Buckling

Buckling - Wikipedia, the free encyclopediaPage 1 of 10 Buckling From 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 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 ■ 1 Columns ■ 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 ■ 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 Further reading ■ 20 External links Columns The ratio of the effective length of a column to the least radius of gyration of its cross section is called the slenderness ratio (sometimes expressed with the Greek letter lambda, λ). This ratio affords a means of classifying columns. Slenderness ratio is important for design considerations. All the following are approximate values used for convenience. ■ https://en.wikipedia.org/wiki/Buckling 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. 03/08/2015 ≈ 0. For both ends pinned (hinged.Buckling . A column under a concentric axial load exhibiting the characteristic deformation of buckling In 1757. 2. For one end fixed and the other end free to move laterally.50. slender. https://en. free to rotate). the introduction of the slightest lateral force will cause the column to fail by buckling. = maximum or critical force (vertical load on column). = column effective length factor. The value of K is given in most structural handbooks.Wikipedia. 3.org/wiki/Buckling 03/08/2015 . An ideal column is one that is perfectly straight. The maximum load. and free from initial stress. The critical load is directly proportional to the second moment of area of the cross section.0. doubling the unsupported length quarters the allowable load. the free encyclopedia ■ Page 2 of 10 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. 1. The restraint offered by the end connections of a column also affects the critical load. and substituting A·r2 for I. The dividing line between intermediate and long timber columns cannot be readily evaluated. If the ratio is greater than 10. while keeping the material thick enough to prevent local buckling. Examination of this formula reveals the following interesting facts with regard to the load-bearing ability of slender columns. However. causes the column to be in a state of unstable equilibrium.0. For both ends fixed. but a long column loaded in the same manner will fail by buckling (bending). sometimes called the critical load. The closer together the inflection points are. it is called an axial load. 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. that is. mathematician Leonhard Euler derived a formula that gives the maximum axial load that a long. A short column under the action of an axial load will fail by direct compression before it buckles. it can be seen that this arbitrary limit would vary with the species of the timber. the above formula may be rearranged as follows. it is considered a long column (sometimes referred to as a slender column). = modulus of elasticity. 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). This bears out the well-known fact that a tubular section is much more efficient than a solid section for column service. ideal column can carry without buckling. whose value depends on the conditions of end support of the column. = 0. The intermediate-length column will fail by a combination of direct compressive stress and bending.wikipedia. ■ If the load on a column is applied through the center of gravity (centroid) of its cross section. homogeneous. the following formula results. If the connections are perfectly rigid. = unsupported length of column. is the effective length of the column. 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. For a given size column. The strength of a column may therefore be increased by distributing the material so as to increase the moment of inertia. For one end fixed and the other end pinned. Using the Euler formula for hinged ends. 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 boundary conditions have a considerable effect on the critical load of slender columns.699. 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. as follows. = area moment of inertia of the cross section of the rod. = 2. the buckling effect being so large that the effect of the axial load may be neglected. the higher the resulting capacity of the column. The formula derived by Euler for columns with no consideration for lateral forces is given below. if lateral forces are taken into consideration the value of critical load remains approximately the same. Since K depends on the modulus of elasticity and the allowable compressive stress parallel to the grain. where The eccentricity of the axial force results in a bending moment acting on the beam element. Elasticity and not the compressive strength of the materials of the column determines the critical load. Another bit of information that may be gleaned from this equation is the effect of length on critical load. The boundary conditions determine the mode of bending and the distance between inflection points on the deflected column. 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. = 1. A load at any other point in the cross section is known as an eccentric load. This formula typically produces a conservative estimate of Fmax. Noselli and Zaccaria. the free encyclopedia where A demonstration model illustrating the different "Euler" buckling modes. leading to multiple bifurcations. An example of a single-degree-of-freedom structure is shown in Fig. there are no elements subject to compression. and is the slenderness ratio.. where the critical load is also indicated.ing. 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. and it is impossible to obtain an ideal column. 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. but recently Zaccaria. The model shows how the boundary conditions affect the critical load of a slender column. the Euler formula on its own has little practical application for ordinary design. appropriate safety factors are introduced into these formulae. and B is the first zero of the Bessel function of the first kind of order -1/3.4 has two buckling loads (one tensile and one compressive). Note that the single-degree-of-freedom structure shown in Fig.Buckling .ing. Watch a movie (http://www. allowing only relative sliding between the connected pieces.wikipedia. Noselli and Misseroni (2011)[4] have shown that buckling and instability can also occur in elastic structures subject to dead tensile load.Wikipedia. in that case a discontinuity of the constraint's curvature is introduced. Page 3 of 10 is the allowable stress of the column. One such formula is the Perry Robertson formula which estimates the critical buckling load based on an initial (small) curvature.3. Buckling under tensile dead loading Usually buckling and instability are associated to compression. which is equal to 1. and cross-sectional area .html) for more details. In fact. with density . even a single-degree-of-freedom system (see Fig.it/~bigoni/tensile_buckling.it/~bigoni/multiple_bifurcations.org/wiki/Buckling 03/08/2015 .unitn. all of which embody the slenderness ratio. The instability and buckling in tension are related to the presence of the slider.4. Notice that the columns are identical. Misseroni. vertical column. The two circular profiles can be arranged in a 'S'-shaped profile.html) for more details. the junction between the two rods. 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. Young's modulus . Flutter instability https://en. 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. For design. 2: Elastic beam system showing buckling under tensile dead loading. Self-buckling A free-standing. Since structural columns are commonly of intermediate length.. In both cases. Consequently. 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.unitn. 2.3) 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').86635086. I is the second moment of area of the beam cross section. 2012[5]). Bigoni. Watch a movie (http://www. Constraints. a number of empirical column formulae have been developed to agree with test data. as shown in Fig. curvature and multiple buckling Fig. apart from the boundary conditions. Recently.it/~bigoni/flutter. Various forms of buckling 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.ing. Fig.Buckling . 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.wikipedia.[6] For instance. These comprise two types of pitchfork bifurcation. Archetypal rigid link models with a single degree of freedom (SDOF) used to demonstrate basic buckling phenomena (see bifurcation diagrams below). and their linearized version is Assuming a time-harmonic solution in the form we find the critical loads for flutter ( where and ) and divergence ( ). https://en.unitn. equilibrium is defined as a stationary point with respect to the degree(s) of freedom of the structure.6 (upper part) together with the divergence instability (lower part) consisting in an exponential growth.[8] Using the concept of potential energy. All can be classified as forms of bifurcation.Wikipedia. We can then determine whether the equilibrium is stable. the free encyclopedia Page 4 of 10 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. The pitchfork bifurcations are the most commonly studied forms and include the buckling of columns and struts. point of inflection or saddle point (for multiple-degree-of-freedom structures only) – see animations below. sometimes known as local buckling. Fig. are rigid and connected through two rotational springs of stiffness k1 and k2. Bigoni and Noselli (2011)[7] have experimentally shown that flutter and divergence instabilities can be directly related to dry friction. This two-degree-of-freedom system does not display a quasi-static buckling. sometimes known as Euler buckling. exhibiting flutter and divergence instability. of linear mass density ρ. 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'). The two rods. 4: A one-degree-of-freedom structure with a 'S'-shaped bicircular profile exhibiting multiple bifurcations (both tensile and compressive). which is well known to be relatively safe (both are supercritical phenomena) and the buckling of shells. we note that the equations of motion are Fig. or unstable.html) for more details. the buckling of plates. which is well-known to be a highly dangerous (subcritical phenomenon). the so-called 'Ziegler column' is shown in Fig. but becomes dynamically unstable. All cases start at the position corresponding to q=0. one saddle-node bifurcation (often referred to as a limit point) and one transcritical bifurcation.5. if it is a maximum. Flutter instability corresponds to a vibrational motion of increasing amplitude and is shown in Fig. . a two-degree-of-freedom system subject to a follower load (the force P remains always parallel to the rod BC). if the stationary point is a local minimum.org/wiki/Buckling 03/08/2015 . 5: A sketch of the 'Ziegler column'. watch the movie (http://www. To see this. Two saddle-node bifurcations (limit points). the structure undergoes a large deformation into a different stable configuration which is not adjacent to the original configuration. the structure deforms into a buckled configuration which is adjacent to the original configuration. if the load is increased infinitesimally beyond the critical load. 6: A sequence of deformed shapes at consecutive times intervals of the structure sketched in Fig.org/wiki/Buckling 03/08/2015 .5 and exhibiting flutter (upper part) and divergence (lower part) instability. but will not suddenly collapse. Link-strut with rotational spring. In structures experiencing limit point instability.Wikipedia. An example of this type of buckling is a toggle frame (pictured) which 'snaps' into its buckled configuration.Buckling . the Euler column pictured will start to bow when loaded slightly above its critical load. the free encyclopedia Truss with spring tie (model shallow tied arch). Animations of the variation of total potential energy (red) for various load values. Asymmetrically supported link-strut. Transcritical bifurcation (asymmetric buckling point). In Euler buckling. Subcritical pitchfork bifurcation (unstablesymmetric buckling point). Supercritical pitchfork bifurcation (stable-symmetric buckling point). Fig. Page 5 of 10 Link-strut with transverse translational spring.[9][10] the applied load is increased by a small amount beyond the critical load.wikipedia. https://en. in generic structural systems with the indicated bifurcation or buckling behaviour. P (black). For example. and can fail by buckling. Radiant heat from the sun is absorbed in the road surface.wikipedia. the wheel spontaneously fails into a characteristic saddle shape (sometimes called a "taco" or a "pringle") like a three-dimensional Euler column.org/wiki/Buckling 03/08/2015 . More on sun kink Cause Sun kink in rail tracks 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: . If spoke tension is increased beyond a safe level. This is normally a purely elastic deformation and the rim will resume its proper plane shape if spoke tension is reduced slightly. causing it to expand. often pulling the underlain railroad ties (sleepers) along. a phenomenon called sun kink. Derivation of buckling force function: The linear thermal expansion due to heating of the track is found using where ■ ■ ■ ■ ΔL = thermal expansion of the rail L = length of the rail/track α = coefficient of thermal expansion ΔT = increase in temperature https://en. It can be considered as a loaded column that has been bent into a circle. forcing adjacent pieces to push against each other. the free encyclopedia Page 6 of 10 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 of spokes.Wikipedia. Surface materials Buckling is also a failure mode in pavement materials.Buckling . rail tracks also expand when heated. since asphalt is more flexible. the pavement can lift up and crack without warning. described as running over a speed hump at highway speeds. If the stress is great enough. primarily with concrete. Going over a buckled section can be very jarring to automobile drivers. Similarly. It is more common for rails to move laterally. 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). off CSX tracks. off CSX tracks. such as displacement and rotation. North Carolina. This function must satisfy the most important boundary conditions. 2002 Amtrak Auto-Train derailment. Maryland. July 6. the free encyclopedia Page 7 of 10 According to Hooke's law the extension due to a force (in the rail) is where ■ ■ ■ ■ ■ ΔL = extension of the rail/track F = the force extending a rod. 2002 Amtrak Capitol Limited derails. due to the difficulty in deciding the constant K. here the induced force in the rail E = modulus of elasticity of rail material (steel) A = cross section of rail L = length of rail Putting these together gives or Accidents These accidents were deemed to be sun kink related (more information available at List of rail accidents (2000–2009)): ■ ■ ■ ■ April 18. 2010 CSX train derails off tracks in Waxhaw. near Crescent City.wikipedia. maximum buckling load often is approximated using energy conservation.org/wiki/Buckling and refer to the first and second derivatives of the displacement. Energy conservation yields: 03/08/2015 . July 29. This way of calculating the maximum buckling load is often referred to as the energy method in structural analysis.Buckling . Therefore. In this method. Florida. the more accurate the result.Wikipedia. near Kensington. The more accurate the displacement function.[11] Energy method Often it is very difficult to determine the exact buckling load in complex structures using the Euler formula. 2012 WMATA Metrorail train derails off tracks near Hyattsville. Maryland. where is the displacement function and the subscripts https://en. July 8. The first step in this method is to suggest a displacement function. regardless of beam configuration or loading. the free encyclopedia Page 8 of 10 Flexural-torsional buckling Occurs in compression members only and it can be described as a combination of bending and twisting of a member. and equal-leg single angles. c) cross section in center with lateral-torsional buckling If an appropriate value of Cb is not given in the table. if the compression flange buckles laterally. and the flexural load increases to a critical limit. failure takes place due to bending or torsion rather than direct compression of the material. (kip-in) Plastic buckling Buckling will generally occur slightly before the calculated elastic buckling strength of a structure. never less.org/wiki/Buckling 03/08/2015 . and the bottom side is in tension. the structure will bow significantly and the material of the column will diverge from a linear stress-strain behavior. structural tees. The tangent modulus is a line drawn tangent to the stress-strain curve at a particular value of strain. lateral torsional buckling at I-beam with vertical force in center: a) longitudinal view. This lower rigidity reduces the buckling strength of the structure and causes at a load less than that predicted by the assumption of lineal elastic behavior. For cantilevers or overhangs where the free end is unbraced. A more accurate approximation of the buckling load can be had by the use of the tangent modulus of elasticity. In wide-flange sections. This mostly occurs in channels. The reason for this factor is to allow for nonuniform moment diagrams when the ends of a beam segment are braced. (kip-in) = absolute value of maximum moment at centerline of the unbraced segment. The modification factor (Cb) Cb is a modification factor used in the equation for nominal flexural strength when determining lateral-torsional buckling. the beam will fail due to lateral buckling of the compression flange. (kip-in) = absolute value of maximum moment at three-quarter point of the unbraced segment. the cross section will also twist in torsion. and the modulus of elasticity decreases as stress increases. (kip-in) = absolute value of maximum moment at quarter point of the unbraced segment. Cb is equal to 1.. If the beam is not supported in the lateral direction (i. The conservative value for Cb can be taken as 1. double-angle shapes. since the shape and cross sections are very critical. perpendicular to the plane of bending). Lateral-torsional buckling When a simply supported beam is loaded in flexure. Plots of the tangent modulus of elasticity for a variety of materials are available in standard references. The stress-strain behavior of materials is not strictly linear even below yield. Dynamic buckling https://en.wikipedia. in place of the elastic modulus of elasticity.Wikipedia. Cb is always equal to or greater than 1. due to non-linear behavior of the material.e. and significantly so as the stresses approach the yield strength. When a slender member is subjected to an axial force. the top side is in compression. resulting in a failure mode known as lateral-torsional buckling. it can be obtained via the following formula: where = absolute value of maximum moment in the unbraced segment. When the compressive load is near the buckling load. b) cross section near support. And it must be considered for design purposes. A table of values of Cb for simply supported beams are shown in Figure 1. but in some cases it may be excessively conservative. Et.Buckling . L the length of the cylinder. caused for example by steam cooling within the pipe and condensing into water with subsequent massive pressure drop. where R is the radius and t is the thickness of the cylinder for various values of L/R. D. Journal of the Japan Society of Mechanical Engineers 19: 41. M. "The Shape of the Tallest Column". (21 April 2011).. Robert M. Observations on eigenvalue buckling analysis within a finite element context.org/10. 8. Maximum buckling occurs near the impact end at a wavelength much shorter than the length of the rod. T. and Gere.doi. Design rules for calculation of the required wall thickness or reinforcement rings are given in various piping and pressure vessel codes. W. is the elastic wave speed. Julius (1936). Society for Industrial and Applied Mathematics 29: 547–554. Buckling of Bars..2010.it/~bigoni/multiple_bifurcations. Theory of Elastic Stability...org/10... Christopher J. Plates.0732). https://en. (1 December 2007). Earls. P. (2012). Wiley. Thompson.0732 (https://dx.php Lindberg. and at a stress many times the buckling stress of a statically-loaded column.1137%2Fs0036141097314537).wikipedia. 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. See also ■ ■ ■ ■ ■ Perry Robertson formula Rail stressing Stiffening Wood method Yoshimura buckling References 1. doi:10. pp. Noselli. 5. If cut-outs are present in the cylinder. (2012). Steven J.Buckling . 11. J. Dynamic Pulse Buckling. "Structures buckling under tensile dead load" (http://www. 1961.pdf) (PDF). doi:10.unitn.it/~bigoni/paper/zaccaria_bigoni_buckling_tension.unitn. 6.1137/s0036141097314537 (https://dx. ISBN 978-0471859918. is the length of the rod. unsupported column (rod) used as a drop hammer.[12] Buckling of thin cylindrical shells subject to axial loads Solutions of Donnell's eight order differential equation gives the various buckling modes of a thin cylinder under compression. D.0505). This can happen in a long. (1973). 107–109.com/2012/07/excessive_heat_probable_cause_in_gr. Martinus Nijhoff Publishers. critical buckling loads as well as pre-buckling modes will be affected.ing. 2 ed. Jones. D. ISBN 978-1560328278. Bigoni. "Effects of the constraint's curvature on structural instability: tensile buckling and multiple bifurcations" (http://www. McGraw-Hill. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Bigoni. which is in accordance with the small deflection theory gives much higher values than shown from experiments. Proceedings of the Royal Society A. this same formula holds for thin cylindrical shells of thickness . M. Springer. D. S. H.2011. Cambridge University Press. G. 3.. C.1098%2Frspa. Annual Stability Conference (New Orleans. D. A general theory of elastic stability. and Shells.. and is the smaller lateral dimension of a rectangular rod. 9. Austria: J. But this analysis.html). Wein. (2007). 11–56. Cox. http://dcist. Zaccaria. 12. the column can sustain a much higher load than its static (slowly applied) buckling load.2010.org/10.. ISBN 978-3-662-24075-5. (2011).unitn. Presence or absence of reinforcements of cut-outs will also affect the buckling load. Bigoni.it/~bigoni). Proceedings of the Royal Society A 467 (2130): 1686–1700.Wikipedia. Buckling of pipes and pressure vessels subject to external overpressure Pipes and pressure vessels subject to external overpressure. Noselli. 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. Florence. K.doi..ing. E. G. D.0505 (https://dx. Hunt.. (1915).. 2. the free encyclopedia Page 9 of 10 If a column is loaded suddenly and then the load released. D.2011.org/wiki/Buckling 03/08/2015 .doi. Journal of the Mechanics and Physics of Solids 59: 2208–2226. 4. Die Knickfestigkeit von Stäben und Stabwerken [The buckling resistance of members and frames] (in German). Ratzersdorfer.1098/rspa. Misseroni. Bigoni.ing. Misseroni. Proceedings of the Structural Stability Research Council. A. 297–298. CRC. L. G. ISBN 9781107025417.1098/rspa. Zaccaria.1098%2Frspa.. Maeve McCarthy (1998). Kato. "Mathematical Investigation on the Mechanical Problems of Transmission Line". D. J. doi:10. LA). Because the buckle wavelength depends only on and . (1987). "Experimental evidence of flutter and divergence instabilities induced by dry friction" (http://www. 7. risk buckling due to compressive hoop stresses. pp. G. 10. Further reading ■ Timoshenko. So it is customary to find the critical buckling load for various structures which are cylindrical in shape from pre-existing design curves where critical buckling load Fcr is plotted against the ratio R/t. Noselli. 4.unitn.edu/DigitalCollection/1970/AFFDLTR70-025. 1945.ing.. additional terms may apply. Inc.pdf) by W. T. Thermoplastic Continuum Mechanics.tw/t_support/tech_pds/files/Tech%20Note-Lateral%20Torsional%20Buckling..php?title=Buckling&oldid=673499880" Categories: Elasticity (physics) Materials science Mechanical failure modes Structural analysis Mechanics ■ This page was last modified on 28 July 2015.org/w/index.pdf Retrieved from "https://en. Italy) (http://ssmg.com.midasuser.iit. Wikipedia® is a registered trademark of the Wikimedia Foundation.it) ■ http://www.wikipedia. ■ Dhakal Rajesh and Koichi Maekawa (October 2002). the free encyclopedia Page 10 of 10 ■ Nenezich.wikipedia. [1] (http://ascelibrary.) ■ Willian T.libdb. United States.njit.F. 2004.Buckling . Segui (2007).E.org. By using this site. ■ Analysis and design of flight vehicle structures. Journal of Aerospace Structures. at 17:22. PhD Thesis.htm ■ Laboratory for Physical Modeling of Structures and Photoelasticity (University of Trento.Bruhn External links ■ The complete theory and example experimental results for long columns are available as a 39-page PDF document at http://lindberglce. you agree to the Terms of Use and Privacy Policy. M.com/tech/buklbook. ■ The Stability of Elastic Equilibrium (http://contrails.aip. Chris Carson. https://en. “Steel Design” Fourth Edition. "Reinforcement Stability and Fracture of Cover Concrete in Reinforced Concrete Members”. ■ Text is available under the Creative Commons Attribution-ShareAlike License.edu:8888/getpdf/servlet/GetPDFServlet?filetype=pdf&id=JSENDH000128000010001253000001&idtype=cvips&ident=freesearch.Wikipedia.org/wiki/Buckling 03/08/2015 . Vol. Koiter. a non-profit organization.