Reinforcement minimization of beams with axial forces

Within the confines of absolute minimum weight design of plastic structures, it has generally been assumed that the relationship between material (cost) and the vector of internal actions is linear (or piecewise linear) that is, sandwich approximations, or essentially constant lever-arm assumptions for reinforced materials, have been employed. Where both axial load and moment act on the one cross-section, or moments are relatively large, such idealization of cost function is no longer consistent with actual behaviour. A nonlinear cost function model must be substituted. The cost-function for an idealized reinforced section under bending moment and axial load (tensile or compressive) is derived herein. Two examples of encastre reinforced beams with axial forces are considered. When the applied load consists of a transverse central point load the optimum design is found to accord with earlier results employing a simplified cost function. When the applied lateral load is uniformly distributed earlier results are only approached in the limit as the lateral load approaches infinity.