Problem $7.$

Consider a beam with a solid rectangular section as shown below. The beam has length $L,$ Young's modulus $E,$ and density $.$ The beam is built$-$in on the left end and supported by two linear springs on the right end with spring constants $k_y$ and $k_z$ for the $y-$ and $z-$directions respectively. The beam is loaded as shown below by a distributed partial$-$span load of magnitude $f_0[\frac{N}{m}]$ and a point force on the right end of magnitude $F[N]$ at the location shown. Gravity is also acting along the negative $z-$direction.

$($a$)$ Formulate the boundary value problem$($s$)$ present $($i$.$e$.,$ find all of the necessary governing differential equations and boundary conditions$).$

$($b$)$ Assuming that $a>b$ and recalling that the moment of inertia of a rectangular beam of height $h$ for bending about the thickness, $t,$ axis is given by $h\frac{t^3}{12},$ evaluate all geometric quantities present in part $($a$).$

$($c$)$ Assuming that $f_0=0$ and $k_y=k_z=0,$ solve for the deformation of the beam.$|$