Bending Stress from Section Modulus (σ = M/S)

This is how beams are actually selected in the field. Work out the maximum moment, divide by the allowable stress, and you have the section modulus you must buy: S ≥ M/σ_allow. A 20 000 ft·lbf moment at an allowable 24 ksi needs S = 240 000 in·lbf ÷ 24 000 psi = 10 in³ — flip to the table, find the lightest shape with S above 10 in³, done. In SI the same check on a 10 kN·m moment through a section with S = 8.333 × 10⁻⁵ m³ gives σ = 10 000 ÷ 8.333 × 10⁻⁵ = 120 MPa.

Enter S as a plain number in m³ (1 in³ = 1.6387 × 10⁻⁵ m³) and M in newton-metres; the answer comes back in pascals. The trap is that passing the stress check is not the same as passing the design. A beam sized purely on S may still deflect visibly, may buckle sideways if its compression flange is unbraced, and may crush its web where it lands on a bearing plate. Stress, deflection, lateral-torsional buckling and bearing are four separate checks, and in long shallow spans deflection usually governs first.