A new mixed finite element method for the Timoshenko beam problem. (English) Zbl 0779.73059

The Timoshenko beam problem, with shear force and rotation as primal variables and displacement as Lagrange multiplier, is examined under the general hypotheses of Brezzi’s theorem concerning existence and uniqueness in saddle point problems. A new finite element approximation is proposed and shown to be convergent for various combinations of interpolations for the three variables.
The paper is organized as follows. In section 2 the differential equations governing the problem are presented, a mixed variational formulation in terms of nondimensional variables is derived, and its analysis is shown. In section 3 a finite element approximation is described and its convergence for a rather general family of finite element interpolations is proved. Numerical results in section 4 on equal-order linear and quadratic elements (shear discontinuous on element interfaces) are obtained by the Galerkin and the perturbed Galerkin methods. We believe the analysis presented here may prove itself useful in the analysis of mixed variational formulations with more than two variables.


74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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