Name: critical buckling load in pin-ended columns
Uploaded: 2024-05-23T13:00:07.000Z
Duration: 1 min 21 s
Channel: JoVE

Considérons une colonne PQ, reliée par des broches aux deux extrémités, avec une charge axiale centrée appliquée à l’extrémité P. Le flambement se produit si cette charge dépasse la charge critique.
Pour calculer la charge critique, imaginez le poteau comme une poutre verticale. Considérons maintenant un point O sur une courbe élastique de la poutre à une distance x de l’extrémité libre P et ayant la déviation y par rapport à la verticale.
Le moment de flexion au point O peut s’écrire comme la dérivée seconde de sa déviation par rapport à la distance x. En réarrangeant les termes, on obtient une équation différentielle du second ordre, exprimant une solution en fonction des fonctions sinus et cosinus.
La première condition aux limites exige que le coefficient B soit nul. La deuxième condition exige que le coefficient A ou le terme sinus soit nul.
Si l'on rend le terme sinus nul, on obtient l'expression de la charge axiale, la valeur la plus basse étant la charge critique ; c'est la formule d'Euler.
En substituant la formule d'Euler dans l'équation différentielle, on obtient une équation pour la courbe élastique après flambement.

formule d'euler pour les colonnes à broches

Euler&#39;s Formula for Pin-Ended Columns

En ingénierie des structures, la stabilité des poteaux soumis à des charges axiales de compression est une considération critique, appelée flambage. Un exemple typique implique une colonne PQ, qui est reliée par des broches aux deux extrémités et soumise à une charge axiale centrée F appliquée à une extrémité, avec une force de réaction de F' = -F à l'autre extrémité. Ici, il est crucial de comprendre que lorsqu’une charge appliquée dépasse la charge critique, un flambage se produit lorsque le système devient instable.

Pour calculer la charge critique, considérez la colonne PQ comme une poutre verticale. Considérons le point O, situé sur la courbe élastique de la poutre, à une distance x de l'extrémité libre P. Avec l'application de la charge, le point O est dévié d'une distance y par rapport à sa position verticale d'origine. À ce stade, le moment de flexion au point O peut être décrit par la dérivée seconde de sa déflexion, y, par rapport à x, symbolisant un pivot vers la compréhension du comportement de la poutre sous contrainte.

<img alt="Equation 1" src="/files/ftp_upload/15676/15676_Equation_1v2.svg" style="text-align: center; height: 40px;">

Où f est défini comme,

<img alt="Equation 2" src="/files/ftp_upload/15676/15676_Equation_2.svg" style="text-align: center; height: 30px;">

Cette équation a une solution générale ayant des termes sinus et cosinus. Les valeurs limites du système donnent les coefficients de la solution.

<img alt="Equation 3" src="/files/ftp_upload/15676/15676_Equation_3.svg" style="text-align: center; height: 24px;">

La solution nécessite que le terme sinus soit nul, donnant l’expression de la charge critique. Cette expression est connue sous le nom de formule d’Euler.

<img alt="Equation 4" src="/files/ftp_upload/15676/15676_Equation_4v2.svg" style="text-align: center; height: 36px;">

La substitution de la formule d'Euler dans l'équation différentielle donne l'expression de la courbe élastique du poteau après flambage.

Ici, il est important de noter que la formule d’Euler est dérivée de l’hypothèse selon laquelle avant chargement, la colonne est parfaitement droite, homogène et isotrope et que la charge axiale est parfaitement appliquée le long de l’axe vertical.

Formule d'Euler pour les colonnes à broches

Considérons une colonne PQ, reliée par des broches aux deux extrémités, avec une charge axiale centrée appliquée à l’extrémité P. Le flambement se produit si cette charge dépasse la charge critique.
À...

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critical buckling load in pin-ended columns

Critical buckling load in pin-ended columns is a key idea in structural engineering. It explains when a column under compressive axial load becomes unstable and starts to buckle. A pin-connected column with a centric load F at one end has a reaction force of F' = -F at the other end. When the applied load rises above the critical load, buckling begins.

To find the critical load, the column PQ can be treated like a vertical beam. Point O is chosen on the elastic curve of the beam at a distance x from the free end P. When the load is applied, point O deflects by a distance y from its original vertical position. The bending moment at point O is then related to the second derivative of y with respect to x, which helps describe how the beam changes shape under stress.

The resulting equation has a general solution with sine and cosine terms. The boundary values of the system determine the coefficients in that solution. The sine term must be zero, and that condition gives the expression for the critical load. This expression is known as Euler's formula.

After Euler's formula is substituted back into the differential equation, the elastic curve of the column after buckling can be found. The formula is based on several assumptions. Before loading, the column must be perfectly straight, homogeneous, and isotropic. The axial load must also be applied perfectly along the vertical axis.

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key terms and concepts

Learn how pin-ended columns buckle under compressive axial load and how Euler's formula gives the critical load and elastic curve after buckling.

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Critical Buckling Load in Pin-Ended Columns

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