Buckling

Resistance of materials

In the field of the Resistance of the materials, the buckling is the tendency which has a beam requested in longitudinal compression to bend, and thus to become deformed in a direction perpendicular to the force applied.

Take for example a flexible plastic rule. If you hold this rule by the ends and draw, the force which you exert will have to produce in the rule of the constraints equal to the mechanical resistance to the traction of the plastic which composes it before it breaks. On the other hand, if you hold this rule between the palms of your hands and push, this one, instead of taking the request nicely in a longitudinal way, will end up folding and will break much more easily than that requested in traction. It is this phenomenon which one names buckling.

Buckling occurs all the more easily as the beam is long and of weak section. The tendency to buckling also depends on the type of fastener of the beam. Even if the term beam is employed here, certain assumptions of the theory of the beams (assumption of small displacements) must be abandoned so that the model provides a credible result. We must accept a Théorie of the second order because displacements are not infinitesimal.

Thus, the critical load from which there is risk of rupture by buckling can be calculated by the formula of Euler :

$F= \ frac {\ pi^2e I} {l_k^2}$

where

$E$ is the modulus Young material;
$I$ is the quadratic Moment of the beam;
$l_k$ is the length of buckling of the beam;

The factor $l_k$ represents a length equivalent to that of a rotulée-rotulée beam. It is about the distance separating two points from inflections from the beam. Thus,

for a beam rotulée with the two ends, $l_k = 1 \ times L$ , the length of the beam;
for a beam fixed with the two ends, $l_k = 0,5 \ times L$ ;
for an embedded-rotulée beam, $l_k = 0,7 \ times L$ ;
for a embed-free beam, $l_k = 2 \ times L$ .

Calculation in practice

This problem is seriously considered in the cases of the dimensioning of Pilier S in Civil engineering and of rods in mechanics, elements necessarily big length and subjected to compression.

In practice however, it is not the formula of Euler which is used to calculate the dimensioning of a beam. One defines usually a geometrical parameter, λ , called slenderness ratio:

$\ lambda = \ frac {l_k} {\ rho}, \ quad \ text {with} \ quad \ rho^2 = \ frac {I} {S}$

where ρ is the radius of gyration of the beam and S the section of this beam.

One can then define a critical slenderness ratio, $\ lambda_c$ , which depends only on the properties of materials:

$\ lambda_c^2 = \ frac {\ pi^2e} {\ sigma_e}$

where $\ sigma_e$ is the Limite rubber band material;

One can then determine the critical load $F_c$ applicable on a beam by comparing his value of twinge $\ lambda$ with the value of $\ lambda_c$ .

If $\ scriptstyle \ lambda \ Leq 20$ , the beam is in simple compression: $F_c = \ sigma_e * S \ quad$

If $\ scriptstyle 20 < \ lambda \ Leq \ lambda_c$ , one then uses the experimental formula of Rankine: $F_c = \ frac {2* \ sigma_e * S} {1 + \ left (\ frac {\ lambda} {\ lambda_c} \ right) ^2}$

If $\ scriptstyle \ lambda > \ lambda_c$ , one then uses the formula of Euler, which can be rewritten in the form: $F_c = \ frac {\ sigma_e * S} {\ left (\ frac {\ lambda} {\ lambda_c} \ right) ^2}$

Geology

In Geology, one finds this same phenomenon with small scales. The compression of an important continental mass causes with large scales (local and/or regional) the formation of an assembly line. But on a whole continent scale (small scales), it occurs an elastic strain which creates a series of " creux" and of " bosses" secondaries.

For example, the alpine collision, in Western Europe, is responsible for the formation of other secondary reliefs whose importance decreases as one moves away from the Alps: Massif Central (" bosse")and Limagne (" creux"), the the Sologne (hollow), Armorican Solid mass and the Alps Mancelles (Bump, although also related to the opening of the Atlantic), the the North Sea (hollow, whose other explanatory factor are also the opening of the Atlantic), the Pays of Caux and the Pays of Bray (bumps), marshes and polders of the area of Calais and the North of the Picardy (hollow), the Boulonnais, the Artois the the Ardennes and the Eiffel (Bumps), the Flanders (hollow).

One could describe the Himalayan by-effect in the same way: plates of Tibet (hollow), Altaï (bump), lake Baïkal (hollow).

See too

Resistance of the materials

External bond

For more complete calculations of buckling…

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