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

Consider a cable fixed at two supports subjected to uniform loading. Determine the maximum tension in the cable.
For analysis, consider the origin at the cable&#39;s center due to its symmetry.
Recall the cable&#39;s shape equation for distributed load and substitute the known constant loading value.
Next, integrate the equation, and by applying the boundary conditions at the origin, the constant C2 is determined.
By taking the first derivative of the cable&#39;s shape equation, the slope can be determined. Applying the boundary conditions for the slope at the origin, C1 is obtained.
By substituting the integration constants and position coordinates of the support and rearranging the equation, the horizontal tensile force is obtained.
Recall the slope equation, and substitute the x-coordinate value at the support where the angle is maximum.
Cabel&#39;s tension changes with the angle, which is the maximum near support.
Applying the trigonometric relation and rearranging the terms, the maximum tension equation is obtained.
Finally, by substituting the horizontal tension equation and known values, the maximum tension in the cable is obtained.&#160;

cable: problem solving

When dealing with a cable that is fixed to two supports and subjected to uniform loading, it is crucial to determine the maximum tension in the cable. This process can be broken down into several key steps, as outlined below:
<img alt="Cable static equilibrium diagram with forces, w=5kN/m, L=10m, h=1m; load analysis." src="/files/ftp_upload/14310/14310_Figure_1v3.svg" style="text-align: center; height: 300px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium, cable tension diagram, w=5 kN/m, L=10 m, h=1 m, structural analysis.">
Analyze the problem: Begin by understanding the given scenario and the conditions of the cable. Identify the supports, the type of loading, and any other relevant information.
Determine the cable's shape equation: Use the principles of equilibrium and the cable's properties to establish the shape equation that describes the cable's curve. This equation relates the cable's shape to the applied load.
<img alt="Mathematical integration formula, y=(1/Fh)∫(∫w₀dx)dx, used in analytical research." src="/files/ftp_upload/14310/14310_Equation_1v2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Integral equation diagram, depicting nested integrals in mathematical analysis.">
Integrate the equation: Integrate the shape equation to obtain a function that represents the shape of the cable. This integration process allows you to determine the constants in the equation.  By applying the boundary conditions at the origin, the value of one of the integration constants can be determined.
<img alt="Static equilibrium formula, y=(1/F_H)(wx²/2+C₁x+C₂), mathematical diagram for education." src="/files/ftp_upload/14310/14310_Equation_2v2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Structural analysis equation: \( y = \frac{1}{F_H} \left( \frac{wx^2}{2} + C_1x + C_2 \right) \).">
Find the slope: Take the first derivative of the cable's shape equation to determine the slope of the cable at any given point. Apply the boundary conditions for the slope at the origin to obtain the value of another integration constant.
Calculate the horizontal tensile force: By substituting the integration constants and the position coordinates of the support into the shape equation. Rearrange the terms to find the horizontal tensile force acting on the cable.
<img alt="Static equilibrium equation, \( F_H = \frac{w_0 L^2}{8h} \); mechanical force formula diagram." src="/files/ftp_upload/14310/14310_Equation_3v2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation \( F_H = \frac{w_0L^2}{8h} \), formula diagram for force calculation.">
Determine the angle: Use the slope equation to calculate the angle of the cable at various points. Find the location along the cable where the angle is at its maximum, usually near the supports. Utilize trigonometric relationships to express the maximum tension in terms of the horizontal tensile force and the angle of the cable.
<img alt="Angle equilibrium equation, θmax=tan⁻¹(w₀L/2FH), static equilibrium analysis diagram." src="/files/ftp_upload/14310/14310_Equation_4v2.svg" style="text-align: center; height: 35px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium formula: θmax=tan^(-1)(w₀L/2Fₕ), mathematical equation representation.">
Find the maximum tension: Substitute the horizontal tension equation and the known values into the maximum tension equation. This will allow you to calculate the maximum tension in the cable.
<img alt="Maximum Shear Tension Equation, T_max: w0L/2 sqrt(1 + (L/4h)^2), formula representation." src="/files/ftp_upload/14310/14310_Equation_5v2.svg" style="text-align: center; height: 40px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation, \(T_{\text{max}} = \frac{w_0 L}{2} \sqrt{1 + \left(\frac{L}{4h}\right)^2}\).">

Cable: Problem Solving

When dealing with a cable that is fixed to two supports and subjected to uniform loading, it is crucial to determine the maximum tension in the cable. ...

Education

JoVE Core

Mechanical Engineering

Unit 1

Internal Forces

KEY TERMS AND CONCEPTS

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

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

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

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 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

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Cable Tension Under Uniform Loading

Description

Transcript

Tags