Problem $1$: Conjugate beam method $(32$ pts$)$

For the beam shown in Figure $1$:

a$)$ Calculate the reactions of supports B and D; $(2$ DCL$+5)$ pts

b$)$ Draw diagrams V and M; $(2+3)$pts

Use the conjugate beam method to:

c$)$ Illustrate the conjugate beam; $(4$ pts$)$

d$)$ Determine vertical displacement at point $\backslash ($ A $\backslash )(\backslash (\backslash$Delta $\backslash$mathrm$\{$A$\})\backslash )$; $(8$ pts$)$

e$)$ Determine the slope at point B $(\backslash (\backslash$theta $\backslash$mathrm$\{$B$\}\backslash ))$; $(3$ pts$)$

f$)$ Determine the vertical displacement at point $\backslash (\backslash$mathrm$\{$C$\}(\backslash$Delta $\backslash$mathrm$\{$C$\})\backslash ).(5$ pts$)$

El is constant along the beam. $\backslash (\backslash$mathrm$\{$E$\}=29000\backslash )$ ksi and I $=50$ in$4$

$50$ kip