Euler Buckling formula 
Dear all, above shows euler buckling formula for different k values to find critical buckling load. May I know if there is a formula to calculate the displacement highlighted in red ?
Thank you for reading and have a nice day !!
 A: First-order instability theories
With a linear instability theory, the amplitude of the displacement is not determined, since it comes as the eigenfunction of an eigenvalue problem.
Example: hinge-hinge beam
Let's see it for the hinge-hinge beam on the left side of your picture.
Writing the equilibrium equation in the deformed configuration, neglecting shear internal actions, we get
$F y(x) = M(x) = - EJ y''(x)$,
being $y(x)$ the equation for the deformed configuration, $F$ the external force applied, $M(x)$ the internal bending moment, and $EJ$ bending stiffness. Rearranging and providing the boundary conditions, we get
$y''(x) + \dfrac{F}{EJ} y(x) = 0, \qquad y(0) = 0 \quad, \qquad y(\ell) = 0$.
The solution reads
$y(x) = A \sin\left(\sqrt{\dfrac{F}{EJ}} x\right)$
with $\sqrt{\dfrac{F}{EJ}} \ell = n \pi$, with $n \in \mathbb{Z}$, and amplitude $A$ not determined. The value of the external force $F_n$ for instability to occur is $F_n = \dfrac{n^2 \pi^2 EJ}{\ell}$, being $F_1 = \dfrac{\pi^2 EJ}{\ell}$ the minimum value. Thus, you can rearrange the deformed configuration as
$y(x) = A \sin\left(\sqrt{\dfrac{F}{EJ}} x\right) = A \sin\left( \pi\dfrac{x}{\ell}\right)$
and the internal bending moment reads
$M(x) = -EJ y''(x) = F A \sin\left(\sqrt{\dfrac{F}{EJ}} x\right) =  F A \sin\left( \pi\dfrac{x}{\ell}\right)$.
Anyway the amplitude is not determined by linear theories alone.
