Shaft Torsional Shear Stress and Angle of Twist
The shear stress in a shaft under torque, and how far it twists.
Example
You enter
- Torque T (lb-in) 12000
- Outer diameter d (in) 1.5
- Length for twist L (in, optional) 24
- Shear modulus G (psi; 11.5e6 steel) 11500000
You get
- Max shear stress 18108 psi
- Theta (deg) 2.9
Details, formula, and sources
Polar moment J = pi(d^4-di^4)/32, max shear tau = T r/J, and twist theta = T L/(J G) for a solid or hollow circular shaft. A 1.5 in steel shaft at 1,000 lb-ft over 24 in -> 18,100 psi, 2.9 deg; a 2 in shaft cuts both far (the d^3/d^4 leverage). A design aid, not the engineer of record.
J = pi(d^4 - di^4)/32; tau = T (d/2)/J (= 16T/(pi d^3) solid); theta = T L/(J G).
The standard circular-shaft torsion relations (mechanics of materials), by name.
The torsion formulas are published free in any mechanics-of-materials reference. The engineer of record governs the design.
Estimate. AHJ and licensed professional govern.
Field names used by the API: T_lbin, d_in, L_in, G_psi, tau_psi, theta_deg
- Circular section J = pi(d^4 - di^4)/32; tau = T r/J; theta = T L/(J G)mechanics of materials
- Pure torsion no bending or axial load; elastic and prismaticmechanics of materials
- No stress concentration keyways, shoulders, and holes raise the local stress; not includedscope of this tile