Seismic Analysis of Lattice Towers › obj › s4 › f2 › dsk1 › tape8 › PQDD_0024 › NQ50201.pdfMohamed Abdel Halim Khedr October 1998 Department of Civil Engineering and

Seismic Analysis of Lattice Towers

by

Mohamed Abdel Halim Khedr

October 1998

Department of Civil Engineering and Applied Mechanics McGill University Montreal, Canada

A thesis subsnitted to the Faculty of Graduate Studies and Research

in partiai fuisument of the requirements for the degree of Doctor of Phüosophy

O Mohamed Abdel Halim Khedr, 1998

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in the absence of s p d c guidelines for the seismic anaiysis of ~suppor t ing

telecomrnunication towers, designen may be tempted to apply simplified building code

approaches to these stmctures. However, these towas respond to earthquake~ in a

different fashion than that of shear buildings. The objective of this research is to propose

simplified mahods for the seisrnic analysis of self-supporting telecommunication towers.

The author studied the specific problem of self-supporthg lattice

telecornmunication towers ushg numerical simulations and applying the modal

superposition method and the response spectnun technique on ten existing towers, typical

of rnicrowave towers usualy erected in Canada. The analyses are carrieci out using a set of

45 strong motion horizontal accelerograms to snidy the horizontal effects. Vertical

dynamic effêcts are studied using two approaches: the fht considering the same

horizontal accelerograms in the vertical direction &er reducing their amplitudes to 75%,

the second using a distinct set of 55 vertical accelerogtams.

As a fim stage, simple regression analyses are paformed on the results to yield

eanhquake amplification factors for the base shear and the total vertical reaction. These

factors are presmted as functions of the tower's largest flexuraî period or largest axid

period of vibration as appropriate and peek ground 8cccleration at tower site. They can be

used by designers to estirnate the expected lm1 of dynamic forces in seKsupporting

telecommunication towen due to an emhquakc.

As a second *e, a simplified rcnic method is proposeci to estimate the membcr

forces in self'-supportmg tdecommuniuuion lanice towm due to both horizontai and

verticai earthquake excitations. It is assumed that the lowest thnt fluemi modes of

vibration in sufnciem to estmiffe the structure's respoase to horizontai excitation

accurstely, whüe ody the first axial mode wiii reflect the actwl behavior of towem in

response to vertical excitation. An acceleration profile dong the hdght of the tower is

defked using the spectral acceleration values correspondhg to the lowest three flexural

mode shapes or the lowest axial mode as appropriate. The mass of the tower is calculateci

and then lumped at the leg joints. A set of equivalent static l a t d or vertical loads can be

determined by simply multiplying the mass by the acceleration. The tower is then analyzed

staticaliy under the &kt of these forces to evaluate the member forces. The maximum

error associated with the proposed simplified static method is found to be 25% in the

e x t m m cases 4th an average enor of &7%.

The effe* of incluâing antennae clusters is also addressed in the d y s i s and

findings are summMsed in the thesis.

îhe study is extended to include transmission line towers in order to ohplfi the

nspoase of coupled tower-cable system by replacing the cables with an equivalent mass.

Seved frrquency analyses were perfoxmed on the system in order to acbieve a better

understanding of the behavior of the coupleci system, however, it was not possible to

simplify this interaction in the way desired. Several obstrvations are presemed which may

help in fùrther saidies.

SO-,

Fate de directives spccl6ques a l'andyse sismique des pylônes de

télécommunications, les concepteurs peuvent être tentés d'appliquer les approches

SimpMées proposées dans codes pour les structures de bâtiments aux pylônes de type

autoporteur. Une mise en garde s'impose toutefois car ces pylônes ne prément pas le

même type de réponse Pismique que le modèle simplifié de bâtiment élevé avec

déformation latérale en cisaillement. L'objectif de cette recherche est de fournir aw

concepteurs de pylônes de télécommunications des outils appropriés à l'analyse sismique

simplifiée.

L'auteur a étudié le problème spécifique aux pyldnes de ttlécomrnunications

autoponairs a treillis en acier à l'aide de simulations numériques par superposition modale

et par analyse spectrale sur dix modèles daaillés de pylônes txistmts, typiques de

l'expérience canadienne. Ces structures prdsanent un comportement essentiellement

linéaire et élastique dans la gamme des efforts considérés. Les analyses ont utilisé les

accelérogrammes de 45 tremblements de terre réels pour les effas horizontaux. Pour les

efféts verticawc, dew séries de soIlicitations ont été considérées : la première consiste à

utiüser les mêmes accélérograrnmes que pour les effets horizontaux mais avec une

amplitude réâuite à 75%, et la seconde est composée de 55 accélérogrammes verticaux

réels.

Dans une première étape, l'analyse des résultats des réactions à la base des pylônes

a pennis d'établir des facteurs d'amplification du cioailement la base et de la

composante dynamique de la réaction verticale. Ces d c w fàctcurs sont exprimés en

fonction de la période naturelle de vibration du pylône m modes t r a n s v d et axhi, et en

fonction de l'arrilération maximaie à la base de la stnichirr. L'auteur propose l'utüisition

de ces &eus comme des indicateurs de réponse seulement, i.e. pour estimer le niveau

probable des effets dynamiques d'ensemble sur la stnicturc.

En seconde étape, I'adyse des résultats s'est fate au niveau détaillé des efforts

daas les membrures principales, soit les montants, les diagonales principales ainsi que les

contreventements horizontaux. L'auteur propose maintenant une méthode &plinée, par

analyse statique avec forces équivalentes aw effets d'inertie, pour estimer les effons dans

les membrures individueiles. Une courbe enveloppe des accélérations maximaies le long du

pylône est définie à partir des accélérations spectrales correspondant aux friquences des

trois premiers modes de vibration transversaux, pour les effas latéraux. a du premier

mode de vibration axial, pour les effèts verticaux. La méthode proposée présuppose donc

la comiaismce de ces fréquences naturelles. La masse de la structure est discrétisée sur

les noeuds des montants principaux et un profil de charges équivalentes aux eEas d'inertie

est obtenu en multipliant diredement le profil de masse avec le profil d'accélération.

L'analyse statique usuelle sous charges équivaientes permet ensuite d'obtenir les forces

dans les membrures. La comparaison des résultats de l'analyse simplifiée avec ceux de

l'analyse des déîailiée des da pylônes étudiés indique des variations dans l'évaluation des

forces internes de l'ordre de * 7% en moyenne avec des écarts aiiant jusqu'à 25%.

L'étude a égpiement considéré l'effet de la masse de groupes d'antennes sur la

réponse sismique des pylônes et les principales conclusions sont résumées dans la thèse.

L'auteur a tenté d'étendre l'application de sa méthode simpiifiée au problème des

pylônes de lignes aériennes de transport d'ha&, mais en vain. L'idée de base consistait

à évalua s'il était possible de remplacer l'effet des câbles de la ligne par une masse

équivalente m pylône. Après avoir f i t plusieurs anaiysts de Wqumcm sur des modèles

couplés &les-pylônes, l'auteur n'a pu généraliser ses rtaihau. Qudques obsawtions

prtscntdes dans la th- pourront servir de point de dCput pour des recherches finues.

The author is indebted to Professor Ghyslaine McClure for her valuable

suggestions, long discussions, support, encouragement, and understanding to the -dent

persod life and feelings. This work wodd have never been completed in such a way

without such supervision. It is near impossible to describe how luclcy 1 am for having her

as my supervisor, many thanks.

The financial assistance provided by the Natural Science and Enghee~g Research

Council of Canada (NSERC) in the form of post graduate scholarship to the author and

research gant is greatiy appreciated.

The hancial assistance provided by Le Fonds pour la Formation de Chercheurs et

l'Aide à la Recherche (Fonds FCAR) in the form of post graduate scholmhip to the author

is greatiy appreciated .

The author appreciates the contribution of both Hydro-Quebec and Marioni, Cyr

and Associates in this work by providing the drawings and' daailed data for the

telecommunication and transmission towers used in this snidy.

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABSTRACT i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOMMAIRE iÜ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNO W LEDGMENTS v

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TABLE OF CONTENTS vi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LlST OF FIGURES x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF TABLES x k

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF SYMBOLS xx

CHAPTER t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTRODUCTiON 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2Objectives 3

1.3 Organization of T u t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

cHAPTER2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -RATURE REWEW 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 6

2.2 Dynrmic Raponse of =supportin& Littice Towen ................. 6

............................................ 2.2.1 Respome(owhd 6

2.2.2Sas~rclspollse . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 9

2.3 Dynamic Response of Trinsmission Line Structures .................. 14

2.4 Seirmic Response of Towe~haped Structuru ...................... 21

............................ 2 . 4 l S c t s n Ù c ~ l l ~ ~ o f o ~ h 0 r t t 0 ~ ~ 1 ~ 21

........................ L 4 . 2 S c i s m r * c r # p o l l ~ e o f i ~ t l ~ t o n r r s 23

. . . . . . . . . . . . . . . . . . . . . . . 2.5 Design Code Approaches for Sdsmic Analysis 25

. . . . . . . . . . . . . . . . 2 . 5 ~ 1 ~ 4 p p r 0 0 ~ h c s f o r d i f f ~ r t n t ~ p e s o f ~ c l ~ m 25

. . . . . . . . . . . . . . . . . . . . . . 2mS.2StondordscmdcodaofprOctjce fort- 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions 32

CHAPTER 3

METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 Telecommunication Towers Used in the Study . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Numerid Modeling of Tdecommuniution Towers . . . . . . . . . . . . . . . . . . 45

3.3 Classification of Tekcommunication Towers . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Transmission Line Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Modeling and Analysis of Transmission Towers . . . . . . . . . . . . . . . . . . . . . 55

3.6 Earthquake Records Used in The Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

CHAPTER 4

EARTHQUAKE AMPLIFICATION FACTORS FOR

TELECOMMUNICATION TOWERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 58

4.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 The Use of NBCC 1995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Horhontd Earthquake Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5 Vertical Earthquake Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

...................................................... 4.6 Discussion 90

..................................................... 4.7 Conclosions 93

vii

CHAPTER 5

. . . . . . . . . . . . . . . . . . . . . . . . . . PROPOSED STATIC METHOD OF ANALYSIS 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Thwmtk.l Background 94

. . . . . . . . . . . . . . . . . . . . . 5 .I . I Equation of motiou and rc~ponse spectnrm 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Modal analysis 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . 1.3 Seismic rtsponse of MlWF systems 97

. . . . . . . . . . . . . . 5.2 Proposed Metbod for Horizontal Earthquake Excitation 99

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Verification of the Proposed Method 1 1 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Vertical Earthquake Excitation 116

. . . . . . . . . . . 5.5 Verification of The Proposed Vertical Acceleration Proffie 119

. . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Effect of Concentrated Mass of Antennae 120

CEIAPTER 6

. . . . . . . . . . . SEISMIC CONSIDERATION FOR TRANSMISSION TOWERS 131

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction 131

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Mathematical Modeling 131

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 M o d d ~ g o f t o w a s 132

62.2 Modeling of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3 Methods of Analysis 135

6 3.1 Nonlinear static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

63.2 Frcqvencyanolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.4 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.5 Equivrknt Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.6 Conclusions ................................................... 146

C-R 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUMMARY AND CONCLUSIONS 148

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Summry 148

7.2 Earthquake Ampli&rtioa Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Simplified Methods of Analysb 150

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Anaiysis of Transmission Liae Touen 151

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Rtcommeodations for Future Work 152

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statement of Originality 153

............................................. REFERENCES 154

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EARTEiQU-AKE RECORDS 159

APPENDIX B

VERlFlCATION OF TEE PROfOSED EXPRESSIONS FOR TEE

EARTEiQUAKE AMPLIFICATION FACTORS 205 . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX C

VERTICAL EARTHQUAKE AMPLIFKATION FACTORS FOR

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TELECOMMUNICATION TOWERS 212

. C 1 Intmductioa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

C.2 Vertid Earthquake Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

C.3 Verticai Earthquakc Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

C.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

APPENDIX D

VEIUFICATION OF THE PROPOSED STATlC METHOD . . . . . . . . . . . . . . . . 223

LIST OF FIGURES

Fig. 3.1 Layout of tower TC1 (dimensions are in m)

Fige 3.2 Layout of tower TC2 (dimensions are in m)





Fig. 3*7 Layout of tower TC7 (dimensions are in m)


Fige 3.9 Layout of tower TC9 (dimensions are in m)

Fig. 3.10 Layout of tower TC 10 (dimensions are in rn)

Fig. 3.1 1 Definition of the mein symbols used in categorizing the lanice towers

Fig. 3.12 Layout of tower TR 1 (dimensions are in m)

Fig. 3.13 Layout of tower TR2 (cîimensions are in m)

Fig. 3.14 Layout of tower TR3 (dimensions are in m)


Fig. 3.16 Layout of tower TRS (dimensions are in m)


Fige 4.1 NBCC 1995 acceleration design spectrum evaluated for 5% damping

Fig. 4.2 Maximum base shear values for tower TC 1

Fig. 4.3 Maximum base shear values for tower TC2

Fig. 4.4 Maximum base sbear velues for tower TC3


Fjg. 4.6 Maximum base shear values for tower TC5 69

Fig. 4.7 Maximum base shear values for tower TC6 70




Fig. 4.1 1 hdaximum base shear values for tower TC l O 74

Fb. 4.12 Base shear amplification factors 76

Fig. 4.13 Total vertical dynamic reaction for tower TC I 78

Fig. 4.14 Total vertical dynarnic reaction for tower TC2 79

Fig. 4.15 Total vertical dynaMc reaction for tower TC3 80

Fig. 4.16 Total vertical dynamic reaction for tower TC4 8 1

Fig. 4.17 Total vertical dynamic reaction for tower TC5 82




Fig. 4.21 Totd vertical dynamic ration for tower TC9 86

Fig. 4.22 Total vertical dynamic reaction for tower TC 10 87

Fig. 4.23 Vertical reaction ampLification factors 89

Fig. 4.24 Acceleration response specva evaiuated for 3% damping 91

Fig. 4.25 Maximum base shear vs. peak ground acceleration in TC3 92

Fig. 4.26 Maximum dyiwiic vertical reaction vs. peak ground d e r a t i o n in TC3 92

Fig. 5.1 Concept of the proposeci method 100

Fig. 5.2 Acceleration profiles for towa TC9 103

xi

Fig. 5.3 Cornparison between member forces obtained fiom dynamic analysis and the

proposed methoâ I O 3

Fig. 5.4 Lateral mode shapes and mass distribution for Group Al 1 06

Fig. 5.5 Lateral mode shapes and mass distribution for Group A2 106

Fig. 5.6 Lateral mode shapes and mass distribution for Group B 1 07

Fig. 5.7 The a, coefficients for Group A l 1 08

Fig. 5.8 The a, coefficients for Group A2 1 09

Fig. 5.9 The a, coefficients for Group B 110

Fig. 5.10 Member forces for tower TC I 113

Fig. 5.1 1 Member forces in tower TC I using NBCC specuwn 115

Fig. 5.12 Proposed vertical acceleration profile 118

Fig. 5.13 Member forces in tower TC 1 under vertical excitation 1 20

Fig. S. 14 Tower TC3 with antemae 122

Fig. 5.15 Member forces including the efféct of antenna mass for tower TC3 using

NBCC 1995 for Z,<Z, ratio in the horizontal direction 122

Fig. 5.16 Mernber forces including the effea of antenna mass for tower TC3 using

NBCC 1995 for Z,<Z, ratio in the vertical direction 123

Fig: 5.17 Effixt of including heavy antennae on tower TC 1 124

Fig. 5.18 Effect of includhg heavy antennae on tower TC2 125

Fig. 5.19 Effect of includhg heavy antennae on tower TC3 125

Fig. 5.20 Wèct of including heavy antennae on tower TC5 126

Fig. 5.21 Wkct of including heavy antennae on tower TC6 126

Fig. 5.22 E f f i of inchidhg heavy antennae on towa TC7 127

Fig. 5.23 EEéct of including heavy antennae on tower TC8 127

Fig, 5.24 Effect of including heavy antennae on tower TC9 128

Fig. 5.25 Efféct of including heavy antennae on tower TC 10 128

Fig. 6.1 Components of transmission line section on double-circuit lattice towers 13 3

Fig. 6.2 Mesh convergence for the lowest transversal mode 134

Fig, 6.3 Ratio between calculatecl naturai penods of the cwpled line system and the

bare tower for tower TR2 with Condor 139

Fig. 6.4 Ratio between dculated natural penods of the coupled line system and the

bue tower for tower TR2 with Curlew 140

Fig. 6.5 Ratio between dculated natural penods of the coupled Iine system and the

bare tower for tower TR2 with Bersfon 141

Fig. 6.6 Ratio between calculated natural penods of the coupled line system and the

bare tower for tower TR6 with Condor 142

Fig. 6.7 Ratio between calculated naturai periods of the coupled Line system and the

bue tower for tower TR6 with Curlew i43

Fig. 6.8 The lowest two dculated longitudinal modes of tower TR6 144

Fig. A 1.1 Eart hquake record Long Beach, NS 1 W (L 1 ) 160

Fig. Al.2 Earthquake record Long Beach N39E (L2) 161

Fig. Al.3 Earthquake record Lower Caüfomia SOOW (L3) 162

Fig. A1.4 Earthquake record San Femando N61 W (L4) 163

Fig. A1.5 Earthquake record San Femando West (LS) 164

Fig. A1.6 Earthquake record San Feniando S37W (L6) 165

Fig. A1.7 Earthquake record San Feniando S90W (L7) 166

zrüi

Fig. A 1.8 Earthquake record San Fernando N 1 5E (L8)

F~B. AL9 Earthquake record San Fernando S38W (L9)

Fig. ALI0 Earthquake record San Fernando SOOW (L 10)

Fig. Al. 1 1 Earthquake record Near E. Cost of Honshu NS (L 1 1 )

Fig. Al.12 Earthquake record Near E. Coast of Honshu (L 12)

Fige A 1.13 Earthquake record Michoacan SOOE (L 13)

Fig. A1.14 Earthquake record Michoacan NOOE (L 14)

Figo A 1.15 Earthquake record Michoacan N90W (L 1 5 )

Fig. A2.1 Earthquake record Imperia1 Vaiiey SOOE (NI)

Fig. A23 Eanhquake record Kem County S69E (N2)

Fig. A2.3 Earthquake record Kem County N21E (N3)

Fig. A2.4 Eanhquake record Borrego Mountain N S W (34)

Fig. A2.5 Earthquake record Borrego Mountain N33E (NS)

Fig. A2.6 Earthquake record San Fernando S90W (N6)

Fig. A2.7 Earthquake record San Femando N90E (N7)

Fige A2.8 Earthquake record San Feniando N90E (N8)

Fig. A2.9 Earthquake record San Femando SOOW (N9)

Fig. M. IO Earthquake record San Fernando N3 7E (N 10)

Fig. M. 11 Earihquake record Near E. Coast of Honshu NS (N11)

Fige At. 12 Eanhquake record Near S. Coast of Honshu EW (N 12)

Fig. AZ.13 Earthquake record Monte Negro NOOW (N13)

Fig. A2.14 Earthquake remrd Michoacan SOOE (N14)

Fig. A2.15 Earthquake remrd Michoacan NWE (NI 5)

Fig. A3.1 Earthquake record Parkfield N65 W (H 1)

Fig. A33 Earthquake record Parffield NUE (H2)

Fig. A3*3 Earthquake record San Francisco S80E (H3)

Fig. A3.4 Earthquake record San Francisco S09E (H4)

Fig. A3.S Earthquake record Helena Montana SOOW (H5)

Fig. A3.6 Earihquake record Lytle Creek S25W (H6)

Fig. A3.7 Earthquake record Oroville N53 W (H7)

Fig. A3.8 Eanhquake record San Fernando S74W (Hg)

Fig. A3.9 Earthquake record San Fernando S21W (Hg)

Fig. A3.10 Eanhquake record Nahanni Longitudinal (H 10)

Fig. A3. I 1 Earthquake record Centeral Honshu TR (H 1 1 )

Fig. A3.12 Earthquake record Near E. Coast of Honshu NS (H 1 2)

Fig. A3.13 Earthquake record Honshu NS (H13)

Fig. A3.14 Earthquake record Monte Negro NOOW (H14)

Fig. A3.1 S Earthquake record Banja Luka N90W (H 1 5)

Fig. B.1 Maximum base shear vs. peak ground acceleration in TC1



Fig. B.4 Maximum base shear vs. peak ground acceleration Ui TC5

Fig. BOS Maximum base sheer vs. peak ground acce1emtion in TC6

Fig. B.6 Maximum base shear vs. Peak gound acceleration ia TC7

Fig. B-7 Maximum base shear vs. peaic grouad accelerafion in TC1

Figo B-8 Maximum base shear vs. peak ground -leration in TC9

Fig. B.9 Maximum base shear vs. peak ground acceleration in TC 1 O 208

Fig. B.10 Maximum dywnic vertical reaction vs. peak ground acceleration in TC 1 209

Fig. B. 11 Maximum dynarnic verticai reaction vs. peek ground acceleration in TC2 209

Fig. B.12 Maximum dynamic vertical reaction vs. peak ground acceleration in TC4 209

Fig. B. 13 Maximum dynamic vertical reaction vs. peak ground acceleration in TC 5 2 10

Fig. B.14 Maximum dynamic vertical reaction vs. peak ground acceleration in TC6 2 10

Fig. B. 15 Maximum dynamic vertical reaction vs. peak ground acceleration in TC7 2 10

Fig. B. 16 Maximum dynamic vertical reaction vs. peak ground acceleration in TC 8 2 1 1

Fig. B.17 Maximum dynamic vertical reaction vs. peak ground acceleration in TC9 2 1 1

Fig. B. 18 Maximum dynarnic vertical reaction vs. peak ground acceleration in TC 1 O 2 1 1

Fig. C. 1 Total vertical dynarnic reaction for tower TC 1 215

Fig. C.2 Total vertical dynamic reaction for tower TC2 215

Fig. C.3 Totai vertical dynamic reaction for tower TC3 215

Fig. C.4 Total vertical dynamic reaaion for tower TC4 216




Fig. C I Total vertical dynamic reaction for tower TC8 217

Fig. C.9 Total vertical dynamic r d o n for tower TC9 217

Fig. C. 10 Total vertical dynarnic reaction for tower TC 1 O 218

Fig. C.11 Vertical r d o n amplification factors 220

Fig. Cl2 Maximum dynaniic vertical reaction vs. peaL ground d e r a t i o n in TC3 220

Fig4 C.13 Maximm dynamic vertical reaction w. peak gnwid accdaation in TC l O 220

Fig. C.14 Cornparison between the expressions for vertical amplification factor

Fig. D.1 Member forces in tower TC2








Fig. D.9 Member forces in tower TC2 using NBCC spectrum

Fig. D.10 Member forces in tower TC3 using NBCC s p m m

Fig. D. 1 1 Member forces in tower TC5 ushg NBCC spectrum

Fig. D.12 Member forces in tower TC6 ushg NBCC spectrum

Fig. D.13 Member forces in tower TC7 using NBCC spectnirn

Fig. D.14 Member forces in tower TC8 using NBCC spectrum

Fig. D.15 Member forces in tower TC9 using NBCC spe*n>m

Fig. D. 16 Member forces in tower TC 10 using NBCC spectm

Fig. D.17 Member forces in tower TC2 under vertical excitation

Fig. D.18 Member forces in tower TC3 under venical excitation

Fig. D.19 Member forces in tower TC5 under vertid excitation

Fig. D.20 Member forces in tower TC6 under vertid excitation


Fig. D.22 Member forces in tower TC8 under verticai excitation

Fig. D.23 Member forces in tower TC9 under venical excitation

Fig. D.24 Member forces in tower TC 10 under vertical excitation

LIST OF TABLES

Table 3.1 Characteristics of the telecommu~cation towers studied

Table 3.2 Classification of towen according to Sackmann (1996)

Table 3.3 Prediction of the lowest three flexural modes

Table 3.4 Characteristics of the transmission towers studied

Table 3.5 Earthquake records used in the study

Table 4.1 Base shear values using the NBCC 1995 design spectrum

Table 4.2 Base shear values (in kN) using the NBCC 1995 static approach

Table 4.3 Linear regression analysis for the maximum base shear

Table 4.4 Linear regression analysis for the maximum vertical reaction

Table 6.1 Cable data

Table 6.2 Cases used in the parametric snidy

Table C.1 Earthquake records used as vertical input

Table C.2 Linear regression analysis for the total vertical reaction

LIST OF SYMBOLS

length of tmss pane1

peak horizontal ground acceleration

cross sectional ana of diagonals

coeflicients used in evaluating the horizontal acceleration profile (i= 1,6)

cross sectional area of main legs

peak vertical ground acceleration

viscous damping constant

equivdent taper ratio

foudation factor

conductor horizontal tension

second moment of atea of the main leg sections at the top

second moment of area of the main Leg sections at the base

!smhess

generalized m e s s

shear codcient at the tower base

tower height

excitation factor for mode i

mass

total mass of the tower

generaiized rnaw

tower mass at position x

intemal bending moment at position x

Mi genenlwd mass for mode i

MW Japan meteorological agency scale

ML local magnitude

Ms surface wave magnitude

R coefficient of correlation

S dimensionless seismic response factor

Sa spectral amleration value correspondhg to the fùndamental axial penod

Sm pseudo-acceleration corresponding to flexural natural period i

Ta bdamental axial period of vibration

r, fundamental flexural period of vibration

u relative displacement

u relative velocity

zï relative acceleration

v zona1 velocity ratio

V peak ground velocity

v(x) intemal sheer force at position x

V I elastic base shear

yh base shear

Y' total maximum vertical reaction

W dead load

w d conductor's weight per unit leagth

woaaii: overhead ground wire weight per unit length

x relative position

the base acce1eration

acceleration-related seismic zone

velocity-related seismic zone

sbear paramet a

mode shape i

undamped circulas fiequency of vibration

damped circular fiequency of vibration

viscous damping ratio

CHAPTER 1

mTRODUCTION

1.1 Introduction

Lattice towen are widely used today as supporthg stmctures, nmely in

telecommunication network systems and in overhead power lines. They are classified into

two main uttegories: guyed towers, also called guyed masts or sirnply masts, and

self-supporthg towers. The present study focuses on the latter.

Self-supporting telecommunication towers are three-legged or four-legged space-

trussed structures with usual maximum height of 120 m to 160 m. These towers consist of

main legs and horizontal and transverse bracings. Main legs are typically composed of 90°

angles (in four-legged towers), 60' xhifflerized or cold-formed angles (in three-legged

towers), or tubular round sections. Vaxious bracing patterns are used but the most

common ones are the chevron and the cross bracing. Classical steel transmission towers

are four-legged selfkupporting lattice structures. Their main legs are also usudy

composed of 90" angles and tubular sections whiie built-up composite sections are less

fiequent. The bracing patterns used in these towers are s d a r to those used in

self-supporthg telecommunication lattice towers.

Self-supporting telecommunication towers are designeci to resia environmentai

loads such as wind and ice accretion on their componmts (in cold cümates), usually

without considering earthquakes. However, some of these towers rnay be crucial

structures in a telecommunication mtwork and th& owner rnay requin tbrt t h y remain

operatiod or at least h v e a severe earthquake, especidy for towers focated in high

risk seismic areas. The 1994 edition of the Canadian standard CSA S37 Antennar. Towers

and Antenm-Supprting Sn~cmres (CSA 1994) introduced a new appendix (Appendk

M) titled "Seisrnic Analysis of Towers", in order to raise the awareness of the

telecornmunication inctustxy on this important issue. Specific recommendatioas for lattice

selfisupporthg telecommunication towers are that, whenever necessary, a somewhat

detailed dynamic anaiysis is io be perfomed using modal superposition. The base

acceleration shouid correspond to the values prescribed by the National Building Code of

Canada (NBCC 1995) for the tower site. These recommendations are very general,

however, and the tower designer is lefl without any specific guidance to assess whether or

not a detailed dynamic d y s i s is mly necessary. It would therefore be desirable to rely

on a shplified, static rnethod of analysis to get an estimate of the relative importance of

the seismic response of the tower. if the accuracy of such a method can be proven,

detailed dynamic analysis may even become unnecessary in the rnajonty of cases. The

design procedure would then include a relatively simple additional step to estimate seismic

effects, which could then be compared to extreme wind or combined wind and ice effécts.

The main design loads of overhead transmission line towers are conductors weight

and environmental loads such as wind and ice or a combination of bo& acting on the

cables and directly on the towers. Severai exceptional loads such as cable breekages and

ice-shedding effms are also comidered in some cases, using equivalent static loads w

methods. However, earthquake effêcts are not considered in tower design, even in high

nsk seismic areas. Nonetheless, thae are a few reports (Pierre, 1995 and Kempner, 19%)

of some transmission tower damages during recent earthquakes. Ahhough in most cases

damages were due to laree movements of the tower foundation, it remains relevant to

d e t e d e the level of dynamic forces these structures are subjected to during earthquakes.

It is noted that the niain difference between the seisrnic behavior of classical transmission

line towers and self-supporting telecornmunication towers arises from the dynamic

interaction between the tower and the cables suppotted. If it is possible to shpliQ this

interaction, aseisrnic design of transmission towers could be based on analysis methods

similar to those for self-supporting telecommunication towers.

1.2 Objectiva

The aim of this study is to achieve the foliowing objectives:

1. To propose a simplifiai static method that can be used in evaluating the member

forces in self-supponing telecommunication towers due to both vertical and horizontal

earthquake excitations. This is done by proposing a representative acceleration profile,

which is combined to the rnass profile dong the height of the towers. The produa of these

two profiles yields a profile of lateral inertia forces. The structure is then analyzed

aatically under the effect of these forces.

2. To assess the sensitivity of the towers to the vemcal wmponent of the

earthquake accelerations.

3. To evaluate the relative importance of the dynamic interaction between the

cables and their supporthg towers in uansmislon I i systems.

4. To adapt the proposed sïrnplified static method for transmission line towers, if

feasible.

1.3 Orynization of T u t

Chapter Two: A literature review is presentecl, which includes the work concernai with

gened dynamic anaiysis of both telecommunication and transmission towers as

well as seismic analysis of such structures. The review also covers seisrnic analysis

of other tower-shaped structures (such as offshore towers and intake outlet

towers), and relevant code approaches and design recommendations.

Chapter Three: A brief description of the essential structural chacteristics of the

telecommunication and transmission towen used in the study is presented. The

main recornmendations and conclusions reached in a prelimlliary investigation

(Sackmann 1996) on teiecommunication towers are presented as these results are

integrated with the m e n t work. The data base of the eanhquake records used in

the current study is also briefly discussed.

Chapter Four: Earthquake amplification factors for telecommunication towers are

presented for both horizontal and vertical excitations. Thea factors are meant to

be a first step in assessing the global level of forces expected to develop in

telecornmunication towers as a resdt of earthquake ground motions.

Chapter Five: In this chapter the scientific background of the proposed simplifieci method

is briefly revieweâ, after which the basic concept of the proposed methoci is

explained. The simplified method's coefficients for the horizontal earthquake

excitation are then given based on the classification explained in Chapter 3. The

applicability of the proposed static method is then verifid through numerical

cornparison baween tower member forces obtained using dynamic ruialysis and

those using the proposeû rnethoâ. Selected earthquake inputs are used in this

comparison in addition to the NBCC response spectrum (NBCC 1995). Coefficients

for the vertical earthquake excitations are dso presented, which are used to

calculate the member forces due to vedcal excitation. A comparison between the

forces obtained firom dynamic analysis and those obtained using these coefficients

is also presented for selected earthquake records. The presence of antennae is also

studied to show the change in tower behavior when heavy antenna clusters are

added. The addition of these antemae is accounted for through two generic cases.

Chapter Sv<: This chapter deds with transmission line towers. The study provides more

insight in the relative effect of the cables and the towers on the dynamic behavior

of the iine system.

Chapter Seven: In this chapter the findings and conclusions of this research are highlighted

in addition to the limitations of the work. Suggestions for relevant funre work are

also included.

CHAPTER 2

LITERATURE: REVIEW

2.1 Introduction

Earthquake loads are not routinely considered in the design of telecommunication

and transmission line structures even in high seismicity regions. This neglect is tolerated

without proper justification or complete understanding of the issue. in areas with IOW

seismicity this may be justified as wind effects will kely govem member design. However,

in areas with high seismicity, stresses in tower members or connections and tower

movements due to earthquake loading may indeed exceed the effects of other environ-

mental loads.

Most of the published work on lattice towers is devoted to the analysis of these

structures under wind. For transmission lines, some exceptional loading cases were dso

studied including cable breakage, ice shedding and galloping conductors. A review of the

relevant publications on dynamic wind and earthquake effects on lattice towers and tower-

shaped structures is presented in the following sections. Design code approaches are also

reviewed which pertain to buildings and safety-related nuclear structures.

2.2 Dyoamic Responr of SelC-supporthg Latticc Towm

2.2.1 Response tu wind

Chiu and Taoka (1973) were among the nrst to present a rational experimental and

theoretical shidy on the dynamic response of self-supporthg lanice towers under r d and

simulatecl wind forces. A thne-legged, 46 m t d sdf-supportecl lanice tower was inSrni-

mented to study its dyaunic respnse to winâ forces. The tower was also ideaüzed as a

space tmss with masses lumpeâ at the horizontai panel points. Cornpuison of the

measured dynamic properties of the tower indicated agreement with the calculateci values.

The validity of tbe cornmon assumption of uncoupled motion between the two principal

horizontal directions was also con£irmed. The study showed that the tower response to

wind is essentially govemed by the hdamental mode of vibration. The average damping

for the fundamentai period was fond to be 0.5?6 of the critical viscous damping value,

which is considered very low.

More recently, Venkateswarlu et al. (1994) conducteci a numencal study of the

response of microwave lattice towers to random wind loads. The dyiamic response was

predicted using a stochastic approach, and a spectral analysis method (fhquency-domain)

was proposed for calculathg the dong-wind nsponse and the resulting gust response

factor. The gust response factor is defined as the ratio of the maximum expected wind

load effect in a specifd time period to the corresponding mean value in the same t h e

period. A fiee-standing four-legged tower of 101 m height was used as a case study. The

variation of the gust response factor dong tower height was calculateci with and without

the contribution of the second and higher laterai modes of vibration of the tower, and it

was found that the maximum contribution of these higher modes to the ps t response

faor was only about 2%. The gust response &or obtained using the proposed stochas-

tic method varied between 1.55 and 1 S8 dong the height. Vdws dculated using the

formulae recornrnended by the Indian (IS1875-1987), Australian (AS 1 l70-2-1989),

British (BS 8100-1986) and American (ASCE 7-88-1990) standards, were found to be

2.03,2.21, 1 -93 and 1.89, respectively. Cornparhg these r d t s , it was concluded that the

standards values are consemative with Merence in the order of 2CE-4 to 40%, at least for

the case study considered.

In two very inportant papen on the dong-wind response of lanice towers,

Holmes (1994,1996) proposed closed-form expressions for the gus response factor for

both the shearing force and the bending moment dong tower height. The genenc tower

used in this study was idealized with linear taper and d o m solidity ratio u, that the drag

coefficient was kept constant. The mass per unit height of the tower, m(r), was assumed

to Vary with the tower elevation, z, according to the foliowing exponential relation:

nt@) = mO(l - & @ y ) (2- 1)

where

m.= rnass per unit length of the tower at the base

h = total height of the tower

k and y are empincal constants determineci so that m(z) best fits the actual mass

distribution of the tower.

In this study, only the effect of the lowest flexural natural mode of the tower was consid-

ered, which was assumed to take the following exponential shape, pi(s):

m(r> =($Y (2.2)

where p is al= a constant determined so that pi(=) best fits the calculated mode shspe.

Using the previous assumptions, expressions for the gust response factors for the shearing

force and the bending moment were obtained in a closed forrn.

It should be noted that three response components were hcluded in the denvation of the

previous expressions namely, mean, background and resonant response. The reader may

refer to Holmes (1994,1995) for a complete explmation of the t«ms used.

These expressions wae then compareâ to those rccommended in the Australian standard

AS 1170.2-1989 and were found to be in agreement. The advantage of the proposed

expressions over the currentiy used is the inclusion of more factors to account for the

effects of various parameters associateci with characteristics of the wind and the structure.

An expression for the aerodynamic damping of the tower, due to the relative motion

between the tower and the surrounding air, was also derived as a ratio of critical viscous

damping. In addition, a closed-form expression for the deflection of the top of the tower

was proposed combining three components of deflections (namely the mean, background

and resonant components). The effects of the tower height, taper ratio, and mean velocity

on the gust response factors were also studied. Finaily, the work was extendeâ and simpii-

fied to predict an &ective static load distribution, including the mean, background fluctu-

ating and resonant components of the wind.

2.2.2 Seismr'c raiporue

One of the nrst publications discussing earthquake effects on antenna-supporthg

lanice towers was authored by KOMO and K i m m (1973). The study aimeci maidy at

coîlecting information on tower vibration mode shapes, n a t d âequencies and darnping

properties. A r d case study of a tower that was instrumentai wben the 1%8 Off-Tokachi

earthquake occurred was presented. The coiiected data was analyzcâ and compand with

numerically sinilulateci resuhs obtained Born a simplined stick mode1 of the tower with

lumped masses and a viscous damping ratio of 1% in di modes. In some members, it is

interesting to note that the earthquake forces were found to exceed those due to wind.

This was confhed by observation of local damage and pemment deformations at the

tower base &a the emthquake.

More rezently, Milnis (1994) studied the seismic response of self-supporthg

telecommunication t o m ushg modal superposition analysis. The aim of this prrliminary

study was to improve understanding of the response of these towers to euthquakes. Six

towers with height ranging ffom 20 m to 90 m were modeled: bare towers only, i.e.

without antennae, attachmmts, anciliary components etc. Three earthquake records were

selected as the base excitation. A detaiîed iinear dynarnic analysis was performeâ using

modal superposition, and it was concluded that the use of the lowea four lateral modes of

vibration provided suflicient accuracy. The f'requency of the fbst axial mode of the towers

was found to be in the range of 11 to 43 Hz, which was either not present in the ffequency

content of the earthquake records used or corresponded to smail amplitudes of input

accelerations. As a resdt, the effects of the vertical component of the earthquakes proved

negligible.

A bst attempt to propose an equivaient static method for the adysis of lattice

seif-supporting telecornmunication towers was made by Mvez (1995). The method was

based on modal superposition, conside~g the &ect of the lowest three flacural modes of

vibration of the tower. As self-supporthg towers behave essentially as cantilever beams,

Wvez suggested the use of naturai fiepuencies and mode shapes expressions deveiopcû

for prismatic d e v e r s . The &e*s of taper ratio and shear deformations were included

by means of correction factors to the classical solution for prismatic Euler cantilevers. The

proposed expression for the naturai frequency of mode i, J, is:

where

L = tower height

Ho= flexural rigidity at the tower base

m.= mass per unit length at the tower base

The parameter L (essentially a dimensionless fiequency) is defined as:

Â1 = ÀbFc,Fc,

where

hb= freguency parameter for prismatic cantilever

F* taper correction factor

Fa= shear correction factor.

The expression for the corresponding flexural mode shapes, +,(x), is

4, (x) = cosh((A ,XIA!) - cos(A,xIL) - a, [sinh(Àlx/L) - sin(rl,x/L)]

w here

The base excitation simulated by Galvez was a sinuraidai wave with maximum amplitude,

üe quai to the peak ground acceleration defined by the National Building Code of Canada

(NBCC 1995) for the tower site. The acceleration profile of the tower response, u(r, t), for

a sinusoida1 ground motion is given by:

~(1, 0 = Ùg z:, #I(~)~(&Mn, a,, t ) (2-7)

w here

y(L,) = 2o,/À,

R = forcing fiequency of the sinusoidal input wave

CO, = 2nJ

ci(S2, al, t)= dynamic amplification bction for mode i

For R t o, and using the frequency ratio #?, = au,.

h 0(R, O,, t ) = - (sin w ,t - 8, sin W) 1 -Bf

At resonance in mode i, R = o, ,

a(C2, cul, t ) = +(sincu,t + o i t cos wlt) (2.1 1)

The base excitation is assumed to be in resonance with each of the lowest three flexural

modes considered, and the modal accelerations for i=l, 2, 3 are calculated. Using the

Square Root of Sum of Squares (SRSS) method, these relative modal accelerations are

combineû and the acceleration profle dong the tower is eslimateci with eq. (2.7). D d e d

dynamic analysis using a total of 45 base accelerograms was used to validate the method

for three existing towers with heights of 90, 103 and 12 1 m. Based on these results, simpli-

fied acceleration profiles were proposecl depending on the AN ratio (peak ground accel-

eration to velocity ratio) of the accelerograms. The inertia force distribution was simply

found by multiplying the acceleration profile by the mass profile. The structure was then

analyzeâ under the eEéct of these quivalent "static" inertia forces. Although simple, the

method did not always give good esthates for the intenial forces. For the main legs in

generaî, the methoâ yielded consemative values accurate enough for prelimlliary design.

However, the results were not systematicllly reliabie and conse~8tive for other chgona1

and horizontal members. The range of Merences baween the force predictions and the

dynamic d y s i s rewihs for horizontal members was betwan -70% and +45%, and for

cross bracings it was in the range of -35% to +25%. The method was M e r iimited to the

tower geomeûy used in the study, Le. a taper ratio (change in width dindeci by taper

height) less than 1: 14.5, a d a total length to tapered length ratio less than 1.15.

A draft of the Amencan TS 13 National Earthquake Hazard Reduction Program

(NEHRP) for non-building structures was released for comments in 1996, whae a simple

design equation (eq. 2.12) for seKsupporting telecommunication towers was suggested. It

was recommended that seKsupporting telecommunication towers be designed to resist an

earthquake lateral force, Y, appiied at the centroid of the tower and calculatecl using the

followhg equation:

where

V= Iateral force

Sal = site specific design spectrai acceleration at nominal penod of 1 second

I = importance &or (I=1 .O for standard towers, 1.25 for essential or post-critical

towers)

W= total weight of tower including aii attacbments

R = response modification b o t

T= hdamentai pend of the tower in seconds.

This equation was meant to resemble the maximum base shear equation used in most

building codes. Howmr, the bais on which the equation was developed is not clear. Ale0

ody the bdamental mode of vibration is considerd in this approach, which is not

accurate for this type of structure: it was first demonstrateci in Mihis (1994) and later

verified in Galvez (1995) and a v e z and McClure (1995) that the contributions of both

the second and third f l e d modes are usually significant.

2.3 Dynamk Response of Transmission Line Structures

The study of the wmplex dynarnic problem arising fiom the coupled behavior of

the tower-cable syaem attracted several researchers. Some of them investigated the

dynamic loads on transmission towers due to galloping of the conductors (Baenziger et al.

1994). conductor breakage (McClure and Tinawi 1987 and McClure 1989), ice shedhg

from the cables (Jamaleddine et al. 1993) and the fiee vibration of the coupled system

(Ozono et al. 1988 and Ozono and Maeda 1992). However, moa of the published work

on seismic analysis of transmission lines involves either the tower or the cable alone

without considering the coupled tower-cable problem.

Long (1974) was among the first to publish on the seisMc response of transmis-

sion towers. more or less at the same period as the pioneering midies on the d y d c

response of telecommunication towers to wind and earthquakes (Section 2.2). Long

neglected completely the effécts of the overhead conducton. The study was later extendeci

to evaluate the forces exerted by the conductors on the tower. The lanice transmission

tower mode1 was divided into two parts: The top part consistecl of the prismatic part and

the cross anns supporting the conducton, and was ideaiizeâ as a flexible unifonn cantiie-

ver. while the bottom part was simply assumed to be a rigid body. The absolute

displacement of the fl&e cantilever p d o q u ( ~ t ) , was then appronmited by the

foliowhg equation:

where

z(t) = gound displacement

hdx) = deflection curve for n o 4 mode of vibration k

/dt) = displacement response to the ground motion of a simple oscülatory system

in mode R

rn = rmiss per unit length

1 = length of flexibie cantilever portion

Eï = flexurai rigidity

qt) = ground acceleration

c&) = deflection due to static uniform looding =k(f)' - +( f )3 + +(fl2

AL = dimensioniess fiequency, positive root of the equation 1 + cosh A cos l = O

for mode k

k = mode nwnber.

Eq. (2.13) is therefore the summation of the horizontal ground displacexnent, the displace-

ment nsponse of the structure to the ground motion using modal aipaposition, and a

correction to the defieaion d t i n g Born the diffêrcnce of acceIdon loadin8s of

ground motion and fia niration. The defiedon at the top of the towa was evaluated

using eq. (2.13), assuming that the maximum values of cach of the thne tams in the

quation ocarncd sim-usiy. A nspoiw spectrum was used to cvaluate the

&um value of the response f'unction At), and the rnaxhurn values of the ground

displacement s, *t), and accelerations, a), were obtained Eom the earthqualce records.

Mer ail these caiculations for a case study of a 43 m transmission tower, it was concluded

that the entire tower moved rigidly with the ground and that no amplification of stresses

was produced by the ground motion. The second part of the study aimed at caiculating the

force exened by the conductors on the tower due to the earthquake excitation, assurning

compatibility of tower motions with conductors motions. Three orthogonal earthquake

directions were considered namely, transverse, longitudinai and vertical. The forces calcu-

lated in the three cases were found to be very small and could be resisted safely. It should

be noted that the tower used in the study was a relatively rigid one havhg a lowest

fiequency of vibration of about 5 Hz.

Kotsubo et al. (1985) perfonned dynamic rneasurements on three transmission

towen before and after installation of the conductors. The purpose of their study was to

determine the effkcts of the conduaon on the dynarnic characteristics of the towen. The

three towers used were two strain towers (with conducton directly anchored to the

tower) with heights of 92.5m and 68.5111, and a suspension tower with height of 92.2m.

The results were published for the case of the suspension tower only. The natural frequen-

cies and modes of vibration of the tower were calculated using both a plane tmss model

and a space truss model. Ambient vibration measurements for the tower were taken kfore

the installation of the cables. The naturai frequencies, modes of vibration and damping

prophes were extracted from these rneasufernents using FFT anaîysis. Mer the Unrlla-

tion of the ables, forced vibration tests ushg an exciter were carried out. The exater was

set up on the third ann from the top of the tower. It was obsemd tht then wae no

signincant changes in the natural fkquencies and the modes of vibration of the tower

before and &er the cable suinging, which suggested that the dyhamic interaction between

the cables and towers is uisignificant for suspension towers. The damping ratio of the

tower was found to be in the range of 0.2 to 2.096 of the critical viscous damping. The

eanhquake responses were then calculated using the plane truss model and the space truss

model ignoring the presence of the cables. For the plane ûuss model. the responses were

calculated for both the longitudinal and the transverse direction to the transmission be. It

was concluded that it is sufficient to model the tower as a plane truss.

In a more recent study conduaed by Li et al. (1991) models for long-span trans-

mission line systems under earthquake effects were presented. The study included the

derivation of mass and aifiess matrices for the tower-cable coupled syaem for the Ion@-

tudinai and transverse directions. For the vertical direction the mass of the conductors was

calculated and lumped at the appropriate joints. For each of the three principal directions a

dynamic analysis was carried out using three earthquake records namely Qian'an (China),

El Centro (USA) and Ninghe (China). The analyses were done for the foiiowing t h e

cases for cornparison:

I- The discretized model of the tower without the conductors;

11- The discretized model of the tower with the mass of the conductors lumped at relevant

tower joints;

III- The coupled tower-conduaor model.

It was found that for the venical ground motion the seismic response of mode! ïï is -ta

than that of model 1. For both the lated and longitudinal ground motions, the r-nse of

mode1 ïïï was pater than that of modd ïï, which in nirn was gr- thea that of modd 1.

It was concluded that the & i s of the conductors on the seismic responx of their

nipponing tower are not negligible and should be taken into consideration.

Li a al. (1994) studied the seismic response of high voltage overhead transmission

liaes. In the study the tower was discretized as a lurnped rnass multi-degne-of-fidom

system in the horizontal direction and assumed rigid in the vertical direction. It was Mer

assumed that the supporthg towers vibrate in phase with each other. Each cable span was

divided into five equal straight segments, the mass of each being lumped at its ends. The

ground motion was assumed acting in the longitudinal direction of the line. The equation

of motion of the coupled tower-conductor system was presented using the previous

assumptions. A numerical example was presented in which a 55 m height tower with

conducton spannuig 400 m was analyzed under the effe* of horizontal earthquake excita-

tion. Three earthquake records were used in the study, namely El Centro (1940) repre-

senting a sofi site, San Femando/Pacoima Dam (1971) representing a medium-st3f site

and Olympia (1965) representing a stiff site. Two models were used in the analyses: one

conside~g the presence of the conductor (model 1) and one negiecting the presence of

the conductors (model II). However, for model II it was not mentioned whether or not

the mass of the conductors was included in the analysis. Displacements and shear forces

were compared for the two models and it was found that neglecting the presence of the

cables could resuit in an underestimation of up to 66% (in case of the P a c o h Dam earth-

quake) in the shear force evaluated at the tower base. It was -ore concluded that the

tower cable interaction greatly affects the seismic response of the towers and neglcaiDg

their presence may lead to unde prediction of intemai forces in the tower membas.

In a pretimuisry study, El Attar et al. (1995) investigated the respoIlSe of

transmission hes under the effect of vertical seismic forces. The tower used in the study

was modeled by plane mss elements while the cables were modeled ushg two-node

straight elements taking geomeaic non-linearities Uito consideration. Damping of the

supporthg towers was assumed to be 2% of the critical viscous damping (h ali modes)

while that of the cables was taken as only 1%. The tower done was subjected to a

horizontal sinusoida1 ground acceleration of 0.28 g representing Viaona (Canada) in

accordance to the NBCC 1990 recommendatiom. From this analysis the maximum

displacement of the tower at the top level was determined, which resulted mainly fiom the

contribution of the first mode of vibration. However, this conclusion should not be gener-

alized as it contradicts most of the published work in this area of interest (Mikus 1994,

a v e z 1995 and a v e z and McClure 19951, which suggests that at least the lowea tfuee

modes of vibration should be inciuded in the analysis. This conclusion might be suitable

for short and stiff towers in which the higher modes are not likely to be triggered by an

earthquake. The cable was subjected to the vertical component of two earthquakes, San

Fernando for low A N ratio and Parkfeld for high A N ratio, after being scaled to a peak

ground acceleration of 0.21 g representing of the horizontal component prescribed for

Victoria. The vertical displacement of the cable at mid span was cdculated and it was

found that the displacement resuiting from the low A N earthquake was more than four

Urnes the response due to the high AN eenhquake. The e f f i of the change in the

damphg ratio of the cable on its response was also investigated and it wes found more

pronounced in records with low AN ratio as a change in the damping value fom L% to

4% resulted in a decrease of 32% in the displacement at mid span in cornparison to a 22%

decrease for the high AN ratio. The study did not report any anaiysis of on the wupled

tower-conductor system.

Ghobara a al. (1996) proposed a simpliiied technique to investigate the effca of

rnultiplasupport excitation on the response of transmission lines. In this study, the towers

were modeled as space tmsses wMe the conductors were modeled using linkages of

two-node straight elementq duly accounting for geometric non-hearities. In modehg the

ground motion dong the transmission line three factors were identified namely, wave

travel effkct resulting from finite sped of seismic wave, incoherency effect redting fiom

refldon and refhction of seismic waves and finaily site effect. However, only the first

two factors were accounted for stochastidy in the study. A numerical example illustrated

the suggested technique in which the maximum lateral displacements of the cable dong the

span, maximum force in tower members and maximum tension in cables were evaluated

for different wave velocities and considering incoherency and wave travel effects. Three

conclusions were drawn fiom the study: Firstly, conside~g the same ground motion for

all supporting towers does not produce the worst case for design. Secondly, although the

velocity of wave propagation has a signifïcant effect on the lateral displacement of trans-

mission lines, the incoherency of seisnic wave is more significarit. Thirdly, the increase in

cable tension due to lateral gound motion is smaü.

As a veq aude approximation, Kempner (1996) aiggested d y z i n g transmission

towers staticaüy under the ef£èct of a single lateral force acting at the tower's center of

u s , using the same equation as presented in the case of oelf-supporthg telecornrnunica-

tion tower (eq. 2.12).

The definitions of the t m s are the same as for the case of seKsupporting towers (see

section 2.2.2), except that W is the tower dead load without including the weight of the

supported wires. The same lateral force is applied in both the longitudinal and transverse

directions. If it is found that eaIthquake loading is ükely to govern the design, a more

detailed laterai force disüibution or modal anaiysis is suggested.

2.4 Seismic Response o f Tower-shaped Structuns

Due to the scarce information available in the üterature available on seismic analy-

sis of lattice seW=supporting towers, the search is duected towards other srnichires that

behave essentidy as cantiievers, namely offshore towers and intake-outlet towers. The

a h of this search is to gain insight of the apptoaches used in anaiyzing such structures

under seismic excitation and to h d if a simplified method for anaiysis is avdable.

2.4.1 S h * c rrsponse of offshore tawer~

P e ~ e n and Kaul(1972) studied the response of offshore towers to strong motion

earthquakes. In their work, the response spehun method of d y s i s was useâ and

comparai with th& proposed stochastic method. In this proposed method, a mean

ergodic Gaussian proass of finite duration was used as the stochastic model for the

horizontal ground acceleration. The a h of the study was to detmnine the transverse shear

distribution and the ovaturning moment dong the height of the tower without investigat-

hg the individual member forces. The towers wae idealhi as stick models with seven

joints dong the height on which the mess of the tower was lumped. A wndaised stifFiiess

ma& comsponduig to the laterai dispîacements of the model was cvaluateâ, and firom

the mass and &ess matrices of the model, the eigenproperties of the towers (fiquen-

cies and mode shapes) were predicted. The distributions of the transverse shear and

overturning moment were then calcuiated using the response spe- of the eanhquake

excitation considering the contribution of the lowest three flemal modes. The results

were found to be comparable to those obtained with the more ngorous aochastic random

vibration analysis.

Anagonstopoulos (1982), in his work on modal solutions for the earthquake

response of offshore towers, concluded that modal superposition gives good eshates of

the overaii response of the towers. For some members, however, the estimated value of

the bending moment was in an error by about 6û%, yet the dinerence in total stress was

less than 13% which can be reduced by increasing the number of modes in the summation.

Due to the uncertainties in the earthquake loadiig, it was suggested to use more e h -

quake excitations instead of increasing the number of modes in the analysis. It was also

concluded that the inclusion of the lowest three modes in each of the three principal struc-

tural directions would be adequate for design purposes.

In the work reported by Chan (1987), response spectnim techniques for multi-

component seismic andysis of offshore platfonns were evaluated. Two plaâorms were

modded taking into account the added mass of water. Thee components of earthquake

input were considered, two horizontal components with the ratio 0.67 : 1 .O and a vertical

component with 0.5. The study aimed at evaluating the techniques used for modal combi-

nation as weU as seisrnic component combination des. The member forces and stresses

calcuiated using Merent combination niles for both the modal ammation and DEsmic

components were cornparcd with those obtained using d*ailed direct intwtion d y s i s .

The different modal combination d e s studied were the Square Root of Sum of Squms

(SRSS), the Complete Quadratic Combination (CQC), and the American Petroleum insti-

Nte (API) method. For difEerent directional seismic inputs, the SRSS and the Multi

Component Quadratic Combination (MCQC) d e s were used. It was concluded that ail of

these combination d e s gave comparable results, and the CQC-SRSS rule was recom-

mended because of its conservatism. As part of this study, Chan aiso checked the error

resulting frorn neglecthg the effea of higher modes (above the eleventh mode) in the

analysis. He concluded that because al1 lower modes are horizontal, the vertical forces

could be underestimated by a tnincated analysis which in mm would Sêct the support

design.

2.4.2 SeiSmic rsponsr of intokeiutlet towm

Valliappan et al. (1980) inveaigated the e f f i of earthquakes on the intake tower

of Magrove Creek dam in Australia, using both dynamic and pseudo-static analyses. The

design spectrum approach was uscd as a bais of the pseudo-static analysis considering

only the lowest flexural mode of vibration. The mode shape used was that reported in

Clough and P e ~ e n (1993) in the form of a cosine hction. The structure was analyreci

aatically under the effe* of inertia forces resuiting fiom multiplying the acceleiation

profile due to the kst mode shape by the mass. Detailed dynamic analysis was also

perfonned and the results obtained fiom both analyses were comparcd. From this

cornparison, it was concluded that the pseudo-static d y s i s conside~g the lowest

flmiral mode is only an approximate solution. Howmr, this conclusion rnight change if

higher modes were included.

A simplifieci methoâ for seismic analysis of iatake-outlet towers was later devel-

oped by Chopra and Goyal (1991). The rnethod was used to estimate the maximum forces

in these towers using the design earthquake spectrum. A sirnplified stepby-step procedure

based on the Stodola and Rayleigh methods for the caldation of the lowest two natural

penods was suggested. It was demonstrateci that eonside~g the lowest two f l e d

modes of vibration is accurate enough for the prelirninary design phase. The procedure cm

be summarized in the following six steps:

1 . Definition of a smooth design spectnun suitable for the site of the tower.

2. Calcuiation of the added mass associated with both the inside and outside water.

3. Definition of the anictwal properties of the tower:

a. Mass per unit height, m&)

b. Flexural rigidity, E&) and shear rigidity, GX(z)A(z)

c. Modal damping ratio, 5.

4. Calculation of the lowest two natural periods of the tower using the proposed sirnplified

procedure.

5. Caiculation of the lateral force distribution for each mode of vibration using a general-

ized singîe-de-of-fieedo approach as follows:

a. Detemine the pseudo-acceleration ordinate S. from the design spectnun com-

spondii to period T. and damping ratio 5. .

b. Calculate the gmeralized mass M. and the generalized excitation tem Lm using

where

H, = tower height

#.(z) = lateral displacements of the tower in the nb vibration mode.

c. Caldate the equivaîent lated foras&') using the foiiowing expression:

6. Calculation of the umxhum shear and bending moment at any section dong the tower

height using the SRSS modal combination method.

It is noted that the method for estimating the lowen two natural periods is accurate if the

variation in the tower cross-sectionai properties cm be expressed in a closed form. Since

self-supporthg lanice towers u d y have irregular changes in their cross-sectional

propenies, the use of this method will only give crude estimates for the natural periods.

Also, a cornputer program was suggested for the implementation of the propod proce-

dure, which means that it is not such a "simplified" procedure.

2.5 Design Code Approaches for Seismic Anilysis

Merent design code approaches for the aaalysis of structures under h q u a k e

loads need to be reviewed. In addition to the few avdable approaches for the analysis of

towers under seismic effects, the recommendations for two other types of structures

m e l y Safi-related nuclear structures and buildings are presented.

2.5.1 Co& appr0lu:h.e~ for ùiffmwnt ypcs ufslnictum

The ASCE 1986 standard on seisniic anaîysis of Safi-relatecl nuclear structures

suggests acceptable anaiysis mthods and provides the mahodology and the input ground

25

motion to be used in calculating the response of such structures. The standard defines two

methods for specifyuig the seismic input, nameIy design spehun and input ground

motion time history. The horizontal component of the earthquake spectral ordinates

(absolute acceleration Sa, spectrai velocity S, and spectral displacement S d ) are obtained

by applying dynamic amplification factors to the corresponding maximum values of

ground motion (acceleration a, velocity v, and displacement d) obtained from the response

spectmm. These amplification factors depend on the amount of damping and are given as

ratios of &,a, S;v, and Sdd. The standard requires the use of two equal horizontal earth-

quake components in orthogonal directions. Two thirds of the horizontal component value

is used as the vertical cornponent of the input. If tirne histories are used. three different

earthquake records should be used in three orthogonal directions. These records must be

selected so as to represent the site conditions.

The standard recognizes four methods for the analysis of such structures: the direct t h e

integration method the response spectrum method, the wmplex frequency method and

the equivalent static method. The first tiuee methods are weU documented in textbooks

(Bathe 1982, Gupta 1992. and Clough and Penzîen 1993) and need not be reviewed here.

As for the equivalent static method, the standard resvias its use to cantilever models with

uniform mass distribution. Multidegree-of-fieedom models (MDOF) of cantilevers with

non uniform mass disuibution can be analyzeà using the static method only if they have a

dominant lowest mode of vibration. In this case, the equivaient static load is detennined by

multiplying the structure's mass profile by a unifonn acceleration profile of magnitude

equal to 1.5 times the peok accekration of the response specmm. For cantilever m c -

nues with uniform mas6 values of 1 .O and 1- 1 applied to the peak specarl aderation rn

used to determine the tower base shear and base moment respectively. The justification of

these values is not presented in the standard. The total response for the thm components

of seismic input is then calculated using the SRSS combination d e .

The usual approach suggested in building codes for seismic anaiysis of regdar

builduigs is to evaluate a global base shear value (Paz 1994). The base shear is then

distributed dong the height of the structure assuming that the lowest mode of vibration is

dominant and that the lateral displacement varies linearly with height. The National Build-

hg Code of Canada (NBCC 1995) specifies the minimum design base shear by the foiiow-

ing equation:

where

R = force modification factor

U = calibration factor = 0.6

V,= equivalent lateral seismic force.

The value of V. is given by the €oUowing equation:

Y, = v s m

where

v = zona1 velocity ratio derived fiom the probabilistic study of the ground motion

S = seismic response factor which is a bction of the fuadamental penod of the

structure and the relative values of the velocity zone, Z, and the acccleration

zone, 2,

I = importance fàctor

F = foundation factor, which depeads on the soi1 conditions at the site.

This base shear force is assumed to wunteract distributecl seismic forces dong the height

of the structure, given by:

where Fi is a concentrateci force applied to the top of the building to acwunt for the effêct

of higher modes and can be calailateci using the fouowing equation:

wMe W and h are the floor weight and elevation, respectively and n is the number of

stories.

It should be noted that the code recornrnends dynarnic analysis using the response

spectrum method and modal techniques for buildings with imguiar stiffness andior mass

distributions.

2.5.2 Smdords and c& of piwctice for lmvcrs

It is important to review the approaches foîlowed and the recornmendatious set

forth in the M i n t standards for earthquake design of towers to stand at the level of

knowledge available to the designers.

in its 1994 edition, the Canad'i Standard Antenms, Tawers and Antemvl

Suppomng S ~ t n r c ~ e s CSA S37-94 has introduced a n m appendix (Appendix M, not a

mandatory part) which addressed the issue of seismic anaiysis of lattice telecommunication

towers, both &supporthg and guyed. In this appendix it is statcd that since most

self-supporting telecommullication towers are typicaliy of high fiequaicy cornpand to

dominant fiequencies of earthqualces, seh ic effects are not likely to be signifiant. The

appendix therefore recommends that a fiequency analysis of the tower be perfomed to

allow for the identification of the tower's sensitive âequency range. If this range coincides

with the fkquency content of the dynemic loading, detailed dynamic analysis should be

performed. If seismic analysis is required, modal superposition should be used for self-

supporthg towers only with modal viscous damping ratios between 1% to 3%. It also

suggests to use accelerograrns or earthquake spectra based on the seismicity levels

prescribed by the NBCC for the tower site.

The Australian Standard AS 3995-1995 Design of steel lanice towers and m a s

also contains an informative appendix (Appendix C) which includes geaerd guidelines for

earthquake design of such structures. In this appendix it is stated that seKsupportiag

lanice towers with height up to 100 m with no significant mass concentrations need not be

designed for earthquake effects. For towers with signifiant mass and height of more than

100 m or lesser height but with significant mass concentrations, it is suggested that they

may be exposed to base shears and overtumhg moments approaching ultimate wind

actions. However, this standard does not offer any guidance as to how to estimate the

tower response.

The Eurocode 8 Design Provisions for Earth~uak Resistunce ofStmctures ENV

1998-3 (1995 draft) devotes a wmplete pari to towers, mas& and chimeys (Part 3). This

part comains a description of basic design rcquirements, seismic action, modehg consid-

erations and methds of aiialysis. It is emphasized that the design pbüosophy of this code

is !O maintain the fhction of the structure ad to prevmt any âanger to nearby b d d i n g

or fadites. However, the code does not provide protection against damage of nonstnic,

tural components. The code suggests several methods to describe seismic input: elastic

response spectnrm, smoothed design spectrum and tirne history representations using

either amficial accelerograms or recorded strong motions. A complee section is devoted

to mathematical rnodeling which hcludes the rocking and translation stifhess of the

foundation. For e l d e a l transmission towers, the code sptcif idy rquires that the

mode1 include a line section of at ieast thra towers (four spans) in order to obtain an

acceptable evaluation of the coupled tower-cable systun.

The code also provides a simplified method of analysis using design spectra in which the

base shear is calculated using the foîlowing expression:

Ft = sd(7)xk, w, (2.21)

this value is then âistributed dong the tower height using the following expression:

where

K= weight of the P mass

h l ~ elevation of the i* mass fiom the base

S s design spectrum ordinate conesponding to the hdamental period

T= fùndamental period of the tower.

However, the code clearly stipulates that this approach is limited to unimportant structures

with height less than 60 m. The code also ailows the use of both modal superposition

analysis and non-linear analysis.

The Unifonn Building Code UBC 1997 devotes a specific section (No. 1632) to

earthquake resistant design of non-building structures. Howeva, for telecomm~cation

towers, tbis section does not speafy any procedure different nom the one used for build-

ing structures. It is suggested that when an approved national standard provides a seismic

design procedure for a certain type of non-buiiding structure, such procedure may be used

under the foiiowhg conditions: the seismic zones and occupancy categones must wnfonn

with the UBC, and the total base shear and overtuming moment calculated must be

gram than 8û% of the values obtahed using the UBC approach. At this point it çhould

be noted that the philosophy of the UBC is to Meguard u@mt major structural failrres

and loss of Ive, not to l h i t h a g e or mazntainfunctiun, wwch is completely different

than the philosophy b e h d the provisions for seismic design of towers in which the main

concern is to maintain the towers' hctionaiity.

In a recent report (EU, 1998) released by the seismic cornmittee of the Electronic

industry Alliance / Telecommunication Industry Association EIA/TIA several recomrnen-

dations about the seismic d y s i s of steel antenna towers are made. The objectives of the

cornmittee were as follows: To define a methodology for use in seismic analysis of towers,

to identiS, simple but conservative assumptions to maLe the analysis easier, to provide

acceptable methods for more ngorous anaiysis techniques and to identify tower chanicter-

istics which inâicate if seismic analysis shodd be performed. At present, however, these

objectives are not al met in a satisfactory way. The cornmittee recommends to follow the

same approach as used in the Uniform Building Code UBC 1997, which is htended for

bdâing structures and is not suitable for towers. The total design base shear value is

caldatesi using the foliowing expression:

where

V= total design base shear

Z= seismic zone factor

I= importance factor equal to 1.25 for important or hazardous facilities and 1 .O for

special or standard fadities

W= total dead load

S= site coefficient

T= fiindamental period of the tower in seconds.

Mer obtaining the base shear it is distributeci dong the tower height in the same manner

as prescribed in the UBC 1997. The EIA/TIA report dso contains a study performed in

order to obtain a threshold criterion that can be used to determine whether a seismic

analysis is vuly needed or not. To this end, a cornparison between base shear values

obtained from wind efkcts and seismic effects was done. From this study it was concluded

that it is unlikely for seismic effects to control over wind. It shouid be noted, however,

that seismic calculations were &ed out using the UBC base shear equation which makes

this approach questionable.

2.6 Concl\isions

From this literature review it can be sem thrit seisrnic analysis of self-supporthg

telecommu.nication towers has received linle attention. Also, the work done in other fields

cannot be applied diredy to sessupporting lattice towers. Since the designers are left

without much guidance to assess if a detailed dynamic anaiysis is requked, earthquake

effbcts are usualiy ignored. For short towers and low ri& scismic a m this may be accept-

able. However, in high ri& areas and for taIl towers the designer should be able to

perfonn at least a simple -tic anaiysis as a quick design check. Therefore, a shplified

static method is proposed in this thesis. The method is based on modal superposition and

the response spectrum approach. It is anticipatecl that the proposed method will give

reliable estimates of the member forces and in most cases pedomiing a detaiied dynamic

analysis wiii become unnecessary.

Analyses of transmission towen under seismic excitation were conducted for

paninilar systems such as very @id towers, suspension towers with relatively light

conductors and very long spans with heavy conductors. The researchers were divided

among themselves into two groups. The first one neglected the tower-wnductor imerac-

tion effects, and even went further by not including the mass of the conductors. The

second group recommended the performance of a detailed anaiysis of the towa-conductor

syaem. As lanice transmission towers are similu to self-supporting la& telecommunica-

tion towers part of this work is devoted to vying to h d an equivalem added mas to

replace the effect of the conductors. If this proved feasible, the work could be extendeci to

investigate the applicabiity of the proposed simplified mahod for telecommunication

towen to transmission towers. if not, this work could suggest oome novel approach

tailored to overhead iine systems.

3.1 Teleconmunication Towm Used in the Study

Ten existing three-legged self-supporthg lanice steel telecornmunication towers

are used in this mdy with heights ranging from 30 m to 120 m: they are representative of

the range of towers usually erected in Canada. Table 1 lias some important charactens-

tics of these towers including their total mass and calculated natural periods corre-

sponding to their fundamental flexural and axial modes of vibration in ail1 air. It should

also be noted that these values are based on caiculations made for the bare conditions,

without including the mass of the antennae and other non-structural attachments.

The geometric layouts of the ten towers are illustrateci in Figs. 3.1 to 3.10; it

should be noted that secondary or redundant memben are not shown on the figures,

however their mass are included in the analysis.

Table 3.1 - Characteristics of the telecommunication towers studied

Base Top Fundamental Fundamental Tower Height Width Width Totai Mass Flexural Penod Axial Perid

(m) (m) (m) [kg) (s) ( s) TC 1 30 5.0 1.2 3,400 0.23 0.029 TC2 42 7.8 1.8 9,900 0.33 0.047 TC3 66 10.8 1.8 27,000 0.54 0.071 TC4 103 21.8 1.5 48,000 0.55 0.085 TC5 76 12.1 1.6 3 2,200 0.59 0.070 TC6 90 14.8 1.8 42,300 0.67 0.088 TC7 83 10.4 2.4 27,000 0.69 0.073 TC8 90 11.9 1.8 36,000 O. 76 0.08 1 TC9 55 6-1 1.2 10,800 0.79 0.064 TC10 121 14.4 2.0 66,200 1.20 O. 122

Fig. 3.1 Layout of towa TC 1 (dimensions ire in m)


36




39

Fig. 3 -6 Layout of tower TC6 (dixnemiions are in m)

Fig. 3.7 Layout of tower TC7 (dimensions an in m)

41



43

Fig. 3.1 O Layout of tower TC 1 O (dimensions are in m)

3.2 Numerid Modding of Telecommunication Towm

The towm are modelai as hear elastic three-dimensional structures with beam

elements for the main legs and truss elements for the diagonal and horizontal members.

The supports are idealized as pinned on rigid foundation. The mass of the main legs and

their tributary members (bracing members spannhg between the legs without any interne-

diate joint) are lumped at the corresponding leg joints. For tniss memben spanning

between the legs with intemediate joints, the mass is lumped at the appropriate leg joints

so that the member itself is assumed massless. Al1 secondary or redundant mernbers are

removed fiom the &ess mode1 since they do not take any load in a Ihear analysis.

However, the$ mass is calculated and lumped at the C O K ~ S Q O ~ ~ ~ ~ leg joints.

3.3 Classificatioa of Tclecommunicrtion Towtn

In this study the classification of the towers adopted by Sackmann (1996) is

followed, where the towers are classified with respect to three main characteristics

namely, the quivalent taper ratio D, the shear coefficient at the base K, and the a/L

(length of truss paneYtower height) ratio of the largest panel.

Fig. 3.1 1 Deihition of the main symbols used in categoriPn8 the lanice towers

The equivalent taper ratio is given by:

where

Ir= second moment of area of the main leg sections at the top

Io= second moment of a r a of the main leg sections at the base.

In calculathg the second moment of area of the main leg section either at the base or at

the top, the following expression can be used:

I = i~~ x&

the symbols used are defined Fig. 3.1 1.

The shear coefficient of the tower at the base is caculated using the shear coeffi-

cient for prismatic beams caldated for the elements of the tower at the base and using the

foiiowing expression:

in the previous expression the shear parameter depends on the ratio of the cross-

sectional areas of the leg and diagonal memben and the length and depth of the tniss

element, as foiiows:

Ar - 1 for cross-braced trusses, and 9 = 2 z ( a )

AL C 3 = SAD(=) for chevron-braced tasses

Two groups of towers were identified (labeleû as A and B) for which a predictor

of the lowest three Bexural modes of vibration was proposeci. The most important

parameter used in classi@mg the groups was the a/L ratio of the largest element. From the

tower data used in the study a k t of dL = 0.1 is the bounâary ktweai the two groups.

Group A was m e r subdivided into two subgroups Al and A 2 Subgroup Al has an

equivalent taper ratio D = 0.1 to 0.2, while subgroup A2 has an equivalent taper ratio D =

0.2 to 0.3. The prediction of the mode shapes is presented in a polynomial fom. Tables

3.2 and 3.3 show the different classification parameters and the mode shapes suggestcd for

the two groups. It shodd be noted that tower TC4 was not included in this prediaion as it

has a profound taper ratio (width at the top to width at the base ratio) of 14.5 which is

higher than that of the remaining nine towers considerd in this study having taper ratios in

the range of 4.2 to 8.2.

Table 3.2 - Classifcation of towers according to Saclmiann (1996)

Table 3.3 - Prediction of the lowest three flexural modes

91 x2.2

Group Al k -2.9x2 + 3.v + .7x4 b .&r + 5 . 4 ~ ~ - 21.99 + 1 6 . 7 ~ ~ - 41 x23

G ~ o u ~ A2 k - 4 . 4 ~ ~ + 5 . 1 9 + .3x4 1 .Sx + 7.3x2 - 3 1 .2x3 + 23 .4x4

41 xU Group B k -2. lx + 1 .4x2 + 1 .7x3

3.&+ 1.Q-51x3 +81x4-34xs

In the ment study, the classification adopted by Sackmann (1996) is followed but

some modifications were made for the lowest flexufal mode shape. This is done because

Sackmann's prediction of the lowest mode is an average for the two groups and it was

found beneficial to use a specific prdction for each group as will be dimssed later.

3.4 Transmission Line Towen

An investigation is carrieci out to assess the appiicaôiiity of the proposed simpli-

fied static method to transmission towers, focusing on the effea of the conductors on

tower bebavior. This study aims at h d i q an equivalent mass that can replace the mass of

the conducton in order to reflect the rd behavior of the tower-conductor coupled

system. To this end, a fraluency anaiysis of the tower-conductor system is perfonned

using the wmputer progr&n ADINA. A frequency andysis of the tower alone is perfomed

and the equivaimt mass of conductor is determineci. Potential dynamic interactions

between the cables and the tower are evaluated. Six existing transmission towers are

studied for power lines of voltage level rmging from 120 kV to 450 kV. Table 3.4 shows

the basic dimensions and the lowest flexural periods of vibration in both the longituciinal

and transverse directions for the six towers.

Table 3.4 - Characteristics of the transmission towers studied

Height Base Top Total Fundamental Fundamental Tower Width Width Mass FIexural Period Flexural Period

(m) (m) (m) (kg) Transverse (s) Longitudinal (s) TRI 48.15 16.50 2.00 13,500 0.298 0.287 TR2 41.60 7.50 1.35 5,500 0.300 O. 298 TR3 48.50 10.50 1.50 6,500 0.42 1 0.416 TR4 34.90 2.00 1.30 5,ooo 0.462 0.461 TR5 58.00 11.60 2.00 13,000 0.472 0.464 TR6 36.50 1.95 1.30 4,500 0.482 0.48 1

Fig. 3.12 Layout of tower TRI (dimensions m in m)

49

Fig. 3.13 Layout of tower TR2 (dllnensiom are in m)

Fig. 3.14 Layout of tower TR3 (dimensions

10.3

are in m)


Fig. 3.16 Loyout of tower TRS ( M o n s are in m)

Fig. 3.17 Layout of tower Ta6 (dimensions are in m)

The layouts of the six towers includiag the basic dimensions, section categories are

iuustrated in Figs. 3.12 through 3.17. It should be noted bat member subdivisions and

other attachments are not shown in these figures.

3.5 Modding and Anaiyds of Transmission Tonen

Transmission towers are modeled as three-dimensional lanice structures, in the

sarne way as the telecommunication towers (discussed in seaion 3.2). The main legs are

modeled as â m e elements while al1 other elements an modeled using tmss elements. In

modeling the cable, however, two-node tensionsnly ûuss elements with prestresshg

force are used. The tower main legs are assumed to be pinned on rigid foundation. Theû

mass and that of their tributary members are lurnped at the corresponding leg joints includ-

ing the mass of truss members having intermediate joints. When modeling the coupled

tower-conductor system, the mass of each cable element is cdcdated and divided equally

on its two end nodes. However, when replacing the conductor with an equivalent mass,

this rnass is lwnped at the appropriate cross-am joint or at the top of the tower for the

case of the overhead ground wire.

The coupled towerconductor system problem involves geometric non-Linearity

due to large displacements of cable joints and therefore requins mon computation effort

than the tower alone. The modal rnethod of analysis is no longer vaüd and a stepby-step

procedure is used. However, after removhg the cables, the tower alone is treated in the

sarne mamer as telecornmunication towets.

3.6 Earthquake Records Used in The Study

A set of 45 strong motion earthquake records obtained âom 23 different events, as

listed in Table 3.5. is used in the present study. This set of accelerograms was previously

used by Tso et al. (1992) in an investigation of the importance of the peak ground accel-

eration to the peak ground velocity ratio (AN) as an indicator of the dynamic characteris-

tics of an earthquake.

The data set was compiled from earthquakes that occurred in many places in the

world to include a vast range of seismological conditions. The effect of localized soi1

conditions is not included in the present audy as ail sdected accelerograms were recorded

on rock or stiff soil sites. It should also be noted that both near-field and far-field records

were included in this set, and dl records have a peak ground acceleration equal to or

p a t e r than 0.04 g.

Three groups of records were identified in accordance with their maximum AN

ratio, namely low, for records with AN < 0.8 @/S. intermediate, for records with 0.8 g 5

AN < 1.2 g/rn/s and high, for records with A N > 1.2 gWs. A complete description of the

earthquake component, magnitude, maximum ground acceleration, maximum ground

velocity, source distance and soil conditions can be found in Too et al. (1992). A graphi-

cal representation of the 45 accelerograms together with their response spaxra (evaluated

for 3% damping) is presented in Appendix A.

These records are used as both horizontal and ver t id base accelerations. The

records are not scaled or adjusted in any way in the analyses. Howwer, in the case of

vertical aderations, the acceldon values are reduced to 75% of th& originai values

to be consistent with the recommendations set forth by muiy design codes for

safety-related nuclear structures and building codes (ASCE 4-86 and NBCC 1995). A

distinct set of 55 vertical earthquake components coilected from 17 different events is alw

used in this study, a list of earthquake names, magnitudes and dates can be found in

Appendix C.

Table 3.5 - Earthquake records used in the shidy

Earthquake Mapitude No. of records Date Long Beach, California M d . 3 2 10/03/1933 Lower Califoda M L ~ . 6 1 30/12/1934 Helena, Montana M ~ 4 . 0 1 3 1/10/1935 Imperia1 Vailey, California M ~ 4 . 6 1 18/05/1940 Kem County, California San Francisco, Caüfornia Honshu, Japan Parkfield, Califomia Bonego M a , California Near E. Coast of Honshu Lytle Creek, CalSomkt San Fernando, California Central Honshu, Japan Near S. Coast of Honshu

Mp7.6 ML=S .4

M ~ 4 . 4 ML=S .6 ML+. 5

Japan MU=7.9 ML4. 4 M t 4 . 6

Mw=5. 5 Japan M d 7 . 0 9

Near E. Coast of Honshu, Iapan &=S. 8 1 1 1/05/1972 Near E. Coast of Honshu, Japan MM=7.4 1 1 7/06/1973 Near E. Coast of Honshu, Iapan &=6.1 1 1611 111974 Oroviile, Caiif'omia ML=S. 7 1 1/08/1975 Monte Negro, Yugoslavia M&. 4 1 9/04/ 1979 Monte Negro, Yugoslavia ML=7. O 1 1 5/04/1 979 Banja Luka, Yugoslavia Mp6.1 1 131081198 1 Michoacan, Mexico -8.1 5 19/09/1985 Na)ianni N.W.T., Canada Nk6.9 1 23/12/1985

MJMA = lapan mctcOrologicai a g a q scdc ML = Id magnitude Ms = surfact wave magnitude

CHAPTER 4

EARTHQUAKE AMPLCFlCA'MON FACTORS FOR

TELECOMMUNICATION TOWERS

4.1 Introduction

Earthquake amplification factors for self-supporting lattice telecommunication

towers are suggested based on a numerical modeling study perfomed on the ten towers

useci, each being subjected to the set of 45 accelerograms. The dynamic analysis is carried

out using modal superposition method with a unifonn damping ratio of 3% of the critical

viscous damping for al1 modes considered in the analyses as mentioned before. Each towa

is analyzed twice under the effect of the same earthquake, once applying the accelerogram

in one of the two principal horizontal directions, and then considering 75% of the same

accelerogram acting in the vertical direction to be consistent with the NBCC 1995. From

these simulations the value of the base shear and total vertical reaction are obtained.

Simple regression analyses are performed on the results nom which horizontal and vertical

amplification factors are found.

These factors are presented as fùnctions of the tower's largest flacutal period or

liugest axial period of vibration as appropriate. When multipüed by the tower mas, these

factors can be used by designers to estimate the expected level of base shear and vertical

reaaion developbd in ~e~supporting telecommunication towers due to an earthquake

event. Vertical earthquake amplification factors are presented in Appendix C using a

distinct set of 55 vertical eaRhqMke records.

4.2 Metbod of Andysis

The modal superposition method is used for the analysis, as irnplmented in the

commercial software SAP90. The number of modes considered varied with each tower, the

main concem being to ensure that at least 90% of the total mass is panicipating in the

horizontal direction and 85% of the total mass is participating in the vertical direction. The

damping ratio is taken as 3% of the critical viscous damping and kept constant in al1

modes included in the analysis. It is important to note that the results reported in this

chapter are those of the dynamic analysis without including the static response due to

seKweight .

4.3 The Use of NBCC 1995

It is paramount to realize that there are important differences between the behavior

of buildings and that of self-supporting telecommunication towen. While mon buildings

respond to horizontal earthquakes essentially in their lowest lateral mode of vibration, it is

not the case for self-supporthg towers whose lowest three flexural modes are usually

significanti y excited.

To illustrate this point, the ten towers used in the study were dynamically analyzed

for their base shear values ushg the NBCC 1995 design speanim shown in Fig. 4.1, which

has three zones Za<Z, Z.=Z and Zt>ZY, where 2. is the acceleration-related seismic zone

and 2, is the velocity-related seismic zone. The modal superposition method was used in

these analyses and two sets of base shear values were dcuîated, the fh cons ide~g ody

the lowest flexuraî mode while the second included the effects of the lowest three flexural n

modes. It shouid be noted that although the NBCC design speanun is giwn for 5%

damping, it was used in this study for illustration purposes ody. The base shear values

obtained tiom these analyses are given in Table 4.1 for a peak horizontal ground velocity v

= 1 mis. As expected, including only the lowest flemal mode greatly underestimates the

maximum base shear, the percentage of error ranging from 14% to 70°h . This confirms

that it is essential to include at least the lowest thm flexural modes in the analyses in

order to capture the response of these structures under horizontal excitation.

O t

O 0.5 1 1.5 7

Penod, s

Fig. 4.1 NBCC 1995 acceleration design specvum evaiuated for 5% damping

Table 4.1 - Base shear values using the NBCC 1995 design spectnim

Base shear @cNl Base shear [kNl considering thne f i e d modes conside~g the lowest flexufal mode

Tower Z A ~ V ZA=Zv ZA>ZV ZAC& &=ZV ZA>&

To fiirther prove the inability of static methods used in builcihg codes to predict

the seisrnic response of towers. the base shear values were also calcuiated for the ten

towers ushg the static approach of the NBCC 1995. The lateral elastic seisrnic force, Y,, is

given by the foilowing expression:

Y, = VSIFW

w here

V,= elsstic base shear in N

v = zona1 velocity ratio

S = dimensionless seismic response factor

F = foudation factor

W = dead load in N.

In these caicuiatioas the importaace factor, foudation factor and zonai velocity ratio were

taken as Umty. The results are given in Table 4.2. Comparing these base shear values with

the resuits obtained fiom the response spectmm analysis presented in Table 4.1, it is clear

that the static code expression consistently overestimates the tower base shear values.

Table 4.2 - Base shear values (in kN) using the NBCC 1995 static approach

ZA<& ZA=Zv ZA>ZV Tower W S VI S Ys S v' TC1 3.3E+04 2.100 70 3 .O00 100 4.200 140

From the previous discussion it is concludeâ that it is necessary to perfom

dynamic analysis in order to hclude the lowest thme flexural modes as the base shear

expressions recornmended by building codes consider the fbdarnental lateral mode only.

It is dso evident that the building code static approach is not suitable for

tdecommunication towers. Moreonr, most building codes do not account for the effects

of the vertical ground motion, which may be important for telecommunication towen.

Therdore, it is necessary and useful to derive earthquake ampüfication fàctors spaaficaiiy

for seKsupporting tel~~~mrnunication towers.

4.4 Horizontai Earthquake Excitation

in order to study horizontal force ampiification factors, each tower is subjected to

each of the 45 earthquake records selected. From this anaiysis, the resulting shear force at

the base of each tower is recordeci. The base shear vaiues are then plotted versus the peak

ground acceleration, peak ground velocity ancl W y A N ratio. The purpose is to tuld the

correlation, if any, between the base shear and each of these characteristics of the ground

motion.

The results obtained for the ten towers are shown in Figs. 4.2 to 4.11. It is seen

that the ten towers follow the same trend and that strong correlation exists between the

base shear values and the peak ground acceleration of the earthquake. Furthemore, the

relation follows a bear trend. Therefore, linear regression anaiyses were petformeci on the

results obtained for each tower in order to find a relation between the base shear values

and the peak ground acceleration. It is observed, however, that the correlation of the base

shear does not seem to be as good with the peak ground velocity (Figs. 4.2. b to 4.1 1 .b)

and that practidy no correlation is found with the AN ratio (Figs. 4.2.c to 4.1 1 .c).

For each tower, the value of the maximum base shear is divided by the tower mass

to obtain a dimensionless factor relating the peak ground acceleration and the dynamic

magnification values. Regession analyses an then performed on each of the three

predefined groups of accelerograms narnely, low, intermediate and hi& A N ratios, as

weli as on the entire set. The resuhs of these analyses are !ummuhd in Table 4.3.

Table 4.3 - Linear repsion analysis for the maximum base shear

Low A N Inter. A N Hi& A N Entire Set Tower Slope R2 Slope R' Slope R' Slope R'

1.71 0.67 1.63 0.69 1.40 0.70 1.84 0.91 1.67 0.84 1.37 0.96 1.58 0.67 1.48 0.89 1.00 0.70 1.59 0.55 1.61 0.80 1.45 0.93 1.44 0.70 1.58 0.83 1.00 0.78 1.3 1 0.70 1.36 0.93 0.86 0.56 1.69 0.80 1.47 0.85 0.96 0.87 1.21 0.63 1.21 0.87 0.80 0.81 1.08 0.86 1.07 0.87 0.76 0.67 1.20 0.89 1.04 0.83 0.67 0.92

R value: d c i e n t of correlation

Fig. 4.2 Maximum base shear values for tower TC 1

Fig. 4.3 Ibximum base shear values for tower TC2

Fig. 4.4 Maximum base shear vahies for towa TC3

67

Fig. 4.5 hdaximum base shtar vaiues for tower TC4


Fig. 4.7 Maximum base shear vaiues for towa TC6

Fig. 4.8 Maximum base s h w vaiues for tower TC7

Fig. 4.9 Mhxhum base shear values for tower TC^

AN. pimls

Fig. 4.10 Maximum base stiear vaiues for towa TC9

Fig. 4.1 1 M d w n base shear values for tower TC 10

For each group of redts the horizontal force amplification factor (maMmum base

shear / tower mass x peak ground acceleration) is plotted versus the fundamental flewral

period of vibration of the corresponding tower, as show in Fig. 4.12. It is noticed that the

plotted data foUows a descending trend in which the tower with srnallest fundamental

flexural period bas the largest amplification factor. Penomiing h e u regression analysis

once again, relations between the horizontal force amplification factors and the

fundamental flexural period of vibration are obtallied. Multiplying these factors by the

mass, M, and the peak ground acceleration, A, the following expressions are obtained for

estimating the maximum base shear:

Vh = M x A x(1.94-0.74~ T f ) forlow ANratio,

C ; i = M x A x ( 1 . 9 0 - 0 . 7 7 x T f ) forintemediate ANratio,

C i , = M x A x ( 1 . 6 0 - 0 . 9 0 x T f ) forhigh ANratioand,

V h = M x A x ( 1 . 7 8 - 0 . 8 2 x T f ) fortheentiresa

where

= base shear, N

M = total mas of the tower, kg

A = peak horizontal ground acceleration, m/s2

7 j = fundamental Bexurai period of vibratioq S.

More discussion of these results follows in section 4.6.

(a) low A N (b) Intermediate AN A

Lowest flexural period, s

O 02 0.4 0.6 0.8 1 1.2 1.4

Lowest fiexu~al pcrioâ, s

(d) Entire set

O 0.2 0.4 0.6 0.8 1 12 1.4

Lowest flcxurai pcriod, s

L

3 1.6 Y s . 12 3 2 0.8

a

Fig. 4.12 Base shear ampüfication factors

--.. - Eq. (4.5) - +- Eq. (4.10)

- - .- -

4.5 Vertid Eirthquake Excitation

The towers are also analyzed considering the 45 earthquakes acting in the vertical

direction. The values of rnaxhnurn vertical r a d o n at the tower base are plotted versus

the peak ground acceleration, peak ground velocity and AN ratio as before. Figs. 4.13 to

4.22 show the results obtained for the ten towers, and it is seen that a similar trend is

followed by al1 towers. As in the case of maximum base shear the correlation between the

maximum vertical reaction and both the peak ground velocity and AN ratio does not

appear very strong. However, excludhg the results obtained for the record with peak

ground acceleration 1.05 g (Nahanni earthquake), the relation between the vertical

reaction and the peak ground acceleration follows a straight line.

Linear regression analyses are then performed and Table 4.4 swnmarizes the

results for the ten towers. Similar to the maximum base shear, the maximum verticd

reaction is dividexî by the pr0hCt of the tower mass by peak ground acceleration in order

to yield a dimensionless factor representing the importance of the dynamic ampiification of

the response.

Table 4.4 - Linear regression analysis for the maximum vertical reaction

Low A N Inter. A N Hipsi A N Entùe Set Tower Slope R2 Slope R' Slope R2 Slope R2 TC1 0.60 0.99 0.61 0.99 0.55 0.97 0.58 0.97 TC2 0.70 0.96 0.65 0.99 0.66 0.98 0.67 0.98 TC3 0.79 0.94 0.82 0.93 0.84 0.92 0.82 0.93 TC4 0.88 0.90 0.91 0.88 0.90 0.93 0.90 0.91 TC5 0.83 0.92 0.80 0.86 0.89 0.88 0.84 0.88 TC6 0.95 0.82 1.00 0.75 1-04 0.89 1.01 0.84 TC7 0.93 0.79 0.90 0.81 0.92 0.84 0.91 0.83 TC8 0.88 0.89 0.92 0.85 0.86 0.86 0.88 0.86 TC9 0.73 0.96 0.76 O.% 0.77 0.94 0.76 0.95

TC10 1.25 0.58 1.29 0.72 1.27 0.66 1.27 0.70 R value: coefficient af corrclaition

Fig. 4.13 Total verticai dynarnic reaction for towa TC 1

Fig. 4.14 Totai vertical dyaamic reaction for tower TC2

i 300 1 -a

f : I

g zoo - .LI

6 1 >

si- .a 300 - @

g

Fig. 4.15 Total vaticai dynamic reaction for tower TC3

Fig. 4.16 Total vert id dynemic readon for tower TC4

Fig. 4.17 Total vatical dynsmic reaction for tower TC5

C l

I m w @ & , O ""A O , &, l 1

Fig. 4.18 Toul vertical dynsmic rcaction for tower TC6

100 t n n

Fig -4.19 Total vertical dyManc reaction for tower TC7

- zoo .-

6 >

Acc. cmiS"2

Fig. 4.20 Total verticai d y d c r d o n for tower TC8

Fig. 4.2 1 T otnl venicai dyniimic reaction for towa TC9

Fig. 4.22 Total vertical dynamic reaction for tower TC10

The vertical force amplification factor is plotted versus the fundamental axial

period of the tower, for each of the four einhquake record groups, in Fig. 4.23. Contrary

to the maximum base shear response (see Fig. 4.13), the data foiiows an ascending trend

in which the tower with lowen fundamental axial period of vibration has the smallest

amplification factor. Linear regression analyses are perfomed folowing the same

reasoning as in the base shear case, in order to define relations between the vertical force

amplification factor and the fiindamental axial period of vibration of the tower. Mer

multiplying these factors by the tower mass, M, and the peak ground acceleration, A, the

following expressions are obtained for estimating the maximum vertical reaction:

Vv = M x A x (0.36 + 6.76 x T a ) for low AN ratio.

b: = M x A x (0.3 1 + 7.58 x T a ) for intemediate A/V ratio.

Z\=MxA x(0.31 +7.68xTa) forhighrnratioand,

Vv = M x A x (0.32 + 7.45 x T a ) for the entire set

where

Vv= total maximum vertical reaction, N

A = peak horizontal ground acceleration in m/t?

Ta = fundamentai axial period of vibration, S.

It should be noted that when applying these expressions to calculate Y , the peak

ground acceleration should not be multiplied by 75% as this constant is aiready included in

the results. A M e r discussion of these results is presented in d o n 4.6.

(a) Low AN (b) Intermediate A N 1.4 ! 1

1.3 - I

a 1.3 ,' L

1

J 1.2 - / 0 1.2 L /- - l ,'

/- 1 - /= 3 1.1 - Q 1.1 b

E 1 - y' . 8 5 .i I L J . C. . -0 = 0.9 - a 0.9 1

O *= 0.8 - f 0.8 !- I .O C) ,y+ E 0.7 - s" - Eq. (4.9) < 0.6 1 */ 4 Eq. (4.1 1) 1

0.5 * 1 0.5 L

First Avial Period. s First Axial Period, s

Fig. 4.23 Vertical reaction amplification factors

4.6 Dbcussisn

From the previous redts it is seen that dyaaaric force amplifications foUow

opposite trends for vertical and horizontal excitations, as a bction of hdamental period

of vibration. This point is clarifieci by studying the average acceleration response spectrum

evaluated for the four series of records for 3% damping ratio, and normaiized with respect

to the peak ground acceleration (Fig 4.24). This figure shows three distinct regions. The

first one is characterized by increasing spectral accelerations in the short period (h~gh

fiequency) range of 0.05 to approximately 0.15 s (4 to 20 Hz). The second one covas a

range of periods of 0.15 to 0.33 s (3 to 6 Hz) for which the spectral accelerations are

maximum and rem& more or less constant. FWy, the third region shows decreasing

spectral accelerations for longer periods of 0.33 to 1.25 s (lower frequencies 0.8 to 3 Hz).

This is in agreement with the kdings for the total dynamic vertical reaction resulthg fkom

the excitation of the fundamental axial modes of the tower, and the base shear resuiting

tiom the fhdmenta! flexurai modes of the towers respectively. It should be noted that

several design spectra follow the sarne ascending-descending trends around approximately

a period of 0.25 s (4 Hz), as reportai by Gupta (1992).

In general, the results indicate that ciass@ing the accelerograms with respect to

their AN ratio does not contribute to improve the accuracy of the estimateci values of

maximum base shear or vertical reaction. As these expressions are oniy for estimating the

level of the dynamic forces developed in towers due to horizontal and vertical earthquake

excitation, it is recommended that only the expressions for the entire set be used. The

dues of the maximum base b a r and totai vertical reaction estimatd using eqs. (4.5)

and (4.9) are dram in Figs. 4.25 anû 4.26, respectively, with the vahies obtaimd fkom

Fig. 4.24 Acceleration response spectra evaiuated for 3% damping

detailed dynamic analyses for TC3. Similar graphs are presented Ui Appendix B for the

nine remaining towers. in order to obtain an upper bound to the expected level of dynamic

forces, one standard deviation is added to the numerical factors of eqs. (4.5) and (4.9) to

yield the following expressions:

Vh = M X A x (1.91 -0.66~ 7") (4.10)

V, =Mx A x (0.36+8.01 x Ta) (4.11)

These upper bound values of the amplification &ors obtained are drawn in Figs. 4.25

and 4.26, for comp~soa with those obtained using the average parameters of the

regressions in eqs. (4.5) and (4.9), respectively.

Fig. 4.25 Maximum base shear vs. peak ground acceleration in TC3

Fig. 4.26 Maximum dyMmic vertical reaction vs. pet& ground acceleration in TC3

4.7 Conclusions

In this chapter two expressions were proposed to estimate the values of the

maximum base shear and total verticai reaction of self-supporting telecommunication

lattice towers, baseci on the tower mass and fundamentai periods (lateral and axial) and on

the peak horizontal acceleration at the tower site. These expressions are valid for towers

with regular geometry and heights up to 120 m. They have not been vedied for taüer

towers but it is expected that the trends should be similar. Furthemore, it is important to

realize that these expressions are not proposed for detailed member design purposes but

only as an approxirnate method to assess the seisrnic sensitivity of towen in the

preliminas, design phase.

These simple seismic response îndicators may help tower designers decide whether

dynamic loads are ikely to infiuence the 6naî design and consequently whether more

refined anaiysis (either static or dynamic) is necessary.

As a fiirther aep in the seismic analysis of telecornmunication towers, the next

chapter proposes a simplified equivaient static method of analysis. The method will deal

with both horizontal and vertical eanhquake excitations.

CHAPTER 5

PROPOSED STATIC METHOD OF ANALYSIS

5.1 Thcoretid Background

As was mentioned in Chapter 1, this study is based on the modal analysis method

and the response spectrum approach. A quick review of the equation of motion, the

response spectrum approach and modal analysis theory is presented hereafter for

completeness.

5.1.1 Equatio~ of d o n and m o n s e spectmm

The equation of motion of a discrete parameter single-degree-of-fieedom system

(SDOF) under the effect of base excitation is given by:

mtï(t) + cti(t) + ku(t) = -rny,(t) (5.1)

where

ÿs = the base acceleration

m = rnass

k = stifhess

c = viscous darnping constant

4u and u = relative acceleration, velocity and displacement respeaively, with

respect to the moving base, and are bctions of t h e t.

This equation can be solved using Duhamel's integrai, and the solution is given by:

*) = -1-1: -m)$(*-C-)gin~)~(t - ~ ) d f ma, (5.2)

where

C is the viscous damping ratio =

cu and c u ~ = undamped and damped circular fiequency of vibration respectively, in

rads.

For small darnping ratios (ECO. 10, cuo = a>), this equation can be written in the fom:

u(t) = 1 y,(r)e)e-C@") sin o(t - r)& (5.3)

Taking the first derivative of (5.3) with respect to tirne:

ii(l) = Ioÿ,(r)e<MY cos* - s)dr - < J;j,(r)e-CNw) s i n w - r)dz (5.4)

Substituting the previous two equations into the forced equation of motion given by:

üf(t) = - W u ( t ) - cu2#(t) (5.5)

The expression for the absolute acceleration, rit , is then given as:

The absolute maximum values of the quantities given by (5.3), (5.4) and (5.6) are the

spectral relative displacement S&,T), the spectral relative velocity S,,(c,ï), and the

spectral absolute acceleration S&,T), respectively. It is now beneficiary to introduce a

new spectral value, the s p d pseudo-velocity S ' i , 2):

S&, ?) = I 1; y,(z)e-Ce) sin CO(t - r)dZ 1,

The relative spectral displacement is gben by:

tams of the pseudo-velocity as:

Sa(& 0 = SP(€Y n (5.9)

which is called the pseudo-acceleration and is denoted as Sp(&!'). The pseudo-

acceleration caa be used to estimate the maximum elastic force, f, developed d u ~ g the

earthquake using the relation k = ah:

f s = k Sd(CY 9 = m Spa(€. I) (5.10)

5.1.2 Modor anaîysis

The total displacement, u, of any linear system can be evaluated using the sum of

the modal components as follows:

U(X, t ) = u&, t ) + u&, t ) + UJ(X, t ) + ..... (5.11)

One can rewrite the previous equation so as to separate the two variables in the following

fonn :

where &) is a fùnction of position only, while Hi) is a fiindon of the only. The main

objective of this separation of variables is to tnuisform the continuous or the muiti-degree-

of-fieedom (MDOF) system to a set of uncoupled algebraif quations reprrsentiPg SDûF

systems. The total response of the system can be obtained by solving these equations using

the mode shapes of the total structure. Throughout the course of this work the cantilever

models were assumed to be continuous rather than lurnped parameter systems which

meaus that the mode shapes and mass distributions are given in c l o d form expressions.

Therefore, it is oniy necessary to fùrther discuss the seismic response of continuous

system.

5. 1.3 S&mk mpollse of conbiutous systems

The equation of motion of a continuous system cari be expressed in the following

fonn:

fdx, t) +f~(x, 1) +fs@, 1) = 0 (5.13)

The t e m in the previous expression are inertia force, damping force and elastic force

respectively. Restrictiag the displacement of this system tc a single shape, mode shape i

for instance, as describeci in section 5.1.2, and applying a Wtual displacement of the form:

du = )i @y& (5.14)

equation (5.13) is reduceà to:

fi,6y +/D& +f~.& = 0

where

f ~ j = I ; f i ( ~ t)#i(x)rfc (5.16)

~ D J = !;fo(x, t)#i(xw (5.17)

fss = Jifs(x, t)h(x)mc (5.18)

are the generalized inertia force, the generalired damping force and the generalwd elestic

force for mode i respectively. Both damping and elastic forces depend only on the relative

motion of the systern. However, imrtia forces depend on the total accelerstion and are

given by:

fi& 11 = m(xM i W ! O + m(x)ys(t) (5.19)

Substituting (5.19) into (5.16) the generalized inertia forces can be expresscd as follows:

fi, =fit) 1; #i(x)n,(x)#i(xW +y&) jk m(x)#i(xm =$(tWi +jJs(t)Li (5.20)

in which Mi and Li are the generalized mass and the excitation factor for mode i. The

expressions for the generaiized damping force and generaiized elastic force c m be written

in the foliowing foxms:

f~~ = Cd (5.2 1 )

~ S J = K a (5.22)

where

Cl= generalized dampiug of mode i

K= generaüzed stiffness of mode i.

Substituthg (5.20), (5.2 1) and (5.22) into (5.1 9, the equation of motion of the general-

ized SDOF representhg mode i can be expressed as:

M,y(r) + C d t ) + KA!) = -L&(t) (5.23)

dividing by Ml, substituthg Ci = 2o1(,A4, and neglesting the sign of the right-hand side,

the equation of motion of the SDûF system can be written in the foilowing form:

Using the response spectmm technique, the maximum acceleration profle of mode i is

given by:

and the contribution of mode i to the maximum elastic force acting on the structure is

given by:

It should be noted that the previous discussion is me for both discrete and continuow

systems with djustments in definitions For disçrete systans the mass, StifFness, and moâe

shapes are given Ui ma& and vector fom, while in continuous systems they are

expressed as hctions of the coordinate qstem used.

For continuous cantilevers, the modal shear force (Y , ) and bending moment (Mfi) at any

relative position x are given by:

L, VI&) = I; m(x) O&) M; Spo(Cr, Ti) d~ ( 5 -27)

and

These modal contributions can be combined using the SRSS method to yield an

estimate of the total maximum response. Considering the three lowea f l d modes, the

total response can be expressed as follows:

and

MAX) = v f ~ p , M ~ ( x ) ~

w here

V(x) = intemal shear force

MAX)= internai bending moment.

5.2 Proposeâ Method For HorizonW Euthquake Excitation

The proposed method is based on the definition of a horizontai acceleration profile

that c m give the same value of shear or bendhg moment at each tower seaion as the

value obtained tiom the SRSS combination of the thne lowest f l d modes. Fig. 5.1 is

a schematic representation of the concept of the proposed method.

For the evaluation of the acceleration profile, the self-supporthg tower is assumed to be a

continuous pismatic cantilever systmi with rigid base^ The mass profile and the lowest

three flexural modes of the tower are therefore assumed to be known in closed form. The

prediction of the fiexwal mode shapes and the mass profile is presented in the work done

by Sackmann (1996).

Using eqs. (5.29) and (5.30) the transverse shear and the bending moment can be written

in the fonn:

and

from which the expression for the acceleration profile a(x) can be extracteci. Mer investi-

gating several towers, it was found that using the acceleration profle obtained fkom

matching values of the bending moment, eq. (5.32), rather than the shear force, eq. (5.3 1).

gives better preâictions for the leg member forces. However, eq. (5.3 1) is more accurate

for diagonals and horizontah. Since the seismic forces developed in horizontals and diago-

nals are usually low and therefore not Uely to govern th& design, it was decided to use

a@) obtained fiom eq. (5.32). Accordingly, the expression for the acceleration profile a(x)

is given by:

where the arefiicients al to a6 depend on the lowest three flexurai mode shapes and the

mess profile of the tower. The values suggested for these coefficients are disaissed in

detaii later.

In order to dernonstrate the potential of the proposed methoci, results obtained for

Tower TC9 are reported hereafler. It should be noted that the coefficients al to as of eq.

(5.33) were caicuiated using the actuai mode shapes and mass proiiie of the tower.

The tower is analyzed using three differeat earthquake accelerograms acting

horizontaîiy dong one p~cipa l direction. The earthquake records used are San Fernando

(1971) N61W for low AN ratio, California (1952) S69E for intemediate AN ratio, and

Parkfield (1966) N65W for high A N ratio. The tower was fkst analyzed using the

sohare SAP90 utiiizing the response spectnun option and including the lowest thra

flexural modes of vibration.

The acceleration profiles were then calculated ushg eq. (5.33) for each eanhquake

record and are given in Fig. 5.2. An examination of this figure shows that the resulting

acceleration profiles do not have the same shape for ail accelerognuns. For the San

Fernando and Caüfomia earthquakes (low and intemediate A N ratio) the acceleration

profile does not change sign dong the tower height. However, the acceleration profile of

the Parkfield earthquake (high AN ratio) is characterized by a change in sign. The values

of the equivdent lateral forces are obtairied by multiplying the maos and the acceleration

profiles. The tower is then analyzed statidy under the effect of these lateral forces. A

cornparison between the member forces obtained using response spectnim d y s i s and the

proposed method is given in Fig. 5.3. It should be noted that for Tower TC9, mernbers 63

through 69 are leg members. Mer m e m h cire typicai horizontais and diagonals.

E San Fernando

Height, m

Fig. 5.2 Acceleration profiles for tower TC9

100 !i San Fcmindo. M

Mernber index

Fig. 5.3 Cornparison between member forces obtained from dywnic analysis

and the proposed rnethod

These resuks show that the forces predicted in the main legs by the proposed method are

in agreement with those obtained using dynamic anaiysis. For some of the lightly loaded

members (other tban the leg members) the dinirence in the two sets of predicted forces is

as high as 30°h. However, the design of these members is not iikely to be govemed by this

load-

To generalize the procedure, the coefficients a, to ab which are fùnctions of both

the mode shapes and the mass distribution of the tower, should be assigned values based

on the general configuration of the towers. To do so, it is suggested that the mode shapes

proposed by Sackmann (1996) for the Merent tower categories be used in calailahg

these coefficients, except that some modifications are made to the lowest flexural mode.

T hese changes are made because of the fact that Sackmann (1 996) used the same predic-

tion for the lowest flexural mode of the three groups. In order to obtain the maximum

possible accwacy it was found beneficiary to use a Merent prediction for each group,

which is still within the bounds suggested by Sackmann (1996). The lowest flexural mode

of the three categories defined in Tables 3.2 and 3 -3 is as foilows:

((x) = x2.* for group Al (5.34)

#(x) = xU for group A2 (5.35)

#(x) = xZofor group B (5.36)

A clos&-form expression for the mass distribution is also necessary to evaluate the

ai coefficients for each category. The mass distribution profiles are found from curve

fitting the mass distributions of the ten towers used in this study taking into ac«wit oniy

the shspe of the maps distribution, i.e. ushg normaiized fuictions. The following expres-

sions are used to evaluate the ai coefficients:

m(x) = 1 - 1.24 x + 0.37 x2 for group Al (5.37)

m(x) = 1 - 1.54 x + 0.87 x2 for group A2 (5.38)

m(x) = 1 - 0.94 x + 0.24 x2 for group B (5.39)

Figs. 5.4 to 5.6 show the lowest three flexural modes of the three categones on the lefi

ordinate while the mess dimibution used in the analysis is plotted versus the right

ordinate. The abscissa represents the relative height x (considering total height equals

unW.

Using the above expressions and equating the bending moment, MAX) of eq. (5.32), at

dEerent levels of the cantilever model, the ai coefficients were evaluated ushg the Mathe-

matica software. The coefficients are plotted in Figs. 5.7 to 5.9.

In summary the following seven seps are suggested to find the equivalent member forces:

1. Calcuiate the lowest three flexural penods of vibration of the tower.

2. Evaluate the mass distribution dong the tower's height and lump the mass at leg

joints.

3. Define a design spectrum suitable for the tower site.

4. Determine the corresponding pseudo-acceleration values for each of the three

lowest f l e d naturai penods and damping ratio ( S',, S' and Sm).

5. Calculate the acceleration profde a@) using eq. (5.33) and the appropnate a,

coefficients using Figs. 5.7 to 5.9.

m & 1 a d 3 + made 2 f mass

4-5 I l -0.5 O 0.2 0.4 0.6 0.8 1

Relative height, x

Fig. 5.4 Laterd mode shapes and mas distribution for Group Al

Relative height, x

Fig. 5.5 Laterd mode shapes and rnass distri ion for Group A2

Relative height, x

Fig. 5.6 Lateral mode shapes and mass distribution for Group B

Relative height, x

l I

-0.2 I L ! 1 I l I O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Relative height, x

Fig. 5.7 The cl, coefficients for Group Al

Relative height, x

Relative height, x

Fig. 5.8 The ai d t i e n t s for Group A2

Relative height, x a)

O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

b) Relative height, x

Fig. 5.9 ïhe ai coefficients for Group B

110

6. Muitipiy the lumped mass at each level by the conespondhg horizontal a&

eration d u e to find the quivalent lateral hertia force profile.

7. Analyze the structure statidy using thes lateral inertia forces as external

loading.

5.3 Verification of the Proposed Metbod

The ten towers used in the study are used to v e w the proposed simplified

method. This is achieved using response specmim analyses petfomed on the towers under

the eEm of the 45 earthquake records separately. The cornputer program SAP90 is used

in this part implementing the option SPEC. The earthquakes are assumed to act dong one

of the principal horizontal directions.

Each tower is classified into groups Al, A2 or B and the corresponding set of ai

coefficients is chosen. The spectral acceleration values corresponding to the lowest three

flexural periods of the tower under investigation are obtained fiom the response spectrum

of the earthquake record used. A set of horizontal forces is then calculated foilowing seps

5 to 7. The intenial forces obtained fiom the equivalent static analysis are then compared

with those obtained using the SPEC option in SAPPO. Fig. 5.10 shows a cornparison

b ~ e e n the results obtained Born both analyses and the percentage of error for each leg

member force. These results are for tower TC1 subjected to three selected earthquakes

&y: Michoacan, Mexico 1985, NOOE component (L14). Kern County, California

1952, N21E component (N3) and San Fanando, Caüfomia 197 1, S74W cornponent

(H8). It shodd be noted that these records an selected to repr*rcat the three

AN categories mmely low, intermediate and high. Sirniiar groups of figures are presented

for the other towers in Appendix D.

As it can be s e m from these figures, for the same tower, the percentage of error

Mers fiom one earthquake record to another. For the same earthquake record, the

percentage of error ciiffers fiom one tower to another. Ushg Merent earthquake records

for the same tower resulted in error due to the use of different spectral acceleration values

for the lowest three flexural modes. It was noticed that as the spectral acceleration values

corresponding to the second and third flexural penods hcreases in cornparison to that of

the first f l e x d period, the percentage of error also inaeases which is usuaîiy the case for

records with high AN ratio. This is due to the faa that each of the lowest three flexurai

modes contribute to the final inertia force profile, hence the tower nsponse and the degree

of this contribution depends on the spectral acceleration values assigned to each of these

modes. As the a c m c y in predicting the second and third mode shapes is less than that of

the lowest flexural mode, assigning the former two modes higher spectral acceleration

values wiii result in an increase in the percentage of emor. For the same earthquake, the

ciifference in the percentage of error from one tower to the other is due to percentage of

error between the tower aaual mode shapes and the mode shapes predicted for its

category, the latter having been used in evaluathg the af coefficients.

5 i 6 SPEC 1

h 4

rr' i 3 s

1

O

a) Earthquake record L14 35 r 1 6

l

SPEC 1 5 - I Rops I4 Y

b) Earthquake record N3 200 , 10

c) Earthquake record H8

Fig. 5.1 O Member forces for tower TC 1

Another swce of error is the mas distribution used in evaluating the a, coefn-

cients. As it can be seen fiom Figs. 5.4 to 5.7, the mass distriution along the tower haght

is approxhated as a smooth average curve. However, the w a l mass distribution is

rather discontinuous with heigbt, with discontinuities at the location of horizontal panels.

As the percentage of error varies from one earthquake to another and fiom one

tower to another, it was found beneficiary to investigate the percentage of error when

considering a smoothed response speanim. To this end, the NBCC 1995 response

spectrum was used in the analysis. Each of the towers used in the study was analyzd

assuming the three regions defined in the code, namely Z.cZ,, Z.=Z, and Z,%. The

towers were analyzed using the response spearum method and the proposed static

method of analysis, and the member forces were evaluated. A comparison between the

forces calculateci using both methods was made in Fig . 5.1 1 for tower TC 1. The results

obtallied for other towers are presented in Appendix D. From this comparison it is

concluded that the maximum expected error using the proposed static method with a

smoothed response spe*rum similar to the one used in NBCC will not exceed 25% in the

extreme cases, and the average error is 17% (Figs D.9 to D. 1 6). The maximum e m r was

found to be with tower TC9, this ciifference is due to the fact that thio tower has a mas

distribution which is not typicaî of the other towers studied.

Fig. 5.1 1 Member forces in tower TC 1 uskg NBCC spearum

5.4 Vertid EarthquiLr Excitation

in a preLimiaary study perfonned by Mihis (1994), it was concluded that the

tower's axial modes of vibration are not likely to be excited and that vertical earthquake

accelerations have linle efféa on the tower's response. In that study, however, the towen

used were subjected to only t h earthquake records. It was decided therefore to mm-

ine these hdings with the towers used in this research and the set of 45 earthquake

records after scaling them to 34 of their onguial values as mentioned in Chapter 4. From

these simulations it was found that although vertical earthquake accelerations do not

produce forces of the same order of magnitude as those due to horizontal accelerations,

the additional induced forces in the main legs rang& form 0.8 to 2.4 times the forces due

to the tower sdf-weight.

From this study it was also found that only the firn axial mode of vibration is likely

to be excited by vertical earthquake accelerations, at h a for the range of frrquency

content of the accelerograms considered.

The modal method of d y s i s and the response spearum techniques are also used

for the study of the vemcal earthquake excitation. A vertical acceleration profile is

proposed which depends on the tower's first axial mode and rnass distniution.

The fundamental axial mode shapes of the towers used in this mdy are found to

be very close to each other. Therefore an average function was selected, fiom m e

fitting, to represent the mode shape. The lowest axial mode is then given by the folowing

expression:

#(x) = 1 . 4 ~ + 0.4&r2 - 1 . ~ + 0.28r4 (5 .40)

where

4 = first axial mode shape

x = dimensionless coordinate measured fiom tower base.

An average profile for the mass distribution of the towers used in this stdy was

also adopted to represent the mass distribution of typical towers. The foiiowing is the

expression used:

ni@) = 1 - 1.11~-o.21x2 +0.57x3

w here

ni@) = mass at position x.

It should be noted that contrary to the study of horizontal earthquake excitation, the

towers were not divided into three groups. This is due to the faa that the fundamental

ouial mode shape of the towers were found to be very close to each other. Hence, eq.

(5.4 1) is simply an average for the mass distributions of the towers used in the study.

The excitation factor and generalized m a s defined in eq. (5.20) for the first axial

mode were calculateû, using the previous two expressions, and found to be L = 0.25 and

M = 0.17.

The vertical acceleration profle for the tower is then evaiuated using eq. (5.25)

with the value of the generaiized rnass and excitation fmor for i=l only. The resulting

vertical acceleration profile is given by:

a(x) = S. x (2.0% + 0 . 7 0 ~ ~ - 1 .709 + 0.41x4) (5.42)

where

a(r) = verticai acce1eration at position x

S. = spectral acceldon value.

Fig. 5.12 shows the vertical acceleration profile evaluated for a unit value of S..

O 0.2 0.4 0.6 0.8 1

Relative height, x

Fig. 5.12 Proposed vertical acceleration profile

The vertical acceleraîion profle takes the shape of the fundamental axial mode multiplied

by 1.47 (the ratio of the excitation factor to the generalized mas) as impüed in its deriva-

tion. If in the funire vertical design spectra and peak vertical ground accelerations are

included in design codes, this acceieration profle should be verified as more axial modes

may be significantly excited.

The solution steps for the vertical euthquaLe excitation do not dEer much from

the horizontal case, however they are prcsented here for completeness:

1. Caldate the fundamentai al perbd of vibration of the tower.

2. Evaiuate the mass distribution dong the tower's kght and lump the mas at leg

joints.

3. Detemine the corresponding pseudo-acceleration value (S.) for the natural

period evaluated at step 1 and the appropnate damping ratio.

4. Calculate the venical acceleration profile a(x) using eq. (5.42) or Fig. 5.12.

S. Multiply the lumped mass at each level by the conesponding vertical accelera-

tion value to h d the equivaient inertia force profile, acting in the vertical

direction.

6. Analyre the structure staticdy using these vertical inertia forces as extemal

loading.

5.5 Verifkation of the Proposcd Vertical Acceleration Profile

To verify the proposed acceleration profile, the towers used in the study were first

analyzed dynamicdy considering the lowea axial mode of vibration and assuming a

constant spectral acceleration vaiue equal to unity. Then the towers were analyzed again

using the proposed static method of analysis and considering the spectral acceleration

equai to unity. This procedure was chosen since only the lowest axial mode is considered.

Therefore, the response of the towers to different earthquake records can simply be qua1

to the response resulting fiom the unit spectral acceleration multiplied by the correspond-

hg spearal acceleration of the earthquake record.

A cornparison between the forces developed in the main leg manbers for towa

TC 1 using both methods is presented in Fig. 5.13; s d a r figures are presented in A p p d -

ix D for the rest of the towers. From these figures it is seen that the proposed acceleration

profile yields excellent results for ail the towers with a maximum ciifference of *100/o and

an average error of 2% only.

Fig. 5.1 3 Member forces in towa TC 1 under vertical excitation

5.6 Eff~ct of Coaccntrated Mms of Antennae

The results presented in the previous sections were evaluated considering the bare

towers without accounting for the effkcts of antenna clusters. The coefficients a, to as

presented for the horizontai earthquake excitation as weil as the expression for the vertical

acceleration profile, eq. (5.42), were obtained ushg the mode shapes of the ban towers.

This approach was justifid due to the fa* tht the arrangement, position, weight and

number of antennae vary sigmficantly firom one tower to mother. As the mode shapes and

mass distribution will chaage to various degrees with the addition of a new antenna it was

found i m p d c a l to consider al1 the possibilities and inchide them in the proposed

metbod. Therefore another approach was adopted to account for the presence of antennae

in the anaiysis. It is proposeci to use the a, coefficients (or eq. (5.42) for vertical excita-

tions) defïned for bare towers and account for the presence of the antennae by simply

multiplying its mess by the comsponding acceleration while taking its eccentricity into

consideration.

After hcluding the antennae masses, the telecornmunication towers were andyzed

using the proposed approach. The member forces evaluated for towen TC3 (shown in

Fig. 5.14 with antennae arrangements) are shown in Figs. 5.1 5 and 5.16 ushg the NBCC

1995 spectrum for Z.<Z in the horizontal and vertical directions respectively. These

figures show agreement between the forces evaluated ushg the two approaches with a

difference of 7% in average in the case of horizontal excitation and only 3% in the case of

vertical excitation. It was found that using the same approach for towers with heavy

antennae, specidy if' the anteme are located at the top of the tower, will result in ovetly

consemative results. This is due to the tact that the mode shapes and mass distributions

used in evaluating the a, coefficients and eq. (5.42) will change sigdcantly. However, this

concem should have less and less importance in the fUtm as antennae used in modem

digital transmission have a vexy srnail diameter and are lightweight comparai to conven-

tional rnicrowave d m s .

Fig. 5.14 Tower TC3 with antennae

Fig. 5.15 Member forces incIuding the effect of anten~ mess for tower TC3

ushg NBCC 1995 for ZaY ratio in the horizontal direction

Fig. 5.16 Member forces Uicluding the effect of anteana mass for tower TC3

using NBCC 1995 for Z,Z, ratio in the vertical direction

At this point it was found necessary to fùrther study the change in member forces

and base shear due to horizontal earthquake excitations when adding heavy antemue. As

was mentioned earlier, the location, orientation and mass of these anteme depend on the

importance and the main function of the tower. Two approaches were used to account for

the presence of these antennae. The first approach is by lumping an eccentric mass with a

value rmging fiom 5% to 100/o of the tower's total mass at the tip of the tower. The

second approach involves distributhg 10% of the tower' s total mass evenly on one Ieg of

the tower's top prismatic part. The eccentricities of these added masses will ailow to study

torsional effècts. The NBCC 1995 design spectm with its tbree regions was used again in

these analyses. To this end, the four cases studied are as follows:

1. The bare tower

2. The tower with a lmped mass at the top quai to 5% ofits total mass

3. The tower with a lumped mess at the top equal to 1% of its total mass

4. The tower with a m a s quai to 100/o of its total mass distributed dong the top

prismatic part.

For each of these four cases the base shear vaiues, the main leg forces and the forces in

selected diagonal and horizontal members are caicuiated. A cornparison between these

redts and those conespondhg to the bare tower case was then pesformed. Figs. 5.17 to

5.25 show the results obtained using the NBCC spectnim for Z.<Z.

0.9 :g .--. 2 s . 0.77

3 3 0.63 0.7 2 :::: 0.7 Ill E 3 0.77

8 -3 0.57 -3 0.63

4 0.5 a 0.57

0.5 0 3 7 = 0.43

0: E ~ a s c 2 m-3 0 3 0.37

0.1 P c r u ~ 4 023

B.S. O. 1

Fig. 5.1 7 Effkct of including heavy antennee on tower TC I

0.4 1 I I t

0.36 m a s e 2 I 1

0.31 0.36 / 0.21 11) m a s e 3 i 0.31 i

SCWJ i 0.21 - 0.07 5 ,

B.S. a 0.07

0 2 4 6 8 10 12 O 0.5 1 i -5 2

Force ratio Force ratio Main Legs D i q d s and f-iorizoatlls

b

Fig. 5.18 Effm of includuig heavy antennae on tower TC2

Foret na0 DLgoii.LdHorEtontils

b)

Fig. 5.19 Effkct of ÛtcIuding heavy antenaae on tower TC3

lil S.S. !

Forcc ratio Main Legs

Force rab0 Diagonds and Horizontils

b)

Fig . 5 -20 Effect of including heavy antennae on tower TC5

Fig. 5.2 1 Efféct of inctuding heavy amcanie on tower TC6

0.67 1) D

0.76 1

Bcasc2 0.37 1 m-3 0.55 , I Bca~c4

- B.S. ) 0.13 - I

t

O 5 1 O 15 20 O O S 1 1.5 2 2.5

Force ratio Force d o Main Legs Diagonais and Horiunitah

b)

0.96 0.93 0.89 0.86 0.83 0.79 0.76 0.73

f 0.69 WJ -6 0.66 & u 0.63 i+ .g 0.59 0 0.56 =

0.52 0.45 0.38 0.32 025 0.18 O. 12 0.05 B.S.

Fig. 5.22 E f f ' of induding heavy antennae on tower TC7

Fig. 5.23 E f f i of includiag heavy antennae on towa TC8

0.94

0.87

0.8

0.72

0.65

0.57 a ea 0.5

f 0.42 - - e 0.35 C

PC 3.27

0 2

O. 13

o. 1

0.06

0.03

B.S.

036 0.48 0.38 028 0.18 0.08 B.S.

O 1 2 3 4 5

Force ratio MaiJl L q s

O 0.5 I 1.5 2

Force ntio Diqods and Hor i zods

Fig. 5 2 4 Effkct of including heavy antennse on tower TC9

Force ntio -LeOs

b)

Ag. 5.25 Effect of inchidhg heavy antennae on tower TC10

From this study it was concludeci that adding the equivdent of antennae masses of

up to 10% of the tower's total mass consistently reduces the maximum base shear value as

a global response indicator. The average reduction is about 20% when compared to the

corresponding bare tower case. This finding is contrary to the intuitive belief (and miscon-

ception) that adding mass wül necessarily increase the inertia forces and hence the base

shear. The explanation for this reduction is that the addition of sigdcant masses to the

tower, and especially near its top, will tend to increase the natural penod of the tower

which in mm wiil reduce the s p w a l acceleration values corresponding to these modined

penods, as it can be seen on the NBCC 1995 spectrurn in Fig. 4.1. When compering leg

mernber forces (Figs. 5.17a to 5.25a). it is found that members near the base carry smaller

forces than the bare tower case. This reduction in leg axial forces near the base was found

to be 15% on average. However, this reduction becomes less significant when moving up

towards the tower's prismatic part near the location of the antennae. At some elevation,

fkom 0.5 to 0.7 of the height, the memben start carrying larger forces than in the bare

case. At the top level for case 3 (a lumped mass of 1% of the tower's mass) the leg

memben carry as much as 20 times the force calculated in case 1 . The same cornparison

was camied out considering some diagonal and horizontal memben (Figs. 5.1 7b to 5 .Zb)

but no clear trend could be identifid in the results near the base, except that the maximum

change never exceeded f 10°! when compared to the bare tower case. It is emphrisized

that the increase in the forces developed in the diagonal and horizontal members adjacent

to the location of the antennae is very signifiant.

From the previous discussion it foUows that including extremely heavy lilltennae

(up to 10% of the tower seSweight) to the top ptisniatic part of telecomm~c~tion

towers has two & i s : fkom a global perspective, it consistently reduces the base shear

values of the tower, but it increases the member forces l o d y near the antenna positions

and this increase tends to propagate downwards with the increase in the lumped mass

value.

The previous study shows that the proposed horizontal acceleration profile will

work well for towers having iightweigbt antemae, of mass of 5% or less of the tower

mas. However, for towers with heavy antennae (above 5%) a corrected evaluation of the

lowest three fl& modes, mass distribution and acceleration profile should take place

prior to the application of the proposed quivalent static method.

This study was not extended to cover the effects of heavy antennae clusters under

vertical excitation. The conciusions of such a case can be drawn ffom the information

available about the fundamental axial period of the towers used and the NBCC 1995

response spectmm presented in Fig. 4.1. From Table 3.1 it is seen that the fundamental

axial penod of the towers used ranges between 0.03 and 0.12 s, which corresponds to the

horizontal part of the NBCC 1995 spectmm (up to 0.2 s). This means that shifting the

tiuidamental axial penod within that range wili not nsult in a change in the comsponding

spectral acceleration value. Therefore, adding antennae to the tower and hence increasing

the total mass wüi resuit in a proportional increase in the total vertical reaction of the

tower. In addition, there wiU be a proportionai increase in the forces developed in kg

memben dong the tower's height as weii as in diagonals and horizontais near the

locations of niass concentration.

cluPTER 6

SEISMIC CONSIDERATIONS FOR TRANSMISSION TOWERS

6.1 introduction

Transmission tower designers lack simpMed methods for seisinic analysis.

However, before such simplified design methods are proposed, a better understanding of

the dynamic behavior of the wupled tower-cable system should be achieved. To gain more

insight into the dynamic response of a transmission h e system and to be able to simpliQ

its response it was found important to perfonn fiequency analysis on such systerns. As

presented in Chapter 2, most of the few published works in this area are devoted to the

response of the cables only. This study, however, is also devoted to the response of the

towers in the tower-cable system. The objective is to evaluate the feasibitity of simpüfyuig

the anelysis of the tower-cable system by replacing the cables by an quivalent rnass and

stifbess.

6.2 Mathematid Modeüiig

The line mode1 used in this onidy consists of six spans of cables attacheci to five

suspension towers. Towen TR2 and TR6 are 6rst used in this type of anaiysis in order to

investigate the possibüity of simpiifjhg the d y s i s , other towers wiiî be used in the

verification stage if successful. Two types of cables are inchdeci: the conducton and the

overheaâ grouad wires, the latter bang directiy anchored to the tower at its peak whüe

the conductors are suspended to the tower cross amis with iiuulator strings. The lmgth of

the M a t o r strings varies in accordance 4th the voitage of the he, typidy betweetl 1.4

13 1

to 4.0 rn for voltage of 120 to 735 kV, respectively. The cable ends which are not

attache. to the suspension tower in the end spans an asmed to be fixed. Fig. 6.1 shows

the typical components of a transmission iine segment used in this saidy. The following

two paragraphe address in more d*ails some modeling considerations for both the towers

and the cables.

61 1 M d i n g of towur~

The transmission towers are modeled as th-dimensional structures in which the

main legs are thtee-diniensionel fiame elements, waile tmss elanents are used for

modeling al1 the other horizontal and diagonal members. This choice of elanmts is used to

simulate, as closely as possible, the acnial behavior of the towers. It should dso be noted

that the model used in the analysis of the coupled tower-cable system is, in faa, a

simplified reduced model of the tower for which a detailed three-dimensionai model was

constnicted. The insulator strings are modeled using two-node truss elements.

6.2-2 Moddng of cables

The cables in each span are rnodeled ushg 20 two-node tension-only tmss

elements with initial prestre~r. The cross section of each cable is assumed cornant. The

material used for the cable elements is assumed hear elastic tension-only. The choice of

the number of tmss elements to be used in the finite element mesh is based on a

convergence test of the lowest modes of nirations of a single span cable; this approach

was used by McClure and Tinawi (1987). Fig. 6.2 shows a cornparison between the results

obtaineâ for the lowest transverse mode usirig diffaent munôers of elemeatq and

dcient acairacy is obtained with 20 elewats.

No. of elements

Fig. 6.2 Mesh convergence for the lowest transversal mode

fhree types of conductors were used in the parametenc study perfomed narnely,

CONDûR 54/7 ACSR (Aluminurn Conduaor Steel Reidorced), CURLEW 5417 ACSR and

BERSFORT 48/7 ACSR. Table 6.1 contains some characteristics of these conductors in

addition to those of the overhead ground wire (O.H.G.W.). The horizontal tensions used

were 10, 15, 20 and 25 kN for CONDOR , 25, 30- 35 and 40 IrN for both CURLEW and

BERSFORT. These values are specified at O°C. For the overhead ground win the

horizontal tension, H~KG.w. was calcdateâ using the foiîowing relation:

WOU.G. W. HoH-G.~. = L d . x

where

Hcond = conductor horizontal tension

WOJXW = overhead ground wire weight per unit length

w- = conductoc's weight per unit length.

Table 6.1 - Cable data

Cable Diamet er, E Weight, Cross Rated mm composite, N/m sectional teasile

MPa area,mm2 stren&kN C O W R 27.7 67,225 14.93 455 127 CURLEW 31.6 68,325 19.38 592 165

BERSFORT 35.6 67,600 23.23 747 180 O.H.G.W. 12.7 172,400 7.44 97 114

6.3 Methods OC Analysis

The computer program ADïhiA (Automatic Dynamic Incremental Nonlinear

Analysis) ADINA R&D (1997), is used in the anslysis of transmission lhe systems. SAP90

could not be used as it can only perform linear analysis. in order to achieve the goai of this

study two types of adysis are perfonned, which are:

1. A non linear static d y s i s under self weight to obtain the static equilibrium

position of the cables.

2. A frequency anaiysis in the initial configuration, to obtain the lowest flexural

modes of the towers and those of the entire cable-tower system.

more details on these analyses follow.

In this part of the simulation, the mode1 comprises five towers with six spans of

cables, and is analyzed under the action of self weight. Due to the geomehically noniinear

response of the suspendeci cables, the lanematics formulation used to obtain the stiffness

ma* of the system is that of large displacements and smaîi strains. The itauion mthod

used to solve the nonlinear problem is the fÙii-Newton 4 t h rdoda t ion of the stiffness

matrix at each itemion The convergence derion selected is baseâ on the displacement

vector nom. The tolerance used was 1.0'. The t h e b a i o n used to apply the

gravitational load is lineu with 100 t h e increments of 0.01 such that at time ~ 1 . 0 the

auil equiiibrium coafiguration of the line is reached.

63.2 Frequency anuiysik

Mer obtaining the cable equilibrium c o ~ ~ t i o n , the naturd periods and mode

shapes of the towen in the tower-cable system are calculated. To detennine the number of

modes sufFcient to capture the sigdcant response of the towers, a linear dynamic

analysis of the six towers (without cables) used in the study was penomed using three

sekcted horizontal earthquake records each having a Werent A N ratio. From this

prelunurary investigation, it was concluded that only the lowest two flexural modes are

participating in the response. It is noted that these towers are generally more ngid than

telecommunication towers for which the third flexural mode of vibration was also found

significant. Therefore, it was concluded that only the lowest two flexural modes in each of

the two main orthogoaal directions (longitudinal and transverse) should be obtained for

the coupled tower-cable system. The method used in fiequency analysis is the subspace

iteration. To obtain the required eigenvdues, a small frequency interval around the

corresponding bare tower fiequency was specified in order to reduce the calculation

effort.

6.4 Pyimetric Study

Eighteen cases were considered in perfonning the fkequency anaiyses, as

summarued in Table 6.2, in which the foilowing parameters an studied: span, cable

tension and conductor type. The cable span (distance between towers) varied from 300 to

400 m, the cable tension varied from 10 to 40 kN and three conductor types were used in

the simuiations. The type of the overhead ground wûe was kept the same, i.e. Grade 180

galvanized steel 12.7 mm in diameter, and its tension was caldated using eq. 6.1 .

Table 6.2 - Cases used in the parametric study

Case Conductor Area E Span Sag Tension initial arain no. type rn' MPa m m kN e

1 350 22.99 10 3.3E-04 2 350 15.28 15 4.9E-04 3 Condor 4.6E-04 6.7E+10 350 1 1.45 20 6. SE-04 4 350 9.15 25 8.2E-04 5 300 1 1.22 15 4.9E-04 6 400 19.97 15 4.9E-04 7 350 11.89 25 6.2E-O4 8 350 9.9 30 7.4E-04 9 Curlew 5.9E-04 6.8E+10 350 8.49 35 8.E-04 10 3 50 7.42 40 9.9E-04 1 1 300 7.27 30 7.4E-04

As was mentioned earlier only towers TR2 and TR6 are used in the ûequency

anaiysis. AU 18 cases are studied for tower TR2 wMe ody the ikst 12 (i.e. Condor and

Curlew) cases are used for tower TR6. The lowest two flexural paiods of viiration in

both the longitudinal and transverse directions are obtained and compareci to the

cotfesponding perioâs of the ban suspension tower.

For each conductor type the results are pooled WO groups: the first group for a

aven tension but with different spans (Figs. 6.3a to 6.7a), and the second group for a

given spm but with different tensions (Figs. 6.3 b to 6.m). These figures show the ratio of

the naturai penods calculated in the coupled tower-able system to the corresponding

values calculateci for bare towers. It is seen from Figs. 6.3b to 6.7b that for a given span

length changing the cable tension (in the range considered) has no significant effect on the

natural period of the syaem. This can be explained as the towers considered in the study

are suspension towers with balancd longitudinal loads. Since the conduaors are attacheci

to the towers' cross arms with insulator strings that can rotate fieely, the change in the

conductor's tension is not directly felt by the tower. Only the change in the tension of the

overhead ground wire is &ecting slightly the periods of the system.

However, varying the span length while keeping the cable tension constant resuited

in a significant change in the periods, as shown in Figs. 6.3a to 6.7a. The period incressed

(in most of the cases) with the increase in span, but this effect did not follow a predictable

trend. Another obsewation is that for a given span length there is no change in the

calculateâ natural periods fiom one type of conductor to another. This obsenation may

not be generalized since the physical characteristics (stifhess and mas) of the conductors

used in the simulations are of the wne order, and only quai and level spans an

considered. The lowest two longitudinal mode shapes of tower TR6 in the coupled systmi

are shown in Fig. 6.8 where it can be observed that the hsulator strings suspendllig the

conductors to the tower rotate while the conduaors' position does not chnge. This

means that the mas of the conductors d w s not participate in the longitudinnl mode

shppes. This obsavation, however, is not me when examinhg the tranmcfse m d a .

a) initial tension = 1 5 kN

125 r

1.15 - ' I 1st longitudinal O

.CI 1 ~LSttT8llsvent a II . a 2nd longindina; 1.1 - I 1 A2ndmnsvcr~

A 1

A

5 1 O 15 20 25 30

Tension, kN

b) Span length = 350 m

Fig. 6.3 Ratio between calcuiated naturai paiods of the coupled iine syotem and the bare

tower for towu TR2 with Condor

a) Mial tension = 30 kN

1 1

20 25 30 3 S 40 45

Tension, kN

b) Span length = 3 50 m

Fig. 6.4 Ratio between calculated naturai periods of the coupled lim system and the bue

tower for tower TR2 with Cwiew

a) initial tension = 30 kN

1 . ! ! I 1

20 25 30 35 JO 45

Tension, kN


Fig. 6.5 Ratio baween calculateci nstural periods of the coupled line system and the bue

tower for tower TR2 with B d o r t

a) Initial tension = IS kN

1.15 1 4

v.7 -- -

14 16 18 20 22 24 26

Tension, kN


Fig. 6.6 Ratio between caldated natunl periods of the coupled iine system and the bore

tower for tower TR6 with Condor

a) Initial tension = 30 kN

20 25 30 35 40 45

Tension, kN


l 1 st longihidind Istmsv«sc 2nd longitudinal

A f a d t ~ ~ v c r s c

Fig. 6.7 Ratio between calailateci naturai penods of the coupled h e system and the bue

tower for tower TR6 with Curiew

Fig. 6.8 The lowest two ccilcuiated longitudinal modes of tower TR6

144

This conclusion canot be generalized to angle or anchor towers where the

conductors are directly connecteci to the tower. For the transverse direction there is no

clear explanation to the bebavior of the system. Therefore, it is beiieved that at this

preliminary stage only the response in the longitudinal direction can be simpüfisd.

6.5 Equivaient M m

From the previous discussion, it is concluded that only the mass of the overhead

ground wire a f f i s the longitudinal modes of the suspension towers. However, Figs. 6.3

to 6.7 indicate that the trend foiiowed by the lowea two longitudinal modes is not linear.

Therefore, a single value for the participating mass will not satisfy all the cases and an

average should be obtained. The approach followed to obtain this equivalent cable mass

for the lowest longitudinal mode can be summarized in the foliowing steps:

1. Obtain the bare tower mode shape for the lowest iongitudinal mode

2. Assuming that the mode shape of the tower is not affected by the presence of

suspended cables and knowing the mass distnbution and fiequency of the bare tower,

calculate the generalized Stifbess of the tower using the foiiowing relation:

where

km = generaiized stiflness

mm = generalized mass, eq. (5.20).

3. Assuming that the genaalllsd &ess of the tower in the system is uacbanged

and knowing the new fiequency of the system in the same longitudinal mode, cildate the

generalized mass of the tower in the system.

4. Knowing the generalited mass and the period, daJite the parîicipating mass

of the overhead ground wire using eq. (6.2).

This approach was applied to the 30 cases considered in the parametric study, 18 for

tower TR2 and 12 for TR6, nom which it was wncluded that about 25 to 35 % of the

overhead ground wUe mass is panicipating in the longitudinal direction. Performing a

frcquency analysis of the tower alone afler replacing the overhead gound wke by 25 to

35% of its mass and neglecting the presence of the conducton showed that for ail the

cases considered this percentage range yielded acceptable results specialiy for the lowest

mode.

6.6 Conclusions

For the cases studied it can be seen that for suspension towen with level and qua1

spans the change in the cable tension had no significant effects on the period of the system.

It is also seen that only the mass of the overhead ground wire affects the longitudinal

modes of vibration of the tower. It is found that 25 to 35% of the overhead ground wire

mass contributes in the longitudinal modes of the system. In the transverse direction,

however, there is no trend foUowed that can enable us to assess the contribution of the

cables' mass to the transverse modes of the system.

At this point it was concludeci thai the response of the coupled tower-cable system

cannot be simpiified in such a mamer that wouid d o w the extension of the proposecl

simplified method for ~e~supportiag te~ecommur~ication towas to transmission line

towers.

It should be noted that these conclusions do not mean that the system is uncoupleci

or has only little coupiing. Therefore, in the case of seismic effects it could be m e that

there is oniy little coupling between the cables and the towers if the tower displacements

are smaiî, however, this need to be quaiifid in mother study.

CHAPTER 7

SUMMARY AND CONCLUSIONS

7.1 Summriry

The aim of this study was to provide tower designers with suitable tools to

evaluate the overail seismic response of self-supporthg lattice telecommunication towers

without perfonning detailed dynamic analysis. Ten existing three-legged seif-supporthg

steel telecommunication towers with height ranghg from 30 to 120 m were used, which

cowr the usual range of such towers erecteà in Canada. A set of 45 earthquake records

categorized with respect to th& maximum A N (peak ground acceleration 1 peak ground

velocity) ratio in three categories narnely: low, intermediate and h;gh was used as

horizontal seismic input. The same set of records was also used as vertical input after

reducing its amplitudes to % when studying the vertical response of telecornmunication

towers. The main findings and contributions can be SUM18fiZed in the following points:

Seismic amplification factors for base shear, eq. (4. IO), and total vertical

reaction, eq. (4.1 l), for self-supporthg lanice telecommunication towers are

proposed in a closed fom.

A simplifiecl quivalent static method is propo~ed to estimate tower member

forces due to horizontal and vertical earthquake excitation.

The effect of including heavy antenna clusters on the base shear and member

forces in self-supporthg telecommuniation towers is investigated.

An investigation of the eequency characteristics of transmission line towers is

d e d out. The objective was to investigate the fe8~1iility of simplifjhg the

response of transmission towers in the coupled tower-cable system and then to

extend the proposed simplified quasi-static method of analysis to cover this

application. Several obsewations are made from this study, but the main

conclusion is that the naturai fnquencies of the towers couici not be simplified

in a manner suitable for the direct application of the proposed equivalent static

method to transmission line towers.

The following sections recapitulate in some detail the major findings and

contributions of this research. Suggestions of fùture work and a statement of origuiality

are also presented.

7.2 Earthquake Amplification Factors

It is demonstrateci that the seismic behavior of self-supporthg lattice

telecommunication towers ciiffers signiticantiy from that of buildings. This is illustrateci

through a cornparison between the maximum base shear values obtained ushg the NBCC

1995 static approach and those obtained using the response spectrum method with the

NBCC earthquake speca~m. The building code static approach consistently overesthtes

the base shear values.

New simple expressions are proposed to estimate the maximum base shear, eq.

(4.10), and vertical r d 0 9 q. (4.11), due to horizontal and vertical earthquake

excitations. These expressions are based on the totaJ msss of the tower, the lowest flaniral

(or axial, as appropriate) pcriod of the tower and the peak ground acceleration spscified

for the tower site. The base reactions obtained fiom these expressions are not meant to be

redistributed dong the tower beight, as in the building code static approach. They are

dsmic response parameters that should indicate whether or not earthquake effécts may

govem the design. if seismic effects are likely to be significant based on these maximum

base reaaions, a more detailed, yet still simple methocl is proposed tu approximate these

effects on member forces.

7.3 Simplifieci Mcthods of Andysh

The research resulted in proposing two simple equivalent static methods for

seismic analysis of self-supporthg lattice telecommunication towers subjeaed to base

excitations in the horizontal and vertical directions.

In the first method, a horizontal acceleration profile is defined based on both the

response spectrum approach and modal d y s i s method, including the effect of the lowest

three fiexural modes of vibration of the tower. The main idea is to define an acceleration

profde which wiU produce the same e f k t on the tower as the combined effect of the

lowest three flexural modes. The proposed acceleration profiie has yielded very g o 4

results in estimating member forces with an average error of *7% and a maximum error

of es% when compared to results obtaiaed fiom nsponse specûum analysis.

In the second method, a verticai acceleration profle is proposed which ody

inciudes the lowest axial mode of vibration of the tower. The use of this acceleration

profile has dso produced very good resuits with a maximum enor of ody f 1% and an

average error of f2% ody.

The & k t s of the presence of heavy antenna clusters near the top of the towers are

also investigated both locally, on individual member forces, and globally on base ractions.

The main hdings are as follows:

Added mass consistently reduces the base shear (within the range of values

considered) with an average reduction of 22% compareci to the corresponding

bare tower case.

Leg member forces near the base are also reduced by 15% on average.

At relative heights of 0.5 to 0.7, force amplifications begin to appear in the leg

members.

Diagonal and horizontal members close to the antenna attachments experience

very large force amplifications. A factor of 20 is cainilated when 1% of the

tower mass (a very large value meant to be an upper bound) was lumped at the

top.

* Diagonals and horizontais near the base experience a change in intemal force

which does not exceed MY?? when wmpared to the bare tower case.

The proposed equivalent static method was verified on bare towers with height

ranging fiom 30 to 120 m. It is suitable for towers with lightweight antennae (less than 5%

of tower self-weight), however, towen with heavy antennae (mas > 5% of tower mass)

cannot be anaiyted ushg this method without modifications to the ai coefficients.

7.4 Anrlyris of Tranamurion Lint Towen

The following conclusions were drawn for suspension towen and cannot be

generaüzed to cover otha types oftmmission towers or line parameters:

The change in the cable tension had no significant dect on the fundamental

periods of the systan.

8 û d y the mnss of the overhead ground wire has an &ect on the fiequency

characteristics of the tower-cable system. The replacement of the ovethead

ground wire by 25% to 35% of its mass produced satisfactory results speciaiiy

for the lowest longihidinal fle>airal penod.

8 The response in the transverse direction could not be simplifieci in the same

manner. This is due to the fact that there is no clear trend in the results that

could enable us to assess the con~bution of the cables' mass to the transverse

modes of the system.

@ The response of the tower-cable system could not be simplifieci in a way

permitting the application of the proposed equivalent static method for seismic

response to transmission line towers.

7.5 Recommtndations for Future Work

The work presented in this thesis has covered several aspects related to the

simplification of the seismic analysis of self-supporthg lanice telecommunication towers,

but more research is still n d e d to M e r investigate the following topics:

1) The effkct of including heavy antennae and accessories on the prdction of the

mode shapes md naturai fiequencies of vibration of telecommunication towers.

2) The effe*. of includhg the torsionsa1 modes of vibrations on the seismic

response of telecommunication towas.

3) The effect of foundation flexibility on the sejsmic response of

telecommunication towers.

4) The applicability of the proposed method to four-legged towers.

5) A comprehensive study on the seismic analysis of transmission line towers.

Statemeot of Originality

To the best of author's knowledge, this work constitutes the first wmprehensive

research on the simplified seismic analysis of self-supporting telecornmunication towers.

The foilowing sumrnarize the three main onguial contributions of the thesis:

1) Two simple seismic response indicators are proposeci which can be used by

tower designers to estimate the maximum base shear and the total vertical

reaction of self-supporthg telecommunication towers due to earthquake

excitations.

2) A simplified equivalent static method is proposed for the prediction of member

forces in self-supporthg telecommunication towen due to horizontal and

vertical earthquake excitations. The originaiiity of the method lies in the

definition of an acceleration profile adapted to the essential dynamic

charactaistics ( n a d fiequencies and mode shapes) of the towers.

3) A better understanding of the effect of heavy antenna clusters on the seismic

response of telecornmunicittion towers.

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APPENDIX A

EARTaQUAKE RECORDS

Al: EARTHQUAKE RECORDS WITH LOW A N RATIO

A2: EARTBQUAKE RECORDS WlTH LNTERMEDIATE A N RATIO

A3: EARTHQUAKE RECORDS WITH HIGH A N RATIO

l

-1.5 ' I l

O 5 10 15 20

The, s

Period., s

Fig. Al . 1 Earthquake record Long Beach, NS I W (L 1)

Event: Long Beach earthquake Compaem: N5 1 W Station: LA. Subway Termllial, Los Angeles, Cal.. Peak Acccleration: 95.63460 cm/# at 7.50 s Peak Velocity: -23.6861 3 cmls at 13.62 s

7 - 1

I

+ - 3% damping 1

p, 1

, \ I 17 ! i

O 0.2 0.4 0.6 0.8 1 1.2

Period, s

Fig. A1 -2 Earthquake record Long Beach N39E (22)

Event: Long Beach earthquake Compnent: N39E Station: L.A Subway Termid, LOS Angeles, Cd. Peak Acceleration: 62.3281 1 cm/2 at 3.36 s PeakVe1oQty:-17.34293ds at 13.32s

- 3% darnping j

Penod, s

Fig. A1.3 Earthquake record Lower California SOOW (L3)

Event: Lower California earthquake Compnent: SûûW Station: EL Centro hperial Vaky Peak Acceleration: - 1 56.82 1 12 cm/s2 at 3.3 2 s Peak Velocity: -20.85573 cmls at 2.90 s

Fig.AI.4 Earthquake record San Femando N6 1 W (L4)

Event : San Femando earthquake Componen~: N6 1 W Station: 2500 Wdshire Blvd., Basement, Los Angeles, Cal. Peak Acceleration: 98.74722 cm/$ Peak Velocity: 19.30336 cm/s

The, s

Fig. A1 -5 Earthquake record San Femando West (L5)

Event: San Fernando earhquake Component: West Station: 3 550 Wdshire Blvd., Basement, Los Angeles, Cal. Peak Acceleration: - 1 29.8 1620 ds' Peak Velocity: 21.58354 cm

Fig. A1.6 Earthquake record San Fernando S37W (L6)

Event : SM Fenando earthquake Component: S37W Station: 222 Figueroa Street, 1st. flwr, Los Angeles, Cal. Peak Acceieration: - 126.88940 d s 2 Peak Velocity: -1 8.63307 an

-1.5 I

O 5 10 15 20

Time, s

1

- , - 3% damping i '\; '1 !

i i I

- -

O 0.2 0.4 0.6 0.8 1 1.2

b) Period, s


Event: San Fernando earthqualre Component: SWW Station: 3470 Wilshke Blvd., Sub-basment, Los Angeles, Cal. Peak Acceleration: 1 1 1.843 80 cm/* Peak Velocity: 18.56407 a n / s

Period, s

Fig. Al. 8 Earthquake record San Fernando N 1 SE (La)

Evmt: San Fernando d q u a k e Compona*: Nl SE Station: 4680 Wdshire Blvd., Basement, Los Angeles, Cd. Peak Accdefation: 1 14.976 1 O cm/s' Peak Vdocity: 21.53514 cm

- 3% damping

O 0.2 0.4 0.6 0.8 1 1.2

b) Period, s


Event: SM Fernando earthquake Compomm: S38W Station: 445 Figueroa Street, Sub-basement, Los Angeles, Cal. Peak Acceleration: -1 16.963 50 cm/$ Peak Velocity: -17.3 1090 cm

Period, s

Fig. Al. 10 Earthquake record San Fernando SûûW (L 10)

Event : San Fernando earthquake Component: SûûW Station: Hoiiywood Storage, Basement, Los Angeles, Cd. Peak Acceleration: 1 03.784 19 cm/$ Peak Velocity -16.96477 cm/s

Fig. A 1.1 1 Earthqhe record Near E. Cost of Honshu NS (L 1 1 )

Event: Near E. Cost of H o d u earthquake Component: NS Station: HK003 Peak Accelefation: -22 1-50 Mn/$ Peak Velocity: 33.40 cm/s

Fig. Al. 12 Earthquake record Near E. Coast of Honshu (L 12)

Event: Near E. Coast of Honshu earthquake Component: NS Station: HKW Peak Acctleration: -200.9 cm/$ Peak Vtlocity: 27.5 cmls

5 - - 3% damping ;

Period, s

Fig. Al. 13 Earthquake record Michoacan SOOE (L 13)

Event: Michoacan earthquake Component: SûûE Station: AZIH Peak Gcceleration: 1 O 1 -30 cm/$ P d Vdocity: - 15 -86 cm/s

Fig. A 1.14 Earthquake record Michoacui NOOE (L 14)

Record: Michoacan earthquake Cornponeai: NOOE Station: TEAC Peak AccclCtgfim: -5 1.30 cm/# Peak Velocity: 7.38 m/s

7

- 3% damping i

Fig. Al. 1 5 Earthquake record Michoacan N90W (L 1 5)

Evm: Micholun earthquake Component: NWW Station: CUMV Peak Acceleration: 3 8 -83 m/s2 Peak Veiocity : -1 1 .O 1 mis

Fig. AZ. 1 Earthquake record Imperia1 Valley SûûE (N 1 )

Eveat : Imperid Valley earthquake Component: SûûW Station:El Cmtro site Imperid Vdey Irrigation D i h a Peak Acceleration: 34 1.70508 cm/$ Peak Vdocity : 3 3 -4428 1 cm/s

- 3% darnping

/ I I 1 I !

O 0.2 0.4 0.6 0.8 1 1.2

Period, s

Fig. A2.2 Earthquake record Kem County S69E (N2)

Event: K a n County earthquake Component: S69E Site: Taft Lincoln school tunnel Peak Accehtion: 1 75 9404 cm/!? Peak Veiocity: - 17.72147 cmis

O O .2 0.4 0.6 0.8 1 1.2

b) Period, s

Fig. A2.3 Earthqurke record Kern County N2 1 E (N3)

Event: Kern County earthquake Component: N21E Station: No. 095 Peak Acceieration: 152.70538 cm/s' Palr Vdocity: -1 5.7254 cmls

-0.5 I

O 5 t O 15 20

Time, s

O 0.2 0.4 0.6 0.8 1 1.2

b) Period, s

Fig. A24 Earthquake record Borrego Mountain N57W (N4)

Event: Borrego Moumain earthquake Component: N57W Station: San ûnofie power plant Peak Accc1eration: -45.53455 cm/? Peak Velocity: 4.20 1 86 d s

Fig . A25 Earthquake record Borrego Mountain N33E (NS)

Evait: Bonego Mountain d q u a k e Companent: N33E Station: 280,33 22 OSN, 1 17 33 17W Peak Acctleration: 40.02722 crn/s2 Peak Velocity: -3.67305 cm/s

- 3% damping 1

Fig. A2.6 Earthquake record SM Fernando SWW (N6)

Event: San Fernando earthqualce Component: SWW Station: 3838 LankefShim Blvd., basement, Los Angeles, Cd. Peak Acceleration: 143.6254 crn/s2 Peak Velacity: 14.87&41 cm/s

V

O 0.2 0.4 0.6 0.8 1 1.2

Period, s

Fig. A2.7 Earthqde record SM Feniando N9ûE (N7)

Event: San Femando earthquake Componcat: N9ûE Station: HoUywOOd Storage P.E. Lut, Los Angdes, Cd. Peak Gcceleratioa: -206,99030 cm/s2 Peak Velocity: -2 1.13652 cmfs

3% damping

b) Period, s

Fig. A2.8 Earthquake record San Fanrndo N90E (N8)

Event: San Femando d q u a k e Compomm: N90E Station: 3407 6th Street, basement, los Angeles, Cal. Peak Acceldon: - 16 1 .WSW cm/s2 Peak Velocity: -16.59729 cm/s

The, s

O 0.2 0.4 0.6 0.8 1 1.2

b) Period s

Fig. A 2 9 Earthquake record SM Fanrado SOOW (N9)

Evenit: San Fernando arthquake Component: SOOW Station: M t h Park Obsmntory. moon rwm, Los Angdes, Cal. Peak Acctleration: - 1 76.89980 cm/$ Peak VdoQty: -20.48 128 d s

-3 O 5 10 15 20

Time, s

- 3% damping 1

Fig. A2.10 Earthquake record San F m d o N37E (N 10)

Event: San Fernando earthquake Compoacnt: N37E Station: 234 Figueroa Street, basement, Los Angeles, Cd. Peak Accelcration: 195.65 120 cm($ Peak Velocity: 16.72186 cm/s

Period, s

Fig. AZ. 1 1 Earthquake record Near E. Coast of Honshu NS (N 1 1)

Evevnt: Near E. Coast of Honshu earthquake Componmt: NS Station: KT036 Peak Accelcration: -69.1 cm/s2 Peak Velocity: -7.2 an/s

- 3% darnping I

0.2 0.4 0.6 0.8 1 1.2

Penod, s

Fig. A2.12 Earthquake record Near S. Coast of Honshu EW (N 12)

Event: Near S. Coast of Honshu earthquake Component: EW Station: H K W Peak Acceleration: 76.1 cm/$ Peak Velocity: 6.8 cm/s

-2 1

O 5 10 15 20

Time, s

Fig. A2.13 Eanhquake record Monte Negro NOOW (N13)

Eveat: Monte Negro earthquake Component: NûûW Peak Accelcration: 1 70.0 cm/s" Peak Velocity: 19.4 d s

Fig. A2.14 Earthquake record Michoacan SOOE (N14)

Event: Michoacan canhqU8ke Component: SûûE Station: SUCH Peak Acceleration: 1 03.12 a& Peak Velocity : - 1 1.6 1 a n l s

!

- 3% damping i I

Fig. A2.15 Earthquake rewrd Michoacan NWE (N 15)

Event: Michoacan earthquake Compomnt: N9ûE Station: VILE Peak Acceleration: - 1 20.87 cm/s2 Peak Velocity: 10.5 1 cmls

- 3% damping

Fig . A3.1 Earthquake record P d d d N6S W (H 1)

Event: Parffield earthquake Component: N65W Station: Tedior, California No. 2 Peak Accekation: -264.35 cm/!? Peak Velocity: -14.51 cm/s

I

- 3% darnping 1

Fig. A3.2 Earthquake record Parkfield NSSE (H2)

Event: P d e l d earthquake Component: N85E Station: Thoiaine, Shandon, California Array No. 5 Peak Gcctlerstion: 425.68 cm/$ Peak Velocity: - a n / s

- 3% damping

Period, s

Fig. A3.3 Eartbquake record San Francisco S8OE (H3)

Event: San Francisco earthquake Component: S80E Station: San Francisco Golden Gate park Peak Acceleration: - 102.80 cm/$ Peak VdOaty: -4.6 1 cm/s

O 0.2 0.4 0.6 0.8 1 1.2

b) Period, s

Fig. A3.4 Earthquake record San Francisco S09E (H4)

Event: Sm Francisco eafthquake Component : Sû9E Station: San Francisco State building, basamnt Peak Acceleraîion: -83 -8 1 c d 8 Peak Velacity: -5.05 cm/s

Period, s

Fig. A3.5 Earthquake record Helena Montana SûûW (HS)

Event: Helena Montana d q u a k e Component: SûûW Station: Heîena, Montana C m U college Peak Acceleration: 143.7 1 cm/$ Peak velocity: 7.21 cm/s

-2 O 5 10 15

The, s

- 3% darnping

O 0.2 0.4 0.6 0.8 1 1.2

Period, s

Fig. A3.6 Earthquake record Lytle Creek S25W (H6)

Event: Lytle Creek earthquke Component : SZS W Station: N/A Peak Acceleration: 1 94.4 1 cm/s2 Peak vdocity: -9.64 d s

- 1 ,

O 7 4 6 8 10 12

Time, s

I

- 3% darnping j

Fig. A3.7 Earthquake record ûrovüle N53 W (H7)

Event : Orovüle earthquake Component: N53 W Station: Oroviiie Dam, Colifornia, Seismograph station, ground level Peak Acceleraîion: -82.5 cm/$ Peak Velocity: -4.44 cm/s

Time, s

O 1 l I

O 0.2 0.4 0.6 0.8 1 1.2

Period, s

Fig. A3.8 Earthquake record San Fernando S74W (Hg)

Event: San Fernando earthquake Component: S74W Station: Pacoirna Dam, California Peak Acctkation: 1054.95 cm/$ Peak Velocity: -5 7.74 cm/s

-2 ' I

O 5 10 15 20

Time, s

Period, s

Fig. A3.9 Earthquake record San Fernando S21 W ( ' 9 )

Event: San Fanudo earthquake Composent: S2 1 W Station: Lake Hughes, Array station 4, California Peak Acceleration: -1 43.5 1 cm/s2 Peak Vdocity: -8.53 cm/s

-10 c I 1 I

! -15 '

1

O 5 10 15 20

The, s

Fig. A3.10 Earthquake record Nahanni Longjtudinal (Hl 0)

Event: Nahami earthquake Cornponent : Longitudinal Station: Peak Accelemtion: 1 100.0 cm/+ Peak Velocity: 46.2 cm/s

T h e , s

- 3% damping j

O 0.2 0.4 0.6 0.8 1 1.2

b) Period, s

Fig . A3.11 Earthquake record Centeral Honshu TR (H 1 1)

Event: Centeral Honshu earthquake Component: TR Station: CB030 Peak Acceleration: 148.10 c d 8 Peak Velocity: 5.9 d s

- 3% damping

Fig. A3.12 Earihquake record Near E. Coast of Honshu NS (H 12)

Evevnt: Near E. Coast of Honshu earthquake Component: NS Station: H K W Peak Acceleration: - 142.8 cm/s2 Peak Veloaty: 6.0 d s

Time, s

Fig. A3.13 Earthquake record Honshu NS (H 13)

Event : Honshu urthquake Component: NS Station: No. 2 Peak Acceleration: -265 .O cm/s2 Peak Velocity: 1 1.1 cm/s

1

Il - 3% darnping '

Period s

Fig. A3.14 Earthquake record Monte Negro NOOW (H 1 4)

Evait: Monte Negro earthquake Component: NûûW Station: Albatros Hotel basernent Peak Acceleration: 42.0 cm/$ Peak Velocity: 1.6 cm/s

O 0.2 0.4 0.6 0.8 1 1.2

b) Period, s

Fig. A3.15 Earthquake record Banja Luka N90W (H 15)

Event: Banja Luka earthquake Component: NWW Station: Banja Luka-4 seismological station Paek Acceleration: 73 .O cm/$ Peak Velocity: 3 -2 cm/s

VEIUFICATION OF THE PROPOSED EXPRESSIONS FOR THE

EARTHQUAKE AMPLIFICATION FACTORS

Fig. B. 1 Maximum base shear vs. peak grouad acceleration in TC1

Fig. B.2 Maximum base shear vs. pedr ground accderation in TC2

- Eq. (4.10)

Fig. 8.4 Maximum base shear vs. peak ground acceleration in TC5

- - -

- Eq. (4.10)

Fig. B.5 Maximum base shear m. perk gruund accelcration in TC6

-

-

- Eq. (4.1 O)

Fig. B.7 Maximum base shear vs. peak ground acceIeration in TC8

Fig. B. 8 Maximum base shear vs. p& ground acceteration in TC9

Fig. B.9 Maximum bise shm vs. pctL ground acccicnfion in TC10

Eq. (4.11) I l Fig. B. 10 Maximum dynamic vertical reaction vs. peak ground acceleration in TC 1

Fig. B. 1 1 Maximum dynamic vertical reaction vs. peak grwnd acceleration in TC2

v - Eq. (4.9)

O 2 4 6 8 10 12

Acc., dsA2

Fie. B. 12 MzWmum dynimic vertical d o n m. pmk @ adcfltion h TC4

Fig. B. 13 Maximum dynamic vertical reaction vs. peak gound accelaation in TC5

Fig. B. 14 Maximum dynamic vertical reaction W. peak ground acceImtion in TC6

f

;t 100 Eq. (4.11)

O

Fig. B.15 hilritimum dynamic vertid d o n M. peak ground acceicrafion in TC7

-

-

-

Eq. (4.11)

Fig. B. 16 Maximum dynamic vertical reaction vs. peak ground acceleration in TC8

- Eq. (4.9) - Eq. (4.1 1)

Fig. B. 17 Maximum dynamic vertical reaction vs. peak ground acceleration in TC9

Fig. B.18 Maxjmum dynumc vaticai renction W. peak punâ acceîcfation in TC10

APPENDIX C

VERTICAL EARTBQUAKE AMPLIFICATION FACTORS FOR

TELECOMMUNICATION TOWERS

C. I Introduction

Earthquake amplification factors for both base shear and vertical reaction of

self-supporthg telecornmunication towers were presented in Chapter 4. These vertical

amplification factors were obtained by reducing the horizontal accelerograms to 75% of

their origuial values and then used as vertical input. This approach was chosen as most

building codes do not contain specific uiformation on the peak vertical groud

accelerations.

It was also found usefiil to present expressions for esthating the vertical

eanhquake amplification factors using vertical earthquake records. The motivation for

presenting these expressions is the fact that vertical earthquake records generally have

hi& frequency content when compared to the horizontal records. The expressions

presented in this appendix should replace the expressions presented in Chaptw 4, Eqs.

(4.9) and (4.1 l), when information about the peak vertical ground accelerations is

indudeci in codes or othenvise available.

C.2 Vertical Earthqualrc Records

In the present snidy, as not al1 the correspondhg vertical components of the

previous set of horizontal accelerograms were available, a distinct set of 55 vertical

earthquake records wiiected from 17 diierent events were rlso considereâ. The majority

of these events were included in the study @ormed by Tso a al. (1992). Table C. 1 lins

the events m e ? magnitude, number of componens and date. It shouid be noted that

wntraiy to the horizontal earthquake records, these recorde wae not classified in

accordance to th& A/V ratios.

Table C. 1 - Earthquake records used as vertical input

Earthquake Magnitude No. of records Date Long Beach, California 6.4 1 10/03/ 1933 Lower California 5.6 1 30/12/1934 Helena, Montana 6 1 3 1/10/1935 hpenai Valley, California 6.6 1 1 81051 1940 Santa Barbara, California 5.5 1 30/06/ 194 1 Borrego Valley, California 6.6 1 21/10/1942 Imperia1 Vailey, California 5.8 1 23/01/1951 Kem County, California 7.5 4 2 1/07/1952 San Francisco, California 5.3 5 22/03/1957 Park Field, California 6.1 5 271061 1 966 Bonego Mount@ California 6.5 12 %/Ml 1968 Lytle Creek, California 5.3 7 1 2/09/ 1970 San Fernando, California 6.6 9 9102/ 1 97 1 Oroville, California 5.7 1 1 IO81 1 975 Michoacan, Mexico 8.1 2 19/09/1985 Nahanni, N.W.T., Canada 6.9 1 23/12/1985 Elrnore Ranch 6.2 2 2411 111987

C.3 Vertical Earthquik Excitation

The towers are also analyzed considering the vertical earthquake set acting in the

vertical direction. The values of maximum venical reaction at the tower base are plotted

versus the peak ground accderation. Figs. C. 1 to C. 10 show the resuits obtained for the

ten towen used in the study ris a fùnction of peak vertical ground accderation.

O 50 100 150 200

Acc., crn/sA2

Fig. C. 1 Total vertical dynamic reaction for tower TC 1 50 1

Fig. C .2 Total vertical dynamic reaction for tower TC2

O 50 100 lm 200

Acc., d s A 2

Fig. C.3 Total vertical dynamic reaction for tower TC3


Fig. C S Total vertical dynarnic reaction for tower TC5

O 50 1 0 150 200

Acc., d s A 2


Acc., cm/sA2



O 50 Io0 IU) 200

Acc., d s A 2

Fig. C.9 Totd v d d dynamic reaction for towa TC9

O 50 100 1 50 200

Acc., cni/sA2

Fig. C. 10 Total vertical dynamic reaction for tower TC10

As seen fiom these figures, the relation between the maximum vertical reaction and the

peak vertical ground acceleration foiiows a linear trend. Therefore, linear regression

analyses are perforrned to correlate the total venical reaction to the peak ground

acceleration and the results are sumrnarized in Table C.2 for the ten towers.

Table C.2 - Linear regression analysis for the total vertical reaction

Tower Slope R' TC1 0.94 0.92 TC2 1.29 0.73 TC3 1-05 0.77 TC4 1.63 0.78 TC5 1.47 0.78 TC6 1.73 0.76 TC7 1.69 0.60 TC8 1.66 0.71 TC9 1.45 0.79 TC10 1.81 0.73 k Codlicicnt o f ~ ~ d a t i o n

In keeping with the procedure foilowed in Cbpter 4, the numimm vertical reaction, Y, is

divided by the tower mas and peak ground vemcal acceleration, M X A, in orda to yield

a dixnensionîess factor. These factors are then plotted versus the fundamental axial p&od

of the tower for each earthquake record (Fig. C. 1 1). Contrary to the maximum base shear

response, the data foiiows an ascending trend in which the tower with lowest fiuidamental

axial period of vibration has the unallest amplikation factor. Linear regression analyses

are performed on the entire set and the foliowing expression is obtained for estimating the

maximum vertical reaction:

Vv = M x A, x (0.85 + 9.37 x T a ) (c.1)

where

K, = total maximum vertical reaction, N

A. = peak vertical ground acceleration, m/s2

Ta = fundamental axial period of vibration, S.

The values of the maximum total vertical reaction estimated using Eq. (C. 1) are

shown in Figs. C. 12 and C. 1 3 , for towers TC3 and TC 1 O, respectively. in order to obtain

an upper bound to the expected level of maximum vertical reaction, one standard

deviation is addeà to the numerical factors of eq. (C. 1) to yield the following expression:

Vv =Mx Av x (0.97+ 1 0 . 9 7 ~ Ta) (C.2)

This upper bomd expression for the maximum vertical reaction values is shom in

Fig. C. 1 1 for cornparison with eq. (C. 1).

- Eq. (C.l) - Eq. (C.2)

Lowest axial period, s

Fig. C. 1 1 Vertical reaction amplification factors

Peak ground acceleration, m/sA2

Fig. C. 1 2 Maximum dynamic vertical reaction vs. peak ground acceleration in TC3

O 0.5 1 1.5 2

Peak ground accelaation, m/sA2

Fig. C. 1 3 MarOmum dyMmic vertical readon vs. peak ground accelemtion in TC l O

C.4 Discussion

Two expressions for the evaluation of the vertical reaction of telecornmunication

towers under vertical seismic excitations are included in this appadix. Fig. C. 14 shows a

cornparison between these expressions and eqs. (4.9) and (4.1 1) after multiplying them by

4/3. This was done as eqs. (4.9) and (4.1 1) were originally evahiated &er reducuig the

peak ground acceleration to 314 of the original value. From this figure it is seen that

amplification factors predicted by eqs. (CA) and (C.2) mceed those predicted by eqs.

(4.9) and (4.11) by about 40%. This was expected as acnial vertical earthquake record

possess higher fiequency content and therefore will result in more amplification.

- - Eq. (C. 1 ) - - Eq. (4.9)

I

0.02 0.04 0.06 0.08 0.1 0.12

Lowest axial perioà, s

Fig. C. 14 Cornparison between the expressions for vertical ampüfication factor

It should be noted that the use of 314 of the peak ground acceleration in the vertical

direction greatly overestimates the expected peak vertid ground accelemtion which is

usuaiiy with lower intensities. Thenfore, even though the expressions presented in

Chapter 4 underestimate the level of dynamic ampiifidon, they are used with higher

intensities of ground acceieration than the actual vertical earthquakes possess.

APPENDIX D

VERlFICATlON OF THE PROPOSED STArrC METHOD

a) Earihquake record L14 120 r r l 3

b) Earthquake record N3 350 1

Fig. D. 1 Member forces in towa TC2

a) Earthquake record L 14 150 :

5 1 53 55 57 59 61 63 65 67 501 503 505

Mcmbcr index

b) Earthquake record N3 1200 i 2

i

i l SPEC ,


Fig. D.2 M e m k forces in tower TC3

a) Earthquake record L 14 300 1 - 4

Mcmber index

b) Earthquake record N3



a) Earthquake record L 14 350 r 1 1 10

M a n ber index


Fig. D.4 Manber forces in for tower TC6

a) Earthquake record L 1 4 400 : ! -5

1

27 28 29 30 31 32 33 34 35 36 37 38 39

Mcmk index

b) Earthquake record N3

27 28 29 30 31 32 33 34

Mmber indar

c) Earthqdce record H8

Fig. D.5 Member forces

35 36 37 38 39

ia tower TC7

a) Earthquake record L 14 400 r 1 5

I



Fig. D.6 Member forces in for towa TC8

a) Earthquake record L 14

b) Earthquake record N3 300 r

C) Earthquake m o r d H8

Fig. D. 7 Member forces in tower TC9

a) Earthquake record L14 350 r 4

b) Earthquake record N3 2000 -5

i l

SPEC ! - j -10


Fig. D.8 Member forces in tower TC IO

SPEC Rop. Di&rrrnccr ! 6

Member index

1

rcn i SPEC Rop. W ~ n c c .

Fig. D.9 Member forces in tower TC2 ushg NBCC spectmm

Fig D. 10 Member forces iu tower TC3 ushg NBCC spectrum

SPEC œ

3 100

5 <r

B 50

O

Mcmbcr index

SPEC 9

Fig. D. 1 1 Member forces in tower TC5 ushg NBCC spectrum

SPEC 1 = I

Fig. D. 12 Membet forces in towa TC6 using NBCC specmim

27 28 29 30 31 32 33 34 35 36 37 38 39

Mcmbcr index

SPEC 1 s

Fig. D. 13 Member forces in tower TC7 using NBCC specrnun

LL SPEC I

Mm bct index

Fig. D. 14 Member forces in tower TC8 using NBCC spectnim

63 65 67 69 71 73 75 77 79 81 03 85 87 89 91 93

Mcmber index

63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93

Mmkr index

Fig. D. 15 Member forces in tower TC9 using NBCC spe-

1 SPEC / -6

Fig. D. 16 Member forces in tower TC 1 O using NBCC specûum

2.5 SPEC P m p d DitlErtncc œ Ezm

5 2 d

1.5 B 6 1 r

0.5

O 37 39 41 43 45 47 49 51 53

38 40 42 44 46 48 50 52 54

Fig. D. 17 Memba forces in tower TC2 under vertical excitation

5 1 53 55 57 59 61 63 65 67 501 503 505 507 Mcmber index

Fig. D. 1 8 Member forces in tower TC3 under vertical excitation

Fig. D. 19 Manber foras in tower TC5 under vertid excitation


8 SPEC Proposad Differcncc 1 6

7 + I

5 6 d 3 5

4 4

5 3

2 ; 3.5

1

O 27 28 29 30 31 32 33 34 35 36 37 38 39

Fig. D.2 1 Member forces in tower TC7 under vertical excitation

Fig. D.22 Member forces in towa TC8 under verticai excitation

63 65 67 69 71 73 75 33 79 81 83 85 87 89 91 93 M e m k index


Fig. D.24 Mernber forces in tower TC IO unds vertical excitation

Documents

Seismic Analysis of Lattice Towers › obj › s4 › f2 › dsk1 › tape8 › PQDD_0024 › NQ50201.pdfMohamed Abdel Halim Khedr October 1998 Department of Civil Engineering and