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ADVANCED NUMERICAL MODELLING IN TUNNEL DESIGN THE EXAMPLE OF A MAJOR PROJECT IN THE UK

Thomas, Alun Howell, Kandi, Eld, Czegldi, dm & Wolf-Gyri, Mnika

Mott MacDonald Magyarorszg Kft., H-1138 Budapest, Npfrd u. 22., Hungary

Keywords: Numerical Modelling; Sprayed Concrete Lining (SCL); calibration

INTRODUCTION On major projects more and more numerical models are used in design. Advanced numerical modelling techniques were developed by Mott MacDonald as part of the design process for an important project which is situated in a densely populated and sensitive urban environment. The 2D numerical modelling process was carried out by the program FLAC and the 3D was carried out by FLAC 3D which both employ an explicit finite difference formulation for the analysis of continua. This paper presents the state-of-the-art numerical modelling that has been used in the design of some of the major new stations of a project in the UK. Many of the stations include complex arrangements of large diameter sprayed concrete lined (SCL) tunnels in close proximity to each other and existing infrastructure. To obtain realistic results from a numerical model both the ground and the tunnel need to be modelled carefully. To ensure the predictions are reliable the numerical model can be calibrated against monitoring data from real tunnels.

THE GROUND This section describes the main features of sophisticated numerical models used in the design of a new metro line. A considerable amount of research has been done into the behaviour of stiff overconsolidated clays and the influences on numerical modelling (Thomas 2003, Van der Berg 1999). As well as the initial conditions (K0 and pore pressure distribution), plasticity, nonlinear elasticity and even anisotropy have been found be important. The constitutive model for the ground replicates the nonlinear relationship between ground stresses and elastic strains as well as its plastic behaviour. The properties of the ground strata vary with depth as do the in situ stresses and pore pressures (see an example in Figure 1). In general the data is taken directly from geological and geotechnical investigations. As the pore pressure is not hydrostatic, moreover at some points it changes quite suddenly and the value of the relationship between lateral horizontal pressure and the vertical pressure, K0, is also changing in the overconsolidated clays, the numerical modelling can be complex. The built-in programming language in FLAC is used to set the depth-varying profiles. Short and long-term analyses were carried out using the undrained and drained conditions during the lifetime of the tunnel. Short-term analyses are performed for simulating construction steps, while long-term analyses are carried out to examine the effects of consolidation. In a short-term analysis the clay strata were modelled as undrained materials, since negligible water flow will occur during the period of construction due to their low permeability. The fluid bulk modulus was set to 2GPa. The undrained shear strength values were used for cohesion along with friction angles of zero to produce a Tresca failure criterion. The granular strata e.g.: Made Ground, sands or gravels were modelled as drained materials, with a complete dissipation of excess pore pressures during

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the timescale of each advance (due to their relatively high permeability). To simulate this, the fluid bulk modulus was set to zero (i.e. the pore fluid will not support any part of the total stress changes) and the effective stress parameters c and were used to produce a Mohr-Coulomb failure criterion.

Figure 1 Depth-varying pore pressure and K0 profiles

Most strata have nonlinear small strain stiffness constitutive models. This was achieved by updating the tangent shear and bulk moduli according to the level of octahedral shear strain. A relationship by the Mott MacDonald team, based on the work of Jardine et al. (1986), and fitted to the laboratory test data, normalised to mean effective stress p.

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Figure 2 Non-linear stiffness strain relationship for Jardine and Brick models

Following results of the calibration study (see later) this relationship was modified according to the equation below, to avoid unrealistically low stiffness around tunnels.

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6.0'* pAGs ii = where A*i is a function of a. (1)

For convenience all moduli have been related to a selected reference modulus, Gshh0. This is defined by an empirical constant, A*hh0, that varies with depth.

THE SPRAYED CONCRETE LINED (SCL) TUNNELS The construction sequence has an important influence on the behaviour of SCL tunnels so all parts of the construction sequence were modelled to make the simulation more realistic. For example, if a large span cavern (e.g. 11m wide) is excavated as a full face excavation, the model may predict lower volume losses and lower bending moments with a more uniform distribution, than a model that replicates the multi-stage subdivision of the face that is truly needed to achieve the required control of ground movements.

Figure 3 An example of construction sequence steps and lining stiffnesses

Sprayed concrete is an unusual construction material as its stiffness varies with age and it can exhibit significant creep (Thomas 2003). Both phenomena were incorporated into the design. The sprayed concrete lining was modelled by one-dimensional linear elastic beam elements attached directly to the periphery of the excavated grid. There is a full moment connection at the nodal links between the adjacent beam elements. It is reasonable to assume that the joints are good quality by the use of steel fibre reinforcement. Tests on one recent high profile project confirmed that joints can be formed with the same strength as the main body of a sprayed concrete lining. To simulate the increase in stiffness of the sprayed concrete with time, the stiffness is different at each of the construction stages. The stiffness is set according to the age of the lining and includes an allowance for creep. The excavation and lining installation times are taken from the cycle time estimations based on construction programme. The properties of SCL lining have been derived from the J2 curve (EN 14487-1:2005 (E)) for a concrete aged up to 4 hours and Chang and Stille (1993) thereafter (see Figure 4). The concrete stiffness has been divided by a factor of 1.5 or 2.0 to account for creep, in accordance with Eurocode 2.

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Figure 4 Sprayed concrete stiffness against age

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The incremental application of load in a 2D model replicates the three dimensional effect seen in a real tunnel whereby the load comes onto the lining as the excavation face advances. These values are based on an assessment of the load development from 2D and 3D numerical models and calibration against monitoring data from real tunnels and it gives a conservative simulation of the actual development of load on SCL tunnels. The same process of incremental application of load has been applied to temporary load-bearing structural walls, using forces normal to the wall that are reduced at the same rate as the application of load from the ground during the relaxation process of the permanent walls. In addition, to better replicate the results found in three dimensional analyses a temporary supporting force at the base of top heading has also been applied in two dimensional analyses. This force makes a correction for the spreading effect of the elephants foot at the base of the concrete (normally the structural members in 2D models act as knife edge loads which can lead to particular problems with models using a small strain stiffness relationship). SCL construction of large diameter tunnels often requires temporary partition of the face during construction. This can result in a temporary ring (e.g. side drift see an example in Figure 5 and enlargement arrangements) that includes several larger radius walls or invert in the primary lining. The resulting geometry subsequently features very tight radius corners. This shape, when modelled in numerical modelling programs, such as FLAC, appears to attract large bending moments near the tight radius corner. Furthermore, in general the lining is 100 to 1000 times stiffer than the ground so the linings tend to attract load. It is believed that in reality, these bending moments do not occur, and local ground arching spreads the load. Representing this phenomenon by using plastic hinges in the models produces more realistic results. All joints in the real tunnel are designed as fully structural joints. The hinges are only an artifice for the 2D numerical modelling to account for 3D effects. The use of plastic hinges produces a prediction of worst case for ground movements, and since the moment capacity of the lining is limited to its actual capacity this provides a good check on the overall stability of the tunnel. The angle of rotation at each hinge in the model is also checked against allowable limits.

Figure 5 An SCL tunnel under construction

RESULTS Results of numerical calculations, whether they are extracted from final or intermediate stage are first checked to see if they look realistic. The first check comes right after setting up the initial conditions of the model. At this stage manual check is quite simple, because the stresses in the soil are undisturbed and the soil properties extracted from the model must be the same as the input data. Later on checking becomes more complicated and we can only rely on measured field data of earlier built similar tunnels (calibration) or results of earlier calculations in similar circumstances to decide whether our results are reasonable or not, because hand-calculations are far too difficult and time consuming.

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Checking plasticity is one way to make sure that the model works properly. Incidence of some plasticity in the top strata especially if it is a weak soil or surcharge is applied on it or around the tunnels mostly above is reasonable (see Figure 6). In case of tunnels close to each other, like at an underground station the soil between the tunnels often reaches its yield limit. The extent of yielding can be limited by choosing an appropriate excavation sequence with a short distance to closing the ring of the lining (Thomas 2003).

Figure 6 Plasticity and lining displacement

Bending moments and axial forces in the circumferential direction are extracted from the FLAC model, then they are increased by a factor of 1.35 and plotted on an interaction diagram with the concrete lining capacity concrete factored by 1.5, steel factored by 1.15 in accordance with the Eurocodes. This way we can ensure that the assumed lining is adequate to carry the loads calculated by the model.

Figure 7 Typical bending moments

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Figure 8 Typical axial forces

A disadvantage of FLAC is that it takes into consideration the prefix of bending moments and axial forces according to the local coordinate system, not to the global one, this way the plots of these results are not consistent (see Figure 8 and Figure 8). Pore pressure and stress plots (see an example in Figure 9) are also of interest, not just from the final stage but at all stages. The way they vary can indicate immediately if the model predicts realistic behaviour or not.

Figure 9 Pore pressure plot at the end of short-term analysis

3D models can help in such cases that can not be modelled in 2D, such as junctions. In these cases the spatial effects can not be simplified to any plane, so realistic results can only be obtained from a 3D model. From a 3D model the same results are extracted as from a 2D one, i.e. structural forces, displacements (see an example in Figure 10), ground stresses, pore pressure, etc. The safety of the lining design is checked using the bending moments and axial forces from the model.

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Figure 10 3D junction lining displacements

CALIBRATION Calibration is fundamental to all numerical modelling. This ensures that computer simulation will produce realistic results. The modelling procedure was calibrated against monitoring data for both the lining (see an example in Figure 11) and the ground from real tunnels in similar ground, as well as 3D numerical models. The real tunnels are a previous Mott MacDonald project, the Heathrow Express Terminal 4 station where extensive monitoring was installed during construction (Thomas 2003, van der Berg 1999). This calibration process led to a refinement of the relaxation process which is used to capture the 3D stress redistribution effects in the 2D plane strain numerical models, the selection of the optimal ground properties (K0 profile) and an assessment of the relative importance of consolidation.

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Figure 11 Lining displacement profiles from the calibration study

Relaxation Stress relaxation (see Figure 12) is a method to simulate 3D stress redistribution effect by replacing the soil stresses along the periphery of the excavation with a set of equivalent gridpoint forces in the

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2D model. In many of the numerical simulations a relaxation of three steps 100%-50%-0% is used, but this is not suitable for multistage excavations. In the calibration study, the relaxation factors for each excavation are varied until the volume loss, surface settlement and lining displacements match field data. In most of the models relaxation of five steps is needed as there is a Top Heading and Invert excavation sequence. The study showed that a relaxation of 100-80(or 60)-40-20-0% was the best (see in Figure 13).

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Figure 12 Relaxation factor vs volume loss for a Top Heading

Ground (K0) Ground properties were defined from laboratory tests or field data, but sometimes it is necessary to refine them because of the limitations of the numerical modelling of the ground behaviour. One example is K0. Experience has shown that if the high values of K0 estimated in the site investigation are used in a numerical model, then the horizontal movements of tunnel linings are overestimated. The K0 value can vary in overconsolidated clays, but in some cases this can be replaced by an alternative profile with a constant value for the whole layer. During the calibration study using constant K0=1.2 together with the right relaxation proved more realistic (Figure 13). The aim of this model was to examine the effect of building two tunnels at the same level symmetrically a few metres (horizontal distance) to the centre of an existing one. Two stages were used for the calibration. The first stage contained a short-term construction period of the platform tunnel, while the second stage is after construction of the concourse tunnel. After stage 1 both models matched perfectly but the effect of the high K0 values become more pronounced after the second tunnel was built. The model with constant K0=1.2 agrees best overall.

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Figure 13 Surface settlement results from the calibration study

The other variable ground property with depth is pore pressure. If the pore pressure profile is hydrostatic above the tunnel but in deeper sections of the ground is different, replacing it with hydrostatic pore pressure profile for the whole model can be acceptable, because the difference in pore pressure under the tunnel causes very small changes in the stresses and strains. This only applies of the change in pore pressure starts more than 1 diameter below the tunnel.

Consolidation Calculations on partial consolidation for a period of 6 months or less showed small changes in stresses and strains both in the SCL lining and in the soil (see Table 1). The conclusion is that

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consolidation can be neglected in case of construction phases with a time difference less than 6 months to simplify modelling.

Table 1 Comparing model results before and after partial consolidation for 6 months Stage Model 01 Model 02 Model 03

Before 130 130 130 Bending moment [kNm] After 130 130 123

Before 1190 1230 1137 Axial force [kN] After 1280 1418 1332 Before 0.87 0.78 0.72 Volume loss [%] After 0.95 0.86 0.79 Before 32.1 22.8 22.3 Surface settlement

[mm] After 33.1 23.2 22.7

CONCLUSION To meet the requirements of state-of-the-art numerical modelling there are many things to take into consideration for SCL tunnels in soft ground. First of all calibration of the model against monitoring data of already built, similar tunnels to get as realistic results as possible. During the calibration process the ground properties and construction sequences are refined and it is decided which are the most sensitive parameteres to be taken into consideration. The next step is to build up a model using the right properties and factors obtained from the calibration and the basis of design. Optimization of the mesh density is one of the best options to decrease running time of the model. In FLAC special theories i.e. Jardine model for clays, depth varying pore pressure and K0 profile can be easily defined using the built-in programming language. Following the above mentioned principles the numerical modelling process will be relatively easy to perform and the model produces realistic results. These models can then provide valuable information on the stability and safety of the proposed design. They can be used to explore changes in excavation sequence and design. They can also be used to check the impact on adjacent structures in the sensitive urban environment..

ACKNOWLEDGEMENT The authors gratefully acknowledge the contribution of many other colleagues in this work, notably Brian Lyons and Yu Sheng Hsu. .

REFERENCES Jardine, R. J., Potts, D. M., Fourie, A.B. & Burland, J.B. (1986), Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction, Geotechnique 36, No. 3, pp. 377-396

Chang, Y. & Stille, H. (1993), Influence of early-age properties of shotcrete on tunnel construction sequences, Shotcrete for Underground Support VI, pp. 110-117.

Thomas, A. H. (2003), Numerical modelling of sprayed concrete lined (SCL) tunnels, Department of Civil & Environmental Engineering, University of Southampton

Van der Berg, J. P. (1999), Measurement and prediction of ground movements around three NATM tunnels, School of Engineering in the Environment, University of Surrey