# Euler's Column Formula

## Buckling of columns

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Columns fail by buckling when their critical load is reached. Long columns can be analysed with the Euler column formula

*F = n π ^{2} E I / L^{2} (1) *

*where *

*F = allowable load (lb, N)*

*n = factor accounting for the end conditions*

*E = modulus of elastisity (lb/in ^{2}, kPa)*

*L = length of column (in, m)*

*I = Moment of inertia (in ^{4}, mm^{4})*

### Factor Counting for End Conditions

- column pivoted in both ends :
*n = 1* - one end fixed, the other end rounded :
*n = 2* - both ends fixed :
*n = 4* - one end fixed, one end free :
*n = 0.25*

### Example - A Column Fixed in both Ends

A square aluminum column with length *5 m* is fixed in both ends. The Moment of Inertia of the column is *4 10 ^{8} mm^{4 } (4000 10^{-8} m)*, the Modulus of Elasticity of aluminum is

*69 GPa (69 10*and the factor for the column fixed in both ends is

^{9}Pa)*4.*

The Euler buckling load can be calculated as

*F = 4 π ^{2} (4000 10^{-8} m^{4}) (69 10^{9} Pa) / (5 m)^{2} *

* = 4354 N*

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