Problem description: The pin$-$jointed planar truss shown in the figure below is to be made of three steel two$-$force members and is to support vertical loads $P_B=3$kips at joint $B$ and $P_C=2$kips at joint $C.$ The lengths of members $AB$ and $BC$ are $L_1=36in.$ and $L_2=24in.,$ respectively. Joint $C$ is free to move vertically. For the steel truss members, the allowable stress in tension is $({}_T{)}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=22$ksi, the allowable stress in compression is $({}_C{)}_{\text{a}\text{b}\text{r}\text{w}}=-10$ksi, and the weight density is $0\mathrm{.}284l\frac{b}{i}n.{?}^3.$ You are to consider truss designs for which the vertical member $AC$ has lengths varying from $L_3=30in.$ to $L_3=50in.($a$)$ Show that, if each member has the minimum cross$-$sectional area that meets the strength criteria stated above, the weight $W$ of the truss can be expressed as a function of the length $L_3$ of member $AC$ by a function that is similar to the one plotted iexample $2\mathrm{.}11$ Fig $1($b$).($Hint: Use the law of cosines to obtain expressions for the angle at joint A and the angle at joint $C.)($b$)$ What value of $L_3$ gives the minimum$-$weight truss, and what is the weight of that truss?

To fulfill the project objectives, you will need to write and submit a report. The suggested format, content and weight $\%$ distribution are as follows:

$(1)5\%$ A description of the structure and the nature of the given problem, along with a brief plan to the solution.

$(2)60\%$ Lay out detailed steps to reach final solution, which should include the following:

a$)12\%$ Draw a FBD for each member and list respective knowns and unknowns.

b$)12\%$ Write equilibrium equations based on the FBDs and solve for support reactions

c$)12\%$ Determine member forces based on the equilibrium of joint $A,B$ and C $.$ Weight densities are used to determine the weight for all members $($or the truss$).$ Neglect weights of all pins and end fixtures.

d$)12\%$ Use the allowable stresses given by the problem to determine the minimum required cross section area of each member, and then the weight of all the members.

e$)12\%$ Give the math expression of the final solution, which is the truss weight varies as a function of $L_3$

$(3)10\%$ Using a computer program $($the type of the language is at your discretion$)$ to find the optimal $L_3$ at which $W$ is minimal. You may set the search range of the length of L$3$ between $30"$ to $50".$

$(4)10\%$ Plot $W$ vs $L_3$ in the given range of $L_3,$ indicating the minimum value of $L_3$ on the plot.

$(5)5\%$ Discussions and conclusions

$(6)10\%$ Format and neatness

Objectives:

The project is to implement the optimal design of a frame based on Allowable Stress of the members' properties. The theories are covered in Section $2\mathrm{.}8,$ Chapter $2$ of Mechanics of Materials, Introduction to Design. The relevant textbook example $2\mathrm{.}11$ may be referenced. Prior to solving, read and understand the question descriptions. When acceptable solutions are reached you will be writing a computer program to pick the best $/$ optimal solution among the acceptable ones to complete the implementation of Optimal Design.

Problem description: The pin$-$jointed planar truss shown in the figure below is to be made of three steel two$-$force members and is to support vertical loads $P_B=3$kips at joint $B$ and $P_C=2$kips at joint $C.$ The lengths of members $AB$ and $BC$ are $L_1=36$ in$.$ and $L_2=24in.,$ respectively. Joint $C$ is free to move vertically. For the steel truss members, the allowable stress in tension is $({}_T{)}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=22$ksi, the allowable stress in compression is $({}_C{)}_{\text{a}\text{l}\text{k}\text{w}}=-10$ksi, and the weight density is $0\mathrm{.}284$