Analyse the same thin-walled square tube two ways — as a closed section (an intact tube) and as an open section (the same tube with a longitudinal slit). Adjust the size, wall thickness and torque to see the torsion constants, twist angles and stiffness ratio update in real time, and feel why a closed box beats an open one by a huge margin.
Parameters
Section size (mid-line) b
mm
Mid-line side length of the square section
Wall thickness t
mm
The thinner the wall, the bigger the open-closed gap
Applied torque T
N·m
Shear modulus G
GPa
Modulus of rigidity. About 79 GPa for steel
Member length L
m
Results
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Closed torsion constant J_closed (mm⁴)
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Open torsion constant J_open (mm⁴)
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Stiffness ratio J_closed/J_open
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Closed-section twist (deg)
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Open-section twist (deg)
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Section-type verdict
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Closed vs open — shear flow and twist animation
Left: the closed section (an intact tube); right: the open section (a tube with a longitudinal slit). The closed section's shear flow circulates around the wall and it barely twists. In the open section the loop is broken, the shear reverses in each wall, and it twists far more.
Torsional stiffness ratio vs width-to-thickness b/t
Closed-section torsion constant (Bredt's formula, with enclosed mid-line area A_m=b² and perimeter p=4b) and open-section torsion constant (sum of thin rectangles). The closed-section torsion constant exceeds the open-section one by a factor of about 0.75·(b/t)².
Twist angle φ (T: torque, L: member length, G: shear modulus, J: torsion constant) and the closed-section shear stress τ (Bredt's formula).
What is Thin-Walled Torsion?
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What is special about "thin-walled torsion"? Isn't it just twisting a bar?
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In short: whether the cross-section has a hole through it — whether the loop is closed or open — changes the torsional strength by orders of magnitude. For a solid round bar your intuition works fine. But for tubes and shapes built from thin walls, whether the section is "closed" (a box or a pipe) or "open" (an I-shape, a channel, a slit tube) changes the torsional stiffness by hundreds of times. That is the most interesting — and most important — point of thin-walled torsion.
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Hundreds of times?! The very same thin square tube can differ that much?
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It can. Keep the sliders at section size b=100mm and wall thickness t=5mm on the left. The closed-section torsion constant is 5.0×10⁶ mm⁴, but for the open section — the same tube with one longitudinal slit — it falls to 16,667 mm⁴. The stiffness ratio is exactly 300. Picture a drinks can: an intact can hardly twists, but cut one lengthwise groove with a can opener and you can twist it with your fingers. That is the closed-to-open change itself.
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Why does just one cut make it so weak?
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The key is the "shear flow". In a closed section, the shear stress produced by the twist circulates all the way around the closed loop of the wall. That circulating flow generates a large resisting torque with only a modest, nearly uniform stress — that is the world of Bredt's formula τ=T/(2A_m·t). Cut a slit and the loop is broken, so the shear flow can no longer make a full lap. Each wall then has to twist independently as a flimsy flat strip, and the shear stress merely reverses across the small thickness. That makes the resistance orders of magnitude smaller.
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So wherever there is torsion, you should always use a closed section?
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In practice, that is exactly what is done. Car frames, aircraft wing boxes, bridge box girders, bicycle frames, drive shafts — almost everywhere torsion matters, the section is a hollow box or tube. Open sections such as I-beams and channels, by contrast, are excellent in bending but notoriously poor in torsion. So when an open section sits where twisting loads act, it must be handled with care — stiffened, or paired into a closed cell.
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How does "warping", in the title, relate to all this?
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Good question. Warping is when a twisted cross-section does not stay plane and rotate as a flat face, but deforms with out-of-plane bumps along the axis. Open sections such as I-beams can actually carry part of the torque through "warping torsion" — the resistance generated when that warping is restrained. So a real open section is a little better than the pure Saint-Venant torsion this tool considers. Even so, the conclusion that the closed section wins by a wide margin holds firm. Learn that big gap with this tool first, then add warping torsion in detailed design.
Frequently Asked Questions
For a thin-walled square tube, the closed-section torsion constant is J_closed = b³·t and the open-section torsion constant is J_open = (1/3)·Σ(b·t³), so their ratio is J_closed/J_open = 0.75·(b/t)². For a mid-line side b=100mm and wall thickness t=5mm the stiffness ratio is 0.75·(100/5)² = 300 — the closed section is 300 times stiffer than the open one. The larger the width-to-thickness ratio b/t (the thinner the wall), the more steeply this gap widens, and a factor of several hundred is normal.
In a closed section the shear stress forms a continuous shear flow that circulates all the way around the closed loop of the wall. Described by Bredt's formula τ = T/(2·A_m·t), this circulating shear flow generates a large resisting torque with only a modest, nearly uniform stress. In an open section the loop is broken, so the shear can no longer circulate around the perimeter; each thin wall twists almost independently as a flat strip, and the torsional resistance collapses.
Cutting a single longitudinal slit breaks the closed loop of the shear flow. Shear stress can no longer circulate around the perimeter, so the section behaves like a collection of flimsy flat strips, and the torsion constant drops from b³·t down to (4/3)·b·t³. The reduction is a factor of 0.75·(b/t)² — hundreds of times for a typical thin wall. A drinks can that twists easily once you cut it lengthwise is the everyday version of this effect.
Warping is the phenomenon in which the cross-section of a twisted member does not stay plane but deforms with out-of-plane bumps along the axis. Open sections such as I-beams and channels carry part of the torque not only by Saint-Venant torsion but also by warping torsion (the resistance generated when warping is restrained). This tool compares only the Saint-Venant torsion constants, but the conclusion that open sections are poor in torsion still holds. A detailed design also considers the warping constant I_w.
Real-World Applications
Automotive body and chassis: Torsional stiffness of the body shell strongly affects handling, ride comfort and crash safety. Side members and cross members use closed-section box tubing, and areas that take large torsion — wheel houses, suspension mounting points — are kept as closed cells wherever possible. Where a door opening structurally "opens" the section, stiffeners and ring frames make up for the drop in torsional stiffness.
Aircraft wing box: The wing carries its torsion load (the aerodynamic pitching moment from the lift distribution) through the "wing box", a closed section enclosed by the front and rear spars and the upper and lower skins. If that box were an open section, the wing would twist in flight and lose aerodynamic control. Even access panels are designed to keep the section closed — for example by thickening the opening edges so the torsion path is not cut.
Bridge box girders: Curved bridges and bridges under eccentric loads develop a torsional moment in the girder. Because I-girders (open sections) are weak in torsion, torsion-dominated bridges use steel box girders or concrete box girders (closed sections). A single box girder carries both torsion and bending efficiently — the major advantage of a closed section.
Machine elements and everyday structures: Drive shafts and propeller shafts are hollow tubes (closed sections) precisely to secure torsional stiffness and strength at low weight. Bicycle frame tubes, scaffolding poles and conveyor roller shafts use tubes for the same reason. By contrast, an L-shaped angle on a shelf or a channel-shaped rail is meant for bending, and you must remember it is weak in any direction that applies torsion.
Common Misconceptions and Pitfalls
The biggest misconception is assuming that if the cross-sectional area (or the second moment of area) is the same, the torsional strength is the same too. In thin-walled torsion, two sections with exactly the same area can differ in torsion constant by hundreds of times depending on whether the loop is closed or open. An I-beam and a box tube with the same bending stiffness (I) have completely different torsional stiffnesses (J). "Strong in bending = strong in torsion" simply does not hold for thin-walled sections. Evaluate bending and torsion separately.
Next, applying this tool's J_closed = b³·t directly to a thick-walled section. Bredt's formula and J = b³·t are approximations that assume the wall thickness t is small compared with the section size b — the "thin-walled" assumption. For a thick box where the width-to-thickness ratio b/t falls below roughly 10, the assumption that the stress is uniform through the thickness breaks down, and the real torsion constant drifts away from the thin-wall value. For thick-walled or solid sections, use the dedicated formulas or FEM. This tool is appropriate in the thin-walled regime (b/t ≳ 10).
Finally, assuming that the torsional stiffness of an open section is fixed by the Saint-Venant torsion constant alone. A real open-section member also carries torque through "warping torsion" when end restraints prevent warping. A short open section with firmly fixed ends can be stronger in torsion than this tool's J_open suggests. Conversely, for a long open section the warping-torsion contribution is small and the behaviour approaches this tool's comparison. When you work with open sections, remember that a full analysis must include the member length, the end restraints and the warping constant I_w.
How to Use
Enter the square tube's outer dimension (bNum, range 20–200 mm) and wall thickness (tNum, range 1–10 mm)
Set the applied torque (trNum, range 10–5000 N·m) and shear modulus G (gNum, range 70–85 GPa for aluminium, 79–82 GPa for steel)
The simulator calculates J_closed using the closed thin-walled formula (4A²t/∮ds) and J_open using Σ(bt³/3), then reports twist angle in degrees and the stiffness ratio to reveal how much torsional rigidity is lost when the tube wall opens
Worked Example
Square aluminium tube: outer side b = 60 mm, wall thickness t = 2 mm, torque T = 200 N·m, G = 72 GPa. Closed section: enclosed area A = 56² = 3136 mm², J_closed = 4(3136)²(2)/240 ≈ 262,700 mm⁴, twist ≈ 0.018°. Open section (same tube slit longitudinally): J_open = 4(60)(2³)/3 ≈ 3,200 mm⁴, twist ≈ 1.44°. Stiffness ratio = 82:1, showing closed sections dominate torsional performance in aerospace and automotive chassis applications.
Practical Notes
A slit or discontinuity in the tube wall reduces J by 80–95%; always verify welds and seams in drive shafts and hollow structural members
Thicker walls yield cubic gains in J_open but only linear gains in J_closed, making closed sections inherently superior for high-torque transmission
Use G = 80 GPa for steel, 72 GPa for 6061-T6 aluminium, and 45 GPa for titanium alloys to match material datasheets