# Euler Buckling Formula

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## Theorem

- $F = \dfrac {\pi^2 E I} {\paren {K L}^2}$

where:

- $F$ = maximum or critical force (vertical load on column)
- $E$ = modulus of elasticity
- $I$ = area moment of inertia of the cross section of the rod
- $L$ = unsupported length of column
- $K$ = column effective length factor, whose value depends on the conditions of end support of the column, as follows:

- For both ends pinned (hinged, free to rotate), $K = 1.0$
- For both ends fixed, $K = 0.50$
- For one end fixed and the other end pinned, $K \approx 0.699$
- For one end fixed and the other end free to move laterally, $K = 2.0$

- $K L$ is the effective length of the column

## Proof

## Also known as

The **Euler buckling formula** is also known as the **Euler column formula**.

## Source of Name

This entry was named for Leonhard Paul Euler.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$)