Single Finite Sine Transform Method for Exact Bending Analysis of Simply Supported Kirchhoff Plate under Parabolic Load

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.