ASME B106.1M Sizing & Standard Stock Recommendation
Iterative multiaxial fatigue sizing with Marin modification factors and Langer first-cycle yield verification.
Haigh Stress Diagram & Fatigue Envelope
Alternating stress (σ_a') vs Mean stress (σ_m') superimposed against ASME, Goodman, Soderberg, and Langer yield boundaries.
Fatigue Failure Criteria Comparison Matrix
Comparative sizing envelopes and reserve safety factors across ASME B106.1M, Goodman, Soderberg, and Gerber models.
| Failure Criterion | Governing Equation | Req. Diameter | Fatigue FOS (n_f) | Status |
|---|---|---|---|---|
DE-ASME EllipticActive Elliptic curve bounded by Se and Sy. Most realistic for ductile steels under fluctuating torsion & bending. | (\sigma_a' / S_e)^2 + (\sigma_m' / S_y)^2 = 1/n^2 | 31.094 mm | 2.00 | SAFE |
DE-Goodman Linear boundary from Se to Sut. Standard practice for brittle components and conservative machine design. | \sigma_a' / S_e + \sigma_m' / S_{ut} = 1/n | 32.289 mm | 1.79 | SAFE |
DE-Soderberg Linear boundary from Se to Sy. Ensures complete protection against first-cycle yielding without extra check. | \sigma_a' / S_e + \sigma_m' / S_y = 1/n | 34.877 mm | 1.43 | SAFE |
DE-Gerber Parabolic curve from Se to Sut. Best statistical fit for pure ductile laboratory test specimens. | n \sigma_a' / S_e + (n \sigma_m' / S_{ut})^2 = 1 | 30.093 mm | 2.20 | SAFE |
Langer Static Yield Checks maximum combined von Mises stress (σ_a' + σ_m' ≤ Sy / n_y) to prevent localized plastic deformation. | \sigma_a' + \sigma_m' = S_y / n_y | 31.533 mm | 1.92 | SAFE |
Torsional Rigidity & Whirling Critical Speed
Elastic angular twist compliance and Rayleigh-Ritz fundamental rotational resonance exclusion zone (±20% Nc).
Marin Endurance Modification Factor Breakdown
Iterative derivation of fully corrected shaft fatigue endurance limit: S_e = k_a · k_b · k_c · k_d · k_e · S_e'
Calculation Summary
| Shaft Profile | Solid Bar |
| Steel Material | AISI-1045 |
| Tensile Strength (S_ut) | 585MPa |
| Yield Strength (S_y) | 310MPa |
| Fatigue Failure Model | ASME-ELLIPTIC |
| Geometric Notch | profile-keyway |
| Alternating Bending (M_a) | 100N·m |
| Mean Torque (T_m) | 200N·m |
| Design FOS (n_d) | 2 |
| Shaft Span (L) | 800mm |
| Min Required Outer Diameter (D_o) | 31.094mm KEY |
| Recommended Stock Bar Size | 32.000mm |
| Reserve Fatigue FOS (n_f) | 1.92 |
| Langer Static Yield FOS (n_y) | 1.92 |
Mathematical Models & Equations
d = \left[ \frac{16 \cdot n}{\pi} \sqrt{ \left( \frac{A}{S_e} \right)^2 + \left( \frac{B}{S_y} \right)^2 } \right]^{1/3} \quad \text{where } A = \sqrt{4(k_b M_a)^2 + 3(k_t T_a)^2}, \; B = \sqrt{4(k_b M_m)^2 + 3(k_t T_m)^2}Verification Calculation Log
- 1Applied Nominal Torque: T = 200.00\text{ N·m} (Mean T_m = 200.00\text{ N·m}, Alternating T_a = 0.00\text{ N·m})
- •Bending Moment Components: Alternating M_a = 100.00\text{ N·m}, Mean M_m = 0.00\text{ N·m} (Stress concentrations: k_b = 2.00, k_t = 1.60 for profile-keyway)
- •Material Properties: S_{ut} = 585.0\text{ MPa}, S_y = 310.0\text{ MPa}, S_e' = 0.5 S_{ut} = 292.5\text{ MPa}
- •Marin Fatigue Modification: S_e = k_a k_b k_c k_d k_e S_e' = (0.833)(0.858)(0.814)(1.000)(1.00)(292.5) = 170.3\text{ MPa}
- •Solve Required Solid Shaft Diameter (DE-ASME Elliptic, n = 2): d = 31.094\text{ mm}
- •Recommended Standard Stock Size: 32.000\text{ mm} (Standard stock safety factor n_f = 2.09, Static yield factor n_y = 2.09)
- •Torsional Rigidity Check (L = 800\text{ mm}): \theta = \frac{T L}{G J} = 1.260^\circ \rightarrow 1.575\text{ deg/m} (Limit: 1.00\text{ deg/m} - FAIL)
- •Rayleigh-Ritz Critical Speed: Fundamental Whirling Speed N_c = 5850\text{ rpm}, Operating Speed = 0\text{ rpm} (SAFE)
Shaft Fatigue Engineering Rules & Guidelines
Engineering Application Notes
ASME B106.1M & Shigley Shaft Design Guidelines: 1. Multiaxial Fatigue Stress States: Rotating shafts undergo fully reversed bending stress (σ_a) and steady or pulsating torsional shear (τ_m). Always use ASME Elliptic or DE-Goodman criteria for sizing. 2. Stress Concentration Mitigation: Generous fillet radii (r/d ≥ 0.10) at bearing and gear shoulders reduce stress concentration factors (Kf) by over 30%, drastically extending fatigue life. 3. Keyway Corrections: Profile end-milled keyways cut through the highest stress fibers. Standard ASME practice applies Kf = 2.0 and Kfs = 1.6 to alternating and mean load components. 4. Rayleigh-Ritz Critical Speed Exclusion: Ensure machine operating speed is kept outside the ±20% resonance envelope (0.80 Nc to 1.20 Nc) to avoid catastrophic whirling failure. 5. Angular Torsional Compliance: For precision machine tool drives, keep torsional twist below 0.25°/m (0.08°/ft) to prevent servo oscillation and gear transmission chatter.
Related Calculators
Applicable Standards
Related Formulas
Related Engineering Articles
ASME B106.1M Shaft Fatigue & Dynamics Calculation Report — Calculation Report
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ASME B106.1M Shaft Fatigue & Dynamics Calculation Report
Design Inputs
- Shaft Configuration: Solid Bar
- Steel Material Grade: AISI-1045
- Tensile Strength (S_ut): 585 MPa
- Yield Strength (S_y): 310 MPa
- Design Safety Factor (n_d): 2.00
- Geometric Notch / Keyway: profile-keyway
- Alternating Bending (M_a): 100 N·m
- Mean Torque (T_m): 200 N·m
- Shaft Span (L): 800 mm
Calculated Results
| Minimum Required Outer Diameter (D_o): | 31.0938 mm |
| Recommended Commercial Stock Bar: | 32.0000 mm |
| Reserve Fatigue FOS (n_f): | 1.92 |
| Langer Static Yield FOS (n_y): | 1.92 |
| Corrected Endurance Limit (S_e): | 170.3 MPa |
Verification Steps
- 1. Applied Nominal Torque: T = 200.00\text{ N·m} (Mean T_m = 200.00\text{ N·m}, Alternating T_a = 0.00\text{ N·m})
- •Bending Moment Components: Alternating M_a = 100.00\text{ N·m}, Mean M_m = 0.00\text{ N·m} (Stress concentrations: k_b = 2.00, k_t = 1.60 for profile-keyway)
- •Material Properties: S_{ut} = 585.0\text{ MPa}, S_y = 310.0\text{ MPa}, S_e' = 0.5 S_{ut} = 292.5\text{ MPa}
- •Marin Fatigue Modification: S_e = k_a k_b k_c k_d k_e S_e' = (0.833)(0.858)(0.814)(1.000)(1.00)(292.5) = 170.3\text{ MPa}
- •Solve Required Solid Shaft Diameter (DE-ASME Elliptic, n = 2): d = 31.094\text{ mm}
- •Recommended Standard Stock Size: 32.000\text{ mm} (Standard stock safety factor n_f = 2.09, Static yield factor n_y = 2.09)
- •Torsional Rigidity Check (L = 800\text{ mm}): \theta = \frac{T L}{G J} = 1.260^\circ \rightarrow 1.575\text{ deg/m} (Limit: 1.00\text{ deg/m} - FAIL)
- •Rayleigh-Ritz Critical Speed: Fundamental Whirling Speed N_c = 5850\text{ rpm}, Operating Speed = 0\text{ rpm} (SAFE)