# Dynamic Response of Layered Medium

## Document Type

Thesis - University Access Only

1994

## Degree Name

Master of Science (MS)

## Department / School

Mechanical Engineering

## First Advisor

Hamid R. Hamidzadeh

## Abstract

Ming-Her Shen

Dynamic response of a multi-layer elastic medium subjected to harmonic surface loading is considered. In development of the analytical solution, the threedimensional theory of elastodynamic is utilized for derivation of the governing partial differential equations for each layer. These equations are solved in the Fourier domain by employing the Double Complex Fourier Transform technique. In the analysis, each layer of the medium is assumed to be extended infinitely in the horizontal x, z directions and has uniform depth in they direction and is considered to be linearly elastic, homogeneous, and isotropic. Utilizing the Integral Fourier Transform, displacements and stresses at any point in each layer can be determined in terms of boundary stresses for each layer. Also, the analysis provides the relation between stress and displacement vectors for the top and bottom of each layer in matrix notation. By satisfying the compatibility of displacements and stresses for each interface, a propagator matrix relating displacements and stresses at the top of the medium to the bottom interface will be obtained. This relates displacement and stress vector on the top surface to the bottom interface by eliminating similar information at each intermediate interface. Analytical inverse transform of the result into the original physical coordinates is a difficult procedure for each layer. Therefore, for multilayered medium, the analytical procedure is not considered. This propagator matrix embodies all of the essential mechanical properties of the medium, and can be used to derive solutions to problems of elastodynamics, such as soil-structure interaction. Particularly, the displacements on the surface can be computed for any given exciting surface distribution. The method developed in this paper is ideally suited for a computer to numerically evaluate the results. The Fourier transform inversion can be accomplished by means of the Fast Fourier Transform (FFT). The method has the advantage of enforcing interface continuity conditions automatically. The technique is applied to a layered system, in which it can predict local as well as global behavior with equal ease.

## Library of Congress Subject Headings

Materials -- Dynamic testing
Elastic analysis (Engineering)
Fourier transformations -- Computer programs

application/pdf

## Publisher

South Dakota State University

COinS