DYNAMIC GUST RESPONSE FACTORS FOR TRANSMISSION LINE ...

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




6.4.7.4 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



7.4.5 Groundv^dre Gust Response Factor 93

7.4.6 Tower Stress 96










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




8.5 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.

o o CD CO

1.80 1.'/J 1.70-1.65-1.60-1.55-1.50-1.45-1.40-

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- I—1—I I 11111—

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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

loads.

19

<N <

KKI

lU

0.1

C 0.01

0.001

0.0001 0-0001 0.001

n[Hz]

Figure 3.5 Spectrum of Longitudinal (alongwind) Wind Velocity Fluctuations (Thomas George, 1996)

A structure will not respond fully to the impact of a gust whose size is only a

small fraction of the size of the structure. A gust, to be fully effective, must

have sufficient spatial extent to envelop both the structure itself and the flow

20

patterns on the windward and leeward sides, which are responsible for the

maximum loads on the structure. A correlation function can be defined

which accounts for the fact that wind gusts are not likely to act

simultaneously over the full extent of a large structure (Vellozzi et al., 1968).

The gust correlation function can vary fi-om unity for completely correlated

flow to zero for uncorrelated flow.

Wind loads on a structure can be derived fi-om the effects of a mean

wind speed plus the effects of the associated fluctuating wind speed. The

response of the structure depends upon the mean wind speed, the correlation

between gust size and structure size, and the correlation between gust

fi-equencies and structural frequencies of vibration.

3.1.2 Statistical Peak Factor

The equations in which gust factors for wind sensitive structures

appear may be written in the form

Y„„=Y + gG,=Y[l + G] (3.3)

where

Y = the response quantity of interest, a function of time, t;

Ymax = maximum expected or design value of Y;

Y = mean value over some period of time, T, of Y;

Gy = the root mean square of the deviations in Y from its mean;

g = the multiple of c^ needed to produce the maximum or peak

value of Y; and

G = the gust factor = g^Y

21

In terms of wind, a random variable normally is used to represent the

wind velocity, wind pressure, or wind response of a structure, and any of

these can be represented by the variable Y in Equation 3.3. Furthermore,

the random process is usually considered to be normally distributed and its

frequency content is assumed to be represented by a spectrum S(n). The

standard deviation of any one of the quantities of interest can be determined

fi-om its spectrum.

In the past, the time period over which the mean Y and the root mean

square Oy have been calculated were of the order of 30 seconds to one hour,

but the new ASCE 7-95 standard is cast in terms of a 3-second gust. Thus,

gust factors will have to be reduced in order not to have much larger design

loads or expected peak quantities than in the past.

A key factor in Equation 3.3 is the peak factor, g, which in the past has

been given typical values between 3.0 and 4.0, based on extreme value theory

for random processes (Davenport, 1964). However, the theory used to

compute g contains some assumptions about the process and the time of

averaging, T, that make it break down if Y and ay for the process are taken

only over a duration of T = 3 seconds. A realistic value of g needs to be

smaller than the extreme value theory predicts, an5rway. These aspects of

the problem of determining a new g are discussed below.

3.1.2.1 Extreme Value Theory for g

The theory of Davenport (1964) results in the following equations for

the mean and the mode of the extreme value distributions of the given

normally distributed random variable, x(t), which is represented by the

X — "x reduced variate, h = :

<^.

0.5572 for the mean h„„ =V21n(nT)-t- , ' (3.4)

^ V21n(nT)

22

for the mode h„„ = V21n(nT) (3.5)

where

n m^

\ m / \ o /

or the square root of the ratio of the second to the zeroeth

moment of the spectrum, and is the frequency at which most of

energy of the spectrum is concentrated; and

T = is the period over which the record is taken.

Thus, nT is an indication of the number of cycles in the time period, T, of the

dominant frequency component. All of the above relationships require the

assumption (Rice, 1944-45) that the number of maxima, N, during T:

N = J ^ ^ T (3.6)

be large.

In applying Equations 3.4 and 3.5 to wind effects, Davenport (1964)

took the duration, T, to be 1 hour or 3600 seconds and assumed the range of

interest of n to be fi-om approximately 0.03 to 3.0, giving a range of nT fi-om

approximately 100 to 10,000. Thus, the means and modes came out as shown

in Table 3.2.

Taking the values of the mean (or even the mode) in Table 3.2 for the

multiplier g shows why g is usually chosen in the range fi-om 3.0 to 4.5 in

Equation 3.3 for long duration T such as 3600 seconds. Such a long duration

allows time for the peak to occur, with many of the waves in the spectrum

reaching a maximum simultaneously.

In order to depict the results of this analysis in simplified time history

form, four components of a representative wind spectrum, along with the

assumed spectrum, are shown added together in Figure 3.6. The four

components are taken at the frequencies shown in Table 3.3, where the

23

spectral values are also shown. Two of the frequencies are above the peak of

the spectrum, one is at the peak, and the fourth is below the peak. The

portion of the time history shown lasts only for 100 seconds, or for one period

of the longest-period wave. The figure illustrates how the different waves

combine in producing the overall peak of the time history. The comparison

between the deviation of the peak firom the overall mean and fi-om the 3-

second mean is also shown.

Table 3.2 Extreme Values Calculated by Davenport (1964) with T = 3600

n = dominant,

frequency

nT

Mean, hmax,(=g)

Mode (hmax)

0.0277 Hz

100

3.225

3.035

0.277 Hz

1,000

3.872

3.717

2.77 Hz

10,000

4.426

4.292

Table 3.3 Frequencies and Spectral Values Selected for Simplified Time History

Frequency (Hz)

0.01

0.05

0.10

0.5

Spectral Value (1/Hz)

0.1

0.4

0.2

0.03

24

0.8 n

0.6-

0.4 -

02 -

0 -

-0.2-

-0.4 -

-0.6-

-0.8 -

f^ \£ \ /CNi V ^ W 7 CM \ F a ^ 'A\ TJ ^ V Jr° " l l Jr

TIme.t sec.

sin(w1t)

sin(w2t)

— - - sin(w3t)

sm^w •ti^

Figure 3.6 Representative Components of Spectrum for Frequencies in Table 3.2

It can be seen fi-om Figure 3.6 that the overall peak of the limited

record shown occurs when there is an approximate combination of the peaks

of the individual curves. Thus all of the frequency components contribute to

the peak, although not in exact relation to their individual peak amplitudes.

The same happens when many more frequency components are included in a

time history. Over a longer period of time, of course, there are more times

when the peaks of the individual curves can combine for the absolute largest

peak value of the entire record. However, the increase in the peak with an

increase in duration is not large. This is why a large increase in nT produces

only a small increase in g in the relationships above.

For the record shown T is 100 seconds and n can be taken as 0.05 Hz,

the frequency of the wave with the largest spectral amplitude, so nT is 5, and

Equations 3.4 and 3.5 predict the mean and the mode of the peaks to be 2.12

and 1.79 times the standard deviation, respectively.

Now, if the largest 3-second "gust" in Figure 3.6 is considered, it

contains the absolute peak of 0.621. It is seen, however, that the number of

maxima during this interval is not "large," thus violating the assumption by

Rice (1944-45) cited in connection with Equation 3.6. Furthermore,

25

Equations 3.4 and 3.5 break down for T = 3 seconds, because then nT is less

than one, the natural log of nT is negative, and the square roots in these

equations do not exist.

By the criterion that the natural log of nT should not be negative, or

the product of n and T should not be less than one, the extreme value theory-

for g developed by Davenport (1964) cannot be used for a spectrum with

dominant frequency of 0.05 Hz for a duration of the averaging time, T, less

than 20 seconds. Thus, it is still valid for a fastest mile wind of up to 180

mph, but it is not valid for a 3-second averaging time.

3.1.3 Gust Response Factor (GRF)

ASCE 7-95 defines gust response factor as the factor that accounts for

the additional loading effects due to wind turbulence over the fastest-mile

wind speed. It also includes loading effects due to dynamic amplification for

flexible structures, but does not take into account cross-wind deflection,

vortex shedding or instability due to flutter or galloping (ASCE, 1995). In

short, the GRF can be defined as the static equivalent of the dynamic

response of the structure to the fluctuations in the mean wind speed

resulting fi-om turbulence.

3.1.3.1 Parameters Affecting the GRF

Parameters that affect the gust response factor can be listed as follows:

critical damping ratio;

fundamental frequency of the structure;

height of the structure;

width of the structure;

basic wind speed; and

type of Terrain.

26

Sensitivity of GRF was studied by Mehta and Kancharala (1986) for the

following building and wind characteristics:

Building Height = 500 fl;.

Building Plan Dimensions = 75 x75 ft.

Fundamental Frequency of Vibration = 0.346 Hz.

Critical damping Ratio =0.015

Basic Wind Speed = 100 mph (Fastest-mile)

Type of Terrain = Suburban Exposure B

The results obtained are shovm in Figures 3.7 through 3.10. The points with

a star in the figures represent the values for the assumed building and wind

characteristics. These figures indicate the sensitivity of the GRF to various

parameters.

3.2 Flexible Structures

ASCE 7-95 defines flexible structures as those which have a ratio of

height to least lateral dimension equal to or more than five or those

structures with a natural frequency of less than 1 Hz. The GRF plays an

important role in the behavior of flexible structures. Whereas the response of

rigid structures can be determined easily compared to flexible structures, the

response of flexible structures is quite complex as the djmamic response of

these structures can dominate the structural behavior. In case of a rigid

structure, due to the high overall stiffness of the structure in the along-wind

direction, the dynamic response can be ignored. For a flexible structure on

the other hand, in order to account for dynamic amplification of the loads and

to design the structures, the design engineer must have knowledge of the

GRF.

27

oc o h-o < l i

en 2: O Q. (f)

1.5

1.3

cr 1.2 J —

Z) O I

1.0 • I . I I \ I I J I

0 0.005 0.010 0.015 0.020 0.025 0.030

DAMPING RATIO

Figure 3.7 Sensitivity of GRF to Damping Ratio

l.6r-

Q: L 5 Q

2 1.4

if)

o Q-(O UJ CC

\-(f) ID O

1.3

1.2

l . l -

.0 O

J L J I L 1 I L J 0.2 0.3 0.4

FUNDAMENTAL FREQUENCY {Hz)

Figure 3.8 Sensitivity of GRF to Fundamental Frequency

2S

6 r -

^ 1 5

o «S 1 . 4 -UJ

O I 3 a. '•'-' en Ixl cr

CO Z)

1.2

I.I

.0 0

1 J I \ I I I

50 100 150 2(X) 250 300

WIDTH OF BUILDING ACROSS WIND FLOW (ft)

Figure 3.9 Sensitivity of GRF to Width of the Building

l.6r-

S 1.5

2 UJ in

1.4-

2 1.3 UJ

cr K l.2h Z)

.0 J L

60 I I I I I

70 80 9 0 100

BASIC WIND SPEED (mph)

110 J I

120

Figure 3.10 Sensitivity of GRF to Basic Wind Speed

3.3 Structural Response

When wind forces act on a structure, the reaction of the structure is

called its "response." This response in case of a transmission line structure is

produced by wind on the tower and wind on the conductors and the

groundv^re. The design of a transmission line structure is based on the peak

loads of an extreme wind on all three of these components. It is the peak

value of the response that is needed for the design. Peak response is the

summation of mean and fluctuating responses as shown in Figure 3.11. For A _

a time period, T, the peak response can be estimated b y R = R + g o R

(3.7)

where A

R = peak response;

R = mean response;

g = statistical peak factor;

CR = RMS of the fluctuating response about the mean response.

ESTIMATED

MEAN RESPONSE

TIME, t

Figure 3.11 Response Model

30

3.3.1 Mean Response of Conductors

Mean response of the conductors is obtained fi-om the mean wind

pressure acting at the height of the conductors. The effective height of the

conductor is calculated as the height of the attachment to the tower less two-

thirds of the sag of the conductor. This effective height is at the center of

pressure of the conductors. The mean wind pressure on the conductors is the

product of kinetic energy of the wind and the force coefficient of the

conductor. The equation for mean wind pressiu-e is

P = —p V C f (3.8) 2

where

P = mean wind pressure;

p = mass density of air (0.0024 slugs/fl;^);

V = mean wind speed; and

Of = conductor force coefficient.

The mean response of the conductors, R, can now be expressed as:

R = P L d (3.9)

where

L = conductor span; and

d = conductor diameter.

Inspection of Equation 3.8 shows that the kinetic energy of the wind per unit

volvmie is converted into a pressure through the force coefficient, Cr. The

force coefficient is a function of the Reynolds Number, the angle of attack,

shape of the conductor, and the roughness of the conductor.

.S.3.2 Fluctuating Response of Conductors

The response of a conductor to fluctuating wind depends on its

dynamic characteristics as well as turbulence in the wind. To determine the

31

fluctuating response of a conductor, the frequency domain approach is

usually employed. In this method fluctuations in the wind and in the

conductor response are represented by spectra. The area under each

spectrum is equal to the mean square of the fluctuations. Several steps

involved in this method are summarized in Figure 3.12. These steps are as

follows:

Step I. Transformation of the gust spectral density function, Su(0, into the

force spectral density function, Svif), through the aerodynamic

admittance function, x^(0.

Step II. Determination of the response spectral density function SR(f) by

multiplying the force spectral density function, SF(0, by the

mechanical admittance function, H2(f).

Step III. Calculation of the mean square value of the response, OR , fi-om the

area under the response spectrum.

Step IV. Calculation of the peak fluctuating response by multiplying the root

mean square value of the response or standard deviation, OR. by the

statistical peak factor, g.

3.4 Changes in ASCE 7-95

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 three-second gust. The three-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) as shown in Figure 3.4.

32

Gust Spectrum

Force Spectrum

<o

o oc

CO ^"

Response Spectrum

' • " " • ' * LA / V ^ W * . ^ X p o * ^

^'. •':*;*v';*x'x'Xvl\ ^ ' y ^

/ifcii^ y^xv>:]x::;xi:i:ix;;::i::::::::::-;-:":iX;::T^

> ' > . x - x - : - > > : - : - : - : : : : : : : ;:•:•:••••••••••••••• i - -

togf

Aerodynamic Admitlarx»

logt

HAechanical Admittance

M

Figure 3.12 Elements of Response Spectrum Analysis

^^

The map showing three-second design wind speeds proposed for ASCE

7-95 is given in Figure 3.13. The values for hurricane regions such as

Florida take into account the different gust patterns for hurricane regions as

compared to those for other regions. Note that winds of the order of 140

miles per hour (mph) are shown for much of Florida and the largest values go

as high as 150 miles per hoiur. A 3-second gust speed of 130 mph compares to

110 mph for the fastest mile winds of ASCE 7-88.

The values of the coefficients used to model the winds at different

heights above ground have also been changed in ASCE 7-95. The model used

for this "wind profile" is called the "power law" and has the form

-il/d Vz = V33 z

33 (3.10)

where Vîs the three-second wind speed at an arbitrary height, z, V33 is the

three-second wind speed at an reference height of 33 feet (the height at

which the basic design values in Figure 3.13 are taken), and — is the

a

powerlaw exponent, which depends on the terrain roughness or exposure at

the site of interest. The "hats" on V and V signify 3-second gust values in

ASCE 7-95. Corresponding mean hourly values have a bar over the quantity.

Fastest mile values do not appear in the standard.

A related part of the power law is the height above which the wind

speed is considered to be constant (no longer slowed down by the terrain),

which is called the "gradient height," Zg. This height also depends on the

exposure of the site, but its values have not changed fi*om ASCE 7-88 to

ASCE 7-95. The values of a for both the old and new standard are presented

in Table 3.4, along with several other parameters to be discussed

subsequently.

34

c o

I I a ^ 0) Q. CL o

O) a.

f £

_. . . .^^- ._^ r> «D «D (O o •* iS r* i/» lo

9'm > o I/) m F >

c

llo

« o

- J

O CV K CM C«l

• o

Ric

o

slan

ds

an S

am

1 2 E C ^

l § 5 ? ? PC Q- O X

« 8 o > ffl •> S- o e •>

•o-o S-° • c c-e C a

c4 JO 2 o c © e 8 * -S • c «

.2 .S .c • £ ^ o s c i>

1 2 « 5

|1 l 8

11 Q. «

3 01

• i 3

I? c a ^ ! • =

I-a « ~ c

.2 _ oi

S & « n *

CO

<u

a CO

o C o o a>

CO

CO bo C

•rH

cn

ID

W o CO <

oo O

a

o

a CO

: ^ TD 0)

a CO T3

c

s CO

Q oo CO

OQ

CO r-H

CO <V

be

^s

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:

ôoa/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. = ^ ^-îrf ^ (6.1)

_ (l + gêEVB^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^

\ô 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

VOîn 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 fiâctions 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 fiôm 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;



(tip deflection = 5.8028 x lO'^ ft per lb

due to 1 lb force at conductor level)

Groundv rire: Span length, L 650 ft


Sag, S 6.25 ft;


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, pîj. 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

= 0.2717

65

Damping factor, C^ = 0.000048 VoCf'(fw(d/12))

= 0.000048 (142.84) (1.0)

/(0.2717(0.11892)

= 0.2122

Conductor term, R^ = (1/Cw) [0.0113 (f^ zJWQy^l^^zJL)]

= (1/0.2122) [(0.0113)x(0.2717x

54.6/142.84)-5/3(54.6/650)]

= 0.1949

Conductor Gust Response Factor, G^ = (1 + gg e E V B^ + R^ )fKy'^

= [1 -H (3.6)(0.75)(0.3093)x

V0.2973 +0.1949 ]/(1.546)2

= 0.6635

Note: For this example the gust response factor for the tower, which is

dominated by the resonance term, is much larger than that for the

conductors, where the resonance term is greatly reduced by the large

aerodynamic damping. A check of the separation factor for this case shows

that the dynamic part of the GRF for the tower is B = VBt + Rt = 1792 and

that for the conductors is A = VB , + R^ = 0.702 in this example. Thus, the

ratio of B/A in Table 6.2 is 2.55, and the corresponding value of e from

Equation 6.12 is 0.77. The conclusion is that even though the frequencies of

the conductors and the tower are separated from 0.2717 to 0.9210 (for a ratio

of 3.39), Davenport's value of 0.75 for the separation factor has acceptable

accuracy. Alternatively, for even greater accuracy, the value of B/A could be

calculated and the value of e evaluated fi-om Equation 6.12.

66

6.4.5 Groundwire Gust Response Factor

On the similar lines as for the conductor, the groundwire gust response

factor is calulated as follows:

Backgroimd term, Bg^ = 1/[1 -i- 0.8(L/Ls)]

= 1/[1-H 0.8(650/220)]

= 0.2973

Frequency, fgw, (simplified formula) = Vl/S = V 1/6.25

= 0.400

Damping factor, Cgw = 0.000048 VoCf^(fgw(d/12))

= 0.000048 (142.84) (1.2)

/(0.400(0.0313)

= 0.6572

Resonance term, Rg^ = (1/Cw) [0.0113 (fg^ ZQ^ôY^'^^Zofh)]

= (1/0.6572) [(0.0113)x(0.400x

54.6/142.84)-5/3(54.6/650)]

= 0.0330

Groundwire Gust Response factor, Gg^

= (1 + gg e E V Bgw -»• Rgw) /Kv2

= [1 + (3.6)(0.75X0.3093)x

VO.2973 + 0.0330 ]/(1.546)2

= 0.6192

6.4.6 Tower Stress

In order to calculate a particular response quantity due to the 3-second

gust design wind by Davenport's method, the influence coefficients for that

quantity for wind on the tower, wind on the conductors, and wind on the

groundwire, 6t, Ow> Og^, respectively, must be used in the general equation:

67

Maximum response = Ot Pt Gt + 0^ Pw Gw + Ogw Pgw Ggw (6.15)

where pt, p ^ and pg^ are the static wind forces on the tower, the conductors,

and the groundwire, respectively, associated vrith the 140 mph windspeed.

In other words, if the 10-minute average VQ of Davenport's method is used in

computing pt, Pw, and pg^, then the corresponding gust factors Gt, G^ and

Ggyf must be the Davenport gust factors vrithout dividing by Ky2. On the

other hand, if the gust factors computed in the previous section with the Kv2

factors are used, then the value of VQ used in computing the forces must be

factored back up to the corresponding 3-second gust level by multiplying by

Ky. Each influence coefficient is the effect on the response of a unit value of

the static force considered.

6.4.6.1 Conductor Contribution

For a conductor at height z, the static force imposed on the tower by

the conductor is

Pw = (Paii^2) ^z2 Cf L d (6.16)

A where the V^. ^^ ^ ® three-second gust wind speed at height z for use with the

gust effect factor calculated above. The three conductors at different heights

have slightly different values of V2, and thus of p^, but these value are very

close for a separation height of only 8 feet (see Figure 5.1). Accordingly, for

simplicity the value of p^ at the level of the middle conductor is used for all

three conductors. Since the middle conductor is 19 feet below the top of the

tower, the value of V2 for exposure C in feet per second is, by the power law,

68

A A Vz = [z/33]l/a(88/60)Vref = [(84-19)/33] 1/^(88/60) 140 = 226.21 ft s

and the corresponding value of three times p^ is, from Equation (6.16),

3pw =3(pair/2)Vz2CfLd = 3(0.0024/2)(226.21)2(1.0)(650)(0.11892)

= 14,2401b.

Next, the influence coefficient for the bending moment at any level below the

middle conductor due to the three conductor forces is the moment at the level

considered due to a one-pound force at the middle conductor. This moment is

one pound times the distance from the middle conductor down to that level,

Omoment(z) = (1 lb) [(h-19 ft)-z]. (6.17)

The associated maximum bending stress is

Ostress(z) = Omoment ^VSxCz) (6.18)

where Sx(z) is the section modulus of the pole's hollow circular cross section

at the level z. This quantity varies with height because of the taper of the

pole. For the spun-cast concrete pole, the influence coefficient for maximum

bending stress due to wind on the conductors is almost constant along the

lower one-third of the height and is close to the value at the base, which is

Ostress(z=0) = (1 lb) [(84-19)-0 ft]/(0.9640 ft^) = 67.43 psf^b

= 0.4682 psi/lb.

69

Finally, the maximum stress at the base of the tower due to the wind

on the three conductors is

Maximum base stress, a^ = 6^ p^ G^

= (0.4682 psi/ lb)(14,240 lb)(0.6635) = 4,424 psi

6.4.6.2 Groundwire Contribution

In every concrete pole considered, the groundvrire is attached at 0.5

feet below the top of the pole. Using the same procedure as for the

conductors, except that the height, wire diameter, and force coefficient are

different, the force on the groundwire is found from Equation (6.16).

Vz = [(84-0.5)/33]l/'7(88/60)140 = 234.45 ft/s

and the corresponding value of pg^ is

Pgw = (Pair/2) Vz2CfLd

= (0.0024/2X234.45)2(1.2X650)(0.03125) = 1,611 lb.

Next, the influence coefficient for the bending moment at any level below the

groundwire due to the force pg^ is the moment at the level considered due to

a one-pound force at the groundwire level,

Omoment(z) = d lb) [(h-0.5 ft)-z] (6.19)

and the associate maximum bending stress is again given by equation (6.18).

As for the conductors, the influence coefficient for maximum bending stress

70

due to wind on the groundwire is almost constant along the lower one-third

of the pole's height and is close to the value at the base:

Ostress(z=0) = (1 lb) [(84-0.5)-0 ft]/(0.9640 ft^)

= 86.62 psf/lb = 0.6015 psi/lb.

Finally, the maximum stress at the base of the tower due to the

groundwire is

Maximum base stress, Og^ = 6g^ pg^ Gg^

= (0.6015 psi/ lbXl,611 lb)(0.6192) = 600 psi.

Even though the groundwire has a greater moment arm than the

conductors, it produces much less stress than the three conductors because it

has a smaller diameter and there is only one.

6.4.6.3 Tower Contribution

Wind forces act all along the height of the tower, and their combined

effect on the bending stress may be calculated by taking a number of

individual segments along the height, determining the force on each

segment, and then computing the moment and stress at any section in a

manner similar to the calculation for the conductor forces. This procedure

has been carried out for the spun-cast concrete pole by means of a computer

program, using 20 segments. As shown above, the 10-minute average wind

speed, in ft^s, at the mid-height of each segment, z, is calculated by the wind

profile as follows:

Vz = (z/33)l/a(VrefX88/60). (6.20)

71

The force on the segment at each height is also influenced by the outside

diameter, D(z), and the force coefficient, Cf. The force on the element at

height z is

P(z) = (paiiy2)Vz2Cf(h/20)D(z).

Adding up the moments of all of these elemental forces above the level of

interest, the moment at that level is found. For the present example the

bending moment and resulting stress are largest at the base and come out to:

Mt(z=0) = 258,550 Ib-ft; and

ot(z=0) = Mt(0)/Sx(0)

= 258,550/0.9640 = 1,863 psi

where the gust response factor is included.

6.4.6.4 Total Stress

Finally, the total maximum stress is the sum of the tower, conductor,

and groundwire effects, or

atotal(z=0) = at(z=0) -i- aw(z=0) + agw(z=0)

= 1,863-»• 4,424 + 600 = 6,887 psi.

Obviously, wind on the three conductors contributes much more than wind

on the tower or on the groundwire in this case.

72

6.4.7 Tower Deflection

Calculation of tower deflections by Davenport's method requires a

computer model of the tower's stiffness properties similar to the one used

herein to determine the natural frequencies and mode shapes. A static

analysis still is sufficient, however. Three unit loads are applied to the

model at distances down fi-om the top of the tower of 11, 19, and 27 feet,

respectively, and the resulting deflection at the tip is the influence coefficient

for wind on the conductors. Then, distributed loads determined from the

wind profile are applied to the model for the effect of wind on the tower.

These loads are distributed such that the wind velocity at the reference

height is unity, and other velocities are scaled to it by the power law (using

the new ASCE 7-95 coefficients). The resulting deflection at the tip is then

an influence coefficient for wind on the tower that must be multiplied by Vo2.

The different forces on the tower at different elevations depend on the tower

diameters as well as the vnnd profile, but these forces can all be represented

in terms of the reference velocity, VQ.


For the contribution of the three conductors, the influence coefficient

fi-om the finite element computer model of the spun-cast concrete pole is

Odefl. = 5.8028 X 10-4 ftyib. (6.21)

Multiplying this coefficient by the conductor force, by the gust response

factor, and by three for the three conductors, the maximum tip deflection due

to wind on the conductors is,

^max = Odefl. Pw ^w

73

= 3(5.8028x10-4 ftyibX14,2401bX0.6635) = 5.483 ft.


For the groundwire contribution, the influence coefficient fi-om the

finite element computer model of the spun-cast concrete pole is

Odefl. = 3.2392 x lO'^ ftlb. (6.22)

Multiplying this coefficient by the groundwire force and by the groundwire

gust effect factor, the maximum tip deflection due to wind on the groundwire

is,

^max = 0(jefl. Pgw ^gw

= (3.2392 X 10-4 ftabXl,6111b)(0.6192) = 0.323 ft.


For the contribution of wind on the tower, the influence coefficient

fi-om the finite element computer model of the spun-cast concrete pole is

A Odefl. = 1.3216 X 10-5 ftA^ref^- (6.23)

This coefficient is multiplied by the square of the 3-second gust windspeed at

the reference height of 33 feet::

A Vref = (140mph)(88/60) = 205.3 ft/sec.

Thus, the maximum tip deflection due to wind on the tower is, multiplied by

the tower gust response factor:

74

A Xmax = Odefl. (Vref)^ Gt = (1.3216x10-5X205.3)2(1.0445) = 0.582 ft.

6.4.7.4 Total Deflection

Finally, the total maximum tip deflection including wind on the tower,

wind on the three conductors, and wind on the groundwire, is

^max = (^max^t "•" (^max^w "*" (^max^gw = 0.582 + 5.483 -»-

0.323 ft = 6.388 ft.

Once again, wind on the three conductors contributes much more to the total

result than wind on the tower or groundwire.

75

CHAPTER 7

SOLARI-KAREEM'S DESIGN MODEL

(ASCE 7-95 COMMENTARY)

7.1 Introduction

The rational model for designing wind sensitive structures presented

in the commentary to ASCE 7-95 is based on the work of Solari (1992a, b)

with modifications by Ashan Kareem. The model is designed for general

structures and requires interpretation, insight, and additional information

for useful application to transmission line structures.

7.2 Notation

Some of the ASCE 7-95 notation is the same as presented previously,

including the symbols GRF for gust response factor, b for building width, h

for building height, and a and Zg for the wind profile parameters. The new

symbols for wind-sensitive structures are defined below.

b, b = multipliers in converting fi-om a 3-second gust speed to a mean

hourly wind speed;

Cfx = drag coefficient;

d = horizontal width of the structure (perpendicular to the wind

direction);

E = modulus of elasticity of the tower material;

GRF = gust response factor;

g = peak factor, taken as 3.5 [corresponding to gg in Davenport];

I(z) = moment of inertia of the single pole tower at height z;

I- = turbulence intensity factor at the equivalent height, z;

K = combined wind profile and mode shape factor in the deflection

response expression;

76

Lj = integral length or scale of turbulence at the equivalent height, z

[corresponding approximately to Lg in Davenport];

/ = multiplier in the equation for I-;

mi = modal mass for the first mode in the response expression for Xj axJ

N i = reduced frequency associated with n^, L- , and V ;

n^ = fundamental natural frequency of the structure, in Hz

[corresponding to ft in Davenport];

Q = RMS background response factor [corresponding approximately to B

in Davenport];

R = RMS resonant response factor [corresponding approximately to R in

Davenport];

R/ = resonant response contribution for the wind spectrum when the

subscript Z = n and for the size factor in a particular direction when /

= b, h, or d;

Sx(z) = section modulus of the single pole tower at height z;

X(z) = lateral tower deflection at height z;

'z = equivalent height of the structure, normally taken as 0.6h for a

building [corresponding to ZQ in Davenport];

d,a = wind profile exponents for 3-second gust and mean hourly winds;

p = fraction of critical damping of the structure [corresponding to ^ in

Davenport];

e = exponent in the equation for I ;

<))(z) = fundamental mode shape of the structure;

r| = nondimensional variable in the size effect equations for R;

^(z) = mass per unit length of the structure;

^ = exponent for the approximate first mode shape in the final response

equations.

77

7.3 Equations

The gust response factor in the ASCE 7-95 Conmientary has been

modified as following for a wind sensitive structure:

GRF = i±Ml£>§!±Z (7.1) 1 + 71^

where the turbulence intensity, I , and the background response term, Q,

are given by the same equations as for a rigid structure (see section 3.2), the

peak factor, g, is normally taken as 3.5, modification in the form of

separation factor,e, (as in case of the Davenport model) with the value of

0.75, and the resonance response term, R, is a combination of several factors,

as follows:

R = iR„R,R,(0.53 + R,) (7.2)

In Equation 7.2, p is the fraction of critical damping and the spectrum term

Rji is given by:

R „ = _ L 1 6 5 N , _ (7.3) " (l + 10.302N,f'

where

N, = ^ ! ^ (7.4) V.

78

is the reduced frequency of the structure. Equation 7.2 corresponds

approximately to Equation 6.10 in Davenport's method.

The other three terms R , Rb, and Rd in Equation 7.2 account for the

size effects in the three directions of the structure. Each Rj follows the form

originally proposed by Vellozzi and Cohen (1968):

Ri = -(^](l-e"^) forTi>0

= 1 for r| = 0

(7.5)

where for the vertical direction, or when i = h.

4.6nih Tl = — = J —

for the lateral direction, when i = b.

4.6nib T[= — ^

and for the longitudinal direction, when i = d.

15.4n,d r|=——

The numerical factors 4.6 and 15.4 in the last three expressions have been

modified by Kareem from those in Vellozzi and Cohen (1968) and even those

in Solari (1992a, b) to take into account recent data and the averaging time

of the wind. The wind velocity V in these equations is the mean hourly

79

wind speed at the equivalent height, z, which is found from the 3-second

reference wind speed at 33 feet (fi-om the map of Figure 3.13), Vj-ef, by

V, = bf v33y ref (7.6)

where all symbols with bars over them pertain to mean hourly values.

Values of, a, b and other parameters in the equations of the modified ASCE

7-95 Commentary are given in Table 3.4.

The three size effect or correlation functions given by Equation 7.5 are

plotted in Figure 7.1. When the different functions are multiplied to produce

the overall resonance term, R, in Equation 7.3, it is found that R becomes

small for structures with any dimension large in comparison to the integral

length scale, L .

-c>-Rb=Rh - x - R d -A-(.53+.47RcD

n1(h, b, or cD^z

Figure 7.1 Size Effect Functions in the ASCE 7-95 Commentary Method

80

The gust response factor determined by Equation 7.1 is geared to take

into account the fact that the 3-second gust speed with which it is used is

likely to be close to the maximum wind speed. This is done by including the

second term in the denominator. Thus, if the radical in the numerator is

equal to unity, as in the case in which Q 2 = 1.0 and R2 = 0, then G = 1.0.

These values of Q and R are for no dynamic response and a small structure

(one completely engulfed by the average gust). Anytime there is no dynamiic

response (R = 0), Equation 7.1 simply reduces to the "complete analysis"

equation for a rigid structure. Equation 3.11, unreduced by the factor 0.9.

Besides the equations for the gust effect factor, the Commentary of

ASCE 7-95 presents expressions for evaluating the maximimi along wind

displacement and the RMS and peak accelerations of a wind sensitive

structure. Of these, only the displacement expression is considered here,

since accelerations are normally of concern only in regard to the comfort of

occupants of a structure or equipment in a structure, and thus would not

apply for a power pole. The maximum along wind displacement is given as

X_(z) = (z) - |p,,,V/KbhC^

[mi(27mi)'] GRF (7.7)

In the form shown (re-ordered from the form in the ASCE 7-95 Conmientary),

this expression is seen to be a standard modal response equation with only

one term, that is, the term representing the contribution of the fundamental

mode. The response consists of three basic elements: the fundamental mode

shape,<})(z), the term in brackets { }, which is usually called the "participation

factor" for the first mode, and the "dynamic amplification factor," GRF.

81

In the ASCE 7-95 Commentary, a simplified form of the first mode

shape is suggested:

<Kz) = ^z^^

Vny (7.8)

where the exponent ^ can be varied to make the first mode for a vertical

cantilever either convex or concave, as shovm in Figure 7.2. This

simplification is reasonable for tall slender buildings, the structures for

which the Commentary is primarily intended. If a building behaves like a

"shear building," then the mode shape of the type on the right in Figure 7.2 is

appropriate, and ^ vnll be less than 1.0. A shear building is one in which the

floors are very rigid in comparison to the columns framing into them and

rigid connections make the columns have vertical tangents at the floor levels.

If, at the other extreme, the building acts like a bending beam, then the mode

shape on the left in Figure 7.2 is correct, and ^ will be greater than 1.0. Real

buildings fall in between these extremes, so that a value of ^ = 1.0 is not

unreasonable, a value that produces a straight line first mode.

For a uniform bending beam the modal coefficient should be ^ = 2.4.

For both the tapered poles of Figure 5.1, a trial and error fit between

Equation 7.8 and the first mode shape determined by the finite element

analysis showed that the closest fit was with the value ^ = 1.8.

The middle term in Equation 7.7, called the participation factor, is

often written as the modal force divided by the modal mass and the square of

the circular natural frequency for the mode considered:

82

T 1

(2/H)

(2^)^^

Figure 7.2 Variations in the Fundamental Mode Equation 0(z) = (z/h)^ with c

cS3

f .. \ 1st Mode Participation Factor =

\^^ J m, (7.9)

where

0)1 = 2 7U fi (= n^) is the fundamental circular natural fi-equency (in

radians per second);

Fi = J F(z) (j)(z)2 dz is the first term in a series expansion of the forcing

function, F(z), in terms of the (orthogonal) natural

modes of the structure; and

mi = j m(z) (j)(z)2dz is the first term in a series expansion of the mass

distribution, m(z), in terms of the natural modes.

With the 3-second gust form of the wind profile:

V(z) = b ' z ^

Vzy V(z) (7.10)

the force per unit length on the vertical structure at height z is given by

1 ,-., ,2. „ 1 F(z) = -p.„V(z)^bC^=-p b3

vz;

-|2

33 bC, (7.11)

For Exposure C,b = 1.0. Then if b is constant along the height, the

simplified form of Equation 7.8 for the first mode shape makes F^ become:

F,=JF(z)(l)(z)'dz =

1 )Pai.V/bC^

(z^«h^^) j Z -Z d2

84

therefore F,= -Jp^,V,^bC^

[[2(d + )+l]

and with z = 0.6 h, this simplifies to:

Fi = JF(z)<t)(z)Mz = '-1 .2>

P.i.V/bC^ J[(0.6r[2(d + )^1]

fl^ \^j

pV/bhC, K

where

K = [(0.6r(2[d^^]^l)]

(7.12)

It turns out that with the values of d for the different exposures in Table 3.4,

this function is almost independent of d and decreases rapidly with . Also,

to a good approximation the factor "2" can be dropped wherever it appears,

resulting in the expression for K in the ASCE 7-95 Commentary:

(d+^+i) (7.13)

85

Figure 7.3 shows how K in Equation 7.12 varies with d and ^ for the four

exposures and Figure 7.3 shows how closely the values of K are when using

the different Equations 7.12 and 7.13 for Exposure C. The parameter K as

given by Equation 8.13 is included in Equation 7.7 for the maximum

displacement of a structure.

7.4 Example Calculations for the Spun-Cast Concrete Pole

In this section the spun-cast concrete pole considered as an example of

Davenports method is analyzed by the method of the modified ASCE 7-95

Commentary. The input data are the same except for the following changes.

First, the ASCE 7-95 Commentary method does not give an equation for

aerodynamic damping of the conductors. A value of P = 0.4 is assumed below

for the conductors. Second, the effective height of the structure is taken as

0.6h instead of 0.65h as in the Davenport method. Finally, the quantity Ky

is not needed, the turbulence intensity I- is used in place of the exposure

factor E, and the vrind field parameters are taken from Table 3.4. Once

again, calculations are carried out separately for wind on the tower, wind on

the conductors, and wind on the groundwire, and some terms such as the

equivalent height, z, the wind parameters at that height (turbulence

intensity, I-, integral scale, L-, and ten-minute average, V-) are calculated

separately for the tower and the conductors. As before, the maximum tower

stresses and deflections are determined in addition to the gust response

factors. In the ASCE method for wind on the tower, the tower deflections are

determined directly from given equations but the analyst must develop his or

her own method to find the stresses.

86

K Factors

1 -1

n,8 -

0.6-

0.4-

0.2 -

n -U ^

s

1 1

0.2

- 1

0.6

- 1 1 1 1 1 1 1 1 1 1 1 1 1 1

— ^ CO

Modd Exponent

2.2

-

2.6

- 1 1

CO

A B C

— - - D

Figure 7.3 Variation of Factor K with Wind Profile Exponent, a, and Mode Shape Exponent, ^

Eq. (8 .13 )

1 — I — I — I — I — I — h -

o oo CM O O ^ ^

CM CM

OO CM

Modd Exponent

Figure 7.4 Comparison of Equations 8.12 and 8.13 for Factor K, and Exposure C

87

7.4.1 Summary of input data

Data for spun-cast concrete pole is same as that for Davenport's model,

however parameters for wind-field are different for modified ASCE 7-95

Commentary method.

Tower: Height, h 84 ft

Diameter, D 1.0567 ft at the top

Taper of diameter out from the top 0.018 ft/linear ft

Mean thickness of the WEJI 0.25 ft

Fundamental frequency, n^ 0.9210 Hz.

Fraction of Critical Damping, P 0.03


Weight density of material, pt 150 Ib/ft^

Modulus of elasticity of material, Et 7.8083 x 10^ psf

Conductors: Span length, L 650 ft


Sag, S 13.542 ft

Fraction of Critical Damping, Pc 0.40

Weight per linear foot 1.63 lb/ft



(tip deflection = 5.8028 x 10"^ ft per lb

due to 1 lb force at conductor level)

Groundwire: Span length, L 650 ft


Sag, S 6.25 ft

88

Fraction of Critical Damping, pw 0.40

Weight per hnear foot 0.273 lb/ft



(tip deflection = 3.2392 x lO'^ ft per lb

due to 1 lb force at groundwire level)

A

Wind Field: 3-Second reference gust speed, Vj.gf = 140 mph

= 205.3 ft/sec

Mass density of air, pîj. 0.0024 slugs/ft^

Exposure Category C (open country)

(see Table 3.4 for b,b,c,l,d,a,e)

7.4.2 General Calculated Values

The follov^ng values are applicable for the tower, the conductors, and

the groundwire :

For Tower:

Average outside diameter (at 42 ft): = 1.0567 -i- 0.018 x 84/20

= 1.8127 ft

Avg. hollow core diam. (at 42 ft): = 1.8137-2x0.25

= 1.3127 ft

Avg. mass per unit length

= 150 7c/4(1.81272-1.31272)/32.2

= 5.717 Ib-ft/sec

Circular natural frequency = 2nni= 2p(0.9210)

= 5.787 rad/sec

Equivalent height of the tower, z = 0 . 6 x h = 0.6x84 ft

= 50.40 ft

89

For Wind:

Turbulence Intensity, IT

Integral scale of the wind, L^

Mean hourly wind at eff. ht. z, V?

Reduced frequency for tower, N^

= c(33/z)l/6 = 0.2(33/50.4)1/6

= 0.1863

= l/(z/33)e = 500(50.4/33)1/5

= 544.2 ft

= b(z/33)a Vref

= 0.65 (50.4/33)1/6-5 205.3

= 142.4 ft/sec

= niLz/V^

= 0.9210(544.2)/142.4 = 3.519


Calculations for tower gust response factor follows:

Background term, Q 2

Spectral response term, R^

Vertical size effect term, R :

Tih = 4.6nih/Vz

= 4.6(0.9210)(84)/142.4

= 2.498

Lateral size effect term, R :

= 1/(1 +0.63 [(b-Hh)/Lz](^-63)

= 1/(1+0.63 [(1.813-I-

84)/544.2]0-63)

= 0.8356

= 7.465 Ni / ( I + 10.302 Ni)5/3

= 7.465 (3.519)/(1 + 10.302(3.519)5/3

= 0.06322

= (l/Tih)-(l/i1h2)(l-e-2T1h)

= 1/2.498 - (0.5/[2.498]2)

( 1 . e2(2.498))

= 0.3207

= (l/Tib)-(l/Tib2)(l-e-2Tlb)

90

Tib = 4.6 nib/Vz = 1/0.05392 - (0.5/[0.05392]2)x

( 1 . e2(0.05392))

= 0.9650

= (l/Tid)-(l/Tid2)(l-e-2T1d)

= 1/0.1805-(0.5/[0.1805]2)x

(l.e-2(0.1805))

= 0.8897

= (l/Pt)RnRhRb[(^-53 + 0.47 R^]

= (l/0.03)(0.06322)(0.3207)x

(0.09650)[0.53 + 0.47(.8897)]

= 0.6184

Tower Gust Response Factor, Gt = [1 + 2g l^ e V Q 2 + R2]/ (I + 7 l^)

= [1 + 2(3.5)(0.1864)(0.75) x

= 4.6(0.9210)(1.813)/142.4

= 0.05392

Longitudinal size effect term, Rj:

Tid = 15.4nidA^z

= 15.4(0.9210)(1.813)/142.4

= 0.1805

Resonance term, R2

V0.8355 + 0.6184]/(l + 7 (0.1864))

= 0.9458


The conductor gust response fcator is calculated as follows:

Equivalent height of the conductor, "z = h^ - (2/3)(S)

= 63-(2/3)(13.542)

Fundamental frequency, nj

Turbulence Intensity, If


= 55.97 ft

= Vl/S = V 1/13.542

= 0.2717 Hz.

= c(33/z)l/6

= 0.2(33/55.97)1/6 =0.1831

= l/(z/33)£ = 500(55.97/33)1/5

91

Mean hourly wind at z, Vf


Reduced frequency, Nj


Vertical size effect term, Rj :

Tih = 4.6 nihA^z

= 4.6(0.2717)(0.11892)/145.0

= 0.001027

Lateral size effect term, Rb:

Tib = 4.6 nib/Vz

= 4.6 (0.2717)(650)/145.0

= 5.613

Longitudinal size effect term, R j:

Tid = 15.4nidA^z

= 555.7 ft

= b(z/33)a Vref

= 0.65 (55.97/33)1/6.5 205.3

= 145.0 ft/sec

= 1/(1 +0.63 [(b + h)/Lz]0-63)

= 1/(1+0.63 [(650 +

0.1198)/555.7](^-63)

= 0.5898

= niLz/Vf

= 0.2717 (555.7)/145.0

= 1.0415

= 7.465 Ni /(I + 10.302 Ni)5/3

= 7.465 (1.0415)/(l+10.302(1.0415)5/3

= 0.1460

= (l/Tih)-(l/Tih2)(l-e-2Tlh)

= 1/0.001027 - (0.5/[0.001027]2)

(l.e-2(0.001027))

= 0.9888

= (l/Tib)-(l/Tib2)(l-e-2Tlb)

= 1/5.613 - (0.5/[5.613]2)x

( 1 . e-2(5.613))

=0.1623

= (l/Tid)-(l/Tid2)(l-e-2Tld)

= 1/0.002785 (0.5/[0.002785]2)x

92

= 15.4(0.2717)(0.11892)/145.0 ( i . e-2(0.002785))

= 0.003438 = 0.9969

Resonance term, R 2 = (l/Pc)RnRhRbt(^-53 + 0.47 RdJ

= (l/0.40)(0.1460)(0.9888)x

(0.1623)[0.53 + 0.47(.9969)]

= 0.05860

Conductor Gust Response Factor, G^

= [1 + 2g Iz e V Q 2 + R 2 ] / ( I + 7 I^)

= [l + 2(3.5)(0.1831)(0.75)x

V0.5898 + 0.0586]/(l + 7(0.1831))

= 0.7775

7.4.5 Groundwire Gust Response Factor

On the similar lines as for conductors, the groundwire gust response

factor is calculated as follows:

Equivalent height of the groundwire, z = h^ - (2/3)(S)

= 83.5 - (2/3)(6.25)

Fundamental frequency, n^

Turbulence Intensity, I7


Mean hourly wind at z, Vg-

= 79.33 ft

= Vl/S = Vl76::25

= 0.4000 Hz.

= c(33/z)l/6

= 0.2 (33/79.33)1/6

= 0.1730

= l/(z/33)£ = 500 (79.33/33)1/5

= 595.9 ft - A

= b(z/33)aVref

93


Reduced frequency, N^


Vertical size effect term, Rb:

Tib = 4.6 nih/Vz

= 4.6(0.4000)(0.13125)/152.7

= 0.0003765

Lateral size effect term, Rb:

Tib = 4.6nib/Vf

= 4.6 (0.4000)(650)/152.7

= 7.831

= 0.65 (79.33/33)1/6-5 205.3

= 152.7 ft/sec

= 1/(1 + 0.63 [(b + h)/Lz]0-63)

= 1/(1+0.63 [(650 +

0.0312)/595.9](^-63)

= 0.6004

= n i Lz/Vj

= 0.4000 (595.9)/152.7

= 1.560

= 7.465 N i /(I + 10.302 Ni)5/3

= 7.465 (1.560)/(l+10.302(1.560)5/3

= 0.1127

= (l/Tih)-(l/Tlh2)(l-e-2r|h)

= 1/0.0003765 (0.5/[0.0003765]2) x

( 1 . e-2(0.0003765))

= 1.057

= (l/Tib)-(l/Tlb2)(l-e-2Tlb)

= 1/7.831-(0.5/[7.831]2)x

(l .e-2(7.831))

= 0.1196

= (l/Tid)-(l/Tld2)(l-e-2Tld) Longitudinal size effect term, Rd:

rid = 15.4nid/Vz = 1/0.001260 - (0.5/[0.001260]2)x

= 15.4(0.4000)(0.13125)/152.7 ( i . e-2(0.001260))

= 0.001260 = 1.0035

Resonance term, R 2 = (l/pg^)RnRhRb[0.53 + 0.47 Rd]

94

= (l/0.40)(0.1127)(1.057)x

(0.1196)[0.53 +0.47(1.0 35)1

= 0.03566

Groundvnre Gust Response. Factor, Gg^

= [1 + 2g Iz eVQ2 + R2y(i + 7 I )

= [l + 2(3.5)(0.1728)(0.75)x

V0.6004 + 0.03566]/(l + 7(0.1728))

= 0.7799

Note: By this method the gust effect factor for the tower is not dominated by

the resonance term and it is not as much larger than the factors for the

conductors and groundwire, as in the Davenport method, even though the

resonance terms for the conductors and groundwire are so small as to be

negligible.

It should also be noted that in applying the ASCE method to

determine tower deflections and stresses, the combined mode shape and wind

profile coefficient, K, is needed. From Equation 7.13, K for the tower is found

with d = 1/9.5 and = 1.8 (the closest value for the tapered spim-cast

concrete pole), and K for the conductors is taken as 1.0.

Tower: Kt = (1.65)a/(a +^+1)

= (1.65)1/9-5/(1/9.5+ 1.8 + 1)

= 0.3628

Conductors and Groundwire: K^ = Kg^ =1.0

95

7.4.6 Tower Stress

Once again the total tower stress is calculated as the sum of

contributions from wind on the conductors, wind on the groundwire, and

wind on the tower. The contributions from wind on the conductors and on

the groundwire are taken as the same as in Davenport's method. The

influence coefficients at any height of interest, z, are applied and are used

with the peak value of the wind force, p^G^K^ or Pgw^gw^^gw, o

determine the stress at that height. There is the question, however, of what

value to take for K^ for the conductors and groundwire. Since K is designed

to account for the combined vertical variations of wind speed and first mode

deflection, whereas the conductors and groundwdre are considered to be

single-degree-of-freedom pendulums at a single elevation, it would appear

that K^ is not needed. Therefore, in the following K^ and Kg^ are taken as

1.0.

A less direct way of determining the contribution to the stress from

wind on the tower must be employed, since the ASCE method gives the

deflected shape rather than the wind pressure as a function of height. In

principle, the bending stiffness, EI, times the second derivative of the

deflected shape could be used to find the moment, and from it the stress.

However, in the ASCE method, the deflected shape is taken to have the form

of the first mode, which is simplified to the expression shown in Equation

8.8, and the second derivative of this expression is uniquely zero. Thus,

either a different form of the deflected shape must be used in this second

derivative approach or some other approach must be developed. In what

follows, the second derivative approach is used and the chosen deflected

shape is that of the first mode of a uniform-section (prismatic) cantilever

beam.

96


Considering all three conductors to act at the height of the middle one,

the peak force applied to the tower is

3pwGwKw = 3[(pair/2) %^ Cf L dJO^Kw

= 3[(0.0024/2)(217.07)2(1.0)(650)(0.11892)](0.7775)(1.0)

= 10,195 Ih

The influence coefficient for the conductor force on the maximum bending

stress at the base (z = 0) is the same as in the Davenport method:

Qstress(z=(^) = 0.4682 psi/lb.

Finally, the maximum stress at the base of the tower due to the conductors is

Maximum base stress, <5yf{z = 0) = 6w[3pwG^Kw]

= (0.4682 psi/lb)( 10,195 lb) = 4,775 psi.

7.4.6.2 Groundv^dre Contribution

The peak conductor force applied to the tower is

PwGwKw = [(Pair/2)Vz2CfLd]GwKw

= [(0.0024/2)(225.19)2(1.2)(650)(0.0313)](0.7799)(1.0) = 1,1581b.

The influence coefficient for the groundwire force on the maximum bending

stress at the base (z = 0) is the same as in the Davenport model:

Qstress(z=(^) = 0.6015 psi/lb.

97

Finally, the maximum stress at the base of the tower due to the groundwire

is

Maximum stress, Cg^îz = 0) = OgwCPgw^ gw gw^

= (0.6015 psi/lb)(l,158 lb)

= 696.4 psi.


Instead of computing wind forces at all heights along the tower and

adding their effects on the moment at a given height, as in the Davenport

method above, in the ASCE method the moment at any height z is found from

the product of the bending stiffness at that height, EI, and the second

derivative of an approximation of the deflected shape. The deflection

response in the ASCE 7-95 Commentary method is

Xt(z) = ({)i(z) {[(l/2)pairVz2 b h Cfi, ]/[mi (27rni)2]} Gt Kt (7.14)

where the first mode shape, (|>i(z), is represented by (z/h)s, which is

normalized to unity at the top of the tower. Finding the corresponding

second derivative always gives a zero result with this expression for (|)i(z), no

matter what the value of the coefficient ^. Thus, an alternative expression

for <|)i(z) is introduced, also normalized to unity at the top of the tower. The

approximation is the first mode shape of a uniform cantilever:

(j)l(z) = {sin(aiz/h) - sinh(aiz/h) + [sin(ai) + sinh(ai)]/[cos(ai) + cosh(ai)] x

[cosh(aiz/h) - cos(aiz/h)]}/2.725 (7.15)

98

where aj = 1.875 radians and the factor 2.725 is used to make (|)i(z=h) equal

to 1.0. Differentiating the expression in Equation 8.15 twice gives

<t)l"(z) = (ai/h)2 {-sin(aiz/h) - sinh(aiz/h) +1.362[cosh(aiz/h) +

cos(aiz/h)]}/2.725. (7.16)

Then the moment at any height z is given by:

Mt(z) = <t)i"(z)EI(z){[(l/2)pairV22 b h Cf ]/[mi (27cni)2]}GtKt (7.17)

and the stress at any height z is simply determined from this moment and

the section modulus of the tower:

Maximum stress at height z = at(z) = Mt(z)/Sx(z). (7.18)

For the present example, the maximum stress due to wind on the

tower generally occurs in the lower part of the pole but not right at the base.

The quantities in Equations 7.17 and 7.18 at the base are as follows.

(t)i"(z=0) = second derivative of first mode shape at the base

(Equation (7.16))

= 0.0004982

E = modulus of elasticity = 7.808 x 10^ lb/ft2

Do = outside diameter at the base = 1.0567 + 0.018(84)

= 2.569 ft

99

Dj = inside diameter at the base = 2.569 - 2(0.25)

= 2.069 ft

l(z=0) = cross-section moment of inertia at the base = (7c/4)(Do4-Di4)

= 1.2381 ft4

Vz = 3-second gust speed at the reference height, z, of 50.4 ft

= 214.7 ft/sec

mi = modal mass = pi(z)<l)2(z)dz = î(z)(z/h)2^dz = JIQ /h2^ z2^dz

= îo/h25[h2^+1/(2^+1)] = ôh/(2^+l) = 5.717(84)/[(2 x 1.8) + 1]

= 104.4 Ib-s2/ft2

G = gust response factor = 0.9458

K = mode/profile shape factor = 1.65^/(a + ^ + 1)

= 1.651/9-5/(1/9.5+ 1.8+1)

= 0.3628

Sx(z=0) = cross section modulus at the base = I(z=0)/(D(z=0)/2)

= 0.9640 ft3-

Equation 7.17 thus gives:

Mt(z=0) = (0.0004982)(7.808x 108)(1.2381){[(l/2)(0.0024)(214.7)2x

(1.8127)(84)(0.8)]/[104.4(5.787)2]}(0.9458)(0.3628)

= 318,500 Ib-ft

and from Equation 7.18 for the maximum stress at the base,

at(z=0) = Mt(z=0)/Sx(z=0) = (318,400/0.9640)/144

= 2,294 psi

100

7.4.6.4 Total Stress

Finally, the total maximum stress is the sum of the conductor,

groundv^re, and tower effects, and assuming that the base value is the

maximum value,

^total(z=0) = Ow(z=0) + Ogw(z=0) + Ot(z=0) = 4,775 + 696 + 2,294

= 7,765 psi.

Once again, wind on the three conductors contributes much more than vrind

on the tower and wind on the groundv^dre. Another observation is that the

total stress computed here with the ASCE 7-95 Commentary method is 1.5

percent smaller than that obtained earher v dth Davenport's method. This

close agreement is surprising in light of the assumptions about the base

curvature and other effects made in applying the ASCE method. The

components do not match, however.


Calculation of tower deflections by the ASCE method again uses the

influence coefficient approach for the effects of wind on the conductors and

wind on the groundvdre. The same computer-calculated influence

coefficients as used in the Davenport method are employed again for the

contributions of the conductor and groundwire forces to the tower deflection .

For the effect of vrind on the tower, the ASCE method directly utilizes

Equation 7.7, where the simplified formula for the first mode shape ^(z)

given by Equation 7.8 with ^ = 1.8 may be used this time. The same values of

Gt and Kt as introduced above are carried over.

101


For three conductors, the peak force applied to the tower is again

10,195 lb and the influence coefficient for the conductor force on the

deflection at the top is the same as in the Davenport method:

edefl.(z=h) = 5.803 x 10-4 ftlb.

Thus, the maximum deflection at the top of the tower due to the three

conductors is

Maximum tip deflection, X^(z = h) = 6^[p^G^K^]

= (5.803x10-4 ft/lb)( 10,195 lb)

= 5.916 ft.


For the groundwire, the peak force applied to the tower is again 1,158

lb and the influence coefficient for the conductor force on the deflection at the

top is the same as in the Davenport method:

edefl.(z=h) = 3.239 x 10*4 ft/lb.

Thus, the maximum deflection at the top of the tower due to the groundwire

is

Maximum tip deflection, Xgw(z = h) = Og tp gw ^ gw^ gw

= (3.2392x10-4 ft/ lb)(l,158 lb)

= 0.375 ft.

102


From Equation 7.7, the maximum tip deflection of the tower due to

wind on the tower is

Xt(z=h) = (l)i(z=84) {[(l/2)pairVz2 b h Cf ]/[mi (27tni)2]} Gt Kt

= (84/84)l-8{[(l/2)(0.0024)(214.7)2x(1.8127)(84)(0.8)]/

[104.4(5.787)2}(0.9458)(0.3628)

= 0.661ft.

7.4.7.4 Total Deflection

Finally, the total deflection at the top of the tower due to conductor,

groundwire, and tower contributions as determined by the ASCE method is

Total tip deflection = X^(z=h) + Xgvv(z=h) + Xt(z=h)

= 5.916 + 0.375 + 0.661

= 6.952 ft.

This result is 7.8 percent larger than that found with the Davenport method

in Chapter VI.

The various results obtained by the ASCE method in this chapter and

those obtained by the Davenport method in Chapter VI are sunmiarized and

discussed further in Chapter IX.

103

CHAPTER 8

SIMILTS MODEL (1976, 1980)

8.1 Introduction

The model of Vellozzi and Cohen (1968) was later modified by Simiu

(1976, 1980) in a way that was incorporated into the design standard ASCE

7-88 (ASCE, 1988). The model utilizes graphs for the determination of

various parameters and is formulated in terms of mean hourly winds and the

metric (SI) system of units. Like the new ASCE 7-95 Commentary method, it

is designed for traditional buildings and related structures and does not

always work directly for slender structures like single power poles. In

particular, the parameters needed to look up quantities may be off the scales

of the graphs. Also, adaptation of the model to conductors is not practical. In

the development of this model, it is assumed that for the large number of

buildings or structures of practical interest, the fundamental mode shape is

linear and the response is dominated by the fundamental mode. Hence,

Simiu's model may be applied to typical tall structures for which the ratios of

higher frequencies to fundamental frequencies are not unusually low, i.e., nil

m > 2, where n is the fundamental frequency in the nth mode (Simiu, 1976).

The two papers published in 1976 and 1980 by Simiu are closely

related but do not have exactly the same notation or content. For example,

no gust response factor is introduced in the 1980 paper, although there is one

in the 1976 paper, and some of the quantities change. The development

herein follows the 1976 paper most closely, with some simplifications in the

notation.

104

8.2 Notation.

Relatively few of the symbols in the papers by Simiu (1976, 1980) are

the same as those presented previously. The symbols needed for the pole

example are defined below. Symbols such as ^ , J , and Y ^ require

reference of graphs shown in Figure 8.1 through Figure 8.3 and Table 8.1.

'a(z) = along-wind deflection at height, z, meters;

B = width (diameter) of the structure perpendicular to the wind

direction, meters;

S = background response term [roughly analogous to Q 2 in Davenport];

(E = background response quantity shown graphically in Figure 8.1;

C D = mean drag coefficient of the structure;

C^ = windward pressure coefficient, taken as 0.6 for a cylinder;

C L = leeward pressure coefficient, taken as 0.4 for a cylinder;

D = depth (diameter) of the structure in the along-wind direction, meters;

f^ = reduced fundamental frequency in the vertical direction, (niH)/u*;

g = peak factor calculated from Eq. 8.11, but taken as 3.5 in Simiu

(1980) [corresponds to gg in Davenport and g in ASCE 7-95];

GF = gust response factor;

H = height of the structure, meters;

J = function depending on ZQ/H, zd/H, and H in Eq. 8.12;

J = function for part of J depending on ZQ/H and z^fH. in Figure 8.2;

m = mass per unit length of the tower, assumed to be constant,

kilograms/meter;

N i = along-wdnd size effect or correlation factor [corresponding to Rd in

ASCE 7-95]

105

ni = fundamental natural frequency of the structure, Hz [corresponding

to ft in Davenport];

p = wind speed retardation factor from Simiu (1980), given as 1.0 for

Exposure C;

(2 = resonance response term [roughly analogous to R2 in Davenport];

Uo(10)= mean hourly wind speed at the standard height of 10 meters,

meters/second;

u* = finction velocity, meters/second;

Ti = wind speed related to u* by Eq 8.8;

Ax = four times the along-wind dimension of the structure, D, meters;

Yji = parameter depending on B/H, ZQ/H, zd/H, and f in Figure 8.3 ;

zd = zero plane displacement, taken as zero for Exposure C;

ZQ = roughness length from Table 8.1, given as 0.07 for Exposure C;

5 = fraction of critical damping of the structure;

V = frequency at which most of the energy of the spectrum is

concentrated, used in calculating the statistical peak factor, g;

T) = reduced frequency for the along-wind size effect or correlation

parameter, N i

8.2.1 Relevant Graphs and Tables from Simiu, 1976

The follov^dng graphs and tables are reproduced here and will be used

for the calculations of the gust response factor by Simiu's method.

106

inRmiioKir9frPM».-T9>inMiih.-*iiaim.^x.ii mpnr r t •mmniDPie^* vBurfe-«f .Baniiiiiii:ianrt.-<..v«iiiiu cr •

• •flllMlllliiiiiirinac.'iiiiuccrknfMnM • • • • H i l l r i i iur.*: -BltwC.r ' .^MiMIMH

Itiitirti:!-:: r< t.* Nr frr Tr.r/mggmn

iiiiHui fr-^5^|iBin:r?fcBiiif IIIIIIIIP:I] •.

•nmBSi»r«i(wrranii i i iuninii i fni ' f i>-Minir« v i - a n i •twgwggt»)R3»erMiiiiniimniiinircyCTiBniryt.-Btll

___"»J?«r!'»iB»r-c»««MniiiiiimiiramtPHiiiiiiiiiiie»B»ti „_.™™.™^-._ ._ . ._ . . , . .

•am»B5'sr^l!iilitr.t»PBiinillirimilinBtjrB»pn:irftrrritniIllllllllllWiP».ri«iiwr».iitKr!?ni^^

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§SSSî§ia&iiiii£Usisi£iiîs§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 ;

•f \ \ «

y \Jv l> , yv.,

p|5i.

'V_l t t

- J " - J

1 • • 1

X t j V ^

I U J l 2 ? ^ ^ H ^ y • ' ='4=

£rssgS£5s5

f - - 4 ? * j * ^ * '

i i j

•-1

4 :

3 i

T H -4 T

u —* -*-*-

h W ^

fli'

M-r- . r-i-

^ muuii

UUIIIII UIM

IIII IH

Ullll

lUIIIIII

T _

' ; J '

1 • - ' " • * •

Imi

Tl''iu,^'ii

«r5r»nâ»Brs'«»a»?Sc*lxr.c>.iciissa».a^

10' . « * ' . « • .<» .<«jk«> iW» •>-. • - ! . .

10 -5

^

Br«BaBiiakBBaBRe«»kB»» iri.xnB£k^BBb« eirBHK>ecB£K&>>:*«BisBeB»r'.r«BSf-Bk'.?*r: «BesRrseesir?EaKsc6ipEa^-«s^.;«asi;«rs»>>2r<:»*?a?«s»s«£>k5eseeA^s>B^Ta£iK5c^:-

H4:i^Ma|s|:T|ia|^j|^m:m?|^âmj^saj^^ | J . J '

10

i i i£3^ f^^fHH.^

I X

-_•_ . l - j - ^ ! - ' - I—14 j I : . ^ - i - : - i J- -I . } ' 4-. ; I I a =

Figure 8.3 Function Y 11

los

Table 8.1 Values Zo/H, Zd/H Corresponding to Various Yu Curves

Curve

A

B

C

D

E

F

G

H

I

r J

J'

K

K'

L

L'

M

M'

N

N'

0

0'

Zo/H

1.3 X 10-5

3.4 X 10-5

8.3 X 10-5

1.1 X 10-4

1.9x10-4

4.7 X 10-4

1.0 X 10-3

1.6x10-3

2.2 X 10-3

2.2 X 10-3

3.4 X 10-3

3.4 X 10-3

5.4 X 10-3

5.4 X 10-3

8.0 X 10-3

8.0 X 10-3

1.3 X 10-2

1.3 X 10-2

1.8 X 10-2

1.8 X 10-2

2.7 X 10-2

2.7 X 10-2

Zd/H

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.06

0.

0.04

0.

0.15

0.

0.10

0.

0.20

0.

0.45

0.

0.30

Note: Zd/H is of the order of 0.1 or less and Zo/H < IO-2, in determining Yji may be assumed zd/H = 0.

109

8.3 Equations

8.3.1 Gust Response Factor

The gust response factor in Simiu's model (1976) is given by

G F = l + g 1.23V« + «

which is a combination of the following two equations;

(8.1)

.1/2

a(z) = 1.23

fsT^ (8.2)

GF = l + g. a'(z) 1/2

a(z) (8.3)

The quantities on the right-hand side of Equation 8.1 are determined as

follows. First, the background response term, (E, is found from the ratio of

the zero plane displacement to the building height, zj/H, and the parameter

in Figure 8.1.

= (i-^)af H

(8.4)

Next, the resonant response term, ^, is found with a considerably greater

number of steps:

« = ( C ^ W + 2 C W C L N I - H C ^ L )

(Cw-fCL)2

r 7 \ nil 11 (8.5)

In this expression C^ and C L are the windward and leeward pressure

coefficients, respectively, Ni is the correlation coefficient between the two

110

faces, fj is the reduced fundamental frequency with regard to the vertical

direction, (niH)/u*, ni is the fundamental frequency in Hz, ^ is the fraction

of critical damping in the first mode, and Y^^ is a function depending on

B/H, ZQ/H, and f that is graphed in Figure 8.3.

The along-wind correlation coefficient, Ni, in Equation 8.5 is the

counterpart of R^ in the ASCE 7-95 Commentary method and depends on the

along-wind reduced frequency, T), by the same type of formula as Rj depends

on its corresponding reduced frequency, r|.

N i = along-wind size effect or correlation factor

1 1 = l - l ( l - e - 2 ^ n n V Tl Tl

(8.6)

where

r|=(3.85niAx)ii (8.7)

and

11 = 2 .5u, In ^ H ^

v ^ o y - 1 (8.8)

In Equation 8.8, the fi:iction velocity, u*, is related to the reference wind

speed, Uo(lO), through the retardation factor, p (depending on the exposure),

by:

u* = 0.0806 p Uo(lO). (8.9)

Next, the peak factor, g, can either be taken as 3.5 or 3.6, as in the

Davenport and ASCE 7-95 Commentary methods, respectively, or calculated

111

as follows. For the calculation, determine v, the frequency at which most of

the energy of the spectrum is concentrated:

V = 4i

(fl-h^) n,. (8.10)

Then the statistical peak factor is determined from the well-known formula

for an averaging time of one hour = 3600 seconds (Davenport, 1964):

g = V2 ln3,600v->- , ^'^'^'^ (8.11) V2 ln3,600v

Finally, the quantity J in the expression for GF in Equation 8.1, is found

from the following equation:

j ^ j ^ ( 1 0 0 ^ 3 0 z , ^ 3 0 z - d ) ^3^2)

H'

where zd = zero plane displacement; Zo= roughness length; and H = height of

the structure. The first term on the right is found from Figure 8.2, entering

with values of ZQ/H and zj/H.

8.3.2 Maximum Along-Wind Displacement

Mean and maximum along-wind deflections are determined in Simiu's

two papers by different methods. For the purpose of this report, the

maximum value is found simply as the gust response factor times the mean

value. Simiu (1976) computes the mean value at height z as:

112

a(z) = 0.3CD ( 2 ^

u. JB(z/H) (8.13)

where Cj) is the mean drag coefficient, m is the mass of the building per unit

height, u*, ni, h, J, and H are as defined above, B is the width of the

structure, and the shape of the deflection is taken as linear over the height of

the structure. Then the maximum displacements are found from:

amax(z) = (GF) a(z). (8.14)

Since the shape of the maximum deformation is linear, there is no curvature

in the tower and no stresses can be evaluated without assuming some

alternative shape, as was done for the ASCE 7-95 Commentary method.

8.4 Example Calculations for the Spun-Cast Concrete Pole

In this section the Spun-Cast pole is considered as an example of

Simiu's method. The input data is explained below. First, a conversion is

needed from the given 3-second gust reference wind speed to the

corresponding mean hourly wind speed. This is similar to employing the

quantity Ky in the Davenport method. Second, calculations are carried out

only for wind on the tower, since Simiu's method is not amenable to

determining the response of the conductors (what value of H would be

meaningful, for instance?). Third, only the maximum tower deflections, not

the maximum tower stresses, are determined since the deformed shape is

linear and has no curvature.

113

8.4.1 Summary of input data

Following parameters for the tower and the wind field are same as

those used in the previous methods.

Tower: Height, H 84 ft = 25.60 m

Average diameter, D = B = 1.8127 ft = 0.5525 m

Fundamental frequency, f = 0.9210 Hz.

Fraction of critical damping, ^ = 0.03

Force coefiicient, Cf = 0 . 8

Weight density of the tower material = 1 5 0 lb/ft3

Mass per unit length, m

= (150 X 1.8127^2 X 7c/4)lb/ft

= (387.11)lb/ft

= (387.11x3.281x0.4536)

= 576.11 kg/m

Modulus of elasticity of material, E^ = 7.81 x lO^psf

Conductors: (not considered)

Wind Field: 3-Second gust speed, V f 140 mph

Wind speed conversion factor, Ky 1.657 (from 3-sec to mean hourly)

Exposure Category C (open country)

8.4.2 General Given or Calculated Values:

The following calculated values are particular to Simiu's model.

Meanhourly wind speed, Uo( 10) = Vj-ef Ky = 140 mph/1.657

= 84.49 mph = 37.47 m/s

114

Roughness length, ZQ (Exposure C or II in Table 1 of Simiu [1980])

= 0.07 m

Retardation factor, p (Exposure C or II in Table 1 of Simiu [1980])

= 1.00

Ratio of roughness length to tower height, ZQ/H = 0.07/25.60

= 0.002734


The tower gust response factor is calculated as follows:

Background term, (E = (1 - z^/H) S = (1 - 0) 7.6

= 7.6

so, S from Fig. 8.1 (ZQ/H = 0.002734 and B/H s 0) is 7.6

Also, Z(i = 0 for Exposure C

Resonance term, ^

C^ = windward coeff. = 0.6

C L = leeward coeff. = 0.4

u, = friction velocity = 0.0806 p Uo( 10)

= 0.0806(1X37.47) = 3.020 m/s

u = 2.5 u. [In (H/zo) - 1];

= 2.5 (3.020) [ln(25.60/0.07) - 1] = 37.01 m/s

Ax = eff. long. dim. = 4 D = 4 (0.5525) = 2.21 m

J] = alongwind reduced frequency = (3.85 ni Ax)/ Ti;

= (3.85 X 0.9210 x2.21)/37.01 = 0.2217

Ni = along-wind corr. factor = (1/TI) - {l/2T]^)il - e' Tl)

= (1/0.2217)-(l/(2x0.22172))(l-e-2x0.2217) = 0.8672

115

fj = reduced fundamental frequency with respect to vertical

direction = niH/ u*

= 0.9210x25.6/3.020 = 7.8072

Yî = parameter from Simiu's Fig. 8.3 with B/H = 0,

?! = 7.81, and ZQ/H = 0.002734 (curve J ) = 0.0025

« = [(Cw2 + 2CWCL N I + CL2)/(CW + CL)2](7U fi/4^) Y ^

= [(0.62 + 2(0.6)(0.4)(0.8672) + (0.42)/(0.6 + 0.4)2](7c x

7.8072)/(4x0.03)](0.0025)

= 0.4784

Statistical peak factor, g

= V2 ln(3600 n) + 0.577/ V2 ln(3600 n)

= V2 ln(3600 X 0.08562) + 0.577/ V2 ln(3600 x 0.08562 = 4.152

where n = [ TF/iS + «) ] ni

= [V 0.4787/(6.6-1-0.4784)] (0.9210)

= 0.08562

Quantity J = J -i- (100 + 30 z^ + 30 z^^)/R^ = 16 + 100/(25.60)2

= 13.15

where z^ = 0 and J = parameter from Simiu's Fig. 8.2 with

Zd/H = 0 and ZQ/H = 0.002734 is 16

Tower Gust Response Factor, GF = 1 + g[1.23>/(B -i- R )/J]

= 1 + 4.152 [1.23 >/(7.6 + 0.4784)/13.15]

= 2.10

116


It is shown in this subsection that Simiu's method does not provide

useful design approach to wind on the conductor and groundwire. First of

all, value for the parameters H and B must be selected. This parameter H is

the height of the structure, which is normally assumed to be a building and

the parameter B is the width of the structure perpendicular to the wind. If

the vertical size of the conductor, that is, diameter d, is used for H, and the

horizontal extent of the conductor, that is, the span L, is used for B, then the

ratio B/H needed to enter the graphs for (E and Y-^^ are for greater than the

largest value of B/A = 3.0 considered in these graphs. Thus, it is impossible

in this case to complete the calculations. Incidentally, Simiu's graphs are not

generated by equations, and it would be dangerous to attempt to extrapolate

them beyond the ranges presented.

The only other logical choice for the parameter H might be the height

at which a conductor or groundwire is attached to a pole or the center of wind

pressure. This choice would avoid the problem of B/H falling outside the

range considered in the graphs. However, it presents a risk in using the

graphs, since H is intended to be the height of a vertically oriented structure

and how this assumption affects the curves cannot be assessed. It is likely

that curves are based on an analysis that considers wind on the structure

from the ground up to the height H, not just at the height H.

In light of this difiiculty in utilizing the graphs for wind on the

conductor and groundwire, Simiu's model is considered not to be applicable to

these components. The determination of a gust response factor is needed in

determining deflections and stresses, so the contributions of wind on the

conductors and groundwire to these response quantities cannot be evaluated

with confidence.

117


With Simiu's method (1976), tower deflections are determined as a

function of nondimensional height, (z/Ho). The mean deflection is given by

the following equation:

a(z) = 0.3CD / 2 \

u, €

JB(z / H) a(z)

= 0.3 (0.8) (3.0202/(576.11 x 0.92102)) (13.15) (0.5525 ) (z/H)

= 0.03254 (z/H).

and the mean tip deflection due to wind on the tower (at z = H) is 0.03254 m.

Finally, the peak value of the tip deflection is this number times the gust

response factor:

amax(z = H) = (GF) a(z) =2.10(0.03254) = 0.06834 meters

= 0.2242 ft.

8.5 Summary of Simiu's Model

Simiu's method (1976, 1980) is designed for hourly mean wind speed.

However, the model can be applied to any averaging time using Durst (1960)

or Krayer and Marshall's graphs (1992). Simiu's model assumes linear first

fundamental mode shape and uses graphs for most part of the calculations

for the gust response factor. It should be noted that the method of Simiu

does not provide a useful design approach for transmission line structures

because the graphs incorporated into the method are not adaptable to wind

on the conductors. The graphs are set up in terms of the tower height, H, a

parameter that refers to the size of a vertically oriented structure (assumed

to be like a building). Use of conductor diameter, d, for H would place the

ratio B/H well beyond the range considered in the graphs, and use of the

attachment height of the horizontally oriented conductor would likely be a

118

misuse of the graphs. Assessing this use is not feasible since Simiu does not

present the basis for the graphs.

119

CHAPTER 9

DISCUSSION OF RESULTS AND SENSITIVITY STUDY

9.1 Introduction

In this chapter the results obtained by Davenport's model and the

modified ASCE 7-95 Commentary method in Chapters VII and VIII,

respectively, for the prototype spun-cast concrete pole are evaluated and

corresponding results are given for the static-cast concrete pole. By following

the detailed equations and numerical calculations presented in the earlier

chapters, the reader can see what is involved in attempting to use each

method for a pole design. When that understanding is combined with a

comparison of the results obtained, a rational choice can be made.

9.2 Comparison of Spun-Cast Concrete Pole Results By Davenport and

modified ASCE 7-95 Method

Table 9.1 summarizes the results of the two methods considered for the

baseline spun-cast concrete pole of Figure 5.1. The following observations

may be made from Table 9.1.

1. All of the response results appear to be rather large for the 84-foot high

spun-cast concrete pole. A maximum stress of the order of 7,000 psi. is 83

percent of the ultimate design stress, f ', of 8,000 psi. and thus is large for

a working stress design, but it is perhaps acceptable for the extreme wind

considered. A tip deflection of more than 6 feet is also large for an 84-foot

pole.

2. The results from the Davenport and modified ASCE 7-95 Commentary

methods are fairly comparable, lending confidence to the validity of both

methods. The modified ASCE results are slightly larger. Either method

can be utilized.

120

3. Wind on the three conductors considered (650 feet long and 1.43 inches in

diameter) clearly dominates over wind on the tower and wind on the

groundwire in producing both stress and deflection.

Table. 9.1 Summary of Results for the 84-Foot Spun-Cast Concrete Pole

Calculated Quantity

Gust Response Factor

- for Tower

- for Conductors

- for Groundwires

Maximum Stress, psi.

- from Wind on Tower

- from Wind on Conductors

- from Wind on Groundwire

Total

Maximum Deflection, ft.




Total

Davenport

Model

1.045

0.664

0.619

1,860

4,420

600

6,890

0.58

5.48

0.32

6.39

Modified

ASCE 7-

95

Method

0.946

0.778

0.780

2,295*

4,775

790

7,765

0.66

5.92

0.38

6.95

Percent

Difference

-9.5

-1-17.2

-1-26.0

-t-23.1

-1-7.9

-I-31.1

-t-12.8

+13.6

-H7.9

-f-16.1

+ 8.8

Note: * Second derivative of first mode shape of uniform cross section pole used.

121

4. All of the gust response factors are less than or close to 1.0, as expected for

the 3-second gust wind speeds with which they are used.

5. The gust response factors determined by the modified ASCE 7-95 method

are smaller than those determined by the Davenport model for the tower,

but are larger for the conductors and groundwire. The approximately 10

percent difference in the tower gust response factors is due to a much

larger resonance term in Davenport's model than in the modified ASCE

method. For the wires the resonance terms are much less important than

the background for the tower in both methods (and, in fact, are negligible

in the ASCE method), so the larger gust response factors in the modified

ASCE method are inherent in the methodology for the background

response.

Table 9.2 shows that the background terms for the tower by the two

methods are very close, but there is a 61% difference in the terms for the

conductors. This difference for the conductors is related to the general forms

of Equation 6.6 in the Davenport model and Equation 3.13 in the ASCE

method. The forms are similar in that they both employ a ratio of conductor

length to integral scale in the denominator, but the multiplying coefficients

and exponents on this term are different and the value of the integral scale

are quite different. As a result of using an integral scale of only 220 feet (see

Table 6.1), the R* in the Davenport method is much smaller than the Q2 in

the ASCE method, which uses a calculated integral scale of 555.7 feet.

Table 9.2 also shows that the resonance terms for both the tower and

conductors by ASCE method are of the order of 70% smaller than those by

the Davenport method. For these calculations. Equation 6.9 and 6.10 are

used in the Davenport method and Equation 7.2 is used in the ASCE method.

Both depend in part on a reduced frequency, but the reduced frequency is

defined differently in each case, and the values differ by a factor of 10. By

the ASCE method, the reduced frequency for the conductors is niLz/Vz, which

122

Table 9.2 Comparison of Background and Resonance Contributions to the GRF in the Davenport and ASCE Methods

Background

Term

-for tower

-for conductors

Resonance Term

-for tower

-for conductors

Davenport

Method

(B*)

0.8748

0.2973

(R*)

2.336

0.1946

ASCE 7-95

Method

(Q2)

0.8356

0.5898

(R2)

0.6184

0.0586

%

Difference

-4.5

-1-60.6

-73.5

-69.9

gives 1.042 for the case considered, whereas in the Davenport method the

reduced frequency for the conductors is fwZo/Vo, which is 0.1039. These

differences are due to Davenport's use of Zo, the effective height, in defining

the reduced frequency, while the ASCE method defines it in terms of the

integral scale, Lz, which is much larger. These differences are adjusted

somewhat in the equations for R* and R2, but other factors also play a part,

including how the reduced frequency enters the equations, the damping

factor, and whether or not a size effect factor is used. In the ASCE method,

size effect factors in all three directions are incorporated into R2 and it would

be much larger than R* from the Davenport method if the same damping

were used but the size factors were omitted. However, the size factors are so

small (a multiplier of 0.1623 comes from Rb) as to make the final value of R2

much smaller than R*. Then the higher damping assumed in the ASCE

method makes R2 still smaller.

The lesson from this discussion is that the details of the two methods

considered reveal significant differences in many places. In the case

123

considered, in spite of the fairly close agreement in the overall GRF values,

they do not agree in the different contributions. In this case the resonance

contribution to the GRF is much larger in Davenport's method for both the

tower and conductors, but it is only large enough in comparison to the

background contribution to make Davenport's GRF larger for the tower.

As noted earlier, a "separation factor," e, of 0.75 is included in the

ASCE method calculations in the same way as proposed by Davenport in his

method. The total stress results of the modified ASCE 7-95 Commentary

method would differ from those of the Davenport method by an additional 15

percent, approximately, if the separation factor had been omitted in applying

the ASCE method.

The deflection results shown in Table 9.1 do not include the effects of

cracked section behavior in the prestressed concrete member. Such an

analysis would be extremely difficult to carry out. More detailed information

about the reinforcement would be needed, including its area, spacing, and

pretensioning stress, and the dynamic properties of the member would

change due to cracking. In fact, there would be a nonlinear type of d)mamic

response, with the section properties var3dng from those of a cracked to those

of an uncracked section in different portions of each cycle of motion and in

different degrees at different positions over the height of the tower. It can be

said, however, that cracking in the concrete would increase the calculated

deflections.

124

9.3 Comparison of Static-Cast and Spun-Cast Concrete Pole Results

In Table 9.3 results of the type presented in Table 9.1 for the spun-cast

concrete pole are included for the 84-foot static-cast concrete pole of Figure

5.1 as well as for the spun-cast pole. Data and results are again shown for

both design methods considered. The purpose of considering the static-cast

pole is to see if significant differences in the results might have occurred if a

different realistic set of pole properties had been adopted as an example

structure.

It can be seen in Table 9.3 that both types of poles follow the same

general pattern of results by the Davenport and the modified ASCE 7-95

Commentary methods. The cross section properties of the two tj^jes of poles

are different and hence the stresses and deflections are different. It is

interesting that under the same wind, the maximum stress is smaller in the

static-cast pole but the tip deflections is larger. For example, wind on the

conductors again dominates the total stresses and deflections. However, by

either method, there are differences in the results for static-cast and spun-

cast poles. These differences are caused by a combination of factors, that is,

differences in fundamental frequency, cross-section at various heights, drag

coefficient, and modulus of elasticity. The larger section modulus of the

static-cast pole at the base seems to help make its maximum stress smaller,

whereas the lower strength concrete makes its modulus of elasticity and thus

its deflections larger.

125

Table 9.3 Comparison of Results for the 84-Foot Static-Cast and Spun-Cast Concrete Poles

Calculated Quantity

Gust Response Factor

- for Tower

- for Conductors

- for Groundwires

Maximum Stress, psi.




Total

Maximum Deflection, ft.




Total

Davenport Model

Spun-

Cast

1.045

0.664

0.619

1,860

4,425

600

6,890

0.58

5.50

0.32

6.39

Static-

Cast

1.173

0.664

0.619

2265

2520

385

6600

2.08

8.30

0.48

10.9

Modified ASCE

7-95 Method

Spun-

Cast

0.946

0.778

0.780

2295*

4770

790

7765

0.66

5.92

0.38

6.95

Static-

Cast

0.999

0.778

0.780

3510*

2180

445

5100

1.33

8.96

0.55

10.8

Note: * Second derivative of first mode of uniform cross section pole used

126

9.4 Sensitivity Parameters

In this section, several parameters that affect the behavior of the

power line system considered but which have been kept constant in the

results so far are varied in order to determine the type and degree of their

effects. In the case of each varied parameter one value above and one below

the value for the prototype pole are included, always within a realistic range.

The parameters specified as fixed and as variables in the sensitivity study

are sununarized in Table 9.4. The values shown there represent the middle-

valued "baseline" structures.

Only parameters whose influences on the response are not obvious are

varied in the sensitivity study. These include: the tower height, h; the

conductor span, L; the tower damping factor, ^^•, the conductor damping factor

cond, and groundwire damping factor, ^ . The conductor damping factor,

cond.» 2^^ ^ ® groundwire damping factor, ^g^ are not varied in the case of

Davenport's model because they are calculated from the wind speed and the

wire diameter.

For other parameters which might vary in the real world, the effects

are more obvious. For example, if the pressure coefficients for the tower, Cft,

or the conductor, Ccond., increase, the corresponding forces increase in direct

proportion; therefore these parameters are not varied below.

The same wind field and terrain factors are also adopted for all cases.

They are those for the examples above, that is, a 3-second speed of 140 miles

per hour and a site with exposure C.

127

Table 9.4 Parameter Values for the Baseline Structures

Parameter

Varied Parameters

Height, h, ft.*

Conductor Span, L, ft

Tower Damping Factor, i*

Conductor damping factor, ^ cond.*

Groundwire damping factor, ^ g^

Fixed Parameters

Mean Tower Diameter/Side, ft

Fund. Tower Freq., fi, Hz

No. of Conductors

Conductor Diameter, d, ft.

Groundwire Diameter, d, ft.

Force coefficient, Cr

Spun Cast

Concrete Pole

84

650

0.03

0.4

0.4

1.8127

0.9210

3

0.11892

0.03125

0.80

Static Cast

Concrete Pole

84

650

0.03

0.4

0.4

1.679

0.6886

3

0.11892

0.03125

1.60

* Varied in the modified ASCE method only (calculated as a single values according to conductor size and wind speed in the Davenport method).

128

9.5 Sensitivity Results

All of the results obtained in the sensitivity study of both the poles

considered, first by the Davenport model and then by the modified ASCE 7-

95 Conunentary method, are shown in Appendix A. As noted at the bottom of

the table the maximum combined stress may not be the base stress or at the

same height as any individual maximum stress from wind on the tower, the

conductors or the groundwire. The maximum stress occurring at each height

of the tower may or may not occur at the same height as individual maximum

stresses. The computer program calculated the maximum combined stress at

each level considered and took the largest of all the maximum.

The results of the sensitivity studies are presented in graphical form

for the spun-cast pole in this subsection. Corresponding graphs are

presented in Appendix B for the static-cast pole.

Figure 9.1 shows the effects of tower height, h, and conductor span, L,

on the combined tip deflection and the combined stress due to wind on the

tower, the conductors, and the groundwire. Both Davenport and modified

ASCE 7-95 Conunentary results are included. The tower height is varied

from 70 to 100 feet, and the conductor span is varied from 550 to 750 feet

while the tower damping factor is maintained at the baseline value of 0.03

and the conductor and groundwire damping factors for the modified ASCE 7-

95 Commentary Method are maintained at the baseline value of 0.4. As

expected, for both the models, an increase in height of tower or span of

conductors increases both the total tip deflection and the stress. For the

tallest tower and longest span the maximum tip deflection is of the order of

12 feet and the maximum base stress in the concrete is of the order of 8,000

psi. It is also seen that the tower height has more effect on the tip deflection

than on the base stress. The graphs for deflections by the modified ASCE

and the Davenports model appear to lie on top of each other because the

results are very close. However, the two methods give distinct stresses.

129

Effect of Tower Height and Span of Conductors on Total Deflections due to Wind on Tower, Conductors, and

Groundwire

o

1 o

•Span=550ft. ASCE

O Span=550ft. Daven.

— -A — Span=650ft. ASCE

— -X — Span=650fl. Daven.

— X - Span=750ft. ASCE

— O - Span=750ft. Daven.

84

Tower Height, ft.

100

Effect of Tower Height and Span of Conductors on Total Stresees due to Wind on Tower, Conductors, and

Groundwire

10000 -r

M Q.

o is to

•Span=550ft. ASCE

0 Span=550ft. Daven.

— -A — Span=650ft. ASCE

— -X — Span=650ft. Daven.

— X - Span=750tt. ASCE

— O - Span=750ft. Daven.

70 84

Tower Height, ft.

100

Figure 9.1 Combined Response Sensitivity to Tower Height and Conductor Span for Spun-Cast Concrete Pole (Exposure C, Vref=140 mph, 5tower=0.03, ASCE Method; cond = 0.4, ^ = 0.4)

130

The effect of tower height on the maximum stress is seen from the

second figure to be essentially linear. The maximum stress for the 100 ft.

pole is approximately 25 percent greater than for the 70 ft. pole, indicating

that in spite of the increase in the wind velocity with height, the rate of

increase in stresses is lower than the rate of increase in height, 100/70 =

1.43. This result may be attributed to the larger cross section at the base of

the pole when it is used for a greater height. Nevertheless, the fairly simple

linear variation in stress shows that an expected trend occurs in spite of the

complexities of the calculations.

A similar conclusion can be made from the maximum deflection results

in first part of Figure 9.1, where a set of smooth lines with upward curvature

appear. Here, the expected result from a simple model of a concentrated

conductor force on a cantilever beam would be an increase in deflection in

proportion to the height cubed. The graphs do show this type of increase but

at a slower rate, as the ratio of tip deflection for the 100 ft. pole to that for the

70 ft. pole is approximately 2.2, whereas the length ratio cubed is (100/70)^ =

2.92. Although these curves look smooth and fairly simple, one would have

to look carefully into the details of each computational method to understand

them completely.

Figure 9.2 shows the effects of tower height, h, and tower damping

factor, t> ^^ h® tip deflection and the stress. The tower height is again

varied from 70 to 100 feet, and the tower damping factor is varied from 0.01

to 0.05 while the conductor span is maintained at the standard value of 650

ft. For the modified ASCE 7-95 Commentary method the conductor and

groundwire damping factors are fixed at the baseline values of cond = 0.4,

^ _0 4 As expected, an increase in the tower damping decreases both the

tip deflection and the base stress, but not by very much. In fact, the decrease

is hardly detectable for the total tip deflection and is only about 12 percent

131

c o -.0

I o

11 X -1 0 6 -9 --8 -

6 5 4 + i

0.01

Effect of Tower Height and Tower Damping on Total Deflections due to Wind on Tower, Conductors, and

Groundwire

-X

-o

=5=

•A •X

: ^

Ht.=70fL ASCE

• Ht=70fl Daven.

— -A — Ht.=84ft. ASCE

— -X — HL=84fL Daven

— X - Ht.=100ft. ASCE

— O - Ht=100ft Daven.

0.03

Damping in Tower

0.05

Effect of Tower Height and Tower Damping on Total Stresses due to Wind on Tower, Conductors, and

Groundwire

a. <n <n 9

W

Ht=70ft ASCE

•Ht.=70fl Daven.

— -A — Ht.=84ft. ASCE

— -X — Ht.=84ft DAven

— X - Ht.=100ft. ASCE

— O - Ht.=100ft Daven.

Figure 9.2 Combined Response Sensitivity to Tower Height and Tower Damping Ratio, towerfor Spun-Cast Concrete Pole(Exposure C, Vre(=140 mph. Span = 650 ft, ASCE Method; cond = 0.4, ^ = 0.4)

132

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 Â^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 ' aân

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:

SponîSO 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, î.,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 ônd. 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

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145

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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.

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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

•A •

• - X -

- - X -

_ . © -

•Spa

-Spp

i «660 f t

=55c n SapnêO a

Bpa i=«60 f t

Span=760 !>.

Sr« = - • 3 ^ .

ASCE

Davan

ASCE

Daver

ASCE

Za-er

Towar Haiffht ft

(2) Tip Deflection

Effect ofTower Height end Spnn of Condnotor on Streeee* dne to Wind on Tower only

— 0 ^ 6 o « - = " " "- ASCE 0 SpBr=''6? fl Ijivan

| . • 'A- - Sapn=6fiO ft A.SCE • - X Spu'^'i')' !V Davan _ .X- Spii-^s'Vft ASCE - - C - •Spana7>0ft Davan

(3) Stress (a) Wind Loads on Tower

Effect of Tower Height end Spen of Condnctor on (ORDc

0- — ^ _ - - - g -

0<36 -" — — :r

O — ; « • - = " " ' A8CE

6 -i.c-^*".: n ASCE X • Sp«ni««0 *. Ow>an X - •Spa-'^'Wft ASCE O - S-a- = *'-Oft ••>'•'

- c « . r Ha>«M

(1) Conductor Gust Response Factor, (GRF)c

Effect of Tower Height and Span of Conductor on Deflection, due to Wind on Conductor! onlr

'.6 -

e n -

= '' -i. •• •

70

. ff * ?=

' • ' • - :

- .= .^^__^^-i=s^^^^^

84

Towar Ha i fh t ft

- : -

• , »

— • Srii-=»6C ". ASCE

O Sr9.'>=^5-- * I;avar

• -A- • S.-.n=6»0 "- ASCE

• -X- • Spa-s*"^- ". D a v . '

- X - 5 - . - = - - : f V=^CE

— 0 - •Spii.- = "5C ft ! • • • -

(2) Tip Deflection

Effect of Tower Hel«ht and Span of Condnctor on Streeaae dne to Mad on Conduoture only

360C -

a 3000 4

i 2600 X : •

•; 2 0 0 0 ^

•. = :- = '-.-.:.-f"-" " :- A^

84

T m n r HaicÔ- •

. . , . . - Q

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100

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0 8pa^««^: f. D«»an

• • -X • • 9p«r"S*C (>. Davar

- X - S p » - « ' ^ " ". ASCE

— - 0 — • 8par*"r~3 1 D a m

(3) Stress (b) Wind Loads on Conductors

Figure B.3 Response Sensitivity to Tower Height and Conductor Span Separated by Load Component for Static-Cast Concrete Pole (Exposure C, Vrcr = 140 mph, ^t.,uvr=0.03, ASCE Method; Ccnd = 0.4)

16^

EfTeoi of Tower Height and Span of Groundwire on (ORRff*

1 76 ^ * *

k 0 7

50.66 06

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• x -

- • X -

_ - o -

- S p a n c u o n ASCE

- Spacvs6fi0 ft. Deven

Sapn=«SOft ASCE

Spana660 ft Daven

•Span^SOft^ ASCE

•6f>an=760ft Dawn

84

Tower Haîht. fl

100

(1) Groundwire Gust Response Factor, (GRF)^v

Effect of To»er Helgrfat and Span of Oroundwlre on DeQection* due to Wind oa Oroundwlre only

• - -A-

- • • X -

- • X -

- - 0 -

-Sp«n=S60ft ASCE

-Span=550ft Daven

SapnxieOft ASCE

S<>an<«60 ft. Daven

-Span=7MA ASCE

-Span=7S0ft Daven

100

(2) Tip Deflection

Effect of Tower H e i ^ t a n d 8 p « a of Qrooadwlre o a Stre«««« due to >^^d o n Gronndwlrc only

SpaaaSeOft ASCE

SpanaCeOft Davan

Sapoi<60 ft. ASCE

Spana«SO ft Davfn

Spen«7Kin ASTE

Spen=<M>n D«v»ii

100

(3) Stress (c) Wind Loads on the Groundwire

Figure B.3 (contd.) Response Sensitivity to Tower Height and Groundwire Span Separated by Load Component for Static-Cast Concrete Pole (Exposure C, Vrof 140 mph, q,„.o,=0.03, ASCE Method; cmd = 0.4)

166

Effect o f T o w e r H«l|rbt and Tower Damping on <ORF)t

001

l - a -1—*~ I - - - A - -

|..-x--;- -x-I - - 0 -

-Hi=70f» ASCE

- Ht='0 f\ Drnvrn

Hl=ft4 ft ASCE

Hl=«4 ft D«vvn

•Ht=100ft ASCE

•Hl = l<X)ft Dev»n

0 03

Dampuif in Tower

0 0 6

(1) Tower Gust Response Factor, (GRF)t

Effect o f T o w e r Height and Tower Damping on Deflectiona due to Wind on Tower only

001 0 03

Dampin( in Tower

0 06

. - - A -

• -x-- - X -

- ^ -

-Ht=70f t ASCE

-HoTO ft. Davan

Hti«<ft . ASCE

Hta«4 ft. Davan

-Htz iooft ASCE

•HlslOO ft Davan

(2) Tip Deflection

Effect o f T o w e r Height and Tower Damping on Streaaea due to >Mnd on Tower oniv

0 01 0 00

Damptng in Tower

0 05

•A-

• • -x

- X -

- o-

-H i»70f t ASCE

-Htz70f t Dav«>

H i :«4n ASCE

Ht=84 ft Davan

•Ht=100ft ASCE

-HtslOOn Deven

(3) Stress Wind Loads on the Tower

Fj;rure B.4 Response Sensitivity to Tower Height and Tower Damping Ratio, flower, Separated by Load Component for Static-Cast Concrete Pole (Exposure C, Vn.i=140 mph. Span = 650 ft, ASCE Method; qcnd = 0.4)

167

APPENDIX C FORTRAN CODE FOR SENSITIVITY STUDY

(1) Spun-Cast Concrete Pole By Davenport's Model

C PROGRAM TO CALCULATE C (I)—GUST RESPONSE FACTOR FOR SPUN-CAST CONCRETE POLE BY THE c PROCEDURE GIVEN BY DAVENPORT C (ID-DEFLECTIONS & STRESSES

CHARACTER* 12 INP,OUT,EXPOSE C STRUCTURE PROPERTIES

DIMENSION ADT(100),DIAI(100),AI(100) DIMENSION AHT(10,1),NSPC(10,1)ASCC(10,1),ASCW(10,1) DIMENSION AZTT(10,1),FREQT(10,1)

C DEFLECTION PARAMETERS DIMENSION Z(100),PHIZ(100),FCC(10,1),FCT(10,1),FCW(10,1)

C BENDING MOMENT PARAMETERS DIMENSION AMT( 100) AMC(100),AMW(100),TM(100) DIMENSION SUM(100),PZ(100),VZ(100)

C STRESSES DIMENSION STRT(100),STRC(100),STRW(100),TSTR(100)

C ON-SCREEN INPUT WRITE(*,*)'INPUT FILE ' READ(*,444)INP WRITE(*,*)OUTPUT 2 FILE-RE AD(*,444)OUT

444 F0RMAT(A12)

0PEN(5,FILE =INP,STATUS = OLD) 0PEN(6,FILE = OUT.STATUS = NEW) WRITE(6,*)'INP FILE....=',INP

WRITE(*,*)EXPOSURECATEGORY....=',EXPOSE READ(*,444)EXPOSE WRITE(6,*)EXPOSURECATEGORY....=',EXPOSE

C READING INPUT DATA

WRITE(6,*)' UNITS : FT,LB (UNLESS MENTIONED OTHERWISE*'

READ(5,*)NH,NSP,NSC,NSW,NZT,NFT,NFC,NFW,NFREQ WRITE(6,1)NH,NSP,NSC,NSW,NZT,NFT,NFC,NFW,NFREQ

1 F0RMAT(1X,9I3)

WRITE(6,*)'NH=# OF HEIGHTS OF TOWER WRITE(6,*)' ' WRITE(6,*)' # HEIGHT(FT)' DO 121I=1,NH READ(5,*)N,AHT(N,1)

168

WRITE(6,*)N,AHT(N,1) 121 CONTINUE

WRITE(6,*)NSP=# OF SPANS OF CONDUCTOR" WRITE(6,*)* • WRITE(6,*)' # "SPAN(FT)' D0 122I=1,NSP READ(5,*)N,NSPC(N,1) WRITE(6,*)N,NSPC(N,1)

122 CONTINUE

WRITE(6,*)'NSC=# OF SAGS OF CONDUCTOR' WRITE(6,*)' ' WRITE(6,*)' # SAG(FT)' DO 140 I=1,NSC READ(5,*)N,ASCC(N,1) WRITE(6,*)N,ASCC(N, 1)

140 CONTINUE

WRITE(6,*)'NSW=# OF SAGS OF GROUND WIRE' WRITE(6,*)' • WRITE(6,*)' # SAG(FT)' DO 141 I=1,NSW READ(5,*)N,ASCW(N, 1) WRITE(6,*)N ASCW(N, 1)

141 CONTINUE

WRITE(6,*)'NH=# OF DAMPING CASES IN TOWER' WRITE(6,*)' • WRITE(6,*)' # %0F CRITICAL' D0 123I=1,NZT READ(5,*)N,AZTT(N,1) WRITE(6,*)N,AZTT(N,1)

123 CONTINUE

WRITE(6,*)NH=# OF FLEX.COEFF. FOR WIND ON TOWER.' WRITE(6,*)' ' WRITE(6,*)' # FLEX. COEFF' DO 139I=1,NFT READ(5,*)N,FCT(N,1) WRITE(6,*)N,FCT(N,1)

139 CONTINUE

WRITE(6,*)'NFC=# OF FLEX.COEFF. FOR WIND ON COND.' WRITE(6,*)' ' WRITE(6,*)' # FLEX. COEFF' DO 138 I=1,NFC READ(5,*)N,FCC(N,1) WRITE(6,*)N,FCC(N,1)

138 CONTINUE

WRITE(6,*)'NFW=# OF FLEX.COEFF. FOR WIND ON GW WRITE(6,*)' •

169

WRITE(6,*)' # FLEX. COEFF" D0 142I=1,NFW READ(5,*)N,FCW(N,1) WRITE(6,*)N,FCW(N, 1)

142 CONTINUE

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.

READ(5,*)DT,TAPT,WTH,AET,RH0T,DC,DGW,CFT,CFC,CFW,JDIV READ(5,*)VREF,ALPHA,ZG,AKALZ,EPSI,RH0AIR,AKV,G

START CALCULATIONS

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)

1200 FORMAT(lX,F9.3,2X,F9.4,2X,F9.4,2X,F9.4) 1222 CONTINUE

WRITE(6,*)'II-CALCULATI0NS FOR GUST RESPONSE FACTORS' C C EQUIVALENT HT OF THE TOWER = 0.65H

EQHT=0.65*AHT(L,1) E=4.7*(AK**0.5)*((33.0/EQHT)**(1.0/ALPHA)) BC=1.0/(1.0+(0.8*NSPC(M,1)/ALZ)) BT=1.0/(1.0-»-(0.375*AHT(L,l)/ALZ)) VBAR= 1.605 *((EQHT/ZG)**(1.0/ALPHA))*88.0/60.0*VREF/AKV

C FOR CONDUCTORS FREQC= SQRT(1.0/ASCC(M,1)) ZHIC =0.000048*VBAR/(FREQC*DC)*CFC AAC= 0.0113*EQHT/(ZHIC*NSPC(M,1)) ABC=(FREQC*EQHTAÂR)**(-1.66666667) RC= AAC*ABC

C FOR GROUND WIRE FREQW= SQRT(1.0/ASCW(M,1)) ZHIW =0.000048*VBAR/(FREQW*DGW)*CFW AAW= 0.0113*EQHT/(ZHIW*NSPC(M,1)) ABW= (FREQW*EQHTArBAR)**(-1.66666667) RW= AAW*ABW

C FOR TOWER BB =0.0123/AZTT(N,1) BA=(FREQT(L,1)*EQHTAÂR)**(-1.66666667) RT= BB*BA

GEFC= (1+(G*EPSI*E*SQRT(BC+RC)))/(AKV**2) GEFW= (1+(G*EPSI*E*SQRT(BC+RW)))/(AKV**2) GEFT= (1+(G*EPSI*E*SQRT(BT+RT)))/(AKV**2)

171

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

1211 CONTINUE C SUMMING UP MOMENTS @ BASE

SUM1=0.0 D0 556J=JDIV,1,-1 SUM1=SUM1+(PZ(J)*((J-0.5)*DIV))

556 CONTINUE AMTO=SUMl*GEFT STRO=AMTO*DB/(2* 144*AIB) WRITE(6,110)DMMM,DMMM,DMMM,AMTO,STRO

C SUMMING UP MOMENTS @ HT Z' DO 555 K=1,JDIV SUM(K) =0.0 D0 1233J=JDIV,K,-1 IF(J.GT.K)THEN SUM(K) =SUM(K) + (PZ(J)*((J-(K+l)-t-0.5)*DIV)) ELSE SUM(K) =SUM(K) + (PZ(J)*((J-K+0.5)*DIV)) ENDIF

1233 CONTINUE AMT(K) =SUM(K)*GEFT STRT(K)=AMT(K)* ADT(K)/(2* 144* AI(K)) WRITE(6,110)Z(K),VZ(K),PZ(K),AMT(K),STRT(K)

110 FORMAT(1X,F9.2,1X,F7.2,1X,F7.2,1X,F10.3,1X,F8.3) 555 CONTINUE

WRITE(6,*)' FOR WIND ON CONDUCTORS ' VBARC=(((AHT(L,l)-19.0)/33.0)**(l/ALPHA))*VREF*88.0/60.0

172

FC=0.5*RHOAIR*(VBARC**2)*DC*NSPC(M,1)*GEFC*CFC*3.0

AMCO=FC*(AHT(L,1)-19.0) SIGMA0=AMC0*DB/(2*AIB* 144.0) DMMM=0.0 WRITE(6,*)'^ • WRITE(6,*)'...Z AMC STRESS" WRITE(6,*)'_FT. FT.LB. PSI. ' WRITE(6,357)DMMM,AMCO,SIGMAO

LDIV=((AHT(L,1)-19.0)/DIV)+1.0 WRITE(6,*)'LDIV =',LDIV DO 124 K =l,LDIV-2 AMC(K) =FC*(AHT(L,1)-19.0-Z(K)) STRC(K) =AMC(K)*ADT(K)/(2*AI(K)* 144.0) WRITE(6,357)Z(K),AMC(K),STRC(K)

357 FORMAT(1X,F9.2,2X,F10.3,2X,F10.3) 124 CONTINUE

AMC(LDIV-1)=FC*(AHT(L,1)-19.0-Z(LDIV-1)) STRC(LDIV-1)=AMC(LDIV-1)*ADT(LDIV-1)/(2*144*AI(LDIV-1)) WRITE(6,357)Z(LDIV-1),AMC(LDIV-1),STRC(LDIV-1)

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)

WRITE(6,*)' ' WRITE(6,*)'...Z TOWER CONDUCTOR GW TOTAL..STRESS " WRITE(6,*)" MOMENTS MOMENTS MOMENTS PSI.' WRITE(6,*)' . •

173

WRITE(6,112)DMMM,AMTO,AMCO,AMGO,TMO,TS WRITE(6,*) AMT(JDIV)=0.0 DO 125K=1,JDIV

TM(K) =AMT(K) +AMC(K)+AMW(K) TSTR(K) =TM(K)*ADT(K)/(2*144*AI(K))

WRITE(6,112)Z(K),AMT(K),AMC(K)AMW(K),TM(K),TSTR(K) 112 FORMAT(1X,F9.4,1X,F10.3,1X,F10.3,1X,F10.3,1X,F14.3,1X,F10.3)

WRITE(6,*) 125 CONTINUE

C C III—-DEFLECTION CALCULATIONS C

VREFD=VREF*88.0/60.0 DEFT=(VREFD**2)*FCT(L, 1)*GEFT DEFC=FC*FCC(L,1) DEFW=FGW*FCW(L,1) TOTAL=DEFT+DEFC+DEFW

WRITE(6,*)'DEFLECTI0N AT TOP DUE TO WIND ON TOWER=",DEFT

WRITE(6,*)'DEFLECTI0N AT TOP DUE TO WIND ON CONDUCTOR='J)EFC

WRITE(6,*)'DEFLECTI0N AT TOP DUE TO WIND ON GW =",DEFW

WRITE(6,*)'T0TAL DEFLECTION OF THE TOWER =',TOTAL

224 CONTINUE 223 CONTINUE 2222 CONTINUE

END

174

(2) Spun-Cast Concrete Pole By ASCE 7-95 Commentary Method

C PROOGRAM TO CALCULATE C (I)—GUST RESPONSE FACTOR FOR SPUN CAST CONCRETE POLE BY THE C PROCEDURE GIVEN IN ASCE 7-95 AND C (II)-DEFLECTIONS & STRESSES

CHARACTER* 12 INP,OUT,EXPOSE

C STRUCTURE PROPERTIES DIMENSION ADT(100),DIAI(100),AI(100)AHT(10,1),NSPC(10,1) DIMENSION ASCC(10,1),ASCW(10,1)^TT(10,1),AZCC(10,1)AZCW(10,1) DIMENSION FREQTdO.l)

C DEFLECTION PARAMETERS DIMENSION Z(100),PHIZ(100),PHI2Z(100),FCC(10,1),FCW(10,1)

C BENDING MOMENT PARAMETERS DIMENSION AMT(100),AMC(100),AMW(100),TM(100)

C STRESSES DIMENSION STRT(100),STRC(100),STRW(100),TSTR(100)

C ON-SCREEN INPUT WRITE(*,*)'INPUT FILE' READ(*,444)INP WRITE(*,*)'OUTPUT 2 FILE' READ(*,444)OUT

444 F0RMAT(A12) 0PEN(5,FILE=INP,STATUS ='OLD') 0PEN(6,FILE=0UT,STATUS ='NEW)

WRITE(6,*)'INPUT FILE =',INP WRITE(*,*)'EXPOSURECATEGORY...=',EXPOSE READ(*,444)EXPOSE WRITE(6,*)'EXP0SURE CATEGORY =',EXPOSE WRITE(6,*) WRITE(6,*) WRITE(6,*)'- UNITS : FT,LB WRITE(6,*)' WRITE(6,*) READ(5,*)NH,NSP,NSC,NSW,NZT,NZC,NZW,NFC,NFW,NFREQ WRITE(6,*)NH,NSP,NSC,NSW,NZT,NZC,NZW,NFC,NFW,NFREQ

WRITE(6,*)'NH= # OF HEIGHTS OF TOWER' WRITE(6,*)' ' WRITE(6,*)'# HEIGHT(FT)' D0 121I=1,NH READ(5,*)N,AHT(N,1) WRITE(6,*)N,AHT(N,1)

121 CONTINUE 175

WRITE(6,*)'NSP= # OF SPANS OF THE CONDUCTOR' WRITE(6,*)' ' WRITE(6,*)"# SPAN(FT)' DO 122 1=1,NSP READ(5,*)N,NSPC(N,1) WRITE(6,*)N,NSPC(N,1)

122 CONTINUE

WRITE(6,*)'NSC= # OF SAGS OF THE CONDUCTOR' WRITE(6,*)' • WRITE(6,*)'# SAG(FT)' DO 138 I=1,NSC READ(5,*)N,ASCC(N,1) WRITE(6,*)N^CC(N, 1)

138 CONTINUE

WRITE(6,*)NSW= # OF SAGS OF THE GROUND WIRE' WRITE(6,*)' ' WRITE(6,*)'# SAG(FT)' DO 139 I=1,NSW READ(5,*)NASCW(N,1) WRITE(6,*)N,ASCW(N,1)

139 CONTINUE

WRITE(6,*)'NZT= # OF DAMPING CASES IN TOWER' WRITE(6,*)' ' WRITE(6,*)'# %0F CRITICAL DAMPING-DO 123 I=1,NZT READ(5,*)N,AZTT(N,1) WRITE(6,*)N,AZTT(N,1)

123 CONTINUE

WRITE(6,*)'NZC= # OF DAMPING CASES IN CONDUCTOR' WRITE(6,*)' ' WRITE(6,*)'# %0F CRITICAL DAMPING* D 0 124I=1,NZC READ(5,*)NAZCC(N,1) WRITE(6,*)N,AZCC(N,1)

124 CONTINUE

WRITE(6,*)NZW= # OF DAMPING CASES IN GROUND WIRE" WRITE(6,*)' ' WRITE(6,*)'# %0F CRITICAL DAMPING' DO 140I=1,NZW READ(5,*)N,AZCW(N,1) WRITE(6,*)N,AZCW(N,1)

140 CONTINUE

WRITE(6,*)'NFC= #0F FLEXIBILITY COEFF.FOR' WRITE(6,*)' WIND ON COND.' WRITE(6,*)' • WRITE(6,*)'# FLEXIBIUTY COEFF'

176

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.

READ(5,*)DT,TAPT,WTH,AET,RH0TAM0DEXP,DC,DGW,CFT,CFC,CFW,JDIV READ(5,*)VREF,BBAR,ABAR,BHAT,AHAT,G,EPSI,ZMIN,CC,AL,RH0AIR

START CALCULATIONS

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)

WRITE(6,*)' • WRITE(6,*)'I—-CALCULATION OF TOWER CROSS SECTION PROPERTIES" WRITE(6,*)" ' AVGDT= DT+(TAPT*AHT(L,l)/2.0) DMH =AVGDT -(2.0*WTH) AVGAT=3.14159*((AVGDT**2)-(DMH**2))/4.0 OMEGAT= FREQT(L,1)*2*3.14159 ADB= DT+(TAPT*AHT(L,1)) DIB=ADB-(2.0*WTH)

AIB=3.14159*((ADB**4)-(DIB**4))/64.0

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

K=l@ BOTTOM D0 11K=1,JDIV

DIV=AHT(L,1)/JDIV PHIZ(K) = K*DIV/AHT(L,1) Z(K) = K*DIV ADT(K)= ADB-(TAPT*Z(K)) DIAI(K)=ADT(K)-(2.0*WTH) AI(K)=((ADT(K)**4)-(DIAI(K)**4))*3.14159/64.0

WRITE(6,1200)Z(K),ADT(K),DIAI(K),AI(K) 1200 FORMAT(lX,F7.2,3X,F7.4,3X,F7.4,3X,F7.4) 11 CONTINUE

C C II—-CALCULATIONS FOR GUST RESPONSE FACTORS C

C -—GEF FOR TOWER C EQUIVALENT HT OF THE TOWER = 0.6H

EQH = 0.6*AHT(L,1) IF(EQH.LT.ZMIN)THEN

178

c c

EQH=ZMIN ELSE EQH=0.6*AHT(L,1) ENDIF

TIRBULENCE INTENSITY AIZ =CC*((33.0/EQH)**(1.0/6.0))

INTEGRAL SCALE OF TURBULENCE ALZ =AL*((EQH/33.0)**EPSI)

BACKGROUND TURBULENCE QSQ =1.0/(1.0+(0.63*(((AVGDT + AHT(L,1))/ALZ)**0.63)))

VREFFTPS = VREF*88.0/60.0

VREFZBAR = BBAR*((EQH/33.0)**ABAR)*VREFFTPS

REDUCED FREQUENCY ANl =FREQT(L,1)*ALZ/VREFZBAR

RESONANCE AT REDUCED FREQUENCY AA =7.465 * ANl BB =(1.0+(10.302*AN1))**1.6667 RN=AA/BB

ETAH =4.6*FREQT(L,1)*AHT(L,1)ATIEFZBAR ETAB =4.6*FREQT(L,1)*AVGDTATIEFZBAR ETAD =15.4*FREQT(L,l)*AVGDTAaiEFZBAR

RH =(1/ETAH) - (1/(2*(ETAH**2))*(1-(EXP(-2*ETAH)))) RB =(1/ETAB) - (1/(2*(ETAB**2))*(1-(EXP(-2*ETAB)))) RD =(1/ETAD) - (1/(2*(ETAD**2))*(1-(EXP(-2*ETAD))))

RSQ=RN*RH*RB*(0.53+(0.47*RD))/AZTT(N,1)

GEF=(1.0-»-((2.0*G*AIZ*0.75)*SQRT(RSQ+QSQ)))/(1.0+(7.0*AIZ))

WRITE(6,*)' ' WRITE(6,*)'GUST EFFECT FACTOR.FOR TOWER =',GEF WRITE(6,*)' '

GEF FOR CONDUCTORS FREQC =SQRT(1.0/ASCC(M,1)) * 19 FT. IS DEDUCTED FOR THE MIDDLE CONDUCTOR @ 19FT FROM TIP OF THE TOWER

EQHC=AHT(L,1)-(0.66667*ASCC(M,1))-19.0 179

AIZC =CC*((33.0/EQHC)**(1.0/6.0)) ALZC =AL*((EQHC/33.0)**EPSI)

BACKGROUND TURBULENCE QSQC =1.0/(1.0 + (0.63*(((DC -»- NSPC(M,1))/ALZC)**0.63)))

VREFZBARC = BBAR*((EQHC/33.0)**ABAR)*VREFFTPS

REDUCED FREQUENCY ANIC =FREQC*ALZC/VREFZBARC

RESONANCE AT REDUCED FREQUENCY RNC =7.465 * AN1C/(1+((10.302*AN1C)**1.6666667))

ETAHC =4.6*FREQC*DC/VREFZBARC ETABC =4.6*FREQC*NSPC(M,1)/VREFZBARC ETADC =15.4*FREQC*DC/VREFZBARC

RHC =(1/ETAHC) - (1/(2*(ETAHC**2))*(1-EXP(-2*ETAHC))) RBC =(1/ETABC) - (y(2*(ETABC**2))*(l-EXP(-2*ETABC))) RDC =(iyETADC) - (1/(2*(ETADC**2))*(1-EXP(-2*ETADC)))

RSQC=RNC*RHC*RBC*(0.53+(0.47*RDC))/AZCC(J,1)

GEFC =(1.0 -h((2.0*G*AIZC*0.75)*SQRT(RSQC+QSQC))) & /(1.0+(7.0*AIZC))

WRITE(6,*)' • WRITE(6,*)'GUST EFFECT FACTOR.FOR CONDUCTORS....=',GEFC WRITE(6,*)' •

C --—GEF FOR GROUND WIRE FREQW =SQRT(1.0/ASCW(M,1))

C * 0.5 FT. IS DEDUCTED FOR THE GW. @ 0.5FT. FROM TIP OF C THE TOWER

EQHW=AHT(L,l)-(0.66667*ASCW(M,l))-0.5 AIZW =CC*((33.0/EQHW)**(1.0/6.0)) ALZW =AL*((EQHW/33.0)**EPSI)

BACKGROUND TURBULENCE QSQW =1.0/(1.0 + (0.63*(((DGW + NSPC(M,1))/ALZW)**0.63)))

REF. WIND SPEED AT EQ.HT. OF STRUCTURE

VREFZBARW = BBAR*((EQHW/33.0)**ABAR)*VREFFTPS WRITE(6,*)'MEAN HOURLY WIND SPEED AT

REDUCED FREQUENCY 180

ANIW =FREQW*ALZW/VREFZBARW

RESONANCE AT REDUCED FREQUENCY RNW =7.465 * AN1W/(1+((10.302*AN1W)**1.6666667))

ETAHW =4.6*FREQW*DGW/VREFZBARW ETABW =4.6*FREQW*NSPC(M,1)/VREFZBARW ETADW =15.4*FREQW*DGW/VREFZBARW

RHW =(1/ETAHW) - (1/(2*(ETAHW**2))*(1-EXP(-2*ETAHW))) RBW =(1/ETABW) - (1/(2*(ETABW**2))*(1-EXP(-2*ETABW))) RDW =(1/ETADW) - (1/(2*(ETADW**2))*(1.EXP(-2*ETADW)))

RSQW=RNW*RHW*RBW*(0.53+(0.47*RDW))/AZCW(ND,1)

GEFW =(1.0 +((2.0*G*AIZW*0.75)*SQRT(RSQW+QSQW)))/(1.0-»-(7.0*AIZW)) WRITE(6,*)' ' WRITE(6,*)'GUST EFFECT FACTOR.FOR GW =',GEFW WRITE(6,*)' •

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)

(AT Z=0 Z^=0) P=((1.875/AHT(L,l))**2)/2.7245 PHI2Z0=P*( 1.3622*2.0)

C0NSTANT=(RH0AIR*AVGDT*AHT(L,1)*CFT*(VHAT**2)*AK*GEF)/ 1 (2*AMODMASS*(OMEGAT**2))

STRESSES @ BASE AMTO=PHI2ZO*AET*AIB*CONSTANT STRO=(AMTO*ADB)/(2*AIB*144.0) WRITE(6,*)' ' WRITE(6,*)'.Z MOMENT STRESS" WRITE(6,*)" Z=0 @ BOT. LB.FT PSI." WRITE(6,*)" ' WRITE(6,1300)DMMM,AMTO,STRO

DO 112K= 1,JDIV

Q=SIN(1.875*PHIZ(K)) R=COS(1.875*PHIZ(K)) S=SINH(1.875*PHIZ(K)) T=COSH(1.875*PHIZ(K))

181

PHI2Z(K)=P*(-Q-S+(1.3622*(T+R))) AMT(K)=PHI2Z(K)*AET*AI(K)*C0NSTANT STRT(K)=(AMT(K)*ADT(K))/(2*AI(K)* 144.0) WRITE(6,1300)Z(K)AMT(K),STRT(K)

1300 FORMAT(1X,F9.2,2X,F10.2,2X,F9.2) 112 CONTINUE

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

AMCO= FC*(AHT(L,1)-19.0) SIGMAZ =(AMCO*ADB)/(2.0*AIB* 144.0)

WRITE(6,*)' • WRITE(6,*)'Z MOMENT STRESS' WRITE(6,*)'Z=0 @ BOT. LB.FT. PSI. ' WRITE(6,*)' " WRITE(6,1300)DMMM,AMCO,SIGMAZ

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)

222 CONTINUE

WRITE(6,*)'-—FOR WIND ON GROUND WIRES ' AKW=1

VHATW=BHAT*((EQHW/33)**AHAT)*VREFFTPS FW=0.5*RHOAIR*(VHATW**2.0)*NSPC(M,1)*DGW*AKW*GEFW*CFW

WRITE(6,*)'F0RCE ACTING AT THE TOP OF TOWER DUE TO ' WRITE(6,*)'GW. WIND LOADING IN LB =",FW

AMGO= FW*(AHT(L,l)-0.5) SIGMAZW =(AMGO*ADB)/(2.0*AIB*144.0)

WRITE(6,*)" WRITE(6,*)'Z MOMENT STRESS' WRITE(6,*)'Z=0 @ BOT. LB.FT. PSL' WRITE(6,*)' WRITE(6,1300)DMMM,AMGO,SIGMAZW

182

DO 2223 K=l,JDIV P =JDIV-K AMW(K) =FW*(Z(P)-0.5) STRW(K) =AMW(K)*ADT(K)/(2*AI(K)*144.0) WRITE(6,1300)Z(K)AMW(K),STRW(K)

2223 CONTINUE

C ADDING STRESSES DUE TO WIND ON TOWER,CONDUCTORS,AND GROUND WIRE

TMO=AMTO+AMCO+AMGO SIGMAZO=TMO* ADB/(2*AIB* 144.0) WRITE(6,*)' * WRITE(6,*)'Z TOTAL MOMENT STRESS' WRITE(6,*)'Z=0 @ BOTTOM LB.FT. PSI.' WRITE(6,*)' ' WRITE(6,1400)DMMM,TMO,SIGMAZO

D0 555K= 1,JDIV TM(K)=AMT(K) +AMC(K)-i-AMW(K) TSTR(K)=TM(K)*ADT(K)/(2*AI(K)*144.0)

WRITE(6,1400)Z(K),TM(K),TSTR(K) 1400 F0RMAT(1X,F7.2,3X,F13.2,3X,F11.2) 555 CONTINUE

WRITE(6,*)' • WRITE(6,*)'III—CALCULATION OF MAXIMUM DEFLECTIONS' WRITE(6,*)' '

WRITE(6,*)'—-WIND ON TOWER ' XMAX=0 WRITE(6,*)' • WRITE(6,*)'.Z XMAXZ' WRITE(6,*)' Z=0 @ BOTTOM" WRITE(6,*)' ' WRITE(6,1500)MMM,XMAX

1500 FORMAT(1X,I3,3X,F10.3) D0 111K=1,JDIV

XMAXZ =CONSTANT*(PHIZ(K)**AMODEXP) P=((1.875/AHT(L,l))**2)/2.7245 WRITE(6,*)Z(K),XMAXZ

111 CONTINUE

XMAXZ=CONSTANT*(PHIZ(JDIV)**AMODEXP)

WRITE(6,*)'DEFL. DUE TO WIND ON TOWER =',XMAXZ

WRITE(6,*)'—FOR WIND ON CONDUCTORS & GROUNDWIRE--183

DEFC=FC*FCC(L,1) WRITE(6,*)'DEFLECTI0N AT TOP DUE TO WIND ON COND....=',DEFC DEFW=FW*FCW(L,1) WRITE(6,*)'DEFLECTION AT TOP DUE TO WIND ON GW =",DEFW

TOTAL=DEFC+DEFW+XMAXZ

WRITE(6,*)T0TAL DEFLECTION =',TOTAL

226 225 224 223 2222

CONTINUE CONTINUE CONTINUE CONTINUE CONTINUE END

184

APPENDDC D

INPUT DATA FOR CONCRETE POLES

(1) Inpu t Data for Spun -Cast Concrete Pole by Davenport 's Model

NH NSP NSC NSW NZT NFT NFC NFW NFRE (Ht.) Spans (Sag)c. (Sag)g (Damp) (Flex)t (Flex)c (Flex)g Q

t (Freq)t 3 3 3 3 3 3 3 3 3

# Ht 1 70.0 2 84.0 3 100.0

# Spans 1 550.0 2 650.0 3 750.0

# Sag-Cond Avg.values

1 10.0833 2 13.5417 3 19.3750

# Sag-GW Avg values

1 4.500 2 6.250 3 8.25

# Damping in Tower

1 0.01 2 0.03 3 0.05

# Flex.Coeff. wind on Tower.

1 8.1486E-06 2 1.3216E-05 3 2.2554E-05

# Flex.Coeff. wind on Cond.

1 4.2719E-04 2 5.8028E-04 3 8.4330E-03

185

#

1 2 3

#

1 2 3

ST 1.056 7

Flex.Coeff. wind on GW. 2.6298E-04 3.2392E-04 4.3943E-04

Natural Freq. of Tower Hz. 1.144760 0.921003 0.740436

TAPT WTH 0.018 0.25

FOR EXPOSURE "C" VREF ALP Zg

HA 140 7.0 900.0

AET 7.808 3E08

AK

0.005

RHOT 150.0

ALZ

220

DC 0.118 92

EPSI

0.75

DGW 0.031 25

pAIR

0.002 4

CFT 0.8

AKV

1.546

CFC 1.0

G

3.6

CFW 1.2

186

(2) Input Data for Spun -Cast Concrete Pole by ASCE 7-95 Commentary Method

NH NSP NSC NSW NZT NZC NZW NFC NFW NFRE Q

# Ht 1 70.0 2 84.0 3 100.0

# Spans 1 550.0 2 650.0 3 750.0

# Sag-Cond Avg.values

1 10.0833 2 13.5417 3 19.3750

# Sag-GW Avg values

1 4.500 2 6.250 3 8.25

# Damping in Tower

1 0.01 2 0.03 3 0.05

# Damping in Conductor

1 0.20 2 0.40 3 0.60

# Damping in GW

1 0.20 2 0.40 3 0.60

# Flex.Coeff. wind on Cond.

1 4.2719E-04 187

2 3

#

1 2 3

#

1 2 3

DT

1.056 7

5.8028E-04 8.4330E-04

Flex.Coeff. wind on GW. 2.6298E-04 3.2392E-04 4.3943E-04

Natural Freq. of Tower Hz. 1.144760 0.921003 0.740436

TAPT WTH

0.018 0.250

FOR EXPOSURE "C" VREF BBA ABA

R R 140 0.65 0.153

846

AET

7.808 3E08

BHA T 1.0

RHO T

150.0

AHA T 0.105 263

AMO DEX P 1.8

G

3.5

DC

0.118 92

EPSI

0.20

DGW

0.031 25

ZMIN

15

CFT

0.80

CC

0.20

CFC

1.0

AL

500.0

CFW

1.2

pAIR

0.002 4

188

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