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Ideal Pinned Column Buckling Equation and Calculation, Euler's Formula

Mechanics of Materials Menu

For the ideal pinned column shown in below, the critical buckling load can be

calculated using Euler's formula:

Open: Ideal Pinned Column Buckling Calculator

Where:

E = Modulus of elasticity of the material

I = Minimum moment of inertia

L = Unsupported length of the column (see picture below)

 Note that regardless of the end condition, the critical load depends not on the material

strength, but rather the flexural rigidity, EI. Buckling resistance can be increased by

increasing the moment of inertia.

When a column buckles, it maintains its deflected shape after the application of the

critical load. In most applications, the critical load is usually regarded as the maximum

load sustainable by the column. Theoretically, any buckling mode is possible, but the

column will ordinarily deflect into the first mode. A column will buckle when the load

P reaches a critical level, called the critical load, Pcr .

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7/27/2019 Ideal Pinned Column Buckling Calculation and Equation - Engineers Edge

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Ideal Pinned Column (Pinned - Pinned)

For columns with different types of support, Euler's formula may still be used if the

distance L is replaced with the distance between the zero moment points. See

"Effective Length Constant Table" below.

This length is called the effective length Le and is illustrated in below. Thus the critical

load equation becomes:

The slenderness ratio is an important parameter in the classification of compression

members, and is represented by the equation:

Where:

r = Radius of gyration

I = Moment of inertia

A = Area cross section

If the slenderness ratio > (greater than) critical slenderness ratio, then the column is

treated as a long column and the Euler buckling formula is applicable.

If slenderness ratio is < (less than) the critical slenderness ratio, the column is treated

as a short column.

In short columns, failure may occur by compression without significant buckling and at

stresses exceeding the proportional limit. For this condition, Johnson's formula is

applicable:

 

7/27/2019 Ideal Pinned Column Buckling Calculation and Equation - Engineers Edge

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For columns that fail subsequent to the onset of inelastic behavior, the constant of 

 proportionality must be used rather than the modulus of elasticity (Engesser formula).

The constant of proportionality, Et, is the slope of the stress-strain diagram beyond

the proportional limit, termed the tangent modulus. Note within the linearly elastic

range, E = Et.

 

Effective Length Constant Table

Column BoundaryEffective Length Constant **

C = Le / L

Free - Free 1.0

Fixed - Free 2.0

Fixed - Pinned .707

Fixed - Guided 1.0

Fixed - Fixed .5

Guide - Free 2.0

Guided - Pinned 2.0

Guided - Guided 1.0

Pinned - Free 1.0

Pinned - Pinned 1.0

** In practice, the Effective Length Constant is increased by a factor approximately

10 - 20 %

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