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CivilComp Proceedings
ISSN 17593433 CCP: 88
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and M. Papadrakakis
Paper 206
A New Consistent Mass Matrix for Timoshenko's Flexural Model J.E. Laier and C.C. Noronha
Department of Structural Engineering, Engineering School of São Carlos, University of São Paulo, Brazil J.E. Laier, C.C. Noronha, "A New Consistent Mass Matrix for Timoshenko's Flexural Model", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 206, 2008. doi:10.4203/ccp.88.206
Keywords: Timoshenko's flexural waves, first and secondspectra bifurcations, velocity dispersion, numerical simulation.
Summary
The objective of this paper is to present a new consistent third order mass matrix to
solve Timoshenko's flexural wave equation. In addition, the matrix mass elements
are given by single terms, which are classically formed by six terms.
Timoshenko's beam theory predicts the existence of two possible modes for flexural wave propagation [2] as it is well known. The lower mode is in very good agreement with the exact elastodynamic solution. The higher mode, which corresponds to the second spectrum of natural frequencies, agrees with the second exact solution at long wavelengths [3], but as wavelengths shorten it can diverge considerably. The numerical integration of the flexural wave equation by using the finite element method via a semidiscretized technique introduces additional velocity dispersion and spurious wave motions as it has been established previously [4]. The proposed two node finite element formulation with no shearlocking effect for Timoshenko's flexural wave motion complements the well known stiffness matrix that takes into account the contribution of the shear deformation and the new consistent mass matrix [1]. In order to obtain the thirdorder of convergence the present formulation considers initially the Taylor expansion for displacement and bending rotation (the element length is the space variable increment). The new consistent mass matrix is than obtained by annulling the terms of the Taylor expansion less than thirdorder. On the other hand, the velocity dispersion analysis is studied by examining the discrete equilibrium equation of motion (equilibrium of a generic node) worked in terms of the numerical wave motion solution [5]. The wave propagation of a typical WF shape beam crosssection is considered as a numerical application. By examining the main results (numerical wave numbers and eigenvector components) one can observe that the proposed consistent mass matrix has a velocity dispersion (global error) similar to the classical one. The new proposed mass matrix is a good mathematical tool to study Timoshenko's flexural wave propagation problems as it presents elements expressed by single terms. References
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