An incremental approach to the solution of snapping and buckling problems

The solution of plate and shell problems by an independent specification of slopes and middle surface displacements is attractive due to its simplicity and ability of reproducing shear deformation. Unfortunately elements of this type are much too stiff when thickness is reduced.

In an earlier paper a derivation of such an element was presented1 which proved very successful in ‘thick’ situations. Here a very simple extension is made which allows the element to be economically used in all situations.

The improved flexibility is achieved simply by reducing the order of numerical integration applied to certain terms without sacrificing convergence properties. The process is of very wide applicability in improvement of element properties.

Reduced integration technique in general analysis of plates and shells

SUMMARY An assessment of flat triangular plate bending elements with displacement degrees-of-freedom at the three comer nodes only is presented, with the purpose of identifying the most effective for thin plate analysis. Based on a review of currently available elements, specific attention is given to the theoretical and numerical evaluation of three triangular 9 degrees-of-freedom elements; namely, a discrete Kirchhoff theory (DKT) element, a hybrid stress model (HSM) element and a selective reduced integration (SRI) element. New and efficient formulations of these elements are discussed in detail and the results of several example analyses are given. It is concluded that the most efficient and reliable three-node plate bending elements are the DKT and HSM elements.

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A study of three‐node triangular plate bending elements

SUMMARY Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.

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Finite element formulations for large deformation dynamic analysis

An octree-based fully automatic three-dimensional mesh generator is presented. The mesh generator is capable of meshing non-manifold models of arbitrary geometric complexity through the explicit tracking and enforcement of geometric compatibility and geometric similarity at each step of the meshing process. The resulting procedure demonstrates a linear growth rate with respect to the number of elements and can be easily integrated with any geometric modeller through a set of geometric operators.

Automatic three‐dimensional mesh generation by the finite octree technique

Compatible triangular plate elements for normal and in-plane displacements, discussing nine degree of freedom element, strain energy, simply supported and clamped plates

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A rapidly converging triangular plate element

A formulation and computer program is developed for the geometrically nonlinear dynamic analysis of shells of revolution under symmetric and asymmetric loads. The nonlinear strain energy expression is evaluated using linear functions for all displacements. Five different procedures are examined for solving the equations of equilibrium, with Houbolt's method proving to be the most suitable. Solutions are presented for the symmetrical and asymmetrical buckling of shallow caps under step pressure loadings and a wide variety of other problems including some highly nonlinear ones.

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Nonlinear dynamic analysis of shells of revolution by matrix displacement method

Formulations and solution procedures for nonlinear structural analysis

The prediction of nonlinear structural behavior by the finite element method wherein buckling does not occur has received considerable attention and, with it, reasonable success has been achieved. However, the post-buckling problem has been less actively pursued probably because of the inherent numerical difficulties encountered. This note reviews very briefly the numerical methods currently being used for pre- and post-buckling analysis and presents a self-correcting approach based on load and displacement incrementation which is shown to be efficient, reliable, and easy to program. Numerical solutions are presented which demonstrate the effectiveness of the method.

Displacement incrementation in non-linear structural analysis by the self-correcting method

This paper presents an assessment of the solution procedures available for the analysis of inelastic and/or large deflection structural behavior. A literature survey is given which summarizes the contribution of other researchers in the analysis of structural problems exhibiting material nonlinearities and combined geometric-material nonlinearities. Attention is focused at evaluating the available computational and solution techniques. Each of the solution techniques is developed from a common equation of equilibrium in terms of pseudo forces. The solution procedures are applied to circular plates and shells of revolution in an attempt to compare and evaluate each with respect to computational accuracy, economy, and efficiency. Based on the numerical studies, observations and comments are made with regard to the accuracy and economy of each solution technique.

Walter E. Haisler

Papers

A rapidly converging triangular plate element

Nonlinear dynamic analysis of shells of revolution by matrix displacement method

Formulations and solution procedures for nonlinear structural analysis

Displacement incrementation in non-linear structural analysis by the self-correcting method

Evaluation of Solution Procedures for Material and/or Geometrically Nonlinear Structural Analysis