sample Calculation 5.8 is used. The equation for branch OA can be written as in Table 5.16,
b"
b.u
b..
bu.
b. u
b..
Cu
c..
CU.
K,
K,
K,
where Ox, 011' O~J OZ/ all, 0% are the unknown rotations
and deflections at the origin. Similar sets of equations can be written for branches OB and 00'.
Inverting the above equations the set given in
Table 5.17 is obtained.
The equations for the branches OB and 00' are
similarly inverted. For equilibrium, the sum of the
individual moments and forces at the origin must
be zero, and the equation of Table 5.18 is obtained
accordingly.
Table 5.18
EI*Ou EI*O. EI*o.
La.u La" Lbn
La uu La u• Lb"
La.. Lb..
Len
E*Io u EI*o. Constant
Lb. u Lb.. LK, = 0
Lb uu Lb u• LK, = 0
Lb. u Lb.. LK 3 = 0
LCzlI Leu 2:K.. = 0
LC uu LC u• LK, ~ 0
LC u
LK, = 0
Mter the above equations are solved, the rotationR
and deflections obtained are substituted in the inverted equations for each of the three branches, and
the moments and forces are obtained.
Sample Calculation 5.11 gives a detailed calculation of the system in Sample Calculation 5.8, in
accordance with the above procedure. Junction
point 0, the rotations and deflections of which are
to be determined, is selected as the free end of the
three branches OA, OB, 00'. (This selection is
important when intermediate restraints in any of
the branches are involved; the equations for the
restraints are placed before the rotation and deflection equations and eliminated first in order to simplify the inversion.) Each branch is solved first
....
0
'"
I II •
I'
-
Myo
Hzo
Fxo
0
0
0
1.000 00
0
0
0
'0172"
0
0
0
H'c
85
2
t
•
•
82
0
0
0
82 /8
0
- 1. 00 00
-33
3
-
425 2. 94 •
4
/;
- 99 '1
.
2
Q_-
-4/31109 2
31.
18017
c;- iR7~71 -f
_._--
r: +
3'
80 /
r
I. 00 00 -
5<, 642 01 -f
+
r
0
f'
[5q;1%
r
15.'
7/134
"
lioN 05 1/5 -/4
2 liOZQ"<;
•
0
- 2 04 02
.
95 % -
rr:4"-~ J. 49/
I
,
1&
6'1< 4
0
0
-/8/ Z(,89G>
~
I
'0 I
0
0
0
0
10
o
Q.
0
o
£
0
0
0
0
a
0
0
0
Q
~/8/1z,
0
0
0
a
0
0
0
1-22(,11057 t5
0
0
0
0
0
.
0
- 03
0
"
-90,3
0
0
0
0
0
0
"8 70
0
0
+1'18
"
'0.,.1'1'8
I
-/BI ;;:;
0
/7.3f
I 31 -IBI Zbl196
.../838,,39128 -9(),3 ~lsl +3'-B~!5
0
~
,
'0
, I
10
Q.
W#~
I:i
Q.
+ .2519
0
o
0
:z
0
0
0
10
Q.
0
0
o
0
0
0
Q
Q
Q.
10
10
-18/Tu8 q(,
Q
£.
a
I
0
43~q
~ 'lot. 152 65
01
0
0
Qt:02
0
0
0
0
0
1/067 1~3
4411/2 J 906. 132 .5 "/0:' 7/~
211 93 -435 34 M -6/2 ., I
!ol
lo!
51/9~441
101
1-/44/1/5/1"'"
01
101
I
1
f/33
0
12~<.Bl
10 1
9314 +19
0
0
0
195 14
-Ini z(,lj,9(;,
0
r 25 977
0
.,
o.
."
10
I0
01
10
I
10
-
I. 000'00 -
...,
. -...
0
0
I
101
10l
I O!
o
- 30
!L
-3
o
t'l
'NIB
0
Q
Q
-/
-
0
7.58176~
~,
- 25 (,%5:
1 65&; 0 +?Iq
~'944,3
--PIPING FLEXIBILITY AND STRESS ANALYSI::
EQUATIONS
i"
CALC.
CHl!:CKEO
PATE
1/- 23-53
-<
(j)
( j)
.. 17
1-18/IZ(8,%
I
10
I
10
I
10
I
!0
..
:z
....
....
(j)
o
+ 4- 0 31 -597
2 7Z0'3
'5
Q
101
101
'"
0
0
0
01 I
1+"I7~zl-/3515<;717" +1?.31913Ib51
101
'"
Q
"".
oI I
aI I
I
I
0
"l
52175
0
-234 39&7 "Z14 194 M
rllo{ 85 7 +214 /94
-/8/ 2689(;,
-102
-52913'
7 -f87
....
C'"2
o
0
0
0
t'l
(j)
C'l
I
- 52
-t
12
ElJTER eElOW
0
01
101 I
I 9/1051-18715.331'21
-
---
0
0
0
iDlrB13z8991
1:
0
0
rio 9q31/3 r 2 00/17"
-
o
0
4B41o~
0814- • ZI. 000:00
g
+
101
0
I
0
-
o
0
0
0
- " 25 /;
r
DNSTANT5
0
0
+ q 24< 77-
I
0
1000 dZ
locoJ',
0
0
0
683
(,145 ,7(;,
5
/000 d",
0
{J:z.
()(J
"8 7 r
1/8/ 64q~. - a 2 ,{,(J -
:!
THE M.W KELLOGG CO
5
0
0
0
- 4- 3
7
GB< 28 -
BRANCH
0
r
22
- 2 /258
0
A- 0
0
0
7509
0
/000
0
/
000-00 _
I 1--82
OS33
50
e
0
1000
I
0
0
0
87 71
0
0
0
0
0
0
-
0
0
0
+
8>(
88 0/ -/81 Z q.
1034 ~ 1"2/3 82/ 20
9
0
. ,
0
0
/()OO
ZO
F"YC
40 .5
0
I
..
'11445
FORM E·Z
CALC. NO. $ / /
~
ii,..
1'1 -
Hxo
H.
123~O
1.!OOOlOD
0
H1CL- 171~81
0
" •- 1004 04
q3
- 1294 -
/oool),.
ho
Fzo
75 q'Z
34ZS8 - 7 If,qo'54 -181 \268 '16
.33do3 - 14 !53d43 1"33 )43/;32 +7B7'Cl8S 87
2!lo.2Q -29 135< 15 + /2 IzMj_'X
."
I
oz oq
0
0
0
-4 02 09
+11 37 /0
0
0
I. ood.ao
0
0
:3 O~II
0
+IS 28/
2 +
231 95
0
0
II •
2' -
Z31,
I ",-
F.,
Hzo
•
0
-4
I
3
+
22 53
0
+'
-
0
lEI +
53 +
""
772 8
0
10
-II
,..,., ,q
-
S •
20,155
13\83
-
,
o!
0
0
10
o
,
I
0
WH
0
0
0
0
0
+781 ,~O: '5
10
'0
10
17'Z 3J
0
0
3
.,
- "-12'10 q3
I
..L I
-r:~ -:;-'BilZ41%
10
0
0
0
0
10L
-
-10/0082./1
/0/
'+201/80888;
o I
'-3r4Q!Z03!771
1
.£
.£
-o 1
~
.£
o. .,
0
0
--,
0'-0 BRANCH
THE M.w. KELLOGG CO I
.
r .,. ,
f=
o
0
£
r
C
.£
o
~
o
o
o
o
o
£
.£
£.
Eo
Eo
Q.
o
o
2,
N
Q
Q
0
0
£.
£.
0
0
0
-IBI ~9'Z1
0
0
0
0
0
0
0
0
0
0
2"
.,
17.. EQUATIONS
fi;:f
4'CALC.
CHI:CKEO'-':;f
IDATE
c.
//- 23 ·.53
"l
t""
t'l
-~
1;::1
t""
...,
~
~
o
0
32
PIPING FLEXIBILITY AND STRESS ANALYSIS
eY///Y/.c;
o
,..2 903 o -2", 95- 00 -r5,3b b55fJ -IS < 17
0
210
+1C,;J2 3<
3 70 -!lO .'0 7'
9 -4'2 7' 14 -4r..tJ1 38 ZZ 0
-2' 0 I
'4 -IBI leo 9(;1
1. +309 103 <5 -44l4 441 22 t83 318 20 _lflOI .'5 40 -1Z 00 87 NO
G' I, 00 00 +14 3/3 '1 -t7 00 /7 +-/3 ~ 75 +135189 to
5~ +
~42l
r
o
£
0
0
0
-25 10 90 ~b06
0
+907~ 03 34
0
-58/ 41«
I . U4 +
-3G 53. 7("
14 '0
0
2-
.£
0
IZ.
0
10
o
1&51%
I
I
o
,,.
."
I
£.
51 ... 698!5"z 27 ~/48 .33~i7
0
0
0
-181 I£. 9,
o
o
-48 Sf:> /2. +/11 7' 85 +2c;33 '/S 93
0
0
Q
0
0
o
0
0
0
Q
o
-S58 oz 2' -2 245 q, +JG '72 24
0
-9077 24 Iq
o
-/7 1- 30 -G 213!' GG +8 257 29 7;9 33 01 flSZ' GO 70 - 3170~571
o
L
047 ("" -45 12. q9 .,.2(,78 84 4(, - 59 33 01 -7550 G3'I .q - 3 7D7 57 -18/ 2
59, 2(, +2 l44Biof
5'1- 1.0 00 + <IN '3 -.3'" 17734 flO 25 70 ~/o( ~,o~+
",,~+ 2 ~?O fr ,.. 223 25
G 1+1/77
o
0
-
I
101
E\.,lTER BELOW
0
IlIlJol dO 12351/'11-2015711331+
I &.,,..
I
ON5TANT5
o
o
%
-/8.
1:±J
0
0
/ 2
I 0
o
iO
--, 0
- lSI
10
£
o
I
t814-l59
4 +"G4""1~t':"J-=-''''-fcf':'p.~I'E''A
I
!
-181 Z<, q
-/8
I
175 2
3 0 +
+
0
0
/2 1<74 13
0
77114'
222
I 14~,
I. 00 00 -34 'J27i74 ,50 07~ 12 +
H1!lL+ 10J:?ZB! /q -30 72.2144- .,.,01 In~§£
3'1-
I
+ 1/ '<='IGI7
I
3/ 0
0
-18
loooo~
0
I
- 50iCI..9Q4
0
0
jooot7z
0
I
.:'"4f?~§Jrl
looacy I /oootz:;
looof}
o
>:z
>t""
~
(fJ
( fJ
-J"~
o
o
o
-II/~
.,.
~113B
-IOGI2~93
1;::1
~
>-3
~
<:'l
t'l
:z
t"l
E.
E.
:::::
-2G1810
~z,
>:z
>t""
- 2.2. 17Q.21/;
j
o
~54
o
o
o
-9 «25k0;
,8.Ll!Oz!4,';
~
~32
>t""
~
("')
>t""
=:::
t"l
:::j
o
I:'
FORM E-2
CALC, NO. 5,//
-0-
162
DESIGN OF PIPING SYSTEMS
M.,
M
Mz
Fx
Fy
Fr
/DOD
(JX
100 0 8
/~ 5 x
/000 8z..
l/oooS
loocoy
.1:1~10~'/I"'~:I'~i~~:;'iIV~~"':I:;"':19!'~ /~'!~ 'lSI'~"~"i'i~'f'i'~'/~ ~ ~ ~ o~:~i~' :~i l/ .~i:i ~ ~ ~
i'7 DO /I 23JI '91 IJ '2lA
7./,
- J I~ S _ ,. 7/4/1. 21 114
p.=="_~±c:l--t.:o++--:+..;O++--+..;o'++""f'0,-+-+~",,,'\,/.:./,,,,:t+."""'f4,,O::;:.''!m;;;''
-1.00000
0
.,.4987
.,.7%
0
5
0
o
*'
}.gaf3.r'i!~ 150 J~J 1. 10. $~'7"
-6'1. lU'..lIV 1-1'1; '"
"'68 '31
-37. 2.5
?16'~'II~/~~~,,~,,~,~~-~,;~.~~~'Jl:L~,!t
~N<.7r.V?:Z6~fj;
r%:0~
0 0 0 0 0 0
2fl. 98/ I ~/16"
-4 '18 7
2 ~"41 422.i4d'27U29
I~ 7/2
[31024
"!16JJ9~
-[,22380{
I
1. -1.00 00
r-I.OOOO
0
00 to
/~ V/J
-
rHooooo
. .
.
M,
Mv
• 1.00000
o
Mz
-
IaJ.:
211 'iJ -43 ,34216'; -SI
t"
f7' 2~'110 -J)2 9~
14; T87.C8j 9
21(93- .351/ "46. 244 ·7.'" 7 -13./90 ~t"2.W
""/412/[94
0
<13 6q 714t.i ~3 7/j:J,tW
-/l60f.
.,.fOOl
f/41219
I-rooooo -I 6%9021 Z ~ "J6!00I: 11-6.23 O/fI41Z1l94Y'f:.,-6J4.92
F
c ..... c.
SIX EQUATIONS
INVERSION Of
Fx
•
515% 8 lzM48 q4~5. '" l2i..54.! 2
0
l. 00
-
HECKt<C
QATE
I_;?
'"
Fz
13'301o.i -14 rn 37'.# 3/
t4n ~t2 87\3~-Z45(,;.JI- Mk-2·
f II8ZO:'l9-2IZ8i b -Z338Q -
1/9 13 7'- 9l '5972 -,sf'l'(,7
;:
• 3313/
I'Loao 00
-
o
0
0
o
1.00000
0
m :f73 0
o
v'"
0
%/5
478 £8
-SOl
o
(.-/
-n 730
- 'JoZ 918 71 41.
91
,;,'TZ (JJ i\:J -BIZ lOd/!3 -1I8z48(,·
0
'781 "(JOIS"/h
0
0
-9585, (,
tlZ/&Vl
~
1-
l2a /4 - ., 180 '4 -.18 71..
-
f 4
51
-
os.. &"
lH4
19-
.,z~ 1" II 948 Z 'f" 191605 '3
-s ~
-Zl:Sd~ ~:~f/':
~
~ 'UYN
0
0
tJ
1Z3tJ7673n3IzSolIg/fll'13
0
t-Zl3't
,/",
V'"
I::
1-1.00 00
;[-1.0000
- 1,00 00,..
V../, Yh
II-tOGDOD
-
WI 3-31,,177
'1'10 ~70 101970
t" ~rZ"4. I'/'h~
',J + 8 7Z3 5 ./~ 'S9
f 8 08' 7 r ~ r
2/1
f
.3S7
6lJ
0
-21 'S4 9- 4>204
1105"6
r ., 000 00 14 '/3 79 -Z'} ()06 17
0'- 0
'jt--
/3 '8 75",,"
l&.~lo.!rZ&-65 I
.IS?
V.::i'Sk5'9 To
~
.35'11391.,..
PR/lNCH
THE MW KELLOGG COl PIPING 1~~~~~b';lTcJf'},~~ €JBHM~ALYSIS
::~t;.·;;,:::·<
[f~~ ~oiJI
fOOO
e1lt
M xo'
..
1" /(,.7Z.04
l1'1.. 0
to tOl-Sf,. 24
/of . .
"'
/of
/of
M
~
4307.11
4307.11
t
.,. lSSa.
•
llot "
'0
IotZA
M' •
o'
Eo""z '" 0
,,,
F ••
'0
Eo F)(. 0
F
•
Fyo'
"'1=0
FZA
z.
Fr-o'
r. F~ co
f-2SSB.81
- 12.10
+4304.'2+Z(.47.1!i'
-
"47./5
- 11 BZAB
-1/
4
+ SIS. 9
--
I
.~
- SS2
-t-435.<&.9
- .45
-
1000
+ ,?S5B.BI
I. 1,(,.
I. ' "
+ 0.9
1000 1i x
Oz
1000
812.70
+-4304.. 9.~ -=.!.!8Z.4~
+988. It;
1" U4'.I3
-
+2?39.41j
- 22.3 .49
--
"AS
-
- 536.99
+110.0(,.
.,. 3'.01
I- 13.27
-S(,."4Z'0-b_
+-70528--:19
+17 GoO.
fSl8Ao
-9"3. 2
r 13.2
-42755.
t 255.1'.'
rS48.53
t-S48.5.3
+
·11&..10
-11&..10
- "2.24
- 8Z 934.19
-B2~_~
.0
•
+1097. f.)~~ __ -=-1.3_~.S3
-53c:;.9~_
-1I~.lq
-S
-11"-./0
.<,
•-23/.53
.01
+412. OS
t-13.27
" .27
.,.17/7.
-7JS'.1f.
t- ~.'"
.14
.14
- 14.IZ
t-
• ."
..
.3.
.3.
tt-
J"
-17.
". S 0.52
• 'f.f. 9'
+944S.37
- 944S.3
. S9
+ 3".32
+133.8S
T 3GG. 32
,1.~~.
~
.J(J~
• 18
.,.1/408.08
•• -/008·/2
-4.71
.99
,.Hld2<1..1.J ".ZZtJSt." ,
-23.40
70
0.7Z .&5 - <#1 .S5 - ·57
• !3 - 2.f9 •- ,. • ••• - 34. -'l/l. -3"". '8
- 20. 9~
• tt
.,.~90fS.1f
-338J./7
t9
13
/cooo y
/OOOOz.
t-
10000y
100119"
1
/0008z
/ooOrSK
..
... - , ,*' .,. •
/0 28' : , ~4 3"
-
•-
-
-•
•
H'
•
.(,()
-
•
-
•
"",
••
-
.."
.,
ONSTANTS
'"
•
-
...
•-
• ,•
, •-
•-
.
--
-
, -•
lS1
C~i>'PING h.t"'StLIIY B STRESS j\N~tYSIS ~
6R EQUATIONS
163
:r.
Z,
-~ ~
" --/""
., " '" """" "
-
..-
Oi+S 4/01
0
0/ 128 41+
- 1.000 00
THE MW KELLOGG C
..
, ."" .,
'" - '"
-
-
~
f/"<f.9~7.~
-/7 686 /
S5'l
f2 -/ 16t
/0
27 <5
qiJ% ,
4/8 / r
II<
+
78 ,~ Jot 81 HOI UJ 179 .71 IS 17<
o,85M'J Iq ".iJ6::, 97J 3 -:'15 ZIt 8' +47/
45 _
2 J2
'11'4/4J .,.
1/
" t" .,. 4/2 os -3
2 31 +
• 60 +-T 402 1/
- I 801 <5.
88
1/ ~9 l "0 G7J 2 S Z? .,.
0
S.,.4 JJ PO
4< 55 r
7<
1.000 00.
_Z
001 2
0/3 28
"0 17 I ,,:/~2 85 73 -:00 91 o -iJJ8~.i1/2 i:.()16 19/
,Hl4! B+
947
1 ' , SI8 40 _ 9<3 92 t- 13 2 -4t
'17
83
t.
~ 1 3J 28
3/6 .,.
135 9.
1<301.
0
0/
2 tIS.
~ ./7 14 45 +
/
67 0 t/S /5
- I. 00 00
00 8~
Ot 05
'f t6
·.l51 () 86
181 M ."
.00 )4(;
~ J61 9&.
/091 IRZ
Il.
988 18
t0
0/
01 +
9 H
I
0
120
H
504
280
300 93
l
to, 45 -/f,J
08Z 18
• 1.00
I
00
00 38
/81 90+
hi HI 4/ :-6DO 59<
I
5
1s.<"
5
/82 !B +
J6 ,42 00 72
0
03 47
0
0
74
2 - 42
2 142 I.•
38 2;
"8 -30 ,>6 98
q2 06+
I ~+
~~
I, 000 00 on 5/
I F15.1.1-l
1.7 ~Ol. /55 2 _ 65417<
G tJ3 85,
Jt
- 1.000 00 -
#T'
\
I-1-2144.J'
~"2005.
S
r 4200'. 7?-_
+"(,..'3
I-n689.8
"1"4.<#"
879."0
+('~.(,3
C"'L •
wf 'Uc>
AT'
r
.::J..~4tJ§~
+ 20 o.4G.
.3~
+- 14./2
- 14./2
-:.S~' ~-~4!-
+/9993':73
.,.-i99t i3·.-7j-·
+ 14. Il
<7
E
,JS
".12 _/In".,
+82.U
+
+ 90.0
.,. 90. 0
0l-t MW KELLOGG col PIPING FLEXI~IL1TY AND STRESS ANALYSIS
-of
.24-
+
+182.83
- G.2.Z4
+&.2.24
r .14+ • 14-
r,
r
.<
+SI8.4-0
'.9-r;,>-
• 3,":...~
-5'3".9
. ,.
ZSS.c.~
- 3'.01
1-3&..01
5
30'4,"
- 30(;.4./1
,
+ 219. S3
' 2 / . S'
-G58.05
-
+ 41Z. 05
+ 55.<;"
7.08
- 9'3. 2
1 ~87"9.4S
.0 I
•
#- Z~S:,,'
1,.20
2 tJ67{,.13
';:338315. S:f~
+219.53
6.20
.,. 5'939.:.2.2
rIlO.O~
-
+4".44-
- 6.20
ioAS
.,. S9 .39. 1l.
~.4
-46'.4-4
+4&'(,.44C..20
-/7'8(".81
+14-.31
+~08 J. 02
• •. 4S •
•
•
•
Of. .43
-5'4/.43
.,. 988./6
-fOI7,z.4",
+ 'l01U. ,_~=
I. (;. 4;.
--- ._-- Z-,-·.'!._~. .. 1:!p:99... __ ~1?'8'·.l!.L_
+a239.49
·466-:-~ .,. 219.5;';_ -3383/5:5-2-
- 8Q.?-'i-- -22'39.4-9
8
--5"41.4>3
."
1.43
-
-
";-i;; 5"
~
---~-S2
CONSrAIVTS
-~~{!.-
+15114.42..,. IS'II". 42,
2474.':'7
- I."~
- 435. "9
+ 43'S.-~9
-Slf.J.7_~
-~/8i..·.48
1000 b r
/000 b Y
i- s/s.97
"'2,,47,IS
.,. 2..!f~Il .. ~!- ...::....~'47. 15
?7.4S
+ /4.3/
,
'"
=
18 9t
<I
38 I".,
62 tI
.. •
lOt ~5
3< 3.
FORM E'I
~:~i~-/?Et'"C CALC. N-S.II
F M J
CALC NO S.II
164
DESIGN OF PIPING SYSTEMS
10008
"r.85254-
.
/0006
.. -6.S-Z6fB
.,ZSS; . I
+ 264Z15
, 0
- 39/./4
i-36 'I.
"'367/.98
M
+/672.0
-1-142S.48
. -
,.
/000
CONSTANT
>'
.. -1.S7737
.
43S.69
1.66
-I
4/497.08
Z.62
,, 435:69 ,.-
515:97
fOE!.]7/. 90
- 17275.18
/0000. ..
t)F.Z4-1SO
- 2647./S - SI5'.97
- 391./4- 1/7275.78 -6877/.90 - 4/497-08
.
S:S2 27.4S
-,. 124.Z3
u.;: 70 - 1182.
.,. 7717.08
- 735;74 ,. 26/4.46
/.66
+6/09.5i
.-
2.62-
~4.$1
-
16104.00
- 90/72.1.6
22..57
- 04s'4-.z31"
, 46Z.9Z.
- 6563.73.-/7686.8/ . 6563,87
f 90172.66
f
_.
M.
•
M
/000
·+/~3.28663
+2558.81
"4307./1
M ..
0'
.
/0006 >
--.ISZB6
0'
f2558.8/
"21 1.49
+15114.42
- 23/0.39
+
t2SSB.8/
i 2/81. 9
-1--/.5'1/4.42
-
988J6
-
2239.49 ""
- 23/0.39
~.Z9.
-
.OB -
2S:84 -
-
f
6.45" -
6.~~ -
-
BIZ.70
692.86
2246./3
f
r
988./6
.54/.43
343.'34:" -+ 3533.47
-
2239.49 -
466.44 t
- 644826 f298494:08 144425.84
859.10
of
-
466.44
- 2305-
- 23/9.04
1:+ /376.3/
+/36 .68
346.28.,.338315:.52.
27.01
S9o.SZ r
oJ.
ZI9.S.3
346.[8 -3383/5.s2
2/9.53
-I-
3.05
42.60 -
3064.!1
{}
.,.
929.27
925.98
24.38
:EM
of
988./6
-
ISI.05" -38763.68
+3
ZS5'.66 ... 5"36.99076.06 +511
.34 +
S; ./10 -56
988./6
ISI.O$' -38763.£.8
f
2.s-5."6 $36. 9 .,.
.,. 34076.06 ,.51145:34 -
.5'6.80 -56
54/.43 oJ. 6081.02
82.76 -3.,960583
+
+-
M%A.
+2647.15
1-2256.80
Mzs
_ 22S6.80
+
Mzo'
-1/82.48
_1008.11
..j..
-2647./5
-
f
..j.
5939.72
.,.
S93!1.7Z
36.0/
- B0I4.34J
-8021.7.
.01
-12337.0
-IZ346.84
36.01
+
7.08 f
110.06
943.67 -/o4QZ. /
4Z.01
+
/!J.Z7
20.93 .,.7052819
-
03 I.
.J. 20357./4
11.44
IM
!THE MW KELLOW col PIPING FLEXI~ILlTY AND STRESS ANALYSIS
/000 f} •
+.85254-
+
F8
Fi o'
515.
43
7
.
-
1000 ()
.15286
+ zz. 9-
4
4Z.33
.
/000 B
=-6.526/8
,
-
265. fi6
,
.
,
668-:48 +
,. 255.66 ,.
548.53
/I
-
5.52 44.71
-
- 4?5./.44 -
+
FYA -
, "-
4-
f:!:!!.-
37/·44
-
2
Z
.45
.40
'f,
ZA
168
Fz.o·
.0 -+ 11057.89
"
6.45 4.99
7.08 +
46.ZI
+
.76
/01 .30
+
1.66
.42
-,
,.
-
1·<;'6 -J-
14. 'SI /.42
2.,20
+
fi:,,6.44
- 536.9
116./0
+
-
+
.67 +
<;3.81
,.
O.OB
-
+
90.08 466.44- - 536.99 116./0 -/1.30 -/- 35
.49 -154-74.58 - 8679.62. ,
7/.'30
+ 3504.49 -1<;474.58 - 857'9.62
6.20 :t
•9
-
Z-7.01
4.13
Z.~~
-~ :L-~
254.30
36.01
235.01
2/9.5?1
33.56
219.53
3 .5
,
110.06
18 •
-/-
-
.
000
o M J-I
w.
It",
,.
4
98.
-
.22 +
879.60 +
864.96
-
1.16
+ 19993./3
4.12.
ZZ.27 +199 3./3
.36
13.27 4-
.14
.66
,
-
U!M>§
- 878.99
-
,
+/114-.86
.57 + 2.020.46
,
-
.36
3
+
-
9
- 99.95"
100.31
179.2.0
- 17 .85
+ 2 91
366.32. 4- 27 .49
-
U.
THE MW. KELLOW COl PIPING FLE'ltlILiTY AND STRESS ANALYSIS
"
"~':KlO
AT" " ~
III
-/.01
66.63
05·10 +8445.37
. til 4-
23.00
.17
- 2
-
14.12.
62.24- +
ez.95. 7fD - 1"- .85
86.60
- 628. aD
+ B(, .02
.14-
14./~_
237.32
l- 62.9:20
62.2.4
9B.18 -829340 IE!.
22.27
:l5.S2:
-
-82934.19
-/-
-
r
CONSTANT
.,. 3G.01
14.1Z. -J62.24 66.63
- c.35.01 I- 2.95.76 + /3 .85 105.10 - 445.3
+
CALC NO.5 1/
S
<= -1.'S7737
116.10 415.97 - Z239.4'3
548.53 - 439.89 , 342..33 - 668.48 + 73111.72 + 11057.89
U;
F'B
-
1000
000
'" i-/33.Z8663 =-95.Z4450
AL
Hl':O<'lD
ATE,.
-
·67
J-2
CALC NO 5·1/
FLEXIBILITY ANALYSIS BY TIlE GENERAL ANALYTICAL METHOD
and checked as if it were fixed at both ends using
the known thermal constants EI* 6 z1 EI* 6 11J EI*11:.
For the unknown rotations arit! dcflections (multiplied by 1000) the downward operations are carried
out as shown on Form E-2. For the upward solutions the prime equations on Form E-2 are transferred to Form E-4. As F, is expressed in terms of
the unknown rotations and deflections only, eq. 6'
is mUltiplied by the coefficient for F, in the 5' equation and added to the same, whereby F y is obtained.
F z is obtained by using the values for F z and FIll
and so on. It should be noted that symmetry about
the diagonal is obtained. Lack of it indicates an
error. Since the results from every column except
the uppermost left-hand one are used to produce
symmetry, only the latter column need be checked.
Since the OA and OB branches are symmetrical,
only the solution of the equations for OA branch
are shown; the difference in signs is easily visualized.
The results of the moments and forces by the inversion of the three branches are now listed on
Form J and added together, satisfying equilibrium
conditions. Thus a set of six equations for the unknown rotations and deflections are obtained. These
are now solved on Form E-1, and the moments of
each branch are obtained by substituting the values
in the individual inverted equations as shown on
Form J-I and Form J-2. The figures shown in
brackets are the results from Sample Calculation 5.8
given for comparison.
The above example is naturally solved in much
less time when done as in Sample Calculation 5.8.
However, if two morc branches were connected at
165
where
O*x = 0'% - zO'" + yO'z
o*y = a'v - xO': + zO/x
0*: = 0': - yO/x + xO'v
and x, y, z are the coordinates at point I.
The above problem can also be solved by the reinversion procedure which makes it possible to
handle almost any piping system, no matter how
complex it may be, without solving sets of equations
of more than six unknowns.
To illustrate this procedure, the system calculated
in Sample Calculations 5.8 and 5.11 is used in Calculation 5.12. From Calculation 5.11 (Form J) the
inverted equations for branches OA and OB are
added togethcr as shown on Form J, and these
equations are now inverted again as indicated on
Forms E-2 and B-1. The result from this reinversion shown on Form E-4 represents a set of equations
for a fictitious line which replaces the two branches.
The coefficients in thcse equations, multiplied by
EI* X 10-·, are the shape coefficients for the fictitious line and can accordingly be added to the shape
eoefficients for branch 00' which are given in Calculation 5.7, as shown on Form D--4, and the problem
can be solved with six equations, shown on Form E-l ~
since it is now reduced to an equivalent two-anchor
problem. The figures shown in brackets are the
rcsults from Calculation 5.8, given for comparison.
By using these moments and forces, the rotations
and deflections can be computed at point 0, and the
point 0 the standard procedure would require the
solution of 24 equations; the inversion solution
would be less time consuming.
The inversion procedurc is especially suitable
where many branches are joined at one point. A
system shown in Fig. 5.15 which represents 42 equations when solved by the regular method is solved
by the inversion procedures in the following manner:
The rotations and dcflcctions at 0 are assumed to
G
E
H
F
be Ox, BYI Oz., Ox, 0," 0: and at point I arc 0' XI Of 111 0' t,
f/ x, b'll, o'z_
Thc system OAB is set up as twelve equations
with A as the fixed end. The six equations for the
B-branch are eliminated first and the rcmaining six
equations are inverted. The systems OeD, I EG,
and IFIl are treated thc same way. A set of twelve
equations can now be set up with the following
unknowns:
FIG. 5.15
Pipe line with eight pointa of fixation.
166
DESIGN OF PIPING SYSTEMS
moments and forces for branches OA and OB are
obtained in the same way as in Calculation 5.11.
Applying this procedure to th,tsystem shown in
Fig. 5.15, the inverted equations for OAB and OeD
are added and reinverted. The same is done for
1EF and 1GH. The reinverted coefficients arc added
to the shape coefficicnts for 01 and thc six equations
are solved.
5.20
Cold Springing
and for a multi plane line it is unlikely that a point
will exist where all three moments are zero.
It is generally preferable to make the final joint
as close as possible to one end of the line so that all
but a small part of its flexibility is available in the
longer part of the line. As no pulling is done of the
shorter line, its free end should be located and preferably clamped in its design position. By so doing
it is established that the longer line has to be pulled
in such a way that its end rotations are zero, and
The Piping Code's rcquirement that cold springing shall be governed by sound judgmcnt leavcs the
method of cold springing to the erection supcrintendent unless a dcfinite procedure is set up by the
designer which will assure that the correct rotations
and deflections are obtained at the closing joint.
The deflections are conveniently taken care of by
pulling the pipes together the amount specified. The
rotations at the closing joint, however, are more
difficult to match because they are a funetion of
couples to be applied rather than single forces. For
a single plane line the closing joint can be selected
at a point where the moment is zero; even so, it
should be remembered that each free end of the line
must be sprung exaetly the correct amount in order
to make the free end rotations the same. Such a
point may also be inconveniently loeated, however,
1000 (lY
/000 8 z
+ 4307.//
oJ 2s"S8.BI
+2S58.81
+5117."2
+2(.47./S"
-2&47./S
o
o
+ 88.16
16
t-2239.49
-2239.49
+4&.6.44
o
o
+8"4.2l
I' zsS8
8/
+/5114.42.
+ ZSSlL
+ '7. G2
-I"/SII4.42
+02.84-
+2&.47./5
- (.4 ./S
r 988./&
- 9 8 "
o
"
A
B
F
be prescribed since thc nceessary pulling of the
longer line can be computcd.
Sample Calculation 5.13 illustrates a procedure
for cold springing which has been succcssfully carried out by The M. W. Kellogg Company on several
occasions. In this calculation, the main steam line
of Sample Calculation 5.8 is assumed to be cold
sprung 100%. In order to protect the turbine, the
final joint is located at the throttlc valvc, point 0,
permitting the valve to be clamped down in its design position. Hence, all the cold pull is taken up
by the portion of the line 00'.
The branch 00' has been computed for the full
expansion in Sample Calculation 5.7. Accordingly
the moments and forces required for 100% cold pull
at point 0 are obtained by multiplying the results
1000 8 x
+ 43'07./1
"
accordingly, a convenient cold spring procedure can
'1$/5.97
-5"/ .9
+Z239. 9
-2239. 49
o
o
- 4 S."
+435.'9
- 4"". 44
- /. "&.
/.""
1000 S Y
1000
-43S.~'i
- I. 6("
+ 35.&.9
(,
CO/VSTANrs
-- ."32
- 90/72.'~
t-219.S3
+21.53
+43'/.06
338 /5.S2
+3383/5.52
- 3(.. 01
t-3l>.O/
-5t.(,42.01
-SI-'-42.0/
-!13Y~
o
- 4't..44
z
•
.<
rS939.72
+ zss. (,t;
-S3'.99
.,
+2SS;"
-S3(..99
44
"'511.32
-1073. 8
o
rZS .l>"
I-t548.S3
-11"./0
-82 '34.19
+ 4853
- '2. 24
-11(../0
+"2.24-
-
+/0
- 2!!2. 20
a9
+1187
2
r2SS:'"
of
/1 32
.06
-II&..
t-219. ~
- 3C. __~
+219.53
1-3(..0/
'f 439. 0(,
Jt
+515.'17
-SI5.97
o
-S3t..
-1073.98
o
1000
o
0
-II .I
- 232.20
- ~2.24
+"2.24-
o
o
r
+-/4-./2
of
-
0.0
0.0
..:!:.J§..9" L6
f-14.IZ
- 14.IZ
/4.12
o
r~~."3
""':"3
r/33.Z
2934./9
-1(.S8t.B.
8
+ /9993,13
+1
993.13
t-39986 ,2~
+944S.3?-_
- 94-4S.3 7
o
--
rnt: MW KELLOGJ col PIPING FLEXltllLiTY AND STRESS ANALYSIS
,.
>I~':>(IO
ATr>.n-Z;'_
M J
CALC NO Sol
r
._---_..
1000 (h;
000 By
-
loooh x
0
0
0
-
0
0
0
0
0
+
'"
-
0
/000 3 r
{ooo!;
/000 f}z.
II • 8 G/422 +- 5 II
I. oolo
9 I""
0
DOD
z
000
F.
lOCO
DOD
F ... ,IOOo
Fz/rooo
QNSTANT5
000 dO
10
•
IIGo
E\..lTER. ~now
00
0
0
H.
3 32 -
"'l
t'"
t'l
(}
(}
2 do 22
- 3 04'- 3
0
0
0
0
l.ioodOO
0
0
0
0
0
0
0
4
r .. 271/8Bl5/
t
I
3
•
1/
7 44 +
I
-
0
0
f/32 0
0
II 32 -
0
0
•
•+
0
0
0
:q'/
41 3
01
0
/ 073 98
0
0
0
44 ,
0
3' +
04 0 ,
07' I••
00
09 41
0
01 -
G /7 -
33.
0
0< -
2'0 '0
0
0
0
• /
-
Ir +
4' -
I
,
0
0
22 01 +
0
/80 I •
0
0
0
0
L ,
I"
I
.' I
J
0
0
•
,
-1/3
02
/,///, 0
9
08 18
-
"'4 +
0
()
+
I
7 45 -
0
0
0
0
-
0
+
/63 /8 +
w
,.
0
7 IS
0
0
0
-
"" " -+
1.000'00
0
0
q 6
0
0
0
•
0
0
00
(mi,
•
"-
0
0
0
IZ 00
n
~2
0
a
- / 00 00
0
0
()
0
()
()
()
0
0
()
()
"()
()
0
0
10 .(),3 +
0
/(, 12
"
0
0
0
()
()
07~ 3
28 2
0
0
()
I-'IPING FLEXIBILITY AND ~ TRI:SS ANALYSIS
-
a
/& 2l
2
•
1
Uq
0"
- 10 24 01
- 27 '51 73
/
52
0
00-
149 75
0
()
-
d
00 00
0
-
+ 7 t
I
0
0
0
0
()
"0
0
a
00
()
v//,
()
CALC.
CHI:CKED
OATE 12.-03' S!J
FORM e-2
CALC. NO.
(IJ
>-3
4 1m
-/60 9(,,4
-
;;t'"
>0<
I::::
>0<
0
0
f-
;;:2
( IJ
-11:>3
0
0
000 00
0
0
0
0
0
40
I 2
do
0
:3 4
-
•
-
0
0
0
+
0
0
(}
- 1 00" 00
'"
0
0
0
::J>0<
0
0
0
0
0
+
0
X
I::::
t'"
02
- 1/3 2
dOO 00
0
3 1'0
I
0"
-
0
0
-
"
--
0
0
0
SO IN
1.0 00
co +
"
a
"
0
197 '0
0"
"
()
5' I -
0
0
173 00
to :3
0
(}
0
00 00
0
143 32 -
•
00 00
0
0
0
0
0
5
-
0
0
0
0
4</85 98
zz
-
oz A4 f-
,~<
-
THE MW KELLOGG CO
•
zz -
'mo
mo
0
0
0
" l. 000 00 +
0
f-
0
0
r +- /I 8
r. 00
(}
97
.5_/i?
=
t 'l
<:'l
t'l
Z
t'l
:::=
;;t'"
;;-
Z
;;t'"
>0<
>-3
( ")
;;.
t'"
ao::
t'l
::l
::;
o
~
168
DESIGN OF Pll'ING SYSTEMS
/000 OJ< 1/000 By
~
1.00000 -
/000 {)z /000 0"
/000 61;
/000 $",
'1
o
M~ 1000
:.x-:::: r'-
+ , ~
o
o
o
I-1.000 00
1
o
0
°
~.O
0
0
v....
1-1.0000
0
-
0
1.0000
0
~ .M3104r .!o9:14t
z '/dOO
t11317J
4z· iO
tJ
0
0
. III f
o
"
(/
"
0
(1
0
()
()
(1
0
0
o
0
1-°"-+-1--1-"~Hf---+"O+-j--I-"0+-1
40dj
+!,,2f,
d,2 i
",-/1-1-+--+'
a
tJ
()
0
~.ooo
t . 0 51.
d
-"I+7A-t/_/t:"-;+-1-+'~:++--+-"~++/_/t:"-:+-1
+. tJO
"
-
00
0
-
o:t.,'/i.dJ8
0
0'//0(10
t;/~ '/j
f.IO'A
;/I'I;IOIOIO i i:i; ~I~ ~I:l vl-+/.i{4~,'I~oOI<,19I- -I"f-':
H;r//()(}(! ~ /000
Ny /000
JdbOI..~..,.. • I/~
of
0
0
()
P.
0
()
0
m#4~
~~~
+. 2
+. i
I "1'3 ~~
0
0
"
r'- /.l,:~11
./3. 9
tJ
0
.,"- /,
- 1.0000 of • 73 M
- .I7J
d
0
r 10000
0
o
d
°
d
d
(J
t)
0
tJ
(]
(J
0
tJ
tJ
r'-/.'.3
rl·looOOOf-.d7~3-.I.:1J
0
1"&:3~90
0
V/
o
- 1.00 00
I-lOOOOQ
o
I-
o
c .....c.
, ,
,INVERSION OF SiX EQUATIONS
If.
8R1tIVCH
0 0'
+
O-A 0-8
-IKEINVE reD)
sJ-
0
,1.1
+
'283.68
4./9
+-
,0
>:
BRANCH
0-0'
-
0-
f?"
VE
+
::33
2 • .5.
,3
-f-
0
CJ
,oS
-
CALC. NO.SI:2'
Fz
5.9
of-
23/.9
FORM E-4
ECO<":O
C ....TE
_
Fx
"Ix
+
o
-fl.
e
5.
o
o
4-oZ9.
o
oZ
0
90.
'7
4
4
14.4/
334.e.!38 -
(; G:..13
0
o
o
I/w1G.17
o
II
23:~,:fi
EO
~
-r ?9.oo
BRANCH
0-0'
0REI
0-8
ve
eo
'/////// '///////,
r
3+
-r
40.09 +
43.
_ 0-0'
-
o
?
75-9
1083
9
G .3<:: - (;/(;84./8
15R. ,yew
o -Q'
O-A
5.9?
>?9fj,8t?
G4.91 2.10
4G<?300. <:>
? ..5.? t
GIC.?2
+-
G
'//
11129.9
8 6.G5
Q~
RE/NYC;
t7
9:?
'//////
23/99h.!3!...!2
0
23/9
.55
69 526.2 - I 8339.2'
0-6
38GLSiJ.
0
(RE''''IIERTl!P)
:E
1-7a 087.2.5 -/48339.Z7
BRANCH
0-0'
O-A
0
B
II
c:;. 3.5/
143 .3
'REINl/ERT'Eo
.,.,
'r MW. KEWIV'f"
CO" PIPING FLEX!-,'~L1TY AND STRESS ANALYSIS
, I [Ie.
~.JI :>UHHATION OF' ~HAPE COEFFICIENTS FOR REtNVERS10H
.....c.
'HCOllD
'"
1/79110.9:::>
M 0-4CALC NO Q.le
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
at point C, these two rotations can be controlled.
An F,-force at point C will control the rotation
of this calculation, shown on Form A, Calculation
No. 5.7, by the ratio of the cold to hot modulus of
elasticity. In practice, howcver,~it is not feasible
to apply a moment at a point. Therefore, three
unknown forces are introduced into the system to
replace the moments. The points of application of
these forces are rather arbitrary; since the purpose
is to prevent rotations at the pulled end (point 0)
their magnitude will deerease as their distance from
that end inereases. However, note that this last
rule applies also as far as the other fixed end of the
line (point 0') is eoncerned, so that the forces are
best located near the center of the line.
In Sample Calculation 5.13 an Fv-force is applied
in the riser at a convenient hanger location, denoted
as F v7' This force will create rotations around the
x- and z-axes, and by introducing another F v-force
around the y-axis.
The calculation is carried out as follows: The
shape coefficients from 0' to 7 are summed on
Form D-3, and the equations for the y-deflections
at point C and 7 and for the z-deflection at point C
are computed on Form J. The simultaneous equations shown on Forms E-2 and E-3 are arranged as
follows: the three equations for the known cold pull,
the latter given on Form A, Calculation 5.8, arc
first entered; the constants are obtained by multiplying t!., = -0.75671 ft, t!.v = -0.54900 ft,
t!., = -0.12524 ft by EJ/144 = 231,577,080. The
three equations for the zero rotations are then
entered, and finally the three equations for the unknown deflections, 01/7, aye, o:e, are entered. The
,
,
Z
3
4
5
•
N,
1+
253
Fx
Nr
~
-
1.000 00 .,.
,,-
6335+
[-65;$.73'/ ;::...
•
::r. .,.
+/~
0'" 52 _
5.35-
25' DO
o.
-1.0000_
N+
(J
,.
1,[ .,.
SS ZZ..
til" 4 ()~J dt.,.
DO OS.,.
9215-
3 +1#
z 6Zt
o
-
.z 11/ -4798
-
1.00
20 #
[+Zq351.],
c
c
,C
-
Q,
n
-It> !ld.f. t.9 _
.,..,., 01 8 ~
4.It ,.,
d
0
+<U
1.
__
c
Ii.
/3 39 I
fZ-/2l'"
I
00 -19
- 5
4
I
11"
25
-/08
.,.
z
12'
c
c
'521 '1-11 ~~ 45
..
• , .,. 7 1M
11:.,.
~t
ONSTANTS
~z
Fr
91.,.
1~ 2 ~ J 342 SB - 7 :70:: lJ
..
Z J' -/~ /8/ "+30 I tJ
"-/9- '" 2D ~ 8 '15 It.
o
-40
0
+
94 'i'
-
/, '53
.,.
I
'j
~131
I
.,.Z
0#0
.
"
.r: 417 ~e DJ
I
(}
.t:1IoJ'd18., ~ Iq,J oJIN9.l..52/1",...
0
-n~, IqlJl ,.Y18#J 0 ~ ,Ict..:(.l ~
0
11: :J~ I&i '1J -/JIJ '/~ /9 -341 f1~ 2' -/1'1 ~
- 1.0000'"
39. /8 T
loa II
~
F,,; .,.
f,...
,J'/. M
2$
43 Z -44ltuoo 4fIJI 18'
J~
-2 UJi'z
.
l
2 01 "
$/11'
3.,.
H.
0 of
Z
[+ 4C5.fJ?} 5 .,.
,os. 3/1
-/.
+ I II
oJ-
-54-
,
0
(J
0
I~ ..,./slll4.51 -'571 l.-/So /3 ~
- 1.00000 r
Z Sa
~
~/l14 .
.!q G;
1/8 8.,. 9' Z5
-22"
117 II
'Z3 '"
·S51. 1"
•
.. ,
/I
dZ
, If., ..
"
(J
o~
31
-2 4/ 0 ;;L.~ 47'
=t: +~ I!1J 7& -14 ~
l. 000 00 :/'h
rf 271.1.<7 F,
HE MW KELLOGG CarPING fLOl61UTV 6 STRESS ANALYSIS ' ...L . A,~,
.
169
6-EQUATIONS
o:~/
. . r: ._.
115
FORM E-I
CALC.N· /2.
DESIGN OF PIPING SYSTEMS
170
~
x
y
%
M,
My
I
·• ,.~.
.. •
2 0"2
M,
I
0
+0
II
/8
.u
Z5
0
.q
•
0
0
.q
..
+A~v
s.
-cq
tU
-Vo
v >c
vz
NEAflJEIi:$
..
IN X-PLANt!
,
y-
#
•
[ +A u
r
YOb
0
0
,
'U o
.cv
-,
,
.6. -
840 20
, /88 /7
t
79 20
95/ 01
I 076 94
0
+
/ 34
,
• {'8759278
.8 ,
0
/ 0{,8 67
I 2/G 5 +eq
4/4 {,4
_s.,.},_clq
0'- 2
x +~tH:>+ct
0
-cU o
-.,302 9'/'1 79 +C KZ';
3 /47
57 38I'J48
~<
- 34
" M48 63
6717 q ·""b·C~q
+ 90 2/6 43
- /9 2/1 95
-ev
•+
-cU o
U~VDO
,
J
.81
- 98 953 / -':'0'0 - 25
-IIG 35 /5 -S"'b-C.z'l
/2 492 95
r 4/3 84/ 23 .C,
Y
4
.s. ,
'Uo
-
-5.
,
z +SI>t>+cfu.
I: ... e x /<
-
0
-0"
Z5 q2
- 3 69 86 +B%.'t 72
- 87 Gq /3
- /9 80
08
-Vo
863 /8 +6,
0'- Z • 98 (,7 6IJ
x +uO(l+vo<) .,. /58 /45 70 -cu."
y "'5.u.+ ClV + /57 024 85 .cvo
59 41 +Bt:K
-Vo
793 96 ... B u,
0
<Cv
0
.cq
/ /70 {,4
IRO 7 -cu
- 1 /70 04
25
- 2I 506
102 31
/35 29
425 09 -54
233 58 -cu
,
75 92 .8.
/78 M
458 /7 -cq
1M 42
-
~z
•
,•
75 92
><q
0
0
+Axr. +
93 + B xx
.0
, d-2 ++ /91< 71
0
y +>
-SA
+ 2/ 63
0
,
-cu
[ .A
+8
0
• 53 87 t-A
OC2 + /4 27
• .u • 22 22 'U o
y .v
5-7
• 22 92 +cv
z
"b
Z-S
L +-Au; + 58
Fv
F,
I
93
+
~.u+c:v
o
- 64 {,542
,250 081 M +C vI:
I. +Cyv
+ /68 082 92
x u .... +c 1 'o' fiB! /21 45
y t5bl>+c- +256 '~ 06
0'-2
% +Uo<>+Voo
1: +en.
0' TO 7
PIPING -FLEXIBILITY ANDSTRESS ANALYSIS
. PLANE SUMMATION OF SHAPE COEFFICIENTS
THE MW KELLOGG col
F ••
F.,
Fz.
., 7'I.J '1~
-,
M"
.,. 75.'12
M
t5f.0~ ~,f:li~ .
Fv,
SUMMA TtON
-JoZ/fjjl t!./j1MI.
"0
M"
.. S8.41
My.
0
7?...JL .""
.
.9'
...
·MiS/.ll .,. 7 'J.tU.
0
0
0
3<J Zlt.!7. Z 68591. 1/ -834.40
0
•
.,. 7,771,78 '!3~ ~ ~
Fyc
.",
",
!~2.
JU1
Myc
he
f,Mt.'l8. -N8J1. ".J).4ZJ8
40 -25t2.34
, ,.
." . .{iii'?'
u.9." -,U11il..''-4(Jl.'c1J1.iif.
IJI,,(JOZ.~I
"
f{M~;;
0
-, , lUd~~a
5f ·f
'1
0
,
~7
- 13.'10 -834.40
.,,7 ~. 9~
~.qO
,2
-3J.7.$F. "
+ .9J
.,. 793. 'i'~ __.S!-.. ~
·¥.'1'l8~
y5'l.41
-473. 90 ~?1fd!l9Z.
!.g,v,!/,.;...£l
f({1 0.8
F,c
M~.
~3 .?SF
0
- 75.92
0
t-UM4.Z7
-g-- 14,'&<'
~
,.(.8. 911
0
.,.(,85'91.11
--
=¥!'YG d88·70 .;;;~
0
0
-/o56al(. ... 793.9(. :.3~:?J~3!4. Z5ttJ~{£ -IO/-t.lJ.l6 (.BJ6'!S.
..
0
0
..(fil.1.33 .. 3/. J'f
0
14 9.,
:l,5~
0
-4188.m 1'8.2AJ~ -!Af:U? 1125,210.3_ -4~tj§_,_8}_ ~3~NJ
0.2
+/8
Ip'ill~'"
0
-/50.901.5
0
t/88.78 +
9J
•
:.:!~-
0
0
- Z69a.5
+I/tH<..-d.j
-7.5.9Z.
-4)7'U(J
'J 0/
/?,;8"iiif,. ·"l.E..u. i"i6~
"'~:E:i
- - -p-
0 __
-7~, ~Z
0
-
-/50901.,57
~~- rI9!!...?E.. t~~04~
:f4§3I.1 -7i-....9...? ./~? ~11A4{.·.J9 lLr.~~
0
0
0
..0
• ~7:c.o'l.2$
203LU - 9tJ.S4
fJ.J.:L ~§.9t ·~77'1.t -75, <7';_ -G"1-Yt/] -759
0
!!:§.5fiJ.L
--
.. -
tTHE MW KELLOW col PIPING
Fyc
Mzc_
+~ 876 43
oM -
CALC.NO.5./3
-.Jt;.'i9.86 .,. :1. 6 -3t.'l9-8
?~ -jt/I'1.8r.. tZfO,tJ8l
-2(;...1'1 -t,G79.U. C ?G." -';;IlZ~ -';;:
-7/l.'~
,.
Sd.'!Z
02 .
-Utili t3152. "
~l.i2. "
Mzc
L
Fv,
MI:~"
Uk7-
1. .7$r. .,.54.
""58.4/
--"----
"UJ,J.,. ",,-¥.m> ,-#,fM! ~..:Il
,
Mzo
CI-II:CI</iO.~
~r'
o ... YII: --~"'I-
-- ~~-.,. .:.!1;J. ~~1.L -.¥.2liJ,
. ?(~ -Uf!l..4- 35 13 "(l!ft) -I 7'1.ztt
f
FROM
FLEXI~ILlTY AND STRESS ANALYSIS
0
+SO 01.57
-~9:1.
l!.!.§!f.:!&
~57.!1:. !]?~
=.!M4 77.7~ --"
- _.
-"Le.
"eco"o
m
- ",-
M
J
C LC NO
"3
r
•
, l'
F,.
F ••
e40 "
I. o 0
F,o
".
- m
J5 •
'"
4'857 75 •
iZ 08' ez F,o - I JI 49 8334/ 70J 75
11
ZJI
4c' 30 iZ •
I 518
J
2
•
'9853
- '" ", -r •
" -- /e
IFyo -
3<,
I. 00 00.
,7c 10 I /70 J2 3 •
148
le5
jiJ
848
.
.
,
zo •
"
97-
IJ 413 52
•
7c 0" 17 •
oJ 80 +
""040 JO +
08 -
M,o
7773 7J •
en IB •
497 35-
'"
0
0
JJ 4.3
0
"".
'9'
J~
-
i.870 +
28 70 -
97 -
on 7i •
09/
5J •
"'
0
F,G
F"
-
J89J -
m 97-
0" 97.
38' Ie •
(;(;98 ..
JO' OJ •
e 44 '7 •
575 ge.
1.9l9 ~
FZG
$S~9
147 'e
984 fO
sec cz
""
"G
'zc
ON5TANTS
0
0
0
+ 175 123' '9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
III o •
24 '9 ...
55.35 ~
5595 /51
2.77Z .3 OJ ...
/4' aoIBc 89 -
"e 70
00 ,0 5253 lIB 55 e8
13 9/ -
Oc. 04 ...
8~ 75
8797 ...
-
94 ZI •
.J 905 /8
Z 2485/
ISO 90.
m
OJ
0
34 '0
18' 7
0
0
,.
C. 2.91
,-
..
10
14' J4
~8JG
-
0
0
0
0
0
0
0
0
EIJTER. eELOW
,C, 9J
-
'"
... Z,51
--
J7
,/7 c7 -
7(; 91 +
lie Ii' 14 40
+
2778 -
24 37
~
4750 -
2. 57 t
71 4' /58 ee -
0
0
0
17.39 09 40 •
2.048 +
52 If
6.3 70 ...
5737(i,
0
0
0
+305 SO
07e 88 1608 57 143 75 .3J 74 -
c9Z <c
4 OB 74
Z :;53 e/
7:; 90
/1
19 93
0
0
S /98 e8 -
Z7 ZS 84 4780 +
0>7 e9 /87 ,4 -
0
0
0
0
0
0
0
--
I :%09
7
•
'i: •
14' -
IMXo :
J 0/8
0
0
l.J ao
0
0
~8
-
0
•
J 12. ...
,-
-
7e'
0'" 7Z IOU; cl •
0' 7 •
909 19 •
834 •
•
0
•
0' 09 •
05 -
264 ...
44
S /5 •
/f8+
28J 01 -
/7/2. ...
7/5 ..
J 97 -
S 00 ..
2>, 89.
06 70 2. d 2:22
0
-
1. 00000 3245
"-
"5 •
S57J ..
0
2J 20
Z 50 ..
.
o •
~ (;30
0
,,-
-
0
Z 23 723 ..
Zf>f> 8 ZS ..
" - I. 00 00 ~
h
17 4 4
/0 '0' 2e •
0
G •
-
483 4 5 -
/ 9/ '''~
..
7J -
088 48 •
St.74 55 0
2ZZS 931 ...
S 48 5/ •
9, •
0' •
2. 97 ...
I •
e'
I. 000 00 - 4 / 5, 7J •
-
IMzo
"
05 •
2. 7C. +
-
THE M.W KELLOGG CO
Oc'
.,.-
/
'"' 27-
.
/7 I') ...
J &7 ...
IJ II 7C5 77'
/6 OU 4 •
1 43/ ,e •
•
0
33/0
S 3/8 45
0
43 57
0
12 4/ -
'4 •
487 ..
/4- so
.5 95 -
5 ge
It, '4
SJ 5
Z of) -
548 ...
99'
, e ZZ 90
310
Z. 775 7C
e 49 <c. IS 74 8
9 JZ 7Z 580 Zi. /0 '3713
7c
474 4/89 l.J 94 +
/I 49
4e
USI 4' •
J' Is,
/ 57 ..
1418
I 29 J 04 ...
5 I
I It. 2 OJ
48 Z 39 ..
5 Z JS
804 7/ 1858 59 •
/27 %
S /54 2. ~ J8 887
851 59
If C..H 59- J 475 ZS •
90 z
544 ...
e5 9 422 0/ -
'0 •
•
"
.
".
0' •
•
HI-'INl:i FLEXIBILITY AN[ . ~ I RESS ,NALYSIS
12 EQUATIONS
0
0
0
0
0
0
0
0
0
0
92~6 99
/ 387 85
0
0
0
0
0
0
0
23 7~
242 -
-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
CALC.
CHt:CKEC
D....TE
-2.3-53
0
0
s
.c=...
"
....
'25
'"
0
9J
+ 127 I3S ,82
.. /Z4 7.,
0
I (;71 94
8290 +
- ,ee
i9 •
28 8 r •
J' - l. 00 00.
IFzo =
M'o
c8Z /J
caz
71
• 29 DO? "
• C, zo8~
.. 2.13 87 08
e'
e.870
Jc8 70
0
- ,05 OJ
- m 10
"17 '" Be27
- S 7SZ.
~
><
>
2:
>
t'"
><
if,
....
if,
c=
><:
...,;
:::
·
·- ".,,_ " :l
OJ
• /0 878 ,
0
•
t'l
t'l
2:
t'l
:::l
19 577 iZ
>
«-
>
•• ,4320 z7 >2:
- 28 SJJ 7c
-;///0
-
5Z< 0
- 52' '0
0
- 20 98/ 0'
• 38
•
•
-
'"
S;
::l
(j
>
17
7 51 8
I 3403
38 e5
25
:::
t'l
8 C.48,B
I:i
·«- "
--
t'"
•
. 51
4~ CJ
FORM E-2
CALC. NO. 5.13
t'"
o
-...,
172
DESIGN OF PIPING SYSTEMS
--j
I
7
fTl
~
r-
5GJ
GJ
U
--
$
A
fTl
.- " ,
:>:
.-
IFri --
7J
•-
F"
'" ~
" •
·
"" -
F :c
'"
J
"·
JaJ -
JG 08
.; 44 -+
J 32. T
3'
'"
0
f-'='-
- Z3f S17
[TI
X
~
,;;;-l
,-<
OJ>
'"C..,.
0
0
,
0
,
0
0
0
0
G 4(J$
1. /2.8 73
I~ 4S~
I
·
0
0
~
0
2.31
(1,/
/004
0
0
1"
1/
.·
0
.,
1.00
;:.
,
,,,
14 l!l
'"84 • 228' ""
0
- c. 3"
,
857 3. 81
- <5 ," 53
"B - ." .,8Z -- ZZS
,
90
'5' "
'"
,
, "
'"
Z.956
0
-- 9457'0
0
GO
,
,
'"44
490
0
-,
,!- IS
•
,0
0
0
- /02f;"
7 "
0
71
,,
'C. • 75 .91 07
,
"
L
, " •- 7S 897 07 ., ," OS
·- , .,00'- l479
m
,
"
'" "
IF'i •• .{ 8.' SJ" ••
" , " '" "
, "
9 ,
- ." "
,
-- '" "
- '"t:.J "
,0
"
"4
-., •• 74 0(;7 - •• ,
,,,
3" " - .0 '" J"
• 1.00
.,
IZ "8 " •
'" " + " ",
"
"
&0655 0'
•
IF'C + J
" - ," "
""•
fO 13
209l.J
Z
0
,
,
,
0
1.000 00
C
U
U
, .<
,
0
IS I()
+
0
08
,
,
,"
'"
.
, " ·'"
,'""
~
S 7;;;.
'53
", ,
~~7
::'8 59 ·231
0
0
.
0
- l:J1
1& ZSl 0'
0
ZoB 80G
I 4i!.7
~
0
/4 /9
32. .98
$G
0
0
0
0
0
0
0
loS 278
~
T
.,
0
0
0
0
0
0
0
0
- lJI 577 OS
0
0
0
Z8J
JOZ
ZJf 577 OS
00 -
,:; ZlB
8 441-
8JZ
I
,.~
"'0
0
Z(j)
"'-l
JJ
[TI
(j)
(j)
J>
z
J>
C(
:>:
I~)'1 -.6(;4
1.000 00
'" "
II
(j)
(j)
0°"
>X ,
"0
,.
" 00
~"
,0
"",
~
:>:
~"
-
1.000 00
-.580
'"
.,
12
,...,
n
r O
n"
."
pr;
~'"
'"
"lo~c
_ ." .J
.175
,
CONSTANTS
··- "
- , "
·-- ". ""
'" "
·• ,
·• .,.'" ,.
I~~ ~
I ' ~ Ii!.
"gO IS
2. 851
ZO .'0
2. S8t:1 'S
2. sso <5
53
- 1l.7 41>4 31
31
• "4I 405
J41 '5
88
- '7 14' "
- •• ,,, JO
·
8 .$6::' 10
•
443
·
0
• "" .:.:3 ""
lS :lio 50
1"/41 8/7 2,1
".
S84 II
,- 7 0042.5
MllI
%
• ".; 88658
• 3<8 35' ,.
"
-
- S3
·•
"
'" "
.co
.'
G55 (;of
(; ,,55
FLEXIBILITY ANALYSIS BY TIlE GENERAL ANALYTICAL METHOD
M yo
M.,
- I. ODD 0() -
I
T
Mt.o
F X7
14/
- ?JI 83 'of'?f.
h/7 G4f::%%
r
I
t
o
0
I.DOOO
"'l 2 -28 G4~Z3
of 3 10b'l!/.
- 2 3S1l7~ "'18~5 !l'J +- I
';;0
-
/
-
Z3J~0
II-I. 00 00
173
~r0
fO}
/
to G"li
0
32 74
3/9
-/(;(,24'10- 33280
f-+-f-+++_3::5f2!"'(,';i'~~7?9 71< 4 t ~ 752 /6
I
- 734113 r ({ 441: 2 f(, 1>88
• Z 715 7
0
'if 8/
_
1-'0000
- 1.00000_
'6 11M-' fl'Z -J5a550
+4 ,x/IM- 3JZ. '/]-? 151,)
1:-10000
0
-
fJ,
0
- 4 ?JI/&, f J'188'6 1 f, 70 I)
- 2 f8J 21 +5. 57atb o{-IO 70. /3
r -, 00000
126',50 rIO 7ofl./2 1 Z8 3116'.5
THE MW KELLOGG col PIPING FlEX~eILiTY AND STRESS ANALYSIS
rlZoM rc.e"" e-
+
I.S4
-734.9 .3e
"" {:N!49.eZ
+
&.8.588.47
r/ol1(# . lIP
-813ZZ.3o
• 73 9.:3?
- fJ. l'
- 3
9.3Z
+ (,.449. Z'Z
3
9.3Z
- 8 :>,.
'~.82
324!3.G4 ..
+
(,,002.01
B .BZ
'0
-0
-
e'(;4 <?
T
<p86CJB.47
•
090
+ CP9/
D
.<?G - 0
-Z.:3199.,3
IG
·0
.90
·0
or 8 ZG£;> • .!5t:D
"'::>
or Z44/.?e.
"0
e
-
GGB .BZ
/5 9.73
.584,.25
.3:34. 1
0
10,9oB. Z(2J .. 0
+
..,. G9/7C'.
FORM E-4
CONSTANT(M~rclV-f £'Z)
-1v./34.e4
c
'~;~~'D··"
6"
" c
r90eG.OG
+
/. 6f, 03
0
r -100000
£5
71Si10 • 2 ,78] 21
37. 59
l.00 DOt
<f:. n:
-;?/9.1
+ 1009 3 . .58
/908
+- (0'1oG.84
w
G
-./7£j~55G3
- -2./1
c
- . 5.8090.9
.. -<:;.97*
6
l5
THE MW KELLCXXJ CO PIING LE I ILiTY AND STRESS ANALYSIS
-. GC>4
I.
0
M
3 51" -7.98*
•
("UO
AT'
H.
M J
CALC NO .6./3
174
DESIGN OF PIPING SYSTEMS
downward solution of these nine equations is now
carried out, and the upward solution is done for the
known constant column only to obtain the moments
and a check. The 4',5',6',7', g', 9' equations are
now entered on Form &-4, and the solution gives
the moments and forces expressed in the unknown
deflections at points C and 7. If the three moments
are equated to zero, the deflections at points C and 7
are obtained as shown on another Form J. These
deflections are now entered on Form E-3, and the
forces at C, 7, and 0 arc obtained, using the second
line for the upward solution on Forms B-2 and B-3.
The line ean now be cold sprung either by measuring the forces or preferably by measuring the required movements at the points of application of
the forces. Thus the hanger in the riser is lowered
7-h} in.; the force required is supplied by the weight
of the riser. The solid hanger at C is lowered 7 in.
and as the force is positive this point is held in
place. Finally a z-stop is provided at point C preventing the line moving more than 2i in. in the
minus z-direction. The forces to be applied at 0
are all less than 2000 lb.
In order to execute the above procedure properly
it is desirable to have the line supported on adjustable constant support hangers to minimize the weight
effects. Although minor adjustments due to errors
in fabrication may be necessary, the method has
proved to be helpful to the erecting crew, as all trial
and error efforts are eliminated, thereby resulting in
great time saving.
The calculated stress for this condition due to the
six forces, can be shown to be 16,750 psi. This
stress will remain until the final joint is finished and
the six restraints imposed on the line are removed.
After the hangers at points C and N and the restraint
at point 0 have been adjusted, the stress will be in
accordance with Sample Calculation 5.9.
5.21
Weight Loading
The problem of determining the effects on piping
"ystems due to weight loading is similar to that of
thermal expansion; the inherent flexibility of the
piping is expressed by the same shape coefficients,
and accordingly the coefficients of the unknown reactions in the equations of a given line are identical
in either case. However, the constant terms of the
equations for thermal expansion are a function of
the fictitious end displacements only, while those
for weight loading are also functions of the weights
involved. These load constants are denoted by the
Jetter R for the rotation equations and T for trans-
lation equations with subscripts indicating the component. Written in the conventional manner, the
equations for a space pipe line with two ends are
shown in Table 5.19.
Table 5.19
Defor-
M. Mu M.
A.. A" A..
A., Au.
A..
Load
F.
B..
Bu.
B..
C..
Fu F. Constants
B.u B..
R.
B., Bu.
R,
B.u B..
R.
T.
C" C..
Cuu Cu.
Tu
T.
C..
mation
Constantg
EI*O.
EI*OIl
EI*O.
EI*o* :J:
EI*o*1I
EI*o*.;
In determining the weight reactions, the line usually is assumed fixed at the ends and, accordingly.
the deformation constants are zero. However, any
known end deformation can always be superimposed on the weight effects. On the other hand, if
the weight reactions have been determined, the
deformations at any point of the pipe line can be
calculated by using the above system of equations
with the shape coefficients and load constants
summed from the fixed end.
Weight loading is either concentrated or uniform.
The load constants for weights such as valves,
counterweights, constant support hangers, or true
vertical pipe line members, assumed to be concentrated at a point N, are calculated according to
Table 5.20.
For uniform loads such as the weights of pipe, its
insulation and contents, the load constants are someTable 5.20
Load
].[:J:N
:ALN
FUN
Constants
A..
A"
A..
Au.
A..
Bn
B"
B..
B. u
B.,
B,u
C. u
Cuu
Cu,
R.
Ru
R.
T.
Tu
A;I;';
B..
B"
B..
T.
where A u, Au, B:: 1I1 etc. = summations of the shape
coefficients taken from the
fixed end 0' to the point of
load application, N.
F fiN = concentrated weight load
(minus for pipe weights, plus
for counterweights).
].[.N =
-FlINZ N .
M:N =
+FIINXN.
IN, ZN =
horizontal coordinates of the
point of load application.
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
what more involved although they are computed
along similar lines. For each member of the pipe
line, they arc calculated in accoraance with Table
5.21. The summation for all members of the pipe
line gives the load constants of the equations in
Table 5.19.
Table 5.23 Shape Coefficients for Weight; General
Formulas for Straight Members
%
AI'I(!
An
An
Bn
An
Bn
Bn
B..
F lIw
B ..
B vv
B..
Cn
C vv
Cv•
A v•
Bn
Bn
where
w li
Constants
R,
Rv
R.
T.
Tv
T.
Wv
D. v
D vv
D..
En
E vv
E: lJ
~
,
a
W/>o =
,
Load
A. n
An
L'
Wb
Table 5.21
M..
101>6 =
b
y
x-plane
+a
= unit load, lb/ft (always negative),
-
The expressions for the shape coefficients for uniform loading depend on the plane in which the member is calculated (Table 5.22).
Table 5.22
x Plane
y Plane
D..
+Wb
Dvv
E;;v
0
0
0
+w.
0
+w.
E vv
+Wbb
+10,,0
+w.
+10
+10
+10100
-Cl'Wll
0
D..
E..
+c;.ow"
z Plane
0
0
0 10
00
where elf = distance of working plane to coordinate plane.
Tbese shapc coefficients are given in Table 5.23
for straight members and in Tablcs 5.24, 5.25, 5.26,
5.27, 5.28, and 5.29 for circular members.
It is of utmost importance in the calculation of
the w-constants that the propcr direction be used.
The direction always points from the free cnd to thc
fixed end, and should bc indicatcd by an arrow. The
direetion detcrmines the angle a betwcen a straight
24
. 2a
+kQ£'f'
-sm
24
W"
= W'bO
w.
-
w.
=
WI""
= +kQ 24
w..
= W'u"
W.
=
aWb
+ bWb
L'
-kQ - COs a
6
L'
+kQ-sina
6
L'
the member, ft.
A,u, Au, B ZlI1 etc. = summations of shape coefficients
for concentrated loads for all mem-
bers from the fixed end up to but
QL'
.
-SlnaCOBa
= W'b<J -
1IICX
X, Z, = coordinates of center of gravity of
not including the member under
consideration. Thus these summations for the first member wiII
always be zero.
DZI/ 1 DUll, Dill = shape coefficients for uniform loading, ft:!.
E Zlil E lIlI , E 11I = shape coefficients for uniform loading, ft 4,
-k
w..
M,w
-Fvwi)
.. f'lb
'f =
_ +F
- Momen t 5 a t ongm,
lr
•
ltD
-kQ- sina
6
X
of a member whose length is L, ft.
FlIlD = WilL, lb (always negative).
Jt
175
+ aw" - bwl'
y-plone
y
a
-a
b
X
z-plone
L'
+kQ- cos a
6
L'
W'ab"'"
-kQ24sinacoBa
w'
2
+kQ24 cos a
Q(1
=
L'
Woo
= +W'ab
W••
=
+ bWa
+w' Ga - aUla
member and the POSltlVC horizontal axis. If the
membcr is assumed rotating around the end to which
the direction arrow is pointing, this angle is positive
if the member rotates in the countcrclockwise dircction, negative if it rotates in the clockwise direction.
For a circular member, the direction is either counterclockwisc or clockwise. Thc anglc <P is always positive. The angle a is always positive measured
counterclockwise from the negative vertical axis to
the radius whcre angle <P begins. The angles <P and a
are illustrated in Figs. 5.16 and 5.17 for counterclockwise and clockwise directions rcspectively.
Form W v shows a convenient way to calculate the
load constants and also a key to the relationship
betwcen the shape coefficients and the load constants
in accordance with Table 5.19.
In Sample Calculation 5.14 a single plane system
is computed for W v of - 280 Ib/ft. The nccessary
176
DESIGN OF PIPING SYSTEMS
Table 5.2·j.
Shape Coefficients for Uniform Loading: General Formulas for Circular Members
·....For Weight of I\:femhers in the x-Plane
Counterclockwise Direction
w, = -QkR' {<I>[sin(a + <1» + sin aJ + 2[cos(a + <1» - cos all
w',. = -QkR'\~[1 + COS2(~ +<1»] _ iHsin2(a + <1» - sin2aJ + Icos(a +<1» - cosalsina)
w'" = +QkR' {~[4> + sin 2(a + <I»J + i[cos 2(a + <1» - cos 2aJ + [sin(a + <1» - sin al sin a}
Clockwise Direction
w,
= +QkR' {<I>[sin(a - <1>). + sin aJ - 2[cos(a - <1» - cos all
w',. = +QkR'{~[l + COS2(~ - <1»] + ~[sin2(a _ <1» - sin2aJ - [cos(a-<I» -cosalsina}
w'" = +QkR'\~[<I> - sin2(a - <I»J + i[cos2(a -<1» - cos2al + [sin(a-<I» -sinalsina}
Table 5.25
Shape Coefficients for Uniform Loading: General Formulas for Circular :Members
For x-Wind Acting on l\Icmbers in the x-Plane
or z-Wind Acting on l\lembers in the z-Plane
or Weight of :l\Iemhers in the y-Plane
Counterclockwise Direction
w, = -QR' \ 1.3 [<I> cos (a + <1» + ~ cos a - sin(a + <1» + 1.25 sin a - 0.25 sin (a + 2<1»]
+k [~cos a - sin(a + <I» + 0.75 sin a + 0.25 sin (a + 2<1»]}
w, = +QR' \ 1.3 [<I> sin (a + <1» + ~ sin a + cos(a + <I» - 1.25 cos a + 0.25 cos(a + 2<1»]
+k [~sin a + cos(a + <1» - 0.75 cos a - 0.25 cos(a + 2<1»]}
W'llIl =
<I>' - 1 + cos ep )
+ 1.3QR4. ( "2
Clockwise Direction
w, = +QR' {1.3 [ <I> cos(a - <1» + ~ cos a + sin (a - <1» - 1.25 sin (l + 0.25 sin (a - 2<1»]
+k [~cos a + sin (a - <1» - 0.75 sin a - 0.25 sin (a - 2<1»]}
w, = -QR'\1.3[<I>Sin(a - <1»
+~sin" - cos(a - <1» + 1.25cosa - 0.25 cos (a - 2<1»]
+k [~sina - cos(a - <I» + 0.75 cosa + 0.25 cos (a - 2<1»]}
w'", = + 1.3QR'
C;' -
I + cos <I>)
_~
__l
i
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
Table 5.26
Shape Coefficients for Uniform Loading: General Formulas for Circular l\'fcmbcfs
For \\'cight of 1\lembcrs in the z-Planc
Counterclockwise Direction
w. = +QkR' 1<I>[cas(" + <1» + cas ,,] - 2[sin(" + <1» - sin "ll
w'., = +QkR'l~[l - caS2(~+4»J + i[sin2(" + <1» - sin 2,,) - [sin(" +<1» - sin,,] cas,,}
w'•• = +QkR'1~[4> - sin 2(" +<1») - i[cas2(" +4') - cas2a] + [cas(" +4,) - cas,,] cas,,}
Clockwise Direction
w. = -QkR'14>[cas(" - <1» + cas,,) + 2[sin(" - 4» - sin "ll
w'., = -QkR' l~
[1 - cos 2(~ - 4»J - ~[sin 2(" - 4') - sin 2,,) + [sin(" - <1» - sin ,,) cas ,,}
w'•• = +QkR'1~[4' + sin 2(" - <1»]- ~[cas2(" - <1» - cas 2,,) + [cas(" - <1» - cas,,]cas"l
Table 5.27 Shape Coefficients for Uniform Loading: Formulas for 90° and 180 0 Circular Members
For \Veight of Members in the x-Plane
w,
W'ba
W'bb
+OA2920kQR'
-0.39270kQR'
-0.13315kQR'
,,= 90·
D
D
-OA2920kQR'
+0.17810kQR'
+0.39985kQR'
,,= 90·
t;
+OA2920kQR'
-0.1781OkQR'
+0.39985kQR'
0
t;
-OA2920kQR'
+0.39270kQR'
-0.I3315kQR'
" = 180·
tJ
-OA2920kQR'
-0.39270kQR'
-0.13315kQR'
" = 270·
tJ
+OA2920kQR'
+0.17810kQR'
+0.39685kQR'
" = 270·
,,= 0·
~
-OA2920kQR'
-0.1781OkQR'
+0.36985kQR'
~
+OA2920kQR'
+0.39270kQR'
-0.13315kQR'
+4.00kQR'
-2.35919kQR'
+2A9740kQR'
-4.00kQR'
+2.35919kQR'
+2A9740kQR'
-4.00kQR'
- 2.35919kQR'
+2A9740kQR'
+4.00kQR'
+2.35919kQR'
+2A9740kQR'
-0.78MOkQR'
+OA9740kQR'
+0.78540kQR'
+OA6740kQR'
-O.78540kQR'
+OA9740kQR'
+O.78540kQR'
+OA6740kQR'
Shape
,,=
0·
a = 180
,,=
0·
" = 180·
D)
D)
0·
((J
((J
,,= 90·
E-,
ex = 270 0
B
" = 270·
,,= 90·
V
V
a = 180 0
,,=
°
°
°
°
177
DESIGN OF PIPING SYSTEMS
178
Table 5.28 Shape Coefficients for Uniform Loading: Formulas for 90° and 180 0 Circular Members
For x-Wind Acting on l\lembers in the x-Plane
or z-Wind Acting on l\tcmbcrs in the z-Planc
or Weight of Members in the y-Planc
Wu
w,
WI "1'
D
D
+0:21460QR'(1.3 + k)
+QR'(0.09204 - 0.50k)
+0.30381QR'
+QR'(0.09204 - 0.50k)
+0.21460QR'(1.3 + k)
+0.30381QR'
Cl
Cl
+QR'(0.09204 - 0.50k)
-0.21460QR'(1.3 + k)
+0.30381QR'
+0.21460QR'(1.3 + k)
-QR'(0.09204 - 0.50k)
+0.30381QR'
-0.21460QR'(1.3 + k)
-QR'(0.09204 - .0.50k)
+0.30381QR'
a = 270·
tJ
tJ
-QR'(0.09204 - 0.50k)
-0.21460QR'(1.3 + k)
+0.30381QR'
a = 270·
~
-QR'(0.09204 - 0.50k)
+0.21460QR'(1.3 + k)
+0.30381QR'
a=
O·
~
-0.21460QR'(1.3 + k)
+QR'(0.09204 - 0.50k)
+0.30381QR'
a=
O·
D!
D)
+ 1.57080QR' (1.3 - k)
-2.00QR'(1.3+ k)
+3.81524QR'
+1.57080QR'(1.3 - k)
+2.00QR'(1.3 + k)
+3.81524QR'
-1.570S0QR'(1.3 - k)
+2.00QR'(1.3 + k)
+3.81524QR'
-1.57080QR' (1.3 - k)
-2.00QR'(1.3 + k)
+3.81524QR'
Shape
a=
O·
a= 90·
a= 90·
a = 180 0
a = 180·
a = 180·
a = 180 0
a=
O·
(]
(]
a= 90·
E::,
- 2.00QR' (1.3 + k)
-1.57080QR' (1.3 - k)
+3.81524QR'
a = 270·
[5,
+2.00QR'(1.3 + k)
-1.57080QR'(1.3 - k)
+3.81524QR'
a = 270 0
V
V
+2.00QR'(1.3 + k)
+1.57080QR'(1.3 - k)
+3.81524QR'
-2.00QR'(1.3 + k)
+1.57080QR'(1.3 - k)
+3.81524QR'
a= 90·
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
Table 5.29 Shape Coefficients for Uniform Loading: Formulas for 90 and 180 0 Circular Members
Q
For Weight of Members in the z-PIanc
Shape
w.
1O ' a b
1O'aa
·0
Q
-0.42920kQR'
+0.178IOkQR'
+0.36685kQR'
a= 90·
Q
+0.42920kQR'
-0.39270kQR'
-0.13315kQR'
a= 90·
[3
+0.42920kQR'
+0.39270kQR'
-0.13315kQR'
a = 180·
[3
-0.42920kQR'
-0.178IOkQR'
+0.36685kQR'
a = 180 0
+0.42920kQR'
+0.17810kQR'
+0.36685kQR'
a = 270·
tJ
tJ
-0.42920kQR'
-0.39270kQR'
-0.13315kQR'
a = 270·
~
-0.42920kQR'
+0.39270kQR'
-0.13315kQR'
a=
O·
~
+0.42920kQR'
-0.178IOkQR'
+0.36685kQR'
a=
o·
D)
0
+0.78540kQR'
+0.46740kQR'
a = 180·
DJ
0
-0.78540kQR'
+0.46740kQR'
a = 180·
((J
0
+0.78540kQR'
+0.46740kQR'
0
-0.78540kQR'
+0.46740kQR'
a=
a=
O·
a= 90·
ex = 270
0
a = 270·
a= 90·
(0
0
+4.00kQR'
+2.35619kQR'
+2.46740kQR'
E:J
-4.00kQR'
-2.35619kQR'
+2.46740kQR'
V
V
-4.00kQR'
+2.35619kQR'
+2.46740kQR'
+4.00kQR'
-2.35619kQR'
+2.46740kQR'
179
DESIGN OF PIPING SYSTEMS
180
z, x, or y
z, X, or y
a
a
b
b
a.
__~
~~
FIG. 5.16
~_y,
'-~'--'------.....-
z, orx
FIG. 5.17
Angles qi and a in the counterclockwise direction.
t
I
.75
8/58.5
"0••
Fx! I~
3
I
526.4
18.50
.73
2.28
a
TEMP.
PRESSURE
PIPE
INSUL.
forces at the free end obtained from the equation
sheet, and any restraint in the line between the free
end and point N. L:M, is the sum of the moment at
the free end referred to the origin, obtained from the
equation sheet, and the moment at the origin caused
by any rcstraint. Since no restraint is included in
the example L:M" L:F.. L:F" are the reactions
from the cquation sheet. XN, YN are the coordinates
of point N. L:F"w is the sum of the uniform load
3.
3/.00
R
h
k
M"N= L:(M,+M,w)+ L:F,YN- L:(F"+F"w)XN
where L:F. and L:F" are the sum of the reacting
I-B
0
·z
Angles qi and a in the clockwise diredion.
reaction at the fixed end is the difference between
the total load of 29,150 lb and the F"-force of 19,864
lb or 9286 lb.
The moment at any point N of the line is calculated in accordance with the following general
formula:
data is given on Form A-2, which also gives the data
for Sample Calculations 5.15 and 5.16. The shape
coefficients for concentrated loads are as usual calculated on Form D-2. The shape coefficients for
uniform loading, in this case W a , w' ab, w' aa, are calculated according to formulas given in Table 5.29
for the circular member 2-4 and in Table 5.26 for
circular members 4-5 and 6-7, and entered on Form
W". The constants for the straight members 5-6
and 7-8 are computed from the formulas on Form
W" and are entered also on that form. For member
1-2 the constants are zero. The sums of all load constants are entered as constants on the equation
sheet, Form E-l, on which the summation coefficients
from Form D-2 also are entered. Solution of the
equations gives the reactions at the free end, the
moment M, referred to the origin. The vertical
MEMBERS
;-
//50F
20 RS/.
Z42.3
-'"
'"ill
.37.7
co~nEN
TOTA
Z O.
w MATERIAL 5 (fl.
UNIT WINO LOAD ON
PROJECTION OF CYLINDRICAL SURFACES
• 60 LBS/FT 2
••
y, z, or x
-~
'\
Y
V-z-srop
0
~V'"
•
F,
"
'\J
X
2
5·
II
Z
S
35·
I
I
6
"0'
42·
-
if-
,,
B,
POINT
Sr:
5p:
S,+Sp:
~,
Sh:
~ POINT
~ 51:
"
2
/390 PSI.
200
1!i90
2200
{
"".
"".
"".
PSI.
5 z:
PSI.
Sy+ SP+ GREATER OF
S. OR Sz' Z,05 PSt
~4/3 Sh : 29'30 PSI.
700
740
27.38
nI,
A
MOMENTS {FT-LBJ
AND FDRCESILBl
ACTING ON RESTRAINTS
X WIND {CALC. NO. 5.151
'Z WIND (CALC.NQ 5.16)
WEIGHT {CALC. NO.5.14l
POINT
M.
M
M,
r.
r
r
{
{
e
I
4
+26,000.
-1/82.5.
-1Z750.
-/9.300.
+59875,. - 7.950.
- l800, + I,Boo.
-8440. 19 e Q
-30,625. -Za9So.
f 2.700. t
Z 110.
- '80 + /80.
!THE MWKELL0G3
col PIPING ORIGINAL
FLE !!'!.I~I' Y AND STRE";;,? ANALYSIS
.
DATA AND RESULTS
.,.7
o.
' .
••An
"'". -
o·
10.
M k
ALe
5-14
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
1-2
2-4
4-5
•
aDo
a
/8
-
b
e
228
100
228
/00
/00
/850
/8 50
/00
/4 50
0
0
75
300
13Z f>/
25 n
II
55 !if)
4 !if)
0
-
5b
q
eq
0
390
72 /5
0
u
u.
cu
v
v.
300
- 450
0
- 83 Z5
cv
SOb+CZq
cu.
0
0
ev.
5.u,1"C I V
a
0
Uoo+VH
4
••
1560
0
0
104 03
0
0
/04
889 85
0
0
0
0
22t,79 13
10'" 75
900 2Z (,79 13
/343 78 2S 858
5bbTC. U
f"
100
0
150
",'
228
a
a
a
5
54
a
7·8
W ~. <'1)
/00
/0
0
R
L
b·7
5'b
,.;
I Q
SHAPE
L
_.
Z
Pl.At<JE
MEM8fR
a
0
924
9/3
/18 5/
0
-
z
COORDINATE
DATA
0
1337 Z7
0
0
I Z9Z 55
/ 457 za
.3 /48 98
STRAIGHT
~
SHAPE AND
4
0
Q
51 .... oc
DIRECTION
b
0
0
0
L
cos oc
.ill ,
c
x
PLANE
co~
0
I'ORMUl.AS
I'OR
s ... R. .... a .. T "'l!... tUtl:l~
.. - ' kQL cos C\
w
"+ 'kOL':.rN""
Ww
y
~IN Loo;
• kQL3
nkQL""
~
$llo.lco.CO~<ll
I'OF<"'Ul."5
cu ... VEO
I'OR
CON5TANTS
+w,
+w_
""l!... aER
see TJlet.ES
5.25 t 5,2:8
229 Z
0
0
0
0
/M bl
- 59 84
0
/80 Db
-
/494
0
Il 5'" /b
0
a
..
35 7aO 80
5 als 3
.,
•••
7010lD
;55 ANAl <SIS
MEMBER DATA
•
- 837 8
0
10
3'" 'f7
0
0
589Z 98 .3 04a Z
0
0
0
0
Z 215 14
715 9b
/6900 79 IZ 951 5
Z 8/0 53 /3 9'"
THE M.W KELLOGG CO e. No r! :fr~;II'JlI ~},""~F 5
MEMBER NO Z-4
IN
PLANE
210 33
n
••
- 1/4 .5
0
0
1 01
10/7 75
SOZ8 14
/0 Sf!
8b
3.7 9Z
5.
- /lZ13""
/892
181 71
0
29
73~ Z3
8
2374 14
0
-
0
- 80
- ?:7
30 92
7.
9 •
- 259 9b
/04 5 - 713 52 0
0
0
0
0
0
If> 'f7
Z4 49
- 8M
- 34
25 %
1373
0
481
-
.
/053
/298
/4/ lJ8 -
- 1= u
/00
100
7
-
2:
-
5./4
CURVED MEMBER DATA
~
Z.28
270·
•
Q
1.00
R
/8.50
•
w y 1-280
F w·w L
F_ w
-/627!.
M_ w =-iF yw
kQR;" il4.43C..I2 GR'
M,w
0
kQR" 2'Z00.1, QR'
M zw o::+xF yw
M. w
0
STRAIGHT MEMBERS
FOR CURVED
.14159
Rii'L
SB./2
e
w·u~" + ~kQL"
+3W...
_cw,~
bw_
+c.w"
.w'""
+w u "
•
,
:=.eE T ....8LE5
w .. + ~ kOL co~ ...
5.2G t- .5.29
w'd'" - ~kaL SINO( (05 <:(
w"'~"+t .. Io:OL" co:.1:O(
Wb .. - 'kOL' Sn. C(
w't" .. =- ,,!,kQL" ~, ... ",eot;",
w'bb'" + ;.lkQL
M..
A"
M....... ·A u
M:w'A' l
A,_
A",
0
-"2 q 2{,3.3J
S. '24- &. ,5:27
W
~/'-tZ7.J. ,"0
e"
B __
Fyw·B.".
Mzw~A l
FyW 'Byy
Au
B"
M.w'Al<z
Mzw·A u
Bu
1<0,' 8,.
Bu
Mlw"B z•
e,v
B.
c,
JEw,aXY
M.... ·B"v
FYWMC y
en
e"
c._
- 5~..5O
/85.
,.,..83.25
Fyw·C. y -/354 '717.
Yw' B1!:v
c_,
+"10,3
"/026, 75
-/6708 9/9.
~GZq 2~3,3
-awb
oW b
SEE. TABLES
F
.bw...
·""'''b
• w .. b
A..
)tw·A. v
-51744.48
51 ... ·Cl
An
M. w
oW,
,~
-aw...
0
+w'.....
+658.%3. '18
"'658963.98
+w .....
+bwb
+IN't>~
+w'b~
+wb...
'tWbb
-~
,
.w,
OW.
0
, ..... l>
y....."
0
0
0
0
~
Wv
-Z80
LOAD CONSTANTS
R,
0
0
0
0
ow_
oW,
0
v· ...."
.cw"
v· w'" '1-/6168,4$4..
7 .48
"
..,..n.07/ (,39.
'"
T.
1 ..,.174838 9so.
V'WDl>
v· W... ..,.W.... -/84,5099A Tv
1-20/218830.
+wC,l
~<:.w...
•
0
oWbb
+W.ll>
r"W. "'.·w.. b "'17~/9..J.7J2
.. w""
. "'..... 6SllP6J.9A
0
v'
Mxw-S;w:t.
MlW·B zz
F"yw·C z
•
ANALY IS
rrHE MW.KELLOGGCO Plt~~Dr ~~t~NTSA~'bRS '~t SGHT
y·wl>.
·62~26HJ
T.
GW
c.A.1.c..,,-~
C"'l.CK~,t;:'~:.:lI
"'T~
-2 ~
F
M W
AL . NO. .I
1111
DESIGN OF PIPING SYSTEMS
182
MEMBER No4~S
IN Z
PLANE
COORDINATE
SHAPE AND
a
DIRECTION
b
~W·
,
PLANE
~
0
k
Q
0
L
1.00
cos '"
R
/8. so
2.28
0
~kQL3
~INl,",
kQR.)
1h.!<QL"
cos C\
kQR"
z
0
FOk .... U ... A!>
<'Or-l
•
'W.
see T",el.E:.s
' kQL cos 0<
5.25
-+ 'kOL',s,NQ<,
t 5.2'8
-
M,w =-:z.F yw
M,w
0
1/.30
'
"
Mz:w z;+xF yw
M,w
STRAIGHT MEMBERS
'5'.'58."
-cw,..
"CoW"
-'ow"
w'uv = ... ~kaL ..
F wow L
F w
/64
4<1JG.l2 ~'"
'7,(.l68.1. GR,'
.o.')"'...
'W,
l-l80
wy
•
·MOS?
R{>'L
CONSTANTS- ,,"OR CURVE.D
cu<>.vr.:o ...eM!>""
~05
~
S'N"CQ511l
F'ORMUl-A$ F'OR
"''';'''flo"R:)
"
51"l QC
k
Q
-11.37
c
MEMBER DATA
CURVED
~
x
STR""QHT
w.
w
y
MEMBER DATA
STRAIGHT
DATA
+,..,'""
+w .."
.-t~kaLco~oo<
w
Z
",'<1.'0. -
~kQL
.W,
SEE. TAe.\...ES
S,NOICO:;Ol,
5.2G .t- 5.29
+1(;/.68
+"',,'0
,5ee TA61.E5
5.24 It- 5,27
""'0",=- :h,kQLo4 ~'N"COSO<
•
.w.
.. - .zkQL' SIN <X
",,'bb" -
M. w
F w
Mzw
A"
M~ .... 'AH
Mzw'A~t
A"
A,<
8"
Mn".·A~y
Mzw'Ayz
FyI" '6 yy
A"
Au
B"
+Sf.9S8.~8
F vw 'B~y
+ 1,35 .51
Mzw'A zz +1.447451.
M..... ·A"z
8u
B"
- 55. So
F"..... Szv 1"/75. GOZ.
Fyw'C xy
+w'l>'"
+w'bb
+wb'"
"Wbb
, " ""y -280 LOAD CONSTANTS
+"",~
0
y'wl>
"'y'''''"
0
0
0
0
~
0
0
0
,W,
'",
0
"
y·w. Hy'W~ -""S:270.
+cwy .W"I> -5-';0.1'
y·cw. "','W"'b +/54 ()4~ T,
0
- ?(;~4o.'1.
nlb/.68
C,
- 55.50
+'7~ 7115.88 <WOI> .Wuy +w...... 294~.()9
Mzw·B zv -305020 . Fyw"C yy - 75,0054<14-, y'Wtt y·W... ...,.......... -8N 0(,5. T
'8,
8"
M"w'B"y
Bn
B"
C"
~ZW·BH
M~w·B~z
THE MWKEL.LOGGCO
+WI>.. .cw...
0
Y ·~w
0
, .....1> ..
FYW'CYI
WE GHT
O ...... f:
"Z577 783.
'8S 415/4/.
-
T,
~::~~KE-lr" N.~
~I~I'!." fL 0');!i~'!Y AND STRE~~ ANALySIS
LOAD CONSTANTS FOR
--
R.
0
0
of-
"'W a ... + 943.09
.. bwl>
..'". ..
83.25
C"
"'/!iSb./G
Mzw"B zx i<fi5.S2~ 499.
M~w' Box
-3/(;4,00
B"
0
+w·"... +-2.943.09
-.l.Wb
'~kQL" l5IN~a.
A..
-a.w...
-5$0./6
-S"io./6
+w'''b
w ....... + z+ k:QL .. co:.:to<
w.
0
+ low...
1Z·23-'5~
879 67~
F. M
.AL . NO,5,/-I
.\
MEMBER Nof·'
IN Z
PLANE
COORDINATE
DATA
STRAIGHT
/.00
~
/.00
7· c.o
7.J.1"
SIN 0<
SHAPE AND
a
-12.98
Q
DIRECTION
S
b
- /..3.73
L
~' x
PLANE
•
..
~kQL.\
nkQL
4
cos '"
~IN~o;:<.
/,3qtJl
'00
$'N"'Oll«
FO ...MU"' ... 5
'OR
~""':')
cU","v/iO
0\
see T ... eLE5
"' 'kQL)5'N"'uy '" + AkQL'"
W,
•
'0'
0
... "' ...
.. - .z.kQL'cos
"'Ofl ..... U1..A5
STR ... ,QM'"
w.
y
0
-/2,98
c
ME.MBER DATA
k
5.25
5.£:8
k
Q
•
R
R ~'L
F w
Mow
kQR~
QR'
M. w
kQR 4
QR'
""OR CURVE.D
'W.
+:l.w",
ow,
-bw y
.w
w.
w, "+~\;QLco~",
Z
.....~b .. - l!ikQL
S'Ne< COSe<
.W,
SEE. TAe.LE.~
5.20;;
.t- 5.29
, 41.96
.
M. w
"W ..
A"
Bw
M.w·A ..
"':w· A .:
;:~ .. '5 • .,.
Aw
A"
Ow
Myw'A~y
Mzw'Ay:
A"
M,w'A~:<
Bu
M.w·B..
B.,
'vI.w·B. y
B"
M~ ... ·B.z
.. ..
-
~Wt
-
~ w·e
~5Zf?_~jI
-J.w...
f-S44.64
+w'""
+ 45.7J
""'",.1
r590. .37
• "-5.JZ
-510.79
b
• 'owl>
"",'lOt
+wb.l.
"<Web
z .t
~'-~~~ -0 -
"-,'W"
"'y ....."
0
0
0
0
0
0
.W,
+W~
"~
., -280
0
F'vw • B: y
- 834 389.
0
y'Wy
-2291. '1'1
"'t.4IS . 0?-=-:: F.,.....C.~~
·C. y . . ;ffi7" :2 iii.
M::.w·B u
0
.C"' y
1--
LOAD CONSTANTS
- - - - 'CT-----
- 0-
-r .3 9 2.~_10
B~,.
t"'2761/,'If.
MEMBERS
-cw' 4
.c:.W y
.",' t~
B"
r IGI. ?A
Mzw'A u +4:454,18G.
A"
M %ow
-""'10
1-------
A..
0
Mzw:+xF yw
STRAIGHT
..
.-bw..
w·...... +l~kQL~ <;o~~o<
•
•
-2/28.
=-z.Fyw
• w'-'v
+",,'lIb
.w.
~,kOLl 5", ~
SEE TABLES
w.
5.24 ~ 5.27
W·IO ... '" - ,qkQL'" :>, ... "'co . . '"
l
w'bb '" + '~kQL A. SIN "",
Mzw +27.62/.44
,,",w
2/28
- 280
F w '" wyL
~
-.4G99
CONSTANTS
... " ... 6Cfl
,
~ "7/1
+. ;290
w,
CURVED MEMBER DATA
-55'
-.8192
+.57.3'
~
'" 41. qG
·w ..
- 1/ 749.
~WJ"
-~/O.19_
-~z
"'3.G08.648.
0
"y'CW, ... ·w4t -1'143021. 1T:l~" 44" laS. 8~
~39. 08G547.
!590, 37
Ow
"3&'507. 15 • .... bl> "Wu • ,
"'392.10
-r:56.382~
Fyw·C.,..,. -G7047.2/5. "'f'''~tK'wu • ,
-/~5..J04 . T
M~w·8.y "'10.830,.36
·""0.. -<;"'", 0
C"
B"
. ......
. .... ..
8"
~
l,lzw'B u
THE MW KELLOGG CO
0
T,
Xll!l~I!Y AND 51.REiS ANALYSIS
~~'t:~><t.t" H.5.
F.,. .... ·(yt
PI~I,:!-~/L
LOAD CONSTANTS FOR
..~·WD.
WE GHT
y'C"""
c ...... ro; 12·23- S3
FCi?M W
CAL~, '10
.t'
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
MEMBER No6 - 7
IN Z
PLA.NE
SHAPE AND
DIRECTION
COORDINATE
DATA
..
- 25.9G
- 27.'<
0
b
c
<EJl
STRAIGHT
.~
MEMBER DATA
W y
1- 280
k
2. 211
0'
F
Q
tu.... oc
Q
1.00
t>
L
c:.os (>(
~kQL~
~INI.~
'6.lb
M"w =-z,F vw
M .. w
0
F w
R ~: L
f, .50
kQR~ !J44fto/1.
R
<AI
QR J
..
wyL
-,]
Mr:w a:+xF w
"
PLANE
MEMBER DATA
c>:
lh.kOL'"
• - ".4'
0
7
CURVED
k
CONSTANTS
e
CURVED
~OR
Mz. w
~9JI.O
STRAIGHT MEMBERS
-cwu,
y
-bw" .
+w·...."
+w ... v
.. + ~ kaL"cQ~ «
w
,
w'.. o" - i4kQL
W ....... +;r.ok.QL
:;,EE TA8LES
SINe< cos 0\
w'bh= . 'hkQL
+5805./8
+bw...
.t- 5.29
r /928. 23
- 42/0./9-
+w·...b
CO~;rc>(
4
+w.b
w" = .!;IcOL ~ SIN <X
w·".. =- ilkQL 4 ~l>I«C06'"
,
5.2(i;
-awb
5.24 4- 5.27
+w'bl
SI"l"QI,
M ltw
M:r.w
Mn.'·A~~
Mzw"A~z
A"y
M~w'A~y
Ayz,
Mzw'A :
A,,:
Az.z.
F w
""3/.93/.09
a
F W -6
"fo 1GB. 88
Bu
X R.
a
-.379(;..80
B tv
°
+490.75
M:r.w~Au r5392 522 F.,.."xS:.
M.w"A,,:r.
-.Jt;.Qt:...22
01-2/08.
-186 2AO
-3'28.76
w., -280
"w\:lb
LOAD CONSTANTS
R.
o
o
o
0
-4-W y
-4-Wa
a
.,.223. "2
-"2
y·w..
~/4
'R;z
+CWy +w"b -42/0./4
y'''''v
0
Bu
.. /.1/0.73
Bh
4t10. 7S
Cvv
.,..j?7?Q.7" H"DO +WUy +W,u. +2/08.96
,,"/$ ~10 182 Fyw"C yy -/24.53.3 90/ y'Wbbf"y'''''u~i'''Y''''''''' -S90 S09 Tv
C;w.v
0
b"~'~'~B••+-----kMT,~w~·~B,,-.rr~47.'~8.~"'~~~tJ~J~8"'F~,w
=:C·<"'-'+'~/:':".3.'::':17.~7~
°
y'Cw Wy'Wn +//78 839 T"
.,.
Mzw·Bt.
Bt. t
M:.... ·S u
C..,z.
+Wt». ~c;w....
0
Fvw.C z
y·"'b.. y"W
0
-109,4542.30
AND S'"E~"S ANALY.b
THE MWKELlOGGCO "'"'N'.
"LOAD"L-X<H'UTY
CONstANTS FOR WEIGHT'
~.-lEMBER NO 7-8
IN
Z
PLANE
SHAPE ANa
DIRECTION
)1
4
-8.24
Q
b
- 34.94
L
c
0
.l.kQL~
5/2,21 cos ~
F'O ..... UL ...S ""OR
cU",""'f.O ...." ... Gen· CONSTANTS
'. • . i-kQLc.oso<
.. i'
t:0149
5Ee T ... e.LE~
5.25 t
I kOL'$IN ~
5.~8
,
w
R
w·o...
.. -t ~-kaL~c03 '"
5Ee. TA~LE5
,,- :.kOL 3 $1'" 0:
-
",'bb"
5.2cP
to .5.29
Q"
G~'
kQR·
M:. w !"i4294,82
""OR CuRVE.D
+w u
+w.
• w.
- 23.72
e STRAIGHT MEMBERS
+~WU
c ..... ,~
_bw"
-tcw y
+bw...
-4-w·.. b
+w·.. b
-awb
+w·b.l
.w.
seE TABLES
~kQL ... :>1"'0< co,; co
+ ·.:.kOL
RI-:L
kQR~
+w .....
+w",y
w·.. b" - iikOL 511>10-;(0:';<,\
w·a.... +l'~l(OL.. co~zO(
w.
•
Q
W'Uy" -t hkQL'"
,
5.24 4- 5.27
. ..
-t"'b...
51 .. '""
+ 24,t91-.82-
'
-aw..
- /95.4.5
~/. qlJ
+w·" ...
+
"''''' ..
T' 7~{' .80
+bwb
"wob
,. "'y - 2BO LOAD
CONSTANTS
B"
• w.
.w•
0
M~""Au
Mzw'A"z
Fyw·B~y
.wl> ~.w..
0
A"
A"
6.,
0
0
M,w'A>;y
Mzw·A y:
.Fyw ·6 yy
0
0
0
a
• w.
+w•
0
~,
y'w.
+cW y +Wdb +766.80
y'CW, W.·W"D -214704- T.
Bu
.4"w-a••
B"
M.w·e • .,.
8.,
M"w~B"t.
-2948.4
.,. 750. 71
Ml.w·A n +4854105 Fyw~8:.y -221.3 393
C yv
Bn
.,..591.21
-9521.74
M: w w8:" "'/4f:S09_1tr! Fyw'C~v .,.. 28.07J898
An
+ 199.80
B"
<,
.,. 750. 7/
Mz .... ·S zy +- JB.z3
8"
FywwC.,.v _IOJ.2:J7,754
Mzw·B u
Fyww(
B"
THE MWKELLOGGCO
., .?.r o14.fJ4
C <
a
a
~
0
"'y ....I> .. y"'"
0
PIPI,:!~ _.L~~I,.t~l! Y AND S I.~~ S A¥AlY IS
-1>2'47,354
"'42,3'8303
-187.82
,........ +52590 T
-cw"
LOAD CONSTANTS FOR WE GH
R.
- 23.72
y''''a +6642
+w"" -tWa ..
•• ....bl> y·w...
"Wbl>
",wb..
t
0
?~3
- 181·82
",
+w'bt
M,w
A"
M.w·A.. :.
F,w
.,.ez,§.78
-
M,w
A.,
A.,
w, - 280
F w '" w..,L
F w -2948.4
M. w =-z:F yw
M. w
0
Mz. w =+xF yw
CURVED MEM8ER DATA
k
«
-.9925
M W
AL . NO. S.I
51'H'<.COli« ,..1210
0
w
51N 0<
/.00
10.53 cos '" •. /21"1
/94.59 SIN""" "'.985/
• - 8.24 lh.kQL""
"
FORMULAS FO~
~T~""'Q ... "" .... r: .... D£<l:'lo
PLANE
y
STRAIGHT MEMBER DATA
«
k
/.00
- 97'
COORDINATE
DATA
F
.
CI-le.CKE.
o ..........--rz-
..
y
~84 94-G &~O
T,
F
MW
.Al . NO..5./
183
,
,s.
Jf ';104)
I-sa 5'16.741 41/, t.:5J -1-/6. (;J&
'18,712. 351 ~?4f. (,04 IH
-4015J41J4
.3 -201r4~J,ZI7
2lS589S& +71J'f,78448/
f5Jo,8g2~
Hz
_Fx
ONSTA.NTS
210 Jj
I. 000 00
_ I 0"0 If."
z -55,/57.7.
-
4
5
G
t'"
-+
Fy
zn 2.'1'"
tJ4
37 ~
I
z·Zf05588~
+/9S] 19 -19 9" 1/
z foSS 0/515 w 12 ~s Ii>
-
-
7205
8/
24q9G-
"3 4/9 64
,"""
3''''''
-;,
70"
9/297
I. +54- 71.5 39 - a 47813
- 1.00000 +
24' 1/
IF... - I 79~ J' .,. 4
II
.5 dS 130 80
fJ(,(, 0l.7
- G ''"84118
-S3~~~
-3 3 It'..~
-.3 /7 10
~ "29 79 ()
-
-!3,IS'o'S'1
.,.90 0921860
"S7'49f~
L 00 00
F'f .,./9 I&:'~ 78
4
F y~ • .,./98' 64,8
FYI" -+ ql8~. 0
:l': .+2fI5o.8
I~
-
F r", • .. t9/S0. 8
1.00
00
I
s
-
1.00000
I
-
1.00000
I
PIP~
THE MW. KELLOGG CO
CONVERSiON TO
cOCe!' Rl1L.ES
x
:
~~ ~
SE::' Ee. S'f
F",
"
F
Fz
M",
Sf::
F",sl
&_k.c
R' "E. n
&.Ee
CR
~
-
~
7
8.88 40./6 -
/ 7'J5.,3';; -
+ /98G4.78 ..
5
-
10.80 IG.84 -
/ 795.3(; -
17 5.3';; -
7.60
2'3.71
. .....
FORM
15:~~"11' N,.
6- EQUATIONS
8
POINT
fL(x'[;I\.ITY 6 STRESS ANAlY(,IS V·L.
5.1G
-
10.61
1 7'Xd(.,
9/.&'0'"
I';; !!JIG.40 .. 13 JI!L~O + 10
"o
E-I
CAlt~·5.flf'
Z
a
18.50
f-
a
18.50
+
17%.3';;
782.7.60 -
/8.50
+
18.50
J.OO
/795.3(,,-
/795.3';; -
/7%.3"
J09.2.0 -
B ~4t;.CO -
928G.()O
+F .z
-F
R"Eh
5'E
M
WHICIiE.Vf.R IS GREATER M
.Rs
+Fz'X
-F","z
R' -
~:(!-~c)
M
- 2/G 2(;4 - 184.133
53340 f 30234
of-
of 128 565
- 3435.9
+/41 G9i!
- 12407
-101 75.3
IS(/; 7/2 - 10/ 7.5-' - 5901
+ 19 049
o
- 332/4
..,./G(/;633 rl4481/
o
.., 2~ 970 + 4305<'3
39/1.5
//7293
558G
/7/ '5/
o
f"/5C 25/
T-544913
-f
~884
-M
IPE
Z
.OZ.ZBO
.3/.00
52(",4
/.12.
/ 3'JZ
~
'+5'
""45 '=51::
THE MWKELLOGG COl eleING':6~~\\'NTrNtN~T~lRsWtNALYSIS
lIH
c .......c.
.........
1-<.
0
D"T~
.. _
C.
FORM
I
.0Z Z80
1.00
I 3(;5
•I
o 5./
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
from the free end to point N. I:M,w is the sum of
the moments at the origin caused by I:F uw· F uw
and M ,ware obtained from Form·'W u· The calculation of the moments at the various points, and the
maximum stresses, are shown on Form F-L
Table 5.31
Shape Coefficients for z-Wind: General
Formulas for Straight Members
z
=
L'
±kQ - coello:
G
5.22 Wind Loading
The analysis of the reactions in piping systems due
to wind loading is similar to that for weight. The
wind is assumed to produce a load which is uniformly
distributed over members perpendicular to the wind
direction, and for other members, uniformly distributed over the projection of the member in a
L'
L -_ _
~y
x-plane
w' gg = ±kQ 24 cos 3 0:
when cos a is plus, lower
signs apply when cos a
is minul'l.
L-
~_y
x-plane
Oa
L'
,
W"V = HQz:j
= W'Ul'
Wu •
=
x
L'
+kQij sin a
W.
+ awu - bw,
'-
._z
=
W.
I
tV "b
L'
±kQ- cos:! a
= =F'
L'
=FkQ-sin 2 ac080:
W'bb =
±kQ- sin 3 a
Wbb
+w'bb + bWb
24
L'
24
Note: upper signs apply
G
kQL4.
L'
:::FkQ- sin 2 a
G
W'ba =
y_plane
x
g
Note: upper signs apply
L'
-kQ-cosa
G
Wu
+ bW
= +w'gb
Wgb
Wag = +w'O.:I - aw"
Table 5.30 Shape Coefficients for x-Wind: General
Formulas for Straight Members
%
185
when gin 0: is plus,
lower signs apply when
l'Iin IX is minus.
24 SlnaCOS a
'2
L'
'-
z
y-plane
w' aa = ±kQ 24 cos 3 a
L'
y
-kQ- cos IX
G
+"/ b + bWa
Wab
=
W,,<1
= +10'nO -
11
L'
+kQ6 sin 0:
aWa
Note: upper signs apply
when cos a is plus, lower
signs apply when cos a
is minus.
,
D'
'----+-x
W u.
= kQ-
x-plane
wuv
= W',.v
24
+ aw,. - bw•.
y
,
W ba =
_.-x
z-plane
24
J.., 4
,
'-
L4 Sill
. :2 aCOSO'
:::r.kQ -~
•
W bb
= ±kQ 24 sin 3 a
Wba
+w' b<'l - alL'b
WbQ
+w' btl + bWb
Note: upper signs a.pply
when sin a is plus,
lower signs apply when
sin a is minus.
plane perpendicular to the wind direction. The
equations given in Table 5.19 are applicable.
The formulas for the shape coefficients for wind
loading for straight members are given in Table 5.30
for the x-wind and in Table 5.31 for the z-wind. For
curved members, the formulas are listed in Tables
5.25, 5.28, 5.32, 5.33. 5.34 and 5.35 for both the
x-wind and the z-wind. The projected length is
denoted by L'. Thus Fzw = wz,L' and F: w = w:L'
for wind loading along the x- and z-axes respectively.
Forms IV, and IV, are used for the computation of the
load constants.
186
DESIGN OF PIPING SYSTEMS
Table 5.32 Shape CoefIicicnts for Uniform Loading: General Formulas for Circular Members
For z-'''ind Acting on I\Icmhcrs in the y-Planc
or x-W'ind Acting on Members in the z-Planc
Note: Signs prefixed to formulas below to be selected as follows:
Use upper signs when arc <1) lies in the I and/or II quadrant.
Use lower signs when arc <!l lies in the III and/or IV quadrant.
Split the arc into two members when 1> lies in I and IV or II and III quadmnts.
Counterclockwise Direction
<I>
±QkR' [ 4" C1 + 2 COS" a) +
0
sin 2 (a +<1»
.
.
8
- cos a smCa+<I» + ~ sm 2a
]I[
J
I
±QkR'l!iIcosCa+<I» - cosa]'1
10'bb
±QkR' (Hsin'Ca+<I» - sin'al - MsinCa+<I» - sinaj + co~a ['I> + sinCa+<I»cosCa+<I» - BinCa+<I»cosa j)
Clockwise Direction
,I>
sin 2Ca-<l»
lV, = 'f'QkR' [ 4" (I + 2cos'a) 8
+ cosasinCa-<I» - fsin2a
J
lV'.o = ±QkR'l!iIcosCa-<I» - cos a]'}
tv'" = ±QkR' {;\Isin'Ca-<I» - sin'a] - ![sinCa-<I» - sinal _ co~a [<I> - BinCa-<I»cosCa-<I» + BinCa-<l»cosal}
Tahle 5.33
Shape Coefficients for Uniform Loading: General Formulas for Circular I\fcmbcrs
For z-\Vind Acting on :Mcmhcrs in the x-Plane
or x- 'Vind Aeting on :Memhcrs in the y-Plane
Note: Signs prefixed to formulas below to be selected as follows:
Use upper signs when arc 1l lies in the I and/or IV quadrant.
Use lower signs when arc (I> lies in the II and/or III quadrant.
Split the arc into two members when 1) lies in the I and II or III and IV quadrants.
]I[
]I
Counterclockwise Direction
QkR
tOa='=F,'
o_.J
' [<I>
C
. 0)
sin 2(a+<I».
C')
3'
4"1+2sIn~a8
+smacosa+'il-sSIllkU
I
w'o' = ±QkR'IUsinCa+<I» - sina]'1
lV'"
±QkR' (Mcos'Ca+<I» - cos' aj - ![cosCa+'I» - cos aj -
Si~ a [<I> - sinCa+<I»cosCa+<I» + sin a cosCa+<I»[)
Clockwise Direction
tL'1l ='
. 2a) + sin2Ca-<l».
( ' ) + "if3Sill
' 2a
±QkR
. ' [<I>
4" CI +2sm
8
-SIllacosa-'l>
J
w',. = ±QkR'l!IsinCa-<I» - sina]'1
10'. . =
±QkR' (i[cos'Ca-<I» - cos'aj- !IcosCa-<I» - cosaj + si~a [<I> + sinCa-<l»cosCa-<I» - sinacosCa-<l»J}
FLEXIBILITY ANALYSIS BY TIlE GENERAL ANALYTICAL METHOD
The pipe configuration previously given in Sample
Calculation 5.14 is calculated for a wind load of
60lb/ft in both the x- and z-directions, and shown
as Sample Calculations 5.15 and 5.16 rcspectively.
A stop in the z- direction at point 4 is included. The
load constant for this stop equation is calculatcd on
Form S,. Suffieient explanation for thc computation is given on the form.
The moments at any point N are computed in
accordance with the following general formulas:
For x-wind:
Af"N= L;M,+ L;M,w+ (L;F x+ L;Fxw)YN- L;F,;rN
where L;F., L;Fv are the sums of thc rcacting forces
at the free end (obtained from the equation sheet),
plus any restraint in the line betwecn the free end
and point N. The sum of the moment at the free
end rcferred to the origin (obtaincd from the equation sheet), and the moment at the origin caused by
any restraint is L;Mx. L;Fxw is the sum of the wind
load from the free end to point N. L;M,w is the sum
of the moments at the origin due to EF zw' F:r;w and
M,w are obtained from Form lVx. The calculation
is shown on Form F -1. Since no restraint is included
in thc cxample, L;M" L;F., and L;Fv are the reactions from the equation shect.
For z.-wind:
M'xN = L;M x + L;Mxw - (L;F x + L;Fxw)YN
M'vN = L;Mv + L;Mvw + (L;F, + L;F,w)XN
0
Table 5.34 Shape Coefficients for Uniform Loading: Formulas for 90° and 180 Circular l\lembers
For z-Vlind Acting on l\fcmbers in the y-Planc
or x-Wind Acting on l\lembcrs in the z-Plane
w,
W'bo
W'bb
+0.1781OkQR'
-0.16667kQR'
-0.04793kQR'
a= 9O'
D
D
-0.39270kQR'
+0.16667kQR'
+0.33333kQR'
a= 9O'
~
+0.39270kQR'
-0.16667kQR'
+0.33333kQR'
a = 18O'
~
-0.1781OkQR'
+0.16667kQR'
-0.04793kQR'
180 0
tJ
tJ
-0.17810kQR'
-0.16667kQR'
-0.04793kQR'
+0.39270kQR'
+0.16667kQR'
+0.33333kQR'
-0.39270kQR'
-0.16667kQR'
+0.33333kQR'
+0.17810kQR'
+0.16667kQR'
-0.04793kQR'
+2.35619kQR'
-1.33333kQR'
+ 1.57080kQR'
-2.35619kQR'
+ 1.33333kQR'
+ 1.57080kQR'
-2.35619kQR'
-1.33333kQR'
+ 1.57080kQR'
+2.35619kQR'
+1.33333kQR'
+ 1.57080kQR'
E:J
L5
+0.42920kQR'
-0.33333kQR'
+0.57080kQR'
+0.42920kQR'
+0.33333kQR'
+0.57080kQR'
Q
Q
-0.42920kQR'
-0.33333kQR'
+0.57080kQR'
-0.42920kQR'
+0.33333kQR'
+0.57080kQR'
Shape
a=
0:: =
0'
a = 27O'
a = 27O'
a=
0'
a=
0'
a = 18O'
a = 18O'
a=
0'
a= 9O'
a = 27O'
a = 27O'
a= 9O'
q
q
D)
D)
((]
((]
187
DESIGN OF PIPING SYSTEMS
188
Table 5.35 Shape Coefficients for Uniform Loading: Formulas for 90° and umo Circular Members
For z-Wind Acting on .Members in the x-Plane
or x-Wind Acting on l\lcmbcrs in the y-PIanc
Shape
a~
0°
a~
gOo
a~
90°
a = 180 0
a = 180 0
x = 270 0
Wo
woo
woo
-0.39270kQR'
+0.16667kQR'
+0.33333kQR'
+0.17810kQR'
-0.16667kQR'
-0.0479:1kQR'
t'I
t'I
+0.17810kQR'
+0.16667kQR'
-0.04793kQR'
-0.39270kQR'
-0.16667kQR'
+0.33333kQR'
tJ
tJ
+0.39270kQR'
+0.16667kQR'
+0.33333kQR'
-0.1781OkQR'
-0.16667kQR'
-0.04793kQR'
D
D
a~
0°
q
q
a=
0°
D)
a = 270 0
-0.1781OkQR'
+0.16667kQR'
-0.04793kQR'
+0.39270kQR'
-0.16667kQR'
+0.33333kQR'
-0,42920kQR'
+0.33333kQR'
+0.57080kQR'
a = 180'
DJ
-0,42920kQR'
- 0.33333kQR'
+0.57080kQR'
a = 1800
((J
+0,42920kQR'
+0.33333kQR'
+0.570S0kQR'
a~
0'
(0
+0,42920kQR'
-0.33333kQR'
+0.57080kQR'
a~
90'
Q
+2.35619kQR'
+ 1.33333kQR'
+ 1.570S0kQR'
a ~ 270 0
t5,
-2.35619kQR'
-1.33333kQR'
+ 1.57080kQR'
a = 270 0
Q
Q
-2.35619kQR'
+ 1.33333kQR'
+ 1.570S0kQR'
+2.35619kQR'
-1.33333kQR'
+ 1.570S0kQR'
a~
90 0
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
MEMBER No./~2
'"
SHAPE AND
DIRECTION
DATA
&
b
11 t
0
y
I
PLANE
MEMBER DATA
STRAl4HT
COORDINATE
PLANE
Z
k
I
Q
L
-tkQL~
I
.3.00
4.50
h:kQL4
3.38
.~
""8.50
I. !JO
0
- 1.50
0
:;T~f,'h"'...';."'' '.:..:::'.'.. t'l!:R''''
w. .,-tkQL <06<'<
.. '"t-kQL SIN 0<.
w'"" ...... kQL
<1: ............ :,> FOR
CURveD ... ~ ... eCR:'>
· ..
SEe
+ qo
~
:iIN 0(
+ 1
0
51 .... ~<>(
+ I
C::O"Ee-:
0
1J,r 0< (0""
0
cos '"
CURVED MEMBER DATA
w.
k
Q
R
kQR
F •• .. W.L
F .w 1+180
~
.j>
5.'2'5 t5.Z8
L'·LRO~l" PL"'~~"
".
..""W.. .
Y.y
~bw",
M w
M
A.
_A.
A••
M
_A
,
A
M~·A J
B•
M
-6 •
...c.w"
-a.w.
"bw..
Y••
... w·.....
+W' .. b
-aw b
-4.50
YW,
~
'f"w· b ..
+wt>...
y"
B..
0
0
0
F"w· B.."
0
6y<
+w.
.w.
0
An
M2W ·A z
F"w· B....
B..
""..'w" "",,'' '...
Au
0
Y.,
M w.An
F" .... B t
W.'W...
0
"".'WI>
B,
lOu
'f"W".
+w....
"~
0
yw.
0
F.w·C••
C.
-c:;w...
0
"~
M w"e
Mtw·B.
F..w·C.
...• ..:W..
0
...• • •bf,
lOu
~(w.,.
'f"W~t>
0
W.'W•• ""•• W
w...(W
....."N>
W~'W
+B~.'2S
qqs.
T
I
'"' ..'" -
I
+
T
MEMBER DATA
STRAIGHT MEMBER DATA
k
~
&
DIRECTION
b
°
°0
Q
L
..LkQL'
cos. ~
SIN'~
k
Q
R
kQR
+ Q.25
nkQL"
co:."""
k:QR
1",0«0)0<
L'.~"o;:'" PL"'::'~"'" " 0
If'
.fi,
y
Z
0
.
2.28
•
4
2G7..... OR 4
<::UR"~O
w''''b= +~kQL" 51"'01<::0)'"
w·........ :!:h:l.:QL" co:; oc
W, ... :; ~kQL~SIl-l toe.
y
•
w·b..- :;J....kQL'I &'''' ,",cos'"
w'bb - :!nl<: QL 5lN ""
5:~3
OEO
T"e.I.l!~
;'.~2
..
t 5.s5
YWb
4- 5.'4-
+6tQ5.Qa
+ 2'220.00
...
M w
M. w
A.
Au
B..
Mzw·A."t
F~w'B""
0
A••
6 ••
Y ••
-A.
M
A••
M
A
·A
Mtw·A. z
•
Au
Mvw'A J
M w.An
BYo
B,
M
-8 •
B
M
M2W·6~.
6,
w.e
M.w·B.
B •
Bu
M w' •
Mtw~8u
THE MW KELLOG:; CO
-2 535.00
F. w
-61 605.
- 4.50
... Q2 408.
-5 .50
+ ff3q Gq
B..
F..... ·B~
-q
•
O.
e.
q.OO
+ 1'1 980.
+ A3.25
F..... ·C"
+184815.
lOu
F.. w·C...
e..
F"w'C,,~
YW.
f8.50)
$ STRAIGHT MEMBERS
-c.w",
+CW y
-a.w"
.. w ......
.,w....
-.lwb
+w· b ..
4Wb...
y~
0
..
0
'
0
.ow,
0
.... 8Cf02/.83
+-8q,02/.8S
, " w .. +GO
~ewy
w.·elY
°
+Wbb
·'52 442.51
·/52 0442.51
LOAD
CONSTANTS
.. w'bl:>
0
o. I
0
0
0
.w, +"/C=Js.qa
R
w.·..
0
b .... 37115 R,
+w... +w.... ~ ~ +15 "l-42.Sf
W,,'W... W.'W w....\>I> ....' lJ46 55"0. T l<
0
.c.w...
"~ 1"81102/.83
W.oCW..
... ~ +5.3413/0. T
0
...
..•
W.·W...
GO
M•• ..... z.F"""
M••
0
M. w "-)iF" ...
M. w I:OS35.00
........ b
w•• W .,. W.'W..
- 4.50
,.
Y
F. w _WIlL
F. w +Z220.00
.. w' .. 1>
0
F. w ·8 yo
+ 3.00
w.
+270
3./4/SQ
0.'
OR .... UL ... ll
R
.. f. .. tU:Rll
~TR ... ~":.uT" ~/~B"'~IUI
tkQL co(><'(
w.
Wy
-+ kQl.: :;INe<
w'"y = + I<QL'
~
.j>
1.00
IA.Sf) RpoL
144,%.11
C.ONSTANTS FOR CURVED
Y• •
+lJ.w....
SE' TAe-I.f!liio
5.25 (. 5.~8
Y. y
-bw"
..w".
~~~ :.c;.~~w,:.~e:~'t'c:~'i~. . . ~'i~ u.:,':.~~ .,,,-.. :> w"... ~, .... Oil
+w ..."
~bw ..
seo TAtll.~:lo
W.
'
-t~ kQL cos '"
PLANE
+ 49QS.
F· RM W
0 5.IS
C ... LC.
COORDINATE
DATA
0
CURVED
- 270
• Goa
R.
O.I!.
•+ ,"oB
T. I
T
0
PIPING_"L~~.I."lLi Y AND S'RE~~. ANALY IS
LOAD CONSTANTS FOR X-WIND
::;IN 0<
CONSTANTS
R
- 4.50
- 2,.70
MEMBER No.2·4
PLANE
Z
'"SHAPE AND
•
LOAD
+'o
O.
6.
F ..... ·C ...
~.
0
M zw ·6z •
Mtw·B n
•
THE MW KELLOG:; CO
W"
• 75
• 10.3. 38
13
.. w·bb
+Wbb
• 83.2.5
,"
."
83.2S
0
Au
6"
""w.....
.ow,
+W"'b
Mzw·A."z
M w.
L
$ STRAIGHT MEMBERS
-cw",
B
B•
0
+w"'-v
Sec TAel..e:lo
- ±:ZkQL cos '"
5.~3 t 5.35
w .... b= ;~.kQL" $11>10(0$"
w'..... -::' kOLA. cos <X
w, -+'lOkQL~St"'Zox
OEO 'Ae.l..e:~
5.~2 4- 5 •.,4W·b.. ":;' kQL eo,,,, ,",cos",
w'bb'" :thl<: QL" SIN ""
M. w
F. w
W.
•
•
...w'ILV
~o':. :'0.t~~w.:.'.::e:~~:;:;:.'r.~·'w~';.~".:,':.t~ ....... & w .. «:o. " ....... 0"
y
.
00. 4
Io;QR"
1+ ,"0
My. .. +iF. w
M
0
M•• ... -y F.. w
M. w + 270
R 'P.L
OR'
CONSTANTS FOR CURVED
TAOoL.es.
189
+W"b
0
W~''''''
0
piPING _"L~~I."lLi Y AND,';;TRE~~. ANALY IS
LOAD CONSTANTS FOR X- WIND
T
CALc.
"'
K<
. ,.
- -
.... 300 164.
I+Q25B Q38.
1+6'-65 B18.
I
F· RM W
05.15
\
DESIGN OF PIPING SYSTEMS
190
MEMBER No4~5
PLANE
Z
DATA
•
DIRECTION
b
4
,\}V
PLANE
.
0
"Q
0
L
0
~ kQL~
5' .... ·""
- 5. 305
0
hkQL'"
co~z'"
"''''''l.A;>
... c.-...."
\1:: ....
'0.."""<:0 "'C"'BCR,.
-tI<QL co::. ...
$"
w.
~
w.
= .f.tkQL "n""
"Q
CONSTANTS
ow.
TAe.. es,
5.25 (.5.28
..w~
.
·
l
S.~3
:;:fil<.QL "''''''''0:'''
w""b
....
:!'IeQL4 co:.",
.... "'
w, ..... ~kQL.)5IN~o:.
w'b.,"':;: kOL 4 :!I'N''''COSD<
... 'bt>" :."1.. < QL s'''' '"
,
M,w
...""w.. .
-cw""
-bw..
+CW y
..w .....
........... '0. .. ,. .... Me .. !> ..... ,,,.
........ "L"''' ... 0......................."'.
w. - :t.:. kQL"c.05
see TAeLe~
y
."
..w""
ow,
.w,
TABl.e~
5.~2
4- S.~4
-aw..
.. bw...
t 5.35
+ 461. q"
.. w' .. o
.. w·....
"w... t>
.. w ....
-awt>
0
..bwt>
0
.. w't> ...
... 8.3'1'1. Zq
'8 qq.2.q
.. w·bb
-124'1.88
24'1.88
"Wb<l.
... 3,377./6
A.
A ..
0
0
0
M..... A~
t,A,w·A~.
F.w·B~.
0
0
0
An
A ..
M,w· A ".
B..
• w.
ow•
0
F ..... ·B y •
w•• w.
w.'w~
,
A
M ..... A
Aa
,
M.w·A n
B •
w.8 y •
M
+
135,51
+
451 (,39.
B"
+ 155~. 16
Mzw·B p
+ 5'255,401.
B ,
B,
- «.<0
M .... 6
M'w· 8 •
B ,
Bn
M...... 8 I
M tw ·8 u
•
0
Y
/3.73
':>T .... 'C. .. T
w.
,
<
"Q
.
MEMBER
STRAl14HT
1.00
~I'" tx
'.00
L
7."0
3
~ kQL 13.IG
hkQL" 13Q.OI
~
~O_."
0
w••(", W._"'. .
0
,
·
Se.
·
· :!;.I.:
::;:. IeOL 4 ~,,,'''''OSl<
"obl> ..
QL >ON
... ·P..
•
SEe
ow.
4- S,~4
M,w
Au
F,w
Bn
M,... ·A.
M'W·AH
F.w·B~~
'"
M.w·A"t
B..
N·
A
A
,
A"
+,,,.
<P
.c::w,.
B..
.. w·....
+W... t>
.. w ....
-awo
+ 49./0
+ 31'3.80
...
.Wb<l.
0
0
0
0
0
M ... ·S ••
M~w·B,. +
F.w·C .. +8861 258.
M..... B.
M..... ·B u
Bn
THE MW KELLOGG CO
F.w·C..
C"
F"w·'.t
- 8S'-
S~9.
-597.73
0
+W,04
M. w ·6,
+Whb
,~
+""...
-22QI.49
+
to "qo. 82-
0
W,'W"
C.
ow'bb
y~
105.88
."
..bwb
0
ow.
LOAD CONSTANTS
R
0
'., + 49.10
w,'W" ... 2946.
........ - 591·7J
W.oCWy
"'
0
"0
R.
W,'''OD
. . ,w.
w.·,'"
w.
76,41
0
•• w.,w
0
-cw",
w~.w
- G74./4
+637.32
53·50
•
+w·b ...
,w.
' w,
......w. w.'w"
... 1415.08
-a.wol
OW'''b
e..... 528 957.
3q 2./0
+20123'3.
G.B
-cw....
C..
12"2,573.
,
.lJ,w....
F ......
+
N
·bw..
M w.An +821 733.
B ..
i"M·/5.0B
B,
OR'
QR'
... l'"
.... 0>.1 " p ........ t
M w· A ,
B
B ,
M o·B
B ,
R;P~ L
+I>wol
F...... 8 ...
ZB
w,
·60
F. w "W.L
F. w
~373.80
M. w ..... :iF"",
M. w
0
M,w =-9 F"w
M. w 3132.27
DATA
FOR CURVED ~ STRAIGHT MEMBERS
t 5.35
• 513'1.27
F· RM W
NO ./5
oW ...
A.
M
.
"0
ow,
TAe.Le~
~.~2
T
+W'u.
TA6Le=-
5.~3
+ 3",Q 522.
., .
'"' . .
•
-.8JQ2.
"'/q 623 G12
c",-c
CURVED MEMBER
R
9".513'5' ... ·'"
9".'-'10 "OR
GO~:C<
+.3Z.QO I<QR·
l' .... O~l'"
,>.I .. co~"
M,.
A ..
...~
_
74.993 T,
+839 .29
...... +503957. T
0
+Wolt>
-.4699
.. hkQL~ ~"''''':;O~ '"
woo - .!:...·IeOL 4 co:. '"
:;tkQL.)f,u"l«
w,
w·,.\.>
ow...
"cw..
...... <. .............
I<QL'>':'O$
W~''''DD
... I 476,008
-1249.88
- 55"
(u .. ""o ... ~ ... 13C<l'" CONSTANTS
ow.
sc:e TAe. ... es
5.25 t 5.28
ow,
:. -t IeQL ':;06 ..
t>~ ... " .
•
ow, ..
-N~ OCW.
" ... t"lo:QL'",''''''
w •• " .. r::,l<QL 4
'0" ." ...... 0 ........ " .0 .................... l v.... l" ~,"" .. " ww" .. f,O' . . . . . . . .
±.:..
, W. ·
+4GI.9(;
W"W~
Z77/8. R,
DATA
<"OOl ... v .... :. "0"
~c'"
... .: ... IH:: ....
w,
.:;"" .......
R
0
ow,
c::o!> <><
0
0" ..........
PLANE
+ 52. 997.
F.w·C~t
- 13.73
,
... . ....
-
LOAD CON5TANTS
R.
PIPING ~'LEX.IJ'lLITY AND STRE"? ANALYSIS
LOAD CONSTANTS FOR X-WIND
COORDINATE
5~
';:
->S
y~
0
+W.
+'~"ib. 16
0
F~"".Bt> .. qqO 651. w~·w",
+22 (,88.13
""
C..
"'''''
F.w·C •• ... 4·4.4'· 264. "".'''''." -N•• W
·cw",
0
C.
... A3.25
C"
z PLANE
DATA
'"SHAPE
AND
• -12.'M
b
DIRECTION
.~
B"
- IA7.432.. F.w·C.
THE MW KELLOGG CO
MEMBER t-105-b
+"36.60
"Wbb
,. w." 60
F.w
B..
M~""
M w.A
10.COI
FOR CURVED ~ STRAIGHT MEM6ERS
w· ... '" .... l<QL"
'-0" .0 ..... 0 ..... ...,0 ,O"' ... V~ .......... H
,o~
F. w .. W.L
F...... I "G~b
M. w ..... iF"w
M wlo
M,. ....... y F",w
M z;w tt.3377.tG
R
'''''''O~'''
"04"'''",,,,, "0"
+305.Glo87
<P
1.00
• c;o
w,
DATA
18·50 Rq,oL
kQR'" 144%./2 GR'
\o:OR" 12""10(,8./ QR'
L' ' ....
H
~~ .. "
.. 0
.. 0 . . . P .........
ox
y
5TR .. ' .... T
CURVED MEMBER
2..1:8
•
:'1'" <><
co~
<
,
5
.
MEMBER DATA
STRAIGHT
COORDINATE
'"SHAPE AND
oWo~
- JS aH T, ... IGp87, Qro7.
~~
0
..... w~t
0
PIPING ,'L ,~.IJ'!L1TY AND STRE~?, ANALY IS
LOAD CONSTANTS FOR X- WIND
+ 13,sQ 63G.
+~90.dZ
.... .. 11449
O"".t~
R.
.
.:. ... '-.:.
"' "'
T
.. lIq7 253.
T
".
FORMW
NO 5./S
C
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
.... EMBER NO G·7
PLANE
'N Z
•
SHAPE AND
OIFlEC TION
~)
7
PLANE
"'Ill
~
,. 00
if>
:>11'.1 0<
Q
R
0
0
5''''''''
kQR~
;;
~ kQL'
/8.50 Rp-L
/4.436.12 OR'
- 23.18
hkQL"
eo~Zo(
kOR"
2{,7,068.(
0
It,I""O~ ..
ro ..
.. O"l ... U ..... " >'0"
<:'-''''''''0 ... t. ... Bc .. ~ C.ONSTANTS
.... 1: ... 6" .. "
SEe
•w.
.w.
TAe.I..E~
5.25 (.5.28
<>LU" , .. 0 ....... "" . . . . . """l.It
~7 kQL"c05
see
0<
TABLe::!>
5.?13
;l.. k.QL'" 5'''''''0'' '"
t 5.35
Y
w .. b -
•
w" .. "" !' kQL~ co~ or.
w.
+'~kQL S""zo;r.
>e' T .... Bt.e~
5.~2 4- 5.~4W b... " ... : kQL 4 e.'N·Q\~05"
.
M._
A.,
M..... A.
A..
Mzw·A.~
An
A..
M w.A
Mzw'Av~
A
An
,
M,w .. -y F"w
M ~w ttl1.97f:,BZ
QR 4-
_cw..
~bw".
+cw y
+Wv,v
•w.
+bw ...
_3.W•
"w'o."
... w·...
• w .. 1>
+w.
.. 8'15.33
'tw....
_.1.w"
'tw· b ...
+ 772.20
F...... B••
8 ..
F."".6 ...
.bWb ·2457G.8/
.w'bb .. 6751.18
.Wbb -/7825 . .33
LOAD CONSTANTS
"'232,42.. 77
- 14Q7/.78
... 8270.99
,0<
yO<
'0<
0
0
0
w. +60
0
0
R.
0
0
w ••"""
+ 13/0. 73
B ..
"".'w..
0
0
0
w•• w~ + 5'3 720. R,
'12.5. F.w·B t • +101, 14fO. w.·w..
+1310.13
+ 251(;,3. II
+w.." .w..... ow... -/78ZS.J3
C ..
M:w·B z• .. 'l.~ 5(,2 757. F.w·C •• + tq 43O,QS4. w•• w"_ w.'W... w•• w~~ _IOGQ,52o T.
-c.w.. 0
.w~ ..
C,
+4CfO.7s
+8270. 'Ill
- 3_"28. 7CB"
Mzw ·6z ...8822 /2.4. F.w·C.. - 2.802- 128. ",.o<:w" 0
+4QCO 25'i, T
""'"w·6.
M.w·B n
..........
Bn
C..
eU"
PLANE
x
0
;;
Z
MEMBER DATA
STRA1~HT
DATA
•
k
Q
L
- B.N
-34.H
0
~ kQL~
- '4.44
"hkQL·
1.00
1.00
10.53
/94.59
5/2.27
0
<
~TA: .... :h""HUT" '::'1l:~B~r:<l'lIo
w. '" -~ kQL C06'"
Wy - + kQL $11-1'"
w·.... _ of" kQL
F"O<:t .... UL ... ~ FOR
cuotVIl:O ... r:"'Br:R!!o
s.e TA&l.E&.
5.25 (,5.Z8
A.
Mw-A"
~IN
ex
S,N~'"
...
kQR"
L··~"o':::·... P .. ,,::,~'"
C.ONSTANTS "OR
• w.
+w.
GR'
_ 0
10.4-5
~c:w
-bw..
..cw"
...
..W ....
+w.
-+bw...
-aw&
-+w·.. b
.. w ......
"W...b
'Wo
+ ,Q'.G9
-tw .....
+j5,Q.53
-0I."'t>
- G 1.52
+1518·01
-tw· b ...
..w\:>,;.
... 21 907.
" .w
Bn
t
627.
.0<
yo<
, 0<
0
0
0
0
0
0
M w'A
Ay<
An
B..
M w"B
• GO
.. W.L·
"627
" .w
M yw ..... :i.F"w
M w
0
M,w "-9 F"w
M. w +2.1,'01
..e.w....
... w..
B
.1
CURVED ~ STRAIGHi" MEMBERS
0
B..
w,
".w
OR
+W.
M w·e •
NO
if>
By.
'A •
FORM W
-
RT-L
F.w·B.."
M
I
~
Mzw·A"t
A y,
Mzw 'A v1O
A,y
.,
c",-c
HC:CK£.O
Q
:!::hk QL'" $11'1 or;
M,_
Au
T
0
k
R
-./2/9
.'1851 kQR'"
eo~E
1+ (;5/6 255.
0
CURVED MEMBER DATA
-.99Z5
eos ...
,+ 41 '124 I'll.
-+w·.."
Se. TA(~L~=W. "" "±tkQL cos or;
S.~3 t 5.35
W:"b- +~.kQL· $lN"'CO~ '"
w·....... - :t:l:;;:k:aL~ cO$. or.
e.El! "ABl.e~
W. .. :;:tkQL Sl~ Zo(.
5.~2 4- 5 •.,4w·b ... - +-'-kQL SON «<OS"
....·bb -
M W
_en-
~
'>10«0)"
~~~ :.o~ ... ~w~~e.:~~';;~"""""'W':."l~ ".:::..~"s S'CoN;> W"",H 6 ...... OR
y
"W"' b
W.'c."'. W,'W....
F.w'C ••
COORDINATE
b
.~w"
PIPING JL ''';t):l~L1 Y AND STRE,S ANALY IS
LOAD CONSTANTS FOR X- WINO
THE MW KELLOffiCO
SHAPE AND
DIRECTION
+4;101,79/.
1'"~03.5
B"
MEMBER NO. 7·8
'N Z . PLANE
I
.w. .w.
R I
.w. + a'ls.n
+w.
M w.A ,
B
M ... ·8 •
B ,
M .... 6
B ,
Mzw·A n
0
w
"w·..".
F.w
B••
+IGe.BB
aT
M
+4W ......
...... \:>..
+ /18.16.82,
W"L
".w .. +772.20
.73304
8"
L·<,"g~'.. P~.. ;'~"'Go .. C
/2.87
"OR CURVED ~ STRAIClHT MEMBERS
;.Ic QL 5'''' ""-
M,w
w. 1+ 60
".w
M yw
iF"...,
DATA
G05 0<
'" +tkQL ~1"0l
.. I>b· :.
Z.28
L
:::-tkQL.:ot>'"
:
k
Q
Z
w.
CURVED MEMBER
- 2.7. 4S
W,~ '" "}.;.l<QL'"
ro" .0 .... 0 ........0 .Q ..... "L~ .....n t v ......... '" .. " _ .. .... G..... 0"
<:o~ _
DATA
~
- 25. 'Ho
'1M" .... "
w.
k
b
!IT<l .. 'C.HT
. w.
MEMBER
STRAIGHT
COORDINATE
DATA
191
w. + 60.
.ow.
-c;;~97."5
.w·bb + .-s"oo.85
+Wbb -GIQG.Aa
LOAD CONSTANTS
R,
0
R
"'.'W" w.·w...
0
+/9/.GQ
+ Q'.21
0
w,'W o ~II
F~W·e1 +34
0/. R. +47(;2 971.
I. W,·W..
~
Bu
C ••
7.
t 2063.90 +w.... .. w...... .W~b -blq(;.BO
M:..... Bz" +13.0B3.079 F.w-C•• +2b 3740GS. w.'W"_ "'.'W "'.'''0.. ~7180.r:l T l< 1+ 3~ oB5 33G.
-cw... 0
.w~ +1518·01
C,
B.
-95Z/.7#
+ 750.71
.....w". + 91 OBI. T + 10566 75~.
0
M..... ·Bz +113:445804. F.w·e. - 5.970 lJI. w.o<:wu
F.w~By.
+/99.BO
Mzw·A n
377 0'9.
•
.w.
",
w
B •
B..
Co<
Myw ·8 ..
M.w·B n
F"w·C...
THE MW KELLOQ; CO
..(W"
""'... b
w.·<W w.'W...
0
T
0
PIPING J~~~!P~LITY AND STRE'~. ANALY IS
LOAD CONSTANTS "OR X- WIND
c",-c.
'"', "' - -
F· RM W
NO ./
,
f~ -1J.76~!l$3. -"9J~ ~/an"
-t~.P.jl~.
- z£980930
nj'M.3H. -/l.Ctl'13~ r2~OO,$S5
2
~ -4$22 3 _U
.,.Z~~. ~S#l9SC
- U:.JZf. 030
4
5
•
M.
,
I
Fx
,
,
"00 -, 1 .0 ,., ,, , liS• -I
2'0
- I. 000
-5" o29~
,
~
3
- 1.000 00,
-2
X 1(//1010
I~
F xw (1lJr.)~""'80''''O
;;,
0
- Zlo~. 5G
0
- 2703.04
0
0
-S7 054
1
-/3 oe2 ~
"0
"//Z 1698 700
V/,
-2 0" So
12<
"
1/
n.-
~,
," 17M....
.u
0] 0'
1.00 00
-
-~~~
. :1:7 3~~::¥.
135 73 80
""
'';
/8J"
•
~
= -4809.Go
:f.
"125
.... o.
10. S6 ~
IZ "0
1
"
I> ;S4 ,'<,
,c"
ONSTANTS
Fy
22 1
-
1.00 00
5
>
- ..
000 00
•
>
THE MW KEllOGG CO
To
(CNVERSIDN
CODE
[~. ,
•y
'>
F.
F
F,
S,- Et:: • 5'£
Sf
"
U
P':::;l
~c _ Ec
.C
".
R<
E<
"'Eh
a STRESS ANALTSIS C"I.-.
6~EQUAT10NS
- • 8.88 ,
POIt\'T
~,,:!LE_S
R' - [I,
PIP\NG fL(XI61t.I1Y
7
29.71 - ZIOC..(. - /47!J.t. - /83.1 - /83./ 7.(.0
40.J~
- 1.000 00
6
5
/0.80 -
/(;.84 707.4
/83.'
FORM E-I
CALC. N-·5.15
••
fo::.~i;2?f·~
-
-
15.1t;
10.C./
333.C.
/83.1
-
.
-
4
3
/8.50
0
303.0
/83./
0
•
·-
/8.50
/4/;;.0
/83.1
.
•-
1
Z
18.SO +
0
18.50
•
Z 703.0
/83.1
Z-c. 171 0
..;I 387 +
ZZ 784 -
2.5!JOI
8109
-
;:.s?~.o-
/83.1
-
:J.OO
r~'
~F .z
OR
Si • S-'L
5'c
:-F"oy
M,
.....Hle HEvCl: IS GREATER ..~
of z·x
-f ~'z
IS}:. •
R·
:? -(1-j
M
c)
-~-~
~F,·y
~l' •
-F x
M,
0
R' -CAL.CUL,..TEO
REACTIONS BA5E:> ON 5' .... co
M)DULUS I Ell
0:050<
.-
5t 02~ -
84 si, +
I col.e. 2:8 92.4 ~
32. /22 43 ~S!J '"
I 39l -
/0 445 ~
, -•
14 /45 -
9013 -
56'
/5 '903 -
1/ 9/2. of-
.3 539
2. 774 ~
.3 387
Zt;141
0
+
825>0 -
9023",
/0 Z3B -
I 977 4 2.09 -
-
J 381
30 (,ZJ
"0'
M',
Y
Z
PLAIJE
M
M·
.M·
z(:O.!>oc.
"', -M\ -M'"
~M·"
:M"
l:
oM""
.M'-
~
L
M'o
~ ~~'''' '"
.~~~
"C.O$",
.M'
~M''l
~
l
PIPE
31.0
M.
~S'N I<
M<
f-12 l
52.co.4
/J
Z
.02.2130
I
.'8
Sb"f M
S'b"f.8M b
25t.=fMt
S ,5 .<5
5··5'·~4::'·;51;
THE MWKELLOGG CO
~IPING
FlEXI.~ILlTY AND STRESS ANALYSIS
MOMENTS AND STRESSES
192
<Ac
"'
OA
-
FORM . I
C
0 S,IS
FLEXIBILITY ANALYSIS BY TilE GENERAL ANALYTICAL METHOD
MEMBER NO. 1·2
PLANE
Z
STRAIGHT MEMBER DATA
COORDINATE
'"SHAPE AND
~
DIRECTION
b
e
:I j
DATA
k
/
+-/8.50
Q
L
/
3.00
-!-kQL,)
4.50
-
/" $"0
0
+-/8.50
"y - 1·50 nkQL
~OOlU"'l."':; ~OR
PLANE
'OT ...... >CoI-lT
·
Wu
•
>:
+hka~
3.·'8
FOR"'Ul.A!> FOR
... E ... OtRS
SEE
S'N IX
,
k
Q
t
co:) ""
0
R
Rt·L
SIN"o<
+/
kQR~
<:.0.0"'",
0
kQR+
oR'
!>'NO<COZ
0
L'.~l'\g.~fi :~D ';~"'C;;"'''
.w.
• wu
TAeLE.5
5.25 ... 5.28
>( "'''''''lO ...... N
.. 0<> C c. ........ ~
5EE TACll..e~
:! J.kQl)COfl"e>t
w.
5.33 .. S.!lS
~~\>••~ .. hkQL4.o'>l~CO!ozol
f~ ... O""""
c
w'.. "
..
·
;!:.n\..QL... cos· ....
:;tkQL:sl"'~"
SOE TA&..l!'i5
5.32 If. 5.~4
~.t.kQL:SIOl ... coso<.
W'bt" :!:tkQL+s,.. 0<
'0
M....
M
A ••
M._·A,.
A.
.
...
0
"tw'u,",
+ ':. ~Hl
+w...,'"
'1'/0./.3
tw....
-"a,Wb
+bwb
~_.
-ll.W...
..w·.....
+w'l:>&
+W'bt>
... ,.
+wbb
tWbL
•• ,.
.w.
,.w.
.w.
F~ .... BH
F, ... ·B yz
A..
5"
M ....A..,
M...... A.,
e ••
e ••
B ••
M ...··B••
B
M. w'6 n
B •
M ......S ...
My...·D ••
-
0
R.
+ 270
0
0
R.
0
"",'W..
0
"",,·w..
0
0
0
Ftw·B u
0
0
0
c ..
'c;w...
0
..~
F",w'C. ..
"','CW-
0
W~'Wl>l
.C"".,
.wa..
0
C ,
..
w.·e .... "',.w••
Ftw,C YJ
C,
w,'~
F.,w·C..,.
R.
SHAPE AND
DIRECTION
~
b
T.
t .......
w........ w......
Wu
II:
SIN IX
Q
/.00
;;
CO$ ..:
R
R~ .. L
e
0
0
l.!,..kQL)
/P.50
SIN",,:
Y
+ 11.78
:!; .1. kQl!COllo ",
'; .hkQ~ SIIHO,COS;QI:
:;.J..kQt.:.s'N~O<
;i4kQL:SIOl 0< COSO<
:!:..J..kQL+51.. l>I.
W't>b"
M••
~ 4/,079
M"w
A••
+ .3.00
A.
M.....A....
A.
M.w·A. y
<10 I2J
237 M W·Ah
+W ... "
• w.
SEE TAeLes
5.33 .} S.?>S
-tbw..
0
0
A
0
+- 3. ijn-
0
M w·A.
D
+W.
.
·e..
.
F ••
+3487.17
.~
8"
4.50
+/5 692
.w.
.. '''1"
• w,
Fr...
8
z.
en
A..
M.w·A .."
B ..
Myw·A.,
B
F tw ·6 u
M.w·B••
B••
M ... ·B...
B
Ftw·C u
C ,
M"w..B.y
M.. w ·6......
e..
B •
My..."Dyt
HE MW KELLOGG CO
c ••
F"w'Cy<
+-72./5
0
f.-:;~ .-,~-
Frw·B z. .,. 25/,599
A••
.
-
t"",''''1>
SEe TA&..I!'~
5.32 If. 5.~+
0
+ 4.50
M".... ~8 ..t .,./84 1 856
+44(;.,899
"'44(;,,899
tw· .....
<l 6'G.:-'~ W"l'''i
t
w. ·
M.~ r+yF,,,,,,
6331:
~c;w
'" +i'kOL3!>llo.ll>l.
w'uv'" + "4k a~
w·b...
OR'
kQR'
k.QR+
w·••• ± hkQL: cos'"",
y
180·
58./
e
w.
W·...b ..
w,
1+ GO
F. w • w.,L'
F. w
+3487.17
Mow
4/079
c.O:''''C(
GR'
IJIJ /' M __ '" -j(F tV.;
L' ... ~o.>tc:Tt.o ... f .. c;." .. 0.
5'>.10«0'
My .... I
58,/2
D
... 0" ., p ....... e
FOI\ ... Ul. ... 5 fa"
5TRAlc.:jHT MEMBERS
e.u" .. ~o ... t:MIlE<lS CONSTANTS FOR CURVED
SEE TA8Le:.~
' WU
+.l..W..
..
"'90 ,,~q
D
0
5.25 (.5.2B
+W.
-b ......
+- 974
D
"C::""'".
0
kQL
~ORMUl.""~. U!;L UPI'
I!~~ .. o'ql. e W'Nc.
c.
..
p~
~.
•
+608
FORM W
.16
Ie
.
CURVED MEMBER DATA
~
k
2.28
270·
L
~
T.
.
C .. fCMfC
0
;:;-:;"-kQL1 C:05l>1.
w. -
+ c;.OA
c .....c.
0
fOIt ... ul,,"S fOR,
ST ..... IG .... "
"'f:"'Bf<l~
Pl.ANE
+ /0./3
k
Q
.Lb, "
Z
STRAIGHT MEMBER DATA
T.
0
~
+w~.
LOAD CONSTANTS
0
-.
J'W'
·w~ "'60
'4.~0
+-270
0
THE MW KELLOGG CO PIP~'bG,.tlflJ~~~N\tNfo~T~~W,:i}ALY;I'
'"
"few"
.w.
M ... ·A>y
COORDINATE
DATA
0
0
~CW ..
H,.7.
• w ... 1>
5 ..
MEMBER NO.2-4
PLANE
Z
w ... -1i,i=.w
M,w 1-3330
.,bw...
.. w·... b
A.,
M ...... B ••
M
• w.
M~w·A ...
B••
-b ..... y
5 ..
"-
M...... B n
+~w
F ..
M ....A ••
A.
0
+ 4.50
v ....
·
•
y
1'0 ......... 1. ... :;, lJl>~
M"w .. -toVF"w
M ...... 1-270
a.0..
..
·I~ (,O
F. w • Wzt...'
F. w 1+180
~
CONSTANTS FOR CURVED ~ STRAI~HT MEMBERS
CU""~D
.... £ ... I'I£R:I
'::-:;;-kQL)cosc<
w. '" +tkQt...:.olto.l"!
. . . ·.,v
4
w,
CURVED MEMBER DATA
90'
/
~
193
-awl:>
tbWb
-tw"t,...
.w·l>b
",wb..
t"'bb
LOAD CONSTANTS
R.
t723,749
R.
+S69/,739
,'Wa
0
0
ow•
.,. 90 (DGq
Wo·W.. .,.S4t/O /40
0
0
0
,
R.
0
b•
- - w..w•......
"",·tw- 0
T.
z
+c.w"
Wz.'<W
+Wn
,.w••
..
..........
~
0
T.
0
+W..
• w......
~LE~!§lIT V AN.9~~TRE?$. ANALYSIS
LOAD CONSTANTS FOR Z-WIND
---
·w .. 1"60
+9747
-/.584820
~-
·c;w...
+W~.
/34J.78
F",w· C .." +4685/189 w........
C.
tWa ..
0
0
.w......
+W.b
..... ,..
WL'W..
0
_:~w..
44'- 899
2(; Bt.JW
c ......c.
~,~
T·
~.
T.,
J.i.J/ (;.84 785
.
FORM W
So
DESIGN OF PIPING SYSTEMS
194
MEUSER No.f·S
'"
SHAPE AND
DIRECTION
4
,~
DATA
~
0
b
0
L.
w. ·"'
z
1.00
co~ <K
R
18·50
R~.L
kQR Il
kQR+
0
kQL~
SIN~""
"9
-11·37
- 5-47
hkQL 4
c.O~20<
&T~"'IG"'T
FOR
CURveo .... r ... flE"s
seE TAfH.. e.:s
"'E:"'f1I<I~
.wu
.w.
- 127
+.a.w..
- ,??
-bw v
w· v"' +r'.kOt.:
::f~':;;~1.c w~"':: iOfH:lJl. .... ~. \1'1>'- VI''' R G'GoNfi w... , ...
·
W, ·
x
y
+ l.kQl!coo L " ,
.........b ..
:;: hkQ~ :I'''' CJl,cothc
w·.... -
±hkQ~ cos·lll.
5.3::5 4- 5.~S
:;.LkQL:~'"·,,
see
+hkQL.:~.. ",co!>""
5 . .32
w'b_ ..
.w.
SEe TAell...e:s
w.
.w,
TAeI...£~
4- ~ ~+
W'bb" :!:..LkQL+~I'" 01
M••
-3708,66
+101.03
-,%q",
A..
M~ ... ·A~~
A.
0
0
M.... ·A...
...
1/77(;,.8&;
0
M w.A ••
0
A
.,.107.'13
w
A.
M .... r:..,
A..
M.w·A••
8 ••
M.......A n
M.w·B.~
M w.B...
8 ••
e., -GOoi261
.
+ 880
.w......
"w&b
+W • •
-lI,wb
.. t1""b
'tW'!>6
..... 'bl>
"wb.
+w/>I>
0
W:..·W"
0
'"
Wa'W~
0
0
0
F.w·B n
0
0
0
C ..
-(W...
0
~WD.
.
C,
27202.60
l=".w·C ..,
0
...........
.. c.w~
.. w.~
0
.....
W.'"".
.w~
/8,443,3"3
..w.~
·w:.. +60
LOAD CONSTANTS
- 522
-31320
-127
R.
-/028,525
_ 7"20
R.
"/~I2,374
R, I
w•• (Ww... (W
F.w·C",.
+ 72. /5
0
_:Lw..
0
:...w.
0
~c.wv
+w·... b
0
.,. 72.15
.,. 48918
F.w·C ••
M",w·!:)v:.. "'849,700
+w'-'"
-e""",
,bw..
,'WI
C
8 •
'RRO
.. .w... ,.
,.
'
5TRA1C4HT MEMBERS
0
0
-tw'""
.w.
8 •
F ..... B .
1.4..... ·6....
-
8 ••
M..w·S". +)283 "'2
-885.35
F .....
8
M"w~B~'JI
'I- ti18
8 ..
8,.
A..
8
"'/271 07{,
F,.
"
T.
T.
0
.. w.~ ';RRO
w.'WH W"W~
+52800
T.
FORM W
.s.I€>
C
MEMBER No.5-'
Z
PLANE
STRAIGHT MEMBER DATA
CURVED MEMBER DATA
~
~
k
k
1.00
-55·
Q
51"l to:
Q
/.00
-.8192
t
C050( +.5736
R
R~~L
L
7·60
kQRl>
i" kQL:) 7.1. /6 SIN r ...,
OR'
I.. I<Q.L 4 1 .01 c.O.!>·'X
k.QR"
QR<
L",,~oJfr;;
to
"("4'"''
OF
510.1 .. CO:l
'"
SHAPE AND
~
DIRECTION
b
,~.
•
PL.ANE
w.
0
"9
- 12.98
- /.3.73
FOR ...... 1.. ...
.w
.w.
seE "A8Le.~
u
5.25 ot- 5.28
w. · !
J(
1O'C. ... S
..... , .....
5.33 4- 5.~S
·
:;.LkQl~IN·«
M ••
~'"2Go.88
...
A••
+/31.52
A.
M.... A.~
-S23.fJ/.
A.
0
M.w·A. v
sEe
+hkQL.:sI'I "'co.:;....,
W'l>b" !/ikQL"'sl'" ..
0
w
M w.A.y
TAel...E~
5 ..32
.. bw a
"'S.918.88
F ••
0
8"
0
~ + /23.90
M .... A~
+ 733 • .349.
.
F..... B ••
B
F..... B • - 83,475
M..waS... +.5PJ'I. 44-5.
Y<
0
""'W,,
0
0
0
~WD'
-2518.
~/83. 0'"
C,
R.
1 ~1I94, O(;~.
R.
1"'f:~7 .J.5t!i'.
R. I
w.·.....
w.·c"". 0
.c.w. .. w. .
0
w..',w .... w••
0
T. I
..
T. I
~
.. w~.
.. w~~ -/49./9
01-32230.
W
Mv...·B.".. -.(083 5/0. f='rw·C.", <1'/4,'97,217, we·..... w.' . . Wl'W~ -835'
8"
LOAD CONSTANTS
0
0
F ..... C"'.
·w:..+60
0
-(W...
C ,
MvwoB.....
~
C ..
F ..... C ••
-804,9/
.
.w. ",
-59.93
""........
-3.596.
.w.
'" - 4/. 96
F ..... B u
Mvw·B yw
e ..
"wbb
0
.
M.w-8.",
""'·I>b
+"'b.
w.·w~
Mv... ·A v:..
8
.. b""b
0
M.w·A ..
8 ••
-a.Wt>
.. w'b.
0
8,.
M.... ·B..
.. .........
0
A..
8
,~
a
~w·"",,
"w"l>
w.'Wy
A..
B..
+ 45&.
-804.9/
-3"7,039.
- 183. 06
0
_:twa
"w·.~t:;
'"
4- !5.~+
M ... ""c. ~F.w
M v... t'"S!1/8.88
MEMBERS
..,.1.19,0/
-/..39·/9
.. w ..."
".
SEE "TAeLe5
.lkQl!coezo::
W·...b .. + hkQ~ .)11ol O<;C05~OC
W,
PL .... '
-tw'", ...
W'b ....
w
M ......... +YF. w
M. w 1""l:i26O.8f
~<;w
w·...... ±n:kQL: C05· ...
y
w,
'60
F. w .. w. L'
F.
1+456
FOR CURVED t;. STRAI~HT
~ol..W",
..
- 41.%
1-544.''1_Dw",
"CW v
~8~.2.8f.
- 59.03
+;..kot
<;w~... ~ ~OR~"'L""~,U$( upp
X
.. 0 .... "
5 fOR
c .. ~Ct<~O
c ..... "'~o .... E .... BERS CONSTANTS
.. - -i:kQL1 c05 t<
= +~kat!510.1t<
w' v"
~;'.':'":'i,\;."
c
fO ...............5 fO~
8"l"I't"'ICoI<lT .... r: .... 'nR;,
Wu
Z
-12.98
- /3,73
.,. 22 ti29,32S
c .. <-c.
rrHE MW KELLOGG CO
PIPING _~LEX.I~'!-t y AN'y_~ TRE?~. ANAL'.,,,,
LOAD CONSTANTS FOR I-WIND
COORDINATE
DATA
• <'0
71/35.16 M w '" -'iF'lW
//. .30
.... w If. 776.8b
EO Lf"'G.YH 0 ..
... '" .. L ..... C
CONSTANTS FOR CURVED
5.25 t. 5.28
'ttkQL.:~uu ..
'-
W,
F,w .. w~ L'
F. w • +f.78
.61081
M. w ",+yF 2W
1/•.10
6331.63 M. w ·3708.'6
a.'
aR'
L· ... ItOJfC
~I"'''''CO'
rOR ....... 1. ... 5
-.zkQLl co.5p<
Wu
Q
"t
c
I'c'lI.n) ... ,,!'> FOR
Pl.ANE
CURVED MEMBER DATA
305k
2.28
STRAIGHT MEMBER DATA
~
k
:SIN 0(
Q
COORDINATE
PLANE
Z
rrHE MW KELLOGG CO PIPn~o'"~bX~~HN\tNf'o~T~~W'I~~ALY'i1"
T.
. --
c",<-c:.
CH!Ct<
0
1~"~644, 801.
FO M W
5.16
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
ME~8ER NO,,"-?
Z
'"
5t-lAPE AND
DIRECTION
~l
-25.'96
Q
511'.1
- 27.45
L
co~:~
c
0
I'O"' ..u.. 1. .... s
- 8.41
-Z3. OB
w,
0(
HQL'
h.I<QL
C.O~~D<
rOR ...UI.. ... fI 1'01'1
cU"'''l!O .... f .... 8fI:lS
CONSTANTS
SlWt"""
510,1OCCO:l
I'O~
·'" T.::kOt...:&l .... "'·
~QLlC050<
+w.
+w,
SEE TAeL..E:e.
5:25 4.5.28
w· ,," +;.kQ.~
O ... "'UL .... ~. U';;II; UP!> "S'C.'Hi w .., ....
~~1.c w;...
P<. ...
SEE TAel-~$
w. " ... .LkQL?co~tc<
5.33 ¢. S.~S
W''''b .. :;: hkQL'" :lIN o<.CO!t'lllC
• ........... ± nkQl,....
wb
y
·
M ••
A••
M..... ·A••
A,
++kQL~'~~'"
- , .,
M......A. y
-2QO
+.lw..
+wb
F ••
+813. (,7
B,.
.Z'
M .... ·A. n
-,
M ,..8 .....
B
.
M.,..w·B""
B
- JOt. 57
M.. . w·B.,... ~20"S,~31.
+w u ,
~a.wb
.. bwb
,.
z. .w.
'"
+w,
0
w,,·w.
H.·W~
~30/.S'1
.w.
0
DIRECTION
~
IPS?' •
b
c
Y
z
-8.24
-·4.94
0
-B.l4
-.34.94
FOflUV ......S FDK
5TI'I'AO<;OUY ... "( ••UU:A",
Pl-ANE
~ 245' 378.
"""'1'0',,
0
0
0
F'lw·6 u
0
0
0
C ..
-ew...
0
.-wt>.
F."".C ••
"",·t"'"
0
"'t·... w
C"
.I:..W"
.w••
0
·
F""".C v <
cn
F.""C n
·
/O.s3 co=:> to<
IlCkQL' 19459 SINl't>:
nkQ[' 512.27 c.o~:rt>:
-525(;,0. R.
-290
W~'W~ -17. 400.
R,
+w.
.w,
TAeI..E.~
I' LL w''''co I'O"lMULA,5, us," UP?
• .. 0"1 C • , ! .lkQl]coo'e< R SEE. ......"....,
w.
:; hlo:Q~ ",W Ol.COSzllC 5.3:5 4-5.~5
•
TAeLe~
y
Wb
:;'..L,kQL:!>"'.j~"
·
w'b~ •
+i4kQL:s,~ 0< COSt>:
!.L.kQL+s,... t>:
Mow
- 22075.0'1
A,
rt69.50
A..
M... ·A •• -3141
A,
0
M.... ·A• .,.
0
"a.
A..
M ....·A ..
6 ..
M.w·B ••
~
Mow·B o.. .
6 ..
+ 45.
M..w-S,,* ~<f. ?9l.
M ....A ••
0
F ••
B..
.
0
F.",.B u
My.... By"l: -25IS,970.
THE MW KELLOGG CO
FORM W
tC
5./
w.
F. w
• w,L'
F. w
1+ (;0
+'3/.80
M. w "' ot-VF .....
M. w ~2 ()1S.lJ'.
M .... " -'Si.F .....
OR.
0,.-
l::D Lf ... GOY" 0"
M w
... ON ::::. p ....... c
·
•
F"l:W,CVl
-tw' ....
~ 195. 'IS
• 67"7.9(,
+$12.27
+W""
- ~431.lof.
+Z3.7Z
+~w....
-/9;.13
-b""v
.bw&
t-G31.80
• 45.97
.,.29.044.
..
.w.
",w.
_ 4A 3 . ' A
+w,
B
0#-151. 'IS
M ....A. + '70/ 056. F,_oB r - 305 33~. Wt·W~
0
A..
Bn
F..w·B u
0
M.,...... A y"
-ew..
B ,
CO'
""t·,.. . .
M .... B\'.
F.w·C. r
.ew"
B
C
M.,..w· B""
0
-"8.3.28
u
f
kQR"
L·.,.... o ... fe
,wb
4-
A,
t-.3Q. 803, 05~.
.
RIf!.L
ow.
SEE TAe.t...E~
5.32
!5,~4·
.,. 520C:;:.03
T,
eo.",C.
c"CCoteo
kaR~
",,'bb ..
M,.
R,
R
W·...b c
:tnkQL cosloo.
+ (,.47 541.
0
Q
"'co,~!i.-
w'•• ..
-3240,873.
0
,.SZ06.#
CONSTANTS FOR CURVED t;. STRAI~HT MEMBERS
5.25 (. 5.28
..
·
LOAD CONSTANTS
-87G
CURVED MEMBER DATA
~
k
-121
5'o.IIX'0'
SEE
-W::. ~"O
"",..::w~
0
T,
<'''''.~
.. w.... +W~ "'\/J~~ -14J.1M~'
35371.72
+28 787417. IV,·.... "".'W.. W l ·"". -8834~8. T,
L
fORMU"'AS fOA
cv"'''c:o "'E ... aEAS
w.
- ";"kQl!cos go;
w, '" +ikQL:Sl .... t>:
w',," +r'..kOt
.0
"wbb
<w.
STRAIGHT MEMBER DATA
~
k
-97I
S'N eo:
Q
I
-9US
'"
0
0
.. W·bb
MEM8ER No.7·S
COORDINATE
DATA
...
.. c.w..,
."' ......
PIPING JLEXIBIJ,.r Y AN9.?TRES.S·ANALYSIS
LOAD CONSTANTS FOR Z'WIND
PLANE
SHAPE AND
~_.:.w
.w ..'"
THE MW KELLOGG CO
Z
+GO
_1""..
-5l!.1644.
F~ .... ·B.~
B
+910,3/9. F...... B. t
8;:",
B
M"'w· 8 ". HZ,962,1J8.
.,.1.7Q,3.34-
+w·b./l
0
0
M.,.......A .....
+w'.....
..w·,~t;.
t-6842.96
M .....·A.·.
"'" 7.$Z8. 4- 2404'.2
1"bWa
A.
A..
-bw.,.
"WI>.
0
-"90.2~
R~·L
-
M w
A..
B••
f
/8.50
ow.
4-
M w.A".
A
'1"133.03
M.w·B .....
/.00
R
~
...w...'"
seE TAe.t...es
5.32
5.~4
0
6"
M...... B••
B..
2.28
.7.J304
M. w .. +yF:z w
1.30$6
kQR 3
G~'
6JJU.J M KW ~/877'.s
kQR4>
~p'
/lUiS.lG M 'oN ~ -'5<I="w
L'."~O~t;
T(.O ... t; .. QT,.., 01'
13.S~
'- 0'" Z PL. ..... ~
M vw ~'84Z.9'
FOR CURVED ~ STRAI~HT MEMBERS
f*-kQL+511'~ t>:
-1A779.5
~13 9. 87
w.
F. w • wf;L'
F. w 1"'8/3..'7
83'
k
Q
c:os 10l.
w·b... +t.ik Q L ~~ ~ cost>:
W'bb"
CURVED ME.MBER DATA
~
~
~T"""IQ"T .... II':M""R~
Wu
z
MEMBER DATA
b
•y
7
PLANE
6TRAIGHT
k
COORDINATE
DATA
PLANE
0
~CW ..
0
-aw...
+w·.~b
+w·.....
+W&b
+w...
-a.wb
+bwb
+w'b./l
"""1>&
••
+W•
+w'bl>
,.
+wbb
·wz. t- fiO
-/93.13
0
l'WI
-11,588
0
+23.72
0
...... ·w~ +'.23
0
R,.
1-.3724 272.
Rv
10#-<#/87 /4.3.
0
0
0
0
.w~
0
""r·...w
........
LOAD CONSTANTS
0
.w.
""",.'' ' ~.w.~
."'.
-ew ....
R. I
T.I
0
T,
0
-"f3s8
.w~
C
"\/J~~
+5 I 0.2
Fltw'C u .,.35.50/,000. Wlt'~ w&..... w.,,,,, -3
PIPlliG. FLEXlBH1 AN9.?TflESS ANALY51S
LOAD CONST NTS FOR Z'WIND
./4
e" ...c.
CH~Cot
0
.470.
. s.Tit 1-3I.Sef.
FORM W
.
.IG
195
NOTES
END TO STOP
R
M
w
T
++-31 (,
2207 .0
.....=....J.ll~7J.!L..5.Q
001
GOB
7BS
~'~~_IL
-
370B.6&
- So B24./3
.. Z'l 7
A.
...
•
3
A
M ... ·h
A
B
!:h~~
8 ••
M~
M .... • ••
M ..·B.
~:-oL!!.u~
C••
F -C
M .....·A.
A ••
M ..... ·A. ,
~
•
,
y
STOP IN X DIRECTION
+ 72
-~~
-
.pe
EI(I..-z
l
_vEl ElK
E I B.
M ... ·Ar.<.
pe
M"··d~~.
£1(' .. .le.-Me
~
8 ••
M ...·B.
,
En
UATION
f.;-;-A..Y.L...
M ....·A••
•
..:; E'r e
~
•
_B_., .
M ... ·A~)'
A..
EIO
--WT.
.
CONSTANT FOR X STOP E
A.
A ••
U ...... ·A.
+5f 'BS Jq
+ 56 17
tU9
-.lEI 6•
... /1. E.I II
EIS
•
E19.
-699'1422
0'•
q, 73'1.
+ '6853'13.
+
.I~
,
, -/-S. 5.M 472
2
3
0
0
- .JJG.71'4 r.!.2G2, 0/9
6,4CJ 717
-I. G47. 08!!
,483.,,2..1. -l '22. 51/
-5<
Z -",Sf.
8 ',f•• ISI;
- 'JUG!lG
-Jlll2J 5 (J
r. '? 'bl. ,),f~ -f-J":?ZS
4 -11, 232 l& -ztl ~ '6 oJ.
...;'3/5$/J.10 ·ZlU/7./!Y.
5
"
,
~
-
Fz
My
Mx
jl80 00
do 7.58 7~
z
•- .4 $ -• 18'" 1597
•
0
0
0
I. 000 00
~ 9/~
..
fl-7 ()() /f8
_0
0
700' -10 II. 2B
• I
+
65 61 +2
9
0
0
0
165 I 695 B4
(}'EJ.!J~
1.000 00 -I-~
'5 -12 921'
- q 945 ~-2 I 66 'H, '7 >
3 "'70 10' G i-28 J7
-
-
-,
I'
104<
•• 1#,
$3
'NJ
4
fo/9 !()4 2
~5
dl ;1/ 26
'/ ~2
4'
-. "
09 <2
".,~
,
ZS
2 67
8/1 $0
F z'w (r"r) .+,~'1' .(;f
-8E
91 +38 ot/ 08
1.00 00 -_
Z WINO
8'
9$6 21
143 7<'
·.1- 00'
-
I
Ot
8/1 2.
'JI 1M <<<
-. 139J!9~
I- +15 3~/9
- 00000
4/ '5 "4'
/. '5 ~
.//0
SS 4/1 0<
I.
- 3 ~/l-H
5
rz. · -
8/1.$0
.. - 03(;41·44
_ 1750.70
'" ·
~~
I~
,,
rZ4
rZ' ·
ONSTANTS
FZ 4
,
-Gc4G.G4
- ,.
000 00
•
,
I
THE MW. KEllOGG CO
PtPI/iG ~LEXIBILITY B S.TRESS. ANALYSIS c ... ~
6-EQUATIONS
196
- 1.000 00 ,.-...-;
.
ro'.:.~~i1?
-
FORM E-I
CALC. N-·S.
FLEXIBILITY ANALYSIS BY THE GENERAL ANALYTICAL METHOD
CONVERSION TO
cooe RULES
OINT
,
,y -
k.
Eh
S·'" Ec .S'e
.
,Eh
S,.
&.
Rl'"
if·C:
Rc
Ec
,.,
P5'
OR.
F
F
F
-
•
+F ·z
M,
~ - · W=-E"
S'E
M
WI-lICHEVE!i: IS GRfAT£
+F,,·x •
~.
-F.... ::
M
~4- (1-]<)
•
+F."y
&
go •
7
8
8.!B 40.J~ -
$11.50 .107$9 +
'8"
.9 -
JZSg
of'
5
- "It;10.80
.84 29.71 -
4
15.1t; -
U
IO.C.1
.
1090.00 ...
/7'J.70 ..
/i,4,
B6M-
1009:. -
IG35
5'39 +
.3 J4S +-
10/;77 +
...
Z. IOj ....
-
II ~(i,5
4 7928 02Z of'"
IG 524 -
4744 -
SOl -
72
•
d 740 ...
J Je.<,-
274
-
J .374 -
581 -
-
Z
3
18..50
0
•
0
/8.50 ...
0
/ .50
In.sa +
/570.80 +
2/0 +
/150.80
+
,., •
•
.3 197
1.00(,5 ...
/
•
0
/8.50
..
17(; .00 ZOOli5 ""
197
-
0
.
03.00
zo 734
.5 2Si!.
Z1004 ..
Z.j9~
Z7 8Z8 0
+
48 3u) -
511;93
JZ 708
29~O +
.32390
J2 'jOS ~
Z7828 -
19.30J -
19303
/9 799 -
- -,
•
R'. CA1.CULATf.D
RE"ACTIONS BA'I!:O oN SIlo.! eIt
HoT MODULUS. I E..
COf,lX
,
M"
-M
Y
<
M'
• M,
-M'T.
M'
>:
.M'l'
PLANE
M•
.M
xco.!!>.
-M'.
JrSIN.
,M'
M'
"cosO(
M'b
M'
... M'I<
+M''Z
,p
Z
f .. 12 l
3/.0
5ZC-.4
P
5 • M
1I.!!ollo.llll.
M,
.02280
/
"
5' -f,8M b
25t."'fMt.
5 +S +45
5 "+5'
-., -5',
THE MWKELLOGG CO.l.
~40
440,;4
PIPING. F.L~~I.H.ILlTY AND STRESS ANALYSIS
MOMENTS AND STRESSES
where "L,F, is the sum of the reacting force at the
free end, obtained from the equation sheet, and any
restraint in the line between the free end and point N.
"L,M, and "L,M yare the sums of the moments at the
free end referred to the origin, obtained from the
equation sheet, and the moments at the origin caused
by any restraint,. "L,F.", is the sum of the wind load
from the free end to point N. M,,,, and M yw are the
sum of the moments at the origin due to F.",. M '''''
My"" and F. w are obtained from Form lV,. Since
a F,-stop is included at point 4- in the example the
F,,-force from the equation sheet and its moment at
the origin must be included from point 5 to the fixed
end. The calculation is shown on Form F-l.
c ....l-c. #. """.
H
•
,-
-
FOR'"
l
-I
S.lb
References
1. The M. W. Kellogg Co. (by D. B. Rossheim, A. R. C. Markl,
H. V. Wallstrom, and E. Slezak), Design of Piping Systems,
1st edition, 1941 (out of print-superseded by 2nd
edition, 1956).
2. H. V. Wallstrom, "General Analytical Method," Heating,
Piping and Air Cond., Vol. 19, No.5, pp. 69-74 (1947).
3. L. H. Johnson, <lSolution of Pipe Expansion Problems by
Punched Card !\.hehines," digest in Mech. Engr., No.
53-F-23, p. 1020 (Dec. 1953).
4. 'V. Hovgaard, HStresses in Three-Dimensional Pipe
Bends," Trans. ASME, Vol. 57, FSP-57-12, pp. 401-476
(1935).
5. W. Hovgaard, HFurther Studies of Three-Dimensional
Pipe Bends," Trans. ASME, FSP-50-13, Vol. 59, No.8,
pp. 647-650 (1937).
For a further discussion of piping analysis, see Appendix D,
Page 359, "A Matrix Method of Piping Analysis and The
Usc of Digital Computers."
CHAPTER
'.
6
Flexibility Analysis by Model Test
P
ROGRESS in the physical sciences has been
marked by the constant use of experiments
as a means of pioneering observation and a
confirmation of reasoning and mathematical prediction. Following the establishment of basic "laws,1J
the experimental approach has proved invaluable
for quantitative measurement of the physical constants which implcmcnt applied science and make
possible the practice of cngineering.
General design principles and specific assumptions
can often be verified or their qualitative significance
determined by simple expcriments. Although a
more refincd approach is usually necessary for
quantitative measurement, significant data are often
achieved with minimum complexity by taking
advantage of basic phenomena such as yiclding or
failure. Where the relative influencc of the variables
involved is not cstablished, or when prototypes are
used for providing designs for mass production,
full-size specimens, where economically feasible, are
favored for positive avoidance of errors. However
in many fields, such as structures, increasing knowledge of fundamentals and significant improvements
in instruments for accurate measurement make it
possiblc to employ scale models with increasing
confidence for the direct solution of problems, particularly where ovcrall rathcr than highly localized
influences arc undcr study.
6.1
stress intensification factors for corrugated and
creased bends and corrugated tangents; also, to
load-deflection tests of both piping assemblies and
large scale models of piping systems made of small
pipe. These combined experiences with the experimental approach led to the routinized solution of
piping flexibility by Model Tcst which is presented
in this chapter.
6.2 The Routinized Model Test
Much of the early urge for the evaluation of piping
expansion cffects by model test was inspircd by the
difflculty in handling othcr than simple problems
by the analytical methods then availablc. Experience in structural and other fields had demonstrated
that rcliable results could be achicved from scale
models; however, expense limited their use to occasional important problems. Economic widespread
application required, first, a comprehensive analytical
development of the gcneral irregular frame in space
for organization and evaluation of the accuracy of
the model test results, and second, rugged precision
equipmcnt and organized test methods for obtaining
reproducible results with reasonable expenditure of
timc. That these requirements havc bcen mct by
the Kellogg Modcl Test Laboratory is attested by
the demands for its services on critical piping,
particularly for public utility installations. Although the horizon of economic analytical solutions
has been broadened by programmed machined calculations, the IVfodel Tester continues in its usefulness for complex problems. Most important it
provides an independent check method which can
parallel manual or programmed machine calculations when double assurance as to the accuracy of
design is desired.
Evolution of the Kellogg Gencral Analytical
The Experimental Approach
As a pioneer contributor to high-temperature
piping design, The M. W. Kellogg Company, by the
employment of strain gages, made early use of
experimental verification of the load-deflection relations of so-called cxpansion bends and of the primary
and secondary stresses in curved pipe. Later effort
was dcvoted toward establishing flexibility and
t
198
1
FLEXIBILITY ANALYSIS BY MODEL TEST
Method, in addition to benefits previously described,
provided the long-needed measuring stick which,
along with visualization and detail'treatment, must
underlie dependable experimental solutions; in addition it accelerated the development of suitable test
equipment. It was appreciated that extreme accUracy in model dimensions or reproduction of cross
section was unnecessary in view of the 12~% thickness tolerance of seamless pipe, and of the largely
indeterminate degree of fixation at terminal ends or
intermediate restraints, and therefore, that solid
rod models would give adequately accurate answers
where curved piping did not predominate.
A stiff adjustable mounting frame soon demonstrated superiority over individualized support
because of the absence of harmful deflection and the
rapidity whieh could be achieved in accurately
positioning models, applying movements, and reading measuring equipment. The desirability of a
rigid arrangement with fixed model ends was appreciated; however, its accomplishment was not immediately aehieved, so that initial approaches resorted
to the application of loads and measurement of movements at the free ends of a model, fixed at one end.
Fred G. Hill brought this concept to its highest
development in an apparatus described in 1941 [1].
The free end of the model was fitted with a fixture
consisting of three moment levers and four needle
pointer position indicators aligned with a fixed
reference ring. In operation, the entire model was
displaced in the three coordinate directions by a
micrometer movement device attached to the fixed
end. The free end was then returned to its original
position with respect to the reference ring by means
of shot-filled buckets attached to the arms of the
moment levers. This device was purchased by the
General Electric Company, but was never extensively
used. None of such designs appeared satisfactory
for sufficiently accurate measurement of rotations,
or for reproducible or rapid results.
Much of the practicable advance toward the model
testing of involved structures was pioneercd by
Beggs [2J. In papers dating back to 1922, he describes mcthods and equipment for applying accurate deflections and for measuring reactions. His
HDeformeter" provided a precise mechanical means
for accomplishing the former; and his micrometer
movement movable crosswires microscope provided
a useful measuring instrument in locations where it
could be applied. His use of lapped hydraulic jacks
for applying simultaneous multilocation loading of
varying magnitude is well known and such devices
are frequently employed in testing apparatus. It is
199
also believed that Professor Beggs first made use of
the principle of successively releasing and weighing
individual reactions by providing an amount of
freedom, small in comparison with the restraining
effect being measured, and then taking the reaction
off the supporting structures and onto the weighing
fixtures within the movement limit established. He
successfully employed this approaeh on individual
tension and compression effents and in various
combinations to suit morc complex restraints.
Over a period of years, many models of involved
structures, such as bridges, buildings, support
frames, floating dry docks, machine frames, etc.,
were successfully tested. Such tests were relatively expensive due to the cost of the model, the
specialized equipment, and the number of manhours required.
The M. W. Kellogg Company elosely followed
much of Professor Beggs' work, and were endeavoring
to adapt his general principles of limited freedom
weighing of reactions to the routine model testing
of piping, when this end was accomplished by
Harold W. Semar [3]. It involved the use of small
plungers and struts to confine a fixture attached to
the model to 0.001 in. to 0.002 in. free movement at
individual measurement locations. Each load was
weighed by a movable ealibrated spring gage which
opposed the fixture reaction passed to it through the
strut and plunger. Each reaction was balanced by
adjustment of the spring until the plungers were
moved to the center of their free travel as indicated
by a dial gage, thus assuring that all load was off the
frame and on the spring, and that the reading would
be taken at the same relative location each time.
Movement was aecomplished by a micrometer feed
mechanism which was carefully set up along the
direction of the resultant expansion.
This method of load measurement was incorporated into The M. W. Kellogg Company's first
produetion model test equipment, which included
an arrangement of vertical posts and horizontal
arms, the latter adjustable in elevation and in plan.
Models could be mounted by simple fixtures to
measuring heads supported by displacement heads
which applied three-dimensional translation by
means of two slides, one of which could be rotated
about the axis of the other. In spite of certain
limitations, this arrangement was in successful
continuous productive operation for several years.
It established general appreciation of the advantages
offered by the model test approach in the visualization of complex configurations and in the economic
study and resolution of over-stiff runs in a system;
200
DESIGN OF PIPING SYSTEMS
and it marked the successful establishment of the
present Model Test Laboratory.
6.3
The Kellogg Model Test
The Kellogg Model Test Method parallels The
M. W. Kellogg Company General Analytical Solution, substituting routinized tests on scale models for
organized mathematical operations in evaluation of
the shape coefficients and their summation into
deflection and rotation equations for simultaneous
solution of the terminal and other restraining forces
and moments.
In establishing corresponding
stresses, including the critical locations, both approaches involve the same conventional structural
calculations; although, in some c3:ses, strains and
hence stresses are obtained directly from electrical
gages either cemented directly on the model or on
reusable fixtures which are temporarily attached to
the model. Horizontal and vertical deflections used
in designing supports or in checking critical clearances, and which can be obtained in the analytical
method by supplementary calculation, are readily
measured on the model. The approach basically is
unlimited with relation to problem complexity; only
available equipment restricts the Kellogg Model
Test Method. Test equipment is presently available
for as many as 15 points of complete fixation.
In the model test, the terminal ends are usually
fixed, although completely hinged, guided, or other
partial end restraints can be provided; and similarly,
any degree or manner of intermediate restraint can
be constructed to represent solid hangers or other
types of supports, stops, guides, etc. In testing,
displacements are applied to the ends or at intermediate restraints, which are representative of the
expansion of the piping or external movements, and
which are related to an initially assumed fixed
origin. The forces and couples resulting from these
displacements, as affect"d by the terminal and intermediate restrictions, are read directly on the load
measuring devices.
As the end and intermediate restraints for the
model and the piping system which it represents are
assumed to be the same, and since both are structures obeying the eonventional load-deflection
relationship, their mutual force and moment relationships can be expressed as a simple ratio of
their respective dimensional and elastic propcrties and corresponding load-deflection relationship,
as follows:
where P = force; ill = moment; E' = modulus of
elasticity; I = moment of inertia; II = amount of
expansion (or displaccment) of each I'frce" end with
reference to some fixed point; and L = length. The
subscripts m and p refer to the model and the pipe,
respeetively.
l\1easured deflections are similarly eOllverted in
scale and corrected for free thermal expansion.
Thus, the end reactions and deflections of the actual
piping are directly proportional to the ratio of its
cross-sectional stiffness to that of the model; directly
proportional to the ratio of the expansions (or displacements) applied; and inversely proportional to
the exponential ratio of their lengths.
The load-deflection relationship of a straight
circular tube, according to the elastic theory and
neglecting seeondary effeets, is identical with that
of a solid circular rod, ei ther curved or straight, of
equal moment of inertia. In curved pipe, ovalization of the cross section occurs, which results in its
increased flexibility in bending at the expense of
augmented local stresses, i.e. stress intensification
as discussed in Chapter 3. Torsional eharaeteristics
are essentially unaffected.
In designing scale models of piping systems, it has
been impracticable, as a routine procedure, to duplicate the flexibility of the curved members by the usc
of tubing as a result of the exponential relationship
between the wall thickness and pipe diameter in the
determination oCthe flexibility factor. Achievement
of the desired relative flexibility eharacteristics often
results in a tubing model of impracticable proportions which, combined with the dimensional limitations and tolerances of commercial tubing, make the
correlation of several branches of different sizes and
thicknesses quite infeasible by this means. For
this reason and for economic considerations, rod,
rather than tubing, is used for models. Rod sizes
are available so that variations in runs and branch
diameter and thickness, 01' in material or temperature, are readily reproduced to refleet the
products of the moment of inertia and the modulus of
elasticity of the corresponding parts of the piping
system. The length scale is selected large enough to
promote accuracy within the dimensional limits of
the test frame. The basic rod size and movement
range a~e established to secure reactions within the
range of accuracy of the instruments.
For curved members of easy radius of five diameters or more, flexibility factors are usually dose to
1
FLEXIBILITY ANALYSIS BY MODEL TEST
201
uni ty, and tests made with solid rod models result in
end reactions which are correct or which contain a
small safety margin. If stress in'tensification factors
are omitted in such cases, resulting combined stresses
will, in general, reasonably approximate analytical
results in which both the flexibility and stress intensification factors have been applied to the curved
members. Weld-ells and similar short radius fittings,
where they comprise only a limited part of the total
developed length, may also be simulated by rod
models with satisfactory predietion of stress.
For more precise representation of curved pipe,
where bends and weld-ells have significant effect,
their flexibility may be simulated in alternate ways
of varying accuracy and suited to different configurations. For single-plane bends with in-plane
loading, or in general where torsional effects of the
curved members are minor, reduced rod size at the
bends can accurately represent relative flexibility of
curved and straight pipe. Where both bending and
torsional effects have significant influence, special
devices are required to provide the reduced stiffness
for bending on two axes, while maintaining undiminished resistance to torsional deflection on the
third axis. This is accomplished by insertion of
units which have been carefully calibrated against
analytical considerations. As the use of snch special
devices increases the test time required, their employment is usually restricted to cases in which the
influence of the flexibility factors is more or less
critical.
The model weight appears in all the readings and,
so long as there is no shift in weight reaction between
the restraints, is cancelled out by the use of differences to determinc the loading corresponding to the
movement range. Where an appreciable weight
shift occurs, counterweights must be used.
6.4 The Kellogg Model Test Laboratory and
Equipment
The model test apparatus proper consists of a
complex but readily adjustable rigid supporting
framework, to which removable units for accurately
applying end or intermediate displacements are
attached; the load measuring instruments are
mounted on these units and the model, in turn, is
attached to the load measuring heads by means of
special holding fixtures.
The movement heads are specially designed and of
precision mailUfacture, to secure individual movement along three perpendicular axes by means of
hand scraped ways and micrometer screws which
minimize rotational and axial backlash. The load
FIG. 6.1
The model test laboratory.
measurement heads represent the first complex
application of electrical strain gages, and also their
ini.tial use on a permanent installation demanding
consistent accuracy of long duration. They consist
of a floating fixture, carefully designed and precisely
manufactured to develop the required six restraining
reactions in three mutually perpendicular planes.
To minimize interaction, this element is mounted
through flexible struts to stiff constant-stress cantilevers. Loads are measured by paired electrical
strain gages; each load measurement circuit is
provided with an individual bridge to minimize resistance variations due to switching, a novel arrangement specifically developed for this equipment.
The measuring heads are wired to a console on
which specialized measuring instruments of both
self-balancing and manual balancing types are
mounted for reading loads directly. Local stresses
may be evaluated by means of model-monnted
strain gages or through the use of instruments
incorporating these gages. A variety of mechanical
and electrical instruments for measurement of
specialized loading, deflections, and for calibration
purposes is also provided.
All models are fabricated in the laboratory which
is equipped with special rod and tubing benders and
equipment for welding, brazing, burning, and local
stress relieving, as well as with machine tools, and
special cutoff and grinding equipment.
202
DESIGN OF PIPING SYSTEMS
FIG. 6.2
Load reading instrument console.
A general view of the laboratory is shown in
Fig. 6.1. A model of a central station main steam
system is shown mounted in the testing frame; on
the right side of the photograph may be seen the
load-reading console which is shown in greater detail
in Fig. 6.2. The cables from the load measuring
instruments pass into the overhead enclosure and
through the rectangular duct to the console. Above
the console may be scen a photograph of the measuring instrument with the cover removed.
6.5
and bored riser simulated by a ilr" rod. At the
lower end, the riser branches into two 10.50" OD X
1.70" lines which continue to the two stop valve:;
from which extend the four 6.625" OD X 0.932"
wall turbine leads to the steam chest. The model
counterparts are of .~-t" and ~t" diameter rod
respectively, and rigid blocks represent the valves.
The weight of the header and riser is carried by a
solid hanger tentatively located at the point of zero
vertical deflection. Variations from this position
serve to redistribute the end reactions between the
superheater elements and the turbine inlet nozzles.
In order to prevent rotation of the header about
other than its longitudinal axis, four guides are
incorporated in the design as indicated at J, K, L
and M. Two stops are also required as shown at
o and I to protect the turbine nozzles. All of these
intermediate restraints are simulated on the model
by tie rods.
The model for this system can be seen mounted in
the testing apparatus in Fig. 6.'1. The solid hanger
on the riser is suspended from a small unidirectional
load measuring unit attached to the arm above the
superheater tubes. In this test, moments which are
ordinarily transferred mathematically from the
measured end reactions to the junction with the
single riser were checked experimentally by means
of strain gages cemented to the model and read with
a standard indicator. This procedure is occasionally
desirable to avoid accumulative errors.
As the end reactions occurring in a single superheater terminal element are disproportionately
small as compared to the other values being measured, it is necessary to group a number of these
together in banks. Readings are taken at each end
Moments and Forces for Operating Condition:
No Cold Spring
M.
M,
.'01:
F•
F,
r~lb
r~lb
r~lb
Ib
Ib
A.
B.
C
+ 625
D ...
+3475
E (one tube) ....
F (one tube) ".
G (one tube) ...
/I.
I. ..
+
+
- 50
-900
-250
-565
+ 35
0
- 45
-1700
-3050
-2650
-4725
-360
-360
Typical Model Tests
Figure 6.3 is a sketch of the model of the main
steam system for a utility power plant designed for
1990 psi steam pressure at 1050 F. As in thc model
shown in Fig. 6.1, the last banks of tubes of the
superheater have been extended through the boiler
arch, becoming a part of the pilling system. Five
sizes of 2t% chrome I% moly piping are proportionally simulated in the model, viz: 2.125" OD X
0.375" wall superheater terminal element tubes
represented by 0.067" diameter rod, a 14.00" OD X
2.65" wall distributing header represented by -H-"
diameter rod, and a 14.00" OD X 2.25" wall, forged
Piping System. of Fig. 6.3
Table 6.1
Location
J ......
K .. ..........
L ....
M ..
0 ....
-1800
+ 75
-
20
5
15
- 395
+
+
30
60
+ 165
- 755
- 135
- 275
- 365
+ 30
+ 30
+ 25
+ 275
+ 15
+ 15
+ 15
f,
Ib
-135
+545
+290
-8M)
+ 1
0
- 3
-1275
+
45
-1050
-1805
-150
+225
- 30
!
1
FLEXIBILITY ANALYSIS BY MODEL TEST
203
9
tI .L"
4
__
TUBE HANGER MOVEf.ttNT
/
GUIDE K
~Il
Ie
o
BOILER ROOF MOVEMENT
~~"71
2S"
"
~
'6
FREE THERMAL
EXPANSION
(t POINT 7
FREE THERMAL
SOUD HANGER
EXPANSION
e POINT 17
1O.50 OD. 1.70"W.T.
M
!J7
46
46
5
47
44.63'
FIG. 6.3
Central statioq main steam system operating at 1050 F and 1990 psi pressure.
bank and at the .middle bank; and the maximum
average stress in ten or twelve tubes is thus found.
As the individual tubes are small and highly flexible,
such determinations have been quite safe. Where,
however, the tubes are not all of the same configuration, strain gages are cemented, by means of special
attachments, to representative tubes to check the
results obtained. The end reactions, stresses, and
deflections obtained for this piping system, for the
operating and cold spring conditions, are tabulated in
Tables 6.1, 6.2, and 6.3, respectively.
Typical of the piping systems for which the model
tester offers a clearcut solution at an important
saving in engineering cost is the 1250 psi main steam
line shown in the drawing of Fig. 6.5. The 12"
Schedule 160 piping connects two boilers with two
75,000 KVA turbines; and a 12" crossover connection permits the operation of either turbine from
either boiler. The two leads to each turbine are 8"
Schedule 160, and horizontal restraints prevent transverse movement at the stop valves. The operating
temperature of the piping is 900 F, and the material
carbon-moly steel. l A free-floating system of six
FIG. 6.4
The model test set-up for the system of Fig. 6.3.
IUnder present pract.ice, carbon-moly would not be recommended for service Ilt this temperature.
DESIGN OF PIPING SYSTEMS
20J
points of fixation is involved, carried entirely on
spring supports. Compensating spring hangers are
assumed to match the weight of lhe piping without
restraining its flexural movements; accordingly, they
involve negligible stress and are ignored in the test.
The model of this piping system is shown in the
testing frame in Fig. 6.6. The main run of 12"
piping is represented by it" diameter rod and the
EXTRANEOUS
MOVEMENTS FOR
PQINTS e,F.
8" turbine leads by i-'l.lI diameter rod, and the
model length scale is t" ~ 1'0", with the stiffness
or EI ratio ~ 1 : 1,580,000. The flexibility factors
of the long radius bends were nearly unity and hence
were neglected. The factor for converting the
model forces to those of the actual installation is
4578, and that for the moments 9157. The full-scale
maximum end forces and moments, as determined
("SUPERHEATER OUTLET
·ZO
_
HEAD_ER~-.~
'4.50'(14.'7'
--:;.,--,,o/;..
•
'f;1'/,'<:J
~
EXTRANEOUS
MOVEMENTS FOR
POINTS B,E
- ;r,i
:/,o,-yO
t .., ./
2.33 1
'co .27393"
50.58'
N
'"
t2
M
SCH.160
0
!
:t
. '"
;;,
0
0. N
'"
34.00'
2!1.16'
34.00'
54.50'
2.96'
F
!'L
~
N,
9.00'
(
g,
3.44'
c
STOP
H
----~,==:.:..:S:T~O:P=G=:::;l~
,-,'!'-_
j_ _~_+__
,r
.
~(~/-
19.33'
B/~
,&
-/-,1'-1 ¥-,;
-,L9.00'
FIG. 6.5
?
/
1lain steam system operating al 1250 psi, supplying two 75,000 KVA power generation unit.<;.
205
FLEXIBILITY ANALYSIS BY MODEL TEST
from the model test, acting on the superheater
header connections and the turbine nozzles are
shown in Table 6.4. Tests for alternate operating
conditions show that this system is most highly
stressed when both turbines are operating on steam
Table 6.2 Piping System of Fig. 6.3
from their respective boilers and the crossover
valves are open. This is the condition recorded in
Table 6.4; the maximum stress of 10,250 psi appears
in the 8// turbine nozzle B.
For the determination of uniformly distributed
weight load effects, tubular models which can
alternately be filled with mercury and emptied have
been used. Distributed loading can also be achieved
:tvloments and Force.o:; before Operation:
100% Cold Spring
M,
M.
f~lb
Location
... - 825
A ..
B..
.... +2400
- 75
D...
-4625
E (one tube) ... - 25
F (one tube) ...
5
G (one tube) ... + 20
II ............
I. ............
J ....
C...
F,
F.
F,
Table 6.4 Piping System of Fig. 6.5
f~lb
ft-Ib
Ib
Ib
Ib
110ments and Forces for Operating Condition:
+ 75
+1200
+350
+ 750
45
0
+ 55
+2275
+4050
+3525
+6275
+ 480
+ 480
+ 525
40
80
+ ISO
+ 485
40
- 40
35
- 220
+1005
+ 365
- 365
20
20
20
+1695
+180
-725
-385
+885
- 1
0
+ 4
No Cold Spring
-
-
M,
-
- 60
+2200
+2400
K-.......
+200
-300
L.......
M......
0
+
40
M,
ft-Ib
Location
-34,500
A ..
B .......... -21,050
-1l,050
C
D
-33.600
- 4400
E.
F .......... -14,250
G
II..
F,
M.
M,
F,
F.
ft-Ib
f~lb
Ib
Ib
Ib
+4600
+3250
+3050
-6650
-2400
+2300
- 9550
+ 390
- 880
- 410
- 480
+ 580
+ 170
+2020
-1390
+1750
-1940
+ 70
+1760
- 320
-1320
-1l60
+ 360
+ 800
-1010
+ 440
+ 570
-13,750
-9000
+ 10,350
+ 6450
+ 6150
"" 10,250 psi at Point B.
Ma.·dmum stress
Allowable stress range ... 20,300 psi.
Maximum stress in 10" line
- 6850 psi at Point 59.
Maximum stress in tubes
8150 psi at Point G.
Maximum stress in 6" turbo leads 4500 psi at Point D.
Allowable stress range
20,200 psi.
Table 6.3 Piping System of Fig. 6.3
Deflections (inches)
From Cold to Hot Position
Location
8
I
40
41
46
47
48
61
18
o
50
51
57
58
59
70
80
90
II
L
M
0"
011
0:
-0.23
0
+ 4.33
+4.60
+4.95
+8.33
+8.62
+ 7.59
+0.27
0
+3.58
+3.90
+4.10
+8.06
+8.35
0
0
0
+4.69
+ 1.05
+3.95
-1.09
-1.09
-3.54
-3.58
-3.58
-4.71
-4.08
-2.30
-1.31
-1.31
-3.60
-3.83
-3.83
-4.66
-4.16
+2.84
+2.84
+2.84
0
+2.27
+0.53
H.1l
+2.11
+ 1.98
+1.31
+0.88
+0.85
+0.79
0
+1.81
+1.81
+ 1.61
+0.91
+0.46
+0.38
+0.38
0
+ 1.52
-1.52
0
0
0
From Design to
100% Cold Spring Position
Location
o;r:
oj,.
0:
8
0
+0.15 +0.01
I
0
+0.15 +0.01
40
-4.72 +2.60 +0.15
41
-4.61 +2.64 +0.43
46
-4.57 +2.64 -0.70
47
-8.61 +3.78 -0.67
48
-B.51 +3.53 -0.61
61
-7.24 +3.55
0
18
+0.53 +0.38 +0.0·1
o +0.53 +0.38 +0.04
50
-4.19 +2.66 +0.2·1
51
-4.13 +2.89 +0.55
57
-3.94 + 2.B9 -0.65
58
-8.34 +3.73 -0.57
59
-8.24 +3.61 -0.57
70
0
-4.15
0
80
0
-4.15
0
90
0
-4.15
0
II
-4.70
0
0
L
-1.06 -4.15
0
M
-3.96
0
0
FIG. 6.6 The model t.cst set-up for the system of Fig. 6.5.
206
DESIGN OF PIPING SYSTEMS
FIG. 6.7
Models of marine piping.
FLEXIBILITY ANALYSIS BY MODEL TEST
FIG. 6.8
Models of power plant piping.
207
203
DESIGN OF PIPING SYSTEMS
FIG. 6.9
!\Iodcls of petroleum and petrochemical process piping.
i
I
sA
FLEXIBILITY ANALYSIS BY MODEL TEST
by attaching small weights along the horizontal runs
of the model, with a larger weight for each riser in
proportion to its length.
~
The model test solution offers particular advantages for the irregular contours with numerous
skewed members resulting from the extreme space
limitations of ship installations. Figure 6.7 shows
models of various main steam and auxiliary piping
systems among the many marine piping assemblies
which have been analyzed.
Power plant piping occupies the major capacity
of the Kellogg Model Test Equipment. This is due
to the complexity of the main steam, reheat, and
other piping systems of large utility installations,
the need for economy in materials and fabrication
for the expensive large-size alloy lines involved, and
the extreme care which is considered advisable in
avoiding damaging reactioIlS on large turbines and
boilers. Figure 6.8 shows various models of utility
power plant piping.
Petroleum, petrochemical and chemical process
plants frequently involve piping of large diameter
and many points of end fixation or intermediate
restraint, often for extremes of pressure and temperature. The model test is well adapted to these more
involved problems and offers considerable advantage
in the rapid study of tentative designs where plot
plans must be established to meet the usual tight
schedules. Figure 6.9 is comprised of photographs
of models of process plant piping under test.
i
i
!
L
209
References
1. Fred !\l. Hill, "Solving Pipe Problems-A Mechanical
Method for Cases Involving Temperature Expansion/'
Meeh. Eng., Vol. 63, No.1, pp. 19-22 (1941).
2. G. E. Beggs, "An Accurate Mechanical Solution of Statically Indeterminate Structures by the Use of Paper I\lodels
and Special Gages," Proc. Amer. Concrete Inst., Vol. 18,
pp.58-82 (1922); IlThe Use of t\'1odels in the Solution of
Indeterminate Structures/' J. PrQ1lklin lnst., Vol. 203,
pp. 375-386 (March 1927).
3. Harold W. Semar, "The Determination of the Expansion
Forces in Piping by Model Test," J. Appl. Mechanics,
Trans. ASME, Vol. 61, p. A-21 (1939).
4. L. C. Andrews, "Analyzing Piping Stresses by Tests of
Models/' Heating, Piping and Air Cond., Vol. 17, No.8,
pp. 425--429 (1945) j "Methods of Making Piping Flexibility Analyses-The Model Test Method/' Heating,
Piping and Air Cond., Vol. 19, No.8, pp. 73-77 (1947);
"Model Test Analysis of Steam Piping," Combustion,
Vol. 20, No. 10, pp. 53-56 (1949); "Piping Flexibility
Analysis by Model Test," Trans. ASME, Vol. 74, No.1,
pp. 123-133 (1952).
5. S. W. Spielvogel, "Model Test Checks Pipe Stress Calculation," Power, Vol. 10, pp. 68-69 (1941).
6. S. Berg, H. Bernhard, and K. Th. Sippell, "Ermittlung del'
Auflagerreaktionen warmbetriebener Rohrleitungen durch
Modellversuche/' Z., VDI, Vol. 83, No.9, pp. 281-285
(1939).
7. Joseph D. Conrad, IlModel Tests Solve High-Pressure
Pipe Problems," Power, Vol. 84, No. 10, pp. 58-61 (1940).
8. Joseph D. Conrad, "Models Help Determine_ Pipe
Stresses," Westinghouse Engineer, Vol. I, No.1, p. 22 (1941).
9. John F. O'Rourke, uHow Model Tests Cut Piping Design
and Fabrication Costs," Power, Vol. 97, No.9, pp. 90-92
(1953).
CHAPTER
7
Approaches for Reducing Expansion Effects:
Expansion Joints
T
when properly incorporated into the piping system,
for satisfactory service life and safe operation. The
too-frequent easy approach of complete dependence
011 catalog or sales information without adequate
understanding and sound application engineering
can lead to disastrous results.
HE other chapters of this volume are concerned essentially with various design aspects
of specific piping configurations or details;
this chapter is devoted to the problem of fitting
piping into an allotted space or maintaining a configuration within acceptable process or other criteria) where conventional design is inadequate.
7.J
7.2
Introduction
Sources of Excessive Expansion Effects
A design involving expansion joints as a substitute for a conventional piping arrangement is sometimes advantageous or necessary for one or more of
the following reasons:
A. \Vhere space is inadequate for a conventional
piping arrangement of sufficicnt flexibility without
overstress.
B. \Vhere minimum pressure drop and absence
of turbulence is essential for process, economic, 01'
operating reasons.
C. \Vhere the reactions are excessive and involve
possible damage to the terminal equipment, or for
economic structure or foundation design.
D. Where it is dcsirable to isolate mechanical
vibrations.
E. \Vhere economics favor other than a conventional piping arrangement; particularly low-pressure
large-diameter piping.
F. \Vhere equipment spacing indicates excessive
area or building volumc.
G. Where layout was inadequatcly planned and
lacks sufficient dimensional provision for cxpansion, so
that conventional stiff design cannot be accomplishcd.
Making adequatc provision for expansion of piping in a confined space can introduce various complexities in order to augment the flexibility of a
HstifT" piping system, such as semi-rigid, non-rigid,
or frce movement arrangements. In Chapter 5 the
calculation of semi-rigid piping systems involving
hinge points is illustrated by examples in which the
stresses are obtaincd for the rigid members, as well
as the rotations at the hinges for joint design.
Non-stiff piping systems involve expansion joints
or joining fittings, such as articulation devices or
hosc, in varying degree, in order to provide for
expansion movement with less or no stress. Hosc,
special fittings, etc. are usually restricted in size
and confined to individual short piping runs; expansion joints, on the other hand, are used with sufficient frequency and sometimes unavoidably in
involved installations, so that their consideration is
necessarily a part of piping systcm design.
Accordingly, this chapter provides a detailed de~cription of the types of commercial expansion joints,
and completes the presentation of expansion joint
movement calculations with illustrative examples.
Included is a discussion of design, fabrication, in::;pcction, handling, and installation a.spects to assist
ill obtaining suitable joints having adequate capacity
J
7.3
Approaches for Heducing Expansion Effects
For purposes of this discussion, piping systems or
runs can be classified with respect to the mechanics
210
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
211
by which they provide for thermal expansion movements, as follows:
Stiff: If without hinge points and capable of deflection and rotation only through strain resulting
from bending, direct, or shcar strcss as applied to
the cross section.
Semi-rigid: If otherwise stiff, but provided with
Stiff
Somi-rigld
one or more hinge points. A system might be stiff
in certain planes and semi-rigid in others within
strength limits of the hinge details.
Non-rigid: If continuously capable of transmitting
direct load and shear, but not bending, so that no
moment due to expansion effects exists anywhere in
the system (requires 3 hinge points in each plane).
Free movement: If continuously incapable of transmitting any load (except by friction or bellows
resistance) whether direct, shear, or bending.
A simple illustration of these classifications is
given in Fig. 7.1
Within the same space and configuration, expansion effects can obviously be lessened by resort to
lesser rigidity and may be eliminated by free movement design. In the following, starting with COnventional stiff piping layout, the factors influencing
required expansion capacity and the variables attendant to each classification of rigidity will be
examined and compared.
The importance of adequate layout to assure plan
and elevation arrangements favorable for piping
flexibility cannot be overemphasized [1]. In general,
piping flexibility is a function of the ratio of the
developed length to the straight-line distance between the points connected, but it is also obviously
sensitive to the relative location of those points. All
advantage should be taken when establishing the
layout, to locate nozzles and elevations to balance
the vertical expansion of equipment against that of
the piping as far as possible, in order to minimize
horizontal equipment expansion which the piping
must absorb. Dead-ended lines, which create restraints when cold, should be avoided with two-way
flow or cireulating lines. Where possible, small
vessels, such as headers, separators, etc., should be
allowed to float with the piping, utilizing flexible
supports if necessary; or, if they arc on resting supports, they should be allowed to slide on the supports
or deflect. Excessive reactions or overstress at par-
ticular locations can often be countered by properly
selected partial restraints.
Stiff Piping Systems. Aside from judicious layout and lowering of end or intermediate restraints,
the stress and reactions in stiff piping systems can
be lowered by a reduction in piping diameter: lesser
Non-flgid
FIG. 7.1
Freo Movoment
Classification of piping systems.
thickness reduces reactions and stresses in terminal
equipment only, not stresses in the piping. More
than one diameter of pipe can sometimes be used to
advantage to secure greater flexibility but the effects
of follow-up elasticity must not be overlooked.
Several smaller-size pipe runs ean also be used in
multiple and arranged to deerease or increase the
stiffness in selected planes. Sample Caleulation 5.2
of Chapter 5 illustrates the manner in which size
variations are handled in the flexibility calculations.
Another means for redueing stiffness and reactions, but usually not stress, is to substitute corrugated pipe for straight pipe in locations subject to
relatively high bending or direct effects. As pointed
out in the discussion of its properties given in Chapter 3, corrugated pipe possesses improved bending
and axial flexibility although the torsional flexibility
is unchanged.
Semi-rigid Piping Systems. This category includes systems which have a limited number of
articulated joints so that thermal expansion effeets
are taken partly by flexure of the pipe and partly by
movement of the joints. With this approach, only
rotation occurs at the hinge points, which must be
sealed by packing or bellows. Longitudinal pressure stress is transmitted through hinge lugs and
pins, ties, or a similar structure, which bridges the
sealing element. Moment due to expansion will
vary through most of the piping but may be zero in
certain portions of the system if more than a single
hinge is used. A typical single-plane problem of
stress evaluation of the stiff members employing
conventional analysis is given in Sample Calculation 5.6 of Chapter 5. The use of semi-rigid design
in single-plane systems is limited only by availability
of a suitable sealing element at the hinge point and
212
DESIGN OF PIPING SYSTEMS
by practical hinge design problems. Semi-rigid de··
sign is applicable to space configurations, but usually
involves limitations because ofllide moment effects
on hinge lugs, and is therefore limited to very low
pressure design of limited diameter. However, when
ground or packed universal movement joints art
used (limited to small size), space configurations are
entirely practical.
Such construction is useful in lowering the moment effects transmitted to terminal equipment and
in reducing the overall stress level, while at the same
time retaining the desirable features of self-support
and self-sufficiency for carrying longitudinal pressure
load. Size, pressure, and weight loads limit the
maximum size as dictated by hinge capacity. For
the usual pipe runs, there is generally no problem in
obtaining hinged joints with adequate rotation
capacity.
Non-rigid Piping Systems. A non-rigid piping
system is capable of transmitting longitudinal pressure load without separate structures and is entirely
free of thermal expansion forces and moments other
than the minor friction Or bellows resistance of the
expansion joints. A one-plane non-rigid piping system must have a minimum of three hinge points
which may be at terminal or intermediate locations.
Multiple non-rigid systems would require three additional hinge points for each added plane when the
joints are capable of movement in one plane only,
and are limited in application due to the eare which
must be exereised to avoid overload of the hinge lugs
by lateral effects. With universal joints, a nonrigid space system requires only three points of
artieulation but speeial problems in supporting and
braeing are likely to be encountered. The thermal
expansion design of non-rigid systems involves
merely the dimensional evaluation of joint rotation
as covered later in this chapter.
Non-rigid design is exceptionally useful for large-
transmitted from the piping, usually as a direct load
near its origin to external anchors or structures, or
may be carried by terminal equipment. If there is
an intervening elbow, the pipe must sometimes
transmit not only the unbalanced pressure load, but
also the moment which it introduces. Obviously,
such a system is illcapable of transmitting forces or
moments through the joints other than those required to move the expansion joints. Unbalanced
pressure load cannot be effectively transmitted across
an expansion joint without possible interference with
free axial movement. It can be balanced with additional compensating expansion joints, as will be
described later, but sueh an arrangement is expensive, so that usually unbalanced pressure loads and
frietion or bellows loads are transmitted to external
ties or anehors. This practice restricts the location
of expansion joints to positions where loads ean be
carried without exeessive eost. Where unbalanced
pressure loads are carried into equipment, care
should be exercised to assess the design provisions,
not only of the equipment, but also the anchor bolts,
foundation, etc.
\Vhere more than one expansion
joint is used to provide for a greater degree of expansion or to take care of movement in more than one
direction, the intervening portions of piping must be
supported and carefully guided to avoid damage to
the joints.
Free movement may be accomplished with flexible
hose instead of expansion joints. Both nonmetallic
and all-metal hoses are available with limitations
on pressure, size, and service flow. They can be pro-
vided with a flexible sheath capable of taking longitudinal pressure load, thus avoiding longitudinal
pressure unbalance. Corrosion and fatigue problems, however, rule against the use of hose in most
permanent installations.
Free movement systems are useful on low-pressure
piping to elosely connect rigid equipment, or to pro-
the layout provides suitable locations for the hinge
tect equipment from piping e.xpansion reactions.
\Vhen substantial pressures are involved, their use
points to keep rotations within economic expansion
is limited because of the cost of the anchors or the
joint capacity.
undesirable pressure reactions on connected equipment.
diameter low- or moderate-pressure service, where
This type of piping system is at-
tractive because it avoids the usc of expansion joints
with a considerable range of transverse movement
and the need for rugged external anchors to take
care of the end pressure load.
Free Movement Piping Systems and Runs.
Free movement piping systems and runs describe
piping in which all thermal expansion movement is
unrestrained, while longitudinal load except for friction or bellows effects is not carried through the
expansion joints. The longitudinal pressure load is
7.4
Packed Typc Expansion Joints
There are basically three types of eommercial
packed expansion joints: slip (axial), swivel (angular), and ball (universal) joints. Of these, only the
slip type has been extensively used for thermal expansion; hence only this type merits full description.
Connection and articulation devices will be men-
tioned only in general terms, as details are readily
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
found in manufacturcr's litcrature, in handbooks [2J,
and in the third article of reference [3J. Many
special packed joints havc becn designcd for specific
usage.
The slip type expansion joint is essentially a pair
of telescoping cylinders, and is basically similar to a
number of common eonnecting devices, such as the
mechanical (gland type) joints used for connccting
cast-iron pipe, and compression sleeve couplings
used on plain-end cast-iron or steel pipe. These
latter joints are usually sealed by a single packing
ring and, in addition to limited axial movement,
will accommodate a small amount of angular movement. They are not suited to absorb thermal expansion in other than mild services.where movements
are small and infrequent. The slip type expansion
joint requires an ample stuffing box and smoothly
finished sliding surfaces with controlled dimensions
and tolerances to be capable of satisfactory performanee in severe services, lind must be limited to purely
axial movement or rotation about the pipe axis for
satisfactory freedom from binding. A typical joint
is shown diagrammatically in Fig. 7.2 which illustrates a "single-ended" unit. When greater axial
movement capacity or "traverse" is required than
is desirably incorporated in one cylinder, a "doubleended" joint may be obtained having two stuffing
boxes in a common body usually supplied with
brackets for anchoring.
The outstanding feature of slip type expansion
joints is that large movements can be accommodated
readily, and usually with economy, since a substantial proportion of the cost lies in the packing gland
and related parts which are independent of the
movement capacity. Also, the poeketing created
when used in the horizontal position is of a minimum
amount and may even be entirely eliminated by
providing a single drain on the barrel.
The significant limitation of packed joints is the
difficulty of establishing and maintaining a seal.
While ground surfaces and piston rings are occasionally used, more generally the seal is dependent
upon packing with limitations on the contacting
fluid and its temperature. Packings arc not covered
herein, and arc governed by the same general considerations affecting pumps, valves, etc.
The considerable amount of force required to
overcome packing friction and its effect on anchor
requirements must not be overlooked. To minimize
this some of the better designed joints are provided
with means for lubricating the packing just prior to
start-up or shutdown movement, or periodically for
joints subject to continual movement in service.
213
Slip Pipe"
lubrkolion FittingJ
Drain moy bo provil$ed 10 eliminate pocketing in horilonlol lines
FIG. 7.2
A conventional slip t.ypc expansion joint.
Maintenance must be anticipated such as tightening
and occasional repacking, the extent of which varies
widely with the service. Excess friction and binding
can be minimized by even and minimum compression
of packing; most slip joints must be depressured for
repacking although plastic packing material is commercially available which may be used alone or in
combination with other packing and can be forced
into the gland during operation; piston rings or
other details are sometimes provided to obtain sufficient tightness to allow repacking under pressure.
The force required to overcome packing friction and
operate the joint is a function of the packing characteristics and the stuffing box pressure; for design
purposes, a value as high as 2000 pounds per inch
of diameter should be used, according to some manufacturers. This figure undoubtedly is conservative
for usual materials and moderate pressure, and is
largely intended to provide some allowance for
installation and operation effects-a rather futile
objective, however} as any margin it provides is in-
adequate to compensate for badly designed or poorly
constructed guides, or for sticking due to corrosion
or deposits. With proper installation and favorable
operation,. it can be said that one-half of the above
figure has been ample for many designs within the
usual service range up t0300 psi and 750 F maximum.
Slip joints are preferably provided with an adequate amount of internal guiding. With such provision, external guiding of the pipe adjacent to the
joint is usually not essential nor desirable, since the
temperature differences of supports may cause misalignment. Intermediate guides are not provided if
the connecting pipes are short, with no possibility of
buckling, and not subjected to lateral expansion
movement. If, on the other hand, significant lateral
movement is present, a double guide, consisting of
two sets of double-acting stops, is usually necessary;
it must be designed and installed to assure straightline movement if jamming is to be avoided.
214
DESIGN OF PIPING SYSTEMS
Although the foregoing speeifieally applies to
joints of the axial movement type, it is also applicable for the most part to rotary motion types.
Aetually, any sliding joint ean usually take rotation
either alone or in combination with movement along
the pipe axis. A sliding joint designed for rotary
motion alone may be provided with two internal
shoulders limiting axial motion to the clearanee between, and enabling the joint to transmit the longitudinal pressure reaction. Rotary slip joints may
be self-sealing, at satisfactory pressure levels, if the
longitudinal pressure reaction is carried through the
packing. Certain eommonly used fittings may serve
as rotary joints. The most elementary of these are
ordinary serewed fittings, eommonly used in groups
Other nonmetallic materials also are in use and must
not be overlooked for speeial applieations. The
(metal) bellows joint is not without drawbaeks associated with its inherent thin wall. The major hazard
of a bellows is blowout with sqdden large-seale release of the pipe eontents. This possibility ean be
satisfaetorily minimized only through adequate design as to stress level and seam details, etc.; propel'
selection of material; quality eontrol of materials and
fabrication; and careful planning, supervision, and
inspection of the installation.
In this respeet, re-
quirements are somewhat more exacting than for
sliding joints. The situation is reversed with regard
to servicing~ since bellows joints require no attention
other than hinge lubrieation on hinged joints.
An
of three or morc for taking expansion in connection
occasional inspection for possible corrosion or other
to building-heating risers, ete. Substantial movements can be accommodated in this manner provided the offsets are adequate to minimize rotation
so that the threads ean be kept tight, and provided
the service fluid and temperature are favorable.
Under the tenus revolving, swing, or swivel joints,
are included fittings which permit angular (rotary
or hinge) motion about one axis only. Ball joints
permit universal motion and consist of spherical contaet surfaces in a ball-and-socket arrangement; sueh
joints are available with either packed or ground
joints. Both the swivel and ball type joints may
have a wide range of angular movement and can be
used in groups of two or three or more to take care
of extreme amounts of space movement. For example, a U-shaped arrangement of five or six feet in
length with three joints provides ample range to
take eare of the relative motions of two ships moored
together when the auxiliary steam lines are connected. Such joints are usually limited to 6 in. size
and to Iow- or moderate-pressure service.
damage, and periodie measurements of the joint
position are desirable to assure that movements of
the pipe have remained within a suitable range.
The force to compress or extend a light-gage eommereial bellows joint may be on the order of 50 to
300 pounds per ineh of diameter, mueh less than for
the slip joint. In the absenee of experimentally derived data this foree may be estimated as suggested
in Subsection 7.5g of this seetion. The longitudinal
pressure reaction of an expansion bellows varies with
the design, but for most praetieal purposes ean be
based on the area at the mean diameter of the bellows, and will normally be higher than for a slip
joint.
Exeept for its inability to aecommodate axial rotation or twisting, the bellows type expansion joint
is more versatile than the paeked joint, sinee indi-
7.5
for equalizing and supporting the bellows, whieh
will be discussed in Subsection 7.5c. There is also
a wide variety of external eonstraints which adapt
vidual assemblies may combine axial movement, it
substantial amount of angular rotation or llcocking , 1!
and also lateral movement or "offset." This versa-
tility leads to a wide variety of bellows shapes and
construction features, as well as auxiliary devices
Bellows Type Expansion Joints
7.Sa Discussion. In the bellows joint the seal
between adjoining pipe ends is effected by means of
a highly flexible membrane. With the need for paeking eliminated it is frequently termed a Hpackless"
joint. The principal problem of paeked joints, that
of maintaining tightness is avoided since the bellows
provides a positive leakproof seal. The bellows is
generally of all-metal eonstruetion with fabrieation
possible from any eommereially available and weldable material. Bellows made of rubber are also
available and find important, although restricted,
the joints to a particular movement or combination
of movements, which merit detailed treatment in
Section 7.6.
7.5b Bellows Details.
The expansion bellows
has appeared in a variety of shapes, a representative
selection of whieh is shown in Fig. 7.3. Bellows eontour usually represents an attempt for an optimum
compromise between the opposing requirements of
flexibility and eapacity to withstand internal pres-
application in low-temperature water piping, principally circulating water, where their corrosion and
sure. In the commercial expansion joint the bellow!:i
abrasion resistanee are noteworthy features [3J.
and are therefore necessarily made from corrosion-
corrugations usually are formed of thin sheet metal,
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
JL
resistant material. Deterioration of any kind is an
all-important factor in view of the high strain level
generally utilized to meet economro and dimensional
restrictions.
Bellows corrugations fall into the following general classifications as to ,configuration and unit
extent: (I) flat disc; (2) formed disc; (3) formed
individual element; and (,4) formed assembly, The
flat and formed discs require exterior welds for joining into elements and interior welds for further combination into a bellows assembly, Flat discs are not
illustrated, the upper six details of Fig, 7,3 being all
variations of formed discs, The first two of plate,
usually -h in, to i in, thick, have found occasional
use mainly on low-expansion vacuum and exhaust
steam lines when conventional joints were unavailable, or expensive, They are usually designed at a
moderate stress-range level to secure fatigue life
comparable to that of the piping, and to avoid stress
corrosion, The flued-head design has given good
service in a number of applications, but occupies
considerable space and is expensive for the movement which it provides, The next four details are
formed discs of light-gage sheet steel. The "corrugated," HU-shaped~' and Clrounded" contours are
successive stages for somewhat improved capacity
for internal pressure,
In order, their shape reduces the number of bellows per linear foot and total extension obtainable
with a given bellows length, The stress for a given
deflection per element is at the same time somewhat
reduced by the curved contour,
The next two details are individual elements
formed from a hoop of light-gage material and thus
eliminate exterior circumferential welds, They are
sometimes made with longitudinal welds also, but
this is not desirable, The U-shape is otherwise similar to the preceding contours, The toroidal elements can be made to larger dimensions and of
heavier material with full emphasis on capacity to
withstand pressure without external support. Lightgage multi-element toroidal bellows formed as a unit
assembly from a single cylinder are also available,
The last two details are again a compromise between flexibility and internal pressure capacity, but
arc formed as a unit assembly from a single lightgage sheet metal cylinder, Sharp corners can be
avoided and favorable contour obtained,
Early preference for close pitch elements, due to
the greater number possible in a given space and
consequent greater movement capacity for equal
dimensions, has largely disappeared in favor of better
pressure capacity and a lesser amount of critically
215
U-5hcped
Flo! (Conical)
Buill up of Plate
Built up of Flue<! He<ld~
flot (Collicol)
Corrugated
Light Gogo Sheet
Rounded
U-Shoped
Ught Goglt Sheet
Toroidal (Cirwlor) or
Semi Toroldol (ellipticoQ
U-5hoped
light ~go Strip
N\f\
U-Shoped
Rounded or 5-Shoped
Formed from light Gogo Cylinder
FIG. 7.3
Various shapes of bellows.
located welding for shaped contours, Close pitch
elements afford less self-cleaning and thus are more
prone to collect sediment which may interfere with
compressive movement and inflict damage, The
open contours also provide better accessibility for
inspection and possible repairs, which are usually
limited to those of a minor nature, since when extensive, the usual result is early failure,
None of the bellows contours is self-draining for
joints in a horizontal position; the rounded and
toroidal shapes also will not completely drain in a
vertical position. Drain connections in the bellows
involve serious stress intensification and possible
weld flaws, and should be avoided,
Bellows are available in layer construction to the
U-shaped and rounded contours of Fig, 7,3, thus
providing increased total thickness for internal pressure, The exact behavior of this layer cOIlStruction
is not yet established, With a close fit and absence
of relative movement at the contact surfaces, the
action would approach that of a structure of the
combined thickness, However, if the contact is such
that movement along the contact surfaces occurs
freely, then a movement capacity equal to that of
an individual layer is obtained for a given stress
range, The load to compress the bellows is a multiple of that for an individual layer, The behavior
is probably nearer that of free movement along the
DESIGN OF PIPING SYSTEMS
216
~
FJO. 7.4
~
Plain bcllO\vs expansion joints.
contact surfaces, in which case it is proper co speculate on the distribution ·of the pressure effeet between layers. With hydraulie forming, the layer
contact is likely to be occasional or absent when
pressure is removed. It is likely that movement
occurs between the layers as temperature changes,
with the pressure reaction between layers offering
only limitcd rcsistance in proportion to degree of
contact and unit load.
The outer layers are usually vented by small diameter holes at the ends, out of the area of high
bending stress, to promote detection of initial rupture or leakage of the inside layer and to minimize
trapping moisture in initial assembly or due to minor
service weepage with possible collapse. Since the
life of the joint is still limited to that of the thin
inner layer, the additional layers do not extend the
service period over that Axpected from a single layer
joint insofar as cyclic movement is concerned; they
do add strength against pressure cffects however,
and tend to minimize the effect of a failure.
7.5c Support and Protection of Bellows.
Simple unsupported bellows (see Fig. 7.4), sometimes referred to as of non-equalizing type, are least
expensive and are used where the service is not too
severe and in locations where ample fixation and
guiding is provided in the piping system. They may
consist of single or multiple assemblies as needed for
capaeity, the number of elements in a single bellows
being limited by considerations of lateral buekling
or llsquirming" as it is sometimes called. On unsupported open-type (U-shaped) corrugations the
pressure limitation generally recommended by most
manufacturers is 30 psi, with higher pressures used
on more favorable contours, altpough \vithout ex-
ternal support only the toroidal contour is used for
significant pressure. Closely spaced discs, usually
corrugated, with heavy end pieces, also withstand a
straint to assure distribution of the axial movement
between the assemblies, and to remove the weight
of the spool pieces from the bellows.
:Means to equalize extreme expansion movement
in a bellows assembly, and in addition to provide
some measure of support to the elements against
internal pressurc loading, is generally considered
desirable since it guards against lateral distortion
(squirming). For corrugated-type bellows, this usually consists of rings suitably contoured to fit in the
spaces between the convolutions, as shown on the
joint of Fig. 7.5. By such arrangement, compressive deflections are definitely limited for each individual element. The support rings may be connected
externally to further insure some equalization of
element movement in all positions; this provision
may be combined with other motion-constraining
or .overall limiting devices described in Section 7.6.
As a support for the bellows against internal pressure
effects, the equalizing rings occupy a role similar to
that of the casing of a pneumatic tire in containing
its inner tube. Obviously, they are most effective
for this purpose in the fully eompressed position.
Rings are sometimes east and sometimes fabricated
by welding. They are usually split into 180 sections and assembled with bolted joints; where used
for substantial pressure one-piece construction is
0
necessary.
The expansion bellows is fragile in comparison
with the pipe and is often covered for protection
against damage during installation or subsequent
service. External sleeves at the same time provide
some measure of operator protection in the event of
a blowout, but must be arranged so as not to inter-
fere with the joint movement and should be removable for inspection of the bellows.
A sleeve on the inside offers protection against
flow erosion, and also against corrosion where maintenance of the corrosion-products film affords protection. The use of an internal sleeve, even though
it may slightly reduce the inside diameter, usually
reduces the pressure drop through an expansion
joint by reducing turbulence. At the expense of
requiring a larger bellows, full-flow area may be had
reasonable degree of pressure when closely spaced,
probably by intersupport of the discs.
Angular and offset movements, as well as axial,
may be accommodated, although two or more bellows assemblies are usually used for offset since, if a
spool picce of sufficient length separates tbe assemblies, the offset attained by this means greatly reduces the required bellows length. Such double
bellows joints should be provided with external con-
Exlel'1ded
FIG. 7.5
Fully Compress.cd
Sclf-cqualizing expansion joint
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
217
by recessing the sleeve; in some cases this construc-
tion may be desirable in order to minimize erosion
of the sleeve itself. An internal sleeve further assists
in keeping the flow in the line away from the bellows
when the joint is installed in a vertical position and
the sleeve is sealed at the top, and may be further
improved in effectiveness by the use of a purge
medium continuously supplied at slightly higher
than line pressure to the space between the bellows
and the sleeve. This has been found to be the most
favorable arrangement with respect to minimizing
contact with the flow or entrance of solids or fluid
whether the line is subject to up-flow or downflow.
Other sleeve arrangements may be necessary or
preferred where other factors are involved. A number of arrangements and details of attachment are
shown in Fig. 7.6. Sufficient clearance must be provided to permit the design movements of the joint;
however, the annular clearance should be kept to
the minimum possible to restrict entry of foreign
Bleed
Connection
Simplo Welded·in
Sleevo
Recened Sleeve With
Bleed Connedion (permib
cocking or offset movement)
material and to minimize purge flow requirements,
if used. Although not often practicable, it is nevertheless desirable that the sleeve be easily removable
for possible replacement or access to the bellows for
cleaning or inspection.
7.5d Fabrication of Bellows Joints. Due to
the fact that expansion joints are used primarily in
free movement piping systems, there is a tendency
to accept lesser design and fabrication quality for
the flanges, necks, etc.
This is, of course, in error
for types of joints which must transmit end load,
and it is questionable generally to allow lower
quality than required for the connecting pipe stub.
Minor flaws can jeopardize the life of the relatively
expensive expansion-joint assembly.
For the bellows assembly, where cyclic strains of
extreme magnitude arc commonly accepted in order
to attain large movement capacity with minimum
space requirements and cost, the aim should be to
achieve a construction quality (particularly of wcldments) equal in fatigue performance to that of the
base material. This poses a challenging problem for
the welding which must be used in all bellows, except
for those which can be made from seamless tubing
or shells. For weld quality which will least affect
the cyclic life obtainable from the base material, the
following measures are of benefit:
1. Minimize the extent of welding.
2. Locate welds away from areas of maxImum
bending stress.
3. Avoid corner, fillet, blind root, or similar welds
Two-pie<c Overlapping SI(WlV(I
(permih small gap fOf wckifl9
movement)
FIG. 7.6
"""'"
Removable Sleeve
fOr Flanged Joint
Internal sleeve arrangements.
of undetermined root quality. Use only butt welds
where possible.
4. Minimize heat-affected zones.
5. Use full heat treatment on ferritic materials,
and on austenitic welds an homogenizing anneal.
6. Avoid root oxidation on welds, and emphasize
elimination of inclusions. Inert gas shielded welds
are preferred for this and to minimize undercuts.
7. Weld procedure should emphasize soundness
and physical properties as nearly identical to the
parent metal as possible. Skilled operators must
secure welds to this procedure.
8. Weld surfaces must be as smooth as that of
the sheet steel, which should be pickled and No. 1
finished, and preferably ground.
9. Use all applicable nondestructive examination
toward assuring the quality of welds, including
pressure and movement tests.
10. Careful fit-up is essential. No interruption
in the assembled surface is tolerable.
To comment specifically on the details of Fig. 7.3,
the conical or flat-plate weld detail as shown is considerably superior to edge fillet welds which have
sometimes been used with poor results; the outer
218
DESIGN OF PIPING SYSTEMS
Bollows materiol
Ven Stoned oyor
£[ongo
8ellows matorial
leQ! welded 10 pipe
stub and supported
by rotaining bond
essary for satisfactory quality deposited metal with
smooth contour and absence of undercuts is difficult
to achieve on the shapes involved. Separate reinforcement is sometimes applied over edge welds, but
usually necessitates fillet or resistance wclds for
attachment and hence only serves to shift the critical location. With edge-welded discs, eonnection
to the necessarily heavier end pieces results in critical welding as illustrated in the lower detail of
Fig. 7.7.
With the U-shapcd and toroidal bellows elements,
the outer circumferential welds can be avoided.
....
ssss~
FIG. 7.7
Heavy end colTI,Igolion.
High bending slrem15
in weld are nol avoided
wilh this deslg'.,
Bellows attachment details.
weld may also be made against a chill ring and increased in depth or back welded if access is possible.
However, the welds are so locatcd that they are
subject to maximum cyclic bending; consequently
very minor flaws will rapidly propagate, unless
stresses of a low order are maintained. The f1uedhead detail permits butt welds; those at the inner
diameter can usually be baek welded; at the outer
diameter back welding is possible if the bellows
width is not too great, othenvise these welds must
be deposited against less desirable chill rings. In
the past, fabricated plate-typc bellows were prefen'cd, since bettcr manual welding quality could be
obtained than on thin sheet material, and thcy are
less readily eritically damaged by stress corrosion.
Such joints have been used at stress ranges comparable to expansion stresses in piping and much
lower than now common for commercial light-gage
bellows.
The ncxt four illustrations iu Fig. 7.3 show lightgage formed sheet discs assembled with the boundaries in fiat contact, which permits seam (resistance)
welding or arc, gas, atomic hydrogen, or inert gas
arc welding.
Excellent wclds have been obtained
particularly where extra precautions have been exercised to minimize root oxidation; however, there is
generally a signifie:.mt stress raising effect at their
location. Resistance welds cause a surface depression, sometimes sharp, and may not be uniform,
particularly at the inner edge; fusion welds are subject to material variations even when deposited by
automatic highly controlled inert arc means and, in
any casc, involve a blind root; in addition, there is
generally a strcss raiser at the toe of the wcld. The
discs of rounded edge contour can be butt welded;
however, the high degree of control and fit-up ncc-
The inner welds remain, so that they serve only to
halve the number of critical welds. Elements are
sometimes made up of several sectors joined by
longitudinal welds. If such welds are carefully
ground flush, inspected, and heat treated, they are
tolerable; unfortunately, however, they are often
left as welded with heavy reinforcement and introduce a significant stress raiser. For joints of toroidal
contour the inside circumferential welds can be located away from the zone of maximum bending by
forming the edge over intervening contoured pipe
spacers, or completely eliminated by hydraulic forming of several toroidal elements from a single eylinder.
The last two details can be formed from a lightgage cylindcr by rolling or by stretch forming under
internal hydraulic pressure or pressure as trans-
mitted through eompressed rubbcr; hydraulic pressure cxpansion into external dies is most commonly
used. Drawn or spun seamlcss shclls of othcr I.han
small size are expensive, so that such cylinders are
usu~lly rolled from sheet metal and havc a longitudinally welded seam. To avoid wrinkles and kinks
and failure during forming, the thickness and contour at the weld must match that of the sheet metal.
The forming operation assists in the location of
flaws and assessment of ductility, provided it is fol-
lowcd by a re-examination employing applicable
no.ndestructive means. As in the case of the toroidal
element the end which attaches to the heavier pipe
stub ean be formed and extended to permit the
attachment weld to be located favorably as to stress
level; t.he weld can be further protccted against
bending by the use of external reinforcement, not
welded, but shrunk or clamped as shown in the
center detail of Fig. 7.7, or the bellows assembly can
be further lengthened and rollcd ovcr end flanges
in the manner of Van Stone construction as shown
in the upper detail of Fig. 7.7, thus eompletely
avoiding circumferential welds.
From the above discussion of details, it should be
clear that the commercial constructions which are
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
most likely to develop fatigue life comparable to
the sheet material and to have generally reproducible performance are those which~avoid welds completely in zones of high bending stress or which
involve a single longitudinal flush weld madc prior
to drastic forming opcrations, which is then carefully inspected and heat treated.
7.5e Establishing Purchasing Requirements
for Bellows Joints. Expansion joints are usually
supplied by a specialty manufacturer. Stock items
are available, particularly with respect to axial
movement joints of IPS standard sizes, but not in
production quantities comparable to that of fittings
and valves. More complex movement types and
large sizes are custom made assemblies utilizing
standardized details for bellows contour, and general details to minimize tool and fixture costs.
Economy is achieved by use of the supplier's
standards, insofar as they are consistent with the
quality needed. In the purchasing of expansion
joints for other than simple installations, it is essential that the supplier be informed of all requirements
and conditions under which the joint must function.
This may be accomplished through data sheets and
specifications, although involved cases may require
drawings. The data sheet can convey design re-
219
9. Installed length and any other space limitations.
10. Limit stops or other constraints required.
11. All loads on constraints (extraneous to those
produced by the joint itself).
12. Materials selection, applicable specifications,
and thermal treatment requirements.
13. Fabrication requirements, particularly as to
welding.
14. Testing and inspection, including nondestructive examination.
15. Marking and shipping requirements.
16. Temporary positioning devices to secure joint
in desired position during installation and which
also protect it during shipment; lifting lugs as required to facilitate erection handling.
17. Applying codes, specifications, and drawings.
18. Information to be furnished by vendor.
Special services will involve additional variables.
This list is not intended to encourage complex and
unnecessary, or unduly restrictive, requirements, or
facturer, and also in assuring that all variables are
insistence on minor details which will only increase
cost without proportional benefit; instead it is intended for use as a checkoff design and purchasing
aid in order to avoid overlooking the essentials.
The final requirements will represent a careful balancing of the cost of various desirable features
against an evaluation of their necessity as dictated
by the hazards of joint leakage or failure and attendant financial loss due to maintenance and loss
of productive capacity. Consideration must be
given to the size and importance of the plant, as
well as its dependence on the particular piping for
continued operation. Experience will dictate the
receiving consideration.
degree to which manufacturer's ratings arc reliable;
quirements as to movements, service conditions, etc.
The specification is necessary wherc a level of acceptable fabrication, inspection, and tests must be
established. Drawings are required when jntricate
movements or special design features arc present.
The following check list will be found a convenient
reference when preparing information for the manu-
1. Flowing medium.
2. Design pressure and temperature. Metal temperatures are sometimes much lower than maximum
flow temperatures.
3. Movement demands, normal operation, and
extreme, as required for starting-up, shutting down
or by upset or emergency conditions.
4. Cyclic effects, frequency of movement, pressures and temperature variations and desired life.
Include not only those attendant to complete periods
of operation, but also those that occur during
where in doubt, the check method for associating
life with stress range which is included in this section
should be used.
Expansion joints are obtainable both with and
without flanges, so that flanges can be omitted at
the expansion joint when considered undesirable.
Similar to valves or other components, which may
require servicing, some users prefer flanges rather
than cutting out and rewelding. Flanges are desirable on alloy lines where stress relief is necessary
operation.
and in general provide for more accurate installation.
It is important to be specific as to cocking (angu-
5. Type of bellows and method of joining to body
(for bellows joints) which is acceptable for service.
6. Dimensions and details of end connections
(flanged or welding).
7. Other connections (drains or bleeds) in body.
8. Sleeves internal or external, or both.
lar) and offset (linear) dimensions. Unless the joint
is to be preset to obtain the total motion required
by movement of the joint on both sides, with installation in a neutral position, the entire cocking
or offset will occur on one side of the center line.
In the interest of economics, it is advisable to preset
L__..~._
220
DESIGN OF PIPING SYSTEMS
the joint for installation so that when assembled
into the piping system, full capacity will be utilized.
This step is, of course, unnecessary for ocCasiO~lS
where the required range is about equally divided
on both sides of the center line.
7.5f Materials and Deterioration. A knowledge of the flowing medium is important from the
standpoint of potential corrosion or erosion. On
light-gage bellows elements, even mildly corrosive
conditions may seriously affect service life in view
of the high stress levels present with conventional
movement ratings. Condensate corrosion during
standby periods, or when metal temperatures are
below the dew point in service is a. notorious contributor to bellows joint failures. Careful flushing
at shutdown will avoid mueh corrosion trouble, particularly where complete drainage is not possible;
reliance on drainage alone is apt to be ineffective.
Knowledge of flow temperatures does not give complete assurance as to metal temperatures, which may
be much lower, depending on exposure or degree of
insulation, with increased condensation corrosion
hazard.
The non-expansion parts. of a joint should be of
materials comparable to that required for the connecting pipe. As mentioned earlier, opinion exists
that for free movement piping systems, minimum
requirements for design, materials, fabrication, and
inspection can be used. However, when the relative
investment in an expansion joint is given consideration, it seems unwise to risk its utility by any compromise in quality of its components. Packings
for sliding joints, as already mentioned, are ·a special subject.
Deterioration of bellows material is influenced by
its thin sheet material form and severe demands involving cyclic strain. Cold work, variations in
analysis or structure, thermal history, inclusions,
and segregation, all contribute to increased sensitivity. Corrosion, particularly as associated with
stress and fatigue, may result from contact with
ordinarily noncorrosive media, so that weight~loss
data or conventional corrosion tests are not reliable
guides. Where a material of assured resistance to
such attack is not economically available, it is necessary to reduce the stress range to purely elastic
action (within twice the yield strength), and in extreme cases to no more than twice the basic allowable stress. Superficial overall corrosion and initial
traces of concentrated attack or pitting are sufficient
to cause accelerated fatigue failure. The possibility
of intergranular corrosion on austenitic steels, as
associated with ehromium depletion at the grain
boundaries, should be avoided by the use of stabilized
materials, with maximum resistance to other accel-
erated attack generally obtained if the composition
is completely austenitic.
Strength, ease of forming, and weldability are also
important considerations. Early expansion joints
were made mostly of copper, but this material is
now largely supplanted by the stainless steels.
These, because of their inherently greater strength
can be used in lighter thicknesses in contact with a
much .wider range of fluids and at higher temperatures arid pressures.
7.5g Fatigue Basis for Predieting Bellows
Life. As a rule, the manufacturer's ratings for bellows joints are not clear eut and do not indicate the
number of cycles to failure. Ordinarily, with the
high stress range attendant to the manufacturer's
rated maximum movements, the number of useful
cycles is apt to be only a small fraction of the
7000 cycles established for pipe in the 1955 Piping
Code rules. Manufacturers are inclined to cite similar service applieations in justification of their
ratings; such statements must be discounted in the
absence of direct data of actual service, or test data,
including the number of cycles.
It is believed, however, that a rational approach
to the bellows design problem is attainable and that
the operation of a bellows, although usually in the
plastic range, does not differ greatly from that of
any loc~lized areas of stress intensification of ordinary piping systems for which design criteria are
now available in the Piping Code, as outlined and
discussed in Chapter 2. Application of this stress
range approach to the fatigue performance of bellows material must be accompanied by an attempt
to recognize and evaluate the effect of local stress
raisers which cannot be eliminated.
Fatigue testing of actual joints is the soundest
approach for establishing the characteristics of a
basic approach or of a specific design. Since no
simple relation has evolved for interconversion of
movement and pressure capacity, incorporation of
detailed rules in the Code would be premature in
tbe present state of knowledge. A basic design
should be verified by tests to establish sufficient data
for extrapolation to other sizes. Since tests of each
specific size of each-design are impractical, there remains the necessity for translating test results to
other sizes of joints.
Until available tests are sufficiently numerons to
establish close parameters for a particular configuration or a more accurate general formula, the following approximate approach has been found to give
\
I
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
reasonably useful results. Expansion joint tests [4, 5J
presently available, while not numerous, nevertheless are in sufficient number to slillw that in general
the results parallel those obtained from tests on
other piping components in the relation of strain
range to number of cycles to failure. Such tests
show that
I
N,,-
En
where N = number of eycles to failure.
E = total range of unit strain due to movement and pressure.
n = a constant for the material used.
The exponent n usually ranges from 3.3 to 4 on
stainless steel bellows, usually nearer the lower figure. Therefore, it is reasonable to assume a value
of 3.5 for stainless steel bellows elements. In tests
of carbon-steel piping components the range was
found to vary from 4 to 6, with an n value generally
m the vieinity of 5.
In keeping with the inclusion of a safety factor
on stress for other parts of a piping system, it would
seem desirable to provide a safety factor of at least
2 on stress (or strain) in the bellows at rated eonditions. This safety factor on stress in turn means a
margin of (2)3.' = 11 on the number of cycles to
failure, since the change is rapid with stress variation. In addition, a minimum performance should be
established for cases where the number of cycles in
operation is expected to be rather low, and it is suggested that the same minimum number of 7000 cycles
used for piping also be applied to the joints. This
will keep the expected performance reasonably well
in line with the remainder of the piping system designed in accordance with the stress range as discussed in Chapter 2. Where cyclic performance
tests are possible, they should be made at the rated
movements and pressure, with both varied over the
range for each cycle.
The following approximate relation can be used
in assessing or extrapolating test data for probable
performance of stainless steel bellows:
(7.1 )
800,000)3.'
(7.2)
8R
where N s = number of cycles which the joint should
be expected to endure safely in service.
This relation affords a means of combining the
effect of extreme emergency movements or other
movements which can be combined with nOfmal
operation to a single equivalent condition. For example, if the extreme movement produces a stress
range S 1 and is expected to occur at most N 1 times,
the fractional part of the fatigue life used for this
condition would be
N,
(7.3)
eo~~oor
and the design number of cycles for the normal
movement must be increased by this fraction to
secure the equivalent number of cycles at normal
movement.
This also makes it apparent that the provision of
design movement in the joint for installation tolerances or similar reasons need not be considered as
having any significant bearing on the fatigue performance of the joint, since they provide for an
initial, not a repetitive condition.
The range of stress in a single-layer bellows may
be approximated by eqs. 7.4 to 7.7. These expressions are based on simple beam analogues similar to
that presented in Chapter 3, Fig. 3.15, with the
constants somewhat increased. The second term
represents the effect of pressure and should be kept
within the Code allowable stress 8 h , although joints
of the type covered by eq. 7.4 have successfully operated at higher stress levels. The pressure term in
eqs. 7.6 and 7.7 ignores pressure bending stresses,
which although not entirely proper provides fair
correlation with available fatigue tests. Despite
the drastic simplification involved in these approximations they generally yield a reasonable estimate
of the strain range for purposes of estimating performance using a criterion such as eq. 7.l.
For flat disc bellows:
3EItJ.
pw2
8R = w2Nd + 2f
(7.4)
where N = number of cycles to failure.
Sn = calculated range of stress, psi, as given
byeqs. 7.4 to 7.7.
For U-shaped bellows without equalizing rings:'
As noted above, the design serviee rating should not
1A more exact evalua.tion of the stresses in a U-sha.ped
bellows may be found in reference (6}.
be assumed higher than about 10% of this or
'S'-
Ns = (
221
8 11 =
2
+ -pw
hO·'w1.5Nd
21 2
1.5EttJ.
(7"'
\
DESIGN OF PIPING SYSTEMS
222
For U-shaped bellows having equalizing rings
which provide support against internal pressure only
along inner edge:2
"""'-
1.5EUJ.
pw
Sa = ho"w1.5Nd + - t
(7.6)
For modified toroidal bellows having minor axis
of ellipse about 0.8 to 0.9w.:
Sa = 1.5Ett>. + pw
w'Nd
t
(7.7)
For true toroidal bellows, Sa may be found from
[7, 8) and the torus membrane formula.
SR = range of stress due to expansion and pressure,
psi.
t = bellows thickness, or thickness of longitudinal
weld seam with reinforcement, whichever is
larger, in.
ti = total movement range, extension and compression, plus equivalent axial movement, in.,
as given by eqs. 7.8 to 7.11.
E = modulus of elasticity at 70 F, psi.
w = bellows width, in.
h = pitch of half-corrugation, in.
N d = number of active bellows discs or halfcorrugations.
p = internal pressure, psi.
The equivalent axial movement corresponding to
the angular rotation on universal, or hinged type,
joints may be determined by
t>. = DOj2
(7.8)
where D = mean bellows diameter, in.
o = total angular rotation, radians.
The equivalent axial movement for a single-bellows offset joint, based on the most severely affected
corrugations (those at the ends), is found from:
t>. = 3Dh,jL
(7.9)
where D is as above and
h, = offset range or total lateral displacement
of one end of the joint with respect to
the other, in.
L = overall length of bellows, in.
For a joint having a double bellows separated by a
spool piece the equivalent axial movement becomes:
3Dh,
t>. =L
-=--+-/::-'[(7Cljc:'-L):-+---:1)
(7.10)
where L = overall length, end to end, of bellows,
including spool piece, in.
I = length of spool piece, in.
'Limit use of Eq. (7.6) aod E':. (7.12) t, w/3 :> h :> w.
The force, F, in lbs, necessary to deflect the bellows an amount t>., can be stated as follows:
For flat disc bellows:
F =
"EDt't>.
,
W Nd
(7.11)
For V-shaped corrugations:'
4EDt't>.
- 3ho. 5W 2 .5 N d
F -
(7.12)
For toroidal expansion joints, consult [7, 8).
7.51. Testing and Quality Control of Bellows
Joints. Expansion joints must properly be classed
with special rather than mass-produced equipment,
with regard to the extent of inspection and testing
during manufacture and installation and of other
quality controls applicable to the fabrication details
and materials involved. In common with specialized equipment, the advisable degree of control is
also related to the reliability of the producer.
Structural tests on bellows expansion joints include independent pressure and movement tests,
and also combined pressure and movement tests.
Tests at I! times the design pressure have been
opposed as unnecessary by manufacturers employing hydraulic forming who reason that the bellows
has already been adequately tested by the forming
pressure. This position ignores the support provided by the external dies during forming which
may be absent or considerably different on the actual
joint, particularly in the extended position which
should be the joint position for the test. Adequate
pressure tests are good insurance against possible
damage during and subsequent to hydraulic forming,
such as in flanging the ends, full heat treatment, etc.
All possible assessment of weakness is highly dcsirable, since once assembled, access is restricted.
Proof tests often cannot be applied after installation
due to weak elements elsewhere, the variable of
liquid load, etc. For pressure tests, whether shop
or after installation, unless the joints are of hinged,
tied, or pressure balanced type, the end load must
be taken by a test frame or other structure.
Tightness tests at operating pressure are routinc
with each cycle of operation or period of maintenance; low and intermediate pressure air tests are
often used at various stages of fabrication for the
same purpose. Such tests essentially only cheek
for leaks.
Movement or flexure tests are desirable, particularly in combination with pressure in assessing the
structural capacity of expansion joints l and assuring
that gross fabrication flaws are not present; if re-
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
peated a few times, the latter aim is more definitely
assured. When extended to an appreciable number
of cycles, the fatigue strength of.that specific joint
and also to a lesser extent of the general design· is
verified, but the joint tested is saerificed.
Flexure elements, in view of their critical nature,
are properly subjected to all nondestructive examination applicable to the material analysis. Usually,
radiographic examination is of limited value due to
the thin material, and ultrasonic examination is not
sufficiently developed for general use on the configuration involved. Magnetic powder examination
is applicable to ferritic materials, and the various
crack detection aids, such as fluorescent penetrant
oil viewed under ultraviolet illumination, penetrant
dye suspensions, and volatile liquids as absorbed by
chalk powder may be used on all materials. It is
preferable that the examination follow a few flexings and where practical be performed at both extreme and neutral positions since tensile stress
would render flaws more readily detectable. The
examination should cover the sheet material, the
welds, and parts to which the bellows is directly
attached.
l
7.6 Expansion Joints with Built-In
Con-
straints
The movement of non-stiff piping systems must
be controlled within limits which do not exceed the
working capacity of the flexible elements which it
contains. This may be accomplished to suit particular situations by:
1. Restraints at desirable locations in the piping
system but not on the flexible members.
2. Details of the flexible-member assembly which
limit the motions of individual elements, or assemblies.
3. Details of the flexible-member assembly which
are of sufficient strength to restrain movement of
both the piping system and its flexible units.
In addition flexible members may be preset to
establish the initial desired bellows position as a
dimensional guide in erection, or else can act as
rigid units exerting positive control of the initial
desired position. The details which limit or control
the extreme range and installed position, or which
serve as a guide for installation dimensions when
either permanently or temporarily a part of the
flexible member assembly, fall within the designation "built-in constraints" as employed herein.
This section is devoted to their examination. Bellows joints, due to their morc widespread usage, afC
used as a principal vehicle for discussion.
223
Flexible elements are subject to damage with any
unanticipated movement which exceeds the margin
provided over their operating capacity. Such movements may originate with foundation settlement,
failure or distortion of structures, yielding of piping
due to temporary overloads, and unexpected piping
deflection such as that due to circumferential thermal
gradients. Distortions which involve lengthy periods of time, such as creep, have the same effect
except that they can be anticipated and the ranges
of movement adjusted occasionally.
Exce.."Sive movement may disengage packed joints
or render them inoperative by distortion or rupture.
Bellows assemblies may be damaged by kinks, etc.
with subsequent rapid fatigue failure, or may suffer
direct rupture, particularly at welds; also the overall
joint may be damaged by distortion or fracture of
stops, hinges, etc.
Expansion joint failure may cause direct or contingent damage resulting from the whipping effects
of the suddenly released piping, or by injury to
personnel, economic loss by fire, and loss of contents.
Packed axial-type joints usually involve some inherent restraint, which can also easily be provided
on the swivel type when desired. Except for highly
specialized types, bellows joints are capable of only
limited torsional resistance, which is undependable
since buckling cannot be closely predicted. In general, bellows joints are dependent for protection
against undesired change in position on the piping
system stiffness, the external restraints provided to
limit movement of the piping system, or on their
built-in constraints. In the following paragraphs,
general types of joints will be examined for suitable
constraint arrangements to accomplish one or more
of the alternate objectives which have been mentioned.
In assessing the desirability for and favorable
choice of built-in constraints, the piping system
should first be examined as a complete frame.
Weight of the metal components, contents, insulation, and attachments must be taken care of externally where the joint design provides limited or
no capacity for such loading; wind, earthquake, or
possible shock effects arc similarly treated. Chapter 8, while intended primarily for stiff piping, also
provides useful assistance for non-rigid piping support and restraint selection and design. Next, the
piping system should be explored as to possible
locations for the installation of suitable guides,
stops, etc., to define and limit movement at the
expansion joints, and simultaneously 1 on the basis
of preliminary analysis, as to the selection of the
224
DESIGN OF PIPING SYSTEMS
flexible joint types and their number. Usually both
the number of joints and their individual number
of elements will be minimized iIl"' the interests of
safety, minimum maintenance, and cost. The final
step is the settlement of general constraint requirements as associated with the movement tolerances
to be applied to t he operating movement range of
each joint.
Built-in constraints for packed joints can usually
be provided simply and without appreciable cost,
if the basic design is rugged and capable of carrying
the additional loading without distortion and attendant binding or leakage. Some of the commonly
used arrangements which are capable of wide variation are as follows:
Axial motion may be limited by an external structure, sueh as tie rods, or by providing a single
exterior shoulder or set of lugs on the male member
and two confining internal shoulders on the easing.
The rotary angular movement of swivel joints may
be confined by lugs on the body or on the elosure
cover. Internal provision can consist of slotting
the male member to a width defining the range of
motion and providing interior lugs on the easing
located properly within, and with respect to, the lugs.
Ball-and-soeket joints are usually capable of transmitting end-pressure load without. auxiliary construction, and swivel (rotary) joints, with arrangements which have been described, can function in
the same manner. An objection to the use of swivel
joints is the 90° change of direction with attendant
pressure drop. A special design of swivel joint has
been in service for a number of years in the main
steam line of a moderate-size high-pressure steam
generating unit, and employs metallic packing which
is self-sealed by the end reaction of the internal
pressure.
For bellows joints, external constraints for limiting movement during scrvice usually fall within one
of the following basic types.
I. Limit rods which confine single bellows assemblies to a desired range of rotation or lateral movement. (See Fig. 7.8, where the lateral movement
type only is illust.rated.)
2. Limit rods in one or more sets, which establish
the range of axial, angular, and lateral movements
for universal type expansion joints. (See Fig. 7.9.)
The rods also serve to minimize the influence of
weight effects of intermediate pipe runs. Similar
details are required when multiple bellows assemblies
are used on axial joints.
3. Hinges which limit bellows to bending effects
to secure purely angular movement; the range of
movement may be established by integral stops,
details of the hinges, or by external limit rods.
(See Fig. 7.10.)
4. Piston arrangement for confining bellows movement within close limits to a purely axial direction.
(See Fig. 7.11.)
Tie rods may be of any structural shape of sufficient cross section to carry the tensile and compressive loading without yielding or buckling. Where
bending or offset motion is involved, the end connections of the rods arc preferably provided with
spherically turned surfaces, hinges, or other det.ails
which will minimize friction. Care should be exercised to assure that the lug plates and their attachment to the adjacent piping are adequate to avoid
local overstressing of the pipe wall with consequent
excessive deflection and unpredictable performance
of the tie rods. Stops are provided on both sides of
the lug plates and are carefully positioned to estabSwivel or Hinged Connection
\
Tie Roch
FIG. 7.8
Tied expansion joint.
Limit Rods
/
Ill.
;11"
It\
"11
\11/
:Iv.
·u·
FIG. 7.9
limit Rods
Universal type expansion joint.
Hinge pin provided
wllh fiNing for servico
lubrication
Welding end construction
Ulown. If £longed. hinges
ore lrSuolly boIled 10 the
flanges.
FIG. 7.10
Hinged expansion joint.
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
lish movement range as related to the bellows
assembly.
When joints are installed in a1iorizontal position,
full-length rods of sufficient strength to provide support for the floating pipe sections are essential, unless other support (external to the joint) is provided.
Such support is to be approached with caution,
however, so as to avoid imposing unanticipated
lateral deflection on the bellows.
The use of hinged expansion joints to accommodate substantial amounts of expansion was initiated
and developed by The M. W. Kellogg Company for
large-diameter piping in fluid catalyst cracking
units and similar applications. Figure 7.10 illustrates a typical design, although many variations
of hinge and other details have been employed. In
an extreme detail, hinges are integrated in complete
cylinders surrounding the joint with sufficient clearance only for the necessary motion; this arrangement
is for maximum distribution of the hinge reactions
around the circumference. The hinges may be attached to the backs of conventional bolting flanges
Of, for welding end construction, to special rings as
shown on the illustration. With hinges provided
on opposite sides of the bellows and installed so that
their hinge pin axes will eoincide with eaeh other,
and essentially with the midpoint of the bellows
element, the maximum movement capacity will be
realized. Machined construction and assembly with
reasonably close tolerances is required. The limitation of this design is that the entire end load due
to pressure, weight, and other effects, unless relieved
by counter weights or springs, must pass through
FIG. 7.11
225
the hinge pin; the hinges and pins must also resist
transverse loading due to wind and other effects,
unless adequate external provisions oppose their
effect.
Axial-movement-type expansion joints are quite
similar in detail to their counterpart in packed
joints, with the substitution of a bellows seal for
the packing. When constructed with machined
parts, this type of design is obviously expensive and
can be justified only for special cases where installed
location and general system arrangement make it
difficult to achieve effective guiding by other means,
or where a large amount of axial movement must
be provided and guiding of the bellows assembly
against lateral buckling is necessary. With an internal arrangement it is also possible to secure maximum confinement of the bellows against abrupt release of contents, and sometimes an auxiliary packed
joint is provided to further minimize this hazard
particularly for toxic content services. Sometimes
joints of this type arc fabricated to reasonably close
forming tolerances, without machining; however,
without lubrication, binding is much more likely to
occur.
In combination, constraints and auxiliary expansion joints can be used to balance end loading due
to internal pressure at an expansion joint location.
This is most readily achieved where a joint is installed adjacent to an elbow, as shown in Fig. 7.12.
In effect, the elbow now becomes a tee with an
identical auxiliary expansion joint as an extension
of the run and with the primary and auxiliary joints
connected by tic rods or similar arrangement. It
Axial movement type expansion joints.
226
DESIGN OF PIPING SYSTEMS
A
~connodion$
\ f-
)
.K'!'~ r
,,,,
TIe Ro<h
J
Blind flongo
>
FIG. 7.12 Pressure balanced expansion joint.
can readily be seen that the end pressure load is
now carried across both joints through the tie rods
and that there is no unbalanced pressure load acting
on the elbow. There will, however, be an unbalanced force acting on the elbow equal to the sum of
the elastic forces required to compress one bellows
and simultaneously extend the other, which should
not be overlooked for the moment loading it may
introduce in the piping. Similar arrangements can
be worked out for packed joints. For joints in
straight runs of piping, designs utilizing larger concentric joints have been devised to effect pressure
balancing but are usually so expensive as to be impractical. Aside from their cost, pressure balancing
by auxiliary expansion joints usually involves the
drawback of dead ends with no flow, unless circulating lines are provided; Subsection 7.5j has pointed
out the hazards of condensation corrosion for metal
temperatures below the flow dewpoint.
Apart from the above types of built-in constraints,
the movement range of joints is sometimes controlled by arrangements and connections to independent structures; for example, limit stops may be
provided which can be temperature adjusted to
establish the maximum compression or extension of
a joint, or, similarly, to limit lateral movement.
Attention is again called to Chapter 8 for the many
types of external restraints which can be used to
control the position of piping at expansion joints.
As previously indicated, the structural capacity
necessary for built-in constraints is dependent on
whether auxiliary external limit stops or other external aids are provided.
To assure proper installation, it is more effective
and often economic to provide preset means for
establishing the installation position of the joints,
which may be consolidated with the built-in constraints for limiting service movements 'where such
afC of sufficient strength, otherwise by a separate
temporary structure.
Where the size of the line and other details permit, it is preferable that these preset restraints be
adequate to provide strength comparable to the
attached piping. In such cases, the installation can
be accomplished without special precautions for protection of the expansion joint and the restraints
released only after piping erection is entirely completed, and just before starting up the job. In the
usual assembly of a rigid piping system, weld shrinkage, the necessity for pulling ends into alignment, as
well as pull-up effeets of flange joints, all combine to
establish an erection internal stress. When suddenly
imposed on a preset expansion joint by removal of
the preset ties, whipping of the line may occur unless
restraining means are employed; more important
deflections and rotations comparable to the internal
loading will appear at the joint, which will in some
degree obviate the preset objectives. This can be
minimized by thermally unloading the piping before
the preset ties are released in the manner described
in Chapter 3.
Built-in constraints and preset ties should be installed under adequate engineering supervision fully
familiar with the dual objective of accurate positioning and protection of the relatively fragile
elements involved. Usually this is more adequately
accomplished by the specialist manufacturer of the
expansion joints, when the necessary facilities are
at hand.
7.7
Establishing Expansion Joint Movement
Demands
The amount and direction of expansion affects the
selection of the type of expansion joint and its constraints which, in turn, control the movement capacity needed in the jomt. Similar to stiff piping systems, design capacities must be established so as to
cover normal operation and all routine conditions.
such as starting-up and shutting down, as well as
all possible occasional and emergency eonditions.
Movements of connected vessels or other equipment
and the effect of connected structures must be ineluded in addition to the expansion of the pipe; in
general, any situation must be examined which may
affect movement at the joints. Prediction of service
life must distinguish between the normal and the
occasional movement demands as covered in Subsection 7.5g. To provide for fabrication and erection deviations, as well as for service distortion of
the piping assembly during its life, the maximum
range of movement of the joint should provide some
excess capacity over that required for thermal expansion effects alone; the amount is dependent in
some degree on the joint design and its built in constraint details. Therefore, setting the final purchase
conditions for an expansion joint requires a broad
knowledge of the process operations which dictate
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
the expansion movements, a thorough familiarity
with installation procedures, and a fair comprehension of the structural details of the system as they
may affect deformation of the pipe immediately or
over its life period.
The approach for determining thermal expansion
movements is dependent upon the type of piping
system involved. An important consideration, as
already pointed out earlier in this chapter when the
systems of varying degrees of stiffness were described, is whether or not the flexure of the pipe is
used to provide in any measure for the thermal
expansion of the system. If bending is present, it
is necessary to obtain the deflection and rotational
displacements by methods given in Chapters 5 or 6.
Where piping flexibility is not a contributing factor,
the task then is to obtain from the original dimensions, and the expanded position, the axial and lateral movements, as well as the angular displacements, at each joint location. This section is
devoted to outlining a convenient approach for obtaining this change in line position with temperature.
No recognized nomenclature applying to expressing the movements of expansion joints presently
exists. Whereas misunderstanding is unlikely for
sliding joints, there has been confusion at times in
properly understanding the movement ranges of
bellows expansion joints. For this reason the conventions shown in Fig. 7.13 are followed.
Tolerances for probable deviations in conventional
erection may represent a substantial part of the cost
of the joint. When such incremental cost is sufficiently large, consideration should be given to more
precise and morc expensive installation, as offering
possible greater economy. Tolerances may be minimized by:
1. Favorable location and structural details, and
a planned and adequately presented erection procedure.
2. The usc of expansion joint prepositioning
structural arrangements, adequate for rigid assembly of the entire piping system without deflection
of the joint, as described in Section 7.6.
Many j Dints present individual problems; in general, it is more economical to depend entirely on
tolerances for installations which are not too complex or when an available margin of excess movement capacity exists or can be had for little extra
cost, and to resort to both tolerances and prepositioning devices, etc., only where critical service or
economics justifies both. Plain bellows joints (nonequalizing) without limit stops and with open-type
corrugations will stand some deformation without
227
affecting their performance. However, offset should
be avoided on single corrugation assemblies, and in
any case should be limited, to avoid local areas of
appreciably reduced radius, or kinks. It is necessary to provide for some deviation from the desired
installation dimensions even when the joint is rigidly
restrained during erection of the piping system,
since the effects of cold pull and weld shrinkage will be
taken up by the joint when the temporary structure
is removed, although these effects may he minimized
by thermal unloading as previously mentioned.
Pressure or temperature deflection, weight and
transient effects, and minor irregularities are difficult to predict. Movement eapacity is usually not
provided for creep deformation of hot lines due to
thermal or weight loadings and similar long-time
changes in dimension, on the assumptions that expansion joints may involve periodic replacement
due to corrosion or fatigue failure and that dimensions will be periodically checked and adjusted if
necessary.
In the case of packed joints the extreme movement limits are needed for establishing the clearances
within the joint and, in the case of bellows joints, arc
Ulldcll~ed Position
~-- A. =Axiol Compreuion
~--8=AllgUIOr
FIG. 7.13
Rotation (radiclld
Action of expansion bellows under
various movements.
DESIGN OF PIPING SYSTEMS
228
rI
Sample Calculation 7.1
Theofcli«ll Installed
Pcnilion
Given Data
.~
t...lVVV\IVVV\.
rvvvvvv\IVl
Normal·
Ronge
IA:::;:" ["--i)
I
~owonccs and Tolerclfl(O$
PLAN
Extreme limits
of Movement
FIG. 7.14
Diagram illustrating range of axial movements of
an expansion joint.
l
Ay,
• b
;..;
y
~Xb
SUPPORTT.::::::::of-Tt----::.
TRUE DISTANCE FROM CENTER-
Sign convention for
rotationll + rotation
Indicahn joint will
open on lid. marked
with an asterisk
~30.92+9.022
c
132.2'
Cellfen of
Hingo Pins
"
FIG. 7.15
, "
Diagram of general three-jointed
hinged-joint system.
ELEVATION
needed for establishing the maximum movement
Unit Expansions
and constraint settings. Figure 7.14 diagrammatically illustrates the range of movements including
tolerances for a simple joint. Sample Calculations
7.1 and 7.2 illustrate the calculation of joint movements including addcd allowances and tolerances.
@400 F, e ~ 0.00229 in.jin.
@700 F, e = 0.00482 in.jin.
Expansion CaI.culalions
Axial:
12(1 J.l2 XO.00229 +39.1 XO.OO482) ~2.57"
Where movements of appreciable range, not evenly
distributed with respect to the center line, are involved, presetting the joints to utilize both sides of
Offset:
= 1.86"
12X32.2XO.OO482
the center line is recommended.
In Sample Calculation 7.1 the calculation of a
universal-type expansion joint is illustrated. While
essentially self-explanatory, it may be of benefit to
point out that the expansions have been calculated
on a coordinate system such that the basic coordinate lies along the axis of the joint, disregarding the
orientation of connected piping or vessels, a procedure which is usually advantageous in simplifying
the computations.
The movement determinations for a three-hinged-
joint system are readily handled by the method outlined below and illustrated in Sample Calculation 7.2.
Such a system may be treated as a linkage assembly
of rigid members; thus the rotations are limited to
the joints and for any given layout are simply a
function of the termiual displacements of the system
and the incremental expansion change in length of
Axial
Compression
Calc. expansion range
Allowance (10%)
Installation tolerance
2.57
11
0.26
11
Exten!lion
Cocking Offset
(Each Side of <1<)
1.86"
0°
0°
0°
0.19
0.5 "
0"
0"
0.5"
Total
3.33"
0.5"
0°
2.55 11 •
Precock or pre-offset
.. .
...
0°
1
Design movement
3:}"
·~s
,,,
0°
Ijll:t:
11
0.5 "
"j
*Uequired on one side of ¢. if no pre-offset.
tPre-offset may be ! of the total movement range (not
including tolerances) = ~ (1.86 + 0.19) ="=I I".
:t:Mnxirnum movement from ¢. with 1" pre-offset =
2.55 - 1 "'" 1.55 ~ Ii".
I
l
APPROACHES FOR REDUCING EXPANSION EFFECTS: EXPANSION JOINTS
the members, or of position, if vessel movement
must be included. While a trigonometric solution
is indicated, it cannot be accomplished with sufficient
accuracy on the slide rule, and the use of logarithms
is time consuming. Hence, the following formulas
were developed which are amply accurate, giving
slide rule solutions to within a few minutes for small
rotation angles (say about 5° or less). The notation
is in accordance with Fig. 7.15, which shows a general three-hinge system.
If Ll x is the effective displacement of joint b relative to a, it may be defined as:
llx = .6 x b -
Expansions (ft)
(treated as given data in this sample calculation)
Case I
1060
O.OOSIS
70
0
xe . ..
0.IS4
0.0
-0.034
0
0
0
0.145
0.111
-0.006
0
0
11= .....
Ll,. ........
Case I
A. ~ -0.034 + 0 - 0.IS4 ~ -0.21S'
Ll, = -0.006 - 0.111 - 0.145 = -0.262'
X'Yo - XoY' = 9.S5 X 16.05 - 12.65 X 1.75 = 136
(-0.21S X 9.S5) + (-0.262 X 1.75)
ISO
136
X -;;:-1.100
<Po =
~
::~
.
•
~
l'
(-0.21S X 12.65) + (-0.262 X 16.05)
ISO
136
X -;;:= -2.94°
<P' =
<Po' ~ 1.10 + 2.94 ~ 4.04°
Case I!
YO
A. = 0
I
x'
t>, ~ 0.275 + 0 + 0 = 0.275'
v.
X
<Po
Dimensions (ft)
Xo ~ 12.65
x, = 9.S5
x = 22.5
O.27,~
Cal£ulations
--<::
"'~
-;,,..
AYO
. ........
6 11/, •• ,-
Given Data
ot>-
. .........
yeo ....
61/a.
Sample Calculation 7.2
7' ,,-
Case I!
T (Line Temp. F) ...
e (Unit Exp. ft/ft) ..
xc
d xa -
229
= 0 + (0.275 X 1.75)
136
ISO = 020°
X"
.
<P' = 0 + (0.275 X 16.05) X ISO ~ I.S60
Yo = 16.05
136
y, = 1.75
y ~ 17.S0
"
<Po' = -0.20 - I.S6 = -2.06
0
Summary
Rotations
Rotation Range
Design
Pro\'ideu
PrePositions Each Side ¢.
cock
Tolerance
Required Provided*
(Nominal)
:l1lce
- - - - - - - - - ----- ----- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4.0
-1.10
0.11
-0.60
0.5)
+0.5
S.O
2.43
a
4.0
+0.70
0.5
0.02
+0.20
----- - - - - - - - - - - - - - - - - - - - - ----- - - - - - - - - - - - - - - - - - - - - - - 4.0
1+ 3 .04
+4.04
0.40
0.5)
-1.0
S.O
7.71
ab
4.0
[-3.06
-2.06
0.21
0.5
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - ·1.0
-2.94
0.29
-2.44
0.5\
II
S.O
+0.5
6.2S
b
4.0
+2.30
0.19
+I.S6
0.5J
\I!
---_._,-. -
Exp.
Case
Jt.
Calculated
AlIo\\'-
hi
lr~
i
i
Noles: All rotations given in degrees.
+
sign indicates rotation such as to open joint on side indicated on the sketch by an asterisk.
·1'he same movement capacity wa.." selected for all three joints on the assumption that the ndvantages of uniformity offset the
extra mat-crinl requirement.
DESIGN OF PIPING SYSTEMS
230
References
.6.v is similarly defined:
1. F. E. Wolosewick, "Expansion Joints and their Application," Petroleum RefinCT, Vol. 29, No.5, pp. 146-150
where e is the unit expansion which applies to the
piping between a and b. The rotation of joint a in
degrees is:
The rotation of joint b in degrees is:
cPb =
AxXa
+ AvYa X180
-
XbYa -
XaYb
7r
The rotation of joint ab is found from:
cPab + cPa + cPb ~ 0
Dimensions and expansion must be in consistent
units.
(1950).
2. S. Crocker, Piping Handbook, McGraw-Hill Book Co.,
New York, 1945.
3. J. E. York, uJoints to Permit Movement," Healing and
Ventilatino. Vol. 46, No.1, pp. 85-88 (Jan. H149)i Vol. 46,
No.2, pp. 93-07 (Feb. H149)j Vol. 46, No.3, pp. 87-91
(:-.lar. 1949).
4. F. J. Feely, Jr. and W. M. Goryl, IIStress Studies on Piping
Expansion Bellows," J. Appl. Mechania, Vol. 17, No.1,
p. 135 (1950).
5. W. Samans and L. Blumberg, IlEnduraoce Testing of
Expansion Joints," ASME Paper No. 54-A-103 (1954).
6. F. Salzmann, "Ueber die Nachgiebigkeit von Wcllrohrexpansionen," Schu:eiz. Bauzlg., Band 127, Nr. 11, pp. 127130.
7. R. A. Clark, "On the Theory of Thin Elastic Toroidal
Shells," J. Math. and Phys., M.LT., Vol. 29, pp. 146-178
(1950).
8. N. C. Dahl, "Toroidal Shell Expansion Joints," J. Appl.
Mechanics, Vol. 20, No.4, pp. 497-503 (1953).
1
CHAPTER
8
Supporting, Restraining, and Bracing
the Piping System
A
THOUGH a piping system may properly be
described as an irregular space frame, it
differs from eonventional structures in that
frequently, due to its slender proportions, it may not
be self-supporting or it may need to be restrained or
braced against eertain effects.
Service temperatures can introduce sufficient
thermal stress or lower the material strength so as
to require supplementary structural assistance.
Limiting the line movement at specific loeations may
be desirable to protect sensitive equipment, to con-
needed to correct failures, sagging, leakage, eqUlpment damage, difficult maintenance, etc.
The analysis of thermal and structural effects in
piping is of limited value unless paralleled by support
design sufficiently complete to assure realization
of the flexibility analysis assumptions. Injudicious
or over-use of supports or lack of advantageous
restraints and braces can create an overload hazard
instead of giving protection to sensitive equipment Of, for satisfactory performance, can require
wind, earthquake, or shock loading. In the absence
needlessly long runs of pipe. While accurate calc).llations are usually not cconomically feasible for
average piping, much can be accomplished toward
of significant thermal expansion, such as in water
economic and satisfactory design by approximations
service, conventional structural practices and the
use of available standard hardware are entirely
adequate and economic.
For other piping where service temperatures
introduce sufficient dimension change and reduction
in material strength, adequate design of supports,
and reasoning when applied by personnel of adequate
trol vibration, or to resist external influences such as
engineering background and experience.
Except for idealized counterbalancing as approached by counterweights, all supports involve
some degree of restraint; conversely, many restraints
and braces unavoidably resist gravitational effects,
so that it is logical and convenient to combine their
treatment ill this chapter. While they involve a
restraints, and braces requires a satisfactory grasp
of localized loading and thermal gradient effects on
high-temperature pressure shells, and reasonable
understanding of the thermal changcs attendant to
service requirements, including emergency and
auxiliary conditions. This latter background is
more readily available during the initial stages of a
project. The planning of pipe supports, restraints,
and braces simultaneously with the establishment of
layout configurations, also offers the added advantage of a more cconomic installation. When relegated to the status of a job-end finish-up item or left
solely to the erector, only conventional structural
treatment can be expected, with later changes
considerable expenditure, pipe supports, restraints,
and braces have received insufficient attention in the
literature from either design or economic aspects.
This chapter will attempt to present general knowledge and opinions which have guided the support of
average piping, and also certain information for use
ill combination with Chapters 4 and 5 when more
careful analysis is necessary.
8.1
A discussion of the problems involved in the provision and design of supports and restraints can be
231
L
Terminology and Basic Functions
232
DESIGN OF PIPING SYSTEMS
presented effectively only after the terms used to
describe them are clearly defined and their functions
are clearly understood.
.~
In the absence of an authoritative text, the
terminology adopted in general for some time by
The M. W. Kellogg Company has bcen used throughout this volume. It is summarized for the reader's
convenience in the following glossary with the hope
that it will receive general acceptance and contribute
to clarity of thinking on the subject at large as well
as in this Chapter.
Restraint. Any device which prevents, resists, or
limits the free thermal movement of the piping.
Support. A device used specifically to sustain a
portion of weight of the piping system plus any
superimposed vertical loadings.
Brace. A device primarily intended to resist displacement of the piping due to the action of any
forces other than those due to thermal expansion
or to gravity. Note that with this definition, a
damping device is classified as a kind of brace.
Anchor. A rigid restraint providing substantially
full fixation (i.e., encastre; ideally permitting
neither translatory nor rotational displacement of
the pipe on any of the three reference axes). It is
employed for purposes of restraint but usually
serves equally well as restraint, support, or brace.
Stop. A device which permits rotation but prevents
translatory movement in at least onc direction
along any desired axis. If translation is prevented
in both directions along the same axis, the term
double-acting stop is preferably applied.
Two-axis Stop. A device which prevents translatory
movement in one direction along each of two axes.
A two-axis double-acting stop prevents translatory
movement in the plane of the axes while allowing
such movement normal to the plane.
Limit Stop. A device which restricts translatory
movement to a limited amount in one direction
along any single axis. Paralleling the various
stops there may also be: double-acting limit stops,
two-axis limit stops, etc.
Guide. A device preventing rotation about one or
morc axes due to bending moment 01' torsion. l
Hanger. A support by which piping is suspended
from a structure, etc., and \vhieh functions by
carrying the piping load in tension.
Resting or Sliding Support. A device providing suplAlthough some users employ the term "guide" loosely
to cover both translatory and rotational restraint and bracing,
it is felt that the distinguishing terminolob,)' used herein
promotes clarity of presentation.
port from beneath the pIpIng but offering no
resistance other than frictional to horizontal
motion.
Rigid (Solid) Support. A support providing stiffness in at least one direction, comparable to that
of the pipe.
Resilient Support. A support which includes one 01"
more largely elastic members (c.g., spring).
Constant-effort Support. A support which is capable
of applying a relatively constant force at any displacement within its useful operating range (e.g.,
counterweight or compensating spring device).
Damping Device. A dashpot or other frictional
device which increases the damping of a system,
offering high resistance against rapid displacements caused by dynamic loads, while permitting
essentially free movement under very gradually
applied displaccments.
Frequently, the detail used at a specific location
performs several functions, e.g., stop and guid<"
guidc and brace, or support and anchor. In such a
case the common practice is to designate the detail
for convenience by only one of the terms, whichever
best fits its primary function. For example, a
resisting couple is provided by a pair of parallel
stops; thus the combination may properly he referred
to as a guide.
In Chapter 2, attention has been directed to the
fact that piping systems, for reasonably economic
design, must operate over a wide range of stress
between ambient and service temperature as a
result of combined thermal expansion and pressurc
strains; the introduction to this chapter commented
on the frequent lack of structural capacity on the
part of unsupported piping to carry weight effects
simultaneously with pressure at the service temperature. Supports, restraints, and braces are there-fore desirable to reduce weight, wind, and, where
possible, expansion and transient effects, so that the
piping system stress range is not excessive for the
anticipated cycles of operation, thus avoiding fatigue
failure. Ideally it would be desirablc to providc
essentially continuous support, i.e., render the pipin~
weightless, and to provide restraints or braces
wherever stress reduction can be accomplished,
From the practical considerations of cost and general
arrangement requirements, however, supports, ett'.,
are limited to locations favorable to their installation, or where their use is offset by proportionate
piping cost reductions.
As pointed out in Chapter 2, piping systems may
function under one or more operating conditions as
J
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
dictated by changes in feed or end products, and as
influenced by service variables, in particular, cyclic
operation or the alternate use of-spare equipment;
they may further involve auxiliary operations such
as starting-up, shutting down, solvent cleaning,
pressure testing, etc., or unanticipated operating
upsets or emergencies due to equipment leaks,
power or equipment failures, etc. Each may involve
different temperatures for individual piping systems
or parts of systems as dictated hy the location of
valves; in addition the necessity for sudden depressuring or removal of contents may require abrupt
establishment of flow with extreme velocities and
reactions, sometimes accompanied by pulsations.
To avoid actual rupture, the design must give due
consideration to such possibilities, and must provide
reasonably adequate support for each set of circumstances. Particular emphasis is directed at conditions involving lengthy duration, high temperature,
and frequent occurrence. This usually dictates that
supports be most completely effective in normal
operation, since for shutdown or other lower temperature conditions the short-time structural
strength is at a higher level and provides greater
latitude for lack of support without damage to the
piping.
8.2
Layout Considerations to Facilitate Sup-
port
Initial layout study of equipment, building, and
structure location and elevation is essential for effective design j over-all economics and appearance are
further improved where the primary piping and its
associated supporting structures and their intereffects are simultaneously subjected to study and
planning. In addition to establishing the general
arrangement and over-all design conditions, early
decisions must be reached on the types of support
structures for intercommunicating piping and their
233
are axiomatic, yet they are surprisingly often ignored
by designers, with awkward and expensive supports
the result. Wherever a number of lines are to pass
through a given space in approximately the same
direction, they should preferably be carried at the
same elevation (assuming horizontal pipe runs), and
parallel to each other, to building walls, to column
lines, or to equipment axes. The piping thus forms
Hbanks" or "racks" which are easily supported on a
common beam or stanchion if overhead, or on sleepers
if the lines are placed just above the ground. Generally, any given line must be routed away from its
minimum-run course in order to include it in the
rack; however, the cost of the attendant extra pipe
and fittings is usually offset by reduced support
costs. Appearance, while secondary to utility, is
also enhanced by regular arrangement and avoidance
of unnecessary skewed or irregular runs of piping;
eye appeal and ease of supporting are usually complementary. Where there is much piping to be run
in an area, as is the case in the majority of process
plants, the most practicable scheme is to establish
specific elevations for groups of lines running in a
given compass direction (say north-south) while
setting other elevations for groups running in a
transverse direction (east-west). This arrangement
avoids interferences and also permits future piping
additions without undue difficulty or unsightliness.
The practice is exemplified in the oil refinery piping
shown in Fig. 8.1, also in the section through a
typical pipe rack in Fig. 8.2.
The decision whether to support a specific line or
group of lines from a building or structure and
thereby eliminate independent support structures is
based on the relative cost of the additional pipe as
compared with the additional supports. Detailed
comparisons are usually not readily accomplished
so that judgment on the part of the designer is necessary. Important lines are occasionally run skewed at
elevation, access provisions for maintenance of equip-
a sacrifice in appearance when significant savings in
ment, cleaning and inspection requirements, and
selection of main support fixture types. Ample
space must be provided for large devices such as
counterweights.
The two cardinal principles in routing lines for
piping or support costs can be made. For large lines
it is advantageous aside from process requirements
to make runs as short and direct as possible so that
the piping tends to be self-supporting. The necessary flexibility for thermal expansion must be
maintained regardless of the nature of the restraint
imposed by the pipe supports; (i.e., whether added
deliberately for the control of thermal expansion
stresses and reactions, or developed unavoidably in
the course of supporting and bracing against other
loads).
Control of thermal expansion by the use of re-
economic support, restraint, and bracing are:
1. Group pipe lines so as to minimize the number
of structures needed solely for pipe supports, restraints, or braces.
.
2. Keep lines located close to possible points of
support, etc., i.e. either to grade or to structures
which arc to be provided for other purposes.
From the standpoint of economy, both of these
straints can serve to:
DESIGN OF PIPING SYSTEMS
234
a. Sectionalize portions of a piping system for
isolation from the influences of the remainder of the
sion will contribute additional resistance to thermal
system.
a. Supports or braces can be used which offer
negligible restraint. Constant-effort supports to take
weight, and damping devices to resist dynamic loads
assure such freedom along all axes. Hangers and
jointed struts offer resistance on one axis and frce
movement on the other axes.
b. Supports or braces can be located at or near
neutral points, i.e., points where little or no thermal
Ioo.c..
b. Protect against overstress such weak spots as
locally reduced size runs of pipe, and sensitive local
or terminal components.
c. Control expansion direction so as to forestall
undesirable displacement at specific locations of the
piping.
Supporting and braeing the piping against loads
originating from sources other than thermal expan-
FIG. 8.1
expansion unless:
movement occurs along the one or more desired axes,
thus minimizing additional restraint.
A pipe bank in a large unit of an oil refinery.
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
The significant general eonsiderations affeeting
the routing of piping for favorable support may be
summarized as follows:
.....
1. The piping system should be self-supporting
insofar as praetieable and eonsistent with flexibility
requirements.
2. Exeess flexibility may make additional supports or restraints necessary to avoid movement and
vibration in such amplitude as to arouse personnel
apprehension. This situation is apt to occur on
vertical lines where only one point of support is
needed to sustain the weight.
3. Free movement expansion joint systems involv-
ing appreeiable unbalanced thrusts from pressure
should be avoided unless such forees ean be taken on
substantial struetures, or at grade.
4. Piping prone to vibrate, sueh as eompressor
suction or discharge and driver exhaust lines, should
235
be routed for support independent from other piping
and lightly braced structures and buildings. Routing
should permit the use of resting or similar supports
offering resistance to motion and providing some
damping eapaeity, rather than hanging supports.
5. The pipe line should be sufficiently close to the
point of support or restraint so that the structural
connection can have adequate rigidity and details
can be simple and economical.
6. Piping from upper connections on vertical
vessels is advantageously supported from the vessel
to minimize relative movement between supports and
piping; hence, such piping should be routed next to
the vessel and supported close to the connection.
7. Piping in structures should be routed beneath
platforms, near major structural members at points
favorable for added loading, to avoid the necessity
of making these members heavier.
A--
por1ion of in~ulalion simply cuI owoy
on modorato tomporature lines
Sa" nOM pia'''' ao .,,,, m.mb<n \
... ~.
~
I
----H /
A
L.
I
\
.
\
r..
(! ... "J. -
\fIClvQ~od temperature lines placed on
shOflI
I
permit1ing insulation
to dear
~Slf\ldurOI 'leol "onmion'
FlO. 8.2
~
Section through typical outdoor overhead pipe rack shO\.. . ing arrangement of north-south runs at two elevation:; llDd
east-west runs nt an intermedintc elevation.
236
DESIGN OF PIPING SYSTEMS
8. Sufficient space should be allotted so that the
proper support assembly details may be accom.~
modated.
9. Access clearance must be provided in order that
support fixture parts requiring maintenance can be
serviced.
8.3 The Elements of the Supporting System:
Their Selection and Location
In the design arrangement studies, the principal
piping is located for satisfactory flexibility and association with the equipment, buildings, and structures
to snit over-all functional requirements and maximum utilization for reactions from the supports and
restraints. In the final stages of this preliminary
engineering the stiffness of the individual piping
systems is examined with due consideration to the
selected restraints. Later when the plans and
elevations of piping, details of vessels) structures,
buildings, foundations and trenches, and setting
plans of pumps, compressors, etc. arc available, the
final selection and location of the supports, restraints,
and braces can be accomplished. This section will
be confined to discussing decisions as to the basic
types and locations of these supports, etc. Subsequent sections are devoted to details of these types.
Advantageous location involves consideration of
the piping proper, the structure to which the load is
transmitted, and the space limitations within which
the assemblies must operate. Preferred points of
attachment to thc piping are:
I. On pipe rather than on piping components such
as valves, fittings, or expansion joints. Under
highly localized loading, flanged or threaded joints
may leak and valve bodies may distort with resulting
seat leakage or binding. Attachments to heavy
components, however, may be acceptable and even
desirable where the effect can be properly provided
for.
2. On straight runs rather than on sharp-radius
bends or welding elbows, since these are already
subjected to highly localized stresses on which thc
local effects of the attachment would be superimposed. Furthermore, attachments on curved
pipe which extend well along the length or circumference of the bend will seriously alter the flexibility
of such components.
3. On pipe runs whieh do not require frequent
removal for cleaning and maintenance.
4. As close as practical t.o heavy load concentrations such as vertical runs} branch lines, motor
operated or otherwise heavy valves, and minor
vessels such as separators, strainers, etc.
Undesirable effeets of piping reactions on foundations, other-purpose structures, buildings, or vessels
can be minimized by locating supports, etc. so as to:
1. Apply loads to eolumn and beams near mainmember intersections to minimize bending effects.
2. Avoid the introduetion of unnecessary torsion
or lateral bending effects.
3. Avoid the introduction of moments or transverse loading to slender members (sueh as wind
bracing) and particularly to compression members
where instability controls the design.
4. Confine connections to independent structures
or foundations when dealing with piping subjeet to
pulsating flow or transmitted mechanieal vibration
unless a careful and comprehensive analysis is made
to assure that the struetures, buildings, etc., are of
adequate strength with nonresonant natural frequency and sufficient stiffness to control amplitude
within the bounds required by psychologieal effeet
and general comfort of personnel.
5. Provide anchors and extremely flexible and
nonresonant intervening pipe runs (e.g., expansion
joints) to maehinery introducing mechanical vibrations in order to isolate the effect by redueing
transmissibility.
In general the advantageous use of other-purpose
foundations, structures, ete. is dietated by their
ability to assume the additional loading with little
or no additional seetion. Recognition must be given
to the need in many cases for rigidity as well as
stress-carrying capacity of the structure used for the
attachment of supports, restraints, and braces.
Connection to relatively flexible struetures or tooslender members of those struetures must be avoided
if their deflection prevents their assumption of
loading in the desired degree because of the relative
stiffness of the pipe. The final ehoice of whether to
provide a separate structure depends on comp4·rative
costs.
Ample clearance must be available for the piping,
including that required for expansion movements of
the pipe and for the supporting elements and their
proper functioning. Fireproofing and insulation
interferences, if overlooked, afe a source of erection
difficulties, troublesome restraint, excessive maintenance, and poor appearance.
The introductory paragraph of this section calls
attention to the necessity for early establishment of
the significant restraints to the free movement of the
piping system under thermal expansion, so that the
general arrangement of equipment structures and
principal piping systems can be established with
proper appreciation of the piping flexibility, and so
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
that subsequent ehanges ean be avoided. It is
further pointed out that the final seleetion and location of the supporting system elements must necessarily await completion of the plans and elevations
of the piping, details of vessels, structures, buildings,
foundations, and trenehes, and setting plans of
pumps, compressors and other equipment. With
their availability an over-all view of the supporting
system problems is obtained, and since usually the
basic functions of restraint, support, and bracing
are advantageously established in that order, the
following discussion will be so arranged, first covering
rigid piping and then followed by an examination of
the problems peculiar to semi-rigid, non-rigid, and
free movement piping systems.
Restraints. Restraints to thermal expansion are
unavoidable in the terminal connections of the piping
system to equipment, vessels, etc., and the preceding
section has shown that additional restraints may be
desirable to control the position, stress, or reactions
at selected locations. The net effect of each added
restraint is a function of its location, and the direction and degree of limitation imposed. Unavoidable
restraints may occur at supports and braces and are
either minimized in relative magnitude by selection
or by location so that their effect can properly be
neglected, or else their influence must be provided
for by the flexibility of the piping system.
RehClolor
237
FIo. 8.4 Typical stops employing tie rods or jointed struts.
Total restraint in the form of terminal connections
to vessels or equipment for the most part establishes
the boundaries of individual legs of a piping system,
whereas that accomplished by direct anchorage to a
structure or foundation is limited in application to
locations where the structural subdivision of individual runs is considered desirable. In general each
added restraint reduces the inherent flexibility;
however, where sufficient margin in the stress range
16V 0.0. Sch. 80 pipe
is available, additional anchors may be desirable to
increase the line's self-supporting capacity, to define
the behavior of complex piping systems under
alternate and involved operating conditions, to
protect runs of lesser stiffness due to reduced section
or higher temperature, to isolate mechanical vibrations, and to change the natural frequency to mini-
c
mize amplitude and avoid resonance.
Turbintl
MomontJ (ft. kip,) and Forces (kip,) Adiog at Point A
SlOp 01 Point B
M.
Nol included
+~7.7
Included
FIG. 8.3
My
M,
F.
Fy
F,
Max. Slto" (PSi)
Magnitudo Poinl
-27.5 +38.9 -.120 -1.97 +1.77
11600
C
-27.9 + .c.l - 5." -.730 + .71 +2.«
12400
0
Use of a stop for the control of thermal
expansion reactions.
The control of thermal expansion reactions or
movements can often be achieved advantageously
without intermediate anchorages by the judicious
use of stops and guides. Frequently it is desired to
limit only a certain component of a terminal reaction
or of the displacement at some point. In such a case
the use of full fixation would produee unnecessary
restraint, in many cases intolerable, unless the piping
can be made more flexihle hy the addition of loops,
etc. A typical application is shown in Fig. 8.3,
DESIGN OF PIPING SYSTEMS
238
forbo
two'''''''''"''
oxh
"""
rostrairrt
~",.""..~.
Fin and Sl"~ for
Small Amounts of
ExpaMion Oiometrally
Clips for Small linn
~S¥
floor Dock.. or Roof
FIG. 8.5
Typical sliding type stops.
which shows a vertical double-acting restraint placed
in a high-temperature reheat steam lead in order to
acting but also they can be made to suitably close
clearances.
Sliding guides and stops frequently may fit more
compactly into a structure. Particularly for vertical
lines, they may be incorporated into the platform
steel at pipe openings where they offer no reduetion
in headroom or other obstruction to passage. In
addition if their application is such as to add frictional resistance to vibrational effects, some advan-
tage may be gained. For the most part, however,
sliding devices are apt to be unsatisfactory for
resisting vibratory loading because of the fabrication difficulties attendant to producing minimum
clearances.
Supports.
The piping system with its terminal
anchors and partial restraints must now be explored
for its adequacy in carrying without distress all
gravitational loading including the weight of the
lower the thermal expansion moment reaction on the
pipe, insulation, contents, fittings, valves, strainers,
turbine. The accompanying table indicates for such
a line a typical set of results of calculations made
with and without the stop.
etc., or any additional weight which may be involved.
While most of these loads are maintained both in
Two basic arrangements of restraints are used, viz.,
one condition. Maximum loads are usually apt to
occur at service temperatures and, becauseof reduction of the material strength, must be considered
directly in the high-temperature design. In the
case of large gas or vapor lines, provision for support
when filled with liquid may be necessary as, for
example, to provide for hydrostatic testing. 2
The spacing of supports on a single horizontal pipe
line in open country is dependent only on the
strength of the pipe. Within the boundaries of a
those such as the typical tie rods and jointed struts
shown in Fig. 8.4, wherein the structural connection
is relatively remote from the pipe attachment, and
those wherein little separation is needed between the
terminal parts, shown in Fig. 8.5. While the details
of these devices are treated in subsequent sections,
the selection of the basic arrangement to be used is
in large measure dependent on the layout, the location of the restraint, and the purpose or combined
purpose for which it is being introduced.
In general tic rods and jointed struts are preferred
for single-acting and double-acting restraints respectively whenever sufficient room is available to
provide adequate length so that the arc of motion
will not deviate sufficiently from a straight-line
path to cause unnecessary restraint. The amount of
deviation may be found as indicated in Fig. 8.6.
The principal merits of tie rods and jointed struts
are their low frictional resistance making them
positive acting and nonjamming. It should be kept
in mind that generally tie rods and single-acting
service and airstream, others may be present only at
process unit, on the other hand, support spacing is
largely determined by the spacing of conveniently
located columns. Commonly, the spacing of support racks must provide for the weakest pipe, although larger spans are sometimes accepted for small
2For infrequent tests, it is sometimes economic to erect
temporary supports rather than design permanent supports
for the purpose.
Tolol movement tongll
devices are only suitable for restraints when sufficient
constant load exists at the point to overcome any
thermal reaction either in the initial or the selfsprung state of the piping, or under any variation
from normal operating conditions.
For example, a
hanger rod will function properly as long as the
weight load exceeds any uplift due to thermal
expansion. Hinged-jointed or ball-jointed struts
are the ideal restraints; not only are they double
0= Deviation from tlto;ght line
h'
""'2i
FIG. 8.6
Motion of tie rod or jointed strut.
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
lines if sag and attendant pocketing of the particular
small lines are not objectionable. Small lines can be
assisted across long spans by prllviding them with
intermediate supports attached to adjacent larger
lines; a group of such lines may also be tied together
so as to become chords of a simple truss. Often,
however, the most practical solution is simply to
increase the pipe size to the point of being selfsupporting over the required span.
In checking the suitability of support spacing for
pipe lines on a horizontal run, the nomographs ment.ioned subsequently in this section are useful for
most purposes. For critical-service piping, the
flexibility check for expansion stress can be extended to include weight effects where necessary by
using methods given in Chapter 5. As discussed
previously in this section, general considerations in
locating supports are that they be placed at points
suitable for the connections to the pipe (no interference with valves, risers, etc.) and to the structure
(in respect both to attachment details and to loading requirements).
Allowable spans for horizontal lines are principally influenced by the need to:
1. Keep stresses within suitable limits. (Instability may be a factor in the case of large thin-walled
pipe. )
2. Limit deflections (sagging), if necessary for:
a. appearance,
b. avoiding pockets,
c. avoiding interferences.
3. Control natural frequency (usually by limiting
the span) so as to avoid undesirable vibration.
In most cases, an adequate estimate of the stress is
readily obtained from the simple beam relationship:
(8.1)
S = 1.2(wl'jZ)
where S = maximum bending stress, (psi.)
Z = section modulus, in. 3
I = pipe span, ft.
w = total unit weight, lb per ft.
For convenience this formula is given in nomographic
form in Chart 0-16 of Appendix C. It is based on
a maximum moment of /.11 = 110wl2, and represents
a compromise between M = ,\w12 for a beam with
fixed ends and M = !w12 for a free-ended beam, as
representative of average runs. Values to suit other
end conditions can be obtained by the use of the
correction factors given in Chart C-18. Overhang
at changes of direction may be beneficial from
a structural standpoint; if provided in optimum
amount, the maximum moment in a line continuous over a series of e'lual spans can be held to that
239
of fixed-end conditions. However, substantial overhang is best avoided on lines prone to vibration.
Major concentrated loads such as produced by
valves, pipe risers, branches, etc., should be at or
near a point of support. The effect of significant
concentrated loads, not located at supports, may be
approximated from eq. 8.1, by multiplyiug the stress
by the factor 2P jwl where P is the concentrated
load in pounds and other symbols are as previously
defined.
Deflection under weight effects is generally of secondary importance in piping just as it is in structures. In fact, some piping designers are inclined
to disregard deflection entirely and to consider the
limiting weight stress as the only criterion. In most
process units, however, the deflection of the line
should be kept within reasonable bounds in order
to minimize pocketing and to avoid possible interference in congested areas due to sagging. Appearance, too, will be a factor in many cases. A practical
limit for average piping in process units is a deflection on the order of ~ in. to 1 in. For piping in
yards or for overland transmission lines a value of
It in. or greater is generally acceptable. For power
piping a deflection limit as small as! in. is specified
by some designers.
Perhaps the most important reason for limitjng
deflection is to make the pipe stiff enough, that is,
of high enough natural frequency, to avoid large
amplitude response under any slight perturbing
force. Although Chapter 9 treats this subject more
fully, it can be stated here, as a rough rule, that for
average piping a natural frequency of 4 cycles per
second will be found reasonably satisfactory. For
pulsating lines from compressorsJ etc., values of
8 cycles per second or higher may be desirable depending on the characteristics of the compressor.
The deflection for a given span may be approximated by the beam rclation:
0= 17.1(wl'jEl)
(8.2)
where I = moment of inertia, in. 4
I = pipe span, ft.
o = deflcction, in.
E = modulus of elasticity, psi.
w = total unit weight, Ib per ft.
Chart 0-17 of Appendix C gives a graphical solution for this equation. Similar to the stress formula,
it is based on M = -h-wI2; factors for other conditions of constraint are included in Chart C-18.
When lines are pitched to facilitate drainage, the
supports may be spaced so as to completely elimi-
DESIGN OF PIPING SYSTEMS
K = 600 for free ends.
Other symbols are as previously defined
except that the weight does not include
the contents, since the pipe empties as
it drains.
The gradient of supports determined by this formula provides that the slope of the deflected line
will not be upward in the direction of drainage but,
will be horizontal or downward. To obtain positive
drainage with a given minimum piteh, the support
gradient must be further increased by the amount
of the minimum piteh. Pitching may also be needed
to vent a hot pump suction line baek to the source
in order to avoid vapor binding.
The advantageous arrangement of support is related to the degree of restraint which can be tolerated, or to the extent and direction of the movements
to be allowed at each location. The fundamental
types are characterized as rigid, resilient, and constant effort, each of which is capable of wide varia-
tion in details and of two basic arrangements,
suspended and resting.
Rigid supports of the suspended arrangement involve solid hangers, while the resting arrangement
may function as a sliding contact or be provided
with rollers or rockers; for special cases, the support
structure may be flexible or of simple- or multiple-
FIG. 8.7 Typica.l rod hanger assemblies.
however, involves considerable added expense of
hinged design to secure movement in onc or two
directions, while maintaining constant elevation.
Solid hangers eliminate friction and sticking be-
supports and is of limited effectiveness with flowing
media which cling in substantial amounts to the
pipe wall. Hence, it is becoming a widespread
practice to avoid pitching by setting up a regular
plant procedure for washing or blowing down the
movement range in proportion to their length, require higher support frames, and involve greater
usage of space; however, they are a preferred choice
where the general plant arrangement permits their
nate pocketing due to sag of the piping. Pitching,
tween the pipe and support,' but are limited in
pipe as dictated by safety, corrosion prevention, or
contamination requirements. Pitched lines are thus
use, particularly on extreme high-temperature 01'
limited to occasional applications where they may
undesirable, Some typical hanger assemblies are
shown in Fig, 8.7, Resting supports, although they
be used either generally or in connection with specific
pieces of equipment. Where substantial pitch is
desired, hanging type supports are generally needed
in order to maintain reasonable uniformity in supporting structures.
other critical service where unassessable restraint is
involve friction, either sliding or rolling, are widely
used and are generally satisfactory, probably due
to the friction load resulting from the weight usually
being low as compared with the thermal expansion
The minimum pitch of supports required to avoid
pocketing due to sag is given by the following
formula:
effects; the reduction of friction by using rollers and
h = Kwl'/EI
(8.3)
ing support assemblies arc shown in Fig. 8.8.
Rigid supports are satisfactory for systems involv-
where h = gradient of supports in feet/IOO feet of
length, and
K = a constant, depending on constraint.
K = 116 for fixed ends, and
ing lengthy horizontal runs with little vertical ex-
rockers is not as reliable as by using hangers, due to
possible wear and lack of luhrication. Typical rest-
3It should be noted, however, that freedom of movement
renders hangers unsuitable for the support of piping subjed-ed.
to shock loading, Le., blow down lines.
_________#Ia
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
2'1l
pansion differential. Much process piping fits this
description since it consists of sUl?J'orted horizontal
runs with vertical runs to vessels, with the expansion
of the vertieal runs largely offset by the change in
length of the vessel shell. For the most part the
foree to produee movement at a sliding contact is
readily available except for unusually heavy lines
or those operating at a temperature where only a
fraction of their room-temperaturc strcngth can be
allowed.
Rigid supports are improperly used under certain
conditions, viz:
1. Where the restraint of expansion movement at
the support location is significant and cannot bc
absorbcd by elastic action of the line within allow'
able stress limits, i.c. the line will yield under each
temperature cycle.
2. Where the line deflection involves reactions
against the support of such magnitude that reasonably free movement of the pipe at the support is not
Ample lenglh 10 be
provided 100 that Ihero will
bo no dClngcr of
diulngClgemenl
····A.~i·. 'A:: .,,,"}
BCI~ El~w­
(fool CCl~' inlegralty)
assured, Le. on stiff lines.
3. For multiple supports on vertical runs.
4. For horizontal run supports adjacent to vertical run connections unless little or no support is
required offstream, at which time the line moves up
from the support.
5. For vessel-attached supports where they cannot be located at a point of negligible differential
expansion or where an intervening loop (i.e. line
flexibility) cannot be provided to take care of the
expansion differential.
FIG. 8.8 Typical resting support assemblies.
For such conditions where a substantial increase
or decrease of support reaction under line position
change can be tolerated, resilient supports offcr an
economic choicc. They are advantageous on long
runs of pipe where even reaction distribution is not
easily attained, also for piping systems subject to
rapid changes in temperature or uneven temperatures with attendant bowing. Usually resilient
supports involve single helical springs incorporatcd
into simple suspcnded (hanger), or resting (usually
sliding type) supports, although in multiple arrangement springs are paralleled to increase load capacity
or arranged in series to incrcase the total travel for
a given load variation. Structural members are
occasionally substituted for springs on large lines
with limited movement and usually consist of flat
plates, bars, or rods, as cantilevers.
\\There uniform support reaction must be maintained over a movement range beyond the load increase limit which can be economically maintained
with resilient design, eonstant effort supports must
be used.
Two designs are used: so-called compen-
sating spring devices, and counterweights. The
former involves one or more springs whose motion
is magnified by leverage or similar mechanical advantage, and is available in standardized units in a
wide range of sizes, each of which can bc adjusted
for an individual load range. The design and manufacture is usually sufficiently refined so that reliable
load measurement indication can be incorporated.
They should also be provided with means for adjustment of position to avoid use of their movement
capacity for this purpose. Such adjustment of position is not only necessary at initial installation, but
also with any subsequent permanent change in the
line contour.
Counterweights are capable of variation over a
wide range of mechanical advantage at the cxpense
of greater movement of the weight and are usually
custom designed for specific installations, since their
use is occasional and largely confined to loads or
242
DESIGN OF PIPING SYSTEMS
movements beyond the range of compensating spring
devices. They are advantageou;; in that the load
is closely controlled and is quitc indcpcndent of
movement, although they are subject to friction
(which, however, can be held low by suitable design
and adequatc lubrication).
Compensating spring devices and countcrweights
are usually of thc suspcnded (hanger) arrangcment,
but are occasionally furnished to a resting arrangement due to clearance requirements, and as such involve greater complexity and proportionate expense.
Braces. Having provided for desirable restraint
of thermal expansion and for adequate support of
gravitational effects, the next step is to assure suitable bracing for other loading which may be anticipated. Some sources of such other loading include:
wind on exposed lines; flow or mechanical vibrations
transmitted from pumps, compressors, turbines, or
other process equipment; earthquake; water hammer; impact due to sudden establishment of flow
(as on relief valves), gunfire, or vehicle movements;
vibration and impact due to high-velocity release
of compressible flow at reducing valves or to atmosphere; and surging of compressible gas or two-phase
(gas plus solid or liquid in suspension) flow.
Protection of a piping system against such influences ean be accomplished by:
1. rvIinimizing the influence at its source, i.e. snubbers, pulsation bottles, elimination of unbalance, etc.
2. Controlling the resulting dcflection of the line
by limit devices or vari"ble restraint.
3. Controlling the resulting movement of the linc
by damping, i.e. energy dissipation.
4. Opposing deflcction or rotation by rigid attachments.
5. Modifying the natural frequency of the line or
supporting structures.
When a disturbing influence can be eliminated or
minimized at its source within economic means, such
an approach is desirable since it avoids less positive
correction measures with possible trial and error for
final solution, as discussed in detail in Chapter 9 in
connection with vibrations. The same is true of
other than vibration disturbances, for example, if a
line is subject to impact through its attachment to
a structure which supports equipment and whose
function involves suddcnly applied loading, it may
be desirable to provide independent support.
Control of disturbing influences within a piping
system is often entirely or substantially accomplished
by the thermal expansion restraints and gravitational supports in combination with the inherent
stiffness of the line. Restraints control pipe move-
ment at their location; in combination with supports
they serve to modify the natural frequency of individual spans. Sliding supports, in addition, often
contribute damping through friction at contact surfaces. Where additional provision is necessary to
protect the line, braces are provided which further
alter the natural frequency by limiting or preventing
spccific movement, or by providing damping.
Deflections and rotations can be prevented by
stops and guides respectively or they can be controlled to a desircd range by limit stops. Stops or
guides can be used to change the natural frequency
of individual spans, and are preferred where they
do not create excessive restraint. :Moderate restraint (and a minor degree of damping) can be
obtained through the usc of spring stops, usually in
opposed pairs, and is of some value in limiting deflection for loads of short duration such as wind
pulses or earthquake.
Appreciable damping is secured from hydraulic
shock absorbers, which are available at an attractive price in the standard sizes used for trucks,
automobiles, and railway cars. Larger size.s are
usually designed around standard sizes of hydraulic
cylinders. Braces employing dry friction damping
are usually limited in size and are apt to be unpredictable in operation.
Restraints, Supports and Braces for Non-Stiff
Piping Systems. Additional supporting system
problcms are introduced by non-stiff piping design
incorporating the various kinds of expansion joints;
in what follows, these are discussed for each of the
three systems of reduced rigidity described in
Chapter 7.
Semi-rigid systems with hinged expansion joints
are closely related to stiff design in supporting system requirements. It is sometimes necessary to
minimize weight effects on the hinges; this usually
requires counterweights or spring hangers. With
two joints and intervening pipe runs free of bending,
this problem is present in grcater degree. Sometimes
restraints are required to minimize torsion on the
hinges.
In non-rigid piping systems all members are free
of thermal bending momcnt, and the strength of
the assembled system is largely dcpendent on the
strength of the hinges at the joints. Rcstraint is
necessary to control the position and limit the expansion movement at the individual hinged joints;
it is provided by tie rods built into the joints and,
further, by external stops. For large-diameter or
otherwise heavy runs, it may be advisable to relieve
the hinges of the principal weight effects; this can
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
be accomplished by various support details, but
usually eonstant-effort hangers ar~ required because
of the large movements involved. Non-rigid piping
is highly dependent on braeing for proteetion against
effects such as those of wind vibration. Hinges are
usually not eapable of assuming lateral loading and
if there is out-of-plane expansion, they also should
be protectcd against torsion. Bracing cmploying
pin- or ball-joint solid struts is favorcd wherc no
interference with the system movement is created;
otherwise, hydraulic snubbers are widely used for
control of vibration and also to minimize deflections
on transient loading.
Free movement piping systems, with thcir lack of
rigidity and their inability to transmit weight or
longitudinal pressure loading, are completely dependent on external means for adequate and safe
operation. Restraints are necessary to maintain
position and alignment at slip joints, to assure free
movements (prevent jamming), and to maintain the
relative position of individual runs, thus limiting
the required range of movement for slip and bellows
expansion joints. A primary function is to resist
unbalanced pressure loads without excessive deflection, which is accomplished with minimum complexity and cost where the location of the necessary
anehors is carefully associated with the necessary
contour of the line and proximity to grade or substantial structures capable of assuming the reaction.
Support selection and functioning is less critical
since lateral movement is absent or minimized; however, free movement piping systems require morc
accurate support installation than other systems in
order to avoid jamming in slip types and fouling of
internal sleeves or limit rods on bellows types.
Rigid supports usually suffice for horizontal runs
while vertical runs are generally self supporting from
the terminal or other point of anchorage. It is generally not necessary to provide a support adjacent
to an anchored horizontal slip joint, and details of
the nearest supports on the slip pipe should provide
for accurate adjustment during installation so that
the support will not produce any moment on the
stuffing box. Support of the adjaeent pipe is needed
for most types of bellaws joints.
Braces are a lesser factor since essentially each
run is an entity and the guides or stops necessary for
alignment usually suffice for other than unusual
transient effects.
8.4
Fixtures
Fixtures refer to that part of a pipe support assembly which can perform the dual functions of
243
transmitting the reactions from pipe to structure
and of controlling the movements of a piping system.
This is in eontrast to the pipe attachment by which
the fixture is connected to the pressure wall, and
the structure which receives the loading from the
fixture; these latter parts are covered in Sections 8.5
and 8.6 respectively. Even though this division is
often artificial since two or all of these parts are
sometimes consolidated as a single unit, it is useful
in emphasizing the three basic functions to be performed.
It also introduces a convenient separation
for preparation of engineering detail and for ordering. The integral attachment, in partieular, is necessarily a part of the piping on alloy or critical
construction and is morc advantageously engineered
by the pressure part designer. The structure similarly is usually considered a part of the structural
design and fabrieation. The fixture is a specialty
which is selected and ordered when the details of
the pipe support assembly are established. This
functional division encourages the interchangeable
use of attachments with various fixtures and simi-
larly with various structures so that standardization
is made practicable.
In the preceding section, selection of the support
assembly details has been presented as related to
individual effeets on the piping system, namely, restraint, support, or brace. Fixtures, in themselves,
can be classed as rigid, resilient, constant effort, and
damping, each of whieh is applicable in some degree
to the three basic functional objectives (restraining,
supporting, or bracing). It should be emphasized
that while fixtures are in themselves secondaryelements (i.e., not part of the pressure container), their
maloperation or failure can in many cases endanger
or actually rupture the pressure piping by either
the direet loss of support or the accompanying
shock.
It is obvious that the general design and arrangement of fixtures in combination with attachments
and structures are subject to almost unlimited variation, so that no attempt will be made here to present
standardized designs with dimensional data or stress
and deflection calculations. The elements are for
the most part not complex, so that routine analysis
is entirely adequate; dimensions are susceptible to
service requirements and individual preferences, and
can easily be worked up for items whieh are frequently used; also, many fixtures are readily available as pre-engineered subassemblies. The presentation in this section is therefore directed at general
background and suggestions to assist with the design (and to improve application) of fixtures for use
244
DESIGN OF PIPING SYSTEMS
FIG. 8.9 A device which serves as a guide against
axial rotation and as n two-axis double-acting stop
against lateral translation.
as restraints, supports, and braces, and will follow
the descriptive classifications of rigid, resilient, constant effort, and damping, in the order named.
Rigid Fixtures. A rigid fixture is one whieh
allows insignificant defleetion along its prineipal axis
and may otherwise limit deflection along other axes
or rotation along any axis. Rigid-fixture action may
be provided by any type of support assembly-viz.,
anchor, guide, or st.op-and will be discussed in that
order.
Sinee an andwr must provide essentially complete
fixation (i.e., full eonstraint against three deflections
and three rotations), the fixture part of the assembly
must neeessarily be integrated with the attaehment
and sometimes also functions as the structure, the
attachment being usually the critical part. For
this reason, anchors have been treated largcly in
,he succeeding section on pipc attachments with
various important types illustrated in Figs. 8.18k,
?n, and p.
It is desirable to emphasize that since
anchors carry substantial loading, and must oftcn
develop the full strength of the attached pipe, and
further, may involve temperature considerations in
complete restraint against reversible loading can
also be attained by a minimum of twelve tensile tie
rods in opposed pairs, or by six struts capable of
tensile and compressive loading so disposed along
and collaborating with both the intervening pipe
and a suitable external structure to reduce all movements to negligible values. Similar results can be
achieved by combined gnides and stops which fnnetion along all three axcs. In these designs of anchors the fixtures can be identified as separate
members.
A guide, from the definition of Section 8.1, restricts
rotation (i.e. introdnces a moment reaction). Where
the degree of restraint is snch that rotation is totally prevented, the gnide may be properly described as a rigid fixture. It will be noted that the
guide fixture may be lugs or circular sleeves if no
pipe attachment is used and the fixture bears directly on the pipe wall or on '! eoncentric cylinder
attached to the pipe in the manner of a skirt to remove the reaction of the guide from direct influence
on the pipe wall. If pipe attachments are providcd
in the form of lugs, trunnions, rings, ears, etc., the
guide fixtures may again be lugs, rings, etc., or tie
rods, or struts. Usually there will be a translatory
movement of the pipe line through guides. Hencc,
with thc close clearances required for doublc-acting
constraint, frictional resistance and the need for
antifriction or lubrication contact surfaces is accct:ttuated; otherwise minimum area of contact is
advisable, preferably line eontact. Tie rods or
struts must be of sufficient length to minimize lateral motion (as covered in greater detail in Seetion 8.3). The most widely used gnides are those
which resist axial rotation thus taking out torsional
moment. A practical and serviceable detail for this
purpose is shown in Fig. 8.9. It should be noted,
however, that in addition to its function as a guide,
this device restrains lateral translatory movements,
thus also performing as a two-axis double-acting sfop.
A stop, as defined in Section 8.1, limits translatory
considerable degree, their design must not only provide adequate statie and fatigue strength, but also
sufficient rigidity together with a satisfactory load
movement, total stops entirely preventing specific
distribution and avoidance of unnecessary stress
or cables, jointed struts or links, and sliding can·
tact brackets in contact with shoes, plates, trunnions,
concentration attendant to contour and thermal
gradients. Favorable geometry cannot be overemphasized, and is best provided by surfaces of
revolution, usually at reduced cost. Dependence
on friction straps or bolted joints in locations affected by temperature change is best avoided, with
preference to integral or welded construction. The
foregoing relates to conventional anchor assemblies;
deflections, thus fitting the description of a rigid
fixture. Various types of stop fixtures are tie rods
or lugs on a pipe wall. Tie rods and struts, which
may be favored as stops due to their freedom from
frictional effects and greater reliability, deserve
special comment. Tie rods or cables offer many
additional advantages such: as low cost, ease of
installation and adjustment, versatility and minimum space requirements insofar as the fixture
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
proper is conccrned. These advantages are lost
with struts, which must usually be jointed or
hinged to take care of lateral deflections. Joints of
struts are prcferably spherical contact surfaces
which may be incorporated between bolted or
threaded unions as illustrated in Fig. 8.10, thus
permitting universal angular motion in degree
dictated by the contact radius, with essentially no
clearance except for the lubrication film. Pin joints
restrict motion to one plane, and involve some
clearance for satisfactory operation and lubrication.
On the average they are somewhat less expensive
but the saving may be offset by the necessity for
more accurate installation and by their lesser
versatility.
For assurance of unrestrained motion in the
desired directions, the contact surface of sliding
devices must be initially parallel and, during
operation, free of local thermal effects or lack of
rigidity in the pipe wall or structure which will
permit distortion. In addition to structural reinforcement, concentrated wear of the pressure pipe
surface, suehas at a location of line contact, should
be avoided on large or otherwise critical lines by
protective wear plates. Temperature effects require
stable attaehments such as trunnions with adequate
length or other p,ovision for heat dissipation.
Resilient Fixtures.
Resilient fixtures find
widest use as supports and occasional use as braces,
but they are rarely useful as restraints. They can be
incorporated in any manner of arrangement, suspended or resting, of single or dual action along one
or more axes. Resilience is almost universally
obtained through the use of helical springs, although
plates or other sufficiently flexible structural members can be used for special designs where loads are
high and movement requirements low. Sprinll;s are
~ lb
IUl
1'Gi1
;::::1
l~
i~
FIG. 8.11
/)
r
7)
~
Typical variable-load spring fixture.
usually single coil units although heavy loads may
necessitate the usc of multiple springs because of
availability and manufacturing limitations.
The type of variable-load spring-support mechanism, which has been widely accepted and is now
available as a standard design, is shown in Fig. 8.11.
It consists of a coil spring loaded in compression,
enclosed with a cylindrical cover, and provided with
a load and deflection indicator. Supporting devices
of this type are commercially available.
It is not considered necessary to cover herein the
subject of spring design. However, it is desirable to
discuss the application of a spring of given characteristics to the support of piping. The principal
consideration is the variation in supporting effort
for a given amount of displacement. This load
variability may be defined as the absolute value of:
Operating load - Shutdown load
Operating load
-"
Values of variability customarily used range from
25% to 50%. The higher the variability, the more
nearly the spring approaches a solid support,
consequcntly the more restraint it applies to the
thermal expansion of the piping. On the other hand,
the smaller the variability, the bulkier the spring
becomes for given load and travel requirements. If a
variability of less than 25% is required, it is apt to be
more practicable to use constant support devices.
The load indicator when observed successively at
Details of ball joint used on jointed struts.
movement and load ranges. For most designs the
load can be adjusted in operation, which is a desirable feature; the overall support design should also
provide means for an adequate range of position
adjustment so that the proper pipe elevation can be
maintained independently of the load adjustment.
Ground spherical
surfaeo
Annular 5fXlco packed
with graphite lubricant
,
-..,
,I
~>l--{'l-+- -I-'f-r-j--.+----,-
,
ambient and operating conditions serves to verify the
Dotoib of Ball Joint
U~ on Jointed SfTvtI
FIG. 8.10
245
246
DESIGN OF PIPING SYSTEMS
Cogo which procompro,""
lho spring and odablishol
il1 maximum extension
Conneding rod
Typltol Sway Broce Mechanism
limil rods which precompron Ihlll
spring Gnd olloblilh ill maximum
oldoluion
Another Momon!,m Using Two Springs
Sway brace employing
prccomprened and
ROlhtonco
limited oelion spring
Plain lprill9
OHerod
without limited odion
by S,do,
l
~"----_:_:==,,_-
Connecting rod il ulually ~
+ Ocfllldion
adi~llld so thai operating
pOlition is 01 this point
FIG. 8.12
Zero
Deflection
Typical sway brace details and load deflection characteristics.
Resilient fixtures are of limited use as restraints
for controlling thermal expansion stress or reactions
but find widespread application as supports. Occasionally they are uscd for bracing when solid ties
or struts cannot be used due to the restraint which
they impose. As braces, load variability characteristics may advantageously be higher than for
supports. The bracing effect is obtained by the
increase in spring load resulting from deflection,
which it is usually desirable to limit in extent thereby
producing the device generally known commercially
as a sway brace. Such devices may employ either a
single resilient fixture or two opposed to each other as
illustrated in Fig. 8.12 which also shows the advantageous effect of precompression (and limiting of the
spring travel) upon the load-deflection characteristic.
Constant Effort
Fixtures.
bell crank which is so arranged that the nsmg
characteristic of the load vs. displacement of the
spring is compensated for by a reduced lever arm
due to the changing position of the bell crank. The
basic idea, illustrated in Fig. 8.13, has been widely
exploited; some of the modifications which have been
satisfactorily used are shown in Fig. 8.14. While, as
might be suspected, the load characteristic of this
type hanger is not perfectly flat, a close enough
approximation is usually provided, the normal
Small chong.: in effedive lever arm
o
Constant effort
fixtures are of two general types, spring loaded and
weight loaded. While these types arc quite interchangeable, advantages of negligible weight, com-
Spring
-~ Lorge change in elleclive lever
Ofm
pactness, lower cost, and availability as a completely
engineered product lie generally on the side of the
spring-loaded types and since their range of capacities is presently so broad, the weight-loaded types
are rendered unnecessary in all but the most extreme
cases of load or movement.
The spring-loaded constant support fixtures
consist of a spring actuated through a lever such as a
Relalively
Can~IQnl
load
FIG. 8.13
Basic idea of the spring loaded constant
support mechanism.
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
variation being only on the order of one per cent.
In some designs, auxiliary (or booster) springs are
added to improve the charactcris1<ic. In general this
type of device is quite dependable, although all the
designs available are not as readily adjusted as
might be desired.
Constant load support may be obtained simply by
the use of a counterbalancing weight. Since it is
seldom practical to use a weight equal to that of the
pipe, it is customary to use a mechanism which will
multiply as well as invert the weight force. TwC'
different designs have been considered practical:
the lever type and the cable type. As noted before,
counterweights are used primarily for heavy load or
large travel applications. The support of the extra
weight of the counterweight itself may be a substantial item, although somewhat reduced on the
cable and sheave type by the elimination of the
beam. The cable type also permits greater freedom
in the location of the weights; however, slightly
greater maintenance is involved. Typical details of
both types of counterweights are shown in Fig. 8.15.
The design of either device involves standard
structural practice and needs no elaboration here
Mechanical advantage
obtained by lever
\
- ----".:;0-----""-.,
~~~=~ JT~~:~ :
3J H~~J!'
FIG. 8.14
I
Various type:; of const-ant tlUpport hangl!fli.
save to reiterate the need for ample and easily made
adjustment, also for a generom; provision for excess movement.
Damping Fixtures.
Damping of vibration:-)
involves the dissipation of energy, which is accomplished in substantial degree only by friction or
..
.~-;~•.::~.;..~;
i.1.
:/
Cable
y
9
Trovel 01
weight
Piptl attachment
V
lever Beam Type
-------'\--------~
Coble
:
Ti
'_:::.:: t·~,
-I \
, /
M"h"kol od,o",o,.
ob10;ood by
Coble 0,,1i Sheave Type
,,
,,
T
-~
---~-
Trovolof
(il:F:ii::t{iI weight
I'7l')'7.77777r
FIG. 8.15 Typical counterweights.
L
247
DESIGN OF PIPING SYSTEMS
218
welds and of hcat-affccted zones adjacent to welds
on the pipe or attachment. Without restraint,
temperature stress is not present; for example, a flat
Action line of
Pipo atlachmenl
rC1training elfect
link
_.-+ .
diameter of successive circumferential elements again
link end
level of oil in re~rvcHr_=~ .J
Connecting
11ruclurc
FIG. 8.16
-
----- -----
Commercial railroad car type shock absorber
utilized as a damping device.
hydraulic snubbing. Single or opposed springs do
not perform this function hut simply store up an~
rct~lrn energy under varying load with only a
trifling amount dissipated in the process.
While dry friction absorbers have not proven
sufficiently reliable for untended simple installations,
hydraulic units such as those widely used on trucks,
railway cars, and automobiles give satisfactory
service with only occasional maintenance and are
available as standard mass produced items. A
typical installation is shown in Fig. 8.16. Larger
size units, if required, may be assembled from
standard all-purpose hydraulic cylinders and conventional hydraulic valves and fittings. Hydraulic
shock absorbers may operate with any fluid; however,
most of such cquipment involves hydraulic oil with
a relatively constant viscosity over the working
range of temperatures.
8.5
rectangular plate heatcd all along onc cdge to prodncc a linear gradient across its full width will
become curved in its plane so that the length of each
element parallel to the hcated cdge is proportional
to its temperature; similarly an opcn-end cylindcr
suhjected to uniform hcat input around thc circnmferencc at one end to produce a linear gradient
along its full length will become a cone with the
I)ipc Attuchnlcnts
The attachment componcnt of a support or restraint assemhly usually introduces stress into the
pipc wall as a result of the structural loading which it
transmits, and also due to the localized heat loss and
the thermal gradient which it causes. Inadequate or
faulty dcsign or fahrication can rcsult in failurc of
the pressure wall with consequent energy release and
tire hazards, particularly on heavy wall thickncss,
air hardening analysis, or otherwise sensitive ma~
tcrials. In the design of pipe attachmcnts it is
essential to apprcciate thc significance of tcmperature gradients and their potential for causing
distortion and cracking, particularly of blind root
proportional to their individual tcmperaturc. Whcn
heat input ccases, the plate and cylinder rcturn to
their original dimensions and shape. Any interruption in the uniformity of the gradient, however,
whether due to heat input, heat loss, thermal conduct·ivity, or discontinuity (i.e. any influence which
opposes free expansion) results in stress whose
magnitude is a function of the character of thc
gradient intcrruption and the individual stiffnesses
of the adjoining sections. With a uniform gradient
the resulting stress is distributcd along its length to
the same pattern; with a sharp change in gradient,
highly localized stresses arise in sufficient magnitude
to maintain continuity of the structure.
From the
foregoing it is evident that at clcvatcd tcmperature,
possible distortion and fatiguc failure at pipc attachments is related to the magnitude of thcrmal stress
alld that this should he cOlltrolled insofar as practicahle by favorablc COlltOur (surface of revolution
rather than flat surface), and minimum heat flow
and attendant gradient.
In providing pipe attachments which have no
significant adverse cffect Oil the strength of thc
pressurc wall, it is essential to control carcfully thc
magllitude and distribution of the structural bcnding
strcss introduced into the pipe wall by thc attachment in additioll to stresses due. to the previously
dcscribcd thcrmal gradient influences. When such
local stresscs are evaluated they should be treatcd in
thc catcgory of sccondary or localizcd stresscs. As
discusscd in Chaptcr 2, the allowable limit for such
stresses when due to sustained loadings eaI1I1ot
reasonably be set at the limit for sustained primary
stresses Sh; instead it is recommended that a limit
of 2Sh be used for desigll purposcs. This limit has
long been used hy Thc M. W. Kellogg Company,
and is based on the secondary stress levcls inhcrcntly,
though not exprcssly, emhodied in thc Prcssure
Vcssel Code rulcs. Wherc the local strcsscs include
the cffect of thcrmal rcactions the allowablc stress
rangc used should be the same as for thc dcsign of
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
249
Sheor lugs
the plpmg for such effects. Local strcsses for
trunnion type attachmcnts can bc approximated by
the approach outlined in Chapter 3, Section 3.14
for nozzle loadings on cylindrical shells. Lug attachments can be similarly approximated by assuming
an appropriate equivalent circular loading.
Aside from favorable stress distribution, pipe
attachment design may involve emphasis on minimum heat loss, protection of insulation, heat dissipation to prevcnt cxeessive local temperature of
connecting steel or concrete members, adaptability
for connection to full fireproofed membcrs, suitability for use on alloy or otherwise sensitive materials, and satisfactory service at high or sub-
.....elded 10 pip.
zero temperatures.
Piping systems are inherently susceptible to
position changes and distortion so that attachments should provide some margin of clearance and
strength for lateral loading components in excess of
normal design range. Fr~ction loading magnitude is
difficult to predict accurately; for single supports
(steel on steel, unlubricated) a friction factor of
from 0.20 to 0.50 is used, as influenced by the design
details and service, while for a line on a number of
successive identical supports this factor is usually
reduced 50% except for end supports. The frictional
effect of a bank of lines is usually less than that of a
single line due to the fact that it is unlikely that all
lines in the bank will move at one time. Hence, a
factor of 0.10 to 0.15 is often used for the lateral
shear on racks.
Pipe attachments fall into two basic classifications:
(1) non-integral and (B) integral with the pipe wall.
Non-integral attachments include the type of
details by which the reaction between a pipe and
support structure is distributed by contact. Typical
details of such attachments, including clamps,
slings, cradles or saddles, and clevises, are shown in
Fig. 8.17. For heavy loads these are sometimes
used in mnltiple. Only the clamp is suitable for
vertical lines; even so, it is usually necessary to
have wclds or projections on the pipe or to locate the
clamp below fittings or flanges to prevent slippage.
For usc as a stop, the clamp may assume the form
of a sectional ring of angle iron or othcr shape. The
conventional clamp made of flat bar with appreeiable gap is really a close-fitting two-pieee sling and
is incapable of developing signifieant contact
pressure, since the ears are easily deformed under
light bolt takeup. The friction effect and attendant
load capacity can be cnhanccd by more rigid design,
by thc use of heavicr bar stock, or by reinforcing
gussets at the ears.
Crodlo Of Saddl.
FIG. 8.17 Typical non-integral attachments.
Cradles or saddles are often used for supportti_ the
pipe simply resting in place. Such attachments can
be made double acting by the use of tie rods encircling the pipe; if a ring or band attachment
extends around the entire circumference, wedge~
are sometimes used.
Clamps, slings, and cleviscs are widely used and
for moderate service appear as standard hardware
in many shapes and of both cast and wrought
matcrials. With reasonable width and the 180 or
greater contact inherent in their design, bending
stress in the pipe is minimizcd. Clamps for substantial load and saddles or bases are standardized
only to the extent of their usage by individual
fabricators and are usually manufactured as required. Non-integral attachments offer advantag;e~
in that their procurement and fabrication call be
entirely independent of the piping, and ill that
freedom in their location simplifies piping details,
fabrication, and erection, thus reducing east.
Further, on alloy piping the absence of welding
eliminates the need for alloy attachments and alloy
0
250
DESIGN OF PIPING SYSTEMS
welds with attendant heat treatment requirements,
thereby aecelerating erection. Their drawback is
t,hat in applications involving the support of vertical
pipes, and for restraints or braces, they cannot
(ineluding stools and trunnions), rings, and skirts.
Illustrative examples of these appear in Fig. 8.18.
In addition the various non-integral attaehments
previously described are sometimes made integral
by welding.
The upper seven illustrations in Fig. 8.18 show
representative applications of ears to vertical and
horizontal runs. Ears which are normal to the pipe
surface are apt to introduce a fair amount of bending
in the pipe wall, although when they are loeated on
maintain effectiveness at elevated temperature since
the intial eompression is rapidly relaxed.
Integral pipe attaehments must be used for
services involving high temperature or relatively
severe load, and further, on restraints or braces
where two-direction action is desirable in a single
member. The simplest means for integration is
the use of spot or fillet welds at the edge of elamps
or saddles. However, such welds are subject. to
the vertical axis of an elbow a reasonable component
of the load is tangential to the elbow surface. On a
horizontal run the tangential ears, 8. 18g, are favored
to minimize bending, although for heavy loads on
large lines a welded-on sling attaehment may be
preferred in order to minimize the structural
importance of the attaehment welds. Obviously,
ears can be used in many other variations. When
used tangentially on vertical pipe, ears resemble a
seetion of a cylindrieal skirt with the intersection
angle and cireumferential spread selected to control
bending effects as in 8.18c. In general, distortion
failure unless the parts are of similar expansion
characteristies, tightly fitted and attached with
welds of adequate proportions. More effective
design requires that the struetural and pressure
parts be unified for favorable load distribution and
adequate heat flow so that welds will not be subjeet
t.o a concentration of structural or thermal stress.
Six basie types of integral attaehments are in common use: ears, shoes, lugs, cylindrical attachments
if1fj
~"
-
." I
h
I
m
Cylindrk(lllug
$1001
"
Trurlrlion
q
Ring
p
o
Skirts
FJO. 8.18
Typicnl int~gral atL'lchmcnts.
1
SUPPORTING, RESTRAINING, AND BRACING TIlE PIPING SYSTEM
and weld failure is much less likely for ears in a
tangential location than for radial ears; and furthermore, assessment of their effect oitthe pressure shell
is also relatively simple. Ears are usually limited to
unidirectional loading, although they can be designed
for lateral loading in their plane. Perpendicular or
out-of-plane loading in any magnitude is best
avoided.
A shoe or lug transmits load through one or more
web plates which often are welded intermittently or
at only the ends, a practice not always to be desired
since it accentuates the temperature difference
between the lug and shell. Unless heavily insulated,
lugs composed of flat sections are subject to distortion under high-temperature service and should be
avoided. Adopting a surface of revolution contour
by making the attachment of a length of pipe serves
to ameleriorate somewhat the serious intersection
problem created by the temperature gradient.
Hence, cylindrical lugs and trunnions are frequently
used in high-temperature service.
Base, stool, and a number of other terms are
applied to pipe attachments which serve directly
as a resting support or rigid anchor of piping to a
structure; a familiar example, the standard base ell,
is illustrated in Fig. 8.8.
Rings, or combinations of rings integral with the
pipe and other type attachments, are used on long
pipe spans where the support reactions are high;
also for attachments on lines subject to collapsing
pressure, and in general on braces or restr.aints for
extreme loading or concentrated effects since they
afford a maximum opportunity for favorable load
distribution.
Skirts offer the best approach for the introduction
of severe axial loading into a pipe, since distribution
around the entire circumference is attained, provided the skirt is of sufficient length. Bending is
minimized where the skirt angle is kept to a minimum. For acute conditions, the skirt can be of two
courses, the first a cylinder attached to the pipe and
the second a cone for the necessary spread. The
attachment welds must receive special consideration
if thermal gradients of considerable magnitude arc
unavoidable. Where fillet weld sizes become excessive the top edge of the skirt is often cut to a
serration pattern to increase the weld length; it is
advisable to avoid sharp corners since they promote
::itress concentration and weld cracks; a wave contour
offers the ultimate advantage in this direction;
Fig. 8.180 shows this detail. Excessive fillet weld
sizes can also be avoided and stress flow lines improved. For extreme thermal gradient effects,
251
vertical slots in the skirt at the top have been advantageous in the case of large pressure vessels and
can be applied to piping as well; the slots should not
interrupt the weld, to avoid stress intensification.
Favorable contour of pipe attachments to minimize the level of stress and to avoid unnecessary
stress concentration is essential in proportion to the
service and loads involved; however, in many cases
conventional structural details are more costly than
equivalent pressure equipment details which use
pipe or other surfaces of revolution. An anchor
lug of pipe or a skirt anchor may seem unorthodox
from the structural engineer's viewpoint; however,
when dealing with radically reduced strength and
severe temperature gradients, loads must be introduced into pressure shells in such a manner as to
avoid unnecessary intersection stresses if the attachment welds and shell are to remain intact.
It must be remembered that fatigue plays an
important role in piping system design and is of
equal importance on attachments. In general,
integral pipe attachments should be subject to the
same requirements as to materials, design (in particular allowable stress), fabrication, and inspection as
the pressure pipe to which they are attached.
Every advantage should be taken in design detail
and the generous use of insulation to minimize heat
loss where excessive temperature gradients would be
harmful; in general this applies to all attachments on
lines in high-temperature service. Various expedients are resorted to in extreme designs for
minimizing heat out-flow. Internal insulation in
skirts, etc. is quite generally desirable; conductive
material such as steel chips is sometimes substituted
for the insulation near the pipe shell to bypass part
of the heat flow around the intersection. Increased
metal thickness locally at the intersection reduces
the unit heat flow in this area and increases the
attachment strength.
8.6
Structures and Structural Connections
The load from pipe supports, restraints, and
braces may be transmitted to other piping, to
pressure vessels, to buildings (roof, wall, or frame),
to a platform or other access framework, to principal
or secondary equipment support structures or
foundations, or to structures provided specifically
for this function. The structure generally serves to
transmit the piping load directly or through other
structures to a foundation and thence into the soil,
although occasionally a reaction system may be
balanced within either a single structure or a combination of structures.
DESIGN OF PIPING SYSTEMS
252
SCIOOndgry member which
Is (frong enough 001
not wffidently sliff
relotiyo kJ the pipo
to lake Cloy oppr.aablo
proportion of the load
FIG. 8.19
Showing jointed strut attached to an
inadequate structure.
The ability of a strueture to eombine with a pipe
attachment and fixture to provide an adequate pipe
support assembly depends upon its capacity to
carry the imposed load without overstress and,.
most important, without excessive deflection under that
load. If exeessive deflection is required to develop a
reaction equal to the loading, the piping system must
either move this amount by sagging or other movement, or, if sufficicntly stiff, transfer part or all of
the load to adjacent restraints, supports, and braces,
or to thc tcrminals of the line. A typical example
is shown in Fig. 8.19, wherein a large pipc is to be
braeed laterally against wind load by a jointed strut
which, in turn, is attached to a structural member.
The deflection of the structure (under the total wind
load transmitted by the jointed stl'llt A) is large
enough to permit essentially the same deflection of
the pipe at point B as that which would occur without this restraint. Hence, the support is quite
ineffective, even though alone it may be capable
after sufficient deflection of taking the required
load without overstress.
In the interest of engineering economics, the usual
flexibility analysis of piping systems neglects weight,
wind, and transient effects, on the assumption that
their influence will be minimized by supports and
favorably placed restraints and braces which do not
significantly affect the overall stiffness. Where
restraints are included in the analysis, the usual
assumption is infinite stiffness in the direction of
their intended reaction. The degree of deflection
that can be tolerated is reliably established only by
complete analysis; however, reasoning and rough
approximations afford sufficient guidance for average
design. Excessive deflection of a support may
substantially increase the strain range of the piping.
Of importance is the strain range to which the
piping is subjected over its complete cycle of operation and, in particular, the avoidance of repetitive
yielding of the pipe. Yielding attendant to initial
or occasional adjustment of supports is usually of
insufficient frequency to affect fatigue life.
It is common practice to assume that support
deflection is not significant so long as the initial and
service weight distribution attained does not result
in sagging, restrict expansion movement, or give
other evidence of distress. As to restraints and
bracing, the degree of deflection which can be tolcrated can usually be approximated by simple beam
calculations for single-plane or other not too complex
piping systems; although whenever warranted, the
method of Chapter 5 can be used. For existing
systems the deflection under a known load can be
measured at critical locations and compared with
strain gage measurements. If the load can be
roughly estimated, such as wind load at a known
velocity, the load in the restraining member can be
measured with a dynamometer, strain gage, or
hydraulic jack; or the relative deflection of the line
can be observed with and without tllc restraint in
place, the differcnce indicating the restraint effectiveness. Similarly the resistance offered by individual
supports or braces to free expansion can be studied
by freeing one location at a time and comparing
the relative deflection.
The degree of deflection which can be tolerated is
also related to the effectiveness of the supports,
restraints} and braces at the service temperature
where content weight, thermal expansion movement
of the line, and extension of the support members all
affect the distribution of load and magnitude of
reactions. At the same time the structural strength
of the piping material is radically reduced fol'
elevated temperature service. Transient loads such
as wind, earthquake, etc., are of secondary importance as compared with more frequent or sustained
loading conditions.
In the interest of economics} existing structures are
SlJl'PORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
253
used wherever m·ailable and suitable. An intermediate structural connection may be involved or the
,upport fixture may be direetly -attached. In its
,implest form, the structural connection may bc a
,imple plate or angle clip. Where the pipe is locatcd
at an appreciable distance from the support structure,
the connection may assume sizeable proportions. In
general, such auxiliary structures are of conventional
design.
The usc of pressure equipment as support structun'-.s for connected piping is often advantageous in
minimizing under thermal change the relative
movement of the :mpport versus the piping, also in
avoiding differential movement of the support
structure and pipe under vibration, wind, and
similar effects. \\Then structural connections are
located on vessels or other equipment where dimensional change due to expansion may occur, they
must be designed to permit that expansion, avoiding appreciable restraint and thc stresses attendant
thereto. To accomplish this, the three following
alternatives may be applied, singly or in combination:
1. Maintain essentially the same temperature in
the attachment as in the shell (a practicable solution
only when the bracket connects to a single vessel clip).
2. Allow flexing of bracket members, of the shell,
or of both within their stress capacity.
3. Provide articulation by the use of jointed or
sliding members.
It is important also that the local stresses in the
shell be investigated in order to design shell attachments properly.
To illustrate the application of these principles, a
number of typical bracket details are shown in Fig.
S.20. They represent designs which have been successfully employed by The M. W. Kellogg Company for
a number of years in the support of piping from
vessels. Figure 8.20a illustrates the first alternative,
wherein a member is cantilevered out from a single
vessel clip of sufficient length to distribute the loading on the shell. Satisfactory eontrol of the temperature gradient is obtained, where necessary, by covering the point of attachment to the vessel with
insulation. Heavier loads require a knee brace which,
when fixed to the shell, becomes an example of the
second alternative, resulting in the bracket of Fig.
S.20b. It is only suitable for moderate temperatures
(say not exceeding 650 F) because of the rigid
attachment and limited flexibility of the members.
For higher temperatures a detail such as that of Fig.
S.20e is employed. This represents a typical application of the third alternative, with only one member
rigidly affixed to the shell while the other simply
(0)
Moy be bolted 10 ollow
slighl omounl of
ortit\llolion
(h)
Not welded. bearing
«Jnlod of angle
on pad only
FIG. 8.20
Typical brackets serving as connecting structure~
for the support of piping from a vessel.
bears on a shell reinforcing pad, sliding as required
to accommodate the expansion.
The foregoing brad<ets are suited for supports
involving little transverse load (e.g. for attachment
of a rod hanger). For higher transverse loads similar
brackets are used but with greater breadth thus
making it necessary to provide for the circumferential
as well as for the longitudinal differential expansion
of the vessel.
Flat roof buildings permit ready support of piping
on sleepers; direct support at grade has the same
advantage but offers obstruction to free access.
Trenches avoid this disadvantage but involve substantial expense plus drainage and corrosion problems, and may contain hazardous gas pockets for
explosive vapors.
Where other-purpose structures are subjected to
significant piping weight or restraint loads, it is
essential that the situation be anticipated during the
progress of design engineering to avoid late changes
with eonsequent undesirable details. All possible
advantage should be taken of the inherent stiffness
DESIGN OF PIPING SYSTEMS
254
c
Chollnell bock 10 back
,'
,
,
,:
,
,,
, 1/-';"
~,
"""
.~
l :r''-NO
:~
~l
for SlIIpend«l Pjping
for Resting Piping
FIG. 8.21
Typical single column or pole type
supporting structures.
of a structure to minimize stress and deflection, for
example, important reactions on beams or columns
should be near main member intersections or at
bracing tic-in points; torsional effects should also be
tontrolled by favorable location. Light roof trusses,
frames, or columns may deflect due to local distor.tion; their use as support structurcs should be
avoided particularly for lines subject to pressure
pulsations. Compressor suction and discharge lines
cause many problcms in the vibration of buildings
and are best supported or restrained to massive
foundations or heavy concrete structures wherever
available; otherwise, isolated supports to grade are
to be preferrcd.
The interconnecting piping between interrelated
process units and similar piping within an individual
unit between vessels, exchangers, pumps, and other
equipment may be, for economics of supporting
structures as well as improved appearance and access,
often collected into a parallel arrangement or "pipe
alley" as shown in Fig. 8.1, with the support elevation usually sufficiently above grade to promote
freedom of movement for operation or access, although locations at grade or in trenches may be
preferred to snit special considerations. Utility, yard
transfer, or similar piping more often involve routing
a single line or a few lines which must cross roads at
a sufficient elevation to permit free vehicular pass~ge
or must go through tunnels, culverts, sleeves, or else
be buried under the roadway. For such interconnecting piping where elevated location is desired, usc
is made where feasible of existing structnres by
attachment through roofs, and by utilizing building
columns as common members in support frames or
for the attachment of supplementary cantilever
support strnctures. Elsewhere individual support
structures are required.
Individual support structures at grade are almost
invariably of concrete and of simple sleeper or saddle
contour. Elevated individual support strnctures
may be single columns made of pipe (illustrated in
Fig. 8.21) although such are usually limited to fairly
small lines. Support bents usually involve the use of
structural shapes, as shown in Fig. 8.2, or else reinforced or unreinforced concrete. When attached, the
material selection is influenced by that of the existing
structure. For independent supports the economie
choice depends on the number of lines and weight to
be carried, height above grade, local material and
labor costs and availability, and requirements relative to fire resistance.
Steel constrnction is preferred where supports may
require relocation with future plant additions, or
removal for access to facilitate equipment repairs or
replacement; they also have an advantage where
space limitations are imposed or extreme loads
carried. The fireproofing of steel usually makes it
compare unfavorably in cost with concrete, and
further, if the beam is also encased, complicates
individual support bracket details. Unreinforced
concrete is usually limited to low or massive structures such as saddles, sleepers, etc. Reinforced concrete cost, when cast in place, is greatly dependent
on the cost of forms and their placement, and the
equipment available for elevated pouring in limited
quantity. Standardized design with reusable forms
makes the first point much less critical but pouring
costs are less easily pared. Precast reinforced concrete promises to offer an economic solution provided
satisfactory details for individual support attachments of variable size can be economically realized.
For concrete snpports or fireproofing the attachment
and fixture must be of such dimension and detail as
to allow for heat dissipation where the line temperature is above 400 F, to avoid deterioration by
calcining.
The sloping of lines is more advantageously accomplished from a standpoint of both economics and
appearance by variation of the pipe support fixtures
rather than the support strnctures.
8.7
Erection and I\fuintcnance of the Supporting, Restraining, and Bracing System
It is desirable that pipc support connecting structures be in position, and support fixtures be available
SUPPORTING, RESTRAINING, AND BRACING THE PIPING SYSTEM
before the assembly of the piping system is initiated,
in order to minimize the expense of temporary struc-
tures and ties and to simplify rigging. Usually the
need for temporary supports and rigging and staging
details is left entirely to the field engineers sinee in
the design phase the order of equipment arrival and
ereetion proeedure is not adequately established.
With eompetent eonstruetion erews, this is the most
eeonomie way of handling the problem in the majority
of eases, provided only that struetural eonneetions
and fixtures are made available to the field so they
can use them effieiently. There will be many instances, however, where at the eost of only a little
extra effort in planning, provisions can be made for
rigging, support of temporary staging for ereetion
equipment or personnel, or making up bolted joints,
etc.; or indications can be given as to how a support
can be modified or temporarily braced to earry increased loading.
No detrimental effeet results from minor plastie
deformation of the pipe from overload or pulling into
alignment during ereetion, exeept for speeial materials which are of limited duetility or are sensitive to
work hardening, or where such deformation leaves
residual stresses whieh may aceelerate loealized or
overall corrosive attaek. It is a good rule for the
ereetion forees to qnestion on all alloy materials
whether sueh deformation is undesirable, and for the
designers to make a practice of warning wherever
such eannot be tolerated. In sueh speeial cases final
make-up pieees are required and must be arranged
for in the initial plans; templates must be obtained
when the piping ereetion is eomplete to the point of
the make-up pieee; support during ereetion must be
such that yielding does not occur.
With tbe completion of erection of the piping and
supports, the support fixtures must be adjusted to
avoid sagging and to attain proper distribution of the
load between supports, also to effect the desired
functioning of the restraints and braces. In the
absence of controlled prcspring, the degree of residual
fabrication strain is not known, so that successive or
cumulative yielding may occur as supports, restraints,
or braces are adjusted. It is desirable to align the
pipe first at the more critical locations and then adjust
intermediate stations to suit, repeating this sequence
until the line position remains stable relative to the
supports.
If the fabrication involves stress relief or other
post-heat-treatment of field welds, supports shonld
be adjusted and auxiliary supports provided if
necessary to minimize stresses at the successive
heated areas. For final elosure welds, not only sup-
255
port stress effeets but also those introdueed ill
obtaining alignment by bolting up flanges, and by
weld shrinkage, are present and may induee plastie
deformation at the weld. Where heat treatment is
speeified instead of, or in addition to, stress relief,
the higher temperatures involved further aeeentuate
these effects so that extreme care is desirable and
usually additional supports are necessary. For this
purpose eounterweights or spring supports are preferable; rigid supports ean be used but they must be
sueeessively adjusted to an effeetive position during
heating and eooling and, partieularly, while at
temperature.
Fabrication residual stresses can be effectively re-
duced by thermal unloading at locations away from
girth welds in particular, and in general by avoiding
all locations where plastic deformation is undesirable
due to the reduced duetility attendant to biaxial or
triaxial stress distribution and to abrupt metallurgical, structure or eontour changes. Such unloading,
whieh is referred to in Chapters 2 and 3, is similar to
and follows the same procedure as loeal stress relief.
It can be employed at pumps, turbines, or other
sensitive equipment, in order to reduce fabrication
stress influence toward misalignment. It is often
applied by the use of gas or oxy-acetylene heating at
relatively rapid heating and cooling rates without
adverse aftereffects where sensitive materials, the
presence of flaws, or corrosive service are not involved. In general, the same precautions as are
exercised in hot forming in the same temperature
range, are sufficient. The use of sueh thermal unloading is to be encouraged, since only minor expense
is involved. It is possible to secure an essentially
presprung eondition if the thermal unloading is
applied when the line is first heated as in warming up
before initial serviee, thns preventing yielding and
ereep at undesirable loeations.
Heating for thermal unloading involves the same
precautions and general approaeh as for local stress
relief. Complete eireumferential areas should be
brought to temperature nniformly and without undue
heat concentration by moving torches continuously
and preferably by using two torches at opposite
locations in the circumference. \Vhen available, ring
gas burners, gas burner mumes, or other stress relief
equipment is advantageously employed. Temperatures can usually be sufficiently controlled by heatsensitive pellets, paint, or by the use of surface or
optical pyrometers.
The eventual distortion of the pipe line under
service and cyeles of temperature is largely dependent on the dimensional stability achieved by pre-
256
DESIGN OF PIPING SYSTEMS
spring or by thermal unloading, and by the effectiveness of the support and restraint adjustment. Cyclic
overstress, with necessary adjustment on each
thermal cyele, and oecasional yielding resulting from
upset conditions, ete., will change the line contour
and modify the reactions at supports and restraints.
Temporary periods of uneven temperature, particu-
larly during heating up, may cause bowing, etc.,
which disappears when equilibrium conditions are
again established.
It is always desirable that the performance of
supports, restraints, and braces be observed during
initial heating up to see if they perform as intended.
Adjustment is needed if unanticipated restraints
oceur which will distort the line or damage the supports. When equilibrium temperature is reached the
supports should be readjusted to the most favorable
position. The operators and maintenance forces
should appreciate the importance of observing the
action of these devices during each period of major
temperature change as well as during service.
Periodic adjustment of supports may well avoid
fatigue failure, unnecessary distortion, leakage] or
other distress. In Section 2.6 of Chapter 2, the significance of calculated deflections is shown to be
as a range of movement and not as an absolute
position. This carries with it the understanding
that supports must be adjusted to suit the immediate
working position of the line.
The foregoing is at least equally applicable to
average minimum engineered pipe systems as to
critical ones. With the former, the deflections, the
support and restraint reactions, and also the stresses
are established for the design of both the piping and
its supporting system by thumb rule or simplified
analyses subject to appreciable error. Observation
of the behavior of the line and supports is necessarily
an essential part of this approach since early correction of observable inadequacies can avoid later
extensive direct and contingent damages to the
piping and connected equipment.
For critical piping it is desirable to denne clearly
the installation and subsequent adjustment requirements, and where at all possible to send a design
engineer thoroughly familiar with the basic and
installation requirements, to assist with and observe
the adequacy of the installation. This is particularly
important on stiff or large high-temperature piping
or where critical materials are involved. In particu-
lar, measures for. prestress should be properly
executed, and the adjustment of special support
and restraint fixtures properly accomplished.
Stops should be adjusted so they will react in the
required degree at service temperature and, if required, also under ambient conditions. It should
be assured that the stop restricts only movement
normal to the contact surfaces, which should be
smooth and reasonably parallel.
Many supports or restraints involving sliding or
moving parts are dependent on the maintenance care
given to them for dependable operation. The
designer may easily make the mistake of placing
too much reliance on such maintenance; delicate
mechanisms are easily put out of order and should
be avoided. In all cases the designer should give
some thought to the consequences should a particular
device fail to function as planned. If consequences
are serious, a more foolproof detail should be sought.
Frictional resistance, when critical, can be combated by going to antifriction devices such as rollers
or self-lubricating details. Self-adjusting features
can often be worked in. An overall appraisal of
this nature can greatly increase a system's reliability
in service..
_ _ _ _ _ _ _ _ _ _efl
CHAPTER
9
Vibration: Prevention and Control
T
HE aim of this chapter is to summarize the
essentials of struetural and acoustie vibrations as applied to piping systems, aid the
piping engineer in establishing design practices which
will minimize the occurrence of objectionable or
damaging piping vibration such as may occur under
structural piping oscillations. On the other hand,
there does not appear to be any gcnerally recognized
up-ta-date text on flow vibration, so that it was
considered desirable to summarize fundamcntal information relative to this important subject. When
detailed treatment of suhjects considered is readily
available, only final pertinent rcsults are given,
whereas an attempt is made to provide necessary
analysis and discussion of specific subjects whose
coverage in the literature appears to be limited.
Certain basic texts may now be briefly mentioned.
These should provide the interested reader with a
gencral acquaintance with the mechanics of vibration and further constitute very useful reference
material. The texts by Den Hartog [I] and Timashenko [2J are best known for their thorough engineering treatment of thc fundamentals of mechanical
resonance, and indicate steps to control or eliminate
excessive vibration when it develops in service.
9.1
Introduction
The deleterious effects of vibration are often not
properly assessed. Failures actually caused by vibration are sometimes attributed to other causes
while, on other occasions, harmless oscillations of
perceptiblc amplitude have given rise to undue
alarm.
The possihility of fatigue fracture and the effect
of fatigue on the life of piping and the superposition
of vibration stresses have already been discussed in
Chapter 2. Among other undesirable effects to be
considered by the piping designer are: flow pulsa-
and structural vibration and contain numerous ex-
amples of oscillatory motion caused by reciprocating
and rotating machinery, self-excited vibrations, etc.,
as well as a comprehensive hihliography on these
problems. Rayleigh's Theory of Sound [3J is the
tions leading to noisy operation and increased flow
turhulence with attendant higher heat transfer rates,
classical text on acoustic, as well as structural, vibration and is recommended, with Love's treatise [4],
pressure drops, corrosion or erosion, and impairment
of operation of flow machinery, valves, and other
to those interested in dctailed derivations and proofs
of mathematical formulas. A further valuable the-
components; damage or leakage at critical joints
and scals; psychological effects of a safe but vibrating piping system,l etc.
In this chapter no attempt will be made to provide
a complete treatise on vibration. While relatively
little has been published strictly on piping vibration,
there is ample covcrage of the general subject of
mechanical vibration which is directly applicablc to
oretical work on acoustic vibrations is that of
Morse [5J. Analogies hetwcen mechanical, electrical,
and acoustic oscillatory systems are presented admirably by Olsen [6]. while Stoker [7J provides an
excellent introduction to the modern theory of nonlinear vibration, such as self-excited oscillations.
For routine design, Marks' [8J and Kent's [9J mechanical engineering hand hooks include a brief summary of formula.s used in engineering applications.
Vibration and shock isolation are treated by Crcde
[IOJ and by Ryder and Gatcombe [llJ.
lExperience would seem to indicate that an amplitude as
small as fir in. in Illrge-size piping (say over 12 in. diameter) is
:Jufficient to cause alarm if the piping is in an enclosed building,
und i in. if in an open structure.
257
DESIGN OF PIPING SYSTEMS
258
Basic concepts of vibration and a gencral discussion of vibration prevention and control are given
in the next section. The sections following include
a more extensive engineering treatment of the subject of structural and acoustic vibrations, with information for the convenience of the piping engineer
and stress analyst in setting up and solving problems which require detailed investigations. This is
followed by an illustration of thc application of
basic vibration concepts and derived equations in
the design analysis of a sample piping system. The
last section of this chapter discusscs piping vibration
from the point of view of diagnosis and correction of
existing conditions, i.e. "trouble shooting".
9.2
Fundamental Considerations in Piping
Vibration
9.2a Definitions. It is appropriate to define a
few terms which are fundamental in any discussion
of vibration theory and practice.
1. The period of vibration, T, (seconds) designates
the time of one complcte oscillation which is repeated in every respect.
2. The frequemy of oscillation, f, (cycles per
second) is equal to the reciprocal of the period of
vibration. The angular frcquency, w, (radians per
second) = 2"fT.
3. The number of degrees of freedom equals the
number of independent quantities defining the position of a system. Thus, a system consisting of a
mass attached to a massless spring and constrained
to unidirectional motion has one degree of freedom,
since thc systcm configuration is completely defined
by the deflection of the spring. A simply supported
flexible beam or pipe, on the other hand, has an infinite number of degrees of freedom because of the
f1exibiiity of each element relative to adjoining ones,
requiring an infinite number of element deflections
to describe the position completely.
4. A principal mode of vibration is a free vibration
(see 9.2b, "Types of Vibration") of a- system vibrating at a definite frequency. Thc number of
principal modcs is cqual to the number of degrces
of freedom. Frequcncics of thc principal modes of
oscillation are callcd natural frcquencies. The lowest natural frequency is called the fundamcntal frequency and corrcsponds to thc fundamental mode
of vibration. A beam, or pipe, has an infinite num-
ber of principal modes. Howevcr, the importance
of thc fundamental frequency is by far the greatcst.
5. Damping can be dcfined as thc reduction of
vibration amplitude through action of frictional
forces.
It is a cure for resonant vibrations, whereas
nonresonant vibrations are not much reduced by
friction dcvices. An examplc of a typical friction
dampener is the shock absorber used in some piping
systcms to limit the amplitude of possible resonant
vibrations.
6. From the practical point of view the most important problem in connection with vibration is the
phenomenon of resonance. Resonance occurs when
a system (mechanical or acoustic) is excited periodically with a frequency at or very near the natural
frequency of the system. If the damping (internal
or external) of the system is small, then the system
will rcspond to excitation, even by a small impulse
at the resonant frequency, with large amplitudes of
vibration, leading to large deflections in the casc of
structural vibrations or large pressure surges in
acoustic systems. These, in turn, are accompanied
by high repeated stresses which arc likely to cause
damage by fatigue failurc of the pipe or components.
7. In an clastic system, periodic application of a
force as distinct from a static force, may lead to
vibratory deflcctions (amplitude) equal to, larger
than, or smaller than static deflcctions. The ratio
between the maximum vibratory amplitude and the
static deflection is called the magnification factor
and is a function of thc ratio of forcing frequency
to natural frcquency and the amount of damping
present in the system.
9.2b Types of Vibration. Three main types of
oscillation must be carefully distinguished: free.
forced, and self-excited.
In free vibration, the system is excited by an cxternal transient impulse (persisting for only a short.
time) and the system vibrates under no external
force. In real systems, some damping is present
and the systcm oscillations will subside unless
another transient impulse is imparted.
In forced vibration, the system oscillates under
the external cxcitation of a periodic perturbing force.
A primary source of excitation might be the unbalance of rotating machinery (e.g. clectric motors,
turbines, compressors, fans, or pumps). Other frequently encountercd sources of forced piping vibration arc thc periodic variation of fluid pressures and
acceleration of masses within the reciprocating
devices.
Self-excited oscillation is a complicated phenomenon. The system vibrates under no periodic external
forces and the vibration persists, due to internal
energy sources, even in the presence of damping.
Among well-known examples of self-excited vibration are the humming of telegraph wires, chattering
of lathe tools, flow surging of fans or blowers, air-
_ _J
VIBRATION: PREVENTION AND CONTROL
plane wing flutter, eombustion and flow instabilities
in furnaces and boilers, etc. In piping systems,
vibrations of self-excited charaCter have been encountered usually in association with flow instabilities, surging of compressors due to unsuitable
operating characteristics, vortices due to steady
wind, pulsating gas-solid streams, etc. Analysis of
vibrations of this type is usually difficult and often
requires experimental investigation.
9.2c Sources of Periodic Excitation. Rotating
machinery invariably constitutes a major source of
mechanical vibration, due to the inevitable mass
unbalance existing in the rotating parts of the machine (see Subsection 9.5b). Unless rotating machinery is very carefully balanced or supported on
an elastic foundation for the purpose of vibration
isolation, forced vibration with a frequency equal
to the rotating speed may be expected in nearby
structures. If the rotating speed is in the neighborhood of the natural frequency of an adjoining
structure, resonant vibration will be induced, leading
to possible failures of piping or components.
A compressor of the reciprocating type is a source
of periodic pressure excitation at a frequeney (cps.)
equal to the rotational speed (rps.) multiplied by
the number of cylinders for single aetion and twice
the number of cylinders for double action for any
given stage. If this frequency approaches the
acoustic frequency of the connected piping system,
acoustic resonance in the form of large periodic
pressure surges will appear. Apart from possible
adverse effects on machinery and its operation, these
pressure pulsations can be transmitted direetly to
the foundations and buildings, and via bends acted
on by periodically variable forces, or through the
connecting pipe itself to other vessels, structures,
or foundations. A piping system vibration frequency equal to the speed of rotating machinery or
the pulses of reciprocating devices is a clear indication of the source of excitation, which can be corrected or minimized by balancing of rotors, inclusion
of vibration dampeners, pulsation snubbers, or other
teehniques described in this chapter, as well as in
Chapter 8.
Another source of periodic excitation of exposed
piping systems involves the action of wind. If air
strikes at right-angles to the axis of a cylinder of
diameter D. (ft) at a steady wind vclocity U (ft/sec),
then there result periodic forces of frequency f
(eycles/sec).
f = SUID. = 0.18UID.
(9.1)
where S = the Strouhal nnmber (= 0.18 for a cyl-
L
259
inder). These aerodynamic forces are due to vortex
motions around the cylinder (Von Karman vortices,
cf. [12]), and act at right angles to the direction of
the wind. Their magnitude is usually relatively
small and essentially equal to the dynamic pressure
acting on the projected area of the cylinder, but if
the frequency is in the neighborhood of a natural
frequency of a piping system the piping will be set
in resonant vibration with perhaps fairly large amplitudes. An example of this phenomenon is the socalled "humming" of telephone wires. Actual cases
of pipe vibration of this character have been observed [13].
The sources of mechanical or acoustic excitation
deseribed so far are more or less of a systematic
nature and can be expected and taken into account
in design. Unexpeeted sources of excitation may
exist in an installation or may develop in the course
of operation. Inasmuch as such excitation sources
cannot be provided for in advance, they must be
dealt with as they oecur.
An example of an apparently self-exeited vibration is that which appeared in oil refine"y fluid
catalytic craeking plants in the form of structural
vibrations, as well as pressure surges, and which was
traced to the gas-solid stream in the catalyst carrier
line. Changes in the line configuration and, partieularly, the catalyst injection detail greatly influenced these vibrations. Other similar difficulties
are occasionally encountered in other process equipment and remedies usually involve trial-and-error
changes in details of the fluid injection meehanism.
9.2d Vibration Prevention and Control.
Elimination or isolation of sources of vibration is
unquestionably the most desirable solution to a
vibration problem. However, it is often not possible to accomplish this objeetive completely. A
slight unbalance of rotating parts will probably persist. Some pressure pulsations due to flow machinery, wind or earthquake effects, etc. should be
expeeted by the piping engineer. Self-excited vibrations are diffieult to predict analytically, and the
designer may have to rely largely on field experience
and data in estimating probable frequencies of
excitation.
Since the piping designer has numerous other considerations which determine a piping system layout,
it is not suggested here that an elaborate vibration
analysis of all standard piping systems be carried
out; nevertheless, the engineer will usually be justified in spending the time needed to insure that the
fundamental natural frequency of a piping system
bearing pulsating flow (e.g. piping directly con-
260
DESIGN OF PIPING SYSTEMS
nected to reciprocating compressors) will not be in
the neighborhood of a forcing frequency. Proper
choice and spacing of supports and braces (guides
and damping devices, Chapter 8), as well as gaspulsation smoothing devices (Seetion 9.8), may be
added in the original design at little initial cost, and
sound judgment in making simplifying assumptions,
is much less expensive than correcting trouble when
and control is the determination of system natural
frequencies. This section summarizes the morc
encountered in the field.
Following establishment of design eriteria relative to allowable piping stresses and defleetions, the
designer should review available information on
probable foreing frequencies and estimate natural
frequencies (struetural and acoustic) of critical
piping. Formulas for the determination of natural
frequencies of simple systems arc given in Sections
9.3 through 9.8 and their use is illustrated in Section 9.9.
Rnowing the possible forcing frequencies (e.g.
from data on rotating and reciprocating maehinery),
an attempt should be made by the designer to prevent resonance of the piping system. The upper
limit of free pipe length (Le. the lower limit of
natural frequency) is usually governed by considerations other than vibration. Moreover, shifting
natural frequencies toward the lower end as compared with the exciting frequency has the disadvantage of not completely eliminating possible vibration
during start-ups and shutdowns of maehinery. The
other approach, that of introduction of additional
intermediate fixed or elastic supports for the purpose of shifting the natural frequency of piping
toward the high side, appears to be a more appropriate method of eliminating vibration although
less economical and conflicting with requirements
of thermal expansion.
Wherever it is not possible to follow either of the
two methods described above, and the natural frequency of the piping system remains dangerously
close to that of the exciting force, considerable
attention must be devoted to isolation by gas pulsation dampeners, elastic foundations, balancing of
rotating machinery, and· provision of adequate
damping dcvices (shock absorbers) at strategic
points in the s,Ystem. So-called dynamic dampeners
which couple a secondary vibratory system to the
main piping system, the former being "tuned" so as
to reduce the amplitude of vibration of the main
system at or near resonance, do not appear to be
practicable for piping. It should be emphasized
that solutions given in this chapter apply only to
relatively simple piping and support configurations.
Unfortunately, no general analytical treatment is
available for dealing with vibration problems, and
as well as experience, is always required.
9.3 Structural Natural Frequency Calculations
The primary prerequisite of vibration prevention
important results pertaining to structural frequencies
of various simple configurations.
9.3a The Spring-Mass Model. The simplest
and most fundamental of all mechanical oscillatory
systems is the one degree of freedom system consisting of a mass attached to a massless spring and
eonstrained to unidirectional motion (Fig. 9.1). Let
k = spring constant
= force in lb required to elongate or
compress the spring by 1 It
m = mass in slugs
_
weight (lb)
- acceleration due to gravity (ft/sec')
W
g
Then the undamped angular natural frequency W n
of the system, expressed in radians per sec l is
Wn
=!
(9.2)
and the frequency fn in cycks per second, eps, is
~
fn = W n =
FE.
2'1r
271" '\j:;,
(9.3)
Hence, the undamped natural period Tn, in seconds,
is
1
2".
2".
T = - = - = -(9.4)
fn
n
W
n
Vk/m
If the load (lb) is designated as IV = mg it is seen
that the static defleetion '" in feet of the spring
under load will be
IV
mg
Ost =
-
k
=-
k
Substituting this into eq. (9.3) gives another expression for the natural frequency fn, also in cps.
fn =
~ r;; = 0.906
2".
'\It V".
(9.5)
VIBRATION: PREVENTION AND CONTROL
Equations 9.3 and 9.5 are of great usefulness, primarily because of their immediate generalization to
eonfigurations far more compicX' than the simple
system for which they were derived.
W=mg
FIG. 9.2
Structure supporting II weight.
Thus, Fig. 9.2 shows a structure supporting a concentrated weight W (lb). Assume that the weight
of the structure can he neglected as compared with
W, and that it is desired to estimate the natural
frequency of vibration of the mass in a vertical
direction. By virtue of eqs. 9.3 or 9.5 the vibration
problem is reduced to a purely structural one. To
use eq. 9.3, calculate the concentrated force (lb)
rpquired to deflect point 0 vertically 1 ft (i.e. the
spring constant k(lb/ft) in the vertical direction at
o and also recognized as an influence coefficient in
structural deflection theory). With W = mg given,
fn is found by substitution into eq. 9.3. Alternately,
the static vertical deflection at point 0 under the
conccntrated weight W can be found and in calculated from eq. 9.5.
261
eq. 9.5, since in the former case the notions of spring
constant or the purely elastic cffect and notion of
mass or inertia effect are separate, while in eq. 9.5
the two notions are combined; as a result, eq. 9.3
lends itself more readily towards generalization.
This is illustrated by the eoncept of effective mass,
whieh will be introduced in the next subscction.
9.3b Frequency and :Mass Effectiveness Factors for Different End Constraints. Between
supports or anchors, a pipe is a beam with uniformly distributed mass. Each restrained length
possesses an infinite number of degrees of freedom,
hence vibration may occur in an infinite number of
modes singly or in combination. In practice, the
fundamental mode is of main interest) but occasionally the second mode may also bc of some
importance.
Thc gcncral cxpression for thc natural frcquencies
of beams with uniform mass distribution, sueh as
pipes, is of thc following form:
in = a~1~~3 = ~2~~I
(9.71
"
where a is a coefficient which varies with the end
conditions and mode. In order to follow thc piping practiee used in the remainder of this book, thc
numerical constant a is left with the dimension
(ft/in.)(ft/scc2 )J.j and the symbols in cq. 9.7 have
the following units2 :
W = total wcight, lb.
weight pCI' ft of pipe (including pipe contento
and insulation), Ib/ft.
E = modulus of clastieity, Ib/in 2
I = moment of inertia, in. 4
L = length of pipe, ft.
fn = natural frequcncy, cps.
Wu =
FIG. 9.3
End weight on n cantilever.
Example. Given a concentrated weight W = mg
being supported at the end of a massless cantilever
of Icngth L, structural moment of inertia I, and
matcrial modulus of elasticity E (see Fig. 9.3). It
is required to estimate the natural frequency of
vibration of the systcm.
The vertical (static) deflcction of 0 under the
weight W is, from structural theory,
WL3
ii" = 48 EI
Values for wU, I, and E can be found in Appendix C.
Tablc 9.1 gives values of a for the eases indicatcd.
The concept of lleffective mass" is a convenient
approaeh by which a distributcd mass is treatcd as
an equivalent concentrated mass. Thus, for a mas::;less eantilevcr of Icngth L (ft) with concentrated
cnd load P (Ib) the natural frcquency is givcn by
eq. 9.6. For uniform wcight distribution, the fundamental mode of vibration is:
using dimensions of W (lb), L (ft), E (lb/in?),
I (in 4 ), and ii" (ft). From eq. 9.5,
in = U\/~ = 0.265\/WL3
(El = 0.130 ~EI
in = 0.906 \/48WiJ
WL 3
(9.6)
In general, eq. 9.3 is of greater usefulness than
l
0.265
fEi
I EI
(see Table 9.1 and eq. 9.7)
2The constant a is nondimensional for a consistent set of
dimensions.
262
DESIGN OF PIPING SYSTEMS
Table 9.1
Mode of Vibration
Suppert
Frequency Coefficient a for
Pipe or Uniform Bar
~f.....~_=
__
-_
Fundamental (1st)
0.265
Cantilever
Second Mode
1.66
Fundamental (1st)
0.743
Second Mode
2.97
Fundamental (1st)
I.I6
Second Mode
3.76
Both ends simply supported
One end simply supported, other end fixed
Fundamental (1st)
§jl '----_/ ~
1.69
Both ends fixed
4.64
Second Mode
Hence it appears that the effective end weight of
a uniform weight distribution is
0.13
W,,, = ( --"
0.260
)2 W = 0.25W = tw
(9.S)
Consequently, the fundamental natural frequency
for a cantilever with a total weight W uniformly
distributed and a concentrated load P at the end is
- 013
in .
=
~ (t W EI
+ P)L3
0.13J
L
2
EI
I
(9.9)
(9.9a)
P
:jw y + L
The factor of effectiveness, t, can also be derived
in a manner which may be generalized to the case of
nonuniformly distributcd loading. Thus, if the
total weight of the cantilcver is Wand L is its
length, then considering only the infinitesimal mass
dm bctween x and (x + dx) (Fig. 9.4), the spring
FIG. 9.4
Elementary muss on cantilever.
constant k = 3Ellx3, the infinitesimal mass dm =
(WI Lg) dx, and the ratio
3EILg
3EI
(L)3
3
dm = Wx dx = Q~ dX) L3 ;k
the infinitesimal mass at x may be considered equivalent to an effective infinitesimal mass at the end
(x = L), multiplied by a weight factor (xIL)3. If
the effective infinitesimal masses at x = L is then
added, the total effective mass is
WJL(:.)3 dx = ~ W = W,,,
gL,
L
4 g
g
or
W,,, = tWo
Of course, this principle of addition of effective
masses is not exact but, as was seen, the approximation for uniformly distributed load is quite
reasonable.
On this approximate basis the procedure can also
be generalized to the case of a non-uniform mass
distribution on cantilever beams. Thus for a mass
as shown in Fig. 9.5, the effective end weight would
be
W1L'+L> (X)3
(L, + L,)4 - (L,)"'
TV"f = dx =
3
W
L 2 L,
L
4L 2 L
(9.10)
j
VIBRATION: PREVENTION AND CONTROL
263
Thus, for this mode, the dynamic deflection curve
may be taken as:
w
y(x, t) = f(x) sin wnt
being the natural frequency for the particular
mode. If x varies between x = 0 and x = L, the
potential and kinetic energies are respectively:
Wn
FIG. 9.5
Distributed mass on cantilever.
Potential energy = !E sin' wnt1L I(x)[f" (x)]' dx
with a natural frequency, cps,
lEI
(9.12)
fn = 0.13 \fWd}
2
For a concentrated load TV at x = L" L, --> 0, and
eq. 9.10 reduces to
TV,rr = TV
(?y
(9.11)
Apart from unusual cases, eqs. 9.10 and 9.11 will
yield results sufficiently accurate for engineering
estimates.
9.3c Variable Stiffness and Variable Mass.
There exists, since Rayleigh [3], a systematic procedure for the approximate calculation of natural
frequencies of structures with non-uniform stiffness
and non-uniform mass distribution. With this
method a reasonable form of the deflection curve
during oscillation is assumed and then maximum
kinetic and potential energies are calculated and
equated.
The result is an expression for the natural (undamped) frequency. If, by chance, the exact form
of the vibratory deflection curve for a particular
mode has been assumed the resulting expression for
the natural frequency for that particular mode will
be exact. In general, at least for the fundamental
mode, any reasonable approximation to the actual
deflection curve will yield results good enough for
practical purposes. A reasonable deflection curve
is one which satisfies at least the major boundary
conditions of the structure (e.g., end conditions for
beams).
IGnetic energy = ~ wn cos' wntlL TV(x)[f(x)]2 dx
2 g
0
(9.13)
so that
1
Max. potential energy = !E
L
I(x)[f" (x)J' dx
Max. kinetic energy = ~ wn'lL TV(x)[f(x)j' dx
2 g
0
from which
Wn
Egl
1
=
L
I(x)[f"(x)]' dx
(9.14)
L
TV(x)[f(x)]' dx
By analogy with eq. 9.2, the numerator and denominator in eq. 9.14 represent an effective spring constant and an effective mass (or load) respectively.
---,
FIG. 9.7
Cantilever.
For the fundamental mode, the approximate form
of the deflection curve for a cantilever, Fig. 9.7, is
f(x) = 28 sin' 1TX
4L
(9.15)
where 8 = maximum deflection (at end).
This satisfies the conditions for zero deflection and
FIG. 9.6
Mass distribution.
Thus, suppose a reasonable form of a deflection
curve for a particular mode for a beam with various
end conditions is f(x), the weight distribution is
TV(x) (Fig. 9.6), and the stiffness is expressed by a
variable moment of inertia I(x), x being measured
from a certain origin.
zero slope at x = 0 and a condition for zero moment
at x = L. It does not satisfy an end shear condition.
Likewise, for simply supported beams and fixedend beams the deflection curves may be taken respectively as
f(x) = /j sin "{
and
f(x) = /j sin' "{
where /j = maKimum deflection (at the center).
264
DESIGN OF PIPING SYSTEMS
For a cantilever, it was seen that the uniform load
may be taken as equivalent to i concentrated end
load. Likewise, on the basis of the assumed deflection curves:
For a simply supported beam, by analogy with
eq. 9.14 the effective center load (equivalent to uniform load), after cancelling the constant 0, is
lVJ:L sm. -dx =.W
W,,, = -
L
2 1rX
1
L
0
(9.16)
Cantilever
fn = 0.13
~ (!-W EI
+ I'lL"
=0.13RSi'I
L'
1
I'
-w
+-L
4 u
(9.9; 9.90 I
Simply supported beamfn = 0.525 ~ (! W EI
+ P)L 3
For a fixed-end beam, the equivalent ccnter load is
lVJ:L .
4 7TX
3
W'''=r; 0 sm r;dx=a W
(9.17)
For beams, the stiffness is usually constant along
the span and the form of the mass distribution is
frequently as follows:
uniform load + end load
and
=
BI
0.525
L2
+-I'
1
-W y
2
I
Fixed end beam
L
(9.1S)
RI
f" = 1.03 '\j (ilW + P)L3
for cantilevers
= 1.03JPi'I
2
uniform load + center load for simply supported
or fixed end beam.
L
Thus, let:
fn = natural frequency of pipe (1st mode), cps.
W = total weight of pipe, lb.
W u = weight of pipe per ft of length (including contents and insulation), Ib/ft.
I' = end load for cantilever or center load for simply
supported or fixed-end pipe (beam), lb.
L = length of pipe, ft.
E = modulus of elasticity of pipe material, Ib/in.'
I = moment of inertia of pipe section, in. 4
Values of wy , E and I for pipe can be found in
Appendix C. Then, for beams with loading conditions as stated above, the following formulas may be
used for the first modc of vibration:
".
3
I'
·w.. +8"
L
(9.19)
9.3d Combined Bending-Torsion. Consider
the cantilever in Fig. 9.8 and assume its mass can
be neglected. Then, the static dcflec:tion at point 0
for a wcight Pis:
'J
L b3
L L
L a3
0= [ 4 s
- + 4S+ 144~
P (9.201
EI a
Eh
OJ a
where La and Lb = the two arms of thc configurations, ft.
E = common modulus of elasticity,
Ib/in 2
G = torsional shearing modulus,
Ih/in 2
I a and h = respective moments, of inertia of
the section, in. 4
J a = polar moment of inertia about
the center of gravity of the circular section a or the "equivalent"
value for a noncircular section.
m. 4
o = deflection, ft.
The effective spring constant, Ibl ft, for point 0 is
Eh/4SLb"
k =
FIG. 9.8
Configuration in bending torsion.
1+
(~) (~J + 3 ~~: (~:)
(9.21)
VIBRATION: PREVENTION AND CONTROL
265
and the natural frequency is then
E1o/PLb'
fa = 0.13
1 + ~ (La)' + 3 E10 (La)
(9.22)
l,
GJ a L b
I a Lb
l,
If the mass of the piping (beam) cannot be neglected, and denoting by Wva and Wvb the weight per
linear foot ([b/ft) of length J"a and Lb respectively,
E1o/W,flLb'
fa = 0.13
a
b
FIG. 9.11
b
2E1o
Eh
fa = 0.13
,(La),
P
"41L'va "4 + ZWyb + "4
0.13
= --2
Lb
Two members: - angle 0 between legs.
idcalized as in Fig. 9.10. Analogous to eqs. 9.22
and 9.23 the following expressions can be written:
10 (La)'
E10-(La)
1+- +3
I L
GJ L
a
o
1 10 (La)' +3E1o
1+--- -(La)
32I L
2GJ L
a
(9.23)
10 (La)'
E10-(La)
1+
--+3
I L
GJ Lb
a
W'fl=tlV~+!Wb+P
0.13
where lV a and lVb are the weights of legs La and Lb.
Equation 9.23 is necessarily an approximation,
and in particular the effect of lVb is a somewhat
crude approximation. A closer approximation can
be obtained by energy methods (see Rayleigh's
method described above). However, in most practical cases, additional refinements are not warranted.
The foregoing results may be applied, in an approximate manner, to estimate the fundamental
natural frequency of a particular pipe bend shown
in Fig. 9.9. For purposes of cstimating the natural
frequency of vibration (in a mode perpendicular to
the plane of the paper), the configuration may be
g.9
Schematic of pipe bend.
p
o
b
= --2
Lb
P
3
(La)
L b + 8Wva L b + Wvb
(9.24)
1 10 (La)'
3E1o
1+-- +
-(La)
-
32Ia Lb
2GJ a L b
Here again in is in cps; h, la' and J a have units
of in.'; E and G, Ib/in.'; La and Lb' ft; IVa and
IV b, lb; Wva and Wvb lb/ft; and load P lb.
9.3e Approxinlate Natural Frequencies of
Pipe Bends with Two Members (Vibration Perpendicular to Plane of Bend). Consider a pipe
bend as shown in Fig. 9.11, having both lcgs the
same in diameter, thickness, and material. The mass
distribution is considered to be due only to the
weight of the pipes.
For the usual pipe materials, the ratio of bending
rigidity to torsional rigidity, EI/GJ = 1.3 = I.
With this approximation, the general expression for
the fundamental natural frequency of vibration perpendicular to the pipe bend will be of the form
lEI
f=a'l/WL 3
}<'IG.
a
2E1o
a
b
b
(9.25)
wherejisincpsIEisinlb/in.2,lisinin.\ lV = total
weight of bend in Ib, L = L, + L 2 = total length
of bend in ft l and a is a numerical factor depending
upon the ratio L,jL 2 and the angle o.
Consider t\'..·o extreme cases:
a. L 2 = 0, L, = L
In this case the bend reduces to a fixcd-end beam
and a = 1.69 constant for all angles.
b. L, = L 2 = !L
FIG. 9.10
ldclllizcd configuration for Fig. 9.9.
In this case (L, = L 2 ), if 0 = 0, the bend reduces
DESIGN OF PIPING SYSTEMS
266
l:t=L/2
FlO. 9.12
Two-member bend 0 = 1r/2.
to two parallel cantilevers and " = 1.06. In the
other extreme, 0 = ", the bend reduces again to a
fixed-end beam and" = 1.69. Consider now the
case,O = ,,/2 (Fig. 9.12). It can be shown that the
deflection y, (in ft) at the point 0 due to a unit load
at that point, in a direction perpendicular to the
plane of the bend, is given by
Obviously curves of " vs. 0 for all other length
ratios L,/L" must be situated between the two
extreme cases (a) and (b) discussed above and
shown in Fig. 9.13. Even though it is not exactly
correct, a simple interpolation between the two
extreme cases for any given length ratio is good
enough for engineering estimates. Thus the two
extreme cases plus simple interpolation define the
values of " for any length ratio and any angle 0 to
within reasonable approximation.
J
f (C.p.I.) = ex
£1
WI'
e
Yo =
48/El
I/L~~
+
I/L;'
1-
I,
..
1+ (El/GJ)(L2I Ltl
1-
•
1+ (El /GJ)(L,/L 2 )
II = l(l,=O)
1.69 1------'--'-'--'-------:"
(9.26)
1.60
For El/GJ = 1 as assumed and L,/L 2 = 1, the
result reduces to
5 L3
15 L 3
y = X 144 = - ,
384 El
8 El
(9.27)
The effective end weight for a cantilever due to
uniformly distributed mass is approximately .Ii of
the total weight of the cantilever. For a fixed-end
beam the effective weight at the center due to uniformly distributed load is % of the total weight of
the beam.
Clearly the coefficient of effectiveness for the 90'
bend of equal pipe lengths is situated between .Ii
and %. If then the effective weight is taken as
lV'fl = %; total weight of bend, the result can not
be greatly in error. With this value of effective
weight at the junction point and the expression for
Yo given above, it is found that 0: = 1.2. Thus for
the bend of equal lengths:
o
a
"2
"
"
1.06
1.2
1.69
If a curve of " vs. 0 is drawn through these three
points it appears as shown in Fig. 9.13.
In the other extreme case (L 2 = 0; L, = I),
" = 1.69 = Constant for all 0, as already remarked
above.
" =
This "curve,"
i.e., the horizontal line
1.69, is also shown in Fig. 9.13.
1.,(0
1.20
1.06 L_--~
+---~~-~--_--~-o
o
"
FIG. 9.13 Approximate natural frequencies of pipe bend
with two legs (vibration l~crpendicuJn.r to plane of bend).
9.3! Plates and Radial Mode in Pipe. The
spring constant (for deflection caused by a concentrated load at center) of a simply supported circular
plate is
Et 3
(9.28)
k = 261-,
r
where k = spring constant, Ib/ft.
r = radius of plate, ft.
t = thickness of plate, ft.
F: = modulus of elasticity, Ib/in.'
and
lJ = Poisson's ratio, assumed to be equal to
0.3.
For a concentrated load P (lb) at the center of
the plate, large with respect to the plate weight, the
natural frequency in cps is then
In =..!:-
2"
3
261Et g = 14.6!
Pr'
rEt
r "\J P
(9.29)
VIBRATION: PHEVENTION AND CONTHOL
The effective conccntrated wcight (at center) of a
total uniformly distributed load lV (lb) on a simply
supported circular plate may be taken as
Weer =
tw
(g. 30)
Using the above units, the natural frequency of a
circular plate clamped at the edge and acted on by
a concentrated load P at the center is (assuming
J' = 0.3)
t/Et
fn = 23.3; '\Ii>
(9.31 )
For a uniformly distributed load lV on a clamped
circular plate, the effective concentrated load (at
center) may be taken as:
lVelf = !lV
(9.32)
A high-frequency periodic pressure variation in a
pipe might induce a structural vibration in a radial
mode, which will result in alternate hoop tension
and compression. The fundamental natural frequency of a pipe, with a free length equal to, say,
at least 5 diameters, in a radial mode is [4]:
fn =
6~5 ~
discussion the reader is referred to standard references on the subject.
The motion of a spring-mass combination when
excited with a periodically varying force (P sin wt)
is governed by the differential equation:
mx+kx = Fsinwt
(9.34)
x being the displacement of the mass or deflection
in spring, relative to the system configuration for
P = o. Although no damping is assumed so far, a
small amount of damping may always be supposed
to exist. Because of this effect the initial transients
will eventually he damped out, so that the remaining
Hsteady state" solution is of the form x = R sin wt.
Substituting this solution into eq. 9.34 gives
(-mw 2 + k)R = P, or
since W n =
vk/m or
R
1
P/k
(9.33)
where D = mean diameter of pipe, in., E is inlb/in?,
and p = density of material, Ib/in 3 However, for
typical cases the frequency fn (cps), as given by
eq. 9.33, is so high that resonance of this type is
hardly ever likely to occur.
Ovalling M odc. Another mode of vibration (observed on large stacks at certain wind speeds) is
what is often referred to as availing. In this mode
of vibration the circular cross section assumes an
elliptical shape with the major and minor axes alter-
Now, P/k represents the sialic deflection of the
spring under the force P, while R is the maximum
dynamic deflection under the periodic force P sin wt.
That is P/k = X,t, R = Xci,n. The absolute value
of the ratio of maximum dynamic deflection (maximum amplitude) to static deflection is defined appropriately as the magnification factor, thus:
M.F. =
For steel pipe, the natural frequency (cps) corresponding to this mode may be estimated from the
expression
where D = diameter, in.
l = thickness, in.
Apart from large vessels or stacks, the natural frequency of ordinary pipe in an ovalling mode is high.
Structural Resonance and IHagnification
Factors
Since resonance is the most important phenomenon in vibration, a few simple derivations of funda-
mentals will be given, although for an exhaustive
I Xci,n
I
X"
(9.35)
Of J from the previous results,
nating in perpendicular directions once per cycle.
9.4
267
M.F. =
11- (w~/wn2) I
(9.36)
A plot of M.F. vs. the ratio of forcing frequeney w
to undamped natural frequency W n (i.e. w/w 11 ) is
sometimes referred to as a resonance curve (for zero
damping). This curve is shown in Fig. 9.14, labeled
o. For this extreme case (Le. uo damping), the
, =
magnification factor becomes infinite when the forc-
ing frequency w coincides with the undamped natural frequency wn •
In the presence of a finite amount of viscous
damping, the differential equation takes the form
mx + ex + kx = F sin wt, c being a coefficient of
viscous damping. It is simpler to consider the
(ultimately equivalent) differential equation
mx + ex + lex = Fc iINt
(9.37)
DESIGN OF PIPING SYSTEMS
268
A' + (cAlm) + wn ' = 0, wn' = k/m, m"" 0, from
which
M.f.
Critical damping occurs when the expression under
,=0
the radical is zero, That is:
6.0
Cc =
(9,39)
2mw n
By the use of eqs. 9.38 and 9.39, eq, 9,37, after
dividing by k, may be written in the form:
5.0
x + -2, i; + x = (F)
. = OIlLe .
-')
W
k
r=t
'.0
t
elW!
Wn~
""
n
The steady state solution for x is of the form:
3.0
.
x = Re iw /
R being in general a complex quantity, the absolute
valne (modulus) of which represents maximum displacement (spring deflection), i.e. IRI = adyn. From
the differential equation:
2.0
1.0
R
1
w. (w)'
-
X" = 1 + 2,-, -
0
0
1.0
FIG. 9.14
Wn
w
2.0
W;;
Resonance (frequency response) curves.
Wn
so that the magnification factor is:
M.F.
where
eiwt = CGS wi + i sin wi
and
i= v=l
1
(9,40)
The effective damping depends not only upon the
viscous friction coefficient but also upon the mass
and spring constant. The overall effect is expressed
in terms of a viscous damping coefficient or damping
ratio:
(9.38)
where Cc is defined as critical damping, pertaining to
the particular system.
The critical damping of an effective spring-massdashpot system is defined as follows: If less damping
is present the system will damp out in an oscillatory
manner when disturbed by a passing transient; for
damping greater than critical the system will damp
out in a nonoscillatory manner, as illustrated in
Fig. 9.15.
The critical damping for a given system is immediately determined from the differential equation
as follows:
For free vibration the differential equation is:
mx + eX + kx = 0. This differential equation is
satisfied by the expression x = Ae" provided that
A is a root of the quadratic mI.'
CA
k = 0, or
°
Plots of M.F. vs w/w n for several values of , > are
shown in Fig. 9.14, in addition to the "pure" resonance curve for, = 0,
The maxima of the lvl,F. are given by:
. /
'
2,vl-,'
(9.4l)
and the maxima occur at:
j
+ +
1
1 :0; (l'vLF.) max =
W
VI - 2,'
(9,42)
,
,
C>Ce
+-+--+-''-Tim.
FIG. 9.15
+---...::::::==-_nme
Oscilla.tory and nonoscillatory damped motion.
269
VIBRATION: PREVENTION AND CONTROL
M.F.
Equation 9.41 holds only over the range
(9.43a)
The corresponding range of frequency ratios is
o -<~
<1
w -
System with
relatively
high noluro!
S)"lem with relatively
frequency
Sow nclurel frequency
relative 10 (I
rololivo 10 the K1me
9ivon forcing
frequency W 1
forcing frequency
(9.43b)
n
1
For t >~,
V2
O+--+--~-+-----_
W
(M.F.) max = 1 at- = 0
o
W"
Thus, the maximum magnification factors in the
presence of damping occur at frequencies somewhat
below the natural frequeney. It is to be notcd that
magnification disappears (i.e. M.F. max :1> 1) at a
damping ratio t = 1/ v'z which is less than critical,
since at critical damping, t = 1.
From Fig. 9.14 it is also observed that the maxima
of the M.F. fall off sharply with an increase of the
damping ratio t from zero. This is seen more clearly
from a plot of thc maxima of M.F. vs. t, as sh"wn in
Fig. 9.16. The usual effectiveness of hydraulic
shock absorbers or any other equivalent form of
viscous damping is based upon this phenomenon.
M.F.
mo,
7.0
Eqvotiom
6.0
{M.F.)mox=
1
.~
2tv'-t'
h>t>O
'.0
..0
J.O
1.0
1.0
O+------+-~------~-
o
FIG. 9.16
n'
1.0
2.0
l"
!\faxima of magnification factor VB. damping ratio.
FIG. 9.17
.,,
.,,
M.F. US a function of w/w no
Generally little inherent damping exists in structures. Structural damping (due essentially to
hysteresis effects in metals) varies with the mode of
vibration, but for fundamental modes (for cantilevers
say) the damping is of the order of a few per cent
at the most, i.e., t = 0.01-D.03. Thus in the absence
of any other damping, such low values of t yield
very high magnifieation factors in the neighborhood
of resonance. Hence, thc resonance curvc (t = 0)
should be used for estimating resonance magnification factors (M.F.) in structures. In practical
estimates a probable maximum M.F. of 50 is sometimes used.
While compromises may be necessary, it is desirable, if possible, to design a system so that the
frequencies of possible external excitation fall
outside the frequency band cqual to say one-half the
natural frequency on either side of resonance, as
shown in Fig. 9.17. In other words, eithcr the
systcm is designed so that the frequency range w of
the expected extcrnal excitation satisfies the inequality w/w n > ~., or elsc the system is designed
so that w/w" <!. In the first case the system is of
relatively low natural frequency (i.e. relative to thc
frequency of the disturbing force), while in the
second case the system is of relatively high natural
frequency.
The first alternative is usually the more economical
but there are limitations. If this alternative is
chosen, the system will always pass through a
resonance condition during starting or stopping.
However, for continuous operation of relatively long
duration this limitation is perhaps not of considerable importance. A more serious limitation may
consist in a conflict with strength or deflection
requirements. For ordinarily, in piping, the natural
frequencies can be reduccd essentially only by an
increase in free length. If structural and relatcd
requiremcnts do not permit the choice of the first
270
DESIGN OF PIPING SYSTEMS
alternative, then the system must be designed on
the basis of the second altcrnative, that is by
increasing the natural frequencies .:ufficiently. Again
this can be achieved in piping essentially only through
a redtWtion in free length by the introduction of
additional supports, rigid or clastic. If neither
alternative is possible, artificial damping must be
introduced, in the forni of bracing devices which
dissipate energy through friction, (hydraulic shock
absorbers, etc.).
The results given above are based upon the use of
the spring-mass-dashpot system as a model. This
model, being a system of one degree of freedom,
necessarily exhibits only one resonance peale
A piping system or even a simply supported beam is,
in reality, a system possessing an infinite number of
degrees of freedom and such a system will exhibit,
theoretically at least, an infinite number of resonance peaks. However, in general the first (fundamental) mode with the lowest natural frequency is
the most important one in any system, and the
resonance behavior in the neighborhood of the
first peak in any system is essentially the same as
that for the simple spring-mass model.
ge5 Damping of Structural Vibrations
9.5a Hydraulic Snubbers. 3 In vibration surveys
the amplitude and frequency of vibration is determined at a point (or various points) of a vibrating
structure. If the amplitude of vibration is excessive
and if this amplitude can be reduced only by the
installation of a hydraulic snubber, it is of importance
to estimate the maximum force the snubber must
transmit and the degree of damping required to
reduce the amplitude of vibration to a tolerable
magnitude.
In this discussion it is assumed that the vibration
is of the forced type and the damping characteristic
of the snubber is of the purely viscous type. It is
necessary then that the following quantities be
known or be estimated or measured:
1. The amplitude of vibration,
l
ROJ ft
from vibration
2. The frequency of vibration,
survey
w, Tad/sec
3. Natural frequency of Vibrat-j
ing structure, W n
from calculations
4. Spring constant of vibrating or experiment
structure, k, Ib/ft
(at point where amplitude is measured)
!The terms "hydraulic snubber" and Hhydraulic shock
absorber" are both used herein tQ denote the kind of damping
device described in Chapter 8.
If
R = amplitude of vibration with damping, ft,
\ =
viscous damping coefficient (nondimensional),
F s = maximum force to be transmitted by snubber,
lb,
Then, we have, from the relations given in the
text,
1
R/R o =
-;:::=======
J2
1 - (w/w )2
(9.44)
'I + [ 2\ (w/w n )
"\j
n
or
\ =
~ 11 - (w/w n )21 ...; (R o/R)2 _ 1
(9.45)
ll - (w/w n )21"';1 - (R/Ro)'
(9.46)
2
F s = kRo
w/w n
Thus, k, Ro, w, and W n being known, by hypothesis,
the maximum sllubber force F s can be determined
from eq. 9.46 for any given amplitude ratio R/R o of
damped to undamped vibration. The required
damping coefficient \ for a desired amplitude ratio
is given by eq. 9.45, or if \ is known, the amplitude
ratio is determined by eq. 9.44.
If \ is not known, eq. 9.45 suggests an experimental method whereby this quantity can be
determined. Consider a simple system whose
natural frequency W n is known , subject it to a
forced vibration of frequency w/w n and measure the
amplitude R o; then connect a hydraulic snubber to
the point where the amplitudc is measured; the
system is now subjected to forced vibration under
the same conditions as before and measurements
are obtained for the reduced amplitude R. The
damping coefficient \ of the snubber can then be
determined from eq. 9.45.
A study of eqs. 9.44, 9.45, and 9.46 shows that
viscous damping is really effective only in tbe neighborhood of resonance (w/w n = 1). This can also be
shown from Fig. 9.14; equation 9.44 represents the
reduction of the M.F. obtained by reading along a
vertical in Fig. 9.14 from the no damping (\ = 0)
curve to that for a given \. Outside the vicinity of
resonance (say when w/w n ::; ~- or w/w n ~ !)I high
damping coefficients are required to reduce the
amplitude to a reasonable degree, Le. very large
maximum snubbcr forces may be needed. This point
will be further illustrated numerically in Section 9.9.
From the above expressions is also obtained:
Fs = kR
o
11 - (w/w n )21
'1+ [1 -
"\j
(w/wn)2J
2\w/w n •
2
(9.47)
271
VIBRATION: PREVENTION AND CONTROL
Unbolollcod Man m
Suppose now that n identical snubbers are connected
at the same point. This is equivalent to an increase
of 1 by the factor n, which changes the expression
MouM
f-t====",===;t/
Rolor
forFsto:
F ~ kR
s
o
11 - (w/wn)'1
'1+ [1 -
"\I
(9.48)
(w/wn)'J'
2nl w/w n
FIG. 9.18
It is seen that the total maximum snubber force
increases considerably less than proportionally with
the number of snubber units. Indeed as n ---> oc, the
max. total snubber force tends to a finite limit value
given by:
(9.49)
A.n infinitely stiff shock absorber (I ~ 00) is a rigid
connection; hence, eq. 9.49 represents the force
transmitted to the ground when the point in question
is rigidly connected. The above is true from a
dynamic point of view. However, shock absorbers
of sufficiently large 1 may still be used to allow
flexibility of piping equilibrium positions in view of
thermal expansion as well as serve to counteract
resonant vibration amplitudes.
From the preceding equations the following, perhaps more useful, relations ean also be obtained:
If R n = amplitude for n shock absorbers
(R t = amplitude for 1 shock absorber)
(Ro = amplitude without shock absorbers)
F n = total max. force transmitted by n shock
absorbers
(F 1 = total max. force transmitted by 1 shock
absorber)
Then:
~: = [1 - (1 - :')
GJf'
Rotating machinery on elastic foundation.
is the rotational speed, then the centrifugal force
F* (lb) due to the unbalanced rotor mass m
(Ib sec'/ft) at radius T (ft) is:
F* = mrw 2
(9.53)
Let k (lb/ft) be the overall spring constant of the
system. Then the static deflection of the spring
under force F* is
(9.54)
Since the undamped natural frequency W n (rad/sec)
of the mass mo (lb sec'/ft) is
Wn
= Vk/mo
we have
(9.55)
and
(9.56)
When the rotational speed is equal to the undamped
natural frequency, the static deflection and the force
F* will be independent of the natural frequency
0*" = (m/mo)T
(9.57)
In the absence of damping, the maximum dynamic
deflection is
(9.58)
(9.50)
and the maximum dynamic force in the spring is
.!..) (Rl)'
J-11 (9.51)
Ro
Fmax = komnx = mOWn omnx = 11 _ (w/w n )21
Fn
Rn
Substituting the expression for 0" from eq. 9.56
into eq. 9.58 and cq. 9.59 gives:
FI
R1
R n = ~ [1 _ (1 _
Rt
n
n'
So that
-=n-
2
(9.52)
Thus if RdRo :<:; i say, then Fn = = F t from eq.
9.50 and R n = Rdn by eq. 9.52.
9.5h Elastic Foundations for Rotating Machincry. The installation of rotating machinery on
elastic foundations is a fairly standard means for
preventing vibration. The basic features of such an
installation may be illustrated on the basis of the
simple spring-mass model, Fig. 9.18. If w(rad/sec)
mOWn 20s t
omax =
?n
(w/w n)'
?no Til _ (w/wn)'1
F m" =
[I _ (w/wn)'1
(9.59)
(9.60)
(9.61 )
or, in view of equations 9.55 and 9.57
Om,,/o", = 11 - (wn/w)'I-t
(9.62)
Fm,,/F' = 11 - (w/wn)'I-t
(9.63)
DESIGN OF PIPING SYSTEMS
272
A plot of Fm.x/F* and om.x/o*" VB. (w/w n ) is shown
in Fig. 9.19. The solid curves represent Fmax/F*
while the dashed curves represe~t omax/o*".
From the above figure it is observed that if minimizing the deflection is of primary interest, then
(w/w n ) ought to be as small as possible in which
casc Fmax tcnds to F*. But if thc reduction to a
minimum of the maximum force transmitted to the
foundation through the spring is of primary consideration, then (w/w n ) should be as high as possible in which case Omo.x tends toward 0* st. The
term elastic is appropriatc, of course, only to the
second case. It is also seen from eq. 9.63 or Fig. 9.19
that F m . . ::s; F* (centrifugal force) corresponding to
a magnification factor less than unity, only when
w/wn ~ ,12. The above results are based upon the
spring-mass system with no damping present. If
the system is such that thc rotating spccd and natural frequencies are near to each other and for various reasons the ratio w/w n cannot be changed, or if
a range of rotational speeds must be provided for,
then viseous damping must be introdueed by means
of dashpots (hydraulic shock absorbers, etc.), as
shown by the dotted outline in Fig. 9.18.
In this case the force transmitted to the foundation is the sum of the forces in the springs and in
thc dashpots and the following results can be
verified:
F mBx
(9.64)
[1 - (:JJ + (2 :J
~::: 1[1 - (:JJ + (21~)'r;, (9.65)
F*
=
I ,
I I
I I
I I
I I
,.
FmQII;
I
I
3.0
,
I'
I
I
I
,
I
2.0
,,
I
1.0
I
I
I
I
/ -~--=:-:----------cw~
o ¥'--;-,
if
,.
1.0
{2
2.0
3.0
-w,
F_
F'
,=0
1.0 1""""----1-'''1.;;::-----''--
o 0:----;,';;.o:-ffr.\2~------"'::-
""
1
FIG. 9.20 Fma.x/F· VS. w/w n for damped vibration.
=
the damping ratio 1 having bccn dcfincd previously
(see Scction 9.4).
Families of curves of Fmnx/F* and omnx/o*at VB.
w/w n with 1 as parameter are shown in Figs. 9.20
and 9.21 respectively. It is seen from Fig. 9.20 that
for w/w n < vi!, the maximum force transmitted to
the foundation is greater than the centrifugal force
mrw' (i.e. the magnifieation is greater than unity)
and it is in this range only that damping is beneficial.
It is also seen from Fig. 9.21 that in so far as deflection in the springs is concerned, damping is
always beneficial.
The ahove results, even though based on a one
degrec spring-mass-dashpot system, may be applied
to morc complex systems provided the effective incrtia (including possibly a distributed mass in the
"springs"), and the effective spring constant, can
,=0
1.0
o
t= co __
....,"---_~-----L---
o
FIG. 9.21
1.0
OmfJr./O· VB. wJw n for damped vibration.
------_.
,
VIBRATION: I'HEVENTION AND CONTHOL
273
be properly estimated. Finally, the above results
illustrate also the essentials of the behavior of shock
absorbers, at least of the viscous type.
For air, if T is the absolute temperature in degrees
R, the speed of sound is approximately
9.6
The natural frequencies for a tube with one end
closed and one end open form an odd harmonic
seale (even harmonics arc absent):
Acoustic Natural Frequency Calculations
Rigorous detennination of aconstic natural frequencies of a piping system is usually difficult.
However, approximate estimates can be made on
the basis of results applying to a few simple configurations.
The piping engineer may find it difficult to interpret and to apply some of the material in the classical
texts on acoustic vibrations. Furthermore, there
does not appear to be an engineering textbook
which would give an up-ta-date summary of flow
vibration analysis and applications as well as a bibliography equivalent to the treatment of mechanical
vibrations by Den Hartog II] and Timoshenko [2J.
The following sections will present a number of
C = 1120V7'/518 = 49.3VT
C
3C
(9.66)
5C
In = 4£; 4£; 4£; etc....
(9.07 )
For a tube H open" at both ends, the natural periods are the same as for a tube HclosedJl at both cnds.
As was pointed out above, it is not always ell-tiy to
decide whether a physically open end may be considered open in acoustic considerations.
In view of
this, the value of the length £ in the above equations
may differ somewhat from the actual physical
length.
derivations of flow vibration relations and a dis-
cussion of the general analytical approach to be
pursued by the piping engineer in dealing with
acoustic oscillations. Section 9.9 gives an example
illustrating the possible application of derived formulas. Subjects adequately treated in the literature
(e.g. water hammer) will be covered very briefly.
Several references dealing with specific acoustic vibration problems of possible interest to the piping
engineer are included in the list of references at the
end of this chapter.
9.6a The Organ Pipe and Resonators. The
organ pipe is a tube with a large length/diameter
ratio, so that the motion of the fluid (say air) inside
the tube is essentially one-dimensional. The major
acoustic unknowns are the end conditions. Usually,
two extreme end conditions are considered, closed
and open ends. At a closed end the pressure variation is a maximum and such a point is referred to as
a node. At open ends the velocity is a maximum
and such points are denoted as loops. The fact that
a pipe end is geometrically open does not mean that
there is always a loop at that end. The designations
closed and open are, in general, strictly applicable in
an acoustical sense only.
The longitudinal natural frequencies, for a tube
with both ends closed, form a harmonic scale:
Period (sec)
2£ 12£ 12£
C' 2 C; 3" C etc....
C
2C
3C
A=Nock Area
--------'-'-,
FIG. n.22
Resonator.
Another frequently encountered acoustic element
is the Helmholz resonator. A resonator is essentially
a chamber with a "neck," the chamber volume being
large as compared to the neck (Fig. 9.22).
A
resonator is considered as a simple spring-mass
system. It is supposed that the air in the neck
vibrates as a solid mass while the air in the chamber
is alternately compressed and rarefied. Based on
these assumptions, the fundamental natural period
is found to be (ef. Rayleigh [3J, vol. ii, eh. xvi):
Period (sec) =
or
JvL
c"'llil
27r
C
Frequency (cps) In = 27r
(A
(9.68)
"'llVi
where C = speed of sound in fluid, ft/see.
V = volumc of chamber, ft3 .
£ = length of neck, ft.
A = cross-sectional area of neck, fe.
Equation 9.68 holds for a "long" neck Helmholz
resonator, so long as:
£» ~ V 7rA
(9.68a)
Frequency (cps) = 2£' 2£' 2£ etc....
A more general result given by Rayleigh [3J is:
where C = velocity of sound in the fluid, ft/see.
£ = length of the tube, ft.
In = ~ ~~
(9.691
DESIGN OF PIPING SYSTEMS
274
where
A
(9.69a)
II = .
£+ ~V;X
has been designated by Rayleigh as an "aeoustie
conductivity."
Thus,
i _~
I
A
• - 2'11'\j 1'(£ + ~V'll'A)
(9.70)
If £» ~V;X then the result reduces to that given
by eq. 9.68. On the other hand, if £« ~V;X then
in = ~J~
211'
I~
(9.71)
v'\l 11'
This is the case of a "cavity resonator."
For a cireular opening of diameter d
A = 'II'd'/4
length Xis greater (by at least a faetor of 2 or 3) than
a representative dimension of the chamber (say the
greatest linear dimension). Othenvise the results
given in eqs. 9.68 to 9.74 are in error.
9.6b Special Cases of Multiple Resonator
Formulas. A piping system with enlargements
(pulsation dampeners, vessels, ete.) eonstitutes a
multiple resonator system. General expressions for
the natural frequeneies of multiple resonator systems
are given in Appendix B.1 4 Simple types of multiple
resonators will now be eonsidered. On the basis
of the results pertaining to simple and multiple
resonators it is possible to estimate, at least approximately, the pertinent acoustic frequencies of the
resonator type of a relative complex piping system
with enlargements. This will be illustrated numerically in Section 9.9.
Consider then the multi-resonator of Fig. 9.23
where the J./s arc the "acoustic conductivities":
the cavity resonator equation becomes
Jl2 =
in=~~
(9.72)
The last result holds provided, t« d/4, where t is
the eavity wall thiekness. Finally, for a spherical
cavity resonator, the result reduces to:
in = £.
~ = 0.22 QD'\l'j)
~
(9.73)
2'11'D'\l;. 'j)
d and D being the diameters of the cavity opening
in = ~
V,
VI
A3
£3 + ~V'II'A3
while the Us, A's and V's are the neck lengths, neck
areas, and chamber volumes respectively.
The above system possesses two degrees of freedom
and can resonate in two distinct modes. The corresponding frequencies arc given by:
X= C = 2'11'.JY(;
in
Il
(9.74)
All results given above hold so long as the wave
No<k
I
I
A,
1-'1' L1
V,
VI
and spherical chamher respectively, provided of
course, d/ D «1. In general, in a Helmholz resonator,
a representative dimension of the neck is supposed
to be much smaller than a representative dimension
of the chamber or cavity An important quantity
in connection with the above results is the II wave
length"
The Il'S being in ft, the V's in ft 3 , C in ft/sec and
in in cps. This is the result as given in Appendix B
in a slightly different form.
The smaller frequency or fundamental is given by
the (-) sign under the radical, the larger frequency
or harmonic corresponding to the (+) sign.
A few special cases may be of particular interest:
a. Suppose 1" = 0 (say A, = 0). The multiple
resonator consists now of two completely separateci
resonators an'l the result must reduce to the simple
4S ee also [24}.
Nock
Nod<
V,
A,
(B.8)
VI V,J
Chamber
Chamber
J
113 = --=~=
I~ [Ill + 112 + llZ + 113 ± J(1l1 + 112 _ 112 + 1l3)' + 41l, 'l
211' \j 2
£2 + ~v;Az
v,
A,
P". L,
FH.... 9.23 Two-chamber resonator system with both ends open.
VIBRATION: PREVENTION AND CONTROL
resonator expressions. Indeed from eq. B.8:
r;;
C
r;:;
iJ
J. = 211' '\IV; ;
h = 211' '\IV;,
(9.75)
b. Suppose 1', = 1'2 = 1'3 = 1'; V, = V 2 = V. Then
from eq. E.8:
J. = ~
211'
G,
(9.76) .
'\Iv
275
The meaning of this result is that if the resonator is
filled with fluid, a disturbance is then introduced
through an opening in a chamber, and the opening
is then closed, the fluid will vibrate in the closed
resonator with a frequency given by eq. 9.78.
e. Suppose A, = 0, so that 1', = 0, while L 2 is
large so that 1', »1'2. Assume also that V 2 » V,
(Fig. 9.26). From eq. E.8 the following approximate result is found:
I, = ~
hence,
211'
It is seen that the relation between harmonic and
fundamental resonator frequencies is not the same
for multiple resonators as in organ pipes.
r;;,
'\IV;
(9.79)
h=~ r;:;
211' '\IV;,
But the fundamental is h, i.e.
h«I,
1',
v,
1',
v,
A, L,
Fro. 9.24 Two-cho.mber resonawr system
with one end closed.
c. Suppose 1'3 = 0(A 3 = 0); 1', = 1'2 = 1'; V, =
V 2 = V (i.e. one end is closed as shown in Fig. 9.24).
Then from eq, E.8
I = ~
211'
J(3 V5) 1:
±
2
or
V
C ~I'
I, = 0.62211'
V
(9.77)
c~- '
12 ~ 162. 211" V'
whence
h- = 2.6
I,
d. Suppose A, = A 3 = 0, whereby 1', = 1'3 = 0;
Then from eq. B.8
1'2 = 1', as in Fig. 9.25.
c~
(9.78)
1= 211" '\IV; + V;
I'
v,
FIG. 9.25
v,
Closed two-chamber resonator system.
FIG. 9.26
Acoustic coupling of resonator system
with vcry long neck.
This result is of practieal interest, for it may be
applied to a piping system with a number of'relatively small enlargements ending with a long pipe
terminating in a large vessel. The result derivcd
here shows that the lowest resonator frequency of
the system may be found by considering merely the
long pipe and large vessel as a simple /lIang neck"
resonator and, for the purpose of calculation of this
lowest frequeney, disregarding the rest of the system. Likewise the higher resonator frequencies may
be calculated disregarding the long pipe and large
vessel. In other words, the system is essentially
uncoupled into systems of low and systems of high
resonator frequencies. If the resulting low frequency
is low enough as compared to the higher frequencies
of the system, this procedure is justified. A further
approximation in the same direction is as follows:
The system is decoupled into simple systems of one
degree of freedom each and the uncouplcd frcquencies are thus ealeulated. In general, the frequeneies
of the coupled system separate the frequencies of the
uncoupled system and vice versa. In this way, it is
possible to establish the approximate vicinity of
intermediate frequencies and upper or lower bounds
of an extreme frequency by simple calculation.
Applications to estimating resonator frequencies of
a piping system are given in Section 9.9.
DESIGN OF PIPING SYSTEMS
276
It is assumed in linear acoustic theory that the
diameter to length ratio of any segment is small.
The following result can be verified from classical
volume=V
Fro. 9.27
nn
acoustic theory:
Chamber with multiple necks.
f. Finally, consider the case of a chamber with
several neck inlets (Fig. 9.27). Let the number of
Hnecks" be n with conductivities J.Ll J.L2, .•• Jln·
Then, it can be shown that
.------=!2
I~l
(9.80)
f "21l""\f + ~2 +V ... + ~"
l
provided the necks are not spaced too closely.
In general, then, a complex piping system may be
considered as consisting primarily of a combination
of organ pipes and simple resonators. No explicit
formulas for the natural periods of a complex system
can be given here. For a detailed study the reader
is referred to Rayleigh [3J and l'vlorse [5J. If the
system can be considered as consisting solely of
organ pipes, then the method given in Subsection 9.6c
and Appendix B can be used to determine natural
periods, while for resonators the formulas given
above may be used. Sample calculations of pulsations from the discharge of a gas compressor cyl-
Let n segments meet at a joint; the area, length
and phase angle of the ith segment shall be ai, LSi,
and 13, and let f be one of the acoustic natural frequencies of the overall piping system which may
consist of (say) s joints. Then at the joint in question the following relation holds:
I: ± ai tan (,,:2 foL Ls,L + l3i)
'~l
0
=
(9.81 )
Such relations hold for each joint. L is any reference length, but preferably a length giving a reasonable value of fo (such as an over-all length of the
main pipe branch) as a first estimate of the system
frequency.
The (+) sign is to be used for the segments with
flow tOlVards the joint. The (- ) sign is to be used
for segments with flow from the joint.
The following additional end conditions are also
to be satisfied:
At intake points
For node 13 = 0
For loop 13 = ,,/2
inder are given in Section 9.9. A coincidence of this
pulsation frequency with a natural acoustic frequency of the piping, organ pipe or resonator type,
should be avoided.
9.6c Piping Systems with Branches and Enlargements. The following tenninology will be
used in dealing with calculation of acoustic frequencies of branched piping systems (see Fig. 9.28):
A branch of a piping system is a continuous pipe,
straight or curved with constant or variable area.
At discharge points
For node 13 = (,,/2)[2m - (fIfo)],
m = 1 for fundamental.
m = 2 for 1st harmonic, etc.
For loop 13 = (,,/2)[m - (fIfo)],
m = 1 for fundamental.
m = 3 for 1st harmonic.
m = 5 for 2nd harmonic, etc.
The steady-state flow rate in a branch is constant.
A joint is a point where several branches meet or
a point where the pipe area of a branch changes
abruptly.
Joint
A segment is a part of a branch with constant cross
section.
Let C = speed of sound in the gas, ft/sec.
fo = CI4L = fundamentalacousticfrequency,
cps, of a simple organ pipe of
length L with a node at one end
and loop at the other end.
L s = length of pipe measured from the beginning of each separate branch, ft.
a = area of a segment meeting at a joint, ft 2 .
13 = phase angle corresponding to segment,
radians.
:-~0
'~,
Joint
~'
1 Bronth
Joint
o;,,/:
Joint
Joint
t
,%
2 Bron<hoi
t::=v=
,
S"moo' ,
FIG. 9.28
Branched piping terminology.
(9.82)
277
VIBRATION: PREVENTION AND CONTROL
Relations 9.81 for each joint plus relations 9.82
are sufficient to determine all ,,'s as well as I, the
fundamental or higher acoustic'irequency of the
system, which is of greatest interest.
In order to apply the above theoretical results to
any piping system, consider the system as eonsisting
of one main branch of length L with one intake and
one discharge point and additional seC<Jndary
branches each with one intake. Then calculate the
frequency 10 for the main branch. The general system thus eonsists of a number of intakc points but
only one discharge point.
The use of these relations will be illustrated in
Section 9.9 numerically. A graphical method which
is the counterpart of the foregoing analytical method
is given by Warming [14]. For a branched system
the analytical method appears to be less time consuming. A brief hydrodynamic derivation of the
above results is given in Appendix B.2.
The equations for the acoustic natural frequencies
of piping systems, considered as consisting of organ
pipe elements, are in general somewhat cumbersome
except for fairly simple systems. A system with
several branches and enlargements requires the solution of a simultaneous system of transcendental
equations and this can be quite time consuming. In
such cases it might