Simple formulas for the dynamic stiffness of pile groups

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1 EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 22; :1 6 [Version: 22/11/11 v1.] Simple formulas for the dynamic stiffness of pile groups Reza Taherzadeh 1, Didier Clouteau 1,, Régis Cottereau 2 1 Laboratoire MSSMat, École Centrale Paris, CNRS UMR 8579, Grande Voie des Vignes, Châtenay-Malabry, France 2 International Center for Numerical Methods in Engineering (CIMNE), Universitat Politècnica de Catalunya, Jordi Girona 1-3, 834 Barcelona, Spain SUMMARY Simple formulas are derived for the dynamic stiffness of pile group foundations subjected to horizontal and rocking dynamic loads. The formulations are based on the construction of a general model of impedance matrices as the condensation of matrices of mass, damping and stiffness, and on the identification of the values of these matrices on an extensive database of numerical experiments computed using coupled Finite Element-Boundary Element (FE-BE) models. The formulations obtained can be readily used for design of both floating piles on homogeneous half-space and endbearing piles, and are applicable for a wide range of mechanical and geometrical parameters of the soil and piles, in particular for large pile groups. For the seismic design of a building, the use of the simple formulas rather than a full computational model is shown to induce little error on the evaluation of the response spectra and time histories. Copyright 22 John Wiley & Sons, Ltd. key words: Soil impedance matrix; pile group foundation; design formulas; lumped-parameter models; hidden variables models 1. INTRODUCTION Whatever the mode of vibration, the dynamic stiffness of a pile group cannot be computed by simply adding the stiffnesses of the individual piles. Depending on the mechanical and geometrical parameters of the soil and piles, the dynamical behavior of each pile can be heavily influenced by that of its neighbors [1]. Among other phenomena, it is clear that the dynamic resonance of the soil constrained within a cluster of piles cannot be modeled when the complex dynamic interaction between these piles is neglected. The main approach to solve this strongly coupled problem is the use of full numerical models, taking into account the soil and the piles with equal rigor. This is however a computationally very demanding approach, in particular for large number of piles, and has only been attempted at, to the knowledge of the authors, by Kaynia [2], using the Boundary Element (BE) method. All other numerical methods in the literature seem to include some simplifying assumptions. Correspondence to: didier.clouteau@ecp.fr Copyright 22 John Wiley & Sons, Ltd.

2 2 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU For example, the axisymetrical Finite Element (FE) model [3], the Ring-Pile model [4], or the closely-spaced plates model [5] can be used, when the geometrical layout of the pile group allows for it. The latter two approaches consist in grouping the piles in concentric circles or soil-pilestripped upright plates, respectively, both allowing for an easier evaluation of the interaction effects. Another approach consists in replacing the pile group by a single equivalent upright beam [6]. In any case, these approaches are not adapted to the needs of civil and structural engineers, who need to design pile foundations with little recourse to computational tools. A more interesting approach for design purposes consists in providing analytical formulas, whose structure is usually derived from physical considerations, and with tabulized parameters, depending on the geometrical and mechanical parameters of the soil and piles. The simplest type of such approaches is based on Winkler s spring model for the soil, for which radiation damping and inertial effects are neglected [7, 8, 9]. A relatively simple method was proposed by Gazetas and Dobry [18] for estimating the damping characteristics of horizontally loaded single pile in layered soil. Following Wolf s approach [1] for the modeling of soil-structure interaction (SSI), other researchers [11, 12, 13, 14, 15, 16] have replaced the soil-pile system by a one degree-of-freedom (DOF) mass, with a damper and a spring. Inertial effects and radiation damping are therefore taken into account to some extent but the general dynamical behavior, and in particular the interaction between the different piles, is heavily simplified. To improve these models, Dobry and Gazetas [17] proposed an approximate formulation accounting for the interaction between the piles, by modeling the waves emanating from each excited pile. The additional term is therefore based on the computation of the propagation of a wave, supposed to be cylindrical, emanating from a single excited pile in a homogeneous domain. The method was further refined by Gazetas and co-workers [19, 2, 21] to attempt to model multiple reflections within the pile group in layered soil. However, a few attempts have been made at accurately modeling the large pile group foundation, in particular for the complex frequency-dependance of end-bearing pile foundations. (Konagai [6] provides formulas valid only for sway, Mylonakis and Gazetas [21] provide formulas valid for all movements but only for a group of nine piles and Nikolaou et al. [4] provide for kinematic pile bending for a group of twenty piles). Despite the significant progress in pile dynamics [22], there is therefore still a need for simple engineering procedures for their design, following the example of the code provisions developed for the seismic design on spread footings [23, 24]. This paper aims at providing such formulas, to be used for both small and large pile groups, as well as for both floating pile groups on homogeneous half-space and end-bearing pile groups. The main novelty of this paper is that the formulas are valid over a range of parameters larger than formulas previously available in the literature (see above references). They can be used for large numbers of piles. This is made possible by the use of a very general dynamic model for the representation of stiffness impedance matrices, the hidden state variable model (Sec. 3.1). The parameters appearing in this model are then fitted using an extensive database of full coupled FE-BE computations of soil-pile systems (Sec. 3.2). The sway and rocking of the foundation are accounted for, in a large range of parameters of the soil and piles, and the formulas are given independently for floating (Sec. 4.1) and end-bearing piles (Sec. 4.2). Further, for the seismic design of a building, the use of the simple formulas rather than a full computational model is shown to induce little error on the evaluation of the response spectra and time histories (Sec. 5). The reader interested in a fast use of the formulas can refer directly, for a floating pile group on homogeneous half-space (respectively, an end-bearing pile group) to Table II (resp., Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

3 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 3 Figure 1. Definition of pile group foundation. Table IV), the coefficients of which are to be used in Eq. (9) and (8) (resp., Eq. (14) and (1)). In these equations, both the dynamic stiffness and the frequency are normalized, as described in Sec THE IMPEDANCE MATRIX OF A PILE GROUP: DEFINITION AND NOTATIONS In this section, we introduce the main notations and define the impedance matrix of a pile group. The normalizations, that will be used throughout the paper, of both the impedance and the frequency, are also introduced Notations In all formulas, the indices s and p will refer to the soil and the piles, respectively. When considering two layers of soil, the top layer will still be denoted s while the bedrock will be denoted b. E s, ν s, G s, and ρ s (respectively, E p, ν p, G p, ρ p, and E b, ν b, G b, ρ b ) hence denote Young s modulus, Poisson s ratio, the shear modulus, and the unit mass of the soil (respectively, of the piles, and of the bedrock). V s and β s (resp., V b and β b ) denote the shear wave velocity and the hysteretic damping of the soil (resp., of the bedrock). All the piles in a group are supposed identical, with a diameter d, a length l p, an inertial moment I p, and they are separated from each other by a distance s (see Fig. 1). They are rigidly attached to a mass-less square cap with a half-width B f, which is supposed to have no contact with the soil. Further we define L = (E p I p /E s ).25, closely related to the critical pile length defined by several authors [25, 26] and an equivalent radius for the cap R f = 2B f / π. Two cases will be considered in this paper: (1) floating pile groups on homogeneous halfspace, and (2) end-bearing pile groups. In the former case, the soil is a homogeneous half-space, while, in the latter case, the soil is composed of a layer of thickness H, resting over a bedrock, in which the tips of the piles are embedded (l p > H) Impedance matrix The impedance matrix or dynamic stiffness matrix Z(ω) of a pile group relates the vector of forces and moments applied on the rigid cap at the top of the piles to the resulting vector of Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

4 4 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU displacements and rotations at the same point. Since the rigid cap is supposed to be square, symmetry considerations ensure that the impedance matrix, in the basis of the rigid body movements of the pile cap, is not full. More precisely, only the diagonal terms, and the coupling terms between the sway along one of the axes of symmetry and the rotation around the other one, are non-null. Besides, the two horizontal axes of symmetry are equivalent, so that the corresponding terms in the impedance matrix are equal. Hence, only five elements should be considered: horizontal, rocking, pumping, torsion, and horizontal-rocking coupling. Finally, as is usually done in earthquake engineering, we disregard the pumping and torsion terms as less significant, and therefore write the impedance as [ ] Zh (ω) Z Z(ω) = hr (ω), (1) Z hr (ω) Z r (ω) where Z h (ω), Z r (ω) and Z hr (ω) are, respectively, the horizontal, rocking, and coupling elements. To simplify the comparisons between different soils and/or foundation sizes, it is customary in foundation design [27, 29] to normalize the impedance matrix by a constant static stiffness matrix K, given here by [ ] Gs R K = f G s Rf 3. (2) It is also classical to use a dimensionless frequency a = ωr f /V s, where ω is the circular frequency. The impedance matrix is finally written Z(a ) = K 1/2 Z(a )K 1/2, (3) where the tilde symbol refers to normalised quantities. Note that the imaginary part of the impedance matrix is sometimes plotted in the literature after a division by ω. While this is interesting in the simple cases where that imaginary part remains more or less linear, such as for circular rigid foundations over homogeneous soils, it is not relevant here, so that that custom will not be followed in the plots of this paper. 3. PRINCIPLE OF THE COMPUTATION OF SIMPLE FORMULAS As described in the introduction, there is an engineering need for simple formulations of the impedance matrix of large pile groups. The goal is to avoid using complicated and timeconsuming computational models, while retaining their accuracy. The usual approach to the derivation of these simple formulations consists in two steps: 1. choose a structure for the formulation, usually based on physical considerations and simplifying assumptions 2. set the values of the parameters for a range of soil and pile group characteristics, usually provided in a tabulized form. In the literature (see the introduction for references), the first step is basically a decision of the designer, which, however, conditions in a large part the quality of the formulation. We therefore choose here a more rational approach, in which no specific structure is chosen a priori, and make use of the so-called hidden variables model of the impedance matrix [35, 36]. The identification of the parameters of the final formulas is then performed by regression from a Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

5 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 5 database of FE-BE computations. Note that the hidden state variable model that is chosen here might appear more mathematically than previous formulations found in the literature. However, remember that this structure is just an intermediate state that allows to find the final formulas that will eventually be used by the engineers. In section 4, we will see some analogies between the formulas proposed, and mechanical systems created as sets of spring, dampers and masses. The differences with lumped parameters is that with the hidden state variables model, the equivalent mechanical model comes out naturally as a consequence of the regression, rather than being chosen a priori The hidden variables model The construction of the hidden variables model of an impedance matrix is based on the supposition that, besides the n Γ physical DOFs on which the impedance is defined (typically the rigid-body modes of the cap of the pile group), there exists n I additional DOFs that represent some internal resonance phenomena inside the soil and the pile group. The resonance modes corresponding to these DOFs cannot be physically identified, as only their influence on the impedance matrix is observable, so the DOFs are referred to as hidden, or inner. With respect to the n = n Γ + n I DOFs, matrices of mass M, damping D, and stiffness K can then be identified, and the dynamic stiffness matrix S(a ) is defined as S(a ) = (K a 2 M) + ia C. (4) The impedance matrix corresponding to the hidden variables model is then the condensation on the n Γ physical DOFs of the stiffness matrix S(a ). More specifically, introducing the block decomposition of Eq. (4), ] ([ [ ]) KΓ K c MΓ M c [ SΓ (a ) S c (a ) S T c (a ) S I (a ) = the impedance matrix is defined as K T c K I ] a 2 M T c M I [ ] CΓ C + ia c C T, (5) c C I Z(a ) = S Γ (a ) S c (a )S 1 I (a )S T c (a ). (6) As the hidden variables are not necessarily physical DOFs, but rather state variables in the background of the physical model, the matrices M, D, and K are really generalized mass, damping and stiffness matrices and do not correspond a priori with the classical mass, damping and stiffness matrices, or to those obtained through the application of some modal reduction technique. Another equivalent form of the hidden variables model can be derived [35], where the hidden parts of the matrices are diagonal, and with no coupling in mass. In that case, the impedance can be written n h Z(a ) = (K Γ a 2 (ia C l c + K l M Γ ) + ia C Γ c)(ia C l c + K l c) T (ki l a2 ml I ) + ia c l I where C l c and K l c are the l th columns of C c and K c, and m l I, cl I and kl I are the diagonal elements of M I, C I and K I. The main interest of this hidden variables model is its generality. Its structure makes it suitable for the representation of any type of impedance matrix, provided that an appropriate number of hidden variables is used. Note that the numerical identification of the matrices M, l=1 (7) Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

6 6 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU C, and K is entirely performed from the knowledge only of the impedance matrix, and that the number of hidden variables can be automatically chosen based on a precision criteria for the approximation of the impedance matrix [39, 35, 36]. Contrary to the lumped-parameter models of the impedance matrix [28], in which the identification of the mechanical elements yields negative values of the springs, dashpots, and/or masses, in the hidden state variable model, the causality and stability of the soil impedance matrix are directly related to the positivity of M, K and C. In other words, in comparison with lumped parameter models, the diagnosis of unphysical models is very natural. In the next section, the numerical method that is used to derive the reference impedance matrices, and to identify the parameters of the formulas, is described. The methodology for the identification of the hidden variables model of a given impedance matrix is also described in App The reference FE-BE model We suppose, for the reference computations, that both the soil and the piles behave linearly and that the contact between the piles and the soil is continuous in all directions, without any slippage or gap. The elastodynamic equations are therefore linear. The numerical approach used to derive the reference results for the calibration of the simple formulations is based on an efficient FE-BE coupling technique that is described in detail in [3, 31] and is briefly recalled below. The soil is separated into two blocks: one, bounded and containing the piles, which is modeled by the FE method, and the other, surrounding the previous one, which is modeled by the BE method (see Fig. 2). Within the FE block, the piles are modeled as Bernouilli beam elements. The two blocks are then assembled using the Craig-Bampton coupling technique [32], so as to lower the computational cost, which may reach high levels for large pile groups. This numerical model was already validated for stiffness problems taken from the literature (in particular [2]) and the results are given in [31]. However, these validation results only concerned floating pile groups on homogeneous half-spaces so that we present here a comparison, on a particular example, of the FE-BE model with the BE approach described in [34]. We therefore consider a 4 9 pile group embedded in a soil with two layers (see Fig. 2). The piles have a Young s modulus of 25 GPa, a diameter of d = 1.3 m and are separated by s = 2.6 m. The first layer of soil is H = 9.5 m-thick, and is formed of a very soft saturated organic clay with S-wave velocity V s = 8 m/s, unit mass ρ s = 1.5 Mg/m 3 and Poisson s ratio ν s =.49. The lower layer of soil is a stiff sand with S-wave velocity V d = 3 m/s, unit mass ρ d = 2 Mg/m 3 and Poisson s ratio ν d =.4, in which the piles penetrate 6 m. In both layers the hysteretic damping is taken as β s = β d =.5. As seen on Fig. 2, the agreement between the results in the two numerical approaches is very good. It should be noted that the frequency-dependance of pile groups is particularly sensitive to the number of piles and to its character of floating or end-bearing. The dynamic stiffness of single piles and pile groups with a small number of piles is nearly independent of frequency [33], while that of larger pile groups may show large variations with frequency. Likewise, the behavior of end-bearing pile groups is much more erratic with frequency than that of floating pile groups on homogeneous half-space. These physical results are retrieved with the FE-BE approach and an example of such comparison is shown in Fig. 3. These results were obtained considering the sample number 4 in Tables I and III. Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

7 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 7 Dynamic stiffness [N/m] 2 x x 19 Damping [N/m] Figure 2. FE model (left) of the 4 9 pile group within a block of soil and comparison of the real (right, up) and imaginary (right, down) parts of the horizontal impedance computed using the FE-BE model (solid line) and the BE model (dashed line) [34]. 3 6 Normalized Real Part [ ] Normalized Imaginary Part [ ] Dimensionless Frequency [ ] Dimensionless Frequency [ ] Figure 3. Real (left) and imaginary (right) parts of the normalized horizontal impedance matrix, for floating (solid line) and end-bearing (dashed line) pile groups. It is also interesting to note, on Fig. 3, for the end-bearing pile group, that the imaginary part of the impedance (it is also true for the rocking term, not shown here) present a small and almost constant value below some cut-off frequency, which is the resonance frequency of the top layer of soil. Indeed, for very low frequencies, surface waves cannot build up in that top layer and take energy away from the foundation, so that the radiation damping is very low. Above that cut-off frequency, a large peak can be observed on the imaginary part (with the real part almost cancelling), indicating a resonance within the soil, that tends to soak energy away from the foundation. 4. COMPUTATION OF SIMPLE FORMULAS FOR PILE GROUPS In this section, we present the derivation of the simple formulas in the cases of the floating pile groups on homogeneous half-space and end-bearing pile groups, and using the ideas Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

8 8 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU 1 5 Normalized Real Part [ ] Normalized Imaginary Part [ ] Dimensionless Frequency [ ] Dimensionless Frequency [ ] Figure 4. Real (left) and imaginary (right) parts of the horizontal impedance matrix for different pile separations: s/d = 2 (solid line), s/d = 2.5 (dashed line) and s/d = 3.5 (solid-dashed line). The figures correspond to a pile group with E p/e s = 3 and R f /l p = Normalized Real Part [ ] Normalized Imaginary Part [ ] Dimensionless Frequency [ ] Dimensionless Frequency [ ] Figure 5. Real (left) and imaginary (right) parts of the rocking term of the dynamic stiffness matrix for different pile separations: R f /l p =.7 (solid line), R f /l p =.65 (dashed line) and R f /l p =.55 (solid-dashed line). The figures correspond to a pile group with E p/e s = 375 and s/d = 2. discussed above. Depending on the type of pile group, and on the type of element of the impedance matrix, more or less hidden variables are necessary to describe its behavior, and, correspondingly, more or less parameters are needed in the formulas Floating pile groups We first consider floating pile groups embedded in a homogeneous half-space. In that case, the variation of the dynamic stiffness with the frequency is rather smooth, as seen in Fig. 4 and 5. More precisely, the dynamic stiffness always has a parabolic variation while the damping coefficient is approximately linear. The parabolic decrease of the real part seems to indicate that a mass remains entrapped between the piles and vibrates in-phase with the cap. The hidden variables model predicts in all cases in the database (described in Table I) a two-dofs system, one for the sway and one for the rocking, and with no hidden variables. Note Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

9 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 9 Table I. The database of soil-pile group systems used to derive the simple formulations for floating pile groups on homogeneous half-space. The range of parameters is 25 E p/e s 15, 2 s/d p 3.6 and.55 R f /l p 2 and constant hysteric damping β s =.5. Sample Piles E p d E s B f l p s V s [-] [GPa] [m] [GPa] [m] [m] [m] [m/s] that the coupling term is negligible. On Fig. 6, a schematic drawing of a system corresponding to such impedance is presented. The superstructure is subjected to the seismic horizontal force f s. The elements of the normalized impedance can therefore be written Z h (a ) = ( k h a 2 m h ) + ia c h Z r (a ) = ( k r a 2 m r ) + ia c r Z sr (a ) =, (8) where the values of k h, c h, m h, k r, c r and m r depend on the case considered. Remember that the definition of the normalized frequency a is given in Sec. 2.2 and that the normalized values in these formulas (8) must be scaled by the static stiffness to yield the actual value of the impedance matrix, as described in Sec In previous works, the leading parameters for this type of pile groups were identified to be the ratio of Young s moduli E p /E s and the normalized separation of the piles s/d [2, 37], or the factor L = (E p I p /E s ).25 related to the active pile length [38, 25, 26]. We decide here to use as leading parameters the normalized radius of the foundation R f /l p and a normalized active pile length ratio L /s. We therefore provide equations of the parameters χ { k h, c h, m h, k r, c r, m r } Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

10 1 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU Figure 6. A schematic drawing of a simple model for floating pile group. in the form χ = λ ( Rf l p ) λ1 ( ) λ2 L, (9) s with the values of λ, λ 1 and λ 2 being provided for each of the parameters. A multiple regression analysis was then conducted with respect to the two quantities R f /l p and L /s, and lead to the values described in Table II. The regression coefficient R is also indicated in the same table to provide an indicator of the accuracy of the regression analysis. In general terms, the formulas in Table II corroborate the observed results that, for short separations between the piles, and for weak soils, both normalized dynamic stiffness and damping increase. Note that, as indicated by the zeros in Table II, the influence of the ratio R f /l p on the horizontal impedance is negligible, while it is rather important for the rocking term. Besides the uniform presentation in Table II, the reader may also find an expanded, non-normalized, version of the same formulas in App. 6, for easier reading End-bearing pile groups We then consider end-bearing pile groups. As stated earlier, their dynamical behavior is much more complicated than that of floating pile groups on homogeneous half-space. The structure of the approximation for the impedance matrix is therefore difficult to guess a priori and we use the hidden variables model in a very general setting. Note that, as the coupling term is negligible in the cases considered, the hidden variables model was identified independently on the horizontal and rocking terms of the impedance matrix. The identification of the hidden variables model for all the cases in the database described in Table IV suggest the consideration of three hidden variables for the sway and none for the rocking. Besides, no coupling in the stiffness for the first hidden variable and no coupling in the damping for the two others seemed to be necessary. The chosen structure for the end-bearing Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

11 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 11 Table II. Coefficients for the horizontal and rocking elements of the impedance of a floating pile groups on homogeneous half-space. ( ) λ1 Rf ( χ = λ L ) λ2 l p s λ λ 1 λ 2 R [%] k h c h m h k r c r m r = m h leq/ pile groups is therefore written, as a special case of Eq. (7) for the hidden variables model, Z h (a ) = ( k h a 2 a m h ) + ia c h + 2 c2 1 k 2 2 k 2 3 ( k 1 a 2 m1)+ia c1 ( k 2 a 2 m2)+ia c2 ( k 3 a 2 Z m3)+ia c3 r (a ) = ( k r a 2 m r ) + ia c r, Z sr (a ) = (1) and represented as a set of mass, springs, and dampers in Fig. 7. The previous observation for the coupling with the hidden variables can be translated in Fig. 7 by the fact the mass m 1 is linked to the foundation by a dashpot while the masses m 2 and m 3 are linked to it through springs. This fact arises from the presence of the cut-off frequency of the top layer of soil that was discussed in Sec More physical remarks can be made in the different frequency ranges defined by the resonance frequencies a α of the masses m α representing the hidden variables. In the low frequency range (a a 1 ), a first-order expansion gives Z h (a ) = k + ia ( c h + c 2 + c 3 ). (11) It is worth noticing that the slope of the imaginary part c + c 1 + c 2 + c 3 is not small since it allows to quickly reach the level of the hysteretical damping. In the range of resonance of mass m 1 (a a 1 2ζ 1 a 1, with ζ α = c α /(2 k α m α )), and supposing that all the resonance frequencies are far enough from each other (a a 2 2ζ 2 a 2 and a a 3 2ζ 3 a 3 ), one has [ Z h (a ) = k h + k 1 k a a 2 2 k a 2 ] 3 3 a2 a 2 3 a2 [ ( ) a 2 2 ( ) + ia c h c 1 + c 2 a 2 2 ] 2 a c 3 3 a2 a 2 3 (12) a2 which means that around a 1, the mass m 1 has the same displacement as the foundation, so that there is no damping contribution from c 1. For a a 2, masses m 2 and m 3 are also linked to the foundation but the dashpots c 2 and c 3 introduce some damping. The equivalent slope Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

12 12 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU Figure 7. A schematic drawing of a simple model for end bearing pile group. around a 1 tends to c + c 2 + c 3 = β eq /a 1 which is actually small as expected to model the sole hysteretical damping β eq. Usually c 1 a 1 β eq. In the range of resonance of the mass m 2 (an equivalent formula can be derived for mass m 3 ), a large imaginary part is brought on by k α 2ζ α, which corresponds to the peaks observed in Fig. 3. Finally, at high frequency (a a 3 ), one has ( ) Z h (a ) = k h c2 1 a 2 m h + ia c h (13) m 1 which classically corresponds to all the masses m 1, m 2 and m 3 being fixed. One can see c 1 as the radiative damping which occurs only above a 1 since for this frequency we have shown that the damping is only β eq /a 1. Thus this model reproduces the cut-off frequency at the resonance frequency of the layer. Once the structure of the approximation has been decided, a multiple regression analysis is performed on the same leading parameters as before, plus the ratio (ρ b V b )/(ρ s V s ) to yield the formulas presented in Table IV. Note that several coefficients appear as zeros in the table, which means that the parameters modeled do not have any influence on the formula. Note also that, as before, the formulas are presented in a non-normalized manner in App. 6 for easier reading. The general formulas for the parameters are χ = λ ( Rf H ) λ1 ( L s ) λ2 ( ) λ3 ρb V b. (14) It is particularly interesting to note that, although the formulas were derived from rather mathematical considerations (the hidden variables model and a regression analysis), they yield a very good evaluation of the resonance frequencies of the soil layer. Indeed, the first fundamental frequency of the soil layer ω s 1 = 2πV s /(4H) and k 1 /m 1 = 1.4V s /H (see App. 6 for non-normalized formulas) coincide. Likewise, the second fundamental frequency of the soil ρ s V s Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

13 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 13 Table III. The database of soil-pile group systems used to derive the simple formulations for end-bearing pile groups. The range of parameters is 125 E p/e s 75, 2.8 s/d p 4.4, 1 R f /H 2.1 and 3 V b /V s 8 and constant hysteretic damping β s =.5. Samples Piles E p d E s B f l p H V s V b s [-] [GPa] [m] [GPa] [m] [m] [m] [m/s] [m/s] [m] layer ω s 2 = 6πV s /(4H) is very well approximated by the third resonance of the simple model k3 /m 3 = 5.1V s /H. 5. IMPACT OF THE FORMULAS ON THE EVALUATION OF DESIGN QUANTITIES In this last section, we discuss the accuracy of the proposed formulas on two practical cases. More particularly, the accuracy of the predicted transfer functions, spectral acceleration on top of a building and relative displacement between top and bottom of the building, using the proposed formulas, is demonstrated. In a second test, we compare the accuracy of our proposed formula with another one from the literature Case 1 For this validation, a 1 1 end-bearing pile group is used, with piles with d p = 1 m, l p = 22 m, s = 5 m, and connected by a 1.1 m-thick, rigid, cap with B f = 25 m. The mechanical properties of the piles are E p = 3 GPa, ν p =.25, and ρ p = 25 kg/m 3. This pile group stands in H = 2 m-thick soil layer, with properties E s = 6 MPa, ν s =.4 and ρ s = 175 Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

14 14 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU Table IV. Coefficients for the horizontal and rocking elements of the impedance of an end-bearing pile group. χ = λ ( Rf H ) λ1 ( L ) λ2 ( ) λ3 ρb V b s ρ sv s λ λ 1 λ 2 λ 3 R [%] k = k h k 2 k c h = c + c m h = m k c m k c m k c m k r c r m r = m leq/ kg/m 3. The mechanical properties of the underlying half-space are E b = 1.5 GPa and ν b =.3 and ρ b = 2 kg/m 3. The real and imaginary parts of the impedance are shown on Fig. 8, both as computed using the numerical FE-BE model, and using the simple formulas of Eq. (1). The agreement between the two approaches is good, in particular for the shaking term, considering the important variability in frequency. Note that the pile group considered here was not used for the regression analysis that determined the parameters in Table IV We now turn to the observation of the accuracy of the proposed formulations for the estimation of engineering quantities of interest. We therefore consider a 6 m high-building (2 floors), with floors of 22.5 m 22.5 m, and 6 columns 6 columns. The slab weight per unit area is 5 kg/m 2 and the characteristics of the beams and columns are, respectively, EI = 5.1 MN.m 2 and EI = 1 MN.m 2. We first consider the estimation of transfer functions in two different cases: (1) using the entire, 6 6, impedance matrix computed from the FE-BE model, and considering both the kinematic and inertial interaction, and (2) using only the horizontal and rocking elements of the impedance matrix computed with the proposed formula (1) and neglecting the kinematic interaction. For both cases, the displacement field is decomposed on a basis that contains the rigid body modes of the building (l m ), which coincide with those of the foundation, and the flexible modes of the building on a rigid basis (φ n ): u(ω,x) = m c m (ω)l m (x) + n α n (ω)φ n (x) = [ c α ] { L Φ } (15) Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

15 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 15 Real part [N/m] 1 x Imaginary part [N.m] 2 x Real part [N/m] Imaginary part [N.m] 2 x x Figure 8. Comparison between the real (up) and imaginary (down) parts of the horizontal (left) and rocking (right) elements of the impedance matrix for a 1 1 end-bearing pile group computed using the simplified formulas (1) (dashed line) and the FE-BE model (solid line). where L is the matrix of the rigid body modes of the structure and Φ is the matrix of the eigenmodes of the structure clamped at its base. The response of the structure, taking into account soil-structure interaction, is then computed using the following formula ([ Z(ω) ] + (1 + 2iβ) [ Λ ] [ ω 2 MΓ M ΩΓ M ΓΩ I ]) { c(ω) α(ω) } { } Z(ω)c (ω) = where the diagonal matrix Λ contains the squares of the lowest circular frequencies of the structure on fixed base and I is the identity matrix arising from the orthogonality of the eigenmodes with respect to the mass matrix. Γ stands for the rigid body modes and Ω for the eigenmodes on fixed base, while c is the kinematic interaction. The differences between the two models with respect to this formulation are the impedance matrix Z(ω) and the kinematic interaction factor takes equal to u i (ω) with having null components but a unitary for the sway term. Besides, it is worth noticing that the simplified model does not correspond to the physical model sketched on Fig. 7 subjected to an uniform acceleration a i. Indeed, inertial forces are not applied on mass m 1, m 2 and m 3 since these masses are in the soil and have their inertial forces already balanced in the soil. The resonance frequencies of the soil are computed at f s 1 = 1.55 Hz and f s 2 = 4.6 Hz. Assuming a horizontal harmonic base motion at the bedrock, the horizontal transfer function at the free surface and at the top of the building are represented on Fig. 9. It clearly shows the effect of the soil-structure interaction, as well as the ability of the formulas (1) to compute the resonance frequency of the coupled system (the peaks of the dotted and dash-dotted lines on Fig. 9 coincide almost exactly). We then consider two real recordings of earthquakes, with different frequency contents (see Fig. 1) and peak ground accelerations (PGA) at about.3 g. On Fig. 11 a comparison is given of the spectral acceleration on top of the building computed in the two cases considered earlier of the FE-BE model supposing inertial and kinematic interaction and the simple formulas (1) neglecting the kinematic interaction. On Fig. (12), a comparison is given of the time histories of the relative displacements between the top and the base of the building. On both figures, the agreement between the two approaches is very good. (16) Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

16 16 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU Transfer Function [ ] Figure 9. Transfer function at ground surface free field (solid line), of the structure without SSI (dashed line), and of the structure with SSI (dashed-dotted line), all computed with the FE-BE approach, and transfer function of the structure with SSI computed with the proposed formulas (dotted line). The figures corresponds to a structure resting on a 1 1 end-bearing pile group. Acceleration [m/s 2 ] Acceleration [m/s 2 ] Time [s] Time [s] Spectral acceleration [m/s 2 ] Frequncy [Hz] Figure 1. Ground acceleration (left) and 5%-damped response spectra (right) recorded in Aegion (Greece) in 1995 (top and solid line), and in Friuli (Italy) in 1976 (bottom and dashed line) Case 2 In this example, we compare our simplified formulations 1 for the impedance of the endbearing pile group introduced in Sec. 3.2 with the simplified formulas proposed in [34, 21]. We use an equivalent with of R f = 8 m, because the formulas in Table IV are proposed for circular or square foundation. Fig. 13 shows this comparison, along with the value of the BE solution. Our formulas seems to behave at least as well as the previously available one. Remember that its range of application, in in particular in terms of the numbers of piles is much larger. Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

17 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS Spectral acceleration [m/s 2 ] Spectral acceleration [m/s 2 ] Figure 11. Comparison of the acceleration response spectra at the top of the building for the Friuli earthquake (left) and the Aegion earthquake (right), using the complete FE-BE model (solid line) and the simple formulation (dashed line). The figures corresponds to a structure resting on a 1 1 end-bearing pile group Relative displacement [m] Relative displacement [m] Time [s] Time [s] Figure 12. Comparison of the relative displacements between the top and the base of the building for the Friuli earthquake (left) and the Aegion earthquake (right), using the complete FE-BE model (solid line) and the simple formulation (dashed line). The figures corresponds to a structure resting on a 1 1 end-bearing pile group. 6. CONCLUSION Simple formulations have been derived for the dynamic stiffness matrices of pile group foundations subjected to horizontal and rocking dynamic loads. These formulations were found using a large database of impedance matrices computed using a FE-BE model. They can be readily employed for design of large foundations on piles and are shown to yield very accurate values of the estimated quantities of interest for building design. The formulations have been derived both for floating pile groups on homogeneous half-space and end-bearing pile groups in a homogenous stratum. They can be used for large pile groups (n 5), as well as for a large range of mechanical and geometrical parameters of the soil and the piles. They provide a first step towards code provisions specifically focused on pile footings. Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

18 18 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU 3 x x Dynamic stiffness [N/m] 1 Damping [N/m] Figure 13. Real (left) and imaginary (right) parts of the horizontal impedance matrix, for a 36 end-bearing pile group computed using the simplified formulas (1) (solid line) and BE solution (dashed line) and simplified analytical solution of [8] (dotted line). REFERENCES 1. Wolf JP. Foundation analysis using simple physical model. Prentice-Hall: Englewood Cliffs, New-Jersey, Kaynia AM. Dynamic stiffness and seismic response of pile groups. Research Report R82-3, Masshachussetts Institute of Technology, ; Waas G, Hartmann HG. Seismic analysis of pile foundations including soil-pile-soil interaction. Proceedings of the 8th World Conference on Earthquake Engineering San Francisco, July 1984; Takemiya H. Ring-Pile analysis for a grouped pile foundation subjected to base motion. Structural Engineering/Earthquake Engineering 1986; 3(1): Ohira A, Tazoh T, Dewa T, Shimizu K, Shimada M. Observation of earthquake response behaviours of foundations piles for road bridge. Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, July 1984; 3: Konagai K, Ahsan R, Maruyama D. Simple expression of the dynamic stiffness of grouped piles in sway motion. Journal of Earthquake Engineering 2; 4(3): Crouse CB, Cheang L. Dynamic testing and analysis of pile group foundation. In Dynamic Response of Pile Foundations-Experiment, Analysis and Observation, Nogami T. (ed). ASCE: New-York, 1987; Mylonakis G, Nikolaou A, Gazetas G. Soil-Pile-Bridge seismic interaction: kinematic and inertial effects. Part I: soft soil. Earthquake Engineering and Structural Dynamics 1997; 26(3): Hutchinson TC, Chai YH, Boulanger RW, Idriss IM. Inelastic seismic response of extended pile-shaftsupported bridge structures. Earthquake Spectra 24; 2(4): Wolf JP. Soil-structure-interaction analysis in time domain. Prentice-Hall: Englewood Cliffs, New-Jersey, Levine MB, Scott RF. Dynamic response verification of simplified bridge-foundation model. Journal of Geotechnical Engineering 1989; 115(2): Spyrakos CC. Assessment of SSI on the longitudinal seismic response of short span bridges. Engineering Structures 199; 12(1): Harada T, Yamashita N, Sakanashi K. Theoretical study on fundamental period and damping ratio of bridge pier-foubdation system. In Proceedings of the Japan Society of Civil Engineers 1994; 489(1-27): Chaudhary MS, Parakash S. Dynamic soil structure interaction for bridge abutment on piles. In Geotechnical Special Publication 1998;2: Spyrakos CC, Loannidis G. Seismic behavior of a post-tensioned integral bridge including soil-structure interaction (SSI). Soil Dynamics and Earthquake Engineering 23; 23(1): Tongaonkar NP, Jangid RS. Seismic response of isolated bridges with soil-structure interaction. Soil Dynamics and Earthquake Engineering 23; 23(4): Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating pile groups. Geotechnique 1988; 38(4): Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

19 SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS Gazetas G., Dobry R. Horizontal response of piles in layered soil. Journal of Geotechnical Engineering 1984; 11(1): Gazetas G, Makris N. Dynamic pile-soil-pile interaction. Part I: Analysis of axial vibration. Earthquake Engineering and Structural Dynamics 1991; 2(2): Makris N, Gazetas G. Dynamic pile-soil-pile interaction. Part II: Lateral and seismic response. Earthquake Engineering and Structural Dynamics 1992; 21(2): Mylonakis G, Gazetas G. Lateral vibration and internal forces of grouped piles in layered soil. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 1999; 125(1): Pender M. Aseismic pile foundation design analysis. Bulletin of the New Zealand National Society for Earthquake Engineering 1993; 26(1): ATC: Tentative provisions for the development of seismic regulations of buildings: cooperative effort with the design profession, building code interests and the research community. Washington, D.C, NEHRP, National Earthquake Hazards blackuction Program. Recommended provisions for the development of seismic regulations for new buildings. Washington, D.C, Randolph M. Response of Flexible Piles to Lateral Loading. In Geotechnique, 1981; 31(2): Poulos HG, Hull TS. The role of analytical geomechanics in foundation engineering. In Foundation Engineering: Current Principles and Practices, ASCE, 1989; 2: Gazetas G. Analysis of machine foundation vibrations: state of the art. In Soil Dynamics and Earthquake Engineering, 1983; 2(1): Wolf JP. Consistent lumped-parameter models for unbounded soil: physical representation. In Earthquake Engineering and Structural Dynamics, 1991; 2(1): Sieffert JG, Cevaer F. Handbook of impedance functions. Surface foundations. Ouest Éditions, Clouteau D, Taherzadeh R. Soil, pile group and building interactions under seismic loading. In Proceedings of the First European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland, September 26, in CDROM. 31. Taherzadeh R, Clouteau D, Cottereau R. Identification of the essential parameters for the lateral impedance of large pile groups. In Proceedings of the 4th International Conference on Geotechnical Earthquake Engineering, Thessaloniki, Greece, June 27, in CDROM. 32. Craig RJ, Bampton M. Coupling of substructures for dynamic analyses. AIAA Journal 1968; 6(7): Miura K, Kaynia AM, Masuda K, Kitamura E, Seto Y. Dynamic behaviour of pile foundations in homogeneous and non-homogeneous media. Earthquake Engineering and Structural Dynamics 1994; 23(2): Gazetas G, Hess P, Zinn R, Mylonakis G, Nikolaou A. Seismic response of a large pile group. In Proceedings of the 11th European Conference on Earthquake Engineering, Paris, September 1998, in CDROM. 35. Cottereau R, Clouteau D, Soize C. Construction of a probabilistic model for impedance matrices. Computer Methods in Applied Mechanics and Engineering 27; 196(17-2): Cottereau R, Clouteau D, Soize C. Probabilistic impedance of foundation: Impact of the seismic design on uncertain soils. Earthquake Engineering and Structural Dynamics 28; 37(6): Gazetas G. Seismic response of end-bearing single piles. International Journal of Soil Dynamics and Earthquake Engineering 1984; 3(2): Poulos HG. Behavior of laterally loaded piles: PART II - group piles. Journal of the Soil Mechanics and Foundations Division, ASCE, 1971; Cottereau R. Probabilistic models of impedance matrices. PhD thesis, Ecole Centrale Paris, France, ( Nikolaou S., Mylonakis G., Gazetas G. and Tazoh T. Kinematic pile bending during earthquake: analysis and field measurement. Geotechnique 21; 51(5): APPENDIX A In this appendix, the practical methodology for the construction of the reduced matrix S(a ) = K a 2 M + ia C is introduced. Three main steps are identified: The impedance of the FE-BE model is computed. More specially a set of values {Z(a l )} of the impedance matrix at a finite number of frequencies (a l ) 1 l L is computed. The set of values {Z(a l )} is interpolated to yield a matrix-valued rational function in the form a N(a )/q(a ), which approximates the behavior of the impedance matrix {Z(a )} of the model. The function a N(a ) is a matrix-valued polynomial in (ia ), Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6

20 2 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU and the function a q(a ) is a scalar polynomial in (ia ). Many methods can be used to achieve that goal. The identification of the matrices K, C and M from the polynomials a N(a ) and a q(a ) is then performed. This step does not involve any approximation and is further detailed in [39]. APPENDIX B In this appendix, we present an extended version of the formulas presented in Tables II and IV, in a non-normalized form. For the case of the floating pile groups on homogeneous half-space the coefficients appearing in the non-normalized version of Eq. (8) are ( k h = 6.8G s R L ).3 f s and c h = 5 GsR2 f V s m h =.4ρ s Rf 3 L s ( L s ) 1.6 ( ).6 k r = 8G s Rf 3 lp ( L R f ( c r = 5 GsR3 f V Rf s l L ).2 p s m r =.7G s Rf 4l ( L ).4 p s s ).4 The range of parameters is 25 E p /E s 15, 2 s/d p 3.6 and.55 R f /l p 2 and constant hysteric damping β s =.5. For the case of end-bearing pile groups, the coefficients appearing in the non-normalized version of Eq. (1) are ( ).5 Rf ( k = k h k 2 k 3 = 1G s R L ).35 f H s c h = c + c 1 = GsR2 f H L ρ b V b V s R f s ρ sv s m h = 1 2 ρ shrf 2, (19) R k 1 = 2.6G 2 f s H c 1 = 1.9 GsR2 f V s (HR f ) 1.5, (2) m 1 = 1.4ρ s Rf 2H ).35 s ) s L ( Rf k 2 = 1.25G s R f H ( c 2 =.4G s Rf 2 ρ b V b ρ svs 2 m 2 =.8ρ s Rf 2H s ρb V b L ρ sv s R k 3 = 16.1G 4 ( f L ) 3 ρsv s s H 3 s ρ b V b c 3 = 3 Gs R 4 ( f L ) 3 ( ) 1.5 ρsv s V s H 2 s ρ b V b R m 3 =.6ρ 4 ( f L ) 3 ρsv s s H 1 s ρ b V b L ρb V b ρ sv s (17) (18), (21), (22) Copyright 22 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 22; :1 6