$(25$ points$)$ Considering bending only, the displacement field for the thin beam shown in figure is

$u(x,y)=\frac{Py}{6EI}(6Lx-3x^2-v{y}^2)$

$v(x,y)=-\frac{P}{6EI}[3L{x}^2-x^3+3v{y}^2(L-x)]$

$w(x,y)=0$

where $P$ is applied force, I denotes the second moment of inertia, $L$ represents the length and $E$ and $v$ stand for Young's modulus and Poisson's ratio, respectively. Determine $($a$)$ the maximum absolute vertical deflection at centroidal axis $(y=0)$ and its location $(5$ points$).$

$($b$)$ the entire stress field, the maximum normal stress and its location $(20$ points$).$