Ok, this took a while, so I hope it helps clear it up a little.
Keeping in mind that the chimney does break in half, let's model it as two rigid halves connected by a joint. The joint will be allowed to rotate freely to simulate breaking point. Keep in mind that this will greatly exaggerate the result. So the first image attached is an animation showing the simulated tower collapsing. Tendency to snap in half is absolutely clear. Second image is the plot of the force required to support the tower at the base. The initial spike is from the model settling in. From there, the force stays fairly level for a while, and then decreases as the tower leans and collapses. Finally, right before the tower crashes into the ground, something interesting happens. The force goes negative. That means that the bottom of the tower tries to "hop" into the air, and my constraint, requiring that the tower stays planted, is not letting it.
It's not perfectly clear to me why this happens. I suspect, the top half pressing down on the far end of the bottom half causes this, but I'm not sure how the top half ends up moving in such a way as to cause that. Either way, this is purely an outcome of tower segments being fairly rigid and interacting with each other.