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

Considérons une poutre PQ simplement supportée de longueur L, portant une charge ponctuelle F au centre. Quelle sera la déviation au centre du faisceau ?
Ici, les réactions aux deux extrémités de la poutre sont égales et chacune représente la moitié de la charge centrale.
Considérons le segment PC et choisissez n’importe quel point situé à une distance x de l’extrémité P. En ce point, le moment dû à la réaction au point P est la charge au point P multipliée par la distance. Une dérivation partielle de l’équation du moment par rapport à la charge à l’extrémité P est la moitié de x. 
Une analyse similaire peut être effectuée pour le segment QC du faisceau.
En utilisant le théorème de Castigliano, la déflexion au point C est déterminée par les dérivées partielles de l'énergie de déformation due à la charge appliquée.
Cette équation peut être simplifiée en considérant les deux segments du faisceau PC et QC.
En effectuant l’intégration sur la moitié de la longueur de la poutre, on obtient une expression de la déviation au centre de la poutre.

théorème de castigliano : résolution de problèmes

La déflexion d'une poutre simplement appuyée qui supporte une charge ponctuelle centrale peut être analysée à l'aide des principes de la mécanique des structures, notamment en appliquant le théorème de Castigliano. Ce théorème relie le déplacement au point d'application de la charge aux dérivées partielles de l'énergie de déformation dans la structure. La poutre simplement supportée avec une charge ponctuelle en son centre a des forces de réaction symétriques au niveau des supports, chacun supportant la moitié de la charge. Le moment de flexion en tout point le long de la poutre est dérivé de ces réactions, calculé sur la distance du support le plus proche.

Le théorème de Castigliano indique que la déflexion au point où la charge est appliquée est déterminée en différenciant l'énergie de déformation de la poutre par rapport à la charge. L'énergie de déformation est calculée sur la base du moment de flexion le long de la poutre, intégré sur sa longueur. Pour cette poutre, l'énergie de déformation due à la flexion est calculée à partir du carré de l'expression du moment de flexion, intégré sur la moitié de la longueur de la poutre.

<img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="text-align: center; height: 106px;">

Puisque la poutre est symétrique, cette valeur est doublée pour tenir compte de la totalité de la poutre et la déflexion au centre de la poutre est calculée. Il dépend de l'ampleur de la charge, du cube de la longueur de la poutre et est inversement proportionnel au produit du moment d'inertie et du module élastique de la section transversale de la poutre.

<img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="text-align: center; height: 42px;">

Théorème de Castigliano : résolution de problèmes

Considérons une poutre PQ simplement supportée de longueur L, portant une charge ponctuelle F au centre. Quelle sera la déviation au centre du faisceau ?
Ici, le ...

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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...

key terms and concepts

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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