Geometrically Nonlinear Structural Analysis by Direct Stiffness Method

Castigliano's theorem is employed to derive the general equilibrium equations governing geometrically nonlinear structural behavior. Three different formulations are presented for the solution of these basic relations, namely, Newton-Raphson, incremental, and initial-value. The application of these solution procedures is considered and the significance and interrelation of various terms in each formulation is pointed out. In each formulation, a particular matrix, termed the nonlinear stiffness matrix, is seen to appear. The individual coefficients of this nonlinear stiffness matrix are presented for pin-jointed bar and plane-stress triangular elements. The nonlinear stiffness coefficients are seen to be simple and easy to apply since they are given in terms of the geometry of the undeformed element and require no coordinate transformation. A physical interpretation is made of the nonlinear stiffness matrix through comparison with the geometric stiffness matrices used in incremental approaches. Application of the new formulation is made to highly nonlinear frame and shell problems.