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

Si consideri una colonna PQ, collegata a pin ad entrambe le estremità, con un carico assiale centrico applicato all'estremità P. L'instabilità si verifica se questo carico supera il carico critico.
Per calcolare il carico critico, immaginare la colonna come una trave verticale. Ora, consideriamo un punto O su una curva elastica della trave a una distanza x dall'estremità libera P e avente la deflessione y dalla verticale.
Il momento flettente nel punto O può essere scritto come la derivata seconda della sua deflessione rispetto alla distanza x. Riorganizzando i termini, si ottiene un'equazione differenziale del secondo ordine, che esprime una soluzione in termini di funzioni seno e coseno.
La prima condizione al contorno richiede che il coefficiente B sia zero. La seconda condizione richiede che il coefficiente A o il termine seno siano pari a zero.
Ponendo il termine seno zero si ottiene l'espressione del carico assiale, dove il valore più basso è il carico critico; questa è la formula di Eulero.
Sostituendo la formula di Eulero nell'equazione differenziale, si ottiene un'equazione per la curva elastica dopo instabilità.

formula di eulero per colonne pin-ended

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

Nell'ingegneria strutturale, la stabilità delle colonne sotto carichi assiali di compressione è una considerazione critica, descritta come instabilità. Un tipico esempio riguarda una colonna PQ, collegata tramite perni ad entrambe le estremità e soggetta ad un carico assiale centrico F applicato ad un'estremità, con una forza di reazione F’ = -F all'altra estremità. In questo caso è fondamentale comprendere che quando un carico applicato supera il carico critico, si verifica un'instabilità poiché il sistema diventa instabile.

Per calcolare il carico critico, immagina la colonna PQ come una trave verticale. Consideriamo il punto O, situato sulla curva elastica della trave, ad una distanza x dall'estremità libera P. Con l'applicazione del carico, il punto O viene deviato di una distanza y dalla sua posizione verticale originaria. A questo punto, il momento flettente nel punto O può essere descritto dalla derivata seconda della sua deflessione, y, rispetto a x, che simboleggia un perno verso la comprensione del comportamento della trave sotto sforzo.

<img alt="Equation 1" src="/files/ftp_upload/15676/15676_Equation_1v2.svg" style="text-align: center; height: 40px;">

Dove f è definito come:

<img alt="Equation 2" src="/files/ftp_upload/15676/15676_Equation_2.svg" style="text-align: center; height: 30px;">

Quest’equazione ha una soluzione generale con termini seno e coseno. I valori al contorno del sistema danno i coefficienti della soluzione.

<img alt="Equation 3" src="/files/ftp_upload/15676/15676_Equation_3.svg" style="text-align: center; height: 24px;">

La soluzione richiede che il termine seno sia zero, dando l'espressione del carico critico. Questa espressione è nota come formula di Eulero.

<img alt="Equation 4" src="/files/ftp_upload/15676/15676_Equation_4v2.svg" style="text-align: center; height: 36px;">

Sostituendo nuovamente la formula di Eulero nell'equazione differenziale si ottiene l'espressione della curva elastica della colonna dopo l'instabilità.

Qui è importante notare che la formula di Eulero deriva dal presupposto che prima del carico la colonna sia perfettamente diritta, omogenea e isotropa e che il carico assiale sia applicato perfettamente lungo l'asse verticale.

Formula di Eulero per colonne pin-ended

Si consideri una colonna PQ, collegata a pin ad entrambe le estremità, con un carico assiale centrico applicato all'estremità P. L'instabilità si verifica se questo carico supera il carico critico.
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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.

Critical Buckling Load in Pin-Ended Columns | JoVE K12

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Critical Buckling Load in Pin-Ended Columns

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