In science, buckling is a mathematical instability, leading to a failure mode. Theoretically, buckling

is caused by abifurcation in the solution to the equations of static equilibrium. At a certain stage

under an increasing load, further load is able to be sustained in one of two states of equilibrium: a

purely compressed state (with no lateral deviation) or a laterally-deformed state.

Buckling is characterized by a sudden sideways failure of a structural member subjected to

high compressive stress, where the compressive stress at the point of failure is less than the

ultimate compressive stress that the material is capable of withstanding. Mathematical analysis of

buckling often makes use of an "artificial" axial load eccentricity that introduces a secondary bending

moment that is not a part of the primary applied forces being studied. As an applied load is

increased on a member, such as a column, it will ultimately become large enough to cause the

member to become unstable and is said to have buckled. Further load will cause significant and

somewhat unpredictable deformations, possibly leading to complete loss of the member's load-

carrying capacity. If the deformations that follow buckling are not catastrophic the member will

continue to carry 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 structure beyond that which

caused the member to buckle will be redistributed within the structure.

Contents

  [hide] 

1 Columnso 1.1 Self-buckling

2 Buckling under tensile dead loading 3 Constraints, curvature and multiple buckling 4 Flutter instability 5 Various forms of buckling 6 Bicycle wheels 7 Surface materials 8 Cause 9 Accidents 10 Energy method 11 Flexural-torsional buckling 12 Lateral-torsional buckling

o 12.1 The modification factor (Cb) 13 Plastic buckling 14 Dynamic buckling 15 Buckling of thin cylindrical shells subject to axial loads 16 Buckling of pipes and pressure vessels subject to external overpressure 17 See also 18 References 19 External links

Columns[edit]

A column under a concentric axial load exhibiting the characteristic deformation of buckling

The eccentricity of the axial force results in a bending moment acting on the beam element.

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. Slenderness ratio is important for design considerations. All

the following are approximate values used for convenience.

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 are 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.

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 buckling (bending), the buckling effect being so

large that the effect of the axial load may be neglected. The intermediate-length column will fail by a

combination of direct compressive stress and bending.

In 1757, mathematician Leonhard Euler derived a formula 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, homogeneous, and free from initial stress. The maximum load, sometimes called the critical

load, causes the column to be in a state of unstable equilibrium; that is, the introduction of the

slightest lateral force will cause the column to fail by buckling. The formula derived by Euler for

columns with no consideration for lateral forces is given below. However, if lateral forces are taken

into consideration the value of critical load remains approximately the same.

where

 = maximum or critical force (vertical load on column),

 = modulus of elasticity,

 = area moment of inertia,

 = unsupported length of column,

 = 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),   = 1.0.For both ends fixed,   = 0.50.For one end fixed and the other end pinned,   = 0.699....For one end fixed and the other end free to move laterally,   = 2.0.

 is the effective length of the column.

Examination of this formula reveals the following interesting facts with

regard to the load-bearing ability of slender columns.

1. Elasticity  and not the compressive strength of the materials of

the column determines the critical load.

2. The critical load is directly proportional to the second moment

of area of the cross section.

3. The boundary conditions have a considerable effect on the

critical load of slender columns. The boundary conditions

determine the mode of bending and the distance between

inflection points on the deflected column. The inflection points

in the deflection shape of the column are the points at which

the curvature of the column change sign and are also the

points at which the internal bending moments are zero. The

closer together the inflection points are, the higher the resulting

capacity of the column.

A demonstration model illustrating the different "Euler" buckling modes. The

model shows how the boundary conditions affect the critical load of a slender

column. Notice that each of the columns are identical, apart from the boundary

conditions.

The strength of a column may therefore be increased by distributing the

material so as to increase the moment of inertia. This can be done

without increasing the weight of the column by distributing the material

as far from the principal axis of the cross section as possible, while

keeping the material thick enough to prevent local buckling. This bears

out the well-known fact that a tubular section is much more efficient

than a solid section for column service.

Another bit of information that may be gleaned from this equation is the

effect of length on critical load. For a given size column, doubling the

unsupported length quarters the allowable load. The restraint offered by

the end connections of a column also affects the critical load. If the

connections are perfectly rigid, the critical load will be four times that for

a similar column where there is no resistance to rotation (in which case

the column is idealized as having hinges at the 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 cross sectional area,

the above formula may be rearranged as follows. Using the Euler

formula for hinged ends, and substituting A·r2 for I, the following formula

results.

where   is the allowable stress of the column, and   is the

slenderness ratio.

Since structural columns are commonly of intermediate length, and

it is impossible to obtain an ideal column, the Euler formula on its

own has little practical application for ordinary design. Issues that

cause deviation from the pure Euler column behaviour include

imperfections in geometry in combination with plasticity/non-linear

stress strain behaviour of the column's material. Consequently, a

number of empirical column formulae have been developed to

agree with test data, all of which embody the slenderness ratio. 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 initial (small)

curvature. 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 Fmax given by:

where Fe is the Euler maximum load and Fc is the maximum

compressive load. This formula typically produces a

conservative estimate of Fmax.

Self-buckling[edit]

A free-standing, vertical column, with density  , Young's

modulus  , and cross-sectional area  , will buckle under its

own weight if its height exceeds a certain critical height:[1][2][3]

where g is the acceleration due to gravity, I is the second

moment of area of the beam cross section, and B is the

first zero of the Bessel function of the first kind of order -

1/3, which is equal to 1.86635086...

Buckling under tensile dead loading[edit]

Fig. 2: Elastic beam system showing buckling under tensile

dead loading.

Usually buckling and instability are associated to

compression, but recently Zaccaria, Bigoni, Noselli and

Misseroni (2011)[4] have shown that buckling and instability

can also occur in elastic structures subject to dead tensile

load. An example of a single-degree-of-freedom structure

is shown in Fig. 2,[where?] where the critical load is also

indicated. Another example involving flexure of a structure

made up of beam elements governed by the equation of

the Euler's elastica is shown in Fig.3. In both cases, there

are no elements subject to compression. The instability

and buckling in tension are related to the presence of the

slider, the junction between the two rods, allowing only

relative sliding between the connected pieces. Watch

a movie for more details.

Constraints, curvature and multiple buckling[edit]

Fig. 3: A one-degree-of-freedom structure exhibiting a tensile

(compressive) buckling load as related to the fact that the

right end has to move along the circular profile labeled 'Ct'

(labelled 'Cc').

Buckling of an elastic structure strongly depends on the

curvature of the constraints against which the ends of the

structure are prescribed to move (see Bigoni, Misseroni,

Noselli and Zaccaria, 2012[5]). In fact, even a single-degree-

of-freedom system (see Fig.3) may exhibit a tensile (or a

compressive) buckling load as related to the fact that one

end has to move along the circular profile labeled 'Ct'

(labelled 'Cc').

Fig. 4: A one-degree-of-freedom structure with a 'S'-shaped

bicircular profile exhibiting multiple bifurcations (both tensile

and compressive).

The two circular profiles can be arranged in a 'S'-shaped

profile, as shown in Fig.4; in that case a discontinuity of the

constraint's curvature is introduced, leading to multiple

bifurcations. Note that the single-degree-of-freedom

structure shown in Fig.4 has two buckling loads (one

tensile and one compressive). Watch a movie for more

details.

Flutter instability[edit]

Structures subject to a follower (nonconservative)

load[clarification needed] may suffer instabilities which are not of the

buckling type and therefore are not detectable with a static

approach.[6] For instance, the so-called 'Ziegler column' is

shown in Fig.5.

Fig. 5: A sketch of the 'Ziegler column', a two-degree-of-

freedom system subject to a follower load (the force P

remains always parallel to the rod BC), exhibiting flutter and

divergence instability. The two rods, of linear mass density ρ,

are rigid and connected through two rotational springs of

stiffness k1 and k2.

This two-degree-of-freedom system does not display a

quasi-static buckling, but becomes dynamically unstable.

To see this, we note that the equations of motion are

and their linearized version is

Assuming a time-harmonic solution in the form

we find the critical loads for flutter ( ) and

divergence ( ),

where   and  .

Fig. 6: A sequence of deformed shapes at

consecutive times intervals of the structure

sketched in Fig.5 and exhibiting flutter

(upper part) and divergence (lower part)

instability.

Flutter instability corresponds to a

vibrational motion of increasing amplitude

and is shown in Fig.6 (upper part) together

with the divergence instability (lower part)

consisting in an exponential growth.

Recently, Bigoni and Noselli (2011)[7] have

experimentally shown that flutter and

divergence instabilities can be directly

related to dry friction, watch the movie for

more details.

Various forms of buckling[edit]

Buckling is a state which defines a point

where an equilibrium configuration

becomes unstable under a parametric

change of load and can manifest itself in

several different phenomena. All can be

classified as forms of bifurcation.

There are four basic forms of bifurcation

associated with loss of structural stability

or buckling in the case of structures with a

single degree of freedom. These comprise

two types of pitchfork bifurcation,

one saddle-node bifurcation (often referred

to as a limit point) and one transcritical

bifurcation. The pitchfork bifurcations are

the most commonly studied forms and

include the buckling of columns and struts,

sometimes known as Euler buckling; the

buckling of plates, sometimes known as

local buckling, which is well known to be

relatively safe (both are supercritical

phenomena) and the buckling of shells,

which is well-known to be a highly

dangerous (subcritical phenomenon).[8] Using the concept of potential energy,

equilibrium is defined as a stationary point

with respect to the degree(s) of freedom of

the structure. We can then determine

whether the equilibrium is stable, if the

stationary point is a local minimum; or

unstable, if it is a maximum, point of

inflection or saddle point (for multiple-

degree-of-freedom structures only) – see

animations below.

Archetypal rigid link models with a

single degree of freedom (SDOF) used

to demonstrate basic buckling

phenomena (see bifurcation diagrams

below). All cases start at the position

corresponding to q=0.

Truss with spring tie (model shallow

tied arch). 

Link-strut with rotational spring. 

Link-strut with transverse

translational spring. 

Asymmetrically supported link-

strut. 

Animations of the variation of total

potential energy (red) for various load

values, P (black), in generic structural

systems with the indicated bifurcation

or buckling behaviour.

Two saddle-node bifurcations (limit

points). 

Supercritical pitchfork bifurcation

(stable-symmetric buckling point). 

Subcritical pitchfork bifurcation

(unstable-symmetric buckling

point). 

Transcritical bifurcation (asymmetric

buckling point). 

In Euler buckling,[9][10] the applied load is

increased by a small amount beyond the

critical load, the structure deforms into a

buckled configuration which is adjacent to

the original configuration. For example, the

Euler column pictured will start to bow

when loaded slightly above its critical load,

but will not suddenly collapse.