Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 60

Fig. 2.1 Forces on the wire
of a strand

U i Q i

Q i sin

ʱ i

S i F i

Q i S i

U i F i F i cos ʱ i

strand-
axis

ʱ i

ʱ i

outer forces on the single wire in the wire layer i of a strand. The division of the
strand tensile force and torque in the wire forces S i and U i will be described later
on. For the present, S i and U i will be presupposed as known.
Both outer forces S i and U i on a wire must be in balance with the inner forces,
the wire tensile force F i and the wire shear force Q i . The forces on a wire of a wire
layer i are shown in Fig. 2.1 . From these, using r i for the lay angle, both of the
following equations can be derived

F i ¼ S i Q i sin a i
cos a i

ð 2 : 1 Þ

and

U i ¼ F i sin a i Q i cos a i : ð 2 : 2 Þ

The shear force Q i of a wire of layer i is caused by the bending and torsion of
this wire, of course geometrically limited by the rope extension. As was first
presented by Berg ( 1907 ), the shear force of a wire in layer i is

Q i ¼ sin a i
r i

: M b ; i cos a i M tor ; i sin a i ð 2 : 3 Þ

with the wire winding radius r W,i = r i , the bending moment M b,i around the bi-
normal and the torque M tor,i around the wire axis. With this the tensile force in a
wire of the wire layer i is

sin 2 a i
r i cos a i

:

S i
cos a i

F i ¼ M b ; i cos a i M tor ; i sin a i ð 2 : 4 Þ

According to Berg ( 1907 ), the portion of the strand torsion moment caused by a
wire of the wire layer i is