Symmetry analysis of an elastic beam with axial load

TL;DR

This paper applies Lie symmetry methods to derive a closed-form solution for an elastic beam with axial load and variable properties, providing an analytical approach to a complex PDE and validating it against numerical solutions.

Contribution

It introduces a novel application of Lie symmetry analysis to solve a complex, variable-coefficient PDE governing elastic beams with axial load.

Findings

Closed-form solutions derived using Lie symmetry methods.

Analytical solutions validated against numerical results.

Applicable to beams with spatially varying properties.

Abstract

We construct the closed form solution of an elastic beam with axial load using Lie symmetry method. A beam with spatially varying physical properties such as mass and second moment of inertia is considered. The governing fourth order partial differential equation with variable coefficients which is not amenable to simple methods of solution, is solved using Lie symmetry. We incorporate boundary conditions and then compare with the numerical solution.