Torsion of a Circular Shaft
Torsion of a Solid Circular Shaft
Power-transmitting shafts carry torque, which produces shear stress and an angular twist. This worksheet computes the peak surface shear stress and the total angle of twist of a solid circular shaft.
From the torsion formula τ = T·r/J and the twist φ = TL/(GJ), where J is the polar second moment of area and G the shear modulus.
Inputs
T := 800 N·m=800 Nm
d := 40 mm=40 mm
L := 1.2 m=1.2 m
G := 80 GPa=80000 MPa
Polar second moment of area
For a solid circle J = πd⁴/32.
J := pi·d^4/32=2.51e-07 m^4
Shear stress and angle of twist
tau_max := T·(d/2)/J=63.66 MPa
phi := T·L/(G·J)=0.05
Results
The surface shear stress must stay below the shaft material’s allowable shear; the angle of twist is often limited for precision drives (a common rule of thumb is about 1° of twist per metre of length). The twist here is reported in radians.
Assumptions & limitations
- Solid prismatic circular section, linear-elastic material.
- Pure torsion, no bending or axial load combined.
- St. Venant torsion; stress concentrations at keyways ignored.