How tall is the tallest Jell-O structure possible?

Question

Given Jell-O with a particular consistency, how tall a structure could be built out of it before it collapses?

Answer 1

If you're asking about a specific structure, for example a column, then the answer will depend on the details of the structure. Different structures have different failure modes. On the other hand if you are just thinking about a pile of Jell-O then the maximum height will be reached when the pressure is roughly the same as the yield stress.

Any solid material has a yield stress at which it will start to flow, deform or break depending on the material (I imagine Jell-O will break). As you start to pile up your Jell-O the Jell-O at the bottom experiences a pressure from the weight of all the Jell-O above it. As the height increases the pressure increases until it reaches the yield stress. At this point the material starts flowing and the structure slumps until the pressure falls back below the yield stress.

The pressure is given by:

$$ P = \rho h g $$

where $\rho$ is the density of Jell-O (approximately the same density as water), $h$ is the height and $g$ is the gravitational acceleration. The maximum height is therefore approximately:

$$ h \approx \frac{\sigma}{\rho g} $$

where $\sigma$ is the yield stress of the Jell-O.

However I have absolutely no idea what the yield stress of Jell-O is, and in any case it's presumably dependant on exactly how it was prepared.

Answer 2

In general, the optimal shape for supporting your own weight is something that looks a lot like the Eiffel tower: it is wide at the bottom, and narrow at the top (although the major design criteria for the Eiffel tower were the need to withstand wind stress, not simply its weight).

I did a calculation earlier for a "wall of ice" where some of the ideas in play are explored: https://physics.stackexchange.com/a/127992/26969

According to this article a "tall tower" that can support some excess weight $P$ at the top has to have a weight of steel $W=P(e^\mu-1)$ where $\mu=\frac{\rho h}{\sigma}$. The exponential relationship holds in the case of Jell-o as well. In principle this means you can build a tower of any height - but the total amount of material needed, which grows exponentially, quickly becomes unmanageable.

Of course if you have a shape with constant cross section, the calculation is easier:

$$\sigma = \rho g h$$

For a certain height $h$ the stress $\sigma$ will be equal to the yield stress, and there's your answer.