On locking-free shear deformable beam finite elements
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This is an elementary exposition of beam finite element models. An overview of existing and new displacement finite element models of the Euler-Bernoulli and shear deformation beam theories is presented. A locking-free finite element model using the form of the exact solution of the Timoshenko beam theory is developed. The element contains the Euler-Bernoulli beam element as a special case, and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams. The procedure is extended to develop a locking-free finite element model of the simplified third-order theory of beams. The element contains the Euler-Bernoulli and the locking-free Timoshenko beam finite elements as special cases. The element stiffness matrix of the unified beam bending element is 4 × 4 and yet has the superconvergence character. The advantage of this element is that it can be used to obtain exact nodal displacements and forces (from the equilibrium equations of the element) in a frame structural analysis using one finite element per a structural member. This cannot be achieved with the traditional equal interpolation, reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used.
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