Page 337 - Civil Engineering Formulas

268 CHAPTER TEN

The distance from the low point C to the left support is
H q o
a cosh 1 f L 1 (10.65)
q o H
where f vertical distance from C to L, ft (m). The distance from C to the right
L
support R is
H q o
b cosh 1 f R 1 (10.66)
q o H
where f vertical distance from C to R.
R
Given the sags of a catenary f and f under a distributed vertical load q , the
L
o
R
horizontal component of cable tension H may be computed from
q o l q o f L q o f R
cosh 1 1 cosh 1 1 (10.67)
H H H
where l span, or horizontal distance between supports L and R a b. This
equation usually is solved by trial. A first estimate of H for substitution in the
right-hand side of the equation may be obtained by approximating the catenary by
a parabola. Vertical components of the reactions at the supports can be com-
puted from

q o a q o b
R L H sinh R R H sinh (10.68)
H H

Parabola
Uniform vertical live loads and uniform vertical dead loads other than cable
weight generally may be treated as distributed uniformly over the horizontal
projection of the cable. Under such loadings, a cable takes the shape of a
parabola.
Take the origin of coordinates at the low point C (Fig. 10.3). If w is the
o
load per foot (per meter) horizontally, the parabolic equation for the cable
slope is
w o x 2
y (10.69)
2H

The distance from the low point C to the left support L is
l Hh
a (10.70)
2 w o l