Determination of the first transition of mode shapes for buckling and free vibration problems of uniform simply supported beams on variable two parameter elastic foundations through the concept of equivalent uniform Winkler foundation

摘要

Two parameter foundation models represent accurately the foundation characteristics compared to the simple, single parameter model (Winkler model). The widely used two parameter foundation model is the Pasternak foundation model. Further, beams on elastic foundation exhibit an interesting phenomenon of changing mode shapes (from the first mode to the second mode and so on) for both buckling and free vibration problems at specific foundation stiffnesses parameter(s). While, evaluating the foundation stiffness parameter for beams on Winkler foundation, for both the buckling and vibration problems is easy, in the case of the two parameter uniform or variable foundations, the procedure is more involved. Further, most of the practicing engineers are very familiar with the Winkler foundation than the two parameter foundations. Hence, it will be very useful and elegant if one obtains an equivalent uniform Winkler foundation to represent the uniform or variable two parameter elastic foundations, for example, the Pasternak foundation, such an attempt is made in this paper. The efficacy of the concept of the equivalent uniform Winkler foundation, in determining the first transition stiffness parameter(s) of mode shapes for the buckling and free vibration problems of beams on either a uniform or a variable Pasternak foundation is clearly demonstrated.