## 410 Cable Suspension Systems

Single cables, such as those analyzed in Art. 4.9, have a limited usefulness when it comes to building applications. Since a cable is capable of resisting only tension, it is limited to transferring forces only along its length. The vast majority of structures require a more complex ability to transfer forces. Thus it is logical to combine cables and other load-carrying elements into systems. Cables and beams or trusses are found in combination most often in suspension bridges (see Sec. 15), while for buildings it is common to combine multiple cables into cable systems, such as three-dimensional networks or two-dimensional cable beams and trusses.

Like simple cables, cable systems behave nonlinearly. Thus accurate analysis is difficult, tedious, and time-consuming. As a result, many designers use approximate methods or preliminary designs that appear to have successfully withstood the test of time. Because of the numerous types of systems and the complexity of analysis, only general procedures will be outlined in this article, which deals with cable systems in which the loads are carried to supports only by cables.

Networks consist of two or three sets of parallel cables intersecting at an angle. The cables are fastened together at their intersections. Cable trusses consist of pairs of cables, generally in a vertical plane. One cable of each pair is concave downward, the other concave upward (Fig. 4.15). The two cables of a cable truss play different roles in carrying load. The sagging cable, whether it is the upper cable (Fig. 4.15a or b), the lower cable (Fig. 14.15d), or in both positions (Fig. 4.15c), carries the gravity load, while the rising cable resists upward load and provides damping. Both cables are initially tensioned, or prestressed, to a predetermined shape, usually parabolic. The prestress is made large enough that any compression that may be induced in a cable by superimposed loads only reduces the tension in the cable;

FIGURE 4.15 Planar cable systems. (a) Completely separated cables. (b) Cables intersecting at midspan. (c) Crossing cables. (d) Cables meeting at supports.

thus compressive stresses cannot occur. The relative vertical position of the cables is maintained by vertical spreaders or by diagonals. Diagonals in the truss plane do not appear to increase significantly the stiffness of a cable truss.

Figure 4.15 shows four different arrangements of cables with spreaders to form a cable truss. The intersecting types (Fig. 4.15b and c) usually are stiffer than the others, for given size cables and given sag and rise.

For supporting roofs, cable trusses often are placed radially at regular intervals. Around the perimeter of the roof, the horizontal component of the tension usually is resisted by a circular or elliptical compression ring. To avoid a joint with a jumble of cables at the center, the cables usually are also connected to a tension ring circumscribing the center.

Cable trusses may be analyzed as discrete or continuous systems. For a discrete system, the spreaders are treated as individual members and the cables are treated as individual members between each spreader. For a continuous system, the spreaders are replaced by a continuous diaphragm that ensures that the changes in sag and rise of cables remain equal under changes in load.

To illustrate the procedure for a cable truss treated as a continuous system, the type shown in Fig. 4.15d and again in Fig. 4.16 will be analyzed. The bottom cable will be the load-carrying cable. Both cables are prestressed and are assumed to be parabolic. The horizontal component Hiu of the initial tension in the upper cable is given. The resulting rise is fu, and the weight of cables and spreaders is taken as wc. Span is l.

The horizontal component of the prestress in the bottom cable Hib can be determined by equating the bending moment in the system at midspan to zero:

where fb = sag of lower cable wt = uniformly distributed load exerted by diaphragm on each cable when cables are parabolic

Setting the bending moment at the high point of the upper cable equal to zero yields

FIGURE 4.16 (a) Cable system with discrete spreaders replaced by an equivalent diaphragm. (b) Forces acting on the top cable. (c) Forces acting on the bottom cable.

FIGURE 4.16 (a) Cable system with discrete spreaders replaced by an equivalent diaphragm. (b) Forces acting on the top cable. (c) Forces acting on the bottom cable.

Thus the lower cable carries a uniform downward load wc + wt, while the upper cable is subjected to a distributed upward force w.

Suppose a load p uniformly distributed horizontally is now applied to the system (Fig. 4.16a). This load may be dead load or dead load plus live load. It will decrease the tension in the upper cable by AHu and the rise by Af (Fig. 4.16b). Correspondingly, the tension in the lower cable will increase by AHb and the sag by Af (Fig. 4.16c). The force exerted by the diaphragm on each cable will decrease by Aw,.

The changes in tension may be computed from Eq. (4.117). Also, application of this equation to the bending-moment equations for the midpoints of each cable and simultaneous solution of the resulting pair of equations yields the changes in sag and diaphragm force. The change in sag may be estimated from

where Au = cross-sectional area of upper cable Ab = cross-sectional area of lower cable

The decrease in uniformly distributed diaphragm force is given approximately by

And the change in load on the lower cable is nearly p - Aw H + 16AbEfb2/312)P (4123)

In Eqs. (4.121) to (4.123), the initial tensions Hiu and Hib generally are relatively small compared with the other terms and can be neglected. If then fu = fb, as is often the case, Eq. (4.122) simplifies to

The horizontal component of tension in the upper cable for load p may be computed from w - Aw

The maximum vertical component of tension in the upper cable is

The horizontal component of tension in the lower cable may be computed from w + w + p — Aw-Hb = Hlb + AHb = Wc W + W,-W Hi (4.128)

The maximum vertical component of tension in the lower cable is

In general, in analysis of cable systems, terms of second-order magnitude may be neglected, but changes in geometry should not be ignored.

Treatment of a cable truss as a discrete system may be much the same as that for a cable network considered a discrete system. For loads applied to the cables between joints, or nodes, the cable segments between nodes are assumed parabolic. The equations given in Art. 4.9 may be used to determine the forces in the segments and the forces applied at the nodes. Equilibrium equations then can be written for the forces at each joint.

These equations, however, generally are not sufficient for determination of the forces acting in the cable system. These forces also depend on the deformed shape of the network. They may be determined from equations for each joint that take into account both equilibrium and displacement conditions.

For a cable truss (Fig. 4.16a) prestressed initially into parabolic shapes, the forces in the cables and spreaders can be found from equilibrium conditions, as indicated in Fig. 4.17. With the horizontal component of the initial tension in the upper cable Hiu given, the prestress in the segment to the right of the high point of that cable (joint 1, Fig. 4.17a) is Tiu1 = HJ cos aRu1. The vertical component of this tension equals Wi1 — Wcu1, where is the force exerted by the spreader and Wcu1 is the load on joint 1 due to the weight of the upper cable. (If the cable is symmetrical about the high point, the vertical component of tension in the cable segment is (Wi1 — Wxu1)/2.] The direction cosine of the cable segment cos aRu1 is determined by the geometry of the upper cable after it is prestressed. Hence Wi1 can be computed readily when Hiu is known.

With Wi1 determined, the initial tension in the lower cable at its low point (joint 1, Fig, 4.17c) can be found from equilibrium requirements in similar fashion and by setting the

## 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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