Beam Bending Stress

Bending stress in a beam

The normal stress produced by bending comes from σ = M·c/I = M/W. Here M is the bending moment, c the distance from the neutral axis out to the extreme fibre, I the second moment of area and W = I/c the section modulus. Take M = 50 kN·m and W = 500 cm³, and you get σ = 50·10³ / (500·10⁻⁶) = 100 MPa. With structural steel ASTM A36 at f_y = 250 MPa and a safety factor of 1.1, the admissible stress lands around 230 MPa. Concrete in bending runs much lower, with admissible σ ≈ 0.15·f_ck.

Keep your units consistent: M in N·m, W in m³, σ in Pa. The Brazilian codes NBR 8800 (steel) and NBR 6118 (concrete) work by limit-state design with partial factors rather than one global FS. What this tool offers is a fast allowable-stress check, the kind you reach for during preliminary sizing.

Applications

Sizing rolled I/W profiles, channels and tees in the early stages. Verifying flexure in reinforced-concrete beams under NBR 6118. Checking timber joists and glulam. And comparing section moduli when you have to pick between steel profiles per NBR 8800.

FAQ

What is the section modulus W? W = I/c packs the geometry into one number. A bigger W means lower bending stress for the same M. Profile tables hand you W_x and W_y straight off.

Top or bottom fibre? When the section is symmetric (I, rectangular) σ_top = σ_bot, so it doesn't matter. With an asymmetric section like a T or channel, take the larger c on the tension side, or whichever side the moment sign points to.

Why doesn't this give ultimate capacity? The elastic σ = M/W rests on linear-elastic behaviour. Plastic capacity is a different animal: it relies on the plastic modulus Z (Z > W) and the yield moment M_p = Z·f_y, which NBR 8800 §F governs.