$(2)$ Consider a bar with a length $L,$ consisting of a homogeneous, isotropic, linearly elastic

material with Young's Modulus $E$ and Poisson Ratio

$u.$ The bar has a rectangular cross$-$

section with a constant width $b,$ but its height $h$ is tapered: $h(x)=b(1+\frac{x}{L}).$ The bar

is fixed at both edges and along its entire length the bar is loaded by a distributed axial

force $F_x(x)=F_0(1+\frac{x}{L}).$

$($a$)$ Is this problem statically determinate? Briefly explain your answer.

$($b$)$ Use the approach via the ordinary differential equation $($ODE$)$ to solve for the dis$-$

tribution of the normal strain ${}_{}(x,y,z)$ in the bar.

$($c$)$ Determine the distribution of ${}_{yy}$ and ${}_{xz}$ in the cross$-$section at $x=\frac{3}{32}L.$

$($d$)$ Determine which cross$-$section has the largest $x-$displacement.