Tower of building blocks

Why does a tower of building blocks (cubes) fall?

Theoretically, as long as the center-of-mass is above the blocks’ bottom faces, and as long as one does not shake the tower too much, it should not fall. This means that, with sufficient care, one can build a tower infinitely high. @Bernhard has suggested the link confirming this.

Nevertheless, any real tower of building blocks collapses, if it is too high.
I am interested in the reasons of this collapse, the mathematical model for this instability, and the experimental evidence confirming this or that model of collapse.

Possible explanations
Scenario 1
The collapse of the tower could be due to the fact that the supporting surface is not perfectly horizontal. In this case there exists a limiting height where the center of mass is not supported anymore and the tower flips over.

The condition of stability in this case:
$$
tanalpha < frac{a}{h} = frac{1}{N},
$$
where $s$ is the side of a cube, $h=Na$ is the height of the tower of $N$ cubes. For small angles ($tanalphaapproxalpha$) we get
$$
N<frac{1}{alpha}=frac{180^circ}{pi alpha^circ}approx frac{60^circ}{alpha^circ},
$$
which means that for incline of $1^circsim 10^circ$ the critical height of tower ranges from 60 to 6 cubes. In my experiments it is about 20-30 cubes, which corresponds to an angle of $2^circ-3^circ$ – not too big to exclude on the basis of visual observations, but also not too small to be a routine thing in home conditions.

Scenario 2
Alternative scenario is imperfections of the blocks themselves: if their faces are not perfectly parallel. It seems that in that case the tower does not necessarily turn over, but a part of it may simply slip. I.e., one could distinguish different scenarios experimentally.

Scenario 3
@BillWatts proposed the self-buckling instability as a possible explanation. In this case the critical height its (in length units):
$$
l_{max} = left(7.8373frac{EI}{rho g A}right)^{frac{1}{3}}
$$
The cubes that I am experimenting with are wooden cubes with side $d=4.5$cm and mass ranging from 30 to 50 gramms. Taking the Young’s modulus of the wood as $10$GPa, the equation above predicts the tower height of 17.5 meters, i.e. nearly 400 cubes, whereas in practice it collapses at 20-25 cubes. Still, this hypothesis cannot be ruled out, since the equation above was derived for a solid wooden tower, rather than disjoint cubes – i.e., the effective Young’s modulus can be much smaller.

Scenario 4
@RogerWood proposed an estimate of instability arising from the fact that the blocks are not be perfectly aligned. While I do not believe this to be a correct explanation, it is the most likely cause of colapse of teh towers built by a child.

Scenario 5
@GTos suggests that one necessarily provides a small torque when putting the last cube, which might be crucial for higher towers.

Some experimental evidence
My testing cubes have for smooth faces and two faces with a carved pattern. When stacking the cubes by smooth faces, the towers typically collapse at 20 cubes. However, when the cubes are stacked by the rugged faces, I can exhaust all the 28 cubes available. Ruggedness of the cube faces may affect friction between the cubes, but also the effective Young modulus. I have repeated this test in different locations, and putting cube sin random order, so I believe that it allows to exclude the first two scenarios, and speaks in favor of scenario 3.

Update
This article addresses a very similar system, but the experimental methods are somewhat different from what one could do with already made wooden cubes that I have.

Note for moderators: this article was originally published in the American Journal of Physics, but the version cited here is from the open access HAL archive (french equivalent of arXiv).

Update from 14/12/2020

• If we go on with the buckling theory, then here is yet another very interesting article, which discusses the buckling of a column as a second order phase transition.
• @AndersSandbreg has brought to my attention a book-stacking experiment, which demonstrates how different scenarios could be in play.
• As could be seen from the above, the major difficulty in confirming/rejecting the buckling hypothesis is not knowing the Young modulus of a composite object, such as a tower of cubes.
• • One solution could be to measure Young modulus from the frequency of oscillations of towers of different height (which requires some technical prowess, but should be doable).
• • Another solution is to associate the tower collapse with the critical slowing of the oscillations (see the paper cited above).