Stresses in Beams

Abstract

The stresses on a beam’s cross section give rise to the axial force, shear force, and bending moment discussed in Chap. 5. The axial force P is exerted by a uniform distribution of normal stress. A distribution of normal stress that exerts a couple causes the bending moment M, and the shear force V is exerted by a distribution of shear stress. The distributions of stress associated with the bending moment and shear force are derived and discussed in this chapter. They are described in terms of a Cartesian coordinate system 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. The beam’s cross section is assumed to be symmetric about the y axis. Analysis of the normal stress due to bending begins by considering a beam subjected to couples M at the ends. From the geometry of deformation and the stress–strain relations, the normal stress distribution is determined to be described by the linear equation σx = − Ey/ρ, where ρ is the radius of curvature of the beam’s axis. By applying equilibrium, it is shown that the radius of curvature is related to the bending moment by 1/ρ = M/EI, where I is the moment of inertia of the beam’s cross section about the z axis. As a result, the distribution of normal stress due to the bending moment is given by σx = − My/I. Examples of the determination of the stress distributions in beams with different cross section and types of loading are presented, followed by a discussion of issues in beam design. Composite beams, consisting of bonded beams of different materials, are analyzed, followed by beams consisting of elastic–perfectly plastic material. The coverage of the normal stress distribution closes with a discussion of beams having cross sections that are not symmetric about the y axis. To analyze the distribution of shear stress on a beam’s cross section, an equation called the shear formula is derived. Consider a horizontal line across the cross section at a height y′. Denote its length by b. Let A′ be the part of the beam’s cross-sectional area above y′, and let the height of the centroid of A′ be denoted by \( {\overline{y}}^{\prime }. \) Define \( Q={\overline{y}}^{\prime }{A}^{\prime }. \) Then the shear formula τav = VQ/bI gives the average value of the shear stress across the width of the horizontal line. Applications of the shear formula are presented, including built-up beams consisting of prismatic beams of the same material that are connected along their lengths and beams having thin-walled cross sections.