Bending of helically twisted cables under variable ... - Pfisterer

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The above equation is also applied for wires in a cable configuration that are subject to bending
[Isaachsen, 1907, Ernst, 1933, Wyss, 1956, Schiffner, 1986, Wang, 1990]. The main prerequisites are
that the cross-sections of the individual wires remain plane during bending and that the material of the
wire is linear elastic, following Hooke’s law. The first had already been experimentally observed earlier
[Czitary, 1962]. The second assumption was met by suitable selection of the external loads.
Bending stiffness
It is known that the moment in the specific support cross-section is proportional to the curvature of the
cross-sectional axis at that position. The proportionality factor is (EJ), also called bending stiffness or
just stiffness, as opposed to elongation stiffness. The stiffness is easy to calculate for a homogeneous
body with known Young’s modulus, which also yields the distortion as a result of external loads. The
conditions in a cable are different since in this case the individual wires are not permanently fixed in
position but, depending on the load, may change position relative to each other.
To determine the stiffness of a cable, it is therefore necessary to know the internal state of displacement
in the cable, as already described and quantified in Section 2.3.
Wire and secondary stiffness
As is known from the strength of materials, the bending curve is determined by considering the balance
of moments in each cross-section of the considered support, by equating the external stress moment M
to the internal moment, calculated from the normal stresses acting on each position of the considered
cross-section. In case of a “component” cross-section as in a helically twisted cable, the external
moment M is for this purpose distributed over the individual cross-sectional components of the cable,
i.e. the individual layer wires (Md,L) and the core wire (MK), Fig. 2.12.