Deflections of Beams

Abstract

Let v be the deflection of a prismatic beam’s neutral axis (the line through the centroid of the cross section) relative to the x axis, and let θ be the angle between the neutral axis and the x axis. The objective of this chapter is to determine v and θ as functions of x for a beam with given loads. For small deflections, the deflection and slope are related by dv/dx = θ. It is shown that the deflection satisfies the second-order differential equation v″ = M/EI, where primes denote differentiation with respect to x, M is the bending moment and I is the moment of inertia of the beam’s cross section about the z axis. Determining a beam’s deflection using this differential equation requires three steps: (1) Determine the bending moment as a function of x in terms of the loads and reactions acting on the beam. This may result in two or more functions M1, M2 … , each of which applies to a different segment of the beam’s length. (2) For each segment, integrate the second order equation twice to determine v1, v2, … . If there are N segments, this step will result in 2N unknown integration constants. (3) Use the boundary conditions on the deflection and slope to determine the integration constants and, if the beam is statically indeterminate, the unknown reactions. If the distribution of the bending moment in a beam is determined in terms of singularity functions as described in Sect. 5.4, the second-order equation can be integrated to determine the beam’s deflection in terms of singularity functions. For beams with more complex loads, this approach reduces the number of integration constants that must be determined and results in a single equation for the deflection throughout the beam, greatly simplifying the analysis. In some cases in which the information required about a beam’s deflection and slope are limited, the moment-area method can be applied. Let A and B be two positions on a beam’s axis with axial coordinates xA and xB, and let θA and θB be the values of the slope at those positions. The first moment-area theorem states that the change in the slope from A to B is θB − θA = AAB, where AAB is the area defined by the graph of M/EI from xA to xB. The second moment-area theorem states that the change in the deflection from A to B is vB−vA=xBθB−xAθA−x¯ABAAB,$$ {v}_B-{v}_A={x}_B{\theta}_B-{x}_A{\theta}_A-{\overline{x}}_{AB}{A}_{AB}, $$ where x¯AB$$ {\overline{x}}_{AB} $$ is the axial position of the centroid of the area defined by the graph of M/EI from xA to xB. The chapter ends with a discussion of superposition of beam deflections. The differential equation governing the beam deflection is linear, which means that solutions can be summed to obtain new solutions. By superimposing tabulated solutions for the deflection, the deflection due to many different types of loading can be obtained.