Local buckling behavior of welded stub columns with normal ...

Journal of Constructional Steel Research 119 (2016) 144–153

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Journal of Constructional Steel Research

Local buckling behavior of welded stub columns with normal and highstrength steels

Gang Shi ⁎, Kelong Xu, Huiyong Ban, Cuocuo LinKey Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China

⁎ Corresponding author at: Department of Civil EngBeijing 100084, PR China.

E-mail address: [email protected] (G. Shi).

http://dx.doi.org/10.1016/j.jcsr.2015.12.0200143-974X/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

Article history:Received 14 July 2015Received in revised form 15 November 2015Accepted 8 December 2015Available online 19 December 2015

High strength steels (HSSs) are increasingly applied in structural engineering due to their benefits in terms ofme-chanical performance and economy. Compared with normal strength steel (NSS) axial compression members,HSS members possess more critical local buckling behavior since its component plates may be designed asbeingmore slender, and different mechanical properties of HSS, in terms of increased yield strength and reducedductility, may result in different local buckling behaviors. In this paper, finite element (FE) analysis is performedto investigate the local buckling behavior of welded box section and I-section stub columns under axial compres-sion with both NSS and HSS being incorporated, where initial geometric imperfections and welding-induced re-sidual stresses are accurately simulated. Through this work, variation rules of post-buckling ultimate stress andlocal buckling stress of the axial compression members with steel yield strength and width-to-thickness ratioare clarified. By comparing the FE analysis results and existing test results with the corresponding designmethods in ANSI/AISC 360-10, Eurocode 3 and GB 50017-2003, it is confirmed that the design methods for thelocal buckling behavior of welded box section and I-section stub columns under axial compression need to bemodified. New design formulas are proposed to take the influence of the steel strength into consideration.

© 2015 Elsevier Ltd. All rights reserved.

Keywords:Local bucklingAxial compression memberSteel strengthFinite element analysisDesign guidance

1. Introduction

Generally steels with a nominal yield strength fy ≥ 460 MPa arecalled high strength steels (HSSs). Compared with normal strengthsteel (NSS) structures, HSS structures exhibit many advantages, includ-ing higher safety, less material usage, lower energy consumption andmore environmental friendliness [1]. Nowadays, HSS has been widelyused in building and bridge structures, such as the Landmark Tower inJapan, the Reliant Stadium in U.S., the Sony Center in Germany, theStar City Complex in Australia, the Bird's Nest Stadium in China, andthe Millau Bridge in France [1,2].

With an increase of the yield strength of steel, ultimate bearing ca-pacity of the axial compression members becomes more sensitive totheir stability behavior rather than the strength. Some experimental in-vestigations have been conducted to investigate the effects of variousgeometric parameters, configuration details and material propertieson the local buckling behavior of HSS axial compression members.Nishino et al. [3] carried out local buckling tests of welded box sectionstub columns fabricated from A514 steel (fy = 690 MPa). Usami andFukumoto [4,5] tested the overall, local and interactive buckling behav-ior of square and rectangular box section columns made from SM58steel (fy = 460 MPa) and HT80 steel (fy = 690 MPa). Rasmussen and

ineering, Tsinghua University,

Hancock [6,7] studied the overall and local buckling behavior ofBISALLOY 80 steel (fy = 690 MPa) columns with I, box and cruciformsections. Clarin [8] experimentally investigated the longitudinal residualstresses and local plate buckling of a series of welded box section col-umnsmade fromWELDOX 700 steel (fy= 700MPa) andWELDOX1100steel (fy=1100MPa). Shi et al. [9,10] carried out an experimental studyon the local buckling behavior of welded box section and I-section stubcolumns fabricated from Q460 (fy = 460 MPa) and Q960 (fy =960MPa)HSSs, and equations in accordancewith various national stan-dards for predicting the buckling strength and ultimate strength werecompared with the test results. In addition, some numerical studieshave also been conducted by using finite element (FE) method. FEmodels validated against test results were respectively established byTang et al. [11] with ABAQUS and Shi et al. [12] with ANSYS to analyzethe local buckling behavior of HSS stub columns.

Based on the literature review, most of the existing research is fo-cused on experimental studies, while very limited information is avail-able regarding accurate numerical analyses. Furthermore, all the afore-mentioned experiments only concerned the ultimate loads of speci-mens except those by Nishino et al. [3], which measured both the ulti-mate loads and the local buckling loads.

A numerical study is conducted in this paper to investigate the localbuckling of welded square box section (labeled as B specimen) andI-section (labeled as I specimen) axial compressionmemberwith differ-ent steel grades. Using the validated FEmodelingmethod, FEmodels aredeveloped for B specimens and I specimens under axial compression to

Fig. 1. Typical FE models: (a) B specimen; (b) I specimen.

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perform parametric analyses, where the initial geometric imperfectionsand the residual stresses of specimens are accurately considered. Basedon the FE analysis results, the variation rules of post-buckling ultimatestress and local buckling stress with steel strength and width-to-thickness ratio of component plates are elucidated, and are further com-pared with corresponding design methods in accordance with differentspecifications for steel structures. Improved design approaches are alsoproposed accordingly.

2. FE analysis

2.1. FE model validation

FE modeling was performed using general-purpose finite elementsoftware, ANSYS. The columnmodelwasmeshed by using the 4-node fi-nite strain shell element, SHELL181 with functions of linear eigenvaluebuckling analysis, nonlinear large strain buckling analysis and input ofinitial stresses. A 40 mm thick rigid end plate established at either endof the columns was meshed by the 3-D structural solid element,SOLID95, which was used to transfer the load and to prevent the stressconcentration. Only translational degrees of freedomof the nodes sharedby both the column elements and the end plate elements were coupled.

Fig. 1 shows typical FE models, where each component plate wasmeshed into 22 elements along the width direction, and the elementsize along the axial direction was around 20 mm. It was found throughmesh refinement trial that when the element number along the width

Fig. 2. Geometric imperfection model

directionwas increased up to 44 and the element size along the axial di-rectionwas reduced to 5mm, the deviation of the numerical resultswaswithin 1%. Therefore the former mesh method was demonstrated to beaccurate enough and it was applied in the FE analysis. Because all thespecimens concerned herein were stub columns which could not failby overall buckling, both ends of the specimens were fully fixed exceptthe axial directional translation of the top end for loading application.

The shape of the local initial geometric imperfection was defined byusing the first eigenvalue buckling mode, as shown in Fig. 2. Accordingto GB 50205-2001 [13], the amplitudes of initial geometric imperfectionwere h0/200 for the component plates of B specimens and the webs of Ispecimens, and B/100 for the flanges of I specimens. The local initialgeometric imperfections were applied in the FE models by using thecommand UPGEOM to update the geometry of the model.

The welding-induced residual stress was also considered in the FEmodeling. The distributions and calculation formulas of residual stressesproposed by Ban et al. [14] were employed in this paper, as shown inFig. 3. To simplify the distribution pattern applied in the FE meshes, astep shape calculated according to the stress equilibrium, drawn in dot-ted lines in Fig. 3, was employed in this work. The simplified residualstresses were applied in the FE models by using the commandINISTATE, which implies that the 5 integration points of each elementpossess the same value of initial stress for representing the residualstress.

Four steps were included in the FE analysis in this paper. Firstly, theFE model of a stub column specimen with end plates was established

: (a) B specimen; (b) I specimen.

Fig. 3. The distributions and calculation formulas of residual stress: (a) B specimen; (b) I specimen (unit of formulas: MPa).

Fig. 4. Schematic of maximummid-surface strain method.

146 G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

without any imperfection. The constraints and a unit compressive loadin the middle of the top end plate were applied, and then a static solu-tion was conducted to obtain the stiffness matrix of themodel. Second-ly, an eigenvalue buckling analysis was carried out to obtain the firstorder buckling mode which was adopted as the initial deformed shapeof the FE model. Thirdly, the unit compressive load was replaced witha yield load of the stub column. After the local initial geometric imper-fections and residual stresses were applied in the model, a nonlinearbuckling analysis was conducted using the arc-lengthmethod. Fourthly,the maximum value of the time of load step was extracted and multi-plied by the yield load to obtain the post-buckling ultimate capacity ofthe specimen. According to the ‘maximum mid-surface strain method’[15,16], the time of load step, when the mid-surface strain on the con-vex point reached themaximumcompressive strain,was also extracted,and the corresponding loadwas defined as the local buckling load. Typ-ical load–strain curve to determine the local buckling loadwas shown inFig. 4.

Based on the existing test results, the FE modeling method abovewas validated in references [9,10,12]. The initial local geometric imper-fections and the residual stresses were applied according to the actualmeasurement data reported in the literature. The comparison betweenexisting test results and FE analysis results in different groups were ex-hibited in Fig. 5, indicating that the FE model was accurate enough toprovide the post-buckling ultimate load and the local buckling load ofwelded stub columns.

Fig. 6 shows a comparison of typical load–lateral deflection curves,which was in good agreement. Based on the above comparisons, itwas demonstrated that the FE modeling approach was validated as suf-ficiently accurate and reliable to investigate the local buckling behaviorof weld stub columns.

2.2. Geometric parameters

Referred to practical engineering, section types of specimens for theFE analysis herein included welded square box section and I-shape sec-tion as shown in Fig. 7. The plate thickness of B specimens was 10 mm

withwidth-to-thickness ratios (h0/tw) ranging from 10 to 70. The flangethickness andweb thickness of I specimenswere 14mmand 10mm, re-spectively, with the width-to-thickness ratio of the flange (b/t) rangingfrom 5 to 35, andwidth-to-thickness ratios of the web (h0/tw) being 20,30 and 40. The length of each specimen (L) was 1.2 B for B specimens,and the larger value of 1.2 B and 1.2 H for I specimens, which werelong enough for the appearance of local buckling. The geometries of allthe specimens are listed in Tables 1 and 2.

2.3. Material properties

The von Mises yield criterion was adopted for all the steel materialsin this paper. A quadruple-linear isotropic hardening material modelwas used for 235 MPa NSS and 460 MPa HSS, and a trilinear isotropichardening material model was applied to 690 MPa and 960 MPa HSS,

Fig. 5. Comparison between test results and FE analysis results.

Fig. 6. Comparison of typical load–lateral deflection curves [9].

147G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

as shown in Fig. 8. The material parameters defined as nominal valuesare listed in Table 3 according to the references [17–19], where fy de-notes the nominal yield strength, fu is the ultimate strength, εy meansthe yield strain corresponding to fy, εst denotes the strain at the onsetof the strain hardening, and εu is the ultimate strain corresponding tofu. The values of Young's modulus E, and the Poisson ratio were set tobe 2.06 × 105 MPa and 0.3 respectively according to the reference [20].

Fig. 7. Section types of specimens: (a)

2.4. FE analysis results

2.4.1. B specimensThe FE analysis results of the 13 B specimens are plotted in Fig. 9.

Fig. 9(a) and (b) shows the variation of the post-buckling ultimatestress and the local buckling stress respectively of B specimens withthe width-to-thickness ratio for steels with nominal yield strengths of

Square box section; (b) I-section.

Table 1Geometries of B specimens.

Specimen L/mm B/mm h0/mm tw/mm h0/tw

BZ-1 120 120 100 10 10BZ-2 180 170 150 10 15BZ-3 240 220 200 10 20BZ-4 300 270 250 10 25BZ-5 360 320 300 10 30BZ-6 420 370 350 10 35BZ-7 480 420 400 10 40BZ-8 540 470 450 10 45BZ-9 600 520 500 10 50BZ-10 660 570 550 10 55BZ-11 720 620 600 10 60BZ-12 780 670 650 10 65BZ-13 840 720 700 10 70

Fig. 8. Constitutive model for steel.

148 G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

235MPa, 460MPa, 690MPa and 960MPa. In Fig. 9, σu denotes the post-buckling ultimate stress, which was the ultimate load (Fu) divided bythe cross-sectional area (A), and σcr represents the local buckling stress,which was the local buckling load (Fcr) divided by the cross-sectionalarea (A).

It could be found from Fig. 9(a) that, for specimenswith an identicalsteel yield strength, their post-buckling ultimate stress decreased withan increase of the width-to-thickness ratio, and the decreasing rate be-came more rapid when the steel yield strength was higher. For speci-mens with the same width-to-thickness ratio, the post-bucklingultimate stress increased with an increase of the steel yield strength,and the increasing rate became slower when the width-to-thicknessratio was larger.

It could be found from Fig. 9(b) that the variation of the local buck-ling stresswith thewidth-to-thickness ratio and the steel yield strengthwas similar to that of the post-buckling ultimate stress. For specimenswith the same steel yield strength, when the width-to-thickness ratio

Table 2Geometries of I specimens.

Specimen L/mm B/mm H/mm b/mm h0/mm t/mm tw/mm b/t h0/tw

IZ-1-20 240 310 228 150 200 14 10 5 20IZ-2-20 240 366 228 178 200 14 10 6 20IZ-3-20 248 422 228 206 200 14 10 7 20IZ-4-20 281 478 228 234 200 14 10 8 20IZ-5-20 315 534 228 262 200 14 10 9 20IZ-6-20 348 590 228 290 200 14 10 10 20IZ-7-20 516 870 228 430 200 14 10 15 20IZ-8-20 684 1150 228 570 200 14 10 20 20IZ-9-20 852 1430 228 710 200 14 10 25 20IZ-10-20 1020 1710 228 850 200 14 10 30 20IZ-11-20 1188 1990 228 990 200 14 10 35 20IZ-1-40 480 310 428 150 400 14 10 5 40IZ-2-40 480 366 428 178 400 14 10 6 40IZ-3-40 480 422 428 206 400 14 10 7 40IZ-4-40 480 478 428 234 400 14 10 8 40IZ-5-40 480 534 428 262 400 14 10 9 40IZ-6-40 480 590 428 290 400 14 10 10 40IZ-7-40 516 870 428 430 400 14 10 15 40IZ-8-40 684 1150 428 570 400 14 10 20 40IZ-9-40 852 1430 428 710 400 14 10 25 40IZ-10-40 1020 1710 428 850 400 14 10 30 40IZ-11-40 1188 1990 428 990 400 14 10 35 40IZ-1-60 720 310 628 150 600 14 10 5 60IZ-2-60 720 366 628 178 600 14 10 6 60IZ-3-60 720 422 628 206 600 14 10 7 60IZ-4-60 720 478 628 234 600 14 10 8 60IZ-5-60 720 534 628 262 600 14 10 9 60IZ-6-60 720 590 628 290 600 14 10 10 60IZ-7-60 720 870 628 430 600 14 10 15 60IZ-8-60 720 1150 628 570 600 14 10 20 60IZ-9-60 852 1430 628 710 600 14 10 25 60IZ-10-60 1020 1710 628 850 600 14 10 30 60IZ-11-60 1188 1990 628 990 600 14 10 35 60

was larger than the data point marked by ‘O’, the difference betweenthe local buckling stress and the ultimate stress, which was commonlyknown as the strength reserve, began to appear. And the differencewasmore evidentwith an increase of the steel yield strength. Therefore,the post-buckling strength should be appropriately utilized in the de-sign of HSS specimens to make full use of its advantage in terms of in-creased strength.

2.4.2. I specimensThe FE analysis results of the 33 I specimens are shown in Figs. 10

and 11. Fig. 10(a) and (b) represents the variation of the post-buckling ultimate stress of I specimens with the width-to-thicknessratio for steels with nominal yield strengths of 235 MPa, 460 MPa,690 MPa and 960 MPa. Fig. 11(a) and (b) shows the variation of thelocal buckling stress with the width-to-thickness ratio. In Fig. 10, σuf

and σuw denote the post-buckling ultimate stress of the flange and theweb, respectively. In Fig. 11, σcrf and σcrw mean the local buckling stressof the flange and the web, respectively.

It could be found from Fig. 10 that the variation of the post-bucklingultimate stress of I specimenswith thewidth-to-thickness ratio and thesteel yield strength was similar to that of B specimens. In addition,Fig. 10(a) shows that for I specimens with small width-to-thicknessratio of the flange, thewidth-to-thickness ratio of theweb hasmore sig-nificant effects on the post-buckling ultimate stress of the flange whenthe steel yield strength was higher. For the flange with same width-to-thickness ratio, the amplitude of variation of the post-buckling stresswas approximately within 20% accounting for the effect of the web con-straint. And the constraint effect became more significant, when theflange width-to-thickness ratio was less than 15. For the web withsame width-to-thickness ratio, the amplitude of variation was smaller(within 10%).

It could be found from Fig. 11 that the variation of the local bucklingstress of I specimens with the width-to-thickness ratio and the steelyield strength was similar to that of B specimens. In addition,Fig. 11(a) shows that the width-to-thickness ratio of the web possessessignificant effects on the local buckling stress of the flange, where theamplitude of variation of the local buckling stress with the web con-straint effect was more than 30%. However, Fig. 11(b) exhibits that the

Table 3Material parameters of steel.

Steel strength/MPa fy/MPa fu/MPa εy/% εst/% εu/%

235 235 370 0.11 2.5 20460 460 550 0.22 2.0 14690 690 770 0.33 – 8960 960 980 0.47 – 5.5

Fig. 9. Variation of plate stress with the width-to-thickness ratio for different steel grades.

149G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

width-to-thickness ratio of the flange has slight effects on the localbuckling stress of the web.

3. Design guidance

3.1. Current design approaches

The FE analysis results were compared with the corresponding de-sign methods in ANSI/AISC 360-10 [21], Eurocode 3 [22] and GB50017-2003 [20]. For the convenience of comparison, the equivalentwidth-to-thickness ratio λpf for the flange of I specimens was definedin Eq. (1). The equivalent width-to-thickness ratio λpw for the web of Ispecimens and the component plate of B specimens were uniformly de-fined in Eq. (2), since the support conditions of these two kinds of plateswere both simply supported on their longitudinal edges.

λpf ¼ffiffiffiffiffiffif yσE

s¼ b=t

28:4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= f y

q ffiffiffiffiffiffiffiffiffiffiffiffiffi0:425

p ¼ b=t

18:5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= f y

q ð1Þ

λpw ¼ffiffiffiffiffiffif yσE

s¼ h0=tw

28:4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= f y

q ffiffiffi4

p ¼ h0=tw

56:8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= f y

q ð2Þ

Fig. 10. The variation of post-buckling ultimate stress of I specimen

In ANSI/AISC 360-10, the effect of local buckling behavior is consid-ered through reduction factors Qs and Qa, which are adopted slenderunstiffened elements and slender stiffened elements, respectively. Thepost-buckling ultimate stress of plate (σu) is determined by Eq. (3).

σu= f y ¼ Q ¼ Q sQa ð3Þ

Reduction factorQs applied for theflange of I specimens is calculatedby Eq. (4).

Q s ¼

1:0;bt≤ 0:64W

1:415−0:65W

bt

� �;0:64W b

bt≤ 1:17W

0:90Wb=t

� �2

;1:17W bbt

8>>>>>><>>>>>>:

ð4Þ

where W ¼ffiffiffiffiffiffiEk ff y

q, 0:35≤k f ¼ 4ffiffiffiffiffiffiffiffiffi

h0=twp ≤ 0:76.

Reduction factor Qa applied for the component plate of B specimensand the web of I specimens is determined by the effective area method,as given in Eq. (5). And the effective width he is calculated by Eq. (6) forthe flange of B specimens and Eq. (7) for the web of B specimens and Ispecimens. As described in Eqs. (6) and (7), there is a slight difference

s with the width-to-thickness ratio for different steel grades.

Fig. 11. The variation of local buckling stress of I specimens with the width-to-thickness ratio for different steel grades.

Fig. 12. Design equations on post-buckling ultimate stress of component plate of Bspecimen.

150 G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

between these two equations with a different consideration of imper-fections in ANSI/AISC 360-10. Eq. (6) is used in this paper as it is moreconservative.

Qa ¼Ae

Að5Þ

he ¼ 1:92tw

ffiffiffiffiffiEf y

s1−

0:38h0=tw

ffiffiffiffiffiEf y

s" #≤ h0 ð6Þ

he ¼ 1:92tw

ffiffiffiffiffiEf y

s1−

0:34h0=tw

ffiffiffiffiffiEf y

s" #≤ h0 ð7Þ

In Eurocode 3, the post-buckling ultimate stress of plate (σu) is de-termined through the effective area factor (ρ), which is calculated byEq. (8) for the flange of I specimens, and by Eq. (9) for theweb of I spec-imens and the component plate of B specimens.

ρ ¼λpf−0:188

λpf2 ;λpf N 0:748

1:0;λpf ≤ 0:748

8<: ð8Þ

ρ ¼λpw−0:22

λpw2 ;λpw N 0:673

1:0;λpw ≤ 0:673

8<: ð9Þ

In GB50017-2003, there are nodetailed instructions for determiningthe post-buckling ultimate stress of plate. Limit values of the width-to-thickness ratio are specified in the Chinese code to prevent local buck-ling occurring before presence of the overall buckling. For the flange ofI specimens, the width-to-thickness ratio is not allowed to exceed thelimit value. For theweb of I specimens or the component plate of B spec-imens, when the width-to-thickness ratio exceeds the limit value, thepost-buckling ultimate stress can be determined by using the effective

width of 40twffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= f y

q, as expressed in Eq. (10).

σu ¼40tw

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= f y

qh0

f y ¼ 0:704λpw

f y ≤ f y ð10Þ

In GB 50017-2003, the local buckling stress of plate is determinedby Eq. (11). However, in ANSI/AISC 360-10 and Eurocode 3, no

design methods are provided for the calculation of the local bucklingstress.

σcr ¼f y

1þ f y2:02σE

� �2 ;σE ¼ χkπ2E

12 1−υ2ð Þ b=tð Þ2ð11Þ

3.2. B specimen

3.2.1. Post-buckling ultimate stressThe FE analysis results and existing test results for the post-buckling

ultimate stress of the component plates of B specimens are comparedwith the design curves in Fig. 12. The horizontal ordinate is the equiva-lent width-to-thickness ratio λp calculated by Eq. (2). The existing testsincluded 6 stub columns with square box section fabricated from235 MPa NSS [23], 10 stub columns fabricated from 460 MPa HSS [5,9,24], 15 stub columns fabricated from 690 MPa HSS [3,4,6], and 4 stubcolumns fabricated from 960 MPa HSS [10]. The steel yield strength ofthe test specimens were obtained from the tension coupon tests. Itwas found that the design results in accordance with ANSI/AISC 360-10 and Eurocode 3 were higher than almost all the data points, whichwere not safe enough. The method in GB 50017-2003 resulted in

Fig. 14. Design methods on post-buckling ultimate stress of web of I specimen.

151G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

overestimations when λp was relatively small, but underestimationswhen λp was larger. Based on the design formula in GB 50017-2003, amodified equation Eq. (12) was proposed and plotted in Fig. 12 bymeans of a solid line, which coincided well with the FEA results andthe existing test results.

σu= f y ¼ 0:605

λp0:6 ≤ 1:0 ð12Þ

3.2.2. Local buckling stressThe FE analysis results and the existing test results for the local buck-

ling stress of the component plates of B specimens are compared withthe design curves in Fig. 13. The existing tests included 4 stub columnswith square box section fabricated from 460 MPa HSS [9], 4 stub col-umns fabricated from 690 MPa HSS [3], and 4 stub columns fabricatedfrom 960 MPa HSS [10]. It was found that the design results from GB50017-2003 were higher than a majority of the data points, whichwas not safe enough. Based on the design formula in GB 50017-2003,amodified equation Eq. (13)was proposed, and thedesign curve is plot-ted in Fig. 13, which fits the trend of the data better than that in GB50017-2003.

σ cr= f y ¼ 1

1þ λp2=1:17

� �1:1 ð13Þ

3.3. I specimen

3.3.1. Post-buckling ultimate stressThe FE analysis results for the post-buckling ultimate stress of the

web and the flange of the I specimens are compared with the designcurves in Figs. 14 and 15, respectively. For the web of the I specimens,the deviation of the post-buckling ultimate stress was approximatelywithin 10% with the effects of the constraint of flanges with differentwidth-to-thickness ratios and of various steel grades, however noevident rules could be found. For the flange of the I specimens, the con-straint effects of webs with different width-to-thickness ratios and thatof steel grades on the post-buckling ultimate stress of flanges were sig-nificant when λpf was relatively small, but became slight when λpf waslarger. Generally, the flange with small width-to-thickness ratios is notprone to local buckling. Therefore, the post-buckling ultimate stress ofthe web and the flange can be separately designed.

Basically, design results from ANSI/AISC 360-10, Eurocode 3 and GB50017-2003 overestimated the post-buckling ultimate stress when λpwor λpf was relatively small, but underestimated it when λpw or λpf was

Fig. 13. Design equations on local buckling stress of component plate of B specimen.

larger. Modified equations for the design of post-buckling ultimatestress of the web and the flange of the I specimens were proposed asEqs. (14) and (15), respectively. The design curves are plotted inFigs. 14 and 15 bymeans of a solid curve, which are in better agreementwith the FE analysis results compared with the existing approaches.

σuw

f y¼ 1:3

1þ λpw2

1:5

� �0:55 ≤ 1:0 ð14Þ

σuf

f y¼ 0:72

λpf0:35 ≤ 1:0 ð15Þ

In order to verify the accuracy of the proposed formulas Eqs. (14)and (15), a comparison was conducted between the design resultsand the existing test results, as shown in Fig. 16. The test specimens in-cluded 25 I-section stub columns: 3 fabricated from 235 MPa NSS [25],12 fabricated from 460 MPa HSS [9,24], 6 fabricated from 690 MPaHSS [6], and 4 fabricated from 960 MPa HSS [10]. Fig. 16 indicates thatthedesign results are lower thanmost of the test results and the averageratio between the design values and the test values is 0.89, which is safeand reasonable.

3.3.2. Local buckling stressThe FE analysis results and the existing test results for the local buck-

ling stress of the web and the flange of I specimens are compared withthe design curves in Figs. 17 and 18, respectively. For the web of the I

Fig. 15. Design methods on post-buckling ultimate stress of flange of I specimen.

Fig. 16. Comparison between the design results and the test results.

Fig. 18. Design methods on local buckling stress of flange of I specimen.

152 G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

specimens, the value of local buckling stress changed little under the ef-fect of the constraint of flanges with differentwidth-to-thickness ratios.The existing tests included 6 specimens fabricated from 235 MPa NSS[15] and 3 specimens fabricated from 960 MPa HSS [10]. For the flangeof I specimens, the constraint effect of webs with different width-to-thickness ratios on the local buckling stress of flanges was significantfor the HSS specimens, but not for the NSS specimens. The existingtests included 6 specimens fabricated from 235 MPa NSS [15], 3 speci-mens fabricated from 460 MPa HSS [9] and 3 specimens fabricatedfrom960MPaHSS [10]. However, adopting conservative design conceptto consider the effect and simplify the design process, the local bucklingstress of the web and the flange was separately calculated through anapproximate design method.

Basically, the design results fromGB 50017-2003 coincidedwell withthe data pointswhenλpw orλpfwas relatively small, but notwhenλpworλpf was larger, which was conservative and may lead to a waste of steel.Based on the design formula in GB 50017-2003, modified formulasEqs. (16) and (17) were recommended for the calculation of the localbuckling stress of theweb and the flange of the I specimens, respectively.As shown in Figs. 17 and 18, the curves obtained from Eqs. (16) and (17)respectively fitted well with a majority of the data points.

σ crw

f y¼ 1

1þ λpw2

2:25

� �1:3 ð16Þ

Fig. 17. Design methods on local buckling stress of web of I specimen.

σcrf

f y¼ 1

1þ λpf2

1:5

� �1:5 ð17Þ

4. Conclusions

In this paper, to investigate the local buckling behavior of welded boxsection and I-section axial compression members, FE analysis was con-ducted by using the validated FE models. The effects of steel yieldstrengths and plate width-to-thickness ratios on the post-buckling ulti-mate stress and the local buckling stress were identified. The FE analysisresults were further compared with the design results in accordancewith various national standards, ANSI/AISC 360-10, Eurocode 3 and GB50017-2003. Based on this work, the following conclusions can be made:

1) With the consideration of geometric initial imperfections andwelding-induced residual stresses, the FE model established in thisstudy could accurately describe the local buckling behavior of stubcolumns, including the post-buckling ultimate bearing load andthe local buckling load.

2) When only the width-to-thickness ratio of steel plate increased, thepost-buckling ultimate stress and the local buckling stress of thecomponent plate of B specimens, and of theflange andweb of I spec-imens decreased. And the decreasing rate became more rapidlywhen the steel yield strength was higher. When only the steelyield strength increased, the post-buckling ultimate stress and thelocal buckling stress of the component plate of B specimens, and ofthe flange and web of I specimens increased. And such increase be-came slower when the width-to-thickness ratio of steel plate waslarger.

3) For the flange of I specimens, the web constraint effect on the post-buckling ultimate stress and local buckling stress was significant.While for the web, the flange constraint effect was limited.

4) The design results from ANSI/AISC 360-10, Eurocode 3 and GB50017-2003 are not consistentwell with the FEA results and existingtest results. New modified formulas on the post-buckling ultimatestress and the local buckling stress were proposed, which was indi-cated to be more accurate and conservative.

Acknowledgments

The authors gratefully acknowledge the financial support providedby the National Natural Science Foundation of China (No. 51478244),and the Excellent Young Scientist Programme of the National NaturalScience Foundation of China (No. 51522806).

153G. Shi et al. / Journal of Constructional Steel Research 119 (2016) 144–153

References

[1] G. Shi, F.X. Hu, Y.J. Shi, Recent research advances of high strength steel structuresand codification of design specification in China, Int. J. Steel Struct. 14 (2) (2014)873–887.

[2] G. Shi, H.Y. Ban, Y.J. Shi, Y.Q.Wang, Recent research advances on the buckling behav-ior of high strength and ultra-high strength steel structures, Proceedings of Shang-hai International Conference on Technology of Architecture and Structure (ICTAS2009) (Vol. 1); Beijing: China Civil Engineering Society, 10 2009, pp. 75–89(China: Shanghai).

[3] F. Nishino, Y. Ueda, L. Tall, Experimental investigation of the buckling of plates withresidual stresses, tests methods for compression members. Test methods for com-pression members, ASTM Special Technical Publication No.419, American Societyfor Testing and MaterialsAmerican Society for Testing and Materials, Philadelphia,PA 1967, pp. 12–30.

[4] T. Usami, Y. Fukumoto, Local and overall buckling of welded box columns, J. Struct.Div. 108 (ST3) (1982) 525–542.

[5] T. Usami, Y. Fukumoto, Welded box compression members, J. Struct. Eng. 110 (10)(1984) 2457–2470.

[6] K.J.R. Rasmussen, G.J. Hancock, Plate slenderness limits for high strength steel sec-tions, J. Constr. Steel Res. 23 (1) (1992) 73–96.

[7] K.J.R. Rasmussen, G.J. Hancock, Tests of high strength steel columns, J. Constr. SteelRes. 34 (1) (1995) 27–52.

[8] M. Clarin, High Strength Steel: Local Buckling and Residual Stresses, Luleå Universityof Technology, Sweden, 2004.

[9] G. Shi, W.J. Zhou, Y. Bai, C.C. Lin, Local buckling of 460 MPa high strength steelwelded section stub columns under axial compression, J. Constr. Steel Res. 100(2014) 60–70.

[10] G. Shi, W.J. Zhou, C.C. Lin, Experimental investigation on the local buckling behaviorof 960MPa high strength steel welded section stub columns, Adv. Struct. Eng. 18 (3)(2015) 423–437.

[11] L.R.B. Tang, M. Mahendran, Behavior of high strength steel compression members,Proceedings of 10th Nordic Steel Construction Conference 2004, pp. 53–64(Denmark: Copenhagen).

[12] G. Shi, C.C. Lin, Y.Q.Wang, Y.J. Shi, Z. Liu, Finite element analysis on the local bucklingbehavior of high strength steel members under axial compression, Advanced Mate-rials Res (CEABM 2011), 243–249 2011, pp. 1477–1482.

[13] GB 50205-2001 Code for Acceptance of Construction Quality of Steel Structures,China Planning Press, Beijing, 2001 (in Chinese).

[14] H.Y. Ban, G. Shi, Y.J. Shi, M.A. Bradford, Experimental investigation of the overallbuckling behaviour of 960 MPa high strength steel columns, J. Constr. Steel Res.88 (2013) 256–266.

[15] B.K. He, Experimental study of local buckling of the centrally compressed members[J], J. Xi'an Inst. Metall. Constr. Eng. 41 (1) (1985) 20–34 (in Chinese).

[16] S.C. Tillman, A.F. Williams, Buckling under compression of simple andmulticell platecolumns, Thin-Walled Struct. 8 (2) (1989) 147–161.

[17] G. Shi,W.J. Zhou, Y. Bai, Z. Liu, Local buckling of steel equal angle members with nor-mal and high strengths, Int. J. Steel Struct. 14 (3) (2014) 447–455.

[18] H.Y. Ban, M.A. Bradford, Flexural behaviour of composite beams with high strengthsteel, Eng. Struct. 56 (2013) 1130–1141.

[19] BS EN 10025-6, Eurocode 3: Hot Rolled Products of Structural Steels: Part 6: Techni-cal Delivery Conditions for Flat Products of High Yield Strength Structural Steels inthe Quenched and Tempered Condition, BSI, London, 2004.

[20] GB 50017-2003, Code for Design of Steel Structures, China Planning Press, Beijing,2003.

[21] ANSI/AISC 360-10, Specification for Structural Steel Buildings, AISC, Chicago, 2010.[22] BS EN 1993-1-5, Eurocode 3: Design of Steel Structures: Part 1–5: Plated Structural

Elements, BSI, London, 2006.[23] L.J.Y. Richard, N.E. Shanmugam, S.L. Lee, Behavior of thin-walled steel box columns

under biaxial loading, J. Struct. Eng. 115 (12) (1989) 3076–3093.[24] I. Kaname, Y. Toshitaka, An experimental study on the ultimate strength and behav-

ior of members made of high strength steel with low-yield ratio, Mem. Fac. Eng.Kumamoto Univ. 45 (1) (2001) 1–13.

[25] A.J. Davids, G.J. Hancock, Compression tests of short welded I-sections, J. Struct. Eng.112 (5) (1986) 960–975.