Library / Mechanical / MEC-35
Mechanical
Bending stress
Stress in a beam at distance y from the neutral axis.
- Ref
- MEC-35
- Rev
- v1
- Field
- Mechanical
- Result
- σ [MPa]
M is the bending moment in N·m, y the distance from the neutral axis to the point of interest in mm, and I the second moment of area of the section in mm⁴. The stress varies linearly with y: zero on the neutral axis and largest at the outer fibre, tension on one side and compression on the other.
Stress in a beam at distance y from the neutral axis. It is zero on the axis and greatest at the outer surface.
Calculator
- Section modulus I/y80,000 mm³
- Stress halfway to the surface (y/2)6.25 MPa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The defaults suit a 50 × 100 mm bar bent about its strong axis: a 1 kN·m moment gives 12.5 MPa at the outer edge, 50 mm from the neutral axis.
- Stress at the neutral axis is zero, so material near the middle contributes little to bending strength. This is why I-beams concentrate material in the flanges.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Bending moment | 1 to 1e7 N·m | 0.0125 | 125,000 | 125,000 | +10% | +10 % |
| I Second moment of area | 1,000 to 1e10 mm⁴ | 0.005 | 50,000 | 50,000 | +10% | −9.09 % |
| y Distance from neutral axis | 1 to 500 mm | 0.25 | 125 | 124.8 | +10% | +10 % |
Sweep of Second moment of area
| Second moment of area (mm⁴) | Bending stress (MPa) |
|---|---|
| 1,000 | 50,000 |
| 7,499 | 6,668 |
| 56,230 | 889.1 |
| 421,700 | 118.6 |
| 3,162,000 | 15.81 |
| 2.371e7 | 2.108 |
| 1.778e8 | 0.2812 |
| 1.334e9 | 0.03749 |
| 1e10 | 0.005 |
Sweep one variable
Variables and how each one contributes
Bending stress
What the equation gives you. Everything else feeds it.
Bending moment
The bending moment at the section, in N·m. Directly proportional: doubling the moment doubles the stress.
Distance from neutral axis
The distance from the neutral axis to the point where the stress is wanted, in mm. Directly proportional: the stress is twice as large twice as far from the axis, so use the outer surface for the maximum.
Second moment of area
The second moment of area of the cross-section, in mm⁴. Inverse: a stiffer section gives a lower stress. A 10 % rise in I lowers σ by about 9 %.
Worked example with the values above
- Convert the moment: M = 1,000 N·m = 1,000,000 N·mm.
- Numerator: M y = 1,000,000 × 50 = 5×107.
- σ = 5×107 / 4,000,000 = 12.5 MPa.
Assumptions and limits
- Elastic bending of a straight beam with a symmetric section, in a plane of symmetry.
- Plane sections stay plane, and the section is away from points of load application or sudden changes of shape.
Common mistakes
- Using the total depth where y is the half-depth to the extreme fibre.
- Mixing the units of I (mm⁴) with M in N·m. The formula converts M to N·mm.
Statistics and uncertainty
σ ∝ M y / I with I often found from a depth cubed, so an error in section depth is magnified: a 1 % error in the depth of a rectangle gives about a 2 % error in the peak stress (3 % in I, less 1 % from y). The moment is usually the least certain input.
Where it is used
- Checking a beam, boom or gun barrel under a bending moment.
- Finding whether a section stays elastic.
Maths you need
- Second moment of area: I = bh³/12
- Moments: force times distance