A Generalization of the Heuristic Gradient Projection for 2D and 3D Frame Optimization

This paper introduces a generalization of the heuristic gradient projection (HGP) method for solving 2D and 3D frames. The main objective is to minimize the frame weight by means of size, topology and shape optimization considering stress constraint activation. HGP can give a specific iterative equation for each element cross section and loading type and consequently reach the optimum solution in a relatively smaller number of iterations compared to general heuristic recursive equations. However, the solution of frames with combined loads applied on the elements might converge slowly or oscillate around the constrained optimum value. Many approaches were investigated for the generalization of the HGP. However, the emphasis was always directed towards axial and bending loads. Although other types of loads may have an effect on the problem, like shear and torsion stresses in shafts or 3D frames. These types of loads are introduced into the optimization problem with more general algorithm. Weighting factors are utilized to give a weight to each stress type applied on each element. This factor is used to change the power of the HGP iterative formula for each element in the frame, which changes the power of the recursive formula according to the contribution of each loading type applied on the element. The proposed technique shows more accurate results in activating the stress constraints than previously developed HGP when dealing with combined loads, and keeps the advantage of the HGP in finding the optimum solution in a relatively small number of structural analyses. In the case studies several sample applications were solved to highlight the robustness of the proposed method.