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316 views18 pagesWinkelman Paper
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© © All Rights Reserved
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/ 18
Sag-Tension Computations and Field
Measurements of Bonneville Power
Administration
PAUL F. WINKELMAN
"ASSOOATE MEMBER AEE
NY TRULY flexible material of uni-
form weight, which a conductor
approximates, will conform to a catenary
‘when suspended between two supports.
However, many forms of sagging compu-
tations are based on the simpler parabolic
equation or on @ compromise between the
ccatenary and the parabolic equation as
shown for some of the basic catenary
equations in Fig. 1. For example, the
value of y (level span sag) for a catenary
may be written as
pattem) ESE
+
wing
aC
‘The first term in the series is « para
bolic sag based on uniform weight along
‘the chord similar to the cable loading on
a suspension bridge. The fist two terms
of the series give a close approximation
of the true y value as expressed by either
‘the hyperbolic or exponential form of
the catenary equation. In all cases,
the sag from the series equation will
be Jess than that for a true catenary.
It is interesting to note that any inflex-
bility ia a cable will also result io a sag
somewhat less than the catenary sag.
For heavy loading areas with compara-
tively ow stringing tensions in long spans,
and particularly in steep inclined spans,
the diference betmeen the eatenary and
the parabolic curve can be appreciable.
Thus, it is important to know the
limitations of methods used in computing
sags and tensions in the office and in
making measurements in the field
‘This matter takes on additional impor-
tance in view of trends toward reduced
clearances and less “builtin” protection
1s reflected in proposed National Elee-
tic Safety Code (NESC) revisions for
high voltage transmission
With the service area of the Bonneville
Power Administration (BPA) in. the
Pacific Northwest extending from the
Rocky ‘Mountains across two major
mountain ranges to the Paciic Ocean,
& considerable portion of our over 8,000
tiles of transmission lines is constructed
jn some of the most rugged mountain
terrain in the United States and in
areas of severe icing. BPA became ex-
posed very early in its construction
Program to the problems of stringing
large conductors over steep, inclined
spans at relatively low tensions. As a
result, there evolved methods of sag-
tension calculations and field measure-
ments that are a blending and simplifica-
tion of old methods,
This paper will describe the BPA
method of computing sag-tension data
and also some of the controls used in
applying these data to the fel.
Sag-Tension Computations
‘A conductor, whether suspended be-
tween towers or supported on the ground,
vill change in length due to changes in
‘temperature and tension. At constant
temperature and within the elastic Himit,
its change in length, AL, for a given
change in tension, AT, follows Hooke's
law:
. Lar.
(Area) Modulus of Elasticity)
‘Hooke's law may be graphically illis-
trated by tension-strain curves. The
slope of these curves for copper and
‘aluminum changes with tension within
the working limit of the conductor.
Furthermore, the slope of these curves
for bimetallic conductor such as ACSR
(cluminum cable steel reinforced) also
changes with temperature
On the other hand, a conductor of «
siven uniform weight suspended os a
catenary between two towers will also
follow the law of the catenary; that is,
‘tension and sag are a function of slack
(Gifterence between curve length
‘and straight line length C between
au: 3 fs aan ate
4 uns Learnt nes 200-0)
entheompteB) + omy
supports). This law may be graphically
illustrated by catenary function curves
in which per-eent slack, 100(Z~C)/L,
is plotted as abscissa with tension and
sag as ordinate,
‘A. most convenient and accurate
‘method of computing sags and tensions
for a conductor suspended as a catenary
for dilferent loadings and temperatures
is a semigraphical method. In this
‘method the catenary function curves
{or a given span length are superimpoied
over the tension-strain curves of the
conductor plotted for each temperature
concerned. This method is used by
[BPA and is similar to that used by Var-
‘ey! for level spans and Ehrenburg? for
steep inclined spans.
‘The BPA method involves the use of
new catenary function tables for level
dead end (DE) spans. These tables
and other simplifications reduce com-
putations and possible errors.
‘Tenston-Sreann CURVES
‘These curves are developed from
repeated tension-strain test data taken
‘at approximate stranding temperature
for the conductor concerned. For ACSR
PRECISION METHOD k
ACh agg
‘conductor, tension-strain curves are re-
quired for both the complete conductor
and the steel core. The procedure sum-
‘marized here is similar to that shown in
reference 1
‘As shown in Fig. 2, with per cent strain
(100AL/L) plotted as abscissa and ten-
sion in pounds as ordinate, certain loads
are held for a period of time to stabilize
the conductor. A large percentage (tests
showed 907 in one case) of the elongation
occurring during these holding periods
is caused by strand setting, The re-
mainder is creep. After each hold period
the conductor is unloaded and then
reloaded. A curve drawn through the
points at the end of each hold period is
called the virtual initial tension-strain
curve. . The duration of hold periods
shown in Fig 2is not necessarily adequate
for all types of stranding or material.
‘However, later field prestressing (tension
‘and time) for sagging is correlated by
BPA with the elongation represented by
‘this virtual initial tension-strain curve,
For an ACSR conductor, the steel
core is subject to the same type of re-
peated loading, The steel core is loaded
‘to match the initial elongation of the
APPROXIMATE METHOD
Bob
wie Hen
me
ACSR conductor at each hold period
‘and then held for the same period of
time. For example, at the first hold
‘period the steel core is stressed in pounds
to equal Ta,E,/eE, With elongation,
AL, and sample length, Z, made the same
for both the ACSR conductor and the
steel core stress-strain tests, then
TL ant
aE” aE,
an
for T and T, intially equal to zero
T and Tytension for conductor and steel
E and Ey=modulus of elasticity for com
ductor and steel core
and armcrosssection area for conductor
‘aad steel core
Plotting both the ACSR conductor
and the steel core data to the same scale
as in Fig. 2, one can obtain the virtual
initial and ‘final tension-strain curves
for the aluminum strands of the ACSR
‘conductor by direct subtraction,
There are now tension-strain curves
for the sluminum and steel components TENSION ~ POUNDS.
Se FOR suerus cours
we
Fis.
ff the ACSR conductor at test tempera-
‘tures. By shifting the aluminum and
steel curves horizontally to right or left
by the product of the temperature change
desired and the coefficient of expansion
of the components and then adding the
two shifted curves together, one obtains
the tension-strain curves, initial ond
final, for the complete ACSR conductor
for any temperature desired.
Fig. 3 shows tension-strain curves
constructed in this manner for ACSR
Pheasant conductor at stringing tempera-
tures. Final curves are constructed for
maximum working tension (MWT) of
£8,000 pounds and 16,700 pounds at 0 F
(degrees Fahrenheit). These are the
‘MWT values used by BPA for this con-
uctor for wood pole and steel construc-
tion, respectively. Note that the inter-
section of initia! and final curves increases
from the MWT value on the 0 F curve
to progressively higher values on the
Percent straw (SE 100)
92 F and 120 F curves. This is the
result of the aluminum strands taking a
larger percentage of the total load at 0 F
than at the higher temperatures. Thus,
st the higher temperatures the aluminum
strands are still at final modulus of
‘elasticity for tensions in excess of MWT.
As shown in the insert on Fig. 2,
the intersection of the initial and final
ccurves for MWT value must be first
located on the component aluminum and
steel curves with the curves. shifted
horizontally to the temperature for the
MWT loading, such as 0 F for heavy
loading. Then, in subsequent shifting
of components for other temperatures,
both the initial and final curves shift
together.
Carmnany Fuxcrion Cunves
Curves in Fig. 4 give for a certain span
Jength the catenary relationship between
percent slack plotted as abscissa and,
3
Tension-zan curves for ACSR Pheasant conductor, rated strength 44,800 pounds, test temperature 76 F
sag and tension plotted as ordinate.
‘The catenary function curves are plotted
to the same scale as the aforementioned
tension-strain curve. Both the slack,
L=C, of the catenary function curves
and the change in conductor length,
AL, for the tension-strain curves are
plotted as abscissa in per cent of condue-
‘tor length—stressed in the former and un-
stressed in the latter case. The resulting
‘error from using stressed length is very
small, but greatly simplifies the com-
‘putation of the catenary function curves.
A practical seale for the abscissa is:
Linch = 0.02% slack oF 0.02% strain
A tension curve is computed for each
type of conductor loading for which
sag-tension data are desired. The sup-
port tension (Tq) curve is the value of
tension in the catenary at the support.
‘The effective tension (T,) curve is the
‘Value of tension for the catenary that 1 T
5 soa
3
Loon Z|
44 7
+ x bj
| Lege Z ge
y Ze
S é
percent strum (A 1003
3. Tension-tzan curves for ACSR Pheasant conductor, singing temperatures
tne vee. tusmtt ww
TO alert
eo uP cai COMPUTATIONS OF DATA FOR
CATENARY FUNCTION CURVES
T
fe @ ve at, ww
° cea
finy*3803 185 hey = S05! LOS An; = 2943 LBS
FOR TENSION AND SAG FACTORS SEC TaBLE T
FE Stack (TABLE vaLvE)_p.<090 440 aalo SEO SEES)
Te: tne en eae Fra ata
ce) un Tens io
tn, sao sc a 6 ves) an) a
iy aR Teen
TS fice com va...) se asccanazlas
hg) FEN ease in
ste cour. we wae_.s| 23 sa] es
ey cerecine Tex ci)
4000 lew, onsets on} ase
a a io
i
8
reel | >
i
TENSION — POUNDS
ss ses]
a a
PERCENT stack (L=2 100)
Fis. 4. Catenary function curves, ACSR Pheasant conductor, 1,800-400t pen satisfies Hooke’s law. For example, if
the unstressed length of a conductor sus-
pended as a catenary is stressed uniformly
throughout its length by the T, value for
the catenary, the conductor will elongate
to the same length as that of the con-
ductor suspended in the catenary. 7,
may be considered the average tension
‘acting on the catenary length of the
‘conductor and is always less than (Tat
Hy/2.
‘The catenary function tables developed
by BPA provide direct, simple and
accurate method of determining the
data for plotting the catenary function
‘curves for any level DE span
DERIVATION OP EQUATIONS FOR
(Catenany FoNcrion Tanues
Precise equations in Fig. 1 may be
written for the level span in Fig. 5 as
Tolows
Sacmy=si/o( ot 1) o)
reat, Ea 2
@
Support tension, Tam T=H1 cooh =F (3)
Sonncaadaasnnnscenie a on co
wk | ERZSERSSRSESEPSSRSESERSSE ‘stressed conductor (4¢,) when suspended
a] | SESSESSgU ASHES ESSEEEEgE Becomes
neeaessecses == areal 2)
for conductor of cross-section area (0)
: and modulus of elasticity (E).
HEGEL
Hl | nmeaniiGg
ae8g3gege:!
Table 1, Catenary Functions
ag fe
ef td [ree
1h
Eft m.
1%
ah ™ oe
a
As this is an expression of Hooke's
Jaw and the effective tension is the value -
of tension that satisfies Hooke’s law,
or very nearly
fe
rot fru
where ¢ is the length of stressed cable
in the catenary
Substituting equations 2 and 3 and
integrating: Fig. 5. Level span eatenary
Conductor length between suppor: 2¢seL
then
Ge Sack ~100(1-=2) a 109-3202
6)
‘Devavoruent oF CaTENARY FUNCTION
Tases
By use of hyperbolic tables, the value
of Z corresponding to each per-cent slack
selected for the first column of Table I
is found from equation 5 by trial and error
‘method and interpolation. The National
Bureau of Standards, United States
Department of Commerce, publishes
place “Tables of Circular and Hyper-
cach selected per-cent slack, itis possible
to derive sag and tension factors appli-
cable to any span length by expressing
equations 1, 3, and 4 in terms of the
Picante reondcgy erred
‘the unknowns (span length A and con-
ductor weight w). Thus,
sagmyn§ (cous 2-1)
from equation 1
4
sae (ooh 2-1)
sephora ean Z-1) (6)
(eoan2)
suppor tenon, Tan aw S82)
from equation 3
Ta coon
support tension factor ”
‘aw 22
5 cmhZ 1
effective tension, Ty. x
we it a 4z aaa]
from equations 2 and 4
eflective tension factor
coz 1
az tiene ©
If the support tension factor is de-
termined first from equation 7, the
fective tension factor can be ‘more
conveniently found by expressing equa-
tion 8 as follows:
ffetive tension factor
mB] eztsupport tension factor) —
‘a
acs
2] ©
nism tion my eee
othe
Hew (support tension factor ~sag factor)
(20)
Or from approximate equation under
item 2 in Fig. 1,
an
Z cance «
Fi, 6. Fi. 4
tupetimpoted over
Fis. 3
Percent stack (4° 190)
6 Table I Table for Finding Horizontal Tension from Seg
0.008 0.0%
S007 aam8
0.004 6.010
seat 1.962
0.005, oon
ame 1
0.000 oo
2.308 ott
0.007 os
2.708 ase
oor oe
0.018 oon
0.08 oon
a0 0.890
naa oe
0.08 ee.
0.08 0.025
wat
# ‘8(sag—correction)
‘The corrections small; aclose approxi
sation of this correction w14\/3847,
‘The then computed value of Hf can be
quickly used to check the approximation,
1m summarizing, Table Tis computed
from equations 6,7, and § after finding
values of Z from equation 5 for selected
Values of percent sack.
aa)
CALCULATING CaTENaRY FuNcTION
Curves From Taste I
For successive table values of per-
cent slack, the following computations
are required for a given level span length
1, TaeAw(support tension factor)
2. Ta—TemAw(eupportefective tension
factor)
i
mat aati}
g
re oa dn a
SPaM-FEET
8, Tem Au(effective tension factor)
4. Sage A(sag factor)
The first two are usually only for
the maximum design loading. The
T, curve for maximum design loadi
is most accurately plotted by subtracting
the TT, values, step 2, from the Tm
value.” A T, curve, step 3, is computed
for each type of conductor loading for
which sag-tension data are required,
such as bare conductor at 0 wind for
stringing charts, and ice and
loadings or combinations thereof for
clearance and overload studies, The
value of w must correspond to the total
weight per unit length of the conductor
{or loading conditions concerned. Only
fone sag curve is required for a given
span,
‘saa-reer
Fig. 7. Stinging chart—dead end spans
For the usual problem where « given
-MWT is the starting point, three points
are sufficient for the Tq and T, curves
‘at maximum design loading, One uses
nly the point on the former curve that
intersects the MWT ordinate and the
corresponding point on the T, curve
directly below this intersection (see
Fig. 6).
‘The MWT starting point, as shown on
Fig. 6, can actually be computed without
plotting the two maximum design lo
‘curves ifthe per-cent slack corresponding
to MWT can be determined with rea-
sonable accuracy. One such convenient
means is by use of Martin's tables.*
Knowing MWT, one calculates Aw/
‘MWT and interpolates for the length
factor; then % Slack=100 (length
factor —1)/length factor, for MWT start
ing point on Tm curve.
The Tw-T, value computed from
Table 1 by interpolation for this cal-
culated per-cent slack will locate T,
for maximum design loading. This
‘method of determining the MWT starting
point is particularly more accurate when
the maximum design loading curves are
relatively flat
‘The number of computations required
for the bare T, curve and the sag curve
‘can be easily determined after working
several spans. Usually five to six points
are adequate.
Sao-Tauston Courutarions For LEvaL
DE Sraxs
For a given MWT at specified design
loading, the catenary function curves,
Fig. 4, are superimposed over the tension-
strain curves, Fig. 3, as shown in Fig. 6.
‘With the tension scales aligned, the
catenary function curves are shifted
horizontally until the initial tension-
strain curve for MWT temperature
intersects the 7, curve for maximum
design loading at’a point directly below
the MWT starting point on the Tq
As the eflective tension satisfies
Hooke’s law, it follows that the inter-
section of the tension-strain curve for
any temperature with any T, curve will
determine the effective tension and per-
‘cent slack in the span for the respective
temperature and loading concerned. ‘The
sag is read off the sag curve at a point
vertically below or above this intersee-
tion. For other than short spans, the
‘horizontal tension can be most accurately
determined from the known sag by equa-
tion 12 or by Table II. In Table 11,
equation 12 is written in the form of Fig, 8. Nomostaph for detemining level
‘span equivalent oF nonlevel spans
For short spans in the order of 500
feet or less, where a considerable change
in tension results'in a very small change
in sag, it is more accurate to calculate
sag by equation 12 from the known
tension (71). For short spans, T, and
HE ae equal for all practial ‘purposes
‘and the sag correction i equation 12
is negligible.
Exaurce:
Given: ACSR Pheasant conductor
MWT=16,100 pounds at 1/2 tach ice, 8
Pounds per square foot wind, 0 F
‘Span length = 1,500 feet
Conducter weight in pounds per foot:
11211635 for bare conductor, no wind
1172.06 for 1/2 inch ie, no wind
1-324 for 1/2 inch ice, 8 pounds per
square foot wind
Find: 1. Initial stringing sags and hor-
zontal tensions at
OF, 30, 60F, OF, 20F
2. ‘Final sags and hovzoatal tensions
1/2 inch ice, no wind, 32 F
inch ie, no wind, 150 F
For computations of data and the
catenary function curves plotted from
‘these data, see Fig.4. For the remainder
of the computations necessary for final
solution, see Fig. 6.
Sac-Tenston Cowrurarions von Sraxr
InccaNeD DE Seaxs
Data for catenary function curves
are computed by the Bhrenburg method.*
‘The Tm at upper support, Ty, and sag
values" of the inclined catenary are
Fig. 9. Derivation of field
seszing equation
Let P represent point of
tangeney which line of sight
maker with citenory end Sy
represent the diference be.
tween Sand Sy then Sis the
42g of span PO and the slope
%
of on 8 Gem 6, Fis. 1)
SSeS
S45 PEE idling of tap:
trol hat bees nour sides eu one
bull the sum ofthe two perallel ides
transpoteand solve for Ss
square of span lengths
Proportional to sage
Substitute SS, for Sand sinplify
165-5)"
OBS = 16S
Solve quecatic equation ot completing the
square
US-O40P=O40— G0
US-G401=2VR
itv)
C4)
plotted against per-cent slack. These
curves are then used with the teasion-
strain curves in the same manner as in
‘the example for level span computations.
In addition to its simplicity and
accuracy, the method just illustrated
provides a flexible means of analyzing
many of the incidental conductor prob-
Jems related to new construction,
line maintenance, and line revisions such
as prestressing, crossing clearances, con-
ductor overloads, effects of creep, ete.
‘The tension-strain curves and catenary
function curves accumulated in the
preparation of stringing charts for new
construction become valuable tools for
subsequent design problems,
Field Measurements
‘Ruumve Span
As shown by DE chart in Fig. 7,
‘the change in tension for a given tempers.
comme i 5+
Feeviy
La
ture change varies with span length
This can be explained by the slack-
tension relations of a catenary. Where
DE spans are at approximately the
same horizoatal tension, a given per-
cent change in slack eauses an increasing
change in tension as span lengths become
shorter. Now, in a series of suspension
spans that have been sagged to the same
Rorizontal tension with all suspension
strings plumb, the tension in each span,
regardless of length, will change esen-
tially the same with changes in tempera-
ture due to small movements of the sus-
pension strings along line. The sus
Pension strings swing to decrease the
slack change in short spans and increase
the slack change in long spans. ‘This
movement of the suspension string is so
small that the horizontal force exerted
by the string on the conductor is nor-
rally negligible.
‘A DE span that gives the same change
jn tension from changes in loading and
‘temperature as that in a series of sus-
pension spans between two anchor or
DE tomers is called the ruling span of
the series of suspension spans. ‘The
ruling span equation is based on the
slack-tension relations of n” spans, each
of rating spa length (4), matching the
slacketeasion relations of a number of
sunequal spans of the same total length.
‘Thus, by equating total length,
Whew Art AvtbAe
Using the first term of approximate
slack equation for a level span from Fig. 1
and equating slack for a given horizontal
tension (H):
PN AMAL tAn!)
Ero ca
WARRAMANSbAat
Substituting for m’A,
Avbdet bald
At
obaat
[EAE eae
ra ee A Wp Avt se
‘Where the spans in a series of suspen- sion spans vary considerably in length, the
tension in the extremely long and short
spans will not be fully equalized to that
of the ruling span and considerable error
can result in desiga clearances, AS
shown by the tension curves of the DE
chart in Fig. 7, the extremely long spans
will have greater cold-weather sag and
smaller hot-weather sag than anticipated;
the opposite will be true for the short
spans. Clearances to both ground and
overhead crossings can be affected.
The problem can be avoided by dead
ending.
BQurvanenr Span
Stringing charts give sags and hori-
zontal tensions for sagging level spans.
For nonlevel spans, the sag is obtained
‘from the stringing chart for an equivalent
level span. The equivalent level span
is equal to:
10 (laid.
Nomogaph for
determining con-
trol for conduc
tor aesing
Fis. 11. Gigho.
Conte! tector
venus segsing
conocton 09 on —|— cogent on
Psa ot coret nero bunve ra. ©
so 2 a
sol if
re
Senor ore ase —
1. VAC or approximately A4+BY/44 for determining the value of the equivalent
suspension spans.
2. 2C—A from reference 3, oF approxi-
mately A+B4/A for DE spans,
span in the fel.
Tusony or Eguivanent Leva Span
Eguanions
‘These approximations are based on To assure plumb suspension strings
subwituting A+BY2A for C. Note at each mupporting tower the conduct
that the approximate equation for the in each span of a serie! of suspension
cquivalest level DE span gives four spans must be sagged fora given tempera
‘times the correction to be added to span ture at the same horizontal tension
length (4) as is required for the equiv- regardless of spaa length (A) or difference
alent level suspension span.
in support elevations (B). The sags
‘The nomograph shown in Fig. 8 is for ruling span charts accomplish this
constructed from the approximate forms for level spans as these sags for a given
of the equivalent level span equations temperature are based on the same hori-
‘and provides a convenient method for zontal tension for any span length. Now,
9 ven a
‘the approximate equation for the sag of
‘a nonlevel span under item 9 in Fig. 1,
ea Ase
see \(z) «
mae.
a
‘This can be written as
Tac
speorrection (13)
wivacy
oO correction
‘The diference of this correction for #
level and a nonlevel span is negligible.
‘The correct sag and horizontal tension
for a nonlevel span is then obtained from
‘the ruling span chart by looking up the
sag for the equivalent level span, VAC.
Stringing chart sags include the correction
for level spans.
Level spans sagged by ruling span
charts will have a support tension at
maximum design loading equal to MWT.
in spans of ruling span length. For
longer or shorter spans the support ten-
sion at maximum design loading is
respectively greater or smaller than
MWT value. On the other hand, DE
charts are designed to give a support
tension in level spans equal to the MWT
value for maximum design loading re-
ardless of span length. As a result,
the stringing tension for a given tempera:
TE SUACL OF SHEAVES +E SLAEE mn SUSPESiON CLAMS
vera
ture changes in a DE chart with span
length. The equivalent level span (2C
A) provides the additional sag re-
quired in a nonlevel DE span to approxi-
mate the MWT value at the upper
‘support for maximum design loading.
It also permits the use of Table I for
‘computing the catenary function curves
to determine the sags of an inclined span.
However, the correct horizontal tensions
for these sags are not those shown on the
DE chart or by the catenary function
curves, but can be computed from
‘equation 13,
‘DE spans on extremely steep hillsides
may be computed by the Ehrenburg
‘method? and the stringing data sub-
mitted to the field in tabulation form.
‘Sacemc Equation
‘The conductor is strung to the correct
sag by the commonly wed eld sagging
equation derived in Fig. 9.
Visi
s-(44)
for checking sag; or
tav3—Vie
for target shots; or
fered
4
10
for angle shots in which Bis (+) or (—)
when support ahead is higher or lower
and ¢ is (+) or (—) when angle is above
or below the horizontal
Exnoxs oF Fret Saccinc Eguanion
‘The field sagging equation is based on a
parabola and involves several approxima
tions when applied to a catenary. With
reference to Fig. 0, these
41. Slope (4S:/PQ) is approximate for
catenary.
2. Sag (S) in a catenary ies tomard upper
support from midpoint in an inclined span.
3. Sags of catenares are only. apprani-
mately proportional to the square of span
lengths
‘The only theoretically accurate sag
rade by the field sagring equation is one
in which h=1 or the control factor (Si/S)
is equal to 1.00 (see Fig. 10). Fig. 11
shows that the error in sagging « given
span increases as the point of tangency of
the ine of sight on a conductor approaches
either support from true sag position
(h=1) oF as the control factor decreases.
Furthermore, the sagging error fora given.
span changes in direction when the point
of tangency of the line of sight is om the
uphill or downhill side of the true sag
position, The curves in Fig. 11 also
show that the sagging error increases
with increasing span length and slope wea foa fap extn}
(B/A) and with decrease in H/w. This
‘error can be appreciable for low stringing
‘tensions on steep inclined spans.
This analysis of the sagging error was
made as a result of complaints from the
field that the conductor was not properly
sagging in some rough terrain in the Cas-
cade Mountains. Even though insulator
offsets and sag corrections were correctly
used, the conductor in one sagging span
would tend to go high and in another
sagging span low, etc, when sagging
several spans in a series of suspension
spans.
‘The resulting analysis showed the im-
portance of controlling the sagging setup.
‘The nomograph in Fig. 10 provides such
1 means of control. This nomograph is
constructed from equation:
Sao
control factor $ m1 —S
‘This is derived directly from the follow-
{ng equation taken from Fig. 9:
Gao
Inaulator offsets and 149 comections, long form
165
By means of the nomograph, the
“agger” can check in a moment the
adequacy of his transit setup as this
control factor is determined by data al-
ready available to the sagger. See
sample calculations in Fig. 10.
‘The contro factor curve in Fig. 10, used
to control the adequacy of the sag, is
plotted from the intersection of the
error curves (for 2,800-foot span and
H/wof 4,575 feet only) in Fig. 11 with the
1/2% accursey ine, Thus, a sagging set-
up giving a control factor that falls on
this curve results in a sagging exror of
1/2% for span length and H/w value
noted, In most cases the maximum
feror would be considerably less than
1/2%, as seldom would a sagger crowd
this control factor curve and, further-
sore, span lengths would normally be
shorter than 2,600 feet. Note also that
the breae in the curve limits the control
factor to 0.75. A control factor curve
"1
bbe plotted for any accuracy limit
esired and for any span length and
Hw value desired. However, the single
control factor curve in Fig. 10 as proved.
adequate.
InsuLator Orrse1s AND Sac
Connections,
A conductor strung in rough terrain
over frictionless sheaves tends to run downhill, that is, the conductor will
‘come to equilibrium with more than chart
sagin the lower spans and less than chart
sag in the upper spans. The stringing
sheaves will always swing toward the
span with the horizontal tangeney of the
comducter, actual or projected, at the
higher elevation. This is
toveard the uphill side
As shown graphically by
difference in horizontal tension
adjacent spans with the conductor at
rest in sheaves is equal to w(s — >.
tension (see Hs—Hy in vector diagram in
Fig, 12, Detail .1) that causes the insi-
lator string to swing eff plumb along fine
Furthermore, the horizontal tension in
cach span becomes less as one strings
downhill and more as one strings uphill,
With the conductor properly attached
to the suspension clamps and at chart
sag in each span, the difference in hori-
zontal tension (H,) between adjacent
spans will be equal to zero and the sus-
pension strings will hang plumb along
line. This condition of equilibrium could
only have been accomplished in a given
‘number of suspension spans between DE
‘towers or temporary snubs if the correct
length of conductor hed been previously
sagged into the sag section when the
conductor was in the sheaves. Expressed
mathematically in Fig. 12:
Zalack ia sheaves=Z slack in suspension
‘This is the key equation in computing.
insulator offsets and sag corrections,
From an assumed horizontal tension in
the first span in the sag-section, one can
compute the horizontal tension and the
resulting slack in cach successive span
for the conductor in the sheaves. If the
summation of these slicks figured for the
‘conductor in the sheaves is greater of less
than that figured for the conductor
the suspension clamps (with horizontal
tension from stringing chart the same in
each span), then a higher or lower
horizontal tension should be used for the
second and final slack computations.
Both the long and short form methods
rive the correct value of H for each span
afterthefirst assumption. In most cases,
‘computations made within 20 F of
sagging temperature are sufficiently
ccurate, Small changes in sagging tem-
perature have little effect om the result.
‘The maximum sag profile is sufficiently
accurate for determining the value of
ye = 3 regardless of sagving tem-
perature.
Lone Fors Merion
The procedure previously outlined is
that followed by the long form method in
Fig. 13. Horizontal tension for condue-
tor in sheave is assumed in Column 7
and corrected to H’ in Columns § to 11
Tn the process, one is basically equating
the summation of slack in the sheaves
to that in the suspension clamps. The
equivalent tension (Uf,) is the tension
that would give the same total slack for
the spans in the sag-section as that eal-
culated for each span by the assumed
values of H in Column 7. This total
slack in terms of 2%(slack)/te? is calcu:
lated in Column 10.
The slack diflerence in each span for
the H’ minus H, value of horizontal
tension is determined in Columns 12
‘to 5 by using the first term of the approx-
‘imate slack equation in Fig. 1 for am
inclined span, w'A‘/24CH?, converted to
inches. This difference in slack in each
span is called an offset. Column 15
also makes allowances for changes in
conductor length, based on the modulus
of elasticity, that occur when the tension
in each span is changed from H" to Hy
Column 16 makes final adjustment to
zero for summation of individual off-
sets in each span, The final insulator
‘offset to be measured in the field at
each tower is a summation given in
Column 18 of the final individual of-
sets computed for each span in Column
17.
‘The sags for the conductor in sheaves
are computed in Columns 19 to 24 by
equation 13 with the correction simplified
to that for a level span. This correction
is made equal to w'A*/384 H" by equating
B to zero and C to A. The sag correc-
tions given to the field are computed in
Column 26 and are the differences be-
‘tween the sags with the conductor in
b '
He
BELLAS
Heeee
HB
Fig, 14. Inulator
offsets and 12g cor
‘ections, short form OLYMPIA~ABERDEEN ime _I ance) 2a
ACSR "IBIS" coNoUCTOR "
ern a wo
| ELEVATION DIFFERENCE © DISTANCE IN FEET BETWEEN THE HIGHEST AND Lowest. Sign change conforms to Note J, Fig. 14,
I OFFSET PER POUND (LONG SPANOI/ NG)
‘ Substituting S* in slack equati
SECTION INVOLVED WHEN TWICE THE B
IFFERENCE 15 EOUAL TO OR LESS THAN THE ELEVATION DIFFERENCE erelspan,
SRG |RAING] Lone WAX WORRNGIELEY ns
section |'Seum| Stan Se ore OFrsers sce SS
eer fer ave om no ornate 4845 buses
- aoack) = SS
= Hoa(stack)
7] as)
Fig. 15. Insulator fet ansh
in equation 14 is the change in
slack or offset in inches for a given
ns for slack (Fig. 1) and sag span per pound change in horizontal ten.
sheaves and those with the conductor in equat
‘suspension clamps. equation 11. Thus, sion. ‘Therefore, in’ order to change
wat the tension in a given span by the dif-
‘Snort Fora Merion sack —A = 2A tee alsauecurt cae arte
‘The computations are considerably 5, conductor in the suspension clamp and
reduced in the short form method shown : the sheave or H, ~ H, one changes the
slack in inches by K(H, — H). The
in Fig. 14 by the use of the diflerentiated
offset is computed by this equation in
form of the first term of the level span ea
‘orveere
vie
Fig. 16. Innultor effets and seg conection field data
GENERAL: The insulator string on the clipped structure adjacent to new section being sagged should be restrsined from longitudinal movement
‘along line when tagging and plunb marking the conductor
‘OFFSETS ere measured from e point vertically below the insulator string attachment in the direction shown by arrows in the ebove sketch, These
offes are to be marked before any clipping-in
‘SAG CORRECTIONS are to be added algebraically tothe chart 9g to obtein the segs while
13
the sheaves We int
Fig, 17. Inlator sag computations
Nomenclature:
HeHorizontal component of conductor
tension
Vy oF Vie Vertical load t support
We Weisht of conductor per unit length
Wee Conductor weight from low point to
DE assenbly
Wee DE assembly height
LSE extembly lensth
sewleand L
For senispan contting of conductor and dead
fend stenbly with verical load at support
‘equel to Vs
Z of vertical forces =O
Van Wet We
1S of moments about point P=
Vine—Hde— We m0
Wa We
Te _ Meteo
en
a #
For dead end assembly in temispan replaced
by conductor length Ls with verical loed at
suppor equal to Vi ~
Vem Wetwe
Venom Hd wn 2 m0
Veer wg Whore
a a
lncrease in conductor s49 due to weight of
dead end estenbly=d.—dy
oe
Column 7 of the short form for assumed
or trial Hy — H and in Column 9 for
the corrected H, — H. The process of
‘equating the summation of slack in the
sheaves to that in the suspension clamps
is accomplished in the short form by
correcting H, — H in Columa 8 Mod-
ulus of elasticity correction is ealeulated
in Column 40, Column 11 makes final
adjustment to zero for summation of
individual offsets. Column 13 gives the
insulator offset to be measured in the
field at each tower and is @ summation
of individual offsets listed in Column
2
‘The sag correction is computed by
equation 15 in Column 14. As differen
tials are only accurate for changes ap-
proaching zero, equations 14 and 15 used
in the short form to compute changes in
slack and sag are only accurate for small
‘changes in Hand slack. The long form
should be used for severe offsets. (See
Note 4 on short form, Fig. 14.)
Line ANALYSIS
Note 3, Fig 14, gives the minimum
value of insulator offsets and sag cor-
rections that are sent to the field. The
2uinch limitation for the individual offset
is used in Fig 15 to determine the
sections of a transmission line where
insulator offsets and sag corrections may
be required. Insulator offsets and sag
corrections computed for these sections
may or may not be sufficient to send to
the field, The selections made by Fig.
15 are based on the longest span requiring
‘half the full “Elevation Difference” of
the sag-section for a 2inch offset. Ac-
tually, the offset for the longest span of a
sag-section is practically always based on
less than half this full “Elevation Dif-
ference.” Furthermore, the longest span
always requires the largest offset to cor-
rect for a given difference in horizontal
tension (H, — H, Fig. 14)
‘Use ne rem Prep
The form used to submit insulator
offsets and sag corrections to the field is
shown in Fig. 16,
Tm any sagging operation, with or
without offsets, it is important to use
ball or roller bearing sheaves properly
‘maintained and lubricated to hold friction
to a minimum. The quality of sheaves
‘and the experience of the crew in working
the conductor to balance out sheave
friction have much to do with the length
fof conductor that can be properly
sagged. Furthermore, all conductors in
‘@ sagesection should be treated uni-
formly with respect to tensions ap-
plied and duration of these tensions
Guring stringing, prestressing, and final
sagging. The average and| maximum
lengths of conductor sagged by BPA for
NESC heavy loading are as follows:
Com
Avenue Maximo
‘The conductor is sagged in control
spans at or near each end of the sag-
section but not less frequently than at 1-
mile intervals. Where possible, the
longer, more level spans are selected for
control
spans. Sagring and plumb-
of the conductor should be
done the same day as overnight tempera.
ture changes and winds may move the
conductor over the sheaves. After
plumbmarking, the suspension stings
may be attached to the conductor at any
time, measuring the offets from the
plumbmark as required
Isutator Saos 18 Stoxr DE Spans
In short DE spans, especially for 230
kkv and higher construction where DE
insulator string assemblies are quite
heavy, allowance should be made for the
sag of the DE assembly. BPA includes
this increase in say caused by the DE
assemblies (rise of DE assemblies minus
rise of conductor in the same length,
4, — dy shown in Fig. 17) whenever its
value is 5% or more of the conductor
sag. The method of computation is
shown in Fig. 17, This same method
may be used to figure sags and tensions
for a conductor with concentrated loads
such as wave traps, catenary lights or
‘markers, taps, ete.
‘The conductor in a DE span is usually
brought to sag with a DE assembly
attached on one end only. To allow for
the reduction in sag that will occur when @
DE assembly is attached to the other
‘end of the conductor, the conductor sag
is actually increased by approximately
twice the value of sag increase (2, —
4). For example, when the conductor
sag should be increased by 1 foot with
‘both DE assemblies in place, the increase
uring sagging operations with the DE
assembly on one end only should be
approximately 2 feet. This extra foot of
sag during sagging operations (or 2
feet total increase) may be considered
as providing the additional length (Z, —
Z,) shown in Fig. 17,
‘Stack m Swot Deap Exp SPANS
Experience has shown that it is desir-
able to provide at Jeast 3 inches of slack
in a DE span for attaching the conductor
and DE assemblies to the supporting
towers. Normally this amount of slack
can be provided and adequate clearance
‘maintained in short DE spans, such as
approaches to substations, by increasing
‘the sag. Furthermore, short spans
should usually be slacked to prevent
‘excessive cold weather tensions. Turn-
‘buckles or similar arrangement should be
included in the DE hardware when oa a
CREEP STRAIN =e + F(TIME)
SH
ee
E
i
ve ~ ones
Fig. 18, Typical creep sesn vers tine coves
design clearance will not permit adequate Fonstant. It is the value of creep strain
soc for ‘at unit time (1 hour for time sale
eee, ‘hours, ete.) for a given slope curve.
Conpucror Cras srsconstant that controls the slope of the
Permanent sag increase of a bare
conductor from nonelastic stretch is
‘caused by:
1. Short time loading such as from ice
land wind loads in which the difference of
initial and final modulus of elasticity is
involved:
2. Long time leading—commonly called
creep
For a given time, conductor ereep
increases with increase in tension and
‘temperature. It will also vary with type
of material and stranding. For the same
conditions, creep will be greatest in
aluminum’ conductors, followed by
copper, and then by steel conductors.
‘The creep for ACSR normally falls
between that for aluminum and copper
conductors.
Initially creep is quite rapid with a
rapid decrease in creep rate. Strand
setting contributes a considerable part of
the total conductor elongation during
this period. ‘This primary creep is
followed by a second stage in which the
creep and decrease in creep rate are both
very slow. With sufficiently high tension
and temperature the second stage can be
followed by a rapid increase in creep
rate to final conductor failure.
Experimental creep strain versus time
data at normal tensions and temperatures
closely follow a straight line on a log-
log scale as shown in Fig. 18. Ac-
cordingly, creep strain (¢) can be rep-
resented by equation:
c= F(time)* as)
‘creep curve.
This straight line plot on log-log
seale permits accurate extrapolation of
creep from relatively short time tests
However, ASTM (American Society for
‘Testing Materials) designation E 139.
58T recommends that test data cover
approximately 10% of the period for
which creep is to be determined.
Tests on various metals at a given
temperature! showed that increasing
creep tensions shifted the creep strain
versus time curves vertically upward
with no appreciable change in slope.
Im other words, Fin equation 16 increased
with tension and 1 remained unchanged
as shown for the curves in Fig 18,
‘More recent testst on ACSR and stranded
allaluminum conductors showed that
‘this substantially constant slope condi-
tion held for tensions varying from 10%
to 60% RS (rated strength) of the
Toble HL, Creep R
conductors. Furthermore, the creep
strain versus time curve for tension at
20% RS had substantially the same
slope for room temperature (67 F to
75 F) as for 200 F, Thus a convenient
creep table, independent of conductor
tensions and normal operating tempera-
tures, can be prepared for extrapolating
creep measured in the field to any time
period desired such as 10, 20, 30 years,
ete. (see Table TTT). This assumes that
the variations in conductor tension and
temperature during period of creep
‘measurements will be typical of the
variations during the extrapolated period.
However, creep will cause some reduc-
tion in the conductor tension depending
(on DE or ruling span length and H/w.
‘The effects of this reduced tension on the
extrapolated value of creep can be
evaluated,
Conductor creep in a given trans-
‘mission line will be dependent on the
everyday stresses and operating tempera-
tures occurring. These will vary with
time and location. Creep studies for
ifferent conductor types are being made
in selected spans by BPA (presently on
ACSR of two, three, and four layers of
aluminum). Two adjacent suspension
spans close to ruling span length are ex-
cellent for such creep studies. Spans are
selected in areas subject to infrequent
icing to simplify the problems of evaluat-
ing that part of conductor elongation
‘caused by icing loads. Readings should
be taken after any severe icing to isolate
these readings from subsequent creep
readings. Sag, conductor temperature,
‘movement of supports, and wind velocity
and direction are measured.
It is most convenient to express the
creep or permanent sag increase in terms
of a temperature increase above ambient,
Such creep measurements for ACSR.
Pheasant with everyday stresses aver-
aging 20% RS, showed creep in the order
15 F temperature increase for 1,000
hours. On the basis of Table TIT for
795-MCM (thousand circular mils) ACSR
conductor of similar stranding and per
a
79S MEM ACSR, 5441/7 ST
10 Yous 30 Yours 60 Yours cent steel the ereep in 10 und 310 years
would be (2.20) (15) oF 33 F and (2.70)
(15) or 41 F respectively. However,
‘data would be desirable for a longer time
period for such extrapolation,
Allowance for ereep during sageing ean
be made either by including the ercep
correction in the stringing charts or by
reading the sag from an uncorrected chart
‘at ambient temperature minus the estab-
lished temperature correction for creep,
Prestnessine For CoNTROLLING CREEP
Controlled laboratory creep tests on
stranded aluminum alloy conductor raise
some question on the long time benefits
of prestressing, Prestressing reduced
subsequent creep during these tests as
‘would be expected; however, the slope of
thecreepcurvewas increased. This would
indicate that the advantage of prestress-
ing would gradually become lost. Tt is
teresting to note from reference 5 that
cold working also increases the slope ofthe
creep curves for aluminum and aluminum
alloys. As mentioned previously BPA
Drestresses primarily to stabilize the con-
ductor to the virtual initial modulus of
elasticity curve.
Conclusions
Refinements of sag-tension computa:
tions in the office can produce no better
‘end results than permitted by the approxi-
‘mations and tolerances in field measure-
‘ments, Therefore, an important part of
the design process is to provide simple,
practical instructions and adequate con.
trols for these measurements in the field.
‘Summary of Nomenclature
A,B,C=horizontal, vertical, and slope dis-
E-E,=modulus of elasticity, conductor and
steel core
‘Hewhorizontal component af conductor
‘tension, Fig. 1
‘Hem horizontal component of conductor ten-
son in suspension clamp, Fig. 13
Henequivalent horizontal component of
‘Conductor tension in sheaves, Fig. 13
Hi’ mcorrected horizontal component of
‘conductor tension in sheaves, Fig. 13
Kewiii/Hymchangein slack or offset in
inches for a span per pound change
in horizontal tension, Fig. 14
Lewconductor length between” supports,
Fig. 1
Lemeonductor length replacing Ly in eate-
ary, Fig. 17
Lym dead end assembly length, Fi
AWT = maximum working tension
RSerated strength
‘Smsag distance in vertical plane measured
from line between supports t0 the
point on the ‘conductor with B/A
Slope, Fig. 1
sag distance at point of tangency of
‘conductor and lin of sight, Fig. 10
Tconductor tension at any ‘point on
featenary, Fig. 1
Temvalue of T for the eatenary that sati-
fies Hook's law
Tnvalue of T for the catenary at the sup-
Se
ort
Zews/H=pararoeter
2.0, cross section area, conductor and stee}
dem vertical rise of Le, Fig. 17
2pm vertical rise of Ly. Fig. 17
emereep strain, Fig. 18
Ihevertial distance from conductor support
to transit for measuring sag. Fig 8
fmconductor length measured from low
point of eatenary to any point x31
Fig.
Gamunstressed length of £
| Svertical distance from conductor support
to target for measuring sag, Fig. ®
weweight of conductor per unit length,
‘eneral; Fig. 1
yew Tor bare conductor, no wind, in
pounds per foot
wpew for conductor and ice, no wind, in
‘pounds per foot
yew in plane of resultant for conductor,
ice, and wind in pounds per foot
sauaug.gnoumcatenary co-ordinates, Fig. 1
References.
Coureransona (ook), Theodore Versey. ets
‘Sem Compeay of Amerie, Patburghs Pa, 1057,
‘Borenburg. ATEE Tramctions, wah 86
3, See Caremsarions ny rmx Une op Maxroy
Tantee (oot) mee” S. "Martin, “Copperead
‘Sie Compeny, Ganeport, Pa 158),
4, Mucnamncs on Ceser vox Sraveronst
‘Anateen Fonmyh Marin. Procedingy, hosrcas
‘Secs of Ci Eopaeers, New York, No val,
8, ota, pp 10-88.
5. A Marnoo or Anavrrino Comer Dara, BG.
Siurm, €: Dement, FM. Howell
Preteens,
eee
Discussion
HH. Rode (Aluminum Company of
‘America, Pittsburgh, Pa.): The author has
made an excellent contribution to an under:
standing of the problems involved in the
calculation of sags and tensions of ACSR
and application of the resulls to Geld use
jn the installation of tansmission line
conductors,
‘The method described is essentially the
sameas used by Alcoa and utilizes the stress.
strain curve for the conductor in liew of
‘numerical values for modulus of elasticity.
‘The tension scale of the tension-strain
curves and’ catenary function curves. is
in pounds, whereas our usual practice is
to use pounds per square inch. The
advantage of using tension in pounds is
that the values are obtained directly, but
the disadvantage is that teasion-strain
diagrams are required for each individual
conductor size. When the scale is in
‘pounds per square inch, the same diagrams
‘and also the same catenary function curves
for bare conductor may be used for all con-
ductor sizes of the same type of stranding:
ie. same proportions of sted and aluminum.
‘This greatly reduces the number of curves
required for making calculations for a
variety of conductor ies, ‘The method
described by the author is convenient for
those dealing with a small number of con-
ductors, but pounds per square inch seems
‘more practical for those working with a large
‘number of conductors
‘The author calls attention to two factors
that will cause permanent increase in sag
from nonelastic stretch. One isi
‘radual nonelastic stretch over a long period
(of time with no significant increase in load.
‘This is commonly reerred to as creep. The
permanent increase in sag resulting from
these causes is not cumulative, and which-
‘ever produces the greater sag will control
the design.
In the case of a transmission line conduc
tor, certain factors. that infuence creep
are’ to some extent self-corrective. The
rate of creep increases as the tension in-
creases and. conversely decreases at the
tension is reduced. Since the tension de-
creases as sag increases as a resultof creep,
this coodition tends to correct itself. The
rate of creep also increases as the tempera:
ttre increases, but in the case of a trans
tission line conductor the tension decreases
as temperature increases, and again the
increased rate because of higher tempera-
ture is to some extent counteracted by the
reduced tension
Tn the ease of ACSR, the creep results
primarily from creep of ‘the aluminum
stands; and, since the proportion ofthe load
6
cartied by the aluminum decreases as tem-
perature increases becauce of difference in
‘coefficient of expansion, the rate of creep
fat higher temperature will be further re.
duced.
‘Span length also hasan influence om creep,
‘As the length of conductor increaces because
of creep, the tension will decrease. The
rate at which the tension decreases ill be
greater in shore spant than in long spans
and the rate of creep will, therefore, be
reduced more rapidly in short spans. Con
sideration ofcreepin design is therefore more
‘moportant on long spans than on short spans.
B, M. Pickens (Anaconda Wire & Cable
Company, Hastings on Hudson, N. ¥.):
In connection with creep we would like to
know if BPA has checked the sags in any
of their lines that have been installed for
several years to se ifthe sag has increased
‘more than expected?
‘We note that the prestressing practiced is
to stabilize the conductor to the virtual
initial “modulus “of elasticity. Approxi-
‘mately what tension is used, in per cent of
‘ated strength, and for what lengeh of time
is the conductor maintained at this tension?
‘The graphical method for calculating sag-
tension values is very complete and shows
the difference between the tension causing
‘longation of the conductor which isan aver, zi tae 73.03 soe
SERIE TE OBA TOR 8 a a
‘ge tension, and the tension at the support
‘which isa maximum tension. They afe not
the same. "This is sometimes overlooked in
calculating long spans. Asan be seen from
the several curves, Figs 3. 4, 5, and 6 the
sraphical solution’ requires a lot of work.
‘The use of a digital computer can reduce
the time required for some of the computa:
tions. Using the program described im an
ATEE paper, the results given in Table 1V
were obtained for an 1,800-foot span which
can be compared sith results shown in
Fig. 5 of this paper.
We believe this is a very close check be-
tween a graphical and a computer method,
particularly since ‘the stress-strain curves
Used were from different tests. The conn
puter solution can be made in afew minutes
‘hile the graphical solution probably takes
ours.
Revensce
1, SkoTenon Catcianon Process rox
Brottat" Consoras ats Pilon A/EE
Tranieion, 9 Tl (Powe nd Apparatal ond
Sgn Wate aah Tost econ,
LH. J. Cook (B. C. Engineering Company,
Lid, “Vancouver, B.'C., Canada): The
author is to be congratulated for a very
comprehensive résumé of the problems of
conductor stringing and sagging, and the
solutions to these problems, Included in
bis paper are a farily of curves showing the
relationship between time and conductor
creep for various designs. I would be most
interested to learn whether he has any sim:
ilar data on the relationship between time
‘and conductor creep for varying conductor
temperatures,
The author's comments regarding the
correct ‘relationship between Gesign’ pro-
‘cedures and accuracies and the approxima.
tions and tolerances in field measurements
‘emphasizes one of the most critical problems
‘of “transmission design. My company's
procedure has been to. produce complete
sagging data well ahead of ttringing. The