Machine Shaft Design Basics

Concept

Machine shafts are rotating or stationary members, typically with a circular cross-section, used to transmit power or motion. Designing a shaft requires considering both static and dynamic (fatigue) loading, as shafts are subjected to fluctuating bending and torsional stresses. The design process involves identifying critical locations where stress concentrations occur, such as steps, keyways, and grooves, and applying endurance limit modifiers to estimate the fatigue strength of the shaft material.

Formula & Method

The endurance limit ($S_e$) at the critical location of a machine part is estimated from the ideal, rotating-beam endurance limit ($S_e'$) by applying various modifying factors (Marin factors):

$S_e = k_a k_b k_c k_d k_e k_f S_e'$

For a solid circular shaft subjected to alternating bending ($M_a$) and steady torsion ($T_m$), the Goodman failure criterion can be used to determine the required diameter $d$ to achieve a design factor of safety $n$:

$\frac{1}{n} = \frac{16}{\pi d^3} \left( \frac{K_f M_a}{S_e} + \frac{\sqrt{3} K_{fs} T_m}{S_{ut}} \right)$

Variables & Units

• $S_e$ = Endurance limit of the specific part, in MPa (or psi).
• $S_e'$ = Rotary-beam test endurance limit, in MPa (or psi).
• $k_a$ = Surface condition modification factor.
• $k_b$ = Size modification factor.
• $k_c$ = Load modification factor.
• $M_a$ = Alternating bending moment, in N·m.
• $T_m$ = Midrange (steady) torsional moment, in N·m.
• $K_f, K_{fs}$ = Fatigue stress-concentration factors for bending and torsion, respectively.
• $n$ = Factor of safety.

Worked Example

Problem: Calculate the required diameter of a rotating solid steel shaft ($S_e = 200 \text{ MPa}$, $S_{ut} = 600 \text{ MPa}$) subjected to a constant torque of $150 \text{ N·m}$ and a completely reversed bending moment of $250 \text{ N·m}$. Assume $K_f = 1.6$, $K_{fs} = 1.4$, and a required factor of safety $n = 2.0$.

Calculation:

1. Identify the given parameters: $S_e = 200 \times 10^6 \text{ Pa}$, $S_{ut} = 600 \times 10^6 \text{ Pa}$, $M_a = 250 \text{ N·m}$, $T_m = 150 \text{ N·m}$, $K_f = 1.6$, $K_{fs} = 1.4$, $n = 2.0$.
2. Substitute the values into the modified Goodman equation for shaft design: $\frac{1}{2.0} = \frac{16}{\pi d^3} \left( \frac{(1.6)(250)}{200 \times 10^6} + \frac{\sqrt{3}(1.4)(150)}{600 \times 10^6} \right)$
3. Calculate the terms in the parenthesis: Bending term: $\frac{400}{200 \times 10^6} = 2 \times 10^{-6}$ Torsion term: $\frac{363.7}{600 \times 10^6} \approx 0.606 \times 10^{-6}$ Sum $= 2.606 \times 10^{-6}$
4. Solve for $d^3$: $d^3 = 2.0 \times \frac{16}{\pi} \times (2.606 \times 10^{-6}) = 26.54 \times 10^{-6} \text{ m}^3$
5. Calculate $d$: $d = \sqrt[3]{26.54 \times 10^{-6}} = 0.0298 \text{ m} = 29.8 \text{ mm}$

Engineering Meaning

Shaft design is an iterative process. Engineers must define the layout to mount gears and bearings, which creates stress concentrations. By using endurance modifiers, engineers account for real-world conditions (like surface roughness and size) that reduce the material's ideal fatigue limit. The Goodman criterion ensures the shaft can withstand the combined cyclic and static loads without failure.

Engineering Check

Ensure that the endurance modifying factors accurately reflect the final manufacturing processes (e.g., machined vs. forged surfaces). Verify that the fatigue stress-concentration factors ($K_f$) correctly account for the notch sensitivity of the selected material, not just the geometric stress-concentration factor ($K_t$).

Explicit Exclusions

This foundational article excludes the calculation of dynamic resonance and critical speeds (whirling). It also excludes detailed deflection analysis and the design of hollow shafts.