How to use this calculator
Enter the bending moment at the section you want to check, usually the maximum moment from a beam analysis. Then enter the second moment of area about the bending axis and the distance from the neutral axis to the outermost fiber. Pick a material, or type a yield strength, to get the factor of safety.
The flexure formula
Stress varies linearly through the depth: zero at the neutral axis, maximum at the top and bottom fibers. For a positive (sagging) moment the top is in compression and the bottom in tension.
| Symbol | Meaning | SI | US |
|---|---|---|---|
| M | Bending moment at the section | N·m (kN·m) | lbf·ft (kip·ft) |
| I | Second moment of area about the bending axis | mm⁴ (cm⁴) | in⁴ |
| c | Distance from neutral axis to extreme fiber | mm | in |
| S | Elastic section modulus, I/c | mm³ | in³ |
| σy | Yield strength of the material | MPa | ksi |
Worked example
Steel I-beam, SI units (the calculator’s default values)
M = 20 kN·m, I = 1,846 cm⁴ = 1.846 × 10⁻⁵ m⁴, c = 100 mm, σy = 250 MPa.
- σmax = (20,000 × 0.100) / 1.846 × 10⁻⁵ = 1.083 × 10⁸ Pa = 108.3 MPa.
- S = I / c = 1.846 × 10⁻⁵ / 0.100 = 1.846 × 10⁻⁴ m³ = 184,600 mm³.
- n = 250 / 108.3 = 2.31.
In US units, M = 10 kip·ft on a W8×10 (I = 30.8 in⁴, c = 3.945 in) gives σ = 120 × 3.945 / 30.8 = 15.4 ksi.
Assumptions and limits
- Linear elastic material: the result is only valid below yield.
- Bending about a principal axis of the section, with plane sections remaining plane.
- The beam is not too short and deep, and has no sudden changes in section near the point checked, where stress concentrations would raise the stress.
- Shear stress, lateral-torsional buckling and local buckling of thin flanges are not checked.
Frequently asked questions
Where do I get M?
From a beam analysis. For a simply supported beam with a uniform load, Mmax = wL²/8 at midspan. The simply supported beam and cantilever calculators give it directly.
What if my section is not symmetric, like a T-beam?
The top and bottom fibers are at different distances from the neutral axis, so the stresses differ. Run the calculation twice with ctop and cbottom; the larger c gives the larger stress.
What factor of safety should I use?
It depends on the application and code. Allowable stress design for steel beams often works out to a factor of about 1.67 on yield, while machine parts with uncertain loads commonly use 2 to 4.
Should I use S or Z?
S (elastic modulus) gives the moment at first yield. Z (plastic modulus) gives the full plastic moment, which steel codes use for the strength of compact sections. This calculator uses S because it reports stress.
For education and preliminary checks only. Have a qualified engineer verify any design before it is built.