Torsional Constant Calculator

Enter a circular shaft's outer diameter (plus inner diameter for a hollow shaft) to get the torsional constant J = πD⁴/32, along with the resulting shear stress, angle of twist, and torsional stiffness.

Quick Facts

Solid shaft
J = πD⁴ / 32
D is the outer diameter; J equals the polar moment of inertia for a circular section.
Hollow shaft (tube)
J = π(D⁴ − d⁴) / 32
D is the outer diameter and d is the inner diameter.
Angle of twist
θ = T·L / (G·J)
T = torque, L = length, G = shear modulus.
Max shear stress
τ = T·r / J
r is the outer radius (D/2); stress peaks at the outer surface.

Your Results

Calculated
Torsional Constant (J)
-
J = πD⁴/32 (or π(D⁴−d⁴)/32), in mm⁴
Maximum Shear Stress
-
τ = T·r / J, at the outer surface, in MPa
Angle of Twist
-
θ = T·L / (G·J), over the shaft length
Torsional Stiffness
-
k = G·J / L, in N·m per radian

Ready

Enter the shaft geometry, torque, length, and shear modulus, then press Calculate.

Formula and Method for the Torsional Constant

The torsional constant, usually written J, measures how much a shaft's cross-section resists twisting under an applied torque. For a circular cross-section it is identical to the polar moment of inertia: a solid round shaft of outer diameter D has J = πD⁴/32, and a hollow round shaft (a tube) with outer diameter D and inner diameter d has J = π(D⁴ − d⁴)/32. This calculator uses J together with the applied torque T, shaft length L, and shear modulus G to also report the maximum shear stress, the angle of twist, and the torsional stiffness of the shaft.

How the calculation works

First J is found from the shaft geometry using the solid or hollow formula above. The maximum shear stress at the outer surface follows from the torsion formula τ = T·r / J, where r = D/2 is the outer radius — shear stress is zero at the center and increases linearly to a maximum at the outer surface. The angle of twist over the shaft's length follows from θ = T·L / (G·J), where G is the shear modulus (modulus of rigidity) of the shaft material. Dividing torque by twist angle gives the torsional stiffness, k = G·J / L = T/θ, which behaves like a rotational spring constant in units of N·m per radian.

Common mistakes

  • Using diameter instead of radius, or vice versa: the formula J = πD⁴/32 uses diameter; the equivalent radius form is J = πr⁴/2 — mixing the two understates or overstates J by a factor of 16.
  • Confusing torsional constant with polar moment of inertia for non-circular sections: for circular (solid or hollow) shafts J equals the polar moment of inertia, but for square, rectangular, or open thin-walled sections the torsional constant is smaller than the polar moment of inertia and requires a separate, often empirical, formula.
  • Mixing units: keep diameters in millimeters, torque in newton-meters, length in meters, and shear modulus in gigapascals as entered here — the calculator converts internally to consistent SI units (meters and pascals) before computing stress and twist.

Real-world applications

  • Drive shaft and axle design uses J to size a shaft so peak shear stress stays below the material's allowable shear strength with a safety margin.
  • Machine design and coupling selection use torsional stiffness to predict how much a shaft will wind up under load and to avoid excessive backlash or resonance.
  • Mechanical engineers use the angle of twist to check that rotating equipment such as turbines, gearboxes, and propeller shafts stays within allowable deflection limits.
  • Comparing a solid shaft to an equivalent-size hollow shaft shows how tubes cut weight while keeping torsional stiffness high, since removing low-stress material near the center barely reduces J.

Frequently Asked Questions

What is the torsional constant (J) and how is it different from polar moment of inertia?
The torsional constant J describes a cross-section's resistance to twisting. For solid or hollow circular shafts, J is numerically identical to the polar moment of inertia: J = πD⁴/32 for a solid shaft or J = π(D⁴ − d⁴)/32 for a hollow shaft. For non-circular sections (square, rectangular, I-beams), the torsional constant is smaller than the polar moment of inertia and needs its own formula because the section warps out of plane under torsion.
How do I calculate the torsional constant for a hollow shaft?
Use J = π(D⁴ − d⁴)/32, where D is the outer diameter and d is the inner diameter, both in the same units. For example, a tube with a 50 mm outer diameter and 30 mm inner diameter has J = π(50⁴ − 30⁴)/32 ≈ 534,071 mm⁴, versus about 613,592 mm⁴ for an equivalent solid 50 mm shaft, at noticeably lower weight.
How does the torsional constant relate to angle of twist and shear stress?
Once J is known, the angle of twist is θ = T·L / (G·J) and the maximum shear stress at the outer surface is τ = T·r / J, where T is torque, L is shaft length, G is the material's shear modulus, and r is the outer radius. A larger J means less twist and lower stress for the same applied torque.
Why do engineers prefer hollow shafts for high torsional stiffness?
Material near the center of a circular shaft carries very little shear stress and contributes little to J, since J scales with the fourth power of radius. Removing that low-stress core to make a tube cuts weight substantially while only slightly reducing torsional stiffness, which is why drive shafts, axles, and aircraft structures often use hollow rather than solid circular sections.

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