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

Bending of helically twisted cables
under variable bending stiffness
due to internal friction, tensile
force and cable curvature
Konstantin O. Papailiou
DISS ETH No. 11057

DISS ETH No. 11057
Bending of helically twisted cables
under variable bending stiffness
due to internal friction, tensile
force and cable curvature
DISSERTATION
for obtaining the title of
DOKTOR DER TECHNISCHEN WISSENSCHAFTEN
[DOCTOR OF TECHNICAL SCIENCES]
of the
EIDGENÖSSISCHE TECHNISCHE HOCHSCHULE ZURICH
presented by
Konstantin O. Papailiou
Dipl. El.-Ing. TH Braunschweig
Dipl. Bau-Ing. Universität Stuttgart
born on 3.7.1946
in Athens, Greece
Accepted on application by:
Prof. Dr. H.-R. Meyer-Piening, Supervisor
Prof. Dr. G. Oplatka, Co-Supervisor
Dr. V. Esslinger, Co-Supervisor
1995

For my father

Summary
The bending of helically twisted cable is a severe loading case for these important structural
components. During bending of such cables, slippage takes place between the individual wires of the
cable, which is greatly influenced by the frictional forces acting at the inter-wire contacts. Because of
this slippage, the cross-sections of the cable do not necessarily remain plane, which has to be
considered in the formulation of the bending stiffness of the cable.
The main subject of this work is to develop, based on generally accepted principles for bending of
cables, an adequate model to quantify the bending behaviour of multilayered helically twisted cables.
This model leads, under consideration of internal friction, to a variable effective bending stiffness which
depends on the cable curvature and the tensile load acting on the cable. With this information it is now
possible to calculate the deflection, curvature and the various components of stress in the wires of the
cable analytically. With the newly developed “cable scanner”, the catenary and curvature of the cable
axis can be determined by scanning the cable surface. The agreement between theory and
measurement is quite good.
The model is extended to cover also ACSR (Aluminium conductor, steel reinforced), a conductor which
is widely used in overhead transmission lines, and is validated through corresponding measurements.
An important application of this work is the calculation of dynamic bending stresses in the individual
wires of such conductors, under wind induced (aeolian) vibrations. These calculations consider the
ordinary alternating bending stresses plus the secondary tensile stresses (“zusatz” stresses) caused by
interlayer friction between wires, which together constitute the longitudinal fatigue stress. The theory
resolves a known discrepancy between measured stresses and stresses predicted by simpler
conductor models in use today, which assume a constant bending stiffness. It also settles the long
outstanding uncertainty on the proper choice of effective bending stiffness for vibrating conductors. This
opens up a new way of determining the stress limits for such conductors under vibrating conditions.

Zusammenfassung
Die Seilbiegung stellt einen wichtigen Betriebszustand von Spiralseilen dar. Während des
Biegevorganges verschieben sich die Drähte des Seiles gegenseitig, wobei die Reibung zwischen den
einzelnen Drähten eine entscheidende Rolle spielt. Durch die Biegeverformung bleiben auch die
Querschnitte für das gesamte Seil nicht notwendigerweise eben. Dies muss in der Formulierung des
Ansatzes für die Biegesteifigkeit des Seiles berücksichtigt werden.
Unter Beachtung von allgemein anerkannten Grundsätzen für die Seilbiegung, wird im Rahmen dieser
Arbeit ein Modell für mehrlagige Spiralseile vorgestellt, dass auf einer, durch die innere Reibung
veränderlichen und von der Seilkrümmung und von der Seilzugkraft abhängigen, effektiven
Biegesteifigkeit basiert. Damit werden die Seillinie, die Seilkrümmung und die verschiedenen
Komponenten der in den einzelnen Drähten des Seiles herrschenden Spannungen berechnet. Mit dem
neuentwickelten “Seiltomographen” werden durch Abtasten der Seiloberfläche, die Seillinie und die
Seilkrümmung experimentell ermittelt. Dabei ist eine recht gute Übereinstimmung zwischen Rechnung
und Messung festzustellen.
Dieses Seilmodell wird auch auf Aluminium-Stahl-Verbundseile für die Stromübertragung erweitert und
durch entsprechende Messungen bestätigt. Anschliessend wird damit, als wichtige Anwendung, die
Ermittlung der Wechselbiegespannungen, die in den einzelnen Drähten von solchen Leiterseilen bei
winderregten Schwingungen entstehen, durchgeführt und mit der heute üblichen Bemessungspraxis
verglichen. Durch die Berücksichtigung der von der inneren Reibung herrührenden Zusatzspannung,
welche zusammen mit der Biegespannung die schwellende Längsspannung in den Seildrähten ergibt,
wird eine bekannte Diskrepanz zwischen den gemessenen und den mit einfachen Seilmodellen, die auf
einer konstanten Biegesteifigkeit basieren, berechneten Wechselbiegespannungen geklärt, sowie die
Unsicherheit über den für diesen Betriebszustand gültigen Ansatz für die effektive Biegesteifigkeit des
Seiles behoben. Damit kann schliesslich ein neuer Weg zur Festlegung der zulässigen
Schwingungsbeanspruchungen für Leiterseile prinzipiell aufgezeigt werden.

Preamble
I completed this thesis in the course of my professional engagement at the Sefag AG company in
Malters/Lu. For this reason, I am grateful to Mr. K.H. Pfisterer, the main Sefag shareholder, for having
enabled me to carry out these demanding studies.
There are two sides to working on a thesis whilst professionally employed:
On the one hand, one embarks on this from a certain age, which should be accompanied by some
maturity and therefore the necessary discernment of the essential. The assertiveness required in
professional life for the pursuance of goals certainly also helps to further the task at hand. In many
cases one also has a better perspective on the practical problems, again helping with the execution of
the work.
Also the stress accompanying such work, performed mainly in spare time, is significant - both
personally and for the family in particular. Only the anticipation of achieving the target you set yourself
and the expectation, or rather the hope, of making a personal contribution to an interesting and
user-oriented engineering problem constantly encourage you to “stick with it”.
This was exactly my experience over the years in which I was engaged with this project. And for this
reason I wish to once again thank all those who helped me with the completion of the work behind this
thesis, now that it is finished.
Special thanks first of all to my “PhD supervisor”, Prof. Dr. Meyer-Piening, for his willingness to support
this work, for making the infrastructure of the Institute for Light Construction and Cableway Technology
(ILS/ETH) available, especially for carrying out the experiments, but particularly for his always
constructive criticism and valuable comment – helping me find my way out of many a dead end.
I am very grateful also to Dr. V. Esslinger of the EMPA and Prof. Dr. G. Oplatka of the ILS, who were
always available for discussions, made valuable suggestions and in this sense were always at my side
in this work, also acting as co-supervisors.
A special word of thanks goes to my friend Prof. Dr. M. Leider. Not only was I able to benefit from his
earlier work, constituting a milestone on this topic, but he was also constantly available in word and
deed, making many valuable contributions.

And another special word of thanks to my “fatherly” friend and erstwhile boss Dr. W. Bückner and my
CIGRE colleague for many years, Prof. Dr. R. Helms of the BAM [Federal Institute for Materials
Research and Testing], Berlin, who introduced me to this topic as a young engineer and fostered my
interest in working on it. Let me also include my old acquaintance Jim Poffenberger in this list who,
through his well-known equation, animated me to study the bending stiffness of conductors in depth.
Also deserving my praise are the staff of the ILS and EMPA [Swiss Federal Laboratories for Materials
Testing and Research], Dübendorf, Division for Fatigue and Service Strength, who were in charge of
the test rig and the measurements and were always ready to support me in the course of the work,
especially Messrs. Kopanakis, Roth, Siebenthaler, Vaczlavic, Bollier, Müller and Schuler of the ILS as
well as Sauter, Beisswanger and Heiniger of the EMPA. My thanks also to Dr. Klaffke of the BAM for
carrying out the friction measurements so competently.
Mr. Frank Schmidberger, who wrote the software programs for this project and also assisted me in
many aspects, both in thinking and in helping, more than deserves my gratitude.
Last but not least, allow me to express my sincere thanks to my dear wife Margarita at this point. Not
only has she always encouraged me to persevere even though family life often bore the brunt but, as a
graphic artist and publisher, she personally handled the entire graphic content – also in the course of
her other professional obligations – and coordinated the pressure of work, shouldering it as well.
Thanks, darling.
All in all, the author would also like to strongly recommend a “lifetime job” such as this even to the “older
generation”, into which group the author is by now obliged to count himself – not by dint of being the
“perennial student”, but because of a yearning for never-ending learning. It enlivens the spirit, creates
intensive contacts to many interesting people and, given a bit of luck as I appear to have had, yields
results that can be directly applied to practical engineering challenges.
Malters, October 1994

Table of Contents
Summary………….…………….…………..…………………………………………….
Preamble…………...……………………………………………………………………..
1. Introduction ...................................................................................................................................... 1
1.1. Status and problem definition.................................................................................................. 1
1.2. Developments to date ............................................................................................................. 5
1.3. Project objectives .................................................................................................................... 6
1.4. Limitations ............................................................................................................................... 7
1.5. Project structure ...................................................................................................................... 7
2. Basic principles ............................................................................................................................... 8
2.1. The tensile stress .................................................................................................................... 8
2.2. The secondary stress ............................................................................................................ 11
2.3. The bending stress ................................................................................................................ 14
2.4. The bending stiffness ............................................................................................................ 20
2.5. The cable bending moment ................................................................................................... 25
2.6. The cable state diagram ........................................................................................................ 32
2.7. The single layer helically twisted cable as a calculation example ........................................ 37
3. Multi-layer cables .......................................................................................................................... 45
3.1. Wire resting points ................................................................................................................ 45
3.2. Tensile force in the outer layer of multi-layer cables............................................................. 46
3.3. Tensile force in the inner layer of a two-layer cable ............................................................. 49
3.4. Tensile force in the inner layers of helically twisted cables with more than two layers ........ 58
3.5. The state diagram for multi-layer cables ............................................................................... 59
4. Calculation method ....................................................................................................................... 65
4.1. Bending of a cable under tensile load ................................................................................... 65
4.2. Analytical calculation of the cable catenary .......................................................................... 71
4.3. Determining the catenary with the finite element method ..................................................... 75
4.4. The SEIL program ................................................................................................................. 77

5. Measurements............................................................................................................................... 85
5.1. The cable scanner ................................................................................................................. 86
5.2. The test set-up ...................................................................................................................... 92
5.3. The test cables ...................................................................................................................... 98
5.4. The material properties ......................................................................................................... 99
5.5. The friction coefficients ....................................................................................................... 100
6. Analysis ....................................................................................................................................... 103
6.1. Test results for the steel cable ............................................................................................ 103
6.2. Test results for the ACSR conductor .................................................................................. 108
6.3. Sensitivity analysis .............................................................................................................. 110
6.4. Influence of the bending stiffness........................................................................................ 111
6.5. Effect of the sample length .................................................................................................. 114
7. Applications ................................................................................................................................. 117
7.1. Wire stresses in vibrating conductors ................................................................................. 117
7.2. Practical approach .............................................................................................................. 122
7.3. Effect of the friction corrosion and the static pre-stress ...................................................... 127
8. Prospects .................................................................................................................................... 130
Annexure I ........................................................................................................................................... 132
Annexure II .......................................................................................................................................... 135
Annexure III ......................................................................................................................................... 137
Annexure IV ........................................................................................................................................ 139
Annexure V.......................................................................................................................................... 141
References .......................................................................................................................................... 143
List of symbols .................................................................................................................................... 148
List of figures ....................................................................................................................................... 155
Resumé ............................................................................................................................................... 159

1. Introduction
1.1. Status and problem definition
1
This project deals with the bending characteristics of tensioned multi-layer helically twisted cables. The
cables under investigation were cables comprising of a straight wire core and concentric layers of
helically twisted round wires, alternately right and left lay, Fig. 1.1.
Fig. 1.1 Basic construction of a helically twisted cable
Helically twisted steel cables have many applications for conveyance, as components of a variety of
constructions.
So-called stranded conductors used in overhead lines for power transmission are a special application
of helically twisted cables. In most cases, these cables comprise of two materials, a so-called core (i.e.
the core steel wire and one or two steel wire layers), mainly for absorbing the mechanical loads acting
on the cable, plus a number of helical layers of aluminium wires to carry the current. In addition, a single
component overhead conductor made of AlMgSi alloys, the so-called Aldrey conductor, is often found in
Switzerland, with very good results under the local operating conditions.
The importance of these overhead conductors for reliable operation of the electrical transmission
network is well known. They are utilised in high voltage transmission lines worldwide, for thousands of
kilometres.

2
Their acquisition value is estimated at several billion Swiss Francs [Papailiou, 1989] and the annual cost
of maintenance to electrical transmission network operators and electricity suppliers is correspondingly
high.
Fig. 1.2 Overhead conductors on a 400 kV high voltage transmission line at Cap Sounion
There can be no doubt that overhead conductors have a significant role to play in the reliability of the
electrical grid, since they must guarantee the uninterrupted transmission of power over large distances
and, in many cases, for decades. The fact that they are subject, under certain circumstances, to wire
breakages, even conductor breakages in the worst case and thereby serious interruptions, is therefore
a cause for concern. The main reason for such breakages is wire fatigue through wind-induced
so-called aeolian vibrations (Aeolos was the ancient Greeks’ god of the winds).
This problem has been occupying high voltage transmission engineers since almost the start of the 20 th
century, when high voltage transmission networks were massively expanded [Bückner, 1988]. The
tremendous economical importance, together with the physical complexity of the problem, are clearly in
evidence by the fact that several international organisations such as CIGRE (Congrès International des
Grands Réseaux Eléctriques) [Recommendations, 1979, Final Report, 1988, Report on Aeolian
Vibration, 1989], the IEEE (Institute of Electrical and Electronic Engineers) [1966] and EPRI (Electrical
Power Research Institute) [1979] are devoting a great deal of attention to the subject to this day.

3
The economic importance lies in the fact that, especially with high voltage transmission lines, i.e.
400 kV lines in particular, the conductors are the most expensive single component and also the most
important factor determining the overall cost of the line, since the choice of conductor material, cross
sectional shape and area and the conductor tension are determining factors for the design of the rest of
the transmission line components, i.e. the fittings, insulators, towers and foundations.
There have been many unsuccessful attempts at solving this problem analytically because the
overhead conductors, even though they appear to have a “simple” construction, are quite difficult to
master mathematically, especially with respect to their mechanical behaviour [EPRI, 1979, Pg 53]. For
this reason we have resorted to measuring the vibration stress on overhead conductors live in the field,
using specially designed recorders [Papailiou, 1987], for over 30 years now. This trend has clearly
intensified over the past years, assisted by the introduction (thanks to modern electronics) of high
performance and compact recorders, Fig. 1.3, which greatly simplify storage and analysis of the
measurements, to the point that such measurements are largely standardised today [CIGRE, 1994] and
are being carried out worldwide.
Suspension clamp
Cable/conductor
Fig. 1.3 Modern conductor vibration recorder in measurement position (dimensions in mm)
89

4
This device measures the deflection amplitude of the vibrating conductor at a short distance (3
1/2" = 89 mm) from the suspension point, which is assumed to be solidly clamped, Fig. 1.3, and
converts this amplitude to a conductor bending stress. This conversion is based on the assumption of a
simple quasi-static conductor model [Poffenberger and Swart, 1965]: the short conductor section (89
mm length) on which the deflection is measured is considered as a clamped cantilever influenced by the
conductor tension and the displacement caused by the vibration amplitude. The bending stiffness
assumed for this conductor cantilever is the sum of the bending stiffnesses of the individual wires (also
wire stiffness) in the conductor. This assumption supposes that the individual wires of the conductor are
each bent around their own longitudinal axis. In this way the strain (elongation) can ultimately be
calculated for the outer layer of wires of the conductor.
This model has the advantage of simple application, but its validity is often in doubt. It has, in particular,
often been found (Hondalus, 1965, Claren and Diana, 1969, CEA, 1986, Ramey, 1987) that the wire
stresses calculated in this way only represent a trend and do not always reflect the actual stresses in the
individual wires of the conductor. There appears to be a large discrepancy between the stresses
calculated as outlined above and the actual stress in the wires, especially in the inner layers, where
wires are often found broken.
In addition, the stresses calculated in this way must often be compared with the service strength
characteristics of the conductor in order to estimate its service life. The latter are S-N curves found in
the laboratory (CIGRE, 1985) or design limits derived therefrom (EPRI, 1979). A further problem arises
from this comparison, demanding a better model for bending of conductors.
This is because most of the S-N curves for conductors in the laboratory are measured on conductors
excited to produce characteristic standing waves and then measuring the maximum bending amplitude
of the conductor. Using the simple conductor cantilever model described above, these maximum
bending amplitudes are then converted to bending stress at the conductor clamp. It is unfortunately not
possible to directly compare these bending stresses with the bending stresses found from
measurements in the field by converting the measured bending amplitude of the conductor near the
suspension (clamp). We are even explicitly warned against this (EPRI, 1979, Pg 60) because it is
known that the assumption of a constant bending stiffness of the conductor represented as a
homogeneous cantilever does not take sufficient cognisance of the conditions as the conductor bends,
which may lead to large uncertainties in the conversion of measured displacements (in the conductor as
a whole) to stresses in the individual wires of the conductor.

5
For all these reasons, the first and most important step towards the solution of the described problems
should be the study of conductor bending, including the effect of variable bending stiffness caused by
the construction of the conductor and concomitant internal friction. This also defines the problem to be
solved with this project.
1.2. Developments to date
Virtually since the day cables were introduced as construction elements, engineers were preoccupied
with their bending behaviour, since they soon recognised that this stress mode is of particular
importance to the reliability in service of the cables. For this reason, many papers have been written on
this topic, contributing to better comprehension of this problem. A full overview of all such work would
exceed the scope of this project. Instead, only those projects are briefly covered below that specifically
addressed the bending stiffness of cables and conductors, as does this project. This is because this
parameter is decisive to the understanding and solution of cable bending problems and it was
addressed in the technical literature at an early stage already.
The literature overview below distinguishes between work on cables for conveyance, which deals
mainly with steel cables, and work on overhead lines, which naturally focuses on conductors.
The first significant work on cables was that carried out by Reuleaux [1861] and Isaachsen [1907],
which also included the first meaningful attempts at calculating the bending stress and catenary of
cables. The work by Ernst [1933] is particularly significant, since he was probably the first to quantify the
effect of inner friction on cable bending behaviour. The work of Leider [1973, 1975] was a further
milestone towards understanding the internal cable conditions, especially in view of the secondary
stress, which he defined in detail. More recent work like that of Schiffner [1986] and Wang [1990], who
refined and developed earlier theories, also bears mentioning. Also to be mentioned in this connection
is the work by Hruska [1951], Czitary [1962], Zweifel [1969], Wiek [1973], Costello [1983] and Raoof and
Hobbs [1988], who dealt with this problem in depth, producing valuable methods and solutions.

6
From early years, overhead transmission engineers have also worked on directly establishing the
bending stiffness of conductors. In these studies, the effective mean stiffness was often calculated from
measurements of the maximum sag of a test conductor clamped at both ends and under constant
tension, assuming that the conductor behaves like a homogeneous beam with constant bending
stiffness [Pape, 1930, Monroe and Templin, 1932, Sturm, 1936, Morisson, 1962, McConell and Zemke,
1980]. In parallel, [Bückner, 1960, Helms, 1964, Scanlan and Swart, 1968, Möcks and Swart, 1969,
Brand, 1972, CEA, 1986, Ramey, 1987] carried out some work in which they, by means of
measurements with strain gauges on the outer layer of wires of a conductor, attempted to calculate the
internal stress conditions in the conductor or of individual wires.
The ongoing interest of engineers in the bending stiffness of cables is clearly documented by new
papers by haulage cable users [Raoof and Huang, 1992, Wiek, 1993] and others in the field of
conductors [AIF, 1991, Zeitler, 1994]. But researchers in the experimental mechanics field [Lanteigne,
1985, Goudeau and Cardou, 1993] are also endeavouring to come to grips with this interesting problem.
1.3. Project objectives
Based on the above, the objective of this project can be defined as follows:
A cable model is developed and existing analytical approaches are augmented to allow the bending
process in a tensioned helically twisted cable to be mathematically defined and predicted. In this
process, a variable bending stiffness is introduced and analytically defined, caused by internal friction in
the cable, and dependent on the external stress on the cable. A measurement method is designed and
implemented to verify the theory, allowing the cable sag and curvature and ultimately the variable
bending stiffness along the cable to be determined with good accuracy. This enables significant insights
to be gained into the inner cable state, especially the relative displacements of the individual wires in the
cable and the stresses prevailing in these. Despite the complexity of the problem, the theory is kept
simple and is explained in a straightforward manner, with a view to extension in ongoing research and
useful practical applications. With this in mind, the programs developed within the framework of this
project will be written user friendly enough to be used as tools in the design of cables or conductors.

1.4. Limitations
7
Although the original problem definition stems from cable vibrations and therefore constitutes dynamic
stressing of the cable, the stress is considered “quasi static” in this project. This approach proved
successful in earlier studies [Poffenberger and Swart, 1965, EPRI, 1979, CEA, 1986]. In this sense, this
project thus does not cover the problem of fatigue in overhead conductors. This is quite a complex
process and a quantitative solution does not appear within reach at present. Based on the new insights
gained here into the “internal” cable mechanisms dependent on bending stiffness as a function of
friction, cable curvature and tensile force on the cable, including the displacement behaviour of the
wires in the cable construction, this project nevertheless hopes to provide a foundation for solving
conductor fatigue problems. And finally, this project does not intend to deal with problems of clamping
and force application through the cable suspension and clamps. This would substantially complicate an
already complex task and render a closed overall solution impossible. The results of this project should,
however, contribute significantly to clarification of this topic as well.
1.5. Project structure
To achieve the abovementioned objectives, this project was structured as follows:
The model for single-layer cables is developed in Chapter 2, taking into account the results of earlier
work and the variable bending stiffness of the cable, as a function of the curvature, internal friction and
tensile force. The developed fundamentals are extended to multi-layer helically twisted cables and the
(non-linear) Moment-Bending characteristic is introduced in Chapter 3. This forms the basis for the finite
element model for the calculation of the bending of tensioned cables in Chapter 4. Chapter 5 deals with
the newly developed measuring method (“cable scanner”), for experimental verification. Chapter 6
provides a comparison between measurement and calculation and a sensitivity analysis of the effect
various parameters have on the interpretation of the results. In Chapter 7, the findings of this project are
transferred to the important practical field of vibrations of conductors and possible influences are
discussed on the dimensioning of such conductors.

2. Basic principles
2.1. The tensile stress
8
This Section deals with the tensile stress on the cable. This is the basic stress on the cables
investigated in this project. In all the tests, the first stress to be applied to the cable under no load is a
tensile force. This tensioning has a significant effect on the bending stress, i.e. the resultant stresses in
the cable.
The book by Feyrer (1990) deals with this topic in detail and only the essential aspects are therefore
repeated here.
The special conditions pertaining to stressed reinforced conductors (especially ACSR overhead
conductors) were discussed in detail especially by Ziebs (1970). In this case also, the full derivation of
the formulae is dispensed with here.
Fig. 2.1 Geometry of a wire belonging to layer L of a helically twisted cable
Fig. 2.1 shows a wire in an arbitrary layer L of a helically twisted cable defined by its cross-sectional
centre, also showing the full lay length ℓL of the unwound wire.

This figure also shows the definition of the lay angle
consideration, comprising of nL round round wires wires of the the same same diameter in the cables investigated here.
Fig. 2.2 shows the components of an external force S tensioning a helically tw twisted isted cable, as they act on
individual individual wires, wires, ignoring ignoring the transverse force, force, caused by by bending bending and torsion of the the wire (caused (caused by the
external external tensile tensile force). force). This This transverse transverse force force is very small small because the lay lay angle and layer radii of the
cable change very y little under tensile load [Feyrer, 1990].
Fig. 2.2 Distribution of the tensile force to individual wires
The wire tensioning force Zd,L in every individual wire of an arbitrary layer L
of the cable is therefore:
The total longitudinal tensile force in the cable is given by the sum:
wires,layers
where the summation also includes the core wire ( (βK = 0).
In the determination of the tensile forces Z Zd,L in the individual wires, it is assumed that the cable
cross-sections sections remain remain the the same same as as a a tensile tensile force force S, S, lengthening lengthening an an arbitrary arbitrary length length of of cable cable ℓs by ∆ℓs
all the wires are in the linear-elastic elastic region, i.e. Hooke’s law law is applicable. Finally, the effects of the
tangential force Ud,L are are assumed very very small. The The changes in the the lay angle angle and the transverse
transverse
contraction of the wire cross-sections sections are ne negligible in this context since the resultant stresses amount
to to only only 1% 1% [CEA, [CEA, 1986] 1986] or or 2% 2% [Feyrer, [Feyrer, 1990] 1990] of of the the total total tensile stress in the the wire. The corresponding
formulae are summarised in Annexure I.
9
the definition of the lay angle βL and the coil diameter dL of the layer under
and also lengthening the associated wire length ℓd,L by ∆ℓd,L, is applied. It is furthermore assu
, is applied. It is furthermore assumed that

Figure 2.3 explains these relationships as wires and cable are stretched under the influence of a tensile
force, with no change in the cable cross-section.
cross
Fig. 2.3 Wire and cable elongation
The elongation of a wire εd,L is smaller than the associated cable elongation εs by a factor of cos
Since the wire tension Zd,L is is the the same same for for all all wires wires in in the the same same layer, layer, Hooke’s law states states that:
or
where Ed,L is Young’s modulus for a wire in position L and with cross-sectional area Ad,L. .
The combination of (2.1), (2.2), (2.3) and (2.4) gives the tensional force Zd,L on an arbitrary wire in layer
L, , as a function of the total cable tension S:
10
by a factor of cos 2 βL

The tensional stress in the wire is then:
where:
is the strain stiffness of the cable.
2.2. The secondary stress
The The derivations derivations below are based on on the the work work by Leider (1973, (1973, 1975), 1975), who who has has made major contributions
to to the the bending bending of cables. For For a a better better overview, overview, they they are are applied applied in in part part to a a single single layer, layer, 7-wire 7 helically
twisted cable – i.e. a cable with a core wire surrounded by a helical layer of 6 wires. It is further assumed
that the wires in the layer do not touch each other circumferentially – an assumption which is generally
true for the cables without fillers investigated here and for ACSR high voltage overhe overhead ad conductors.
This This can can be be achieved achieved through through a a slight increase increase of of the diameter of of the core wire wire over over that of the wires in
the layer [Leider, 1975, Feyrer, 1990].
The tensile forces ZL in in the the individual individual wires of of the cable layer layer create a radial force force on on the layers
underneath, or the core wire, corresponding to a radially oriented distributed load pL. p . This load is:
where ρL is the curvature radius of a wire helix of the layer and rrL
is the coil radius of this layer. Fig. 2.4
shows the parameters in the cable.
11

Fig. 2.4 Determining the radial pressure of the wires
The radial force exerted by a wire element with a wrap angle ddα
is:
Considering the geometric relationships, Fig. 2.4, we find:
The curvature radius ρL and the coil radius rL r are related as follows [Hütte, 1955]:
By substituting (2.8), (2.10) and (2.11) in (2.9) we get:
where φ is the incremental angle of the helix, Fig. 2.4.
In In the the case case of of a a relative relative displacement displacement between between the the wire element element of of the the layer layer wire wire under under consideration
considera
and the core wire, e.g. through bending, the radial force dNL dN creates a friction force dRL, , Fig. 2.5:
12

Where µ L is is the the coefficient coefficient of of friction friction between between the the wire element element under under consideration consideration and and its subsurface,
e.g. the core wire of the cable.
Maintaining intaining the the static static balance balance of of forces forces in in the the axial axial direction of of the wire we obtain, Fig. 2.5:
Where dZL is is the the maximum maximum difference difference in in tensile tensile force a wire wire element element can resist resist through through friction.
friction.
Fig. 2.5 Force relationships in a wwire
element in layer L
Integrating (2.14) over φ and inserting the boundary condition
yields
Where Zd,L is is the the tensional tensional stress stress on on the the wire wire alone, alone, in acc. with (2.5), which is the same same for all wires wires in
layer L.
ZL, however, is the maximum (total) tension ZZL(φ)
) that can be countered by friction force at every point φ
on the wire. Apart from tension force Zd,L Z countering only tension stress therefore, friction generates a
variable tension force component along the wires of layer L, provided the relative movement is
sufficient:
13

It It appears appears suitable suitable for for further further derivations derivations to relate this this secondary secondary force to the wire cross cross-sectional cross area
in order to arrive at the so-called called secondary secondary stress. This is a a pure tension tension stress which is is assumed
constant over the wire cross-section section but it it also also varies along the wire as as a function function of of the wire rotation
angle φ.
Where σd,L is the constant tensile stress in the wire as calculated in (2.6) above.
Note that this secondary (tensile) stress only occurs if the cable is bent by the action of an external
transverse transverse force. force. The The curvature caused by this force stretches the wires wires in the the individual layers layers (the
(the
wires wires of of the outer outer layer and and the core wire wire in in a a single single layer cable) differently, attempting attempting to displace these
relative relative to each other. In this this it it is is assumed assumed that that the above equation, which was originally derived for a
straight cable, is also applicable to a bent cable with only negligible errors - because the change in the
lay angle βL is small during bending and therefore negligible [Wang 1990].
2.3. The bending stress
When a cable is bent, the elongation of the individual wires in a layer L is comprised of two different
components [Leider, 1973], Fig.2.6.
The “normal” bending elongation εb,L b,L is caused by bending g the wire around its own neutral axis. Its
maximum value at the surface of the wire can be approximated using the so-called so called “Reuleaux
equation”:
where δd,L is the diameter of a wire in layer L and ρ is the curvature radius of the cable axis.
If the wire cross-section section under consideration – or rather, its centre – lies outside of the neutral axis of the
cable, it experiences an additional bending elongation εzus,L, , proportional to the curvature of the cable
axis κ = 1/ρ and the distance hd,L d,L of the wire axis from the neutral axis of the considered cable
cross-section section (= cable axis), Fig. 2.7. We calculate:
14

In In this we we assume assume that that the the wire wire in in this this layer layer does not not shift shift relative to the layer underneath and that
initially βL = 0.
Analogously for the associated bending stresses:
Fig. 2.6 Wire stresses as the cable bends
(a) wires not displaced (b) wires displaced
Here, dL is the average diameter of the wire layer L and φ is the “position angle” of the wire cross-section
cross
in the considered cable cross-section section Fig. Fig. 2.7, 2.7, which which can can only assume discrete values values in in this context,
differing from each other by 2π/nL in each case, depending on the position of the nnL
individual wires in
the cable cross-section. section. It is sufficient in this respect to co consider the range - ≤ ϕ ≤ + since the
conditions in the wire repeat periodically outside of these limits. ϕ is also the incremental angle of the
helix described by the centre of the wire cross-section, cross the so-called called strand rotation angle, Fig. 2.4. It
determines determines the the position position on the the cable axis and can assume any value, contiguous between 0 and 2 22π.
The “exact” calculation alculation of the bending stress σb,L, , which in its most simple form is approximated by the
Reuleaux Reuleaux equation equation (2.20), (2.20), has has been been carried carried out out by by various various authors already already [Leider, [Leider, 1977, Czitary, 1962,
Wiek, 1973, Schiffner, 1986, Wang, 1990] and need not theref therefore be repeated here.
15

Fig. 2.7 Geometry of the cable cross cross-section
It It can can be calculated calculated that that the average value value of the the “exact” calculated bending stress equals about about 90% 90% of
of
the Reuleaux stress. The latter can therefore be used - for better clarity in the following considerations,
especially especially for for the the calculation calculation of the bending bending stiffness. stiffness. Of course course the the computer program, Section 4.4,
uses uses the complete complete equation equation for for bending bending stress stress [Leider, 1977], 1977], which which includes not only the change in
magnitude, but also the the change in direction of of the curvature curvature vector, leading leading to to different bending bending stresses
in the individual wires of the same cross cross-section (see Fig. 7.3a).
Finally, Finally, it it should should be noted noted that that the the torsional torsional stress in the bending wires wires amounts to only a few perc percent, if
any, any, of the Reuleaux stress [Schiffner, 1986] and is therefore ignored, since since it is insignificant in the
context of this project.
The The relationship relationship (2.19) (2.19) is is initially initially applicable applicable only only to to a a parallel parallel bundle bundle of of wires, wires, as as shown schematically in
Fig. 2.6. 6. It is necessary to adapt (2.19) when considering a helically twisted wire.
When changing from a straight wire (βL (β = 0) to a helically twisted one, it must additionally be considered
that, as derived in Section 2.1, the elongation of the wire as calculated in (2.3) is smaller than that of a
straight wire by a factor of cos 2
βL. . The secondary stress (2.21) must therefore be adjusted
correspondingly:
This This stress stress is is applicable applicable to to a a cable cable configuration without without displacements. For For correct correct representation of
of
the e conditions, conditions, including including internal internal friction, friction, as the cable is bending, the the displacement status of the wire
must be included in the stress calculations.
16

As As is well well known known from earlier observations observations [EPRI, 1979], and and also also intuitive, intuitive, the wires in a cable beg begin to
move on their underlying layer when a certain curvature is exceeded - Fig. 2.8.:
Fig. 2.8 Wire displacement during bending
The position on the cable cross-section section where the displacement starts starts and the affected length of of the
specific wire depend on whether the secondary stress in the non-displaced displaced wire, as calculated in (2.22),
exceeds the maximum value that can be resisted by friction (2.17).
By equating the abovementioned equations (2.17) and (2.22) for ϕ → π/2 /2 and solving for the curvature,
we find:
Thus σzus,L (2.22) > σzus,L (2.17), which is valid also for all wires of the considered cross
therefore therefore assumed that that the the specific specific individual individual wire wire will slide over its entire length length once this this “final”
“final”
curvature is exceeded.
The displacement starts at position ϕϕ,
, however, where the slope of (2.22) exceeds the slope of (2.17) for
the the first time, Fig. 2.9. As As shown shown earlier earlier [Leider, [Leider, 1973], this this is the case almost almost exactly where ϕ = 0, (also
see Fig. 2.7). The curvature associated with the start of slipping is:
With With partial partial slippage slippage therefore, the secondary stress stress in in a a wire wire is is governed in sections by the two curves
of of equation equation (2.17) (2.17) and and (2.22), (2.22), Fig. 2.9. 2.9. The bending bending region region where both stress conditions “coexist” along
the wire is called the transition region. This region will be dealt with later.
17
(2.17), which is valid also for all wires of the considered cross-section. It is

Fig. 2.9 Curves Curves for the secondary stress, depending on the displacement displacement status of the wire
Fig. 2.9 (centre curve) shows the wire in the transition region, since both stress var variants, iants, (2.17) and
(2.22), occur occur along the the same same wire. wire. The The secondary secondary stresses acc. acc. to (2.22) (2.22) at at the initial curvature curvature (2.24)
and at the final curvature (2.23) are also shown.
In any event, even if slippage occurs over the entire length of the wire as the curvature ( κ > κe,L)
increases, the secondary stress is not relieved but is merely limited to σzus,L,max, , i.e. the maximum
secondary secondary stress allowed allowed by by friction friction acc. acc. to (2.17), (2.17), thus remaining “frozen” at this value value in in the slipping
wire, Fig. 2.10.
The individual idual wire associated with a certain angle ϕ in the cross-section section under consideration does not
slip slip unless its its secondary secondary stress stress in in acc. acc. with with (2.22) exceeds exceeds the the maximum maximum transferable frictional
secondary stress after (2.17). The curvature at which displaceme displacement nt starts for this particular wire can be
found by equating (2.17) and (2.22):
18

Fig. 2.10 Maximum bearable secondary stress with complete wire slippage
This This shows shows that that every every wire wire starts starts to to slip slip after after a certain certain curvature curvature which which depends depends on its its posi
position in the
cable cross-section section and is defined by the angle ϕ. . This means that the individual wires of the considered
cross-section section do not necessarily begin to slip at the same curvature.
For ϕ → 0, (2.25) transits to (2.24) and to (2.23) for ϕ → π/2, Fig. . 2.11; i.e. the wires at the “level” of the
cable axis (ϕ = 0) are the first and the wires at the cable “crest” (ϕ ( = π/2) /2) are the last that are slipping.
Fig. 2.11 Curvature where wire slippage begins (function of ϕ)
19

2.4. The bending stiffness
As mentioned mentioned in in the the introduction introduction to to this project, the determination determination of of the bending stiffness stiffness is is of cardinal
cardinal
importance to the bending of cables. Even though the bbending
ending stiffness is a “defined” (auxiliary)
quantity, quantity, it it significantly significantly facilitates facilitates the understanding understanding of the bending bending processes processes because because it is is a
combination of material and cross-sectional sectional values. Its Its use is is therefore therefore common in technical literature
and many studies, some already cited, hav have e the purpose of determining this. In this light, the conditions
as cables bend will be investigated below in more detail.
Bending equation
The The natural natural equation equation for for the the elastic curve of of a support support is is the the basis for for its its distortion distortion and is known from the
strength of materials:
where:
M : bending moment acting on the support cross cross-section
y(x) : bending curve of the support (here, the catenary)
κ= 1/ρ : curvature of the neutral axis
ρ = 1/y”: curvature radius of the neutral axis
(EJ) : bending stiffness ness of the support cross cross-section
20

21
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.

Fig. 2.12 Distribution of the cable bending moment over the individual wire wires
For the equilibrium condition we have:
Since these observations refer to the centre of the wire cross cross-sections sections we must, as shown in Fig. 2.12,
multiply the normal stresses by cosβ cosβL to find the internal wire moments.
The abovementioned natural equati equation on for the bending curve may now be applied to the entire cable.
Using 2.26, we find:
The The very small small bending bending moment moment causing causing a curvature transverse to to the cable bending “plane” is
neglected in this case [Wang, 1990].
As previously discussed in Section 2.3, a wire is subjected to two normal stress components, σb,L and
σzus,L, , as as the the cable bends. bends. The bending bending stiffness of the wire correspondingly correspondingly also also has has two components.
These stiffness components may be calculated as follows:
Corresponding to the two components ponents of of the the wire stress, viz. the bending stress and and the secondary
stress and Fig. 2.12, the wire moment M d,L is given by:
22

Where Ad,L is the cross-sectional sectional area, yd,L y is the cross-sectional ordinate and hd,L is the distance of the
considered wire in layer L from the cable axis (see also Fig. 2.12).
Equation Equation 2.27 2.27 may may also be be written in the following manner manner for for an an individual wire in the the layer:
layer:
By comparing (2.28) and (2.29) and by solving the first integral of (2.28) for the cross-section cross section of the wire
with σb,L as in (2.20), we obtain (EJ) (EJ)min,d,L, , the wire stiffness affecting how an individual wire bends
around its own transverse axis:
The corresponding cable stiffness, denoted (EJ) (EJ)min, , is obtained by summing the individual wire
stiffnesses acc. to (2.30):
where EK is Young's modulus and δδK
is the diameter of the core wire.
This This would would be be the the only only stiffness stiffness of of the the cable cable if the individual wires wires were lying lying on top of each other without
friction.
The friction forces between the individual wires however, analogous to elongation and stress, Section
2.3, give rise to secondary stiffness (EJ) (EJ)zus,d,L as in (2.32).
Since Since the the secondary stiffness stiffness arises from the secondary stress, which which is constant over the
cross-section section of the wire, it is not actually nnecessary
ecessary to integrate in order to determine the second part in
(2.28). The secondary stiffness of an individual wire in layer L is therefore:
23

Depending Depending on on the the slip slip in in the considered wire, the secondary stresses stresses from from (2.17) or or (2.22) must be
inserted in this equation, as well as the distance hhd,L
of the centre of mass of the cross-section cross of the
wire wire from from the the neutral neutral axis axis of of the the bent bent cable cable acc. to (2.19). (2.19). This secondary secondary stiffness stiffness is dependent on on the
angle ϕ in both cases, i.e. it varies as ϕ varies aries along an individual wire or yields different results for the
different wires of the same cable cross cross-section, depending on the ϕ “position” of each wire under
consideration.
The secondary stiffness acc. to (2.22) for an individual wire and when the wires have not slipped (called
Region I) is therefore given by:
and by (2.17) when the wires are fully displaced (called Region II):
Summing Summing equation equation (2.33) (2.33) over over all all wires wires and layers yields the the secondary secondary stiffness stiffness of of the the cable cable with
undisturbed wires:
In In this this case, case, the the cable cable has has the the maximum maximum bending bending stiffness, stiffness, called called the the cable cable stiffness (EJ) (EJ)max, (EJ) calculated
from from the the sum sum of of the the wire wire stiffness stiffness acc. acc. to to (2.31) (2.31) and and the the secondary secondary stiffness stiffness acc. to to (2.35):
(2.35):
This result is similar to the case where the ca cable ble is considered a homogeneous, rigid beam and its area
moment moment of inertia is is calculated calculated using using Steiner’s theorem. theorem. This This is also a a plausibility plausibility check check on the
correctness of the approach used here to obtain the secondary stiffness.
Equation (2.34), however, for or the the first time time describes describes an additional additional contribution to to wire stiffness, not
considered considered to to date date in in cases cases of of fully fully slipped slipped wires. This additional stiffness, approaching zero as the
cable curvature increases, arises from the maximum secondary stress allowed by friction forces in the
wires wires acc. to to (2.17), (2.17), and and opposes opposes the external moment of the the load, load, in in addition addition to to the the wire wire stiffness stiffness acc.
to (2.31).
24

This “remanent” secondary stiffness (EJ) zusII is calculated by the summation of (2.34) over all wires in
the cable. le. The functions in (2.34), however, have no closed solution for (EJ) (EJ)zusII and the summation
must be done numerically (see also Section 2.7).
For the summations in the two equations for stiffness (EJ) (EJ)min in acc. with (2.30) and (EJ) (EJ)zusI in acc. with
(2.33), 3), a simple equation can be found for each, however:
The summation in each case includes the nL n identical wires in a layer of the cable. The corresponding
cable cable stiffnesses are simple simple to to calculate from the summation summation of of all all the the layer layer stiffnesses stiffnesses using the
equations above, Section 3.5.
The following identity is a special case of (2.38):
This is valid for ϕi = 2 /nL i = 1,…..,n 1,…..,nL and nL > 2, which is always true for helically twisted cables.
This This relationship, relationship, hitherto hitherto known known as as an an empirical empirical rule [EPRI, [EPRI, 1979, p.15], p.15], is proven in Annexure II.
II.
2.5. The cable bending moment
Based on the known stiffness ffness (EJ) and curvature κ in the cable, the natural equation for the elastic curve
can be used to calculate the associated cable bending moment M = (EJ) κ.
By using the relationship derived in the previous paragraph:
25

this results in:
In this equation, Mmin is is the the wire wire bending bending moment moment of of the the cable, as a result of the the wires wires bending around
their own axis and Mzus is is the secondary bending bending moment moment of of the the cable, cable, caused caused by internal internal friction
friction
between wire layers and dependent on the slippage experienced by the individual wires.
The The entire entire internal internal cable cable moment moment M is is therefore the the total of of the wire moment and the secondary
secondary
moment. moment. These These relationships relationships are are illustrated illustrated in in Fig. Fig. 2.13. 2.13. The two (curvature) Regions I I and II II are are shown
shown
separately, respectively where e there there is is no no slippage anywhere along along the the wires wires and where the wires have
have
slipped slipped along along their their entire length. length. These These two conditions conditions of the cable are characterised by the two different
bending stiffnesses acc. to (2.36) and (2.37) and are explained in more det detail ail in the next paragraph.
Region I Transition Region II
Fig. 2.13 Moments - Bending (M-B) B) curve of a single layer cable
26

Average transition curvature
Referring to Fig. 2.13, we may make the approximating assumption that, ign ignoring oring the transition region
which lies between the initial curvature κa and the final curvature κe, , as shown in Section 2.3, the
stiffness changes instantaneously at the average transition curvature κm. . This approximation
significantly improves the clarity of of the the following explanations explanations and is therefore introduced at at this point.
The index L is is furthermore furthermore dispensed dispensed with in the following, because the cable considered considered in this
paragraph has only one layer.
Table 2 below shows the effective bending stiffnesses and bending moments of the cable under these
assumptions, in the two Regions I and II, Fig. 2.13.
Region I
(no wire slippage)
Region II
(complete wire
slippage)
(EJ) min + (EJ) zusl = (EJ) max
(EJ)min + (EJ) (EJ)zusII = (EJ)min +MR/κ (EJ)minκ + MR
Table 2.1 Bending stiffnesses and bending moments in Regions I and II
The moment MR was provisionally introduced as an auxiliary parameter. This moment will gain physical
meaning further down only. We have:
Considering (2.34), we obtain:
If, in (2.43), the secondary force Zd d
(EJ)Total
where Z(ϕ) is the total and Zd the “pure” wire tensile force, independent of ϕ, we find:
27
(EJ)maxκ
MTotal
, which is a function of ϕ, , is replaced by (Z(ϕ) (Z( - Zd),

We may derive the mean transition curvature κm from the following consideration: In the simplified
presentation presentation (Fig. (Fig. 2.13), 2.13), the the cable moments in Regions Regions I I and and II should, per definition, be the same for
this curvature, i.e. using Table 2.1:
This yields the average transition curvature:
Figure 2.14 .14 below below shows shows the the curve curve of of cable cable stiffness across the curvature. This figure shows the "exact"
curve curve for for bending bending stiffness, stiffness, i.e. i.e. taking taking into account account a a finite finite transition transition region region between between curvatures κa and
κe as in (2.24) and (2.23). For comparison, the figu figure re also shows the approximate stiffness curve
discussed discussed above, above, which which features a a sudden transition from the undisturbed condition to to fully fully slipped
slipped
cable wires at the average transition curvature κm as in (2.46).
This figure is of particular rticular practical signi significance since it quantifies the variable bending stiffness of a
cable cable as as caused by by internal friction friction for for the first time. time. This This dependence of the bending stiffness on the
curvature of the cable will be used later again, in the calculation of the cable catenary, characterising the
results of this project.
The associated transition moment MMm
is calculated from (2.45) as:
The intersection MR of the M-B B line line with with the the moment moment axis axis is is found from from Fig. 2.13 as follows:
28

Fig. 2.14 Schematic curve of bending stiffness (EJ)(κ) (EJ)( across the
cable curvature for a single layer cable
But M R is given by (2.43):
MR is the residual friction moment or the maximum moment fric friction tion can resist, which remains “frozen” in
the the cable after after all all the the wires have have slipped, slipped, continuing to oppose the external load, because κ > κm
(Region II):
Region I Transition Region II
The moment MII above the average transition curvature κm is therefore a combination of the wire
moment Mmin and the residual friction moment MMR.
29

The moment moment below the transition curvature curvature however, however, i.e. i.e. in Region I I (no wire slippage) equals equals the
maximum possible cable moment:
The plausibility of the above considerations is strengthened through the following observations at the
boundaries:
Using Using (2.24) (2.24) and and (2.23), (2.23), the the starting starting and and end curvatures of the the wire displacement displacement were were determined for for a
cable cross-section, section, based based on the balance balance of of forces forces on the individual affected wire elements:
Based on simple considerations, the average transition curvature κm, , was also derived from Fig. 2.13
(2.46):
Substituting the MR component of a single wire from (2.43) yields:
But for the single wire e considered here, (2.30) and (2.33) also state:
The associated “pure” wire tensioning force (i.e. without the bending component ZZzus)
) therefore:
30
Start of wire displacement
End of wire displacement

To determine the curvature at which the wire starts slipping, we must set ϕ = 0 in (2.52) (see
Section 2.3).
Using the series expansion:
we find, for small values of ϕ and ignoring higher order terms that are small:
Under the same assumptions, we also find:
Substituting this, together with (2.53) 53) and (2.54) in (2.52), yields:
and
Analogously, the transition curvature for the end of the wire slippage at ϕ = /2, i.e. for the entire length
of the considered wire, can be found from (2.52):
or, since sin ( /2) = 1:
31
(equation (2.24)) q.e.d.
(equation (2.23)) q.e.d.

Fig. Fig. 2.15 2.15 finally shows shows the the stiffness stiffness Regions I, transition transition and II, II, i.e. i.e. the curve of initial initial and final curvature
radius ρa and ρe, and of the average age transition curvature radius ρm as a function of the tensile stress in
the wire, σd.
Region II
Fig. 2.15 Initial, final and average transition curvature radius as a function
of the “pure” tensile stress in the wire
2.6. The cable state diagram
The moment-bending curve (M-B B curve) curve) derived for the cable in the previous paragraphs, paragraphs, Fig.2.13, is of
cardinal cardinal significance both both for for later later considerations considerations in the the course course of this project project and for fundamental
considerations regarding the internal condi condition tion of bending cables. It may also be described as the state
diagram diagram of of the the cable. cable. For For this this reason, reason, a a few few additional additional deliberations deliberations will hereinafter be made regarding
this depiction of cable characteristics, introduced here for the first time.
32
Region I

Fig. 2.16 Cable bending processes in the M-B M diagram
The M-B B diagram, diagram, Fig. Fig. 2.16, 2.16, allows allows us us to to easily easily determine the curvature status status (and therefore also the
stress status) and the associated effective bending stiffness of the cable under any external load, i.e.
any external bending moment.
If, for instance, the cable experiences an external loading moment MMb,
, then this state diagram shows
that the cable will assume a curvature κb. To reach the “end” point Mb/κb, , the cable can only assume deformation conditions along the MM-B
curve
if the original condition was friction-free free (initial loading). loading). The The cable therefore at at first reacts to to an an external
external
load load with its initial initial stiffness, stiffness, corresponding corresponding to to the the cable with with undisturbed wires wires and maximum cable
stiffness (EJ)max – see see (2.36). This This stiffness remains up to the average transition transition curvature curvature κm, which, as
described described above, is is the approximate approximate transition point between the undisturbed cable condition and a
cable with fully slipped wires. From this point onwards, the external bending moment is opposed only by
the minimum stiffness, or wire stiffness (EJ) (EJ)min as in (2.31), which could also be called the final stiffness
of the cable.
Region I
Region II
If, therefore, bending of the cable has caused wire slippage along th the e entire length of the cable, then
further bending involves only the wire stiffness in (2.31).
33

During During bending in in Region Region II II (full (full wire wire slippage) slippage) therefore, the the cable reacts reacts to to further loading as if the
individual wires are slipping relative to each other with no friction. But in this region the residual friction
moment MR (see (see (2.43)), (2.43)), which which friction friction forces forces “froze” “froze” into the cable structure, will also act to oppose the
external load.
At the final curvature κb therefore, the cable exhibits an average stiffness (EJ)b with a magnitude
appearing constant during the entire bending process, from (0/0) to (Mb/κb). ). This average cable stiffness
over the entire bending process is:
which is therefore independent of the final curvature κb.
Hysteresis
The M-B B diagram diagram derived derived in in the previous paragraphs was initially explained in the the first first quadrant of of the
M-B B coordinate coordinate system, system, since since this this is of of particular importance to to the the bending bending stress in a cable. In the
following deliberations, the M-B B diagram will be exten extended ded to the other quadrants as well. In this way, the
cable cable status can can be be clearly clearly illustrated illustrated also also as the the load load is reduced, or for cyclical stresses.
Two cases must be differentiated in this analysis, Fig. 2.17(a) and (b):
Case (a)
The final curvature κb reached as a result of the external load MMb
is smaller than the transition curvature,
Fig. 2.17a:
therefore:
κb < κm
Mb = (EJ)Iκb = (EJ)maxκb
i.e. i.e. the the conditions conditions in in the the cable cable are are always always described described by the straight line with the slope (EJ) = (EJ) (EJ)max –
both for increasing and decreasing loads.
34

Fig. 2.17 Cable curvature in the MM-B
diagram
(a) κb < κm : no hysteresis
(b) κb > κm : hysteresis; left: first stress, right: subsequent stress
Case (b)
The final curvature κb reached from the friction friction-free state, following external stress Mb, , is larger than the
transition curvature:
κb > κm
Above Above the transition transition curvature, curvature, the the cable can only assume states that that are on the straight line in
Fig. 2.17b, with the slope (EJ) min.
The balance of moments is now:
M b=M R+ ( EJ ) min κ b
where MR is is the the max. max. possible possible friction friction moment (remanent friction moment) in in acc. acc. with (2.43).
The above can now be used to construct the M-B M B diagram for a cable for a complete cycle of increasing
and decreasing load.
35

Fig. 2.18 Hysteresis in the M-B B diagram as a cable bends
Departing from the origin (M = 0, κ = 0), the cable follows the straight line � with the slope of (EJ) (EJ)max
(undisturbed cable) until the transition curvature κm is reached, then following the straight straig line � with
the slope of (EJ) min. . In this process, the maximum friction moment M MR, , applicable at the break point,
remains “frozen” in the cable.
If the stress direction is now reversed at the maximum stress (M (Mb/κb) ) (stress reduction), the cable
initially follows the straight line � with the slope of (EJ)max (EJ) and running parallel to straight line �, i.e. it
opposes the stress relieving moment with the maximum bending stiffness (EJ) max. . In this process, the
“frozen” remanent friction moment M MR must first be relieved along the curvature path κm. m
From From this this point point onwards, onwards, which which can can be be viewed viewed as as a new “origin” for for the cable, the the process process during stress
stress
increase is repeated analogously, but in reverse order (straight line � and �). ). If the stress is again
reversed, the cable follows straight line � to get to the original origin again, thereby closing the
hysteresis loop (in the illustrated case).
36

2.7. The single layer helically twisted cable as a calculation example
The deliberations ons and and equations equations derived derived above above will now be applied to to a a single single layer layer conductor
conductor. The
aluminium/steel aluminium/steel conductor conductor 35/6 35/6 in in acc. with with DIN 48204, 48204, Fig. Fig. 2.19, 2.19, is is used in the example because because it it has
a simple and clear construction – rendering it suitable for “manual” calculations. The conductor data of
interest here are:
− 1 core wire of steel δ = 2.7 mm, ASt A = 5.73 mm 2 = Ad
− 6 layer wires of aluminium δ = 2.7 mm, AAl A = 34.38 mm 2 = 6 Ad
− Lay angle β = 10° (estimation)
− Average layer diameter dm = 5.4 mm = 2 rm
− Coefficient of friction µ = 0.1 (assumption)
− Young’s modulus ESt St = 210 000 N/mm 2 , EAl = 70 000 N/mm 2
Fig. 2.19 Cross-section section of an ACSR conductor 35/6 (see also Fig. 2.7)
Internal conductor state
The question of how far the wire displacement has progressed must first be resolved. This may be
solved using the two equations (2.24) and (2.23):
37

with:
and:
with:
The tensile stress in a wire in the aluminium layer, σzug can now be calculated if the tensile force on the
conductor is known – using (2.6).
Fig. 2.20 shows the two curvature radii for the beginning of wire slippage ρa, , or for the start of complete
wire slippage ρe as a function of the tensile stress in the wi wire re in acc. with (2.60) and (2.61). For the
selected selected conductor, conductor, therefore, this diagram diagram shows shows the state of of wire wire slippage slippage in the conductor as a
function function of any any tensile tensile stress. stress. The same same figure figure also shows the the average average transition transition curvature curvature radius
ρ m=l/κ m in acc. with (2.46).
Start, end and transition
curvature radii [m]
Fig. 2.20 Limiting curvature radii for an a ACSR conductor 35/6
38
Aluminium wire tensile stress [N/mm2]

Wire stiffness
Independent of the magnitude of the wire slippage, the first component of the stiffness, the wire stiffness
(EJ)wire, , has a constant value. Applying equation (2.31), we get:
EJwire
EJwire
Secondary stiffness
In Region I (no wire wire displacement), displacement), the the secondary stiffness of of an an individual wire depends on the position
of the individual wire in the conductor cross cross-section. section. This is represented by the factor sin 2 ϕ in (2.33).
The stiffness of an individual wire along the axial directi direction on of the conductor is therefore a function of this
factor, since the angle ϕ assumes values of ϕ = 0 to ϕ = 2π along the lay length of a wire.
If we now sum these stiffness values over the wires, mutually displaced by ϕ = 60° within the same
conductor cross-section section (Fig. 2.19), the sum remains remains constant for for every every conductor conductor cross cross-section,
independent of the position of the individual wires, since:
Wires
The general proof of this was discussed in Section 2.4 for an arbitrary numbe number of wires nL L in a wire layer
L and is given in Annexure II.
Wires
No. of layer wires
This yields a constant value for the secondary stiffness EJ EJzusI in Region I in acc. with (2.33):
Alum. wires
39

The The situation situation is is different different in in Region Region II (complete wire wire slippage). slippage). Depending on on the the angle
angle ϕ which
determines its position in the conductor cross-section, cross section, the secondary stiffness of an individual wire
changes here, acc. to the function (e µ sin βϕ -1)sinϕ - see (2.34).
Using Using the the results results of of this this function function as as applied applied to the the individual individual wires, wires, it is not possible to find a closed
solution for the secondary stiffness component zusII and the summation must therefore be
carried out numerically.
Since the function (e µ sin βϕ -1) has thus far only been defined in the 1
/2 ≤ ϕ ≤ π/2), /2), the symmetry of the axes (of the secondary seconda stress) to the straight line requires that ϕ = π/2:
(EJ) zusII is always positive, since the functio
and 1 st quadrant, Fig. 2.21.
Fig. 2.21 Curve of the secondary stress over a full lay length ( (ϕ = 0 to 2π)
The The secondary stiffness stiffness in in Region Region II II in in acc. acc. with with (2.34) (2.34) is is calculated by summation summation over all all the the wires, wires, of
of
the function e µ sin βϕ -1) sinϕ. . For the conductor under consideration here, we obtain a value of 0.062
which is used in the equations below.
This yields:
with:
C = 0.062 . 5.755 . 0.9848 . (5.4 / 2) = 0.95 mm 3
40
1) has thus far only been defined in the 1 st and 4 th
and
quadrants (- π
is always positive, since the functions (e µ sin βϕ -1) and sinϕ always have the same sign in the 4
always have the same sign in the 4 th
. For the conductor under consideration here, we obtain a value of 0.062

The The secondary secondary stiffness therefore depends on the two (variable) values of curvature and tensile stress
in the aluminium wires. The secondary stiffness (EJ) (EJ)zus has correspondingly been plotted in Fig. 2.22 as
a function of the curvature. The parameter for the family of curves is tensile stress σzug in the aluminium
wires of 10, 20, 30 and 40 N/mm 2 .
Fig. 2.22 Secondary stiffness of ACSR conductor 35/6;
1: σzug = 10N/mm 2 ess of ACSR conductor 35/6;
; 2: σ zug = 20N/mm 2 ; 3: σzug = 30N/mm 2 ; 4: σzug = 40N/mm 2
The The curves curves clearly clearly show show that, as as physical physical considerations lead lead us us to to expect, expect, the secondary stiffness
increases as the tensile stress increases in the aluminium wires, i.e. as their radial pressure on the steel
core increases – i.e. the conductor slips at larger curvatures.
In In the transition region, both displacement displacement conditions conditions occur occur at at the same same conductor conductor cross cross-section –
depending on the position of the
41
Secondary stiffness
Curvature
individual wire in the conductor cross-section section as defined by the
angle ϕ. . Here Here the the two two curves of of the secondary stress given by (2.22) and and (2.17) for angles ϕ between
- π/2 and +π/2 /2 have, apart from the trivial solution ϕ = 0, two further intersections (see ee also Fig. 2.10).

As As the curvature curvature increases, increases, these these move move outwards, outwards, until until the the entire entire wire wire is slipping. Figure 2.23 shows the
secondary stress curve for an ACSR 35/6 conductor with a curvature in the transition region.
Secondary stress [N / mm^2]
Fig. 2.23 The The secondary secondary stress stress in in the the transition transition region; region; the the two two vertical vertical lines indicate the
the
intersections of (2.17) and (2.22)
After the transition region, the secondary stiffness with complete wwire
ire slippage finally settles to (2.61)
and (2.64):
Secondary stress curve
This stiffness limit is therefore independent of the tensile stress.
It is furthermore clear that the influence of the parameters σzug and κ increases as the secondary
stiffness (EJ)zusI moves towards secondary stiffness (EJ) (EJ)zusII. (EJ)zusII decreases continuously until it
reaches reaches its its limit limit after after the the transition transition region region in in acc. acc. with with (2.65), (2.65), whereafter whereafter it it approaches approaches a limiting value of
zero (Fig. 2.22) as the conductor curvature κ increases in acc. with (2.64).
42
Region of wire slippage

Conductor bending moment
The The deliberations deliberations in in Section Section 2.5 2.5 with with respect to to the conductor bending moment moment will be explained here here by
by
applying them to the ACSR 35/6 conductor and quantifying them.
In Region I (no wire displacement), displacement), the secondary bending moment increases proportionally to to the
curvature, curvature, since the the secondary secondary stiffness stiffness is is constant constant in this region, region, as already noted in (2.63), Fig. 2.24.
For Region I (no wire displacement) therefore:
The secondary moment at at the the start of the the wire wire displacement may be found by by substituting substituting the value
found in equation (2.60) for the curvature:
Bending moment [Nm]
Secondary bending moment
Fig. 2.24 Secondary bending moment in an ACSR conductor 35/6 as a function
of the conductor curvature for different values of tensile stress in the wire;
1: σzug = 10N/mm 2 ; 2: σzug zug = 20N/mm 2 ; 3: σzug = 30N/mm 2 ; 4: σzug = 40N/mm 2
43
Curvature

This shows that, based on the proportionality between tensile and radial stress, the wire starts slipping
at increasing bending stress as the tensile stress increases in the wire.
The The rate rate of of increase of the secondary secondary bending bending moment moment decreases decreases towards the end end of the transition
region until it reaches a constant value with fully slipped wir wires es (Region II), which is dependent on the
actual actual tensile tensile stress. stress. This This value value remains remains constant constant even even with increasing curvature curvature and is calculated as
(2.64):
From this point on, only the wire moment
opposes the external load and the (total) Moment-Bending Bending curve is therefore proportional to (EJ) (EJ)min, in
Region II, Fig. 2.25. The small circles mark the positions (M (Mm /κm ) for the individual M-B B curves of this
conductor.
Total bending moment
Fig. 2.25 Total bending moment in an ACSR conductor 35/6 as a function of
the conductor curvature for different values of tensile stress in the wire;
1: σzug = 10N/mm 2 ; 2: σzug zug = 20N/mm 2 ; 3: σzug = 30N/mm 2 ; 4: σzug = 40N/mm 2
44
Curvature

3. Multi-layer cables
45
In the previous sections of this project, the theory of variable bending stiffness caused by internal friction
was derived for single-layer cables only, i.e. for cables comprising of a core wire and a top layer of wire.
This has the advantage that the relationships remain clear because the construction of the cable is
relatively simple and the effort required for “manual calculations” – for checking – is still justifiable. The
following sections deal with multi-layer helically twisted cables, which are quite common in practise and
therefore of special practical significance.
This will be dealt with in several steps, for better comprehension. Since, in multi-layer cables, the
individual wires rest on single points of the layer underneath (with the exception of the innermost layer,
which rests on the core wire), the number of points of contact must first of all be determined. This is
followed by looking at a helically twisted cable with a core wire and two layers and calculating the
variable tensile force in these two layers. Finally, the variable tensile forces caused by interior friction
are determined for each layer in the general case of a helically twisted cable with an arbitrary number of
layers of the cable. Knowledge of these tensile forces then forms the basis for subsequent calculation of
the variable bending stiffness of multi-layer cables.
3.1. Wire resting points
With multi-layer cables, the wires of the outer layer (Index a) rest on the wires of the layer underneath at
spots only (Index i), Fig. 3.1. In this process, the wire helix changes to a “polygon chain” between the
points of contact, especially in the case of high tensile forces and “soft” wire material.
A wire of the outer layer which is parallel to the cable axis (lay angle of the outer layer βa = 0) crosses ni
wires of the inner layer over the distance li (lay length of the inner layer as measured along the axis of
the cable). If the outer wire now spirals (βa ≠ 0), ni more or fewer points of contact are created per turn of
the outer wire, depending on whether the lay angle βa and βi (lay angle of the inner layer) have opposing
or equal directions; ni (la / li) contact points are now created over the lay length la of the outer layer.

A wire of the outer layer therefore touches the layer underneath at b ba,i points nts of contact per lay length
la = πda / tanβa:
The following convention is defined for the sign:
different inner and outer lay directions: directions +
same inner and outer lay directions: -
Analogously, one wire of the inner layer touches the layer above at bi,a contact points per lay length li:
with the same signing convention as above.
The The helically helically twisted twisted cables cables under consideration consideration here always have have alternating right and left lays, i.e. the
+ sign is always applicable in the above equations.
3.2. Tensile force orce in the outer layer of multi multi-layer cables
In the case of the single-layer layer cable dealt dealt with in in Chapter 2, 2, the the outer layer rests on the core wire
everywhere. Equation (2.15) gives the tensile force Za Z in this outer layer:
where Zd,a is the tensile force alone of the outer layer acc. to (2.5). ZZa
is again dependent on the angle
ϕa, although – for better clarity – we will write Z( Z(ϕ) = Z below.
46

In the case of the multi-layer layer cables dealt with in this section, however, the outer layer rests on the layer
underneath at ba,i points of contact, see Fig. 3.1.
Since Since the the tensile tensile force in in the the wire wire can only change at these resting resting points, points, the tensile force force in in the the outer
outer
layer changes step-wise, wise, each step hhaving
a height of µaDN DN DNa. DN . Because, if a wire element in the outer
layer (lay angle βa) ) is deflected by an angle Dϕa by the tensile force Za, , it produces a normal force DN DN a,
acting on the layer underneath (refer also Chapter 2.2) with a magnitude of:
The balance alance of forces on on a wire element in the outer layer in a point of contact between this this element and
and
the inner layer, Fig. 3.1, again produces:
Fig. 3.1 Balance of forces on a wire element in the outer layer of a two-layer two layer cable at the
contact point with ith the layer underneath
Considering Dϕa = 2π/ba,i, we find:
By repeated application of the equilibrium conditions (3.5), the tensile force of the n th wire element in the
outer layer is found as:
47

The Zn,a values values are are the vertices of the “staircase” tensile force curve in the outer layer, Fig.3.2, and they
are points on an exponential function that differs only by a factor Ka K from the function which describes
the wire tension when it rests on the wire undern underneath eath everywhere (3.3). The wire tension Za Z in the outer
layer is therefore:
Fig. 3.2 Curve of the wire tension in the outer layer of a multi-layer multi cable
The factor Ka is obtained by equating (3.7) and (3.8):
Taking the logarithm and considering that ϕa = n 2π/ba,i, for the n th wire element under consideration
yields:
48
wire element under consideration

But for 20 points of contact (ba,i = 20) already, corresponding to a small helically twisted cable
measuring approx. 20 mm in dia diameter, factor Ka = 0.993 and the difference between the two
exponential exponential functions functions (3.3) and (3.8) (3.8) is is less than than 0.02 0.02 %. The “normal” “normal” exponential function acc. acc. to to (3.3)
(3.3)
can thus be used to calculate the tensile force in the outer layer, i.e. KKa
can be set to 1 without risk of
significant error.
3.3. Tensile force in the inner layer of a two two-layer cable
The tensile force in the inner layer of a two-layer two layer cable is next calculated. For better clarity, this is done
in several steps.
General case: Spot contact and variable tensile force in the outer layer
The The general general case case of of a a helically helically twisted twisted cable is first discussed here, comprising of of a core core wire and and two
layers, layers, one one outer outer and and one one inner inner [Leider, [Leider, 1974]. 1974]. In In this case, case, the outer layer is assumed to to be be resting resting on
on
the he inner inner layer on on contact points points (idealised). (idealised). The The inner inner layer, however, rests rests on the core core wire
everywhere - as explained for single single-layer cables in the previous chapters.
With the two-layer layer helically twisted cable, the tensile force in the inner layer ZZi,k,
, which is in contact with
the core wire everywhere, is given by (2.15):
Here, Zd,i is the tensile force in the inner layer on its own.
With With these these cables, the the inner inner layer layer tensile tensile force is also affected affected by by the outer outer layer. As explained in the
previous section, section, a a friction friction force force acts acts on on the the outer outer wire wire at at every every point point of contact contact between outer outer and and inner
layer, Fig. 3.3, with a value (2.13) of:
49

According According to to the law of action action and reaction, a force of of equal equal magnitude magnitude but but opposite opposite direction direction acts on
the inner wire. Furthermore, the normal force DNa exerted by the tensile force in the outer layer is
transferred to the core wire through the wire of the inner layer. A friction force is therefore created at that
point of contact between inner layer and core wire, Fig. 3.3, of magnitude:
If the lay angles of the outer and inner layer
helically twisted twisted cables cables examined examined here), the displacement of these these wire wire layers layers will will be in in different
different
directions, directions, resulting resulting in the summation of of the the two two friction forces forces (3.12) and (3.13) acting on the the inner inner wire,
wire,
Fig. 3.3. The effect of the lay angle on the magnitude of this total friction force may be neglected [Wang,
1990].
Fig. 3.3 Effect of the tensile force in the outer wire on the inner wire of a two-layer two layer cable;
wires sketched in parallel, for clarity
This produces a stepped curve for the tensile force in the inner layer ZZi,
, similar to the tensile force in the
outer outer layer layer as as derived in the previous section, Fig. 3.4, 3.4, with with the the following following magnitude magnitude of the step at each
outer layer/inner layer point of contact:
By substituting DNa from (3.6) and ZZa
from (3.8), the following step magnitudes are found:
50
ay angles of the outer and inner layer βa and βi have opposite signs (as is always the case with the
Outer wire
Inner wire
Core wire

Here, the factor Fa is the combination of all the constants in this formula.
Fig. 3.4 Wire tension in the inner layer of a two two-layer cable
Note that, through the addition of the DZi,a share at the contact points between the inner/outer layer, the
outer outer layer layer at these these points points increases increases the the tensile tensile force in the inner inner layer. layer. This occurs in in steps steps at angles of
of
ϕ = 2π/b a,i .
This shifts the tensile force force curve curve for the inner inner layer (and this is is the change in the the inner layer tension
caused caused by its its contact with the core wire everywhere) between between the points of contact contact with with the the outer outer layer
to fictitious “larger” angles ϕ. . Thus, for the “construction” of the final “tensile force vs ϕϕ”
curve for the
inner layer, those ϕ values values corresponding corresponding to the current tensile force in the observed wire element of the
inner inner layer layer must must be be used. used. This This process process is illustrated in Fig. 3.4. It It is evident there that, that, contrary to th the
outer outer layer layer (Fig. 3.2), 3.2), the the tensile tensile force force in in the inner layer layer changes changes between the individual inner/outer layer
contact points as well, given by the function
51

First First approximation: Continuous Continuous contact and and variable tension tension in the outer outer laye layer laye
With very small angles Dϕ ≈ dϕ, , i.e. for many contact points bba,i
between the outer and inner layers, the
sine sine in in (3.4) (3.4) may be replaced replaced by the the angle angle itself. A wire wire element in the outer layer therefore exerts the
following force inwards:
It must also be considered that a wire element of length ddld
corresponds to a length dls = lLdϕ/2π along
the cable axis, whereby lL is the lay length of the considered layer, Fig. 3.5:
Fig. 3.5 Length relationships between wire and cable elements
For the outer layer thus:
and for the inner layer:
52
Cable axis

Thus la dϕa/2π = li dϕi/2π must must be be true true for for the same same cable section, i.e. i.e. for for the same lengths lengths along along the
cable for the outer layer and inner layer ddls,a
= dls,i = dls. . The change in wire lengths caused by the
polygon polygon chain chain effect mentioned mentioned in Section Section 3.1 may be neglected, neglected, since it it is is very small, small, especially for
layers with many wires. Therefore, if also ddϕ
i = dϕ for simplification:
and (3.17) becomes:
na wires of the outer layer now press inward across the entire cross-section cross section of the cable, with a total
force of:
This force is absorbed by ni wires of the inner layer in the ssame
cross-section section of the cable. For a single
wire in the inner layer thus:
The The balance of forces forces on the wire element, Fig. 3.3, yields the the following following for the tensile force force change change dZ
dZi
of an inner layer wire element, at every point of contact with the outer layer:
Here ma = µa + µi = 2µ if µa = µi
Now the tensile force Za in in the the outer outer layer is is approximated approximated in (3.24) by the exponential function:
53
i = µ (which is often justified).

This means that the outer layer is assumed to be resting on the inner layer everywhere (not just in
spots), spots), which, as shown in in the the previous previous section, section, is is a good approximation for conventional conventional cables.
cables.
Therefore:
and:
With y = Zi (ϕ) and y' = dZi(ϕ)/dϕϕ,
equation (3.27) corresponds sponds to an inhomogeneous differential
equation of type:
with the forcing function s(ϕ). ). This means:
In the above, a(ϕ) ) is a constant, independent of ϕ. The forcing function s(ϕ), ), on the other hand, is
dependent on the angle ϕ and describes the effec effect t of the friction forces originating in the outer layer on
the tensile force Zi in the inner layer.
Using the abbreviations:
the description is simplified to:
54

The solution of the inhomogeneous differential equation (3.28) is by the so so-called called variation of constants:
Therefore:
Using the previously applied boundary condition:
Zi(ϕ = 0) = ZZd,i
the integration constant Ci is found as:
And the final solution of the differential equation (3.28) becomes:
To determine the tensile force Zi,i in a deeper layer ( (Index ndex i,i), (3.24) must be correspondingly modified:
using:
55
(forcing function)

By applying the Zi just just achieved acc. to to (3.37), (3.37), the the entire entire calculation process could now, in principle, be
repeated in the differential equation (3.38) and ZZi,i
be determined therefrom.
Second approximation: Continuou Continuous contact and
constant tension on the outer layer
It is easily recognised that substituting Z i and Za to calculate the tensile force Zi,i of a deeper layer in acc.
with (3.38), results in a correspondingly complex forcing function.
For this reason, reason, the the tensile tensile force force in in the the outer outer layer layer will will now now be be assumed approximately constant with Z ZZa
= Zd,a for all angles ϕ in (3.24). Thus, in (3.32) and (3.33):
i.e. both a(ϕ) and s(ϕ) ) are constants in this special case, independent of ϕ, and the different differential equation
(3.28) simplifies:
with the solution:
Here, Index c indicates the assumption of a constant tensile force in the outer layer.
The boundary condition
yields:
56

The final solution of the differential equation (3.41) is thus:
Comparing the two curves Zi and Zc,i c,i in Fig. 3.6, as calculated from equations (3.37) and (3.45), with the
“exact” staircase form as in Fig. 3.4, very good agreement is found in both cases - confirming the
approximations introduced here. (3.45) can the therefore refore be used for further calculations without significant
errors – i.e. Zi = Zc,i may be substituted for the tensile force of the inner layer.
Fig. 3.6 Wire tension Zi in the inner layer of a two two-layer cable according to the different
approximations:
Zi (Fig. 3.4) Spot contact and variable tensile force in the outer layer
Zi (3.37) Continuous Continuous contact contact and variable variable tensile force in the the outer layer
Zc,i (3.45) Continuous Continuous contact contact and constant constant tensile force force in the outer layer
57
(Fig. 3.4)

3.4. Tensile Tensile force in in the the inner inner layers layers of of helically twisted cables with more than than two
two
layers
To To analyse analyse cables with more than two two layers, the individual layers layers are indexed from the outside in, using
n, n-1, n-2, .... .
Analogous to (3.31a) we find:
Furthermore, the forcing function Cz,n z,n-2 for layer (n-2) 2) is calculated, using (3.31c), (3.38) and (3.41), as:
Referring to (3.39), the so-called called friction factor m mL of any wire layer L of the cable is:
Equation (3.47) shows that the forcing func function tion of each layer also contains the forcing functions of the
layers above, whereby a constant tensile force Z d is always assumed for all layers above the layer under
investigation.
Since the outer layer has no forcing function, Cz,n C must be set to 0 and Cz,n-1 becomes:
Correspondingly, the wire tensions in the individual layers are:
58

Layer n:
Layer n-1:
Layer n-2:
etc.
Since Since (3.47) is recursive, the formulae (3.50) are relevantly simple simple to to “program”, making making further
further
calculations easier.
3.5. The state diagram for multi multi-layer cables
The state diagram (M-B B diagram) for multi multi-layer layer cables can now be developed, using the results from
single-layer layer cables cables in in Chapter 2 2 and and the the successful successful general derivation in in this Chapter of the tensile force
in multi-layer cable wires.
Analogous to the single-layer layer cable, the following relations – Index L – (some already defined in Chapter
2) are applicable to each layer of the multi-layer multi cable, where the Index d,L refers to wire d of layer L and
the summation – Index d – is is in each case case carried carried out out over over all the the wires wires in the same layer:
layer:
59

To determine the stiffnesses, moments and curvatures for each layer of a multi multi-layer layer cable by the above
formulae (3.51) to (3.58), the tensile forces of the individual wires of the layer under consideration, each
with its position angle ϕ applicable at the cable cross-section, cross section, must be formed and totalled. The
corresponding corresponding values values for the the complete cable are then found by adding adding up the values of the t individual
wire layers of the cable – Index L – shown in (3.51) to (3.58) as follows:
60

This can now be used to draw the MM-B
diagram for a multi-layer layer cable, as shown in curve 0 in Fig. 3.7,
which represents a four-layer layer cable as used in th the e later measurements. This curve is the result of
successive overlaying of the M-B B curves for the individual cable layers (L 1, L2, L3, L4) ) as shown in Fig.
3.7.
Fig. 3.7 The M-B B diagram for a four four-layer cable
Using the M-B diagram
The M-B B state state diagram is is particularly particularly useful for for evaluating evaluating the the effects of the the bending stiffness on the
bending bending characteristics characteristics of of a a cable. cable. The The different different possibilities will be explained below, below, with reference to
Fig. 3.7a-e.
61

Assumption 1: Undisturbed isturbed cable configuration
In In this case, case, the bending stiffness is is calculated calculated using Steiner’s Steiner’s theorem or, acc. to Section 2.4 and
(3.51a), the maximum bending stiffness (EJ) (EJ)max. This corresponds to curve 1 in the M-B B diagram, Fig.
3.7a.
Fig. 3.7a M-B diagram with undisturbed cable configuration, curve 1
Assumption 2: Fully slipped cable configuration
In In this this instance, instance, the the bending bending stiffness stiffness is calculated calculated by by adding the the individual individual wire stiffnesses as as minimum
wire stiffness (EJ) min acc. to (2.31), curve cur 2 in the M-B diagram, Fig. 3.7b.
Fig. 3.7b M-B B diagram with fully slipped cable configuration, curve 2
62
κ

Assumption 3: Only the outer layer is slipping
An An “intuitive” “intuitive” assumption sometimes sometimes made made [Gopalan, 1986] is is that only the the outer layer
layer slips when the
cable bends, whilst the remaining (inner) layers do not slip. Curve 3 in Fig. 3.7c shows this. It is evident
that this assumption differs from the actual M-B M curve of the cable, curve 0, , above a certain stress
moment, describing a stiffer cable than the reality, especially for greater curvatures.
Fig. 3.7c M-B B diagram when only the outer layer of a cable slips, curve 3
Assumption 4: All layers slip simultaneously
In this case we assume [Ernst, 1933] that the cable can only assume two states during bending:
a) State I:
the wires have not slipped, thus (EJ) (EJ)I = (EJ)max
b) State I I:
at a critical stress moment MMm
– corresponding to a critical transition curvature κm
in the cable slip simultaneously; the cable resists furt further her increase of the moment only with the
wire bending stiffness, thus (EJ)I (EJ) I = (EJ)min. A total residual friction moment MR as in (3.59) also
resists further bending in multi-layer multi cables.
It is clear that this assumption, as illustrated by curve 4 in Fig. g. 3.7d, approaches the actual bending
behaviour of the cable fairly closely – as illustrated by curve 0 in the M-B B state diagram.
63
m – all the wires

Fig. 3.7d M-B B diagram diagram under under the the assumption assumption that that all all layers slip slip simultaneously, curve 4.
Assumption 5: The cable cable maintains maintains a a constant constant bending bending stiffness stiffness throughout
throughout
the bending process
This This assumption assumption is is occasionally occasionally made made [Claren, [Claren, 1978] 1978] to simplify the the required calculations. It It is is illustrated
by line 5 in Fig. 3.7e. This “average” stiffness, obviously lying between (EJ) (EJ)min and (EJ) (EJ)max, is often
determined determined through through tests tests [Sturm, [Sturm, 1936, Morrisson, Morrisson, 1962, Mc Mc Connel and and Zemke, Zemke, 1980, AIF, 1991,
Zeitler, 1994] and its magnitude is given as a multiple of the wire stiffness (EJ) min. . It is, however, only
valid for a specific end curvature κbb
and a specified initial condition and does not constitute a specific
cable constant as such.
Fig. 3.7e M-B B diagram with an average constant stiffness, curve 5
64

4. Calculation method
4.1. Bending of a cable under tensile load
Considering Considering the the deliberations deliberations in in the the previous sections sections of this this paper, paper, particularly particularly the the last last section, on the
M-B B diagram diagram of of the the cable and and on on the the cable hysteresis, it is is opportune at this point to discuss some
fundamental damental characteristics characteristics of of a a cable cable subjected subjected to increasing increasing and and decreasing load with simultaneous
tensile tensile and and transverse transverse load. load. The The transverse transverse force force V is introduced at the the center of the the cable, causing a
bending displacement y at this point, Fig. 4.1.
Fig. 4.1 Static system with load and directions
The following assumptions were made:
- the cable is under a constant tensile load S at all times.
- the the transverse transverse force force V V increases increases and and decreases continuously continuously in the loading Phases 0, 1 and 2
and the unloading Phases hases 3, 4 and 5 respectively.
- the cable stiffness is dependent on bending – as derived in Chapter 2.4.
- the the influence influence of of the the transverse transverse forces forces (in (in the the bearings) bearings) on the local local transverse transverse compression
(between the cable layers) caused by the tensile force and therefore affecting the internal
friction conditions in the cable, has been ignored.
The The individual individual phases phases of of the the bending bending process process will will now be be qualitatively qualitatively described described and and illustrated in
diagrams.
65

Phase 1
The The relative position position of of the the wires wires in the cable does not change, change, the the slope slope of the curve in the
force-displacement diagram is “proportional” to the stiffness (EJ) (EJ)max.
The area under the curve in Fig. 4.2a represents the total work performed by the transverse force,
comprising the work W1B to bend the individual wires and the work W W1D to stretch the wires lying outside
of of the the neutral neutral axis; the latter latter work work being being contributed contributed by by the secondary stress (Chapter (Chapter 2.2).
2.2).
Fig. 4.2a Bending of a cable under tensile t load; illustration of Phase 1
Phase 2
After After the the limiting limiting curvature curvature has has been been exceeded, the the wires begin to shift shift in their relative position and
there there is is no no further further increase in in strain. strain. The The slope of the curve curve is therefore less steep and is now
“proportional” to the stiffness (EJ) min min, Fig. 4.2b.
The The area area under under the the curve curve again again represents represents the the work work performed performed by the the transverse force, separated separated in
this case into the work W2B to continue bending the individual wires and the work WW2R
to overcome the
friction in n changing the relative position of the wires.
66

Fig. 4.2b Bending of a cable under tensile load; illustration of Phase 2
The The work work performed performed by by the transverse force in Phases 1 and 2 2 (loading) (loading) is is stored in part (W (W1B, (W W1D and
W2B) in the cable as elastic strain and the remainder WW2R
has been converted to frictional heat.
Phase 3
This This phase phase (Fig. (Fig. 4.2c) represents the the beginning of of the the transverse force reduction (load decrease)
whereby part of the stored energy is released again.
By reducing g the bend, the associated portion of the bending work W W3B is, on the one hand, released
again.
Reversing Reversing the direction of motion, however, however, also also results results in in reversal reversal of the friction forces. The strain built
up in Phase 1 is therefore initially reduced agai again, n, without relative movement of the wires. The slope of
the curve is again “proportional” to (EJ) max, , as in Phase 1. The associated strain energy W 3D = W1D is
released.
67

Fig. 4.2c Bending of a cable under tensile load; illustration of Phases 3 and 4
Phase 4
The The strain strain from from Phase Phase 1 1 is is fully fully relaxed. relaxed. Further Further reduction reduction of the the curvature curvature initially initially increases the strain in
the opposite direction. The slope of the curve therefore remains “proportional” to (EJ) (EJ)max
Strain energy W4B is released by the ongoing reduction of the curvature.
The strain energy W4D can be provided by the released bending energy WW4B.
Only the energy W4BRest = W4B - W4D 4D is therefore released externally.
Phase 5
The cable layers cannot absorb further strain. The wires begin to reverse their relative position - in the
opposite direction as in Phase 2. The slope of the curve is “proportional" to (EJ) (EJ)min only.
68
max.

Because of the relative movement of the wires, the released bending energy W W5B is partly used for
overcoming friction W5R. . The externally available energy is therefore only WW5BRest
= W5B
Fig. 4.2d Bending Bending of of a cable under tensile load; illustration of of Phase Phase 5 and the residual
energy WBRest
The end of Phase 5 is marked by the disappearance oof
f the transverse force. Due to friction forces
though, though, the the curvature curvature does does not not return to zero zero as the load disappears, but retains a residual value. The
deformation deformation energy energy is is correspondingly correspondingly not not released released entirely, but both both the Phase Phase 4 strain energy W W4D and the bending energy WBRest remain stored in the cable.
69
- W5R.
If If the the transverse force now increases increases again, again, a Phase 0 must must be allowed for for before before Phase 1 1 can can restart.

Phase 0
The strain energy W0D = W4D stored in Phase 4 is released in Phase 0. But, as opposed to Phase 3, the
energy energy is is not not available externally, but but is converted converted to bending work (see Fig. 4.2e). 4.2e). The The transverse
force, therefore, only needs to provide the energy difference WW0BRest
= W0B - W0D. . The slope returns to
“proportional” to (EJ)max.
Fig. 4.2e Bending of a cable under tensile load; illustration of Phases 0 and 1:
left - initial load, right – subsequent loads
In the force-displacement diagram diagram therefore, therefore, a cable subjected to to a transverse force initially initially traverses
traverses
the curve 1 - 2 - 3 - 4 - 5 once and thereafter the curve 0 - 1 - 2 - 3 - 4 - 5 for each subsequent cycle, Fig.
4.2f.
Initial load
The area of this “hysteresis” curve corresponds to the “s “self-damping” damping” of the cable in one full load cycle
and and is is proportional proportional to to the friction energy converted to heat. heat. This corresponds corresponds to to the sum of W W2R + W5R. As As is is evident from from Fig. Fig. 4.2f, 4.2f, the the cable cable is “permanently” “permanently” deformed after the first first stress stress release. This
This
deformation deformation corresponds corresponds to to a a permanent permanent curvature curvature as shown in in Figure Figure 2.17b, 2.17b, which which in turn results in
the associated curvature-dependent dependent stresses stresses (bending (bending stress stress and secondary stress) being “frozen” into
the cable wires.
70
Subsequent loads

Initial load
Fig. 4.2f Basic force-displacement
displacement characteristic during cable bending
4.2. Analytical calculation of the cable catenary
In this section, an attempt is made to analytically calculate the catenary of a twisted cable with variable
bending bending stiffness. stiffness. This This method method is is based based on on the fundamental fundamental work produced produced by Ernst (1933) on this
topic. topic. Czitary (1962) (1962) has has also also dealt dealt with with this this topic at at length. length. The results results from the previous sections sections in in this
paper, describing the quantitative e determination of the moments moments-bending bending characteristics (M (M-B diagram)
of the cable are in addition applied here.
Two Two goals goals are are hereby targeted. On On the one hand, hand, the analytical analytical solution solution can can be be used to check check the the finite
finite
element solution of the problem, introduced in the next chapter. On the other hand, it is intended to
establish a reasonable scope of application of analytical (closed) solutions.
The method is again centered red around the cable state diagram (M (M-B B diagram) derived in Section 2.6, but
in its simplified ified form (Fig. (Fig. 4.3), 4.3), i.e. i.e. an an approximation approximation is made that a a cable cable under tensile and transverse
load can assume only two “discrete” bending stiffness conditions.
71
Subsequent loads

Fig. 4.3 M-B B diagram for the analytical calculation
Before Before transverse loading (always (always under under the the assumption of of a prior tensile tensile load kept constant constant over the
entire procedure), the inter-layer layer friction friction forces forces are are preventing movement of the wires, i.e. the bending
stiffness (EJ) max of the cable is at its peak. This also represents the initial slope of the M-B M B characteristic,
Fig. 4.3. If, after exceeding a mean transition moment Mm M which is a function of the tensile load (Section
2.5), the friction is overcome, then all the wires si simultaneously multaneously slide over each other along their entire
length and the cable is now at its minimum (wire) stiffness (EJ)min. (EJ) In this region, the M-B B characteristic
has a slope “proportional” to (EJ)min. .
The maximum tolerable friction moment MR M (Section 2.5), , however, remains “frozen” into the twisted
cable as a residual moment when the condition changes from (EJ)max (EJ) to (EJ)min, , continuing to oppose,
together together with with the wire wire moment, moment, the other other instantaneous instantaneous external loads exceeding the mean transition
moment Mm, Fig. 4.3.
Region I
These moments MR and Mm, , including the associated mean transition curvature κm, , are needed for the
analytical analytical solution solution to follow and are are for the first time quantified quantified here in Chapter 2 for single single-layer single cables
and Chapter 3 for multi-layer cables. s.
72
Region II

To simplify the calculation, it is useful to place the yy-axis
axis of the coordinate system specifically on the
transition between the two stiffness conditions (EJ) (EJ)max and (EJ)min in the cable catenary curve and to
“define” the x-axis as the asymptote of the cable, see Fig. 4.4 (x & y axes dash-dotted).
dash
Fig. 4.4 Coordinate system and stress, for the analytical calculation
Then, for the external bending moments
and
for Region I (Index I)
for Region II (Index II)
The The corresponding corresponding internal internal bending moments resulting resulting from the the specific stiffness are obtained obtained simply
from the aforementioned state diagram of the cable (Fig. 4.3) for:
and for:
x < 0 → κ < κm
x > 0 → κ > κm
73
x < 0 : MI = S yI
x > 0 : MII = S yII
: MI = (EJ)max κI
: MII = MR + (EJ)min κII
(4.1)
(4.2)
(4.3)
(4.4)

The The balance balance of of moments moments on on the the cable cable yields yields the the differential differential equations of the catenary, i.e.:
for Region I (Index I)
or
and for Region II (Index II)
or
The The solution of of these these differential differential equations equations and and the the determination determination of of the integration constants through
consideration consideration of of the boundary boundary and and transition conditions have been described by by Ernst Ernst (1933) (1933) and are
are
shown in Annexure III in abbreviated form. The analy analytically tically calculated catenary is subsequently
transformed into the xy (bold) (bold) coordinate system system in in Fig. Fig. 4.4 4.4 to enable a a direct comparison with the
measurements and the FE calculation which is to follow and which also uses the latter coordinate
system.
In this s way, way, we we can can compute compute the the catenary catenary analytically. This This derivation derivation assumes, assumes, however, that the cable
has two different bending stiffnesses (EJ) max and (EJ) min along its axis as the transverse load is applied,
i.e. from V = 0 to V = Vmax, , at constant tensile force S. But, as explained in Section 4.1 already, this is not
the case in practice since the cable exhibits the maximum bending stiffness (EJ) (EJ)max along its entire
length length at the the beginning beginning of of transverse loading (V (V = 0). As the stress stress increases, the wires, st starting from
the the point where where the the transverse transverse force force is is applied, are slipping over over an ever increasing region of the cable,
causing the cable stiffness to change to (EJ) min in that region.
74

75
The final stiffness condition with two different stiffness Regions I and II of the cable (Fig. 4.3), used in
the analytical derivation, is only achieved once the maximum transverse load V = Vmax has been
reached.
The analytically calculated maximum sag of the cable is therefore always greater than the actual sag,
since the cable behaviour is in reality more "stiff" than analytically assumed.
Stated differently, this means that the analytical solution, even if it includes a regionally variable
stiffness, does not account for the non-linearity of the bending process caused by this variable cable
stiffness and the step-wise application of the transverse load.
4.3. Determining the catenary with the finite element method
As shown in Sections 4.1 and 4.2, it is not possible to find a closed analytical solution to the problem due
to the variable cable stiffness caused by internal friction, ultimately resulting in a physical non-linearity in
the force-displacement curve for the cable.
For this reason, the finite element method (FEM) was chosen to compute the cable catenary and a
dedicated FE program was written for this purpose. The technical literature [Papailiou, Müller, Roll,
1982] may be referenced for details about the FEM with a view to applying the program to similar cable
problems. The method used is briefly as follows:
If the static and dynamic equilibrium conditions are satisfied at all the vertices of the mesh, the FEM can
be applied, based on the following set of equations:
[F] = [KG] [u] (4.7)
Where [KG] is the stiffness matrix, [F] is the force vector (known forces) and [u] is the displacement
vector (unknown deformation).
To determine the stiffness matrix [KG] of the total system, in this case the cable, the element stiffness
matrix [Ke] must first be found. For the 2D flexural elements used here, this matrix is determined using a
well known method [ANSYS, 1988]. The matrix elements essentially comprise the bending stiffness,
Fig. 4.9, and the strain stiffness of the cable, acc. to (2.7).

76
In the next step, for an “arbitrary” position of the element in the so-called global coordinate system, the
element stiffness matrix must be transformed using the following equation [ANSYS, 1988]:
[K] = [T] T [K e] [T] (4.8)
At this point, the geometric non-linearity of the problem (“stress stiffening”) is additionally taken into
consideration.
“Stress stiffening” is the additional stiffening of a structure, caused by the instantaneous deformation
status in the structure. This effect is caused by coupling between longitudinal and transverse
displacements and should be taken into account in structures such as the cables described here, i.e.
structures with small cross sectional areas compared to their lengths and including a bending stiffness
much smaller than the strain stiffness.
In the FE calculation, this effect is taken into account through a so-called incremental stiffness matrix
[Se], which is added to the normal element stiffness matrix [Ke].
Refer to [ANSYS, 1988] and [Cook, Malkus, Plesha, 1989] for the method for establishing this.
The overall stiffness matrix [K G] is created by overlaying the element stiffness matrices, whereby the
stiffness values of the two elements adjoining a vertex are totaled. The associated matrices are
compiled in Annexure IV.
Using this, the system of equations (4.7) describing the unknown deformations can in principle be
solved.
From the deformations which have now been calculated (the cable catenary), the curvature is
subsequently determined for the center of each element, using the following procedure:
A cubic approach is used in the FE model for the displacement curve over the element length:
y = a + bx + cx 2 + dx 3 (4.9)
The calculated displacements and rotations at the two vertices of the element (Index 1 and 2) are now
entered into this equation. This yields a system of four equations by which the coefficients of the
polynomial in equation (4.9) can be determined.

The curvature e at the center of each element can thereby be computed:
Thereafter, this curvature and the stiffness stiffness-bending bending characteristic of the cable, Fig. 4.9, is used to
determine determine the the bending bending stiffness for each each element, to be applied applied in in the element stiff stiffness stiff and the
incremental incremental stiffness stiffness matrices. matrices. It It is is used there to replace the the constant constant bending bending stiffness (EJ) max, used at
the the start start of of the the calculation. calculation. Using Using these new new stiffness matrices, the calculation just described may be
repeated step by step until the sag sag of of the cable is within a freely selectable convergence convergence limit limit over the
entire entire length length of of the cable. The The entire calculation calculation process process is described within within the the framework framework of of the SEIL
SEIL
computer program in the next section.
4.4. The SEIL program
The measurements described cribed in in the the following following chapters chapters serve serve to to corroborate the theory theory on on the the variable
variable
bending bending stiffness stiffness of of cables cables as introduced in this paper. The The SEIL program was was written written for this purpose.
It has a modular structure as follows:
a) The input module: the the cable cable data, data, experimental experimental data data and and calculation calculation assumptions are are entered
entered
here.
b) The computation module: the the catenary is computed here as well as as the derived derived parameters such
as cable curvature, wire tensions, etc.
c) The output module: all the parameters of interest and their curves are shown here on the screen
or printed and plotted.
d) The comparison with measurements: this part offers the option of comparing the measurement
data data obtained with with the the evaluation evaluation programs still still to to be described described with the calculated re
results to obtain
a direct measure of the agreement between calculation and measurement.
The The individual individual modules modules of of the the SEIL program will be be briefly discussed in the following sections.
77
(4.10)

Input module
The cable data input, Fig. 4.5, is self explan explanatory, atory, with the exception of the wire rotation angle ϕ, which
determines determines the the position position of of the examined wire in a layer, relative relative to to the applied force force (at (at x x = = 0) in each
case. For ϕ = 90° in the single layer cable in Fig. 4.6, for instance, the wire is at th the e crest of the cable
catenary, catenary, diametrically opposite opposite the the point where the force is applied. applied. The wire tensions are then also
calculated calculated and and displayed in the output module with with respect to to this “reference” wire (wire no. 1).
1).
Number of wires: 24 [ - ]
Young’s modulus for the wires: 180645 [ N / mm^2 ]
Fig. 4.5 SEIL program; cable data input
Coefficient of friction µ: 0.5 [ - ]
Wire rotation angle ϕ (1): 90 [ ° ]
Fig. 4.6 Definition of the wire rotation angle ϕ in the SEIL program
There There are are several several options options when entering entering the the test parameters and and the calculation assumptions:
It is possible, for instance, to select either a specific transverse load or a specific max. sag at mid span,
Fig 4.7.
78
Dataset no. 1
Cable layer no. 4
Wire diameter: 3.54 [ mm ]
Lay angle β: 13.97 [ ° ]

79
Dataset no. 1
Tensile force: 280 000 [ N ]
Load type: (x) Transverse force
( ) Path
( ) Distributed load
Transverse force, path, q: 40 000 [ N ], [ m ], [ N/m ]
Cable length: 0.5 [ m ]
Number of elements: 100 [ - ]
Spacing: 10 [ - ]
Number of steps: 10
Fig. 4.7 SEIL program; input of the test and calculation parameters
The number of discrete elements for the cable is selectable as is the ratio between the first element
(transverse force application in the bending region) and the last (tensile force region) by entering a
“spacing value”. This renders the elements in the bending region proper smaller than in the tensile
region, leading to a better and more realistic match between the measured and calculated catenary.
By selecting the number of load steps, we can also specify in how many steps to apply the transverse
load (or transverse displacement), thereby significantly affecting the results - because of the
non-linearity of the force-displacement curve shown in Section 4.1. When executing the SEIL program,
we can also select whether to calculate the initial stress or subsequent stressing – as shown in Fig. 4.2f.
Finally, it is possible to determine the effect of the selected bending stiffness approach, Fig. 4.8, on the
calculated result and compare this with the considerations discussed in Fig. 3.7a-e (see Section 6.4).

( x )E*I = f ( k )
80
Stiffness for catenary:
( ) E*I = f ( k ) (simplified)
( ) E*I = Elmin
( ) E*I = Elmax
( ) E*I = Eldef
( ) E*I = f ( k ) (only A-L r.)
Eldef: 0 [ N*mm^2 ]
Fig. 4.8 SEIL program; entering the different options for the bending stiffness
Computation module
Based on the cable data entered and the cable tensile load, the cable stiffness is first determined as a
function of the cable curvature; Fig. 4.9, curve 1, shows an example of such a characteristic curve for
the wire rope used here.
Then, depending on the number of load steps, the transverse load V is divided into equal steps (10, for
instance).
The first load step is therefore �
�
the second is 2
�� ��
and the last is 10 �
��
For the first load step (e.g. �
�� ), it is initially assumed that the maximum bending stiffness (EJ)max –
curve 3 in Fig. 4.9 - is applicable to the entire cable. Using this assumed bending stiffness, the finite
element method described in Section 4.3 is applied to a finite element model comprising of 100 discrete
2D rod elements to represent the cable to compute the cable catenary and, based on that, the curvature
for each element. This is used to determine the bending stiffness of each cable element, based on the
stiffness-bending characteristic, Fig. 4.9, curve 1, which was created at the start of the calculations.
Using these “updated” element stiffnesses, the catenary and the local curvature is again computed and
compared to the catenary of the iteration step. If the difference in displacements between two iteration
steps is less than 10 -8 mm for all element vertices, the iteration stops and the current values for the
catenary are saved for further processing. Otherwise, the iterations continue until the above condition
for termination is met.
= V.

Stiffness [N*m^2]
Fig. 4.9 Stiffness-bending bending characteristic; 1: (EJ)(κ); 2: (EJ) min; 3: (EJ) max
Sag [ m ]
Fig. 4.10 Comparison of FE and analytical calculations: both cable
catenaries are virtually identical
81
Overall stiffness
Sag
Curvature

82
The same process is now repeated for the second load step, e.g. 2 �
, whereby the algorithm always
starts from the last computed bending stiffness curve along the cable. The third load step follows, e.g.
3 �
, etc. until the full transverse load V has been reached.
��
The computation module also contains the analytical solution for the catenary as described in
Section 4.2, which may then be included in the corresponding diagram for comparison.
The results of the FEM calculation only agree with the analytical calculations if the transverse load is
applied to the cable in a single step in the FEM calculation - an option that is included in the SEIL
program. In this way, the accuracy of the FEM calculation can also be checked, as demonstrated in Fig.
4.10. As shown here, the calculated catenaries using the FEM and the analytical methods are virtually
identical.
Output module
Once the calculation has been successfully completed, an OK message is shown on the screen, also
showing the number of iterations required. Typical values for 100 elements and 10 load steps amount to
a total of about 100 iterations, with a computation time of about 1 minute on a 486/33 PC. Thereafter,
the output can be requested. Fig. 4.11 demonstrates the numerous output possibilities of the SEIL
program.
The input data may be printed for the record. The command “print results” displays salient results in
tabular form, Fig. 4.12. The command “plot results” offers a selection of various graphs. This allows
presenting up to ten datasets in one graph.
The measurement results as well as the analytical solution can be superimposed in the cable catenary
diagram. The comparison between theory and measurements is also a very important function of the
SEIL program This comparison is the subject matter of Chapter 6.
The flow chart, Fig. 4.13, serves to illustrate the basic method of calculations in the SEIL program.
��

Fig. 4.11 The output options for the SEIL program
Cable stiffness Wire stiffness
[N*mm^2]
83
Max. sec. stiffness
[N*mm^2]
Total stiffness
[N*mm^2]
Core wire 1'853'579 0 1'853'579
Core – layer 1 9'952'834 64'019'413 73'972'247
Core – layer 2 26'188'622 567'012'033 593'200'655
Core – layer 3 50'510'909 2'243'933'918 2'294'444'827
Core – layer 4 82'943'444 6'203'208'119 6'286'151'562
Moments Transition (rho)
[m]
Transition (k)
[1/m]
Moment m -.)
[Nm]
Moment R-.)
Layer 1 1.844 0.542 431.112 34.727
Layer 2 5.293 0.189 379.160 95.031
Layer 3 12.277 0.081 309.304 136.590
Layer 4 33.057 0.030 190.163 119.772
Max. sag = 21.009 [mm]
Display
Print
Plot ►
Secondary stiffness
Secondary bending moment
Total stiffness
Total bending moment
Sag curve
Curvature curve
Stiffness curve
Bending moment curve
Force-displacement curve
Tension curve
Bending stress
Secondary stress
Total stress
Layer
Fig. 4.12 SEIL program; results in tabular form
[Nm]

84
Fig. 4.13 Flow chart of the SEIL program calculations
α
Input
Transverse load V = 0
(EJ) = (EJ)max = const.
V = V + ∆V
Stiffness-bending
curve (EJ)(κ)
Calculation of the stiffness
matrix [K]
Solving the system of equations
[F] = [K][u]
after element vertices
displacements [u]
Determining the curvature of all
element centers
All element vertices
displacement between
two iterations <
convergence limit
V = Vmax
Output
ω

5. Measurements
85
The theory postulated in the previous chapters of this project on the stress and strain situation in
helically twisted cables, including the effects of internal friction and the resultant variable bending
stiffness, will now be tested through measurements in the laboratory. The selected measuring
procedure and concomitant test setup as well as the EDP based measurements and analyses will be
described. This is followed by comparing and discussing the measured and calculated results in
Chapter 6.
A number of examples have been described in the technical literature, which has in part already been
cited here in another context, of measurements designed to check the bending characteristics of
cables. Most of the work on steel cables, primarily for application in conveyance technology, addresses
the so-called forced bending (the cable is bent over a cable drum, assuming its curvature, Fig. 5.1a),
measuring the stress in the individual wires of the cable by means of strain gauges [Schmidt, 1965,
Wieck, 1973, Windsperger, 1982, Schiffner, 1986, Wang, 1990].
The work of Wang [1990] is one of the more recent contributions in this area and, similar to this project,
deals also with the free bending of a cable – the cable is subjected to a transverse force such that the
cable is free to assume its curvature around the point where the force is introduced. He attempts to
approximate the catenary and determines the curvature therefrom by measuring the cable sag at four
points near the point where the transverse force is applied. Wyss [1956] also reports on measurements
of elongation and sag in cables subjected to transverse stress that are free to assume any curvature.
Work in the field of overhead conductors (mostly ACSR) naturally deals almost exclusively with free
bending, like this project. In almost all cases, to find the bending stiffness, the maximum sag under
constant tensile stress is determined as a function of the transverse force and from this, a “global”
bending stiffness is computed [Sturm, 1936, Morisson, 1962, McConnell and Zemke, 1980, AIF, 1991,
Zeitler, 1994]. In some of the studies in this field, the bending elongation (stress) was determined using
strain gauges near where the force was applied [Möcks and Swart, 1969, Poffenberger and Metha,
1970, Brandt, 1972, Ramey, 1987] and the results compared to those predicted by theory.

(a)
Fig. 5.1 Cable bending: (a) forced and (b) free bending
An An analysis analysis of of these these publications publications and and their measurement procedures procedures led led to to the conclusion that that a
quantum quantum leap in in measurement technology technology would be necessary to accurately accurately verify the theories
theories
expounded nded here. The measuring sensors available today and the PC PC-based based measuring processes were
ideally suited for this task. And so, within the framework of this project, the “ “cable able scanner” was created
– a symbiosis of measuring hardware and evaluation softwa software re which will be described in the following
section.
5.1. The cable scanner
The The basic basic premise premise for for the the measurement measurement of of the deformed cable condition condition rests on determining determining the
curvature curvature of of the the cable cable axis, axis, since since all all the the other other parameters calculated in the previous ch chapters ch (cable
catenary, catenary, bending bending stress, stress, secondary secondary stress, stress, bending stiffness, stiffness, bending bending moment, etc.) are functions of
this this “key parameter”. The The conductor curvature curvature can can also assist in the the indirect indirect experimental determination
of the wire stresses in the inner layers ayers – a parameter that is of particular interest (see also Chapter 7)
and and that that is is hardly hardly measurable measurable with with the strain gauge based measuring techniques techniques used so far. But, since
it is practically impossible to directly measure the curvature of the cable axis without destroying the
cable structure, the following measuring procedure was defined.
First, using contact-free free measurements, the the cable cable surface surface is surveyed near the point point where where the
transverse force is to be introduced, i.e. at the point where the bending stiffness of the cable results in
significant bending.
86
(b)

A A LDS LDS (laser (laser distance sensor) is is used for this purpose. This is is a distance sensor based on the
triangulation principle, Fig. 5.2 [Leuze, 1990], as follows:
Measuring range
Cable
Fig. 5.2 Operating principle of the LDS sensor
A A modulated laser beam beam is is generated by by an an infrared infrared laser diode and and directed directed onto onto the object object to to be
be
measured (the cable) via lenses. The diffuse reflection from the measured surface is projected onto a
high high resolution resolution analog analog position sensor via via a a system of lenses with an aperture aperture that determines the
measuring measuring range. range. The The sensor sensor generates generates two position position dependent dependent currents. Hybrid electronics eliminates
stray light and computes the e position of the diffuse reflection on the the sensor. The The power power of the laser diode
is automatically adjusted internally, to match the reflectivity of the measured object.
The The measured measured change change in in distance distance between between the the measured object object and the LDS sensor sensor is is conv
converted to
proportional analog quantities.
Laser diode
Optics
87
Optics

The The sensor sensor is is mounted mounted on on an an XYZ XYZ table table to to change position in in 3 3 dimensions, dimensions, Fig. Fig. 5.3. Resident Resident software
[Beisswanger, [Beisswanger, 1992] drives the three stepper stepper motors motors on the XYZ XYZ table (with (with feedback) to ensure tha that the
laser is always optimally positioned for the measurement (about 50 mm from the cable surface) and
processes processes the the measurement measurement data data it it receives receives from from the the laser laser amplifier. The laser moves moves to to and and fro in this
process. After each optimal positioning, the cab cable le surface perpendicular to the cable axis is scanned
800 times, Fig. 5.4.
XYZ table
Loading rig
Fig. 5.3 Operating principle of the cable scanner
The The measurement measurement data data for for this this cross cross section section are saved saved and the the sensor moves along along the cable axis to
the the next next “cross “cross section”. section”. The The distance distance between sections is is freely selectable selectable and was set set at at 3 mm for
these measurements. In this way, 52 measurements cover a measuring zone of 3 x 52 = 156 mm. This
zone was selected slightly asymmetrical to the po point int of application of the transverse force, to measure
as much as possible of the bending zone to one side of said force.
The REDU program processes these data further to find the catenary of the cable axis.
A data file containing the 52 sections with 80 800 0 points each is at first created. The points for an individual
section define the surface of the individual wires in the outer cable layer, Fig 5.4.
88
Cable

89
To define the bending line of the cable axis, i.e. the catenary, the cable centre is determined for each
individual section by detecting the individual wires in the section as follows:
1 The diameter of a wire in the outer layer and the outside diameter of the previous layer (or the
core wire diameter in a single layer cable) must be entered. By also entering the lay angle of the
outer layer, the elliptical shape of the wire cross section perpendicular to the axis of the cable is
considered, although for conventional cable values the deviation from the ideally circular is
about 1.5 % and therefore immaterial to this analysis.
2 Only the first 30 points in the section are at first analysed. These points are approximated by a
3 rd degree polynomial. Based on this polynomial, the centres of curvature of the two end points
and of the centre point of the considered 30 point section are determined. A fictitious wire centre
point is computed using the average coordinates for these three points. To evaluate the
polynomial approximation, the difference is computed between the distance of each of the 30
points to the fictitious wire centre point and the specified wire radius and the root mean square
is calculated for this variation for all the points.
3 This procedure is repeated for all 800 points of the cross section, Fig. 5.4, by sliding the window
of 30 points over the dataset (the scanned cable surface), point by point. The error curve always
has a local minimum if the window is over a single wire.
4 The centre of the cable cross section is determined from the two fictitious wire centres that
exhibit the smallest error.
5 The cable cross-section can now be constructed using the parameters determined above,
Fig. 5.5. In this way, we obtain 52 cable cross-sections, each with a centre point. These
cross-sections are shown in Fig. 5.6 for a scanned cable.
6 The catenary is thereafter approximated by a 4 th degree polynomial through the centres of the
52 cross-sections, which can then also be superimposed in the diagram with the calculated
catenary, Fig. 5.10. An optimising algorithm for minimising the squares of the errors is also
applied. By differentiating twice, the cable curvature can also be determined – a parameter
needed especially for determining the stress. The program additionally searches for the
“significant” 5 – 8 points for the envelope of the cable surface view, where the sawtooth line also
shown indicates the position of the highest point of the individual wires in the outer layer in plan
view, Fig. 5.7.

Fig. 5.4 Single cross-section section (800 measured points)
Fig. 5.5 Construction of the cable cross-section section and determination of the cable axis, using the
scanned cable surface data; note the gap between wires
Fig. 5.6 Graphic representation of the measured 52 cross cross-sections (scans)
90

Fig. 5.7 Determining the catenary (= curvature along the cable axis)
By By applying a a hidden hidden line line algorithm (program (program SEIL3D), SEIL3D), the possibility furthermore exists of showing the
scanned scanned cable length in in three three dimensions and viewing viewing this from from various perspectives, Fig. 5.8.
Fig. 5.8 Three-dimensional nsional representation of the scanned length of cable
91

5.2. The test set-up
92
To better optimise the “cable scanner”, this was initially installed and commissioned in a test rig at the
EMPA Dübendorf, Fatigue / Service Strength Department, Fig. 5.9.
Fig. 5.9 Test rig for single-layer cables
In this rig, tensile forces up to 8 kN and sags at the centre of the cable up to 30 mm could be
continuously adjusted. Only single layer cables (1 core wire and 6 layer wires) with approx. 10 mm
diameter and 3.34 m sample length were analysed here. The simple cable construction enabled fine
adjustment to match the hardware and software components described earlier and served as
preparation for the actual measurements.

It It was was soon soon evident evident that, that, in in these these relatively relatively thin thin cables, the theoretically expected influence of internal
internal
friction on bending stiffness was very small and therefore not reliably measurable.
This is also clear from the calculation of the cable catenary, Fig. 5.10. The difference between the two
calculated catenaries for (EJ) min, , top curve, and (EJ)( (EJ)(κ), ), bottom curve, is insignificant. This is
immediately immediately evident evident if if the associated bending bending stiffness stiffness curve curve is included included in this analysis, Fig. 5.11. This
shows that for a load of S = 8 kN and V max = 2.8 kN, the single-layer cable (7x3.26 mm) is in fully slipped
condition near the clamps.
The The measured values values for for the catenary, also shown in the diagram, diagram, Fig. Fig. 5.10, 5.10, show show very good
agreement with the calculated value values, however.
Sag [m]
Fig. 5.10 Measured and calculated catenary for the single layer steel cable
1: (EJ)(κ); 2: (EJ)min; ; + : measurement
93
Sag

Stiffness EJ [N*m^2]
Fig. 5.11 Bending stiffness of the steel cable 7 x 3.26 at S = 8 kN and Vmax = 2.8 kN
1: (EJ)(κ); 2: (EJ) min
The The conditions conditions for for the the choice choice of the the required parameters for the actual actual measurements were therefore
clear:
It had to be a multi-layer layer cable (since (since such cables are more common in practice practice than
than single layer
cables) cables) that that was was capable capable of of taking taking high high tensile tensile forces. forces. This creates creates high radial forces which significantly
impede impede slipping slipping between between the the wire wire layers layers and therefore therefore measurably measurably increases the bending stiffness stiffness due
due
to inner friction. The test cable should should also have have a high yield yield strength to prevent (partial) plastic
deformation of the cable cross-section section with conventional loads, which would in in turn turn complicate
complicate
interpretation of the results.
Stiffness curve
The actual measurements were carried out on a test rig, shown schematically in Fig. 5.12, of the ILS
(Institute for light-weight weight construction and cableway technology) at the ETH in Zurich.
94

Load bearing frame with 300 kN tension transducer
Lifting cylinder (max. stroke = 250 mm)
Cable φ 32 mm
Transverse force transducer / 2x 20 kN
EMPA measuring frame
Fig. 5.12 Functional diagram of the ILS test rig
95
Stroke max. = 60 mm
Hyd. group
Hyd. accumulator
(tensioning)

This machine can handle tensile forces up to 300 kN and sags in the centre of the cable up to 60 mm,
which lead to transverse forces of approx. 60 kN in the cables tested here. A hydraulic system
maintains the tensile force during the measurement.
The free ee length of the cable clamped in the test rig was 1 000 mm and the two clamps were arranged for
the test set-up up to correspond to a “cable “cable-beam”, beam”, firmly clamped on both sides and simultaneously
stressed with a tensile and a transverse force – see also Fig.4.1.
Fig. 5.13 shows some details of the measuring arrangement.
Fig. 5.13 Details of the test rig
The The transverse transverse force was was applied in in the centre of the clamped cable, cable, via a hydraulic cylinder and a
saddle with a selected radius smaller than the "free" radius assumed by the cable due to its bending
stiffness. stiffness. This This was an an attempt to to minimise the effect the the application application of of the the force force has on the inner
displacement conditions of the cable. The saddle also supported two piezo piezo-electric electric load cells and
carried the e aluminium frame for the LDS sensor, Fig. 5.14.
96

Fig. 5.14 Force introduction at the centre of the cable; top – LDS sensor
In In addition addition to to the the measurement data data from from the the “cable scanner” scanner” described earlier, the the tensile tensile force,
force,
transverse force and and transverse transverse displacement displacement at the point where the the force force was was introduced was plotted
on an X-Y Y plotter and the associated data registered on a PC, Fig. 5.15.
Fig. 5.15 Measuring and analysis equipment on the test rig
97

5.3. The test cables
A four-layer helically twisted steel cable was chosen as the test sample.
98
Table 5.1a shows the cable data and Fig. 5.16a shows the construction of the cable. This cable will
hereinafter be referred to as S32.
To complete the link to overhead conductors, a typical ACSR conductor of type Cardinal was in addition
analysed. These conductors are frequently used for 400 kV overhead lines. Fig. 1.1 Table 5.2b shows
the data and Fig. 5.16b shows a cross section to illustrate the construction of the Cardinal conductor.
Layer
No. of
wires
Wire φ
[mm]
Young’s modulus
[N/mm^2]
Core 1 3.72 197'183.0 —
1 6 3.54 180'645.0 14.22
2 12 3.54 180'645.0 13.69
3 18 3.54 180'645.0 13.99
4 24 3.54 180'645.0 13.97
Table 5.1a Data for the steel cable S32
Layer
No. of
wires
Wire φ
[mm]
Young’s modulus
[N/mm^2]
Core 1 3.34 210'000.0 —
1 6 3.34 176'800.0 6.06
2 12 3.32 58'900.0 11.99
3 18 3.32 65'360.0 11.80
4 24 3.32 64'800.0 13.10
Table 5.1b Data for the ACSR conductor Cardinal
ß
[°]
ß
[°]

(a)
Fig. 5.16 Cross Cross section section of (a) (a) S32 and (b) (b) Cardinal Cardinal (schematic: (schematic: in in reality, reality, the the wire wire cross cross-sections
cross
shown shown here here are elliptical and the the layer layer wires wires exhibit gaps gaps along along the circumference)
5.4. The material properties
Even though Chapter 6 shows that the results are relatively independent of Young's modulus for the
wires, wires, tensile tests tests were were carried carried out out on on individual individual (straightened) wires taken from from each each layer layer of of the two
test samples S32 and Cardinal. The sample lengths were 100 mm for all the tests and the tests were
carried out in accordance with DIN 51210 51210 at the Swiss Federal Aircraft Manufacturer Manufacturer Emmen (F+W).
In In all of the the tensile tests, tests, the the wires wires were were tensioned tensioned to approx. approx. 1% elongation five five times in in succession succession and
and
then tensioned to breaking point.
As expected, the linear elastic region of the stress-strain stress strain curve for the wires was slightly extended due
to to straightening straightening of of the the originally originally helical helical layer layer wires wires and to work work hardening, hardening, to to the following values:
values:
approx. 1 400 N/mm 2 for S32 (steel wires)
approx. 1 300 N/mm 2 for the Cardinal steel wires
approx. 160, 170, 180 N/mm 2 , depending on layer, for the Cardinal aluminium wires
Young's modulus for the wires (found from the slope of the recorded σ-ε characteristics) remained
constant constant even during during repeated repeated tensioning, tensioning, whereas whereas the originally straight core core wires wires exhibited exhibited clearly
clearly
larger elasticity values [E-values].
99
(b)

The input values for the SEIL program were finally chosen as:
Steel cable : E core
Steel cable : E layer
Cardinal : Ecore
Cardinal : Esteel layer
Cardinal : Ealum. layer
5.5. The friction coefficients
The friction coefficient between the different wire layers of the cable/conductor is a significant factor in
the the theory theory of of variable variable bending bending stiffness stiffness of of cables/conductors cables/conductors expounded here, as is easily realised by
inspection of the various equations derived in Chapters 2 and 3. The measurement of this coefficient of
friction is described in this section.
Various Various tests were were carried carried out out for for this this purpose at at the laboratory for for dry dry friction friction and vibration wear of the
Bundesanstalt für Materialprüfung, Berlin (BAM), as summarised below.
Friction and wear wear tests tests were carried out out on on crossed steel and aluminium wires subjected to oscillating
sliding movement. The test material comprised of 5 cm lengths of single wires, cut from the S32 cable
and Cardinal conductor under investigation. The samples were use used as delivered.
Fig. 5.17 shows the tribology system and the parameters used in these tests:
Fig. 5.17 Tribology system
100
= 200.000 N/mm 2
= 180.000 N/mm 2
= 210.000 N/mm 2
= 180.000 N/mm 2
= 65.000 N/mm 2

Test parameters
Oscillating width dx
(double amplitude)
Oscillating frequency
Normal force FN
No. of cycles N
Crossing angle
Temperature T
Rel. humidity U
=
=
=
=
=
=
=
60 - 80 µm
0.5 Hz
20 N
0 - 2000
25°
23° C
30 - 50 %
Fig. Fig. 5.18 5.18 explains explains the the measurement measurement and and shows a a typical typical curve curve of the coefficient of of friction µ.
101
No. of cycles N. 1000
Fig. 5.18 Typical curve for the coefficient of friction µ (right hand scale) vs. the number of cycles N

Triggered by the dx signal, a 150 point force-displacement curve is measured after every second cycle
(i.e. 1000 times in the course of these these tests, tests, with with N N = 2000). 2000). From From the the region of this so so-called friction
force hysteresis overlaid in Fig. 5.18, the coefficient of friction µ is computed for each ten measuring
points (n = 10) using the average values of the friction forces +F +FRm1 and -FRm2 in accordance with the
following equation:
In all the tests, the curve of the coefficient of friction µ exhibited a fuzzy two-phase phase behaviour that has
also been documented by others [Waterhouse, 1994]:
Phase 1: low coefficient of friction
Phase 2: high coefficient of friction
The The results results for for the the coefficients coefficients of of friction should should therefore rather rather be considered considered as tendencies.
The The evaluation evaluation of these these tests tests yielded the the following following coefficients of friction as inputs for the calculations in
the SEIL program:
- for S32 :
- for Cardinal :
It It should should be be noted that that the higher values values for for the the steel steel cable cable S32 S32 (compared (compared to earlier data [Wang, 1990]
and and measurements measurements [Leider, [Leider, 1975] 1975] can can be explained by the surface surface condition condition of the tested steel wires:
these these wires are are coated coated with grey, thick thick deposits (zinc dust dust paint) that that are are also also not not homogeneously
homogeneously
distributed. distributed. These These relatively relatively high coefficients coefficients of friction friction are not totally totally unwelcome, unwelcome, however, since they
they
enhance the effect of internal friction ion on on the bending stiffness of the cable, cable, which which is is the objective objective of this
project.
102
µ = 0.3-0.4
µ = 0.3-0.7
In In the friction friction measurements measurements on on the the Cardinal Cardinal wires, wires, the the friction friction coefficient coefficient values for aluminium wire
pairings (Al/Al and Al/St) of approx. µ = 0.5 compare well with other work [CEA, 1986] and can be
explained explained by by “indentation” “indentation” of of the the relatively relatively soft aluminium wires; the the somewhat lower values for steel
wire pairing (layer 1 / core wire) of approx. µ = 0.3, are comparable to values found in the literature.

6. Analysis
6.1. Test results for the steel cable
103
As explained in Chapter 5 already, the major measurements were carried out on cables clamped on
both sides with a free length of one metre between, Fig. 4.1. In this chapter, we shall discuss the
measurements on the four-layer helically twisted steel cable S32, Chapter 5.3.
Table 6.1 shows the test series that were performed. In these tests, the cable tension S was kept
constant whilst the transverse force Vmax was continuously increased up to a maximum value of about
10 – 15 % of the tensile force. Before the actual measurement, the cable was tensioned ten times,
without transverse stress, up to the maximum possible tensile force of 300 kN, to encourage even
“settling” of the wires in the cable structure.
S [kN] 280 140 80 40
Vmax [kN] 40 20 10 5
Table 6.1 Tensile and transverse forces during the measurements on the steel cable S32
All the test series were carried out at least twice to document the repeatability of the results. The
differences between the two measurement series were within the measuring accuracies in almost all
instances.
In each case, the catenary was measured near the transverse force application point and also the
force-displacement curve at the centre of the cable. These were then compared to the calculations. Fig.
6.1 below shows the catenary for the maximum tensile force / transverse force combination. The
crosses in this figure represent the corresponding SEIL program calculation results, for comparison.

Sag [m]
Fig. 6.1 Measured and calculated catenary for S32 for S = 280 kN and
Vmax = 40 kN; 1: variable stiffness (EJ)( (EJ)(κ); 2: minimum (wire)
stiffness (EJ)min; ; 3: maximum (cable) stiffness (EJ) (EJ)max; + : measurement
It is evident from this figure igure that that the the agreement agreement between between the the FE FE calculations calculations (input (input data: data: see see Fig. 4.5, 4.6
and and 5.1a) 5.1a) and the the measurements measurements is is very very good when applying the variable cable stiffness stiffness (EJ)( (EJ)(κ)
calculations introduced here (curve 1). The result of calculations based on the generally accepted
simplifying assumption of constant bending stiffness (EJ) min, curve 2, or (EJ) max, , curve 3) is, however,
quite different from the measurements.
Note that the fact that the catenary based on variable stiffness lies between that for maximum (no wires
have have slipped) and and minimum (all wires slipped) slipped) stiffnesses lends credibility to to these results.
The curve of bending stiffness along the sample cable length is shown in Fig. 6.2.
The curve shows that wire slippage has already affected a large part of the cable at this load. In the
centre of the cable in particular, at x = 0, all the wires in the cable have already slipped except those in
the inner layer; the bending stiffness of the cable is 820 Nm 2 at this point, which is about ten times the
wire stiffness (EJ)min.
Figures Figures 6.3, 6.3, 6.4 6.4 and and 6.5 6.5 show show the catenary for other other tensile tensile / / transverse force force combinations in Table 6.1.
These figures always show the catenaries calculated with (EJ) (EJ)min and (EJ)max and d also the measured
values.
104
Sag

Stiffness EJ [N*m^2]
Fig. 6.2 Bending stiffness of S32 at S = 280 kN and Vmax = 40 kN;
1: (EJ)(κ); 2: (EJ)min; ; 3: (EJ) (EJ)max
Sag [m]
105
Stiffness curve
Sag
Fig. 6.3 Measured and calculated catenary for S32 at S = 140 kN and
Vmax = 20 kN; 1: (EJ)(κ); ); 2: (EJ)min; (EJ) 3: (EJ)max

Sag [m]
Fig. 6.4 Measured and calculated catenary for S32 at S = 80 kN and
Vmax = 10 kN; 1: (EJ)(κ); ); 2: (EJ) min; 3: (EJ) max
Sag [m]
106
Sag
Sag
Fig. 6.5 Measured and calculated catenary for S32 at S = 40 kN and
Vmax = 5 kN; 1: (EJ)(κ); ); 2: (EJ) (EJ)min; 3: (EJ)max

The agreement between theory and measurements is very good at these loads as well.
As explained in detail in Section 4.1, the variable bending stiffness (EJ)(κ) ) assumed here gives rise to a
non-linear force-displacement characteristic characteristic or hysteresis hysteresis as the transverse load load on the cable is
is
increased increased and and reduced reduced again. again. This is also also used used here here to verify verify the theoretical conclusions. Fig. 6.6 bel below
shows the measured and calculated (curve 1) hysteresis of the cable S32 at S = 280 kN = constant and
V = variable = 0 kN – 40 kN – 0 kN. kN. Once again, again, there is is good agreement agreement between both the absolute
magnitude of the maximum sag at the centre of the cabl cable e (which is also influenced by the sample length
and and the the stiffness stiffness of the the support) support) and and the shape of the hysteresis curve. curve. In this this respect, it is is evident evident that
that
the measured trajectory of the force force-displacement curve does not agree with the calculations as well we in
the first 1/3 rd of of the the rising rising part part as it it does does for for falling loads, where it is virtually identical identical with the
calculations. calculations. The The explanation explanation might be a settling process of the wires in the the cable cable structure that that is not
yet fully completed as the load is inc increased again – as also described elsewhere [Raoof und Huang,
1992]. 1992]. In In the the relaxation relaxation phase, phase, however, however, the the cable cable starts from a a fully stressed condition, condition, in which its
behaviour behaviour is is obviously obviously very very well well described described by by the the theory theory of of variable variable bending stiffness expo expounded expo here.
This This might might also also explain explain the the somewhat somewhat larger larger variances between measurement and calculation calculation at at lower
cable cable tensions, as well well as the the tendency for “higher” catenaries found in the measurements.
measurements.
107
Force Force-displacement
Sag [m]
Fig. 6.6 Hysteresis in S32 at S = 280 kN and Vmax = 40 kN; 1: (EJ)(κ);
2: (EJ) min; 3: (EJ) max ; measurements shown in bold

108
Fig. 6.6 also clearly shows that the cable hysteresis “disappears” under the conventional assumption of
constant bending stiffness (EJ)min, curve 2, or (EJ)max, curve 3 and that the maximum sag does not
agree with the measured values in both cases.
6.2. Test results for the ACSR conductor
One of the goals of this project is to investigate the bending properties of reinforced conductors such as
the aluminium-steel conductors (ACSR) that are typically used in high voltage overhead lines. A number
of supplementary tests were for this reason carried out to verify that the approach of variable bending
stiffness is applicable to this special type of helically twisted cable as well. The tests were similar to
those described above for the all-steel cable, carried out on an ACSR conductor code-named Cardinal,
Section 5.3.
Due to the lower breaking strength of Cardinal as compared to the S32 steel cable, the tensile forces
had to be limited to lower values. The relevant measurements were then carried out with a tensile force
of 40 kN – approx. 25 % of the nominal Cardinal breaking strength – which also approximates the
tensile forces in this conductor when used on overhead lines. The maximum transverse force at the
centre of the conductor was limited to 4 kN - approx. 10 % of the conductor tension. It was noted in this
respect that the aluminium wires of the conductor were still in the elastic region of the σ - ε characteristic
for this stress combination, Section 5.4.
Figures 6.7 to 6.9 again show the comparison between calculations and measurements for Cardinal.
Figure 6.7 shows the measured and calculated (using (EJ)(κ)) catenary for Cardinal for S = 40 kN and
V max = 4 kN, also showing the catenaries for constant bending stiffnesses (EJ) min and (EJ) max resp.
Again, the good agreement with the measurements is striking. Figure 6.8 shows the (symmetrical)
stiffness curve for half the conductor sample length (0.5 m) and Figure 6.9 shows the measured and
calculated force-displacement curve (hysteresis).
Reasonable agreement between FE calculations and measurements is apparent also for this type of
helically twisted cable (overhead line reinforced conductors: ACSR), which strengthens the general
validity of the conductor model postulated here and the chosen calculation procedure.

Sag [m]
Fig. 6.7 Measured and calculated catenary for Cardinal at S = 40 kN and
Vmax = 4 kN; 1: (EJ)(κ); ); 2: (EJ) (EJ)min; 3: (EJ)max
Stiffness EJ [N*m^2]
109
Sag
SStiffness
curve
Fig. 6.8 Bending stiffness for Cardinal at S = 40 kN and Vmax V = 4 kN;
1: (EJ)(κ); 2: (EJ)min; ; 3: (EJ)max (EJ)

Fig. 6.9 Hysteresis for Cardinal at S = 40 kN and Vmax = 4 kN; 1: (EJ)(κ);
2: (EJ)min; 3: (EJ)max ; measurements shown in bold
6.3. Sensitivity analysis
Having Having documented documented the the good agreement between between theory and measurements, a a sensitivity analysis analysis was
subsequently performed ormed to to evaluate evaluate the effect of the various parameters in the calculations, such as the
cable/conductor data, the stresses and the calculation assumptions.
This This analysis analysis shows shows that that the the wire wire diameter diameter and Young's Young's modulus modulus for for the the wires, wires, provided they they vary within
a a range range given given by by the the manufacturing manufacturing tolerances, tolerances, have have little little effect effect on on the the calculated calculated results. Other values,
such such as as the lay angle, angle, have more of an an influence, even though this parameter is generally not quoted in
the manufacturer’s data sheets. This fact could also be used to achieve certain cable properties, such
as as reducing reducing the the maximum maximum curvature curvature and and therefore the maximum wire tension tension at a predetermined load,
by by judicious judicious choice choice of of the the lay lay angle. angle. This This angle angle is not not prescribed prescribed in the cable cable standards standards but it is merely
required to specify a range.
110
Force Force-displacement curve
Sag [m]

111
As expected, the coefficient of friction has a decisive influence on the calculated results. This is
somewhat disappointing because, as shown in Section 5.5, this parameter is difficult to quantify. The
coefficient of friction is furthermore likely to change over the service life of the cable, due possibly to
corrosion or to abraded material at the contact points between wires. This knowledge of the large
influence the coefficient of friction is nevertheless valuable, since the theory expounded here allows us
to quantify this and manage its effect on cable stresses – by the choice of suitable lubricating agents, for
instance.
Apart from the cable data, the external forces applied in the course of the tests may also vary within
certain ranges, thereby affecting the results of the measurements. Due to the test rig, the tensile force,
assumed constant in the theoretical analysis, may vary up to ± 5 %. Despite special arrangements
(liberally dimensioned hydraulic reservoir of the tensile test machine) this was in evidence during the
tests. The effects of these fluctuations on the results are of minor significance, however.
6.4. Influence of the bending stiffness
The bending process in tensioned cables was discussed in Section 3.5 and illustrated in Fig. 3.7a-e, on
the basis of the M-B diagram and with different assumptions for the bending stiffness of the cable.
Using the SEIL program, these relationships will be demonstrated by example of the test series. In the
data entry for the SEIL program, it is possible to choose between different stiffnesses, Fig. 4.8.:
(EJ)(κ) Variable stiffness, as a function of the curvature (Fig. 3.7)
(EJ)min Minimum (wire) stiffness (Fig. 3.7a)
(EJ) max
Maximum (cable) stiffness (Fig. 3.7b)
(EJ)(a.-L.) Stiffness if only the outer layer (a.-L) slips (Fig. 3.7c)
(EJ)(ver) Stiffness if all the layers slip simultaneously after a
certain transition curvature (Fig. 3.7d), (ver = simplified)
(EJ)(def) The cable stiffness is defined as constant with a value between
(EJ)min and (EJ)max (Fig.3.7e), (def = defined)

Figure Figure 6.10 6.10 shows shows the the curves curves for for these these different stiffnesses as a function of of the the cable cable curvature, at at a
tensile force of S = 280 kN, for S32.
Stiffness [N*m^2]
Fig. 6.10 The different bending stiffnesses for S32, as a function of
the curvature at S = 280 kN; 1: (EJ)(κ); 2: (EJ)min; 3: (EJ)max;
4: (EJ)(a.-L.); L.); 5: (EJ)(ver); 6: (EJ)(def)
Figure 6.11 shows the catenary of of the the cable cable for these different assumptions of of stiffness. The measured
values from the test with S = 280 kN and VVmax
= 40 kN are also shown.
It is evident from this figure igure and from Fig. 6.12 that, that, as as expected and as qualitatively qualitatively described described in
Section 3.5, , curve curve 1 (variable (variable stiffness) predicts the the bending behaviour of the cable best and and curve curve 5
(stiffness (stiffness based based on on the the assumption assumption that that all all wires will slip slip after a a certain transition curvature) comes a
close second. Figure 6.1 also shows the catenary as determ determined ined using the analytical calculations in
Section 4.2 for comparison.
Figure Figure 6.12 6.12 finally finally shows the bending moment moment at at the clamps as a function of of the the curvature curvature at at the the same
point, with the different assumptions for bending stiffness as a parameter. It was shown in Fig. 3.7
already already how this diagram, diagram, computed computed by by the SEIL SEIL program with the data for for cable parameters and
and
stresses stresses as as inputs, may also also be constructed with the knowledge of the internal internal friction moments of the
cable.
112
Overall stiffness
Curvature [1/m]

Sag [m]
Fig. 6.11 Catenary for S32 with S = 280 kN and Vmax = 40 kN, with different
stiffness assumptions; 1: (EJ)( (EJ)(κ); 2: (EJ) min; 3: (EJ) max; 4: (EJ)(a.-L.);
5: (EJ)(ver); 6: (EJ)(def); 7: analytical; + : measurement
Bending moment [Nm]
Overall bending moment
Fig. 6.12 Bending moment for S32 with S = 280 kN and Vmax = 40 kN with different
stiffness assumptions; 1: (EJ)( (EJ)(κ); 2: (EJ)min: (EJ)max; 4: (EJ)(a.-L.);
5: (EJ)(ver); 6: (EJ)(def)
113
Sag
Curvature [1/m]

6.5. Effect of the sample length
114
Following the good agreement between the above measurement results and the calculations, a
concluding supplementary measurement was performed to establish to what extent the relatively short
sample length of one metre (i.e. clamped length = approx. 3 x lay lengths of the outer layer) influenced
the measurement results. The S32 and Cardinal test samples were subjected to a further measurement
in a cable testing machine of the ILS for this purpose. This machine (see Fig. 6.13) can clamp samples
of up to 3 m in length (approx. 9 lay lengths), applying a tensile force of up to 300 kN.
A defined deflection, or transverse force, was applied to the samples by means of two coil springs, Fig.
6.14. The samples were guided in the machine on three steel rollers to again create a static load system
as in Fig. 4.1. Since this was only a check measurement, only the force-displacement curve for the
centre of the samples (= force introduction) was recorded using the calibration curve of the spring and
an inductive linear transducer.
The measurement results, i.e. the force-displacement curve, are shown in Fig. 6.15 for S32 at a tensile
force of 140 kN and in Fig. 6.16 for Cardinal at a tensile force of 80 kN. The results of the FE
calculations for these stresses, subject to the assumption of variable bending stiffness (EJ)(κ), curve 1,
are also shown in the Figures, for comparison. Again we note the good agreement with the theory,
especially in the light of the rather large discrepancies between the measurement and calculations
based on the conventional assumptions of constant bending stiffness (EJ)min, curve 2, or (EJ)max, curve
3.

115
Fig. 6.13 300 kN tensile testing machine for measurements
on approx. 3 m long samples
Fig. 6.14 Force introduction at the centre of the sample,
using coil spring and linear transducer

116
Force--displacement
curve
Fig. 6.15 Hysteresis in S32 at S = 140 kN and Vmax = 10 kN;
1: (EJ)(κ); 2: (EJ)min; ; 3: (EJ)max (EJ) ; measurements shown in bold
Force Force-displacement curve
Fig. 6.16 Hysteresis for Cardinal at S = 80 kN and Vmax = 10 kN;
1: (EJ)(κ); 2: (EJ)min; ; 3: (EJ)max (EJ) ; measurements shown in bold
Sag [m]
Sag [m]

7. Applications
7.1. Wire stresses in vibrating conductors
In In this this Chapter, Chapter, we we shall shall try try to to transfer transfer the the results results of of this this work work into practise. A method should in
particular particular be be found found how how to to apply apply the the knowledge knowledge gained gained on the the bending behaviour behaviour of conductors to the
determination determination of wire stresses caused caused by by wind wind induced induced conductor conductor vibration. vibration. This, as mentioned in
Chapter 1, is of special significance to the service reliability rel of overhead lines.
For this purpose, the quasi-static static theory theory of bending postulated here is is extended to to the two dynamic
(periodic) (periodic) alternating alternating bending bending patterns patterns in in conductor conductor vibration, vibration, as applied in some some earlier earlier work work [EPRI,
[EPRI,
1979, CEA, 1986] and as theoretically theoretically fleshed out by Hagedorn [1980] and Hartmann and and Kern Kern [1991].
In this respect, the following assumptions are introduced in a first approximation:
Only Only the the very very small small deflections deflections at at the the clamp clamp corresponding corresponding to the conditions during during conductor vi vibration
are are considered; considered; the large large static static sag sag defined defined by by the conductor conductor catenary catenary is is ignored. It is is therefore
assumed assumed that that the relevant relevant (dynamic) (dynamic) stresses and and deformations deformations may be be superimposed over the
corresponding static values, both of conductors that are iin
n an original straight condition and of
conductors conductors that that have have already already been been (statically) (statically) deformed. deformed. The The presence presence of a a decay length
length lA is
furthermore furthermore assumed, assumed, after which the conductor experiences only a tensile force, but but no no moment moment M.
M.
Fig. 7.1 Definition of f the decay length of the bending moment at the clamp
117

This This is is based based on on earlier earlier measurements measurements [Helms, [Helms, 1964], showing that a a decay length exists,
corresponding approximately to the lay length of the outer layer of tthe
he conductor, Fig. 7.1.
In In the the following following it it is is assumed that that the conductor is originally rigidly clamped, extending horizontally on
one one side side and and “stressed” “stressed” with with a a displacement corresponding to to the the measured measured bending amplitude Y YYb
of
the the conductor at this this point. point. The The measuring sensor is commonly commonly set set up up to to measure measure at a point x b, 89 mm
(31/2”) ”) from from the last last point point where where the the clamp clamp touches the conductor, Fig. 7.2. 7.2. In this respect it must be
noted that double amplitudes are generally meas measured ured and registered in the overhead line field and that
“half” the applied amplitude must be entered to make the results comparable.
Fig. 7.2 Method for measuring the bending amplitude on a conductor
With these assumptions and under this stress, we now calculate the conductor length near the clamp,
using using the the methods methods proposed proposed in the preceding preceding sections. The The results results include, among other, the stresses
in in the the different different wire wire layers. layers. This This is because because the the SEIL SEIL program used used has the option option of of calculating the
stresses ses both both along the the length length of each individual wire wire in the axial axial direction of of the conductor and and also for
all the wires in the same conductor cross cross-section, section, especially at the clamp. The components of the wire
stress, i.e. the tensile stress, bending stress and secondary stress, can also be shown separately,
which which helps helps in in identifying identifying each each individual individual component’s component’s contribution to to the total stress stress of of the the considered
wire.
These These typical stresses stresses are shown shown in Figures 7.3 to 7.5 for for the outermost layer layer (layer (layer 4) of t tthe
Cardinal
conductor calculated in Section 7.2.
118

Bending stress [N/mm^2]
Bending stress curve
Fig. 7.3a Bending stress curve for the individual wires in the Cardinal cross cross-section
at the clamp; ϕ = π/2: /2: “top” wire, see Fig. 4.6; the horizontal line represents the Reuleaux
stress, see Sec. 2.3.
Bending stress [N/mm^2]
Bending stress curve
Fig. 7.3b Bending stress along the first layer of a wire in the 4
point x = 0 is the position of the clamp
th Cardinal layer;
0 is the position of the clamp
119

Secondary stress [N/mm^2]
Secondary stress curve
Fig. 7.4a Secondary stress of the individual wires in the Cardinal cross-section cross section at the clamp;
ϕ = π/2: /2: “top” wire, see Fig. 4.6
Secondary stress [N/mm^2]
Secondary stress curve
Fig. 7.4b Secondary stress along the first layer of a wire in the 4
point x = 0 is the position of the clamp
th Cardinal layer;
0 is the position of the clamp
120

Total stress [N/mm^2]
Fig. 7.5a Total stress in the individual wires in the Cardinal cross-section section at the clamp;
ϕ = π/2: /2: “top” wire, see Fig. 4.6; the horizontal line represents the “pure”,
constant tensile stress in the wire
Total stress [N/mm^2]
Total stress curve
Fig. 7.5b Total stress in the first lay length of a wire in the 4
position x = 0 is at the clamp; the horizontal line represents the “pure” constant
tensile stress in the wire
th layer of Cardinal;
0 is at the clamp; the horizontal line represents the “pure” constant
121
Total stress curve

After After the the wire stresses stresses have have been been calculated calculated using using the the calculation calculation procedure introduced here, a critical
vibration vibration amplitude amplitude could now now be be calculated calculated for every every overhead overhead conductor by by comparing, in a first
estimate estimate of of the the service service reliability reliability of of the the conductor, conductor, the the maximum maximum calculated calculated stress in the aluminium
wires with the fatigue strength of such wires of approx. 35 N/mm 2 (axially, normal stress), as
experimentally determined [CEA, 1986].
7.2. Practical approach
The The above procedure is is now now further explained with reference to the overhead conductors (type
Cardinal) which were used for the measurements in this project. These conductors, as mentioned
earlier, are often used particularly for high voltage transmission lines.
In a thought experiment, a 360 mm mm length length of of Cardinal conductor (which closely closely approximates the lay
length la of the outer aluminium wire layer of this cconductor)
onductor) is clamped on one side and stressed with a
tensile force S and with a predetermined displacement ymax y at the free end, Fig. 7.6. The tensile force S
on the conductor is chosen as 20 % % of of the calculated calculated Cardinal Cardinal breaking strength of 152 kN, i.e. about
30.5 kN. kN. This This tensile tensile stress stress is probably probably close to the average tensile stress stress in in this this type of conductor in a
practical application. The predetermined displacement yymax
at the free end of the considered conductor
length is chosen as 1.45 mm, such that tthe
sag of the conductor is 0.14 mm at a distance of 89 mm from
the the clamp. clamp. According According to to current current overhead overhead line line design design practise, this value at this point (where practical
measurements measurements are are also also taken) taken) and and at at this this average average tensile tensile stress stress corresponds to the conse conservative limit
for the double bending amplitude Y b of 0.28 mm [EPRI, 1979].
Fig. 7.6 Principle for calculating alternating bending stresses in the wires
of vibrating overhead conductors
122
Figure 7.7 shows the curve for the bending stiffness of this conductor in the configuration chosen here,
Fig. Fig. 7.6. 7.6. It It turns turns out out that that only only the the wires in in the the outer outer layer of of the the conductor conductor have slipped fully at this stress.

Stiffness EJ [N*m^2]
Fig. 7.7 Bending stiffness curve for Cardinal, Ca “stressed” as in Fig. 7.6
Table Table 7.1 7.1 shows shows the the maximum maximum value value for for the different wire wire stresses in the individual layers of this
conductor. conductor. The The maximum longitudinal longitudinal fatigue stress stress in the aluminium aluminium wires wires (layers 2, 3, 4), which is
relevant in this analysis, lysis, is is the the sum sum of of the the wire wire bending bending stress and the the secondary stress and and amounts amounts to
to
approx. 31 N/mm 2 . . It It occurs occurs in in the penultimate penultimate layer layer (just under under the outer layer) and its value is still
still
below the above cited fatigue limit for aluminium wires of 35 N/mm 2 . For this conductor therefore, the
agreement agreement between between the the theory theory expounded expounded here here and and the the conventional conventional practical method is good. Both
Both
methods lead to comparable results.
Wire stresses
- - - - - - - - - - - - - - - - - - - - -
Tensile str.
[N/mm^2]
Core wire 165.59 0.00 14.92
Layer 1 137.86 24.84 12.15
Layer 2 44.44 15.99 3.64
Layer 3 49.38 26.62 4.05
Layer 4 48.47 20.73* 3.90
Table. 7.1 Wire stresses in the individual Cardinal layers when stressed as in
Fig. 7.6; *: this layer has slipped
123
Stiffness curve
Secondary str.
[N/mm^2]
Bending str.
[N/mm^2]
Total stress
[N/mm^2]
180.51
174.84
64.07
80.05
73.10

124
The above is summarised in Fig. 7.8 below. The figure shows the maximum wire stresses at the clamp
and as a function of the bending amplitude at 89 mm from the clamp – for all Cardinal aluminium layers;
layer 4 is the outer layer. These curves show the physically realistic conversion between bending
amplitude and wire stresses. For comparison, the figure also shows the curve for the traditionally
applied conversion equation according to Poffenberger-Swart [1965], Annexure V, which assumes a
constant bending stiffness of the conductor, based on the minimum (wire) stiffness. This shows that the
conversion in use today considerably underestimates the wire stresses arising from alternating
bending, which in turn requires the application of a massively lower fatigue limit for conductors – approx.
8.5 N/mm 2 - [EPRI, 1979], as compared to the experimentally determined fatigue strength of individual
wires - approx. 35 N/mm 2 - [CEA, 1986]. These two limits are also shown in Fig. 7.8, for comparison.
It is further clear from this figure that, at a vibration amplitude of 0.14 mm for the Cardinal conductor
considered here, individual aluminium wires in the third layer almost reach the fatigue limit of approx.
35 N/mm 2 . Table 7.1, on the other hand, shows that only the outer aluminium layer of the conductor has
slipped in this case – indicated by an *. The “pure” bending stress of the wires in this layer – i.e. the
bending stress caused by bending around the wire axis – is only 3.9 N/mm 2 .
The decisive fact here is that the secondary stress must also be considered when calculating the wire
stress at this point - which was neglected in the Poffenberger-Swart approach. This thus far “forgotten”
secondary stress is identical in the fully slipped outer layer of the conductor to the maximum secondary
stress this layer can resist and can be calculated as 20.7 N/mm 2 according to 2.17, Table 7.1.
The longitudinal fatigue stress (bending stress and secondary stress) in the individual wires of the outer
Cardinal layer is therefore approx. 25 N/mm 2 . The static tensile stress in the wires must be added to
this, although it is of minor importance in this context [CIGRE, 1979, CEA, 1986]. It amounts to
48.5 N/mm 2 for the wires in the outer layer, Table 7.1.
This statement on the “forgotten” secondary stress is in agreement with various experimental
investigations in which dynamic measurements of the conductor elongation were carried out and which
on occasion noted a discrepancy between the stress calculated in acc. with Poffenberger-Swart and the
actually measured wire stress [Hondalus, 1965, Claren and Diana, 1969, CEA 1986, Ramey, 1987]. To
date, this fact was, as noted above, "corrected" by defining a much lower fatigue limit for the conductor
than for the individual wire, viz. 8.5 N/mm 2 instead of 35 N/mm 2 .

Longitudinal fatigue wire stresses [N/mm 2 ]
Max. recommended bending
amplitude for Cardinal [EPRI, 1979]
Layer 2
Fig. 7.8 Longitudinal fatigue wire stresses in the individual Cardinal aluminium layers,
as as calculated with with the the SEIL SEIL program program at at the the conductor conductor clamp, clamp, as a a function function of
the bending amplitude YYb/2
at 89 mm distance from the he clamp; the wire bending
stress of layer 4 is shown in bold, for compar comparison.
125
Poffenberger-
Swart equation
(layer 4)
Layer 3
Layer 4
Design limit for alum. wires
[CEA, 1986]
Design limit for ACSR conductors [EPRI, 1979]
Bending amplitude [µm]

The The trend trend determined here here for for the the Cardinal Cardinal conductor conductor and shown shown in in Fig. 7.8 7.8 is confirmed also in the very
comprehensive comprehensive study study by Kommenda Kommenda and Swart Swart [1967], in which almost 2000 measurements were
performed on seven different overhead conductors to determine the relationship between conductor
bending amplitudes and wire bending stress: the conversion using Poffenberger-Swart Poffenberger Swart yields higher
and and lower lower values of of wire wire stress for for larger and smaller bending amplitudes respectively. respectively. This
This
non-linearity ty was evident evident already already in in the cited cited measurements measurements by Kommenda Kommenda and and Swart [1967] and has
now, now, for for the the first first time, time, been been quantified quantified within within the the framework framework of of this work. It is is becoming increasingly
significant in terms of the methods applied today for estimating tthe
he service life of overhead conductors
[Bückner, [Bückner, Helms, Helms, Papailiou, Papailiou, 1985, 1985, 1988], 1988], since since not only only the maximum bending stress, stress, but but also the
entire entire measured collection collection of of bending bending amplitudes under load, load, must must be converted to wire stresses as
realistically as possible.
Fig. Fig. 7.9 7.9 schematically schematically illustrates illustrates the the longitudinal longitudinal wire wire fatigue fatigue stresses for a a vibrating overhead
overhead
conductor conductor in in the the case case where where only only the the outer outer wire wire layer has slipped, slipped, both in accordance with
Poffenberger-Swart Swart (a) and in accordance with the conductor model developed here (b). Since the
Poffenberger-Swart Swart equation (see also Annexure V) assumes a constant stiffness (EJ) min – sum of the
bending stiffnesses of the individual wires of the conductor – it yields a softer conductor than the actual
(see also Fig. ig. 3.7b), 3.7b), which which in in turn turn yields yields a a larger curvature curvature and larger larger wire bending stress than in reality.
This effect was compensated for in part because the Poffenberger
Poffenberger-Swart Swart calculation ignores the
secondary stress which, as already shown, has the effect of a longitudinal fatigue stress in the wires,
also also when the wires have have fully fully slipped. slipped. It is also evident from Fig. 7.8 7.8 that that the the different different bending bending stresses
in in the the individual wire wire layers layers can can be be calculated and shown (see also also Fig. 7.8 7.8 and and Table Table 7.1) if if the
effective ive variable bending stiffnesses in the conductor are considered.
Fig. 7.9 Schematic illustration of the wire stresses during conductor vibration:
(a) in acc. with Poffenberger-Swart Poffenberger with (EJ)min = constant;
(b) in acc. with the SEIL model with (EJ)(κ) (EJ)( = variable
126

127
7.3. Effect of the friction corrosion and the static pre-stress
A possibility was discussed in the preceding section how to apply the conductor model developed within
the framework of this project to the alternating bending stresses in the individual conductor wires, which
is of primary concern in practise. As explained in Chapter 1 already, these are caused by wind-induced
conductor vibrations.
Since the last-mentioned stresses are definitive for fatigue in the wire material (which in extreme cases
may lead to wire and conductor breakages) it is opportune in this section to make a few observations on
the commonalities and differences between the processes in the conductor as described and
experimentally confirmed here and the actual physical processes as conductors vibrate.
Friction corrosion
As explained already in the introduction (Chapter 1), it has been credibly documented [EPRI 1979,
CIGRE 1988] that fatigue in overhead conductors can be attributed to friction corrosion at the points of
contact between the wires of the individual layers. It is interesting to note that wire breakages at the
point of contact between two wire layers are about ten times more frequent than breakages at contact
points between two wires of the same layer [Cardou and Cloutier, 1989].
This is understandable since the circumferential forces at the points of contact between wires of the
same layer are proportional to the sine of the lay angle of that layer, which is significantly smaller than
the radial forces between two adjoining layers (Chapter 2) and which, in addition, add up from “outside”
to “inside” (Chapter 3). Neglecting these contact “lines” between two adjacent wires in the same layer
therefore appears justified in the calculation of the bending stiffness.
The friction corrosion process occurring during conductor vibration is rather complicated and depends
on several parameters. Apart from the pressure at the points of contact and the amplitude of the stress
vibration (and the relative movement it causes between two wires that touch), it essentially also
depends on the properties of the intermediate layer (surface roughness, own stress, environment,
temperature, etc.) [Esslinger, 1994].

128
Even if the conductor model postulated here cannot account especially for the last-mentioned effects
and if, therefore, the determination of the wire (and therefore conductor) service life based on the
calculated wire stresses does not at first appear quantifiable, the model, as shown in the previous
section, is capable of predicting the stress vibration amplitude in the wires as the conductor vibrates, as
well as the radial forces and relative displacement at the wire contact points. This may be considered
the necessary basis for a better estimate of the friction corrosion phenomenon and therefore the service
life of the conductor [Raoof, 1994].
The capability of the SEIL program to easily run through various construction and load parameters for
different conductors, allowing a relative evaluation of the friction corrosion behaviour and therefore
service life, should not be underestimated.
In this respect, the first calculation “checks” with the SEIL program, applied to the fairly detailed
documentation (CEA report [1986]) of long-term vibration tests on an ACSR overhead conductor
code-named Bersimis, showed satisfactory agreement with the measurements. The wire stresses in the
individual conductor layers were in this case calculated using the SEIL program at the different load
levels. The alternating bending amplitude, measured at a distance of 89 mm from the conductor
“clamp”, which in this special case consisted of a simple bearing support without clamping, was chosen
as the load parameter. Using these stresses and the S-N curves for the aluminium wires given in the
same paper (determined under simulated friction corrosion conditions), the number of load cycles to
failure were determined. This number was then compared to the actual number of load cycles found in
the long term vibration test on the entire conductor before the first wire breakage. In almost all the
cases, both results were of the same order of magnitude which, considering the well known deviation in
long term tests and the abovementioned additional (apart from the wire stress) influences on the friction
corrosion process, may be considered a useful first approximation.
Static pre-stress
A further point to be highlighted here is the different original position of the conductor in these
measurements compared to its stress in an overhead line. The development in time of the maximum
wire stress at the clamp is to this end discussed further, with reference to the idealised characteristic of
the wire material, Fig. 7.10.

Fig. 7.10 Schematic curve of alternating conductor bending be in the σ-ε diagram
The The process process of of stringing stringing the the conductors conductors and fitting them in in the suspension clamp clamp causes causes high conductor
conductor
elongation elongation due to to static static conductor conductor bending bending which which exceeds the limit of of elasticity of of the aluminium wire wire in
almost all cases (point �).
129
If If this this point point is is considered considered the the new new static static position position of of rest of the conductor, it is possible that, in the first
half cycle, the wire cannot assume a higher stress – it moves towards point � on the characteristic.
Swinging back, it now follows the load relaxation path for the material (virtually parallel to the
proportionality line), passes the static null position at �, reaches point � at the opposite cycle peak
and, as long as the vibration amplitude remains within these limits, follows the � - � - ��
characteristic,
which which is is in in the the linear linear region. region. For For the the conductor conductor vibrations vibrations under consideration here, the assumptions
assumptions
made made in in the the conductor model, model, of of wire wire material material in in the the linear linear elastic elastic region region and and a a horizontal original
position of the conductor, Fig. 7.1, may therefore be considered satisfied.

8. Prospects
130
In the last chapter of this project we will attempt to draw conclusions on what has been achieved and
make proposals for further work in this field, based on the experience gained here.
To begin with, we note that the goals as set out in the introduction have largely been achieved. A model
for the bending behaviour of multi-layer helically twisted cables could be defined that is simple and clear
but nevertheless addresses the essential characteristics. It was possible to quantify the bending
stiffness as a function of internal friction, constituting the fundamental basis of the calculations within the
framework of this project. Based on this, it was possible to develop equations describing the
cable/conductor sag at specific external loads and also the curvature of its axis, the relative
displacement of the wires and, finally, the various stresses, i.e. the tensile stress, the secondary stress
and the bending stress in the individual wires of the cable/conductor.
The newly introduced “cable scanner” offers the possibility of measuring the cable/conductor catenary
and especially the curvature of its axis by using the measuring techniques and software available today
to scan the cable/conductor surface.
After finding fairly good agreement between calculation and measurements, a long standing uncertainty
in the approach to bending stiffness and a discrepancy between the measured and calculated wire
stresses under aeolian vibration conditions could be explained by extending the conductor model to
include these conditions. This should result in a re-check of the permissible stresses in such
conductors.
With this in mind, the following continuing studies might be profitable in the future:
The first requirement would be to check the theory postulated here by more accurate, especially higher
local resolution, stress measuring techniques than is possible with today’s strain gauges. The speckle
interferometry method would be well suited for such measurements. The first tests carried out at the ILS
of the ETH to this end [Michel, 1993] showed promising results.

131
Since the relationship between the parameters of the cable/conductor (dimensions, lay length, material
characteristics) and wire stresses can now be quantified using the model developed here, the design of
cables/conductors could now be optimised to account for these wire stresses. In this respect, the
coefficient of friction and the lay length have the greatest effect and should enjoy priority in further
studies. Detailed studies of the friction coefficient are of special interest, e.g. with reference to their
dependence on load, the cable/conductor design, the wire surface, lubrication and the age of the
cable/conductor.
A further important subject that bears future investigation is the behaviour of the conductor in the
suspension clamp. It would be of great academic interest and practical significance to apply or adapt the
conductor model developed in the course of this project to the conductor section in the clamp.
Furthermore, the method described to determine the permissible vibration stress in the ACSR
conductor considered here should also be applied to other overhead conductors and fleshed out
through additional investigations.
An attempt could also be made to apply the basics of conductor hysteresis during bending, as derived
here, to calculate self-damping - complementing the existing experimental results.
It would finally be of particular practical interest if it were possible to use the results of the calculations of
internal cable/conductor conditions as expounded here (and in particular the longitudinal fatigue
stresses in the individual wires in vibrating conductors) to estimate the service life of such cables based
on the service life of the constituent individual wires, which is far easier to determine.

Annexure I
The The tensile stress stress in in the the wire, wire, calculated calculated from from the the external external tension on the the cable in acc. with (2.6), feeds
directly into the calculation of the secondary stress in acc. with (2.17) and therefore has a decisive
influence on the calculation of the cable curvature.
As As mentioned mentioned in in Section 2.1 2.1 already, a a number of of simplifying assumptions assumptions were made made there there in in the
determination of the tensile stress in the wire. TThese
hese simplifications are listed and quantified in this
Annexure, Annexure, but but largely largely without detailed detailed derivation derivation of the the equations, equations, which which are are also given in Feyrer [1990].
Influence of the transverse contraction
As a tensile force S is applied to the cable, it is not only elongated (see also Fig. 2.3), but the transverse
contraction of the wires also reduces its diameter, Fig I.1.
(Fig. 2.3)
Fig. I.1 Elongation of a wire under tensile stress on the cable, also considering the
transverse contraction
132

If a uniform transverse contraction coefficient of ν = 0.3 is assumed for all wires, then the transverse
contraction of the coil radius rL of layer L of the cable becomes:
The elongation ∆ld,L of a wire in layer L when the cable is stretched by ∆ls is:
After a number of substitutions we find that the tensile stress σd,L in a wire in layer L is:
The difference between this “exact” equation compared to (2.6) lies in the factors νLsin 2
here, in addition. For conventional lay angles of βL = 10°...15°, these factors are virtually unity and
therefore do not significantly affect the value of the tensile stress in the wire.
Bending and torsion stresses with tensile loads
When a tensile tensile force force is is applied applied to to the the cable, cable, the change in the spatial spatial curve curve of the wire (see also
Fig. 2.4) induces bending and torsion stresses
2.4) induces bending and torsion stresses σL,Z and τL,Z in the individual wires of the cable.
These can be determined from the change in the curvature or in the helical shape of the wire after the
tensile load has been applied as follows:
133
2 βL which occur

Considering Considering the the above explanations on the influence of the transverse contraction contraction and Figure Figure I.1, the
parameters that changed under the tensile stress, i.e. the t coil radius rL,Z and the lay angle
considered wire can be calculated:
The The bending bending and and torsion torsion stresses in in the wires under “pure” tensile tensile stress can can be calculated using the
above equations. Substituting the numerical values for the cables used yields rather small values for
these these stresses stresses as compared compared to to the tensile stress stress in in the the wires in in acc. with (2.6), which can generally be
be
ignored [Ziebs, 1970, Wang, 1990].
134
In this context, the friction-dependent dependent secondary tensile stress as calc calculated ulated by Andorfer [1983] and
which which occurs occurs with with tensile tensile fatigue fatigue loads loads need not not be considered, considered, since its its effect is is significant only only for
tensile stresses in individual wires in stranded wire ropes.
and the lay angle β L,Z of the

Annexure II
The following identity is implicit in the derivation of equation (2.38) in Section 2.4:
for ϕi = 2π/nLi with i = 1,....,nL and nL
The angle ϕi denotes the position of each of the n L wires of a layer L in a helically twisted cable, see
Fig. 2.7.
To prove this relationship, take nL vectors (n (nL>2) >2) of the same magnitude and directions ϕi forming a
closed regular n-sided sided polygon and therefore vectorially adding up to zero, Fig. II.1.
Fig. II.1 Vectorial l presentation of the position of the wires in a cable cross cross-section, section, for a
layer with nL = 6 wires
135
L>2, >2, which is applicable to helically twisted cables in general.

It follows that:
If the angle is doubled on both sides of the equation:
Therefore, according to the trigonometric identity:
i.e.
and:
Substitute this in:
to yield:
136
for
2i > nL
for 2i ≤ nL

Annexure III
The The general general solutions solutions for for the the differential differential equations equations (4.5b) and and (4.6b) (4.6b) derived derived in Section 4.2 are:
The integration constants C1 to C4 C are determined from the following llowing boundary and transition
calculations (see Fig. 4.4):
And, in addition, (2.47) gives the relationship between MMm
and MR as:
therefore:
and:
137

The position of the catenary in the x-y
coordinate system is ultimately determined at x = a, the abscissa
of of the the point point of of application application of of the the transverse load load V, V, which also also coincides coincides with with the maximum maximum bending
bending
moment and maximum curvature of the cable, Fig. 4. 4.4. At this point, the angle α of the tangent to the
cable catenary satisfies the condition that the balance of forces be maintained, i.e.:
or:
This finally yields the equation determining the distance a:
With this equation, all the parameters of interest can now be determined, e.g. the maximum curvature at
the point where the force is applied:
and the maximum bending moment at the same point:
And finally, by simple coordinate transformation, the parameters α and a calculated above may be
transformed into the traditional xy (bold) coordinate system, (see Fig. 4.4).
138

Annexure IV
The The calculation calculation of of the the cable/conductor cable/conductor catenary catenary using using Finite Elements Elements (FEM) was explained in principle
in Section 4.3. This Annexure deals with a few more details on this, especially the stiffness matrices
used.
Figure Figure IV.1 IV.1 below below schematically schematically shows shows a a bending bending element in in the the global coordinate coordinate system, system, as used in
the SEIL program. The figure igure also also shows shows the relevant parameters for transformation from from local local to to global
g
coordinate system.
139
Fig. IV.1 SEIL bending element in the global coordinate system

The The following following matrices matrices are used used in the the Finite Finite Element calculations: The The local stiffness matrix [K [Ke ]:
The incremental stiffness matrix [S e ], where S is the (axial) tensile stress in the cable/conductor:
The transformation matrix [T]:
140

Annexure V
As mentioned in Chapter 7, the Poffenberger-Swart Poffenberger Swart equation [Poffenberger and Swart, 1965] is
generally used these days to convert the conductor bending amplitudes measured near the suspension
clamp clamp to to bending bending stresses, stresses, in in the the outer wire wire layer, layer, at at the clamp. These are are of primary primary interest interest for for the
evaluation evaluation of of the the service service reliability reliability of of the the conductor. conductor. Since a fundamental purpose of this this work is is to vverify
v
and and extend extend this this conversion, conversion, see see also also Fig. 7.8, 7.8, it is is briefly briefly discussed discussed below for completeness.
The Poffenberger-Swart Swart equation is:
σb,PS = (PS) Yb
Where σb,PS is the bending stress in the wires of the outer wire layer, YYb
is the measured double bending
amplitude at a distance xb from the clamp (generally, xb x = 89 mm) and (PS) is the conversion factor.
where:
d = wire diameter of the outer wire layer
E = Young’s modulus for the wire material in this layer
S = the conductor tension
xb = the measuring distance
(EJ) min = the minimum wire stiffness of the conductor, calculated as follows:
141

For single component conductors:
For ACSR:
142

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List of symbols
Latin symbols
148
a Abscissa of the point of application of the transverse cable stress (Fig. 4.4)
a, b, c, d Constants for the cubic FE approach
A Al
Ad
Ad,L
ASt
(AE)s
b a,i
bi,a
C CI CII
C e,a C e,i C e,n
Area of an aluminium wire
Area of a wire for a single layer cable
Area of a wire in layer L
Area of a steel wire
Strain stiffness of the cable
Number of the points of contact between wires of the outer/inner layers
Number of the points of contact between wires of the outer/inner layers
Constants in the calculation example with the 35/6 cable (Section 2.7)
Constants in the solution of the differential equation for the
Cc,i Cz,i Cz,n Ci wire tensioning force in multilayer cables (Chapter 3)
da
di
dm
E Al
Ed
Ed,L
E K
ESt
(EJ) b
(EJ)(κ)
(EJ)d,L
(EJ) Draht
(EJ)Gesamt
(EJ)K
(EJ) L
(EJ)max
Coil diameter (= circle segment diameter) of the outer layer
Coil diameter of the inner layer
Coil diameter of a single layer cable
Young's modulus of an aluminium wire
Young's modulus of a layer wire of a single layer cable
Young's modulus of a wire in layer L
Young's modulus of the core wire
Young's modulus of a steel wire
Average bending stiffness of the cable at curvature κb
General representation of the variable (curvature-dependent) bending stiffness
of a cable; the (κ) symbol may be omitted, for clarity.
Bending stiffness of a wire in layer L
Minimum bending stiffness (wire stiffness) of the cable
Total bending stiffness of the cable
Bending stiffness of the core wire
Bending stiffness of layer L
Maximum bending stiffness of the cable

(EJ)max,L
(EJ)min
(EJ)min,d,L
(EJ) min,L
149
Maximum bending stiffness of layer L
Minimum bending stiffness of the cable
Minimum bending stiffness of a wire in layer L
Minimum bending stiffness of layer L
(EJ)zus(κb) Secondary stiffness of the cable at the max. curvature κb
(EJ)zus,d,L
(EJ)zusI
(EJ) zusI,L
(EJ)zusII
(EJ)zusII,e
(EJ)zusII,L
Secondary stiffness of a wire in layer L
Secondary stiffness of the cable in Region I (no wire slippage)
Secondary stiffness of layer L in Region I
Secondary stiffness of the cable in Region II (full wire slippage)
Secondary stiffness at the end of the transition region
Secondary stiffness of layer L in Region II
(EJ)(a.-L.) Cable stiffness if only the outer layer slips
(EJ)(ver) Cable stiffness if all layers slip simultaneously after a certain transition
curvature
(EJ)(def) Cable stiffness with an assumed constant value between (EJ) min and (EJ) max
Fa
F N
FRm1, FRm2
Factor in (3.15)
Normal force in the friction tests
Friction forces in the friction tests
[F] Force vector for the FE analysis
hd,L
hL
Distance of a wire in layer L from the cable axis
Distance from the cable axis, of a layer wire in a single layer cable
[KG ] Overall stiffness matrix
[K e] Element stiffness matrix
Ka
Exponential factor in (3.9)
L Length between clamps of the sample cable in the tensile tester
lA
l a
l d,L
li
li,i
lL
l n
dls
dls,a
dls,i
∆ld,L
∆lS
Decay length for the alternating bending stresses in the outer layer
Lay length of the outer layer
Wire length in the direction of the axis of a wire layer L
Lay length of the inner layer
Lay length in layer i,i
Lay length in layer L
Lay length in layer n
Length of a wire in the direction of the cable axis
Length of a wire in the outer layer in the direction of the cable axis
Length of a wire in the inner layer in the direction of the cable axis
Elongation of a wire in the direction of the wire axis
Elongation of a wire in the direction of the cable axis

ma
mi
mL
150
Friction summation factor for the outer layer
Friction summation factor for the inner layer
Friction summation factor for layer L
M Bending moment effective in the cable cross-section
Mb
Md,L
MGesamt
MI
M II
MK
Mm
M m,L
Mmax = MSeil
Mmin = MDraht
MR
MR,L
M zus
MzusII,a
MzusII,e
Maximum stress moment
Wire moment of a wire in layer L (proportional)
Total bending moment
Total bending moment in Region I
Total bending moment in Region II
Core wire moment
Average transition moment of a single layer cable
Average transition moment for layer L
Maximum (cable) bending moment, so-called cable moment
Minimum (wire) bending moment, so-called wire moment
Residual friction moment
Residual friction moment for layer L
Secondary bending moment
n Sequential index
na
n i
ni,i
nL
n n
Secondary bending moment at the start of wire displacement
Secondary bending moment at the end of wire displacement
Number of wires in the outer layer
Number of wires in the inner layer
Number of wires in layer i,i
Number of wires in layer L
Number of wires in layer n
N Number of cycles in the friction tests
dNa
∆Na
dN L
PL
rL
rm
dRL
∆Ra
Radial force of a (differential) wire element in the outer layer with assumed
continuous resting of the outer layer on the inner layer
Radial force of a (finite) wire element in the outer layer with “spot” contact
between outer and inner layer
Radial force of a (differential) wire element of layer L
Radial pressure of a wire in layer L on the core wire
Coil radius (= circle segment radius) of layer L
Coil radius for a single layer cable
Friction force of a (differential) wire element of layer L
Friction force outer/inner layer

∆Ri Friction force inner layer/core wire
S Cable tensile force (S = const. > 0)
[Se] Incremental element stiffness matrix
s(φ) Forcing function
151
ds Length of a wire element in the direction of the wire
[T] Transformation matrix for the FE analysis
[u] Displacement vector for the FE analysis
U Relative humidity during the friction tests
U d,L
Circumferential force for a wire in layer L
V Transverse stress
Vmax
W0B
W0D
W 0BRest
W1B
W1D
W 2B
W2R
W3B
W3D
W4B
W 4D
W4BRest
W5B
W5R
W5BRest
W BRest
x b
Maximum transverse stress
Bending energy in Phase 0
Strain energy in Phase 0
Residual bending energy in Phase 0
Bending energy in Phase 1
Strain energy in Phase 1
Bending energy in Phase 2
Friction energy in Phase 2
Bending energy in Phase 3
Strain energy in Phase 3
Bending energy in Phase 4
Strain energy in Phase 4
Residual bending energy in Phase 4
Bending energy in Phase 5
Friction energy in Phase 5
Residual bending energy in Phase 5
Residual bending energy stored in the cable after a full cycle
(stress increase and decrease)
Measuring distance for the vibration measurement (Chapter 7)
y(x) Bending curve of the cable (i.e. the cable axis), the catenary
yd,L
yI
yII
Y b
Cross section ordinate of a wire in layer L
Catenary of the cable in Region I
Catenary of the cable in Region II
Double bending amplitude at distance x b = 89 mm from the clamp during cable
vibration
Z(ϕ) Total wire force is dependent on ϕ; to improve readability, symbol ϕ is not
always shown

Z0
Z1
Za
Z c,n
Zd
Z d,a
Zd,i
Zd,L
Zi
Zi,i
Z i,k
Z L
Zn
Zn,a
Zzus,L
dZa
dZ i
dZi,i
dZ L
∆Zi,a
152
Initial wire force (total) when bending a wire element
Final wire force (total) when bending a wire element
Wire force (total) when bending a wire in the outer layer
Wire force (total) when bending layer n and assuming constant (i.e.
independent of ϕ) wire forces in all layers further out
Tensile force (constant) in a layer wire of a single layer cable
Tensile force (constant) in a wire of the outer layer
Tensile force (constant) in a wire of the inner layer
Tensile force (constant) in a wire of layer L
Wire force (total) when bending a wire in the inner layer
Wire force (total) when bending a wire in layer i,i
Wire force (total) when bending the inner layer, which is in continuous contact
with the core wire
Maximum (total) tensile force during bending that can be resisted due to
friction at every point on the wire – function of ϕ
Wire force (total) when bending
Step height of the tensile force in the outer layer, caused by its spot contacts
with the inner layer
Secondary force in a wire of layer L, dependent on angle ϕ
Maximum tensile force difference in a wire element in the outer layer that can
be resisted by friction, assuming the outer layer makes continuous contact
with the layer underneath
Maximum tensile force difference in a wire element in the inner layer that can
be resisted by friction, assuming the outer layer makes continuous contact
with the layer underneath
Maximum tensile force difference in a wire element in the layer i,i that can be
resisted by friction, assuming the outer layer makes continuous contact with
the layer underneath
Maximum difference in tensile force a wire element can resist through friction
(constant)
Step height of the tensile force in the inner layer, caused by its spot contacts
with the outer layer

Greek symbols
153
α Angle between tensile force and transverse stress
dα Angle of the helical line of a wire element, so-called wrap angle
β Lay angle of a single layer cable
β a
β i
βi,i
βL
βn
Lay angle of the outer layer
Lay angle of the inner layer
Lay angle of the layer i,i
Lay angle of layer L
Lay angle of layer n
δ Diameter of a layer wire of a single layer cable
δd,L
δk
ε b,L
εd,L
εs
ε zus,L
Diameter of a wire in layer L
Diameter of the core wire
The “normal” bending elongation the wire experiences when bending about its
own neutral axis
Wire elongation with pure tensile stress
Cable elongation with pure tensile stress
Secondary bending strain of a wire in layer L
κ Curvature of the cable (i.e. cable axis) κ = 1/ρ
κa
κa,L
κb
Cable curvature at the end of the wire displacement
Initial curvature of the wire displacement in layer L
Maximum cable curvature under a stress M b
κd,L(ϕ) Curvature of a wire in layer L at the start of wire slippage (depending on position
κe
κ e,L
κ m
κ m,L
κI
κII
angle ϕ of the wire in the cross-section)
Cable curvature at the end of the wire displacement
Final curvature of the wire displacement in layer L
Average transition curvature
Average transition curvature of layer L
Cable curvature in Region I
Cable curvature in Region II
µ Friction coefficient, basic
µa
µi
µi,i
µL
µn
Friction coefficient for the outer layer
Friction coefficient for the inner layer
Friction coefficient for layer i,i
Friction coefficient for layer L
Friction coefficient for layer n

ρ Curvature radius of the cable axis
ρ a
ρe
ρL
ρ m
σb,PS
σd
σb,Reuleaux
σd,L
σ b,L
σ ���
���,�
154
Curvature radius of the cable axis at the start of wire displacement
Curvature radius of the cable axis at the end of wire displacement
Curvature radius of the helix line of a wire in layer L, before bending
Average curvature radius of the cable axis
Alternating bending stress in acc. with Poffenberger-Swart with (EJ)min
Tensile stress in a wire in the layer of a single layer cable
Wire bending stress in acc. with Reuleaux, so-called Reuleaux stress
Tensile stress in a wire in layer L
Wire bending stress in a wire in layer L
Maximum secondary stress in a wire of layer L that can be resisted after full
wire slippage
σ L(ϕ) Total stress in a wire in layer L during bending, dependent on angle ϕ; to
σzus,L
improve readability, the symbol ϕ not always shown
Secondary force in a wire of layer L (dependent on angle ϕ)
ϕ Rotation angle of the helix curve of a wire, the wire rotation angle or position
ϕa
ϕi
angle of a wire in the cable cross-section
Rotation angle of the helical curve of a wire in the outer layer
Rotation angle of the helical curve of a wire in the inner layer
dϕ Angle element for the (differential) wire element
dϕ a
dϕ i
Angle element for the (differential) wire element in the outer layer
Angle element for the (differential) wire element in the inner layer
∆ϕa Angular difference between two contact points a wire in the outer layer has
with the inner layer

List of figures
155
Fig. 1.1 Basic construction of a helically twisted cable .................................................................... 1
Fig. 1.2 Overhead conductors on a 400 kV high voltage transmission line at Cap Sounion ........... 2
Fig. 1.3 Modern conductor vibration recorder in measurement position (dimensions in mm) ......... 3
Fig. 2.1 Geometry of a wire belonging to layer L of a helically twisted cable .................................. 8
Fig. 2.2 Distribution of the tensile force to individual wires .............................................................. 9
Fig. 2.3 Wire and cable elongation ................................................................................................. 10
Fig. 2.4 Determining the radial pressure of the wires .................................................................... 12
Fig. 2.5 Force relationships in a wire element in layer L ................................................................ 13
Fig. 2.6 Wire stresses as the cable bends (a) wires not displaced (b) wires displaced ................. 15
Fig. 2.7 Geometry of the cable cross-section................................................................................. 16
Fig. 2.8 Wire displacement during bending .................................................................................... 17
Fig. 2.9 Curves for the secondary stress, depending on the displacement status of the wire ....... 18
Fig. 2.10 Maximum bearable secondary stress with complete wire slippage .................................. 19
Fig. 2.11 Curvature where wire slippage begins (function of ϕ) ....................................................... 19
Fig. 2.12 Distribution of the cable bending moment over the individual wires ................................. 22
Fig. 2.13 Moments - Bending (M-B) curve of a single layer cable ................................................... 26
Table 2.1 Bending stiffnesses and bending moments in Regions I and II ........................................ 27
Fig. 2.14 Schematic curve of bending stiffness (EJ)(κ) across the cable curvature for a single layer
cable .................................................................................................................................. 29
Fig. 2.15 Initial, final and average transition curvature radius as a function of the “pure” tensile stress
in the wire .......................................................................................................................... 32
Fig. 2.16 Cable bending processes in the M-B diagram .................................................................. 33
Fig. 2.17 Cable curvature in the M-B diagram (a) κb < κm : no hysteresis (b) κb > κm : hysteresis; left:
first stress, right: subsequent stress.................................................................................. 35
Fig. 2.18 Hysteresis in the M-B diagram as a cable bends .............................................................. 36
Fig. 2.19 Cross-section of an ACSR conductor 35/6 (see also Fig. 2.7) ......................................... 37
Fig. 2.20 Limiting curvature radii for an ACSR conductor 35/6 ........................................................ 38
Fig. 2.21 Curve of the secondary stress over a full lay length (ϕ = 0 to 2π) .................................... 40
Fig. 2.22 Secondary stiffness of ACSR conductor 35/6; 1: σzug = 10N/mm 2 ; 2: σzug = 20N/mm 2 ; 3:
σzug = 30N/mm 2 ; 4: σzug = 40N/mm 2 .................................................................................. 41
Fig. 2.23 The secondary stress in the transition region; the two vertical lines indicate the
intersections of (2.17) and (2.22) ...................................................................................... 42
Fig. 2.24 Secondary bending moment in an ACSR conductor 35/6 as a function of the conductor
curvature for different values of tensile stress in the wire; 1: σzug = 10N/mm 2 ; 2: σzug =
20N/mm 2 ; 3: σzug = 30N/mm 2 ; 4: σzug = 40N/mm 2 ............................................................. 43
Fig. 2.25 Total bending moment in an ACSR conductor 35/6 as a function of the conductor curvature
for different values of tensile stress in the wire; 1: σzug = 10N/mm 2 ; 2: σzug = 20N/mm 2 ; 3:
σ zug = 30N/mm 2 ; 4: σ zug = 40N/mm 2 .................................................................................. 44
Fig. 3.1 Balance of forces on a wire element in the outer layer of a two-layer cable at the contact
point with the layer underneath ......................................................................................... 47
Fig. 3.2 Curve of the wire tension in the outer layer of a multi-layer cable .................................... 48

156
Fig. 3.3 Effect of the tensile force in the outer wire on the inner wire of a two-layer cable; wires
sketched in parallel, for clarity ........................................................................................... 50
Fig. 3.4 Wire tension in the inner layer of a two-layer cable .......................................................... 51
Fig. 3.5 Length relationships between wire and cable elements ................................................... 52
Fig. 3.6 Wire tension Z i in the inner layer of a two-layer cable according to the different
approximations: ................................................................................................................. 57
Fig. 3.7 The M-B diagram for a four-layer cable ............................................................................ 61
Fig. 3.7a M-B diagram with undisturbed cable configuration, curve 1 ............................................ 62
Fig. 3.7b M-B diagram with fully slipped cable configuration, curve 2 ............................................ 62
Fig. 3.7c M-B diagram when only the outer layer of a cable slips, curve 3 .................................... 63
Fig. 3.7d M-B diagram under the assumption that all layers slip simultaneously, curve 4. ............ 64
Fig. 3.7e M-B diagram with an average constant stiffness, curve 5 ............................................... 64
Fig. 4.1 Static system with load and directions ............................................................................ 65
Fig. 4.2a Bending of a cable under tensile load; illustration of Phase 1 .......................................... 66
Fig. 4.2b Bending of a cable under tensile load; illustration of Phase 2 .......................................... 67
Fig. 4.2c Bending of a cable under tensile load; illustration of Phases 3 and 4 .............................. 68
Fig. 4.2d Bending of a cable under tensile load; illustration of Phase 5 and the residual energy
W BRest ................................................................................................................................. 69
Fig. 4.2e Bending of a cable under tensile load; illustration of Phases 0 and 1: left - initial load,
right – subsequent loads ................................................................................................... 70
Fig. 4.2f Basic force-displacement characteristic during cable bending ......................................... 71
Fig. 4.3 M-B diagram for the analytical calculation ....................................................................... 72
Fig. 4.4 Coordinate system and stress, for the analytical calculation ............................................ 73
Fig. 4.5 SEIL program; cable data input ............................................................................................. 78
Fig. 4.6 Definition of the wire rotation angle ϕ in the SEIL program .............................................. 78
Fig. 4.7 SEIL program; input of the test and calculation parameters ............................................. 79
Fig. 4.8 SEIL program; entering the different options for the bending stiffness ............................. 80
Fig. 4.9 Stiffness-bending characteristic; 1: (EJ)(κ); 2: (EJ)min; 3: (EJ)max ..................................... 81
Fig. 4.10 Comparison of FE and analytical calculations: both cable catenaries are virtually identical
........................................................................................................................................... 81
Fig. 4.11 The output options for the SEIL program .......................................................................... 83
Fig. 4.12 SEIL program; results in tabular form ............................................................................... 83
Fig. 4.13 Flow chart of the SEIL program calculations .................................................................... 84
Fig. 5.1 Cable bending: (a) forced and (b) free bending .............................................................. 86
Fig. 5.2 Operating principle of the LDS sensor .............................................................................. 87
Fig. 5.3 Operating principle of the cable scanner .......................................................................... 88
Fig. 5.4 Single cross-section (800 measured points) ..................................................................... 90
Fig. 5.5 Construction of the cable cross-section and determination of the cable axis, using the
scanned cable surface data; note the gap between wires ................................................ 90
Fig. 5.6 Graphic representation of the measured 52 cross-sections (scans) ................................ 90
Fig. 5.7 Determining the catenary (= curvature along the cable axis) ........................................... 91
Fig. 5.8 Three-dimensional representation of the scanned length of cable ................................... 91

157
Fig. 5.9 Test rig for single-layer cables .......................................................................................... 92
Fig. 5.10 Measured and calculated catenary for the single layer steel cable 1: (EJ)(κ); 2: (EJ)min; + :
measurement .................................................................................................................... 93
Fig. 5.11 Bending stiffness of the steel cable 7 x 3.26 at S = 8 kN and Vmax = 2.8 kN 1: (EJ)(κ); 2:
(EJ)min ................................................................................................................................ 94
Fig. 5.12 Functional diagram of the ILS test rig ............................................................................... 95
Fig. 5.13 Details of the test rig ......................................................................................................... 96
Fig. 5.14 Force introduction at the centre of the cable; top – LDS sensor ....................................... 97
Fig. 5.15 Measuring and analysis equipment on the test rig ............................................................ 97
Table 5.1a Data for the steel cable S32 ........................................................................................ 98
Table 5.1b Data for the ACSR conductor Cardinal ....................................................................... 98
Fig. 5.16 Cross section of (a) S32 and (b) Cardinal (schematic: in reality, the wire cross-sections
shown here are elliptical and the layer wires exhibit gaps along the circumference) ....... 99
Fig. 5.17 Tribology system ............................................................................................................. 100
Fig. 5.18 Typical curve for the coefficient of friction µ (right hand scale) vs. the number of cycles N .
......................................................................................................................................... 101
Table 6.1 Tensile and transverse forces during the measurements on the steel cable S32 .......... 103
Fig. 6.1 Measured and calculated catenary for S32 for S = 280 kN and V max = 40 kN; 1: variable
stiffness (EJ)(κ); 2: minimum (wire) stiffness (EJ) min; 3: maximum (cable) stiffness (EJ) max;
+ : measurement ............................................................................................................. 104
Fig. 6.2 Bending stiffness of S32 at S = 280 kN and Vmax = 40 kN; 1: (EJ)(κ); 2: (EJ)min; 3: (EJ)max
105
Fig. 6.3 Measured and calculated catenary for S32 at S = 140 kN and Vmax = 20 kN; 1: (EJ)(κ); 2:
(EJ)min; 3: (EJ)max ............................................................................................................. 105
Fig. 6.4 Measured and calculated catenary for S32 at S = 80 kN and V max = 10 kN; 1: (EJ)(κ); 2:
(EJ)min; 3: (EJ)max ............................................................................................................. 106
Fig. 6.5 Measured and calculated catenary for S32 at S = 40 kN and Vmax = 5 kN; 1: (EJ)(κ); 2:
(EJ)min; 3: (EJ)max ............................................................................................................. 106
Fig. 6.6 Hysteresis in S32 at S = 280 kN and Vmax = 40 kN; 1: (EJ)(κ); 2: (EJ)min; 3: (EJ)max ;
measurements shown in bold ......................................................................................... 107
Fig. 6.7 Measured and calculated catenary for Cardinal at S = 40 kN and Vmax = 4 kN; 1: (EJ)(κ);
2: (EJ)min; 3: (EJ)max ......................................................................................................... 109
Fig. 6.8 Bending stiffness for Cardinal at S = 40 kN and Vmax = 4 kN; 1: (EJ)(κ); 2: (EJ)min; 3:
(EJ)max ............................................................................................................................. 109
Fig. 6.9 Hysteresis for Cardinal at S = 40 kN and Vmax = 4 kN; 1: (EJ)(κ); 2: (EJ)min; 3: (EJ)max ;
measurements shown in bold ......................................................................................... 110
Fig. 6.10 The different bending stiffnesses for S32, as a function of the curvature at S = 280 kN; 1:
(EJ)(κ); 2: (EJ)min; 3: (EJ)max; 4: (EJ)(a.-L.); 5: (EJ)(ver); 6: (EJ)(def) ............................. 112
Fig. 6.11 Catenary for S32 with S = 280 kN and Vmax = 40 kN, with different stiffness assumptions; 1:
(EJ)(κ); 2: (EJ)min; 3: (EJ)max; 4: (EJ)(a.-L.); 5: (EJ)(ver); 6: (EJ)(def); 7: analytical; + :
measurement .................................................................................................................. 113
Fig. 6.12 Bending moment for S32 with S = 280 kN and Vmax = 40 kN with different stiffness

158
assumptions; 1: (EJ)(κ); 2: (EJ) min: (EJ) max; 4: (EJ)(a.-L.); 5: (EJ)(ver); 6: (EJ)(def) ..... 113
Fig. 6.13 300 kN tensile testing machine for measurements on approx. 3 m long samples ......... 115
Fig. 6.14 Force introduction at the centre of the sample, using coil spring and linear transducer . 115
Fig. 6.15 Hysteresis in S32 at S = 140 kN and Vmax = 10 kN; 1: (EJ)(κ); 2: (EJ)min; 3: (EJ)max ;
measurements shown in bold ......................................................................................... 116
Fig. 6.16 Hysteresis for Cardinal at S = 80 kN and V max = 10 kN; 1: (EJ)(κ); 2: (EJ) min; 3: (EJ) max ;
measurements shown in bold ......................................................................................... 116
Fig. 7.1 Definition of the decay length of the bending moment at the clamp ............................... 117
Fig. 7.2 Method for measuring the bending amplitude on a conductor ........................................ 118
Fig. 7.3a Bending stress curve for the individual wires in the Cardinal cross-section at the clamp;
ϕ = π/2: “top” wire, see Fig. 4.6; the horizontal line represents the Reuleaux stress, see
Sec. 2.3. .......................................................................................................................... 119
Fig. 7.3b Bending stress along the first layer of a wire in the 4 th Cardinal layer; point x = 0 is the
position of the clamp ....................................................................................................... 119
Fig. 7.4a Secondary stress of the individual wires in the Cardinal cross-section at the clamp; ϕ = π/2:
“top” wire, see Fig. 4.6 .................................................................................................... 120
Fig. 7.4b Secondary stress along the first layer of a wire in the 4 th Cardinal layer; point x = 0 is the
position of the clamp ....................................................................................................... 120
Fig. 7.5a Total stress in the individual wires in the Cardinal cross-section at the clamp; ϕ = π/2: “top”
wire, see Fig. 4.6; the horizontal line represents the “pure”, constant tensile stress in the
wire .................................................................................................................................. 121
Fig. 7.5b Total stress in the first lay length of a wire in the 4 th layer of Cardinal; position x = 0 is at the
clamp; the horizontal line represents the “pure” constant tensile stress in the wire ..... 121
Fig. 7.6 Principle for calculating alternating bending stresses in the wires of vibrating overhead
conductors ....................................................................................................................... 122
Fig. 7.7 Bending stiffness curve for Cardinal, “stressed” as in Fig. 7.6 ....................................... 123
Table. 7.1 Wire stresses in the individual Cardinal layers when stressed as in Fig. 7.6; *: this
layer has slipped ................................................................................................................................. 123
Fig. 7.8 Longitudinal fatigue wire stresses in the individual Cardinal aluminium layers, as calculated
with the SEIL program at the conductor clamp, as a function of the bending amplitude Y b/2
at 89 mm distance from the clamp; the wire bending stress of layer 4 is shown in bold, for
comparison. ..................................................................................................................... 125
Fig. 7.9 Schematic illustration of the wire stresses during conductor vibration: (a) in acc. with
Poffenberger-Swart with (EJ) min = constant; (b) in acc. with the SEIL model with (EJ)(κ) =
variable ............................................................................................................................ 126
Fig. 7.10 Schematic curve of alternating conductor bending in the σ-ε diagram ........................... 129
Fig. I.1 Elongation of a wire under tensile stress on the cable, also considering the transverse
contraction ........................................................................................................................... 132
Fig. II.1 Vectorial presentation of the position of the wires in a cable cross-section, for a layer with
nL = 6 wires ...................................................................................................................... 135
Fig. IV.1 SEIL bending element in the global coordinate system .................................................. 139

Resumé
Name Konstantin O. Papailiou
159
Domicile Hellbühlstrasse 37, CH-6102 Malters / Lu
Date of birth 3 July 1946
Place of birth Athens / Greece
Marital status married, two children
Education
1963 – 1968 Electrical Engineering studies, TH Braunschweig,
Degree: Dipl.-Ing. (Elec. Eng.)
1969 - 1973 Civil Engineering studies, Stuttgart University,
1973 – 1974 Military service
Professional career
Degree: Dipl.-Ing. (Civil Engineering)
1975 - 1985 Section Head, foreign countries and technology of the Gesellschaft für
elektrische Anlagen (GEA), D-Fellbach
since 1986 Managing Director, Sefag AG, CH-Malters
Professional organisations
- Working group 11 (Mechanical aspects of overhead conductors and fittings)
and Working group 03 (Insulators) of Study Committee 22 (Overhead lines)
of CIGRE, Paris
- Technical commission 36 (Insulators and Bushings) of the SEV, Zurich
- Technical committee 36A (Insulated bushings) of the CENELEC, Brussels
- Senior member of IEEE, New York
- VDE, VDI, SMG