Alain Cardou's blog

I have been asked to give more details on Dr Cloutier's contribution to the field of overhead electrical conductors (power transmission lines) mechanics. I just received the attached document, written by one close colleague of his, Dr. Claude Hardy, who retired from IREQ, Hydro-Quebec research branch. A nice tribute to Louis Cloutier's legacy.

My friend and former colleague, Dr. Louis Cloutier, passed away on February 26. A long time Hydro-Quebec engineer, among other achievements, he will be remembered for his major contribution to the solution of electric power transmission mechanical problems, most notably to Aeolian vibration control, and related conductor fatigue.

A new paper by Chen et al.: Experimental research on bending performance of structural cable. In: Construction and Building Materials, 15 October 2015, Vol. 96, pp. 279-288. Equivalent bending stiffness has been obtained applying a force at midspan of simply supported, about one meter long, specimens. A number of tensile force levels have been applied (including zero). Non-linear force-deflection curves are shown.

Single strand cable bending, stick-slip, analytical models require that a choice be made between two contact modes between adjacent wires: either radial (between layers) or lateral (between same layer wires). In most recent models (e.g. Papailiou’s) radial contact is selected. A “lateral contact” model has been presented by Panetti in 1944 and can be found in the proceedings of the Turin Royal Academy of Science. A translated version from Italian is proposed in the attached file.

Some 2014 papers on the mechanics of helical strands: cables and electrical conductors

A 2014 paper on cable FEA modelling

One approach to helical strand (cable or overhead electrical conductor) bending analysis puts the emphasis on friction rather than on elastic curved rod behavior. Based on Coulomb’s laws of friction, it leads to stick-slip models where strand  bending stiffness varies with imposed curvature (Papailiou, 1995). Such an approach can be traced back to a Ph.D. thesis by H. Ernst, in 1933. This work is in fact often referred to in cable analysis reports. An English translation (with a short presentation) of the dissertation analytical part is attached.

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.

Single strand cable (spiral) bending and OEC (Overhead Electrical Conductor) bending are somewhat similar problems. This is the reason why the following new paper is noteworthy within the context of this blog. It emanates from a Slovak team: S. Kmet, E. Stanova, G. Fedorko, M. Fabian, J. Brodniansky. Title : “Experimental investigation and finite element analysis of a four-layered spiral strand bent over a curved support”. Published in “Engineering Structures”, Vol. 57, December 2013, pp. 475-483.

I just discovered (thanks to Dr K.O. Papailiou) this interesting 2007 paper on helical strand mechanical modelling. It is an up-to-date (and apparently quite exhaustive, except for non-English language contributions) review on the subject by T. Manvel Raj and N.S. Parthasarathy, of India. Its title: “A Complete Review on Friction Models of Composite Cables”. It was published in a Russian Academy of Applied Mechanics and Sciences journal, the International Journal of Mechanics of Composite Materials and Constructions, Vol. 13, No 3, pp. 356-384. Here is the Abstract:

Stick-slip model (based exclusively on Coulomb’s laws of friction) for conductor bending may be improved by taking into account actual inter-layer contact conditions: elliptical contact areas with tangential elasticity and micro-slip regions. Starting with Papailiou’s model, this has been done by Paradis and Légeron. Work was presented in 2011 at the 9th International Symposium on Cable Dynamics, held in Shanghai (China).  Model has been applied to variable curvature case using FEA (in a Matlab environment). Numerical results are compared with Papailiou’s experimental  results.

Bending of cables and, for that matter, of any helical strand system (such as overhead electrical conductors) is a challenging solid mechanics problem, with geometry and multibody frictional contact aspects. As seen in several recent papers, one workable approach is based on Coulomb’s laws of friction.  An earlier contribution by Lehanneur (1949), published in French, has generally been  ignored. The attached report presents a translation into English (including a presentation) of this most interesting work.

Effect of tornadoes on electrical power lines has been discussed by Sébastien Langlois in his M.Eng. thesis : “Design of Overhead Transmission Lines subject to Localized High Intensity Wind”. McGill University, Montreal, QC, Canada, 2007 (available on-line on the “Theses Canada” portal). Work presents various national and international standards (ASCE, Cigré, IEC etc.), historical cases, as well as models for numerical simulation.

An interesting presentation of the state of the question by  Dr Louis Cloutier, in 2008.

In the review paper by Spak & al. (January 2013) on cable bending mechanical models, one class of models was not specifically mentioned, in which authors attempt to take into account the tangential elasticity at points of contact between cable layers. There are at least two noteworthy contributions:

Cable  and Overhead Electrical Conductor mechanical models are closely related. This can be seen from a recent review (January 2013) by Spak, K., Agnes, G. and Inman, D. : Cable Modeling and Internal Damping Developments, in the ASME journal Applied Mechanics Reviews, Vol. 65. The 18 pages paper, with its 86 references, has an interesting section on bending stiffness, with a rather (comparatively) extensive presentation of Papailiou’s stick-slip model. Use of the Coulomb spring-slider model for internal damping prediction is also presented.

In elementary mechanics textbooks, power lines are generally given as a typical example of a so-called “flexible cable”. However, in the design of these critical elements of a power grid  it is often necessary to take into account the conductor bending stiffness, or flexural rigidity. While relatively low compared with a bar having an equivalent solid cross-section, it does have some influence on a conductor response to wind excitation, in particular from the fatigue strength point of view.