engleski

Computational aspects of configuration dependent interpolation in non-linear higher-order 2D beam finite elements

In non-linear 3D beam theory with rotational degrees of freedom (see e.g. Simo, 1985.) configuration-dependent interpolation may be utilized to provide a result invariant to the choice of the beam reference axis (see e.g. Borri and Bottasso, 1994.) or invariant to a rigid-body rotation (see e.g. Crisfield and Jelenić, 1999.). For 2D beam elements, the latter issue vanishes, and such elements are more illustrative for the study of accuracy of the configuration-dependent interpolation in higher-order elements. Since the approximate character of the finite-element method stems from the introduction of interpolation functions for the field variables and their variations, the actual choice of the interpolation functions is of a great importance for the accuracy of the finite element method. The most commonly used standard Lagrangian interpolation procedure (with reduced integration used to eliminate the shear locking effect) is satisfactory if our attention is limited to finding the results for the field variables at nodal points in linear analysis. However, if we want to obtain the exact field distribution this kind of interpolation is unable to do so, even in linear analysis. To obtain the exact solutions in linear analysis simply using the right interpolation function and without applying reduced integration, a different kind of interpolation, called the linked interpolation is needed, in which the unknown displacement function do not depend only on the nodal displacements, but also on the nodal rotations (see e.g. Jelenić and Papa, 2011.). Marco Borri and Carlo Bottasso have developed a so-called fixed-pole formulation which effectively generalizes the idea of linked interpolation to non-linear two-noded beam elements. This interpolation is called the helicoidal interpolation by the authors and is based on the fact that the tangent to the beam centroidal axis and the normal of the cross section follow the same transformation rule (see e.g. Borri and Bottasso, 1994.). This kind of interpolation can also be obtained starting from the condition of constant Reissner’s strain measures (see e.g. Reissner, 1972.). In this paper higher-order configuration-dependent interpolation (i.e. nonlinear in the field variables) has been introduced and the results compared with the results from the literature that include the helicoidal interpolation of Borri and Bottasso as well as the results when the interpolation functions are obtained by taking a linear distribution of the Reissner strain measures.

non-linear beam theory; linked interpolation; configuration-dependent interpolation

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