Figure $3($a$)$: fixed$-$fixed column for exercise $11\mathrm{.}3.($b$)$ FBD if complete column; $($c$)$ FBD at distance $x$ from top.

An elastic column $AB$ of length $L$ is fixed at both ends $($meaning that end rotations are precluded$)$ and compressed by an axial load $P$ at the top end $A.$ Axis $x$ origin at $A,$ positive down. The elastic modulus is $E$ and minimum second$-$moment of inertia of the cross section is I. See Figure$\mathrm{.}4($a$).$ The kinematic boundary conditions are $v^{\text{'}}(0)=0,v(0)=0,v^{\text{'}}(L)=0,$ and $v(L)=0.$ Using LPB analysis, find the first two critical loads $P_{cr}=P_{cr1}$ and $P_{cr2}>P_{cr1},$ and verify that the effective length for the first critical load is $L_{\text{e}\text{f}\text{f}}=\frac{1}{2}L$

Partial answers. ODE obtained via FBD of Figure$\mathrm{.}4($c$)$: EIv ${?}^{\text{'}\text{'}}+Pv=R_{A}x+M_A$; characteristic equation: $[2(1-cosL)-LsinL]=0$; first critical load $P_{cr1}=\frac{4{}^{2}EI}{L^2}=39\mathrm{.}48E\frac{\text{I}}{L^2}.$

$($Second one is not $4$ times $P_{cr1},$ as would be the case for a pinned$-$pinned column.$)$