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Bull Earthquake Eng (2008) 6:213–239 DOI 10.1007/s10518-007-9055-0 ORIGINAL RESEARCH PAPER

Behaviour of deep immersed tunnel under combined normal fault rupture deformation and subsequent seismic shaking

Ioannis Anastasopoulos · Nikos Gerolymos · Vasileios Drosos · Takis Georgarakos · Rallis Kourkoulis · George Gazetas

Received: 22 June 2007 / Accepted: 25 November 2007 / Published online: 18 December 2007 ? Springer Science+Business Media B.V. 2007

Abstract Immersed tunnels are particularly sensitive to tensile and compressive deformations such as those imposed by a normal seismogenic fault rupturing underneath, and those generated by the dynamic response due to seismic waves. The paper investigates the response of a future 70 m deep immersed tunnel to the consecutive action of a major normal fault rupturing in an earthquake occurring in the basement rock underneath the tunnel, and a subsequent strong excitation from a different large-magnitude seismic event that may occur years later. Non-linear ?nite elements model the quasi-static fault rupture propagation through the thick soil deposit overlying the bedrock and the ensuing interaction of the rupture with the immersed tunnel. It is shown that despite imposed bedrock offset of 2 m, net tension or excessive compression between tunnel segments could be avoided with a suitable design of the joint gaskets. Then, the already deformed (“injured”) structure is subjected to strong asynchronous seismic shaking. The thick-walled tunnel is modelled as a 3-D massive ?exural beam connected to the soil through properly-calibrated nonlinear interaction springs and dashpots, the supports of which are subjected to the free-?eld acceleration time histories. The latter, obtained with 1-D wave propagation analysis, are then modi?ed to account for wave passage effects. The joints between tunnel segments are modeled with special non-linear hyper-elastic elements, properly accounting for their 7-bar longitudinal hydrostatic pre-stressing. Sliding is captured with special gap elements. The effect of segment length and joint properties is explored parametrically. A fascinating conclusion emerges in all analysed cases for the joints between segments that were differentially deformed after the quasi-static fault rupture: upon subsequent very strong seismic shaking, overstressed joints de-compress and understressed joints re-compress—a “healing” process that leads to a more uniform deformation pro?le along the tunnel. This is particularly bene?cial for the precariously de-compressed joint gaskets. Hence, the safety of the immersed tunnel improves with “subsequent” strong seismic shaking!

I. Anastasopoulos · N. Gerolymos · V. Drosos · T. Georgarakos · R. Kourkoulis · G. Gazetas (B ) School of Civil Engineering, National Technical University of Athens, Heroon Polytechniou 9, 15780 Zografou, Athens, Greece e-mail: gazetas@ath.forthnet.gr

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Keywords Immersed tunnel · Fault rupture propagation · Dynamic analysis · Earthquake · Soil–structure interaction · Fault rupture–soil–tunnel interaction

1 Introduction The objective of this paper is to investigate the behaviour of a deep immersed tunnel under the combined action of a major fault rupturing underneath, and subsequent strong seismic shaking. The presented research is part of a feasibility study for a railway immersed tunnel in the Rion–Antirrion straits, in Greece. The construction of the tunnel will be a technological challenge, due to the combination of great water depth (67 m; presently the world record), very high seismicity, and deep alluvial soils. At the narrowest crossing of the Corinthian Gulf trench, which is characterised by an extensional slip rate of about 1 cm/year, the tunnel will unavoidably cross at least one active fault. It must be capable of safely undertaking the ensuing permanent ground deformation. Moreover, in the 100-year design life of the tunnel, strong seismic shaking from another earthquake, originating at a different major fault located not directly under the site, is a distinct possibility given the numerous active faults in the Corinthian and Patraicos Gulfs, as well as in their neighboring regions. Such seismic shaking will ?nd the immersed tunnel already “injured” from the faulting-induced deformation of the ?rst seismic event (which will have occurred perhaps several 10s of years earlier). It is crucial for a successful design to ensure that the permanent tensile deformation due to the normal-fault rupture, and the superimposed subsequent dynamic deformation during shaking will not jeopardize the water-tightness of the tunnel at any moment. Figure 1 sketches the two loading situations studied in the paper.

2 A future Rion–Antirrion railway link The feasibility of a railway link of the Rion–Antirrion straits in Greece is being explored (Fig. 2). The straits will be crossed at approximately 300 m east of the recently built cablestayed road bridge. At this narrowest point, the under-water length is about 2.5 km, with a maximum depth of 70 m, nearly constant for about 1 km (Fig. 3). A hybrid tunneling solution has been developed combining a 1-km central immersed tunnel at the deepest section of the crossing, with two bored “approach” tunnels (of about 8 km total length) at the two sides. Designed to accommodate two-way rail traf?c, the immersed tube will be about 24 m wide and 11 m high. While more design details can be found in Anastasopoulos et al. (2007a), it is emphasised here that the huge hydrostatic loading necessitates that walls be 1.5 m thick. The longitudinal cross section of the central immersed part is depicted in Fig. 3, along with an indicative soil classi?cation and NSPT blow counts, based on the exploration that had been conducted for the neighboring bridge. The soil consists of alternating layers of sandy gravel to gravel, silty sand and clay, of medium density and stiffness. The geotechnical exploration reached a maximum depth of 100 m below the seabed, without encountering bedrock. A detailed geophysical tomography (Tselentis et al. 2004) revealed that limestone bedrock (V p > 3, 500 m/s) lies at about 800 m depth. Geophysical data and earlier shear wave velocity Vs measurements were utilised to derive six idealised soil pro?les. The Rion–Antirron straits, the narrowest crossing of the Corinthian Gulf trench, is associated with a tensile tectonic environment of N–S direction, expressed by a sequence of normal faults of E–W strike, dipping towards the axis of the trench (Fig. 2). Being a very active seismic zone, it has been the source of ?ve Ms > 6 earthquakes in the last 30 years. The most

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(a) FAULT RUPTURE DEFORMATION : QUASI–STATIC

Tensile opening : De-compression Additional Compression

Fault Rupture h

+

(b) SUBSEQUENT SEISMIC SHAKING : DYNAMIC

De-compression + Re-compression

accelerationsα i (t)

i

i+1

bs

Fig. 1 Problem de?nition—the two critical types of seismic loading that may take place in the life of the tunnel: (a) quasi-static deformation due to normal fault rupture underneath the tunnel and (b) subsequent strong seismic shaking in the form of shear wave incident at an angle βs

recent event was the Ms 6.4 Aegion 1995 earthquake (Bernard et al. 1997; Gazetas 1996), with an epicentral distance of 28 km from the site. The straits are also affected by more distant seismogenic zones.

3 Seismic design of the proposed immersed tunnel Until today, no immersed tunnel is known to have ever been “hit” by an underneath rupturing fault. On the other hand, two immersed tunnels have been subjected to fairly strong seismic shaking: the Bay Area Rapid Transit (BART) tunnel in California, and the Osaka South Port (OSP) immersed tunnel in Japan. Both tunnels behaved exceptionally well, sustaining no measurable damage. BART, built in the 60s, was one of the ?rst underground structures designed against seismic loading and equipped with special 3-dimensional joints (Kuesel 1969; Douglas and Warshaw 1971; Bickel and Tanner 1982). It survived the 1989 Loma Prieta Ms 7.1 earthquake with no damage, sustaining only a small relative displacement between

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Lake Trihonis

Active Fault Potentially Active Fault 0 10 20 (km)

(M 5.5,

2007)

Psathopyrgos Fault

Antirrion Rion

Pa tra s

s Patraiko

Patraikos Fault

Gulf

Hel Aegion i (M ki Fau 7, 1 l 868 t )

Aegion Fault (M 6.4, 1994)

Gulf of Corin th

Pyrga ki–M amou ssia F ault

PELOPONNESUS

Fig. 2 Rion–Antirrion straits, along with active and potentially active faults, and the magnitudes of some recent earthquakes

0

RION

ANTIRRION

20

40

Bored T

Immersed Tunnel

unnel

De pth (m )

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T un ne

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50

120

0 50

NSPT

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NSPT

50

GW S ML 4+500 5+000 5+500

CH CL

6+000

Length (km + m)

Fig. 3 Longitudinal section of the proposed tunnel, along with indicative soil classi?cation and NSPT blow counts, based on available geotechnical data

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217

Gina Profile (compressed)

Shear Key

Omega Profile Tendon Coupler Tendon

Fig. 4 Schematic illustration of the aseismic design of immersion joints: gina gasket are compressed during installation, providing the necessary water-tightness; omega seals and tendons with couplers are used as secondary protection measures; shear-keys are installed to restrict lateral and vertical differential displacements between consecutive segments

end segments and approach structures (PB 1991). OSP performed also exceptionally during the 1995 MJMA 7.2 Kobe earthquake. While the above tunnels are located at shallow depths, we note the recent trend of installing immersed tunnels at great depths, such as the currently under construction Bosporus Crossing immersed tunnel in Istanbul (56 m deep, 1.5 km long) and the South Korean Car Tunnel (47 m deep, 3 km long). The proposed design of the studied tunnel is similar to that of the Osaka South Port tunnel (Fig. 4). The precast concrete segments, ?oated over a pre-excavated trench, will be lowered with the help of special sinking rigs. Once the two segments gain contact, the water between them will be drained, and the Gina gasket will be compressed due to one-sided hydrostatic water pressure, achieving the necessary water-tightness. Given the 650 kPa (6.5 bar) pressure, the use of the largest available Gina gasket is unavoidable: under compressive loading, rubber gaskets may fail due to tensile lateral strains (a Poisson type effect; see Kelly 1997). Although the largest available Gina section is estimated to be capable of undertaking the loading, special analysis and testing will be a prerequisite for acceptance. Secondary lines of defense include the “omega” seal and longitudinal tendons. But since none of them can sustain substantial shear, “shear keys” are also constructed once the segments have been connected. The scope of this paper is to analyse the behaviour of the tunnel under a sequence of: (a) normal fault-induced displacement, and (b) subsequent strong seismic shaking. The behaviour of the tunnel subjected to a variety of seismic shaking scenarios has been investigated in Anastasopoulos et al. (2007a). It was shown that the tunnel could safely resist a variety of soil-ampli?ed strong excitations, of PGA as large as 0.60 g, PGV as large as 80 cm/s, and containing long–period pulses. In the present paper we parametrically investigate three possible segment lengths: 70, 100, and 165 m. While the modern economic trend (e.g. Marshall 1999) is to use longer segments, the combination of such depth (67 m) with the vivid tectonic environment make the technical feasibility the foremost priority of the design. The type of Gina gasket is also parametrically investigated, as the longitudinal deformation of the tunnel depends on its properties, and ensuring its impermeability is critical.

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8

20 cm 2 5 cm

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6

(Kiyomiya, 1995)

Load (MN/m)

36 cm

Gina Type - A 4

Gina Type - B

40 cm

2

72 cm

0 0 10 20 30 40

Compressive Displacement (cm)

Fig. 5 Hyperelastic force–displacement backbone curves of rubber joints used in the analysis: Type A refers to the largest available GINA gasket, estimated based on half-scale tests of Kiyomiya (1995); Type B is a hypothetical logical projection

Fault-induced extension and differential settlement will unavoidably decompress at least some of the joints. Additionally, during a subsequent strong seismic event, that may occur years later, the longitudinal vibration of the tunnel will subject the immersion joints to cyclic (dynamic) re-compression and de-compression. The magnitude of the total de-compression is critical for the design of the tendons. If such de-compression is signi?cant, the tendons will have to engage, undertaking large tensile forces to ensure water-tightness. The problem is that such tensile loads will eventually be undertaken by the concrete, leading to unavoidable cracking and loss of water-tightness. Two types of joints (Fig. 5) are parametrically investigated. Type A refers to the idealised behaviour of the largest currently available Gina pro?le, while type B is a hypothetical double-sized Gina-type gasket. The behaviour of this hypothetical gasket provides wider deformation limits, permitting signi?cant additional compression and de-compression. The hyper-elastic load-displacement backbone curves of the two types of gasket are estimated on the basis of test results of half-sized models (Kiyomiya 1995). Another problem parameter is the shear-key “allowance”: if large enough, some differential transverse displacement between consecutive segments will be allowed; its reduction will lead to a more-or-less “?xed” connection. Although its effect has already been shown to be minor (Anastasopoulos et al. 2007a), it is parametrically investigated for completeness. Two extreme values are explored: 5 and 20 mm.

4 Methodology of analysis The ?nite element code ABAQUS (2004) is utilised to perform non-linear quasi-static and dynamic transient analysis of the tunnel. The layout of the model is depicted in Fig. 6. The immersed tunnel is modeled as a series of beams connected to the soil through interaction springs and dashpots. Tunnel segments are modeled with special beam elements that take account of shear rigidity. As discussed in detail in Anastasopoulos et al. (2007a), each

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u nn el

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in ion Jo Immers

t

dT Bore

z

y

x

e nt S egm ) m (bea Bore d Tu n nel

gs & Sprin ots p Dash

gs & Sprin ots p Dash

Immersion Joint z x

Immersion Joint

Segment (beam)

Segment (beam)

Segment (beam)

Fx Gina

Immersion Joint

Fx

?x Friction Segment (beam) ?x ?x

Fy Fz k

“slider” c Fy Passive + Friction ?ky ?y

?y ?y ?z

Shear Key Allowance

placement -induced dis Step 1 : Fault

xi-1 /Cα

t

xi /Cα

t

xi+1 /Cα

t

Fy

Step 2 : Asynchronous Seismic Shaking

Fig. 6 Finite element modeling layout: in the ?rst step, the fault-induced displacements are applied to the tunnel quasi-statically. Then, in a second step we apply the asynchronous seismic excitation

immersion joint is modeled with special 64-node frames representing the perimeter of the tube–collar connection. Adjacent frames are connected with each other with non-linear hyperelastic springs, simulating the behaviour of the segment joints. The analysis is conducted in three steps: ? In “Step 0” the hydrostatic pressure is applied statically to the end of each segment, to simulate the initial hydrostatic compression. The details of the installation procedure, along with a discussion on time-dependent stress relaxation effects can be found in Anastasopoulos et al. (2007a). In “Step 1” the fault-induced displacement pro?le (computed separately with 2-D planestrain fault rupture propagation analysis) is applied quasi-statically on the tunnel supports.

?

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?

At the ?nal “Step 2”, the model is subjected to dynamic excitation. The ground motion is ?rst “propagated” through the alluvial deposit to take account of soil ampli?cation effects. The resulting ground motion (at seabed) is then applied along the tunnel supports with a time lag.

The reverse situation, i.e. dynamic excitation (Step 2) followed by fault-induced displacement (Step 1), is not reported herein. Ground shaking of the un-deformed structure has been explored in detail in Anastasopoulos et al. (2007a), concluding that the dynamic excitation does not yield signi?cant permanent displacements of the joints. This means that if Step 2 was followed by Step 1, the results would be practically the same with Step 1 analysed alone. 4.1 Soil–tunnel interaction parameters Even for an embedded tunnel within an idealised elastic medium, the selection of appropriate spring coef?cients is not always straight forward. St.John and Zahrah (1987) derived approximate expressions for the spring stiffnesses [expressed as force over displacement per unit length of the tunnel (kN/m/m)], as functions of the incident wavelength. For the two horizontal stiffnesses (x and y), their solution is based on integration of Kelvin’s fundamental solution for a “point” load acting within an in?nite elastic medium, assumed homogeneous and isotropic. For the vertical spring stiffness (z), they utilised the fundamental solution of a surface-loaded half-space: Flamant’s problem (see Poulos and Davis 1974; Davis and Selvadurai 1996). Although with the introduction of wavelength they managed to circumvent the singularity of a plane-strain solution, an additional uncertainty was introduced. For the problem investigated herein, the adoption of such a solution would not be easily justi?able for the following reasons: (i) Since the tunnel is not embedded, Kelvin’s full-space solution is not even approximately applicable; half space solutions can be seen to be more appropriate. (ii) The rigidity of tunnel segments with respect to the near-surface (relatively soft) soil will lead to an almost uniform translation in the horizontal (x or y) direction, rather than sinusoidal deformation; hence, the rigid-foundation solution is more appropriate. (iii) The singularity of the plane-strain solution is “suppressed” due to soil non-homogeneity: the soil modulus is not constant, but increases with depth (Gazetas 1983). It is therefore reasonable to make use of published elastodynamic solutions for a rigid long rectangular foundation resting on half-space. For the problem investigated herein, the approximate expressions for non-homogeneous half-space proposed by Gazetas (1991) are utilised. Using as a starting point the synthesized generic soil pro?les (Anastasopoulos et al. 2007a), the effective shear modulus pro?le is eventually modeled as: z m G(z) ≈ G o 1 + α (1) B where G o is the shear modulus at z = 0; 2B (=23.5 m) is the width of the tunnel; and α, m soil-model parameters. To take account of soil non-linearity, G o , α, and m were obtained by curve ?tting an equivalent-linear G = G(z) pro?le (i.e. the pro?le of the shear modulus in the last iteration of the equivalent-linear wave propagation analysis), rather than the initial G max pro?le. The distributed vertical and horizontal springs are then obtained as the static vertical, lateral, and axial stiffnesses of a very long tunnel. Expressed as stiffnesses per unit length, the Winkler moduli kz , k y , k x , (in kN/m/m) in terms of Poisson’s ratio (ν ≈ 0.50) are: 0.73 kz ≈ (2) G o (1 + 2α)m 1?v

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221

ky ≈

2 2 Go 1 + α 2?v 3 1? B L

m

(3)

m

kx ≈ k y ?

0.2 0.75 ? v

1 Go 1 + α 2

(4)

These expressions are considered valid for all frequencies—a reasonable simpli?cation for translational modes of vibration (e.g. Gazetas 1991). The effect of embedment is ignored, and “added” indirectly only in the lateral (y) direction (Fig. 6), by increasing the relevant spring coef?cient, k y , by the elastic sidewall resistance: k y ≈ E ?ll h/2B (5)

where E ?ll , the (average) Young’s modulus of the back?ll, and h, the “effective” depth of embedment. This approach bears similarities with the one of Vrettos (2005), whose spring stiffnesses are also independent of wavelength or frequency. Vertical, cz , lateral, c y , and longitudinal, cx , dashpot coef?cients, re?ecting hysteretic and radiation damping within the half-space, are obtained similarly (Gazetas 1991). In view of the strong soil in-homogeneity and the relatively low dimensionless frequency parameters, ao = (2π/T )B/Vso , that are of prime interest here, these coef?cients play a minor role in the response and are not further discussed. As illustrated in the schematic representation of Fig. 6, in the longitudinal (x) direction the behaviour of the soil-tunnel interface is approximated with a simple slider of friction coef?cient ?x . In the transverse (y) direction, the “interface” is a little more complex, with sliding accompanied by passive type deformation of the back?ll. Therefore, the “equivalent friction coef?cient”, ? y , is estimated through 2-D plane-strain analysis of the tunnel crosssection. Immersion joints (connecting adjacent tunnel segments) are modeled with non-linear hyper-elastic springs. In the longitudinal (x) direction, the springs simulate the behavior of the Gina gasket. Their hyper-elastic restoring force–deformation backbone curves (Fig. 5) are calibrated with the results of half-size model tests by Kiyomiya (1995). In the transverse (y) and vertical (z) directions such gaskets cannot transfer shear; the drift of one segment relative to the other depends solely on shear-key allowance. Thus, the behavior of the joint is simulated through special “gap” elements. Such elements are initially inactive, becoming very stiff after closure of the gap (i.e. the shear-key allowance), depending mainly on the stiffness of the concrete section in the area of the shear key.

5 Analysis of fault rupture propagation We ?rst analyse fault rupture propagation through the overlying soil deposit. The location and magnitude of a potential dislocation at the ground surface depend not only on the type and magnitude of the fault rupture, but also on the composition of overlying soils, and the presence of a structure (e.g. Slemmons 1957; Brune and Allen 1967; Berrill 1983; Taylor et al. 1985; Buwalda and St. Amand 1955; Kelson et al. 2001; Sanford 1959; Hors?eld 1977; Roth et al. 1981; Cole and Lade 1984; Lade et al. 1984; Bray 1990, 2001; Bray et al. 1994a,b; Lazarte and Bray 1995; Johansson and Konagai 2004; El Nahas et al. 2006). Most importantly, the likelihood of a fault rupture of (vertical) magnitude h to propagate all the way to the ground surface (i.e. at the seabed) through a soil deposit of depth H mainly depends on the bedrock offset ratio h/H and soil ductility (Bray 1990; Cole and Lade 1984; Lade et al. 1984;

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Anastasopoulos et al. 2007b). For normal faults and for dip angles α ranging from 45? to 60? , the required h/H for fault outcropping is of the order of 1% (Anastasopoulos et al. 2007b). If h/H does not exceed this critical value, the fault does not outcrop at the ground surface (seabed), and the soil deforms in a more-or-less quasi-elastic manner (Cole and Lade 1984; Lade et al. 1984; Anastasopoulos et al. 2007b). Such deformation can be expected to be much less severe than a distinct fault scarp. The importance of realistic estimation of the maximum probable h/H is evident. Geotechnical exploration in Rion–Antirrion only reached a maximum depth of 100 m below seabed, without encountering bedrock. To shed more light in the tectonic environment of the area, and to gain better knowledge of the alluvium thickness H , a detailed geophysical investigation (with passive tomography) was conducted (Tselentis et al. 2004), with a network of 70 seismographs capturing seismic waves originating from small magnitude earthquakes. Tomography imaging was used to locate potentially active faults. The main conclusions of the geophysical investigation are as follows: (i) No seismic activity was detected at depths less than 1.5 km, verifying the hypothesis that faults within the straits do not outcrop on the seabed. (ii) As depicted in Fig. 7a, ?ve rupture zones were detected within the straits, with direction practically perpendicular to the tunnel axis. (iii) Limestone bedrock (V p > 3, 500 m/s) was detected to lie at depths of the order of 800 m (Fig. 7a). Based on these results, we analyse the central part of the straits down to 800 m depth where “sound” bedrock was detected. The analysis is conducted in plane-strain, with the ?nite element (FE) code ABAQUS. The FE method has proven successful in analysing fault rupture propagation through soil, provided that certain conditions are satis?ed: the use of a re?ned mesh, a suitable nonlinear constitutive law for the soil, etc. (e.g. Bray et al. 1994a,b). The model is displayed in Fig. 7b. At the H = 800 m deep base rock a normal fault of dip angle α ruptures with an offset (downward displacement) of vertical amplitude h. The total width of the model is B = 4H = 3, 200 m, following the recommendation of Bray (1990). At the central 1,600 m of the model, the discretisation is ?ner, with the quadrilateral elements being 20 × 20 m (width × height); at the two edges the mesh is coarser (40 × 20 m). The offset is applied to the left part of the model in small consecutive steps. Based on the ?ndings of a thorough literature review (Anastasopoulos 2005), an elastoplastic constitutive model with Mohr–Coulomb failure criterion and isotropic strain softening was adopted and encoded in ABAQUS. Such models have been successfully employed in modeling the failure of embankments and cut slopes (Potts et al. 1990, 1997). Strain softening is introduced by reducing the mobilised friction angle ?mob and the mobilised dilation angle ψmob with the increase of plastic octahedral shear strain: ?mob = ?p ?

? p ??res P γoct , γ fP P for 0 ≤ γoct < γ fP P for γoct ≥ γ fP

ψmob

?res , ? ?ψ 1 ? p = ? 0,

P γoct γ fP

? P , for 0 ≤ γoct < γ fP ? ? for γ P ≥ γ P

oct f

(6)

where ? p and ?res the peak mobilized friction angle and its residual (or critical state) value; ψ p the peak dilation angle; γ fP the plastic octahedral shear strain at the end of softening. Soil behaviour before yielding is modeled as linear elastic, with a secant modulus G S increasing

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Distance (km)

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2000

Vp (m/s)

3000

4000

RION

(b)

B = 4H = 3200 m

An tirr ion

ANTIRRION

Fau lt

Hanging wall

a h

Fig. 7 (a) Mapping of active faults in the area of the crossing, based on geophyical tomography (Tselentis 2004), (b) Finite element discretization for the plane strain analysis of fault rupture propagation through the 800 m soil sediment

with depth, according to Eq. 1 with m = 1. Model parameters are calibrated through direct shear test results, and an approximate scaling method is employed to take account of scale effects, as described in more detail in Anastasopoulos and Gazetas (2007). The effectiveness of the FE modeling methodology was initially qualitatively veri?ed against published case histories (Gilbert 1890; Slemmons 1957; Witkind et al. 1962; Brune and Allen 1967; Stein and Barrientos 1985; Taylor et al. 1985) and experimental results (Hors?eld 1977; Cole and Lade 1984). Quantitative validation was provided through successful predictions of centrifuge model tests in sand (Anastasopoulos et al. 2007b). Given the uncertainties in the soil pro?le, two idealised soils were parametrically investigated: (a) (b) dense cohesionless soil: ? p = 45? , ?res = 30? , ψ p = 15? , ψres = 0? , and γ y = 1.5% loose cohesionless soil: ? p = 30? , ?res = 25? , ψ p = 5? , ψres = 0? , and γ y = 3.0%

where γ y is a parameter related to soil ductility. The dense soil reaches “failure” at relatively low strains (“brittle” behaviour), while the loose soil is capable of undertaking larger strains before yielding (“ductile” behaviour).

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To be on the conservative side, and despite the more optimistic geophysical results, h d = 2 m was set as the design bedrock offset (vertical component). It is noted that this offset will not necessarily take place during a single seismic event, but may also be the result of aseismic slip accumulation over the years. The dip angle was also conservatively assumed to be α = 45? . As already discussed, for small values of h/H the rupture does not reach the seabed. With H = 800 m, a bedrock offset of 2 m corresponds to h/H = 0.25%. The required h/H for outcropping of the fault rupture is an increasing function of soil ductility, expressed through γ y . Even for the “brittle” soil with γ y = 1.5%, an h/H in the order of 0.75% is required for the fault to break-out (Anastasopoulos et al. 2007b).

6 Effects of fault rupture on the tunnel To illustrate the phenomenon, characteristic pictures of the propagating fault rupture through a deposit of an idealised dense sand layer is given in Fig. 8: four snapshots of deformed mesh with superimposed plastic strains, for bedrock displacement h ranging from 1 to 6 m (darker regions denote higher plastic strains). Observe that for bedrock displacements up to h = 4 m (i.e. h/H = 0.50%), the rupture does not emerge on the surface (seabed). Only when the imposed bedrock displacement exceeds 6 m (i.e. h/H = 0.75%) does the dislocation (barely) emerge. For the design displacement of 2 m, the rupture is clearly far from reaching the seabed (model surface). The situation is even better with the idealised loose sandy soil. Figure 9a, b portray the pro?les of vertical, y, and horizontal displacement, x, along the seabed for the (conservative) dense sandy soil and h d = 2 m. The horizontal distance, d, is measured from the point of application of the bedrock offset. Figure 9c depicts the slope of the ground surface β, a useful response parameter in assessing the damage potential to overlying structures. The horizontal strain, εx , along the ground surface (positive values are for tension) is depicted in Fig. 9d. Observe that the ground surface deforms smoothly (without a scarp), as the rupture cannot outcrop. The “bending” deformation, expressed through the distortion angle β (Fig. 9c), reaches its peak (0.26%) at d ≈ 200 m. On the other hand, the horizontal (tensile) deformation achieves its maximum value, εx = 0.25%, at distance d ≈ 500 m (Fig. 9d). The computed displacement pro?les ( x and y) are used as the “input” displacement on the tunnel supports. The tensile deformation causes decompression of the joints. The bending deformation de-compresses some of the immersion joints and possibly further compresses

h=1m

h=2m

h=4m

h=6m

Fig. 8 Snapshots of deformed mesh and plastic strain for a fault rupture α = 45? in dip, and bedrock displacements h = 1, 2, 4, and 6 m, for the idealised dense sandy soil

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Horizontal distance d (m)

Horizontal distance d (m)

Fig. 9 Rupture propagation analysis results for a fault dipping at α = 45? and bedrock displacement h d = 2 m, for the idealised dense sandy soil: (a) vertical displacement y, (b) horizontal displacement x, (c) angular distortion β, and (d) horizontal strain εx at the seabed

some other. Since the maximum bending and tensile deformations do not occur in the same location, two tunnel-fault rupture relative positions are identi?ed (Fig. 9): (1) (2) Position 1: the center of the tunnel coincides with the location of max β, and Position 2: the center of the tunnel coincides with the location of max εx .

Predicting which is the “worst-case” scenario is not straightforward. Therefore, both positions are investigated. As depicted in Fig. 10, the normal fault-induced deformation leads to signi?cant decompression of most immersion joints. When the tunnel is at Position 2 relative to the propagating fault rupture (tunnel center at max εx ), the tectonic deformation leads to decompression of all joints (Fig. 10a, c, d). In contrast, with the tunnel at Position 1 (tunnel center at max β), joints near the left edge of the tunnel (Fig. 10b) are subjected to increased compression due to sagging deformation of the seabed. The thickness of the rubber gasket does not appear to affect substantially the de-compressed pro?le. Typical results of a complete analysis are portrayed in Fig. 11, in terms of longitudinal joint deformation δx , sliding displacement x , bending moment M y , and axial force N . Figure 11a corresponds to segment length L = 70 m with type A gasket, while Fig. 11b to L = 100 m with type B gasket. While L = 165 m was also examined, the results are not shown herein: with such a large length the tunnel cannot sustain the total distress without several joints experiencing net tension—a precarious situation indeed. Similarly, the “allowance” in the shear-keys plays only a minor role; thus, results are given only for a single value (5 mm) of this “allowance”. In all cases, the presented results correspond to the position “2” scenario, which proved to be critical. The following conclusions are drawn: 1. The application of the fault-induced tensile displacement de-compresses all the joints. The de-compression (δx ) is largest near the middle of the tunnel, and is more pronounced when the segment length is 100 m.

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40 30

Initial Hydrostatic Compression

(a)

δx (c m)

(b)

δx (c m)

40 30 20 10

Initial Hydrostatic Compression

20 10

Position 2

Position 1

0 0 200 400 600 800 1000

0 0 200 400 600 800 1000

Horizontal distance d (m)

Horizontal distance d (m)

(c)

δx (cm)

40 30 20 10

Position 2

(d)

δx (cm)

40 30

Initial Hydrostatic Compression

Initial Hydrostatic Compression

20 10 0

Position 2

0 0 200 400 600 800 1000

0

200

400

600

800

1000

Horizontal distance d (m)

Hydrostatic Fault-induced

Horizontal distance d (m)

Fig. 10 Fault-induced decompression of the immersion joints, starting from the initial hydrostatic compression: (a) 70 m segments with Type B gasket, at Position 2 (max tensile displacement), (b) 70 m segments with Type B gasket, at Position 1 (max bending displacement), (c) 70 m segments with Type A gasket, at Position 2, and (d) 100 m segments with Type B gasket, at Position 2

2. The initial slippage x is greater near the middle of the tunnel, where the maximum tensile deformation εx takes place. Increasing the segment length only marginally increases the maximum sliding displacement. 3. At ?rst, the axial force (N ) exhibits an initial hydrostatic “pre-stressing” of 160 MN (Step 0). Then, application of the fault-induced deformation (Step 1) causes longitudinal decompression of the tunnel, reducing N signi?cantly. For the 70 m segments, N drops from 160 MN to barely 30 MN near the middle. The increase of segment length makes things worse: the initial compression is completely lost, and even some (small) tensile stressing develops (at point “C”).

7 Methodology of dynamic analysis Figure 12 illustrates schematically the methodology of dynamic analysis. In a ?rst sub-step, the ground motion is applied at the bedrock and “propagated” through the soil deposit to compute the free-?eld (i.e. ignoring soil–structure interaction) ground motion at the seabed. The latter is then applied to the already deformed (due to fault-induced deformation) tunnel supports (springs and dashpots), as described in the sequel. Free-?eld acceleration time histories are computed trough 1-D wave propagation analysis, conducted with two methods: (i) making use of the widely accepted equivalent linear approximation—code SHAKE (Schnabel et al. 1972), and (ii) applying the nonlinear constitutive model “BWGG”, developed by Gerolymos and Gazetas (2005) and encoded in the numerical code NL-DYAS. While the ?rst method is most popular, it has certain limitations,

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227

(a)

A

A

B B

C C

70 m segment length – Type A gasket

Joint Decompression δx (cm) A B C 1 16 10 Segment Slippage ? x (cm) 0 7.5 5 Axial Force N (MN) -130 -30 -50 Bending Moment My (MNm) -70 40 5

(b)

AA BB CC

100 m segment length – Type B gasket

Joint Decompression δx (cm) A B C 4 21 14 Segment Slippage ? x (cm) 1 -9 -2 Axial Force N (MN) -100 -80 10 Bending Moment My (MNm) -150 -10 140

Fig. 11 Joint deformation, sliding displacement, axial forces and bending moments due to fault-induced deformation for (a) 70 m segments and (b) 100 m segments. Fault at Position 2, type B gasket, and 5 mm shear-key allowance

especially in the case of very deep soil deposits (as in our case). Such disadvantages can be overcome with NL-DYAS, which is capable of reproducing even some of the most complex non-linear characteristics of cyclic behavior, such as cyclic mobility and liquefaction. More details on the 1-D wave propagation analysis can be found in Anastasopoulos et al. (2007a). The computed free-?eld time histories are then applied to tunnel supports, but not simultaneously. For long structures, the ground motion differs from one support to another: in the arrival time, in amplitude, and in frequency content. Such variations are due to: (a) the “wave passage” effect (seismic waves do not propagate only vertically), (b) local soil conditions (variation of the soil pro?le along the alignment), and (c) random “geometric” incoherence (due to re?ections, refractions, and superposition of incident seismic waves stemming from random ground heterogeneities). An elaborate review on the subject has been published by Hashash et al. (2001). To take account of the “wave passage” effect, the computed free-?eld acceleration time histories are applied on tunnel supports (springs and dashpots) with a time lag, following the methodology of EC8—Part 2 (2002). At distance xi along the axis of the tunnel, the seismic excitation will arrive with a time lag (Fig. 12): ti = xi /Cα (7)

Vertically propagating seismic shear waves would appear traveling along the ground surface (the seabed) with Cα → ∞. Waves propagating at incidence angle βs = 0? from the vertical, appear traveling with ?nite Cα :

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L = 980 m

a

x

a

xi

a

xj

a

L

t

xi /Cα

t

xj /Cα

t

L /Cα

t

Cα : apparent wave velocity ? at distance xi , time lag ti = xi/C α

0.6 0.6

0.50 g

0.6

0.63 g

a (g) 0 0.48 g

0 5 10 15

0

0

-0.6

-0.6 0 5 10 15

-0.6 0 5 10 15

t (sec)

t (sec)

t (sec)

00

250

γ

SHAKE NL-DYAS

γ

Depth (m)

G G?

ξ

100

1300

Dense Sand

Loose Sand

0 250 500 750 1000 1250

Vs (m/sec)

0.6 0.6 0.6

JMA Kobe 1995

0.24 g

0

Aegion 1995

Lefkada 2003

0.24 g

0

a (g) 0 0.24 g

-0.6 0 5 10 15

-0.6 0 5 10 15

-0.6 0 5 10 15

t (sec)

t (sec)

t (sec)

Fig. 12 Dynamic analysis methodology and results for the worst-case scenario (“stiff” soil pro?le with Vs = 1, 300 m/s below 100 m depth). The bedrock acceleration excites the 1-D soil column; wave propagation is analysed alternatively with the equivalent linear method (SHAKE) and the truly nonlinear code NL-DYAS to derive the acceleration at the seabed (characteristic such time histories are shown). Then, the acceleration time histories are imposed with a time lag on the spring-dashpot supports of the tunnel

Cα = Vs / sin βs

(8)

where Vs is the shear wave velocity of the soil near the surface. βs depends on the distance to the source and the ray path, but also and crucially on the shear wave velocity contrast between the near-surface soil layers and the underlying ground: larger contrast increases the refraction of seismic waves, decreasing βs , thanks to Snell’s law. Field records in Japan have shown that Cα may range from 1,000 to 2,000 m/s (Kiyomiya 1995; Okamoto 1984). For the 1971 San Fernando and the 1979 Imperial Valley earthquakes, O’Rourke et al. (1982) estimated Cα to range from 2,100 to 5,300 m/s. Following the recommendations of EC8—Part 2 (2002), and to be on the conservative side, we assume Cα = 1, 000 m/s.

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229

Regarding local soil conditions, the soil is assumed to be uniform along the tunnel: a reasonable simpli?cation at this stage of knowledge. The effect of geometric incoherence was investigated in the early stages of the study applying both the coherency loss functions of Abrahamson et al. (1991), and the theoretical coherency loss model of Luco and Wong (1986). The effect of such incoherence was found to be insigni?cant compared to the wave passage effect, especially in view of the conservative choice of Cα = 1, 000 m/s. 7.1 Analysis of soil response Consistently with the Greek Seismic Code (EAK 2000) we selected A = 0.24 g for the basement excitation. Four real earthquake records were selected and downscaled for analysis (Table 1): (a) the JMA record of the 1995 MJMA 7.2 Kobe earthquake (Fukushima et al. 2000); (b) the Rinaldi record of the 1994 Northridge Ms 6.8 earthquake (Trifunac et al. 1998); (c) the record of the 1995 Aegion Ms 6.2 earthquake (Gazetas 1996); and (d) the record of the 2003 Lefkada Ms 6.4 earthquake (Gazetas et al. 2005; Benetatos et al. 2005). This ensemble of ground motions ensures that the excitation is rich in long periods and strong pulses (a critical parameter for inelastic systems), and that near-fault effects are taken into account. More details on the seismic environment and the selection and scaling of the ground motions can be found in Anastasopoulos et al. (2007a). Figure 12 illustrates 1-D wave propagation results for three of the records and for one of the worse-case scenarios: “stiff” soil pro?le with Vs = 1, 300 m/s below 100 m depth. The Aegion and the Kobe JMA records produce peak ground acceleration (PGA) in the order of 0.50 g at the seabed, while the Lefkada accelerogram yields 0.63 g. Five more possible soil pro?les were parametrically investigated. From all of the examined soil pro?le and seismic excitation scenarios, PGA and PGVs were found to range between 0.40 and 0.63 g and 0.42 and 0.81 m/s, respectively (Table 1).

8 Combined effects of fault rupture and subsequent seismic shaking Typical results of a complete analysis are portrayed in Figs. 13–16, in the form of time histories of longitudinal accelerations (ax ), axial forces (N ), longitudinal sliding displacements ( x ), and axial joint deformations (δx ), which develop at various “points” along the tunnel during strong seismic shaking. Quasi-static deformation and “distress” is assumed to have already taken place in a preceding event of fault rupturing underneath the site of the tunnel (Step 1). Each of these ?gures presents results for two different values of segment length L: 70 and 100 m. Results for L = 165 m are not shown: with such a large length the tunnel would not be able to sustain the total developing stressing without several joints experiencing net tension— a precarious situation indeed. Only results for the (scaled-down) Kobe JMA excitation are shown in these ?gures, since the other motions lead to a qualitatively similar behaviour, and the conclusions drawn herein are not altered. The “allowance” in the shear-keys proved to play only a minor role; thus, results are given only for a single value (5 mm). Finally, only the most critical location of the emerging fault rupture, position “2”, is examined here. Figure 13 depicts the time histories of longitudinal acceleration ax which develop in the middle of three characteristic tunnel segments (B, C, and D), and contrasts them with the ground (seabed) excitation (point A). Observe that the response of the tunnel differs from point to point. The peak acceleration is ampli?ed from 0.48 g at the base to approximately 0.60 g near the terminal segments (at the two edges of the tunnel, B and D). Near the center, however, the (longitudinal) acceleration is cut-off to about 0.30 g, implying sliding. This

123

230

123

Original PGA (g) 0.82 0.79 0.54 0.43 33 0.24 52 0.24 164 0.24 81 0.24 24 50 23 18 PGV (cm/s) PGA (g) PGV (cm/s) Scaledc Seabedd PGA (g) 0.48 0.53 0.50 0.63 PGV (cm/s) 64 81 48 42

Table 1 Strong motion records used in the analysis

Record:

Record data

Ms

E-dista (km)

F-distb (km)

Kobe JMA

7.2

22

1.5

Rinaldi

6.8

9

2

Aegion 1995

6.2

11

4

Lefkada 2003

6.4

14

14

a Epicentral distance b Distance to fault c Input for 1-D soil response analysis d Output from 1-D soil response analysis, used as input to the tunnel-soil model

Bull Earthquake Eng (2008) 6:213–239

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231

0.60

0.6 0.3 0 -0.3 -0.6

(a)

0.6 0.3

A (base)

0.6 0.3 0 -0.3

B

C

0.6 0.3 0 -0.3 -0.6

E

ax (g) 0

-0.3 -0.6 0

0.48 g

5 10

-0.6 0 5 10

0

5

10

0

5

10

t (sec) B C

E

A

(b)

ax (g)

0.6 0.3 0 -0.3 -0.6 0

A (base)

0.6 0.3 0 -0.3

B

0.60

0.6 0.3 0 -0.3 -0.6

C

0.6 0.3 0 -0.3 -0.6

D

0.48 g

5 10

-0.6 0 5 10

0

5

10

0

5

10

t (sec)

B

C

D

A

Fig. 13 Longitudinal acceleration time histories for: (a) 70 m segments and (b) 100 m segments. (Kobe JMA excitation, fault at Position 2, type B gasket, and 5 mm shear-key allowance)

differential longitudinal response is attributed to the non-uniform (along the axis) faultinduced displacement and the ensuing tendency of the (spring-restrained) tunnel segments to “readjust” under the subsequent dynamic loading. This is explained in detail in the next section. If the fault-induced displacement did not pre-exist (Step 1), the dynamic response of the tunnel would be nearly uniform (Anastasopoulos et al. 2007a). The increase of L does not appear to have a signi?cant effect on such accelerations. The sliding displacement x of tunnel segments is seen in Fig. 14. In the examined direction (x), the tunnel segments have already slipped before application of the asynchronous dynamic excitation (Step 2), due to the fault-induced longitudinal deformation (Step 1). This initial slippage is greater near the middle of the tunnel, where the maximum tensile deformation εx takes place, and increases with segment length (from 7 cm for L = 70 m, to about 11 cm L = 100 m). The increase of L also leads to increased additional dynamic relative displacements. While for L = 70 m the additional sliding due to the dynamic excitation reaches 10 cm, for L = 100 m the maximum relative dynamic displacement exceeds 16 cm (location B). In all cases, the residual slippage is invariably signi?cantly lower than the maximum value. Figure 15 illustrates the axial forces N in characteristic cross-sections of the immersed tunnel. At ?rst, the axial force exhibits the initial hydrostatic “pre-stressing” of about 160 MN (Step 0). Then, the application of the fault-induced deformation (Step 1) causes longitudinal decompression of the tunnel, reducing the axial force (N ) signi?cantly. The re-adjustment of the segments during dynamic shaking (further explained in the sequel), leads to a

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(a)

? x (cm)

30 20 10 0 -10 0

A

30 20 10 0 -10

B

17 cm

30 20 10 0 -10

C

30 20 10 0 -10

D

5

10

0

5

10

0

5

10

0

5

10

t (sec)

A

B

C

D

(b)

? x (cm)

30 20 10 0 -10 0

A

30 20 10 0 -10

B

27 cm

30 20 10 0 -10

C

30 20 10 0 -10

D

5

10

0

5

10

0

5

10

0

5

10

t (sec)

A

B

C

D

Fig. 14 Longitudinal sliding displacements for: (a) 70 m segments and (b) 100 m segments. (Kobe JMA excitation, fault at Position 2, Type B gasket, and 5 mm shear-key allowance)

re-distribution of the longitudinal axial force. Segments with higher axial forces tend to get un-loaded, while those subjected to tension (smaller compression) tend to attract higher compression. At the end of shaking, this re-distribution leads to a nearly uniform distribution of the axial force: about 70 MN for L = 70 m, and 60 MN for L = 100 m. In the latter case, a maximum net tension of about ?10 MN is observed during shaking. The longitudinal deformation (δx ) of the immersion joints is perhaps the most crucial response parameter for the seismic safety of an immersed tunnel. Their time histories during seismic shaking, and after the fault rupture has already had its effect, are portrayed in Fig. 16, for the 70 m-segment tunnel. We compare the response of the tunnel equipped with the Type A gasket, versus Type B gasket. Notice that in the longitudinal direction, the Gina gasket experiences an initial hydrostatic compression (Step 0) of 17 and 28 cm, for Type A and B gasket, respectively (Fig. 10). The previous application of the fault-induced tensile displacement opens-up all the joints, with those near the middle of the tunnel experiencing the greatest tension (see also Fig. 10). For Type A gasket, the 17 cm of hydrostatic compression are practically completely lost at the central part of the tunnel (point C), where the remaining compression amounts to a mere 2 cm. The situation near the terminal segments A and D is less critical, with the remaining compression of the gasket being 15 and 6 cm, respectively. During subsequent seismic shaking, the gaskets experience alternating cycles of de-compression and re-compression. While the dynamic compression is acceptably small, there is some tensile deformation in gaskets A and C—a rather precarious situation.

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t (sec)

233

(a)

N (MN)

0 0 -50 -100 -150 -200

2

4

6

8

10 0 -50 -100 -150

0

2

4

6

8

10 0 -50 -100 -150

0

2

4

6

8

10

A

B

C

F*

Dynamic

-200

F*

Dynamic

-200

F*

Dynamic

A

B

C

t (sec)

(b)

N (MN)

0 0 -50 -100 -150 -200

2

4

6

8

10 0 -50 -100 -150

0

2

4

6

8

10 0 -50 -100 -150

0

2

4

6

8

10

A

B

C

F*

Dynamic

-200

F*

Dynamic

-200

F*

Dynamic

A

B

C

* :fault–induced displacement

Fig. 15 Axial force and bending moment time histories for: (a) 70 m segments and (b) 100 m segments (Kobe JMA excitation, fault at Position 2, type B gasket, and 5 mm shear-key allowance)

The use of Type B rubber gasket improves the performance substantially. One of the reasons, of course, is the 28 cm of initial hydrostatic compression. Much greater margins for de-compression are thus available. The fault-induced de-compression of the joints still leaves a respectable minimum compression of 12 cm in the critical central part of the tunnel (point C). As expected, the largest decompression is observed at joints near the middle of the tunnel, where the fault-induced tensile deformation of the seabed is maximum; near the end segments unloading of gaskets is substantially less. During seismic oscillation, the rubber gaskets experience cycles of de-compression and re-compression. The total dynamic-plusstatic compression does not, in any case, exceed the acceptable limits of the Type B gasket: δx,max ≈ 29 cm. More importantly, the gasket remains always compressed, maintaining its water-tightness: δx,min > 10 cm, in all joints. A re-distribution of joint deformation is readily observed: over-stressed joints tend to de-compress, under-stressed to re-compress, so that the safety of the immersed tunnel improves after strong seismic shaking. This is a fundamental conclusion, observed in all the parametric cases. Moreover, it is completely explainable theoretically. Its practical signi?cance and its generality should not be underestimated: a detrimental static de-compression of a few of the joints (arising from a rupturing fault, or perhaps from differential settlements, or from any other cause) may be relaxed due to the tunnel oscillations during a strong seismic shaking!

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30 20

(a)

δx (cm)

A

15 cm 10 cm

30 20 10 0

B

30 20 10

C

30 20 10

D

10 0 0

-0.5 cm

7 cm 5 cm

2 cm

4 cm

6 cm 8 cm

0

5 10

5

10

0

0

-0.5 cm

0

5

10

0

5

10

t (sec)

A

B

C

D

30 (b) 28 cm A

20 cm

30 20

B

17 cm 16 cm

30 20

C

16 cm

30 20

D

17 cm 19 cm

20

δx (cm)

10 0 0 5 10

12 cm

10 0 0 5 10

10 12 cm 0 0 5 10

10 0 0 5 10

t (sec)

A

B

C

D

Fig. 16 Longitudinal joint deformation for: (a) 70 m segments with Type A gasket and (b) 70 m segments with Type B gasket (Kobe JMA excitation, fault at Position 2, and 5 mm shear-key allowance)

9 Dynamic re-adjustment of tunnel segments: a “Healing” process As already cited, during the asynchronous dynamic oscillation of the tunnel, a redistribution of total deformation among joints is observed. This section sheds further light into this interesting phenomenon. As depicted in Fig. 17, the application of the fault-induced deformation leads to de-compression of the immersion joints. When the tunnel is at Position 2 relative to the propagating fault rupture (tunnel center coinciding with the location of maximum εx ), the tectonic deformation leads to de-compression of all joints (Fig. 17a, c, d). On the other hand, when the tunnel is placed at Position 1 relative to the fault rupture (center of the tunnel coinciding with the location of max. β), most joints de-compress, except those near the left edge of the tunnel (Fig. 17b). At this location, the sagging deformation of the seabed seems to be the most important, imposing some further compression onto the joints. More speci?cally, in the case of the 70 m segment tunnel with the (“thick”) Type B rubber gaskets at Position 2 (Fig. 17a), the joints de-compress from their initial hydrostatic compression of 28 cm to a minimum of 12 cm (a de-compression of 16 cm) near the center of the tunnel (where maximum tensile deformation occurs). This de-compression is reduced signi?cantly near the left edge of the tunnel (δx ≈ 26 cm), but not to the same extent at the right (δx ≈ 18 cm). As shown in Fig. 17c, the thickness of the rubber gasket does not substantially affect the de-compressed pro?le. But while the de-compression is practically the same, in the case of the (slimmer) Type A rubber gasket, the initial hydrostatic compression is only 18 cm. As a result, the joints near the center almost loose all of their pre-stressing (δx ≈ 2 cm). This

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235

(a)

δx (cm)

40 30

(b) 40

30

δ x (cm)

Position 2

20 10 0 0

20 10 0

Position 1

200

400

600

800

1000

0

200

400

600

800

1000

Horizontal distance d (m)

Horizontal distance d (m)

(c)

δ x ( c m)

40 30 20 10 0 0

Position 2

(d) 40

30

δ x ( c m)

200 400 600 800 1000

20 10 0 0

Position 2

200

400

600

800

1000

Horizontal distance d (m)

Hydrostatic Fault Dynamic-Max

Horizontal distance d (m)

Dynamic-Min Residual

Fig. 17 Dynamic redistribution of the immersion joint deformation: (a) 70 m segments with Type B gasket, at Position 2 (max tensile displacement), (b) 70 m segments with Type B gasket, at Position 1 (max bending displacement), (c) 70 m segments with Type A gasket, at Position 2, and (d) 100 m segments with Type B gasket, at Position 2

is threatening for the water-tightness of the tunnel. Increasing L to 100 m (combined with a Type B gasket) improves the safety margins: δx,min ≈ 6 cm near the center of the tunnel (Fig. 17d). This effect can be easily explained: while for L = 70 m the imposed deformation is “absorbed” by a total of 13 joints, in the case of the 100 m segments the same deformation is transmitted to only nine joints. Obviously, this decrease in the number of joints unavoidably increases the “opening” of each joint. Observe that the total de-compression (of all joints) remains practically the same: δx,total ≈ 1.5 m. An insight into the aforementioned dynamic re-adjustment of the tunnel segments can be developed with the help of Fig. 18, which illustrates the “history” of longitudinal deformations of six characteristic (Type B) joints along the 70 m segment tunnel, subjected to fault rupturing at Position 2. Initially, when only the hydrostatic compression has been applied (Step 0), all joints are compressed by exactly the same amount, reaching δx ≈ 28 cm. Application of the fault-induced dislocation (Step 1) leads to a differential de-compression of the joints, as already explained in Fig. 17. At Step 2, the dynamic oscillation tends to redistribute the joint compressions and relocate the tunnel segments. Observe that joint 1, which had experienced the least de-compression, now experiences the greatest de-compression during dynamic oscillation. On the contrary, joint 7 at the center of the tunnel, with the largest fault-induced de-compression, is re-compressed the most during shaking. This surprisingly favorable performance is attributable to the hyper-elastic behaviour of the immersion joints, combined with the “ability” of the tunnel segments to slide over the seabed. Let us now look into the performance of joints 1 and 7 in more detail. As depicted in Fig. 18, the fault-induced stressing only slightly de-compresses the rubber gasket of joint 1 (from 28 to 26 cm): point a1 in the graph. At the same time (Step 1), joint 7 is signi?cantly de-compressed

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236

1 2 3 5 7 9 11

Bull Earthquake Eng (2008) 6:213–239

Step 0 :

Hydrostatic Compression

30

Step 1 :

Fault-induced Displacement

Step 2 :

Dynamic Oscillation

a1

25

1

2

20

b1

3 11

δx (cm)

15

b7

9 7 8 8

a7

10

Load (MN/m)

Load (MN/m)

1 Khyper Kel 0 10 b1 20

7 Khyper b Kel a7 7 0 10

6 4

6 4

5

2

0

a1

30 40

2

0

? x (cm)

2

? x (cm)

8

20

30

40

0 0 4 6 10

t (sec)

Fig. 18 Longitudinal joint deformation for the 70 m segment tunnel, equipped with the “thick” Type B rubber gasket, and 5 mm shear-key allowance, at Position 2 (maximum tensile deformation) relative to the fault rupture. While the fault-induced displacement in Step 1 “opens” the joints, the asynchronous dynamic shaking tends to relocate the segments and allow for a redistribution of the gasket deformations: observe the substantial homogenization of the residual compression of the joints

to 12 cm: point a7 in the graph. Now look into the hyper-elastic load-deformation curves of the two joints (see also Fig. 5). Joint 1, after application of the fault-induced displacement remains suf?ciently compressed to lie on the stiffer hyper-elastic regime, where the stiffness is K hyper ≈ 50 MPa. In stark contrast, joint 7, having de-compressed signi?cantly, lies on the initial soft (elastic) branch, of K el ≈ 3 MPa. As a result, during dynamic shaking, joint 1 will be an order of magnitude stiffer than joint 7! Since the joints are connected in series, the stiffer ones simply force the softer to compress. In other words, over-compressed joints (stiffer) force the under-compressed joints (softer) to re-compress. Of course, such a re-compression leads in turn to hyper-elastic stiffening, making the latter stiffer. Finally, an “equilibrium” is reached, with all joint compressions being substantially “homogenised”. At the end of shaking, the stiffer joint 1 exhibits a residual de-compression δx ≈ 20 cm (point b1 in the graph), while the softer joint 7 is re-compressed to δx ≈ 13 cm (point b7 ). It must be pointed out that such a behaviour would hardly be possible if it were not for tunnel sliding: if the segments were “?xed” on the seabed, the stiffness of the joints would play no, or very minor, role. 10 Conclusions Several conclusions of practical signi?cance can be drawn from the presented study (and some additional parametric results not shown here for the sake of brevity):

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237

(1)

(2)

(3)

(4)

(5)

A properly designed immersed tunnel (suitable thick elastic gaskets; relatively small segment length; shear keys with suf?cient “allowance”; un-stressed tendons) can safely resist: ? a normal earthquake fault rupture with a dislocation (offset) of 2 m in the basement rock, 800 m underneath the tunnel ? a subsequent strong ground shaking arising from a different signi?cant seismic fault not crossing the site, but producing peak acceleration of at least 0.50 g ? the cumulative effect of the above two events, even in the worst possible sequence: fault rupture followed by shaking. The initial hydrostatic compressive force is independent of segment length, or of total number of joints. Increasing the total number of the joints (or decreasing segment length) leads to increased total initial hydrostatic compressive deformation of the tunnel. The initial compression of the Gina gaskets must be large to ensure its water-tightness by avoiding net tension, but strong enough to avoid lateral tensile rupture. The dynamically-induced longitudinal deformation of the immersion joints depends on segment length L and the thickness of the Gina gasket. Increasing L unavoidably increases both the fault-induced and the dynamic deformation of the immersion joints. Since the tunnel segments are signi?cantly stiffer than the Gina gaskets, they tend to behave as rigid blocks and most of the imposed deformation is “absorbed” in the joints. Obviously, decreasing the number of joints, increases their deformation. Increasing the thickness of the Gina gasket leads to greater initial hydrostatic compression: from 17 cm for the Type A gasket, to 29 cm for Type B. Since the tectonic deformation is mainly tensile, this increase of the initial compressive deformation leads to higher safety margins. During seismic shaking, the dynamic re-compression and de-compression of the joints is not that sensitive to gasket thickness. During subsequent (asynchronous) dynamic shaking, the tunnel segments tend to readjust by sliding longitudinally over the seabed. This is accompanied by a redistribution of the longitudinal joint deformation leading to a more uniform pro?le of compression along the length. This surprising “healing” behaviour is attributable (to a substantial extent) to the hyper-elasticity of the rubber gaskets, combined with the “capability” of the segments to slide on the seabed. Thus, while the fault displacements lead to a detrimental de-compression different from joint to joint, upon subsequent seismic shaking the most seriously de-compressed joints become more ?exible compared to joints with minor de-compression. Since the joints are connected in series, the stiffer ones will force the softer to re-compress—a bene?cial effect of great signi?cance.

Although this research was prompted by the needs of a speci?c engineering project, the methods of analysis, the proposed design concepts, and hopefully many of the conclusions of this study are suf?ciently general to be applicable in other immersed tunneling projects.

Acknowledgements The authors would like to acknowledge the ?nancing of this research project by the Greek Railway Organization (OSE). We acknowledge the invaluable help over the years from Dr. Takashi Tazoh of the Institute of Technology, Shimizu Corporation, Japan.

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