Influences of type of wave and angle of incidence on seismic bending moments in pile foundations

Influence of type of wave and angle of incidence on seismic bending moments in pile foundations ∗ J.M. Zarzalejos, J.J. Azn´arez, L.A. Padr´on and O.Maeso Instituto Universitario de Sistemas Inteligentes y Aplicaciones Num´ericas en Ingenier´ıa (SIANI) Universidad de Las Palmas de Gran Canaria Edificio Central del Parque Cient´ıfico y Tecnol´ogico Campus Universitario de Tafira, 35017, Las Palmas de Gran Canaria, Spain {jmzarzalejos,jjaznarez,lpadron,omaeso}@siani.es 2014 Abstract When analysing the seismic response of pile groups, a vertically–incident wavefield is usually employed even though it does not necessarily correspond to the worst case scenario. This work aims to study the influence of both the type of seismic body wave and its angle of incidence on the dynamic response of pile foundations. To this end, the formulation of SV, SH and P obliquely–incident waves is presented and implemented in a frequency–domain boundary element–finite element code for the dynamic analysis of pile foundations and piled structures. Results are presented in terms of bending moments at cap level of single piles and 3 ×3 pile groups, both in frequency and in time domain. It is found that, in general, the vertical incidence is not the most unfavorable situation. In particular, obliquely–incident SV waves with angles of incidence smaller than the critical one, situation in which the mechanism of propagation of the waves in the soil changes and surface waves appear, yield bending moments much larger than those obtained for vertically–incident wavefields. It is also shown that the influence of pile–to–pile interaction on the kinematic bending moments becomes significant for non–vertical incidence, especially for P and SV waves. 1 Introduction The dynamic response of single piles and pile groups is a topic of great interest that has been extensively studied during the last forty years. However, some aspects of the problem are not yet well understood and hence require more investigation. Among others, the characterisation of the excitation is usually simplified considering only a vertically–incident wavefield. When a stratified medium is considered, and provided that the stiffness of the different layers increases with depth, the refraction processes which take place as the incident waves travel through the layers tend to make them impinge on the ground surface almost vertically. Nevertheless, the seismic actions are, in general, composed of a combination of waves which propagate with an angle of incidence not necessarily vertical [1]. The main aim of this paper is to investigate the influence of the type of incident wave and of its angle of incidence on the dynamic response of pile foundations embedded in a homogeneous halfspace. To this end, bending moments at pile heads will be estimated and compared both in frequency and in time domain. ∗This is the peer reviewed version of the following article: J.M. Zarzalejos, J.J. Azn´arez, L.A. Padr´on and O.Maeso, Influences of type of wave and angle of incidence on seismic bending moments in pile foundations, Earthquake Engineering and Structural Dynamics 43: 41-59, 2014, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/eqe.2330/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. 1 The response of single piles of different properties to obliquely incident body waves was studied for the first time in the pioneer work by Mamoon and Ahmad [2] by means of the kinematic interaction factors of the foundations. This paper was the starting point for the following work by Mamoon and Banerjee [3], where the response of single piles and pile groups to travelling SH waves was investigated. Makris and Badoni [4] also studied this problem for Rayleigh and obliquely–incident SH waves making use of a Winkler–type approach. However, the most comprehensive study can be found in the work by Kaynia and Novak [5], where kinematic interaction factors of different configurations of pile foundations were obtained both for obliquely incident body waves and for Rayleigh waves. In addition, Gazetas et al. [6] presented one of the first approaches to the determination of bending moments at pile heads. This work makes use of a 3–D frequency–domain boundary element–finite element formulation to quantify the influence of type of wave and angle of incidence on bending moments at pile heads. To this end, frequency response functions for the bending moments developed at pile heads of single piles and pile groups are presented for different types of waves and angles of incidence. Special attention is given to the SV case in order to investigate the influence of the arising surface waves. Inertial interaction is also taken into account by the existence of a superstructure. Envelopes of maximum bending moments are also shown for a type 1 elastic response spectrum for ground type C as defined in Part 1 of Eurocode 8 [7]. It is shown that the consideration of non–vertical wavefields and, especially, of obliquely–incident SV waves, yields much larger bending moments than those corresponding to the usually considered vertically–incident S waves. 2 Formulation of incident fields propagating with generic angles of incidence The objective of this section is to present the governing equations for a planar wavefront propagating through an elastic halfspace with a generic angle of incidence. They will be presented in a format suitable to be used inside a classic boundary element code as the incident field. When a wave reaches the free surface of the halfspace, other reflected waves must appear in order for the free–field boundary conditions to hold. The displacements in the ith direction (xi, i= 1,2,3) of any point of the halfspace can be expressed as the superposition of the effects of the incident and reflected waves. Since the problem is harmonic, and dropping the repeated eiωt terms, the displacements can be written as ui= n X j=1 bj iAje−ikj(s(j)·r)(1) being uithe component in the ith direction of the total displacement, nthe total number of waves of the problem (i.e., the incident plus the reflected waves), bj ithe component in the ith direction of the vector containing the direction cosines of the displacements produced by the jth wave, Ajthe amplitude of the jth wave, kjthe wave number of the jth wave, and s(j)·r the dot product of the unit vector defining the direction of propagation of the jth wave (i.e., vector s) and the vector containing the coordinates of the point where the displacements are calculated. Note that while sdenotes the direction of propagation of the wavefront, bexpresses the direction of the particle displacements, and that P and SV waves are associated to in-plane motion in the x2−x3plane while SH waves produce horizontal displacements in the x1direction. Also, the direction of the incident wave is defined by the angle θo, measured in the x2−x3plane with respect to the horizontal axis x2(note that, in most seismological papers, θois defined with respect to the vertical axis x3). As it is known, when an SH wave reaches the free surface, the reflected wave is a single SH wave. Conversely, if the incident wave is a P or an SV one, then after the process of reflection there appears both a P and an SV wave (see figure 1). 2 Free surface Incident wave Reflected waves Figure 1: Incident and reflected plane waves in elastic half-space. Definition of angles of the incident and reflected field. The amplitudes of the reflected waves, shown in table 1, are obtained by assuming unit amplitude incident waves, and applying equilibrium and compatibility conditions at the traction– free surface [8,9]. The angles of these reflected waves are obtained taking into account that the boundary conditions are independent of the x2direction, being θ1(angle of the reflected wave of the same type of the incident one) equal to θoin all cases, θ2=cos−1[κ−1cos(θo)] for a reflected P wave due to an incident SV wave, and θ2=cos[κcos(θo)] for a reflected SV wave due to an incident P wave, with κ=cs/cp<1, cs=pµ/ρ and cp=pλ+ 2µ/ρ the velocities of S and P waves, µand λthe Lam´e constants, and ρthe density of the medium. Table 1: Amplitudes of the incident and reflected waves of the problem. Inc. wave Waves A SH SHinc 1 SHref 1 SVinc 1 SV SVref κ2sin (2 θo) sin (2 θ2)−cos2(2 θo) κ2sin (2 θo) sin (2 θ2) + cos2(2 θo) Pref −κsin (4 θo) κ2sin (2 θo) sin (2 θ2) + cos2(2 θo) Pinc 1 P Pref κ2sin (2 θo) sin (2 θ2)−cos2(2 θ2) κ2sin (2 θo) sin (2 θ2) + cos2(2 θ2) SVref 2κsin (2 θo) cos (2 θ2) κ2sin (2 θo) sin (2 θ2) + cos2(2 θ2) In the case of an incident SV wave, the angle of incidence causing the cosine of the reflected P wave to have a unit value is called critical angle, having the expression given by eq. (2), depending only on the Poisson’s ratio of the soil νs. θcr = cos−1cs cp = cos−1s1−2νs 2 (1 −νs)(2) If θo< θcr, the meaning of the reflected P wave has to be reviewed. The direction of propagation of this wave is given by eq. (3). Substitution of eq. (3) into the general expression of the displacements of the reflected P wave given by eq. (4), allows to obtain, after performing 3 some transformations, the alternative expression shown in eq. (5). cos (θ2) = 1 κcos (θo) ; sin (θ2) = −ir−1 + 1 κ2cos2(θo) (3) "u2 u3#P ="cos (θ2) −sin (θ2)#Aref pe−ikp[cos(θ2)x2−sin(θ2)x3](4) "u2 u3#P ="cos (θo)/κ ip−1 + cos2(θo)/κ2#Aref peξ x3e−ikscos(θo)x2(5) with ξbeing a real constant given by ξ= +kpr−1 + 1 κ2cos2(θo) (6) Eq. (5) corresponds to a wave that propagates horizontally, in the x2direction, with a velocity that depends on the angle of the incident SV wave. Such a wave produces horizontal and vertical out–of–phase motions (u2and u3) with an amplitude decreasing with depth, being this the reason why the sign of the square root in eq. (3) must be negative. Note also that, as Ainc pis a complex quantity, this wave is not necessarily in phase with the incident SV wave. Therefore, this reflected wave can be understood as a surface wave, influencing the dynamic response of structures submitted to the incidence of SV waves with angles of incidence below the critical one, as will be shown later. 3 Problem statement 3.1 Problem definition This paper is focused on the analysis of the influence of the angle of incidence and type of wave on the dynamic response of piled structures. The problem is sketched in figure 2. For the sake of simplicity, the superstructure is studied as a single rigid slab supported by massless flexible and inextensible columns, and founded on a square pile group excited by SH, SV or P waves travelling in the x2–x3plane, as shown in figure 2and according to the coordinate system defined in section 2. This model of the superstructure can represent both a single–degree–of–freedom system (like a one–storey shear building) or an equivalent system defining the behaviour of a multimodal structure oscillating according to a specific mode of vibration. Incident wave (SH, SV or P) Horz. motion due to: SV and P waves SH waves Figure 2: Problem definition. 4 The dynamic response of the structure can be defined by its rigid base fundamental period T, the height hof the resultant of the inertia forces for the mode under study, the mass m participating on the mode and the corresponding structural damping ζ. The horizontal stiffness of the structure is k= 4 π2m/T2, with a hysteretic damping given by a complex stiffness of the type K=k(1 + 2 i ζ). The structure is founded on a 3 ×3 square pile group embedded in a viscoelastic halfspace. Pile groups are defined by the length Land diameter dof the piles, the centre–to–centre spacing sbetween adjacent piles, the pile cap mass mo, its moment of inertia with respect to a horizontal axis going through its center of gravity Ioand by a parameter b measuring half of the width of the foundation. If soil–structure interaction effects are taken into account, the behaviour of both the pile cap and the superstructure can be defined by that of a four–degree–of–freedom system. These degrees of freedom represent horizontal, vertical and rocking movements at the rigid pile cap (uc,uc x3and φ, respectively) and inter–storey drift of the superstructure (udef ). Note that, as the columns are inextensible, rocking motion of pile cap and slab are identical. 3.2 Physical properties of the system The geometric and material properties of the piles are: diameter d= 1 m, Young’s modulus Ep= 2.2·1010 N m−2, Poisson’s ratio νp= 0.25 and density ρp= 2500 kg m−3. These values can be considered as typical for reinforced concrete. The soil properties are: Young’s modulus Es= 2.2·108N m−2, Poisson’s ratio νs= 0.4, critical angle 65.9◦and density ρs= 1750 kg m−3. With these properties, a velocity of the S waves in the soil of cs= 210 m s−1is obtained. Structural (ζ) and soil (β) damping ratios are equal and have a value of 0.05. The previous values give rise to the following mechanical and geometric dimensionless relations both for soil and foundation: Young’s modulus ratio Ep/Es= 102, soil–pile densities ratio ρs/ρp= 0.7, and slenderness ratio of the piles L/d = 15. Besides, a ratio between pile separation and diameter s/d = 5 is used when studying the 3 ×3 piled foundation. Two different configurations have been considered for the superstructure, a first one with a period of 0.159 s in which the vibrating mass, of 3.5·105kg, is located at a height of 10 m and the mass and moment of inertia of the foundations are 8.75 ·104kg and 1.75 ·106kg m2, respectively; and a second configuration with a period of 0.317 s, a vibrating mass of 7 ·105kg at a height of 20 m, being the mass of the foundation 1.75 ·105kg and its moment of inertia 1.4·107kg m2. The preceding values lead to aspect ratios h/b = 2 and 4, to a ratio between the stiffnesses of structure and soil h/ (T cs) = 0.3, to a structure–soil mass ratio m/ 4ρsb2h= 0.2 and to a foundation–structure mass ratio mo/m = 0.25. The values chosen for the last three parameters are considered to be representative of typical constructions (see, e.g., [10,11,12]). In addition, the moment of inertia of the foundation is characterised by the 5% of the m h2factor. 4 Methodology 4.1 Numerical tool In this work, a frecuency–domain boundary element–finite element formulation [13] is used to tackle the problem, being the Boundary Element Method (BEM) employed to model the soil as a homogeneous, isotropic, viscoelastic, semi–infinite region and the Finite Element Method (FEM) used to model piles as Euler–Bernoulli beams and superstructures as multi–storey buildings composed of vertical columns and horizontal rigid slabs. The system of equations arising from the application of the BEM to the soil is coupled with that resulting from using the FEM to model the motion of pile groups and superstructures, leading to a single system of equations describing the behaviour of the entire problem. The loads acting in the problem consist of harmonic plane wavefronts propagating through the soil. The validity of the equations of the BEM–FEM model forces these expressions to be 5 written in terms of the field diffracted by the foundation and by the structure. Putting this diffracted field as the difference between the total and the incident fields [14], the equations of the coupled model in terms of the former for every domain Ω (see [13]) can be expressed as: Hss us−Gss ps− np X j=1 Gspjqsj+ np X j=1 Υsj Fpj=Hss us I−Gss ps I(7) being Hss,Gss and Gspjthe influence coefficients, usand psthe displacements and tractions of the total field, npthe number of piles in the domain, qsjthe tractions along the pile–soil interface, Υsj a three–component vector representing the contribution of the axial force Fpjat the tip of the jth load line, and us Iand ps Ithe displacements and tractions of the incident field, which can be found using formulae described in section 2. This way, the right hand side of eq. (7) is known. 4.2 Definition of the input seismic motions The input to the system is given by free–surface horizontal accelerograms generated making use of SIMQKE [15] to be compatible with a specific elastic response spectrum. In this work, the elastic response spectrum is taken from Part 1 of Eurocode 8 [7]. According to the system properties, a type 1 elastic response spectrum for ground type C and 5% damping is used. In order to achieve statistical significance, a set of three non–correlated accelerograms, with base– lines corrected according to [16], is used in this work. The magnitude of the design ground aceleration in terms of reference peak ground acceleration is ag= 0.25 g (with peak ground acceleration for ground type C PGA = 0.288 g) and a duration of t= 30 s. A general incident field as that described in section 2will originate, with the exception of the SH case, not only horizontal but also vertical displacements at the reference point in the free surface. Therefore, the horizontal accelerogram described above will be associated, in general, with a vertical component. In the case of incident SV waves, the vertical peak ground acceleration (PGAv) corresponding to the different angles of incidence is shown in Table 2for ν= 0.4. Note that, for θ= 57.7 ° , the relationships between vertical and horizontal excitations would be in the order of the value recommended in Eurocode 8 [7] when defining a realistic vertical elastic response. On the other hand, for incident P waves, the corresponding vertical peak ground acceleration values would range from PGAv= 0.563 g for θo= 55 ° , to PGAv= 2.015 g for the almost vertical θo= 80 ° . Table 2: Vertical peak ground acceleration (PGAv) corresponding to different angles of SV incident waves. ν= 0.4. Horizontal PGA= 0.288 g. angle ( ° ) 90 80 70 66 θcr = 65.9 65 60 57.5 55 PGAv(g) 0 0.040 0.058 0.012 0 0.040 0.167 0.256 0.389 4.3 Computation of time–history results The abovementioned BEM–FEM numerical model can be used to obtain the frequency response functions relating different magnitudes of interest, such as bending moments or lateral deflections, to the seismic excitation. The following section of this article shows several results obtained from the estimation of time–history moments at pile heads. These values in time domain of the bending moments are calculated by taking the inverse Fourier transform of the product H(ω)X(ω), where X(ω) is the Fourier transform of the free–field horizontal accelerations at ground surface used to define the excitation. Thus, H(ω) has to be defined as the ratio between the bending moments (M(ω)) and the free–field horizontal accelerations at ground surface (¨uff (ω)) at point (0,0,0). Therefore, it is of interest to analyse the evolution of the factor |uff |/Ainc with the angle of incidence (i.e., 6 with θo) since H(ω) will strongly depend on it. Figure 3shows the variation of this factor for different values of the Poisson’s ratio of the soil. It can be seen that the free–field displacements are independent of the angle of incidence for SH waves, while the variation is smooth and dependent on the Poisson’s ratio of the soil for P waves. However, a different trend is observed in SV waves. The displacements observed for angles comprised between 0 and 50 degrees are significantly smaller than those corresponding to more vertical incident waves, although the variation is smooth in the range between 0 and 45 degrees (with null values of the displacement for such angles). This produces the SV frequency response functions, once normalised by the free–field horizontal displacements, to have values of different orders of magnitude for angles of incidence smaller or greater than about 50◦. Even more, the frequency response functions will not be defined for 45◦since |uff (θo= 45◦)|/Ainc SV = 0. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 10 20 30 40 50 60 70 80 90 |uff| / Ainc θo SH P - νs= 0.2 P - νs= 0.3 P - νs= 0.4 SV - νs= 0.2 SV - νs= 0.3 SV - νs= 0.4 Figure 3: Variation of the modulus of the free–field horizontal displacement with the angle of incidence in the plane x2x3for SH, P and SV incident wavefronts. The green shadow represents the selected range of angles of incidence for the chosen value of νs= 0.4. On the other hand, actual soils are seldom homogeneous. On the contrary, they are usually layered, with the stiffness of the strata increasing with depth. For this reason, it is generally assumed that the probability of arrival to the surface of significantly non–vertical incident S or P plane waves is reduced because of the processes of refraction that take place in the interfaces. Therefore, in order to make the results for obliquely incident waves relatively comparable, only angles of incidence comprised between 55 and 90 degrees (zone defined by a green shadow in figure 3) will be considered from this point forward. 4.4 Validation of the BEM-FEM Code The aim of this subsection is the validation of the proposed formulation through a set of comparison results. First, kinematic interaction factors, in terms of displacements and rotations, are presented against those previously published by Kaynia & Novak [5]. The properties of piles and pile groups are: Ep/Es= 100, ρp/ρs= 1.5, β= 0.05, L/d = 20, νp= 0.25, νs= 1/3 and s/d = 5. Figure 4shows the comparison for single piles and 3 ×3 pile groups under incident SV waves, while figure 5shows the case of single piles and 4 ×4 pile groups under incident P waves. In both cases, a very good agreement is reached. In second place, the frequency response functions for the bending moments at the pile head for a single pile problem are presented both by the BEM-FEM coupled model and by multidomain BEM model [17]. The main properties of the problem are: Ep/Es= 100, ρs/ρp= 0.7, L/d = 20, β= 0.05, νp= 0.25 and νs= 0.4. Rotation and vertical displacement at pile head are not allowed whereas horizontal displacement is permitted. The results are represented in figures 6 and 7for different angles of incidence (θo) of an impinging SV wavefront. In figure 6, real and 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 |u| / |uff| 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 |ϕ| d / |uff| ao 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 ao θo= - BEM-FEM code θo= - BEM-FEM code θo=70 - BEM-FEM code θo=90 - BEM-FEM code 50 30 θo= - Kaynia & Novak [5] θo= - Kaynia & Novak [5] 30 50 θo=70 - Kaynia & Novak [5] θo=90 - Kaynia & Novak [5] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 o o o o o o o o Figure 4: Absolute value of the frequency response functions for horizontal displacements (top) and rocking (bottom) of single piles (left) and 3×3 pile groups (right) under SV incident waves. Comparison against results by Kaynia & Novak [5]. imaginary parts of the bending moments at the pile head, presented in a dimensionless form, are shown against the dimensionless frequency (defined as ao=ω d/cs) for three angles of incidence more vertical than the critical one. These dimensionless moments are obtained dividing the bending moments by the free–field horizontal displacement and by a cross–coupled term of the stiffness of an Euler–Bernoulli beam, linked to the relationship between displacements at one end of the beam with the bending moments developed in such a point. More precisely, and despite several other angles have been compared, results for 90, 75 and 60 degrees are shown for the sake of brevity. In addition, figure 7presents results for angles of incidence around the critical one (i.e., for 65, 65.8 and 66 degrees, being the critical angle θcr = 65.9◦). A very good agreement between both methodologies can be seen, especially in the low–frequency range, with maximum errors up to a 10%. Once the BEM–FEM model has been proven to be a valid tool for this problem, and taking into consideration that this approach is far less costly than the more rigorous BEM–BEM one and, as a consequence, allows to perform parametric analyses easier, such a tool will be used from now on. 5 Results 5.1 Influence of angle of incidence on bending moments at single pile heads In order to discern the general trends and the most interesting features of the problem at hand, this section addresses again the seismic response, in terms of bending moments at the pile head, of a single pile embedded in a halfspace. The main properties, boundary conditions at the pile head and type of incident wave (SV wave) are the same that were used in the previous section 4.4. Since the influence of pile-to-pile interaction on the kinematic response of pile groups for a vertically-incident wavefield is small in the low and intermediate frequency ranges [18], this problem constitutes a good first approach. This way, a parametric analysis is performed on this simpler problem to investigate the influence of type of wave and angle of incidence on the seismic response of the system. The conclusions drawn from this study will be useful for interpretation 8 0.8 0.9 1 1.1 1.2 1.3 |u| / |uff| 0 0.1 0.2 0.3 0.4 0.5 |ϕ| d / |uff| aoao 0 0.1 0.2 0.3 0.4 0.6 0.8 1 1.2 1.4 0 0.05 0.1 0.15 0 0.1 0.2 0.3 0.4 0.5 θo= - BEM-FEM code θo= - BEM-FEM code θo=70 - BEM-FEM code 50 30 θo= - Kaynia & Novak [5] θo= - Kaynia & Novak [5] 30 50 θo=70 - Kaynia & Novak [5] o o o o o o Figure 5: Absolute value of the frequency response functions for horizontal displacements (top) and rocking (bottom) of single piles (left) and 4 ×4 pile groups (right) under P incident waves. Comparison against results by Kaynia & Novak [5]. -120 -100 -80 -60 -40 -20 0 20 0 0.2 0.4 0.6 0.8 1 /(|uff| EpIp) ao L2 Re(M) /(|uff| EpIp) L2 -120 -100 -80 -60 -40 -20 0 20 0 0.2 0.4 0.6 0.8 1 ao Im(M) BEM-FEM, θ0=90o BEM-FEM, θ0=75o BEM-FEM, θ0=60o BEM, θ0=90o BEM, θ0=75o BEM, θ0=60o Figure 6: Real and imaginary parts of the transfer functions of the bending moment at the single pile head. Incident SV waves. 90, 75 and 60 degrees. of the results corresponding to a pile group. 5.1.1 Frequency response functions Figure 8a represents the values of the modulus of the dimensionless bending moment for a discrete set of dimensionless frequencies (ao= 0.01, 0.05, 0.1, 0.2, 0.3 and 0.4) against the angle of incidence. Results are obtained for angles between 55 and 90 degrees in steps of 2.5◦with additional values around the critical angle (65.8, 66 and 67 degrees). An inflection point can be seen for θo=θcr, which can be understood taking into account the change in the mechanism of propagation of the waves in the soil produced in such an angle. Below this angle, the variability of the bending moments is much greater than after it, where the moments notably increase with θoand ao. An alternative representation of figure 8a is shown in figure 8b, where the frequency response functions for the bending moment at the pile head are plotted against the dimensionless frequency for a set of angles of incidence (55, 60, 65, 65.8, 66, 70, 80 and 90 degrees). It is seen that 9 0 20 40 60 80 100 120 140 160 0 0.1 0.2 0.3 0.4 |M| L2/ (EpIp|uff|) ao Kinematic interaction 0 0.1 0.2 0.3 0.4 ao h/b=2 0 0.1 0.2 0.3 0.4 ao h/b=4 θo=55º θo=60º θo=65º θo=70º θo=80º Figure 15: Frequency response functions for the bending moment at the head of the corner pile. 3×3 pile group. Incident P waves. Angles of incidence between 55 and 80 degrees. their tendencies. As in the single pile case, the maximum moment resisted by a pile is presented by means of a set of grey bands. In this instance, three ranges of maximum moments resisted by the section of a pile made of C 25/30 reinforced concrete with three different amounts of reinforcement and subjected to an axial force variable between 0 and 500 kN are obtained [19] and represented. This way, the lower value corresponds to a reinforcement of 10 bars of S 500 steel of 25 millimetres of diameter, while the second one is obtained with 16 bars of S 500 steel of 32 millimetres and the higher value is reached with 26 bars of S 500 steel of 32 millimetres of diameter. The shape of the envelopes is similar independently of the superstructure’s aspect ratio. The tendency followed by the piles located in the inner side of the pile cap is the same than in the single pile, with a minimum around the critical angle. The magnitude of the bending moments in the outer piles is much greater than that developed in the inner piles of the group. Besides, the moment estimated in the outer piles varies almost linearly with the angle of incidence, with a maximum value for 55◦and a minimum for vertical incidence. Finally, the effect associated with inertial interaction is more noticeable in the inner piles because the magnitude of the moments produced by kinematic interaction is much smaller, being constant the contribution of inertial interaction for all the piles of the group. With regard to incident SH waves, the envelopes of maximum bending moments at pile heads, taking into account the three accelerograms defined in section 4, can be seen in figure 17. It is shown that the bending moments are nearly independent of the angle of incidence for the inner piles. For the outer piles, however, there is a significant variation with the angle of propagation of the incident wavefield. There are noticeable differences between the values developed only due to kinematic interaction and those arising from the problem including the building. As the magnitude of the bending moments due to the kinematic interaction is much smaller for SH than for SV waves, the influence of the superstructure becomes more apparent in this case. The absolute magnitude of the inertial interaction is expected to be approximately constant in every case, being this the reason why the influence of the superstructure is greater for SH waves only in relative terms. Besides, the higher dissimilarities that can be observed in the low– frequency range between the corresponding frequency response functions for incident SH waves can also explain the aforementioned differences between the values for kinematic interaction and for the problem including inertial interaction, since the time–history is dominated by the low frequencies. However, and with the only remarkable case of the pile located at the corner of the group, there is no significant difference between the results obtained for an aspect ratio of 2 and those for a value of the aspect ratio of 4. In general terms, the efforts in the outer piles of the group (i.e., the green and the black piles) tend to decrease, up to about a 23%, as the angle of incidence becomes more vertical, while the opposite trend is verified for the inner piles (i.e., the red and the blue piles), with increments up to about a 268%. It is also worth pointing out that the results are much smaller than the corresponding values for the non–vertical incident 16 0 0.5⋅106 1⋅106 1.5⋅106 2⋅106 2.5⋅106 3⋅106 Mmax (N m) Kinematic interaction h/b=2 h/b=4 0 0.5⋅105 1⋅105 1.5⋅105 2⋅105 2.5⋅105 55 60 65 70 80 90 Mmax (N m) 55 60 65 70 80 90 55 60 65 70 80 90 θ0θ0θ0 26∅32 16∅32 10∅25 26∅32 16∅32 10∅25 26∅32 16∅32 10∅25 Figure 16: Envelopes of maximum bending moments. Incident SV waves. Angles of incidence between 55 and 90 degrees. The grey zones represent the maximum moment resisted by the section of the concrete pile with C25/30. Three diferent amounts of reinforcement (indicated in the graph) are used. SV waves. 0 1⋅105 2⋅105 3⋅105 4⋅105 55 60 65 70 80 90 Mmax (N m) θ0 Kinematic interaction 55 60 65 70 80 90 θ0 h/b=2 55 60 65 70 80 90 θ0 h/b=4 Figure 17: Envelopes of maximum bending moments. Incident SH waves. Angles of incidence between 55 and 90 degrees. Finally, figure 18 shows the envelopes for incident P waves, considering the three accelerograms defined in section 4, where only the maximum values for each aspect ratio are presented anew. The results exhibit a very similar trend to those concerning incident SV waves. Summarising, the outer piles present similar values, decreasing nearly linearly with the angle of incidence, while the inner piles are submitted to almost the same value of the bending moment with independence of the angle of incidence. The magnitude of these moments is comprised between that of SV and SH incident waves, being lower than the former and higher than the latter. For this reason, the influence of the presence of the superstructure is, in this case, more noticeable than in the SV case, but less important than in the SH case. 6 Conclusions The influence of the type of incident seismic waves and of its angle of incidence on the dynamic response of pile foundations is investigated in this paper by means of a BEM–FEM coupled methodology. The formulation implemented for the study, the methodology used in the anal17 0 0.2⋅106 0.4⋅106 0.6⋅106 0.8⋅106 1⋅106 55 60 65 70 80 Mmax (N m) θ0 Kinematic interaction 55 60 65 70 80 θ0 h/b=2 55 60 65 70 80 θ0 h/b=4 Figure 18: Envelopes of maximum bending moments. Incident P waves. Angles of incidence between 55 and 80 degrees. The grey zone represents the maximum moment resisted by the section of the concrete pile with C25/30 reinforced with 10∅25mm steel bars. yses, and the definition of the problem at hand are first briefly presented. Then, different configurations are studied. In order to highlight some basic features of the problem, the case of a single pile subjected to obliquely–incident SV waves is addressed in the first place. Afterwards, a 3 ×3 pile group subjected to SH, SV or P obliquely–incident waves is studied. Here, not only the effect of kinematic interaction, but also that of the inertial interaction is analysed by considering the presence of a superstructure. The most important conclusions that can be drawn from this study are:  In the case of single piles subjected to SV waves, the largest bending moments arising at the pile heads does not necessarily occur for vertical incidence. In the frequency response functions, an inflection point can be seen at the critical angle due to the change in the mechanism of propagation of the waves in the soil produced at such an angle. For angles of incidence more horizontal than that, the variability of the bending moments with the angle of incidence is much larger than for more vertical angles, situation this last one in which bending moments increase with frequency and angle of incidence, being this increase more moderated at low frequencies.  For low frequencies, the largest moments arise for angles of incidence more horizontal than the critical one. For this reason, the envelopes of bending moments in time domain for single piles subjected to synthetic accelerograms show that piles designed to withstand safely the bending moments developed under vertical incidence would fail for angles of incidence more horizontal than the critical one.  The central piles of a group subjected to SV waves keep the trends observed in the frequency domain in the single pile, but the outer piles withstand much larger bending moments due to the kinematic restriction imposed by the rigid cap. Besides, the bending moments at the heads of these outer piles increase as the angle of incidence becomes more horizontal, both for SV and P incident waves. The kinematic bending moments are independent of pile–to–pile interaction only for vertical incidence.  The envelopes of maximum bending moments in time domain for pile groups subjected to SV and P waves show trends similar to those already described above for the frequency domain. In this case, and contrarily to what happens for single or central piles, the maximum bending moments developed at the outer piles increase almost linearly as the angle of incidence becomes more horizontal.  The maximum bending moments developed in time domain at the heads of piles in a group subjected to SH waves increase, as expected, with the presence of a superstructure. However, for the cases studied here, the large kinematic bending moments due to non– vertical P and SV waves imply a much smaller relative importance of the inertial interaction 18 in the case of incident P waves, and an even smaller relative influence when SV waves are considered. Besides, for P and SV waves, the contribution of inertial interaction is more important in the central piles, where the kinematic bending moments are smaller. For SH waves, on the contrary, inertial interaction has similar effects on all piles in the group. When observing the frequency response functions, the effect of inertial interaction becomes significant only around the fundamental frequency of the system.  In the case of incident SH waves, the presence and properties of a superstructure have a much larger effect on the frequency response functions for the bending moments than the angle of incidence. In fact, maximum bending moments in the outer piles are independent of the angle of incidence while, for central piles, bending moments are maximum for vertical incidence. On the contrary, for incident SV waves, the angle of incidence is the most important parameter. As a summary, this paper highlights the importance of considering the characteristics of the seismic excitation when estimating the internal forces in pile foundations. In particular, this paper studies the influence of the type of wave and its angle of incidence on the estimation of the bending moments developed in the pile heads of deep foundations. The propagation of non– vertical waves, especially of obliquely–incident SV waves with angles of incidence smaller than the critical one, results in bending moments much larger than those obtained for vertically–incident wavefields. It has also been shown that the influence of pile–to–pile interaction on the kinematic bending moments at pile heads (insignificant for vertical incidence) increases significantly as the angle of incidence becomes more horizontal, in particular for P and SV waves. ACKNOWLEDGEMENTS This work was supported by the Ministerio de Ciencia e Innovaci´on of Spain (Subdirecci´on General de Proyectos de Investigaci´on) through research project BIA2010–21399–C02–01 and co–financed by the European Fund of Regional Development (FEDER). It was also supported by the Agencia Canaria de Investigaci´on, Innovaci´on y Sociedad de la Informaci´on (ACIISI) of the Government of the Canary Islands, through research project ProID20100224 (co-financed by FEDER) and through the research fellowship TESIS20100084, whose recipient is Jos´e M. Zarzalejos, and by FEDER funds. The authors would like to thank for this support. References [1] O’Rourke MJ, Bloom MC, Dobry R. Apparent propagation velocity of body waves. Earthquake Eng Struct Dyn 1982; 10:283–294. [2] Mamoon SM, Ahmad S. Seismic response of piles to obliquely incident SH, SV and P waves. J Geotech Eng, ASCE 1990; 116:186–204. [3] Mamoon SM, Banerjee PK. Response of piles and pile groups to travelling SH-waves. Earthquake Eng Struct Dyn 1990; 19:597–610. [4] Makris N, Badoni D. Seismic response of pile groups under oblique-shear and Rayleigh waves. Earthquake Eng Struct Dyn 1995; 24(4):517–532. [5] Kaynia AM, Novak M. 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