Learn: Angle of twist
A shaft can remain safe in stress yet twist too much for its function. Angle of twist measures the relative rotation between the two ends of a shaft segment under torque.
What the formula is saying
Torque acting over a longer length accumulates more twist. The product GJ is torsional rigidity, so greater material stiffness or a more effective section reduces rotation.
Read the symbols in plain language
- T
- Torque
Torque. The same torque acting over a longer segment produces more relative rotation.
N·mUse N·m as the base unit shown here. Use torque in N·m, length in m, G in Pa and J in m⁴. The calculated angle is in radians; multiply by 180/π to express it in degrees.
- L
- Shaft length
Shaft length. The same torque acting over a longer segment produces more relative rotation.
mMetres measure length; 1 m = 1000 mm.
- G
- Shear modulus
Shear modulus. Multiply material shear stiffness by the polar geometric property.
PaOne pascal is one newton per square metre. 1 MPa = 10⁶ Pa.
- J
- Polar second moment
Area-weighted squared distance from the shaft centre: the sum of the two perpendicular centroidal second moments.
m⁴The fourth power of metres is used for a second moment of area; 1 m⁴ = 10¹² mm⁴.
- θ
- Result to find
Angle of twist. Dividing load-length by rigidity gives the twist in radians, not degrees.
rad
Sort out the units first
Use torque in N·m, length in m, G in Pa and J in m⁴. The calculated angle is in radians; multiply by 180/π to express it in degrees.
Assumptions before calculating
Treat the material as homogeneous and linearly elastic, with small strains. Use properties for the actual temperature and loading direction; the calculation is a model of behaviour before yielding, not a failure test.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find θ and explain the result in the stated output unit.
- T · Torque
- 5000 N·m
- L · Shaft length
- 2 m
- G · Shear modulus
- 79000000000 Pa
- J · Polar second moment
- 0.00000982 m⁴
Accumulate torque over length
The same torque acting over a longer segment produces more relative rotation.
(5000) × (2) = 10000 N·m²Find torsional rigidity
Multiply material shear stiffness by the polar geometric property.
(79000000000) × (0.00000982) = 775780 N·m²Calculate the relative angle
Dividing load-length by rigidity gives the twist in radians, not degrees.
(10000) ÷ (775780) ≈ 0.01289025239 rad
Does this worked answer make sense?
Twist is proportional to length and torque. Doubling G or J halves the angle, and reversing torque reverses the rotation sign.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- T · Torque
- 2500 N·m
- L · Shaft length
- 1.5 m
- G · Shear modulus
- 27000000000 Pa
- J · Polar second moment
- 0.00002 m⁴
Accumulate torque over length
The same torque acting over a longer segment produces more relative rotation.
(2500) × (1.5) = 3750 N·m²Find torsional rigidity
Multiply material shear stiffness by the polar geometric property.
(27000000000) × (0.00002) = 540000 N·m²Calculate the relative angle
Dividing load-length by rigidity gives the twist in radians, not degrees.
(3750) ÷ (540000) ≈ 0.006944444444 rad
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- T · Torque
- 4000 N·m
- L · Shaft length
- 2.5 m
- G · Shear modulus
- 80000000000 Pa
- J · Polar second moment
- 0.000012 m⁴
Find: Learn: Angle of twist
A hint, not the answer
Torque acting over a longer length accumulates more twist. The product GJ is torsional rigidity, so greater material stiffness or a more effective section reduces rotation.
Use torque in N·m, length in m, G in Pa and J in m⁴. The calculated angle is in radians; multiply by 180/π to express it in degrees.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Accumulate torque over length
The same torque acting over a longer segment produces more relative rotation.
(4000) × (2.5) = 10000 N·m²Find torsional rigidity
Multiply material shear stiffness by the polar geometric property.
(80000000000) × (0.000012) = 960000 N·m²Calculate the relative angle
Dividing load-length by rigidity gives the twist in radians, not degrees.
(10000) ÷ (960000) ≈ 0.01041666667 rad
Avoid the common trap
Do not report radians as degrees. Use shear modulus G rather than Young’s modulus E. A shaft with twice the length does not have the same twist under the same torque.
When this method applies — and when it does not
Assume a prismatic circular shaft with constant torque, G and J along the segment. For stepped shafts, compute each segment separately and sum signed rotations. Warping restraint and noncircular sections are not covered.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Angle of twist. Use the relevant elastic-response, equilibrium, constitutive-relations or shear-and-torsion module; the lesson states its particular sign convention.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.