In
scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
,
buckling is a mathematical instability, leading to a
failure modeStructural failure refers to loss of the loadcarrying capacity of a component or member within a structure or of the structure itself. Structural failure is initiated when the material is stressed to its strength limit, thus causing fracture or excessive deformations...
.
Theoretically, buckling is caused by a
bifurcationBifurcation means the splitting of a main body into two parts.Bifurcation or Bifurcated may refer to:*Bifurcation , the division of issues in a trial for example the division of a page into two parts....
in the solution to the equations of
static equilibriumA standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
. At a certain stage under an increasing load, further load is able to be sustained in one of two states of equilibrium: an undeformed state or a laterallydeformed state.
In practice, buckling is characterized by a sudden failure of a structural member subjected to high compressive stress, where the actual compressive stress at the point of failure is less than the ultimate compressive stresses that the material is capable of withstanding. For example, during earthquakes, reinforced concrete members may experience lateral deformation of the longitudinal reinforcing bars. This mode of failure is also described as failure due to
elastic instabilityElastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads.Single degree of freedomsystems:...
. Mathematical analysis of buckling makes use of an axial load eccentricity that introduces a moment, which does not form part of the primary forces to which the member is subjected. When load is constantly being applied on a member, such as column, it will ultimately become large enough to cause the member to become unstable. Further load will cause significant and somewhat unpredictable deformations, possibly leading to complete loss of loadcarrying capacity. The member is said to have buckled, to have deformed.
Columns
The ratio of the effective length of a
columnA column or pillar in architecture and structural engineering is a vertical structural element that transmits, through compression, the weight of the structure above to other structural elements below. For the purpose of wind or earthquake engineering, columns may be designed to resist lateral forces...
to the least
radius of gyrationRadius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis....
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
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...
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.
 A short concrete
Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...
column is one having a ratio of unsupported length to least dimension of the cross section not greater than 10. If the ratio is greater than 10, it is a long column (sometimes referred to as a slender column).
 Timber
Wood is a hard, fibrous tissue found in many trees. It has been used for hundreds of thousands of years for both fuel and as a construction material. It is an organic material, a natural composite of cellulose fibers embedded in a matrix of lignin which resists compression...
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 stressIn continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
parallel to the grain, it can be seen that this arbitrary limit would vary with the speciesIn biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...
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 gravityIn physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...
of its cross section, it is called an
axialAxial may mean:* Along the same line as an axis of rotation in geometry* A type of modal frame in music* One of several anatomical directions in an animal body* Axial age, the period from 800 to 200 BC in China, India and the western world...
loadStructural loads or actions are forces, deformations or accelerations applied to a structure or its components.Loads cause stresses, deformations and displacements in structures. Assessment of their effects is carried out by the methods of structural analysis...
. A load at any other point in the cross section is known as an
eccentricIn mechanical engineering, an eccentric is a circular disk solidly fixed to a rotating axle with its centre offset from that of the axle ....
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 (
bendingIn engineering mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically...
), the buckling effect being so large that the effect of the direct load may be neglected. The intermediatelength column will fail by a combination of direct compressive stress and bending.
In 1757,
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
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
equilibriumA standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
; 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
forceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
(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 loadbearing ability of slender columns.
 Elasticity
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
and not compressive strength of the materialsIn materials science, the strength of a material is its ability to withstand an applied stress without failure. The applied stress may be tensile, compressive, or shear. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a...
of the column determines the critical load.
 The critical load is directly proportional
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
to the second moment of areaThe second moment of area, also known as the area moment of inertia, moment of inertia of plane area, or second moment of inertia is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the crosssectional plane...
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 and the distance between inflection points on the deflected column. The closer together the inflection points are, the higher the resulting capacity of the column.
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 wellknown 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 (hinged at the ends).
Since the moment of inertia of a surface is its area multiplied by the square of a length called the radius of gyration, the above formula may be rearranged as follows. Using the Euler formula for hinged ends, and substituting A·r
^{2} 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 strut behaviour include imperfections in geometry in combination with plasticity/nonlinear 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 factorsFactor of safety , also known as safety factor , is a term describing the structural capacity of a system beyond the expected loads or actual loads. Essentially, how much stronger the system is than it usually needs to be for an intended load...
are introduced into these formulae. One such formular is the
Perry Robertson formulaThe Perry Robertson formula is a mathematical formula which is able to produce a good approximation of buckling loads in long slender beams, and is the basis for the buckling formulation adopted in EN 1993....
which estimates of the critical buckling load based on an initial (small) curvature. The Rankine Gordon fomular is also based on eperimental results and surgests that a strut will buckle at a load Fmax given by:
where Fe is the euler maximum load and Fc is the maximum compresive load. This formular typically produces a conservative estimate of Fmax.
Selfbuckling
A freestanding, vertical column, with density
, Young's modulus
, and radius
, will buckle under its own weight if its height exceeds a certain critical height:
where
g is the acceleration due to gravity,
I is the
second moment of areaThe second moment of area, also known as the area moment of inertia, moment of inertia of plane area, or second moment of inertia is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the crosssectional plane...
of the beam cross section, and
B is the first zero of the
Bessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of the first kind of order 1/3, which is equal to 1.86635...
Buckling under tensile dead loading
Usually buckling and instability are associated to compression, but recently Zaccaria, Bigoni, Noselli and Misseroni (2011) have shown that buckling and instability can also occur in elastic structures subject to dead tensile load.
An example of a singledegreeoffreedom structure is shown in Fig. 1, 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. 2.
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.
Flutter instability
Structures subject to a follower (nonconservative) load may suffer instabilities which are not of the buckling type and therefore are not detectable with a static approach. For instance, the socalled 'Ziegler column' is shown in Fig.3.
This twodegreeoffreedom system does not display a quasistatic buckling, but becomes dynamically unstable.
To see this, we note that the equations of motion are
and their linearized version is
Assuming a timeharmonic solution in the form
we find the critical loads for flutter (
) and divergence (
),
where
and
.
Flutter instability corresponds to a vibrational motion of increasing amplitude and is shown in Fig.4 (upper part) together with the divergence instability (lower part) consisting in an exponential growth.
Recently, Bigoni and Noselli (2011) have experimentally shown that flutter and divergence instabilities can be directly related to dry friction, watch the
movie for more details.
Limit point vs bifurcation buckling
Bifurcation buckling is sometimes called Euler buckling even when applied to structures other than Euler columns. As 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.
In structures experiencing limit point instability, if the load is increased infinitesimally beyond the critical load, the structure undergoes a large deformation into a different stable configuration which is not adjacent to the original configuration. An example of this type of buckling is a toggle frame (pictured) which 'snaps' into its buckled configuration.
Bicycle wheels
A conventional
bicycle wheelA bicycle wheel is a wheel, most commonly a wire wheel, designed for bicycle. A pair is often called a wheelset, especially in the context of ready built "off the shelf" performanceoriented wheels....
consists of a thin rim kept under high compressive stress by the (roughly normal) inward pull of a large number of spokes. It can be considered as a loaded column that has been bent into a circle. As such, if spoke tension is increased beyond a safe level, the wheel spontaneously fails into a characteristic saddle shape (sometimes called a "taco" or a "
pringlePringles is a brand of potato and wheat based snacks originally developed by Procter & Gamble. Pringles are sold in more than 140 countries and have yearly sales of more than...
") like a threedimensional Euler column. This is normally a purely elastic deformation and the rim will resume its proper plane shape if spoke tension is reduced slightly.
Surface materials
Buckling is also a failure mode in
pavementRoad surface or pavement is the durable surface material laid down on an area intended to sustain vehicular or foot traffic, such as a road or walkway. In the past cobblestones and granite setts were extensively used, but these surfaces have mostly been replaced by asphalt or concrete. Such...
materials, primarily with concrete, since
asphaltAsphalt concrete is a composite material commonly used in construction projects such as road surfaces, airports and parking lots. It consists of asphalt and mineral aggregate mixed together, then laid down in layers and compacted...
is more flexible. Radiant heat from the
sunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
is absorbed in the road surface, causing it to
expandThermal expansion is the tendency of matter to change in volume in response to a change in temperature.When a substance is heated, its particles begin moving more and thus usually maintain a greater average separation. Materials which contract with increasing temperature are rare; this effect is...
, forcing adjacent pieces to push against each other. If the stress is great enough, the pavement can lift up and crack without warning. Going over a buckled section can be very jarring to
automobileAn automobile, autocar, motor car or car is a wheeled motor vehicle used for transporting passengers, which also carries its own engine or motor...
drivers, described as running over a
speed humpA speed hump is a rounded traffic calming device used to reduce vehicle speed and volume on residential streets. Humps are placed across the road to slow traffic and are often installed in a series of several humps in order to prevent cars from speeding before and after the hump...
at highway speeds.
Similarly,
rail tracksThe track on a railway or railroad, also known as the permanent way, is the structure consisting of the rails, fasteners, sleepers and ballast , plus the underlying subgrade...
also expand when heated, and can fail by buckling, a phenomenon called
sun kinkSun kink refers to a condition that can occur on hot days in rail tracks. The phenomenon is caused by what is properly termed as buckling.The buckling force in the track due to warming up is a function of the rise in temperature only and is independent of the track length: F = E A \alpha_L \Delta...
. It is more common for rails to move laterally, often pulling the underlain railroad
tieA railroad tie/railway tie , or railway sleeper is a rectangular item used to support the rails in railroad tracks...
s (sleepers) along .
Energy method
Often it is very difficult to determine the exact buckling load in complex structures using the Euler formula, due to the difficulty in deciding the constant K. Therefore, maximum buckling load often is approximated using energy conservation. This way of deciding maximum buckling load is often referred to as the energy method in structural analysis.
The first step in this method is to suggest a displacement function. This function must satisfy the most important boundary conditions, such as displacement and rotation. The more accurate the displacement function, the more accurate the result.
In this method, there are two equations used (for small deformations) to approximate the "inner" energy (the potential energy stored in elastic deformation of the structure) and "outer" energy (the work done on the system by external forces).
where
is the displacement function and the subscripts
and
refer to the first and second derivatives of the displacement. Energy conservation yields:
Flexuraltorsional buckling
Occurs in compression members only and it can be described as a combination of bending and twisting of a member. And it must be consider for design purposes, since the shape and cross sections are very critical. This mostly occurs in channels, structural tees, doubleangle shapes, and equalleg single angles.
Lateraltorsional buckling
When a simple beam is loaded in flexure, the top side is in compression, and the bottom side is in
tensionIn physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...
. When a slender member is subjected to an axial force, failure takes place due to bending or torsion rather than direct compression of the material. If the beam is not supported in the lateral direction (i.e., perpendicular to the plane of bending), and the flexural load increases to a critical limit, the beam will fail due to lateral buckling of the compression flange. In wideflange sections, if the compression flange buckles laterally, the cross section will also twist in torsion, resulting in a failure mode known as
lateraltorsional buckling.
Plastic buckling
Buckling will generally occur slightly before the theoretical buckling strength of a structure, due to plasticity of the material. When the compressive load is near buckling, the structure will bow significantly and approach yield. The stressstrain behaviour of materials is not strictly linear even below yield, and the modulus of elasticity decreases as stress increases, with more rapid change near yield. This lower rigidity reduces the buckling strength of the structure and causes premature buckling. This is the opposite effect of the plastic bending in beams, which causes late failure relative to the
EulerBernoulli beam equationEuler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the loadcarrying and deflection characteristics of beams. It covers the case for small deflections of a beam which is subjected to lateral loads only...
.
Dynamic buckling
If the load on the column is applied suddenly and then released, the column can sustain a load much higher than its static (slowly applied) buckling load. This can happen in a long, unsupported column (rod) used as a drop hammer. The duration of compression at the impact end is the time required for a stress wave to travel up the rod to the other (free) end and back down as a relief wave. Maximum buckling occurs near the impact end at a wavelength much shorter than the length of the rod, at a stress many times the buckling stress if the rod were a staticallyloaded column. The critical condition for buckling amplitude to remain less than about 25 times the effective rod straightness imperfection at the buckle wavelength is
where
is the impact stress,
is the length of the rod,
is the elastic wave speed, and
is the smaller lateral dimension of a rectangular rod. Because the buckle wavelength depends only on
and
, this same formula holds for thin cylindrical shells of thickness
.
Buckling of thin cylindrical shells subject to axial loads
Solutions of Donnel's eight order differential equation gives the various buckling modes of a thin cylinder under compression. But this analysis, which is in accordance with the small deflection theory gives much higher values than shown from experiments. So it is customary to find the critical buckling load for various structures which are cylindrical in shape from preexisting design curves where critical buckling load F
_{cr} is plotted against the ratio R/t, where R is the radius and t is the thickness of the cylinder for various values of L/R, L the length of the cylinder. If cutouts are present in the cylinder, critical buckling loads as well as prebuckling modes will be affected. Presence or absence of reinforcements of cutouts will also affect the buckling load.
Buckling of pipes and pressure vessels subject to external overpressure
Pipes and pressure vessels subject to external overpressure, caused for example by steam cooling down and condensating into water with subsequent massive pressure drop, risk buckling due to compressive
hoop stressesCircumferential stress is a type of mechanical stress of a cylindrically shaped part as a result of internal or external pressure.The classic example of circumferential stress is the tension applied to the iron bands, or hoops, of a wooden barrel...
. Design rules for calculation of the required wall thickness or reinforcement rings are given in various piping and pressure vessel codes.
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