Sensitivity analysis and optimization of truss/beam components of arbitrary cross-section II. Shear stresses

This paper presents a general approach for detailed analysis and design optimization of arbitrary cross-sections of truss/beam built up structures. The approach allows arbitrary shape parametrization of 2-D cross-sections, as long as the coordinates of the contour vertices and their velocities are available, and is well suited for integration with existing CAD modelers and FEM analyzers. It leads to an inexpensive 2-D size/shape optimization in an alternative to costly 3-D shape optimizations, virtually impossible for real-life built up structures. Any composite multi-contour cross-section is first discretized with elementary triangles. Direct integration on the surface, using closed-form formulas, allows computation of the cross-section axial properties. Numerical integration on the boundary, along the line segments used to describe the contour, allows the computation of the shear properties. The power-series method is used to obtain the equilibrium equations and their governing linear warping system. The design sensitivities are calculated by the direct differentiation method requiring only backward substitutions on the triangular stiffness matrix. Numerical tests extensively verify the accuracy and the practical use of the formulation and implementation.