Name: beam deflection from central load
Uploaded: 2024-05-23T13:00:13.000Z
Duration: 1 min 14 s
Channel: JoVE

Considere una viga PQ simplemente apoyada de longitud L, que lleva una carga puntual F en el centro. ¿Cuál será la deflexión en el centro de la viga?
Aquí, las reacciones en ambos extremos del haz son iguales y cada uno es la mitad de la carga central.
Considere el segmento PC y elija cualquier punto a una distancia x del extremo P. En este punto, el momento debido a la reacción en el punto P es la carga en el punto P multiplicada por la distancia. Una diferenciación parcial de la ecuación de momento con respecto a la carga en el extremo P es la mitad de x. 
Se puede realizar un análisis similar para el segmento QC de la viga.
Usando el teorema de Castigliano, la deflexión en el punto C está determinada por las derivadas parciales de la energía de deformación debida a la carga aplicada.
Esta ecuación se puede simplificar considerando los dos segmentos de haz PC y QC.
Al realizar la integración en más de la mitad de la longitud de la viga, se obtiene una expresión de la deflexión en el centro de la viga.

teorema de castigliano: resolución de problemas

<div>
 La deflexión de una viga simplemente apoyada que soporta una carga puntual central se puede analizar utilizando principios de mecánica estructural, particularmente aplicando el teorema de Castigliano. Este teorema relaciona el desplazamiento en el punto de aplicación de la carga con las derivadas parciales de la energía de deformación en la estructura. La viga simplemente apoyada con una carga puntual en su centro tiene fuerzas de reacción simétricas en los apoyos, cada uno de los cuales soporta la mitad de la carga. El momento flector en cualquier punto a lo largo de la viga se deriva de estas reacciones, calculadas sobre la distancia desde el soporte más cercano.
 
 El teorema de Castigliano indica que la deflexión en el punto donde se aplica la carga se determina diferenciando la energía de deformación de la viga con respecto a la carga. La energía de deformación se calcula en función del momento flector a lo largo de la viga, integrado en su longitud. Para esta viga, la energía de deformación debida a la flexión se calcula a partir del cuadrado de la expresión del momento flector, integrada a lo largo de la mitad de la longitud de la viga.
 
 <img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="text-align: center; height: 106px;">
 Dado que la viga es simétrica, este valor se duplica para tener en cuenta toda la viga y se calcula la deflexión en el centro de la viga. Depende de la magnitud de la carga, el cubo de la longitud de la viga, e inversamente proporcional al producto del momento de inercia y el módulo elástico de la sección transversal de la viga.
 
 <img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="text-align: center; height: 42px;">
</div>

Teorema de Castigliano: resolución de problemas

Considere una viga PQ simplemente apoyada de longitud L, que lleva una carga puntual F en el centro. ¿Cuál será la deflexión en el centro de la viga?
Aquí, el ...

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Castigliano&#39;s Theorem: Problem Solving

beam deflection from central load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam.

For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support.

Castigliano's theorem says the displacement at the load point is found by taking the partial derivative of the strain energy with respect to the load. Strain energy is the energy stored by the beam as it bends. It is calculated from the bending moment along the beam and integrated over the beam length.

Because the beam is symmetric, the strain energy is first found over half the beam and then doubled for the full length. The center deflection depends on the load, the cube of the beam length, and the beam's moment of inertia and elastic modulus. It is inversely proportional to the product of those two material and cross-section properties.

Beam Deflection from Central Load

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castigliano&#39;s theorem: problem solving

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

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Learn how Castigliano's theorem is used to find the center deflection of a simply supported beam with a central point load using strain energy.

Beam Deflection from Central Load | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Beam Deflection from Central Load

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