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Calculation of sag and tension 

Dr. Mohsen niasati


Calculation of sag and tension

Supports at the same level

Figure below shows a wire AOB of length l supported at two towers A and B and are spaced L unit apart.

Let O is the lowest point of the wire. Consider a length OP of the curve length s.

If w=weight/unit length, H=tension at point O and T=tension at point P, the tension T can be resolved into horizontal and vertical components as :

Then

In triangle shown in figure below,

ds represents very small section and therefore we have :

or

and

Substituting the value of tan(θ), we get

and

Integrating both sides, we have

Where A is the integration constant. With initial condition at x=0, s=0 we find A=0. Therefore

and

At x=L/2, s=l/2, we obtain

Expanding the sinh(wL/2H) and ignoring higher order terms, we get

From above equations, we can get

Integrating both sides, we obtain

Where B is the integration constant and can be obtained with initial condition at x=0, y=o. Thus, we get B=–H/w. Therefore

It is equation of the sag that is called a catenary.

 

Source:

Electric Power Generation, Transmission and Distribution By S. N. Singh (pages 229,230,231)