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Engineering Structures 28 (2006) 1367–1377 www.elsevier.com/locate/engstruct

Numerical simulation for large-scale seismic response analysis of immersed tunnel

Jun-Hong Ding a,? , Xian-Long Jin a,b , Yi-Zhi Guo a , Gen-Guo Li c

a High-Performance Computing Center, Shanghai Jiao Tong University, Shanghai, 200030, China b State Key Laboratory of Vibration, Shock & Noise, Shanghai Jiao Tong University, Shanghai, 200030, China c Shanghai Supercomputer Center, Shanghai, 201203, China

Received 2 June 2005; received in revised form 5 December 2005; accepted 5 January 2006 Available online 23 March 2006

Abstract This paper presents a three-dimensional numerical simulation method for large-scale seismic response calculation based on a newly built immersed tunnel in Shanghai. The whole analytical model consists of the surrounding soil, the tunnel segments, and the detailed ?exible joints. Both the number of nodes and the number of elements exceed one million. This model takes account of nonlinear material behavior such as soil, non-re?ecting boundary de?nition, and soil–tunnel interaction. Final calculation has been performed on the Dawning 4000A supercomputer using the ?nite-element code LS-DYNA 970 MPP. The result provides a global understanding of this immersed tunnel under seismic loading and reveals the weak parts. It also provides relevant data and references for the aseismic design of immersed tunnel and ?exible joints in the future. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Immersed tunnel; Soil–tunnel interaction; Seismic response; Supercomputer; FEM; Non-linearity

1. Introduction The Waihuan Tunnel is a newly built immersed tunnel under the Huangpu River opened to traf?c recently in the city of Shanghai, China. Having a large cross-section size (42 m × 9.55 m), it is a 2888 m-long bi-directional immersed tunnel with eight lanes. This tunnel is ranked the biggest in Asia and the third biggest in the world at present (Figs. 1 and 2). Meanwhile, the local soft clayer soils are distributed widely and in a complicated way in urban areas, having a direct in?uence on the seismic response of an immersed tunnel located in the soil. The study of the dynamic behavior of long underground structures, such as transportation tunnels or pipelines, to seismic waves is an important engineering problem in dynamic soil–structure interaction. Stamos and Beskos [1] pointed out that seismic wave diffraction by underground structures was

? Corresponding address: High Performance Computing Center (Room 602), Department of Mechanical Engineering, Shanghai Jiao Tong University, Huashan Road 1954, Shanghai 200030, China. Tel.: +86 21 62932256; fax: +86 21 62932177. E-mail address: junhong ding@hotmail.com (J.-H. Ding).

a complex problem, which can only be solved accurately, economically and under realistic conditions with the aid of numerical methods such as the ?nite-element method (FEM) and the boundary-element method (BEM). Hashash et al. [2] summarized several approaches in quantifying the seismic effect on an underground structure, and pointed out that it was most appropriate to utilize three-dimensional models for analyzing axial and bending deformations of such long underground structure. Regarding the immersed tunnel, Kiyomiya [3] outlined several seismic design methods and countermeasures to earthquakes in Japan including the multi-mass-spring model, quasi-three-dimensional ground model, and ?nite-element model. However, he also pointed out that memory capacity and time cost were two bottlenecks to the large calculation model in the past. Ingerslev and Kiyomiya [4] provided guidance on the magnitude to be selected for the seismic loads and how to apply them to an immersed tunnel. It was also suggested that all immersed tunnels should be designed for seismic effects appropriate to their location. There are also several examples involving the previous practice of such analysis. Youakim et al. [5] investigated

0141-0296/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2006.01.005

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Fig. 1. General plan of the immersed tunnel project site.

nonlinear material behavior, large deformation effects, and the nonlinear nature of the contact in the nonlinear analysis of tunnels embedded in clayey or sandy soil using twodimensional (2-D) FEM analysis. Taylor et al. [6] described the seismic retro?t strategy, the primary results obtained by numerical analyses, using 2-D FLAC and ABAQUS software for the upgrade of the George Massey Tunnel. Kozak et al. [7] utilized ?nite-element analysis to study the soil–structure interaction behavior due to earthquake ground motion in the Alameda Tubes seismic retro?t project, considering wave passage, scattering, and re?ection effects. Hashash et al. [8] conducted a seismic retro?t study on assessing the vulnerability of the tubes to seismic shaking with three-dimensional soil–structure interaction models of the tunnels and portal buildings, using the beam element for the tunnel segments and the so-called ‘spoke’ wheel for the joints. Development of the computer has been promoting FEM to be a more effective approach to solving earthquake-engineering problems. With the appearance of high-performance computer and its increasing application, it is possible that threedimensional dynamic FEM can be a chance for the large-scale seismic response analysis of an immersed tunnel with its supercomputing and mass storage capability. This paper provides a large-scale numerical simulation method for seismic response analysis and its corresponding application to a recently built immersed tunnel in Shanghai. The structure of the paper is as follows: the methodology used for analysis is brie?y introduced in Section 2 (such as the explicit time integration scheme and various material modeling); the analytical model is described in Section 3 (such as the CAD model and ?nite-element model); the results and the corresponding discussion are shown in Section 4; the conclusion and direction for potential future work are given in Section 5. 2. Methodology This simulation was performed with the Mpp970 version of LS-DYNA on a Dawning 4000A supercomputer. This supercomputer has a peak speed of 10 trillion ?oating-point operations per second and uses an array of 2192 AMD 64-bit

Fig. 2. Typical cross-section of the immersed tunnel (Unit: mm).

Opteron processors. LS-DYNA is a general-purpose ?niteelement code for analyzing the large deformation dynamic response of structures. The essential ingredient determining the solution properties is the use of an explicit time integration scheme, which is a slight modi?cation of the standard central differential method (CDM). 2.1. Basic equations of CDM Generally, the motion equation of a deformed body for nonlinear dynamic behavior [9] can be described as: M x(t) = P ? F + H ? C x ¨ ˙ (1)

where M is the global mass matrix, P accounts for the global load vector (nodal load, body force, surface force, etc.), C is the damping matrix, H is the global hour-glass resisting force vector handling the hour-glass deformation modes, and F is the assembly of equivalent nodal force vectors from all the elements

n

F=

m=1 Vm

B T σ dV

(2)

where B is the strain-displacement matrix, σ is the stress vector, and V is the volume in the current con?guration. The explicit CDM is adopted to solve the motion equation by time integration [9,10]: ˙ x(tn ) = M ?1 [P(tn ) ? F(tn ) + H (tn ) ? C x(tn?1/2 )] ¨ ˙ ¨ x(tn+1/2 ) = x(tn?1/2 ) + ( tn?1 + tn )x(tn )/2 ˙ ˙ x(tn+1 ) = x(tn ) + tn x(tn+1/2 ) (3) (4) (5)

where x(tn ) is the nodal acceleration vector at time tn , x(tn+1/2 ) ¨ ˙ is the nodal velocity vector at time tn+1/2 , and x(tn+1 ) represents the nodal displacement vector at time tn+1 .

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After the nodal location and the nodal acceleration at time tn are acquired together with the nodal velocity at time tn+1/2 , the nodal displacement at time tn+1 can be calculated by Eq. (5). In the time domain, the displacement, velocity, and acceleration of each discrete point can be calculated through such integral recursive formulae. CDM is suitable for solving complicated problems such as wave propagation [11] due to the fact that, when some nodes are ?rst disturbed, neighboring nodes and then other nodes will behave gradually with time, which is consistent with the features of wave propagation. However, it is necessary to guarantee a small time step for modeling wave propagation, because the explicit CDM is conditionally stable. In fact, the time step size depends on the minimum natural period of the whole mesh to guarantee the calculation stability of the central difference method. Hence, during the solution, a new time step size is determined by taking the minimum value over all elements: t n+1 = α · min{ t1 , t2 , t3 , . . . , t N } where α is a scale factor, N is the number of elements, and is the critical time step size. 2.2. Parallel algorithm for contact interface In the soil–tunnel analytical model, some parts that contact each other will possibly have extrusion and sliding behaviors under seismic loading. To handle this problem, it ?nally comes down to the search for contact objects and the calculation of contact forces as a nonlinear boundary condition. Such a contact problem is often solved by the symmetrical penalty method [9], which consists of placing normal interface springs between all penetrated nodes and the contact surface. First, the penetration between the slave node n s and the master segment si is judged within each time step. If there is no penetration, there is no treatment; otherwise, a normal contact force f s will be calculated: fs = mki n i (7) (6) t

Friction in LS-DYNA is based on a Coulomb formulation, and the maximum frictional force Fv is de?ned as: Fv = ?| f s | (10)

where ? is the coef?cient of friction and f s is the normal force at slave node n s . The possible frictional force f ? of the tn+1 time step can be obtained through the friction force f n of time step tn . Meanwhile, since the value of f ? should not exceed the maximum frictional force Fv , the real friction force f n+1 of the next time step will be treated differently according to the contrast between | f ? | and Fv : f n+1 = f ? f n+1 = Fv f ? /| f ? | (| f ? | > Fv ). (| f ? | ≤ Fv ); (11)

The time cost of common calculations using explicit algorithms such as LS-DYNA depends on the element size and wave speed. However, when it comes to parallel calculation, it is the parallel algorithm involving contact that affects the calculation ef?ciency. Hallquist et al. [12] developed a standard algorithm in LSDYNA for problems with contact. According to Eq. (1), this algorithm can be extended and described as follows: 1. Determine the mass matrix M (diagonal matrix). 2. Sub-divide the ?nite-element mesh into groups, adjust the numbers of elements in the groups for load balancing, then assign the shared nodes to the groups for contact checking. For each time step: 3. Perform a calculation to determine (P ? F + H ? C x), ˙ calculate the contact force vector Rc (including the normal contact force and the tangential frictional force), and add the contact force vector Rc to the global load vector P to form a new one P (within each group). 4. Compute acceleration (within each group): x = M ?1 (P ? ¨ F + H ? C x). ˙ 5. Adjust the accelerations of shared nodes between the groups (communicate between processors): ¨ ¨ x = x g1 + x g2 + x g3 + · · · . ¨ ¨ 6. Compute the velocity and the displacement (within each group). 7. Continue with step 3. 2.3. Material model for soil Several soil layers close to the tunnel mainly in?uence the seismic response of the underground tunnel. Here, the soil’s behavior is assumed to be governed by an elastic–plastic constitutive relation based on the Drucker–Prager criterion. Considering the possible in?uence by the hydrostatic stress component on the yield strength of soil and rock material, the Drucker–Prager criterion [13] is described on the basis of modi?cation of the von-Mises yield criterion: F = α J1 + (J2 )1/2 ? k = 0 (12)

where m is the amount of penetration, n i is the normal to the master segment at the contact point, and ki is the stiffness factor for the master segment, ki = f K i A2 i . Vi (8)

Here, f is a scale factor for interface stiffness, and K i , Ai and Vi are the bulk modulus, the volume, and the surface area of the element that contains si , respectively. Meanwhile, each of the four nodes ( j = 1, 2, 3, 4) that comprise si adds a contact force, fm = φ j (ξc , ηc ) f s

j

(9)

where φ j (ξc , ηc ) is the value of the shape function of si at the contact point coordinates (ξc , ηc ). Similarly, the treatment will then be applied to the master nodes and the slave surfaces again.

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where J1 is the ?rst invariant of the stress tensor, and J2 is the second invariant of the stress deviator tensor. The material constants α and k can be calculated as: 2 sin φ α= √ ; 3(3 ± sin φ) 6c cos φ k= √ 3(3 ± sin φ) (13)

2.5. Material model for steel shear key The material model called “MAT PLASTIC KENEMATIC” in LS-DYNA [14] can be used to model the material behavior of the steel shear key among the ?exible joint components. This model is suitable for modeling isotropic and kinematic hardening plasticity with the operation of including rate effects. Varying a parameter called β between 0 and 1 can specify isotropic, kinematic, or a combination of isotropic and kinematic hardening. Meanwhile, the strain rate is accounted for by using the Cowper–Symonds model [9,16], which scales the yield stress by a strain rate dependent factor σy = 1 + ε ˙ C

1/ p

where ? is the angle of internal friction and c is the cohesion value. Here, the positive and negative signs indicate the tensile and compressive conditions, respectively. The ?ow rule de?ning the direction of the plastic ?ow is given by ˙ ?G εp = λ ˙ ?σ (14)

˙ where G is a plastic potential and λ is a positive scalar quantity de?ning the amplitude of plastic ?ow. For non-associative plasticity, the plastic potential is selected so that its derivative with respect to the stress tensor yields ?G 1 = αψ δi j + √ si j ?σi j 2 J2 (15)

(σ0 + β E p εeff )

p

(18)

where δi j is the Kronecker delta, si j is the stress deviator, and αψ is de?ned by a given dilation angle ψ and relations analogous to Eq. (13). Moreover, a Drucker–Prager criterion with elastic–perfectly plastic material response is considered with respect to the hardening behavior. The surrounding soil layers in the study can be modeled with the use of the card “MAT DRUCKER PRAGER” in LS-DYNA [14]. 2.4. Material model for rubber As two kinds of ?exible joint components, GINA gasket and vibration isolation bearing are both made of rubber material. Rubber shows nonlinear features in both geometric and material behavior. The Mooney–Rivlin model is a model that is widely used for simulating rubber material in many ?nite-element analysis software products [15]. In this model, the strain energy density function [9] is used to describe incompressible rubber material behavior as follows:

?2 W = C10 (I1 ? 3) + C01 (I2 ? 3) + C(I3 ? 1) + D(I3 ? 1)2

where p and C are the Cowper–Symonds strain rate parameters, and ε is the strain rate. Therefore, the current radius of the yield ˙ surface σ y is the sum of the initial yield strength σ0 plus the p p growth β E p εeff . Here, εeff is the effective plastic strain, and E p is the plastic hardening modulus, de?ned as: Ep = E tan E E ? E tan (19)

where E tan is the tangent modulus and E is the Young’s modulus. 2.6. Material model for cable The material model called “MAT CABLE DISCRETE BEAM” in LS-DYNA [14] can be selected to modify the material behavior of prestressed steel cable acting as another kind of ?exible joint components. This material can be used as a discrete beam element. The force, F, generated by the cable is nonzero if and only if the cable is in tension. The force is given by: F = max(F0 + K L, 0.) (20)

where F0 is the initial tensile force, L is the change in length, and K is the stiffness. In Eq. (14), L can be calculated by: L = L cur ? (L ini ? Doff ) (21)

(16) where I1 , I2 , and I3 are invariants of the right Cauchy–Green Tensor; and C10 and C01 are two material constants that can be obtained through the calculation based on uniaxial stretch or shear experimental data. Here, C and D are two constants decided by C10 and C01 , which can be described as: C = 0.5C10 + C01 ; C10 (5υ ? 2) + C01 (11υ ? 5) D= 2(1 ? 2υ) (17) where υ is Poisson’s ratio. In LS-DYNA, this model has been incorporated and can be used with the card “MAT MOONEY RIVLIN RUBBER” [14].

where L cur is the current length, L ini is the initial length, and Doff accounts for the offset used for F0 . Here, the stiffness K is de?ned as: E · Ac (22) K = (L ini ? Doff ) where E is the Young’s modulus and Ac is the cross-sectional area. The offset Doff can be input as a positive length for an initial tensile force in this study. 2.7. Non-re?ecting boundary It is inevitable that the ?nite boundary of the ?niteelement model will cause the seismic waves to be re?ected

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and then superimpose the waves, because such a soil–tunnel model is achieved by truncating the in?nite domain via an arti?cial boundary. Therefore, it needs to properly de?ne a ?nite computational domain with the appropriate de?nition of an arti?cial boundary. There are typically three approaches to model such a boundary, including the boundary-element method, the non-re?ecting boundary condition, and the use of in?nite elements [17]. Non-re?ecting boundaries are used on the exterior boundaries of the surrounding soil model of an in?nite domain to prevent arti?cial stress wave re?ections generated at the model boundaries from reentering the model and contaminating the results. In LS-DYNA, this approach is de?ned by providing a complete list of boundary segments, to which viscous normal shear stresses will be applied. It is useful for limiting the size of models in the case of handling geomechanical problems, and this approach can be described as: σnor = ?ρcd vnor ; σshear = ?ρcs vtan (23)

Fig. 3. The three-dimensional model of the whole immersed tunnel.

where ρ, cd and cs are the material density, the dilatational wave speed, and the shear wave speed of the transmitting media, respectively. The magnitude of these stresses is proportional to the particle velocities in the normal, vnor , and tangential, vtan , directions. This approach is consistent with the normal viscous boundary presented by Lysmer [18], which is accomplished by adding concentration dashpots on the nodes at the arti?cial boundary to modify radiation damping of the in?nite domain. In LS-DYNA, this approach has also been incorporated and can be used with the card “BOUNDARY NON REFLECTING”. 3. Finite-element modeling 3.1. Solid model First, the EDS-Unigraphics software has been used to construct the three-dimensional solid model describing the shape of the immersed tunnel, the surrounding soil, and the ?exible joints between adjacent tunnel segments. This step is necessary, and prepared for the following ?nite-element modeling process. This immersed tunnel is composed of 13 segments, including two connecting roads, two opening segments, two grating segments, and seven immersed segments, as well as ?exible joints at eight different places (from J1 to J8) (Fig. 3). Among them, seven immersed segments include: E1 (108 m); E2 (104 m); E3 (100 m); E4 (100 m); E5 (108 m); E6 (108 m); and E7 (108 m). Apart from those tunnel segments and ?exible joints, the Huangpu River model and the surrounding soil model (300 m deep into the local bedrock) have also been constructed according to the geological exploration data. With a total 755 solid parts, the ?nal soil–tunnel model can not only portray the serpentine shape of this tunnel and surrounding soil as a whole (Fig. 4) but also describe special parts in detail such as the ?exible joints (Fig. 5).

Fig. 4. The surrounding soil with multi-layers (explosive view of some upper soil layers).

Fig. 5. The three-dimensional model of the tunnel segment with ?exible joints.

3.2. Finite-element model Based on geometrical modeling, the ?nite-element model can then be constructed by utilizing the preprocessor Hypermesh. The ?nite element model of this immersed tunnel has been built with the eight-node hexahedron solid element, except the beam element for the prestressed cable. Fig. 6 shows the mesh model of the tunnel segment with a detailed description of the ?exible joints between two adjacent segments. The surrounding soil is meshed by the four-node tetrahedral element, because the three-dimensional CAD model of the surrounding soil is divided into 21 layers with various geometrical con?gurations. Both the linear hexahedron

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Fig. 6. The mesh of an the immersed segment with ?exible joints.

Fig. 7. An example of the mesh of one surrounding soil layer. Fig. 8. The mesh of the whole soil–tunnel system: (a) global mesh effect; (b) detailed mesh at one entrance. Table 1 Mesh details of the global ?nite-element model Component Tunnel segments Flexible joints Surrounding soil Total Element 688,283 18,495 497,176 1,203,954 Node 880,748 40,387 93,397 1,014,532 Table 2 Selected material models in LS-DYNA for simulation Name Tunnel segment Concrete shear key Steel shear key GINA gasket Isolation bearing Prestressed cable Surrounding soil Material model for simulation *MAT *MAT *MAT *MAT *MAT *MAT *MAT ELASTIC ELASTIC PLASTIC KINEMATIC MOONEY-RIVLIN RUBBER MOONEY-RIVLIN RUBBER CABLE DISCRETE BEAM DRUCKER PRAGER

elements and the linear tetrahedral elements have the following degrees of freedom at each node: translations, velocities, and accelerations in the nodal x, y, and z directions. The distance between the boundary of the surrounding soil and the tunnel contour on the same side is more than four times the section width of this immersed tunnel. Fig. 7 shows one typical layer of soil with the feature of tunnel penetration and the whole threedimensional soil–tunnel mesh modeling. The ?nal three-dimensional size of this soil–tunnel system model is 1670 m × 1110 m × 296 m. As shown in Table 1, the number of nodes and elements reaches 1,014,532 and 1,203,954, respectively. The global ?nite-element model of this soil–tunnel system and a detailed view of one entrance of the tunnel are illustrated in Fig. 8. To deal with contact, the card “CONTACT SINGLE SURFACE” has been selected for the interaction setting between the ?exible joint components and the tunnel segments, and the card “CONTACT SURFACE TO SURFACE” has been adopted for the interaction setting between the tunnel segments and the surrounding soil. Furthermore, the addictive “AUTOMATIC” contact option has been employed for both

contact settings in order to improve the calculation of stability and ef?ciency. 3.3. Material property data LS-DYNA currently contains over 200 constitutive models to cover a wide range of material behavior. Several appropriate material models adopted for this seismic simulation are shown in Table 2. The elastic material model is assumed for tunnel segments and concrete shear key, with material parameters shown as follows: Young’s modulus, 31.5 GPa; density, 2.5 g/cm3 ; Poisson’s ratio, 0.167. Material parameters of steel shear key include: Young’s modulus, 210 GPa; density, 7.85 g/cm3 ; Poisson’s ratio, 0.3; yield stress, 235 MPa; tangent modulus, 1.18 GPa. The Mooney–Rivlin material model is assumed for the GINA gasket and isolation bearing with material parameters such as isolation bearing including: density, 1.1 g/cm3 ; Poisson’s ratio, 0.499; C10 , 0.293 MPa;

J.-H. Ding et al. / Engineering Structures 28 (2006) 1367–1377 Table 3 Material parameters of several upper soil layers surrounding tunnel Soil layer Drab yellow silty clay Gray sandy silt Gray mucky silty clay Gray sandy silt Gray mucky clay Gray silty clay Gray sandy silt Dynamic shear modulus (MPa) 15.38 16.64 31.37 68.74 31.35 40.42 56.74 Poisson’s ratio 0.40 0.35 0.45 0.35 0.45 0.45 0.40 Density (g/cm3 ) 1.93 1.86 1.76 1.86 1.75 1.72 1.76 Cohesion (kPa) 19 7 14.1 8.4 12 12.5 9.8

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Angle of internal friction (? ) 20.6 24.8 13.5 19.4 15.3 7.4 10.2

Fig. 9. A Tangshan seismic wave record acceleration time history. Fig. 10. The resultant displacement in E6 and E7.

C01 , 0.189 MPa. Moreover, each beam for cables has been given an initial prestress of 30 kN. As has been mentioned, the seismic response of a tunnel is mainly affected by the layers of surrounding soil closer to the tunnel. Material property data for seven layers of soil through which this tunnel is excavated are illustrated in Table 3 as an emphasis. 3.4. Boundary condition and calculation A Tangshan seismic wave record in 1976 has been applied as ground motion at the bedrock (Fig. 9). According to the exceeding probability of 10% in 50 years and the designed 7-degree preventive intensity of the tunnel, amplitude modulation has been applied to make the maximum ground acceleration value of this seismic wave equal 1 m/s2 . The seismic wave has been input at the bedrock surface under two conditions: one is that the seismic wave will propagate along the longitudinal direction (Y -direction in Fig. 3), while the transverse (X-direction in Fig. 3) and vertical (Z -direction in Fig. 3) displacements are constrained; the other is that the seismic wave will propagate along the transverse direction, while the longitudinal and vertical displacements are constrained. Meanwhile, the lateral boundary surfaces of the surrounding soil have been modeled with non-re?ecting characteristics in both of these two cases, in spite of the large size of the soil model. There are two domain decomposition methods supported in LS-DYNA 970 MPP, including RCB (the Recursive Coordinate Bisection algorithm) and GREEDY (a simple neighborhood

expansion algorithm). RCB, the default method, has been used in this seismic response simulation because of its generally better performance. Each time it costs about 45 h for the calculation of each case with 32 CPUs in the Dawning 4000A supercomputer. 4. Results and discussion Final calculations have produced plenty of data describing the response of the immersed tunnel under seismic excitation, where the results can be post-processed in several styles. For example, Fig. 10 shows the resultant displacement distribution in the segments E6 and E7 at time 10.7 s as the seismic wave propagates along the longitudinal direction. Such graphs provide a direct and global understanding of the tunnel segments’ deformation, however they cannot re?ect the seismic response of the ?exible joints in detail. The ?exible joints between segments are important but weaker parts of the earthquake-resistant design of an immersed tunnel. Not only must they have superior anti-seepage properties, but they are also observed to prevent unacceptable deformation under seismic loading. Hence, more attention should be paid to seismic response analysis of the ?exible joints. The function performed by the ?exible joints depends on the inner component parts such as the GINA gasket and the steel/concrete shear key. Here, the relative displacement between the tunnel segments, the axial force of the cable, the amount of compression of the GINA gasket, and that of the isolation bearings are all measured for analysis.

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Fig. 12. Examples of relative displacement time histories at different ?exible joints: (a) position of J3-Point 3; (b) position of J7-Point 4.

Fig. 11. Analytical targets and measuring positions in each ?exible joint: (a) relative displacement between segments; (b) axial force in the cable; (c) compression of the GINA gasket, (d) compression of the isolation bearings.

As shown in Fig. 11, four points at the outside corner along the tunnel axis direction are chosen to evaluate the degree of deformation between adjacent tunnel segments. Using the same method, the maximum compression of the GINA gaskets at eight ?exible joints can also be measured (Fig. 12). Fig. 12(a) shows the variation in relative displacements with time between adjacent tunnel segments at the position

of Point1 in the ?exible joint J3, when the seismic wave is propagating along the longitudinal direction. Furthermore, Fig. 12(b) shows the variation in relative displacement with time between adjacent tunnel segments at the position of Point1 in the ?exible joint J7, when the seismic wave is propagating along the transverse direction. These two ?gures indicate explicitly that the two ?exible joints are at the tensile and compressive stages, respectively. In the tensile stage, Fig. 13(a) and (b) describe the maximum relative displacements between adjacent segments where there are ?exible joints as the seismic wave propagates along the longitudinal and transverse directions, respectively. In addition, Fig. 14 illustrates the maximum axial force in the prestressed cables in each ?exible joint under the same conditions. The maximum compression amounts (D) of the GINA gasket between the adjacent tunnel segments (from J1 to J8) are measured one by one and shown in Table 4 (original width: 15 cm). Furthermore, the compression amounts of 24 isolation bearings (original width: 8 cm) at each point of the ?exible joint are also measured. The maximum compression amounts of these 24 isolation bearings are selected as the maximum values (d) listed in Table 4. There is another calculation model, designed by using the mass-spring system method, for comparison; see Fig. 15. This is provided by the Shanghai Tunnel Engineering & Rail Transit Design and Research Institute (STEDI). In the mass-spring

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Fig. 13. Maximum relative displacement between adjacent tunnel segments: (a) seismic wave along the transverse direction; (b) seismic wave along the longitudinal direction. Table 4 Compression measuring results of GINA gaskets and isolation bearings Joint number Seismic wave along X-direction D (mm) d (mm) 15.7 21.2 20.7 18.1 14.6 13.7 16.2 14.6 3.84 5.22 7.17 3.92 6.37 5.83 5.19 5.62 Seismic wave along Y -direction D (mm) d (mm) 14.1 19.3 22.9 17.4 6.46 6.74 5.57 5.13 5.71 4.35 5.83 4.37 6.21 6.78 5.93 5.36

Fig. 14. Maximum cable tension at eight ?exible joints: (a) seismic wave along the transverse direction; (b) seismic wave along the longitudinal direction.

J1 J2 J3 J4 J5 J6 J7 J8

model, the surrounding soil is divided into several strips and each strip is considered as a lumped mass. Each strip is connected to each another by the springs K 2 and K 5 , and connected to the bedrock by the springs K 3 and K 6 at the same time. Here, the tunnel is simpli?ed as several beams, and each beam is connected to each another by the springs K 7 , K 8 , and K 9 . Meanwhile, the tunnel is connected to the soil by the linear springs K 1 and K 4 . In addition, each of the springs mentioned above is coupled with a viscous damper. The result acquired using this mass-spring model is also shown here. At the location of segment connections where

Fig. 15. The spring-mass model of the soil–tunnel system.

there are ?exible joints, Fig. 16 illustrates the maximum relative displacements of adjacent tunnel segments when the seismic wave is propagating along the longitudinal direction, and the maximum relative transverse displacements when the seismic wave is propagating along the longitudinal direction.

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other ?exible joints such as J3 are also vulnerable and worthy of more attention. Through the analysis, it is concluded that the application of eight ?exible joints plays a remarkable role in decreasing the inner stresses on parts and increasing the whole structural aseismic capability. Nevertheless, strict measures should also be taken to control the deformation and displacement of tunnel segments to ensure the waterproof capability of the ?exible joints effectively. Future work will include a more in-depth analysis of the current abundant results, research on the in?uence on the ?nal results of the choice of different material models, and a study of modi?ed RCB decomposition to get better load balance and overall performance, according to the particular distribution of contact zones in the analytical model.

Fig. 16. Maximum displacements in all the ?exible joints.

Acknowledgements By comparison, it is found that, apart from joints J1 and J8 on each end of the tunnel, J3, J5, and J6 are the other three ?exible joints exhibiting more relative displacement between adjacent segments. The reason for the behavior of J3 lies in the fact that the top of segment E2 is slightly higher than the riverbed, and is thus exposed to the water. Furthermore, compared with the tunnel segments from E1 to E4, E5 and E6 have longer segment length and show a straighter con?guration. Most importantly, the ?exible joint J5 is located at the deepest position in the surrounding soil from the point of view of the whole immersed tunnel. All these reasons have direct in?uences on the relatively noticeable behavior of joints J5 and J6 under earthquake conditions. 5. Conclusions This paper presents a novel and reliable simulation method for estimating the seismic response of an immersed tunnel in the city of Shanghai. The three-dimensional large-scale solid model of a soil–tunnel system is constructed using Unigraphics; the corresponding ?nite-element model uses Hypermesh. Meanwhile, several important and necessary factors such as material nonlinearity and contact nonlinearity are also taken into consideration. The analytical model put forward in this study fully depicts the real three-dimensional con?guration of the immersed tunnel and geological features of the construction site. The calculation has been accomplished successfully using LS-DYNA MPP on the “Dawning” supercomputer. Finally, there are some detailed and meaningful results with respect to the ?exible joints and the tunnel segments, as well as an overall understanding of the behavior of this tunnel. In general, the seismic response numerical simulation presented in this study proves that explicit FEM combined with parallel computing is a feasible and effective approach for dealing with such a large analytical model for seismic response simulation. Furthermore, compared with the massspring system method, new results obtained through the approach described in this paper not only accord with the conclusion that attention should be paid to the ?exible joints on the two ends (J1 and J8) of this tunnel, but also that The authors gratefully acknowledge the Shanghai Tunnel Engineering & Rail Transit Design and Research Institute for support and cooperation, the Earthquake Administration of Shanghai Municipality for professional guidance in seismic analysis, and the Shanghai Super Computing Center for offering access to the Dawning 4000A supercomputer with LSDYNA 970 MPP. The authors also wish to express special thanks to Dr. Li-Jun Li, Dr. Yuan Cao, and Hong Chen for helpful advice. This research is supported ?nancially by the National Natural Science Foundation of the People’s Republic of China (Serial Number: 60273048). References

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