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# Shell Finite Elements Evaluation Tests

Thin shell problems encountered in design are often difficult to analyse because of the complexity of the structural geometry and the complex nature of the constitutive equations. In addition, the engineer involved with designing this class of structure can find that the intuition and experience gained from considering more conventional configurations cannot be employed. In order to gain confidence that a specific design is adequate to the task, recourse is often made to experimental validation where scale models or structural components are tested. Building up confidence in a design is always important and in many cases as, for example, in the nuclear industry, this is vital. Where such factors as reliability are concerned this requires a high degree of detailed knowledge of the structural deformation and stress patterns which the test process will provide.

**The finite element method can replace these test procedures providing confidence in the accuracy of the method for predicting the structural performance can be maintained.** Unfortunately, the basic approximate nature of the theory applying to thin shell analysis conflicts in certain aspects with the approximations inherent in the finite element method. This clearly creates doubt in the mind of potential users and some action is required to overcome the problem.Furthermore, there are standard QA and validation requirements associated with the general use of finite element methods for the solution of structural analysis problems. Thus some form of validation procedure is required for finite elements which purport to simulate the behaviour of thin shells.

The classical approach to validating individual finite elements rests upon the concept that the element should satisfy a variational principle and exhibit other mathematical properties. However, in the case of many finite elements, and particularly those developed for the analysis of thin shells, use is made of intuitive and heuristic arguments which ignore any appeal to an underlying variational principle.By its very nature this approach leads to elements with attractive properties which results in their extensive use in the major finite element systems. In this situation, confidence that the elements will always give accurate results clearly cannot rest upon the satisfaction of a variational principle and an alternative approach must be sought to provide a suitable validation procedure. This must involve some form of numerical testing to stand in place of the variational principle and provide an indirect sufficiency test upon which the validation of an individual element can be based.

In the case of non-shell elements a two stage procedure is normally recommended. First the element is subject to single element tests which numerically validate the elements convergence properties.Secondly, assemblages of elements are tested in a benchmark mode to asses the elements ability to provide adequate results within an acceptable level of efficiency. For thin shells this two stage procedure can be followed in outline though single element tests are difficult to construct for doubly-curved structures. In consequence, the first must now involve the use of small element assemblages or patch tests. These patch tests must now form a sufficient set as to provide a rational substitute for direct theoretical validation that a finite element exhibits the required convergence properties. The creation of such tests must rest upon the requirement that all the characteristic solution modes for a thin shell are recovered at least to the limits of accuracy afforded by classical theory. While it is inevitable that patch tests refer to a specific shell shape it is reasonable to assert that a finite element for general shell analysis should satisfy tests for the simplest shells of constant Gaussian curvature which may be zero, positive or negative.The work reported in this document discusses tests for zero and positive Gaussian curvature shells.

Traditionally, patch tests concern sets of linearly independent strains or linearly independent constant curvature changes. In the case of zero Gaussian curvature surface such tests can be created and in the present report the discussion for this class of structure focuses on the cylindrical configuration. Unfortunately this relatively simple state of affairs does not appear to prevail in the case of shells with non-zero Gaussian curvature and recourse has to be made to independent but non-constant strain and curvature changes. These do, however, represent the characteristic solution modes of membrane and in extensions l bending states. As in the zero Gaussian curvature case a specific surface has to be selected to represent the infinite of potentially doubly configurations.For the present set of tests a spherical shell is used and the question of the influence of not having equal values for the principal curvature nor torsion of the surface is left open. In addition no attempt is made to explore the edge effects and discontinuities. Nevertheless, the tests presented do provide a starting point for a rational discrimination method for evaluating shell finite elements though much further work is required to establish a complete understanding of the problem and associated complete range of tests.

In the remaining sections we first present the various co-ordinates systems for presenting the results in forms suitable for the majority of finite element systems. The theoretical basis for the various evaluation tests are given in section 3 for the spherical shell only since the justification for the cylindrical case represent a simple application of the cylindrical shell equations. Section 4 tabulates the tests for both spherical and cylindrical shells whilst section 5 identifies the limitations of these tests and indicates where further work would be advantageous.

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# About

A.Morris

First Published 1986

Softback Report - 25 Pages