Sensitivity analysis for sectional stiffness of anisotropic beams: The direct and adjoint methods

This paper concerns with the determination of sectional stiffness matrix and its sensitivity for anisotropic beams with initial curvatures. The governing equations of a three-dimensional beam is formulated to a Hamiltonian system, and the reduction of the Hamiltonian matrix leads to a set of singular, linear equations of the sectional warping and compliance matrix. The left and right null spaces of the singular linear system are explicitly constructed, and the existence and uniqueness of sectional warping and compliance matrix are justified. The direct and adjoint methods for sensitivity analysis of sectional stiffness matrix are developed. The existence and uniqueness of solutions for singular equations resulting from the direct and adjoint methods are proved formally. Numerical examples are presented to validate the proposed methods. The predictions of sensitivities from the direct and adjoint methods are shown to agree well with results of the complex-step method.