7.11 : Cable Subjected to Concentrated Loads

Flexible cables are commonly used in various applications for support and load transmission. Consider a cable fixed at two points and subjected to multiple vertically concentrated loads. Determine the shape of the cable and the tension in each portion of the cable, given the horizontal distances between the loads and supports.

For the analysis, the cable is assumed to have the following properties:

1. Flexible, allowing it to change shape under the influence of applied loads.
2. Inextensible, meaning its length does not change under tension.
3. A negligible weight that implies that the cable's self-weight does not significantly impact its behavior.

The cable consists of several straight-line segments, with each segment subjected to a constant tensile force. In order to determine the reaction forces at the supports, a free-body diagram of the cable can be drawn. However, in the given case, the number of unknown reaction components typically exceeds the number of available equilibrium equations. An additional equation is required. Consider point D on the cable at a known distance from the supports to obtain an additional equation. By drawing a free-body diagram of segment AD and using the moment equilibrium equation at point D, another equation that helps solve the system can be derived.

The vertical distance from support A to each concentrated load can be determined with the known reaction forces. This can be achieved by recalling the equilibrium equation, which states that the sum of the vertical forces acting on the cable must equal zero. Once the vertical distances have been determined, the tension in each cable segment can be calculated. The tension is at its maximum when the segment has the largest inclination angle.