Normally when structural members are in compression it’s a good thing. They will not fail except by crushing (exceeding their compressive yield strength), and fatigue does not occur for elements in compression. However if the geometry of the member is such that it is a “column” then buckling can occur. Buckling is particularly dangerous because it is a catastrophic failure that gives no warning. That is, the structural system collapses often resulting in total destruction of the system and unlike yielding failures, there may be no signs that the collapse is about to occur. Thus design engineers must be constantly on vigil against buckling failure.
The classic analysis for buckling comes to us from Euler.The fundamental analysis is for a pinned-pinned joint.For this case,the critical load,i.e.,the load at which the column will collapse is given by
Pcr = π2 E I = π 2 E A
where, E is the modulus of Elasticity, I is the second moment of area,
L is the length of the column, A is the cross - sectional area of the column,
and Sr is the slenderness ratio Han expression of the aspect ratio of the columnL
The slenderness ratio is defined as
S = L
where, k is the radius of gyration, and is defined as
k = √ I/A = √ smoa/A
where, again I is the second moment of area, and A is the cross-sectional area of the column. This comes
from the definition of radius of gyration, I = k2 A.
The critical load can also be represented as a critical load unit by presenting it as a ratio of the critical
load to the cross-sectional area,i.e.,
Pcr = π2 E