Stress Analysis of a Square Elastic Body Under Biaxial Loading Using Airy Stress Functions

TL;DR

This paper develops analytical solutions for stress distributions in square elastic bodies under biaxial loads using Airy stress functions, providing insights into boundary conditions and stress variations.

Contribution

It introduces closed-form solutions for stress fields in square bodies under biaxial loading, extending previous work with series expansions satisfying boundary conditions.

Findings

Stress solutions match experimental photoelastic patterns.

Biaxial loading results in spatially varying principal stress differences.

Superposition approach effectively models combined loading scenarios.

Abstract

This study presents an analytical investigation of stress distributions in square-shaped elastic bodies subjected to concentrated compressive loads under uniaxial and biaxial conditions. By employing the Airy stress function method, we derive closed-form solutions that satisfy the governing biharmonic equation and the prescribed boundary conditions along the edges of the square domain. The stress components are expressed as series expansions, with coefficients determined to enforce boundary constraints. In the uniaxial compression case, the resulting stress fields exhibit strong agreement with photoelastic fringe patterns previously observed in experimental studies. For biaxial loading, the solution represents a superposition of two orthogonal compression scenarios, producing spatial variations in the principal stress difference depending on the location within the domain.