Rail accelerations can be used on the defect detection and health monitoring of railway vehicle and track components; therefore, mathematical models that predict this response are of interest for reproducing its behaviour in a wide range of situations. The numerical track models based on the Timoshenko beam theory introduce a non-physical response, which is especially noticeable in the rail accelerations. It is due to the lack of dynamic convergence of the Timoshenko finite element (FE). This paper addresses this phenomenon employing an enhanced formulation of the Timoshenko FE that includes internal degrees of freedom (iDoF). The iDoF shape functions are derived from the Timoshenko beam dynamic governing equations. Firstly, the formulation is presented, and its performance is compared with a similar Timoshenko FE formulation. Secondly, the proposal is assessed in the dynamic modelling of railway track structures. The use of iDoF efficiently corrects the non-physical response of rail accelerations by improving the FE dynamic convergence. Subsequently, a filtering criterion for accelerations is proposed, which removes the remaining non-physical response while guaranteeing the conservation of coherent frequency content. Finally, practical cases are simulated for which the proposed methodology is proved to be more efficient and reliable than the standard approach.