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The Torsional Constant Calculator computes the torsional constant (also called the polar moment of inertia \(J\) for circular sections, or the St. Venant torsion constant \(J_t\) for non-circular sections) for three common cross-sectional shapes: solid circle, hollow circle, and rectangle. This property is fundamental to determining the shear stress and angle of twist in members subjected to torque.
For circular cross-sections, the torsional constant equals the polar moment of inertia, and the classical formulas are exact:
$$J_{solid} = \frac{\pi d^4}{32}$$
$$J_{hollow} = \frac{\pi (d_o^4 - d_i^4)}{32}$$
Circular sections are the only shapes where cross-sections remain perfectly plane during torsion. This is why the \(\tau = Tr/J\) formula applies exactly only to circular shafts.
For rectangular cross-sections, the situation is more complex. Rectangular sections warp during torsion — the cross-section does not remain plane. The exact solution involves an infinite series, but an excellent approximation widely used in practice is:
$$J_{rect} = k \cdot a \cdot b^3$$
where \(a\) is the longer side, \(b\) is the shorter side, and the coefficient \(k\) depends on the aspect ratio \(b/a\). The approximation used here is:
$$k \approx \frac{1}{3} - 0.21 \frac{b}{a}\left(1 - \frac{b^4}{12a^4}\right)$$
This formula is accurate to within 0.5% for all aspect ratios and is cited in Timoshenko's and Roark's classic references.
The torsional constant is central to shaft design, structural analysis of non-circular members, and determining the torsional stiffness \(GJ/L\) of any member. Engineers use it when analyzing torsional vibrations in machinery, lateral-torsional buckling of beams, and combined loading in structural frames.
Hollow circular sections are particularly efficient in torsion. The material near the center of a solid shaft contributes very little to \(J\), so removing it (creating a tube) achieves nearly the same torsional resistance at substantially reduced weight. This is why drive shafts, helicopter rotors, and structural tubes are hollow.
The calculator applies the appropriate formula based on the selected shape:
Solid circle:
$$J = \frac{\pi d^4}{32}$$
Hollow circle:
$$J = \frac{\pi (d_o^4 - d_i^4)}{32}$$
Rectangle (a ≥ b):
$$J \approx k \cdot a \cdot b^3, \quad k = \frac{1}{3} - 0.21\frac{b}{a}\left(1 - \frac{b^4}{12a^4}\right)$$
The approximation coefficient \(k\) ranges from about 0.141 (for a square, \(b/a = 1\)) to 0.333 (for very thin rectangles, \(b/a \to 0\)). All dimensions are in mm, giving \(J\) in mm⁴.
A larger torsional constant means greater resistance to twisting. For circular sections, \(J\) grows with the fourth power of diameter, so a small increase in diameter dramatically increases torsional rigidity. Hollow sections with a wall thickness ratio (\(d_i/d_o\)) of 0.6–0.8 offer an excellent balance between weight and torsional performance. Rectangular sections are inherently less efficient in torsion than circular ones — a square has only about 84% the torsional constant of a circle with the same area.
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A 50 mm diameter solid shaft has J ≈ 613,592 mm⁴ (61.4 cm⁴). This value would be used with τ = Tr/J to find shear stress, or with θ = TL/(GJ) to find the angle of twist.
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A 100 mm × 50 mm rectangular bar has J ≈ 1,737,000 mm⁴ using the St. Venant approximation with k ≈ 0.139. Despite the larger area compared to the 50 mm shaft above, rectangular bars are less efficient in torsion due to stress concentrations at corners.
The torsional constant \(J\) (or \(J_t\)) is a geometric property of a cross-section that quantifies its resistance to torsion. For circular sections, it equals the polar moment of inertia \(J = \pi d^4/32\). For non-circular sections, it is the St. Venant torsion constant, which accounts for warping effects and is always less than the polar moment of inertia.
Only for circular cross-sections (solid and hollow). For all other shapes, the St. Venant torsion constant is less than the polar moment of inertia because non-circular sections warp during torsion. Using \(J = I_x + I_y\) (polar moment) for a rectangular section would overestimate its torsional resistance.
Only circular cross-sections have the axial symmetry needed to keep cross-sections plane during twisting. For rectangles (and all other non-circular shapes), different parts of the cross-section must displace axially to satisfy equilibrium and compatibility, causing the warped shape. This warping changes the stress distribution fundamentally.
The formula \(J = k \cdot a \cdot b^3\) with \(k \approx 1/3 - 0.21(b/a)(1 - b^4/12a^4)\) is accurate to within about 0.5% for all aspect ratios. It is widely used in engineering practice and cited in references such as Roark's Formulas for Stress and Strain and Timoshenko's Theory of Elasticity.
For a hollow shaft, \(J = \pi(d_o^4 - d_i^4)/32\). A thin-walled tube retains most of the torsional constant of the solid shaft at greatly reduced weight. For example, a tube with \(d_i/d_o = 0.8\) retains 59% of the solid shaft's \(J\) with only 36% of the material weight — making hollow sections highly efficient.
This calculator covers three standard shapes. For open thin-walled sections like I-beams, the torsional constant is approximately \(J \approx \Sigma (b_i t_i^3 / 3)\), summing over each rectangular element (flanges and web). Open sections have very low torsional stiffness compared to closed (hollow) sections, which is why I-beams are poor at resisting torsion.
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