Simple formulas for estimating a lumped parameter model to reproduce impedances of end-bearing pile foundations

The paper presents simple formulas to evaluate parameters of a lumped system reproducing the frequency-dependent dynamic stiffness of end-bearing pile foundations. The model can be implemented in commercial finite element software and allows performing inertial soil-structure interaction analyses of structures considering the coupled roto-translational, vertical and torsional behaviour of the soil-foundation system. Pile groups arranged in a square layout are considered; the soil profile is constituted by a homogeneous deformable soil layer overlying a bedrock where piles are socked for a fixed length. Formulas are calibrated with a nonlinear least square procedure, based on data provided by an extensive non-dimensional parametric analysis of the soil-foundation systems, performed with a Winkler's type model, previously developed by the authors, which assumes soil and piles to behave linearly. Firstly, the suitability of the adopted numerical tools in capturing the dynamic stiffness of end-bearing foundations is proven. Then, capabilities of the proposed formulas in estimating parameters of the best lumped systems are shown, and, for some case studies, comparisons of non-null terms of the impedance matrix obtained through the best lumped system, the one computed through the formulas, and the impedances resulting from the dynamic analyses, are presented. Finally, some applications of the proposed formulas in the framework of the seismic soil-structure interaction analysis of bridges are shown to demonstrate the capability of the adopted lumped system and the formula efficiency in assuring a reliable evaluation of the superstructure seismic response, with respect to that obtained from a more rigorous approach.