Internal Forces and Moments in Beams

Abstract

Consider a straight, horizontal beam with arbitrary supports. Let a cartesian coordinate system be oriented with its origin at the centroid of the beam’s cross section, the x axis extending to the right along the beam’s axis, and the y axis upward. Assume that the external forces and couples on the beam and the reactions due to its supports exert a two-dimensional system of forces and couples in the x‐y plane. Let a plane be passed through the beam perpendicular to its axis, and consider the free-body diagram of the part of the beam to the left of the plane. The distribution of stress on the plane can be represented by an equivalent system consisting of two components of force and a couple. One of the force components is the axial force P perpendicular to the cutting plane, which is assumed to act at the centroid of the cross section and be positive in the positive x direction. The other force component is the shear force V tangential to the cutting plane, assumed to be positive in the negative y direction (downward). The couple is the bending moment M, assumed to be positive in the counterclockwise direction about the z axis. If the external loads on the beam and the reactions due to its supports are known, the equilibrium equations can be applied to the free-body diagrams on either side of the cutting plane to determine the internal forces and moment P, V, and M. Graphs of V and M as functions of x are called the shear force and bending moment diagrams, respectively. It can be shown that the shear force and bending moment in a beam subjected to a distributed load w satisfy the differential equations dV/dx = − w, dM/dx = V. In principle, these equations can be used to determine the shear force and bending moment as functions of x. The first equation is integrated to determine V, and then the second equation is integrated to determine x. In doing so, the effects of point forces and couples must be accounted for. A substantial complexity is that each time the equation describing the distributed force on the beam changes, or at each location where a point force or couple acts, a new free-body diagram must be used or new solutions of the two differential equations must be obtained. It would be convenient if the loading on an entire beam could be expressed by a single function. This can be done using what are called singularity functions. These are defined and it is shown how point forces are represented by delta functions, couples are represented by dipoles, and distributed forces are represented by Macaulay functions. Examples of using singularity functions to determine the shear and bending moment diagrams are presented.