Size and shape optimization of flexural structures with explicit sensitivity analysis

Abstract

While optimum design of flexural systems is a well stablished topic of structural optimization most applications only consider size variables. Also, sensitivity analysis of the stiffness matrix with respect to shape variables has been usually carried out by finite differences. In this paper an approach to size and shape optimization of flexural systems is presented using explicit differentiation of the stiffness matrix and the vector loading with respect to size and shape variables. Several examples starting from a simple one and proceeding to cases with a quite large number of variables are included in the paper. tion problem is solved considering the internal dimensions of the cross-sectional areas of the bar as design variables. In this paper an application of size and shape optimization of flexural systems will be presented. SENSITIVITY ANALYSIS APPROACHES First order sensitivity of a structural response ‘y with respect to design variable xi can be written as OPTIMUM DESIGN OF FLEXURAL SYSTEMS It is well know that expression (2) may be solved by two approaches. In the direct differentiation method du is obtained dxi Optimum design of flexural systems is a well stablished topic on structural optimization. Plastic and elastic analysis, with linear and non linear programming have been already considered. References [l-6] represent an overview of results in different problems and typologies including frames, grillages and beams. While the number of applications is large, all of them are related with size optimization. The design problem is posed usually in terms of the cross sectional areas of the bars which are considered design variables. Other mechanical parameters, as inertia modulus I or strength modulus W are linked to area A by using the following expressions I = u, A”1 W = a,,. AtIF (1) Sometimes, design optimization is made in several steps, by carrying out multilevel optimization techniques [7-81. In that approach, the optimum values of the areas are first identified then a second optimizathrough the equation An alternative approach starts up by introducing (3) into (2) and giving A vector, ussually called adjoint variable h is dellned as the result of a system of linear equations