Torsional Constant Calculator (J for round, tube, rectangle, box and open sections)

Understanding the torsional constant calculator — concept and principle

The torsional constant J tells you how much a shaft or beam resists twisting because of its cross-section shape. This calculator finds J for a solid round bar, a round tube, a solid rectangle, a thin-walled closed box and a thin-walled open section, then gives the torsional rigidity GJ and the twist per metre under a torque. It is used in machine design and structural engineering.

The true torsional constant is only 55 % of the polar moment for this 2 : 1 rectangle.

How it is calculated

Solid round: J = π D⁴ ÷ 32; tube: J = π (D⁴ − d⁴) ÷ 32

Solid rectangle a × b (a ≥ b): J = (a b³ ÷ 3) × [1 − (192 ÷ π⁵)(b ÷ a) Σ tanh(nπa ÷ 2b) ÷ n⁵], n = 1, 3, 5 …

Thin closed box: J = 4 Am² t ÷ s (Bredt)

Thin open section: J = U t³ ÷ 3

Twist rate = T ÷ (G J)

Am is the area enclosed by the wall centreline, s its perimeter, U the developed length, t the wall thickness.
G is the shear modulus in GPa and T the torque in N·m.

The principle

When a round shaft twists, every cross-section rotates as a rigid disc and shear stress grows linearly from the centre, so J equals the polar moment of inertia. Non-circular sections warp out of plane, and their corners carry little stress, which makes them less stiff than the polar moment suggests. Saint-Venant solved the rectangle exactly with the series above. Closed thin tubes are very stiff because shear flows all the way around the wall, while the same material slit open must resist twisting through its thin wall thickness alone, which is why an open section is dramatically weaker in torsion.

Worked example

Default inputs: solid rectangle 100 mm × 50 mm, G = 79 GPa, torque 500 N·m.

a = 100, b = 50, so ab³ ÷ 3 = 100 × 125,000 ÷ 3 = 4.167×10⁶ mm⁴.
The series correction gives J = 2.859 × 10⁶ mm⁴ = 285.9 cm⁴ = 2.859 × 10⁻⁶ m⁴ (J ÷ ab³ = 0.229, the textbook coefficient for a/b = 2).
Polar moment for comparison: 100 × 50 × (100² + 50²) ÷ 12 = 5.208 × 10⁶ mm⁴.
GJ = 79×10⁹ × 2.859×10⁻⁶ = 225.8 kN·m².
Twist = 0.5 kN·m ÷ 225.8 kN·m² = 0.002214 rad/m = 0.1269° per m.

How to read the result

A larger J means less twist for the same torque and length. For the rectangle, using the polar moment would overestimate stiffness by more than 80 %. Compare section types for the same material: a closed tube is far stiffer in torsion than an I-beam or channel of similar weight. Multiply the twist per metre by the member length to get the total angle.

Common mistakes

Using the polar moment of inertia for non-circular sections.

Keep in mind: These are elastic stiffness values for uniform members. Stress, buckling and connection design need separate checks under the applicable codes and a qualified engineer's review.

Frequently asked questions

Why is J different from the polar moment for a rectangle?

Non-circular sections warp, so their corners carry little shear and the real stiffness is lower.

What is the shear modulus of steel?

About 79–80 GPa for common structural and carbon steels.

Why is a slit tube so weak in torsion?

Without a closed loop, shear cannot circulate around the wall, so only the thin wall thickness resists twisting.

Sources: Beer & Johnston, Mechanics of Materials, torsion of non-circular members (coefficients c₂ for rectangular bars) · PTC Creo Simulate Help, "Beam Section Property Calculations" (thin-walled closed and open section formulas)

Checked 9 Nov 2026. PTC Creo Simulate Help, “Beam Section Property Calculations” (closed thin-wall J = 4A_m²t/U, open thin-wall J = ⅓Ut³; J = I_p only for circles). Rectangle coefficients checked against Beer & Johnston’s table of c₂ (J = c₂ab³: 0.1406 at a/b = 1, 0.229 at 2, 0.263 at 3, 0.312 at 10) as reproduced in IUST e-book notes on torsion; the series is the standard Saint-Venant solution.

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