Analysis of free vibrations of stepped timoshenko beam on elastic foundation

The problem of beam on elastic foundation is encountered frequently on foundation of buildings, highways, railways and geotechnical applications. Therefore a great number of studies have been done on this subject. The beams subjected to static or dinamic loads, might have constant or variable cross-section. Stepped beams are created in order to save weight or satisfy various engineering requirements. In this study, free vibrations of a cantilever beam that consists of two pieces, has cross-section changes abruptly and rests on elastic foundation are examined within the framework of Timoshenko beam theory. The solution is also made for Euler- Bernoulli beam to compare the results. The Winkler foundation model is used as foundation model. After removing the coupling on displacement and rotation in the beam governing equations, closed form solutions of the differential equations are obtained by using trigonometric and hyperbolic functions. Boundary and continuity conditions are used to obtain homogenous equations. Natural frequencies of the system and mode shapes related to these frequencies are obtained from the numerical solution of these homogenous equations. Numerical results are also for the case in which there is no foundation. These results are compared with the results of Timoshenko and Euler- Bernoulli beams and given in the related tables and figures.