Question

Beam ABC in the figure is composed of segment AB of length L and uniform section stiffness EI joined to segment BC of length L and uniform section stiffness 2EI. The beam is fixed at C (x=2L) and loaded by a downward concentrated force, P, applied at its free end A (x=0).
You will use Castiglano's theorem to determine the vertical downward deflection vA and the rotation (slope) ϑA of the beam at its free end A .

1.
Obtain symbolic expressions for the integrands in the two expressions below, which give the complementary energy in the two segments of the beam, 𝔼∗AB and 𝔼∗BC, in terms of P, x, and EI ( enter this as EI without the multiplication sign). NOTE that you do not need to carry out the integration, just write the integrands: e.g., if the expression for 𝔼∗AB were to be 𝔼∗AB=∫L0(ax+b)dx, you would just write a∗x+b in the answer box.
2.
Use Castigliano's theorem to obtain a symbolic expression for the downward vertical deflection vA of the free end A, in terms of P, L, and EI ( enter this as EI without the multiplication sign).
3.
Use Castigliano's theorem to obtain a symbolic expression for the rotation ϑA (the slope) of the beam at A in terms of P, L, and EI ( enter this as EI without the multiplication sign). Hint: You will have to introduce a dummy load that is work-conjugate to the desired rotation ϑA