Scaling laws for displacement of elastic beam by energy method

Abstract

This paper introduces the Engesser's first theorem to the area of structural similarity. General derivation of the scaling laws is described and applied to problem of symmetrical bending of elastic beam under simultaneous application of various loading types. The applied loads considered are concentrated forces, distributed forces, and self-weight load. New sets of complete similarity conditions and scaling laws are established, especially for the problem of nonlinearly elastic beams. It has been shown that the complete similarity condition can be achieved by a geometrically similar model or distorted model satisfying an additional condition. The scaling laws are verified by the problems of a nonlinearly elastic cantilever beam and linearly elastic 2-D frame, which the displacement solutions are determined by analytical and numerical approaches, respectively. These scaling laws are found to accurately predict the displacement of the prototypes in all cases.