The structure shown in Figure $1$ is supported by fixed support at A and roller support at $B$ and D$.$ An internal hinge at $C$ connects the elements of this structure. A distributed load with the intensity of $q=10k\frac{N}{m}$ is acting from A to C $.$

a$)$ Calculate the bending moment at A and $B(M_A$ and $M_B)$ using the method of conjugate beam.

$($Hint: first of all, find $M_B$ using the equilibrium of the original beam. Then, consider the condition for equilibrium of the conjugate beam for $M_A)$

b$)$ Calculate the reaction of all supports.

c$)$ Draw the bending moment diagram of the structure.

d$)$ Find the following quantities using the method of conjugate beam:

i$.)$ Rotation at point $B({}_B).$

ii$.)$ Vertical displacement at point $C({}_{cy}).$

iii.$)$ Rotation at point C $($at the left and the right of the hinge$)({}_{C^{-}}{}_{c^{+}}).$

iv$.)$ Vertical displacement at points $E({}_{Ey}).$

v$.)$ Rotation at point $E({}_E).$