3D response analysis of a shield tunnel segmental lining during construction and a parametric study using the ground-spring model

Tunnelling and Underground Space Technology 90 (2019) 369–382

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3D response analysis of a shield tunnel segmental lining during construction and a parametric study using the ground-spring model Pattanasak Chaipanna, Pornkasem Jongpradist

T

⁎

Department of Civil Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Thung Khru, Bangkok, Thailand

A R T I C LE I N FO

A B S T R A C T

Keywords: Ground-spring model Shield tunneling Soil-tunnel interaction 3D FEM Lining

In this research, an analysis method aiming as a tool to investigate the behavior of a tunnel lining during construction is developed based on a nonlinear ground-spring model in conjunction with a finite element method. A nonlinear ground-spring model that considers yield pressures is proposed and implemented into the numerical analysis. All relevant components of the shield tunnel construction process including the TBM, jack thrust force, shield tail wire brush, segmental lining, key-segment and ring and segment joints and proper interactions are considered in the analysis model. The reliability of the developed method is verified by comparing the analysis results with the full-scale test and field measurements of a previous study. The developed analysis method could reasonably reproduce the responses of a segmental lining at various stages with special attention during construction. A series of parametric analyses are conducted to highlight the robustness of the developed method to capture the effect of the key factors on the structural behavior of the segmental lining during construction. The effects of the tunnel depth, forces from eccentric jack thrust together with shield tail wire brush and position of key segment on the induced lining stresses can be well captured.

1. Introduction Shield tunnel construction, a rapidly developed construction technique, has been increasingly employed in urban development due to several merits such as low impact on the surrounding structure, good applicability in a wide range, allowing for a long boring route and any tunnel depth (Guglielmetti et al., 2007; Koyama, 2003). A segmental lining is a vital component of the shield tunnel, which supports against various kinds of loads during construction as well as the surrounding soil pressure, water pressure and a localized load from any future activities throughout the service stage. Generally, the segments are staggeringly assembled within the Tunnel Boring Machine (TBM) and connected together by tightened curve bolts to making up the tunnel ring. Normally, a tunnel ring consists of several standard segments, two special segments and one key-segment (standard + special + key). The reinforced concrete adopted in casting the tunnel segment is generally designed according to the standard load cases of demoulding, storage, embedded ground condition and grouting processes (DAUB, 2013). During the tunnel construction, various loads act on the tunnel lining, including the jack thrust force, shield tail wire brush pressure, grouting pressure and ground pressure. The hydraulic jack pushes against the lining edge to drive the TBM. The already congested

underground space has obligated the excavation of the tunnel at greater depths and unfavorable conditions, leading to higher load levels. In this context, there has been an increase in the seriousness and the frequency of lining damage during the construction stage, as reported in previous studies (e.g., Han et al., 2017; Sugimoto, 2006; Yang et al., 2017), particularly during driving along a curve alignment (Sugimoto, 2006). Segment cracking usually takes place at the key and adjacent segments (Yang et al., 2017). In addition to the contact deficiency during the installation of the segments (Blom et al., 1999; Burguers et al., 2007; Waal, 1999; Mo and Chen, 2008), the great jack thrust force is also a reason for vast segment damage and cracks, which appear between jack pads and under jack pads in the tunnel axis direction (Blom, 2002; Conforti et al., 2017; Liao et al., 2015; Sugimoto, 2006). The jack thrust also forces key-segment squeezing through the gap between adjacent segments, which causes chips in the edge or spalls at the corner (Blom et al., 1999; Cavalaro and Aguado, 2012; Fuente et al., 2017). Furthermore, during TBM driving along the curve alignment, not only the hydraulic jack but also shield tail wire brush will press on the lining. The pressing of the wire brush induces segment dislocation and damage (Mo and Chen, 2008; Yang et al., 2017). Various load scenarios are suggested to be considered in the current design guidelines (BTS and ICE, 2004; ITA-WG2, 2000; JSCE, 2007). These also include the jack

⁎ Corresponding author at: Department of Civil Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit, Bang Mod, Thung Khru, Bangkok 10140, Thailand. E-mail address: [email protected] (P. Jongpradist).

https://doi.org/10.1016/j.tust.2019.05.015 Received 13 September 2018; Received in revised form 18 April 2019; Accepted 18 May 2019 Available online 23 May 2019 0886-7798/ © 2019 Elsevier Ltd. All rights reserved.

Tunnelling and Underground Space Technology 90 (2019) 369–382

P. Chaipanna and P. Jongpradist

involving various considered stages, tunnel depths and curved trajectory driving of TBM. Another merit is that a large number of soil elements can be disregarded and the difficulties of boundary modelling can be diminished. The analysis model thus composes of FE meshes of segmental linings with or without other structural components (such as TBM). The linings are connected with contacts and springs and subjected to relevant loads. Large movements or deformation may be expected during construction, particularly along the curve alignment. In the current study, the applied load scenario considers the initial stage prior to the occurrence of such large deformation.

thrust during construction. However, the suggestion does not cover how to take into account the extremely great jack thrust during driving along the curve alignment. This is also true for the extra loads due to the pressing of shield tail wire brush. This deficiency is attributed to the lack of an analysis method, which is capable of considering these effects for assessing the lining structural response. On the other hand, for a tunnel in the service stage, the safety of existing tunnels may be affected by adjacent new construction or changes in the environment. The basement excavation can induce stress relief in the ground above the tunnel and lead to the damage of the tunnel lining (Chang and Hwang, 2001). The flood water pumping out of the tunnel could also induce the damage to the tunnel lining (Empel et al., 2006). This effect also indicates the necessity to have an analysis method, which is capable of accurately reproducing the structural response of segmental lining due to extra loads from any possible future activity during the design process or the assessment for countermeasures. This paper develops an analysis method, which is capable of reasonably reproducing the structural responses of the segmental lining, based on the ground-spring model in conjunction with a 3D-FEM. Finite element software ABAQUS version 2016 (ABAQUS Inc., 2015) is used to model and analyze the segments, segment and ring joints, TBM, shield tail wire brush and hydraulic jack. The nonlinear ground-spring model is derived from the hyperbolic function taking into account the soil yield stresses and initial soil subgrade modulus. The method is flexible and accounts for any kind of load. The performance of the developed method is validated against previous full-scale test and field measurements. Parametric analyses are then carried out to prove the robustness of the developed method to reflect the main influence factors on tunnel lining behavior during construction, including the tunnel depth and eccentric jack thrust force.

2.2. Ground-spring model The ground springs are used to update the soil pressures according to the deformation of the structure with which the soil interacts (lining and TBM in this study). The soil pressure is changed according to the deformation of the tunnel lining and its direction to the limited values as the passive or active pressures, if the soil yields under inwards or outwards movement with respect to the lining, respectively. The most common ground reaction curve (relation between the soil movement and pressure) is linear with a slope of the subgrade modulus (kr) as shown in Fig. 1a. Previous works considering this type of linear ground spring include Blom et al. (1999), Mo and Chen (2008), Li et al. (2015, 2014) and Wang et al. (2011) for cases without consideration of yield stresses and Arnau and Molins (2011) for case with consideration of yield stresses. However, soil behavior is essentially nonlinear. Moreover, in some works, the ground springs represent only far-filed soil interaction. Consequently, soil elements are still needed to enclose the tunnel lining as performed by Mo and Chen (2008). The non-linear ground spring model has been adopted for TBM-soil interaction by Sramoon et al. (2002) and Sugimoto et al. (2007). Despite the nonlinear interaction between the segmental lining and soil is well realized, there exists no study attempting to consider the nonlinear ground-spring model in analysis of the tunnel lining. In addition, the yield stresses considered in all previous studies were assumed rather than derived from the soil behavior. In this paper, a nonlinear ground-spring model is developed and implemented into the FEM. The spring elements and initial pressures are applied on the outer surface of the structure that contacts the soil. Both normal (Fig. 1b) and shear (Fig. 1c) springs are taken into account in the ground-spring model. A hyperbolic function is employed to describe the interaction between the lining deformation and soil pressure. The characteristic of the hyperbolic function is prescribed by the yield stress of the soil and the coefficient of the subgrade reaction, which represent the limit and initial slope of the function, respectively. The radial deformation is decomposed into vertical (δv) and horizontal (δh) deformations. The vertical and horizontal deformations are introduced to compute the vertical (Eq. (1) or (2)) and horizontal pressures (Eq. (3) or (4)), respectively. The vertical and horizontal pressures are multiplied by the carried areas to calculate the vertical and horizontal forces, respectively. Then, the vertical and horizontal forces are combined in the radial direction to generate the nonlinear ground-spring response.

2. Development of the analysis method 2.1. Concepts and framework The structural mechanisms and phenomena involved in the transfer of forces within segmental lining are essentially three dimensional (3D) behavior. Currently, a three dimensional finite element analysis (3DFEA) is progressively developed. In particular, the contact and interaction features allow users to conveniently model the contact between two surfaces, for example, segment-to-segment and TBM-to-segments contacts. Although the full modelling considering all elements involved (i.e., surrounding soil, TBM, segmental lining, segment and ring joints, jack thrust force) is possible as successfully performed for tunnel construction (Alsahly et al., 2016; Blom et al., 1999; Kavvadas et al., 2017; Mo and Chen, 2008; Ninić and Meschke, 2017), excavation above the tunnel (Chen et al., 2016; Shi et al., 2017; Zhang et al., 2017) and pile under loading adjacent to the existing tunnel (Haema et al., 2017; Lueprasert et al., 2015, 2017; Schroeder et al., 2004), it becomes inconvenient if series of parametric study must be conducted to investigate or assess the segmental lining responses under various conditions. It is also well known that the analysis results of segmental lining are different to those of continuous lining although the concept of equivalent flexural stiffness is introduced to consider the effect of segment and ring joints. In particular, the localized deformation and stress concentration due to the existences of joints and key-segment cannot be reproduced. When these items are of concerns, the segmental lining together with appropriate interactions are essentially considered in the analysis. When the attention is paid on the structure response, the ground-spring model can be used to represent the soil-to-structure interaction (Alsahly et al., 2016; Blom et al., 1999; Li et al., 2015, 2014; Sramoon et al., 2002; Sugimoto et al., 2007; Wang et al., 2011) and thus the soil elements are not modelled. The main advantage of using ground-spring model to represent the surrounding soil is the convenience of analytical mesh preparation for any tunnel situations 370

kr δ v ⎤ / invert − σv, o) tanh ⎡ σv, a = (σvcrown ,a ⎢ σ − σ crown / invert ⎥ v, a ⎣ v, o ⎦

(1)

kr δ v ⎡ ⎤ / invert − σv, o) tanh ⎢ crown / invert σv, p = (σvcrown ,p ⎥ σ − σ v, o ⎦ ⎣ v, p

(2)

kr δh ⎤ σh, a = (σh, a − σh, o) tanh ⎡ ⎢ σh, o − σh, a ⎥ ⎣ ⎦

(3)

kr δh ⎤ σh, p = (σh, p − σh, o) tanh ⎡ ⎢ σh, p − σh, o ⎥ ⎣ ⎦

(4)

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The shear spring is divided into axial and tangential directions. The response of the axial and tangential springs are expressed in Eqs. (5) and (6), respectively.

kδ τaxial = τmax tanh ⎡ s axial ⎤ ⎢ τmax ⎦ ⎥ ⎣ τtangent = τmax tanh ⎡ ⎢ ⎣

ks δtangent ⎤ τmax ⎥ ⎦

(5)

(6)

In Fig. 1b, the initial slope of the hyperbolic function can be described by the coefficient of subgrade reaction, kr, which is defined as the ratio of the applied pressure and displacement (kN/m2/m). In practice, the coefficient of subgrade reaction depends on the elastic property of the soil and the tunnel diameter. The empirical coefficient of subgrade reactions are suggested in several past studies (e.g., Waal, 1999; JSCE, 2007; RTRI, 2008; Ge, 2002). Moreover, the studies of the coefficient of subgrade reaction are also carried out by the analytical method (Morgan, 1961; Winkler, 1867; Muir Wood, 1975). In this study, Morgan’s equation is adopted to estimate the coefficient subgrade reaction, as expressed in Eq. (7). For the shear spring, the ks is assumed to be one-third of that of the radial spring, as expressed in Eq. (8) (Arnau and Molins, 2011; Wang et al., 2011).

kr =

3E R (1 + v )(5 − 6v )

(7)

ks =

1 kr 3

(8)

The yield pressure also depends on the direction of the movement and location of the lining, as shown in Fig. 2. To estimate the yielding pressure on the tunnel crown, Terzaghi’s formula has been used to describe the earth pressure acting on the tunnel crown. Fig. 3a explains the assumption of the Terzaghi’s formula, which considers the plane strain condition. When the soil mass above the tunnel gradually moves downward while both sides of the soil mass remain stationary, the

Fig. 1. Conceptual modification of the nonlinear ground-spring model: (a) conventional pressure-deformation relation of the lining; (b) model of the radial spring response; (c) model of the shear spring response.

where σv, o (=γh) and σh, o(=γhK o) are at-rest vertical and horizontal earth pressures, σv, a and σv, p are the active and passive vertical pressures at tunnel crown and invert. The parameter K o is the at-rest horizontal coefficient of lateral earth pressure, which can be computed from 1 − sin φ (Jacky, 1944).

Fig. 2. Schematic of the tunnel movement and soil pressure: (a) upward movement; (b) downward movement; (c) lateral movement. 371

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Fig. 3. Terzaghi’s arching effect on the vertical pressure at the tunnel crown: (a) active situation; (b) passive situation.

relative movement within the soil is resisted by a shear resistance along both sides of the soil mass. The remaining pressure acting on the tunnel crown is referred to as Terzaghi loosening earth pressure and is adopted as the active pressure for the nonlinear ground spring. The boundary of the soil mass has a slope of 45° + φ/2 from the horizontal plane, and the strip of soil mass is 2B1 (see Eq. (9)) in width. On the other hand, when the tunnel attempts to move upward, as shown in Fig. 3b, an upward pressure is required to overcome the weight of the soil mass and the shear force between both sides. This pressure is used to define the passive pressure in the ground-spring model. The soil mass is 2B2 in width (see Eq. (10)), and the yielding zone slope is 45° − φ/2 with respect to the horizontal plane.

φ π B1 = Ro cot ⎛ + ⎞ 4⎠ ⎝8

(9)

φ π B2 = Ro cot ⎛ − ⎞ 4⎠ ⎝8

(10)

σvinvert = Nc cD + Nq γHD + ,p

2H 2H B1 (γ ′ − 2c / B1 ) ⎛1 − exp−K tan φ B1 ⎞ + P0exp−K tan φ B1 2K tan φ ⎝ ⎠

(11)

σvcrown = ,p

2H 2H B2 (γ ′ + 2c / B2 ) ⎛exp K tan φ B2 − 1⎞ + P0exp K tan φ B2 2K tan φ ⎝ ⎠

(12)

σh, a = γhK a − 2c K a

(15)

σh, p = γhKp + 2c Kp

(16)

where h is considered depth and Ka and Kp are the coefficients of the lateral active and passive earth pressures, respectively. The limiting tunnel-soil shear stress (τmax ) can be defined as the Mohr-Coulomb failure criteria, which can be expressed as following equation:

τmax = α + σn′ tan ψ′

(17)

where α is the adhesion between the soil and lining, σn′ is the effective soil normal stress, and ψ is the friction angle between soil and lining. In this study, α and ψ are assumed to be equal to c and φ, respectively. Fig. 4 shows the configuration of the ground spring, whereas the symbols r, t and z are the radial, tangential and axial axes, respectively. Based on the above derivation, ground reaction curves of ground springs at various depths and angular positions can be formulated and applied to the corresponding springs via the ABAQUS input file (see ABAQUS 2016 Documentation (ABAQUS Inc., 2015)).

where γ is the unit weight of soil, H is the burden depth, K is the lateral earth pressure coefficient, P0 is the ground surface pressure and c is the soil cohesion. The yielding soil pressure below the tunnel invert is calculated following the bearing capacity equation. Eq. (13) expresses the active pressure at the tunnel invert in the case of the soil uplifting the tunnel (σvinvert , a ). On the other hand, in the case of the tunnel pushing down the soil, a passive pressure (σvinvert , p ) is reached and expressed in Eq. (14) (ALA, 2001).

σvinvert = Ncv cD + Nqv γHD ,a

(14)

Here Ncv is the vertical uplift factor for clay, Nqv is the vertical uplift factor for sand, Nc, Nq and Nγ are the bearing capacity factors, γ is the soil unit weight, H is the tunnel invert depth and D is the tunnel diameter. For the lateral aspect, Rankine’s theory of active (σh, a ) and passive, (σh, p ) earth pressures are used to prescribe the horizontal yield pressure, which is expressed as:

Here, B1 and B2 are the width of the strips of the soil mass as active and passive pressures, respectively, R0 is the outer radius of the tunnel and φ is the internal friction angle of soil. The equation of the active crown (σvcrown , a ) and passive (σv, p ) vertical pressures at the tunnel crown can be respectively given by

σvcrown = ,a

1 Nγ γD 2 2

2.3. Finite element model with ground springs By implementing the ground springs to represent the interaction between surrounding soil and the structures, only the segmental lining and the involved structures are necessary to be modelled. Fig. 5 shows the example of 3D-FEA model used in the simulation of tunnel construction in this study as referred to the case study of the Botlek Railway Tunnel (BRT) (Blom, 2002). The model consists of tunnel rings, TBM and slot elements to represent the hydraulic jacks. The number of rings

(13) 372

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Fig. 4. Ground-spring attachment.

wall in the longitudinal tunnel-axis. Each jack pad interacts with the segment(s) with a surface-to-surface contact. Moreover, the shield tail wire brush is considered in the model as solid elements, as shown in Fig. 7b. The wire brush is attached to the shield tail and interacts with the outer surface of segments by the surface-to-surface contact. Young’s modulus and Poisson’s ratio of the wire brush are 4.00 × 104 kN/m2 and 0.2, respectively (Mo and Chen, 2008). The TBM and wire brush models include 1176 shell elements (4 node reduced integration, S4R) and 1112 solid elements (C3D8R) and 3086 nodes, respectively. The jack pad model includes 448 of the C3D8R elements and 1260 nodes.

and relevant structures to be considered in the analysis depends on the scenario described in the next section. For simulation of lining during the construction, 9 rings have been considered on the basis of the relevant loads, i.e., face pressure, grouting pressure, shield tale wire brush pressure and soil pressure (Mo and Chen, 2008). The segmental lining model includes 12,672 elements and 21,600 nodes. The segment arrangement and numbering are also depicted in the figure. The first tunnel ring, which supports the jack thrust force, is named “R1”. A ring includes one key-segment, two special segments and five standard segments. The key-segment is modeled as wedge shape. The key-segments of odd and even segment ring are respectively set at 12.86° and −12.86° measured from the tunnel crown in a clockwise direction. The segment numbering within a ring in a clockwise direction starts at the key-segment named “S1”. The tunnel segments are modeled as solid elements having eight nodes (C3D8R). The material parameters used in the 3D FEA model are listed in Table 1. Details of segment assembly and joints are illustrated in Fig. 6, of which standard and special segments have eight circumferential springs and four longitudinal springs representing ring and segment bolts, respectively. For key-segments, only the longitudinal springs are assigned in the proposed model. A linear elastic spring with coupling stiffness, kv is assigned for both the circumferential and longitudinal springs representing the joints. A surfaceto-surface contact is used to simulate the interaction between segments representing the packing material. This contact is modelled as hard contact which composes the normal and tangential behaviors, using the Lagrange multiplier formulation (Oden and Martin, 1985). By assigning the non-penetration and allowing the separation to the contacts, the normal compression and tension are supported by the contacts and springs, respectively. Both the contacts and the springs are responsible for the shear interaction.

2.5. In-situ load Various kinds of load can be flexibly applied to the segmental lining and structures modeled depending on the considered situation and scenario. The soil pressure is directly calculated from the vertical and horizontal soil pressures, as shown in Fig. 8a. The vertical soil pressure on the crown is equal to the overburden pressure, while the horizontal soil pressure is defined as the vertical soil pressure multiplied by the coefficient of lateral earth pressure. The effective earth pressure and hydrostatic pressure are introduced in the calculation for the cohesiveless soil, whereas the total earth pressure is for cohesive soil. Finally, the vertical and horizontal pressures are transformed into normal and shear loads, respectively, acting on the tunnel, as shown in Fig. 8b. Note that the water pressure results to only the normal load. The first tunnel ring is not loaded in the radial direction. The first half of R2 is supported by a wire brush. The second half width of R2 and full width of R3, R4, R5 and R6 are loaded by the grouting pressure. The applied grouting pressure is calculated in the same fashion of soil pressure using reduced soil parameters (c, ϕ and E) by half (Blom et al., 1999). It is assumed that the grouting material is partially hardened. Therefore, the stiffness of springs representing the grouting material are reduced to half (Blom et al., 1999). The last three tunnel rings (R7, R8 and R9) are loaded by soil pressure. In engineering practice, the cutter face pressure can be varied in the range between the active and passive pressures depending on the discharge rate of the TBM, which in turn depends on the TBM advance rate. Although any magnitudes of cutter face pressure can be applied in the analysis, the value in this study is computed from the water pressure

2.4. Model of the TBM Fig. 7a shows the section of the TBM. The TBM is modeled using shell elements, which were assigned with elastic properties. The weight of the TBM is taken into account by applying the density of the TBM shell. The chamber wall is created to support the jack thrust force. The TBM is pushed by means of jack thrust forces, which are defined by slot connector elements. The slot elements link the jack pads and chamber 373

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Fig. 5. Relevant components of the shield tunneling analysis.

analytical model are not limited to only the ones presented above. For example, the induced load from nearby construction may be calculated from Midlin’s solution (Mindlin, 1936).

Table 1 Full-scale test parameters (Blom, 2002). Full-scale tested parameter

Magnitude

Radius of the tunnel lining, R Thickness of segment, t Width of ring, w Modulus of elasticity of concrete, EC Contact length longitudinal joint, lt Shear coupling stiffness, kv Friction coefficient between segments, νt Measured compressive strength

4.525 m 0.400 m 1.500 m 40 × 104 MN/m2 0.170 m 1.0 × 102 MN/m 0.40 64 MPa

3. Analysis results and discussion 3.1. Verification of the proposed method As previously mentioned, there are several components involved in an analysis of a tunnel lining during construction, particularly the presence of several complex interactions. Thanks to comprehensive laboratory test and field measurements of the tunnel lining for Botlek Railway Tunnel (BRT) project performed by Blom (2002), a systematic and reliable verification of the proposed analysis method can be done. This section presents the verification of the proposed analysis method, which is divided into three subsequent categories as follows: Case I emphasizes a reasonable model of the segments and their interactions. The model of segments, the surface-to-surface contact between adjacent segments and rings as well as segment joints are verified. The laboratory loading test on three connected rings of a segmental lining was adopted. In Case II, the ground-spring model and the computed surrounding soil load, which are divided into radial and tangential

and at-rest soil pressure due to the lack of monitored data. The pressure acts on the cutter face, which increases with the depth. The cutter face pressure is shown in Fig. 8c and can be computed by the following equation (Kanayasu et al., 1995);

L1 = γ0 HK + Hw γw + 20 (kPa)

(18)

where γ0 is the unit weight of soil, γw is the unit weight of water, K0 is the coefficient of the at-rest earth pressure, H is the depth of the tunnel axis, and Hw is the water table height. The loads applied to the 374

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Fig. 6. The finite element model of the segmental lining and its coupling.

surrounding shield pressure, jack thrust and shield tail wire brush are added (from Case II). 3.1.1. Case I: Verification of the tunnel segments and their interactions The full-scale load test on the 3 ring lining for the Botlek Railway Tunnel was performed by Blom (2002) at the Stevin Laboratory at the Delft University of Technology. Three vertically staggered arranged rings were used in the test. A ring consisted of 8 segments (5 standards + 2 specials + 1 key), as shown in Fig. 9a. The top and middle rings are called Ring1 and Ring2, respectively. The radial ovalization pressure and axial forces of hydraulic jacks were applied on the outer faces of the linings and upper edge of the lining in Ring1 (Fig. 9b), respectively. The analysis model with the same configuration as the full-scale test reproduced to simulate the tested event. Table 1 tabulates the properties and parameters of the lining adopting in the analysis. Note that all values are exactly the same as those utilized in Blom (2002). Fig. 10 shows the comparison of the radial displacements of lining Ring1 and Ring2, which are obtained from the full-scale test, the frame analysis previously performed by Blom (2002) and the proposed analysis method in this study. The radial displacement is plotted against the circumferential angle (θ), which is measured from the tunnel crown in a clockwise direction. The segment joint positions are also shown by the vertical dotted lines. The results obtained from the frame analysis and the proposed analysis method in this study are similar to those of the full-scale test. The maximum radial displacement occurs at the tunnel invert. As expected, the magnitude of inward displacement is larger than that of the outward displacement due to the applied radial pressure. The comparison of the tunnel invert displacement between Ring1 and Ring2 indicates that the displacement of Ring1 is larger than that of Ring2 owing to the existence of the segment joint at the tunnel invert for Ring1. The computed radial displacements at the springline (90° and 270°) in this study is conformable to those of the full-scale test result, while radial displacements at the tunnel crown and invert are slightly

Fig. 7. (a) Section of TBM and (b) Interaction between the lining and TBM.

components, are additionally implemented in Case I and verified. This case referred to the field measurement of the segmental lining in the BRT project at the ring far away from the excavation face on which the construction loads have no influence. The results from the rings at the excavation face are used for Case III, in which the TBM model, 375

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Fig. 8. Load model schemes; (a) conventional load, (b) cylindrical load, and (c) facial load.

Fig. 9. The finite element model for the full-scale test: (a) model of the tunnel lining; (b) loading configuration.

Fig. 10. Comparison of the radial displacement obtained from this study with the experimental measurements and frame analysis: (a) Ring1; (b) Ring2.

greater than those of the full-scale test result. The difference probably originated from two causes, i.e., (1) the low stiffness of the lining spring (segment and ring springs), which are obtained from Blom (2002), and (2) the pretension of lining spring is not considered in the simulation.

Fig. 11 illustrates the tangential bending stresses obtained from the full-scale test, the frame analysis and the proposed analysis method. As can be observed in the figure, despite some minor discrepancies, the results from the frame analysis and this study generally agree with 376

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Fig. 12. Geological profile of the BRT (adopted Blom, 2002). Table 2 Parameters for analysis with the ground-spring model. Ground-spring parameter Bulk unit weight, γ (kN/m3) Saturated unit weight, γsat (kN/m3) Friction angle, ϕ (°) Earth pressure coefficient at rest, K0 Resistance coefficient, ks (kN/m3)

16.0 18.0 34.75 0.43 5091

ground-spring model is considered a reasonable representation of the soil-tunnel interaction. 3.1.3. Case III: Verification of the shield tunnel construction model The field measurement of the BRT project is also used to verify the proposed analysis method for reproducing the lining responses during construction. The TBM and construction loads are also included in the analysis model. The computed axial stresses along the tunnel perimeter during tunnel construction, which is extracted from the middle section of R1 (which supports the jack thrust force), are compared to the field measurement, as shown in Fig. 14. The distribution of the results from the proposed analysis method is in a good accordance with the field measurement. The stress magnitudes from both analysis and field measurements are in the same range. The axial stresses induced in the lower half lining are generally larger than those of the upper half lining. The distribution of the principal stress vectors on the mid plane of the lining along the tunnel axis is shown in Fig. 15. The vectors depict the directions and relative magnitudes of the principal stresses. The legend representing the ranges of the magnitude is also provided in the figure. Only lining R1, R2 and R3 are of interest and illustrated in the figure. Moreover, a magnified figure of a segment at the invert of R1 is also shown on the left of the figure. It is seen that the minor principal stresses, σ3 (negative value: compression) on the observed plane are generally direct along the tunnel axis, whereas the major principal stresses are directly along the circumferential direction. The σ3 vectors in R1 are larger than those in R2 and R3 in order, indicating the effect of the jack thrust forces. From the magnified figure, it is clearly seen that the largest σ3 vectors occur at the front edge of R1 along the thrust pads. Notably, the major principal stress vectors (circumferential direction) change from compression (negative) in R3 to less compression or even tension (positive) in R1. The induced tensile stresses occur mainly in the zones of the front edge, particularly between the thrust pads, where longitudinal cracks have been frequently reported (Liao et al., 2015; Sugimoto, 2006). This finding implies that for analyses with greater jack thrusts (such as at a greater depth or curve alignment), larger tensile stresses are expected and consequently cracks are potentially exhibited. This behavior indicates the effectiveness of this proposed analysis method to reflect the potential crack occurrence due to construction loads. Fig. 16a and b present a shedding of tangential and axial stresses, respectively, which are taken from a plane at mid-thickness. As already

Fig. 11. Comparison of the tangential bending stress obtained this study with the experimental measurements and frame analysis: (a) Ring1; (b) Ring2.

those of the full-scale test. The result from the proposed analysis method tends to slightly underestimate the peak tangential bending stresses, which generally occur at the mid-length of the segments. By comparing the tangential bending stress at the invert of Ring1 and Ring2, it is clearly seen that the one at Ring2 (at mid-length of the segment) is much larger than that of Ring1 (the joint). Note that the variation at the key-segment (as taken place in the test) can be observed only in developed method. The models of the segments and their interactions used in the proposed analysis method is considered to be effective. 3.1.2. Case II: Verification of the ground-spring model The aim of this section is to verify the nonlinear ground-spring model and the assigned surrounding pressure strategy. A reference case is obtained from the field measurement of BRT project (Blom, 2002). The tunnel configuration of the reference case is shown in Fig. 12, and the soil parameters are tabulated in Table 2. The model used in the analysis refers to Fig. 6 by disregarding the TBM, shield components and relevant construction loads. The loads applied in the analysis comprise the radial pressure and shear stress. The axial constraints are applied at the end of the tunnel ring R1 (front edge) and R9 (rear edge). The results are obtained from R6. The inside and outside tangential stresses, tangential bending moment and tangential normal force are depicted in Fig. 13a–d, respectively. From the figures, the results from the proposed analysis method generally agree with the field measurement rather than those of the frame analysis (which generally tend to show underestimation). The proposed analysis method based on the 377

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Fig. 13. Result of the verification of the ground-spring model: (a) Inside tangential stress (kPa); (b) Outside tangential stress (kPa); (c) Tangential bending moment (kN m/0.75 m); and (d) Tangential normal force (kN/0.75 m).

edge of the adjacent segments, which is one of most common failure patterns found during construction (Blom, 2002; Sugimoto, 2006). The high stress appears in R1, especially in the tunnel invert zone because it supports the jack thrust force. Comparing the key-segment and its neighbor segments, the stress that occurs in the key-segment is greater. This finding implies that if the key-segment is located at the invert zone, high induced stress is expected. From the above verifications, it can be concluded that the selected models of the segments, joints, interfaces, TBM and the procedures to apply relevant loads in the analysis are appropriate to reproduce the lining behavior during tunnel construction. The developed groundspring model reasonably represents the soil-tunnel interaction. The tunnel lining responses reflecting to the observed behaviors in an actual condition can be reasonably reproduced by the analysis method developed in this study. Thus, the developed analysis method is proven to be sufficient to capture the lining behavior during tunnel construction and under other changes in the environment for future investigation.

Fig. 14. Distribution of axial stress due to the jack thrust force during tunnel construction.

mentioned, the tensile (positive value) tangential stresses occur in the R1 between and under the jack pads. It is seen in Fig. 16a that the maximum tangential compressive stress occurs at the interface between the key-segment and adjacent segments of R9, which is caused by oval soil pressure. However, as shown in Fig. 16b, the tensile axial stress exhibits in the adjacent segments. This effect is attributed to the hydraulic jack that pushes the tapered key-segment. Consequently, the key-segment pulls the adjacent segments following the jack thrust. This phenomenon indicates that the key-segment causes a chipping of the

3.2. Parametric analysis To highlight the robustness of the developed analysis method with application to engineering practice, the ability to capture the effect of key factors on the structural behavior of tunnel linings during construction, such tunnel depth and jack eccentric thrust force, is investigated. A series of parametric studies are conducted, covering a significant range of parameters, as listed in Table 3.

378

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Fig. 15. Distribution of the principal stress vector in the middle plane of the tunnel lining.

Fig. 17. Maximum tangential and axial stresses at various tunnel depths.

the effect of the tunnel depth on the segment stresses for both the service and construction stages; the tunnel depths are set at 12.5, 25.0 and 50.0 m. In Fig. 17, the maximum tangential and axial stresses are plotted against the tunnel depth (h). The arrow and R(X) sign in the circle indicate the point and the ring number at which the maximum stress occurs, respectively. Here, the subscript “+”, “−”, “22”, and “33” denote the maximum tensile, maximum compressive, tangential stress, and axial stress, respectively. Similarly, the superscript “const” and “serve” denote the construction stage and serviceability stage, respectively. It is clearly seen that for the service stage, the σ−serve 22 (maximum hoop stress) has always been much larger than the σ−serve 33 (maximum axial stress). The degree of stress that increases against depth can be clearly serve noticed for σ−serve 22 but insignificantly for σ−33 . This effect is because major loads during the service stage (such as ground and water pressures) act in the radial direction. The σ−serve 22 occurs at the inner surface of the lining at a position near the tunnel springline due to the deformed oval shape, while the σ−serve 33 exhibits at the key-segment. For the construction stage, an increase in the tunnel depth results in const const const increases in all of σ+const 22 , σ−22 and σ−33 . It is clearly seen that the σ−33 const becomes larger than σ−22 for all depths during construction, by means of the concentrated jack thrusts. The degree of the increasing in σ−const 33 against depth is larger than that in σ−const 22 . This increase indicates that the main consideration should be given to the axial stress in the lining during construction and the construction loads play a more important role for tunnels at great depths. The σ−const 33 in tunnels at depths of 12.5 and 25 m appear at the invert owing to the jack thrust force at the invert is relatively greater. However, the σ−const 33 in a tunnel at a depth of 50 m appears in the key-

Fig. 16. Distribution of the stresses in the middle plane of the tunnel lining: (a) tangential stress; (b) axial stress.

Table 3 Parameters used in the shield tunneling analysis. Parametric study

Value

Bulk unit weight, γ (kN/m ) Saturated unit weight, γsat (kN/m3) Friction angle, ϕ (°) Modulus of elasticity of soil, Es (kN/m2) Tunnel depth, H (m) Addition jack thrust force about x-axis (%) Addition jack thrust force about y-axis (%) 3

16.0 18.0 30.0 50,000 12.5, 25.0, 50.0 −50, −25, 0.0, 25, 50 −50, −25, 0.0, 25, 50

3.2.1. Effect of the tunnel depth The tunnel depth directly causes the greater loads acting on the tunnel, particularly during the tunnel construction. This section shows 379

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segment. For such a deep level, the jack thrust forces at the tunnel crown and invert are not that different. In the case of a tunnel at a depth of 12.5 m, the σ−const 22 occurs on the outer surface at the springline of R3. For a tunnel at a shallow depth (cover/diameter, C/D < 2) (Vu et al., 2017), the horizontal pressure (soil and grouting pressures) is slightly greater than the vertical pressure, which causes a vertical oval deformation of the tunnel lining. Further investigation reveals that the positive bending moment also appears at the springline location. Moreover, the upward movement of the lining is also captured. This phenomenon is known as buoyancy (Blom et al., 1999; Vu et al., 2017), as can happen with shallow tunnel. This result indicates that the proposed analysis method in this study can also capture the lining movement behavior of a shallow tunnel as well. The transverse aspect of tunnel stress is explained by the radial pressure. In the case of a deep tunnel (C/D > 2), the radial pressure is subdivided into a uniform and an ovalizing pressure (Blom, 2002; Muir Wood, 1975). The ovalizing pressure has a great effect in the zone that the soil pressure is applied (R7, R8 and R9). For a tunnel at a depth of 25 m, the ovalizing pressure effect is more pronounced than a uniform pressure, although the σ−const 22 occurs on the outer surface of the keysegment. On the other hand, for a very deep tunnel, the effect of uniform pressure is greater than the ovalizing pressure. The σ−const 22 appears on the inner surface at the springline of R7. Fig. 17 also illustrates the maximum tangential tensile stress of the const construction stage, σ+const 22 . The σ+22 increases with an increasing tunnel depth. A formula of Macginley and Choo (2014) is employed to compute the tensile strength of the concrete segment, which is equal to 4243 kPa. As shown in the figure, the σ+const 22 is greater than the tensile strength when the tunnel depth approximately exceeds 40 m. 3.2.2. Effect of the eccentric jack thrust force In the case of TBM driving along the curved alignment, the varying hydraulic jack initially thrusts against the lining to steer the moments. The jack thrust force can vary from 25% to 90% of the maximum installed thrust (Grübl and Thewes, 2005). The vertical and horizontal curved alignments are considered in this simulation. The vertically curved alignment is divided into the upward (X−) and downward (X+) directions. Similarly, the horizontal alignment includes turn left (Y+) and turn right (Y−) directions. The simulations of the starting curve alignment are carried out by reducing the jack thrust forces from the straight alignment (25% and 50%) at the concave side. These reduced forces are added to the jack thrusts at the convex side at the corresponding mirror positions. Fig. 18a shows the maximum tangential and axial stresses due to the eccentric jack thrust. The tunnel axis level is set at 25 m below ground surface. The results of eccentric jack thrust about the x- and y-axes illustrated in Fig. 18a shows that the values of σ−const 33 increased in correspondence with the eccentric forces. Meanwhile, the σ−const 22 remains almost constant and always appears at R1 for all eccentric jack thrust forces. The σ−const 33 during the horizontal driving is larger than that during the vertical driving. For horizontal driving, the σ−const 33 between turn left and turn right driving are different due to the existence of the key-segment. The σ−const 33 in the case of upward rotation becomes smaller than that of the downward rotation (at the same tunnel depth) despite the fact that the force at the invert is larger than that at the crown. This effect is because the σ−const 33 occurs at the key-segment of the downward move rotation, which implies that the key-segment should not be located at the tunnel invert where the jack thrust force is greatest. Fig. 18b shows the induced maximum tensile stresses due to the const eccentric jack thrust force. The σ+const 33 and σ+22 increase with an increase in eccentric jack thrust force for both driving about the X- and Y-axes. The σ+const 33 always occurs at the edge between the key and adjacent const segments. The σ+const 22 is greater than the σ+33 for all eccentric jack thrust force. The maximum tensile stress always appears at R1. Most of σ+const 22 appears at the tunnel invert, except the downward rotation where the const σ+const 22 appears at the key-segment. The σ+22 of driving about the Y-axis

Fig. 18. Maximum axial and tangential stresses with various eccentric jack thrust forces: (a) compressive stress; (b) tensile stress.

Fig. 19. Contacted pressure with various eccentric jack thrust forces patterns.

is greater than that about the X-axis and exceeds the tensile strength when the eccentric jack thrust forces reaches 50%. Furthermore, this analysis also simulates the contact pressure between the shield tail wire brush and outer segment surface, as shown in Fig. 19. The contact pressure indicates that the wire brush squeezes the segment, which is a cause of segment dislocation, and in another case, the null contact pressure implies the possible gap between the wire brush and segment. The contact pressure increases with an increase in 380

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Table 4 Maximum inner forces of joints under different conditions. Parametric case

Segment joint

Ring joint

Tensile force (NSJ) Force (kN) Service stage at tunnel depth = 25.0 m 0.04

Shear force (QSJ)

Tensile force (NRJ)

Shear force (QRJ)

Position

Force (kN)

Position

Force (kN)

Position

Force (kN)

Position

S4-S5

4.1

S5-S6

–

–

6.6

313°

8.8 19.5 38.6

R1, S8-S1 R1, S8-S1 R1, S8-S1

1.1 1.1 2.0

R1-R2, 339° R1-R2, 339° R1-R2, 339°

14.4 12.1 22.6

R2-R3, 4° R1-R2, 4° R1-R2, 4°

R1, R1, R1, R1, R1, R1, R1, R1,

4.4 0.9 1.2 1.4 0.8 0.7 1.3 1.6

R1-R2, R1-R2, R1-R2, R1-R2, R1-R2, R1-R2, R1-R2, R1-R2,

19.7 15.3 11.8 14.7 15.0 12.2 10.1 14.3

R1-R2, R1-R2, R1-R2, R1-R2, R1-R2, R1-R2, R1-R2, R1-R2,

Construction along straight alignment varying tunnel depths H = 12.5 m 0.14 R6, S8-S1 H = 25.0 m 0.07 R6, S8-S1 H = 50.0 m 0.15 R6, S8-S1

Construction with eccentric jack thrust forces at tunnel depth = 25.0 m Turn right (Y-50) 0.10 R4, S1-S2 39.1 Turn right (Y-25) 0.08 R7, S5-S6 27.4 0.13 R6, S8-S1 16.7 Turn left (Y+25) Turn left (Y+50) 0.15 R6, S8-S1 15.1 Upward (X-50) 0.11 R6, S8-S1 14.1 Upward (X-25) 0.11 R6, S8-S1 12.3 Downward (X+25) 0.09 R6, S8-S1 24.7 Downward (X+50) 0.09 R7, S5-S6 31.0

S8-S1 S8-S1 S1-S2 S1-S2 S8-S1 S8-S1 S1-S2 S1-S2

4° 21° 339° 339° 339° 339° 339° 339°

4° 4° 4° 4° 4° 4° 4° 4°

segment joint of R6. The shear forces (QSJ and QRJ) increase as the depth of tunnel increases and their maximum values always appear in the joint near the crown between the key segment and the adjacent segment of R1 for QSJ and in joint at the tunnel crown of R1 for QRJ. This explains the spalling of key segment during the construction as reported by Yang et al. (2017).

the eccentric force of the jack thrust. The huge contact pressure appears in the key-segment. When the TBM attempts to turn left and right (rotate about positive and negative Y-axis), the distribution of contact pressures is seen along the right and left side, respectively. In the case of upward driving of the TBM, the distribution of contact pressures can be observed along the tunnel crown, and the maximum value occurs at the key-segment. Therefore, when the TBM starts driving, not only may the eccentric jack thrust forces damage the key-segment but also the contact pressures from wire brush may also facilitate key-segment dislocation, as reported by Mo and Chen (2008). For the downward driving case, the contact pressures distribute along the tunnel invert.

4. Conclusion The nonlinear ground-spring model with a hyperbolic function, which takes into account the soil-tunnel interaction, is developed and implemented in a finite element method for analyzing the behaviors of a tunnel segmental lining. The analysis model includes all relevant components of the segments, shield and their interactions and loads covering service and construction stages. Each relevant component is systematically verified by comparing the analysis results with full-scale test data and field measurements from a previous study. The performance of the developed analysis method is examined by series of parametric studies. The conclusions can be drawn as follows:

3.2.3. Induced inner forces of the segment and ring joints The induced inner forces of the segment and ring joints are also investigated in the study. Since there are many joints in the model, only the maximum tensile (N) and shear (Q) forces of segment and ring joints and their locations are discussed and shown in Table 4. The tensile and shear forces in the joints represent the possibility of the opening and the offset of joints, respectively (Gong et al., 2019). The offset of joints may result in joint squeezing and eventually the segment spalling (BTS and ICE, 2004). The symbols NSJ, QSJ, NRJ and QRJ denote tensile and shear forces of the segment and ring joints, respectively. NSJ acts in the tangential direction and QSJ lies on the radial-longitudinal plane. Similarly, NRJ acts in the longitudinal direction and QRJ lies on the radial-tangential plane. It is clearly seen that the values of Q are generally greater than those of N. NRJ are greater than NSJ for all cases. On the other hand, QSJ are mostly greater than QRJ. Under service stage, no tensile force (NRJ) is observed from the ring joints. This agrees well with the longitudinal stresses in the segments which are always under compression as observed in the previous section. During construction, small NRJ are seen and the maximum values always appear in the joints between the front rings (R1-R2) at the tunnel crown. For segment joints, under service stage, very small NSJ are noticed and the maximum value occurs at the joint near the tunnel invert. This is attributed to the ovalizing pressure in radial direction. Under construction stage for straight alignment, larger tensile forces (compared to those of service stage) of segment joints are observed. As the tunnel is constructed at greater depth, the maximum NSJ occurs in the key segment joint of R6 that is far from the excavation face and subjected to the grouting pressure in the radial direction. Note that R6 is the last ring that the grouting pressure is applied and there is no ring joint for the key segments. During driving along curve alignment, the maximum NSJ still mostly happen in the key

1. Good agreement between the analysis results and full-scale test data and field measurements are obtained, indicating the reliability of the developed analysis method. 2. The effect of the tunnel depth and eccentric jack thrust force on the induced lining stresses can be well captured in the parametric study, highlighting the robustness of the developed analysis method. These two factors are believed to be the key influencing parameters on the damage of a segmental lining during construction. The method is acceptable for use as a tool for analyzing the behavior of a tunnel lining during construction. 3. During tunnel construction at great depths, the jack thrust has an enormous effect on the tunnel lining. The influence of the jack thrust is not only in the axial direction but also in tangential direction. The jack thrust introduces tensile stress in the tangential direction, which may cause a longitudinal crack in the segment, particularly between the jack pads. During the lining design process, one should properly consider modeling the jack thrust forces as individual components. 4. The key segment plays a crucial role in the lining responses, and the analysis of shield tunneling must consider the existence of the key segment. Besides the tunnel segmental lining responses during construction 381

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presented in this article, the developed analysis method can be applied to analyze the impact on the existing segmental lining due to adjacent structure/construction loads such as pile load, shallow footing, ground surcharge load and un-loading excavation. The analysis of impact from pile load has currently been being performed by the authors. We expect to report the results in the near future.

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