Applications of the finite Fourier sine transform to two-dimensional elastic beam problems

We use the finite Fourier sine transform method in order to determine the stress and displacement distributions in the two-dimensional elastic beams. We study on the two plane-stress problems of elastic beams in Cartesian coordinates in which the Saint-Venant approximation can be considered. In the first problem, a simply supported rectangular beam under transverse sinusoidal load on its top side and in the second problem, a long rectangular beam subjected uniform tension at each lateral side are considered. These problems are formulated using Airy stress function method as the fourth-order biharmonic equations. The prescribed boundary conditions for each problem are also written in terms of Airy stress function. These biharmonic equations are reduced to the fourth-order, linear, constant coefficient ordinary differential equations by applying the finite Fourier sine transform.The computed solutions of these ordinary differential equations are inverted to find the solutions of the biharmonic equations and then, the solutions of the orginal problems are obtained. These solutions coincide with the results in [20]. In order to understand the nature of the solutions, better and to clarify the results, the computed solutions are plotted by using Matlab software.