R-function method and variational method for the bending problem of functionally graded plates with fixed supports and complex shapes

In engineering practice, analytical solutions for the bending problem of functionally graded plates are usually available only when the boundary conditions are simple. When using numerical methods like the variational method to solve the problem, trial functions can generally be constructed only for simple-shaped boundaries. In contrast, the R-function method can be effectively used to address problems with complex boundary shapes. This study integrates the R-function theory with the variational method to investigate the bending problem of functionally graded plates with complex boundaries. By employing the R-function theory, complex regions can be represented as implicit functions, which facilitates the construction of trial functions that satisfy complex boundary conditions. The paper elaborates on the variational principle and R-function theory, derives the variational equation for the bending problem of functionally graded plates, and validates the feasibility and accuracy of the method through numerical examples of rectangular, U-shaped, and L-shaped functionally graded plates. The results are compared with those from other literature and the finite element method (FEM) using ANSYS, showing good agreement.