$($B$)-$ Theory: $(5\%)$

Beginning with the strong form for a $1$D bar

$AE\frac{d^{2}u}{d{x}^2}=0$

where $E$ is Modulus of Elasticity $($from calibration above$),$ A is cross$-$sectional Area, and $u$ is displacement,

derive the weak form, assuming linear weight function tilde$(u)=N_1(x)u_1+N_2(x)u_2,$ and element length $L=1.$

$($Note: Show each step in the derivation fully$).$ The essential and natural boundary conditions are

$u{|}_x=0=0$

$\frac{du}{dx}{|}_x=L=\frac{S_{11}}{E}$

$1\mathrm{.}03$

where $S_{11}$ is the stress at the final increment in the single linear element $($reduced integration$)$ simulation in

$($A$).$ Following the weak form derivation, using the Galerkin Method, determine $u_2$ for the boundary

conditions above. Compare the results of this force driven analysis to the displacement driven analysis in

$($A$).$ For this questions please use: A $=$ cross sectional area $=1$m$^2$; E $=$ youngs modulus $=29672\mathrm{.}28989$MPa; s$11=344\mathrm{.}015$MPa