Shaft Design Basics: Fatigue, ASME Sizing & Stress Analysis

When a shaft transmits rotational power, the applied torque ($T$) generates torsional shear stress ($\tau$). In circular solid cross-sections, shear stress is zero at the neutral axis and reaches its maximum value at the outer surface radius:

$$\tau_{\max} = \frac{T \cdot r}{J} = \frac{16 T}{\pi d^3}$$

For complete derivations and hollow shaft equations, refer to our [Torsional Shear Stress Formula Reference](/formulas/torsional-shear-stress).

Simultaneously, transverse forces from mounted components (spur/helical gears, belt pulleys, and sprockets) create transverse bending moments ($M$). As the shaft rotates, surface fibers alternate continuously between peak tension and compression, producing fully reversed cyclic fatigue stress:

$$\sigma_b = \frac{M \cdot c}{I} = \frac{32 M}{\pi d^3}$$

Real transmission shafts incorporate geometric transitions including bearing seating shoulders, retaining ring grooves, and drive keyways. These geometry changes create local stress concentration zones where local stresses can be 1.5× to 3.0× higher than nominal values.

The [ASME B106.1M Standard](/standards/asme-b106-1m-shaft-design) establishes standardized criteria for sizing transmission shafting: 1. **Allowable Shear Stress Limits**: Without keyways, design allowable shear stress is limited to $\tau_{\text{allow}} = \min(0.30 S_y, 0.18 S_u)$. 2. **Keyway Penalty Factor**: When keyways are cut into the shaft, ASME B106.1M mandates a **25% reduction** in allowable stress ($\tau_{\text{allow, keyway}} = 0.75 \tau_{\text{allow}}$). 3. **Combined Fatigue & Torsion**: Sizing under combined loads uses maximum shear stress (Tresca) or distortion energy (DE-Goodman) criteria, as detailed in our [ASME Shaft Sizing Formula](/formulas/asme-shaft-sizing).

Figure 1: Typical transmission shaft schematic showing pulley forces, bearings, shoulders, keyway stress concentration zones, and loading vectors.