DYNAMIC GUST RESPONSE FACTORS FOR TRANSMISSION LINE STRUCTURES by RAJESH SHIMPI, B.S.E. A THESIS IN CIVIL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN CIVIL ENGINEERING Approved August, 1996
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DYNAMIC GUST RESPONSE FACTORS FOR
TRANSMISSION LINE STRUCTURES
by
RAJESH SHIMPI, B.S.E.
A THESIS
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
CIVIL ENGINEERING
Approved
August, 1996
ACKNOWLEDGMENTS
The author expresses sincere thanks to his advisor and committee
chairman, Associate Professor William P. Vann, for his encouragement and
guidance throughout the course of this thesis. Special appreciation is also
extended to Professor Kishor C. Mehta and Assistant Professor Partha P.
Sarkar for their earlier direction of author's work and their interest as other
members of the thesis committee.
Financial support from the Institute for Disaster Research (IDR),
Department of Civil Engineering, Texas Tech University is gratefully
acknowledged.
The author shall ever remain indebted to his sisters Neeta and
Abhilasha, brother-in-law Deepak, and his friend Aditi Samarth for their
love and moral support throughout his graduate program.
Finally, the author would like to express 'thanks' to his parents, to whom
he dedicates this thesis.
11
TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
ABSTRACT vii
LIST OF TABLES viii
LIST OF FIGURES ix
1. INTRODUCTION 1
2. STATEMENT OF THE PROBLEM 3
2.1 Objectives and Scope 5
3. STATE OF KNOWLEDGE 7
3.1 Wind Engineering 7
3.1.1 Wind Characteristics 7
3.1.1.1 Wind Speed 8
3.1.1.2 Variation of Wind Speed vsdth Height 8
3.1.1.3 Effect of Averaging Time on Mean Wind Speed 12
3.1.1.4 Atmospheric Turbulence 17
3.1.2 Statistical Peak factor (g) 21
3.1.2.1 Extreme Value Theory for'g' 22
3.1.3 Gust Response Factor (GRF) 26
3.1.3.1 Parameters Affecting the GRF 26
3.2 Flexible Structures 27
3.3 Structural Response 30
3.3.1 Mean Response of Conductors 31
3.3.2 Fluctuating Response of Conductors 31
3.4 Changes in ASCE 7-88 32
4. DESIGN OPTIONS FOR DYNAMICALLY SENSITIVE STRUCTURES 39
5. POLE, CONDUCTOR, AND GROUNDWIRE DESIGN DATA 42
5.1 Concrete Poles 42
111
5.1.1 Static-Cast Concrete Poles 42
5.1.2 Spun-Cast Concrete Poles 43
6. DAVENPORT'S MODEL (ASCE, 1991) 47
6.1 Introduction 47
6.2 Notation 48
6.3 Equations 50
6.4 Example Calculations for Spun-Cast Concrete Pole 61
6.4.1 Sununary of Input Data 62
6.4.2 General Calculated Values 64
6.4.3 Tower Gust Response Factor 65
6.4.4 Conductor Gust Response Factor 65
6.4.5 Groundwire Gust Response Factor 67
6.4.6 Tower Stress 67
6.4.6.1 Conductor Contribution 68
6.4.6.2 Groundwire Contribution 70
6.4.6.3 Tower Contribution 71
6.4.6.4 Total Stress 72
6.4.7 Tower Deflection 73
6.4.7.1 Conductor Contribution 73
6.4.7.2 Groundwire Contribution 74
6.4.7.3 Tower Contribution 74
6.4.7.4 Total Deflection 75
7. SOLARI AND KAREEM'S MODEL (ASCE 7-95) 76
7.1 Introduction 76
7.2 Notation 76
7.3 Equations 78
7.4 Example Calculations for Spun-Cast Concrete Pole 86
7.4.1 Summary of Input Data 88
7.4.2 General Given and Calculated Values 89
IV
7.4.3 Tower Gust Response Factor 90
7.4.4 Conductor Gust Response Factor 91
7.4.5 Groundv^dre Gust Response Factor 93
7.4.6 Tower Stress 96
7.4.6.1 Conductor Contribution 97
7.4.6.2 Groundwire Contribution 97
7.4.6.3 Tower Contribution 98
7.4.6.4 Total Stress 101
7.4.7 Tower Deflection 101
7.4.7.1 Conductor Contribution 102
7.4.7.2 Groundwire Contribution 102
7.4.7.3 Tower Contribution 103
7.4.7.4 Total Stress 103
8. SIMIU'S MODEL (1976, 1980) 104
8.1 Introduction 104
8.2 Notation 105
8.2.1 Relevant Graphs and Tables from Simiu 1976 106
8.3 Equations 110
8.3.1 Gust Response Factor 110
8.3.2 Maximum Alongwind Displacement 112
8.4 Example Calculations for a Spun-Cast Concrete Pole 113
8.4.1 Summary of Input Data 114
8.4.2 General Given and Calculated Values 114
8.4.3 Tower Gust Response Factor 115
8.4.4 Conductor Gust Response Factor 117
8.4.5 Tower Deflection 118
8.5 Summary of Simiu's Model 118
9. DISCUSSION OF RESULTS AND SENSITIVITY STUDY 120
9.1. Introduction 120
9.2. Comparison of Spun-Cast Concrete Pole Results by Davenport's Model and ASCE 7-95 Commentary Method 120
9.3 Comparison of Spun-Cast and Static-Cast Concrete
Pole Results 125
9.4 Sensitivity Parameters 127
9.5 Sensitivity Results 129
10. CONCLUSIONS AND RECOMMENDATIONS 139
10.1 Summary 139
10.2 Conclusions 141
10.3 Recommendations 143
BIBLIOGRAPHY 145
APPENDIX A. TABLE OF SENSITIVITY STUDY RESULTS FOR
SPUN-CAST AND STATIC-CAST CONCRETE POLES 149
B. SENSITIVITY STUDY GRAPHS FOR STATIC-CAST POLE 162
C. FORTRAN CODE FOR SENSITIVITY STUDY 168
D. INPUT DATA FOR CONCRETE POLES 185
VI
ABSTRACT
Transmission line structures are flexible, line-like, wind-sensitive
structures used for distribution of electricity. Dynamic wind loads on these
structures result from two components: wind loads on the tower and wind
loads on the conductors. Various approaches are available for the calculation
of the gust response factor. The Gust response factor (GRF) is the static
equivalent of the dynamic loads acting on the transmission lines. The ASCE
7-95 Commentary Method (1995) has a procedure to evaluate the GRF based
on the new 3-second gust wind speeds adopted in the code. This procedure is
for general categories of structures. Davenport's model (1979) is tailored
exclusively for transmission lines and is flexible v^th any averaging time.
Simiu's model, which again is not developed for transmission line structures,
uses graphs for the major part of the GRF calculations. In this study,
Davenport's model is used as a reference model for the calculation of GRF
and foundations of approaches put forward by ASCE 7-95 and Simiu are
studied. All these methods are considered in evaluating the loads on
representative transmission line systems using Static-Cast and Spun-Cast
concrete poles. Sensitivity studies are carried out for understanding the
effects of different parameters in the Davenport and ASCE 7-95 methods and
modifications are suggested in the ASCE 7-95 method.
VU
LIST OF TABLES
3.1 Values of Power Law Exponent and Gradient Height based on 3-sec. Averaging Time in ASCE 7-95 11
3.2 Extreme Values Calculated by Davenport (1964) with T = 3600 24
3.3 Frequencies and Spectral Values Selected for Simplified Time History 24
3.4 Values of Wind Parameters in ASCE 7-88 and ASCE 7-95 36
5.1 Properties of Static-Cast and Spun-Cast Concrete Poles 45
5.2 Results of SPRINT Analysis for Static-Cast
and Spun-Cast Concrete Poles 45
6.1 Parameters for Use in Davenport's Equations 52
6.2 Separation Factor, e, for Different Ratios of B/A 59
8.1 Values of vHy
and ^z ^ ZA. corresponding to Various
Yji Curves 109
9.1 Summary of Results for 84-Foot Spun-Cast Concrete Pole 121
9.2 Comparison of Background and Resonance Contributions to the GRF in the Davenport and ASCE Methods 123
9.3 Comparison of Results for the 84-Foot Static-Cast,
and Spun-Cast Concrete Poles 126
9.4 Parameter Values for the Baseline Structures 128
A.l Sensitivity Study Results of Spun-Cast Concrete Pole 150
A.2 Sensitivity Study Results of Static-Cast Concrete Pole 156
VUl
LIST OF FIGURES
3.1 Typical Wind Speed Record 9
3.2 Typical Profiles of Mean Wind Speed and associated Gradient Height 13
3.3 Idealization of Gust Spectrum Plot over an Extended Range (Davenport, 1972) 14
3.4 Influence of Averaging Time on the Mean Wind. Speed (after Durst, 1960, and Krayer and Marshall, 1992) 16
3.5 Spectrum of Longitudinal Wind Velocity Fluctuations 20
3.6 Representative Components of Spectrum for
Frequencies in Table 3.2 25
3.7 Sensitivity of GRF to Damping Ratio 28
3.8 Sensitivity of GRF to Fundamental Frequency 28
3.9 Sensitivity of GRF to Width of the Building Ratio 29
3.10 Sensitivity of GRF to Basic Wind Speed 29
3.11 Response Model 30
3.12 Elements of Response Spectrum Analysis 33
3.13 Basic Design Wind Speed Map Proposed for
ASCE 7-95 Using 3-Second Gust Speeds 35
5.1 Typical Properties of Concrete Poles 44
6.1 Davenport's Background Response Terms as Function of The Size Ratio 54
6.2 Davenport's Gust Response Factor for the Tower (Simplified Equation) 55
6.3 Davenport's Gust Response Factor for the Conductors (Simplified Equation) 55
6.4 Spectra of Wind Speed, Conductor Response, and Tower Response 58
7.1 Size Effect Functions in the ASCE 7-95 Commentary Method 80
IX
7.2 Variations in the Fundamental Mode Shape Equation
rzV (|)(z)= — with^ 83
7.3 Variations of Factor K with Wind Profile Exponent, a, and Mode Shape Exponent, ^ 87
7.4 Comparison of Equation 8.12 and Equation 8.13 for Factor K and Exposure C 87
8.1 Function S (Simiu, 1976) 107
8.2 Function J (Simiu, 1976) 107
8.3 Function Y^^ (Simiu, 1976) 108
9.1 Combined Response Sensitivity to Tower Height and Conductor Span for Spun-Cast Concrete Pole 130
9.2 Combined Response Sensitivity to Tower Height and Tower Damping Ratio, tower for Spun-Cast Concrete Pole 132
9.3 Response Sensitivity to Tower Height and Conductor Span Separated by Load Component for Spun-Cast 134 Concrete Pole
9.4 Response Sensitivity to Tower Height and Tower Damping Ratio, tower, Separated by Load Component for Spun-Cast Concrete Pole 137
9.5 Sensitivity of Davenport's Aerodynamic Damping in the Conductor and Groundwire to Span 138
B.l Combined Response Sensitivity to Tower Height and Conductor Span for Static-Cast Concrete Pole 163
B.2 Combined Response Sensitivity to Tower Height and Tower Damping Ratio, tower for Static-Cast Concrete Pole 164
B.3 Response Sensitivity to Tower Height and Conductor Span Separated by Load Component for Static-Cast Concrete Pole 166
B.4 Response Sensitivity to Tower Height and Tower Damping Ratio, tower, Separated by Load Component for Static-Cast Concrete Pole 167
CHAPTER 1
INTRODUCTION
A nation-wide system of electric power supply involves transmission
lines as an integral part of the network. The basic function of transmission
lines is to transmit electricity fi-om power plants. Therefore, continuous,
uninterrupted, and efficient functioning of transmission lines is needed in
order to balance demand-supply requirements. For meeting this demand,
transmission lines should be structurally reliable. At the same time, the
transmission tower and the conductors attached to it should function as a
single unit. Therefore, a great deal of effort and a high standard of design
must be enforced to avoid structural failure that may result due to a critical
loading condition.
Transmission line structures are more sensitive to d3Tiamic loads than
most type of structures. The most common and important dynamic loads
result fi*om wind on the tower, conductors, and ground wire. A typical
transmission line consists of a series of towers with conductors and
groundwires spanning between each pair of consecutive towers. Conductors
are highly flexible line-like structures with uniformly distributed mass along
the span (Davenport, 1979).
Wind loading on transmission lines consists of three parts. First, some
wind loads act directly on the transmission tower itself. Second, the
conductors are subjected to wind loads and in turn, these loads are
transmitted to the tower. Third, wind loads on a groundwire are transmitted
to the tower in the same way. Wind on the conductors is invariably the most
critical of the three loadings. However, all three parts are important in
ascertaining the overall effects of wind loads on transmission towers.
When wind loads act on wires, it is recognized that a wind gust of
maximum intensity does not act simultaneously on the entire span between
towers. Due to this spatial effect, net wind forces are reduced. At the same
time, wind speeds vary in time, and because of these "gust fluctuations,"
towers and conductors can be subjected to resonance. Thus, spatial
variations of wind gusts and fluctuating components of the gusts have
opposite effects on the response of transmission line structures.
In order to avoid complex calculations in structural dynamics a single
'factor' can be assessed to account for dynamic effects resulting fi-om gust
fluctuations. Several analytical models have been developed in the past to
calculate this dynamic factor which, when multiplied by the static response,
gives the maximum dynamic response of the structure. This dynamic factor
is usually referred to as the "Gust Response Factor," and this terminology,
with the acronym GRF, is used throughout this manuscript. The subject of
this thesis is to study different analytical models for determining the GRF
and to recognize the most critical parameters influencing the structural
behavior through a sensitivity study.
CHAPTER 2
STATEMENT OF THE PROBLEM
Since transmission line structures are unique in being very wind
sensitive and having strong loads applied to them through wind on long
flexible wires as well as wind on the towers or poles themselves, special
methods of analysis are needed for their design. In the past, the method of
Davenport (1979, ASCE 1991) has been the most accepted one, and it has
been used in conjunction with standard wind maps giving expected fastest
mile winds. With the advent of wind maps based on a 3-second gust in ASCE
7-95, the question of adapting the Davenport method to these maps arises.
Also, the Commentary to ASCE 7-95 presents a new method for analyzing
wind sensitive structures which might be as appropriate as Davenport's
method for transmission lines, or more so. In order to evaluate which method
is best for transmission line structures, and to understand the assumptions,
questions, and complexities involved in each method, a study of these two
methods and any other available "rational analysis" methods (ASCE 1995) is
needed.
2.1 Methods Considered
As mentioned in the first chapter, various analytical approaches are
currently in practice for calculation of the gust response factor (GRF). There
are three models of "rational analysis" for determining the GRF and thus the
design wind pressure for a d3niamically wind sensitive structure. These
models are as follows:
1. Simiu's Model (1976, 1980), based upon Vellozzi and Cohen's Model
(1968).
2. Davenport's Model (1979, EPRI 1987, ASCE 1991); and
3. Solari and Kareem's Model (ASCE 7-95, Commentary).
The model of Simiu (1976, 1980) was developed for general
categories of structures and was an update and modification of the model
of Vellozzi and Cohen (1968). Vellozzi and Cohen's approach formed the
basis for the ANSI A-58.1 (1982) and ASCE 7-88 (1988) design standards.
Vellozzi and Cohen's and Simiu's formulations are distinct fi-om the other
two models (Davenport and ASCE 7-95) discussed in this manuscript in
that they rely in part on information in graphs and thus are not as
adaptable to computer calculations.
The model of Davenport (1979) is specialized to transmission line
structures. This model grew out of Davenport's (1962) earlier analysis of
"line like structures" and has been referenced and adopted in a number of
other publications, including ASCE's "Guidelines for Electrical Transmission
Line Structural Loading" (1991). The model is based on a 10-minute average
wind speed, but adaptations of it to a fastest mile wind have been published
(EPRI, 1987; ASCE, 1991). It separates tower, conductor, and groundwire
responses, thus giving independent gust response factors for the tower and
the wires. Then based on the differences in the natural frequencies of the
tower and the wires, a separation coefficient 'e' is used. Furthermore,
Davenport's model assumes that even though the ground wires and
conductors are located at different heights on the tower, loads on the wires
are fully correlated, i.e., all lines experience peak responses at the same time.
This assumption probably overestimates the total peak forces that the tower
'receives' fi-om the conductors and groundwires.
The model of Solari (1992a, b) has been modified for presentation in
the Commentary to ASCE 7-95 by Kareem. This model is for general
structures and has the distinction of dealing directly with 3-second gust
wind speeds. Nevertheless, it utilizes mean hourly speeds in portions of
its treatment, since it too is based on frequency response concepts and
probabilistic peak factors which cannot be used directly with a 3-second
duration.
The results one gets for design wind pressures will vary according to
which one of these design models is used. Davenport's model and Solari-
Kareem's model are treated in the greatest detail herein, mainly because
Davenport's model is so well tailored to transmission lines and has been used
so extensively in their design, and because an understanding of the new
model of Solari and Kareem is desired for comparison. Also, these two
models avoid the problem of using graphs, and so are more amenable to
computer usage than the models proposed by Vellozzi and Cohen (1968) and
by Simiu (1980).
2.2 Objectives and Scope
The general objectives are as follows :
1. To study the three models introduced in the preceding section:
a. Simiu's Model (1980);
b. Davenport's Model (ASCE 1991);
c. Solari-Kareem Model (ASCE 7-95 Commentary Method) with a 3-
second gust speed.
in order to understand the foundations of the equations and to make
recommendations about their practicality for transmission line systems.
2. To compare results for the gust response factor (GRF), deflections, and
stresses obtained by the Davenport and ASCE 7-95 models for typical
transmission line structures.
3. To suggest modifications for the Davenport Model and the ASCE 7-95
Commentary Method with regard to their determination of the gust
response factor (GRF), deflections, and stresses in transmission line
structures.
4. To carry out sensitivity studies using parameters such as height of the
tower, percentage of critical damping in the tower and the conductors, and
span of the conductors.
5. Based on the sensitivity study results, to identify basic parameters that
are influential in the calculation of the gust response factor (GRF) and
which may merit additional attention in the future for more accurate
solutions.
CHAPTER 3
STATE OF KNOWLEDGE
3.1 Wind Engineering
3.1.1 Wind Characteristics
Although wind loads play a major role in the design of buildings, the
nature of wind itself is a subject with which engineers are generally not very
familiar. This situation is due partly to the interdisciplinary nature of the
subject and partly because of the lack of emphasis usually given to wind
engineering in engineering curricula. As a result, design for wind forces has
tended to become compartmentalized; the estimation of design wind loads is
often delegated to others and divorced from the analysis and design of the
building itself. Indeed, to a few engineers, destructive winds are little more
than unpredictable acts of God capable of little or no scientific explanation.
When designing any building to resist wind forces, one of the chief
factors that has to be taken into account, affecting both cost and safety, is the
design load that is likely to be imposed on the building by the wind. It is not
therefore surprising that as urban areas continue to grow and more
sophisticated analyses and designs of buildings are achieved, more attention
is given to design wind speed and attendant loads. It should be said at the
outset that buildings should not be designed for the "highest recorded speed"
at a site, but should be designed to resist wind speeds that are likely to occur
with specific probabilities.
The movement of air near the surface of the earth is generally
described in terms of a wind velocity vector having both magnitude and
direction. The scalar quantity used to describe wind speed must be defined
with respect to averaging time, ground terrain, and height above ground.
Wind speeds can be described in terms of peak wind, mean wind, fastest-mile
wind, 3-second gust or annual extreme fastest-mile wind. Each of these
terms has a unique meaning and serves to describe one particular aspect of
wind.
3.1.1.1 Wind Speed
Movement of air parallel to the ground is generally termed as "wind"
for engineering purposes. Typical wind speed record is shown in Figure 3.1.
Wind speed varies in space and time. It consists of a mean wind speed and
fluctuations about the mean.
1. Mean wind speed is the mean value of a wind speed record taken over
some time interval. Wind gusts are fluctuations about the mean value.
It is common to refer to a mean wind speed as mean hourly, 10-minute or
1-minute average wind speed. It should be noted that a 10-m standard
height above ground (flat terrain) is used in these standard
measurements.
2. Ppflk wind speed is the maximiun instantaneous value of the wind speed
that is recorded. Most commonly used anemometers have response times
of one to three seconds. Hence, a peak wind speed is generally a 3-second
gust.
3.1.1.2 Variation of Wind Speed with Height
3.1.1.2.1 Gradient Wind and Gradient Height. Natural and man-
made obstructions retard the movement of air close to the ground. At some
height above the ground, the movement of air is independent of these ground
obstructions. This unobstructed wind speed is termed the "gradient wind
speed," and the lowest height at which the air movement is not retarded is
termed the "gradient height."
The wind speed above the gradient height may be considered to be
constant. The variation of wind speed with height below the gradient height
is strongly influenced by the terrain roughness (surface obstacles). This
125inpli -
^ 1 2 i *
iO in
Figure 3.1 Typical Wind Speed Record (Simiu and Scanlan, 1986)
variation or profile can be defined by the Power Law or the Logarithmic Law.
For engineering purposes, the wind speed profile is usually used in the
Power Law form (Davenport, 1960) where at any height above ground the
wind can be represented as -ll/a
for 0 < = z < = z g (3.1) V = V z
z 8 J
and Vz = Vg
where
for z > z g
V2 = wind speed at any height, z, mph,
Vg = gradient wind speed, mph,
z = height above ground, ft,
Zg = gradient height, fli,
1/a = power-law coefficient.
The values of gradient height, Zg, and power-law exponent 1/a depend on the
ground surface roughness. Surface roughness is the cumulative drag effect of
all obstructions to the wind. The roughness is characterized by the density,
size, and height of buildings, trees, vegetation, rocks, etc., on the ground.
Surface roughness will be minimum over water and maximum over a large
city.
The power law is used in both the American National Standard ASCE
7-95 and in the National Building Code of Canada (NRCC, 1980). Values of
gradient height, Zg, and power law exponent, 1/a, from ASCE 7-95 are shown
in Table 3.1 for different exposures or types of terrain.
Davenport (1960) took wind data from 19 different locations around
the world and determined the power-law coefficient, 1/a, at each location.
The variation of 1/a at the different locations was attributed solely to the
variation in terrain roughness. The values of 1/a varied fi-om 1/10.5 for
10
Table 3.1 Values of Power Law Exponent and Gradient Height based on 3-second Averaging Time in ASCE 7-95
Exposure*
A B C D
PowerLaw Exponent, d
1/5 1/7 1/9.5 1/11.5
Gradient Height Zg (fii)
1500 1200 900 700
* refers to Exposure Categories in ASCE 7-95
Exposure A: Large city centers with at least 50% of the buildings having
a height in excess of 70 feet;
Exposure B: Urban and suburban areas, wooded areas, or other terrain
with numerous closely spaced obstructions having the size
of single-family dwellings or larger;
Exposure C: Open terrain with scattered obstructions having heights
generally less than 30 feet. This category includes flat open
country and grasslands;
Exposure D: Flat, unobstructed areas exposed to wind flowing over large
bodies of water.
11
coastal waters to 1/1.6 at the center of a large city. He also found that the
gradient height, Zg, varied from 885 ft over flat open country to 2020 ft over a
large city. Some typical profiles for mean wind speed at and associated
gradient height for the same gradient wind speed of 146 mph are shown in
Figure 3.2.
3.1.1.3 Effect of Averaging Time on Mean Wind Speed
Different definitions of wind speed have major implications in the
determination of wind loading. The same wind record provides different
mean wind speeds depending on the averaging time used. Various national
standards around the world use different definitions of wind speed, e.g., the
National Building Code of Canada (NRCC, 1990, 1990a) uses a mean hourly
wind speed, the American National Standard ASCE 7-95 (1995) uses a 3-
second gust speed, and the British (BSI, 1972) and Australian (SAA, 1989)
Standards utilize a 2-second gust speed.
The mean wind speed values are higher for shorter averaging times
and vice-versa. The main reason for this is that short gusts of high wind
speed last for very short periods of time. A wind record which is to be used
for calculation of a mean wind speed and an RMS value of wind speed should
be long enough to reflect the effects of low frequency components of
mechanical turbulence generated by the terrain roughness and short enough
for stationarity. Davenport (1972) has developed a power spectral density
plot over an extended time history as shown in Figure 3.3. This plot provides
a background for choosing the averaging time interval for mean wind speed.
This spectrum has two distinct t57pes of air flow: (a) macrometeorological or
climate fluctuations, and (b) micrometeorological fluctuations or gusts.
These fluctuations are separated by a stationary time interval which is called
the spectral gap which varies between 10 minutes and 1 hour. Based on this
spectral gap, mean values averaged over 10 minutes to 1 hour are optimum
12
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for stability. In this study by Davenport, the wind speeds were averaged over
record length of 12 minutes.
Inasmuch as wind speed magnitudes are a function of averaging
period, there is an obvious requirement for data on mean wind speeds
averaged over various periods of time ranging fi-om one hour down to a few
seconds. This requirement has, to some extent, been met by the work of
Durst (1960) and Hollister (1970). On the basis of a statistical analysis of
wind records from Cardington and, for shorter periods than 5 seconds, from
the data of Ann Arbor (Sherlock and Stout, 1937; Sherlock, 1952), Durst
obtained the results as shown in Figure 3.4.
The most striking change in the wind design provisions fi-om ASCE 7-
88 to ASCE 7-95 is fi-om a basic design wind speed that represents a fastest
mile wind to one that represents a 3-second gust. The 3-second gust speed is
considerably greater than the corresponding fastest mile wind, having a ratio
that varies with the averaging time used in determining the fastest mile
wind. This ratio is different in hurricane and non-hurricane regions (Krayer
and Marshall, 1992).
The effect of averaging time on the measured wind speed is shown in
Figure 3.4. There the ratio between the mean wind speed measured over an
arbitrary time interval, V , and the mean hourly wind speed VsgQO ^ given
by the lower curve for a non-hurricane region (Durst, 1960) and that for a
hurricane region is given by the upper curve (Krayer and Marshall, 1992).
As an example, in a non-hurricane region a 90 mph fastest mile wind would
have an averaging time of 40 seconds and a ratio to the mean hourly speed of
1.30, while the three-second speed has a ratio to the mean hourly speed of
1.53, giving a ratio between the three-second gust speed and the fastest mile
speed of 1.18. The corresponding ratio for a 90 mph fastest mile wind in a
hurricane region is 1.21. For a 120 mph fastest mile wind the ratio of the
three-second gust to the fastest mile speed is 1.15 in a non-hurricane region
15
and 1.18 in a hurricane region. Thus, three-second gust winds are of the
order of 20 percent larger than corresponding fastest mile winds.
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m
- I—1—I I 11111—
10 - i — I — I — I 1 1 1 1 1 —
100 -1 1—I I M i l l r
1000 I I I I T I -
10000 GUST DURATION. SEC
Figure 3.4 Influence of Averaging Time on the Mean Wind Speed (after Durst, 1960, and Krayer and Marshall, 1992)
16
3.1.1.4 Atmospheric Turbulence
Examination of the wind record in Figure 3.1 shows that wind speed at
a point in space fluctuates. The fluctuating part of wind is termed as
turbulence. The wind speed over a given time interval can be considered as
consisting of a mean component and a fluctuating component. The mean
wind speed (based on, e.g., a 10-minute record) increases with height, but the
amplitude of the fluctuating component remains essentially constant with
height. There is, however, a tendency for the amplitude of the fluctuations to
be larger near the ground over rough terrain. Turbulence induced by the
interaction of the moving air with obstacles is referred to as "mechanical"
turbulence. Convective turbulence caused by mesometeorological conditions
(e.g., an unstable atmosphere) is called meteorological turbulence.
The analysis of atmospheric turbulence is characterized by the
following quantities:
1. Turbulence Intensity,
2. Integral Scales of Turbulence,
3. Spectra of Turbulent Velocity Fluctuations.
3.1.1.4.1 Turbulence Intensity. The expression for turbulence
intensity is
i(.)=^Bi> (3.2) U(z)
where
U(z) = mean wind speed at elevation z; and
7u^(z) = root mean square of the fluctuations in wind speed; u.
Turbulence intensity is the intensity of turbulence in the wind flow
and is denoted as I( z). It indicates the relative amplitude of the fluctuations
compared to the mean wind speed. It usually varies with exposure category
17
and height above ground level. Of the four exposures, Exposure A ( ASCE 7-
95) has the highest turbulence intensity at the reference height considered
and Exposure D the lowest. The turbulence intensity reduces for a particular
structure with height in any exposure category.
In statistical terminology, I( z) is referred as a coefficient of variation
(standard deviation divided by mean). A decrease in turbulence intensity
with height is expected because at greater heights, while both the mean and
RMS values of wind speed increase, the increase in the RMS value is less
because of the reduced effect of the shearing action of the terrain roughness
(Jan, 1982).
3.1.1.4.2 Integral Scales of Turbulence. The spatial size of a gust
acting on a building or structure is called the Integral Scale of Turbulence.
The chances of a small building or a structure being engulfed by a gust is
higher than for a tall or massive building or structure.
Technically speaking, the integral scale of turbulence is a measure of
the average size of the turbulent eddies. The eddy wave length is a measure
of eddy size and is defined as A. = U/n, where U = wind speed,
n = fundamental natural fi-equency of periodic fluctuations, and k=2K/X is the
eddy wave number.
In all, there are nine integral scales of turbulence, corresponding to
the three dimensions of the eddies associated with the longitudinal,
transverse, and vertical components of the fluctuating velocity, u, v, and w.
These quantities are defined as: Lux, Luy, Luz; Wx, Wy, Lvzl and Lwx, Wy,
Lrsvz. If the direction of wind flow is taken along X direction, then the
integral scales in the Y and Z directions associated with the u-component of
velocity (along the X direction) are about one-third and one-half the integral
scale in X direction, respectively (Simiu and Scanlan, 1986).
18
3.1.1.4.3 Spectra of Turbulent Velocity Fluctuations. Since the wind speed
fluctuates randomly, its fluctuating properties need to be considered in
statistical terms. A complete representation of the fluctuating component of
wind is the gust spectrum, which gives the distribution of the mean square
speed over the frequency domain. The gust spectrum is helpful in
determining the dynamic response of a structure. The wind speed spectrum
illustrated in Figure 3.5 is obtained from wind measurements in an open
field in Lubbock, Texas. Its general shape is t5T)ical of the winds measured
at other locations. The graph in Figure 3.5 indicates that the wind speed
fluctuates at all frequencies between 0.0005 and 5 cycles per second (Hz)
The corresponding periods of the fluctuations are from 2000 to 0.2 seconds.
The graph also illustrates that there is much more energy in the spectrum at
a frequency of 0.05 Hz than at a frequency of 0.5 Hz. The energy at
frequencies larger than 1.0 Hz is negligible.
In dynamic analysis of a structure subjected to gust loading, significant
dynamic amplification of response can occur at the resonance frequency, i.e.,
when a natural frequency of vibration of the structure falls in the range of
strong wind fluctuation. For example, if a structure has a frequency of
vibration of 0.1 Hz (a fundamental period of 10 seconds), there can be
significant dynamic amplification of the response, because the fluctuating
component of the wind has a fair amount of energy at that frequency, as
shown in Figure 3.5. On the other hand, if the natural frequency of vibration
of the structure or one of its component is higher than 1 Hz (the fundamental
period is less than 1 second) the dynamic amplification of the response will
be neghgible because the energy in the wind speed spectrum at these
frequencies, as shown in Figure 3.5, is extremely small. This consideration of
natural frequencies justifies the apphcation of wind loads as quasi-static
loads on most structures and structural elements, rather than as dynamic
Since the basic wind speeds in ASCE 7-95 are greater than in ASCE 7-
88, the recommended values of the gust response factor, GRF, are lower. In
fact, the three-second wind speed is not far below the maximum
instantaneous wind speed in a record, and if a structure is fairly stiff or
"rigid" (that is, its fundamental natural frequency is high relative to the
frequency content of the wind), then the maximum response of the structure
will basically be a static response to the peak wind pressure associated with
this three-second wind speed. Then the gust response factor should be less
than 1.0, depending on the size of the structure. This value is lower than the
typical value of 1.3 to 1.5 when using a fastest mile wind.
Table 3.4 Values of Wind Parameters in ASCE 7-88 and ASCE 7-95
Expo.
A
B
C
D
old a
1/3
1/4.5
1/7
1/10
new A
a 1/5
1/7
1/9.5
1/11.5
A
b
0.64
0.84
1.00
1.07
a
1/3.0
1/4.0
1/6.5
1/9.0
b
0.30
0.45
0.65
0.80
c
0.45
0.30
0.20
0.15
Kft:)
180
320
500
650
e
1/2.0
1/3.0
1/5.0
1/8.0
^min
(ft)
60
30
15
7
There are two provisions in ASCE 7-95 for determining the gust
response factor for "rigid" structures. In the first provision, called the
"simpUfied method," no detailed calculations are required, and GRF is simply
taken as 0.8 for Exposures A and B and 0.85 for Exposures C and D. This
option is appropriate for relatively small structures which can be completely
engulfed by the size of a 3-second gust. In the second provision, called the
"complete analysis," the GRF is calculated taking into account the turbulence
intensity and integral scale of the wind and the size of the structure, as
follows:
36
GRF = 0.9 (1^7I,Q) (1 + 71,)
(3.11)
where z is the so-called equivalent height of the structure, Ij is the
turbulence intensity in the wind at that height:
^ooa/6 (3.12)
I-=d z 33
I z )
and Q represents the background root mean square (rms) response of the
structure to the wind as affected by the ratio of the structure's size to the
integral scale of the wind:
Q^ = 1
1 + 0.63 V
b-t-h ^0.63^
(3.13)
with b and h representing the width and height of the structure, respectively,
and Lj representing the integral scale of the wind (a measure of the spatial
extent of the gusts):
U =t .33>
(3.14)
The values of c in Equation 3.12 and I and e in Equation 3.14 depend on the
exposure and are given in Table 3.4. Typically, z is taken as 0.6(h) for a
building in ASCE 7-95.
37
Note that the GRF fi-om Equation 3.11 is insensitive to I-, which is
generally in the range of 0.15 to 0.25 during strong wind events. If I, is
equal to 0.143, for example, then 71- is equal to 1.0, and Q must be at least
1.22 for the GRF to exceed 1.0.
Note also that the frequency of the structure is not considered in these
equations. In Equation 3.11, the gust response factor is lowered by a
reduction in correlation of wind-induced loads which act over larger surfaces,
but the dynamic response of the structure is not considered. The factor 0.9 in
this equation is used to calibrate its results to those of ASCE 7-88 for rigid
structures.
When a structure is considered to be dynamically sensitive to the
wind, on the other hand, ASCE 7-95 states that any established "rational
method" may be used to evaluate GRF. In this case the gust response factor
accounts for the dynamic characteristics of the structure as well as the size
effect. In principal this statement allows the use of amy old or new method of
dynamic wind analysis, but the value of GRF must relate to the three-second
wind speed if ASCE 7-95 is being used and Figure 3.4 is employed to
determine the basic design wind speed.
38
CHAPTER 4
DESIGN OPTIONS FOR DYNAMICALLY SENSITIVE
STRUCTURES
The gust response factor, GRF, is a factor on the static wind pressure
or force to be used in designing a structure. Its role is seen most
fundamentally in the equation for the total force on a structure:
Force = -pV^' * A * GRF * C (4.1)
where p is the mass density of the air, V^ is the wind speed at height z, A is
the projected area exposed to the wind, and Cf is the force coefficient. If V2
in Equation 4.1 represents a three-second gust, it will be larger than the
corresponding fastest mile wind, ten-minute average wind, or mean hourly
wind. Thus, to obtain a design force comparable to that for one of these other
reference v dnd speeds, the gust response factor in Equation 4.1 must be
smaller when using a three-second reference wind speed, V2. In particular,
to make the total force the same when using a three-second gust, V3.sec, as
when using a fastest mile wind, V ^ , with everything else equal, the ratio of
the two gust response factors would have to be:
GRF ( \7 \
3-sec
G^^fin
V fin
V 3-sec>/ (4.2)
where the subscript "fin" stands for "fastest mile wind." At the reference
height of 33 feet, for example, and for a fastest mile wind of 90 mph in a
hurricane zone, the ratio in Equation 4.2 comes out to be (1/1.21)2 = 0.683.
Thus, if the GRFfm as determined by the previous standard were a typical
value such as 1.4, the value of GRFs.gec would be (0.683)(1.4) = 0.956 to
39
produce the same design force. This result shows in another way that gust
response factors associated with the 3-second gust winds of ASCE 7-95 may
be as low as 1.0 or less.
One option in the design of a dynamically sensitive structure under
ASCE 7-95 is to use an established design method based on a different
reference wind speed (fastest mile, 10-minute average, or mean hourly) and
simply convert the resulting GRF according to Equation 4.2. Such a
manipulation was presented in the ASCE Guidelines for Electrical
Transmission Line Structural Loading" (ASCE, 1991), where a value of the
wind speed correction factor Ky = 1.21 was used to convert fi-om the 10-
minute average wind utilized in the method of Davenport (1979) to results
for a fastest mile wind.
A further theoretical consideration in regard to the three-second wind
speed is that three seconds is too short a duration for the probability-based
peak factors common to all the well-established methods of dynamic analysis
to be valid. In other words, one cannot estimate a peak dynamic response as
the mean (static) response during the three seconds plus gg = 3.5 or so times
the RMS response during that era. All of the design methods currently in
use rely on the concepts of frequency response analysis, and the ratio of the
peak in the time history to the RMS value of that record as estimated by
extreme value statistical techniques presented by Rice (1944) and Davenport
(1964). These techniques assiune, however, that the time over which the
mean and RMS quantities are calculated is of sufficient duration for a
number of peaks to occur so that a probabihty distribution of those peaks can
be formulated (Davenport, 1964). In three seconds, these assumptions
cannot be satisfied, so some longer time period has to be considered in
accounting for the dynamic response of the structure. Then a conversion to
the three-second basis can be made.
40
In some national standards and codes a time period of one hour is
used. However, this duration is inappropriate in a hurricane region because
of the rate of movement of the wind field. Studies of hurricane events
(Krayer and Marshall, 1992) indicate that no more than ten minutes should
be used as an averaging time. Either a ten-minute (600 seconds) duration or
the variable 30 to 40 second duration of a fastest mile wind in the hurricane
range (90 to 120 mph) would be appropriate for a dynamic analysis using the
frequency response method and a probabilistic peak factor.
41
CHAPTER 5
POLE, CONDUCTOR AND GROUNDWIRE DESIGN DATA
Wood, concrete and steel single poles are commonly used for
transmission lines. In this study, only concrete poles are considered since the
use of wood poles is less prevalent in the industry because of the cost of the
wood and low fiber stresses. Also, most of the wood poles and some steel
poles used in practice are guyed and hence require separate analysis. The
supports have a range of pole heights and conductor spans as described
below. The conductors are generally of the three-phase type, meaning there
are three conductors on each line, not bundled. The three conductor locations
considered on the tower are 11 ft., 19 ft., and 27 ft. below the tip of the pole.
Groundwires are at one-half foot from the tip. The conductors are fi-om 1.0 to
1.5 inches in diameter and weigh fi-om 1.0 to 1.6 pounds per linear foot,
depending on the electrical load they must carry, and a have force coefficient
of 1.0. The overhead ground wires are typically 3/8-inch in diameter and
weigh about 1/4 pound per linear foot and have a force coefficient of 1.2.
5.1 Concrete Poles
5.1.1 Static-Cast Concrete Pole
The concrete poles considered are of two types and are approximately
115 feet long and stand 70 to 100 feet above the ground. One type is a "static
cast" pole, which has a tapered-square outside shape that includes a solid
cross-section in the upper 40 feet and a round hollow opening below that.
The tip is 1.0964 ft. square and the outside dimensions taper outward by one
inch per 6 feet toward the bottom. This pole has a force coefficient of 1.6 for a
90 degree wind angle of attack.. The hollow portion has a least wall
thickness (at the sides) of 4.5 inches (see Figure 5.1). 6,000 psi concrete is
commonly used for this static-cast pole.
42
5.1.2 Spun-Cast Concrete Pole
The other type of concrete pole is a round "spun cast" pole that is cast
by placing 7-wire prestressing strands in a fixture along with wet concrete
and then rotating the fixture about the longitudinal axis so that the concrete
is thrown by centrifugal force toward the outside, where it solidifies before
the prestressing forces are released. The concrete thickness is at least 3
inches, and it can go up to 4 or 6 inches for the most heavily loaded poles.
The tip diameter is 1.0567 ft. and tapers 0.216 inches per linear foot outward
toward the bottom. This pole has the force coefficient of 0.8. This pole can be
as long as 130 feet and stand 85 or more feet above ground. 8,000 psi
concrete is commonly used for this spun-cast concrete pole.
The insulators by which the conductors are considered to be connected
to both towers are the same types of porcelain insulators as used throughout
the US. Insulators are typically 8 ft long for 230 KV lines and 5 to 6 ft long
for 138 KV lines. Sometimes they are braced.
Typical spans between supports are 550 to 750 feet for 230 KV lines
(with an average of 650 ft), and 350 to 550 feet for 138 KV lines (with an
average of 450 ft). Sometimes the poles for a given span are much higher
than for other spans because of the clearances required.
These poles are "wind sensitive," since their fundamental frequencies
are close to or below 1.0 Hertz. According to ASCE 7-95, any structure with a
fundamental frequency below 1.0 Hz should be considered to be wind
sensitive. Typical pole dimensions of each type analyzed are shown in Figure
5.1. These are considered to be prototype or "baseline" examples of the two
poles.
Fundamental frequencies and flexibiUty coefficients were calculated
for different spans and heights. Flexibility coefficients are helpful in
deflection calculations presented in example calculations of the spun-cast
concrete pole. The natural frequencies and mode shapes for both the poles
43
7f
O O T
CP
^''~•M
" (MJNJIMUM) r
I M UJ
±
8000
150
0= v0^67
-TAPER =
Spun-Cast Concrete Pole
>^ n-o(^G/isa^ 7^
o
±. m
^^ 4-5
fMlKIJMUM)
/ ^
I
. I
f c - ' 6 0 0 0
Static-Cast Concrete Pole
g
O
Ii
h X
X
Figure 5.1. Typical Properties of Concrete Poles
44
were computed using the finite element program CDA/SPRINT, with careful
modeling of the tapering of each pole and are shown in Table 5.1.
Table 5.1 Properties of Baseline Static-Cast and Spun-Cast Concrete Poles (Units: Ft. and Lb.)
Height
70, 85,
100
Span
550
650
750
Sag of
GW
4.5
6.25
8.25
Sag of
Cond.
10.0833
15.5417
19.3750
Damping
in Tower
0.01
0.03
0.05
Damping
in Cond.*
0.20
0.40
0.60
Damping
in GW.*
0.40
0.40
0.40
* Damping values are for the ASCE 7-95 Commentary Method. Damping in Conductors and Ground Wires is calculated in case of Davenport's Model.
45
Table 5.2 Results of SPRINT Analysis for Concrete Poles
Type of
Concrete
pole
I. Static-
Cast
70 ft.
84 ft.
100 ft.
II. Spun-
Cast
70 ft.
84 ft.
100 ft
Natural
Frequency in
Hz.
0.886914
0.688640
0.551696
1.144760
0.921003
0.740436
Flex, coeff for
Wind on
Tower
2.2896e-05*
4.2000e-05*
6.5234e-05*
8.1486e-06*
1.3216e-05*
2.2554e-05*
Flex, coeff for
Wind on GW.
3.6221e-04**
4.7841e-04**
6.3604e-04**
2.6298e-04**
3.2392e-04**
4.3943e-04**
Flex, coeff for
Wind on 3-
Conductors
6.0487e-04***
8.7893e-04***
1.2500e-03***
4.2719e-04***
5.8028e-04***
8.4330e-04***
* For calculating deflections at the top of tower due to wind on tower, multiply flexibility coefficient by square of the reference wind speed at 33 feet in ft/sec to get the tip deflection in feet.
** , *** For calculating deflections at the top of tower due to wind on ground wire and conductors, multiply flexibility coefficient by force on the groundwire or conductors of the tower to get the tip deflection in feet.
46
CHAPTER 6
DAVENPORT'S MODEL (ASCE, 1991)
6.1 Introduction
The load determination model developed by Davenport (1964, 1979) is
now well established in the transmission line industry, having been
incorporated into ASCE's guidelines for the design of transmission line
systems (ASCE, 1991) and discussed in some detail by other references such
as EPRI, 1987. Therefore, a detailed development of the underlying theory
will not be presented here, but the equations and their assumptions will be
given along with an example for comparison with Simiu's model and the
ASCE 7-95 Commentary model considered.
The key relationships for Davenport's model, as well as for the other
two models, are between the spectra of the wind and the dynamic structural
response and between the root mean square (RMS) value of the dynamic
response and the peak response. The estimate of the area under the
response spectnma is used to calculate the RMS response, and a statistical
"peak factor" is then used to determine the expected maximum instantaneous
value, or design value, of this d5niamic response. The peak dynamic response
is then related to the static response (under the mean wind) in developing a
gust factor to be applied to the static response.
Another key point is that Davenport's model deals expUcitly with the
effects of the wind on the conductors or "wires" and wdnd on the supporting
tower or pole. Other models must be adapted to account for these distinct
effects. Davenport even incorporates a "separation coefficient" related to how
the conductor and tower effects combine. In general this coefficient depends
on the degree of separation between the fundamental frequencies of the
structure and the conductors, but Davenport suggests a fixed value of 0.75,
based on typical degrees of this separation. Davenport's model was originally
47
formulated with respect to a 10-minute averaging time for the wind, but it
has been adjusted to a fastest mile wind in the ASCE Guidelines (ASCE,
1991) and can be adjusted to other reference winds with proper care.
6.2 Notation (ASCE. 1991)
The following symbols are used in the Davenport's Model
Bt, Bw = dimensionless term for the area under the response spectrum due to
the quasi-static "background" wind loading on the structure (t for
tower), conductors (w for wires);
Cf = force coefficient for the conductors;
D(z) = pole diameter at height, z, feet;
d = conductor diameter, feet;
E = exposure factor evaluated at the effective height of the conductors or
structure, ZQ;
f|., f^ = fundamental frequency of the free-standing structure in the
transverse direction, of horizontal sway of the conductors, in Hertz;
gg = statistical peak factor dependent on the frequency characteristics of
the response (the moments of the response spectrum) and the 10
minute sampling interval of the wind, taken as 3.5 to 4.0 with a
suggested "typical value" of 3.6;
Gt, Gw = gust response factor for wind on the structure (tower), on the
conductors (wires);
h = total height of the structure above ground;
Ky = ratio of the reference wind speed used (such as fastest mile wind) to
the 10-minute average vrind speed in open country (exposure C) at
the 33-ft reference height;
L = span of the conductors between supporting structures;
48
Lg = transverse integral scale of the wind turbulence;
Rt, Rw = dimensionless term for the area under the response spectrum due to
the partial resonance of the structure, conductors;
S = conductor sag at midspan;
Sx(z) = pole section modulus at height, z, feet cubed;
V = design vnnd speed at the 33-foot reference height, in mph;
VQ = 10-min average wind speed at the effective height of the structure
and conductors (note that for VQ one effective height is assumed for
the system as a whole);
X = along-wind deflection;
Zg = gradient height of the atmospheric boundary layer;
ZQ = effective height above ground of the structure (0.6h) or conductors
(2/3 the height of the structure from the ground up to the attachment
points of the insulators, if used, minus one-third the sum of the
insulator length and the conductor sag) (Note: while different ZQ
values are defined for the conductors and the tower, a single value of
0.6 times the tower height has been used in published examples
[Davenport, 1979, EPRI, 1987]);
a = power law coefficient;
e = approximate coefficient for the separation of the conductor and
structure response terms in the general gust factor equations, taken
as 0.75;
K = surface drag coefficient for determining the exposure factor, E;
c = tower stress;
^t' ^w = fi*action of critical damping for structure, conductors (due to
aerodynamic damping for conductors).
49
6.3 Equations
The Davenport equations, using the above symbols, are as follows. For
the gust response factors.
_(l-hg,eEVB,-hR,)
K. G. = ^ ^-^irf ^ (6.1)
_ (l + g^eEVB^Ti:)
K, Gt=- ^ T V -• (6.2)
In Section 2.5.1 of the ASCE Guidelines, simplified versions of these
equations are given. Taking the suggested values of gg = 3.6, e = 0.75, and
Ky =1.2 (this last for a single fastest mile wind speed of 70 miles per hour),
and assuming that the resonance terms R^ and R can be neglected, the
equations simplify to:
G, = 0 . 7 - h l . 9 E ^ (6.3)
G, = 0.7-hl.9EVB^. (6.4)
As mentioned earlier, the separation factor, e, is unique to Davenport's
formulation and is taken as 0.75, based on the fact that conductor and
structure fundamental frequencies are usually separated by 0.5 to 1.0 Hz.
Some judgment could be used in adjusting this factor for a smaller or larger
frequency separation. The statistical peak factor, gg, is based on studies by
Rice (1946) and Davenport (1964) for stationary records of a given duration,
and a value of 3.5 or 3.6 is accepted in all of the methods discussed herein.
50
The exposure factor, E, is related to the type of terrain at the site and the
effective height of the structure, as follows:
E = 4.9>/ic" ^33^
\^o y (6.5)
where k is the surface drag coefficient. Its values for exposures A, B, and C
are given in Table 6.1.
As indicated above, Ky is the conversion factor fi-om the results for a
10-minute average wind speed to another basic design wind speed. In the
ASCE Guidelines, the alternate design wind speed is the fastest mile wind,
for which the following empirical equation is used for Ky:
Ky = 0.81V0.09 (6.6)
where V is the fastest mile wind speed. The approximation is satisfactory
only for values of V between 20 and 110 mph. This empirical formula or one
like it (EPRI, 1987) is needed because the averaging time varies for fastest
mile winds. For converting to another basic wind speed such as a 3-second
gust, a different value of Ky must be determined from the one of the curves of
Figure 3.4. The Ky for converting from Davenport's 10-minute basis to a
three-second gust basis in a hurricane zone is a fixed value of 1.546.
The remaining terms in Davenport's equations are the dimensionless
background and resonance terms for the RMS response. They are made
dimensionless essentially by dividing the total response (static due to the
mean wind plus dynamic due to fluctuations about the mean) by the static
response. Thus in each of the Equations 6.1 and 6.2, the static part is
represented by the constant "1.0" in the initial term and the dynamic part is
51
represented by the second term. Making the second term dimensionless is a
process that depends on the ratio of the standard deviation of the wind speed
to the mean wind speed, called the "turbulence intensity" of the wind. The
turbulence intensity could appear in the equations and be given
representative values in Table 6.1 for different exposures. Instead, in
Davenport's equations the relationship between the mean wind and the
standard deviation is taken care of by the exposure factor, E, and empirical
equations for the background contribution, B, and the resonance
contribution, R.
Table 6.1 Parameters for Use in Davenport's Equations
Exposure
Category
B
C
D
Power Law
Coefficient, a
4.5
7.0
10.0
Gradient
Height, Zg (ft.)
1200
900
700
Surface Drag
Coefficient, k
0.010
0.005
0.003
Turbulence
Scale, Lg (ft.)
170
220
250
The empirical equations for the two quasi-static background terms are
as follows:
B„. = r r \
(6.7)
l-hO.8 vLsy
B.= r u\
(6.8)
1-1-0.375 vLsy
52
In each case the value of B depends entirely on the ratio of the length of the
slender member (span L for the conductor and height h for the tower) to the
turbulence scale, Lg, of the wind. Davenport's values of Lg are given in Table
6.1 for different exposures. Both B terms are unity for a very short span or
very short tower, but generally they drop somewhat below unity for typical
conductor spans and tower heights, as shown by the plots in Figure 6.1. For
example, for an open country exposure C, Lg fi-om Table 6.1 is 220 feet. If the
conductor span is 450 feet, then B^ is only 0.379, and if the tower height is
80 feet, then Bt is 0.880.
It may be noted that the nondimensional backgroimd terms B^ and B^
in Davenport's formulation correspond to the term Q 2 in ASCE 7-95. The
three expressions have somewhat similar forms, but they do not correlate
perfectly because of the existence of other terms in the equations for G where
they appear. In particular, the gust response factor, gg = 3.6, the separation
factor, e = 0.75, and the exposure factor, E, multiply - B^ in Davenport's
Equation 6.3, whereas 2gg = 7 and the turbulence intensity, Ij multiply
VO^in the ASCE 7-95 Equation 6.1. The effects of these different
representations of the "size effect" for transmission line structures are among
the differences in method to be examined in this study. For comparison, Q 2
is shown in Figure 6.1 along with B^ and B .
53
Background Response Terms
I I H I I I I I I I I I I I I I I H I I I I I I I I H I I I I I I I I MM MM < > ^ C N J c O G q > p ^ C M I ^ c O O ^ d ^ oi CO rj iri "d i < o d c >
LAs or h/Ls
Figure 6.1 Davenport's Background Response Terms as Functions of the Size Ratio (Exposure C, L=450', H=80')
In Davenport's simplified gust response Equation 6.4, by taking the
effective height of the tower, ZQ, as two-thirds times h, both E and Lg are
fixed for a given exposure and G can be represented in a single plot versus h
with family curves for the different exposures B, C, and D. This plot is given
below as Figure 6.2. On the other hand, the conductors may have
independent values of span, L, effective height, ZQ, (due to tower height
variations), and exposure, so curves for G , from the simplified Equation 6.3
must be plotted versus L for different exposures with ZQ as a family
parameter. This type of plot is given below as Figure 6.3.
The remaining terms in Davenport's detailed gust response Equations
6.1 and 6.2 are the resonance response terms R^ and R , which are given by:
54
1.70
a o » -
12 bJ if) Z o Q. in UJ
oc
tn O UJ
a:
u
I 60
I 50
1.40
1.30
1.20
1.10
t.OO
^ B - * ^
C ^ a ^
• * ~ o ^
EXPOSUf
1
\Z CATEGORY
40 60 eO 100 120 140 160 160 200
TOTAL STRUCTURE HeCKT (FEET)
Figure 6.2 Davenport's Gust Response Factor for the Tower (Simplified Equation)
I 50
XT. O
1.20
S UJ ifi z o 0 . ifi UJ
3 O
UJ
I 10
1.00
.90
.80 200 400 600 eOO 1000 1200 1400 1600 1600 2000
DESIGN WIND SPAN (FEET)
Figure 6.3. Davenport's Gust Response Factor for the Conductors (Simplified Equation)
•^^
0.0113 rr . \-"'r Cz„
V u (6.9)
"•-k 0.0123 (6.10)
where ^y^ and ^ ^ e the fi^actions of critical damping in the conductors and
the tower, respectively, L is the span of the conductors, f^ and f are the
fundamental natural frequencies of the conductors and the tower,
respectively, ZQ is the effective height, VQ is the windspeed (10-minute
average) at the effective height ZQ:
r. \ V„ = 1.605
\^g J
I/O
^88^^"^ 60 V o u y \ ^ A , ; K.
(6.11)
gmd the terms f ZoA g ^^^ t o o ^ ® called "reduced fi-equencies" of the
conductors and the tower, respectively. The derivations of these equations
are given in Davenport (1979), following results by Manuzio and Paris
(1964), Castanheta (1970), and Ohtsuki (1967). In Equation 6.11, the first
part converts the wind speed fi-om the reference height to the effective height
of the structure, ZQ, the middle part converts from miles per hour (mph) to
feet per second (fps), and the Ky factor converts fi^om the fastest mile
windspeed V to a 10-minute average windspeed.
Graphical representations of various spectra are helpful in
understanding both the background and resonance response terms B and R.
In Figure 6.4 fi-om Davenport (1979), part (a) represents the spectrum of the
wind, part (b) represents aerodynamic admittance functions, and part (c)
56
represents the spectrum of the conductor response, and the spectrum of the
tower response. The area under the wind spectrum is the mean square of the
fluctuating component of the wind speed, and the total area under each
shaded curve is the mean square of the respective response. The differences
between the dashed lines and the solid lines in part(c), disregarding the
narrow superimposed peaks, represent the effects of the aerodynamic
admittance functions, which depend primarily on size effects. Each shaded
area is composed of two parts, the resonance peak area, E^R, which occurs in
the vicinity of the fundamental frequency and is strongly dependent on the
damping factor, and the area for the background or quasi-static response,
E^B, which differs from the wind spectrum area only as affected by the
aerodynamic admittance function. Together these areas constitute the total
mean square of the fluctuating response. The peak dynamic response then is
taken as ggC times the RMS response, or g eEVB -i- R .
The so-called "separation factor" is defined by the approximation:
(A2 + B2)1 /2 = E (A -I- B )
where A and B are of similar magnitude and e = 0.75. This equation is used
to combine the two mean square values into one simpler expression without
squares or square roots. In other words, it allows A and B to be added
directly even though, as probabilistic quantities, they actually should be
combined as the square root of the sum of their squares. The above
expression can be written as follows in terms of A, B, and e:
57
Spectra, f SJ^ ) /V
S^ ( f ) = Power Spectral Density at Frequency f
- 2 /3
Frequency
(a) Spectra for Horizontal Wind Velocity (Horizontal Turbulence)
Admittance
ODnductor
f f. c 't Frequency
Admittance
Tower
J .
f f i c 't Frequency
(b) Admittance Functions for Conductor and Tower
Conductor
(c) Response Spectra for Tower and Conductor
Figure 6.4 Spectra of Wind Speed, Conductor Response, and Tower Response (EPRI Report, 1987)
5K
A J I + m =" r A
\
e = (6.12)
If we assimie different ratios of B/A, the values of the separation factor , e,
shown in Table 6.2 are obtained.
Table 6.2 Separation Factor, e, for Different Ratios of B/A
B/A
1.0
1.2
1.5
2.0
3.0
5.0
10.0
50
oo
Separation Factor, e
0.7071
0.7100
0.7210
0.7445
0.7906
0.8500
0.9140
0.9800
1.000
Table 6.2 shows that if the dynamic portions of the GRF values for the tower
and conductors have a ratio between 1.0 and 3.0, then his recommended
value of e of 0.75 is justified. This ratio is examined in the studies to follow.
It may be noted that the resonance peak in part (c) of Figure 6.4 for
the conductor spectrum is shown quite a bit farther to the left; than that for
the tower spectrum, since the fundamental frequency of the tower is expected
59
to be considerably higher than that of the conductors. The tower frequency
should be determined by a detailed structural analysis.
Davenport (1979) recommends that the sway frequency of the
conductors be calculated from the pendulum formula with the effective
length of the pendulum taken as two-thirds the sag:
^ 12^JV2// (6.13)
Here g is the acceleration of gravity, in feet per second squared, and S is the
conductor sag, in feet. This formula is approximately equal to Vl/S (EPRI,
1987).
Damping levels are nearly always difficult to estimate in structures.
Estimates of the tower damping factor generally range fi-om 2 to 5 percent,
although values fi-om 4 to 8 percent are mentioned in the ASCE Guidelines
(ASCE, 1991). The structural damping in the conductors should be equally
small or smaller, but the aerodynamic damping of the conductors in a strong
wind is considerable. Davenport (1979) uses the following equation to
estimate the aerodynamic damping of the conductors and neglects the
structural damping by comparison.
f ^j r^ \
^^ = 0.000048 v„c,
vUci/12), (6.14)
where d is the diameter of the conductor in inches.
It may be worth noting that in analyzing field data fi-om the Moro Test
Site in Oregon, Kadaba (1988) found conductor aerodjmamic damping factors
ranging from 0.2 to more than 0.6, whereas Equation 6.14 generally gives
values in the range fi-om 0.2 to 0.4. Equation 6.9 shows that R^ is strongly
60
dependent on , so this estimate is important. If Equation 6.14 is used to see
what the aerodynamic damping of a single pole tower is, a value of the order
of 0.01 is obtained.
6.4 Example Calculations for a Spun-Cast Concrete Pole
Sample calculations by Davenport's model (ASCE, 1991) are presented
in this subsection for the 84-foot tapered spun-cast concrete pole of Figure
5.1. Its material properties and fundamental natural frequency are given in
that figiu-e. The outside diameter at the top is 1.0567 feet, and it tapers
outward at the rate of 0.018 feet per foot of length. The mean thickness the
wall is 0.25 feet. The wind drag coefficient is assumed to be 0.8. The
damping factor for the tower is assumed to be 0.03 (0.02 fi-om structiu-al
damping and 0.01 fi-om aerodynamic damping). The three conductors are
attached at distances of 11, 19, and 27 feet, respectively, fi-om the top. The
groundwire is attached at a distance of 0.5 feet from the top. It may be noted
that the fundamental frequency of the pole is calculated without the
conductors and groundwire attached and with an assumption of perfect fixity
at the base. Some realistic flexibility of the foundation would make the
frequency less. The pole is assumed to be in open country (Exposure C) in a
part of Florida where the design 3-second gust speed is 140 mph fi-om Figure
3.13. The span of each conductor is 650 ft;, its diameter is 0.11892 feet, and
its sag is 13.5417 ft;. The span of the groundwire is 650 ft, its diameter is
0.0313 feet, and its sag is 6.25 ft;. The force coefficients for the conductors
and groimdwire are taken as 1.0 and 1.2, respectively.
The calculations for the gust effect factor, maximum tower deflection,
and maximum tower stress are easily performed with these data, either by
hand, by a spreadsheet, or by a computer program. All three methods have
been used for checking purposes.
61
The first calculations by Davenport's method (ASCE, 1991) are used to
determine the gust effect factors for wind on the pole, wind on the
conductors, and wind on the groundwire. Then the calculations are extended
to determine the associated maximum fiber stresses and tip deflections under
the design wind. Fiber stress is assumed to be the normal design criterion
for the poles; tip deflection is added to help provide understanding of the
associated calculation steps and the overall structural behavior. In some of
the methods considered in this study, deflections are readily determined and
stresses take a certain amount of extra work and insight. In Davenport's
model stresses are readily determined and deflections require the extra work.
Deflections are calculated using uncracked concrete section properties.
Accurate accounting of cracked section properties would require a separate
analysis at each level of the tapered pole as well as more detailed information
about the pre-stressing strands and their tensions than is currently
available. Also, how cracked section properties would combine with
uncracked section properties would depend on the moments at different
levels of the pole and would thus vary from case to case. Finally, during
dynamic response the pole would be oscillating between cracked and
uncracked stages, and the effects of these changes would vary with the
amplitude of the motion, making the frequency and mode shape analysis
non-linear (amplitude dependent) as well as complicating the frequency
domain analysis
6.4.1 Summary of Input Data
The following data are basically the same for all methods, but change
slightly according to the parameters required by the method. The spun-cast
pole considered is the "baseline" or reference case for the sensitivity studies
of Chapter IX, where variations in tower height, conductor span, and tower
damping are examined.
62
Pole: Height, h 84 ft,
Diameter, D 1.0567 ft at the top
Taper of diameter out from top 0.018 ft/linear ft
Mean thickness of the wall 0.25 ft
Fundamental frequency, ft 0.9210 Hz.
Fraction of Critical Damping, Ct0.03
Force Coefficient, Cf 0.80
Weight density of material, p 150 Ib/ft^
Modulus of elasticity of material, Et 7.8083 x 10^ psf
Flexibility coefficient:
(tip deflection = 1.3216 x lO'^ ft per ft/sec
due to 1 ft/s wind on tower)
Conductors: Span length, L 650 ft
Diameter, d 0.11892 ft
Sag, S 13.542 ft;
Force Coefficient, Cf 1.0
Flexibility coefficient:
(tip deflection = 5.8028 x lO'^ ft per lb
due to 1 lb force at conductor level)
Groundv rire: Span length, L 650 ft
Diameter, d 0.0313 ft
Sag, S 6.25 ft;
Force Coefficient, Cf 1.2
FlexibiUty coefficient:
(tip deflection = 3.2392 x lO'^ ft; per lb
63
due to 1 lb force at groundwire level)
Wind Field: 3-Second gust speed, V, f 140 mph
Wind speed conversion factor, Ky 1.546
Mass density of air, p^ij. 0.0024 slugs/ft^
Site Exposure Category C (open country)
(see Table 6.1 for Lg, K, a, and Zg)
6.4.2 General Calculated Values
The following values are applicable for the tower, the conductors, and
the groundwire :
For Tower:
Average outside diameter: = 1.0567 -»- 0.018 x 84/20
= 1.8127 ft r^^h-r
Average hollow core diam. = 1.8127 - 2 x 0.25
= 1.3127 ft
Circular natural frequency = 27cft = 27r(0.9210)
= 5.787 rad/s
Equivalent height of the tower, ZQ = 0.65 x h
= 0.65x84 ft =54.60 ft
For Wind: A
10-minute average wind speed (mph) = Vj-ef/Ky = 140/1.546
= 90.56 mph
Exposure Factor, E, for height ZQ = 4.7 VK~(33/ZO)1/«
= 4.7 VOOOB (33/54.60)1/7
= 0.3093 64
Reference 10-minute windspeed, VQ,
= 1.605(zo/zg)l/a (88/60)(^ref^v) in ft s
= 1.605(54.6/900)l/7(88/60)(90.56)
= 142.84 ftys
6.4.3 Tower Gust Response Factor
Tower gust response factor is calculated as follows:
Background term, Bt
Resonance term, Rt
= l/[l-h0.375(h/Lg)]
= 1/[1 + 0.375(84/220)]
= 0.8748
= (1/Ct) [0.0123 (ft ZoA o)- ^ ]
= (1/0.03) [0.0123 X
(0.9210(54.6)/142.84)-5/3]
= 2.3358
Tower Gust Response factor, Gt = (1 + gg e E V Bt + Rt )/Kv2
= [1 -I- (3.6)(0.75)(0.3093)
xVO.8748 + 2.3358 ]/(1.546)2
1.0445
6.4.4 Conductor Gust Response Factor
Calculations for the conductor gust response factor follows:
Background term, B^ = 1/[1 + 0.8(L/Lg)]
= 1/[1-I-0.8(650/220)]
= 0.2973
Frequency, f , (simplified formula) = Vl/S = V 1/13.5417
n„n..,.-.—^ n,,-:.,. MmMii,ii,imirnH.-.-Miiiir-.-.^- •••iiiiiMrrmiDfTrr^-nn " • - • K M I I I H m " r = r * S s S " 5 :
M 10
Figure 8.1 Function
10 W-2 V"
Figure 8.2 Function J
107
2S \0 ss 40
U
10 -1
ii:u£i&ifiAElfijUi£EiiiUE§SSS^i§ia&iiiii£Usisi£ii^is§illiiillillSll >^f l i , *CK;» f taCK*B^»>««<Mj»» i t : ; *«WaB>«B»»B<. t lS i«4Be« I .SB»:3^«#CBVaKB*9»BMK*V«» i a r ;
over the range of damping factors considered. These relatively small effects
of a large percentage change in the tower damping factor are caused by the
fact that forces from wind on the conductors dominate each response, and
these forces are sensitive to the conductor damping, not the tower damping.
In the three vertically arranged segments of Figure 9.3, the effects of
tower height, h, and conductor span, L, are broken down according to the
three types of v dnd loading considered: wind load on the tower in part (a),
wind load on the conductors in part (b), and wind load on the groundwire in
part (c). How the variations in h and L affect the individual gust response
factors (the top curves) as well as the contributions to tip deflection and base
stress are shown.
The trends for deflection and stress due to wind on the three
components are all basically the same as for their combined effects as seen in
Figure 9.1. The new features of Figure 9.3 are the gust response factors for
the individual components at the top of these segments. First of all, it should
be noted that the vertical scales for the gust response factors have narrow
ranges, showing that the physical dimensions h and L do not have much
effect on these quantities. Secondly, it is seen that while the GRF increases
slightly with tower height for all three components in the modified ASCE
method, the GRF decreases slightly with tower height for the conductor and
groundwire in the Davenport model. The decrease with height in the
Davenport model is due to a decrease with height in Davenport's exposure
factor, E, according to Equation 6.5, which appears in the numerator of
Equation 6.1. In the modified ASCE method, there is a similar decrease with
height in the turbulence intensity, Ij, according to Equation 3.12, but I
appears in both the numerator and the denominator of modified ASCE
expression for the GRF, Equation 8.1. Thus there is a slight increase with
133
ECbct ofTower Height and Span of Conductor on<GRK)t
I 1
106
1 •
O M
0.9
0.85
0 8
- O — S p i n = r i ! « j ft ASCE
O fipan=5.^0 ft Dav«n i
' -A - ' Sp>n=>i60ft ASCE ]
' - X - • Sp>n«6fi0 ft Daven i
•X- •Spar , - ' ' f . j ft ASCE
I— O - -l>pan=7&0ft Davan >
70
lower Heighl i\
(!) Tower Gust Response Factor, (GRF)t
Effect of Tower Height and Span of Conductor on Deflectiona due to ^A^d on Tower only
- O — S|)iin=,S5U n ASCE
- ^ — S p a n = 5 5 0 f t Davan
A - ' i i p a n ^ U ) ft ASCE
X ' • S p a n ^ e O ft Davan
• X - -Span=750 f t ASCE
• O - -Span^TSO ft. Davan
84
Tower Haight It
(2) Tip Deflection
Effect ofTower Height and Span of Conductor on Streacei doe to NVind on Tower only
— 0 ^ & D » r i = 5 3 ' ft ASCt
— ^ — b p a r . ^ j b ' ' f". ^ avpii
•A- SipnifibO r>. AsCE
• • -X- • S p h r s D i : ft :. avar
- - X - Spsn-TS.'. f: ASCE
— -O— • bpan-TaO f ' a^an
84
Towar Helfhl ft
(3) Stress (a) Wind Loads on Tower
Effect of Tower Height and Spaa of Coaduotor OB(QRF)C
0 76
0 46
ir-. - r r r r -n-rr r -^
- . : - .8::;_
I — O — B p a n s f t S O f t ASCE
' o Spans660 ft Davan
I - - - A " S p a n ' 6 6 0 f t ASCE |
; • - X - • Span«««0 ft Davan '
I - -X- ' S p « n « 7 6 0 f t ASCE
I— - o - • Span°760 ft Davan
84
Towar Haighl ft
(1) Conductor Gust Response Factor, (GRF)c
Effect of Tower Height and Span of Coadoetor on De0ection» due to ^ b d on Condaotors oaly
- O — S p a n « 6 6 0 f t ASCE
- ^ — 6 p « n » 8 S C f t Davan '
- A ' ' Spana690 f t ASCE
• X - B p a n ^ 6 0 f t Davan
•X- -SpansTOOn ASCE
- O — • S p a n = 7 6 0 ft DB%an
(2) Tip Deflection
Effect ofTower Height and Spaa of Cooduotor oa Daflaotlona due to Wind on Condnotore oaly
- O — S p a n s 6 6 0 n ASCE
- ^ — S p o n » 6 5 0 ft Davan
•A- ' 6 p a n ' 6 6 0 n ASCE
- X - • Spans660f t Uavan
X - - S p a n ^ T M f t ASCE
O - • S p a n s 7 6 0 ft Davan
Tower n''i,fn; ft
(3) Stress (b) Wind Loads on Conductors
Figure 9.3 Response Sensitivity to Tower Height and Conductor Span Separated by Load Component for Spun-Cast Concrete Pole (Exposure C, Vror = 140 mph, Ciowe,=0.03, ASCE Method; amd = 0.4)
134
Effect of Tower Height and Span of Groundwire OD (GRF)fw
"»» \f——.
ATS"
^ ^ •J
0.65 i
I:::: Oti •
r 0
. _ . • • _ - - $ . . - • - • • -
-v:-v.-.::_:::r;5:;;_;;,,,, 84
Tow«- Haifhi. fi
• . : _• : .- _ . J!
. : : - : : _ ; X
)(M
•A--
• X • •
- X -
- o-
-S,-»»p=f550n ASCK
-Sp«n£&5<.<0 L)*^' ! !
S p a n i i j ^ n ASCt:
Spon^iSO ft Devon 1
Spans7r«r i ASCE '
•Sfiens'SOn Dev»n |
(1) Groundwire Gust Response Factor, (GRF)KW
Effect of Tower Hei|^ht and Span of Groundwire on Deflectiona due to ^Mnd on Groundwire onlv
— O — 6 p « n « * 5 0 n ASCt
O 6p»n=.S50 n Om-m,
• • - A - - 6pi in««60n ASCE
- • X- • Rr«na«60 '.1 I>iv*i>
- - X - Spw.sK*; ft .VS-.T
- -O— - S M M I T W n 'iav^n
(2) Tip Defllection
Effect ofTower Heiiri^t «nd Span of Groaadwire on Streaa** due to Wind on Groundwire onK
»60
ivso
' 660
660
460
70
«
,^^ —a -o •X
. - • A -
• • - X -
- -x-- o -
-Spana6Mft .ASCE
- 8 p a n « 6 M I t Oavva.
• Span-«60 i t ASCR
Spana6Mfl Oa«tn
•Span*7K>ft ASCE
•S|Mia'7m ft. Daven
S4
TowTT Height, ft
100
(3) Stress <c) Wind Loads on the Groundwire
Figure 9.3 (contd.) Response Sensitivity to Tower Height and Groundwire Span Separated by Load Component for Spun-Cast Concrete Pole (Exposure C, Vrei=140 mph, ^i.,uvv=0.03, ASCE Method; c.md = 0.4)
height in the GRF in the modified ASCE method but a sHght decrease in the
Davenport model.
In Figure 9.4, the effects of tower height, h, and tower damping factor,
^t, on the gust response factor, the tip deflection, and the base stress are
isolated for wind on the tower only. Here the damping factor has a stronger
effect in reducing each response than in Figure 9.2. This result is because
each response to tower loads is directly affected by the tower damping, but
these effects were overwhelmed by conductor loads in Figure 9.2.
Finally, in Figure 9.5, plots of the sensitivity of the aerodynamic
damping factors ^g^ and ^ond. to the conductor span, L, are shown. Different
sags also come into play for the different spans because the sag affects the
frequency of the wire and thus the damping by Davenport's formula. It can
be seen that as the span increases each damping factor increases fairly
significantly. All of the values for the groundwire are considerably greater
than for the conductors because of the large difference in diameter.
.Vi
001
Effect of Tower Height and Tower Damping on ( G E H .
A - -
- X -
- x -- - o -
- Ml .70(1 ASCE
-M I -70B Oe«wi
HI . e 4 1 Asce
H | . « 4 I 0*>«n
-Hl.«100« ASCE
•HI-1001LO*vwi
003
OampiiiK in Tower 0.09
(1) Tower Gust Response Factor, (GRF).
Effect of Tower Height and Tower Damping on Deflection*
due to Wind on Tower only
- A
• - X • -
- -x-
- - o -
- H i = 7 0 f t ,\SCE 1
- H i =70 ft Dnv»n
Hi =.04 ft \S(T.
H I <44 fi <)*\'«n
•Ht=100ft ASCE
• Hi X100 ft Uax'wi
(2) Tip Deflection
Effect of Tower Heif^t and T«wer Damping on Stresses due
to ^Mnd on Tower only
4000
3600
•4 3000 -. I A jawo £ 3000
l & O O •
b
1000
001
—o—Ht.«7o (I <\sci:
— O Hl.s70 n. Dawn
. . ^ • - Hl.««4ft ASCE
• - .X- • Hi.««4 ft. D«v«i
" i_.x--Hi«ioor. vsrE
• — - — . - . j . . . - . . ^ . . , . . . ; ^ i - . O - . H t = 1 0 0 f t Davn
- • • X -
0.03
Dcmpinf in Tow«r
006
(3) Stress Wind Loads on the Tower
Figure 9.4 Response Sensitivity to Tower Height and Tower Damping Ratio, lowc, Separated by Load Component for Spun-Cast Concrete Pole (Exposure C, VrH-140 mph, Span = 650 ft, ASCE Method; ...nd = 0.4)
137
Davenport Aerodynamic Damping for Conductors
adyn
amic
Dam
pin
g
a) <
028
026
0.24
0.22
0.2
0 18
0 16
0 14
0 12
0 1
550
t ^ ^ 4
—O— 70 n Pole
—A— 84 fl Pote
—X— 100 ft Pote
650
Span in Ft
750
Davenport Aerodynamic Damping for Ground-Wire
0 4 i
550
—X-
- 70 n Pote
-84 n
- icon
Pote
Pote
650
Span in FI
750
Figure 9.5 Sensitivity of Davenport's Aerodynamic Damping in Conductor and Groundwire to Span
i. s
CHAPTER 10
CONCLUSIONS AND RECOMMENDATIONS
10.1 Summary
The main objective of this study was to evaluate carefully the available
methods for designing transmission line structures as "wind sensitive
structures" in conjunction with the new 3-second gust wind speeds of ASCE
7-95 (ASCE, 1995). The key term in such a design is the gust response
factor, or GRF, which depends on the characteristics of the vidnd field and the
dynamic properties of the structure. The GRF must then be appropriately
combined with other design calculations to produce design deflections and
stresses. These quantities are important for the structural survivability and
serviceability of transmission line systems. The methods or models
considered included one published by Davenport (1979, 1991), one by Simiu
(1976, 1980), and one in the Appendix to the design standard ASCE 7-95.
The Davenport model, which was exclusively developed for
transmission line structures, was adapted herein to the 3-second gust
reference wind speed and then used as a guideline model to compare the
results from the other two approaches. Both of the other two methods were
originally developed for the calculation of gust response factors for general
types of slender or flexible structures and required some adaptation for
application to transmission line structures. Simiu's model resorts extensively
to graphs in the calculation of gust response factors. It was found that the
validity of adapting the model to transmission line structures, particularly in
accounting for wind on the conductors and ground wires, is highly
questionable because of the parameters chosen by Simiu and ranges of those
parameters in the graphs. Since these wires contribute more to the total
stresses and deflections than wind on the tower, and since the method cannot
139
be coded into a computer program, the method was used only in a limited
way to determine deflections due to wind on the tower.
Extensive calculations were carried out using both the Davenport
method and a modified version of the ASCE Commentary method to
determine gust response factors, tip deflections, and maximum stresses in a
typical concrete transmission pole supporting three conductors and a
groundwire. Calculations due to wind on the pole, the conductors and the
groundwire were included. These calculations were presented so that reader
can understand the intricacies involved in the use of the different models.
The SPRINT finite element program was used to model the example pole
with beam elements, taking into account the taper of the pole. The pole's
natural frequencies, natural modes of vibration, and flexibility coefficients
for wind on the tower, wind on the conductors, and wind on the groundwire
were determined with this program. The flexibility coefficients were then
used in the calculations by the different design methods.
The detailed step-by-step calculations clearly revealed the various
assumptions made in appl3dng the Davenport and ASCE methods to
transmission line systems. In the Davenport method, for example, a
separation factor is incorporated to account for the differences in frequency
between the tower, the conductors, and the groundwire, and an empirical
equation is used to arrive at a value of the aerodynamic damping in the
conductors and groundwires. Not so clear is the assumption in this method
as well as in the ASCE method that the peaks of the forces exerted on the
tower by the three conductors occur simultaneously. A key assumption in the
ASCE method is that the first mode shape of the pole can be modeled by a
simplified equation, but a different form of the mode shape must be used to
determine stresses due to wind on the pole.
In developing the equations for each method considered, various terms
were examined carefully. In Davenport's model, the separation factor, e, was
140
studied and his value of 0.75 was accepted since this value appears to be
close to what will occur in practice. In a similar way, in the ASCE method
the factor K that accounts for the relationship between the wind profile and
the first mode shape of the structure was examined and its expression in the
code was compared to the exact expression. Also, adaptation of this factor for
conductors and groundwires was considered.
Another benefit of the detailed calculations is that the results show the
effects of various parameters and of different aspects of each formulation.
Direct comparisons between the results from the Davenport and modified
ASCE models were presented for gust response factors, tip deflections, and
maximum stresses, broken down by contributions from wind on the tower,
wind on the conductors, and wind on the groundwire.
Results were given not only for the example problems for which
detailed computations were shown, but also for a second representative
concrete pole with different properties and for ranges of several properties of
each pole, such as tower height, conductor span, and tower damping. These
results were presented in a sensitivity study by means of tables and graphs.
10.2 Conclusions
In the basis of this work, the following conclusions are drawn.
1. Either the Davenport method as presented in ASCE's "Guidelines for
Electrical Transmission Line Structural Loading" (ASCE, 1991) or the
modified Solari-Kareem method as presented in the Commentary of ASCE
7-95 (ASCE, 1995) can be effectively used in the design of single pole
transmission line supports in conjunction with 3-second gust wind speed
maps. The modified ASCE method incorporates a "separation factor" as
proposed by Davenport.
141
2. The Simiu method (1976,1980) is not safe to apply to the design of
transmission line structures unless a means can be proven to utilize
Simiu's graphs for wind on the conductors and ground wires.
3. All three methods considered can be adapted to a wind speed averaged
over any time up to one hour with the help of either the Durst curve for
non-hurricane regions or the Krayer and Marshall curve for hurricane
regions. This is particularly important in that it means the Davenport
method can be used with 3-second gust wind speed maps.
4. For the example pole considered the gust response factors (GRFs)
determined by the Davenport and modified ASCE methods agree fairly
closely. They are generally less than or close to 1.0. The GRF values as
determined by the Davenport method are slightly higher than those
determined by the modified ASCE method for the tower, but slightly
lower for the conductors and groundwire.
5. Although the final GRF values determined by the Davenport and modified
ASCE methods are fairly close, the contributions from individual
background and resonance components do not agree closely. The
resonance terms for the conductors differ by as much as 70 percent
between the two methods. A thorough understanding of the various
rather complex equations and the differences in specified parameter
values is needed in order see where such differences between the two
methods come from.
6. The sensitivity studies show that changes in GRF, deflection, and stress
with tower height and conductor span generally follow expected patterns,
but that damping in the tower has a very small effect on the response.
One unexpected trend is that an increase in tower height slightly
increases the conductor GRF by the ASCE method but it slightly
decreases the GRF by the Davenport method.
142
7. The parameter K in the ASCE method is given by an approximate formula
in the standard, and it is insensitive to the power law exponent, d .
However, it is very sensitive to the first mode shape exponent, ^, which
was found to be approximately 1.8 for the poles considered.
10.3 Recommendations
The following recommendations are made, based on the results above.
1. The Davenport method should be used for transmission line design
because it has been specifically geared for 'line-like structures," it has
been accepted in the transmission line industry over time, and it does not
require the assumption of a first mode shape in calculating stresses.
2. A separation factor, e, between the contributions of wind on the tower and
wind on the conductors and groundwire should be used with any method
and should be evaluated more thoroughly in the future, either through
analytic studies or experimentation.
3. Since aerodynamic damping in the conductors and groundwire is very
important to the survival of a transmission line system, a more exact way
of estimating this quantity than Davenport's equation should be explored,
either theoretically or experimentally.
4. An expression for the second derivative of the first mode shape other than
the simplified one used in the standard is needed to calculate stresses in
the ASCE method. Also, a better approximation than the one adopted
herein appears to be needed.
5. For the conductors the ASCE method term for the background response,
Q2, needs further study. Davenport gives different equations for the
background terms for the tower and for the conductors, but in the ASCE
method a separate expression is not given since the method was not
designed for slender horizontally oriented structiu'es.
143
6. A new model for determining gust response factors for transmission
structures should be attempted using aerodynamic admittance functions
and ARM A (auto-regressive moving average) models to generate
appropriate wind loading time histories. Then these time histories should
be coupled with a finite element d3mamic analysis program from which
gust response factors can be extracted.
144
BIBLIOGRAPHY
American National Standards Institute, (1992), ANSI 05.1-1992: American National Standard for Wood Poles - Specifications and Dimensions, 26 pp.
American Society of Civil Engineers, (1990), "Design of Steel Transmission Pole Structures," 2nd Ed., ASCE Manuals and Reports on Engineering Practice No. 72, 103 pp.
American Society of Civil Engineers, (1991), "Guidehnes for Electrical Transmission Line Structural Loading," ASCE Manuals and Reports on Engineering Practice No. 74, 139 pp.
American Society of Civil Engineers, (1988), "ASCE 7-88: Minimum Design Loads for Buildings and Other Structures, Section 6, Wind Loads."
American Society of Civil Engineers, (1995), "ASCE 7-95: Minimum Design Loads for Buildings and Other Structures, Section 6, Wind Loads."
American Society of Civil Engineers, (1995), "Commentary for ASCE 7-95: Section 6, Wind Loads," pp. 6-30 to 6-58
ANSI A58.1, (1988), "Guide to the Use of The Wind Load Provisions of ANSI A58.1."
Cartwright, D.E. and Longuet-Higgins, M.S. (1956), "Statistical Distribution of the Maxima of a Random Function," Proceedings of the Royal Society, A, Vol. 237, pp. 212-232.
Davenport, A.G. (1961), "The Application of Statistical Concepts to the Wind Loading of Structures,' Paper No. 6480, Proceedings of the Institution of Civil Engineers. Vol. 22, pp. 449-472.
Davenport, A.G. (1962), "The Response of Slender, Line-Like Structures to a Gusty Wind," Paper No. 6610, Proceedings of the Institution of Civil Engineers. Vol. 23, pp. 389-408.
Davenport, A.G. (1964), "Note on the Distribution of the Largest Value of a Random Function vrith Apphcation to Gust Loading," Paper No. 6739, Proceedings of the Institution of Civil Engineers. Vol. 28, pp. 187-196.
145
Davenport, A.G., (1964b), "The Buffeting of Large Superficial Structures by Atmospheric Turbulence," Annals, New York Academy of Sciences, Vol. 116, pp. 135-159.
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Davenport, A.G. (1967), "Gust Loading Factors," Journal of Structural Engineering, ASCE, Vol. 93, No. ST3, pp. 11-33.
Davenport, A.G. (1979), "Gust Response Factors for Transmission Line Loading," Proceedings. 5th International Conference on Wind Engineering, J. E. Cermak, Ed., Ft. Collins, CO: Permagon Press, pp. 899-909.
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Greenway, M.E., (1979), "An Analytical Approach to Wind Velocity Gust Factors," Journal of Industrial Aerodynamics. Vol. 5, pp. 61-91.
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146
Kadaba, R.R., (1988), "Response of Electrical Transmission Line Conductors to Extreme Wind Using Field Data," PhD. Dissertation submitted to the Department of Civil Engineering, Texas Tech University, 142 pp.
Kareem, A. (1985), "Lateral-Torsional Motion of Tall Buildings to Wind Loads," Journal of the Structural Division. ASCE, Vol. I l l , No. 11, pp. 2479-2496.
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147
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148
APPEND DC A
TABLE OF SENSITIVITY STUDY RESULTS
FOR SPUN-CAST AND STATIC-CAST POLES
In the following table, results for the gust response factor, tip
deflection, and maximum stress for all of the cases considered are presented.
The first page for each combination of parameters give these quantities for
wind on the tower and wind on the conductors as well as the totals for these
components plus wind on the groundwire. The separate results for wind on
the groundwire are shown on the second page for each combination of
parameters. The first table is for the spun-cast pole and the second is for the
static-cast pole.
149
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APPENDIX B
SENSITIVITY STUDY GRAPHS FOR THE STATIC-CAST POLE
The following graphs present results for the gust response factor, tip
deflection, and maximum stress for the static-cast concrete pole considered.
These curve are the counterpart of Figures 9.1 to 9.4 for the spun-cast pole.
Basically the same trends may be observed.
162
Effect of Tower Height and Conductor Span on Total Deflections due to Wind on Tower, Conductors, and
Groundwire
. . . A -• - -X-
- - X -- . 0 -
-6pan=650 a. ASCE
-Span=660 ft.. Daven
Span=«50 ft. ASCE
Span=660 fl. Daven -Span=7B0ft. ASCE
•Span=760 ft. Daven
T'^v.er Heiffht, f
Effect of Tower Hei^^t and Conductor Span <m Total Stresses due to Wind on Tower, Conductors, and Groxindwire
8000 T
4000
Span=660 ft. ASCE
8pan=660 ft Daven.
Span^GO fl. ASCE
Span^:660 ft. Daven
6pan=7B0 ft. ASCE
Spens^60 ft. Daven
70 84
Towpr He)»;^t. ft
100
Figure B.l Combined Response Sensitivity to Tower Height and Conductor Span for Static-Cast Concrete Pole (Exposure C, Vre.(-140 mph, 6„we.,=0.03, ASCE Method; t-umi = 0.4, c,w = 0.4)
163
Effect ofTower Height and Tower Damping on Total Deflections due to \ ^ n d on Tower, Conductors, and
Groundwire
c o
c : o>
a
'- 15 f 10 T
5 4-
0.01
"'X x -0
ht=70 ft. ASCE
Ht=70 ft. Daven
Ht=84 ft. ASCE
Ht=84 ft Daven
Ht=100ft ASCE
ht=100 ft Daven
0.03
Damping in Tower
0.05
Effect of Tower Height and Tower Damping on Total Stresses due to V^^d on Tower, Conductors, and Groundwire
— a Ht=:70 ft. ASCE
- - -A- -
• - - X -- X -- - 0 -
n*/—i\j It. L /wcn. Ht=84 ft. ASCE Ht?*4 ft. Daven
-Ht=100ft. ASCE -Ht=100ft. Daven
0.03
Damping m Tower
0.06
Figure B.2 Combined Response Sensitivity to Tower Height and Tower Damping Ratio, qi.,wci for Static-Cast Concrete Pole(Exposure C, VrH-140 mph, Span = 650 ft, ASCE Method; ^c.nd = 0.4, ^ w = 0.4)
164
Effect of Tower Hcl«lit u d 8p«B of Condnetor on (ORDt
1 5 -
13 •.
i 1 I 0 -
OT -
06 -
-^— Spans650 ft Dsvvn i
-A-- S«c-=>=E.: ft ASCE •
•X - Sp«-=»;s^ r. D«>f
r - 5tf -=760(l ASCX
•o - -So- ="!.: n. o»»-
u Toww HeiftiL ft
(1) Tower Gust Response Factor, (GRF)t
Effect ofTower Helcht and Spaa of Condnetor on Deflection! dne to ^ d on Tower only
WRITE(6,*)NFREQ=# OF FREQUENCIES OF TOWER FOR DIFFERENT HEIGHTS'
WRITE(6,*)' • WRITE(6,*)' # FREQ.INHz' DO 143 I=1,NFREQ READ(5,*)N,FREQT(N,1) WRITE(6,*)N,FREQT(N,1)
143 CONTINUE
C DT,TAPT,WTH,AET,RHOT,DC,DGW,CFT,CFC,CFW—STRUCTURE C PARAMETERS C VREFALPHA,ZG,AK,ALZ,EPSI,RHOAIR,AKV,G-DAVENPORT"S C PARAMETERS C JDIV = NO. OF DIVISIONS DESIRED FOR BM. AND STRESS CALCS.
LL=0.0 D0 2222L=1,NH DO 223 M=1,NSP DO 224 N=1,NZT LL=LL+1.0 WRITE(6,*)'CASE NO =",LL t
WRITE(6,*)"HEIGHT OF THE TOWER =',AHT(L,1)
WRITE(6,*)'SPAN OF CONDUCTOR =',NSPC(M,1) WRITE(6,*)'DAMPING IN TOWER =',AZTT(N,1) WRITE(6,*)'FLEXI. COEFF.FOR WIND ON T0WER...=',FCT(L,1) WRITE(6,*)'FLEXI. COEFF.FOR WIND COND =",FCC(L,1) WRITE(6,*)'FLEXI. COEFF.FOR WIND GW =',FCW(L,1) WRITE(6,*)'FREQUENCY OF TOWER IN Hz....=',FREQT(L,l) WRITE(6,*)'SAG OF CONDUCTOR IN FT ='ASCC(M,1) WRITE(6,*)'SAG OF GROUND-WIRE IN FT =',ASCW(M,1)
C
C WRITE(6,*)'I—-CALCULATION OF TOWER C/S PROPERTIES'
DB =DT+(TAPT*AHT(L,1)) DIB=DB-(2*WTH)
170
AIB=:((DB**4)-(DIB**4))*3.14159/64 DMMM=0.0 WRITE(6,*)' • WRITE(6,*)'....Z DIA HOLLW ...MOMENT WRITE(6,*)Z=0@BOTTOM OUTSIDE DIA OF INERTIA' WRITE(6,*)' FT. FT. FT. FT^4' WRITE(6,*)' • WRITE(6,1200)DMMM,DB,DIB,AIB
C Z=0 @ BOTTOM DO 1222 K=l,JDIV DIV=AHT(L,1)/JDIV Z(K) =DIV*K PHIZ(K) =Z(K)/AHT(L,1) ADT(K) =DB - (TAPT*Z(K)) DIAI(K)=ADT(K)-(2*WTH) AI(K)=((ADT(K)**4)-(DIAI(K)**4.0))*3.14159/64 WRITE(6,1200)Z(K),ADT(K),DIAI(K),AI(K)
c WRITE(6,*)'in-—CALCULTIONS FOR BMS. AND STRESSES"
C WRITE(6,*)' FOR WIND ON TOWER '
WRITE(6,*)' WRITE(6,*)'...Z VZ F MOMENT STRESS' WRITE(6,*)' FT FT/S LB LB.FT. PSI.' WRITE(6,*)' •
C Z=0 @ BOTTOM
C CALCULATION OF WIND SPEED @ HT.'Z' ON TOWER USING POWER LAW D0 1211K=1,JDIV AA=((Z(K)-(DIV/2))/33.0)**(1.0/ALPHA) BB=VREF*88.0/60.0 VZ(K)=AA*BB
C WIND FORCE ACTING ON ONE DIVISION @ HT'Z' ON TOWER IF(KNE.1)THEN PZ(K)=0.5*RHOAIR*(VZ(K)**2.)*CFT*((ADT(K)+ADT(K-1))/2)*DIV ELSE PZ(K)=0.5*RHOAIR*(VZ(K)**2.)*CFT*((ADT(K)+DB)/2.)*DIV ENDIF
C FOR WIND ON GROUND WIRES VBARG=(((AHT(L,l)-0.50)/33)**(l/ALPHA))*VREF*88.0/60.0
FGW=0.5*RHOAIR*(VBARG**2.0)*DGW*NSPC(M,1)*GEFW*CFW WRITE(6,*)F0RCE AT THE TOP OF TOWER =',FGW WRITE(6,*)'DUR TO WIND ON GW.' AMGO=FGW*(AHT(L, l)-0.5) SIGMA0W=AMG0*DB/(2*AIB* 144.0) DMMM=0.0 WRITE(6,*)' • WRITE(6,*)'...Z AMGW STRESS" WRITE(6,*)"_FT LB.FT PSI " WRITE(6,357)DMMMAMGO,SIGMAOW
DO 1244K=1,JDIV P = JDIV-K AMW(JDIV)=0.0 AMW(K) =FGW*(Z(P)-0.5) STRW(K) =AMW(K)*ADT(K)/(2*AI(K)* 144.0) WRITE(6,357)Z(K),AMW(K),STRW(K)
1244 CONTINUE
ADDING MOMENTS AND STRESSES TMO=AMCO+AMTO+AMGO TS=TM0*DB/(2*AIB* 144.0)
DO 137I=1,NFC READ(5,*)N,FCC(N,1) WRITE(6,*)N,FCC(N,1)
137 CONTINUE
WRITE(6,*)'NFW= #0F FLEXIBILITY COEFF.FOR' WRITE(6,*)' WIND ON GW.' WRITE(6,*)' ' WRITE(6,*)'# FLEXIBIUTY COEFF' DO 141 I=1,NFW READ(5,*)N,FCW(N.l) WRITE(6,*)N,FCW(N,1)
141 CONTINUE
WRITE(6,*)'NFREQ= #0F FREQUECIES OF TOWER * WRITEre,*)" FOR DIFFERENT HEIGHTS WRITE(6,*)" ' WRITE(6,*)'# FREQUECY IN Hz' DO 142 I=1,NFREQ READ(5,*)N,FREQT(N,1) WRITE(6,*)N,FREQT(N,1)
142 CONTINUE
C VREF,BBARABAR,G,AHAT,BHAT,CTAL,EPSI,ZMIN,CCAL,RHOAIR C ACSE 7-95 PARAMETERS C DC,CFT,CFC,FCCAET,RHOT,RHOAIR,TAPT,OMEGAT—STRUCTURE C DT,AMODEXPO,WTH PARAMETERS C JDIV = NO. OF DIVISIONS DESIRED FOR BM. AND STRESS CALS.
LL=0 D0 2222L=1,NH DO 223 M=1,NSP DO 224 N=1,NZT DO 225 J=1,NZC DO 226 ND=1,NZW LL=LL-hl WRITE(6,*)1 WRITE(6,*)' ' WRITE(6,*)"CASE # =",LL WRITE(6,*)' •
WRITE(6,*)'HEIGHT OF THE TOWER =',AHT(L,1)
WRITE(6,*)'SPAN OF THE CONDUCTOR =',NSPC(M,1) WRITE(6,*)'SAG OF CONDUCTOR FT.=',ASCC(M,1)
177
WRITE(6,*)'SAG OF GW FT.=",ASCW(M,1) WRITE(6,*)"% OF CRITICAL DAMPING IN TOWER =',AZTT(N,1) WRITE(6,*)'% OF CRITICAL DAMPING IN CONDUCTOR..=',AZCC(J,l) WRITE(6,*)'% OF CRITICAL DAMPING IN GW =',AZCW(ND,1) WRITE(6,*)'FLEXIBIUTY COEFF. FOR WIND ON COND =',FCC(L,1) WRITE(6,*)'FLEXIBIUTY COEFF. FOR WIND ON GW.. =",FCW(L,1) WRITE(6,*)'FREQUENCY OF TOWER Hz =',FREQT(L,1)
DMMM=0.0 WRITE(6,*)' ' WRITE(6,*)' Z DIA HOLLOW MOMENT OF' WRITE(6,*)'Z=0@BOTTOM OUTSIDE DIA. INERTIA " WRITE(6,*)" FT. FT. FT. FT'^4 " WRITE(6,*)' ' WRITE(6,1200) DMMM,ADB,DIB,AIB
WRITE(6,*)'- ' WRITE(6,*) IV—-BENDING MOMENT AND STRESSES CALCULATIONS" WRITE(6,*)" '
WRITE(6,*)"—FOR WIND ON TOWER " VHAT =BHAT*((EQH/33.0)**AHAT)*VREFFTPS AK =(1.65**AHAT)/(AHAT+AMODEXP+1.0) AMODMASS=fRHOT/32.197)*AVGAT*AHT(L,l)/((2.0*AMODEXP)+1.0)
WRITE(6,*)'—FOR WIND ON CONDUCTORS ' AKC=1 WRITE(6,*)'K =',AKC VHATC=BHAT*((EQHC/33)**AHAT)*VREFFTPS FC=0.5*RHOAIR*(VHATC**2.0)*NSPC(M,1)*DC*AKC*GEFC*CFC*3.0 MULTIPUED BY 3 FOR THREE CONDUCTORS
C LOCATING THE DIVISION WHERE CONDUCTOR IS ATTACHED LDIV=JDIV-(19.0/DIV) D 0 222K=1,LDIV-1 P =JDIV-K AMC(K) =FC*(Z(P)-19.0) STRC(K)=AMC(K)*ADT(K)/(2*AI(K)*144.0) WRITE(6,1300)Z(K),AMC(K),STRC(K)