Exercise $4.$ Figure $19$ shows a beam with inertial product $EI$ in the shape of a quarter ring with radius $R$ on the end loaded with a vertical force $F.$

Figure $19$: Exercise $4.$

a$.$ Determine the angular deformation on the end of the beam with Castigliano's theorem.

b$.$ Determine the horizontal deflection on the end of the beam with Castigliano's theorem.

c$.$ Determine the vertical deflection on the end of the beam with Castigliano's theorem.

$$ solutions are below, order $($c$),($b$),($a$))()()($