Optimal shape design of a frame structure for minimization of maximum bending moment

In practical designs, the maximum bending moment in a given flexural structure always has an importance in estimating both the structural displacement (stiffness) and stress (strength) performance. It is of particular interest to designers to develop an appropriate procedure to minimize the maximum moment through an optimal design of the structure. For this purpose, this paper presents a heuristic optimization algorithm, called the evolutionary shift method, for minimization of the maximum bending moment by virtue of the structural shape or geometry optimization. First, the sensitivity analysis of the bending moment in a flexural member is investigated with respect to a general nodal coordinate by using the heuristic adjoint method. An explicit formulation of the moment derivative is derived in detail. Then, on the basis of the design sensitivity, the nodal positions are shifted in the steepest descent direction (the negative of the gradient) to reduce the maximum moment most effectively. Consequently, the optimal solution is achieved gradually. Finally, three classical examples are presented to demonstrate the validity of the sensitivity establishment and the capability of the proposed optimization algorithm for solving the maximum-moment minimization problem. Results show that the structural shape optimization can reduce the maximum moment enormously, and deserves more attention in engineering applications.