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

Теорема Кастильяно полезна для определения перемещений и вращений в конструкциях, претерпевающих упругие деформации.
Он гласит, что частная производная энергии упругой деформации относительно данной нагрузки эквивалентна смещению в точке приложения конкретной нагрузки.
Аналогично, предположим, что структура подвержена парам. В этом случае угол поворота конструкции в точке приложения пары рассчитывается путем частичного дифференцирования энергии деформации относительно приложенной пары.
Для конструкций, подвергающихся скручиванию из-за медленно приложенных крутящих моментов, угол скручивания участка вала может быть рассчитан путем взятия частичной производной энергии деформации по отношению к приложенному крутящему моменту.
Если балка с неравномерным сечением подвергается деформации из-за нагружения, энергия деформации выражается в виде интеграла. Здесь прогиб в точке приложения нагрузки рассчитывается путем дифференцирования энергии деформации по отношению к нагрузке до интегрирования.

Теорема Кастильяно

Теорема Кастильяно анализирует смещения и вращения в упругих конструкциях. Она связывает производную энергии упругой деформации с приложенными силами или моментами, что позволяет рассчитывать деформации. Теорема утверждает, что частная производная полной энергии деформации системы по отношению к определенной нагрузке приводит к смещению в точке приложения нагрузки. Этот принцип применим как к силам, так и к моментам.

<img alt="Equation 1" src="/files/ftp_upload/15688/15688_Equation_1.svg" style="text-align: center; height: 38px;">

Энергия деформации, которая измеряет работу, совершаемую внешними нагрузками во время упругой деформации, имеет решающее значение для этих расчетов. В таких условиях, как балки с неоднородным поперечным сечением или сложная нагрузка, прогиб в любой точке можно определить с помощью теоремы Кастильяно путем дифференцирования энергии деформации по отношению к нагрузке.

<img alt="Equation 2" src="/files/ftp_upload/15688/15688_Equation_2.svg" style="text-align: center; height: 34px;">

Аналогичным образом, для элементов конструкции, таких как валы, подвергающихся деформации кручения из-за приложенного крутящего момента, угол скручивания можно определить с помощью того же метода. Энергия деформации из-за кручения учитывает свойства материала и геометрические характеристики вала.

<img alt="Equation 3" src="/files/ftp_upload/15688/15688_Equation_3.svg" style="text-align: center; height: 34px;">

С помощью теоремы Кастильяно прогнозируется поведение конструкций при различных условиях нагрузки, что помогает проектировать более эффективные и безопасные системы. Эта теорема улучшает понимание реакций конструкций и облегчает оптимизацию конструкций.

Теорема Кастильяно полезна для определения перемещений и вращений в конструкциях, претерпевающих упругие деформации.
В нем говорится, что частичный...

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Mechanical Engineering

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KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

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.

mechanical engineering

Using Strain Energy to Find Deflection

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