Bending Stress (σ = Mc/I)

Bend a beam and the top fibres shorten while the bottom fibres lengthen; somewhere between runs a neutral axis that does neither, and the stress grows linearly with distance c from it, peaking at the outermost fibre. Galileo posed the problem in Two New Sciences in 1638, sketching a cantilever built into a wall — but he assumed the beam pivoted about its bottom face, with the entire section in tension, and so overestimated a rectangular beam's strength by a factor of three. Antoine Parent in 1713 and Coulomb in 1773 put the neutral axis where it belongs, at the centroid, and the formula settled into the shape used ever since.

A 10 kN·m moment on a 50 × 100 mm rectangle (I = 4.167 × 10⁻⁶ m⁴, c = 0.05 m) gives σ = 10 000 × 0.05 ÷ 4.167 × 10⁻⁶ = 120 MPa. Two traps. First, I must be in m⁴ and M in N·m here — mixing in⁴ with N·m produces a number wrong by seven orders of magnitude. Second, c is measured from the neutral axis, which for a symmetric section is the mid-depth (half the total depth, not the full depth) and for an unsymmetric one must be located by finding the centroid first. The formula also assumes pure bending of a straight, prismatic, elastic beam; sharp notches, holes and welds concentrate stress well above what it predicts.