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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:

Curved beam loaded at 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:

enter image description here

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.

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    $\begingroup$ 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). $\endgroup$ – hazzey Jun 23 '16 at 13:55
  • $\begingroup$ 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. $\endgroup$ – David Jun 23 '16 at 14:08
  • $\begingroup$ Related (and would be the direct answer if it used a fixed connection instead of pinned): engineering.stackexchange.com/q/10229/33 $\endgroup$ – hazzey Jun 23 '16 at 14:44
  • $\begingroup$ @David - bending stresses on a cantilever at the point of force application are zero... $\endgroup$ – AndyT Jun 23 '16 at 15:03
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Your assumptions are incorrect. The breaking point will almost always be at the fixed point.

Here are the internal forces in your structure with loading at two different positions:

enter image description here

Axial force is always greatest at the support, and its value is the same regardless of where the load is applied. Bending moment is also greatest at the support, but the shorter lever arm in the second case leads to lower bending moments.

Therefore, the most critical point will be at the support, regardless of where the load is applied.

The only possible wrinkle is if you're dealing with reinforced concrete (RC) or some other material where the resistance to compression is far greater than to tension. An RC structure might actually fail at some other point, since the compression may (up to a point) actually help since it will reduce the tension suffered due to bending. Therefore, it's possible that the critical point in such a structure is somewhere with sufficient bending moment, and not enough compression to mitigate it.

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  • $\begingroup$ Thanks - I am indeed interested in a case where the compressive strength is much larger than the strength in tension. Is there any way how this problem could be tracked analytically, or at least with some form of qualitative argument? $\endgroup$ – David Jun 24 '16 at 6:25
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    $\begingroup$ @David: That's very relevant information which you should really add to your original question. I'm working on how to edit this answer given this new information. $\endgroup$ – Wasabi Jun 24 '16 at 19:26

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