Reading and references:

For each of the following exercises create the specified files with
your answer(s). Submit the requested files using the
course **submit** utility.

We will set up the solution using the "method of joints". We write the free-body diagrams for each of the five vertices (joints) and require that the sums of the horizontal and vertical force components vanish. Also, we require that the net torque on the entire structure about some point vanishes. For this, we choose joint A, but any point works. We get a set of linear equations in the unknown forces, which are easily solved by a systematic approach.

Rather than solving the problem systematically "by hand", as given by
MATHalino, we will use the Python **numpy.linalg** package. So we
will write down the full set of linear equations, convert them to a
matrix problem, and write Python code to read the matrix and vector,
solve the linear system of equations, and write the answer. We do
this to develop experience with the process, so we can be prepared to
solve more complicated problems, such as a real 3D bridge with more
trusses.

So, first we set up the linear system of equations. We will take the convention that the forces are all tensions. Remember that tension in a rod pulls inward from either free end. If a tension turns out to be negative, it is a compression, meaning it pushes outward at the free ends. Here are the equations we need to solve. Please check them. See the free-body diagrams in the reference solution to follow the setup:

Balance clockwise torque about A \[ 1000 \times 5 + 3000 \times 15 + 4000 \times 10 - 20 R_E = 0 \] Balance net force on joint A \[ F_{AB} \cos(60) + F_{AC} = 0 \\ R_A + F_{AB} \sin(60) = 0 \] Balance net force on joint B \[ -F_{AB} \cos(60) + F_{BD} + F_{BC} \cos(60) = 0 \\ -F_{AB} \sin(60) - F_{BC} \sin(60) - 1000 = 0 \] Balance net force on joint C \[ -F_{AC} - F_{BC} \cos(60) + F_{CD} \cos(60) + F_{CE} = 0 \\ F_{BC} \sin(60) + F_{CD} \sin(60) - 4000 = 0 \] Balance net force on joint D \[ -F_{BD} - F_{CD} \cos(60) + F_{DE} \cos(60) = 0 \\ -F_{CD} \sin(60) -F_{DE} \sin(60) - 3000 = 0 \] Balance net force on joint E \[ -F_{CE} - F_{DE} \cos(60) = 0 \\ -F_{DE} \sin(60) - R_E = 0 \] Note that the angle 60 is in degrees, not radians!

For this exercise, you are asked to convert these equations into a
matrix form **S t = w**, where **S** is an 11 X 9
matrix, **t** is a column vector with 9 unknowns and **w** is a
column vector with constant weights. If it isn't clear how to go from
equations to matrix multiplication, see the example in the first
triangulation box in the Gaussian-elimination handout linked above.
So we can check your work, please arrange the matrix so the columns of
the matrix (and the elements of the solution vector **t**) go in
this order:
\[ R_A, F_{AB}, F_{AC}, F_{BD}, F_{BC}, F_{CD}, F_{CE}, F_{DE}, R_{E} \]
and the rows are in the same order as the equations above. Create a
file called **intruss** with 11 lines, one
for each row of the matrix. On the **i**th line, put
the **i**th row of the matrix, followed by the right-hand-side
value **w[i]** for that row. This strategy makes it easy to
eliminate an equation (see below).

Later in this assignment, you will need to delete two rows of this
file in order to get just nine equations with nine unknowns. Please
submit the file **intruss** with the nine
rows that you actually use below to get the solution.

Put your code in a file **truss.py** to hand in.

Of course, you will want to check your answers against the MATHalino answers.

(b) How much load can the bridge take? Suppose the bridge members can withstand a compression of 6000 pounds before buckling and a tension of 3000 pounds before snapping. With the load at points B and D kept the same as shown in the figure, how much load at point C can the bridge take before it fails? Which member fails first? Put your answers in the solution file. Also, explain how your solved the problem.

Put all your answers in the **solution** file. Then submit it.