Henryk Stolarski

Shear and membrane locking in curved C0 elements

Abstract Locking phenomena in C 0 curved finite elements are studied for displacement, hybrid-stress and mixed formulations. It is shown that for a curved beam element, shear and membrane locking are interrelated and either shear or membrane underintegration can alleviate it. However, reduced shear integration tends to diminish the membrane-flexural coupling which characterizes curved elements. Locking can also be expected in certain types of mixed formulations, the hybrid-stress formulations avoids locking for beams (but not for shells). Methods for avoiding locking are explored and alternatives evaluated.

Limitation principles for mixed finite elements based on the Hu-Washizu variational formulation

Abstract In a pioneering paper Fraeijs de Veubeke introduced a limitation principle for finite elements based on the Hellinger-Reissner (two-field) variational formulation. In this paper, a similar principle is presented in the context of the Hu-Washizu (three-field) variational formulation.

Locking and shear scaling factors in C° bending elements

Abstract Two types of locking are identified in linear field C ° elements: locking caused by shear resistance in constant curvature deformation and in linear curvature deformation, respectively. Locking of the first type can be ameliorated by decoupling constant curvature bending from shear, which in some elements can be achieved by selective reduced integration. Once the modes are decoupled, a scaling factor on the shear stiffness can be used to alleviate the second type of locking. These findings are explained in terms of a linear displacement, linear rotation beam element and applied to the improvement of a triangular plate element. It is shown that the resulting triangular plate element is free from locking and performs excellently in a wide variety of problems.

On the equivalence of mode decomposition and mixed finite elements based on the Hellinger—Reissner principle: part I: theory

Abstract A theorem is presented which defines the conditions under which an equivalent mixed finite element based on the Hellinger-Reissner principle exists for a given mode decomposition element. Mode decomposition elements are a family of elements in which parasitic stresses are removed by a suitable projection to enhance the rate of convergence of elements which tend to lock; so far they were outside of any standard finite element formulation. Elements that, by virtue of this theorem, are found equivalent to appropriate mixed elements can be then analyzed by techniques which exist within the framework of Hellinger-Reissner formalism.

A Kirchhoff-mode method for C0 bilinear and serendipity plate elements☆

Abstract A new formulation for the C0 quadrilateral and Serendipity plate elements is presented. It is an extension of the mode-decomposition approach that has been successfully applied in evaluating the C0 triangular linear plate element. In contrast to the triangular element, the concept of an equivalent discrete Kirchhoff configuration is utilized. The elements are applied to several examples and the performance of these elements is shown to be quite good.

Bending and Shear Mode Decomposition in C° Structural Elements

ABSTRACT ABSTRACT Various formulations for ° structural finite elements are discussed. By means of two examples, a linear beam and a triangular plate element, it is shown that the proper decomposition of the deformed configuration into its bending and shear modes is essential for the effective performance of C° elements. It is also shown that reduced integration can only be successful if it is equivalent to this decomposition.

An alternative formulation of the geometric stiffness matrix for plate elements

Abstract A simplified formulation of the geometric stiffness matrix for plate elements is presented. In this formulation the transverse displacement is defined along the element boundary but not for the element interior as with the usual formulation. As such the formulation is particularly suitable for use with hybrid stress or discrete Kirchhoff methods which are also based on boundary approximation of the transverse displacement. The simplicity, computational economy and accuracy obtained with the formulation compare favorably with the usual order formulation.