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Sag-tension Calculations

A CIGRE Tutorial Based on

Technical Brochure 324

Dale Douglass, PDC

Paul Springer, Southwire Co

14 January, 2013



1/14/13 IEEE Sag-Ten Tutorial

Why Bother with Sag-Tension?

• Sag determines electrical clearances, right-of-

way width (blowout), uplift (wts & strain), thermal

rating• Sag is a factor in electric & magnetic fields,

aeolian vibration (H/w), ice galloping

• Tension determines structure angle/dead-end/broken wire loads

• Tension limits determine conductor system

safety factor, vibration, & structure cost

2




Sag-tension Calculations – Key

Line Design Parameters

• Maximum sag – minimum clearance to

ground and other conductors must be

maintained usually at high temp.

• Maximum tension so that structures can

be designed to withstand it.

• Minimum sag to control structure uplift

problems & H/w during “coldest month” to

limit aeolian vibration.

3




Key Questions

• What is a ruling span & why bother with it?

• How is the conductor tension related to the

sag?

• Why define initial & final conditions?

• What are typical conductor tension limits?

• Modeling 2-part conductors (e.g. ACSR).

4




What is a ruling span?

5

Strain Structure Suspension Structure




( )max

2

3 Average Average RS S S S ≈ + −

S1 S2 S3

RS

6

S +----+S +S

S +----+S +S = RS

n21

3n

32

31



The Ruling Span

• Simpler concept than multi-span line

section.

• For many lines, the tension variation with

temperature and load is the same for the

ruling span and each suspension span.

• Stringing sags calculated as a function of

suspension span length and temperature

since tension is the same in all.

1/14/13 IEEE Sag-Ten Tutorial 7




The Catenary Curve

• HyperbolicFunctions & Parabolas

• Sag vs weight & tension

• Length between supports• What is Slack?

8



The Catenary – Level Span

Sag D

H - Horizontal Component of Tension (lb) L - Conductor length (ft)

T - Maximum tension (lb) w - Conductor weight (lb/ft)

x, y - wire location in xy coordinates (0,0) is the lowest point (ft)

D - Maximum sag (ft) S - Span length (ft)

y(x) ≈

D (sag at belly)

D ≈

Max.

Tension H

(S/2, D)(end support)

≈ S +

≈ S +


Span




Catenary Sample Calcs

for Arbutus AAC

20.7453 60012.064 3 678

8 2780

D ft ( . m)⋅

≅ =

⋅

w=0.7453 lbs/ft Bare Weight H=2780 lbs (20% RBS)

S=600 ft ruling span

600 0.7453 8 12.064600.647

24 2780 3 600

2 2 2

2 2 L 600 1 + 600 1 + ft

⋅ ⋅≅ ⋅ = ⋅ = ⋅ ⋅

2

2

8 12.0640.647

3 600

Slack = L - S = 600 ft ⋅

⋅ = ⋅

( )0.64712.064 (3.678 )

8

3 600Sag = ft m

⋅ ⋅=

10

Notice that 8inches of slack

produces 12 ft of

sag!!



Catenary Observations

• If the weight doubles, and L & D stay the

same, the tension doubles (flexible chain).

• Heating the conductor and changing the

conductor tension can change the length &

thus the sag.

• If the conductor length changes even by a

small amount, the sag and tension can

change by a large amount.





Conductor Elongation

• Elastic elongation (conductor stiffness)

• Thermal elongation

• Plastic Elongation of Aluminum

– Settlement & Short-term creep – Long term creep

L H

L E Aε

∆ ∆= =

⋅

A A

LT

Lα

∆= ⋅ ∆

12




Conductor Elongation

Manufactured Length

Thermal

Str ain

ElasticStrain

Long-time

CreepStrainSettlement&1-hr

creep

Strain

13




Sag-tension Envelope

GROUND LEVEL

Minimum Electrical

Clearance

Initial Installed Sag @15C

Final Unloaded Sag @15C

Sag @ Max Ice/Wind Load

Sag @ Max Electrical

Load, Tmax

Span Length

14



Simplified Sag-Tension Calcs


w=0.7453 lbs/ft Bare

H=2780 lbs (20% RBS)

S=600 ft

( )( )12.8 6* 167 60 600.647*(1.00137) 601.470 L 600.647 1 + e ft ≅ ⋅ − − = =

Slack = L - S = 1.470 ft

( )1.470 18.1878

3 600 D = ft ⋅ ⋅ =

L = 600.647 ft

L-S = Slack = 0.647 ft

D = 12.064 ft

795kcmil 37 strand Arbutus AAC @60F

Now increase cond temp to 167F

2 20.7453 6001844

8 8 18.187

w S H lbs

D

⋅ ⋅= = =

⋅ ⋅

15



Simplified Sag-Ten Calcs (cont)


( )1844 2780601.470* (0.999786) 601.341

0.6245*7 6 L 601.470 1 + ft

e

− ≅ ⋅ = =

Slack = L - S = 1.341 ft

( )1.34117.37

8

3 600 D = ft

⋅ ⋅=

795kcmil 37 strand Arbutus AAC @60F

Increasing the cond temp from 60F to 167F, caused

the slack to increase by 130%, the tension to drop by

from 2780 to 1844 lbs (35%) & sag to increase from

12.1 to 18.2 ft (50%).

After multiple iterations, the exact answer is 1931 lbs

16



Numerical Calculation





Tension Limits and Sag

Tension at 15C

unloaded initial- %RTS

Tension at max

ice and windload - %RTS

Tension at max

ice and windload - kN

Initial Sag at

100C - meters

Final Sag at

100C - meters

10 22.6 31.6 14.6 14.6

15 31.7 44.4 10.9 11.0

20 38.4 53.8 9.0 9.4

25 43.5 61.0 7.8 8.4

18



Modeling Non-Homogeneous

Conductors• Typically a non-conducting core with outer

layers of hard or soft aluminum strands.

– Core shows little plastic elongation and a

lower CTE than aluminum

– Hard aluminum yields at 16ksi while soft

aluminum yields at 6ksi.

– For Drake 26/7 ACSR, alum is 14/31 ofbreaking strength




IEEE Sag-Ten Tutorial 20

Given the link between stress and strain in each component as shown in equations (13),the composite elastic modulus, E AS of the non-homogeneous conductor can be derived by

combining the preceding equations:

The component tensions are then found by rearranging equations (17):

AS AS

A A AS A

A E

A E H H

⋅

⋅⋅= (18a) and

AS AS

S S AS S

A E

A E H H

⋅

⋅⋅= (18b)

Finally, in terms of the modulus of the components, the composite linear modulus is:

AS

S

S

AS

A

A AS A

A E

A

A E E ⋅+⋅= (19)

S S

S

A A

A

AS AS

AS AS

E A

H

E A

H

E A

H

⋅=

⋅=

⋅≡ε (17)

Component Tensions – ACSRCIGRE Tech Brochure 324

1/14/13




Linear Thermal Strain - Non-Homogeneous A1/S1x ConductorFor non-homogeneous stranded conductors such as ACSR (A1/Syz), the composite

conductor’s rate of linear thermal expansion is less than that of all aluminium conductors

because the steel core wires elongate at half the rate of the aluminium layers. The

composite coefficient of linear thermal expansion of a non-homogenous conductor such

as A1/Syz may be calculated from the following equations:

+

=

AS

S

AS

S S

AS

A

AS

A A AS

A

A

E

E

A

A

E

E α α α (20)

Linear Thermal Strain – ACSRCIGRE Tech Brochure 324

1/14/13




For example, with 403mm2, 26/7 ACSR (403-A1/S1A-26/7) “Drake” conductor, the

composite modulus and thermal elongation coefficient, according to (19) and (20) are:

MPa E AS 746.468

8.65

1906.468

8.402

55 =

⋅+

⋅=

66 1084.186.468

8.65

74

190105.11

6.468

8.402

74

55623 −− ⋅=

⋅

⋅⋅+

⋅

⋅−= e AS α

Example Calculations – ACSRCIGRE Tech Brochure 324

1/14/13

35% higher than alum

alone

20% less than alum alone



Experimental Conductor Data& Numerical Sag-Tension

Calculations

Paul Springer

Southwire





Experimental Plastic Elongation Model

• Conductor composite (core component +conductor component) properties are non-linear

and poorly modeled by linear model

• By the 1920s, the experimental model wasdeveloped:• Changes in slack from elastic strain, short-term creep, and

long-term creep are determined from tests on finished

conductor

• Algebra used to compute sag and tension• Graphical computer developed to solve the enormously

complicated problem

• Modern computer programs are based on the graphical

method1/14/13



Early work station – analog computer

Alcoa Graphical Method workstation 1920s to 1970s1/14/13 IEEE Sag-Ten Tutorial 25



Stress-Strain Model – Type 13 ACSR

Initial Modulus

Core Initial Modulus

Aluminum Initial Modulus

10-year Creep Modulus

Aluminum 10-year Creep

26



Stress-Strain Model – Type 13 ACSS

27



IEEE Sag-Ten Tutorial

28

Modeling thermal strains

• Almost all composite conductors exhibit a “knee

point” in the mechanical response

• At low temperature, thermal strain (or sag with

increasing temperature) is the weighted average

of the aluminum and core strain• Above the knee point temperature, thermal sag is

governed by the thermal elongation of the core

• Thermal strains cause changes in elastic strains.The computations are iterative and extremely

tedious – but an ideal computer application




SAG10 Calculation Table

From Southwire SAG10 program 29




Summary of Some Key Points

• Tension equalization between suspension spansallows use of the ruling span

• Initial and final conditions occur at sagging andafter high loads and multiple years

• For large conductors, max tension is typicallybelow 60% in order to limit wind vibration & uplift

• Negative tensions (compression) in aluminum

occur at high temperature for ACSR because ofthe 2:1 diff in thermal elongation between alum& steel

30




General Sag-Ten References

• Aluminum Association Aluminum Electrical Conductor Handbook Publication No. ECH-56"• Southwire Company "Overhead Conductor Manual“

• Barrett, JS, Dutta S., and Nigol, O., A New Computer Model of A1/S1A (ACSR) Conductors, IEEE Trans., Vol.PAS-102, No. 3, March 1983, pp 614-621.

• Varney T., Aluminum Company of America, “Graphic Method for Sag Tension Calculations for A1/S1A (ACSR)and Other Conductors.”, Pittsburg, 1927

• Winkelman, P.F., “Sag-Tension Computations and Field Measurements of Bonneville Power Administration, AIEEPaper 59-900, June 1959.

• IEEE Working Group, “Limitations of the Ruling Span Method for Overhead Line Conductors at High Operating

Temperatures”. Report of IEEE WG on Thermal Aspects of Conductors, IEEE WPM 1998, Tampa, FL, Feb. 3,1998

• Thayer, E.S., “Computing tensions in transmission lines”, Electrical World, Vol.84, no.2, July 12, 1924

• Aluminum Association, “Stress-Strain-Creep Curves for Aluminum Overhead Electrical Conductors,” Published7/15/74.

• Barrett, JS, and Nigol, O., Characteristics of A1/S1A (ACSR) Conductors as High Temperatures and Stresses,IEEE Trans., Vol. PAS-100, No. 2, February 1981, pp 485-493

• Electrical Technical Committee of the Aluminum Association, “A Method of Stress-Strain Testing of AluminumConductor and ACSR” and “A Test Method for Determining the Long Time Tensile Creep of Aluminum Conductorsin Overhead Lines”, January, 1999, The aluminum Association, Washington, DC 20006, USA.

• Harvey, JR and Larson RE. Use of Elevated Temperature Creep Data in Sag-Tension Calculations . IEEE Trans.,Vol. PAS-89, No. 3, pp. 380-386, March 1970

• Rawlins, C.B., “Some Effects of Mill Practice on the Stress-Strain Behaviour of ACSR”, IEEE WPM 1998, Tampa,FL, Feb. 1998.

31



The End

A Sag-tension Tutorial

Prepared for the IEEE TP&CSubcommittee by Dale Douglass

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