Timoshenko describes the buckling behaviour of the free thin ring under a constant external radial load. In the case of any small starting deformation there develops under critical loading the buckling of the circular ring as is shown in (Image 5.3.3.7.2.3.2-1).

The bending moment M at any point of the circular ring can then be developed from the equilibrium conditions and geometric linkages as follows:

The deflection w of the ring in the radial direction is thereby described by the differential equation:

their general solution

Then the critical buckling load is determined for general buckling wave numbers k as:

The various failure figures for k = 2,3,4 are shown in (Image 5.3.3.7.2.3.2-2). With the smallest non-trivial solution k = 2, the critical buckling load of the free pipe is

Often the factor (1-ν^{2}) is included in the equation (5-4) (Formel 5.3.2.6.1.2.1) for considering prevented lateral expansion of the long pipe. For a full-walled free pipe of length "1" the general formula can then be written as: