Abstract
In this paper, a nodal integration method (NIM) is presented to deal with the static and dynamic problems of Timoshenko beam. In the present method, linear-shape functions are employed to approximate the displacement field, and smoothing domains based on the nodes are further formed for computing the stiffness matrix. Through a smoothing operation, the shear locking is effectively avoided and the computation gets much simpler. For static problems, the upper bounds for a set of benchmark examples are obtained by nodal integration. For dynamic problems, while keeping the shear stiffness matrix the same as NIM, integration based on elements is adopted to construct the bending stiffness matrix to improve the stability and diminish singular modes caused by pure nodal integration. Results computed in this way prove to be much better than pure nodal integration method for free vibration and forced vibration problems. Numerical examples indicate that very accurate results can be obtained when a reasonable number of nodes is used. Both computational efficiency and accuracy are achieved by above formulations.
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