Bending Stress Calculator
The maximum bending stress in a beam from the bending moment and the cross-section — rectangle, circle or a section you enter — with the second moment of area, the section modulus and the utilization against an allowable stress.
Bending puts the outer fibres of a beam in tension on one face and compression on the other, and the stress grows linearly from the neutral axis: σ = M y ÷ I, where y is the distance to the outer fibre and I the second moment of area.
How the bending stress calculator works
Bending puts the outer fibres of a beam in tension on one face and compression on the other, and the stress grows linearly from the neutral axis: σ = M y ÷ I, where y is the distance to the outer fibre and I the second moment of area. I ÷ y is the section modulus, so σ = M ÷ Z. Compare with the material’s allowable stress for the utilization.
Formula: σ = M × y ÷ I; rectangle I = b h³ ÷ 12; circle I = π d⁴ ÷ 64
Worked examples
| Inputs | Maximum bending stress (MPa) | Note |
|---|---|---|
| 10 kN·m on a 100 × 200 mm rectangle | 15 | 15 MPa |
| 5 kN·m on a 100 mm round bar | 50.93 | 50.9 MPa |
| Custom section: I = 1,000 cm⁴, y = 100 mm, 20 kN·m | 200 | 200 MPa — over |
FAQFrequently asked questions
Why is depth so much more important than width?
I grows with the cube of depth but only linearly with width: doubling the depth of a rectangular beam cuts its bending stress by four (and its deflection by eight). That is why joists stand on edge and I-beams put their material far from the center.
What allowable stress should I use?
Steel design codes work from a yield stress (S275, S355, or 36–50 ksi in the US) with safety and load factors; timber and aluminium use published characteristic values. The default of 165 MPa is a common allowable for mild steel — confirm against your code.
Where is the maximum moment?
At mid-span for a simply supported beam under a uniform load (wL²/8), at the wall for a cantilever (wL²/2). This calculator takes the moment as given; the beam reactions calculator finds it.
Does this cover shear and deflection?
No — bending stress is one of three checks. Shear governs short deep beams, and deflection (serviceability) often governs long ones even when stress is fine.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Institute of Standards and Technology — the US measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.