Abstract
A new concept of explanation of the shear locking phenomenon occurring in the Timoshenko beam is presented. Instead of element matrix analysis the global expression of the equilibrium equation for a whole beam is considered. Based on the element stiffness matrix the equilibrium conditions for a regular discretized beam are derived in the form of one difference equation that converges to the analytical differential formulation for the continuous beam. It is clearly shown that application of the linear shape functions leads to an overly stiff element in the Mindlin elements because this approximation converges to the solution of a different equilibrium equation. From the stiffness matrices of exactly and reduced integrated linear and quadratic elements the difference equilibrium conditions are derived and they are compared with the exact equations of the bending beam problem. For each type of element the improved one is elaborated that assures the exact solution for beams irrespective of beam thickness and discretization (number of elements). The numerical analyses for various boundary conditions, beam load, discretization and thickness ratios are given.
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