ASME Shaft Sizing Formula (ASME B106.1M)
Standard ASME B106.1M governing equation for transmission shaft diameter sizing under combined steady torque and reversed bending fatigue loads.
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Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| d | Shaft diameter | mm | in | The minimum required outer diameter of the circular solid transmission shaft. |
| N_sf | Safety factor | dimensionless | dimensionless | Design Factor of Safety (FOS) to account for operating uncertainties and fatigue. |
| S_y | Material yield strength | MPa | psi | The material tensile yield strength limit. |
| M | Bending moment | N·m | in·lb | The peak bending moment acting on the critical shaft section. |
| T | Torque | N·m | in·lb | The steady or alternating torsional torque acting on the shaft section. |
Step-by-Step Derivation
- 1
Under combined bending and torsion, a rotating shaft experiences normal bending stress \sigma_b = 32M / (\pi * d³) and torsional shear stress \tau = 16T / (\pi * d³).
- 2
Applying Maximum Shear Stress theory (Tresca criteria), the maximum shear stress at the critical outer fiber is: \tau_{max} = \sqrt{(\sigma_b / 2)^2 + \tau^2}.
- 3
Substituting \sigma_b / 2 = 16M / (\pi * d³) yields: \tau_{max} = \frac{16}{\pi * d³} * \sqrt{M^2 + T^2}.
- 4
In accordance with ASME B106.1M, setting \tau_{max} equal to the allowable design shear stress \tau_{allow} = S_y / (2 * N_sf) gives: S_y / (2 * N_sf) = \frac{16}{\pi * d³} * \sqrt{M^2 + T^2}.
- 5
Solving explicitly for shaft diameter d yields the governing ASME sizing equation: d = [ (32 * N_sf / (\pi * S_y)) * \sqrt{M^2 + T^2} ]^(1/3).
Worked Example Calculation
Determine the required solid transmission shaft diameter for a machinery shaft ($S_y = 250\text{ MPa}$) transmitting a continuous torque of $350\text{ N}\cdot\text{m}$ and experiencing a peak transverse bending moment of $180\text{ N}\cdot\text{m}$. Use a design safety factor $N_{sf} = 2.0$.
- •Convert input units to consistent SI units: S_y = 250 N/mm², M = 180,000 N·mm, T = 350,000 N·mm, N_sf = 2.0.
- •Compute equivalent combined moment root: √(M² + T²) = √(180,000² + 350,000²) ≈ 393,573 N·mm.
- •Apply the ASME sizing equation: d = [ (32 * 2.0 / (π * 250)) * 393,573 ]^(1/3).
- •Evaluate the interior cubic term: d³ = 0.081487 * 393,573 ≈ 32,071 mm³.
- •Calculate the cubic root: d = 32,071^(1/3) ≈ 31.8 mm.
Engineering Assumptions
- •Homogeneous, isotropic, elastic material behaving in accordance with Tresca maximum shear stress yield criteria.
- •Steady torque and fully reversed cyclic bending load per standard rotating-beam shaft assumptions.
- •Pure circular solid cross-section without initial residual stresses.
Design Limitations
- •Static Tresca approximation does not fully account for fatigue notch sensitivity or surface finish factors (for severe cyclic fatigue, apply ASME DE-Goodman with Marin factors).
- •When keyways are cut into the shaft, ASME B106.1M requires reducing allowable stress by 25% or applying appropriate stress concentration factors ($K_t \approx 1.6 - 2.0$).
- •Deflection and critical speed limits must be checked independently to avoid resonance or excessive gear tooth misalignment.
Academic References & Standards
Design of Transmission Shafting
Standard code for sizing transmission shafting under combined loading.
Shigley's Mechanical Engineering Design, 11th Edition
Rotating shaft design under combined fatigue and static loads.
Roark's Formulas for Stress and Strain, 9th Edition
Torsional and flexural properties of circular sections.