SEISMIC STRAIN WEDGE MODEL FOR ANALYSIS OF SINGLE PILES UNDER LATERAL SEISMIC LOADING

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1 SEISMIC STRAIN WEDGE MODEL FOR ANALYSIS OF SINGLE PILES UNDER LATERAL SEISMIC LOADING Aslan Sadegh Hokmabad, Al Fakher and Behzad Fatah Centre for Buld Infrastructure Research (CBIR), School of Cvl and Envronmental Engneerng, Unversty of Technology Sydney (UTS), Sydney, Australa School of Cvl Engneerng, College of Engneerng, Unversty of Tehran, Iran ABSTRACT One of the most effectve methods of analysng a sngle ple and ple groups under lateral loadng s Stran Wedge Model (SWM). SWM has a number of advantages n comparson wth tradtonal p-y curves, but ths model could tradtonally only be used to analyse ples under monotonc loads. In the present paper, SWM has been modfed to consder dynamc lateral loadng. Based on ths new method, called Sesmc Stran Wedge Model (SSWM), a computer code has been developed for lateral analyss of ples. Usng ths computer code, some case studes have been analysed and the results show good agreement wth test data. Ths paper ntroduces SSWM as a smple and powerful soluton to analyse ples under lateral sesmc loadng. INTRODUCTION Lateral vbraton of ples s an mportant consderaton n the desgn of pled structures subjected to dynamc exctatons due to earthquake, wnd, operaton of machnes and waves n offshore envronment (Das & Sargand, 999). The sesmc response of ple foundatons s a complex process nvolvng nertal nteracton between structure and ple foundaton, knematcs nteracton between ples and sol, and the non-lnear response of sol to strong earthquake motons (Fnn, 005). Several methods have been developed for dynamc response analyss of ple foundatons as lsted n Table. Dependng on how ples and sols are modelled, developed methods may be broadly classfed nto contnuum-based approaches and Wnkler (or subgrade reacton) approaches. In the contnuum-based approaches, sol s modelled as contnuum meda. These approaches were developed to acheve a drect and complete analyss of ples under dynamc condtons and are essentally of a numercal nature such as Fnte Element Method (FEM) or Boundary Element Method (BEM) (Caro et al., 005). Moreover, new procedures to couple BEM and FEM for dynamc analyss of ples have been ntroduced (Padron et al., 007). Fnte element analyses of ples have been carred out by several researchers such as Kuhlemeyer (979), n whch the ples are represented by axsymmetrc elements, and energy-absorbng boundares are employed to represent the far feld. Consderng the obvous lmtatons of the fnte element method for modellng boundares at nfnty, these analyses represent consderable achevements n charactersng the dynamc behavour of ples (Das & Sargand, 999). Boundary element formulaton has been used by some researchers such as Banerjee et al. (987), for the dynamc analyss of ples. Boundary element method offers advantages over other methods prmarly because of ts ablty to take nto account the three dmensonal effects of sol contnuty and boundares at nfnty. However, the major problem wth ths method s the accuracy of the numercally constructed dynamc solutons snce the convergence of the sem-nfnte ntegral s dependent on the frequency parameter (Das & Sargand 999). Generally, for these methods sgnfcant computatonal efforts are requred and large systems of equatons have to be solved (Caro et al., 005). Due to complexty and unavalablty of the requred sol propertes, the applcaton of Wnkler approaches s more common. As an example, the load-transfer method models the ple as an elastc member and the sol s vewed as dstrbuted sprngs and dashpots, wth constant or frequency dependent stffness, concentrated at a fnte number of nodes. The sprng constants are obtaned from analytcal calculaton or expermental data. The major advantage of ths approach les n ts ablty to smulate nonlnearty, heterogenety, and hysteretc degradaton of the sol surroundng the ple by smply changng the sprng and dashpot constants (Das & Sargand, 999). In contrast, the man drawback of the load-transfer approach s dealsaton of the sol contnuum wth dscrete sol reactons (Allotey & El Naggar, 008). In other words, the shear transfer between the sprngs s one of the obvous mssng fundamental mechansms n Wnkler models (Fnn, 005). In order to overcome to the man drawback of Wnkler methods, Stran Wedge Model Australan Geomechancs Vol 46 No March 0 3

2 (SWM) s developed. SWM creates relatonshp between the three-dmensonal responses of sol-ple to the beam on elastc foundaton parameters (Ashour et al., 998). However, applcaton of SWM analyss has been lmted to ples under monotonc loads. In ths study SWM s modfed n order to nclude dynamc loads. Table Summary of the lateral sesmc analyss of ples methods Methods Reference Remarks Equvalent Cantlever Beam on Elastc Foundaton Novak s Analyss Method Davsson & Robnson (965); Nar et al. (969) Braud & Tucker (984) Novak (974) Ple s reduced to an equvalent cantlever; No nformaton can be obtaned on the moment, stresses, and dsplacements along the length of the ple; Only good for ntal analyss. Presents the sol nertal force on the Wnkler sprngs stffness, and the ple nertal force and sol dampng as an addtonal force at each node. A new governng equaton of Beam on Elastc Foundaton (Includng nertal force and dampng) s solved wth the Fnte Deference Method. Ths approach derves stffness and dampng constants for ples, and thus enables the use of the smple lumped-parameter approach. Column of Sol Parmelee et al. (964); Idress & Seed (968) Sol mass s lumped at dscrete ponts along the depth of the each layer and these masses lnked by sprngs and Dashpots connected n parallel; Earthquake effect s presented as a horzontal acceleraton at the base; It s assumed that sol and ple have the same dsplacement n each node (the ple s moved essentally wth the sol) whch s not acceptable n layered sols. P-y Curves Sprngs and Dashpots to model sol Contnuum-Base Approaches Matlock (970); Reese et al. (974) El Naggar & Bently (000); Tavaraj (00); Gerolymos & Gazetas (006) Kuhlemeyer (979); Banerjee et al. (987); Brown & She (990) Ple s modelled as an elastc member and the sol s modelled as a seres of nonlnear sprngs (p-y curves); Ths s a sem-emprcal method that accounts for dynamc loads by modfyng the statc p-y curves based on emprcal data. (dfferent researchers suggests dfferent curves) Ple s modelled as an elastc member, and sol s modelled as a seres of sprngs and dashpots (dfferent models are suggested); These models, based on the determnaton of the near- and far-feld responses, can be grouped nto coupled and uncoupled models. Sol s modelled as a contnuum meda and dfferent numercal approaches such as the Fnte Element Method (FEM) or Boundary Element Method (BEM) are used to solve the governng equatons. METHOD OF ANALYSIS To develop the general formulaton of SSWM, three methods ncludng Horzontal Slce Method, pseudo-dynamc Method, and Stran Wedge Model are used and combned. Horzontal Slce Method presents the dea of dvdng passve sol nto horzontal slces (Shahghol et al., 00). The Pseudo-dynamc Method s used to calculate the addtonal earthquake force appled at each slce or each depth along the ple (Nmbalkar et al., 006) and the Stran Wedge Model s employed to determne the sol resstance aganst the lateral loadng by the three-dmensonal passve wedge developed n the sol n front of the ple (Ashour et al., 998). In ths secton, each of these three methods s descrbed and then the formulaton of SSWM s presented.. HORIZONTAL SLICE METHOD The Horzontal Slce Method (HSM) was ntally developed for sesmc stablty analyss of renforced slopes and walls (Shahghol et al., 00). HSM s based on lmt equlbrum method assumng a falure surface wth a falure 3 Australan Geomechancs Vol 46 No March 0

3 wedge dvded nto a number of horzontal slces (Fgure). Because of the horzontal slces, the falure surface does not ntersect the horzontal renforcements; accordngly the renforcements have no drect nfluence on nterslce forces. Accordng to Nour et al. (006), equlbrum of force and moment can be satsfed for each slce of the whole sldng wedge. In Fgure b, FH, FV, FH + and FV + are the nter-slce forces, N and S are the normal and shear forces upon base of the slce, T s the tensle force of the renforcement, K h and K v are the sesmc acceleraton coeffcents n horzontal and vertcal drectons, and W s the weght of the slce. Falure surface FV y FH L Renforcement elements S T K h W (+K v )W N L + x (a) (b) FV + FH + Fgure : Horzontal Slce Method: (a) dvdng sol n to horzontal slces, (b) Forces actng on a sngle horzontal slce ncludng renforcement (after Shahghol et al., 00). PSEUDO-DYNAMIC METHOD Pseudo-statc methods are commonly used for solvng varous desgn problems assocated wth ple foundatons subjected to earthquake motons (Fnn, 005). The pseudo-statc methods consder the dynamc nature of earthquake loadng very approxmately (Kramer, 996), and do not consder the effects of tme and body waves travellng through the sol durng the earthquake. However the pseudo-dynamc method developed by Steedman & Zeng (990) for sesmc slope stablty analyss can consder the phase dfference due to the fnte shear wave propagaton. In the pseudo-dynamc method t s assumed that the shear modulus s constant wth depth along the ple. Therefore, the followng equatons can be wrtten: / V s = ( G / ρ) () / V p = [ G( ν ) / ρ( υ )] () T = π / ω = 4L / V s (3) where, V s s shear wave velocty (m/s), G s shear modulus of sol (N/m ), ρ s densty of sol (kg/m 3 ), V p s the prmary wave velocty (m/s), υ s Posson s rato, T s perod of lateral shakng (s), and ω s angular frequency of base shakng (rad/s). In pseudo-dynamc method, only the phase and not the ampltude of the acceleraton vares along the ple. In addton, n ths method t s assumed that both the horzontal and vertcal vbratons, wth acceleratons a h and a v, respectvely, start at exactly the same tme, and there s no phase shft between these two vbratons resultng n a crtcal condton for desgn (Nmbalkar et al., 006). However, the sesmc acceleraton s consdered as beng harmonc snusodal, whch s one of the lmtatons of the orgnal pseudo-dynamc method proposed by Steedman & Zeng (990). As shown n Fgure, f the base of the slope s subjected to harmonc horzontal and vertcal sesmc acceleratons wth ampltudes a h and a v, the horzontal and vertcal acceleratons at depth x below the ground surface at tme t can be express as: L x ah( x, t) = ah snω( t ) Vs (4) L x av( x, t) = av snω( t ) Vp (5) Australan Geomechancs Vol 46 No March 0 33

4 In the SSWM, the above acceleratons are employed to calculate the addtonal earthquake force appled at each depth along the ple. x Crest of the slope L Falure surface a v (x,t) Base (toe level) a v a h (x,t) a h Fgure : Pseudo-dynamc method for slopes.3 STRAIN WEDGE MODEL Non-lnear sprngs (p-y) are wdely used to analyse sngle ples under statc lateral loads. However, the p-y curve method as descrbed by Resse and Vanmpe (00) does not consder the sol contnuty and ple characterstcs such as stffness, cross secton and head condtons. SWM overcomes these lmtatons and allows the assessment of the p-y response of laterally loaded ples based on the three-dmensonal sol-ple nteracton and ts dependence on both sol and ple propertes (Norrs & Abdollaholae, 985). In the SWM, the sol resstance aganst the lateral loadng s determned by the three-dmensonal passve wedge developed n the sol n front of the ple. As shown n Fgure 3, the geometry of the moblsed passve wedge s charactersed by the wedge base angle (β m ), the passve wedge depth (h), and the wedge fannng angle (φ m ). All of these parameters are functons of sol horzontal stran at each loadng step. Fgure 3: Passve wedge of sol as developed n front of the ple n the SWM (Ashour et al., 998) 34 Australan Geomechancs Vol 46 No March 0

5 SWM analyss mantans both the local stablty of the sol sublayer (horzontal slce) and the global stablty of the ple and the passve sol wedge. Accordng to Heteny (946) for the global stablty of the system, the governng equaton of beam on elastc foundaton (Equaton 6) should be satsfed. 4 d y d y EI( ) + P ( ) + E ( y) = 0 4 x s dx dx where, EI s ple stffness (N. m ), y s ple deflecton at depth x (m), x s vertcal dstance between any pont n sol mass and external borders of sol mass (m), P x s appled axal load at the ple head (N), and E s s modulus of subgrade reacton (N/m ). On the other hand, the local stablty s performed by dvdng the sol layer nto thn sublayers and the equlbrum of forces actng on each sublayer (slce) should be satsfed. As shown n Fgure 4, the exstng loads are horzontal stresses n front of the wedge (Δσ h ), ple sde shear stresses (τ), statc lateral sol-ple nteracton load (p s ), and the wedge sde forces (F). As explaned and dscussed by Ashour et al.(998), t s assumed that the sde forces of the wedge act wth an angle equal to the moblsed effectve stress frcton angle of the sol (φ m ). Snce the fannng angle of the wedge s equal to the moblsed effectve stress frcton angle of the sol, the wedge sde forces are perpendcular to the ple lateral movement drecton. Moreover, the nter-slce shear forces between varous layers are not consdered n the local stablty of the sol sublayers, however, the effect of neghbourng sol layers n the SWM s consdered n calculatng the depth of passve wedge. The complete formulaton of the SWM s explaned by Ashour et al. (998 & 004). (6) Δσ h B C F φ m φ m F τ P s τ Fgure 4: Geometry and equlbrum of each sublayer n SWM 3 SEISMIC STRAIN WEDGE MODEL (SSWM) The Sesmc Stran Wedge Model (SSWM) s developed by consderng the earthquake horzontal acceleraton (a h ) and vertcal acceleraton (a v ) at the base of the sol-ple system (Fgure 5). Usng pseudo-dynamc method, acceleraton at each depth along the ple at any partcular tme can be obtaned from Equatons (4) and (5). It s assumed that all the assumptons of SWM are vald n SSWM, so as lke as SWM the sol medum n the passve wedge represents the sol response to the lateral loads n the SSWM. As shown n Fgure 5b, the sol mass of the th sublayer can be calculated as: L3 = tan( βm)( h x) (7) BC D V L + ) = 3 ) H (8) m = ρ V (9) BC + D m = ρ L3 H (0) where, L 3 s the horzontal length of wedge (m), β m s the passve wedge angle (rad), h s the depth of passve wedge (m), V s the volume of the th sublayer (m 3 ), BC s front wdth of the sol wedge (m), D s the ple dameter (m), m s the mass of the th sublayer (kg), H s the thckness of the sol sublayer (kg), and ρ s the densty of the th sublayer (kg/m 3 ). Australan Geomechancs Vol 46 No March 0 35

6 x 3 BC L 3 L n a v (x,t) D a h (x,t) H a v a h Fgure 5: Sesmc Stran Wedge Model (SSWM): (a) horzontal and vertcal sesmc acceleratons at each depth, (b) Slce of wedge at depth x. Usng the concept of horzontal slce method (HSM), the addtonal horzontal force that s generated by the earthquake n the th layer (P dh ) s: a ( x, t) W g P K W h P dh = dh h () = () where, K h s the sesmc acceleraton coeffcent n the horzontal drecton, g s the acceleraton due to gravty, and W s the weght of sol n the th layer. As shown n Fgure 6, for the equlbrum of the th layer, local stablty should be satsfed by ncludng the nduced earthquake load (P dh ). Δσ h B C P dh τ P s τ Fgure 6: Local equlbrum of a sol-ple sublayer n the presence of sesmc loads. The crtcal condton occurs when P dh s n the same drecton as the statc lateral sol-ple nteracton load (Fgure 5). Thus: P = P + P s dh (3) 36 Australan Geomechancs Vol 46 No March 0

7 P s = ( Δσ ) BC S + τds (4) h where, P s s the statc lateral sol-ple nteracton load, P s the total lateral sol-ple reacton (statc plus sesmc), Δσ h s the horzontal stress n the passve wedge n front of ple, and S and S are ple shape factors. For a crcular ple cross secton S and S are equal to 0.75 and 0.5, respectvely, and for square ple cross secton both S and S are equal to.0 (Braud et al., 984). Smlarly, the addtonal vertcal force nduced by the earthquake n the th sublayer s obtaned by: a ( x, t) W (5) g P = K W (6) v P dv = dv v where, P dv s the vertcal sesmc force n the th sublayer (N) and K v s the sesmc acceleraton coeffcent n the vertcal drectons. As shown n Fgure 5, the upward vertcal acceleraton s the crtcal drecton snce t reduces the vertcal effectve stress n the sol as explaned below: m Pdv ( Δσ vd ) m = H γ (7) = A BC+ D m Kv γ L3 ( ) H ( Δ ) = (8) σ vd m Hγ BC+ D = L3 ( ) m Δ vd ) m = = ( σ ( K ) H γ v where, (Δσ vd ) m s the total (statc and sesmc) vertcal effectve stress (N/m ). A pseudo-statc analyss assumes no phase change n the body waves travellng through the sol medum s occurred. Under ths condton, acceleratons a h (x,t) and a v (x,t), n all of the sublayers are assumed to be constant, thus: ah( x, t) ah lm ( pdh ) = lm( W ) = W = KhW (0) Vs Vs g g m m m av ( x, t) av lm ( Δσ vd ) m = lm ( ) Hγ = lm ( ) Hγ = ( Kv ) Hγ () Vp Vp g Vp g = = where, K h and K v are the sesmc acceleraton coeffcents n the base n the horzontal and vertcal drectons. To compare the results of earthquake modellng wth pseudo-statc and pseudo-dynamc method, a case study s presented. The algorthm of soluton wth the SSWM used to develop a computer code s descrbed n the followng secton. 4 LATERAL ANALYSIS OF PILES (LAP) COMPUTER CODE Lateral Analyss of Ples (LAP) computer code was ntally developed by the authors to analyse the lateral behavour of sngle ples and ple groups under statc loadng (Fakher et al., 009). LAP allows the use of dfferent methods such as SWM to analyse ples under lateral statc loadng. In ths study, LAP has been modfed to analyse ples under sesmc lateral loadng usng SSWM. A detaled flowchart of the calculaton process s presented n Fgure 7. The flowchart s a modfcaton of the orgnal one developed by Ashour et al. (004) for SWM. In Fgure 7, γ s the unt weght of sol (kn/m 3 ), φ s the effectve angle of nternal frcton of sol (rad), ε 50 s the horzontal stran n the sol n 50 percent stress level, S u s the undraned shear strength of sol (N/m ), D s the ple wdth or dameter (m), y 0 s the targeted deflecton of the ple head (m), G s the shear modulus of sol (N/m), υ s the Posson s rato of the sol, M s the appled moment at the ple head (N.m), P x s the appled axal load at the ple head (N), P t s the appled lateral load at the ple head (N), T s the perod of lateral shakng (s), Δt s the tme steps (s), N t s the number of tme steps, σ v0 s the vertcal effectve stress (N/m ), SL s the horzontal stress level n sol at SWM (N/m ), h s the depth of passve wedge of sol (m), P 0t and Y 0t are the ple head lateral load and dsplacement respectvely at tme t, P 0T and Y 0T are the maxmum ple head lateral load and dsplacement respectvely durng the tme T. = (9) Australan Geomechancs Vol 46 No March 0 37

8 START Input DATA Sol statc propertes (γ, φ, ε 50, S u and layer thckness) Ple Propertes (EI, D, L, y 0) Sol dynamc propertes (G, υ) Loadng (M, P x, P T, a h, a v) (h) new and (P 0) new NO IF (h) SWM = (h) BEF (Y 0) SWM = (Y 0) BEF Yes Dvdng sol layers nto thn sol sublayers () Assume lateral stran (ε) n sol n front of the ple, Δt=T/N t, t= Δt Fnal (converged) geometry of the passve wedge (φ m, β m, BC n each sol sublayer and h) and the assocated E and E s profle, P 0t, and Y 0t of the ple Based on a h, a v, and Eq. (4),(5) calculate a h(x,t), a v(x,t) and P dh and (Δσ dv) m at tme t t=t+δt Yes IF t T Based on ε (assumed), ε 50, σ vo(ashour et al. 98) calculate Δσ h, SL, φ m, h (assume h n the frst tral), BC, and E s n each sublayer (.e. Es profle along the ple) at Δσ h = σ d and ε Usng the current profle of Es, the laterally loaded ple s analysed by Eq. 3 under an arbtrary plehead lateral statc load (P T). The ple head deflecton (Y 0) and depth of passve wedge ( h) assumed usng beam on elastc foundaton analyss are compared to those of the SWM analyss. Y = max( Y 0 t ), 0 t T 0 T 0 T = max( P 0 t ), 0 t T P STOP NO Fgure 7: Flowchart of calculaton process for Sesmc Stran Wedge Model for analyss of a sngle ple. 5 VERFICATION AND PARAMETRIC STUDY In order to verfy the performance of SSWM, centrfuge test results on a sngle ple carred out by Fnn & Gohl (987) are compared wth the LAP outcomes. In the prototype model, a sngle ple wth a dameter of 0.57 m and a length of.9 m was subjected to a base moton wth the maxmum horzontal acceleraton of 0.58g, and the ple head acceleraton and the maxmum moment along the ple were measured. The ple was embedded n sand wth frcton angle of 30 degrees, dry unt weght of 4.7 kn/m 3, and sol vod rato of Detals of the test and the prototype model were explaned further by Tabesh (997). The ple s analysed by LAP ncludng SSWM. SSWM, as a new method for analysng ples under lateral dynamc loads, can use both pseudo-statc and pseudo-dynamc methods. Fgure 8 presents the results of ths analyss for four sets of data ncludng laboratory measurements, pseudo-dynamc analyss, pseudo-statc analyss, and the analyss performed by Tabesh (997) usng SAPAP computer program. As shown n Fgure 8, the results of analyss by LAP usng SSWM for both pseudo-statc and pseudo-dynamc are n good agreement wth the measured maxmum moment along the ple and the locaton of the maxmum moment. However, pseudo-dynamc method estmates larger maxmum moments along the ple n comparson wth pseudo-statc method. In the pseudo-statc method, horzontal and vertcal acceleratons, a h (x,t) and a v (x,t), n all of the sublayers are assumed to be constant (Equatons 0 & ), but n pseudo-dynamc method phase dfference due to fnte shear wave propagaton s consdered (Equatons 4 & 5). Therefore, pseudo-dynamc method results are more 38 Australan Geomechancs Vol 46 No March 0

9 crtcal and realstc analyss results. It should be mentoned that SSWM uses a smple method to consder the sesmc behavour of the system by ncludng the mnmum number of parameters. Parametrc study s performed on the same ple wth the same propertes as reported n the verfcaton exercse, to nvestgate the effect of base acceleraton on the ple head dsplacement. Varous weghts on ple head and dfferent base acceleratons are ncluded n ths parametrc study as compared n Fgure 9. Fgure 8: Measured and calculated values of maxmum moment along the ple. Fgure 9: Effect of base acceleraton on the ple head dsplacement usng the SSWM. As shown n Fgure 9, a larger horzontal base acceleraton (K h ) causes a larger ple head dsplacement. Based on Equaton (4), when the horzontal base acceleraton (K h ) ncreases, the horzontal sesmc acceleraton at each depth at tme t ncreases, resultng n an ncrease of earthquake horzontal force n each layer (P dh ) whch n turn causes larger ple head dsplacement. Larger upward vertcal base acceleraton (K v ) ncreases the ple head dsplacement as well. Accordng to Equaton (5), when the vertcal base acceleraton (K v ) ncreases, the vertcal sesmc acceleraton at each depth at tme t ncreases resultng n an ncrease n the vertcal earthquake force n each layer (P dv ) (Equatons 5 & 6). Therefore, accordng to Equaton (9), the Australan Geomechancs Vol 46 No March 0 39

10 sol effectve vertcal stresses at each depth decrease. Based on SWM formulaton provded by Ashour et al. (998), reducton n the sol effectve vertcal stresses results n decreasng of the calculated stffness of the sprngs modellng the sol reacton aganst the lateral loadng whch n turn causes larger ple head dsplacement. However, the nfluence of the vertcal base acceleraton on the ple head dsplacement s less than the horzontal base acceleraton. For example n the present parametrc study for the ple head weght of 300 kn, ncreasng the horzontal sesmc acceleraton coeffcent from 0. to 0.3 ncreases the ple head dsplacement from.5 cm to 4.5 cm (.8 tmes), however, ncreasng the vertcal sesmc acceleraton coeffcent from zero to 0., for the same case, nduces less than 0 percent ncrease n the ple head dsplacement. 6 CONCLUSION In ths study, a new method called Sesmc Stran Wedge Model (SSWM) s proposed for analysng ples under lateral sesmc loads. The man formulaton of ths method s based on SWM and the addtonal Earthquake force s calculated usng the Pseudo-dynamc Method. Verfcaton of the model was performed wth expermental data, whch shows good agreement between the predcted and measured maxmum moment along the ple. However, further verfcatons wll be helpful to examne the model to predct the ple head movement for combnaton of vertcal and horzontal loadng. In addton, accordng to the parametrc study performed to nvestgate the effects of base acceleraton on the ple head dsplacement, a larger horzontal base acceleraton (K h ) causes a larger ple head dsplacement. Increasng n the upward vertcal base acceleraton (K v ) has the smlar nfluence on the ple head dsplacement; however, vertcal base acceleraton has less mpact on the ple head dsplacement compared to that of the horzontal base acceleraton. The SSWM approach presented here provdes a smple and powerful method for solvng the problem of the sesmc analyss of laterally loaded ples. The SSWM overcomes the man drawback of the tradtonal p-y approach, whch does not consder stress transfer between the sprngs. In addton, the SSWM can consder the effect of phase dfference n sesmc travellng through the sol durng the earthquake exctaton. It should be noted that nput data, requred n the SSWM, s can be obtaned from the basc sol propertes that are typcally avalable to the desgners. 7 REFERENCES Allotey, N. and El-Naggar M.H. (008) Generalzed dynamc Wnkler model for nonlnear sol-structure nteracton analyss, Canadan Geotechncal Journal, 45(4), Ashour, M., Norrs, G. and Pllng, P. (998) Lateral loadng of a ple n layered sol usng the stran wedge model, Journal of Geotechncal and Geoenvromental Engneerng, 4 (4), ASCE, Ashour, M., Pllng, P. and Norrs, G. (004) Lateral behavor of ple groups n layered sols, Journal of Geotechncal and Geoenvromental Engneerng, 30 (6), ASCE, Banerjee, P.K., Sen, R. and Daves. T.G. (987) Statc and dynamc analyss of axally and laterally loaded ples and ple Groups. In: Sayed, S.M. (Ed.). Geotechncal Modelng and Applcatons, -43. Braud, J.L. and Tucker, L.M. (984) Ples n sand: a method ncludng resdual stresses, Journal of Geotechncal Engneerng, 0(), Brown, D.A. and She, C. (990) Three dmensonal fnte element model of laterally loaded ples, Computers and Structures, 0(), Caro, R., Conte, E. and Dente, G. (005) Analyss of ple groups under vertcal harmonc vbraton, Computers and Geotechncs, 3(7), Das, Y.C. and Sargand, S.M. (999) Forced vbraton of laterally loaded ples, Internatonal Journal of Solds and Structures, 36(33), Davsson, M.T. and Robnson, K.E. (965) Bendng and Bucklng of Partally Embedded Ples, Proc. of 6th Internatonal Conference on Sol Mechancs and Foundaton Engneerng, Canada, 3 (3-6), El-Naggar, M.H. and Bentley, K.J. (000). Dynamc analyss for laterally loaded ples and dynamc p-y curves, Canadan Geotechncal Journal, 37(6), Fakher, A., Hokmabad, A.S. and Saeed-Azzkand, A. (009) Assessment of lateral load-transfer methods of ples by full scale n-stu tests, Proc. of the nd Internatonal Conference on New Developments n Sol Mechancs and Geotechncal Engneerng, Ncosa, Cyprus, Fnn, W.D. (005) A study of ples durng earthquakes: Issues of desgn and analyss, Bulletn of Earthquake Engneerng, 3, Sprnger, Australan Geomechancs Vol 46 No March 0

11 Fnn, W.D. and Gohl, B. (987) Centrfuge Model Studes of Ples under smulated earthquake Lateral Loadng, In Dynamc Response of Ple Foundatons-Experment, Analyss, and Observaton, Geotechncal Spec. Publcatons,, ASCE, Gerolymos N. and Gazetas, G. (006) Development of Wnkler model for statc and dynamc response of casson foundatons wth sol and nterface nonlneartes, Journal of Sol Dynamc and Earthquake Engneerng, 6(5), ELSEVIER, Heteny, M., (946). Beams on elastc foundaton. Scentfc seres, The Unversty of Mchgan Press, Unversty of Mchgan Studes, Ann Arbor. Idrss, I.M. and Seed, H.B. (968) Sesmc response of horzontal sol layers, Journal of Sol Mechancs and Foundatons, 94(4), ASCE, Kramer, S.L. (996) Geotechncal Earthquake Engneerng, Prentce Hall, Inc., Upper Saddle Rver, New Jersey. Nar, K., Gray, H. and Donovanm, C. (969) Analyss of ple group behavour, In Performance of deep foundatons, Amercan Socety for Testng and Materals, Specal Techncal Publcaton 444, Nmbalker, S.S., Choudhury, D. and Mandal, J.N. (006) Sesmc stablty of renforced-sol wall by pseudo-dynamc method, Geosynthetcs Internatonal, 3(3), Thomas Telfords Ltd, 9. Norrs, G. and Abdollaholae, P. (985) Laterally loaded ple response: studes wth the stran wedge model, State of Calforna, Department of Transportaton, Report No. CCEER-85-. Nour, H., Fakher, A. and Jones. C.J.F.P. (006) Development of horzontal slce method for sesmc stablty analyss of renforced slopes and walls, Geotextles and Geomembranes,4 (3), Novak, M. (974) Dynamc stffness and dampng of ples, Canadan Geotechncal Journal, (4), Matlock, H. (970). Correlatons for desgn of laterally loaded ples n soft clay, Proc. of nd Annual Offshore Technology Conference, Dallas, paper OTC-04, Padron, L.A, Aznarez, J.J. and Maeso, O. (007) BEM FEM couplng model for the dynamc analyss of ples and ple groups, Engneerng analyss wth boundary element, 3(6), Parmelee, R.A., Penzen, J., Scheffey, C.F., Seed, H.B. and Thers, G.R. (964). Sesmc effects on structures supported on ples extendng through deep senstve clays, Insttute of Engneerng Research, Unversty of Calforna, Berkeley, Rep.SESM64-. Reese, L, C. and Van Impe, W.F, (00) Sngle Ples and Ple Groups Under Lateral Loadng, A.A Balkema, Rotterdam, Brookfeld. Reese, L.C., Cox, W.R. and Koop, F.D. (974) Analyss of laterally loaded ples n sand, 6th Annual Offshore Technology Conference, Austn Texas, (OTC080), Shahghol, M., Fakher, A. and Jones, C.J.F.P. (00) Horzontal slce method of analyss, Techncal note. Geotechnque, 5(0), Steedman, R.S. Zeng, X. (990) The nfluence of phase on the calculaton of pseudo-statc earth pressure on a retanng wall, Geotechnque, 40(), 03. Tabesh, A. (997) Lateral sesmc analyss of ples, PhD. Thess, supervsor: Poulos, G., Unversty of Sydney, Australa. Thavaraj, T. (00) Sesmc analyss of ple foundatons for brdges, Ph.D. Thess, Unversty of Brtsh Columba, Vancouver, B.C., Canada. Australan Geomechancs Vol 46 No March 0 4

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