Seismic history analysis of asymmetrical adjacent buildings with soil-structure interaction consideration

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1 Universiy of Wollongong Reserch Online Fculy of Engineering - Ppers (Archive) Fculy of Engineering nd Informion Sciences 2011 Seismic hisory nlysis of symmericl djcen buildings wih soil-srucure inercion considerion Mehme Uz Universiy of Wollongong, meuz@uow.edu.u Muhmmd N. S Hdi Universiy of Wollongong, mhdi@uow.edu.u hp://ro.uow.edu.u/engppers/3386 Publicion Deils Uz, M. E. & Hdi, M. N S. (2011). Seismic hisory nlysis of symmericl djcen buildings wih soil-srucure inercion considerion. 8h World Conference on Erhquke Resisn Engineering Srucures (pp ). Wessex, UK: WI Press. Reserch Online is he open ccess insiuionl reposiory for he Universiy of Wollongong. For furher informion conc he UOW Librry: reserch-pubs@uow.edu.u

2 Erhquke Resisn Engineering Srucures VIII 225 Seismic hisory nlysis of symmericl djcen buildings wih soil-srucure inercion considerion M. E. Uz & M. N. S. Hdi School of Civil, Mining nd Environmenl Engineering, Universiy of Wollongong, NSW, Ausrli Absrc he seismic response hisory nlysis of muli-sorey symmeric djcen buildings wih soil-srucure inercion (SSI) during impc is invesiged in his sudy. he coupled muli-degree of freedom modl differenil equions of moion for he wo wy symmeric sher buildings re derived nd solved using sep by sep soluion by he fourh-order Runge-Ku mehod wih impc nd wihou impc. he SSI forces re modelled in he form of he frequencyindependen soil springs nd dshpos. A numericl exmple of wo-wy symmeric four-sorey djcen buildings under he vriion of he imporn SSI sysem prmeers such s he lrge nd smll SSI effecs is invesiged under he exciion of he 1940 El Cenro Erhquke. his sudy shows h he lerl orsion response of boh buildings is ffeced when impc kes plce wih he djcen buildings. he firs hree modl response hisories of ech building re significnly reduced due o he effec of he smll SSI. Furher, i is lso observed h he roof wis of he ligher buildings is decresed for he lrge SSI effec compred o he smll SSI effec. Finlly, in incresed eccenriciies, he impc response of buildings is significnly severe. Keywords: erhquke engineering, impc effecs, soil-srucure inercion, symmery. 1 Inroducion Providing he required seprion disnce is no lwys possible. Mny reserchers hve invesiged he pounding problem of seismic nlysis of wo wy symmeric buildings wih soil-srucure inercion under he wo WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press doi: /eres110191

3 226 Erhquke Resisn Engineering Srucures VIII direcionl ground moions [1 5]. In hese sudies, o del wih he nonproporionl dmping of he SSI sysems, equivlen modl dmping ws clculed o fcilie he modl response hisory nlysis. he equivlen modl dmping ws esimed by eiher qunifying he dissiped energy in he soil [6] or mching he pproximion pproches norml mode soluion wih he rigorous soluion for cerin srucurl locion [3]. For engineering pplicions wihou he need for clculing he compliced equivlen modl dmping, simple nd rel vlued modl response hisory nlysis hs been developed [7]. Moreover, he effecs of boh he SSI sysems nd pounding on he coupled buildings under srong erhqukes hve no been invesiged fully. he im of his sudy is o conduc comprive sudy in order o invesige he SSI effec ono djcen buildings considering he effec of pounding. Bsed on he fourh order Runge-Ku mehod, MALAB progrm is developed o solve he equions of moion for he SSI sysems of coupled buildings subjeced o pounding effecs under he exciions of erhquke ground ccelerion nd numericl simulions presened re used. he issue of he pounding of bidirecionl symmeric djcen buildings wih he SSI sysems is no fmilir wih he complex-vlued seismic nlysis procedures in he frequency domin [8, 9]. A response hisory nlysis procedure considering of boh effecs is proposed using he MDOF modl equions of he moion by wching he soluions of equions of moions for he whole SSI sysems nd pounding on he djcen buildings in he following numericl exmples. 2 heoreicl bckground of sysem model In order o consider he effecs of he lrge nd smll SSI sysems, wo-wy symmeric coupled buildings re modelled in his sudy. o vlide he equion of moion for he whole SSI sysem compring o he muli-degrees of freedom modl equions of he moion, modl nlysis sudy is conduced herein. he inercion forces he soil-srucure inerfce re simuled using frequency-independen spring nd dshpo se in prllel [10]. he recngulr dimensions of foundions for boh buildings re convered s circulr fooings in order o dop he frequency independen spring nd dshpo se. he simplified model of N- nd S- sorey coupled buildings resing on he surfce of n elsic hlf-spce is shown in Figure 1. For i= (1,2,.,N) nd j=(1,2,.,s), m,k i xi,c xi,k yi,c yi,ixi nd yi I re he mss, he elsic srucurl siffness, dmping coefficiens nd momens of ineri of he reled floor bou he xes hrough he cenre of mss (CM) nd prllel o he x nd y xes for Building A nd Building B, respecively. he subscrips i nd j in Figure 1 re he sorey number of he buildings h denoe 1, 2,., N for Building A nd 1, 2,.,S for Building B. Moreover, he subscrips nd superscrips of nd b symbolize Building A nd Building B, respecively. For he horizonl componen of ground moion ssumed o be uniform over he bse of he buildings, he ol number of degrees of freedom is 3N+5 for N-sorey building nd herefore 3N+5 nd 3S+5 equions re required for boh Building A nd Building B, WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

4 Erhquke Resisn Engineering Srucures VIII 227 Figure 1: Elevion of dynmic model for symmeric djcen sher buildings. respecively. Equions for Building B re he sme s shown below for Building A ssocied wih he number of sorey (j) of Building B. Hence, only he equions of Building A re expressed here. rnslion in he longiudinl (x) nd rnsverse (y) direcions nd roion bou he CM of hese equions my be expressed s shown in Eqn. (1). As n exmple, for N- sorey Building A, he ol number of degrees of freedom, 3N+5 re obined s 3N equions of dynmic equilibrium of ech floor of he supersrucure for he rnslion in he x nd y direcions nd roion bou he cenre of mss nd 5 degrees of freedom due o inercion he foundion. he 3N equions of dynmic equilibrium of ech floor of Building A my be expressed s p M x C x K x F 0 ic x i x i xij p M y C y K y F 0 ic y i y i yij (1) r M f C x e C y C f K x e K y 2 i x i y i R ic x i y i p K R ic Fij 0 WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

5 228 Erhquke Resisn Engineering Srucures VIII where M, C x, K x, Cy nd K y re he N N sub-mrices of mss, dmping nd lerl siffness in he x nd y direcions of Building A, respecively. x ic, y ic nd i re he ol displcemens of cenre of mss of he floors in he longiudinl nd rnsverse direcions, nd he ol wis of he floors bou he P vericl xis (z) in Building A, respecively. Fxij denoes he pounding forces P in he x direcion wih he help of he non-liner viscoelsic model [11]. Fyij P nd F ij hve been considered by he Coulomb fricion model [12, 13]. x i, yi nd ic re he displcemen vecors wih respec o he bse in he x nd y direcions of he cenre of resisnce (CR) nd he wis of he floors wih respec o he bse. Moreover, wo wy symmeric buildings re modelled s he CR being no coinciden wih he CM long he wo horizonl plne xes. he sic eccenriciies of he cenre of resisnce from he cenre of mss (e nd f) in he x nd y xes re he sme for ech floor deck, lhough he CR my vry from sorey o sorey. Hence, he CR ssocied wih he djcen buildings is ssumed o lie eccenriciies e,f for Building A nd e,f b b for Building B. he rdii of gyrion (r nd r b ) of ny rigid floor decks re bou he cenre of mss for ech building. nd in Eqns. (1) nd (2) re he orsionl siffness mrix K R K M defined bou he CR nd he CM, respecively. Furhermore, C x, Cy nd in Eqn. (1) re he dmping mrices for Building A, ssumed o be C R proporionl o he siffness mrices s defined in Eqn. (2). K K e K f K 2 2 M R y x Cx K x, Cy Ky nd C R K R in which is consn vlue in erms of he rio of he coefficien nd siffness of he buildings. he displcemen vecors in he reled direcions of Building A wihou SSI effecs cn be defined by Eqn. (3). xic xi f ic (3) yic yi e ic where x ic nd y ic re displcemen vecors of degrees of freedom of supersrucure bou he CM. x ic,yic nd ic cn be expressed in view of he ' ' following relionship of xic nd yic vecors, which re he degrees of freedom of he supersrucure defined s in Eqn. (4)., xicx if ic, xic xif ic, yicy ie ic, yic yie ic x ic xo 1 xg 1 o hixic nd x ' ic xo 1 o hixic y ic yo 1 yg 1 o hi yic nd y ' ic yo 1 o hiyic i o 1 ic (2) (4) WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

6 Erhquke Resisn Engineering Srucures VIII 229 where x,y, o o o nd o re he degrees of freedom he bse ssocied wih rnslions nd rocking bou he x nd y xes, respecively. o is he wis bou he z xis. Afer subsiuing in Eqns. (2), (3) nd (4), nd rerrnging ino o Eqn. (1), more concise form for he 3N 3N sub-mrices of he supersrucure resing on rigid bse on he lef upper corner of M,C nd K cn be wrien herein. Wih reference o Figure 1, he equion of moion for he whole foundion sysem for Building A cn be wrien for he rnslion in he x nd y xes, wis bou he z xis nd rocking bou he x nd y xes, respecively s shown in Eqn. (5) [10]. m x x 1 M x P 0 o g o ic x m y y 1 M y P 0 o g o ic y rm r 1 M o o i N xi o i ic x i0 h M y Q 0 N yi o i ic y i0 h M x Q 0 where I xi nd I yi re momens of ineri of he i h floor bou he xis hrough he CM nd prllel o he longiudinl nd rnsverse direcions, respecively. m o is he mss of he foundion of Building A. hi nd h j re he column vecor composed of he sorey heighs of Building A nd Building B hroughou he foundion o ech floor, respecively. Erhquke ground ccelerions in he x nd y direcions re shown s x g nd y g, respecively. Px,Py,,Qx nd Qy re he inercion forces of Building A bsed on frequencyindependen soil springs nd dshpos s shown in Eqn.(5) [4]. he definiions of spring nd dshpo consns of he sic impednce funcions re clerly presened wih vrious subscrips [10]. Firs ssume h wo buildings remin in he liner elsic rnge nd hence hey do no yield under erhquke exciion. In such cse, he equion of moion in Eqn. (6) for he couple buildings wih he whole inercions such s he SSI, orsionl coupling nd he pounding involved responses of djcen buildings modelled wih elsic sysems ech floor level s sysem is P M 0 U C 0 U K 0 U F P b b b b b b P b 0 M U 0 C U (6) 0 K U F P b b b where M, C, K, M, C nd K re he mss, dmping nd siffness mrices P b of couple buildings, respecively. Moreover, F, P nd P re vecors conining he forces due o impc beween floors wih msses m,m i j nd loding of he djcen buildings in h order (see Eqns. (7), (8) nd (9)). U, U, U b, U, b U nd b U re he vecors of ccelerion, velociy nd displcemen of he sysem respecively. (5) WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

7 230 Erhquke Resisn Engineering Srucures VIII M NN NN NN M NN 2 NN NN r M M mo M 1 M 0 mo r 1 M 0 0 rmo 0 0 N 0 hi M xi 0 i0 N hi M yi i0 3N53N5 (7) where 0 nd 1 re he N 1 column vecors whose elemens re equl o zero nd one, respecively. K Kx 0N N f Kx Kx1 0 Kxf 0 Kxhi K y e K y 0 K y 1 K ye K yhi 0 K K f K e K 1 e K h f K h M x y M y i x i K 1 Kx 1 0 f Kx Kxhi K 1 K y 1 e K y 1 1 K yhi 0 symm. K 1 K M1 e Ky hi f Kxhi symm. K h i K y h i 0 K hi Kxh i Cx 0N N f Cx Cx 1 0 Cx f 0 Cx h i C y e C y 0 C y 1 C y e C y hi 0 C M Cx f C y e C M1 e C y hi fcx hi C 1 Cx 1 0 f Cx Cx h i C C 1 C y 1 e C y 1 1 C y hi 0 symm. C 1 C M 1 e C yhi f C x hi symm. C hi C y hi 0 C hi Cxhi M 1 xg ' P x ic Fxij M 1 y g ' 0 y P ic Fyij P F ij P,U,F mo 1 M x g i P xo mo 1 M yg y o 0 o hi M yg o hi Mx g o (3N5) 1 (3N5) 1 (3N5) 1 (8) (9) P P For simuling he pounding force during impc F xij, F ij i 1,2,..N; j 1,2,..,S, he nonliner viscoelsic model is used beween he sorey levels of he wo djcen buildings, which is bsed on he following formul in Eqn. (10) s boh pproch period nd resiuion period of collisions [11, 14]. WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

8 Erhquke Resisn Engineering Srucures VIII 231 p xij for F 0 3 ij 0; for nd F c p 2 xij ij ij ij 3 ij ij 0 ij 0 ; for nd ; (10) p 2 xij F ij 0 mm i j cij 2 ij ; m m e 2 e e ' ' ' ' ij xic xjc D; x x ; ij ic jc i j ij 0 (11) where ij nd ij in Eqn. (11) is he ol relive displcemen nd velociy beween boh buildings wih respec o he foundion, respecively. On he oher hnd, he pounding forces in he rnsverse direcion F P yij hve been clculed by he Coulomb fricion model [12, 15]. 3 Properies of he srucures he dimensions of boh Building A nd Building B re recngulr in pln wih 20m 15m nd 25m 20m, he lrger pln dimensions being prllel o he longiudinl direcion (x) for ech building, respecively. he rio of he bse mss o he floor mss of he buildings is 3 for ech building. he following bsic vlues describing he srucurl chrcerisics in ble 1 hve been used: ble 1: Srucurl chrcerisics of buildings. Sorey no Heigh of floor level, h,h i j (m) Building A 6 m 10 k 10 i kg i N/m 8 Building B 6 m 10 k 10 j m j N/m 1F F F F F Moreover, for he rnslion in he x nd y xes, wis bou he z xis nd rocking bou he x nd y xes, he dimensions of recngulr bse of djcen buildings cn be convered ino n equivlen circulr bse hving he sme re s he pln of ech building bsed on he formuls deermined by Richr e l. [10]. Hence, he clculions of he rdius of bse mss deermined by Richr e l. [10] re used wih considering rnslions, roion nd rocking direcions, 8 WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

9 232 Erhquke Resisn Engineering Srucures VIII herein. he heigh of ech sorey is 2.85 m in boh buildings. he momen of ineri of he rigid body for ech building bou he cenroidl xes prllel o he x nd y xes re evlued by replcing ech floor wih disc of rdius (r o ). he rnslionl siffness in he rnsverse direcion nd he orsionl siffness bou he cener of mss for ech sorey for ech building re proporionl o he siffness in he longiudinl direcion of he sme sorey nd given by he following formul in Eqn. (12). k yi k yj k k i j y y 2 2 (12) k k rk rk he rios, y nd xi xj xi re ken s 1.32 nd 1.69 for boh buildings. he consn of proporionliy is evlued on he bsis of 2% of criicl dmping in he fundmenl mode of supersrucures in boh buildings. he densiy of soil medium, nd Poisson s rio, re ken o be 1922 kg/m 3 nd 0.333, respecively. In order o exmine he effeciveness of he rigorous mehod for he whole SSI sysems wih using he direc inegrion mehod o solve he equions of moion, wo soil ypes re invesiged in he rnge of he sher velociies, v of 65 m/sec (sof soil) nd 300 m/sec (hrd soil) hve been s specificlly chosen for his sudy. Cse I (v s =65 m/sec) nd Cse II (v s =300 m/sec) re creed for he SSI sysems resing on he sof nd hrd soils in order o invesige he seismic response of he djcen buildings under lrge nd smll SSI effecs, respecively. hese cses re subjeced o he NS nd EW componens of he 1940 Elcenro erhquke record long he x nd y xes, respecively. Bsed on dimensionless frequency, 0 resuling from 0o fr o /vs 1.5, he mximum wve frequencies of boh f mx 2.1 Hz, 9.7Hz of Building A, 1.8 Hz nd 7.8 Hz of Building B for Cse I nd Cse II, respecively. Mos of he energy of he reled ground moion is frequencies less hn 1.8 Hz. Hence, by using frequency independen spring nd dshpo se for he coupled buildings, he SSI effecs of he chosen cses cn be conduced effecively. Furhermore, ech coupled building hs been modelled s reference buildings resing on rigid bse wih similr supersrucure. he iniil gp, D, beween he buildings hs been ken s 0.04 m. In order o consider he effec of pounding ono he coupled buildings, when he conc of he buildings in he longiudinl direcion hs been deeced, he pounding forces in he rnsverse nd vericl direcions hve been pplied. For he response ime hisories of he whole SSI sysem of he coupled buildings, he rigorous mehod which uses he direc inegrion mehod o solve he equion of moion shown in Eqn. (1) is denoed s rigorous (Rig). 4 Resuls of fixed buildings for he SSI effecs he pounding of he djcen buildings modelled s elosoplsic muli degree of freedom lumped mss sysems wih eiher elsic srucurl behviour is sudied under he creed eiher Cse I or Cse II. Firsly, in order o invesige he SSI b xj WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

10 Erhquke Resisn Engineering Srucures VIII 233 sysems on he behviour of he coupled buildings wih he lrge nd smll SSI effecs, he ol response hisories of Cse I nd Cse II bsed on he deformion vecors of boh supersrucures of he wo buildings modelled s elsic sysems in Figure 2 nd Figure 3, respecively. Figure 2: ol response ime hisories of Cse I he fourh floors of he djcen buildings modelled s elsic sysems under he 1940 Elcenro erhquke. I cn be seen from Figure 2 h boh buildings cme ino conc six imes bsed on he x direcionl displcemens of he fourh floors during he erhquke. Due o collisions in x direcion nd he effec of orque force, he concs beween he buildings in he y xes re developed, lhough he pounding forces re no severe compred o he highes conc poins in he x direcion. While he ligher nd more flexible Building A compred o Building B is subjeced o more wis bou is z xes he op floor levels he lowes period of he ground moion, he roions of he op floor of Building B increse fer he conc beween he buildings. I cn be clerly noed from Figure 3 h he number of conc poins nd he sensiiviy of pounding forces in boh direcions re significnly incresed beween he buildings s modelled elsic sysems. I shows he impornce of he SSI effecs on he seismic response hisories of he djcen buildings under he wo direcionl 1940 Elcenro Erhquke. he roof wis of boh buildings is considerbly decresed for Cse I compred o h of Cse II. Figure 4 nd Figure 5 show he deformion in he rnslions, roions in he x nd y xes nd wis bou he vericl z xes nd of he foundions of he buildings modelled s elsic sysems considering Cse I nd Cse II, respecively. By compring Figure 4 nd Figure 5, i cn be seen h he vlues of he responses he foundion for Cse II (hrd soil) re considerbly reduced by reson of he smll SSI effec. In order o invesige he effec of he buildings WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

11 234 Erhquke Resisn Engineering Srucures VIII Figure 3: ol response ime hisories of Cse II he fourh floors of he djcen buildings modelled s elsic sysems under he 1940 Elcenro erhquke. Figure 4: ol response ime hisories of Cse I he foundions of he djcen buildings modelled s elsic sysems under he 1940 Elcenro erhquke. modelled s elsic sysem on he SSI effecs, he deformion prmeers wih considering pounding beween he buildings nd he SSI forces re conduced here. WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

12 Erhquke Resisn Engineering Srucures VIII 235 Figure 5: ol response ime hisories of Cse II he foundions of he djcen buildings modelled s elsic sysems under he 1940 Elcenro erhquke. 5 Conclusion he vlues of he pounding force nd he number of impcs re lrger in he Cse II compred o Cse I. he resuls of furher invesigion show h he responses bsed on deformion vecors of supersrucures for ech building re significnly reduced by incresing sher wve velociy, while he SSI forces re incresed he foundion of he buildings. A high sher wve velociy, he op floor deformions of couple buildings re slighly on he conservive side. Finlly, in incresed sher wve velociy, he impc response of buildings is significnly severe. References [1] Hdi, M.N.S., Uz, M.E. Improving he dynmic behviour of djcen buildings by connecing hem wih fluid viscous dmpers. in 2nd Inernionl Conference on Compuionl Mehods in Srucurl Dynmics nd Erhquke Engineering, COMPDYN Islnd of Rhodes, Greece: Insiue of Srucurl Anlysis & Seismic Reserch Nionl echnicl Universiy of Ahens, 280, [2] Sivkumrn, K.S., Blendr,. Seismic nlysis of symmeric mulisorey buildings including foundion inercion nd p-[del] effecs. Engineering Srucures 1994; 16(8): [3] hmbirjh, B., Chn Weng,. & Seng-Lip, L. Modl dmping for orsionlly coupled buildings on elsic foundion. Erhquke Engineering & Srucurl Dynmics 1982; 10(5): WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press

13 236 Erhquke Resisn Engineering Srucures VIII [4] hmbirjh, B., Chn Weng,. & Seng-Lip, L. Vibrion of symmericl building-foundion sysems. Journl of Engineering Mechnics 1983; 109(2): [5] Uz, M.E., Hdi, M.N.S. Dynmic nlyses of djcen buildings conneced by fluid viscous dmpers. in Sevenh World Conference on Erhquke Resisn Engineering Srucures ERES VII. Limssol, Cyprus: Wessex Insiue of echnology, , [6] Novk, M., Hifnwy, L.E. Effec of soil-srucure inercion on dmping of srucures. Erhquke Engineering & Srucurl Dynmics 1983; 11(5): [7] Jui-Ling, L., Keh-Chyun,. & Edurdo, M. Seismic hisory nlysis of symmeric buildings wih soil srucure inercion. Journl of Srucurl Engineering 2009; 135(2): [8] Uz, M., Hdi, M.N. Invesiging he effecs of pounding for inelsic bse isoled djcen buildings under erhquke exciions. in Proceedings of he 21s Ausrlin Conference on he Mechnics of Srucures nd Merils. Melbourne, Vicori, Ausrli: he Neherlnds: CRC Press, pp , [9] Hdi, M.N.S., Uz, M.E. Inelsic bse isoled djcen buildings under erhquke exciion wih he effec of pounding. in he 5h Civil Engineering Conference in he Asin Region nd Ausrlsin Srucurl Engineering Conference 2010 CECAR 5/ASEC Sydney, Ausrli, , [10] Richr, F.E., Hll, J.R. & Woods, R.D. Vibrions of soils nd foundions. Prenice-Hll, Englewood Cliffs; [11] Jnkowski, R. Erhquke-induced pounding beween equl heigh buildings wih subsnilly differen dynmic properies. Engineering Srucures 2008; 30(10): [12] Chopr, A.K. Dynmics of srucures. heory nd pplicions o erhquke engineering. Englewood Cliffs: Prenice-Hll; [13] Wriggers, P. Compuionl conc mechnics wih 12 bles. Springer; [14] Hdi, M.N., Uz, M. Bse isoled djcen buildings considering he effec of pounding nd impc due o erhqukes. in Inernionl Congress on Advnces in Civil Engineering. rbzon, urkey: Eser Ofse Mbcilik, pp , [15] Jnkowski, R. Experimenl sudy on erhquke-induced pounding beween srucurl elemens mde of differen building merils. Erhquke Engineering & Srucurl Dynmics 2010; 39(3): WI rnscions on he Buil Environmen, Vol 120, 2011 WI Press