ELASTO-PLASTIC BUCKLING BEHAVIOR OF H-SHAPED BEAM WITH LARGE DEPTH-THICKNESS RATIO UNDER CYCLIC LOADING

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1 SDSS Rio STABILITY AND DUCTILITY OF STEEL STRUCTURES E. Batista, P. Vellasco, L. de Lima (Eds.) Rio de Janeiro, Brazil, Setember 8 -, ELASTO-PLASTIC BUCKLING BEHAVIOR OF H-SHAPED BEA WITH LARGE DEPTH-THICKNESS RATIO UNDER CYCLIC LOADING Tao WANG*, Kikuo IKARASHI* * Deartment o Architecture and building Engineering, Tokyo Institute o Technology s: anntao@gmail.com, ikarasahi@arch.titech.ac.j Keyords: Couled Buckling, Vertical Buckling, Plate Slenderness ratio, Plastic Deormation Caacity. Abstract. Plastic deormation caacity o thin-alled structure is not only aected by the late slenderness ratio but also by the other dimensional arameters such as asect ratio, sectional area. This aer resents an exerimental study on H-shaed Beams subjected to cyclic loading in large range o deormation ith large deth-thickness ratios hich are over or comarable ith the limited values o those restricted in the current standards (Re. [] and []). The geometric arameters are relevantly identiied to evaluate the lastic deormation ratio, and the ne divisional rank or lastic deormation caacity o H-shaed beam is roosed. INTRODUCTION According to the standards or steel structures in many countries, late slenderness ratio (idththickness ratio or deth-thickness ratio) is assumed to be o major arameter to estimate the lastic deormation caacity o thin-alled steel structures. U to no many exerimental and analytical researches have been reorted aimed to reveal the relationshi beteen lastic deormation caacity and late slenderness ratio. These results have no been incororated in all sorts o standard and recommendations o steel structures. Consequently, late slenderness ratio is rigidly restrained in order to revent the elastic buckling []. In Jaan, the division or late slenderness ratio o H-shaed beam is rovided ith resect to the lastic deormation caacity as a general rovision hich is only alicable in a limited region here the deth-thickness ratio is relatively small []. It is commonly knon that couled local buckling may dominate the H-shaed beam s behavior hen the deth-thickness ratio is large. On the other hand, recent studies subjected to H-shaed Beam ith large deth-thickness ratio had shon the ractical use o thin late as the earthquake-resisting element even though the late slenderness ratio is over the limited value in some cases [-5]. Hoever, no standardized method is currently available or estimating the lastic deormation caacity o such beams. This aer is intended to develo an easy method to estimate the lastic deormation caacity o H- shaed beam ith large deth-thickness ratio, and aims at revealing the ost buckling behavior under the cyclic loading situation by contrast ith those under the monotonic loading situation. CYCLIC LOADING TEST Figure is the loading model subjected to the local buckling test here the lateral direction as cramed in the test arrangement so as to revent the overall buckling. Secimens used or testing ere SS grade, H-shaed elded steel beams. The dimension and cross-section constants or each secimen (H-D-b-t -t ), given a number is listed in Table, in hich seven tyes o lates ith dierent thickness are adoted. They are the same sort o material but a little dierent rom each other based on the tension test due to their dierent thickness. The mechanical roerty o each late is listed in Table. 377

2 Table : Secimens list secimen section(nominal value) (surveyed value) H-D-b-t -t d/t b/t l No. H No. H No.3 H No. H H No.6 H No.7 H No.8 H No.9 H No. H Table : aterial roerty ixed end t (mm) y (N/mm ) σ u (N/mm ) E (kn/mm ) ε u (%) Elng. (%) Y.R. (% ) l=, 5 γ stiener Figure : Loading model δ y 3δ δ δ Q δ b d t mδ t D marks l λ =l/d λ =l/b A = dt A =bt d /t b/t β σ y, σ y Table 3: Parameter deinition meaning Length o beam Asect ratio o eb Asect ratio o lange Sectional area o eb Sectional area o lange Web deth-thickness ratio Flange idth-thickness ratio Bending incline (β=) Yield stress o eb, lange t A t b A d D dt / N(cycle) Figure : Loading rogram E/ σy No. No.3 No., 9 No. No.7 No.6, No.8 P-I- P-I- P-II....6 bt / E/ σ y Figure 3: Plate slenderness ratio Cyclic loading rogram o this test is shon in Figure, in hich δ y and δ are the dislacement related ith y And given by: δ = l y y y 3EI + GA δ = l 3EI + GA () () In hich EI and GA are lexural rigidity and shear rigidity. The other geometric arameters and marks used in this study are deined in Table 3. y And denote the bending moment at the end late on the yielded condition hich can be reresented by, y = Qy l = σy t( d t) + σ y bdt (3) 6 378

3 = Q l = σ y t( d t ) + σ y bdt () According to the Standard or Limit State Design o Steel Structures o Jaan [], the divisional curves ith three ranks or idth -thickness ratio and deth-thickness ratio are shon in Figure 3. The relationshi beteen idth-thickness ratio and deth-thickness ratio about each secimen is also shon by lots in the same igure. 3 EXPERIENTAL RESULTS AND COPARISON 3. Hysteresis Curve and Post Buckling Behavior Figure sho the hysteresis curves o three o the secimens ith dierent tyical behaviors or No.3, and No.7 resectively. The horizontal axis and vertical axis sho the dimensionless numbers or the dislacement (δ) and load(q). In the elastic region the relationshi beteen δ and Q is deined as, Q/ Q = δ / δ (5) s E s According to Re.[6], Q s can be reresented by, sb( sb ) Qs sb > = sb Q otherise Where, Q s denotes the yield shear load taking account o bending eect. Q is the yield load on ure shear condition, denotes the ull lastic moment born on eb, sb denotes the ratio o shear to bending on the yield condition. They are given by, Q = A σ / 3 (7) y y (6) = t d σ (8) λ / β sb = (.5 + A / A ) 3 Each hysteresis curve in Figure exhibits a stable behavior until the local buckling haens. For each secimen, the load dros don immediately the local buckling haens, and a racture is observed several cycles later, then the beam collases in the end. As the hotos o collased beams shon in Figure 5, the buckling modes can be divided into three tyical tyes: (a) shos eb buckling tye (W.tye); (b) shos couled buckling tye (C.tye); (c) shos vertical buckling tye (V.tye). The mode o lange buckling tye is not discussed due to large deth-thickness ratio o eb used in this test. Comared ith the other to tyes, it can be observed rom the Figure (c) that the degradation incline o V.tye is extremely shar. Vertical buckling may be induced by eb local buckling hen langes are not strong enough to sustain the buckled eb. As a consequence, the bending rigidity o beam quickly alls don once the buckled eb makes a dent in lange. The vertical buckling does not haen even though eb bucking dominates in case o Figure (a) because the bending burden on lange is relatively small or α (Eq.()) is large, hence the vertical buckling is revented. On the other hand, the (9) 379

4 vertical buckling does not haen in case o Figure (b) because the deth-thickness ratio o eb is relatively small or (d/t )/(b/t ) is small. As a discussion result, the degradation incline o non-v.tye is about -7% o the elastic incline; the necessary conditions or non-v.tye are about α>.5 or (d/t )/(b /t )<6 (arox). Where, α is the ratio o shear to bending stress, equal to or ure bending, can be exressed by, Q/Q s A α = + β λ () 6 A No.3 : α =.9 racture - max δ/ E δ s (a) No.3 (a) No.3, Web buckling tye (W.tye) : α = δ/ δ E s (b) (b), Couled buckling tye (C.tye) No.7 : α = (c) No.7 δ/ E δ s (c) No.7, Vertical buckling tye (V.tye) Figure : Load- dislacement hysterisis curve Figure 5: Buckling mode 3. Skeleton curve and restriction on late slenderness ratio According to Re.[7], the skeleton curve can be deined as shon in Figure 6 on the right side transormed rom hysteresis curve on the let side by joining the reneed oints beore reaching the maximum load. Ater the maximum load, the skeleton curve is treated as raing line. Based on the rule deined in Figure 6, all hysteresis curves o the secimens (aendix Figure A) are transormed into the skeleton curves (Figure 7). The maximum load o No. and No.3 shon in Figure 7 38

5 are over the lastic shear load Q s even though their deth-thickness ratios are lager than the limited values, indicating that the restriction or deth-thickness ratio in Re. [] and [] is conservative. According to Re.[5], imroved method or restriction o idth (deth)-thickness ratio had been roosed. The restriction curve is exressed as ollos: BTW E / σ y BTF = BTW BTW E / σ y Where, α is given as Eq. (), BTW is given by Eq. () belo: ().9 α < / 6 BTW = α / 6 α / E / σ y 3. α > / BTW and BTF are the restrictive value o deth and idth-thickness ratio. They are roosed by taking account o the eect o late slenderness ratio and the other geometric arameters comrising late sectional area and asect ratio based on the calculation o buckling strength. As a samle shon in Figure 8 here the rectangular lots are the value on restriction curves obtained rom Eq. (), circle lots are the real idth (deth)-thickness ratio o each beam, the ratio o the value at circle lots to rectangular lots on each situation can be reresented by, () d / t b/ t WF = BTW = (3) BTF Q/Qs max δ/eδ s μ+ δ/eδs Q/Q s No.3 No. No.9 No. No.7 No.8 No. No.6 aximum load μ: lastic deormation ratio No. Hysteresis curve Skeleton curve Figure 6: Deinition o skeleton curve 6 8 δ/ E δ s Figure 7: Skeleton curves (all secimens) d/t Q max /Q s 8 6 α =.3 α =.38 α =.9 No.3 No.7 BTW, BTF Eq. () 5 5 Figure 8: Restriction curve (Eq. ()) b/t Figure 9: aximum load and 38

6 Where, WF is reerred to as normalized late slenderness ratio. Substituting orm Eq. () into Eq. (3), WF is given by, ( d/ t ).63( b/ t) 5( b/ t ) WF = + 3 3/ 3 BTW ( E / σ y) BTW Figure 9 shos the relationshi beteen WF and maximum load Q max. It can be observed rom the igure that Q max > Q s hen WF<. Thereore, the normalized late slenderness ratio WF deined in Eq.() can be regarded as a arameter to restrict the beam dimension to revent elastic local buckling. /3 () PLASTIC DEFORATION RATIO AND NORALIZED SLENDERNESS RATIO In Figure 6, μ denotes the lastic deormation ratio at the oint o maximum load, it is a arameter deined to evaluate the lastic deormation caacity in this study. As shon in Figure 3, the divisional curve o P-I- is to ensure μ>, P-I- is to ensure μ>, P-II is to ensure μ> and Q max > Q s. Figure (b) and (c) sho the relationshi beteen WF and μ obtained rom a number o FE analysis subjected to monotonic loading here the same models as Figure as adoted. The idththickness and deth-thickness ratio about each beam is shon in Figure (a). The asect ratio o eb o each beam is set at <λ =l/d<. According to the buckling modes, the results are roughly divided into to arts in Figure (a). Figure (b) shos the lots in case o (d/t )/(b/t ) <3, or (d/t )/(b/t )>6 here a large art o their buckling modes are single buckling modes or non-c.tye buckling modes. Figure (c) shos the lots in case o 3<(d/t )/(b/t )<6 hich are assumed to be the couled buckling modes. According to the lots shoing the relationshi beteen WF and μ in both Figure (b) and (c), μ tends to become larger as WF becomes smaller. Thereore, WF can be reasonably regarded as a major arameter to estimate the lastic deormation ratio. As described earlier (Chater 3.), the load alls don immediately ater the buckling, such henomenon is indeendent to the buckling style in case o cyclic loading situation. But the load does not all don immediately ater the single buckling in case o monotonic loading situation. Consequently, there is some disersion in the Figure (b). But the disersion in Figure (c) is small due to its couled buckling modes. To minimize the disersion and evaluate the lastic deormation caacity saely, the normalized slenderness ratio WF exressed by Eq. () is modiied as given by ( d / t ) 5σ y 36 σ = + BTW E BTW σ y y ( b / t ) (5) d/t (d/t )/(b/t )<3, or >6 3<(d/t )/(b/t )<6 μ μ b/t WF WF (a): Object o deth and idth-thickness ratio (b): (d/t )/(b/t ) <3 or >6 (c): 3 <(d/t )/(b/t )<6 Figure : Plastic deormation caacity and 38

7 Where, α is given as Eq. (), BTW is given as Eq. (6).. α < / 6 BTW (6) = 5.8.6α / 6 α / E / σ y.9 α > / Figure (a) shos the relationshi beteen μ and (Eq. (5)). Each black lot denotes the case hen late slenderness ratio is over the restricted value as shon in Figure (b). According to the osition o these black lots in Figure (a), the restriction can be alleviated in case o <. The circle lots denoting the results under cyclic loading in this test are smaller than the other cases. They are distributed in the loer limit. To estimate the lastic deormation caacity reasonably, the loer limit o μ must be adoted due to the disadvantageous loading situation. According to the result shon in Figure, the divisional rank or lastic deormation caacity is roosed as ollos: Rank P-I-(μ>): <.75; Rank P-I-(μ>): <.85; Rank P-II (μ>): <. μ P-I- P-I- P-II WF=.75 WF=.85 WF= Ex(cyclic) 8 Ex(cyclic) Ex 3) (monotonous) Ex 3) (monotonous) FE(monotonous) FE(monotonous) CONCLUSIONS (a): Imroved division (b): current division Figure : Division or lastic deormation caacity This aer resents an exerimental study o the local buckling behavior subjected to H-shaed beam. The current standards or steel structures (re.[] and []) are shon to be generally conservative. According to the hysteresis curve o each secimen, the vertical buckling induced by eb local buckling as ound to be the major eect on the deterioration. The aroximation o the necessary conditions or non-vertical buckling tye as given. Comared ith exeriment and a number o numerical analysis under monotonic loading situation, the lastic deormation ratio o the beams under cyclic loading is relatively small in case o beam ith non-c.tye buckling modes. The geometrical arameters such as asect ratio, sectional area as ell as the late slenderness ratio are identiied to evaluate the lastic deormation ratio. Imroved division or lastic deormation caacity is roosed using normalized late slenderness ratio. REFERENCES: [] Architectural institute o Jaan, Design Standard or Steel Structures, 5. [] Architectural institute o Jaan Standard or Limit State Design o Steel Structures,. [3] Toshiro SUZUKI, Kikuo IKARASHI and Yasuhiro TSUNEKI, A Study on Collase ode and Plastic Deormation Caacity o H-Shaed Steel Beams under Shear Bending, Journal o Structural and Construction Engineering, Arch, inst. o Jaan, 7, 85-9,.9 d/ t 3 E/ σy P-I- P-I- (μ >) (μ >)....6 P-II (μ >) b/ t E / σ y 383

8 [] Kikuo IKARASHI and Tao WANG, A ethod or Evaluation o Elastic Buckling Strength o H- Shaed Steel ember under Bending-Shear and Axial Force, Journal o Structural and Construction Engineering, Arch, inst. o Jaan, No.63, 37-6, 7.3 [5] Kikuo IKARASHI and Tao WANG, Evaluation o Width-Thickness Ratio Limitation o H-Shaed Beam embers Based on Couled Buckling Analysis, Journal o Structural and Construction Engineering, Arch, inst. o Jaan, No.69, 77-8, 8.7 [6] Jaan Society o Civil Engineers, Guidelines or Stability Design Steel Structures, 5. [7] Architectural institute o Jaan, Evaluation Procedures or Perormance -Based Seismic Design o Buildings -Calculation o Resonse and Limit Strength, Energy Balance -Based Seismic Resistant Design, Time History Resonse Analysis-, 9. APPENDIX: Table A: Test result Q/Q s No. Secimen No. No. No.3 No. No.6 No.7 No.8 No.9 No. α WF μ Buckling mode W V W W C V V C W C δ/ δ E s Q/Q s No. No.3 No δ/ E δ s Q/Q s δ/ δ δ/ E s E δ s No.6 No δ/ δ E s Q/Q s δ/ E δ δ/ δ s E s No.8 No.9 No δ/ δ δ/ δ δ/ δ E s E s E s Figure A: Hysteresis curves (all secimens) 38