Geometric and Material Nonlinear Analysis of Frames with Nonuniform Cross-Sections

The postbuckling response of structures with multi winding loops is characterized by the appearance of multi adjacent equilibrium paths, which often makes the iterations difficult to converge to the desired path. In this study, some key issues for tracing the postbuckling response of a structure using an incremental-iterative approach are discussed. Concerning the finite element equations used, it is essential that the corrector used for recovering the element forces from the element displacements be made as accurate as possible, and that the predictor for computing the structural displacements under given load increments, which are approximate by nature due to linearization involved, be accurate to the level not to misguide the direction of iterations. As for the incremental-iterative scheme, it is required to be: (1) numerically stable in passing the limit points, (2) self-adjustable for the load increments, and (3) automatic in reversing the loading direction. To demonstrate the ideas involved, some examples with highly complicated post-buckling responses will be solved in this study.
Referring to the material nonlinear analysis, the quasi-plastic-hinge approach proposed by Attalla et al. (1994) is adopted in this study. This approach is formulated using the total form expressions, in which numerical integrations and the finite difference method are employed to obtain the inelastic flexibility coefficients. Such a procedure is computationally inefficient in general. To overcome the above drawback, Leu and his co-workers made some improvements in the quasi-plastic-hinge approach, in which the elastic-plastic flexibility coefficients were determined explicitly. In this study, the formulation of the quasi-plastic-hinge approach for uniform and nonuniform cross sections will be conducted. To reduce to complexity involved in the formulation for nonuniform cross sections, an improved approach is proposed. To compare different formulations for the inelastic analysis of frames with uniform and nonuniform cross sections, including the proposed approach, three benchmark problems of planar steel structure will be used as the basis and thoroughly studied. By the numerical studies, the reliability of the quasi-plastic-hinge approach for uniform and nonuniform cross sections will be verified.