Shaft design and torsion analysis are central topics in machine design, mechanics of materials, and mechanical power transmission. This cheat sheet helps students connect torque, shear stress, angle of twist, diameter, material strength, and design factors in one reference. It is useful for sizing circular shafts, checking whether a shaft is safe, and understanding how details such as keyways and shoulders affect stress.
Key Facts
- For a solid circular shaft, the polar moment of inertia is J = pi d^4 / 32.
- For a hollow circular shaft, the polar moment of inertia is J = pi (do^4 - di^4) / 32.
- Maximum torsional shear stress is tau_max = T c / J, where c is the outer radius of the shaft.
- For a solid circular shaft, maximum shear stress can be written as tau_max = 16 T / (pi d^3).
- Angle of twist is theta = T L / (J G), where theta is in radians when consistent units are used.
- Power, torque, and angular speed are related by P = T omega, with omega in rad/s.
- A basic allowable stress check is tau_max <= tau_allow, where tau_allow is commonly based on yield strength divided by a factor of safety.
- Stress concentrations are included by using tau_peak = Kt tau_nominal for elastic stress estimates near shoulders, keyways, grooves, or holes.
Vocabulary
- Torque
- Torque is the twisting moment applied to a shaft, usually measured in N m or lb in.
- Polar moment of inertia
- Polar moment of inertia, J, measures a circular section's resistance to twisting and depends strongly on shaft diameter.
- Shear stress
- Shear stress is the internal stress acting tangentially across a material due to torsion or transverse loading.
- Angle of twist
- Angle of twist is the rotation between two shaft cross sections caused by applied torque.
- Factor of safety
- Factor of safety is the ratio between a failure limit and the allowed working stress or load.
- Stress concentration factor
- Stress concentration factor, Kt, is a multiplier that estimates increased local stress near geometric discontinuities.
Common Mistakes to Avoid
- Using diameter instead of radius for c is wrong because tau_max = T c / J requires the distance from the center to the outer surface.
- Mixing units such as mm, m, N mm, and N m in one calculation gives incorrect stress or twist values because torsion formulas require consistent units.
- Using J = pi d^4 / 64 for torsion is wrong because that expression is the area moment of inertia for bending, not the polar moment for a solid circular shaft.
- Ignoring stress concentrations at shoulders, keyways, and grooves is unsafe because local peak stress can be much higher than the nominal torsional stress.
- Checking strength but not angle of twist can lead to a shaft that does not fail but is too flexible for gears, bearings, couplings, or precision motion.
Practice Questions
- 1 A solid steel shaft has diameter d = 40 mm and transmits torque T = 500 N m. Find the maximum torsional shear stress using tau_max = 16 T / (pi d^3).
- 2 A solid shaft has length L = 1.2 m, diameter d = 30 mm, torque T = 250 N m, and shear modulus G = 80 GPa. Find the angle of twist in radians.
- 3 A motor transmits P = 10 kW at 1200 rpm through a shaft. Find the torque using P = T omega and omega = 2 pi rpm / 60.
- 4 A shaft has a shoulder and keyway near the same location. Explain why the designer should not rely only on the nominal torsional stress when judging safety.