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This paper analyzes the free vibration frequencies of a beam on a Winkler–Pasternak foundation via the original Timoshenko–Ehrenfest theory, a truncated version of the Timoshenko–Ehrenfest equation, and a new model based on slope inertia. We give a detailed comparison between the three models in the context of six different sets of boundary conditions. In particular, we analyze the most common combinations of boundary conditions deriving from three typical end constraints, namely the simply supported end, clamped end, and free end. An interesting intermingling phenomenon is presented for a simply-supported (S-S) beam together with proof of the ‘non-existence’ of zero frequencies for free-free (F-F) and simply supported-free (S-F) beams on a Winkler–Pasternak foundation.

In this paper the free vibration frequencies of beam on one‐parameter (Winkler) elastic foundation are analyzed via original Timoshenko‐Ehrenfest theory as well as two truncated versions of the Timoshenko‐Ehrenfest beam theory. The traditional version of Timoshenko‐Ehrenfest beam theory is contrasted with the truncated version that lacks the fourth‐order time derivative, as well as with the recently developed version that incorporates slope inertia effect. Advantages and disadvantages of each approach are indicated in the context of free vibrations of the beam on Winkler foundation. In addition an intriguing intermingling phenomenon is discussed in detail, and its implications on three theories.

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