Angle of Twist Calculator

The angle of twist is calculated using ϕ = TL / (JG), where T is the applied torque (N·m), L is the shaft length (m), J is the polar moment of inertia (m⁴), and G is the shear modulus of the material (Pa). The result is in radians, which can be converted to degrees by multiplying by 180/π.

Torque and angle of twist have a directly proportional (linear) relationship, assuming the material stays within its elastic limit. Doubling the applied torque doubles the angle of twist. This relationship holds true as long as the shear stress does not exceed the material's shear yield strength.

In the elastic region, the torque vs. angle of twist graph is a straight line whose slope represents the torsional stiffness (JG/L) of the shaft. Once the material begins to yield, the curve becomes nonlinear, indicating permanent deformation. Engineers use this graph to determine safe operating torque limits.

The angle of twist is dimensionless in the SI system when expressed in radians (rad), since it is the ratio of arc length to radius. It is commonly also expressed in degrees (°) for practical engineering purposes. When using SI units for T (N·m), L (m), J (m⁴), and G (Pa), the formula directly yields the angle in radians.

For a solid circular shaft, J = πd⁴/32, where d is the diameter. For a hollow circular shaft, J = π(d_o⁴ − d_i⁴)/32, where d_o and d_i are the outer and inner diameters respectively. A larger J means the shaft is stiffer and will twist less under the same torque.

Yielding begins when the shear stress at the outer surface reaches the shear yield strength (τ_y) of the material. The maximum shear stress is τ = Tc/J, where c is the outer radius. Setting τ = τ_y and solving for T gives the maximum torque before yield, which you can then substitute back into ϕ = TL/(JG) to find the maximum allowable twist angle.

For a solid 100 mm diameter bar, J = π(0.1)⁴/32 ≈ 9.817 × 10⁻⁶ m⁴. Using ϕ = TL/(JG) = (10,000 × 3) / (9.817×10⁻⁶ × 26×10⁹) ≈ 0.1176 rad, which equals approximately 6.74°. Note: Omni's original example used G = 80 GPa (steel value); for aluminum the correct G is approximately 26 GPa.

The shear modulus G, also called the modulus of rigidity, characterizes a material's resistance to shear deformation. A higher G means the material is stiffer and will twist less under the same torque. Steel (G ≈ 80 GPa) is much stiffer than aluminum (G ≈ 26 GPa), so a steel shaft will twist roughly 3× less than an aluminum shaft of identical geometry under the same torque.