February 16, 2006
To:
AISI Committee Members
Subject: Progress Report No. 1
Direct Strength Design for Cold-Formed Steel Members With Perforations
Please find enclosed our first progress report summarizing the initial research work on
expanding the capabilities of the Direct Strength Method to cold-formed steel members
with perforations. Our efforts leading up to this report have been focused in three areas:
1) conducting a thorough literature review of experimental and analytical studies
focused on cold-formed steel members with holes
2) developing finite element modeling guidelines to ensure the accuracy of our
computer modeling of members with holes
3) comparing existing experimental stub column and long column data to DSM
predictions
We look forward to your comments regarding this ongoing research.
Sincerely,
Cris Moen
moen@jhu.edu
Ben Schafer
schafer@jhu.edu
Summary of Progress
Preliminary Finite Element Work:

Evaluated ABAQUS shell elements for
use in elastic eigenbuckling
problems, then selected the S9R5 nine node element

Determined finite element meshing density and aspect ratio limits to be
used when modeling cold-formed steel members with holes and with
rounded corners
Parameter Studies:

Completed a parameter study which evaluated the effects of a single
SSMA slotted hole on the elastic buckling behavior of an intermediate
length SSMA 362S162-33 cold-formed steel channel

Completed a parameter study which evaluated the effects of circular hole
diameter on the elastic buckling behavior of an intermediate SSMA
362S162-33 length cold-formed steel channel
Comparison of DSM Predictions to Experimental Test Data

Compiled stub column and long column experimental data

Produced finite element models for all specimens in ABAQUS

Evaluated the buckled shapes of all specimens and determined the critical
local, distortional, and global buckling loads for use in the DSM

Performed a detailed comparison of DSM strength predictions and
experimental tests
1 Introduction
This progress report summarizes results from the first research phase
focused on extending the Direct Strength Method to cold-formed steel members
with holes. The primary goal of this phase is to use existing test data coupled
with finite element analysis to develop design procedures for conventional
members with holes using the Direct Strength Method (DSM).
Before starting any modeling or analysis, a preliminary literature review
was conducted to find experimental data from tests on cold-formed steel
compression members with holes and to become familiar with the state-of-the-art
design and analysis practices in this field. This literature review is presented in
Section 2.
Section 3 describes the preliminary work for performing an elastic
eigenbuckling analysis in ABAQUS, a commercial finite element program. The
accuracy limits of the ABAQUS S4, S4R, and S9R5 element types are defined,
meshing strategies for modeling holes and rounded corners are developed, and
comparisons are made between finite element and finite strip eigenbuckling
solutions.
Section 4 presents a study which evaluates the effects of a Steel Stud
Manufacturer’s Association (SSMA) slotted hole and a circular hole on the elastic
buckling characteristics of intermediate length compression members. In Section
4.3, ABAQUS finite element results are compared to the “unstiffened element”
1
approximation for predicting the local buckling strength of cold-formed steel
members with holes.
Section 5 provides a detailed comparison of the tested strengths of coldformed steel stub columns and long columns specimens with holes to the
predicted strengths determined using finite element analysis and the DSM.
Section 6 concludes this report by summarizing additional Phase I work required
before we head off into the more theoretical challenges in Phases II and III of this
project.
2 Literature Review
A plethora of relevant research exists on the strength of thin plates and
cold-formed steel members with holes.
References 1 through 5 provide
experimental data and predicted strengths for compression members with a
channel cross section and a single perforation. This experimental data is the
primary focus in Section 5 of this report.
References 6 and 7 present
experimental data and predicted strengths for two connected channel sections,
each with a single perforation. References 8 through 11 contain experimental
data and predicted strengths for compression members with multiple
perforations.
References 12 and 13 provide a summary of the design and
research state-of-the-art for perforated thin-walled structures. References 14 and
15 summarize existing experimental data for cold-formed steel members with
holes and describe the development of effective width design equations.
2
References 16 through 21 provide experimental and predicted strengths of thin
perforated plates.
3
3 Preliminary FEM Studies
Before setting out to calculate the elastic buckling behavior of cold-formed
steel members with finite element methods, a study was performed to determine
a) the most accurate ABAQUS thin shell element to use in this eigenbuckling
analysis, and b) to define limits on the element mesh density and aspect ratio.
Three of the ABAQUS elements available for an elastic buckling analysis
are the S9R5, S4, and S4R elements. The S9R5 element is a doubly curved thin
shell element with nine nodes. The S4 and the S4R elements are four node
general purpose shell elements valid for both thick and thin shell problems.
3.1
Modeling Thin Plates in ABAQUS
3.1.1 Stiffened Plate
A series of elastic buckling analyses were performed in ABAQUS on a thin
plate simply supported on all sides, referred to in this study as a stiffened plate.
The plate is loaded uniaxially in this study. The buckled shape of a stiffened
plate is shown in Figure 3.1.
4
Figure 3.1 Buckled Shape of Stiffened Plate
The theoretical buckling load for a stiffened plate is:
N cr 
kEt 3 2
12b 2 1  2 
Ncr has dimensions of force per unit length and b is the width of the plate, as
shown in Figure 3.1. E is the modulus of elasticity of the plate material,  is the
Poisson’s ratio, and t is the thickness of the plate.
The buckling coefficient k is:
 mb n 2 a 
k 


mb 
 a
2
where b is the length of the plate as shown in Figure 3.1, and m and n are the
number of buckling half-wavelengths in the a and b directions, respectively. In
Figure 3.1, m  4 and n  1 .
5
The first method used to evaluate the accuracy of the S4, S4R, and S9R5
elements was to compare the theoretical versus calculated k as a function of
element aspect ratio. The garland curve in Figure 3.2 allows comparison of the
theoretical k to buckling coefficients calculated in ABAQUS by varying the plate
length to width ratio,
6
1
5.5
plate buckling coefficient, k
a
. The element aspect ratio is held constant at 8:1.
b
S4
S4R
S9R5
Theory
0.8
0.6
5
0.4
4.5
0.2
4
0
3.5
0
0
0.2
1
0.4
2
0.6
3
0.8
1
4
5
a/b
Figure 3.2 Accuracy of ABAQUS S9R5, S4, and S4R Elements for A Simply
Supported Plate with Varying Plate Aspect Ratios, Element Aspect
Ratio Held Constant at 8:1
The S9R5 element performs accurately for the element aspect ratio
considered, with a maximum error of 1.2 percent. The S4 and S4R elements are
not as accurate in this case, with maximum errors of 24.8 percent and 20.6
percent respectively. These results are consistent with the nature of each type of
6
element.
The S9R5 element uses a quadratic shape function to estimate
displacements and can therefore capture the half-sine wave of a buckled plate
buckling half-sine wave with as little as one element. The S4 and S4R elements
use linear shape functions to estimate displacements, and therefore require at
least three elements to coarsely estimate the shape of a half sine wave.
The idea of using the number of elements required to model a halfbuckling wave seems to be a more useful indicator of mesh density and model
accuracy than just considering the element aspect ratio. Figure 3.3 implements
this idea by demonstrating the improvement in modeling accuracy for a stiffened
plate as the number of S9R5 elements per half-wavelength increase.
It is
observed that for one S9R5 element per half-wavelength the modeling error is 2.1
percent, and for two elements the error reduces to 0.1 percent.
7
4.2
1
buckling coefficient, k
4.15
S9R5
Theory
0.8
4.1
0.6
4.05
0.4
4
0.2
3.95
0
0
0.2
0.4
0.6
0.8
1
3.9
0
1
2
3
4
5
Number of S9R5 elements per half wavelength 
6
Figure 3.3 Accuracy of S9R5 Elements as a Function of the Number of Elements
Provided Per Buckled Half-Wavelength, Simply Supported Plate (See
Figure 3.1 for buckled shape and loading conditions)
3.1.2 Unstiffened Plate
It is common design practice in the cold-formed steel industry to calculate
the local buckling strength of plate with a hole by approximating the plate as two
plates, both simply supported on three sides and free to displace on the fourth
side. This type of plate will be referred to as an unstiffened plate in this study.
The buckled shape of an unstiffened plate is shown in Figure 3.4.
8
Figure 3.4 Buckled Shape of Unstiffened Plate (See Figure 3.1 for plate
dimensions and loading nomenclature)
The theoretical buckling coefficient k for an unstiffened plate can be solved for
by using the following equations:

   2 

 2
m 2 2 
m 2 2 
tanh(

b
)







 tanh(  b)
a2 
a2 

1/ 2
 m 2 2 m 2 1/ 2 
  2 
k 
ab
 a

1/ 2
 m 2 2 m 2 1/ 2 
   2 
k 
a
ab


For unstiffened plates, the buckling half wavelength,  , is always the length of
the plate itself. Figure 3.5 compares the theoretical to predicted k versus the
number of S9R5 elements provided along the length a of the plate. The element
aspect ratio is held constant at 8:1. The S9R5 element produces an error of 1.9
percent with two elements along the length of the plate, and an error of 0.5
percent with four elements along the length of the plate.
9
2
1
S9R5
Theory
1.8
buckling coefficient, k
1.6
0.8
1.4
0.6
1.2
1
0.4
0.8
0.6
0.2
0.4
0
0.2
0
0
0
0.2
0.4
0.6
0.8
1
2
3
4
5
Number of ABAQUS S9R5 elements per half wavelength 
1
6
Figure 3.5 Accuracy of S9R5 Elements as a Function of the Number of Elements
Provided Per Buckled Half-Wavelength, Unstiffened Plate (See Figure
3.4 for buckled shape, Figure 3.1 for loading conditions)
3.1.3 Conclusions
Even before initiating this study, it was expected that the S9R5 element
would be an accurate and versatile performer when it comes to elastic buckling
analyses. The evidence presented supports this initial expectation, and therefore
the S9R5 will be the element of choice for the rest of the studies in this report.
The guidelines for meshing density along the length a of a plate are summarized
below :
10

Use at least two S9R5 elements per buckling half-wavelength in areas
where the finite element model most resembles a stiffened plate (e.g.,
web, flanges of a channel section)

Use at least four S9R5 elements per buckling half-wevelength in areas
where the finite element model most resembles an unstiffened plate
(e.g., web plate near the hole, flange stiffeners)

Limit the S9R5 element aspect ratio to a maximum of 8:1
Additional meshing guidelines pertaining to the meshing density for the width
b of plates with holes will be presented in Section 3.2.
3.2 Modeling Holes in ABAQUS
3.2.1 Description of Work
This
study
will
attempt
to
establish
some
minimum
meshing
requirements for plates with holes by studying the convergence of the
eigenbuckling solution for a web plate from an SSMA 362S162-33 structural
channel. The meshing of the holes in this study was performed with MATLAB
code that creates layers of S9R5 elements radiating from the hole to the edge of
the plate. The number of element layers is varied for an industry standard SSMA
slotted hole, a circular hole, and a square hole.
Figure 3.6 shows the meshing layout with four layers of elements surrounding an
SSMA slotted hole, a rectangular hole, and a circular hole.
11
Figure 3.6 Finite Element Meshes for Slotted, Rectangular, and Circular holes
(Four layers shown here)
All plates in this study are modeled as stiffened plates. Figure 3.7 shows the
variation in buckling load of the first plate buckling mode, a half sine wave, as
the number of layers of elements around the hole increase.
1.2
1
0.8
0.8
0.6
0.6
0.4
N
cr,hole
/N
cr,no hole
1
circular hole
square hole
SSMA slotted hole
0.4
0.2
0.2
0
0
0
0
0.2
2
0.4
0.6
0.8
4
6
8
10
12
Number of element layers around hole
1
14
16
Figure 3.7 Convergence of the Plate Bucking Load as the Number of Element
Layers Around Hole Increases for Simply Supported Plates (See
Figure 3.6 for examples of plates with holes)
12
The ABAQUS buckling load per unit length of plate, Ncr , varies widely for one
and two layers of elements but become relatively stable at four layers of elements
and beyond. It is interesting to note that for the SSMA slotted hole the plate is
softer for a smaller number of layers, which is contradictory to the trends
presented by the plates with circular and square holes.
3.2.2 Conclusions
The following meshing guidelines are recommended for the use of S9R5
elements surrounding a hole:

Provide at least four layers of elements surrounding the hole to ensure an
accurate eigenbuckling solution

Although the aspect ratio is more difficult to define for the radial mesh,
limit the maximum approximate element aspect ratio of the radial
elements to 8:1
3.3 Modeling Rounded Corners in ABAQUS
3.3.1 Description of Work
A parameter study was conducted to determine the effect of the number
of elements used to model the rounded corners of an SSMA 600S162-68 structural
channel. The ABAQUS S9R5 is a curved thin shell element so it was expected to
perform well, even with as little as two elements. The number of elements was
varied from one to five around each corner of the channel, with the
13
corresponding element aspect ratio varying from 5 to 22. Figure 3.8 compares a
typical channel corner modeled with one element to that of a corner modeled
with three elements.
It is interesting to note that just one S9R5 element
successfully follows the radial geometry of the corner. This is made possible by
the fact that the S9R5 element uses a quadratic function to define its shape.
Figure 3.8 Channel Corner Modeled as a) One S9R5 Element, b) Three S9R5
Elements (SSMA 600S162-68 Shown)
Figure 3.9 plots the elastic buckling load associated with the minimum local Pcr
and distortional Pcrd versus the number of elements used to mesh around the
corner of the channel.
14
1
1
Local
Distortional
0.9
0.8
0.8
0.7
0.6
cr
P /P
y
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
0
1
0
0.2
2
0.4
0.6
0.8
1
3
4
5
Number of elements around corner
6
Figure 3.9 Variation in Pcr and Pcrd versus the Number of S9R5 Elements Used
to Model each Channel Corner for an SSMA 600162-68 Channel,
L=1220 mm (See Figure 3.8 for examples from ABAQUS)
3.3.2 Conclusions
This study shows that the accuracy of the eigenbuckling analysis for coldformed steel channels is not affected by 1) the number of S9R5 elements around
the corner, and 2) the aspect ratio of the elements used to model the corners. It is
recommended that the element aspect ratios used to model around corners
should be less than 20:1 to ensure accuracy of the model.
Future work will
compare eigenbuckling analyses with straight line ABAQUS models to analyses
with rounded corner ABAQUS models.
15
3.4 Comparison of ABAQUS and CUFSM
This portion of the preliminary modeling work will focus on verifying
that the ABAQUS finite element model, including loads, boundary conditions,
and element mesh produce results consistent with the finite strip method used in
CUFSM.
A member with an SSMA 362S162-33 structural channel section is
modeled in ABAQUS. This member does not have a hole and is modeled at
three different lengths corresponding to the local buckling and distortional halfwavelengths as predicted by CUFSM. In addition, a finite element model is
created to calculate the flexural-torsional buckling of a long member in
ABAQUS. The results of the ABAQUS and CUFSM analyses are presented in
Table 3.1.
Table 3.1. Comparison of ABAQUS Eigenbuckling Results versus CUFSM
Buckling Mode
Local
Distortional
Flexural-Torsional
Buckling
HalfWavelength
(mm)
69
447
2438
Member
Pcr/Py
Pcr/Py
Length
(mm)
(CUFSM) (ABAQUS) % D ifference
69
0.26
0.28
-7.3%
447
0.63
0.64
-0.5%
2438
0.18
0.17
0.6%
It is observed that the global and distortional results are consistent between
ABAQUS and CUFSM, although there is a 7.3 percent difference between the
local buckling predictions.
16
To study the possible cause of this local buckling difference, the same
member was analyzed in ABABQUS with member lengths equaling one to five
times the local buckling wavelength. This was done to evaluate the effect of the
ABAQUS boundary conditions on the model, since they are expected to become
less pronounced as the length of the member increases. This trend is observed in
Figure 3.10, where
Pcr
approaches the CUFSM results as the number of local
Py
half-wavelengths increase.
1
1
ABAQUS
CUFSM
0.9
0.8
0.8
0.7
0.6
crl
P /P
y
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
0
1
0
0.2
2
0.4
0.6
3
number of half wavelengths
0.8
4
1
5
Figure 3.11 Comparison of Pcr for a 362S162-33 Channel Calculated With
ABAQUS and with CUFSM, Length of Member Varies from 1 to 5
times the Local Half-Wavelength of 69 mm (No holes in these
members)
17
The results are still somewhat unexpected though, because intuitively the
ABAQUS results should be “stiffer” than the CUFSM results for the shortest
member length corresponding to one half-wavelength.
Also, the ABAQUS
results to do not converge to the CUFSM results, but instead exceed them as the
member becomes longer.
A detailed investigation of the transverse shear
stiffness of the ABAQUS S9R5 element will be completed in future work to
resolve these unexpected convergence results.
18
4 Parameter Studies on Channels with Holes
4.1
Influence of an SSMA Slotted Hole on Elastic Buckling
This study investigates the effects of the industry standard SSMA slotted
hole on the elastic buckling behavior of an intermediate length cold-formed steel
member. The typical compression member in this study has a length L of 1220
mm and is modeled with an SSMA 362162-33 structural channel cross section. A
single slotted hole is centered in the web and moved in increments of
x
along
L
the member’s length, where x is the distance from the edge of the member to the
centerline of the hole. Figure 4.1 defines the dimension nomenclature for the
channel sections evaluated in this report.
Figure 4.2 defines the dimension
nomenclature for the hole shapes considered in this report.
summarizes the dimensions of the SSMA 362162-33 cross section.
19
Table 4.1
Figure 4.1 Channel Cross Section Dimension Nomenclature (Dimensions are
out-to-out)
Figure 4.2 Hole Dimension Nomenclature (Slotted hole shown, circular and
rectangular holes similar)
20
Table 4.1 SSMA 362S162-33 cross section dimensions
SSMA Designation
362S162-33
H
(mm)
92.1
B1
(mm)
41.3
B2
(mm)
41.3
D1
(mm)
12.7
D2
(mm)
12.7
R
(mm)
1.9
t
(mm)
0.88
Of special interest in this study are the new buckling mode shapes created
by the addition of the hole, which may have potential consequences when
predicting member strength with the DSM. The first new mode type is a mixed
local-distortional mode where the distortional buckling is located primarily near
the hole. The mode will be referred to as DH in this report. Another common
buckled shape is an antisymmetric distortional hole mode. This mode will be
referred to as DH2 in this report. The occurrence of these new distortional mode
types is directly related to the local reduction in bending stiffness of the web near
the hole.
The reduction in web bending stiffness increases the distortional
tendencies of the flanges. In addition, the typical local buckling mode will be
referred to as L, the pure distortional mode as D, and the global mode as GFT,
GF or GT. Figure 4.3 summarizes the common types of buckled mode shapes for
the member in this study.
21
Mode Shape Label
Pcr/Py
Local
0.282
DH
0.307
DH2
0.514
D
0.650
GFT
0.614
Buckled Shape
22
Figure 4.3 Buckling Mode Shape Summary, 362S162-33 Channel with SSMA
Slotted Hole, L=1220 mm, Hole located at Midlength of Member
(Refer to Figure 4.5 for a comparison of the elastic buckling loads)
Figure 4.4 shows the locations of the holes considered in this study, as
well as the DH buckling mode for this member.
Figure 4.4 SSMA Slotted Hole Location and Mixed Local-Distortional Buckling
x
(DH), 362S162-33 Channel, L=1220 mm , =0.09,0.28,0.47,0.66,0.84
L
4.1.1
CUFSM Boundary Conditions
Two different sets of boundary and loading conditions are considered in
this study.
The first set of conditions is consistent with those assumed in
23
CUFSM. Warping is allowed at the ends of the member and restrained at the
midlength of the member. Also, the cross section is restrained transversely at the
member ends to preserve the original shape of the cross section. The member is
loaded at both ends with a uniform pressure using consistent nodal loads.
1
1
0.9
D (CUFSM)
0.8
0.8
GFT (CUFSM)
0.7
0.6
D
GFT
DH2
DH
L
cr
P /P
y
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
0
0
0
0.1
0.2
0.2
0.4
0.3
0.4
x/L
0.6
L (CUFSM)
0.8
0.5
0.6
0.7
1
0.8
Figure 4.5 Influence of SSMA Slotted Hole Location on Pcr for a 362S162-33
Channel, CUFSM Boundary Conditions, L=1220 mm (Refer to Figure
4.3 for buckled shape summary)
Figure 4.5 plots the variation in elastic buckling load for local, distortional,
and global modes as the slotted hole moves along the length of the member.
Pcr is not affected by the hole, and Pcrd shows only small deviations from the
predicted load of the pure distortional mode calculated in CUFSM.
interesting observation is that Pcre decreases from the Pcr
, no _ hole
One
value as the hole
moves towards the end of the member. With the hole near the member end, a
24
unique mode develops containing a mixture of local, flexural-torsional and
distortional buckling, shown in Figure 4.6. The warping of the cross-section due
to the flexural-torsional buckling seems to cause local and distortional buckling
at the hole, reducing the member’s torsional stiffness.
Figure 4.6 Mixed Mode Caused by SSMA Hole At Member End- Local,
Distortional and Flexural-Torsional Buckling, L=1220 mm
4.1.2 Modified boundary conditions
The boundary conditions are now modified to evaluate their effect on the elastic
buckling mode shapes and Pcr as defined in Section 4.1.1.
The modified
conditions are modeled to be consistent with plates welded to the ends of the
cold-formed steel member.
No warping or edge rotation is allowed at the
member ends, although both strong axis and weak axis member rotation about
the centroid of the cross section is allowed.
Finally, a torsional restraint is
provided at one end, and is left free at the other.
Figure 4.7 presents the eigenbuckling results from ABAQUS for the
member with modified boundary conditions. There is little variation in the local
and distortional buckling loads, which is consistent with the CUFSM boundary
25
condition results in Figure 4.5. The DH modes are again present, although the
additional stiffness from the end constraints restrict this mode until the holes
moves away from the ends of the member. This not the case for the DH2 mode,
which occurs at a relatively constant load for all positions considered along the
member.
1
1
0.9
0.8
0.8
0.7
0.6
D
GT
DH2
DH
L
cr
P /P
y
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
0
0
0
0.2
0.1
0.2
0.4
0.3
0.6
0.4
0.8
0.5
0.6
1
0.7
x/L
Figure 4.7 Influence of SSMA Slotted Hole Location on Pcr for a 362S162-33
Channel, Modified Boundary Conditions, L=1220 mm (Refer to Figure
4.8 for buckled shape of GT mode)
The most interesting observation from Figure 4.7 is that the buckling
mode of the controlling global mode, a pure torsional mode (GT) in this case, is
not affected by the location of the hole.
26
Figure 4.8 GT Mode With Hole at Member End (Torsionally Free at One End)
Comparing this result to the global flexural-torsional buckling results of Figure
4.5, there seems to be a reduction in the Euler buckling load Pe only when
warping at the member ends is allowed. Additional work is planned to evaluate
the interaction of warping deformations and holes on the strength of coldformed steel members.
4.2 Influence of a Circular Hole on Elastic Buckling
This study investigates the effects of the diameter of a single circular hole
on the elastic buckling behavior of an intermediate length cold-formed steel
member. The typical compression member in this study has a length L of 1220
mm and is modeled with an SSMA 362162-33 structural channel cross section.
The hole is located at the midlength of the member is and centered transversely
in the web.
Figure 4.9 shows the change in hole diameter sizes considered in this
study, as well as the DH buckling modes for this member. Figure 4.10 shows the
typical buckled mode shapes (L, DH, DH2, GFT, and D) for this study.
27
Figure 4.9 Variation in Circular Hole Diameter for 362S162-33 Channel, Mixed
Local-Distortional Buckling (DH) Shown, L=1220 mm
28
Mode Shape Label
Pcr/Py
Local
0.274
DH
0.332
DH2
0.456
D
0.629
GFT
0.627
Buckled Shape
Figure 4.10 Buckling Mode Shape Summary, 362S162-33 Channel with Circular
Hole, L=1220 mm, Hole Depth to Web Width ratio of 0.47
29
1
1
0.9
0.8
0.8
0.7
0.6
D
GFT
DH2
DH
L
cr
P /P
y
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
0
0
0
0.2
0.2
0.4
0.4
0.6
b/H
0.6
0.8
0.8
1
1
Figure 4.11 Influence of Circular Hole Diameter on Pcr for a 362S162-33
Channel, CUFSM Boundary Conditions, L=1220 mm (Refer to Figure
4.10 for buckled shapes)
Figure 4.11 plots the variation in
Pcr
for local, distortional, and global modes as a
Py
function of the hole diameter divided by web width,
b
. It is observed that the
H
buckling loads Pcr , Pcrd , and Pcre vary little with hole diameter. This trend is
consistent with the SSMA slotted hole location study conducted in Section 4.1 of
this report.
It is interesting to note that the buckling load associated with the
distortional mode near the hole, DH, shows a decreasing trend as
30
b
increases.
H
As the size of the hole increases, the restraining effect of the web on the flanges
reduces.
It is also noted that an antisymmetric distortional hole mode, DH2,
develops as the size of the hole increases beyond a
b
of approximately 0.2. This
H
type of distortional mode does not typically influence the strength of this channel
section because it is related to a buckling load higher than that of the first
distortional mode. Adding the hole has made this DH2 mode more prevalent,
and it is currently unclear how the presence of this mode affects the strength of
the member. The trend of the DH2 data is unexpected since the DH2 mode
reaches a minimum critical buckling load at a
b
H
of approximately 0.5.
Intuitively, it would seem that Pcr for this DH2 mode should decrease as
tends toward unity.
b
H
Future work will attempt to define the theoretical
mechanics underlying the DH and DH2 modes, and will also attempt to isolate
the effects of these modes on the strength of cold-formed steel members with
holes.
4.3 Unstiffened Element Approximation Study
Appendix 1 contains a preliminary study which evaluates the elastic
buckling predictions of a plate with a hole using finite element methods to the
current “unstiffened element” design approximation. Future work is planned to
31
evaluate this approximation when compared to finite element buckling
predictions of intermediate length SSMA channel members with holes.
32
5 Preliminary Comparison of DSM to Experimental Data
The goal of this study is to evaluate the effectiveness of the Direct Strength
Method in predicting the strength of cold-formed steel members with a single
perforation. The results of five sets of experimental programs from the literature
(1,2,3,4,5) are compared with DSM predictions. Table 5.1 provides a summary of
the experimental programs considered here.
Table 5.1 Summary of Experimental Data
Reference
Author
Publication Date
1
Ortiz-Colberg
1981
2
3
4
5
Pekoz and Miller
Sivakumaran
Abdel-Rahman
Pu
1987
1994
1998
1999
Types of Specimens
Stub Column
Long Column
Stub Column
Stub Column
Stub Column
Stub Column
Cross Section
Lipped Channel
Lipped Channel
Lipped Channel
Lipped Channel
Lipped Channel
End Conditions
Fixed-Fixed
Weak Axis Pinned
Fixed-Fixed
Fixed-Fixed
Fixed-Fixed
Fixed-Fixed
Stub column and long column results will be compared separately in this
study because of the differences in tested boundary conditions and controlling
buckling modes between the two specimen types. Table 5.2 summarizes the stub
column specimen data, including cross section and hole dimensions, tested
ultimate load Ptest , tested specimen yield stress Fy , specimen yield force Py , g
(calculated with the gross cross sectional area), and the modulus of elasticity E
for each specimen considered. Table 5.3 summarizes the same information for
the long column specimen data.
33
Table 5.2 Stub Column Experimental Data
Study and Specimen Name
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Abdel-Rahman 1998
Abdel-Rahman 1998
Abdel-Rahman 1998
Abdel-Rahman 1998
Abdel-Rahman 1998
Abdel-Rahman 1998
Abdel-Rahman 1998
Abdel-Rahman 1998
Pu 1999
Pu 1999
Pu 1999
Pu 1999
Pu 1999
Pu 1999
Pu 1999
Pu 1999
Pu 1999
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Siva 1988
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
Pekoz Miller 1994
S4
S7
S6
S8
S5
S3
S14
S15
A-C
A-S
A-O
A-R
B-C
B-S
B-O
B-R
C-2.0-1-30-1
C-2.0-1-30-2
C-2.0-1-30-3
C-1.2-1-30-1
C-1.2-1-30-2
C-1.2-1-30-3
C-0.8-1-30-1
C-0.8-1-30-2
C-0.8-1-30-3
A2
A3
A4
A5
A6
A7
A8
B2
B3
B4
B5
B6
B7
B8
1-12
1-13
1-17
1-19
2-11
2-12
2-14
2-15
2-16
2-24
2-25
2-26
Member
L
t
Material
E
nu
(mm) (mm) (GPa)
305 1.2 203
305 1.3 203
305 1.3 203
305 1.3 203
305 1.3 203
305 1.3 203
305 1.9 203
305 1.9 203
425 1.9 203
425 1.9 203
475 1.9 203
475 1.9 203
250 1.3 203
250 1.3 203
300 1.3 203
300 1.3 203
370 2.0 203
370 2.0 203
370 2.0 203
360 1.2 203
360 1.2 203
360 1.2 203
360 0.8 203
360 0.8 203
360 0.8 203
200 1.6 205
200 1.6 205
200 1.6 205
200 1.6 205
200 1.6 205
200 1.6 205
223 1.6 205
265 1.3 210
265 1.3 210
265 1.3 210
265 1.3 210
265 1.3 210
265 1.3 210
265 1.3 210
276 1.9 203
276 1.9 203
456 0.9 203
456 0.9 203
276 1.9 203
276 1.9 203
456 0.9 203
456 0.9 203
456 0.9 203
456 0.9 203
456 0.9 203
456 0.9 203
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
Hole Dimensions
Hole Type
a
b
radius
Circular
Circular
Circular
Circular
Circular
Circular
Circular
Circular
Circular
Square
Oval
Rectangle
Circular
Square
Oval
Rectangle
Square
Square
Square
Square
Square
Square
Square
Square
Square
Circular
Square
Circular
Square
Circular
Square
Oval
Circular
Square
Circular
Square
Circular
Square
Oval
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular
(mm)
19.1
38.1
31.8
44.5
26.4
12.7
26.4
38.1
63.5
63.5
114.3
114.3
38.1
38.1
101.6
101.6
26.9
26.6
26.6
26.5
26.5
26.5
26.4
26.4
26.4
16.5
16.5
33.0
33.0
49.5
49.5
102.0
29.0
29.0
58.0
58.0
87.0
87.0
102.0
70.0
70.0
57.0
57.0
65.0
65.0
57.0
57.0
57.0
57.0
57.0
57.0
(mm)
19.1
38.1
31.8
44.5
26.4
12.7
26.4
38.1
63.5
63.5
63.5
63.5
38.1
38.1
38.1
38.1
26.9
26.8
26.7
26.5
26.5
26.4
26.5
26.4
26.5
16.5
16.5
33.0
33.0
49.5
49.5
38.0
29.0
29.0
58.0
58.0
87.0
87.0
38.0
41.0
41.0
40.0
40.0
38.0
38.0
40.0
40.0
40.0
40.0
40.0
40.0
(mm)
9.5
19.1
15.9
22.2
13.2
6.4
13.2
19.1
31.8
--31.8
--19.1
--19.1
--------------------8.3
--16.5
--24.8
--19.0
14.5
--29.0
--43.5
--19.0
-------------------------
33
H
Cross Section Dimensions
B1
B2
D1
D2
R
(mm)
89.0
89.2
89.0
89.0
89.0
88.9
89.3
89.3
203.0
203.0
203.0
203.0
101.5
101.5
101.5
101.5
100.0
100.0
100.0
98.4
98.4
98.4
97.6
97.6
97.6
92.1
92.1
92.1
92.1
92.1
92.1
92.1
152.4
152.4
152.4
152.4
152.4
152.4
152.4
92.0
92.0
152.0
152.0
92.0
92.0
152.0
152.0
152.0
152.0
152.0
152.0
(mm)
41.1
41.4
41.0
41.0
41.1
41.0
42.5
42.5
41.6
41.6
41.6
41.6
41.6
41.6
41.6
41.6
52.0
52.0
52.0
52.0
52.0
52.0
52.0
52.0
52.0
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
37.0
37.0
34.0
34.0
37.0
37.0
35.0
35.0
35.0
35.0
35.0
35.0
(mm)
37.8
37.9
37.7
37.6
37.6
37.5
37.8
37.9
41.6
41.6
41.6
41.6
41.6
41.6
41.6
41.6
52.0
52.0
52.0
52.0
52.0
52.0
52.0
52.0
52.0
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
41.3
37.0
37.0
34.0
34.0
37.0
37.0
35.0
35.0
35.0
35.0
35.0
35.0
(mm)
12.5
12.7
12.5
12.5
12.5
12.3
12.9
12.9
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.0
16.0
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.7
12.0
12.0
8.0
8.0
12.0
12.0
8.0
8.0
8.0
8.0
8.0
8.0
(mm) (mm)
12.8 2.5
13.0 2.5
12.9 2.5
12.8 2.5
12.8 2.5
12.8 2.5
12.8 2.5
12.9 2.5
13.0 3.8
13.0 3.8
13.0 3.8
13.0 3.8
13.0 2.5
13.0 2.5
13.0 2.5
13.0 2.5
16.0 4.0
16.0 4.0
16.0 4.0
16.0 2.8
16.0 2.8
16.0 2.8
16.0 2.0
16.0 2.0
16.0 2.0
12.7 3.2
12.7 3.2
12.7 3.2
12.7 3.2
12.7 3.2
12.7 3.2
12.7 3.2
12.7 2.6
12.7 2.6
12.7 2.6
12.7 2.6
12.7 2.6
12.7 2.6
12.7 2.6
12.0 2.4
12.0 2.4
8.0
2.4
8.0
2.4
12.0 2.4
12.0 2.4
8.0
2.4
8.0
2.4
8.0
2.4
8.0
2.4
8.0
2.4
8.0
2.4
Yield Stress and Force
Fy
P y,g
(MPa)
324.8
334.5
355.2
355.5
342.1
342.1
326.9
328.3
385.0
385.0
385.0
385.0
319.0
319.0
319.0
319.0
306.1
231.6
237.6
192.9
192.9
192.9
171.3
171.3
171.3
340.6
340.6
340.6
340.6
340.6
340.6
340.6
262.6
262.6
262.6
262.6
262.6
262.6
262.6
358.0
358.0
309.0
309.0
366.0
366.0
302.0
302.0
302.0
302.0
302.0
302.0
(kN)
74.1
76.9
81.7
81.7
78.9
78.9
114.6
115.1
214.6
214.7
214.7
214.7
81.2
81.2
81.2
81.2
134.3
101.6
104.3
51.8
51.8
51.8
31.0
31.0
31.0
101.8
101.8
101.8
101.8
101.8
101.8
101.8
84.6
84.6
84.6
84.6
84.6
84.6
84.6
121.4
120.8
61.9
61.9
123.5
123.5
61.7
61.0
61.7
62.4
62.4
61.7
Experimental Data
P test,1
P test,2
(kN)
62.9
56.3
61.4
60.5
62.5
64.5
109.4
106.8
114.3
114.3
117.3
111.8
58.1
59.2
57.4
56.7
104.9
81.2
81.3
41.6
42
41.9
20.5
20.1
20.5
85.8
84.7
81.7
81.6
78.1
77.6
72.6
54
53.2
53.4
51
47.1
47
51.6
114.5
104.8
24.3
26.2
98.7
98.3
26.6
25.9
26.0
27.0
27.0
27.8
(kN)
----------------121.5
123.8
119
117.3
55.3
53.6
54.8
57.1
-----------------------------------------------------------------------
Table 5.3 Long Column Experimental Data
Study and Specimen Name
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
Ortiz-Colberg 1981
L2
L3
L6
L7
L9
L10
L14
L16
L17
L19
L22
L26
L27
L28
L32
Member
L
t
(mm) (mm)
1600 1.2
686 1.2
1600 1.2
1600 1.2
991 1.2
988 1.2
993 1.2
1295 1.9
1298 1.9
686 1.9
1143 1.9
1143 1.9
686 1.9
686 1.9
1600 1.9
Material
E
nu
(GPa)
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
203 0.3
Hole Dimensions
Hole Type
a
b
(mm) (mm)
Circular
12.7 12.7
Circular
25.4 25.4
Circular
25.4 25.4
Circular
38.1 38.1
Circular
25.4 25.4
Circular
38.1 38.1
Circular
12.7 12.7
Circular
25.4 25.4
Circular
38.1 38.1
Circular
38.1 38.1
Circular
38.1 38.1
Circular
25.4 25.4
Circular
25.4 25.4
Circular
25.4 25.4
Circular
25.4 25.4
Cross Section Dimensions
Yield Stress and Force
radius
H
B1
B2
D1
D2
R
Fy
P y,g
(mm) (mm) (mm) (mm) (mm) (mm) (mm)
(MPa)
(kN)
6.4
89.2 41.2 37.7 12.7 12.9 2.5
315.2
71.9
12.7
89.2 41.2 37.7 12.7 12.9 2.5
295.9
67.5
12.7
89.2 41.2 37.7 12.7 12.9 2.5
317.9
72.5
19.1
89.2 41.2 37.7 12.7 12.9 2.5
313.8
71.6
12.7
89.2 41.2 37.7 12.7 12.9 2.5
302.1
68.9
19.1
89.2 41.2 37.7 12.7 12.9 2.5
291.7
66.5
6.4
89.2 41.2 37.7 12.7 12.9 2.5
295.9
67.5
12.7
89.2 41.2 37.7 12.7 12.9 2.5
331.7
115.2
19.1
89.2 41.2 37.7 12.7 12.9 2.5
331.7
115.2
19.1
89.2 41.2 37.7 12.7 12.9 2.5
355.2
123.3
19.1
89.2 41.2 37.7 12.7 12.9 2.5
322.1
111.8
12.7
89.2 41.2 37.7 12.7 12.9 2.5
315.9
109.7
12.7
89.2 41.2 37.7 12.7 12.9 2.5
333.1
115.7
12.7
89.2 41.2 37.7 12.7 12.9 2.5
291.7
101.3
12.7
89.2 41.2 37.7 12.7 12.9 2.5
330.3
114.7
Experimental Data
P test,1
P test,2
(kN)
(kN)
37.8
--50.5
--37.8
--37.6
--41.8
--44.9
--42.7
--76.5
--66.7
--94.3
--89.0
--85.0
--97.4
--99.6
--59.2
---
Table 5.4 provides the minimum and maximum nondimensional cross
sectional ratios for the short column specimens, and Table 5.5 summarizes the
same information for the long column specimens.
Table 5.4 Summary of Nondimensional Ratios for the Short Column Specimens
D/t
min
max
H/t
6.3 46.3
20.0 172.7
B avg /t
H/B
19.3
65.0
1.9
4.9
t
(mm)
0.8
2.0
b/H
L/H
0.1
0.6
1.7
3.7
Fy
(MPa)
171.3
385.0
Table 5.5 Summary of Nondimensional Ratios for the Long Column Specimens
min
max
D/t
H/t
B avg /t
H/B
6.6
10.3
46.2
71.6
20.4
31.7
2.3
2.3
t
(mm)
1.2
1.9
34
b/H
L/H
0.1
0.4
7.7
17.9
Fy
(MPa)
291.7
355.2
ABAQUS eigenbuckling analyses were conducted for each specimen
considered in the study. Member boundary conditions and loading conditions
were modeled to be consistent with the actual experimental conditions. For each
model the local, distortional, and global buckling modes required for the DSM
calculations were manually selected from the buckled modes in ABAQUS.
Because of the large number of mixed distortional hole modes, a maximum of
three possible DSM distortional modes were selected for each specimen. The
first mode selected is typically a mixed local mode with distortional deformation
near the hole, designated as DH. The second mode is typically a mixed mode
exhibiting local buckling and antisymmetric distortional buckling near the hole.
This type of mixed mode is designated as DH2. Finally, the distortional mode
most resembling that of a pure distortional mode from a specimen without a hole
was recorded and designated as D. It is important to note that for all DSM
predictions in this study, Pcrd is chosen as the lowest distortional buckling load
from the DH, DH2, and D modes.
5.1 Stub Columns
The yield strength of the cold-formed steel stub column specimens in this
study is calculated by using the equation:
Py 
35
Fy
A
Fy is the measured yield stress of a tensile coupon cut from the web of the
channel member.
A is typically the gross cross-sectional area of the member,
although for a member with a hole the area can also be calculated as Anet . Anet
accounts for the removal of a portion of the web due to the presence of the hole.
In this study, the influence of both Ag and Anet will be considered when
evaluating the ability of DSM to predict the strength of cold-formed steel
members with holes. Figure 5.1 plots the test to predicted ratio for the stub
columns, assuming that Pcrd is the minimum load of the three distortional D
modes chosen. The mean and standard deviation of the short column data is
provided in Table 5.6.
36
1.8
1
Gross Area
Net Area
1.6
0.8
1.4
/P
0.8
test
1
P
n
1.2
0.6
0.4
0.6
0.2
0.4
0.2
0
0
0
0
0.2
1
0.4
0.6
0.8
1
5
6
2
3
4
hole depth/member length (H/L)
Figure 5.1 Test to Predicted Ratio for Stub Column Tests, Minimum D mode
taken for Pcrd
Table 5.6 Statistical Properties of Stub Column Test to Predicted Ratios
Gross Area
N et Area
Mean
1.044
1.182
Standard
D eviation
0.164
0.156
Figure 5.2 shows the correlation between the test to predicted strength
ratio,
Ptest
b
, versus the hole depth to web width ratio,
. When Py ,net is used to
H
Py
normalize the tested strength values,
Ptest
is greater than one for 8 out of the 51
Py
37
specimens. When Py , g is used to normalize the tested strength values,
Ptest
for
Py
all of the stub column specimens is less than one. Therefore it seems that Py , g is
a better upper bound for predicting the strength of the stub column specimens in
this study.
1.4
1
Py =Ag*Fy
1.2
Py =Anet*Fy
0.8
1
P
test
/P
y
0.6
0.8
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.1
0.4
0.6
0.2
0.3
0.4
hole depth/web width (b/H)
Figure 5.2 Variation in the Stub Column Strength
Width
0.8
0.5
1
0.6
Ptest
with Hole Depth to Web
Py
b
( Both Py , g and Py ,net are used to normalize Ptest )
H
Figure 5.3 compares the ratio of Pcr
experimental model to Pcr without a hole,
38
determined from the ABAQUS
Pcr
Pcr
, hole
, no _ hole
, as a function of hole depth
Pcr ,hole
b
.
is greater than one for the majority of the specimens
H Pcr ,no _ hole
to web width,
and grows larger as
2.5
b
increases .
H
1
Stub column data
Unstiffened element approx.
0.8
0.6
1.5
0.4
1
P
crl,hole
/P
crl,no hole
2
0.2
0.5
0
0
0
0
0.2
0.1
0.4
0.6
0.2
0.3
0.4
hole depth/web width (b/H)
0.8
0.5
1
0.6
Figure 5.3 Influence of Hole Depth on Stub Column Local Buckling
Strength Pcr ,hole and Comparison to the “Unstiffened Element”
Approach
One explanation for the increase in
Pcr
Pcr
, hole
is that the hole causes the
, no _ hole
length of the local buckling half-wavelength to decrease, which in turn causes an
increase in the force required to initiate buckling. For small holes or wide webs,
the length of the half-wavelength is not affected, but for large holes the lack of
web material at the hole location does not allow a local wave to form. This
39
phenomenon would reduce the lengths of the local waves along the member.
Another explanation may be that
Pcr
Pcr
, hole
is related to boundary condition
, no _ hole
effects. Future work will compare the results presented here to ABAQUS models
without holes in order to resolve the uncertainty regarding these trends.
Figure 5.3 also compares the stub column data to the “unstiffened
element” approach for predicting the local buckling strength of members with
holes. This method models the area around the hole as two unstiffened plates,
one on either side of the hole. The “unstiffened element” curve is derived by
calculating the ratio of the buckling stress for two unstiffened plates to the
buckling stress of a simply supported plate:
f cr , hole
f cr , ss

0.43H 2
1

4   H  b)  
2

2
It is observed by comparing the stub column strength data to the “unstiffened
element” approximation that for small holes or wide webs the approximation is
conservative, although for large holes or narrow webs the approximation is
extremely unconservative.
40
2.5
1
0.8
0.6
1.5
0.4
1
P
crd,hole
/P
crd,no hole
2
minimum D mode
pure D mode
0.2
0.5
0
0
0
0
0.2
0.1
0.4
0.6
0.2
0.3
0.4
hole depth/web width (b/H)
0.8
0.5
1
0.6
Figure 5.4 Influence of Hole Depth on Stub Column Distortional Buckling
Strength Pcrd ,hole , Both Minimum DH and pure D Modes Considered
Figure 5.4 compares Pcrd ,hole calculated using ABAQUS to Pcrd , no _ hole as a
function of hole depth to web width,
b
. The data does not seem to indicate a
H
strong correlation between hole size and distortional buckling, although it can be
observed that the first distortional buckling mode near the hole occurs in most
instances at a much lower load than the pure distortional modes. Additional
work is required to evaluate if these lower DH modes affect the strength of coldformed steel compression members.
41
Figure 5.5 compares the tested strength values of the specimens to the
DSM local column curve using Py ,net . All but five of the stub column specimens
meet or exceed the limits of the DSM design equation, and the data trend seems
to follow the column curve.
1.4
1
L buckling controls
D buckling controls
DSM Pnl
1.2
0.8
0.6
0.8
P
test
/P
y,net
1
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.5
0.4
1
1.5
0.6
0.8
2
2.5
1
3
0.5
Local Slenderness, (P y,net/Pcrl)
Figure 5.5 Comparison of the Stub Column Strengths to the DSM Local Buckling
Curve, Py ,net Assumed
Figure 5.6 compares the tested strength values to the DSM local column
curve when normalized by Py , g . The data has shifted closer to the DSM curve
now and the general trend seems better than with the Py ,net normalization.
Specimens with predicted strengths controlled by distortional buckling are
denoted separately on the graph from specimens with strengths controlled by
local buckling.
42
1.4
1
L buckling controls
D buckling controls
DSM Pnl
1.2
0.8
0.6
0.8
P
test
/P
y,g
1
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.5
0.4
1
0.6
1.5
0.8
2
2.5
1
3
0.5
Local Slenderness, (P y,g/Pcrl)
Figure 5.6 Comparison of the Stub Column Strengths to the DSM Local Buckling
Curve, Py , g Assumed
Figure 5.7 compares the tested strength of the stub column specimens to
the DSM distortional and local column curves as normalized by Py ,net . Figure 5.8
provides the same comparison, although for this graph Py , g is used for the
normalization of the tested strength.
Specimens controlled by distortional
buckling and local buckling are differentiated on the plots.
43
1.4
1
D buckling controls
L buckling controls
DSM Pnl
1.2
0.8
DSM Pnd
0.6
0.8
P
test
/P
y,net
1
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.5
0.4
1
0.6
1.5
0.8
2
0.5
(Py,net/Pcrl)
2.5
1
3
0.5
,(Py,net/Pcrd)
Figure 5.7 Comparison of the Stub Column Strengths to the DSM Local and
Distortional Buckling Curve, Py ,net Assumed
1.4
1
D buckling controls
L buckling controls
DSM Pnl
1.2
0.8
DSM Pnd
0.6
0.8
P
test
/P
y,g
1
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.5
0.4
1
0.6
1.5
0.5
(Py,g/Pcrl)
0.8
2
2.5
1
3
0.5
,(Py,g/Pcrd)
Figure 5.8 Comparison of the Stub Column Strengths to the DSM Local and
Distortional Buckling Curve, Py , g Assumed
44
Figure 5.7 shows that Py ,net overpredicts strength in most cases when
compared to the DSM column curves. In Figure 5.8, the test data is closer to the
DSM predictions when normalized with Py , g , although the trend of the data does
not seem consistent with the DSM column curves.
Figure
5.9
compares
Pcr
calculated
using
ABAQUS
assuming
experimental end conditions to the Pcr calculated using ABAQUS assuming
CUFSM boundary conditions.
3
1
0.8
2
1.6 Factor
Suggested by DSM
Design Guide
0.6
1.5
0.4
1
0.2
P
crd,test bc
/P
crd,cufsm bc
2.5
0.5
0
0
0
0
0.2
1
0.4
0.6
2
3
4
web width/flange width (H/B1)
0.8
1
5
6
Figure 5.9 Influence of Fixed-Fixed Experimental Boundary Conditions versus
CUFSM Boundary Conditions on the Local Buckling Strength
Predictions from ABAQUS for the Stub Column Data with Holes
45
The distortional boost varies widely around the DSM Design Guide
recommended value of 1.6, although the major conclusion from this plot is that
the boundary conditions have a large influence on the distortional buckling loads
for stub columns. This influence can be observed in Figure 5.10, which provides
a comparison of the displaced shape of two ABAQUS stub column models, one
with experimental end conditions and one with CUFSM boundary conditions.
The warping restraint of the experimental boundary conditions significantly
restrains the flanges from distorting, even with the presence of a hole.
Figure 5.10 Comparison of ABAQUS Stub Column Models with a) Experimental
and b) CUFSM Boundary Conditions (Siva 1998, Spec. B-5), Notice
Warping at Ends of CUFSM Model
46
Figure 5.11 plots the local buckling boundary condition magnifier,
function of the hole depth to member length,
Pcr
,test _ bc
Pcr
,CUFSM
, as a
b
. The dimensions for the x-axis
L
 b  H 
of this plot were determined by assuming the coupled effect of     .
 H  L 
1.6
1
1.4
0.8
1
0.6
0.8
0.4
0.6
P
crl,test bc
/P
crl,cufsm bc
1.2
0.2
0.4
0.2
0
0
0
0
0.2
0.05
0.4
0.6
0.1
0.15
0.2
0.25
hole depth/member length (b/L)
0.8
0.3
1
0.35
Figure 5.11 Local Buckling Experimental versus CUFSM Boundary Condition
 b  H 
Magnification as a Function of     for Stub Columns
 H  L 
The primary conclusion drawn from this figure is that the fixed-fixed
boundary conditions cause an increase in the buckling load as 1) the hole size
increases or 2) as the member length of the stub column decreases.
47
The
magnification is not as large or as varied as in the distortional buckling boost
show in Figure 5.9, but still is significant as
b
exceeds 0.2.
L
5.2 Long Columns
Only a small amount of test data exists in the current literature for
intermediate and long column cold-formed steel compression members
containing a single perforation. The fifteen long column specimens in this study
were taken from the Ortiz-Colberg study conducted in 1981 (1).
Figure 5.12
plots the test to predicted ratios for the long column specimens, assuming that
Pcrd is the minimum load of the three distortional D modes chosen. The mean
and standard deviation for the long column data is provided in Table 5.7.
48
1.8
1
Gross Area
Net Area
1.6
0.8
1.4
/P
0.8
test
1
P
n
1.2
0.6
0.4
0.6
0.2
0.4
0.2
0
0
0
0
0.2
0.4
0.6
0.8
5
10
15
hole depth/member length (H/L)
1
20
Figure 5.12 Test to Predicted Ratio for Long Column Tests, Minimum D mode
taken for Pcrd
Table 5.7 Test to Predicted Ratio for Long Column Specimens
Gross Area
N et Area
Mean
1.137
1.236
Standard
D eviation
0.094
0.094
Figure 5.13 plots the change in Pcr as the hole depth to web depth,
b
,
H
increases. It is clear from this figure that the hole does not affect Pcr when
considering the long column specimens in this study.
49
2.5
1
0.8
0.6
1.5
0.4
1
P
crl,hole
/P
crl,no hole
2
0.2
0.5
0
0
0
0
0.2
0.1
0.4
0.6
0.2
0.3
0.4
hole depth/web width (b/H)
0.8
0.5
1
0.6
Figure 5.13 Influence of Hole Depth on Long Column Local Buckling
Strength Pcr ,hole
Figure 5.14 compares the change in Pcrd as the hole depth divided by web
width,
b
, increases. The buckling load for both the minimum distortional mode
H
near the hole, DH, and that for the pure distortional mode D are plotted.
50
2.5
1
0.8
0.6
1.5
0.4
1
P
crd,hole
/P
crd,no hole
2
minimum D mode
pure D mode
0.2
0.5
0
0
0
0
0.2
0.1
0.4
0.6
0.2
0.3
0.4
hole depth/web width (b/H)
0.8
0.5
1
0.6
Figure 5.14 Influence of Hole Depth on Long Column Distortional Buckling
Strength Pcrd ,hole , Both Minimum DH and pure D Modes Considered
The impact of increasing hole size on the pure distortional mode is small,
although the buckling load for the lowest mixed distortional hole mode shows a
decreasing trend with increasing hole size. This trend is corroborated by the DH
curve in Figure 4.11 from the circular hole parameter study results.
Figure 5.15 plots Pcre for the long column specimens as a function of the
hole length divided by member length,
a
. Pcre values that control the design of
L
the column are differentiated from the other long column data. Pcre is associated
with weak axis flexural buckling for all long column members in this report.
51
2.5
1
0.8
0.6
1.5
0.4
1
P
cre,hole
/P
cre,no hole
2
Long column data
Pnecontrols design, P ne=Pnl
0.2
0.5
0
0
0
0
0.2
0.02
0.4
0.6
0.8
0.04
0.06
0.08
hole length/member length (a/L)
1
0.1
Figure 5.15 Effect of Hole Length on the Euler Buckling Load Pcre , Py , g Assumed
in DSM Calculation ( Euler buckling always occurs in weak axis
flexure, holes are always located at member midlength)
It is observed in the figure that as hole length increases, Pcre decreases. The range
of
a
and the number of specimens are both small, but it seems that additional
L
work is warranted to evaluate if this trend continues for larger
a
and for
L
members with multiple web perforations.
Figure 5.16. compares the tested strength of the long columns which are
controlled by local buckling against the DSM local column curve.
52
1.4
1
Local buckling controls
DSM Pnl
1.2
0.8
0.6
0.8
P
test
/P
ne,g
1
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.5
0.4
1
0.6
1.5
0.8
2
2.5
1
3
0.5
Slenderness, (P ne/Pcrl)
Figure 5.16. Comparison of Long Column Strengths to the DSM Local Buckling
Curve, Py , g Assumed
It is difficult to make any firm conclusions with the small number of long column
specimens , although the data does at least look to be of similar magnitude as
that predicted by the DSM curves. It should be noted that Pnd did not control for
any of the long column specimens. It is expected that for intermediate and long
column specimens with larger
b
than those considered in this study, that
H
distortional buckling will be more likely to control the design.
53
1.4
1
Global buckling controls, P ne=Pnl
All Long Column Specimens
DSM Pne
1.2
0.8
0.6
0.8
P
test
/P
y,g
1
0.4
0.6
0.4
0.2
0.2
0
0
0
0
0.2
0.5
0.4
1
0.6
1.5
0.8
2
1
2.5
3
0.5
Slenderness, (P y,g/Pcre)
Figure 5.17 Comparison of Long Column Strengths to the DSM Global Buckling
Curve, Py , g Assumed, Euler Buckling occurs at Weak Axis Flexure,
Holes at Midlength of Members
Figure 5.17 compares the tested strength values of the Euler buckling controlled
specimens to the DSM global column curve.
The tested strengths are
significantly higher than the DSM predicted strengths, although the general
trend of the data seems to follow the design curve.
Figure 5.18 plots the local buckling boost
depth divided by member depth,
Pcr
,test _ bc
Pcr
,CUFSM
as a function of hole
b
. The boost is determined by comparing the
L
ratio of Pcr calculated in ABAQUS using experimental end conditions and the
Pcr calculated in ABAQUS using the CUFSM boundary conditions.
54
1.8
1
Long Columns
Stub Columns
1.6
0.8
1.2
0.6
1
0.4
0.8
0.6
P
crl,test bc
/P
crl,cufsm bc
1.4
0.2
0.4
0.2
0
0
0
0
0.2
0.05
0.4
0.6
0.1
0.15
0.2
0.25
hole depth/member length (b/L)
0.8
0.3
1
0.35
Figure 5.18 Local Buckling Experimental versus CUFSM Boundary Condition
 b  H 
Magnification as a Function of     for Long Columns and Stub
 H  L 
Column Specimens
The experimental end conditions and the CUFSM boundary conditions produce
very similar results for the long column specimens.
The experimental end
conditions are weak axis pinned-pinned and warping restrained at the member
ends, where as the CUFSM boundary conditions are pinned-pinned but with
warping allowed. Therefore, it can be concluded that warping deformations do
not influence the local buckling loads for the long column specimens in this
study.
55
6 Challenges and Future Work
Finite Element Modeling of Cold-Formed Steel Members:

More work is needed to resolve the differences between rounded corner
vs. straight line finite element models. This work will include a detailed
investigation of the S9R5 transverse shear formulation and its effect on
our elastic buckling predictions.
“Effective Width Approximation” vs. Finite Element Strength Predictions:

Appendix 1 presents a preliminary study which compares the “effective
width” approximation for a plate with a hole to an equivalent finite
element solution More work is planned to expand this study to evaluate
the accuracy of the approximation on full scale cold-formed steel members
with holes modeled in ABAQUS
Extending DSM to cold-formed steel members with holes:

More intermediate and long column experimental data is necessary to
calibrate the existing DSM column curves for members with holes and to
understand what types of mixed distortional modes are present in the
post-buckling state. Future experimental work is planned to evaluate the
strength of intermediate length compression members with holes.

There are many mixed distortional modes created by the addition of a
hole, and we need better tools for identifying and categorizing these
distortional modes.
56
57
Appendix 1
Elastic Buckling of Plates
The critical bucking stress for a simply supported plate subjected to compression in
one direction is given by
2
t 
f cr  kCm   , (1)
 w
where, t and w are the thickness and width of the plate, respectively; k is the
buckling coefficient, and is dependent on the type of loading, boundary condition, and
length to width ratio of the plate; Cm is the material constant given by
 2E
. (2)
Cm 
12 1  v 2
For a simply supported plate of large length to width ratio (i.e., l/w greater than 4)
with uniform compression along the length, a value of k = 4 can be used with very little
loss in accuracy [Yu, 2000]. For a plate with 3 sides simply supported and 1 side free,
under uniform compression and large l/w ratio, k = 0.425.


Elastic Buckling of Plates with Holes
In cold-formed steel structural members, holes may be provided in webs or flanges
of the member for functional requirements like piping, electrical cables, ducts and other
utilities. Openings may also be required to accommodate the transverse member, which
may be structural or non-structural. The presence of holes alters the stiffness and strength
of the members.
Consider a simply supported plate, under uniform compression along the length,
with a central hole of width wh (Figure 1). The width of the plate at the hole is (w – wh).
Further, due to the presence of the hole, the boundary conditions, for the portion of the
plate at the hole, change to simply supported on 3 sides and free on 1 side (S3F1).
58
wh
Simply Supported
Simply Supported
w
Figure 1: Simply supported plate with circular hole under uniform longitudinal
compression.
Using these parameters, the buckling coefficient for a S3F1 plate can be expressed
as
f cr, w/ h  w  2
(3)
k w/ h 
  ,
Cm  t 
where the critical buckling stress for a S3F1 plate is given by,
2
f cr, w/ h
 t 
 0.425Cm 
 . (4)
 w  wh 
Thus, by re-substitution, kw/ h can be expressed as
2
 w 
 , (5)
k w/ h  1.7
w

w
h 

and its variation with respect to k = 4 is shown in Figure 2. Substituting wh = 0 in Eq. (5)
gives the buckling coefficient for a S3F1 plate of width w/2.
59
10
8
k
6
4
2
0
0
0.2
0.4
0.6
0.8
1
w h /w
Figure 2: Buckling coefficient for plate with and without hole.
Hole Specifications
The hole size and shape vary depending on the manufacturers. Special shapes of
holes can be requested, however, the oval hole shape shown in Figure 3 is very
commonly used for webs of structural studs. These holes are centered at 610mm
along the length of the stud, and centrally located on the width of the web.
Finite Element Analysis
This study investigates the effect of holes on the critical buckling of the web plates
of studs subjected to pure compressive load along the length of the plate. The web of the
studs is considered to be simply supported along the edges that intersect with the flanges.
The buckling analysis is performed using the commercially available finite element
software ABAQUS.
l = 101.6mm
305mm
R = 19.05mm
610mm
610mm
w = 38.1mm
(a)
(b)
60
Figure 3: (a) Oval hole used in structural studs; (b) Location of the oval hole on the stud
web.
Element Type
In this study, the general purpose three-dimensional, stress/displacement, reduced
integration with hourglass control, shell element S4R (available in ABAQUS), is used to
model the plates. S4R has 4 nodes (quadrilateral), with all 6 active degrees of freedom
per node. S4R allows transverse shear deformation, and the transverse shear becomes
very small as the shell thickness decreases.
Loading and Boundary Conditions
The accuracy of the elastic buckling analysis using finite element method depends
on the mesh size, and the accuracy of the model to simulate the actual loading and
the boundary conditions. To minimize the local effect, it is necessary to use
consistent nodal loading to simulate the uniformly distributed compressive load.
This is achieved by applying concentrated loads which are proportional to the
tributary area associated with the corresponding node. For e.g., if P is the total
uniformly distributed load to be applied to a plate discretized into 4 elements, the
load should be applied as shown in Figure 4.
Finite Element Mesh
The coefficient of buckling k for a simply supported plate under uniform
compression is 4.0. This is used to determine the fineness of the finite element mesh
in the finite element analysis (FEA); the mesh size that yields the k value close to
4.0 for a simply supported plate without hole, is used to determine the buckling load
and the buckling coefficient of the same plate with hole. The length of the plate is
taken as 4w, where w is the plate width.
Section Properties: Plate Sizes
The width of the structural stud sections given in the Steel Stud Manufacturers
Association (SSMA) catalog range from 63.5mm to 304.8mm [SSMA, 2001]. The
minimum thickness ranges from 33mils to 97mils (design thickness of 0.879mm to
2.583mm).
P/8 P/4 P/4 P/4 P/8
61
Figure 4: Consistent distributed load proportional to the tributary area for linear S4R
elements.
Analysis Results
Buckling analyses is performed for simply supported web plates of the studs
subjected to pure compressive loading along the length. Figure 5a shows the first
buckling mode for the simply supported web plate without opening of a 362S162-43 stud.
This is the typical local buckling mode, where the plate forms crests and troughs of half
wavelength equal to the width of the plate. Figure 5b shows the first buckling mode for
the simply supported web plate of a 362S162-43 stud with central opening.
w = 92.075mm
l = 4w
Figure 5a: First buckling mode for the web plate of 362S162-43 (w = 90.075mm).
62
Figure 5a: First buckling mode for the web plate of 362S162-43 (w = 90.075mm).
The buckling stress for plate without (w/o) and with (w/) hole for plates of 43mils (t
= 1.14554mm) thickness are listed in Table 1a. The data in Table 1.1 is shown in Figure
6. The k values from the FEA results for plate without hole are in good agreement with
those calculated using the plate theory, i.e., Eq. (1). The k values for plate thickness of
97mils (t = 2.58318mm) show the same trend (see Figure 6b and Table 1b).
As seen from Figures 6a and 6b, the value of buckling coefficient for a plate with 1
central standard hole is different (generally smaller) from that of the plate without hole.
The reduction in the k value is a function of the ratio of the width of the hole to the width
of the plate; for very small holes, as expected, the k value is same as that of plate without
hole. However, for larger width of the hole, values of k have a tendency to increase.
Further investigation is required to determine the factors influencing the k values for
plates with holes.
63
10
Plate Theory w /o Hole
Plate Theory w / Hole
FEA w /o Hole
FEA w / Standard Hole (t = 1.14554)
8
w h = 38.1
k
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
w h /w
Figure 6a: Comparison of coefficient of buckling for t = 1.14554mm (first mode).
10
Plate Theory w /o Hole
Plate Theory w / Hole
FEA w /o Hole
FEA w / Standard Hole (t = 2.58318)
8
w h = 38.1
k
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
w h /w
Figure 6b: Comparison of coefficient of buckling t = 2.58318mm (first mode).
References
1. Yu, W., Cold-Formed Steel Design, 3rd Edition, John Wiley & Sons, Inc., NY, USA.
2. SSMA, Product Technical Information, ICBO ER-4943P, Steel Stud Manufacturers
Association, Chicago IL, USA.
64
Table 1a: Comparison of the plate theory results with the FEA results for plate (43mils) with and without openings.
fcr (MPa)
(1)
200P-43
225P-43
250S162-43
350S162-43
362S162-43
400S162-43
550S162-43
600S162-43
800S162-43
1000S162-43
1500P-43
(2)
50.8
57.15
63.5
88.9
92.075
101.6
139.7
152.4
203.2
254
381
(3)
44.346
49.889
55.432
77.605
80.377
88.692
121.951
133.038
177.384
221.729
332.594
(4)
0.750
0.667
0.600
0.429
0.414
0.375
0.273
0.250
0.188
0.150
0.100
k
Plate
Theory
w/ Hole
(5)
373.241
294.906
238.874
121.875
113.614
93.310
49.354
41.471
23.328
14.930
6.635
(6)
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
(7)
2538.038
1128.017
634.509
158.627
140.514
101.522
39.657
31.334
15.018
8.782
3.482
64
(8)
27.200
15.300
10.625
5.206
4.947
4.352
3.214
3.022
2.575
2.353
2.099
Ratio
of k
(8)(6)
FEA
w/o Hole
(9)
6.800
3.825
2.656
1.302
1.237
1.088
0.804
0.756
0.644
0.588
0.525
(10)
373.255
293.687
238.404
121.535
112.476
92.742
49.239
41.037
23.048
14.748
6.555
(11)
4.000
3.983
3.992
3.989
3.960
3.976
3.991
3.958
3.952
3.951
3.952
FEA w/
Standard
Hole (t =
1.14554)
(12)
628.679
400.183
258.269
110.292
100.91
81.612
42.501
35.812
20.663
13.626
6.536
k (w.r.t.w)
Plate
Theory
w/o Hole
k (w.r.t.w)
FEA
k (w.r.t.w)
wh/w
(wh= 38.1)
w/t
(t = 1.14554)
w (mm)
Stud ID
Plate Theory
(13)
6.738
5.428
4.325
3.620
3.553
3.499
3.445
3.454
3.543
3.651
3.940
Ratio of k
(13)(11)
(14)
1.684
1.363
1.083
0.907
0.897
0.880
0.863
0.873
0.897
0.924
0.997
Table 1b: Comparison of the plate theory results with the FEA results for plate (97mils) with and without openings.
fcr (MPa)
(1)
200P-97
225P-97
250S162-97
350S162-97
362S162-97
400S162-97
550S162-97
600S162-97
800S162-97
1000S162-97
1500P-97
(2)
50.8
57.15
63.5
88.9
92.075
101.6
139.7
152.4
203.2
254
381
(3)
19.666
22.124
24.582
34.415
35.644
39.331
54.081
58.997
78.663
98.328
147.493
(4)
(5)
0.750
0.667
0.600
0.429
0.414
0.375
0.273
0.250
0.188
0.150
0.100
1897.920
1499.591
1214.669
619.729
577.726
474.480
250.965
210.880
118.620
75.917
33.741
(6)
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
4.000
(7)
12905.858
5735.937
3226.465
806.616
714.511
516.234
201.654
159.332
76.366
44.657
17.704
65
Ratio
of k
(8)(6)
(8)
(9)
27.200
15.300
10.625
5.206
4.947
4.352
3.214
3.022
2.575
2.353
2.099
6.800
3.825
2.656
1.302
1.237
1.088
0.804
0.756
0.644
0.588
0.525
FEA
w/o Hole
(10)
1842.66
1474.151
1215.971
612.877
568.615
467.558
247.56
208.12
117.005
74.897
33.38
(11)
3.884
3.932
4.004
3.956
3.937
3.942
3.946
3.948
3.946
3.946
3.957
FEA w/
Standard
Hole (t =
1.14554)
(12)
2783.803
1930.275
1254.317
550.381
504.851
407.626
214.468
181.466
105.782
70.238
33.289
k (w.r.t.w)
k
Plate
Theory
w/ Hole
k (w.r.t.w)
Plate
Theory
w/o Hole
FEA
k (w.r.t.w)
wh/w
(wh= 38.1)
w/t
(t = 2.58318)
w (mm)
Stud ID
Plate Theory
(13)
5.867
5.149
4.131
3.552
3.495
3.436
3.418
3.442
3.567
3.701
3.946
Ratio of k
(13)(11)
(14)
1.511
1.309
1.032
0.898
0.888
0.872
0.866
0.872
0.904
0.938
0.997
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67
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70