Problem MT$2-1(50$ points$)$ Rigid bar ACE, which has total length $2$L$,$ is supported by three vertical elastic bars at points $\backslash (\backslash$mathrm$\{$A$\},\backslash$mathrm$\{$C$\}\backslash )$ and E $.$ Bar ACE is horizontal at temperature $\backslash ($ T$\_\{0\}\backslash )$ and in the absence of any applied forces. The three vertical members are all of length H $,$ and are made of the same material with Young's modulus $\backslash ($ E $\backslash )$ and linear coefficient of thermal expansion $\backslash (\backslash$alpha $\backslash ).$ Member AB has cross sectional area $\backslash ($ A $\backslash ),$ member CD has area $\backslash (2$ A $\backslash ),$ and member EF has area A$/2.$

Let the temperature of members AB and $\backslash ($ E F $\backslash )$ increase from $\backslash ($ T$\_\{0\}\backslash )$ to $\backslash ($ T $\backslash ),$ while the temperature of bar CD does not change. Also assume that member $\backslash ($ A C E $\backslash )$ does not change length during this thermal loading.

a$.$ Determine the stresses in the three vertical members.

b$.$ Determine the rotation of bar ACE.

Express your answers in terms of the parameters given: $\backslash ($ A$,$ E$,\backslash$alpha$,\backslash$mathrm$\{$~L$\},\backslash$mathrm$\{$H$\},$ T$\_\{0\}\backslash )$ and $\backslash ($ T $\backslash ).$