Free vibration of flexural-shear plates

In this paper, multi-storey buildings with narrow rectangular plane configuration (narrow buildings) are treated as one-step or multi-step cantilever flexural-shear plates in analysis of free vibration. The governing differential equations for free vibration of flexural-shear plates with variably distributed mass and stiffness are established and reduced to Bessel's equations by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness and mass along the height of the plates. The general solutions of one-step flexure-shear plates are derived and used to obtain the frequency equation of multi-step cantilever flexural-shear plates. A new exact approach is presented which combines the transfer matrix method and closed form solutions of one step flexural-shear plates. Two numerical examples demonstrate that the calculated natural frequencies and mode shapes of the narrow buildings are in good agreement with the corresponding experimental data. It is shown that it is possible to regard a building with rigid floors as a cantilever flexural bar that is a special case of a cantilever flexural-shear plate. Thus, the proposed methods in this paper are suitable for the calculation of free vibrations of narrow buildings and common shear-wall buildings.