Integrated topology and size optimization for frame structures considering displacement, stress, and stability constraints

The challenge for practical application of frame structural optimization had previously been investigated by many works, while the mechanical performance requirements such as the displacement, stress, and stability requirements, were often considered separately within optimization, hindering their practical applications. For this purpose, an integrated topology and size optimization strategy of frame structures, in which the structural weight is taken as the objective with the constraints regarding the displacement, stress, as well as stability, is presented in this paper. Different from former researches, each beam is assigned with a topology variable representing the presence of the beam and a size variable correspond to the cross-sectional geometric properties. To achieve an optimized design with standard members, by cooperating the ordered multi-material SIMP (solid isotropic material with penalization) interpolation with the normalized Heaviside functions, the continuous size design variables are projected onto the discrete standard sizes conformed to standard library. Moreover, the comprehensive measure, including the stress relaxation, the pseudobuckling mode treatment scheme, the aggregation constraint, and varying constraint limit schemes, is employed to deal with the multiple constraints in the optimization model. Then, the sensitivities of the objective and constraint functions with respect to topology and size design variables are derived, respectively, and the proposed integrated optimization problem is solved by a nested optimization algorithm. Finally, several numerical examples are presented to demonstrate the feasibility of the proposed approach.