Euler's Column Formula
Calculate 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 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}, Pa (N/m^{2}))
L = length of column (in, m)
I = Moment of inertia (in^{4}, m^{4})
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 = π^{2} 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 Ibeam 7 x 4 1/2 x 5.80 with a Moment of Inertia i_{y} = 5.78 in^{4}. The Modulus of Elasticity of aluminum is 69 GPa (69 10^{9} Pa) and the factor for a column fixed in both ends is 4.
The Moment of Inertia can be converted to metric units like
I_{y} = 5.78 in^{4} (0.0254 m/in)^{4 }
= 241 10^{8} m^{4}
The Euler buckling load can then be calculated as
F = (4) π^{2} (69 10^{9} Pa) (241 10^{8} m^{4}) / (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
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