Three different formulas give the shear stress from torsion. For one section they can differ by orders of magnitude, and which one applies depends on how the wall is built rather than on its size or its material.
So far, what any textbook says. What is rarely spelt out is what happens when two of the three apply to the same section, because there they do not agree, and the difference always goes the same way.
The three formulas
- Cauchy, τ = T·r / Iₚ. Circular sections only, and there it is exact: the one case where plane sections stay plane. Stress grows linearly with radius, least on the inside and greatest at the outer face.
- Bredt, τ = T / (2·Aₘ·t). For a closed thin wall. The torque is carried by a shear flow circulating around the enclosed area. Aₘ is the area enclosed by the wall’s mid-line, not by its outer face; confusing the two is a common error.
- Saint-Venant, τ = T·t / J with J = (1/3)·Σb·t³. For an open thin wall. With no closed circuit, the flow has to turn back on itself across the thickness, which is why thickness enters cubed.
Where two apply, they disagree
Take a circular tube. Cauchy applies and is exact. Bredt applies too: there is a closed wall and a flow running round it. But Bredt assumes the stress is constant through the thickness, and Cauchy knows it grows with radius. So Bredt reports an average where Cauchy reports the maximum.
| t / rₘ | Example (rₘ = 50 mm) | τ Bredt / τ exact | Bredt lands below 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 % |
In all four cases Bredt lands below. For a tube with a 5 mm wall over a 50 mm mean radius, still thin-walled by the usual rule of thumb, the real stress comes out 4.5% higher than the computed one. With a 10 mm wall, 8.2%.
Bredt’s error does not split either way: it always lands on stresses lower than the real ones.
A 4.5% gap does not stand out in a check. It reads as passing.
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 another order. A 100×100 mm square tube with a 5 mm wall, slit lengthwise, keeps its area, its weight and almost all of its bending inertia.
| Closed | Slit | Factor | |
|---|---|---|---|
| J [mm⁴] | 4,286,875 | 15,833 | 271 |
| τ [MPa] | 11.08 | 315.79 | 29 |
Torsional stiffness falls by a factor of 271 and the stress multiplies by 29. A C-channel and a square tube of the same weight do not behave the same way in torsion.
What Stabileo does with this
It 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 rather than choosing one silently.
The centroid, the shear centre and the core get the same treatment: derived step by step and in view, on the real polygon of the section rather than on a per-shape formula.
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 use Bredt as well, know that you will land low.
- Is warping restrained? Then Saint-Venant alone is not enough.
And in any case, it is worth recording which theory was used alongside the value.