An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

Abstract

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

Keywords

• bending
• nonclassical boundaries
• anisotropic thin plates
• analytical solutions
• improved Fourier series solutions