Bending Stress Calculator (σ = M/Z = M·y/I)
By Santaji GadepatilinUpdated November 3, 2026
A bending moment stretches one face of a beam and compresses the other, with the stress greatest at the outer fibres. This bending stress calculator applies σ = M ÷ Z (equivalently M·y ÷ I) to find that peak stress and compares it with the material's yield strength — the surface strain it produces is what strain gauges measure on loaded beams.
How to use the bending stress calculator
Enter the bending moment, the section’s elastic section modulus Z in cm³ and the material’s yield strength. The calculator returns the maximum bending stress at the outer fibre and the factor of safety against yielding. To size a section, enter the stress you will allow and clear Z.
Bending stress formula
σ = M × y ÷ I = M ÷ Z · FoS = Sy ÷ σ
Example
An IPE 200 (Z = 194 cm³) carrying a 15 kN·m moment, S235/A36-type steel with 250 MPa yield:
- σ = 15,000 N·m ÷ 194 × 10⁻⁶ m³ = 77.3 MPa
- Factor of safety = 250 ÷ 77.3 = 3.23
Moment capacity of IPE sections (Sy = 250 MPa)
| Section | Z (strong axis) | Moment at first yield | With FoS 1.5 |
|---|---|---|---|
| IPE 100 | 34.2 cm³ | 8.6 kN·m | 5.7 kN·m |
| IPE 120 | 53 cm³ | 13.2 kN·m | 8.8 kN·m |
| IPE 140 | 77.3 cm³ | 19.3 kN·m | 12.9 kN·m |
| IPE 160 | 109 cm³ | 27.2 kN·m | 18.2 kN·m |
| IPE 180 | 146 cm³ | 36.5 kN·m | 24.3 kN·m |
| IPE 200 | 194 cm³ | 48.5 kN·m | 32.3 kN·m |
| IPE 240 | 324 cm³ | 81.0 kN·m | 54.0 kN·m |
| IPE 300 | 557 cm³ | 139.2 kN·m | 92.8 kN·m |
Section moduli are nominal values from European IPE tables.
Notes
- Slender beams can fail by lateral-torsional buckling before reaching the yield stress; design codes reduce the capacity of unrestrained beams.
- Combine bending with axial or shear stresses (for example with the von Mises criterion) when both act together.
Frequently asked questions
What is the bending stress formula?
σ = M·y ÷ I = M ÷ Z, where M is the bending moment, y the distance from the neutral axis to the outer fibre, I the second moment of area and Z = I/y the elastic section modulus.
Where is bending stress highest?
At the top and bottom surfaces, farthest from the neutral axis, at the section with the largest bending moment — mid-span of a simply supported beam, or the fixed end of a cantilever.
What is the difference between elastic and plastic section modulus?
The elastic modulus Z gives the moment at which the outer fibre first yields. The plastic modulus (larger, by about 15% for I-beams and 50% for rectangles) gives the moment when the whole section has yielded; steel design codes use it for compact sections.
How do I get the bending moment?
For a simply supported beam with a central load it is PL/4; with a uniform load wL²/8; for a cantilever with an end load PL. The beam deflection calculator gives these moments too.