# Bracket (beam) problem

1. Sep 13, 2009

### Cranky

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

a frame is composed as it is shown in the figure, each of length L. bending stiffness of AO is EI, of OB is 3EI. Force P is acting in the middle of AO, i.e at L/2. the question is to calculate the rotation "theta" at O.

2. Relevant equations

3. The attempt at a solution
the problem is statically undetermined.
to calculate the rotation we have to use castigliano's teorem and take the partial derivative with the respect to the ficticious moment M0 at point O, putting the M0=0:
theta=d/dMo(complementary elastic energy) at M0=0.

in order to get W we have to have 2 beams A0 and OB: (by superposition principle)
W= W(ao)+W(ob)

W(ao)=L/2EI*I (M1^2 dx)
W(ob)=L/6EI *I (M2^2 dx)

where I= integral from 0 to L
M1 and M2 internal moments in beams AO and BO respectively.

I am stuck on equillibrium equations:
in X dir: Rax-Rbx=0
in Y dir: Ray+Rby-P=0
and then there should be moment equillibrium also:
p.A: Mo+PL/2+M1 (??)=0 (by connecting points A and B)?
p.B Mo-M2=0 (???)
is there an internal moment near points A and B?

I'm mixed up about the equillibrium equation.Will be greatful for any help and suggestions
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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