Finite Element Modelling and Design of Cold-Formed Steel Sections - VERSIÓN EXTENDIDA

7/23/2019 Finite Element Modelling and Design of Cold-Formed Steel Sections - VERSIN EXTENDIDA

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Department of Civil Engineering

Sydney NSW 2006

AUSTRALIA

http://www.civil.usyd.edu.au/

Centre for Advanced Structural Engineering

Finite Element Modelling and Design of

Cold-Formed Stainless Steel Sections

Research Report No R845

Maura Lecce, BASc, MASc

Kim JR Rasmussen, MScEng, PhD

April 2005


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Centre for Advanced Structural Engineering

http://www.civil.usyd.edu.au/

Finite Element Modelling and Design of Cold-Formed

Stainless Steel Sections


Maura Lecce, BASc, MASc

Kim Rasmussen, MScEng, PhD

April 2005

Abstract:This report describes the numerical investigation of cold-formed, thin-walled stainless steel

sections subject to distortional buckling under compression. Austenitic alloy 304 and ferritic

alloys 430 and 3Cr12 were considered. A finite element model calibrated to the data

gathered in a recent experimental programme (Lecce and Rasmussen 2005) shows thatmaterial anisotropy can be ignored and that an accurate calibration model can be achieved

provided nonlinear yielding and enhanced corner properties are included in the model. FE

analyses of more than 570 simple lipped and lipped channels with intermediate stiffeners

covering a distortional buckling slenderness range 0.47 d 3.64 reveal that enhanced

corner properties may become significant for stocky sections with a large corner area (d


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Finite Element Modelling and Design of Cold-Formed Stainless Steel Sections April 2005



Copyright Notice

Department of Civil Engineering, Research Report R845Finite Element Modelling and Design of Cold-Formed Stainless Steel

Sections

2005 Maura Lecce, Kim JR Rasmussen

[email protected]

[email protected]

This publication may be redistributed freely in its entirety and in its original

form without the consent of the copyright owner.

Use of material contained in this publication in any other published works must

be appropriately referenced, and, if necessary, permission sought from the

author.

Published by:


The University of Sydney

Sydney NSW 2006

AUSTRALIA

April 2005

This report and other Research Reports published by The Department of Civil

Engineering are available on the Internet:

http://www.civil.usyd.edu.au


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Table of Contents

Table of Contents ..................................................................................................................3

List of Tables .............................................................................................................................4

List of Figures ............................................................................................................................5

Notation......................................................................................................................................7

1 Introduction ......................................................................................................................9

2 Numerical Investigation ...................................................................................................9

2.1 Scope of Numerical Study.......................................................................................9

2.2 Model Calibration ...................................................................................................9

2.2.1 General...........................................................................................................9

2.2.2 Elastic Perfectly Plastic Material Model ..........................................................12

2.2.3 Nonlinear Plastic Material Model based on Flats Only....................................12

2.2.4 Nonlinear Plastic Material Model based on Flats Only with Initial Anisotropy

13

2.2.5 Nonlinear Plastic Material Model with Enhanced Corner Properties ..............152.2.6 Calibration Model and All Experimental Tests ................................................15

2.3 Simple Lipped Channels vs. Lipped Channels with Intermediate Stiffeners........19

2.4 Modelling Parameters: Further Investigation........................................................21

2.4.1 Investigating Anisotropy ..................................................................................22

2.4.2 Investigating Imperfection Magnitude and Element Type...............................24

2.5 Modelling Parameters: Further Investigation........................................................27

2.5.1 General..............................................................................................................27

2.5.2 Boundary Conditions and Initial Imperfections ...............................................27

2.5.3 Material Properties ...........................................................................................28

2.5.4 FE Test Results.................................................................................................29

2.6 Conclusions of Numerical Investigation ...............................................................333 Evaluation of Current Design Practices..........................................................................33

3.1 General and Scope.................................................................................................33

3.2 Effective Width Approach ....................................................................................34

3.3 Direct Strength Method.........................................................................................44

4 Design Recommendations ..............................................................................................49

4.1 Ultimate Limit States Design Criteria ...................................................................49

4.2 EWA Recommendations .......................................................................................49

4.3 Recommended Direct Strength Design Curves.....................................................51

5 Conclusions ....................................................................................................................54

6 References ......................................................................................................................55

Appendix A..............................................................................................................................57Appendix B ..............................................................................................................................82


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List of Tables

Table 2. 1. Summary of Model Calibration Results ................................................................18

Table 3. 1. Experimental Tests and EWA Predicted Strengths for Simple Lipped Channels 35Table 3. 2. Summary of FE Test to EWA Predicted Strengths for Simple Lipped Channels .36

Table 3. 3. Experimental Tests and EWA Evaluation of Lipped Channels with Intermediate

Stiffeners.........................................................................................................................38

Table 3. 4. Summary of FE test to EWA Predicted Strengths for Lipped Channels with

Intermediate Stiffeners ...................................................................................................39

Table 3. 5 Summary of Test to Current Cold-Formed Carbon Steel DSM Predicted Strengths

for Simple Lipped Channels ...........................................................................................47

Table 3. 6. Summary of Test to Current Cold-Formed Carbon Steel DSM Predicted Strengths

for Lipped Channels With Intermediate Stiffeners.........................................................48

Table 4. 1. Proposed Factors for EWA AS/NZS 4673: Simple Lipped Channels ...............49Table 4. 2. Proposed Factors for EWA EC3 Part 1-4/1-3: Simple Lipped Channels...........50

Table 4. 3. Proposed Factors for AS/NZS 4673 EWA: Lipped Channels with Intermediate

Stiffeners.........................................................................................................................50

Table 4. 4. Proposed Factors for EWA EC3 Part 1-4/1-3: Lipped Channels with


Table 4. 5. Summary of Test to Proposed DSM Predicted Strengths......................................53

Table 4. 6. Proposed Factors for DSM of Stainless Steel Sections: Simple Lipped Channels

........................................................................................................................................53

Table 4. 7. Proposed Factors for DSM of Stainless Steel Sections: Lipped Channels with



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List of Figures

Figure 2. 1. Test Specimens 304D1a (left) and 304D1b (right) for Calibration Model ..........10

Figure 2. 2. Boundary Conditions for Model Calibration FE Analyses ..................................10

Figure 2. 3. Close-up Images of FE Mesh for Model Calibration Analyses from FEMGV 6.4........................................................................................................................................11

Figure 2. 4. Model Calibration Build-up (inward flange movement) (ABAQUS image at

advanced stages of buckling)..........................................................................................11

Figure 2. 5. Model Calibration Build-up (outward flange movement) (ABAQUS image at

advanced stages of buckling)..........................................................................................12

Figure 2. 6. NLP_ISO Mises Stress Distributions at Ultimate Load for Inward (two left) and

Outward (two right) Flange Movement of a Simple Lipped Channel............................13

Figure 2. 7. Experimental and Calibration FE Load vs. End Shortening Curves for Simple

Lipped Channels.............................................................................................................16

Figure 2. 8. Experimental and Calibration FE Load vs. End Shortening Curves for Lipped

Channels with Intermediate Stiffeners ...........................................................................17Figure 2. 9. von Mises Membrane Stresses for Simple lipped Channel (top row) and Lipped

Channel with Intermediate Stiffeners (bottom row) at Maximum Load........................19

Figure 2. 10. Progression of Stress Distributions for a Simple Lipped Channel .....................20

Figure 2. 11. Progression of Stress Distributions in a Lipped Channel with Intermediate

Stiffeners.........................................................................................................................21

Figure 2. 12. Investigating Anisotropy b/t=54.........................................................................22

Figure 2. 13. Distribution of von Mises stresses for Isotropic (left) and 50% Anisotropy

(right) Analyses. (post-buckling end deflection approximately 3mm)..........................23

Figure 2. 14. 50% Anisotropy with (left) and without (right) Orientation (last increment,

post-buckling).................................................................................................................23

Figure 2. 15. Investigating Anisotropy b/t=106.......................................................................24Figure 2. 16. Investigating Anisotropy b/t=26.5......................................................................24

Figure 2. 17. Effects of Imperfection Magnitude for 304D1...................................................25

Figure 2. 18. Study on Imperfection Values and S4R/S4 Elements.......................................26

Figure 2. 19. Boundary Conditions of FE Tests ......................................................................28

Figure 2. 20. FE and Experimental Results: 304 Simple Lipped Channels.............................29

Figure 2. 21. FE and Experimental Results: 430 Simple Lipped Channels.............................30

Figure 2. 22. FE and Experimental Results: 3Cr12 Simple Lipped Channels.........................30

Figure 2. 23. FE and Experimental Results: 304 Lipped Channels with Intermediate Stiffeners

........................................................................................................................................31

Figure 2. 24. FE Test and Experimental Results: 430 Lipped Channels with Intermediate

Stiffeners.........................................................................................................................31

Figure 2. 25. FE and Experimental Results: 3Cr12 Lipped Channels with Intermediate

Stiffeners.........................................................................................................................32

Figure 2. 26. Austenitic 304 and Ferritic 430 and 3Cr12 Test Results....................................33

Figure 3. 1.Aeff/Agvs. dfor Alloy 304 (Aeff determined by AS/NZS 4673) ...........................40

Figure 3. 2. 304 Test to AS/NZS 4673 Predicted Strengths vs.Aeff/Ag....................................40

Figure 3. 3. 430/3Cr12 Test to 4673 Predicted Strengths vs.Aeff/Ag .......................................41

Figure 3. 4. 304 Test to EC3 Part 1-4/1-3 Predicted Strengths vs.Aeff/Ag ...............................41

Figure 3. 5. 430/3Cr12 Test to EC3 Part 1-4/1-3 Predicted Strengths vs.Aeff/Ag ....................42

Figure 3. 6. 304 Test to AS/NZS 4673 Predicted Strengths vs. Percent Corner Area.............42Figure 3. 7. 430/3Cr12 Test to EWA AS/NZS 4673 Predicted Strengths vs. Percent Corner

Area ................................................................................................................................43


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Figure 3. 8. 304 Test to EWA EC3 Part1-4/1-3 Predicted Strengths vs. Percent Corner Area

........................................................................................................................................43

Figure 3. 9. 430/3Cr12 Test to EC3 Part 1-4/1-3 Predicted Strengths vs. Percent Corner Area

........................................................................................................................................44

Figure 3. 10. 304 Test Data Compared with Current DSM Curves for Cold-Formed Carbon

Steel ................................................................................................................................45Figure 3. 11. 430 Test Data Compared with Current DSM Curves for Cold-Formed Carbon

Steel ................................................................................................................................45

Figure 3. 12. 3Cr12 Test Data Compared with Current DSM Curves for Cold-Formed Carbon

Steel ................................................................................................................................46

Figure 4. 1. Proposed DSM Distortional Buckling Design Curve for Cold-Formed Austenitic

Stainless Steel Sections ..................................................................................................52

Figure 4. 2. Proposed DSM Distortional Buckling Design Curve for Cold-Formed Ferritic

Stainless Steel Sections ..................................................................................................52


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Notation

Ac = corner area

Aeff = effective area (total effective area also represented byAe,t)

Ag = gross areaBf = overall flange width

Bl = overall lip width

Bw = overall web width

COV = coefficient of variation

D = dead load

Eo = initial elastic modulus

Es = secant modulus

Et = tangent modulus

Fm = ratio of mean to nominal cross-sectional properties

L = column length; live load

Lcr = critical distortional buckling half-wavelengthLC = longitudinal compression

LT = longitudinal tension

Mm = ratio of mean to nominal material properties

Pn = design strength

Pu = distortional test ultimate load (also Pu,t)

Pu,FE = finite element distortional test ultimate load

Pu,T = experimental distortional test ultimate load

Pu,sc = stub column ultimate load

Vm = COV of F

Vm = COV ofM

b = element widthd = section depth

di = depth of intermediate stiffener

di,w = width of intermediate stiffener

e = parameter used in the modified Ramberg-Osgood equation

f = stress

fcr = critical buckling stress

fn = design strength

fy = yield strength

fy,c = predicted corner yield strength

fy,f

= specified yield strength of the flats (virgin material)

fu = ultimate strength

fu,f = specified ultimate capacity of the flats (virgin material)

k = plate buckling coefficient

kf = plate buckling coefficient of the flange

m = parameter used in modified Ramberg-Osgood equation

n = Ramberg-Osgood parameter

r = centerline radius

ri = inner corner radius

t = thickness

= reliability index for ultimate limit states design criteria

= straine = engineering strain

tp = true plastic strain

0.01 = strain, 0.01%


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0.2 = strain, 0.2%

= resistance factor

= buckling slenderness

d = distortional buckling slenderness

l = local buckling slenderness

= normal stress

e = engineering stress

t = true stress

u = ultimate stress

y = yield stress

0.01 = 0.01% proportionality stress

0.01,c = 0.01% proportionality stress of corners (cold-worked)

0.01,f = 0.01% proportionality stress of flat (virgin material)

0.2 = 0.2% proof stress

0.2,c = 0.2% proof stress of corners (cold-worked)

0.2,f = 0.2% proof stress of flat (virgin material)

= imperfection

d = measured or recommended imperfection at the flange-lip junction

l = measured or recommended imperfection at the centre of the web element

= shear stress

d = distortional buckling reduction factor

d,f = distortional buckling reduction factor for the flange

d,w = distortional buckling reduction factor for the web


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1 Introduction

The purpose of this report is to present the numerical investigation of cold-formed stainless

steel sections, based on a recent experimental progamme on cold-formed simple lipped and

lipped channels with intermediate stiffeners made from austenitic 304 and ferritic 430 and

3Cr12 alloys (Lecce and Rasmussen 2005). The experimental and finite element (FE) test

data gathered were used to evaluate current design procedures available for stainless steel.

However, since design codes usually evolve along with cold-formed carbon steel codes, these

were examined also. Finally, design recommendations are made for the design of stainless

steel sections failing in the distortional buckling mode.

2 Numerical Investigation

2.1 Scope of Numerical StudyNumerical studies were carried out to increase the number of data points, or test points, from

which to draw conclusions and recommendations regarding the distortional buckling

behaviour of stainless steel channel sections. First, a model calibration study was conducted

using the 19 experimental distortional buckling tests by considering the experimentally

measured material and geometric properties. Further numerical studies with respect to

anisotropy, imperfections and element type (S4R vs S4) were carried out to confirm that the

calibration model is valid for a greater range of section geometries and material properties.

Following this, a total of 270 simple lipped channels with a distortional buckling slenderness

range of 0.47 d3.64 and 306 lipped channels with intermediate stiffeners with a

distortional buckling slenderness range of 0.47 d 3.27 were tested by finite element

analyses. To study the effects of enhanced corner properties, typical brake-pressed r/t ratios

of 1 and 2.5 were chosen.

The FE package ABAQUS (2001), Version 6.4, was used for the numerical analyses

and input files were created using the engineering software FEMGV6.4-02 (FEMSYS 2002).

Specific modelling issues are described in the following subsections.

2.2 Model Calibration

2.2.1 GeneralAll data used to develop the calibration model are reported in Lecce and Rasmussen (2005).

Fixed-end boundary conditions with a uniform displacement applied to one end was used in

the model and represented the actual experimental conditions. To save computational time,

symmetrical failure mode was assumed about the mid-web axis and only half of the cross-

section was modelled. This assumption is valid because the experimental tests developed

essentially symmetrical distortional buckling deflections as shown in Figure 2.1. The

boundary conditions are shown in Figure 2.2 and are given with respect to the ABAQUS 1-2-

3 axes which correspond to the x,y and z axes. Over 9000 elements, typically 5mm by 5mmsquare were used for the model calibration described here. Images of the mesh from different

perspectives are provided in Figure 2.3. The FE models representing the other experimental

tests are similar. Average measured geometric dimensions for test specimens 304D1a and


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304D1b were used in the model (Lecce and Rasmussen 2005). Two sets of analyses were

carried out; one for a positive (+ve) imperfection value, which would trigger inward flange

movement and another for a negative (-ve) imperfection value which would trigger outward

flange movement (as experienced in the experimental test). In the following discussions

these will be referred to as inward model and outward model. The imperfection amplitude,

0.25mm, is equal to the average of the absolute imperfection values measured at mid-heightof the flange-lip junction of all 304D test specimens (304D1a, 304D1b, 304D2a and

304D2b), a method adopted by Hasham and Rasmussen (2002). (The imperfection sign

convention used in Lecce and Rasmussen (2005) is opposite to that used in FE modelling).

Figure 2. 1. Test Specimens 304D1a (left) and 304D1b (right) for Calibration Model

Figure 2. 2. Boundary Conditions for Model Calibration FE Analyses


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Figure 2. 3. Close-up Images of FE Mesh for Model Calibration Analyses from FEMGV 6.4

For each set of analyses the material model was constructed by considering nonlinear

stress-strain hardening, initial anisotropy, and enhanced corner properties. The discussions in

the following subsections make reference to Figures 2.4 and 2.5.

0

25

50

75

100

125

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

End Shortening (mm)

Load(kN)

304D1 Test

PP_(1.23)

NLP_ISO_(1.00)NLP_ANISO=5%_(1.01)

NLP_EC_ISO_(1.03)

NLP_EC_ANISO=5%_(1.04)

Imperfection=+0.25mm

(flanges move in)

(Pu,FE/Pu,T)

Figure 2. 4. Model Calibration Build-up (inward flange movement) (ABAQUS image at

advanced stages of buckling)


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0

25

50

75

100

125

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

End Shortening (mm)

Load(kN)

304D1 Test

PP_(1.14)

NLP_ISO_(0.94)

NLP_ANISO=5%_(0.95)

NLP_EC_ISO_(1.00)

NLP_EC_ANISO=5%_(1.01)

Imperfection=-0.25mm

(flanges move out)

(Pu,FE/Pu,T)

Figure 2. 5. Model Calibration Build-up (outward flange movement) (ABAQUS image

at advanced stages of buckling)

2.2.2 Elastic Perfectly Plastic Material ModelThe first material model considered was an elastic perfectly-plastic material, labelled PP and

the plastic yield stress was equal to the experimentally measured 0.2(242MPa) determined

from the longitudinal compression coupons of the flats (virgin material). Unsurprisingly this

simplistic material model offered a poor match to the experimental test, as shown in Figures

2.4 and 2.5. Studies in the past have also shown that an elastic perfectly-plastic analysis for

stainless steel leads to erroneous results (Rasmussen et al. 2003). The ultimate FE load toultimate experimental test load ratios, Pu,FE/Pu,T, (given in the legends of Figures 2.4 and 2.5)

are 1.23 and 1.14 for inward and outward models, respectively. Evidently by simply

changing the sign of the imperfection, different strengths can be obtained. In this example

the ultimate FE load, Pu,FE, for the inward model is 7.4% greater than the outward model.

Extensive studies (Silvestre and Camotim 2004) on this phenomenon for cold-formed carbon

steel have shown that, for simple lipped channels the elastic post-buckling response leads to a

greater ultimate load if the flanges move inwards and the same result is found here with the

PP model.

2.2.3 Nonlinear Plastic Material Model based on Flats OnlyThe next model included stainless steel material nonlinearity. The true stresses, t, and true

plastic strains, tp, as required by ABAQUS, were derived from the longitudinal compression

engineering stress-strain data of the 304 flat material using the following equations:

( )eet += 1 (1)

( )o

tetp

E

+= 1ln (2)

where e and e are the engineering stresses and strains. For this model, only the flat

properties were considered and applied to the entire cross-section. This nonlinear plastic

(NLP) hardening model assumes an initially isotropic (ISO) yield surface described by the


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von Mises criterion and expands, or hardens, isotropically. The resulting load vs. end

shortening NLP_ISO curves for inward and outward models correspond to Pu,FE/Pu,Tratios of

1.00 and 0.94 in Figures 2.4 and 2.5, respectively. Clearly the NLP_ISO model is a

significant improvement from the previous PP model and the imperfection sign used in the

model becomes less important in terms of ultimate load. Nevertheless, the shape of the load-

end shortening curve is better described by the outward model. It is interesting to examinethe von Mises stress distributions for the NLP_ISO model at ultimate load and these are

shown in Figure 2.6 for inward and outward flange movement (the contours are scaled with

respect to the ultimate load and deflections are amplified).

Figure 2. 6. NLP_ISO Mises Stress Distributions at Ultimate Load for Inward (two left)

and Outward (two right) Flange Movement of a Simple Lipped Channel

At ultimate load, the section with outward flange movement has achieved greater

deflections compared with the model with inward flange movement. As shown in the

contours, both models show high stresses at the lip but the model with outward flange

movement shows higher stresses at the flange-web corner and is likely a consequence of

attaining greater deflections. Furthermore it is evident that the stresses in flange-lip region of

the inward model is greater and consequently so is the ultimate load.

2.2.4 Nonlinear Plastic Material Model based on Flats Only with Initial

AnisotropyAnother material characteristic considered was the anisotropy. By using ABAQUS built-in

modelling tools anisotropy was incorporated in the initial yield surface according to the Hill

criteria described by:

[ ] 2/1212213223222112113323322 222)()()()( NMLHGFf ++++++= (3)

where F, G, H, L, M and N are given by:

+=

+= 211

233

222

211,0

233,0

222,0

2

0 11121111

2 RRRF


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+=

+=

222

211

233

222,0

211,0

233,0

20 111

2

1111

2 RRRG

+=

+=

233

222

211

233,0

222,0

211,0

20 111

2

1111

2 RRRF

=

=

223

223,0

20 1

2

3

2

3

RL

=

=

213

213,0

20 1

2

3

2

3

RM

=

= 212

212,0

2

01

2

3

2

3

RN

(4)

where 0 is the reference yield stress and 0 is the reference shear yield stress, 300 = .

The yield criterion only has this form when the principal axes of anisotropy are the axes of

reference. That is, for cold-rolled sheets, the principal axes lie in the direction of rolling,

transversely in the plane of the sheet and normal to this plane (Hill 1950). The 11 is the

stress in the direction of rolling, 22is transverse to the direction of rolling and 33is normal

(or through-thickness) anisotropy. By default ABAQUSassumes the direction of rolling is in

the global x-1 axis and maintains this alignment unless the user defines otherwise. ABAQUS

requires the user to define the following stress ratios to satisfy the Hill criteria:

;00.1;05.1;00.10

3333

0

2222

0

1111 ======

RRR

00.1;00.1;01.10

2323

0

1313

0

1212 ======

RRR (5)

For the example considered here, the reference stress of the 304 material, 0is equal

to the 0.2value (242MPa) given by the longitudinal compression coupon tests of the flats.

Since the test specimens were loaded in the longitudinal direction (with respect to rolling),

the 11value is equal to 0, givingR11= 1.00. The 22value is equal to the transverse yieldstress (254MPa) of the compression coupons and thus R22 = 1.05. The through thickness

anisotropy was assumed to be unity (ie; 33=11=0) which is reasonable for thin plates; thus

R33 = 1.00. The reference shear yield stress, 300 = and 31212 = where 12 is the

diagonal yield stress (244MPa) and thus R12 = 1.01. The ratios R13=R23= 1.00. These

strength ratios are defined under the *PLASTIC card in the ABAQUSinput file. The material

model assumes that initial anisotropy remains constant and the plastic hardening, or

expansion of the yield surface occurs isotropically. For a 3D model with multiple surfaces, it

is important to define element orientation to ensure that the anisotropy is aligned correctly for

every element and avoid erroneous results. In ABAQUS, this can be accomplished by using

the *ORIENTATION card following the element definition. For the sections modelled in

this study, it was important to map the local element longitudinal 11 direction axis to

coincide with the global z-3-axis (direction of loading) and the transverse direction 22

normal to the 11 direction. (Failure to define the orientation can lead to significant errors,

and this is demonstrated in a separate study presented in Section 2.4).


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The results of the nonlinear plastic model with initial anisotropy, NLP_ANISO=5%,

amounted to less than 1% difference in ultimate load compared to the initially isotropic

model, NLP_ISO and the Pu,FE/Pu,T ratios are 1.01 and 0.95 in Figures 2.4 and 2.5,

respectively. These results suggest that material anisotropy has little effect on the ultimate

strength of the sections tested and may be ignored. However, to confirm that this is true also

for sections of different material thicknesses and higher degrees of anisotropy furtherinvestigation was carried out and is described in Section 2.4.

2.2.5 Nonlinear Plastic Material Model with Enhanced Corner PropertiesThe basic isotropic nonlinear plastic hardening model was modified to include the enhanced

corner properties, which were applied strictly to the corner geometry of the section. The true

stress and true plastic strain properties were derived from the corner coupon stress-strain data

using Eqns. 1 and 2. The NLP_EC_ISO Pu,FE/Pu,T ratios are 1.03 and 1.00 for +ve

imperfection and ve imperfections, respectively (c.f. Pu,FE/Pu,T= 1.00,Pu,FE/Pu,T= 0.94 for

NLP_ISO). Both sets of NLP_EC_ISO results show good agreement with the test result but

the load versus end shortening behaviour is better described by the outward model whichsimulates the actual flange movement of the test. Overall, the sign of imperfection seems to

be less important for the inelastic stainless steel material model compared to the elastic

plastic material model. Finally initial anisotropy was included (NLP_EC_ANISO=5%) and

again results show that the effect of anisotropy is negligible.

2.2.6 Calibration Model and All Experimental TestsThe above discussion shows that material nonlinearity and enhanced corner strengths govern

the ultimate section strength of a stainless steel section and initial anisotropy can be

neglected. Thus the NLP_EC material model was used to evaluate all experimental tests.

The imperfection sign does not have a great impact on the ultimate load but rather shows

more influence in the post-ultimate range as seen in the experimental tests. Nevertheless, theimperfection sign used for the models simulated the actual flange movement observed during

the tests. For sections which developed both inward and outward flange movement along the

specimen length during the test, the imperfection sign assumed in the model was generally

positive. The load vs. end shortening curves obtained from ABAQUSare shown along with

the experimental plots in Figures 2.7 and 2.8 for simple lipped channels and channels with

intermediate stiffeners, respectively. The ultimate loads are summarized in Table 2.1.


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0

20

40

60

80

100

120

0.0 0.5 1.0 1.5 2.0 2.5

End Shortening (mm)

Load

(kN)

304D2a (in)

304D2b (in)

FE_imp=+0.25mm (in)

0

5

10

15

20

25

30

35

40

45

0.0 0.5 1.0 1.5 2.0 2.5End Shortening (mm)

Load

(kN)

430D1a (in/out)

430D1b (in/out)

FE_imp=+0.10mm (in/out)

0

5

10

15

20

25

30

35

40

45

50

0.0 0.5 1.0 1.5 2.0 2.5

End Shortening (mm)

Load(kN)

430D2 (out)

FE_imp=-0.10mm (out)

0

5

10

15

2025

30

35

40

45

0.0 1.0 2.0 3.0

End Shortening (mm)

Load(kN)

430D3a (in/out)

430D3b (in/out)


0

20

40

60

80100

120

140

160

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN

)

3Cr12D2 (in/out)

3Cr12D1 (in/out)


Figure 2. 7. Experimental and Calibration FE Load vs. End Shortening Curves for

Simple Lipped Channels


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0

20

40

6080

100

120

140

160

0.0 1.0 2.0 3.0 4.0 5.0 6.0

End Shortening (mm)

Load

(kN)

304DS1a (out)

304DS1b (in)

FE_imp=+0.22mm (in)


0

10

20

30

40

50

60

70

0.0 1.0 2.0 3.0 4.0 5.0

End Shortening (mm)

Load

(kN)

430DS1 (in/out)

FE_imp=-0.23mm (in/out)

0

10

20

30

40

50

60

70

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN)

430DS2 (in)

FE_imp=+0.23mm (in)

0

10

20

30

40

50

60

70

0.0 1.0 2.0 3.0 4.0 5.0 6.0

End Shortening (mm)

Load(kN)

430DS3 (out)


0

10

20

30

40

50

60

70

80

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN)

430DS4 (in)

FE_imp=+0.23mm (in)

0

20

40

60

80

100

120

140

160

180

0.0 1.0 2.0 3.0 4.0 5.0 6.0

End Shortening (mm)

Load(kN)

3Cr12DS2 (in/out)

3Cr12DS1 (in/out)


Figure 2. 8. Experimental and Calibration FE Load vs. End Shortening Curves for

Lipped Channels with Intermediate Stiffeners

Note that the 304DS1a and 304DS1b tests were the only set of experimental twin testswhere one specimen exhibited inward flange movement and the other exhibited outward

flange movement. The test and FE curves for these tests provide a clear example of the

agreement that can be achieved if the correct flange movement is simulated (see Figure 2.8).

From the plots in Figures 2.7 and 2.8 and Table 2.1, it is evident that very good

agreement between the calibration model and experimental tests was achieved, with a mean

Pu,FE/Pu,T ratio of 0.99 and a coefficient of variation (COV) of 0.0263. The same set of FE

analyses were conducted using S4 elements but the differences in ultimate load were less than

0.5% (and at least double the computational time) thus confirming that S4R elements were

adequate for the analyses of the experimental tests.


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Table 2. 1. Summary of Model Calibration Results

da P u,T P u,FE P u,FE/P u,T

Specimen ID mm kN kN

304D1a 102 0.99

304D1b 101 1.00

304D2a 104 0.99

304D2b 104 0.99

430D1a 39 1.02

430D1b 39 1.02

430D2 -0.10 45 43 0.96

430D3a 40 0.94

430D3b 39 0.96

3Cr12D1a 138 1.02

3Cr12D1b 139 1.01

304DS1a -0.22 132 138 1.04

304DS1b +0.22 134 133 0.99

430DS1 -0.23 60 58 0.96

430DS2 +0.23 62 60 0.97

430DS3 -0.23 64 64 1.00

430DS4 +0.23 72 69 0.96

3cr12DS1a +0.16 163 0.98

3cr12DS1b +0.16 161 0.99

mean 0.99

stdv 0.026

COV 0.0263

+0.10

+0.10

160

aimperfection amplification applied to the critical distortional buckling

mode. Negative sign means outward flange movement (opposite sign to

experimental data)

+0.36

101

103

40

38

140

-0.25

+0.25


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2.3 Simple Lipped Channels vs. Lipped Channels with

Intermediate StiffenersBy FE modelling, it is easy to plot the stress contours and visualize the development of

stresses and deflections of simple lipped channels compared with those of lipped channelswith intermediate stiffeners. To examine this, two sections are considered: the 304D1a/b

simple lipped channel modelled in Section 2.2 with overall dimensions of Bw= 106mm,

Bf= 90mm, Bl= 12.8mm, (d = 0.96) and the 304DS1a/b section with similar overall

dimensions of Bw= 122mm, Bf= 90.6mm, Bl= 12.8mm (d = 0.85). The section length

(800mm) and fixed-end conditions are identical for the two sections. For demonstration

purposes, the material model does not include enhanced corner properties. The von Mises

membrane stress distribution, at ultimate load, is depicted in Figure 2.9 for sections with

inward flange movement.

Figure 2. 9. von Mises Membrane Stresses for Simple lipped Channel (top row) and

Lipped Channel with Intermediate Stiffeners (bottom row) at Maximum Load

From these different views, one can see that the highest membrane stresses are

developed around the flange/lip area for both section types and that intermediate stiffener

provides an obstruction to the spread of stress to the flange-web corner.

Figures 2.10 and 2.11 show the evolution of the mid-surface stresses for the two

different cross-section types through seven images which correspond to the points in the

accompanied load vs. end shortening graph. The contours are scaled with respect to the

stress state of the last increment considered.


21/194




Figure 2. 10. Progression of Stress Distributions for a Simple Lipped Channel

0

20

40

60

80

100

120

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN)


22/194




Figure 2. 11. Progression of Stress Distributions in a Lipped Channel with Intermediate

Stiffeners

The first four images show the progression of stresses leading up to the ultimate load

(fifth image) and the last two show the post-peak stress development. By comparing Figures

2.10 and 2.11, it is clear that the simple lipped channel develops a higher concentration of

stresses at the flange-web corner whereas the stresses in the channel with intermediate

stiffeners involve a much larger area. This suggests that the latter is much more effective atdistributing the load and therefore more of the cross-section reaches higher loads. This

agrees with experimental results where channels with intermediate stiffeners exhibited greater

material nonlinearity at ultimate load. Overall, the section corners, including intermediate

stiffeners, are responsible to carry higher loads and for this reason it becomes important to

consider the enhanced corner properties in the evaluation of the section strength.

2.4 Modelling Parameters: Further InvestigationThe calibration model developed in Section 2.2 provided excellent agreement experimental

tests and cover a distortional buckling slenderness range of 0.76 d1.10. This section

will confirm that the FE model also works for a greater range of sections by investigating

further the effects of anisotropy, imperfection value and element type (S4R vs S4).

0

20

40

60

80

100

120

140

0.0 1.0 2.0 3.0

End Shortening (mm)

Load(kN)


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2.4.1 Investigating AnisotropyFrom the calibration model, it was evident that material anisotropy had little influence on the

ultimate strength. However, the actual measured anisotropy was relatively low (R22=1.05)

and the question remains whether a higher degree of anisotropy would also have negligible

influence. Three sections with thicknesses 1.96mm, 1mm and 4mm were considered with

overall geometric properties of experimental test 304D1a/b. The corresponding b/t ratios are54, 106 and 26.5. The imperfection value remained constant (-0.25mm) and the thicknesses

were varied [t=1.96mm, (b/t=54); t=1mm, (b/t=106); t=4mm, (b/t=26.5)]. For each section,

two anisotropy values were checked; the measured experimental anisotropy of 5% and an

exaggerated anisotropy of 50%.

The load versus end shortening curve for b/t=54 is given in Figure 2.12. Evidently,

there is negligible difference between the isotropic (ISO) curve and the 5% anisotropy curve

(ANISO=5%) with a marginal difference in ultimate load of approximately 1%. If the

material anisotropy is exaggerated to 50% (ANISO=50%), the maximum increase in ultimate

load is less than approximately 4%. Figure 2.13 displays the von Mises stresses at the mid-

surface of the shell elements for an isotropic model (left) and a model with 50% anisotropy

(right). The comparison is made in the post-ultimate range where the axial compression is

approximately 3mm and the load has dropped to approximately 65% of the ultimate. The

contour plots show, as expected, that the transverse stresses developed are greater in the

anisotropic model. For the same b/t ratio, material orientation was omitted from the analysis

[see cyan curve for ANISO=50% (No orient.), Figure 2.12] resulting in a significantly

different behaviour with an ultimate load 17% greater than the correctly oriented model. The

von Mises stress distributions in the post-ultimate state for the models ANISO=50% and

ANISO=50% (No orient.) is given in Figure 2.14 and shows the significantly different stress

distributions. This example is given to demonstrate the importance of correct element

orientation when anisotropy is applied.

0

20

40

60

80

100

120

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN)

ISO_t=1.96mm

ANISO=5%_t=1.96mm

ANISO=50%_t=1.96mm

ANISO=50% (No orient.)_t=1.96mm

Figure 2. 12. Investigating Anisotropy b/t=54


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Figure 2. 13. Distribution of von Mises stresses for Isotropic (left) and 50% Anisotropy

(right) Analyses. (post-buckling end deflection approximately 3mm)

Figure 2. 14. 50% Anisotropy with (left) and without (right) Orientation (last

increment, post-buckling)

Figures 2.15 and 2.16 show the results for b/t=106 and b/t=26.5, respectively. The

exaggerated anisotropy generally has the effect of improving material ductility with little

strength benefits.


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0

5

10

15

20

25

30

35

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN)

ISO_t=1mm

ANISO=5%_t=1mm

ANISO=50%_t=1mm

Figure 2. 15. Investigating Anisotropy b/t=106

0

50

100

150

200

250

300

0.0 2.0 4.0 6.0 8.0

End Shortening (mm)

Loa

d(kN)

ISO_t=4mm

ANISO=5%_t=4mm

ANISO=50%_t=4mm

Figure 2. 16. Investigating Anisotropy b/t=26.5

Overall, from a practical design-engineering viewpoint, the simple isotropic

hardening plasticity model suffices for statically loaded stainless steel members. However,

anisotropy is affected by material deformation and cold working and more accurate material

modelling may become important for members under cyclic loading. This was shown by

Olsson (1998) and Gozzi (2004) who developed a material model to account for, among other

phenomenological behaviour, the effects of stainless steel material anisotropy and its

evolution under repeated loading.

2.4.2 Investigating Imperfection Magnitude and Element TypeAnother modelling issue to consider in greater detail is the imperfection magnitude used. For

the model calibration described in Section 2.2 (section 304D1a/b), the magnitude of the

imperfection (-0.25mm) was equal to the average of the absolute measured imperfectionvalues at the flange-lip junction located at mid-height. However, to investigate the sensitivity

to imperfections two other magnitudes were considered including the average of the absolute

maximum measured imperfections at the flange-lip junction (-0.32mm) and a calculated


26/194




imperfection value according to the adopted Walker (1975) equation given by

d = 0.3t(0.2/cr)0.5

, where cris the critical elastic distortional buckling stress obtained from

the elastic buckling analysis (0.2= 242MPa,cr= 263MPa, t = 1.96mm; d= -0.56mm).

Figure 2.17 shows the load vs. end shortening results of test specimen 304D1 (used in

the model calibration of Section 2.2) and the FE analyses. Evidently, even if the imperfection

value is doubled, the difference in ultimate load is marginal. It should be noted here that thedistortional buckling slenderness for this fixed ended model is d = (0.2/cr)

0.5 = 0.96.

0

20

40

60

80

100

120

0.0 1.0 2.0 3.0 4.0

End Shortening (mm)

Load(kN)

304D1

NLP_EC_imp=-0.25mm(average at mid-length)

NLP_EC_imp=-0.32mm(max average)

NLP_EC_imp=-0.56mm(walker equation)

Figure 2. 17. Effects of Imperfection Magnitude for 304D1

Of the three imperfection values considered, only the Walker imperfection was

calculated from the material and distortional buckling properties which makes the Walker

equation particularly useful when experimentally measured imperfections are unavailable.

Other methods to estimate initial geometric imperfections have been suggested by Schaferand Pekoz (1998). In their work, two types of imperfections are defined; Type 1: l for local

buckling of the web, and Type 2: d for distortional buckling of the flange-lip and are both

directly proportional to the section thickness. In the suggested probabilistic method,

l = 0.14t and l = 0.66t (d = 0.64t and d = 1.55t ) respectively correspond to 25% and

75% probability that the imperfections will be less than these maximums and will be referred

to as Schafer_25% and Schafer_75%. Type 1 imperfection magnitudes were comparable

with the experimentally measured imperfections of the flange-lip corners and thus only

= 0.14t (Schafer_25%) and = 0.66t (Schafer 75%) were considered. According to these

equations, the imperfections are independent of the section slenderness and increase for

increasing section thickness. Therefore, a stockier section, which is conceivably lesssusceptible to develop larger initial imperfections, is penalized. This is unlike the Walker

equation where the imperfection amplitude is proportional to the square root of the

distortional buckling slenderness value so that, keeping all other dimensions the same the

imperfections decrease for increasing section thickness.

Three pin-ended sections with l = 3.122 (local buckling critical),d = 1.203 and

d = 0.637 (distortional buckling critical) were modelled. Both S4R and S4 elements were

considered and the results are presented in Figure 2.18.


27/194




0

1

2

3

4

5

6

7

0 2 4 6 8End Shortening (mm)

Load(kN

)

S4R_Walker_imp=0.44mm

S4R_Schafer_25%_imp=0.07mm


S4_Walker_imp=0.44mm

S4_Schafer_25%_imp=0.07mm


l =3.122

d=2.240

t=0.5mm

0

10

20

30

40

50

60

0 2 4 6 8 10End Shortening (mm)

Load(kN)







l =1.028

d=1.203

t=1.5mm

0

50

100

150

200

250

0 2 4 6 8 10 12 14End Shortening (mm)

Load(kN)







l =NA

d=0.637

t=4.0mm

Figure 2. 18. Study on Imperfection Values and S4R/S4 Elements

The Walker and Schafer_25% imperfections produced similar results for all sections

whereas the Schafer_75% became significantly conservative for the stockier section (see plotwhere d = 0.637). The Walker equation is suitable for all sections and conveniently takes

into account the section slenderness and material strength without being overly conservative.


28/194




Finally, the results show that there is no significant advantage to using the S4 elements rather

than the S4R.

2.5 Modelling Parameters: Further Investigation

2.5.1 GeneralThe sections tested by FE analyses were proportioned to fail by distortional buckling and the

geometries generally fell within the limitations outlined by the cold-formed carbon steel code

NAS Appendix 1 (2004). The distortional buckling slenderness values ranged from

0.47d3.64 for simple lipped channels and 0.47 d3.27 for lipped channels with

intermediate stiffeners. Section thickness ranged from 1mm to 8mm and r/t ratios of 1 and

2.5 were considered. The corner area to gross area, Ac /Ag ranged from 1.57% to 47%.

Section geometry and material properties are given in Appendix A.

2.5.2 Boundary Conditions and Initial ImperfectionsThinWall (Papangelis and Hancock 1995) finite strip elastic buckling analyses, which

assumes pinned ends, were conducted for each section to determine the critical elastic

buckling stress and buckling half-wavelength. This information was used to initially

construct an ABAQUSmodel consisting of a column seven half-wavelengths long with fixed-

end boundary conditions and intermediate pins at each half-wavelength at the flange-web

corners. The boundary conditions prevented overall buckling, whilst allowing distortional

buckling to develop. The critical distortional failure occurred at the middle of the fourth half-

wavelength (half the total column length) essentially under pin-ended conditions and this was

confirmed by the excellent agreement found with the critical buckling stress obtained by

ABAQUS with that determined by ThinWall. To save computational time, distortionalbuckling symmetry was assumed for all tests and this allowed one quarter (one half of the

total length and one half of the cross-section) of the model to be analysed with the

appropriate boundary conditions. Figure 2.19 shows an image of a channel with intermediate

stiffeners created in FEMGV6.4-02. As shown in the image, three and a half critical

distortional buckling lengths are modeled and at the fixed end, rotations about all axes are set

to zero and only displacements in the 3-axis (z-axis) direction are allowed. At every

distortional buckling critical length, pins are placed at the web-flange corner and

displacements in the 2-axis are set to zero. At the half-length edge, symmetry is used and the

boundary conditions include zero displacement in the 3-axis direction, and zero rotation

about the 1-axis and 2-axis. The number of elements used varied for each test from

approximately 3000 to over 20,000 elements, with a coarser mesh at the fixed end and a finermesh at the half-length end where critical distortional buckling failure occurs.


29/194




Figure 2. 19. Boundary Conditions of FE Tests

Initial imperfections were included in the nonlinear post-buckling Static Riks analysis

as an amplification of the critical elastic distortional buckling mode. The imperfection

amplitude, d, was based on the Walker equation (1975) where d= 0.3t(fy/fcr)0.5, and fcr is

the critical elastic distortional buckling stress obtained from the FE elastic buckling analyses.

2.5.3 Material PropertiesThe material properties are based on the longitudinal compression properties for the 304, 430

and 3Cr12 material specified in the AS/NZS 4673 (2001). Enhanced corner properties werecalculated for r/t ratios of 1 and 2.5, where r is the centreline corner radius and tis the section

thickness, and are based on the AS/NZS 4673 (2001) model for predicting corner strength.

The design corner yield strength to design flat yield strengthfy,c/fy,f for the 304 material was

2.34 and 1.85 for r/t=1.0 and r/t=2.5, respectively whereas the fy,c/fy,f ratios for the 430 and

3Cr12 material was approximately 1.77 (r/t=1.0) and 1.56 (r/t=2.5). The full-range stress-

strain curve proposed by Rasmussen (2003) was used to construct stress-strain data needed

for the material modelling according to the following equations:

>+

+

+

=

y

m

yu

y

u

y

y

n

y

ffforff

ff

E

ff

ffforf

f

E

f

2.0

2.0

0

002.0

(6a)

)5(0375.01

1852.0

=

n

e

f

f

u

y (6b)

en

EE

/002.01

02.0

+= (6c)

u

y

f

fm 5.31+= (6d)


30/194




0E

fe

y= (6e)

True stress and true plastic strains were then calculated using Eqns 1 and 2. Three

sets of FE analyses were carried out for each alloy; the first set used material properties of the

flats only for the entire cross-section and the second and third sets used enhanced materialproperties of the corners according to a ratio of r/t=1.0 and r/t=2.5, applied to the corners

only.

2.5.4 FE Test ResultsAll FE test results are presented in Appendix A. The FE data points have been plotted in

Figures 2.20 to 2.25 in terms offu/fyversus d, wherefuis the ultimate average stress obtained

by FE or experimental tests,fyis the specified minimum yield strength of the flat (or virgin)

material and d is the distortional buckling slenderness given by d = (fy/fcr)0.5

. The FE

results are labeled flats, r/t=1and r/t=2.5 representing the material properties used inthe model and Test represents the experimental test results.

The experimental test sections, which had an r/t ratio of approximately 2.5 are in good

agreement with the FE r/t=2.5 test results for all alloys. By comparing the FE results for

alloy 304 with those for alloys 430 and 3Cr12, it is evident that the enhanced corner

properties have the greatest impact on 304 stainless steel sections and this is unsurprising

since the corner strength enhancement was greater for this alloy. (The corner to flat yield

strength ratio for the 304 stainless steel was 2.34 and 1.85 for r/t=1.0 and r/t=2.5,

respectively, whereas the ratios were approximately 1.77 and 1.56 for r/t=1.0 and r/t=2.5 of

the ferritic stainless steel). For simple lipped channels, notable strength improvements (up to

12% for alloy 304 and 6% for alloys 430 and 3Cr12) exist for fairly stocky sections (d1 exhibit little to no strengthimprovements. (See also Tables A.13-A.15 in Appendix A).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu/fy

304 Test

304 Flats

304 r/t=1

304 r/t=2.5

Figure 2. 20. FE and Experimental Results: 304 Simple Lipped Channels


31/194




0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu/fy

430 Test

430 Flats

430 r/t=1

430 r/t=2.5

Figure 2. 21. FE and Experimental Results: 430 Simple Lipped Channels

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu

/fy

3Cr12 Test

3Cr12 Flats

3Cr12 r/t=1

3Cr12 r/t=2.5

Figure 2. 22. FE and Experimental Results: 3Cr12 Simple Lipped Channels


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0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu/fy

304 Test

304 Flats

304 r/t=1

304 r/t=2.5

Figure 2. 23. FE and Experimental Results: 304 Lipped Channels with Intermediate

Stiffeners

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu/fy

430 Test

430 Flats

430 r/t=1

430 r/t=2.5

Figure 2. 24. FE Test and Experimental Results: 430 Lipped Channels with

Intermediate Stiffeners


33/194




0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu/fy

3Cr12 Test

3Cr12 Flats

3Cr12 r/t=1

3Cr12 r/t=2.5

Figure 2. 25. FE and Experimental Results: 3Cr12 Lipped Channels with Intermediate

Stiffeners

Furthermore, the enhanced corner properties have a greater influence on channels

with intermediate stiffeners and the maximum strength enhancement is 25% for the austenitic

alloy with r/t=2.5 and approximately 16% for the ferritic alloys. Again, the effect of

enhanced corner properties is prevalent for stockier sections with a significant corner area.

(See also Tables A.28-A.30 in Appendix A). It should be noted that the data points which

have reached capacities beyond the yield strength, i.e.,fu/fy> 1.00, have been plotted but havebeen ignored in the development of direct strength design equations.

All results have been superimposed on one plot shown in Figure 2.26. From this plot

one can see that the austenitic 304 band of results are generally lower than the ferritic 430

and 3Cr12 results apart from those sections which have developed greater strengths due to

enhanced corner properties (approximately d


34/194




0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

d

fu/f

y

Austenitic 304

Ferritic 430 and3Cr12

Figure 2. 26. Austenitic 304 and Ferritic 430 and 3Cr12 Test Results

2.6 Conclusions of Numerical InvestigationSeveral conclusions can be made from the numerical investigations including the following:

- Stainless steel material nonlinearity must be accounted for,

-

Numerical analyses should be based on the compressive material properties for

the longitudinal direction,

-

Anisotropy can be ignored for statically loaded members,

- Distortional buckling flange movement, instigated by the imperfection sign, may

be ignored as it does not have significant consequences for the ultimate load,

-

Section corners and intermediate stiffeners carry significant loads and the

accuracy of a model can be improved by including the enhanced corner properties,

-

Enhanced corner properties can cause strength increases of approximately 5% or

greater for sections with approximately d


35/194




whether current design guidelines for distortional buckling are unconservative or overly

conservative.

The experimental and finite element data presented in Section 2.5 were used to

evaluate the effective width approach (EWA) of the current cold-formed stainless steel codes.

However, because cold-formed stainless steel codes usually evolve along with those for cold-

formed carbon steel, the EWA and Direct Strength Method (DSM) approaches available inthe AS/NZS 4600 (1996), NAS (2001) and NAS Appendix 1 (2004) were also considered.

3.2 Effective Width ApproachEWA for Simple Lipped Channels

Distortional buckling is treated in the AN/NZS 4673 (2001), ASCE (2002) and the EC3

Part 1-4/1-3 (2004) under the elements section of the codes and the design strength is

calculated according to the EWA. The section capacity is based on local plate instability with

allowance for post-buckling strength development. In the Australian and North American

design standards, a flange is partially stiffened ifIs/Ia< 1.0 whereIsis the moment of inertia

of the lip andIais the adequate moment of inertia required for the flange element to behaveas an adequately stiffened element. The lip effective width, treated as an outstand

compression element is reduced according to the ratio Is/Ia, and the flange element plate

buckling coefficient, kf 4, used to find the flange effective width, takes into account the

distortional buckling. The AS/NZS 4673 (2001) and ASCE (2002) design rules for

distortional buckling are identical and for simple lipped channels are essentially the same as

those provided in the cold-formed carbon steel codes AS/NZS 4600 (1996) and the NAS

(2001).

The EC3 Part 1-4 (2004) for stainless steel distortional buckling refers to the

procedures outlined in the cold-formed carbon steel code, EC3 Part 1-3 (2004), but prescribes

more conservative plate buckling strength curves to take into account stainless steel

nonlinearity (see Appendix B). Aside from this, the design procedures followed are inaccordance with EC3 Part 1-3 (2004). The EC3 Part 1-4/1-3 approach to distortional buckling

differs from that found in the Australian/New Zealand and North American codes in that

rather than using anIs/Iareduction factors and a reduced flange plate buckling coefficient, a

distortional buckling slenderness reduction factor, d is used to reduce the area of the edge

stiffener (edge stiffener includes the effective lip and one half of the effective flange elements

which have been already reduced for local buckling). The critical buckling stress of the edge

stiffener, which is required for the calculation of the dfactor, can be found a) by equations

based on the theory of an elastic foundation (Timoshenko and Gere 1961) and is given by

fcr=2(KEIs)0.5/As where K is the spring stiffness of the edge stiffener and Is and As are

geometric properties of the stiffener, or b) from a rational elastic buckling analysis, such as

that provided by ThinWall or ABAQUS. Both methods of determiningfcrare considered andthe former will be referred to as the Traditional method and the latter will be referred to as the

Alternative method. The value of d is optionally refined iteratively (provided that d< 1)

and is done here only for the EC3 Traditional method. From a design perspective, the EC3

Alternative method is easier to use, particularly when the section geometry becomes

complicated.

The design procedures for simple lipped channels are outlined in Appendix B and

gives reference to the appropriate code clauses. The section effective areas and predicted

loads for all tests are also provided in Appendix B (detailed material and geometric properties

are listed in Appendix A). The results of the EWA for Australian/New Zealand, North

American and European codes for the 304, 430 and 3Cr12 alloys are summarized in Tables3.1 and 3.2 for experimental and FE tests, respectively. The mean test to predicted load

ratios, Pu,t/Pn, standard deviations (stdv) and coefficient of variations (cov) are shown for

each set of results. Two different predicted design strengths have been calculated; the first


36/194




set of predicted strengths ignores enhanced corner properties (EC Prop.) and is based on the

total effective area times the yield strength of the flats (Pn=fy,fAe,t) and fall under the heading

Without EC Prop.. The second set of predicted design strengths includes enhanced corner

properties and is based on the total effective area of the flats times the yield strength of the

flats plus the effective area of the corners times the yield strength of the corners (Pn=fy,fAe,f +

fy,cAe,c) and the results fall under the heading With EC Prop.. It should be clarified that thecodes considered do not permit the use of enhanced corner strengths unless a section is fully

effective and is not subject to heat treatment and thus the predicted capacities excluding

enhanced corner properties are apt. However, sections that do actually benefit from enhanced

corner properties (ie experimental and FE r/t=1 and FE r/t=2.5) may partially offset the

detrimental effects of material nonlinearity, making the design strength predictions seem

reasonable. If enhanced corner properties are considered, then the assessment is directed at

determining if the carbon-steel based codes are valid and safe enough to account for stainless

steel material nonlinearity.

Table 3. 1. Experimental Tests and EWA Predicted Strengths for Simple LippedChannels

P u,t P n P u,t/P n P n P u,t/P n P n P u,t/P n P n P u,t/P n P n P u,t/P n P n P u,t/P n

Test ID kN kN kN kN kN kN kN

304D1a 102 101 1.01 121 0.84 86 1.19 100 1.02 95 1.08 112 0.91

304D1b 101 101 1.00 120 0.84 86 1.18 100 1.01 95 1.07 112 0.90304D2a 104 101 1.03 121 0.86 86 1.21 100 1.04 96 1.09 114 0.92

304D2b 104 101 1.03 121 0.86 86 1.21 100 1.04 96 1.09 114 0.91

mean 1.02 0.85 1.20 1.03 1.08 0.91

stdv 0.014 0.012 0.018 0.015 0.008 0.007

cov 0.014 0.014 0.015 0.015 0.008 0.008

430D1a 39 39 1.00 42 0.91 32 1.20 35 1.11 34 1.15 37 1.04

430D1b 39 39 0.99 43 0.90 32 1.22 35 1.12 34 1.14 37 1.04

430D2 45 41 1.10 45 1.00 34 1.33 37 1.22 35 1.28 38 1.17

430D3a 40 38 1.04 42 0.94 32 1.24 35 1.14 34 1.16 37 1.06

430D3b 39 38 1.01 42 0.92 32 1.21 35 1.11 37 1.05 40 0.96

mean 1.03 0.93 1.24 1.14 1.16 1.06

stdv 0.044 0.041 0.051 0.047 0.083 0.075

cov 0.042 0.044 0.041 0.041 0.071 0.0713Cr12D1a 138 138 1.00 155 0.89 115 1.20 127 1.09 123 1.12 137 1.00

3Cr12D1b 139 138 1.01 155 0.90 115 1.21 127 1.10 123 1.13 137 1.01

mean 1.00 0.89 1.21 1.09 1.13 1.01

stdv 0.005 0.004 0.006 0.006 0.006 0.005

cov 0.004 0.005 0.005 0.005 0.005 0.005

With EC

Prop.

Without EC

Prop.

With EC

Prop.

Without EC

Prop.

With EC

Prop.

Without EC

Prop.

AS/NZS 4673 (2001), ASCE

(2002), (AS/NZS 4600 (1996)

NAS (2001))

EC3 Part 1-4/1-3 (2004)

Traditional Method

EC3 Part 1-4/1-3 (2004)

Alternative Method

Referring to the experimental results and predicted design strengths in Table 3.1 it can

be seen that for all alloys all codes are conservative as long as enhanced corner properties are

ignored. In fact, the EC3 Part 1-4/1-3 (2004) Traditional method is overly conservative. The

discrepancy between the mean Pu,t/PnEC3 Traditional and Alternative strength predictions

lies in the model to determine the critical elastic buckling stress. In the Traditional method,

the critical buckling stress does not account for the fixed-end conditions whereas the critical

buckling stress used in the Alternative method is obtained from the ABAQUSelastic buckling

analyses of the fixed-ended experimental test models. As explained in Lecce and Rasmussen


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(2005) fixed-ends have the effect of increasing the critical distortional buckling strength

compared to a pin-ended section. Therefore, the distortional buckling slenderness is greater

for the Traditional method evaluation leading to a greater reduction for distortional buckling.

If enhanced corner properties are included in the strength prediction, then all codes become

unconservative except for the EC3 Part 1-4/Part 1-3 (2004) Traditional method.

Considering the FE results shown in Table 3.2 the codes are noticeably lessconservative or unconservative compared with the experimental results shown in Table 3.1,

whether enhanced corner properties are ignored in the design strength predictions or not. The

inconsistency between experimental and FE Pu,t/Pn results are due to the differences in end

conditions of the experimental and FE tests. Furthermore, the FE tests cover a much larger

range of cross-sections resulting in a greater spread of data.

Table 3. 2. Summary of FE Test to EWA Predicted Strengths for Simple Lipped

Channels

Without EC

Prop.

With EC

Prop.

Without EC

Prop.

With EC

Prop.

Without EC

Prop.

With EC

Prop.

Alloy/FE Test

Series

statistical

variablesP u,t/P n P u,t/P n P u,t/P n P u,t/P n P u,t/P n P u,t/P n

mean 0.84 0.84 0.94 0.94 0.96 0.96

304/"flats" stdv 0.069 0.069 0.038 0.038 0.054 0.054

COV 0.082 0.082 0.041 0.041 0.057 0.057

mean 0.86 0.80 0.97 0.91 0.98 0.92

304/"r/t=1" stdv 0.085 0.068 0.045 0.037 0.053 0.051

COV 0.099 0.085 0.047 0.040 0.054 0.056

mean 0.88 0.78 0.99 0.91 0.99 0.90

304/"r/t=2.5" stdv 0.102 0.072 0.046 0.032 0.055 0.052COV 0.117 0.093 0.046 0.035 0.055 0.058

mean 0.89 0.89 1.02 1.02 1.05 1.05

430/"flats" stdv 0.048 0.048 0.061 0.061 0.093 0.093

COV 0.054 0.054 0.060 0.060 0.089 0.089

mean 0.90 0.86 1.04 1.00 1.06 1.03

430/"r/t=1" stdv 0.054 0.048 0.057 0.059 0.094 0.089

COV 0.060 0.056 0.055 0.059 0.089 0.087

mean 0.90 0.83 1.05 0.99 1.07 1.00

430/"r/t=2.5" stdv 0.071 0.057 0.059 0.060 0.089 0.091

COV 0.080 0.070 0.056 0.061 0.084 0.092

mean 0.90 0.90 1.00 1.00 1.03 1.03

3Cr12/"flats" stdv 0.047 0.047 0.047 0.047 0.068 0.068

COV 0.053 0.053 0.047 0.047 0.066 0.066

mean 0.91 0.87 1.01 0.98 1.04 1.00

3Cr12/"r/t=1" stdv 0.054 0.047 0.048 0.047 0.062 0.064

COV 0.059 0.054 0.047 0.048 0.060 0.064

mean 0.91 0.84 1.06 0.99 1.07 1.00

3Cr12/"r/t=2.5" stdv 0.057 0.043 0.049 0.054 0.082 0.086

COV 0.062 0.051 0.046 0.055 0.077 0.087

AS/NZS 4673 (2001),

ASCE (2002), (AS/NZS4600 (1996) NAS (2001))

EC3 Part 1-4/1-3 (2004)

Traditional Method

EC3 Part 1-4/1-3 (2004)

Alternative Method

Looking at the results for the FE 304 study in Table 3.2, it is clear that all codes fail to

adequately predict the section capacity. As long as enhanced corner properties are ignored,

the Pu,t/Pnratio marginally improves for r/t=1.0 and even more so for r/t=2.5 when compared

to the Pu,t/Pn of the flats. As seen in Section 2, the sections tested which actually benefit fromthe enhanced corner properties are relatively few and therefore the Pu,t/Pnratios improve only

marginally.


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For the ferritic 430 and 3Cr12 alloys, the North American and Australian codes

overestimate the section capacities even if enhanced corner properties are ignored and result

in mean Pu,t/Pnratios less than 1.00. The EC3 Part 1-4/1-3 (2004) methods work reasonably

well for the ferritic stainless steels and becomes increasingly conservative for r/t=1.0 and

r/t=2.5. For example, the mean Pu,t/Pnratios for the 430 alloy, listed in Table 3.2 under the

EC3 Alternative method, are 1.05, 1.06 and 1.07 for the flats, r/t=1 and r/t=2.5 test sets,respectively, provided enhanced corner properties are ignored. Nevertheless, the mean test to

predicted strengths are marginally larger than unity and to satisfy limit states design criteria, a

more conservative resistance factor than currently specified would be required.

EWA for Lipped Channels with Intermediate StiffenersThe AS/NZS 4673 (2001) and ASCE (2002) EWA for lipped channels with intermediate

stiffeners are identical and are similar to the AS/NZS 4600 (1996) and NAS (2001). For

these codes, the intermediate stiffener of a partially stiffened flange element (where kf< 4) is

completely ignored and the flange element is designed as a simple edge-stiffened element.

The justification for ignoring the intermediate stiffener in cases where kf


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Table 3. 3. Experimental Tests and EWA Evaluation of Lipped Channels with


P u,t P n P u,t/P n P n P u,t/P n P n P u,t/P n P n P u,t/P n P n P u,t/P n P n P u,t/P n

Test ID kN kN kN kN kN kN kN

304DS1a 132 119 1.11 142 0.93 119 1.11 142 0.93 143 0.92 180 0.73

304DS2b 134 120 1.12 142 0.94 120 1.12 142 0.94 144 0.93 181 0.74

mean 1.11 0.94 1.11 0.94 0.93 0.74

stdv 0.0098 0.0083 0.0098 0.0083 0.0089 0.0069

cov 0.0088 0.0089 0.0088 0.0089 0.0096 0.0093

430DS1 60 51 1.17 57 1.05 51 1.17 57 1.05 65 0.92 73 0.81

430DS2 62 52 1.20 58 1.07 52 1.20 58 1.07 67 0.92 76 0.81

430DS3 64 57 1.12 63 1.01 57 1.12 63 1.01 68 0.93 77 0.82

430DS4 72 57 1.27 62 1.15 57 1.27 62 1.15 72 1.00 81 0.88

mean 1.19 1.07 1.19 1.07 0.94 0.83

stdv 0.0623 0.0570 0.0623 0.0570 0.0378 0.0340

cov 0.0525 0.0532 0.0525 0.0532 0.0401 0.0408

3Cr12DS1a 163 152 1.07 171 0.95 152 1.07 171 0.95 174 0.93 205 0.80

3Cr12DS1b 161 152 1.06 171 0.94 152 1.06 171 0.94 175 0.92 205 0.79

mean 1.07 0.95 1.07 0.95 0.93 0.79

stdv 0.0099 0.0085 0.0099 0.0085 0.0097 0.0081

cov 0.0093 0.0090 0.0093 0.0090 0.0104 0.0102

NAS (2001)EC Part 1-4/1-3 (2004)

Alternative Method

With EC

Prop.

Without EC

Prop.

With EC

Prop.

Without EC

Prop. With EC Prop.Without EC

Prop.

AS/NZS 4673 (2001), ASCE

(2002), (AS/NZS 4600 (1996))

When the FE test results are considered (see Table 3.4) it is clear that all codes are

moreunsafe than those provided for the simple lipped channels and the necessity of capturing

the stainless steel material behaviour becomes even more apparent. The AS/NZS 4673(2001) and NAS (2001) provide essentially the same results and overall both are comparable

to the results given by the EC3 Part 1-4/1-3 (2004). The spread in data is also significantly

larger for channels with intermediate stiffeners. For example, the AS/NZS 4367 (2001)

evaluation of the 304/flats results have a mean Pu,t/Pn=0.84 and COV=0.082 for simple

lipped channels and Pu,t/Pn=0.78 and COV=0.143 for channels with intermediate stiffeners.


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Table 3. 4. Summary of FE test to EWA Predicted Strengths for Lipped Channels with


Without ECProp.

With ECProp.

Without ECProp.

With ECProp.

Without ECProp.

With ECProp.

Alloy/FE Test

Series

statistical

variablesP u,t/P n P u,t/P n P u,t/P n P u,t/P n P u,t/P n P u,t/P n

mean 0.78 0.78 0.78 0.78 0.78 0.78

304/"flats" stdv 0.112 0.112 0.110 0.110 0.083 0.083

COV 0.143 0.143 0.142 0.142 0.107 0.107

mean 0.83 0.77 0.83 0.77 0.83 0.75

304/"r/t=1" stdv 0.154 0.111 0.153 0.110 0.126 0.073

COV 0.185 0.144 0.184 0.143 0.152 0.098

mean 0.85 0.75 0.84 0.75 0.86 0.73

304/"r/t=2.5" stdv 0.169 0.100 0.169 0.101 0.150 0.079

COV 0.199 0.133 0.201 0.135 0.175 0.108

mean 0.85 0.85 0.85 0.85 0.84 0.84

430/"flats" stdv 0.066 0.066 0.068 0.068 0.056 0.056

COV 0.077 0.077 0.080 0.080 0.067 0.067

mean 0.88 0.84 0.87 0.84 0.86 0.81

430/"r/t=1" stdv 0.087 0.070 0.089 0.071 0.072

Finite Element Modelling and Design of Cold-Formed Steel Sections - VERSIÓN EXTENDIDA

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