# Vertical City - How High?

While asking some other questions related to vertical cities, I've found out that the cities can't just expand upward to infinity. There is a limit to how high up a tower can go. There were some conflicting opinions on just where that limit was, though, so I decided to make a separate question out of it.

In a near-future, how high could we build vertical cities?

While I'd like this question to be useful to everyone, I do also have a few specific cases for my own writings. I am considering writing a novel set in a vertical city. This city isn't just one tower, but an entire forest of towers, each trying to be above the last. Their bases would be huge (think around five city blocks square or more), and they would have skyways on multiple levels connecting them to each other. (I don't know if this would provide more stability or not - there are a lot of them. They are the primary mode of travel between towers.) Each tower is more or less self-contained; that's where everyone sleeps, eats, and works. The streets below are virtually unused.

Each of the successive tallest buildings of the world has been designed to the limits of current materials and architectural knowledge. As our science advances, that limit will slowly go up.

Thucydides link is very enlightening in that regard. The current record holder, the Burj Khalifa, could probably not have been built 10 years earlier.

As the buildings grow taller, the complexities and especially costs increase. All of the tallest skyscrapers are giant "mine is bigger than yours" projects of hubris built at ridiculous cost. Dubai pretty much went bankrupt trying to build the Burj Khalifa, which now is named after the Emir of Abu Dhabi who financed the rest of the construction.

Super skyscrapers don't make any economic sense and that is probably going to be the biggest limitation in building a city of them. Unless there is really no space left to build, you could house many times more people if the buildings are of a more normal height.

Check out the diagram of the Burj Khalifa here to see just how small the upper half of the building is. Your interconnected walkways would have to be quite long (and stretchy, to deal with the buildings' sway in the wind).

Here's one complication already happening: The ground may sink under the foundations

Frankly, the rectangular mile high blocks built closely together you see in sci-fi movies need serious handwaving or sci-fi technology.

## Edit: some numbers.

To guess a number, let's take Shanghai Tower which has its roof at 561 meters, only about 30m lower than Burj Khalifa, which has 244m of uninhabited spire. It als has 20% more floor space, a broader top and is in the middle of a high-rise district, so overall a better fit for a vertical city.

Now we can start guessing some improved potential:

• locating the city on bedrock in a mostly earthquake-free zone should allow a whole city of this height.
• locating the city away from the common hurricane paths should allow an extra 10%.
• Increased support from the walkways should balance out against increased storm strength from global warming.
• Improved construction and materials would probably add 10% per decade. A little slower than the past two decades, but in absolute height it's more progress than ever before.

So in 20 years it's about 30% taller: 730m. In 50 years it might be 60% more, about 900m. Round it to 1 km for a nice number, since this is extremely rough guessing anyway.

• Care to hazard an estimate for the actual question here: "In a near-future, how high could we build vertical cities?"? – Samuel Nov 9 '15 at 17:55
• "Dubai pretty much went bankrupt trying to build the Burj Khalifa" - But how much of this was construction cost vs engineering cost? A lot of technology is very expensive to build once, but to build again, just a fraction of the cost since the R&D work needs only to be done once. – colmde Nov 10 '15 at 8:39
• @colmde - since I worked on the project, lol, the construction cost was much, much, much more than the PD&E cost percentage-wise. But you are correct of course that a second Burj Khalifa would be much cheaper to build due to the lessons learned, etc. – Mikey Jul 26 '17 at 20:21

Actually no: you cannot just arbitrarily add floors to an existing structure (which is essentially using a building as part of a foundation of a new building). the foundation must be properly designed for the expected mass and loads of the structure.

• That doesn't really tell me how tall a building can be, just that you need to plan it to be that tall first. If you make the base for it, is there practically no limit to the height? – Thomas Reinstate Monica Myron Nov 9 '15 at 3:52
• @Tommy - no. It absolutely depends on the place where you build (earthquakes, soft soil, etc), on the weather (high winds, hurricanes, etc), and on the material science employed. With our current technology we are pretty much hitting our limits on how high we can make our towers. A lot of technology and newer, stronger materials would have to go into building something significantly taller. You would need sci-fi technology and materials to make mega-towers possible, and a very stable world. – AndreiROM Nov 9 '15 at 4:28
• So what is the limit that we are hitting? That's what I'm trying to find out. :) – Thomas Reinstate Monica Myron Nov 9 '15 at 4:38
• – ratchet freak Nov 9 '15 at 9:15

The limit is dictated by economics rather than technology. If you assume that everyone in the building wants to arrive and leave at ground level at least once per day, you get to a point where adding sufficient lift (elevator) capacity to support an extra storey has negative economic benefit because of the nonproductive area it consumes on lower storeys.

One way around that is a vertical city where the denizens of the upper levels do not leave on a daily basis. Instead, there are shops, parks, services, all provided on those levels by people living there. There's still an economic limit, since all the things that they consume still have to be transported up from ground level, but I've read that a mile-high vertical city is not impossible and does not require exotic materials.

By the way, a tall skycraper (with mass dampers) is less vulnerable to earthquakes than a much shorter one or a regular house.

• My vertical city is like that - everything is in it. In fact, it produces everything it needs. Food isn't transported up at all. – Thomas Reinstate Monica Myron Nov 9 '15 at 23:19

Such cities are often depicted in science fiction. With the right materials there is no reason why such a city could not exist, or go as high as you'd like it to.

There are several things to keep in mind however:

Geology

Such a city would be incredibly heavy. If build on anything other than solid bedrock it would most likely collapse, piece by piece. Earthquakes would also work a number on it, since these sort of upward reaching and interconnected towers would be far more rigid than a stand alone building.

Weather

If the planet this city exists on is prone to high winds, powerful storms, etc, these cities would once again be threatened. The effects could be somewhat downplayed by technological advances.

In all honesty, if the story is set in the future, where technology is sufficiently advanced, these towers could easily go as high as kilometers, and kilometers, with the old towers being connected together to become the foundation of even larger mega-structures.

"you cannot just arbitrarily add floors to an existing structure (which is >essentially using a building as part of a foundation of a new building)"

and he/she is right - in a real world scenario.

However, since this is a science fiction setting, imagine powerful force fields, being used to reinforce those towers, or parts of the buildings themselves being filled-in in order to turn them into massive support columns. Buildings which once scraped the skies being confined within the core of the mega-tower, and becoming underground slums that never see the daylight.

I'm not saying this makes sense with our current technology, or even technology that might be developed within the next century. It just has to sound plausible enough

• That's what I thought - interconnecting the old towers into massive bases would provide support. The answer to my comment on this question said otherwise though: worldbuilding.stackexchange.com/a/25477/6620 (Sorry, I don't know how to link in comments.) – Thomas Reinstate Monica Myron Nov 9 '15 at 1:48
• The question is about the near future, so I don't understand why so much of this answer talks about sci-fi possibilities that might not be developed within the next century. – sumelic Nov 9 '15 at 5:51
• @TommyMyron the timestamp is a permalink to the comment – ratchet freak Nov 9 '15 at 9:20
• @Sumelic - It all depends on the universe of the author, doesn't it? If you invent a new "technology" that allows you to get away with it, you're golden. If not, then you're stuck with "normal" construction means. Simple as that. – AndreiROM Nov 9 '15 at 12:27

Assume that volume scales linearly with height (i.e. we'll roughly approximate the building shape as a cube).

The mass/weight of your constructions will scale roughly as the cube of height (we'll assume the volume of the building scales linearly with height - meaning the x-y footprint goes up as the height does). However the structural strength only scales as the square of volume. So no matter how strong our materials, we eventually get too big to support a bigger building.

Increased materials strength can improve your results but it can't change the shape of the curve - no matter how strong the material. If the building exceeds the size above, it will always collapse under its own weight.

## Materials

To look at this from a materials perspective it helps to have an idea of the pertinent equations.

The force on the building is the weight (mass * acceleration) of the portions of the building above the current floor.

$$W = a_{gravity} \cdot m = a_{gravity} \cdot \rho \cdot V = a_{gravity} \cdot \rho \cdot l^3$$

$$F_{compression} = A \cdot \sigma_{compression} = l^2 \cdot \sigma_{compression}$$

At max load, the maximum compressive force your structure can support is equal to its weight, so you get this:

$$a_{gravity} \cdot \rho \cdot l^3 = l^2 \cdot \sigma_{compression} \rightarrow l_{max} = \frac{\sigma_{compression}}{a_{gravity} \cdot \rho}$$

V = volume ($m^3$)
A = Area ($m^2$)
l = distance ($m$)
$\sigma$ = material strength (Pascals - $Pa$)
g = acceleration due to gravity ($\frac{m}{s^2}$)
$\rho$ = density ($\frac{kg}{m^3}$)

Maximum measured compressive strength of any material (diamond) comes in at something between 100 & 300 GPa. So make the following assumptions

$g = 9.8 \frac{m}{s^2}$
$\sigma = 300 GPa$
$\rho = 2,500 \frac{kg}{m^3}$

Solve for $l$:

$$l_{max} = \frac{300,000,000,000 \frac{kg}{m \cdot s}}{9.8 \frac{m}{s^2} \cdot 2,500 \frac{kg}{m^3}} = 12,244,898 m$$

12 million meters (12,000 km) is a pretty tall building. Your limitation on height would not be due to materials constraints (if you use the right materials).

Bear in mind that this structure would be solid at the base with no room for anything other than structure, the material would have to be diamond, and the crust of the planet it was on would sag so the actual height would be substantially smaller. In fact the shear amount of materials involved would probably be the equivalent of a planetoid (that's not a moon!).

## Foundation

But what sort of foundation can hold up that kind of mass. Although the Earth's crust seems rigid, the reality is it floats on top of the Earth's mantle and is not rigid enough to support even its own mass - it has to float on the Earth's mantle.

The highest points on Earth all exist in the Himalayan Mountain range. In that range, one continental crust is subducting under another. It has pushed the top plate over 5 miles high but this lofty altitude can only be supported by the crust sagging 60 miles or more deep beneath it.

What this means for your building project is that even a sufficiently advanced society are likely to be unable to create large structures of more than 5 miles or before the crust starts sagging beneath the weight of the structure.

## Otherstuff

Typical engineering for ground structures (like buildings) uses a factor of safety of 10x. Meaning you design the building for 10x the load its supposed to support.
This ignores any other forces such as wind, dynamic stresses, buckling, earthquakes, etc.

• You're growing vertically upwards so actually weight scales linearly and strength doesn't scale at all.... – Tim B Nov 18 '15 at 21:14
• Yes, that's the square / cube problem. Strength scales as the square of linear distance (area) while weight scales as the cube of the linear distance (volume). So the limiting case is the "first floor" is solid structural material - which is the case I used above. – Jim2B Nov 18 '15 at 21:59
• Nope. It scales as a cube if expanding in 3 dimensions. We only expanding in one though so it is linear. A tower twice as high...does it weigh twice as much as or eight times? – Tim B Nov 18 '15 at 23:12
• Animals expand in 3 dimensions which is where the square cube stuff comes in, this is a different case though. – Tim B Nov 18 '15 at 23:13
• But you notice at the very beginning that I specify an assumption of a square building shape - thus we do suffer the square / cube problem. Either way it doesn't matter. The amount of area required to support a height goes up linearly with that height (because it is still holding up a volume of material). So the equation remains the same even if the size of the building does not. – Jim2B Nov 19 '15 at 4:57