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

45

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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


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


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


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




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


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


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




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


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


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


for 5.1c ≤λ Pne = yP658.02c ⎟

⎠⎞⎜

⎝⎛ λ


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


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




6.0

y

crd6.0

y

crd PPP

PP

P25.01 ≤⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

where λd = crdy PP


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


for 5.1c ≤λ Pne = yP658.02c ⎟

⎠⎞⎜

⎝⎛ λ


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


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




6.0

y

crd6.0

y

crd PPP

PP

P25.01 ≤⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

where λd = crdy PP


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



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


Pn l

/Pne





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


Pn l

/Pne





Pcre=100Py, λc=0.10

Pynet=0.80Py



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


Cap at Pynet (Options 3, 4, 5)

Replace Py with Pynet in strength and all slenderness calculations (Option 2)

Pynet=0.80Py


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


λl=(Pne/Pcrl)

0.5

Pn/P

y

0 0.5 1 1.5 2 2.5 30

0.5

1


λl=(Pne/Pcrl)

0.5

Pn/P

y

0 0.5 1 1.5 2 2.5 30

0.5

1


λl=(Pne/Pcrl

)0.5

Pn/P

y

0 0.5 1 1.5 2 2.5 30

0.5

1


λl=(Pne/Pcrl

)0.5

Pn/P

y


0 0.5 1 1.5 2 2.5 30

0.5

1


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


λl=(Pne/Pcrl)

0.5

Pn/P

ne

0 0.5 1 1.5 2 2.5 30

0.5

1


λl=(Pne/Pcrl)

0.5

Pn/P

ne

0 0.5 1 1.5 2 2.5 30

0.5

1


λl=(Pne/Pcrl)

0.5

Pn/P

ne

0 0.5 1 1.5 2 2.5 30

0.5

1


λl=(Pne/Pcrl)

0.5

Pn/P

ne


0 0.5 1 1.5 2 2.5 30

0.5

1


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


λd=(Py/Pcrd)0.5

Pn/P

y

0 0.5 1 1.5 2 2.5 30

0.5

1


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


λd=(Py/Pcrd)0.5

Pn/P

y

0 0.5 1 1.5 2 2.5 30

0.5

1


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


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


Anet/Ag

Pte

st/P

n

0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5


Anet/Ag

Pte

st/P

n

0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5


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


λl=(Pne/Pcrl

)0.5

Pte

st/P

n

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


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


λl=(Pne/Pcrl)

0.5

Pte

st/P

n

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


λl=(Pne/Pcrl)

0.5

Pte

st/P

n

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


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


λd=(Py/Pcrd)0.5

Pte

st/P

nd

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


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


λd=(Py/Pcrd)0.5

Pte

st/P

nd

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


λd=(Py/Pcrd)0.5

Pte

st/P

nd

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


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


λc=(Py/Pcre)0.5

Pte

st/P

ne

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


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


λc=(Py/Pcre)0.5

Pte

st/P

ne

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


λc=(Py/Pcre)0.5

Pte

st/P

ne

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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 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


true plastic strain

true

stre

ss, k

si


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


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


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


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


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


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


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


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


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


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


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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


true plastic strain

true

stre

ss, k

si


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!


02


xσ 20 ty ≤≤



02


xσ 20 ty ≤≤


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

Documents

Progress Report 4 final.pdf