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# Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods

T.A. Elgohary1, L. Dong2, J.L. Junkins3, S.N. Atluri4

Department of Aerospace Engineering, Texas A&M University, College Station, TX. Student Fellow, Texas A&M Institute for Advanced Study.
Corresponding author, Department of Engineering Mechanics, Hohai University, China, dong.leiting@gmail.com
Department of Aerospace Engineering, Texas A&M University, College Station, TX. Founding Director, Texas A&M Institute for Advanced Study
Center for Aerospace Research & Education, University of California, Irvine. Fellow & Eminent Scholar, Texas A&M Institute for Advanced Study, Texas A&M University, College Station, TX.

Computer Modeling in Engineering & Sciences 2014, 98(6), 543-563. https://doi.org/10.3970/cmes.2014.098.543

## Abstract

In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of [Kondoh and Atluri (1986)]. By using the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton’s method (and the Newton-Raphson iteration method that is widely used in nonlinear structural mechanics), which necessitates the inversion of the Jacobian matrix, fails to pass the limit load as the Jacobian matrix becomes singular. Although the so called arc-length method can resolve this problem by limiting both the incremental displacements and forces, it is quite complex for implementation. Moreover, inverting the Jacobian matrix generally consumes the majority of the computational burden especially for large-scale problems. On the contrary, by using the presently developed Scalar Homotopy Methods, convergence near limit loads, and in the post-buckling region, can be easily achieved, without inverting the tangent stiffness matrix and without using complex arc-length methods. The present paper thus opens a promising path for conducting post-buckling and limit-load analyses of nonlinear structures. While the simple Williams’ toggle is considered as an illustrative example in this paper, extension to general finite element analyses of space frames, plates, shells and elastic-plastic solids will be considered in forthcoming studies.

## Keywords

Elgohary, T., Dong, L., Junkins, J., Atluri, S. (2014). Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods. CMES-Computer Modeling in Engineering & Sciences, 98(6), 543–563.

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