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

Castigliano의 정리는 탄성 변형을 겪고있는 구조물의 변위와 회전을 결정하는 데 유용합니다.
주어진 하중에 대한 탄성 변형 에너지의 편도함수가 특정 하중의 적용 지점에서의 변위와 동일하다는 것을 나타냅니다.
마찬가지로, 구조가 커플의 대상이 된다고 가정합니다. 이 경우, 커플의 적용 지점에서 구조체의 회전 각도는 적용된 커플에 대한 변형 에너지를 부분적으로 구별하여 계산됩니다.
천천히 가해지는 토크로 인해 비틀림이 발생하는 구조물의 경우, 샤프트 단면에 대한 비틀림 각도는 적용된 토크에 대한 변형 에너지의 편도함수를 취하여 계산할 수 있습니다.
단면이 균일하지 않은 빔이 하중으로 인해 변형을 겪는 경우, 변형 에너지는 적분으로 표현됩니다. 여기서, 하중 적용 지점에서의 처짐은 적분 전에 하중에 대한 변형 에너지를 구별하여 계산됩니다.

카스티글리아노의 정리

카스티글리아노의 정리는 탄성 구조의 변위와 회전을 분석합니다. 탄성 변형 에너지의 미분을 적용된 힘 또는 모멘트와 연관시켜 변형을 계산할 수 있습니다. 이 정리는 특정 하중에 대한 시스템의 전체 변형 에너지의 부분 미분으로 인해 하중이 적용되는 지점에서 변위가 발생한다고 명시합니다. 이 원리는 힘과 모멘트 모두에 적용됩니다.

<img alt="Equation 1" src="/files/ftp_upload/15688/15688_Equation_1.svg" style="height: 38px;">

탄성 변형 동안 외부 하중에 의해 수행된 작업을 측정하는 변형 에너지는 이러한 계산에 매우 중요합니다. 단면이 균일하지 않거나 복잡한 하중을 받는 빔과 같은 조건에서는 하중에 대한 변형 에너지를 차별화하여 카스티글리아노의 정리를 사용하여 임의 지점에서의 처짐을 유도할 수 있습니다.

<img alt="Equation 2" src="/files/ftp_upload/15688/15688_Equation_2.svg" style="height: 34px;">

마찬가지로, 적용된 토크로 인해 비틀림 변형을 겪는 샤프트와 같은 구조 요소의 경우 동일한 방법을 사용하여 비틀림 각도를 결정할 수 있습니다. 비틀림으로 인한 변형 에너지는 재료의 특성과 샤프트의 기하학적 특성을 설명합니다.

<img alt="Equation 3" src="/files/ftp_upload/15688/15688_Equation_3.svg" style="height: 34px;">

카스티글리아노의 정리를 통해 다양한 하중 조건에서 구조물의 거동을 예측하여 보다 효율적이고 안전한 시스템 설계에 도움을 줍니다. 이 정리는 구조적 반응에 대한 이해를 향상시키고 구조 설계의 최적화를 촉진합니다.

Castigliano의 정리는 탄성 변형을 겪고있는 구조물의 변위와 회전을 결정하는 데 유용합니다.
그것은 부분적으로 ...

Education

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JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Energy Methods

에너지 방법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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