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

Гибкие кабели используются для поддержки и передачи нагрузки в различных приложениях.
Рассмотрим кабель AB, закрепленный в двух точках, испытывающий множественные вертикальные сосредоточенные нагрузки.
Определите форму троса и натяжение в каждом участке троса, зная горизонтальные расстояния между грузами и опорой.
Для анализа предполагается, что кабель является гибким, нерастяжимым и имеет незначительный вес.
Кабель состоит из нескольких прямолинейных сегментов, на каждый из которых действует постоянное растягивающее усилие.
Нарисуйте схему свободного тела троса для определения сил реакции на опорах. При этом число неизвестных компонент реакции превышает уравнения равновесия.
Рассмотрим точку D на кабеле на известном расстоянии и нарисуем схему свободного тела отрезка AD. Используя уравнение равновесия момента в точке D, получается дополнительное уравнение.
Вспоминая уравнения равновесия и используя силы реакции, получают расстояние по вертикали между опорой А и каждой сосредоточенной нагрузкой.
Аналогичным образом можно получить напряжение в каждом сегменте, которое является максимальным, когда сегмент имеет наибольший угол наклона.

<ol>
	<li>Бир, Ф..; Джонстон, Э.Р.; Мазурек, Д.Ф; Кромвель,.Д. и Селф, Б.. (2019). Векторная механика для инженеров \u2012 Статика и динамика. Нью-Йорк: Макгроу-Хилл. С. 403-404.</li>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 381.</li>
</ol>

Кабель, на который действуют концентрированные нагрузки

Гибкие кабели часто используются в различных приложениях для поддержки и передачи нагрузки. Рассмотрим кабель, закрепленный на двух точках и подвергнутый нескольким вертикальным концентрированным нагрузкам. Определите форму кабеля и натяжение в каждом участке кабеля, учитывая горизонтальные расстояния между нагрузками и опорами.

Для анализа предполагается, что кабель обладает следующими свойствами:

<ol style="list-style-type:lower-alpha;">
	<li>Гибкий, что позволяет ему изменять форму под влиянием приложенных нагрузок.</li>
	<li>Бесстыковый, что означает, что его длина не меняется под действием натяжения</li>
	<li>Незначительная масса, что подразумевает, что собственный вес кабеля не влияет значительно на его поведение.</li>
</ol>

Кабель состоит из нескольких прямолинейных отрезков, причем каждый отрезок подвергается постоянной растягивающей силе. Чтобы определить реакционные силы на опорах, можно построить схему свободного тела кабеля. Однако, в данном случае, количество неизвестных реакционных компонент обычно превышает количество доступных уравнений равновесия. Требуется дополнительное уравнение. Рассмотрим точку D на кабеле на известном расстоянии от опор, чтобы получить дополнительное уравнение. Используя схему свободного тела отрезка AD и уравнение моментов равновесия в точке D, можно получить еще одно уравнение, которое поможет решить систему.

Вертикальное расстояние от опоры A до каждой концентрированной нагрузки можно определить с известными реакционными силами. Это можно сделать, используя уравнение равновесия, которое гласит, что сумма вертикальных сил, действующих на кабель, должна быть равна нулю. После определения вертикальных расстояний можно рассчитать натяжение в каждом сегменте кабеля. Натяжение достигает максимума, когда сегмент имеет наибольший угол наклона.

<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;" />

Гибкие кабели обычно используются в различных приложениях для поддержки и передачи нагрузки. Рассмотрим кабель, закрепленный в двух точках и подвергнутый ...

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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

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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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Finding Tension in a Loaded Cable

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