Shadlou, M. & Bhattacharya, S. Ge ´otechnique [http://dx.doi.org/10.1680/geot.13.P.107] 1 Dynamic stiffness of pile in a layered elastic continuum M. SHADLOU  and S. BHATTACHARYA† A set of formulas for the dynamic stiffness of a pile (spring and dashpot coefficients) to use in inertial interaction analysis is proposed, utilising elastodynamic solutions. The method is based on solving a Lagrangian system of coupled equations for the pile and the soil motions for a range of vibration frequencies and also by considering the vertical, radial and angular stresses on the pile–soil interface. The solution extensively uses Bessel functions of the second kind and results are compared with finite-element models and field pile load tests. A dimensionless frequency related to the well-known active length of pile is proposed to separate inertial and kinematic interactions. A formula is also proposed for estimation of the active length of a pile in a two-layered soil. A specific depth is introduced beyond which soil layering does not have any appreciable effects on dynamic stiffness. It is commonly (rather arbitrarily) assumed that the first natural frequency of soil strata differentiates radiation (geometric) damping from hysteretic (material) damping for both types of interactions of the pile–soil system. In contrast, this paper proposes a new formulation based on relative pile–soil stiffness and frequency of the pile head loading to differentiate these two classes of damping behaviour. The application of the formulation is shown through an example. KEYWORDS: dynamics; piles; soil/structure interaction; stiffness INTRODUCTION One of the important parameters for dynamic analysis of a soil–pile system using the beam-on-dynamic Winkler foun- dation (BDWF) method is the dynamic stiffness of soil–pile elements. To obtain the bending moment in a pile affected by inertial or kinematic loading, and to analyse the behav- iour of a superstructure supported on a pile embedded in a layered soil, the accuracy of estimation of the pile–soil stiffness is very important (e.g. see Mylonakis et al. (1997) for linear soil–pile interaction effects and Liyanapathirana & Poulos (2010) for liquefied ground). The popular plane strain assumption developed by Novak (1974) and Gazetas & Dobry (1984a) is based on the translational vibration of a cylindrical rigid disc, embedded in a solid continuum. Owing to the nature of the formula- tion, the dynamic stiffness of each layer is represented by a unique value related to soil and pile properties, and the effects of shear distortion between each layer are ignored. Kavvadas & Gazetas (1993) proposed a spring coefficient for one- and two-layer soils considering kinematic inter- action obtained from back analysis of the pile bending moment from the finite-element model. Damping coefficients are usually represented by hysteretic or radiation coeffi- cients. The radiation damping formulation (see Table 1) proposed by Gazetas & Dobry (1984a, 1984b) is the main assumption in many research investigations (e.g. see Mylo- nakis et al. (1997), Liyanapathirana & Poulos (2010) and Dezzi et al. (2010)). In the current research, this parameter will be evaluated for three-dimensional (3D) soil–pile dy- namic interactions, and a new formulation will be proposed for one- and two-layer soils. The plane strain dynamic stiffness proposed in the current research can be distin- guished from other research because of the consideration of 3D effects in the interaction phenomenon. The main motiva- tion behind the study is to develop a robust, but also numerically efficient framework for computation of dynamic Winkler coefficients of pile–soil elements. BRIEF LITERATURE REVIEW OF DYNAMIC STIFFNESS OF PILES Because the analysis of pile behaviour using the modulus of subgrade reaction approach requires knowledge of its variation along the pile, Biot (1937), Vesic (1961), Davisson & Gill (1963) and Bowles (1997) proposed a constant subgrade modulus for each layer of soil. Broms (1964) presented a model for subgrade reaction varying with depth. Based on the one-dimensional (1D) wave propagation ideali- sation (Lysmer & Richart, 1966; Berger et al., 1977), Gazetas & Dobry (1984b) developed a simple model for radiation damping coefficient comparable with the plane strain case of a single pile embedded in different strata (Novak, 1974; Nogami & Novak, 1977; Dobry et al., 1982; Dobry & Gazetas, 1988; Makris & Gazetas, 1992). Yoshida & Yoshinaka (1972), Roesset (1980), Kavvadas & Gazetas (1993), Mylonakis (2001) and Sica et al. (2011) proposed other types of dynamic stiffness characterised by a function of elastic modulus of each layer. Kagawa & Kraft (1980) developed plane strain dynamic stiffness valid for one-layer soil and validated the spring coefficient proposed by Yoshida & Yoshinaka (1972), and the radiation damping model proposed by Berger et al. (1977). Tokimatsu & Nomura (1991) and Cubrinovski & Ishihara (2004) proposed a reduc- tion factor depending on pore-water pressure generation for considering dynamic resistance from liquefiable strata. A summary of the aforementioned research findings is shown in Table 1. ELASTODYNAMIC FORMULATION Theory and assumptions A single pile of length L p and diameter D p modelled by Euler–Bernoulli beam theory and surrounded by an n-layered, isotropic, linear elastic soil is considered for the Manuscript received 1 July 2013; revised manuscript accepted 23 January 2014. Discussion on this paper is welcomed by the editor.  Department of Civil Engineering, University of Bristol, Bristol, UK. † Department of Civil and Environmental Engineering, University of Surrey, UK.