Engineering Mechanics

Chapter 15 : Equilibrium of Strings „„„„„ 323

Fig. 15.1.
Let w= Uniformly distributed load per unit length,
l= Span of the cable, and
yc= Central dip of the cable.
Now consider any point (P) on the string. Let the coordinates of this point be x and y with
respect to C, the lowest point of the string as origin. Now draw the tangent at P. Let θ be the inclina-
tion of the tangent with the horizontal as shown in the figure.

We know that the portion CP of the string is in equilibrium under the action of the following
forces :

1. Load (w.x) acting vertically downwards,


2. Horizontal pull (H) acting horizontally at C, and


3. Tension (T) acting at P along the tangent.
   Resolving the forces vertically and horizontally,
   T sin θ= w x ...(i)
   and T cos θ = H ...(ii)
   Dividing equation (i) by equation (ii),

tan

wx

H

θ= ...(iii)

We know that tan

dy

dx

θ=

∴

dy w x

dx H

=

or.

wx

dy dx

H

=

Integrating the above equation,

2

2

wx

yK

H

=+

where K is the constant of integration. We know that at point C, x = 0 and y = 0. Therefore K = 0.

or

2

2

wx

y

H

=
This is the equation of a parabola. It is thus obvious, that the shape of a string, carrying a
uniformly distributed load over its horizontal span, is also a parabola.

15.3.TENSION IN A STRING

The determination of tension in the string or cable is one of the important criterion for its
design. As a matter of fact, the tension in a string depends upon the magnitude and type of loading as