$1-$ The figure below shows a uniform cross section semicircular curved beam with a mean radius $R=510mm,$ a moment of inertia I equal $815{10}^{6}m{m}^4,$ about in axis out of the page directed through the centroid of an area section, and a modulus of elasticity $E=205$ GPa. Using Castigliano's second theorem, determine:

a$)$ the allowable magnitude of the force P if the beam is not to deflect $($extend$)$ by more than $0\mathrm{.}1$ mm horizontally at the point where the force is applied $($point $A).$

b$)$ the magnitude and direction of the corresponding vertical deflection