# What is the practical limit on the number of storeys in residential buildings built with monolithic reinforced concrete?

I have recently bought a flat in a 25-storey building. I wonder what is the practical limit for the number of storeys in a mass-produced residential buildings built from monolithic reinforced concrete?

Can we expect the number of storeys in typical buildings of this type to rise in the forthcoming years or this is the reasonable limit of the technology? In all references I saw so far it was claimed that this technology has no limits on the height of the buildings. But I doubt this because all skyscrapers I know were built from steel.

If monolithic reinforced concrete has no limitations, why have very tall buildings not been built with this technology?

• Wherever that limit is, it's far beyond the economic break-even level for ("pure") steel frame. I guess it would be possible to build a 50-store reinforced concrete building, but it would cost half as much to build one of the same size with a steel frame. So the practical limit here is of economic nature: steel frame requires higher startup cost, so it's impractical in low buildings, but its cost scales much better with height. – SF. Feb 8 '15 at 12:15
• They have. In fact the tallest building in the world, Burj Khalifa, just so happens to be made of concrete. So there's that. – Mr. P May 22 '16 at 18:33

Any limit is going to be hard to quantify. There are a lot of factors that have to be weighed when choosing the basic material type.

The short answer is that the limit has already been chosen for each building. This was done during the design by the architects and engineers that worked on the building. Some of these decisions might have depended on the technologies that were available at the time that the building was designed.

Some of the factors that would have been taken into account:

• Cost of steel versus concrete - The relative price of the materials has changed throughout history.
• Strength of concrete available - It used to be the case that concrete was limited to about 4,000 psi (27.6 MPa) compressive strength. Modern high strength concrete can be higher than 10,000 psi (69 MPa).
• Strength of steel available - Steel strengths have increased from 36 ksi (248 MPa) to 50 ksi (345 Mpa) and even 100 ksi (689 MPa).
• Area of wall and column space required to support the upper floors - Buildings are heavy. As the building gets taller, there is more weight pressing down on the bottom floors. This increased force requires more area of material. At some point, the usable space on the bottom floors is reduced more than is acceptable. Per unit of area, steel is stronger than concrete, so it will take less area to support the same load.
• Stiffness of the building - Very tall buildings sway as the wind blows on them. How much they move is controlled by the weight and stiffness of the building.
• Future creep (shortening) of the building - Both steel and concrete creep. That is they compress over time if a constant force is applied. The amount of creep is affected by age, strength or the material and forces acting on the material. In very tall buildings, this shortening needs to be accounted for in the design. A lighter building will need to accommodate less creep.
• Seismic (earthquake) design - Steel is a ductile material. Concrete is a brittle material. In locations where high seismic forces are expected, steel may be required. It has the ability to undergo extreme deflections without complete failure.
• Quality control - Concrete will be poured onsite, and steel is typically fabricated offsite under controlled conditions. The anticipated quality of the end product or amount of oversight required to ensure a quality product are both a cost consideration.

There are a lot of factors that go into the design of skyscrapers. Each item above has a cost associated with it. The end result is at least partially controlled by the estimated price.

Modern skyscraper designs sometimes include a concrete core that goes all or most of the way to the top. This shows that there isn't much of a height limit to concrete construction as long as you are ok with a reduced usable volume.

In all references I saw so far it was claimed that this technology has no limits on the height of the buildings.

This statement is more or less true.

hazzey's answer has already done a good job of summarizing the actual limitations of building height - i.e., the factors that, in any real application, control the decision of how many storeys to build a building. However, there is still the question of how high a structure could be, assuming we were able to ignore all of these other factors.

If we make a simplifying (and very naive) assumption that the only limitation of the height of a structure is the compressive strength of the concrete itself, and also that the only load being carried by the concrete is the load resulting from the weight of the vertical monolithic concrete column above (there are no live loads, or load transfers; the building is essentially a massive block of reinforced concrete), the calculation is fairly straightforward.

1. Unit weight of concrete: $$\gamma_c=150\frac{\text{lbf}}{\text{ft}^3}$$
2. Compressive strength of concrete (high performance concrete): $$f'_c=20,000\frac{\text{lbf}}{{\text{in}}^2}$$
3. Stress carried by concrete at the bottom: $$f=H_{c}\gamma_c$$
4. Set $f=f'_c$ and solve for maximum height: $$H_{max}=\frac{f`_c}{\gamma_c}=\frac{20,000\text{psi}}{150\text{pcf}}=19,200\text{ft}$$

This is so high (3.64 mi, or 5.85 km) that the acceleration due to gravity would be noticeably different at the top of the structure; the unit weight of concrete at the top would be roughly be 99.82% of what it is at the bottom - that is, about 149.73 pcf.

Additionally, the incredible stress applied to the concrete would result in appreciable strains. One equation for the modulus of elasticity of high strength concrete (from ACI) is:

$E_c=40,000\sqrt{f'_c}+1\times 10^6 \text{psi}=6,657\text{ksi}=45.9\text{GPa}$

According to Hooke's Law, the maximum strain at the bottom of the structure would be around 0.3%:

$\varepsilon_{max}=\frac{f'_c}{E_c}=0.3\%$

To find the strain across the entire structure height, we simply integrate:

$$\int_{0}^{H_c}\frac{f(z)}{E_c}\text{d}z=28.8\text{ft}$$ where $f(z)=\gamma_cz\cdot g(z)$ (gravity, $g$ is a function of height $z$).

This means the reduced height of the structure after taking into account concrete strain would be around 19170 ft (3.63 mi, or 5.84 km).

According to this article from Contruction Week Online, at 92 storeys (423 m, or 1388 ft) Trump International Hotel and Tower is currently the world's tallest concrete building (by their definition), and it is the 9th tallest building in the world. This is around 7% of the height possible (as defined by the simplified analysis above). Although the simplified analysis ignores all sorts of practical considerations and includes no safety factors, it is at least somewhat instructive as to what might be possible using high performance reinforced concrete.

• I would say this is calculating an upper bound for the height: We do not expect that it is possible to build that high (so it's not "the highest possible") - but to be able to build "not higher than" that. Which is a very useful information to understand this kind of problem. (+1) – Volker Siegel Mar 3 '15 at 20:02
• This assumes a constant section which one might argue is a very limiting choice to make. Allow the structure to be wider at the base than at the top and you'd be approaching infinity unless you introduce some more practical concerns. We most certainly could reach space, but the real question is at what cost? ;) – Mr. P May 22 '16 at 18:42
• @Mr.P would it really be infinity though? Seems like the stress at the bottom of a cone or pyramid shape would eventually crush the concrete. But you're correct that it could be a lot higher than this- I should update my answer using that idea. – Rick supports Monica May 22 '16 at 21:50
• @Rcik Teachey: Well, I figure that whenever we're approaching the compressive limit we can just flare the base a bit more and thus spread the force over an even larger area and we're good to go again. However if we introduce even the tiniest bit of reality the main problem would really be the tensile forces needed to resist the angular momentum trying to fling the whole thing off into space once we pass the geo-stationary layer. But before that we'd probably run into other problems, like suffocating all of humanity on the co2 released producing our cement :) – Mr. P May 23 '16 at 7:57
• Also the tallest concrete building in the world is Burj Khalifa, it has been the tallest manmade structure since 2007 (when it wasn't even close to completion). – Mr. P May 23 '16 at 8:13