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 semi-discretized 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 shear-locking 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 third-order 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 third-order. 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 cross-section is considered as a numerical application. By examining the main results (numerical wave numbers and eigen-vector 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.

1

J.E. Laier, "Mass lumping, dispersive properties and bifurcation of Timoshenko's flexural waves", Adv Eng Software, 33, 605-10, 2007. doi:10.1016/j.advengsoft.2006.08.018

2

N.G. Stephen, "Second frequency spectrum of Timoshenko beams", J. Sound Vibr, 80(4), 578-82, 1982. doi:10.1016/0022-460X(82)90501-6

3

J.G. Stepheson and J.C. Wilhot Jr., "An experimental study of bending impact waves in beams", Experimental Mechanics, 5(1), 16-21, 1965. doi:10.1007/BF02320899

4

Z.P. Bazant, "Spurious reflection of elastic waves in nonuniform finite element grids", Comp Meth Appl Mech Eng, 16, 91-00, 1978. doi:10.1016/0045-7825(78)90035-X

5

C.M. Wu and B. Lundberg, "Reflections and transmission of energy of harmonic elastic waves in bent bar", J. Sound Vibr, 190(4), 645-59, 1996. doi:10.1006/jsvi.1996.0083

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