DYNAMIC CHARACTERISTICS OF STRUCTURES ON PILES AND FOOTINGS

Transcription

1 4 th International Conference on Earthquake Geotechnical Engineering June 5-8, 007 Paper No. 167 DYNAMIC CHARACTERISTICS OF STRUCTURES ON PILES AND FOOTINGS Andrea MARAVAS 1, George MYLONAKIS, Diitri L. KARABALIS 3 ABSTRACT In thi paper, novel analytical olution are preented for ingle degree-of-freedo (SDOF) ocillator founded on footing and pile on copliant oil. Firt, eact forula for the fundaental natural period of the above tructure, encopaing the frequency dependence of the variou ipedance ter, are derived. Second, cloed-for olution for the correponding daping coefficient are derived. It i hown that the coon approiation of neglecting higher-order ter involving product of daping coefficient i unneceary and potentially inaccurate for highly-daped oiltructure yte. To addre the iue of coupled waying-rocking ocillation at the pile head, the reference yte i tranlated to the depth below the pile head where the reultant oil reaction to the pile i applied, to enure a diagonal ipedance atri. Third, the aount of radiation daping generated fro a ingle pile and a footing are copared. To thi end, the concept of tatically and geoetrically equivalent SSI yte i introduced. It i hown that a tructure founded on a pile ay generate twice the aount of radiation daping of a iilar tructure on a pread footing. Reult are provided in ready-to-ue graph and chart that elucidate the alient feature of the proble and can be directly ipleented in deign. The paper copleent and etend the einal tudie in the ubject by Parelee, Veleto, Bielak and their co-worker. Keyword: Soil-tructure interaction, pile, footing, natural period, radiation daping INTRODUCTION Knowledge on the ubject of dynaic Soil-Structure Interaction (SSI) ha been derived ainly fro tudie of tructure on at foundation during the lat forty year. The eiic repone of tructure on pile foundation ha received coniderably le reearch attention. More iportantly, the reult of thee effort have not yet lead to etablihed deign ethod and/or code proviion, uch a the iple ethod developed for tructure on urface foundation (ATC-3, NEHRP-03, EC-8). Therefore uch i yet to be learned on the ubject before a coprehenive undertanding i developed on the role of baic proble paraeter on the eiic repone of pile-upported yte. The goal of thi article are: (1) to review available ethod on the ubject with ephai on the deign-oriented olution by Veleto and co-worker (1974, 1975, 1977) and Wolf (1985), () to preent a novel olution for deterining the fundaental natural period and effective daping of a iple ocillator upported on a urface foundation, (3) to etend the olution to encopa pile foundation, (4) to preent reult for typical tructural yte on copliant ground, and (5) to copare radiation daping generated fro a urface foundation and a pile foundation. 1 Reearcher, Departent of Civil Engineering, Univerity of Patra, Eail: anarav@upatra.gr Profeor, Departent of Civil Engineering, Univerity of Patra, Greece, Eail: ylo@upatra.gr 3 Profeor, Departent of Civil Engineering, Univerity of Patra, Eail: karabali@upatra.gr

2 STRUCTURE ON SURFACE FOOTING The claical approach for elatodynaic analyi of oil-tructure interaction ai at replacing the actual tructure by an equivalent iple ocillator upported on a et of frequency-dependent pring and dahpot accounting for the tiffne and daping of the oil ediu. Thi odel ha been adopted by everal reearcher, including Parelee (1967), Veleto et al (1974, 1975, 1977), Jenning & Bielak (1973), Wolf (1985) and ore recently Avile et al (1996, 1998). A brief overview of available ethod leading to cloed-for olution i preented below. Baed on thee procedure, a novel, accurate and traightforward chee for analyzing the proble i preented. Claical Solution The yte tudied i hown in Figure 1. It involve a iple ocillator on fleible bae repreenting a ingle torey tructure, or a ulti torey tructure after a pertinent reduction of it degree-of-freedo (e.g., conidering that the a i concentrated at the point where the reultant inertial force act). u u1 h k K u r C C u4 u3 (a) K C K (b) Figure 1. (a) Structure idealized by a tick odel, (b) Reduced ingle degree-of-freedo odel The tructure i decribed by it tiffne k, a, height h, and daping ratio ζ, which ay be either vicou or linearly hyteretic. The foundation conit of a rigid urface circular footing of radiu r reting on a hoogeneou, linearly elatic, iotropic halfpace decribed by a hear odulu G, a denity ρ, Poion ratio ν, and hyteretic daping ratio ζ. Foundation tiffne i odeled by frequency-dependent pring K and K repreenting tiffne in tranlational and rocking ocillation, repectively. Following Veleto & Nair (1974) and to enure unifor unit in all tiffne ter, K i epreed by a tranlational vertical pring acting at ditance r fro the center of the footing. Daping i odeled by a pair of dahpot C, C, attached in parallel to the pring, repreenting energy lo due to hyteretic action and wave radiation in the oil ediu. In the preent forulation, the influence of foundation ebedent and foundation a i neglected. The dynaic ipedance forula K ( ω) along any degree of freedo of the yte i defined according to the K ( ω) = K + iωc = K(1+ iζ ) (1) in which K i the real part of the ipedance, ω C i the correponding iaginary part, ω i the cyclic ecitation frequency, and i (= -1) the iaginary unity.ζ i an energy lo paraeter, which i analogou (yet not identical) to the vicou daping coefficient of a iple ocillator.

3 ζ I( ) ( ω ) = K ωc Re( K ) = K () For the odel in Figure 1a, the foundation pring and dahpot can be epreed by the forula propoed by Veleto & Meek (1974): C K = a K, K = a K (3) Kr Kr = χ, C = χ (4) V V where V denote the propagation velocity of ditortional wave in the halfpace and K i the tatic horizontal tiffne of the foundation defined by 8 K = Gr (5) ν α, α, χ and χ are dienionle factor that depend on Poion ratio for the halfpace aterial, and the dienionle frequency ωr a 0 = V (6) Under eiic ecitation, the yte diplace and deflect a hown in Figure. The tranlation of the a relative to ground i copoed of three part: (1) horizontal tranlation due to waying otion of footing u, () horizontal tranlation due to rocking otion of footing, u and (3) horizontal deflection of the colun, u c. Baed on thee definition, the ipedance of the yte i defined a: ( P K = K 1+ iζ (7) u + u h+ u where K and ζ denote the apparent tiffne and daping coefficient at elevation h. c ) P u u uc h k C r K C K Figure. Deflection diagra for oil-tructure yte

4 The repone of the oil-tructure yte depend on the echanical propertie of the foundation, the oil, the upertructure and the characteritic of the ecitation. Thee are uarized in the following dienionle paraeter (Veleto et al 1974, 1975, 1977): (i) The wave paraeter σ V σ = (8) f h c where f c = k/ /π denote the natural frequency of the fied bae tructure. (ii) The relative a denity for the tructure and the oil γ γ = (9) πρ hr (iii) The daping ratio ζ of the tructure for fied bae condition. (iv) The lenderne ratio (h/r). (v) The Poion ratio ν of the oil. (vi) The hyteretic daping ratio ζ of the oil. Solution by Veleto and co-worker (1974, 1975, 1977) The ai of thee olution i to connect the propertie of the oil-tructure yte ( T, ζ ) with the propertie of the fied bae tructure (Τ, ζ), o that the influence of oil-tructure interaction on the dynaic behavior of the tructure can be elucidated. Thi connection i epreed by the following pair of equation (Veleto 1977): 3 k Kh ν π γ 1 ν a h T = T = T K K + a ( h σ ) ν a r r (10) ( ) 3 ζ = ζ T 0 + ζ (11) T whereζ repreent the daping of the tructure (aued to be of vicou nature) and ζ 0 the radiation daping of the footing. The latter i given by (Veleto and Nair, 1975) ( ) 4 3 π γ T ( ν) χ r 3(1 ν) χ ζ 0 = 3 T + σ a( a + ia 0χ) h a( a + ia 0χ) (1) The derivation i baed on etting the reonant period and peak peudo-acceleration of the actual elatodynaic yte equal to that of an equivalent iple ocillator. More dicuion i given below. Solution by Wolf (1985) The yte conidered by Wolf i identical to that hown in Figure 1 and. The ain difference with the Veleto approach i that frequency-independent oduli defined by the value a = 1, χ = 0.575, a = 0.15, χ = 0.15 are adopted for the foundation. Alo, the repone of the yte i deterined by directly olving a et of three iultaneou governing equation for degree of freedo u 1, u 3, and u 4 hown in Figure 1.

5 The propertie of the replaceent ocillator in thi olution are given by ω k kh = ω c 1+ + K K (13) ω ω ω ω ζ = ζ + 1 ζ + ζ + ζ ωc ωc ω ω (14) In the above equation, ω = Kr / h, ω = K /, ω = k/ define the uncoupled cyclic c natural frequencie of the yte under rocking ocillation of the bae (upertructure aued rigid), waying ocillation of the bae, and ocillation of the upertructure (foundation aued rigid), repectively. Note the ipler for of Eqn (14) a copared to Eqn (1). Notwithtanding the theoretical ignificance and practical appeal of the above ethod, they both can be criticized on the following iportant apect: (a) Both ethod neglect product of daping ratio (ζ i ζ j ) a negligible higher order ter. Thi approiation i quetionable for highly-daped oil-tructure yte. (b) The effective daping in the Veleto approach arie fro an approiate procedure leading to an epreion containing iaginary ter (Eqn 1). Thi liit ignificantly it uitability for practical application. (c) Structural daping in the Veleto olution i trictly of vicou nature. (d) Frequency dependence of foundation pring and dahpot in the Wolf approach i neglected. (e) Structural daping in the Wolf olution i trictly of hyteretic nature. (f) Both olution eploy rather cople procedure involving either equivalence of repone of different dynaic yte (Veleto), or olution of iultaneou linear equation (Wolf). (g) In both olution, foundation a and rotational inertia are neglected Propoed eact procedure In thi ection a iple eact olution to the proble hown in Figure 1 and i preented. The olution contain no approiation in the derivation of the fundaental natural period and effective daping of the yte. Furtherore, the eact frequency-varying foundation ipedance ay be eployed. Mention ha already been ade that the total horizontal deflection of the yte can be decopoed a u of the three odular diplaceent hown in Fig, i.e., ut = uc + u + u (15) Thi iplie that the aociated copliance can be viewed a cople pring attached in parallel and, thereby, the dynaic ipedance of the yte can be epreed through the well-known uation rule h 1 = + + K K K r k (16) in which the aociated ipedance are cople valued and frequency dependent. Subtituting each cople ipedance ter in Eq. (16) by it repreentation according to Eq. (1) yield the eact daping and natural frequency of the yte a (Marava, 006)

6 ζ ζ ζ + + ω ( 1 4 ) ( 1 4 ) c ( ζ ω ζ ω + ζ + ) ζ = ω ζ ω ζ ( 1+ 4 ) ( 1 4 ) c ( ω + ζ ) (17) 1+ 4ζ 1+ 4ζ 1+ 4ζ ω = + + ω ( 1+ 4ζ ) ω ( 1+ 4ζ ) ωc 1+ 4ζ ( ) 1 (18) The above olution are eact in the ene that no approiation apart fro thoe involved in nuerically evaluating the foundation ipedance are eployed. Given the frequency-dependent nature of foundation ipedance, an iterative procedure i generally required to deterine ζ and ω fro thee olution (Veleto & Nair 1974). Note that oitting the product ζ i, the above forula duly reduce to thoe in Eqn (14) and (13), repectively. Paraetric analyi and coparion with claical ethod Coparative graph for the variation of the ratio T / T (invere of ω / ω c ) and the effective daping of the yte ζ veru the lenderne ratio h/r are preented in Figure 3. Uing the propoed eact procedure, the influence of the relative a ratio γ and the aterial hyteretic daping ratio ζ on the period and effective daping of the oil-tructure yte i preented in Figure 4. For the paraetric analyi, value of factor α, α, χ and χ correpond to a Poion ratio of It i evident fro Figure 3(a,b) that the reult fro the ethod of Veleto are in relative agreeent with thoe obtained by the propoed eact procedure. Alo in Figure 3(c,d), we oberve that reult fro the ethod of Wolf are quite different fro thoe of the propoed procedure. Thi i ainly due to the auption of frequency-independent pring and dahpot adopted by Wolf...0 h/r=5 h/r=3 h/r= h/r=1 Τ Τ Eact procedure Veleto (1977) ζ Eact procedure Veleto (1977) h/r= h/r= (a) ζ = 0.05, ν = 0.45, γ = 0.15, ζ = 0 (b) ζ = 0.05, ν = 0.45, γ = 0.15, ζ = 0 h/r=1 0.5 h/r=0.33 ω ωc h/r=5 Eact procedure Wolf (1985) h/r= ζ Eact procedure Wolf (1985) h/r=0.33 h/r=1 h/r= h/r= (c) ζ = 0.0, ν = 0.45, γ = 0.15, ζ = 0.05 (d) ζ = 0.0, ν = 0.45, γ = 0.15, ζ = 0.05 Figure 3. Coparion of propoed eact olution with thoe of Veleto (1977) and Wolf (1985)

7 Τ Τ γ = ζ 0.1 γ = (a) ζ = 0.0, ν = 0.45, h/r = 1, ζ = 0.05 (b) ζ = 0.0, ν = 0.45, h/r = 1, ζ = h/r=5 0.5 Τ Τ h/r=1 ζ 0.1 ζ = i ζ = (c) ζ = 0.0, ν = 0.45, γ = 0.15 (d) ζ = 0.0, ν = 0.45,, h/r = 5, γ = 0.15 Figure 4. Paraetric reult uing the propoed eact procedure. (a) Period of SSI yte a function of γ, (b) Effective daping of SSI yte a function of γ, (c) Period of SSI yte a function of oil daping ratio ζ, (d) Effective daping of SSI yte a function of ζ The above obervation i jutified by the fact that for low value of ratio (i.e., low value of ω c ) the reult of the two ethod are nearly identical. It i alo apparent that the reult obtained by Wolf loe accuracy with decreaing value of h/r. Figure 4(a,b) how that the variation in relative a ratio γ affect ignificantly the period and daping of the oil-tructure yte. More pecifically, increaing γ lead to ore fleible yte and higher value of daping ratio. The hyteretic daping ratio of oil ζ, doe not affect uch the yte period, epecially for tall tructure (h/r=5), a hown in Figure 4(c). On the other hand, it affect coniderably the effective daping of the yte, a hown in Figure 4(d). STRUCTURE ON PILE FOUNDATION The cae of a tructure on a ingle pile foundation i invetigated net. The proble i treated uing a variance of the ethod for pread footing preented in the previou ection. The aount of energy radiated by a pile- or a footing-upported tructure are copared via by the concept of tatically equivalent SSI yte, introduced in thi work Soil-pile-tructure yte and ethod of analyi The yte tudied i hown in Figure 5. It conit of the linear elatic SDOF tructure decribed in previou ection, founded on a ingle fleible, circular olid pile of Young odulu E p, diaeter d, a per unit length p, and length L, which i conidered to be greater than the pile effective length L e. Accordingly, the pile can be conidered infinitely long. The oil i conidered a linearly elatic hoogeneou, iotropic halfpace, a decribed above. Soil-pile yte can be repreented by three dynaic ipedance K, K rr, and K r, correponding to waying, rocking, and cro-waying-rocking of the pile head, repectively. In thi tudy, analytical epreion for the dynaic ipedance are ued, a derived by Novak, (1974) and Mylonaki (1995)

8 u u1 P h k h k k u4 K rr u3 K K r L EpIp P e K I = 8 K (a) (b) (c) Figure 5. (a) Model of pile-upported-tructure, (b) Ditribution of oil reaction due to horizontal loading, (c) Reduced odel with two dynaic ipedance 3 K 4 EpIpλ, Kr EpIpλ, Krr EpI p d = = = λ (19) 1/4 k pω iωc λ + = (0) 4EI p p where λ i a wave nuber paraeter, I p i the oent of inertia of the pile cro ection, and k and c are the oduli of ditributed pring and dahpot along the pile ued to odel oil reaction. The latter are given by -1/4 ζ k k, 6aop = δe c = ρvd + (1) ω where a op =a o (where the footing radiu r in Eq. (6) i replaced by the pile diaeter d) and δ i the dienionle Winkler factor, given a function of pile-oil tiffne ratio E p /E (Dobry et al, 198) Ep δ = 1.67 E () Under horizontal loading the oil react in the anner hown in Figure 5(b). The reultant of the ditributed reaction i applied at depth e below the pile head. Since the reference yte i anchored at the pile head, a cro waying-rocking ipedance ter K r i neceary for odeling the copliance of the foundation. Thi ter i not copatible with the analyi of the pread footing preented earlier. In order to overcoe thi proble, the reference yte can be tranlated to a depth e, where the total oil reaction i applied. In thi anner, the cro ipedance K r vanihe and the ipedance atri of the pile becoe diagonal, a hown in Figure 5(c). Thi tranforation i approiate, a it require the pile to be rigid between the depth z = 0 and z = e. However, thi introduce little error, ince e i uually all copared to the overall pile length. The tranfored ipedance K and are given by the epreion e K rre

9 where K = K, K = K K e+ K e, K = 0 e rre rr r re r (3) K 1 e = K = λ (4) i the aforeentioned eccentricity. The tranforation hown in Figure 5(c) allow the uage uing the eact procedure developed for the analyi of tructure on urface footing. Thu, the natural frequencie ω i of the oil-pile-tructure yte required by Eq. (17) and (18)) are coputed by Marava (006) a K ω =, ω = Krr ( h+ e) (5) where ( 4 ) ( ω ) ( ω ) 1/4 3/8 3 p p p K = E I k + c co φ 4 (6) 3/4 1/8 1 1 rr p p p ( 4 ) ( ω ) ( ω ) K = E I k + c co φ 4 4 (7) ωc φ = Arc tan k pω (8) The correponding daping ratio ζ i are given by the epreion (Marava 006) in which φ i defined by Eq. (8) ζ = tan φ, ζ = tan φ 4 4 (9) Reult of Paraetric Analye The repone of the oil-pile-tructure yte depend on the propertie of the pile, the upporting oil, the upertructure and the ecitation. Thee propertie are included in above dienionle paraeter, a in the cae of the tructure upported on urface footing. In Figure 6, reult obtained with the ue of the propoed eact procedure are preented. Specifically, Figure 6(a) preent the influence of the lenderne ratio (h/d) on the natural period of the interacting yte. Evidently, the behavior i iilar to the behavior of the oil-footing-tructure yte preented earlier. The ae obervation hold for the effective daping, a hown in Figure 6(b). The influence of pile-oil tiffne ratio E p /E, on the propertie of the yte i ignificant, a hown in Figure 6(c,d). For relatively fleible pile (low value of E p /E ), the oil-pile-tructure yte i ore fleible and diipate larger aount of energy.

10 Τ Τ h/d = h/d = (a) ζ = 0.0, ν = 0.45, γ = 0.15, ζ = 0.05, ρ p /ρ = 1.40, E p /E = 100 (b) ζ = 0.0, ν = 0.45, γ = 0.15, ζ = 0.05, ρ p /ρ = 1.40, E p /E = 100 ζ E p /E = Τ Τ ζ E p /E = (c) ζ = 0.0, ν = 0.45, γ = 0.15, ζ = 0.05, ρ p /ρ = 1.40, h/d = 5 (d) ζ = 0.0, ν = 0.45, γ = 0.15, ζ = 0.05, ρ p /ρ = 1.40, h/d = 5 Figure 6. (a) Syte period a function of h/d, (b) Syte daping a function of h/d. (c) Syte period a function of E p /E, (d) Syte daping a function of E p /E Coparion of the two SSI yte In thi ection, the concept of tatically and geoetrically equivalent interacting yte i introduced, in an effort to copare the daping of yte on pile or urface footing foundation. To achieve thi, the lateral tiffne of the two SSI yte hould be coparable at the elevation of a. Accordingly, the two yte are geoetrically equivalent, i.e., h/d=h/r, if the ratio of Young oduli for the oil (Marava 006): ( ) ( f ) 1/4 1/4 3 ( )(1 ) 1 4 / E πδ E v v h d p = ( p) ( ) 1/4 1/4 16 p E 16 E δ Ep h ( p) π E d (30) In the above equation, ybol f denote footing and ybol p pile. Becaue a pile i uch tiffer ( f ) than a footing, the upporting oil of the footing ut have a odulu of elaticity E quite larger ( p) than the odulu of elaticity of the oil around a pile E. Uing the propoed eact procedure, the effective daping of the two SSI yte i copared for different ratio h/d and E p /E, a hown in Figure 7a and 7b. Furtherore, coparative graph of radiation daping for the two foundation type are preented in Figure 7c and 7d. It i obviou that a tructure on a pile ay eperience a uch a three tie the aount of daping generated fro the ae tructure on a pread footing (Figure 7a). Thi obervation i jutified in view of the towdienional nature of wave propagation around a pile, which reult to uch higher energy diipation.

11 ζ Pile Footing ζ ζ 0.06 Pile Footing σ ( f ) (a) ζ = 0.0, ν = 0.45, h/d = 1, ζ = 0.05, ρ p /ρ = 1.40, E p /E = 1000 (b) ζ = 0.0, ν = 0.45, h/d = 5, ζ = 0.05, ρ p /ρ = 1.40, E p /E = () σ f ( f ) ζ (p) ζ 0.6 Pile Footing ζ Pile Footing ζ (f) (a 0 ) p (a 0 ) p (c) ν = 0.45, ζ = 0.05, ρ p /ρ = 1.40, E p /E = 1000 (d) ν = 0.45, ζ = 0.05, ρ p /ρ = 1.40, E p /E = 1000 Figure 7. (a, b) Effective daping of a tructure founded on pile and footing, (c) Radiation daping of a pile and footing due to tranlational ocillation, (d) Radiation daping of a pile and footing due to rocking ocillation CONCLUSIONS A novel analytical procedure for deterining the dynaic characteritic of iple tructure founded on urface footing and pile wa preented. Uing the propoed ethodology, the influence of coon auption on the coputation of the echanical propertie of uch yte wa elucidated. Reult were provided in ready-to-ue graph and chart that elucidate the alient feature of the proble and can be directly ipleented in deign. By introducing the concept of tatically and geoetrically equivalent SSI yte, the aount of radiation daping generated fro a ingle pile and a footing were copared. The ain concluion of the tudy are: (1) The propoed olution i ipler, ore accurate, and ore general than the claical ethod by Parelee, Veleto, Bielak, Wolf and co-worker. () The coon approiation of neglecting higher-order ter involving product of daping coefficient ay be inaccurate for highly-daped SSI yte. (3) The propoed analyi can eaily incorporate ebedded foundation, by tranlating the reference yte to the depth below the urface where the reultant oil reaction i applied. Thi enure a diagonal foundation ipedance atri and greatly iplifie calculation. (4) A tructure founded on a pile ay generate 100% ore radiation daping than a iilar tructure on a pread footing. The difference get ore pronounced with high-frequency, quatty tructure on tiff oil.

12 REFERENCES Avile, J. & Perez-Rocha, L.E. Evaluation of interaction effect on the yte period and the yte daping due to foundation ebedent and layer depth Soil Dynaic & Earthquake Engineering. 15(11), 7, 1996 Avile, J. & Perez-Rocha, L. E. Effect of foundation ebedent during building-oil interaction, Earthquake Engineering & Structural Dynaic, 7(1), , 1998 Dobry, R., Vicente, E., O'Rourke, M. J., & Roeet, J. M. "Horizontal tiffne and daping of ingle pile", J. Geotech. Engng Div., ASCE, 108(3), , 198 Eurocode EC-8 Structure in eiic region, Part 5: Foundation, retaining tructure, and geotechnical apect, Bruel: Coiion of the European Counitie, 1990 Gerolyo, N., Gazeta, G. & Mylonaki, G. Fundaental natural period and effective daping of pile-upported bridge pier, Proc., 11 th European Conference on Earthquake Engineering, Balkea, Rotterda, 1998 Jenning, P.C. & Bielak, J. "Dynaic of building-oil interaction", Bulletin of the Seiological Society of Aerica, 63(1) 9-48, 1973 Marava, A. Dicrete odel for dynaic oil-tructure interaction of tructure on rigid urface or pile foundation, MS Thei, Univerity of Patra, Greece, 006 (in Greek) Mylonaki, G. Contribution to the tatic and eiic analyi of pile-upported bridge pier, Ph.D. Diertation, State Univerity of New York, Buffalo, 1995 NEHRP Recoended Proviion for Seiic Regulation for New Building and other Structure, Building Seiic Safety Council, Wahington, D.C., 1997 Novak, M. Dynaic tiffne and daping of pile, Canadian Geotechnical Journal, 11, , 1974 Parelee, R. Building-foundation interaction effect, Journal of Engineering Mechanic Diviion, ASCE, 93(EM), , 1967 Seto, A, Kappo. A, Pitilaki, K. Seiic Repone of Long R/C Bridge: Effect of Coupled Ground Motion Variability and Soil-Foundation Interaction, Proc., 1 th European Conference on Earthquake Engineering, 00, 1 page on CD-ROM Stewart, JP, Fenve, GL, Seed, RB. Seiic Soil-Structure Interaction in Building I: Analytical Method, Journal of Geotechnical and Geoenvironental Engineering, ASCE, 15(1), 1999 Veleto, A.S. & Meek, J.W. Dynaic behavior of building-foundation yte, Earthquake Engineering & Structural Dynaic, 3(), , 1974 Veleto, A.S. & Nair, V.V. Seiic interaction of tructure on hyteretic foundation, Journal of Structural Engineering, ASCE, 101(1), , 1975 Veleto, A. S. Dynaic of Structure-Foundation Syte, in: Hall, W. J. (ed.), Structural & Geotechnical Mech., Prentice-Hall, 1977 Wolf, J. P. Dynaic Soil-Structure Interaction, Prentice Hall, 1985