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| '''Flexural rigidity''' is defined as the [[force]] [[Couple (mechanics)|couple]] required to bend a non-[[Stiffness|rigid]] structure to a unit [[curvature]] or it can be defined as the resistance offered by a structure while undergoing bending
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| ==Flexural rigidity of a bar==
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| {{main|Euler-Bernoulli beam equation}}
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| In a [[beam (structure)|beam]] or [[cylinder (geometry)|rod]], flexural rigidity (defined as EI) varies along the length as a function of x shown in the following equation:
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| :<math>\ EI {dy \over dx}\ = \int_{0}^{x} M(x) dx + C_1</math>
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| where <math>E</math> is the [[Young's modulus]] (in Pa), <math>I</math> is the [[second moment of area]] (in m<sup>4</sup>), <math>y</math> is the transverse displacement of the beam at '''x''', and <math>M(x)</math> is the [[moment (physics)|bending moment]] at ''x''.
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| Flexural rigidity has SI units of [[pascal (unit)|Pa]]·m<sup>4</sup> (which also equals [[newton (unit)|N]]·m²).
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| ==Flexural rigidity of a plate (e.g. the lithosphere)==
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| {{main|Plate theory}}
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| The thin lithospheric plates which cover the surface of the Earth are also subject to [[lithospheric flexure|flexure]], when a load or force is applied to them. On a geological timescale, the lithosphere behaves elastically (in first approach) and can therefore bend under loading by mountain chains, volcanoes and so on.
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| The flexure of the plate depends on:
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| # The plate thickness (usually referred to as [[mechanical thickness of the lithosphere]]).
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| # The elastic properties of the plate
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| # The applied load or force
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| As flexural rigidity of the plate is determined by the Young's modulus, [[Poisson's ratio]] and cube of the plate's elastic thickness, it is a governing factor in both (1) and (2).
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| Flexural Rigidity<ref>{{cite book
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| |author=[[Lev Landau|L.D. Landau]], [[Evgeny Lifshitz|E.M. Lifshitz]]
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| |year=1986
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| |title=Theory of Elasticity
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| |edition=3rd |volume=Vol. 7
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| |publisher=[[Butterworth-Heinemann]]
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| |page=42
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| |isbn=978-0-7506-2633-0
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| }}</ref>
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| <math>D = \dfrac{Eh_e^3}{12(1-\nu^2)}</math> | |
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| <math>E</math> = Young's Modulus
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| <math>h_e</math> = elastic thickness (~10–15 km)
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| <math>\nu</math> = Poisson's Ratio
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| Flexural rigidity of a plate has units of [[pascal (unit)|Pa]]·m<sup>3</sup>, i.e. one dimension of length less from the one for the rod, as it refers to the moment per unit length per unit of curvature, and not the total moment.
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| I is termed as moment of inertia.J is denoted as 2nd moment of inertia/polar moment of inertia.
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| ==See also==
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| *[[Lithospheric flexure]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Flexural Rigidity}}
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| [[Category:Solid mechanics]]
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| {{Classicalmechanics-stub}}
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| [[de:Biegesteifigkeit]]
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| [[es:rigidez flexural]]
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