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In this scenario, an underwater city is a collection of domes interconnected by tunnels 17 feet wide and 17 tall. The largest of the city's domes is 71,000 square feet in area and 137 feet tall. The city is situated 500 feet beneath the surface.

When you descend deeper into water, it becomes imperative that you shrink the size of the glass windows to make sure that they don't crack under pressure.

In a dome of that size at 500 feet below sea level, how big and how thick must each window be?

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    $\begingroup$ stupid question, but -- why would you want windows down there? there's no sunlight anymore at such depth, so you'd just see blackness. I think your inhabitants would be a lot more satisfied with having good-quality screens on the wall where they can display whatever nature they want $\endgroup$ – subrunner Jun 8 '16 at 5:36
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    $\begingroup$ What you've missed is that the interface between the window and wall is more significant than the size of the window. $\endgroup$ – Separatrix Jun 8 '16 at 7:40
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    $\begingroup$ @Aify we have tags specifically for designing structures, this question is fine for the site. $\endgroup$ – James Jun 8 '16 at 13:40
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    $\begingroup$ @Aify You are assuming the user has a certain level of engineering knowledge that he may or may not have. (I personally wouldn't know where to even begin answering this) None of us are an expert in all the topics on WB. A glass domed habitat under the ocean is a perfectly acceptable (and not wholly uncommon) setting for us to build. The depth details the user selected are his business and the answers it has generated could well be used by future people building an underwater dome to calculate what they need. $\endgroup$ – James Jun 8 '16 at 18:04
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    $\begingroup$ @subrunner Well if the domes have lights on the outside, you could still see quite easily whatever they light up, $\endgroup$ – Ryan Jun 9 '16 at 15:55
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The main issue here is the geometry of your windows; while 500' below the surface isn't that far, it's still a considerable amount of force over a large area. Your city, from a quick-and-dirty engineering standpoint, is just a series of pressure vessels.

So let's say you want a comfortable 1 atm of pressure inside the city. In reality, you'd want a higher pressure inside, to help counter-act the pressure on the outside. According to various groups, it looks like divers (breathing air) should only go down to 50-60 m (~164'-196'), somewhere around 1/3 the depth of your city.

Engineering toolbox has a nifty thick-walled pressure vessel stress calculator. If any of these stresses exceed the stresses of your "glass," your connecting tubes will break! Oh, we should also mention that there are different types of glass: borosilicate glass (pyrex), common fiberglass (E-Glass), S-glass, soda-lime, even Star-Trek inspired magnesium aluminate, and many others! Each of these have their benefits and weaknesses. Let's assume you're using a 'safety glass', which is supposed to withstand 15000 psi of pressure. Using the tool from engineering toolbox, the minimum thickness allowed is: 4 inches (otherwise the hoop stress is too much). This has no safety factors involved, so you likely want this thickness to be much higher, at least 8 inches.

You can perform a similar calculation for your domes.

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  • $\begingroup$ 500 feet is almost exactly 150 meters, which is three times 50-60 meters. Did you confuse a unit somewhere? $\endgroup$ – a CVn Jun 8 '16 at 19:52
  • $\begingroup$ @MichaelKjörling Yes, and I fixed it. $\endgroup$ – PipperChip Jun 8 '16 at 19:53
  • $\begingroup$ Don't forget transparent aluminum! First proposed in Star Trek, it is now a real thing: magnesium aluminate. Much stronger and harder than glass. $\endgroup$ – Paul Chernoch Jun 8 '16 at 22:41
  • $\begingroup$ @PaulChernoch I added it, but I literally can't add every transparent material. The purpose of mentioning the many types of glass is to show that merely saying "glass" could mean one of many materials. $\endgroup$ – PipperChip Jun 9 '16 at 15:33
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For every 33 feet deeper in the water, the pressure increases by about 14.5 psi [reference]. Also, the atmospheric pressure at sea level is about 14.7 psi [reference].

At 500 feet below sea level, the pressure would be

$$ \left(\frac{500}{33} \times 14.5\right) + 14.7 = 234.39~\text{psi}$$

Fully tempered glass in the US is generally rated above 65 megapascals (9,400 psi) in pressure-resistance, while heat-strengthened glass is between 40 and 55 megapascals (5,800 and 8,000 psi) - Reference

As incredible as it seems, apparently, just a couple of inches should be easily strong enough.

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  • $\begingroup$ Until that one person gets drunk and throws something at it :-P $\endgroup$ – AndreiROM Jun 8 '16 at 13:40
  • $\begingroup$ A couple of inches long, wide, thick, or two inches in all of the above? $\endgroup$ – JohnWDailey Jun 8 '16 at 13:41
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    $\begingroup$ You've confused compressive strength with structural. Your calculation shows only that, for very small, thick windows the glass will not crush to powder. By analogy, consider that concrete has typical compressive strength in the 2,000 - 5,000 psi range, and 15 psi overpressures are considered adequate to destroy any concrete building other than a bunker. $\endgroup$ – WhatRoughBeast Jun 8 '16 at 14:29
  • $\begingroup$ @WhatRoughBeast: Yes I was expecting something like that. I tried searching the internet for pressure versus thickness x area type of unit for glass but there just wasn't any information on that.\ $\endgroup$ – Youstay Igo Jun 8 '16 at 15:23
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The simpler solution would be to not keep the pressure in your city the same as the surface pressure, that way the difference between the water pressure and the air pressure would be minimal. The pressure difference between the upper areas (+137ft) and lower areas will be more pronounced than at the surface, but should be fine.

You cannot breathe a regular air mixture either, 500' is past the point of oxygen being toxic, and nitrogen will be causing problems too. There are a number of mixtures with helium that would likely be recommended.

The main reason you don't want to keep surface pressure is that any leak is going to cause a big problem FAST.

Granted, people coming and going will need to deal with decompression etc.

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