Bending: Encyclopedia - Bending

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 clothes 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.

Bending - Stress in a beam

Beam bending is analyzed with the Euler-Bernoulli beam equation. The classic formula for determining the bending stress in a member is:

• σ 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

This equation is valid only when the stress at the extreme fiber (i.e. the portion of the beam furthest from the neutral axis) is below the yield stress of the material it is constructed from. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the design of steel structures.

See also

• Deflection
• Contraflexure

Category: Continuum mechanics

Adapted from the Wikipedia article "Bending", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki