The planar rigid bicycle model is one of the most popular models used in vehicle dynamics. It has widely been used in studying vehicle handling characteristics and designing steering control system for vehicles. This paper analyses a modified dynamic model called the "Elastic Bicycle Model." This model improves upon the classical bicycle model by taking into account the flexibility of the vehicle frame by using concepts from the Euler beam theory. Complete set of the resulting dynamic equations of this model are presented. Non-dimensional versions of the equations are used to investigate the steady state response of the model. Finally, the results of the response study obtained by modeling a small truck with an elastic model and the classical bicycle model are presented. These include the steady state solutions as function of different parameters as well as a transient solution in response to a saw-tooth steering input and a step input. Octave® has been used for simulation purpose. Differences are observed in the response of a truck using the rigid bicycle model and the elastic bicycle model.