Progress Report 4 final.pdf
Transcript of Progress Report 4 final.pdf
July 30, 2007 To: AISI Committee Members Subject: Progress Report No. 4
Direct Strength Design for Cold-Formed Steel Members with Perforations
Please find enclosed the fourth progress report summarizing our research efforts to extend the Direct Strength Method to cold-formed steel members with perforations. The research focus during this period was on developing tools and extending ideas to advance a DSM approach for members with holes. The elastic buckling and nonlinear behavior studies of cold-formed steel columns with holes continued. Tested strengths from the column experiments in Progress Report #3 were compared against preliminary DSM predictions. A nonlinear finite element procedure was formalized and tested, and a new method for predicting and modeling residual stresses was developed. The influence of holes and the “net section” idea are integrated into the DSM strength prediction equations and evaluated against tested data and simulations. With the completion of this report, a preliminary framework for extending the Direct Strength Method to members with holes is now in place. Sincerely,
Cris Moen [email protected]
Ben Schafer [email protected]
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Summary of Progress The primary goal of this AISI-funded research is to extend the Direct Strength Method to cold-formed steel members with holes. Research begins September 2005 Progress Report #1 February 2006 Accomplishments:
• Evaluated the ABAQUS S9R5, S4, and S4R thin shell elements for accuracy and versatility in thin-walled modeling problems
• Studied the influence of element aspect ratio and element quantity when modeling rounded corners in ABAQUS
• Developed custom MATLAB tools for meshing holes, plates, and cold-formed steel members in ABAQUS
• Determined the influence of a slotted hole on the elastic buckling of a structural stud channel and classified local, distortional, and global buckling modes
• Investigated the influence of hole size on the elastic buckling of a structural stud channel
• Performed a preliminary comparison of existing experimental data on cold-formed steel columns with holes to DSM predictions
• Conducted a study on the influence of the hole width to plate width ratio on the elastic buckling behavior of a simply supported rectangular plate
Papers from this research: Moen, C., Schafer, B.W. (2006) “Impact of Holes on the Elastic Buckling of Cold-Formed Steel Columns with Application to the Direct Strength Method”, Eighteenth International Specialty Conference on Cold-Formed Steel Structures, Orlando, FL. Moen, C., Schafer, B.W. (2006) “Stability of Cold-Formed Steel Columns With Holes”, Stability and Ductility of Steel Structures Conference, Lisbon, Portugal.
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Progress Report #2 August 2006 Accomplishments:
• Evaluated the influence of slotted hole spacing on the elastic buckling of plates (with implications for structural studs)
• Determined the impact of flange holes on the elastic buckling of an SSMA structural stud
• Conducted a preliminary investigation into the nonlinear solution algorithms available in ABAQUS
• Compared the ultimate strength and load-displacement response of a rectangular plate and an SSMA structural stud column with and without a slotted hole using nonlinear finite element models in ABAQUS
• Calculated the effective width of a rectangular plate with and without a slotted hole using nonlinear finite element models in ABAQUS
Progress Report #3 February 2007 Accomplishments:
• Conducted an experimental study to evaluate the influence of a slotted web holes on the compressive strength, ductility, and failure modes of short and intermediate length Cee channel columns
• Studied the influence of slotted web holes on the elastic buckling behavior of cold-formed steel Cee channel beams and identified unique hole modes similar to those observed in compression members
• Demonstrated that the Direct Strength Method is a viable predictor of ultimate strength for beams with holes
Progress Report #4 July 2007
Accomplishments: • Completed experimental program including tensile coupon tests, elastic
buckling study of specimens, and DSM strength comparison • Developed nonlinear finite element approach including a prediction
method for residual stresses and completed preliminary finite element studies of the 24 column specimens
• Proposed DSM approach for columns with holes, compared the options against the column database, conducted preliminary simulations to explore column strength curves
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Table of Contents Summary of Progress ................................................................................................................... 2 1 Introduction .......................................................................................................................... 5 2 Experimental Program ........................................................................................................ 8 3 Nonlinear Finite Element Modeling of Structural Stud Columns with Holes .......... 22 4 Preliminary DSM Prediction Methods for Columns with Holes ................................ 54 5 Future Work........................................................................................................................ 77 References .................................................................................................................................... 79 Appendix A ‐ Tensile Coupon Plastic Strain Data ................................................................. 80 Appendix B ‐ Corrections to Progress Report #3.................................................................. 101 Appendix C Predicting Manufacturing Residual Stresses and Plastic Strains in Cold‐Formed Steel Structural Members .......................................................................................... 102
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1 Introduction
The research work presented in this progress report represents a continuing
effort to develop a general design philosophy that relates elastic buckling
behavior to the ultimate strength of cold-formed steel members with
perforations. The general framework for this philosophy is being developed
around the Direct Strength Method (DSM), which uses the local, distortional, and
global elastic buckling modes to predict the ultimate strength of cold-formed
steel members (NAS 2004, Appendix 1).
The final objective of this research project is to extend DSM to cold-formed
steel columns and beams with holes, which is being met through research goals
defined in three phases:
Phase I
1. Study the influence of holes on the elastic buckling of cold-formed
steel members.
2. Formalize the identification of buckling modes for members with
holes.
3. Compare existing experimental data on members with holes to the
current DSM specification.
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Phase II
1. Experimentally validate DSM as a rational analysis method for cold-
formed member with holes
1. Formalize the relationship between elastic buckling and ultimate
strength for members with holes
Phase III
1. Increase our understanding of post-buckling mechanisms for members
with holes through non-linear finite element models and laboratory
testing
2. Modify the current DSM specification to account for members with
holes
3. Develop tools and method that engineers may use for easy application
of DSM to members with holes
Research summarized in Progress Report #1 addressed the Phase I goals for
cold-formed steel compression members with elastic buckling studies that
evaluated the influence of holes on thin plates and cold-formed steel channel
studs. Progress Report #2 continued the elastic buckling research by studying
the influence of flange holes in SSMA structural studs and the impact of slotted
web hole spacing on the performance of an SSMA structural stud. The report
also presented preliminary nonlinear finite element model results of thin plates
and cold-formed steel compression members with holes. Progress Report #3
produced an elastic buckling study that evaluated the influence of holes on the
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elastic buckling of beams. The Direct Strength Method was evaluated for beams
with holes. Also in Progress Report #3, an experimental study of 24 column
specimens was conducted to evaluate the influence of slotted web holes on the
compressive strength and failure modes of short and intermediate length
columns.
This report, Progress Report #4, completes the column experimental
program with the tensile coupon test results and an elastic buckling study of the
24 column specimens. The influence of holes on the local, distortional, and
global buckling modes are discussed, and the elastic buckling behavior
employed to make strength predictions using the Direct Strength Method. The
measurements and results from these experiments are used to develop and
evaluate a nonlinear finite element modeling approach that will be an essential
tool for extending the Direct Strength Method to members with holes. This
report presents the preliminary validation results of this modeling approach,
including a proposed prediction method for residual stresses and the
implementation of contact boundary conditions in ABAQUS. Five potential
DSM formulations for columns with holes are presented and compared to the
column test database compiled in Progress Report #1 and nonlinear simulations.
This document summarizes research that wrestles with more mature ideas than
the past reports, suggesting that progress is being made in the development of a
DSM approach applicable to members with holes. The Phase III work, finalizing
the DSM prediction framework for columns and beams with nonlinear modeling
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and the development of simplified methods for predicting elastic buckling, will
commence for the final year of the project.
2 Experimental Program 24 cold-formed steel column specimens were tested in January 2007 at
Johns Hopkins University. The motivation for this testing was to evaluate the
influence of holes on column strength and failure mechanisms, and to obtain
detailed measurements and load-displacement data that could be used to
validate nonlinear finite element models. The testing procedures, specimen
dimensions and experiment results can be found in Progress Report #3, although
the tensile coupon tests were still ongoing at the time of publication. The full set
of specimen yield stresses is now complete and is provided in this section. This
material testing data is used in conjunction with the specimen elastic buckling
behavior (calculated in ABAQUS) to compare Direct Strength Method (DSM)
predictions to tested specimen compressive strengths. This comparison
completes the experimental program.
2.1 Materials Testing Table 2.1 summarizes the steel yield stress fy determined from tensile tests
on coupons taken from the web and flanges of each column specimen. The
tensile coupon thickness tw and minimum width wmin used to convert the tensile
yield force into an engineering stress are also provided.
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Table 2.1 Column specimen yield stresses obtained from tensile coupon tests
tw wmin fy tf1 wmin fy tf2 wmin fyin. in. ksi in. in. ksi in. in. ksi
362-1-24-NH362-2-24-NH362-3-24-NH362-1-24-H 0.0390 0.4945 55.9 0.0391 0.4963 59.3 0.0391 0.4968 58.5362-2-24-H 0.0368 0.4886 52.9 0.0390 0.4950 58.8 0.0391 0.4945 59.5362-3-24-H 0.0394 0.4945 55.6 0.0394 0.4927 N/C 0.0394 0.4947 56.4362-1-48-NH 0.0392 0.4985 59.4 0.0393 0.4965 59.7 0.0392 0.4975 59.9362-2-48-NH 0.0393 0.4990 59.2 0.0394 0.4975 59.3 0.0393 0.4970 59.2362-3-48-NH 0.0389 0.4930 58.0 0.0391 0.5000 58.9 0.0390 0.4930 60.1362-1-48-H 0.0391 0.4998 59.5 0.0393 0.4985 58.2 0.0394 0.4991 58.1362-2-48-H 0.0390 0.4992 58.8 0.0391 0.4961 60.6 0.0391 0.4975 59.8362-3-48-H 0.0401 0.4990 57.8 0.0400 0.4957 58.0 0.0397 0.4978 59.1600-1-24-NH600-2-24-NH600-3-24-NH600-1-24-H 0.0414 0.4899 61.9 0.0422 0.4940 63.6 0.0428 0.4964 60.3600-2-24-H 0.0427 0.4964 57.8 0.0384 0.4874 55.6 0.0424 0.4938 61.8600-3-24-H 0.0429 0.4966 59.7 0.0431 0.4954 58.0 0.0430 0.4960 62.6600-1-48-NH 0.0434 0.4985 58.7 0.0436 0.4955 62.3 0.0434 0.4965 59.3600-2-48-NH 0.0435 0.4985 N/C 0.0430 0.4970 63.4 0.0430 0.4970 63.3600-3-48-NH 0.0436 0.4995 60.4 0.0432 0.4955 N/C 0.0433 0.4965 61.9600-1-48-H 0.0429 0.4970 60.3 0.0426 0.4980 63.0 0.0429 0.4970 60.8600-2-48-H 0.0429 0.4994 61.8 0.0428 0.4962 62.1 0.0431 0.4977 62.2600-3-48-H 0.0430 0.4992 60.7 0.0434 0.4961 59.7 0.0430 0.4977 64.0
NOTE: N/C TTest was not completed successfully
0.4955
0.0438 0.04380.0432
0.4985 53.3
0.4950 60.6
57.40.4975 54.7
SpecimenWeb East FlangeWest Flange
0.0368 0.03720.0302
0.5000 55.90.4950 59.7
The statistics in Table 2.2 demonstrate that the two cross-section types, 362 and
600, have similar a mean yield stress, 58.1 ksi versus 60.9 ksi. The shapes of the
stress-strain curves for each cross-section type are different though, the 362
specimens have a gradually yielding stress-strain curve and the 600 specimens
have a flat yield plateau. Refer to Progress Report #3, Section 2.3.4.3 for a
detailed discussion of the differences in material behavior between these two
specimen groups.
Table 2.2 Yield stress statistics
mean STDVksi ksi
362S162-33 58.1 1.6600S162-33 60.9 1.5
yield stress, fyStud Type
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The steel yield stresses summarized here are now used to calculate strength
predictions using the Direct Strength Method.
2.2 Preliminary DSM Predictions
The basic premise of the Direct Strength Method is expressed for a column
as Pn=f(Pcrl, Pcrd, Pcre, Py) where Pn is the nominal column strength, Py is the axial
load at yielding, and Pcrl, Pcrd, Pcre are the axial loads at which elastic local,
distortional, and global (flexural, torsional, flexural-torsional) buckling occur
(NAS 2004 Appendix 1). Pcrl Pcrd, and Pcre are calculated with an eigenbuckling
analysis in ABAQUS and includes the influence of the slotted holes and tested
boundary conditions. Pcrd is the elastic buckling mode that most closely
resembles the pure distortional (D) buckling mode and may also contain local
web buckling (D+L) caused by the slotted hole. This D+L mode was shown in
Progress Report #1 to be a more accurate indicator of compressive strength than
the distortional hole mode DH (where the flanges distort only near the hole) for
the structural studs and slotted hole size considered here. Section 4 of this
report examines the issue of net vs. gross cross-sectional area. For this
preliminary analysis, the gross cross-sectional area is used to calculate Py .
2.2.1 Finite Element Modeling
The elastic buckling behavior of the 24 column specimens is obtained with
eigenbuckling analyses in ABAQUS (ABAQUS 2004). All columns are modeled
with S9R5 reduced integration nine-node thin shell elements. Refer to Progress
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Report #1 for a detailed discussion of the S9R5 element (Moen and Schafer 2006).
Cold-formed steel material properties are assumed as E=29500 ksi and ν=0.3.
The centerline Cee channel cross-section dimensions input into ABAQUS are
calculated using the out-to-out dimensions and flange and lip angles at the
midlength of each column specimen (see Progress Report #3, Table 2.3 and Table
2.4 corrected in Appendix B). Element meshing is performed with a custom-built
Matlab program (Mathworks 2006) written by the first author. Figure 2.1
provides an example of a typical column mesh, where the longitudinal mesh
spacing is 1 in. and holes are defined with a series of element lines radiating from
the opening. Two elements model the rounded corners here because S9R5
elements can be defined with an initial curved geometry. Refer to Section 3.3 of
Progress Report #1 for more information on modeling rounded corners with
S9R5 elements. The maximum element aspect ratio is limited in all
eigenbuckling finite element models to 8:1. Refer to Progress Report #1 for the
background mesh studies that results in this aspect ratio limit.
1 in. (Typ.)
Figure 2.1 Typical finite element mesh in ABAQUS
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Each column specimen is loaded with a compressive axial force. The axial
force is applied as a set of consistent nodal loads in ABAQUS to simulate a
constant pressure across the bearing edge of the specimen. The nodes on the
loaded column face are coupled together in the direction of loading (1 direction)
with an ABAQUS “pinned” rigid body constraint. This constraint ensures that
all nodes on the loaded face of the column translate together, while the rotational
degrees of freedom remain independent (as in the case of platen bearing). Figure
2.2 summarizes the loading and boundary conditions used in the eigenbuckling
analyses.
1
2
3
ABAQUS “pinned “ rigid body reference node constrained in 2 to 6 directions, ensures that all nodes on loaded surface move together in 1 direction
Nodes bearing on top platen constrained in 1, 2 and 3
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Figure 2.2 Boundary and loading conditions implemented in the ABAQUS eigenbuckling analyses
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2.2.2 Elastic Buckling Results The local (L), distortional (D), and global buckling (G) buckling behavior of the
24 column specimens are presented in this section. Special attention is paid to
the influence of the holes on the critical elastic buckling load and the buckled
mode shapes.
2.2.2.1 Local and Distortional Buckling The local and distortional buckled shapes for specimens with and without
slotted holes are compared in Figure 2.3 to Figure 2.6. The addition of the slotted
web hole terminates web local buckling in all 362 specimens and decreases the
quantity of local half-waves in the 600 specimens. These observations are
consistent with the elastic buckling studies performed by Moen and Schafer
(2006), where it was shown that a hole can increase or decrease the number of
buckled half-waves (and the critical elastic buckling load) of rectangular plates
and cold-formed steel structural studs. To isolate the influence of the hole and
tested boundary conditions on elastic buckling, the ABAQUS finite element
eigenbuckling results are compared to CUFSM finite strip results in Table 2.4.
(CUFSM relies solely on the member cross-section to evaluate elastic buckling
and is not capable of modeling warping fixed end boundary conditions or the
discontinuity of a hole). ABAQUS predicts slightly higher local critical elastic
buckling loads (Pcrl) than CUFSM, although holes and fixed-fixed boundary
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conditions are shown to have a minimal influence on Pcrl for the specimen
lengths, cross-section geometries, and hole size considered in this study.
The ABAQUS distortional buckled shapes in Figure 2.3 to Figure 2.6
reveal that slotted web holes create distortional hole modes (DH) and mixed
modes with both local buckling and pure distortional modes (D+L). The
presence of these unique hole modes is noted in previous work summarized in
Progress Reports #1 and #2. Figure 2.3 displays an asymmetric distortional hole
mode for a structural stud where the hole has been cut slightly offset from the
web centerline. For both the 362 and 600 specimens with holes, local buckling
mixes with the pure distortional mode. Table 2.4 demonstrates an increase in the
distortional critical elastic buckling load, Pcrd, from CUFSM to ABAQUS for the
24 in. specimens. This boost is attributed to the fixed-fixed tested boundary
conditions which modify the distortional half-wavelengths for these short
specimens. Pcrd for the 600 specimens is most sensitive to the fixed-fixed
boundary conditions. The mixing of local buckling and the pure D mode for the
specimens with holes does not influence Pcrd .
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Table 2.3 Comparison of ABAQUS and CUFSM critical elastic buckling values
Length t Pcrl, ABAQUS Pcrl, CUFSM ABAQUS/CUFSMPcrd, ABAQUS
(pure D mode)
Pcrd, CUFSM ABAQUS/CUFSM
in. in. kips kips kips kips362-1-24-NH 24.099 0.0385 4.86 4.88 1.00 8.85 7.10 1.25362-2-24-NH 24.098 0.0385 4.75 4.76 1.00 8.41 7.10 1.18362-3-24-NH 24.098 0.0385 4.97 4.99 1.00 8.84 7.10 1.25362-1-24-H 24.099 0.0391 5.86 5.70 1.03 9.24 9.50 0.97362-2-24-H 24.099 0.0383 5.41 5.30 1.02 10.32 9.00 1.15362-3-24-H 24.099 0.0394 5.71 5.60 1.02 9.49 9.30 1.02
362-1-48-NH 48.214 0.0392 5.15 5.17 1.00 9.67 9.05 1.07362-2-48-NH 48.301 0.0393 5.17 5.19 1.00 9.64 9.00 1.07362-3-48-NH 48.191 0.0390 5.09 5.11 1.00 9.52 8.98 1.06362-1-48-H 48.216 0.0393 5.29 5.16 1.03 9.08 9.00 1.01362-2-48-H 48.232 0.0391 5.22 5.09 1.03 8.97 8.86 1.01362-3-48-H 48.197 0.0399 5.68 5.53 1.03 8.98 8.94 1.00
600-1-24-NH 24.100 0.0436 3.45 3.48 0.99 6.76 5.30 1.28600-2-24-NH 24.103 0.0436 3.43 3.48 0.99 6.66 5.30 1.26600-3-24-NH 24.099 0.0436 3.43 3.48 0.99 6.64 5.30 1.25600-1-24-H 24.101 0.0421 3.26 3.41 0.96 7.01 4.90 1.43600-2-24-H 24.099 0.0412 3.22 3.02 1.06 6.75 4.90 1.38600-3-24-H 24.101 0.0430 3.46 3.21 1.08 7.34 5.00 1.47
600-1-48-NH 48.255 0.0435 3.46 3.49 0.99 5.17 5.10 1.01600-2-48-NH 48.250 0.0432 3.37 3.40 0.99 5.69 5.10 1.12600-3-48-NH 48.295 0.0434 3.41 3.44 0.99 5.66 5.10 1.11600-1-48-H 48.089 0.0428 3.40 3.34 1.02 5.07 5.10 0.99600-2-48-H 48.253 0.0429 3.39 3.33 1.02 4.97 5.10 0.98600-3-48-H 48.060 0.0431 3.43 3.37 1.02 5.03 5.10 0.99
* cross section dimensions vary slightly also, resulting in slightly different CUFSM predictions between specimens, see Table 3.1
Local Buckling Distortional Buckling
Specimen Name
DIMENSIONS*
Table 2.4 Critical elastic buckling load statistical comparison
Mean STDEV Mean STDEV600s 1.01 0.03 1.19 0.18362s 1.01 0.01 1.09 0.10no holes 0.99 0.00 1.16 0.09holes 1.00 0.02 1.12 0.1948" specimens 1.01 0.02 1.03 0.0524" specimens 1.01 0.03 1.17 0.15
Distortional BucklingABAQUS/CUFSM
Local BucklingSpecimen Group
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Local Buckling (L) Distortional Buckling
Hole terminates web local buckling asymmetric
distortional hole mode with local buckling at the hole
Hole shifted off centerline web
DDH+LD
Figure 2.3 Local and distortional buckled mode shapes for short (L=24 in.) 362 specimens
Local Buckling (L) Distortional Buckling
Hole changes number of half-waves from 5 (NH) to 6 (H)
distortional and local buckling at the hole
D DH+L D
Figure 2.4 Local and distortional buckled mode shapes for short (L=24 in.) 600 specimens
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Local Buckling (L) Distortional Buckling (D)
Hole terminates web local buckling
Holes cause mixed distortional-local mode
D+LD
Figure 2.5 Local and distortional buckled mode shapes for intermediate length 362 specimens
Local Buckling Distortional Buckling
Holes change number of half-waves from 8 (NH) to 12 (H)
Holes cause mixed distortional – local mode
D DH+L
Figure 2.6 Local and distortional buckling mode shapes for intermediate length 600 specimens
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2.2.2.2 Global Buckling The global (Euler) critical elastic buckling load Pcre is calculated for all
intermediate length column specimens using eigenbuckling analyses in
ABAQUS and for all short specimens using CUTWP. CUTWP is a MATLAB-
based program that employs the classical cubic stability equation to solve for the
global buckling modes (Sarawit 2006). An accurate prediction of Pcre with hole
and boundary condition effects is important for the intermediate columns since
the local buckling DSM prediction is a function of Pcre. For the short specimens,
Pcre does not influence the strength predictions (because it is so large compared to
Py) and therefore Pcre can be estimated (with less computation effort) using
CUTWP.
Research in Progress Report #1 demonstrated that isolated slotted holes in
362S162-33 structural studs did not influence Pcre. This observation is confirmed
again in Figure 2.7, where the global buckling load of the intermediate length
362-1-48-H specimen (including hole effects) is compared to the theoretical
buckling load Pcre from CUTWP (without hole effects). The difference between
the hole and no hole predictions is less than 2%. The controlling global mode
predicted by both ABAQUS and CUTWP is flexural-torsional buckling. For the
the intermediate length 600-1-48-H specimen, the slotted web holes are found to
decrease Pcre by 10% in the same study. This reduction is corroborated with an
ABAQUS eigenbuckling analysis of the 600-1-48-H specimen in ABAQUS, but
with the holes removed. Future research is needed to understand which cross-
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section shapes and at what column lengths web holes influence global buckling
behavior. This effort will be a major focus of the continuation of the project as
practical design rules are developed.
362-1-48-H 600-1-48-H
Pcre,ABAQUS=29.5 kips Pcre,ABAQUS=56.3 kips
Global Buckling
Holes cause a 10% reduction in Pcre
Pcre,CUTWP=30.0 kipsPcre,CUTWP=62.5 kips
Figure 2.7 Comparison of global mode shapes for intermediate length 362 and 600 specimens (global mode shapes in ABAQUS include local interaction)
2.2.3 Comparison of Column Strengths to DSM Predictions
Table 2.5 provides the specimen geometry and DSM prediction details for
each specimen and Table 2.6 presents the average test-to-predicted ratio for
different specimen groups. The DSM predictions are consistently conservative
for both hole and no hole specimens. The strengths of all 362 specimens are
controlled by local buckling, although distortional buckling was also observed in
testing near peak load (refer to Progress Report #3, Appendix A). The
distortional and local strength predictions, Pnd and Pnl, are also similar for the 600
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specimens. DSM predicts that most of these specimens will experience a
distortional failure, although the presence of the hole in the 600-24-X-H columns
lowers Pcrl enough to predict a local failure mode. The strength prediction
calculated using the distortional hole mode, Pnd(DH+L), is much lower than the
predicted strengths in this study and is judged too conservative to consider for
the specimen geometry and hole size in this study. A future study is planned to
compare the current AISI effective width prediction method to the DSM results
reported here.
Table 2.5 Comparison of Preliminary Direct Strength Method predictions to experiments
L t Hole Type Fy Ag Py,g Pcre PcrlPcrd
(DH+L)
Pcrd
(pure D mode)
Pne PnlPnd
(DH+L)
Pnd
(pure D mode)
Controlling Buckling
ModePtest
Test to Predicted**
in. in. ksi in 2 kips kips kips kips kips kips kips kips kips kips362-1-24-NH 24.099 0.0385 --- 55.1 0.280 15.46 109.44 4.86 --- 10.55 13.64 8.49 --- 9.85 L 10.48 1.235362-2-24-NH 24.098 0.0385 --- 55.1 0.282 15.56 112.45 4.75 --- 10.18 13.68 8.46 --- 9.72 L 10.51 1.243362-3-24-NH 24.098 0.0385 --- 55.1 0.285 15.72 112.16 4.97 --- 10.70 13.76 8.65 --- 10.00 L 10.15 1.174362-1-24-H 24.099 0.0391 Slotted 57.9 0.282 16.36 119.26 5.86 6.38 9.24 15.44 9.41 7.98 9.55 L 10.00 1.063362-2-24-H 24.099 0.0383 Slotted 57.1 0.275 15.72 112.77 5.41 5.68 10.32 14.83 8.92 7.38 9.84 L 10.38 1.164362-3-24-H 24.099 0.0394 Slotted 56.0 0.293 16.41 130.56 5.71 6.63 9.49 15.57 9.38 8.14 9.69 L 9.94 1.060
362-1-48-NH 48.214 0.0392 --- 59.7 0.284 16.93 30.46 5.15 --- 9.67 13.41 8.21 --- 9.93 L 9.09 1.107362-2-48-NH 48.301 0.0393 --- 59.2 0.283 16.75 29.48 5.17 --- 9.64 13.20 8.14 --- 9.86 L 9.49 1.166362-3-48-NH 48.191 0.0390 --- 59.0 0.281 16.59 29.56 5.09 --- 9.52 13.11 8.06 --- 9.76 L 9.48 1.176362-1-48-H 48.216 0.0393 Slotted 58.6 0.283 16.57 29.95 5.29 5.65 9.08 13.15 8.18 7.55 9.54 L 8.95 1.094362-2-48-H 48.232 0.0391 Slotted 59.7 0.282 16.83 29.71 5.22 5.78 8.97 13.28 8.20 7.69 9.56 L 9.18 1.120362-3-48-H 48.197 0.0399 Slotted 58.3 0.289 16.84 36.18 5.68 6.21 8.98 13.86 8.68 7.98 9.57 L 9.37 1.079
600-1-24-NH 24.100 0.0436 --- 58.7 0.420 24.66 244.48 3.45 --- 6.76 23.64 10.19 --- 10.04 D 11.93 1.188600-2-24-NH 24.103 0.0436 --- 58.7 0.418 24.53 234.92 3.43 --- 6.66 23.48 10.12 --- 9.94 D 11.95 1.202600-3-24-NH 24.099 0.0436 --- 58.7 0.416 24.45 218.42 3.43 --- 6.64 23.34 10.08 --- 9.91 D 12.24 1.236600-1-24-H 24.101 0.0421 Slotted 61.9 0.404 25.02 239.34 3.26 3.11 7.01 23.95 10.06 6.65 10.30 L 12.14 1.207600-2-24-H 24.099 0.0412 Slotted 58.4 0.396 23.10 238.45 3.22 2.92 6.75 22.18 9.54 6.20 9.72 L 11.62 1.219600-3-24-H 24.101 0.0430 Slotted 60.1 0.411 24.72 242.63 3.46 3.27 7.34 23.69 10.21 6.80 10.49 L 11.79 1.155
600-1-48-NH 48.255 0.0435 --- 60.1 0.418 25.13 61.80 3.46 --- 5.17 21.20 9.52 --- 8.79 D 11.15 1.269600-2-48-NH 48.250 0.0432 --- 63.4 0.413 26.18 59.64 3.37 --- 5.69 21.79 9.59 --- 9.43 D 11.44 1.213600-3-48-NH 48.295 0.0434 --- 61.2 0.415 25.39 60.16 3.41 --- 5.66 21.28 9.49 --- 9.27 D 11.29 1.218600-1-48-H 48.089 0.0428 Slotted 61.4 0.411 25.25 56.27 3.40 3.20 5.07 20.93 9.38 6.78 8.72 D 11.16 1.280600-2-48-H 48.253 0.0429 Slotted 62.0 0.410 25.46 53.04 3.39 3.19 4.97 20.82 9.35 6.79 8.66 D 11.70 1.351600-3-48-H 48.060 0.0431 Slotted 61.5 0.416 25.56 55.76 3.43 3.21 5.03 21.10 9.47 6.83 8.73 D 11.16 1.278
L=Local Buckling, D=Distortional Buckling* * The predicted strength is the minimum of Pne, Pnl, and Pnd ( pure D mode). The Pnd (DH+L) distortional hole strength prediction is provided only for comparison.
TESTS
Specimen Name
ELASTIC BUCKLINGSPECIMEN DIMENSIONS AND MATERIAL PROPERTIES DSM PREDICTIONS
Table 2.6 Preliminary Direct Strength Method comparison statistics
Specimen Group Mean STDEV
600s 1.23 0.05362s 1.14 0.06no holes 1.20 0.04holes 1.19 0.0648" specimens 1.20 0.0924" specimens 1.18 0.06
Test to Predicted Ratio
21
2.2.4 Conclusions An eigenbuckling analysis was performed in ABAQUS for each of the 24
structural stud column specimens in this experimental study. The slotted web
holes did not influence the local, distortional, and global critical elastic buckling
loads (Pcrl, Pcrd, Pcre) for the specimen geometry and hole size. This result
produces similar (and conservative) Direct Strength Method predictions for
specimens with and without holes.
Although the critical elastic buckling loads were not significantly
influenced by the presence of holes, it was observed that the elastic buckling
mode shapes were often different for specimens with holes. The slotted holes
damped web local buckling in the 362 specimens and modified the quantity of
local buckling half-waves in the 600 specimens. This conclusion is consistent
with the observed influence of the slotted holes during the specimen tests. For
example, specimen 600-1-48-NH (no hole) exhibited primarily distortional
buckling at peak load while specimen 600-1-48-H (with hole) maintained both
local and distortional buckled half-waves through the peak response (see
Progress Report #3, Section 2.5.2.2). The web holes also created unique mixed
distortional and local modes in the eigenbuckling analyses which were sensitive
to the location of the hole relative to the centerline of the web.
The critical elastic global buckling load was not influenced by the holes in
the intermediate length 362 specimens, but did decrease Pcre of the intermediate
length 600 specimens by 10%. The reduction did not influence the DSM
22
prediction though, since a distortional failure was predicted for these specimens.
It is important to note that while the elastic buckling loads and ultimate strengths
in this experimental study were in general insensitive to the presence of holes,
the post-peak ductility of the columns was often reduced (especially in the 362
specimens) which is an important consideration when designing in active
seismic areas.
3 Nonlinear Finite Element Modeling of Structural Stud Columns with Holes
Nonlinear finite element modeling is a valuable tool for predicting the
behavior of structural members at their ultimate limit state. The primary
complication with these computational predictions is their sensitivity to
modeling assumptions. It is therefore good practice to develop a nonlinear
modeling approach around a set of experiment results, where actual physical
behavior can be compared to computational predictions. In this section, the
completed research on a nonlinear modeling approach for structural stud
columns with holes is presented. The 24 column tests described in Progress
Report #3 are modeled with the experimentally obtained steel stress-strain
curves, initial imperfections (local, distortional, and global) and predicted
residual stresses. A comparison of experimental and computationally predicted
failure modes and ultimate strengths reveals mixed successes to date and
research is ongoing. The development of this modeling approach is important to
23
this project because it will be used to support specific changes to DSM method
for members with holes.
3.1 Element Meshing and Cross-section Dimensions The behavior of the 24 column specimens is predicted with the general
purpose finite element program ABAQUS (ABAQUS 2004). All columns are
modeled with S9R5 reduced integration nine-node thin shell elements. Refer to
Progress Report #1 for a detailed discussion of the S9R5 element.
The centerline Cee channel cross-section dimensions input into ABAQUS
are calculated using the out-to-out dimensions and flange and lip angles at the
midlength of each column specimen as provided in Table 3.1. tavg is the average
of the west flange, east flange, and web thicknesses including the zinc coating
and is used to convert the out-to-out dimensions to centerline dimensions.
tbare,avg is the average of the west flange, east flange, and web thicknesses but
does not include the zinc coating. tbare,avg is the specimen thickness input into the
ABAQUS models. The location of the slotted web holes relative to the centerline
of the web are input into ABAQUS with the offsets provided in Table 3.2.
24
Table 3.1 Specimen dimensions used in ABAQUS finite element models Length
Lavg H B1 B2 D1 D2 F1 F2 S1 S2 RB1 RB2 RT1 RT2 tavg tbare,avgin. in. in. in. in. in. degrees degrees degrees degrees in. in. in. in. in. in.
362-1-24-NH 24.099 3.654 1.550 1.621 0.411 0.431 86.8 86.0 8.4 12.8 0.188 0.188 0.172 0.188 0.0407 0.0385362-2-24-NH 24.098 3.712 1.586 1.585 0.416 0.422 87.6 85.5 11.4 11.6 0.172 0.203 0.266 0.281 0.0407 0.0385362-3-24-NH 24.098 3.623 1.677 1.679 0.425 0.399 86.3 85.4 9.6 9.4 0.188 0.172 0.281 0.281 0.0407 0.0385362-1-24-H 24.099 3.583 1.650 1.595 0.430 0.437 87.6 85.6 11.1 10.9 0.297 0.281 0.188 0.203 0.0421 0.0391362-2-24-H 24.099 3.645 1.627 1.593 0.440 0.391 86.3 85.2 4.4 10.3 0.313 0.313 0.188 0.188 0.0421 0.0383362-3-24-H 24.099 3.672 1.674 1.698 0.418 0.426 87.7 86.1 10.5 10.8 0.266 0.266 0.188 0.188 0.0417 0.0394
362-1-48-NH 48.214 3.624 1.611 1.605 0.413 2.976 85.0 85.6 7.8 10.1 0.172 0.172 0.281 0.281 0.0414 0.0392362-2-48-NH 48.301 3.624 1.609 1.585 0.407 0.421 84.2 84.6 8.0 10.8 0.188 0.172 0.281 0.281 0.0418 0.0393362-3-48-NH 48.191 3.614 1.604 1.599 0.425 0.401 85.3 84.1 9.1 12.2 0.188 0.188 0.266 0.266 0.0403 0.0390362-1-48-H 48.216 3.622 1.602 1.595 0.420 0.412 85.6 84.2 8.5 9.8 0.172 0.172 0.281 0.281 0.0410 0.0393362-2-48-H 48.232 3.623 1.594 1.610 0.425 0.403 85.6 83.8 8.3 11.2 0.172 0.172 0.281 0.281 0.0421 0.0391362-3-48-H 48.197 3.633 1.604 1.610 0.395 0.432 84.1 85.3 9.7 7.3 0.172 0.172 0.281 0.250 0.0403 0.0399
600-1-24-NH 24.100 6.037 1.599 1.631 0.488 0.365 92.5 93.7 1.6 2.1 0.172 0.156 0.250 0.203 0.0466 0.0436600-2-24-NH 24.103 6.070 1.582 1.614 0.472 0.380 91.2 94.1 1.7 2.3 0.203 0.203 0.266 0.266 0.0466 0.0436600-3-24-NH 24.099 6.030 1.601 1.591 0.369 0.483 94.1 91.0 -2.2 3.5 0.156 0.172 0.266 0.219 0.0466 0.0436600-1-24-H 24.101 6.040 1.594 1.606 0.484 0.359 90.4 92.3 1.0 2.0 0.250 0.219 0.172 0.172 0.0460 0.0421600-2-24-H 24.099 6.011 1.608 1.602 0.369 0.500 93.2 88.7 1.8 1.1 0.203 0.234 0.172 0.172 0.0467 0.0412600-3-24-H 24.101 6.032 1.606 1.577 0.360 0.478 93.3 89.3 0.1 4.1 0.250 0.203 0.172 0.172 0.0461 0.0430
600-1-48-NH 48.255 6.018 1.621 1.609 0.486 0.374 90.6 92.8 0.2 1.4 0.172 0.172 0.234 0.219 0.0461 0.0435600-2-48-NH 48.250 6.017 1.596 1.601 0.931 0.357 89.9 91.9 2.0 2.4 0.172 0.172 0.234 0.234 0.0453 0.0432600-3-48-NH 48.295 6.026 1.585 1.627 0.489 0.338 90.0 92.1 2.6 2.3 0.172 0.172 0.266 0.219 0.0452 0.0434600-1-48-H 48.089 6.010 1.598 1.625 0.480 0.388 90.0 92.6 2.5 2.1 0.188 0.156 0.250 0.219 0.0450 0.0428600-2-48-H 48.253 6.017 1.589 1.607 0.476 0.356 88.9 91.2 2.4 1.0 0.172 0.172 0.234 0.234 0.0459 0.0429600-3-48-H 48.060 6.062 1.632 1.588 0.366 0.480 92.3 89.4 0.7 3.6 0.172 0.172 0.219 0.250 0.0461 0.0431
SpecimenOut-to-Out Dimensions Corner Angles Outside Corner Radii Sheet thickness
West East
D1
B1
H
B2
D2
F1 F2
S1 S2
Orientation in testing machine(front view)
EastWest
South
North
L
a a
Section a-aRB1
RT1 RT2
RB2
West East
Figure 3.1 Specimen length and cross-section notation
25
Table 3.2 Slotted hole dimensions and locations X Δh L hole d hole X Δh L hole d holein. in. in. in. in. in. in. in.
362-1-24-H L/2 -0.096 4.003 1.492362-2-24-H L/2 0.104 4.000 1.502362-3-24-H L/2 -0.024 4.005 1.493362-1-48-H (L-24)/2 0.087 3.999 1.500 (L+24)/2 0.112 4.001 1.494362-2-48-H (L-24)/2 0.047 4.001 1.496 (L+24)/2 0.091 4.003 1.494362-3-48-H (L-24)/2 -0.042 4.000 1.493 (L+24)/2 -0.062 4.003 1.491600-1-24-H L/2 -0.090 4.002 1.498600-2-24-H L/2 0.105 4.001 1.491600-3-24-H L/2 0.104 4.001 1.493600-1-48-H (L-24)/2 -0.117 4.002 1.494 (L+24)/2 -0.127 3.998 1.497600-2-48-H (L-24)/2 -0.092 4.001 1.499 (L+24)/2 -0.100 4.002 1.498600-3-48-H (L-24)/2 0.121 3.999 1.497 (L+24)/2 0.127 4.003 1.494L is the average length of the specimen
Specimen
North
X
Lhole
hhole
Front view
Δh (+ shift shown)
Figure 3.2 Slotted hole location and dimension definitions
Element meshing is performed with a custom-built Matlab program
(Mathworks 2006) written by the author. The typical mesh is consistent with that
used in the eigenbuckling analyses described in Section 2.2.1 (see Figure 2.1),
26
where the longitudinal mesh spacing is 1 in. and holes are defined with a series
of element lines radiating from the opening. Two elements are used to model the
rounded corners and the maximum element aspect ratio is limited in all finite
element models to 8:1.
3.2 Boundary Conditions and Application of Load The ABAQUS boundary conditions are modeled to simulate the friction-
bearing end conditions used in the column tests. These end conditions allowed
deformation of the cross-section at the bearing ends under load (slipping) and lift
off of the bearing ends for some specimens, especially for large deformations
beyond peak load (Refer to Progress Report #3, Section 2.5.2.3). Contact
modeling is employed in ABAQUS to capture the behavior of these bearing
conditions. A master analytical rigid surface is defined to represent the top and
bottom platen as shown in Figure 3.3. Each surface is assigned a reference node
located at the loading line (refer to Figure 3.8 for a definition of the loading line).
The rigid surfaces simulate fixed-fixed conditions by restraining the reference
node degrees of freedom, and the specimen is loaded by applying an imposed
displacement to the bottom surface reference node. Top and bottom node-based
slave surfaces are defined to simulate the bearing end of each specimen. The
tributary bearing area is defined at each node in the slave surface to ensure that
contact stresses are simulated accurately in ABAQUS.
27
1
2
3
45
6
ABAQUS Analytical Rigid Surface (Typ.)
Restrain rigid surface reference node in 2 to 6 directions
Restrain rigid surface reference node in 1 to 6 directions
Apply imposed displacement of surface in 1 direction
Assign friction contact behavior in ABAQUS between rigid surface and specimen
Figure 3.3 Contact boundary condition as implemented in ABAQUS
A Coulomb friction model is enforced in ABAQUS between the master
and slave surfaces by defining a static and kinetic coefficient of friction, μs and μk,
for steel-on-steel contact. The assumed values for μs and μk are 0.7 and 0.6 in this
study, which are based on recommendations by Oden and Martin (1985). Slip
occurs in the model once the shear stress at the contact interface exceeds μsfn,
where fn is the normal contact stress at the bearing surface.
The locations of the rigid surfaces are defined to be in contact with the
specimen ends when the first step of the analysis begins. This does not
guarantee perfect contact though, so the ABAQUS command ADJUST is used to
zero the contact surface and avoid numerical instabilities during the first analysis
28
step. The ADJUST command modifies the geometry of the specimen to close
infinitesimal gaps, but does not result in internal forces or moments in the
specimen.
3.3 Nonlinear Material Modeling Steel yielding and plasticity is simulated with a classical metal plasticity
model assuming isotropic hardening in ABAQUS. A Mises yield surface is
defined with the true stress and true plastic strain obtained from unixial tensile
coupon tests for each specimen. Three stress-strain curves (west flange, east
flange, and web) were obtained for each specimen (refer to Progress Report #3,
Section 2.3.4). The experimentally obtained engineering stress-strain curves are
converted to true stress and strain and then averaged point-by-point to produce
a yield stress, proportional limit, and plastic strain curve for each specimen. The
plastic strain curve is input into ABAQUS with the *PLASTIC command. The
plastic strain curve for specimen 600-3-48-H is provided in Figure 3.4. The
circular points are those that are input into ABAQUS. For Mises stresses below
the proportional limit, ABAQUS assumes linear elastic behavior with the cold-
formed steel material properties of E=29500 ksi and ν=0.3. The plastic strain
curves (and individual data points) for all 24 specimens are provided in
Appendix A.
29
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
true plastic strain
true
stre
ss, k
si
Proportional Limit
Yield Stress
Figure 3.4 Plastic strain curve for specimen 362-3-48-H used in ABAQUS to model nonlinear
material behavior (refer to Appendix A for the details on the development of this curve)
3.4 Initial Geometric Imperfections The ultimate strength and failure mechanisms of cold-formed steel
columns are sensitive to initial geometric imperfections in the shape of their
eigenmodes. In this study, the local (L) and distortional (D) elastic buckling
mode shapes in Figure 3.5 are calculated with an ABAQUS eigenbuckling
analysis and then superimposed on the specimen nodal geometry. The
magnitude of the imperfection shapes are scaled to specimen measurements.
The short column specimens are evaluated with an imperfection combination of
L+D. The intermediate length columns are evaluated with two different types of
imperfection magnitudes, L+D and L+(4xD), to evaluate the influence of
distortional imperfection magnitude on ultimate strength. It was observed in
Section 2.2.2 that the presence of holes can change the shape of the L and D
30
modes. To ensure that both hole and no hole specimens are modeled with the
same imperfection shapes, the elastic buckling mode shapes for the specimens
with holes were calculated with the holes closed in as shown in Figure 3.6. This
procedure ensures that the load-displacement behavior of both the hole and no
hole specimens are compared on equivalent basis (both will have the no hole L
and D imperfection shapes). Filling in the holes is necessary (instead of
eliminating them completely) because it preserves the nodal numbering and
geometry of the specimens with holes, making it convenient to impose the L and
D modes in ABAQUS.
362-X-24
L D
600-X-24
L D
362-X-48 600-X-48
L DL D
Figure 3.5 Imperfection mode shapes implemented in ABAQUS
31
Hole is filled in with S9R5 elements to produce no hole local buckling shape
Figure 3.6 Slotted holes are filled with S9R5 elements to obtain no hole imperfection shapes
Web deviations and angular variations of the west and east flanges were
measured as part of the experimental program detailed in Progress Report #3.
These measurements are used to obtain the modal amplifications factors
provided in Table 3.3.
Table 3.3 Local and distortional imperfection magnitudes implemented in ABAQUS
in. in.362-1-24-NH 0.038 0.092362-2-24-NH 0.054 0.049362-3-24-NH 0.036 0.070362-1-24-H 0.052 0.126362-2-24-H 0.058 0.055362-3-24-H 0.044 0.086
362-1-48-NH 0.071 0.028362-2-48-NH 0.080 0.036362-3-48-NH 0.057 0.045362-1-48-H 0.084 0.028362-2-48-H 0.066 0.033362-3-48-H 0.050 0.033
600-1-24-NH 0.061 0.053600-2-24-NH 0.075 0.057600-3-24-NH 0.096 0.038600-1-24-H 0.062 0.039600-2-24-H 0.089 0.061600-3-24-H 0.087 0.090
600-1-48-NH 0.071 0.030600-2-48-NH 0.077 0.044600-3-48-NH 0.073 0.025600-1-48-H 0.095 0.031600-2-48-H 0.049 0.036600-3-48-H 0.068 0.033
Local (L) Distortional (D)
Modal Imperfection Factors
Specimen
32
The local imperfection factor is the maximum deviation from the average
web elevation (refer to Progress Report #3, Table 2.9) in inches. The distortional
imperfection factor is calculated by first setting the reference flange deviation as
the values of F1 and F2 (see Figure 3.1 and revised Progress Report #3, Table 2.4
in Appendix B of this report) at the midlength of the member (X=12 in. for the
short specimens, X=24 in. for the intermediate length specimens). The distance
between the reference flange tip and the flange tips at the other measured cross
sections along the length (X=6, 18 in. for the short specimens, X=12, 18, 30, 36 in.
for the intermediate length specimens) are calculated as shown in Figure 3.7. The
maximum deviation (either positive or negative) is taken as the distortional
amplification factor D.
The sign of the distortional amplification factor is determined for each
ABAQUS model to ensure that the elastic distortional buckling mode coincides
with the observed failure mode of the specimen (flanges buckling in or out). It
should be noted the finite element models in this study consider the measured
angles at the midlength of the columns, not the baseline geometry with 90 degree
corners commonly assumed in the modeling of cold-formed steel members.
Since the flanges were measured with corner angles varying by as much as eight
degrees from a right angle, the modeled cross section can also be considered a
different type of distortional imperfection in addition to the variation in angle
along the specimen length.
33
Reference cross section at the midlength of the member
Deviation from reference cross section (measured at X=6, 18 in. for the the short specimens and X=12, 18, 30 and 36 in. for the 48 in. long specimens).
B1 B2
θ1 θ2
( )iiBD θsinmax(= where i=1 or 2
Figure 3.7 Definition of notation used to calculate the distortional imperfection factor D The initial out-of-straightness of each column specimen was measured in
the MTS machine under a small preload before the start of each test. This global
imperfection is superimposed on the nodal geometry for each specimen finite
element model as shown in Figure 3.8. The magnitude of the global
imperfection, Δg, is provided in Table 3.4.
aa
Section a-aSpecimen COG (Typ.)
Δg (+ shown)
Loading Line
Figure 3.8 Definition of out-of-straightness imperfections implemented in ABAQUS
34
Table 3.4 Out-of-straightness imperfection magnitudes Δg
in.362-1-24-NH -0.024362-2-24-NH 0.004362-3-24-NH 0.038362-1-24-H -0.012362-2-24-H 0.034362-3-24-H -0.023362-1-48-NH 0.047362-2-48-NH -0.028362-3-48-NH 0.012362-1-48-H 0.066362-2-48-H 0.013362-3-48-H -0.003600-1-24-NH -0.063600-2-24-NH -0.141600-3-24-NH 0.063600-1-24-H -0.078600-2-24-H 0.076600-3-24-H 0.069600-1-48-NH -0.036600-2-48-NH -0.087600-3-48-NH -0.049600-1-48-H -0.098600-2-48-H 0.072600-3-48-H 0.020
Specimen
3.4.1 Residual Stresses and Equivalent Plastic Strains The manufacturing of structural stud columns involves the coiling and
uncoiling of sheet steel and the roll-forming of the cross-section. These processes
impart residual stresses and permanent plastic strains that contribute to the
initial state of the member when developing a nonlinear finite element model.
Appendix C provides a thorough literature review and description of the
manufacturing process as it relates to residual stresses. It also describes the
development of a prediction method based on elementary mechanics that
produces residual stress and strain distributions which can be readily input into
ABAQUS as part of a member’s initial state.
The residual stress prediction method assumes that residual stresses occur
in the flat regions of the cross-section from coiling and in the corners from coiling
and roll-forming. The coiling residual stresses are largest when the sheet
35
thickness t is large (>0.068 in.) and the yield stress is low (<40 ksi). The members
in this study have a relative low sheet thickness (~0.040 in.) and high yield stress
(~ 60 ksi) and therefore the method predicts that coiling residual stresses are zero
in the specimen cross-sections. Therefore only the roll-forming of the corners is
considered when calculating residual stresses and plastic strains.
Initial stresses are defined in ABAQUS based on the element local
coordinate system defined in Figure 3.9. The stresses are defined through the
thickness starting from section point 1 (SNEG). Stresses in the 1 direction are
defined as longitudinal and stresses in the 2 direction are referred to as
transverse.
SNEG
2
1SPOS
Element normal
Figure 3.9 ABAQUS element local coordinate system for use with residual stress definitions
36
Residual stresses and plastic strains are applied only to the corner
elements in the specimen finite element models. The equivalent plastic strain
distribution is shown in Figure 3.10, where εp is defined as:
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
zp r
t2
1ln32ε
rz is the centerline corner radius, which is typically different for each of the four
corners of each specimen (since the outside radii RT1, RT2, RB1, and RB2
measurements vary).
εp
εpSNEG
SPOS
Figure 3.10 Equivalent plastic strain distribution at the corners of the cross-section
The transverse residual stress distribution is provided in Figure 3.11 and
the longitudinal distribution in Figure 3.12. The distributions are based on the
yield stress σyield listed in Table 2.5 for each specimen.
37
SNEG
SPOS
-0.50σyield
+σyield -σyield
+0.50σyield
2
Figure 3.11 Transverse residual stress distribution applied at the corners of the cross-section
SNEG
SPOS
+0.05σyield
1
-0.05σyield
-0.50σyield
+0.50σyield
Figure 3.12 Longitudinal residual stress distribution applied at the corners of the cross-section
The transverse stress distribution has the special property that it is self-
equilibrating for both moment and axial force, meaning that total force and
moment through the thickness is zero. This self-equilibrating characteristic
ensures that no deformation (or redistribution of stress) will occur in ABAQUS in
the initial state. The longitudinal stress distribution is only self-equilibrating for
axial force and will impose a small bending moment in the member. The
deformations associated with this bending moment are infinitesimal and very
small redistributions in stress are observed (±0.1 ksi) in the initial state.
38
The number of element section points through the thickness dictates the
accuracy of the residual stress distribution. If only a small number of section
points are used, the discontinuity in stress at the middle thickness cannot be
modeled accurately and excessive transverse deformations of the cross-section
will occur. Figure 3.13 demonstrates the decrease in unbalanced through-
thickness transverse moment, MUB, as the number of section points increase
(sheet thickness is assumed as t=0.040 in. and σyield=60 ksi). As the number of
section points decrease, the residual stress approaches 0.50My, where My is the
yield moment of the sheet steel per unit width defined as:
yieldy
tM σ6
2
=
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
# of element through-thickness section points
MU
B/My
Figure 3.13 Influence of section points on the unbalanced moment (accuracy) of the transverse residual stress distribution as implemented in ABAQUS
39
55 section points are used in the specimen finite element models for this study as
a compromise between model accuracy and computational cost. ABAQUS
recommends not to use more than 250 section points for the S9R5 element
(private communication).
3.5 Nonlinear solution methods The primary solution algorithm used in this study is the ABAQUS default
Newton Raphson iteration scheme with added artificial damping (*STATIC,
STABILIZE in ABAQUS). Previous research summarized in Progress Report #2
demonstrated this solution method to be a robust solver of highly nonlinear
problems such as the compression of a thin plate with a hole. The method
encourages solution convergence by adding artificial damping to locations in the
model that are experiencing large changes in deformation between load steps.
Results from the artificial damping method are compared to those calculated
with the Modified Riks Method (*STATIC, RIKS in ABAQUS), an energy-based
nonlinear solution method also discussed in Progress Report #2. ABAQUS
automatic time stepping was enabled for both artificial damping and the
modified Riks solutions. The time stepping parameters for the two methods are
summarized in Table 3.5.
Table 3.5 Solution step parameters implemented in ABAQUS
*STATIC, STABILIZE *STATIC, RIKSSuggested initial step size 0.01 0.005Maximum Step Size 0.01 0.01Total analysis "time" 1 ---Number of Increments 100 300
Solution Step Parameters
40
3.6 Discussion of Results The nonlinear finite element results for the 24 column specimens are now
compared against the tested results. A key observation while performing the
modeling was that the predicted specimen ultimate strengths were dependent
upon the predicted failure mode. In other words, different ultimate strengths
were observed for the same type of specimen, depending upon the failure mode
predicted for the column by the model. Another important and related
observation was that different predicted failure modes for the same specimen
could be produced by changing the primary modeling decisions, especially
imperfections, residual stresses, boundary conditions, and the nonlinear solution
method. The results presented here attempt to isolate and highlight these
sensitivities so that confident conclusions regarding the viability of the proposed
modeling approach can be made.
3.6.1 Ultimate Strength and Failure Mechanisms The specimen failure modes predicted in ABAQUS are summarized in
Figure 3.14 to Figure 3.17. A majority of the modes resemble the experiment
results (those marked with an asterisk in the figures), while some modes are only
observed in the finite element predictions. When comparing the tested ultimate
strengths to finite element predictions in Table 3.6, it is observed that for the
group of models that predict a failure mode consistent with experiments, the
test-to-predicted ratio is 0.99, while when including all of the models the test-to-
41
predicted ratio is 0.90. This result suggests that physically reasonable failure
modes are an indicator of finite element model prediction accuracy, and that
further work is still required.
Table 3.6 Comparison of ABAQUS nonlinear results to tested strengths Experiments
L+D L+4xDkips kips kips
362-1-24-NH 10.48 9.73 --- 1.08 ---362-2-24-NH 10.51 10.13 --- 1.04 ---362-3-24-NH 10.15 10.11 --- 1.00 ---362-1-24-H 10.00 10.40 --- 0.96 ---362-2-24-H 10.38 11.29 --- 0.92 ---362-3-24-H 9.94 9.96 --- 1.00 ---362-1-48-NH 9.09 11.86 10.41 0.77 0.87362-2-48-NH 9.49 11.79 10.34 0.80 0.92362-3-48-NH 9.48 11.43 8.93 0.83 1.06362-1-48-H 8.95 11.17 N/C 0.80 N/C362-2-48-H 9.18 11.13 N/C 0.82 N/C362-3-48-H 9.37 11.27 9.37 0.83 1.00600-1-24-NH 11.93 10.58 --- 1.13 ---600-2-24-NH 11.95 12.75 --- 0.94 ---600-3-24-NH 12.24 12.00 --- 1.02 ---600-1-24-H 12.14 12.57 --- 0.97 ---600-2-24-H 11.62 12.90 --- 0.90 ---600-3-24-H 11.79 12.67 --- 0.93 ---600-1-48-NH 11.15 14.09 13.87 0.79 0.80600-2-48-NH 11.44 14.13 14.06 0.81 0.81600-3-48-NH 11.29 13.22 13.13 0.85 0.86600-1-48-H 11.16 13.58 13.19 0.82 0.85600-2-48-H 11.70 13.37 10.98 0.88 1.07600-3-48-H 11.16 13.58 11.89 0.82 0.94Note: shaded values have failure mode consistent with experimentsN/C Not completed, imperfection caused excessive element distortion error in ABAQUS
Specimen Ptest
Test to PredictedPtestABAQUS models
Imperfections Imperfections
L+D L+4xD
42
362-2-24-NH362-3-24-NH
362-1-24-NH362-2-24-H362-1-24-H
362-3-24-H
* *
* Failure mode observed in experiments
Figure 3.14 Summary of ABAQUS predicted failure modes for the short 362 column specimens
600-2-24-H600-3-24-H
600-1-24-H600-2-24-NH600-3-24-NH
600-1-24-NH
**
* Failure mode observed in experiments
Figure 3.15 Summary of the ABAQUS predicted failure modes for the short 600 column specimens
43
362-1-48-NH362-2-48-NH362-3-48-NH
362-1-48-NH 4xD362-2-48-NH 4xD362-3-48-NH 4xD
362-1-48-H362-2-48-H362-3-48-H
362-1-48-H 4xD362-2-48-H 4xD362-3-48-H 4xD
**
* Failure mode observed in experiments 4xType A 4 times the distortional imperfections are applied
Figure 3.16 Summary of the ABAQUS predicted failure modes for the intermediate length 362
column specimens
600-1-48-NH 4xD600-1-48-NH600-2-48-NH600-3-48-NH600-2-48-NH 4xType A600-3-48-NH 4xType A
600-1-48-H600-2-48-H600-3-48-H600-1-48-H 4xD
600-2-48-H 4xD600-3-48-H 4xD
*
* Failure mode observed in experiments 4xD 4 times the distortional imperfections are applied
Figure 3.17 Summary of the ABAQUS predicted failure modes for the intermediate length 600
specimens
44
Four of the six short 362 specimens exhibit a predicted mixed local-
distortional mode consistent with experiments. The finite element models even
develop the distinct distortional deformation at the hole for the short 362
specimens after peak load. The common failure mode for four out of the six
short 600 specimens was local web buckling followed by post peak flange
yielding and distortion. This was also consistent with experiments.
The intermediate length 362 specimens modeled with the L+D
imperfections experience localized buckling of the web and yielding of the
flanges. This failure mode is not consistent with the experiments, which was
observed as initial local buckling and dominant distortional buckling at peak
load before failing in flexural-torsional buckling. The ultimate strength
predictions for these specimens are on average 30% higher than the tested
results. The 362 column specimens modeled with L+4xD imperfections (4 times
the measured distortional imperfections) produced deformation patterns and
ultimate strengths more consistent with experimental results for two out the six
specimens, although again none of the specimen finite element models predict
the flexural-torsional failure.
The ultimate strength and failure modes for the intermediate length 600
specimens modeled with L+D imperfections were generally inconsistent with
experimental failure mode of web local buckling switching to dominant
distortional buckling after peak load. The L+4xD imperfection modeling trial
45
produced two 600 specimens with holes consistent with experiments, suggesting
that the distortional imperfections magnitudes (D) should have been derived
based on the maximum distortional imperfections in the cross-section instead of
using the midlength specimen cross-section dimensions as a baseline.
Additional work is planned to determine if a torsional imperfection will enable
the global buckling mode or if the friction coefficient between the platens and the
specimen was lower in the experiments than what was assumed in the finite
element models.
3.6.2 Modeling Issues Preliminary tests with the contact boundary conditions revealed solution
inconsistencies for both artificial damping and modified Riks solution methods.
The models commonly completed the solution without convergence errors, but
the internal specimen compressive forces were not consistent with the applied
displacements (either very large or very small). The inconsistency between
imposed displacements and internal forces commonly occurred in the first five
steps and was sensitive to the initial step size chosen for the model. This issue
was resolved in the artificial damping algorithm by applying additional
damping to the contact formulation (*CONTACT CONTROLS, STABILIZE in
ABAQUS) and by using the step size parameters in Table 3.4. These issues were
addressed in the modified Riks Method with the time stepping parameters listed
in Table 3.4, which were obtained by trial and error.
46
3.6.3 Influence of Residual Stresses and Imperfections The ultimate strengths of the column specimens are expected to be
sensitive to the residual stresses, equivalent plastic strains, and initial geometric
imperfections imposed in the finite element models to define the initial state.
This sensitivity is expressed in Figure 3.18 by comparing the load-displacement
response of specimen 600-2-24-NH with different initial states. It is observed that
imperfections decrease ultimate strength for this specimen, while the corner
equivalent plastic strains and residual stresses increase ultimate strength. The
boost from the corner plastic strains is expected, since the increase in apparent
yield stress from corner strain hardening stiffens the cross-section (commonly
known as the cold-work of forming effect). The increase in strength from the
residual stresses is not intuitive, although it is noted in Figure 3.19 that the
failure mode changes with the addition of residual stresses to one consistent with
experiments. As was concluded in Section 3.6.1, different failure modes for the
same specimen may correlate to substantially different (higher or lower) ultimate
strengths. Compression residual stresses are dominant on the inside of the
corner and tension on the outside of corner, and therefore the inside of the corner
will yield first under a compressive load. This uneven yielding through the
thickness results in out of plane deformations that are consistent with the
outward flaring of the flanges in the depicted failure mode.
47
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
axial displacement, in.
colu
mn
axia
l loa
d, k
ips
No ImpsImpsImps+Plastic StrainImps+Plastic Strain+RSExperiment
Figure 3.18 Influence of imperfections, residual stresses, and plastic strains on the load-displacement
response of specimen 600-2-24-H
L+D Imps L+D Imps
Initial plastic strain at corners
L+D Imps
Initial plastic strain at corners
Residual stressesat corners
ExperimentNo imperfections, residual stresses, or plastic strains
Figure 3.19 The influence of imperfections, residual stresses, and plastic strains on the failure mode
of specimen 600-2-24-H
48
3.6.4 Influence of Solution Method The artificial damping solution algorithm consistently provided physically
realistic results to the highly nonlinear problems in this study. Figure 3.20
compares the load-displacement response of specimen 600-2-24-NH calculated
with the artificial damping algorithm (*STATIC, STABILIZE with *CONTACT
CONTROLS, STABILIZE) to the same problem solve with the modified Riks
method (*STATIC, RIKS). The ultimate strength of the artificial damping
solution is 6% greater than the Riks solution and exhibits a stiffer, more ductile
response over the range of the test. The difference between the failure modes
predicted by the two methods contributes to this difference, the artificial
damping algorithm mode being more consistent with experiment observations.
Cross-section slipping at the bottom platen is observed in the Riks solution in the
post-peak range (as noted in Figure 3.20), while no slipping is observed in the
artificial damped solution. The two solutions provide upper and lower bounds
to the experiment load-displacement curve, and it is therefore difficult to
conclude with certainty which one is a better predictor of specimen behavior.
49
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
axial displacement, in.
colu
mn
axia
l loa
d, k
ips
Artificial DampingModified RiksExperiment
Artificial damping predicts stronger, more ductile response than the Riks Method
Cross section slips on platen in Riks Method here, no slip observed with artificial damping
Figure 3.20 Comparison of artificial damping and modified Riks solutions for
specimen 600-2-24-NH
3.6.5 Contact Modeling Observations The friction-bearing end conditions were chosen for this experimental
program based on the existing resources in our structures lab and the ease with
which the specimens could be aligned and tested. These boundary conditions
were expected to behave as fixed-fixed, although during the test slipping of the
cross-section and lifting off of the specimens were observed for some specimens
in the post-peak region of the load-displacement curve. To evaluate the
influence of the contact boundary conditions, the load-displacement response of
specimen 600-2-24-NH is evaluated against the fixed-fixed boundary conditions
described in Figure 3.21. The results of the two simulations in Figure 3.22 are
almost identical until displacements well into the post-peak range, suggesting
that no slipping of the cross-section is occurring in the model with friction
contact modeling. There is evidence that the slipping is restrained by the
50
artificial damping in the contact model since the Riks solution for the same
specimen and with the same friction coefficient in Figure 3.20 predicts slipping in
the post-peak range.
1
2
3
45
6 Use ABAQUS “pinned”rigid body to constrain all bearing edge nodes to a reference node at the section center of gravity
Apply imposed displacement of surface in 1 direction
Restrain bearing edge nodes in 1, 2 and 3 directions
Restrain rigid body reference node in 2 to 6 directions
Fixed-Fixed
Figure 3.21 Fixed-fixed boundary conditions in ABAQUS
0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
axial displacement, in.
colu
mn
axia
l loa
d, k
ips
Contact SurfaceFixed-Fixed
The failure response with friction-bearing boundary conditions is similar to that with fixed-fixed boundary conditions
Figure 3.22 A comparison of ABAQUS nonlinear solutions considering fixed-fixed and contact
boundary conditions for specimen 600-2-24-NH
51
The ABAQUS contact modeling depicted in Figure 3.23 was successful at
simulating the lift off of the specimen lips for a 600 specimen in the post peak
range.
ABAQUS contact modeling captures lift off of lip for 600 specimen
Figure 3.23 The ABAQUS contact boundary conditions successfully predict the lift off of the flanges
for a short 600 specimen test in the post-peak range
3.7 Conclusions The experimental results from 24 structural stud column specimens were
used to evaluate a proposed nonlinear finite element modeling approach
proposed by the author. The initial state of each specimen was simulated in
ABAQUS, including geometric imperfections, residual stresses and plastic
strains. The tested boundary conditions and applied loading were modeled by
employing contact modeling in ABAQUS to simulate the friction bearing
52
surfaces of the top and bottom loading platens. The comparison of experimental
results and finite element predictions highlighted a distinct connection between
ultimate strength and failure modes. When the predicted failure mode of the
specimen was consistent with experimental observations, the predicted ultimate
strength was consistent with the test strength.
When evaluating the solution sensitivity to modeling assumptions, it was
observed that geometric imperfections decreased strength, while residual
stresses and plastic strains at the cross-section corners increased strength. The
artificial damping and modified Riks solution methods provided an upper and
lower bound on the load-displacement response of the specimen considered, and
the contact boundary conditions with artificial damping performed as fixed-fixed
boundary conditions well into the post-peak range. The overall performance of
the finite element modeling approach was to overpredict the strength of the
specimens by 10%, although when the specimen failure modes matched those
observed in experiments, the tested and predicted strengths were within 1% on
average.
3.8 Future Work Research is currently being conducted to develop greater confidence in the
proposed nonlinear modeling approach for both short and intermediate length
structural stud columns with holes. A study is planned to evaluate the influence
of the friction-bearing boundary conditions on the tested ultimate strength. The
53
evaluation of contact stresses for both friction-bearing and fixed-fixed boundary
conditions will be compared using the laboratory-measured friction coefficient.
It is hoped that this study will identify the limitations of the contact boundary
conditions for others interested in using these conditions in their own tests. The
experiments will also be modeled with more conventional finite element
assumptions (perfect cross section geometry, probabilistic imperfection
magnitudes, no residual stresses) to evaluate them against common benchmarks.
The sensitivity of the intermediate length specimens to torsional imperfections
will be evaluated. A detailed study is planned to determine how the proposed
residual stress and plastic strain distributions influence failure modes and
ultimate strength. With the experience gained from this addition research, we
will be able to use this powerful computational tool with confidence to develop
and test our DSM design assumptions for members with holes.
54
4 Preliminary DSM Prediction Methods for Columns with Holes
Our research results (through the first four progress reports) demonstrate
that the current Direct Strength Method (without modification) is a viable
predictor of ultimate strength for columns with holes when the controlling
failure mode (local, distortional, or global) occurs in the slenderness ranges
associated with elastic buckling type failures. As column slenderness (again
either local, distortional, or global) decreases into the inelastic buckling and
yielding failure regions, the Direct Strength Method is observed to often
underpredict column strength. This underprediction is related to the reduction
in column cross sectional area at the hole (net vs. gross area influence). In this
section, five options for extending DSM to columns with holes are presented and
discussed. All of the options use the critical elastic buckling loads (Pcrl, Pcrd, Pcre)
calculated including the influence of the holes. Option 1 and 2 were studied in
Progress Report #1 ( Refer to Table 5.6 and Table 5.7 of PR#1) and are the most
straightforward of the options to implement. Options 3 through 5 are new
proposed modifications to the DSM equations that limit the column capacity to
the net cross section as the column slenderness (local, distortional, or global)
decreases. The DSM equations for each option are provided with illustrative
plots. The column data collected in Progress Report #1 and the column
experiment results in Progress Report #3 are used to compare the five options.
Beam data will be studies in the next progress report. The test-to-predicted ratio
55
statistics are provided for each method and future work is described that will
lead us soon to an important project milestone, a DSM approach for columns
with holes.
4.1 Proposed DSM Prediction Options Five DSM strength prediction options for columns with holes are
presented here.
Option 1: Include hole(s) in Pcr determinations, ignore hole otherwise This method, in presentation, appears identical to currently available DSM expressions
Flexural, Torsional, or Torsional-Flexural Buckling The nominal axial strength, Pne, for flexural, … or torsional- flexural buckling is
for 5.1c ≤λ Pne = yP658.02c ⎟
⎠⎞⎜
⎝⎛ λ
for λc > 1.5 crey2c
ne P877.0P877.0P =⎟⎟⎠
⎞⎜⎜⎝
⎛
λ=
where λc = crey PP
Py = AgFy Pcre= Critical elastic global column buckling load … (including hole(s)) Ag = gross area of the column
Local Buckling The nominal axial strength, Pnl, for local buckling is
for λl 776.0≤ Pnl = Pne
for λl > 0.776 Pnl = ne
4.0
ne
cr4.0
ne
cr PPP
PP
15.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛− ll
where λl = lcrne PP Pcrl = Critical elastic local column buckling load … (including hole(s)) Pne is defined above.
Distortional Buckling The nominal axial strength, Pnd, for distortional buckling is
for λd 561.0≤ Pnd = Py
for λd > 0.561 Pnd = y
6.0
y
crd
6.0
y
crd PPP
PP
25.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−
where λd = crdy PP
Pcrd = Critical elastic distortional column buckling load … (including hole(s))
56
Option 2: Include hole(s) in Pcr determinations, Use Pynet everywhere The only change in this method is to replace Py with Pynet
Flexural, Torsional, or Torsional-Flexural Buckling The nominal axial strength, Pne, for flexural, … or torsional- flexural buckling is
for 5.1cnet ≤λ Pne = ( ) ynetPcnet2
658.0 λ
for λcnet > 1.5 creynet
cnetne PPP 877.0877.0
2=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
λ
where λcnet = creynet PP
Pynet = AnetFy Pcre= Critical elastic global column buckling load … (including hole(s)) Anet = net area of the column
Local Buckling The nominal axial strength, Pnl, for local buckling is
for λl 776.0≤ Pnl = Pne
for λl > 0.776 Pnl = ne
4.0
ne
cr4.0
ne
cr PPP
PP
15.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛− ll
where λl = lcrne PP Pcrl = Critical elastic local column buckling load … (including hole(s)) Pne is defined above
Distortional Buckling The nominal axial strength, Pnd, for distortional buckling is
for λdnet 561.0≤ Pnd = Pynet
for λdnet > 0.561 Pnd = ynet
6.0
ynet
crd6.0
ynet
crd PPP
PP
25.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−
where λdnet = crdynet PP
Pcrd = Critical elastic distortional column buckling load … (including hole(s))
57
Option 3: Cap Pne and Pnd, otherwise no strength change, include hole(s) in Pcr determinations This method puts bounds in place and assumes local buckling interaction happens at ‘capped’ Pne Flexural, Torsional, or Torsional-Flexural Buckling The nominal axial strength, Pne, for flexural, … or torsional- flexural buckling is
for 5.1c ≤λ Pne = ynet.y PP658.02c ≤⎟
⎠⎞⎜
⎝⎛ λ
for λc > 1.5 ynet.crey2c
ne PP877.0P877.0P ≤=⎟⎟⎠
⎞⎜⎜⎝
⎛
λ=
where λc = crey PP
Pcre= Critical elastic global column buckling load … (including hole(s)) Py = AgFy Pynet = AnetFy Ag = gross area of the column Anet = net area of the column
Local Buckling The nominal axial strength, Pnl, for local buckling is
for λl 776.0≤ Pnl = Pne
for λl > 0.776 Pnl = ne
4.0
ne
cr4.0
ne
cr PPP
PP
15.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛− ll
where λl = lcrne PP Pcrl = Critical elastic local column buckling load … (including hole(s)) Pne is defined above
Distortional Buckling The nominal axial strength, Pnd, for distortional buckling is
for λd 561.0≤ Pnd = Py ≤ Pynet
for λd > 0.561 Pnd = ynet.y
6.0
y
crd6.0
y
crd PPP
PP
P25.01 ≤⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−
where λd = crdy PP
Pcrd = Critical elastic distortional column buckling load … (including hole(s))
58
Option 4: Cap Pnl and Pnd, otherwise no strength change, include hole(s) in Pcr This method puts bounds in place and assumes local-global interaction happens at full Pne
Flexural, Torsional, or Torsional-Flexural Buckling The nominal axial strength, Pne, for flexural, … or torsional- flexural buckling is
for 5.1c ≤λ Pne = yP658.02c ⎟
⎠⎞⎜
⎝⎛ λ
for λc > 1.5 crey2c
ne P877.0P877.0P =⎟⎟⎠
⎞⎜⎜⎝
⎛
λ=
where λc = crey PP
Pcre= Critical elastic global column buckling load … (including hole(s)) Py = AgFy Ag = gross area of the column
Local Buckling The nominal axial strength, Pnl, for local buckling is
for λl 776.0≤ Pnl = ynetne PP ≤
for λl > 0.776 Pnl = ynetne
4.0
ne
cr4.0
ne
cr PPPP
PP15.01 ≤⎟⎟
⎠
⎞⎜⎜⎝
⎛⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛− ll
where λl = lcrne PP Pcrl = Critical elastic local column buckling load … (including hole(s)) Pne is defined in Section above. Pynet = AnetFy Anet = net area of the column
Distortional Buckling The nominal axial strength, Pnd, for distortional buckling is
for λd 561.0≤ Pnd = Py ≤ Pynet
for λd > 0.561 Pnd = ynet.y
6.0
y
crd6.0
y
crd PPP
PP
P25.01 ≤⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−
where λd = crdy PP
Pcrd = Critical elastic distortional column buckling load … (including hole(s))
59
Option 5: Cap Pnl with transition, cap Pnd, include hole(s) in Pcr determinations This method puts bounds and transition in place, assumes local-global interaction at full Pne
Flexural, Torsional, or Torsional-Flexural Buckling The nominal axial strength, Pne, for flexural, … or torsional- flexural buckling is
for 5.1c ≤λ Pne = yP658.02c ⎟
⎠⎞⎜
⎝⎛ λ
for λc > 1.5 crey2c
ne P877.0P877.0P =⎟⎟⎠
⎞⎜⎜⎝
⎛
λ=
where λc = crey PP
Pcre= Critical elastic global column buckling load … (including hole(s)) Py = AgFy Ag = gross area of the column
Local Buckling The nominal axial strength, Pnl, for local buckling is
for λl 1lλ≤ Pnl = maxnP l
for 21 lll λ<λ<λ Pnl = ( )12
12maxnmaxn PPP
ll
lllll λ−λ
λ−λ−− λ
for λl 2lλ≥ Pnl = ne
4.0
ne
cr4.0
ne
cr PPP
PP
15.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛− ll
where λl = lcrne PP Pnlmax = ynetne PP ≤
λl1 = maxnne PP776.0 l
λl2 = 25.1
ne
maxn
maxn
neP
P3549.133.3
PP
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛ l
l
Pλl2 = ne
4.0
22
4.0
22
P1115.01 ⎟⎟⎠
⎞⎜⎜⎝
⎛
λ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛
λ−
ll
Pcrl = Critical elastic local column buckling load … (including hole(s)) Pne is defined in Section above. Pynet = AnetFy Anet = net area of the column
Distortional Buckling The nominal axial strength, Pnd, for distortional buckling is
for λd 561.0≤ Pnd = Py ≤ Pynet
for λd > 0.561 Pnd = ynet.y
6.0
y
crd6.0
y
crd PPP
PP
P25.01 ≤⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−
where λd = crdy PP
Pcrd = Critical elastic distortional column buckling load … (including hole(s))
60
The DSM global prediction curves are compared for the five options in
Figure 4.1. Option 2, which replaces Py with Pynet, was evaluated in Progress
Report #1 (also in Section 2.2.3 of this report) and reduces Pne for all values of λc
(also in Section . Option 3 proposes a cap on Pne for lower values of global
slenderness (where inelastic buckling and yielding occur) to account for the
reduction in column cross-sectional area from a hole.
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
global slenderness, λc or λcnet
Pne
/Py o
r Pne
/Pyn
et
No change to DSM Global Curve (Options 1, 4, and 5)
Cap DSM Global Curve at Pynet(Option 3)
Replace Py with Pynet in all strength and slenderness calculations (Option 2)
Pynet=0.80Py
Assumptions for this plot:
Figure 4.1 Summary of preliminary DSM global column strength curves where Pynet=0.8Py
Pnl is a function of the global strength prediction Pne in the DSM
formulation. Therefore, any modifications to Pne equations (Options 2 and 3) will
also influence the Pnl column prediction curve as shown for a short column in
Figure 4.2. Option 4 proposes to cap Pnl at Pynet to account for the loss in strength
61
from the reduced cross-sectional area of the column. Figure 4.3 demonstrates
how the local buckling prediction curves change when a larger hole (0.60Pynet
instead of 0.80Pynet) is assumed for the same short column. Option 5 is based on
preliminary nonlinear finite element studies shown in Figure 4.4 which suggests
Pnl decreases from Pynet before reaching the transition from the cap to the local
prediction curve. Future simulation work is planned to evaluate this transition
area with different cross-sections and hole sizes.
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
local slenderness, λl
Pn l
/Pne
Cap and transition at Pynet (Option 5)
No change (Option 1)
Cap at Pynet (Option 4)
Cap Pne at Pynet (Option 3)
Pcre=5Py, λc=0.44
Pynet=0.80Py
Assumptions for this plot:
Replace Py with Pynet in all strength and slenderness calculations (Option 2)
Figure 4.2 Summary of DSM local column strength curves, plot assumes λc=0.44 and Pynet=0.80Py
62
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
local slenderness, λl
Pn l
/Pne
Cap and transition at Pynet (Option 5)
No change (Option 1)
Cap at Pynet (Option 4)
Cap Pne at Pynet (Option 3)
Pcre=5Py
Pynet=0.60Py
Assumptions for this plot:Replace Py with Pynet in all strength and slenderness calculations (Option 2)
Figure 4.3 Summary of preliminary DSM local curve options for a short column with a large hole,
λc=0.44 and Pynet=0.60Py
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
λl=(Pne/Pcrl)
0.5
Pte
st/P
ne
Preliminary simulation results suggest a transition from Pynetto Pnl curve may be warranted
Option 4 – cap Pnl
Option 1 – No change to curve
Figure 4.4 Simulation results suggest a transition may be required from Pynet for the DSM local
strength curve
63
The influence on Pnl from a change in global column slenderness can be observed
by comparing Figure 4.4 for a short column to Figure 4.5 for a stub column
(λc=0.10). Option 2 and Option 3 are the same for a stub column because Pne is
capped at Pynet for both as shown in Figure 4.1.
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
local slenderness, λl
Pn l
/Pne
Cap and transition at Pynet (Option 5)
No change (Option 1)
Cap at Pynet (Option 4)
Cap Pne at Pynet (Option 3)
Pcre=100Py, λc=0.10
Pynet=0.80Py
Assumptions for this plot:
Replace Py with Pynet in all strength and slenderness calculations (Option 2)
Figure 4.5 Summary of preliminary DSM local column strength curves for a stub column, λc=0.10
and Pynet=0.80Py
The proposed distortional column strength curve in Figure 4.6 is independent of
the local and global curves and is proposed with a complete replacement of Py
for Pynet when calculating Pnl and λd or just a cap at Pynet (Option 3, 4 and 5).
64
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
distortional slenderness, λd or λdnet
Pnd
/Py
No change (Option 1)
Cap at Pynet (Options 3, 4, 5)
Replace Py with Pynet in strength and all slenderness calculations (Option 2)
Pynet=0.80Py
Assumptions for this plot:
Figure 4.6 Summary of preliminary DSM distortional column strength curves, plot assumes
Pynet=0.80Py
4.2 Comparison of Proposed Methods to Column Data Tested column strengths are now evaluated against the five preliminary
DSM prediction methods discussed in the previous section. Figure 4.7 to Figure
4.11 compares 78 column tests again local, distortional, and global DSM
predictions, and also denotes specimens that are predicted to fail by full yielding
of the column cross section. Figure 4.7 demonstrates that the majority of the 78
column data points are controlled by a local buckling or yielding type failure. In
general the DSM local predictions (red circles) tend to be lower than the tested
strengths (filled blue circles) for all five options. Option 1 (no change to DSM
method) does not predict the strength of the yielded (stocky) specimens
accurately. This is direct evidence that the “net section” reduction for stocky
65
specimens does exist. Option 2 (using Pynet throughout) is the only method that
is a conservative predictor for all data points, although this comes at a price of
being too conservative in some cases, particularly at high slenderness. Options 4
and 5 are better predictors of the locally controlled specimens but underpredict
strength for two stub columns with large holes. The influence of hole size on the
DSM local predictions is expressed in Figure 4.12, where it is shown that a Pynet
cap on strength is prudent (compare Option 1 to Options 2 to 5).
The DSM distortional and global predictions are conservative for all
methods as shown in Figure 4.10 and Figure 4.11, although more data points are
needed to complete a thorough comparison. Future is work is planned to
continue the evaluation of these DSM strength predictions for distortional-
controlled and global-controlled columns using nonlinear finite element
simulations.
66
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 1 - Py everywhere
λl=(Pne/Pcrl)
0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 2 - Pynet everywhere
λl=(Pne/Pcrl)
0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 3 - cap Pne, Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 4 - cap Pnl, Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
y
DSM Local PredictionLocal Controlled TestDSM Yield ControlYield Controlled Test
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 5 - transition Pnl, cap Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
y
Options 4 and 5 work well on average, but underpredict strength for two stub columns with very large web holes (Pynet=0.65 Py)
Pynet is needed to accurately predict strength
Figure 4.7 Comparison of preliminary DSM local prediction options versus tested data
67
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 1 - Py everywhere
λl=(Pne/Pcrl)
0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 2 - Pynet everywhere
λl=(Pne/Pcrl)
0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 3 - cap Pne, Pnd
λl=(Pne/Pcrl
)0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 4 - cap Pnl, Pnd
λl=(Pne/Pcrl
)0.5
Pn/P
y
DSM Local PredictionLocal Controlled TestDSM Yield ControlYield Controlled Test
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 5 - transition Pnl, cap Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
y
Figure 4.8 Comparison of preliminary DSM local prediction options versus tested data, stub columns only (λc<0.20)
68
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 1 - Py everywhere
λl=(Pne/Pcrl)
0.5
Pn/P
ne
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 2 - Pynet everywhere
λl=(Pne/Pcrl)
0.5
Pn/P
ne
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 3 - cap Pne, Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
ne
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 4 - cap Pnl, Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
ne
DSM Local PredictionLocal Controlled TestDSM Yield ControlYield Controlled Test
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 5 - transition Pnl, cap Pnd
λl=(Pne/Pcrl)
0.5
Pn/P
ne
Figure 4.9 Comparison of preliminary DSM local prediction options versus tested data normalized with Pne
69
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 1 - Py everywhere
λd=(Py/Pcrd)0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 2 - Pynet everywhere
λdnet=(Pynet/Pcrd)0.5
Pn/P
y
DSM Dist. PredictionDist. Controlled TestDSM Yield ControlYield Controlled Test
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 3 - cap Pne, Pnd
λd=(Py/Pcrd)0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 4 - cap Pnl, Pnd
λd=(Py/Pcrd)0.5
Pn/P
y
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5Option 5 - transition Pnl
, cap Pnd
λd=(Py/Pcrd)0.5
Pn/P
y
Figure 4.10 Comparison of preliminary DSM distortional strength predictions against tested column data.
70
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
λc=(Py/Pcre)0.5
Pn/P
y
Option 1 - Py everywhere
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
λcnet=(Pynet/Pcre)0.5
Pn/P
y
Option 2 - Pynet everywhere DSM Global Prediction
Global Controlled Test
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
λc=(Py/Pcre)0.5
Pn/P
y
Option 3 - cap Pne, Pnd
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
λc=(Py/Pcre)0.5
Pn/P
y
Option 4 - cap Pnl, Pnd
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
λc=(Py/Pcre)0.5
Pn/P
y
Option 5 - transition Pnl, cap Pnd
Figure 4.11 Comparison of preliminary DSM global strength predictions versus tested column data
71
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
Option 1 - Py everywhere
Anet/Ag
Pte
st/P
n
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
Option 2 - Pynet everywhere
Anet/Ag
Pte
st/P
n
Local ControlledYield Controlled
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
Option 3 - cap Pne, Pnd
Anet/Ag
Pte
st/P
n
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
Option 4 - cap Pnl, Pnd
Anet/Ag
Pte
st/P
n
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
Option 5 - transition Pnl, cap Pnd
Anet/Ag
Pte
st/P
n
Figure 4.12 Influence of reduced cross sectional area (from the hole) on test-to-predicted ratios for DSM local and yield -controlled specimens
72
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 1 - Py everywhere
λl=(Pne/Pcrl
)0.5
Pte
st/P
n
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 2 - Pynet everywhere
λl=(Pne/Pcrl
)0.5
Pte
st/P
n
Local ControlledYield Controlled
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 3 - cap Pne, Pnd
λl=(Pne/Pcrl)
0.5
Pte
st/P
n
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 4 - cap Pnl, Pnd
λl=(Pne/Pcrl)
0.5
Pte
st/P
n
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 5 - transition Pnl, cap Pnd
λl=(Pne/Pcrl)
0.5
Pte
st/P
n
Figure 4.13 Comparison of local buckling controlled test-to-predicted ratio using the preliminary DSM strength prediction options
73
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 1 - Py everywhere
λd=(Py/Pcrd)0.5
Pte
st/P
nd
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 2 - Pynet everywhere
λdnet=(Pynet/Pcrd)0.5
Pte
st/P
nd
Distortional ControlledYield Controlled
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 3 - cap Pne, Pnd
λd=(Py/Pcrd)0.5
Pte
st/P
nd
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 4 - cap Pnl, Pnd
λd=(Py/Pcrd)0.5
Pte
st/P
nd
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 5 - transition Pnl, cap Pnd
λd=(Py/Pcrd)0.5
Pte
st/P
nd
Figure 4.14 Comparison of distortional buckling controlled test-to-predicted ratio using the preliminary DSM strength prediction options
74
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 1 - Py everywhere
λc=(Py/Pcre)0.5
Pte
st/P
ne
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 2 - Pynet everywhere
λcnet=(Pynet/Pcre)0.5
Pte
st/P
ne
Global Controlled
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 3 - cap Pne, Pnd
λc=(Py/Pcre)0.5
Pte
st/P
ne
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 4 - cap Pnl, Pnd
λc=(Py/Pcre)0.5
Pte
st/P
ne
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Option 5 - transition Pnl, cap Pnd
λc=(Py/Pcre)0.5
Pte
st/P
ne
Figure 4.15 Comparison of global-controlled test-to-predicted ratios using preliminary DSM strength prediction options
75
Test-to-predicted ratio trends of the five DSM options are compared in
Figure 4.13 to Figure 4.15 for local, distortional, and global controlled specimens.
The predictions of Options 4 and 5 predictions present themselves as the best
predictors of strength in terms of following visible trends when compared to the
column test data. This is confirmed in Table 4.1, which compares the test-to-
predicted statistics for each of the five DSM options.
Table 4.1 Summary of test-to-predicted statistics for the preliminary DSM prediction options
average STDEV average STDEV average STDEV average STDEV1 use Py everywhere 1.03 0.11 1.09 0.16 1.14 0.11 0.84 0.082 use Pynet everywhere 1.17 0.08 1.23 0.13 1.27 0.10 1.04 0.123 cap Pne and Pnd 1.15 0.07 1.30 0.04* 1.16 0.10 1.03 0.114 cap Pnl and Pnd 1.07 0.08 1.15 0.10 1.16 0.10 1.01 0.115 transition Pnl and cap Pnd 1.06 0.08 1.15 0.10 1.16 0.10 1.03 0.11
* This standard deviation was calculated with only three points and is not necessarily representative of the method
Test-to-Predicted Ratio by Failure ModeDescriptionDSM
Option Local Distortional Global Yielding
4.3 Conclusions Five preliminary options are proposed here to extend the Direct Strength
Method to columns with holes. The efficacy of Options 1 and 2 were evaluated
previously in Progress Report #1, and are now compared to three new options
that represent the cumulative knowledge of the research conducted to date. It is
hypothesized that the “net section” strength reduction does exist for specimens
with low slenderness ratios (local, distortional, or global).
The modifications to the DSM global strength predictions are based on the
observation that as global slenderness approaches zero, Pn will equal Py for
columns without holes and Pynet for columns with holes. For globally-controlled
specimens with high slenderness, the influence of the hole is included in the
76
calculation of Pcre and the global strength limit of Pn=0.877Pcre is then applicable
to columns with and without holes. The motivation for the changes to the local
DSM strength predictions are best described for a stub column, first with high
local slenderness. In this case, Pnl is larger than Pcrl but much less than Py. If Pcrl
reflects the hole and the predicted Pn is less than Pynet then the axial capacity
exists to develop through to Pnl. For a stub column with low local slenderness,
the capacity Pn=Py but should be limited to Pynet for members with holes. The
same philosophy described for local buckling also applies to the proposed
modifications to the distortional-controlled DSM predictions.
It is concluded here that Option 1 (no changes to DSM) is not viable
because of its inability to predict the “net section” reduction. The most straight-
forward option for implementation into the current specification is Option 2
(Pynet everywhere) if the conservative predictions can be tolerated. Option 3 has
a conservative bias and the lowest variability which does warrant some merit.
Options 4 and 5 offer the lowest bias and low variability; these are both viable
candidates. Option 5 is the least designer-friendly, which is also an important
consideration when deciding on a method.
More exploratory work is needed to develop the final framework for this
method. The nonlinear finite element tools described in Section 3 will be an
essential tool, since they can generate more data points in slenderness ranges
where the existing column database is lacking.
77
4.4 Future Work The near term focus to bring this work to completion will be on nonlinear
finite element parameter studies like the one shown in Figure 4.4. Exploring all
regions of the DSM column curves will allow us to formulate the method to
minimize prediction bias and variability. We also plan to formally define the
general design method (COS perspective) and to develop DSM design guidelines
for specific members with specific holes (COFS perspective).
5 Future Work The research summarized in this report represents the most mature ideas,
observations, and solutions to date. Our efforts have led to the development of a
preliminary Direct Strength Method approach for columns with holes, with a
similar method for beams to come in the near future. The continued
development of a nonlinear finite element approach will be essential, since
simulation will be the primary tool for testing our DSM approaches from now
on. The mixed success of the finite element modeling in this report warrants a
step back to study in more detail the influence of residual stresses, initial
imperfections, and contact boundary conditions as they relate to column
strength. Our determined focus will also continue on the prediction of elastic
buckling of cold-formed steel members with holes, since this is the foundation of
the Direct Strength Method. An important step to bringing this method to the
design community will be the simplified procedures for calculating the critical
78
elastic buckling loads that include the influence of holes. The development of
formal elastic buckling definitions and an automated modal identification
process are also planned.
79
References ABAQUS (2004). ABAQUS/Standard Users Manual version 6.5. ABAQUS, Inc., www.abaqus.com, Providence, RI. Moen, C. and Schafer B.W. (2006). “Impact of holes on the elastic buckling of cold-formed steel columns with applications to the Direct Strength Method.” 18th Int’l Spec. Conf. on Cold-Formed Steel Structures, Orlando, FL. North American Specification (NAS). (2004). 2004 Supplement to the North American Specification for the Design of Cold-Formed Steel Structures. American Iron and Steel Institute, Washington, D.C. Oden, J. T., and J. A. C. Martins (1985). “Models and Computational Methods for Dynamic Friction Phenomena,” Computer Methods in Applied Mechanics and Engineering, vol. 52, pp. 527–634. Sarawit, A. (2006). "CUTWP Thin-walled section properties" December 2006 update <www.ce.jhu.edu/bschafer/cutwp> website referenced in July 2007.
80
Appendix A - Tensile Coupon Plastic Strain Data The true plastic stress-strain curves provided here for each specimen were input
in ABAQUS using the command *PLASTIC. Refer to Section 3.3 of this report for
details on their development. These curves were created by first averaging the
three engineering stress-strain curves for each specimen (west flange, east flange,
and web) and then transforming the average curve into true stresses and strains
with the following equations:
)1()1ln(
ootrue
otrue
εσσεε
+=+=
εtrue and σtrue are the true stress and strain and εo and σo are the engineering
stress and strain in the above equations. The tables in this appendix provide just
the plastic component of the true strain since this is what is required in
ABAQUS:
yieldtruep εεε −= , where Eyield
yield
σε =
81
Specimen 362-1-24-NH, 362-2-24-NH, 362-3-24-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=55.1 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 33.05
0.001 46.100.002 51.880.007 60.300.012 64.890.017 68.370.027 73.970.037 78.120.047 81.270.057 83.830.067 86.16
82
Specimen 362-1-24-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=57.9 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 34.2
0.001 50.00.002 56.10.007 64.40.012 68.30.017 72.00.027 78.60.037 82.50.047 86.20.057 88.70.067 91.00.077 92.90.087 94.60.097 96.20.107 97.5
83
Specimen 362-2-24-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=57.1 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 27.7
0.001 45.10.002 53.30.007 62.80.012 67.10.017 71.20.027 76.70.037 81.30.047 84.70.057 87.40.067 89.70.077 91.60.087 93.40.097 94.80.107 96.2
84
Specimen 362-3-24-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=56 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 31.7
0.001 47.80.002 53.60.007 61.10.012 64.70.017 68.40.027 74.80.037 78.60.047 82.20.057 84.60.067 86.90.077 88.80.087 90.40.097 91.90.107 93.1
85
Specimen 362-1-48-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=59.7 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 37.8
0.001 51.60.002 56.60.007 64.30.012 68.20.017 72.00.027 78.10.037 82.10.047 85.90.057 88.30.067 90.80.077 92.50.087 94.30.097 95.70.107 97.1
86
Specimen 362-2-48-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=59.2 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 35.0
0.001 50.90.002 56.50.007 64.30.012 68.30.017 72.10.027 78.20.037 82.30.047 86.10.057 88.50.067 90.90.077 92.80.087 94.50.097 96.00.107 97.3
87
Specimen 362-3-48-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=59 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 34
0.001 500.002 560.007 640.012 680.017 720.027 780.037 820.047 860.057 880.067 910.077 920.087 940.097 960.107 97
88
Specimen 362-1-48-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=58.6 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 33.2
0.001 50.10.002 56.10.007 64.20.012 68.20.017 72.00.027 78.00.037 82.20.047 85.70.057 88.40.067 90.70.077 92.70.087 94.40.097 95.80.107 97.2
89
Specimen 362-2-48-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=59.7 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 37.3
0.001 51.50.002 56.70.007 64.20.012 68.20.017 72.00.027 78.10.037 82.40.047 85.80.057 88.50.067 90.80.077 92.70.087 94.40.097 95.90.107 97.2
90
Specimen 362-3-48-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=58.3 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 37.5
0.001 50.60.002 55.60.007 62.90.012 67.10.017 70.90.027 76.50.037 80.80.047 84.10.057 86.80.067 89.10.077 90.90.087 92.60.097 94.10.107 95.4
91
Specimen 600-1-24-NH, 600-2-24-NH, 600-3-24-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=58.7 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 37.5
0.001 54.70.002 58.30.007 60.00.012 61.50.017 64.00.027 70.20.037 74.40.047 77.50.057 80.00.067 81.90.077 83.50.087 84.90.097 86.10.107 87.2
92
Specimen 600-1-24-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=61.9 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 35.0
0.001 58.90.002 61.40.007 62.30.012 62.90.017 63.60.027 69.40.037 74.40.047 77.80.057 80.50.067 82.60.077 84.40.087 85.90.097 87.20.107 88.4
93
Specimen 600-2-24-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=58.4 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 33.0
0.001 54.00.002 57.20.007 58.90.012 61.50.017 64.40.027 70.50.037 74.80.047 78.00.057 80.40.067 82.40.077 84.10.087 85.50.097 86.70.107 87.9
94
Specimen 600-3-24-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=60.1 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 34.9
0.001 57.50.002 60.10.007 60.80.012 61.80.017 63.60.027 69.80.037 74.10.047 77.60.057 79.90.067 82.00.077 83.70.087 85.10.097 86.50.107 87.6
95
Specimen 600-1-48-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=60.1 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 44.4
0.001 58.90.002 60.30.007 61.00.012 62.00.017 64.80.027 70.60.037 75.00.047 78.30.057 80.80.067 82.80.077 84.50.087 86.00.097 87.30.107 88.5
96
Specimen 600-2-48-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=63.4 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 41.9
0.001 62.10.002 63.10.007 63.60.012 64.00.017 64.50.027 69.80.037 75.00.047 78.50.057 81.20.067 83.50.077 85.30.087 87.00.097 88.40.107 89.6
97
Specimen 600-3-48-NH
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=61.2 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 36.0
0.001 55.60.002 60.00.007 61.70.012 62.00.017 62.50.027 68.50.037 73.40.047 76.90.057 79.40.067 81.60.077 83.30.087 84.90.097 86.20.107 87.4
98
Specimen 600-1-48-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=61.4 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 37.5
0.001 57.70.002 60.90.007 61.90.012 62.40.017 64.30.027 70.40.037 75.00.047 78.40.057 81.00.067 83.10.077 84.80.087 86.30.097 87.70.107 88.9
99
Specimen 600-2-48-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=62 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 37.4
0.001 57.60.002 61.60.007 62.60.012 63.30.017 64.10.027 69.40.037 74.20.047 77.80.057 80.60.067 82.90.077 84.80.087 86.40.097 87.80.107 89.1
100
Specimen 600-3-48-H
0 0.02 0.04 0.06 0.08 0.1 0.120
10
20
30
40
50
60
70
80
90
100
Avg. Yield Stress=61.5 ksi
true plastic strain
true
stre
ss, k
si
true plastic strain, εp
true stress, σtrue
ksi0 32.4
0.001 57.10.002 61.30.007 62.20.012 62.60.017 64.50.027 70.40.037 75.20.047 78.60.057 81.10.067 83.20.077 84.90.087 86.50.097 87.80.107 89.0
101
Appendix B - Corrections to Progress Report #3 Some of the values in Progress Report #3, Table 2.4 were reported in error. The
corrected values are highlighted in the table below.
X S1 S2 X F1 F2 X F1 F2 X F1 F2 X F1 F2 X F1 F2
in. degrees degrees in. degrees degrees in. degrees degrees in. degrees degrees in. degrees degrees in. degrees degrees362-1-24-NH 12 12.767 8.367 6 82.600 84.500 12 86.033 86.833 18 84.533 87.000362-2-24-NH 12 11.367 11.567 6 86.800 84.800 12 87.600 85.467 18 86.400 83.700362-3-24-NH 12 9.567 9.433 6 85.700 85.000 12 86.300 85.400 18 85.600 83.000362-1-24-H 12 11.130 10.930 6 83.200 83.970 12 87.600 85.600 18 84.330 86.430362-2-24-H 12 4.367 10.267 6 86.000 85.133 12 86.333 85.167 18 84.400 84.500362-3-24-H 12 10.533 10.833 6 85.200 86.333 12 87.700 86.133 18 87.667 89.033362-1-48-NH 12 7.800 10.100 12 85.100 85.600 18 84.300 85.000 24 85.000 85.600 30 84.000 85.200 36 85.300 85.700362-2-48-NH 12 8.000 10.800 12 85.500 84.900 18 84.800 85.100 24 84.200 84.600 30 84.800 85.300 36 85.200 84.900362-3-48-NH 12 9.100 12.200 12 86.900 84.000 18 85.800 83.900 24 85.300 84.100 30 86.400 83.400 36 86.100 83.700362-1-48-H 12 8.500 9.800 12 86.500 84.800 18 86.600 85.000 24 85.600 84.200 30 85.500 85.100 36 86.400 84.400362-2-48-H 12 8.300 11.200 12 86.800 84.800 18 86.500 84.200 24 85.600 83.800 30 85.500 84.100 36 86.700 83.800362-3-48-H 12 9.700 7.300 12 85.300 85.200 18 84.700 86.100 24 84.100 85.300 30 84.400 84.700 36 85.200 85.000600-1-24-NH 24 1.567 2.133 6 90.567 92.033 12 92.467 93.733 18 91.433 93.767600-2-24-NH 24 1.733 2.333 6 91.000 92.033 12 91.167 94.067 18 91.467 93.333600-3-24-NH 24 -2.167 3.500 6 93.700 89.767 12 94.067 91.033 18 92.733 89.667600-1-24-H 24 0.967 2.033 6 89.000 91.000 12 90.400 92.267 18 91.200 92.600600-2-24-H 24 1.800 1.100 6 94.433 90.900 12 93.233 88.733 18 91.967 89.000600-3-24-H 24 0.100 4.100 6 93.500 90.000 12 93.300 89.300 18 90.100 86.300600-1-48-NH 24 0.167 1.400 12 91.033 92.933 18 90.833 92.700 24 90.600 92.800 30 91.333 92.900 36 91.667 93.200600-2-48-NH 24 2.000 2.367 12 90.767 91.900 18 90.233 92.300 24 89.900 91.867 30 90.967 92.000 36 91.467 92.767600-3-48-NH 24 2.600 2.300 12 90.000 92.100 18 89.200 91.900 24 90.000 92.100 30 90.700 92.600 36 90.900 92.500600-1-48-H 24 2.533 2.100 12 90.933 92.167 18 91.000 92.767 24 90.000 92.633 30 91.000 92.000 36 91.100 92.967600-2-48-H 24 2.400 1.000 12 89.000 90.700 18 89.200 91.000 24 88.900 91.200 30 89.600 91.600 36 90.200 92.200600-3-48-H 24 0.667 3.633 12 93.067 89.400 18 93.000 89.500 24 92.300 89.433 30 93.467 89.900 36 93.467 89.600
NOTE: X is the longitudinal distance from the south end of the specimen
Specimen
Appendix C - 102
Appendix C Predicting Manufacturing Residual Stresses and Plastic Strains in Cold-Formed Steel Structural Members
C.1 Introduction
Thin cold-formed steel members begin as thick molten hot steel slabs. The
slabs are hot-rolled and then cold-reduced before coiling and shipment to
manufacturing plants. Once at a plant, the sheet is unwound through a
production line and plastically folded to form the final shape of a structural
member. This manufacturing process imparts residual stresses and plastic
strains through the sheet thickness that influence the load-displacement response
and ultimate strength of cold-formed steel members.
A general method for predicting the manufacturing residual stresses and
plastic strains is proposed in this appendix. The procedure is founded on
common industry manufacturing practices and basic physical assumptions. A
primary motivation for the development of this method is to define the initial
state of a cold-formed steel member for use in a nonlinear finite element analysis.
The method is intended to be accessible to a wide audience including
manufacturers, design engineers, and the academic community.
Common sheet steel manufacturing and cold-forming practices are
introduced to identify the dominant processes influencing residual stresses and
plastic strains. The structural mechanics employed are defined for every step,
and the prediction method is verified with measured residual strains from
Appendix C - 103
experimental studies. A procedure for implementing the predicted residual
stresses and strains into a nonlinear finite element model is also discussed.
C.1.1 Manufacturing Process
C.1.2 Sheet Coil Production
Steel sheet coils used in cold-formed steel manufacturing start as
rectangular steel slabs (pictured in Figure C.1) as thick as 200 mm (8 in.) heated
to 1200ºC (2200ºF) (US Steel 1985).
Figure C.1 A rectangular steel slab is heated in preparation for the roughing mill which will reduce the slab thickness by up to 80 percent. (courtesy Mittal Steel)
Each slab is moved through a roughing mill made up of four sets of steel rollers
(roughing stands) to reduce the thickness to a minimum of 40 mm (1.57 in.). The
steel sheet then enters a finishing mill where the thickness is again reduced, this
time to a minimum of approximately 2 mm (0.079 in.). The sheet is cooled to
540ºC (1000ºF) on a long, flat conveyor belt with water jets and then rolled into
Appendix C - 104
coils with outer diameters ranging from 1220 mm (48 in.) to 1830 mm (72 in.)
(Figure C.2).
Figure C.2 After the finishing mill, the sheet is coiled hot and then cooled before pickling and cold-reducing. (photo courtesy Don Allen, CFSEI)
As the coils cool to room temperature, scale (oxide or rust) forms on the
surface of the sheet which can inhibit the zinc galvanizing bond and shorten the
life of roll-former dies. This scale is removed with a process called pickling,
where the cold coil is unrolled, flattened with a tension leveler, and then moved
through a hydrochloric bath. Depending upon the final desired thickness, the
steel sheet thickness is then further reduced by cold-rolling. This process applies
intense pressure through the thickness of the sheet with a series of rollers, and
increases the steel temperature by up to 200ºC (400ºF) from friction between the
sheet and the rollers. This cold-working decreases the steel’s ductility and
Appendix C - 105
increases its stiffness in the rolling direction. The steel sheet is placed through a
continuous hot-dip galvanizing line where it is annealed and cleaned before
entering a bath of molten zinc (Figure C.3).
Figure C.3 The steel sheet is annealed and cleaned before entering a bath of molten zinc along the finishing line. (courtesy Mittal Steel)
Annealing restores ductility to the steel that was lost in the cold-reducing
process. The sheet is passed through a dryer to solidify the zinc coating and is
then recoiled and moved to storage until it is transported to a manufacturer for
use in the production of cold-formed steel structural members (Figure C.4).
Appendix C - 106
Figure C.4 Sheet coil storage (courtesy Mittal Steel)
C.1.3 Cold-Forming Methods
C.1.3.1 Continuous Roll-Forming
Once the steel sheet coil arrives at the manufacturer, it is cut into smaller
width coils by unrolling the sheet through a coil slitter. The sheet widths are
determined based on the dimensions of the member cross sections to be
produced. The individual sheets are rolled back into coils and moved to the
production line. A typical roll-forming manufacturing line consists of a coil
spindle, a set of steel forming dies adjusted for a specific cross section (i.e., Cee,
angle, Zee, etc.), and hydraulic shears that cut the structural members to their
final lengths.
C.1.3.2 Press Braking
Discrete lengths of sheet steel can be cold-formed into a specific cross
section by press-braking. Each individual steel sheet is cut to the final member
length and then oriented on the lower bed of the press-braking machine. A stiff
overhead die is lowered to cold-bend the sheet to a specific cross-sectional
Appendix C - 107
geometry. Any residual curvature of the sheet from coiling is removed before
press-braking with a tension leveler or equivalent flattening system.
C.1.4 Sources of Residual Stresses
The manufacturing process described in Section C.1.1 identifies several
potential sources of residual stresses and strains in cold-formed steel structural
members. Manufacturing of the sheet coils, coiling and uncoiling, heating and
cooling, and cold-forming of the cross section (both cold-rolling and press
braking) are all contributors to the initial stress state. In the simplified prediction
method proposed here, the coiling and uncoiling of the sheet steel after
annealing and the cold-forming of the cross section are assumed to be the
dominate contributors to residual stresses and plastic strains. This assumption is
supported by existing theoretical work (Hill 1983, Ingvarsson 1975, Dat 1980,
Kato and Aoki 1978, Quach et al 2006a) and by recent computational results
(Quach et al 2006a, Quach et al 2006b).
C.1.5 Types of Residual Stress Distributions
Three types of through thickness residual stresses are discussed in the
presentation of this prediction method: bending, membrane, and yield-release
residual stresses. The shape of each stress distribution is provided in Figure C.5.
Laboratory measurements of released surface strains have confirmed the
presence of membrane and bending strains (refer to Section C.1.4). Yield-release
stresses are a result of plastic bending followed by elastic springback. They
Appendix C - 108
induce zero net bending and axial force through the thickness and are difficult to
measure experimentally. Yield-release residual stresses are predicted
theoretically by Hill (1983) and Shanley (1957) and have been measured in
thicker steel sheets by Key and Hancock (1993).
Bending Membrane Yield-Release
Sheet thickness
Figure C.5 Types of through-thickness residual stresses
C.1.6 Measured Residual Stress Data
Several experimental studies have measured the longitudinal membrane
and bending residual strains in roll-formed and press braked steel members
including Ingvarsson (1975), Dat (1980), Weng and Peköz (1990), de M. Batista
and Rodrigues (1992), Key and Hancock (1993), Kwon (1992), and Bernard (1993).
The strains are measured with strain gauges at the outer fibers of thin strips or
coupons that are cut from the cross section of a completed member. The strip
curvature is translated from outer fiber strains into membrane and bending
residual stress distributions described in Figure C.5. Strains are converted to
stresses by multiplying by the steel yield stress σyield. This residual strain data
was originally collected and statistically summarized by Schafer and Pekoz
Appendix C - 109
(1998) for unstiffened and stiffened cold-formed steel cross-sectional elements.
The mean and standard deviations in Table C.1 and Table C.2 reflect most of this
original data (except for Ingvarsson and Key and Hancock) in addition to more
recent measurements on press-braked and roll-formed Cees published by Young
(1997) and Abdel-Rahman and Sivakumaran (1997). The data set is organized
into two groups: flats (Cee flanges, lips, and webs as well as decking) and cold-
formed corners. Positive membrane stresses are tensile stresses and positive
bending stresses cause tension at y=-t/2 (see Figure C.7 for coordinate system).
The entire residual stress data set is provided in Appendix A.
Table C.1 Residual stresses in roll-formed structural members
Mean STDEV Mean STDEVCorners 5.7 10.1 32.0 23.8 23Flats 1.8 10.7 25.2 20.7 120
ElementResidual stress as %σyield No. of
SamplesMembrane Bending
Table C.2 Residual stresses in press-braked structural members
Mean STDEV Mean STDEVCorners 4.3 7.4 28.7 19.8 11Flats 0.2 2.8 13.2 19.7 108
ElementResidual stress as %σyield No. of
SamplesMembrane Bending
The statistics demonstrate that the average membrane stresses are small
relative to the steel yield stress for both roll-formed and press-braked members.
Bending residual stresses in the flats are lower in press-braked members than in
roll-formed members. It is hypothesized that this difference is caused by the
flattening of the steel sheet before press-braking which eliminates the influence
Appendix C - 110
of residual coiling curvature present in roll-formed structural members. Figure
C.6 summarizes the press-braked and roll-formed residual bending stress data as
histograms. This data is used in Section C.3.5 to validate the proposed prediction
method for roll-formed structural members.
Roll-formed
Flats
Press-braked
Corners
-0.5 -0.25 0 0.25 0.5 0.75 10
10
20
30
40
Obs
erva
tions
-0.5 -0.25 0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
P(X
<x)
bending residual stress (σy/σyield)-0.5 -0.25 0 0.25 0.5 0.75 10
10
20
30
40
Obs
erva
tions
-0.5 -0.25 0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
P(X
<x)
bending residual stress (σy/σyield)
-0.5 -0.25 0 0.25 0.5 0.75 10
2
4
6
8
10
Obs
erva
tions
bending residual stress (σy/σyield)-0.5 -0.25 0 0.25 0.5 0.75 10
2
4
6
8
10
Obs
erva
tions
bending residual stress (σy/σyield)
Figure C.6 Histograms and approximate cumulative distribution functions for bending residual stress measurements in corner and flat cross sectional elements
Appendix C - 111
C.2 Assumptions and Definitions
C.2.1 Stress-Strain Coordinate System and Notation
The stress-strain coordinate system and geometric notation used in the
forthcoming derivations are defined in Figure C.7. Although the forming
direction is depicted for the roll-forming method, the direction of the sheet as it
comes off the coil is also applicable to press-braked members. The x-axis is
referred to as the transverse direction and the z-axis as the longitudinal direction.
Forming direction A
A
sheet steel coil
roller dies
xy
z
z
xy
rz
t
Section A-A
Elevation View
rx
Figure C.7 Stress-strain coordinate system with relation to the manufacturing process
Appendix C - 112
An elastic-plastic material model is assumed, where σyield is the yield stress of the
steel sheet before it has been cold-formed. The yield strain εyield and σyield are
related by the steel elastic modulus E:
Eyield
yield =εσ
C.2.2 Calculating Equivalent Plastic Strains
Equivalent plastic strains are used in nonlinear finite element programs
such as ABAQUS to define the initial material state of a member that has
previously experienced plastic deformation (ABAQUS 2004). The multi-axial
engineering plastic strains at a point (εx, εy, εz) are converted to true principal
strains and then mapped to a uni-axial stress-strain curve from a tensile coupon
test using the von Mises yield criterion equation (Ugural and Fenster 2003):
( ) ( ) ( ) ( )[ ] 2/1213
232
2213
2 εεεεεεε −+−−−=yp
where )1ln(1 xεε += , )1ln(2 yεε += , )1ln(3 zεε +=
Equivalent plastic strain, εp, accounts for increased yield stress and lower
ductility in regions of a structural member that have been plastically deformed.
The equations for εp are written in terms of true strains to accommodate the large
deformation material models employed in nonlinear finite element analyses.
Figure C.8 demonstrates how εp is expressed in a true stress-strain diagram.
Appendix C - 113
pε
trueσ
trueε
yieldσ
Cold-formed steel
Virgin steel
Figure C.8 Definition of equivalent plastic strain as related to a uniaxial tensile coupon test
C.2.3 Governing Assumptions The following assumptions are employed to derive the prediction method:
1. Plane sections remain plane before and after coiling and cold-forming.
If plane sections did not remain plane, the strains through the sheet thickness would no longer be proportional to y and a relationship between the radius of curvature and strain could not be defined. For a detailed discussion of residual stresses in bending refer to Shanley (1957).
2. The sheet thickness remains constant before and after cold-forming.
This assumption is supported by R. Hill’s sheet bending theory for thin sheet bending (Hill 1983), which demonstrates that the sheet thickness remains constant before and after forming if no tension is applied in combination with the radial forming. Dat’s thickness measurements of cold-formed channel cross sections found that corner thicknesses were consistently five percent less than the web and flange thicknesses (Dat 1980). Thus, modest thinning does occur but is ignored here.
3. The sheet neutral axis remains constant before and after cold-forming.
Appendix C - 114
R. Hill’s sheet bending theory requires a shift in the through-thickness neutral axis towards the inside of the corner as the sheet plastifies (Hill 1983). This shift is calculated as six percent of the sheet thickness (t) when assuming a radius of 2t. A neutral axis shift of similar magnitude is observed in the nonlinear finite element model results for thin press-braked steel sheets presented in Quach et. al (2006b). This small shift is ignored here to simplify the derivations. 4. The steel stress-strain curve is elastic-perfectly plastic.
The simplification of the steel stress-strain behavior allows for an accessible hand calculation of residual stresses and strains. More detailed stress-strain models that include isotropic and kinematic hardening are available but require some minimum level of computational effort. 5. Plane strain behavior causes longitudinal residual stresses from cold-
forming in the transverse direction. The plane strain assumption implies that each point in the x-y plane will remain in the x-y plane after forming of the corner. 6. Coiling and uncoiling of the steel sheet after annealing produce residual stresses and plastic strains in roll-formed structural members. The residual curvature from coiling is assumed to be locked into the roll-formed member; the possibility of flattening before rolling is ignored. Also, the uncoiling, coiling, slitting, and recoiling of the sheet after the coil arrives at the manufacturer is ignored. It is assumed if the coil is uncoiled and coiled again with the same spindle rotation direction (both clockwise or both counterclockwise) then the residual curvature and stresses are not affected. 7. The steel sheet is fed from the top of the coil (convex residual curvature) into the roll-forming bed (Figure C.9a).
This assumption is consistent with measured bending residual stress data (see Section C.1.4) and manufacturing setups suggested by roll-forming equipment suppliers (www.bradburygroup.com). The author did observe the alternative setup in Figure C.9b (sheet steel unrolling from the bottom of the coil) at a roll-forming plant, suggesting that the direction of uncoiling is a source of variability in measured residual stress data.
Appendix C - 115
Sheet has residual CONCAVE curvature coming off the coil
Sheet has residual CONVEX curvature coming off the coil
(a)
(b)
Roll-forming bed
Figure C.9 Roll-forming setup with sheet coil fed from the (a) top of the coil and (b) bottom of coil. The orientation of the coil with reference to the roll-forming bed influences the direction of the
coiling residual stresses.
8. Coiling, uncoiling, and flattening of the steel sheet after annealing produce residual stresses and plastic strains in press-braked structural members. Removal of the residual coiling curvature (for example, flattening with a tension leveler) is common industry practice for press-braked members. The uncoiling and coiling during the slitting of the large width coil is ignored here as it was for roll-formed members. 9. Membrane residual stresses are zero.
Membrane residual stresses have been measured by several researchers ( Ingvarsson 1975, Dat 1980, Weng and Peköz 1990, de M. Batista and Rodrigues 1992, Key and Hancock 1993, Kwon 1992), and Bernard 1993), although the magnitudes are commonly small relative to bending residual stresses. A physical cause of these stresses has not been pinpointed, although possible sources include the neutral axis shift associated with plastic deformation and measurement error. Membrane residual stresses are ignored here. 10. Residual sheet stresses in the y (thickness) direction are zero.
R. Hill’s sheet bending theory predicts a maximum stress of -0.10σyield (compression) in the y direction during corner cold-forming at the neutral axis of the sheet. These stresses are assumed to reduce to zero after release of the imposed radial
Appendix C - 116
displacement and any residual stress in the y direction is not considered in the predictive model. 11. The full residual curvature from coiling is locked into the final structural member after cold-forming. This assumption is supported by the fact that typically the weak axis bending stiffness of the structural member is more than 2000 times the bending stiffness of the individual sheet. For thicker sheets originating from the inner core of the coil, the curvature has been observed to cause a permanent bow of the final member. Manufacturing tolerances (out-of-straightness limit of L/384) typically prevent these members from entering into service.
C.3 Roll-formed structural members - predicting residual stresses and plastic strains
The prediction method proposed here assumes that two separate
manufacturing processes contribute to the through-thickness residual stresses in
roll-formed members: (1) sheet coiling, uncoiling, and flattening and (2) cross
section roll-forming. Algebraic equations for predicting the through-thickness
residual stress and equivalent plastic strains in corners and flats are derived and
then summarized in flowcharts (see Figure C.12 and Figure C.13). Predictions
are compared to the measured residual strains discussed in Section C.1.6.
Residual stress and strain predictions for common industry sheet thicknesses and
yield stresses are provided for quick reference at the end of this section.
C.3.1 Sheet Coiling, Uncoiling, and Flattening
C.3.1.1 Residual Stresses - Sheet Coiling, Uncoiling, and Flattening
The coiling of the sheet steel after annealing may cause plastic
deformation through the sheet thickness depending upon the sheet thickness t,
Appendix C - 117
radial location of the sheet in the coil rx, and steel yield stress σyield. The
prediction equations are derived with basic beam bending mechanics and
employ the well known relationship between strain and curvature:
x
z
ry1
=ε
rx is the radial location of the sheet within the coil as defined in Figure C.7, and εz
is the engineering strain through the thickness y in the coiling direction z. y
varies from -t/2 to t/2, where t is the sheet thickness. For portions of the sheet
that are near the outside of the coil, yielding of the steel will not occur.
Substituting εz=εyield and y=t/2 (outer fiber strain) into the strain-curvature
relationship defines the radial threshold between elastic and fully plastic
deformation:
yieldx
trε2
≤ yielding of steel sheet occurs
yieldx
trε2
> steel sheet remains elastic
The radial yield threshold for common sheet thicknesses with σyield =345 MPa (50
ksi) is calculated in Table C.3 and demonstrates that longitudinal residual
stresses from coiling are more likely to occur as sheet thickness increases. Also
note that as the steel yield strain (stress) increases, more of the sheet in the coil
remains elastic. This means that coiling residual stresses are less likely to exist in
higher strength steels.
Appendix C - 118
Table C.3 Coil radius yielding threshold for industry standard sheet thicknesses
trx
(yielding threshold)
mm mm0.84 ≤2461.09 ≤3231.37 ≤3991.73 ≤5112.46 ≤726
εyield =345 MPa /203395 MPa = 0.0017
trx
(yielding threshold)
in. in.0.033 ≤9.70.043 ≤12.70.054 ≤15.70.068 ≤20.10.097 ≤28.6
εyield =50 ksi /29500 ksi = 0.0017
As-shipped outer coil radii range from 610 mm (24 in.) for thicker sheet steel to
915 mm (36 in.) for thinner sheet steel to control the transportation weight. Inner
coil radii range from 255 mm to 305 mm (20 in. to 24 in.) as dictated by the
equipment at the manufacturing plants.
For sheet steel where the coil radius rx is smaller than the yielding
threshold, the elastic core c through the thickness is defined as:
t
+εyield
-εyield
c z
y
trc yieldx ≤= ε2 The steel is assumed to behave as an elastic-perfectly plastic material, and
therefore the stresses through the thickness after coiling are:
Appendix C - 119
+σyield
-σyield
y
zc
yieldσ+ 22tyc
≤≤
xryE−
22cyc
≤≤−
yieldσ− 22cyt
−≤≤−
=),( xcoilz ryσ
As the sheet is uncoiled, the imposed radial deformation on the sheet is removed
and elastic uncoiling occurs. The uncoiling results in a change in through-
thickness stress:
),( xuncoilz ryσ
z
y
[ ]I
yrMry xcoilx
xuncoilz
)(),( −=σ
Appendix C - 120
where ( ) ⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛= 2
2
31
2)( yieldxyieldx
coilx rtrM εσ , 31
121 tI ⋅⋅=
The portion of the uncoiled sheet that experiences plastic deformation maintains
a permanent radius of curvature:
EIrM
r
rrx
coilx
x
xx
uncoil
)(11)(
−=
On the coil (R)Residual curvature after uncoiling (Runcoil)
Flattened as sheet enters the roll-formers
Change in curvature locks in bending residual stresses in final member
DETAIL A
DETAIL A
Figure C.10 Bending residual stresses exist in roll-formed members from residual coiling curvature
Figure C.10 demonstrates that the curved sheet is pressed flat as it enters the roll-
forming line, locking in bending residual stresses:
),( xflatten
z ryσz
y
Appendix C - 121
IyrMry x
flattenx
xflatten
z
)(),( =σ where ⎟⎟⎠
⎞⎜⎜⎝
⎛=
)(1)(
xuncoil
xx
flattenx rr
EIrM
This type of residual stress does not exist in press-braked members since the steel
sheet is mechanically leveled by yielding before cross section cold-forming (See
Section C.1.3.2). The total through-thickness longitudinal residual stresses from
coiling, uncoiling, and flattening in roll-formed members are then:
( )xz ry,σ+ + =
y
z
Coiling Uncoiling Flattening
+σyield
-σyield
y
zc ),( xflatten
z ryσ z
y
),( xuncoilz ryσ
z
y
),(),(),(),( xzxflatten
zxuncoilzx
coilz ryryryry σσσσ =++ , yieldxz ry σσ ≤),(
C.3.1.2 Equivalent Plastic Strain - Sheet Coiling
Equivalent plastic strain accumulates during the coiling of the sheet steel
but does not occur during uncoiling or flattening of the sheet. The plastic strain
distribution through the sheet thickness at any radial location rx less than the
elastic-plastic threshold is:
Appendix C - 122
c z
y
),( xpz ryε
=),( xpz ryε
2cy −≤
2cy ≥yield
xry ε−
0 otherwise
yieldxry ε−
Converting εz to true strain:
)1ln(,
pztruez
εε +=
The assumption of constant sheet thickness leads to εy,true=0 and the plane strain
conditions leads to εx,true=0. εz,true is thus the only strain component that
contributes to the equivalent plastic strain. The Cartesian coordinate system is
equivalent to the principal axes for this problem, resulting in the following
principal strains:
0,1 == truexεε , 0,2 == trueyεε , )1ln(,3pztruez εεε +==
Substituting for the principal strains and simplifying leads to an equation for
through-thickness equivalent plastic strain from coiling and uncoiling:
Appendix C - 123
y
c),( x
coilingp ryε
( )),(1ln32),( x
pzx
coilingp ryry εε +=
C.3.2 Cross Section Roll-Forming
A set of algebraic equations are derived here to predict the transverse and
longitudinal residual stresses and plastic strains created by roll-forming of a
cross section. These roll-forming stresses and strains are assumed to exist only at
the location of the formed corners and should be added to the coiling stresses
and plastic strains to obtain a complete prediction of the initial state of the
member cross section. The primary variable influencing the roll-forming
residual stresses and strains is the steel yield stress σyield.
C.3.2.1 Residual Stresses - Cross Section Roll-Forming
The forming of a rounded corner in a thin steel sheet is assumed to occur with
a gradual decrease in the radius of curvature as depicted in Figure C.11.
Appendix C - 124
Radius decreases as corner is formed
Figure C.11 Cold-forming of the steel sheet occurs as bend radius decreases
The engineering strain in the steel sheet, εx, and the bend radius, rz, are
related for both small and large deformations with the well-established equation
for beam bending:
yrx
z
ε=
1
This geometric relationship is valid for elastic and plastic bending of the steel
sheet. For the small corner radii common in the cold-formed steel industry (rz
=2t), the steel sheet yields through its thickness during the cold-forming process.
This is demonstrated by calculating the magnitude of the elastic core c associated
with the manufactured corner radius rz:
t
+εyield
-εyield
c x
y
Appendix C - 125
zyieldz rrc ε2)( =
When substituting 2t for rz and using the following assumption for εyield:
0017.06.203
345===
GPaMPa
Eyield
yield
σε
c is calculated as 0.6% of the sheet thickness implying full yielding through the
thickness. Even for radii up to 8t, it can be calculated that the majority of the
sheet cross section has yielded. Assuming an elastic-perfectly plastic stress-
strain curve, the steel sheet will reach the following fully plastic stress state as the
corner approaches its final manufactured radius:
-σyield
+σyield
x
y
yieldbendx y σσ −=)(
02
≤≤− yt=)(ybend
xσ
20 ty ≤≤
yieldbendx y σσ +=)(
After the sheet becomes fully plastic through its thickness, the engineering strain
continues to increase as the radius decreases. When the final bend radius is
Appendix C - 126
reached and the imposed radial displacement is removed, an elastic springback
occurs that elastically unloads the corner. The change in stress through the
thickness from this elastic rebound is described by the following equation:
+σyield
-σyield
t/2 Fp
Fp
x
y
( )I
yMybendzrebound
x
−=)(σ , 31
121 tI ⋅⋅=
where 2
tFM Pbendz = ,
21 t
F yieldP
⋅⋅=
σ
After substitution and simplification the equation becomes:
ty
y yieldreboundx
σσ
3)( = ,
22tyt
≤≤−
The final transverse stress state is then the summation of the fully plastic stress
distribution through the thickness and the unloading stress from the elastic
springback of the corner:
Appendix C - 127
+ =
+1.5σyield
-1.5σyield-0.5σyield
+σyield
-σyield
-σyield
+σyield
+0.5σyield
x
y
)()()( yyy xreboundx
bendx σσσ =+ ,
22tyt
≤≤−
σx is the approximate transverse residual stress through the thickness from the
cold-forming of the corner. This stress is nonlinear through the thickness and
self-equilibrating, meaning that axial and bending sectional forces are absent in
the x- direction after forming.
The transverse residual stresses will also have components in longitudinal
direction if plane strain conditions are assumed. The plane strain assumption
implies that each point in the x-y plane will remain in the x-y plane after forming
of the corner. A relationship between the stresses in the transverse and
longitudinal directions develops from this assumption:
xz νσσ =
The Poisson’s ratio, ν, is commonly assumed as 0.30 for steel deformed elastically
and 0.50 for steel deformed plastically. The longitudinal residual stresses
through the thickness, σz , are determined based on these assumptions:
Appendix C - 128
-0.05σyield
+0.05σyield
+0.50σyield-0.50σyield+ =
+1.5σyield
-1.5σyield
-σyield
+σyield
0.50 0.30 z
y
)()()( yyy zreboundxelastic
bendxplastic σσνσν =+ ,
22tyt
≤≤−
σz is self-equilibrating for axial force through the thickness but causes a residual
longitudinal moment that can be observed as residual strain in laboratory
measurements.
C.3.2.2 Equivalent Plastic Strain - Cross Section Roll-Forming
Transverse plastic strains occur as the corner is cold-formed:
x
y
z
px r
y=ε
Appendix C - 129
The equivalent plastic strain associated with the cold-forming of the corner is
calculated with the von Mises yield criterion method described in Section C.2.2.
Converting εx to true strain:
)1ln(,
px
p
truexεε +=
The assumption of constant sheet thickness before and after loading leads to
εy,true=0 and the plane strain conditions leads to εz,true=0. εx,true is thus the only
strain component that contributes to the equivalent plastic strain. The Cartesian
coordinate system is equivalent to the principal axes for this problem, resulting
in the following principal strains:
)1ln(,1pxtruex εεε +== , 0,2 == trueyεε , 0,3 == truezεε
Substituting for the principal strains and simplifying leads to an equation for
equivalent plastic strain at a cold-formed corner:
y
)1ln(32),(
zz
bendp r
yry +=ε , 22tyt
≤≤−
It should be noted that in this plastic strain distribution, the infinitesimal elastic
core at the center of the sheet is ignored.
Appendix C - 130
C.3.3 Prediction Method Summary
Figure C.12 and Figure C.13 summarize the residual stress and strain
prediction method for roll-formed members.
Start
Yielding on the Coil?
Flat or Corner?
Yielding on the Coil?End
Flat Corner
Yes
No
Yes
No
End
End
yieldx
trε2
≤
yieldx
trε2
>
Yes, yields on coil
No, remains elastic
yieldx
trε2
≤
yieldx
trε2
>
Yes, yields on coil
No, remains elastic
[ ]I
yrMry xcoilx
xuncoilz
)(),( −=σ
( ) ⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛= 2
2
31
2)( yieldxyieldx
coilx rtrM εσ
31121 tI ⋅⋅=
EIrM
r
rrx
coilx
x
xx
uncoil
)(11)(
−=
IyrMry x
flattenx
xflatten
z
)(),( =σ
⎟⎟⎠
⎞⎜⎜⎝
⎛=
)(1)(
xuncoil
xx
flattenx rr
EIrM
yieldσ+ 22tyc
≤≤
xryE−
22cyc
≤≤−
yieldσ−22cyt
−≤≤−
=),( xcoilz ryσ
yieldσ+ 22tyc
≤≤
xryE−
22cyc
≤≤−
yieldσ−22cyt
−≤≤−
=),( xcoilz ryσSheet Coiling
Sheet Uncoiling
Sheet Flattening
Corner Bending
Corner Rebound
Sheet Coiling
Sheet Uncoiling
Sheet Flattening
)()( yy bendxplastic
bendz σνσ =
ty
y yieldreboundx
σσ
3)( = 22
tyt≤≤−
No residual stresses!
yieldbendx y σσ −=)(
02
≤≤− yt=)(ybend
xσ 20 ty ≤≤
yieldbendx y σσ +=)(
yieldbendx y σσ −=)(
02
≤≤− yt=)(ybend
xσ 20 ty ≤≤
yieldbendx y σσ +=)(
50.0=plasticν
)()( yy reboundxelastic
reboundz σνσ = 30.0=plasticν
trc yieldx ≤= ε2
Figure C.12 Flowchart summarizing the prediction method for residual stresses in roll-formed members
Appendix C - 131
Start
Yielding on the Coil?
Flat or Corner?
Yielding on the Coil?End
Flat Corner
Yes
No
Yes
No
End
End
yieldx
trε2
≤
yieldx
trε2
>
Yes, yields on coil
No, remains elastic
yieldx
trε2
≤
yieldx
trε2
>
Yes, yields on coil
No, remains elastic
Sheet Coiling
Corner Bending
Sheet Coiling
No equivalent plastic strains!
( )),(1ln32),( x
pzx
coilingp ryry εε +=
=),( xpz ryε 2
cy −≤
2cy ≥yield
xry ε−
0 otherwise
yieldxry ε−
)1ln(32),(
zz
bendp r
yry +=ε22tyt
≤≤−
,
trc yieldx ≤= ε2
Figure C.13 Flowchart summarizing the prediction method for equivalent plastic strains in roll-formed members
Appendix C - 132
C.3.4 Employing the Prediction Method in Practice The prediction method input parameters required to calculate corner
residual stresses and equivalent plastic strains are the sheet thickness t, yield
stress σyield, steel modulus of elasticity E, and the corner bend radius rz. All of
these parameters are typically known or can be estimated by a researcher or
designer. The coiling, uncoiling, and flattening residual stresses (σcoiling , σuncoil ,
and σflatten ) and coiling plastic strains are influenced by t, σyield, E, and rx , the
radial location of the sheet in the coil prior to roll forming. rx varies through the
sheet coil, and therefore the residual stresses and strains will also vary from
member to member. The range of inner and outer coil radii provided in Section
C.1.2 provides a lower and upper bound on rx. Also, the probability that a
structural member will be manufactured with a certain rx can also be quantified
by relating the radial location to linear length along the sheet coil. The details
on how to quantify rx in an average sense are derived here and then
implemented in the prediction method to calculate mean residual stress and
strain distributions for common industry parameters t and σyield .
The relationship between coil radius rx and linear location of the sheet S
within the coil can be described using Archimedes spiral (CRC 2003) :
( )22)( innerxx rrt
rS −=π
Appendix C - 133
The spiral maintains a constant pitch with varying radii, where the pitch is the
thickness of the steel sheet t as shown in Figure C.14 and L is the total length of
sheet in the coil. rinner and router are the as-delivered inside and outside coil radii.
rx
StartEnd
t LS =0=S
S
Figure C.14 Coil coordinate system and notation
Archimedes spiral can be used to describe the probability that the steel sheet will
come from a certain radial location in the coil. The random variable R is defined
as the compliment to the deterministic quantity rx. The probability that R is less
than or equal to rx , P(R≤rx), is defined as a cumulative distribution function
(CDF) by normalizing S with L:
( ) [ ]22)()(innerxxRx
x rrtL
rFrRPLrS
−==≤=π
where ( )22innerouter rr
tL −=
π
Appendix C - 134
The plot of this CDF in Figure C.15 has an increasing slope with increasing rx ,
demonstrating that more of the total sheet length is located towards the outside
of the coil. (Note in this figure that rx is made dimensionless by dividing through
by the inner coil diameter rinner.)
1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
rx/rinner
P(R
≤ r x)=
S(r x)/L
Figure C.15 Relationship between the coil radius rx and linear location S in the coil described as the cumulative distribution function for rx
The probability that R is equal to rx , P(R=rx ), is the probability density function
(PDF) of rx. The PDF equation is calculated by taking the derivative of FR(rx):
( )tL
rrfdr
rdFrRP xxR
x
xRx
π2)()( ====
FR(rx) and fR(rx) are now used to evaluate the coiling, uncoiling, and flattening
residual stresses and strains in an average sense.
Appendix C - 135
The total through-thickness residual stress distribution from coiling,
uncoiling, and flattening in rolled formed members is defined as σz. The
equations for σcoiling , σuncoil , and σflatten (summarized in Figure C.12 flowchart) can
be used to develop a prediction for σz:
flattenz
uncoilz
coilingzyieldxz ytEr σσσσσ ++=),,,( ,
An expression for the mean predicted through thickness residual stresses follows
as:
∫= outer
inner
r
r xyieldxzxRyieldz drytErrfytE ),,,()(),,( ,, σσσσ
Thus for any given E, t, and σyield , and using fR(rx) from Archimedes spiral, the
mean predicted residual stresses from coiling, uncoiling, and flattening can be
determined. The variability in the prediction model is quantified with the
equation:
{ } [ ]∫ −= outer
inner
r
r xyieldzyieldxzxRzyieldg drytEytErrfytEs 2,,,
2 ),,(),,,()(:),,( σσσσσσ
sg is the standard deviation from the mean predicted residual stress and can be
used to set bounds on the estimate. With the same methodology, equations are
derived for the mean value and prediction variability of coiling equivalent plastic
strain εp :
∫= outer
inner
r
r xxcoiling
pxRcoiling
p dryrrfy ),()()( εε
{ } [ ]∫ −= outer
inner
r
r xcoiling
pxcoiling
pxRcoiling
pg dryyrrfys22 )(),()(:)( εεε
and for the elastic core c of the sheet associated with coiling:
Appendix C - 136
∫= outer
inner
r
r xyieldxxRyield drrcrfc ),()()( εε
{ } [ ]∫ −= outer
inner
r
r xyieldyieldxxRyieldg drcrcrfcs 22 )(),()(:)( εεε
These general equations provide a way to estimate the coiling, uncoiling and
flattening residual stresses and strains in roll-formed members, although because
of the definite integrals, they take some time to implement and solve. To make
these predictions directly accessible for the range of common steel sheet
thicknesses and yield stresses, approximations for the mean values and
variability of σz and εp are provided here in metric units (σyield as MPa, t as mm,
E=203.5 GPa, rinner =254 mm, router= 610 mm):
=coilingpε̂
tyield 191<σyieldt σ64 1032.21043.5 −− ×−×
yieldt σ74 1096.61033.2 −− ×−× tt yield 335191 ≤≤ σ
=zσ̂tyield 103<σyieldσ97.0
yieldt σ55.0157 − tt yield 285103 ≥≥ σ
{ }=zgs σ:ˆ2
tyield 87<σ
( )225.08.97 yieldt σ−
0
tt yield 16387 <≤ σ( )275.02.65 yieldt σ+−
tt yield 391163 ≤≤ σ
Appendix C - 137
{ }=coilingpgs ε:ˆ2
tyield 182<σ( )241037.1 t−×
( )274 1003.71065.2 yieldt σ−− ×−× tt yield 376182 ≤≤ σ
and in US units (σyield as ksi, t as in., E= 29500 ksi, rinner =10 in., router= 24 in.):
=zσ̂tyield 380<σyieldσ97.0
yieldt σ55.0578 − tt yield 1051380 ≤≤ σ
=coilingpε̂
tyield 703<σyieldt σ51060.10138.0 −×−
yieldt σ63 1080.41093.5 −− ×−× tt yield 1235703 ≤≤ σ
{ }=coilingpgs ε:ˆ2
tyield 670<σ( )231048.3 t−×
( )263 1085.41073.6 yieldt σ−− ×−× tt yield 1387670 ≤≤ σ
The approximate elastic core c is calculated using the estimated plastic strains:
yield
yield
coilingp
e
tc
ε
εε
+−=
⎟⎠⎞
⎜⎝⎛
1ˆ
ˆ2
3 where tc ≤ˆ
{ }=zgs σ:ˆ2
tyield 320<σ
( )225.0360 yieldt σ−
0
tt yield 600320 <≤ σ( )275.0240 yieldt σ+−
tt yield 1440600 ≤≤ σ
Appendix C - 138
C.3.5 Comparison of predicted residual stresses to measured data
The prediction method presented here provides a through-thickness
residual stress distribution with bending and yield-release components for roll-
formed structural members. The membrane component is predicted as zero (see
Figure C.5). The measured data reported in Section C.1.4 are made up of
longitudinal residual strains that are converted into bending and membrane
residual stresses. A logical point of comparison is then the bending residual
stress since it is a common parameter between the measured and predicted stress
distributions.
The comparison will be conducted with the 18 roll-formed (RF) specimens
from the residual stress database (Appendix A). The radial location rx from
which each specimen originated is approximated. The distribution of the set of
predicted radii is then compared to the theoretical distribution for rx derived
with Archimedes spiral in Section C.3.4.
The location of the specimen in the coil, rx, is approximated by minimizing
the sum of the mean squared errors (MSE) for the {1,2,i,…n} measurements taken
around the cross section in each of the 18 roll-formed specimens:
⎟⎠⎞⎜
⎝⎛∑
=
n
iir
MSEx 1
min
where
2
,⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
iyield
predictedi
measuredi
iMSEσ
σσ
Appendix C - 139
(Both corner and flat measurements are included in the minimization.) This
minimization is essentially a linear regression, where the radial location rx is
chosen to maximize the likelihood that all MSEi will be zero (Fritz et al. 2006a,
Fritz et al. 2006b). The predicted residual stress distribution contains both
bending and yield-release components and therefore the bending component
must be isolated to compare with the measured values. The total predicted
longitudinal residual stress distribution is integrated to calculate the sectional
moment through the thickness:
∫−
=2
2
t
tzx ydyM σ
Mx is then converted into a predicted outer fiber bending residual stress:
I
tM xpredicted
i
⎟⎠⎞
⎜⎝⎛
= 2σ
Figure C.16 demonstrates the minimization technique for de M. Batista and
Rodrigues (1992) Specimen CP1. The radial location that minimizes the
prediction error is 1.60rinner in this case, and is summarized in Table C.4 for all 18
roll-formed specimens considered. The radial location is undefined for the three
Bernard specimens since the bending residual stresses in the flats are predicted
to be zero. These three specimens are cold-formed steel decking with a thin sheet
thickness t ranging from 0.75 mm to 1 mm (0.028 in. to 0.039 in.) and a relatively
high yield stress σyield ranging from 600 MPa to 650 MPa (86 ksi to 94 ksi). In this
Appendix C - 140
case, the coiling and uncoiling of the steel sheet will occur elastically. Measured
bending residual stress magnitudes for the three Bernard specimens are on
average 0.03σyield which is consistent with predictions.
1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
7
8
rx/rinner
Σ M
SE
i
Figure C.16 The sum of the mean squared errors of the predicted and measured bending residual stresses for de M. Batista and Rodrigues (1992), Specimen CP1 is minimized when rx=1.60rinner
Appendix C - 141
Table C.4 Radial location in the coil that minimizes the sum of the mean square prediction error for roll-formed structural members
Researcher Specimen rx/rinner
de M. Batista and Rodrigues 1992 CP2 1.20de M. Batista and Rodrigues 1992 CP1 1.60
Weng and Peköz 1990 RFC13 1.80Weng and Peköz 1990 RFC14 1.10Weng and Peköz 1990 R13 1.45Weng and Peköz 1990 R14 1.30Weng and Peköz 1990 P3300 1.95Weng and Peköz 1990 P4100 1.50Weng and Peköz 1990 DC-12 2.30Weng and Peköz 1990 DC-14 1.60
Dat 1980 RFC14 2.00Dat 1980 RFC13 2.45
Bernard 1993 Bondek 1 UNDEFBernard 1993 Bondek 2 UNDEFBernard 1993 Condeck HP UNDEF
Abdel-Rahman and Siva 1997 Type A - Spec 1 1.55Abdel-Rahman and Siva 1997 Type A - Spec 2 1.55Abdel-Rahman and Siva 1997 Type B - Spec 1 1.25
rinner=254 mmUNDEF zero bending residual stresses are predicted, rx is undefined
The statistical distribution of these predicted radial locations are now
compared to the theoretical distribution defined by Archimedes spiral with the
H. Kolmogorov-Smirnov Goodness of Fit test (Reference ???). Consider the
following null hypothesis that the random variable for the coil location R is
described by the statistical distribution derived from Archimedes spiral:
H0: The cumulative distribution function (CDF) of R is FR(rx)
To test the null hypothesis, an approximate CDF, FR(rx)*, is derived from the rx
predictions in Table C.4 (Bernard specimens excluded) and compared against the
corresponding theoretical CDF FR(rx) in Table C.5.
Appendix C - 142
Table C.5 Test MSE Approximate Theoretical
Prediction CDF CDFi rx/rinner FR(rx)*=i/15 FR(rx)1 1.10 0.07 0.042 1.20 0.13 0.093 1.25 0.20 0.114 1.30 0.27 0.145 1.45 0.33 0.226 1.50 0.40 0.257 1.558 1.55 0.47 0.289 1.6010 1.60 0.60 0.3111 1.80 0.73 0.4512 1.95 0.80 0.5613 2.00 0.87 0.6014 2.30 0.93 0.8615 2.45 1.00 1.00
rinner=254 mm (10 in.)
The maximum value of the difference between the approximate and theoretical
CDF is:
{ })()(max ,,*
ixRixR rFrFD −=
Figure C.17 compares the approximate and theoretical CDFs and demonstrates
that the maximum difference, D=0.29, occurs when rx/rinner=1.60.
Appendix C - 143
1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
rx/rinner
P(R
≤rx)
D=0.29
)( xR rF
*)( xR rF
Figure C.17
The KS test says if:
αDD ≤ then the hypothesis, H0, is significant at the -percent level
For n > 4 and < 50%, the approximate relation between significance level and
the parameter D is:
)1(2 2
2 +−≈ nDe αα
A statistical rule of thumb is that when > 5%, the prediction is consistent with
the theoretical distribution it is being compared to. Substituting D=D=0.29 into
the above equation, the significance level of the test is calculated as
approximately 14%. This result implies that the predictions for rx are consistent
with that described by Archimedes spiral. Future work is needed to make
stronger conclusions regarding the relationship between coil location and
Appendix C - 144
residual stresses. Experiments are needed to confirm the relationship between
residual bending stresses and the location of the sheet steel in roll-formed
structural members.
C.3.6 Characterization of Measurement Error in Roll-Formed Members
The predicted radial locations in Table C.4 are now used to derive a model
of measurement errors that can be used to characterize the bias and variability in
experimental work on residual stresses. The bending residual stresses in the 18
roll-formed structural members discussed in Section C.3.5 are calculated with the
predicted radial location rx. The difference between the predicted and measured
residual bending stresses is calculated at all flat and corner locations with the
equation:
iyield
predictedi
measuredi
ie,σ
σσ −=
The measurement error histogram for the flat cross-sectional elements in
Figure C.18 demonstrates that the mean experimental error μe is near zero with a
standard deviation se=±0.15σyield. The corner element error histogram in Figure
C.19 demonstrates a negative bias of μe=-0.16σyield and a standard deviation
se=±0.19σyield. Although it is difficult to make definitive conclusions with the
small amount of corner data (a total of 23 measurements), it is hypothesized that
the increased measurement accuracy in the flats is due to the relative ease of
placing strain gauges and sectioning when compared to the corner regions.
Appendix C - 145
-1 -0.5 0 0.5 10
5
10
15
20
25
30
35
40
(σmeasured-σpredicted)/σyield
Obs
erva
tions
yielde σμ 03.0+=
yieldes σ15.0±=
Figure C.18 Histogram of bending residual stress prediction error (flat cross-sectional elements) for 18 roll-formed specimens
-1 -0.5 0 0.5 10
5
10
15
20
25
30
35
40
(σmeasured-σpredicted)/σyield
Obs
erva
tions
yielde σμ 16.0−=
yieldes σ19.0±=
Figure C.19 Histogram of bending residual stress prediction error (corner cross-sectional elements)
for 18 roll-formed specimens
Appendix C - 146
C.4 Press-braked structural members - predicting residual stresses and plastic strains Press-braked members are assumed to originate from sheet steel that has
been coiled after annealing, uncoiled, and then leveled before being cut to length.
The leveling process removes the residual coiling curvature by bending the steel
sheet in the opposite direction of the coiling curvature. The imposed reverse
curvature is released after yielding the steel, leaving a flat sheet that is then
press-braked into its final form. This is different from roll-formed members,
where the residual coiling curvature is locked into the structural member as the
sheet passes through the roll-forming line (see Figure C.10)
C.4.1 Residual Stresses - Sheet Coiling, Uncoiling, and Leveling The general procedure for calculating residual stresses from coiling,
uncoiling, and leveling is summarized in Figure C.20. The coiling and uncoiling
stresses (σcoil and σuncoil, Step 1 and 2 of Figure C.20) in press-braked members are
assumed to be the same as those in the roll-formed members presented in Section
C.3.1.1. The amount of reverse curvature required to produce a flat sheet (Step 4,
Figure C.20) can be determined iteratively or evaluated with the Quach et. al.
(2006a) closed form treatment of residual stresses and equivalent plastic strains
from coiling, uncoiling, and leveling.
Appendix C - 147
+
+
+
=
=
=
Coiling Uncoiling
Apply opposite curvature to yield and flatten
Longitudinal residual stress in flat sheet
),( xuncoilz ryσ
),( xcoilz ryσ
Remove imposed reverse curvature
y
z
z
z
(1) (2) (3)
(3)(4) (5)
(5)(6) (7)
Figure C.20 Longitudinal stresses associated with the coiling, uncoiling, and flattening of steel sheet
used in press-braked structural members An example of the through-thickness longitudinal stresses resulting from coiling,
uncoiling, and leveling before press-braking is provided in Figure C.21.
Appendix C - 148
-1 -0.5 0 0.5 1-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
σz/σyield
y/t
(1) After coiling
(3) After uncoiling
(5) Leveling with reverse curvature
(7) Final residual stress distribution
Figure C.21 Residual stresses from coiling, uncoiling, and flattening of a steel sheet: rx=430 mm (17 in.), t=2.46 mm (0.097 in.), σyield=345 MPa (50 ksi)
For a specific structural member, the originating coil location of the sheet rx is
unknown. Monte Carlo simulation is used to determine the residual stresses in
an average sense. The radial location rx is treated as a random variable with the
PDF described in Section C.3.4 and 2000 trials are conducted for five common
sheet thicknesses: 0.84 mm (0.033 in.), 1.09 mm (0.043 in.), 1.37 mm (0.054 in.),
1.73 mm (0.068 in.), and 2.46 mm (0.097 in.). The modulus of elasticity E is
assumed as 203.5 GPa (29500 ksi), rinner=254 mm (10 in.), and router=610 mm (24
in.). Average residual stress distributions for steel sheets that have been coiled,
uncoiled, and leveled in preparation for press-braking are provided in Figure
C.22. The residual stress distributions are self-equilibrating for both axial force
and moment (see Figure C.5), and stress magnitudes are zero when t=0.84 mm
(0.033 in.) and 1.09 mm (0.043 in.).
Appendix C - 149
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5
-0.25
0
0.25
0.5
y/t
σz/σyield
y/t=b (typ.)t/tmin b
σz/σyield
at y/t=bσz/σyield
at y/t=0.501.64 0.4 -0.01 0.032.06 0.36 -0.02 0.072.94 0.33 -0.07 0.18
tmin=0.84 mm (0.033 in.)
Figure C.22 Mean residual stress distributions resulting from coiling, uncoiling, and flattening of
thin steel sheet with σyield=345 MPa (50 ksi)
C.4.2 Equivalent Plastic Strain - Coiling and Leveling
The through-thickness equivalent plastic strain in press-braked members
is assumed to be the same as for roll-formed members since the initial coiling of
the sheet is commonly the dominant source of plastic strain.
C.4.3 Cross Section Press-Braking
Residual stresses and plastic strains caused by press braking are assumed
to be the same as those found in the corners of roll-formed members. Refer to
Section C.3.2 for the appropriate prediction equations.
C.5 Acknowledgements
The development of this residual stress prediction method would not
have been possible without accurate information on the manufacturing process
of sheet steel coils and cold-formed steel structural members. Thanks to Clark
Appendix C - 150
Western Building Systems, Mittal Steel USA, and the Cold-Formed Steel
Engineers Institute (CFSEI) for their major contributions to this research,
especially:
Clark Western Bill Craig Ken Curtis Tom Lemler Joe Wellinghoff Mittal Steel Ezio Defrancesco Jean Frasier Narayan Pottore CFSEI Don Allen
Appendix C - 151
Appendix C ( References) ABAQUS (2004). ABAQUS/Standard User’s Manual Version 6.5. ABAQUS, Inc., www.abaqus.com, Providence, RI. Abdel-Rahman, N. and Sivakumaran, K.S. (1997). “Material properties models for analysis of cold-formed steel members.” ASCE Journal of Structural Engineering, 123(9), 1135-1143. Batista, E. de M. and Rodrigues, F.C. (1992). “Residual stress measurements on cold-formed profiles.” Experimental Techniques, 16(5),25-29. Bernard, E.S. (1993). Flexural Behaviour of Cold-Formed Profiled Steel Decking. Ph.D. Thesis, University of Sydney, Australia. CRC (2003). Standard Mathematical Tables and Formulae. CRC Press, New York, NY. Dat, D.T. (1980). The Strength of Cold-Formed Steel Columns. Cornell University Dept. of Structural Engineering Report No. 80-4, Ithaca, NY. Fritz, W, Jones, N.P., and Igusa, T. Hill, R.(1983). The Mathematical Theory of Plasticity. Oxford University Press, New York. Ingvarsson, Lars (1975). “Cold-forming residual stresses, effect of buckling.” Proceedings of the Third International Specialty Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, 85-119. Kato, B. and Aoki, H. (1978). “Residual stresses in cold-formed tubes.” Journal of Strain Analysis, 13(4), 193-204. Key, P.W. and Hancock, G.J. (1993). “A theoretical investigation of the column behavior of cold-formed square hollow sections.” Thin-Walled Structures, 16, 31-64. Kwon, Y.B. (1992). Post-Buckling Behaviour of Thin-Walled Sections. Ph.D. Thesis, University of Sydney, Australia. Quach, W.M., Teng, J.G., and Chung, K.F. (2006a). “Residual stresses in steel sheets due to coiling and uncoiling: a closed-form analytical solution.” Engineering Structures, 26, 1249-1259. Quach, W.M., Teng, J.G., and Chung, K.F. (2006b). “Finite element predictions of residual stresses in press-braked thin-walled steel sections.” Engineering Structures, 26, 1609-1619.
Appendix C - 152
Schafer, B.W. and Peköz, T. (1997). “Compuational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses.” Journal of Constructional Research, 47, 193-210. Shanley, F.R. (1957). Strength of Materials. McGraw-Hill Book Company, Inc, New York, NY. Ugural, A.C. and Fenster, S.K. (2003). Advanced Strength and Applied Elasticity, 4th Edition. Prentice Hall, Upper Saddle River, NJ. US Steel (1985). The Making, Shaping, and Treating of Steel, 10th Edition. Herbick & Held, Pittsburgh, PA. Weng, C.C. and Peköz, T. (1990). “Residual stresses in cold-formed steel members.” ASCE Journal of Structural Engineering, 116(6), 1611-1625. Young, B. (1997). The Behavior and Design of Cold-Formed Channel Columns. Ph.D. Thesis, University of Sydney, Australia.
Appendix C - 153
Appendix C (Appendix A– Residual Stress Measurements)
membrane bendingde M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 -0.08 0.42de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 -0.09 0.43de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 0.09 0.60de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 0.04 0.69de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 0.03 0.66de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 -0.07 0.51de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 0.05 0.56de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 0.08 0.62de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF stiffened 295 1.52 0.14 0.38de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 0.05 0.15de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 0.07 0.24de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 -0.10 0.24de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 0.07 0.10de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 0.25 0.46de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 0.33 0.35de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 -0.11 0.25de M. Batista and Rodrigues 1992 CP2 Lipped Channel RF edge stiffened 295 1.52 -0.13 0.11de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 0.00 -0.02de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 0.00 0.02de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 -0.04 0.11de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 -0.05 0.18de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 -0.03 0.20de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 0.04 -0.05de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 0.02 -0.07de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 0.02 -0.05de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB stiffened 327 1.52 -0.02 -0.03de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.00 0.00de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.00 0.02de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.00 0.00de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.00 -0.02de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.01 -0.02de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.01 -0.11de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.01 -0.13de M. Batista and Rodrigues 1992 CP3 Lipped Channel PB edge stiffened 327 1.52 0.03 -0.17de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 0.09 0.61de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 0.03 0.57de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 0.16 0.54de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 -0.03 0.45de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 -0.41 0.47de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 -0.02 0.66de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 -0.07 0.47de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 0.00 0.48de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF stiffened 290 2 0.04 0.75de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.03 0.14de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.20 0.28de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.19 0.29de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 -0.03 0.05de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.16 0.19de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.26 0.34de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.00 0.21de M. Batista and Rodrigues 1992 CP1 Lipped Channel RF edge stiffened 290 2 0.09 0.19
residual stress as%fyResearcher Test Type RF,PB Element fy(MPa) t(mm)
Appendix C - 154
membrane bendingWeng and Peköz 1990 RFC13 Lipped Channel RF stiffened 355 2.4384 -0.01 0.25Weng and Peköz 1990 RFC13 Lipped Channel RF stiffened 355 2.4384 0.00 0.23Weng and Peköz 1990 RFC13 Lipped Channel RF corner 355 2.4384 0.06 0.46Weng and Peköz 1990 RFC13 Lipped Channel RF edge stiffened 355 2.4384 0.01 0.26Weng and Peköz 1990 RFC13 Lipped Channel RF edge stiffened 355 2.4384 0.00 0.29Weng and Peköz 1990 RFC14 Lipped Channel RF stiffened 380 1.905 0.00 0.53Weng and Peköz 1990 RFC14 Lipped Channel RF stiffened 380 1.905 0.03 0.59Weng and Peköz 1990 RFC14 Lipped Channel RF corner 380 1.905 0.09 0.68Weng and Peköz 1990 RFC14 Lipped Channel RF edge stiffened 380 1.905 0.05 0.43Weng and Peköz 1990 RFC14 Lipped Channel RF edge stiffened 380 1.905 -0.01 0.39Weng and Peköz 1990 PBC14 Lipped Channel PB stiffened 250 1.8034 0.00 0.24Weng and Peköz 1990 PBC14 Lipped Channel PB stiffened 250 1.8034 -0.02 0.24Weng and Peköz 1990 PBC14 Lipped Channel PB corner 250 1.8034 0.03 0.50Weng and Peköz 1990 PBC14 Lipped Channel PB edge stiffened 250 1.8034 -0.02 0.22Weng and Peköz 1990 PBC14 Lipped Channel PB edge stiffened 250 1.8034 0.00 0.25Weng and Peköz 1990 R13 LC web indented RF stiffened 345 2.1844 0.01 0.52Weng and Peköz 1990 R13 LC web indented RF stiffened 345 2.1844 0.00 0.47Weng and Peköz 1990 R13 LC web indented RF corner 345 2.1844 0.06 0.65Weng and Peköz 1990 R13 LC web indented RF edge stiffened 345 2.1844 -0.01 0.33Weng and Peköz 1990 R14 LC web indented RF stiffened 343 1.905 0.04 0.47Weng and Peköz 1990 R14 LC web indented RF stiffened 343 1.905 0.04 0.48Weng and Peköz 1990 R14 LC web indented RF corner 343 1.905 0.05 0.67Weng and Peköz 1990 R14 LC web indented RF edge stiffened 343 1.905 0.01 0.28Weng and Peköz 1990 P3300 Lipped Channel RF stiffened 384 2.667 0.00 0.21Weng and Peköz 1990 P3300 Lipped Channel RF stiffened 384 2.667 -0.01 0.21Weng and Peköz 1990 P3300 Lipped Channel RF corner 384 2.667 0.02 0.39Weng and Peköz 1990 P3300 Lipped Channel RF edge stiffened 384 2.667 -0.01 0.22Weng and Peköz 1990 P4100 Lipped Channel RF stiffened 355 1.905 -0.01 0.19Weng and Peköz 1990 P4100 Lipped Channel RF stiffened 355 1.905 0.00 0.17Weng and Peköz 1990 P4100 Lipped Channel RF corner 355 1.905 0.00 0.40Weng and Peköz 1990 P4100 Lipped Channel RF edge stiffened 355 1.905 0.01 0.22Weng and Peköz 1990 DC-12 2 LC RF stiffened 305 2.667 -0.01 0.23Weng and Peköz 1990 DC-12 2 LC RF stiffened 305 2.667 -0.01 0.21Weng and Peköz 1990 DC-12 2 LC RF corner 305 2.667 -0.03 0.46Weng and Peköz 1990 DC-12 2 LC RF edge stiffened 305 2.667 0.01 0.28Weng and Peköz 1990 DC-12 2 LC RF edge stiffened 305 2.667 -0.02 0.30Weng and Peköz 1990 DC-14 2LC RF stiffened 309 1.905 0.00 0.26Weng and Peköz 1990 DC-14 2LC RF stiffened 309 1.905 -0.01 0.29Weng and Peköz 1990 DC-14 2LC RF corner 309 1.905 0.05 0.48Weng and Peköz 1990 DC-14 2LC RF edge stiffened 309 1.905 0.01 0.26Weng and Peköz 1990 DC-14 2LC RF edge stiffened 309 1.905 -0.01 0.32Weng and Peköz 1990 P11 Lipped Channel PB edge stiffened 231 2.9972 -0.01 0.36Weng and Peköz 1990 P11 Lipped Channel PB edge stiffened 231 2.9972 -0.03 0.40Weng and Peköz 1990 P11 Lipped Channel PB corner 231 2.9972 0.04 0.57Weng and Peköz 1990 P11 Lipped Channel PB edge stiffened 231 2.9972 0.00 0.35Weng and Peköz 1990 P16 Lipped Channel PB stiffened 221 1.6256 -0.07 0.37Weng and Peköz 1990 P16 Lipped Channel PB stiffened 221 1.6256 -0.02 0.37Weng and Peköz 1990 P16 Lipped Channel PB corner 221 1.6256 0.03 0.56Weng and Peköz 1990 P16 Lipped Channel PB edge stiffened 221 1.6256 0.00 0.36
Element fy(MPa) t(mm)residual stress as%fy
Researcher Test Type RF,PB
Appendix C - 155
membrane bendingDat 1980 PC14 Lipped Channel PB lip 275 1.8542 -0.03 0.33Dat 1980 PC14 Lipped Channel PB corner 275 1.8542 0.01 0.05Dat 1980 PC14 Lipped Channel PB edge stiffened 275 1.8542 -0.03 0.10Dat 1980 PC14 Lipped Channel PB edge stiffened 275 1.8542 -0.01 0.19Dat 1980 PC14 Lipped Channel PB edge stiffened 275 1.8542 -0.03 0.18Dat 1980 PC14 Lipped Channel PB corner 275 1.8542 0.06 0.25Dat 1980 PC14 Lipped Channel PB stiffened 275 1.8542 0.04 0.43Dat 1980 PC14 Lipped Channel PB stiffened 275 1.8542 -0.05 0.39Dat 1980 PC14 Lipped Channel PB stiffened 275 1.8542 -0.04 0.36Dat 1980 PC14 Lipped Channel PB stiffened 275 1.8542 0.16 0.20Dat 1980 PC14 Lipped Channel PB stiffened 275 1.8542 0.01 0.43Dat 1980 PC14 Lipped Channel PB stiffened 275 1.8542 0.07 0.50Dat 1980 PC14 Lipped Channel PB corner 275 1.8542 0.02 0.11Dat 1980 PC14 Lipped Channel PB edge stiffened 275 1.8542 0.02 0.13Dat 1980 PC14 Lipped Channel PB edge stiffened 275 1.8542 -0.01 0.21Dat 1980 PC14 Lipped Channel PB edge stiffened 275 1.8542 -0.01 0.07Dat 1980 PC14 Lipped Channel PB corner 275 1.8542 -0.03 -0.01Dat 1980 PC14 Lipped Channel PB lip 275 1.8542 0.01 0.49Dat 1980 RFC14 Lipped Channel RF lip 275 1.8542 0.00 0.24Dat 1980 RFC14 Lipped Channel RF corner 275 1.8542 0.06 0.06Dat 1980 RFC14 Lipped Channel RF edge stiffened 275 1.8542 0.00 0.12Dat 1980 RFC14 Lipped Channel RF edge stiffened 275 1.8542 0.02 0.27Dat 1980 RFC14 Lipped Channel RF corner 275 1.8542 0.00 0.02Dat 1980 RFC14 Lipped Channel RF stiffened 275 1.8542 0.02 0.42Dat 1980 RFC14 Lipped Channel RF stiffened 275 1.8542 -0.02 0.45Dat 1980 RFC14 Lipped Channel RF stiffened 275 1.8542 -0.04 0.48Dat 1980 RFC14 Lipped Channel RF stiffened 275 1.8542 -0.02 0.28Dat 1980 RFC14 Lipped Channel RF corner 275 1.8542 0.09 0.07Dat 1980 RFC14 Lipped Channel RF edge stiffened 275 1.8542 0.00 0.25Dat 1980 RFC14 Lipped Channel RF edge stiffened 275 1.8542 -0.01 0.12Dat 1980 RFC14 Lipped Channel RF corner 275 1.8542 0.08 0.06Dat 1980 RFC14 Lipped Channel RF lip 275 1.8542 0.01 0.16Dat 1980 PBC13 Lipped Channel PB lip 268 2.286 -0.04 0.36Dat 1980 PBC13 Lipped Channel PB corner 268 2.286 0.25 0.37Dat 1980 PBC13 Lipped Channel PB edge stiffened 268 2.286 0.03 0.13Dat 1980 PBC13 Lipped Channel PB edge stiffened 268 2.286 0.02 0.14Dat 1980 PBC13 Lipped Channel PB edge stiffened 268 2.286 -0.06 0.30Dat 1980 PBC13 Lipped Channel PB corner 268 2.286 0.04 0.28Dat 1980 PBC13 Lipped Channel PB stiffened 268 2.286 0.05 0.59Dat 1980 PBC13 Lipped Channel PB stiffened 268 2.286 -0.02 0.55Dat 1980 PBC13 Lipped Channel PB stiffened 268 2.286 -0.02 0.51Dat 1980 PBC13 Lipped Channel PB stiffened 268 2.286 -0.02 0.50Dat 1980 PBC13 Lipped Channel PB stiffened 268 2.286 0.01 0.53Dat 1980 PBC13 Lipped Channel PB stiffened 268 2.286 0.03 0.50Dat 1980 PBC13 Lipped Channel PB corner 268 2.286 -0.01 0.20Dat 1980 PBC13 Lipped Channel PB edge stiffened 268 2.286 0.05 0.25Dat 1980 PBC13 Lipped Channel PB edge stiffened 268 2.286 -0.01 0.18Dat 1980 PBC13 Lipped Channel PB edge stiffened 268 2.286 0.04 0.15Dat 1980 PBC13 Lipped Channel PB corner 268 2.286 0.04 0.27Dat 1980 PBC13 Lipped Channel PB lip 268 2.286 0.07 1.06Dat 1980 RFC13 Lipped Channel RF lip 268 2.286 0.12 0.41Dat 1980 RFC13 Lipped Channel RF corner 268 2.286 0.43 0.27Dat 1980 RFC13 Lipped Channel RF edge stiffened 268 2.286 -0.02 0.12Dat 1980 RFC13 Lipped Channel RF edge stiffened 268 2.286 0.01 0.21Dat 1980 RFC13 Lipped Channel RF corner 268 2.286 0.04 0.04Dat 1980 RFC13 Lipped Channel RF stiffened 268 2.286 0.03 0.49Dat 1980 RFC13 Lipped Channel RF stiffened 268 2.286 -0.02 0.43Dat 1980 RFC13 Lipped Channel RF stiffened 268 2.286 -0.01 0.40Dat 1980 RFC13 Lipped Channel RF stiffened 268 2.286 0.01 0.47Dat 1980 RFC13 Lipped Channel RF corner 268 2.286 0.05 0.16Dat 1980 RFC13 Lipped Channel RF edge stiffened 268 2.286 -0.04 0.17Dat 1980 RFC13 Lipped Channel RF edge stiffened 268 2.286 0.06 0.02Dat 1980 RFC13 Lipped Channel RF corner 268 2.286 0.25 0.18Dat 1980 RFC13 Lipped Channel RF lip 268 2.286 0.35 -0.01
Element fy(MPa) t(mm)residual stress as%fy
Researcher Test Type RF,PB
Appendix C - 156
membrane bendingBernard 1993 R410-1 IST-Deck PB edge stiffened 653 0.585 0.01 0.17Bernard 1993 R410-1 IST-Deck PB edge stiffened 653 0.585 -0.01 0.10Bernard 1993 R410-1 IST-Deck PB stiffened 653 0.585 0.00 0.12Bernard 1993 R410-1 IST-Deck PB edge stiffened 653 0.585 0.01 0.01Bernard 1993 R410-1 IST-Deck PB edge stiffened 653 0.585 0.00 0.13Bernard 1993 R410-1 IST-Deck PB edge stiffened 653 0.585 0.00 0.13Bernard 1993 R412-1 IST-Deck PB edge stiffened 653 0.585 0.01 -0.01Bernard 1993 R412-1 IST-Deck PB edge stiffened 653 0.585 -0.01 -0.03Bernard 1993 R412-1 IST-Deck PB stiffened 653 0.585 0.00 -0.04Bernard 1993 R412-1 IST-Deck PB stiffened 653 0.585 0.01 -0.05Bernard 1993 R412-1 IST-Deck PB edge stiffened 653 0.585 -0.01 -0.02Bernard 1993 R412-1 IST-Deck PB edge stiffened 653 0.585 0.00 -0.04Bernard 1993 R412-1 IST-Deck PB edge stiffened 653 0.585 0.02 -0.05Bernard 1993 R412-2A IST-Deck PB edge stiffened 653 0.585 0.00 -0.04Bernard 1993 R412-2A IST-Deck PB stiffened 653 0.585 0.00 -0.02Bernard 1993 R412-2A IST-Deck PB stiffened 653 0.585 0.01 -0.04Bernard 1993 R412-2A IST-Deck PB stiffened 653 0.585 0.01 -0.03Bernard 1993 R412-2A IST-Deck PB edge stiffened 653 0.585 0.01 -0.01Bernard 1993 R412-2A IST-Deck PB edge stiffened 653 0.585 0.01 -0.04Bernard 1993 R412-2A IST-Deck PB edge stiffened 653 0.585 0.01 -0.04Bernard 1993 R412-2B IST-Deck PB edge stiffened 653 0.585 0.00 0.12Bernard 1993 R412-2B IST-Deck PB edge stiffened 653 0.585 0.02 0.05Bernard 1993 R412-2B IST-Deck PB stiffened 653 0.585 0.01 0.07Bernard 1993 R412-2B IST-Deck PB stiffened 653 0.585 0.01 0.12Bernard 1993 R412-2B IST-Deck PB stiffened 653 0.585 0.02 0.07Bernard 1993 R412-2B IST-Deck PB edge stiffened 653 0.585 0.02 0.09Bernard 1993 R412-2B IST-Deck PB edge stiffened 653 0.585 0.02 0.08Bernard 1993 R412-2B IST-Deck PB edge stiffened 653 0.585 0.02 0.08Bernard 1993 Bondek 1 Decking RF lip 578 1.008 0.04 0.00Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 -0.04 0.06Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 0.19 0.17Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 -0.01 -0.13Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 0.02 0.03Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 -0.11 -0.09Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 -0.18 0.29Bernard 1993 Bondek 1 Decking RF stiffened 578 1.008 -0.03 0.05Bernard 1993 Bondek 1 Decking RF lip 578 1.008 0.12 -0.46Bernard 1993 Bondek 2 Decking RF lip 608 1.008 -0.03 0.10Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 -0.02 0.30Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 0.28 0.19Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 -0.12 0.04Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 -0.08 0.03Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 -0.42 0.43Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 0.12 0.23Bernard 1993 Bondek 2 Decking RF stiffened 608 1.008 0.35 -0.07Bernard 1993 Condeck HP Decking RF lip 651 0.752 0.02 0.10Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 0.02 0.00Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 -0.06 -0.12Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 -0.02 0.06Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 0.00 -0.13Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 -0.03 -0.02Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 -0.10 -0.10Bernard 1993 Condeck HP Decking RF stiffened 651 0.752 0.13 0.03
Researcher Test Type RF,PB Element fy(MPa) t(mm)residual stress as%fy
Appendix C - 157
membrane bendingKwon 1992 CH1 Lipped Channel PB stiffened 590 1.1 -0.01 0.00Kwon 1992 CH1 Lipped Channel PB stiffened 590 1.1 0.00 -0.01Kwon 1992 CH1 Lipped Channel PB stiffened 590 1.1 0.02 0.05Kwon 1992 CH1 Lipped Channel PB edge stiffened 590 1.1 0.01 0.02Kwon 1992 CH1 Lipped Channel PB edge stiffened 590 1.1 -0.01 0.00Kwon 1992 CH1 Lipped Channel PB edge stiffened 590 1.1 -0.01 0.07Kwon 1992 CH2 LC+ISt PB stiffened 590 1.1 0.02 -0.06Kwon 1992 CH2 LC+ISt PB stiffened 590 1.1 0.00 -0.07Kwon 1992 CH2 LC+ISt PB stiffened 590 1.1 -0.01 0.01Kwon 1992 CH2 LC+ISt PB stiffened 590 1.1 0.01 0.05Kwon 1992 CH2 LC+ISt PB edge stiffened 590 1.1 -0.01 0.03Kwon 1992 CH2 LC+ISt PB edge stiffened 590 1.1 -0.01 0.01Kwon 1992 CH2 LC+ISt PB edge stiffened 590 1.1 0.01 0.07
Abdel-Rahman and Siva 1997 Type A - Spec 1 Lipped Channel RF edge stiffened 385 1.91 0.05 0.07Abdel-Rahman and Siva 1997 Type A - Spec 1 Lipped Channel RF edge stiffened 385 1.91 0.01 0.06Abdel-Rahman and Siva 1997 Type A - Spec 1 Lipped Channel RF corner 385 1.91 -0.02 0.08Abdel-Rahman and Siva 1997 Type A - Spec 1 Lipped Channel RF corner 385 1.91 0.02 0.67Abdel-Rahman and Siva 1997 Type A - Spec 1 Lipped Channel RF stiffened 385 1.91 0.03 0.10Abdel-Rahman and Siva 1997 Type A - Spec 1 Lipped Channel RF stiffened 385 1.91 0.04 0.16Abdel-Rahman and Siva 1997 Type A - Spec 2 Lipped Channel RF edge stiffened 385 1.91 0.04 0.08Abdel-Rahman and Siva 1997 Type A - Spec 2 Lipped Channel RF edge stiffened 385 1.91 0.05 0.11Abdel-Rahman and Siva 1997 Type A - Spec 2 Lipped Channel RF corner 385 1.91 -0.01 0.05Abdel-Rahman and Siva 1997 Type A - Spec 2 Lipped Channel RF corner 385 1.91 -0.03 0.65Abdel-Rahman and Siva 1997 Type A - Spec 2 Lipped Channel RF stiffened 385 1.91 0.02 0.13Abdel-Rahman and Siva 1997 Type A - Spec 2 Lipped Channel RF stiffened 385 1.91 0.03 0.12Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF edge stiffened 320 1.22 0.02 0.07Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF edge stiffened 320 1.22 -0.02 0.11Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF corner 320 1.22 -0.01 0.14Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF corner 320 1.22 0.04 0.37Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF edge stiffened 320 1.22 0.01 0.08Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF edge stiffened 320 1.22 0.03 0.07Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF corner 320 1.22 0.00 0.35Abdel-Rahman and Siva 1997 Type B - Spec 1 Lipped Channel RF stiffened 320 1.22 0.02 0.25
Young 1997 L48 - Spec. 1 Lipped Channel PB lip 545 1.476 -0.03 0.01Young 1997 L48 - Spec. 1 Lipped Channel PB edge stiffened 545 1.476 0.02 -0.01Young 1997 L48 - Spec. 1 Lipped Channel PB edge stiffened 545 1.476 0.02 0.06Young 1997 L48 - Spec. 1 Lipped Channel PB edge stiffened 545 1.476 0.00 0.02Young 1997 L48 - Spec. 1 Lipped Channel PB stiffened 545 1.476 -0.02 0.02Young 1997 L48 - Spec. 1 Lipped Channel PB stiffened 545 1.476 0.00 0.06Young 1997 L48 - Spec. 1 Lipped Channel PB stiffened 545 1.476 0.00 0.06Young 1997 L48 - Spec. 2 Lipped Channel PB lip 545 1.476 -0.01 -0.04Young 1997 L48 - Spec. 2 Lipped Channel PB edge stiffened 545 1.476 0.02 0.07Young 1997 L48 - Spec. 2 Lipped Channel PB edge stiffened 545 1.476 -0.01 0.00Young 1997 L48 - Spec. 2 Lipped Channel PB stiffened 545 1.476 0.01 0.07Young 1997 L48 - Spec. 2 Lipped Channel PB stiffened 545 1.476 0.01 0.06
Researcher Test Type RF,PB Element fy(MPa) t(mm)residual stress as%fy