3279: Main Span

Main Span

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Title text: Wind stress? Don't be silly. When has wind stress ever been a problem for a suspension bridge?

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Explanation

The "main span" of a bridge refers to the longest section between two support towers. Maximizing this is often a useful feature of a design because it can reduce the engineering needed to build the support towers and provide a wider uninterrupted navigable space below the bridge. Typically, the main span of a suspension bridge has only suspension cables to hold up that section of the bridge; the largest such span to date is just over 2000 m long, though there are plans for spans up to 3300 m. In the design depicted, a spherical, lighter-than-air balloon is used to provide an upward force in order to hold up this suspension bridge, allowing the main span to exceed the theoretical maximum length (estimated in the comic as 3–4 km) that would be stable under the forces of tension and compression alone. It appears that the design essentially uses the balloon as a replacement for a central support tower (perhaps because the water there is so deep), providing the same function of holding up the bridge.

The diagram has a number of unreadable annotations, but there is an equation for the diameter of the balloon which is likely:

$d=\sqrt[3]{\frac{6n}{\pi }\frac{(M_{1}l+M_2)}{({\rho }_c-{\rho }_h)}+S_1}$

This is the formula for the diameter of a sphere, $d=\sqrt[3]{(6/\pi )V}$, combined with some other terms: $M_1$ is likely the mass per length of span, $l$ would be length of the span held up by the balloon, $M_2$ would be another contributor to the mass (such as the weight of the balloon and its hardware), and ${\rho }_c$ and ${\rho }_h$ would be the densities of cold and hot air, respectively. Dividing the total mass by the difference in densities gives you the volume $V$ needed inside the balloon to provide the buoyancy necessary to counter the weight of the bridge. The multiplier $n$ and the added value $S_1$ might be safety factors, or $n$ might be for the case of multiple spans held up by the balloon.

Using $M_1=20$ tons/m (estimated value for a long suspension bridge), $l=4$ km, ${\rho }_c$ and ${\rho }_h$ for dry air at 20°C (room temperature) and 95°C (typical in a hot-air balloon), negligible balloon mass $M_2$, and negligible safety factors $S_1$ and $n$ (we're on xkcd!), the balloon would need to be approximately 850 m in diameter — nearly a kilometer.

The title text is a reference to the Tacoma Narrows Bridge from 1940. There's a well-known video showing it collapsing as a result of wind stresses that matched the bridge's natural resonance frequency. This is often used in physics classes to demonstrate resonance, and in engineering classes as a cautionary tale. Note, however, that in that case it might technically have been more aeroelastic fluttering than resonance. In the comic scenario, though, the suspicion is that the wind is catching the much larger surface area of the balloon and tugging it out of position, resulting in stress to the bridge.

While using a single balloon to support a structure at this scale would be incredibly impractical, if not outright impossible, designs that are similar to a degree have been proposed and built. Most similarly, Pont de Singe (Monkey Bridge) was an art installation near Manchester, UK, where a short bridge was suspended by helium balloons, although guests were not allowed to use it. Additionally, pontoon bridges, which float on water rather than in air, are somewhat common. More speculatively, active support, using active methods rather than passive strength to support structures too large to exist otherwise, has been proposed in speculative settings, but no major structure has been built with active support.^{[actual citation needed]}

Transcript

[A diagram of a suspension bridge across a cross-section of a body of water is shown. The bridge has three spans. The left and right spans are held up using normal pillars that bury into the ground under the shallow portions of the water. The middle span has no pillar, and is held up by a large balloon tethered to the top of the tower by about a dozen cables. There are two-headed arrows above the main cables to the left and right of the left tower, to the left of the middle tower, and to the left and right of the cables from the balloon to the middle tower. There are mostly unreadable annotations to the right of the left tower (diagonally above the arrows), to the left of the middle tower, and to the left and right of the balloon. Above the balloon is a line segment whose length matches the width of the balloon with the annotation "d" in the middle. To the right of the balloon is an equation for d, written in small gray text.]

[Caption below diagram:]

How to get past the 3-4 km limit on suspension bridge main spans

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Discussion

Surely anchor lines from the balloon to the sea floor perpendicular to the bridge would sort the wind issue. Drummertv (talk) 16:27, 31 July 2026 (UTC)

I added a transcript, but I left the d=... formula vague. If someone can fill it in that would help. We don't use MathJax in transcriptions, do we? Barmar (talk) 16:30, 31 July 2026 (UTC)

I read this with my brother, and as soon as we saw the title text I cracked up (we live in Tacoma.) RadiantRainwing (talk) 18:40, 31 July 2026 (UTC)

Ah... So it resonated with you! 92.23.10.248 20:02, 31 July 2026 (UTC)

(Imaginary upvote) RadiantRainwing (talk) 23:03, 31 July 2026 (UTC)

I saw that video three times in three separate classes in uni. Fephisto (talk) 14:35, 3 August 2026 (UTC)

I didn't...but then literature and linguistics don't really provide much opportunity for that. I saw it at school though – I'm pretty certain it was in music when we were talking about resonant frequencies and how they interfere with each other, but the same teacher taught me geography, so there's a small chance it was to do with that instead. Weird crossover of relevance that I never considered before. Yorkshire Pudding (talk) 16:58, 3 August 2026 (UTC)

Strange that the title text is specifically "Wind stress," which is specific to the wind interacting with the surface of the water rather than the other stresses such as pulling on that balloon or of course Tacoma Narrows. 99.70.40.149 (talk) 20:45, 31 July 2026 (please sign your comments with ~~~~)

Bridges are extremely heavy. I'm actually curious just how absurdly huge a balloon would have to be to support that much weight. It'd almost certainly be bigger than in the comic. 206.193.5.5 21:35, 31 July 2026 (UTC)

I just logged in to ask the same question. How many Hindenburgs would this require? I'm assuming that upsidaisium and unobtanium remain synonyms, and hence unavailable for the project. 205.175.118.2 21:58, 31 July 2026 (UTC)

Let the total mass of the span to be supported be 100,000 tonnes - I read that the current maximum is around 90,000 tonnes. Let the mass to be supported by balloon be 10,000 tonnes, with the remaining 90,000 tonnes to be supported by cables. Let the balloon design to support the bridge be that of the dirigible Hindenburg, containing 200,000 cubic meters of hydrogen. 833 cubic meters of hydrogen is needed to lift 1 tonne; 1 Hindenburg can lift 240 tonnes. To lift the specified span, 42 Hindenburgs containing hydrogen are required; if the lifting gas is helium, 46 Hindenburgs will be required. 2605:59C8:160:DB08:B038:6886:5D34:C618 01:08, 1 August 2026 (UTC)

If the load on the balloon is 50,000 tonnes (more in keeping with the scenario as shown), you'd need 208 (hydrogen) - 229 (helium) Hindenburgs. 2605:59C8:160:DB08:B038:6886:5D34:C618 13:22, 1 August 2026 (UTC)

If the length of the bridge deck be set to 5 km, I make out the balloon's radius to be ca. 350 m. A sphere with that radius has a volume of 1.8 X 10^8 cubic meters - or about 900 Hindenburgs of lifting gas at 200,000 m^3 per dirigible. If this is correct, the scale of the balloon in the cartoon is accurate; if anything, it's overengineered. 2605:59C8:160:DB08:A820:FB4F:7559:1B42 09:56, 2 August 2026 (UTC)

Is there really is a theoretical limit? I could not find any source for that claim. 200.86.125.34 00:33, 1 August 2026 (UTC)

Hang a cable (rope, chain, etc) vertically. Hang more. At some length the cable will rip from its own weight. The maximum length before failure is hardly affected by diameter so the maximum length is fairly consistent for a specified material (steel, nylon, cotton). The current winners are expensive synthetic ropes (or spider webs?). IIRC this length, for typical rope-stuff, is something less than a mile or a KM. How to convert from hanging rope to bridge-rope is beyond me. --PRR (talk) 22:46, 1 August 2026 (UTC)

"The maximum length before failure is hardly affected by diameter" - to see that, consider that if you can hang one cable of diameter D, area A = πD^2/4, and make it length L before its own weight breaks it at the top, then you can hang *two* identical cables next to each other. Then just glue them together along their length (using a very light glue), and you have 1 cable of area 2*A. What you *can* do to optimize the cable length is vary the diameter along the length - the top of the cable is carrying the most weight, so it needs the largest cross section, and the bottom of the cable is carrying nothing, so it can have zero cross section. (If you do the math, you'll find that the cable diameter along its length is an exponential function, and it can be infinitely long, if you ignore a few minor details.) 163.116.145.66 17:12, 4 August 2026 (UTC)

This reminds me of some levels in the Polybridge games, in which anchor points are visually attached to the basket of unsuspecting hot air balloons 2600:1014:B090:633:0:15:E4DC:901 00:50, 1 August 2026 (UTC)

I was about to say. My first reaction to this comic (after the scoff) was “Someone’s been playing Poly Bridge again.” HaruruChanDesu (talk) 01:51, 1 August 2026 (UTC)

I ran in to the wiki JUST to mention this exact thing XD

Makes you wonder, are we gonna actually mention this being either intentionally or unintentionally a Poly Bridge reference in the main article? 71.236.133.105 18:34, 1 August 2026 (UTC)

This XKCD gets uploaded right as I begin playing Polybridge again? What are the odds! Tux1 (talk) 06:24, 2 August 2026 (UTC)

At this point, "bridge crime comics that could be read as Poly Bridge references" is big enough to be a category. There's the dangling bridge one, there's the jumps one... 2601:1C2:97C:F550:BC90:66AF:8E4D:9FAE 06:31, 3 August 2026 (UTC)

An alternative to a balloon would be a space elevator. That presumably results in a different set of problems. 87.75.45.201 14:11, 1 August 2026 (UTC)

I am glad someone finally mentioned a very carefully unbalanced space elevator. Maintaining exactly the right connection height and lift force with perturbations is left as an exercise to the reader. 208.198.19.195 14:35, 3 August 2026 (UTC)

That looks like a balloon shaped by some ropes plus its own envelope, and no such balloon I ever saw is as spherical as the depiction in the comic. And yes, I am aware of how pettily pedantic this sounds! It's just that xkcd accuracy standards are so high that I am spoiled. 69.191.176.41 14:52, 3 August 2026 (UTC)

It's a schematic (as evidenced by the annotations), so obviously it's an idealised depiction. 82.13.184.33 15:09, 3 August 2026 (UTC)

To stop leakage of helium, hydrogen, or hot air, an envelope material might be expensive, and a sphere minimizes area per volume. At km-scale area, this might be more cost effective and simpler to design than other shapes. --99.48.45.17 19:32, 6 August 2026 (UTC)

You could search for images of American Civil War balloons. Some of them were quite spherical. Also as noted below shockingly small if you are used to hot air balloons. IIRC there is a replica of the Intrepid flying somewhere. Although the reenactors use He instead of H2. Also they usually don't put shot in the muskets. 76.180.39.133 18:06, 7 August 2026 (UTC)

The balloon diameter would be much smaller if filled with hydrogen. What could go wrong? 192.5.18.7 (talk) 14:53, 4 August 2026 (UTC) (please sign your comments with ~~~~)
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