Using the governing differential equations $($equilibrium$,$ constitutive equation, and compatibility$)$

for the truss$/$bar element, resolve the pinned$-$pinned single bar truss structure subjected to a

distributed axial load $w_x(x)=\frac{2w_{0}x}{L}$ shown in the figure below.

Part $1(20$ points$)$: Obtain an expression for the axial displacement field $u(x)$ and the internal

axial force $N(x).$

Part $2(10$ points$)$: Obtain the expression for the normalized displacements tilde$(u)(\stackrel{}{x})$ and widetilde$(N)(\stackrel{}{x})($just

like problem $1)$ using:

tilde$(u)(\stackrel{}{x})=\frac{\frac{EA}{w_{0}L}}{b}ar(u)(\stackrel{}{x})$

tilde$(N)(\stackrel{}{x})=\frac{\frac{EA}{w_{0}L}}{b}ar(N)(\stackrel{}{x})$

Since,

$\frac{?}{\text{b}\text{a}\text{r}}(u)(\stackrel{}{x})=\frac{u(x)}{L}\frac{?}{b}$

$ar(N)(\stackrel{}{x})=\frac{N(x)}{AE}$

The fields tilde$(u)(\stackrel{}{x})$ and widetilde$(N)(\stackrel{}{x})$ can be obtained directly from $u(x)$ and $N(x)$ by using the following

formulae:

tilde$(u)(\stackrel{}{x})=(\frac{EA}{w_{0}L})(\frac{u(x)}{L})=(\frac{EA}{w_0{L}^2})u(x)$

widetilde$(N)(\stackrel{}{x})=(\frac{EA}{w_{0}L})(\frac{N(x)}{AE})=(\frac{1}{w_{0}L})N(x)$

Once you get these expressions, plot widetilde$(u)(\stackrel{}{x})$ and widetilde$(N)(\stackrel{}{x}).$

Part $3(10$ points$)$: Obtain the support reactions. Verify that they are in equilibrium with the

external distributed load.