Optimum design of structures with stress and displacement constraints using the force method

A new structural analysis and optimization algorithm is developed to determine the minimum weight of structures with the truss and beam-type members under displacement and stress constraints. The algorithm combines the mathematical programming based on the sequential quadratic programming (SQP) technique and the finite element technique based on the integrated force method. The equilibrium matrix is generated automatically through the finite element analysis while the compatibility matrix is obtained directly using the displacement–deformation relations and the single value decomposition (SVD) technique. By combining the equilibrium and compatibility matrices with the force–displacement relations, the equations of equilibrium with the element forces as variables are obtained. The proposed method is extremely efficient to analyze and optimize the truss and beam structures under stress and displacement constraints. The computational effort required by the force method is found to be significantly lower than that of the displacement method. The effect of the geometric nonlinearity in the structural optimization problems under the stress and displacement constraints were also investigated and it is illustrated that the geometric nonlinearity is not an important issue in these types of problems and hence, it does not affect the final optimum solution significantly. Four examples illustrate the procedure and allow the results to be compared with those reported in the literatures.