Calculator Inputs
Formula Used
This calculator uses the thin-walled closed-section torsion relation based on the median (mid-thickness) perimeter:
- J = 4 Am2 / Σ(s/t)
- Am is the area enclosed by the median line of the tube.
- s is each wall segment length along the median line.
- t is the thickness for the corresponding segment.
For a rectangular tube with different side thicknesses, the sum is: Σ(s/t) = (bm/ttop) + (bm/tbottom) + (hm/tleft) + (hm/tright), where bm and hm are median dimensions.
How to Use This Calculator
- Enter outer width B and outer height H in your chosen input unit.
- Select uniform thickness or enter different thickness per side.
- Pick an output unit to display J and the reference polar moment.
- Optionally add shear modulus G to compute torsional rigidity GJ.
- Optionally add torque T to estimate twist per length θ′.
- Press Calculate to show results above this form.
Example Data Table
| Outer B | Outer H | Thickness | Mode | Estimated J | J Unit |
|---|---|---|---|---|---|
| 120 mm | 80 mm | t = 4 mm | Uniform | 3.238e+6 | mm⁴ |
| 200 mm | 120 mm | t = 6 mm | Uniform | 1.906e+7 | mm⁴ |
| 150 mm | 100 mm | Top/Bottom 4 mm, Left/Right 6 mm | Individual | 7.350e+6 | mm⁴ |
| 10 in | 6 in | t = 0.25 in | Uniform | 1.014e+2 | in⁴ |
| 0.30 m | 0.20 m | t = 0.01 m | Uniform | 1.265e-4 | m⁴ |
Example results are rounded and intended for quick comparison.
Understanding the torsional constant
Rectangular hollow sections resist torsion by shear flow around a closed perimeter. The torsional constant J links torque to twist, via θ′ = T/(GJ). Unlike the polar moment (I_x + I_y), J reflects the true shear distribution in non‑circular sections and should be used for torsion checks.
Typical hollow rectangle dimensions
These tubes are common in frames, racks, and equipment supports because they deliver high stiffness with moderate mass. Typical outer sizes range from 50×30 mm to 300×200 mm, with wall thickness often 2–12 mm. Aspect ratios B/H often fall between 1.0 and 2.5. Early design usually compares several sizes before final load combinations are finalized.
Thin‑walled closed‑section approach
This calculator applies the thin‑walled closed‑section relation. Each wall segment is treated as a plate carrying nearly uniform shear flow, and the segment compliances are combined. The governing expression is J = 4 A_m² / Σ(s/t), where A_m is the area enclosed by the median line and s/t is evaluated for each side.
Median dimensions and mean area
Median geometry is estimated at mid‑thickness. With side‑specific thicknesses, b_m = B − (t_left + t_right)/2 and h_m = H − (t_top + t_bottom)/2, giving A_m = b_m h_m. The sum term is Σ(s/t) = b_m/t_top + b_m/t_bottom + h_m/t_left + h_m/t_right.
Thickness impact on J
Thickness has a strong effect because each term contains 1/t. If you increase all thicknesses, Σ(s/t) decreases and J rises, while the median dimensions change only slightly. Example: B = 120 mm, H = 80 mm, t = 4 mm gives J ≈ 3.24×10⁶ mm⁴; larger thickness values increase J. Unequal thickness lets you stiffen one direction without adding mass everywhere.
Unit scaling and conversions
Units matter because J scales with length to the fourth power. Converting from mm⁴ to m⁴ multiplies by 10⁻¹², and converting from in⁴ to mm⁴ multiplies by 25.4⁴ ≈ 4.16×10⁵. Use the output selector to keep geometry, loads, and material data consistent.
From J to stiffness and twist
After J is known, torsional rigidity is GJ. Typical shear modulus values are about 79.3 GPa for steel and 26 GPa for aluminum alloys. Enter G to obtain GJ in N·m². Enter torque T to estimate twist per length; higher T increases θ′, while higher G or J reduces θ′.
Applicability and limitations
The thin‑walled method is most reliable when walls are slender, commonly t/b_m < 0.1 and t/h_m < 0.1, and when corner details are not dominant. Very thick tubes, cutouts, or strongly varying thickness may require more detailed analysis. The reference polar moment is shown for comparison only.
1. What is the torsional constant J used for?
J converts applied torque into twist through θ′ = T/(GJ). It is the correct property for torsion in non‑circular sections, while I_x + I_y is only a reference metric.
2. Why does the calculator use the median line?
For thin-walled closed sections, shear flow is assumed to act along the wall mid-thickness. Using median dimensions gives a practical estimate of the enclosed area A_m and the segment lengths in Σ(s/t).
3. Can I enter different thicknesses on each side?
Yes. Choose “Different thickness per side” and enter t_top, t_bottom, t_left, and t_right. The tool then evaluates Σ(s/t) using b_m for top/bottom and h_m for left/right.
4. What does Σ(s/t) mean?
It is the sum of each wall’s segment length divided by its thickness. Larger segments or thinner walls increase Σ(s/t), which lowers J because the section becomes more torsionally compliant.
5. How do I compute twist per meter?
Enter shear modulus G and an applied torque T. The calculator reports θ′ in rad/m and degrees per meter. If torque is left blank, you still get GJ for stiffness comparisons.
6. When is this thin-walled method not suitable?
It can be inaccurate for very thick tubes, strong corner effects, cutouts, or highly irregular thickness changes. If t is not small compared with b_m or h_m, consider a more detailed torsion analysis.
7. Why show the polar moment (I_x + I_y) at all?
Many engineers recognize it from bending calculations, so it helps with quick comparisons. For rectangular torsion, do not substitute it for J; use J for twist and stress checks.