A beam$-$column, having a constant $EI$ along its length, is subjected to moment $M_a$ at both ends, while under a known axial compressive load $P.$ The orientations of $M_a$ are shown in the figure. Using the $4^{th}$ order differential equation of beam$-$column to find

$($a$)$ the moment $M$ at the mid$-$span $($i$.$e$.,x=\frac{L}{2})$ when the load ratio $\frac{P}{P_E}$ is equal to $0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}6,0\mathrm{.}8,0\mathrm{.}95,0\mathrm{.}99,$ respectively, where $P_E=\frac{{}^2}{L^2}EI.$

$($b$)$ In the Canadian Standard CSA$/$S$16,$ Clause $13\mathrm{.}8\mathrm{.}4,$ the factor, $U_1=\frac{1}{1-(\frac{P}{P_E})},$ is used to account for the second$-$order effects due to the deformation of a beam$-$coumn between it ends. Please compare the obtained moment from $($a$)$ with the moment $U_1{M}_a$ for the above $\frac{P}{P}E$ ratios.