Axisymmetric analytical stiffness matrices with Green-Lagrange strains

Abstract

Stiffness matrices based on the non-linear Green-Lagrange definition seem complicated, but for the case of a linear displacement ring-element with triangular cross-section, closed form final results are listed, directly suited for coding in a finite element program. These analytical secant and tangent element stiffness matrices are obtained by 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. As an example of application, numerical results for a circular plate problem show the indirect severe errors that may result from a linear strain model. It is difficult to predict the indirect errors that follow from the erroneous displacement field, and the explanations behind such predictions are attempted. 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 presented resulting stiffness matrices are especially useful in design optimization, because analytical sensitivity analysis can then be performed.