Long Columns with Central Loading (Buckling)
A column failure is always sudden, total, and unexpected, and hence dangerous. There is no advance warning. A beam will bend and give visual warning that it is overloaded; but not so for a column.
The relation between the critical load and the column material and geometry is developed with reference to the figure shown. We assume a bar of length l loaded by a force p acting along the centroidal axis with different end conditions.
(1)
Equation (1) is called the Euler column formula,
where
P_{cr} = critical load (N)
C = end condition constant (see Table)
E = modulus of elasticity (N/m^{2})
I = second moment of area (m^{4})
l = shaft length (m)
Table (Theoretical End Restraint Coefficients)
Illustration 
End Conditions 
C 
(a) 
Both ends pinned 
1 
(b) 
Both ends built in 
0.5 
(c) 
One end pinned, one end built in 
0.707 
(d) 
One end built in, one end free 
2 
(e) 
One end built in , one end fixed against rotation but free 
1 
(f) 
One end pinned, one end fixed against rotation but free 
2 
