In the figure below, a linearly increasing distributed load with maximum strength $\backslash ($ w$\_\{\backslash$max $\}=14\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ is applied to a beam. The total length of the beam is $\backslash ($ L$=18\backslash$mathrm$\{$~m$\}\backslash )$ and the distances between points are $\backslash ($ a$=8\backslash$mathrm$\{$~m$\}\backslash ),$ and $\backslash ($ b$=10\backslash$mathrm$\{$~m$\}\backslash ).$

A$.$ Calculate the support reactions. Include the relevant free$-$body diagram.

B$.$ Determine the internal shear force and bending moments as functions of position $\backslash ($ x $\backslash ),$ where $\backslash ($ x $\backslash )$ is the distance along the beam measured from point $\backslash ($ A $\backslash ).$ Include any relevant free$-$body diagrams.

C$.$ Sketch the internal shear force $\backslash ($ V$($x$)\backslash )$ and bending moment $\backslash ($ M$($x$)\backslash )$ along the entire beam. Calculate and label the values of the shear force and bending moment at the end points of each piecewise function. Label the values and location of any minimum or maximum values as well.