alma mater studiorum - università di bologna modeling of

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ALMA MATER STUDIORUM - UNIVERSITÀ DI BOLOGNA

FACOLTA’ DI INGEGNERIA

CORSO DI LAUREA IN INGEGNERIA CIVILE

DISTART
Dipartimento di Ingegneria delle Strutture, dei Trasporti,
delle Acque, del Rilevamento, del Territorio

TESI DI LAUREA

in

PROGETTO IN ZONA SISMICA

MODELING OF CONDUCTORS IN ELECTRICAL
EQUIPMENT & SENSITIVITY STUDIES

CANDIDATO: RELATORE:
Benassi Andrea Chiar.mo Prof. Ing. Tomaso Trombetti

CORRELATORI:
Prof. Ing. Andrei M. Reinhorn
Dr. Ing. Stefano Silvestri

Anno Accademico 2008/09

Sessione III

ABSTRACT
It has been reported that the presence of flexible conductors between electrical equipment during
an earthquake might be responsible for generating destructive forces at the top of such apparatus.
The purpose of this study is to investigate the effect of interaction between two equipment items
connected by a cable conductor, through a finite-element model describing the dynamic behavior
of flexible conductors interconnecting electrical equipment items.
The first part of this study consists in modeling previous experimental tests of a cable conductor
only, already present in literature. The sensitivity of the modeling to the various parameters
affecting the solution is specifically checked. The initial compression in the cable coming from
the static loads is found to be the most significant parameter to evaluate the accuracy of the
model for the dynamic analysis. If the analytical and actual values are close, it is reasonable that
the finite-element model will provide an adequate approximation of the dynamic behavior of the
cable. The biggest errors related to the initial compression are directly caused by overestimated
values of the bending stiffness. Compression can shift the natural frequencies, changing the
dynamic properties of the cable. The model implementing an adequate variable bending stiffness
reproduces the experimental results with an average error of 4.8%: the model simply considering
a constant bending stiffness shows an average error of 16.2%.
The second part of this report aims to model the dynamic behavior of a flexible conductor
interconnecting two electrical equipment items modeled through equivalent beams: a very large
range of interconnected systems can be analyzed simply changing the structural properties of
these beams. Some ground motions are applied to the base of the structure, and different initial
shapes of the cable are assumed. The cable is redistributing the horizontal forces from the more
affected element to the less affected. In the vertical direction both the equipment items are more
severely tested in the interconnected configuration, due to the vertical inertia provided by the
cable. The moment at the base of the two cantilevers generated by the axial force in the cable has
the same order of magnitude of the moment due to the seismic response of the equipment itself.
It is suggested to design flexible connections so that the range of natural frequencies at which
they are likely to be excited are different from those of the equipment they are interconnecting,
in order to avoid the risk of dynamic interaction and resonance between them.
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AKNOWLEDGEMENT
The author greatly appreciates the essential support of Boneville Power Administration and the
California Energy Commission (California Institute for Energy and Environment for the
California Energy Commission) for the financial support provided to University at Buffalo – the
State University of New York during the development of this project. This entire work has been
developed and completed at the State University of New York at Buffalo, under the supervision
of local professorial staffs and making use of its laboratories and facilities.
The support of Prof. M. Sivaselvan from University of Colorado at Boulder is also gratefully
acknowledged. Special thanks are due to Mr. J.-B. Dastous of Hydro-Québec Research Institute
for providing a large number of information on his previous studies and results, and for
invaluable assistance in the use of his subroutine for the finite-element program FEAP. The
writer expresses his sincere appreciation to Prof. A. Filiatrault of State University of New York
at Buffalo, for his assistance in looking insight some specific issues regarding the cable
dynamics. Special thanks must be also given to all the graduate students of State University of
New York at Buffalo that supplied information and data about the qualification tests and other
studies on the high-voltage substation electrical equipment items.

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TABLE OF CONTENTS
ABSTRACT ................................................................................................................................................................ III
AKNOWLEDGEMENT ................................................................................................................................................. V
CHAPTER 1 ...................................................................................................................................................... 1
INTRODUCTION ......................................................................................................................................................... 1
1.1 Introduction .................................................................................................................................................... 1

1.1.1 Background ....................................................................................................................................................................... 1

1.1.2 Purpose of the Study ........................................................................................................................................................ 2

1.1.3 Organization of the Present Work .................................................................................................................................. 2
CHAPTER 2 ...................................................................................................................................................... 5
GENERAL DISCUSSION ABOUT FLEXIBLE CABLES ....................................................................................................... 5
2.1 Introduction .................................................................................................................................................... 5
2.2 Historical Development of Tension-Loaded Structures .................................................................................. 8
CHAPTER 3 .................................................................................................................................................... 13
CASE STUDIES AND NUMERICAL EXAMPLES ........................................................................................................... 13
3.1 Evaluation of Internal Forces and Spans in a Pre-tensioned Cable Subject to a Vertical Concentrated Load
at the Middle of its Span .................................................................................................................................... 13

3.1.1 Glossary ........................................................................................................................................................................... 13

3.1.2 Cable Stress and Strain................................................................................................................................................... 14

3.1.3 Forces (H e V) Transmitted to the Ends ........................................................................................................................ 18

3.1.4 Numerical Example ........................................................................................................................................................ 19

3.1.5 Parametric Study: Influence of the Vertical Load P ..................................................................................................... 21

3.1.6 Table resuming the Influence of the Vertical Load P on Ntot, H and V ....................................................................... 24
3.2 Evaluation of Internal Forces and Shape of an Inextensible Cable Subject to its Dead Weight: the Catenary
Case .................................................................................................................................................................... 25

3.2.1 Glossary ........................................................................................................................................................................... 25

3.2.2 Evaluation of the Shape and the Internal Tension ....................................................................................................... 25

3.2.3 Forces (H e V) Transmitted to the Ends ........................................................................................................................ 29

3.2.4 Numerical Example ........................................................................................................................................................ 30

3.2.5 Parametric Study: Influence of the Length of the Cable 2L......................................................................................... 31
(cid:3)
3.2.6 Table resuming the Influence of the Length of the Cable 2L on (cid:1) , (cid:2) , (cid:3) , f and .................................................. 35
(cid:1) (cid:2) (cid:2) (cid:4)
3.3 Interaction between “Beam Effect” and “Cable Effect” in a Simply Supported Element undergoing Uniform
Vertical Distributed Load .................................................................................................................................... 36

3.3.1 Glossary ........................................................................................................................................................................... 36

3.3.2 Evaluation of the Axial Stress Arising in the Element .................................................................................................. 36

3.3.3 Numerical Example ........................................................................................................................................................ 40

3.3.4 Parametric Study: Influence of the Span L ................................................................................................................... 43
(cid:3)
3.3.5 Table Resuming the Influence of the Span L on (cid:4), f, N, ........................................................................................... 46
(cid:4)

3.3.6 Parametric Study: Influence of the External Load (cid:5) .............................................................................................. 47
(cid:5)(cid:6)(cid:7)
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(cid:3)
3.3.7 Table Resuming the Influence of the External Load (cid:5) on (cid:4), f, N, ...................................................................... 49
(cid:5)(cid:6)(cid:7) (cid:4)
3.4 Influence of the Axial Tension on the Natural Frequencies of a Simply Supported Beam ............................ 50

3.4.1 Glossary ........................................................................................................................................................................... 51

3.4.2 Evaluation of the Natural Frequencies under the Influence of an Axial Load ........................................................... 51

3.4.3 Numerical Example ........................................................................................................................................................ 53

3.4.4 Parametric Study: Influence of the Axial Force N ........................................................................................................ 54

3.4.5 Table Resuming the Influence of the Axial Force N on the Natural Frequencies (cid:6) ................................................ 55
(cid:8)
CHAPTER 4 .................................................................................................................................................... 57
FLEXIBLE STRANDED CONDUCTORS FOR ELECTRICAL PURPOSES ........................................................................... 57
4.1 Introduction .................................................................................................................................................. 57
4.2 Glossary ........................................................................................................................................................ 59
4.3 Common Seismic Design of Substation Equipment ...................................................................................... 60
4.4 Bending of Stranded Conductors .................................................................................................................. 62
4.5 Experimental Investigation on the Dynamic Behavior of Flexible Conductor ............................................... 64

4.5.1 Test Parameters.............................................................................................................................................................. 65

4.5.2 Test Specimens ............................................................................................................................................................... 67

4.5.3 Static Tests ...................................................................................................................................................................... 68

4.5.4 Frequency-Sweep Tests ................................................................................................................................................. 69

4.5.5 Sine-Start Tests ............................................................................................................................................................... 70

4.5.6 Conclusions ..................................................................................................................................................................... 70
4.6 Constant Bending Stiffness Studies .............................................................................................................. 71

4.6.1 Catenary Cable ................................................................................................................................................................ 72

4.6.2 Cable with Flexural Stiffness and Inertia ...................................................................................................................... 74
4.7 Variable Bending Stiffness Studies ............................................................................................................... 77

4.7.1 Bending Stiffness Calculation ........................................................................................................................................ 78

4.7.2 Internal Moment Calculation ........................................................................................................................................ 79

4.7.3 Implementation in a Finite-Element Formulation ....................................................................................................... 80

4.7.4 Time Integration Method .............................................................................................................................................. 80

4.7.5 Modeling of Experimental Tests .................................................................................................................................... 81

4.7.6 Results ............................................................................................................................................................................. 83
4.8 Framework of the Present Study .................................................................................................................. 84
4.9 Scope of the Present Study ........................................................................................................................... 86
4.10 Further Studies ........................................................................................................................................... 87
CHAPTER 5 .................................................................................................................................................... 89
MODELING OF CONDUCTORS IN ELECTRICAL EQUIPMENT & SENSITIVITY STUDY .................................................. 89
5.1 Bending Stiffness .......................................................................................................................................... 89

5.1.1 Constant Bending Stiffness ............................................................................................................................................ 90

5.1.2 Variable Bending Stiffness ............................................................................................................................................. 91

5.1.3 Comparisons ................................................................................................................................................................... 92
5.2 Loading Steps ............................................................................................................................................... 95

5.2.1 Dastous’ Model ............................................................................................................................................................... 96

5.2.2 Der Kiureghian’s Model ................................................................................................................................................. 99
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5.2.3 Comparisons ................................................................................................................................................................. 100
5.3 Initial Conditions......................................................................................................................................... 103

5.3.1 Compression in the Cable ............................................................................................................................................ 103

5.3.2 Central Sag .................................................................................................................................................................... 109
5.4 Parameters ................................................................................................................................................. 113

5.4.1 Rotational Lumped Damping ....................................................................................................................................... 115

5.4.2 Rayleigh Damping ......................................................................................................................................................... 117

5.4.3 Half-Step Residual Tolerance ....................................................................................................................................... 120
5.5 Element Formulation .................................................................................................................................. 123

5.5.1 Timoshenko Linear Formulation ................................................................................................................................. 124

5.5.2 Euler-Bernoulli Cubic Formulation .............................................................................................................................. 125

5.5.3 Hybrid Formulation ...................................................................................................................................................... 125

5.5.4 Comparisons ................................................................................................................................................................. 126
5.6 Time Integration Method ........................................................................................................................... 127

5.6.1 HHT-alpha Method ....................................................................................................................................................... 128

5.6.2 Energy-Conserving Algorithm ...................................................................................................................................... 129

5.6.3 Comparisons ................................................................................................................................................................. 130
5.7 Convergence of Solution ............................................................................................................................. 131

5.7.1 Time Stepping ............................................................................................................................................................... 132
5.8 Modeling of Conductor Only ...................................................................................................................... 135

5.8.1 Sine-Start Tests ............................................................................................................................................................. 138

5.8.2 Frequency-Sweep Tests ............................................................................................................................................... 144
CHAPTER 6 .................................................................................................................................................. 149
INTERCONNECTED EQUIPMENT: SENSITIVITY STUDY ........................................................................................... 149
6.1 Conductor & Two Flexible Posts ................................................................................................................. 149

6.1.1 Tests Setup .................................................................................................................................................................... 151

6.1.2 Static Step ..................................................................................................................................................................... 159

6.1.3 Sine-Start Tests ............................................................................................................................................................. 162

6.1.4 Base Motion Tests ........................................................................................................................................................ 180
CHAPTER 7 .................................................................................................................................................. 191
CONCLUDING REMARKS........................................................................................................................................ 191
CHAPTER 8 .................................................................................................................................................. 197
REFERENCES .......................................................................................................................................................... 197
CHAPTER 9 .................................................................................................................................................. 201
APPENDIXES .......................................................................................................................................................... 201
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