Shape optimization of frame structures through a hybrid two-dimensional analytical and three-dimensional numerical approach

In this work, we propose a method for the shape optimization of frame structures using a mixed analytical–numerical approach. The goal is to achieve a uniform-strength frame structure, ensuring optimal material utilization and weight minimization. The optimization is performed in two calculation steps. The first step uses an analytical model based on the Timoshenko beam theory, where appropriate mathematical steps give a uniform-strength shape of the entire structure. Depending on the type of cross-section analyzed, the exact uniform-strength profile of each element is derived by solving for three parameters related to the forces and moments acting on the element. These parameters are obtained by solving a nonlinear system of equations, which includes the external and internal kinematic constraints of the structure, as well as equilibrium equations for each element. However, the solution obtained using the one-dimensional theory is limited in areas affected by boundary effects, such as the interconnection regions between elements and those near the supports, for a decay distance at least equal to the characteristic diameter of the section. To address this limitation, the second optimization step involves incorporating solutions that account for a triaxial stress field. This is typically carried out by discretizing the structure using the finite element method. The frame geometry obtained from the previous analytical solution is constructed, and the regions affected by boundary effects are optimized using the Biological Growth Method (BGM). This is an iterative, bio-inspired method modeled on the growth of trees, which increases trunk diameter in proportion to the loads experienced. The method is applied simultaneously to all regions where three-dimensional effects are significant, with the aim of achieving uniform strength in areas influenced by boundary effects. An important aspect of applying the BGM is maintaining the topology of the initial mesh, which is ensured through the use of mesh morphing techniques. The results of the two-step optimization process are shown on simple geometries involving few elements, and on more complex geometries of mechanical interest.