Higher order tapered beam finite elements for vibration analysis

In this paper, explicit for mass and stiffness matrices of two higher order tapered beam elements for vibration analysis are presented. One possesses three degrees of freedom per node and the other four degrees of freedom per node. The four degrees of freedom of the latter element are the displacement, slope, curvature and gradient of curvature. Thus, this element adequately represents all the physical situations involved in any combination of displacement, rotation, bending moment and shearing force. The explicit element mass and stiffness matrices eliminate the loss of computer time and round-off-errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. Comparisons with existing results in the literature concerning tapered cantilever beam structures with or without an end mass and its rotary inertia are made. The higher order tapered beam elements presented here are superior to the lower order one in that they offer more realistic representations of the curvature and loading history of the beam element. Furthermore, in general the eigenvalues obtained by employing the higher order elements converge more rapidly to the exact solution than those obtained by using lower order one.