Structural optimization with frequency constraints using the finite element force method

A structural optimization algorithm is developed to minimize the weight of structures with truss and beam-type members under single- or multiple-frequency constraints. The cross-sectional areas of the structural members are considered as the design variables. The algorithm proposed combines the finite element technique based on the integrated force method with the mathematical programming technique. The equilibrium matrix is generated automatically using the finite element analysis, and the compatibility matrix is obtained directly using the displacement-deformation relations and the single value decomposition technique. When combining the equilibrium and the compatibility matrices with the force-displacement relations, the frequency eigenvalue equations are obtained with element forces as variables. Three structures, composed of truss and frame-type members, are studied to illustrate the procedure, and the results are compared with the literature. It is shown that, in structural problems with multiple frequency constraints, the analysis procedure (force or displacement method) significantly affects the final optimum design. The structural optimization based on the force method results in a lighter design. The proposed structural optimization method is efficient to analyze and optimize both truss and beam-type structures.