# Fragmentation of a curved cantilever beam under compression

Suppose we have a curved beam of a rectangular cross-section $A$, Young's modulus $E$, and a constant radius of curvature $R$. We now expose the beam to a vertical point load of magnitude $P$ at its free end:

What I now want to know is where the beam will snap (I am assuming perfectly brittle material behaviour, and that the beam material is far stronger in compression than in tension). For this simple case, this is relatively straight forward (the beam will snap very close to the point of force application), but I want to know what happens when $P$ is shifted along the negative x-axis closer to the clamped end of the beam. For clarity:

Etc. pp.

Intuitively, as $P$ is shifted along the x-axis, the dominant type of stress will shift from bending to compression; if $P$ is applied very close to the clamped point, it's essentially a buckling problem, while it's similar to the bending of a straight cantilever beam when $P$ is applied at the free end. Now what happens in between, and how does this transition take place?

My gut feeling is that as $P$ is shifted (for clarity, I don't mean to imply any dynamic loading here), the point at which the beam will snap will increasingly move away from the point of force application, so if we were to start loading the beam at its far end until it snaps, then move the point of force application to the new far end and so on, we would end with snapped fragments of the beam which increase in length. However, I have thus far failed to come up with a sensible model that would allow me to predict whether this is what is actually happening, and would be grateful for any hint.

• Why in your first case do you say that it will snap at the point of load? In this situation, bending moment seems to be the largest force. Moment is related to distance from a load, therefore the highest moment is away from the load (at the fixed base).
– hazzey
Jun 23, 2016 at 13:55
• That's certainly what would happen for a horizontal cantilever beam - in the present case, however, most of the load close to the fixed base is compressive, and not in bending (if we moved the point of force application very close to the base, the beam would eventually buckle rather than snap due to bending stresses). Close to the point of force application, the quarter-circle approximates a horizontal cantilever, and that's why my guess is that the bending stresses are maximum around the point of force application in the first case. Jun 23, 2016 at 14:08
• Related (and would be the direct answer if it used a fixed connection instead of pinned): engineering.stackexchange.com/q/10229/33
– hazzey
Jun 23, 2016 at 14:44
• @David - bending stresses on a cantilever at the point of force application are zero... Jun 23, 2016 at 15:03

Your assumptions are incorrect. The breaking point will almost always be at the fixed point.