Circular-Shaft Angle of Twist
Calculate elastic twist of a uniform circular shaft.
Try this formula
Calculate signed angle of twist or choose another supported quantity.
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Try this formula
Calculate signed angle of twist or choose another supported quantity.
Use decimals or scientific notation, such as 1.2e3. Calculations update as you type.
Signed angle of twist
Enter the known values to see the result.
Use small strains and a homogeneous, linearly elastic material. Loads are static or slowly applied; remain below the proportional limit. The shaft is circular, and T, J, and G are constant along the length. The equation returns radians; define the positive torque and twist directions consistently.
Difficulty
Intermediate
Read time
5 min
Prerequisites
Basic algebra, mechanics, and consistent SI units
Source
Donney Fan — MECH 360 — Mechanics of MaterialsReference equation: Torsion — Angle of Twist. FormuLab wrote new explanations, assumptions, applications, and numerical examples.
Definition
A longer shaft twists more for the same torque. A larger polar area moment or shear modulus raises torsional stiffness, reducing the angle of twist.
Variables & Units
| Symbol | Meaning | Units |
|---|---|---|
| Signed angle of twist | ||
| Signed internal torque | ||
| Shaft length | ||
| Polar second moment of area | ||
| Shear modulus |
Real-World Applications
Shaft alignment
Estimate relative angular displacement between shaft ends.
Torsional stiffness
Compare shaft dimensions or materials.
Worked Example
Problem
A 1 m shaft carries 500 N·m, with J = 0.00000025 m⁴ and G = 80 GPa. Find its twist.
Given
Solution
Final Answer
Contents
Info
Tools
Source & review
Reference equation: Torsion — Angle of Twist. FormuLab wrote new explanations, assumptions, applications, and numerical examples.
Assumptions
- • Use small strains and a homogeneous, linearly elastic material.
- • Loads are static or slowly applied; remain below the proportional limit.
- • The shaft is circular, and T, J, and G are constant along the length.
- • The equation returns radians; define the positive torque and twist directions consistently.
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FormuLab is a study reference. Verify formulas, units, assumptions, and course-specific conventions before relying on them.