If you were searching for how to calculate the torsional constant of I-beams and many other structural profiles, search no more! When studying the behavior of beams subjected to torsion, we usually focus on circular bars. Still, sometimes it's necessary to go beyond, as some non-circular members can experience torsional loads (an example of this is the wing span of an aircraft). For this reason, we created this tool.
Keep reading to learn more about the torsional constant, what to consider when calculating it, and the formulas behind this calculator. 🙋 Once you get the torsional constant, you can use it in an angle of twist calculation.
What is the torsional constant?
Besides shear modulus and beam length, the torsional constant (also known as the torsion constant) is one of the properties that describe the torsional stiffness of non-circular beams (similar to the polar moment of inertia). Opposite to shear modulus, it is a geometrical property, as it only depends on the form and dimensions of the cross-section.
To calculate the angle of twist of a beam under torsion using the polar moment of inertia, the cross-section of the member must remain plane and undistorted, which only occurs in circular sections. Fortunately, we can obtain analog expressions of the polar moment of inertia for different geometries using advanced solid mechanics procedures. The torsional constant is the property that allows us to calculate the angle of twist of non-circular bars:
φ = T·L / (K·G)
In this equation:
- φ — Angle of twist;
- T — Torque applied to the beam;
- L — Shaft length;
- G — Shear modulus of the shaft material; and
- K — Beam torsional constant, the one we get with this calculator.
The torsional constant in this formula has the same function as the polar moment in a circular beam, but we term it here with the symbol K to differentiate the two. For a solid circular bar, the torsional constant equals the polar moment of inertia; for any other shape, it is generally smaller.
Important things to consider when using this calculator
- Consistent units. The torsional constant is a purely geometric property with units of length⁴. Enter every dimension in the same unit you select (mm, cm, m, in, or ft); the result will be in that unit raised to the fourth power (e.g. mm⁴ or in⁴).
- Metric and US (imperial) systems. Pick millimeters, centimeters, or meters for the metric system, or inches and feet for the American system. The formulas are identical — only the units change.
- Rectangle orientation does not matter. The calculator automatically treats the longer side as the long dimension and the shorter side as the thickness, so you can enter width and height in any order.
- Thin-walled open sections. The I-beam option uses the thin-walled open-section approximation. It is accurate when the wall thickness is small compared with the flange width and web height. The same idea extends to channels, angles, and T-sections by summing (1/3)·b·t³ over each rectangular element.
- The torsional constant is not the polar moment. Using the polar moment of inertia for non-circular sections overestimates the torsional stiffness, because non-circular cross-sections warp out of plane under torsion.
Torsion constant formulas
This calculator supports the following cross-sections:
| Cross-section | Torsional constant formula |
|---|---|
| Solid circle (diameter d) | K = π·d⁴ / 32 |
| Hollow circle / tube (outer D, inner d) | K = π·(D⁴ − d⁴) / 32 |
| Rectangle (long side a, short side b) | K = a·b³·[1/3 − 0.21·(b/a)·(1 − b⁴/(12·a⁴))] |
| Ellipse (semi-axes a, b) | K = π·a³·b³ / (a² + b²) |
| I-beam (thin-walled, flanges bf×tf, web hw×tw) | K = (2·bf·tf³ + hw·tw³) / 3 |
For a solid circle and a hollow circle, the torsional constant equals the polar moment of inertia, because circular sections do not warp. For all other shapes, the constant is derived from solid-mechanics solutions (Saint-Venant torsion theory) and is smaller than the equivalent polar moment.
Example: How to calculate the torsional constant of a rectangle cross-section beam
Suppose we have a rectangular steel beam with a long side a = 100 mm and a short side b = 20 mm. We apply the rectangle formula:
K = a·b³·[1/3 − 0.21·(b/a)·(1 − b⁴/(12·a⁴))]
- Compute the ratio b/a = 20/100 = 0.20.
- Compute b⁴/(12·a⁴) = 20⁴ / (12 · 100⁴) = 160,000 / 1,200,000,000 ≈ 0.000133.
- Bracket term: 1/3 − 0.21 · 0.20 · (1 − 0.000133) ≈ 0.33333 − 0.04199 ≈ 0.29134.
- Multiply: K = 100 · 20³ · 0.29134 = 100 · 8,000 · 0.29134 ≈ 233,072 mm⁴.
So the torsional constant of this rectangular beam is about 2.33 × 10⁵ mm⁴. Switch the unit selector to inches and enter the same shape in inches to obtain the result directly in in⁴.
FAQs
What is the torsional constant?
It is a geometric property of a cross-section that quantifies its resistance to twisting. Together with the shear modulus and length, it determines the angle of twist of a non-circular beam under torque through φ = T·L / (K·G).
Is the torsional constant the same as the polar moment of inertia?
Only for circular sections (solid or hollow). For any non-circular shape, the torsional constant is smaller than the polar moment of inertia because the cross-section warps out of plane during torsion.
How do I calculate the torsional constant of an I-beam?
For a thin-walled I-beam, sum (1/3)·b·t³ over each rectangular element. With two flanges (width bf, thickness tf) and one web (height hw, thickness tw): K = (2·bf·tf³ + hw·tw³) / 3. Select the "I-Beam (Thin-Walled)" option above to do this automatically.
What units does the torsional constant have?
Length to the fourth power. If you enter dimensions in millimeters, the result is in mm⁴; in inches, the result is in in⁴. Always keep every input in the same unit.
Can I use the torsional constant to find the angle of twist?
Yes. Once you have K, plug it into φ = T·L / (K·G), where T is the applied torque, L is the shaft length, and G is the shear modulus of the material.