Design of Machine Elements: Failure Theories, Fatigue and Shaft Sizing
Design philosophy and failure theories
Design applies a factor of safety to a chosen strength: n = S/σ. For ductile materials use maximum shear stress (Tresca) or von Mises; for brittle materials use maximum principal stress (Rankine). Von Mises matches ductile test data better under combined loading.
Fatigue and joints
- S–N curve: the endurance limit is the stress below which steel life is effectively infinite (roughly 0.5 S_ut at zero mean stress).
- Goodman: σ_a/S_e + σ_m/S_ut = 1/n; Soderberg replaces S_ut with S_y and is more conservative.
- Stress concentration K_t multiplies nominal stress; use K_f in fatigue calculations.
- A bolted joint carries preload first; external load is shared between bolt and members by their stiffness.
- A keyway weakens the shaft, so design a keyed shaft at about 75% of the torsion-only diameter strength, i.e. enlarge d.
Worked example
A solid shaft transmits 500 N·m with allowable shear stress 40 MPa. Find d. τ = 16T/(πd³), so d = (16T/(πτ))^(1/3) = (16 × 500 000/(π × 40))^(1/3) = (63 662)^(1/3) ≈ 39.9 mm. Choose the next standard size, 40 mm; with a keyway, increase to about 45 mm.
Traps
- Mixing N·m with d in mm; convert torque to N·mm first.
- Using S_ut where the endurance limit S_e belongs.
- Ignoring keyways and shoulders when sizing shafts.
- For a hollow shaft, using d³ instead of (d_o³ − d_i³).