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

Si consideri un cavo fissato a due supporti soggetti a carico uniforme. Determinare la tensione massima nel cavo.
Per l'analisi, considerare l'origine al centro del cavo a causa della sua simmetria.
Richiama l'equazione di forma del cavo per il carico distribuito e sostituisci il valore di carico costante noto.
Successivamente, integrare l'equazione e, applicando le condizioni al contorno all'origine, viene determinata la costante C2.
Prendendo la derivata prima dell'equazione di forma del cavo, è possibile determinare la pendenza. Applicando le condizioni al contorno per la pendenza all'origine, si ottiene C1.
Sostituendo le costanti di integrazione e le coordinate di posizione del supporto e riorganizzando l'equazione, si ottiene la forza di trazione orizzontale.
Richiama l'equazione della pendenza e sostituisce il valore della coordinata x in corrispondenza del supporto in cui l'angolo è massimo.
La tensione di Cabel cambia con l'angolo, che è il massimo supporto vicino.
Applicando la relazione trigonometrica e riorganizzando i termini, si ottiene l'equazione della massima tensione.
Infine, sostituendo l'equazione della tensione orizzontale e i valori noti, si ottiene la massima tensione nel cavo.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica \u2012 Statica e Dinamica. Hoboken, New Jersey: Pearson Prentice Hall. pp 384-385, 393.</li>
</ol>

cavo: risoluzione dei problemi

Quando si tratta di un cavo fissato a due supporti e sottoposto a un carico uniforme, è fondamentale determinare la tensione massima nel cavo. Questo processo può essere suddiviso in diversi passaggi chiave, come indicato di seguito:
<img alt="Figura 1" src="/files/ftp_upload/14310/14310_Figure_1v3.svg" style="text-align: center; height: 300px;" />
Analizziamo il problema: Si inizia comprendendo lo scenario dato e le condizioni del cavo. Si identificano i supporti, il tipo di carico e qualsiasi altra informazione rilevante.
Determinare l'equazione della forma del cavo: Utilizzare i principi dell'equilibrio e le proprietà del cavo per stabilire l'equazione della forma che descrive la curva del cavo. Questa equazione mette in relazione la forma del cavo con il carico applicato.
<img alt="Equazione 1" src="/files/ftp_upload/14310/14310_Equation_1v2.svg" style="text-align: center; height: 35px;" />
Integrare l'equazione: Integrare l'equazione della forma per ottenere una funzione che rappresenti la forma del cavo. Questo processo di integrazione consente di determinare le costanti dell'equazione. Applicando le condizioni a contorno all'origine, è possibile determinare il valore di una delle costanti di integrazione.
<img alt="Equazione 2" src="/files/ftp_upload/14310/14310_Equation_2v2.svg" style="text-align: center; height: 35px;" />
Trovare la pendenza: Calcolare la derivata prima dell'equazione della forma del cavo per determinare la pendenza del cavo in ogni punto. Applicare le condizioni al contorno per la pendenza all'origine per ottenere il valore di un'altra costante di integrazione.
Calcolare la forza di trazione orizzontale: Sostituendo le costanti di integrazione e le coordinate di posizione del supporto nell'equazione della forma. Riarrangiare i termini per trovare la forza di trazione orizzontale agente sul cavo.
<img alt="Equazione 3" src="/files/ftp_upload/14310/14310_Equation_3v2.svg" style="text-align: center; height: 35px;" />
Determinare l'angolo: Utilizzare l'equazione della pendenza per calcolare l'angolo del cavo in vari punti. Trovare la posizione lungo il cavo in cui l'angolo è massimo, di solito vicino ai supporti. Utilizzare le relazioni trigonometriche per esprimere la tensione massima in termini della forza di trazione orizzontale e dell'angolo del cavo.
<img alt="Equazione 4" src="/files/ftp_upload/14310/14310_Equation_4v2.svg" style="text-align: center; height: 35px;" />
Trovare la tensione massima: Sostituire l'equazione della tensione orizzontale e i valori noti nell'equazione della tensione massima. Questo permetterà di calcolare la tensione massima nel cavo.
<img alt="Equazione 5" src="/files/ftp_upload/14310/14310_Equation_5v2.svg" style="text-align: center; height: 40px;" />

Cavo: Risoluzione dei Problemi

Quando si ha a che fare con un cavo fissato a due supporti e sottoposto a un carico uniforme, è fondamentale determinare la tensione massima nel cavo. ...

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Mechanical Engineering

Ingegneria Meccanica

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KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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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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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

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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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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...

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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

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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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