Influence of pile inclination angle on the dynamic properties and seismic response of piled structures.

Influence of pile inclination angle on the dynamic properties and seismic response of piled structures ∗ Cristina Medina, Luis A. Padr´on, Juan J. Azn´arez, Orlando 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 {cmedina, jjaznarez, lpadron, omaeso}@siani.es, web: http://www.siani.es 24 October 2014 Abstract This paper aims to contribute to clarify whether the use of battered piles has a positive or negative influence on the dynamic response of deep foundations and superstructures. For this purpose, the dynamic response of slender and non-slender structures supported on several configurations of 2×2 and 3 ×3 pile groups including battered elements is obtained through a procedure based on a substructuring model which takes soil-structure interaction into account. Results are expressed in terms of flexible-base period and maximum shear force at the base of the structure. Moreover, modified response spectra considering soil-structure interaction effects are provided for different rake angles. It is shown that an increment of the rake angle can result in beneficial or detrimental effects depending on the structural slenderness ratio. Keywords: Inclined piles, Piled foundations, Soil-structure interaction, Effective period, Effective damping, Substructure model, Seismic response 1 Introduction The dynamic behaviour of buildings is affected by kinematic and inertial effects associated to soilstructure interaction (SSI). Their influence on the fundamental period and damping of soil-structure systems have been broadly investigated for shallow foundations [1–6] as well as for embedded foundations, either considering only inertial interaction (e.g. [7, 8]) or taking also into account the modified5 foundation input motion defined by kinematic interaction [9–13]. A few studies [14–23] analysing the effects of SSI on the dynamic characteristics of pile-supported structures can also be found in the scientific literature . Furthermore, up to the author’s knowledge, only Gerolymos et al. [24] and ∗Draft of the paper originally published in Soil Dynamics and Earthquake Engineering 2015; 69:196-206. http://dx.doi.org/10.1016/j.soildyn.2014.10.027. This work is released with a Creative Commons Attribution Non-Commercial No derivatives License. 1 Giannakou et al. [25] have analysed the influence of using deep foundations with inclined piles on the dynamic response of the structure they support.10 In recent years, inclined piles have recovered their popularity. Indeed, several studies has shown the beneficial role of battered piles on the seismic response of the structure [24, 26–28]. However, further research is needed to be able to elucidate in which cases the presence of raked piles is beneficial or detrimental. The aim of this work is to evaluate the influence of the rake angle on the dynamic response of15 shear structures founded on square pile groups comprising inclined piles and embedded in homogeneous viscoelastic half-spaces subjected to vertically incident S waves. The analysis is addressed through a simple and accurate procedure [23] based on a substructuring model in the frequency domain that takes into account kinematic and inertial interaction effects. A boundary element-finite element (BEMFEM) formulation [29–31] has been used to compute the impedance functions and the kinematic20 interaction factors. Results for several configurations of 2 ×2 and 3 ×3 pile groups including battered elements are obtained. The seismic response of the superstructure is presented in terms of the effective period and the maximum shear force at the base of the structure per effective earthquake force unit Qm. Moreover, results in terms of effective period and damping are used to build modified response spectra25 for different values of the rake angle. 2 Methodology The dynamic behaviour of linear shear structures supported on pile groups and subjected to vertically incident plane S waves is analysed in this paper by using a three-degree-of-freedom (3DOF) system as the one depicted in Fig. 1a. This system is defined by the foundation horizontal displacement ucand30 rocking ϕc, together with the structural horizontal deflection u. Ld s b s/2 x y Figure 1: (a) Problem definition (b) substructure model of a one-storey structure and (c) equivalent single-degree-of-freedom oscillator. The structure is considered to be founded on a square regular group of piles embedded in a 2 homogeneous, viscoelastic and isotropic halfspace. Pile heads are constrained to a rigid square cap of negligible thickness and mass mo, which is free of contact with the ground surface. The moment of inertia of this pile cap is denoted by Io. All piles have identical geometrical properties defined35 by length Land sectional diameter d. Several configurations of pile groups have been considered in this study. Each one of them is defined by number of piles, foundation halfwidth b, centre-to-centre spacing between adjacent piles sand rake angle of piles θ. It is worth noting that some vertical piles are included in 3 ×3 pile groups for the purpose of maintaining symmetry with respect to planes xz and yz.40 The superstructure consists of massless and axially inextensible columns that support the structural mass m, which is situated at the height hof the resultant of the inertia forces for the mode of vibration under study. The moment of inertia of the vibrating mass, which is distributed over a square area, is denoted by I. Its dynamic behaviour, corresponding to fixed-base condition, is characterized by the structural stiffness kand its viscous damping ratio ξ.45 The 3DOF system dynamic response, considering kinematic and inertial interaction effects, can be studied through a substructuring model in the frequency domain such as that represented in Figure 1b. This model consists of a building-cap structure supported on springs and dashpots representing the soil-foundation stiffness and damping in the horizontal (kxx, cxx), rocking (kθθ, cθθ) and crosscoupled horizontal-rocking (kxθ, cxθ) vibration modes, respectively. The whole system is subjected to50 the horizontal (ug) and rocking (ϕg) motions measured at the massless pile cap level when subjected to free-field motion at the surface ugo. In this paper, a BEM-FEM coupling model [28–31] is used to compute translational Iu=ug/ugo and rotational Iϕ=ϕgb/ugokinematic interaction factors, as well as impedance functions at each frequency ao, which are usually written as Kij =kij + iaocij, where kij and cij are the mentioned55 frequency-dependent dynamic stiffness and damping coefficients, respectively, i=√−1 is the imaginary unit. The dimensionless excitation frequency is defined as ao=ωb/cs, being ωthe excitation circular frequency, cs=pµs/ρsthe speed of propagation of shear waves in the halfspace, and µsand ρsthe soil shear modulus of elasticity and mass density, respectively. Following other authors [2, 3, 8, 12] and in order to characterize the soil-foundation-structure60 system, other dimensionless parameters, covering the mean features of SSI problems, has been used. These are: (1) structural slenderness ratio h/b; (2) fixed-base structure damping ratio ξ; (3) dimensionless fixed-base natural frequency of the structure λ=ωn/ω; (4) foundation-structure mass ratio mo/m; (5) wave parameter σ=csT/h (that measures the soil-structure relative stiffness); (6) mass density ratio δ=m/(4ρsb2h) between structure and supporting soil; (7) Poisson’s ratio νs; and (8)65 damping ratio ξsof the soil. A hysteretic damping model of the type µs=Re[µs](1+2iξs) is considered in this study for the soil material. The dimensionless parameters used to characterize the pile foundation are: pile spacing ratio s/d, pile-soil Young’s modulus ratio Ep/Es, size of the square pile group, embedment ratio L/b, pile slenderness ratio L/d, dimensionless frequency ao, soil-pile densities ratio ρs/ρpand rake angle θ.70 A simple and accurate procedure developed by Medina et al.[23] is used in this paper to determine the dynamic characteristics of an equivalent single-degree-of-freedom (SDOF) oscillator (Fig. 1c) which reproduces, as accurately as possible, the response of the 3DOF system shown in Fig. 1b within the range where the peak response occurs. This response is expressed in terms of Q=|ω2 nu/(ω2ugo)|, which represents the ratio of the shear force at the base of the structure to the effective earthquake75 3 force. The equivalent SDOF system can be defined by its damping ratio ˜ ξand its undamped natural period ˜ T. The effective period ˜ T/T =˜ λ=ωn/˜ωncan be found as the root of Eq. (1), being ˜ωnthe undamped natural frequency of the equivalent oscillator. The effective damping ˜ ξcan be obtained from Eq. (2). 1−1 λ2−1 λ2α2 xx(λ)−1 λ2α2 θθ(λ)= 0 (1) ˜ ξ=Iu+h bIϕ−1"ξ′ ˜ λ2+1 ˜ λ2ξxx α2 xx(1 + i2ξxx)+ξθθ α2 θθ(1 + i2ξθθ)# (2) where,80 ξ′=ω ωn ξ(3) α2 xx =σ21 16π2 h b 1 δ˜ kxx (4) ξxx =˜cxx 2˜ kxx (5) α2 θθ =σ21 16π2 h b 1 δRe b2 (h+D)2˜ KθθD(6) ξθθ = Im hb2 (h+D)2˜ KθθDi 2Re hb2 (h+D)2˜ KθθDi(7) being ˜ Kxx =Kxx/(µsb) = ˜ kxx + i˜cxx and ˜ KθθD=1 µsb3Kθθ −K2 θx Kxx (8) b2 (h+D)2= h b2 −2h b˜ Kθx ˜ Kxx + ˜ Kθx ˜ Kxx !2  −1 (9) where D=D(ω) = −Kxθ/Kxx represents the virtual depth of the point at which the soil-foundation interaction must be condensed to obtain a diagonal impedance matrix. Finally, the maximum shear force at the base of the structure per effective earthquake force unit Qmis obtained as85 Qm= Max  1 ω2 ω2 n˜ T T2−1−i2˜ ξω ωn ˜ T T  (10) 4 3 Results The procedure explained above is applied in this section to the study of the influence of using deep foundations with inclined piles on the seismic response of the superstructure. Such influence is measured here in terms of the effective system period ˜ T/T, the maximum shear force at the base of the structure per effective earthquake force unit Qmand the elastic response spectra.90 Table 1: Values for the dimensionless parameters in the cases under investigation νsξsEp/Esρp/ρsL/b L/d s/d ξ δ 1/σ mo/m h/b 2×2 3 ×3 0.4 0.05 1030.7 2 7.5 3.75 2.5 0.05 0.15 0 −0.5 0 1,2,5,1015 7.5 5 30 15 10 Results for different soil-foundation-structure systems as described in Section 2, are studied in the frequency range of interest for seismic loading (ωd/cs<0.5, according to Gazetas et al. [32]). The dimensionless parameters corresponding to these configurations are listed in Table 1. These values are representative for typical buildings and soils [12, 33] and are related to those studied in Medina et al. [23], whose aim was analyzing the influence of the pile foundation on the dynamic95 response of the soil-foundation-structure system, and where several values of the embedment ratio L/b were considered. In contrast, the present work aims to analyse the effect of inclined piles on the structural response (by means of an additional parameter θrepresenting the rake angle) and, therefore, an intermediate value of this parameter L/b = 2 has been chosen as representative. Regarding the structural slenderness ratio h/b, the range of values taken into account is similar to those considered100 in previous studies [2, 12, 23, 25]. The varying values of the pile spacing ratio s/d are chosen in order to make the different results more comparable among each other by keeping the foundation halfwidth bconstant for configurations with different number of piles. The foundation halfwidth is defined as b=sfor 2 ×2 pile groups and b= 3s/2 for 3 ×3 pile groups. Four different rake angles have been considered: θ=0°(vertical piles), 10°, 20°and 30°.105 3.1 Effective period Figs. 2 and 3 present ˜ T/T as a function of 1/σ for different rake angles θ, which illustrates the influence of the rake angle on the system effective period for the different configurations of 2 ×2 and 3 ×3 pile groups under study. Discrete points to be read on the right axis provide a zoomed view in those cases in which it is necessary.110 For short and squat buildings (h/b = 1), in which the horizontal displacement is the controlling factor, the system period decreases for higher rake angles. This is because an increment of the rake angle leads to an increase of the horizontal stiffness due to the contribution of the pile axial stiffness to withstand the lateral loads. In order to illustrate this effect, Fig. 4 present the impedances of three different 2 ×2 pile groups with L/d = 7.5 (left column), L/d = 15 (central column) and L/d = 30115 (right column). The stiffness values kij are represented with solid lines to be read on the left axis, whereas the damping values cij depicted with dashed lines to be read on the right axis. 5 1.5 2.0 2.5 3.0 T ˜/T h/b=1 L/d=7.5 s/d=3.75 θ=0º θ=10º θ=20º θ=30º h/b=2 h/b=5 1.6 1.8 2.0 2.2 T ˜/T (Zoom) h/b=10 1.5 2.0 2.5 T ˜/T L/d=15 s/d=7.5 1.6 1.8 2.0 T ˜/T (Zoom) Zoom θ=0º θ=10º θ=20º θ=30º 1.0 1.5 2.0 2.5 0.0 0.1 0.2 0.3 0.4 T ˜/T 1/σ L/d=30 s/d=15 0.0 0.1 0.2 0.3 0.4 1/σ 0.0 0.1 0.2 0.3 0.4 1/σ 0.0 0.1 0.2 0.3 0.4 0.51.4 1.6 1.8 2.0 T ˜/T (Zoom) 1/σ Figure 2: Effective period ˜ T/T for different 2 ×2 pile groups. Ep/Es= 1000 and ξs= 0.05. Solid lines to be read on left axis. Dotted lines to be read on right axis when a zoomed view is needed. In the case of slender structures (h/b = 10), the effect of the rake angle on the system period depends on the variation of the rocking stiffness as well. An increment of the rake angle generally leads to a decrease of the rocking impedance (second row in Fig. 4). This results from the fact that120 vertical impedance of single piles experiences a reduction when piles are inclined. Exceptionally, in those cases with little spacing between adjacent piles (left column in Fig. 4) the pile-soil-pile interaction effect takes predominance over that of inclination and the vertical impedance of each pile increases with the rake angle since the distance between the pile tips widens with depth. Thus, in those cases in which the increase of the rake angle leads to a reduction of the rocking stiffness (L/d = 15 and125 L/d = 30), the system period experiences an increase with θ. Accordingly, a reduction of the system period results from the increase of the rake angle when L/d = 7.5 since the rocking stiffness increases in this case. 6 1.5 2.0 2.5 3.0 T ˜/T h/b=1 L/d=7.5 s/d=2.5 h/b=2 θ=0º θ=10º θ=20º θ=30º h/b=5 1.1 1.2 1.3 1.4 1.5 T ˜/T (Zoom) h/b=10 1.5 2.0 2.5 T ˜/T L/d=15 s/d=5 1.1 1.2 1.3 1.4 1.5 T ˜/T (Zoom) 1.0 1.5 2.0 2.5 0.0 0.1 0.2 0.3 0.4 T ˜/T 1/σ L/d=30 s/d=10 0.0 0.1 0.2 0.3 0.4 1/σ 0.0 0.1 0.2 0.3 0.4 1/σ 0.0 0.1 0.2 0.3 0.4 0.5 1.1 1.2 1.3 1.4 1.5 T ˜/T (Zoom) 1/σ Zoom θ=0º θ=10º θ=20º θ=30º Figure 3: Effective period ˜ T/T for different 3 ×3 pile groups. Ep/Es= 1000 and ξs= 0.05. Solid lines to be read on left axis. Dotted lines to be read on right axis when a zoomed view is needed. 3.2 Maximum structural shear forces Figs. 5 and 6 depict the system response of structures founded on 2 ×2 and 3 ×3 pile groups, in130 terms of the maximum shear force at the base of the structure per effective earthquake force unit Qm. Dotted lines to be read on the right axis provide a zoomed view in those cases in which it is necessary. For short and squat buildings (h/b = 1 and h/b = 2), the increment of the rake angle results in lower values of the maximum shear force at the base of the structure. This effect is due to several concurrent factors: an increase of the horizontal damping cxx, a reduction of the translational kinematic135 interaction factor, (which predominates for non-slender structures) and an increase in the horizontal stiffness of the foundation which leads to a reduction of the effective period which, in turn, entails an increment of the dissipated energy which contributes to reduce Qm. In the case of slender structures (h/b = 10), an increase of the rake angle leads to slightly greater values of Qmdue to the reduction of the rocking damping cθθ and to the increment of the overturning140 moment, which is the controlling factor in these cases. 7 0.15 0.20 0.25 0.30 0.5 1.0 1.5 2.0 kxx/(µs d) x 10-2 L/d=7.5 s/d=3.75 θ=0º θ=10º θ=20º θ=30º 0.3 0.6 0.9 1.2 0.5 1.0 1.5 2.0 L/d=15 s/d=7.5 0.3 0.6 0.9 1.2 0.5 1.0 1.5 2.0 (ωd/cs) cxx/(µs d) x 10-2 L/d=30 s/d=15 0.04 0.05 0.06 0.025 0.050 0.075 kθθ/(µs d3) x 10-4 0.1 0.2 0.3 0.2 0.4 0.6 0.5 1.0 1.5 0.5 1.0 1.5 (ωd/cs) cθθ/(µs d3) x 10-4 -1.5 -1.0 -0.5 -1 0 1 kθx/(µs d2) x 10-2 -5 -2 1 0.5 3.0 5.5 -4 0 4 1.5 5.0 8.5 (ωd/cs) cθx/(µs d2) x 10-2 -2.00 -1.25 -0.50 0.25 0.0 0.1 0.2 0.3 0.4 0.50.0 0.5 1.0 1.5 kzz/(µs d) x 10-2 ωd/cs -4 -2 0 2 0.0 0.1 0.2 0.3 0.4 0.50 1 2 3 ωd/cs -4 -2 0 2 0.0 0.1 0.2 0.3 0.4 0.50 1 2 3 (ωd/cs) czz/(µs d) x 10-2 ωd/cs Figure 4: Impedance functions of different 2 ×2 pile groups. Ep/Es= 1000 and ξs= 0.05. Solid lines to be read on left axis. Dashed lines to be read on right axis. Fig. 5 allows to show the extent to which kinematic interaction influences the system dynamic response. To this end, results involving both kinematic and inertial interaction or only inertial interaction are represented. It can be seen that the ability of the foundation to filter the seismic input has significant effects on the variation of Qm. All configurations under study show a reduction of145 the translational kinematic interaction factor Iufor higher rake angles, as illustrated in Fig. 7, that presents kinematic interaction factors for three different 2×2 pile groups with L/d = 7.5 (left column), L/d = 15 (central column) and L/d = 30 (right column). Generally, Iϕincreases significantly with 8 5 10 15 20 Qm h/b=1 L/d=7.5 s/d=3.75 h/b=2 Inertial interaction only θ=0º θ=10º θ=20º θ=30º h/b=5 Kinematic and inertial interaction θ=0º θ=10º θ=20º θ=30º 12 14 16 Qm (Zoom) h/b=10 5 10 15 Qm L/d=15 s/d=7.5 12 14 Qm (Zoom) 0 5 10 15 0.0 0.1 0.2 0.3 0.4 Qm 1/σ L/d=30 s/d=15 0.0 0.1 0.2 0.3 0.4 1/σ 0.0 0.1 0.2 0.3 0.4 1/σ 0.0 0.1 0.2 0.3 0.4 0.510 12 14 Qm (Zoom) 1/σ Zoom θ=0º θ=10º θ=20º θ=30º Figure 5: Maximum structural response value Qmfor different 2 ×2 pile groups. Ep/Es= 1000 and ξs= 0.05. Dotted lines to be read on right axis provide a zoomed view. θfor large pile-to-pile separation ratios such as s/d = 5,7.5,10,15, except for small rake angles [28]. However, for small pile-to-pile separation (L/d = 7.5 and s/d = 2.5 or s/d = 3.75) the rotational150 kinematic interaction factor Iϕof inclined piles is smaller than the one corresponding to vertical piles for all rake angles within the range under study. A minimum cap rotation can be achieved by inclining piles a small rake angle as θ=1°or θ=3°[28]. It might accordingly be inferred that a minimum value of the maximum shear force at the base of the structure Qmcould be reached for these rake angles. However, this does not occur (results not shown155 for the sake of brevity) because even though the rotational kinematic interaction factor Iϕincreases with the rake angle, as shown in Fig. 7, so does the horizontal damping cxx (see Fig. 4) which leads to a reduction of Qmas the rake angle θincreases. Regarding the relationship between the geometric and mechanic properties of the foundation, one could think that the geometric point where the extension of the raked pile axes meet above160 the cap hp=s/(2 tan θ) could be close to the center of stiffness of the pile group, computed as 9 0 3 6 9 12 15 18 0.0 0.2 0.4 0.6 0.8 Se/ag T (s) h/b=1 L/d=7.5 1/σ=0.2 Fixed base θ=0º θ=10º θ=20º θ=30º 0.0 0.2 0.4 0.6 0.8 1.0 T (s) h/b=2 L/d=7.5 1/σ=0.4 Figure 13: Elastic response spectra of the motion of the mass corresponding to the 1940 El Centro Earthquake a 2 ×2 pile group with s/d = 3.75 being T= 0.44 s.Ep/Es= 1000 and ξs= 0.05. extension of the raked pile axes meet above the cap hp) above or below unity does not imply a change of trend in the structural response presented in terms of maximum shear force at the245 base of a structure when a fixed pile-cap connection exists. This is due to the fact that, in that case, the height Dof the centre of stiffness of the pile group is not related to hp. •In most cases, a reduction of the spectral acceleration can be observed as the rake angle increases. However, the more slender the superstructure, the less systematic and significant are the beneficial effects of rake angle on the structural response.250 5 Acknowledgements This work was supported by the Subdirecci´on General de Proyectos de Investigaci´on of the Ministerio de Econom´ıa y Competitividad (MINECO) of Spain and FEDER through research project BIA2010-21399-C02-01 and also by the Agencia Canaria de Investigaci´on, Innovaci´on y Sociedad de la Informaci´on (ACIISI) of the Government of the Canary Islands and FEDER through research project255 ProID20100224. C. Medina is a recipient of a fellowship from the Program of predoctoral fellowships of the University of Las Palmas de Gran Canaria (ULPGC). The authors are grateful for this support. 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