Fourier-Bessel Transform Method for Finding Vertical Stress Fields in Axisymmetric Elasticity Problems of Elastic Half Space Involving Circular Foundation Areas

Page:

207-216

DOI:

https://doi.org/10.18208/ama_a.550405

OPEN ACCESS

Abstract:

In this work, the Fourier-Bessel transformation method was used to determine the vertical stress fields in axisymmetric elasticity problems of elastic half space involving circular foundation areas subject to uniformly distributed loads. A stress-based formulation of the elasticity problem was adopted. The biharmonic stress compatibility equation was solved using the variable separable technique to obtain a general solution for the bounded stress-functions as Fourier-Bessel integrals. Egorov expressions for the vertical stress fields defined in terms of harmonic functions were used to obtain the vertical stress fields. The load distribution was similarly transformed by the Fourier-Bessel transformation. Enforcement of the boundary condition of the equilibrium of the internal vertical stress at the *z* = 0 plane and the applied load yielded the unknown parameter of the bounded Fourier-Bessel transform integral, and thus, the full determination of the bounded stress function $\Omega(r, z)$. The vertical stress fields $\sigma_{z z}(r, z)$ were obtained from the bounded stress potential function using Egorov expressions for $\sigma_{z z}(r, z)$ Evaluation of the integration problem yielded mathematical expressions for the vertical stresses at any point in the elastic half space. The vertical stresses at any point under the center of the circular foundation were also determined, and tabulated. The mathematical expressions for vertical stresses obtained using Fourier-Bessel transform method were identical with solutions in the technical literature.

Keywords:

*Fourier-Bessel transform method, axisymmetric elasticity problem, circular foundation, elastic half space, vertical stress field, stress potential function*

1. Introduction

2. Research Aim and Objectives

3. Statement of Research Problem

4. Theoretical Framework / Governing Equations

5. Methodology

6. Results

7. Discussion

8. Conclusions

Appendix

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