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Using Castigliano's Theorem

  1. Apr 18, 2010 #1
    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

    [tex]\delta=\frac{1}{(EI)} \int_{0}^{L}M* \frac{\partial M}{\partial P} dx[/tex]

    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:

    [tex]\frac{L^3 P - L^2 P Xq}{48EI}[/tex]

    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.
     
  2. jcsd
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