Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 74

Fig. 2.7 Moving trihedral
(tangent, main normal and
binormal) of a space curve

main normal

normal plane

osculating plane

tangent

binormal

tangent plane

With this, for the space curve for a wire in a straight stranded wire rope, the
equations of Andorfer ( 1983 ) are in parameter form

x ¼ r S sin u S
r W cos u 0 þ m u S ½ ð Þ sin u S þ sin u 0 þ m u S ð Þ cos b cos u S

y ¼ r S cos u S
þ r W cos u 0 þ m u S

ð 2 : 40 Þ

½ ð Þ cos u S sin u 0 þ m u S ð Þ cos b sin u S

z ¼ h S

2 p u S r W sin u 0 þ m u S

ð Þ sin b :

The Eq. ( 2.37 ) for the bending stress can only used for the strand center wires
of stranded ropes. For the lay wires in the strands this simple equation is not valid,
because the curvature plane turns around the wire axis against the wire. Deter-
minant for the change of bending stress is therefore not only the change of the
curvature radius q but also the turning angle c k so that the maximum bending stress
occurs in another fibre of the wire. Leider ( 1977 ) presented firstly this fact in case
of bending a strand. Schiffner ( 1986 )—respecting this—calculated the wire
bending and torsion stresses by changing the space curve in a stranded rope under
the action of the wire rope tensile force. Depending on the small rope elongation
and diameter reduction under rope tensile forces these stresses are also small
(Fig. 2.7 ).
The effect of the turning angle c k on the wire bending stress can be demon-
strated for the case when a strand is bent over a sheave. For the straight strand the
curvature radius of a lay wire is q 0 = r W /sin 2 a. For the bent strand Wiek ( 1981 )
and with a small correction Leider ( 1977 ) have derived the curvature radius q of
lay wires for the different position of the wire element in relation to the sheave
axis. As turning angle Leider ( 1977 ) has used the angle between the main normals
but Schiffner ( 1986 ) found, that the angle between the osculating plane before
changing and the main normal after changing is correct for the turning angle