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I was thinking about this question from quite a while back about height limits of a brick and mortar tower. One answer pointed out that, due to the high compressive strength of brick, one could build a tower 7.4 km high without taper before individual bricks fail in compression. So it seems highly likely that a high brick structure will fail in another way. One candidate for a failure (the other two that come to mind would be lateral forces and accumulated errors in the masonry) is buckling pr more precise self buckling. Wikipedia gives a formula for self buckling but for the purpose of this question the simpler case of critical stress to cause buckling is more interesting (because the formula is simpler):

$$\sigma = \frac{F}{A} = \frac{\pi^2 E}{(l/r)^2}$$

The crucial element is, I think, Youngs' modulus $E$, which represents stiffness in tension. You often hear the statement that the mortar or rather the connection brick - mortar - brick has practically no strength in tension. Intuitivly, a structure build from several parts should be worse at resisting buckling than a monolith - So the formula given above for buckling would give an upper limit to the maximum stress. However quite slender brick structures, like catalan vaults or tall chimneys, where built and withstood the test of time.

My ultimate question is: $E$ is a material constant. Strictly speaking a brick tower is a a structure made of bricks and mortar. How is the effective $E$ determined in this case?

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    $\begingroup$ $E$ does not represent strength. It represents stiffness. Also your "simpler formula" is for a (massless) tower with a downward force applied to the end, which is not what "self buckling" means. $\endgroup$ – alephzero Jan 5 at 16:36
  • $\begingroup$ You are right about E. pertinent to my question is that allowable stress (external load) and critical height (self buckling) scale with E. Now that I think of it, understanding the difference between buckling and self buckling would probably answer my question. $\endgroup$ – mart Jan 5 at 16:50
  • $\begingroup$ I will think about this some more, possibly to retract or change the question, but not before tomorrow - unless someone wants to answer in the meantime of course. $\endgroup$ – mart Jan 5 at 16:54
  • $\begingroup$ That answer says granite with no joints will get to 7.4km high. Brick is nowhere near as strong as granite, and the mortar between the bricks is not as strong as the bricks. $\endgroup$ – achrn Feb 5 at 13:53
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As in @alephzero's comment, the formula you refer to is wrong.

The analysis of a column of nonhomogenous, composite material can be a bit complicated.

Empirically the ancient masons had some secret codes and formula's as to how to treat tall masonry or arches and big domes.

In Iran, they have even built ancient masonry towers designed to shake per demand. My parents took me to see it when I was a child. Then later the government stopped the show to preserve the structure as national heritage.

There is a Wikipedia page on buckling under self-weight. self-buckling .

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  • $\begingroup$ If you read the history of medieval cathedral-building in Europe, the "ancient masons' secret formula" often amounted to "if it falls down, build it again with thicker walls". It could take many years to build those large structures, and there sometimes the original designer had moved on to start several more building projects before the first one fell down, leading to a chain of emergency design changes! $\endgroup$ – alephzero Jan 5 at 18:33
  • $\begingroup$ yes. but lets give credit where it's due. they left the foundation sit for 7 years to settle, so the glorious structure built later on top last centuries. they invented primitive cement. they inventel tose long lasting tiles. $\endgroup$ – kamran Jan 5 at 18:50

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