Engineering · Strength of materials

Torsional shear stress in a circular shaft

In a linearly elastic circular shaft under torsion, shear stress increases in proportion to radius.

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Formula

τ (r) = \frac{T r}{J}

τ in Pa; T in N·m; r in m; J in m⁴

Variables & units

Variables & units
Symbol	Meaning	Unit
$τ(r)$	Torsional shear stress at radius r	Pa
$T$	Torque about the shaft axis	N·m
$r$	Radial distance from the axis	m
$J$	Polar second moment of area of the circular section	m⁴

LaTeX

\tau(r)=\frac{T r}{J}

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Derivation & background

The Saint-Venant solution applies to straight solid or annular circular shafts in linear-elastic torsion. For a solid circle J=πR⁴/2; for a tube J=π(R⁴−Ri⁴)/2. Stress is zero at the center and largest at the surface. Noncircular sections warp and do not follow this simple formula.

Exam blueprint

Validity range

Derivation steps

In a linearly elastic circular shaft under torsion, shear stress increases in proportion to radius.

1State the model assumptions and list quantities with units.
2Substitute into τ(r) = Tr/J; convert quantities to SI first.
3For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa.

Rearrangements

Base equation for the target quantity

\tau(r)=\frac{T r}{J}

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Torsional shear stress in a circular shaft” and check its unit and magnitude.

For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: τ in Pa; T in N·m; r in m; J in m⁴.

Exam context

Estimate elastic torsional stress in drive shafts.

Formula cluster

Strength of materials

In a linearly elastic circular shaft under torsion, shear stress increases in proportion to radius.

E = σ/ε = FL₀/(A₀ΔL)

Young's modulus

M = F·r·sin(θ)

Torque

M = EIκ ≈ EI·w″

Euler–Bernoulli beam moment–curvature relation

Worked example

For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa.

Applications

Estimate elastic torsional stress in drive shafts.

How do you calculate with Torsional shear stress in a circular shaft?

Here is how to work through a typical Torsional shear stress in a circular shaft (τ(r) = Tr/J) task step by step:

1
Task
Work through the numerical example for “Torsional shear stress in a circular shaft” and check its unit and magnitude.
Solution path
For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa.

Frequently asked questions about Torsional shear stress in a circular shaft

Where is τ greatest in a solid circular shaft?+

Since τ(r)=Tr/J grows linearly with r, shear stress is zero at the shaft axis and reaches its maximum at the outer surface r=R. This is for an ideal circular shaft. A hollow shaft has a different J and therefore a different stress and capacity. The distribution applies to linear-elastic torsion of straight circular shafts and uses the polar area moment J. Radius r is measured from the shaft axis. Do not transfer it to arbitrary noncircular sections, where warping and different shear-stress distributions occur. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

How do I calculate J for a solid shaft?+

For a solid circle of radius R, J=πR⁴/2. In terms of diameter d, J=πd⁴/32. Use radius or diameter with the matching form: because of the fourth power, confusing them has a large effect. J describes section geometry and has units m⁴. The distribution applies to linear-elastic torsion of straight circular shafts and uses the polar area moment J. Radius r is measured from the shaft axis. Do not transfer it to arbitrary noncircular sections, where warping and different shear-stress distributions occur. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Why does the formula not apply to a rectangle?+

A noncircular section warps under torsion, so shear stress does not follow a simple linear radial distribution. The polar area moment alone does not give torsional rigidity. Rectangles and open sections require other torsion constants and models, especially when warping is restrained. The distribution applies to linear-elastic torsion of straight circular shafts and uses the polar area moment J. Radius r is measured from the shaft axis. Do not transfer it to arbitrary noncircular sections, where warping and different shear-stress distributions occur. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

How do I check the result and units for Torsional shear stress in a circular shaft?+

Write the quantity and unit from the variable list next to every value: τ in Pa; T in N·m; r in m; J in m⁴. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa. 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. The distribution applies to linear-elastic torsion of straight circular shafts and uses the polar area moment J. Radius r is measured from the shaft axis. Do not transfer it to arbitrary noncircular sections, where warping and different shear-stress distributions occur. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Which model limit matters for Torsional shear stress in a circular shaft?+

The background states this starting condition: The Saint-Venant solution applies to straight solid or annular circular shafts in linear-elastic torsion. For a solid circle. 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. The distribution applies to linear-elastic torsion of straight circular shafts and uses the polar area moment J. Radius r is measured from the shaft axis. Do not transfer it to arbitrary noncircular sections, where warping and different shear-stress distributions occur. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

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Which formula describes Torsional shear stress in a circular shaft?

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Which common mistake happens with Torsional shear stress in a circular shaft?

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Subject references and further reading

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.

[1]Engineering LibreTexts, Beam Bending, Buckling, and Torsion, circular-shaft torsional stress.

More formulas: Mechanics, strength and materials

\Delta\vec p=\int_{t_1}^{t_2}\vec F_{\mathrm{ext}}dt, \qquad \Delta\vec L=\int_{t_1}^{t_2}\vec\tau_{\mathrm{ext}}dt

Linear and angular impulse–momentum balance

\sum_i \vec F_i=\vec 0,\quad \sum_i M_{i,O}=0

Static equilibrium conditions

\sigma_\mathrm{m} = \frac{F}{A}

Average axial normal stress

\nu=-\frac{\varepsilon_\mathrm{quer}}{\varepsilon_\mathrm{axial}}

Poisson’s ratio

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Quanta Study editorial team

Symbol

Meaning

Unit

τ(r)

Torsional shear stress at radius r

T

Torque about the shaft axis

N·m

r

Radial distance from the axis

J

Polar second moment of area of the circular section

m⁴

Exam blueprint

Validity range

Derivation steps

In a linearly elastic circular shaft under torsion, shear stress increases in proportion to radius.

1State the model assumptions and list quantities with units.
2Substitute into τ(r) = Tr/J; convert quantities to SI first.
3For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa.

How do you calculate with Torsional shear stress in a circular shaft?

Here is how to work through a typical Torsional shear stress in a circular shaft (τ(r) = Tr/J) task step by step:

Task

Work through the numerical example for “Torsional shear stress in a circular shaft” and check its unit and magnitude.

Solution path

For a solid shaft of radius 20 mm under T=100 N·m, J≈2.51×10⁻⁷ m⁴ and maximum stress is about 7.96 MPa.

Frequently asked questions about Torsional shear stress in a circular shaft

Where is τ greatest in a solid circular shaft?+

How do I calculate J for a solid shaft?+

Why does the formula not apply to a rectangle?+

How do I check the result and units for Torsional shear stress in a circular shaft?+

Which model limit matters for Torsional shear stress in a circular shaft?+