Bending stress formula
When a beam bends, one face stretches and the other compresses, creating bending stress that is highest at the surfaces. The flexure formula gives that peak stress from the bending moment and the beam cross-section.
Variables
| σ | Bending stress | psi or Pa |
| M | Bending moment | lb·in or N·m |
| c | Distance from neutral axis to outer fiber | in or m |
| I | Area moment of inertia | in⁴ or m⁴ |
Rearranged
σ = M / S, where S = I / c
Worked example
A beam carries a 5,000 N·m moment, with I = 8 × 10⁻⁶ m⁴ and c = 0.1 m.
Apply the flexure formula: 5,000 × 0.1 / 0.000008 = 62.5 MPa.
Bending stress is zero at the neutral axis and greatest at the outer fibers, where c is largest. Because the section modulus S equals I divided by c, the formula simplifies to stress equals moment over section modulus, which is how beams are usually sized.
Sizing a beam section?
See the Section Modulus Formula and the Steel Beam Properties Chart.
The flexure formula
The flexure formula, stress equals moment times c over I, comes from assuming plane sections stay plane as a beam bends. Strain grows linearly with distance from the neutral axis, so stress does too, peaking at the outer fibers. This is why deep beams are efficient: putting material far from the neutral axis raises I and lowers stress.
Using section modulus instead
Since the outer-fiber distance c and the moment of inertia I always appear together as I over c, engineers combine them into the section modulus S. The formula then reads stress equals moment divided by S, so picking a beam becomes a matter of finding a section whose S keeps the stress below the allowable value.
FAQ
What is the bending stress formula?
Bending stress equals the moment times the distance to the outer fiber divided by the moment of inertia, sigma = Mc / I, or equivalently M / S.
Where is bending stress highest?
At the outer fibers, farthest from the neutral axis, where c is largest. It is zero at the neutral axis itself.
What is the section modulus?
It is I divided by c, a single property that combines the cross-section geometry so bending stress is simply moment over section modulus.