Thin elastic plates subject to vibration in their own plane

Abstract

Whereas analytic and experimental investigations of plates subject to lateral vibrations have been rather thorough, the present study is an analytic determination of the various critical frequencies of vertically cantilevered thin elastic rectangular plates vibrating freely within their own planes. Within the restrictions imposed by excluding any motion perpendicular to the face of the plate, the upright edges are free to move in the other two directions, as is the top horizontal edge. Three different base conditions are imposed:

• A clamped lower edge; • A lower edge which is freely vibrating transversely in the plane of the wall where the vertical fibers of the wall are fixed at their roots; and • A horizontally freely pulsating lower edge where the vertical fibers of the wall are fixed at their roots.

The first two conditions are considered in relation to plate vibrations which are essentially vertical while the first and third conditions are each employed with essentially horizontal plate vibrations. In every case the effect of a uniform load placed along the upper edge is studied.

Critical frequencies and associated amplitude coefficients are obtained for various ratios of base length to wall height.

The solution, which is presented in tabular and graphic forms, is obtained by using the method of iteration on the Rayleigh-Ritz energy procedure.

It is concluded that, for a wall with a clamped base vibrating in accordance with the given stipulations, the fundamental period is proportional to the square root of the face area of the wall. When the base of the wall is vibrating there is only one critical period, and it varies with the height of the wall. The factor of proportionality should take into account the material of which the wall is composed.

For designing unframed walls, subjected to dynamic loads in their plane, where the applied shear is to be taken as some constant times the dead load at the base of the wall, the recommended lateral force requirements of the Seismology Committee of the Structural Engineers Association of California, as set forth in 1959, seem adequate as modified above.