Lower bounds to fundamental frequencies and buckling loads of columns and plates

A general derivation of expressions for lower bounds to fundamental frequencies and buckling loads is given for the class of structures governed by linear elastic theory in the prebuckling state. These expressions involve two Rayleigh quotients both of which are upper bounds for the fundamental frequency under a prescribed load. The displacement trial functions must satisfy force and kinematic continuity but no other conditions are required. Thus, if appropriate high order base functions are used, the finite element procedure can be used to systematically narrow the difference between the upper and lower bounds.The theory is illustrated with several column and plate problems. The finite element method is applied to uniform and nonuniform columns with a representative set of boundary conditions. Elementary trial functions are used to show that reasonable bounds can also be obtained for plates subjected to known states of stress. Since the lower bound is obtained with a variation of the classical technique of Rayleigh, these results indicate that the method may be suitable for conservatively estimating buckling loads and fundamental frequencies of engineering structures.