The Flexure Problem for Rectangular Beams with Slits

Abstract

Experimental and approximate theoretical studies, made at the United States Forest Products Laboratory, Madison, Wisconsin, show that the mean shearing stress in the neutral plane of a rectangular beam with symmetrical horizontal slits or checks extending along its lateral faces is less than that given by the usual engineering formulas; that is, there is a definite ``two‐beam'' action which relieves the shearing stress in the portion of the beam weakened by the slits. This paper presents an exact solution of the flexure problem for a beam of such section and confirms the main features of the distribution of shearing stress found in the earlier investigation. The solution of the flexure problem requires the solution of Laplace's equation within the plane region coincident with a cross section of the beam, subject to certain boundary conditions. This solution is obtained by using the Schwartz‐Christoffel transformation of a polygon onto a half‐plane and solving the transformed boundary value problem. The exact solution is set up as an infinite series. To simplify the calculations two approximate solutions are also obtained, one of which is used in treating a numerical example.