Assuming a rigid beam and fixed and uncrushable end points, the beam could fail at any point though an unevenness or weakness in it's structure.
I think that's the point of this topic. You might as well have started with the classic "Assume a spherical cow ..."
For example, back in the late 1970's, a problem was presented and a variety of four outcomes were presented to the poll respondents. The problem as that a ball was descending down a spiral track. The variety of outcomes all had to do with what would happen to the ball's trajectory once it left that spiral track. The correct choice, that the ball would proceed straight out and fall into a trajectory consistent with free-fall was not the winner. Rather, the choice that the ball would continue to follow a spiral trajectory as it fell was the winner. And completely wrong.
The question is what the "common sense" solution would be. Even though that would invariably be
wrong.