Opening your learning space
Starting the app …
Opening your learning space
Starting the app …
A prismatic circular shaft with constant torque twists according to torque, length and torsional rigidity.
Free to browse · flashcards or a multiple-choice quiz · import from Essential
Formula
φ in rad (dimensionless); T in N·m; L in m; G in Pa; J in m4
| Symbol | Meaning | Unit |
|---|---|---|
| Relative rotation of the shaft sections | rad | |
| Constant torque | N·m | |
| Shaft-segment length | m | |
| Shear modulus | Pa | |
| Polar second moment of area | m4 |
\varphi=\frac{T L}{GJ}The formula integrates twist rate dφ/dx=T/(GJ) for constant T, G and J. Variable sections or materials require piecewise integration. It assumes linear-elastic Saint-Venant torsion; supports, notches and noncircular sections can introduce additional deformation. Radians are dimensionless ratios of arc length to radius.
The formula integrates twist rate dφ/dx=T/(GJ) for constant T, G and J. Variable sections or materials require piecewise integration. It assumes linear-elastic Saint-Venant torsion; supports, notches and noncircular sections can introduce additional deformation. Radians are dimensionless ratios of arc length to radius.
A prismatic circular shaft with constant torque twists according to torque, length and torsional rigidity.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Angle of twist of a circular shaft” and check its unit and magnitude.
For T=100 N·m, L=1 m, G=80 GPa and J=2.51×10−7 m4, φ≈0.00498 rad or 0.285°.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: φ in rad (dimensionless); T in N·m; L in m; G in Pa; J in m4.
Formula cluster
A prismatic circular shaft with constant torque twists according to torque, length and torsional rigidity.
For T=100 N·m, L=1 m, G=80 GPa and J=2.51×10−7 m4, φ≈0.00498 rad or 0.285°.
Estimate shaft torsional stiffness and elastic rotation.
Here is how to work through a typical Angle of twist of a circular shaft (φ = TL/(GJ)) task step by step:
Task
Work through the numerical example for “Angle of twist of a circular shaft” and check its unit and magnitude.
Solution path
For T=100 N·m, L=1 m, G=80 GPa and J=2.51×10−7 m4, φ≈0.00498 rad or 0.285°.
Each shaft segment contributes twist per unit length. With constant T, G and J, these contributions add, so φ is proportional to L. A longer shaft is less torsionally stiff with other properties fixed. The relation is linear-elastic and does not describe large plastic twist. This relation assumes a circular prismatic shaft, linear elasticity and constant torque, shear modulus and polar area moment along the segment. Use radians for φ. If a property changes, sum segment twists or integrate T/(GJ); joint and support flexibility are separate effects.
G is the shear modulus and describes elastic shear deformation. For an isotropic linear-elastic material, G=E/[2(1+ν)] when Young’s modulus E and Poisson’s ratio ν are known. This relation assumes isotropy. Use directional elastic properties for anisotropic materials instead of applying it blindly. This relation assumes a circular prismatic shaft, linear elasticity and constant torque, shear modulus and polar area moment along the segment. Use radians for φ. If a property changes, sum segment twists or integrate T/(GJ); joint and support flexibility are separate effects.
Divide the shaft into segments where T, G and J can be treated as constant. Compute φi=TiLi/(GiJi) and sum signed twists. Since J scales with R4 for solid circles, thin sections often dominate total rotation. Shoulder notches need separate assessment. This relation assumes a circular prismatic shaft, linear elasticity and constant torque, shear modulus and polar area moment along the segment. Use radians for φ. If a property changes, sum segment twists or integrate T/(GJ); joint and support flexibility are separate effects.
Write the quantity and unit from the variable list next to every value: φ in rad (dimensionless); T in N·m; L in m; G in Pa; J in m4. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For T=100 N·m, L=1 m, G=80 GPa and J=2.51×10−7 m4, φ≈0.00498 rad or 0.285°. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. This relation assumes a circular prismatic shaft, linear elasticity and constant torque, shear modulus and polar area moment along the segment. Use radians for φ. If a property changes, sum segment twists or integrate T/(GJ); joint and support flexibility are separate effects.
The background states this starting condition: The formula integrates twist rate dφ/dx=T/(GJ) for constant T, G and J. Variable sections or materials require piecewise. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. This relation assumes a circular prismatic shaft, linear elasticity and constant torque, shear modulus and polar area moment along the segment. Use radians for φ. If a property changes, sum segment twists or integrate T/(GJ); joint and support flexibility are separate effects.
Quanta exam set
Curated exam set for "Angle of twist of a circular shaft":
Question (front)
Which formula describes Angle of twist of a circular shaft?
Answer in your set
Question (front)
How do you rearrange Angle of twist of a circular shaft for Base equation for the target quantity?
Answer in your set
Question (front)
Which common mistake happens with Angle of twist of a circular shaft?
Answer in your set
+ 8 more cards: units, variables, derivation, example, exam task
The formula set contains 11 cards. After importing, review them and start practicing; your ratings determine later review intervals.
This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.
Create a curated FSRS exam set for φ = TL/(GJ): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
Curated formula set · LaTeX · FSRS spaced repetition · import from Essential