Engineering Mechanics

Chapter 15 : Equilibrium of Strings „„„„„ 331

15.7. LENGTH OF A STRING

It means the actual length of a string or cable required between two supports, when it is loaded
when it is loaded with a uniformly distributed load and hangs in the form of a parabola. Here we shall
discuss the following two cases :

1. When the supports are at the same level, and


2. When the supports are at different levels.

15.8.LENGTH OF A STRING WHEN THE SUPPORTS ARE AT THE SAME LEVEL

Consider a string ACB supported at A and B at the same level, and carrying a uniformly distrib-
uted load as shown in Fig 15.9. Let C be the lowest point of the cable.

Fig. 15.9.
Let w= Uniformly distributed load per unit length of the span
l= Span of the cable, and
yc= Central dip of the cable
We have already discussed in Art. 15.4 that the cable hangs in the form of a parabola,
and the equation of the parabola is given by the relation,

2

2

wx

y

H

=

Differentiating this equation with respect to x,

2

2

dy wx wx

dx H H

== ...(i)

Now consider a small portion PQ of length ds of the string
as shown in Fig. 15.10. Taking the length of the are PQ equal to the
length of the chord PQ, we find that

(^) ds=+dx^22 dy
2
1
dy
dx
dx
⎛⎞
=+⎜⎟
⎝⎠
Substituting the value of
dy
dx
from equation (i) in the above equation,
2
1
wx
ds dx
H
=+⎛⎞
⎜⎟
⎝⎠
Fig. 15.10.