Torsional Shear Stress in Shafts
Calculates the maximum and radial torsional shear stress (τ = T·r/J) in solid and hollow circular shafts under torsional moments.
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Primary Mathematical Expression
Design Schematic
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| T | Applied torque | N·m | in·lb | The torsional moment load twisting the bar. |
| r | Radial distance | mm | in | Radial distance from center axis to the point of interest. |
| J | Polar moment of inertia | mm⁴ | in⁴ | The geometric torsional resistance property of the cross-section. |
| \tau | Torsional shear stress | MPa | psi | The resulting shear stress at radius r. |
Step-by-Step Derivation
- 1
A circular bar is subjected to pure torsion, causing cross-sections to rotate relative to each other about the longitudinal axis.
- 2
Shear strain \gamma is proportional to radial distance r: \gamma = r * \phi, where \phi is the twist rate per unit length.
- 3
Assuming elastic behavior, Hooke's Law for shear applies: \tau = G * \gamma = G * r * \phi.
- 4
The resisting internal torque must balance the external torque: T = \int \tau * r * dA = \int (G * r² * \phi) * dA = (G * \phi) * J.
- 5
Substituting G * \phi = T / J back into the stress equation yields the torsional shear stress formula: \tau = T * r / J.
Worked Example Calculation
A solid circular shaft of radius 15 mm is subjected to a torque of 450 N·m. The polar moment of inertia J is 7.95 \times 10^4 mm⁴. Calculate the maximum torsional shear stress.
- •Identify input parameters: T = 450 N·m = 450,000 N·mm, r = 15 mm, J = 79,500 mm⁴.
- •Apply the Torsional Shear Stress Formula: \tau = T * r / J.
- •Compute: \tau = (450,000 * 15) / 79,500 = 6,750,000 / 79,500 = 84.9 MPa.
Engineering Assumptions
- •The bar has a circular cross-section (solid or hollow).
- •The material is homogeneous, isotropic, and obeys Hooke's Law in shear.
- •Cross-sections remain plane and do not warp during twisting.
- •Stress is within elastic limit (no plastic deformation).
Design Limitations
- •Not applicable to non-circular cross-sections (which undergo warping, requiring Prandtl stress function analysis or FEA).
- •Underestimates stress if twisting exceeds the elastic limit of the material.
- •Does not account for stress concentrations at keyways, fillets, or diameter changes (apply K_t factors per ASME B106.1M).
- •For combined loading, use ASME combined stress equation (see ASME Shaft Sizing formula).
Academic References & Standards
Shigley's Mechanical Engineering Design, 11th Edition
Textbook covering torsional stresses and shafts.
Design of Transmission Shafting
Standard code for sizing transmission shafting.
Torsional deformation and stress formulas.
textbook
Hollow steel bars for machining of cutting tools
Hollow shaft dimensions.