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Encyclopedia > Buckling
Mechanical failure modes
Buckling
Corrosion
Creep
Fatigue
Fracture
Melting
Thermal shock
Wear
Yielding
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In engineering, buckling is a failure mode characterised by a sudden failure of a structural member subjected to high compressive stresses, where the actual compressive stresses at failure are smaller than the ultimate compressive stresses that the material is capable of withstanding. This mode of failure is also described 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. Buckling can refer to: the buckling of stressed materials in engineering, or Buckling (fish), a form of smoked herring Category: ... For the hazard, see corrosive. ... Creep is the term used to describe the tendency of a material to move or to deform permanently to relieve stresses. ... In materials science, fatigue is the progressive, localised, and permanent structural damage that occurs when a material is subjected to cyclic or fluctuating strains at nominal stresses that have maximum values less than (often much less than) the static yield strength of the material. ... For fractures in geologic formations, see Rock fracture. ... In physics, melting is the process of heating a solid substance to a point (called the melting point) where it turns into a liquid. ... Thermal shock and thermal loading refer to the disfuntion (and perhaps, crack) of a material due to the heating, especially non-stationary and non-uniform. ... In materials science, wear is the erosion of material from a solid surface by the action of another solid. ... Yield strength, or the yield point, is defined in engineering and materials science as the stress at which a material begins to plastically deform. ... Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ... Structural failure refers to loss of the load-carrying capacity of a component or member within a structure or of the structure itself. ... This article is about engineering. ...

A column under a centric axial load exhibiting the characteristic deformation of buckling.

The ratio of the 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. All the following are approximate values used for convenience. Image File history File links A column exhibiting the characteristic deformation of buckling under a centric axial load. ... For other uses, see Column (disambiguation). ... The radius of gyration is a defined measure for the dimension of an object, a surface, or an ensemble of points. ...

• A short steel column is one whose slenderness ratio does not exceed 50; an intermediate length steel column has a slenderness ratio ranging from 50 to 200, 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 not greater than 10. If the ratio is greater than 10, it is a long column.
• 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 dividing line between intermediate and long timber columns cannot be readily evaluated. One way of defining the lower limit of long timber columns would be to set it as the smallest value of the ratio of length to least cross sectional area that would just exceed a certain constant K of the material. Since K depends on the modulus of elasticity and the allowable compressive stress parallel to the grain, it can be seen that this arbitrary limit would vary with the species of the timber. The value of K is given in most structural handbooks.

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 the column to be in a state of unstable equilibrium; that is, any increase in the load, or the introduction of the slightest lateral force, will cause the column to fail by buckling. The Euler formula for columns is Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ...

$F=frac{pi^2 EI}{(Kl)^2}$

where

F = maximum or critical force (vertical load on column),
E = modulus of elasticity,
I = area moment of inertia,
l = unsupported length of column,
K = column effective length factor, whose value depends upon the conditions of end support of the column, as follows.
For both ends pinned (hinged, free to rotate), K = 1.0.
For both ends fixed, K = 0.50.
For one end fixed and the other end pinned, K = 0.70.
For one end fixed and the other end free to move laterally, K = 2.0.

Examination of this formula reveals the following interesting facts with regard to the load-bearing ability of columns. In physics, force is anything that can cause a massive body to accelerate. ... In solid mechanics, Youngs modulus (also known as the modulus of elasticity or elastic modulus) is a measure of the Stiffness of a given material. ... The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ...

1. Elasticity and not compressive strength of the materials of the column determines the critical load.
2. The critical load is directly proportional to the second moment of area of the cross section.
3. The boundary conditions have a considerable effect on the critical load of slender columns. The boundary conditions determine the mode of bending and the distance between inflection points on the deflected column. The closer together the inflection points are, the higher the resulting capacity of the column.
A demonstration model illustrating the different "Euler" buckling modes. The model shows how the boundary conditions affect the critical load of a slender column. Notice that each of the columns are identical, apart from the boundary conditions.

The strength of a column may therefore be increased by distributing the material so as to increase the moment of inertia. This can be done without increasing the weight of the column by distributing the material as far from the principal axes 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 solid section for column service. Look up elastic in Wiktionary, the free dictionary. ... Strength of materials is materials science applied to the study of engineering materials and their mechanical behavior in general (such as stress, deformation, strain and stress-strain relations). ... In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ... The second moment of area, also known as the area moment of inertia and less precisely as the moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ... Image File history File links Download high-resolution version (1632x1224, 289 KB) Model produced as part of a University project to demonstrate the different Euler buckling modes and the effect of boundary conditions on the critical buckling load. ... Image File history File links Download high-resolution version (1632x1224, 289 KB) Model produced as part of a University project to demonstrate the different Euler buckling modes and the effect of boundary conditions on the critical buckling load. ...

Another bit of information that may be gleaned from this equation is the effect of length upon critical load. For a given size column, doubling the unsupported length quarters the allowable load. The restraint offered by the end connections of a column also affects the critical load. If the connections are perfectly rigid, the critical load will be four times that for a similar column where there is no resistance to rotation (hinged at the ends).

Since the moment of inertia of a surface is its area multiplied by the square of a length called the radius of gyration, the above formula may be rearranged as follows. Using the Euler formula for hinged ends, and substituting A·r2 for I, the following formula results.

$sigma = frac{F}{A} = frac{pi^2 E}{(l/r)^2}$

where F / A is the allowable stress of the column, and l / r is the slenderness ratio.

Since the structural column is generally an intermediate-length column and it is impossible to obtain an ideal column, the Euler formula has little practical application for ordinary design. Consequently, a number of empirical column formulae have been developed to agree with test data, all of which embody the slenderness ratio. For design, appropriate safety factors are introduced into these formulae. Factor of safety (FoS) can mean either the fraction of structural capability over that required, or a multiplier applied to the maximum expected load (force, torque, bending moment or a combination) to which a component or assembly will be subjected. ...

## Self-buckling of columns

A freestanding, vertical column of circular cross section, with density ρ, Young's modulus E, and radius r, will buckle under its own weight if its height exceeds a certain critical height:

$h_{crit} = left(frac{2.5Er^2}{rho g}right)^{1/3}$

## Buckling of surface materials

Similarly, railroad tracks also expand when heated, and can fail by buckling. It is more common for rails to move laterally, often pulling the underlain railroad ties (sleepers) along with them. Railroad or railway tracks are used on railways, which, together with railroad switches (points), guide trains without the need for steering. ... Ferroconcrete sleepers A variant fastening of rails to wooden sleepers A railroad tie, cross tie, or sleeper is a rectangular object used as a base for railroad tracks. ...

## Energy method

In many situations, it is very difficult to determine the buckling load in complex structures using the Euler formula, due to the difficulty in deciding on the constant K value. Therefore, maximum buckling load often is approximated using energy conservation. This way of deciding maximum buckling load is often referred to as the energy method in structural analysis.

The first step in this method is to suggest a displacement function. This function must satisfy the most important boundary conditions, such as displacement and rotation. The more accurate the displacement function, the more accurate the result.

In this method, there are two equations used to calculate the inner energy and outer energy.

$A_{inner} = 0.5int EI(w_{xx}(x))^2dx$
$A_{outer} = 0.5P_{crit}int (w_{x}(x))^2dx$

where w(x) is the displacement function. Energy conservation yields:

Ainner = Aouter

## Lateral-torsional buckling

A demonstration model illustrating the effects of lateral-torsional buckling on an I-section beam.

When a beam is loaded in flexure, the compression side is in compression, and the tension 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 fail due to lateral buckling of the compression flange. In wide-flange sections, if the compression flange buckles laterally, the cross section will also twist in torsion, resulting in a failure mode known as lateral-torsional buckling. Image File history File links Ltb. ... Image File history File links Ltb. ... This article or section does not adequately cite its references or sources. ... Physical compression is the result of the subjection of a material to compressive stress, resulting in reduction of volume. ... Tension is a reaction force applied by a stretched string (rope or a similar object) on the objects which stretch it. ...

## Plastic buckling

Buckling will generally occur slightly before the theoretical buckling strength of a structure, due to plasticity of the material. When the compressive load is near buckling, the structure will bow significantly and approach yield. The stress-strain behaviour of materials is not strictly linear even below yield, and the modulus of elasticity decreases as stress increases, with more rapid change near yield. This lower rigidity reduces the buckling strength of the structure and causes premature buckling. This is the opposite effect of the plastic bending in beams, which causes late failure relative to the Euler-Bernoulli beam equation. The elementary Euler-Bernoulli beam theory is a simplification of the linear isotropic theory of elasticity which allows quick calculation of the load-carrying capacity and deflection of common structural elements called beams. ...

## Dynamic buckling

If the load on the column is applied suddenly and then released, the column can sustain a load much higher than its static (slowly applied) buckling load. This can happen in a long, unsupported column (rod) used as a drop hammer. The duration of compression at the impact end is the time required for a stress wave to travel up the rod to the other (free) end and back down as a relief wave. Maximum buckling occurs near the impact end at a wavelength much shorter than the length of the rod, at a stress many times the buckling stress if the rod were a 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

σL = ρc2h

where σ is the impact stress, L is the length of the rod, c is the elastic wave speed, and h is the smaller lateral dimension of a rectangular rod. Because the buckle wavelength depends only on σ and h, this same formula holds for thin cylindrical shells of thickness h.

Source: Lindberg, H. E., and Florence, A. L., Dynamic Pulse Buckling, Martinus Nijhoff Publishers, 1987, pp. 11-56, 297-298.

Compressive stress is the stress applied to materials resulting in their compaction (decrease of volume). ... The elementary Euler-Bernoulli beam theory is a simplification of the linear isotropic theory of elasticity which allows quick calculation of the load-carrying capacity and deflection of common structural elements called beams. ...

## References

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

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