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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 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 (241 10-8  m4) (69 109 Pa) / (5 m)2 

 =  262594 N

 = 263 kN

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