Torsional Constant Calculator

The Saint-Venant torsional constant, usually written J, describes how a cross-section resists uniform elastic twisting. In the basic torsion relation, angle of twist is TL/(GJ), where T is torque, L is member length, and G is shear modulus. A larger J therefore produces less twist for the same material, length, and torque. The constant has units of length to the fourth power, such as mm⁴ or m⁴. For a solid circular section, J equals the polar second moment of area. For noncircular sections those quantities are not interchangeable because warping changes the shear-stress distribution. This calculator addresses a solid rectangle using the approximation J = bt³[1 − 0.63(t/b) + 0.052(t/b)⁵]/3, where b is the longer side and t is the shorter side. It automatically orders the entered dimensions, so rotating the rectangle does not change the result. The approximation behaves sensibly from thin strips through square sections. Because the shorter dimension is cubed, thickness strongly controls torsional rigidity. Doubling only the short dimension can increase J dramatically, while increasing the long dimension has a nearly linear effect for a thin rectangle. For a square, the result is approximately 0.1407 times the side length to the fourth power. The formula is intended for solid, uniform rectangles rather than hollow tubes, open channels, I-sections, or built-up members. Maintain one length unit throughout. Dimensions entered in millimetres produce J in mm⁴. To convert mm⁴ to m⁴, multiply by 10⁻¹²; converting the dimensions first is often less error-prone. Torsional rigidity is GJ, not J alone, so material shear modulus must be included when calculating actual twist. Maximum stress also requires the appropriate rectangular-section stress coefficient and should not be estimated using the circular-shaft formula. Saint-Venant theory assumes a prismatic member, linear elastic material, small deformation, and loading sufficiently far from local disturbances. Noncircular sections warp during torsion, and restrained warping can add stresses not represented by this constant. Thin-walled open sections can also be sensitive to buckling. Use section-specific standards, finite-element analysis, or a qualified engineer for complex geometry and critical structures. This calculator is best suited to coursework, quick section comparisons, preliminary shaft or bar sizing, and checking a rectangular torsion calculation before more detailed analysis.