Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 75

A a þ B b þ C c
A 2 þ B 2 þ C 2

c k ¼ arcsin p ð 2 : 41 Þ

ð Þ ð a 2 þ b 2 þ c 2 Þ

with

A ¼ y 0 0 z 0 0 z 0 0 y 0 0

B ¼ z 0 0 x 0 0 x 0 0 z 00

C ¼ x 0 0

y 0 0 y 0 0 x 0 0

and

a ¼ y 0 x 0 y 00 y 0 x 00 Þ z 0 z 0 x 00 x 0 z 00 ð ð Þ

b ¼ z 0 y 0 z 00 z 0 y 00 Þ x 0 x 0 y 00 y 0 x 00 ð ð Þ

c ¼ x 0 z 0

x 00 x 0 z 00 Þ y 0

y 0

z 00 z 0 y 00

ð ð Þ:

The equation for the osculating plane is

A ð X x 0 Þ B ð Y y 0 Þ C ð Z z 0 Þ 0

and for the main normal

X x
a

¼ Y y
b

¼ Z z
c

:

X, Y and Z are the coordinates of the centre of the moving trihedral for the space
curve for which the bending stress is considered. The parameter equations x 0 , y 0
and z 0 present the space curve before changing, and x, y and z afterwards.
The maximum change of bending stress resulting from the space curve change

is

:

r b ¼ d

1
q cos w max c k

Þ 1
q 0 2 E

ð cos w max ð 2 : 42 Þ

The turning angle w max for the virtual fibre with the maximum stress change is
determined by

0
@

1
A :

sin c k
cos c k q
q 0

w max ¼ arctan ð 2 : 43 Þ

Following Schiffner ( 1986 ), the calculation of the torsion stress has to be
adjusted on the space curve with the winding