Keywords: axisymmetric, analytical FE, Green-Lagrange strains, anisotropy, stiffness matrix.

Axisymmetric finite element analysis is of major importance in mechanical and civil engineering. Pressure vessels, shafts, plates, shells, cylinders, etc. can often be modeled axisymmetric, and some of these models can not adequately be limited to strains that are linear depending on displacement gradients. The present paper is a follow up on a similar paper [1], restricted to plane problems.

Separating the dependence on the material constitutive parameters and on the stress-strain state from the dependence on the initial geometry and the displacement assumption, we obtain analytical secant and tangent element stiffness matrices. For the case of a linear displacement ring-element with triangular cross-section, closed form results are listed, directly suited for coding in a finite element program. As an example of application, numerical results for a circular plate problem show the indirect errors that may result from a linear strain model.

The nodal positions of an element and the displacement assumption give six basic matrices that do not depend on material and stress/strain state, and thus are unchanged during the necessary iterations for obtaining a solution based on Green-Lagrange strain measure. The resulting stiffness matrices are especially useful in design optimization, because this enables analytical sensitivity analysis.

Why these efforts to give an alternative stiffness evaluation, when well established numerical integration and fast computers can do the job? For research within optimal design the sensitivity analysis is of major importance. We need to answer questions like: Change in response for change in material parameters? Change in response for change in boundary shape? With the presented matrices these questions can be answered analytically without the difficult choice of a proper finite difference approach. For sensitivity analysis robustness is more important than computer-time, so saving computer-time is not the motivation.

The present formulation follows the basic matrix approach in [2] for small strains. This means that the element geometry, the element orientation, the element nodal positions, and the displacement assumption are described by a few basic matrices that do not depend on material and stress/strain state. Material parameters and displacement gradients together give factors, so that linear combinations of the basic matrices determine the needed stiffness matrices.

As stated in textbooks, like in [4] and in [3] a rigid rotation of a linear strain element will give rise to an erroneous compressive strain dilatation , which is not small ( results in ). However, pure rigid rotations in the linear strain models are not actual due to an erroneous displacement field and the linearity is then maintained. It is difficult to predict the indirect errors that follows from the erroneous displacement field, and in the present paper numerical results for a circular plate problem show this. Thus understanding of the results from linear analysis must be based on the understanding of the erroneous expansive dilation, that element rotation give rise to when linear strain models are applied.

1

Pedersen, P.
Analytical stiffness matrices with Green-Lagrange strain measure. Int. J. Numer. Meth. Engng. (to appear), 2004. doi:10.1002/nme.1174

2

Pedersen, P., and Cederkvist, J. Analysis of non-axisymmetric vibrations and the use of basic element matrices. Computers and Structures 14:255-260, 1981. doi:10.1016/0045-7949(81)90011-0

3

Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. Concepts and Applications of Finite Element Analysis. New York, USA: Wiley. 719 pages, 2002.

4

Crisfield, M. A. Non-linear Finite Element Analysis of Solids and Structures, volume 1 and 2. Chichester, UK: Wiley. 345 and 494 pages, 1991, 1997.

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