Name: finding tension in a loaded cable
Uploaded: 2023-09-22T11:49:27.000Z
Duration: 1 min 28 s
Channel: JoVE

フレキシブルケーブルは、さまざまなアプリケーションでのサポートと負荷伝送に使用されます。
ケーブルABが2点に固定されており、複数の垂直集中荷重が発生するとします。
ケーブルの形状とケーブルの各部分の張力を決定し、負荷とサポートの間の水平距離を把握します。
分析のために、ケーブルは柔軟性があり、伸びず、重量が無視できると仮定します。
ケーブルはいくつかの直線セグメントで構成されており、それぞれが一定の引張力を受けます。
ケーブルの自由体線図を描画して、サポートの反力を決定します。ここでは、未知の反応成分の数が平衡方程式を超えています。
既知の距離にあるケーブル上の点 D を考慮し、セグメント AD のフリーボディ図を描きます。点Dでのモーメント平衡方程式を使用すると、追加の方程式が得られます。
平衡方程式を呼び戻し、反力を使用することにより、支持体Aと各集中荷重との間の垂直距離が得られます。
同様に、各セグメントの張力を得ることができ、これはセグメントの傾斜角が最大の場合に最大になります。

<ol>
	<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。403-404ページ。</li>
	<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。381ページ。</li>
</ol>

集中荷重にさらされたケーブル

伸縮性のあるケーブルは、さまざまな応用でサポートと負荷伝達に一般的に使用されています。2つのポイントで固定され、複数の垂直集中荷重にさらされるケーブルを考えます。荷重とサポート間の水平距離が与えられた場合、ケーブルの形状とケーブルの各部分の張力を求めます。

<img alt="Figure 1" src="/files/ftp_upload/14307/14307_Figure_1v2.svg" style="text-align: center; height: 350px;" />

解析のために、ケーブルは以下の特性を持つと仮定されています：

<ol style="display: flex; flex-direction: column; gap: 10px; list-style-type: lower-alpha;">
 <li>応力を受けて形状を変化させることができる柔軟性</li>
 <li>張力下で長さが変化しない伸縮性</li>
 <li>無視できる重量で、ケーブルの自重がその挙動にほとんど影響を与えないこと</li>
</ol>

ケーブルはいくつかの直線セグメントで構成され、各セグメントは一定の引張力にさらされます。サポートでの反力を求めるためには、ケーブルの自由体図を描くことができます。しかし、この場合、未知の反力成分の数は通常、利用可能な平衡方程式の数を超えます。追加の方程式が必要です。追加の方程式を得るために、サポートから既知の距離にあるケーブル上の点Dを考えます。セグメントADの自由体図を描き、点Dでのモーメント平衡方程式を使用することで、システムを解くのに役立つ別の方程式が導かれます。

<img alt="Equation 1" src="/files/ftp_upload/14307/14307_Equation_1v2.svg" style="text-align: center; height: 13px;" />

 サポートAから各集中荷重までの垂直距離は、既知の反力によって計算できます。これは、平衡方程式によって得られます。この方程式は、ケーブルに作用する垂直な力の合計がゼロになることを述べています。垂直距離が求められると、各ケーブルセグメントの張力を計算できます。セグメントの傾斜角が最大のとき、張力は最大になります。

<img alt="Equation 2" src="/files/ftp_upload/14307/14307_Equation_2v2.svg" style="text-align: center; height: 13px;" />

<img alt="Equation 3" src="/files/ftp_upload/14307/14307_Equation_3v2.svg" style="text-align: center; height: 13px;" />

フレキシブルケーブルは、サポートや負荷伝送のためにさまざまなアプリケーションで一般的に使用されています。ケーブルが2点に固定され、...

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

Cable Subjected to Concentrated Loads

finding tension in a loaded cable

Flexible cables can carry vertical point loads between two fixed supports. The cable is treated as flexible, inextensible, and nearly weightless. Under these conditions, it changes shape to form straight-line segments, and each segment carries a constant tensile force.

To analyze the cable, a free-body diagram is drawn for the whole cable. This helps identify the reaction forces at the supports. In many cases, the number of unknown reaction components is greater than the available equilibrium equations, so an extra equation is needed.

A point such as D, at a known distance from the supports, can provide that extra equation. By isolating segment AD and taking moments about point D, another equilibrium relation is obtained. The vertical distance from support A to each concentrated load can then be found from the reaction forces. Using the condition that the sum of the vertical forces on the cable equals zero, the remaining cable geometry can be determined.

Once the vertical distances are known, the tension in each cable segment can be calculated. The tension changes from segment to segment because the cable slope changes. The tension is largest in the segment with the greatest inclination angle.

Finding Tension in a Loaded Cable

sign convention

Beam Forces: Tension, Shear, and Bending

Beam analysis depends on the internal forces and moments inside a structure. These include normal force, shear force, and bending moment. Engineers use the method of sections to cut the beam and find these values. A sign convention helps show whether each result is positive or negative. Normal force acts perpendicular to the beam’s cross-section. It can pull the material in tension or push it in compression. Tension is taken as positive, while compression is taken as negative. This makes it...

normal and shear force

Beam Loads and Internal Forces

Beam loads create internal forces inside a structure. Weight, pressure, and other external forces can all act on a beam. These internal forces affect the beam’s stability and strength. Engineers study these forces to find their size and direction. A common approach is the method of sections. In this method, an imaginary cut is made at a chosen point on the beam, and the forces at that section are analyzed. One internal force is the normal force, also called the axial force. It acts...

bending and torsional moments

Forces That Bend and Twist Materials

Bending and torsional moments describe how materials respond to force. These ideas are important in structural engineering because they help explain how beams, rulers, and screws change shape under load. A moment depends on both the size of the force and the distance from the point of interest to the force’s line of action. A bending moment forms when an external force makes a structural element bend. If the element bends upward, the top side is in compression and the bottom side is in tension.

internal loadings in structural members: problem solving

internal-loadings-in-structural-members-problem-solving

Beam Internal Force Analysis at Point A

Beam internal force analysis shows how normal force, shear force, and bending moment act inside a structural member. These internal loadings help engineers check whether a beam can support the external forces applied to it. The example uses a beam OC that weighs 5 kN and is inclined at 53.13° with the horizontal. The beam is supported at both ends. The goal is to find the internal loadings at point A. The first step is to find the reaction force at point C. This is done by taking the moment...

beams

Beam Types and Support Conditions

Beams are key parts of structural engineering and construction. They support loads that act at different points along their length. Engineers use beam shape and support conditions to understand how a beam will perform. Beams can be classified by geometry. Straight beams are the most common and have a constant cross-section along their length. Tapered beams change in cross-section from one end to the other, while curved beams have a curved shape. Beam cross-section also matters. Common shapes...

shear diagram

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...

bending moment diagram

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

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...

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

shear-and-bending-moment-diagram-problem-solving

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...

cable subjected to concentrated loads

Flexible cables can carry vertical point loads between two fixed supports. The cable is treated as flexible, inextensible, and nearly weightless. Under these conditions, it changes shape to form straight-line segments, and each segment carries a constant tensile force. To analyze the cable, a free-body diagram is drawn for the whole cable. This helps identify the reaction forces at the supports. In many cases, the number of unknown reaction components is greater than the available equilibrium...

cable subjected to a distributed load

Finding Cable Shape Under Distributed Load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation. The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations...

cable subjected to its own weight

Cable Sag Under Its Own Weight

Overhead power transmission cables sag under their own weight, and that shape affects how they carry electricity across long distances. To study this behavior, the cable is analyzed with a generalized loading function. This function treats the load along the cable's arc length instead of its projected length, which gives a more accurate picture of the cable's response. The analysis then focuses on a small cable segment. A free-body diagram is drawn to show the forces acting on that segment.

cable: problem solving

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 to analyze a flexible cable with vertical concentrated loads, use equilibrium and moments to find support reactions, cable shape, and segment tension.

Finding Tension in a Loaded Cable | JoVE K12

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mechanical engineering

Finding Tension in a Loaded Cable

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