Name: using strain energy to find deflection
Uploaded: 2024-05-23T13:00:12.000Z
Duration: 1 min 18 s
Channel: JoVE

Le théorème de Castigliano est utile pour déterminer les déplacements et les rotations dans les structures subissant des déformations élastiques.
Il stipule que la dérivée partielle de l’énergie de déformation élastique concernant une charge donnée est équivalente au déplacement au point d’application de la charge particulière.
De même, supposons que la structure soit soumise à des couples. Dans ce cas, l’angle de rotation de la structure au point d’application d’un couple est calculé en dérivant partiellement l’énergie de déformation par rapport au couple appliqué.
Pour les structures subissant une torsion due à des couples appliqués lentement, l’angle de torsion d’une section de l’arbre peut être calculé en prenant la dérivée partielle de l’énergie de déformation par rapport au couple appliqué.
Si une poutre de section transversale non uniforme subit une déformation due à une charge, l’énergie de déformation est exprimée en termes d’intégrale. Ici, la déflexion au point d’application de la charge est calculée en différenciant l’énergie de déformation par rapport à la charge avant l’intégration.

théorème de castigliano

Le théorème de Castigliano analyse les déplacements et les rotations dans les structures élastiques. Il relie la dérivée de l'énergie de déformation élastique aux forces ou moments appliqués, permettant le calcul des déformations. Le théorème stipule que la dérivée partielle de l'énergie de déformation totale d'un système par rapport à une charge spécifique entraîne le déplacement au point où la charge est appliquée. Ce principe s'applique à la fois aux forces et aux moments.

<img alt="Equation 1" src="/files/ftp_upload/15688/15688_Equation_1.svg" style="text-align: center; height: 38px;">

L'énergie de déformation, qui mesure le travail effectué par les charges externes lors de la déformation élastique, est cruciale pour ces calculs. Dans des conditions telles que des poutres avec des sections transversales non-uniformes ou des chargements complexes, la déflexion en tout point peut être dérivée à l'aide du théorème de Castigliano en différenciant l'énergie de déformation par rapport à la charge.

<img alt="Equation 2" src="/files/ftp_upload/15688/15688_Equation_2.svg" style="text-align: center; height: 34px;">

De même, pour les éléments structurels tels que les arbres subissant une déformation en torsion due au couple appliqué, l'angle de torsion peut être déterminé en utilisant la même méthode. L'énergie de déformation due à la torsion tient compte des propriétés du matériau et des attributs géométriques de l'arbre.

<img alt="Equation 3" src="/files/ftp_upload/15688/15688_Equation_3.svg" style="text-align: center; height: 34px;">

Grâce au théorème de Castigliano, le comportement des structures sous diverses conditions de chargement est prédit, contribuant ainsi à la conception de systèmes plus efficaces et plus sûrs. Ce théorème améliore la compréhension des réponses structurelles et facilite l'optimisation des conceptions structurelles.

Théorème de Castigliano

Le théorème de Castigliano est utile pour déterminer les déplacements et les rotations dans les structures subissant des déformations élastiques.
Il y est indiqué que le ...

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Castigliano&#39;s Theorem

using strain energy to find deflection

Castigliano's theorem uses strain energy to find deflection and rotation in elastic structures. Strain energy is the work done by external loads during elastic deformation. The theorem links that energy to the movement caused by the load.

The key idea is simple. The partial derivative of the total strain energy with respect to a specific force gives the displacement at the point where that force acts. The same method also works for a moment, where the derivative gives the rotation at the point of loading.

This approach is useful for beams with non-uniform cross-sections and for structures with complex loading. In these cases, the deflection at any point can be found by differentiating the strain energy with respect to the load. The method is also used for shafts in torsion, where applied torque causes elastic twisting.

For torsional deformation, the strain energy depends on the material properties and the geometry of the shaft. Using Castigliano's theorem, the angle of twist can be calculated from that energy. This helps predict how structures respond under different loads and supports safer, more efficient design.

Using Strain Energy to Find Deflection

strain energy

Load-Deformation and Stored Strain Energy

Strain energy is the energy stored in a material when it deforms under load. It is a key idea in materials science and structural engineering. A loaded rod fixed at one end shows this clearly. When an axial force pulls on the free end, the rod stretches and stress builds inside it. As the axial force increases, the rod elongates more. This creates a direct link between the applied force and the amount of deformation. Engineers can show this relationship with a load-deformation plot. The plot...

strain-energy density

Energy Stored in Loaded Rods

Energy stored in a rod under axial load helps show how a material will respond to stretching and breaking. When the rod elongates, it stores strain energy as potential energy inside the material. Strain energy density is the amount of that energy per unit volume. In the elastic region, stress and strain have a linear relationship that follows Hooke's Law. The strain energy density in this region is found from the area under the stress-strain curve up to the elastic limit. This recoverable...

elastic strain energy for normal stresses

elastic-strain-energy-for-normal-stresses

Strain Energy in Stressed Materials

Strain energy in stressed materials describes the energy stored when a material deforms under load. It is an important idea in materials science and engineering. For axial loading with a uniform stress distribution, the stress is the internal force divided by the cross-sectional area, and the strain follows from the stress through the modulus of elasticity. When the stress is uniform, the strain energy density can be found from the product of stress and strain. That density is then integrated...

elastic strain energy for shearing stresses

elastic-strain-energy-for-shearing-stresses

Shear Energy Stored in Twisted Shafts

Shear energy stored in twisted shafts depends on elastic strain energy in a material. This energy is stored when a solid is deformed within its elastic limit. It comes from internal work done against the cohesive forces inside the material. When a material carries shearing stress, it also develops shearing strain. The strain energy density is the energy stored per unit volume. Within the elastic limit, stress is proportional to strain through the modulus of rigidity, so the strain energy...

impact loading

Impact Loading on a Fixed Rod

Impact loading on a fixed rod happens when a moving object strikes a stationary rod that has a uniform cross-sectional area and is fixed at one end. The impact makes the rod deform and begin to develop stress. This setup shows how a sudden hit can store energy in a structure. As the rod moves back toward its original shape, the stress rises to a maximum. At that point, the energy from the moving object has changed from kinetic energy into strain energy. Kinetic energy is the energy of motion,...

impact loading on a cantilever beam

Dropped-Weight Bending of a Cantilever Beam

Dropped-weight bending of a cantilever beam shows how impact loading changes a beam’s shape and stress. The beam has a circular cross-section and is fixed at one end. An object is dropped on the free end from a height h, so its potential energy from gravity becomes kinetic energy at impact. After the object hits the beam, the beam bends and develops a bending moment along its length. The bending is not uniform. The fixed end carries the highest strain energy, which is the energy stored in the...

castigliano&#39;s theorem

Castigliano's theorem uses strain energy to find deflection and rotation in elastic structures. Strain energy is the work done by external loads during elastic deformation. The theorem links that energy to the movement caused by the load. The key idea is simple. The partial derivative of the total strain energy with respect to a specific force gives the displacement at the point where that force acts. The same method also works for a moment, where the derivative gives the rotation at the point...

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...

key terms and concepts

Learn how Castigliano's theorem uses strain energy to calculate deflection, rotation, and angle of twist in beams and shafts under elastic loading.

Using Strain Energy to Find Deflection | JoVE K12

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Using Strain Energy to Find Deflection

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