Consider a buckled simply-supported column of length L under an external axial compression force F, as shown in the left schematic below. The transverse displacement of the buckled column is represented by w.

        

The right schematic shows the forces and moments acting on a cross-section in the buckled column. Moment equilibrium on the lower free body yields a solution for the internal bending moment M,

Recall the relationship between the moment M and the transverse displacement w for an Euler-Bernoulli beam,

Eliminating M from the above two equations results in the governing equation for the buckled slender column,

The governing equation is a second order homogeneous ordinary differential equation with constant coefficients and can be solved by the method of characteristic equations. The solution is found to be,

where . The coefficients A and B can be determined by the two boundary conditions , which yields,

The coefficient B is always zero, and for most values of m*L the coefficient A is required to be zero. However, for special cases of m*L, A can be nonzero and the column can be buckled. The restriction on m*L is also a restriction on the values for the loading F; these special values are mathematically called eigenvalues. All other values of F lead to trivial solutions (i.e. zero deformation).

The lowest load that causes buckling is called critical load (n = 1).

The above equation is usually called Euler's formula. Although Leonard Euler did publish the governing equation in 1744, J. L. Lagrange is considered the first to show that a non-trivial solution exists only when n is an integer. Thomas Young then suggested the critical load (n = 1) and pointed out the solution was valid when the column is slender in his 1807 book. The "slender" column idea was not quantitatively developed until A. Considère performed a series of 32 tests in 1889.

The shape function for the buckled shape w(x) is mathematically called an eigenfunction, and is given by,

Recall that this eigenfunction is strictly valid only for simply-supported columns.

Note:	1.	Boundary conditions other than simply-supported will result in different critical loads and mode shapes.
	2.	The buckling mode shape is valid only for small deflections, where the material is still within its elastic limit.
	3.	The critical load will cause buckling for slender, long columns. In contrast, failure will occur in short columns when the strength of material is exceeded. Between the long and short column limits, there is a region where buckling occurs after the stress exceeds the proportional limit but is still below the ultimate strength. These columns are classfied as intermediate and their failure is called inelastic buckling.
	4.	Whether a column is short, intermediate, or long depends on its geometry as well as the stiffness and strength of its material. This concept is addressed in the columns introduction page.