Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 188

Fig. 3.7 Secondary tensile
stress in a half wire winding,
uniformly bent strand

+

zi zi zsi

ˀ

ˀ 0 /2

winding angle ˕

the strand elements can therefore occur over some lay lengths. Schmidt ( 1965 ) found
that these displacements cannot be calculated using a single equation. Leider ( 1973 ,
1975 ) arrived at the same result. Therefore he calculated the secondary tensile stress
using an approximation method which follows the rope elements that move step by
step over the sheave. His method has the simplification that the rope running over the
sheave remains straight right up to its contact point with the sheave and that the wires
and strands in the rope piece lying on the sheave cannot be displaced.
Using a comparable method but without the two simplifications, Schiffner ( 1986 )
calculated the secondary tensile stresses in the wires of a stranded rope. The rope as
a chain of rope elements runs step by step over the sheave whereby the step width
meets the length of the rope element. The calculation starts with as realistic a
bending line of the free rope as possible. The different tensile stresses in the rope's
cross-section produce a bending moment. With this bending moment and the
moment from the outer tensile rope force, a new bending line is calculated for the
free rope. Using this new bending line, the calculation is repeated until the bending
line calculated for the rope coincides with last bending line of the free rope.
In Fig. 3.8 , the secondary tensile stress has been drawn for the wire rope moving
over the sheave, together with the bending stress in an outer wire (u S = 0; u W = 0).
The calculated stress and the stress course correspond to the one which had been
measured by Schmidt ( 1965 ), Wiek ( 1973 ) and Mancini and Rossetti ( 1973 ).
With his method, Schiffner ( 1986 ) is not only able to calculate the secondary
tensile stress but also the bending line of the free rope both before and after the rope is
bent over a sheave. Figure 3.9 shows such bending lines of a rope before and after
moving over a sheave with increased scale in the cross direction. As is to be seen in
this figure, the wire rope is not only deformed in the sheave plane but also perpen-
dicular to that. The deflection difference of the rope ends in the sheave plane for the
rope running on or off is a criterion for the necessary bending work and for the
bending stiffness caused by the friction, Schmidt ( 1965 ), Schraft ( 1997 ).
One effect of the secondary tensile stress is, for example, that cross lay ropes
6 9 37-FC with thin wires have lower endurance than wire ropes 6 9 19-FC with
thicker wires even though their bending stress is 40 % higher, Woernle ( 1929 ). In
fibre ropes running over sheaves there is practically no bending stress in the fibres.
The endurance of these ropes mainly depends on the secondary tensile stress,
Feyrer and Vogel ( 1992 ) and Wehking ( 1997 ).