# Using Castigliano's Theorem

1. Apr 18, 2010

### Phoenix70

1. The problem statement, all variables and given/known data

Use Castigliano’s second theorem to calculate the equation for the deflection as a function of
position in a simply supported beam with a load P applied in the center of the span. Assume that E and I are constant along the length of the beam. Compare your result with that found in
deformable solid books. 30 points. Hint: a dummy load in an arbitrary location would be the way
to start.

2. Relevant equations

$$\delta=\frac{1}{(EI)} \int_{0}^{L}M* \frac{\partial M}{\partial P} dx$$

3. The attempt at a solution

Well, I've been at this all night, and well into the morning now, and it seems simple enough so I must be missing something. My class just changed professors mid-semester, and now we've got homework due on materials we haven't yet covered in class so my grasp of this is a little shaky.

I'm a bit confused as to what is meant by the term dummy load, and where that would be located and how it would factor into the moment equation when you take a cut of the beam.

So far, I've tried placing a load Q, at an arbitrary location Xq from the left hand side of the beam. I allowed the location of this load to float at the right hand of the cut I made in the beam, such that the resultant moment at the cut wouldn't have contributions from the dummy load, but that might not be right. I've tried a couple different variations on this, but for the section from 0 to L/2, I keep getting:

$$\frac{L^3 P - L^2 P Xq}{48EI}$$

Obviously, somewhere there's a mistake in the assumptions I'm making to set this up, as the answer does not match a textbooks. Any help or pointers in solving this would be greatly appreciated.