Show that in the case of a simply supported beam, if a couple M is applied at one end, the slope at the point of application of the couple is MZ/3EZ, where Z is the length of the beam. Using...

Show that in the case of a simply supported beam, if a couple M is applied at one end, the slope at the point of application of the couple is MZ/3EZ, where Z is the length of the beam.

Using Castigliano’s theorem, calculate the vertical deflection at the middle of a simply supported beam carrying a uniformly distributed load of w N/m over full span.