What is the maximum displacement at which a series of blocks can be displaced vertically?

Question

This is a really simple question that I've been thinking. The statement is as following:

Consider you have a series of blocks, each of length $l$, width $w$, height $h$, and mass $m$ wich is evenly distributed across the entire block. There is the force of gravity. Imagine some surface that is totally flat, in which we place our block in $n=1$, where $n$ is the index for the number of blocks used.

We are going to define some quantities $\delta l_n$, $\delta w_n$, for each $n$ with one really simple condition:

• $\delta l_n$ is defined as the displacement along the axis of $l$ the most upper block in the system has with the block that's beneath it, with the condition that the entire system of blocks is at equilibrium (no blocks fall).
• The same thing happens with the axis along $w$.
• Obviously, the displacements do not act on $n=1$ (only one block) and $n=0$ (no blocks).

With these conditions, I want to ask two questions:

• Is there any function that relates $n$ to the displacement at $n$?
• If so, how does the height $h$ and the mass $m$ of the blocks affect the final displacement?

Here I post a diagram for orientation:

As I have already said, with "maximum" I refer to the maximum length $\delta l_n$ and $\delta w_n$ can take without gravity letting the system of blocks collapse.

Answer 1

I remember such a problem, the displacement of cubes placed one on another was only in one direction. The total displacement could be arbitrarily large, and maximum displacements formed the harmonic series, if I remember correctly. Maybe I'll be able to add details later.

EDIT (Dec 7,2024): See discussion and references at https://math.dartmouth.edu/~pw/papers/maxover.pdf