KSCE Journal of Civil Engineering (2015) 19(6):1626-1636 Copyright ⓒ2015 Korean Society of Civil Engineers DOI 10.1007/s12205-015-0406-0
Geotechnical Engineering
pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205
TECHNICAL NOTE
Numerical Modeling of the Dynamic Behaviour of Tunnel Lining in Shield Tunneling Saba Gharehdash* and Milad Barzegar** Received July 5, 2013/Revised March 18, 2014/Accepted September 2, 2014/Published Online February 19, 2015
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Abstract This paper studies the impact of metro train operation on the shield tunnel lining and its soft foundation. A complex elasto-plastic 3D dynamic finite difference model is used by fully considering the joints to show the dynamic response of the shield tunnel buried in soft soil under the vibrating load. The simulation result for joint was compared with the result when the joint was not considered. Through the formulation of computationally efficient numerical models, this paper examines the dynamic behaviour of these two particular types of lining structure. The differential Eq.s governing the vibration of a curved beam is discretized by the FLAC3D code. Numerical results demonstrate that an operating metro train induces significant dynamic response in the structure of the lining of the shield tunnel and its soft foundation. Of two horizontally symmetric the one near the joint is more severe in its dynamic response than that of the one far from the joint; the nearer the zone of the foundation soil to the lower half of the segment-ring, the more severe the dynamic response. The dynamic response influenced by joints is more severe than the response not influenced by joints, showing that the non-joint assumption is somewhat impractical. Keywords: shield tunneling, finite difference method, dynamic response, vibrating load, joint ··································································································································································································································
1. Introduction With the rapidly growing population in metropolitan areas, mass rapid transit systems built underground have emerged as an effective transportation tool for relieving the saturated ground traffic in different parts of the world. For more than one-hundred years researchers have been formulating models of train induced vibration. The early models consider only discrete parts of the system, for example, Winkler’s track model consists of a single infinite beam supported on an elastic foundation (Winkler, 1867). With the advent of modern computer technology, models considering multiple elements of the system have been developed. Forrest and Hunt (2006a, b) proposed a three-dimensional analytical model for studying the train-induced ground vibration from a deep underground railway tunnel of circular cross-section. The tunnel is assumed to be an infinitely long, thin cylindrical shell, whereas the surrounding soil is modeled by means of the wave Eq.s for an elastic continuum. Concerning the vibrations due to trains moving in underground tunnels embedded by multi layers of soil deposits, numerical methods, such as finite element method, that are capable of simulating the tunnel structure and variations in soil layers appears to be most favored by engineers. However, traditional finite elements suffer from the drawback that the geometric radiation effect of the half space cannot be
properly modeled. Thus, other schemes have to be incorporated to simulate such an effect. However, even with current modeling technology, simplifying assumptions are needed. These simplifications are often decided based on available computational power or engineering intuition, and in many cases the inaccuracy introduced by these simplifying assumptions remains unquantified. In the past decade, the trend in the literature is towards the development of numerical models, such as fiNite-element Models Or Coupled finite-Element, boundary-Element (FE-BE) models. Finite-element models of the soil require transmitting boundary conditions to correctly simulate wave propagation and prevent reflections at mesh boundaries. The approximate boundary conditions that may be used must be placed in the far field, resulting in significant computational requirements, especially at high frequencies where fine meshes are required (Sheng et al., 2005). By coupling Finite-Element and Boundary-Element modeling methods together (FE-BE), limitations in the use of either method can be overcome. Boundary elements are well suited to the analysis of infinite media. However, when boundary elements are applied to thin structures such as tunnels both faces of the structure must be discretised, and numerical problems result. Hence a coupled approach, using finite elements for the tunnel and boundary elements for the soil, is preferred. Degrande et al. (2006b) used a 3D periodic coupled finite element-boundary
*Graduated Student, Dept. of Mining and Metallurgy, Amir Kabir University of Technology, Tehran 158754413, Iran (Corresponding Author, E-mail:
[email protected]) **Graduated Student, Dept. of Mining and Metallurgy, Amir Kabir University of Technology, Tehran 158754413, Iran (E-mail:
[email protected]) − 1626 −
Numerical Modeling of the Dynamic Behaviour of Tunnel Lining in Shield Tunneling
element formulation to study the dynamic interaction between a tunnel and a layered soil due to a harmonic excitation on the tunnel invert. In addition, Gardien and Stuit (2003) presented a finite element based modular model for predicting the vibrations induced by underground railway traffic. Such a model consists of three sub-models: the static deflection model, the track model and the propagation model. Although two-dimensional models offer reasonable qualitative results and computation times 10002000 times shorter than three-dimen-sional models (Andersen et al., 2006), they are unsuitable for predicting underground-railway vibration as they can neither account for wave propagation along the track nor accurately simulate the radiation damping of the soil (Gupta et al., 2007, 2008, 2009, 2010). Three-dimensional modelling has hugely expensive computational requirements; hence the current trend is towards numerical methods that utilise the invariance in the tunnel’s longitudinal-axis direction (Kuo, 2010). The vibration effects created by operating a train in a variety of tunnel structures have been studied using a coupled periodic Finite Element-Boundary Element (FE-BE) model (Beckers, 2010). Dynamic Response Characteristics of Joint have been investigated not only on the dynamic stability evaluation of tunnel structures but also on the tunnel construction design of high speed railways (Haibing et al., 2010). The seismic behavior of shield circular tunnels and surrounding granular soils subjected to cyclic loading of a vibrating machine foundation have been studied by Hatambeigi et al. (2011). Wang (2011) used a fully coupled three-dimensional multi-body finite element model coupled with a perfect matching layer approach to study the ground vibrations created by a high speed maglev train running in a tunnel. Xingwei (2011) presented a study on dynamic response of saturated soft clay under the subway vibration loading. From the review given above, we realize that the previous studies on ground-borne vibrations associated with the underground railway traffic are voluminous. However, most of these studies were performed for a specific case, rather than on the fundamental effects of soil and tunnel support structures on ground vibrations. The number of parameters involved in describing the underground environment makes the formulation of a comprehensive model of vibration from underground railways a virtually impossible task. To fill such a gap, an extensive study will be conducted in this paper to identify the key parameters in lining design that may affect the ground vibrations caused by trains moving through the underground tunnels. For this reason, the modelling focuses on aspects of the vibration generation and propagation problems, for example wheel-rail interaction for vibration propagation in multi-layered soil. Particular attention is paid to the assumptions that are inherent in these models, and any work that has been done to quantify these assumptions. The purpose of this paper is express of new approach about importance of joints in lining of shield tunneling when loading is dynamic. So, we gave a new numerical model to shield tunneling based on a dynamic finite difference method which was programmed by C++ language, and have run on FLAC3D software. Vol. 19, No. 6 / September 2015
2. Finite Difference Simulation Model (Static Simulation) The finite difference method becomes a significant and general numerical tool to analyze of geotechnical works because of ability to considering of ground heterogeneity, non-linear soil behavior and soil-structure interaction. At this model the excavation diameter regarded 9 m with a cover height of 11 m. Cover-to-Depth ratio (C/D) was 1.2, where C and D are the cover depth and diameter of the tunnel. The model has dimensions of a 80 m length, 75 m width, and 30 m depth, consisting of 55000 3D brick elements, 786 shell and 64260 nodes. The depth of the tunnel at its centre was taken to be 10 m. The soil was considered to be formed by horizontally layered soil with the material properties listed in Table 1. The element sizes shown in Fig. 1 were generated to meet the requirements for simulating the highest frequency of loading considered (20 Hz), namely, the maximum element size (L) and minimum mesh extent (R) were selected such that L ≤ λs/6 and R = 0.5λs, where λs is the shear
Fig. 1. (a) 3-D finite Difference Mesh, (b) Representation of the Different Shield Tunneling Components in the Model
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Table 1. Physical and Mechanical Properties Various Soil Layers Along the Tunnel Layer
Soil type
I II III
SC-SM GC-GM CL-ML
Thickness m 10 10 10
Water Plasticity Earth pressure Undrained Young's Friction Poisson's content index coefficient at rest shear strength Modulus angle ratio ν w Ip Ko Cu E ϕcu (%) kPa MPa degree 44 17 0.43 21 8 0.3 21 30 15 0.36 -z 10 0.28 13 35 21 0.61 -z 10 0.34 20
wave length of soil corresponding to the highest frequency considered (Yang et al., 1996). In terms of the accuracy and efficiency of computation, modeling with such meshes appears to be most economic, which thus will be adopted throughout the numerical studies. Principal material has been assumed was Clay and Silty that illustrated in Table 1. The model is regarded sufficiently large to allow any possible failure mechanism in order to develop and avoid any influence from the model boundaries. All translational and rotational degrees of freedom were restrained at the bottom of the model. Translations in transverse direction and rotations were restrained at the vertical faces of the model. Except of boundaries in the model, the number of degree of freedoms in nodes are 3 translations and 3 rotations. The problem of varying shear strength in depth due to soft material was also analyzed in this model. It is assumed in the model based on plasticity, the dilation angle is near to the friction angle. The soil layers were regarded elastoplastic material in conformity with the Mohr-Coulomb failure criteria. The initial stresses were calculated through a first calculation of the soil deformation under specific gravity. An undrained analysis was carried out by using the undrained parameters of the soil as shown in Table 1. A horizontal ground surface was assumed with the water table at the ground level. It is assumed that the tunnel was driven by an earth pressure balance TBM in soft soil. Fig. 1 shows various components of the obtained numerical model. According to the TBM specifications, the length of the shield was assumed to be 10 m. The step-by-step excavation process was modeled by repeated rezoning of the element mesh at the cutting face and repeated insertion of elements in the tunnel lining and the tail void grout, respectively. Furthermore, the boundary conditions for the face support and grouting pressure were adjusted based on tunnel advance. Elements of the grout are directly connected to elements of the lining on the inner side and the soil on the outer side. In this paper, it was assumed that the grouting pressure linearly decreased along tunnel axis and finally reached to earth pressure (Zhang et al., 2005). According to some results of actual measurement in Zhang et al. (2005), the grouting pressure is between 0.1 MPa and 0.5 MPa. Grouting pressure was modeled by pore pressure boundary conditions on the grout element nodes at the shield tail with a variation of 15 kN/m2/m over the height. In Table 2, vertical soil pressures for the largest and shallowest depths are presented. In this paper, Pin denotes grouting pressure. In order to present the general circumstances, Pin1 was assumed
Dilation angle ψ degree 20 12 20
Cohesion Ccu kPa 10 11 25
Unit weight γbulk kN/m 3 19.6 19.8 18.2
Table 2. Boundary Pressures for Grout Injection (Zhang et al., 2005) Tunnel Depth (m) σν (kN/m2) Grout pressure (bar)
Top Bottom 10 19 ±149 ±318 ±3.5 ±5.2
Fig. 2. Grouting Pressure Acting on the Segments
to be ±3.5bar, and Pin2 as a pressure on the lowest point of the lining to be ±5.2bar, (Fig. 2). A non-uniform face support pressure was applied to consider of the pressure differences between the tunnel crown and the invert due to the muck in the excavation chamber. The conditioned muck pressure in the excavation chamber was considered as a distributed pressure which be increased with depth linearly. Its gradient was assumed to be equal to 13 kPa/m. The tunnel was excavated from the face of the mesh seen in Fig. 1(α). The segmentation of the tunnel lining was considered in the model. Two model-modes were adopted to establish the numerical model of the lining of segments. For the first mode, the segment lining was established with joints (called Break-Joint). The second mode was established as an unbroken tube-like body without joints (called Non-Joint). The outside and inside diameters of a segment-ring were regarded 8.9 m and 8.2 m. The width and thickness of the segment-rings were 1.5 m and 0.35 m. The joints were arranged asymmetrically. All components were loaded with their self-weight. Segments, grout, building and stratum were simulated with brick elements, while joints between two segments
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Table 3. Material Models and Parameters used in the Analysis Material Type
Constitutive model
Shield Grout Lining
Elastic Mohr-Coulomb Elastic
Thickness m 0.1 0.12 0.35
E Bulk Modulus MPa 210000 40 30000
were simulated with interface elements which its bending moment carrying capacity was determined by a set of rotational springs as depicted Fig. 1(b). The interaction between soil and tunnel lining was also taken into account through a set of normal sub-grade reaction springs. Tables 3, 4 shows the mechanical properties of encountered materials used in the model.
3. Numerical Analysis of Dynamic The finite difference method has proved to be an effective method to analyses the vibration effects created by a train running in a tunnel. The momentum equations are solved both in the spatial domain and the time domain in the finite difference approach. In the spatial domain, the non-linear dynamic finite difference equation is given by: int ext (1) MX·· + CX· + F = F Where M and C are the mass and damping matrices, respectively, Fint and Fext are the internal and external force vectors, respectively, X· and X·· are the nodal velocity and acceleration vectors, respectively. In the time domain, the solution of Eq. (1) is obtained using the implicit central difference scheme, in which the accelerations, velocities, and displacements in every time step are updated as follows (Hallquist, 2006): –1 ext int X·· ( t ) = M [ F ( t )−F ( t )−CX· ( t )] n
n
n
n – 1/2
--- ( ∆tn – 1 + ∆tn )X·· ( tn ) X· ( tn + 1/2 ) = X· ( tn – 1/2 ) + 1 2 · ) X ( t ) = X ( t ) + ∆t X ( t n+1
n
n
(2)
n + 1/2
Where ∆tn = tn + 1 −tn and ∆tn – 1 = tn −tn – 1 are time intervals. Compared with explicit schemes, the use of a lumped-mass matrix means that the implicit integration scheme requires less memory and is less time consuming. However, the central difference method is conditionally stable, which requires that the time step should be less than the critical time step, which can be represented as: ∆tc = e
min ( le /ce )
3.1 Wheel-rail Loads Comprehensive analysis requires realistic models of ground
ϕ (Friction angle) Degree 35 -
C (Cohesion) kPa 600 -
ρbulk (Density) kg/m3 7850 1500 2600
vibration generation and propagation (Kumaran, 2003). The purpose of this contribution is to present a composite method that the existing metro lines dynamic forces are calculated by numerical deterministic method. These calculated dynamic forces can be applied on the tunnel structures of finite difference models as load excitation. 3.2 Simulated Deterministic Loads of Metros The theory of simulated deterministic method (Changshi et al., 1990) is adopted to calculate the metros dynamic wheel-rail forces, which a conversion expression (Eq. (4)) are presented by using FFT algorithm to transform the measured acceleration of rail, and then dynamic expression system (Eq. (5)) based on a vehicle-track coupled model is established to simulate the dynamic wheel-rail interaction forces P(t). Z··o ( t ) =
N ---- – 1 2
∑ ( AN cosnωt + Bn sinnωt)
(4)
n=0
Where Z·· 0 is rail acceleration; N is sampled data number; An, Bn 2π , are FFT coefficient; ω is fundamental frequency, ω = 2π ------ = --------T N∆t T is recorded periodic time, ∆t is sample interval. m1z··1 + c ( z· 1 – z· 0 ) + k ( z 1 – z 0 ) = 0 p( t ) = ( m0 + m1 )g + m0 z··0 + m1 z··1
(5)
Where m0 is bogie and wheel unsprung mass; m1 is vehicle sprung mass; k and c are spring stiffness and damping coefficient; z0 and z1 are displacement of unsprung and sprung masses; z· 0 , z··0 , z· 1 and z··1 are velocity and acceleration corresponding to unsprung and sprung masses respectively; g is gravity acceleration; P(t) is wheel-rail interaction force. According to the experimental train-induced vibration data in Beijing, Fig. 2 shows a calculated result of dynamic wheel-rail force using this method, which use 2 s--m = 1700N. the following metro train parameters, , m 1= 0 2 6 m 6 s 10 N s11050N. ---- , k = 1.46 × ----------- , c = 0.16 × 10 N --. m m m The excitation load generated by moving trains on the unevenness rail is simulated by an excitation load function composed of static and three harmonic forces (Eq. (6)) (Fig. 3). F( t ) = p0 + p1 sinω 1 t + p2 sinω 2 t + p3 sinω 3 t
(3)
Where e indicates the elements of the model, le is the characteristic length of the element, and ce is the wave speed in that element.
Vol. 19, No. 6 / September 2015
ν (Poisson's ratio) 0.17 0.25 0.2
(6)
where P0 is static load; P1, P2 and P3 are vibration load accordance to the unevenness vector height, αi, pi = M0 α iω 2i where M0 is unsprung mass, ω i is radian frequency, ω i = 2π ---ν- , where υ is Li 3 illustrate train velocity and Li is unevenness wavelength. Fig. the load of trains, which parameters are, L1 = 10 m, αi = 3.5 mm; L2 = 2 m, α2 = 0.4 mm; L3=0.5 m, α3 = 0.08 mm p0 = 125 kN,
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Fig. 3. Relationship Curve of Exciting Force and Time (Zhang, 2001) 2
M 0 = 1510N. s---- , ν = 90 km/h. m
Based on the FDM (Finite Difference Method) analysis software of FLAC3D, a 3D FD model of the tunnel and the surrounding soil layers are established to characterize the spatial overlapping positions between the lining and joints. The Rayleigh damping [c] = α[M] + β[K] of the model takes the coefficients of α = 0.578, β = 0.004, which calculated from modal analysis frequencies (1.64 Hz and 2.10 Hz). The special visco-elastic artificial boundary conditions are used to eliminate the reflection of propagating waves. The surrounding soil is divided into three layers. Where α is the mass-proportional damping constant and β the stiffness-proportional damping constant. Given multiple degrees-of-freedom, the coefficients can be found from: βω α ξ k = --------- + ---------k (k = 1, 2,..., n) 2 2ω k
(7)
Where ξk is the critical damping ratio and ω k the angular frequency of the system. Based on the Eq. of free vibrations, two angular frequencies (ω i and ω j) can be obtained. From lab or site test, two critical damping ratios (ξi and ξj) can be obtained (Changshi et al., 1995). Therefore, according to Eq. (2), α and β can be obtained. If ω i = ω j = ω 0, then: ⎧ α = ξ 0ω 0 ⎨ ⎩ β = ξ 0/ω 0
components of the velocity at the boundary. For the sake of simplicity, the small-strain damping ratio D0 was considered constant with depth. In the dynamic analyses, the bottom of the mesh was assumed to be rigid, and the lateral sides were characterized by the viscous boundaries proposed by Lysmer and Kuhlmeyer (1969), parameters a = 1.0 and b = 0.25. In order to perform a dynamic analysis, first, a linear viscoelastic constitutive model for the soil was selected. Then, Plasticity was added which leading to a non-associated visco-elasto-plastic constitutive assumption characterized by a Mohr-Coulomb yield criterion. In this study, joints between liner pieces are also treated as curved beams (Eisenberger and Efraim, 2001). Obviously, the geometrical and constitutive parameters of joints are quite different from those of liner pieces. A curved beam segment having a cross-sectional area A and moment of inertia I about the area’s centroid and radius of curvature R. A curvilinear coordinate system (x, y, z) is defined such that the x coordinate is coincident with the centroidal curved axis and y and z coordinate coincide with the principal axes of the cross section. When the parameters in the governing differential Eq.s change suddenly from one region to another, then domain decomposition becomes a necessity. At the interface of two sub-domains, the geometrical compatibility conditions include the continuity of transverse and axial and displacement, and the rotatory angle of the cross-section, while the internal force conditions include the continuity of the axial force, shear force and moment. Suppose Nk and Nk+1 sample points are used for the discretization of the Kth and the (K+1)th curved beam, respectively. As a result, the curved beam K and K+1 are connected at the point (Nk, K) and (1, K+1), so the interface conditions for the displacements and the internal force are written as follows: (10)
vˆ NK K = vˆ 1K + 1
(11)
ϕˆ NK K = ϕˆ 1K + 1
(8)
NK
NK + 1
j=1
j=1
NK vˆ NK K - + ϕˆ NK K – ∑ c(N1K)jKK vˆ jK⎞ kcK GK A K ⎛ -------⎝ R ⎠ j=1 NK + 1 vˆ 1K + 1 ˆ 1) - + ϕ1K + 1 – ∑ c(1jN vˆ ⎞ = kcK + 1 GK + 1 AK + 1 ⎛ ----------K + 1 jK + 1⎠ ⎝ R j=1 NK
Where σ and τ are the viscous traction stresses in normal and shear directions, ρ is the mass density, Vp and Vs are the p- and swave velocities, and υn and υτ are the normal and shear
NK + 1
EK I K ∑ cNK jNK ϕˆ jK = EK + 1 IK + 1 ∑ cNK + 1NK + 1ϕˆ jK + 1 j=1
(9)
(12)
vˆ NK K⎞ vˆ 1K + 1⎞ ( 1) - = EK + 1 AK + 1 ⎛ ∑ c(N11)jNK + 1uˆ jK + 1 + ----------EK A K ⎛ ∑ cNK jKK uˆ jK + -------⎝ ⎠ ⎝ R R ⎠
where ω 0 is the basic frequency and ξ0 is the damping ratio corresponding to ω 0. In order to eliminate the reflection of scattered waves on the artificial boundary and to simulate the elastic recovery within reasonable limits on the far-field, the quiet-boundary scheme proposed by Lysmer and Kuhlemeyer (1969), involving dashpots attached independently to the boundary, was adopted. The dashpots may provide viscous traction stresses in both normal and shear directions: ⎧ σ = –ρ V pυ n ⎨ ⎩ τ = –ρ Vs υ τ
uˆ NK K = uˆ 1K + 1
(1)
(1)
(13)
(14)
(15)
j=1
where E, G are Young’s modulus and shear modulus for the curved beam material, kc is the shear correction factor, ρ is the density of the beam material, f, g, m are the distributed axial force, radial force and moment along the length of the curved beam, respectively, u is the axial displacement, v is the radial displacement,
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Table 4. Mechanical Property of Interface Normal stiffness coefficient kn (MPa/m) 3×105
Interface type Joint
shear stiffness coefficient ks (MPa/m) 1×105
Tensile strength T (MPa) 100
and ϕ is the rototary angle due to the bending moment. Where the subscripts K, K +1 represent the K-th and (K +1)-th curved beam, respectively. For the tunnel liner addressed in this paper, each segment of curved beam which includes all liner pieces and joints, the following Eq. is obtained: ⎛ Sbb Sbd ⎞ ⎧ Ub ⎫ 2 ⎛ M bb 0 ⎞ ⎧ Ub ⎫ ⎧ Fb ⎫ ⎜ ⎟⎨ ⎟⎨ ⎬−ω ⎜ ⎬=⎨ ⎬ ⎝ Sdb Sdd ⎠ ⎩ Ud ⎭ ⎝ 0 Mdd ⎠ ⎩ Ud ⎭ ⎩ Fd ⎭
(16)
where Ub is the displacements of end points (boundary points) for the curved beams, Ud is the displacements of the domain points for the curved beam segments; Fb is the distributed external force applied on the boundary points of curved beams, Fd is the external force imposed on the domain points of curved beams. Applying the continuity conditions on each interface between different subdomains, then the following continuity conditions for all liner pieces and joints are obtained: Ub = TUd
(17)
Cohesion C (kPa)
Friction ϕ (o)
20
10
Angular joint stiffness (kN m/rad) 300-12000
Subgrade modulus (kN/m2/m) 3750-56250
interface conditions between each sub-domain, we calculate the natural frequency of an open fixed-fixed supported circular curved beam with central opening angle equal to π/3, 2π/3, π, respectively. Besides the continuity condition (13)-(15) between each segment, the first point of the first segment and the last point of the last segment satisfy the following boundary conditions. In this paper, damping ratio for lining, soil and grout is 0.006, 0.03 and 0.05, respectively. Accordingly, the shear wave velocity of the soil is Cs = 77.93 m/s, and the compressional wave velocity is Cp = 145.79 m/s.
4. Results 4.1 Dynamic Response of Segment-rings In order to analyze the dynamic response in different positions of segment-rings, the grid points on certain cross sections of the lining segments were monitored during the overall process. These points were denoted by central angles with central point at
The freedoms of the end points are eliminated by substituting (17) into (16): ⎛ Sbb Sbd ⎞ ⎧ T ⎫ 2⎛ M 0 ⎞ ⎧ T ⎫ = ⎧ Fb ⎫ ⎜ ⎟ ⎨ ⎬Ud −ω ⎜ bb ⎟⎨ ⎬ ⎨ ⎬ ⎝ Sdb Sdd ⎠ ⎩ I ⎭ ⎝ 0 M dd ⎠ ⎩ I ⎭ ⎩ Fd ⎭
(18)
Multiplying (18) by matrix [TT I] , the condensed discrete Eq. of motion for the domain points are obtained as follows: *
2
*
*
( Sd – ω Md )Ud = F d
(19)
⎞⎧ ⎫ * T ⎛ Sd = [ T I ]⎜ Sbb Sbd ⎟ ⎨ T ⎬ ⎝ Sdb Sdd ⎠ ⎩ I ⎭
(20)
⎞⎧ ⎫ * T ⎛ M d = [ T I ] ⎜ M bb 0 ⎟ ⎨ T ⎬ ⎝ 0 Mdd ⎠ ⎩ I ⎭
(21)
⎫ * T ⎧ Fd = [T I]⎨ Fb ⎬ ⎩ Fd ⎭
(22)
It should be noted that since the constraints introduced by (13)(15) are internal constraints for the lining rather than the boundary condition, the solution of the (19) includes rigid body displacement. The response of the domain points subject to load Fb* can be calculated by (19) as follows: *
Ud = H ( ω )Fb *
(23) 2
* –1
H ( ω ) = ( Sd – ω Md )
(24)
where H(ω) is the frequency response matrix. The displacements at the boundary points of the curved beam can be evaluated by (17) and (19). To check the scheme for the implementation of the Vol. 19, No. 6 / September 2015
Fig. 4. Curve of Displacement Response of a Segment-ring: (a) Nonjoint, (b) Break-joint
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the tunnel center. The central angle of each point was accounted for by the counter rotated angle θ from the highest point of segment-rings to the monitored point. Fig. 4 presents the displacement response of segment-rings. It shows that the displacement response of Break-Joint is more severe than that of the Non-Joint. Their discrepancy is maximal at the bottom of the segment-ring (θ = 180o) and minimal at the top of the segmentring (θ = 0o). The curve of the Break-Joint shows that the displacement response of the segment-ring with joint reduces gradually from the bottom to the top. The response in the segment- ring sides is somewhat smooth in the range of about θ = 70o-110o and 250o-300o. But the curve at the right side is smoother than left side because the left side is a joint with severe response, while the right side is a segment with weak response. Figure 5 presents the velocity of response in segment-ring. It shows that the velocity of response in the Break-Joint is more severe than Non-Joint, and the velocity at the bottom of the segment-ring is obviously more severe than top of the segmentring. The peak of response velocity was occurred in the range of θ = 120o-240o. From Figs. 4 and 5 we can see that the dynamic response of the Break-Joint is evidently more severe than Non-
Joint. The severe dynamic response zone of the lining segments is largely distributed in the half below of the segment-ring. The curves of the dynamic response show a small asymmetric pattern because of the asymmetric arrangement of the BreakJoint joints. While the curves for the Non-Joint are almost symmetrical. Figure 6 presents the displacement time-path in two horizontally symmetrical points A and B in the zone on both sides (springline) of the segment lining. Point A is located in close to the joint segment on the right side, and Point B is just located at a joint on the left side. As is shown in Fig. 6, both points have the same pattern of the displacement time-path. But the displacement value for Point B is higher than Point A. It shows that the dynamic response at the joint location is higher than adjust of the joint due to reflection of the dynamic wave at the joint. 4.2 Dynamic Response of Tunnel Foundation The vibration of segments lining will make problems of the tunnel foundation which may affect the safety and stability of the tunnel. From Fig. 7, response in the nearest zone to the segment and the nearest zone to the bottom of the tunnel is more severe. At the bottom of the tunnel, the soil foundation has vibrated in some zones (about at θ = 100°-260o). This fact shows that these zones are most prone to deformation. Figure 7 presents the response rate of shear strain in the soil foundation of the tunnel around the segment-ring. The positions of monitored points are denoted by central angles (as mentioned above). The results of Break-Joint were compared with NonJoint. According to the result, the response rate of the shear strain of soil foundation in the Break-Joint is obviously more severe than Non-joint. The curve of the Break-Joint had a slight bump at the bottom. The response rate of soil foundation is small for the half upper of the tunnel and is high for the half below. (in the range of θ = 100o-260o). The maximal response rate appears near the joint between two normal segments at the bottom of the tunnel.
Fig. 5. Curve of Velocity Response of a Segment-ring
4.3 Time Effects on the Bending Moments and Hoop Forces Figure 8 illustrates the evolution of the hoop force and bending moment during the vibration. It is noted that in aforementioned
Fig. 6. Curve of Displacement Time-path
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Numerical Modeling of the Dynamic Behaviour of Tunnel Lining in Shield Tunneling
Fig. 8. Evolution of: (a) Hoop Force, (b) Bending Moment during the Vibration
studies, the tunnel liner is treated as a homogeneous closed ring or annularity. Obviously, this model is only valid for a monolithic liner structure. However, most tunnel liners are assembled by several liner pieces connected by joints. Since the effective stiffness of joints are quite different from that of the bulk liner, the internal force distribution along an assembled liner is quite different from that of a monolithic liner. Figure 8(a) and Fig. 8(b) show that for θ = 0o, only slight difference of the forces between the homogeneous liner and the piecewise liner is observed. However, for θ = 45o the difference between the homogeneous case and the piecewise case is significant, especially around the joints. The difference between the two cases is significant and the hoop force of the piecewise liner increases greatly around the joints, which makes the piecewise tunnel liner very dangerous. For the θ = 90o, the difference between the two cases is also pronounced: at the joints forces increase suddenly, while for Non-Joint liner, remain nearly constant. Internal forces of the liners for the two cases fluctuated significantly for the θ = 135o, for example, the hoop force with t = 2s is about 2.5 times of that with Break-Joint. Moreover, due to dynamic effects, the responses of the liner segments at the incident side and shelter side are not symmetrical with respect to the 900 and 2700 radial lines anymore. Fig. 8(a) also demonstrates that the difference of the hoop forces between the two cases becomes obvious especially around the θ = 180o. Vol. 19, No. 6 / September 2015
Moreover, the responses of the liner segments at the θ = 45o is larger than that of θ = 135o. The bending moment for θ = 0o, 90o, 180o is 0 in the case of Non-Joint liner, whereas in the case of Break-Joint liner fluctuated wildly between −100 and 100 kNm/ m. The difference of the internal forces and bending moment between the two cases becomes more obvious in regions near the joints, the piecewise liner is subject to much larger internal force and bending moment (Q, M) than the homogeneous liner. 4.4 Shear Distortions due to Vibration Load Shear distortions defined here are the relative horizontal displacements between tunnel crown and invert. It is apparent that the shear distortions are transient in the Break-Joint in compare of Non-Joint (Fig. 9), following similar cycles of ground displacements. The magnitude of shear distortions is highly dependent on the amplitude of shear waves and ground conditions. The peak shear distortion occurs when the amplitude of shear waves reaches to peak amount in duration of vibrating. Near the shear zone, a maximum shear strain approximately 0.25% and 0.8% for two modes non-joint and break-joint were estimated. In addition, the presence of concrete segments and backfill grout will serve as a cushion to make the transition less abrupt. 4.5 Induced Stresses due to Vibration Load in Lining Time histories of the calculated hoop stresses in the lining are
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Diagram of displacement velocity in lining elements over time is shown in Fig. 11. Fig. shows that in non-joint mode, at first, velocity of lining elements will be reached about 2.5 mm/s because of passing of pressure wave. After passing time, because weakening of reflection wave amplitude from earth surface, velocity will be damped quickly. In break-joint mode for lining, the velocity in the lining elements due to pressure wave passing is more than the non-joint mode. During the time, velocity wave contacts with the joint repeatedly and its reflection produces the vibration in lining so that over time the amount it will be reduced. Finally, the lining elements move with the same velocity of the joint. Fig. 9. Time Histories of Lining Shear Distortions at Break-joint and Non-joint Modes
shown in Fig. 10 for modes Break-joint and non-joint modes. Maximum hoop compressive stress is approximately 11 MPa at the break-joint. This amount for non-joint was 3.5 MPa. Fig. 10 also shows that stresses are transient and maximum stress occurs only once during the design vibration load. Stresses in the second-highest load cycle are approximately 7 MPa in Breakjoint and 0.3 MPa, non-joint).
Fig. 10. Time Histories of Hoop Stresses in the Lining at Braekjoint and Non-joint Modes
Fig. 11. Varying of Lining Elements Velocity in Two Modes Nonjoint and Break-joint Over Time
4.6 Results of Investigation on the Axial Stress Distribution in the Invert, the Crown and the Spring-line of the Tunnel Lining After applying the dynamic loading, the axial stress distribution in different parts of the lining on the invert, crown, spring-line were calculated. In Figs. 12 and 13, axial stresses in the lining, without and with joint, during six seconds have been shown. In both modes with reaching of pressure wave to lining elements
Fig. 12. Varying of the Axial Stress in Lining Elements in Non-joint Mode
Fig. 13. Varying of the Axial Stress in Lining Elements in Breakjoint Mode
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Numerical Modeling of the Dynamic Behaviour of Tunnel Lining in Shield Tunneling
and then movement of wave toward below, compressive stress produces in elements. Pressure wave moves along the lining with the velocity that is two times more than velocity in soil. Since the contact between lining and soil is limited, displacement and stress transferring from lining to soil is difficult, and it caused to significant increase in stress along the lining. With reaching of stress wave to earth surface, pressure wave becomes tension wave and spread to the lining. As a seen in case of lining without joint, after passage of time and damping of applied pressure wave to model, stress remains in lining. In lining with joint, distribution of stress in elements depend on stress wave. Generally, stress released by failure of joint. Complete failure occurred in joint, stress reached to zero in points of lining springline. However, in point of invert and crown because of lack of failure in joint and limited stress, few stress remains in lining. 4.7 Distribution of Dynamic Axial Strain Along the Lining Distribution of axial strain along the lining at the different times is shown in Figs. 14 and 15. In lining with non-joint mode after one second, negative elastic strain is produced due to
Fig. 14. Dynamic Axial Strain Distribution Along the Lining at Different Times in the Non-joint Mode
Fig. 15. Dynamic Axial Strain Distribution Along the Lining at Different Times in the Break-joint Mode Vol. 19, No. 6 / September 2015
passing of pressure wave. In following, after 2 seconds causing by reflection of pressure wave and transform to tension wave, with moving toward springline and invert, positive strain near to zero be produce. Strain increases during passing the time with move to crown. As a result, maximum strain is occurred approximately 0.03% near of the lining crown. In the lining with breakjoint mode, after one second, the same state with non-joint mode, the negative strain is created in the lining. With reflection of stress wave, strain on the crown of the lining could not be zero due to existence of the joint, and with movement toward the springline and invert of the lining, strain as a positive strain will increase to maximum amount. The end of three seconds the positive strain reaches to maximum amount. The continual displacement of the joint and the failure of the interface, strain gradually decrease, and after six seconds, practically the lining strain reaches zero.
5. Conclusions A new method for dynamic response of a piecewise lining subjected to vibration load has been proposed in the paper. The new method is based on the curved beam theory for the curved tunnel lining. Based on the derivation and numerical examples presented above, the follow conclusions can be drawn: Due to the stiffness difference between joints and tunnel pieces, the joints can enhance the hoop force and the bending moment of a tunnel liner significantly, which is important for a correct design of a tunnel liner. The wave velocity has a direct influence on the response of the tunnel liner. With increasing velocity, the displacements tunnel liner will increase. Moreover, the increase of displacement can increase the difference of internal forces between a homogeneous liner and a piecewise liner. The dynamic response of the lining of segments due to metrotrain vibrating load are similar for both Break-Joint and NonJoint in the range of the half-upper of the segment-ring, while in the range of the half below of the segment-ring, the dynamic response in the Break-Joint is more severe than the Non-Joint. The severely dynamic response zones of segment lining of the Break-Joint is distributed largely in the range of the half below of the segment-ring, specially, in the range of θ = 120o-240o. The dynamic response of lining segments in the Break-Joint shows an asymmetric pattern for arrangement of joints. In two horizontally symmetrical corresponding positions of lining of segments, the dynamic response of the next of joint is more severe than the beyond the joint. It is different from that of the Non-Joint which is entirely symmetric. The results also show that the Non-Joint assumption is somewhat unreliable. The maximum shear strain of the soil foundation takes place near the joint between two normal segments at the bottom. The foundation soil has evidently vibrated in some zones in the range of θ = 100o-260o at the bottom of the tunnel. The maximum response value occurs near the joint between two normal segments at the bottom of the tunnel.
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In zones such as soil, joint structures and tunnel linings, differential shear in tunnel linings due to vibration load can occur over a soil around the tunnel. For precast concrete segmentally lined tunnels, stiffer backfill grout between the segments and the soil can serve as a cushion to make transition with less abrupt, and is a viable approach to limit shear distortions. Joint displacement is a function of soil movement. Although, the lining without joint in comparison with the lining with joint is able to reduce soil movement, but the joint resistance is not enough to control of soil movement. And in this case, separation occurs between lining and grout that leads to release stress in the lining.
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