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

Considere una columna PQ, conectada por clavijas en ambos extremos, con una carga axial céntrica aplicada en el extremo P. Se produce pandeo si esta carga supera la carga crítica.
Para calcular la carga crítica, imagine la columna como una viga vertical. Ahora, considere un punto O en una curva elástica de la viga a una distancia x del extremo libre P y que tiene la deflexión y de la vertical.
El momento flector en el punto O se puede escribir como la segunda derivada de su deflexión con respecto a la distancia x. Reorganizando los términos, se obtiene una ecuación diferencial de segundo orden, que expresa una solución en términos de funciones seno y coseno.
La primera condición de contorno requiere que el coeficiente B sea cero. La segunda condición requiere que el coeficiente A o el término seno sean cero.
Al hacer el término seno cero, se obtiene la expresión de la carga axial, siendo el valor más bajo la carga crítica; esta es la fórmula de Euler.
Al sustituir la fórmula de Euler en la ecuación diferencial, se obtiene una ecuación para la curva elástica después del pandeo.

fórmula de euler para columnas con extremos pin

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

<div>
 En ingeniería estructural, la estabilidad de las columnas bajo cargas axiales de compresión es una consideración crítica, descrita como pandeo. Un ejemplo típico involucra una columna PQ, que está conectada por pasadores en ambos extremos y sometida a una carga axial céntrica F aplicada en un extremo, con una fuerza de reacción de F’ = -F en el otro extremo. Aquí, es crucial comprender que cuando una carga aplicada excede la carga crítica, se produce pandeo a medida que el sistema se vuelve inestable.
 
 Para calcular la carga crítica, imagine la columna PQ como una viga vertical. Considere el punto O, situado sobre la curva elástica de la viga, a una distancia x del extremo libre P. Con la aplicación de la carga, el punto O se desvía una distancia y de su posición vertical original. En este punto, el momento flector en el punto O puede describirse mediante la segunda derivada de su deflexión, y, con respecto a x, lo que simboliza un pivote hacia la comprensión del comportamiento de la viga bajo tensión.
 
 <img alt="Equation 1" src="/files/ftp_upload/15676/15676_Equation_1v2.svg" style="text-align: center; height: 40px;">
 Donde f se define como,
 
 <img alt="Equation 2" src="/files/ftp_upload/15676/15676_Equation_2.svg" style="text-align: center; height: 30px;">
 Esta ecuación tiene una solución general que tiene términos seno y coseno. Los valores límite del sistema dan los coeficientes de la solución.
 
 <img alt="Equation 3" src="/files/ftp_upload/15676/15676_Equation_3.svg" style="text-align: center; height: 24px;">
 La solución requiere que el término seno sea cero, lo que da la expresión de carga crítica. Esta expresión se conoce como fórmula de Euler.
 
 <img alt="Equation 4" src="/files/ftp_upload/15676/15676_Equation_4v2.svg" style="text-align: center; height: 36px;">
 Al sustituir la fórmula de Euler en la ecuación diferencial se obtiene la expresión de la curva elástica de la columna después del pandeo.
 
 Aquí es importante señalar que la fórmula de Euler se deriva del supuesto de que antes de la carga, la columna debe ser perfectamente recta, homogénea e isotrópica y que la carga axial se aplica perfectamente a lo largo del eje vertical.
 
</div>

Fórmula de Euler para columnas con extremos pin

Considere una columna PQ, conectada por clavijas en ambos extremos, con una carga axial céntrica aplicada en el extremo P. Se produce pandeo si esta carga supera la carga crítica.
Para...

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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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