# Euler's Column Formula

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 I-beam 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*and the factor for a column fixed in both ends is

^{9}Pa)*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

## Related Topics

### • Beams and Columns

Deflection and stress in beams and columns, moment of inertia, section modulus and technical information.

## Related Documents

### Area Moment of Inertia - Typical Cross Sections I

Typical cross sections and their Area Moment of Inertia.

### Area Moment of Inertia - Typical Cross Sections II

Area Moment of Inertia, Moment of Inertia for an Area or Second Moment of Area for typical cross section profiles.

### British Universal Columns and Beams

Properties of British Universal Steel Columns and Beams.

### Concrete Columns

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### HE-A Steel Beams

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### Mass Moment of Inertia

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### Mild Steel - Square Bars Weight

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### Radius of Gyration in Structural Engineering

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### Steel Pipe Columns - Allowable Loads

Allowable concentric loads for steel pipe columns.

### Stiffness

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### Three-Hinged Arches - Continuous and Point Loads

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### W-Beams - American Wide Flange Beams

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### Weight of Beams - Stress and Strain

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### Wood Columns - Safe Loads

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### Young's Modulus, Tensile Strength and Yield Strength Values for some Materials

Young's Modulus (or Tensile Modulus alt. Modulus of Elasticity) and Ultimate Tensile Strength and Yield Strength for materials like steel, glass, wood and many more.