With this angle of twist calculator, you can find the angle of twist of a member subjected to a torque T by entering the variables of the angle of twist formula: shaft length, torque, polar moment of inertia, and shear modulus. It supports both the metric (SI) and the American (US customary) systems of units.
Although the deformations in power transmission shafts are relatively small, an excess of them may cause vibration problems, resulting in noise and improper synchronization. They can also cause significant displacements in larger members connected to them (e.g., gears) because of the radial distance of those members. For these and other reasons, it is necessary to study the relationship between torque and angle of twist and the equation behind it.
The formula for angle of twist calculation
The angle of twist is the relative rotation of one face of a shaft with respect to another face when a torque is applied to that shaft. The angle of twist of one end of a shaft with respect to the other end is given by:
φ = T·L / (J·G)
In the previous equation:
- φ — Angle of twist (calculated in radians);
- T — Internal torque;
- L — Shaft length;
- J — Polar moment of inertia; and
- G — Shear modulus of the shaft material.
The shear modulus and the polar moment of inertia represent the material's reluctance to twist or suffer a shear strain. While the shear modulus is a material property, the polar moment is a geometric property that depends on the shaft cross-section.
For a solid circular shaft of diameter D, the polar moment of inertia equals:
J = (π / 32) · D⁴
For a hollow circular shaft with an internal diameter d, it equals:
J = (π / 32) · (D⁴ − d⁴)
For non-circular shafts, the angle of twist equation also holds, but you must use the torsional constant instead of the polar moment.
Assumptions of the angle of twist equation
The angle of twist equation is not a catch-all formula, as it works under certain assumptions:
- The member (a shaft, usually) is straight, and its cross-section is uniform or invariable;
- Along its length L, the member experiences the same internal torque T; and
- The material is homogeneous and behaves in a linear elastic way along the length L.
The angle of twist units
This calculator will give you the correct result regardless of the input units, because it converts every value to a consistent set internally. Even so, to use the angle of twist equation by hand, you must guarantee dimensional homogeneity between the different variables. To achieve that homogeneity, use the following units:
| Variable | Metric (SI) | American (USCS) |
|---|---|---|
| Angle of twist (φ) | radian (rad) | radian (rad) |
| Torque (T) | newton-meter (N·m) | pound-force inch (lbf·in) |
| Length (L) | meter (m) | inch (in) |
| Polar moment of inertia (J) | m⁴ (input as mm⁴) | in⁴ |
| Shear modulus (G) | pascal (Pa) (input as GPa) | psi (input as ksi) |
The angle of twist equation for variable torque, material, or radius
When the torque, cross-section, or material is not constant along the length of the member, the assumptions above are no longer met. In that case, you must split the member into segments where the torque, polar moment, and shear modulus are each constant, compute the angle of twist of every segment with φ = T·L / (J·G), and add them together:
φtotal = Σ Tᵢ·Lᵢ / (Jᵢ·Gᵢ)
For a continuously varying torque or section, the sum becomes an integral over the length of the shaft.
Shear modulus of typical engineering materials
The shear modulus (also called the modulus of rigidity) measures a material's resistance to shear deformation. The table below lists approximate values for common engineering materials, which are built into the material selector of this calculator.
| Material | Shear modulus G (GPa) | Shear modulus G (ksi) |
|---|---|---|
| Structural steel | 79.3 | 11,500 |
| Stainless steel | 77.2 | 11,200 |
| Cast iron | 41.0 | 5,950 |
| Aluminum | 26.0 | 3,770 |
| Titanium | 41.4 | 6,000 |
| Copper | 44.7 | 6,480 |
| Brass | 37.0 | 5,370 |
| Bronze | 45.0 | 6,530 |
| Magnesium | 17.0 | 2,470 |
| Glass | 26.2 | 3,800 |
FAQs
What is the angle of twist?
The angle of twist is the relative angular rotation of one cross-section of a shaft with respect to another when a torque is applied. It is calculated in radians using φ = T·L / (J·G) and can be converted to degrees by multiplying by 180/π.
How do I calculate the angle of twist of a shaft?
To calculate the angle of twist: 1) Find the internal torque T and the shaft length L. 2) Determine the polar moment of inertia J from the cross-section (J = π·D⁴/32 for a solid circular shaft). 3) Look up the shear modulus G of the material. 4) Apply φ = T·L / (J·G). The result is in radians.
What units does the angle of twist use?
The angle of twist itself is a radian (a dimensionless ratio). To get a meaningful result, the torque, length, polar moment, and shear modulus must all be in a consistent unit system, such as N·m, m, m⁴, and Pa (metric), or lbf·in, in, in⁴, and psi (American).
How can I reduce the angle of twist?
You can reduce the angle of twist by increasing the shaft diameter (J grows with the fourth power of the diameter), shortening the shaft, choosing a material with a higher shear modulus, or lowering the applied torque.
Does this calculator work for hollow shafts?
Yes. Select the "Hollow circular shaft" cross-section and enter both the outer diameter D and the inner diameter d. The calculator uses J = π·(D⁴ − d⁴)/32.