Basic data on mechanical calculation of wires

Basic equations of stresses in the wires’ materials, their sag and span lengths

The wire is attached to supports at points A and B. Let the height of attachment be the same.

The wire is sagging due to its own weight and acquires the form of catenary. 

The tension at each point of the wire is defined by its stretching and is directed to this point tangentially.

The equation for catenary:

(1)

– half of the wire length in the span;

– distance from the lowest point of the wire to «» axis;

– running coordinate.

 

«» axis is the axis of symmentry. The position of «» axis is defined in the following way: let us assume, that there is an ideal block at point B and that at this point B  the wire is thrown over the block and hangs freely/ The length of the sagging wire is selected in such a way so that its weight would balance the stress in the wire (the length is – «»).

(2)

thus

(3)

When we know the stress, it is possible to sort out the tension.

то есть (we divided (3 ) by ).

Thus, when we know the tension at the lowest point of the wire () we are able to sort out any tension at any point of the wire.the difference between.

(the difference between and is insignificant, especially for even terrain).

Let us define the sag of the wire.

Let us write down the value for form (2) equation:

, or The expression for defining the sag is defined by expanding into series the first equation of (1) system. We can insert the value for right away.

, but on the other hand:

Let us equate the right parts of these equations and insert the value (for point «В»)); we get:

We can define the sag.

To define it, it in enough to have the right member of equation

Let us the define the wire length in the span by expanding into series the second equation of (1) system:

(; )

Thus,- wire length in the span.