Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 70

Fig. 2.5 Wire space curve in
a straight spiral rope

z

I

x

r

ʱ

r

˕

tan ʱ

˕

y

Although the moments M b and M tor out of ( 2.6 ) and ( 2.7 ) can be neglected in
calculating the wire tensile force, the bending and torsion stresses resulting from
these moments can be considerable. The stresses come from the change of the
curvature K and the winding T.
The curvature K of a space curve is in parameter form according to ( 2.32 ) with
the curvature radius q

K ¼ 1
q

s

x 0 2 þ y 0 2 þ z 0 2

ð 2 : 33 Þ

Þ 2

ð Þ ð x 00 2 þ y 00 2 þ z 00 2 Þ x 0 ð x 00 þ y 0 y 00 þ z 0 z 00

K ¼ :

Þ 3

ð

x 0 2 þ y 0 2 þ z 0 2

The winding T shows how strongly the space curve differs from the osculating
plane in the neighbourhood of a point. The winding is

x 0 y 0 z 0

x 00 y 00 z 00

x 000 y 000 z 000

T ¼ q 2

Þ 3 :

ð 2 : 34 Þ

ð x 0 2 þ y 0 2 þ z 0 2

For the simple case of a wire in a straight strand or spiral rope with the wire
winding radius r, the curvature radius q is

r
sin 2 a

q ¼ ð 2 : 35 Þ

and the winding

T ¼ sin a cos a
r

: ð 2 : 36 Þ