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The equilibrium approach to the finite element method (FEM) is applied to the equilibrium problem for the Reissner-Mindlin plate. This is achieved using the Southwell stress functions [2], which are exploited to express stress-related quantities. Bending and twisting moments depend on the first derivatives of the stress functions, whereas the transverse shearing forces are determined by their second derivatives.
The stress functions are approximated using two C1 triangular elements, developed by Argyris, and Hsieh, Clough and Tocher, as well as a C1 rectangular element, developed by Bogner, Fox, and Schmit [1]. Each of the two components of the stress functions is approximated separately using the element shape functions and degrees of freedom, which include the values of the Southwell functions, their first-order derivatives, and their second-order derivatives. Consequently, the elements have 42, 24, and 32 degrees of freedom, respectively, for the aforementioned methods.
Additionally, Lagrange multipliers are defined as extra degrees of freedom at the corner nodes of the elements to satisfy equilibrium conditions for point forces acting at these corners. The boundary conditions are formulated in terms of stress-related quantities and are enforced using multi-point constraint elements, which employ Lagrange multipliers as degrees of freedom. This allows the imposition of conditions that are linear combinations of the degrees of freedom of the triangular plate elements. The stress-based formulation for FEM is derived using the principle of minimum complementary energy (or, equivalently, the complementary work equation).
The results obtained using this equilibrium approach are compared to those derived using the displacement-based element method, which involves the use of triangular isoparametric elements with 12 and 22 degrees of freedom. Upper and lower bounds for the strain energy are determined using the dual properties of the two FEM formulations. The relative errors of the approximate solutions are calculated using the Synge method [3].
References:
Ciarlet P.G. The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, 1978.
Fraeijs de Veubeke B, Zienkiewicz O.C. Strain-energy bounds in finite-element analysis by slab analogy. Journal of Strain Analysis, 1967; 2(4):265-271.
doi:10.1243/03093247V024265
Synge J.L. The Hypercircle in Mathematical Physics. Cambridge: Cambridge University Press, 1957.
