KINEMATIC SOIL-STRUCTURE INTERACTION EFFECTS ON MAXIMUM INELASTIC DISPLACEMENT DEMANDS OF SDOF SYSTEMS

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1 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China KINEMATIC SOIL-STRUCTURE INTERACTION EFFECTS ON MAXIMUM INELASTIC DISPLACEMENT DEMANDS OF SDOF SYSTEMS Y.Y. Lin 1 1 Associat Profssor, Dpt. of Civil and Watr Rsourcs Eng., Chayi Univrsity, Chins Taiwan ABSTRACT : This papr prsnts a statistical study of th kinmatic soil-foundation-structur intraction ffcts on th maximum inlastic dformation dmands of structurs. Discussd hr is th inlastic displacmnt ratio dfind as th maximum inlastic displacmnt dmands of structurs subjctd to foundation input motions divid by thos of structurs subjctd to fr-fild ground motions. Th displacmnt ratio is computd for a wid priod rang of lasto-plastic singl-dgr-of-frdom (SDOF) systms with various lvls of latral strngth ratios and with diffrnt sizs of foundations. Svnty-two arthquak ground motions rcordd on firm soil with avrag shar wav vlocitis btwn 180 m/s and 360 m/s ar adoptd. Th ffcts of priod of vibration, lvl of latral yilding strngth and dimnsion of foundations ar invstigatd. Th rsults show that kinmatic intraction will rduc th maximum inlastic displacmnt dmands of structurs, spcially for systms with short priods of vibration, and th largr th foundation siz th smallr th maximum inlastic displacmnt bcoms. In addition, th inlastic displacmnt ratio is narly not affctd by th strngth ratio of structurs for systms with priods of vibration gratr than about 0.3 s and with strngth ratios smallr than about 3.0. Exprssions obtaind from nonlinar rgrssion analyss ar also proposd for stimating th ffcts of kinmatic soil-foundation-structur intraction from th maximum dformation dmand of th inlastic systm subjctd to fr-fild ground motions. KEYWORDS: inlastic dformation dmands, kinmatic intraction, foundation input motion, fr-fild motion, strngth ratio. 1. INTRODUCTION For th analysis of th dynamic rspons of a structur subjctd to arthquaks, th ground motions ar inputtd from th bas of this structur and ar commonly assumd to b th fr-fild motions. Actually, th input motions should b th bas-slab motions (foundation input motions) if thr is a foundation in th structur. Diffrncs btwn th fr-filds motion and th foundation input motion rsult from kinmatic intraction. Th prsnc of stiff foundations on or in soil will caus th bas-slab motion to dviat from th fr-fild motion. Kinmatic intraction rducs foundation motions rlativ to th fr-fild du to th diffrncs in stiffnss btwn th foundation and surrounding soil (Todorovska and Trifunac 1992, Avils and Prz-Rocha1998, Stwart t al. 1998, Avils t al. 2002). For shallowly mbddd foundations, th primary factor contributd to such dviation is th bas-slab avraging ffct in which fr-fild motions associatd with inclind or incohrnt incidnt wav filds ar avragd within th footprint ara of th bas-slab du to th kinmatic constraint of ssntially rigid-body motion of th slab. It has bn shown that kinmatic intraction is most noticabl at high frquncis of xcitation. This fatur may imply that short-priod structurs ar asir than long-priod ons to b affctd by kinmatic intraction sinc stiff structurs ar snsitiv to high frquncy motions. Th purpos of this papr is to discuss th ffcts of kinmatic soil-foundation-structur intraction on th maximum dformation dmands of nonlinar SDOF systms with known rlativ strngth. Rsults of a statistical study of th ratio of maximum inlastic displacmnt rsponss drivd from th foundation input motion to thos drivd from th fr-fild motion ar prsntd. Svnty-two arthquak ground motions rcordd on fr-fild stations locatd on stiff soil ar usd. Th ffcts of dimnsions of foundations, priods of vibration and lvls of latral strngth ar invstigatd. In addition, for dsign practic, xprssions obtaind from nonlinar rgrssion analyss ar also proposd. 2. FOUNDATION INPUT MOTIONS

2 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China Th ffcts of kinmatic intraction ar, in gnral, quantifid by transfr functions H(f), th ratio of th foundation input motion to th fr-fild motion in frquncy domain. Th transfr function is actually a kind of low pass filtrs (Holloway 1958, Hudson 1979). Whn th fr-fild motion runs through foundations, th high frquncy wavs of th fr-fild motion will b filtrd by th foundations. Th foundation input motion can b dtrmind by th following stps. (a) Calculat Fourir transforms of th fr-fild motion; (b) Multiply th amplitud obtaind from Stp (a) by H(f); (c) Prform an invrs Fourir transform with th modifid amplitud, and thn th bas-slab motion can b obtaind. Although th foundation input motion can b drivd from th abov procdur, in this papr it will b approximatly stimatd by conducting an qually-wightd running avrag to th fr-fild motion in tim domain. It has bn shown that doing a running avrag to th fr-fild motion in tim domain is xactly qual to using th following low pass filtr in frquncy domain. H f sin(π ft ) w ( ) = (1) πftw whr f= ω / 2π = th natural frquncy (Hz) of th harmonic componnt of th ground motion considrd, T w = th filtring intrval which dpnds on th siz of foundations as follows. 09b T w = 51(1 ) (2) According to Eq. 2, th only variabl ( T w ) ndd in th approximat running avrag mthod can b acquird from th foundation siz ( b ). Fig.1 prsnts th foundation motions drivd from th fr-fild motion of th 1952 Krn County arthquak. In this figur, squar foundations with widths of 20 and 72 m ar usd. Th two bas-slab motions ar smoothr than th fr-fild motion. Fig.2 dpicts th Fourir amplituds of th foundation input and fr-fild motions. It can b obsrvd that th bas-slab motion has lss high frquncy contnt than th fr-fild on. High frquncy motions dcrass as th dimnsion of foundations incrass. Thrfor, th xistnc of foundations can rstrict th high frquncy ground motions to b importd into structurs. 3. STUDY MODEL Onc th foundation input motion is obtaind, th following trm (inlastic displacmnt ratio, K R ) is implmntd for studying th ffcts of kinmatic intraction on th maximum inlastic displacmnt dmands of structurs. i, b K R = (3) whr K R is dfind as th maximum latral inlastic displacmnt for structurs subjctd to th foundation input motion ( i,b ) dividd by that for structurs subjctd to th fr-fild motion ( i,f ) on systms with th sam mass and initial stiffnss (i.., sam priod of vibration, T). In Eq. 3, i,b and i,f ar computd in systms with constant yilding strngth rlativ to th strngth rquird to maintain th systm lastic (i.., constant rlativ strngth). Hr th rlativ latral strngth is masurd by th strngth ratio (R) dfind as ms i, f a R = (4) Fy in which m= th mass of th systm, S a = th acclration spctral ordinat, and F y = th latral yilding strngth of th systm. Th numrator in Eq. 4 rprsnts th latral strngth rquird to maintain th systm lastic, which somtims is also rfrrd to as th lastic strngth dmand. In this study, i,b and i,f ar calculatd for SDOF systms having lasto-plastic hystrtic bhavior, a viscous damping ratio of 5%, and svn lvls of strngth ratios (R=1, 1.5, 2, 3, 4, 5 and 6). For ach arthquak rcord and ach strngth ratio, th inlastic displacmnt ratio (K R ) is computd for a st of 57 priods of

3 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China vibration from 0.15 to 3.0 s with an incrmnt of 5 s. In addition, th foundations discussd hr ar limitd to b squar and hav width (b ) qual to 20, 43 and 72 m. According to th fr-fild and foundation input motions of Fig.2, Fig.3 shows th displacmnt rspons historis for th systm with vibration priod of s and strngth ratio of 5, rspctivly. In th cas, th dformation trac drivd from th bas-slab motion is smallr than that drivd from th fr-fild motion. Th inlastic displacmnt rspons spctra with R= 1 and 5 for th fr-fild and foundation input motions of th Krn County arthquak ar prsntd in Fig.4. From this figur, th inlastic displacmnt ratio (K R ) can b obtaind by Eq. 3 as shown in Fig EARTHQUAKE GROUND MOTIONS USED IN THE STUDY A total of 72 arthquak acclration tim historis rcordd in th stat of California from 9 diffrnt arthquaks ar usd as th fr-fild motions in this invstigation. Ths ground motions hav th following charactristics: (a). Thy ar rcordd on acclrographic stations whr nough information xists on th gological and gotchnical conditions at th sit that nabls th classification of th rcording sit in accordanc to rcnt cod rcommndations; (b) Thy ar rcordd on firm sits with avrag shar wav vlocitis btwn 180 and 360 m/s.; (c) Thy ar rcordd on fr-fild stations or in th first floor of low-ris buildings with ngligibl soil-structur intraction ffcts; (d) Thy ar rcordd in arthquaks with surfac wav magnituds (M s ) largr than 6.1; and () At last on of th two horizontal componnts had a pak ground acclration (PGA) largr than 40 cm/s STATISTICAL RESULTS Fig.6 dpicts th ratios of S a,foundation /S a,fr-fild which ar drivd from th man normalizd lastic rspons spctra (S a ) computd from th fr-fild and foundation motions (b =20, 43, and 72 m) of th 72 arthquak rcords. It is clar that th kinmatic soil-foundation-structur intraction obviously rducs th maximum lastic acclration rspons of th short-priod (stiff) systms (T<0.3 s for b =20 m, T< s for b =43 m, and T< s for b =72 m). Th gratr th siz of foundations th smallr th acclration rspons bcoms. In othr words, th kinmatic intraction is bnficial to lowris buildings with larg foundations. For instanc, if a structur has a priod of vibration of sc, th maximum acclration rspons can b dcrasd in prcntag of 5% (b =20 m), 19% (b =43 m) and 39% (b =72 m) with rspct to thir fr-fild ons. Long-priod systms ar not snsitiv to th kinmatic intraction. Fig.7 shows th man inlastic displacmnt ratios ( ) corrsponding to all of th 72 ground motions for systms with diffrnt lvls of strngth ratios and diffrnt sizs of foundations. Th man maximum inlastic dformation dmands drivd from th foundation input motions for all systms ar smallr than thos drivd from th fr-fild ons (i.., all < ). This mans that th kinmatic soil-foundation-structur intraction would not incras th maximum inlastic displacmnt dmands of all structurs. As indicatd in th lastic acclration rspons spctra (th prcding paragraph), th rduction of maximum dformation dmands is obvious as systms with priods of vibration lss than about 0.3 s for b =20 m, s for b =43 m, and s for b =72 m. Although incrasing th siz of foundations rsults in th dcras of displacmnt dmands, th kinmatic intraction ffcts can almost b ignord for long-priod (soft) structurs (T s for b =20 m, T s for b =43 m, and T 1.2 s for b =72 m). In addition, for systms with strngth ratios lss than 3.0 and priods of vibration gratr than about 0.3 s (Fig.7), th inlastic displacmnt ratios ( ) ar narly indpndnt of th inlastic hystrtic charactristic of structurs (i.., for R 3. 0 and T 0.3 s is almost idntical to ach othr). Th man inlastic displacmnt ratio is vry important bcaus thy rprsnt what can b xpctd on avrag. Howvr, it is qually important to quantify th lvl of disprsion in th inlastic displacmnt ratio. A common and ffctiv way to quantify th disprsion is through th cofficint of variation (COV), which is dfind as th ratio of th standard dviation to th man. Fig.8 prsnts COVs of th inlastic displacmnt ratios corrsponding to all ground motions considrd hr for various strngth ratios (R) and foundation widths ( b ). It can b sn that disprsion incrass as th dimnsion of foundations and th lvl of strngth ratios

4 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China incras. Th lastic cas (R=) always has th smallst rrors. Disprsion is high for systms with short priods of vibration (T< 0.3 s for b =20 m, s for b =43 m, and s for b =72 m) rgardlss of th latral strngth ratio. Furthrmor, for a givn lvl of strngth ratios, th COVs dcras as incrasing th priods of vibration of structurs. 6. NONLINEAR REGRESSION ANALYSES For arthquak-rsistant dsign, it is dsirabl to hav a simplifid xprssion of th inlastic displacmnt ratios to stimat th maximum inlastic displacmnt dmands of th systms subjctd to th foundation input motion from thos of th systms subjctd to th fr-fild motion (i.., to prdict i,b from i,f ). For th objctiv, th Lvnbrg-Marquardt mthod (Bvington and Robinson 1992) is conductd for obtaining th xprssion. Th proposd quation is givn as follows. i, b 1 = i, f at, = (32R Tw 57Tw ) + 10Tw a (5) whr T = th priod of vibration of systms, R= th strngth ratio of systms, and T w = th filtring intrval as shown in Eq. 2, which is th function of foundation siz ( b ). Eq.5 is plottd in Fig.9. In th short-priod rgion, this quation is primarily controlld by th variabl of natural priod (T ) with an xponntial tndncy. Ths curvs convrg and kp almost constant as priods of vibration of systms incras. Eq.5 corrsponds to a surfac in th -T-R- b spac that provids stimats of man inlastic displacmnt ratios as functions of T, R and b. Sinc th inlastic displacmnt ratios for R 3.0 and T 0.3 s ar almost indpndnt of th strngth ratios (Fig.10a), Eq.5 can b furthr simplifid in th rang as Eq.6 shown in Fig.10b. 7. CONCLUSIONS i, b i, f = T w Through th inlastic displacmnt ratio, a statistical study of th kinmatic soil-foundation-structur intraction on th inlastic displacmnt dmands of lasto-plastic SDOF systms has bn carrid out by a larg numbr of fr-fild and foundation input motions. Th inlastic displacmnt ratio is dfind as th maximum latral inlastic displacmnt of structurs subjctd to foundation input motions dividd by that of th sam structurs subjctd to fr-fild motions. This ratio dpnds on th priod of vibration of systms (T), th strngth ratio of systms (R), and th dimnsion of foundations ( b ). From th invstigation of th study, th following conclusions can b drawn. (1) Th foundation input motions always hav lss high frquncy contnts than th fr-fild ons bcaus of th prsnc of foundations (Fig.2). High frquncy ground wavs ar filtrd by foundations, and only th long-priod motions can b passd into structurs. (2) Kinmatic intraction will rduc th maximum inlastic displacmnt dmands of structurs, spcially for short-priod systms (Fig.7). Th rason is that short-priod systms ar mor snsitiv to th high frquncy xcitation than long-priod ons. Th rduction in displacmnt dmands du to kinmatic intraction is apparnt whn systms hav priods of vibration lss than about 0.3 s for foundations with width ( b ) of 20 m, s for b =43 m, and s for b =72 m. Long-priod systms ar almost not affctd by th kinmatic soil-foundation-structur intraction. (3) Incrasing th dimnsion of foundations ( b ) rsults in a dcras of th maximum inlastic dformation dmands of structurs (Fig.7) bcaus th largr th dimnsion of foundations th mor th high frquncy motions ar filtrd (Fig.2). (4) Cofficints of variation of inlastic displacmnt ratios incras with dcrasing th priods of vibration of systms and with incrasing th width of foundations (Fig.8). Disprsion is larg for systms with priods T 1.6 (6)

5 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China smallr than 0.3 s ( b =20 m), s ( b =43 m), and s ( b =72 m). (5) Simplifid quations hav bn drivd from nonlinar rgrssion analyss for stimating th maximum inlastic displacmnt dmands of systms subjctd to foundation input motions from thos of systms subjctd to fr-fild motions. 8. REFERENCES Avils, J. and Prz-Rocha, L.E. (1998). Effcts of foundation mbdmnt during building-soil intraction. Earthquak Enginring and Structural Dynamics 27, Avils, J., Suarz, M. and Sanchz-Ssma, F.J. (2002). Effcts of wav passag on th rlvant dynamic proprtis of structurs with flxibl foundation. Earthquak Enginring and Structural Dynamics 31, Bvington, P.R. and Robinson, D.K. (1992). Data Rduction and Error Analysis for th Physical Scincs. McGraw Hill, Nw York. Holloway, L.J. (1958). Smoothing and filtring of tim sris and spac filds. Advancs in Gophysics 4, Hudson, D.E. (1979). Rading and intrprting strong motion acclrograms. EERI Monograph Sris, Earthquak Enginring Rsarch Institut, Oakland, California. Stwart, J.P, Sd, R.B. and Fnvs, G.L. (1998). Empirical valuation of inrtial soil-structur intraction ffcts. Rport No. PEER-89/07, Pacific Earthquak Enginring Rsarch Cntr, Univrsity of California, Brkly, California. Todorovska, M.I. and Trifunac, M.D. (1992). Th systm damping, th systm frquncy and th systm rspons pak amplituds during in-plan building-soil intraction. Earthquak Enginring and Structural Dynamics 21,

6 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China Ground acclration (gal) fr-fild foundation, b=20m foundation, b=72m Krn County 1952, 180, Stn.# Tim [s] Magnitud Krn County 1952, 180, Stn.# 135 fr-fild foundation, b=20m foundation, b=72m Frquncy, f [Hz] Displacmnt (cm) Fig.1 Comparison btwn th fr-fild motions and th foundation input motions drivd from th 1952 Krn County arthquak (b =20 and 72 m). - Krn County 1952, 180, Stn.# 135, T=s, R=5.0 fr-fild foundation, b=72m Fig.2 Fourir amplitud of th fr-fild and foundation input motions (b =20 and 72 m) for th 1952 Krn County arthquak. i (cm) fr-fild, R= foundation, R= fr-fild, R=5.0 foundation, R= Pak rspons undr foundation input motion (45 cm) (b =72 m) Pak rspons undr fr-fild motion (13 cm) Krn County 1952, 180, Stn.# Tim [s] 1.2 Fig.3 Displacmnt rspons historis drivd from fr-fild and foundation input motions of th 1952 Krn County arthquak for T= s and R=5.0. (b =72m) Krn County 1952, 180, Stn.# 135 R=5.0 R= 1.2 Fig.5 Inlastic displacmnt ratios ( i, b / i, f ) for th 1952 Krn County arthquak (R= and 5.0). Fig.4 Displacmnt rspons spctra drivd from fr-fild and foundation input motions of th 1952 Krn County arthquak for R= and 5.0. Sa,foundation/Sa,fr-fild b=20m b=43m b=72m Fig.6 Ratios of man lastic rspons spctra drivd from fr-fild motions to thos drivd from foundation input motions.

7 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China COV (b =20m) R= (b =20m) R= COV (b =43m) R= (b =43m) R= COV (b =72m) R= (b =72m) R= 1.2 Fig.7 Man inlastic displacmnt ratios ( i, b / i, f ) against priods for squar foundations with width (b ) qual to 20, 43 and 72 m. 1.2 Fig.8 Cofficints of variation of inlastic displacmnt ratios for all 72 arthquaks.

8 Th 14 th World Confrnc on Earthquak Enginring Octobr 12-17, 2008, Bijing, China b =20m 43m 72m (b =20m, rgrssion) R= 1.2 R=3.0 R= 1.2 Fig.10a Man inlastic displacmnt ratios drivd from all 72 arthquak rcords for systms with T 0.3 s and R 3.0. (b =43m, rgrssion) R= 1.2 (b =72m, rgrssion) R= 1.2 Fig.9 Proposd inlastic displacmnt ratios for computing kinmatic intraction of squar foundations. b=20m b=43m b=72m 1.2 Fig.10b Simplifid inlastic displacmnt ratios proposd for systms with T 0.3 s and R 3.0.