Initially curved compressed beam

For a pin-ended slender beam with unstressed profile $y_0$, bending moment depends on curvature change: $M=EI(y-y_0)$. An axial dead load gives $EI(y-y_0)=-Fy$ in the small-slope model.

With $y_0=Y\sin (\pi x/L)$ and $F_c={\pi }^{2}EI/L^2$, the equilibrium branch connected to zero load has $y=Y\sin (\pi x/L)/(1-F/F_c)$. The pole as $F$ approaches $F_c$ marks breakdown of the small-deflection model, not a physically infinite beam deflection.