$1)(10$ Pts$)$ Consider the given finite element$($s$)$ of variable or constant cross$-$section subjected to distributed axial force $(2\mathrm{.}5$ points each$).$

a$)$ Determine the element stiffness matrix $\backslash ([$K$]^\{$e$\}\backslash )$ for Case$-1$ and Case$-2.$

b$)$ Determine the element nodal force vector $\backslash (\backslash \{$f$\backslash \}^\{$e$\}\backslash )$ for Case$-1$ and Case$-2.$

c$)$ Determine the global stiffness matrix $[\backslash (\backslash$boldsymbol$\{$K$\}]\backslash )$ for Case$-3$ and compare with Case$-2.$

d$)$ Determine the global nodal force vector $\backslash (\backslash \{$f$\backslash \}\backslash )$ for Case$-3$ and compare with Case$-2.$

Case$-1$: single linear element

Case$-3$: two linear elements

Here, $\backslash (\backslash$boldsymbol$\{$p$\}(\backslash$overline$\{\backslash$boldsymbol$\{$x$\}\})=\backslash$boldsymbol$\{$p$\}\_\{\backslash$mathbf$\{0\}\}\backslash )$ is an unknown constant, so the consistent nodal force vector $\backslash (\backslash \{\backslash$boldsymbol$\{$f$\}\backslash \}\backslash )$ is defined as a function of $\backslash (\backslash$boldsymbol$\{$p$\}\_\{0\}\backslash ).$