City University of Hong Kong
Dept. of Physics & Materials Science
AP2104 Mechanics of Solids
Pure Bending of a Beam
Yield Criteria for Ductile Materials under Plane Stresses
Pure Bending of a Beam
1. To examine the stresses at various positions of the beam under a constant load of pure bending.
2. To determine the curvature of deflection of the beam.
1. Pure Bending and Nonuniform Bending
When analyzing beams, it is often necessary to distinguish between pure bending and nonuniform bending. Pure bending refers to flexure of a beam under a constant bending moment. Therefore, pure bending occurs only in regions of a beam where the shear force is zero ( because V = dM/dx ). In contrast, nonuniform bending refers to flexure in the presence of shear forces, which means that the bending moment changes as we move along the axis of the beam. As an example of pure bending, consider a simple beam AB loaded by two couples M1| having the same magnitude hut acting in opposite directions (Fig. 1a). These loads produce a constant bending moment M = M1 throughout the length of the beam. Note that the shear force V is zero at all cross sections of the beam. [pic]
Fig. 1 (a) Simple beam in pure bending. (b) Cantilever beam in pure bending
Another illustration of pure bending is given in Fig. 1b, where the cantilever beam AB is subjected to a clockwise couple M2 at the free end. There are no shear forces in this beam, and the bending moment M is constant throughout its length. The bending moment is negative (M = - M2). The symmetrically loaded simple beam of Fig. 2a is an example of a beam that is partly in pure bending and partly in nonuniform bending, as seen from the shear-force and bending-moment diagrams (Figs. 2b and c). The central region of the beam is in pure bending because the shear force is zero and the bending moment is constant. The parts of the beam near the ends are in nonuniform bending because shear forces are present and the bending moments vary.
Fig. 2 Simple beam with central region in pure bending and end regions in nonuniform bending
2. Deformations of a Beam in Pure Bending
Consider a portion ab of a beam in pure bending subjected to positive bending moments M (Fig. 3a). We assume that the beam initially has a straight longitudinal axis (the x axis in the figure) and that its cross section is symmetric about the y axis. Under the action of the bending moments, the beam deflects in the xy plane (the plane of bending) and its longitudinal axis is bent into a circular curve (curve ss in Fig. 3b). The beam is bent concave upward, which is positive in curvature.
Fig. 3 Deformation of a beam in pure bending
In this bending deformation, o’ is the center of curvature
Cross sections mn and pq rotate with respect to each other about axes perpendicular to the xy plane. Longitudinal lines on the convex (lower) part of the beam are elongated, whereas those on the concave (upper) side are shortened. Thus, the lower part of the beam is in tension and the upper part is in compression. There is a surface in which longitudinal lines do not change in length. This surface, shown by the dashed line ss in Figs. 3 and, is called the neutral surface of the beam. [pic]
Fig. 4 Relationship between the bending moment and the normal stresses
3. Maximum Stresses at a Cross Section
The maximum tensile and compressive bending stresses acting at any given cross section occur at points located farthest from the neutral surface ( Fig. 4 ). Let c1 and c2 denote the...
Please join StudyMode to read the full document