Shear force while bending a metal rod

[skrat]

Homework Statement

Let me first add a picture of the experiment and than try to describe my problem:

So I have two rods (one with a square profile and the other one with a circle profile) and what I do is I apply some force in the middle. The rod bends under the applied pressure.

Ideally I could say that all the force from the weight is concentrated only in one point of the rod. naturally, that is not true - a more realistic model is that the pressure from the weight is somehow distributed over the whole length of the rod. The shear force depends on how far away from the supporter we are or how close to the center we are where the shear force is the biggest. Sadly the shear force does not decrease linearly. Now my problem is to find out a theoretical model to describe how the shear force depends from the distance from the supporter.

Homework Equations

The Attempt at a Solution

I have no idea what to do... :/

 

[haruspex]

You don't say whether the load is in the middle -looks like it is.
Either way, you can calculate the vertical force at each support.
Assuming the rod is pretty much horizontal at the load, you can treat it as though the rod is held rigidly at that point while a support pushes up at each end. This allows you to treat the two sides separately.
At distance x from the load, what is the bending moment due to the support beyond x?

 

[skrat]

You don't say whether the load is in the middle -looks like it is.
Either way, you can calculate the vertical force at each support.
Assuming the rod is pretty much horizontal at the load, you can treat it as though the rod is held rigidly at that point while a support pushes up at each end. This allows you to treat the two sides separately.
At distance x from the load, what is the bending moment due to the support beyond x?

The load is in the middle!
And yes, the idea is to treat the two sides separately.

Well, I checked some old notes and I found and expression that describes how much the rod bends as function of x:
$u(x)=-\frac{F_0l^3}{48EJ}(1-6(\frac{x}{l})^2+4(\frac{x}{l})^3)$ where E is young's module, J moment of inertia and l length of the one half rod.

So... Moment should therefore be M=F(x)*u(x), where F(x) should tell how the load is distributed over the entire rod?

 

[haruspex]

Well, I checked some old notes and I found and expression that describes how much the rod bends as function of x:
$u(x)=-\frac{F_0l^3}{48EJ}(1-6(\frac{x}{l})^2+4(\frac{x}{l})^3)$ where E is young's module, J moment of inertia and l length of the one half rod.

So... Moment should therefore be M=F(x)*u(x), where F(x) should tell how the load is distributed over the entire rod?

Why would the moment be given by that?
The moment at a point is the cross-product of the applied load and the distance from the point to the load. (More generally, the sum of these on one side.) If the upward force at a support is F then that's (l-x)F.
From that you can determine the curvature at point x, and by integrating get the gradient and, eventually, the total vertical displacement at point x. Depending on what you mean by 'how much the rod bends', one of those should match the equation from your notes.

 

[skrat]

Why would the moment be given by that?
The moment at a point is the cross-product of the applied load and the distance from the point to the load. (More generally, the sum of these on one side.) If the upward force at a support is F then that's (l-x)F.

Omg, What was I thinking?...

From that you can determine the curvature at point x, and by integrating get the gradient and, eventually, the total vertical displacement at point x. Depending on what you mean by 'how much the rod bends', one of those should match the equation from your notes.

the expression above, u(x), should be the vertical displacement at given x. Now... I didn't really want to calculate the curvature at point x, I kinda wanted to just say that curvature is constant.

I have to plot a graph that will describe how shear force depends on x (and how momentum depends on x). So...

 

[haruspex]

I kinda wanted to just say that curvature is constant.

Not much chance of that. The curvature will be approximately proportional to the bending moment.
The section on end-loaded cantilevers at http://web.mit.edu/emech/dontindex-build/full-text/emechbk_8.pdf might help.