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

Рассмотрим трос, закрепленный на двух опорах под равномерной нагрузкой. Определите максимальное натяжение кабеля.
Для анализа учитывайте начало координат в центре кабеля из-за его симметрии.
Вспомните уравнение формы кабеля для распределенной нагрузки и подставьте его на известное значение постоянной нагрузки.
Затем интегрируем уравнение, и путем применения граничных условий в начале координат определяется константа C2.
Взяв первую производную уравнения формы кабеля, можно определить наклон. Применяя граничные условия для наклона в начале координат, получается C1.
Путем подстановки констант интегрирования и координат положения опоры и перегруппировки уравнения получается горизонтальная растягивающая сила.
Вспомните уравнение уклона и подставьте значение x-координаты в опоре, где угол максимален.
Натяжение кабеля изменяется с углом, который является максимальной ближней опорой.
Применяя тригонометрическое соотношение и переставляя слагаемые, получается уравнение максимального напряжения.
Наконец, путем подстановки уравнения горизонтального натяжения и известных значений получается максимальное напряжение в кабеле.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 384-385, 393.</li>
</ol>

Кабель: решение задач

При работе с кабелем, жестко закрепленным на двух опорах и подвергнутым равномерной нагрузке, крайне важно определить максимальное напряжение в кабеле. Этот процесс можно разбить на несколько ключевых шагов, описанных ниже:

<img alt="Figure 1" src="/files/ftp_upload/14310/14310_Figure_1v3.svg" style="text-align: center; height: 300px;" />

Анализ проблемы: Начните с понимания данной ситуации и условий кабеля. Определите опоры, тип нагрузки и другую соответствующую информацию.

Определение уравнения формы кабеля: Используйте принципы равновесия и свойства кабеля для установления уравнения формы, описывающего кривую кабеля. Это уравнение связывает форму кабеля с приложенной нагрузкой.

<img alt="Equation 1" src="/files/ftp_upload/14310/14310_Equation_1v2.svg" style="text-align: center; height: 35px;" />

Интегрирование уравнения: Интегрируйте уравнение формы, чтобы получить функцию, представляющую форму кабеля. Процесс интегрирования позволяет определить константы в уравнении. Путем применения граничных условий в начале, можно определить значение одной из констант интегрирования.

<img alt="Equation 2" src="/files/ftp_upload/14310/14310_Equation_2v2.svg" style="text-align: center; height: 35px;" />

Определение угла наклона: Возьмите первую производную уравнения формы кабеля, чтобы определить угол наклона кабеля в любой заданной точке. Примените граничные условия для угла наклона в начале, чтобы получить значение другой константы интегрирования.

Вычисление горизонтальной силы натяжения: Подставьте константы интегрирования и координаты положения опор в уравнение формы. Переставьте слагаемые, чтобы найти горизонтальную силу натяжения, действующую на кабель.

<img alt="Equation 3" src="/files/ftp_upload/14310/14310_Equation_3v2.svg" style="text-align: center; height: 35px;" />

Определение угла: Используйте уравнение наклона для вычисления угла кабеля в различных точках. Определите место вдоль кабеля, где угол достигает максимума, обычно это близко к опорам. Используйте тригонометрические соотношения, чтобы выразить максимальное напряжение в терминах горизонтальной силы натяжения и угла кабеля.

<img alt="Equation 4" src="/files/ftp_upload/14310/14310_Equation_4v2.svg" style="text-align: center; height: 35px;" />

Нахождение максимального напряжения: Подставьте уравнение горизонтального напряжения и известные значения в уравнение максимального напряжения. Это позволит вам вычислить максимальное напряжение в кабеле.

<img alt="Equation 5" src="/files/ftp_upload/14310/14310_Equation_5v2.svg" style="text-align: center; height: 40px;" />

При работе с тросом, который закреплен на двух опорах и подвергается равномерной нагрузке, очень важно определить максимальное натяжение троса. ...

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Mechanical Engineering

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Internal Forces

Внутренние силы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

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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...

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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...

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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...

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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 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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Cable Tension Under Uniform Loading

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