What this calculator does
Bending stress is the internal stress a beam experiences when it bends under load: tension on one face, compression on the other, and zero stress along the neutral axis running through the middle. It is the standard check that a beam is strong enough for the bending moment it carries, distinct from the simple axial stress of a straight pull or push.
The formula is bending stress equals the bending moment times the distance from the neutral axis to the outer edge of the beam, divided by the second moment of area of the cross-section. It is often written as sigma equals M y over I, and the maximum stress always occurs at the outermost fibre, furthest from the neutral axis.
The formula
Multiply the bending moment by the distance from the neutral axis to the extreme fibre (the outer edge where stress is greatest), then divide by the second moment of area of the beam's cross-section. The result is the maximum bending stress the beam experiences at that section.
| Term | Meaning |
|---|---|
| M | Bending moment, the internal turning force the beam resists at the section being checked. |
| y | The distance from the neutral axis to the outer edge (extreme fibre) of the cross-section, where bending stress is greatest. |
| I | Second moment of area of the cross-section, describing its resistance to bending. |
| σ | Bending stress, the resulting internal stress at the extreme fibre. |
The inputs explained
| Field | What to enter |
|---|---|
| Bending moment (M) (N·m) | The bending moment at the section being checked, from the applied loads and the beam's support conditions. |
| Distance from neutral axis to extreme fibre (y) (mm) | The distance from the neutral axis to the outer edge of the cross-section; for a symmetric section this is half the overall depth. |
| Second moment of area (I) (cm⁴) | The second moment of area of the beam's cross-section, which depends on its shape and dimensions. |
When to use it
Checking a beam is strong enough for a known load
Once the bending moment from a load is worked out, comparing the resulting bending stress against the material's allowable stress is the standard strength check before a beam is specified.
Comparing two cross-sections of the same material
Holding bending moment and material fixed and changing only the section's depth and second moment of area shows directly how much a deeper or differently shaped beam reduces peak stress.
Working alongside a deflection check
Bending stress and beam deflection are both consequences of the same load and cross-section, and a beam design is usually checked against both: stress for strength, deflection for stiffness and serviceability.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How bending stress changes with bending moment
A fixed 75 mm distance to the extreme fibre and 8,000 cm⁴ second moment of area, across a range of bending moments.
| Bending moment | Maximum bending stress | Stress in pascals |
|---|---|---|
| 1000 N·m | 0.94 MPa | 937,500 Pa |
| 2000 N·m | 1.87 MPa | 1,875,000 Pa |
| 5000 N·m | 4.69 MPa | 4,687,500 Pa |
| 8000 N·m | 7.50 MPa | 7,500,000 Pa |
| 10000 N·m | 9.38 MPa | 9,375,000 Pa |
| 15000 N·m | 14.06 MPa | 14,062,500 Pa |
How bending stress changes with distance from the neutral axis
A fixed bending moment of 5,000 N·m and second moment of area of 8,000 cm⁴, across a range of distances to the extreme fibre.
| Distance to extreme fibre | Maximum bending stress | Stress in pascals |
|---|---|---|
| 25 mm | 1.56 MPa | 1,562,500 Pa |
| 50 mm | 3.12 MPa | 3,125,000 Pa |
| 75 mm | 4.69 MPa | 4,687,500 Pa |
| 100 mm | 6.25 MPa | 6,250,000 Pa |
| 125 mm | 7.81 MPa | 7,812,500 Pa |
| 150 mm | 9.38 MPa | 9,375,000 Pa |
Questions
How is bending stress different from the stress in the stress calculator?
The general mechanical stress calculator covers simple axial stress: a straight force divided by cross-sectional area, uniform across the section. Bending stress instead varies across the cross-section, from tension on one face to compression on the other, and depends on the bending moment and the section's second moment of area rather than a simple applied force.
Where is bending stress greatest in a cross-section?
At the extreme fibres, the points furthest from the neutral axis, top and bottom of a typical symmetric section. Stress is zero at the neutral axis itself and increases linearly with distance from it, reaching a maximum at the outer edge.
Why does a deeper beam reduce bending stress?
A deeper cross-section has a larger second moment of area, which sits in the denominator of the formula, so for the same bending moment a deeper section spreads the internal stress more effectively and reduces the peak value at the extreme fibre.
What is the neutral axis?
It is the line through a beam's cross-section, usually through its centroid, that experiences no bending stress at all; material on one side is in tension, material on the other side is in compression, and the transition between the two happens at the neutral axis.
For the deflection of the same beam under load, see the beam deflection calculator. For a simple axial force-over-area stress instead, see the mechanical stress calculator.