Uncertainty in long-term behavior and buckling of concrete-filled steel tubular. columns

Transcription

1 APCOM & SCM th Dember, 213, Singapore Uncertainty in long-term behavior and buckling of concrete-filled teel tubular column * Xue Shi, Wei Gao, Yong-Lin Pi, Mark A. Bradford 1 1 School of civil and environment engineering, Univerity of New South Wale, Autralia. Abtract Thi paper preent a long-term and buckling analyi of concrete-filled teel tubular (CFST) column under utained axial compreion by accounting for the uncertaintie of creep and rinkage of the concrete core. The interval of the final rinkage train and final creep coefficient of concrete core are derived from tet reult. An interval analytical model baed on the algebraically tractable age-adjuted efftive modulu method i propoed for the uncertain longterm and buckling analyi of CFST column. An interval finite element model wa developed in thi paper for long-term behavior and buckling analyi. Perturbation method wa employed to determine the two bound of the olution. The reult of the propoed analytical model and finite element model were compared with experimental reult and analyzed. Keyword: creep, rinkage, interval analyi, interval finite element analyi, perturbation method 1. ntroduction Concrete-filled teel tubular (CFST) column have been ued in contruction ince the mid-198 (Schneider, 1998) and bome increaingly popular in both high-rie building and bridge (Sham and Saadeghvaziri, 1997). A CFST tion conit of a teel tube and a concrete core (Fig. 1). Creep and rinkage of the concrete core occur with an increae of time, which influence the longterm behavior of CFST column ignificantly. t i of great importance to corrtly predict efft of creep and rinkage of the concrete core on the long-term behavior of CFST column. Steel tion Concrete Figure 1. Cro-tion of CFST column Experimental and analytical tudie have been performed by pioneer over the pat three dade. Terrey et al. (1994) conducted imilar experiment on circular CFST column axially loaded at an earlier age of the concrete core. The firt tet on quare CFST column wa carried out by Morino et al. (1996)and ix concentrically loaded column, two centrically loaded column and one flexural member were teted. Experiment for imilar cro-tion were implemented by Uy (21) 1

2 applying axial loading to the CFST column at 14 and 28 day of age of the concrete core reptively. Square CFST column ubjted to higher utained load were invetigated by Han et al. (24). Terry et al. (1994), Uy (21) and Han et al. (21) predicted the concrete timedependent behavior by uing the age-adjuted efftive modulu (AAEM) method propoed by AC-29. Cheng et al. (25) introduced a three-dimenional nonlinear laminated element into the long-term modeling and aumed the creep behavior to be decribed by the Kelvin model. All of thee experimental and analytical invetigation treated the creep and rinkage behavior of the concrete core a determinitic phenomena. However, it i noted that the creep coefficient obtained from tet vary ignificantly from one experiment to another. Very different prediction of the time-dependent behavior of CFST column have been reported in different tudie. Thi ow that the uncertaintie of creep and rinkage of the concrete core do exit. To predict the long-term behavior of CFST column reaonably, thee uncertaintie have to be conidered. Uncertain analyi of engineering tructure ha been developed in the lat two dade and probabilitic method are uually ued if ufficient probabilitic information i available to validate the ditribution or probability denity function of random variable. Other non-probabilitic approache uch a interval arithmetic and fuzzy et theory are excellent alternative when the tatitical data of variable are not enough. For the longterm analyi of CFST column conidering the creep and rinkage of the concrete core, probabilitic method require probabilitic ditribution of the final creep coefficient and final rinkage train to be determined firt. Unfortunately, the available tet data for creep and rinkage of the concrete core of CFST tion are quite limited. Hence, it i impoible to derive corrt probabilitic ditribution of the final creep coefficient and final rinkage train. n thi paper, interval are adopted to repreent the uncertaintie. n interval model, only the lower and upper bound are required, which will be determined from the currently available experiment reult for the final creep coefficient and final rinkage train. The age-adjuted efftive modulu method (AEMM) (Bažant, 1972) i ued to decribe the creep of the concrete core and a virtual work method i ued to etabli the differential equation for the long-term-analyi of CFST column that are ubjted to a utained axial uniform compreion. nterval analye are then implemented to predict the uncertain long-term behavior of CFST column and buckling load caued by the variation of the creep and rinkage of the concrete core. Finally, an extenive parametric tudy i carried out to evaluate the influence of time, load level, teel ratio for CFST column. 2. nterval analytical analyi 2.1 nterval linear elatic analyi of long-term behavior of CFST column To predict the long-term performance, interval contitutive model conidering creep and rinkage of the CFST column need to be etablied. The baic aumption adopted for the interval longterm linear elatic analyi of CFST column in thi paper are: (1) deformation of the CFST column are elatic and atify the Euler Bernoulli hypothei; and (2) the teel tube i fully bonded with the concrete core. Then, the linear train of the CFST column can be expreed a u (1) where u i the vertical diplacement along the longitudinal dirtion of the CFST column. Baed on the age-adjuted efftive modulu method (AEMM), the tre in concrete can be expreed a E ( ) E ( u ) (2) c where E i the age-adjuted efftive modulu of concrete, i the rinkage train of concrete and can be given by AS36 (21) 2

3 where t i the loading time, final calculated by final t t 35 t i the final rinkage train of concrete when t. E can be E Ec ( t, ) 1 ( t, ) ( t, ) (4) 3 where i the age at loading, ( t, ) i the aging coefficient and ( t, ) i the creep coefficient that can be expreed a.6 ( t ) ( t, ) [ ].6 final (5) 1 ( t ) where i the final creep coefficient when t. The aging coefficient ( t, ) can be final expreed a (Gilbert, 1988) where with ( t, ) 1 * 1 t 2 t * k1 k ,7 (3) (6) (7) k1.78.4e (8) 1.33,7 k2.16.8e (9) t.118,7 final 1.25 (1) A the concrete core i aumed to be fully bonded with the teel tube, the deformation of the teel and concrete mut be compatible with each other. Conequently, their mhanical membrane train are equal to each other and the mhanical train at the interface between the teel tube and concrete core are the ame. Therefore, the tre in the teel tube can be written a where E i the Young modulu of teel. ' E Eu (11) The differential equation for the long-term analyi of the CFST column can be obtained uing the virtual work method. When the virtual work principle i ued for the long-term equilibrium of the CFST column, it can be tated a W dv dv Pu V c (12) Vc where V i the volume of the teel tube, V c i the volume of the concrete core, ( ) denote the Lagrange operator of imultaneou variation. By ubtituting Eq. (1), (2) and (4), the tatement of the principle of virtual work given by Eq. (13) can be written a L " L ( c ) ( c c ) W A E A E u udy A A P u (13) ntegrating Eq. (13) by part lead to the differential equation of equilibrium for the long-term behavior of CFST column " u (14) and lead to the tatic boundary condition for CFST column a

4 A + Acc P at x=l (15) where L i the length of the CFST column. The eential geometric boundary condition i u = at x = (16) The long-term diplacement of the CFST column can be obtained from Eq. (14) - (16) a P Ac E u x (17) A E Ac E and the train of the teel tube and concrete core can be obtained from Eq. (17) a ' P Ac Ec /[1 ( t, ) ( t, )] u A E A E /[1 ( t, ) ( t, )] c c n thi paper, the final creep coefficient final and the final rinkage train final can be decribed in term of interval variable a [, ] (19) final final final (18) [, ] (2) final final final Baed on the interval arithmetic and the determinitic olution of long-term diplacement, the interval long-term diplacement of CFST column can be obtained from Eq. (17) a P Ac Ec / [1 ( t, ) ( t, )] u x (21) A E A E / [1 ( t, ) ( t, )] c c P Ac Ec / [1 ( t, ) ( t, )] u x A E Ac Ec / [1 ( t, ) ( t, )] and the interval train of the teel tube and concrete core can be obtained from Eq. (18) a P Ac Ec /[1 ( t, ) ( t, )] A E A E /[1 ( t, ) ( t, )] c c (22) (23) P A E /[1 ( t, ) ( t, )] c c A E A E /[1 ( t, ) ( t, )] c c (24) 2.2 nterval buckling analyi The claic equilibrium equation for column can be expreed a 4 2 d v d v E P (25) 4 2 dx dx where x denote the axial coordinate, v i the tranvere defltion, P i the applied axial force, E i the Young modulu and i the ond moment of area. By uing the kinematic boundary condition that v = at x, L and the tatic boundary condition, the olution of Eq. (25) can be obtained a v C in x (26) 1 where i a time-dependent dimenionle axial force parameter defined by 2 P (27) E E where P can be expre a 4 c

5 Conidering the P A E A E A E (28) c c E i an interval variable, Eq. (26) bome A E A E A E v C x in 1 c c c E E c A E Ac E c Ac E and the critical load can be obtain a when = n. E E The interval critical load when P cr can be expreed a c 2 2 c cr 2 P (29) n ( E E ) (3) L n ( E E ) A E L [, ] (31) n ( E E c) Ac EL c c 2 2 L ( A E Ac E ) L ( A E Ac E ) 3. nterval finite element analyi The equilibrium equation of the CFST column can be derived from the principle of virtual work that require T T c V (32) Vc T du { d } { } dv { d } { } dv { du } { p} The relationip between the tre and train of concrete core can be expreed a { } [ D ]{ } (33) c where [ D ] i the interval tre-train matrix for concrete core. Similarly, the relationip between the tre and train of the teel tube i { } [ D ]{ } (34) Strain are determined from diplacement, that i { } [ B]{ u } (35) where [ B ] i the train-diplacement matrix. Subtituting Eq. (33) to (35) into Eq. (32) yield T T T du { du } { [ B] [ D ][ B]{ u } dv [ B] [ D ][ B]{ u }(1 ) dv { P}} V Vc (36) The CFST member i aumed to deform from the previou equilibrium tate defined by {P} and {u} to an incremental equilibrium tate defined by{ PP} and{ u u}. Applying principle of virtual work, we can obtain du( u u ) (37) By uing Taylor erie expanion, Eq. (4) bome ( du ) T ( du ) T { } { u } { } { p} u p (38) Subtituting Eq. (36) to Eq. (38), we have { du } K( u) u { du } P( u ) (39) where load. u i the increment of interval diplacement of the tructure, Pu ( ) T i the increment of Ku ( ) T i the interval tangent tiffne matrix of the tructure and can be expreed a 5

6 T T T KT ( u) [ B] [ D ][ B] dv [ ] [ ][ ] [ ] ( [ ])[ ] V B D B dv B D B dv V c Vc K K K (4) K i the interval K i the interval train tiffne caued by of which K i the linear interval elatic tiffne matrix for teel tubular, efftive tiffne matrix for concrete core, and rinkage. K and K are both dependent on time. The tangent tiffne matrix i updated after each increment of load or each increment of diplacement due to creep and rinkage. 4. Model validation and dicuion 4.1 Determination of interval for the final rinkage train and creep coefficient The empirical value of the final rinkage train and creep coefficient are propoed final in everal experimental tudie. The value of the final rinkage train given by Han et al. (24), Morino et al. (1996), Terrey et al. (1994) and Uy (21) i 43.5, 83.6, 5 and 16, reptively. Correpondingly, the final creep coefficient i.5,.83, 1. and 1.5 reptively. Thee value vary coniderably. To account for thee variation in the long-term analyi of CFST column, the interval of the final rinkage train and creep coefficient of their concrete core can be derived from thee tet reult a final = [43.5, 34] and final = [.5, 1.7] reptively, which are ued in thi tudy. t can be expted that the reult obtained by the interval model propoed in thi paper will contain thee experimental reult, in other word, the experimental reult will fall into the interval bound produced by the propoed model. 4.2 Long-term behavior of CFST column by interval analytical analyi Han et al. (24) carried out long-term tet on CFST quare tion column. The dimenion of the quare tion are 1 mm and the thickne of the quare tion i 2.93 mm. The length of the CFST column i L = 6mm. Young modulu of the teel tube E = MPa and Young modulu of the concrete core E c = 292 MPa. The firt loading time i 28 day after concrete core cating. A central axial load of 36 kn wa applied to the CFST column. final Figure 2. Comparion of creep and rinkage train The analytical interval olution for the train of CFST column i compared with the tet reult in Figure 2. t can be oberved that the interval uncertainty analyi can provide good upper and lower bound for tet reult. 4.3 nterval numerical buckling analyi 6

7 Figure 3. nterval critical buckling load on long-term utained loading Figure 4. Compreion force on concrete core when buckling Fig. 3 ow the interval critical buckling load under long-term utained load obtained by the interval finite element analyi method. Fig.4 ow the compreion force on concrete core f / N with time t, reptively. Young module of the teel and concrete are c cr E 2GPa and E c 3GPa mm..the ize of tion i1mm 1mm 3mm. The length i 1 t can be een from Fig. 3 that when time t increae, the critical buckling load dreae ignificantly due to the efft of the creep and rinkage. The dreae range i [7%, 2%] at t=3day. Fig. 4 ow that, along the time t, the buckling reitance contribution from the concrete i dreaing while form the teel tubular i increaing. Figure 5. Axial train for limit point buckling 7

8 Fig. 5 ow the compreion force on the concrete core with axial train. Buckling i invetigated in a way that the column under an incremental load and the load criterion applie (Zhou, 21). When creep buckling tend to happen, increment of the load bome very mall. From thee figure, it can be een that the axial train i boming maller when buckling happen, in other word, the buckling reitance of the column i dreaed due to the creep and rinkage. 5. Concluion Thi paper preent a theoretical tudy on the uncertain long-term and buckling analyi of concrete-filled teel tubular column. An interval analytical model baed on the algebraically tractable age-adjuted efftive modulu method i propoed to decribe the time-dependent behavior of concrete in CFST column. The olution of thi model i compared with the experimental reult reported by other reearcher, which ow the good agreement. Baed on the energy method, the formulation for elatic buckling of the teel plate in rtangular CFT column under axial compreion are derived. An interval finite element wa developed to decribe the longterm behavior and analyi buckling. The buckling load or buckling time can be evaluated uing thi model. n the future, the propoed model will be further developed to analyze other type of CFST tructure accounting for the uncertaintie in their material and geometric propertie. Reference 1. Schneider, S. P. (1998), Axially loaded concrete-filled teel tube. J. Struct. Eng., 124(1), pp Sham M, Saadeghvaziri MA.( 1997), State of the art of concrete-filled teel tubular column. AC Struct J; 94(5), pp Terrey, P. J., Bradford, M. A., & Gilbert, R.. (1994), Creep and rinkage of concrete in concrete-filled circular teel tube. Proc.,6th nt. Sympoium on Tubular Structure, Melbourne, Autralia, pp Morino, S.; Kawaguchi, J.; & Cao, Z. S. (1996), Creep Behavior of Concrete-Filled Steel Tubular Member. Compoite Contruction in Steel and Concrete 111. Proceeding of an Engineering Foundation Conference, ree, Germany, pp Uy, B. (21), Static long-term efft in ort concrete-filled teel box column under utained loading. AC Structural. Journal 98(1), pp Han, L. H., Tao, Z. & Liu, W. (24), Efft of Sutained Load on Concrete-Filled Hollow Structural Steel Column. J. Struct. Eng 13(9), pp Cheng, X.D., Li, G.Y. & Ye, G.R. (25), Three-dimenional nonlinear analyi of creep in concrete filled teel tube column. Journal of Zhejiang Univerity (Science) 6A(8), pp Bažant, Z.P. (1972), Prediction of concrete creep efft uing age-adjuted efftive modulu method. AC Structural Journal 69(4), pp AS36,(21), Autralian Standard: Concrete Structure, Standard Aociation of Autralia, Sydney. 1. R.. Gilbert, (1988), Time Efft in Concrete Structure, Elevier Applied Science, Amterdam. 11. H.Z. Zhou. (21), Buckling of reticulated laminated veneer lumber ell in conideration of the creep Engineering Structure 32, pp