Problem $2$: $(13$ marks $+3$ BONUS marks$)$

Consider a member AB with weight W held up by a network of metal cables shown in the figure below. All the cables are made of the same material and have the same diameter $\backslash ($ d$=4\backslash$mathrm$\{$~mm$\}\backslash ).$ Assume the material yield stress is $\backslash (\backslash$sigma$\_\{$y$\}=60\backslash$mathrm$\{$MPa$\}\backslash ).$ One of the requirements in the design of this structure is that the stress in the cables must never exceed $\backslash (60\backslash \%\backslash )$ of the cable yield stress. In this problem, we would like to find the maximum allowable weight of member $\backslash ($ A B $\backslash )$ based on the stress allowed in each cable. I will walk you through the solution steps in each of the parts below. You will earn part marks by showing your work and drawing the correct FBDs$.$ Parts j and k are for BONUS marks ONLY.

a$.(2$ marks$)$ Calculate the length of each cable in the system. Provide your numerical results in $\backslash (\backslash$mathbf$\{$m$\}\backslash )$ to two decimal places, in the appropriate location on Blackboard.

b$.(2$ marks$)$ Draw a FBD of the member AB$.$ In order to earn full marks you must have all of the relevant forces and angles shown in your figure.

c$.(1$ mark$)$ Using your FBD from part b$),$ show that the magnitude of the force in AD is equal to magnitude of the force in BC $($ie$.\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$AD$\}\}=\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$BC$\}\}\backslash )).$

d$.(2$ marks$)$ Draw an FBD of cable CD$.$ In order to earn full marks you must have all of the relevant forces and angles shown in your figure.

e$.(1$ mark$)$ Using your FBD for member CD from part d$),$ show that the magnitude of the force in AD is equal to magnitude of the force in DE $($ie$.\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$AD$\}\}=\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$DE$\}\}\backslash )).$

f$.(2$ marks$)$ Provide an explanation for the following: To calculate the max allowable weight of member AB $,$ it is sufficient to calculate the stress in AD and CD only. Note: this problem would be quite long if you had to solve for the stress in every single member.

g$.(1$ mark$)$ Write an equation which relates the stress in AD to the weight W $.$

h$.(1$ mark$)$ Write an equation which relates the stress in CD to the weight W $.$

i$.(1$ mark$)$ Calculate the maximum weight allowable in member AB$.$ Provide your numerical answer in $\backslash (\backslash$mathbf$\{$N$\}\backslash )$ to two decimal places in the appropriate location on Blackboard.

j$.($BONUS$)(2$ marks$)$ Calculate the axial elongation in each member using the fact that the Young's modulus of this material is $\backslash (\backslash$mathrm$\{$E$\}=175\backslash$mathrm$\{$GPa$\}\backslash ).$ Provide your numerical results in $\backslash (\backslash$mathbf$\{$m m$\}\backslash )$ to two decimal places, in the appropriate location on Blackboard.

k$.($BONUS$)(1$ mark$)$ Calculate the vertical displacement of the entire system using the elongation of the cables due to the weight of member AB $.$ Provide your numerical results in mm to two decimal places, in the appropriate location on Blackboard.