Distribution of Stress in Built-In Beams of Narrow Rectangular Cross Section

Abstract

Abstract This paper deals with the problem of the distribution of stress in cantilever beams of narrow rectangular flanged cross section with one end of the beam rigidly built-in. Since an exact solution of this plane-stress problem appears difficult to obtain, an approximate solution is derived by applying the principle of least work. Instead of the linear normal stress distribution of the elementary beam theory, a third-degree polynomial is assumed, and the spanwise variation of this stress curve is determined by means of the calculus of variations. Numerical results are obtained with regard to the stresses at the built-in end of the beam, in their dependence upon (a) the span-height ratio of the beam, (b) the flange area-web area ratio of the beam, (c) Poisson’s ratio of the material, and (d) the distribution of load along the span. It is found that the deviations from the results of the elementary theory may be appreciable when the distance of the center of gravity of the load curve from the built-in end of the beam is less than twice the height of the cross section of the beam.