Elastic postbuckling analysis via finite element and perturbation techniques. Part 1: Formulation (original) (raw)
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Int J Numer Method Eng, 1993
The general theory developed in Part I of this paper for the finite element stability analysis of structural systems, using perturbation expansions in the vicinity of a critical point, is applied here to the analysis of shells of revolution. The discretization of the shell is performed by means of a semianalytical approximation, and the matrices required for the evaluation of critical points and postcritical equilibrium paths are obtained. Two cases are presented: bifurcation in axisymmetric and in asymmetric buckling modes. The derivatives required for an imperfection analysis are also obtained. A technique of switching between two paths using continuation methods is also discussed, in which the switch is performed using derivatives of the perturbation expansion. Results are presented for bifurcation in axisymmetric and in non-axisymmetric modes, and compared with known solutions or with results from changing the path using continuation methods; good correlation is shown. For structures displaying unstable bifurcation, the influence of load and geometric imperfections is evaluated.
Elastic postbuckling analysis via finite element and perturbation techniques. I - Formulation
International Journal For Numerical Methods in Engineering, 1992
Equations for the equilibrium, stability, and critical state of discrete elastic systems are presented with the original set of generalized coordinates and loads preserved in the total potential energy. The new formulation is approximated, using a finite element approach based on interpolation of displacements, in which the derivatives of the potential energy are approximated. The energy analysis is related to the more conventional finite element notation in terms of stiffness matrices, and it is shown how it can be included in existing codes.
An asymptotic-numerical method to compute the postbuckling behaviour of elastic plates and shells
International Journal for Numerical Methods in Engineering, 1993
In this paper, we apply an asymptotic-numerical method for computing the postbuckling behaviour of plate and shell structures. The bifurcating branch is sought in the form of polynomial expansions, and it is determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can easily be computed by using recurrent formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. However, the polynomial expansions have a radius of convergence which limits the validity of the solution to a neighbourhood of the bifurcation point. in the present form, the method should be viewed as a cheap and automatic way of completing a linear buckling analysis. It is illustrated in two examples: a square plate under in-plane compression and a cylindrical shell under pressure.
1995
for introducing me into and teaching me about the field of finite elements and its engineering applications. Many thanks for the superb guidance, continuous encouragement and optimistic attitude which I enjoyed throughout the course of research work as well as the huge amount of freedom I was given in order to lead it to a successful completion. My deep appreciation is directed to Professor J. R. Whiteman for making all necessary arrangements for the completion of this thesis. Special acknowledgements are directed to Dr D. J. Allman of the Defence Research Agency in his role as a technical coordinator and for the interesting discussions and his helpful cooperation. Thanks to Dr M. K. War by and Dr S. Shaw as well as other colleagues at BleOM for the time, companionship and help they have given me. Also, I would like to thank Dr N. Herrmann from the University of Hanover for enabling my trip to the UK in the first place and Dr S. A. Matar for giving me the idea and convincing me to do a PhD. A thousand thanks for the support I have received from Jack, Michael, Giovanni and other friends over the last few years. A special thank you goes to Roshi. A simple thank you is not sufficient for expressing my gratitude towards my family who always stood behind me and without whose support it would never have been possible for me to accomplish this work.
Computer Methods in Applied Mechanics and Engineering, 1993
Beginning with the work of Koiter in 1945, valuable insights into the postbuckling behavior of structures have been gained by Lyapunov-Schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point. Here this methodology is generalized to include nonlinear prebuckling behavior, as well as multiple, not necessarily coincident buckling modes. The expansion of the reduced equilibrium equations is performed about a reference point (which need not coincide with any of the bifurcation points), and applies no matter whether the modes are coincident, closely spaced, or well separated. From a variety of possible decompositions of the admissible space of displacements, two are incorporated into a finite element program. Theoretical considerations, and numerical examples in which asymptotic results are compared to 'exact' results, indicate that one of the decompositions has some important advantages over the other. Examples include a shallow arch, and a beam on elastic foundation problem exhibiting symmetry-breaking modal interaction.
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.
Finite element post-buckling analysis for frames
International Journal for Numerical Methods in Engineering, 1977
Presented in this paper is a direct method of analysis of the elastic non-linear behaviour of frames. Emphasis is given to the method's capability of tracing the post-buckling path from a bifurcation point although the method can also trace the non-linear behaviour of frames with eccentricities. The method is proposed as an alternative to the main methods currently in use, the perturbation method and the incremental method. Conditions for equilibrium and stability are developed from a variational approach to the total potential. A finite element approximation is made and an efficient solution technique for the resulting non-linear equations is developed. Results for three frames are given demonstrating good agreement with solutions generated from other approaches.