Short Answer Section $($Total: $20$ marks$)$

Problem $1$: $(9$ marks$)$

The structure above is subject to two external loads of equal magnitude P$=500$kN$.$ All three members have a circular cross$-$section with area equal to A$=4000$mm$^(2).$ The length between supports A and B is L$=5$m$.$ All members are made of the same material with Young's Modulus E$=200$GPa. For this question, calculate all final numerical answers to THREE decimal places and submit these answers into the appropriate field in Blackboard.

a$)(2$ marks$)$ Draw an FBD of joint C$.$ To earn full marks, you must have all the relevant forces and angles denoted on your FBD$.$

b$)(2$ marks$)$ Calculate the vertical displacement of joint C$.$ Give your final numerical answer in mm$.$

c$)(1$ marks$)$ Calculate the horizontal displacement of joint C$.$ Give your final numerical answer in mm$.$

d$)(2$ marks$)$ Calculate the internal axial stress in member AB$.$ Give your final numerical answer in GPa.

e$)(1$ mark$)$ Which of the members has the highest internal axial stress in magnitude?

f$)(1$ mark$)$ Which of the members has the lowest internal axial stress in magnitude?

Problem $2$: $(11$ marks$)$

Two rubber pads with thickness $($t$=4$mm$)$ are clamped between a set of three steel plates. The rubber pads are square in cross$-$section with side length equal to $160$ mm $.$ If a loading is applied to the steel plates as shown above with P$=30$kN$,$ answer the following questions. For this question, calculate all final numerical answers to THREE decimal places.

a$)(2$ marks$)$ Does the diagram above represent single or double shear? Explain. In order to earn full marks you must provide an explanation, not just either "single" or "double".

b$)(2$ marks$)$ Calculate the magnitude of the average shear stress on the rubber pads. Give your final numerical answer in kPa $.$

c$)(2$ marks$)$ If the rubber is known to have a Young's modulus of E$=0\mathrm{.}25$GPa and a Poisson ratio of $0\mathrm{.}35,$ calculate the magnitude of the shear strain under the stress condition described above. Give your final numerical answer in $\%.$

d$)(2$ marks$)$ Calculate the magnitude of the relative displacement for top surface of the pad with respect to the bottom surface of the pad given the shear stress. Give your final numerical answer in mm$.$

e$)(2$ marks$)$ Draw a diagram which shows how the cross$-$section of the rubber pad deforms under the stress. In order to earn full marks your diagram must be fully labelled with relevant quantities such as shear force, strain, displacement etc.

f$)(1$ mark$)$ If the thickness of the pad doubled, would the shear stress go up$,$ down, or remain the same?

Problem $3$: $(27$ marks$)$

Beam ABCD is tied to steel members BE and CF $.$ Due to forces P$\_(1)$ and P$\_(2),$ the beam deflects as stress is applied to the steel members. Assume P$\_(1)=400$kN and P$\_(2)=520$kN$.$ Due to differences in the steel

manufacturing, the Youngs modulus of BE and CF is $150$ GPa and $200$ GPa respectively. Finally, the cross$-$sectional areas of members BE and CF are $11000$mm$^(2)$ and $10325$mm$^(2)$ respectively. Assume that

beam ABCD is initially horizontal prior to loading.

NOTE: for the answers below, round all final numerical answers to THREE decimal places. Pay attention to the units requested for your final answer.

a$)(1$ mark$)$ Based on the problem setup, do you expect member BE to be in tension or compression?

b$)(1$ mark$)$ Based on the problem setup, do you expect member CF to be in tension or compression?

c$)(3$ marks$)$ Draw FBDs for the following: Beam ABCD, member BE$,$ and member CF$.$ To earn full marks on this you must have all the correct forces and directions on your diagrams.

d$)(3$ marks$)$ Calculate the deformation in member BE$.$ Provide your final numerical answer in mm$.$

e$)(3$ marks$)$ Calculate the deformation in member CF$.$ Provide your final numerical answer in mm$.$

f$)(2$ marks$)$ Calculate the vertical change in position for point A after the deformation in the steel members occurs. Give your final numerical answer in m$.$

g$)(2$ marks$)$ Calculate the vertical change in position for point D after the deformation in the steel members occur. Give your final numerical answer in m$.$

h$)(2$ marks$)$ Calculate the Poisson's ratio for both CF and BE is the shear modulus for CF is $75$ GPa and the shear modulus for BE is $60$ GPa $.$ Give your final numerical answers.

i$)(4$ marks$)$ Calculate the new cross$-$sectional area of member BE after the deformation. Assume a circular cross$-$section. Give your final numerical answer in mm$^(2).$

j$)(4$ marks$)$ Calculate the new cross$-$sectional area of member CF after the deformation. Assume a circular cross$-$section. Give your final numerical answer in mm$^(2).$

k$)(2$ marks$)$ Explain why the solution method used for the above parts only approximates the true loading distribution on beam ABCD.