### Politecnico di Milano

### School of Civil, Enviromental and Land Management Engineering

Master of Science in Civil Engineering

### Analysis of the hydrodynamic forces acting on

### a Submerged Floating Tunnel during a seaquake

Supervisor: Prof. MARTINELLI Luca

FILIPPI Angelo 884714

**Ringraziamenti**

**Ringraziamenti**

*In primo luogo desidero esprimere gratitudine nei confronti del Prof. *
*Martinelli per la grande pazienza e disponibilità mostrata nel guidarmi attraverso lo *
*svolgimento del presente lavoro di tesi. Ringrazio inoltre il team SOLIDeng, per *
*aver messo a mia disposizione quanto necessario per lo svolgimento delle *
*simulazioni numeriche cui il presente elaborato fa riferimento.*

*Un grandissimo ringraziamento va a tutti i colleghi di università, conosciuti in *
*questi anni, che mi hanno permesso di capire l’importanza e i vantaggi del gioco di *
*squadra, e di trovare sempre il lato ’divertente’ di qualsiasi problema. Tra questi *
*ci tengo in particolare a citare Lorenzo ’Pedro’, Marzio, Roberto e Alessio, anche *
*se la lista completa è veramente lunga.*

*Impossibile non ringraziare gli amici che mi hanno accompagnato durante *
*questi anni di università, tra serate di divertimento e ore passate in biblioteca: *
*Alessandro ’Cons’, Daniele ’Dani’, Emanuele ’Manu’, Francesco ’Capi’, *
*Francesco ’Ciga’, Mattia ’Tami’, Michele ’Nok’, Nadir ’Nad’.*

*Una persona, entrata nella mia vita negli ultimi mesi, che va sicuramente *
*men-zionata* *è Laura: il senso di sicurezza che hai saputo trasmettermi, credendo in *
*me sopratutto nei momenti più critici, è stata una base forte su cui questo lavoro *
*di tesi ha fondamento.*

*Un grandissimo grazie va ai miei genitori, mamma Giacinta ’Giaci’ e papà *

*Marino, che, oltre a sopportarmi durante questi anni di studi, ci sono sempre stati, *
*dandomi supporto per ogni minimo problema.*

*Ultima persona, ma sicuramente non per importanza, che cito * *è mio fratello *
*Lorenzo ’Lollo’, che non esito a deﬁnire il mio migliore amico. Spero in futuro di poter *
*fare sempre aﬃdamento su di te, come ho avuto il piacere di fare in questi mesi.*

## Contents

Introduction 1

1 A look to the future: the Submerged Floating Tunnel 3

1.1 Introduction to the Submerged Floating Tunnel . . . 3

1.1.1 The concept of Submerged Floating Tunnel . . . 3

1.1.2 Design principles and construction methods . . . 5

1.1.3 Historical development . . . 7

1.1.4 Strengths and weaknesses . . . 10

1.1.5 Forecasts for the Submerged Floating Tunnel . . . 11

1.2 Design features . . . 13

1.2.1 Fundamental aspects . . . 13

1.2.2 Loading conditions . . . 15

2 Loads acting on a submerged floating tunnel 19 2.1 Wave forces acting on a submerged structure . . . 19

2.1.1 Inertia force . . . 19

2.1.2 Drag force . . . 20

2.1.3.1 Reynolds number . . . 21

2.1.3.2 Keulegan-Carpenter number . . . 21

2.1.3.3 Relative roughness . . . 22

2.1.3.4 Diffraction parameter . . . 23

2.1.3.5 Strouhal number . . . 23

2.1.4 Small and large submerged structures . . . 24

2.2 The linear wave problem . . . 25

2.2.1 Water motion field reconstruction . . . 25

2.2.2 Bernoulli equation and pressure field . . . 29

2.3 Potential flow approach during wave forces evaluation . . . 30

2.3.1 Steady flow term . . . 31

2.3.2 Unsteady flow term . . . 35

2.3.3 Morison approach for hydrodynamic force . . . 36

2.3.4 Hydrodynamic coefficients determination . . . 38

2.3.5 Design rules while considering hydrodynamic coefficients for offshore structures . . . 40

2.4 Diffraction theory . . . 43

3 Reconstruction of the water motion field due to a seaquake 47 3.1 The seaquake phenomenon . . . 47

3.1.1 A brief description of seaquakes effects on ships . . . 48

3.1.2 Seaquake effects on offshore structures . . . 49

3.2 Offshore and onshore earthquake effects . . . 52

Contents

3.4 The idea of transfer function . . . 56

3.4.1 Semi-infinite domain . . . 56

3.4.2 Elastic layer resting over a semi-infinite rigid base . . . 56

3.4.3 Elastic layer resting over a semi-infinite elastic base . . . 58

3.5 Water motion field due to seaquake . . . 61

3.5.1 Hamamoto theory . . . 61

3.5.1.1 Boundary conditions . . . 63

3.5.1.2 Problem linearization . . . 64

3.5.1.3 Motion field derived from Hamamoto theory . . . . 66

3.5.1.4 Hamamoto transfer function . . . 67

3.5.2 Li et al. theory . . . 71

3.5.2.1 Stress field reconstruction . . . 72

3.5.2.2 Seaquake plane problem . . . 73

3.5.2.3 Vertical propagating waves . . . 76

3.5.2.4 Li transfer function . . . 79

3.5.3 Heng et al. theory . . . 80

3.5.4 Water material damping . . . 82

3.5.4.1 Complex number analysis . . . 83

3.6 Flexible ground layer resting above bedrock . . . 87

3.6.1 Governing equation . . . 88

3.6.2 Boundary conditions . . . 89

3.6.2.1 Bedrock kinematic boundary condition . . . 89

3.6.2.3 Seabed dynamic equilibrium boundary condition . . 90

3.6.3 Problem solution . . . 90

3.6.4 Transfer function between bedrock and seabed . . . 93

3.7 Double flexible ground layer resting above bedrock . . . 94

3.7.1 Transfer function between bedrock and seabed . . . 99

4 The hydrodynamic behavior in the Messina Strait 103 4.1 Submerged Floating Tunnel project in Messina Strait . . . 103

4.2 Reconstruction of water motion field around the Submerged Floating Tunnel . . . 106

4.2.1 Seaquake effects in absence of ground layer and with damping neglected . . . 106

4.2.1.1 Sea depth of 40 m . . . 108

4.2.1.2 Sea depth of 182.5 m . . . 114

4.2.1.3 Sea depth of 325 m . . . 119

4.2.2 Seaquake effects in absence of ground layer and considering damping . . . 123

4.2.2.1 Sea depth of 40 m . . . 123

4.2.2.2 Sea depth of 182.5 m . . . 127

4.2.2.3 Sea depth of 325 m . . . 131

4.2.3 Comparisons between undamped and damped case in motion field evaluation . . . 134

4.2.4 Seaquake effects in presence of a single ground layer . . . 136

4.2.4.1 Sea depth of 40 m . . . 139

4.2.4.2 Sea depth of 182.5 m . . . 142

Contents

4.2.5 Seaquake effects obtained by increasing the previous ground

layer width . . . 148

4.2.5.1 Sea depth of 40 m . . . 149

4.2.5.2 Sea depth of 182.5 m . . . 152

4.2.5.3 Sea depth of 325 m . . . 155

4.2.6 Comparison between the original and modified single layer cases158 4.2.7 Seaquake effects in presence of a double ground layer . . . 160

4.2.7.1 Sea depth of 40 m . . . 164 4.2.7.2 Sea depth of 182.5 m . . . 167 4.2.7.3 Sea depth of 325 m . . . 170 4.3 Hydraulic analysis . . . 173 4.3.1 Reynolds number . . . 173 4.3.2 Keulegan-Carpenter number . . . 176 4.3.3 Diffraction parameter . . . 178 4.3.4 Relative roughness . . . 179

4.3.5 Considerations on the hydrodynamic coefficients choice . . . 179

4.4 Drag and inertia forces behavior . . . 182

5 Force analysis on the Submerged Floating Tunnel 191 5.1 Introduction . . . 191

5.2 Modal analysis on the two-bar equivalent 2-D model . . . 193 5.3 Analysis of Morison force in the frequency/period range of interest . 196

Conclusions 209

B Bernoulli equation 215

C Wave forces from potential flow approach 217

D Derivation of Hamamoto seaquake velocity potential 219

D.1 Resolution with separation of variables method . . . 220

D.2 Spatial function Z(z) solution . . . 221

D.3 Temporal function T (t) solution . . . 222

D.4 Velocity potential φHam expression . . . 223

E Derivation of Li et al. seaquake velocity potential 225 F Setting of single flexible ground layer seaquake problem 231 G Hydrodynamic forces for single and double ground layer cases 235 G.1 Single ground layer resting over bedrock . . . 235

## List of Figures

1 The three principal methods for crossing waterways. . . 1

1.1 Concepts of a SFT with different anchoring systems . . . 4 1.2 The visual effects of an SFT in an environment characterized by deep

waters are minimal . . . . 4 1.3 The SFT represented with different anchoring systems. . . 7 1.4 Example of a Submerged Floating Tunnel in an urban area . . . . . 10 1.5 Norway’s fjords, for which the SFT could represent an optimal solution. 11 1.6 Example of Submerged Floating Tunnel with circular section . . . 14

2.1 Vortex shedding phenomena due to wind (the same happens into wa-ter flow) . . . 23

2.2 Infinitesimal cube element on which mass conservation is verified. . . 26 2.3 Example of the effects of a no slip boundary condition on motion field

. . . 27 2.4 Water domain in which linear wave problem is solved. . . 28 2.5 Velocity profile on x-z plane reconstructed from linear wave theory. . 28 2.6 Behavior of potential flow around a cylinder . . . 30 2.7 Pressure distributions, theoretical and experimental, measured by

2.8 Values of drag coefficient CD obtained experimentally for a smooth

circular cylinder in a stationary flow. . . 33

2.9 Drag coefficient in fluids, evaluated experimentally with approximately Re = 104. . . 35

2.10 Limits of application for small and large structures . . . 37

2.11 Measured force Fm(t) and velocity record V (t). . . 39

2.12 Offshore engineering applications. . . 40

2.13 Suggested drag and inertia coefficient values from DNV. . . 42

3.1 Semi-infinite domain. . . 56

3.2 Elastic layer resting over an infinitely rigid base. . . 57

3.3 Flexible soil layer resting on a semi-infinite flexible domain. . . 58

3.4 Transfer function |H2(f )|. . . 60

3.5 Brief graphical description of the seaquake problem. . . 61

3.6 Domain considered in the Hamamoto theory. . . 62

3.7 Graphical representation of a linear progressive wave and its charac-teristics parameters. . . 64

3.8 Hamamoto acceleration transfer function HHam,aand its dependence from the frequency f , such that ω = 2πf . . . 68

3.9 Hamamoto acceleration transfer function HHam,a plotted in order of increasing seawater depth. . . 69

3.10 Cartesian reference coordinate system considered. . . 72

3.11 Scheme of 2-D seaquake problem. . . 73

3.12 Scheme of 2-D seaquake problem. . . 76

3.13 Li acceleration transfer function HHam,aand its dependence from the frequency f , such that ω = 2πf . . . 80

List of Figures

3.14 Analytical model of Submerged Floating Tunnel hit by P-waves at bedrock, as reported in . . . 81 3.15 Hamamoto transfer function module and angle obtained in the

un-damped case. . . 84 3.16 Hamamoto transfer function module and angle obtained for damped

case. . . 85 3.17 Behavior of the angle θ in complex plane. . . 86 3.18 Schematic view of seaquake plane problem when a ground layer

rest-ing above the bedrock is considered. . . 87 3.19 Schematic view of seaquake plane problem when considering two

ground layers above bedrock. . . 94

4.1 Aerial view of the Messina strait. . . 104 4.2 Acceleration time history for a fixed frequency. . . 109 4.3 Module and phase (with respect to bedrock excitation) of

accelera-tion frequency-dependent coefficient, for d = 40 m (NO GROUND LAYER AND NO WATER DAMPING). . . 110 4.4 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 40 m (NO GROUND LAYER AND NO WATER DAMPING). . . 112 4.5 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 40 m (NO GROUND LAYER AND NO WATER DAMPING). . . 113 4.6 Behavior of response and related driving force in complex plane

(Ar-gand). . . 115 4.7 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 182.50 m (NO GROUND LAYER AND NO WATER DAMPING). . . 116 4.8 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 182.50 m (NO GROUND LAYER AND NO WATER DAMPING). . . 117

4.9 Module and phase (with respect to bedrock excitation) of displace-ment frequency-dependent coefficient, for d = 182.50 m (NO GROUND LAYER AND NO WATER DAMPING). . . 118 4.10 Module and phase (with respect to bedrock excitation) of

accelera-tion frequency-dependent coefficient, for d = 325 m (NO GROUND LAYER AND NO WATER DAMPING). . . 120 4.11 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 325 m (NO GROUND LAYER AND NO WATER DAMPING). . . 121 4.12 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 325 m (NO GROUND LAYER AND NO WATER DAMPING). . . 122 4.13 Module and phase (with respect to bedrock excitation) of

accelera-tion frequency-dependent coefficient, for d = 40 m (NO GROUND LAYER WITH WATER DAMPING). . . 124 4.14 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 40 m (NO GROUND LAYER WITH WATER DAMPING). . . 125 4.15 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 40 m (NO GROUND LAYER WITH WATER DAMPING). . . 126 4.16 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 182.50 m (NO GROUND LAYER WITH WATER DAMPING). . . 128 4.17 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 182.50 m (NO GROUND LAYER WITH WATER DAMPING). . . 129 4.18 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 182.50 m (NO GROUND LAYER WITH WATER DAMPING). . . 130 4.19 Module and phase (with respect to bedrock excitation) of

accelera-tion frequency-dependent coefficient, for d = 325 m (NO GROUND LAYER WITH WATER DAMPING). . . 131

List of Figures

4.20 Module and phase (with respect to bedrock excitation) of velocity frequency-dependent coefficient, for d = 325 m (NO GROUND LAYER WITH WATER DAMPING). . . 132 4.21 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 325 m (NO GROUND LAYER WITH WATER DAMPING). . . 133 4.22 Comparison between frequency-dependent acceleration coefficient α (f )

with and without water damping for sea depth of d = 40 m. . . 134 4.23 Comparison between frequency-dependent acceleration coefficient α (f )

with and without water damping for sea depth of d = 182.5 m. . . . 135 4.24 Comparison between frequency-dependent acceleration coefficient α (f )

with and without water damping for sea depth of d = 325 m. . . 135 4.25 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 40 m (SINGLE GROUND LAYER). . . 139 4.26 Module and phase (with respect to bedrock excitation) of the velocity

frequency-dependent coefficient, for d = 40 m (SINGLE GROUND LAYER). . . 140 4.27 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 40 m (SINGLE GROUND LAYER). . . 141 4.28 Module and phase (with respect to bedrock excitation) of

acceler-ation frequency-dependent coefficient, for d = 182.50 m (SINGLE GROUND LAYER). . . 142 4.29 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 182.50 m (SINGLE GROUND LAYER). . . 143 4.30 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 182.50 m (SINGLE GROUND LAYER). . . 144

4.31 Module and phase (with respect to bedrock excitation) of acceleration frequency-dependent coefficient, for d = 325 m (SINGLE GROUND LAYER). . . 145 4.32 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 325 m (SINGLE GROUND LAYER). . . 146 4.33 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 325 m (SINGLE GROUND LAYER). . . 147 4.34 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 40 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 149 4.35 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 40 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 150 4.36 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 40 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 151 4.37 Module and phase (with respect to bedrock excitation) of

acceler-ation frequency-dependent coefficient, for d = 182.50 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 152 4.38 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 182.50 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 153 4.39 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 182.50 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 154 4.40 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 325 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 155 4.41 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 325 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 156

List of Figures

4.42 Module and phase (with respect to bedrock excitation) of displace-ment frequency-dependent coefficient, for d = 325 m (SINGLE GROUND LAYER INCREASED THICKNESS). . . 157 4.43 Comparison between frequency-dependent acceleration coefficient α (f )

with original and increased ground layer for sea depth of d = 40 m. . 158 4.44 Comparison between frequency-dependent acceleration coefficient α (f )

with and without water damping for sea depth of d = 182.5 m. . . . 159 4.45 Comparison between frequency-dependent acceleration coefficient α (f )

with and without water damping for sea depth of d = 325 m. . . 159 4.46 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 40 m (DOUBLE GROUND LAYER). . . 164 4.47 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 40 m (DOUBLE GROUND LAYER). . . 165 4.48 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 40 m (DOUBLE GROUND LAYER). . . 166 4.49 Module and phase (with respect to bedrock excitation) of

acceler-ation frequency-dependent coefficient, for d = 182.50 m (DOUBLE GROUND LAYER). . . 167 4.50 Module and phase (with respect to bedrock excitation) of velocity

frequency-dependent coefficient, for d = 182.50 m (DOUBLE GROUND LAYER). . . 168 4.51 Module and phase (with respect to bedrock excitation) of

displace-ment frequency-dependent coefficient, for d = 182.50 m (DOUBLE GROUND LAYER). . . 169 4.52 Module and phase (with respect to bedrock excitation) of acceleration

frequency-dependent coefficient, for d = 325 m (DOUBLE GROUND LAYER). . . 170

4.53 Module and phase (with respect to bedrock excitation) of velocity frequency-dependent coefficient, for d = 325 m (DOUBLE GROUND

LAYER). . . 171

4.54 Module and phase (with respect to bedrock excitation) of displace-ment frequency-dependent coefficient, for d = 325 m (DOUBLE GROUND LAYER). . . 172

4.55 Reynolds number for d = 40 m (DOUBLE GROUND LAYER). . . . 174

4.56 Reynolds number for d = 182.5 m (DOUBLE GROUND LAYER). . 175

4.57 Reynolds number for d = 325 m (DOUBLE GROUND LAYER). . . 175

4.58 Keulegan-Carpenter number for d = 40 m (DOUBLE GROUND LAYER). . . 176

4.59 Keulegan-Carpenter number for d = 182.5 m (DOUBLE GROUND LAYER). . . 177

4.60 Keulegan-Carpenter number for d = 325 m (DOUBLE GROUND LAYER). . . 177

4.61 Diffraction parameter in the f − Dout domain. . . 178

4.62 Suggested drag and inertia coefficient values from DNV. . . 180

4.63 Drag force for d = 40 m (DOUBLE GROUND LAYER). . . 183

4.64 Drag force for d = 182.5 m (DOUBLE GROUND LAYER). . . 183

4.65 Drag force for d = 325 m (DOUBLE GROUND LAYER). . . 184

4.66 Inertia force for d = 40 m (DOUBLE GROUND LAYER). . . 185

4.67 Inertia force for d = 182.5 m (DOUBLE GROUND LAYER). . . 186

4.68 Inertia force for d = 325 m (DOUBLE GROUND LAYER). . . 186

4.69 Ratio between drag and inertia forces for d = 40 m (DOUBLE GROUND LAYER). . . 187

4.70 Ratio between drag and inertia forces for d = 182.5 m (DOUBLE GROUND LAYER). . . 188

List of Figures

4.71 Ratio between drag and inertia forces for d = 325 m (DOUBLE GROUND LAYER). . . 188

5.1 SFT and anchoring system 2-D model. . . 192 5.2 Equivalent 2-D geometrical model with one bar at each side, simulated

corresponding model in ANSYS (unit in meters). . . 194 5.3 Graphical representations of structure’s modes . . . . 195 5.4 Module and phase (with respect to bedrock excitation) of drag, inertia

and Morison forces, for z = 71.25 m (NO GROUND LAYER WITH WATER DAMPING). . . 199 5.5 Module and phase (with respect to bedrock excitation) of drag, inertia

and Morison forces, for z = 142.5 m (NO GROUND LAYER WITH WATER DAMPING). . . 200 5.6 Module and phase (with respect to bedrock excitation) of drag, inertia

and Morison forces, for z = 213.75 m (NO GROUND LAYER WITH WATER DAMPING). . . 201 5.7 Module and phase (with respect to bedrock excitation) of drag, inertia

and Morison forces, for z = 285 m (NO GROUND LAYER WITH WATER DAMPING). . . 202 5.8 Module and phase (with respect to bedrock excitation) of Morison

transfer function, for z = 71.25 m (NO GROUND LAYER WITH WATER DAMPING). . . 203 5.9 Module and phase (with respect to bedrock excitation) of Morison

transfer function, for z = 142.5 m (NO GROUND LAYER WITH WATER DAMPING). . . 204 5.10 Module and phase (with respect to bedrock excitation) of Morison

transfer function, for z = 213.75 m (NO GROUND LAYER WITH WATER DAMPING). . . 205 5.11 Module and phase (with respect to bedrock excitation) of Morison

transfer function, for z = 285 m (NO GROUND LAYER WITH WA-TER DAMPING). . . 206

C.1 Behavior of potential flow around a cylinder. . . 217

## List of Tables

2.1 Relative roughness for different pipe materials. . . . 22

2.2 Morison coefficients suggested by Clauss, for circular cylinders. . . . . . 41

2.3 Morison coefficients suggested by API and SNAME. . . 41

4.1 Ground layers considered for the Messina Strait model. . . 105

4.2 Morison coefficients suggested by Clauss , for circular cylinders . . . .180

4.3 Morison coefficients suggested by API and SNAME. . . 181

5.1 SFT characteristics for the analyzed case. . . 191

5.2 Anchoring bars properties. . . 192

5.3 Modal analysis result obtained by Shi for the two-bar equivalent 2-D model. . . 194

## Introduction

The SFT (Submerged Floating Tunnel) represents a, relatively, new and innova-tive type of structure, and both a challenge for many fields of the Civil Engineering. In fact, today, there is no structure like this built in the world. The concept consists in a tubular structure submerged in water and, taking advantage of the buoyancy forces, is suspended in it with the usage of floating islands at the sea surface or anchoring bars at the seabed.

The born of this idea (in the ’80s) was due to the necessity of evaluate an alterna-tive for crossing waterways characterized by great depth. In fact, for these cases, the design of a bridge (single or multi-span) can be very trivial, both for constructional and economic reasons. So, from this point of view, SFTs are solutions that provide a good ratio cost - length, with respect of suspended bridges or underground tunnels.

The three principal methods for crossing waterways.

The longitudinal development of this type of structure is significantly magnified by the effect of water buoyancy and, for the same reason, an SFT can be built even in very deep waters, where no other crossing solutions are possible.

More, a submerged tunnel, beyond economic advantages, leads to:

• Reduction of environmental problems: as the underwater element cannot be seen;

• Solve all the transit problems related to the weather conditions;

• From the structural point of view, a superior anti-vibration characteristics. All these factors leads to the dejection of many geographic barriers that obstruct the transportation technologies development, improving the journey’s times and increasing their safety.

Anyway, due to its relatively short age, the concept of SFT still needs to be covered, in order to obtain an overview of its structural behavior to all types of environmental and operational loading. In fact, the main reason for which this type of structure has not been realized, is due to safety concerns. Several researches have been conducted in the last century, especially in Norway, China and Italy.

From the studies, it seems that SFT is a good earthquake-resistant structure, suitable for waterway crossing in high seismicity zones. The interactions between the ground motion and the building happen in three ways: at the structural joints placed at the shore connection, by the cables that acts like filters of the excitation (due to their transversal flexibility), and by the interaction with the water. In this last one, many actions are considered, like seaquakes or wave and current forces.

The main goal of this thesis work is to analyze the hydrodynamic forces as result of the ground shaking due to an earthquake. This process starts by studying what happens in the sea where gravity waves are present, analyzing the classic linear wave problem studied in ocean engineering, that let to compute the motion field in the seawater. Then the approaches to compute the related forces are presented: the theoretical one (diffraction theory) and the practical one (based on the usage of the Morison equation), introducing also the hydrodynamic coefficients (experimentally evaluated) from which the forces can be determined. Later, this theory is adapted to the seaquake phenomenon, introducing also the concept of transfer function and material damping in order to obtain a better representation of what happens in the reality. At last, the reconstruction of motion field and seaquake forces for the Messina Strait practical case is analyzed, by looking how the hydrodynamics coefficients are influenced by the ground motion due to seismic motion.

### Chapter 1

## A look to the future: the

## Submerged Floating Tunnel

### 1.1

### Introduction to the Submerged Floating Tunnel

1.1.1 The concept of Submerged Floating Tunnel

In the last years, in order to solve waterways crossing problems, a new type of infrastructure has been proposed: the Submerged Floating Tunnel. Its employment must be considered as an alternative to other two types of infrastructures, charac-terized by a better design experience, accumulated during many years:

• Bridges: supported by well-developed design approaches and experience, they can be of different kinds (traditional, cable-stayed or suspended), covering different spans ranges according to the type;

• Underground tunnels: in which the crossing happens beneath the ground (usu-ally in presence of shallow waters), they take advantage of favorable geotech-nical conditions;

The final choice between these kind of structures obviously must consider the eco-nomic aspects, but, for the case of the SFT, the reliability of the structure is not easy to be determined, since no Submerged Floating Tunnel has not builted yet.

Briefly, a SFT consists in a hollow prismatic structure (with circular section in the major part of the cases) of concrete and steel, placed underwater, sized in order to contain road tracks and all the related services. Its key feature is provided by the medium in which it is immersed (water), that loads the structure with a buoyancy

force. Considering the weight of the tubular element, the net resulting force is very important, since it determines the static and dynamic behavior. In this context, the tunnel can float with a positive buoyant force, for which tension legs or floating pontoons are usually employed, or with a net negative force, where piers or columns act as foundations for the structure, fixing it to the bottom (the related height’s limitation is about 100 m). The overall density of the tunnel is more or less the same of the water’s one.

Figure 1.1: Concepts of a SFT with different anchoring systems [16].

The Submerged Floating Tunnel has some features that make it a competitive solution for:

• Fixed depth in water of 20 − 50 m, in order to avoid problems related to water pressure;

• Crossing distances of different kilometers;

• In those environment for which the landscaping and natural aspects are very important.

Figure 1.2: The visual effects of an SFT in an environment characterized by deep waters are minimal [16].

1.1. Introduction to the Submerged Floating Tunnel

1.1.2 Design principles and construction methods

The resultant design of this type of structure consists in a mix between the con-cept of offshore structures and the construction theory of immersed tunnels. The two main themes are safety and applicability, so the result must express these two concepts at the same time. A synthetic, but complete, description is provided by Prof. Amol B. Kawade and Miss. Shruti P. Meghe [1].

A first aspect to deal with is the study of the tubular structure behavior in water, because the SFT is subjected to different environmental loads. So, before taking into account safety and operational requirements, some basic principles must be followed:

• Carefully design the buoyancy to weight ratio, that it can be greater than 1 (with optimal values in the range [1.25 − 1.4]) if kept in its position by a tethering system, or it can be less than 1 if suspended to barges, pontoons or floating islands;

• Consider the differences in structural requirements (strength, stiffness and stability) during the constructional and operative stages;

• Provide to the tunnel a shape characterized by a gentle curvature, in order to be protected against hydrodynamic forces due to corrents (hydrodynamic coefficient will be analyzed in the following chapters).

Once guaranteed that the structure “stands alone” (so good that cables are needed to hold it in place), the road track and all related services needed to assuring safety and functionality can be considered in the design process. It must be considered that the SFT is a kind of construction that is subjected not only to continuous/stationary loads (due to wind, waves and sea currents), but also to temporary/transient loads (seaquakes and sea storms).

Under construction, the tunnel is assembled starting from different sealed sections, floated and sunk to their place and, when fixed, the seals are broken. Another option consists into build the sections unsealed, and after welding operations, the water is pumped out.

Schematically, the Submerged Floating Tunnel can be divided into three main components:

• Tubular structure

Usually designed with an external circular, elliptical or polygonal section. The used materials are steel and concrete, keeping into account the corrosion

prob-lematics. As mentioned before, the structure is divided in many sub elements, with length varying between 100 - 500 m. It’s easy to understand how the ex-ternal diameter (controlling the volume) is important from the point of view of tunnel’s net buoyancy.

• Anchoring system

It represents one of the first designed element of a SFT. It can be of various types, like showed in figure 1.3:

– SFT with pontoons

It isn’t affected by water depth effects, but, being positioned at free sur-face level, is sensitive to wind, waves, currents and the possibility of ships collision must be considered. In order to guarantee the system’s safety, the design must be developed in such a way that if one of the pontoons is lost, the structure survives.

– SFT supported on columns

The vertical, compressed, elements are added in order to prevent the structure’s sinking (so the buoyancy to weight ratio is less than 1). Sub-stantially it’s an underwater bridge, and so the stronger limit is repre-sented by the water depth, that must not exceed a few hundred meters. – SFT with tethers to the bottom

Due to the fact that the tethers are in tension during all the lifetime of the structure, cables, as alternative of bars, can be used for sea ground connection. The main differences between the two elements is that only bars can bear compressional loads and, looking to the deformed shape, they are characterized by a straight configuration, while cables shows a catenary type initial shape. Using bars can provide a better lateral stiffness (especially if steel is employed).

– SFT unanchored

The unique anchoring system is present at the tunnel’s ends. It is in-dependent from depth, but obviously the limit is here represented by length, even if its approximated value is already unknown (more studies are necessary).

• Shore connections

The main goal is to connect the flexible tube immersed in water with the rigid part bored in the ground. So, the joints should be able to accommodate the tube’s movements (for example free longitudinal expansion), and at the same time they must guarantee water tightness. In seismic areas, like the ones considered for the seaquake, the joints must be able to care about different displacements between the two shores. This problem represents a separate chapter, object of study of the structural branch of civil engineering.

1.1. Introduction to the Submerged Floating Tunnel

Figure 1.3: The SFT represented with different anchoring systems.

1.1.3 Historical development

Differently from the older underwater tunnels, the first proposals of floating tunnel came out in Europe only in the late nineteenth century. S. Pr´eault was the first one to propose this kind of solution, in the 1860, in order to cross the Bosphorous, as an underwater railway founded on piers, with spans of 150 m, located 20 m below the surface. Next, a solution of crossing the English Channel with a floating tunnel was proposed by Edward Reed, in 1882, but it was rejected by the English Parliament for fear of invasions.

Norway certainly represents a country in which the theory of the SFT has been deeply studied and discussed [2]. Since 1923, Archimede’s Bridge has been consid-ered as solution to cross the fjords in a quick way, better than underground tunnels, whose are limited by great water depths. The most well known evaluated case in Norway is Høgsfjord [4], but it is never been realized due to political reasons.

The change of pace in the analysis of solutions contemplating the SFT starts in the 1969, when Alan Grant made a prize-winning proposal for a 5.3 km Archimedes Bridge between Calabria and Sicily, i.e. over the Messina Strait, characterized by a water depth of 350 m. Anyway, mostly due to the sinking ship issues, at the end of the 20th century the detailed studies was left apart in favor of the much discussed bridge (only in the last years the SFT is come back in the scene).

20 years later, in the 1989, a working group for immersed and submerged floating tunnels was established by ITA (International Tunnel Association), which made it catch the world-wide attention [2], especially in Italy, Norway, China, Japan, USA and other countries. The association holds each year several meetings, where a number of technical papers are also presented and published. Another great job of the Working Group consists in the constant update of a searchable database, with public access, with the aim to remove the mystique and explain the methodology and terminology.

After this notable historic moment, specialists starte to work on this new topic not only on the theoretical and numerical fields, but also on the design and construction aspects with growing interests. A large number of proposal were studied all over the world.

In 2004, a Protocol of Scientific and Technological Cooperation between the Peo-ples Republic of China and the Italian Republic was signed for the Sino-Italian Joint Laboratory of Archimedes Bridge (SIJLAB), which let to renew the global interest and encouraged the SFT development [3]. The group proposed a prototype, in the artificial Qiandao Lake, in China, of the SFT. It was located above a maximum wa-ter depth of 30 m, with the tunnel axis placed 9.2 m below the still wawa-ter level, and long about 100 m (divided into five sub-modules of 20 m). The tubular element was constituted by an internal steel cylinder, a middle reinforced concrete layer, and a protective aluminum coating, for an external diameter of 4.4 m [5]. It was intended to be used for research purposes for some years, and then devoted to pedestrian usage and tourist attraction. Afterwards, much wider and deeper researches were based on this project. Although the project is now in a stand-by situation waiting for financial support, the obtained scientific achievements are profitable for further research or other SFT proposals.

The study of the Submerged Floating Tunnel consists in a multi-discipline ap-proach, due to its nature and location that influence the dynamic behavior. Con-sidering the structure as immersed in water and composed by tubular and cable elements, the subjects analyzed are:

1.1. Introduction to the Submerged Floating Tunnel

• Seismic engineering: by the point of view of both support-transmitted and fluid-transmitted loads;

• Fluid-structure interaction: that leads to the definition of forces due to sea waves and vortex shedding;

• Vehicle-structure interaction: resulting in forces due to traffic loading;

• Phenomena related to impact and explosions: applied to both internal and external events;

• Geotechnical engineering: to study the behavior of seismic signal through dif-ferent layers of flexible soil strata.

From a structural point of view, many researchers have developed different methods, with different instruments and theories, in order to evaluate the effects of an earth-quake on the structure. M. Di Pilato et al. [7] described a numerical procedure for the dynamic analysis of the SFT, implementing hydrodynamic and seismic loads. An interesting review of the researches carried on by the research group is provided by F. Perotti [6]. The response of the system due to seaquakes was analyzed by Shi in her PhD Thesis work [9], and by Martinelli et al. [8]. Dong Man-sheng and Li Man studied the problem of an SFT subjected to seismic loads by using the anal-ogy of the elastic beam on elastic foundation [10], and analyzed the behavior of the submerged tunnel under the combined effects of waves and currents [11].

Regarding the effects of an earthquake by an hydrodynamic point of view, i.e. the seaquake, C. Li et al. [12] have evaluated the motion and stress field in water layer due to a seaquake by adapting the theory used in soil-structure interaction from Wolf [13]. Another attempt to evaluate the effects of a seaquake was made by Hamamoto [14], by imposing more complete boundary conditions, but taking into account only the vertical propagation of seismic P-waves.

The goal of this thesis work is also to compare different methods used for recon-struct the motion field due to a seaquake, that provide a little bit different boundary conditions, and try to understand how the change of one of those changes the final result, in terms of motion and so in terms of forces on the SFT.

1.1.4 Strengths and weaknesses

Due to the fact that it is a totally different type of structure with respect to bridges or underground tunnels, the Submerged Floating Tunnel is characterized by its own features. Summarizing, the pros of a Submerged Floating Tunnel are:

• No environmental impact, due to the underwater positioning; • The possibility to take advantage from water buoyancy;

• It doesn’t create problems for ship passing, if the tunnel is located enough deep;

• The construction methods used let to build the different elements in densely populated area and then moved to the definite site;

• Its relatively easy to remove at the end of its lifetime;

• The cost of this type of structure is only linearly proportional to its length; • Theoretically, it can reach spans of any length;

• Being totally submerged in water, it’s not subjected to wind actions, and the effects of water waves are lower with respect to the same, due to wind, on a bridge;

• The mechanical soil properties are less binding than other types of structures.

Figure 1.4: Example of a Submerged Floating Tunnel in an urban area [17].

Obviously, there are not only positive aspects that must be kept in count, but there are some critical aspects:

• Very little experience is available for this relatively new type of structure; • Due to the unusual location of the tunnel (water), more care in the safety

assessment is needed;

• Even if the construction costs are more or less linear with the length, much resources must be employed for preliminary researches and experiments; • Only very complex studies can include in the model different accidental loads

like ship collisions or terroristic attacks.

Indeed, the numerous pros let to understand why this type of structure has been largely studied during the last years. But, from the other side, the cons are related

1.1. Introduction to the Submerged Floating Tunnel

mostly to safety reasons, so the hard work consists into convince the public on the enormous potentiality of the Submerged Floating Tunnel.

1.1.5 Forecasts for the Submerged Floating Tunnel

Until now, no Submerged Floating Tunnel has been built, even that during these years many proposal has come out.

Among all, in Norway, a 40-bilion $ infrastructure project has been planned in order to solve different problems that bother the efficiency of the road system of the country. In fact the almost 700-mile trip between the cities of Kristiansand in the south and Trondheim in the north typically runs about 21 hours, at an average speed of about 30 miles per hour. The project consists into the substitution of the numerous ferries along the way (seven) with bridges, conventional tunnels and what could be the world’s first “floating tunnel”. This solution would help to split in half the travel time between the two previous mentioned cities.

Figure 1.5: Norway’s fjords, for which the SFT could represent an optimal solution.

A futuristic project has been proposed by the National Advisor Bureau in the UAE in order to create a rail connection to India, from Fujairah to Mumbai, with the aim of improving the bilateral trade between the two nations. In this optic, a Submerged Floating Tunnel could be realized in order to help the passage of the road in sea tracks.

Anyway, the above mentioned project could be seen like a prove of the increasing demand for transporting large number of people and goods over long distances. So, why don’t think to a very long SFT? If one thinks to the velocity that trains can nowadays reach (at least 600 km/h), this kind of transport could represent a great alternative to connect continents.

Obviously, many efforts must still be done in order to assure the maximum relia-bility of the structure to all types of loads and conditions, in order to persuade the public opinion of the efficiency and safety of the Submerged Floating Tunnel.

1.2. Design features

### 1.2

### Design features

1.2.1 Fundamental aspects

Thinking to a new project of Submerged Floating Tunnel means basically to face two very important aspects:

• Placement and length

It is simply given by the distance between the two shores. The SFT is thought for long crossings, so its convenience with respect to other types of structures increases with length. For very long tunnels, the end connections to the shore become a critical aspect of the design, especially when high seismic forces (and high differential displacements) must be considered.

• Depth

This aspect is strictly related to the necessary space for surface navigation; considering important sea-strait crossings, where large ships can navigate, a 30 - 40 m clear depth is deemed to be sufficient.

Regarding the cross section of the tunnel, it can be: • Circular

It has a marked effect on the hydrodynamic and structural behavior. However, it is also decided on the basis of functional requirements; for instance, the number of lanes for cars or railroad tracks and the various types of services that must be considered [3]. The internal diameter should be large enough to accommodate them and to ensure the normal required operation. The external diameter, on the other hand, has the prominent influence on the ratio between water buoyancy and tunnel weight, which is expected to be larger than one. It was detected that the increase of the ratio from 1.25 to 1.4 can lead to impressive improvements of the SFT response to extremely severe sea states. In general, a large ratio can improve structural performances under severe environmental loads; it is worth noting, however, that high net buoyancy forces can result in very high tension forces on the foundations under extreme loading, this is an aspect that must be carefully considered in terms of feasibility and cost. Besides that, the cross-section must have enough strength to be always kept in the elastic region for any loading condition, to avoid any type of cracking in order to meet the water tightness requirement.

• Other shapes

There are some different types, like circular, polygonal or elliptical, rectangular and circular tubes enclosed inside an external shell having streamlined shape

[15]. The configuration depends in a great measure on traffic lanes and related facilities.

Figure 1.6: Example of Submerged Floating Tunnel with circular section [16].

Regarding the material, in FEHRL (1996), it is mentioned that the selection of the materials to be used when building a Submerged Floating Tunnel must be made according to the structural and functional performance which are intended to be ensured, but it must also be a compromise among several factors such as resistance to the marine environment, fabrication, assembly and maintenance issues, time needed for supply, material and constructional cost, etc. Also, it is specified that the possible materials used in tunnel are steel, reinforced concrete, pre-stressed reinforced concrete, aluminum alloys and rubber foam [15]. The most acceptable and reasonable material, having a large and experienced application in other immersed tunnels, is the steel-concrete composite one, which has higher strength and good resistance to fatigue from the steel material, and higher resistance to the corrosion and heavy beneficial weight (especially in SFT) from the concrete material.

Referring to the supporting system, the tunnel body will move upward because of the positive net difference between water buoyancy and tunnel weight if there were no anchoring system. Or, as to the other supporting system (with pontoons floating on the water surface), the tunnel will sink down to the seabed because of dead weight. As a result, the supporting system seems to be essential in the pre-liminary design. The two different load-carrying systems, seabed anchoring system and pontoon, can be seen as alternatives. The latter one is only accustomed to very calm environments. More detailed proposals are based on seabed anchoring system, which can be provided by means of anchoring bars or cables. Until now, the anchor-ing bar solution seems to be more competitive in complex environment, as it will be discussed in detail. When the anchoring bars are selected, the cross-section should be made in detail. Hollow sections seem to be the best choice, because they can have almost neutral buoyancy and higher lateral stiffness efficiency. The bars’ material is steel with high strength and good resistance to fatigue, which is of common usage

1.2. Design features

in offshore engineering.

1.2.2 Loading conditions

There are many external loads that must be considered in SFT design, due to the complex environment, like hydrostatic pressure, buoyancy, current, traffic loading, the seismic motion and so on. Technically, to make it clear, they can be divided in the following groups:

• Permanent loads

Including dead weight of the structural and non-structural components, water buoyancy and hydrostatic pressure;

• Functional loads

Closely related to design function of the SFT, as traffic loads, contempling passage of cars, trucks, trains, or even pedestrian traffic; also, changes of load conditions during different construction periods, and even various ballast conditions must be considered.

Regarding the environmental loads, they consist in the following: • Hydrodynamic loads

The hydrodynamic actions play an essential role in the analysis and design of SFT. They are mainly generated by waves, which are produced by the action of wind on the water surface, and by currents due to different phenomena. Currents are also responsible of vortex shedding phenomena, due to the inter-action with fixed or moving slender bodies. All these types of loads are well known and commonly managed by designers of traditional offshore structures. • “Direct” earthquake loads

The strong ground motion occurring at the seabed propagates into the tunnel through the tunnel shore connection and anchoring system. Due to the fact the tunnel length it cannot be assumed, as in conventional structures, seismic motion cannot be assumed as equal and simultaneous at all support points; the so called multiple-support seismic excitation must be introduced, taking the spatial variability of the ground motion into account.

• Tsunamis loads

Tsunamis, huge devastating wave movements coming from the seabed, are po-tentially caused by the subsea earthquakes or volcano eruptions or other un-dersea explosions. As one characteristic, tsunamis have very long wavelength and small amplitude in open sea. When it arrives close to the shore or in shallow waters, the wave height increases sharply. The effect of tsunamis on a

SFT must be still thoroughly investigated, especially in view of 3-dimensional effects that can arise in sea straits.

• Seaquake loads

The seaquake consists of a hydrodynamic pressure variation propagating from the seabed to the tunnel. Limited studies have been performed on STFs under seaquake loads, but some similar researches on different offshore structure, like large platforms, were performed with the conclusion that seaquake had significant adverse influence on structural safety. This means that it is worth to devote some attention on this load when working on the SFT structural design.

The present thesis work regard a chapter to the analysis of hydrodynamic loads due to waves, with the goal to use the same basic fundamentals in order to observe the effects of seaquake loads on the structure.

At last, there are the so-called accidental loads, that cannot be neglected in the preliminary sketch. They are given in the following:

• External collisions with sinking objects or ships, or impacts with submarines; • Internal impacts of vehicles on the tunnel structure;

• Fire inside the tunnel body: this is also a headache to each tunnel crossing, like immersed or underground tunnels;

• Explosion inside the tunnel body; • Explosion outside the tunnel;

Bibliography

[1] Miss. Shruti P. Meghe Prof. Amol B. Kawade. Submerged floating tunnel. Amrutvahini College of Engineering, Sangamner.

[2] Christian Ingerslev. Immersed and floating tunnels. Procedia Engineering, 4:51 – 59, 2010. ISAB-2010.

[3] H˚avard Østlid. When is sft competitive? Procedia Engineering, 4:3–11, 12 2010.

[4] Rolf Magne Larssen and Svein Erik Jakobsen. Submerged floating tunnels for crossing of wide and deep fjords. Procedia Engineering, 4:171 – 178, 2010. ISAB-2010.

[5] Luca Martinelli, Gianluca Barbella, and Anna Feriani. Modeling of qiandao lake submerged floating tunnel subject to multi-support seismic input. Procedia Engineering, 4:311–318, 12 2010.

[6] Federico Perotti, Gianluca Barbella, and Mariagrazia Di Pilato. The dynamic behaviour of archimede’s bridges: Numerical simulation and design implica-tions. Procedia Engineering, 4:91–98, 12 2010.

[7] M Di Pilato, Federico Perotti, and P Fogazzi. 3d dynamic response of sub-merged floating tunnels under seismic and hydrodynamic excitation. Engineer-ing Structures, 30:268–281, 01 2008.

[8] Luca Martinelli, Marco Domaneschi, and Chunxia Shi. Submerged floating tun-nels under seismic motion: Vibration mitigation and seaquake effects. Procedia Engineering, 166:229 – 246, 2016. Proceedings of International Symposium on Submerged Floating Tunnels and Underwater Tunnel Structures (SUFTUS-2016).

[9] Chunxia Shi. Problems related to the seismic behaviour of submerged floating tunnels. PhD thesis, Politecnico di Milano, 2013.

[10] Dong Man-sheng and Li Man. The dynamic responses of the of the submerged floating tunnel under seismic effect. Procedia Engineering, 166:152 – 159, 2016. Proceedings of International Symposium on Submerged Floating Tunnels and Underwater Tunnel Structures (SUFTUS-2016).

[11] Dong Man-Sheng, Tian Xue-Fei, Zhang Yuan, and Tang Fei. Vibration control of the submerged floating tunnel under combined effect of internal wave and ocean current. Procedia Engineering, 166:160 – 170, 2016. Proceedings of Inter-national Symposium on Submerged Floating Tunnels and Underwater Tunnel Structures (SUFTUS-2016).

[12] Chao Li, Hong Hao, Hongnan Li, and Kaiming Bi. Theoretical modeling and numerical simulation of seismic motions at seafloor. Soil Dynamics and Earth-quake Engineering, 77:220 – 225, 2015.

[13] John P. Wolf. Dynamic Soil-Structure Interaction. Prentice Hall, 1985.

[14] Takuji Hamamoto. Stochastic fluid—structure interaction of large circular float-ing islands durfloat-ing wind waves and seaquakes. Probabilistic Engineerfloat-ing Mechan-ics, 10(4):209 – 224, 1995.

[15] Giulio Martire. The Development of Submerged Floating Tunnels as an inno-vative solution for waterway crossings. PhD thesis, Universit`a degli Studi di Napoli Federico II, 2010.

[16] Norwegian Public Roads Administration. The e39 coastal highway route: An-imation. https://www.vegvesen.no/en/roads/Roads+and+bridges/Road+ projects/e39coastalhighwayroute/film.

[17] Lidvard Skorpa. Developing new methods to cross wide and deep norwegian fjords. Procedia Engineering, 4:81–89, 12 2010.

### Chapter 2

## Loads acting on a submerged

## floating tunnel

### 2.1

### Wave forces acting on a submerged structure

As showed in a remarkable paper on the motion of pendulums, Stokes [3] showed that the expression for the force on a sphere oscillating in an unlimited viscous fluid consists of two terms: inertia, involving the sphere acceleration, and drag, involving velocity.

2.1.1 Inertia force

This type of action is correlated to the flow acceleration’s effects. Now, Newton’s second law of motion states that forces and acceleration are correlated, so from an acceleration in the water domain results a force, independently if the structure is present or not.

By integrating the pressure field around a cylinder, it is possible to distinguish two types of contributes:

• The first one, due to pressure gradient force, that is proportional to the mass of displaced fluid for unit length of the inserted object.

This component is fully equivalent to the Froude-Krilov force, which results from an integration of pressure field in the undisturbed wave domain.

• The second one, known as disturbance force, that considers the fact that the submerged object disturbs the flow.

By considering the element impermeability, the local velocities, and thus ac-celerations, result modified. This happens only if a force is exerted on the fluid, and this force comes only from the considered object.

A general interpretation of those two elements can be that pressure gradient in the fluid implicates an acceleration of the fluid, exerting a kind of buoyancy force on the object (that is the Froude Krilov contribution). The second term is present in order to consider the local pressure gradient caused by the fluid acceleration in the submerged element’s neighboring.

2.1.2 Drag force

By analyzing, again, the forces acting on a vertical cylinder, it is possible to recognize a steady flow term proportional to the squared velocity (for turbulent flow) and the cylinder diameter. It is caused by a constant current, and its direction is the same as the flow one.

In general terms, the drag is a force depending on fluid properties and by its size, shape and object speed.

2.1.3 Non-dimensional parameters

The analysis of the forces experienced by a submerged structure is largely simpli-fied by the usage of some hydrodynamic coefficients. Briefly, the study of drag and inertia forces to which many types of structures are subjected, is carried out by the usage of proper coefficients, in this case drag and inertia coefficients, that vary for different cases (fluid, element’s shape, velocity, and so on...). The advantage of the usage of those coefficients is represented by the fact that it is possible to include all the practical cases in the evaluation of the forces to which a structure is subjected with a single expression.

Like mentioned before, the two considered hydrodynamic coefficients are depen-dent by many variables, that are in some way summarized in non-dimensional param-eters. For example, the Reynolds and the Keulegan-Carpenter numbers determine the importance of drag force on the structure. Another parameter influencing both drag and inertia coefficients is the structure surface’s roughness, described from the relative roughness. The importance of the waves scattered from the structure surface is determined by the diffraction parameter. In the end, regarding vortex induced vibration, the phenomenon magnitude is analyzed by the Strouhal number.

2.1. Wave forces acting on a submerged structure

2.1.3.1 Reynolds number

The Reynolds number is defined as the ratio of inertial and viscous forces: Re = vL

ν (2.1)

where v is fluid velocity, L consists in a representative dimension of the considered scale and ν is the kinematic viscosity. After many experiments in the 1880s, Osborne Reynolds find out that the flow regime (laminar or turbulent) depends mainly on this non-dimensional parameter. At large Re (turbulent flow ), inertia forces are large relative to viscous ones, and thus those last cannot prevent fluid’s random fluctuations. Vice versa, at small or moderate Re (laminar flow), the viscous forces are large enough to suppress these fluctuations and to keep the flow “neat”.

The optimal situation is the one in which laminar, transitional and turbulent flows are defined for proper ranges of Reynolds number, but this is not the case in practice. Many other variables as surface roughness, fluctuations in the flow and so on must be considered in the analysis. Typically, in most practical conditions, that can be considered valid also for the SFT analysis, are:

• Laminar flow: Re ≤ 2300;

• Transitional flow: 2300 ≤ Re ≤ 4000; • Turbulent flow: Re ≥ 4000.

The transitional flow represents a switching between laminar and turbulent flow. It’s important to notice that in such carefully controlled experiments, laminar flow has been maintained at Reynolds numbers of up to 100000.

2.1.3.2 Keulegan-Carpenter number

As reported in the reference paper [4], this non-dimensional parameter determines the relative contribution of inertia and drag forces for bluff objects in an oscillatory flow. It is defined as:

KC = vT

L (2.2)

where v and L are again the velocity and scale terms seen for Reynolds number, while T represents the oscillation’s period.

Typically, looking to the SFT case, the length scale L is associated to the tunnel’s external diameter, ranging from the 4.4 m of the prototype used in the Qiandao Lake

[1] to the 15.95 m proposed for the Messina Strait project [2, 11]. Regarding the
velocity v and the period T , Shi [2] reconstructs the velocity field at the tunnel
depth, recognizing the natural frequency, for the Messina Strait case. The obtained
values of peak velocity and natural frequency, considering a depth of the seawater
layer of 325 m, are respectively of 3 m_{s} and 1.2 Hz (for a oscillation period value
of _{1.2}1 s = 0.83 s). So, typical values of Keulegan-Carpenter number results in this
case of 3

m s·0.83 s

15.95 m = 0.16, that correctly represents order of magnitude of KC for a

submerged tunnel. Later, it will be see how this value shows that inertia forces are dominant on drag ones for the most part of the considered frequency domain (even more true if the KC number decrease more).

2.1.3.3 Relative roughness

The relative roughness is computed by particles average size on the surface of a
certain object interacting with the flow, given by , normalized by the equivalent
cross-sectional diameter of the structure member D. In practice it is equal to _{D}.

This parameter is used in order to express in some way the friction factor into computing the hydrodynamic forces acting on an element in a flow, and it varies with the considered material (here reported in terms of absolute roughness, that is dimensional, and that must be related to the tunnel diameter):

Pipe material Absolute roughness [mm] Drawn brass 0.001524

Drawn copper 0.001524 Commercial steel 0.04572

Wrought iron 0.04572 Asphalted cast iron 0.12192 Galvanized iron 0.1524

Cast iron 0.25908 Wood stave 0.18288 - 0.9144

Concrete 0.3048 - 3.048 Riveted steel 0.9144 - 9.144

Table 2.1: Relative roughness for different pipe materials. [5]

In the evaluation of a certain material roughness, particular attention must be putted on the element’s age. The accumulation of corrosion byproducts and sus-pended particles on the walls can increase the roughness, and so the related frictional forces.

2.1. Wave forces acting on a submerged structure

2.1.3.4 Diffraction parameter

This particular parameter establishes the size of the scattered wave from the
surface of the structure, and it is defined by the ratio of the member diameter D
and the wave length λ, that is: πD_{λ} .

2.1.3.5 Strouhal number

This parameter is named after Vincenc Strouhal, a Czech physicist who experi-mented the vortex shedding phenomena. The Strouhal number is defined as:

St = fsL

v (2.3)

in which Lis again the characteristic length, v is the flow velocity and fsis the vortex

shedding frequency of a resting body.

The vortex shedding is a phenomena consisting in an oscillating flow that takes place when a fluid such as air or water flows past a bluff body at certain velocities, depending on its size and shape. Experimentally, this happens when water flow crosses the body, the fluid velocity around it is slowed down due to viscous effects, forming the so-called boundary layer. At a certain point, this boundary layer can separate from the body, causing non-symmetric vortexes that change the pressure distribution along the surface.

So, the object will tend to move toward the low-pressure zone. It is then possible to say that the frequency at which vortex shedding takes place, fs, is related to the

Strouhal number.

Figure 2.1: Vortex shedding phenomena due to wind (the same happens into water flow) [12].

The main effects of this phenomenon is the presence of a cross-flow action on the considered element, behaving like a vibration, and it is thus subjected to resonance phenomena.

2.1.4 Small and large submerged structures

The main hypothesis made into computing wave forces on a structure consists into considering it fixed in its equilibrium position. Starting from this, the main distinction is made between:

• Small structures: for which the Morison simplified approach is used;

• Large structures: requiring the diffraction/radiation theory, with the first one determining the forces on the structure due to waves in its equilibrium po-sition, while the second one considers the moving structure in water. The knowledge of exciting forces and of the related hydrodynamics coefficients, let to compute the structure’s motion, using the equations of motion for each de-gree of freedom (six) of the structure itself. Later, the drag effects on smaller members of the structure may be included.

The necessity of this distinction is due to the fact that at the present there is no reliable procedure for calculating the wave interaction with a structure for all the possible conditions of interest. For example, by examining the wake phenomenon (region behind a certain object immersed in a flow in which some turbulence and flow separation phenomenon are observed), it is immediate to understand that this case of separate flow is impossible to treat analytically. Not only, but considering the most of practical cases, that are characterized by turbulent flow (high Reynolds numbers), lead to the fact that the irrotational flow hypothesis, with small-amplitude waves and small velocities, used in linear wave theory, is not properly correct.

In case of structures large with respect to the wave length (see the diffraction parameter), wake effects are not important, so inertia forces dominates and accu-rate calculation methods exists (diffraction theory). When the considered body is smaller, wake plays a dominant role on both the drag and inertia force components, and roughness characteristics of the object are also significant, here the simplified approach of Morison equation is applied.

2.2. The linear wave problem

### 2.2

### The linear wave problem

2.2.1 Water motion field reconstruction

A simplistic definition of wave represents it as a perturbation travelling in space and time. In a fluid domain, for example a sea, it can be originated by each pertur-bation interfering with the static equilibrium of the water (ships, earthquakes, living organisms and so on...). A common problem in ocean engineering is the one related to sea waves originated by wind, inducing a friction over the sea surface. This kind of waves are called “gravity waves”, because gravity has in this case an equilibrium restoring function.

Due to the fact that a wave travels in space and time, its description in a 3-D field depends generally on four independent variables (three spatial and one temporal). In order to get a simplified explanation of the problem, plane waves are considered in this chapter, assuming that crests of the wave are infinitely long with constant elevation in the transversal direction. In other words, there is no movement of fluid along the normal to the wave travelling direction.

Fluid mechanics teaches how the fluid motion is completely described by the velocity vector, varying with time and space. In the considered plane problem:

v (x, z, t) = vx(x, z, t) ix+ vz(x, z, t) iz (2.4)

The first assumption introduced, that states that fluid can be considered as incom-pressible, leads to the following mass conservation equation, applied to an elementary control volume:

∂vx

∂x + ∂vz

∂z = 0 (2.5)

In order to discover the water motion field, another hypothesis, of irrotational fluid, is made, that is:

∇ × v = 0 (2.6)

(in which the Nabla operator ∇ is a vectorial differential operator, returning a scalar quantity that is the sum the three directional derivatives of a function) this assumption is more reasonable for inviscid fluid. The vorticity (i.e. ∇ × v = 0) of a fluid element cannot change except through the action of viscosity or some other nonuniform phenomena (here neglected). Thus, if a flow originates in an irrotational

Figure 2.2: Infinitesimal cube element on which mass conservation is verified.

region, it remains irrotational until some non-uniform process alters it.

So, the velocity components can be expressed with the usage of a single scalar function, called the velocity potential φ (x, z, t):

vx = − ∂φ (x, z, t) ∂x (2.7) vz = − ∂φ (x, z, t) ∂z (2.8)

By substituting the two expressions in the mass conservation equation, it can be obtained the so called Laplace equation:

∂2φ ∂x2 +

∂2φ

∂z2 = 0 (2.9)

that is a second-order partial differential equation. Laplace equation is the only needed element letting to describe the motion of waves at high distance from the boundary.

In fact, the motion in the proximity of bottom and surface is also influenced by the initial and boundary conditions. However, their effect vanishes with increasing distance from the boundaries . An example of this can be observed in a one dimen-sional flux in the horizontal direction, with the no slip boundary condition at the wall: the rotational and irrotational regions can be distinguished, as showed in the picture 2.3.

2.2. The linear wave problem

Figure 2.3: Example of the effects of a no slip boundary condition on motion field [13].

Laplace equation is valid in all the showed domain in figure 2.4, but in order to solve it, some boundary conditions are needed:

• Seabed boundary condition, that must be impervious, and therefore the velocity component along z must be null:

vz(x, z = −h, t) = −φz = 0 at z = −h (2.10)

• Kinematic boundary condition along the water surface, imposing that fluid particles located at free surface must remain there during wave travel (this assumption is ok if the wave doesn’t break), and denoting the vertical coordinate of the wave surface as z = η (x, t)

dz = dη (x, t) = ∂η ∂xdx + ∂η ∂tdt (2.11) dz dt = ∂η ∂x dx dt + ∂η ∂t dt dt (2.12)

the following condition is obtained:

vz = ηxvx+ ηt at z = 0 (2.13)

• Dynamic boundary condition at the free surface, referring to forces equilibrium, so expressed with the usage of Bernoulli’s principle and consid-ering that relative pressure at the water interface with atmosphere is null:

η + 1 2g φ 2 x+ φ2z − 1 gφt= 0 (2.14)

boundary conditions are not linear and the shape of free surface is unknown.

Figure 2.4: Water domain in which linear wave problem is solved.

The previous equations are suitable simplified by considering the wave amplitude
much smaller than its length scale for variations in x- direction. So, the simplified
system results:
φxx+ φzz = 0
φz= 0 at z = −h
φz+ ηt= 0 at z = 0
η − 1_{g}φt= 0 at z = 0
(2.15)

this problem can be solved by using the separation of variables method (see appendix A), and this lead to the final solution:

φ = H 2 ω k cosh [k (h + z)] sinh (kh) sin (ωt − kx) (2.16) Where H is the wave height at the free surface, ω and k are respectively the circular frequency and the wave number. From this solution it is possible to evaluate the motion field in x and z directions in all the considered domain, due to the gravity waves.