Buckling of Columns

 Key Concepts:  Structures under compressive loads might fail due to excessive compressive normal stress (termed crushing) or may become unstable and deflect laterally.  If the latter happens the structure is said to "buckle".  Generally long, slender structures, termed "columns", are susceptible to buckling under compressive loads.

In a Nut Shell:  The objective in the analysis of columns undergoing buckling is to predict

the “buckling load” for various types of end supports.  The top figure below depicts a simply supported column of length L and of flexural rigidity EI with compressive load P.  The lower figure shows the buckled shape after lateral deflection.

The differential equation governing the deflection, y(x),  is:

 d2y/dx2 + (P/EI) y  =  0

where  y(x) is the lateral deflection of the column

P  is the compressive load acting on the column

E  is the modulus of elasticity for the material of the column

I  is the moment of area of the X-section of the column

EI  is termed the flexural rigidity of the column

For simply supports  y(0) =  y(L)  =  0 which leads to solution   sin( √(P/EI) L)  =  0
which is satisfied when   √(P/EI) L)  =  n π.