A NUMERICAL MODEL FOR THE NONLINEAR SEISMIC ANALYSIS OF THREE-DIMENSIONAL R.C. FRAMES

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1 October -7, 8, Beijin, China A NUMERICAL MODEL FOR THE NONLINEAR SEISMIC ANALYSIS OF THREE-DIMENSIONAL R.C. FRAMES F. Mazza and M. Mazza Researcher, Dept. of Enineerin Modelin, University of Calabria, 8736, Rende (Cosenza), Italy Research Fellow, Dept. of Enineerin Modelin, University of Calabria, 8736, Rende (Cosenza), Italy Eail: fabio.azza@unical.it, irko.azza@unical.it ABSTRACT: A luped plasticity odel for the nonlinear dynaic analysis of three-diensional reinforced concrete (r.c.) fraes subjected to bi-directional round otion is proposed. The frae ebers are idealized by eans of two parallel eleents, one elastic-perfectly plastic and the other linearly elastic, assuin a bilinear oentcurvature law. An interaction surface axial force-biaxial bendin oent is considered for the end sections of each frae eber. The nonlinear dynaic analysis is perfored usin a step-by-step procedure based on a two-paraeter iplicit interation schee and an initial-stress like iterative procedure. At each step, the elastic-plastic solution is obtained adoptin the Haar-Kàràn principle. A sinle-storey r.c. three-diensional frae subjected to a bi-directional artificial round otion, whose response spectru atches on averae that adopted by Eurocode 8 for a ediu risk seisic reion and a ediu soil, is assued as test structure for the nuerical investiation. The sensitivity of the odel to chanes in input paraeters is investiated. Moreover, a refined fibre odel is considered to validate the proposed nuerical odel. KEYWORDS: Three-diensional structure, nonlinear seisic analysis, bi-directional otion, axial force-biaxial flexure interaction, luped plasticity odel. INTRODUCTION In conventional earthquake desin it is accepted that the structure could undero inelastic deforations under stron round otions. However, these deforations ust be liited in order to avoid local or lobal collapse as well as to liit the structural daae. A nonlinear dynaic analysis to predict the three-diensional structural response is needed for irreular distributions (in plan and/or in elevation) of ass, stiffness and strenth and/or siultaneous horizontal bi-directional round otion (Maliulo and Raasco, 7). Many nonlinear three-diensional odels of reinforced concrete (r.c.) slender frae ebers have been proposed in literature. They can be classified accordin to the level of discretization and odelin. Finite eleent odels require the availability of triaxial concrete stress-strain odels (e.. Kwon and Spacone, ), providin an accurate representation of the nonlinear response, but the coputational effort restricts their application to structural sub-asseblaes. In the last few years, fibre odels, able to follow the detailed stress-strain response at a lare nuber of points over several cross-sections of a frae eber, have becoe popular due to their cobination with adaptive nonlinear analysis, applyin autoatic esh refineent when and where necessary (Izzudin et al., ). This approach represents a ood balance between reliability of the results and coputational efficiency. A representation of the key features of the structural behaviour can be obtained usin r.c. eber-type odels. Specifically, the ultisurface plasticity odel (Powell and Chen, 986) and the nine-sprin odel (Lai et al., 984) represent classical exaples of luped plasticity odels. Recently, a odel based on luped daae echanics and theory of fracture has also been proposed (Marante and Flórez-López, 3). Finally, distributed plasticity odels in which eber inelasticity is onitored at several sections alon the frae eber have also been proposed in literature (e.. Sfakianakis and Fardis, 99). The ai of this work is the forulation of a luped plasticity odel for r.c. frae ebers with axial force-biaxial flexure interaction, which represents a suitable coproise between siplicity and accuracy. A finite eleent code for the nonlinear dynaic analysis of three-diensional r.c. fraed structures subjected to bi-directional round otion is prepared, investiatin the sensitivity of the proposed odel to variations in input paraeters. Moreover, the odel is tested usin results fro a refined fibre odel (SeisoSoft, 8).

2 October -7, 8, Beijin, China. THREE-DIMENSIONAL R.C. FRAME MODELING.. Discretization in space and tie The spatial fraed structure shown in Fiure a is discretized as an n-deree-of-freedo syste (six for each joint), with translational nodal asses k. The dynaic equilibriu equation can be expressed as: M u& ( t ) + Cu& ( t) + f[ u( t) ] p( t) = (..) where M is the ass diaonal atrix, C is the viscous dapin atrix in the classical Rayleih for, f is the elastic-plastic reaction vector (i.e., the vector collectin bendin oents with equilibratin shear forces, elastic axial forces and torsional oents shown in Fi. a) and p represents the external load vector includin the inertial forces correspondin to the horizontal round accelerations alon the lobal axes X and Y: & u (t) = ψ&& (t)cosγ ζ && (t) senγ ; & v (t) = ψ&& (t)senγ+ζ && (t) cosγ (..a,b) referrin to a biaxial round otion rotated at an anle γ (i.e., ψ& & and & ξ round accelerations shown in Fi. a). Fiure Structural discretization of a spatial frae and natural strain odes of a bea eleent Tie interation is carried out by usin an iplicit two-paraeter schee that proves to be optial with reard to nuerical instability, round-off error effects and beat phenoena between spurious solutions (Casciaro, 975). The schee operates by dividin the tie axis in successive intervals Δt=t -t, eployin for each interval the followin recurrin equations: q q + ( α) Δt( s p ) + ( + α) Δt( s p) = ; u = u + ( β) Δtu& + ( + β) Δtu& (..3a,b) where q(=m u& ) is the oent vector and α and β depend on tie step Δt and axiu and iniu vibration periods of the structure... Elastic solution The bea eleent of a three-diensional frae (Fi. a) is described by the coordinates of the two nodes (i and j) associated with the end sections, in the space defined by the lobal Cartesian syste (X, Y, Z). In the local syste (x, y, z) the kineatical behaviour can be conveniently expressed as a cobination of the six natural strain odes shown in Fiure b (Aryris et al., 979): T { εx, χsy, χey, χsz, χez Θx } neue u = =B (..) n, where B ne is a copatibility atrix (Mazza, 7) and u e is a vector collectin the local coponents of the end-section displaceents and rotations shown in Fiure a:

3 October -7, 8, Beijin, China { u,v,w, θ, φ, ψ,u,v,w, θ, φ ψ } T u (..) e = i i i i i i j j j j j, In ters of natural strain odes, the expression for the bea eleent strain enery becoes: L Φ n [ u n ] = ε ξ+ ( χ +χ χ ξ) ξ+ ( χ +χ χ ξ) ξ+ Θ ξ EA xd EI y sy ey ey d EIz sz ez ez d GJ t xd (..3) where EA is the axial stiffness, EI y and EI z represent the flexural stiffness around the local axes y and z and GJ t is the torsional stiffness, while the shear deforations are nelected (i.e., GA * y and GA * z ). This forulation allows the eleent stiffness atrix to be expressed in a diaonal for (Aryris et al., 979)..3. Elastic-plastic solution Each frae eber is idealized by eans of a two-coponent odel, constituted of two parallel eleents one elastic-perfectly plastic and the other linearly elastic, assuin a bilinear oent-curvature law; axial and torsional strains are assued to be fully elastic (i.e., N=N E and M t =M te are assued). Moent-curvature relationships for the plastic hines luped at the end cross sections (i and j), in which inelastic deforations are assued to occur, are deterined by the correspondence to the elastic axial force (N E ) due to static and dynaic (seisic) loads, referrin to a three-diensional axial force-biaxial bendin oent elastic doain. At each step of the analysis the elastic-plastic response of a eneric frae eber, once the initial state and the increental load in the step are known, can be obtained as a solution of the Haar-Kàràn principle, adoptin an initial-stress like iterative procedure (Casciaro, 975). Aon the stress states verifyin dynaic equilibriu conditions (Eq...), the step-end elastic-plastic solution ={ yi, zi, yj, zj } T has the iniu distance fro the elastic solution E ={ Eyi, Ezi, Eyj, Ezj } T, iniizin the copleentary enery: L L Πc[ ] = {( yi Eyi ) + ( yj Eyj) ( yi Eyi ) ( yj Eyj)} + {( zi Ezi ) + 6EIy 6EIz (.3.) L L + ( zj Ezj) ( zi Ezi )( zj Ezj)} + ( N NE ) + ( M t M te ) = in. EA GJ under the constraints represented by the plastic adissibility conditions: M y yi M y ; Mz zi Mz ; M y yj M y ; Mz zj Mz (.3.a,b,c,d) The yieldin oents at the end cross sections (i and j), correspondin to biaxial bendin, are evaluated with respect to the principal axes y (M y (N E )) and z (M z (N E )), accordin to the expression: α α ( M M ) + ( M M ) y py z pz = j t (.3.3) where M py (N E ) and M pz (N E ) represent the yieldin oents correspondin to sinle bendin and the exponent α is variable dependin on the elastic axial force N E (Eurocode, 4: α ). More specifically, if the elastic solution lies in the elastic doain then = E ; otherwise, the elastic-plastic solution () is deterined by the followin three step sequence (Mazza, 7): () yi () () () () () { M,in{ M,Eyi } ; = ax{ M,in{ M, Ezi } = ax (.3.4a,b) pyi pyi zi (k ) (k ) Eyi yi Ezi zi yj = ax Mpyj,in Mpyj,Eyj ; zj = ax Mpzj,in Mpzj,Ezj (.3.5a,b) Eyj yj Ezj zj yi = ax Mpyi,in Mpyi,Eyi ; zi = ax M pzi,in Mpzi,Ezi (.3.6a,b) The solution at the node i (j) is evaluated takin into account the elastic-plastic response at the node j (i). In particular, Equations (.3.5a,b) and (.3.6a,b) are solved iteratively until the difference, in two consecutive loops (k- and k), between the yieldin oents correspondin to biaxial bendin, obtained by the return appin by closest-point projection to the elastic doain, is less than a prefixed error at the end sections i and j. pzi pzi

4 October -7, 8, Beijin, China Finally, the elastic-plastic response of the structure is obtained by the Equations (..3a,b), iteratively solved in ters of the velocity at the step end ( u& ), by referrin to the followin residual iteration schee: ( j) (j) (j+ ) (j) ( j) ( α) Δt{ s p } + ( +α) Δt{ s p } ; u = u Hr (j) (j) r = q q + & & (.3.7a,b) in which the index j is referred to the eneric iteration loop, s=f[u]+cu& and H is the iteration atrix { M + ( + α)( + β) Δt K + ( + α) Δt } H = E C (.3.8) where K E is the elastic stiffness atrix. The iteration loops are stopped when an appropriate easure of the equilibriu error (r (j) ) becoes less than a prefixed tolerance. 3. TEST STRUCTURES A sinle-storey r.c. three-diensional frae with syetric plan is assued as test structure for the nuerical investiation. Two cases are exained, referrin to coluns with square (Fi. a: A c =4c 4c) and rectanular (Fi. b: A c =4c 7c) cross sections and supposin riid irders with elastic behaviour. More specifically, the test structures shown in Fiure are desined accordin to the provisions of Eurocode 8 (3), assuin: hih ductility class (behaviour factor, q=5); ediu-risk seisic reion (peak round acceleration, PGA=.5) and ediu soil class (class C, subsoil paraeter S=.5). The ravity load N v, correspondin to a noralized axial load ν v (=N v /(A c f c )) of the colun equal to., is applied in each joint of the test structure where a luped ass v (=N v /) is considered. A cylindrical copressive strenth (f c ) of 5 N/ for the concrete and yield (f sy ) and ultiate (f su ) strenths of 45 N/ and 54 N/ for the steel are considered. The lonitudinal reinforceent ratio ρ s (=A s /A c ) of each colun is assued equal to %, correspondin to 6φ6 (A s = 3c ) and 8φ6 (A s = 56c ), respectively for square and rectanular cross sections (Fi. 3). Coluns with square cross section Coluns with rectanular cross section Fiure R.c. fraed structures (diensions in c) Table First yieldin axial loads and sinle bendin oents for the critical end zones of a colun Fiure 3 Details of lonitudinal and transverse reinforceents of a colun: critical end zones; central zone Square section Rectanular section N t 437 kn 54 kn N c 586 kn 37 kn N y 4 kn 4 kn N z 4 kn 49 kn M py 85 kn. 358 kn. M pz 85 kn. 57 kn. M py,ax 38 kn. 73 kn. M pz,ax 38 kn. 36 kn.

5 October -7, 8, Beijin, China At the critical end reions of the coluns, whose lenth is assued accordin to Eurocode 8 (4), the spacin of hoops and cross ties is assued to be equal to c, ensurin that the distance between consecutive lonitudinal bars enaed by hoops or cross ties does not exceed c (Fi. 3a). Alon the central reion of the coluns only hoops with a spacin of c are considered (Fi. 3b). The onotonic constitutive law proposed by Mander et al. (988) is assued for the concrete, with the ultiate concrete strain odified accordin to Montejo and Kowalsky (). Stress-strain curves for square and rectanular cross sections of the coluns, distinuishin between unconfined and confined concrete, are considered constructin the elastic doain axial load-biaxial bendin oents. More specifically, in Table the paraeters correspondin to the first yieldin of the lonitudinal reinforceent (ε sy =f sy /E s =.8), for the square and rectanular cross sections of the critical end reions, are reported: tensile axial load (N t ); sinle bendin oents for an axial load equal to zero (M py and M pz, with respect to y and z axes); axiu sinle bendin oents (M py,ax and M pz,ax, with respect to y and z axes) and correspondin axial loads (i.e., N y and N z ); oreover, the copression axial load (N c ) corresponds to a concrete deforation equal to ε c =. ( ε sy ). 3. NUMERICAL RESULTS In order to study the behaviour of the test structures, whose properties are illustrated in the previous section, a coputer code has been prepared for the nonlinear dynaic analysis of r.c. three-diensional fraes subjected to biaxial round otion in the horizontal direction. In particular, all the results have been obtained usin as seisic input an artificial round otion (labelled as EC8.C) alon the lobal axes X and Y (Fi. ). This round otion, havin a duration of seconds, is enerated by usin the coputer code SIMQKE (Gasparini and Vanarcke, 976), so as to be stationary in frequency in the rane of vibration periods.5s s, with a value of PGA close to that of the taret EC8 spectru for ediu soil and ediu risk seisic reion (i.e., PGA=.5.5=.9). In the Rayleih hypothesis, the dapin atrix is assued as a linear cobination of the ass atrix and the elastic stiffness atrix, assuin a viscous dapin ratio equal to 5% with reference to the first (translational) and third (torsional) vibration periods of the structure. Firstly, a sensitivity study was perfored to define the necessary accuracy of the input paraeters for the application of the proposed luped plasticity odel (LPM). The sensitivity of the odel was considered in ters of displaceent tie histories at the top of the coluns, alon the lobal axes X and Y. More specifically, in Fiure 4 the curves for square (Fi. 4a) and rectanular (Fi. 4b) cross sections, assuin three values for the flexural stiffness of the coluns, to account for the different effect of the axial force on the crackin (EI=rEI, where the coefficient r is adopted for reducin the eoetric flexural stiffness EI ) are reported. u X =u Y () u Y () EI y(z) =.47EI,y(z) EI y(z) =.7EI,y(z) EI y(z) =EI,y(z) Luped plasticity odel (p=5%): rectanular section PGA=.5x.5=.9 Luped plasticity odel (p=5%): square section PGA=.5x.5=.9 EI y =.5EI,y ; EI z =.47EI,z EI y(z) =.7EI,y(z) EI y(z) =EI,y(z) Tie (s) Fiure 4 Effect of chane in the flexural stiffness (EI) on the response of the luped plasticity odel (LPM)

6 October -7, 8, Beijin, China The secant stiffness of the equivalent bilinear oent-curvature law for the ravity loads only (N v ), connectin the oriin with the yield point (i.e., EI y(z) =.47EI,y(z) for square cross section, EI y =.5EI,y and EI z =.47EI,z for rectanular cross section), is considered as reference value. As can be observed, the response is sensitive to the choice of the flexural stiffness EI in ters of both axiu displaceent and wavefor and periodicity of the tie history. These effects proved to be ore evident in the case of coluns with rectanular cross section (Fi. 4b), characterized by a hih seisic response (i.e., havin a axiu drift anle of %). Response sensitivity of the LPM for the effect of chane in hardenin ratio (p) of the bilinear oent-curvature law, expressed as a percentae of the eoetric flexural stiffness EI, is also investiated. Specifically, curves of top displaceent for rectanular cross section are reported in Fiure 5, assuin three values of the hardenin ratio (i.e., p=, correspondin to an elastic-perfectly plastic behavior,.5% and 5%). As expected, the effect is liited ainly to axiu values, for seisic intensities stron enouh that yieldin and plastic deforations occurred. Analoous results, which are oitted for sake of brevity, are also obtained for square section. u X () Luped plasticity odel (rectanular section: EI y =.5EI,y ; EI z =.47EI,z ) PGA=.5x.5=.9 p=5% p=.5% p=% Tie (s) Fiure 5 Effect of chane in the hardenin ratio (p) on the response of the luped plasticity odel (LPM) Finally, the nuerical results obtained by usin a refined fibre odel (FM) available in the coputer prora SeisoStruct (SeisoSoft, 8) are used for testin the reliability of the proposed luped plasticity odel (LPM). To this end, tie histories of displaceent at the top of the coluns, alon the lobal axes X and Y (Fis. 6a and 7a,b), and base shear of the ore stressed colun, alon the local axes y and z (Fis. 6b and 7c,d), are plotted for square (Fi. 6) and rectanular (Fi. 7) cross sections. u X =u Y () Base shear T y = T z (kn) PGA=.5x.5=.9 Square section: Luped plasticity odel (EI y =.47EI,y ; EI z =.47EI,z ; p=5%) PGA=.5x.5=.9 Square section: Luped plasticity odel (EI y =.47EI,y ; EI z =.47EI,z ; p=5%) Tie(s) Fiure 6 Coparison between LPM and FM for coluns with square section: top displaceent alon the lobal axis X( Y); base shear of the ore stressed colun alon the local axis y( z)

7 October -7, 8, Beijin, China More specifically, in the FM each colun is odeled with six sub-eleents, two for each critical end reion (with a lenth equal to /8 of the colun heiht) and two for the central reion (with a lenth equal to /4 of the colun heiht). The concrete cross section is subdivided in 4(=x) and 7(=x35) fibres, respectively, for square and rectanular shapes. The nonlinear cyclic behaviour of the concrete is odeled usin the Mander odel, distinuishin between unconfined and confined concrete, for both critical-end and central reions of each colun, whereas the steel is odeled usin the Meneotto and Pinto odel. Moreover, in the LPM the flexural stiffness EI is assued to be equal to the above entioned secant values and a hardenin ratio p=5% of the eoetric flexural stiffness EI is considered. Fro the coparison of results it is evident that the LPM provides a satisfactory estiation of the seisic response in all the exained cases. However, it is worth entionin that the LPM siulates only the essential aspects of the hysteretic behaviour. u X () u Y () Base shear T y (kn) Base shear T z (kn) PGA=.5x.5=.9 Rectanular section: Luped plasticity odel (EI y =.5EI,y ; EI z =.47EI,z ; p=5%) PGA=.5x.5=.9 Rectanular section: Luped plasticity odel (EI y =.5EI,y ; EI z =.47EI,z ; p=5%) Tie (s) Rectanular section: Luped plasticity odel (EI y =.5EI,y ; EI z =.47EI,z ; p=5%) PGA=.5x.5=.9 PGA=.5x.5=.9 Rectanular section: Luped plasticity odel (EI y =.5EI,y ; EI z =.47EI,z ; p=5%) Tie (s) Fiure 7 Coparison between LPM and FM for coluns with rectanular section: (a, b) top displaceents alon the lobal axes X and Y; (c, d) base shears of the ore stressed colun alon the local axes y and z (c) (d)

8 October -7, 8, Beijin, China 4. CONCLUSIONS A luped plasticity odel (LPM) for the nonlinear dynaic analysis of three-diensional r.c. fraes subjected to bi-directional round otion is proposed and a finite eleent code is prepared. The results obtained in this study perit to be drawn the followin conclusions. The nonlinear seisic response of the LPM is sensitive to the choice of different input paraeters (e.., flexural stiffness reduction factor and hardenin ratio of the bilinear oent-curvature law), influencin both axiu value of the response paraeters and wavefor and periodicity of the correspondin tie histories. Fro the coparison with a refined fibre odel (FM), available in the coputer prora SeisoStruct, it can be seen that the LPM provides a satisfactory siulation of the flexural hysteretic behaviour of r.c. frae eleents with axial force-biaxial bendin oents interaction. The LPM is relatively siple and, therefore, can be efficiently incorporated in the nonlinear dynaic analysis of coplex ulti-storey r.c. fraed structures. Further studies are needed to investiate its reliability and accuracy in predictin the nonlinear seisic response, includin the refineents needed to take into account the deradation of the hysteretic capacity. ACKNOWLEDGMENTS The present work was financed by R.E.L.U.I.S. (Italian network of university laboratories of earthquake enineerin), accordin to convention D.P.C. R.E.L.U.I.S. /7/5 (ite 54) research line no. 3. REFERENCES Aryrys, G.H., et al. (979). Finite eleent ethod - the natural approach, Coputer Methods in Applied Mechanics and Enineerin 7/8, -6. Casciaro, R. (975). Tie evolutional analysis of nonlinear structures. Meccanica 3:X, Eurocode (4). Desin of concrete structures - part -: eneral rules and rules for buildins. C.E.N., European Coittee for Standardisation. Eurocode 8 (3). Desin of structures for earthquake resistance - part : eneral rules, seisic actions and rules for buildins, C.E.N., European Coittee for Standardisation. Gasparini, D. and Vanarcke, E. (976). Siulated earthquake otions copatible with prescribed response spectra. Massachusetts Institute of Technoloy, Departent of Civil Enineerin. Izzudin, B.A., Siya, A.A.F.M. and Lloyd Sith, D. (). An efficient bea-colun forulation for 3D reinforced concrete fraes. Coputer and Structures 8, Kwon, M. and Spacone, E. (). Three-diensional finite eleent analyses of reinforced concrete coluns. Coputer and Structures 8, 99-. Lai, S.-S., Will, G.T. and Otani, S. (984). Model for inelastic biaxial bendin of concrete ebers. Journal of Structural Enineerin :, Maliulo, G. and Raasco, R. (7). Seisic response of three-diensional r/c ulti-storey frae buildin under uni- and bi-directional input round otion. Earth. Enineerin and Structural Dynaics 36, Mander, J.B., Priestley, M.J.N. and Park, R. (988). Theoretical stress-strain odel for confined concrete. Journal of Structural Enineerin 4:8, Marante, M.E. and Flórez-López, J. (3). Three-diensional analysis of reinforced concrete fraes based on luped daae echanics. International Journal of Solids and Structures 4, Mazza, F. and Mazza, M. (7). An iterative ethod for the elastic-plastic dynaic analysis of spatial fraed structures (in italian). 4 National Conference A.I.C.A.P.7 (Italian Association Reinforced and Prestressed Concrete), Salerno (Italy), 3-3. Montejo, L.A. and Kowalsky, M.J. (7). Set of codes for the analysis of reinforced concrete ebers. Technical report no. IS-7-, North Carolina State University. Powell, G.H. and Chen, P.F.-S. (986). 3D bea-colun eleent with eneralized plastic hines. Journal of Enineerin Mechanics :7, SeisoSoft (8). SeisoStruct A coputer prora for static and dynaic nonlinear analysis of fraed structures. Available fro URL: http// Sfakianakis, M.G. and Fardis, M.N. (99). Nonlinear finite eleent for odelin reinforced concrete coluns in three-diensional dynaic analysis. Coputer and Structures 4,

Seismic Analysis of Structures by TK Dutta, Civil Department, IIT Delhi, New Delhi.

Seismic Analysis of Structures by TK Dutta, Civil Department, IIT Delhi, New Delhi. Seisic Analysis of Structures by K Dutta, Civil Departent, II Delhi, New Delhi. Module 5: Response Spectru Method of Analysis Exercise Probles : 5.8. or the stick odel of a building shear frae shown in