A torque applied to a circular shaft twists it. The shear stress grows linearly from zero at the center to a maximum at the surface, τ_max = Tc/J, and the shaft rotates through an angle φ = TL/(GJ). Those two formulas, plus P = Tω for power, size most drive shafts.
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How a circular shaft carries torque
Under torque, each cross-section of a circular shaft rotates as a rigid disk. Plane sections stay plane and radial lines stay straight. That makes the shear strain, and in the elastic range the shear stress, proportional to the distance ρ from the center.
The result is the torsion formula τ = Tρ/J. The largest stress is at the outer surface, where ρ = c. J is the polar moment of inertia of the cross-section: πc⁴/2 for a solid shaft, π(c_o⁴ − c_i⁴)/2 for a hollow one.
Angle of twist
The angle one end rotates relative to the other is φ = TL/(GJ), in radians, where G is the shear modulus. GJ is the torsional stiffness. If torque, material or diameter change along the shaft, add φ for each segment, keeping track of the sign of each internal torque.
Shear stress follows the same Hooke’s-law idea as axial stress and strain: τ = Gγ in the linear-elastic range.
Torque from power and speed
Motors are rated by power and speed, not torque. Power is torque times angular velocity, P = Tω, with ω in rad/s. Convert from rpm with ω = 2πn/60.
At a fixed power, a shaft that turns slower carries more torque. That is why the low-speed side of a gearbox needs a thicker shaft than the high-speed side.
These formulas apply to circular shafts only, solid or hollow. Square and rectangular sections warp and need different formulas.
They assume a homogeneous, linear-elastic material and stresses below the shear yield strength.
Near keyways, shoulders and holes, stress concentrations raise the real peak stress above Tc/J.
Key formulas
Formulas used on this page, with the textbook each is cited from
Formula
Expression
Where
Cited from
Torsion formula
τ = T·ρ / J, τ_max = T·c / J
T = internal torque, ρ = radius to the point, c = outer radius
Hibbeler, Mechanics of Materials
Polar moment of inertia (solid)
J = π·c⁴ / 2 = π·d⁴ / 32
c = radius, d = diameter
Gere & Goodno, Mechanics of Materials
Polar moment of inertia (hollow)
J = π·(c_o⁴ − c_i⁴) / 2
c_o, c_i = outer and inner radii
Gere & Goodno, Mechanics of Materials
Angle of twist
φ = T·L / (G·J)
φ in radians, G = shear modulus, L = length
Beer & Johnston, Mechanics of Materials
Power transmission
P = T·ω, ω = 2π·n / 60
P in W, T in N·m, ω in rad/s, n in rpm
Hibbeler, Mechanics of Materials
Worked examples
Example: Stress and twist in a solid steel shaft
Given
A solid steel shaft has a diameter of 40 mm and a length of 1.2 m.
It carries a constant torque of 800 N·m.
Assume G = 77 GPa, a typical textbook value for steel.
Find
The maximum shear stress τ_max and the angle of twist φ in degrees.
Stress varies linearly with radius, so at the inner surface, c_i = 10 mm:τ_min = 23.2 MPa × (10 / 15) = 15.5 MPa
Answer: T ≈ 98.8 N·m, with shear stress of about 23.2 MPa at the outside and 15.5 MPa at the bore. Hollow shafts use material efficiently because the low-stress core is removed.
Common mistakes
Using rpm directly in P = Tω. Convert to rad/s first: ω = 2πn/60. Using rpm gives a torque about 9.5 times too small.
Using the bending formula I instead of J. Torsion uses the polar moment J. For a circle J = 2I, so mixing them up halves or doubles the stress.
Plugging a diameter into the radius form. J = πc⁴/2 uses the radius; J = πd⁴/32 uses the diameter. Pick one and stick with it.
Reporting φ in radians without saying so. φ = TL/(GJ) gives radians. Convert to degrees only at the end, and label the unit.
Applying the torsion formula to a square bar. It only holds for circular sections. Non-circular shafts warp and need their own formulas.
In industry
Frequently asked questions
Why is the shear stress zero at the center of a shaft?
Shear strain is proportional to distance from the axis, and the axis itself does not distort. So in the elastic range the stress at the center is zero and rises linearly to the surface.
What is the difference between G and E?
E relates normal stress to normal strain; G relates shear stress to shear strain. For an isotropic material they are linked by G = E / (2(1 + ν)).
Why are hollow shafts common?
Material near the center carries little stress, so removing it saves a lot of weight for a small loss in strength and stiffness.
Do these formulas work for a stepped shaft?
Yes, segment by segment. Find the internal torque, J and length of each segment, then add the angles of twist. Stress concentrations at the steps need separate treatment.