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For the ideal pinned column shown in below, the
critical buckling load can be calculated using Euler's formula:
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.
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:
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
Boundary |
Effective
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 |
|
