Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 63

S i ¼ F i cos a i :

The strand tensile force is the sum of all wire tensile force components

S ¼ X

n

z i S i ¼ X

n

z i F i cos a i ð 2 : 13 Þ

i¼0 i¼0

In addition to the known symbols, n is the number of wire layers counted from
the inside with n = 0 for the centre wire and z i is the number of wires in the wire
layer i.
For the following, it will be presupposed that the strand cross-section rests
plane if the strand with the length l S is elongated with Dl S by a tensile force. The
elongation can now be calculated and, from this, the tensile force of all the wires.
The tensile force of a wire in a wire layer i is

F i ¼ Dl i
l i

E i A i : ð 2 : 14 Þ

l i is the wire length, Dl i the wire elongation, E i the elasticity module and A i the
cross-section of a wire in the wire layer i. The extension of that wire is

e i ¼ Dl i
l i

: ð 2 : 15 Þ

With l S for the length of the strand, the length of the wire is

l S
cos a i :

l i ¼ ð 2 : 16 Þ

In Fig. 2.3 , the unwound wire about the strand axis is shown before and after
the strand elongation. Therefore, when the failures of higher classification are
neglected, then the wire elongation is

Dl i ¼ Dl S Du i tan a i ð Þ cos a i

or

Dl i ¼ Dl S cos a i Du i sin a i : ð 2 : 17 Þ

The contraction of the winding radius respectively the circumference in relation
to the wire extension—that transverse contraction ratio can also be designated as
''Poisson's ratio'' of the wire helix—is