Sag tension calcs-ohl-tutorial

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


( )max23Average AverageRS S S S≈ + −

S1 S2 S3

RS

6

S + ---- + S + SS + ---- + 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) ≈ 𝒘𝒘𝟐

𝟐𝟐

D (sag at belly)

D ≈ 𝒘𝒘𝟐

𝟖𝟐

Max.

Tension H

(S/2, D) (end support)

𝑳 ≈ S 𝟏 + 𝒘𝟐𝒘𝟐

𝟐𝟒𝟐𝟐 ≈ S 𝟏 + 𝟖𝟖𝟐

𝟑𝒘𝟐


Span


Catenary Sample Calcs for Arbutus AAC

20.7453 600 12.064 3 6788 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 12.064 600.64724 2780 3 600

2 2 2

2 2L 600 1 + 600 1 + ft ⋅ ⋅≅ ⋅ = ⋅ =

⋅ ⋅ 2

2

8 12.064 0.6473 600

Slack = L - S = 600 ft ⋅⋅ = ⋅

( )0.64712.064 (3.678 )

83 600

Sag = ft m⋅ ⋅

=

10

Notice that 8 inches 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 HL E A

ε∆ ∆= =

⋅

A AL T

Lα∆

= ⋅∆

12


Conductor Elongation

Manufactured Length

ThermalStrain

ElasticStrain

Long-timeCreepStrain

Settlement&1-hrcreepStrain

13


Sag-tension Envelope

GROUND LEVEL

Minimum ElectricalClearance

Initial Installed Sag @15C

Final Unloaded Sag @15C

Sag @ Max Ice/Wind Load

Sag @ Max ElectricalLoad, 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) 601.470L 600.647 1 + e ft≅ ⋅ − − = =

Slack = L - S = 1.470 ft

( )1.47018.187

83 600

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 600 18448 8 18.187w SH lbs

D⋅ ⋅

= = =⋅ ⋅

15

Simplified Sag-Ten Calcs (cont)


( )1844 2780601.470*(0.999786) 601.341

0.6245*7 6L 601.470 1 + ft

e −

≅ ⋅ = = Slack = L - S = 1.341 ft

( )1.34117.37

83 600

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 wind load - %RTS

Tension at max ice and wind load - 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 of

breaking 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, EAS of the non-homogeneous conductor can be derived by combining the preceding equations:

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

ASAS

AAASA AE

AEHH⋅⋅

⋅= (18a) and ASAS

SSASS AE

AEHH⋅⋅

⋅= (18b)

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

AS

SS

AS

AAAS A

AE

AAEE ⋅+⋅= (19)

SS

S

AA

A

ASAS

ASAS EA

HEA

HEA

H⋅

=⋅

=⋅

≡ε (17)

Component Tensions – ACSR CIGRE Tech Brochure 324

1/14/13


Linear Thermal Strain - Non-Homogeneous A1/S1x Conductor For 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

SS

AS

A

AS

AAAS A

AEE

AA

EE ααα (20)

Linear Thermal Strain – ACSR CIGRE 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:

MPaEAS 746.468

8.651906.4688.40255 =

⋅+

⋅=

66 1084.186.468

8.6574

190105.116.4688.402

7455623 −− ⋅=

⋅

⋅⋅+

⋅

⋅−= eASα

Example Calculations – ACSR CIGRE 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 was developed:

• 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

method 1/14/13

Early work station – analog computer

Alcoa Graphical Method workstation 1920s to 1970s 1/14/13 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

Presenter

Presentation Notes

Initial vs final, steel vs aluminum


Summary of Some Key Points • Tension equalization between suspension spans

allows use of the ruling span • Initial and final conditions occur at sagging and

after high loads and multiple years • For large conductors, max tension is typically

below 60% in order to limit wind vibration & uplift • Negative tensions (compression) in aluminum

occur at high temperature for ACSR because of the 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, AIEE

Paper 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,” Published

7/15/74. • 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 Aluminum

Conductor and ACSR” and “A Test Method for Determining the Long Time Tensile Creep of Aluminum Conductors in 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&C

Subcommittee by Dale Douglass

Technology

Sag tension calcs-ohl-tutorial