Name: shear change under distributed loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 23 s
Channel: JoVE

分布荷重、2 つの集中荷重、およびいくつかのモーメントが発生しているビームについて考えます。せん断と分布荷重の関係を確立します。
まず、集中荷重や結合モーメントのないビーム上の要素セクションを考え、そのセクションの自由体図を描きます。
平衡を維持するには、断面の右側に作用するせん断力を少量かつ有限の量だけ増加させる必要があります。
分布荷重の合力は、断面の右端から分数距離で作用します。
鉛直力の平衡方程式を使用して、せん断と荷重の関係が得られます。
次に、方程式の両側をΔxで割り、Δxをゼロに近づけることで、せん断力の傾きを決定することができます。これは、分布荷重強度に等しくなります。
最後に、方程式を並べ替え、2つの任意の点間の要素セクション上の分布荷重を積分すると、せん断の変化と荷重曲線の下の面積との間に関係が生じます。

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	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。ニュージャージー州ホーボーケン:ピアソンプレンティスホール。370-371ページ。</li>
	<li>ビール、FP;ジョンストン、ER;マズレック、D.F;クロムウェル、PJおよびセルフ、BP(2019)。エンジニアのためのベクトル力学 \u2012 静力学と動力学。ニューヨーク:マグロウヒル。391-392ページ。</li>
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分布荷重とせん断力の関係

構造解析における分布荷重とせん断力の関係を理解することは、さまざまな荷重条件にさらされる梁を解析するために重要です。分布荷重、2つの集中荷重、およびカップルモーメントがかかる梁の場合を考えてみましょう。

<img alt="Figure 1" src="/files/ftp_upload/14304/14304_Figure_1v2.svg" style="text-align: center; height: 300px;" />

与えられた場合におけるせん断力と分布荷重の関連は、与えられた手順に従って確立することができます。まず、集中荷重やカップルモーメントのない梁の要素部分を考えます。この部分に作用する力を解析するために、自由物体図を描くことができます。図は、梁の長さに沿って作用する分布荷重、節点の右側に作用するせん断力V(x)、および平衡を維持するために追加される微小なせん断力dVで構成されます。

垂直方向の平衡を維持するためには、節点の右側に作用するせん断力に微小かつ有限な量ΔVを加える必要があります。分布荷重の合力w(x)Δxは、節点の右端からの分数の距離で作用します。

<img alt="Equation 1" src="/files/ftp_upload/14304/14304_Equation_1v2.svg" style="text-align: center; height: 13px;" />

垂直力の平衡の方程式を用いると、せん断力と荷重の間の以下の関係が得られます。

<img alt="Equation 2" src="/files/ftp_upload/14304/14304_Equation_2v2.svg" style="text-align: center; height: 32px;" />

次に、式の両辺をΔxで除算し、Δxをゼロに近づけます。

この式からわかるように、せん断力の傾きは分布荷重の強度に等しいことが示されています。

最後に、式を再配置し、2つの任意の点QとRの間の要素部分の分布荷重を積分します。

<img alt="Equation 3" src="/files/ftp_upload/14304/14304_Equation_3v2.svg" style="text-align: center; height: 50px;" />

この積分方程式は、せん断力の変化と荷重曲線の下の面積との関係を示しています。点QとRの間でのせん断力の差は、これらの2つの点の間の分布荷重曲線の下の面積に等しいです。

構造解析における分布荷重とせん断力の関係を理解することは、さまざまな荷重を受ける梁を解析するために重要です。

relation-between-the-distributed-load-and-shear

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Internal Forces

内力

KEY TERMS AND CONCEPTS

主要用語と概念

Relation Between the Distributed Load and Shear

shear change under distributed loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment.

To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear force dV needed to keep the section in equilibrium.

The vertical forces must balance. For that reason, the shear force on the right-hand side changes by a small amount, ΔV. The distributed load over the small length Δx creates a resultant force of w(x)Δx. That force acts at a fractional distance from the right end of the section.

From the equilibrium equation for vertical forces, a relation between shear and load is obtained. When both sides are divided by Δx and Δx is allowed to approach zero, the slope of the shear force curve equals the distributed load intensity. This means the load level at a point controls how quickly shear changes along the beam.

The equation can also be rearranged and integrated over the elemental section between two points, Q and R. This integral form shows that the change in shear force equals the area under the distributed load curve between those points. In other words, the difference in shear force from Q to R is found by measuring the area under the load curve.

Shear Change Under Distributed Loading

sign convention

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Beam Types and Support Conditions

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Beam Shear Force Diagram

Beam shear force diagrams show how shear changes along the length of a supported beam. In beam mechanics, they help students track the internal shear force caused by perpendicular loads. The beam AB in this example is supported at both ends, so the first step is to draw a free-body diagram that shows the external forces and internal reactions. The reaction forces at supports A and B are found with equilibrium equations for force and moment. For the beam shown, the vertical reaction at A is 24...

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Beam Load Analysis with Moment Diagrams

Beam load analysis uses a bending moment diagram to show how bending changes along the length of a beam. This diagram helps engineers and designers check whether a structure can handle applied forces. It is also useful for parts such as shelving arms, which must resist forces and moments from a load. To build the diagram, start by finding the reactive forces and any couple moments on the beam. In some cases, especially when a beam is inclined, these forces must be resolved into components that...

relation between the distributed load and shear

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

relation between the shear and bending moment

relation-between-the-shear-and-bending-moment

Shear and Moment Diagrams in Beam Loads

Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments. The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a...

shear and bending moment diagram: problem solving

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Beam Load Analysis with Shear Diagrams

Beam load analysis uses shear and bending moment diagrams to track the internal forces in a beam. These diagrams are important when a beam carries concentrated loads and a distributed load. They help show how the beam responds so structures can be designed safely and efficiently. The first step is to draw a free-body diagram of the whole beam. A free-body diagram shows the concentrated loads, the distributed load, and the reaction forces at the supports. Use the equilibrium equations, where...

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Cable Tension Under Uniform Loading

Cable tension under uniform loading is found by analyzing a cable fixed between two supports. The problem starts with the support conditions and the load on the cable. From there, the cable’s shape must be described with an equation that matches the curve formed by the load. The next step is to integrate the shape equation to get the cable’s shape function. This makes it possible to solve for constants in the equation. The boundary condition at the origin gives one constant. The slope, which...

key terms and concepts

Learn how distributed load affects shear force in beams, from a free-body diagram to the slope relation and the area under the load curve.

Shear Change Under Distributed Loading | JoVE K12

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mechanical engineering

Shear Change Under Distributed Loading

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