Software for the verification of Timoshenko beam finite elements. (January 2019) - Day, David
Abstract : Software is presented for verifying Timoshenko beam finite elements. A popular fiite element is shown to converge to the wrong answer. An element modification sufficient for convergence is reviewed. Abstract: The classic paper Han et al. Dynamics of transversely vibrating beams using four engineering theories, Journal of Sound and Vibration, 225, 1999, sets up the continuation problem that determines the exact vibrational frequencies and mode shapes of prismatic Timoshenko beams. This article presents a portable implementation of a solver for the problem. In order to reliably compute a complete set of modes, without redundancies, it is crucial to formulate the frequency equation as a continuation problem. The continuation problem is solved for a thickness parameter starting from zero thickness (the Euler beam), and increasing up to the finite thickness of interest. The continuation problem has a bifurcation point, at which a wave number variable passes from real to a purely imaginary value. For finite thickness, the continuation problem for all sufficiently high frequency modes crosses through this bifurcation. Han et al.[7] solve the continuation problem using an unspecified differential equation solver, and dummyTXdummy- do not discuss accuracy. Here a solution of the continuation problem using pseudoarclength continuation is presented. This article presents a description of how to compute this singularity exactly, and develops software that maintains full working precision. Furthermore the details of the computation of the mode shapes from the frequencies are presented for free-free, clamped-clamped and clamped-free boundary conditions. The source code is available under a MIT license from github in the TimoshenkoBeamEigenvalues repository.