Buckling

In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have "buckled". Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress in slender columns.
Buckling may occur even though the stresses that develop in the structure are well below those needed to cause failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformations that occur after buckling do not cause the complete collapse of that member, the member will continue to support 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 buckled part of the structure beyond that which caused the member to buckle will be redistributed within the structure. Some aircraft are designed for thin skin panels to continue carrying load even in the buckled state.
Forms of buckling.
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 and their failure mode. The slenderness ratio is important for design considerations. All the following are approximate values used for convenience.
If the load on a column is applied through the center of gravity (centroid) of its cross section, it is called an axial load. A load at any other point in the cross section is known as an eccentric load. A short column under the action of an axial load will fail by direct compression before it buckles, but a long column loaded in the same manner will fail by springing suddenly outward laterally (buckling) in a bending mode. The buckling mode of deflection is considered a failure mode, and it generally occurs before the axial compression stresses (direct compression) can cause failure of the material by yielding or fracture of that compression member. However, intermediate-length columns will fail by a combination of direct compressive stress and bending.
In particular:
The theory of the behavior of columns was investigated in 1757 by mathematician Leonhard Euler. He derived the formula, the Euler 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, made of a homogeneous material, and free from initial stress. When the applied load reaches the Euler load, sometimes called the critical load, the column comes to be in a state of unstable equilibrium. At that load, the introduction of the slightest lateral force will cause the column to fail by suddenly "jumping" to a new configuration, and the column is said to have buckled. This is what happens when a person stands on an empty aluminum can and then taps the sides briefly, causing it to then become instantly crushed (the vertical sides of the can may be understood as an infinite series of extremely thin columns). The formula derived by Euler for long slender columns is given below.
To get the mathematical demonstration read: Euler's critical load
where
Examination of this formula reveals the following facts with regard to the load-bearing ability of slender columns.
A conclusion from the above is that the buckling load of a column may be increased by changing its material to one with a higher modulus of elasticity (E), or changing the design of the column's cross section so as to increase its moment of inertia. The latter can be done without increasing the weight of the column by distributing the material as far from the principal axis of the column's cross section as possible. For most purposes, the most effective use of the material of a column is that of a tubular section.
Another insight that may be gleaned from this equation is the effect of length on critical load. Doubling the unsupported length of the column quarters the allowable load. The restraint offered by the end connections of a column also affects its critical load. If the connections are perfectly rigid (does not allowing rotation of its ends), the critical load will be four times that for a similar column where the ends are pinned (allowing rotation of its ends).
Since the radius of gyration is defined as the square root of the ratio of the column's moment of inertia about an axis to its cross sectional area, the above Euler formula may be reformatted by substituting the radius of gyration formula_12 for formula_4:
where formula_15 is the stress that causes buckling the column, and formula_16 is the slenderness ratio.
Since structural columns are commonly of intermediate length, the Euler formula has little practical application for ordinary design. Issues that cause deviation from the pure Euler column behaviour include imperfections in geometry of the column in combination with plasticity/non-linear stress strain behaviour of the column's material. Consequently, a number of empirical column formulae have been developed that agree with test data, all of which embody the slenderness ratio. Due to the uncertainty in the behavior of columns, for design, appropriate safety factors are introduced into these formulae. One such formula is the Perry Robertson formula which estimates the critical buckling load based on an assumed small initial curvature, hence an eccentricity of the axial load. 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 formula_18 is the Euler maximum load and formula_19 is the maximum compressive load. This formula typically produces a conservative estimate of formula_20.
Self-buckling.
To get the mathematical demonstration read: Self-buckling
A free-standing, vertical column, with density formula_21, Young's modulus formula_3, and cross-sectional area formula_23, will buckle under its own weight if its height exceeds a certain critical value:
where formula_25 is the acceleration due to gravity, formula_4 is the second moment of area of the beam cross section, and formula_27 is the first zero of the Bessel function of the first kind of order −1/3, which is equal to 1.86635086…
Plate buckling.
A plate is a 3-dimensional structure defined as having a width of comparable size to its length, with a thickness that is very small in comparison to its other two dimensions. Similar to columns, thin plates experience out-of-plane buckling deformations when subjected to critical loads; however, contrasted to column buckling, plates under buckling loads can continue to carry loads, called local buckling. This phenomenon is incredibly useful in numerous systems, as it allows systems to be engineered to provide greater loading capacities.
For a rectangular plate, supported along every edge, loaded with a uniform compressive force per unit length, the derived governing equation can be stated by:
where
The solution to the deflection can be expanded into two harmonic functions shown:
where 
The previous equation can be substituted into the earlier differential equation where formula_36 equals 1. formula_40 can be separated providing the equation for the critical compressive loading of a plate:
where
The buckling coefficient is influenced by the aspect of the specimen, formula_37 / formula_45, and the number of lengthwise curvatures. For an increasing number of such curvatures, the aspect ratio produces a varying buckling coefficient; but each relation provides a minimum value for each formula_35. This minimum value can then be used as a constant, independent from both the aspect ratio and formula_35.
Given stress is found by the load per unit area, the following expression is found for the critical stress:
From the derived equations, it can be seen the close similarities between the critical stress for a column and for a plate. As the width formula_38 shrinks, the plate acts more like a column as it increases the resistance to buckling along the plate’s width. The increase of formula_37 allows for an increase of the number of sine waves produced by buckling along the length, but also increases the resistance from the buckling along the width. This creates the preference of the plate to buckle in such a way to equal the number of curvatures both along the width and length. Due to boundary conditions, when a plate is loaded with a critical stress and buckles, the edges perpendicular to the load cannot deform out-of-plane and will therefore continue to carry the stresses. This creates a non-uniform compressive loading along the ends, where the stresses are imposed on half of the effective width on either side of the specimen, given by the following:
where
As the loaded stress increase, the effective width continues to shrink; if the stresses on the ends ever reaches the yield stress, the plate will fail. This is what allows the buckled structure to continue supporting loadings. When the axial load over the critical load is plotted against the displacement, the fundamental path is shown. It demonstrates the plate's similarity to a column under buckling; however, past the buckling load, the fundamental path bifurcates into a secondary path that curves upward, providing the ability to be subjected to higher loads past the critical load.
Flexural-torsional buckling.
Flexural-torsional buckling can be described as a combination of bending and twisting response of a member in compression. Such a deflection mode must be considered for design purposes. This mostly occurs in columns with "open" cross-sections and hence have a low torsional stiffness, such as channels, structural tees, double-angle shapes, and equal-leg single angles. Circular cross sections do not experience such a mode of buckling.
Lateral-torsional buckling.
When a simply supported beam is loaded in bending, the top side is in compression, and the bottom side is in tension. 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 experience a lateral deflection of the compression flange as it buckles locally. The lateral deflection of the compression flange is restrained by the beam web and tension flange, but for an open section the twisting mode is more flexible, hence the beam both twists and deflects laterally in a failure mode known as "lateral-torsional buckling". In wide-flange sections (with high lateral bending stiffness), the deflection mode will be mostly twisting in torsion. In narrow-flange sections, the bending stiffness is lower and the column's deflection will be closer to that of lateral bucking deflection mode.
The use of closed sections such as square hollow section will mitigate the effects of lateral-torsional buckling by virtue of their high torsional stiffness. 
"C""b" is a modification factor used in the equation for nominal flexural strength when determining lateral-torsional buckling. The reason for this factor is to allow for non-uniform moment diagrams when the ends of a beam segment are braced. The conservative value for "C""b" can be taken as 1, regardless of beam configuration or loading, but in some cases it may be excessively conservative. "C""b" 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. Tables of values of "C""b" for simply supported beams exist.
If an appropriate value of "C""b" is not given in tables, it can be obtained via the following formula:
where
The result is the same for all unit systems.
Plastic buckling.
The buckling strength of a member is less than the elastic buckling strength of a structure if the material of the member is stressed beyond the elastic material range and into the non-linear (plastic) material behavior range. When the compression load is near the buckling load, the structure will bend significantly and the material of the column will diverge from a linear stress-strain behavior. The stress-strain behavior of materials is not strictly linear even below the yield point, hence the modulus of elasticity decreases as stress increases, and significantly so as the stresses approach the material's yield strength. This reduced material rigidity reduces the buckling strength of the structure and results in a buckling load less than that predicted by the assumption of linear elastic behavior.
A more accurate approximation of the buckling load can be had by the use of the tangent modulus of elasticity, Et, which is less than the elastic modulus, in place of the elastic modulus of elasticity. The tangent is equal to the elastic modulus and then decreases beyond the proportional limit. The tangent modulus is a line drawn tangent to the stress-strain curve at a particular value of strain (in the elastic section of the stress-strain curve, the tangent modulus is equal to the elastic modulus). Plots of the tangent modulus of elasticity for a variety of materials are available in standard references.
Crippling.
Sections that are made up of flanged plates such as a channel, can still carry load in the corners after the flanges have locally buckled. Crippling is failure of the complete section.
Diagonal tension.
Because of the thin skins typically used in aerospace applications, skins may buckle at low load levels. However, once buckled, instead of being able to transmit shear forces, they are still able to carry load through "diagonal tension" (DT) stresses in the web. This results in a non-linear behaviour in the load carrying behaviour of these details. The ratio of the actual load to the load at which buckling occurs is known as the "buckling ratio" of a sheet. High buckling ratios may lead to excessive wrinkling of the sheets which may then fail through yielding of the wrinkles. Although they may buckle, thin sheets are designed to not permanently deform and return to an unbuckled state when the applied loading is removed. Repeated buckling may lead to fatigue failures.
Sheets under diagonal tension are supported by stiffeners that as a result of sheet buckling carry a distributed load along their length, and may in turn result in these structural members failing under buckling. 
Thicker plates may only partially form a diagonal tension field and may continue to carry some of the load through shear. This is known as "incomplete diagonal tension" (IDT). This behavior was studied by Wagner and these beams are sometimes known as Wagner beams. 
Diagonal tension may also result in a pulling force on any fasteners such as rivets that are used to fasten the web to the supporting members. Fasteners and sheets must be designed to resist being pulled off their supports.
Dynamic buckling.
If a column is loaded suddenly and then the load released, the column can sustain a much higher load than its static (slowly applied) buckling load. This can happen in a long, unsupported column used as a drop hammer. The duration of compression at the impact end is the time required for a stress wave to travel along the column 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, and at a stress many times the buckling stress of a statically-loaded 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 formula_60 is the impact stress, formula_5 is the length of the rod, formula_62 is the elastic wave speed, and formula_63 is the smaller lateral dimension of a rectangular rod. Because the buckle wavelength depends only on formula_60 and formula_63, this same formula holds for thin cylindrical shells of thickness formula_63.
Theory.
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 determining the constant K. Therefore, maximum buckling load is often approximated using energy conservation and referred to as an energy method in structural analysis.
The first step in this method is to assume a displacement mode and a function that represents that displacement. 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.
The method assumes that the system (the column) is a conservative system in which energy is not dissipated as heat, hence the energy added to the column by the applied external forces is stored in the column in the form of strain energy.
In this method, there are two equations used (for small deformations) to approximate the "strain" energy (the potential energy stored as elastic deformation of the structure) and "applied" energy (the work done on the system by external forces).
where formula_69 is the displacement function and the subscripts formula_70 and formula_71 refer to the first and second derivatives of the displacement. 
Single-degree-of-freedom models.
Using the concept of "total potential energy", formula_72, it is possible to identify four fundamental forms of buckling found in structural models with one degree of freedom. We start by expressing
where formula_74 is the strain energy stored in the structure, formula_75 is the applied "conservative" load and formula_76 is the distance moved by formula_75 in its direction. Using the axioms of elastic instability theory, namely that equilibrium is any point where formula_72 is stationary with respect to the coordinate measuring the degree(s) of freedom and that these points are only stable if formula_72 is a local minimum and unstable if otherwise (e.g. maximum or a point of inflection).
These four forms elastic buckling are the "saddle-node bifurcation" or "limit point"; the "supercritical" or "stable-symmetric" bifurcation; the "subcritical" or "unstable-symmetric" bifurcation; and the "transcritical" or "asymmetric" bifurcation. All but the first of these examples is a form of "pitchfork bifurcation". Simple models for each of these types of buckling behaviour are shown in the figures below, along with the associated bifurcation diagrams.
Engineering examples.
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. It can be considered as a loaded column that has been bent into a circle. If spoke tension is increased beyond a safe level or if part of the rim is subject to a certain lateral force, the wheel spontaneously fails into a characteristic saddle shape (sometimes called a "taco" or a "pringle") like a three-dimensional Euler column. If this is a purely elastic deformation the rim will resume its proper plane shape if spoke tension is reduced or a lateral force from the opposite direction is applied.
Roads.
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. Going over a buckled section can be very jarring to automobile drivers, described as running over a speed hump at highway speeds.
Rail tracks.
Similarly, rail tracks also expand when heated, and can fail by buckling, a phenomenon called "sun kink". It is more common for rails to move laterally, often pulling the underlying ties (sleepers) along.
These accidents were deemed to be sun kink-related ("more information available at List of rail accidents (2000–2009)"):
Pipes and pressure vessels.
Pipes and pressure vessels subject to external overpressure, caused for example by steam cooling within the pipe and condensing into water with subsequent massive pressure drop, risk buckling due to compressive hoop stresses. Design rules for calculation of the required wall thickness or reinforcement rings are given in various piping and pressure vessel codes.
Super- and hypersonic aerospace vehicles.
Aerothermal heating can lead to buckling of surface panels on super- and hypersonic aerospace vehicles such as high-speed aircraft, rockets and reentry vehicles. If buckling is caused by aerothermal loads the situation can be further complicated by enhanced heat transfer in areas where the structure deforms towards the flow-field.
External links.
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