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So, humans have colonised this planet. It is of Earth-mass and volume and so has Earth-like gravity. They are now building a dome to accommodate the population. The question is, how big can they make it?

Background info:

The dome is made from three-inch thick panes of synthetic diamond (built by nanotech). There are two layers of these panes, one on top and the other beneath a steel latticework. This double-layer design is to retain heat, rather like a gigantic thermos. There are no supporting pillars or struts.

What is the maximum size a dome of this nature could theoretically reach?

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    $\begingroup$ Diamond, one of the best thermal conductors, five times better than copper, to retain heat? $\endgroup$
    – L.Dutch
    Commented Jan 27, 2023 at 9:23
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    $\begingroup$ Hence the two layers to form an air pocket. $\endgroup$
    – user98816
    Commented Jan 27, 2023 at 9:40
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    $\begingroup$ There are multiple dome-like projects like these in the world currently. But for inspiration you could look into the Eden Project in Cornwall, UK. $\endgroup$
    – Plutian
    Commented Jan 27, 2023 at 10:59
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    $\begingroup$ This isn't a useful question. The amount of material brought by the colonists isn't infinite and the amount they bring plus their manufacturing capacity compared to the time they have to solve the problem will create a fairly finite solution. But more to the point, what's stopping you from picking a number? The largest public dome on Earth is the Tacoma Dome at 530' diameter, readily allowing for four stories of construction inside. How much room do you need? What are all the other building materials? $\endgroup$
    – JBH
    Commented Jan 27, 2023 at 21:18
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    $\begingroup$ Why a single dome? If the dome is ginormous most of it is wasted space. $\endgroup$
    – Daron
    Commented Jan 28, 2023 at 16:51

4 Answers 4

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There is no theoretical limit based on loading/stress. Geodesic domes get stronger the bigger they get. There may be other limiting factors such as:

  • Availability of materials, though if it's just carbon this won't be a very limiting limit.

  • The lay of the land – whether there a river, canyon, bog that makes it hard to build there

If you are writing a civilisation with diamondoid nanoassembly, that's saying they have extraordinarily high technology. A large dome isn't going to stretch credibility further than making diamonds out of dirt.

The steel will act as a 'thermal bridge' conducting heat out, like metal bolts do in a mundane roof.

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    $\begingroup$ Thanks for this, also +1 upvote for the comment on the steel. $\endgroup$
    – user98816
    Commented Jan 27, 2023 at 10:22
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    $\begingroup$ Already gave you thumbs-up. You could make it better by mentioning air pressure as additional support. $\endgroup$
    – Boba Fit
    Commented Jan 27, 2023 at 13:33
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    $\begingroup$ "There is no theoretical limit..." yes, there is. As the dome size expands, the the weight of the building materials plus any force from wind outside will lead to collapse. As the dome increases, that upper arc flattens out. The limits have everything to do with the building materials. @user98816 provided one of two required materials (the materials used to build the triangles), but did not provide the connecting material. No material is unbreakable, so there most certainly is a theoretical (indeed, a very practical) limit because, no, geodesic domes don't get stronger as they get bigger. $\endgroup$
    – JBH
    Commented Jan 27, 2023 at 21:14
  • $\begingroup$ By "stronger as they get bigger" they probably meant that the strength to weight ratio goes up with the size of the dome (it does approach a limit, so that doesn't violate basic principles). However, I think OP is confused, assuming most of the strength comes from the panes, while in reality it comes from the struts. And the weight of the panes will go up with the surface area of the sphere. The domes strength refers purely to an unclad dome. $\endgroup$
    – IronEagle
    Commented Feb 4, 2023 at 1:08
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This calculation might not be 100% accurate, but let's give it a try:

The maximum size of the dome would be limited by the strength of the material used to construct it. In this case, the dome is made of synthetic diamond, which has a high strength-to-weight ratio and can withstand significant loads.

However, even with synthetic diamond, there would still be limits to the size of the dome due to the laws of physics and the weight of the dome itself. The final size would depend on factors such as the dimensions of the panes, the strength of the steel latticework, and the load that the dome would need to support, such as the weight of the atmosphere inside the dome.

Let's assume the dome has a height of "h" and a radius of "r". The volume of the dome can be calculated as:

V = (2/3)πr^2h

The mass of the dome can be calculated as the product of its volume and the density of synthetic diamond, which is approximately 3.52 g/cm^3:

m = ρV = (3.52 g/cm^3)(2/3)πr^2h

The compressive strength of synthetic diamond is approximately 4.5 GPa. The compressive force on the lowermost layer of diamond blocks can be calculated as:

F = m * g

where "g" is the acceleration due to gravity (9.8 m/s^2 on Earth). The compressive stress on the lowermost layer of blocks can be calculated as:

σ = F / A

where "A" is the cross-sectional area of the lowermost layer of blocks. If this stress exceeds the compressive strength of the diamond, the blocks will fail. Setting the compressive stress equal to the compressive strength and solving for the mass "m", we get:

m = F / (σ / ρ) = (4.5 GPa) * A / (g / ρ)

Finally, we can use the volume equation to calculate the maximum height "h" of the dome:

h = (3m / (2πr^2ρ)) = (3 * 4.5 GPa * A / (2π * r^2 * g * ρ^2))

Note that this is a rough estimate and the actual size of the dome would depend on many other factors, such as the distribution of stress and the presence of any internal loads or stresses.

To get the maximum size of the dome in meters, we'll need to plug in numerical values for the variables. Let's assume a compressive strength of 4.5 GPa and a density of 3.52 g/cm^3 for the synthetic diamond, and an acceleration due to gravity of 9.8 m/s^2.

Using the formula we derived above:

h = (3 * 4.5 * 10^9 Pa * A) / (2 * pi * r^2 * 9.8 * 3.52^2 g/cm^3)

We can simplify this further by assuming a certain cross-sectional area "A" for the blocks. For example, if we assume each block has a square cross-section with sides of length "s", then:

A = s^2

Substituting this into the formula and simplifying:

h = (3 * 4.5 * 10^9 * s^2) / (2 * pi * r^2 * 9.8 * 3.52^2)

This equation can be solved for the maximum height of the dome for a given radius "r" and block size "s". The maximum size in meters will be the maximum height "h" for a given radius "r".

Plugging in the values of r = 10000 m and s = 10 m into the formula:

h = (3 * 4.5 * 10^9 * 10^2) / (2 * pi * 10000^2 * 9.8 * 3.52^2)

h = (3 * 4.5 * 10^11) / (2 * pi * 10^8 * 9.8 * 3.52^2)

h ≈ 473.9 m

So for the given values of r = 10000 m and s = 10 m, the maximum height of the dome would be approximately 473.9 meters.

----------------------EDIT-----------------------------

For a hollow dome it's a bit more tricky to exactly determine a maximum height but we can try:

Formula to estimate the maximum height of a hollow dome:

h = (F_total * t) / (pi * r^2 * E * A)

Where:

h is the maximum height of the dome

F_total is the total force acting on the dome ( total compressive force that the dome is designed to support )

t is the thickness of the dome's walls

r is the radius of the dome

E is the Young's modulus of the material used for the dome's walls

A is the cross-sectional area of the dome's walls

Note that this is a simplified formula that assumes the dome's walls are uniform in thickness and composition and that the load is evenly distributed over the entire structure.

The maximum height of a hollow dome would likely be limited by the weakest point in the structure, which could be influenced by a number of factors such as the composition and thickness of the walls, the size and shape of the interior space, and any internal loads or stresses. This formula is intended to provide a rough estimate of the maximum height of a hollow dome, and a more detailed analysis would be necessary to obtain a more accurate result.

There are several factors that limit the maximum height of a hollow dome, including:

  1. Strength of the material used for the dome's walls: The walls of the dome must be strong enough to support the weight of the dome and any additional loads it may experience, such as wind, snow, or earthquakes. The maximum height of the dome will be limited by the strength of the material used for the walls.

  2. Cross-sectional area of the dome's walls: The cross-sectional area of the walls will affect the amount of stress that can be borne by the walls. A larger cross-sectional area will result in lower stress and a higher maximum height.

  3. Internal loads and stresses: The dome may experience internal loads and stresses due to changes in temperature, humidity, and pressure. These loads and stresses will affect the maximum height of the dome.

  4. Shape and size of the interior space: The size and shape of the interior space will affect the distribution of loads and stresses within the dome.

  5. Design and construction: The design and construction of the dome will play a role in determining the maximum height. Factors such as the thickness of the walls, the size and shape of the interior space, and the presence of any internal loads or stresses will all influence the maximum height.

Let's assume that the synthetic diamond has the same resistance as a natural one and the walls are three-inch thick as OP mentioned ( the thickness will limit the height a lot, if we could increase this value we could achieve higher heights )

A, the cross-sectional area of the dome, can be calculated as follows:

A = (pi * t^2) / 4

where t is the thickness of the dome wall (3 inches or approximately 0.0762 meters). Plugging this value into the equation, we find that

A = (pi * 0.0762^2) / 4 = 0.00451 m^2.

Since we can't know for sure the exact value for F_total in this example, I'm assuming values comparing to the world's tallest building, which could be in the range of millions of tons ( it's built to resist wind loads, seismic forces and the weight of its structure )

Assuming F_total as 100.000.000 tons ( had to go this high because of the thickness of the walls, may be a good idea to increased the thickness )

Using the formula

h = (F_total * t) / (pi * r^2 * E * A)

With the following values:

F_total = 100,000,000 tons = 100,000,000 * 10^6 g

t = 3 inches = 0.0762 m

r = 10,000 m

E = Young's modulus of diamond = 1,000,000 N/m^2

A = cross-sectional area = pi * r^2 = pi * 10,000^2 = 314,159,265 m^2

Plugging in these values, we get:

h = (100,000,000 * 10^6 g * 0.0762 m) / (pi * 10,000^2 m^2 * 1,000,000 N/m^2 * 314,159,265 m^2)

h = approximately 213.82 m

BUT, if we can't get our F_total ( total compressive force that the dome is designed to support ) that high , our height is drastically reduced.

Example:

Using F_total=1.000 tons and other values as follows:

t=0.0762m (thickness of the diamond panes, which is three inches)

r=10000m (radius of the dome)

E=1050GPa (Young's modulus of natural diamond)

A=1 (assumed area)

we can calculate the maximum height of the dome as follows:

h = (F_total * t) / (pi * r^2 * E * A)

h = (1.000 * 0.0762) / (pi * 10000^2 * 1050 * 1)

h = 0.114m

So, with these assumptions, the maximum height of the dome would be approximately 0.114m.

Well, I don't think we can go further than this in especulations and calculations, I hope it helped in some way!

Also is worth saying that i'm not a mathematician or physics specialist, some things may not be 100% accurate.

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  • $\begingroup$ Did you calculate volume for a solid dome? It would be pretty but probably they will want a hollow one. $\endgroup$
    – Willk
    Commented Feb 1, 2023 at 19:06
  • $\begingroup$ Yeah I think I calculated it as a solid dome not a hollow one, well, it's the closest I could get on calculations haha $\endgroup$
    – Archerspk
    Commented Feb 1, 2023 at 20:43
  • $\begingroup$ I gotta think a solid dome would be heavier! You could calculate 2 domes, inner and outer differing in diameter by the thickness of the wall, and subtract the inner from the outer. This would allow a dome of more hugeness. $\endgroup$
    – Willk
    Commented Feb 1, 2023 at 20:52
  • $\begingroup$ I tried calculating things for a hollow dome, don't know for sure if it's 100% correct, but I think it's our best shot right now, if anyone find something wrong please tell me and I'll update the answer with correct details. $\endgroup$
    – Archerspk
    Commented Feb 2, 2023 at 12:27
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    $\begingroup$ I think the tiny dome might be from choosing A=1 in second example. You did walk thru the math which is what I wanted and your 213 meter dome seems plausible for a scifi diamond dome. Bounty awarded! $\endgroup$
    – Willk
    Commented Feb 5, 2023 at 15:16
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There was a question a few weeks ago about the size of a dome on the moon.

The limits for us are probably set by economics and risk analysis rather than by engineering. If you have a giant dome, and a steady supply of raw materials, then the time taken to build a dome will go as (roughly) the square of the dome area. The risks of the dome being penetrated by a meteorite or some other random event go as the square of the area. if all the air escapes, the costs of the damage might go as the square of the area. We are already up to the sixth power of radius, and we could go higher. The advantages of building many small domes are enormous.

You might design one a mile across to win an architectural award, and let your client pick up the tab. This is not entirely silly - think of the glass cupola of the Albert Hall in London.

For more info, ask Tom Scott

https://londonist.com/london/secret/behind-the-scenes-at-the-royal-albert-hall-part-2

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One key parameter is the atmospheric pressure. If it's significantly less than on Earth the mass of the dome might partially be supported by internal pressure. If the pressure is very low like on Mars the entire dome could be supported. One big issue with a dome on Mars is containing the pressure, but in a truly massive dome 15lb/sqin could support a lot of heavy structural elements.

If supported by internal pressure in this way there need not be any limit to the size of the dome based on the mass of the structure of the internal pressure.

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