In this study, the size optimization problem of a beam, in which the beam heights are determined along the beam axis, is considered as a shape optimization problem of a plate in plane elasticity. To achieve the shape optimization, different principles such as structural analysis, sensitivity analysis and mathematical programming are interrelated. The ANSYS that is a reliable finite element package programme is used only for the static analysis. The objective of this optimization problem is to minimize the volume of the beam under the constraints that the maximum value of the von Mises stress in each node do not exceed a predefined value and the beam axis is to remain straight. Design variables are the vertical coordinates of the corner nodal points of design elements. The design sensitivities are calculated through the finite, difference method and linear programming technique is used for obtaining the final shape of a beam. Several examples are solved under different loadings and boundary conditions. The obtained optimum beam volumes and the design variables are given in tables and the optimum beam shapes are plotted. One of the results given is compared with the literature.