GUST RESPONSE FACTORS FOR

TRANSMISSION LINE LOADING

A. G. DAVENPORT
The University of Western Ontario, London, Ontario, Canada

SUMMARY

The objective of the paper is to derive simple factors for estimating the response of a transmission line
system, consisting of towers, conductors and ground wires, to gusty wind. The approach is based on
the statistical methods which takes account of the spatial correlation and energy spectrum of windspeed
and the dynamic response of the transmission line system. The main focus is on the transverse loading.
The influence of the variation in drag coefficient is discussed.
INTRODUCTION

The importance of wind loading (as well as the combination of wind and ice) in the design of transmission
line systems renders the accurate estimation of these loads economically significant. Historically, one of
the imponderables has been the action of gusts. It was recognized that the extreme gusts do not envelope
the entire span, and that some reduction reflecting the spatial extent of gusts might be anticipated; but
at the same time the possibility of some resonant response and dynamic amplification would tend to
aggravate the loads and offset the spatial reductions. The seriousness of both these trends would depend
on the characteristics of the turbulence.

A framework for investigating this problem of the response of structures to turbulent wind was provided
by the statistical methods suggested by the writer' 1 , λ These methods are ideally suited to transmission
line structures and have been expanded on in Europe, Japan and elsewhere' 3 ' 4 ' 5 '"'''. They also found
application in codes for buildings in the form of so-called "gust response factors"™\

The purpose of this note is to derive simplified gust response factors for a tower and line system. These
consider the frequency response characteristics of the conductors as well as the tower and the spatial
characteristics of gusts. The influence of terrain roughness is taken into account. Only transverse load­
ing is considered herein; longitudinal loading is however amenable to the same treatment.

DEFINITIONS

Although it is the ultimate objective to consider the response of a total transmission line system consist­
ing of towers, conductors and overhead ground wires we shall initially consider the simpler system shown
in Figure 1 consisting of only a single conductor and towers. The following defines properties of the
towers, conductor and the wind required in the analysis:

899
 A. G. Davenport

CENTRE OF
WIND PRESSURE

\\\\\\\\\\\\\^^^^

Fig. 1 Definition of transmission line system

Towers Height above ground - ht (Tl)

Fundamental transverse frequency of free-standing tower — ft (T2)
Structural damping - f^ (T3)
Drag coefficient — Cp (T4)
Conductor Typical span — L; (CD
Sag - D; (C2)
Mass/unit length — Mc; (C3)
Fundamental frequency for horizontal sway — fc 1_ flL· (C4)
Cable diameter - d; 2π 2D (C5)
Average cable density — pc; (C6)
Drag coefficient — Cn — generally a function of Reynolds number (C7)
Wind* Effective height (at approximate centre of pressure of structure) — hQ (Wl)
Mean wind speed at effective height - UQ (W2)
Mean wind speed at reference height, zref -Ut ref (W3)
Velocity variation with height, z,
Ufz) _
(--±-) or U(z) = \ / / [ In (z/z0) (W4)
Uref
t u
'ref ref
where Uref is the mean velocity at a reference height zref.
Mean square longitudinal turbulent wind speed fluctuation
=6K
°u Vre}
where K is surface drag coefficient (W5)
Representative terrain roughness factors (W6)

Gradient Height Power Law Exponent Surface Drag Coefft.

Z G (m) a K (z r e f = 10 m)
Open country, 250 .10 .0015
flat shorelines
Farmland, scattered trees 300 .16 .005
and buildings
Woodland, Suburbs 400 .28 .015
 Gust Response Factors for Transmission Line Loading 901

Spectrum of horizontal turbulence at frequency, / ,

'ΑΙξ. _- 28 lui2'3 (W7)

K Uref U2

(valid for higher frequencies f>Uz /z)

Transverse integral scale of turbulence L^; typically Lg « 65 m. (increasing
slightly with height and decreasing with roughness) (W8)
Transverse correlation of turbulence:
Ru (x, x) = exp - \ x - x\lLs (W9)
Transverse narrow band correlation of turbulence:
Ru (x, x;f) = exp -cf\x-x'\ 1U0 (W10)

where typically c = 8

MEAN RESPONSE
Consider some response R of the tower (base bending moment for example). Suppose that the mean
wind forces on the tower and conductor at the reference height hQ are respectively

Jt = 'ApU0^CDt (])
2
pc = '/zp U0 CDc
Let us define influence functions θ^ and 6tc such that the mean response R of the tower in this
wind is given by
R=ettpt + Btcpc (2)
The values of 6tt and dtc clearly depend on the geometry of the tower, its structural properties, its
aerodynamic properties and the form of the velocity variation with height. Since the effects are static
they may be found straight forwardly and are not further elaborated on here.

DYNAMIC RESPONSE

To simply matters we will,in considering the dynamic response, assume that the properties of the wind
field are homogeneous over the height of the line and characterized by the actual properties at the re­
ference height hQ. This height is likely to be close to the centre of mass of the conductor i.e. 2/3 D
below the suspension point. This simplification will not involve significant errors.

In estimating the fluctuating response we make an important observation and assumption concerning
the dynamic properties of the tower and conductors — namely that they are only lightly coupled
dynamically and that the free standing tower frequency fj is much greater than the fundamental fre­
quency of the conductor swing. This assumption means that the tower motion has negligible effect
on the conductor behaviour and that the tower "sees" the wind forces transmitted at the cable suspen­
sion point as quasi-static forces, at frequencies well below fj.

With these assumptions the spectra of the responses are as indicated in Figure 2. We can represent the
mean sqaure fluctuating response of the tower in the form

°R2 =Bc + R +B +
c t R
t (3)

in which the four components of response are as follows:

902 A. G. Davenport

WIND SPEED CONDUCTOR

-2
RESPONSE

Fig. 2 Spectra of wind speed and conductor and tower responses

Bc — the mean square response to the background, quasi-static, wind loads acting on the conductors
at frequencies < fc. These loads are only partially correlated over the spans.
Rc — the mean square resonant response of the conductor at frequencies close to fc.
Bt — the mean square response to the background, quasi-static wind loads acting on the tower, at
frequencies ^ / ^ . These are only partially correlated over the height h.
Rt — the mean square resonant response of the tower at frequenices at or near ft.

This expression is not exact in so far as it neglects the cross-coupled terms due to the correlated wind
loads on the tower and conductor. This is not considered a significant omission and the point is refer­
red to later. Because of their proximity loads on conductors are assumed fully correlated.

With this description of the mean square response σ^ , the peak response R can then be expressed as:

R R+goR (4)
where g is a statistical factor, weakly dependent on the frequency characteristic of the response and
the sampling interval. For present purposes, assuming sampling times of the mean wind of the order
of 10 min., appropriate values of g are in the range 3.5 < g < 4.0.

We now need to evaluate the component factors Bc, Rc,Bt, Rt from the external parameters. The
derivation in detail would follow the procedures described in references 4, 5 and 6. Following these
results we find to a close approximation:

B
c - e
tc 4pc JL
2
l+±- J±
2
7Γ LS

This simplifies, using statement W5 above, to:

B„= 24K(^L\ pc etc (5)

1 + 8L
 Gust Response Factors for Transmission Line Loading 903

Rc = e)c 4pc2 f S (f
c*/ J L _ J L _ JL·
U02 4ξ cfcL 4

Using statements W5 and W7 this simplifies to:

Rc = 24 K M T pc2 6t2c .0113 (khf* Ho. l

- (6)

In this f = total damping

= structural damping + aerodynamic damping.

For the conductor, aerodynamic damping is likely to be significantly greater than the structural damp­
ing which is also difficult to ascertain. For these reasons, we will take

* *aero — 2
Pad >/2PairU0 CD(;d

2 * / c mc U0 2nfcpcll2U0

= J_ Pair U
° Cn (7)
2
π d
Pc fc

We note here that typically

3 3
Pair Ä -0024 slugs/ft and pc « 5 slugs/ft so PairlPc Ä
.00048 and for these values
U
* » . 00048 o r
c
π fcd

= 4.8xl0'5 U0/fcd CD (8)

Turning to the tower,

2
2 2
Ott i

Bt = ett 4pc JL _^L__

U0 1+ .375 JL

Substituting from (W5)

Bt = 24 Kl _ J p f Btt (9)
\ n ' 1 + .375 '
L
S
904 A. G. D a v e n p o r t

Finally —
2 4 2 f S U
R
K
t =θ°tt 4p Pc < « W— « °
U0 4tt .375cfth0

24 κ{ΖΙΐΓ\ ap" θη2 .0123 (flS) "5/3 i _ (10)

U
K' o fr
Generally ?^ 1 - 3 % and aerodynamic damping is less important.

To clean up the algebra we will write:

B* = L (ID
1 +.8 L/Ls
R: = .0113{fshV> k
JL L· (12)
V
Ü0 ' L ic
B* = I (13)
1 + .375 h/Ls

R*t = .0123(^γ/3 J_ (14)

U0 ' tt

and E = 24 Kl i ] (an exposure factor) (15)

\h0 I
A similar expression may be given if the logarithmic profile is used.
Then
a =E
R * hi PI(BÎ +R }
* (16)

The peak response follows as before from equation (4). In this expression all the conductors have been
"lumped" together. They could equally well have been treated separately in which case the term for
the conductor would be replaced by a sum. The addtional complexity would not seem justified in a
simple approach and the improvement in accuracy could hardly be justified. In effect, for purpose of
the simplified analysis, the conductors have been constrained to more in phase with one another.

In the above analysis the possibility of tower-conductor correlation has been ignored. This only arises
when considering the low frequency, quasi-steady loads Bc and Bf. Since the bulk of the cable excita­
tion is at mid span and remote from the tower this is not likely to be a significant component and,
except for very short spans ( « Lg), can be reasonably ignored. Also, the conductors are constrained
so their motion is fully correlated which may slightly exaggerates the real situation. These two effects
as well as being small should also compensate one another. This simplifies the resulting equation.

GUST RESPONSE FACTOR

The peak response is given in equation (4) and can be written as

R = R + g oR
 Gust Response F a c t o r s for Transmission Line Loading 905
e + e + 2 Δ
= tc Pc ttP~c SE Ιθχ p c (B* + R*) + dt) p / (B* +R*)1 ' (17)

If the terms for the conductors and tower in the square root are of similar magnitude we may use the
approximation

(A2 + B) Yz
* e(A+B) where e ^ .75
and rewrite

R=6tcic 11 +geEy/B* + R*I +ettJt I 1 + g e Ey/ß* + R* 1

= Q
tc~PcGc + ettPtGt (18)

where Gc and Gt are "gust response factors" given by

Gc = 1 + geEy/ß* + R* (19)

Gt = 1 -hgeEy/B* + R* (20)

SUSPENDER FORCES

The forces Rg^ in the suspender arms may also be determined from the above. The detailed develop­
ment will give

RSA=pcdL(l+gE y/ß* + R*) (21)

"SPAN FACTOR" APPROACH

Armitt^) has described an approach based on a 2-second gust speed. Denoting this speed by UQ we
can show

= U0(l+g s/TR (II) a) (22)

Hence, ignoring higher order terms,

u*=u02 u+2g ^Τκ(1±)α+...)

= U0 (1+gE) (23)

Following this Armitt give the maximum suspender arm load as

2
= 1//z G
^SA P Uo D d L x span factor

= pcd L(l +gE) (span factor) (24)

Comparing (24) and (21) gives
l+gEy/B*v c + R*
c
"Span factor" = * <25)
1 +gE

Evaluation of this expression for a typical mean gradient wind speed of say 100 ft/sec should lead to
span factors similar to those suggested by Armitt^3). It would also be possible to incorporate in this
906 A. G. Davenport
expression the appropriate reductions for terrain. This would require the numerator to be computed
for the particular terrain category and the denominator for the reference Open country' condition.

It might also be convenient to include the effect of the changes of mean wind speed and turbulence
in terrains differing in roughness from the standard terrain. Denoting the standard terrain by the
subscript "st", the modified span factor taking into account the exposure of the line is then
g j l+gEy/B* + Rt
'span-exposure factor1' = ( _ ) £ °. (26)
U 1+
st sEst

The factor (U/Usf) can be found through the gradient velocity and can be written (hjzj ''ho'zg' St
An alternative expression using the logarithmic profile is (zQ09 In h0/zQ) /(z0'°9 lnhQlz0)st. In
the subscripted bracket the terms affected by terrain, a, zg and zQ are subscripted.

INFLUENCE OF DRAG COEFFICIENT VARIATIONS

The theory for gust response described above is based on the conventional assumption of a constant
drag coefficient. For conductors it is common for Jhere to be a dramatic decrease in the drag coefficient
Cjy at Reynolds numbers in the range 1 to 5 x 10 . Examples of this are shown in Figure 3. These
Reynolds numbers frequently coincide with design wind speeds. This change in drag coefficient affects
both the appropriate drag coefficient to be used in determining the mean pressure, p~c, as well as the
fluctuating component expressed through the gust response factor.

For the mean pressure, the drag coefficient corresponds to that for the Reynolds number, R = _
v
corresponding to the mean wind speed UQ. The fluctuating component is however augmented by
an additional term dependent on the slope of the drag coefficient at that Reynolds number. This
modified drag coefficient, Cn 1 for the fluctuating component is
C = [C
D D +%(dCD Id Re) Re]

2
This is derived from the derivative of the force, pc = Vz p U Cjy (U) with respect to wind speed U.

Since the slope at the 'drag crisis' is negative, this has the effect of reducing the effective drag coef­
ficient influencing the dynamic components of conductor sway. This modified drag coefficient should
be used in estimating both Bc and Rc above. In Rc the drag coefficient also modifies the aerodynamic
damping. As a result the values of Rc are multiplied by the factor (Cp / CD ) and Bc by the square
of this factor. If this factor is less than unity, the overall fluctuating response7 is reduced while the
relative importance of resonant fluctuations is increased.

To illustrate this point consider the average slopes of curves 4 and 5 in Hgure 3 at Re = 3 x . / ^ ( c o r ­
responding to 36 ft/sec and 52 ft/sec for the 1.6" and 1.1" diam. cables respectively). At this Re,
Cun ^ .75. The average slope of these curves, (d CD Id Re), is approximately -2 x 10" 5 . So that
c c

Cß = 1.75 -Vzx2 x 10'5 x 3 x 10*] = .45

and cX lCn =.60

A reduction of this magnitude, of course, has a significant effect in reducing the dynamic loads seen
by the suspender arms and the tower. The apparent differences between the full scale and wind tunnel
results is worthy of further study.
 Gust Response Factors for Transmission Line Loading 907
Source Curve Conductor No of outside Wire/Cable
Diam. in. strands diam.
Wind 1 1.125 24 9
tunnel 2 .770 18 7
3 1.695 42 15
Full scale 4 1.108 16 6.3
estimates 5 1.60 27 10

1.2
\ \
\ \ N
\
I.I \ \
\
\

1.0
1 Ν ®\
© Λ
' S.
s
z 0.9 1
UJ
I N'
^ —*
U.
ÜJ
d>
o \
υ 0.8
\ /
<
Q
< \ / •
/
0.7 \ \ V
\ \ X ,^
v5)
0.6

0.5
2 3 4 5
REYNOLDS NUMBER x IÖ 4 , N R

Fig. 3 Conductor drag coefficients C D

EXAMPLE

To indicate the general character of the gust factors consider the following examples:
Conductor L = 1000 ft; D = 32 ft; d = 0.1 ft; fc = .2 Hz; CD = 1.0
Tower h = 150 ft; hQ = 100 ft; ft = 1 Hz; ff = .01
Wind U0 = 100 ft/sec; Ls = 210 ft (~ 65 m)
Open country a = .16 K = .005
Wooded terrain a = .28 K = .015
Referring to equations 8 and 11 through 15 ;
908 A.*G. Davenport

B* = I = .21

1 + .8 L/Ls

ic = 4.8xlOmS Üjfcd = .24

R* = .0113(fchoIÏÏo)-*l* (hoID^=.07

Bt = 1 = .79
1 + .375 h/Ls
R* = .0123(ftholUo) - 5 / 3 if1 =1.23

E = y/24 K(zreJh0) = .30 (open country)

= .61 (wooded country)

Thus gust response factors are as follows:

Open Country Wooded Country

Conductor: Gc = 1 +geEJ** + R* 1.42 1.61
Tower: Gt = 1 +geEy/ß* + R* 2.12 2.64
"Span factor": Gc/(1 + g E) (with e = 1.0) .76 .71

These results indicate the potentiality for systematic development of gust factors and span factors for
ranges of conductor and tower properties.

ACKNOWLEDGEMENTS

The incentive for this study was provided by on-going design studies for the Pennsylvania Power and
Light Company and by research into transmission line response to wind sponsored by the Electric
Power Research Institute. Special thanks are extended to Messrs. H. Davidson, J. Mozer and T. Di Gioia
of GAI Consultants, E. Goodwin of P.P. and L. and P. Landers of E.P.R.I., for useful suggestions and
comments on this study.

REFERENCES

1. Davenport, A. G., The Application of Statistical Concepts of the Wind Loading of Structures,
Proc. Inst. Civ. Eng., Vol. 19, 1961.

2. Davenport, A. G., The Response of Slender Line-Like Structures to a Gusty Wind, Proc. Inst.
Civil Eng., Vol. 23, 1962.]

3. Armitt, J., Cojan, M., Manuzio, C. and Nicolini, P., Calculation of Wind Loadings on Components
of Overhead Lines, Proceedings IEE, Vol. 122, No. 11, November, 1975, pp. 1242-1252.

4. Manuzio, C , and Paris, L., Statistical Determination of Wind Loading Effects on Overhead Line
Conductors, CIGRE Report 231, Paris, Irance, 1964.

5. Castanheta, M., Dynamic Behaviour of Overhead Power Lines Subject to the Action of Wind,
CIGRE Report 22-08, Paris, France, 1970.
 Gust Response Factors for Transmission Line Loading 909

6. Ohtsuki, A., Studies on Wind Pressure Upon Overhead Transmission Lines and Their Movement,
Paper Presented to Fujikura Electric Wire Manufacturing Company, Ltd., Japan, January 30, 1967.

7. Fujikura Electric Wire Manufacturing Company, Ltd., Movement of Overhead Transmission Line
and Statistical Approach, Japan, January 30, 1967.

8. Davenport, A. G., Gust Loading Factors, J. Str. Div. Proc, ASCE, Vol. 93, 1967.

9. Davenport, A G., Wind Structure and Wind Climate, Proc. Int. Res. Seminar, Safety of Structures
Under Dynamic Loading, June 1977, Norwegian Inst. of Tech. Trondheim.

10. Davenport, A. G., The Prediction of the Response of Structures to Gusty Wind, Proc. Int. Res.
Seminar, Safety of Structures Under Dynamic Loading, June 1977, Norwegian Inst. of Tech,
Trondheim.