Shape optimization of 2D beam-structures to obtain the uniform strength

Abstract

This paper provides a new method to design the optimal shape of 2D frame structures even (or especially) if not statically determined. The optimization here addressed involves obtaining the uniform strength of the whole structure, but the proposed procedure can be followed for other optimization goals. Two types of beam cross-sections are studied, in which the solution in terms of balance, kinematic and beam-shape is found analytically and involves six unknowns for every beam element. For frame structures, some internal and external constraint conditions must be satisfied; using the analytical solution within the boundary conditions, the optimization problem boils down to finding the roots of a non-linear system. This way allows to work inside the optimization workspace, avoiding the use for each iteration of other solvers (e.g. F.E. software) and obtaining the solution with a high-computation speed. Two tests are shown, which result in a uniform-strength behaviour of the maximal stress at every cross section, and a lightness gain higher than 70% by respect to the reference initial structure. This technique can be extended to other types of cross-sections or 3D frame structures.