1 Introduction. Abstract

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1 Abstract This paper presents a three-dimensional numerical model for analysing via finite element method (FEM) the mechanized tunneling in urban areas. The numerical model is meant to represent the typical characteristics of the tunneling process by means of slurry shield tunnel boring machine (slurry shield TBM). Different soil constitutive models already implemented in PLAXIS 3D are employed for assessing the influence of the soil material model on the solution and therefore on the predictions made regarding the tunneling construction process. Further analysis of the numerical results is done in order to assess the model quality and a procedure for choosing the most adequate model is proposed. Keywords: mechanized tunneling, numerical simulation, parameter adaptation. 1 Introduction In July 2010 at the Ruhr-Universität Bochum, Germany, started a new collaborative research center SFB 837 for the research on interaction models in mechanized tunneling. The collaborative project consists of 14 subprojects and it is funded by the German Research Foundation (DFG). This paper is a part of the work within the subproject C2 Methods of system identification for the adaptation of numerical simulation models. Mechanized tunneling is a well established method which allows for tunnel advances in a wide range of soils and rocks. In the last 20 years the use of Slurry shield TBM and Earth pressure balanced shield TBM for excavating tunnels in urban areas became very common for many reasons like higher excavation speed under small overburden, operation in loose ground, and under a water table, the real-time construction of the lining, minimizing at the same time the surface settlements. For the estimation the magnitudes and the distribution of the strains during and 1

2 after the tunnel excavation the choice of a realistic soil constitutive model is very important. In this paper, three different models with increasing levels of complexity are used and their impact on the prediction of the displacements is analyzed. Additional variants of the model with tunnel overburden 1D and 5D are calculated to obtain its influence in the simulation. 2 Description of the Numerical Simulation All numerical simulations are performed with the finite element code PLAXIS 3D, version finite element package developed for three-dimensional analysis of deformation and stability in geotechnical engineering. We used three types of soil constitutive models, which are already implemented in PLAXIS, namely the well known linear elastic perfectly-plastic Mohr-Coulomb (MC) model and two advanced models: Hardening Soil (HS) model and Hardening Soil model with small-strain stiffness (HSsmall). The HS model allows for accounting the plastic collapse (isotropic hardening cap plasticity) as wall as plastic shearing due to deviatoric loading with shear/frictional hardening (deviatoric yielding). For the deviatoric yielding a non-associated and for the cap plasticity an associated plastic flow rule are prescribed. The HS-small model is an extension of the HS model to account for the increased stiffness of soils at small strains, i.e. shear strains lower than ( ). For detailed description of these advanced models see [1], [2] and [3]. The values of the input parameters for the three soil models are listed in Table 1. In order to compare the solutions employing MC model with constant stiffness and the HS type models the Young s modulus used in the MC model is taken to be equal to the one used in the HS model. This way it is believed that we are able to more realistically represent the unloading path during tunneling and it allows us to compare MC and HS models. In this study E (MC) = Eur ref (HS) = kn/m 2. Three dimensional finite element model has been set up after preliminary study of the boundary effects. Because the geometry, material properties, initial and excavation conditions are all symmetric about a vertical plane of symmetry that is parallel to the tunnel axis (i.e. the X-axis), only one-half of the model need to be analyzed (Fig. 1). The model is 60 m long (in the X-axis direction), 40 m wide (in the Y-axis direction) and 45 m deep (in the Z-axis direction). The chosen slurry shield TBM is 9 m long, simulated as a circular plate element, and an area with a total length of 45 m in the Z-direction has be investigated. The tunnel diameter is D=8.5 m and the overburden equals to 1D or 5D. The finite element discretization adopted for the simulation is chosen after studying the effects of the different finite element mesh size on the numerical results, and it is shown in Figure 1. The space occupied by the soil material is discretized using 10-node tetrahedral elements. The number of the elements is about elements with total up to nodes. The shield tunnel lining is often designed using prefabricated concrete ring segments, which are bolted together within the TBM to form the tunnel lining. During erection of the tunnel lining the TBM remains stationary. Once a tunnel lining ring 2

3 has been fully erected, excavation is resumed, until enough soil has been excavated to erect the next lining ring. The lining and the TBM are modeled as a linear elastic plate elements with properties listed in Table 1. The difference between the external diameter of the TBM shield and the newly installed tunnel lining causes a gap between the soil and the lining at the tail end of the TBM (Fig. 2). This gap is filled with grout while the TBM advances. One of the main function of the grout is to prevent large deformations of the surrounding soil and settlements at the ground surface. The gap is modeled as a 16 cm thick soil layer (method 2) where the grout pressure is simulated as an increased water pressure from about 145 kn/m 2 (705 kn/m 2 by overburden equals to 5D) at the tunnel crown to 230 kn/m 2 (790 kn/m 2 by overburden equals to 5D) at the tunnel invert. These values of the grouting pressure are chosen so that there will be a higher (or equal) to the sum of the vertical earth pressure and the ground water pressure (in this case no water table is defined in the simulation). Another way to simulates the grout pressure in PLAXIS is to apply an increasing volumetric strain to this 16 cm thick soil layer. The third way (method 1) to model the action of this grout pressure on the surrounding subsoil is to apply a non-uniformly distributed load directly to the soil elements by deactivating the plate elements in the same place (see Fig. 7). Thereupon it is assumed that the grout has settled enough not to cause additional deformations. The comparison between the calculated displacements at nodes on the ground surface, on the tunnel crown, on the tunnel invert and on the tunnel side wall by calculated following method 1 and method 2 shows a good agreement of the results and no significant differences in the displacements. The advantage to use method 2 is the reduced number of finite elements and the following reduction in the calculation time in about three times. The short calculation time is desirable for the inverse analysis which will be the next step within the subproject C2 and the reason is that the FE-calculation must be run many times during the optimization procedure. The disadvantage of method 1 is that there is no additionally applied pressure from the grouting acting on the tunnel lining, which can lead to a smaller calculated shear forces and bending moments in the lining. The support pressure at the face needed to prevent active failure at the tunnel face is simulated as a non-uniformly distributed pressure, 115 kn/m 2 (675 kn/m 2 by overburden equals to 5D) at the tunnel crown and 200 kn/m 2 (760 kn/m 2 by overburden equals to 5D) at the tunnel invert. The contact between the shield skin (plate elements in our model) with the surrounding ground (soil elements), and between the tunnel lining (plate elements in our model) and the ground is simulated via reducing with 40% the shear strength of the soil at this contact zone. To simulate this behaviour of the soil-structure interaction a special joint elements called in PLAXIS interfaces are applied to the plates on their side in contact with the soil. The excavation process is modeled by means of a quasistatic formulation of the corresponding mathematical model. It results in modeling the excavation process via step-by-step procedure and consequently the advancement of the shield tunneling is applied. In the first step, i.e. first calculation phase, the initial conditions are applied. 3

4 Figure 1: Model geometry and FE discretization. All of the next calculation phases (i.e. excavation stages) are meant to simulate an advance of 1.50 m. The steps which are modeled in a single excavation stage are the following: - excavation of the soil in front of tunnel (deactivation of the finite elements at that place); - applying a support pressure at the tunnel face; - activation of the TBM shield (the next 1.50 m); - applying a back-fill grouting pressure at the back of the TBM; - installing (activation) a new concrete lining ring with width of 1.50 m. In each calculation phase the input for the staged construction is identical, except for its location, which has been shifted in the X-axis direction by 1.50 m each phase. 4

5 Figure 2: Modeling of the back-fill grouting. 3 Evaluating the Influence of the Soil Constitutive Model on the Results Figure 3 demonstrates the results of the shield tunneling simulated with the MC model for the soil material and the effect of uplifting on the ground surface due to unloading during the tunneling process. It is observed that without differentiating between loading and unloading regarding the values of the stiffness modulus the application of the MC model results in an unrealistic lifting or in a less settlements (Fig. 4) of the ground surface associated with unloading due to the excavation of the tunnel. Further it can be observed that the use of HS and HS-small models allows more realistic modeling of the ground surface displacements and in this case there is no uplifting. The HS-small model which incorporates small strain behavior give smaller and more realistic displacements than the displacements calculated using the HS model. Figures 5 and 6 show the results from the benchmark with varying the Young s modulus in the MC model and the comparison with the results if the HS model is used. It was confirmed again that the MC model by using constant stiffness modulus is not able to predict a realistic settlements due to tunnel excavation. 4 Conclusions Three different constitutive models were used for the mechanized tunneling simulation. The well-known linear elastic-perfectly plastic MC model is not well suited for modeling tunnel excavation problems. Therefore advanced models like HS and HSsmall model are required in order to obtain a realistic prediction of the deformations during shield tunneling. In other words the realistic predictions for the deformations during shield tunneling may be done if the employed soil constitutive model takes into 5

6 Figure 3: Vertical displacements on the ground surface in the cross section located at the 18 -th m from the tunnel beginning, overburden equal to 1D. Figure 4: Vertical displacements on the ground surface in the cross section located at the 18 -th m from the tunnel beginning, overburden equal to 5D. 6

7 Figure 5: Vertical displacements on the ground surface in the cross section located at the 18 -th m from the tunnel beginning, overburden equal to 1D. Figure 6: Vertical displacements on the ground surface in the cross section located at the 18 -th m from the tunnel beginning, overburden equal to 5D. 7

8 Soil Parameters Constitutive Model Parameters MC HS HS-small ϕ [ ] ψ [ ] c [kn/m 2 ] E [kn/m 2 ] E ref [kn/m 2 ] E ref oed [kn/m 2 ] Eur ref [kn/m 2 ] G ref [kn/m 2 ] γ [-] p ref [kn/m 2 ] m [-] R f [-] ν [-] ν ur [-] γ unsat [kn/m 3 ] γ sat [kn/m 3 ] R inter [-] Tunnel Lining TBM Parameters Model: linear-elastic d [m] E [kn/m 2 ] γ [kn/m 3 ] ν [-] Table 1: Parameters for the soil models, for the lining and for the TBM. account at least: the non linearity of the stress-strain curve and the stress dependency of the soil stiffness moduli; the different stiffness during loading and unloading. Additionally if the soil model does not take into consideration the nonlinear soil behaviour at small strains this leads to a considerably too soft response, i.e. the calculated deformations are overestimated. Therefore from the three presented in this paper soil constitutive models for simulating the shield tunneling process the most adequate is the Hardening Soil model with small-strain stiffness. 8

9 Figure 7: Different methods to simulate the grouting pressure and a comparison of the calculated deformations (here the surface settlements). The model used is the HS model. Acknowledgments This work was done as part of the C2 project of the DFG Collaborative Research Center 837. M. Datcheva acknowledges the Bulgarian Science Fund under the grant DSAB02/6 for supporting her sabbatical leave to the Ruhr-Universität Bochum enabling her participation in this study. References [1] T. Schanz, P.A. Vermeer and P.G. Bonnier, The hardening soil model: Formulation and verification, Beyond 2000 in Computational Geotechnics - 10 Years of PLAXIS, [2] T. Benz, R. Schwab and P. Vermeer, Small-strain stiffness in geotechnical analyses, Bautechnik, 86: 16-27, doi: /bate [3] R.B.J. Brinkgreve, E. Engin and W.M. Swolfs, PLAXIS 3D Version 2010, Material Models Manual, ISBN-13: , printed in the Netherlands,

10 List of Symbols γ Unit wight [kn/m 3 ] γ unsat Unit wight of the soil with natural humidity [kn/m 3 ] γ sat Unit wight of the water saturated soil [kn/m 3 ] γ 0.7 Level of shear strain at which the secant shear modulus G s is reduced to about 70 % of G 0 (i.e. G s = 0.722G 0 ) [-] ν Poisson s ratio [-] ν ur Poisson s ratio for unloading-reloading [-] ϕ Peak angle of internal friction [ ] ψ Angle of dilatancy [ ] c Cohesion [kn/m 2 ] D Tunnel diameter [m] d Thickness of the concrete tunnel lining [m] E Drained Young s modulus of the soil [kn/m 2 ] E Young s modulus of the concrete lining [kn/m 2 ] E ref 50 Secant stiffness in standard drained triaxial test [kn/m 2 ] E ref oed Tangent stiffness for primary oedometer loading [kn/m 2 ] E ref ur Unloading / reloading stiffness [kn/m 2 ] G ref 0 Shear modulus at very small strains [kn/m 2 ] m Power for stress-level dependency of stiffness [-] p ref Reference stress for stiffnesses [kn/m 2 ] R inter Strength reduction factor for interfaces in PLAXIS [-] R f Failure ratio q f /q a [-] q f Ultimate deviatoric triaxial stress [kn/m 2 ] q a Asymptotic value of the shear strength of the soil [kn/m 2 ] 10