» Moment-Shear-Load Relations
Consider the simply supported beam at right. By isolating an element of the beam with length dx, and applying the equilibrium conditions learned in Statics, relations between MOMENT, SHEAR and LOAD can be derived:

dV dx = q(x)     and     dM dx = V(x)

Convention Note:

• A shear force acting on a positive face in a positive direction is POSITIVE. (A positive shear for on the right face acts up)
• A positive moment causes a beam to bend into a "happy face"-shape.

» Beam Bending
When a beam is subjected to PURE BENDING, it deforms in the manner shown below. Viewed from the side, the deflection takes the form of a circular arc with a radius of R (measured to the neutral axis of the beam). From geometry, the strain in the beam must be:

e(y) = y R

• Above the neutral axis (y > 0), the strain is negative (compressive);
• Below the neutral axis the strain is positive (tensile).
• k = 1/R is the Curvature.

» Bending Stress
Knowing the strain due to bending, from Hooke's Law, the stress due to bending is then:

s(y) = Ee(y) = Ey R = Eyk

By applying the equilibrium conditions and making a couple of simple substitutions, BENDING STRESS can be given as: