ASME Combined Shaft Sizing Equation
Sizing standard for transmission shafting under combined torsional shear and reversed fatigue bending loads.
Primary Mathematical Expression
Design Schematic
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| d | Shaft diameter | mm | in | The minimum required outer diameter of the circular solid shaft. |
| N_sf | Safety factor | dimensionless | dimensionless | Design Factor of Safety (FOS) to account for uncertainties. |
| S_y | Material yield strength | MPa | psi | The material tensile yield strength limit. |
| M | Bending moment | N·m | in·lb | The bending moment load acting on the shaft section. |
| T | Torque | N·m | in·lb | The torsional torque load acting on the shaft section. |
Step-by-Step Derivation
- 1
Under combined bending and torsion, the shaft experiences maximum normal stress \sigma_x and shear stress \tau_{xy}.
- 2
According to the Maximum Shear Stress theory (Tresca criteria), the maximum shear stress is: \tau_{max} = \sqrt{(\sigma_x/2)^2 + \tau_{xy}^2}.
- 3
Substituting bending stress \sigma_x = 32M / (\pi * d³) and torsional shear stress \tau_{xy} = 16T / (\pi * d³): \tau_{max} = (16 / \pi * d³) * \sqrt{M^2 + T^2}.
- 4
Set allowable shear stress to S_y / (2 * N_sf): S_y / (2 * N_sf) = (16 / \pi * d³) * \sqrt{M^2 + T^2}.
- 5
Solving for diameter d yields the ASME combined loading sizing equation: d = [ (32 * N_sf / (\pi * S_y)) * \sqrt{M^2 + T^2} ]^(1/3).
Worked Example Calculation
Determine the solid shaft diameter for structural steel (S_y = 250 MPa) subjected to a bending moment of 180 N·m and torque of 350 N·m. Use a safety factor N_sf of 2.0.
- •Identify input parameters: S_y = 250 MPa = 250 N/mm², M = 180 N·m = 180,000 N·mm, T = 350 N·m = 350,000 N·mm, N_sf = 2.0.
- •Apply the ASME Combined Sizing equation: d = [ (32 * N_sf / (\pi * S_y)) * \sqrt{M^2 + T^2} ]^(1/3).
- •Calculate root term: \sqrt{180,000^2 + 350,000^2} \approx 393,573 N·mm.
- •Compute: d³ = (32 * 2.0 / (\pi * 250)) * 393,573 \approx 0.081487 * 393,573 \approx 32,071 mm³.
- •Take the cube root: d = 32,071^(1/3) \approx 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.
Design Limitations
- •Does not account for dynamic fatigue stress concentration factors due to keyways, shoulders, or surface finishes (requires Gerber/Goodman fatigue criteria).
Academic References & Standards
Design of Transmission Shafting
Standard code for sizing transmission shafting.
Shigley's Mechanical Engineering Design, 11th Edition
Textbook covering rotating shaft design under fatigue and static loads.