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

Castiglianoの定理は、弾性変形を受ける構造の変位と回転を決定するのに役立ちます。
これは、特定の荷重に関する弾性ひずみエネルギーの偏微分が、特定の荷重の適用点での変位と同等であることを示しています。
同様に、構造がカップルに服従しているとします。この場合、カップルの適用点における構造物の回転角度は、適用されたカップルに対するひずみエネルギーを部分的に微分することによって計算されます。
ゆっくりと加えられるトルクによってねじれが発生する構造物の場合、シャフトのセクションのねじれ角度は、加えられたトルクに対するひずみエネルギーの偏微分を取ることによって計算できます。
不均一な断面を持つ梁が荷重によって変形した場合、ひずみエネルギーは積分で表されます。ここで、荷重の印加点でのたわみは、積分前の荷重に対するひずみエネルギーを微分することによって計算されます。

カスティリアーノの定理

カスティリアーノの定理は、弾性構造における変位と回転を分析します。弾性ひずみエネルギーの導関数を、加えられた力またはモーメントに関連付け、変形の計算を可能にします。この定理は、特定の荷重に関する装置の全ひずみエネルギーの偏導関数によって、荷重が適用される点での変位が生じると述べています。この原則は力とモーメントの両方に当てはまります。

<img alt="Equation 1" src="/files/ftp_upload/15688/15688_Equation_1.svg" style="height: 38px;">

弾性変形中に外部荷重によって行われる仕事を測定するひずみエネルギーは、これらの計算にとって重要です。不均一な断面や複雑な荷重を伴う梁などの条件では、カスティリアーノの定理を使用してひずみエネルギーを荷重に対して微分することで、任意の点でのたわみを導き出すことができます。

<img alt="Equation 2" src="/files/ftp_upload/15688/15688_Equation_2.svg" style="height: 34px;">

同様に、加えられた力によりねじれ変形を受ける軸などの構造要素についても、同じ方法を使用してねじれ角度を決定できます。ねじれによるひずみエネルギーは、材料の特性と軸の幾何学的属性を考慮します。

<img alt="Equation 3" src="/files/ftp_upload/15688/15688_Equation_3.svg" style="height: 34px;">

カスティリアーノの定理により、さまざまな荷重条件下での構造の挙動が予測され、より効率的で安全な装置の設計に役立ちます。この定理により、構造応答の理解が深まり、構造設計の最適化が容易になります。

Castiglianoの定理は、弾性変形を受ける構造の変位と回転を決定するのに役立ちます。
それは部分的な...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Energy Methods

エネルギー法

KEY TERMS AND CONCEPTS

主要用語と概念

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

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mechanical engineering

Using Strain Energy to Find Deflection

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