$1.$ A rectangular beam with dimensions width $($b$)$ and height $($h$),$ as illustrated in Figure P$1,$ is subjected to a transverse shear force $($V$).[15$ pts$]$

$($a$)$ Derive the shear formula and sketch the shear stress distribution over the cross section of the beam. $[5$ pts$]$

$($b$)$ Show that the sum of shear stresses over the cross$-$sectional area equals the appl ied shear force $($V$).[5$ pts$]$

$($c$)$ Discuss the limitation of the application of the shear formula. $[5$ pts$]$

Figure PI