NONLINEAR POST-BUCKLING ANALYSIS OF TRUSS STRUCTURES USING AN INCREMENTAL SECANT STIFFNESS APPROACH

Based on the basic principle that the deformational resistance of a structural member is independent of the rigid body motion, an incremental potential energy formulation has been redeveloped for geometrically nonlinear analysis of spatial truss structures by assuming that a truss member undergoes a reversed process of "natural deformation-rigid body motion" implied by the Updated-Lagrangian formulation. A secant stiffness matrix has been derived to solve finite displacement and finite strain problem in a typical incremental step of the nonlinear post-buckling analysis using the proposed energy method. Such a procedure effectively overcomes the issues in the Updated Lagrangian approaches related to the tediously obtained, inconsistent high-order stiffness matrices. Alternatively, the recommended secant stiffness matrix is capable of predicting nonlinear responses that are impossible by using the secant stiffness operator defined in the co-rotated configuration and accurately, yet in a simple manner, recovering member forces as compared with the high-order stiffness matrices. In this sense, to improve the robustness of numerical algorithms for a reliable path tracking as well as iteration efficiency, an incremental secant stiffness approach is presented for solving nonlinear problems in which the proposed secant stiffness matrix plays a common role in both the ‘predictor’ and ‘corrector’ in a typical iterative step. As a result, a direct iteration scheme with the cylindrical arc-length constraint is established to ensure a converged solution in arbitrary regions of the equilibrium path. An automatic loading technique that can suitably adjust the step size is therefore put forward for tracing the path with multiple loops. The results from benchmark examples demonstrate the capabilities of the secant stiffness approach for completely removing the ‘turning back’ of the solution direction, making a fast convergence to correct solutions and reliably indicating the full-range behavior of truss structures.