a$.$ Consider the displacement functions $u,v$ designed for the shear loaded cantilever beam as shown in Fig. $1$

$u=-\frac{P{x}^{2}y}{2EI}-\frac{vP{y}^3}{6EI}+\frac{P{y}^3}{6IG}+(\frac{P{l}^2}{2EI}-\frac{P{b}^2}{8IG})y$

$v=\frac{vPx{y}^2}{2EI}+\frac{P{x}^3}{6EI}-\frac{P{l}^{2}x}{2EI}+\frac{P{l}^3}{3EI}$

Fig. $1$

Calculate all the strains $({}_i),$ stresses $({}_{ij})$ in the beam. Check if the beam is in static equilibrium.