Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 131

X

n w

e T ¼ Dl 1
l 1

¼ 1
A 1

Dl i ; 1
l 1

A i ; 1 : ð 2 : 97g Þ

i ¼ 0

Introduced in ( 2.97f ) the component of the longitudinal force of the wire i in the
strand axis direction—induced by the rope twisting—is

X

n w

S i ; 1 ¼ A i ; 1
A 1

Dl i ; 1
l 1

A i ; 1 E Dl i ; 1
l 1

A i ; 1 E :

i¼0

and the enforced extension of the wire i in strand axis direction is

X

n w

e i ¼ 1
A 1

Dl i ; 1
l 1

A i ; 1 Dl i ; 1
l 1

:

i¼0

With this equation and the relation for the parallel lay strands

tan a i ¼ r i
r n

tan a n ¼ r i
r n

tan a

the enforced extension of the wire i in the strand axis direction is

r

1 2 r i ; 1 x 1 r i ; 1
r n ; 1

e i ¼ X

n w

A i ; 1
A1

tan a r i ; 1 x 1 1

i¼0

r

1 2 r i ; 1 x 1 r i ; 1
r n ; 1

tan a r i ; 1 x 1 þ 1 : ð 2 : 97h Þ

The longitudinal stress of a wire i according to ( 2.20 )—neglecting the strand
contraction and left out the index 1 as there is only one strand layer—is

r long ; i ¼ e i E :

In addition to that the tensile stress from an outer rope tensile force is according
to ( 2.31 )

S
A cos a i cos b :

r t ; i ¼

The both stresses can be added to a resulting tensile stress

r res ; i ¼ r long ; i þ r t ; i :

The calculation is only valid and the rope structure remains intact, if the
resulting longitudinal stresses of all wires are tensile (positive).