Keywords: shell, geometrically non-linear, updated Lagrangian formulation.

Thin plates and shells represent a particular type of component widely used in the construction of structural system that offer large internal space, gymnasiums, cooling towers, space structures, fuselages of aircraft, etc [1]. One basic property with plates and shells is that they have one dimension, i.e., thickness, much smaller than other two. Thin Plate and shell problems are often categorized as large- deflection problems as their deflection under working or extreme loading conditions may have an order of magnitude comparable to that of their thickness. To analysis the load-deflection behaviour of plates and shells, theories of nonlinear nature are required to consider the effect of large deformation. In this regard, a geometric stiffness matrix has to be included, in addition to the elastic stiffness matrix used in linear analysis of shell structures.

There are three approaches to applying finite element methods to shell structures:

1. the shell structure is faceted with flat element,
2. two-dimensional shell theory or Love theory is used to develop a curved-shell element, and
3. the curved–shell elements are formed by degenerating the three-dimensional strain displacement relation[1].

Generally, the curved shell elements can be computationally very expensive in the case of nonlinear analysis of shell structures, because of the complexity of the formulation and the need to compute the curvature information. Flat shell elements are more attractive because of their simplicity and the ease with which they can be built from already existing membrane and plate bending element.

For large deformation analysis by finite element method, the equations of equilibrium of a structure may be derived from two alternative approaches as the total Lagrangian and updated Lagrangian formulations[2,3]. From the theoretical standpoint, both approaches should lead to same results for the same nonlinear problem. The total Lagrangian formulation has found wide applications in problems involving geometrical nonlinearity and elastic stability. The updated Lagrangian formulation may be particularly useful for slender structures such as beams, plates and shells.

In this paper, finite element formulation for geometrical nonlinear analysis of general shells using a three-node flat triangular shell element is presented. The flat shell element is obtained by combining the DKT plate bending element of Discrete Kirchhoff Theory[4] and a membrane element derived from the Linear Strain Triangular (LST) element[1]. The geometrically nonlinear analysis is performed using an Updated Lagrangian Formulation. An incremental iterative method based on the arc length method in conjunction with Newton-Raphson method is implemented in the present study for static analysis. In order to estimate the accuracy of the present formulation in predicting the nonlinear response of large structures, diverse models are analyzed. The present results are in good agreement with the results available in the existing literature and those obtained using the commercial finite element software (ABAQUS). The presented formulation did not show any convergence problems. As results, it can be concluded that the presented procedure is very useful for nonlinear analysis of shell structures.

1

Palazotto, A.N., Dennis, S.T., "Nonlinear Analysis of Shell Structures", AIAA, 1992.

2

Meek, J. L., Ristic, S., "Large Displacement Analysis of Thin Plates and Shells Using a Flat Facet Finite Element Formulation", Computer Method in Applied Mechanics and Engineering, 145(3-4), 285 299, 1997. doi:10.1016/S0045-7825(96)01220-0

3

Meek, J. L., Wang, Y. C., "Nonlinear Static and Dynamic Analysis of Shell Structures with Finite Rotation", Computer Method in Applied Mechanics and Engineering, 162(1-4), 301 315, 1998. doi:10.1016/S0045-7825(97)00349-6

4

Fafard, M., Dhan, G., Batoz, J. L., "A New Discrete Kirchhoff Plate/Shell Element with Updated Procedures", Computers and Structures, 31, 591 606, 1989. doi:10.1016/0045-7949(89)90336-2

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