Méthode des moments d'aires

Cet article est une ébauche concernant le génie mécanique.

Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants.

La méthode des « moments d'aires » concerne la déformation des poutres en flexion, et permet de calculer la pente et la flèche d'une poutre.

La méthode des moments d'aires est une méthode par intégration géométrique permettant de calculer la déformée d'une poutre en la reliant à un diagramme M/EI.

La méthode des moments d'aire

Considérons une poutre rectiligne. Après déformation, sa courbe moyenne a pour équation

y=u_{y}(x)

u_y(x) étant la flèche à l'abscisse x considérée. La pente φ est la dérivée de cette courbe :

\varphi ={\frac {\mathrm {d} u_{y}}{\mathrm {d} x}}

Le rayon de courbure ρ vérifie

{\frac {1}{\rho }}\simeq {\frac {\mathrm {d} ^{2}u_{y}}{\mathrm {d} x^{2}}}

La théorie des poutres nous donne

{\frac {1}{\rho }}={\frac {\mathrm {d} \varphi }{\mathrm {d} x}}={\frac {\mathrm {M} _{\mathrm {f} z}}{\mathrm {E} \mathrm {I} _{\mathrm {G} z}}}

où

M_fz est le moment fléchissant ;
E est le module de Young du matériau ;
I_Gz est le moment quadratique de la section droite par rapport à l'axe Gz.

On a donc :

d\varphi ={\frac {\mathrm {M} _{\mathrm {f} z}}{\mathrm {E} \mathrm {I} _{\mathrm {G} z}}}\mathrm {d} x

La variation de pente entre deux points A et B de la poutre s'écrit :

\varphi _{\mathrm {AB} }=\int _{\mathrm {A} }^{\mathrm {B} }\mathrm {d} \varphi =\int _{x_{\mathrm {A} }}^{x_{\mathrm {B} }}{\frac {\mathrm {M} _{\mathrm {f} z}}{\mathrm {E} \mathrm {I} _{\mathrm {G} z}}}\mathrm {d} x

Cela représente l'aire, comprise entre x_A et x_B, sous la courbe M_fz/EI_Gz :

\varphi _{\mathrm {AB} }=\left\lbrack {\text{aire sous}}\ {\frac {\mathrm {M} _{\mathrm {f} z}}{\mathrm {E} \mathrm {I} _{\mathrm {G} z}}}\right\rbrack _{\mathrm {A} }^{\mathrm {B} }

Les théorèmes de la méthode des moments d'aire

La méthode des moments d'aires est basée sur deux théorèmes dits théorèmes des moments d'aires.

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Voir aussi

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