Bending

File:Beam in Bending.png

Figure 1. A side view of a simply supported beam (top) bending under a distributed lateral load (bottom).

Figure 2. The internal forces and the cross-sectional stress distribution of a beam in bending.

In engineering mechanics, bending (also known as flexure) characterizes the behavior of a structural element subjected to a lateral load. A structural element subjected to bending is known as a beam. A closet rod sagging under the weight of clothes on coat hangers is an example of a beam experiencing bending.

Bending produces reactive forces inside a beam as the beam attempts to accommodate the flexural load: in the case of the beam in Figure 1, the material at the top of the beam is being compressed while the material at the bottom is being stretched. There are three notable internal forces caused by lateral loads (shown in Figure 2): shear parallel to the lateral loading, compression along the top of the beam, and tension along the bottom of the beam. These last two forces form a couple or moment as they are equal in magnitude and opposite in direction. This bending moment produces the sagging deformation characteristic of members experiencing bending.

The compressive and tensile forces shown in Figure 2 induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-Beams) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region.

Stress in a beam

The classic formula for determining the bending stress in a member is: ${\sigma }={\frac {My}{I_{x}}}$

• ${\sigma }$ is the bending stress
• M - the moment at the neutral axis
• y - the perpendicular distance to the neutral axis
• I_x - the second moment of inertia about the neutral axis x

See also

• Second moment of area also known as the second moment of inertia