Three different formulas give the shear stress due to torsion. For the same section they can differ by orders of magnitude, and which one applies depends on the kind of wall the section has.
That much is in any textbook. What is rarely explained is what happens when two of the three apply to the same section: they give different answers, and the difference always goes the same way.
The three formulas
- Cauchy, τ = T·r / Iₚ. Circular sections only, and there it is exact: it is the only case in which plane sections remain plane. Stress grows linearly with radius, from a minimum at the inner face to a maximum at the outer face.
- Bredt, τ = T / (2·Aₘ·t). For closed thin-walled sections. The torque is carried by a shear flow that circulates around the enclosed area. Aₘ is the area enclosed by the mid-line of the wall, not by its outer face; mixing the two up is a common mistake.
- Saint-Venant, τ = T·t / J with J = (1/3)·Σb·t³. For open thin-walled sections. With no closed path, the flow has to turn back on itself across the thickness, which is why thickness appears cubed.
When two apply, they disagree
Take a circular tube. Cauchy applies and is exact. Bredt applies too: the wall is closed and a shear flow runs around it. But Bredt assumes the stress is constant through the thickness, while in Cauchy it grows with radius. So Bredt gives the average and Cauchy gives the maximum.
| t / rₘ | Example (rₘ = 50 mm) | τ Bredt / τ exact | Bredt low by |
|---|---|---|---|
| 0.05 | 2.5 mm wall | 0.976 | 2.4 % |
| 0.10 | 5.0 mm wall | 0.955 | 4.5 % |
| 0.20 | 10.0 mm wall | 0.918 | 8.2 % |
| 0.50 | 25.0 mm wall | 0.850 | 15.0 % |
Bredt is low in all four cases. For a tube with a 5 mm wall and a 50 mm mean radius, which the usual rule of thumb still counts as thin-walled, the actual stress is 4.5% higher than the calculated one. With a 10 mm wall, 8.2%.
The error always has the same sign: Bredt comes out low.
A 4.5% difference does not stand out in a design check, so a member that fails can look as if it passes.
Opening the wall changes the order of magnitude
Between overlapping theories the difference is a few per cent. Between a closed and an open wall it is orders of magnitude. Take a 100×100 mm square tube with a 5 mm wall and slit it lengthwise: it keeps its area, its weight and nearly all of its bending stiffness.
| Closed | Slit | Factor | |
|---|---|---|---|
| J [mm⁴] | 4,286,875 | 15,833 | 271 |
| τ [MPa] | 11.08 | 315.79 | 29 |
Torsional stiffness drops by a factor of 271 and the stress goes up by a factor of 29. A C-channel and a square tube of the same weight behave very differently in torsion.
What Stabileo does with this
Stabileo shows all three, each with its formula, its terms and its value. The ones that do not apply are shown too, with the reason. And when two are valid for the same section, it shows the difference between them instead of quietly picking one.
The centroid, the shear centre and the kern are handled the same way: computed step by step, in full view, on the actual polygon of the section rather than from a formula for each shape.
In short
- Does the wall form a closed circuit? Bredt, with Aₘ measured on the mid-line.
- Is it open? Saint-Venant, and thickness enters cubed: the thickest wall governs.
- Is it circular? Cauchy, and it is exact. If you also use Bredt, expect it to come out low.
- Is warping restrained? Then Saint-Venant alone is not enough.
Either way, record which theory you used alongside the value.