# Elastic instability

﻿
Elastic instability

Elastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads.

ingle degree of freedom-systems

Consider as a simple example a rigid beam of length "L", hinged in one end and free in the other, and having an angular spring attached to the hinged end. The beam is loaded in the free end by a force "F" acting in the compressive axial direction of the beam, see the figure to the right.

Moment equilibrium condition

Assuming a clockwise angular deflection $heta$, the clockwise moment exerted by the force becomes $M_F = F L sin heta$. The moment equilibrium equation is given by

$F L sin heta = k_ heta heta$

where $k_ heta$ is the spring constant of the angular spring (Nm/radian). Assuming $heta$ is small enough, implementing the taylor expansion of the sine function and keeping the two first terms yields

$F L Bigg\left( heta - frac\left\{1\right\}\left\{6\right\} heta^3Bigg\right) approx k_ heta heta$

which has three solutions, the trivial $heta = 0$, and

$heta approx pm sqrt\left\{6 Bigg\left( 1 - frac\left\{k_ heta\right\}\left\{F L\right\} Bigg\right)\right\}$

which is imaginary (i.e. not physical) for $F L < k_ heta$ and real otherwise. This implies that for small compressive forces, the only equilibrium state is given by $heta = 0$, while if the force exceeds the value $k_ heta/L$ there is suddenly another mode of deformation possible.

Energy method

The same result can be obtained by considering energy relations. The energy stored in the angular spring is

$E_mathrm\left\{spring\right\} = int k_ heta heta mathrm\left\{d\right\} heta = frac\left\{1\right\}\left\{2\right\} k_ heta heta^2$

and the work done by the force is simply the force multiplied by the distance, which is $L \left(1 - cos heta\right)$. Thus,

$E_mathrm\left\{force\right\} = int\left\{F mathrm\left\{d\right\} x = F L \left(1 - cos heta \right)\right\}$

The energy equilibrium condition $E_mathrm\left\{spring\right\} = E_mathrm\left\{force\right\}$ now yields $F = k_ heta / L$ as before (besides from the trivial $heta = 0$).

tability of the solutions

Any solution $heta$ is stable iff a small change in the deformation angle $Delta heta$ results in a reaction moment trying to restore the original angle of deformation. The net clockwise moment acting on the beam is

$M\left( heta\right) = F L sin heta - k_ heta heta$

An infinitesimal clockwise change of the deformation angle $heta$ results in a moment

$M\left( heta + Delta heta\right) = M + Delta M = F L \left(sin heta + Delta heta cos heta \right) - k_ heta \left( heta + Delta heta\right)$

which can be rewritten as

$Delta M = Delta heta \left(F L cos heta - k_ heta\right)$

since $F L sin heta = k_ heta heta$ due to the moment equilibrium condition. Now, a solution $heta$ is stable iff a clockwise change $Delta heta > 0$ results in a negative change of moment $Delta M < 0$ and vice versa. Thus, the condition for stability becomes

$frac\left\{Delta M\right\}\left\{Delta heta\right\} = frac\left\{mathrm\left\{d\right\} M\right\}\left\{mathrm\left\{d\right\} heta\right\} = FL cos heta - k_ heta < 0$

The solution $heta = 0$ is stable only for $FL < k_ heta$, which is expected. By expanding the cosine term in the equation, we obtain the approximate stability condition


heta| > sqrt{2Bigg( 1 - frac{k_ heta}{F L} Bigg)}

for $FL > k_ heta$, which the two other solutions satisfy. Hence, these solutions are stable.

Multiple degrees of freedom-systems

By attaching another rigid beam to the original system by means of an angular spring a two degrees of freedom-system is obtained. Assume for simplicity that the beam lengths and angular springs are equal. The equilibrium conditions become

$F L \left( sin heta_1 + sin heta_2 \right) = k_ heta heta_1$

$F L sin heta_2 = k_ heta \left( heta_2 - heta_1 \right)$

where $heta_1$ and $heta_2$ are the angles of the two beams. Linearizing by assuming these angles are small yields

The non-trivial solutions to the system is obtained by finding the roots of the determinant of the system matrix, i.e. for

Thus, for the two degrees of freedom-system there are two critical values for the applied force "F". These correspond to two different modes of deformation which can be computed from the nullspace of the system matrix. Dividing the equations by $heta_1$ yields

For the lower critical force the ratio is positive and the two beams deflect in the same direction while for the higher force they form a "banana" shape. These two states of deformation represent the buckling mode shapes of the system.

ee also

* Buckling

*"Theory of elastic stability", S. Timoshenko and J. Gere

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• elastic instability — A condition in which a compression member will fail to bend before the stated compressive strength of the material is reached …   Aviation dictionary

• elastic — adj 1 Elastic, resilient, springy, flexible, supple are comparable when they mean able to endure strain (as extension, compression, twisting, or bending) without being permanently affected or injured. Elastic and resilient are both general and… …   New Dictionary of Synonyms

• Molecular beam epitaxy — A simple sketch showing the main components and rough layout and concept of the main chamber in a Molecular Beam Epitaxy system Molecular beam epitaxy (MBE) is one of several methods of depositing single crystals. It was invented in the late… …   Wikipedia

• Molecular-beam epitaxy — (MBE), is one of several methods of depositing single crystals. It was invented in the late 1960s at Bell Telephone Laboratories by J. R. Arthur and Alfred Y. Cho.MethodMolecular beam epitaxy takes place in high vacuum or ultra high vacuum (10−8… …   Wikipedia

• Buckling — In engineering, buckling is a failure mode characterized by a sudden failure of a structural member subjected to high compressive stresses, where the actual compressive stress at the point of failure is less than the ultimate compressive stresses …   Wikipedia

• Stranski-Krastanov growth — (SK growth, also Stransky Krastanov or Stranski Krastanow) is one of the three primary modes by which thin films grow epitaxially at a crystal surface or interface. Also known as layer plus island growth , the SK mode follows a two step process:… …   Wikipedia

• solids, mechanics of — ▪ physics Introduction       science concerned with the stressing (stress), deformation (deformation and flow), and failure of solid materials and structures.       What, then, is a solid? Any material, fluid or solid, can support normal forces.… …   Universalium

• Elasticity of cell membranes — A cell membrane defines a boundary between the living cell and its environment. It consists of lipids, proteins,carbohydrates etc. Lipids and proteins are dominant components of membranes. One of the principal types of lipids in membranes is… …   Wikipedia

• Column — For other uses, see Column (disambiguation). National Capitol Columns at the United States National Arboretum in Washington, D.C …   Wikipedia

• Neutron — This article is about the subatomic particle. For other uses, see Neutron (disambiguation). Neutron The quark structure of the neutron. (The color assignment of individual quarks is not important, only that all three colors are present.)… …   Wikipedia