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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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Department of Civil Engineering
Centre for Advanced Structural Engineering
http://www.civil.usyd.edu.au/
Finite Element Modelling and Design of Cold-Formed
Stainless Steel Sections
Research Report No R845
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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Research Report No R8452
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
M.Lecce@civil.usyd.edu.au
K.Rasmussen@civil.usyd.edu.au
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:
Department of Civil Engineering
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
Intermediate Stiffeners ...................................................................................................50
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
Intermediate Stiffeners ...................................................................................................54
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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)
FE_imp=+0.10mm (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)
FE_imp=+0.36mm (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)
FE_imp=-0.22mm (out)
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)
FE_imp=-0.23mm (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)
FE_imp=+0.16mm (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.
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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)
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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
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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.
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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
S4R_Schafer_75%_imp=0.33mm
S4_Walker_imp=0.44mm
S4_Schafer_25%_imp=0.07mm
S4_Schafer_75%_imp=0.33mm
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)
S4R_Walker_imp=0.52mm
S4R_Schafer_25%_imp=0.21mm
S4R_Schafer_75%_imp=0.99mm
S4_Walker_imp=0.52mm
S4_Schafer_25%_imp=0.21mm
S4_Schafer_75%_imp=0.99mm
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)
S4R_Walker_imp=0.76mm
S4R_Schafer_25%_imp=0.56mm
S4R_Schafer_75%_imp=2.64mm
S4_Walker_imp=0.76mm
S4_Schafer_25%_imp=0.56mm
S4_Schafer_75%_imp=2.64mm
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.
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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.
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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)
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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
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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
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
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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
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
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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/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
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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
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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
Intermediate Stiffeners
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
Intermediate Stiffeners
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
Recommended