Q$3.(20$ points$)$ Consider a bar subjected to the uniform line load in the axial direction. This is

the same problem from the "Galerkin Form Example" in the class lectures. Use the same

material properties:

$A=1i{n}^2$;$,L=10in$;$,E=10,000$;$q=20l\frac{b}{i}n$;$,P=100lb$

Steps:

Use the unequal length linear elements shown, with node $2$ located at $x=\frac{L}{3}.$

Determine the nodal displacements and axial stress distribution.

$1,E,A,2,3$

$x=0(1)x=\frac{L}{3}(2)x=L$

$u(0)=0,f(x)=q,$

Next, perform a separate numerical approximation using quadratic shape functions to find

a numerical solution. Use the following trial functions: $u^h(x)=u_a{N}_a(x)+u_b{N}_b(x),$

$N_a(x)=4x\frac{L-x}{L^2},N_b(x)=x\frac{2x-L}{L^2}.$ These two functions are supported over

the entire length $\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]K\left[\begin{array}{c}{u}_a\end{array}\right]$

$u_b=\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]\frac{EA}{L}\left[\begin{array}{ccc}\frac{16}{3}& -\frac{8}{3}& \frac{7}{3}\end{array}\right]$

$-\frac{8}{3}\left[\begin{array}{c}{u}_a\end{array}\right]$

$u_b$

$\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]F=\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]\left[\begin{array}{c}2q\frac{L}{3}\end{array}\right]$

$q\frac{L}{6}+P0.$ Verify that the following global stiffness matrix and force

vector are obtained:

$\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]K\left[\begin{array}{c}{u}_a\end{array}\right]$

$u_b=\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]\frac{EA}{L}\left[\begin{array}{ccc}\frac{16}{3}& -\frac{8}{3}& \frac{7}{3}\end{array}\right]$

$-\frac{8}{3}\left[\begin{array}{c}{u}_a\end{array}\right]$

$u_b$

$\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]F=\left[\begin{array}{cc}{w}_a& w_b\end{array}\right]\left[\begin{array}{c}2q\frac{L}{3}\end{array}\right]$

$q\frac{L}{6}+P$