Dynamic soil-structure interaction in a 3D layered medium treated by coupling a semi-analytical axisymmetric far field formulation and a 3D finite element model

摘要

A strategy for dynamic soil-structure interaction problems involving three-dimensional layered soil is proposed. It is based on coupling an axisymmetric formulation for the regular layered far field region with a general three-dimensional finite element model of the near field via a cylindrical near field / far field interface. Using a virtual work statement, an axisymmetric scaled boundary finite element method (SBFEM) for the elastodynamic analysis of 3D layered continua is derived. Here, a vertical line, which is rotated around the origin of the cylindrical coordinate system is discretized in the finite element sense. A nonlinear differential equation in dynamic stiffness is obtained for each term of the Fourier series describing the circumferential variation of displacements in the far field, which must be solved numerically for each frequency of interest. While this leads to considerable savings in numerical cost compared to the effort associated with calculating a fully coupled stiffness matrix for the 3D problem, the numerical efficiency can be further increased by re-casting the problem using the method of weighted residuals. In doing so, a link with the well-known thin-layer method is established. The latter leads to a standard eigenvalue problem for the calculation of wave numbers and modes. The axisymmetric stiffness formulation obtained using one of the two aforementioned techniques is coupled with the 3D finite element model of the near field via the dynamic stiffness matrix relating nodal forces to nodal displacements at the cylindrical interface. Following a physically motivated approach, individual columns of the latter are obtained by considering situations where a unit displacement is assumed for the corresponding degree of freedom and zero displacements elsewhere and by expanding the corresponding displacement field into a Fourier series. The proposed strategy is applied to various soil-structure interaction problems involving flexible foundations of irregular shape resting on layered ground over rigid bedrock.