First, derive the linear equations for Joint $3($i$.$e$.,$ explain where they come from$).$ Then solve

the linear system of equations using Matlab. To do this, use the built$-$in Matlab function for

LU factorization as follows: $[$L$,$U$,$P$]=$ lu$($A$).$ To solve the two resulting triangular systems,

you may use the $$operator$.$ Use the diary command to record the output from running your

script. Turn$-$in your Matlab script and output for this problem.

$2\mathrm{.}3.$ The following diagram depicts a plane truss

having $13$ members $($the numbered lines$)$ con$-$

nected by $10$ joints $($the numbered circles$).$ The

indicated loads, in tons, are applied at joints $2,$

$5,$ and $6,$ and we wish to determine the resulting

force on each member of the truss.

For the truss to be in static equilibrium, there

must be no net force, horizontally or vertically,

at any joint. Thus, we can determine the mem$-$

ber forces by equating the horizontal forces to the

left and right at each joint, and similarly equat$-$

ing the vertical forces upward and downward at

each joint. For the eight joints, this would give

$16$ equations, which is more than the $13$ unknown

forces to be determined. For the truss to be stati$-$

cally determinate, that is$,$ for there to be a unique

solution, we assume that joint $1$ is rigidly fixed

both horizontally and vertically, and that joint $8$

is fixed vertically. Resolving the member forces

into horizontal and vertical components and defin$-$

ing $=\frac{\sqrt[\phantom{2}]{2}}{2},$ we obtain the following system of

equations for the member forces $f_i$ :

Joint $2$: $\{\begin{array}{c}{f}_2=f_6\end{array}$

$f_3=10$

Joint $3$: $\{\begin{array}{c}{f}_1=f_4+f_5\end{array}$

$f_1+f_3+f_5=0$

Joint $4$:$\{\begin{array}{c}{f}_4=f_8\end{array}$

$f_7=0$

Joint $5$: $\{\begin{array}{c}{f}_5+f_6=f_9+f_{10}\end{array}$

$f_5+f_7+f_9=15$

Joint $6$: $\{\begin{array}{c}{f}_{10}=f_{13}\end{array}$

$f_{11}=20$

Joint $7$:$\{\begin{array}{c}{f}_8+f_9=f_{12}\end{array}$

$f_9+f_{11}+f_{12}=0$

Joint $8$: $\{\begin{array}{c}{f}_{13}+f_{12}=0\end{array}$

Use a library routine to solve this system of linear

equations for the vector f of member forces. Note

that the matrix of this system is quite sparse, so you may wish to experiment with a banded system solver or more general sparse solver, although

this particular problem instance is too small for

these to offer significant advantage over a general

solver