Despite the importance of plates in structural analysis the flexural analysis of plates under parabolic load has not been extensively studied. This paper presents single finite sine transform method for exact bending solutions of simply supported Kirchhoff plate under parabolic load. The governing equation of equilibrium is a fourth order non-homogeneous differential equation in terms of the deflection The considered thin plate problem has Dirichlet boundary conditions at all the edges. This recommends the use of the finite sine integral transform method whose sinusoidal kernel function satisfies the boundary conditions. The sinusoidal function of x used for the sine transform kernel in this paper satisfies the Dirichlet boundary conditions along edges. The transformation simplifies the problem from a partial differential equation (PDE) to an ordinary differential equation (ODE) in the transformed space. The general solution, obtained using methods for solving ODEs is found in terms of unknown constants of integration which are found by using the finite sine transform of Dirichlet boundary conditions along the and edges. The solution in the physical domain space variables is then found by inversion as a rapidly convergent single series with infinite terms. A one term truncation of the single infinite series yields center deflection solution that is only 2% greater than the exact solution. A three term truncation of the infinite series for gave exact center deflections. Bending moments are found using the bending moment deflection relations as convergent single series with infinite terms.
This paper attempts to obtain bending solutions to plates under uniformly distributed and hydrostatic load distributions using Ritz variational methods and basis functions that are found by superposing trigonometric series and third degree polynomials. Two cases of boundary conditions were considered. In one case, three edges were simply supported and the fourth edge was clamped (SSCS thin plate). In the second case, the adjacent edges were clamped and the other edges were simply supported (SCCS thin plate). This work presents first principles, rigorous derivation of the governing Ritz variational functional and the displacement basis functions for the boundary conditions investigated. The solution is presented in analytical form. The obtained results are compared with previous results obtained using Levy series and Ritz methods and found to be in close agreement . The disadvantage of the method is the associated computational rigour, but the benefit is the accuracy of the results. Comparisons of the present results for center deflections and center bending moments with results in the literature show that there is negligible difference. Double series expressions were found for deflections and bending moments for the plate bending problems solved. Evaluation of the double series expressions at the plate center gave center deflection results that differed from the exact solutions by for to for for uniformly loaded thin plates with three simply supported edges and one clamped edge (SSSC). The differences in the center bending moments Mxx were found to vary from for to for In general, the present results yielded reasonably accurate solutions for the plate bending problems studied.
Thin plate bending analysis is an important research subject due to the extensive use of plates in the different fields of engineering and the need for accurate solutions. This article uses the Ritz variational method and a superposition of trigonometric and polynomial basis functions to solve the Kirchhoff-Love plate bending problems (KLPBPs). The unknown displacement function in the Ritz variational functional (RVF) to be minimized is sought as linear combinations of basis functions Fm(x) and Gn(y) that are found by superposing sine series and third degree polynomial functions with the polynomial parameters determined such that all boundary conditions of deformation and force are satisfied. The displacement is thus expressed in terms of unknown displacement parameters Amn which are found upon minimization of RVF with respect to Amn. The minimization process gave a matrix stiffness equation in Amn with the stiffness matrix and force matrix found from Fm(x) and Gn(y) and their derivatives. The algebraic equation is solved, and the deflection and bending moments obtained. The problems considered were clamped (CCCC) plates under uniform and hydrostatic distribution of loads and plates with opposite edges clamped, the rest simply supported (CSCS) under uniformly distributed loading. Comparison of the solutions by Generalized Integral transform method, Levy-Nadai series method, and symplectic eigenfunction superposition confirms that the present results are accurate.