Thin-Walled Tube in Torsion (Bredt)

stressmass-cost

Verified build 4 relations · 1 identities proven · 3 modeling steps · 6 parity samples

Look at anything that resists twist while trying to be light — driveshafts, bicycle frames, aircraft fuselages, racing-car torsion boxes — and you will find a closed thin-walled tube. The solid shaft taught that the middle of a section barely works in torsion; Bredt’s formulas take the lesson to its limit by deleting the middle entirely:

\tau = \frac{T}{2\,A_m\,t} \qquad \theta = \frac{T\,S\,L}{4\,A_m^2\,G\,t} \qquad S^2 \ge 4\pi A_m

Three things to notice, one per formula:

You are buying enclosed area, not metal. The stress depends on the section only through $A_m$ — the area the wall’s median line encloses — and the wall thickness. Shape is almost irrelevant: circle, box, airfoil, all obey the same two-knob formula, because the shear flow $q = \tau t$ runs the perimeter like current in a loop and its torque is $2qA_m$ for any closed curve. Double the enclosed air, halve the stress. This is why a tube beats a rod at equal mass so decisively.
Stiffness loves area even more. The twist falls with $A_m^2$ — and rises with perimeter $S$ , so for a fixed enclosed area the stiffest shape is the one with the least perimeter: the circle, again. A flattened tube (same $S$ , shrinking $A_m$ ) loses stiffness quadratically, which you can feel by twisting a drinking straw before and after stepping on it.
Mathematics polices the knobs. Not every $(A_m, S)$ pair is a section: the isoperimetric inequality $S^2 \ge 4\pi A_m$ — the circle encloses the most area any perimeter can — is a hard existence boundary, and the widget refuses beyond it. The sim draws a stadium (straight-sided oval) that realizes your exact $A_m$ and $S$ ; the refusal fires at precisely the moment no such shape exists to draw. A validity envelope enforced by a 2,000-year-old theorem.

One warning the formulas whisper: all of this is for closed cells. Slit the tube lengthwise — turn the loop into an open ‘C’ — and the shear flow loses its return path; torsional stiffness collapses by orders of magnitude (the solid-shaft page’s failure notes already promised this). Closedness is not a detail. It is the entire product.

Try it

Configuration

Material

T3, bare flat sheet 0.010-0.128 in. thick, AMS 4037 / AMS-QQ-A-250/4 (MIL-HDBK-5J Table 3.2.3.0(b1), p. 3-71)

Bound properties of 2024-T3 aluminum sheet (bare)
G	4 Msi	typical	mil-hdbk-5j
sigma_y	47 ksi	design min.	mil-hdbk-5j
rho	0.1 lb/inch**3	typical	mil-hdbk-5j

4 materials in the database are not listed here: no published value in our cited sources for every property this THING needs.

Materials modeled here: 2024-T3 aluminum sheet (bare) 304 stainless steel 6061-T6 aluminum 7075-T6 aluminum AISI 1045 medium-carbon steel AISI 4340 low-alloy steel (Ni-Cr-Mo) ASTM A36 structural steel (hot-rolled) C26000 Cartridge Brass (70/30) Ti-6Al-4V

Governing relations

\tau = \frac{T}{2\,A_m\,t}

Assumes: closed single-cell thin-walled section — the shear flow q = τt is constant around the cell, and the torque is its moment, T = 2qA_m (Bredt's first formula); wall thin enough that the stress is uniform through it; A_m and S measured on the wall median line; any closed shape — circle, box, airfoil; the section enters only through A_m and t · Valid while: No closed curve encloses this much area with this little perimeter — the circle is the absolute best (isoperimetric inequality, S² ≥ 4πA_m with equality only for the circle). This section cannot exist; there is nothing to analyze. The wall is no longer thin next to the section's size (t exceeds a fifth of the characteristic radius 2A_m/S) — Bredt's uniform-stress picture is degrading: the true stress varies through the wall, the OUTER fiber carrying roughly t/(2r) more than this mid-wall τ (≈9 % at this boundary), so the displayed τ and SF are becoming optimistic.

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.11 "Thin-Walled Tubes": shear flow, Bredt's torsion formulas τ = T/(2tA_m) and the torsion constant J = 4A_m²t/(∮ds/t), validity limits of the thin-wall approximation.

\theta = \frac{T\,S\,L}{4\,A_m^2\,G\,t}

Assumes: uniform wall thickness (the general form integrates ∮ds/t around the cell); linear elastic; ends free to warp (closed thin tubes warp very little — this is the section family where Saint-Venant torsion is most honest)

\mathrm{SF} = \frac{\sigma_y/2}{\tau}

Assumes: maximum-shear-stress (Tresca) criterion — pure shear in the wall, same convention as the torsion-shaft · Valid while: Wall shear stress exceeds the shear yield strength (σ_y/2, maximum-shear-stress criterion) — the wall has yielded and the elastic shear flow stops being the truth.

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y).

m = \rho\,S\,t\,L

Assumes: thin wall — cross-section area ≈ S·t

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

q = t \tau

1. The modeling step: cut a small rectangle out of the wall and balance it along the tube's axis. The longitudinal shear forces on its two cut edges must cancel, so the product q = τt — the shear FLOW — is the same everywhere around the cell, like current in a loop of wire: where the wall thins, the stress rises to keep q constant. Thin spots are hot spots. — equilibrium of a wall element: shear flow is constant modeling step

T = 2 A_{m} q

2. Sum the moment of that flow about any interior point: T = ∮ q·r⊥ ds = q·∮ r⊥ ds. The leftover integral is pure geometry — r⊥ds is twice the area of the sliver swept from the moment center, so the loop integral counts the enclosed area twice, ∮ r⊥ ds = 2A_m, for ANY closed shape. That is why a torsion tube is rated by the area it encloses, not the metal it contains. (The build cannot do line integrals; the test pipeline evaluates ∮ r⊥ ds for an ellipse and a rectangle and gets 2A both times.) — moment resultant: the swept-area identity modeling step

\tau = \frac{T}{2 A_{m} t}

3. Combine the two: Bredt's first formula. Thickness enters only linearly, and enclosed area does the heavy lifting — double A_m at the same wall (growing the perimeter as the shape demands) and the stress halves. Compare the solid shaft, where stress fights for d³: enclosing air is the cheapest strength there is. — Bredt's first formula

\theta = \frac{L S T}{4 A_{m}^{2} G t}

4. Stiffness by strain energy: U = ∮∫ τ²/(2G) dV = (q²L/2G)·∮ds/t, and Castigliano's θ = ∂U/∂T delivers Bredt's second formula — twist grows with perimeter but falls with the SQUARE of enclosed area. (Re-derived by exactly that energy route in the test pipeline.) — Bredt's second formula, by Castigliano modeling step

How it fails

The widget models a perfect closed cell with uniform thin walls. Real tubes leave the model in several directions:

A slit is a catastrophe, not a detail. Cut the cell open and the constant shear flow has no closed loop to run; the section reverts to open-section torsion, with stiffness falling by factors of hundreds to thousands and stress concentrating at the slit tips. A cracked weld seam in a torsion box is therefore not a 10 % problem — it is a different, far worse THING. Inspectability of seams is a design requirement, not paperwork.
Thin walls buckle in shear. Bredt computes the stress the wall must carry; it says nothing about whether a thin panel can carry it. Past a slenderness limit the wall wrinkles diagonally (shear buckling) well below the shear yield stress — the elegant skins of aircraft get stringers and ribs precisely to break the panels into sizes that won’t. The thinner you exploit Bredt, the sooner stability, not strength, takes over.
Thin spots are hot spots. Shear flow is constant, so stress scales as $1/t$ locally — a wall thinned 30 % by corrosion or by manufacturing tolerance runs 43 % hotter exactly there. Drain holes and fastener lines, which interrupt the flow, concentrate it further.
Warping restraint at the ends. Saint-Venant torsion lets cross-sections warp freely; weld a thin-walled tube to a rigid flange and the restrained end develops longitudinal stresses the free-twist model never sees. Closed circular tubes barely warp (the kindest case); boxes and airfoils warp more, and short boxes between stiff bulkheads can carry a meaningful fraction of the torque through warping stresses.
Multi-cell sections share the flow. A wing box with two or three cells splits the shear flow among them — statically indeterminate, solved by twist compatibility, and out of this page’s single-cell scope (and, for now, this catalog’s: it is exactly the kind of problem the roadmap’s cyclic-solver phase exists for).
Joints interrupt the loop. Bolted or bonded seams must transmit the full shear flow $q = T/2A_m$ as a running load (N per meter of seam) — a number this widget hands you directly, and the first thing to check on any built-up box.

Fixed-Fixed Torsion Shaft (Interior Torque)
A solid circular shaft built into a wall at BOTH ends, with a torque applied at an interior station. Equilibrium gives one equation for the two wall reaction torques; the missing equation is compatibility — the twist at the load point is single-valued — and the build solves the coupled 2×2 system exactly. The larger reaction lands on the SHORTER segment, and the material cancels out.
- stress
- mass-cost
Rectangular Shaft in Torsion (Saint-Venant)
Twist a solid rectangular bar and the shear stress does something the round shaft never does: it peaks at the MIDDLE of the long side and drops to exactly zero at the corners. Two cited coefficients c1, c2 — functions only of the side ratio a/b — set the peak stress and the twist, and an equal-area round shaft beats it on both counts. Why square shafts are a bad deal.
- stress
- mass-cost
Shaft in Torsion (Solid, Circular)
The power-transmission workhorse: twist a solid circular bar and shear stress winds around it. Three material properties drive three different outputs — stiffness (G) sets the twist, strength (σ_y) sets the margin, and the stress itself doesn't care what the shaft is made of at all.
- stress
- torque-power
- mass-cost
Shaft under Combined Bending + Torsion
Real shafts never get to choose: the belt that twists them also bends them. Bending and torsion land on the same surface element, Mohr's circle finds the worst plane, and two failure criteria — Tresca and von Mises — disagree by up to 15 % about how bad it is.
- stress
- mass-cost
Cantilever Beam (End Load)
A beam fixed at one end, loaded at the other — the fruit-fly of structures. One widget shows why stiffness (E) and strength (σ_y) are independent axes: swap steel for titanium and deflection goes UP while the safety factor also goes up.
- stress
- mass-cost
Composite Bar (Core + Sleeve)
A solid core inside a concentric sleeve, bonded between rigid end plates and pushed by a centric axial load. The two materials must stretch together, so the load splits in proportion to each member's axial stiffness A·E — and the build solves that coupled 2×2 share exactly. Swap the sleeve's metal and watch the load migrate to the stiffer member.
- stress
- mass-cost

Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- t L
Impact Loading (Falling Mass, Energy Method)
- m_tube m
- t b
- t d
- t h
- t L
Symmetric Two-Bar Truss
- t d
- t L
- theta alpha
Thermal Assembly (Two-Segment Bar Between Rigid Walls)
- t L_1
- t L_2
Cantilever Beam (End Load)
- t b
- t h
- t L
Circular Plate under Uniform Pressure (Clamped vs Simply Supported)
- t a
- t t
- tau q
- tau sigma_allow
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- t r_i
- t r_o
- t w
Fixed-Fixed Beam (UDL)
- t b
- t h
- t L

+ 25 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.11 "Thin-Walled Tubes": shear flow, Bredt's torsion formulas τ = T/(2tA_m) and the torsion constant J = 4A_m²t/(∮ds/t), validity limits of the thin-wall approximation.
Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y).
The classical isoperimetric inequality — every simple closed plane curve satisfies S² ≥ 4πA, with equality only for the circle. See e.g. Blåsjö, V., "The Isoperimetric Problem", American Mathematical Monthly 112 (2005), 526–566.