Part B: Linear Systems and Design Decisions $($will be graded$)$

Consider the structure in Figure $3.$ We will study the forces in the members of this structure

for various angle $.$ Develop the equations of static equilibrium using the method of joints

and write them in a matrix form; the stiffness matrix A should be in terms of $.$

Hint $1$: The supporting structure is a pin at A and a roller at D $.$ While varying the choice

of $,$ the length and position of the members $AB,AE,ED$ and CD stay unchanged, the

length of the members $BC,BE$ and CE adjusts as $$ varies. You only need the angles to

construct your linear system.

Hint $2$: At each joint, you can assume all forces to be pointing out of the joint.

Hint $$ n $n=-2$dots$+$dots

Figure $3$: Structure for lab assignment.

You will write a single MATLAB script that does the following:

Compute all internal forces and reaction forces simultaneously for a particular value of

$.$ You may use the $""$ command; the use of inv$()$ is prohibited. $[$Note that the default

unit for angle is radians in MATLAB$]$

Vary $$ in the range of $5$ to $55($in $1$ increments$)$ and in a single figure, plot the mag$-$

nitudes of all the internal and reaction forces, as a function of $.$ Your figure should be

properly labeled with readable font size and units if applicable, each curve should be dis$-$

tinguishable from the other $($by use of line styles, thickness, colors, etc.$).($Optional: If you

want to make the colors distinguishable, a set of RGBs that are maximally perceptually

distinct is given in Appendix at the end.$)$

$($Bonus$)$ Optimal design: If you have to minimize the maximum internal force, what is

the best choice of $?$ Justify your answer with a MATLAB figure and add it$/$them to the

figure PDF as subplots if multiple plots are needed.