Method of Internal Constraints and its Application

An approximate method that is of both general and systematic character is presented. The method unifies a number of engineering theories in the statics and dynamics of plates and bars, and, in addition, is simple and direct. The method is based on assumptions concerning the components of the elastic displacement. In fact, it is assumed that the components of the elastic displacement must comply with special equations of constraint. These equations of constraint, which determine the elastic displacement, are the basis for the term “ method of internal contraints. ” The method is applied first to the discussion of vibrations of elongated straight rods. The equations of motion and the boundary conditions for flexural vibrations are derived. It is shown that these equations reduce, in the plane-stress case, to the well-known Timoshenko beam equation. The equations of motion and boundary conditions for longitudional vibrations of elongated rods are derived, and the results given by these equations are compared with those given by use of other engineering theo ries and with those given by the exact Pochhammer solution. Finally the method is applied to the discussion of flexural vibrations of rectangular thin plates, and the equations of motion and proper boundary conditions are derived. These equations of motion coincide with the equations of motion derived by R. D. Mindlin, M. ASCE. The present equations reduce, in the static case, to equations that are very similar to those obtained by E. Reissner. However, the present method of deriving the equations is simpler and more direct than are these other methods.