Nonlinear Analysis of R/C Frame Under Static And Cyclic Loadings Lin-Cong Zhoua, Long-Zhu Chenb (Department of Civil Engineering, Shanghai JiaoTong University, Shanghai, 200240, P.R. China) aemail:zhoulcmf@yahoo.com.cn, bemail:lzchen@sjtu.edu.cn Keywords: Nonlinear analysis; R/C Frame; Partial Fiber Model; Cyclic loading Abstract. This paper proposes a reliable and computationally efficient finite-element model (Partial Fiber Model) for the nonlinear analysis of reinforced concrete (R/C) frames under static and cyclic loadingconditionsthatinducemultiaxialbendingandaxialforces.Thebeam-columnmemberis composed of three parts: middle elastic and two plastic regions at the two ends of beam. The plastic regionsarediscretizedintolongitudinalsteelreinforcementandconcretefiberelements.The nonlinear behaviors of the elements are derived from the nonlinear stress-strain relations of the steel andconcretefibers.Theglobalstiffnessmatrixofbeam-columncanbededucedfromthoseof mentionedthreeparts.Numericalexamplesarecalculatedtoprovetheaccuracyandefficiencyof the model. The results of nonlinear analysis show the validity of the model to describe the nonlinear response of frame subjected to static and cyclic loadings. Introduction Structuresinregionsofhighseismicriskwillnotrespondelasticallywhensubjectedtolarge seismicloading.Muchefforthasbeendevotedinthelastthreedecadestodevelopingmodelsof inelastic response of R/C elements subjected to large cyclic deformation reversals. VariousanalyticalmodelswhichhavebeenproposedtopredictthenonlinearresponseofR/C frames can be classified into four groups: (1) Microscopic finite element models. (2) Discrete finite element(member)models.(3)Yieldsurfacemethod.(4)Macroscopicfiniteelementmodels. Memberfiniteelementmodelsarethebestcompromisebetweensimplicityandaccuracyin nonlinear seismic response studies [1]. TheveryfirstinelasticgirdermodelwasproposedbyCloughetal.[2]in1965.Theelement consistsoftwoparallelelements,oneelastic-perfectlyplastictorepresentyieldingandtheother perfectlyelastictorepresentstrain-hardening.Giberson[3]proposedanothermodelin1969.This modelconsistsoftwononlinearrotationalspringswhichareattachedattheendsofaperfectly elastic element representing the girder. Aoyama et al. [4] added a component to the Cloughs model to simulate plastic response of concrete and steel. Otani [5] attached one inelastic rotational spring at each end of the Clough member to represent the fixed-end rotations at the beam-column interface duetoslipofthereinforcementinthejoint.Thedisadvantageofthesemodelsisthatthe axial-flexuralcouplingisneglected.Toovercomethelimitationsofclassicalplasticitytheory,Lai etal.[6]proposedafiberhingemodelthatconsistsofalinearelasticelementextendingoverthe entire length of the reinforced concrete member and has one inelastic element at each end. Yang et al. [7] modified Lais model to four-spring model. The models mentioned above all assume that the inelastic behavior of R/C frames concentrates at theendsofbeamsandcolumns,andmodelthebehaviorbyinsertingzero-lengthplastichingesat theendsofthemembers.Thezero-lengthhingeconceptiscomputationallyconvenientbut theoreticallyinconsistent,becauseitimpliesinfinitestrains.Bazant[8]provedthatzero-lengthis theoreticallyimpossiblewhenhingesundergosofteningdeformation.Theconceptoffinite-length hingesinR/Cbeamshasbeenusedbyseveralresearchers.Thefinite-lengthhingesareoften viewedasanimprovementofzero-lengthhinges,whicharecommonlyusedfortheinelastic analyses of R/C frame structures. Soleimani [9] proposed a model, which assumes that the inelastic deformationsgraduallyspreadfromthebeam-columninterfaceintothememberasafunctionof loadinghistory.Therestofthebeamremainselastic.Averysimilarmodelwasdevelopedby Meyer et al. [10]. Darvall and Mendis [11] proposed a similar but simpler model with end inelastic Key Engineering Materials Vols 340-341 (2007) pp 1387-1392 Online: 2007-06-15 (2007) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/KEM.340-341.1387All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TransTech Publications, www.ttp.net. (ID: 14.139.208.84, National Institute of Technology Rourkela_TRIAL-19/04/15,08:51:58)deformations defined through a trilinear moment-curvature relation. Liew and Tang [12] developed a two-surface plastic hinge model. Takayanagi and Schnobrich [13] divided the element into a finite numberofshortlongitudinalelementsdescribedbyGibersonsmodel.Toaccountforthe interactionbetweenaxialforceandbendingmoment,athreedimensionallimitsurfacewas introducedfortherotationalsprings.FilippouandIssa[14]alsosubdividedtheelementinto different subelements. Each subelement describes a single effect, such as inelastic behavior due to bending, shear behavior at the interface or bond-slip behavior at the beam-column joint. Kaba and Mahin [15] presented a flexibility-based fiber element. Taucer et al. [16] and Spacone et al. [1] extended the method to the formulation of a fiber beam-column element. Petrangeli et al. [17] proposed another advanced fibre model. William[18]analyzedasinglestructureusingdifferentmethods,anddrewsomeconclusions. Thethree-dimensionalfiniteelementmodelisunabletoprovideusableresultsduetonumerical difficulties.Thedegradinghingemodelisthesimplestandmostrobustofthethreeconsidered, makingitthemostpracticalfortheanalysisoflargestructures.Thefiberbeammodelisthebest overall performer withregard to accuracy, but it is not asrobust as the degrading hinge model.In addition, the solution is sensitive to the degree of mesh refinement. This paper adopts the distributed plastic hinge to analyze frames, and refines the fiber element to simulate the properties of plastic parts. The proposed nonlinear model is implemented in a computer program for the nonlinear analysis of R/C frames under static and cyclic loads. Numerical solutions arepresentedfortwostructurestodemonstratethefeasibilityofthemethodofanalysispresented herein. Finite Element Model Inthisstudy,displacement-basedmethodisusedto formulatepartialfiberelement(PFE).Themodel assumes that material nonlinearity takes place attwo ends, leaving elastic part of the member between the twoplasticparts,asshowninFig.1.Thenonlinear behaviorisassumedtooccurwithintheequivalent plastic hinge length, lp. It can be taken as lp=0.08z+0.022dbfy [19]. The plastic parts are modeled as the fiber beam-column element. zyxzi Mi WVMyiii UMMWVzmmymmmU ,u14,u2,u3,u5,u6,u,u7,u9,u,u810 k=1,n(xk,yk,zk)M(x), z z(x)N (x), (x) (x) y y M(x),xyz The element forcesanddeformations and thecorresponding sectionforces and deformations of plastic parts are shown in Fig. 2. These are grouped in the following vectors: Element force vector [F]=[Ui ,Vi ,Wi,Myi ,Mzi,Uj ,Vj ,Wj,Myj ,Mzj]T,Elementdeformationvector{U}={u 1 ,,u10}T , Sectionforcevector[Q]=[N(x),My(x),Mz(x)]T,Sectiondisplacementsvector{V}={u(x),v(x), w(x)}T and Section strain vector {Vs}={(x), y(x), z(x)}T. The strain in the ith concrete and steel fiber can be defined by Eqs. 1and 2 ci z ci y ciy x z x x ) ( ) ( ) ( + = .(1) si z si y siy x z x x ) ( ) ( ) ( + = .(2) Where yi and zi are the distances between the ith fiber and the reference axis. Fig.1 Model of beam Fig. 2 Element and section force and displacement lle p l l pA B Ci m nj1388 Engineering Plasticity and Its ApplicationsThe section forces can be determined by the strain of fiber 22( ) ( )( ) ( )( ) ( )y zy y y yz yz z yz z zN x T T T xM x T T T xM x T T T x ( (= ` ` ( ( ) ) .(3) WhereTy=EciAcizci+EsiAsizsi,Tz=(EciAciyci+EsiAsiysi),T2y=EciAciz2ci+EsiAsiz2si, T=EciAci+EsiAsi, Tyz=(EciAciycizci+EsiAsiysizsi), T2z=EciAciy2ci+EsiAsiy2si, while Eci, Aci andEsi,Asi aretangentmodulusandcross-sectionalareasoftheithconcreteandsteelfiber respectively. Withtheshapefunction[N],sectiondisplacements{V}canbedeterminedbyelement displacements { } | |{ }TV N U = .(4) The section strains are directly related to the section displacements )` )`+)`(((((((

=)`xx wxx vxx uxx wxx vxx ux wx vx uxxxxxxzy) () () (0 0 00 0 0) ( ) ( ) (21) () () (0 00 00 0) () () (2222.(5) Substitute Eq. 4 into Eq. 5, and the matrix notation is { } ||{ } | | | | ( ) { } =T T Tl nB U B B U = + .(6) Where [Bl] and [Bn] are first-order and second-order transformation matrices. Application of the principle of virtual displacements, integration of section stiffness [T(x)] along plastic part yield stiffness matrix [Kp]. || | | || ( )pTplK B T x B dx( = .(7) Where lp is the length of the plastic hinge. The stiffness matrices of plastic parts [KA] and [KC] can be deduced according to Eq.7. The stiffness matrix of elastic part [KB] can be obtained in general way. The stiffness matrices of three parts can be expressed as submatrices A Aii imAA Ami mmk kkk k ((( (( = ((( ,B Bmm mnBB Bnm nnk kkk k ((( (( = ((( ,C Cnn njCC Cjn jjk kkk k ((( (( = ((( .(8) Theequilibriumequationsofthreepartscanbeobtainedonthebasisofstiffnessmatrix.The staticcondensationtechniquecanbeusedtoeliminatethefreedomsofinnernodes,becausethe externalforceoninnernodesiszero.Thentheglobalstiffnessmatrixofspacebeamcanbe obtained Key Engineering Materials Vols. 340-341 1389| |10 0 00 0 0A A A B B Aii im mm mm mn miC C B B C Cjj jn nm nn nn njk k k k k kkk k k k k k (((((((((( + ((((= (((((((((( + .(9) Constitutive Models TheconstitutivemodelofconcreteusedinthisstudyisshowninFig.3[1,20].Inthemodelthe monotonicconcretestress-strainrelationincompressionisdescribedbythreeregions.Thetensile behavior of the model takes into account tension stiffening and the degradation of the unloading and reloadingstiffeningforincreasingvaluesofmaximumtensilestrainaftertheinitialcracking.The modelassumesthattensilestresscanoccuranywherealongthestrainaxis,eitherasaresultof initial tensile loading or as a result of unloading from a compressive state. The model is described in more detail by Mohd Yassin [20] and Spacone et al.[1]. unconfined concreteconfined concreteA0B C( , K f' ) c o NLKJHGFE,IDC,MBA+ fssmfyfmoom+ - - TheconstitutiveruleofsteelusedinthisstudyisshowninFig.4.Themonotonicstress-strain responseofsteelisassumedtobetrilinear.TheinitialstiffnessofthereinforcementisEs,there arises a flat-top yield plateau when the yield stress fy is reached. At a strain of sh, a strain-hardening response with a stiffness of Esh begins. The reloading and hysteretic response of the reinforcement ismodeledafterSeckinwithsomeminorsimplifications.Themodel,describedinmoredetailby Vecchio [21], requires five parameters to be retained in memory for each reinforcement component at each integration point. Numerical Examples Toapprovethecomputationalefficiencyofthe describedmethods,twonumericalexamplesare employed. Exampleone.Areinforcedconcretecanti- leverbeamwitharectangularcrosssectionis modeledbytheproposedmodel.Thelongi- tudinalreinforcementconsistsoffoursteelbars. Thedimensionsandloadingarrangementofthe beamareshowninFig.5.Theparametersare listed in Table 1. Table 1 Material parameter of example one Compressive strength of concrete[MPa] Yield strength of the longitudinal steel [MPa] Area of steel [mm2] Youngs modulus of Concrete[GPa] Youngs modulus of Steel[GPa] 47.92333.72253.3524.57199.96 0 20 40 60 80 100 120 140 16001000200030004000500060007000[20]present methodLoad P(N)Displacement(mm)Fig.5 Load-deflection curves of example one P2540122203(unit:mm)Fig. 3 Concrete constitutive model [1,20]Fig. 4 Reinforced steel constitutive model [21] 1390 Engineering Plasticity and Its ApplicationsThetheoreticalandexperimentalresultsarecomparedinFig.5.Itcanbeseenthattheir corresponding curves agree well. Exampletwo.Thesecondexamplereferstoathree-storeyfour-bayR/Cframe[22]ofFig.6. The frame belongs to a structure with weak columns but strong beams. Most of columns of the first story failed catastrophically, which induced the frame collapsed in Tangshan earthquake in China of 1976. The beams and columns of other stories dropped unbrokenly. The real building is adopted to verify the proposal method. The dimension of the frame is shown in Fig. 7. The frame is excited by the actual acceleration time history recorded at Ninghe during Tangshan earthquake as shown in Fig. 8. The parameters are given in Table 2. 4000433049707 -7EA8 1774*5400=218002600 26007*5400=43000 250c 500250c 500250c 450 250c 450250c 500250c 500300c350300c300300c300350c350300c300300c300350c350300c300300c3005400 5400 0 5 10 15 20-150-100-50050100150Acceleration(cm/s2)Time(s) 0 1 2 3 4 5 6-150-100-50050Displacement(mm)Time(s) SAP2000 Present method Table 2Material parameter of example two Compressive strength of concrete[MPa] Yield strength of the longitudinal steel [MPa] Area of steel [mm2] Youngs modulus of Concrete[GPa] Youngs modulus of Steel[GPa] 7.2372.4253.3522200 The maximum steel strain in the columns of the first storey exceeds the ultimate reinforced strain, whichindicatesthecolumnsaredamaged.Fig.9showstheresultsofpresentedmethodand SAP2000 agree well. Summary A PFE model is presented to analyze the response of a structure subjected to static and time-variant loads.Withthehelpofanalysisofrealbuilding,itisshownthattheresultagreeswellwiththe actuality.TheproposedalgorithmisratherefficientinpredictingtheresponseofnonlinearR/C frames subjected to static and cyclic loadings. From the numerical investigation of two structures it can be concluded that the proposed method is accurate and efficient. Fig. 6 Plan view of plant [mm]Fig. 7 Dimensions of frame [mm] Fig. 8 Ninghe earthquake waveFig. 9 Time history of disp. of example two two Key Engineering Materials Vols. 340-341 1391References [1]E. Spacone, F.F. Taucer and F.C. Filippou: Earthq. Eng. Struct. Dyn Vol. 25 (1996), p. 711 [2]R.W.Clough,K.L.Benuska,andE.L.Wilson:ThirdWorldConferenceonEarthquake Engineering, Vol. 11 (1965), p. 125 [3]M.F. Giberson: J. Struct. Eng. ASCE Vol. 95 (1969), p. 137 [4]H. Takizawa, H. Aoyama: Earthq. Eng. Struct. Dyn Vol.4 (1976), p. 523 [5]S.Otani: J. 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J Vol. 96 (1999), p. 193 [22]H.X.Liu:SeismichazardofTangshanearthquake(Earthquakepublishinghouse,Beijing. 1986)(in Chinese) 1392 Engineering Plasticity and Its ApplicationsEngineering Plasticity and Its Applications 10.4028/www.scientific.net/KEM.340-341 Nonlinear Analysis of R/C Frame under Static and Cyclic Loadings 10.4028/www.scientific.net/KEM.340-341.1387