Torsional Constant Calculator
Calculate the torsional constant K of solid circular, square, rectangular, and elliptical cross-sections. Free online physics calculator with step-by-step breakdowns and charts.
About This Calculator
The Torsional Constant Calculator computes the torsional constant K (also called the torsion constant) for common engineering cross-section shapes: solid circular, solid square, solid rectangle, and solid ellipse. The torsional constant is a geometric property that measures a cross-section's resistance to twisting, used in the fundamental torsion formula φ = TL / (KG), where T is the applied torque, L is the beam or shaft length, G is the shear modulus of the material, and φ is the resulting angle of twist.
Unlike the polar moment of inertia J, which only applies to circular sections (where cross-sections remain plane under torsion), the torsional constant K generalizes the twist analysis to non-circular shapes that experience warping. For circular sections, K = J = πd⁴/32. For rectangles, squares, and ellipses, different formulas based on Saint-Venant's torsion theory are used, all derived from the elastic stress function approach.
Supported formulas
- Solid circle: K = π·d⁴/32 (d = diameter)
- Solid square: K = 9·a⁴/64 (a = side length)
- Solid rectangle: K = a·b³/3 - 0.21·b⁴ + 0.0175·b⁸/a⁴ (a = width, b = height, a ≥ b). Error no greater than 4%.
- Solid ellipse: K = π·a³·b³/(a² + b²) (a = semi-major, b = semi-minor)
The formulas used in this calculator assume a straight, uniform beam loaded only by equal and opposite twisting couples applied at its ends in planes normal to its axis, with stresses remaining within the elastic limit. These assumptions follow the Saint-Venant torsion theory and are standard in mechanical and civil engineering design codes including Roark's Formulas for Stress and Strain.
Applications: The torsional constant is essential in the design of drive shafts, aircraft wing spars, bridge girders under eccentric loading, building frames subjected to seismic torsion, and any structural member where twisting loads must be analyzed for safe deformation limits.
Frequently Asked Questions
What is the torsional constant?
The torsional constant (also called the torsion constant), denoted K, is a geometric property of a cross-section that quantifies its resistance to twisting. It is used in the angle of twist formula φ = TL/(KG), where T is applied torque, L is beam length, G is the shear modulus of the material, and K is the torsional constant. Unlike the polar moment of inertia J, which only applies to circular sections, the torsional constant works for any cross-sectional shape.
How is torsional constant different from polar moment of inertia?
The polar moment of inertia J can only be used to calculate twist in circular cross-sections where cross-sections remain plane and undistorted under torsion. The torsional constant K generalizes this to non-circular shapes like rectangles, squares, and ellipses where warping occurs. For circular sections only, K equals J. For all other shapes, K is less than J and accounts for the reduced torsional stiffness caused by cross-sectional warping.
What are the units of the torsional constant?
The torsional constant K has units of length to the fourth power. In the SI system, it is typically expressed in mm⁴ or m⁴. In the US customary system, it is expressed in in⁴. Our calculator outputs results in mm⁴.
How do you calculate the torsional constant of a rectangle?
For a solid rectangle with width a (larger dimension) and height b (smaller dimension), the torsional constant is K = a·b³/3 - 0.21·b⁴ + 0.0175·b⁸/a⁴. This formula has an error no greater than 4% compared to exact analytical solutions. For example, a 10 mm × 5 mm rectangle gives K = 10×5³/3 - 0.21×5⁴ + 0.0175×5⁸/10⁴ = 284.73 mm⁴.
What is the torsional constant formula for a circle?
For a solid circular cross-section, the torsional constant equals the polar moment of inertia: K = J = π·d⁴/32, where d is the diameter. For a solid circle, K and J are identical because circular sections do not warp under torsion.
What is the torsional constant formula for a square?
For a solid square cross-section with side length a, the torsional constant is K = 9·a⁴/64. This formula gives the exact torsional constant for square sections based on Saint-Venant's torsion theory.
What is the torsional constant formula for an ellipse?
For a solid elliptical cross-section with semi-major axis a and semi-minor axis b, the torsional constant is K = π·a³·b³/(a² + b²). This formula comes from the analytical solution of the Saint-Venant torsion problem for elliptical sections.
Why is the torsional constant important in engineering?
The torsional constant is essential for designing structural members and machine components that experience twisting loads. Engineers use it to calculate the angle of twist of beams, shafts, and structural members under torque, ensuring that deformations stay within acceptable limits. It is critical in the design of drive shafts, aircraft wings, bridge girders, and building frames subjected to torsional loading. The torsional constant directly affects the torsional stiffness k = KG/L of a structural member.