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

Considere uma viga simplesmente apoiada PQ de comprimento L, carregando uma carga pontual F no centro. Qual será a deflexão no centro da viga?
Aqui, as reações em ambas as extremidades do feixe são iguais e cada uma é metade da carga central.
Considere o segmento PC e escolha qualquer ponto a uma distância x da extremidade P. Neste ponto, o momento devido à reação no ponto P é a carga no ponto P vezes a distância. Uma diferenciação parcial da equação do momento em relação à carga no final P é metade de x. 
Uma análise semelhante pode ser realizada para o CQ do segmento da viga.
Usando o teorema de Castigliano, a deflexão no ponto C é determinada pelas derivadas parciais da energia de deformação devido à carga aplicada.
Esta equação pode ser simplificada considerando os dois segmentos de feixe PC e QC.
Realizar a integração na metade do comprimento da viga dá uma expressão para a deflexão no centro da viga.

teorema de castigliano: resolução de problemas

A deflexão de uma viga simplesmente apoiada que suporta uma carga de ponto central pode ser analisada utilizando princípios da mecânica estrutural, em particular aplicando o teorema de Castigliano. Este teorema relaciona o deslocamento no ponto de aplicação da carga às derivadas parciais da energia de deformação na estrutura. A viga simplesmente apoiada com uma carga pontual em seu centro tem forças de reação simétricas nos apoios, cada um suportando metade da carga. O momento fletor em qualquer ponto ao longo da viga é derivado a partir dessas reações, sendo calculado em relação à distância do apoio mais próximo.

O teorema de Castigliano indica que a deflexão no ponto onde a carga é aplicada é determinada pela diferenciação da energia de deformação da viga em relação à carga. A energia de deformação é calculada com base no momento fletor ao longo da viga, integrado ao longo do seu comprimento. Para esta viga, a energia de deformação devido à flexão é calculada a partir do quadrado da expressão do momento fletor, integrado ao longo da metade do comprimento da viga.

<img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="height: 106px;">

Como a viga é simétrica, esse valor é duplicado para levar em conta toda a viga, e a deflexão no centro da viga é calculada. Ela é dependente da magnitude da carga, do cubo do comprimento da viga, e inversamente proporcional ao produto do momento de inércia e do módulo de elasticidade da seção transversal da viga.
<img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="height: 42px;">

Teorema de Castigliano: Resolução de Problemas

Considere uma viga simplesmente apoiada PQ de comprimento L, carregando uma carga pontual F no centro. Qual será a deflexão no centro da viga?
Aqui, o ...

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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

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mechanical engineering

Beam Deflection from Central Load

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