On a Smart Use of 3D-FEM in Tunnelling

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1 On a Smart Use of 3D-FEM in Tunnelling P. A. Vermeer Institute of Geotechnical Engineering, University of Stuttgart, Germany P. G. Bonnier Plaxis B.V., Delft, Netherlands S. C. Möller Institute of Geotechnical Engineering, University of Stuttgart, Germany ABSTRACT: Three-dimensional finite-element analyses are considered for the purpose of predicting settlements above NATM tunnels. First of all computational results from proper step-by-step tunnel installations are shown. This numerical procedure is characterized by successive removal of tunnel front elements in combination with a successive installation of shotcrete lining elements. This step-by-step simulation appears to be extremely computer time consuming, as a steady-state solution with a constant shape of the settlement trough is only reached after many steps of excavation. Depending on the starting conditions, initial settlements are either much to small or much to large. For complex NATM tunnels with staged construction, this leads to extremely long meshes and consequently to excessive computer run times. To avoid long lasting computations, a new fast way of settlement analysis is presented. Instead of the time consuming step-by-step installation, we propose a smart procedure that is referred to as the all-in-one installation. This procedure is validated for various different tunnels and shown to be computationally extremely fast. The new procedure makes 3D FEanalyses fully feasible for the prediction of surface settlements. 1 INTRODUCTION At present tunnels tend to be analysed on the basis of 2D finite element computations, because 3D analyses are considered to be extremely time consuming. As a result 3D analyses are presently the domain of researchers. Consulting engineers will only perform 3D-FEM analyses when facing complex geometries, e.g. tunnel joints or connections to underground stations, but not for straight-ahead tunnelling. Indeed, 3D analysis can be very cumbersome and we better retain existing 2D approaches. However, it is smart to supplement existing 2D approaches by some 3D calculations. For judging possibilities, we should distinguish between the 3 main focuses of tunnel analyses (Vermeer 2001), i.e. A B C tunnel heading stability surface settlements loads on lining but in this paper the focus will only be on surface settlements. For tunnel heading stability the reader is referred to Vermeer and Ruse (2001) and loads on linings are addressed by Vermeer et. al. (2002). First of all a 3D excavation of a circular NATM tunnel will be considered. Hereafter a 3D sequential excavation of a flat NATM tunnel profile will be analysed. 2D-analyses will be performed and computed shapes of settlement troughs are compared to those of 3D analysis. In order to match the depth of 2D settlement troughs to those from full 3D-analyses we will develop a simple combination of 2D and 3D-analysis, which does not need excessive computer run times. 2 3D EXCAVATION OF A CIRCULAR TUNNEL An important concern of tunnelling is the development of surface settlements. A 3D-analysis is needed for a proper prediction of the settlement trough. To investigate the development of the settlement trough from a circular tunnel we divided a block of 100 x 55 x 28 m into 4300 volume elements with a total of nodes (Figure 1). For the parameters of the Mohr-Coulomb (MC) model (Brinkgreve and Vermeer 1995), we took E=42 MPa, ν=0.25, c=20 kpa, ϕ=20 o, ψ=0 and K 0 =1-sinϕ. The NATM tunnel with a diameter of 8 m and a cover of 16m was modelled in a symmetric half with an unsupported excavation length of 2m. Each computational phase consists thus of d=2m of excavation, in which one slice of soil elements is switched off. Within the same phase

The shotcrete lining has a thickness of 30cm, a Young s Modulus of 20MPa and a Poisson s ratio of ν=0.

2 Figure 1. Shadings of vertical displacements after 60m of stepwise circular tunnel excavation Figure 2. Settlement trough after 60m of stepwise circular tunnel excavation a ring of lining elements is switched on to support the previous excavation. The shotcrete lining has a thickness of 30cm, a Young s Modulus of 20MPa and a Poisson s ratio of ν=0. This way of modelling a NATM tunnel will be referred to as step-by-step installation (Wittke 1984). Figure 1 and Figure 2 show the computed settlement trough after 30 excavation phases. The cross section from the steadystate of this trough compares well to the Gaussian shape (Peck 1969) in Figure 3 as measured in practice: 2 x S = S max exp, (1) 2 2i where S is the settlement, S max is the maximum settlement in the middle above the tunnel, x is the horizontal distance to the tunnel axis and i=k z 0 is the horizontal distance from the tunnel axis to the point of inflexion of the settlement trough with K as an trough width parameter and z 0 as the depth from surface to tunnel axis. The longitudinal shape is somewhat peculiar. In fact, a steady-state solution with a constant shape of the trough is only reached after about 35m of excavation. To investigate the reason of the peculiar longitudinal settlement distribution, we varied the initial Figure 3. Shape comparison of the transversal settlement trough between the analytical Gauss curve and the 3D circular tunnel calculation conditions at the very beginning of excavation. Please note that Figure 2 had been obtained by performing an unsupported excavation for d=2m of tunnel for the first computational phase. This analysis also leads to the lower curve in Figure 4. In another analysis the first excavation, but all further steps were unsupported excavations, as also considered in the analysis of Figure 1 and Figure 2. The tremendous influence of the first excavation phase is demonstrated in Figure 4. Depending on the starting conditions, initial settlements are either below or above the stationary solution. In both cases the

3 Figure 4. Settlements above tunnel axis after 80m of stepwise circular tunnel excavation. Upper curve for start with lining and lower curve for initial phase without lining Figure 5. Shadings of vertical displacements after 48m of stepwise top heading excavation disturbance extends over a considerable length of about 35m. Towards the left model boundaries settlements become to large because of the influence of the vertical fixities which are sliding contact bearings. The considerable initial disturbance implies that one has to simulate tunnel excavations over a large length with many excavation phases, in order to arrive at a reliable steady state solution. 3 3D SEQUENTIAL EXCAVATION To investigate the development of the settlement trough from a sequential excavation of top heading and bottom we modelled the Rennsteig tunnel, which is a development between Oberhof and Zella-Mehlis in Germany. For that reason we divided a block of 100x45x23m into 6272 volume elements with a total of nodes (Figure 5). For the parameters of the MC-model, we took E=60MPa, ν=0.25, c=20kpa, ϕ=30 o, ψ=0 and K 0 =1-sinϕ. The parameters for the lining were taken the same as for the circular tunnel. This NATM tunnel has a total diameter from roof to bottom of 9m and is covered by 11m of soil. The distribution of the longitudinal settlement trough after 24 excavation phases of the top heading in Figure 5 is similar to the shape of the Figure 6. Shape comparison of the transversal settlement trough between the analytical Gauss curve and the 3D top heading calculation settlement trough from the circular tunnel in Figure 1 except the settlements in the beginning: For the first computation phase we started with a supported excavation. The cross section from the steady-state of the longitudinal settlement trough perfectly compares to the Gaussian distribution (Figure 6). In this case the shape from the 3D analysis has become more steep. This effect obviously comes from the tunnel profile as the corners of the top heading section receive extremely high bending moments. Therefore the top heading profile leads to much more

4 Figure 7. Developing bottom settlement trough. Each curve shows the transversal settlement trough after a further excavation of 10m Figure 8. Shadings of vertical displacements after 48 m of stepwise bottom excavation deflection than the circular profile and settlements are cumulated in the middle of the settlement trough. Before analysing the settlements from the excavation of the bottom all displacements of the top heading were reset to zero. The following step-bystep installation was similar to the installation procedure of the top heading. After analysing the development of the longitudinal bottom settlement trough we found that its shape did not compare to the top heading settlement trough. As shown in Figure 7 initial disturbances lie below the steadystate level, although the first computation phase was started with a supported excavation. The settlements in front of the proceeding excavation are much smaller than the ones behind the tunnel face. This is realistic because the finished top heading in front does not allow any major deformations whereas the unsupported area of the bottom (see Figure 8) behind causes much larger deformations. As the tunnel face moves towards the left settlements in front become influenced by the model boundary and again get excessively too large. One has to note that the steady-state solution of the bottom settlement was only reached after an excavated tunnel length of 70m which is twice as much as needed for the top heading. Therefore Figure 9. Shape comparison of the transversal settlement trough between the analytical Gauss curve and the 3D bottom calculation computer runtimes have become even larger. The steady-state settlement of S=2mm from the bottom lies far below the calotte value (cf. Sellner 2000) of S=3.04cm. The transversal settlement trough from the steady-state solution of the bottom settlements is shown in Figure 9. For a shallow tunnel like this its shape does not compare to the analytical Gauss curve any more because the maximum settlement appears not to be over the tunnel axis.

5 Figure 10. Shape comparison of the transversal settlement trough from 2D and 3D circular tunnel calculation 2D tunnel lining (Figure 11). It has to be noted that it is important to know how close the 2D analysis gets to soil body failure. As shown in Figure 11 the load displacement curve becomes more and more flat towards the failure load so that the appropriate β-value needs to be determined more precisely. In order to get a proper solution in this case we needed to specify the β-value up to the fourth decimal place. The same method was used to set the β-value of the sequential excavation. The β-value we computed for the top heading was β = 0.441, for the bottom β = It is important to note that the computed β-value for the bottom is totally different from the β-value of the top heading. Therefore it has become clear that the β-value is highly dependent on the geometry of the excavated section. 5 THE FAST SETTLEMENT ANALYSIS Figure 11. 2D-tunnel analysis in which soil elements are gradually removed 4 THE 2D ANALYSIS The section which follows from the steady-state 3D settlement trough can easily be simulated in a 2D analysis. In fact Figure 10 shows that the shape of the 2D transversal settlement trough is identical with the one from the 3D calculation. To simulate the section in a 2D analysis the socalled β-method (cf. Panet and Guenot 1982) is used where the lining is installed after a prescribed amount of unloading. After installation of the lining initial stresses are reduced totally and the lining is loaded. To set the exact β-value the conclusion can be made that the section area A of the 2D settlement trough is equal to the volume per meter V/d which results from an excavation step due to the longitudinal 3D steady-state settlement trough: A = V. (2) d For the present case of a circular tunnel the application of Equation 2 led to a corresponding β-value of β = , i.e. 36,12% of the initial supporting pressure should be retained before installing the To avoid long lasting computations as are needed to get the steady-state settlement solution in a 3D calculation, we developed a fast way of settlement analysis. Instead of performing the time consuming method of step-by-step installation we perform only two phases. The first phase is used to install a more or less complete tunnel. To this end soil elements are switched off and lining elements are switched on up to the steady-state level of the longitudinal settlement trough. The second phase is used to model a single excavation with an unsupported length, i.e. d=2 m for the present example, and all previous displacements are reset to zero. This method will be referred to as all-in-once installation. One will now compute a more ore less circular settlement crater as indicated in Figure 12. Its volume V represents the so called volume loss for a single excavation. It can now be argued that this value should be equal to the volume loss of a single excavation due to the steady-state settlement trough of the step-by-step installation method. In order to predict the settlement of the 3D step-by-step installation of the circular tunnel in a fast 2D analysis we computed the volume loss per meter from the all-in-once installation. This value was then used to set the area of the 2D settlement trough as shown in chapter 4. The settlement we computed for the circular tunnel was S=3.92cm from the fast 2D analysis which compares well to the finding of S=3.81cm from the 3D step-by-step installation including a difference of only 2.9%. The same method was used to predict the steady-state settlement S=2.92cm of the top heading. From the 3D step-by-step installation we generated S=3.04cm which is a difference of only 3.9%.

6 Figure 12. Settlement crater from a single cutting For the bottom settlement we first installed the whole top heading in two computation phases similar to a 2D calculation by using the calculated β- value from the 2D analysis. Before calculating the settlements of the bottom all displacements from the top heading were reset to zero. Although the shape of the bottom transversal settlement trough does not fit the Gaussian distribution the fast 2D settlement analysis did lead to an acceptable result of S=1.83mm. From the 3D step-by-step installation we generated S=2.00mm which is a difference of 8.8%. 6 CONCLUSIONS It has been shown that a full 3D FEM analysis of tunnel excavation needs an extremely large number of excavation steps. Such full 3D analysis are not feasible in engineering practice. 3D analysis of both circular excavation and top heading excavation show a Gaussian settlement trough as observed in practice. However when analysing bottom excavation a basically different cross section is found. On comparing 2D and 3D analyses one observes identical shapes of the settlement trough. In order to arrive at the same settlement one has to select appropriate β-values. It has been shown that β-values can be obtained from a very simple 3D analysis that requires little computational effect. The 3D procedure for evaluating β-values is proposed as a smart use of 3D-FEM. Peck, R.B. (1969). Deep excavation and tunnelling in soft ground. Proc. 7 th International Conference on Soil Mechanics and Foundation Engineering, State of the Art Volume. pp , Mexico City. Sellner, P.J. (2000). Prediction of displacements in tunnelling. Institute for Rock Mechanics and Tunneling, No 9, Technical University of Graz, Graz Vermeer, P.A. (2001). On a smart use or 3D-FEM in tunnelling. Bulletin of the PLAXIS Users Assciation (NL), N0 11, Delft Vermeer, P.A. and Brinkgreve (1995). Finite Element Code for Soil and Rock Analyses. A.A. Balkema / Rotterdam / Brockfield Vermeer, P.A. and Ruse, N. (2001). Tunnel Face Stability (in German). Geotechnik, 24 No 3 pp Bonnier, P.G., Möller, S.C. and Vermeer, P.A. (2002). Bending moments and Normal Forces in Tunnel Linings. To be published in 5th Eur.. Conf. Num. Methods in Geot. Eng., Paris Wittke, W. (1984). Rock Mechanics. Springer-Verlag, Berlin REFERENCES Mair, R.J. and Taylor, R.M. (1997). Bored tunnelling in the urban environment. Proceedings of the 14th Int. Conf. on Soil Mech. and Found. Eng., Vol. 4 pp , Hamburg Panet, M. and Guenot, A. (1982). Analysis of convergence behind the face of tunnel. Institution of Mining and Metallurgy, Proc. Tunneling 82 pp , London

PHASED EXCAVATION OF A SHIELD TUNNEL

5 PHASED EXCAVATION OF A SHIELD TUNNEL The lining of a shield tunnel is often constructed using prefabricated concrete ring segments, which are bolted together within the tunnel boring machine to form