Exact and Approximate Bending Solutions for Rectangular Plates with Elastic Boundary Supports: An Example for the Top Slab of Box Culverts

Rectangular plates with elastic boundary supports are widely employed in various engineering practices, and the challenge lies in the discontinuity of basis functions of classical Fourier series solutions at the boundaries. In this paper, the top slab of box culverts was taken as the research object, and the bending problem of the plate was solved by exact and approximate methods. First, the modified Navier method and Li’s method were generalized and applied. The modified Navier method expands the partial derivatives of the deflection inconsistent with the boundary conditions using the Fourier series, while Li’s method calculates the deflection as the superposition of the double Fourier cosine series and the supplementary functions. The results indicate that the modified Navier method has better convergence compared to Li’s method in the case of using appropriate basis functions. Second, a modified Galerkin method was proposed based on the generalized variational principle, which relieves the limitation of the traditional Galerkin method on the force boundaries and deals with the terms containing Poisson’s ratio in the boundary conditions. Compared with traditional approximate methods, the modified Galerkin method demonstrates higher accuracy. The modified Galerkin method was utilized to solve the bending problem of thin plates with large deflections, taking the top slab as an application scenario.