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

卡斯蒂利亚诺定理可用于确定发生弹性变形的结构中的位移和转角。
该定理指出，弹性应变能关于某一载荷的偏导数等于该特定载荷作用点处的位移。
类似地，假设结构受到力偶作用。此时，结构在力偶作用点处的转角可通过将应变能对所施加的力偶进行偏微分来计算。
对于因缓慢施加的扭矩而发生扭转变形的结构，可通过对应变能关于所施加扭矩的偏导数来计算轴段的扭转角。
如果由于载荷作用导致非均匀横截面梁发生变形，则应变能可用积分形式表示。此处，通过在积分前对应变能对载荷求导，可计算载荷作用点处的挠度。

卡斯提利亚诺定理

<div>
 卡斯提利亚诺定理能够用来分析弹性结构中的位移和旋转。它将弹性应变能的导数与所施加的力或力矩联系起来，从而可以对其中的变形进行计算。该定理指出，一个系统中的总应变能相对于特定荷载的偏导数结果就是荷载作用点处所产生的位移。这一原理同样也适用于力和力矩。
 
 <img alt="Equation 1" src="/files/ftp_upload/15688/15688_Equation_1.svg" style="text-align: center; height: 38px;">
 应变能可以用来测量弹性变形过程中外部载荷所做的功，对于这些计算来说是至关重要的。在横截面不均匀或梁受到复杂荷载的情况下，可以使用卡斯提利亚诺定理来对荷载的应变能进行求导，以此来推导出位于任意一点处的挠度。
 
 <img alt="Equation 2" src="/files/ftp_upload/15688/15688_Equation_2.svg" style="text-align: center; height: 34px;">
 类似地，对于因施加扭矩而发生扭转变形的结构元件（如轴），则可以使用相同的方法来确定其扭转角度。由扭转所引起的应变能则考虑了材料的特性和轴的几何属性。
 
 <img alt="Equation 3" src="/files/ftp_upload/15688/15688_Equation_3.svg" style="text-align: center; height: 34px;">
 通过应用卡斯提利亚诺定理，则可以预测出结构在各种荷载条件下的行为。这有助于设计出更高效、更安全的系统。该定理还加深了人们对结构响应的理解，同时还有助于优化结构设计。
 
</div>

卡斯蒂利亚诺定理在确定经历弹性变形的结构的位移和转角方面具有重要作用。  
该定理指出，偏导数……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Energy Methods

能量法

KEY TERMS AND CONCEPTS

关键术语与概念

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

Description

Transcript