The simply supported slender beam shown below is loaded with distributed load p from C to B and has an overhang from D to B. Applying Castigliano's theorem, find the vertical deflection at point C in terms of p, a, E, and I.
Note: The general equation for deflection using Castigliano's theorem is
$$\delta_{i}=\frac{1}{AE}\int N\frac{\partial N}{\partial P_{i}}dx+\frac{1}{EI}\int M\frac{\partial M}{\partial P_{i}}dx+\frac{1}{AG}\int \alpha V\frac{\partial V}{\partial P_{i}}dx+\frac{1}{JG}\int T\frac{\partial T}{\partial P_{i}}dx$$
p
A
C
D
B
a
a
a