Problem $2$: Use the method of virtual work to determine the smallest moment of inertia $\backslash ($ I $\backslash )$ required for the beam shown, so that its maximum deflection does not exceed the limit of $\backslash (1/360\backslash )$ of the span length. Use both the mathematical and graphical evaluations of the virtual work integral.

$2\mathrm{.}1)$ Determine the real internal moment equations for the beam.

$2\mathrm{.}2)$ Determine the virtual internal moment equations for the beam.

$2\mathrm{.}3)$ Determine the smallest $\backslash ($ I $\backslash )$ so that the maximum deflection does not exceed the limit of $\backslash (1/360\backslash )$ of the span length. Evaluate the virtual work equation by mathematically integrating it$.$

$2\mathrm{.}4)$ Redo $2\mathrm{.}3$ by evaluating the virtual work equation using the graphical procedure.

$2\mathrm{.}5)$ Use the graphical procedure to determine the maximum slope of the beam. Use the value for $\backslash ($ I $\backslash )$ that you found in $2\mathrm{.}3$ and $2\mathrm{.}4.$

$2\mathrm{.}6)$ The loads on the beam represent the expected Dead Load on the system. No Live Load is expected. Using LRFD combination $1,$ you know to multiply the dead loads by $1\mathrm{.}4$ to obtain the design loads $($this includes all forces and couple moments$).$ Under these new design loads, what is the smallest moment of inertia $\backslash ($ I $\backslash )$ required so that the beam's maximum deflection does not exceed the limit of $\backslash (1/360\backslash )$ of the span length. PLEASE PROVIDE DETAILED ANSWERS!!