Euler's formula: both ends hinged

Euler's critical (crippling) load is the smallest axial load at which an initially straight, slender column becomes unstable and buckles sideways.

Assumptions

The column is perfectly straight, of uniform section and homogeneous material, the load is perfectly axial, the material obeys Hooke's law, deflections are small, and the effect of shear and axial shortening is ignored.

Derivation

Let the column AB of length $L$ be hinged at both ends, with the load $P$ producing a small deflection $y$ at distance $x$ from A.

   P
   |  B
   |  /\
   |  \ y
   |  /
   A /
   P
 (A origin, x along column)

The bending moment at the section is $M = -Py$ (the moment of $P$ tends to reduce the deflection, so it is opposite in sign to $y$). Using the differential equation of the elastic curve:

$$EI\frac{d^2y}{dx^2} = -Py \;\Rightarrow\; \frac{d^2y}{dx^2} + k^2 y = 0,\qquad k^2 = \frac{P}{EI}$$

The solution is

$$y = C_1\cos kx + C_2\sin kx$$

Boundary conditions:

• At $x = 0$, $y = 0 \Rightarrow C_1 = 0$.
• At $x = L$, $y = 0 \Rightarrow C_2\sin kL = 0$.

For a buckled shape $C_2 \ne 0$, so $\sin kL = 0$ and $kL = n\pi$. The smallest non-zero load is for $n = 1$:

$$k L = \pi \;\Rightarrow\; \frac{PL^2}{EI} = \pi^2$$ $$P_{cr} = \frac{\pi^2 EI}{L^2}$$

Here $I$ is the least moment of inertia, since the column buckles about the axis of least resistance. The shape is a half sine wave, $y = C_2\sin(\pi x/L)$, with maximum deflection at mid-height.