Three Applications of the Reciprocal Theorem in Soil - Structure Interaction

Using the Reciprocal Theorem and the principle of superposition, this dissertation examines three procedures for obtaining the impedance functions and driving force vectors necessary for a soil-structure interaction analysis. The first procedure employs a volume integral of body forces and the displacement Green’s Functions to obtain the frequency dependent force-displacement relationship (impedance functions) for a rigid embedded foundation and the generalized force vector required to hold the rigid foundation fixed while subjected to incident waves (driving force). Although a formulation based on a volume integral requires more computation than the others based on surface integrals, this procedure proves to be numerically stable and is an excellent tool for linear soil-structure interaction analyses. The second procedure is based on integrating the surface displacements with the traction Green’s Functions and surface tractions with the displacement Green’s Function, over the surface of a rigid foundation. This well-known formulation has numerical instability problems at the resonant frequencies of the medium interior to the foundation surface. This dissertation explains the reason behind the instability using an analytical solution of the Hilbert-Schmidt method and recommends an algorithm which uses the l’Hospitale Rule to obtain the solution at those critical frequencies. The development herein for this method is not meant to be a practical solution, but an academic exercise to explain a long-standing, puzzling problem. The third and final procedure is to design a radiation boundary for a three-dimensional finite element grid using the surface displacements and surface tractions at the discrete artificial boundary, along with the Green’s Functions, to calculate outgoing wave motion at nodes immediately outside of the artificial boundary, thereby eliminating unwanted reflection of the outgoing waves. A procedure is recommended to obtain required data from a commercially available software program. A modal method can be employed to maximize the efficiency of the analysis.