Shafts are rotating machine elements that transmit power while supporting gears, pulleys, sprockets, or couplings. A real shaft usually carries torque and transverse loads at the same time, so it experiences both torsional shear stress and bending normal stress. Good shaft design prevents yielding, fatigue failure, excessive deflection, and vibration problems.
It also ensures that bearings, gears, keys, and shoulders fit safely without creating dangerous weak spots.
The most critical sections are often not where the load is applied, but where the geometry changes, such as shoulders, grooves, snap ring seats, and keyways. These features create stress concentrations that amplify local stress and reduce fatigue life. Engineers combine bending and torsion using failure theories such as maximum shear stress or distortion energy theory, then check deflection and slope so gears and bearings stay aligned.
A safe design balances strength, stiffness, manufacturability, size, weight, and cost.
Key Facts
- Torque from power and speed: P = Tω, so T = P/ω.
- Bending stress in a solid circular shaft: σ = Mc/I = 32M/(πd^3).
- Torsional shear stress in a solid circular shaft: τ = Tc/J = 16T/(πd^3).
- Polar moment of inertia for a solid circular shaft: J = πd^4/32.
- Area moment of inertia for a solid circular shaft: I = πd^4/64.
- Von Mises equivalent stress for combined bending and torsion: σ_vm = sqrt(σ^2 + 3τ^2).
Vocabulary
- Shaft
- A rotating machine member that transmits torque and supports attached components such as gears, pulleys, or couplings.
- Torque
- A twisting moment that causes rotation and produces shear stress in a shaft.
- Bending moment
- An internal moment caused by transverse loads that produces tensile stress on one side of a shaft and compressive stress on the other.
- Stress concentration
- A local increase in stress caused by a sudden change in geometry such as a shoulder, groove, hole, or keyway.
- Deflection
- The elastic displacement of a shaft under load, which can affect gear mesh, bearing life, and alignment.
Common Mistakes to Avoid
- Checking torsion only and ignoring bending, which is wrong because gears and pulleys create transverse loads that can dominate the maximum stress.
- Using the nominal diameter at a shoulder or keyway without a stress concentration factor, which is wrong because local geometry raises the actual peak stress.
- Assuming the maximum stress always occurs at the middle of the span, which is wrong because critical sections often occur near load application points, shoulders, bearings, and keyways.
- Designing only for strength and skipping deflection checks, which is wrong because a shaft can be strong enough but still bend enough to misalign gears or overload bearings.
Practice Questions
- 1 A solid circular steel shaft has diameter d = 40 mm and transmits torque T = 600 N m. Calculate the maximum torsional shear stress using τ = 16T/(πd^3).
- 2 At a critical section, a solid shaft has diameter d = 50 mm, bending moment M = 900 N m, and torque T = 700 N m. Calculate σ = 32M/(πd^3), τ = 16T/(πd^3), and σ_vm = sqrt(σ^2 + 3τ^2).
- 3 A shaft has a gear mounted next to a sharp shoulder and a keyway. Explain why this location may be more critical for fatigue failure than a smooth section with a slightly larger bending moment.