Shaft Torsion Analysis

Calculate shear stress, torsion angle and torque for circular shafts under torsion — solid or hollow cross-section

Inputs

T

Torque

N·m

d

Shaft Diameter

[\tau]

Allowable Shear Stress()

MPa

Formula Interpretation

Polar Moment — Solid Circle

I_p = \dfrac{\pi d^4}{32}

Sum of ρ²Δa over the full cross-section. For a solid circle of diameter d: $I_p = \dfrac{\pi d^4}{32}$ .

Section Modulus — Solid Circle

Z_p = \dfrac{\pi d^3}{16}

Z_p = I_p / r. For a solid circle: $Z_p = \dfrac{\pi d^3}{16}$ . Maximum shear stress occurs at the outer radius.

Polar Moment — Hollow Circle

I_p = \dfrac{\pi (d_2^4 - d_1^4)}{32}

For hollow circle (outer d₂, inner d₁): $I_p = \dfrac{\pi (d_2^4 - d_1^4)}{32}$ . Hollowing removes the low-stress core while retaining most stiffness.

Section Modulus — Hollow Circle

Z_p = \dfrac{\pi (d_2^4 - d_1^4)}{16\,d_2}

For hollow circle: $Z_p = \dfrac{\pi (d_2^4 - d_1^4)}{16\,d_2}$ . Dividing by the outer radius d₂/2 gives the maximum-stress section modulus.

Shear Stress from Torque

\tau = \dfrac{T}{Z_p}

Design formula ⑦: the maximum shear stress $\tau$ is proportional to torque $T$ and inversely proportional to $Z_p$ .

Shear Stress from Angle (Direct)

\tau = G \cdot \dfrac{r\,\theta}{l}

Formula ②: shear stress from the torsion angle. $\tau$ is proportional to $G$ and outer radius $r$ , and inversely proportional to shaft length.

Torsion Angle

\theta = \dfrac{T\,l}{G\,I_p} \quad (\text{rad})

Formula ③: torsion angle $\theta$ in radians. Multiply by 57.3 to convert to degrees.

Stiffness Limit

\varphi \leq 0.25 \,{^\circ\!/\text{m}}

Stiffness criterion: the torsion angle per unit length must not exceed 0.25 °/m (i.e. 1/4° per metre). This is the standard rigidity limit for transmission shafts.

Torque from Power

T = \dfrac{60\,000}{2\pi} \times \dfrac{P \times 10^3}{N} \approx \dfrac{9550 \, P}{N} \quad (\text{N·m})

Formula ④: exact derivation T = P·60/(2π·n). With $P$ in kW and $N$ in rpm, T ≈ 9550·P/N in N·m.

Knowledge Points

Torsional Shear Strain

When a shaft is twisted, any fibre parallel to the axis shears into a helix. The shear strain γ = r·θ/l relates the outer radius r, total torsion angle θ (rad) and shaft length l. The linear distribution means γ = 0 at the neutral axis and γ_max at the outer surface.

Polar Moment of Inertia

I_p = Σρ²Δa measures the resistance of a cross-section to twisting — analogous to the second moment of area for bending. A hollow shaft removes material near the axis (low ρ, small contribution) and is therefore nearly as stiff as a solid shaft at significantly lower weight.

Stiffness Requirement

For transmission shafts, the allowable stiffness is typically 0.25 °/m (or 1/4° per metre) to avoid vibration and misalignment. When this condition governs, it may require a larger diameter than the strength condition alone would demand.

Worked Example

A solid circular shaft of diameter 30 mm, length 500 mm, has a torsion angle of 1° when twisted. Find the shear stress. Given: $82\,\text{GPa}$ .

Step 1 — Shear Stress Directly from Angle (Formula ②)

\tau = G \cdot \dfrac{r\,\theta}{l} = 82\times10^3 \times \dfrac{15 \times \frac{\pi}{180}}{500} = 42.9 \;\text{MPa}

Step 2 — Verify via Section Properties

I_p = \dfrac{\pi \times 30^4}{32} \approx 79{,}522 \;\text{mm}^4

Z_p = \dfrac{\pi \times 30^3}{16} \approx 5{,}301 \;\text{mm}^3

T = \tau \cdot Z_p = 42.9 \times 5{,}301 \approx 227{,}400 \;\text{N·mm} = 227.4 \;\text{N·m}

Step 3 — Verify Torsion Angle (Formula ③)

\theta = \dfrac{T\,l}{G\,I_p} = \dfrac{227{,}400 \times 500}{82{,}000 \times 79{,}522} \approx 0.01745 \;\text{rad} = 1°

Result: τ = $42.9\,\text{MPa}$ . All three approaches give the same answer, confirming the consistency of formulas ①②③.

Extended Knowledge

•For shafts with multiple torque inputs (e.g. a gearbox), draw a torque diagram first: the torque varies along the shaft length and the maximum value governs the design.
•Saint-Venant''s torsion applies only to circular cross-sections. For non-circular sections (rectangular, I-beam, etc.) the shear stress is non-uniform and separate theories (Prandtl''s stress function, finite element) are required.
•In fatigue-loaded shafts, the allowable shear stress [τ] must account for stress concentration factors at keyways, grooves and press-fit interfaces. Combined bending and torsion is analysed using the von Mises or Tresca criterion.