Buckling - Wikipedia, The Free Encyclopedia

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BucklingFrom Wikipedia, the free encyclopedia

In science, buckling is a mathematical instability, leading to a failure mode. Theoretically, buckling is caused by a bifurcation in the solution to theequations 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: anundeformed state or a laterally-deformed state.

In practice, buckling is characterized by a sudden failure of a structural member subjected to high compressive stress, where the actual compressivestress at the point of failure is less than the ultimate compressive stresses that the material is capable of withstanding. For example, duringearthquakes, reinforced concrete members may experience lateral deformation of the longitudinal reinforcing bars. This mode of failure is alsodescribed as failure due to elastic instability. 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 ascolumn, it will ultimately become large enough to cause the member to become unstable. Further load will cause significant and somewhatunpredictable deformations, possibly leading to complete loss of load-carrying capacity. The member is said to have buckled, to have deformed.

Contents

1 Columns1.1 Self-buckling

2 Buckling under tensile dead loading3 Constraint's curvature and multiple buckling4 Flutter instability5 Limit point vs bifurcation buckling6 Bicycle wheels7 Surface materials8 Energy method9 Flexural-torsional buckling10 Lateral-torsional buckling

10.1 The Modification Factor (Cb)11 Plastic buckling12 Dynamic buckling13 Buckling of thin cylindrical shells subject to axial loads14 Buckling of pipes and pressure vessels subject to external overpressure15 See also16 References

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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.

17 External links

Columns

The ratio of the effective length of a column to the least radius of gyration of its cross section is called theslenderness ratio (sometimes expressed with the Greek letter lambda, ). This ratio affords a means ofclassifying columns. Slenderness ratio is important for design considerations. All the following are approximatevalues used for convenience.

A short steel column is one whose slenderness ratio does not exceed 50; an intermediate length steel columnhas 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 column is one having a ratio of unsupported length to least dimension of the cross section notgreater than 10. If the ratio is greater than 10, it is 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 crosssection is equal to or less than 10. The dividing line between intermediate and long timber columns cannot bereadily evaluated. One way of defining the lower limit of long timber columns would be to set it as the smallestvalue of the ratio of length to least cross sectional area that would just exceed a certain constant K of thematerial. Since K depends on the modulus of elasticity and the allowable compressive stress parallel to thegrain, it can be seen that this arbitrary limit would vary with the species of the timber. The value of K is givenin most structural handbooks.

If the load on a column is applied through the center of gravity of its cross section, it is called an axial load. Aload 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 mannerwill fail by buckling (bending), the buckling effect being so large that the effect of the direct load may beneglected. 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 thecolumn to be in a state of unstable equilibrium; that is, the introduction of the slightest lateral force will causethe 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.


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A demonstration model illustrating the

different "Euler" buckling modes. The

model shows how the boundary

conditions affect the critical load of a

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.

Elasticity and not compressive strength of the materials of the column determines the critical load.1.The critical load is directly proportional to the second moment of area of the cross section.2.The boundary conditions have a considerable effect on the critical load of slender columns. The boundary conditions determine the mode ofbending and the distance between inflection points on the deflected column. The closer together the inflection points are, the higher theresulting capacity of the column.

3.

The strength of a column may therefore be increased by distributing the material so as to increase the momentof inertia. This can be done without increasing the weight of the column by distributing the material as farfrom 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 solidsection for column service.

Another bit of information that may be gleaned from this equation is the effect of length on critical load. For agiven size column, doubling the unsupported length quarters the allowable load. The restraint offered by theend connections of a column also affects the critical load. I f the connections are perfectly rigid, the criticalload 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 ofgyration, the above formula may be rearranged as follows. Using the Euler formula for hinged ends, and

substituting Ar2 for I, the following formula results.


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slender column. Notice that each of the

columns are identical, apart from the

boundary conditions.

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 incombination with plasticity/non-linear stress strain behaviour of the column's material. Consequently, a number of empirical column formulae havebeen developed to agree with test data, all of which embody the slenderness ratio. For design, appropriate safety factors are introduced into theseformulae. One such formula is the Perry Robertson formula which estimates of the critical buckling load based on an initial (small) curvature. TheRankine Gordon formula is also based on experimental results and suggests that a strut will buckle at a load Fmaxgiven 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

A free-standing, vertical column, with density , Young's modulus , and radius , will buckle under its own weight if its height exceeds a certain

critical height:[1][2][3]

whereg is the acceleration due to gravity, I is the second moment of area of the beam cross section, andB is the first zero of the Bessel function ofthe first kind of order -1/3, which is equal to 1.86635086...

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 todead tensile load. An example of a single-degree-of-freedom structure is shown in Fig. 1, where the criticalload 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


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Fig. 2: Elastic beam system showing

buckling under tensile dead loading.

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').

Fig. 4: A one-degree-of-freedom

structure with a 'S'-shaped bicircular

profile exhibiting multiple bifurcations(both tensile and compressive).

Fig. 5: A sketch of the 'Ziegler

column', a two-degree-of-freedom

compression. The instability and buckling in tension are related to the presence of the slider, the junctionbetween the two rods, allowing only relative sliding between the connected pieces.

Constraint's curvature and multiple buckling

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 asingle-degree-of-freedom system (see Fig.3) may exhibit a tensile (or a compressive) buckling load as relatedto the fact that one end has to move along the circular profile labeled 'Ct' (labelled 'Cc').

The two circular profiles can be arranged in a 'S'-shaped profile, as shown in Fig.4; in that case a discontinuityof 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(http://www.ing.unitn.it/~bigoni/multiple_bifurcations.html) for more details.

Flutter instabilityStructures 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.[6] For instance, the so-called 'Ziegler column' isshown in Fig.5.

This two-degree-of-freedom system does not display a quasi-static buckling, but becomes dynamicallyunstable. To see this, we note that the equations of motion are


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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 , are rigid and

connected through two rotational

springs of stiffness k1 and k2.

and their linearized version is

Assuming a time-harmonic 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.6 (upper part) together with the divergenceinstability (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


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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.

the movie (http://www.ing.unitn.it/~bigoni/flutter.html) for more details.

Limit point vs bifurcation buckling

Bifurcation buckling[8][9] is sometimes called Euler buckling even when applied to structures other than Eulercolumns. 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 columnpictured 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 largedeformation into a different stable configuration which is not adjacent to the original configuration. An example of this type of buckling is a toggleframe (pictured) which 'snaps' into its buckled configuration.


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Sun kink in rail tracks

Bicycle wheels

A conventional bicycle wheel consists of a thin rim kept under high compressive stress by the (roughly normal) inward pull of a large number ofspokes. 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 wheelspontaneously fails into a characteristic saddle shape (sometimes called a "taco" or a "pringle") like a three-dimensional Euler column. This isnormally 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 pavement materials, primarily with concrete, since asphalt is more flexible.Radiant heat from the sun is absorbed in the road surface, causing it to expand, forcing adjacent pieces to push

against each other. If the stress is great enough, the pavement can lift up and crack without warning. Goingover a buckled section can be very jarring to automobile drivers, described as running over a speed hump athighway speeds.

Similarly, rail tracks also expand when heated, and can fail by buckling, a phenomenon called sun kink. It ismore common for rails to move laterally, often pulling the underlain railroad ties (sleepers) along.

Energy method

Often it is very difficult to determine the exact buckling load in complex structures using the Euler formula,


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due to the difficulty in deciding the constant K. Therefore, maximum buckling load often is approximated using energy conservation. This way ofdeciding 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 asdisplacement 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:

Flexural-torsional 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 considered fordesign purposes, since the shape and cross sections are very critical. This mostly occurs in channels, structural tees, double-angle shapes, andequal-leg single angles.

Lateral-torsional bucklingWhen a simple beam is loaded in flexure, the top side is in compression, and the bottom side is in tension. When a slender member is subjected to anaxial force, failure takes place due to bending or torsion rather than direct compression of the material. If the beam is not supported in the lateraldirection (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 thecompression flange. In wide-flange sections, if the compression flange buckles laterally, the cross section will also twist in torsion, resulting in afailure mode known as lateral-torsional buckling.

The Modification Factor (Cb)

Cb is a modification factor used in the equation for nominal flexural strength when determining Lateral-torsional buckling. The reason for this factor


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Figure 1: Cb values for simply

supported beams

is to allow for non-uniform moment diagrams when the ends of a beam segment are braced. The conservative value for Cb can be taken as 1,regardless of beam configuration or loading, but in some cases it may be excessively conservative. Cb is always equal to or greater than 1, never less.For cantilevers or overhangs where the free end is unbraced, Cb is equal to 1. A table of values of Cb for simply supported beams are shown in Figure1.

If the value of Cb is not given in the table, an equation can be used to find Cb.

where

=absolute value of maximum moment in the unbraced segment, (kip-inch)=absolute value of maximum moment at quarter point of the unbraced segment, (kip-inch)=absolute value of maximum moment at centerline of the unbraced segment, (kip-inch)=absolute value of maximum moment at three-quarter point of the unbraced segment, (kip-inch)

Plastic buckling

Buckling will generally occur slightly before the theoretical buckling strength of a structure, due to plasticityof the material. When the compressive load is near buckling, the structure will bow significantly and approachyield. The stress-strain behavior of materials is not strictly linear even below yield, and the modulus ofelasticity decreases as stress increases, with more rapid change near yield. This lower rigidity reduces thebuckling strength of the structure and causes premature buckling. This is the opposite effect of the plasticbending in beams, which causes late failure relative to the Euler-Bernoulli beam equation.

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) bucklingload. 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 requiredfor 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 awavelength much shorter than the length of the rod, at a stress many times the buckling stress if the rod were a statically-loaded column. The criticalcondition for buckling amplitude to remain less than about 25 times the effective rod straightness imperfection at the buckle wavelength is


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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 .[10]

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 criticalbuckling load for various structures which are cylindrical in shape from pre-existing design curves where critical buckling load F cr is plotted againstthe 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. I f cut-outs are presentin the cylinder, critical buckling loads as well as pre-buckling modes will be affected. Presence or absence of reinforcements of cut-outs will alsoaffect 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 stresses. Design rules for calculation of the required wall thickness or reinforcementrings are given in various piping and pressure vessel codes.

See also

Perry Robertson formulaWood methodStiffening

References

^Kato, K. (1915). "Mathematical Investigation on the Mechanical Problems of Transmission Line".J ournal of the J apan Society of Mechanical Engineers19: 41.

1.

^Ratzersdorfer, Julius (1936). Die Knickfestigkeit von Stben und Stabwerken. Wein, Austria: J . Springer. pp. 107109.2.^Cox, Steven J .; C. Maeve McCarthy (1998). "The Shape of the Tallest Column". Society for Industrial and Applied Mathematics29: 547554.3.^D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni, Structures buckling under tensile dead load. Proceedings of the Royal Society A, 2011, 467,1686-1700. (http://www.ing.unitn.it/~bigoni/tensile_buckling.html)

4.

^D. Bigoni, D. Misseroni, G. Noselli and D. Zaccaria, Effects of the constraint's curvature on structural instability: tensile buckling and multiplebifurcations. Proceedings of the Royal Society A, 2012, doi:10.1098/rspa.2011.0732. (http://www.ing.unitn.it/~bigoni/multiple_bifurcations.html)

5.

^Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, 2012 . ISBN 9781107025417.6.^D. Bigoni and G. Noselli, Experimental evidence of flutter and divergence instabilities induced by dry friction. J ournal of the Mechanics and Physics of7.


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Solids, 2011, 59, 22082226. (http://www.ing.unitn.it/~bigoni)^"Buckling of Bars, Plates, and Shells" By Robert M. J ones8.^"Observations on eigenvalue buckling analysis within a finite element context" by Christopher J . Earls9.^Lindberg, H. E., and Florence, A. L., Dynamic Pulse Buckling, Martinus Nijhoff Publishers, 1987, pp. 1156, 297298.10.

Timoshenko, S. P., and Gere, J. M.,Theory of Elastic Stability, 2 ed., McGraw-Hill, 1961.Nenezich, M.,Thermoplastic ContinuumMechanics, J ournal of Aerospace Structures, Vol. 4, 2004.

The Stability of Elastic Equilibrium (http://contrails.iit.edu/DigitalCollection/1970/AFFDLTR70-025.pdf) by W. T. Koiter, PhD Thesis, 1945.Dhakal Rajesh and Koichi Maekawa (October 2002). "Reinforcement Stability and Fracture of Cover Concrete in Reinforced ConcreteMembers. [1] (http://ascelibrary.aip.org.libdb.njit.edu:8888/getpdf/servlet/GetPDFServlet?filetype=pdf&id=JSENDH000128000010001253000001&idtype=cvips&ident=freesearch.)Willian T. Segui (2007). Steel Design Fourth Edition. United States. Chris Carson.Analysis and design of flight vehicle structures- E.F.Brune

External links

The complete theory and example experimental results for long columns are available as a 39-page PDF document at http://lindberglce.com/tech/buklbook.htmVideo on Buckling under tensile dead loading (http://www.youtube.com/user/RoyalSociety#p/u/0/EKngs1vvcJU)Laboratory for Physical Modeling of Structures and Photoelasticity (University of Trento, Italy) (http://ssmg.ing.unitn.it)http://www.midasuser.com.tw/t_support/tech_pds/files/Tech%20Note-Lateral%20Torsional%20Buckling.pdf

Retrieved from "http://en.wikipedia.org/w/index.php?title=Buckling&oldid=534712086"Categories: Elasticity (physics) Materials science Mechanical failure modes Structural analysis

This page was last modified on 24 January 2013 at 20:52.

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