Home exercise $4\mathrm{.}2$

Given a two$-$span linear elastic beam. Joints A and $B$ are fixed, and simple support is located at joint $C.$ The beam is loaded by a uniformly distributed load with intensity $q$ acting on the left span. A concentrated moment $M_C$ is applied at joint $C$ in the clockwise direction as shown in the figure. The span $AC$ has length $2L$ and bending stiffness $2EI.$ The span $CB$ has length $L$ and bending stiffness $EI.$

Use the slope$-$deflection method to determine the rotation at joint $C$ and bending moment at joints A and B$.(5$p$)$

Determine and draw the distribution of the moment and shear force. $(2$p$)$

Hint. Remember the sign convention for the bending moment.