Long Answer Section: $($Total: $40$ marks$)$

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 on mm$.$

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

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 $.$