Bending stress calculator (σ = My/I)
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Result
For a sagging (positive) moment the fibres above the neutral axis are in compression and those below it in tension; a hogging moment reverses this. The stress is zero at the neutral axis and grows linearly with y.
Find the normal stress caused by a bending moment with the flexure formula σ = M·y/I. Pick a rectangular, solid circular or hollow circular section to have I and the extreme fibre distance c worked out for you, or enter I and y directly for any other shape. The result gives the stress in MPa (or ksi), the section modulus S = I/c and the full calculation, with the units converted along the way.
Step-by-step solution
Units
Section properties
Flexure formula
Key formulas
- Bending stress
- σ = M·y/I
- Rectangle
- I = bh³/12
- Section modulus
- W = I/ymax
How to calculate bending stress
- Find the bending moment M at the section, usually the largest moment from the bending moment diagram.
- Find the second moment of area I of the cross-section about its neutral axis, for example b·h³/12 for a rectangle or πD⁴/64 for a solid circle.
- Find the distance y from the neutral axis to the point you need. The largest stress is at the extreme fibre, y = c, which is h/2 for a rectangle.
- Use consistent units. With M in N·mm, y in mm and I in mm⁴, σ = M·y/I comes out in N/mm², which is MPa.
- Check with the section modulus: σ_max = M/S, with S = I/c.
Questions and answers
Which side of the beam is in tension?
For a sagging (positive) moment the bottom fibres are in tension and the top fibres in compression. A hogging moment, for example at the fixed end of a cantilever, reverses this. The stress is zero at the neutral axis.
What is the section modulus?
S = I/c, the second moment of area divided by the distance to the extreme fibre. It turns the flexure formula into σ_max = M/S, so the section modulus a beam needs is the bending moment divided by the allowable stress.
Where is the neutral axis?
For bending without an axial force, the neutral axis passes through the centroid of the cross-section. For symmetric shapes that is at mid-height; for a T-section, find it with the moment of inertia calculator.
Why must I convert kN·m to N·mm?
Because σ = M·y/I only gives MPa when the units match. 1 kN·m = 10⁶ N·mm, and the calculator shows this conversion in the first step. In US units, kip·ft is converted to kip·in, so the stress comes out in ksi.
Is the flexure formula always valid?
It assumes a straight beam of linear-elastic material, plane sections that remain plane and bending about a principal axis. It does not apply after yielding or to strongly curved beams, and near point loads and supports the local stresses can be higher.
Practise it yourself
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