HIGHER-ORDER LINKED INTERPOLATION FOR 3D BEAM ELEMENTS

In this work a new family of linked interpolation functions for Timoshenko beam elements will be presented. Linked interpolation implies higher order interpolation for the transverse displacements than the rotations. It is well known that a polynomial interpolation of the same order used both for the transverse displacements and the rotations results in a phenomenon known as the shear locking (1). This can be avoided by using the so-called reduced integration. However, this technique applies to the correction of the stiffness matrix and leaves unanswered the question of how exactly the displacement and rotation fields are interpolated. We derive our linked interpolation by consistently providing internal degrees of freedom at a suitable number of internal nodes common for both the displacement and the rotation degrees of freedom, where this number depends on the order of the polynomial describing the applied loading. Exact solutions are obtained by solving the differential equation for such Timoshenko beam problem and expressing it in terms of the nodal values for the displacements and rotations. Thus, linked interpolation provides exact solutions for a general static polynomial loading and in this manner eliminates the problem of shear locking. This methodology is presented on a full 3D Timoshenko beam problem.