Euler's formula: fixed-free column with limitations

Euler's critical load for a column fixed at one end and free at the other is derived below.

Assumptions

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, length $L$, have its lower end A fixed and its upper end B free. Under the load $P$, the free end deflects by $\delta$. Take the origin at the fixed end A, with $y$ measured from the original axis.

      P
      v
   B  )  <- free end, deflection delta
      )
      ) y
      )
   A  fixed

At a section distance $x$ from A, the lever arm of $P$ is $(\delta - y)$, so $M = P(\delta - y)$:

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

The solution is

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

Boundary conditions:

• At $x=0$: $y=0 \Rightarrow C_1 = -\delta$.
• At $x=0$: $\dfrac{dy}{dx} = 0 \Rightarrow C_2 = 0$.

So $y = \delta\,(1 - \cos kx)$.

• At $x=L$: $y = \delta \Rightarrow \delta = \delta(1 - \cos kL) \Rightarrow \cos kL = 0$

The smallest solution is $kL = \dfrac{\pi}{2}$:

$$\frac{PL^2}{EI} = \frac{\pi^2}{4}$$ $$P_{cr} = \frac{\pi^2EI}{4L^2} = \frac{\pi^2EI}{(2L)^2}$$

The effective length is $L_e = 2L$, so this column carries one quarter of the load of a hinged-hinged column of the same length.

Limitations of Euler's formula

1. It applies only to long (slender) columns. In terms of stress, $\sigma_{cr} = \dfrac{\pi^2E}{\lambda^2}$ must not exceed the proportional limit, so $\lambda \ge \pi\sqrt{E/\sigma_p}$ (about 80 to 100 for mild steel). For short and intermediate columns it over-estimates the load.
2. It assumes a perfectly straight column, a perfectly axial load and homogeneous material. Real columns have initial crookedness and eccentricity, so they fail at a lower load.
3. It is valid only within the elastic range (Hooke's law), so $\sigma_{cr}$ cannot exceed the proportional limit.
4. It ignores the effect of direct compression, shear deformation and imperfect end conditions.
5. For very slender columns it is accurate, and for intermediate columns empirical formulas (Rankine, Johnson, Secant, straight-line) are used instead.