Name: cable tension under uniform loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 29 s
Channel: JoVE

2つのサポートに固定されたケーブルが均一な負荷を受けると考えてください。ケーブルの最大張力を確認します。
解析では、ケーブルの対称性によるケーブルの中心での原点を考慮します。
分布荷重のケーブルの形状方程式を思い出し、既知の定数荷重値を代入します。
次に、方程式を積分し、原点に境界条件を適用することで、定数C2が決定されます。
ケーブルの形状方程式の 1 次導関数を取ることで、傾きを決定できます。原点の勾配に境界条件を適用すると、C1が得られます。 
支持体の積分定数と位置座標を代入し、方程式を並べ替えると、水平方向の引張力が得られます。
傾き方程式を思い出し、角度が最大になるサポートの x 座標値を代入します。
Cabelの張力は、最大ニアサポートである角度によって変化します。
三角関数の関係を適用し、項を並べ替えると、最大張力方程式が得られます。
最後に、水平張力方程式と既知の値を代入することにより、ケーブルの最大張力が得られます。

<ol>
	<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。384-385、393ページ。</li>
</ol>

ケーブル：問題解決

2つの支持点に固定され、均一な荷重にさらされているケーブルに取り組む際には、ケーブルの最大張力を確定することが重要です。このプロセスは以下に示されるいくつかの重要なステップに分解することができます：

<img alt="図1" src="/files/ftp_upload/14310/14310_Figure_1v3.svg" style="text-align: center; height: 300px;" />

問題の分析： 与えられたシナリオとケーブルの状況を理解し、支持点、荷重の種類、および他の関連情報を特定することから始めます。

ケーブルの形状の方程式の決定：平衡の原理とケーブルの特性を使用して、ケーブルの曲線を記述する形状の方程式を確立します。この方程式は、ケーブルの形状と適用された荷重を関連付けます。

<img alt="方程式1" src="/files/ftp_upload/14310/14310_Equation_1v2.svg" style="text-align: center; height: 35px;" />

方程式の積分：形状の方程式を積分して、ケーブルの形状を表す関数を得ます。この積分プロセスによって、方程式内の定数を決定することができます。原点における境界条件を適用することで、積分定数の1つの値を決定することができます。

<img alt="方程式2" src="/files/ftp_upload/14310/14310_Equation_2v2.svg" style="text-align: center; height: 35px;" />

傾斜角の算出：ケーブルの形状の方程式の一次導関数を取ることにより、任意の点でのケーブルの傾斜を決定します。原点における傾斜の境界条件を適用することで、もう1つの積分定数の値を得ることができます。

水平引張力の計算：積分定数と支持位置の座標を形状の方程式に代入して、ケーブルに作用する水平引張力を求めます。項を並び替えて、ケーブルに作用する水平引張力を見つけます。

<img alt="方程式3" src="/files/ftp_upload/14310/14310_Equation_3v2.svg" style="text-align: center; height: 35px;" />

角度の測定：傾斜方程式を使用して、様々な点でのケーブルの角度を計算します。ケーブルの角度が最大となる位置（通常は支持点付近）でのケーブルの角度を水平引張力とケーブルの角度の三角関数的な関係を利用して表します。

<img alt="方程式4" src="/files/ftp_upload/14310/14310_Equation_4v2.svg" style="text-align: center; height: 35px;" />

最大張力の算出：水平引張力の方程式と既知の値を最大張力の方程式に代入します。これにより、ケーブルの最大張力を計算することができます。

<img alt="方程式5" src="/files/ftp_upload/14310/14310_Equation_5v2.svg" style="text-align: center; height: 40px;" />

2つのサポートに固定され、均一な負荷を受けるケーブルを扱う場合、ケーブルの最大張力を決定することが重要です。...

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: 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 is the first derivative of the shape equation, gives the cable’s tilt at any point. A second boundary condition at the origin is used to find another constant.

After the constants are set, the horizontal tensile force can be calculated from the support coordinates and the shape equation. The slope equation is then used to find the cable angle at different points. The angle is usually largest near the supports, so that location is used to determine the maximum tension.

Trigonometric relationships connect the maximum tension to the horizontal tensile force and the cable angle. Once the horizontal tension equation and the given values are substituted into the tension formula, the maximum tension in the cable can be found.

Cable Tension Under Uniform Loading

sign convention

Beam Forces: Tension, Shear, and Bending

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Beam Loads and Internal Forces

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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...

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Beam Types and Support Conditions

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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

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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...

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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...

cable subjected to a distributed load

Finding Cable Shape Under Distributed Load

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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 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

Follow the steps to find maximum cable tension under uniform loading using the cable shape equation, slope, boundary conditions, and trigonometry.

Cable Tension Under Uniform Loading | JoVE K12

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mechanical engineering

Cable Tension Under Uniform Loading

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