Critical buckling load analysis (also known as Eigenvalue buckling analysis) examines the geometric stability of models under primarily axial load. Buckling can be catastrophic if it occurs in the normal use of most products. Once the geometry starts to deform, it can no longer withstand even a fraction of the initially applied force.
The element types available for critical buckling are beams, plates, and bricks (and their variations). The element chosen can affect the type of buckling multiplier calculated. For example, beam elements are only a line with the crosssection represented mathematically; beams can only calculate a global buckling. Plate and brick elements can calculate localize buckling since the crosssection is modeled with elements. See Figure 1.
(a) Beam elements give a "global" buckling shape and load multiplier. 
cross section 
(b) Plate and brick elements can give global and local effects. In this example, the flange of the beam buckles locally which may be at a critical load less than the load to cause global buckling. 
Since the critical buckling is calculated by solving an eigenvalue problem (similar to a modal analysis), the results are suitable for thin and slender structures (like Euler columns). No corrections are incorporated for short columns, thick plates (compared to the length and width of the plate), or the yield strength of the material. See Figure 2. Also, this is a linear analysis, so there are no stiffness changes due to the deflection and therefore no large deflection effects such as "Pdelta" (loaddeflection) effects. Consequently, the critical load and buckling mode shape can be determined, but what happens after buckling is not available. (Nonlinear analysis should be performed if postbuckling results are needed.)
Critical Load 
Slenderness Ratio 
Euler Equation Johnson correction calculated results 
Buckling analysis is used to determine if a specified set of loads will cause buckling and to find the shape of the buckling mode. Engineers can then design supports or stiffeners to prevent local buckling. It is useful in situations where a part or assembly is subjected to an axial load or when a model undergoes edge compression.
Loads that do not cause compression will not affect the buckling calculation. It is important to note that when calculating the critical buckling load for beam elements, only the axial component of the loads will be considered. For example, imagine an offset load on the end of a bar (see Figure 3(a)). This can be represented as an axial load and moment on the end of a beam model (Figure 3(b)). Since the moment does not create "axial stress" in the beam elements, the offset load in this analysis produces the same result as a pure axial load. (The moment creates bending stresses; the theoretical compression on one side of the beam element does not affect the buckling calculation.)
(a) Physical model of a bar with a load P offset from the centerline by a distance e. 
(b) Beam model created in FEA. The load eccentricity results in a moment applied to the node. Due to the nature of beam elements and critical buckling, the moment does not affect the results. The wrong element type is used in this example. 
The calculated buckling load multipliers are shown in the Results environment, in the log file, and in the summary file. The buckling load multiplier indicates when the model will buckle. Multiply all of the applied loads on the model by the buckling load multiplier, and this is the theoretical load that causes buckling. Keep in mind that real parts tend to buckle at lower loads than the theoretical value due to imperfections in reallife manufacturing (initial curvature, eccentrically applied loads, and so on). Very small deviations can have enormous effects on the critical load in reality.
If the buckling load multiplier is negative, this indicates that reversing the applied loads (and scaling by the multiplier) will cause the model to buckle. For example, if a pressure of 1000 Pa is applied to the model but this puts it into tension, a buckling multiplier of 0.75 indicates that the part will buckle with a 750 Pa compression load.
Tip 
Critical buckling scales all applied loads by the calculated buckling multiplier. In some situations, users may want to scale the live loads (like pressure) while other loads (like gravity) are not scaled. If the constant loads are significant and therefore need to be included, use this procedure:

Since critical buckling is an eigenvalue solution, the displacement results will show the buckling mode shape, but the magnitude of the displacements are meaningless. Likewise, there are no stress or strain results from a critical buckling analysis.