MECHANICS  THEORY



Column Buckling

Four Basic Column Configurations 

Besides the basic pinnedpinned
(simply supported)
column, there are other types of column boundary conditions. A column can be fixed at
one or both ends, and if one end is fixed, then the other can be free.
Each of these basic column types (shown at the left) can be solved in a similar
fashion as the pinnedpinned column by developing a differential equation and
then solving it using the end conditions. (The derivations are omitted since
they are similar to the
simply support column case presented previously.)
The basic column types are given below with their critical load equation. 



Pinned  Pinned 
Free  Fixed 
Fixed  Fixed 
Pinned  Fixed 








Buckling Equations for Critical Load P_{cr} 



Pinned 
Pinned 
Free 
Fixed 
Fixed 
Fixed 
Pinned 
Fixed 
L_{e} = L 
L_{e} = 2 L 
L_{e}= 0.5L 
L_{e}= 0.7L 

Equivalent (or Effective) Lengths 

Alternate Buckling Equation Formulation


The four buckling equations given above can be summarized into one equation
using the concept of equivalent length (sometimes referred to as effective length). Each of the four buckled column types
have a similar buckling shape for a given part of their length. These similar
shapes are noted in the diagram as Equivalent
Length, L_{e}. Using this
equivalent length, L_{e}, a single buckling equation can be used for
all four column types.





Slenderness Ratio, L_{e}/r

For Short Column (Low Slenderness
Ratio) Failure May
be Compression 

As a column height is reduced, the critical buckling load increases. This relationship
can be seen if the basic Euler formula, P_{cr} = π^{2}EI/L^{2},
is graphed. To make it easier to plot this function, it can be rewritten
using the definition of the radius of gyration, r, which is
r = (I/A)^{0.5} [note,
this is not the radius]
Substituting, gives




Euler Curve for
Various Stiffness Values 

This function is only accurate in predicting buckling for slender columns. While each material is different, a slenderness ratio of 100 or greater will generally fail due to buckling. For columns with a slenderness ratio less than 100, the material may fail in compression and the yield stress must be check in addition to buckling.
As demonstrated with the simulation on the left, the value of E, stiffness, has little effect on the general shape of the Euler curve. 


