7.14:
Cable: Problem Solving
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's center due to its symmetry.
Recall the cable'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'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'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.
7.14:
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:
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.
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.
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.
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.
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.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 384-385, 393.