6b. Bending in Beams  (SSES Ch. 6.1)
Moment-Shear-LoadBeam BendingBending Stress
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Moment-Shear-Load Relations
Consider the simply-supported beam subjected to distributed load q(x) at right. By isolating an element of length dx, and applying the equilibrium conditions, relations between moment, shear and distributed load can be derived:

Note the convention:

  • A shear force is termed POSITIVE if in acts on a positive face in a positive direction (or on a negative face in a negative direction). A positive shear force on the right-face - the positive x-face - acts upward, in the positive y-direction.
  • A positive moment causes a beam to bend into a "happy face"-shape. A moment is POSITIVE if it acts on a positive face about a positive axis (or a negative face about a negative axis). Here, the moment on the right face acts on the +x-face, about the +z-axis, which is out of the plane (screen, paper).

q(x) should be w(x)
and constant over dx.

Beam Bending

When a beam is subjected to pure bending (constant moment), it deforms in the manner shown at right. Viewed from the side, the deflection takes the form of a circular arc with a radius of curvature of R (measured to the neutral axis of the beam). The neutral axis is a line along the beam axis that does not change length and so has zero strain.

From geometry, the strain in the beam must be:

  • For a positive moment, the strain above the neutral axis (y > 0), the strain is negative (compressive);
  • For a positive moment:, the strain below the neutral axis the strain is positive (tensile).
  • Kappa, k = 1/R is the Curvature.

Bending Stress

From Hooke's Law, the stress due to bending is related to the bending strain:

Applying the equilibrium conditions and making a couple of algebraic substitutions, the Bending Stress - a normal stress in the x-direction - is:

  • I is the moment of inertia (the second moment of area) of the beam cross-section;
  • y is measured from the neutral axis which passes through the centroid of the cross-section;
  • M is the moment acting on the beam cross-section.

The Maximum Bending Stress occurs at the top and/or bottom of the beam, where y = ymax = c. The magnitude of the maximum bending stress is:

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Updated: 05/24/09 DJD