Two dimensional analysis

Tunnel excavation is a three-dimensional problem (Swoboda, 1979; Gens, 1995). Fully three dimensional (3D) numerical analysis, however, often requires excessive computational re- sources (both storage and time). Therefore, tunnel excavation is often modelled two dimen- sionally (2D). Various methods have been proposed to take account of the stress and strain changes ahead of the tunnel face when adopting plane strain analyses to simulate tunnel construction.

The Gap method: This method proposed by Rowe et al. (1983) pre-describes the final tunnel lining position and size which is smaller than the initial size of the excavation boundary. Soil movement into the tunnel is allowed until the soil closes the gap between tunnel and initial excavation boundary position. The gap parameter is the difference between the diameters of the initially excavated boundary and final tunnel size.

The convergence-confinement method: This method which is also referred to as the λ-method was introduced by Panet & Guenot (1982). The parameter λ describes the proportion of unloading before the lining is installed. For 0 < λ < 1 the remaining radial stress on the lining is σr = (1 − λ)σr0 where σ0r is the initial stress in the radial direction.

The progressive softening method: This method was developed by Swoboda (1979) for modelling tunnel excavation using the New Austrian Tunnelling Method (NATM). It involves reducing the stiffness of the soil within the tunnel boundary before tunnel excavation is simulated, thus allowing soil to move towards the tunnel boundary. The volume loss control method: This method described by Addenbrooke et al. (1997)

prescribes a volume loss VL (defined in Equation 2.3). Tunnel excavation is simulated over a number of increments. After each increment VL is calculated. As soon the pre- scribed VL is reached the lining is placed. If the analysis only focuses on the ground displacement (and no results of the lining stresses and moments are required) the anal- ysis can be terminated after the required volume loss has been achieved.

The longitudinal-transverse method: Finno & Clough (1985) performed plane strain analyses of both longitudinal and transverse sections of the tunnel to account for stress changes and soil movements ahead of the tunnel face. In a transverse section stress changes were applied prior to tunnel excavation to obtain soil displacement similar to those obtained from the longitudinal analysis. Tunnel construction was then simulated adopting the gap method (see above).

For the last of the above methods it has to be noted that a plane strain analysis of the longitudinal tunnel basically models an infinite slot cut in the soil at some depth below the ground. Rowe & Lee (1992) compared longitudinal settlement profiles from such a plane strain approach with results from 3D analysis and concluded that the longitudinal plane strain approach significantly overestimates the ground displacements and the extent of the plastic zone compared with 3D results.

All of the above methods cause volume loss to develop during the excavation process. The gap method can be seen as a simulation of radial VL along a tunnel shield (with the gap parameter being the gap between cutting bead and the lining). However, as volume

loss also has a tunnel face component it is difficult to determine the gap parameter which accounts for the different sources of volume loss. In stiff clay, with its low permeability, tunnel construction can be considered undrained and the volume loss can be calculated from the surface settlement trough. As outlined above values of VL have been reported for a wide range of tunnelling projects. This measure is therefore suitable as a design parameter and can be directly adopted as done in the volume loss control method.

Using this method Addenbrooke et al. (1997) performed a suite of 2D tunnelling analyses investigating the effects of different pre-yield soil models on the results. These models were

1. linear elastic with Young’s modulus increasing with depth

2. non-linear elastic, based on the formulation by Jardine et al. (1986). Shear stiffness varies with deviatoric strain and mean effective stress while the bulk stiffness depends on volumetric strain and mean effective stress (referred to as J4 in the original publication). This model is outlined in more detail in Section 3.3.1, Page 90.

3. non-linear elastic with shear and bulk stiffness depending on deviatoric strain and mean effective stress level (referred to as L4). The model also accounts for loading reversals. All pre-yield models were combined with a Mohr-Coulomb yield surface.

Addenbrooke et al. (1997) modelled tunnel construction of the Jubilee Line Extension (JLE) beneath St. James’s Park, London. Field data from this greenfield site are reported by Standing et al. (1996). Two tunnels were constructed and both were included in the numerical model. However, only results from the first tunnel (the westbound tunnel, z0 = 30.5m) will be summarized here.

The soil profile consisted of London Clay beneath a top layer of Thames Gravel. The lateral earth pressure coefficient at rest adopted in the clay was K0 = 1.5. This value was within the upper bound of K0 values reported by Hight & Higgins (1995).

Figure 2.12 shows the surface settlement obtained from their analyses together with the field data. Their study demonstrated the necessity of including small strain stiffness into the pre-yield model as the predictions of the linear elastic model are inadequate. The responses of the two non-linear models are similar. However, both models predict a settlement trough which is too wide compared with the field data. As a consequence the maximum settlements are too small as the analyses were performed under volume loss control.

Figure 2.12: Surface settlement troughs ob- tained from different isotropic soil models (L4 and J4: Non-linear elastic, perfectly plastic, after Addenbrooke et al., 1997).

Figure 2.13: Surface settlement predictions obtained from different input parameters for a non-linear elastic perfectly plastic model (after Gunn, 1993).

The fact that FE analyses predict too wide settlement troughs in a high K0-regime has been reported by many authors. Gunn (1993) presented results from analyses with K₀= 1.0. He also applied a non-linear pre-yield model. Figure 2.13 compares the results with the Gaussian settlement trough, showing that the predicted settlement troughs are too wide. Similar conclusions were drawn by Grammatikopoulou et al. (2002) who adopted kinematic yield surface models in their analyses.

The role of K0in the prediction of tunnel construction was highlighted by Gens (1995). He stated that the importance of K0 is often neglected in the published literature. Addenbrooke (1996) compared FE settlement predictions for tunnel construction of the Jubilee Line for both K0= 1.5 and 0.5 with field measurements. He concluded that the low K0cases showed a deeper and narrower settlement trough and consequently were closer to the field data. Similar results were presented more recently by other authors for 2D and 3D analyses (Guedes & Santos Pereira, 2000; Doleˇzalov´a, 2002; Lee & Ng, 2002).

Addenbrooke (1996) performed axisymmetric analyses of a tunnel heading. The results showed a reduction in radial stresses while the hoop stresses around the tunnel boundary increased. Addenbrooke (1996) concluded that at tunnel springline this radial stress reduction causes the lateral stress ratio to reduce while it increases at the tunnel crown and invert. This

Figure 2.14: Layout of zone of reduced K0 (after Potts & Zdravkovi´c, 2001).

Figure 2.15: Surface settlement troughs ob- tained from different anisotropic soil models (AJ4i and AJ4ii: Non-linear elastic, perfectly plastic, after Addenbrooke et al., 1997). change in stress state can be represented in a plane strain analysis by a zone of reduced K0 around the tunnel. The layout of such a zone is shown in Figure 2.14. Apart from this zone the lateral earth pressure coefficient at rest was kept at a global value of K0 = 1.5.

Analyses which adopted such a zone showed an improved settlement profile compared with the global K0 = 1.5 cases. All analyses adopted a non-linear elasto-plastic model (number 2 in the above list). The results also showed that the lateral extent of the K₀-reduced zone did not influence the surface settlement predictions. Potts & Zdravkovi´c (2001), however, pointed out that justifying a K₀-reduced zone from the stress changes in front of the tunnel face is inconsistent with the use of a non-linear elastic model assuming no strain has taken place ahead of the tunnel face.

In order to improve the FE predicted settlement profiles it has been suggested by Lee & Rowe (1989) to include anisotropic soil models in the analyses. In 1996 Simpson et al. presented results from FE analysis of the Heathrow Express trial tunnel. By comparison of results from a linear elastic transversely anisotropic soil model with those of a non-linear isotropic model they showed that soil anisotropy gives better surface settlement predictions for overconsolidated clay. However, only limited details about the applied soil-model were given

and no information was provided about the initial stress profile adopted in their analyses. In their study, Addenbrooke et al. (1997) included soil anisotropy in the 2nd of the 3 pre- yield models listed above. The transversely anisotropic soil parameters were derived from the small strain stiffness formulation of the original isotropic model and by defining ratios of Ev’/Eh’ and Gvh/Ev’ where the index ‘v’ and ‘h’ refer to vertical and horizontal stiffness measures respectively2. Two analyses were performed. The first one adopted anisotropic ratios observed in field measurements reported by Burland & Kalra (1986) (Ev’/Eh’ = 0.625, Gvh/Ev’ = 0.44) while the second one reduced Gvh/Ev’ to 0.2, thus making the clay very soft in shear.

Figure 2.15 shows the results from the analyses. When compared with Figure 2.12 it can be seen that the first parameter set (referred to as AJ4i) in the anisotropic analyses does not improve the settlement profile significantly. The second parameter set (AJ4ii) with its low value of shear modulus Gvh yields a settlement trough which is closer to the field data. Addenbrooke et al. (1997), however, point out that the low value of G_vh adopted in this 2nd parameter set is not appropriate for London clay. This led to their conclusion that unrealistic soil stiffness is required to achieve better settlement predictions when modelling tunnel excavation in plane strain with K0>1.0.

It has been pointed out before that tunnel excavation is basically a 3D process. 3D analyses should therefore be able to predict the surface settlement more precisely. Due to advances in computational power over recent years 3D numerical analysis has become more widely applied. The next section summarizes how to simulate tunnel construction in 3D and presents results from various publications.