tpwrd.2010.2042083

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 4, OCTOBER 2010 2997

Strain Measurements on ACSR Conductors DuringFatigue Tests II—Stress Fatigue Indicators

Sylvain Goudreau, Frédéric Lévesque, Alain Cardou, and Louis Cloutier

Abstract—Fatigue strength of overhead conductors of differenttypes is often presented on the same diagram, a so-called S-N di-agram, the vertical axis being generally taken as the alternatingstress amplitude. In this paper, it is shown that these stress levelsare merely fatigue indicators with inherent limits, and should beused accordingly. Their practical interest is to allow a regroupingof the fatigue results of different types of conductors. It will beshown that these indicators have some inherent limits in their def-inition and can only be used according to their intended purpose.

Index Terms—Aluminium conductor, fatigue, strain measure-ment.

I. INTRODUCTION

I N THE FIRST paper [1], measured strains at several crosssections on the outer layer uppermost and lowermost wires

for two conductors under cyclic bending were reported. The ex-perimental method followed and the mathematical treatment ap-plied onto the experimental data was also summarized. In thelast section of this paper, it was shown that the two alternatingstresses PS and , which are used to express thefatigue severity undergone by a vibrating conductor fitted ona short metallic suspension clamp, are rather different from thealternating stresses that might be computed from the strain mea-sured on the vibrating conductor.

In this paper, first, the development leading to equationsyielding PS and will be summarized and,second, the influence of the parameters governing these expres-sions will be examined. It must indeed be noted that neither ofthem takes the geometric characteristics of the supports into ac-count. Finally, a review of several technical reports and papersreporting measured strains will be presented and a discussionon how these measured strains correlate with the calculated PS

and values will follow.

II. STRESS FATIGUE INDICATORS

The development of the two alternating stresses PS andis summarized in what follows; except for a few

Manuscript received October 02, 2009. Current version published September22, 2010. Paper no. TPWRD-00739-2009.

S. Goudreau, A. Cardou, and L. Cloutier are with the Department of Me-chanical Engineering and the GREMCA Laboratory (Groupe de REchercheen Mécanique des Conducteurs Aériens). Université Laval, Québec City, QCG1V 0A6, Canada (e-mail: [email protected]; [email protected]; [email protected]).

F. Lévesque is with the Department of Civil Engineering, Université de Sher-brooke, Sherbrooke, QC J1K 2R1, Canada (e-mail: frederic. [email protected]).

Digital Object Identifier 10.1109/TPWRD.2010.2042083

Fig. 1. Standing-wave vibration.

Fig. 2. Deflected shape in the clamp region.

steps, this development is identical to the one found in [2] and[3]. These two stresses are used as fatigue severity indicators inorder to regroup all fatigue results of different conductors in asingle S-N diagram. To begin, it is assumed that some distanceaway from a fixed support, a vibrating conductor takes the shapeof a sine wave (Figs. 1 and 2)

(1)

where is the antinode amplitude of the sine-shaped deflec-tion curve, the wavelength, and is the distance from thefixed support beyond which the sine-shaped deflection curveis defined. The fixed support is assumed to be a square-facedbushing.

The bending curvature of the conductor is assumed to begiven by the usual solid beam equation (Fig. 3)

(2)

where is the departure of conductor centerline (Fig. 3) fromthe sine-shaped loop in the vicinity of the bushing. It will

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2998 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 4, OCTOBER 2010

Fig. 3. Deviation of the conductor centerline from the sine-shaped loop nearthe fixed end.

be assumed that the angular amplitude in Fig. 2 is small sothat the coordinate is assumed to be equal to . is the localbending moment and is the equivalent local bending rigidityof the conductor. Since and assuming that the usualbeam bending equation holds, the preceding equation yields

(3)

With the boundary condition that approaches zero for large, the solution to the differential equation is given by

(4)

The slope of the conductor relative to the axis is

(5)

and at 0, is equal to (Fig. 2). The curvatureof the conductor at the mouth of the square-faced bushing is

(6)

As a first approximation, the angle is assumed to be equalto the maximum value of of the sine-shaped deflectioncurve and it is given by

(7)

The curvature of the conductor at the mouth of the square-facedbushing becomes

(8)

It is assumed that each wire of the conductor bends about itsown neutral axis and the alternating stress amplitude at themouth of the square-faced bushing is expressed as

(9a)

(9b)

where is the wire diameter on the outer layer of the conductor(in fact, it can be the diameter of any wire of the conductor),Young’s modulus of aluminium, the conductor minimumbending rigidity , which is the sum of all individual wiresbending rigidity.

Depending on the engineering needs, may be expressed interms of the vibration parameters: frequency and the antinodeamplitude . Using the velocity of traveling waves on a tautstring,

(10)

may be expressed as

(11)

Another expression for can be derived from the local field(Fig. 2) near the bushing mouth. Assuming small deflections,the position of the conductor centerline relative to the axisis

(12)

From (6)

(13)

and, combined with (4), (12) becomes

(14)

Finally, factoring out in (14) and using (6) yields

(15)

where the standard position for evaluating is 89mm from the bushing mouth. The value of is obtainedby measuring the so-called peak-peak bending amplitude

. For a short metallic suspension clamp, thestandard position for measuring is 89 mm from the last pointof contact (LPC) between the conductor and the bed of theclamp.

Combining (9a) and (15) yields another expression for

(16)

This last equation is known as the Poffenberger & Swart stress(PS ).

III. PARAMETERS INFLUENCING THE COMPUTED

Both (11) and (16) seem to indicate that stresses are directlyproportional to wire diameter . So it would be interesting tocheck how sensitive these equations are with respect to .

Assume: 1) that a multilayer conductor is made of wiresof the same material (aluminium), which is the case in ACAR,AAC, AAAC conductors; 2) that all wires are of the same di-ameter ; 3) that all wires act independently in bending (thus

GOUDREAU et al.: STRAIN MEASUREMENTS ON ACSR CONDUCTORS DURING FATIGUE TESTS II 2999

TABLE IGEOMETRICAL AND MECHANICAL PROPERTIES OF ACSR CONDUCTORS

yielding the minimum rigidity ). In the case of (11),becomes

(17)

where is the density of aluminium kg/m .It is found that no transverse dimension of the wire section

appears in the stress equation meaning that the last expression isindependent of wire diameter and, thus, the equation of stressis proportional to a constant which depends only on materialproperties. In the case of aluminium

(18)

In fact, for aluminium-conductor steel-reinforced (ACSR)multilayer conductors, [2] and [3] indicate that the constant

may vary from 0.171 to 0.200 and in the samereports, it is suggested to use a mean value of 0.186 (MPas/mm). It should be remembered that the computation of isbased on the assumption of independent bending of wires.

Now, with the other limit hypothesis that all wires act asif they were welded together (no slip), the bending rigidity is

and is calculated by considering the conductor as a solidbeam. In this case, the overall diameter of the conductor isused instead of the wire diameter in (9a). The resulting com-putation of of conductors tested to build the S-Ndiagram shown in Figs. 4 and 5 is summarized in Table I. Therelationship versus differs by a factor of, at most, 15%from the one computed with . It should be noted, however,that the ACSR conductors in this table do not exactly meet theprevious hypotheses, as they include a 7 steel-wire core.

The alternating stress is used as a fatigue severityindicator in order to regroup all fatigue results of different con-ductors in a single S-N diagram. In Figs. 4 and 5, most datapoints are from [2, Figs. 3.2–13a and b], with some additionaldata points [4], [5]. Each data point is computed with its ownproportionality factor (Table I) by using . In Figs. 4 and 5,

Fig. 4. Fatigue tests of a two-layer ACSR. This is from [2, Fig. 3.2–13a] up-dated with some additional data points.

Fig. 5. Fatigue tests of a three-layer ACSR. This is from [2, Fig. 3.2–13b],updated with some additional data points.

some data points deviate slightly from those shown in [2, Figs.3.2–13a and 3.2–13b] since those data coming from [3] com-puted with a average value of the proportionality factor of 0.186


were used again in [2] and some raw data from the GREMCAlaboratory have been slightly adjusted. Also, some data points ofthe ACSR Drake conductor [10] have been modified accordingto [4]. Looking at Figs. 4 and 5, the fatigue results expressedas , which are practically directly proportional to a value of

, are very consistent. As reported in [2], the S-Ndiagram for three-layer ACSR conductors differs from the oneobtained for the two-layer ACSR conductors; their endurancelimits , defined as the highest alternating stress amplitudewith a zero wire break at 500 Mc, are about 22 MPa and some-what less than 30 MPa, respectively. These S-N diagrams arefor ACSR conductors fitted in short commercial metallic sus-pensions clamps and in metallic bell mouth clamps.

The fact that (11) is practically independent of diameter andof the bending rigidity hypothesis seems to indicate that drawingthe S-N diagram with taken as the ordinate is as good asusing (11).

By comparison, it will now be shown that the value ofyielded by (16) is not directly proportional to the wire diameter.The mathematical development is based on the same assump-tions as before. Let be the normal traction stress computedaccording to (19) (lay angles are neglected)

(19)

and becomes

(20)

Assuming that in the denominator of (16) with the term, the following expression is obtained:

(21)

It is possible to deduce approximately the influence of wire di-ameter using (21). It appears that for a given static traction stress

, the alternating bending stress is not proportional to wirediameter .

In Figs. 6 and 7, while most data points have already been re-ported in [2, Figs. 3.2–24 and 3.2–25], there are some additionaldata points [4], [5]. Considering all of the tested ACSR conduc-tors (with the associated testing conditions) in Figs. 6 and 7, thecorresponding ratio computed by using (16) varies from27.2 to 46.3 (MPa/mm) (Table I). Using (21), differs by,at most, 6% from the one computed using (16).

Instead of using , it is now assumed that the prevailingbending rigidity is and that the distance from the neutralaxis to the outer fiber is . In this case, is at least 2.6times the value computed with (Table I).

Comparing Figs. 6 and 7, one sees that the S-N diagram forthree-layer ACSR conductors differs from the one obtained forthe two-layer ACSR conductors; the vertical axis is the alter-nating stress computed by using . The endurancelimit for three-layer ACSR conductors is about 8.5 MPa.Howewer, for the two-layer conductor case, it is noted that atthe 15-MPa level, the number of cycles of the first wire break

Fig. 6. Fatigue tests of two-layer ACSR. From Fig. 3.2–24 [2] with some ad-ditional data points.

Fig. 7. Fatigue tests of three-layer ACSR. From Fig. 3.2–25 [2] with someadditional data points.

varies from 17 Mc to 150 Mc and because of a lack of fatiguetest results in the range of 10 to 15 MPa, the correspondingendurance limit cannot be clearly determined yet. These S-Ndiagrams are for ACSR conductors fitted in short commercialmetallic suspension clamps and in metallic bell mouth clamps.It should be noted that according to the macroscopic parameter( or ) used to compute leads to different values of

.Using (20), (9b) yields

(22)

To obtain the fatigue S-N curve of a “conductor-clampsystem,” [2] and [6] recommend that fatigue tests be carriedout by using a resonance test bench. With such a bench, theactive length of the conductor is a multiple of the loop lengthof the vibrating conductor. For a given loop length and a givenantinode amplitude (or a given peak-peak amplitudebending deflection depending of the controlled parameter),the alternating stresses and vary differentlyto the traction force. The alternating stress , computed by


TABLE IIEXPERIMENTAL DATA

All data were collected from original papers. Regression lines through experimental data were computed. Data appearing in [2] and [3] are indicated by .There are some minor differences between the computed ratio in [2] and the ratio given in [3].Experimental conditions:

1) regression line through five data points; strain gauges on wires of the outer layer; �� obtained from experimental data � , � ;2) regression line through two data points; strain gauges on wires of the outer layer; �� obtained from ��, � and through (7) and (10);3) regression line through 16 data points; the 16 data points come from curves in the original paper; curves obtained from strain gauge data measured on the

two uppermost and the two lowermost wires of the outer layer; �� deduced from ��, � and through (7) and (10);4) regression line through 10 and more data points; strain gauges on wires of the outer layer; �� obtained from experimental data � , �, � ,

and through (10);5) mean of four data points; one strain gauge on the two adjacent uppermost wires of the outer layer; �� obtained from experimental data � and � ;6) mean of two data points; one strain gauge on the two adjacent uppermost wires of the outer layer; �� obtained from experimental data � and � ;7) regression line through 4 data points; the nearest strain gauge from the Keeper Edge (KE) on the uppermost wire of the outer layer (Drake, wire 1–5;

Bersfort, wire 1–7) and the nearest strain gauge from the Last Point of Contact (LPC) on the lowermost wire of the outer layer (Drake, wire 1–13; Bersfort,wire 1–7); �� obtained from experimental data � and � . In the calculation of the theoretical � �� , �� is computed with 4 aluminiumwires removed from the outer layer and is computed with no aluminium wire removed.

using (22), will vary according to the square root of the tractionforce and the variation of the alternating stress , computed byusing (21), is smaller. For example, consider the AAAC Aster570. It is made of 61 wires (wire dia. 3.45 mm) and its con-struction is 24/18/12/6/1. Doubling the traction force from 20%RTS to 40% RTS multiplies the ratio (22) by a factor

. The ratio given by (16) and (21) changes by a factorof 1.265 and of 1.245, respectively. Seppä [7] and Poffenbergerand Komenda (discussion in [8]) reported that the strain mea-sured in the clamp vicinity of a conductor varies as the squareroot of the applied tension force.

It seems difficult to reconcile the two approaches based on(11) and (16) since the traction force variation and the bendingrigidity variation affect the alternating stress differently. Ob-viously, the difference between (9b) and (16) comes from the

way the curvature at 0 is computed. In both cases, the orig-inating equation is (6)

(23)

In (9b), is obtained from the sine-shaped deflection curveof the vibrating conductor through the following equation:

(24)

while in (16), comes from (13) and it is expressed in termof the amplitude A of the deflection curve. The value A is deter-


mined by measuring the peak-peak displacement at the position

(25)

It is easy to experimentally obtain an empirical relationshipbetween and at a given frequency of excitation butthis does not give the exact value of and it does not providewhich position the equal slope between the deflection curve ofthe local field near the bushing and the sine-shaped deflectioncurve of vibrating conductor occurs. On the other hand, it shouldbe remembered that (23) and (25) are based on the assumptionof a uniform bending rigidity along the length of the conductor.In fact, the bending rigidity is probably high (in the order of

) at a portion of the deflection curve very close to the fixedend due to the clamping action of the bolting of the two halvesof the square-faced bushing (or of the bolting of the keeper for a“conductor-suspension clamp” assembly) and in the asymptoticportion of the deflection curve. Elsewhere, between these tworegions, the bending rigidity may be much lower, as slip betweenwires is easier.

IV. EXPERIMENTAL DATA

A. Alternating Stress Based on

Table II presents a summary of the experimental data reportedby different laboratories in order to compare the measured alter-nating strain (transformed into corresponding alternating stress)to the stress calculated by using (11). Table II shows that withonly three exceptions, the data from tests with the square-facedbushing follow more closely (11) than do data from tests per-formed using the short commercial suspension clamp. As al-ready pointed out, (11) does not take into account the geometryof the clamp and looking at the results of McGill and Ramey([9], [10]), the radius of curvature of the generic clamp seemsto significantly affect the measured alternating strains. Also, inthe case of a short suspension clamp, strain gauges cannot be lo-calized as close to the edge as with a square-faced bushing sincethe exit of the keeper and of the bed of suspension clamp havea relieving curvature which can interfere with the strain gaugeswhen glued too closely to the support.

The experimental data show a large scatter. As reported in[2], Claren & Diana [8] found that strain measurements varieda lot from one wire to the next ( 30%). Seppä [11] reported adeviation of the measured dynamic strain value from the com-puted average of the order of 25%. Looking at [9] and [10], onefinds a variation of about 20% with respect to the mean value.

B. Alternating Stress Based on

Equation (16) will be modified in order to facilitate the com-parison between experimental and theoretical results; to com-pare measured strains, (16) is modified as follows:

(26)

Fig. 8. � (central alternating) function of bending amplitude ACSR Drakewith suspension clamp.

and in order to compare conductors of different sizes, the fol-lowing ratio is used:

(27)

Fig. 8 shows the strains measured at the uppermost po-sition of the ACSR Drake conductor fitted in the “conventionalsuspension clamp” reported by Poffenberger and Swart [13] andin the generic clamps used by McGill and Ramey [9], [10]. Thegeneric clamps have a radius of curvature of 6 in. and 12 in.with (“deep bed”) or without (“shallow bed”) a bed well fittedto the conductor, and the sag angle is 11 . The straight lines cor-respond to the linear theory in (26) computed by usingat three traction force levels. The strain values shown for the

-axis interval [0.9 mm, 1.3 mm] [10] in Fig. 8 were mea-sured on two adjacent wires and as close as technically feasiblefrom the keeper edge (KE). It can be noticed, even with the largescatter for a given generic clamp, that increasing the radius ofcurvature decreased the alternating strain and, except forone case, that the theoretical straight line does not provide agood fit for these experimental data.

Fig. 9 is an enlarged view of Fig. 8 in the low amplitude re-gion. Poffenberger and Swart [13] explained the nonlinearity ofthe experimental data by changes in flexural rigidity caused byinterstrand slippage at higher displacements. The difference inbehavior at 15% and 25% RTS was associated with a “specu-lative possibility that the last point of contact between the con-ductor and the conventional suspension clamp may have shiftedslightly in reducing the tension. If the last point of contact wascloser to the support center line, the corresponding lengtheningof the deflection arm and shifting the gauge would both resultin lower strain values.” The two extreme points (partially filledsquares) on each experimental data set are two experimentalpoints used by Poffenberger and Swart [13] to build Fig. 11.

Fig. 10 shows experimental data from an IEEE Committeereport [14]. It shows measured strain values of an ACSR 583.2kcmils 18/7 conductor fitted in an articulated suspension clampunder a traction force of 30% RTS. The sag angle (determined


Fig. 9. � (central alternating) versus bending amplitude ACSR Drake withthe suspension clamp.

Fig. 10. � (central alternating) versus bending amplitude 583.2-kcmil ACSRconductor with a suspension clamp.

by using the ratio of the vertical load to the tension load) is ap-proximately 4.6 . The peak-peak bending amplitude is mea-sured relative to the clamp. Strain gauges no. 2, 4, and 5 wereglued on the three uppermost wires on the conductor at the lastpoint of contact between the top of conductor and suspensionclamp. Gauge no. 3 was located on the same wire as no.4 at 12.7mm from that gauge, on the clamp side. Gauge no. 1 was locatedon the gauge no. 2 wire at 12.7 mm from that gauge, on the con-ductor span side. As seen in Fig. 10, the scatter of strain valuescan be high; for example, at 0.5 mm, the strain amplitudegiven by gauge no. 4 is 35% higher than the mean value obtainedfrom the remaining four gauges. The upper data point (partiallyfilled square) corresponds to an experimental point used by Pof-fenberger and Swart [13] to draw Fig. 11.

Fig. 11 shows the experimental data for seven conductorsreported by Poffenberger and Swart [13] and those obtainedfor seven conductors by Claren and Diana [8]. Each point cor-responds to the maximum strain value at the maximum dis-placement obtained in any set of data, where interstrand bindingwould be at a minimum. The data (six of seven conductors) re-ported in [13] come from unpublished reports by Rulhman andSwart and by Edwards and Boyd at the publication time of [13].The data from Rulhman and Swart and Edwards and Boyd re-ports are used in [14] and it is reported that “tests carried onunarticulated clamps with various mouth radii showed that thestrain-bending-amplitude relationship was constant with radii

Fig. 11. Strain/displacement factor versus stiffness parameter (exp. pointscomputed by using EImin).

ranging from 1.6 mm (1/16 in) to 152 mm (6 in).” The curvedrawn in Fig. 11 corresponds to the theoretical curve computedwith (27) using . The solid symbols correspond to datawhere the peak-peak bending amplitude exceeds 0.25 mm(0.010 in.) and the clear symbols correspond to less than 0.25mm. The Claren and Diana data [8] were obtained for seven con-ductors fitted with a square-faced aluminum bushing at a zerosag angle. Unfortunately, for those data reported in [13], theexact boundary conditions (sag angle, geometry of the clamps)were not given for each experimental point. Poffenberger andKomenda pointed out that the scatter of the Claren & Diana data[8] may be attributed to not exceeding 0.25 mm (discussionin [8]) and that using data obtained with amplitudes largerthan 0.25 mm would show a better fit with the theoretical curvebased on . A. T. Edwards (discussion in [8]) referring towork reported in [14] mentioned that “it was impossible to pre-dict where the maximum strain would occur. Thus, it was foundto be necessary to install strain gauges on the top three strandsat the clamp just where the conductor leaves the clamp, and justinside and outside the mouth of the clamp. The maximum strainwas chosen from the five separate strain amplitude readings atand about the clamp.”

Komenda and Swart [15] did perform a statistical analysis of39 different laboratory tests and ten data sets measured at dif-ferent field locations. It resulted in 49 different data groups; eachdata group “ ” contains a large number of points (onlythree data groups have ). Each experimental point istransformed according to . For each data group,a mean value is computed and, finally, using

from (26), is computed as . This procedure enablesmapping the nonlinear behavior of data into a straight-line theo-retical curve. Taking into account the 49 data groups, they com-puted a weighted average and a standard de-viation 0.246 using the weight of each data group.The corresponding confidence band is equal to [0.888, 1.077](Table III). They concluded that, on average, (26) is a reason-able approximation of the relationship between strain and

.Unfortunately, no information regarding the type of support

corresponding to each data group was reported in [15]. It wasmentioned in the paper that the strain gauges were located atthe point of maximum strain for the support configuration being


TABLE IIIANALYSIS RESULTS OF THE DATA REPORTED BY KOMENDA AND SWART [15]

studied but no specific information was given on how it was ac-complished. All data were grouped together without regardingthe geometrical characteristics of the support (such as a square-faced bushing) or a short commercial suspension clamp. Theprediction band (Table III), computed from the analysis basedon all data, gives an interval which probably characterizes theeffect of the geometrical parameters of the support, of the posi-tion where the strain gauges are glued, and of the variation fromone wire to the adjacent one if single strain gauge monitoringis used; these effects can also be appreciated by the large varia-tion of individual values of shown in the column identified as

.Now, the precise mathematical meaning of the terms “confi-

dence” and “prediction” band should be recalled. If 100 samplesfrom a population were drawn and a 99% confidence band of themean was computed for each sample, then 99% of these confi-dence bands would contain the true mean of the population. Inpractice, usually one sample is drawn and one confidence bandis computed, so it cannot be said that this single confidence bandnecessarily contains the true mean of the population. But it canbe said that with 99% confidence, the true mean of the popula-tion is contained in this single computed confidence band. Thislast statement implies a notion of frequency: we do not knowwhether this single confidence band contains the true mean ofthe population but we do know that the method used to computethe confidence band yields a true statement in 99% of the cases.On the other hand, the prediction band shown in Table III givesthe interval in which the value of a single future observation isexpected to be in 99% of the cases.

Looking at the other results of Table III, which were obtainedfollowing the same computational approach, several pointsought to be emphasized.

1) The field data probably correspond to a short metallic con-ventional suspension clamp and is lower than 0.983.

2) Looking at the laboratory data of the ACSR Pheasant, theresulting is much less than 1.0 and the 99% confi-dence band does not include the value of 1; this indi-cates probably that significantly departs from 1.0. Isthis behavior related to a specific conductor-support systemor to specific characteristics of the measuring system?(Is the value identical in each data group?)

3) Considering the laboratory data of ACSR Drake, which hasa large number of data groups 22), the resultingis higher than 1.0, and the 99% confidence band does not

include the value of 1. The same question as in theprecedent point arises.

A more refined statistical analysis should take into accountthe clamp parameters and the strain gauge location; this analysisshould be based on data obtained with measured at the stan-dard value (89 mm). Unfortunately, in [15], the valueused for each data group is not reported.

Scanlan and Swart [16] did test an ACSR Pheasant fittedat both ends of the vibrating span with what seems to besquare-faced bushings. A zero sag angle and a rather lowtraction force of 14.3% RTS were imposed. Spanwise dynamic

variation along the vibrating conductor was determinedby measuring with optical means the peak-peak transversedisplacements at 15 positions from the farthest fixed end fromthe shaker inward to midspan. Several tests were carried outat different values of midspan deflection. Using the classicalbeam bending equation, dynamic bending stiffness was thennumerically determined along the 30-in-long half span. Effec-tive dynamic values as high as 56% were found atthe fixed end with an average value of 42% . The generaltrend of the dynamic effective was found to decrease atpositions away from the fixed ends.

Fig. 12 is obtained by grouping the data of [1], [8], [10], andthose reported in [13]. The data shown are those where isgreater than 0.25 mm (0.010 in.) and each point correspondsto the maximum strain value at the maximum displacement ob-tained in any set of data. The curve drawn corresponds to thetheoretical PS (27) computed by using (or ac-cording to [15]). It can be noticed that the experimental data of[1] and [10], which are associated with clamps that differ greatlyfrom a square-faced bushing, depart significantly from the the-oretical curve which shows that clamp characteristics do influ-ence strain value .

In order to show the scatter of strain measured on adjacentwires, strain data measured by McGill and Ramey ([9], [10])on an ACSR Drake conductor are compared in Table IV withstrain values predicted by (26). Strain gauges are glued on thetwo uppermost wires of the outer layer 6.35 mm away from theKE. The mean of all measured strains corresponds to 54% ofthe theoretical value. A scatter as high as 20% compared tothe mean of the two measured adjacent wires was found in eachtest. As previously mentioned, this scatter was also reported in[8] and can be seen in Figs. 10 and 12.


TABLE IVEXPERIMENTAL DATA FROM [10]

Assumed value: 1.295 mm (0.051 in.). The � value reported in [10] for this test is 0.38 mm (0.015 in.). Comparing the strain values of this test with the strain

values of the � � 1.09-mm (0.043 in.) test (line above in Table), it seems that the � value (0.015 in.) is very low. All tests were carried out under the

same testing conditions and should give approximately the same values of � for the 12-in. radius clamp. Checking in [9] and [10], the reported values of

computed stresses, the ratio of these computed stresses is 1.1837 which is approximately equal to the following ratio of � values 0.051 in./0.43 in. Thus, the

value of 0.015 in. seems to be a typographical error. The testing conditions were �� 284.5 mm/s (� � 32 Hz; double amplitude mid-loop � 17.8

mm), sag angle � 11 deg.), 28% RTS, � � 89 mm. The strain data of this test are used in Figs. 8 and 12 with the assumed value of � � 1.295 mm.

The strain values of this test are not used in the computed mean of 54%.

Fig. 12. Strain/displacement factor versus stiffness parameter (exp. pointscomputed using EImin).

V. CONCLUSION

Section II gives the theoretical development of an alternatingstress based either on vibration parameter or on the bendingamplitude, which have been noted and . InSection III, it is shown that it is difficult to reconcile both stressvalues because the traction stress influences each value ac-cording differently and because the use of the two limit bendingrigidities ( and ) has a much stronger influence onthe PS value than on the value . In Sections IV-Aand B, the correlation between the reported experimental dataand the theoretical was examined. It was found that thescatter of experimental strain values is large and that in manycases, the correlation between experimental strains and theoret-ical is weak.

Figs. 4–7 are the fatigue S-N diagrams for two-layer andthree-layer ACSR conductors fitted on short commercialmetallic suspensions clamps and on metallic bell mouthclamps. In these diagrams, the vertical axis is the alternatingstress amplitude or . The two-layer

and the three-layer ACSR conductors show a different patternwith respect to the so-called “Endurance Limit ” (which isoften defined as the highest amplitude with a zero wire breakafter 500 million cycles). As reported in [2], more fatigue testsare needed to better characterize the difference between thetwo- and three-layer ACSR conductor endurance limit. In [2]and [3], selecting the endurance limit for all multilayer ACSRconductors based on the three-layer conductor fatigue diagrameither at equal to 22 MPa or equal to 8.5MPa is suggested.

Dividing one or the other of values by the aluminiumYoung’s modulus , one obtains a characteristic strain levelwhich may be compared with the strain measured on a con-ductor fitted to a particular suspension clamp. Using thiscomparison to determine its specific fatigue performance orthe severity of vibration at the clamp is misleading because, inmany cases, the experimental data do not show a strongcorrelation to both theoretical models for . In fact, becausethe endurance limits of three-layer ACSR conductors havedifferent values depending on the way they are determined, it ismost probable that the equations on which these two modelsare based do not contain the fundamental parameters whichare necessary to characterize the fretting fatigue phenomenoninvolved in wire breaks. It is, of course, recognized that theoriginal intent of defining the alternating stress was to have away of regrouping fatigue results for different conductor typesand test conditions based on the use of macroscopic parameter( or ), thus eliminating the inherent scatter arisingfrom strains measured on an individual wire.

REFERENCES

[1] F. Lévesque, S. Goudreau, A. Cardou, and L. Cloutier, “Strain mea-surements on conductors acsr conductors during fatigue tests I—Ex-perimental method and data,” IEEE Trans. Power Del., vol. 25, no. 4,pp. 2825–2834.

[2] L. Cloutier, S. Goudreau, and A. Cardou, “Chapter 3: Fatigue of over-head conductors,” in EPRI Transmission Line Reference Book: Wind-Induced Conductor Motion. Palo Alto, CA: EPRI, 2006.


[3] C. B. Rawlins, “Chap. 2: Fatigue of overhead conductors,” in Transmis-sion Line Reference Book—Wind-Induced Conductor Motion. PaloAlto, CA: EPRI, 1979, EPRI Res. Project 792.

[4] G. E. Ramey, “Conductor fatigue life research” Jul. 1981, EL-1946,EPRI Res. Project 1278-1, Final Rep.

[5] GREMCA, “Fatigue tests on the 48/7 Bersfort acsr conductor at Ybvarying from 0.3 mm to 0.35 mm” (in French) Dept. Mechan. Eng.,Laval Univ., Quebec City, QC, Canada, Tech. Rep. SM-2007-01, 2007.

[6] CIGRE Task Force B2.11.07, “Fatigue endurance capability of con-ductor/clamp systems—Update of present knowledge,” in ConférenceInternationale des Grands Réseaux Électriques, 2006, Tech. brochureno TBD.

[7] T. O. Seppä, “Self-damping measurements and energy balance of acsrdrake,” in Proc. IEEE Winter Power Meeting, New York, Jan. 31–Feb.5, 1971, pp. 1–8.

[8] R. Claren and G. Diana, “Dynamics strain distribution on loadedstranded cables,” IEEE Trans. Power App. Syst., vol. PAS-88, no. 11,pp. 1678–1690, Nov. 1969.

[9] P. B. McGill and G. E. Ramey, “Effect of suspension clamp geometryon transmission line fatigue,” J. Ener Eng., ASCE, vol. 112, no. 3, pp.168–184, 1986.

[10] G. E. Ramey, “Conductor Fatigue life research eolian vibration of trans-mission lines,” Jan. 1987, EL-4744, EPRI Res. Project 1278-1, FinalRep.

[11] T. Seppä, Transmission Line Vibration IV, Fatigue Tests of ACSR IbisConductor. Helsinki, Finland: Imatran Voima Osakeyhtio, 1968.

[12] A. R. Hard, “Studies of conductor vibration in laboratory span, outdoortest span and actual transmission lines,” CIGRE, Rep. 404, 1958.

[13] J. C. Poffenberger and R. L. Swart, “Differential displacement and dy-namic conductor strain,” IEEE Trans. Power App. Syst., vol. PAS-84,no. 4, pp. 281–289, Apr. 1965.

[14] IEEE comittee report, “Standardization of conductor vibration mea-surements,” IEEE Trans. Power App. Syst., vol. PAS-85, no. 1, pp.10–22, Jan. 1966.

[15] R. A. Komenda and R. L. Swart, “Interpretation of field vibration data,”IEEE Trans. Power App. Syst., vol. PAS-87, no. 4, pp. 1066–1073, Apr.1968.

[16] R. H. Scanlan and R. L. Swart, “Bending stiffness and strain in strandedcables,” presented at the IEEE Winter Power Meeting, New York, 1967.

Sylvain Goudreau received the Bachelor and Master degrees in mechanicalengineering from École Polytechnique de Montréal, Montréal, QC, Canada, in1977 and 1980, respectively, and the Ph.D. degree in mechanical engineeringfrom Université Laval in 1990.

He is a Professor at the Department of Mechanical Engineering at Univer-sité Laval, Québec City, QC, Canada, and Principal Researcher of Groupe deREcherche en Mécanique des Conducteurs Aériens Research Group. He is aRegistered Professional Engineer in the Province of Québec. Before beginninghis Ph.D. studies, he was with the National Research Council of Canada (Institutdu génie des matériaux, Montréal), where he was involved in the developmentof a mechanical testing laboratory and in studies on composite materials. In1988, he joined the Mechanical Engineering Department at Université Laval.His research activities are in the field of mechanical behavior of overhead lineconductors and their related fatigue problems. He is author or coauthor of manytechnical reports and papers on these subjects.

Frédéric Lévesque is a Postdoctoral Fellow at the Department of Civil En-gineering, Sherbrooke University, Sherbrooke, QC, Canada. He received theB.Sc., M.Sc., and Ph.D. degrees in mechanical engineering from the Univer-sity Laval, Québec City, QC, Canada .

His research interests are the mechanics involved in the fatigue of overheadelectrical conductors (contact and fracture mechanics, stress analysis, modeliza-tion, and mitigation of aeolian vibrations).

Alain Cardou received the M.Sc. and Ph.D. degrees in mechanics and materialsfrom the University of Minnesota at Minneapolis.

He is Adjunct Professor and, formerly, Head of the Department of MechanicalEngineering at Université Laval, Québec City, QC, Canada. His general researchinterests are stress and strength analysis, of which he is the author or coauthorof many papers. For several years, in collaboration with some power utilitiesand within the GREMCA research group, he has been working on overheadelectrical conductor fatigue problems.

Dr. Cardou is a Registered Professional Engineer in the Province of Québec,a Fellow of the Canadian Society for Mechanical Engineering, and a memberof the American Academy of Mechanics.

Louis Cloutier received the Ph.D. degree in mechanical engineering fromUniversité Laval, Québec City, QC, Canada, in 1966, and pursued postdoctoralwork in contact mechanics for a year at Cambridge University, Cambridge,U.K.

He held different functions in research laboratories (National ResearchCouncil of Canada and IREQ [Hydro-Québec’s Research Institute]), industries(Gleason Works, Roctest Ltd., and Sogequa, Inc.), and universities (Laval andSherbrooke). His professional experience of more than 40 years led him towork in several projects related to mechanical power transmission, medicalinstrumentation, and electrical power transmission. In the last field, his interestshave been mainly devoted to problems related to transmission-line mechanics:conductors, insulators, spacers, accessories, and more recently, line supports.He is the author or coauthor of several publications in related fields and holdstwo patents. He was Chair Holder from 2004 until his retirement in 2009 of theindustrial chair created by Hydro Québec TransÉnergie in collaboration withthe Natural Sciences and Engineering Research Council of Canada for studiesof the structural and mechanical aspects of overhead transmission lines. He isAdjunct Professor at the Department of Mechanical Engineering of UniversitéLaval.

Dr. Cloutier is a member of l’Ordre des Ingénieurs du Québec, an activemember of several technical and learned societies, a Distinguished Member ofCIGRE, and Fellow of the Canadian Society for Mechanical Engineering.

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