Why bending moment is constant on an end loaded beam

[nevermaybe]

Homework Statement

For the same lab as I posted previously, I have been asked to prove a situation. I think I'm either over thinking it or it really is that complicated.

I've been asked to prove why the bending moment is constant across an end loaded beam (inbetween the knife edges - the two points that hold the beam up). Conversely, it then goes on to ask why does the bending moment depend on point P (arbitrary point along the beam) for a centre loaded beam, and where it would fail if overloaded?

Pictures!

End loaded beam:

http://puu.sh/2mmwW

Centre Loaded Beam:

http://puu.sh/2mmxs

Homework Equations

C = (Y.I)/r

C is the bending moment
Y is the Young's modulus of the material
I is the geometrical moment of intertia of the cross section of the beam
r is the radius of curvature of the neutral surface

The Attempt at a Solution

From my understanding, I and Y will always remain constant on both beams, regardless of how they're loaded. However, the radius of curvature must be the only reason why the bending moment would not be constant. In saying this, I don't know how to go about proving why the radius of curvature would depend on where it's loaded along the beam.

I asked the lab demonstrators, but they seem to either be unfamiliar with the lab, or refused to give too much detail about how to go about this problem.

 

[PhanthomJay]

Instead of looking at curvature, can you focus on the shear- moment relationship V = dM/dx? what is the shear between the supports for case 1? For case 2, where can you say that M is at a maximum value?

 

[nevermaybe]

Thanks for the reply!

Interesting suggestion, I did some research on shear-moment relationship online and it definitely seems viable at explaining the situation. I find it strange that they don't mention shear at all in the lab report/introduction, so I'm not sure if I'm meant to go about it this way.

I'll be heading into the lab tomorrow so I'll see if I can get some more help on this and update the thread!