$(10$ points$)$ The rectangular beam $Fx=0x=ay=by=-bF,x,yFx=0c_{1}6xy=C_{1}x{y}^3+C_{2}xy{\displaystyle \underset{-b}{\overset{b}{}}}{r}_{xy}dy=F$;$x=0.=-(c_{1}3y^2+c_2)0is$ subjected $to$ a transverse force $Fat$ the end $x=0,$ and built$-inat$ the end $x=a,$ and the horizontal boundaries $aty=b$ and $y=-b$ are traction free. Please find the normal stresses and shear stress $(expressedin$ terms $ofF,x,y,$ and $b)$ using the following Airy stress function and boundary conditions, including the one listed below using the shear stress and transverse force $Fatx=0.$

$c_{1}6xy$

$=C_{1}x{y}^3+C_{2}xy$

$0$

${\displaystyle \underset{-b}{\overset{b}{}}}{r}_{xy}dy=F$;$x=0.$

$=-(c_{1}3y^2+c_2)$

Goundau