= Solution
Put $P=-F>0$. With $G=0$, the <Euler elastica> equation is $EI\theta''+P\sin\theta=0$. <Linearization> about the straight configuration gives $EI\theta''+P\theta=0$. Under the <boundary conditions> explicitly stated, $\theta(0)=\theta(L)=0$, a nonzero solution requires
$$
\boxed{P_n=EI\left(\frac{n\pi}{L}\right)^2,\qquad
\theta_n(s)=C\sin\frac{n\pi s}{L},\qquad n=1,2,\ldots.}
$$
Thus \b[with only the stated <boundary conditions>], the first critical compressive force is $\boxed{F_c=-\pi^2EI/L^2}$. The corresponding linear transverse <displacement field>, with $y(0)=0$, is
$$
y_n(s)=\frac{CL}{n\pi}\left(1-\cos\frac{n\pi s}{L}\right),
$$
and $x(s)=s$ to first order. These are the successive small-amplitude <buckling> modes. The nonlinear branches retain the <simple pendulum> equation; if the angular amplitude is $\theta_m$ and $m=\sin(\theta_m/2)$, fitting $n$ half-periods into the rod gives $P=4EI n^2K(m)^2/L^2$, which tends to $P_n$ as $m\downarrow0$. Here $K$ is the <complete elliptic integral of the first kind>, with $m$ its modulus, not its squared parameter.
\b[<Endpoint position compatibility for the Euler elastica> adds a condition when fully fixed clamps have equal transverse height.] If the word “clamped” also imposes $y(L)=y(0)$, then $\int_0^L\theta_n\,ds=0$. With $G=0$, this retains only even $n=2j$, yielding
$$
\boxed{P_j=4j^2\pi^2EI/L^2,\qquad P_{\rm first}=4\pi^2EI/L^2.}
$$
The same exclusion is exact on the nonlinear branches: integrating the elastica equation gives $P\int\sin\theta\,ds=-EI[\theta'(L)-\theta'(0)]$, and an odd number of half-periods reverses the nonzero endpoint slope. Hence odd $n$ changes the endpoint height. The conventional fixed-clamp first <Euler buckling> load is the second boxed value, whereas the source's explicitly listed <boundary conditions> alone give the first. Additional clamped modes with a nonzero transverse reaction require releasing the imposed $G=0$. Load-controlled buckling also permits axial shortening; an exactly inextensible rod whose endpoint separation is fixed equal to its full length has no room to buckle.
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