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# Euler's Column Formula

## Buckling of columns

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

F = n π2 E I / L2                             (1)

where

F = allowable load (lb, N)

n = factor accounting for the end conditions

E = modulus of elastisity (lb/in2, Pa (N/m2))

L = length of column (in, m)

I = Moment of inertia (in4, m4)

### Factor Counting for End Conditions

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

### Note!

Equation (1) is sometimes expressed with a k factor accounting for the end conditions:

F = π2 E I / (k L)2                             (1b)

where

k = (1 / n)1/2    factor accounting for the end conditions

 n 1 4 2 0.25 k 1 0.5 0.7 2

### Example - A Column Fixed in both Ends

An column with length 5 m is fixed in both ends. The column is made of an Aluminium I-beam 7 x 4 1/2 x 5.80 with a Moment of Inertia iy = 5.78 in4. The Modulus of Elasticity of aluminum is 69 GPa (69 109 Pa) and the factor for a column fixed in both ends is 4.

The Moment of Inertia can be converted to metric units like

Iy = 5.78 in4 (0.0254 m/in)4

= 241 10-8  m4

The Euler buckling load can then be calculated as

F = (4) π2 (69 109 Pa) (241 10-8  m4) / (5 m)2

=  262594 N

= 263 kN

### Slenderness Ratio

The term "L/r" is known as the slenderness ratio. L is the length of the column and r is the radiation of gyration for the column.

• higher slenderness ratio - lower critical stress to cause buckling
• lower slenderness ratio - higher critical stress to cause buckling
• slenderness ratios L/r < 40: "short columns" where failure mode is crushing (yielding)
• slenderness ratios 40 < L/r < 120: "intermediate columns" where failure mode is a combination of crushing (yielding) and buckling
• slenderness ratio of 120 < L/r < 200: "long columns" where failure mode is buckling

## Related Topics

• Beams and Columns - Deflection and stress, moment of inertia, section modulus and technical information of beams and columns

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