This applet calculates the critical axial compressive load (and stress) that will cause a column to buckle. 

A column is defined as a relatively slender straight bar of uniform cross-section. A column will buckle when the load P reaches a critical level, called the critical load, P_cr. For the ideal pinned column shown in Figure 1, the critical buckling load can be calculated using Euler's formula:

where E is the modulus of elasticity of the material, I is the minimum moment of inertia, and L is the unsupported length of the column.  Note that regardless of the end condition, the critical load depends not on the material strength, but rather the flexural rigidity, EI. Buckling resistance can thus be enhanced by deploying material so as to maximize the moment of inertia, but not to the point where the section thickness is so small as to result in local buckling or wrinkling.

Figure 1. Ideal pinned column

When a column buckles, it maintains its deflected shape after the application of the critical load. In most applications, the critical load is usually regarded as the maximum load sustainable by the column. Theoretically, any buckling mode is possible, but the column will ordinarily deflect into the first mode.

The ideal column equation is only valid for a column with both ends pinned. Other equations must be used if the end conditions are other than pinned. The end conditions will effect the critical buckling load, P_cr.

Figure 2. Effect of boundary conditions on effective length.

For columns with different types of support, Euler's formula may still be used if the distance L is replaced with the distance between the zero moment points. This length is called the effective length L_e and is illustrated in Figure 2. Thus the critical load equation becomes:

The slenderness ratio is an important parameter in the classification of compression members, and is represented by the equation:

where r represents the radius of gyration:

If the slenderness ratio is greater than the critical slenderness ratio, 

the column is classified as a long column and the Euler buckling formula is applicable.  If the slenderness ratio is less than the critical slenderness ratio, the column is classified as a short column.  In short columns, failure occurs by compression without appreciable buckling and at stresses exceeding the proportional limit. For this condition, Johnson's formula is applicable:

For columns that fail subsequent to the onset of inelastic behavior, the constant of proportionality must be used rather than the modulus of elasticity (Engesser formula). The constant of proportionality, Et, is the slope of the stress-strain diagram beyond the proportional limit, termed the tangent modulus. Note within the linearly elastic range, E = Et.

The foregoing discussion have related to ideal, homogenous, concentrically loaded columns. Actual design requires the use of empirical formulas based on a strong background of testing and experience. See  references for some special purpose formulas.

The column buckling problem is defined by entering the required column geometrical properties (column length, cross-sectional area, and moment of inertia), the desired design safety factor, and the applicable end conditions or supports of the column in the INPUT PARAMETERS section of the input form.

Figure 3. Module input form

Column section properties (area, A, and moment of inertia, I) can be input automatically via the Section Properties module by pressing the Sections... button.

The effective length, L_e, of the column depends on the selected end conditions. When a new end condition is selected from the End Conditions pull-down menu (Note that the beam graphic will automatically update showing the new end conditions), the module automatically enters an effective length constant, C = L_e / L, into the Effective Length Constant input field.  If a different effective length constant is desired, select the User Defined option and enter a value for C in the Effective Length Constant input field.  Default values for effective length constants are shown in Table 1.

Table 1.  Module default values for effective length, L_e

The modulus of elasticity, E, and the yield strength, s_y, of the material are also required inputs and are entered in the INPUT PARAMETERS section. Material can be input automatically via the Material Properties module by pressing the Materials... button.

All input parameters must use consistent units to get correct answers. For example, if the length is entered in m, the area is entered in m², the moment of inertia entered in m⁴, the modulus of elasticity and yield strength in Pa; then the critical load will be displayed in N and the critical stress will be displayed in Pa.

The module will verify that proper material properties were entered. If either the modulus of elasticity or the yield strength are less than or equal to zero a message will be displayed in a separate window. The module will also verify if valid column properties were entered. If the length, cross-sectional area, moment of inertia, design safety factor, or the effective length constant are less than or equal to zero a message will be displayed in a separate window.

• Radius of Gyration
• Effective Length
• Slenderness Ratio
• Critical Slenderness Ratio
• Critical Load (with factor of safety)
• Critical Stress (with factor of safety)
  

Figure 4. Module tabulated results.