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

Der Satz von Castigliano ist nützlich für die Bestimmung von Verschiebungen und Rotationen in Strukturen, die elastischen Verformungen unterliegen.
Sie besagt, dass die partielle Ableitung der elastischen Dehnungsenergie für eine gegebene Last äquivalent zur Verschiebung der jeweiligen Last am Aufbringungspunkt ist.
Nehmen wir in ähnlicher Weise an, dass die Struktur Paaren unterworfen ist. In diesem Fall wird der Drehwinkel der Struktur am Anbringungspunkt eines Paares berechnet, indem die Dehnungsenergie in Bezug auf das angelegte Paar teilweise differenziert wird.
Bei Strukturen, die aufgrund langsam aufgebrachter Drehmomente verdreht werden, kann der Verdrehungswinkel für einen Abschnitt der Welle berechnet werden, indem die partielle Ableitung der Dehnungsenergie in Bezug auf das aufgebrachte Drehmoment genommen wird.
Erfährt ein Balken mit ungleichmäßigen Querschnitten aufgrund von Belastung eine Verformung, wird die Dehnungsenergie in Form eines Integrals ausgedrückt. Hierbei wird die Durchbiegung am Angriffspunkt der Last berechnet, indem vor der Integration die Dehnungsenergie in Bezug auf die Last differenziert wird.

castiglianos theorem

Der Satz von Castigliano analysiert Verschiebungen und Drehungen in elastischen Strukturen. Er setzt die Ableitung der elastischen Dehnungsenergie in Beziehung zu den angewandten Kräften oder Momenten und ermöglicht so die Berechnung von Verformungen. Der Satz besagt, dass die partielle Ableitung der gesamten Dehnungsenergie eines Systems in Bezug auf eine bestimmte Last die Verschiebung an dem Punkt ergibt, an dem die Last angewandt wird. Dieses Prinzip gilt sowohl für Kräfte als auch für Momente.

<img alt="Equation 1" src="/files/ftp_upload/15688/15688_Equation_1.svg" style="text-align: center; height: 38px;">

Die Dehnungsenergie, die die von externen Lasten während der elastischen Verformung geleistete Arbeit misst, ist für diese Berechnungen von entscheidender Bedeutung. Unter Bedingungen wie Balken mit ungleichmäßigem Querschnitt oder komplexer Belastung kann die Durchbiegung an jedem Punkt mithilfe des Castigliano-Theorems abgeleitet werden, indem die Dehnungsenergie in Bezug auf die Last differenziert wird.

<img alt="Equation 2" src="/files/ftp_upload/15688/15688_Equation_2.svg" style="text-align: center; height: 34px;">

Ähnlich verhält es sich mit der gleichen Methode bei Strukturelementen wie Wellen, die aufgrund eines angelegten Drehmoments eine Torsionsverformung erfahren. Die Verformungsenergie aufgrund der Torsion berücksichtigt die Materialeigenschaften und die geometrischen Merkmale der Welle.

<img alt="Equation 3" src="/files/ftp_upload/15688/15688_Equation_3.svg" style="text-align: center; height: 34px;">

Mithilfe des Castigliano-Theorems lässt sich das Verhalten von Strukturen unter verschiedenen Belastungsbedingungen vorhersagen, was bei der Entwicklung effizienterer und sichererer Systeme hilft. Dieses Theorem verbessert das Verständnis struktureller Reaktionen und erleichtert die Optimierung struktureller Designs.

Castiglianos Theorem

Der Satz von Castigliano ist nützlich für die Bestimmung von Verschiebungen und Rotationen in Strukturen, die elastischen Verformungen unterliegen.
Darin heißt es, dass die Teilregelung ...

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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

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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...

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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

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Dropped-Weight Bending of a Cantilever Beam

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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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