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1 Anoyati, G., Mylonaki, G. and Lemnitzer, A. (016) Soil reaction to lateral harmonic pile motion. Soil Dynamic and Earthquake Engineering, 87. pp ISSN Available from: We recommend you cite the publihed verion. The publiher URL i: Refereed: Ye (no note) Diclaimer UWE ha obtained warrantie from all depoitor a to their title in the material depoited and a to their right to depoit uch material. UWE make no repreentation or warrantie of commercial utility, title, or fitne for a particular purpoe or any other warranty, expre or implied in repect of any material depoited. UWE make no repreentation that the ue of the material will not infringe any patent, copyright, trademark or other property or proprietary right. UWE accept no liability for any infringement of intellectual property right in any material depoited but will remove uch material from public view pending invetigation in the event of an allegation of any uch infringement. PLEASE SCROLL DOWN FOR TEXT.

2 Soil Reaction to Lateral Harmonic Pile Motion George Anoyati Pot-Doctoral reearcher, Department of Civil and Environmental Engineering, Samueli School of Engineering, Univerity of California Irvine, Irvine, CA George Mylonaki Chair in Geotechnic and Soil-Structure Interaction, Department of Civil Engineering, Faculty of Engineering, Univerity of Britol, Britol UK Anne Lemnitzer (Correponding Author) Aitant Profeor, Department of Civil and Environmental Engineering, Samueli School of Engineering, Univerity of California Irvine, Irvine CA, ABSTRACT The analytical repreentation of dynamic oil reaction to a laterally-loaded pile uing 3D continuum modelling i reviited. The governing elatodynamic Navier equation are implified by etting the dynamic vertical normal tree in the oil equal to zero, which uncouple the equilibrium in vertical and horizontal direction and allow a cloed-form olution to be obtained. Thi phyically motivated approximation, correctly conforming to the exitence of a free urface, wa not exploited in earlier tudie by Tajimi, Nogami and Novak and lead to a weaker dependence of oil repone to Poion ratio which i in agreement with numerical olution found in literature. The tre and diplacement field in the oil and the aociated reaction to an arbitrary harmonic pile diplacement are derived analytically uing pertinent diplacement potential and eigenvalue expanion over the vertical coordinate. Both infinitely long pile and pile of finite length are conidered. Reult are preented in term of dimenionle parameter and graph that highlight alient apect of the problem. A detailed dicuion on wave propagation and cutoff frequencie baed on the analytical finding i provided. A new dimenionle frequency parameter i introduced to demontrate that the popular plane-train model yield realitic value for oil reaction only at high frequencie and low Poion ratio. 1

3 1. Introduction The degree of accuracy in predicting the lateral repone of a pile ubject to dynamic loading i trongly dependent on the reaction of the urrounding oil to the pile motion. Given it implicity and veratility, the mot well-known model capturing the oil reaction i decribed in the pioneering work of Baranov [1] and reviited by Novak []. The baic model aume no variation in repone along the vertical coordinate, hence treating the oil medium around the pile a a erie of uncoupled incompreible horizontal lice in the analyi. Conequently, the Baranov-Novak model can be viewed a a plane train cae. Although thi model yield oil reaction in cloed form, it ha been hown that reliable prediction are only obtained in the high frequency range. To addre the limitation identified above, the current tudy eek to provide a olution that build upon the work of Tajimi [3], Nogami and Novak [4], and Saitoh and Watanabe [5] which handle the problem in three dimenion. Our pile i conidered a vertical cylinder and the oil i modelled a a continuum, taking into account all three component of oil diplacement under the aumption of zero dynamic vertical normal tree. A imilar aumption wa adopted in earlier tudie [6] [7], however, propoed olution were limited to the kinematic repone of laterally loaded pile. Thi phyically motivated implification i particularly attractive, a it repect the boundary condition aociated with the preence of a tre free oil urface and reduce the number of governing equation to two intead of three a found in the claical elatodynamic theory ( [8], [9]). Contrary to early tudie where the vertical oil diplacement wa et equal to zero ( [10], [3], [4]), in thi work the aumption will be le retrictive: vertical oil diplacement i mall, yet not zero. A hown in the following, thi approach overcome the ingularitie of earlier model in the important cae of incompreible oil. The equation of motion in the oil medium are then olved analytically through pertinent eigen-expanion. Cloed-form olution a a function of pile diplacement amplitude are obtained for the diplacement field in the oil and the oil reaction to lateral pile motion. The oil reaction i expreed in term of a dimenionle oil reaction factor (R ) which depend on pile lenderne, oil material damping, Poion ratio and excitation frequency. The effect of thee parameter on the reaction factor are explored analytically and preented in dimenionle graph. In the dynamic regime, the real part of the reaction factor decribe the tiffne of the oil layer and the imaginary part decribe the correponding damping. The reaction of the oil layer can be directly employed in the olution of the oilpile interaction problem baed on the boundary condition at the pile head and tip, and can be eaily calculated for any given pile diplacement profile [11]. Soil-pile interaction analyi ha alo been undertaken and will be preented in a companion tudy. Empirical expreion for dynamic oil reaction to lateral pile motion have been propoed, among other, by Roeet [1], Dobry et al. [13], Gazeta and

4 Dobry [14] [15], and Mylonaki [16]. Summarie of available information are provided in Tajimi [17], Pender [18], Gazeta and Mylonaki [19], Syngro [0], Guo [1] and more recently by Saitoh and Watanabe [5] and Shadlou and Bhattacharya []. Additional information on the Tajimi model i provided by Akiyohi [3], Veleto and Younan [6], Chau and Yang [4], and Latini et al. [5]. The cope of thi paper i multi-fold: (i) to review the plane train model of Baranov Novak and dicu under which condition it prediction are realitic; (ii) to derive an improved three-dimenional elatodynamic olution with an emphai on the performance at high value of oil Poion ratio; (iii) to explain the wave propagation mechanim that develop in oil under the influence of harmonic lateral pile ocillation; (iv) to extend the olution to the important cae of an infinitely long pile; (v) to provide novel normalization cheme for oil reaction that lead to reult approaching a ingle mater curve, a well a implified formulae that can be ued in application. Upon ucceful derivation, the improved model will be ued in a follow-up paper to analyze the lateral harmonic ocillation of a ingle pile.. Problem Definition The oil pile ytem conidered in thi tudy i decribed in cylindrical coordinate a hown in Fig. 1: a vertical cylindrical pile i embedded in a homogeneou oil layer overlying a rigid bedrock and i ubjected to a harmonic lateral movement w(z, t) = w(z, ω) e iωt, where t i the time variable, ω i the cyclic excitation frequency and i i the imaginary number (i = 1). The oil layer of thickne H i treated a a continuum and i decribed by it Young modulu E, ma denity ρ and Poion ratio ν. In dynamic analye the oil i treated a a diipative material with hyteretic damping β expreed through a complex valued Young modulu E = E (1 + iβ ). Note that for the problem at hand a perfectly bounded interface between pile and oil i aumed, although thi aumption can be relaxed [6]. 3. Model Development The equilibrium of force acting on an arbitrary oil element along the radial and tangential direction i expreed in term of Cauchy tree a: 3

5 r rz 1 r 1 ur r r z r r t 0 (1a) r z 1 u r r z r r t 0 (1b) where u r = u r (r, θ, z, t) and u θ = u θ (r, θ, z, t) are the horizontal and tangential oil diplacement, σ r = σ r (r, θ, z, t) and σ θ = σ θ (r, θ, z, t) are the normal tree acting along r and θ, repectively; τ rz = τ rz (r, θ, z, t) i the hear tre along z and perpendicular to r; τ rθ = τ rθ (r, θ, z, t) i the hear tre along θ and perpendicular to r; and τ θz = τ θz (r, θ, z, t) i the hear tre along z and perpendicular to θ. Note that the lat term in each equation aociated with the oil ma denity ρ i accounting for the inertia of the oil in radial and tangential direction under dynamic excitation. Taking into account tre train relation in cylindrical coordinate (ee Appendix A) and conidering harmonic oil repone of the form u r = u r (r, θ, z, ω) e iωt and u θ = u θ (r, θ, z, ω) e iωt, Eq (1) can be rewritten in term of diplacement in the following form: ru ru 1 r u 1 u r u r u * r r r r r z V r 0 (a) 1 ( ru ) ur 1 ( rur) u u u * r r r r r z V 0 (b) where V = V 1 + iβ i the complex valued hear wave propagation velocity in the oil. In the above equation η i a dimenionle compreibility coefficient which i olely a function of oil Poion ratio and i aociated with the effect of vertical oil diplacement on tree. It i noted that unlike the vertical repone mode where a number of tre-diplacement term are et equal to zero in a correponding formulation [7], in the olution at hand etting σ z equal to zero i the only approximation involved. It i alo worth mentioning that due to thi approximation hear tre τ rz ceae to be zero at the oil urface, yet thi violation typically ha a minor effect on the olution [6], [7], [11]. Additional dicuion i provided in the enuing. Following Graff [8], a degenerate Helmholtz decompoition cheme i applied to uncouple the above et of partial differential equation. To thi end, horizontal and tangential diplacement are expreed in term of two potential function Φ and Ψ a hown below: u r 1 r r (3a) 4

6 u 1 r r (3b) Subtituting Eq (3) in () lead to the following new et of uncoupled differential equation 0 * (4a) z V 0 * (4b) z V where 1 1 i the Laplacian operator. The olution to the above equation can r r r r be obtained uing the method of eparation of variable which allow the tranformation of the partial differential equation into a et of ordinary differential equation which are eaier to handle. In the realm of thi approach, potential function Φ and Ψ can be expreed a product of three modular function i.e., Φ(r, θ, z) = R 1 (r) Θ 1 (θ) Z 1 (z) and Ψ(r, θ, z) = R (r) Θ (θ) Z (z), which yield the following equation: 1 d R 1 dr 1 1 d 1 dz * R 1 dr r dr r 1 d Z1 dz V 0 (5a) 1 d R 1 dr 1 1 d 1 dz * R dr r dr r d Z dz V 0 (5b) The above mathematical repreentation lead to the decompoition of the partial differential equation into three ordinary differential equation in r, θ and z: d R 1 dr n q R 0 (6a) dr r dr r d n 0 d (6b) dz a Z 0 dz (6c) from which it arie that 5

7 q 1 a * V (7) a and n being poitive real number; q ha dimenion of 1 / Length and can be viewed a an attenuation parameter (wavenumber) for radially propagating wave. Upon obtaining the general olution to the above equation ( [8]), the potential function are written a follow: qr qr in B co n A in az B co az A1 In B1 Kn A n 3 (8a) 3 q r q r in co in co az A 4 In B4 Kn A5 n B5 n A6 az B (8b) 6 where I n ( ) and K n ( ) are the modified Beel function of the n-th order and the firt and econd kind, repectively. A i, B i (i = 1,, 3,, 6) are contant to be determined from the boundary condition of the problem. To enure bounded repone at large radial ditance from the pile (r ), contant A 1 and A 4 aociated with the modified Beel function I n ( ) mut vanih. Conidering the direction of pile loading to be along θ = 0, contant A and B 5 mut vanih a well to atify the condition of zero radial and tangential diplacement component, u r and u θ, at θ = π and θ = 0, repectively. Thi eliminate the trigonometric function in( ) and co( ) in Eq (8a) and (8b), repectively. Thi i in accordance with a poitive diplacement u r (u r > 0) in the range π θ π and a negative diplacement (u r < 0) in the range π θ 3π. Likewie, the tangential diplacement i poitive (u θ > 0) in the range 0 θ π and negative (u θ < 0) in the range π θ π. The above are valid for n = 1. The additional condition of zero oil diplacement at the bae of the oil layer and tre-free oil urface (τ rθ = 0) enforce A 3 = A 6 = 0 and co ah = 0 which, in turn, yield a m 1, m 1,, 3,... m (9) H where a m are the eigenvalue of the ytem with a 1 < a < a 3 < a N (N being the total number of mode employed in the analyi). In light of the above, Eq (8) implify to B in a z (10a) m co K1q m r 5 in K1q m r m m A in a m z (10b) 6

8 The differential equation with repect to z (Eq. 6c) i recognized a a Sturm Liouville (S-L) equation with contant coefficient. Since parameter a i not pecified, finding the value for which nontrivial olution exit i part of the S-L theory. Such value are called the eigenvalue of the boundary-value problem and the correponding olution for function Z(z) are the eigenfunction (i.e., the "oil mode"). It i important to note that thee mode are not necearily aociated with dynamic oil repone (i.e., they are merely an orthogonal et of function) and exit even in the tatic cae. Correponding to each eigenvalue a m a unique eigenfunction Φ m (z) exit which i called the m-th fundamental olution. For the current problem the trigonometric function in a m z are the normal mode (eigenfunction) z in a z (11) m m which atify the ordinary differential equation (Eq. 6c) with repect to the patial variable z and the boundary condition of the problem. The firt five mode of vibration are depicted in Fig., where the point of zero diplacement are the node of vibration and the point of maximum vibration are the correponding antinode. The normal mode have the important property of orthogonality which i decribed mathematically a follow H k z m z dz 0, m k m, k 1,, 3,... (1) 0 It i anticipated that the olution hould be obtained by the uperpoition of all particular olution (mode). Thu oil diplacement component u r and u θ hould be expreed a an infinite um of Fourier term including the oil mode Φ m and a term U r,θ m aociated with the patial variable r. Accordingly, U r, mr, z (13) u r, m m m1 The olution will be obtained a a uperpoition of the particular olution ( mode, Eq 15). Since oil mode form an orthogonal et, pile diplacement w can be expreed through a normal-mode expanion imilar to oil repone (Eq. 13), m m m1 W z w z (14) where the frequency varying coefficient W m are meaured in unit of length. Subtituting Eq. (10) into (3), impoing compatibility of diplacement in horizontal and tangential direction 7

9 i.e., u r (d, 0, z, ω) = w(z, ω) and u θ (d/, π, z, ω) = w(z, ω) and taking into account the orthogonality of oil mode, diplacement component u r, u θ are obtained a u r,,z, r co U r m r, W m m z (15a) m1 u r,,z, in U mr, Wm m z (15b) m1 which atify the ymmetry condition u r (r, π, z) = u r (r, π, z) and u r (r, 0, z) = u r (r, π, z), where U r m r d d m m 0 m 1 m m 1 m r r, B K q r K q r A K q r U d mr, d Bm K1 qm r Am m K0 qm r K1 qm r r r (16a) (16b) A m and B m being the dimenionle contant A m K1m m K0m K K K K K m 0 m 1 m m 0 m m 0 m 1 m (17a) B m K1m m K0m K K K K K m 0 m 1 m m 0 m m 0 m 1 m (17b) where m = q m d (Eq. 7) i a dimenionle parameter. Compreibility coefficient Mention ha already been made to the tudy of Nogami and Novak [4] where the vertical oil diplacement u z wa et equal to zero and, thu, the correponding normal train wa ε z = 0. Thi aumption lead to the following expreion for the compreibility factor 1 1 (18) which expree the quare root of the ratio of the contrained modulu M (P-wave modulu) to the hear modulu of the oil material (i.e., η = M G ). A problem ariing from the ue of thi equation i the enitivity to Poion' ratio, a η become infinitely large when ν approache 0.5. Thi behavior i 8

10 puriou (e.g. lead to zero value of wavenumber q in Eq. 7) and ha not been oberved in rigorou numerical olution of uch problem, e.g., [9], [30], [6]. It i important to note that depite that u z i aumed to be zero, the pecific olution i till three dimenional (i.e. not plane train) a the variation of the repone with repect to the vertical coordinate (z) i finite due to the hear coupling among the variou lice. The alternative aumption of σ z = 0, [11], which complie with a tre-free oil urface lead to finite vertical diplacement which yield: 1 (19) Equation (18) and (19) are compared graphically in Fig. 3. Except where pecifically indicated otherwie, the olution preented hereafter are baed on Eq. (19). 4. Soil Reaction At the oil-pile interface, the amplitude of horizontal oil reaction p(z, ω) reulting from the pile motion i expreed a [4], [6], [6], r,0 r,0 0 co in d p z d (0) where σ r,0 = σ r (d, θ, z, ω) and τ rθ,0 = τ rθ (d, θ, z, ω) are the maximum radial normal tre and hear tre, repectively, acting at the periphery of the pile * r,0 co m m m d m1 and G G S W z (1) * r,0 in m m m d m1 T W z () where the dimenionle coefficient S m and T m are given by the following expreion: S B K K K A K K m m m 0 m 1 m 1 m m m 0 m 1 m m (3a) 9

11 T B K K A K K K m m m m 0 m 1 m m m 0 m 1 m 1 m (3b) Subtituting the above expreion into Eq. (0) and noting that the horizontal oil reaction can be written in the alternative form π 0 co θ dθ = π 0 in θ dθ = π,, m m m m1 * * pz G R W z (4) where R m i a complex valued oil reaction factor aociated with the m-th oil mode: R * m m K K K K K K1 m m K0 m K1( m) K1 m m K0 m K1( m ) m 0 m 1 m m 0 m m 0 m 1 m (5) in Eq. (5) i denoted the econdary compreibility coefficient and i expreed a: 1 (6) Note that Eq. (6) correpond to the compreibility factor η adopted by Anoyati and Mylonaki [7] for the axially-load problem. It i alo important to note that in the alternative olution of Nogami and Novak [4] the reaction factor R m i obtained from the above expreion uing η σ = η, where η i given from Eq. (18). In both olution the oil reaction factor depend mainly on oil parameter (i.e., ν, β, V, G ), the excitation frequency ω and on only one pile parameter, which i the diameter d. In the dynamic regime, the complex-valued oil reaction factor can be cat in the following equivalent form R Real R i Imaginary R R 1 i (7) * * * m m m m m where R m (ω) = Real(R m ) i the dynamic torage tiffne and β m R m (ω) = Imaginary(R m ) i the correponding lo tiffne. The dimenionle parameter β m = Imaginary(R m ) Real(R m ) define an equivalent damping ratio, which i analogou to percentage of critical damping in a imple ocillator (Clough and Penzien [31]). R m (ω) can be interpreted a a frequency dependent pring and β m R m (ω) ω a a dahpot attached in parallel to the pring. Note that the damping ratio β m i different from the oil material damping β, and can be alternatively expreed a β m = β + (β r ) m, where (β r ) m, 10

12 i the radiation damping ratio aociated with the m-th propagating mode. β m β i a pecial cae which i valid only for frequencie in the range 0 < ω < ω m. Recalling Eq. (5) oil reaction i expreed a p(z, ω) = π R m G w(z). Thi how that R m can be viewed a a dimenionle factor which modifie the oil tiffne G due to the preence of the pile. In addition, it i clear that R m i mode-dependent which indicate that each mode ha a different influence on the oil tiffne. Expreing the oil reaction in the imple form p = k w [k can be viewed a a Winkler modulu (in unit of force per length quare)] allow u to write k = (π R m ) G, which how that an average depth-independent Winkle modulu multiplied by the profile of diplacement at the oilpile interface yield the oil reaction profile. 5. Wave Propagation When a pile ocillate energy i tranmitted to the oil. Part of thi energy i tored in the oil in the form of dynamic deformation and the remaining part i lot, e.g., being tranformed into heat (material damping) or radiating to infinity in the form of tre wave (radiation damping). An initial dicuion preented below treat the vertical and horizontal direction eparately invetigating the contribution of each mode to the wave propagation, followed by a preentation of the combined wave effect. 5.1 Vertical direction In vertical direction wave propagation i aociated with the term in a m z e iωt. Thi term decribe an ocillating motion of the m-th mode, which i of a tationary nature in pace (i.e., along the vertical axi z) and varie with time t. Uing trivial mathematical procedure, the above term may be rewritten in the alternative form i(e i(amz ωt) e i(amz+ωt) )/. Thi equivalent repreentation how that each exponent repreent a diturbance propagating in the vertical direction. The firt exponent decribe a wave travelling from the oil urface to the bae, while the econd term i aociated with a wave that follow the revere path. Uing the principle of uperpoition thee two oppoite directional wave which have the ame frequency ω, wavenumber a m and amplitude form a tanding wave. Thi wave doe not travel vertically, but tand till and ocillate horizontally. Hereby each mode m form a vertically varying diturbance (dynamic deformation) which varie with time. Thi not a travelling wave, thu no lo of energy in term of radiation damping occur. 11

13 It can be eaily deduced that at depth Z m,k = H(k 1) (m 1) the amplitude of the motion i alway zero for the mode examined and thee point are called the node. Note that the index m refer to the m-th mode and the index k to the number of node which appear in length H of the ame mode. Evidently, only one node exit at z = 0 for the 1-t mode (m = 1) wherea for the -nd mode (m = ), two node appear at z = 0 and z = H 3, and o on (Fig. ). Due to the boundary condition at the bae, the location z = 0 repreent a node in all mode. Analogouly, depth decribed by Z m,k = H(k 1)/(m 1) are called antinode and the wave amplitude i maximum. The ditance between two ucceive node or antinode i equal to the wavelength π a m or 4H (m 1) for each mode. Note that when rewriting the upercript of the exponential function a ia m (z + V m t) or ia m (z V m t), the term V m = ω a m with dimenion of velocity arie, which tand for the phae velocity of the wave aociated with a given mode m. Thi indicate that for a given frequency ω each harmonic wave propagate with a mode-pecific velocity which i different for each mode. Thi reveal the development of a ditortion mechanim called modal diperion a the propagation velocity i not the ame for all mode. In concluion, at each depth in the oil layer influx and efflux of energy occur due to upward and downward travelling wave. For each mode thee oppoite directional wave form a tanding wave, which i characterized by node and anti-node. At the node influx and efflux of kinetic energy i balanced, hence no motion occur. Neverthele, train till develop and change with time. The repone of the oil i computed by uperpoition of N mode (or a uperpoition of N tanding wave). Conequently motion develop at all depth. 5. Horizontal direction In the horizontal direction the wave propagation i aociated with the term U r,θ m (Eq 13, 15) and i mathematically repreented through the Beel function K 0 ( ) and K 1 ( ) and their argument q m (Eq. 7). The variation of q m with frequency i chematically illutrated in Fig 4 and 5 for an undamped (β = 0) and a damped (β 0) medium. In abence of oil material damping (i.e., for an undamped medium a hown in Fig. 4a), Eq. (7) may be rewritten a q m 1 m V (8) 1

14 where ω m = (m 1) ω 1 and ω 1 = πv H are the m-th and 1-t reonant frequencie of the ytem, repectively. For ω < ω m, q m i real valued and decreae with increaing frequency. At ω = ω m, q m drop to zero and a further increae in frequency (ω > ω m ) yield purely imaginary wavenumber q m = i q m, which increae with increaing frequency (q m being a real number). ω m repreent the tranition from propagation to non-propagation and i called the cutoff frequency of the m-th mode or m-th reonant frequent of the ytem. Note that ω 1 coincide with the natural frequency of the depoit in hearing ocillation. No wave propagation (i.e., radiation of energy) i oberved for lower frequencie. A hown later, ω 1 i aociated with a ignificant drop in tiffne and an increae in damping due to the emergence of travelling wave. For ω < ω m, U r,θ m (Eq 13, 15) i reponible for a monotonic decreae in oil repone with increaing horizontal ditance from the pile axi and i not aociated with wave propagation in the medium. For ω > ω m, tre wave aociated with the m-th mode emerge from the oil-pile interface and propagate horizontally in the oil medium. At ω = ω m, q m = 0 and yield an infinite wavelength λ (= π q m ) which for a finite frequency ω m yield an infinite phae velocity V (= ω q m ). Thi indicate that at any reonant frequency ω m there i no patial variation in motion in r direction for the m-th mode. Since U r,θ m (ω = ω m ) = 1 (Eq 15), the ytem undergoe harmonic motion and the contribution of the m-th mode to the oil vibration can be expreed uing the following implified equation: u W i t r, m co, in z e m (9) m m In preence of material damping β, in the oil (i.e., damped medium a hown in Fig. 4b), Eq. (7) may be approximated a follow: q 1 i (30) m m V " imaginary " real part" part" Contrary to the undamped medium, the above equation yield non-real-valued wavenumber q m, even in the frequency range of ω < ω m. Complex valued wavenumber for damped oil caue a monotonic reduction in oil diplacement with radial ditance, which i found to be tronger than in an undamped medium. At ω ω m, Real(q m ) drop to a minimum (non-zero) value. For ω > ω m the trend i revered and Real(q m ) increae with frequency. On the other hand, Imaginary(q m ) alway increae with frequency. A in the cae of an undamped medium, for ω > ω m wave aociated with the m-th mode 13

15 tart to emerge from the oil-pile interface and propagate horizontally in the oil medium, while q m i complex valued intead of purely imaginary. Figure 5 preent the frequency pectrum for the firt five mode for an undamped medium and a medium with oil damping β = Auming that the complex valued q m can be written a q m + i q m, it become evident that for a given frequency q 1 < q < < q 5 and q 1 > q > > q 5, for both, an undamped (β = 0) and a damped (β = 0.05) medium. Undamped medium (Fig. 5): At ω = ω 1, q 1 = 0 while all q 1+m pertaining to higher mode attain poitive real value. Further increae in frequency (ω 1 < ω < ω ) yield a purely imaginary q 1 = i q 1, wherea the ret of wavenumber q 1+m remain real valued. Now the Beel function K( ) of complex argument repreent travelling wave in the radial direction with an in-phae component Re[K( )] and an out-of-phae, Im[K( )] which emanate from the pile periphery and radiate to infinity with continuouly decreaing amplitude. Thee wave control radiation damping. The firt reonance influence excluively the firt mode, while the higher mode contribute only to a monotonic attenuation of oil repone with radial ditance. An additional increae in frequency (ω < ω < ω 3 ) yield q = i q, which indicate that the econd mode contribute additionally to the lo of energy through radiation. Likewie, with increaing frequency higher mode contribute gradually to wave propagation (q m gradually become purely imaginary). The total effect i a uperpoition of all thee wave. Damped medium (Fig. 5): All curve follow the trend decribed in Fig. 4 (b). Note that the larget deviation among the damped and the undamped curve for each mode i oberved cloe to the reonant frequency ω m, and i becoming more ignificant at higher mode. For ω < ω m the effect of material damping i minor and the curve practically coincide. Baed on Eq. (13) and conidering the variation in time the oil repone can be expreed a u r,θ m U r,θ m (r, ω) Φ(z) e iωt. For an undamped oil layer (β = 0) the following cae may be examined: a) for ω < ω m, the term U r,θ m i real valued b) at ω = ω m (reonance), U r,θ m = 1 c) for ω > ω m, the term U r,θ m i complex valued Cae (a): When ω < ω m, the energy tranmitted from pile ocillation to the oil i tored in term of dynamic deformation. 14

16 Cae (b): At reonant frequency ω = ω m, no patial variation in motion in r direction for the m-th mode exit. Since U r,θ m (ω = ω m ) = 1, the ytem undergoe only harmonic motion (i.e., tanding wave in the form of Φ m (z) e iωt ) and the contribution of the m-th mode to the oil vibration can be expreed uing the implified expreion provided in Eq. (9). Cae (c): When ω > ω m, travelling wave develop in the oil and lo of energy occur (radiation damping). Note that in preence of oil material damping (β 0) and for frequencie ω < ω m lo of energy i due to oil material damping. 6. Plane train model The baic aumption of the plane train model i that all derivative with repect to the vertical coordinate are zero, thu no vertical and hear train develop on the plane perpendicular to the pile axi and only an incompreible horizontal oil lice of the oil medium i conidered in the analyi [1]. Thi model can be viewed a mathematically accurate for an infinitely-long pile embedded in a half pace and ubjected to uniform lateral diplacement along it whole length [3]. The Baranov-Novak oil retraining action can be expreed through a complex valued reaction factor R [3]: R * q 4 K1 q K1 K1 q K0( ) K0 q K1( ) q K q K K q K ( ) q K q K ( ) (31) where and q are dimenionle frequency parameter i a 0 and q 1 i (3a,b) and a 0 = ωd V i a dimenionle frequency and η i obtained from Eq. (18). Parameter R i contant with depth and independent of the condition at the boundarie of the oil layer. Therefore, it cannot capture the layer reonance and exhibit an aymptotic behavior for ω 0 (Fig. 8). In the low-frequency range the oil reaction factor decreae rapidly with decreaing frequency and become zero at ω = 0. Accordingly, the model cannot capture tatic tiffne. Thi deficiency ha been identified in earlier tudie [4], [16]. 15

17 Evidently, thi i not the cae for a pile of finite length embedded in a oil tratum overlying a tiff bae. In thi tudy, the oil reaction varie with depth and it variation depend on the boundary condition at the two end of the pile. Furthermore, it i enitive to the natural frequencie of the oil layer. The profile of oil reaction in term of dynamic tiffne and damping will be examined in a companion paper. Note that the preent olution can be reduced to the plane train model [4] by eliminating the variation of oil diplacement component in the vertical direction (i.e., etting a = 0 in Eq 6c and 7). Depite it implicity, the plane train model yield realitic prediction for frequencie beyond cutoff (ee enuing dicuion). Thu, it can be viewed a a pecial cae of the propoed more complete olution, retricted to frequencie beyond cutoff. Thi i an inherent weakne of the plane train model, it aumption being valid only after wave propagation initiate in the medium (a 0 > a cutoff ). Note that oil reaction derived from the Baranov-Novak plane train model i expreed a p(ω) = π G R. Thi i eentially the complex Winkler modulu (dynamic Winkler pring tiffne and dahpot) for laterally loaded pile, and will be invetigated in a follow up paper. 7. Numerical Reult 7.1 Static condition The effect of pile length, or equivalently the oil layer thickne, and the elected oil mode on the tatic oil reaction factor are invetigated in Fig. 6. It i hown that for a given L/d, higher value of R m correpond to higher mode. Thi trend i more pronounced for hort pile (L/d < 10) where the reaction factor pertaining to the 10-th mode i approximately five time higher than the value for the 1-t mode. Comparion with the early olution of Nogami and Novak [4] how that the reult of that tudy are alway higher than thoe obtained from the propoed model. The deviation i tronger with decreaing pile length and higher mode the maximum deviation being oberved for L/d = 5 and m = 10. In Fig. 7 the effect of Poion ratio on the tatic reaction factor i invetigated for two elected mode. It i hown that higher value of v alway correpond to higher R m for the range of pile length conidered. A anticipated, the lower the Poion ratio the better the agreement with the prediction of the earlier tudy ( [4]). 7. Dynamic regime 16

18 Reult for the oil reaction factor in the dynamic regime obtained from the propoed olution are preented in Fig The olution of Nogami and Novak [4] i added for comparion. Different repreentation of the oil reaction factor and frequency are employed, which hed light into the phyic of the problem. Some general trend are oberved: For each mode m, the dynamic oil reaction factor (i.e., real part of oil reaction) decreae with increaing frequency up to the m-th reonance, while for the ame frequency range, damping (i.e., imaginary part of oil reaction) i practically unaffected by frequency and depend olely on the oil material damping (ince only weak travelling wave aociated with the m-th mode develop in the medium). At m-th reonance the dynamic reaction R m attain a local minimum, which i aociated with a ditinct jump in damping due to energy radiation, a horizontally travelling wave emerge in the oil medium. The variation of the real and the imaginary part of dynamic oil reaction with frequency for the firt five mode pertaining to a long pile i hown in Fig. 8. The frequency i normalized by the firt reonant frequency of the ytem, ω 1. A trong dependence of tiffne (i.e., real part of oil reaction) on the ocillation mode i oberved below the reonant frequency, while at the ame frequency range the damping (i.e., imaginary part of oil reaction) i practically unaffected by frequency and i controlled by oil material damping. With increaing frequency the dynamic reaction factor become gradually independent of oil mode, with all curve practically converging to a ingle curve at high frequencie. A anticipated, reult from the tudy of Nogami and Novak [4] are alway higher than thoe of the propoed model. An alternative repreentation of the reult preented in Fig. 8 i hown in Fig. 9, where the dynamic oil reaction factor i normalized by it tatic value (ω = 0), and the lo of energy (imaginary part) i normalized by twice the real part. It i oberved that thi type of repreentation cannot capture the difference between Nogami and Novak [4] and the propoed olution. Thi indicate that the dynamic modifier expreed by the ratio R m (ω) / R m (ω = 0) i eentially identical in the two olution and thu independent of the compreibility of the oil medium in the vertical direction controlled by coefficient η. A perhap better repreentation for frequencie in the range 0 < ω < ω m i illutrated in Fig. 10, where the frequency i normalized by a different value for each mode, namely, the m-th reonant frequency for the m-th propagating mode. It i hown that for dynamic oil reaction all reult collape to a ingle curve. Alo, damping i practically contant and controlled by oil material damping. The combined effect of oil material damping, pile length (layer thickne) and mode of vibration i invetigated in Fig. 11. For a given L/d higher value of oil material damping correpond to higher value of the dynamic oil reaction factor R. Thi trend i revered for damping ince β i not a dahpot 17

19 coefficient but a dimenionle performance index expreed a the ratio of the imaginary part of the complex tiffne [Imaginary(R )] divided by twice it real part ( R) (Eq. 7). Conidering that the imaginary part of the tiffne i practically unaffected by oil material damping for a given pile or oil thickne (i.e., contant L/d or H/d), the behavior of β i governed by the dynamic tiffne R in the denominator. Thu, higher dynamic tiffne curve correpond to lower damping ratio. In addition, the effect of damping i tronger for hort pile (L/d = 5) in the high frequency range and for higher mode (e.g., m = 3). At reonance (ω = ω 1 for the 1-t mode, ω = 5ω 1 for the 3-rd mode) the dynamic reaction R m attain a minimum value, with the tronger drop pertaining to the lowet material damping. Note that for the extreme cae of zero material damping the drop i maximum and R m would reach zero. The effect of mode number on dynamic oil reaction and damping for a hort (L d = 10) and a long (L d = 50) pile in an undamped oil medium i preented in Fig. 1 (a) and (d). Note that different normalization parameter are ued for the frequency below and beyond reonance: a 0 = ωd/v being the familiar dimenionle frequency parameter and a cutoff,m = ω m d/v being the cutoff frequency of each mode or m-th reonance. For frequencie below m-th reonance all curve in Fig. 1 (a) tart from unity, a the dynamic tiffne i normalized with it tatic value (ω = 0), and decreae monotonically with frequency. It i hown that higher mode are aociated with a higher decreae in tiffne. Thi effect i more pronounced for hort pile. Over the ame frequency range, damping i independent of frequency (Fig. 1b) and practically equal the oil material damping, i.e., all damping curve converge to a ingle curve before reonance. Beyond the cut-off frequency, wave tart to propagate in the medium reulting in a udden increae in damping (Fig. 1b). It i hown that the dynamic tiffne become inenitive to the oil thickne H (Fig. 1a). Thi i an anticipated behavior, ince the wave emitted from the periphery of the ocillating pile tend to pread out in a horizontal manner without regard for the vertical dimenion ( [19], [16]). Thi wave radiation pattern explain the very good agreement oberved between the plane train model and the more rigorou olution. Figure 1 (c) and (d) depict the dimenionle oil reaction impedance and ugget that: (i) below cutoff (ω < ω m ) pring and dahpot are bet repreented in the form R m (ω) R m (ω = 0) and β, a function of a 0 a cutoff,m ; (ii) beyond cut-off (ω > ω m ), tiffne i bet repreented in the form R m (ω) and both parameter a a function of the dimenionle frequency function (a 0 a cutoff,m ) 1/. Note that the latter parameter ha been uccefully employed in the repreentation of the vertical oil reaction in high frequencie for the cae of an axially-loaded pile in a homogeneou tratum [7], but ha not been explored in the lateral mode. 18

20 The effect of pile lenderne or oil layer thickne on oil reaction factor of a damped medium i preented in Fig. 13. The numerical reult are baed on the dimenionle frequencie introduced in Fig. 1. Only the firt mode i taken into account. For frequencie below cutoff, longer pile or thicker trata alway correpond to higher value of dynamic oil reaction factor; wherea for the damping coefficient all curve practically converge. For frequencie beyond cutoff all curve converge into a ingle curve and the oil reaction can be well captured by the plane train model. The effect of Poion ratio on dynamic oil reaction and damping ratio in the dynamic regime i invetigated in term of the 1t mode for a hort and a long pile and hown in Fig. 14. For frequencie below cutoff, variation in Poion ratio do not affect tiffne and damping (Fig. 14 a1, b1, c1). For frequencie beyond cutoff, a minor effect of Poion ratio on damping i oberved at high frequencie (Fig. 14 c). On the other hand, the reult in Fig. 14 (a) indicate that the effect i trong on dynamic oil reaction, with higher value of v reulting in higher tiffne. However, an alternative repreentation of oil reaction hown in Fig. 14 (b) (dynamic oil reaction i normalized by tatic value) how that the influence of Poion ratio on pure dynamic tiffne i oberved only in the high frequency range and can be conidered negligible. The trong variation depicted in Fig. 14 (a) are attributed to the effect of Poion ratio on the tatic oil reaction. 8. Simplified expreion for R * (ω) Reult obtained from the propoed model a hown in Fig. 14 (a) are plotted againt prediction from the plane train model in Fig. 15 (a). Evidently, the plane train model yield unrealitic reult for the incompreible cae, the parabolic-like decreaing trend can be attributed to a "trapped ma" effect which i not exhibited in the 3D model [33]. Thi i an inherent weakne of the impler model and it i uggeted to ue reult obtained from Eq. (31) only for Poion ratio le than 0.4. In thi tudy, a implified expreion for the oil reaction factor i preented. Taking the limit ν 0.5 Eq. (31) attain the aymptotic form R * 1 ( ) 4 K K 0 (33) which doe not yield realitic reult (Fig. 15a). Herein, the above expreion i modified to account for Poion ratio. Thi can be achieved by introducing a new emi empirical expreion for the parameter 19

21 i a 0 1 i (34) which encompae the compreibility parameter η and the empirical parameter χ which i a function of the Poion ratio only (Table 1). Reult from Eq (34) and (35) are compared with thoe obtained from the propoed model in Fig. 15 (b). The very good agreement in the range 0.1 < ν < 0.4 and the improved performance over the plane train model for ν = 0.5 can hardly be overtated. 9. Infinitely long pile The above olution can be readily extended to model an infinitely-long pile embedded in a half pace. Although idealized, thi limit cae i ueful ince the olution for infinitely long pile i independent of the thickne of the oil layer (H/d) and cover all flexible pile (actual pile length > active length). In thi light, one may aume that the olution for pile of finite length i ueful only for low value of pile lenderne L/d. A dicuion on active pile length i provided in Randolph [34], Velez et al. [35] and Syngro [0]. For an infinitely long pile no ditinct eigenmode exit. Intead, the olution for oil diplacement and tree i obtained by integrating over all poible value of α. The oil reaction may be written a follow: pz, G * * R W z da (35) 0 where R α i the complex valued oil reaction factor (Eq. 5) R * a K K K K K K1 K0 K1( ) K1 K0 K1( ) (36) where = qd/, and q obtained from Eq. (7). A for the cae of a oil layer R a (ω) = Real(R a ) i the dynamic torage tiffne and β a = Imaginary(R a ) Real(R a ) define an equivalent damping ratio. A three-dimenional repreentation of the dynamic oil reaction and damping ratio a a function of the excitation frequency and the parameter α i hown in Fig 16 and 17. Note that each curve correpond to a oil layer of finite length that i characterized by a fictitiou natural frequency, yet thee natural frequencie are uppreed upon integration in term of Eq. (35). In both graph all natural frequencie are placed on a traight line for which a d = ω d/v. 0

22 10. Concluion An approximate three dimenional olution i developed for the dynamic reaction of a homogeneou half pace and a oil layer over a rigid bae to the laterally ocillating pile. Contrary to the claical elatodynamic equation which cannot be olved analytically, the propoed approach allow a cloed form olution to be obtained both for an infinitely long pile and a pile of finite length. The main finding of thi tudy can be ummarized a follow: a) The main aumption adopted in thi tudy i that the vertical dynamic normal tre σ z i zero. Thi approximation i compatible with the preence of a free urface and lead to a mall, yet finite vertical oil diplacement. Thi overcome the enitivity of earlier model to Poion ratio for nearly incompreible media. b) From the interference of upward and downward traveling diturbance aociated with each mode m in the oil layer a tanding wave emerge. The total repone i a uperpoition of m-th tanding wave, which i a patially varying (tand till and ocillate left and right), but not propagating diturbance (not a travelling), and naturally no lo of energy in term of radiation damping occur. The drop in dynamic oil reaction and udden increae in damping are aociated with the natural frequencie of the oil layer and the emergence of travelling wave in the horizontal direction. The wave propagation phenomenon wa thoroughly dicued in thi tudy and wa invetigated by mean of a frequency pectrum uing the firt five mode. c) A new dimenionle incremental frequency parameter (a 0 a cutoff,m ) 1/ wa introduced for decribing the oil reaction in the high frequency range (a m > a cutoff,m ). It wa hown that thi repreentation allow the reaction factor to exhibit the ame behavior regardle of actual oil layer thickne (or pile length), value of Poion ratio, oil material damping and ocillation mode. d) It wa found that the dynamic oil reaction factor below cutoff frequency i bet normalized by the correponding tatic tiffne (ω = 0) a a function of dimenionle frequency ratio a 0. Beyond the cutoff frequency, the dynamic tiffne can be bet normalized by the a cutoff,m oil hear modulu (no pecial cheme i needed) and i bet expreed a a function of incremental frequency (a 0 a cutoff,m ) 1/ (Fig. 1). The only exception exit when invetigating the effect of v on dynamic oil reaction, where a normalization with the tatic value 1

23 i required over the whole range of frequencie (Fig. 14). Thee propertie tem from the dependence of the olution on the cutoff frequency of each mode and the gradual tranformation of the wave field emitted from the pile with increaing frequency beyond reonance, from threedimenional to two-dimenional (plane train). e) It wa alo oberved that with increaing frequency the plane train olution converge to the more rigorou olution. However, ignificant dicrepancie in tiffne appear, epecially for hort pile, for frequencie below cutoff and for the particular cae of v = 0.5 in the high frequency range. f) A imple, improved expreion for determining dynamic oil reaction baed on an aymptotic form of the claic plane train olution which take into account the compreibility of the oil wa preented in Eq (33) and (34). g) A olution for the dynamic oil reaction factor and the correponding damping ratio were derived for the cae of an infinitely long pile in a half pace. Thi i achieved by expreing the oil repone in term of integral intead of Fourier erie, a in the cae of pile of finite length (Eq. 35). Soil reaction wa then derived a a uperpoition of modular oil reaction for all poible depth of oil layer to form a half pace.

24 Notation A, B, C, D integration contant depending on boundary condition and type of loading d pile diameter E G, G H Young modulu of oil real- and complex-valued hear modulu of oil thickne of oil layer m ma of an infiniteimal oil element in cylindrical coordinate (r dr dθ dz ρ ) p horizontal oil reaction to pile motion q m r t V, V w W m u r, u θ z oil frequency-dependent parameter radial or horizontal coordinate time variable real-valued and complex-valued oil hear wave propagation velocity lateral pile diplacement pile repone Fourier coefficient oil diplacement component (radial, tangential) vertical coordinate Greek Symbol a, a m β β r η, η σ ν ρ σ r, σ θ, σ z τ θz, τ zr, τ rθ Φ, Ψ ω ω m, ω n poitive real-valued number (eigenvalue) oil material damping radiation damping compreibility parameter oil Poion ratio oil ma denity normal tree in oil hear tree in oil potential function cyclic excitation frequency m-th reonant and natural frequency Laplacian operator 3

25 Appendix A In cylindrical coordinate, normal train are written in term of diplacement via the well-known equation r ur r (A.1a) z uz z (A.1b) u r 1 u r r (A.1c) Baed on the aumption of zero normal tre σ z, normal train ε z i written a z 0 r K (A.) and normal tree σ r and σ θ can be expreed in term of normal train ε r and ε θ in the form * r G r (A.3a) * G r r (A.3b) where K 0 = ν (1 ν ) i the coefficient of lateral preure at ret for an elatic material depending olely on Poion' ratio; i alo a function of Poion' ratio, given by Eq. (6) in the preent formulation and Eq. (18) in the work by Tajimi. Shear train are written in term of diplacement a z u z 1 uz r (A.4a) rz r u z u r r z (A.4b) 1 ur u u r r r (A.4c) Shear tree τ in term of hear train γ z G (A.5a) * z (A.5b) * r G r 4

26 zr G (A.5c) * zr In light of Eq. (A.a), the derivative τ rz z and τ θz z attain the alternative form rz * ur z G z z r (A.6a) z * u 1 z G z z r (A.6b) which depend olely on the horizontal diplacement component u r and u θ. Subtituting Eq (A.), (A.3), (A.6) into the equilibrium equation (1), yield the Navier equation (). 5

27 Reference [1] V. Baranov, "On the calculation of an embedded foundation. Voproy Dinamiki i Prochnoti.," polytech.int.riga, Latvia., vol. 14, pp , [] M. Novak, "Dynamic Stiffne and Damping of Pile," Canadian Geotechnical Journal, vol. 11, no. 4, pp , [3] H. Tajimi, "Dynamic analyi of a tructure embedded in an elatic tratum," in 4th World Conference on Earthquake Engineering, Santiago, Chile, [4] T. Nogami and M. Novak, "Reitance of oil to a horizontally vibrating pile," Earthquake Engineering and Structural Dynamic, vol. 5, no. 3, pp , [5] M. Saitoh and H. Watanabe, "Effect of flexibility on rocking impedance of deeply embedded foundation," J. Geotech. Geoenvironmental Eng., vol. 130, no. 4, pp , 004. [6] A. Veleto and A. Younan, "Dynamic oil preure on rigid cylindrical vault," Earthquake Engineering and Structural Dynamic, vol. 3, no. 6, pp , [7] A. Veleto and A. Younan, "Dynamic modeling of repone of rigid embedded cylinder," J. Eng. Mech., vol. 11, no. 9, pp , [8] K. Graff, Wave Motion in Elatic Solid., Dover Publication, [9] A. Eringen and E. S. Suhubi, Elatodynamic: Volume II: Linear Theory, Academic Pre, [10] H. Matuo and S. Ohara, "Lateral earth preure and tability of quay wall during earthquake," in nd World Conference on Earthquake Engineering, Tokyo, Vol I, [11] G. Anoyati, "Contribution to kinematic and inertial analyi of pile by analytical and experimental method," PhD thei, Univerity of Patra, Rio, Greece, 013. [1] J. Roeet, "Stiffne and damping coefficient of foundation," in ASCE Geotechnical Engineering 6

28 Diviion National Convention, [13] R. Dobry, E. Vicente, M. O'Rourke and M. Roeet, "Horizontal tiffne and damping of ingle pile," Journal of Geotechnical and Geoenvironmental Engineering, vol. 108, no. 3, pp , 198. [14] G. Gazeta and R. Dobry, "Horizontal repone of pile in layered oil," J. Geotech. Engrg. Div., vol. 110, no. 1, pp. 0-40, 1984a. [15] G. Gazeta and R. Dobry, "Simple Radiation Damping Model for Pile and Footing," J. Eng. Mech., vol. 110, no. 6, pp , 1984b. [16] G. Mylonaki, "Elatodynamic model for large-diameter end-bearing haft," Journal of the Japanee Geotechnical Society: Soil and Foundation, vol. 41, no. 3, pp , 001b. [17] H. Tajimi, "Seimic Effect on Pile," in Spec. Seion 10, IX, ICSMFE, Tokyo, [18] M. Pender, "Aeimic pile foundation deign analyi," Bull. New Zealand Nat. Soc. for Earthquake Engrg., vol. 6, no. 1, pp , [19] G. Gazeta and G. Mylonaki, "Seimic oil-tructure interaction: New evidence and emerging iue.," Spec. Publ. No. 75, ASCE, pp , [0] C. Syngro, "Seimic repone of pile and pile upported bridge pier evaluated through cae hitorie," Ph.D. Thei, City Univerity of New York, 004. [1] W. D. Guo, Theory and practice of pile foundation, London: CRC Pre, 01. [] M. Shadlou and S. Bhattacharya, "Dynamic tiffne of pile in a layered elatic continuum," Geotechnique, vol. 64, no. 4, pp , 014. [3] T. Akiyohi, "Soil-pile interaction in vertical vibration induced through a frictional interface," Earthquake Engineering and Structural Dynamic, vol. 10, no. 1, pp , 198. [4] K. Chau and X. Yang, "Nonlinear interaction of oil-pile in horizontal vibration," J. Eng. Mech., vol. 131, no. 8, pp ,

29 [5] C. Latini, V. Zania and B. Johanneon, "Dynamic Stiffne and damping of foundation for jacket tructure," in 6th International Conference on Earthquake Geotechnical Engineering, Chritchurch, New Zealand, 015. [6] X. Karatzia, P. Papatylianou and G. Mylonaki, "Horizontal Soil Reaction of a Cylindrical Pile Segment with a Soft Zone," J. Eng. Mech., vol. 140, no. 10, pp , 014. [7] G. Anoyati and G. Mylonaki, "Dynamic Winkler modulu for axially loaded pile," Geotechnique, vol. 6, no. 6, pp , 01. [8] C. R. Wylie and L. C. Barrett, Advanced engineering mathematic, McGraw-Hill, [9] R. Butterfield and P. K. Banerjee, "The elatic analyi of compreible pile and pile group," Geotechnique, vol. 1, no. 1, pp , [30] A. P. S. Selvadurai and R. K. N. D. Rajapake, "On the load tranfer from a rigid cylindrical incluion into an elatic half pace," International Journal of Solid and Structure, vol. 1, no. 1, pp , [31] R. W. Clough and J. Penzien, Dynamic of Structure, New York: McGraw Hill, [3] M. Novak, T. Nogami and F. Aboul-Ella, "Dynamic oil reaction for plane train cae," Journal of the Engineering Mechanic Diviion, vol. 104, no. 4, pp , [33] F. E. Richart, J. R. Hall and R. D. Wood, Vibration of oil and foundation, Prentice Hall, [34] M. Randolph, "The repone of flexible pile to lateral loading," Geotechnique, vol. 31, no., pp , [35] A. Velez, G. Gazeta and R. Krihnan, "Lateral dynamic repone of contraint-head pile," Journal of Geot. Eng., vol. 109, no. 8, pp , [36] G. Mylonaki, "Winkler modulu for axially loaded pile," Geotechnique, vol. 51, no. 5, pp , 001a. 8

30 [37] M. Novak and T. Nogami, "Soil-pile interaction in horizontal vibration," Earthquake Engineering and Structural Dynamic, vol. 5, no. 3, pp ,

31 Table Table 1: Value of empirical parameter χ a function of Poion ratio Poion ratio ν χ

32 FIGURES Fig. 1: Problem conidered: Harmonically excited cylindrical vertical pile embedded in oil tratum overlying a rigid bae Fig. : Firt five mode of oil layer due to lateral pile motion 31

33 Fig. 3: Variation of compreibility factor with Poion' ratio Fig. 4: Schematic repreentation of the frequency pectrum in the m-th mode 3

34 Fig. 5: Frequency pectrum for the five firt mode in an undamped and a damped oil medium 33

35 Fig. 6: Effect of mode and pile lenderne on tatic oil reitance factor 34

36 Fig. 7: Effect of Poion' ration on tatic oil reitance factor 35

37 Fig. 8: Variation of real and imaginary part of oil reaction with frequency for elected mode; Comparion with Nogami and Novak [4] and plane train model; H d = 50, β =

38 Fig. 9: Variation of dynamic oil reaction R m (ω) and damping with frequency for elected mode; Comparion with Nogami and Novak [4]; H d = 50, β =

39 Fig. 10: Variation of dynamic oil reaction R m (ω) and damping with frequency for elected mode; Comparion with Nogami and Novak [4]; H d = 50, β =

40 Fig. 11: Variation of dynamic oil reaction R m (ω) and damping with frequency for elected mode, pile lenderne and material damping 39

41 Fig. 1: (a), (d) Effect of number in mode on oil reaction in dynamic regime (β = 0); (b), (c) Schematic repreentation of variation of dynamic oil reaction and damping ratio with novel dimenionle frequency parameter 40

42 Fig. 13: Effect of pile lenderne on oil reaction in dynamic regime; β = 0.05, v =

43 Fig. 14: Effect of Poion ratio on dynamic oil reaction and damping; β =

44 Fig. 15: Effect of Poion ratio on dynamic oil reaction R(ω); (a) comparion with plane train model; (b) comparion with implified (improved) expreion; β =

45 Fig. 16: 3D repreentation of dynamic oil reaction factor R a (ω) with frequency and (αd); β = 0.01, ν = 0.4 Fig. 17: 3D repreentation of damping ratio β a with frequency and (αd); β = 0.01, ν =