Three alternative versions of Bresse–Timoshenko theory for beam on pure Pasternak foundation

In this study, we analyze the free vibrations of a beam on Pasternak foundation by using three alternative beam theories in contrast to other studies which focus on original Bresse–Timoshenko equations only. Namely, we treat original Bresse–Timoshenko theory as well as two of its truncated versions neither containing the fourth order time derivative in contrast to the original Bresse–Timoshenko equations. The analyses with latter two theories has never being conducted in the literature. Specifically, we deal with the truncated version of the Bresse–Timoshenko theory obtainable by neglecting the fourth-order time derivative, and attendant slope inertia based version that is variationally derived. Apparently for the first time in the literature, these three theories are compared with each other based on several papers available in the open literature. It is conceded that for prediction of the initial spectrum of frequencies the original theory of Bresse–Timoshenko is not needed and either of two theories suffices, with preference given to the truncated model, without the slope-inertia term.