Name: beam deflection from central load
Uploaded: 2024-05-23T13:00:13.000Z
Duration: 1 min 14 s
Channel: JoVE

長さLの単純に支持されたビームPQで、中心に点荷重Fを運ぶとします。ビームの中心でのたわみはどうなりますか?
ここでは、ビームの両端での反力は等しく、それぞれが中心荷重の半分です。
セグメントPCを考え、端Pからの距離xにある任意の点を選択します。このとき、点Pでの反力によるモーメントは、点Pでの荷重に距離を掛けた値です。端点 P の荷重に対するモーメント方程式の偏微分は x の半分です。 
ビームのセグメントQCについても同様の解析を行うことができます。
Castiglianoの定理を使用すると、点Cでのたわみは、加えられた荷重によるひずみエネルギーの偏微分によって決定されます。
この方程式は、ビームPCとQCの2つのセグメントを考えることで簡略化できます。
ビームの長さの半分以上で積分を実行すると、ビームの中心でのたわみの表現が得られます。

カスティリアーノの定理: 問題解決

中心点荷重がかかる単純に支持された梁のたわみは、構造力学の原理、特にカスティリアーノの定理を使用して解析できます。この定理は、荷重適用点での変位を構造内のひずみエネルギーの偏導関数に関連付けます。中心に点荷重がある単純に支持された梁は、支持に対称的な反力を持ち、それぞれが荷重の半分を負担します。梁に沿った任意の点での曲げモーメントは、これらの反力から導出され、最も近い支持部からの距離に基づいて計算されます。

カスティリアーノの定理は、荷重が適用される点でのたわみは、荷重に対して梁のひずみエネルギーを微分することによって決定されることを示しています。ひずみエネルギーは、梁に沿った曲げモーメントに基づいて計算され、その長さにわたって積分されます。この梁の場合、曲げによるひずみエネルギーは、梁の長さの半分に沿って積分された曲げモーメント式の 2 乗から計算されます。

<img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="height: 106px;">

梁は対称であるため、この値は梁全体を考慮して 2 倍になり、梁の中心での偏向が計算されます。これは、荷重の大きさ、つまり梁の長さの 3 乗に依存し、慣性モーメントと梁断面の弾性率の積に反比例します。

<img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="height: 42px;">

長さLの単純に支持されたビームPQで、中心に点荷重Fを運ぶとします。ビームの中心でのたわみはどうなりますか?
ここでは、...

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: 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 displacement at the load point is found by taking the partial derivative of the strain energy with respect to the load. Strain energy is the energy stored by the beam as it bends. It is calculated from the bending moment along the beam and integrated over the beam length.

Because the beam is symmetric, the strain energy is first found over half the beam and then doubled for the full length. The center deflection depends on the load, the cube of the beam length, and the beam's moment of inertia and elastic modulus. It is inversely proportional to the product of those two material and cross-section properties.

Beam Deflection from Central Load

strain energy

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strain-energy density

Energy Stored in Loaded Rods

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Strain Energy in Stressed Materials

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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

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...

castigliano&#39;s theorem: problem solving

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 is used to find the center deflection of a simply supported beam with a central point load using strain energy.

Beam Deflection from Central Load | JoVE K12

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mechanical engineering

Beam Deflection from Central Load

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