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Mechanical
Transverse shear stress in a beam
Shear stress at a cut through a loaded beam.
- Ref
- MEC-23
- Rev
- v1
- Field
- Mechanical
- Result
- τ [MPa]
Elastic beam theory. Q is the first moment of the area above or below the cut about the neutral axis, b the local width, I the second moment and V the shear force. Units here: V in kN, Q in 10³ mm³, I in 10⁶ mm⁴, b in mm give τ in MPa.
Shear stress in a beam depends on the shear force, the amount of material beyond the cut and the width at the cut.
Calculator
- Shear force in N12,000 N
- Shear stress in terms of V/(I/Q·b)22.5 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.
- Shear stress peaks at the neutral axis and is zero at the top and bottom faces.
- A thin web carries high shear stress because b appears in the denominator.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| I Second moment | 0.01 to 1,000 10⁶ mm⁴ | 0.12 | 12,000 | 12,000 | +10% | −9.09 % |
| V Shear force | 0.1 to 500 kN | 0.1875 | 937.5 | 937.3 | +10% | +10 % |
| Q First moment | 1 to 5,000 10³ mm³ | 0.1125 | 562.5 | 562.4 | +10% | +10 % |
| b Width at the cut | 2 to 300 mm | 1.5 | 225 | 223.5 | +10% | −9.09 % |
Sweep of Shear force
| Shear force (kN) | Shear stress (MPa) |
|---|---|
| 0.1 | 0.1875 |
| 0.29 | 0.5437 |
| 0.8409 | 1.577 |
| 2.438 | 4.572 |
| 7.071 | 13.26 |
| 20.5 | 38.45 |
| 59.46 | 111.5 |
| 172.4 | 323.3 |
| 500 | 937.5 |
Sweep one variable
Variables and how each one contributes
Shear stress
What the equation gives you. Everything else feeds it.
Shear force
Shear force at the section. Directly proportional.
First moment
First moment of the area beyond the cut about the neutral axis. Directly proportional.
Second moment
Second moment of area of the whole section. Inversely proportional.
Width at the cut
Width of the section at the cut. Inversely proportional: thin webs carry high shear.
Worked example with the values above
- V Q = 12,000 N × 200,000 mm³ = 2.4×109 N·mm³.
- I b = 5,333,000 mm⁴ × 20 mm = 1.067×108 mm⁵.
- τ = 22.5 MPa.
Assumptions and limits
- Elastic, prismatic beam.
- Shear stress uniform across the width b.
Common mistakes
- Using Q for the wrong side of the cut.
- Mixing mm and m units.
Statistics and uncertainty
Q and I are both computed from dimensions, so cut positions should be consistent; the error is dominated by the shear force estimate.
Where it is used
- Web and bolt shear checks.
- Adhesive bond and weld line stresses.
Maths you need
- Second moment of area: I
- Shear force diagram: V