Torsional Constant Calculator
Calculate the torsional constant (J) for eight structural cross-section shapes, then find the angle of twist and maximum shear stress for a given torque. Pick your section shape, enter the dimensions, and the results update instantly.
Formula
Worked example
A solid steel shaft of radius 25 mm, length 500 mm, G = 80,000 MPa, carrying T = 1,000 N-mm: J = pi x 25^4 / 2 = 613,592 mm^4. Angle of twist phi = 1000 x 500 / (80000 x 613592) = 0.0000102 rad = 0.000584 deg. Max shear stress tau = 1000 x 25 / 613592 = 0.0408 MPa.
What is the torsional constant?
The torsional constant J (also written K in some texts) is a geometric property of a cross-section that quantifies how strongly it resists twisting. It appears in the two fundamental torsion equations: the angle of twist phi = TL / (GJ), where T is the applied torque, L is the shaft length, and G is the shear modulus; and the maximum shear stress tau = Tc / J, where c is the distance from the centroid to the outermost fibre. A larger J means less twist and lower stress for the same applied torque. For circular sections, J equals the polar moment of inertia. For all other shapes, J is strictly smaller than the polar moment because St. Venant torsion theory accounts for warping of the cross-section.
How to use this calculator
Select your unit system (mm or inches), then choose the cross-section shape from the drop-down menu. Only the inputs relevant to that shape are shown. Enter the dimensions and, optionally, the applied torque T, shaft length L, and shear modulus G to get the angle of twist and maximum shear stress as well. Results update as you type. The Show your work panel below the results walks through every arithmetic step with your actual numbers substituted in.
Solid vs hollow sections and open vs closed shapes
The geometry of a cross-section has a dramatic effect on its torsional stiffness. A hollow circular tube of the same outer radius as a solid shaft achieves roughly 90% of the solid shaft's J while using far less material. Closed sections (hollow circles, hollow rectangles) are vastly stiffer in torsion than open sections (I-beams, channels, angles) of comparable wall thickness, because shear flow must travel all the way around a closed path rather than dying out at a free edge. This is why structural hollow sections (RHS, CHS) are preferred when torsion is the dominant load, while I-sections are efficient for bending but relatively poor twisters.
Material shear modulus reference values
The shear modulus G links geometry (J) to structural response (phi, tau). Common values: structural steel, 80,000 MPa (11.6 x 10^6 psi); stainless steel, 77,000 MPa; aluminium alloys, 26,000 to 28,000 MPa (3.8 x 10^6 psi); copper, 44,000 MPa; titanium alloys, 43,000 to 44,000 MPa; engineering polymers, 1,000 to 4,000 MPa. For preliminary design, structural steel G = 80,000 MPa and aluminium G = 26,000 MPa are standard default values.
Torsional constant formulas by cross-section
| Section | Formula for J | Accuracy |
|---|---|---|
| Solid circle | J = pi*r^4 / 2 | Exact |
| Hollow circle | J = pi*(r^4 - ri^4) / 2 | Exact |
| Solid ellipse | J = pi*a^3*b^3 / (a^2 + b^2) | Exact |
| Solid rectangle | J = a*b^3/3 - 0.21*b^4*(1 - b^4/12a^4) | Approx (< 1% error) |
| Thin-walled rectangle | J = 2*t*t1*(a-t)^2*(b-t1)^2 / (a*t+b*t1-t^2-t1^2) | Approx (thin walls) |
| I-section (equal flanges) | J = 2*K1 + K2 + 2*alpha*d^4 | Approx (< 10% error) |
Summary of exact or best-practice approximate formulas for J (St. Venant torsion constant). a and b are outer dimensions; r, ri are radii.
Frequently asked questions
Is the torsional constant J the same as the polar moment of inertia Ip?
Only for circular sections. For a solid or hollow circle, J = Ip = pi*(r_outer^4 - r_inner^4)/2 exactly, because the shear stress distribution is symmetric and the section does not warp. For all other shapes - rectangles, ellipses, I-sections - J is less than Ip because warping reduces torsional stiffness. Using Ip in place of J for a non-circular section will overestimate stiffness and underestimate both the twist angle and shear stress.
Why is the I-section so weak in torsion compared to a box section?
An I-section is an open thin-walled section. When it twists, the flanges and web warp freely at their tips, so shear flow is not forced to travel a closed path around the profile. The torsional constant of an I-section is roughly J = sum(bt^3/3) over each plate, which scales with the cube of thickness. A rectangular hollow section of similar weight routes shear flow around a closed loop, giving a J that can be 50 to 100 times larger.
What units does J have, and how do I convert between mm and inches?
J has units of length to the fourth power: mm^4 or in^4. To convert, use 1 in = 25.4 mm, so 1 in^4 = 25.4^4 = 415,230 mm^4. This calculator shows results in mm^4 when you select the metric unit system, and in in^4 when you select imperial. The stress and moment units also switch automatically.
How accurate are the rectangle and I-section formulas?
The solid rectangle formula (Timoshenko approximation) has an error below 1% for all aspect ratios a/b >= 1. The thin-walled hollow rectangle formula is accurate when wall thickness is small compared to the outer dimensions (t/b < 0.1 is a common rule of thumb). The I-section formula achieves around 4 to 10% accuracy relative to finite-element solutions, which is acceptable for preliminary design.
What is the angle of twist formula and what does it depend on?
The angle of twist in radians is phi = TL / (GJ). Twist is directly proportional to the applied torque T and shaft length L, and inversely proportional to both the shear modulus G and the torsional constant J. Doubling the torque doubles the twist; doubling the shaft length doubles it; switching from steel (G = 80,000 MPa) to aluminium (G = 26,000 MPa) increases twist by a factor of about 3. Choosing a section with a larger J is the structural way to reduce twist.
Can I use this calculator for thin-walled open sections like channels or angles?
For thin-walled open sections not listed here - channels (C), angles (L), T-sections - the approximation J = sum(b_i * t_i^3 / 3) applies, where each term covers one plate of width b_i and thickness t_i. This calculator does not implement those shapes directly, but you can use the solid rectangle mode for a single plate and sum results manually.