Free Vibrations Analysis of Timoshenko Beams on Different Elastic Foundations via Three Alternative Models

The scope of the research is to provide a simpler and more consistent equation for the analysis of the natural frequencies of a beam with respect to the widely used one introduced by Timoshenko in 1916. To this purpose, the free vibrations of a beam resting on Winkler or/and Pasternak elastic foundations are analyzed via original Timoshenko theory as well as two of its truncated versions, which have been proposed by Elishakoff in recent years to overcome the mathematical difficulties associated with the fourth-order time derivative of the deflection. Former equation takes into account for both shear deformability and rotary inertia, while latter one is based upon incorporation of the slope inertia effect. Detailed comparisons and derivations of the three models are given for six different sets of boundary conditions stemming by the various possible combinations of three of the most typical end constraints for a beam: simply supported end, clamped end and free end. It appears that the two new theories are able to overcome the disadvantage of the original Timoshenko equation without predicting the unphysical second spectrum and to produce very good approximations for the most relevant values of natural frequencies. As a consequence, the inclusion of these simpler approaches is suggested in future works. An intriguing intermingling phenomenon is also presented for the simply supported case together with a detailed discussion about the possible existence of zero frequencies for the free–free beam and the simply supported–free beam in the context of different types of foundations.