Guy Towers

STATIC AND DYNAMIC ANALYSES OF GUYED ANTENNA TOWERS

YOHANNA M. F. WAHBA B.Sc.(EIons), M.A.Sc., P.Eag., P.E.

A Dissertation subrnitted to Collcge of Graduate Studies and Research through

Civil and Envitonmental Engineering Program in partial Nfilment of the requirements for the

degree of Doctor of Philosophy at the University of Windsor

Windsor, Ontarîo, Canada 1999

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Canada

Yobanna M. F. Wahba 1999

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Please sign below, and give address and date.

in this investigation, the static response of guyed communication towen is

investigated using two different finite element models (three-dimensional ûuss model.

and an equivalent beam model), and a beam-on-non linear spnngs analogy. A

cornparison between the analytical techniques is presented for different loading Ievels.

Results from the anaiytical models are verified by testing four scale model towers to

collapse. The analytical techniques are also extended to prototype towers and

conclusions are drawn regarding the suitability of the analfical models to the static

analysis of such towers.

A dynarnic testing facility (shake table), suitable for testing of communication

towers, was designed, built, and instrumented at the Structural Engineering Laboratory at

the University of Windsor. This study presents the experimental investigation, testing

facility set-up, construction of models, materials and procedures used for the static, fiee

vibration, fmed vibration, and ultimate load tests of guyed tower models. The facility was

used to test five scaled guyed tower models under fke vibrations and foiwd base motion.

The tests substantiated the theoretical nnite element techniques.

The finite element analysis is appiied to eight prototype towers subjected to fiee

vibration and forced vibration motion. A parametric study on 33 towen is conducted to

investigate the main parameters influencing the fhx vibration of these structures. Based on

this parametric study, three empirical equations were derived to estimate the fundamental

natural fiequencies of the tower. These estirnated fiequencies would be valuable in

establishg the dynamic characteristics of the towers for use by design engineers.

ACKNOWLEDGEMENTS

The author wishes to express his deep appreciation and gratitude to his advisors Dr.

M. K. S. Madugula and Dr. G. R Monforton, for their constant support and valuable

supervision during the development of this research. To them 1 say, "1 am gratefhi - thank

you".

The author wishes to thank Dr. G. Abdel-Sayed, for his help and encouragement

throughout the duration of the author's graduate studies.

Many Thanks are also due to Mr. D. Marshall, for his encouragement and for the

pictues of towers that he supplied me. Also, th& to Messrs. G. Patton, K. DeBelser, R.

Sullivan, of LeBlanc Ltd. for their support and encouragement to complete this dissertation.

The author wishes to acknowledge the financial support provided by the Naturai

Sciences and Engineering Research Council of Canada, The Ministry of Colleges and

Universities d Ontario, and the University of Windsor.

Finally, the author is deeply grateful to his parents and his wife for their great

support, understanàiig, and patience throughout the course of this research; to them 1 Say

''Thanks".

viii

TABLE OF CONTENTS

List of Tables 88.8888.8888. 8m.m88m8m88m.888888888888m88.m88o8.8.8.8a888888m888.w8o888888.m8.88888m8888m...8,a~.888 xiii

1 -4 Contents and Amuigement of the Dissertation ....... .. .. .. ... ... ...... . .. .. . . . . . ..... . . ............ ......* 6

CHAPTER II BACKGROUND AND REVlEW OF LITERATURE mm8888.8m888..a888..... 11

3.3 Finite-Elefnent Approach ......................................................................................... 26 3.3.1 Description of the Finite-Elemeat Program 'ABAQUS' ...................................... 30

............................. 3.3.2 Finite-Element Modeliing of Guyed Communication Towers 30 3.3.2.1 Mast Modelling ........................................................................ 31 (a) 3 D-Truss Model ..................................................................... 31 (b) 3D-Beam Mode1 ..................................................................... 32 (c) Beam Mode1 ............................................................................. -32

.................................................................... 3 -3 .2.2 Guy Modelling -34 ........................................................... 3.3.2.3 Boundary Conditions -34

3 .3.2.4 Material modelling ............................................................ 35 (a) Mast Matenal ............................................................................. -35

............................................................................. @) Guy Cables -35 .................................................................................................. 3.3.3 %tic ~ W S 36

3.3 -4 Free-wxation h a l y sis ..................................................................................... 37 3 03 . 5 Nm-Limar W d c h d y sis ........................................................................... 39 3 -3 -6 Up-to-Collapse Anal~sis .................................................................................... 40

4.10 Expenmental Semp a d Testh3 procedure ............................................................ 61 4.10.1 Stage 1 : Static hads (Elastic Behaviour) ........................................................ 61 4.1 0.2 Stage 2: F~ee-vib~ation Tests .......................................................................... 62 4.10.3 Stage 3 : Forced Vibration Tests ...................................................................... 63 4.10.4 Stage 4: Up-to Collapse Tests ......................................................................... 64

CHAPTER V RESULTS FROM MODEL TOWERS TESTS ..m........................... 98

5.1 Introduction ............................................................................................................. 98

5 -2 Static Loading ......................................................................................................... 99 5.2.1 Load Case 1 .................................................................................................... 99 5.2.2 Lod Case 11 ......................... ... .................................................................... 100

5.3 Free Vibration Tests .............................................................................................. 101

5.5 F o r d Vibration Tests ........................................................................................... 107

5 -7 Discussioa ................... ..., ............................................................................. 112

5.8 summq ..........................*......... ........*...*............................................*.................. 113

CHAPTER VI RESULTS FROM 'TYPICAL PROTOTYPE TOWERS ~..~.~~.~~..~~ 172

6.4 Free vibration Analysis ............................. ......................................................... 180 6.4.1 Natural Frequencies and Mode Shapes of Mast-Guys System .......................... 181

................................ 6.4.1.2 Effect o f Ichg on the Free Vibration of Tower 185 6-42 Namal Frequencies and Mode Shapes of M m ............................................. 186

7.1 s-ary ................................................................................................................. 256

7.2 conclusions ............................................................................................................... 257

7.3 Suggestions for Future Research ........................................................................... ... . 259

REFERENCES .................................... ................... .......................................... 260

Table 4-1 Details of Expecimental Towers

Table 5-1 Guy Forces For Model Tower 1 (Load Case 1)

Table 5-2 Guy Forces for Model Tower 1 (Load Case-2)

Table 5-3 Guy Forces for Model Tower U (Load Case-1)

Table 5-4 Guy Forces for Model Tower II (Load Case-2)

Table 5-5 Guy Forces For Model Tower HI (Load Case-1)

Table 5-6 Guy Forces for Model Tower HI (Load Case-2)

Table 5-7 Guy Forces for Model Tower IV (Load Case-1)


Table 5-9 Guy Forces for Model Tower V (Load Case-1)

Table 5- 10 Guy Forces for Model Tower V (Load Case-2)

Table 5-1 1 Natural Frequencies for Model Tower 1

Table 5- 12 Natural Frequencies for Model Tower lI

Table 5-13 N a t d Frequencies for Model Tower III

Table 5-14 Naturai Frequencies for Model Tower IV

Table 5-15 Nanual Frequencies for Model Tower V

Table 6-1. Details of Prototype Towers Used in the Snidy

Table 6-2. Guy Forces under Design Loads

Table 6-3. Details of Sample Towen Used for Venfication of Results 198

Table 6-4: EfTective Mass Compoaents for the Fist 100 Modes of Tower PV 199

Table 6-5. Effect of Initial Tension on Natural Frequency of Towen 204

Table 6-6. Denvation of Empirical Equation for Tower Naturai Frequency 205

Table 6-7. Venfication of Equation 6-2 on Sarnple Towers 206

Table 6-8. Naturd Frequencies and Mode Shapes for Tower V Under Different Ice

Accretions 207

Table 6-10. Effective Mass for the First 10 Vibrational Modes of Mast (Tuwer V- Guy

Modes Suppressed) 209

Table 6-1 1 : Ratios of Guy to Mast Stiffiesses (&fim) 210

Table 6-12. Cornparison of the Mast Frequencies for Various Initial Tensions 21 1

Table 6-13. Comparison between Calculated and Estimated Frequencies of Towers 21 2

Table 6-14. Verification of Equation 6-9 on Sarnple Towen 213

xiv

List of Figures

Figure 1-1.550 m Guyed Tower 7

Figure 1-2. Close up of a Tall Guyed Tower Mast 8

Figure 1-3. Base for a 450 m Ta11 Guyed Tower 9

Figure 1-4. Typicd Short Guyed Tower for Wireless Applications 10

Figure 3-1. Guy Mode1 in Displaced Position 43

Figure 3-2. Iterative Technique for Non-Linear Behaviour 44

Figure 3-3. Finite Element Truss Mode1 for Mode1 Tower 1 45

Figure 3-4. Typical Mast Section Sub-Mode1 Used to Detemine Equivalent Bearn

Properties 46

Figure 4- 1 . Typical Mast of Mode1 Towers 67

Figure 4-2. Typical Mast of Guyed Tower used for Heights up to 200 m 68

Figure 4-3. Details of a Typical Tapered Mast Base 69

Figure 4-4. Details of a Fully Articulated Mast Base 70

Figure 4-5. Details of a Typical Star Base of a Guyed Mast 71

Figure 4-6. Typical Mast Base of Mode1 Towers 72

Figure 4-7. Profile o f Mode1 Tower 1 73

Figure 4-8. Profile of Mode1 Tower II 74

Figure 4-9. Typical Torsion Resistor 75

Figure 4-10. Torsion Resistor of Model Tower III

Figure 4-1 1. Model Tower III

Figure 4-12. Profile of Model Tower ï I I

Figure 4-13. Profile o f Mode1 Tower N

Figure 4-14. Model Tower IV

Figure 4- 15. Profile of Model Tower V

Figure 4- 16. Model Tower V

Figure 4- 17. Inherent Twist as a Result of the Manufacturing of the Models

Figure 4- 1 8. Guy Tension Adjusters

Figure 4-19. Shake Table and Plan of Test Set-up

Figure 4-20. Shake Table

Figure 4-21. Fundamental Natutal Frequency and Mode Shape of Shake Table

Figure 4-22. Acceleration History o f Shake Table for a 30 Hz Frequency

Figure 4-23. Dia1 Gauges Mounted on Model Tower V

Figure 4-24. Mount Detail of LVDTs and Accelerometers

Figure 4-25. Measurement of Tower Base Accelerations

Figure 4-26. Shake Table Driving Actuator

Figure 4-27. The Actuator Controller and the Data Acquisition System

Figure 4-28. Load Cells for Measuring Guy Tensions

Figure 4-29. Top View of Test Set-up 95

Figure 4-30. Static Load Application Set-up 96

Figure 4-3 1. Mode1 Tower V at Collapse 97

Figure 5-1. Deflection of Model Tower 1 under Load Case 1 (45 N at top loading

point) 130

Figure 5-2. Deflection of Model Tower I under Load Case II (27 N at al1 loading

points) 131

Figure 5-3. Deflection of Model Tower II under Load Case 1 (45 N at top loading

point) 132

Figure 5-4. Deflection of Model Tower II under Load Case II (27 N at al1 loading

points) 133

Figure 5-5. Deflection of Model Tower III under Load Case 1 (45 N at top loading

point) 134

Figure 5-6. Deflection of Model Tower III under Load Case II (27 N at al1 loading

points) 135

Figure 5-7. Deflection of Model Tower N under Load Case 1 (45 N at top loading

point) 136

Figure 5-8. Deflection of Model Tower IV under Load Case II (27 N at al1 loading

points) 137

Figure 5-9. Deflection of Model Tower V under Load Case 1 (27 N at the top three

loading points) 138

Figure 5-10. Deflection of Model Tower V under Load Case II (27 N at al1 loading

points) 139

Figure 5-1 1. Acceleration Response for Free Vibration of Model Tower I 140

Figure 5- 12. Frequency Domain of the Acceleration Response shown in Figure

5-1 1 140

Figure 5- 13. Acceleration Response for Free Vibration of Mode1 Tower II 141

Figure 5-14. Frequency Domain of the Acceleration Response shown in Figure

5-13 141

Figure 5-15. Acceleration Response for Free Vibration of Model Tower III 142

Figure 5- 16. Frequency Domain of the Acceleration Response shown in Figure

5-15 142

Figure 5- 1% Acceleration Response for Free Vibration of Model Tower IV 143

Figure 5-18. Frequency Domain of the Acceleration Response shown in Figure

5-17 143

Figure 5- 19. Acceleration Response for Free Vibration of Model Tower V 144

Figure 5-20. Frequency Domain of the Acceleration Response s h o w in Figure

5-19 L 44

Figure 5-21. First Nahiral Frequency and Mode Shape of the Top Guy (Model

Tower V) 145

Figure 5-22. First Nahinil Frequency and Flexural Mode Shape of the Mast (Model

Tower V) 146

Figure 5-23. Second Naairal Frequency and Flexural Mode Shape of the Mast

(Model Tower V) 147

Figure 5-24. First Torsional Frequency and Mode Shape of the Mast (Model

Tower V) 148

Figure 5-25. First Natural Frequency and Mode shape of the Bottom Guy (Model

Tower V) 149

Figure 5-26. Sinusoidal Deriving Displacement History 150

Figure 5-27. Sweep Test Results for a Frequency of 22 Hz (Tower IV

Accelerometers 1-3) 151

Figure 5-28. Sweep Test Results for a Frequency of 22 Hz (Tower IV


Figure 5-29. Forced Vibration Results For Northridge Time History (Tower IV

Accelerometers L -4) 153

Figure 5-30. Forced Vibration Results For Northndge Time History (Tower N


xix

Figure 5-31. Forced Vibration Results For Nanbu T h e History (Tower III


Figure 5-32. Forced Vibration Results For Nanbu Time History (Tower III

Accelerorneters 5-7) 156

Figure 5-33. Shake Table Acceleration History for Ground Motion modeled after

Nanbu N-S Direction 157

Figure 5-34. Measured and Calculated Acceleration Histories for Shake Table

Motion Shown in Figure 5-3 3 (Tower III- Acc. 1) 158

Figure 5-35. Measured and Calculated Acceleration Histories for Shake Table

Motion Shown in Figure 5-33 (Tower III- Acc. 3) 159

Figure 5-36. Measured and Calculated Displacement Histories for Shake Table

Motion Shown in Figure 5-33 (Tower UI- D 1) 160

Figure 5-37. Measured and Calculated Displacement Histories for S hake Table

Motion Shown in Figure 5-33 (Tower III- D3) 161

Figure 5-38. Load versus Displacement at the top Guy Level (Model Tower 1) 162

Figure 5-39. Load versus Displacement at the top Guy Level (Model Tower III)163

Figure 5-40. Finite Element Displaced Shape of Model Tower III at Failure 164

Figure 5-41. Mode1 Tower III at Failure 165

Figure 5-42. Load versus Displacement at the top Guy Level (Model Tower IV)166

Figure 5-43. Finite Element Displaced Shape of Model Tower IV at Failure 167

Figure 5-44. Load versus Displacement at the top Guy Level (Model Tower V) 168

Figure 5-45. Finite Element Displaced Shape of Model Tower V at Failure 169

Figure 5-46. Mode1 Tower V at Failure 170

Figure 5-47. Close-up of Failed Mast of Mode1 Tower V 171

Figure 6- 1. Profile of Prototype Tower PI 213

Figure 6-2. Profile of Protome Tower PI1 214

Figure 6-3. Profile of Prototype Tower PI11 215



Figure 6-6. Profile of Prototype Tower PU1 2 18



Figure 6-9. Cornparison of Leg Loads and Face Shear under Design Loads for

Prototype Tower P 1 22 1

Figure 6- 10. Cornparison of Deflections under Design Loads for Prototype Tower

PI 222

Figure 6-1 1. Cornparison of Leg Loads and Face Shears under Design Loads for

Prototype Tower PV 223

Figure 6-12. Comparison of Deflections under Design Loads for Prototype Tower

PV 224

Figure 6- 13. Comparison of Leg Loads and Face Shears under Design Loads for

Prototype Tower PVIII 225

Figwe 6- 14. Comparison of Deflections under Design Loads for Prototype Tower

PVIII 226

Figure 6-1 5. An Array o f Seven Towers with Different Heights Connected

Through Catenary Guy System 227

Figure 6-16. Load as a Ratio of Design Loads Vs. Deflection at the Top Guy Level

for Prototype Tower PI 228

Figure 6- 17. Failure Shape as Predicted by the Finite Element Model for Prototype

Tower PI. 229

Figure 6-18. Load as a Ratio of Design Loads Vs. Deflection at the Top Guy Level

for Prototype Tower PV 230

Figure 6-19. Failure Shape as Predicted by the Finite Element Model for Prototype

Tower PV 23 1

Figure 6-20. Load as a Ratio of Design Loads Vs. Deflection at the third Guy

Level fiom the top for Prototype Tower PV 232

Figure 6-21. Failure Shape as Predicted by the Finite Element Model for Prototype

xxii

Tower PVIII 233

Figure 6-22. Failure Shape as Predicted by the Finite Element Mode1 for Prototype

Towet PVIII 234

Figure 6-23. First Twenty Mode Shapes of Prototype Tower PV Modes (1 -4) 235

Figure 6-24. First Twenty Mode Shapes of Prototype Tower PV Modes (5-8) 236

Figure 6-25. First Twenty Mode Shapes of Prototype Tower PV Modes (9-12) 237

Figure 6-26. First Twenty Mode Shapes of Prototype Tower PV Modes (13-16)238

Figure 6-27. First Twenty Mode Shapes of Prototype Tower PV Modes ( 1 7-20)239

Figure 6-28. Variation of Fust Natural Frequency with Height of Tower 240

Figure 6-29. Effect of king on Mode Shapes of Tower PV 24 1

Figure 6-30. First Flexural Frequency and Mode Shape of Prototype Tower PI

(Guy Modes Suppressed) 242

Figure 6-3 1. First Flexural Frequency and Mode Shape of Prototype Tower PI1


Figure 6-32. First Flemiral Frequency and Mode Shape of Prototype Tower PlII


Figure 6-33. First Flexural Frequency and Mode Shape of Prototype Tower P N


Figure 6-34. First Flexural Frequency and Mode Shape of Prototype Tower PV


Figure 6-35. First Flexural Frequency and Mode Shape of Prototype Tower PVI


Figure 6-36. First Flexural Frequency and Mode Shape of Prototype Tower PVII


Figure 6-3 7. First Flexwal Frequency and Mode Shape of Prototype Tower PVIII


Figure 6-38. Variation of First Flexural Frequency of the Tower (Guy Modes

Suppressed) with Height 250

Figure 6-39. Time History of Deflections of Mast for Tower PIV Subjected to El-

Centro N-S Ground Motion 25 1

Figure 6-40. Time Histocy of Guy Stresses for Tower PIV Subjected to El-Centro

N-S Ground Motion 252

Figure 6-4 1. Time History of the Top Guy Stresses for Tower PVi Subjected to

Top Guy Galloping (f= 0.2 17 Hz) 253

Figure 6-42. Time History of the Top Guy Stresses for Tower PVI Subjected to

Top Guy Galloping (f- 0.6 17 Hz) 254

Figure 643 . Time History of the Deflection at the Top of Tower PVI Subjected to

Top Guy Galloping (f- 0.6 17 Hz) 255

Nomenclature

accelerations

atea of guy

displacements

modulus of Elasticity

elastic modulus of guy

naturai frequency, Hz.

shear modulus

guy radius

horizontal displacement at guy levels

total height of tower

second moment of inertia

structure intemal forces for an iteration a

bending stifniess about x axis

torsionai constant

structure stifniess matrix

global elastic stiffiess matrix

global geomeûic stitniess matrix

length of guy

applied torsionai moment on the mast

applied extemal loads

force residuai resulting fiom an iteration a

displacement vector at the nodes

displacement in the three global axes

weight of guy / unit length

angle of inclination of the wind to the guy axis

circular fiequency of the structure

densiiy

displacement of the structure

Poissons's ratio

correction factors for bridge geometry and curvanire

rotations in the ihree global axes

angle of twist

CHAPTER I

INTRODUCTION

1.1 General

Communications are playing an ever-increasing role in our society and the

demand for reliable communications is growing. Due to the ongoing expansion of the

information highway, and the introduction of new technologies such as digital television,

there is a growing need to better utilize the capacity of existing towea and to optimize the

capacity of new structures. Furthemore, the demand for more towers to be erected in

urbanized areas has not only made it dificult to obtain building pennits but also created

the need for far more reliable structures. This also has made the existing ''vertical real

estate" a valuable asset and created a constant need to upgrade them.

Most of the communication failities built today f d into one of two types of

applications: the fitst for use by the communications indusûy (mostly for wireless

communications) and the second for b d c a s t (radio and television) applications.

Fmm a structurai point of view, communication towers can be classined into one of three

types: (i) monopoles, which are cantilevered tubes with heights up to 70 m. (ii) self-

supporting lattice towers, wbich are commonly used for heights up to 120 m, although for

urban sites where the pnce of the land is more valuable, seKsupporting towers for heights

up to 300 m have k e n used, and (iii) guyed towers, which have been utilised for taller

structures up to 620 m. Figure 1-1 shows a pichue of a 550 rn guyed tower, while Fig. 1-2

shows a close up of a tall guyed tower mast and Fig. 1-3 shows the tower base. These

figures represent typical tall (broadcast) tower applications. Figure 1-4 is the photograph of

a typical short (100 m) guyed tower norrnally used for wireless and telecornmunications

applications.

A typical guyed tower consists of a mast that is usually of a constant cross section

and one or more levels of guy cables (that provide lateral support to the structure) anchored

at the ground level. Masts with triangular cross sections are most commonly used in North

America; however, square cross section rnasts are popular in Europe and other parts of the

world. The mast n o d l y consists of a latticed structure made fkom angles, solid rounds or

pipes. Guy cables are made of braided high strength cable +S.

1.2 Need for the Investigatioa

Guyed towea expedence a non-linear behaviour even under working conditions.

These non-linearities result from the changes in support stiflhess with the change in the guy

tension due to applied loads or even original design pre-tensions, the non linear force

deformation relationship of the structure, and the large displacements experienced even

under normal design loads.

ALthough ment advancements in cornputers have allowed fat more larger and

sophisticated techniques to be used, the normal anaiyticai methods used in the design of

these towers are still very simple. However, there has been very little experirnental data that

cm be used to compare the dinerent available analytical procedures.

While the cumnt design practices in North America recommend the minimum

static loads and analysis methods for the design of guyed towers, very few, if any, towers

are checked under dynamic loads. Practicai considerations in the design process require the

designer to understand the effect of different key parameters on the dynamic characteristics

of the structure. A thorough understanding of the f'undamental naturai kquency of the

structure would be helpful in understanding the dynamic behaviour of the structure and in

tum, determine if a m e r rigorous dynamic analysis is justified.

Because of the inherent non-linearity in the behaviour of guyed towers, a non-linear

dynamic anaiysis is usually tequireâ, eqecially for transient loads. Dynamic non-linear

analyses are computationally expensive and redts must be examined catefhily. That is, the

numeric solution may be stable, yet the mathematical mode1 may not represent the acnial

physical state. To date, experimental investigations on guyed towers subjected to forced

vibrations or loaded to failure are unavailable. Therefore, ceseaich in this area is needed.

1.3 Objectives and Scope

The main objectives of the present research work are to:

1. Compare the finite elernent models (beam and tms models) with the analytical methods

(beam on non-linear s p ~ g s ) that are currently w d in the design of guyed

communication towers.

2. Design a dynamic testing facility

3. Obtain experimental data on displacement, accelerations, and guy forces of guyed

communication tower models under static service loads, dynamic loads, and failure

loads.

4. Verification of the analyticai models in item (1) above through the experimental results.

5. hvestigate the parameters influencing the nahual frequencies and mode shapes of guyed

towers.

In this dissertation, a detailed analytical and experîmental study on guyed

communication towers is presented. Analysis is h e d on the finite-element method that

seems to be the only approach capable of including al1 the important factors uifluencing the

structural behaviour. The main panuneters examined in the present research are: total

height, span-to-face width ratio, number of guy levels, guy systems, height of cantilever,

antenna loading, and environmental loaàs.

The scope of this study thus includes the following:

Literature review of available expehntal and theoretical research work, and codes of

practice for guyed communication towea.

Design, fabrication, and instrumentation of a new dynarnic testing facility (shake table)

that cm be used for testing of communication towers.

Experimental investigation of different guyed tower models with various heights,

nurnber of guy levels, and guy systems under static and dynamic loading to mesure

deflections, accelerations, and guy tensions.

Cornparison of the experimental findings with results from different analytical

techniques.

Verification of the analytical techniques used in the static, ultimate failure, and dynamic

analysis of guyed communication towea.

A study of the important panuneters that determine the dynamic properties of guyed

towers.

Application of the analytical models to typical prototype towers.

Recommendations with respect to the analysis procedures to be used for the static and

dynamic analyses.

Derivation of em~incal formulas for the estimation of the naturai kquencies of guyed

1.4 Contents and Arrangement of the Dissertation

The literature review and previous work on guyed towers is ~ufnmarized in Chapter

II. Chapter iïI describes the various numerical modelling techniques used in the analficd

study including the idealkation and modelling of guyed towers as well as the non-linear

static analysis, pushover static analysis, fiee-vibration analysis, and forced vibration

analysis. In Chapter IV, the experimental work conducted on five scale-mode1 guyed

towers is described. This includes the details of the models, instrumentation, loading

system, and the test procedure. Chapter V pmsents the discussion of the results obtained

from the experimentai investigation as well as fiom the numerical analyses. Chapter VI

deals with the application of this study to actual prototype towers. Chapter W gives a

mmmary of this research, the conclusions reached, and recommendations for m e r

researc h.

Figure 1-1.550 rn Guyed Tower

Figure 1-2. Close up of a Ta11 Guyed Tower Mast

Figure M. Base for a 450 m Ta11 Guyed Tower

Figure 1 -4. Typical Short Guyed Tower for Wireless Applications

CHAPTER Il

BACKGROUND AND REVIEW OF LITERATURE

2.1 Geaeral

The structurai complexity of guyed towers has attracted many researchers to study

the behaviour of such structures under various conditions and extensive related research is

available. A recent litemture review included in the Draft Guide for Dynarnic Response of

Lanice Towea (ASCE 1999) has cited over 500 publications on that subject alone,

including guyed and self-supporthg towers. This interest maybe attributed to the fact that

these structures are the tallest in the world as they stand over 600 m above ground, they

have a relatively high rate of failure, and their behaviour is generally non-linear.

This chapter presents the background for the present investigation which is divided

mainly into four distinct parts. The first part is concerned with the static analysis of guyed

towers; the second part with the stability and uitimate load carrying capacity; the third with

the previous experimental work; and the fourth with the dynamic response of guyed towers.

23 Static Analysis of Guyed Towers

The first step in the design or aaalysis of a guyed tower is a static analysis. Early

research in this area was concemed with the elementary design methods. Schott and

Thunton (1956) applied a simplified approach for the analysis of guyed antenna towers. It

was assumed that al1 loads are symmetrical with no twisting induced on the tower. Another

simplification was the assumption that the tower's deflected shape under wind loads would

be a straight line. This later simplification was important for the subsequent application of

the three-moment equation for the analysis of the mast which was treated as a continuous

beam. Graphical methods were later used to account for the different support conditions.

Also, Pocock (1956) performed low speed wind tunnel tests to study the effect of wind

loads on antennas and mast sections and to determine the drag and cross-wind forces.

Cohen and Perrin (1957a) carried out extensive research to ascertain the primary

loads on a guyed tower and to study the effect of the shape and orientation of the tower on

wind resistance. Also, empirical formulas and charts were derived to calculate the drag

coefficients, lift coefficients, and shape factors depending on the individual member and

mast section. In a continuation of the same work, the authors published another paper

(Cohen and Penh 195%) outlinhg the use of the three moment equation for the analysis of

guyed towers considering the mast as a bearn-coIrmui supporteci on linear elastic springs.

Rowe (1958) included the effect of changes in geometry on the analysis of guyed towers; a

simplified non-linear method was introduced, but it was not considered effective as it fded

to include the sag effects of the cables.

Hull (1960) presented a study on the bending analysis and stability of ta11 multi-level

guyed towers. Several approximations were used to simpliQ the problem: the cable was

assumed to be parabolic with only d o m vertical loads acting on it and cable tensions

remained constant tbroughout the length of the cable. The moment distribution method was

also used in the bending analysis. Furthemore, the deflected shape was assumed to be in

the form of a straight line.

Dean (1961) has presented a rather accurate formulation of a cable mode1 that is

suitable for both static and dyaamk analyses. nie differential equations of the cables were

formulated based on a perfectly flexible behaviour. This allowed accounting for the change

in cable tensions due to the movement of the rnast. Later research (Meyers 1963) studied

guyed towers as non-linear sûuctural systems. This research emphasised the stability of the

structure. Chord length of the cable, instead of the arc length, was used in the calculations.

A standard solution was prepared, then a parametric study illusûated the effect of the

various factors on the solution. Some of the secondary eRects such as shortening of the

mast due to bending and axial loads, the forcedisplacement, and the moment-rotation

characteristics of rnast and mast rotation at guy attachment points were included in the

analysis. This was considered to be a comprehensive approach that Iooked into the non-

linear behaviour of the structure with very little approximation.

Subsequently, iterative proceâures were introduced. Livesley and Poskitt (1963)

developed an iterative approach for the analysis of guyed towers subjected to wind loads. In

this approach, bending was considered in the two principal planes of the mast and the

analysis of the guy cabie was based on the parabolic approximation. Axial movement of the

mast was neglected but the effect of axial loads on the flexural stiffness was taken into

account. Aiso, in this appmach, guy eccentricities at the attachment points were included.

The tower movement was considered to be dong the line of action of the forces.

Anothet milestone was the introduction of an analysis method by Odley (1966)

based on an iterative approach that was readily suitable for cornputer programming. This

method analyzed the mast and the guy systems at each guy level separately and an iterative

approach based on the compatibility of the displacements at the guy points was introduced.

Thus, secondary moments at the guy attachent points were included. The readiness of this

approach for programming made it the choice of several commercial programs developed

for the analysis of guyed towers. This model also considered axial loads on the mast as well

as wind and ice loads on the guys. This model was based on an elastic beam supported by

non-linear springs. The sprhg constant at any deflection point was denved based on the

state of tension at the guys.

A procedure was developed to account for torsional effects on the towea by

Marshall (1966). This also included some experimental venfication of the analysis method.

This methd simplified the guy models and restricted its application to shorter towers.

Another method was developed in the same year and it used interactive diagrams to design

in s e v d stages (Müdofslq and Abegg 1966). The aaalysis was carried out in two stages.

First, the tower was anaiysed as a continuous beam on elastic supports and then the tower

was analysed as a beam-column to include the amplification of stresses resulting fiom the

axial loads. The maximum stresses at the guy points and mid spans were presented on the

diagrams for different face widths of towers. This method neglected the displacements at

the support points due to the small displacement-span ratio. The other major drawback of

this method: it was not versatile to account for many special conditions and it could not be

programmeci effectively.

In the 1970'~~ interest in the analysis and design of guyed towers continued.

Livesley (1970) formdated an automated design method for guyed towers mbjccted to

deflection coflstraints. However, the analysis assumed a linear load-deflection relatiowhip.

A computer program was developed by Richelt et al. (1971) to interactively design

towers. Later Bell (1972) developed a structural optimisation system for the design of guyed

towers. The stiffiess maûix method with stability fhctions was used to mode1 the mast,

while guys were modelled as springs. The technique was limited to a certain class of towea

ranging fkom 30.5 to 152.4 m. Torsional effects were neglected and bending in one

direction ody was considered.

Skop (1979) developed a new method for the detennination of the cable spring

constants. The method was more general in that it accounted for arbitrarily loaded guys

containing any n u m k of disrrete masses. Later Rosenthal and Skop (1980) published an

analysis method where the mast was idealised as a two-dimensional structure and the cables

were analysed by the method of unaginary reactions as formulated by Skop and O'Hara

(1970, 1972). No predehed shape was asmmed for the cables. A h , this method

modelled the base as mounted on rotational springs, which allowed a wide range of

boundary conditions to be modelled.

Ekhande and Mdugda (1988) prwented a three-dimensional finite element

formulation for a cable element. A twodimensional example was iilustrated in that paper;

however. the method can be generally used to mode1 any guyed tower system. Later Issa

and Avent (1 991) presented a method to obtain dkctly the forces in al1 individuai members

of the mast by the use of discrete field mechanics techniques.

The current trend with standards including the Canadian Standard for antenna

towen CSA S37-1994 (Canadian Standards Association 1994) has changed to limit States

design in which the stnictwes are analyzed under factored loads or near failure loads

(Wahba et al. 1994). It is also expected that the US Standard for Towea (ANSIIIIAIEIA-

2 2 2 3 will switch to load and resistance factor design based on the Arnerican Mtute of

Steel Construction design specincation (ABC 1994). Also, with the introduction of new

technologies, existing structures are analyzed for the addition of new loads (antenoas).

Generally, the industly is ushg bearn-on-Mngs models (Odley 1966) to analyse ever-

Uicreasing talier and complicated structures.

2 3 Stability of Guyed Towers

The stability analysis of guyed towers is not a simple problem. As the governing

equilibrium equations are not linear homogeneous equations, instability is not a

bifurcation problem. Several design approaches use the member effective length factors

based on unsupported length of the mast between the guy attachment points. What

complicates the matter further is that under horizontal loads, these supports (guy

attachent points) are also moving. One of the advantages of using Limit States Design

is that the tower is analysed under factored loads. The andysis of the structure under

factored loads (at failure) helps the designer to examine any signs of instability at the

ultimate load level, provided that the analysis method is sophisticated enough to account

for large deformations and detect instability. The tower is considered unstable when large

defornations occur with a small increase in the applied loads.

In continuation of the work done by Meyers (1963) on the stability of guyed

towers, Goldberg and Gaunt (1973) published a paper on the stability of multilevel-guyed

towers. in this research, a method for the analysis of towers was formulated and the effect

of parameters afEecting the stability of towers was presented. This study Iooked into the

effect of initial tensions, guy sizes, and spans on the stability. Also, this snidy considered

the stability of the towers under increased horizontal loads. increashg the applied

horizontal loads h m wind would redt in an increase of the applied axial load fiom the

supporthg guys that may evenhially lead to overall buckling of the tower. In their analysis,

buckling of the tower was considered as the point at which no M e r horizontal loads could

be applied to the structure. Later Cbjes and Chen (1979) and Chajes and Ling (1981)

studied the critical buckling load of towers. Formulas and curves were presented for the

detemination of the critical load for guyed towers. However, this research was lllnited to

short single level guyed towers.

Williamson (1973) examined the effect of king on the stability of a special taIl

guyed tower. In this tower the uppemost guy level consisted of an anay of twenty-four

conducting cables senhg as a radiahg element of the antenna system. Also, the effect of

increasing the stiffiess of mast and guys on the relative cost was studied. Later Costello

and Phillips (1983) looked hto the p s t buckling behaviour of guyed towers.

2.4 Experimental Investigation on Guyed Towers

'ïhere is lirnited experimental data published on actual guyed communication

towers or scale mode1 towers especially under transient loads (seismic response) or for

free vibration. Marshall (1966) developed a procedure to detennine the deflections and

guy tensions of guyed towers subjected to direct loads and torsional loads. This study,

which included experimental research, was geared to observing the torsional distortions

which were not contemplated seriously before that time.

Novak et al. (1 978) studied the vibration of towers due to gailoping of iced cables.

The study included: (i) measurements of the aerodynamic properties of the ice-covered

guys on stationary sectional models of cables in the wind tunnel, (ii) theoretical prediction

of the galloping oscillations of the sectional models and their experimental venfication on

vibrating sectional models in the wind tunnel, and (iii) theoretical prediction of vibration

of a tower due to galloping oscillation of its guy.

Nakamoto and Chiu (1985) published the results of instrumentation of a 245 m

guyed tower. The tower had four guy levels. Accelerometers and anemorneters were

placed on the tower. Results fiom full-scale wind velocity and structural response data

fkom this tower were analysed to determine (i) the power-law exponent for wind profile,

and (ii) estimates of the resonant fkequencies and ratios of the critical damping.

Vombatkere and Radhakrishnan (1 986) have published the results of tests

perfomed on 1.9 rn models of guyed towers for wind turbines. Tests were perfomed on

single level guyed towers of a tubular cross section. Also, the effect of the top mass and

the initial tensions on the fist n a d fiequency was presented.

2.5 Dynamic Analysis

2.5.1 Dynamic Responsc under Wind Loads

Publications related to dynamic analysis of guyed communication towers almost

pertain exclusively to wind response. Davenport (1959) and Davenport and Steels (1965)

proposed a linear mode1 to describe the vibration of the guys. This work was later used to

study the dynarnic response of the CFPL tower in London, Ontario. Most of that work

was restricted to the prediction of the dynamic respoase of the structures where the wind

loads are random in nature. McC&y and Hartmann (1972) have published a study on

the dynamics of guyed towers and the parameters used in determination of the

mathematical mode1 used for the anaiysis of guyed towers.

Earlier research published in this area was focused on the dynamic response of

towea due to wind loads. This research questioned the application of a static gust factor

that is applied throughout the height of the structure (Vellozzi 1975). Modal dynamic

analysis using available wind spectra was perforrned to simulate the effect of gusting

winds*

nie complexity of performing a modal dynamic analysis and the requirement of

special analytical skills made it very rare to apply these procedures in a design office.

Later research by Davenport and his collaborators (Davenport and Allsop 1 983; Gersto ft

and Davenport 1986; Davenport and Sparling 1992) concentrated on the development of

patch loading methods that approximate more realistically the dynamic eKects of

turbulence, using a series of patch load patterns. Since these methods require only non-

linear static analysis tools which are readily available, different standards (BSI 1994;

CEN 1997) have adopted different variations of patch loading methods. More recently,

even simpler analysis methods that account for the gusting effects of wind (Gress and

Sparling 1998) were developed to avoid the large number of loading cases required in the

analysis proposed by the patch loading method.

Recently SparLing and Davenport (1998) published a study on the three-

dimensionai dynamic response due to wind turbulence. In this study, the non-linear

dynamic response of guyed towers using stepby-step integration of the goveming

equations of motions was presented.

Augusti et al. (1 986) in theû research on wiad response of guyed towers presented

a model of 200 m guyed mast with three guying levels using equivalent elastic linear

springs for the guy cables. The stifbess of the springs varies with the frequency of

oscillation; however inertia effects of the cables were aot modelled. Later, Augusti et al.

(1990) have presented a model of guyed towers in which the cables were presented by a

mesh of five to twelve cable elements.

2.5.2 Dynimic Response to Transient Laids

Other sources of dynamic loads of guyed towers arise from seismic loads, sudden

ice shedding of guy wires, and sudden rupture of wires. Until recently very little research

was done to investigate these areas. Generally guyed towers have behaved reasonably

well under seismic loads although there have been reports of local failures, and serious

misalignment of towers as a direct result of the Loma Prieta and Northridge earthquakes.

Also, there have k e n reports of catastrophic failures of towea due to sudden ice

shedding and also due to rupture of guyed wires (usudly involving sabotage).

Recently, McClure and collaborators (McClure et al. 1993, McClure and Guevara

1994) have published tesults on the non-linear seismic response due to two horizontal

acceleration histories on t h e different guyed towen canging in heights from 24 m to 342

m.

Sudden ice shedding, (where a major chunk of ice on a guy wire is forced to shed

almost instantaneously) induces the tower to respond dynamically. McClure and Lin

(1994) published results of an anaiysis performed on three guyed towen (24.4, 60.7, and

213 m) with two, four and seven guy levels. Results indicated that the effect of ice

shedding should be m e r examined. especially on ta11 towers (more than 100 m).

CHAPTER III

MODELLING AND ANALYSE

3.1 General

The available anaiysis tools for guyed towers were reviewed in Chapter II.

However, in order to substantiate and assess the different modelling and analysis

techniques, this study has used two approaches for the d y s i s . The first, which is the most

commonly used in the industry, is the treatment of the tower as a conthuous beam resting

on non-linear elastic supports using solution techniques based on linearised slope-deflection

equations. The second approach is a finite element procedure and within this approach three

different models have been suggested. This Chapter describes the various models used in

this research and the analysis methods applied. The different methods of analysis described

herein were applied to the experimental mode! towers of Chapter IV as explained in detail

in Chapter V and on prototype towers as presented in Chapter VI.

3.2 Beam on Non-Linear Elastic Supports Approach

Most of the available commercial programs that are mitien specificaiiy for the

analysis and design of guyed communication towers use the beam on non-linear elastic

supports model. The program used in this analysis is GUYMAST by Weisman

Consultants Inc., Downsview, Ontario, Canada (Guymast 1987). This model is primarily

developed by Odley (1966) and is explained briefly in the following.

3.2.1 Modelling

In this approach the tower is broken into three different numerical models as

follows: (i) every set of guys connected to the tower at the same level is analysed

independently, (ii) a continuous beamîolumn model is used to analyse the mast in the

two orthogonal vertical planes, and (iii) the shaft subjected to twisting moments is

supported by torsional s p ~ g s at the guy levels.

3.2.2 Analysis

The analysis is based on an iterative approach as follows: First, with an arbitrary

set of initial displacements, the guy model is used to obtain the guy stiffEess and guy

loads to be applied to the mast models in order to calculate a new set of mast

displacements which are then fed back into the guy model to obtain a better

approximation of guy stiffhess and loads. This process is repeated until the

displacements calcuiated by the mast model match those used by the guy model in

determining the support stiffness within a specified tolerance.

The stifniess of the guy supporthg system is determined fiom the following

procedure. As shown Ui Figure 3- 1, for any particular value of displacement A, the value

of the horizontal displacement Hg at the guy level is determined as follows (Odley 1966):

1. A value of Hg is assumed

2. The expected length of the guy L, is determined fiom

3. The change in guy length Ag is calculated fiom:

Ha w,' A, =- (sec' 0+-)

A g E , 1 2 ~ :

4. The unstretched Iength is the difference:

L, =L-A,

Where, A, is the cross-sectional area of the guy, Eg is the guy elastic modulus.

Since the unstretched length, Lt, is invariant with the load, the originally calculated length is

compared to that given by Eq. (3-3) and if the values do not agree, a new value is assumed

for H, the horizontal displacement, and the calculations repeated until satisfactory

agreement is reached.

The above procedure is repeated for al1 guys at the same attachment point fiom

which the force at the guy level redting h m a ceriain displacement, A, is calculated.

As mentioned before, the nomal indusîry practice does use commercial programs

based on the above appmh and specifically written for guyed communication towers. The

simplicity of these programs imposes some lhitations on the type of cross-section used,

panel configuration, similarity in ail mast faces, mast pmperties, etc. This approach is

applicable only for static loadhg where mernbers rernain in the elastic region.

3.3 Finite-Element Approach

The finite element method is considered to be a very powerful, versatile, and

flexible tool. Using a finite-element method for analysis makes it possible to model a tower

in a more redistic rnanner and to provide a full description of its structural members and

thus expect a more accunite structural response within the elastic and postslastic stages up

to collapse. The most important advantage of this method is its capability to model various

arrangements of stnichual elements, material properties, and boundary conditions. A

general f i t e element package named ABAQUS (HKS 1995) was used throughout this

study to determine both the static and dynarnic response of guyed towea. A general

description of the program is presented herein dong with the modelling of the different

components of the towers.

The finite-element method is an analysis technique in which the entire structure is

discretized into a nnite numbei of regions (elements) that are interco~ected at certain

points (nodes). With a displacement formulation, the e e s s matrix of each element is

denved and the global stiftiiess matrix of the entire structure can be formulated by the direct

stifiess method. The assembly of these elements to form the whole tower superstructure

is physically equivalent to superimposing these element equations mathematicdly. The

result is a large set of simultaneous equations that can be solved using cornputers.

Nodal forces act at each nodal point, which result in displacements and rotations.

A standard set of simultaneous equations cm be written to relate these physical

quantities. From the potentid energy formulation, the following equation is obtained in a

matrix form:

where np is the potential energy of the system; {U} is the global displacement vector; (P}

is the global load vector; [Ke] is the global elastic stifl'hess matrix; and b] is the global

geometnc stiffiess matrix. The geometric stiffness matrix is included in the analysis to

account for the defomed geometry of the elements and the effect of initial conditions on

the structure. This will account for the non-hear load deflection behaviour.

Differentiating the potential energy of the system with respect to the displacement

and equating the resdt to zero, results in the following:

which cm be simplified into the basic finite element equation relating the globai

displacements and the global loads as follows:

where: K = & + KG

The above equation is in tum solved and thus the displacements and forces can be

determined.

In a iinear elastic problem, loads are applied to a mode1 and the response is obtahed

directiy in one step. This single step analysis cmo t be used for guyed tower as the effect of

the initial conditions resulting h m the pre-stressed guys needs to be included in the

analysis. Thus, even for an analysis in which the response is in the elastic range of a tower, a

non-linear analysis must be used.

In a non-linear finite-eiement anaiysis, several linear steps are taken through an

incrementation scheme. There are several methods to solve non-linear problems. One of

these methods is the well-known Newton's method, which is a numerical technique for

solving the non-linear equiliirium equations. By solviag a series of hear problems, the

non-linear solution of the problem is iteratively obtained. The non-linear response of a

structure to a srnall load increment, AP, is shown in Figure 3-2. Let [Po] be the initial load

at a certain time increment, vol be the initial displacement, and [&] be the initial tangent

stiffaess at Po] and [Po]. Using the stmctwe's tangent s-ess, Ko and AP, the

displacement correction, ci, can be calculated for the structure. Using ch the structure's

configuration is updated to y. Then the structure's intemal forces, Io, are calculated in this

updated configuration. The difference between the total applied load, P, and L can now be

calculated as:

(3-7) Ra = P-1,

where R, is the force residual for the iteration.

if F&, is zero at every degree of fieedom in the model, then point "a" in Figure 3-2

would lie on the load deflection c w e and the structure would be in equilibrium. However,

in a non-linear aiiysis R will never be exactly zero, so an acceptable tolerance value has to

be set. If & is less than this force residual tolerance at al1 nodes, the solution would be

considered in equilibriurn and P, Ii , and ii. would be a valid equilibcium configuration for

the structure under the applied load. However, before accepting the solution, the analysis

should also check that the last displacement correction is small relative to the total

incremental displacement. If the ratio is pa ter than a prescribed percentage (e.g., l%), the

cornputer program performs another iteration by forming a new stifiess ma&, with a

value of tangent stifbess other than Ko, based on the updated configuration, Ui, and repeats

the two checks again. Both convergence checks must be satisfieâ before a solution is said to

have converged for that 1 0 4 incrernent

This procedure is repeated until the increment of displacements or the unbalanced

forces become nul1 or sufficiently close to null. Inctements should be small to ensure correct

modelling of history-àependant effects. The choice of increment size is a matter of

computational efficiency. If the increments are too large. more iterations will be required. In

Newton's method, a large increment can prevent any solution fiom king obtained because

the initial state is far away h m the equilibrium state.

3.3.1 Description of the Finite-Element Program 'ABAQUS'

ABAQUS ()IKS 1995) is designed as a general finite element package for

numerical modelling of structural response in linear and non-linear static and dynarnic

analysis. This computer program runs as a batch application to assemble a data deck bat

describes a problem so that it cm analyse the structure. A data deck for this computer

program contaùis mode1 data and history data. Model data defines a finiteelement model:

the elements. nodes, element properties, material definitions, nodal constraints, and any data

that specify the model itseIf. History data define what happens to the model, the sequence of

events or loaduig steps for which the model's respome is sought.

3 3 3 Finite-Elemeat Modelling of Guyed Communication Towers

A threedimensional finite-element analysis is wd to mode1 guyed commuaication

towers. Three different finite element rnodels weie explored for this analysis. Also,

different types of analyses were applied. h this section, element types used for different

models as weil as material modelling in both the elastic and pst-elastic loading stages are

presented. The results from the mudel presented herein were compared with results

obtained fiom testing five scale-mode1 guyed towers subjected to several loading cases

presented in Chapter N.

332.1 Mast Modelling

This mode1 represents a detailed and accurate modelling of the tower. Each

member (legs, diagonals and honzontals) forming the latticed mast was modelled as two

node three-dimensional truss elements with three degrees of freedom at each node.

Element T3D2 fiom ABAQUS library was chosen. This element had three degrees of

W o m at each node, namely the three displacements (ü1, U2, U3). The element used was

based on updated Lagrangian formulation and does not include higher-order non Iinear

t m o s . A plot of the nnite element mode1 of the mast for Model Tower-I is show in Figure

3-3. The general p d c e for the detailing of these structures is that applied antenna loads

are concentric at the panel points and linle fixity is provideci at the co~ections between the

bracuig and the main tower legs. Thus it can be reasonably assumed that a three-

dimensional tms mode1 is applicable. Due to the large number of members in a guyed

tower, this mode1 would resuit in large number of degrees of fitedom, and in tum requires

larger memory space and more CPU tirne.

(b) 3D-Beam M W

The legs of a guyed mast are continwus over the panel points and typically are

spliced every 6 m or 9 m. This continuity would result in bending moments especially at

the panel points. A three-dimensional two-node generai beamîolumn element, narned B3 1

in ABAQUS library, was adopted to model the mast legs. The element had two nodes with

six degrees of fieedom at each node, three displacements (Ul, U2, U3) and three rotations

(al, @2,<03). Mast bracings (horizontals, diagonds) are modelled as T3D2 (two node3D

t u s ) elements. This model was chosen to study the eEect of the secondary moments on

members forces.

(c) Beam Mode1

In this model, two-node bearn-column elements (B3 1) with six degrees at each

node were used to model the mast. One equivalent element was used for the entire mast

cross section. To account for any changes in the beam pmperties, an element has been

used for every panel in the mast.

In this model it is important to fïnd the beam properties that closely define the

mast behaviour. General bearn elements that accouat for bending stiffness (EI), shear

rigidity (GA), torsional rigidity (GJ), and axial stiflhess (EA) were used. Therefore, a

three dimensional sub-mode1 of every section, (A typical section of a tower is 6 to 9 m) as

shown in Figure 3-4, was created. The beam properties were calculated fiorn the anaiysis

of these sub-models as follows:

1) A moment was applied at the top of the submodel (through tende and

compressive axiai forces applied at the opposite legs), and the deflection at the

top, A, was calculated. Since this mode1 represents a cantilever beam, the

deflection under pure moment is:

M L ~ A=- 2 EI (3-8)

where L is the height of the section, and M is the applied moment. Thus EI was

detemiined

2) A horizontal load, P, was applied at the top of the tower and the sub-rnodel is

anaiysed again for the top horizontal deflection. The calculated defiection was

used to detemine the shear rigidity (GA) h m the following formula using EI as

determined fom step (1):

PL^ PL A=- +- 3EI GA

(3-9)

3) Torsional Moment, Mt, was applied and the torsional rigiâity GJ was estimated

h m the computed angle of twist &.

This process was repeated for every section of the mast where there is a change in

the element pmperties

To account for the eccentricity at the guy connection with the mast, MPC (multi-

point constraint) type, BEAM, was used. This option enforces a constraint between two

different nodes by introducing a rigid beam between these two points. This ensured full

interaction between the guys and the mast and kept the guy attachments eccenhic with

respect to the mast centre.

3.3.2.2 Guy Modelling

Each cable was discretized into several 3-D cable elements. These elements have

the same formulation as the 3D ûuss elements with the exception that the cable material

is modelled as tension-only material. Also, because of the geometnc non-linear anaiysis

used in these models, these elements were capable of handling applied loads even

perpendicular to their axial direction. A convergence test was applied to determine the

optimum number of elements, and it was noted that the overall behaviour was not

criticaily dependant on the number of cable elements. It was concluded that 12 to 24

elements per cable, dependhg on the cable length, was adequate to mode1 the sag.

3.323 Boundary Conditions

Three typical arrangements at the base of the mast are show in Figures 4-3, 4 4 ,

and 4-5. In these support arrangements, dl three displacement components at the base were

resûained; however, only one rotational component, the twisthg angle about the vertical

axis, could be considered restrained. At the guy anchor points, dl three displacement

components were restrained.

31.2.4 Material modeliing

in order to ensure that the validity of the results, the materiai must be suficiently

defined to provide suitable properties for the analfical technique used.

(a) Mast MateriaI

For the elastic andysis, a purely elastic material option was used. In this case the

only input required was the elastic modulus of the material. In the dynamic analysis, mass

properties were usecl and a h damping properties were Uitroduced as determined for the

free vibration test perfomed on the model towers. However, with up-toîollapse anaiysis

the simple plasticity model, i r . perfect plasticity, was used In this model, the yield surface

acts as a failure sudace with no straîn hardening parameters.

(b) Guy Cables

Guy cable materials were modelled in the same manner as the mast material;

however, '"NO COMPRESSION" option was introduced which limits the material to

tension only. Also, in the material definition, the mass of cables was calculated based on

the weight of the cable and not on the nominal diameter.

A cornputet prograrn was written by the author to perform the sub-modelling

anaiysis of the beam model and to determine the mast properties. It also generated the

model for ABAQUS format including nodes, elernents, materials. boundary conditions, etc.

This prograrn allows the analysis of a wide range of towea that define a spectrum in

industry practice.

3.33 Sbtic Analysis

Because of the inherent non-linear behaviour of guyed towers, geometric non-

linearity is included in the f ~ t e element analysis procedure. The non-linear procedure in

the computer program offers two approaches to obtain a convergent solution at minimum

cost Direct user control of increment size is one choice, whereby the user specifies the

incrementation scheme. This is particularly usehl in repetitive analysis where the user has a

very good feel of the problem. Automatic control is an aiternative choice: the user defines a

period of history (a '%tep" in the tenninology of the program) and at the same time specifies

certain tolerances or emr measures. The computer program then autornaticaily selects the

increments to mode1 the step. Genedy, the automatic control is more efficient thm the

repeated user controlied running of the problem to obtain a satisfactory incrementation

scheme. In addition, automatic control is extremely valuable in cases where the tirne or load

incremeat varies widely through the step.

This analysis is performed in iwo steps. In the first step the gravity loads resulting

fiom the self-weight of the tower, in addition to the forces due to the initiai pre-stressing

in the cables, are applied and equilibrium is achieved. Then, in the second step,

environmentai loads (wind and ice loads) are applied. In this step the load is applied

through twenty equal increments.

33.4 Free-Vibra tion Analysis

Guyed towers possess a non-limar behaviour; thus non-linear dynamic analysis is

wananted. However, the nanual îkquencies of the tower determine the dynamic behaviour

of îhe structure. Although the naturai muencies are not used in a direct integration

analysis, yet it is used in the modal analysis, which have been used in the predication of the

behaviour of towers under dynamic wind loads (Vellozzi 1975). It was also recommended

that for the modal analysis under wind loads, the nanual frequencies of the structure be

determined in its displaced position due to mean wind speed. A fke-vibration analysis,

which is a simple eigenvalue extraction problem, is based upon the following generai

equation:

( - W 2 ~ ~ + s ~ * +KU)# = O (3-1 1)

where: M~ is the mass matrix (which is positive definite); C' is the damping ma& (which

is neglected by the computer program during the eigenvaiue extraction); K'' is the stiffiess

mairix; 4 is the eigenvector (the mode of vibration); a> is the fkquency value; and i and j

are degrees of fieedom. This is a linear perturbation problem and depends on the initial

conditions of the structure. Therefore, the stiffhess of the structure as determined at the end

of the previous analysis step is included Ui the eigenvalue extraction. This is important in

order to detect the natural frequencies of the structure under different displaced positions, or

due to various loadings resulting fiom wind or ice, or wind and ice combined.

The FREQUENCY procedure in the cornputer program ( K S 1995) uses an

eigenvaiue techaique to extract the kquencies of the current system and the correspondhg

mode shapes. The users need to specify the number of eigenvalues required. in the

computer program, the eigenvectors are normaiised so that the largest displacement entry in

each vector is unity. If the displacements are negligible, as in a torsional mode, the

eigenvectors are nomalised so that the largest rotation entry in each vector is UN@. In

addition to extracting the naturai fkquencies and mode shapes, the computer program

automatically calculates the participation fector (which indicates the strength of the motion

in the global x-, y-, and z-directions), and the effective mass (which if added for al1 modes

shouid sum to the total mass of the model). Thus, if the effective masses of the modes used

in the analysis add up to a value that is significantly less than the mociel's total mass, this

suggests that modes that have siBnificant participation in a certain direction have not been

extracted. For that purpose, Standards such as W C 1995) requires that at least 90% of the

participating mass of the structure must be accounted for in the modes used in a modal

analysis for each p~cipal horizontal direction. For guyed towers, the number of mode

shqes required to get that mass participation ratio was found to be hi&, as discussed in

detail in Chaptea V and VI.

The d y s i s was performed in two steps. in the first step, the gravity loads

resulting fiom the self-weight o f the tower, in addition to the forces due to the initial pre-

stressing in the cables are applied and equilibrium is achieved. Also, in this step

additional loads resulting fiom ice or wind are applied if the response is required to be

investigated in these pre-set conditions. Then, in the second step, the naturai fiequency

analysis is performed.

3a3m5 Non-Linear Dynamb Aaalysis

ABAQUS (HKS 1995) offers several methods for performing dynamic

analysis of structures in which inertia cffects are important. Direct integration of the system

must be used when non-linear dynamic response is king studied. In this method, the global

equations of motion of the systems are integrated h u g h tirne, which makes it an

expensive and tixne-collsuming solution. This method was used to ver@ the analysis

against the m d dynamic response for the tested tower models. The method used by

the program is an extension of the trapemidal rule. Also, a numerical damping parameter,

a, needs to be defined. This is purely numenc and it varies for O to a maximum level of 6%

when the time incmnent is 40% of the period of oscillation. Therefore, this artincial

damping is never ~bstantial for this application. T i e increments used in this analysis

werc 0.01 second.

The applied loads in this step were d e t e d e d for the measured base motion

accelerometer mounted on the shake table for the mode1 towers tested. This motion was

meanued at a rate of 400 rradings per second and read by the data acquisition system and

then transferred into an ASCII file that is read by the ABAQUS andysis. The AMPLITUDE

option in the cornputer program allows arbitrary time variations of load given throughout

the step. The default of this option is to give the magnitude of the force as a multiple

(hction) of the reference magnitude given on the data liM of the loading. For prototype

towers, actual ground acceleration histories were used for the analysis. The output of the

dynamic analysis includes the dynamic response of the structure due to the prescribed load-

time history in the fom of the shauiing action-tirne histories. These straining actions

include accelerations, dynamic displacements, and dynamic stresses.

This analysis was also perfomwd in two steps: fïrst under static loads of initial

tensions, and the second under dynamic transient loads.

Up-to-collapse analysis was performed to determine the ultimate behaviour and

also the uitimate load capacity of guyed towers. This d y s i s is of particular significance

especially today were most Standards have adopted Limit States Design, which requires

that towers be analysed under loading conditions that are very close to failure levels.

For a non-linear up-to-coilapse analysis, the cornputer program provides the

modified Rücs method (HKS 1995). with automatic incrementation for solving such cases.

By using this method, the cornputer program prints out a load proportionality factor, h, at

each increment during the step, then the current magnitude for the load component, Pm, can

be defined as:

where PO is the magnitude of the load component at the start of the step; Plcf is the

magnitude of this load component as dehed by the user for the step. For dead load, Prcf =

Po, so that the load magnitude remains constant. The Riks algorithm attempts to step dong

the equilibrium path (the load-displacement cwe) by prescribing the path length dong the

curve to be îraversed in each increment. This means that the load magnitude is determined

as part of the solution and the user must oniy specify the fmishing conditions of the step.

This can be done through defïning either a maximum vaiue of h. beyond which the solution

is not of interest or a displacement value at a specined degree of M o m as a nnishing

value. If no finishing condition is specified, the analysis will continue up to the number of

increments specifieâ by the user at the beginaing of the analysis in the non-linear step.

In this anaiysis, the nist step is for initial conditions and the second step is used to

determine the dtimate load as a ratio of the applied loads under service load conditions.

3.4 Analysis of the Tower Models and the Prototype Towers

A convergence study was conducted to choose the finite-element mesh. Pilot runs

were used to detemine the number of cable elements used for each guy. As for the tnrss

model, it would not be of any additional value to break the elements any M e r . For

bearn models, a kam element was used for every panel (which is a very refined mesh).

However, it was verified that a beam element for every section would result in the same

level of accuracy as the refmed model provided that there is no change in material

properties within a tower section (which is a reasonable assumption).

The two fuiite-element models were verified and substantiated by results nom

testing five tower models as described in Chapter IV. Later, a wide range of guyed towers

was examinai as described in Chapter VI. Appendix Al shows a list of the fnite-element

cornputer program input &ta used in the analysis.

Figure 3-1. Guy Mode1 in Displaced Position (Source: Odky 1966)

Load Ka

Displacement

Figure 3-2. lterative Technique for Non-Linear Behaviour (Source: HXT 1995)

Figure 3-3. Finite Element Tmss Model for Model Tower 1

Figure 3-4. Typical Mast Section Sub-Mode1 Used to Determine Equivalent Beam Properties

EXPERIMENTAL STUDY

4.1 Introduction

This chapter is concemed with the testing set-up, construction of models, materials

and procedures used for the static, fke vibration, forced vibration, and ultimate load tests of

five scale-mode1 towers. in order to conduct this experimental investigation, a new testing

facility was designed and buiit by the author. The sizes of the scaled guyed tower models,

mnge of accelerations, guy forces, displacements were first estimated so that the range and

sensitivity of the sensors would be of the same order. A shake table was buiit and

iastrumented in the Structures Laboxatory of the University of Windsor.

4.2 Scope of Erperimental Program

To better understand the behaviour of guyed latticed communications towea in

cesisting horizontal loads, ground excitations, and overloaâs, an experimental progratn was

undertaken. The main objectives of the experimental sndy were: (i) to establish accurate

experimental data to venfy and substantiate the structural response predicted by the

analytical modelling of guyed towcrs; (ii) to obtain data related to the dynamic response of

guyed towers subjected to ground excitations and to ver@ the numerical technique for

dynamic d y s i s ; (iii) to obtain data related to the k-vibration response of such towers;

and (iv) to examine the non-linear up-to-failure response of this type of tower. in order to

achieve these goals and obtain the most meaningfid experimental da@ five 1/20 scale

models of towers were built. This scale was chosen with due consideration to the space

available in the Laboratory, the size of the shake table, and the guy force levels as discussed

in detail in the following sections.

4.3 Description of Tower Models

The experimental program was c d e d out on five 1/20 linear-scale guyed tower

models under horizontal static loadings, ike-vibration conditions, forced vibration

conditions, and up-toîollapse loads. The objective of the experimental investigation

dictated that the model study should be of the bbdùect type". In a "direct" model test, the

model is built such that al1 its properties are as similar as possible to those of the prototype

but on a reduced d e . Although it was not intended in this study to model a particular

design and develop a o n e - t ~ n e comspondence between the model and the prototype

response, it was possible to closely simulate the construction of typical welded prototype

guyed towers. The deteminhg factor used in the choice of model materials was the

practicai aspects involved in fabrication, instrumentation, and testing technique. Mso, a

small scale bad to be used to accomodate the model within the laboratory space; this made it

dificuit with respect to the choice of sizes of sections and materials available.

The five models had the same mast configuration. However, the parameters changed

in the models were tower heights, span between guys, number of guy levels. and the guying

system. The guy anchor radius was kept constant, thus the ratio between the guy d u s and

the tower height varied in the five models.

The towers tested had a ûiangular cross-section made fiom welded steel rods. The

legs of the tower mast were made of 3.1 mm (118 in) solid round bars and the bracing

system was made of 1 .S mm (1/16 in) round bars. The mast had constant properties i.e., face

width, bracing pattern, panel heights, and member sizes. The face width was consaicted as

30.5 mm (1.2 in) and the panels had a constant height of 38 mm (1.5 in). Figure 4- 1 shows

the typical mast of model towers. Figure 4-2 shows a section of an AM radiator tower, in

which the structure is utilised as the radiating antema. These two figures show the

construction similarity between the prototype and model rnasts.

Bases of typical modem guyed towers are aaiculated to allow rotations in the

bending planes. Figure 4-3 shows one of the methods used to provide such articulation by

bringing the tower legs into one point and thus reducing the moment resistance of the mast

base. Figure 4-4 shows a more elaborate and expensive detaii where the tower base is M y

articulated through the concave and convex plates. This detail is normally used on taller

and heavier towers. Figure 4-5 shows a mast base in which the mast legs are connected

through a base of heavy beams (star base) bearing on a smaller cylinder. This detail is

typically used for middle range heights of towers and a similar detail has been adopted for

tower models as shown in Figure 4-6. The mast legs are connected to the upper plates and a

solid cylinder is in-between the plates, allowing rotations about the horizontal axes. The

fiction on the cylinder, resulting fiom the self weight of the mast and the additional axial

loads fiom the prestressed guys, provides partial resistance to rotation about the vertical

axis.

4.3.1 Tower Model I

Three guy levels supported the mast and each level had three guys. The tower had a

total height of 2204 mm and a guy radius (distance tiom the centre of the tower base to the

anchor point) of 1400 mm. The guy levels were connected to the mast legs at 570 mm, 1330

mm, and 2090 mm elevations. A profile of tower model4 is shown in Fig. 4-7. The guys at

the top guy level had a diameter of 0.5 mm (0.018 in) while the lower two levels had a guy

size of 0.3 mm (0.012 in).

4.3.2 Tower Model II

This tower model had a total height of 2432 mm (96 in) and supported by three guy

levels of the same sizes as model I. The guy levels were comected to the mast at 570 mm,

1292 mm, and 2280 mm elevations, thus leaving a top cantilever of 152 mm (6 in). A

@ile of model-II is show in Figure 4-8.

4.3.3 Tower Model III

This model was coasmicted to study the effect of torsion-resistoa, which are anns

made to extend the tower face. Figure 4-9 shows a typical torsion resistor instaiîed on a

ûiarigular tower. Figure 4-10 shows the actual torsion resistor coastructed on tower Model-

m. The tower was identical to Model 11 in al1 other aspects with the exception of the torsion

resistor that was used on the top guy level. Figure 4-1 1 shows a piciure of the model as

consrnicted and the profile is shown on Figure 4- 12.

4.3.4 Tower Model IV

This tower model had a total height of 3048 mm (1 20 in) and mpported by four guy

levels at the elevations shown in tower profile (Fig.4-13). The top two guy levels were

made of 0.53 mm (0.021 in) size guys and 0.45 mm (0.018 in) guys were used for the lower

two guys. The top cantilever was 229 m (9 in). A photograph of tower model-IV is shown

in Figure 4-1 4.

4.3.5 Tower Model V

This is the tallest tower rnodel. It had a total height of 3658 mm (144 in) and

supporteci by four guy levels at the elevations shown in the tower profile (Fig.4-15). The

laboratory ceiling as weli as the dimensions of the shake table limited the height. On the

top two guy levels 0.61 mm (0.027 in) guys were used and 0.51 mm (0.021 in) guys were

used for the other three levels. A photograph of the tallest tower tested (de l -V) is shown

in Fig. 4-16. Table 4- 1 summarises the geometry of the tower models.

4.4 Materials

Material, sirnilar to the prototypes, were used to fabncate the tower models. The

mast steel was solid round bars that were cut to size. The bars were of steel grade CSA

G4O.2 1-300 W , with a modulus of elasticity of 200 000 MPa.

The guys were made of corrosion resistant Type 302 stainless steel wire ropes

(aircraft cables). Al1 the guys used were 1x7 single strand construction. The ropes were

preformed by the manufacturer, to remove intemal stresses and provide easier handling

while minimizing hying. The ropes were dso pre-stretched to provide uniformity in

strength and elongation properties. The ultirnate breaking strengths from the

manufacturer's published data were used in the analysis; however, tests were made to

coafum these numben and also to detemine the equivalent modulus of elasticity for the

cables. The tested modulus of elasticity for the guys was 165 000 MPa.

4.5 Mode! Analysis and Similitude

The development of a mode1 to satidy necessary similitude requirements of a

complex structure such as a ta11 guyed tower within the space and performance limitations

of a shake table can prove clifficuit. Some of these difficulties are discussed in the design of

the shaking table (Section 4 3 . However, the model as constnicted and describeci above

would have a h e m scale 1/20 - 1/30 of typical towers. Thus the prototype towers would

range in heights h m 45 m (150 A) to 110 m (360 ft). The member sizes for the mast

components would range fiom 3 1 mm (1 . î S in) to 95 mm (3.75 in) solid round bars. The

guy sizes would range h m 6 mm (0.25 in) to 18 mm (0.75 in). Considering the above, the

models as consüucted would be considered weight-distorted models of typical short to

medium-height guyed towers used for wireless applications. Taller towers with heights up

to 600 m used prllnarily for broadcast purposes would be even far more difficult to model.

However, one of the purposes of this experirnental program is to establish confidence in the

analytical procedures, which in tum cm be used to analyse taller structures that are difficult

to model and test.

4.6 Construction of the Tower Models

The steel rods for the bracings (horizontals and diagonals) were cut into length. The

main legs were cut into long pieces, and each piece spans the total tower height. The panels

were marked on the legs and a form mis made to hold the legs and the bracing as they were

welded togethet. This helped to d u c e the deformations and twist in the tower due to the

heat h m the welding. However, it was not feasible to entkly eliminate any twist in the

tower construction and a close look at Figure 4-17 show signs of twist in the towea.

The tower was then comeeted to the top plate of the base and held vertically while

adjusting the tensions in the three guys simultaneously with the aid of the length adjustment

tools shown in Figure 4-18. The tensions were monitored through the data acquisition

system and bmught up to the initial tension levels which are nomally specified as 1 O- 1 5%

of the ultirnate guy tension.

4.7 Vibration Excitation System (Shake Table)

There were severai challenges that had to be overcome in order to design and

construct a vibration excitation system that cm be used for light structures in gened. Some

of these challenges were:

1- the size of the structure: Guyed towers are very taU, theu heights may vary anywhere

fiom 30-600 m. Considering that the prefened economical anchor radius for a typical

guyed tower is about 0.6-0.7 of the height of the specific tower (and c m be as small as 0.3),

a relatively large shake table is required to model the tower with the anchonng guys.

2- the natural fiequencies of the towers: The structure is flexible, lightweight, and has

relatively low natural ûequencies. Therefore, the shake table design involved selecting an

optimal stiflhess-to-mass ratio while avoiding frequency interference between the shake

table and the tested structures.

3- memkr sizes: Guyed towers are usually made of a d o m face width and due to the

lateral support of the guys, smail members are used for the legs and the bracing of such

towers. For example, a typical 100 rn guyed tower may have soüd round legs of 51 mm (2

in) diameter. In order to use a (l/20) sale model of such tower, a mast of 5 m height wodd

be nquired with an anchor radius of 3.0 m using legs of oniy 2.5 mm diameter which is

vey srnall for the hancihg and construction of the models.

Thus, the shake table had to be large enough to accommodate the anchorhg points

of the guys and to enable the use of reasonable size scaie models, rigid enough in order to

minimise the interference effects from a vibrating mode1 on the table, and with minimum

mass so as to minimise the driving force requiternents.

4.7.1 Limitations of Tower Models

The similitude laws of a dynarnic model constnicted fiom the same material as the

prototype dictate the scaling factors for the different physicai quantities of the model. The

length scale Si , which is the ratio between the length of the model to that of the prototype

determines the dimensions, displacements, weight, and the naturai fiequency of the model

(Sabnis et al. 1983). Therefore, the resdting kquency of the scaled model is (s$' times

the natumi hqency of the prototype. Thus for a typical guyed tower with a natural

fiequency of 1.0 Hz. the resdting 1/20 d e model of mch tower would have a fkquency

of 20 H z This requUes that the shake table be able to perform with frequencies up to 30

Hz

4.7.2 Configuration

nie resuiting design consisteci of a rectaopuiar 3.5 m x 2.5 m table.. These

dimensions allowed for an anchor radius of 1.40 m for a triangular shaped tower.

Therefore, the maximum anchor radius of the guys that can fit within the shake table (1.4

m) and the ceiling of the Structures Laboratory limited the height of the test towes to 3.6 m.

In order to simplify the table construction, the choice of the supporting material for the

table was limited to either spring supports or to mller bearings. Nine spring supports were

originally used at the support points: however. that proved to be inpracticai as the horizontal

stifniess of the supporting s p ~ g s was aot controlled and it lacked a fail mechanism to

prevent it fiom snapping out of position. The final layout of the table is shown in Fig. 4-19

while the details of its constmction are shown in Fig. 4-20. As shown in Fig. 4-20, the table

was made of a sandwich plate consisting of two 6.5 mm (114 in) steel plates bonded to a

relatively thick 220 mm styrofoarn (plystyrene) core. The two plates were connected

together by strips of steel plates welded dong the perimeter of the table and through bolts

that tie the two plates every 500 mm (20 in). The use of a sandwich plate allowed for high

rigidity while minimishg the rnoving mass thus allowing the use of an available 90 kN (20

kip) actuator to produce kquencies up to 35 Hz. The actuator is displacement-controlled

through a displacement transducer that is used as a feedback device.

The shake table is supported at nine points by low Wction roller bearings to

minimise distortion of the desired table response. The b e a ~ g s are mounted on three rails

allowing a maximum travel of * 75 mm (3.0 in). Each rail is connected to a W360x45 steel

section and fixed to the laboratory flmr.

4.7.3 Analysk and Verification

A finite element mode1 of the table was nrst used to predict the naturai fkquencies

of the system, the required dnving force of the actuator, and the performance of the table

before its fabrication. in order to ensule that the table wouid perfonn for frequencies up to

30 Hz, the first bending fkquency of the table must be higher than 30 Hz Figure 4-21

shows the finite element mode1 used for the analysis of the shake table and its hdamentai

naturai fkquency and mode shape.

The performance of the table was measured through different sinusoidal waves for

the required range of fkquencies. Three accelerometers mounted on the table at the

opposite end to the actuator were used to verify the performance of the table. The purpose

of these accelerometers was to measure the motion of the table to ensure that accelerations

at the different points are uniform with minhum distortion. The readings of these

accelerometers were compared to the expected theoretical waves. Acceptable perfomiance

of the table was verified for fkquencies up to 35 Hz. Figure 4-22 shows the reading fiom

the accelerometers for sinusoidai waves of 30 Hz.

4.8 Instrumentation

4.8.1 Mechanical Dia1 Gauges

Mechanicd dial gauges havhg 0.01 mm mvel sensitivity were used to measure the

tower deflections at the mid-span between guy leveis and at the top two guy attachments

levels of the tower models. The did gauge readings were manualiy taken at each hcrement

of the static load throughout the test procedure. Figure 4-23 shows the dia1 gauges mounted

on the tower model V.

4.8.2 Linear Variable Displacement Transducen, LVDTs

Linear variable displacement transducers (LVDTs), of 150 mm stroke, were used at

five locations dong the height of the towers. Figure 4-1 1 shows the locations of the LVDTs

for the tested tower model III. Mount Details of LVDTs and accelerometers are shown in

Fig. 4-24. The plate attached to the tower leg was only to provide contact surface for the

LVDT,

4.8.3 Accelerometers

Semiconductor acceleration eansducea with built-in amplifier were used at six

locations dong the tower height. Five measured the accelerations in the direction of the

motion and the sixth m e d the acceleration in the perpendicular direction. Aiso, a

seventh accelerometer was mounted on the shake table at the tower base to measure the

input motion to the tower. This base accelerometer is shown in Fig. 4-25. The

accelerometers were of the piezo-resistive type with a built-in temperature compensation

circuit, with a meamhg range of IO. 1 to *50 g's, a fkquency response up to 1400 Hz. One

of the important characteristics of these accelerometers was theV lightweight, which made it

possible to meamre the acceleratiom without distorthg the mass-stiffhess properties of the

system. Each accelerometer had a mass of ody 13 grams. A view of the accelerometer

mounting detail to the tower is show in Fig. 4-24.

The locations of the instruments used on each of the tower model are shown on their

profiles (Figs. 4-7,4-8,442,443, and 4-15).

4.9 Test Equipment

4.9.1 Hydraulic Actuator

A Gilmore 20-kip (89 kN) model 433-20 hydraulic actuator was used as the âriving

force for the forced vibration tests of the tower modeis. One end of the actuator was fixed

to the testing h e while the other end was bolted to the shake table (Fig. 4-26). The

hydraulic actuator system consisted of the actuator, which included a position transducer,

servo valve, and an intemal LVDT. A 76 litres per minute hydraulic pump powered the

system and the signal wes controlled through a digital controller manufactured by

Interlaken The servo-amplifier controller controlled the servo-valve of the acniator, the

rate of flow of oil into the acniator, the static set point, gain and damping. This controller

was also capable of using a predefhed set of hamonic displacements or a userdefined set

of displacement hctiom. It was decided that the input fùnctiom would be displacement

but the ndting accelerations at the base of the tower would be measured for input

accuracy into the numerid model. Figure 4-27 shows the acnuitor controller computer to

the left and the data acquisition system to the right.

4.9.2 Load Cells

Special load cells were custom-made for this testing. These consisted of aluminium

rings with two strain gauges fixed on the inside and outside of the ring to form a half-bridge

configuration. Calibrated weights were used to calibrate the readhg from the load cells and

these were used with an amplification of 2.0 on the data acquisition system to convert the

strain readings into forces in Newtons. Figure 4-28 shows the four load cells at one of the

anchor points and each ~g is connected to one of the cables. These load cells had a

maximum capacity of 400 N and were used to continuously monitor the guy forces through

al1 the static and dynamic tests performed.

4.9.3 Data Acquisition System

During both the static and dynamic tests, the data h m the sensors (LVDTs and

accelerators) were captured by a test control softwarr (TCS) using a MEGADAC 3000

Senes data acquisition unit. The test control software, TCS, is a powerful tool developed

especially for acquiring, reducing, and anaiyzing the dynamic analog data captured using the

MEGADAC. It simplifies the process for collecting, converting, monitoring, plotting, and

rwiewing the MEGADAC test data For dynamic tests, the TCS was adjusted to sarnple the

data at a rate of 400 readings pet second per sensot* As for the static tests, readings were

oniy taken at each load increment The MEGADAC 3000 Series is designed to measure at

rates up to 25,000 samples per second. The OPTIM's Differential Input Modules used with

MEGADAC accept signais directly h m vimially al1 active and passive sensors. In the h t

model tower, the filtet used in the module captured fnquencies up to 250 Hz, while for the

other models, the fïiter used captured fiequencies up to 100 Hz.

4.10 Experimental Setup and Testing Procedure

Each tower model was guyed through adjustable length devices (tumbuckles) and

the load cells monitored the forces in the guys. It was possible to obtain useful data with the

simplest set of instrumentation, described in Section 4.9, when appropriately deployed. The

scope of the measurement program was decided at the beginning. Deflections, accelerations

dong various points of the mast, and the forces in the guys were measured. Stresses in the

mast codd be derived nom the numerical models if it agreed with the measured deflections

and guy forces.

Figure 4-29 shows the general test setup, with the LVDTs mounted on a frame

secured to the main testing fisune and the accelerometers mounted on the moving mast. The

test procedure comprised of static, free vibration, forced vibration, and up-to-collapse tests

as described in the foiiowing stages.

4.10.1 Stage 1: Static Loads (Elastic Behaviour)

During this stage of loadllig, the model towers were tested elasticaiiy under various

loading conditions representing the applied horizontai wind loads. These static

concentrated load tests were important to confinn the existing anaiysis tools commonly

used in the industry to design these structures. Also, it was important to evaluate the finite

element model used. The loads were appBed through horizontal cables attacheci to the mast

at various points dong the height; each cable was connected to a weight pan over a pulley

that is attached to the test fiame. Figure 4-30 shows these cables with the weight pans

attached to them. The mcdels were tested in the elastic range while deflections and guy

forces were measund. Three -tic loading conditions were applied: (1) one concentrated

load at the top; (2) two concenûated loads applied at the top two loading points; (3)

concentrated loads applied at al1 loading points.

The loads were applied through standard caiibrated steel weights placed in the

weight pans. uierements of 9 N (2 lb) were used in this test

4.10.2 Stage 2: FrWibration Tesb

The fixe-vibration tests on the five tower models were perfomed by first mounting

the accelemmeters at the different elevations on the mast. Due to the flexibility and

relatively light weight nature of the structure, it was feasible to caphue at least the nrst five

modes by either appiying small impact loads (hammer test) ot by imposing initiai

displacements that are suddenly released. These loads or displacements were applied at the

top of the mast. in either case the structure was fke to vibrate and the accelerations were

caphired at a rate of 400 readings per second. The test control software (TCS) using the

MEGADAC 3000 &ta acquisition unit capRued the data h m the sensors. The output fiom

the test cases was then fed into a data analysis and display software, DADiSP, @SP

Development Co. 1991) for a fast Fourier transfomi (FFT). The FFT analyzer gave the

spectrum respome of the towers in the fiequency domain from which the naturai

tiequencies of the models were deduced. The DADiSP software also yielded the

magnitudes and phase angles for the mode shape. The naturai fkequencies of the structure

were determined under two conditions (i) initial conditions (undisplaced shape) and (ii)

under a set of horizontal loads resultîng in a displaced condition. This second loading

condition simdated the condition of the structure under static wind or mean wind speed.

4.10.3 Stage 3: Forced Vibration Tests

During this stage of testing, two forced vibrations test were canied out:

1. Sweepsine wave test: This was perfomd by conducting a set of tests that varied the

actuatot excited fkquency over a fieguency range of 5 to 35 Hz. The maximum

displacement magnitude was kept constant while the frequency was varied. The data

acquisition system continuously ncorded and the tests were visually observed. During

these tests, at the resonant kquencies very m n g vibrations were observed.

2. Transient load tests: The displacement histones of three Metent earthquakes were

programmed to the hydraulic actuator and the towers were tested under various

intensities of these displacements. The ground motions were obtained fkom the data

published by Iwan (1997) for Landen Earthquake with a maximum velocity direction

(N80W) and for NorthRdge Earthquake N-S direction, and Nanbu Earthquake;

maximum velocity direction (N49W). The displacement histories were used as the

input to the actuator's controller. It should be noted that these displacements did not

closely simulate the above mentioned earthquakes as only twenty discrete input

fiinctions could be used. However, the resulting applied mast base accelerations were

recorded by the accelerometer mounted on the table and fed to the analysis model as the

input acceleration history.

The primary purpose of these tests was to test the analytical procedures in the prediction

of mast displacements and accelerations under ground motion or any other applied transient

loads.

4.10.4 Stage 4: Up-to Collapse Tests

Finally, each model was tested to failure using a simulated static wind load that

varied with the height simüar to the CSA S37-94 (Canadian Standards Association 1994)

wind profile. Six horizontal loading points were used as show in mode1 tower profiles

(Figures 4-7'4-8,442,443, and 4-15). For al1 models tested, the load was applied in equal

increments. Mer each increment, the load was maintained constant while mast deflections,

and guy forces were recorded. The loads were applied until collapse resulting either fiom

broken cables or fiom mast failms. Figure 4-3 1 shows tower mode1 V at failure.

Table 4-1 Details of Experimental Towers

No of Guy Mode1 Tower Height(m) Comrnents

Levels

I 2.2 1 3

II 2.44 3

ru 2.44 3 Torsion resistor at top level

Figure 4- 1. Typical Mast of Mode1 Towers

Figure 4-2. Typical Mast of Guyed Tower used for Heights up to 200 m

Figure 4-3. Details of a Typical Tapered Mast Base

Figure 4-4. Details of a Fully Articuiated Mast Base

Figure 4-5. Details of a Typical Star Base of a Guyed Mast

Figure 46. Typical Mast Base of Mode1 Towers

A

Cross Section

LVDT'S NO1 ELEV

Figure 4-7. Profile of Mode1 Tower 1

ACCELEROMETER NO 1 ELEV

LOAOING POINTS NO 1 ELEV

& Al - A 6

Cross Section

Figure 4-8. Profile of Mode1 Tower II

Figure 4-9. Typical Torsion Resistor

Figure 4-10. Torsion Resistor of Model Tower III

Figwe 4- 1 1. Mode1 Tower III

Cross Section

Figure 4-1 2. Profile of Mode1 Tower III

Figure 4-13. Profile of Mode1 Tower IV

Figure 4-14. Mode1 Tower IV

80

4 Al- A ï

Cross Section

Figure 4- 15. Profile of Mode1 Tower V

81

Figure 4-16. Mode1 Tower V

82

Figure 4-17. inherent Twist as a Result of the Manufacturing of the Models

Figure 4-18. Guy Tension Adjusters

1 MRECTKJN OF MOT ION

Figure 4-19. Shake Table and Plan of Test Set-up

85

6 mm STEEL PLATE 300 mm STYROFOAM

HYDRAULIC ACTUATOR STYROBOARD STEEL PLATE

///A I

W SEC

Figure 4-20. Shake Table

Figure 4-2 1. Fundamental Natural Frequency and Mode Shape of S h a h Table

Figure 4-22. Acceleration History of Shake Table for a 30 Hz Frequency

Figure 4-23. Dia1 Gauges Mounted on Model To

Figure 4-24. Mount Detail of LVDTs and Accelerometers

Figure 4-25. Measurement of Towet Base Accelerations

Figure 4-26. Shake Table Driving Actuator

Figure 4-27. The Actuator Controller and the Data Acquisition System

Figure 4-28. Load Cells for Measuring Guy Tensions

Figure 4-29. Top View of Test Set-up

95

Figure 4-30. Static Load Application Set-up

Figure 4-3 1. Mode1 Tower V at Collapse

CHAPTER V

RESULTS FROM MODEL TOWERS TESTS

5.1 Introduction

The results of the static and dynamic tests obtained fiom the five mode1 towers

described in Chapter IV are discussed in this chapter. The main objective of the experimental

program was to evduate and demonstrate the accuracy of the various nm&cal anaîyticai

produres used for the anaiysis of guyed towers. In doing so, the data fiom the tests were

also usefiil to study the efféçt of height on the behaviour of guyed communication towers, to

study the influence ofaumber of guy levels torsion resistors, and base supports on the dynamic

characteristics of guyed towers, and to estimate dampiag of the structures.

In this chapter, the experimental and theoretical results for the five tower models are

presented and compareci. The r d t s presented include deflections, guy forces, and failure

loads of the tower modds under static horizontal loads. A h , tbe r d t s presented herein

indude the deflections and acceleration histories under severai dymmic loading conditions.

Furthemore, the naturai fiequenues, mode shapes, and damping characteristics of the tower

models are presented hereia Because of the large amount of data coliected, for brwity, only

results showing typical behaviow are presented.

5.2 Static Loading

In order to study the elastic response of guyed towers due to static loads, five guyed

model towers were tested aud d y s e d using the dinerent numerid techniques. Concentrated

horizontal loads were applied at several locations dong the tower height wMe deflections and

guy forces were measured. These horizontal loads simulate wind loads on the structure.

Figures 4-7, 4-8, 4-12, 4-13, and 4-15 of the tower profiles show the locations of the load

applications points and the measured defledon points for tower models I, II, IiI, IV, and V

respectively .

5.2.1 Load Case I

In this load case, the model towers 1 to IV were loaded at the top (Ioadtog point L1) of

the towers. Results in Figures 5-1, 5-3, 5-5, and 5-7 show the calculateci and the measured

deflections dong the heigbt of the tower under a concentrated load of45 N (10 Ib) for model

towers 1 to [V rrspedively. Note that this load on the model towers represents a concentrated

load of 17.8 kN (4.0 kips) on a 20:l prototype structure. Figure 5-9 shows the redts for

model tower V under thra concentrated loads of 26.7 N (6 Ib) each applied at the top three

loadhg points (L 1 to L3 in Fig. 4- 15). Tables 5-1,5-3,505, 5-7, and 5-9 compare the different

a d y t i d models with the measured cable tensions for this load case. In ail cases, very good

agreement between the experhental and the Werent theoretical models can be observed.

Also, it can be noticed that the results for the two finite element methods show here produce

virtually identical resuits in terms of deflections and guy forces. The beam on non-hear elastic

supports adysis results are in general agreement to the two finite element numencal modeh.

Cornparhg the deflectiom of tower modds 1 and II, both towers have identical mast

configuration and the same guy sizes and initial tensions but the oniy ditfierence is that tower II

is 10% taiier than tower I and the deflections under the same loads are 35% more for tower II

(Figs. 5-1 and 5-3). In cornparison, deflections of tower üX, (which is identical to towers 1 and

II with the exception of the torsion resistor at the top level) are less than half of those of towers

1 and TI (compare Fig. 5-5 with Fig. 5-1 and 5-3).

52.2 Load Cage II

Mts in Figs. 5-2, 54, 5 6 , 5-8, and 5-10 dustrate the d d a t e d and measured

deflections at the deflection measwing points dong the tower. in this load case, concentrateci

loads of 26.7 N (6 Ib) were applied at each of the loadhg points show on the tower profiles.

Tables 5-2, 5-4, 5-6, 5-8, and 5-10 compare the mea~u~ed guy tensions with the caiculated

tensions f?om the three anafysis techniques. In a l cases, good correlation between the

experimental and the theoreticai results can be observeci. However, it should be wted that the

experimental deflections are generaiiy higher than the theoretical deflection by an average of

6%. Also, it can be notiad that the r d t s nom both Wte element procedures are nearly

identical while the beam on non-linear springs anaiysis r d t s are comparable but not with the

same lwel of close agreement as that noticed for load case 1. Furthemore, it can be noticed

that the defledoas for mode1 tower Vary by up to 1% kom both the FE techniques and the

beam on s p ~ g analysis. in this loading case, the cables are stress& up to 500/0 of tfieir uitimate

capacity which is a comparable to the service load level.

From an engineering point of view, the experirnental redts are in good agreement

with the analytical models. Any of the three analysis techniques can be saiisfactorily used to

determine the static behaviour of the structures at these load levels. Furthermore, modeiling

the mast as an equivalent beam, whose properties are based on various submodels of the truss

mast, give r d t s that are nearly identicai to that ofthe 3-dimensional ûuss analysis.

5.3 Free Vibration Tests

In order to detennine the natuml âequencies of the tested towers, severai impact loads

were applied to the tower models, and then the towers were lefl to vibrate fkely. The towers

were excited by a hammer or by inducing an initial displacement that is suddenly released.

Before analyshg the results, the adequacy and quality of the data were verified by visual

examination of the aderation titane histories on the monitor. Figures 5-1 1, 5- 1 3, 5- 1 5, 5- 17,

and 5-19 show typical acceieration t h e histones for tower models I, 4 IV, and V,

respectivdy. The extraction of usefui infomtion invaxiably requires the computation ushg a

Fourier T d o r m . Therefore, the data captured fiom the sensors were fed into a Fast Fourier

Traosform adyser to give the response of the tower in a ne<ruency domah instead of a

tirne domain. Figures 5-12, 5-14, 5-16, 5-1 8, and 5-20 show the fiequency spectra for the

acceleration the histories shown in Figures 5- 1 1, 5-13, 5-1 5,s-17 and 5-19 for the f i e tower

models. The naturai fiequencies of the towers were determineci by the examiaation of the

frequency spectra rdting fiom the acceleration time histories captured from the five

accelerometers rnounted on the tower. Tables 5-1 1 to 5-1 5 summarise the natural fiequencies

and the mode shapes of the five tower rnodels. Good correlation between the experirnental and

theoreticai hdings can be observed. A h , it should be noted that both finite element analyticai

rnodels yielded very sirnilas resuhs.

For the shortest tower, Mode1 Tower I, the computed &st natural fiequency of

vibration using the finiteelement andysis was 15.6 Hz which is quite close to the fiequency of

16.5 Hz rdting from the experimental data coilected (Wahba et al. 1996). Also, the

correspondîng mode of vibration was torsional mode. Furthemore, it should be noted that this

first tower mode1 was @al, as the base support did not provide any restrain agaiast twist of

the rnast. A diierent tower base design was later used for the other towers as shown in Fig. 4-

26. A numerid analysis used to check the effect of the support conditions and restrallriag the

base against twist d t e d in a first flexml mode shape with a natural fkpency of 25.8 Hz

A h , it can be noticed nom Tabie 5-1 1 that the tirst four mast modes wae exgted fiom

dinerent impact tests, and the modes were identi6ied on the &eqpency spectni. The fiequency

spectrum was usefùi in predicting the mainil fiequencies; however the mode shapes were

predicted from the sine-sweep tests and with the help of the numerid analysis as explained in

the sequel.

The experhental resuits shown in Table 5-1 2 show that the value of the fùndarnental

Iiequency for model tower [I is 22 Hz measured versus 22.3 tiom the FE bearn model and 22.7

Eom the tniss model. This 15% reduction in the natural tieqyency compared to model tower

I., is a direct result of the change in height as this is the oniy parameter that was changed

between models II and 1. A h , it can be noticeâ that the natural fiequency of the fint torsional

mode was much higher and it came in order after the second bending mode. This is a direct

result of the fwty of the base against twist. This f i t y is a r d t of the ûiction on the pin

c o h g fiom the weight of the structure and the pre-stressing force from the initial tensions of

the guys.

Table 5-13 shows the r d t s for model tower JII, and masistent results between the

numerical FE models and the experirnental r d t s are evident; however, wmparing the

fundamental natuml 6equency of model tower KiI with that of model tower II, there is a 15%

increase. This increase may be attriiuted to the increase in sMbess of the top guy level as six

guys of the same slle were used instead of only three used on tower II. Aiso, a sign<ôcant

increase in the torsional mode tkequency can be noticed (the torsionai mode frequency is more

than two times that of model II), which is a direct result of the presence of a torsion-resistor on

the top guy level.

Resuits for model tower IV are shown in Table 5-14. h addition to the agreement

between the theoretical and experimental resdts, it can be also noticed that the lowea mast

bencihg fiequency is higher than that of tower LI and comparable to that of tower 1. Although

this tower is 33% tder than that of tower l, the mast spans between guy leveh are comparable-

This dong with the increase in guy stühess due to the use of larger guy sizes on tower N,

kept the lowest nanual fiequencies comparable.

Table 5-15 shows the results of the naniral fkequencies for model Tower V. This

tower was chosen to show the typical mode shapes of the test specimens. Figure 5-21 shows

the fundamental mode shape of the tower, which involves mast and guy motion. Figure 5-22

shows the est f l d mode shape of the mast and it is the seventh mode. The second flexural

mode of the mast is shown in Fig. 5-23 whiie the torsionai mode is shown in Fig. 5-24. Figure

5-25 shows a typical guy mode for the lowest set of guys. It aui be noticPd that this mode

does not include any mast motion. Cornparkg the dominant fieqyaicies with those of model

tower IV, it cm be not id that the fiequency of the fùndarnental rnast mode shape is 22%

lower than that of tower IV. Thme two towers share the same mast properties, number of guy

levels, guy radius, but tower V is 200/0 tder than tower W.

5.4 Sweep-Sine Wave Tests

The objective ofconducthg the sweep-sine wave tests on the five tower models was to

obtain the vibration signature of the acceleration tirne histories at different locations on each

tower. The forcing fùnction tbat was applied to the sbake table, and consequently to the tower

base and anchors, was a sinusoidal wave as shown in Fig. 5-26. The towers were subjected to

the same hction over a range of âequencies mging from 10 to 32 Hz A typical set of

results for one ofthese tests is show in Figures 5-27 and 5-28. The displacement histories and

the fiequency spectra are show for the six accelerometers that are mounted on the tower, and

the seventh, which is mounted on the shake table. The set of results presented in these two

figures is for Tower IV under a forcing sinusoidal wave of 22 Hz. These tests were used to

help iden* the mode Sapes of the tower models after detennining their naturd frequencies.

The aeady state vibration at the first few modes of vibration was measured at the six locations

dong the entire beight of each tower. The identification of the associated mode shape required

the determination of the phase between the signal fiom the six measured locations Simple

addition and subtraction of signais and obsenritlg how the amplitude of the Fourier Transfonn

peaks changed could achieve this, as peaks of signais in-phase wodd krease and those of out

of phase wodd decrease. However, this method is only applicable if the real part of the

fiequency components are either in-phase or outsf-phase. Also, studying the magnitude of the

response of the different semn in the âequency domaui helped in identifying the dominant

mode dupe at that location. Tbis uui be noticed fiom Fig. 5-27, where for accelerometer #3

(A3 in Fig. 4-13), higber fr#luencies d t e d in a kger magnitude than the fiindamentai

fiequency. This is opposite to accelerometer #1 (Al in Fig. 4-13) where the dominant

vibration mode is the fiuidamental fiequency. It should be noted that these tests also helped to

CO* the predicted oahiral fiequencies, as the viirations of the modeis at resonance

fiequencies where observed to be very vigorous and, in s e v d cases, this was strong enough

to break one or two of the cables.

The determination of the mode shap is a very difficuit ta& to extract theoretidy and

even more so experimentally. This is due to (i) close spacing of the Merent nequencies, (ii)

the interaction between the guy modes and the mast modes, (üi) the large number of guy

modes that corne in with the free vibration analysis, and (iv) very small deflections of the mast

as comparecl to the guys (shown on the normalisai mode shape diagrams). For example, Fig.

5-21 shows the first mode shape of model tower V, from which it is clear that this mode is an

interaction between the top guy level and the mast's fjrst bending mode; however, the second

mode shape, shown in Fig. 5-22, is pureiy the first mast bending mode. In order to identify the

mast mode shapes without the Uiterence fiom the Merent guy modes, a model consisting of

one element for each guy was constructeci. This suppressed the guy mode shapes only. Using

this analysis, it was usetiil to go back to the tiill model and pick the rnast modes that would

have similar frequencies to those predicted fiom the singie elernent guy models. Also, the

information regarding the effective mass associateci with each mode helped in establishg the

direction ofvl'bration ofthat mode.

5.5 Forced Vibrations Tests

The model towers were tested under different sets of transient forced vibration

fûnctions. This dernoristrates the use of the analytical procedure in a seismic analysis,

where the time history of acceleration at the base and the anchor points of the tower give

the forcing fùnction. As explained earlier in Chapter IV, the forcing functions were taken

From three earthquake displacement time histories that are fed to the shake table-forcing

actuator. The acceleration time history at the tower base is measured from accelerometer

No. 7, which is mounted on the table at the base of the structure (Figure 4-25). Figures 5-

29 and 5-30 show typical red t s for one of these tests. The results show in these figures

are for model tower IV under a simulation of the Northridge earthquake scaled to a

maximum acceleration of 5.0 g at the tower base (Accelerometer A7 in Figure 5-30). It

should be aoticed that the maximum accelerations at the top of the tower (accelerometer

Al in Figure 5-29) are about 23 g, which is close to five times the maximum input at the

base (accelerorneter No. 7).

Figures 5-31 and 5-32 show another set of results for an acceleration history

resembling the Nanbu Earthquake scaled to a maximum acceleration of 3.5 g as show in

the base acceleration time history (Figure 5-33). Figures 5-34 and 5-35 compare the

measured acceleration histories with the predicted ones at accelerometers 1 and 3

respedvely. In order to compare the numerical results to the experimental values,

damping had to be introduced to the model and detemined by calibration of results to the

measured response.

The concept of structural damping as a fiaction of the critical damping associated

with each mode c m o t be extended to non-linear applications where the equations of

motion of the system are integrated directly, and the fiequencies are aot part of the

solution. Furthemore, the natural fiequencies of the system are constantly changing

because of the non-linearities. Therefore, for t h e domain d y s i s , viscous damping is the

practical choice of damping mode[. From the analysis of the experùnental results under

free vibrations, a mass proportional Rayleigh damping a = 1.007, and a niffhess

proportional Rayleigh damping P = SE-04 were evaluated from the caiibration of the

andytical solution with the experimental results.

The displacement response for the acceleration history shown in Figure 5-33 is

shown in Figures 5-36 and 5-37. The good agreement between the measured and the

predicted deflections can be noticed.

5.6 Upto-Collapse Tests

Findy, each modd was tested to fdure to examine its non-linear response upto-

collapse. In the ôrst three modd towers, five horizontal wncentrated loads were applied dong

the height of the tower pulling it towards one of the anchors. In the remahhg two towers, six

concentrated loads wen used. For al1 the rnodels, the load was incremented at a constant rate.

Afier eech incremeat, the load was maintained constant whiie recording deflections and guy

tensions. After Wure of each tower modei, the appüed load was released slowly. A towa

model was considerd to have M e d d e r when one or more of the cables fàüed (and

wnsequently, the mast) or when the m a t Med to a point that it could not cany any m e r

loads.

For tower I, equd loads were applied at each of the fbe loadhg points. These loads

were increased just above 36 N at each of the loading points. At that point, the second guy

lwel cable ruptureci; immediately der, the lowest guy lewl also broke and the mast became

unstable because ofthe shear at the base and the tower fded. The second guy level f?om the

top broke at a tension of 82 N (181b). This was a sudden and unexpected failure as this load

was about 25% lower than the pubiished breakhg strength of 112 N for the cable. This

reduction in the cable strength can be attriiuted to the method of guy termination (using a

closed loop for the guy t e e t i o n typically reduces the load capacity by 25%). Figure 5-38

compares the load applied at each of the loading points versus the defleztion at sensor Dl.

Also, the analytid resuits are wmpared to the expetimental data. From this cornparison, it

can be noticed that both arialyticai models are in agreement with the experimental r d t s .

Also, both fiaite element models r d t s were the same. The b m n - ~ n - ~ p ~ g ~ model r d t s are

consistent with the FE moâeb up to the guy t'ail~e- Howwer, the bearn-on-springs analysis

fdeû to predict the dtimate cap- of the structure and the discrepuicy between its results

and that âom the FE analysis inmeases as the load approaches the faim level as ihstrated in

the load-deflection m e (Fig. 5-38).

Tower II was not loaded to Mure. The tnaimum loads were appiied in the elastic

range and presented by load case II for this model. Later, the tower mast was salvaged and

modifieci by adding the tomkm resïstor to it. A new set of guy cables was used and the tower

was tested as model tower m.

ln loading tower iil, the horizontal loads were increased with the height of the

structure to simulate the increase in wind pressure with height. The loads applied to the

loading points were incceased with the foIîowirig ratio âom bottom to top: 1, 1.25, 1.25, 1.5,

1.5.2. To dustrate that, if2 newtons were appiied at the lowest level (L6), 2.5 N would be

applied at load points 5 and 4, and 3 N at points 3 and 2, etc. Figure 5-39 shows the deflection

at point D2 versus the load applied at point L2. As the ioad at that point reached 40 N, the

taught guy at the second guy level fiom the top has failed, however the tower was capable of

resisting more loads, another two load increments were applied before the tower was

considered fded. The tower fidure starteci with the second guy level but at ultimate load it

was an overall bucwiog failure between the top two spans. Figure 5-40 shows the deflected

shape of the FE model, whiie Figure 541 shows a photograph of the tower at fdure.

Cornparison of Figs. 5-40 and 5 4 1 shows how the finite element model predicted the faiied

shape. A h , Fig. 5-39 shows that the ânite element model has predicted the fdure in the

second guy level fkom the top as ihstrated in the grapb It can be also noticed tiom the same

graph that the beam-on-springs anaiysis is in agreement with the tinite element models at lower

load levels As the load increases, the Merence between the two di&rent anaiyses is more

apparent and that dEerence reaches 28% at the guy breaking point. After tbat load level, as

the material non-linearities are not coDYdered in the bearn-on-springs analysis, the failure load

codd not be detected fkom this analyticai model.

In Load testing tower model IV, qua1 loads were applieâ at each of the six loading

points. Figure 5-42 shows this load versus the displacements at the highest measured point Dl.

The fdwe was suddm due to the cable rupture at the second guy level. As won as the taught

cable fded, the mast snapped and it was not possible to accurately record the tower failure

load. The last load appüed at each loading level was 53.5 N. Figure 5-43 shows the finite

element model showing the deflected shape at failure. The load deflection diagram also

indiates the same characteristics as noticed in tower iII, but not as apparent since the failure

was primarily a guy failure and not a mast failure, and the maximum deflection reported is at

the guy point and not in the middie ofthe guy span

For the tallest tower tested, model tower V, small load increments of 4.5 N were

applied at the top three loading points. This aiîowed capturing the load deflection behaviow of

this tower. The measured f i e load was 62 N and the predicted fdwe loadsfiorn the finite

element models was 64 N. The chronologid mode of fdure can be summarised as follows:

(0 buckling in the mast top span b e e n the tbird and fourth guy level nsulting in the

formation ofa plastic hhge (Figuns 5-46 and 5-47) fonowed by, (ii) rupture of taught cable at

the top guy level.

Figure 5 4 4 shows the load displment airw for Tower V. Experimental readings in this

test were fiequent aad thus a good ihstration of the bebaviour was available. The redts

confirm the findings of tbe previous tested towers; a good agreement between the experimental

data and the finite element modek, with the beam-on-springs modd not haviog the same level

of agreement at higher load levels. At fdure load the Merence between the iinite element

rnodels and the beam-on-springs models was approximately 12%.

5.7 Discussion

The r d t s presented in this chapter indicate a good agreement, in geaeral, between the

expehental and fiaite-element results in the elastic, fk-vibration, forced vibration, and non-

iinear responses of the tower models. The r d t s indicate that the level of agreement in the

static analysis in both lineu and non-hear static response is vay good considering di types of

experimental erron. These erroa include measurement, loadiag, tnanufac~iring, and boundary

condition enors. It can be observeci that in the elastic range at very low load levels, the 6nite-

eiement solution overestunated the arperimental deflection. This maybe attriiuted to the smaii

resistance in the LVDTs seasors that may have impacted on the vexy mail measured

deflections.

in the fiee-vibration response, the expaimetital hdamental n a d fiqencies of the

tower models w a e clear to detect and were very close to those obtained fiom the finite-

element models. Howevery the proximity of the natural fiequemies of the tower and al1 the

supeduous guy modes made it very difncuh to detect the higher mode shapes either in the

muneriai models or even in the measwed deta Therefon, a higher level of discrepancy

between the measured and cornputeci naniral fiequencies can be scpected at the higher mode

shapes.

The simple constniction of the shake table with even mal1 possibiüty of tiltingy or

yawing in the table movement could be attributed to another source of error that would appear

in the forced vibration response. Furthermore, the small sale used to create the models made

it wiy diflicuit to manufacture and most importantly to test.

5.8 Summary

The structural response of multi-level guyed towers was d e d experimentally and

anaiytically through a fmite-element model. Compatisons of the results were made for the

horizontal defidoos, guy forces, and horizontal acceierations. From this anaîysis, the ultirnate

load capacity of a structwe was predicted dong with its natural fiequencies, mode shapes and

transient dynamic response. The foIIowing is a sumrnaq of the findings that can be drawn î?om

this analysis:

1. Good agreement between the acpaimental and theoreticai results supports the reiiabiiity of

using the fite-eiement aaalysis presented to predkt the elastic response, fiee-viiration

response, forcd viiration response, ad dtimate load-caqhg capacity of guyed towers.

2. For symmetrical m m , the eSUn,alent beam finite element model, explaineci in detail in

Chapter IV, does predict the behaviour of the structure with the same level of accuracy as

the Ml t n ~ s s modei.

3. The beam-on springs model, wbich is currently the most commonly aud widely used in the

industry, does provide a v a y good agreement with the euperimental r d t s at service load

levels. However, as the loads approach failure, this level of agreement deteriorates by 8% - 25% depending on the type offdwe.

4. Height is the major parameter affecting the natural fiequency of the tower. The use of

stiffer guys improves the t'undamental fiwency and the application of torsion resistors

drarnaticaliy improves the torsional fi-equency.

5. The simple shake table designed and constructed by the author as explained in Chapter IV,

was successtiily used in the excitation and testing of the model guyed towers and can be

used for fùture testiog ofligbt flexible structures.

Table 5-1 Guy Forces For Model Tower 1 (Load Case 1)

Level r GUY Number

FE Bearn

Model

N

FE Tmss

Model

0

B eam-On-

Springs

1

Table 5-2 Guy Forces for Model Tower 1 (Load Case-2)

FE Beam

Mode1

0

FE Tmss

Model

(NI

Bearn-On-

Springs

M

- ---

GUY Level

Table 5-3 Guy Forces for Model Tower II (Load Case-1 )

Guy 1 Guy ( Experimentai ( FE Beam FE Tmss

Model

N

Level

Beam-On-

Springs

0

Number 0 Mode1

0I)

Table 5-4 Guy Forces for Model Tower II (Load Case-2)

Beam-On-

Springs

(NI

0.3

t

&Y Level

1

Guy

Number

FE Beam

Mode1

0 0.3

Experimental

(N)

FE Truss

Mode1

0 0.3 I 1 O

Table 5-5 Guy Forces For Model Tower III (Load Case-1)

"""'"""I Level Number Beam-On-

Springs

(NI

FE Beam

Model

N

FE Tmss

Model

(NI

6.2

Table 5-6 Guy Forces for Model Tower III (Load Case-2)

FE Beam

Model

0

FE Tmss

Model

(N)

GUY Level

@Y Number

GUY Level


&Y Number

Experimental

0

20

FE Beam

Mode1

(NI 21.3

FE Tmss

Mode1

(NI

21.3

Beam-On-

Springs

0 21.1


GUY Level

Guy

Number

Experimental

0

FE Beam

Mode1

(NI

FE Tmss

Mode1

0

Beam-On-

Springs

(NI


GUY Level

FE Beam

Mode1

(NI

Guy Number

Experimental

(N)

FE Truss

Mode1

(NI

Beam-On-

Springs

(NI


GUY Level

I*""'" Number

FE Beam FE Tmss Beam-On-

Mode1 Mode1 Springs

Table 5-1 1 Natural Frequencies for Mode1 Tower I

1 Torsional

Bending 1' (x-plane) rn Top Guy modes

Bending 2* (x-plane)

Bending 2M (y-plane)

2M Guy modes

Bending 3" (x-plane)

Bending 3" @plane)

Natural Frequency

q l TRUSS

Table 5-12 Natural Frequencies for Mode1 Tower II

Mode

1 Bending (y-plane) 1 22

Natural Frequency

(Hz)

l Bending lSL (x-plane)

Bending 2nd &-plane) I 34w2

E i 8

22

Top Guy 1" Modes

2"' Guy 1' Modes


BEAM

22.3

34.2

Guy Modes

Torsional Modes

Bending 3m Modes

TRUSS

22.3

27.5

27.7

35.82

40

47

72

27.5

27.7

35.82

42.1

53 -6

72.6

42.1

53 .O

72.6

Table 5-13 Natural Frequencies for Mode1 Tower III

1 Mode I Natural Frequency

EJYP

Bending 3m (y-plane)

Bending 1" mode (x-plane)

Bending la mode @-plane)

Guy Modes


Bending 2w &plane)

Guy-Modes

Guy Modes

Bending 3m (x-plane)

1 Torsional Modes 1114

BEAM TRUSS

25.3

25.3

26

44.6

44.6

47.3

25.73

25.73

28.5

41.1

41.1

42.2 i

25 -73

25.73

28.5

41.1

41.1

42.2

53.5 - 56.8

74

52

63

53.5 - 56.8

74

Table 5-14 Natural Frequencies for Mode1 Tower IV

Mode Natural Frequency

Top Guy Modes

Bending 1" mode (x-plane)

Bendiag 1" mode (y-plane)

EXP

21.6

Guys Modes

Bending 2M Mode (x-plane)

Bending 2m Mode (y-plane)

22.5

22.5

Guy-Modes

Bending 3m (x-plane)

Bending 3m @-plane)

BEAM

22.1

32.8

32.8

TRUSS

22.1

24.8

24.8

51.5

51.5

24.8

24.8

29.0

36.1

36.1

29.0

36.1

36.1

0.0

56.3

56.3

0.0

56.3

56.3

Table 5-15 Naturai Frequencies for Mode1 Tower V

Top Guy Modes

Bendiag 1 mode (x-plane)

Bending 1" mode (y-plane)

Guys Modes

Bending 2d Mode (x-plane)

Bending 2M Mode @-plane)

Torsional

Bending 3m (x-plane) w

Bending 3m (y-plane)

Natural Frequency

EXP

0.0 5.0 10.0 15.0 hflution (mm)

Figure 5-1. Defîection of Mode1 Tower 1 under Load Case 1 (45 N at top loading point)

2500

2000

1 500

IWO

500

O

0.0 5.0 10.0 15.0 20.0 25.0 30.0 Deflection (mm)

Figure 5-2. Deflection of Model Tower 1 under Load Case II (27 N at al1 loading points)

5.0 10.0 15.0 20.0 Deflectlon (mm)

Figure 5-3. Deflection of Model Tower II under Load Case 1 (45 N at top loading point)

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 Deflection (mm)

Figure 5-4. Deflection of Model Tower II under Load Case II (27 N at ail loading points)

O

Figure 5-5

0.0 5.0 10.0 15.0 20.0 Detlection (mm)

'. Deflection of Mode1 Tower III under Load Case top loading point)

f 1 1 1 I 1 I 1 1 I 1 1 I 1 1 1 1 1 1 1 1 1 I 1

- J - - l - - l - - 1 1 1 1 I 1

I I 1 1 1 I 1 1 1 1 1 1 1 t 1

1 t 1 1 I t - - - - - - - - - - I I I I 1 I 1 1 1 1 I 1 1 1 1 I 1 I I I 1 1 I I 1 1 I

' 7 " , - - T "

I 1 t 1 I t 1 I t 1 1 1 I 1 I 1 1 1 1 1 1 I 1 1

kperimentai '+ - Truss FE Model - - - - Beam FE Model

Beam-on-springs, 1 I I I 1 I a

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 Mlectîon (mm)

Figure 5-6. Deflection of Model Tower III under Load Case iI (27 N at al1 loading points)

-5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Defiaction (mm)

Figure 5-7. Deflection of Mode1 Tower N under Load Case 1 (45 N at top loadkg point)

- - - - Beam FE Model

0.0 5.0 10.0 15.0 20.0 25.0 30.0 Defieetion (mm)

Figure 5-8. Deflection of Model Tower IV under Load Case II (27 N at ail loading points)

-5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Defiection (mm)

Figure 5-9. Defiection of Mode1 Tower V under Load Case 1 (27 N at the top three loading points)

- Truss FE Model - - - - &am FE Model

-5.0 0-0 5.0 10-0 15.0 20.0 25.0 30.0 Deflection (mm)

Figure 5-10. Deflection of Model Tower V under Load Case II (27 N at al1 loading points)

0.80 Time (s)

Figure 5- 1 1. Acceleration Response for Free Vibration of Mode1 Tower I

Figure 5-12. Frequency Domain of the Acceleration Response show in Figure 5-1 1

0.00 0.50 1 .O0 1.50 Time (s)

Figure 5-13. Acceleration Response for Free Vibration of Mode1 Tower II

Figure 5-14. Frequency Domain of the Acceleration Response shown in Figure 5-1 3

0.80 Time (s)

Figure 5- 15. Acceleration Response for Free Vibration of Model Tower III

Figure 5- 16. Frequency Domain of the Acceleration Response shown in Figure 5- 15

0.00 0.40 0.80 Time (s)

Figure 5- 17. Acceleration Response for Free Vibration of Model Tower IV

Figure 5- 18. Fregwncy Domain of the Acceleration Response shown in Figure 5-17

0.80 fime (s)

Figure 5-19. Acceleration Response for Free Vibration of Model Tower V

Figure 5-20. Frequency Domah of the Acceleration Response shown in Figure 5-19

Figure 5-2 1. First Natutal Frequency and Mode Shape of the Top Guy (Mode1 Tower V)

Figure 5-22. First Naturai Frequency and Flexwal Mode Shape of the Mast (Mode1 Tower V)

Figure 5-23. Second Naturai Frequency and Flexural Mode Shape of the Mast @hiel Tower V)

RESTART FILS:

EICEîMDDE 29

AmQuS t'ERS1

Figure 5-24. First Torsional Frequency and Mode Shape of the Mast (Mode1 Tower V)

Figure 5-25. First Namal Frequency and Mode shape of the Bottom Guy (Mode1 Tower V)

1 .O0 2.00 Time (sec)

Figure 5-26. Sinusoidal Deriving Displacement History

150

Figure 5-33. Shake Table Acceleration History for Ground Motion modeled after Nanbu N-S Direction

I I I I I I

0.00 0.40 0.80 m e (s)

Figure 5-34. Measured and Calculated Acceleration Histories for Shake Table Motion Shown in Figure 5-33 (TowerIII- Acc. 1)

I I I I I 0.00 0.40 0.80

Tîme (s)

Figure 5-3 5. Measured and Calculated Acceleration Histories for Shake Table Motion Shown in Figure 5-33 (Towerm- Acc. 3)

- Experirnental - - - FE Beam Model

0.40 Time (s)

Figure 5-36. Measured and Calculated Displacement Histories for Shake Table Motion Shown in Figure 5-33 (TowerIII- D 1)

0.40 Time (s)

Figure 5-37. Measured and Calculated Displacement Histories for Shake Table Motion S h o w in Figure 5-33 (TowerIII- D3)

20.00 40.00 Dispiscement (mm)

Figure 5-38. Load versus Displacement at the top Guy Level (Model Tower I)

- I

Broken Cable --- - -

bpenmentaî - FE Tmss Model - - r FE barn Madei

Rem-on-springs Model

40.00 80.00 120.00 Displaœment (mm)

Figure 5-39. Load versus Displacement at the top Guy Level (Model Tower III)

Figure 540. Finite Element Displaced Shape of Mode1 Tower iII at Failure

Figure 5-4 1. Mode1 Tower III at Failure

Operimentai - FETniaa Madal - - - FE Baam Model

Baamsn-spnngs Model

40.00 80.00 Displacement (mm)

Figure 542. Load versus Displacement at the top Guy Level (Model Tower IV)

Figure 543. Finite Elemeat Displaced Shape of Mode1 Tower N at Failure

50.00 100.00 150.00 Displacement (mm)

Figure 5-44. Load versus Displacement at the top Guy Level (Mode1 Tower V)

Figure 5-45. Finite Element Displaced Shape of Mode1 Tower V at Failure

Figure 5-46. Model Tower V at Failure

170

Figure 547. Close-up of Failed Mast of Mode1 Tower V

CHAPTER VI

RESULTS FROM TYPICAL PROTOTYPE TOWERS

6.1 General

The purpose of tbis chapter is to report on the application of the f i t e element models,

described in Chapter Ui and verified from the experimental results as discussed in Chapter V,

to a number of typical towers. This would also help generate information that is usefd to the

designer of guyed antenna towers regarding their structural behaviour. As the common

pradce in North America uses simple analytid models, descnied earlier as beam on elastic

supports, to analyse these towers, this study compares the various finite elernent models to the

beam-on s p ~ g s analogy. Generaiiy, the m e n t North Amencan Standards (CSA S37-94,

ANSVEIA 222-F 1996) do not account for the dynamic charaderistics of the towers in the

design The main objectives of this chapter are: O to compare the finite elemmt models with

the m e n t adysis methods at âifkrent load levels, (ii) to investigate the influence of the major

parameters affecting the natural fiequemies of guyed towers, (ii to generate a database fiom

typical towers that can be used in the development of design methods to account for the

dynamic 104s on these towers, (iv) to deveiop empirical formulas for the prediction of the

n a d fiequencies of guyed towers that can be used in the design process and (v) to

demonstrate the application of the finite element analyticai techniques in the forced vibration

analysis of towers.

6.2 Tower Prototypes

There are many parameters involved in the design of towers. Usually the end user

specifies the required height and the maximum allowed degree of tilt/twia at the antenna

locations that would satie certain swiceability requirements of the communications aetwork

or broadcast pattern The sûucturai designer wrmaüy has the fieedom to change the size of

the masr (face width), nimber of guy levels, radius of the guy anchors, location and number of

torsion resistors, initial guy tensions, etc. Based on the designer's choice for the above

parameters, sizes of structural members of mast and guys are determineci. It should be noted

that these parameters are closely inter-related so that any small change in one of these

parameten may very well require changes to the structural components of the tower. For

example, for the same design requirements a larger (face width) mast supponed by a s d

number of guys can be used instead of a d e r mast with more number of guys to s a w the

same requirements. Tûerefore, it was decided to perfionn this study on typical towers in order

to assure the compatibility of the various components ofthe tower.

Eight towers were chosen for the purpose of this anaiysis. These are existing

towers, the data for which were available from various manufacturers and consultants so

as to be representative of design trends. Care has been taken to ensure confidentiaüty of

al1 the information regarding the designer, manufacturer, orner and location of the tower.

The towers range in height fiom 46 m (150 ft) to 600 m (2000 ft). The reference wind

pressures various from 306 Pa (lowest wind region in Canada) to 1002 Pa (highest wind

region). The ice thicknesses also represent the four ice classes o f 10, 25, 40, and 50 mm.

The tower profles (Figure 6-1 to 6-8) show the geometry including member sizes and

initial guy tensions, antenna loads, design wind speed or pressure, and nominal ice

thickness. The tower height, number of guys, and specified wind and ice loads are also

summuised in Table 6-1. It should be noted that the first five towers represent typical

towers used mainly in the wireless industry, and the other three towers represent the ta11

towers that are used in the broadcast applications.

Various de@ trends were represented in thk group of towers. For example, tower

P W has relatively smd mast face width (2.45 m) and seven guy levels with a typical guy span

(distance between two consecutive guys) of 45 m compared to tower PVI, which bas a larger

mast face width (3 .O m) and but only four guy levels with typical span of 65 rn

A computer program was developed to create the tower models in the format of

the Finite Elernent package ABAQUS, thus generating nodes, elements and boundaty

conditions depending on the type of FE model chosen (tniss or beam model). This finite

element modelling dows for al1 geometcical and structural details of the tower (e.g.

174

changes in cross-section properties, torsion resiston, eccentnc locations of antenoas,

slope in cross section towards the base of the towers, etc). For the Finite Element beam

models, the program also created the sub-models as discussed in Chapter III fiom which

the equivalent beam properties were denved. The source code in Fortran for this program

is provided in Appendk M.

6.3 Static Loading Analysis

The main purpose of this analysis is to compare the fiaite element anaiysis with the

model commody used in Uidustry @eam on non-hear elastic supports). One of the avdable

commercial programs that utilises this d o g y and used in this analysis is GUYMAST

(Weisman Consultants lac. 1997). Both finite element analytical models and the beam-on-

springs model are explaineci in detail in Chapter III. This study was perfomied on the towers

under two different loading conditions: the design loading condition and the up-to-collapse

(dtimate) loading condition Towers PI (46 m), PV (122 m) and PVIII (600 m) were used in

this study to demonstrate the results of the static adysis.

6.3.1 Strtic Design Loading Condition

Loads were de t ehed eccording to the design standard (CSA S37-94; ANSI EIA-

222F-1996) based on the geographical location ofthe tower. T h forces applied to the towers

due to wind and ice loads based on the exposeci wbd a m , drag factor, orientation of the tower

with respect to wind direction, etc were calculated. Six different load combinations were

considered: in the first three, design wind was considered blowing in three diffetent directions

on the bare tower and in the other three load combiitions, fi@ percent of the maximum wind

was considered blowing in the same three wind directions on fiilly iced towers (Wahba et al.

1998~). The loads also considered any eccentricities in the applied loads due to the location of

the antennas or the ciBiirent mountings.

Figures 6-9 and 6-1 1 compare the envelope of forces in the mast (kg loads and face

shears) resulting fiom the three Werent models for Towers PI and PV respectively. It can be

shown âom these two figures, that the resulting leg loads Corn the beam-on-springs model are

within 8% of the Finite element models. Also, the equivalent beam FE model results are

almost identical to the FE truss model. Face shears (which determine the forces in the bracing

members) are shown to be in very close agreement between the three different models. Table

6-2 compares the guy tensions resulthg fiom the three models which demonstrate that they are

in close agreement as weii. However, the guy forces from the beam-on-springs model are

siightly underestimateci (up to 7%). That also CO& the aulier hdings of the analysis of the

tested experimental models.

Figure 6-13 shows the leg louis and fiice shear for the tallest tower (PMII). This

tower was onginally designed using the version C of the "EIA -222" code. However, the

loads for this study were prepared accordhg to the current revision (F) of the Standard. Also,

it is located in an area where ice accretion is of no wncaq thus only the bue condition was

considereâ. In this figure, the Ieg 104s and tace sûears are shown for the wind blowing paraHel

to the plane of one the guy sets. It cm be noticed that the d t s are consistent behueen the

FE model and the beam-on springs model.

Figures 6-10, 6-12, and 6-14 show a cornparison betweea the deflections due to the

three models. It cm be clearly seen that for displacemeats, the beamsn spRogs model is

consistent with the finite element models. However, the twist angle does not show the sarne

level of agreement between the MO dyt ica l models. The Merence cm be attnbuted to the

modelling of the equivaht sprhgs that replace the guy assemblies at each level. The angle of

twist is not show for Tower Vüi as the loads were mostly symmetncal and produced very

Little torsion.

Figure 6-15 shows an array of seven towers connected at the top through a set of

catenary cables that are tensioned. The analysis of this array was feasible only through a finite

element mode1 as it can account for the interaction between towers under loaded conditions.

6.3.2 Static Up-to-Collapst Lords

As explained earlier, the significance of this loading condition is to test the ability of the

analytical models to predia the behaviour at the ultimate loading conditions (near failure). This

is of prime importance for structures such as guyed towers where non-linear behaviour is

exhibiteci. Furthemore, the recent cbange fiom Workhg Stress Design to Limit States Design

requires tbat the towers be anaiysed at or near Mure. By adysing the structure at that level,

and provided that the nght analytical tools are used, the designer should be able to predict any

sigas of Uistability of the sûucture. In order to detect instability of the mast, both P-delta

effkcts must be considered: (i) the chord rotations due to the displacement of the guy points,

and (i) the member m a t u r e due to the instabiility between two or more guy levels. ui the FE

models the foliowing procedure was appüed: (i) non-linear geometrical analysis was generally

applied in which large displacement theory was introduced, (ii) the mast between guy levels

was M e r discreticized into large number of elements (one per panel) to account for the

member mature between guy points Also, matenal non-linearity was introduced in the

modei to account for guy rupture. For the beamsn-sp~gs model, the displacements a? guy

levels were accounted for by the eccentric application of the guy forces at the displaced

position that is determineci from the iterative procedure.

In this study, Towers PI, PV, and PVIII were used io demonstrate the up-to-collapse

behaviow of the structure. The applied loads that were used in the fust step (design loads)

were increased gradually up to failure.

Figure 6-16 shows the percentage of the design load versus the deflection at the top of

Tower PI. The load deflection cum is show for loadiig condition in which the wind is

blowing in the direction of paralle1 to one the guy sets on a bue tower. It cm be noticed that

ultimate load is about five times the design load level for this load case. This is an example

where the design of the structure is governeci more by the deflection cnteria than by the

strength. Therefore, the f w e load was much greater than the design load. The tower

behaves in a more hear marner once the guys go slack @active) and only the taught cable

provides the lateral support. Also, there is close agreement between the FE model and the

beam-on-sp~gs model at the design load; however, the difference hcreases as the load level

increases anci the tower reaches the Mure Iwel. Furthemore, Figure 6- 17 shows the deflezted

sbape for the tower at fidure. It can be noticed that the ddected shape is quite strajght. The

Wure in this case was in the top guy. F i one of the taught cables in the wind direction

fàüed and the point of M u e can be noticed on the load deflection m e (Figure 6-16). Then

the second cable at that top level aiso Med under M e r inaease in load. From the deflected

shape and the load-deflection behaviour of this structure, it can be seen that the fiilure was only

in the guys, while the mast was stable uotil collapse.

Figure 6-18 illustrates the load deflecîion curve for Tower PV. Sidar cuncIusiow

made nom the analysis of Tower PI can be made for this tower except that the ultimate load is

only 7% higher than the design load. The deflected shape show in Figure 6-19 also shows

that the tower's deflected shape is close to a straight line. The failure also took place due to

the rupture of the taught cable at the top guy level, wbich experienced the highest deflection.

As the FE element mode1 was capable of introducing material non-linearities, it was feasible to

detect the fdure load fiom this method.

Figure 6-20 shows the load deflection m e for Tower PVm. This is the tallest tower

analyseci in this study and considered as one of the tdea towers ever built. It should be

pointed out that for these tail multi-Ievel guyed towers it is difncult to keep a straight deflected

shape for the tower. This is also evident fiom Figure 6-14, which shows the deflection of the

tower under design loads. Two types of analysis were perfomed: 6rst, material non-liaeacity

was included in both the mast and the guys, and the second where material non-linearities of

the guys oniy were considered. The percentage of the applied load with respect to design load

versus the deflection at the second guy Id Grom the top is show for both arialyses in Figure

6-20. It can be noticed âom the fidure (ddected) shape (Figure 6-21) that the Mure is in the

mast between the second âom the top to the fourth fiom the top guy levels. As the load

increases the deflection at the second and third guy level, fkom the top, increase rapidly thus

magnifjmg the load-âeflection eff'ects which in tum results in mast instability at the span

between these two guy levels. The ultimate load is only 50% higher than the design loads. in

the second case where only material non-hearity of the guys was coosidered, the mode1 was

able to resist higher loads until the third guy fiom the top reached fdwe and additionai

increase of the load caused the three guys immediately below to fd. Therefore, ththe mast was

completely unsupporteci for more than two-thirds the fidi tower height. The failure shape for

the second case is shown Figure 6-22.

6.4 Free Vibrations Analysis

The main objective of this portion of the study was to investigate the key parameters

that may affect the natural fiquencies and mode dupes of prototype guyed antenna towers. A

sensitivity study was first undertaken to determine the various parameters that may influence

the vibration response (Wahba et al. 1998a). Based on this sensitivity d y s i s , an extensive

study was then wnducted to produce empirical equations for the determination of the lowest

frequency of the structure. A h , &ixt of ice ametion and initiai tensions on the natural

fiequencies of towers was investigated. In this study, the same eight towers (Figures 6-1 to 6-

8) were first used to develop the empincal equations. Later these equations were verified using

an additionai twenty-five prototype towers (Table 6-3) representing a wide range of height (46

m - 600 m). The mode sbapes of interest for the purpose of this study were: (i) the lowest

mode, which is n o d y the top guy wiration mode, c i the 6rst f l d mode of the mast,

and (iii) the first torsional mode of the mast.

6.4.1 Naturd Frequencies and Mode Shapes of Mast-Guys System

In order to excite the viiration modes of the guys as wel as the mast, each cable was

discretized into 12 to 24 elements depending on its length. A free vibration analysis was

perfonned and the lowest twenty natural kequencies and their correspondhg mode shapes

were determined. It was noticed that for the towers analysecl, the lowest naturd fkequency

ahvays corresponded to the first vibrational mode of the top guys. The lowest twenty mode

shapes for tower PV are shown in Figures 6-23 to 6-27 as representative of the Srpical results

for this analysis. Table 6-4 shows the e f f i e mass in the t h e global directions. From this

table, it can be reaiised that the total effective mas in the two horizontal directions for the

Iowest hundred modes of vibration for this tower did not constitute more than 2 8% of the total

structural mass. Furthemore, the total effective mass in the vertical direction constituted less

than 12% of the structural mas. Therefore, fiom the vibration modes show in Figure 6-23 to

6-27 and the effkctive mass for these modes (Table 6-4). it can be concluded that Grst hundred

modes were ody guy modes, some of wbich involveci wupling motion with the rnast.

Results sbown in Table 6-5 show the lowest naturai fiequencies of the eight towers

under Merent sets of initiai tensions. From the sensitivity analysis pafomed on these towers

and d s e d in Table 6-5, the following conclusions were made with respect to the

parameters affectiag the lowest naairal âequencies of the towers:

IV.

Height: Height has the most effect on the lowest natural frequency; it can be

noticed tom Table 6-5, that the height is the most direct factor in detemiiniag the

lowest naturai fiequency regardless of the desiga wind and ice Ioads, number of

guys, size of mas$ etc.

Mast Snfiess to m a s ruth: Table 6-6 shows the "Wm" ratios for the différent

towers, and as can be noticed, it has linle infiuence on the natural fiequency of the

mat-guys system.

Torsion Resistors: These fixtures are usually used at heights just below the

microwave antennas in order to minimise the twist at these locations. Six guys

instead of three are comected to the mast (eight in case of square towers) at that

level. These torsion resistors help d u c e the twist of the mast but their efféct on

the lowest naturai fiequency of the tower is minimal (les than 5%).

Inira/ Guy Tensi'ons: The initial guy tension has an effect on both the se~ceabiîity

and the safety of the towers (Wahba et al. 1996). In the current Canadian Standard

for towers CSA S37-94 (CSA 1994). the initial guy tension may Vary fiom 8 to

I 5% of the ultimate strength of the guy but it is usually specified to be 1 0%. The

initial guy tension of the eight towers ùicluded in this study were varied aom 8-

15% and the e f f i on the naairal fiequencies is show in Table 6-5. As seen in

Table 6-5, a change in the initial tension of the guys anéas the natural fhquencies

of such towers by as much as 35%. It is also clear that the height of the tower is

the prime parameter* which has the largest effect on the lowest natural fiequency of

these towers.

An empiricai formula to daennine the lowest oaturai fiequency of the tower, which

involves the guy motion, would be helpfiil in the initiai design phase to determine the expected

range of fiequencies of the structure. The devdopment of an empirical formula for the

determination of the lowest fiequency of a guyed tower is explaineci as followç:

The natural fresuencies of the eight prototype towers were calcuiated from the FE

model and presented in Table 6-6. It was concluded from the examination of the results that

the height of the tower, H, has the deteminhg effect on the lowest natural fiequency. Ln Table

66 the height of the tower is shown in second colwnn, the fourth column presents the

calculated frequency fiom the model and the fifth colurnn shows the calculated fiequency tiom

the empirical equation that was determined to be function of the height as shown in equation 6-

1.

w here:

fis the natural fiequency of the guyed mast

H is the height of the tower

The constants (CI and C2) in the equation were detedeci by the minimisation of the

summation of square of the Merences. The lowest naturd tiequency, j; in cyc1edsec (Hz) of

the tower, based on initial design tension equal to 1û% of the uitimate guy capacity, can be

183

predicted f?om the following equation as hction of the tower height:

f =34.5*Ha9

w here:

H is the height of the tower in metres

In imperid units Equation 6-2a cm be written as:

f = 1 0 0 * H " ~

where, H is the height of the tower in A.

The maximum difference between the empirical equation (6-2) and the calculated values for the

eight prototype towers was found to be less than 6% as shown in Table 6-6. Thus, Equation

6-2 can be used even at the preliminary design stage, where the height would be the only

parameter.

Equations 6-2 was later applied to additional twenty-five prototype towen and the

results (Table 6-7) were consistent with those shown in Table 6-6. Furthemore, results of

the free vibration analysis of an existing 302 m tower (Saxena et al. 1989; McCafiey and

Hartmann 1970) show the lowest natural fiequency for îhat tower as 0.217 Hz. Applying

the simple equation (6-Za), the expected namal fiequency is 0.20 Hz (9% difference).

Figure 6-28 illustrates the effect of the structure height on the tower's lowest

natural fiequency. The graph shows good agreement between the predicted frequency

fiom the empincal equation 6-2 and with the computed values from the finite element

model.

6.4.1.2 Effect of king on the Fm Vibration of Towers

Most of the previous research on free vibration of towen was aimed at the extraction

of the natural fiequemies and mode shapes on bue towers thus neglecting the effect of icing

on the dynamic properties of the towers (Wahba et al. 1998b). Also, design codes that deal

with the dynamic analysis of towers (e.g. the reconimendations for the dynamic d y s i s of

towers due to wind e f f i s uicluded in CSA-S37 do not include the dynamic &ect of wind on

iced towers.

Table 6-8 compares the tower naturd fkequencies and lower mode shapes, for Tower

PV, calcuiated for ice accretions of O (Le. no ice), 10, 25, 40 mm thichesses. From the

results, it is clear that these types of structures when iced exhibit significantiy reduced low

naturd fhquencies thus making them vuinerable to dynamic effkcts of wind even at low wind

speeds. What is even more signifiant is that for an ice accretion of 10 mm, a 19% reduction

in the nahiral fiequency cm result and thk reduction can reach 45% for the case of40 mm ice.

Although it is not anticipated tbat d o r m ice ametion of this magnitude would be formed

throughout the tower, Smaller ice accretions wodd stii i produce appreciable reductions of the

natural tieqwncies. Furthemore, for the natural fiequencies obtained under iced cases* it was

noticed that icing causes a considerabIe increase in the coupbg action between the mast modes

and the guy modes; this makes the towers more vulwrable to galloping of the guys. Also, in

the absence of ice, the motion in the mast was nepiigiile compared to that of the guys.

However, for the i d condition, not O@ did the number of coupled mast-guy modes increase

but also the efféct ofguy motion on the mast is greatly magmfied. This is shown in Figure 6-29

where the first and second bending modes coupled with guy motion for 40 mm ice are show

This is particularly important for the serviceability of the tower under iced condition and

gusting winds.

It should also be noted that ice accretion on the tower is usudy accompanied by a decrease in

the temperature from the design temperature. This should also be accounted for in the

calculation of the the viirations of the towers under iced conditions. The reduction in the

temperature would result in an increase in the guy tensions which, in tum, inaease the naniral

frrquencies. As a r d t of these two opposite effects on the structure, the change in the

natural fiequencies of the tower may be lower than that show in Table 6-8.

6.4.2 Natural Fnquencies and Mode Shapea of Masts

As stated in Section 1.3, another objective of this study is to identify the parameters

that affect the naturai frequencies and mode sbapes of the masts. Uniike the analysis perfonned

earlier to determine the lowest natural âequency of the tower, this d y s i s suppresses the

modes of vibration ofguys. This is achieved by using the same rnodels as explained in Chapter

Di, with the exception that the guys are modelied as one element per cable. This suppresses

the guy modes and oniy the mest fhdamental modes are excited.

The eight prototype towers under study were adysed using the modified model ofone

element per cable, and the lowest ten vibrational modes were extracted. The fundamental

flexu~al and torsiod modes for the eight towers, their order withui the extracted modes, and

the percentage of the effdve mass to the total mass of the structure were calcuiated and are

presented in Table 6-9. Cornparhg Tables 6-5 and 6-9, it can be concludeci that, for shorter

towers, the Bern id fiequency (with guy modes suppressed) is considerabiy higher than the

lowest fkquency of the structure. The différence is s d e r for taller towers. Cornparhg Table

6-4 and Table 6-9, it can be seen that the mast modes exhibit considerably higher effective mas

and the lowest ten naturai Eequencies excited 80 to 90% of total structure mass. From Table

6-9, it is obvious that the torsional modes are predominant for ta11 towers (taller than 300 m).

The fiindamental ûequencies of guyed towers (guy modes included) depends mostly on

the height of the tower. however, height is not the only factor innuencing the natural

frequencies of the towers with guy modes suppressed. For example, although prototype

Tower PV is 10% taller than tower PIV, it exhibits a higher natural fiequency. The lowest ten

modes with their respective effeaive masses in the three global directions are shown in Table

6-10 for Tower PV (as a representative of the typical results). It cm be noticed that the total

effèctive mas of these ten modes (81% of the total m m ) is much highei than the lowest

hundred modes dnven fiom the tiiU model (1 8% of the total mass) as shown in Table 6-4. Th

fact, both results complement each other: the effdve mass in the horizontal direction for the

guy modes constitutes 18% of the structure total mas, while 8 1% of the total mass is exciteci

tom the siagie cable dernent model. It can be aiso wtid k t the vertical effective mass

component in the single cabie elemmt modd is negügi1e (les that 0.01%). The same

conclusions apply to ail towers analysed. This means that the axial mast modes frequencies

were not excited as the fiequencies of mast axial modes are significantly higher than the

bending or torsionai modes.

Figures 6-30 to 6-37 show the fundamental flexucal fiequencies and mode shapes of

towers PI to PVIII respectively. From these figures it can be noticed tbat the fuadamental

flmd mode shape varies largely corn one tower to another. On closer examination of these

mode shapes, it can be noticed that it is mainiy ulauenced by the relative horizontal component

of the axial stiffness of the guy, Khg, to the mast span flexural stiffness, Km, at the sarne span

between the two guys.

& caa be expressed as:

where:

K h g = Horizontal cornponent of the guy axial stiffhess

E, = Elastic modulus of guy

Lg = Chord length of guy

0 = inclination angle of the guy to the horizontal

Equation 6-3 can be written in a the foilowing fom

where Gr is the guy radius (measured from the centre ofthe tower to the anchor point)

K, the relative mast span stifIiiess, can be expressed as:

w here:

E, = elastic modulus of mast

I, = Second moment of area of mast

Lm = mast span (distance between two consecutive guys).

Closer examination of "IG,,JL" for the different guy levels for the eight towers,

shown in Table 6-1 1, explains the dEerent flemral mode shapes shown in Figures 6-30 to

6-37. For example, prototype tower PI has a (i(he/[<lll) for the top guy (28.6) that is more

than twice than that of the lowea guy (1 1.43). The higher stifbess of the mast support at

the top is reflected in the flexural mode shape of the tower shown in Figure 6- 30. The

same conclusions were h v e d at with respect to prototype Towers PU, PI11 and PV.

Prototype Tower P N has relatively constant "mJKM9 ratios at the different guy levels

and therefore its fundamental flexural mode shape is quite similar to a free-standing

structure. Tower PVI has a relatively sMer lowest guy level(&#L= 3 5.0) compared to

the top level ( K A = 19.3) and that effect can be clearly seen on the mode shape. Also

note the relatively very flexible guy support at the seventh guy level in tower PWI and

how that influenced the mode shape for that tower.

in order to understand the parameters inûuencing the natural frequency of the

towers (with guy modes suppressed), the initial tensions of the towers were increased

fkom the design values of 10% of the ultimate strength to 15%. The results are

sumrnarised in Table 6-12 where it can be noticed that the initial tension has negligible

effect (less than 1%) on the mast flexural modes. However, it has considerable effect on

the torsional modes of the structure. For example in Tower PVIII, by increasing the initial

tensions fiom 10 to 15%, the torsional mode natural frequency increased fkom 0.34 1 Hz

to 0.379 Hz and thus rnoving it from the first (lowest) mast frequency to the third

frequency and higher than the fiequency of the tlexural modes.

Another parameter that was examined was the guy cross-sectional area, which in

tum is translated into the stifiess of the mast supports. increasing the top guy area made

a sigdcant change to the mast natural frequency to Tower PIV only, where it is evident

fkom the mode shape and (Kt&&) for that guy that it is relatively flexible. Increasing the

area of the top guy by 50% increased the natural fiequency of the mast fiom 1.45 Hz to

1.67 Hz (1 5%).

Upon doser examination of the natural frequencies and mode shapes of these eight

prototype towers, it was possible to relate the following parameters as the most influentid

on the lowest flexural natural fiequency of the mast:

i) The tower totai mass, m

ii) The tower stifhess measured by, Kg

where:

anda is the total number ofguys,

H'@ is the elevation of the ith guy levei, and H is the total height of the tower.

Table 6-13 shows these two parameters calculated for each of the eight towers. An

empirical equation, which is a fhction of these two parameters, was Uiitiaiiy estimated

to be of the foiiowing form:

f =C, *(ZkCJC2 *(nt)'' (6-7)

The constants G, C2, and G were determined through the foîîowing tterative

procedure:

I) initial values were assumecf for the constants

2) The fiequencies were estimateci fiom Eqn. (6-7)

3) The summation of squares of the difrences between the calculated tiequencies

(Table 6-9) and the estimated fiequencies (Eq. 6-7) was calculated

4) The constants were calailated based on the minmization of the surnmation of

squares of the difiefences calculatecl in step3.

Based on the above, the equation for the naturai fkquency of the flexu~al mode of a

trianguiar guyed tower (guy modes suppressed) may be calculatecl f?om the followiag

empirical equation:

where:

fis the mast f l d mode âequency (guy modes s u p p r d ) in Hz

C is a constant depending on the uaits used

= 2.58 for m d c units

= 176.5 for imperid units

I is the average moment of inertia of the mast

m is the total mas of the tower in tom or (1000 lb)

Kg is as per Equation (6-6') in whicb,

As is the area of each guy at the ith guy level in mm' (in2)

G, is the Guy radius in rn (ft)

L,, Chord length of the guys at the ith guy level in m (fi)

Table 6- 13 shows the maximum difference (error) between the empirical equation and

the computed values fiom the FE mode1 as 9%. Also, similar results were obtained

when the equation was applied to a sample of twenty five additional towers (given in

Table 6-3) as show in Table 6-14.

A hrther simplification of the equation rnay be achieved by estimating the

naniral fiequency as dependent on the height of the tower only. Following the same

procedure used for Equation 6-8, but using the height as the only vanable, the

approximate flexural mode Eequency (guy modes suppressed) of a viangular mast of a

guyed tower can be estimated fiom the foliowing Equation:

-

where H is the height of the tower in m.

This equation wuld serve as a guide for the flexiity of the structure. It is particularly

usefbl at the initiai design stage where height is the O@ known parameter- Also, as

Standards generally consider structures to be flexible if they exhibit a natural fiequency

of l e s than 1 Hz. Equatioas 6-8 and 6-9 can be used to decide if the tower should be

considered fluaile in which case a more rigorous dynamic analysis would be justified.

Cornparison sbown in Fig. 6-38 shows a good agreement between the calculated

naturai fiequencies of the mast with the predicted fiequencies as per Eq. 6-9.

6.5 Forced Vibration Anrlysis of Guyed Antenna Towen

The prototype towen were analysed under a forced vibration adysis for two different

loaâing cases: (0 the tower base and anchors were subjected to base motion using El Centro

KS acceleration history (HKS 1995), and ( i the top guys of the towers were subjected to

hannonic galioping motion in the vertical plane for two different frequencies, with the fira

coinciding with the guy natural fiequency and the second with the mast natural fiequency. In

order to capture the behaviour of the tower under these forced vibrations, the FE models with

the discretized guys were used.

R d t s of the seismic d y s i s performed on the eight towers have confhned the

eariier results reported by McClure and Guevara (1993 and 1994) that the deflectioos, guy

forces, mast axial loads are wd below their design values. Figures 6-39, and 6-40 show the

time histories of the 6rst five seconds of the deflections of the mast at the top, and the

compressive stresses in the legs of the tower base for Tower P N respectiveiy. It cm be

noticed that the axial compressive stresses are w d below the design strength value of

approximately 200 MPa It shouid be noted that the seismic analysis perfomed assumed the

same phase in the gmund motion between the base and the anchors. This may not be accurate,

as a change of phase between the base motion and the anchors is quite possible especiaiiy on

taler towers where the distance between the Merent supports may exceed 700 a

Galloping of guys on taii towers, in which the guys would wnnally undergo low-

fiequency hi@-amplitude vibrations, has been a fiequent phenornenon. Eariier research

(Novak et al. 1978) has indicated that taIl towers are more susceptible to gailoping guys.

Observations have confimeci this for taller towers (300 m and above). Furthemore, some of

these gaiioping episodes have been recordeci on tape in which the amplitude of vibration

reached 3 m This research studied the e f f i of the guy gdoping on the guy forces and the

mast loads. Two gdoping kequencies of vibrations, coinciding with the lowest guy mode and

mast mode, with the same amplitude of 3 m were used on the td Towers PM, PW, and

PMII. Figure 641 indicates the results for the axial stresses in the top guy stresses, under a

harmonic motion of 3 m with a fiequency o f 0.2 17 Hz ( k t natural frequency of the top guys).

Figures 6-42 aad 6-43 show the top guy stresses and the tower deflections at the top,

respectively, for the same amplitude of vibration but with a ûequency of 0.6 17 Hz (first mast

flexurai fiequency). It shodd be noted that in the second case where the fiequency of

oscillation of the guys coincided with that of the mast, the efféa on the tower is more serious

and the level of stresses in the guys exceeded their dtirnate strength which would in nim r d t

in fdure.

Table 6-1. Detds of Prototype Towers Used in the Study

Tower Wind

Pressure

(Pd

Height

(m)

Ice

Tbickness

(mm)

Guy

Leveis

Tower Description

Torsion resistor at ievel3

No torsion resistors -

Torsion resistor at level4

No torsion resistors

Torsion resistors at leveis 4& 5




Table 6-2. Guy Forces under Design Loads

Tower

PI

PV

Guy Level Guy Forces (kN)

1

2

3

1

2

3

4

5

6

Tmss Mode1

9.1

12.5

15.0

88.7

110

136.2

174.5

175.6

212.1

Bearn Mode1

9.1

Beam-on-spriags

8.9

12.5 12.0

15.0

89.1

1 10.3

136.4

14.7

8 1.8

104.5

132.5

174

175.5

2 10.6

163.6

173.4

212

Table 6-3. Details of Sample Towers Used for Verification of Results

Tower Name Height (ml

3 1 351 1 25 1 T/R at 2nd and 3rd level

3 1 451 25 1 TIR at 2nd and 3rd level

5 30 1 25 T/R at 4th and 5th level

4 45 1 25 TIR at 3rd and 4th level

4 45 1 25 T/R at 3rd and 4th level

4 466 51 T/R at 1st and 4th level 5 341 1 1 T/R at 1st and 5th Ievel

7 1 4 5 1 1 10 1 TIR at 5th and 7th level

5 30 1 10 TJR at ail five guy levels

6 54 1 1 O T/R at 3rd and 6th level

5 43 1 1 10 TIR at lst, 2nd 4th and 5th level 9 1 451 1 10 1 TIR at 8th and 9th level 9 45 1 10 T/R at 3rd 4th and 9th level

1

6 912 O 3 m (face width), 25 m Candelabn

6 418 25 3 m (face width), 13 m

6 1 418 1 13 1 3 m (face width) 7 816 O 3 m (face width) 8 912 O 3 m (face width)

8 1 739 1 13 1 3.7 m (face wvidth) 8 816 O 3.7 m (faœ wvidth)

9 7 39 O 3 -7 m (face width)

9 1 9 1 2 1 O 1 3.7 m (face width)

Table 6-4. Effective Mass Components for the fist 100 modes of Tower PV

MODE NO X-COMPONENT Y-COMPONENT 2-COMPONENT

1 4.29E-04 1.72E-03 7.73E-15

2 1.72E-03 4.29E-04 3.90E-14

3 1.10E-11 1.72E-Il 4.24E-17

1 2.73E-13 1.288-12 4.58E-18

5 3.04E-13 1 .09E-12 1.89E-16

6 1.92E-12 5.22E- 15 1.0 1E-17

7 3 S2E-03 1.12E-03 3.14E-13

8 1.12E-03 3 S E - 0 3 8.18E-16

9 1.62E-14 3 .ME-14 9.44E-14

10 2.88E-04 1 SOE-03 3.18E-14

11 1 SOE-03 2.88E-04 5S5E~lQ

12 1.21E-12 9.14E-14 4.1 lE-03

13 8.37E-14 5.69E-14 3.13E-13

14 7.49E-13 3.15E-12 4.63E-17

15 8.52E-13 2.87E-12 8.53E-17

16 3.32E-12 7.81E-15 1.03E-17

17 2.99E-04 1.24EQ3 1.38E-14

18 1 .ME43 2.99E-04 7.74E-13

19 2.11E-12 5.32E-13 1.17E-13

20 9.94E44 3.1SE-01 4.308113

2 1 3. ISE104 9.94E-04 S.42E-14

22 6.46E-14 3.67E-14 3.11E43

23 2.16E-07 3.22E-06 9.7fE-13

99 5.85E-05 4.52E-06 1.22E-08

100 3.82E-06 1.60E-07 2.25848

TOTAL 1.79E-02 1 .79842 1.17E-02

TOTAL MASS 9.7528748E42 9.7528748842 9.7528748E-02

Effective mms %

Tord mass

Table 6-5. Effect of Initial Tension on Natural Frequency of Towers

Tower

PIV

First Natural Frequencies (Hz)

Initial Tension as % of ultimate strength

10 % 12 % 1s % TT-

Table 6-6. Denvation of Empincd Equation for Tower Natural Frequency

Tower

PI PI1 Pl l l PIV PV PVI PVI I PVI I I

Height (ml

45.7 61 .O 76.2 11 3.7 122 297.5 365.8 591.3

FR€Q(Eq 6-2) (Hz) OlFF DIF FA2 CONSTANTS

1.1 O6 -4.086003 1.67E-05 Cl =34.50 0.854 4,16E-O2 1.73E-03 C2=.90 0.698 -1 -1 7E*O2 1.36E-O4 0.487 -2.28E-O2 5.22E-04 0.457 7.46E-03 5.56E-05 0.205 -1.20E-02 1.45E-04 0.1 70 1.02E-02 1.04E-04 0.1 10 4.55E-04 2.07E-07

Table 6-7. Verifkation of Equation 6-2 on Sample Towers

STS 171.8 0.1 3 ST6 487.7 0.13

I

SV 476.4 0.13 ST8 555.0 0.12

Frequency FEM (Hz)

-- -

Tower Name

Height (m)

Frequency 0%- 6-21

(Hz)

ifference %

Table 6-8. Natural Frequencies and Mode Shapes for Tower V Under

DEerent Ice Accretions

Mode Number

O mm

ice

2 5 m

ice

40mm

ice

Table 6-9. Mast Natural Frequencies of Prototype Towers (Guy Modes

Suppressed)

Prototype

Tower

First Mast

FIexural

Frequency

W)

Mode

Number

Fist Mast

Torsional

Frequency

(W

Mode

Number

% of

effective

mass of 10

modes to I

total mass

Table 6-10. Effective Mass for the First 10 Vibrational Modes of Mast

(Tower V- Guy Modes Suppressed)

TOTAL

Total Mass

Effective mass - % Total mass

Table 6-1 1. Ratios of Guy to Mast Stifhesses @Ch&)

Tower Guy (guy level) Ana Towr PI G1 0.0653 G2 0.079 G3 0.1 58

Tower Pt l G1 0.079 G2 0.1 12 G3 0.1 49

Tower Plll G1 0.0379 G2 0.0653 G3 0.0653 G4 0.228

Tower P1V G1 0.19 G2 0.233 G3 0.233 04 0.284 G5 0.336

Tower PVI Gl 3.04 G2 3.75 G3 4.96 G4 6.83

Tower PVlll G1 1 .O3 G2 1.97 G3 24 G4 1 35 G5 1 -84 G6 287 G7 1.47 G8 3.04 W 1.97

Tower Guy M8d Ag " hight Elevatlon Ineitlr lm/Lh3 Gr/LA2 KhgiKms

37.5 87.5

137.5

525 117.5

1 85

59 119 1 79 ns

66.6 136.6 199.1 266.6 336.6

525 1 17.5 182.5

24s 31 2.5 367.5

206.7 420.8 642.2 856.3

1 47.63 295.27 $429

590.54 767.7

974.39 1196

154.2s 368.32 597.17

826 t ûS4.84 tZï6.29 1 SOS. 1 4 1 733.97 1911.13

Table 6-12. Cornparison of the Mast Frequencies for Various Initial Tensions

Prototype

Tower

Initial Tension as per design

(10% of ultimate capacity)

Increased Initial Tension

(1 5% of ultimate capacity)

First Mast

Flexural

Frequency

OIz)

First Mast

Torsionai

Frequency

(Hz)

First Mast

Flexural

Frequency

(Hz)

First Mast

Torsionai

Frequency

(Hz)

Table 6- 13. Derivation of Empirical Equation for Tower Fundamental

Flewal Frequency (guy modes suppressed)

height mass (m) (tons) 45.72 f .63E+00 60.96 2.45€+00 76.2 4.45€+00

1 13.538 1 ,O1 €+O1 1 18.872 1 .il E+O1 294.437 3.05€+02 364.236 2,98€+02 591.61 7 4.68€+02

GUY stiff.

1 -91 E+OO 1.50€+00 1.56€+00 3.1 9E+OO 7.43€+00 1.79€+01 1.1 5E+O1 9,38€+00

FRÉQ FEM

2.838 2.264 1.498 1.450 1.754 0.61 7 0.463 0.379

FREQ

Table 6-14. Verification of Equation 6-8 on Sample Towers

Frequency Frequency Tower Height FEM Name (m) (Eq. 6-21 (Hz) (Hz)

s"r2 382.5 0.42 0.46 ST3 460.9 0.41 0.37 ST4 442.0 0.42 0.44 STS 471-8 0.58 0.54 ST6 487.7 0.48 0.46 S T ~ 476.4 0.50 0.47

Cross Section

Figure 6-1. Profile of Prototype Tower PI

1- ASP 600 O 61rn T.L 7/8"

A Cross Section

WIND: q - 600 Pa. ICE: Class 41 25 mm

Figure 6-2. Profile of Prototype Tower PI1

214

1 - SRI. 21 0-A4 O TOP T.L. f.ûF4-50A 2- PAL10-17C OlSH O 74.7m T L 1 5/8' 3- SRL 410-C4 O 70.15m T.L UF4-50A 4- PAL10-17C DlSH O 68.0m T.L 1 5/8" 5- SRL 210-A4 O 45.0m T L Wf4-SOA 6- SRL 210-A4 O 56.6m T.L. UW4-SOA

Cross Section

WIND: a = 302 Pa. ICE: Class 1 10 mm

Figure 6-3. Profile of Prototype Tower PI11

215

Y Cross Section

WIND: q - 850 Pa, CE: Cioss III 4Omm

Figure 6-4. Profile of Prototype Tower PIV

216

A Cross Section

WIND: q = 500 Pa. ICE: Closs IV SOmm

Figure 6-5. Profile of Prototype Tower PV

PNTFNNAQ & UNES: 1- 85' Antenna 20" O 0 O TOP T.L (2) 6 1/8"

Cross Section

BASIC WlND SPEED: 75 mph RADIAL CE: 0.0 mm

Figure 6-8. Profile of Prototype Tower PVIII

220

O 10 20 30 40 50 6û 70 Leg Loads (kN)

0.0 2.5 5.0 7.5 10.0 Face Shear [kN)

Figure 6-9. Comparison of Leg Loads and Face Shear under Design Loads for Prototype Tower P I

O 25 50 75 100 125 150 DefieCam (mm)

0.0 0.5 1 .O 1.5 Twisting of Tower (degrees)

Figure 6- 10. Cornparison of Defîections under Design Loads for Prototype Tower PI

O 200 400 600 800 1000 Leg Loads (kN) O 15 30 45 60

Face Shear (kN)

Figure 6-1 1. Cornparison of Leg Loads and Face Shears under Design Loads for Prototype Tower PV

O 200 400 600 800 Deflection (mm)

0.0 0.5 1.0 1.5 2.0 Twisting of Tower (degrees)

Figure 6-12. Cornparison of Deflections under Design Loads for Prototype Tower PV

O 1000 2000 3000 4000 Leg loads (kN)

O Leg Loads (kN)

Figure 6- 13. Cornparison of Leg Loads and Face Shears under Design Loads for Prototype Tower PVIII

O 2 4 6 Leg Loads (khi)

Figure 6-14. Cornparison of Deflections under Design Loads for Prototype Tower PVIII

Figure 6-15. An A m y of Seven Towers with Different Heights Connected Through Catenary Guy System

0.40 0.80 Displacement (m)

Figure 6-16. Load as a Ratio of Design Loads Vs. Deflection at the Top Guy Level for Prototype Tower PI

Figure 6- 17. Failure Shape as Predicted by the Finite Element Mode1 for Prototype Tower PI

0.40 0.80 1.20 Displacement (m)

Figure 6-18. Load as a Ratio of Design Loads Vs. Deflection at the Top Guy Levei for Prototype Tower PV

Figure 6-19. Failure Shape as Predicted by the Finite Element Mode1 for Prototype Tower PV

I I I 0.00 4.00 8.00

Displaœment (m)

Figure 6-20. Load as a Ratio of Design Loads Vs. Deflection at the third Guy Level fiom the top for Prototype Tower PV

Figure 6-2 1. Failure Shape as Predicted by the Finite Element Mode1 for Prototype Tower PVIII

Figure 6-22. Failure Shape as Predicted by the Finite Element Mode1 for Prototype Tower P WI

Figure 6-23. First Twenty Mode Shapes of Prototype Tower PV Modes (1-4)


Figure 6-25. First Twenty Mode Shapes of Prototype Tower PV Modes (9- 12)

Figure 6-26. First Twenty Mode Shapes of Prototype Tower PV Modes (13- 16)


1.00 - - Eqn (62) -- - 0.75 -

3

0.50 - I

0.25 - -

0.00 I I T I I I I I I 1 1

0.0 100.0 200.0 300.0 400.0 500.0 600.0 Height of Towcrr (m)

Figure 6-28. Variation of First Nahiral Frequency with Height of Tower

Figure 6-29. Mode Shapes of Tower PV (40 mm [ce)

241

PACTOR 329.

lb S m 2 n m r T 1 QüEITC!?= 2 .84

St'ERSIOlf: 5.7-7 DATE: 16-JAtf-1999

Figure 6-30. First Flexural Frequency and Mode Shape of Prototype Tower PI (Guy Modes Suppressed)

VEWIûtt: 5.7-7 DATE: 16-3AIf-1999 t:

Figure 6-3 1. First Flexural Frequency and Mode Shape of Prototype Towei PI1 (Guy Modes Suppressed)

Figure 6-32. First Flexural Frequency and Mode Shape of Prototype Tower PIII (Guy Modes Suppressed)

Figure 6-33. First Flexural Frequency and Mode Shape of Prototype Tower PIV (Guy Modes Suppressed)

Figure 6-34. First Flewal Frequency and Mode Shape of Prototype Tower PV (Guy Modes Suppressed)

Figure 6-35. Fim F l e d Frequency and Mode Shape of Prototype Tower PVI (Guy Modes Suppressed)

Figure 6-36. First Flexual Frequency and Mode Shape of Prototype Tower P W (Guy Modes Suppressed)

Figure 6-37. First Flexural Frequency and Mode Shape of Prototype Tower PVI[I (Guy Modes Suppressed)

0.00 100.00 200.00 300.00 400.00 500.00 600.00 HeigM of Tower (m)

Figure 6-38. Variation of First Flexural Frequency of the Tower (Guy Modes Suppressed) with Height

0.00 1 .O0 2.00 3.00 4.00 5.00 me (s)

Figure 6-39. The History of Deflections of Mast for Tower PIV Subjected to El-Centro N-S Ground Motion

0.00 t .O0 2.00 3.00 4.00 5.00 Time (s)

Figure 6-40. Time History of Guy Stresses for Tower PIV Subjected to El-Centro N-S Ground Motion

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 Ttme (s)

Figure 6-4 1. T h e Histocy of the Top Guy Stresses for Tower PVI Subjected to Top Guy Galloping (f= 0.2 17 Hz)

0.00 1 .O0 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Time (s)

Figure 642. The History of the Top Guy Stresses for Tower PM Subjected to Top Guy Galloping (F 0.6 1 7 Hz)

0.00 1 .O0 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Time (sec)

Figure 6-43. Time History of the Dediection at the Top of Tower P M Subjected to Top Guy Galloping (+ 0.6 L7 Hz)

CHAPTER VI1

SUMMARY AND CONCLUSIONS

In this investigation, the static response of guyed communication towers under

service and up-to-collapse loads were investigated and analysed using a beam on non-linear

springs analogy and two finite element models: a three-dimensional tniss, and an equivaient

beam model. A cornparison between the analytical rnodels was presented for different static

loading levels. The analyticai procedures were verified by experimental nsults obtained

fiom tests on five scaled guyed tower models and wbsequently applied to eight prototype

towers. Furthemore, the uitimate load carrying capacity was determined by testing of four

tower models and the resuits were verified against the predicted values fiom the fulte

element procedures. Conclusions were drawn with respect to the suitability of the anaiytical

models under different loading conditions.

A dynamic testing facility (shake table), suitable for testing light structures, was

designed, built, and performance tested. The table was used to test five scale-mode1 guyed

towers under fke vibrations and forced base motion. The test results substantiated the

theoretical finite element predictive procedures. ComparWn of the results was presented

and conclusions were h w n .

The finite element procedures were applied to thirty-three prototype towers

subjected to fke vibration and eight towers subjected to forced vibrations. A study was

conducted to investigate the main parameters affecthg the al!-important lower naturai

frequencies of the mast-guy system and that of the mast. Based on this shidy, empirical

equations were developed to detemiine the fundamentai natural frequencies of the towers

(with and without guys modes suppression). The empincai equations should be valuable in

establishing the dynamic characteristics of the towers for use in design practice.

7.2 Conclusions

The following conclusions are drawn based on the remlts obtained fiom the

theoretical and experimental studies and are classifled as foliows:

a) Ana@icai

8 The finite element models presented (with geometric non lhearities included) in this

research are capable of predicting the guyed tower khaviour upto-collapse.

For symmetticaiiy b d masis, the quivalent beam finite element mode1 predicts the

behaviour of the structure within the same level of accuracy as the full miss model.

The beam-on springs model, which is cmntly the most comrnonly and widely used in

the industry, provides a g d agnement with the experimental results at s e ~ c e load

levels. However, as the loads approaches fidure, this level of agreement deteriorates by

8% - 25% depending on the type of failure.

6) Experimental

r A dynamic testing facility (shake table) that is suitable for testing of guyed towen was

economidly fabricated, and instrumented.

r Good agreement between the experimental and theoretical results supports the reliability

of using the finite-element models presented to predict the elastic response, free-

vibration response, forced vibration response and ultimate load-carrying capacity of

guyed towers.

Heigbt is the major parameter affecthg the naturd frequency of the tower. The use of

M e r guys increases the f'undarnental frrquency and the application of torsion resistors

dramatically increases the torsional kquetlcy.

Mast fkquencies are siBnificantly highu than the lowest tower fmlwncies; however,

the ddynamic khaviour of the tower is largely dependent on the mast fiequency as it

constitutes a high ratio of effective mass

The empirical equations (Eqm. 602,698, and 6-9) derived h m the analysis of prototype

towers, show good agreement with the caicuiated values ami c m be used to predict the

hdamental nahuai frequencies of guyed towers.

N a m frequencies and mode shapes of guyed towers are dependent on the relative

stitniess of the individual guy levels to the mast.

Generally, guyed towers less than 200 m in height need not be considered as flexible

structures and rigorous dynamic d y s i s is not waminted as the mast nanual fkequency

is generally less than 1 Hz.

7.3 Suggestions for Future Research

For &nue research, the following suggestions are made:

1- The seismic loads as well as wind gust factors need to be related to the naturai

fiequencies of the towers. The empkical equations derived fiom this study can be

extended to derive equivalent static loads to be applied for design of towen

2- Experimental research needs to be carried out on more towen. Dynamic models with

mass compensation are recommended so as to keep the natural fiequencies of the

models well below the capacity of the testing facility.

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Appendix A l

SAMPLE ABAQUS INPUT

'HEADING 1246FT TOWER PWI- BEAM MOOEL- FREE VIBRATION ANALYSIS

"BEAM ELEMENTS FOR MAST "DATA CHECK 'PREPRINT, HISTORY=NO, ECHO=NO, MODEt=NO 'NODE ,NS€T=MAST

1 0.00 0.00 0.00 2 0.00 0.00 88.56 3 0.00 0.00 177.12 4 0.00 0.00 265.68 5 0.00 0.00 354.24 6 0.00 0.W 442.80 7 0.00 0.00 531.36 8 0.00 0.00 619.92 9 0.00 0.00 708.4 10 0.00 0.00 797.04 11 0.00 0.00 885.60 12 0.00 0.00 974.16 13 0.00 0.00 1062.72 14 0.00 0.00 1151.28 15 0.00 0.00 1239.84 16 0.00 0.00 1328.40 17 0.00 0.00 1416.96 18 0.00 0.00 1505.52 19 0.00 0.00 1594.08 20 0.00 0.00 1682.64 21 0.00 0.00 1771.20 22 0.00 0.00 1859.76 23 0.00 0.00 1948.32 24 0.00 0.00 2036.88 25 0.00 0.00 2125.44 26 0.00 0.00 2214.00 27 0.00 0.00 2302.56 28 0.00 0.00 239t.12 29 0.00 0.00 2479.68 30 0.00 0.00 2568.24 31 0.00 0.00 2656.W 32 0.00 0.00 2745.36 33 0.00 0.00 2833.92 34 0.00 0.00 2922.48 35 0.00 0.00 3 1 1.04 36 0.00 0.00 3099.60 37 0.00 0.00 3188.16 3 0.00 0.00 3276.72 39 0.00 0.00 3365.28 40 0.00 0.00 3453.84 41 0.00 0.00 3542.40 42 0.00 0.00 3630.96

1 170 -2520.80 -1455.40 7 176.00 1 171 -2274.06 -1312.94 7693.60 1172 -2027.31 -1170.47 8611.20 1173 -1780.56 -1028.01 9328.80 1174 -1533.81 g85.55 10046.40 1 175 -1287.06 -743.09 1076400 1176 -1040.31 400.63 11481.60 1177 -793.56 48.17 12199.20 1178 -546.81 315.70 12916.80 1 179 -300.06 -173.24 13634.40 1180 -53.31 -30.78 14352JO 'ELEMENT, TYPEB31, USET=MAST1

1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 19 19 19 20 20 20 21 21 21 22 22 22 23 23 23 24 24 24 25 25 25 26 26 26 27 27 27 28 28 28 29 29 29 30 30 30 31 31 31 32 32 32 33 33 33 34 34 34 35 35 35 36 3ô 36 37

37 37 38 38 38 39 39 39 40 40 40 41 41 41 42 42 42 43 43 43 44 44 44 45 45 45 46 46 46 47 47 47 48 48 48 49 49 49 50 50 50 51 51 51 52 52 52 53 53 53 54 54 54 55 55 55 56 56 5û 57 57 57 50 58 58 59 59 59 60 60 60 61 61 61 62 62 62 63 63 63 64 64 64 65 65 65 66 66 66 67

'ELEMENT, TfPE=031, ELSET=MAST2 67 67 68 68 68 69 69 69 70 70 70 71 71 71 72 72 72 73 73 73 74 74 74 75

'ELEMENT, TYP€=B31, ELSEMAST3 75 75 76 76 7s n n n 78 78 78 79 79 79 80 80 80 81 81 81 82 82 82 83

83 83 84 84 84 85 85 85 86 86 86 87

'ELEMENT, fYPE=B31, ELSET=MAST4 87 87 û8 88 88 89 ô9 89 90 90 90 91 91 91 92 92 92 93 93 93 94 94 94 95 45 45 96 96 96 97 97 97 98 98 98 99

'ELEMENT, NPE=B31, ELSET=MAST5 99 99 100 100 100 101 101 101 102 102 102 103 103 103 104 104 104 1% 105 105 106 106 106 107 107 107 108 108 108 109 109 109 110 110 110 111

'ELEMENT, TYPE=B31, ELSET=MASTG 111 111 112 112 112 113 113 113 114 114 114 115 115 115 116 116 116 1t7 117 117 118 118 118 119 119 119 120 120 120 121 121 121 122 122 122 123 123 123 124 124 124 125 125 125 126 126 126 127

'€LEMENT, TYPE=B31, ELS€T=MASTï

127 127 128 128 128 129 129 129 130 130 130 131 131 131 132 132 132 133 133 133 134 134 134 135 135 135 136 136 136 137 137 137 138 138 138 139 'ELEMENT, TYPE=B3 1, ELSET=MAST8

139 139 140 140 140 141 141 141 142 142 142 143 143 143 144 144 144 145 145 145 146 146 146 147 147 147 148 148 148 149 149 149 150 150 150 151 151 151 152 152 152 153 153 153 154 154 154 155 155 155 1% 156 156 in

'ELEMENT, lYPE=B3î, ELSET=MASTS 157 157 158 158 158 159 159 159 160 160 160 161 161 161 162 162 162 163

'ELEMENT, TYPE=Cl D2, RS€T=GUYl 500 500 501 501 501 502 502 502 503 503 503 504 504 504 505 505 505 506 506 506 507 507 SOt 508 530 530 531

812 812 813 813 813 814 830 830 831 831 031 832 832 832 833 833 833 834 834 034 835 835 835 036 836 036 837 837 837 838 838 838 039 839 839 84û 840 840 841 841 841 842 842 842 843 043 843 844 860 860 861 861 861 862 862 862 863 863 863 864 064 864 865 865 865 866 866 866 867 867 86'7 868 868 868 869 869 869 870 870 870 871 871 871 872 872 872 873 873 873 874

'ELEMENT, TYPE=CI 02, ELS€T=GUYS 900 800 901 901 901 902 902 902 903 903 903 904 904 904 905 905 905 906 9û6 906 907 907 907 908 908 908 909 909 909 910 910 910 911 911 911 912 912 912 913 913 913 914 914 914 915 915 915 916 930 830 931

1016 1016 1017 1017 1017 1018 1030 830 1031 1031 1031 1032 1032 1032 1033 1033 1033 1034 1034 1034 1035 10% 1035 1036 1036 1036 1037 1037 1037 1038 1038 1038 1039 1039 1039 1040 1040 1040 1041 1041 1041 1042 1042 1042 1043 1043 1043 1044 t 044 1044 1045 1045 1045 1046 1046 106 1047 1047 1047 1 0 4 1060 860 1061 1061 1061 1062 1062 1062 1063 1 O63 1 063 1064 1064 1064 1 O65 1065 1065 1066 t O66 1066 1 O67 1067 1067 1068 1068 1068 1069 1069 1069 1070 1070 1070 1071 1071 1071 1072 1072 1072 1073 1073 1073 1074 1074 1074 1075 1075 1075 1076 1076 1076 1077 ion ion 1078

'ELEMENT, TYPE=ClD2, ELSET=GUYï 1100 800 1101 1101 1101 1102 1102 1102 1103 1103 1 103 1104 1104 1104 1105 1105 1105 1106 1106 If06 1107 1107 1107 1 IO8 1108 1108 1109

lin lin 1178 1178 If78 1179 1179 1179 1180

'MPC BEAM, 508, 21 BEAM, 538, 21 BEAM, 568, 21 BEAM, 610, 41 BEAM, 640, 41 BEAM, 670, 41 BEAM, 712, 61 BEAM, 742, 61 BEAM, ï72, 61 BEAM, 814, 81 BEAM, 844, 81 BEAM, 874, 81 BEAM, 916,105 BEAM, 946,105 BEAM, 976,105 BEAM, 1018,133 BEAM, 1048,133 BEAM, 1078,133 BEAM, 1120,163 BEAM, 115OI 163 BEAM, 1180,163 'MATERIAL, NAME=STEEL 'E WSTIC 29000.00 0.30

'DENSITY 7.33E-07 'MATERIAL, NAME=CABLEl 'ELASTIC 24000.00 0.30

'DENSITY 7.33E-07 'NO COMPRESSlON 'SOLID SECTION, MATERIAMABLEI, ELSET=GUYI 0.846 'SOLID SECTION, MATEFUAl=CABLEl ELSET=GUY2 0.846 'SOLID SECTION, MATERIAL=CAûLEl, ELSET=GUY3 1.350 'SOLI D SECTION, MATERIAL=CABLEl, ELSET=GUY4 1.040 'SOUD SECTION, MATERIAL=CAûLEl, ELSET=GWS 2.710 'SOLID SECTION, MATERIAL=CABLEl, ELSR=GUY6 3.940

'SOLI D SECTION, MATERlAL=CABLEI, ELSET=GUY7 4.130 'BEAM GENERAL, SECTIONISECTION=GENERAL,ELSET=MAST1 ,OENSITY=.843E46 123.840,199308.10, ,1993?3.40,200260.50

l.lo.lo. 29000.00 338.09

'BEAM GENERAL SECTION,SECnON=GENERAt,ELSff =MAST2,DENSIPI=.846E-û6 1 15.44û,18579û.ûû1 ,185794.10,186649.40

1 .,o.Io* 29000.00 361.97

'BEAM GENERAL SECTlONlSECTION=GENERAL,ELSET=MAST3,0ENSllY=.918E46 1 15.44û1185790.00, ,f85794,10,185613.40

1 .lo.,o. 29000.00 699.26

'BEAM GENERAL SECTION,SECTION=GENERAL,ELSET=MAST4,DENSlfY=.875E-06 92.040, t46130,00, , 148133.20, 148449.90

l.lo.lo. 29000.00 453.99

'BEAM GENERAL SECT10NlSECTION=GENERAt,ELSET=MASTS,DENS6 92.0401 148129.601 , 148133.Z01 147293.OO

1 .,o.Io. 29000.00 877.04

*BEAM GENERAL SECTIONlSECTION=GENERAL,ELSET=MAST6,DENSIN=.901 E46 77,910,125389.10, ,125391.70,125384.60

1 .,o.Io. 29000.00 536.33

'BEAM GENERAL SECTIONlSECTI0N=GEN€ELSET=MAST7,DENSIPI=. 101 E-05 n.910,125~.70, ,12539t.70~124150.20

l.,o.lo. 29000.00 lO36.tO

*BEAM GENERAL SECT10N,SECT10N=GENERAL1ELSET=MAST8,DENSITV=.901 E-06 77.9 1 O, i 25339.10, ,125391.70,1253~4.60

1 .,o.,o. 29000.00 536.33

'BEAM GENERAL SECT10NlSECTION=GENERALlELSET=MAST9,DENSITY=. IO1 E-05 i7.910,125388.70, ,lZS9t .7011241S8.20

1 .,o.Io. 29000.00 1036.10

'BOUNDARY 11113 116 500tt3 53011t3 560,1,3 800,113 8301113 860, t13 *INITIAL CONDITIONS, P/PE=STRESS

GUY1 20.33 GUY2 20.33 GUY3 20.44 GUY4 20.43 GUYS 20.44 GUY6 19.90 GUY7 20.19 'STEP, NLGEOM 'STATIC, DIRECT 'DCOAD GUY1 ,GRAV,386.4,0,0,-1 GUY2,GRAV,386.4,0,0,-1 GUY3,GRAV,386.4,0,0,-1 GUY4,GRAV,386.4,OI0,-1 GWS,GRAV, 386.4,0,0,-1 GUYG,GRAV,386.4,0,0,-1 GUY7,GRAVI386.4,0,0,-1 IWASTI ,GRAV,386.4,0,0,-1 MASTZ,GRAV,386.4,0,0,-1 MAST3,GRAV,386,4,0,Ot*1 MAST4,GRAV,386.4,0,Or-1 MAST5,GRAV,386.4,OI0,-1 MASTG,GWV,386.4,0,0,-1 MAST7,GW,386.4,Ol0,-1 MASTB, GRAV,386.4,0,0,-1 MAST9,GRAV,386.4,0,OI-t 'EL PRINT, ELSET=GUY 1 ,FREQ=O 'NODE PRINT, NSET=GUY 1, FREGO 'ENDSTEP 'STEP, NLGEOM 'FREQUENCY 100,,,,250 'EL PRINT, ELSET=GUYl SI 1 'NODE PRINT, NSET=MAST U 'RESTART, WRITE 'ENDSTEP

Appendix A2

BEAM MODELER SOURCE CODE

PROGRAM MODEUER DIMENSION AREA(#),EL(40) DIMENSION NCT(40,2),lVC(40,6), IVC2(15,3) DIMENSION CN(30,3),P(4û,4),0C(4û13),EK(6,6),SK(4û,4û) 1 ,u(solq,F(~,4) REAL CND(20ûûI3),ARE(20) INTEGER ND(2000) l NTEGER ELE(2000,3) INTEGER W(20), NL(20), RG(20), NNG(20)

+,XSAH(20),G(20) REAL ELVG(20),GA(20), GD(20),GH(20), GR(20),MAR(20),AA(20), IT(20)

+lA1G(20),AiA(20),8S(20),GW(20),GAR(20), EMG(2O),FPL(300,7) C W C T E R RL'15,F1LO'I5,HEAD'70,T1'70 WRITErl'(A18)')'ENTER INPUT FILE READ(+,'(Af 5)')FIL WRITE(*,'(A18)')'ENTER OUTPUT FlLE READ(*,'(AlS)')FlLO OPEN(1 S,FILE=FIL) READ(1 SI1(A70)')HEAD READ(15,'(A70)3Tt DO 10 1 = 1,20 READ (1 51*lERR=100) ELVM(I)lTW(I),NL(t)~FW(I)l PH(I)JSAL(lO)

+,XSAD(1O),BW(10),EM(10),X,XSAH(lO) NMS = I

10 CONTINUE C READ GUY GEOMETRY 100 DO20 1 = 1,20

READ (15,',ERR=200)ELVG(I),GA(I),GD(I),GH(I),GR(I),MAR(I),AA(I) +,IT(I),RG(I),AIG(I),AIA(I) NG=I

20 CONTINUE C READ GUY MATERIAL PROPERTIES 200 DO 30 l = 1,NG

READ (1 S,')NNG(I), ELVG(I),GA(I), BS(I),GW(I), GAR(I), EMG(I),THCOF +, UNSTRL, RG(I),AIG(I)

30 CONTINUE REWIND 15

PH(J)=PH(I) XSAL(J)=XSAL(l) XSAD(J)=XSAD(I) ew(J)=sw(i) EM(J)=EM(I) XSM(J)=XSAH(I) CONTINUE CONTINUE READ GUY GEOMETRY

READ(15,'(A45)')Tl DO70 1 = 1,NG READ (15,')~(1),GA(I),GD(1),GH(I),GR(I),MAR(1),M(I)

+O IT(I)lRG(l)l~G(I),AIA(I) DO 60 J =I,NG ELVG(J)=ELVG(I ) GA(J)=GA(t ) GD(J)=GD(I) GH(J]=GH(I) GR(J)=GR(I) MAR(J)=MAR(I) WJ)=AA(I) IT(J)=IT(I) RG(J)=RG(I) AIG(J)=AIG(I) AI A(J)=AI A(1)

60 CONTINUE 70 CONTINUE C RE30 GüY MATERIAL PROPERTES

REA0(15,'(A45)')Tl DO 90 1 = 1,NG READ (15,') NNG{l)lELVG(I)lGA(I)lBS(I)~GW(I)~GAR(I)lEMG(l)~THCOF

+, UNSTRL, RG(I), AIG(I) DO 80 J=I,NG es(J)=ss(r) GW(J)=GW(I) GAR(J)=GAR(I) EMG(J)=EMG(I)

80 CONTlNUE 90 CONTINUE

NP=ELVM(NMS)IPH(I) C REAO FORCES ON TOWER AND CABLES c REAû(I5,'(A45)')Tl c REAû(15,'(A45)')Tl c DO 91 I = I,NP+I c REAû (15,')FPL(I, 1),FPL(I12),FPL(l13)IFPC(I14)lFR(l,~lFPL(l16) c +,FR(llT) cg1 CONTiNUE c READ(1S1'(A45)')T1

c READ(1S,*)AZI,SPD,REFP,TEMPC,RICElDICE,CON,PRFC,PRMI DO 95 1=1 ,NMS PH(I)=PH(1)'12. ELVM(I)=ELVM(1)'12. Fw(I)=MI(1)*12

95 CONTINUE DO 190 I =I,NG ELVG(I)=ELVG(I)*12

110 CONTINUE NP=ELVM(NMS)IPH(l) OPEN(2,FILE=flLO) CLOSE(1S) WR1TE(2,*)nHEADING' WRIT€(2,'(A70)')HEAD WRITE(2,')'"BEAM ELEMENTS FOR MAST' WRITE(2,')"DATA CHECK' WRITE(2,*)"PRE?RINT, HISTORY=NO, ECHO=NO, MODEL=NO' NO(1)=1 DO 115 I=1,1500 CND(I, l)=û.O CND(l,2)=O.O CND(I ,3)=O.O

115 CONTINUE DO 120 I = 1,NP NO(b1) = ND(I)+I CND(I+l, 1) = CND(I,1) CND(I+1,2) = CND(12) CND(I+I,3) = CND(1,3) + PH(1)

120 CONTINUE C WRlTlNG MAST NODES

WRlTE(2,*)"NODE ,NSET=MASf WR1TE(2,'(l5,3F10.2)')(N0(I),(CN0(I, J),J=t ,3), t=l ,NP+l)

C WRfTlNG GUY NODES ELN = 0. DO 150 K = 1,NG XT=MAR(K)'12WN(3.14159/ t ûû*AA(K)) M=MAR(K)'12COS(3. I415~18O*AA(K)) ZT=ELVG(K) XB=GR(K)'12*SIN(3.14159/1BO'GA(K)) YB=GR(K)*l rCOS(3.14lWl ûûeGA(K)) ZB=O.O X=(Xl-XB)M NG(K) Y=(n-VB)/NNG(K) Z=(rnB)INNG(K) WRITE(2,'(At 5,ll)')nNODE ,NSET=GüY,K M =(K-1 )V00 + 500 ND(M)=M CND(M, 1)= XB

CND(M,2)= YB CND(M,3)= ZB DO 140 1 = M,M+NNG(K) IF (CND(1,2).EQ.CNO(I-100,2)AND.CND(I,1).EQ.CND(l-t W,l).AND.CND

+(I,3).EQ,CND(l-100,3)) ND(I)=ND(I-100) IF (ELVG(K).EQ.ELEV) NO(M+NNG(K))=ND(M+NNG(K~70) WRJTE(2,'(I5,3F10.2)')ND(l),CND(I, 1),CND(It2),CND(1,3) ND(I+I)=I + 1 CNO(I+l, l)=CND(I,l) + X CND(1+1,2)=CN0(1,2) + Y CND(I+I ,3)=CND(1,3) + Z

140 CONTINUE XT=MAR(K)*1 2'51 N(3.14 1 5911 80'(AA(K)+120)) YT=MAR(K)' l2'COS(3.l4l59ll8O'(AA(K)+lZO)) A= ELVG(K) XB=GR(K)'l2*SIN(3.14159118û~GA(K)+120)) Y B=GR(K)*12'COS(3.1415911 8ût(GA(K)+120)) ZB=O.O X=(XT-XB)N N G(K) Y=(YT-Y B)l(NNG(K)) Z=(ZT-ZB)/(NNG(K)) M =(K-1)'100 + 530 ND(M)=M CND(M,l)= XB CND(M,2)= YB CND(M,3)= ZB DO 160 I = M,M+NNG(K) IF (CND(1,2).EQ.CNO(I-100,2).AND.CND(l,l).EQ.CND(l*~OO,l)~D.CND

+(i,3).EQCND(i-100,s)) ND(I)=ND(I-100) IF (ELVG(K).EQ. ELEV) ND(M+NNG(K))=ND(M+NNG(K)Xl) WRlTE(2,'(i5,3FlO.2)')ND(t),CND(t, 1),CND(1,2),CN0(i13) ND(l+l)=I + 1 CND(l+t , I)=CND(t,l) + X CND(I+I ,2)=CND(l,2) + Y CND(I+1,3)=CND(1,3) + Z

160 CONTINUE XT=MAR(K}'12°SlN(3.1 4159/180*(AA(K)+240)) YT=MAR(K)'l ZtCOS(3. f4159Il ôû'(AA(K)+24û)) A=€iVG(K) XB=GR(K)'12%1 N(3.1415911 8Oe(GA(K)+24O)) YB=GR(K)ei2COS(3.141591180'(GA(K)+240)) 284.0 X=(XTXBWNG(K) Y=(YT-YB)r(NNG(K)) Z=(mZB)I(NNG(K)) M =(K-1)'100 + 560 ND(M)=M CND(M, 1)= XE

CND(M,2)= YB CND(Mt3)= ZB DO 170 1 = MIM+NNG(K) IF (CNO(I,2). EQ.CNO(I-t00,2).AND.CND(I,I).EQ.CND(I-100,1).AND.CND

+(ll3).EQ.CND(I-100,3)) ND(I)=ND(I-100) IF (ELVG(K).EQ.ELEV) ND(M+NNG(K))=ND(M+NNG(K)-160) WRITE(2,'(15,3f 10.2)')ND(I),CN0(Il 1),CND(It2),CND(I,3) ND(I+l)=I + 1 CND(I+l,l)=CND(I, 1) + X CND(I+1 ,2)=CND(It2) + Y CND(I+1 ,3)=CND(1,3) + Z

170 CONTINUE ELEV=ELVG(K)

150 CONTINUE C ELEMENT DEFINITION FOR MAST

HSEC=O.O 00 190 I=I,NMS ELNB=HSEC HSEC=ELVM(I) Nl=ELNB/PH(I) + 1 N2=ELVM(I)/PH(I) + 1 WRITE(2,'(A30,1 l)')"ELEMENTl VPE=B31, ELSET=MAST',l DO 180 J=Nl,N2-1 ELQJ, l)=ND(J) ELE(J,2)=ELE(J, 1) ELE(Jt3)=ELE(J, l)+l WRITE(2J315)') ELE(J, 1),ELE(J,2),ELE(Jt3)

180 CONTINUE 190 CONTINUE

C ELEMENT DEFlNlTlON FOR GüYS HSEC=O.O DO 220 I=l,NG M=(l-1 )Y00 + 500 WRITE(2,'(A30,1 I)')'*ELEMENT, NPE=C1 D2, ELSET=GUY,l DO 205 J=M,M+NNG(I)-1 ELE(J,1)= J ELE(J,2)=NO(J) ELE(Jt3)=ND(J+1) WRm(2,'(315)') ELE(J8 1),ELE(J,2),ELE(J13)

205 CONTINUE 0 0 210 J=M+3û1M*NNG(1)+29 ELE(J, 1)= J ELE(J,2)=ND(J) ELE(JI3)=ND(J+1) WRlTE(2,'(3I5)') ELE(J, l), ELE(J12),ELE(J,3)

210 CONTINUE 00 215 J=M+6OlM+NNG(I)+59 ELE(J, t)= J

EtE(J,2)=ND(J) UE(J,3)=ND(J+l) WRITE(2,'(3l5)') ELQJ, 1 ), ELE(J,2),ELE(J,3)

215 CONTINUE 220 CONTINUE C MPC DEFINITION FOR THE RIGERS €LN=#. WRiTE(2,'(A4)')'MPC' DO 240 I=1 ,NG IF (ELVG(I).EQ.ELEV) GOTO 230 NI =ELVG(I)IPH(l) + f N2=NO((I-1)'lW + 500+NNGjl)) NEL=2ûûû +3'(1-1) WRITE(2,'(A5,15,al ,i4)')'BEAM,',N2,',',N 1 WRITE(2,'(A5,15,al ,i4)')'BEAM,',N2+30,',', NI WRlT E(2,'(A5,15,aI,i4)~BEAM,',N2+6O,',',Nl

230 ELEV=ELVG(t ) 240 CONTINUE C MATERIAL DEFINITION FOR THE MAS1

WRITE(2,'(A21)')"MATERIAL, NAME=STEEC WRITE(2,'(A8)')'WASTlC' WRITE(2,'(F10.2,F6.2)')EM(l),0.3 WRITE(2,'(A8)')"DENSITT WRITE(2,'(A8)1)7.33€47'

C MATERIAL DEFINITION FOR THE CABLES EtASMG=O. DO 260 I=I,NG IF (EMG(I).EQ.ELASMG) GOTO 250 WRITE(2,'(A21 ,II)')'*MATERIAL, NAME=CABLE', I WRlTE(2,'(Aô)'~ELASTlC' WRITE(2,'(F10.2,F6.2)')EMG(1),0.3 WRlTE(2,'(A8)')"DENSllY' WRIT€(2,'(A8)')7.33E-O7' WRITE(2,'(A15)')"NO COMPRESSION'

250 ELASMG=EMG(I) 260 CONTINUE C SECTION DEFINITION FOR THE CABLES

EiASMG=O. DO 300 I=l,NG

IF (DEMG) 270,280,270 270 K=l

GO TO 290 280 K=K 290 CONTINUE

WRlTE(2,'(A30,11 ,Al 1 ,I1)')"SOLID SECTION, MATERJAL=CABLF +, K,', ELS ET=GüY,l ARE(I)=GAR(I)

WRlTE(2,'(F53)')ARE(l) ELASMG=EMG(I)

300 CONTINUE C SECION DERNITION FOR THE RlGERS C ELN=O. C DO 310 I=I,NG C IF (ELVG(I).EQ.ELR(J GOTO 305 C WRITE(2,'(A58,1 l)')'*BEAM SEC~ON,SECTlON=CIRCULAR,MATERIM=STEEL C +,ELSET=RIGER',I C WRITE(2,'(A3)') '5.0' C W RITE(2,'(AS)') '0,0,11 C305 ELN=ELVG(I) C3t0 CONTINUE C C SECTION DEFlNlTlON FOR THE MAST

EiASMG4. DO 330 I=l,NMS

C C CALCUiATION OF SECTION PROPERTIES C

NM=39 NN=15 NDF=36 NLC=3 E=EM(l)

C DEFINITION FOR AREA DO 305 K=1,12 AREA(K)=XSAL(I)

305 CONTINUE DO 306 K=13,24 AREA(K)=XSAD(I)

306 CONTINUE DO 307 K=25,39 AREA(K)=XSAH(I )

307 CONTINUE CN(1,3) =0.0 00 308 K=t ,5 CN(K,l)=O.O CN(K,2)=0.86603'FW(1)'2.B. CN(K+1,3)=CN(K,3)+PH(l)

308 CONTINUE CN(6,3) 4 . 0 DO 309 K=6,10 CN(K, l)=MI(l)IZ CN(K,2)4.86603'MI(l )B. CN(K+ll3)=CN(K,3)+PH(1)

309 CONTINUE CN(11,3) =0.0

DO 310 K=lI,l5 CN(K, l)=FW(l)n. CN(K,2)r0.86603'FW(l )B. CN(K+l ,3)=CN(K13)+fH(1)

310 CONTINUE DO 31 1 K=1,4 NCT(K, 1)=K NCT(K,2)=NCT(K1 1) + 1

311 CONTINUE DO 312 K=5,8 NCT(K, l )=KI I NCT(K,2)=NCT(K11) + 1

312 CONTINUE 00 313 K=9,12 NCT(K, 1 )=K+2 NCT(K.2)=NCT(Kl 1) + 1

313 CONTINUE DO 314 K=l3,l5,2 NCT(K, 1)=K-7 NCT(K,Z)=NCT(K. 1) - 4

314 CONTINUE DO 315 K=14,16,2 NCT(K, l)=K-12 NCT(K,2)=NCT(K, 1) + 6

315 CONTINUE 0 0 316 K=l7,l9,2 NCT(K. 1)=K6 NCT(K,2)=NCT(K, 1) - 4

316 CONllNUE DO 317 K=18,20,2 NCT(KI l)=K-11 NCT(K,?)=NCT(K, 1) + 6

317 CONTINUE 00 318 K=21,23,2 NCT(K, l)=K-20 NCT(K,Z)=NCT(K, 1) + 11

318 CONTINUE DO 319 K=22,24,2 NCT(K, 1)=K-1 O NCT(K12)=NCT(K, 1) - 9

319 CONTINUE 00 320 K=25,34 NCT(K, 1)=K-24 NCT(KJ)=NCT(K,l) + 5

320 CONTINUE DO 321 K=35,39 NCT(K, t)=K-24 NCT(K,2)=NCT(K1 1) - 10

321 CONINUE DO 322 K=2,5 IVCZ(K, 1) =(K-2)3+1 IVCZ(K,2) =(K-2)3+2 IVC2(K13) =(K-2)3+3

322 CONïlNUE 00 323 K=7,10 IVC2(K1 1) =(K-3)9+1 IVCî(K,2) =(K-3)'3+2 IVC2(K, 3) =(K3)'3+3

323 CONTINUE DO 324 K=12,15 tVCZ(K, 1) =(K-4)3+1 1VC2(K1 2) =(K-4)%2 IVC2(K13) =(K-4)%3

324 CONTINUE DO 325 K=t,11,5 lVC2(Kl 1 ) IVC2(K12) =O IVC2(K13) =O

325 CONTINUE DO 326 K=1,39 IVC(K, 1) =IVC2(NCT(K11), 1) IVC(K,2) =IVC2(NCT(K1 1),2) IVC(K,3) =IVCZ(NCT(K, 1),3) IVC(K,4) =IVC2(NCT(K12), 1) IVC(K,5) =IVC2(NCT(K12),2) IVC(K,G) =IVC2(NCT(K,2),3)

326 CONTINUE DO 328 K=llNDF DO 328 M=l ,NLC P(K,M)=O.O

327 CONTINUE 328 CONTlNUE

P(12,1)=-10. P(24,1)=5. P(36,1)=5. P(11,2)=10. P(23,2)=10. P(35,2)=10. P(fOI3)=10. P(22,3)=5. P(34,3)=5. P(l113)=0. P(23,3)=8.66 P(35,3)28.66

c WRiTE(2,'(Fl0.4)')(AREA(K), K=1 .NM) c WRiTE(2,'(3F10.3)1)((CN(K, J), J=I , W = I ,NN)

c WRITE(2,'(215J')((NCT(KlJ),J=i ,2),K=l ,NM) c WRlTE(2,'(6i5)l((iVC(K,J), J=l,G),K=I ,NM) c WFUTE(2,'(F10.3)')(P(K, l), K=l ,NDF)

CAL DIRCOS(NM,NN,EL,CN,NCf,DC) C CLEARING OF THE MASTER STJFFNESS MATRlX

DO 740 K=l,NDF 00 735 J=I,NDF SK(K,J) =0.0


DO 745 M=I,NM C U STlFF (MIE,AR~DCIEL,EK) C A L GENK (MIIVCINM, EK,SKINDF)

745 CONTINUE C

CAL1 SOLVE (SKIUIP,NDFINLC) C GALL STRESS (NM,NLC,E,U,DC,IVC,F,AR~EL)

FLM=10.'0.8660eFW(1) Y=(U(lI, f)+U(23,1)+U(35,1))13. SI 1 1 =FLM'(4.ePH(l))WI(2.'E'Y) FP=(3û8(4.'PH(i))'3)/(3.'E'Sl 1 1) Y2=(U(f 1,2)+U(23,2)4(35,2))13.

c wnte(2,32 GAREA=(N.V.'PH(l))I(YZ.FP)

C WRITE(2,870) C870 FORMAT(II1, lOX,'RESULTS OF THE ANALY SIS') C WRITE (2,875) Ca75 FQ RMAT(1 ~X,'===-Y-=========U-=' 1 C WRITE (2,880) C88û FORMAT(11/13X,'D.0.F.'15Xt1DISPLACMENTS FOR CASES OF LOADING') C IF (NLC-2) 881,883,885 C881 WRITE (2,882) (K,(U(Kl J),J=I ,NLC),K=1 C882 FORMAT (PX,12,5X,F10.4) C GO 10 890 C883 WRITE (2,884) (K,(U(K,J), J=1 ,NLC),K=l ,NDF) C884 FORMAT(SX,12,5X,F10.4,5X,F 10.4) C G O T O M C885 WRITE (2,886) (K,(U(K,J),J=I ,NLC),K=l,NDF) CM6 FORMAT (SX,l2,!%, F1 0.4,5XIF 1 O.4,5X,FlO.4) Ca90 WRITE (2,891) C89 1 f ORMAT(1/1,3X,'MEMB. NO.',SX,'FORCES FOR CASES OF LOADI NG') C IF (NLC-2) 892,894,896 C892 WRlTE (2,882) (Kl(F(Kl J),J=1 ,NLC),K=1 ,NM) C GO TO 899 C894 WRITE (2,884) (Kl(F(Kl J), J=1 ,NLC),K=l ,NM) C GO TO 899 C896 WRITE (2,886) (K,(F(K,J),J=l ,NLC),K=1 ,NM) C GO TO 899

WRlTE(Z,'(A48,1 1 ,A9,E8.3)lHBEAM GENERAL SECTION,SECTlON=GENERAL +,ELSET=MAST,I,',DENSIW=',DENS WRITE(2,'(F9.3,al ,F9,2,Al ,a101F9.2,A1 ,F9.2)~AREAMl',',1~04*S11 1

+,',',',', 1.04°S122,',',1.05'S133 G(I) = 2.1 7'G(I) GOTO 887

ûû6 DLEN=SQRT(PH(I)"2+W(l)Y) DRTO=(XSAL(I}'PH(I)+XSAH(I)'FW(I)+XSAû(i J'DLEN)/(XSAi(l)'PH(1)) OENS=7.33E-û7'ORTO WRITE(2JA48, I l ,A9,€8.3)')"BEAM GENERAL SECTION,SECTION=GENERAt

+,ELSET=MAST,I,',OENSIN=',DENS WRITE(2,'(F9.3,aIlF9.2,A1,alO,F9.2,A1,F9.2)')AREAM,',',SIl l,',',

+',',S122,',',SI33 887 WRITE(2,'(Aô)')'t. ,O.,O.'

c ELASMG=EMG(I) 330 CONTINUE C BOUNDARY OEFlNlTiONS

WRlE(2,'(AS)')"BOUNDARY' WRITE(2,'(A6)')'lI 1,3' WRITE(2,'(A4)')'1 ,ô' ND(400) = 400 DO 350 I=l,NG 1 F (ND((1-1)'100+50O).EQ.ND((l-2)*100+500)) GOTO 350 WRITE(2,'(l4,A4)') NO((1-1).100 + 500),',1,3' WRITE(2,'(MIA4)') ND((1-1)*100 + W),', 1,3 WRITE(2,'(14,A4)') ND((1-1 )*IO0 + SO),' , 1,s

350 CONTINUE C INITIAL TENSION DEFINITION

WRITE(2,'(A32)')"1N tTlAL CONDITIONS, T(PE=STRESS' DO 370 I=l,NG STRS=IT(I)IARE(I) WRITE(2,'(A3,11 ,F?.2)')'GUY ,l,STRS

370 CONTINUE

C C STEP DEFINITION

WRITE(2,'(A13)')"STEP, NLGEOM' WRlTE(Zl'(Al5)r)"STAnC, DIRECT'

C C GRAVIR LOAD ON CABLES

WRiTE(2,'(Aû)l)'DLOAD' DO 380 I=l,NG WRlTE(2,'(A3,I 1 ,A18)')'GUY ,l,',GRAVl386.4,OIOl-1'

380 CONTINUE DO 381 I=l,NMS WRITE(2,'(AQ,Il ,A18)')'MAS~,l,',GRAV1386.4,0,0,-1 '

381 CONTINUE C ELIMINATING PRiNTOUT FOR THIS STEP

WRITE(2,'(A28)')"EL PRINT, ELSET=G W1, FREQ=O' C WRITE(2,'(A2)')'1E'

WRlTE(Z,'(A29)')"NODE PRiNT, NSET=GUY 1 ,FREQ=O' C WRITE(2,'(A2)')Y 1' C

WRITE(2,'(A8)')I.€NDSTEP' C C FREQUENCY EXTRACTION

WRiTE(2,'(A13)')l.ST€P, NLGEOW WRITE(Z,'(AlO)')"FREQUENCY WRiTE(2,'(A10)')'1001,,,250' WRlTE(2,'(A2l)')"EL PRINT, ELSET=G W 1' WRITE(2,'(A3)')'S 1 1' WRITE(2,'(A22)')"NODE PR1 NT, NSET=MAST' WRITE(2,'(AI)')'U' WRITE(2JA1 S)')"RESTART, WRITE' WRiTE(2,'(Aô)l)'ENDSTEP' STOP END

C SUBROUTINE DlRCOS TO CALCULATE DIRCOS OF MEMBERS SUBROUTINE DIRCOS(NM,NN,EL,CN,NCT,DC) DIMENSION EL(40),CN(30,3),NCT(4û12),DC(40~3) 00 600 M=l,NM B(M) 4.0 DO 580 5=1,3 EL(M) =EL(M)+(CN(NCT(M,2), J)-C#(NCT(M, t)lJ))7

580 CONTINUE EL(M)=SQRT(EL(M)) DO 590 J=1,3 DC(M, J)=(CN(NCT(M12),J)-CN(NCT(M,l),J))EL(M)


RETURN END

C SUBROUTINE STlFF TO CALCUME THE ELEMENT STIFFNESS MATRlX C IN THE GLOBAL DIRECTION

SUBROUTINE STlFF (MIEIAR&DCl&EK) DIMENSION AR~40),DC(40,3),~40),EK(6,6) DO 640 1=1,3 DO 620 J=1,3 K=1+3 L=J+3 EK(I, J)=E*AREA(MrDC(M,I)'DC(M, JyEL(M) EK(I,L)=EK(t, J) EK(KIJ)=EK(Il J) EK(K,L)=EK(Il J)

620 CONTINUE 640 CONTlNUE

RETURN END SUBROUTINE GENK TO GENERATE THE STlFFNESS MATRlX K

SUBROUTINE GENK (M,IVCINMIEK,SK,NDF) DIMENSION IVC(40,6),EK(6,6),SK(40,40) DO 690 1=1,6 \

N *IVC (Ml 1) IF(N.EQ.0) GO TO 690 DO 6ûû J=l,6 K = IVC (M,J) IF ( K.EC2.0) GO TO 680 SK(N,K)= SK(N,K)+EK(I,J)


RETURN END SUBROUTINE SOLVE (SK,U,P,NDF,NLC) DIMENSION SK(40,40),U(40,4),P(40,4) K=O

480 K=K+1 A=SK(K,K) 00 500 J=l ,NDF SK(K, J)=S K(K, J)IA

500 CONTINUE 00 51 5 J=f ,NLC P(Kl J)=P(Kl J)IA

515 CONTINUE N 4

550 N=N+1 IF(N.EQ.W) GO TO 560 IF(N.GT.NDF) GO T0 575 0= SK(N,K) DO 560 J=l,NDF SK(N, J)=SK(N,J)-SK(K,J)'B

560 CONTINUE DO 570 J=l ,NLC P(N, J)=P(N, J)-P(K,J)'B

570 CONnNUE ns IF(N.LT.NDF) GO TO 550

IF(K.LT.NDF) GO TO 48û DO 620 I=I ,NDF DO 610 J=1 ,NLC u(Ii J)=p(I 14


RETURN END StlBROüTiNE SIRESS(NM,NLC,E,U,DC,IVC8FIAREA,€L) OlMENSlON U(4û,4),DC(4û,3),IVC(4û16),F(4Ot4) DIMENSION AREA(4û),EL(40) DO 990 I=1 ,NM DO 985 N=1, NLC F(I,N) =0,0 DO 980 J=1,3 A4.0 K=J+3 A=A+(DC(I, J)'U(IVC(I, K),N)*DC(I1 J)'U(IVC(l, J), N)) 1 *E'AREA(I)IEL(I) F(I, N)=F(I, N)+A

980 CONTINUE 985 CONTINUE 990 CONTINUE

RETURN END

Yohanna M. F. Wahba

1967 Born on the 14* of March in Cairo, Egypt.

1989 Graduated with B. Sc. degree (Honour) in Civil Engineering fiom Cairo University,

Cairo, Egypt.

1989 Joined SEMAB, Manufacnired Steei, Heliopolis, Egypt, as a Design Enguieer

1990 Enrolled in a Master's program in Civil Engineering Depanment, University o f

Windsor, Canada. Also, joined the Department as a Teaching and Research

Assistant.

1992 Graduated with a Master of Applied Science, University of Windsor.

1992 Enroiied in the Faculty of Graduate Studies and Research, University of Windsor,

Windsor, Ontario, Canada, in a program leading to the degree of Doctor of

Philosophy in Civil Engineering.

1995 Passed professional engineering examinations I and II, for regisuation in State of

Michigan, U.S.A.

1996 Registered Professional Engineer, in the Province of Ontario, Canada.

1996 Joined Jay Desai, Consuiting Engineen, West Bloomfield, MI, as a Project Design

Engineer.

1997 Joined LeBlanc & Royle Telcom inc., Oakville, Ontario, Canada, as the Chief

Engineer.

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Guy Towers