It is an easy task to obtain the analytical solution for a simple Euler-Bernoulli beam. Difficulties, however, do arise when a beam structure has a large number of different-type intermediate constraints, such as exterior simple supports, interior hinges, rollers and other elastic supports. The purpose of this paper is to develop a systematic approach to obtain exact solutions for any kinds of complex beam configurations. The main feature of this approach is that the vibration motion is treated as the result of wave scattering around different media. The discovery of wave scattering phenomena around intermediate constraints sheds the light, for the first time, on the fact that any constraint can serve as a waveguide capable of transmitting and reflecting waves in either directions. This new finding enables us to treat a complex beam structure as a stepped beam which can be handled easily by using the wave propagation approach, previously developed by Yong and Lin (1989) for the analysis of piece-wise periodic structures.

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