The reflection method in the bending of beams and plates

The problem of determining the deflection and stress in a plate under transverse loading can be approached by first considering the plate to be a portion of an infinite plate, ignoring the prescribed boundary conditions. The known loads are then applied to the infinite plate and their effects are calculated at those points which correspond to the boundary of the original plate. A system of suitably chosen loads and moments is then applied on the infinite plate at points beyond the boundary of the original plate such that the prescribed boundary conditions are satisfied.

For an exact solution, the number of external loads and moments would have to be infinite. However, in order to deal with the problem numerically, only a finite number of each are considered. Thus, solutions are obtained by satisfying the boundary conditions at only a finite number of points. The method is illustrated for beams and then extended to plates.

Several problems with known solutions are solved and the results compared with the exact values. Also, plots of the deflection and moment along the centerline of a cantilevered triangular plate are presented.

Discussions of the problem of plates with holes and the effect on the solution of various placements balancing loads are also presented.

An IBM 1620 digital computer is used to facilitate calculations