Insight on kinematic bending of flexible piles in layered soil

Soil Dynamics and Earthquake Engineering 43 (2012) 309–322

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Insight on kinematic bending of flexible piles in layered soil Raffaele Di Laora a, Alessandro Mandolini a, George Mylonakis b,n a b

Department of Civil Engineering, Second University of Napoli, Aversa, Italy Department of Civil Engineering, University of Patras, Rio, Patras 26500, Greece

a r t i c l e i n f o

abstract

Article history: Received 10 May 2012 Received in revised form 27 June 2012 Accepted 28 June 2012 Available online 29 August 2012

The behaviour of a kinematically stressed pile in layered soil under the passage of verticallypropagating seismic S waves is investigated by means of rigorous three-dimensional Finite-Element (FE) analyses. Both pile and soil are idealized as linearly viscoelastic materials, modelled by solid elements and pertinent interpolation functions in the realm of classical elastodynamic theory. The system is analyzed by a time-Fourier approach in conjunction with a modal expansion in space. Constant viscous damping is considered for each natural mode, and a FFT algorithm is employed to switch from frequency to time domain and vice versa in natural or generalized coordinates. The scope of the paper is to: (a) provide some rigorous elastodynamic results in both frequency and time domains that can be used as reference cases; (b) elucidate the role of a number of key phenomena and salient dimensionless parameters for the amplitude of pile bending at an interface separating two soil layers of different stiffness; (c) propose a simplified semi-analytical formula for evaluating such moments; (d) provide some remarks about the role of kinematic bending in seismic design of pile foundations, with emphasis on the long-standing issue of establishing an optimal pile diameter to resist such bending. The results of the study offer a new interpretation of kinematic pile bending in terms of the interplay between pile and soil, expressed through dimensionless layer thickness, pile-to-soil stiffness ratio and impedance contrast at the layer interface. A case study from Japan is presented. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction Kinematic pile bending is the outcome of a complex interaction between soil and an embedded pile during the passage of seismic waves travelling horizontally or up and down (after reflection) in the ground. Naturally, kinematic bending develops regardless of the presence of a superstructure, in soil that has not experienced a severe loss of strength such as that induced by soil liquefaction. The problem is not tractable by pure equilibrium considerations and, thereby, did not receive proper attention by geotechnical researchers until pertinent analysis methods were developed in the 1980s [1–3]. Observations in the field [4,5], the laboratory [6], and computer simulations [7–10] have demonstrated that kinematic pile bending may be severe near: (1) interfaces separating soil layers of sharply different stiffness; (2) the pile head in presence of a stiff restraining cap. These observations may explain the concentration of damage at depths where bending arising from loads imposed at the pile head are negligible. On the basis of these efforts a number of simple methods have been developed to quantify the phenomenon and provide rational design tools [2,5,11]. Reviews on the subject have been presented, among others, by Novak [12], Pender [13], and Gazetas and Mylonakis [14].

n

Corresponding author. Tel.: þ30 2610 996542; fax: þ30 2610 996576. E-mail address: [email protected] (G. Mylonakis).

0267-7261/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2012.06.020

At present it seems that certain aspects of kinematic pile bending are well understood while others remain unresolved [15]. The object of this work is to: (a) present a set of harmonic steady-state elastodynamic results obtained by a rigorous FE analysis to be used as benchmark cases; (b) by virtue of a wider set of transient elastodynamic results using real earthquake motions, provide insight into a number of key phenomena associated with pile bending at an interface separating two soil layers; (c) propose a new semi-analytical formula for evaluating such moments; (d) discuss certain practical aspects on seismic design of pile foundations, notably the long-standing issue of establishing an optimal pile diameter to resist kinematic moments; (e) present a case study from Japan involving a kinematically-stressed pile.

2. Review of simplified design methods Considering the pile as a flexural elastic beam, the bending moment at a given elevation can be determined from the strength-of-materials formula:   M ¼ Ep Ip 1=R p ð1Þ where M¼M(z, t) the time-varying pile bending moment at depth z, Ep is the pile Young’s modulus, Ip the pile crosssectional moment of inertia and (1/R)p the time varying pile

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curvature. From Eq. (1), it follows that the evaluation of bending moment along the pile relies on the proper estimation of pile curvature. The simplest approach for assessing the above parameter is to neglect pile-soil interaction and, thereby, assume that (1/R)p equals soil curvature (1/R)s in free-field conditions. This assumption was adopted in the pioneering studies of Margason [16] and Margason and Holloway [17], and was later employed in the NEHRP provisions [18]. Under the convenient hypothesis of one-dimensional seismic excitation consisting of vertically-propagating shear waves, soil curvature may be determined from the expression: ð1=RÞs ¼

aðz,t Þ V 2s

ð2Þ

which is an alternative form of the one-dimensional wave equation for a homogeneous medium. In this formula a(z,t) is the depth-dependent soil acceleration in the free-field and Vs is the corresponding propagation velocity of shear waves. It can be readily recognized that this method cannot be used in a layered profile, as curvature at interfaces separating soil layers of different stiffness tends to infinity (due to the different shear strains above and below the interface), whereas an unbroken pile will always experience finite curvature. Furthermore, applying formulae (1) and (2) slightly above or below the interface may underestimate or overestimate pile curvature depending on the circumstances. Finally, this approach may violate the boundary conditions of the problem, as it will invariably predict finite bending at the pile head even in the absence of a restraining cap. To overcome these limitations, special techniques have been developed for assessing kinematic bending moments in layered soils. Some of these methods are outlined in the ensuing. More comprehensive reviews are provided in Refs. [14,15,19,20,29]. A simple mechanistic model for kinematic pile-soil interaction in two-layer soil was proposed by Dobry and O’Rourke [2], under the assumptions of infinitely thick soil layers and a uniform static shear stress field in the soil mass. In the realm of this model, Dobry and O’Rourke derived an explicit solution for the pile bending moment at the interface:  3=4 M ¼ 1:86 Ep Ip ðG1 Þ1=4 g1  F ð3Þ where G1 is the soil shear modulus in layer 1, g1 the (uniform) soil shear strain in layer 1; F ¼F(G2/G1) is the dimensionless function:  1=4   G2 F ¼ c3 ðc1Þ c2 c þ 1 , with c ¼ ð4a; bÞ G1 The Winkler spring moduli (i.e., the ratios between soil reactions and pile displacements at the same depth) used by Dobry and O’Rourke are k1 ¼3 G1 and k2 ¼3 G2, following early recommendations by Novak et al. [21] and Roesset [22]. Based on a parametric study involving a number of pile-soil configurations analysed using a Beam-on-Dynamic-WinklerFoundation (BDWF) model, Nikolaou et al. [5] suggested a simple approximate formula for estimating the maximum harmonic steady-state bending moment at the interface between two consecutive soil layers under resonant conditions. The expression is based on a ‘‘characteristic’’ shear stress tc, which is proportional to the actual shear stress that is likely to develop at the interface, as function of maximum free-field surface acceleration, as:

tc  as r1 h1 The regression formula proposed in that study is:  0:3  0:65   Ep V s2 0:5 3 L M max ðoÞ ¼ 0:042tc d d E1 V s1

ð5Þ

ð6Þ

where L/d denotes pile slenderness, Ep/E1 pile-soil stiffness ratio, and Vs2/Vs1 the ratio between shear wave propagation velocities in the two soil layers. The above equation indicates that bending moment tends to increase with increasing pile diameter, pile-soil stiffness contrast and layer stiffness contrast. A weakness of Eq. (6) is that it predicts infinite bending moment both for very slender piles (L/d-N) and soils with high layer stiffness contrast (Vs2/Vs1-N), a behavior which, obviously, is not physically realizable. Under transient seismic excitation, the above authors showed that these trends are still valid, but with the peak values of transient bending moments smaller than steady-state amplitudes at resonance by a factor ranging between 3 and 5. An enhanced mechanistic model has been proposed by Mylonakis [11] to overcome some of the limitations of the Dobry and O’Rourke formulation. The model offers the following improvements: (1) both layers are assumed to be thick (though not to an infinite extent); (2) the seismic excitation is truly dynamic, imposed at the base of the soil in the form a harmonic displacement of frequency o; (3) both radiation and material damping in the soil are explicitly accounted for, by using a bed of distributed dashpots attached in parallel to that of the springs; (4) the geometric and inertial properties of the profile are incorporated through the properties of the seismic shear waves propagating vertically in the soil. To quantify the interaction phenomenon, a ‘‘strain transmissibility’’ function was introduced in [11], relating peak pile bending strain and soil shear strain at the interface, (ep/g1). In the static limit (o-0) this function is given by: ("     # )    1=4 1  ep 1 4  2 k1 h1 h1 1 cðc1Þ1 ¼ c c c þ1 3 g1 st 2 Ep d d ð7Þ where k1 represents the Winkler modulus of the first layer and is primarily related to its Young’s modulus E1 [22]. The effect of frequency on strain transmissibility was taken into account through a dimensionless dynamic amplification coefficient F. According to available numerical solutions [11,15,19], this coefficient varies between 1 and 1.5, with an average value of about 1.25. Based on a parametric analysis performed using finite-element code VERSAT-P3D [23], Maiorano et al. [19] proposed two modified procedures for evaluating the maximum kinematic pile bending moment at a layer interface, based on the methods proposed in Refs [5] and [11]. Other contributions [15,19,24–26] reported results from parametric investigations conducted using both BDWF and FE analyses, along with simple procedures and design-oriented expressions for the kinematic moments at the pile head and the interface. Details are provided in the original papers. Notwithstanding the usefulness and practical appeal of the above studies, they all seem to suffer from some noteworthy limitations. First and foremost certain fundamental aspects of kinematic bending such as the distinction between negative and positive bending (defined by points of contra-flexure along the pile) have not been explored, nor has been the interplay of key parameters such as pile-soil stiffness ratio and interface depth. Moreover, only a small number of pile-soil configurations has been examined, corresponding to a limited range of parameter values—mainly relatively stiff piles (Ep/Es Z500) and deep interfaces (h1/d Z10). The physics of the response outside this range is uncertain. Last but not least, the proposed regression formulae have been calibrated using results from approximate Winkler solutions and pseudo-3D FE analyses. In the ensuing, results from a comprehensive numerical study performed by means of a rigorous 3D dynamic FE scheme are presented and discussed, allowing new interpretations of the fundamental mechanism controlling kinematic pile bending in

R. Di Laora et al. / Soil Dynamics and Earthquake Engineering 43 (2012) 309–322

layered soil. In addition, simple formulae are derived to be used in applications.

3. Numerical analyses The problem considered in this work is depicted in Fig. 1. A vertical solid elastic cylindrical fixed-head pile embedded in a sub-soil made of two layers resting on rigid rock. Seismic excitation in the form of vertically-propagating S waves is imposed at the base of the profile. To explore the physics of kinematic interaction, detailed FE analyses were conducted by the authors using the generalpurpose computer platform ANSYS [27], under the assumption of linear viscoelastic behaviour for the soil and pile materials and perfect bonding at the interfaces. The dimensions of the FE model were selected to ensure that free-field response at the boundaries of the mesh is not affected by outward-propagating waves generated at the pile-soil interface (radiation condition). Hence, vertical displacements were restricted at the lateral boundaries to simulate 1-D conditions for vertical S waves. The nodes at the base of the model were fully restrained to represent the rigid rock. The lateral boundaries of the mesh were located 80 pile diameters away from the pile axis in the direction parallel to that of shaking, and 55 diameters in the orthogonal direction.

311

Particular attention was paid in refining the mesh close to discontinuities. To this end, a sensitivity analysis of kinematic pile bending was carried out as function of height and type of finite elements close to the interface separating the two soil layers. Computed bending moments were found to increase with decreasing element size and with increasing accuracy of polynomial interpolation within the elements. Satisfactory accuracy was achieved (results not shown due to space limitations) for both 20-noded and 8-noded elements of vertical size of 0.1 pile diameters. Hence, solid 8-noded linear isoparametric elements were selected to represent the soil and the pile, as they are more economical over the 20-noded option. To accurately determine maximum moments, the mesh was refined 1.5 diameters above and below the interface. 10620 nodes and 8642 elements per mesh were employed. The analysis was carried out in the frequency domain, by superimposing the contributions of a sufficient number of vibrational modes of the pile-soil system (covering frequencies up to 15 times the fundamental one), each having a constant level of viscous damping set to 10%. In this way the drawbacks stemming from use of common energy loss formulations such as Rayleigh damping were avoided. An FFT algorithm was employed to transfer responses from the frequency to the time domain and vice versa. Six different actual recorded earthquake motions, selected from the Italian strong motion Database [28], scaled to 1 m/s2 PGA, were employed as bedrock motions to cover a wide range of excitation characteristics. The main features of these signals are summarized in Table 1. Corresponding time histories and Fourier spectra are plotted in Fig. 2. The response of the model in the freefield compares favourably to results obtained by program EERA [29]. As an example, Fig. 3 shows a comparison in terms of maximum acceleration with depth. Further details are provided in Ref. [30].

4. Parametric studies Response to harmonic earthquake excitation of the pile in Fig. 1 is governed by 16 independent variables (i.e., h1, H, L, d, Ep, E1, E2, rp, r1, r2, b1, b2, np, n1, n2, o) which describe the geometric, material and excitation properties of the problem. As the rank of the dimensional matrix is 3, dimensional analysis [31] suggests that 13 ( ¼16–3) dimensionless ratios suffice to fully describe the response. It should be noticed that because of linearity, rock acceleration ar is a trivial independent variable, as it simply scales pile bending in a proportional manner and, thereby, can be used to normalize the response. Of the aforementioned thirteen ratios, six are known to have first-order influence on the solution, i.e., Ep/E1, E2/E1, h1/d, L/d, H/d, od/Vs1. The appearance of shear wave velocity Vs1 in the last ratio should not come as a surprise, as it is merely a function of E1, n1 and r1. [For the same reason the stiffness contrast between the two layers, E2/E1, can be expressed in the alternative forms G2/G1 and

Fig. 1. Problem under investigation.

Table 1 Earthquake records used as bedrock input motions. Station name

Event

Mw

Distance [km]

Comp.

PGA [g]

PGV [cm/s]

Arias intensity [m/s]

Site class

Vs,30 [m/s]

Tolmezzo–Diga Ambiesta Sturno Borgo–Cerreto Torre San Rocco Tarcento Nocera Umbra–Biscontini

Friuli 06/05/1976 Irpinia 23/11/1980 Umbria-Marche (AS) 12/10/1997 Friuli (AS) 11/09/1976 Friuli (AS) 11/09/1976 Umbria-Marche (AS) 03/10/1997

6.4 6.5 5.1 5.8 5.5 5.0

20 30 10 24 8 7

NS WE WE NS NS NS

0.357 0.321 0.162 0.090 0.211 0.186

22.84 71.96 4.78 3.47 8.12 3.89

0.787 1.393 0.065 0.032 0.108 0.158

A A A B A B

1092 1134 1000 600 843 442

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Fig. 2. Time histories and corresponding Fourier spectra employed in the transient analyses.

Table 2 Parameter cases for analysis of maximum transient kinematic bending strains using the selected ground motions. In all cases, b1 ¼ b2 ¼ 0.1, np ¼ 0.2, n1 ¼ n2 ¼ 0.3, r2/r1 ¼ 1.125 when Vs2/Vs1 ¼1.5 or 2, and 1.25 otherwise.

Fig. 3. Comparison in terms of maximum acceleration with depth between FE solution and results by EERA.

Vs2/Vs1 which will be employed for static and dynamic conditions, respectively.] The remaining dimensionless quantities (i.e., rp/r1, r2/r1, b1, b2, np, n1, n2) have generally second-order influence on pile-soil interaction (although damping may have first-order influence on site response) and, thereby, were kept constant in the analyses. Whereas in harmonic analysis fixing the values of the 13 dimensionless parameters is sufficient for producing perfectly similar systems, in the realm of transient analyses additional considerations are needed, as a real earthquake excitation contains a variety of dimensional Fourier components, oi, leading to a large number of dimensionless excitation frequencies (oid/Vs).

Study

H/d

L/d

h1/d

Ep/E1

Vs2/Vs1

1

30

24

2 4 8 16

300 1000 10000

1.5 2 3 6

2

30

24 40

4 8 16

150 667 1500

1.5 2 3

To address this issue, two parametric studies were conducted, as summarized in Table 2. The first study utilizes the fixed physical parameters H¼30 m, d¼1 m, L¼24 m and E1 ¼50 MPa. A total of 48 different combinations involving dimensionless ratios Ep/E1, Vs2/Vs1 and h1/d ranging, respectively, from 300 to 10,000, 1.5 to 6, and 2 to 16, were employed. The second study considers H¼ 30 m, d¼ 0.5 m, Ep ¼30 GPa, with the dimensionless ratios Ep/E1, Vs2/Vs1, h1/d and L/d varying from 150 to 1500, 1.5 to 3, 4 to 16 and 24 to 40, respectively—a total of 54 cases. Note that the first study employs constant E1 and variable Ep values, whereas the second employs constant Ep and variable E1. As will be shown below, this difference has significant implications in the interpretation of transient response. For the six earthquakes listed in Table 1, a suite of 6  (48þ54)¼ 612 different configurations were analyzed. In all cases, r1 ¼1.6 Mg/m3, b1 ¼ b2 ¼0.1, np ¼0.2, n1 ¼ n2 ¼0.3. Density

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313

contrast r2/r1 was set equal to 1.125 when Vs2/Vs1 ¼1.5 or 2, and 1.25 otherwise.

at the interface. The overall bending moment at the interface can thus be viewed as the superposition of two counteracting moments:

5. Results

(1) a negative moment that the pile would experience in a (hypothetical) homogeneous soil having stiffness equal to that of the first layer; (2) a positive moment due to the restraining action provided by the increased soil stiffness below the interface.

5.1. Static behaviour: Mechanism for development of bending at the interface The role of layer stiffness contrast in the development of kinematic bending in static conditions (o-0) is explored in Fig. 4, under the assumption of constant base acceleration (ar ¼1 m/s2). Evidently, in homogeneous soil (G2/G1 ¼1), a fixed-head pile follows almost perfectly the movement of soil at nearly all elevations. Accordingly, soil and pile curvatures are equal everywhere except near the moment-free pile tip where the former curvature is finite and the latter is zero. On the other hand, a free-head pile experiences (by definition) zero moment at the top and, thereby, pile curvature deviates from soil curvature within an active pile length from the surface. Note that according to the notation employed in this paper, pile curvature and associated bending moment are negative in the homogeneous profile under positive ground excitation. A different behaviour is observed in two-layer soil (G2/G1 41): with increasing stiffness contrast between the layers, the progressively stronger restraint provided by the stiffer layer tends to act as a fixity (clamped boundary condition), which attracts a significant amount of bending moment of opposite sign to reduce pile rotation

For relatively low stiffness contrasts (G2/G1 ¼1.5), this restraint brings the negative moment close to zero—a beneficial effect. For higher stiffness contrasts (G2/G1 ¼2, 4, 10) the overall bending becomes positive, increasing monotonically with increasing (G2/G1)— obviously a detrimental effect. As a result, two distinct points of contraflexure exist along the pile, separating regions of positive and negative bending (Fig. 4). This behaviour is also recognizable in Fig. 5a and b, in which strain transmissibility (ep/g1) is plotted against layer stiffness contrast G2/G1 for free- and fixed-head conditions, respectively. Also shown in the graph are results from the analytical solution in Ref. [11] (Eq. (7)) using a spring modulus k¼ 6 (Ep/E1)  0.125 E1. (The influence of Winkler modulus on pile bending is further discussed in Ref. [15]). Note that as the Winkler solution is derived for thick soil layers, it is in good agreement with the more rigorous results regardless of fixity conditions at the pile head, only for relatively deep interfaces (i.e., h1/d ¼6, 10). The successful comparisons observed for fixed-head piles for all interface depths (Fig. 5b) deserves special mention.

Fig. 4. Development of kinematic pile bending at a soil layer interface as function of stiffness contrast between the layers for static conditions and unit base acceleration (Ep/E1 ¼ 1000, L/d¼ 20, h1/d¼ 10).

Fig. 5. Static strain transmissibility for against layer stiffness contrast and interface depth for a (a) free-head and (b) fixed-head pile. In all cases, Ep/E1 ¼ 1000.

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Specifically, in homogeneous soil, c ¼1; Eq. (7) predicts:   dg ep  ¼ 2z

ð8Þ

where z ( ¼h1) is depth. Recalling that under static conditions

g/z¼ dg/dz¼ (1/R)s, the above solution can be cast in the equivalent forms:       d 1 d 1 ep  ¼ ¼ 2 R s 2 R p

ð9Þ

which indicate that at deep elevations pile and soil curvatures coincide regardless of boundary conditions at the pile head. The above results may explain why the model in Ref. [11] works better for fixed-head piles in presence of a shallow interface (Fig. 5b). Indeed, the zero-rotation boundary condition at the pile head forces strain transmissibility to coincide with that in Eq. (8) for any fictitious interface depth z¼ h1 when G1 ¼G2. [The slight discrepancy between Eq. (8) and the FE results in Fig. 5b could be attributed to different layer densities considered in the numerical simulations.] On the other hand, a free-head pile in homogeneous soil always experiences different curvature near the surface than the free-field soil—hence the less satisfactory performance of Eq. (8) for shallow interfaces in Fig. 5a.

5.1.1. Zero-bending surface The results in Fig. 5 clearly suggest that there are infinite combinations of G2/G1, h1/d and Ep/E1 ratios that induce zero strain transmissibility at the interface, thus inducing zero bending in that location. Accordingly, a locus of points representative of zero transmissibility may be defined as function of the three aforementioned parameters. This locus may be derived directly from Eq. (7), which, however, would provide reliable results only for deep interfaces. By virtue of the herein reported FE analyses, a more accurate formula for a fixed-head pile was derived by the authors using non-linear regression analysis:    9=32 Ep h1 7 ðc1Þð2=3Þ ¼ d crit 20 E1

ð10Þ

Eq. (10) is depicted in Fig. 6, plotted as a surface in (G2/G1) (Ep/E1) (h1/d) space. All points below the surface (marked in grey colour) are representative of negative transmissibility (ep/g1 o0). The opposite is true for points above the surface. Clearly, for the combination of parameters defined in Eq. (10), the point of contraflexure closer to the surface (see Fig. 4 and insert in Fig. 6) is located at the interface and bending is negative everywhere, except for a small region between the two points of contraflexure. For these conditions, the maximum absolute bending moment always occurs at the pile head.

Fig. 6. Zero-transmissibility surface based on FE analyses. Points below the surface (volume marked in grey) indicate negative transmissibility (ep/g1 o 0); the opposite holds for points above the surface. Dark colour on the surface suggests large interface depths and vice versa.

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It should be noticed that for progressively smaller layer stiffness contrasts, the critical interface depth gradually migrates to deeper elevations; for c¼1 it tends to infinity. This is understood given that in homogeneous soil a pile would always experience negative bending along its length. Zero transmissibility can be exhibited only by an infinitely-long pile, due to the (theoretically) infinite soil shear strain developing at infinite depth. Contours of critical interface depth are represented in the 2-D plot in Fig. 7. An alternative representation is offered in Fig. 8a, where Eq. (10) is plotted against FE results by varying layer stiffness contrast for different values of pile-soil stiffness ratio. Also shown in the figure are results from the Mylonakis’ model (Eq. (7)). Setting (ep/g1)¼0 and solving for critical depth the latter model leads to:      h1 1 c2 c þ 1 Ep 9=32 ¼ d crit 5 cðc1Þ E1

ð11Þ

As evident in Fig. 7, the predictions of the proposed regression formula are in excellent agreement with the FE results. It is noted that (h1)crit is always located within the pile active length, which may be approximated by [8,32,33]:    1=4 L Ep ¼ 1:5 d active E1

ð12Þ

assuming values of about 10–15 diameters for pile-soil stiffness contrasts in the range 103–104.

315

5.1.2. Balanced bending surface In the same spirit as above, an infinite set of combinations of G2/G1, h1/d and Ep/E1 ratios can be defined for which pile bending strains at the pile head and the interface are numerically equal (but of opposite sign). Such a configuration might be desirable as it would lead to a more uniform distribution of bending along the pile. By means of the herein reported FE analyses, the following expression was derived for h1/d, to be referred hereafter to as balanced interface depth:    9=32 Ep h1 9 ðc1Þð3=5Þ ¼ ð13Þ d bal 10 E1 On the other hand, Eqs. (7) and (8) may be combined to retrieve the interface depth corresponding to the balanced condition according to the mechanistic model in Ref. [11]:      h1 1 2c4 2c3 þ 3c2 2c þ 1 Ep 9=32 ¼ ð14Þ d bal 5 E1 c4 2c3 þ2c2 c Eqs. (13) and (14) are plotted in Fig. 8b against FE results. It is noted that the balanced interface depth is small compared to active length for large values of layer stiffness contrast (G2/G1 46C8), and large for smaller values of G2/G1. Evidently, interface depths larger than balanced lead to interface bending stronger than head bending and vice versa. Apart from their intrinsic theoretical interest, the expressions in Eqs. (10) and (14) are of practical significance since for a given geotechnical configuration there always exist two pile diameters which make interface bending, respectively, zero and equal to that at the pile head. The first configuration allows bending moments to vanish at the interface, leading to a distribution of internal forces similar to that arising from inertial actions. The overall bending (kinematic and inertial) is thus negligible at the portion of the pile embedded in the second layer. On the other hand, the balanced configuration represents a design optimum for kinematic bending. Accordingly, knowledge of the balanced interface depth allows control of the ratio of head and interface moments, which is important in practical applications. 5.2. Harmonic response: Reference solutions

Fig. 7. Contours of critical interface depth by varying layer stiffness contrast and pile-soil stiffness ratio.

In Table 3, analysis results for selected cases, expressed in terms of bending strain and strain transmissibility, are provided as reference solutions in the frequency domain. The excitation consists of unit rock acceleration having frequency 0.5, 1 and 2 times the fundamental frequency of the site, o1. Evidently peak bending strains always occur at resonance, although strain transmissibility increases with frequency even beyond o1. This is

Fig. 8. Critical interface depth (a) and balanced interface depth (b) as function of layer stiffness contrast and pile-soil stiffness ratio.

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Table 3 Reference solutions for maximum harmonic steady-state kinematic bending strains and associated strain transmissibility. (Parameters employed correspond to study #1 in Table 2.) h1/d Ep/E1 Vs2/Vs1 x1 x/x1 ¼ 0.5 [Hz]

ep

x/x1 ¼ 1 ep/c1 ep

[  10  4]

x/x1 ¼2 ep/c1 ep

[  10  4]

ep/c1

[  10  4]

8

300

2 3 6 1000 2 3 6

1.79 2.49 3.20 1.79 2.49 3.20

0.832 1.161 1.964 0.456 0.840 1.250

0.088 0.123 0.223 0.048 0.089 0.142

4.108 6.130 8.443 2.306 4.447 5.409

0.093 0.132 0.236 0.052 0.096 0.151

1.227 1.690 1.371 0.754 1.282 1.073

0.118 0.206 0.486 0.072 0.156 0.380

16

300

1.41 1.58 1.68 1.41 1.58 1.68

1.956 2.891 4.076 1.246 1.895 2.722

0.110 8.174 0.168 11.095 0.237 14.307 0.070 5.391 0.110 7.453 0.158 9.764

0.118 0.177 0.253 0.078 0.119 0.173

0.916 0.694 0.498 0.720 0.564 0.425

0.189 0.283 0.402 0.149 0.230 0.343

2 3 6 1000 2 3 6

not in accord with the perception of maximum transmissibility at resonance suggested in earlier investigations [19]. This behaviour appears to be counterintuitive, yet may have a simple mechanistic interpretation: considering, as a first approximation, the second layer as an infinitely stiff medium, it is straightforward to show that maximum shear strain at the interface occurs at resonance [34]. Increasing excitation frequency beyond resonance will lead to progressively smaller strains at the interface, which, for an undamped system, may even reach zero at certain frequencies (e.g., o/o1 ¼2). Under such conditions, transmissibility will be infinite since pile bending will always be finite. Note that the presence of a second layer of finite stiffness and damping will not alter these patterns, as the compliance of the second layer will merely shift the resonant frequencies to lower values, while the presence of damping will ensure that interface strains will not vanish altogether. Hence, transmissibility will reach a maximum at o/o1 ¼2 given the small value of soil shear strain at the interface, whereas pile bending will not decrease by the same amount due to the strong variation of soil strain immediately above the interface. In other words, while in the static case g1 is sufficient for describing free-field response (as shear strain varies proportionally with depth), in dynamic conditions frequency controls the distribution of soil strains within the profile and, hence, magnitude of pile bending for a given value of g1. This suggests that the dependence of transmissibility on frequency might simply be the result of a variation in ground response—not a variation in SSI. 5.3. Transient response The simple assumption that a fixed-head pile follows soil movement in homogeneous soil is reasonable for both harmonic steady-state and transient excitations. Although the ratio of pile and soil curvatures decreases with frequency (given the inability of the pile to follow short wavelengths in the soil), for common pile-soil stiffness ratios it approaches unity [30]. With reference to two-layer soil, results of a transient analysis are shown in Fig. 9. Note that soil and pile curvatures are essentially equal along the pile, except close to the interface where soil curvature is infinite. This reinforces the idea of the interface acting as a ‘‘perturbation agent’’ providing additional bending, whose effects vanish within an active pile length above and below the specific elevation. Thereby, the region of increased bending depends on pile diameter and pile-soil stiffness ratio and is thus larger within the softer layer.

Fig. 9. Comparison of maximum time-domain absolute pile and soil curvature for fixed- and free-head conditions in a two-layer soil (Ep/E1 ¼ 300; Vs2/Vs1 ¼ 3; h1/ d¼ 16).

Mention has already been made to interface bending as a superposition of two counteracting phenomena: site response and pile-soil interaction. The dependence of this interplay on interface depth is depicted in Fig. 10a, b. For low layer impedance contrasts (Vs2/Vs1 ¼ 1.5, Fig. 10a), an increase in h1/d from 2 to 4 may result to a decrease in pile bending, despite a possible increase in site response. This is consistent with Fig. 5, as the (negative) transmissibility decreases in absolute value, approaching zero with increasing h1/d. Beyond a certain value, a further increase in h1/d will tend to increase transmissibility (and hence pile bending) because of the continuous increase in the positive bending contribution provided by the interface effect. Particularly interesting are the results depicted in Fig. 10b, referring to a higher impedance contrast (Vs2/Vs1 ¼3): the sharper stiffness discontinuity forces bending strain to increase even at small interface depths. This is anticipated since, with increasing impedance contrast, transmissibility shifts to positive values even for very small interface depths. Furthermore, moving h1/d from 8 to 16, bending increases or decreases as function of frequency content of the excitation signal. Despite the increase in dynamic transmissibility, ground response may decrease for high-frequency signals because of the decrease in the fundamental frequency of the site (due to the deeper soft layer). Note the difference with static response: for a given base acceleration, (i) static free-field shear strain at z ¼h1 would have doubled (as the interface would be twice as deep) and (ii) static transmissibility would have increased leading to a more-than-double bending value at the interface. Similar interpretations can be made for the effect of impedance contrast between the soil layers, shown in Fig. 10c, d: if the interface is sufficiently deep (Fig. 10c, h1/Lactive ¼0.64), an increase in impedance contrast through an increase in Vs2 leads to an increase in pile bending. This holds for modest variations in ground response, as the change in soil stiffness may affect g1. On the other hand, if the interface is sufficiently shallow relative to active pile length (Fig. 10d, h1/Lactive ¼0.27) so that the transmissibility is negative (recall observations in Fig. 4), an increase in impedance contrast will generally lead in lower pile bending, as g1 is not expected to change significantly. Note that if for a given Vs2/Vs1 ratio transmissibility is positive, increasing stiffness contrast may significantly increase pile bending even for a shallow interface (e.g., Fig. 10d, data points for Sturno time history). With reference to Fig. 10e, f, an increase in Ep/E1 is generally accompanied by an increase in pile bending moment.

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317

Fig. 10. Maximum pile bending strain at layer interface for a fixed-head pile as function of interface depth (a), (b), layer impedance contrast (c), (d) and pile-soil stiffness ratio (e), (f) for different rock motions (a), (b): Ep/E1 ¼ 1000; (c), (d): h1/d ¼ 4; (e), (f): h1/d¼ 16, Vs2/Vs1 ¼ 3.

Nevertheless this would always be valid only in the static case. In the dynamic regime the trend would hold when pile Young’s modulus increases under constant soil stiffness (and, hence, constant ground response). Indeed, in the latter case bending strain may decrease with increasing pile-soil stiffness ratio (Fig. 10e). Yet the trend is less than linear, so bending moment will actually increase as it is proportional to pile Young’s modulus. On the other hand, if soil stiffness decreases under constant pile stiffness, ground response will be modified depending on the frequency characteristics of the input motion and, thereby, maximum soil shear strain g1 may increase or decrease, as evident in Fig. 10f. To the best of the Authors’ knowledge, the sensitivity of the analysis to the way Ep/E1 ratio is established has not been recognized in the past. Finally, slenderness L/d has apparently little influence on pile bending (not shown) if the portion of the pile penetrating the second layer is sufficiently long. This condition is usually fulfilled in practice, as 3–4 pile diameters generally suffice to fully mobilize pile bending at the interface. 5.4. Regression formulae A rational regression formula for interface bending must account for both the aforementioned negative and positive contributions discussed above, especially in presence of small to moderate stiffness contrasts between the soil layers. A candidate formula is Eq. (7) which encompasses both effects. A possible

drawback of that expression lies in its difficulty to separate the contributions of the negative and positive bending mechanisms. To overcome this limitation, FE data in terms of absolute pile bending strain, which encompasses site response, is presented in Fig. 11a corresponding to uniform peak acceleration for all signals with reference to a deep interface. Evidently, the correlation is quite satisfactory, especially at large strain amplitudes resulting from resonance and/or strong stiffness contrasts. A simple regression formula was developed by the authors, which can be cast in the simple form: "

ep ¼ w g1

#  1  0:25 1 h1 Ep 0:5  þ ðc1Þ 2 d E1

ð15Þ

where the first term inside the brackets pertains to site response and the second to the stiffening effect due to the presence of the second layer (interface action); w is a regression coefficient. Note that for G2/G1 ¼ c¼1 the latter term disappears and only the negative contribution, due to the first layer, remains. Conversely, for (h1/d)-N the first term vanishes and only the positive contribution remains. The associated coefficient w was found to be close to unity (about 0.93) and, thereby, is of minor importance from a practical viewpoint. Comparing transmissibility derived from FE results and Eq. (7), an amplification factor F of about 1.25 was derived (not shown here in the interest of space). This value is consistent with the recommendations by Mylonakis [11] as explained in Refs. [15,19].

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Fig. 11. Comparison between absolute pile bending strain from FE results and a regression formula for (a) deep and (b) shallow interfaces. All input motions are scaled to PGA¼ 1 m/s2.

According to the aforementioned observations about harmonic response, this amplification factor is associated primarily with soil response rather than pile-soil interaction. Applying Eq. (15) to shallow interfaces for fixed-head piles (Fig. 11b), using w ¼1 seems to provide a less satisfactory performance, due to the interplay between head and interface moment which becomes significant as their distance gets shorter. Eq. (15) performs better for high bending strains and worse for lower ones. Nevertheless, the latter represent conditions close to the zero-bending surface and, thereby, are of minor engineering interest. 6. Discussion Practical estimates of peak kinematic bending moments can be established on the basis of alternative formulae, some of which are discussed below. Following Sica et al. [15] and Maiorano et al. [19], a rational approach can be formulated based on the maximum transient shear strain at the interface, g1,dyn. Knowledge of this parameter requires performing a site response analysis which can be carried out using standard computer software (e.g., EERA [18]). The bending moment can then be computed from the expression:   2Ep Ip ep M max ¼ g F ð16Þ d g1 st 1,dyn 2 where factor F2 is analogous to the frequency-domain factor F introduced in Ref. [11]. Corresponding values are provided in Fig. 12a, plotted against the ratio between the mean excitation frequency [35] and the fundamental natural frequency of the site. All points in the graph correspond to deep interfaces, whereas the value of static transmissibility (ep/g1)st is obtained from Eq. (15) using the value w ¼0.74 ( ¼ 0.93/1.25), 0.93 being the value of dynamic coefficient w and 1.25 the average value of F according to the present work. Evidently, F2 typically ranges between 1 to 1.5 and exhibits a weak dependence on frequency, as observed in Ref. [15]. Alternatively, peak transient bending moment may be estimated on the basis of surface acceleration, as, thereby avoiding a site response analysis. In this case:   2Ep Ip ep as r1 h1 M max ¼ F1,s ð17Þ d g1 st G1 Factor F1,s generally attains values below unity (Fig. 12b). The term (as r h1/G1) on the right side defines an interface shear strain evaluated for a stress reduction factor rd ¼1 [18], thereby being an upper bound to the true shear strain in that elevation.

[Data plotting above 1 in the graph can be attributed to scattering associated with the use of the regression formula in Eq. (15).] Considerable scattering is observed, which can be attributed to dynamic site effects. On the basis of these results a conservative selection for F1,s would be 1. As a third alternative, Mmax can be correlated to rock acceleration, ar, through the analogous expression:   2Ep Ip ep ar r1 h1 F1,r ð18Þ Mmax ¼ d g1 st G1 where F1,r is an pertinent factor (Fig. 12c). Compared to the previous parameter, F1,r exhibits a stronger dependence on frequency which can be attributed to site response. Naturally, the ratio between F1ir and F1,s can be interpreted as a ‘‘site factor’’ in the context of seismic Codes for perfectly rigid bedrock conditions. Fig. 12d, e, f present corresponding results based on the predominant frequency fp obtained from the peak value of the 5%-damped acceleration response spectrum. The observed trends are analogous to those based on the mean frequency fm. Accordingly, knowledge of an ‘‘exact’’ excitation frequency is deemed inessential for interpreting the results. 6.1. Importance of pile diameter A number of investigators have argued over the dependence of seismic performance of kinematically-loaded piles on their diameter [11,16,26,36]. A common perception is that slender piles are preferable over large diameter piles, as they can comply easier to seismically-induced soil deformations. Eq. (15) helps shedding light on this issue. Bending strain at the interface is controlled by a negative contribution which is proportional to pile diameter, and a positive contribution which is independent of it. It follows that for deep interfaces and sharp stiffness contrasts between the layers, an increase or decrease in pile diameter has no effect on induced bending strain [11]. An alternative way of expressing this is that for high values of G2/G1 and h1/d, capacity and demand bending moments at the interface are both proportional to the third power of pile diameter (d3), therefore a change in d does not lead to an increase or decrease in safety. On the other hand, for small stiffness contrasts and shallow interfaces (for instance near the pile head), bending strain is nearly proportional to pile diameter. This, however, does not suggest that large-diameter piles always exhibit a inferior performance over slender ones. Indeed, depending on the characteristics of the problem, use of large-diameter piles may shift the bending profile towards a balanced configuration, thereby reducing kinematic moments (Fig. 8b). Based on the results of this

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319

Fig. 12. Dynamic factors against dimensionless frequencies fm/f1 (a), (b), (c) and fp/f1 (d), (e), (f). Results are relative to cases corresponding to deep interfaces.

study, the effect of pile diameter on interface moments may be beneficial or detrimental depending on the circumstances. The importance of pile diameter on kinematic moments at the pile head has been discussed in a companion article [26].

7. Case study The pile under consideration is part of the foundation of a three-storey building in Japan, which experienced the M8 2003 Tokachi-Oki Earthquake and reportedly suffered damage 20 m below the ground surface [37] (although this could not be verified by the authors). The data presented below were taken from Ref. [37] and presentation material from a mini symposium organized as part of the year 2007 4th International Conference on Earthquake Geotechnical Engineering, Thessaloniki, Greece [4ICEGE; M. Maugeri, Personal Communication]. Pile is made of high-strength pre-stressed concrete; has a length of 28.5 m and a hollow-cylindrical cross-section with an external and an internal diameter of 0.4 and 0.3 m, respectively (i.e., wall thickness of 0.05 m). Soil consists of about 20 m of soft clay underlain by 10 m of sand formations, as shown in Fig. 13a. The pile tip rests on a layer of dense gravel at the depth of 30 m. The earthquake took place on 26/09/2003, with an epicentral distance from the site of 42 km, and was recorded locally by a vertical array consisting of two instruments placed on the soil surface and a depth of 153 m. Fig. 13b shows the acceleration spectra of the recorded accelerograms near the building [available at KiK-net

website (http://www.kik.bosai.go.jp)], at the sandstone bedrock and the soil surface. A large amplification of ground motion intensity between rock and surface is observed, with corresponding PGA’s of 0.06 g and 0.4 g, respectively. Also shown in the figure are the predictions of an equivalent-linear site response analysis performed using computer code EERA [29] based on the shear modulus reduction and damping amplification curves by Vucetic and Dobry [38] and Rollins et al. [39], as shown in Fig. 13c. As can be noticed from the graph, computed and recorded motions are in good agreement, except at low periods (due to the spurious filtering of high frequencies in equivalentlinear analyses). Fig. 13b shows that the predominant period of the rock motion is about 0.9 s, while the fundamental period of the ground is approximately 2 s. Additional analytical results are shown in Fig. 13d, plotted in terms of peak shear strain and shear modulus reduction with depth.

7.1. Simple computation of kinematic moments The pile reportedly experienced damage at a depth of 20 m from ground surface, where a large discontinuity in soil stiffness takes place. Peak soil shear strain in that location was calculated at about 1.7  10  2. Assuming, as a first approximation, that Ep ¼50 GPa (a low-strain estimate for pile Young’s modulus), Poisson coefficients equal to 0.35 and 0.45 for the sandy silt and the clay, respectively, and G1 ¼ 2800 kPa, G2 ¼34500 kPa (as derived from the

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Fig. 13. Case study under investigation. (a) Idealised soil shear wave velocity and mass density profiles; (b) acceleration response spectra of recorded and computed ground motions; (c) soil shear modulus reduction and damping ratio against shear strain (d) computed peak shear strain and shear modulus reduction at the last iteration (Data according to [37].)

analysis), Eq. (15) may be applied with w ¼1 to give:

ep ¼ 1  0:017  ½1=ð2  50Þ þ ð5  107 =8120Þ0:25 ðð34500=2800Þ0:25 1Þ0:5  ¼ 1:63  103 Thus pile curvature is:   1 1:63  103 ¼ 8:2  103 m1 ¼ R p 0:2

ð19Þ

ð20Þ

In light of the nonlinear moment-curvature behavior of the pile cross section depicted in Fig. 14 (which was established for zero axial load given the depth of the interface at hand), the pile curvature estimated by Eq. (20) lies beyond the linear region. A simple iterative procedure was employed to adjust the pile Young’s modulus on a curvature-compatible basis until the procedure converges. After 3 iterations the following estimates were obtained: Ep ¼17.2 GPa, ep ¼2.18  10  3, (1/R)p ¼ 10.9  10  3 m  1, M¼163 kN m. The procedure recommended by Mylonakis [11], using a dynamic amplification factor ^ ¼1.25 yields at last iteration: 2

ep ¼

Fig. 14. Idealised relationship between bending moment and curvature of the pile cross section for zero axial load [37].

1:25  0:0171  ð1:87 :87 þ 1Þ 2  1:874  ð20 = 0:4Þ f½3  ð2:3  8120 = 1:97  107 Þ0:25  ð20 = 0:4Þ 1:87  ð1:871Þ1g ¼ 1:85  10

3

corresponding to a pile bending moment of 156 kN m.

ð21Þ

To apply the Nikolaou et al. [5] procedure in Eq. (6), an estimation of the characteristic shear stress tc is required. Considering, in accordance with the previous methods, the calculated value of peak ground acceleration at the surface (0.27 g), one

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obtains:

tc ¼ 0:27  9:81  1:5  20 ¼ 80 kPa

ð22Þ

Employing a factor Z ¼0.25, corresponding to non-resonant conditions (earthquake predominant period of 0.9 s; soil period of 2 s) for a large number of cycles, peak transient bending moment is estimated at: M ¼ 0:25  0:042  94  0:43  ð28:5=0:4Þ0:3 ð5  107 =8120Þ0:65  ð138=43Þ0:5 ¼ 99:7 kN m ð23Þ The corresponding curvature is 2.32  10  3 m  1 and the bending strain at the outer fiber is 4.64  10  4. Note that no iterations were necessary in the latter calculations, as pile curvature lies within the linear part of the diagram in Fig. 14. The low estimates obtained by this method are probably related to the large interface depth (h1/d¼ 50), which lies outside its range of calibration [5]. The values of peak bending moment predicted by Eqs. (15) and (7) are 78 to 82% of the ultimate moment of 200 kNm (Fig. 14), thus are viewed as satisfactory considering the uncertainties of the problem and the various approximations of the analysis.

8. Conclusions Results from a comprehensive parametric study on kinematically stressed piles in two-layer soil were presented and discussed, highlighting a number of phenomena useful for shedding light into the complex physics of pile-soil interaction. The main conclusions of the study are: 1. Kinematic bending in the vicinity of a layer interface may be viewed as the superposition of two components: a negative contribution imposed by soil curvature in the first (soft) layer, and a positive contribution provided by the restraint effect of the second (stiff) layer. In this context, the interface can be viewed as a bending perturbation agent whose influence vanishes beyond an active pile length above and below the specific elevation. 2. A zero transmissibility (and thus zero moment) condition at the interface can be defined according to Eq. (10) for infinite combinations of interface depth (h1/d), pile-soil stiffness ratio (Ep/E1) and layer stiffness contrast (G2/G1). Likewise, a balanced bending moment condition at the interface can be defined according to Eq. (13) for infinite combinations of the above parameters to ensure equal (absolute) bending at the pile head and the interface. Accordingly, for a given soil profile there always exist two specific pile diameters, which allow control of curvature distribution along the pile. 3. Unlike absolute bending strain which becomes maximum at resonance, strain transmissibility (ep/g1) increases with excitation frequency even beyond the fundamental frequency of the site. This phenomenon has not been identified in the past and reflects the dependence of ground response on frequency, not pile-soil interaction. 4. Small interface depths and stiffness contrasts tend to reduce, in absolute terms, interface pile bending as compared to bending in purely homogeneous soil having stiffness equal to that of the first layer. Conversely, deep interfaces and sharp stiffness contrasts lead to higher bending than in homogeneous soil. This trend holds for both steady-state and transient excitations. 5. Increasing pile-soil stiffness ratio generally amplifies pile bending moments; this is always true for static conditions. In transient dynamic conditions, the trend is also valid if Ep/E1 increases by increasing Ep under constant Es. Contrarily, if Ep/E1

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increases by decreasing Es under constant Ep the change in dynamic soil response may induce higher or lower bending depending on the circumstances. This suggests that two pilesoil configurations having the same governing dimensionless ratios (Ep/E1, h1/d, Vs2/Vs1) may respond very differently to an earthquake motion. 6. For shallow interfaces and low layer stiffness contrasts pile bending strain is negative and nearly proportional to pile diameter. On the other hand, for deep interfaces and sharp layer stiffness contrasts pile bending strain is positive and essentially independent of pile diameter (Eq. (15)). In other words, for high G2/G1 and h1/d both capacity and demand moments at the interface are, by first-order analysis, proportional to the third power of pile diameter (d3), thus a change in d does not increase or decrease safety. Accordingly, no general trend between absolute pile bending and pile diameter can be established and, hence, an increase or decrease in pile diameter may have a beneficial or detrimental effect in seismic performance depending on the circumstances. This is not true for kinematic moments at the pile head where demand increases with the fourth power of pile diameter (d4), thereby an increase in d decreases safety. This behaviour is examined in a companion paper [26]. 7. Bending estimates provided by the regression formula in Eq. (15) and the analytical solution of in Ref. [11] are compatible with observations from a case study involving a pile reportedly damaged during the 2003 Tokachi-Oki event. Both methods provide values which are close to the ultimate moment sustained by the cross section of the pile. On the other hand, the method in Ref. [5] provided only fair predictions of seismic demand on the pile, as some parameters associated with the specific problem lie outside its range of calibration. As a final remark it is fair to mention that the conclusions of the study are valid under the hypothesis of linear or equivalentlinear viscoelastic behaviour for all materials and perfect bonding at the interfaces. Accordingly, some of the findings may require revision in presence of strong non-linearities in the pile and the soil induced by high-amplitude earthquake shaking.

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