If Burj Khalifa falls from 2m height, will there be a dent?

Question

I got this question from my son, if Burj Khalifa would suddenly fall from 2m height, how big would the dent be, if any? Of course we have to work with approximations. I found out that the weight of Burj Khalifa is about 500,000 tons. Physics was long time ago, but maybe it's a fun exercise for you guys.

Answer 1

Crater

If Burj Khalifa falls from 2m height, will there be a dent?

Yes. And my guess is $3$ m.

There are multiple crater scaling laws, valid for different projectile speed ranges and ground properties, but the paper Projectile-shape dependence of impact craters in loose granular media (arxiv, e-print 1, e-print 2) considers the similar situation of heavy cylinders falling upright on sand-like ground and provides an equation (Eq. 4) for the dent (crater) depth:

$$ d = 0.139\sqrt{\frac{m}{\rho_g}} \frac{H^{1/3}}{\mu A^{5/12}}, $$

where $m$ is the mass of the falling object, $A$ its base area, $\rho_g$ the ground density (supposed to be a loose granular material such as sand), $\mu\approx\tan{\theta_0}$ is the coefficient of static friction of the ground grains, where $\theta_0$ is its angle of repose, and $H$ is the height of the fall.

Supposing that the ground is sand and using

• $H =2$ m,
• $m = 45 \cdot 10^7$ kg,
• $\rho_g = 1590$ kg/m$^3$,
• $\mu=\tan{38^\circ}\approx 0.78$,
• $A = 5600$ m$^2$,

yields $d=3.3$ m.

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Collapse

Another interesting question, also raised in the comments, is whether the building deforms significantly in this experiment. I'd say it doesn't significantly $-$ unless it collapses. Does it?

If we consider the movement from the height $H=2$ m to the bottom of the crater, at $d = 3.3$ m, we have a change in potential energy $\Delta U = -mg(H+d)$ and no net change in kinetic energy, so the resisting force of the ground do work $W = -F_gd =-\Delta U$, leading, supposing constant force, to

$$ F_g = \frac{H+d}{d}mg \approx 1.6 mg, $$

so that $F_{\mathrm{res}} = F_g-mg = 0.6 mg$, which implies an acceleration $a\approx 0.6g$.

This doesn't exactly bode well for the building, since $a$ is unlikely to be constant, and its peak value might be considerable above the estimate, but it also doesn't necessarily seals its destiny.

Safety margins for concrete structures should allow for at least about twice the projected load, which would mean $2g$, but that doesn't apply trivially, since this load is mostly supposed to be static.

Comparing to earthquake accelerations, values about $0.6g$ are supposed to cause heavy damage, though some buildings might resist $3g$ (or even a total, i.e., not only vertical acceleration above $4g$). Burj Khalifa is earthquake proof up to magnitude $7.0$, but magnitude doesn't translate directly into acceleration, as other factors, such as depth of the quake, are also important. For instance, a magnitude $9.2$ earthquake can have peak acceleration as low as $0.18g$, while a $6.3$ earthquake as high as $2.2g$. All in all, I can imagine it not crumbling with this fall, but, if a constant $a$ is too bad an approximation, the building chances of surviving might actually be low.

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In time: the value for the area of the building base is a rough estimate obtained from this ground floor plan:

_{source 1, source 2}