# Helical Strand in Bending

Several approaches have been used to deal with the complex problem of helical strands in bending. One of them is based on Love’s curved rods theory. It has first been used by Costello in his classical monograph “Theory of Wire Rope”. It has been extended by Sathikh et al. in several papers. A new contribution from this group is a Ph.D. Thesis by D. Gopinath : “Some studies on bending response of the stranded cable under free bending and constrained bending”, Anna University, Chennai (India), October 2013, 121 pages. On line at http://ir.inflibnet.ac.in:8080/jspui/handle/10603/17169

The main difficulty with this approach is that each wire in the strand yields a set of 6 PDE, coupling the 6 internal forces on a wire cross-section (3 forces and 3 moment components). Hypotheses have to be made on the wire-wire contact conditions. Gopinath follows Costello and Sathikh by assuming no-slip contact (what he calls pre-slip conditions). With this hypothesis, using linearized kinematical conditions, the reverse problem is tractable, and the author obtains a 3x3 stiffness matrix for a given strand, the 3 parameters being the strand extension, torsion and radius of curvature. Letting the first two vanish, one gets a curvature-bending moment linear relationship.

Application is made to a 1x19 steel cable (19 steel wires consisting in 2 layers of 6 and 12 wires with a single core wire). Variation of moment vs radius of curvature is shown in Fig. 5.3. For easier interpretation, it should be redrawn using curvature instead of radius of curvature. Curve is practically linear for small curvatures (large radii). It bends downwards for larger curvatures. The linear part (which has to go through the origin) practically coincides with the stick-slip models straight line (pre-slip regime), corresponding to the strand  maximum bending stiffness (Papailiou, 1995). Deviation between these models occur at high curvature, where wire torsion becomes non-negligible. However, extending the curve into the high curvature domain is questionnable as inter-wire slip occurs.

For example, using the following typical values: strand load strength: 130 kN. Applied axial load: 20% of that strength. Steel-steel coefficient of friction: 0.15. A stick-slip model predicts incipient slip on the outer layer at radius of curvature of about 42 m. Below this value, the stick hypothesis does not hold. Interestingly, wire torsion tends to bring a decrease in strand bending stiffness (ratio moment/curvature). This is confirmed by the reported FEA analysis results.

A note about terminology. Here, free bending means that strand is bent without contact with a pulley or roller. However, since radius of curvature is assumed to be uniform, this is not what is usually considered as free bending. In fact, wherever strand bending is free, curvature varies from point to point. Author assumes that, in practice, such variation is small, for example within the span of a hanging electrical conductor (between two towers). Hypothesis does not hold, however, in a suspension clamp vicinity.