Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 187

For a rope with a constant lay angle of the strands, the displacement angle for
u = 90 is D # = 0.0023 rad or D0 = 0.133
Length of the displacement bow for u = 90 is DL/d = 0.0290

3.1.2.2 Simplified Calculation of the Secondary Tensile Force

When a wire rope is bent, a wire tensile stress will be caused by the friction
between the wires and the strands. This stress is called the secondary tensile stress
and it increases or reduces the normal tensile stress of the wires differently in
different parts of the rope. The sum of all wire tensile forces in a rope cross-section
remains unchanged and results in the wire rope tensile force. Like the bending
stress and the ovalisation stress in the sheave groove, the secondary tensile force is
a fluctuating stress which reduces the endurance of a wire rope.
Already Isaachsen ( 1907 ) presented a first equation to calculate the secondary
tensile stress. Benoit ( 1915 ) and Ernst ( 1933 ) improved it. Schmidt ( 1965 ) eval-
uated a correct equation for the secondary stress in the wire of a uniformly bent
strand. For reasons of symmetry, he noticed the wire can only be displaced
between the inner and the outer point of the rope bow, as shown in Fig. 3.6 . In the
case considered with uniformly bent strands, according to Schmidt ( 1965 ) the
secondary tensile stress in an outside wire is

r ts ¼ r t ð e l sin a ð u 0 u Þ 1 Þ:

ð 3 : 11 Þ

In this equation, r t is the normal tensile stress, l is the friction coefficient, a is
the wire lay angle and u is the wire winding angle. u 0 is the winding angle for that
the secondary tensile force is zero. This angle is a little greater than p/2, Fig. 3.7 .
On the inner layer wires in parallel lay strands, friction forces work in opposite
directions, thus resulting in only a relatively small secondary tensile force,
Schmidt ( 1965 ) and Leider ( 1974 ). In strandswith crossing wire layers, however,
the secondary tensile stress increases for the inner wires from layer to layer. In two
wire layer strands, the secondary tensile stress of the inner layer wires is about
three times and in three wire layer strands—with the same lay angle in an alter-
nating direction and the same wire diameter—five times that found in the outer
wires, Ernst ( 1933 ), Schmidt ( 1965 ) and Leider ( 1974 ). Cross lay ropes only have
a relatively low bending endurance, both for this reason and because of the
pressure in the crossing points.

3.1.2.3 Step-by-Step Calculation of the Secondary Tensile Stress

Equation ( 3.11 ) for the calculation of the secondary tensile stress presupposes that
the rope will be uniformly bent over the whole of its length. In fact, however, the
curvature of the rope changes gradually depending on the distance to the contact
point of the sheave. Very early on, Donandt ( 1934 ) stated that the displacements of