- Email: [email protected]
The failure mechanism of surrounding rock around an existing shield tunnel induced by an adjacent excavation
Computers and Geotechnics 117 (2020) 103236
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
The failure mechanism of surrounding rock around an existing shield tunnel induced by an adjacent excavation
T
⁎
Fu Huanga, , Min Zhanga, Fen Wanga, Tonghua Linga, Xiaoli Yangb a b
School of Civil Engineering, Changsha University of Science and Technology, Changsha, Hunan, China School of Civil Engineering, Central South University, Changsha, Hunan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Upper bound theorem Variational approach Surrounding rock failure mechanism Pit excavation Existing shield tunnel
Foundation pit excavation may disturb the initial stress field and induce the deformation of the surrounding rock mass. Due to the excavation, the originally compact rock mass and lining structure may become loose with adverse impacts on the normal operation of the existing tunnel. A new failure mechanism is constructed on the basis of the deformation characteristics of the rock mass around an existing tunnel induced by an adjacent excavation. Using this failure mechanism, the upper bound solution of the slip surface equation for the rock mass around an existing tunnel is derived in the framework of the upper bound theorem in conjunction with a variational approach. The shape and range of the slip surface are plotted for different parameters. A three-dimensional numerical model is constructed to simulate the deformation of the rock mass around the existing tunnel induced by adjacent excavation. By comparing the analytical results for the slip surface with the failure surface provided by numerical simulation, it is found that the differences between the analytical results and the numerical results are small. This comparison shows that the proposed method is valid for investigating the deformation of the rock mass around the existing tunnel induced by adjacent excavation.
1. Introduction In large cities, the subway network is always concentrated in the center of the city. To facilitate traveling and promote the commercial value of buildings, most high-rise buildings are located close to existing subway lines. Because foundation pit excavation is a crucial process in the construction of high-rise buildings, the surrounding rock deformation induced by foundation pit excavation and the effects of deformation on adjacent subway structures are issues that engineers must confront. Numerous investigations show that foundation pit excavation induces deformation of the surrounding rock, and the deformation may transmit to the adjacent regions. When this happens, the originally compact rock mass and lining structure may become loose, and a cavity may even come into being at the back of the tunnel lining. This deformation of the rock mass causes undesirable contact between the rock mass and lining structure, leading to stress concentration in a local area of the lining structure that may result in cracking and damage of the lining structure. Consequently, the study of the influence of foundation pit excavation on adjacent existing underground structures is an issue with high engineering significance. The effect of foundation pit excavation on adjacent existing tunnels has aroused the interest of some investigators. To investigate the
⁎
responses of existing shield tunnels to adjacent excavation, Liang et al. [1] proposed a new analytical method that considers both the bending and shearing effects of a shield tunnel. Using this method, they studied the tunnel-ground interaction and evaluated tunnel longitudinal deformations subjected to adjacent excavation. Later, Liang et al. [2] modified this analytical method by introducing the Pasternak foundation model with a modified subgrade modulus to evaluate the deformation behavior of existing shield tunnels induced by adjacent excavation. The main advantage of their method is that the shield tunnel is treated as a continuous Euler-Bernoulli beam, which accurately predicts the shield tunnel behaviors associated with adjacent excavation. Based on a 2D finite element method, Zheng and Wei [3] presented a study investigating tunnel deformation and stress redistribution around the tunnel lining due to overlying excavation. They found that the deformation and stress redistribution of the tunnel lining varied with the variation of the location between the pit and the tunnel. Subsequently, Zheng et al. [4] developed a simplified semi-empirical method to assess the responses of a tunnel induced by an adjacent excavation on the basis of finite element analysis in conjunction with a set of collected case histories. Huang et al. [5] used 3D numerical analysis to investigate tunnel deformation due to adjacent foundation pit excavation. Their achievements show that the relative position between
Corresponding author. E-mail addresses: [email protected] (F. Huang), [email protected] (X. Yang).
https://doi.org/10.1016/j.compgeo.2019.103236 Received 7 January 2019; Received in revised form 9 August 2019; Accepted 1 September 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
of the stress concentration in this region at the back of the tunnel lining. Consequently, the stress concentration is the main factor that leads to the deformation of a tunnel lining. However, the transfer process of the ground deformation is concealed, which makes the position and range of the undesirable contact zone at the back of the tunnel lining difficult to determine precisely. Thus, it is necessary to develop a method to reveal the transfer mechanism and deformation mode of the rock mass around an existing tunnel induced by adjacent excavation. This paper presents a theoretical method to investigate the deformation mechanism of the rock mass around an existing tunnel induced by an adjacent excavation. Based on the deformation characteristics of the rock mass around an existing tunnel provided by a numerical simulation technique, a two-dimensional failure mechanism composed of a detached curve is established to determine the undesirable contact zone at the back of the tunnel lining. Using the upper bound theorem, the external rate of work and the rate of internal energy dissipation in the constructed failure mechanism are calculated. The objective function of the detached curve composed of the external rate of work and the rate of internal energy dissipation is obtained, and the upper bound solution of the detached curve equation is derived from a variational approach. Finally, the validation of the proposed analytical method is performed via comparison with a numerical simulation. The aim of this study is to develop an analytical method to determine the shape and range of the undesirable contact zone at the back of an existing tunnel induced by adjacent excavation.
the tunnel and the excavation, tunnel diameter, excavation dimensions, and tunnel protection measures are the key factors that affect the responses of tunnels during the excavation of adjacent foundation pits. Chen et al. [6] conducted a three-dimensional numerical analysis to study the influence of a nearby large excavation on existing twin tunnels in soft soils. By investigating the stress and displacement of the tunnels during different construction stages of the nearby excavation, they obtained an interaction mechanism between the nearby excavation surrounding soils and existing twin tunnels. Finding that the basement-tunnel interaction has been simplified as a plane strain problem in existing research, Shi et al. [7] used a three-dimensional numerical simulation technique to study the influences of excavation geometry, sand density, tunnel stiffness and joint stiffness on tunnel responses. More recently, Shi et al. [8] established a three-dimensional numerical model to investigate the validity of countermeasures that are used to reduce tunnel response due to overlying basement excavation. By analyzing the responses of a tunnel for different parameters of the countermeasures, they found that wall penetration depth, wall stiffness and lining thickness are three key factors that can be used to alleviate basement excavation-induced adverse effects on existing tunnels. Similar to the tunnel-ground interaction induced by adjacent excavation, the soil disturbances resulting from multiline tunneling in soft soil affect the safety of existing tunnels. Based on an actual project in Shanghai, Zhang and Huang [9] employed a three-dimensional finite element numerical simulation method to analyze the deformation of existing subway tunnels induced by an earth pressure balance (EPB) shield machine that drilled in the surrounding strata. Although the influence of a basement excavation on an adjacent tunnel can be studied by means of theoretical analysis and numerical simulations, some scholars found that the centrifuge model test is also an effective way to investigate the tunnel responses induced by basement excavation. Ng et al. [10] designed a three-dimensional centrifuge test to investigate the effects of a basement excavation on an existing tunnel in dry sand. Combining the results of centrifuge tests and numerical analyses, they found that the bending deformation of the tunnel is directly related to the location between the tunnel and the basement. Subsequently, finding that the effects of sand density and basement wall stiffness on nearby tunnels due to excavation are neglected in existing studies, Ng et al. [11] employed a series of three-dimensional centrifuge tests to investigate these effects on the complex basement-tunnel interaction. To estimate the influence of new shield tunneling on an existing underlying large-diameter tunnel, Li et al. [12] designed a centrifuge model test to simulate the whole construction process of a new shield tunnel excavated above an existing tunnel. By studying the displacement and bending moments of the existing tunnel, they found that grouting is an effective measure to alleviate the responses of the existing tunnel induced by the new shield tunneling. Similarly, Jiang et al. [13] also employed three-dimensional centrifuge model tests to investigate the response of an existing horseshoe-shaped tunnel induced by the construction of two perpendicularly undercrossing tunnels. Although the interactions between tunnels and adjacent excavations have drawn the attention of numerous investigators, previous studies mainly focused on the deformations of existing tunnels induced by adjacent excavation. However, the deformation of a tunnel lining is only a phenomenon, what leads to this phenomenon is an issue deserving further study. Unfortunately, the studies of the cause of the deformation for a tunnel induced by an adjacent excavation are rare in existing literatures. By investigating the interaction mechanisms between the surrounding soils and an existing tunnel, it can be found that the essential cause of the deformation for the tunnel is the disturbance of the initial stress field induced by an adjacent excavation. Due to the disturbance of the ground induced by adjacent excavation, ground deformations inevitably occur in the adjacent region. Furthermore, the ground deformation transmitted to the surrounding region of an existing tunnel causes undesirable contact between the tunnel lining and surrounding rock. The undesirable contact would induce the occurrence
2. Engineering background There is a plan to build a basement adjacent to an existing shield tunnel in Shenzhen. The foundation pit of this basement is an irregular polygon that is located at the intersection of two main routes. The length and width of the pit are 150 m and 110 m, respectively. Based on the geological investigation report and the actual conditions of this project, the cut and cover method is selected to excavate the foundation pit. The average excavation depth of this pit is 15 m, and the maximum excavation depth is 16.5 m. The surroundings of this project are complex, with the north, west and east of the pit adjacent to main routes. Moreover, there is an existing shield tunnel in the south of the pit, and the minimum distance between the tunnel segment and the edge of the pit is 16 m. A plan view of the planned pit and the existing tunnel is illustrated in Fig. 1. To guarantee the safety of the surrounding buildings during the excavation of the foundation pit, the retaining structure of the foundation pit should resist the earth pressure resulting from the excavation. Based on the locations of the surrounding buildings, the excavation depth and the geological conditions of the foundation pit, a supporting scheme is adopted for the foundation pit as follows: (1) A diaphragm wall with a thickness of 1.2 m and a pre-stressed anchor with a horizontal spacing of 1.4 m are used to support the south side of the pit (the side adjacent to the existing tunnel). (2) A bored pile with a diameter of 1.1 m and a soil nailing wall are employed to reinforce the north, west and east sides of the pit. As the existing tunnel is adjacent to the south side of the pit, the supporting system composed of a diaphragm wall and a pre-stressed anchor is the focus of this paper. A sectional view of the retaining structure on the south side of the planned pit is shown in Fig. 2. 3. Numerical simulation 3.1. Numerical simulation of deformation for the rock mass around an existing tunnel induced by an adjacent excavation To investigate the deformation characteristics of the rock mass around an existing tunnel induced by an adjacent excavation, a three2
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Office building
Existing tunnel
Planned foundation pit 16m Minimum distance
Diaphragm wall Fig. 1. Plan view of the planned pit and the existing tunnel.
displacement in the undersurface of the model is fixed. A shell element, solid element and cable element are used to simulate the segmented lining, diaphragm wall and pre-stressed anchor, respectively, as shown in Fig. 4. To resist the huge earth pressure, the diaphragm wall needs to be reinforced in an actual project. However, the rebar in the diaphragm wall is difficult to construct by using solid element. Thus, an equivalent stiffness principle is employed to simulate the reinforced diaphragm wall. Based on this principle, the Young’s modulus parameter of the reinforced diaphragm wall is obtained which can be used to simulate the mechanical property of the reinforced diaphragm wall. The mechanical parameters of the structures such as lining, concrete floor slab, diaphragm wall and pre-stressed anchor are illustrated in Table 1. The interface elements are employed here to simulate the contact between
dimensional model is constructed by using the finite difference software FLAC3D [14]. This model is composed of 27,180 elements, and the dimensions of the 3D numerical model are 240, 100, and 6 m in the transverse, vertical, and longitudinal directions, respectively, as shown in Fig. 3. Because this study focuses on the deformation of the rock mass around the existing tunnel, a non uniform mesh is applied in the generation of this model. As shown in Fig. 3, the rock mass adjacent to the existing tunnel is constructed by dense meshes, while the rock mass away from the region of interest is constructed by sparse meshes. The advantages of the non uniform mesh are that fewer elements are used in the model and the computational efficiency is improved. The boundary conditions of the model are assumed as follows: The ground surface is free, the side surfaces have roller boundaries, and the vertical
Fig. 2. Sectional view of the retaining structure for the planned pit. 3
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Fig. 3. Numerical model of an excavation adjacent to an existing shield tunnel.
In addition, the minimum distance between the diaphragm wall and the outer edge of the segmented lining is 16 m. In accordance with the stages of the actual construction process, the excavation simulation procedure is listed as follows, as shown in Fig. 5: (1) The parameters are assigned to the elements of the rock mass. In addition, the elements of the tunnel lining and diaphragm wall are activated. The initial stress field of the model is generated. (2) For the first stage of excavation, the pit is excavated to a depth of 4 m, and the first row of pre-stressed anchors is installed at a depth of 3 m. (3) For the second stage of excavation, the pit is excavated to a depth of 8 m, and the second row of pre-stressed anchors is installed at a depth of 6 m. (4) For the third stage of excavation, the pit is excavated to a depth of 12 m, and the third row of pre-stressed anchors is installed at a depth of 9 m. (5) Finally, the pit is excavated to a depth of 15 m, and the concrete floor slab of the pit is poured. The excavation is completed.
Fig. 4. 3D numerical model of the retaining structure and the shield tunnel segment.
3.2. Numerical simulation results
Table 1 Parameters for diaphragm wall, concrete floor slab, lining and pre-stressed anchor. Parameter type (unit)
Diaphragm wall and concrete floor slab
Lining
Pre-stressed anchor
Thickness (m) Density (kg/m3) Young’s modulus (MPa) Poisson’s ratio
1.2 2500 2.4e4
0.3 2500 2.0e4
/ / 2.1e5
0.167
0.167
/
To obtain the deformation mode of the rock mass around the existing tunnel provided by numerical simulation, the contours of the maximum shear strain rates of the whole model for different excavation stages are plotted, as illustrated in Fig. 6. The decrease in the maximal shear strain rates is shown by gradually varying shades from yellow to dark blue, and the dark blue region indicates that no shear failure occurred. Thus, the boundaries colored light blue and dark blue can be regarded as the boundaries of the shear slip surface. Because the deformation of the rock mass induced by the excavation in stage I and stage II is slight, particular attention has been paid to studying the deformation of the rock mass in stage III and stage IV. Fig. 6 shows that the slip surface extends approximately from the position of four o’clock to the position of one o’clock at the back of the segmented lining in the surrounding rock. Plastic deformation occurs by smooth sliding on the slip surface. Due to the plastic deformation, an approximately parabolic failure block that is constituted by the closed slip surface is formed at the back of the segmented lining in the surrounding rock. Based on the position of this failure block, an undesirable contact zone at the back of the segmented lining can be determined for the existing shield tunnel. The formation mechanism of the undesirable contact zone can be explained as follows. Because of the excavation unloading effect on the foundation pit, the diaphragm wall begins to move toward the pit under the action of the earth pressure. Due to the slight movement of the diaphragm wall, the pre-stressed anchors produce a tensile force to resist this excavation-induced movement. When this tensile force is transmitted to the anchor end of the pre-stressed anchors, tensile (or shear) zones may form in the rock mass around the anchor end. Because
Table 2 The parameters of the rock mass in the framework of the HB failure criterion. γ (kN/m3)
E (MPa)
μ
σci (kPa)
a
mb
s
20.0
55
0.3
1200
0.5
0.4981
0.00267
Notes: E is elasticity modulus; μ is Poisson’s ratio.
the rock mass and the diaphragm wall and the contact between the rock mass and the lining. The rock mass is modeled using the Hoek-Brown model, and the diaphragm wall and concrete floor slabs are modeled with elastic beam elements. Moreover, the mechanical parameters of the rock mass are summarized in Table 2. Based on the project described above in the engineering background, the diameter of the shield tunnel is 11 m, the thickness of the segmented lining for the shield tunnel is 0.3 m, the thickness of the diaphragm wall is 1.2 m, and the depth of the diaphragm wall is 34 m. 4
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
upper bound solution of the equation for the slip surface can be derived in combination with other parameters. 4. Failure mechanism of the rock mass around an existing tunnel induced by adjacent excavation The numerical simulation results indicate that a narrow transition layer of plastic deformation forms in the rock mass around the existing tunnel. Based on the location and shape of this transition layer, a failure mechanism for the rock mass around an existing tunnel induced by adjacent excavation is constructed in a kinematically admissible velocity field. It is assumed that there is a curve f(x) that is used to describe the narrow transition layer of the rock mass extending from an arbitrary location at the back of the existing tunnel lining to another location at the back of the existing tunnel lining. An inverted funnel-shaped failure surface forms that is symmetric about the y-axis. This surface is shown in Fig. 7. The rock mass delimited by the curve f(x) is regarded as a failure block, and plastic flow occurs along the curve f(x) resulting from the smooth sliding of adjacent surfaces between the failure block and the surrounding rock. The failure block can be treated as a rigid block, which implies that there is no internal volumetric strain in the failure block and that all the plastic flow occurs at the slip surfaces. It is clear that the relative sliding of adjacent surfaces between the failure block and the surrounding rock would cause the occurrence of energy dissipation. Using the dissipation of energy and external rate of work in the failure mechanism, the analytical equation of the curve f(x) can be derived from the upper bound theorem in conjunction with a variational approach. Because the tensile (or shear) zones result from the resistance of the pre-stressed anchors, the direction of the velocity vector for the failure block is related to the relative position between the tunnel and the prestressed anchors, and this position can be determined by a numerical simulation as mentioned above. To make this failure mechanism symmetrical and simplify the calculation process, the direction of the y-axis for the coordinate system is set in accordance with the direction of the velocity vector for the failure block. Fig. 7 shows that the velocity vectors of failure block v make an angle β relative to the vertical direction, and s and n are the tangential direction and normal direction for a random point in the slip surface direction, respectively. L denotes the half-length of the bottom of the failure block. 5. Upper bound solution of the failure surface for the rock mass around an existing tunnel induced by adjacent excavation As indicated above, plastic flow occurs along the slip surface f(x) and causes the dissipation of energy. Thus, it is necessary to determine which conditions characterize the transition of the material from an elastic state to a plastic flow state. According to Chen [15], this condition, satisfied in the yield state, is called the failure criterion. Presently, the Hoek-Brown (HB) nonlinear failure criterion is widely used to estimate rock mass strength [16]. However, as pointed out by Hoek and Brown [17], because of the lack of suitable alternatives, the HB nonlinear failure criterion can also be applied to heavily broken rock mass or even engineering soils. Thus, the HB failure criterion is employed here to investigate the deformation of the rock mass. Assuming that the excavation obeys the HB failure criterion and its associated flow rule, the plastic potential that satisfies the HB failure criterion may be written in the following form:
Fig. 5. The excavation procedure: (a) stage I; (b) stage II; (c) stage III; (d) stage IV.
the tensile strength of geotechnical materials is low, these tensile (or shear) zones may extend into the surrounding area. When the tensile (or shear) zones connect to each other in the area adjacent to an existing tunnel, a narrow transition layer of plastic deformation forms in the rock mass around the existing tunnel. Finally, an undesirable contact occurs between the surrounding rock and the segmented lining of the existing shield tunnel. Furthermore, the velocity vectors of the failure block are also illustrated in Fig. 6. The velocity vectors of the failure block clearly form an angle (β) with the vertical direction. β is a significant parameter that is required in the analytical calculation of the slip surface. Fig. 6 shows that the value of β is approximately 50° for stage III and 60° for stage IV. Based on these values of β obtained from the numerical simulation, the
Ω = τ - Aσci ⎛ ⎝
⎜
σn + σtm ⎞ σci ⎠
B
⎟
(1)
where σn is the normal stress, τ is the shear stress, A and B are material constants, and σci and σtm are the uniaxial compressive strength and tensile strength of the rock mass, respectively. The plastic strain rate derived from the normality condition corresponding to the HB plastic 5
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Fig. 6. Contours of the maximal shear strain rates of the rock mass around the tunnel in different excavation stages: (a) stage III; (b) stage IV.
cot θ = f ′ (x )
potential function for a random point on the slip surface can be written in the general form:
⎧ εṅ =
∂Ω λ ∂σ
= - λAB
⎨ ∂Ω γ ̇ = λ ∂τ = λ ⎩ n
(
(3)
On the basis of the angle relationship between the velocity vector v and the tangent of a certain point, the plastic normal strain rate εṅ and plastic shear strain rate γ̇n for this point can be expressed as:
σn + σtm B − 1 σci
)
v
⎧ εṅ = t sin θ ⎨ γṅ = − v cos θ t ⎩
(2)
where λ is a scalar proportionality factor. As shown in Fig. 7, θ is the inclination angle of the tangent for a random point on the slip surface. If f ′ (x ) is the first derivative of the slip surface equation f (x ) , the following equation is obtained:
(4)
where t is the thickness of the slip surface. Substituting Eqs. (2) and (3) into Eq. (4), the normal stress σn and shear stress τ for this point on the slip surface can be written as: 6
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Fig. 7. Failure mechanism of the rock mass around an existing tunnel induced by an adjacent excavation.
ξ [f (x ), f ′ (x ), x ] = PD − Pγ − Pq
1
⎧ σn = σci [ABf ′ (x )]1 − B − σtm ⎨ τ = Aσci [ABf ′ (x )]1 −BB ⎩
Substituting Eqs. (7), (8) and (10) into Eq.(11), the expression of ξ is found to be:
(5)
As pointed out by Fraldi and Guarracino [18], the rate of energy dissipation can be determined by multiplying the stress and strain rates. Consequently, the rate of energy dissipation for a random point on the slip surface is computed from:
{
1
ΔD = σn εṅ + τγṅ = σci [ABf ′ (x )]1 − B (1 − B −1) − σtm
}t
ξ = ∫0
∫0
L
D 1 + f ′ (x )2 dx =
L
∫0 {σci [ABf ′ (x )]
1 1−B
(1 − B −1) − σtm
L
f (x )dx −
∫0
L
v g (x )dx⎤ cos β ⎦t
b2 − x 2
v dx t
L b
L
v
}
v
(6)
L b
(12) whereψ is a functional and can be written as: 1
ψ [f (x ), f ′ (x ), x ] = σci [ABf ′ (x )]1 − B (1 − B −1) − σtm − γf (x ) cos β
(13)
As pointed out above, the load derived from the virtual work equation is either higher or equal to the actual limit load. Thus, to obtain the real upper bound solution, it is necessary to find the values of the variables that make the upper bound solution the least or the most critical numerically or analytically. However, it can be found that ξ is determined by the functional ψ . Thus, seeking the expression of the slip surface f (x ) that makes the objective function ψ reach the extremum is a challenge for this investigation. Fortunately, in simplest terms, ψ is an integral-type functionalwhose extremum can be derived from a variational approach. According to variational theory, the functional ψ has an extremum if and only if the necessary condition presented as follows is satisfied:
} vt dx
(8)
whereγ is the unit weight of the rock mass and β is the angle between the velocity vectors of failure block v and the direction of gravity. Moreover, g (x ) is the equation of the cross section for the shield tunnel and can be expressed as:
g (x ) =
v
L
(7)
∫0
− σtm − γf (x ) cos β
= ∫0 ψ [f (x ), f ′ (x ), x ] t dx − ∫0 γg (x ) cos β t dx + bqv arcsin
The half rate of work for the failure block produced by weight is given by:
Pγ = γ ⎡ ⎣
1
ci [ABf ′ (x )]1 − B
L
Because the failure mechanism is symmetric about the y-axis, half of the energy dissipation and external rate of work are calculated for mathematical simplicity. The half rate of energy dissipation along the slip surface can be obtained by integrating Eq. (6) over the interval [0, L], which can be expressed as:
PD =
{σ
L
− ∫0 γg (x ) cos β t dx + bqv arcsin
v 1 + f ′ (x )2
(11)
∂ψ d ⎡ ∂ψ ⎤ =0 − dx ⎢ ∂f (x ) ⎣ ∂f ′ (x ) ⎥ ⎦
(14)
Eq. (14) is also called the Euler equation of functional ψ . By substituting Eq. (13) into Eq. (14), a nonlinear second-order constantcoefficient differential equation is obtained:
(9)
where b is the radius of the shield tunnel. The rate of work for the supporting pressure produced by the segmented lining of the shield tunnel is:
− γ cos β + σci (AB ) 1 − B [f ′ (x )] 1 − B f ″ (x )
L Pq = bq arcsin ⎛ ⎞ v cos π ⎝b⎠
Solving Eq. (15), the analytical solution of this differential equation can be expressed as:
1
(10)
where q is the supporting pressure of the shield tunnel. The upper bound theorem states that a load that is no less than the actual collapse load can be obtained by equating the external rate of virtual work to the internal rate of energy dissipation. Thus, the construction of an objective function that describes the relationship between the external rate of work and the internal rate of energy dissipation can be considered the first step to derive the upper solution of the slip surface f (x ) . Based on this theory, an objective function that is composed of the external rate of work and the internal rate of energy dissipation is written as:
f (x ) =
2B − 1
1 −1 ⎛ γ ⎞ A B cos β ⎝ σci ⎠ ⎜
⎟
1−B B
1 =0 1−B
(15)
1 B ⎛x cos β − c0 ⎟⎞ − c1 γ⎠ ⎝
⎜
(16)
where c0 and c1 are integration constants. Substituting Eq. (16) into Eq. (13), the expression of ψ has the following form: 1
c B 1 1 1 B−1 ψ = − A− B γ B σci B ⎜⎛x cos β − 0 ⎟⎞ − σtm + γc1 cos β B γ⎠ ⎝ 7
(17)
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Ground surface
Substituting Eq. (17) into Eq. (12), the expression of ξ can be written in the following form: B−1
(
)
1+B
( )
1 1 1 1 c c B ξ = − 1 + B cos β A− B γ B σci B ⎡ L cos β − γ0 − − γ0 ⎢ ⎣ +(γc1 cos β − σtm ) Lv
− γ cos β
(
L 2
b2 − L2 +
b2 2
arcsin
L b
1+B B ⎤
) v + bqv arcsin
⎥ ⎦
L b
v A=0.34 A=0.29 A=0.24 A=0.19
(18)
Diaphragm wall
As shown in Eq. (16), there are two unknown constants c0 and c1 that should be determined before we obtain the final expression of the slip surface f (x ) . Because the upper failure mechanism satisfies the stress and geometric boundary conditions, the values of c0 and c1 can be derived from the equations provided by these boundary conditions. Because the shear stress at the junction of the slip surface and y-axis is zero, a microelement at this position is selected as an analysis object. Based on the equilibrium differential equation of this microelement, the shear stress of the microelement at the mentioned position is given by:
τxy (x = 0, y = −h) =
Pit floor
Fig. 8. Slip surfaces of the rock mass induced by an adjacent excavation for various values of A. Ground surface
1 σn sin 2θ − τn cos 2θ = 0 2
(19)
where sin 2θ and cos 2θ can be derived from a trigonometric function transformation in conjunction with Eq. (3). Substituting σn and τn illustrated in Eq. (5) into Eq. (19), the following value of c0 is obtained:
B=0.50 B=0.60 B=0.70 B=0.90
(20)
c0 = 0
By substituting Eq. (20) into Eq. (16), the expression of the slip surface f (x ) can be simplified to the following form:
f (x ) =
1 −1 ⎛ γ ⎞ A B cos β ⎝ σci ⎠ ⎜
1−B B
⎟
Diaphragm wall
1
(x cos β ) B − c1
Pit floor
(21)
As shown in Fig. 7, the slip surface f (x ) and the equation of the cross section for the shield tunnel g (x ) intersect at point (L,0). Thus, the geometric boundary condition, which is concerned with equations f (x ) and g (x ) , can be written as:
f (L) = g (L)
Fig. 9. Slip surfaces of the rock mass induced by an adjacent excavation for various values of B.
(22)
Ground surface
However, the two constants c1 and L are still undetermined. Therefore, another equation should be found. By equating the external rate of work to the rate of internal energy dissipation, an equation that is used to calculate unknown constants c1 and L is obtained: B−1
−
1 1 1 1 A− B γ B σci B 1 + B cos β
− γ cos β
(
L 2
1+B ⎡ (L cos β ) B ⎤
⎣
b2 − L2 +
⎦
b2 2
arcsin
L b
)
+ (γc1 cos β − σtm ) L + bqv arcsin
L b
=0
tm
ci
tm
ci
tm
ci
tm
ci
Diaphragm wall
(23)
Combining Eqs. (22) and (23), the values of c1 and L are determined by using a numerical method. Substituting c1 and L into Eq. (21), the final upper bound solution of the slip surface f (x ) is obtained.
Pit floor
6. Calculation results and parameter analysis
Fig. 10. Slip surfaces of the rock mass induced by an adjacent excavation for various values of σtm.
Based on the analytical equation of the slip surface f (x ) , the shape of the failure block can be plotted to describe the range of the undesirable contact zone at the back of the existing tunnel. To study the influence of different parameters on the range of the undesirable contact zone, slip surfaces are illustrated in Figs. 8–13 for parameters σci =1.2 MPa, corresponding to b = 5.5 m, A = 0.19–0.34, σtm =σci /100~σci /40, B = 0.50–0.90, γ = 17.5–22.5kN/m3, q = 40–55 kPa, and β = 30–60°. Figs. 8–13 show that the slip surface extends approximately from the position of five o’clock to the position of one o’clock at the back of the segmented lining in the surrounding rock, and the shape of the slip surface approximates a parabola. Furthermore, the parameters A, B, σtm, γ, q and β have significant influences on the range of the slip surface. Fig. 8 shows the slip surface of the rock mass for different valuesof A
when the other parameters are fixed. As shown in this figure, the range of the slip surface clearly increases with increasing values of A. Similarly, Figs. 9–13 illustrate the influences of changing B, σtm, γ, q and β on the range of the slip surface when the other parameters are fixed. The range of the slip surface clearly increases with decreasing values of B, γ and q, whereas it increases with increasing values of σtm. However, in contrast to the other parameters, β is a parameter that affects not only the range of the slip surface but also the position of the slip surface. As mentioned above, β denotes the angle between the velocity vector of the failure block and the vertical direction. Because the velocity vector of the failure block is difficult to observe in actual engineering practice, β can be obtained from the velocity vectors 8
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
simulation under the same conditions, the HB failure criterion is invoked in the numerical simulation process. Moreover, the HB failure criterion used in the FLAC3D program is expressed in terms of the major and minor principal stresses as follows:
kN/m3 kN/m3 kN/m3 kN/m3
σ σ1 = σ3 + σci ⎛mb 3 + s ⎞ ⎝ σci ⎠ ⎜
Diaphragm wall
(24)
whereσ1 and σ3 are the major and minor principal stresses, respectively and mb, s, and a are material constants that are related to the geological strength index (GSI). Because the derivation of the analytical equation for the slip surface requires an assumption that the rock mass is isotropic, the surrounding rock of the numerical model is assumed to be a homogeneous material. As mentioned in 3.1, the mechanical parameters of the rock mass used in the numerical simulation are summarized in Table 1. However, the HB failure criterion used in the upper bound calculation is represented by normal and shear stresses, while the HB failure criterion invoked in the FLAC3D program is expressed in Eq. (24). Several parameters are different in these two forms of the HB criterion. Therefore, the parameters of the HB failure criterion represented by normal and shear stresses should be converted into the parameters represented in terms of the major and minor principal stresses to make the comparison under the same conditions. Hoek and Brown [17] proposed a method to achieve the equivalent conversion of the parameters in terms of these two forms, and Huang et al. [19,20] used this method to complete the comparison between the upper bound calculation and numerical simulation. Based on the method proposed by Hoek and Brown [17], the equivalent parameters are obtained for the two forms of the HB failure criterion, as illustrated in Table 3. Using the parameters presented in Table 3 and the value of β provided by numerical simulation, the upper bound solutions of the slip surface for stage III and stage IV are plotted in Fig. 14. It is clear that the slip surface derived from the upper bound calculation is similar to the boundary of the shear slip surface obtained from numerical simulation. To better compare the slip surface from the upper bound calculation and that of the numerical simulation, the upper bound solutions of the slip surface are superimposed on Fig. 15. The upper bound solutions of the slip surface for stage III and stage IV shown in this figure are clear approximations to the slip surface provided by numerical simulation. The good agreement between the upper bound solutions and the numerical simulation results for the slip surface of the rock mass indicates that the method proposed in this paper is valid. By comparing the upper bound calculation with the numerical simulation, it is found that the numerical simulation technique can be used to construct complex models that are closer to an actual engineering project. However, a complex model means that a significant number of elements are required in the modeling process and that the calculation process is time consuming. Thus, relative to the numerical simulation technique, the method presented in this paper is a more efficient approach.
Pit floor
Fig. 11. Slip surfaces of the rock mass induced by an adjacent excavation for various values of γ.
q q q q
a
⎟
kPa kPa kPa kPa Diaphragm wall
Pit floor
Fig. 12. Slip surfaces of the rock mass induced by an adjacent excavation for various values of q.
Diaphragm wall
Pit floor
Fig. 13. Slip surfaces of the rock mass induced by an adjacent excavation for various values of β.
provided by numerical simulation. Fig. 13 shows the slip surface of the surrounding rock for different values of β when the other parameters are fixed. It can be seen from this figure that the position of the slip surface moves downward with increasing values of β, and the range of the slip surface clearly increases with increasing values of β.
Table 3 Equivalent parameters for the two forms of the HB failure criterion.
7. Comparison with numerical simulation results To evaluate the validity of the upper bound solution of the slip surface presented in this paper, the analytical solution is compared with the results derived from the finite difference software FLAC3D. To make the comparison between the analytical calculation and the numerical 9
A B σci σtm
0.34 0.7 1.2 MPa 0.012 MPa
a mb s σci GSI
0.5 0.4981 0.00267 1.2 MPa 46
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
8. Conclusion Based on the deformation characteristics of the rock mass around an existing tunnel induced by an adjacent excavation, a new failure mechanism is established that can be used to determine the shape and range of the undesirable contact zone at the back of the existing tunnel lining. Using this failure mechanism, an analytical solution of the slip surface for the rock mass around the existing tunnel induced by the adjacent excavation is derived from the upper bound theorem in conjunction with a variational approach. The shape and range of the undesirable contact zone are plotted for different parameters on the basis of the analytical solution of the slip surface. The shape of the slip surface shows that an approximately parabolic failure block forms at the back of the segmented lining of the existing shield tunnel, and this failure block can be regarded as the potential undesirable contact zone between the tunnel lining and surrounding rock. Furthermore, by analyzing the variation in the shape of the slip
Diaphragm wall Stage
Pit floor
Stage
Fig. 14. Upper bound solutions of the slip surface for the rock mass induced by an adjacent excavation in different excavation stages.
Fig. 15. Comparison of the slip surfaces provided by the upper bound theorem and the numerical simulation for different stages: (a) stage III; (b) stage IV. 10
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
surface with different parameters, it is found that the range of the slip surface increases with decreasing values of B, γ and q, whereas it increases with increasing values of σtm and β. To evaluate the validity of the upper bound solution of the slip surface, the analytical solution is compared with the results derived from the numerical simulation technique. Based on the actual project presented in this paper, a 3D numerical model is constructed, and the contours of the maximum shear strain rate of the rock mass are obtained; these contours can be used to describe the shear failure of the rock mass around the existing tunnel induced by the adjacent excavation. The slip surface derived from the upper bound calculation and the contours of the maximum shear strain rate provided by the numerical simulation are compared under the same conditions. The good agreement between the two methods indicates that the proposed method is effective for calculating the slip surface of the rock mass around the existing tunnel induced by the adjacent excavation.
[5] [6]
[7]
[8]
[9] [10]
[11]
[12]
Acknowledgments [13]
This study was supported by the National Natural Science Foundation of China (Grants 51878074 and 51678071) and InnovationDriven Project of Central South University (No. 2019CX011). Their financial supports are greatly appreciated.
[14] [15] [16]
References
[17] [18]
[1] Liang RZ, Xia TD, Huang MS, Lin CG. Simplified analytical method for evaluating the effects of adjacent excavation on shield tunnel considering the shearing effect. Comput Geotech 2017;81:167–87. [2] Liang RZ, Wu WB, Yu F, Jiang GS, Liu JW. Simplified method for evaluating shield tunnel deformation due to adjacent excavation. Tunnel Undergr Space Technol 2018;71:94–105. [3] Zheng G, Wei S. Numerical analyses of influence of overlying pit excavation on existing tunnels. J Cent South Univ 2008;15(S2):69–75. [4] Zheng G, Yang XY, Zhou HZ, Du YM, Jiayu Sun JY, Yu XX. A simplified prediction
[19] [20]
11
method for evaluating tunnel displacement induced by laterally adjacent excavations. Comput Geotech 2018;95:119–28. Huang X, Schweiger HF, Huang HW. Influence of deep excavations on nearby existing tunnels. Int J Geomech 2013;13(2):170–80. Chen RP, Meng FY, Li ZC, Ye YH, Ye JN. Investigation of response of metro tunnels due to adjacent large excavation and protective measures in soft soils. Tunnel Undergr Space Technol 2016;58:224–35. Shi JW, Ng CWW, Chen YH. Three-dimensional numerical parametric study of the influence of basement excavation on existing tunnel. Comput Geotech 2015;63:145–58. Shi JW, Zhang X, Chen YH, Chen L. Numerical parametric study of countermeasures to alleviate basement excavation effects on an existing tunnel. Tunnel Undergr Space Technol 2018;72:145–53. Zhang ZG, Huang MS. Geotechnical influence on existing subway tunnels induced by multiline tunneling in Shanghai soft soil. Comput Geotech 2014;56:121–32. Ng CWW, Shi JW, Hong Y. Three-dimensional centrifuge modelling of basement excavation effects on an existing tunnel in dry sand. Can Geotech J 2013;50(8):874–88. Ng CWW, Shi JW, Mašín D, Sun HS, Lei GH. Influence of sand density and retaining wall stiffness on three-dimensional responses of tunnel to basement excavation. Can Geotech J 2015;52(11):1811–29. Li P, Du SJ, Ma XF, Yin ZY, Shen SL. Centrifuge investigation into the effect of new shield tunneling on an existing underlying large-diameter tunnel. Tunnel Undergr Space Technol 2014;42:59–66. Jiang B, Chen LA, Yang JS, Wang SY, Ng CWW. Effects of twin-tunnel excavation on an existing horseshoe-shaped tunnel considering the influence of a settlement joint. Can Geotech J 2017;54(9):1346–55. Itasca Consulting Group Inc, 2006. FLAC3D Manual, third ed. (FLAC3D Version 5.00). Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975. Zhang DB, Jiang Y, Yang XL. Estimation of 3D active earth pressure under nonlinear strength condition. Geomech Eng 2019;17(6):515–25. Hoek E, Brown ET. Practical estimate the rock mass strength. Int J Rock Mech Min Sci 1997;34:1165–86. Fraldi M, Guarracino F. Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek-Brown failure criterion. Int J Rock Mech Min Sci 2009;46:665–73. Huang F, Zhao LH, Ling TH, Yang XL. Rock mass collapse mechanism of concealed karst cave beneath deep tunnel. Int J Rock Mech Min Sci 2017;91:133–8. Huang F, Ou RC, Li ZL, Yang XL, Ling TH. Limit analysis for the face stability of a shallow-shield tunnel based on a variational approach to the blow-out failure mode. Int J Geomech 2018;18(6):1–11.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
The failure mechanism of surrounding rock around an existing shield tunnel induced by an adjacent excavation
T
⁎
Fu Huanga, , Min Zhanga, Fen Wanga, Tonghua Linga, Xiaoli Yangb a b
School of Civil Engineering, Changsha University of Science and Technology, Changsha, Hunan, China School of Civil Engineering, Central South University, Changsha, Hunan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Upper bound theorem Variational approach Surrounding rock failure mechanism Pit excavation Existing shield tunnel
Foundation pit excavation may disturb the initial stress field and induce the deformation of the surrounding rock mass. Due to the excavation, the originally compact rock mass and lining structure may become loose with adverse impacts on the normal operation of the existing tunnel. A new failure mechanism is constructed on the basis of the deformation characteristics of the rock mass around an existing tunnel induced by an adjacent excavation. Using this failure mechanism, the upper bound solution of the slip surface equation for the rock mass around an existing tunnel is derived in the framework of the upper bound theorem in conjunction with a variational approach. The shape and range of the slip surface are plotted for different parameters. A three-dimensional numerical model is constructed to simulate the deformation of the rock mass around the existing tunnel induced by adjacent excavation. By comparing the analytical results for the slip surface with the failure surface provided by numerical simulation, it is found that the differences between the analytical results and the numerical results are small. This comparison shows that the proposed method is valid for investigating the deformation of the rock mass around the existing tunnel induced by adjacent excavation.
1. Introduction In large cities, the subway network is always concentrated in the center of the city. To facilitate traveling and promote the commercial value of buildings, most high-rise buildings are located close to existing subway lines. Because foundation pit excavation is a crucial process in the construction of high-rise buildings, the surrounding rock deformation induced by foundation pit excavation and the effects of deformation on adjacent subway structures are issues that engineers must confront. Numerous investigations show that foundation pit excavation induces deformation of the surrounding rock, and the deformation may transmit to the adjacent regions. When this happens, the originally compact rock mass and lining structure may become loose, and a cavity may even come into being at the back of the tunnel lining. This deformation of the rock mass causes undesirable contact between the rock mass and lining structure, leading to stress concentration in a local area of the lining structure that may result in cracking and damage of the lining structure. Consequently, the study of the influence of foundation pit excavation on adjacent existing underground structures is an issue with high engineering significance. The effect of foundation pit excavation on adjacent existing tunnels has aroused the interest of some investigators. To investigate the
⁎
responses of existing shield tunnels to adjacent excavation, Liang et al. [1] proposed a new analytical method that considers both the bending and shearing effects of a shield tunnel. Using this method, they studied the tunnel-ground interaction and evaluated tunnel longitudinal deformations subjected to adjacent excavation. Later, Liang et al. [2] modified this analytical method by introducing the Pasternak foundation model with a modified subgrade modulus to evaluate the deformation behavior of existing shield tunnels induced by adjacent excavation. The main advantage of their method is that the shield tunnel is treated as a continuous Euler-Bernoulli beam, which accurately predicts the shield tunnel behaviors associated with adjacent excavation. Based on a 2D finite element method, Zheng and Wei [3] presented a study investigating tunnel deformation and stress redistribution around the tunnel lining due to overlying excavation. They found that the deformation and stress redistribution of the tunnel lining varied with the variation of the location between the pit and the tunnel. Subsequently, Zheng et al. [4] developed a simplified semi-empirical method to assess the responses of a tunnel induced by an adjacent excavation on the basis of finite element analysis in conjunction with a set of collected case histories. Huang et al. [5] used 3D numerical analysis to investigate tunnel deformation due to adjacent foundation pit excavation. Their achievements show that the relative position between
Corresponding author. E-mail addresses: [email protected] (F. Huang), [email protected] (X. Yang).
https://doi.org/10.1016/j.compgeo.2019.103236 Received 7 January 2019; Received in revised form 9 August 2019; Accepted 1 September 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
of the stress concentration in this region at the back of the tunnel lining. Consequently, the stress concentration is the main factor that leads to the deformation of a tunnel lining. However, the transfer process of the ground deformation is concealed, which makes the position and range of the undesirable contact zone at the back of the tunnel lining difficult to determine precisely. Thus, it is necessary to develop a method to reveal the transfer mechanism and deformation mode of the rock mass around an existing tunnel induced by adjacent excavation. This paper presents a theoretical method to investigate the deformation mechanism of the rock mass around an existing tunnel induced by an adjacent excavation. Based on the deformation characteristics of the rock mass around an existing tunnel provided by a numerical simulation technique, a two-dimensional failure mechanism composed of a detached curve is established to determine the undesirable contact zone at the back of the tunnel lining. Using the upper bound theorem, the external rate of work and the rate of internal energy dissipation in the constructed failure mechanism are calculated. The objective function of the detached curve composed of the external rate of work and the rate of internal energy dissipation is obtained, and the upper bound solution of the detached curve equation is derived from a variational approach. Finally, the validation of the proposed analytical method is performed via comparison with a numerical simulation. The aim of this study is to develop an analytical method to determine the shape and range of the undesirable contact zone at the back of an existing tunnel induced by adjacent excavation.
the tunnel and the excavation, tunnel diameter, excavation dimensions, and tunnel protection measures are the key factors that affect the responses of tunnels during the excavation of adjacent foundation pits. Chen et al. [6] conducted a three-dimensional numerical analysis to study the influence of a nearby large excavation on existing twin tunnels in soft soils. By investigating the stress and displacement of the tunnels during different construction stages of the nearby excavation, they obtained an interaction mechanism between the nearby excavation surrounding soils and existing twin tunnels. Finding that the basement-tunnel interaction has been simplified as a plane strain problem in existing research, Shi et al. [7] used a three-dimensional numerical simulation technique to study the influences of excavation geometry, sand density, tunnel stiffness and joint stiffness on tunnel responses. More recently, Shi et al. [8] established a three-dimensional numerical model to investigate the validity of countermeasures that are used to reduce tunnel response due to overlying basement excavation. By analyzing the responses of a tunnel for different parameters of the countermeasures, they found that wall penetration depth, wall stiffness and lining thickness are three key factors that can be used to alleviate basement excavation-induced adverse effects on existing tunnels. Similar to the tunnel-ground interaction induced by adjacent excavation, the soil disturbances resulting from multiline tunneling in soft soil affect the safety of existing tunnels. Based on an actual project in Shanghai, Zhang and Huang [9] employed a three-dimensional finite element numerical simulation method to analyze the deformation of existing subway tunnels induced by an earth pressure balance (EPB) shield machine that drilled in the surrounding strata. Although the influence of a basement excavation on an adjacent tunnel can be studied by means of theoretical analysis and numerical simulations, some scholars found that the centrifuge model test is also an effective way to investigate the tunnel responses induced by basement excavation. Ng et al. [10] designed a three-dimensional centrifuge test to investigate the effects of a basement excavation on an existing tunnel in dry sand. Combining the results of centrifuge tests and numerical analyses, they found that the bending deformation of the tunnel is directly related to the location between the tunnel and the basement. Subsequently, finding that the effects of sand density and basement wall stiffness on nearby tunnels due to excavation are neglected in existing studies, Ng et al. [11] employed a series of three-dimensional centrifuge tests to investigate these effects on the complex basement-tunnel interaction. To estimate the influence of new shield tunneling on an existing underlying large-diameter tunnel, Li et al. [12] designed a centrifuge model test to simulate the whole construction process of a new shield tunnel excavated above an existing tunnel. By studying the displacement and bending moments of the existing tunnel, they found that grouting is an effective measure to alleviate the responses of the existing tunnel induced by the new shield tunneling. Similarly, Jiang et al. [13] also employed three-dimensional centrifuge model tests to investigate the response of an existing horseshoe-shaped tunnel induced by the construction of two perpendicularly undercrossing tunnels. Although the interactions between tunnels and adjacent excavations have drawn the attention of numerous investigators, previous studies mainly focused on the deformations of existing tunnels induced by adjacent excavation. However, the deformation of a tunnel lining is only a phenomenon, what leads to this phenomenon is an issue deserving further study. Unfortunately, the studies of the cause of the deformation for a tunnel induced by an adjacent excavation are rare in existing literatures. By investigating the interaction mechanisms between the surrounding soils and an existing tunnel, it can be found that the essential cause of the deformation for the tunnel is the disturbance of the initial stress field induced by an adjacent excavation. Due to the disturbance of the ground induced by adjacent excavation, ground deformations inevitably occur in the adjacent region. Furthermore, the ground deformation transmitted to the surrounding region of an existing tunnel causes undesirable contact between the tunnel lining and surrounding rock. The undesirable contact would induce the occurrence
2. Engineering background There is a plan to build a basement adjacent to an existing shield tunnel in Shenzhen. The foundation pit of this basement is an irregular polygon that is located at the intersection of two main routes. The length and width of the pit are 150 m and 110 m, respectively. Based on the geological investigation report and the actual conditions of this project, the cut and cover method is selected to excavate the foundation pit. The average excavation depth of this pit is 15 m, and the maximum excavation depth is 16.5 m. The surroundings of this project are complex, with the north, west and east of the pit adjacent to main routes. Moreover, there is an existing shield tunnel in the south of the pit, and the minimum distance between the tunnel segment and the edge of the pit is 16 m. A plan view of the planned pit and the existing tunnel is illustrated in Fig. 1. To guarantee the safety of the surrounding buildings during the excavation of the foundation pit, the retaining structure of the foundation pit should resist the earth pressure resulting from the excavation. Based on the locations of the surrounding buildings, the excavation depth and the geological conditions of the foundation pit, a supporting scheme is adopted for the foundation pit as follows: (1) A diaphragm wall with a thickness of 1.2 m and a pre-stressed anchor with a horizontal spacing of 1.4 m are used to support the south side of the pit (the side adjacent to the existing tunnel). (2) A bored pile with a diameter of 1.1 m and a soil nailing wall are employed to reinforce the north, west and east sides of the pit. As the existing tunnel is adjacent to the south side of the pit, the supporting system composed of a diaphragm wall and a pre-stressed anchor is the focus of this paper. A sectional view of the retaining structure on the south side of the planned pit is shown in Fig. 2. 3. Numerical simulation 3.1. Numerical simulation of deformation for the rock mass around an existing tunnel induced by an adjacent excavation To investigate the deformation characteristics of the rock mass around an existing tunnel induced by an adjacent excavation, a three2
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Office building
Existing tunnel
Planned foundation pit 16m Minimum distance
Diaphragm wall Fig. 1. Plan view of the planned pit and the existing tunnel.
displacement in the undersurface of the model is fixed. A shell element, solid element and cable element are used to simulate the segmented lining, diaphragm wall and pre-stressed anchor, respectively, as shown in Fig. 4. To resist the huge earth pressure, the diaphragm wall needs to be reinforced in an actual project. However, the rebar in the diaphragm wall is difficult to construct by using solid element. Thus, an equivalent stiffness principle is employed to simulate the reinforced diaphragm wall. Based on this principle, the Young’s modulus parameter of the reinforced diaphragm wall is obtained which can be used to simulate the mechanical property of the reinforced diaphragm wall. The mechanical parameters of the structures such as lining, concrete floor slab, diaphragm wall and pre-stressed anchor are illustrated in Table 1. The interface elements are employed here to simulate the contact between
dimensional model is constructed by using the finite difference software FLAC3D [14]. This model is composed of 27,180 elements, and the dimensions of the 3D numerical model are 240, 100, and 6 m in the transverse, vertical, and longitudinal directions, respectively, as shown in Fig. 3. Because this study focuses on the deformation of the rock mass around the existing tunnel, a non uniform mesh is applied in the generation of this model. As shown in Fig. 3, the rock mass adjacent to the existing tunnel is constructed by dense meshes, while the rock mass away from the region of interest is constructed by sparse meshes. The advantages of the non uniform mesh are that fewer elements are used in the model and the computational efficiency is improved. The boundary conditions of the model are assumed as follows: The ground surface is free, the side surfaces have roller boundaries, and the vertical
Fig. 2. Sectional view of the retaining structure for the planned pit. 3
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Fig. 3. Numerical model of an excavation adjacent to an existing shield tunnel.
In addition, the minimum distance between the diaphragm wall and the outer edge of the segmented lining is 16 m. In accordance with the stages of the actual construction process, the excavation simulation procedure is listed as follows, as shown in Fig. 5: (1) The parameters are assigned to the elements of the rock mass. In addition, the elements of the tunnel lining and diaphragm wall are activated. The initial stress field of the model is generated. (2) For the first stage of excavation, the pit is excavated to a depth of 4 m, and the first row of pre-stressed anchors is installed at a depth of 3 m. (3) For the second stage of excavation, the pit is excavated to a depth of 8 m, and the second row of pre-stressed anchors is installed at a depth of 6 m. (4) For the third stage of excavation, the pit is excavated to a depth of 12 m, and the third row of pre-stressed anchors is installed at a depth of 9 m. (5) Finally, the pit is excavated to a depth of 15 m, and the concrete floor slab of the pit is poured. The excavation is completed.
Fig. 4. 3D numerical model of the retaining structure and the shield tunnel segment.
3.2. Numerical simulation results
Table 1 Parameters for diaphragm wall, concrete floor slab, lining and pre-stressed anchor. Parameter type (unit)
Diaphragm wall and concrete floor slab
Lining
Pre-stressed anchor
Thickness (m) Density (kg/m3) Young’s modulus (MPa) Poisson’s ratio
1.2 2500 2.4e4
0.3 2500 2.0e4
/ / 2.1e5
0.167
0.167
/
To obtain the deformation mode of the rock mass around the existing tunnel provided by numerical simulation, the contours of the maximum shear strain rates of the whole model for different excavation stages are plotted, as illustrated in Fig. 6. The decrease in the maximal shear strain rates is shown by gradually varying shades from yellow to dark blue, and the dark blue region indicates that no shear failure occurred. Thus, the boundaries colored light blue and dark blue can be regarded as the boundaries of the shear slip surface. Because the deformation of the rock mass induced by the excavation in stage I and stage II is slight, particular attention has been paid to studying the deformation of the rock mass in stage III and stage IV. Fig. 6 shows that the slip surface extends approximately from the position of four o’clock to the position of one o’clock at the back of the segmented lining in the surrounding rock. Plastic deformation occurs by smooth sliding on the slip surface. Due to the plastic deformation, an approximately parabolic failure block that is constituted by the closed slip surface is formed at the back of the segmented lining in the surrounding rock. Based on the position of this failure block, an undesirable contact zone at the back of the segmented lining can be determined for the existing shield tunnel. The formation mechanism of the undesirable contact zone can be explained as follows. Because of the excavation unloading effect on the foundation pit, the diaphragm wall begins to move toward the pit under the action of the earth pressure. Due to the slight movement of the diaphragm wall, the pre-stressed anchors produce a tensile force to resist this excavation-induced movement. When this tensile force is transmitted to the anchor end of the pre-stressed anchors, tensile (or shear) zones may form in the rock mass around the anchor end. Because
Table 2 The parameters of the rock mass in the framework of the HB failure criterion. γ (kN/m3)
E (MPa)
μ
σci (kPa)
a
mb
s
20.0
55
0.3
1200
0.5
0.4981
0.00267
Notes: E is elasticity modulus; μ is Poisson’s ratio.
the rock mass and the diaphragm wall and the contact between the rock mass and the lining. The rock mass is modeled using the Hoek-Brown model, and the diaphragm wall and concrete floor slabs are modeled with elastic beam elements. Moreover, the mechanical parameters of the rock mass are summarized in Table 2. Based on the project described above in the engineering background, the diameter of the shield tunnel is 11 m, the thickness of the segmented lining for the shield tunnel is 0.3 m, the thickness of the diaphragm wall is 1.2 m, and the depth of the diaphragm wall is 34 m. 4
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
upper bound solution of the equation for the slip surface can be derived in combination with other parameters. 4. Failure mechanism of the rock mass around an existing tunnel induced by adjacent excavation The numerical simulation results indicate that a narrow transition layer of plastic deformation forms in the rock mass around the existing tunnel. Based on the location and shape of this transition layer, a failure mechanism for the rock mass around an existing tunnel induced by adjacent excavation is constructed in a kinematically admissible velocity field. It is assumed that there is a curve f(x) that is used to describe the narrow transition layer of the rock mass extending from an arbitrary location at the back of the existing tunnel lining to another location at the back of the existing tunnel lining. An inverted funnel-shaped failure surface forms that is symmetric about the y-axis. This surface is shown in Fig. 7. The rock mass delimited by the curve f(x) is regarded as a failure block, and plastic flow occurs along the curve f(x) resulting from the smooth sliding of adjacent surfaces between the failure block and the surrounding rock. The failure block can be treated as a rigid block, which implies that there is no internal volumetric strain in the failure block and that all the plastic flow occurs at the slip surfaces. It is clear that the relative sliding of adjacent surfaces between the failure block and the surrounding rock would cause the occurrence of energy dissipation. Using the dissipation of energy and external rate of work in the failure mechanism, the analytical equation of the curve f(x) can be derived from the upper bound theorem in conjunction with a variational approach. Because the tensile (or shear) zones result from the resistance of the pre-stressed anchors, the direction of the velocity vector for the failure block is related to the relative position between the tunnel and the prestressed anchors, and this position can be determined by a numerical simulation as mentioned above. To make this failure mechanism symmetrical and simplify the calculation process, the direction of the y-axis for the coordinate system is set in accordance with the direction of the velocity vector for the failure block. Fig. 7 shows that the velocity vectors of failure block v make an angle β relative to the vertical direction, and s and n are the tangential direction and normal direction for a random point in the slip surface direction, respectively. L denotes the half-length of the bottom of the failure block. 5. Upper bound solution of the failure surface for the rock mass around an existing tunnel induced by adjacent excavation As indicated above, plastic flow occurs along the slip surface f(x) and causes the dissipation of energy. Thus, it is necessary to determine which conditions characterize the transition of the material from an elastic state to a plastic flow state. According to Chen [15], this condition, satisfied in the yield state, is called the failure criterion. Presently, the Hoek-Brown (HB) nonlinear failure criterion is widely used to estimate rock mass strength [16]. However, as pointed out by Hoek and Brown [17], because of the lack of suitable alternatives, the HB nonlinear failure criterion can also be applied to heavily broken rock mass or even engineering soils. Thus, the HB failure criterion is employed here to investigate the deformation of the rock mass. Assuming that the excavation obeys the HB failure criterion and its associated flow rule, the plastic potential that satisfies the HB failure criterion may be written in the following form:
Fig. 5. The excavation procedure: (a) stage I; (b) stage II; (c) stage III; (d) stage IV.
the tensile strength of geotechnical materials is low, these tensile (or shear) zones may extend into the surrounding area. When the tensile (or shear) zones connect to each other in the area adjacent to an existing tunnel, a narrow transition layer of plastic deformation forms in the rock mass around the existing tunnel. Finally, an undesirable contact occurs between the surrounding rock and the segmented lining of the existing shield tunnel. Furthermore, the velocity vectors of the failure block are also illustrated in Fig. 6. The velocity vectors of the failure block clearly form an angle (β) with the vertical direction. β is a significant parameter that is required in the analytical calculation of the slip surface. Fig. 6 shows that the value of β is approximately 50° for stage III and 60° for stage IV. Based on these values of β obtained from the numerical simulation, the
Ω = τ - Aσci ⎛ ⎝
⎜
σn + σtm ⎞ σci ⎠
B
⎟
(1)
where σn is the normal stress, τ is the shear stress, A and B are material constants, and σci and σtm are the uniaxial compressive strength and tensile strength of the rock mass, respectively. The plastic strain rate derived from the normality condition corresponding to the HB plastic 5
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Fig. 6. Contours of the maximal shear strain rates of the rock mass around the tunnel in different excavation stages: (a) stage III; (b) stage IV.
cot θ = f ′ (x )
potential function for a random point on the slip surface can be written in the general form:
⎧ εṅ =
∂Ω λ ∂σ
= - λAB
⎨ ∂Ω γ ̇ = λ ∂τ = λ ⎩ n
(
(3)
On the basis of the angle relationship between the velocity vector v and the tangent of a certain point, the plastic normal strain rate εṅ and plastic shear strain rate γ̇n for this point can be expressed as:
σn + σtm B − 1 σci
)
v
⎧ εṅ = t sin θ ⎨ γṅ = − v cos θ t ⎩
(2)
where λ is a scalar proportionality factor. As shown in Fig. 7, θ is the inclination angle of the tangent for a random point on the slip surface. If f ′ (x ) is the first derivative of the slip surface equation f (x ) , the following equation is obtained:
(4)
where t is the thickness of the slip surface. Substituting Eqs. (2) and (3) into Eq. (4), the normal stress σn and shear stress τ for this point on the slip surface can be written as: 6
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Fig. 7. Failure mechanism of the rock mass around an existing tunnel induced by an adjacent excavation.
ξ [f (x ), f ′ (x ), x ] = PD − Pγ − Pq
1
⎧ σn = σci [ABf ′ (x )]1 − B − σtm ⎨ τ = Aσci [ABf ′ (x )]1 −BB ⎩
Substituting Eqs. (7), (8) and (10) into Eq.(11), the expression of ξ is found to be:
(5)
As pointed out by Fraldi and Guarracino [18], the rate of energy dissipation can be determined by multiplying the stress and strain rates. Consequently, the rate of energy dissipation for a random point on the slip surface is computed from:
{
1
ΔD = σn εṅ + τγṅ = σci [ABf ′ (x )]1 − B (1 − B −1) − σtm
}t
ξ = ∫0
∫0
L
D 1 + f ′ (x )2 dx =
L
∫0 {σci [ABf ′ (x )]
1 1−B
(1 − B −1) − σtm
L
f (x )dx −
∫0
L
v g (x )dx⎤ cos β ⎦t
b2 − x 2
v dx t
L b
L
v
}
v
(6)
L b
(12) whereψ is a functional and can be written as: 1
ψ [f (x ), f ′ (x ), x ] = σci [ABf ′ (x )]1 − B (1 − B −1) − σtm − γf (x ) cos β
(13)
As pointed out above, the load derived from the virtual work equation is either higher or equal to the actual limit load. Thus, to obtain the real upper bound solution, it is necessary to find the values of the variables that make the upper bound solution the least or the most critical numerically or analytically. However, it can be found that ξ is determined by the functional ψ . Thus, seeking the expression of the slip surface f (x ) that makes the objective function ψ reach the extremum is a challenge for this investigation. Fortunately, in simplest terms, ψ is an integral-type functionalwhose extremum can be derived from a variational approach. According to variational theory, the functional ψ has an extremum if and only if the necessary condition presented as follows is satisfied:
} vt dx
(8)
whereγ is the unit weight of the rock mass and β is the angle between the velocity vectors of failure block v and the direction of gravity. Moreover, g (x ) is the equation of the cross section for the shield tunnel and can be expressed as:
g (x ) =
v
L
(7)
∫0
− σtm − γf (x ) cos β
= ∫0 ψ [f (x ), f ′ (x ), x ] t dx − ∫0 γg (x ) cos β t dx + bqv arcsin
The half rate of work for the failure block produced by weight is given by:
Pγ = γ ⎡ ⎣
1
ci [ABf ′ (x )]1 − B
L
Because the failure mechanism is symmetric about the y-axis, half of the energy dissipation and external rate of work are calculated for mathematical simplicity. The half rate of energy dissipation along the slip surface can be obtained by integrating Eq. (6) over the interval [0, L], which can be expressed as:
PD =
{σ
L
− ∫0 γg (x ) cos β t dx + bqv arcsin
v 1 + f ′ (x )2
(11)
∂ψ d ⎡ ∂ψ ⎤ =0 − dx ⎢ ∂f (x ) ⎣ ∂f ′ (x ) ⎥ ⎦
(14)
Eq. (14) is also called the Euler equation of functional ψ . By substituting Eq. (13) into Eq. (14), a nonlinear second-order constantcoefficient differential equation is obtained:
(9)
where b is the radius of the shield tunnel. The rate of work for the supporting pressure produced by the segmented lining of the shield tunnel is:
− γ cos β + σci (AB ) 1 − B [f ′ (x )] 1 − B f ″ (x )
L Pq = bq arcsin ⎛ ⎞ v cos π ⎝b⎠
Solving Eq. (15), the analytical solution of this differential equation can be expressed as:
1
(10)
where q is the supporting pressure of the shield tunnel. The upper bound theorem states that a load that is no less than the actual collapse load can be obtained by equating the external rate of virtual work to the internal rate of energy dissipation. Thus, the construction of an objective function that describes the relationship between the external rate of work and the internal rate of energy dissipation can be considered the first step to derive the upper solution of the slip surface f (x ) . Based on this theory, an objective function that is composed of the external rate of work and the internal rate of energy dissipation is written as:
f (x ) =
2B − 1
1 −1 ⎛ γ ⎞ A B cos β ⎝ σci ⎠ ⎜
⎟
1−B B
1 =0 1−B
(15)
1 B ⎛x cos β − c0 ⎟⎞ − c1 γ⎠ ⎝
⎜
(16)
where c0 and c1 are integration constants. Substituting Eq. (16) into Eq. (13), the expression of ψ has the following form: 1
c B 1 1 1 B−1 ψ = − A− B γ B σci B ⎜⎛x cos β − 0 ⎟⎞ − σtm + γc1 cos β B γ⎠ ⎝ 7
(17)
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
Ground surface
Substituting Eq. (17) into Eq. (12), the expression of ξ can be written in the following form: B−1
(
)
1+B
( )
1 1 1 1 c c B ξ = − 1 + B cos β A− B γ B σci B ⎡ L cos β − γ0 − − γ0 ⎢ ⎣ +(γc1 cos β − σtm ) Lv
− γ cos β
(
L 2
b2 − L2 +
b2 2
arcsin
L b
1+B B ⎤
) v + bqv arcsin
⎥ ⎦
L b
v A=0.34 A=0.29 A=0.24 A=0.19
(18)
Diaphragm wall
As shown in Eq. (16), there are two unknown constants c0 and c1 that should be determined before we obtain the final expression of the slip surface f (x ) . Because the upper failure mechanism satisfies the stress and geometric boundary conditions, the values of c0 and c1 can be derived from the equations provided by these boundary conditions. Because the shear stress at the junction of the slip surface and y-axis is zero, a microelement at this position is selected as an analysis object. Based on the equilibrium differential equation of this microelement, the shear stress of the microelement at the mentioned position is given by:
τxy (x = 0, y = −h) =
Pit floor
Fig. 8. Slip surfaces of the rock mass induced by an adjacent excavation for various values of A. Ground surface
1 σn sin 2θ − τn cos 2θ = 0 2
(19)
where sin 2θ and cos 2θ can be derived from a trigonometric function transformation in conjunction with Eq. (3). Substituting σn and τn illustrated in Eq. (5) into Eq. (19), the following value of c0 is obtained:
B=0.50 B=0.60 B=0.70 B=0.90
(20)
c0 = 0
By substituting Eq. (20) into Eq. (16), the expression of the slip surface f (x ) can be simplified to the following form:
f (x ) =
1 −1 ⎛ γ ⎞ A B cos β ⎝ σci ⎠ ⎜
1−B B
⎟
Diaphragm wall
1
(x cos β ) B − c1
Pit floor
(21)
As shown in Fig. 7, the slip surface f (x ) and the equation of the cross section for the shield tunnel g (x ) intersect at point (L,0). Thus, the geometric boundary condition, which is concerned with equations f (x ) and g (x ) , can be written as:
f (L) = g (L)
Fig. 9. Slip surfaces of the rock mass induced by an adjacent excavation for various values of B.
(22)
Ground surface
However, the two constants c1 and L are still undetermined. Therefore, another equation should be found. By equating the external rate of work to the rate of internal energy dissipation, an equation that is used to calculate unknown constants c1 and L is obtained: B−1
−
1 1 1 1 A− B γ B σci B 1 + B cos β
− γ cos β
(
L 2
1+B ⎡ (L cos β ) B ⎤
⎣
b2 − L2 +
⎦
b2 2
arcsin
L b
)
+ (γc1 cos β − σtm ) L + bqv arcsin
L b
=0
tm
ci
tm
ci
tm
ci
tm
ci
Diaphragm wall
(23)
Combining Eqs. (22) and (23), the values of c1 and L are determined by using a numerical method. Substituting c1 and L into Eq. (21), the final upper bound solution of the slip surface f (x ) is obtained.
Pit floor
6. Calculation results and parameter analysis
Fig. 10. Slip surfaces of the rock mass induced by an adjacent excavation for various values of σtm.
Based on the analytical equation of the slip surface f (x ) , the shape of the failure block can be plotted to describe the range of the undesirable contact zone at the back of the existing tunnel. To study the influence of different parameters on the range of the undesirable contact zone, slip surfaces are illustrated in Figs. 8–13 for parameters σci =1.2 MPa, corresponding to b = 5.5 m, A = 0.19–0.34, σtm =σci /100~σci /40, B = 0.50–0.90, γ = 17.5–22.5kN/m3, q = 40–55 kPa, and β = 30–60°. Figs. 8–13 show that the slip surface extends approximately from the position of five o’clock to the position of one o’clock at the back of the segmented lining in the surrounding rock, and the shape of the slip surface approximates a parabola. Furthermore, the parameters A, B, σtm, γ, q and β have significant influences on the range of the slip surface. Fig. 8 shows the slip surface of the rock mass for different valuesof A
when the other parameters are fixed. As shown in this figure, the range of the slip surface clearly increases with increasing values of A. Similarly, Figs. 9–13 illustrate the influences of changing B, σtm, γ, q and β on the range of the slip surface when the other parameters are fixed. The range of the slip surface clearly increases with decreasing values of B, γ and q, whereas it increases with increasing values of σtm. However, in contrast to the other parameters, β is a parameter that affects not only the range of the slip surface but also the position of the slip surface. As mentioned above, β denotes the angle between the velocity vector of the failure block and the vertical direction. Because the velocity vector of the failure block is difficult to observe in actual engineering practice, β can be obtained from the velocity vectors 8
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
simulation under the same conditions, the HB failure criterion is invoked in the numerical simulation process. Moreover, the HB failure criterion used in the FLAC3D program is expressed in terms of the major and minor principal stresses as follows:
kN/m3 kN/m3 kN/m3 kN/m3
σ σ1 = σ3 + σci ⎛mb 3 + s ⎞ ⎝ σci ⎠ ⎜
Diaphragm wall
(24)
whereσ1 and σ3 are the major and minor principal stresses, respectively and mb, s, and a are material constants that are related to the geological strength index (GSI). Because the derivation of the analytical equation for the slip surface requires an assumption that the rock mass is isotropic, the surrounding rock of the numerical model is assumed to be a homogeneous material. As mentioned in 3.1, the mechanical parameters of the rock mass used in the numerical simulation are summarized in Table 1. However, the HB failure criterion used in the upper bound calculation is represented by normal and shear stresses, while the HB failure criterion invoked in the FLAC3D program is expressed in Eq. (24). Several parameters are different in these two forms of the HB criterion. Therefore, the parameters of the HB failure criterion represented by normal and shear stresses should be converted into the parameters represented in terms of the major and minor principal stresses to make the comparison under the same conditions. Hoek and Brown [17] proposed a method to achieve the equivalent conversion of the parameters in terms of these two forms, and Huang et al. [19,20] used this method to complete the comparison between the upper bound calculation and numerical simulation. Based on the method proposed by Hoek and Brown [17], the equivalent parameters are obtained for the two forms of the HB failure criterion, as illustrated in Table 3. Using the parameters presented in Table 3 and the value of β provided by numerical simulation, the upper bound solutions of the slip surface for stage III and stage IV are plotted in Fig. 14. It is clear that the slip surface derived from the upper bound calculation is similar to the boundary of the shear slip surface obtained from numerical simulation. To better compare the slip surface from the upper bound calculation and that of the numerical simulation, the upper bound solutions of the slip surface are superimposed on Fig. 15. The upper bound solutions of the slip surface for stage III and stage IV shown in this figure are clear approximations to the slip surface provided by numerical simulation. The good agreement between the upper bound solutions and the numerical simulation results for the slip surface of the rock mass indicates that the method proposed in this paper is valid. By comparing the upper bound calculation with the numerical simulation, it is found that the numerical simulation technique can be used to construct complex models that are closer to an actual engineering project. However, a complex model means that a significant number of elements are required in the modeling process and that the calculation process is time consuming. Thus, relative to the numerical simulation technique, the method presented in this paper is a more efficient approach.
Pit floor
Fig. 11. Slip surfaces of the rock mass induced by an adjacent excavation for various values of γ.
q q q q
a
⎟
kPa kPa kPa kPa Diaphragm wall
Pit floor
Fig. 12. Slip surfaces of the rock mass induced by an adjacent excavation for various values of q.
Diaphragm wall
Pit floor
Fig. 13. Slip surfaces of the rock mass induced by an adjacent excavation for various values of β.
provided by numerical simulation. Fig. 13 shows the slip surface of the surrounding rock for different values of β when the other parameters are fixed. It can be seen from this figure that the position of the slip surface moves downward with increasing values of β, and the range of the slip surface clearly increases with increasing values of β.
Table 3 Equivalent parameters for the two forms of the HB failure criterion.
7. Comparison with numerical simulation results To evaluate the validity of the upper bound solution of the slip surface presented in this paper, the analytical solution is compared with the results derived from the finite difference software FLAC3D. To make the comparison between the analytical calculation and the numerical 9
A B σci σtm
0.34 0.7 1.2 MPa 0.012 MPa
a mb s σci GSI
0.5 0.4981 0.00267 1.2 MPa 46
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
8. Conclusion Based on the deformation characteristics of the rock mass around an existing tunnel induced by an adjacent excavation, a new failure mechanism is established that can be used to determine the shape and range of the undesirable contact zone at the back of the existing tunnel lining. Using this failure mechanism, an analytical solution of the slip surface for the rock mass around the existing tunnel induced by the adjacent excavation is derived from the upper bound theorem in conjunction with a variational approach. The shape and range of the undesirable contact zone are plotted for different parameters on the basis of the analytical solution of the slip surface. The shape of the slip surface shows that an approximately parabolic failure block forms at the back of the segmented lining of the existing shield tunnel, and this failure block can be regarded as the potential undesirable contact zone between the tunnel lining and surrounding rock. Furthermore, by analyzing the variation in the shape of the slip
Diaphragm wall Stage
Pit floor
Stage
Fig. 14. Upper bound solutions of the slip surface for the rock mass induced by an adjacent excavation in different excavation stages.
Fig. 15. Comparison of the slip surfaces provided by the upper bound theorem and the numerical simulation for different stages: (a) stage III; (b) stage IV. 10
Computers and Geotechnics 117 (2020) 103236
F. Huang, et al.
surface with different parameters, it is found that the range of the slip surface increases with decreasing values of B, γ and q, whereas it increases with increasing values of σtm and β. To evaluate the validity of the upper bound solution of the slip surface, the analytical solution is compared with the results derived from the numerical simulation technique. Based on the actual project presented in this paper, a 3D numerical model is constructed, and the contours of the maximum shear strain rate of the rock mass are obtained; these contours can be used to describe the shear failure of the rock mass around the existing tunnel induced by the adjacent excavation. The slip surface derived from the upper bound calculation and the contours of the maximum shear strain rate provided by the numerical simulation are compared under the same conditions. The good agreement between the two methods indicates that the proposed method is effective for calculating the slip surface of the rock mass around the existing tunnel induced by the adjacent excavation.
[5] [6]
[7]
[8]
[9] [10]
[11]
[12]
Acknowledgments [13]
This study was supported by the National Natural Science Foundation of China (Grants 51878074 and 51678071) and InnovationDriven Project of Central South University (No. 2019CX011). Their financial supports are greatly appreciated.
[14] [15] [16]
References
[17] [18]
[1] Liang RZ, Xia TD, Huang MS, Lin CG. Simplified analytical method for evaluating the effects of adjacent excavation on shield tunnel considering the shearing effect. Comput Geotech 2017;81:167–87. [2] Liang RZ, Wu WB, Yu F, Jiang GS, Liu JW. Simplified method for evaluating shield tunnel deformation due to adjacent excavation. Tunnel Undergr Space Technol 2018;71:94–105. [3] Zheng G, Wei S. Numerical analyses of influence of overlying pit excavation on existing tunnels. J Cent South Univ 2008;15(S2):69–75. [4] Zheng G, Yang XY, Zhou HZ, Du YM, Jiayu Sun JY, Yu XX. A simplified prediction
[19] [20]
11
method for evaluating tunnel displacement induced by laterally adjacent excavations. Comput Geotech 2018;95:119–28. Huang X, Schweiger HF, Huang HW. Influence of deep excavations on nearby existing tunnels. Int J Geomech 2013;13(2):170–80. Chen RP, Meng FY, Li ZC, Ye YH, Ye JN. Investigation of response of metro tunnels due to adjacent large excavation and protective measures in soft soils. Tunnel Undergr Space Technol 2016;58:224–35. Shi JW, Ng CWW, Chen YH. Three-dimensional numerical parametric study of the influence of basement excavation on existing tunnel. Comput Geotech 2015;63:145–58. Shi JW, Zhang X, Chen YH, Chen L. Numerical parametric study of countermeasures to alleviate basement excavation effects on an existing tunnel. Tunnel Undergr Space Technol 2018;72:145–53. Zhang ZG, Huang MS. Geotechnical influence on existing subway tunnels induced by multiline tunneling in Shanghai soft soil. Comput Geotech 2014;56:121–32. Ng CWW, Shi JW, Hong Y. Three-dimensional centrifuge modelling of basement excavation effects on an existing tunnel in dry sand. Can Geotech J 2013;50(8):874–88. Ng CWW, Shi JW, Mašín D, Sun HS, Lei GH. Influence of sand density and retaining wall stiffness on three-dimensional responses of tunnel to basement excavation. Can Geotech J 2015;52(11):1811–29. Li P, Du SJ, Ma XF, Yin ZY, Shen SL. Centrifuge investigation into the effect of new shield tunneling on an existing underlying large-diameter tunnel. Tunnel Undergr Space Technol 2014;42:59–66. Jiang B, Chen LA, Yang JS, Wang SY, Ng CWW. Effects of twin-tunnel excavation on an existing horseshoe-shaped tunnel considering the influence of a settlement joint. Can Geotech J 2017;54(9):1346–55. Itasca Consulting Group Inc, 2006. FLAC3D Manual, third ed. (FLAC3D Version 5.00). Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975. Zhang DB, Jiang Y, Yang XL. Estimation of 3D active earth pressure under nonlinear strength condition. Geomech Eng 2019;17(6):515–25. Hoek E, Brown ET. Practical estimate the rock mass strength. Int J Rock Mech Min Sci 1997;34:1165–86. Fraldi M, Guarracino F. Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek-Brown failure criterion. Int J Rock Mech Min Sci 2009;46:665–73. Huang F, Zhao LH, Ling TH, Yang XL. Rock mass collapse mechanism of concealed karst cave beneath deep tunnel. Int J Rock Mech Min Sci 2017;91:133–8. Huang F, Ou RC, Li ZL, Yang XL, Ling TH. Limit analysis for the face stability of a shallow-shield tunnel based on a variational approach to the blow-out failure mode. Int J Geomech 2018;18(6):1–11.