Tutorial Presentation on Sag Tension calculation

1. Sag-tension Calculations
A CIGRE Tutorial Based on
Technical Brochure 324
Dale Douglass, PDC
Paul Springer, Southwire Co
14 January, 2013

2. 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

3. 1/14/13 IEEE Sag-Ten Tutorial
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

4. 1/14/13 IEEE Sag-Ten Tutorial
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

5. 1/14/13 IEEE Sag-Ten Tutorial
What is a ruling span?
5
Strain Structure Suspension Structure

6. 1/14/13 IEEE Sag-Ten Tutorial
( )max
2
3
Average AverageRS S S S≈ + −
S1 S2 S3
RS
6
S+----+S+S
S+----+S+S
=RS
n21
3
n
3
2
3
1

7. 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

8. 1/14/13 IEEE Sag-Ten Tutorial
The Catenary Curve
• HyperbolicFunctions & Parabolas
• Sag vs weight & tension
• Length between supports
• What is Slack?
8

9. 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) ≈
𝒘𝒘 𝟐
𝟐𝟐
D (sag at belly)
D ≈
𝒘𝒘 𝟐
𝟖𝟖
Max.
Tension H
(S/2, D)
(end support)
𝑳 ≈ S 𝟏 +
𝑺 𝟐
𝒘 𝟐
𝟐𝟒𝑯 𝟐 ≈ S 𝟏 +
𝟖𝟖 𝟐
𝟑𝟑 𝟐
1/14/13 IEEE Sag-Ten Tutorial 9
Span

10. 1/14/13 IEEE Sag-Ten Tutorial
Catenary Sample Calcs
for Arbutus AAC
2
0.7453 600
12.064 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 12.064
600.647
24 2780 3 600
2 2 2
2 2
L 600 1+ 600 1+ ft
   ⋅ ⋅
≅ ⋅ = ⋅ =   
⋅ ⋅   
2
2
8 12.064
0.647
3 600
Slack = L - S = 600 ft
 ⋅
⋅ = 
⋅ 
( )0.647
12.064 (3.678 )
8
3 600
Sag = ft m
⋅ ⋅
=
10
Notice that 8
inches of slack
produces 12 ft of
sag!!

11. 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.
1/14/13 IEEE Sag-Ten Tutorial 11

12. 1/14/13 IEEE Sag-Ten Tutorial
Conductor Elongation
• Elastic elongation (conductor stiffness)
• Thermal elongation
• Plastic Elongation of Aluminum
– Settlement & Short-term creep
– Long term creep
L H
L E A
ε
∆ ∆
= =
⋅
A A
L
T
L
α
∆
= ⋅∆
12

13. 1/14/13 IEEE Sag-Ten Tutorial
Conductor Elongation
Manufactured Length
Thermal
Strain
Elastic
Strain
Long-time
Creep
Strain
Settlement
&1-hr
creep
Strain
13

14. 1/14/13 IEEE Sag-Ten Tutorial
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

15. Simplified Sag-Tension Calcs
1/14/13 IEEE Sag-Ten Tutorial
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
( )1.470
18.187
8
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 2
0.7453 600
1844
8 8 18.187
w S
H lbs
D
⋅ ⋅
= = =
⋅ ⋅
15

16. Simplified Sag-Ten Calcs (cont)
1/14/13 IEEE Sag-Ten Tutorial
( )1844 2780
601.470*(0.999786) 601.341
0.6245*7 6
L 601.470 1+ ft
e
 − 
≅ ⋅ = =   
  
Slack = L - S = 1.341 ft
( )1.341
17.37
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

17. Numerical Calculation
1/14/13 IEEE Sag-Ten Tutorial 17

18. 1/14/13 IEEE Sag-Ten Tutorial
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

19. 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
1/14/13 IEEE Sag-Ten Tutorial 19

20. 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
AA
ASA
AE
AE
HH
⋅
⋅
⋅= (18a) and
ASAS
SS
ASS
AE
AE
HH
⋅
⋅
⋅= (18b)
Finally, in terms of the modulus of the components, the composite linear modulus is:
AS
S
S
AS
A
AAS
A
A
E
A
A
EE ⋅+⋅= (19)
SS
S
AA
A
ASAS
AS
AS
EA
H
EA
H
EA
H
⋅
=
⋅
=
⋅
≡ε (17)
Component Tensions – ACSR
CIGRE Tech Brochure 324
1/14/13

21. IEEE Sag-Ten Tutorial 21
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
S
S
AS
A
AS
A
AAS
A
A
E
E
A
A
E
E
ααα (20)
Linear Thermal Strain – ACSR
CIGRE Tech Brochure 324
1/14/13

22. IEEE Sag-Ten Tutorial 22
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 74
6.468
8.65
190
6.468
8.402
55 =





⋅+





⋅=
66
1084.18
6.468
8.65
74
190
105.11
6.468
8.402
74
55
623 −−
⋅=





⋅





⋅⋅+





⋅





⋅−= eASα
Example Calculations – ACSR
CIGRE Tech Brochure 324
1/14/13
35% higher than alum
alone
20% less than alum alone

23. Experimental Conductor Data
& Numerical Sag-Tension
Calculations
Paul Springer
Southwire
1/14/13 IEEE Sag-Ten Tutorial 23

24. IEEE Sag-Ten Tutorial 24
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
method1/14/13

25. Early work station – analog computer
Alcoa Graphical Method workstation 1920s to 1970s1/14/13 IEEE Sag-Ten Tutorial 25

26. Stress-Strain Model – Type 13 ACSR
Initial Modulus
Core Initial Modulus
Aluminum Initial Modulus
10-year Creep Modulus
Aluminum 10-year Creep
26

27. Stress-Strain Model – Type 13 ACSS
27

28. 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

29. 1/14/13 IEEE Sag-Ten Tutorial
SAG10 Calculation Table
From Southwire SAG10 program
29

30. 1/14/13 IEEE Sag-Ten Tutorial
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

31. 1/14/13 IEEE Sag-Ten Tutorial
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

32. The End
A Sag-tension Tutorial
Prepared for the IEEE TP&C
Subcommittee by Dale Douglass