The following summary of elementary statics of cables applies to completely flexible and inextensible cables but includes correction for elastic stretch. The formulas derive from the fundamental differential equation of a cable shape
H horizontal component of tension produced by w w distributed load, which may vary with x Two cases are treated: catenary, the shape taken by a cable when the load is uniformly distributed over its length, and parabola, the shape taken by a cable when the load is uniformly distributed over the projection of the span normal to the load.
Table 15.6 lists equations for symmetrical cable. These equations, however, may be extended to asymmetrical cables, as noted later.
The derivation of the equations considered the cable as inextensible. Actually, the tension in the cable stretches it. The stretch, in, of half the cable length may be estimated from
Properties of asymmetrical cables may be obtained by determining first the properties of their component symmetrical elements.
For a parabolic cable (Fig. 15.43), determine point C on the cable, which lies on a horizontal line through a point of attachment. The horizontal distance of C from the support
The portion of the cable between C and the lower support is symmetrical. Its ordinates, slope, length, and cable tension may be computed from the equations in Table 15.6.
For a catenary cable (Fig. 15.44), point C on a horizontal line through the lower support may be located by stepwise solution of the equation y = h cos x/h for a symmetrical catenary.
An initial solution may be obtained by use of a parabola. Substitution in the exact equation then yields more accurate values. When distances l1 and l2 of C from the supports have been determined, the ordinates, slope, length, and cable tension of the symmetrical portion of the cable may be computed.
The portion of the cable from C to the high point is an oblique cable (Fig. 15.45), a special case of the asymmetrical cable. Its properties can be obtained with the equations in Table 15.6 and Eq. (15.4) by treating the oblique cable as part of a symmetrical one.
(See also Art. 4.8.)