## 49 Simple Suspension Cables

The objective of this and the following article is to present general procedures for analyzing simple cable suspension systems. The numerous types of cable systems available make it impractical to treat anything but the simplest types. Additional information may be found in Sec. 15, which covers suspension bridges and cable-stayed structures.

Characteristics of Cables. A suspension cable is a linear structural member that adjusts its shape to carry loads. The primary assumptions in the analysis of cable systems are that the cables carry only tension and that the tension stresses are distributed uniformly over the cross section. Thus no bending moments can be resisted by the cables.

For a cable subjected to gravity loads, the equilibrium positions of all points on the cable may be completely defined, provided the positions of any three points on the cable are known. These points may be the locations of the cable supports and one other point, usually the position of a concentrated load or the point of maximum sag. For gravity loads, the shape of a cable follows the shape of the moment diagram that would result if the same loads were applied to a simple beam. The maximum sag occurs at the point of maximum moment and zero shear for the simple beam.

The tensile force in a cable is tangent to the cable curve and may be described by horizontal and vertical components. When the cable is loaded only with gravity loads, the horizontal component at every point along the cable remains constant. The maximum cable force will occur where the maximum vertical component occurs, usually at one of the supports, while the minimum cable force will occur at the point of maximum sag.

Since the geometry of a cable changes with the application of load, the common approaches to structural analysis, which are based on small-deflection theories, will not be valid, nor will superposition be valid for cable systems. In addition, the forces in a cable will change as the cable elongates under load, as a result of which equations of equilibrium are nonlinear. A common approximation is to use the linear portion of the exact equilibrium equations as a first trial and to converge on the correct solution with successive approximations.

A cable must satisfy the second-order linear differential equation

where H = horizontal force in cable y = rise of cable at distance x from low point (Fig. 4.14) q = gravity load per unit span

### 4.9.1 Catenary

Weight of a cable of constant cross section represents a vertical loading that is uniformly distributed along the length of cable. Under such a loading, a cable takes the shape of a catenary.

To determine the stresses in and deformations of a catenary, the origin of coordinates is taken at the low point C, and distance s is measured along the cable from C (Fig. 4.14). With qo as the load per unit length of cable, Eq. (4.91) becomes qo ds

where y' = dy/dx. Solving for y' gives the slope at any point of the cable:

A second integration then yields y = H h H -1) = Ho + (fS+■•• (4.94)

Equation (4.94) is the catenary equation. If only the first term of the series expansion is used, the cable equation represents a parabola. Because the parabolic equation usually is easier to handle, a catenary often is approximated by a parabola. For a catenary, length of arc measured from the low point is ## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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