POST-LOCAL BUCKLING OF STEEL PLATES IN CONCRETE-FILLED THIN- WALLED STEEL TUBULAR COLUMNS UNDER BIAXIAL LOADING

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1 POST-LOCAL BUCKLING OF STEEL PLATES IN CONCRETE-FILLED THIN- WALLED STEEL TUBULAR COLUMNS UNDER BIAXIAL LOADING Qig Qua LIANG ABSTRACT: This paper studies the post-local ucklig ehaviour of steel plates i cocrete-filled thi-walled steel tuular colums uder iaxial loadig usig the fiite elemet method. Geometric ad material oliear fiite elemet aalyses are udertake to ivestigate the post-local ucklig stregths of imperfect steel plates uder stress gradiets. Based o the results otaied from the oliear fiite elemet aalyses, a set of effective width formulas are proposed for determiig the ultimate stregths of steel plates i cocrete-filled steel tuular colums. The proposed effective width formulas are verified y comparisos with formulas developed y other researchers ad ca e used directly i the desig of steel plates i cocrete-filled steel tuular eam-colums. KEYWORDS: composite colum; effective width; post-local ucklig; steel plates; ultimate stregth.. INTRODUCTION Thi steel plates i cocrete-filled steel tuular (CFST) colums uder iaxial loadig are sujected to stress gradiets. The local uckig ehaviour of plates uder eccetric loadig has ee studied y Walker []. Rhodes et al. [] ivestigated the local ad post-local ucklig ehaviour of plates with iitial imperfectios. Usami [] proposed effective width formulas for predictig the ultimate stregth of simply supported steel plates i compressio ad edig. Shamugam et al. [4] ivestigated the ultimate loads of thi-walled steel ox eam-colums icludig local ucklig effects. The local ucklig ehaviour of steel plates i CFST colums is characterized y the outward ucklig mode. Ge ad Usami [5] performed oliear fiite elemet aalyses o short thi-walled CFST colums ad proposed ultimate stregth formula for steel plates i uiform compressio. Wright [6] derived the limitig width-to-thickess ratios for proportioig steel plates i cotact with cocrete. The ultimate load ehaviour of CFST colums with local ucklig effects has ee studied experimetally y Uy ad Bradford [7], Bridge et al. [8] ad Uy [9]. Liag ad Uy [] ad Liag et al. [,] ivestigated the local ucklig ehaviour of steel plates i composite memers ad proposed desig formulas for determiig the ultimate stregths of steel plates i such memers. Steel plates i cocrete-filled thi-walled steel tuular colums uder iaxial loadig are sujected to stress gradiets. This paper studies the post-local ucklig stregths of steel plates uder edge stress gradiets i cocrete-filled steel ox eam-colums y usig the geometric ad material oliear fiite elemet aalyses. Based o the results otaied from the oliear fiite elemet aalyses, a set of effective width formulas are proposed for determiig the ultimate stregths of steel plates i cocrete-filled steel ox eam-colums. The proposed desig formulas are examied agaist availale desig formulas reported i the literature. Lecturer i Structural Desig, Faculty of Egieerig ad Surveyig, The Uiversity of Souther Queeslad, Australia

2 . FINITE ELEMENT ANALYSIS. GENERAL The fiite elemet code STRAND7 [] was used to ivestigate the post-local ucklig stregths of steel plates i thi-walled CFST eam-colums. The four edges of a we or flage i a CFST eamcolum were assumed to e clamped due to the restrait provided y the cocrete core []. The vo Mises yield criterio was adopted i the oliear aalysis to treat the material plasticity of steel plates. A eight-ode quadrilateral plate/shell elemet was employed i all aalyses. A mesh was used i all aalyses ad was foud to e adequate to yield accurate results for use i practice.. INITIAL IMPERFECTIONS I the preset study, the form of iitial out-of-plae deflectios was take as the first local ucklig mode. The maximum magitude of the iitial geometric imperfectios at the plate cetre was take as w =. t for steel plates i cocrete-filled steel tuular eam-colums. A idealized residual stress patter i a cocrete-filled welded steel tuular colum was adopted i the model i which the compressive residual stress was take as 5 percet of the yield stregth of the steel plate []. Residual stresses were icorporated i the fiite elemet model y prestressig.. MATERIAL STRESS-STRAIN LAW FOR STEEL PLATES A welded steel plate displays a rouded stress-strai form that differs from the tesile test ehaviour of a coupo without residual stresses as reported y Liag ad Uy []. I the preset study, the rouded stress-strai curve of steel plates with residual stresses was modeled usig the Ramerg-Osgood formula [4], which is expressed y ε = + () E 7. 7 where ad ε are the uiaxial stress ad strai respectively, E is the Youg s modulus,. 7 is the stress correspodig to E.7 =. 7E, ad is the kee factor that defies the sharpess of the kee i the stress-strai curve. The kee factor = 5 was used i Equatio () to accout for the isotropic strai hardeig of steel plates [].. STEEL PLATES UNDER EDGE COMPRESSION The post-local ucklig ehaviour of steel plates uder gradiet compressive stresses as show i Figure is studied i this sectio. The stress gradiet coefficiet α is defied as the ratio of the miimum edge stress ( ) to the maximum edge stress ( ). Stress gradiet coefficiets ragig from. to.,.4,.6,.8 ad. were cosidered. Square steel plates (5 5 mm) with iitial geometric imperfectios ad weldig residual stresses were studied. The thickess of the steel plates was varied to give differet /t ratios ragig from to. The yield stregth of steel plates was MPa ad the Youg s modulus was GPa. Load-lateral deflectio curves for steel plates with various /t ratios uder a stress gradiet of α =. 8 are preseted i Figure (a). The figure shows that all steel plates cosidered caot attai the yield stregth ecause of the effects of iitial imperfectios ad stress gradiets. The stiffess, critical local ucklig stregth ad ultimate stregth of steel plates geerally decrease with a icrease i the plate

3 Figure. Clamped steel plates uder edge compressio ad i-plae edig. Load /f y /t = /t = 4 /t = 5 /t = 6 /t = 7 /t = 8 /t = 9 /t = Load /f y α =. α =. α =.4 α =.6 α =.8 α = Lateral deflectio at plate cetre w/t Lateral deflectio at plate cetre w/t (a) () Figure. Load-defectio curves for steel plates uder stress gradiets: (a) α =. 8 ; () /t =. width-to- thickess ratios. However, steel plates with the /t ratios of ad 4 ca attai the same ultimate stress, for these stocky plates udergo yieldig oly. It appears from Figure (a) that the ultimate stregth of a steel plate with a /t ratio of is oly 6.4 percet of its yield stregth. Apparetly, local ucklig sigificatly reduces the ultimate stregth of sleder steel plates. Fig. () demostrates the effects of stress gradiet coefficiets o the load-deflectio ehaviour of steel plates with a /t ratio of. The figure shows that the lateral stiffess of the steel plate is reduced whe icreasig the stress gradiet coefficiet α. It is also see that icreasig the stress gradiet coefficiet α reduces the ultimate stregth of the steel plate. Figure provides the ultimate stregths of steel plates sujected to edge compressio i cocrete-filled steel ox eam-colums. It ca e see from Figure that for steel plates uder the same stress gradiet the ultimate stregth of steel plates decreases with a icrease i the plate width-to-thickess ratio. As expected, icreasig the stress gradiet factor (α ) would reduce the ultimate stregth of a steel plate regardless of its width-tothickess ratio. 4. STEEL PLATES UNDER IN-PLANE BENDING The post-local ucklig stregths of steel plates uder i-plae edig as depicted i Figure are ivestigated here. The geometry ad material properties used i the aalyses were the same as those preseted i the precedig sectio. The stress gradiet coefficiet was varied from. to.4,.6,.8 ad.. The effect of stress gradiet coefficiets o the load-deflectio ehaviour of the steel plate with a /t ratio of 8 is schematically demostrated i Figure 4. It is see that reducig the stress gradiet coefficiet icreases the lateral deflectio of the plate uder the same loadig level. 5. DESIGN FORMULAS FOR ULTIMATE STRENGTH It ca e see from Figure that the ultimate stregth of a steel plate with prescried geometric

4 .4.6 u/f y..8.6 α =. α =.4 α =.6 α =.8 α =. Load /f y α =. α =.8 α =.6 α =.4 α =..4. Fiite elemet aalysis Proposed Equatio () /t Figure. Ultimate stregths of steel plates Lateral deflectio at plate cetre w/t Figure 4. Steel plates uder i-plae edig imperfectios ad residual stresses is a fuctio of the /t ratio, stress gradiet coefficiet (α ) ad the yield stregth ( f y ). A sigle formula is proposed to approximately express the ultimate stregth of steel plates with stress gradiet coefficiets greater tha zero as follows: u u = ( +. 5φ ) ( φ <.) () f y f y where φ = α ad u is the ultimate stress of steel plates uder uiform compressio, which ca e determied y u f y = t () Figure shows that the ultimate stregths of steel plates predicted y the proposed Equatios () ad () are compared well with those otaied from the fiite elemet aalysis. Figure 5 schematically depicts the effective width of a thi steel plate i the post-local ucklig regime uder compressio ad i-plae edig. Based o the results otaied from the oliear fiite elemet aalyses ad the proposed ultimate stregth formulas, effective width formulas for determiig the ultimate stregth of clamped steel plates uder edge compressive stress gradiets i cocrete-filled steel tuular eamcolums are proposed as e 4 7 = for >. t α (4a) e 5 7 = for =. + α (4) t e = + φ (5) ( ) e where e ad e are the effective widths as depicted i Figure 5. Note that for ( e + e ), the steel plate is fully effective i carryig loads ad the ultimate stregth of the steel plate ca e determied usig Equatios () ad (). 4

5 Figure 5. Effective width of steel plate uder compressio ad i-plae edig 6. COMPARISONS WITH EXISTING FORMULAS The proposed desig Equatio () are compared with those give y Liag ad Uy [] ad Ge ad Usami [5] ad Nakai et al. [5] for steel plates uder uiform compressio. It ca e see from Figure 6(a) that the proposed desig formula compares very well with those of Liag ad Uy []. For steel plates with a /t ratio greater tha 6, the ultimate stregth of the plates predicted y Equatio () is etwee those calculated usig equatios y Usami [5] ad Nakai et al [5]. The proposed desig formula yields coservative predictios of the ultimate stregths for steel plates with a /t ratio less tha u/f y.6.4. Proposed Equatio () Liag ad Uy [] Ge ad Usami [5] Nakai et al. [5] u/f y α =.4 α =.6 α =.8 Proposed Equatios (4) ad (5) Usami [] /t (a) () Figure 6. Compariso of the proposed formulas with those preseted y other researchers: (a) uiform compressio; () No-uiform compressio /t The proposed effective width formulas for clamped steel plates uder o-uiform compressio are compared with formulas developed y Usami []. The width-to-thickess ratio parameter β i Usami s equatios was calculated usig the elastic local ucklig coefficiet k for clamped steel plates uder o-uiform compressio, which is proposed ased o the fiite elemet aalysis as k = α + 5.α (6) Figure 6() provides a compariso of the ultimate stregths predicted y the proposed effective width formulas Equatios (4) ad (5) ad those give y Usami [] with ucklig coefficiet k determied y Equatio (6) for clamped plates with stress gradiet coefficiets of.4,.6 ad.8. The iitial geometric imperfectio was take as w =. t ad the compressive residual stress was assumed to e.5 f y for steel plates (5 5 mm). Figure 6() shows that the proposed effective width formulas compare very well with those give y Usami []. 5

6 7. CONCLUSIONS The post-local ucklig ehaviour of steel plates i cocrete-filled thi-walled steel tuular colums uder iaxial loads has ee ivestigated y udertakig the geometric ad material oliear fiite elemet aalyses i this paper. The effects of stress gradiet coefficiets ad width-to-thickess ratios o the ultimate stregths of steel plates i cocrete-filled steel ox colums were ivestigated. Based o the results otaied from the oliear fiite elemet aalyses, a set of desig formulas were proposed for determiig the ultimate stregths of steel plates uder compressio ad i-plae edig. These desig formulas ca e used directly i the desig of steel plates i cocrete-filled thi-walled steel tuular eam-colums ad are suitale for iclusio i composite desig codes. Moreover, these effective width formulas ca e icorporated i advaced aalysis methods to accout for local ucklig effects o cocrete-filled steel tuular colums ad composite frames. 8. REFERENCES. Walker, A.C., Flat rectagular plates sujected to a liearly-varyig edge compressive loadig. I Thi Walled Structures, ed. A. H. Chilver. Chatto & Widus, UK, 967, pp Rhodes, J., Harvey, J.M. ad Fok, W.C., The load-carryig capacity of iitially imperfect eccetrically loaded plates. Iteratioal Joural of Mechaical Scieces Vol. 7, No., 975, pp Usami, T., Effective width of locally uckled plates i compressio ad edig. Joural of Structural Egieerig, ASCE, Vol 9, No. 5, 99, pp Shamugam, N.E., Liew, J.Y.R. ad Lee, S.L., Thi-walled steel ox colums uder iaxial loadig. Joural of Structural Egieerig, ASCE, Vol. 5, No., 989, pp Ge, H.B. ad Usami, T., Stregth aalysis of cocrete-filled thi-walled steel ox colums. Joural of Costructioal Steel Research, Vol., 994, pp Wright, H.D., Local staility of filled ad ecased steel sectios. Joural of Structural Egieerig, ASCE, Vol., No., 995, pp. 8-88, 7. Uy, B. ad Bradford, M.A., Local ucklig of thi steel plates i composite costructio: experimetal ad theoretical study. Proc. Ist Civ. Egrs. Structures ad Buildigs, Vol., 995, pp Bridge, R.Q., O Shea, M.D., Garder, P., Grigso, R. ad Tyrell, J., Local ucklig of square thi-walled steel tues with cocrete ifill. Proceedigs of the Iteratioal Coferece o Structural Staility ad Desig, Sydey, Australia, 7-4, Uy, B. Stregth of cocrete-filled steel ox colums icorporatig local ucklig. Joural of Structural Egieerig, ASCE, Vol. 6, No.,, pp Liag, Q.Q. ad Uy, B., Theoretical study o the post-local ucklig of steel plates i cocretefilled ox colums. Computers ad Structures, Vol. 75, No. 5,, pp Liag, Q.Q., Uy, B., Wright, H.D. ad Bradford, M.A., Local ucklig of steel plates i doule ski composite paels uder iaxial compressio ad shear. Joural of Structural Egieerig, ASCE, Vol., No., 4, pp Liag, Q.Q., Uy, B. ad Liew, J.Y.R., Local ucklig of steel plates i cocrete-filled thi-walled steel tuular eam-colums. Joural of Costructioal Steel Research, 5 (sumitted).. STRAND7. G + D Computig, Sydey, Australia,. 4. Ramerg, W. ad Osgood, W.R., Descriptio of stress-strai curves y three parameters. NACA Techical Note. No. 9, Nakai, H., Kitada, T. ad Yoshikawa, O.A., A desig method of steel plate elemets i cocrete filled square steel tuular colums. Proc. Japaese Society of Civil Egieers, Vol. 56, No. -, 985, pp (i Japaese). 6