Variational Derivation of Truncated Timoshenko-Ehrenfest Beam Theory

The beam theory allowing for rotary inertia and shear deformation and without the fourth order derivative with respect to time as well as without the slope inertia, as was developed by Elishakoff through the dynamic equilibrium consideration, is derived here by means of both direct and variational methods. This formulation is important for using variational methods of Rayleigh, Ritz as well as the finite element method (FEM). Despite the fact that literature abounds with variational formulations of the original Timoshenko-Ehrenfest beam theory, since it was put forward in 1912-1916, until now there was not a single derivation of the version without the fourth derivative and without the slope inertia. This gap is filled by the present paper. It is shown that the differential equations and the corresponding boundary conditions, used to find the solution of the dynamic problem of a truncated Timoshenko-Ehrenfest via variational formulation, have the same form to that obtained via direct method. Finally, in order to illustrate the advantages of the variational approach and its adaptability to the finite element formulation, some numerical examples are performed. The calculations are implemented through a software developed in Mathematica language and results are validated by comparison with those available in the literature.