In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled.^[2] Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.

Buckling may occur even though the stresses that develop in the structure are well below those needed to cause failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformations that occur after buckling do not cause the complete collapse of that member, the member will continue to support the load that caused it to buckle. If the buckled member is part of a larger assemblage of components such as a building, any load applied to the buckled part of the structure beyond that which caused the member to buckle will be redistributed within the structure. Some aircraft are designed for thin skin panels to continue carrying load even in the buckled state.

Forms of buckling

Columns

The ratio of the effective length of a column to the least radius of gyration of its cross section is called the slenderness ratio (sometimes expressed with the Greek letter lambda, λ). This ratio affords a means of classifying columns and their failure mode. The slenderness ratio is important for design considerations. All the following are approximate values used for convenience.

If the load on a column is applied through the center of gravity (centroid) of its cross section, it is called an axial load. A load at any other point in the cross section is known as an eccentric load. A short column under the action of an axial load will fail by direct compression before it buckles, but a long column loaded in the same manner will fail by springing suddenly outward laterally (buckling) in a bending mode. The buckling mode of deflection is considered a failure mode, and it generally occurs before the axial compression stresses (direct compression) can cause failure of the material by yielding or fracture of that compression member. However, intermediate-length columns will fail by a combination of direct compressive stress and bending.

In particular:

• A short steel column is one whose slenderness ratio does not exceed 50; an intermediate length steel column has a slenderness ratio ranging from about 50 to 200, and its behavior is dominated by the strength limit of the material, while a long steel column may be assumed to have a slenderness ratio greater than 200 and its behavior is dominated by the modulus of elasticity of the material.
• A short concrete column is one having a ratio of unsupported length to least dimension of the cross section equal to or less than 10. If the ratio is greater than 10, it is considered a long column (sometimes referred to as a slender column).
• Timber columns may be classified as short columns if the ratio of the length to least dimension of the cross section is equal to or less than 10. The dividing line between intermediate and long timber columns cannot be readily evaluated. One way of defining the lower limit of long timber columns would be to set it as the smallest value of the ratio of length to least cross sectional area that would just exceed a certain constant K of the material. Since K depends on the modulus of elasticity and the allowable compressive stress parallel to the grain, it can be seen that this arbitrary limit would vary with the species of the timber. The value of K is given in most structural handbooks.

The theory of the behavior of columns was investigated in 1757 by mathematician Leonhard Euler. He derived the formula, termed Euler's critical load, that gives the maximum axial load that a long, slender, ideal column can carry without buckling. An ideal column is one that is:

• perfectly straight
• made of a homogeneous material
• free from initial stress.

When the applied load reaches the Euler load, sometimes called the critical load, the column comes to be in a state of unstable equilibrium. At that load, the introduction of the slightest lateral force will cause the column to fail by suddenly "jumping" to a new configuration, and the column is said to have buckled. This is what happens when a person stands on an empty aluminum can and then taps the sides briefly, causing it to then become instantly crushed (the vertical sides of the can may be understood as an infinite series of extremely thin columns).^{[citation needed]} The formula derived by Euler for long slender columns is

$${\displaystyle F_{c}={\frac {\pi ^{2}EI}{(KL)^{2}}}}$$

where

• ${\displaystyle F_{c}}$, maximum or critical force (vertical load on column),
• ${\displaystyle E}$, modulus of elasticity,
• ${\displaystyle I}$, smallest area moment of inertia (second moment of area) of the cross section of the column,
• ${\displaystyle L}$, unsupported length of column,
• ${\displaystyle K}$, column effective length factor, whose value depends on the conditions of end support of the column, as follows.
  • For both ends pinned (hinged, free to rotate), ${\displaystyle K=1.0}$.
  • For both ends fixed, ${\displaystyle K=0.50}$.
  • For one end fixed and the other end pinned, ${\displaystyle K\approx 0.699}$.
  • For one end fixed and the other end free to move laterally, ${\displaystyle K=2.0}$.
• ${\displaystyle KL}$ is the effective length of the column.

Examination of this formula reveals the following facts with regard to the load-bearing ability of slender columns.

• The elasticity of the material of the column and not the compressive strength of the material of the column determines the column's buckling load.
• The buckling load is directly proportional to the second moment of area of the cross section.
• The boundary conditions have a considerable effect on the critical load of slender columns. The boundary conditions determine the mode of bending of the column and the distance between inflection points on the displacement curve of the deflected column. The inflection points in the deflection shape of the column are the points at which the curvature of the column changes sign and are also the points at which the column's internal bending moments of the column are zero. The closer the inflection points are, the greater the resulting axial load capacity (bucking load) of the column.

A conclusion from the above is that the buckling load of a column may be increased by changing its material to one with a higher modulus of elasticity (E), or changing the design of the column's cross section so as to increase its moment of inertia. The latter can be done without increasing the weight of the column by distributing the material as far from the principal axis of the column's cross section as possible. For most purposes, the most effective use of the material of a column is that of a tubular section.

Another insight that may be gleaned from this equation is the effect of length on critical load. Doubling the unsupported length of the column quarters the allowable load. The restraint offered by the end connections of a column also affects its critical load. If the connections are perfectly rigid (not allowing rotation of its ends), the critical load will be four times that for a similar column where the ends are pinned (allowing rotation of its ends).

Since the radius of gyration is defined as the square root of the ratio of the column's moment of inertia about an axis to its cross sectional area, the above Euler formula may be reformatted by substituting the radius of gyration ${\displaystyle Ar^{2}}$ for ${\displaystyle I}$:

$${\displaystyle \sigma ={\frac {F}{A}}={\frac {\pi ^{2}E}{(l/r)^{2}}}}$$

where ${\displaystyle \sigma =F/A}$ is the stress that causes buckling in the column, and ${\displaystyle l/r}$ is the slenderness ratio.

Since structural columns are commonly of intermediate length, the Euler formula has little practical application for ordinary design. Issues that cause deviation from the pure Euler column behaviour include imperfections in geometry of the column in combination with plasticity/non-linear stress strain behaviour of the column's material. Consequently, a number of empirical column formulae have been developed that agree with test data, all of which embody the slenderness ratio. Due to the uncertainty in the behavior of columns, for design, appropriate safety factors are introduced into these formulae. One such formula is the Perry Robertson formula which estimates the critical buckling load based on an assumed small initial curvature, hence an eccentricity of the axial load. The Rankine Gordon formula, named for William John Macquorn Rankine and Perry Hugesworth Gordon (1899 – 1966), is also based on experimental results and suggests that a column will buckle at a load F_max given by:

$${\displaystyle {\frac {1}{F_{\max }}}={\frac {1}{F_{e}}}+{\frac {1}{F_{c}}}}$$

where ${\displaystyle F_{e}}$ is the Euler maximum load and ${\displaystyle F_{c}}$ is the maximum compressive load. This formula typically produces a conservative estimate of ${\displaystyle F_{\max }}$.

Self-buckling

A free-standing, vertical column, with density ${\displaystyle \rho }$, Young's modulus ${\displaystyle E}$, and cross-sectional area ${\displaystyle A}$, will buckle under its own weight if its height exceeds a certain critical value:^[3][4][5]

$${\displaystyle h_{\text{crit}}=\left({\frac {9B^{2}}{4}}\,{\frac {EI}{\rho gA}}\right)^{\frac {1}{3}}}$$

where ${\displaystyle g}$ is the acceleration due to gravity, ${\displaystyle I}$ is the second moment of area of the beam cross section, and ${\displaystyle B}$ is the first zero of the Bessel function of the first kind of order −1/3, which is equal to 1.86635086...

Plate buckling

A plate is a 3-dimensional structure defined as having a width of comparable size to its length, with a thickness that is very small in comparison to its other two dimensions. Similar to columns, thin plates experience out-of-plane buckling deformations when subjected to critical loads; however, contrasted to column buckling, plates under buckling loads can continue to carry loads, called local buckling. This phenomenon is incredibly useful in numerous systems, as it allows systems to be engineered to provide greater loading capacities.

For a rectangular plate, supported along every edge, loaded with a uniform compressive force per unit length, the derived governing equation can be stated by:^[6]