Buckling

For other uses, see Buckling (disambiguation).

In science, buckling is a mathematical instability, lead-ing to a failure mode. Theoretically, buckling is causedby a bifurcation in the solution to the equations of staticequilibrium. At a certain stage under an increasing load,further load is able to be sustained in one of two states ofequilibrium: an undeformed state or a laterally-deformedstate.In practice, buckling is characterized by a sudden fail-ure of a structural member subjected to high compressivestress, where the actual compressive stress at the point offailure is less than the ultimate compressive stresses thatthe material is capable of withstanding. Mathematicalanalysis of buckling often makes use of an axial load ec-centricity that introduces a secondary bending moment,which is not a part of the primary applied forces to whichthe member is subjected. As an applied load is increasedon a member, such as column, it will ultimately becomelarge enough to cause themember to become unstable andis said to have buckled. Further load will cause signifi-cant and somewhat unpredictable deformations, possiblyleading to complete loss of the member’s load-carryingcapacity. If the deformations that follow buckling are notcatastrophic the member will continue to carry the loadthat caused it to buckle. If the buckled member is partof a larger assemblage of components such as a build-ing, any load applied to the structure beyond that whichcaused the member to buckle will be redistributed withinthe structure.

1 Columns

The ratio of the effective length of a column to the leastradius of gyration of its cross section is called the slen-derness ratio (sometimes expressed with the Greek let-ter lambda, λ). This ratio affords a means of classify-ing columns. Slenderness ratio is important for designconsiderations. All the following are approximate valuesused for convenience.

• A short steel column is one whose slenderness ratiodoes not exceed 50; an intermediate length steel col-umn has a slenderness ratio ranging from about 50 to200, and are dominated by the strength limit of thematerial, while a long steel column may be assumedto have a slenderness ratio greater than 200 and itsbehavior is dominated by the modulus of elasticity

A column under a concentric axial load exhibiting the character-istic deformation of buckling

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

of the material.

• A short concrete column is one having a ratio of un-supported length to least dimension of the cross sec-tion equal to or less than 10. If the ratio is greaterthan 10, it is considered a long column (sometimesreferred to as a slender column).

• Timber columns may be classified as short columnsif the ratio of the length to least dimension of thecross section is equal to or less than 10. The dividingline between intermediate and long timber columnscannot be readily evaluated. One way of defining thelower limit of long timber columns would be to setit as the smallest value of the ratio of length to leastcross sectional area that would just exceed a certain

1

2 1 COLUMNS

constant K of the material. Since K depends on themodulus of elasticity and the allowable compressivestress parallel to the grain, it can be seen that thisarbitrary limit would vary with the species of thetimber. The value of K is given in most structuralhandbooks.

If the load on a column is applied through the center ofgravity (centroid) of its cross section, it is called an axialload. A load at any other point in the cross section isknown as an eccentric load. A short column under the ac-tion of an axial load will fail by direct compression beforeit buckles, but a long column loaded in the same mannerwill fail by buckling (bending), the buckling effect beingso large that the effect of the axial load may be neglected.The intermediate-length column will fail by a combina-tion of direct compressive stress and bending.In 1757, mathematician Leonhard Euler derived a for-mula that gives the maximum axial load that a long, slen-der, ideal column can carry without buckling. An idealcolumn is one that is perfectly straight, homogeneous,and free from initial stress. The maximum load, some-times called the critical load, causes the column to be ina state of unstable equilibrium; that is, the introductionof the slightest lateral force will cause the column to failby buckling. The formula derived by Euler for columnswith no consideration for lateral forces is given below.However, if lateral forces are taken into consideration thevalue of critical load remains approximately the same.

F =π2EI

(KL)2

where

F = maximum or critical force (vertical loadon column),E = modulus of elasticity,I = area moment of inertia,L = unsupported length of column,K = column effective length factor, whosevalue depends on the conditions of end supportof the column, as follows.

K

K

K

K

KL is the effective length of the column.

Examination of this formula reveals the following inter-esting facts with regard to the load-bearing ability of slen-der columns.

1. Elasticity and not the compressive strength of thematerials of the column determines the critical load.

2. The critical load is directly proportional to thesecond moment of area of the cross section.

3. The boundary conditions have a considerable effecton the critical load of slender columns. The bound-ary conditions determine the mode of bending andthe distance between inflection points on the de-flected column. The inflection points in the deflec-tion shape of the column are the points at which thecurvature of the column change sign and are also thepoints at which the internal bending moments arezero. The closer together the inflection points are,the higher the resulting capacity of the column.

A demonstration model illustrating the different “Euler” bucklingmodes. The model shows how the boundary conditions affect thecritical load of a slender column. Notice that each of the columnsare identical, apart from the boundary conditions.

The strength of a column may therefore be increased bydistributing the material so as to increase the moment ofinertia. This can be done without increasing the weightof the column by distributing the material as far from theprincipal axis of the cross section as possible, while keep-ing the material thick enough to prevent local buckling.This bears out the well-known fact that a tubular sectionis much more efficient than a solid section for columnservice.Another bit of information that may be gleaned from thisequation is the effect of length on critical load. For agiven size column, doubling the unsupported length quar-ters the allowable load. The restraint offered by the endconnections of a column also affects the critical load. Ifthe connections are perfectly rigid, the critical load willbe four times that for a similar column where there is noresistance to rotation (in which case the column is ideal-ized as having hinges at the ends).Since the radius of gyration is defined as the square rootof the ratio of the column’s moment of inertia about anaxis to cross sectional area, the above formula may berearranged as follows. Using the Euler formula for hinged

3

ends, and substituting A·r2 for I, the following formularesults.

σ =F

A=

π2E

(ℓ/r)2

where F/A is the allowable stress of the column, and l/ris the slenderness ratio.Since structural columns are commonly of intermediatelength, and it is impossible to obtain an ideal column,the Euler formula on its own has little practical applica-tion for ordinary design. Issues that cause deviation fromthe pure Euler column behaviour include imperfections ingeometry in combination with plasticity/non-linear stressstrain behaviour of the column’s material. Consequently,a number of empirical column formulae have been de-veloped to agree with test data, all of which embody theslenderness ratio. For design, appropriate safety factorsare introduced into these formulae. One such formula isthe Perry Robertson formula which estimates the criticalbuckling load based on an initial (small) curvature. TheRankine Gordon formula (Named forWilliam JohnMac-quorn Rankine and Perry Hugesworth Gordon (1899 –1966)) is also based on experimental results and suggeststhat a column will buckle at a load F ₐₓ given by:

1

Fmax=

1

Fe+

1

Fc

where Fₑ is the Euler maximum load and F is the maxi-mum compressive load. This formula typically producesa conservative estimate of F ₐₓ.

1.1 Self-buckling

A free-standing, vertical column, with density ρ , Young’smodulus E , and radius r , will buckle under its ownweight if its height exceeds a certain critical height:[1][2][3]

hcrit =

(9B2

4

EI

ρgπr2

)1/3

where g is the acceleration due to gravity, I is the secondmoment of area of the beam cross section, and B is thefirst zero of the Bessel function of the first kind of order−1/3, which is equal to 1.86635086...

2 Buckling under tensile deadloading

Usually buckling and instability are associated to com-pression, but recently Zaccaria, Bigoni, Noselli and Mis-seroni (2011)[4] have shown that buckling and instability

Fig. 2: Elastic beam system showing buckling under tensile deadloading.

can also occur in elastic structures subject to dead tensileload. An example of a single-degree-of-freedom struc-ture is shown in Fig. 1, where the critical load is also in-dicated. Another example involving flexure of a structuremade up of beam elements governed by the equation ofthe Euler’s elastica is shown in Fig.2. In both cases, thereare no elements subject to compression. The instabilityand buckling in tension are related to the presence of theslider, the junction between the two rods, allowing onlyrelative sliding between the connected pieces. Watch amovie for more details.

3 Constraints, curvature and mul-tiple buckling

Fig. 3: A one-degree-of-freedom structure exhibiting a tensile(compressive) buckling load as related to the fact that the rightend has to move along the circular profile labeled 'Ct' (labelled'Cc').

Buckling of an elastic structure strongly depends on thecurvature of the constraints against which the ends of thestructure 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 ten-sile (or a compressive) buckling load as related to the factthat one end has to move along the circular profile labeled'Ct' (labelled 'Cc').The two circular profiles can be arranged in a 'S'-shapedprofile, as shown in Fig.4; in that case a discontinuity ofthe constraint’s curvature is introduced, leading to multi-ple bifurcations. Note that the single-degree-of-freedomstructure shown in Fig.4 has two buckling loads (one ten-sile and one compressive). Watch a movie for more de-

4 5 VARIOUS FORMS OF BUCKLING

Fig. 4: A one-degree-of-freedom structure with a 'S'-shaped bi-circular profile exhibiting multiple bifurcations (both tensile andcompressive).

tails.

4 Flutter instability

Structures subject to a follower (nonconservative) loadmay suffer instabilities which are not of the buckling typeand therefore are not detectable with a static approach.[6]For instance, the so-called 'Ziegler column' is shown inFig.5.

Fig. 5: A sketch of the 'Ziegler column', a two-degree-of-freedomsystem subject to a follower load (the force P remains always par-allel to the rod BC), exhibiting flutter and divergence instability.The two rods, of linear mass density ρ, are rigid and connectedthrough two rotational springs of stiffness k1 and k2.

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

13ρl

21 (l1 + 3l2) α1 +

12ρl1l

22 cos(α1 − α2)α2 +

12ρl1l

22 sin(α1 − α2)α

22 + (k1 + k2)α1 − k2α2 +

+(β1 + β2)α1 − β2α2 − l1P sin(α1 − α2) = 0,

12ρl1l

22 cos(α1 − α2)α1 +

13ρl

32α2 − 1

2ρl1l22 sin(α1 − α2)α

21 − k2(α1 − α2)− β2(α1 − α2) = 0,

and their linearized version is

13ρl

21 (l1 + 3l2) α1 +

12ρl1l

22α2 + (k1 + k2)α1 − k2α2 − l1P (α1 − α2) = 0,

12ρl1l

22α1 +

13ρl

32α2 − k2(α1 − α2) = 0.

Assuming a time-harmonic solution in the form

αj = Aj e−iΩ t, j = 1, 2,

we find the critical loads for flutter ( Pf ) and divergence( Pd ),

Pf,d =k2l1

·k + (1 + λ)3 ∓ λ

√k(3 + 4λ)

1 + 3λ/2

where λ = l1/l2 and k = k1/k2 .

Fig. 6: A sequence of deformed shapes at consecutive times in-tervals of the structure sketched in Fig.5 and exhibiting flutter(upper part) and divergence (lower part) instability.

Flutter instability corresponds to a vibrational motion ofincreasing amplitude and is shown in Fig.6 (upper part)together with the divergence instability (lower part) con-sisting in an exponential growth.Recently, Bigoni and Noselli (2011)[7] have experimen-tally shown that flutter and divergence instabilities can bedirectly related to dry friction, watch the movie for moredetails.

5 Various forms of buckling

Buckling is a state which defines a point where an equilib-rium configuration becomes unstable under a parametricchange of load and can manifest itself in several differentphenomena. All can be classified as forms of bifurcation.There are four basic forms of bifurcation associated withloss of structural stability or buckling in the case of struc-tures with a single degree of freedom. These comprisetwo types of pitchfork bifurcation, one saddle-node bi-furcation (often referred to as a limit point) and onetranscritical bifurcation. The pitchfork bifurcations arethe most commonly studied forms and include the buck-ling of columns and struts, sometimes known as Eulerbuckling; the buckling of plates, sometimes known as lo-cal buckling, which is well known to be relatively safe

5

(both are supercritical phenomena) and the buckling ofshells, which is well-known to be a highly dangerous (sub-critical phenomenon).[8] Using the concept of potentialenergy, equilibrium is defined as a stationary point withrespect to the degree(s) of freedom of the structure. Wecan then determine whether the equilibrium is stable, ifthe stationary point is a local minimum; or unstable, ifit is a maximum, point of inflection or saddle point (formultiple-degree-of-freedom structures only) – see ani-mations below.In Euler buckling,[9][10] the applied load is increased bya small amount beyond the critical load, the structure de-forms into a buckled configuration which is adjacent tothe original configuration. For example, the Euler col-umn pictured will start to bow when loaded slightly aboveits critical load, but will not suddenly collapse.

In structures experiencing limit point instability, if theload is increased infinitesimally beyond the critical load,the structure undergoes a large deformation into a differ-ent stable configuration which is not adjacent to the orig-inal configuration. An example of this type of bucklingis a toggle frame (pictured) which 'snaps’ into its buckledconfiguration.

6 Bicycle wheels

A conventional bicycle wheel consists of a thin rim keptunder high compressive stress by the (roughly normal) in-ward pull of a large number of spokes. It can be consid-ered as a loaded column that has been bent into a cir-cle. If spoke tension is increased beyond a safe level,the wheel spontaneously fails into a characteristic saddleshape (sometimes called a “taco” or a extquotedblpringleextquotedbl) like a three-dimensional Euler column. Thisis normally a purely elastic deformation and the rim willresume its proper plane shape if spoke tension is reducedslightly.

7 Surface materials

Sun kink in rail tracks

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 sur-face, causing it to expand, forcing adjacent pieces to pushagainst each other. If the stress is great enough, thepavement can lift up and crack without warning. Goingover a buckled section can be very jarring to automobiledrivers, described as running over a speed hump at high-way speeds.Similarly, rail tracks also expand when heated, and canfail by buckling, a phenomenon called sun kink. It ismore common for rails to move laterally, often pullingthe underlain railroad ties (sleepers) along.

8 Energy method

Often it is very difficult to determine the exact bucklingload in complex structures using the Euler formula, dueto the difficulty in deciding the constant K. Therefore,

6 12 DYNAMIC BUCKLING

maximum buckling load often is approximated using en-ergy conservation. This way of calculating the maximumbuckling load is often referred to as the energy method instructural analysis.The first step in this method is to suggest a displacementfunction. This function must satisfy the most importantboundary conditions, such as displacement and rotation.The more accurate the displacement function, the moreaccurate the result.In this method, there are two equations used (for smalldeformations) to approximate the “inner” energy (the po-tential energy stored in elastic deformation of the struc-ture) and “outer” energy (the work done on the system byexternal forces).

Uinner =E

2

∫I(x)(wxx(x))

2 dx

Uouter =PCrit2

∫(wx(x))

2 dx

where w(x) is the displacement function and the sub-scripts x and xx refer to the first and second derivativesof the displacement. Energy conservation yields:

UInner = UOuter

9 Flexural-torsional buckling

Occurs in compression members only and it can be de-scribed as a combination of bending and twisting of amember. And it must be considered for design purposes,since the shape and cross sections are very critical. Thismostly occurs in channels, structural tees, double-angleshapes, and equal-leg single angles.

10 Lateral-torsional buckling

When a simply supported beam is loaded in flexure, thetop side is in compression, and the bottom side is intension. When a slender member is subjected to an axialforce, failure takes place due to bending or torsion ratherthan direct compression of the material. If the beam isnot supported in the lateral direction (i.e., perpendicularto the plane of bending), and the flexural load increasesto a critical limit, the beam will fail due to lateral buck-ling of the compression flange. In wide-flange sections,if the compression flange buckles laterally, the cross sec-tion will also twist in torsion, resulting in a failure modeknown as lateral-torsional buckling.

10.1 The modification factor (Cb)

where

Mmax

MA

MB

MC

11 Plastic buckling

Buckling will generally occur slightly before the calcu-lated elastic buckling strength of a structure, due to non-linear behavior of the material. When the compressiveload is near the buckling load, the structure will bow sig-nificantly and the material of the column will divergefrom a linear stress-strain behavior. The stress-strainbehavior of materials is not strictly linear even belowyield, and the modulus of elasticity decreases as stress in-creases, and significantly so as the stresses approach theyield strength. This lower rigidity reduces the bucklingstrength of the structure and causes at a load less than thatpredicted by the assumption of lineal elastic behavior.A more accurate approximation of the buckling load canbe had by the use of the tangent modulus of elasticity, E ,in place of the elastic modulus of elasticity. The tangentmodulus is a line drawn tangent to the stress-strain curveat a particular value of strain. Plots of the tangent modu-lus of elasticity for a variety of materials are available instandard references.

12 Dynamic buckling

If a column is loaded suddenly and then the load re-leased, the column can sustain a much higher load thanits static (slowly applied) buckling load. This can happenin a long, unsupported column (rod) used as a drop ham-mer. The duration of compression at the impact end is thetime required for a stress wave to travel up the rod to theother (free) end and back down as a relief wave. Maxi-mum buckling occurs near the impact end at a wavelengthmuch shorter than the length of the rod, and at a stressmany times the buckling stress of a statically-loaded col-umn. The critical condition for buckling amplitude toremain less than about 25 times the effective rod straight-ness imperfection at the buckle wavelength is

σL = ρc2h

where σ is the impact stress,L is the length of the rod, c isthe elastic wave speed, and h is the smaller lateral dimen-sion of a rectangular rod. Because the buckle wavelengthdepends only on σ and h , this same formula holds forthin cylindrical shells of thickness h .[12]

7

13 Buckling of thin cylindricalshells subject to axial loads

Solutions of Donnell’s eight order differential equationgives the various buckling modes of a thin cylinder undercompression. But this analysis, which is in accordancewith the small deflection theory gives much higher val-ues than shown from experiments. So it is customary tofind the critical buckling load for various structures whichare cylindrical in shape from pre-existing design curveswhere critical buckling load F ᵣ is plotted against the ra-tio R/t, where R is the radius and t is the thickness ofthe cylinder for various values of L/R, L the length of thecylinder. If cut-outs are present in the cylinder, criticalbuckling loads as well as pre-buckling modes will be af-fected. Presence or absence of reinforcements of cut-outswill also affect the buckling load.

14 Buckling of pipes and pressurevessels subject to external over-pressure

Pipes and pressure vessels subject to external overpres-sure, caused for example by steam cooling within the pipeand condensing into water with subsequent massive pres-sure drop, risk buckling due to compressive hoop stresses.Design rules for calculation of the required wall thick-ness or reinforcement rings are given in various pipingand pressure vessel codes.

15 See also

• Perry Robertson formula

• Stiffening

• Wood method

• Yoshimura buckling

16 References[1] Kato, K. (1915). “Mathematical Investigation on the Me-

chanical Problems of Transmission Line”. Journal of theJapan Society of Mechanical Engineers 19: 41.

[2] Ratzersdorfer, Julius (1936). Die Knickfestigkeit vonStäben und Stabwerken. Wein, Austria: J. Springer. pp.107–109.

[3] Cox, Steven J.; C. Maeve McCarthy (1998). “TheShape of the Tallest Column”. Society for In-dustrial and Applied Mathematics 29: 547–554.doi:10.1137/s0036141097314537.

[4] D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni,Structures buckling under tensile dead load. Proceedingsof the Royal Society A, 2011, 467, 1686-1700.

[5] D. Bigoni, D. Misseroni, G. Noselli and D. Zaccaria, Ef-fects of the constraint’s curvature on structural instability:tensile buckling and multiple bifurcations. Proceedings ofthe Royal Society A, 2012, doi:10.1098/rspa.2011.0732.

[6] Bigoni, D. Nonlinear Solid Mechanics: Bifurcation The-ory and Material Instability. Cambridge University Press,2012 . ISBN 9781107025417.

[7] D. Bigoni and G. Noselli, Experimental evidence of flut-ter and divergence instabilities induced by dry friction.Journal of the Mechanics and Physics of Solids, 2011, 59,2208–2226.

[8] “A general theory of elastic stability” By J. M. T. Thomp-son & G. W. Hunt, Wiley, 1973

[9] “Buckling of Bars, Plates, and Shells” By Robert M. Jones

[10] “Observations on eigenvalue buckling analysis within a fi-nite element context” by Christopher J. Earls

[11] http://dcist.com/2012/07/excessive_heat_probable_cause_in_gr.php

[12] Lindberg, H. E., and Florence, A. L., Dynamic PulseBuckling, Martinus Nijhoff Publishers, 1987, pp. 11–56,297–298.

• Timoshenko, S. P., and Gere, J. M., Theory of Elas-tic Stability, 2 ed., McGraw-Hill, 1961.

• Nenezich, M., Thermoplastic ContinuumMechanics,Journal of Aerospace Structures, Vol. 4, 2004.

• The Stability of Elastic Equilibrium byW. T. Koiter,PhD Thesis, 1945.

• Dhakal Rajesh and Koichi Maekawa (October2002). “Reinforcement Stability and Fracture ofCover Concrete in Reinforced Concrete Members”.

• Willian T. Segui (2007). “Steel Design” Fourth Edi-tion. United States. Chris Carson.

• Analysis and design of flight vehicle structures-E.F.Brune

17 External links• The complete theory and example experimentalresults for long columns are available as a 39-page PDF document at http://lindberglce.com/tech/buklbook.htm

• Laboratory for Physical Modeling of Structures andPhotoelasticity (University of Trento, Italy)

• http://www.midasuser.com.tw/t_support/tech_pds/files/Tech%20Note-Lateral%20Torsional%20Buckling.pdf

8 18 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

18 Text and image sources, contributors, and licenses

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