Improved longitudinal seismic deformation method of shield tunnels based on the iteration of the nonlinear stiffness of ring joints

Accepted Manuscript Title: Improved longitudinal seismic deformation method of shield tunnels based on the iteration of the nonlinear stiffness of ring joints Authors: Jing Zhang, Chuan He, Ping Geng, Yue He, Wei Wang PII: DOI: Reference:

S2210-6707(18)31542-7 https://doi.org/10.1016/j.scs.2018.11.019 SCS 1344

To appear in: Received date: Revised date: Accepted date:

7 August 2018 12 November 2018 13 November 2018

Please cite this article as: Zhang J, He C, Geng P, He Y, Wang W, Improved longitudinal seismic deformation method of shield tunnels based on the iteration of the nonlinear stiffness of ring joints, Sustainable Cities and Society (2018), https://doi.org/10.1016/j.scs.2018.11.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Improved longitudinal seismic deformation method of shield tunnels based on the iteration of the nonlinear stiffness of ring joints

Jing Zhang1, Chuan He1*, Ping Geng1, Yue He1, Wei Wang1

1

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Key Laboratory of Transportation Tunnel Engineering (Southwest Jiaotong University), Ministry of Education, Chengdu,

1*Key

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China. E-mail: [email protected]

Laboratory of Transportation Tunnel Engineering (Southwest Jiaotong University), Ministry of Education, Chengdu,

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E-mail: [email protected]

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*Corresponding Author

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China,

Highlights:

Analytical equations for solving the nonlinear bending and axial stiffness of shield tunnel ring joints were

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



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proposed.

The inherent nonlinearity of ring joints was demonstrated and the 3D surfaces of stiffness of ring joints

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were obtained.

An iterative algorithm for the stiffness of ring joints based on their internal forces was proposed.



The proposed iterative algorithm was integrated with generalized longitudinal seismic deformation method

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

to form an improved method.



The validity of the improved method was verified by comparing the results of longitudinal internal forces, opening and stresses of ring joints based on a longitudinal beam-spring model.

ABSTRACT： Shield tunnels have undergone some prominent damage at joints due to the special structural composition of reinforced concrete segments that are connected by bolts. The generalized longitudinal seismic deformation method can achieve a better balance between the accuracy and efficiency of the calculation than the traditional seismic deformation method by using the time history of ground displacement of free-field as the input for the

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longitudinal beam-spring model of shield tunnels to determine the seismic responses of shield tunnels. However, the bending and axial stiffness of ring joints in this method simply are assumed to be constant values,

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which ignores the nonlinearity of the ring joints due to their discrete structures. In this paper, we demonstrate the inherent nonlinearity of ring joints by analytical derivation and obtain the stiffness of the 3D surfaces of

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ring joints under different loadings. Then, we propose an iterative algorithm for the stiffness of ring joints

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based on their internal forces, and this algorithm is integrated with generalized longitudinal seismic

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deformation method to form an improved method. The validity of the proposed method was verified by comparing the results of the longitudinal internal forces, the opening of joints, and the stresses on the joints

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obtained by generalized method and the improved method based on a Xiamen metro shield tunnel. Compared

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to the generalized methods, the results of the improved method indicated that the maximum joint opening, the compressive stress on the concrete, and the tensile stress on the bolts increased significantly, and these results

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demonstrated that the former methods of seismic design is unsafe.

KEYWORDS:

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shield tunnels; longitudinal seismic calculation; loading states; deformation modes; nonlinear stiffness; ring joints

1. Introduction Shield tunnels are the most commonly used structures in most urban subway construction in China, and are required to be seismically fortified. Because of the special structural composition of reinforced concrete segments connected by bolts, shield tunnels have the feature of relative flexibility suffer less from earthquakes compared to tunnels built by the mining method, and cut-and-cover tunnels (Nakamura et al., 1996, Wang et al.,

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2001, Wang et al., 2009). Even so, some shield tunnels have had some seismic damage, such as cracks in the

concrete, the dislocation of segments, the infiltration of ground water, and damage at the joints, which is the

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most prominent damage (Ayala et al., 1990, Lin et al., 2009). These problems may result in the loss of normal function or have an adverse effect on the durability of the tunnel in the event that its main structure is not

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destroyed. Shield tunnels also must be designed to account for seismic loading, and, because their stiffness in

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the longitudinal direction is much smaller than it is in the cross-section, the ring joints are more prone to large

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deformations and therefore require more attention in the seismic design. The longitudinal seismic analysis theory of a shield tunnel has undergone a transition from the free-field

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deformation approach to the soil-structure interaction approach (Wang et al.,1993, Hashash et al., 2001), and

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now the soil-structure interaction approach, which was established based on the beam-on-elastic foundation theory, is used extensively. Analytical solutions of the internal forces in a tunnel can be acquired using this

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method in which the beam elements are used to simulate the structure of the tunnel, and the foundation spring coefficients are used to represent the soil-structure interaction (Kuribayashi et al., 1974, St. John and Zahrah,

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1987). Also, more advanced analytical methods have been proposed that simulate the effects of the soilstructure interaction by modeling tunnels as cylindrical shells embedded in an elastic half-space (Luco and de Barros, 1994, Kouretzis et al., 2006, de Barros and Luco, 2010). The longitudinal seismic analysis of shield tunnels is difficult because an accurate simulation of the structure of the ring joints is required. In consideration of the low efficiency associated with using some

sophisticated 3D shield tunnel models to conduct the seismic dynamic analysis, the two most practical longitudinal models of shield tunnels are the longitudinal continuous model and the longitudinal beam-spring model. The longitudinal continuous model, proposed by Shiba et al. (1988, 1989), considers the shield tunnel to be a homogenous beam with equivalent stiffness in the longitudinal direction. Its equivalent stiffness takes into account the reduction in the longitudinal stiffness caused by ring joints. The main disadvantage of this

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model is that the longitudinal equivalent stiffness is based on longitudinal bending or axial behavior, but the equivalent bending stiffness actually is not equal to the equivalent axial stiffness. The beam-spring model,

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proposed by Koizumi et al. (1988), models the segment rings and ring joints with beam elements and spring elements, respectively, so that the response of the ring joints can be obtained directly. In this method, it is not

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convenient to determine joint stiffness, so it often is determined by tests (Nishino et al., 1986).

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The longitudinal seismic deformation method commonly is used to determine the longitudinal seismic

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internal force of shield tunnels. In doing so, the two models mentioned above are used, and the deformation of the ground is assumed to be distributed in a sinusoidal manner (Kawashima, 1994). Although the

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assumption of this method that the deformation of the ground is sinusoidal makes it easier to use, it is not

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suitable for use with non-uniform ground that has large contrasts in stiffness because this changes the seismic response of tunnels. A generalized longitudinal seismic deformation method, proposed by He et al. (1999,

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2013), denoted as He method, extends the application of the seismic deformation method to non-uniform ground, which takes the ground displacement time history as the input of the beam-spring model and solves

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the internal forces of shield tunnels in a quasi-static manner (Fig. 1).

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(a) (b) Fig. 1. Schematic diagram of longitudinal seismic deformation method: (a) traditional method; (b) He method (recreated from He at al., 2013)

The ring joints of shield tunnels are comprised of the cross-sections of the joints and bolts, in which

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compressive stress is transmitted only by the joint cross-section concrete, and tensile stress is transmitted

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collectively by the concrete and joint bolts. The contact state and the contact area of the cross-section of a

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joint are both variable depending on the different loading states. Therefore, the evaluation of the stiffness of

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ring joints is a typical, nonlinear problem, even if the nonlinearity of the material and geometric nonlinearity

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are not considered. However, in the literature, the stiffness of the joint spring is assumed to be a constant value in seismic analysis by using the beam-spring model of shield tunnels and ignoring its nonlinearity (Koizumi

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et al., 1988, 2000, He et al., 1999, 2013).

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In this study, our aim was to propose an improved method to amend the above insufficiency as follows. First, the analytical solution to the nonlinear bending and axial stiffness of the ring joints is obtained. Then,

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an iterative algorithm for the nonlinear stiffness of the ring joints is proposed and introduced into He method to form the Improved longitudinal seismic deformation method (hereafter, Improved method), which can simulate the mechanical properties of ring joints more accurately. Then, to verify the effectiveness of the proposed method, we used both He method and the Improved method to calculate the seismic responses of a metro shield tunnel that passes longitudinally through the non-uniform ground in Xiamen.

2. Nonlinearity of the ring joints of a shield tunnel 2.1 Assumptions made to obtain the analytical solutions As mentioned above, evaluating the stiffness of ring joints is a typical nonlinear problem even if the nonlinearity of the material and geometric nonlinearity are not considered. In order to demonstrate this feature,

following assumptions were made in order to obtain the analytical solutions:

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in the analysis, both the concrete lining and the steel bolts are considered to be linear elastic components. The

(1) Plane cross-section assumption: the cross-sections of shield tunnels always are plane before and after

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longitudinal deformation, which means the deformation at any point on the cross-section is proportional to the distance from the point to the neutral axis.

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(2) The discrete longitudinal joint bolts are assumed to be continuously and evenly distributed along the

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circumference of the ring. (The equivalent joint bolts are shown in Fig. 4(b)).

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(3) Compressive stress is transmitted only by the concrete in the compressive area, and, in contrast, tensile stress is transmitted by the concrete and joint bolts together in the tensile area.

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(4) Studies have suggested that the two mainly longitudinal deformation modes of a shield tunnel are

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longitudinal bending and axial deformation, e.g., (St. John and Zahrah, 1987). Therefore, we focused this paper on the analysis of the combined effect of the longitudinal moment and the axial force, and we ignored

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the pre-tightening of bolts and the transverse shear of the shield tunnels.

2.2 Loading states and deformation modes of ring joints

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When subjected to the combined action of the longitudinal bending moment and the axial force, the

loading states of ring joints can be divided into three types, i.e., pure bending (PB), compression bending (CB), and tension bending (TB). The changing of the deformation modes of the ring joints from fully closed (Mode 1) to fully opened (Mode 7) is a natural and continuous process (Table 1). When a joint is partially opened (Modes 3, 4, and 5), its neutral axis (red dashed line) will move within

the full height of the cross-section. When the deformation mode of this joint changes from fully closed (Mode 1) to partially opened (Mode 3), it must undergo the critical deformation mode (Mode 2) in which the stress at the top of the cross-section of the joint is zero. Similarly, when this joint expands from partially opened (Mode 5) to fully opened (Mode 7), it must undergo the critical deformation mode (Mode 6) in which the stress at the bottom of the joint cross-section is zero. Note that each loading state may contain different

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deformation modes, and the possibility of each deformation mode’s occurring at a certain loading state is represented by the symbols, ×, ●, and ○. The following analytical derivation proves the stated premise.

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Table 1. Classification of ring joint deformation modes under different loading states

States

PB CB TB

Note that：

2

× ● ×

× ● ×

4

5

6

7

× ● ×

○ ● ○

● ● ●

○ ○ ●

× × ●

3

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Loading

1

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Deformation Modes

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N

Graphical Representation

1. ×：the impossible deformation modes in this certain loading state;

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2. ●：the possible deformation modes in this certain loading state; 3. ○：the critical deformation modes in this certain loading state;

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2.3 Nonlinear bending stiffness of ring joints Fig. 2 shows that a segment ring is simulated by a beam element in the longitudinal beam-spring model

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of shield tunnels, and the bending mechanical behavior of a ring joint is simulated by a rotational spring element. The angle between the two surfaces, Bb and Cc, of the joint, θ’, is the rotation angle of the spring element in the longitudinal beam-spring model. The bending stiffness of ring joints, Kθ’, can be defined as the bending moment that corresponds to the unit rotation angle of the spring. For ease of derivation below, the angle θ between the middle cross-sections, Aa and Dd, of two adjacent segment rings is introduced as an intermediate variable.

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Fig. 2. Definition of rotation angle and bending stiffness of ring joints

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2.3.1 Mode 1 and Mode 2

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In Modes 1 and 2, the joint is fully closed only if it is in the compression bending state. The stress on the

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joint cross-section that has the distribution shown in Fig. 3 can be calculated using Eqs. (1) and (2). In Mode 2, the bending stress at the top of the cross section is equal to the axial stress, but it has the opposite sign ( Eq.

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as Mc1 and Nc1, respectively.

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(3)), and the longitudinal bending moment and the axial force that correspond to this critical state are denoted

Fig. 3. Stress distribution of ring joints in deformation Mode 1  1   N   M   2   N   M

(1)

N N   N  A   D 2 c  (1   2 )  4  M   M  M 3  Ws  D  (1   4 )  32 

(2)

 1  0   2   N   M  2 N  2 M M c1 

(3)

N c1 D d2 (1  2 ) 8 D

(4)

where σ1 and σ2 are the stresses at the top and bottom, respectively, of the joint cross-section; M and N are the longitudinal moment and the axial force, respectively; and the terms, d and D, are the inner and outer diameters of the segment ring, respectively.

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2.3.2 Mode 3 and Mode 4

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Fig. 4 shows the stress and deformation on the cross-section of the joint in deformation Mode 3. Based on the above assumptions, the deformation compatibility equations and mechanical equilibrium equations of

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M

A

N

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ring joints in Mode 3 can be expressed as Eqs. (5) - (9):

(b)

(c)

(d)

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(a)

Fig. 4. Stress and deformation of ring joints in deformation Modes 3: (a) sketch of compression bending

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state; (b) the joint cross-section; (c) stress distribution on the joint cross-section; (d) deformation on the joint cross-section

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Deformation compatibility equations: c 

ls D   (  x)  2 2 2

l j rx   t  s   ( r  x)  D/ 2 x 2 2 2

Mechanical equilibrium equations:

(5)

(6)

Ec c Ec t  / 2   / 2  (r cos   x)rtd =2 (r cos   x)rtd  N 0 0  D/ 2 x D/ 2 x 

(7)

kr  j  / 2 Ec c  / 2  (r cos   x)rtd  2 (r cos   x)rd  N 0 0  D/ 2 x rx

(8)

Ec c Ec  t  / 2   / 2  (r cos   x)2 rtd  2 (r cos   x)2 rtd  M  Nx 0 0 D/ 2 x  D / 2 x 

(9)

2

2

2

where ls is the width of a segment ring; x is the distance from the neutral axis to the geometric symmetry axis

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of the joint cross-section; x is equal to rsinφ; φ is the angle that represents the location of the neutral axis, where 0 ≤ φ ≤ π/2; εt is the tensile strain at the top of the joint cross-section; εc is the compressive strain at the

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bottom of the joint cross-section; δj is the elongation of the joint bolts at the top of the joint cross-section; r is the distance from the joint bolts to the centroid of the joint cross-section; kr, (kr = nkj/(2πr), is the tensile

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stiffness of the equivalent joint bolts; kj is the tensile stiffness of one joint bolt; n is the number of joint bolts

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in a single ring; Ec is the Elastic Modulus of concrete; and t is the thickness of the segment.

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Combining Eqs. (5) - (9) and introducing the four intermediate variables shown in Eq. 10, i.e., A0, B0, C0,

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and D0, the relationship between the joint rotation angle, θ’, and the longitudinal internal forces, (M, N), can

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be obtained as shown in Eq. (11), which includes a transcendental equation of φ (Eq. (12)). Eqs. (11) and (12) are the joint rotation angle function equation and the neutral axis position equation, respectively.

(10)

  N ls 1 1   Ec t A0 A0     1    2 (  )      2r Ec t kr B0   kr ls B0 B0    '  (r  r sin  ) ( Ec t    A0  N  1 )  ( D / 2  r sin  ) kr ls B0 2r 2 kr B0 

(11)

Nr (ls kr  Ec t )  (C0 B0  A0 D0 )  ( M  Nr sin  )  B0  Nr  D0 Ec t  A0  kr ls  A0  kr ls  B0

(12)

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   A0  cos   sin   ( 2   )   B  cos   sin   (    )  0 2   1 C  (   )(  sin 2  )  3 sin(2 )  0 2 2 4   1 3  D0  (   )(  sin 2  )  sin(2 )  2 2 4

Eq. (11) can be simplified as Eq. (13) in deformation Mode 4, where the neutral axis coincides with the geometric symmetry axis, and the longitudinal bending moment and the axial force that correspond to this

critical state are denoted as Mc2 and Nc2, respectively. N c 2 k r ls ls 1    2 E r 2 t ( E t  k )  c r c  E t 2 r  N c  '  (   )  D kr ls 2r 2 kr

M c 2  Nc 2  (

(13)

 r (kr ls  Ec t )  r 

2 Ec t

4

)

(14)

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2.3.3 Mode 5 Deformation Mode 5 (Fig. 5), in which the neutral axis of the joint cross-section moves down below the

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geometric symmetry axis, has a stress distribution and a deformation similar to those in Fig. 4. By changing the signs of N and φ in the formulas above, the joint rotation angle function in deformation Mode 5 and the

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M

A

N

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neutral axis position equation can be obtained.

(b)

(c)

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(a)

Fig. 5. Stress and deformation of ring joints in deformation Mode 5: (a) sketch of the loading state; (b) the

section

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joint cross-section; (c) stress distribution on the joint cross-section; (d) deformation on the joint cross-

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For the pure bending state, substituting N = 0 into Eqs. (7) - (9) yields Eqs. (15) - (17): 

ls cos   ( / 2   )sin   M Ec I c cos3 

(15)

ls  sin   M Ec I c cos3 

(16)

'

1 2

  cot    ( 

Kj EC AC / ls

)

(17)

where Ic is the moment of inertia of the joint cross-section; Ac is the area of the joint cross-section; and Kj, (Kj = nkj), is the total tensile stiffness of the joint bolts in a segment ring. Note that the equivalent longitudinal bending stiffness (EI)eq of shield tunnels in length ls can be derived based on the pure bending state, as proposed by Shiba et al. (1988). The equivalent longitudinal bending stiffness is shown as Eq. (18), which is precisely the transformation of Eq. (15), and the neutral axis position

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equation is the same as Eq. (17). Thus, the bending stiffness of ring joints derived for the pure bending state is just one special case of their nonlinear stiffness. cos3   Ec I c cos   ( / 2   )sin 

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( EI )eq 

(18)

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For common shield tunnel linings in China, the ratios of the thicknesses, t, of the segments to the outer

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diameters of the segment rings are approximately the same, and it usually is in the range of 0.05 - 0.06 (PRC

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National Standard GB50157, 2003), and the material parameters of the concrete and the bolts basically are

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stable. Substituting the commonly used values into Eqs. (15) - (17), Eq. (17) has a unique solution in the range

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of 0 ≤ φ ≤ π/2, and this solution cannot be 0 or π/2. It can be demonstrated that only the deformation Mode 5 corresponds to the pure bending state, and Modes 4 and 6 are its critical deformation modes. Further, it is

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proven that the position of the neutral axis is unique and that the value of the bending stiffness of the ring

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joints is constant during the pure bending state. Based on the pure bending state, when the tunnel is compressed simultaneously with bending, the area

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of compression increases so that the neutral axis of the joint cross-section moves upwards. Similarly, when the tunnel is tensioned simultaneously with bending, the area of tension increases so that the neutral axis of the joint cross-section moves downwards. Therefore, deformation Modes 1 - 5 correspond to the compression bending state, and deformation Modes 5 – 7 correspond to tension bending state. 2.3.4 Mode 6 and Mode 7 In Modes 6 and 7, the joint is fully opened only if it is being subjected to the tension bending loading

state (Fig. 6). The deformation compatibility equations and mechanical equilibrium equations of ring joints in

(a)

(b)

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deformation Mode 7 are expressed as Eqs. (19) - (24):

(c)

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Fig. 6. Stress and deformation of ring joint in deformation Mode 7: (a) longitudinal deformation on cross-

Deformation compatibility equations:





2  [ kr  j 2 

(kr  j1  kr  j 2 )  (r  r cos  )

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0

(20)

2

( Ec t1  Ec t 2 )  ( D / 2  r cos  ) ]  rtd  N D

ED

0

(19)

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Axial force equilibrium equations: 2 [ Ec t 2 

A



2r

N

( t1  ls / 2   t 2  ls / 2)  j1 / 2   j 2 / 2    D 2r 2  j1 / 2   j 2 / 2  '

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section; (b) stress distribution of concrete; (c) stress distribution of bolts

2r

]  rd  N

(21)

(22)

Moment equilibrium equations:

( Ec t1  Ec t 2 )  ( D / 2  r cos  ) D ]  ( D / 2  r cos  )  rtd  M  N  D 2

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

2 [ Ec t 2  0



(kr  j1  kr  j 2 )  (r  r cos  )

0

2r

]  (r  r cos  )  rd  M  N  r

(24)

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2 [ kr  j 2 

(23)

where εt1 and εt2 are the tensile strain at the top and bottom of the ring joint cross-section, respectively; δj1 and δj2 are the elongations of the joint bolts at the top of the joint cross-section, respectively. Combining Eqs. (19) - (24), the joint rotation angle, θ’, can be expressed as shown in Eq. (25). In deformation Mode 6, the strain at the bottom of the joint cross-section is 0, and the longitudinal bending moment and axial force that correspond to this critical state are noted as Mc3 and Nc3, respectively.

'

M c3 

M  kr r 3

(25)

N c 3 r 2 ls D

(26)

The above analysis indicates that the corresponding bending stiffness is theoretically +∞ in Modes 1 and 2, in which the ring joints are fully closed and the rotation angle is 0. The joint rotation angle, θ’, is proportional to the bending moment in Modes 6 and 7, which means the bending stiffness is a constant value. For general

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cases, the expression of the joint rotation angle, θ’, including the longitudinal internal force, (M, N), is shown

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as Eq. (11), and this equation can be abbreviated as θ’ = f(M, N). The neutral axis position equation (Eq. (12)) can be abbreviated as g(φ) = 0, which is a transcendental equation, which must be solved first before the

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rotation angle and the bending stiffness of ring joints can be determined.

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2.4 Nonlinear axial stiffness of ring joints

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In the traditional longitudinal beam-spring model of shield tunnels, the calculation of the axial stiffness

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of ring joints is based on the single axial loading mechanical behavior shown in Fig. 7. In the single

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compression loading condition (Fig. 7(a)), the openings of the ring joints are 0, so, theoretically, the compressive stiffness of the ring joints is +∞. For the single tension loading condition (Fig. 7(b)), the ring

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joints are opened evenly so that their tensile stiffness can be considered as the total tensile stiffness, Kj, of the

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joint bolts in one segment ring. The longitudinal equivalent stiffness model also uses the same principle to

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calculate the equivalent axial stiffness of shield tunnels (Shiba et al., 1988).

(a)

(b)

Fig. 7. Longitudinal mechanical behavior of shield tunnels under axial force: (a) compression; (b) tension

Similar to Section 2.3, in which the longitudinal moment and axial force were considered at the same time, the nonlinear axial stiffness of the ring joints is calculated in this section.

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(a)

(b)

Fig. 8. Stress distribution on center cross-section and neutral axes’ location of homogeneous beam: (a) pure

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bending; (b) tension bending

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Fig. 8 shows the position of the neutral axis of one beam during pure bending. The original pure bending

M

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neutral axis, shown in red dashed lines in Fig. 8(b), will extend δl and produce tensile stress σx during the tension bending condition. According to the principles of material mechanics, it can be proved that σx is exactly

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equal to the normal stress, σN, induced by axial force N.

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Therefore, the above principle can be used to define the axial stiffness for the CB or TB states, i.e., to define the relationship between the elongation of the neutral axis in the PB state, δl, and the axial force, N, in

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Fig. 9. The axial stiffness of ring joints Kt and Kc are calculated as follows:

(a)

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(b)

Fig. 9. Comparison of ring joints’ deformation between tension bending and pure bending: (a) deformation

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mode 5; (b) deformation mode 7

(1) In Modes 1 and 2, the ring joints are fully closed so that the compressive stiffness of the ring joints

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theoretically is +∞.

A

N

(2) In Modes 3, 4, and 5, the ring joints are partially opened. The TB neutral axis is located below the PB

M

neutral axis, as shown in Fig. 9(a), but the CB neutral axis is located above the PB neutral axis. The elongation of the PB neutral axis, δl, and the axial stiffness of ring joints (compressive Kc or tensile Kt) can be calculated

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by Eqs. (27) and (28):

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 l  x0  x  '

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K c / Kt 

N r  sin 0  r  sin   '

(27) (28)

(3) As shown in Fig. 9(b), the ring joints are fully opened in Modes 6 and 7. The elongation of the PB

A

neutral axis, δl, and the tensile stiffness, Kt, of the ring joints can be calculated by Eqs. (29) and (30):

l 

(r  x0 ) (r  x0 )   j1   j2 2r 2r

(29)

N  r 2  2M  x0 2 Kt  2    kr  r 3

(30)

The above derivation indicates that the bending and axial stiffness of all of the ring joints are nonlinear variables related to the longitudinal bending moment and axial force. Because the expressions of joint rotation

angle, θ’ (Eq. (11)) and axial stiffness Kt/Kc (Eq. (28)) contain transcendental equation, g(φ)=0, the display expression of stiffness cannot be obtained. If the parameters of a specific ring segment are used, the threedimensional stiffness surfaces of ring joints, Kθ’-M-N, Kt/Kc-M-N, can be obtained, and the corresponding stiffness of one certain ring joint can be interpolated by using the internal force combination (M, N).

3. Improved longitudinal seismic deformation method

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3.1 He method

As shown in Fig. 10, the implementation of He method consists mainly of two parts. First, a 3D model

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is built for the free field (without the shield tunnel), and a dynamic analysis is conducted to obtain the time history of the ground displacement at the location of the longitudinal axis of the tunnel. Then, a longitudinal

U

beam-spring model of the shield tunnel is established, and the time history of ground displacement that was

A

N

obtained in the previous step is used as input to solve the longitudinal seismic response of the shield tunnel in

M

the quasi-static way.

The longitudinal beam-spring model in the lower part of Fig. 10 consists mainly of three components,

ED

i.e., (1) beam elements that simulate the segment rings of the shield tunnel; (2) the spring elements that

PT

simulate the ring joints, including axial springs, rotational springs in two directions, and shear springs in two directions; and (3) ground springs that simulate the soil-structure interaction, including horizontal, vertical,

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and longitudinal springs, the stiffnesses of which can be obtained by multiplying the subgrade reaction

A

coefficients by the spring action area.

longitudinal axis of the tunnel

free field dynamic analysis

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ground displacement time history

beam-spring model for GLSDM

z

K y

y

x

(spring elements)

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K z

a ring joint

a segment ring

K ux

(beam element)

Ksy

wi(x,t)

ground spring

vi(x,t)

ui(x,t) ground displacement time history input

M

A

( ktv、kth、ka )

N

U

Ksz

Fig. 10. Calculation diagram of generalized longitudinal seismic deformation method (He method)

ED

3.2 Iteration algorithm for the stiffness of the ring joints

PT

As analyzed above, the relationship between the bending and axial stiffnesses of ring joints Kθ’, Kt , Kc shows complex nonlinear characteristics according to longitudinal bending moment M and axial force N of

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the ring joints. It is necessary to calculate their true values iteratively based on the internal force combination (M, N).

A

We proposed an iterative algorithm for the stiffness of ring joints based on longitudinal internal forces

and this iterative algorithm was integrated with He method. We called the proposed method the Improved method. The stiffness of the ring joints used in the beam-spring model can be updated in each time step according to the longitudinal internal forces M and N, so that the seismic response of the shield tunnel is calculated using the updated true stiffness. The improved calculation flow is shown in Fig. 11, where {K0},

{Kij} and {Kij’} represent the initial value and the values before and after the iteration values of the ring joints’ stiffness, respectively. The calculation procedure of the Improved method is as follows: (1) Establish the longitudinal beam-spring model of the shield tunnel shown in Fig. 10. (2) Establish a three-dimensional free-field model of the engineering site and conduct a dynamic time history analysis to obtain the ground displacement time history of each point at the location of the tunnels’

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longitudinal axis, i.e., u(x,t)、v(y,t)、and w(z,t).

(3) The bending stiffness {Kij}θ’ and the axial stiffness {Kij}t/c of the ring joints in the longitudinal beam-

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spring model are initialized to be {K0}θ’、{K0}t/c, where i represents the calculation time step, and j represents the sequence number of the ring joint.

U

(4) The ground displacement at time step i is applied to the end of the ground springs, and then the

A

N

longitudinal bending moment and the axial force of each ring joint, i.e., {Mij}, {Nij}, are solved and provided

M

as outputs.

(5) Substitute the internal forces obtained from step (4) into the 3D stiffness surfaces, i.e., Kθ’-M-N, Kt/c-

ED

M-N, to interpolate the stiffness of each ring joint, i.e., {Kij’}θ’, {Kij’}t/c.

PT

(6) Compare {Kij}θ’ with {Kij’}θ’ and compare {Kij}t/c with {Kij’}t/c joint by joint. Theoretically, when the two are equal, the iteration calculation in the current time step is completed. In consideration of the balance

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between the accuracy of the calculation and efficiency in engineering, the difference between the two is less than 1% after several repeated attempts, when the calculation results can satisfy the requirements. Therefore,

A

the difference of 1% between the stiffness in two consecutive iteration steps is taken as the convergence criterion.

(7) If step (5) satisfies the convergence criterion, the next time step is applied and solved, and steps (4) (6) are repeated;

(8) If step (5) does not satisfy the convergence criterion, then update the stiffness using {Kij’}θ’, {Kij’} t/c instead of {Kij}θ’, {Kij} t/c and recalculate the internal forces of each ring joint {Mij}, {Nij}. Repeat steps (4) (6) until the convergence criterion is met and then proceed to the next step. (9) The solution is complete when the convergence criterion is satisfied in the last time step.

Establish the longitudinal beamspring model

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Sequential solution in each time step

Yes

Is the initial time step

N

No

A

Replace{Kij}with{Kij’} to assign the stiffness of ring joints

M

Apply the ground displacement in this time step and solve

Call ground displacement in current time step

U

Initialize the stiffness of ring joitns{K0}

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Obtain the ground displacement time history at the tunnel location u(x,t), v(y,t), w(z,t)

No

PT

ED

Output longitudinal moment{Mij} and axial force{Nij}

Meet the convergence criterion

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Substitute the internal forces into the 3D stiffness surface to interpolate the corresponding stiffness of ring joints {Kij’}

Save the current step results and proceed to the next time step

No Yes

ti=tmax

Yes Complete the calculation and output the results

A

Iterative calculation in one time step

Fig. 11. Flow chart of the stiffness iteration and its application

4. Numerical verification 4.1 Engineering background The information and analyses presented in this paper are based on the cross-sea subway shield tunnel of Xiamen Rail Transit Line 2. The tunnel from Haicang Avenue to Dongdu Road that passes longitudinally

through fully-weathered ground to slightly-weathered ground was selected as the object of this research, as shown within the red dashed rectangle in Fig. 12, to verify the validity of the proposed method. Table 2 provides the mechanical properties of the ground in the target interval. Dongdu Road

Haicang Avenue

Datu Island

Main channel （Xiamen Island）

Fully weathered ground

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Modeling field

slightly weathered ground

Fig. 12. Profile view of the shield interval from Haicang Avenue to Dongdu Road

Fully weathered ground －3

Unit weight of volume, γ (kN·m ) Modulus of elasticity, E (MPa)

20.6

Cohesion, c (kPa)

50.00

Friction angle,  (deg) Subgrade reaction coefficient, K(MPa/m)

24

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Table 2. Mechanical properties of the ground

Slightly weathered ground 26.3

25.00

800.00

U

450.00 30

N

60

200

A

Fig. 13 shows the cross-section of a segment ring with outer and inner diameters of 6.7 and 6.0 m,

M

respectively. The segment ring consisted of six precast, reinforced concrete segments that were 1.5 m long

ED

and had elastic moduli of 34.5 GPa. Sixteen M30 bolts with diameters, lengths, and elastic moduli of 30 mm,

A

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530 mm, and 210 GPa, respectively, were set in one single ring.

Fig. 13. Sketch of a segment ring

4.2 Analysis of the stiffness of the ring joints 4.2.1 Bending stiffness The 3D surface of Kθ’-M-N is shown in Fig. 14, where the tensile axial forces are positive, and the compressive forces are negative.

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Kθ’=+∞

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Modes 1 & 2

U

Modes 3 & 4

A

N

Mode 5

Mode 5

Mode 6

Mode 7

A

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PT

ED

M

(a)

(b) Fig. 14. 3D surface graph of bending stiffness Kθ’: (a) compression bending; (b) tension bending The three-dimensional surface of Kθ’-M-N in the compression bending state is composed of two parts,

which are shown in different colors in Fig. 14(a). The gray region in the upper half of Fig. 14(a) represents the +∞ bending stiffness when the ring joints are fully closed (Modes 1 and 2). The cyan region in the lower half of Fig. 14(a) represents the limited bending stiffness value when the ring joints are opened partially (Modes 3, 4, and 5). Fig. 14(b) shows the 3D surface of Kθ’-M-N in the tension bending state, and its shape is expressed in the two-steps style, and the values of the bending stiffness of the ring joints are maintained

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between two specific levels.

It can be seen that the bending stiffness of a ring joint is not a constant value for different loading states,

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and it is related to the internal force combination (M, N) of this ring joint. In addition, Fig. 14(a) shows that the axial compression increases the bending stiffness of the ring joints, and the greater the compressive axial

U

force becomes, the more the degree of stiffness improves. As shown in Fig. 14(b), when the ring joints are

A

N

fully opened (Modes 6 and 7), the values of bending stiffness of the ring joints are located on the lower-step

M

of the two-steps surface.

Taking the results of the calculation of the lining in this paper as an example, the minimum bending

ED

stiffness of the ring joints was 2.14×1010 N·m/rad due to the action of the axial tensile force, and it was

PT

5.57×1010 N·m/rad for pure bending, with the latter being a factor of approximately 2.6 greater than the former. It was apparent that, when the ring joints are fully opened, the axial tensile force has a significant effect on

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reducing the bending stiffness of the ring joints. 4.2.2 Axial stiffness

A

The three-dimensional surface of Kc-M-N shown in Fig. 15(a) is similar to that shown in Fig. 153(a). The

cyan region in the lower half of surface in Fig. 15(a) shows the limited compressive stiffness of the ring joints when they are partially opened (Modes 3, 4, and 5). When the ring joints change from Mode 3 to Mode 2, Kc moves from the cyan region to the gray region in the higher half, where the compression stiffness of the ring joints theoretically was +∞.

Under the state of tension bending, it was observed that the overall characteristics of the surface of KtM-N are similar to the style of the two-steps. With the increase of bending moment, the tensile stiffness of the ring joints increased gradually from the pure tensile stiffness (Kt = 4.239×109 N/m in this paper) in an approximate exponential form to a certain level and then gradually approaches horizontal, and the tensile stiffness finally became stabile at about 3.1×1010 N/m. It was apparent that the tensile stiffness of the partially-

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opened ring joints (Mode 5) was a factor of 7.3 times greater than that fully-opened ring joints (Mode 7).

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Kc=+∞

U

Modes 1 & 2

M

A

N

Modes 3 & 4

(a)

A

CC E

PT

ED

Mode 5

Mode 5

Mode 6

Mode 7

(b)

Fig. 15. 3D surface graph of axial stiffness: (a) compressive stiffness Kc; (b) tensile stiffness Kt

4.3 Results of the Improved method 4.3.1 Calculation model and cases As mentioned earlier, the first step of He method is to calculate the time history of the ground displacement. The finite difference software, FLAC3D, was used to establish the 3D free field model with the

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length and width of 150 m (a total of 100 rings) and a height of 22.4 m, as shown in Fig. 16. The position of the longitudinal axis of the tunnel in the model is indicated by a yellow dotted line. Studies have suggested

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that near-surface seismic waves are mainly shear waves propagated vertically (Kawashima, 1994). Therefore, in this paper, the synthetic seismic wave (Fig. 17) provided by the Earthquake Department of Xiamen is input

U

from the plane XOZ at the bottom of the model, which propagates vertically upward in the direction of OY

CC E

PT

ED

M

A

N

and shakes in the direction of 45° with OZ.

A

free-field boundry

the 50th ring 10.8m slightly weathered ground Y

11.6m

fully weathered ground X

O

direction of shaking 45° Z direction of propagation

Fig. 16. Half calculation model of the simplified ground

4

Acceleration (m/s2)

Amax=0.33g

2

0

-2

time (s)

-4 5

10

15

20

25

30

Fig. 17. Artificial synthetic seismic wave acceleration

35

40

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0

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The longitudinal beam-spring model in Fig. 10 was established using the FEM simulation software, ANSYS. Each segment ring was simulated with one 3D beam element, and each ring joint was simulated with

U

a group of spring elements. Table 3 shows that the bending and axial stiffnesses of the ring joints in Case A

N

were assumed to be constant values based on Shiba et al. (1988); however, the values in Case B were calculated

A

from the 3D stiffness surfaces in Sections 4.2.1 and 4.2.2. Shear behavior of the ring joints was not the focus

M

of the research in this study, so the shear stiffness of the ring joints, Ks, in both cases was assumed to be +∞.

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Table 3. Calculation cases and the stiffness parameters of ring joints Case ID Method

Bending stiffness Kθ’ 5.57×1010 N·m/rad

He method

Case B

Improved method

PT

Case A

stiffness iteration

Axial stiffness Kt / Kc Kt=4.24×109 N/m Kc=+∞ stiffness iteration

Shearing stiffness Ks +∞ +∞

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4.3.2 Envelope results of peak internal forces The longitudinal internal forces of the three-dimensional beam elements in the model of a shield tunnel

A

include the bending moment, My, due to the curvature in the horizontal plane, the bending moment, Mz, due to the curvature in the vertical plane, and the axial force, Nx. Fig. 18 shows the sign convention of internal forces.

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Fig. 18. Longitudinal force and moment of a shield tunnel 12000 Case B max Case A max

10000

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Mz (kN·m)

8000 6000 4000

U

2000 0 10

20

30

40

50

60

70

80

90

100

N

0

Ring number

A

0

M

-4000

ED

Mz (kN·m)

-2000

-6000

-8000

PT

Case B min Case A min

-10000

CC E

0

10

20

30

40

50

60

70

60

70

80

90

100

Ring number

(a)

30000

Case B max Case A max

A

25000

My (kN·m)

20000 15000 10000 5000 0 0

10

20

30

40

50

Ring number

80

90

100

0 -5000

My (kN·m)

-10000 -15000 -20000 -25000 Case B min Case A min

-30000 0

10

20

30

40

50

60

70

80

90

100

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Ring number

(b) 5000

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Case B max Case A max

4000

2000

U

Nx (kN)

3000

N

1000

0 10

20

30

40

50

60

A

0

70

80

90

100

M

Ring number

0

ED

-10000

Nx (kN)

-20000 -30000

PT

-40000

Case B min Case A min

-50000

A

CC E

-60000

0

10

20

30

40

50

60

70

80

90

100

Ring bunmer

(c)

Fig. 19. Envelope results of longitudinal internal forces: (a) Mz; (b) My; (c) Nx

Fig. 19 shows the envelope results of the peak internal forces of both cases. In Fig. 19(a) and (b), the envelopes of the positive and negative bending moments are approximately symmetrical due to the constant bending stiffness of the ring joints in Case A. In comparison, the bending stiffness of each ring joint in Case B, which is calculated from the stiffness surface in section 4.2, is not fixed at each time step, and its maximum

value (+∞) is much larger than that in Case A. Therefore, the peak values of Mz and My obtained in Case B are significantly higher, and the shapes of their envelopes are quite different. Fig. 19(a) and (b) also show that an apparent increase in the peak moment occurs near the 50th ring in both Case A and B, and this can be explained by the significant difference in the stiffnesses of the different grounds. Due to the different bending stiffness of the ring joints in Case B, the relatively higher moments also

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occur simultaneously on the beam elements in the fully-weathered ground. In addition, the peak value of My was larger than Mz, which indicated that the curvature in the horizontal plane of the tunnel was stronger than

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that in the vertical plane.

Fig. 19(c) shows that the axial force envelopes in both cases had the same shape, and all of them increased

U

significantly in the junction of the soft ground and the hard ground and then gradually were attenuated along

N

the longitudinal axis of the shield tunnel in soft ground. In addition, since the maximum compressive stiffness

M

A

(+∞) was significantly higher than the maximum tensile stiffness in both cases, the peak value of the compressive axial force was much larger than the tensile axial force.

ED

We also found that the maximum value of the axial tension obtained in Case B was about 10% to 15%

PT

higher than that in Case A. This was because the nonlinear tensile stiffness of the ring joints increased from 4.24×109 N/m for pure bending to 3.1×1010 N/m for tension bending, an increase of a factor of about 7.3.

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4.3.3 Envelope results of the opening and stresses of the ring joints Compared to the longitudinal internal forces, the results of the opening and stresses of ring joints shown

A

in Fig. 20 directly affect the longitudinal seismic design of shield tunnels. In He method (Case A), the joint opening and stresses due to the bending moment and axial force are calculated based on the analytical equations assuming pure bending and the single-axial loading state, respectively. Then, the two results due to moments and axial force are added to verify the seismic design. In the method we have proposed, due to the nonlinearity stiffness of ring joints, substituting the longitudinal moment, M, and the axial force, N, in the

analytical equations in Section 2 can yield the joint opening and joint stresses. 1.0 Case A (due to moment) Case A (due to axial force) (○+●) Case B

Joint opening δc (mm)

0.8

0.933

0.732

0.6

0.4

0.2

0

10

20

30

40

50

60

70

80

90

Ring number

Case A (due to moment) Case B (due to axial force) (○+●) Case B

12000

1004.79

10000 8000

N

738.74

A

6000

2000 0 10

20

30

40

60

70

80

90

100

(b)

PT

Maximum tensile stress of joint concrete σt (kPa)

50

Ring number

ED

0

M

4000

600 500

CC E A

SC R

14000

100

U

Maximum compressive stress of joint concrete σc (kPa)

(a)

IP T

0.0

Case A (due to moment) Case A (due to axial force) (○+●) Case B

467.53 448.92

400 300 200 100

0 0

10

20

30

40

50

Ring number

(c)

60

70

80

90

100

Case A (due to moment) Case A (due to axial force) (○+●) Case B

350 300

319.30 271.65

250 200 150 100 50 0 0

10

20

30

40

50

60

70

80

Ring number

90

100

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Maximum tensile stress of ring joint bolts σj (MPa)

400

(d)

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Fig. 20. Envelope results of longitudinal deformation and stresses: (a) Joint opening δc; (b) Maximum compressive stress of concrete σc; (c) Maximum tensile stress of concrete σt; (d) Maximum tensile stress of

U

bolts σj

N

In Fig. 20, the joint opening and stresses in Case A due to the bending moment and the axial force are

M

A

noted as ○ and ●, respectively, and the algebraic sum of the two results is noted as ○+●, where the moment is the resultant bending moment of My and Mz. The seismic response envelopes obtained by the Improved method

ED

are not the simple algebraic sums of the seismic responses due to the bending moment and the axial force in

PT

Case A. Although the shapes of the envelopes of the two cases are similar, the results obtained by the Improved method have different degrees of increase and more adverse effect compared to He method.

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With respect to the maximum seismic response, the ring joint opening, δc, the joint concrete compressive stress, σc, and the tensile stress of the bolts, σj, in Case B increased by 27.46, 36.01, and 17.56%, respectively,

A

more than those in Case A. However, the joint concrete tensile stresses, σt, in both cases were approximately equal, differing by only 4.15%. It is the evident that He method may result in making the design unsafe, and, to avoid this result, it is necessary to take into account the nonlinearity characteristics of the ring joints for different loading states in the longitudinal seismic design of shield tunnels.

Conclusion First, the mechanical behaviors of ring joints of shield tunnels are divided into three loading states and seven deformation modes. The relationship of the rotation angle of the ring joints, the longitudinal moment, and the axial force was derived, and analytical equations were proposed for solving the nonlinear bending and axial stiffness of ring joints. Then, we proposed an iterative algorithm for the stiffness of the ring joints based

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on the longitudinal internal forces, and the algorithm was integrated into He method, thereby forming a new seismic calculation method that we called the Improved method. The 3D surfaces of Kθ’/Kc/Kt-M-N were

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obtained based on a real shield tunnel that passes longitudinally through non-uniform ground. The validity of the proposed Improved method was verified by comparing its results of the envelopes of longitudinal internal

U

forces and the opening and stresses of the ring joints to those obtained by He method. The main conclusions

N

are as follows:

M

A

(1) The analytical expressions f(M,N) of the joint rotation angle θ’ or axial stiffness Kc/Kt are implicit analytical expressions that include the transcendental equation g(φ)=0, which indicates that the nonlinearity

ED

of longitudinal bending and the axial mechanical behavior of ring joints are their inherent properties due to

PT

the special structural characteristics of shield tunnels, even if the nonlinearity of the material and geometric nonlinearity are not considered.

CC E

(2) For the compression bending state, the 3D surfaces of the ring joints bending and axial compressive stiffness, Kθ’-M-N, Kc-M-N, are formed by the connection of two parts, on which the stiffness of the ring joints

A

increases from finite values to +∞. Compared with the pure bending and uniaxial loading states, the stiffnesses of Kθ’ and Kc are not constant, and both increased. (3) For the tension bending state, the 3D surfaces of ring joints bending and axial tensile stiffness, Kθ’M-N, Kt -M-N, are expressed as two-steps style, and the stiffnesses of Kθ’ and Kt are limited between two specific levels. When the ring joints are fully opened, the stiffnesses of Kθ’ and Kt are weakened significantly

by compared to the deformation mode, in which the joints are opened partially. (4) The bending and axial stiffnesses of the ring joints may be different for each time step of the Improved method, and they also have a remarkable difference from the former methods. As a result, the amplitude of the bending moments of the ring joints increased significantly, and, simultaneously, the shapes of the envelopes of the bending moment change remarkably. The results of the maximum joint opening, concrete

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compressive stress, and the tensile stress on the bolts obtained by the Improved method have a non-negligible increase compared with He method, which demonstrates 1) that the former methods maybe unsafe for use in

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the seismic design of shield tunnels and 2) that the validity of the proposed Improved method was verified.

N

U

Acknowledgments

A

This research was supported by the National Key R&D Program of China (2016YFC0802201) and the

ED

M

National Natural Science Foundation of China (51578457).

PT

References

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U

Koizumi, A., Murakami, H., & Nishino, K. (1988). Study on the analytical model of shield tunnel in

A

N

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M

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A

Lin, G., Luo, S.P., & Ni, J. (2009). Damages of Metro Structures due to Earthquake and Corresponding Treatment Measures. Model tunneling technology, 46, 36-41+47.

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Hyogoken-Nambu Earthquake and Its Investigation. JSCE, Committee of Earthquake Engineering, 287295. Nishino, K., Yoshida, K., Koizumi, A. (1986). In-situ tests and consideration on shield tunnel in the longitudinal direction. JSCE, 12, 131-140. PRC National Standard. Code for design of metro (GB50157-2003), Ministry of Construction.

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Shiba, Y., Kawashima, K., Obinata, N., & Kano T. (1988). An evaluation method of longitudinal stiffness of shield tunnel linings for application to seismic response analysis. JSCE, 10, 319-327.

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Shiba, Y., Kawashima, K., Obinata, N., & Kano, T. (1989). Evaluation procedure for seismic stress developed in shield tunnels based on seismic deformation method. JSCE, 4, 385-394.

U

St. John, C.M., & Zahrah, T.F. (1987). Aseismic design of underground structures. Tunneling and

A

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Underground Space Technology, 2, 165-197.

M

Wang, J.N. (1993). Seismic Design of Tunnels: A State-of-the-Art Approach. (Monograph). New York: Parsons Brinckerhoff Quade and Douglas Inc., (Chapter 3).

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Wang, W.L., Wang, T.T., Su, J.J., Lin, C.H., Seng, C.R., & Huang, T.H. (2001). Assessment of damage in

16, 133-150.

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Wang, Z.Z., Gao, B., Jiang, Y.J., & Yuan, S. (2009). Investigation and assessment on mountain tunnels and geotechnical damage after the Wenchuan earthquake. Science in China Series E: Technological Sciences,

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Figure captions

Fig. 1. Schematic diagram of longitudinal seismic deformation method Fig. 2. Definition of rotation angle & bending stiffness of ring joints Fig. 3. Stress distribution of ring joints in deformation Mode 1 Fig. 4. Stress and deformation of ring joints in deformation Modes 3

Fig. 6. Stress and deformation of ring joint in deformation Mode 7 Fig. 7. Longitudinal mechanical behavior of shield tunnels under axial force

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Fig. 5. Stress and deformation of ring joints in deformation Mode 5

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Fig. 8. Stress distribution on center cross-section and neutral axes’ location of homogeneous beam Fig. 9. Comparison of ring joints’ deformation between tension bending and pure bending

Fig. 10. Calculation diagram of generalized longitudinal seismic deformation method (He method)

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Fig. 11. Flow chart of the stiffness iteration and its application

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Fig. 12. Profile view of the shield interval from Haicang Avenue to Dongdu Road

Fig. 15. 3D surface graph of axial stiffness

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Fig. 14. 3D surface graph of bending stiffness

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Fig. 13. Sketch of a segment ring

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Fig. 16. Half calculation model of the simplified ground Fig. 17. Artificial synthetic seismic wave acceleration

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Fig. 18. Longitudinal force and moment of a shield tunnel Fig. 19. Envelope results of longitudinal internal forces

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Fig. 20. Envelope results of longitudinal deformation and stresses