$8\mathrm{.}3$ The $\backslash (3-\backslash$mathrm$\{$m$\}\backslash )$ tall wooden vertical column has the dimensions shown in Figure $8\mathrm{.}29.$ There is a fixed collar at the middle of the column so that the column cannot deflect at the middle. Properties of wood are $\backslash ($ E$=11\backslash$mathrm$\{$GPa$\},\backslash$sigma$\_\{$Y$\}=40\backslash$mathrm$\{$MPa$\}\backslash ).$

i$.$ Determine the critical buckling load P $($produced by the top beam$)$ considering $($a$)$ the fixed collar is absent and $($b$)$ the fixed collar is present.

ii$.$ What is the most likely mode of failure, buckling type or compressive type when $($a$)$ the fixed collar is absent and $($b$)$ the fixed collar is present? Your answer must be supported by analysis.

iii. Find the critical length $\backslash ($ L $\backslash )$ at which both buckling failure and compressive failure become equally possible modes of failure.