Stress-Strain Creep and Temperature Dependency of ADSS Sag An

Stress-Strain, Creep, and Temperature Dependency of ADSS (All Dielectric Self Supporting)Cables Sag & Tension Calculation

Cristian Militaru

Alcoa Fujikura Ltd., Spartanburg, SC

Abstract

It has been common in the industry to calculate sag &tension charts for ADSS cables without taking intoconsideration the influence of creep, coefficient of thermalexpansion (CTE). and the difference between the initial andfinal modulus. In some applications where the sag andtension performance of the cable is not critical. thepresentation of data in this manner is appropriate.However, the great majority of applications require veryexact determination of sag and tension, and the influence ofthe above factors is important. There is also confusionbetween the final state (after creep) and the loadingcondition (wind+ice). which are 2 different cases.Following thorough and repeated stress-strain and creeptests, this paper will show that ADSS cable has both aninitial state and a final state, each state having anunloaded (bare cable) and a loaded (ice and/or wind)case with resulting sag 8 tension charts as a function ofcreep and CTE. Additionally, the results of this work werecompared and validated by common industry sag &tensionsoftware, including Sag10 and PLS-CADD.

Y

1Fig.1 - Catenary Curve Analytic Method

.x + k, (1 I), followed by:

Catenary Curve Analytic Methody+ l + y

4-Z = .(f+)(12) which has as solution:

Fig.1 presents an ADSS cable element under the extrinsic(wind and ice) stresses and intrinsic (cable weight)stresses, with a length, On the curve y(x), given by the

formula:

y = sinh&.x+k,) (13) ; integrating rel.(13) results:

y=f.cosh($.*+k,)+k, (14);for x=0 results:

I= I,/=. dr (1); yields: g = ,/q (2)

XIAlso,the equilibrium equations results in:

H, =H, =H (3); V,-V, =dV=w.dl (4)considering rel. (2). the derivative of rel. (4) yields:

dV dl- = W.-g= w - l+y,,,du F

(5); also, the slope in any

point of the catenary curve is defined as the first derivativeof the function Y(~) of the curve:

V=H.tancp=H.%=H.y(6);yields:

dV-= H 4a!x Yz

(7) ; and: g = H.JJ (8) : using rel. (5)

and (8) results: H. y = w . (9) , and then:

integrating rel. (10),

,,=c(15)and: y=O (16),so: k, = k, = 0 (17).w

resulting the catenary curve

y = sinh? (20).a

In Fig.2 the designations are:S= span lengthB=S/2= half span length (assuming level supports)D= sag at mid-spanH= tension at the lowest point on the catenary (horizontaltension) - only for level span case, it is in the center of thespanT= tension in cable at structure (maximum tension)P= average tension in cableL/2=arc length of half-spanI= arc length from origin to point where coordinates are

(x,y)

International Wire & Cable Symposium Proceedings 1999 605

a and C respectively = distance of origin (of supportrespectively) from directdx of catenaryt= angle of tangent at support with directrixk= angle of tangent at point (x,y)w= resultant weight per unit length of cableE = cable strain (arc elongation in percent of span).

n

At the limit shown in Fig.2, for: x =: = B. rel. (18)

becomes:

C=a.o&(21); where:c&z=OS. (22)a

and:sinhX=6.5. (23), so,a

I = jm.dx= jJq.dr (24),or,0 0

[= jcosh; . cln (25), resulting from rel. (25) the cable

length equation: (28). At the limit, for

x =: = B, rel.(26) becomes -1(20

is defined as arc elongation in percent of span:

A Taylors series for cash yields:

;:r-l terms--]- 3 terms- ___- -+

-2 terms formula for rel. (28) results in:

yielding the parabolic equation: (36)

3 terms formula for rel. (28) results in:

D=u.[l++(+)2 +$@-*] (37)

which, using B=S/2 and rel. (19) yields the approximate

catenary formula : D =1 g+&:;31 (38)

For sags larger than 5% of the span, ret. (36) theparabolic equation gives erroneous results, while rel. (38)provides a more accurate solution, the exact solutionbeing given by rel. (28).

Example for an ADSS cable on transmission lines

The following variables, for the same span, have thesame values for any material (ACSR, AAC, EHS, ADSS,etc.) as long as they respect the cetenary equations.Span: S=1400 [ft]; w=1 [Ibs/ft]= constant value; H=9038[Ibs]=assumed value; a=H/w=9038 [fl]; B=S/2=700 [ft];

C=mosh~=9065.12 [fit D=C-a=27.12 [ft];

L=2.a.sinhB=1401.4 Ml; E= i-1 .100=0,1Ma ( 1

$. 100 = 1.9372 [%I; T= IV. c = 9065 [Ibs];

H+T?=9Ocj5 [Ibs]; PC---=

29052 Dbsl: p = 9052 Ws1

s,me of the ADSS characteristics, presenteWd in this

example, are:d=0.906 [in]; ,.f = a.& = 0.6447 [in2];

wc=0.277 [Ibs/ft].Loading Curve Type:B= ADSS w/o ice or wind Cbare unloaded cable):resultant weight: wr=wc=0.277 [Ibs/ft]H= ADSS plus heavy loading: according to NESC:

Regular ice density: rice = 57 [lbs/ft3];

Ice radial thickness: t=0.5 [in]; Temperature: 0 = 0 [OF]:Wind velocity: VW=40 [mph]; NESC factor: k=0.3;Wind pressure: Pw=0.0025,Vw2 =4 ]ps6;Ice weight:

A (d+2.t)* -d2w,, = - .

4 1 4 4.yi, = 0.875 Ws/fil

Wind force: w, = pw ti 2 ) = 0.635 [Ibs/ft]

Resultant weight:

w, =& +wicJ +ww2 + k = 1.615 [Ibs/ft]

606 International Wire & Cable Symposium Proceedings 1999

LoadingI Curve

Loading Resultant cross S=1400 [ft]weight: wr Sectional Stress [psi]

Type [lbs/ft] area:A [in21 +(pj

B 0.277 0.6447(509g+9

H 1.615 0.6447

area:A [in21

B 0.277 0.6447(509g+9

H 1.615 0.6447 @q.g=m

Tensions Limits:a) Maximum tension at Oo F under heavy loading not to

exceed 51.35% RBS: MWT=51.35%RBS [Ibs].

I INote: M& (max. working tension) was selected less thanMRCL, in order for this ADSS cable to cope with limit c)presented below.b) initial tension (when installed) at 600F w/o ice or wind

(bare cable) not to exceed 35% RBS:

TEos, = 35%RBS [Ibsl;

c) Final tension at 600F w/o ice or wind (bare cable) notto exceed 25% RBS:

Guide to Columns:1 and 2 are the same for any span, any material.3,4,5,6 are the same, for the same span for anymaterial: ACSR, AAC, EHS, ADSS, etc.7 and 8 are different, from one material to another(ACSR, AAC, EHS, ADSS, etc.)

This catenary table is transformed in a Preliminary Sag-Tension Graph, in Fig.4. This graph has 2 y axes: leftside: stress [psi], og and OH: B-bare cable: H-heavy load.and right side: sag: D [ft]. Also, it has 2 x axes, strain. E[%] (arc elongation in percent of span) and temperature, 8WI.

Stress-Strain Tests

Stress-strain tests on ADSS cable performed in the labshow (see Fig.3) that they tit a straight line, characterizedby a polynomial function of 1st degree, whereas metalliccables (conductors. OPT-GW, etc.) are characterized by apolynomial function of the 4th degree (5 coefficients).From ail the tests performed, results show, that differencesexist between the initial modulus. E i (slope of the chargecurve) and the final modulus, E f (slope of the dischargecurve) and their values depend upon the ADSS cabledesign. There is also a permanent stretch, E p (alsoreferred to as set"), at the discharge, which depends onthe ADSS design.

Creep Tests

According to the ADSS cable draft standard, IEEE P-12221, the creep test must be performed at a constanttension equal to 50%. MRCL for 1000 hours at roomtemperature of 60 F. in general, for ADSS cables.MRCL=[%MIN...%M~RBS therefore the test is done atT=[%MIN/2...%MAX/2] RBS=ct. (see Fig.3). Consideringa nominal value of MRCL=50% RBS, the defaultconstant tension for the test would be: T=25%. RBS. Thecreep test on the cable design analyzed was performed ata constant tension: T=50%, MRCL=28Yc RBS, because forthis cable: MRCL=56%- RBS. The values ware recordedafter every hour (see Fig.8CreepTest, Polynomial Curveand Fig.G-"Creep Test: Logarithmic Curve). The strainafter 1 hour, defined as initial creep, was 42.89 [pin/in].After 1000 hours (41.8 days) the strain was 314.10 [pinlin].So the recorded creep during the test, defined as strain at1000 [h] minus strain at 1 [h], is 271.41[pin/in]. Theextrapolated value after 87380 hours (10 years, 364days/year) was 1142.69 [pin/in]. Therefore, the IO yearscreep, which is defined as strain at 87380 [h] minusstrain at 1 [h]. was 1100 [pin/in]=O.ll [%I. Other creeptests performed on other ADSS cable designs showed IOyears creep values in the same range. The curves on thestress-strain and tension-strain graphs are identical. Theonly difference is that on the ordinates (y) axis, whengoing from tension [ibs] to stress [psi], there is a divisionby the cross-sectional area of the cable, A [in2]. Thevalues on the strain (x) axis remain the same. For thestress-strain graph (Fig.3) at a tension (stress) equal withthe value for which the creep test was performed, T=50%-MRCL (a=50%. MRCL/A), a parallel to the x axisintersects the initial modulus curve in a point of abscise,0.5. E MRCL, and from that point, going horizontally.adding the IO years creep value of 0.11 [%I, is obtainedthe point on the IO years creep curve corresponding tothat tension for which the creep test was performed.Drawing a line from the origin through that point gives theslope (the modulus) for the IO years creep, E 0 Always,


Anqle: a'>d >p

Slope: ton g >ton d >ton /i

Modulus: E,> Ei > E,

tnOl .A0 iep(10 Y'S)

Fig. 3 - General Stress-Strain Chart for an ADSS cable

Fig. 4 - Preliminary Sag-Tension Graph for an ADSS cable


CREEP: E vs. TIME: t (FITTED LINE)

I I / .:

100 1000 10000 100000

TIME:t 1 hours 11

Fig. 5- Creep test for a particular ADSS cable : Polynomial Curve

log E = 0.2889 .log(t)+ log42.69

TIME: t [ hours ] LOG 87360 h(10 yn) 1 year=364 days

Fig. 6 - Creep test for a particular ADSS cable : Logarithmic Curve


for any ADSS design, the relation between the 3 moduli isEf>Ei>Ec.

Coefficient of Thermal Expansion

The values for CTE (designated here as a) weredetermined by the individual material properties in a mixture

formula: a = ,i, &Ai I,: EIAI (39) where oi, At, Ei are

the CTE, cross-sectional area and modulus of each one ofthe t elements in the ADSS construction respectively. Forthe great majority of ADSS cable designs, the influence ofCTE is smaller than that of creep. Designs with a lownumber of aramid yarn ends (typically for short spans) willyield larger differences in sags due to temperature thandesigns with a high number of aramid yam ends. This isdue to the fact that the aramid yarn is the only element witha negative a , while the rest of the elements have a positivea. To appreciate the impact of the contribution of aramidyam to the ADSS CTE, designs with low number of aramidyam ends have a CTE typically in the range 2.10-6 [l/OF]to 9.10-8 [l/OF

tlwhich is relatively close to aluminum,

CTE=l2.8.10- [l/OF], and sometimes larger than steel,CTE=8.4.10-6 [l/OF]. The CTE for cables with highernumbers of aramid ends are often 100 to 1000 timessmaller, 2.10-8 [l/OF] to 8.10-g [IloF]. and so, for thosedesigns, the influence of CTE on sag is negligible.

Sag-Tension Charts

The well known general equation of change of state:

(40)

shows- that the change in slack Is only equal to thechange In elastic elongatlon + change In thermalelongation, and does not include the change In plasticelongation (the creep). Therefore, the above relation istrue only if the 2 states of the cable are in the same stage,initial or final. When viewing sag charts (Fig.1 1 & Fig.12).this equation will allow a user to go only vertically from onecase to another case, but it will not allow him to gohorizontally (same temperature, same loading conditions,from initial stage to final stage) due to the influence ofcreep. A simplistic way of solving this issue which is stillused in some European countries is the following: the creepinfluence is considered to be equivalent with an off-settemperature, 8creep. given by the ratio (conductor IO yrs.creep-initial elongation)/ CTE. But this is not an exactmethod, because it only calculates an INITIAL sag&tensionchart, with the FINAL sag&tension chart being identicalwith the initial chart, the only thing is that the Initial chart,is moved to align It with the new correspondingtemperature. Therefore, the final sag at temperature 8 isequal with the initial sag at temperature 9+9creep. Themost accurate and exact solution is the graphic method. Inthis method, which was developed by Alwa2, 3 , the stress-strain graph (Fig.3) of the ADSS cable is superimposed onthe ADSS preliminary sag-tension graph (Fig.4), so theirabscissas coincide and the whole system of curves fromFig.3 are translated to the left, parallel with the x axis.up until the initial curve. noted 2, in Fig. 3 (and also in

Fig.7) intersects the curve H on the index mark=11300psi (tension limit a]) the imposed maximum tension atOoF under heavy load. For purposes of this paper, a MWTof 51%RBS was imposed. This MWT, which is less thanthe cables MRCL of 56%RBS, was used to be sure thatneither tension limits b] or c] will be exceeded. Therefore,tension limit a] is the governing condition. Thesuperimposed graphs then appear in Fig.7. The resultantinitial sag at OoF under heavy loading (54.10 ft.) is foundvertically above point a] on curve D. The initial tension at8OoF. bare cable=8750 psi (4352 Ibs) is found at theintersection of curve 2 with curve B, and thecorresponding sag (15.59 ft) is on curve D. The finalstress-strain curve 3a. which is the curve after loading tothe maximum tension (MWT=51%RBS). at OOF, is drawnfrom point a], which is the intersection point of curves 2and H, parallel to curve 3, which is the final stress-straincurve afler loading to MRCL=56%RBS, at OoF. Now, thefinal tension at OoF. after heavy loading =6440 psi (4151Ibs) is found where curve 3a intersects curve B. Thecorresponding sag (18.35 fl) is found vertically on curve D.The next operation is to determine whether the final sagafter IO years creep at 8OoF will exceed the final sag afterheavy loading at OoF. Before moving the stress-straingraph from its present position, the location of OoF on itstemperature scale is marked on Fig.7 as reference pointR. The temperature off-set to the right at 8OoF (Fig.7) in%strain is equal to a80oF~100=0.01992 j%] (41) wherea=3.32.10-8 [l/OF] is the ADSS CTE. Therefore, thestress-strain graph is moved to the right with 0.01992 [%I(Fig.7) until 8OoF on the temperature scale coincides withreference point R (Fig.8). The initial tension at 8OoF=8530psi (4210 Ibs) is found at the intersection of curve 2 withcurve B, and the corresponding sag (18.1 lft) is foundvertically on curve D (Fig.8). The final stress-strain curve3 b. under heavy loading, after creep for IO years at 800F.is drawn from the intersection point of curves 4 and B,parallel to curve 3. The final tension at 800F after creep forIO years=5500 psi (3548 Ibs) is located at the intersectionof curve 3b (or curve 4) with curve B . The correspondingsag (19.13 ft) is found vertically on curve D (Fig.8). Sincethe final sag at 600F after creep for 10 years=19.13 ft(Fig.8) exceeds the flnal sag at OoF after heavyloading=16.35 ft (Fig.7), creep is the governing case.For this case, users of Sag10 will see the flag, CREEP ISA FACTOR. SAG10 will print only the final chart aflercreep (not the final chart afler heavy load). Users of PLS-CADD will see the same results in the chart called FINALAFTER CREEP (see Fig.1 1 & Fig. 12). The final sag andtension at OoF must now be corrected using the revisedstress-strain curve. For this purpose, the temperature axiswill have an off-set of 0.01992 [%] to the lefl (Fig.8) toprovide the values at OoF. Therefore, the stress-straingraph is moved to left (Fig.8) until OoF on the temperaturescale coincides with reference point R (Fig.9). Thecorrected final tension at OoF, bare cable, (after creep for10 years at 8OoF)=5720 psi (3887 Ibs) is found at theintersection between curves 3b and B. The correspondingfinal sag (18.40 ft.) is found vertically on curve D. The finaltension at OoF under heavy loading, (after 10 years creepat 8OoF)=lO900 psi (7027 Ibs) is found at the intersectionbetween curves 3b and H. Its corresponding resultantfinal sag (58.07 fl) its on curve D (Fig.9). When


I.,.:,.. ..I... ,..I./,

Fig. 10- Final Trial. for IZOOF, after adjustment for 10 years creep correction

6 1 2 International Wire & Cable Symposium Proceedings 1999

determining temperatures for calculation of sag andtension performance, the maximum temperature of theADSS cable should be the maximum ambienttemperature plus the heat absorbed by the cable. Areasonable assumption is 12OoF (49oC). Electricalconductors can reach higher values. i.e. 167JF (75oC), or212oF (IOOOC), due to the continuous current rating of theconductor, which does not exist for ADSS cables. Thus. for12OoF, the temperature off-set to the right (Fig.9) to getvalues at 12OoF. in %strain is : a~120oF~100=0.03964 [%I.Therefore, the stress-strain graph is moved to the right withthis value (Fig.9) until 12OoF on the temperature scalecoincide with reference point R (Fig.10). The initial tensionat 12O~F=6311 psi (4069 Ibs) is found at the intersection ofcurve 2 with curve B, and corresponding sag (16.67 ft) is oncurve D. The final tension at 12OoF (after creep for 10 yearsat 6OoF)=5265 psi (3407 Ibs) is found at the intersection ofcurve 3b (or 4) and curve B, and corresponding sag (19.91ft) is on curve D (Fig.10).

Conclusions

Using this ADSS cable characteristics as input data, theoutput in SAG10 is presented in Fig.11, while the outputscreen for PLS-CADD is presented in Fig.12. As can benoticed. the graphical method presented above producesvery similar results in these two programs, as well as inother sag and tension programs on the market. As a note,for different ADSS designs and different span and loadingconditions, there can be many situations when thepermanent elongation after heavy loading (due to thestretch of the cable, E p) is larger than the elongationafter 10 years creep. In these cases, for users of theSAG10 program, the flag CREEP IS NOT A FACTOR isshown, and the final sag printed is the sag after heavy load(no more after IO years creep). Users of PLS-CADD willsee the same result in the chart called FINAL AFTERLOAD. The influence of creep on ADSS cable sags isdifferent from one design to another. As an example, thedifference between the final and initial sag can range from0.5 ft up to 1.2 fl in a span range of 200-600 fl, and from 1.5ft. up to 2.5 fl in a span range of 600-1400 fl, under NESCHeavy loading. For spans over 1600 ft the differences canbe 3-3.5 ft. For spans under NESC Light or Mediumloadings, the creep influence results in sag differences lessthan the numbers listed above. The influence of thecoefficient of thermal expansion of the ADSS cables issmaller than that of creep: as an example, changes in sagdue to temperatures ranging from -2OoF to 12OoF wouldyield 0.5 ft up to 1.75 fl for low aramid yarn countsapplications, and becomes negligible (0.01 ft) for thosedesigns with maximum numbers of aramid yarns.

References

1. IEEE 1222P- Standard for All Dielectric Self-Supporting FiberOptic Cable (ADSS) for use on Overhead Utility Lines - Draft,April 1995

2. Aluminum Electrical Conductor Handbook. chapter 5- thirdedition, 1989

3 . Alcoa Handbook, Section 8:Graphic Method for Sag TensionCalculation for ASCR and Other Conductors-1970

Fig.12 - PLS-CADD Output for this ADSS design

Mr. Cristian Militaru received MS degree (1980-1985) and Ph.D.degree (1990-1995) in Electrical Power Engineering fromPolytechnic University of Bucharest, Romania. He worked for 11years as a Transmission Design & Consultant Engineer in thepower utility industry in Europe, Middle East and SouthEast Asia.Since 1996 he has been employed with Alcoa Fujikura Ltd., USA,as a Development Engineer in the OPT-GW 8 ADSS cable andhardware department. Mailing Address: Alcoa Fujikura Ltd.P.O.Box 3127, Spartanburg, SC 29304-3127.


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Stress-Strain Creep and Temperature Dependency of ADSS Sag An