A fixed$-$end beam is loaded by a uniform load $\backslash ($ q$=15\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash )$ and a point load $\backslash ($ P$=\backslash )30$ kN at mid$-$span. The beam has a length of $4$ m and modulus of elasticity of $205$ GPa $.($Hint: Use the method of superposition and Table H$).$

a$.$ Find reactions at $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ).$

b$.$ Calculate the height of the beam if the displacement at mid$-$span is known to be $3$ mm $.$ Assume that the beam has rectangular cross section with $\backslash ($ h $/$ b$=2\backslash ).$