Stability of a Timoshenko beam resting on a Winkler elastic foundation

The influences of a Winkler elastic foundation modulus, slenderness ratio and elastically restrained boundary conditions on the critical load of a Timoshenko beam subjected to an end follower force are investigated. The characteristic equation for elastic stability is derived. It is found that the critical flutter load for the cantilever Timoshenko beam will first decrease as the elastic foundation modulus is increased and when the elastic foundation modulus is greater than the corresponding critical value, which corresponds to the lowest critical load, it will increase, instead. In particular, if the elastic foundation modulus is large enough, the critical flutter load for the cantilever Timoshenko beam can be greater than that of the Bernoulli-Euler beam. For a clamped-translational or clamped-rotational elastic spring supported beam resting on an elastic foundation, there exists a critical value of the spring constant for each beam. At this critical point, the critical load jumps and the type of instability mechanism changes. The jump mechanisms for beams resting on elastic foundations of different modulus values are different.