Constrained buckling of variable length elastica: Solution by geometrical segmentation (original) (raw)

Effects of the constraint's curvature on structural instability: tensile buckling and multiple bifurcations

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

Bifurcation of an elastic structure crucially depends on the curvature of the constraints against which the ends of the structure are prescribed to move, an effect which deserves more attention than it has received so far. In fact, we show theoretically and we provide definitive experimental verification that an appropriate curvature of the constraint over which the end of a structure has to slide strongly affects buckling loads and can induce: (i.) tensile buckling; (ii.) decreasing-(softening), increasing-(hardening), or constant-load (null stiffness) postcritical behaviour; (iii.) multiple bifurcations, determining for instance two bifurcation loads (one tensile and one compressive) in a single-degree-of-freedom elastic system. We show how to design a constraint profile to obtain a desired postcritical behaviour and we provide the solution for the elastica constrained to slide along a circle on one end, representing the first example of an inflexional elastica developed from a buckling in tension. These results have important practical implications in the design of compliant mechanisms and may find applications in devices operating in quasi-static or dynamic conditions.

Elgohary, Tarek A., Dong, Leiting, Junkins, John L. and Atluri, Satya N. (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, Vol. 98, No. 6, pp. 543-563.

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 ]. 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 1 Department of Aerospace Engineering, Texas A&M University, College Station, TX. Student Fellow, Texas A&M Institute for Advanced Study.

Direct Evaluation of the Post-Buckling Behavior of Slender Structures Through a Numerical Asymptotic Formulation

Direct Methods for Limit States in Structures and Materials, 2014

The analysis of slender structures, characterized by complex buckling and postbuckling phenomena and by a strong imperfection sensitivity, is heavily penalized by the lack of adequate computational tools. Standard incremental iterative approaches are computationally expensive and unaffordable, while FEM implementation of the Koiter method is a convenient alternative. The analysis is very fast, its computational burden is of the same order as a linearized buckling load evaluation and the simulation of different imperfections costs only a fraction of that needed to characterize the perfect structure. In this respect it can be considered as a direct method for the evaluation of the critical and post-critical behaviour of geometrically nonlinear elastic structures. The main objective of the present work is to show that finite element implementations of the Koiter method can be both accurate and reliable and to highlight the aspects that require further investigation.

Interactive buckling analysis with finite strips

International Journal for Numerical Methods in Engineering, 1985

INTR 011 U CTION Thin-walled structural elements subjected to end compression or moments are liable to buckle principally in two types of modes: local and overall. The local mode is characterized by individual plate elements buckling out of plane with the junctions remaining essentially straight. The half wave length of buckling in this case is of thc same order of magnitude as the width of the constituent plates. Overall buckling may be either purcly flexural (Euler type) or lateral torsional. A study of interaction of the two modes is of considerable interest in the design of thin-walled metal structures.

Nonlinear buckling formulations and imperfection models for shear deformable plates by the boundary element method

Journal of Mechanics of Materials and Structures, 2010

This paper presents a nonlinear buckling analysis of shear deformable plates. Two models of imperfections are introduced: small uniform transverse loads and distributed transverse loads, according to the number of half-waves indicated by the eigenvectors from linear elastic buckling analysis. A simple numerical algorithm is presented to analyze the problems. Numerical examples with different geometries, loading and boundary conditions are presented to demonstrate the accuracy of the formulation.

Multisegment Integration Technique for Post-Buckling Analysis of a Pinned-Fixed Slender Elastic Rod

This paper investigates the post-buckling behavior of a slender axially inextensible elastic rod with pinned-fixed end. The set of five first order nonlinear ordinary differential equations with boundary conditions specified at both ends constitutes a complex two point boundary value problem. By using multisegment integration technique, the highly nonlinear boundary value problems are numerically solved. Results are presented in non-dimensional graphs for a range of prescribed loading condition. The secondary equilibrium paths and the post-buckling configurations of the rod are presented.

Plate Buckling Including Effects of Shear Deformation and Plate Bending Curvatures Using the Boundary Element Method

Boundary Elements and other Mesh Reduction Methods XLIV

In this paper, the plate bending curvature was included in the geometrical non-linearity (GNL) effect beyond the deflection derivatives to perform plate buckling analyses. The boundary element method (BEM) was adopted and the formulation employed two integrals related to the GNL effect, with one computed on the boundary and the other on the domain. The eigenvalue problem was solved with the inverse iteration method. Results obtained with different boundary conditions were compared to values in the literature.