On the Application of Numerical Analysis in Tunnelling P. A. Vermeer, S. C. Möller & N. Ruse Institute of Geotechnical Engineering, Pfaffenwaldring 35, 70569 Stuttgart,Germany
[email protected]
At present tunnels tend to be analyzed 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 3DFEM analyses when facing complex geometries, e.g. tunnel joints or connections to underground stations, but not for straight-ahead tunneling. For judging possibilities, we distinguish between the 3 main focuses of tunnel analyses: tunnel heading stability, surface settlements and structural forces in linings. No doubt, tunnel heading stability requires a 3D FE-analysis, as otherwise one can not possibly capture the very significant arching in frictional ground. Here, it will be shown that safety factors can be computed relatively easily by the so-called SSR-FEM. In contrast to stability analyses, settlement analyses tend to require very large 3D meshes, but a so-called smart 3D analysis is proposed to overcome this difficulty. For the computation of structural forces in tunnel linings, a relatively simple 2D analysis is presented which matches the solution of a full 3D analysis reasonably well.
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INTRODUCTION
The uses of underground space are many, varying from simple storage caves and cellars to large complexes for storage of both liquids and solid materials and finally to many different kinds of underground traffic and transport routes. Tunnels built for highways and railroads may be shallow in urban areas or deep underneath major mountain ranges. Such deep tunnels may operate under considerable external groundwater pressure, whereas water supply and sewage tunnels also operate under internal pressure. Both the cut & cover method for very shallow excavations and deep underground excavations call upon the disciplines of soil and rock engineering. As for all construction activities empirical knowledge and engineering judgement play a dominant role, but with the advance of computers and geomechanics numerical analysis is of growing importance. Indeed, forces in support structures of complex three-dimensional underground openings (Fig. 1) need to be assessed numerically. Moreover numerical analyses are used for settlement predictions. On the other hand possibilities of finite element analyses for tunnel heading stability are rarely used. In this paper attention is focussed on stability as well as on settlements and structural forces.
Fig. 1. Complex three dimensional tunnel geometries
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TUNNEL HEADING STABILITY
One of the main problems when constructing a tunnel is to ensure the stability of the tunnel heading. In many European countries this stability is analysed on the basis of the sliding wedge mechanism in Fig. 2, being used for shield tunnelling as well as NATM-tunnelling. No doubt, this method gives an indication about the minimum support pressure needed in shield tunnelling and the factor of safety in NATM-tunnelling, but the model is crude and thus bound to be replaced by finite element analyses. Shield tunnelling in homogeneous soil can also be analysed by means of empirical formulas for the minimum failure pressure pf. Such an equation is of the form
p f = c ⋅ N c + γ ⋅ D ⋅ Nγ + q ⋅ N q
(1)
where c can be a drained as well as an undrained cohesion, γ is the unit soil weight, D the tunnel diameter and q a possible surface load. For undrained situations, the stability numbers Nc, Nγ and Nq depend on the relative cover C/D. For drained conditions, these stability numbers depend on the soil friction angle.
Fig. 2. Analytical wedge model for tunnel heading stability analysis
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ON THE STABILITY NUMBERS
The face pressure Eq. (1) was intensively studied by Vermeer et al. (2002). Despite its restriction to homogeneous soil conditions, the formula is of great practical value as it offers good insight into the face stability problem, at least in combination with proper expressions for Nc, Nγ and Nq. For undrained shield tunnelling, we proposed to use
æCö N c = 5.86 ç ÷ èDø
0.42
and
N γ = 0.5 + C D .
(2)
For drained conditions, we derived
Nc = cot ϕ´ ,
Nγ =
1 − 0.05 , 9 tan ϕ´
Nq = 0 .
(3)
At least for ϕ´ > 20° and a relative cover C/D > 1.5. The face-pressure Eq. (1) not only of great importance for getting insight into the problems of face stability, but also for validation of numerical analyses. Indeed, the non-linear elastoplastic finite element method is a powerful tool, but computer codes need to be validated. Moreover, users will have to train themselves and need benchmark problems with known solutions when using relatively new numerical methods such as the SSR-FEM.
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Fig. 3. Above: Typical strength-displacement curve for assessment of safety factor by SSR-FEM. Below: Shadings of incremental displacements at collapse, showing shell type sliding body.
SHEAR STRENGTH REDUCTION FEM (SSR-FEM)
The most versatile method for stability analysis is the SSR-FEM. This method has been applied to slope stability analysis in twodimensional as well as three-dimensional situations. Numerical comparisons have shown that SSR-FEM is a reliable and robust method for assessing the safety factor and corresponding failure mechanism. One of the main advantages is that both the safety factor and the mechanism emerge naturally from the analysis. In SSR-FEM, the factor of safety is defined as
F =
tan ϕ real c = real tan ϕ min cmin
(4)
where cmin and tan ϕmin are minimum values as needed for equilibrium. These values are obtained by reducing the real shear strength parameters stepwise down to failure in an elastoplastic FE-analysis. The method originates from Zienkiewicz et al (1973) and has been used most recently by Cai & Ugai (2003) for anchored slopes. We have adopted this method for threedimensional stability analyses of tunnel headings (Vermeer et al, 2002). Fig. 3 shows results from a particular tunnel stability problem. Considering homogeneous ground we have compared this method to the face-pressure Eq. (1) to find excellent agreement.
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Relevant data for a stability analysis is also indicated in Fig 4. Using these data we assessed factors of safety for several soil profiles; two resulting factors for the top-heading being shown in Fig. 4 and denoted by the symbol F. It should be noted that all SSR-FEM results were obtained by using a fully threedimensional code (Plaxis 3D Tunnel). However, numerical effort is limited as relatively small meshes can be used as indicated in Fig. 3. This is completely different from 3D settlement analyses, where relatively large meshes are required in order to guarantee good accuracy.
CASE STUDY WITH SSR-FEM
The SSR-FEM is applicable both in shield and NATM tunnelling. For NATM tunnelling, this is well demonstrated by the Rennsteig tunnel in Thuringia. With a length of 8 km it is the longest motorway tunnel in Germany. The excavation of this double-tube tunnel was done by sequential construction of a top heading followed by bench and invert, as indicated in Fig. 4.
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Fig. 4. Soil profile of Rennsteig road tunnel in Thuringia, Germany
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STEP-BY-STEP EXCAVATION
To get a better understanding of mesh requirements for a 3D settlement analysis we modelled a NATM tunnel. The sequential excavations of top-heading and invert were analysed with the elastoplastic Mohr-Coulomb (MC) model and results are shown
Fig. 5. Computed displacements for an NATM tunnel. The 1st figure shows displacements after excavation of top-heading in graded shadings from black to white. The 2nd figure shows the corresponding settlement trough. The 3rd figure shows shadings of displacements after excavation of the invert.
in Fig. 5. The way of modelling NATM tunnels is also known as the Step-By-Step Method where settlements are obtained by a stepwise removal of soil elements and installation of tunnel lining. From this analysis we obtained that meshes have to be larger than the steady-state settlement trough itself. Due to boundary conditions of vertical rollers initial disturbances on the right side of Fig. 5 affect settlements over a considerable excavation length. As a consequence one has to simulate a lot of excavation steps in order to arrive at a reliable steady state solution. Therefore such a three dimensional settlement analysis is very time consuming (Vermeer et al, 2002).
Fig. 6. Settlement trough and ground-response curve from classical 2D analysis with unloading factor β
Usually in practice unloading factors are either chosen according to the experience of the judging engineer or by considering two different conservative cases: e.g. β = 0.2 for settlements, as it will give relatively large settlements and β = 0.7 for structural forces in the lining, as it will give relatively large loads on the lining. No doubt this method is not very satisfactory from a scientific point of view.
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Because of the costs of a full 3D calculation, most tunnels are presently analysed by the use of 2D finite element computations. In practice the most widespread 2D method is the so-called Load Reduction Method (Panet & Guenot, 1982). As indicated in Fig. 6, a prescribed percentage β of initial stress σ0 is left inside the tunnel as a support pressure to take into account the missing three dimensional arching effect. Most settlements (∆s1) will take place in this phase. After installation of the lining this support pressure is removed and some additional settlements (∆s2) occur due to loading of the lining. Indeed, this way of modelling the transverse settlement trough is very efficient, as very low computer run time is required. Moreover, by comparing results we obtained that for appropriate unloading factors β the transverse settlement trough exactly corresponds to the 3D solution. Although the Load Reduction Method is straight forward to use and therefore very common in practice there is still one major shortcoming remaining about it. No secured method has yet been provided for determining proper β-values. Settlements and also loads on linings will be computed directly dependent on this input parameter.
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SMART 3D-ANALYSIS
2D-ANALYSIS To overcome shortcomings of 2D analyses as well as time consuming 3D analyses we developed a smart way of 3D analysis. Based on this new approach we use a full three dimensional model but instead of excavating many slices we only compute two excavations. From the last excavation increment we can extrapolate the entire three dimensional settlement trough. This method has been well described by Möller et al (2003). The final settlement value coming from this smart 3D calculation can now be used to find appropriate unloading factors for a 2D calculation. For a tunnel with sequential excavations we obtained β = 0.3 for top heading and β = 0.7 for invert (Bonnier et al, 2002) giving insight that unloading factors are also strongly dependent on the geometry of excavated tunnel sections.
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STEEPNESS OF SETTLEMENT TROUGH
Besides determining unloading factors for a two dimensional analysis the fast way of tunnel analysis is a useful tool to analyse the settlement trough using different soil models. To compare
Fig. 7. Steinhaldenfeld tunnel in Stuttgart
results of different soil models with measurements from site we modelled the Steinhaldenfeld subway tunnel in Stuttgart, Germany shown in Fig. 7. The ground shown in Fig. 8 consisted mainly of overconsolidated marl, which may be considered to be a hard soil as well as a soft rock. As the 1000m long NATM tunnel was constructed in an urban area settlements were carefully measured. For modelling the settlement trough in practice, the MohrCoulomb (MC) Model is a very common approach as models of higher order are mostly the domain of researchers. Fig. 9 shows results of MC and Hardening Soil (HS) Model (Brinkgreve et al, 2002) compared to measurements from site. It is observed that the MC-model predicts a relatively wide and shallow trough, which deviates significantly from the measured one. However HS does a lot better as it takes into account the different behaviours for primary loading and unloading-reloading of overconsolidated soils.
10 BENDING MOMENTS AND NORMAL FORCES
Fig. 8. Ground profile of Steinhaldenfeld tunnel
Fig. 9. Cross section of observed and computed settlement trough of Steinhaldenfeld tunnel
Fig. 10. Observed and computed longitudinal settlement trough of Steinhaldenfeld tunnel
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An important topic of tunnelling concerns the assessment of structural forces in linings. In order to predict realistic values of bending moments and normal forces a full 3D analysis is needed. To calculate realistic values for a circular tunnel, we divided a block of 100x40x28m into 8840 volume elements with a total of 26809 nodes (Fig. 11). For the parameters of the MC-model, we assumed a Young’s Modulus of E=42Mpa, a Poisson’s Ratio of ν=0.25, a cohesion of c=20kPa, a friction angle of ϕ=20o, a dilatancy angle of ψ=0 and K0=1-sinϕ for the coefficient of initial lateral earth pressure. The NATM tunnel with a diameter of 8m and a cover of 16m was modelled in a symmetric half with an unsupported excavation length of 2m. The shotcrete lining has a thickness of 30cm, a Young’s Modulus of 20MPa and a Poisson’s Ratio of ν=0. Fig. 12 presents resulting normal forces after 80m of ‘step-bystep installation’. Within a single lining segment of d=2m a sharp drop of tangent normal force is observed from 1400kN/m at the front to 200kN/m at the back. This is due to the fact that the unsupported tunnel heading is mostly arching on the front and not so much on the back of the last lining segment. The average tangent normal force, i.e. the solid line, appears to have a steadystate magnitude of 750kN per meter of tunnel length. The steady state is characterized by the horizontal part of the solid line The average normal force following directly after the tunnel heading is 600kN/m and still below the average value of 750kN/m. Further excavation steps will disturb the 3D arching effect around the tunnel heading and therefore lead to additional loading on the lining. In the steady state the arching is predominantly 2D.
Fig. 11. Shadings of vertical displacements after 80m of stepwise excavation. The steady-state settlement of s=4.5cm is reached after an excavation of 35m.
Similar to normal forces bending moments follow a zigzagging pattern as shown in Fig. 13. Considering average values, small bending moments of -3kN/m are found next to the tunnel heading. In the middle of the excavated tunnel bending moments reach an average steady state value of -22kNm/m. Towards the right boundary of the finite element mesh the lining is loaded more heavily giving a bending moment of –43kNm/m. However, this is a numerical effect that relates to smooth roller boundaries.
Fig. 12: The step-by-step installation of the tunnel leads to zigzagging normal forces in the ring direction of the lining (compression is positive)
Fig. 13: The step-by-step installation of the tunnel leads to realistic zigzagging bending moments in the ring direction of the lining
One has to note, that the two thinner lines above and below the average lines in Fig. 12 and Fig. 13 mark the maximums and minimums of results due to a coarser 3D mesh. Indeed, computed structural forces appear to depend significantly on the discretisation of lining elements. Fig. 14 and Fig. 15 show values of the two different mesh coarsenesses in more detail. The finer mesh was generated with three elements per lining segment and the coarser mesh was generated with only one element per segment. Each element thereby contains two Gaussian stress points. It is important to note, that the increase of the maximum values due to the finer mesh is about 40% for normal forces and 10% for bending moments. This large increase relates possibly to the higher flexibility of the finer mesh and obviously very much to the fact, that the Gaussian stress points from the finer mesh are positioned closer to the edges of the segment. This means, that the stress points are more close to the actual maximums and minimums in the front and the rear part of the segment.
There is no doubt, that the zigzagging bending moments and normal forces are realistic, but the cost of a ‘step-by-step’ simulation is difficult to justify for many practical tunnel applications. Therefore it is advisable to perform a 2D analysis in combination with a smart 3D Analysis as discussed in previous sections. After matching a 2D calculation to the 3D solution giving β=0.3 for the present case, we computed bending moments and normal forces as illustrated in Fig. 16. This figure can be used to compare the 2D data to the zigzagging 3D data of Fig. 12 and Fig. 13. It appears from Fig. 16a, that 2D normal forces are below the average value of the ones from the 3D ‘step-by-step installation’. On the other hand, Fig. 16a shows that 2D bending moments appear to match the average ones from a 3D analysis quite well. This average value would seem to be a highly realistic value, as the zigzagging in Fig. 12 and Fig. 13 is most probably excessively large. Shotcrete shows substantial creep and stress-relaxation so that one may expect a considerable damping of all oscillations around the average values.
Fig. 14: Computed normal forces for coarse and fine mesh
Fig. 15: Computed bending moments for coarse and fine mesh
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Fig. 16: 2D and 3D Bending moments and normal forces
10 CONCLUSIONS Examples of numerical applications for tunnel heading stability and settlements have been shown. For analysing tunnel heading stability, the shear strength reduction method appeared to be a powerful tool to obtain both factors of safety and mechanisms of failure. However, as numerical results need to be validated, it is recommended to compare factors of safety from shear strength recuction method as much as possible to findings from the face pressure Eq. (1) with stability numbers as given by Eps. (2) and (3). For computing settlements it has been shown, that transverse settlement troughs can be modelled well with the load reduction method, at least when using an appropriate constitutive soil model. However appropriate values for unloading factors (β) remain uncertain. Therefore reliable settlements may only be modelled three dimensional. A proposed smart 3D analysis can overcome shortcomings of time consuming conventional 3D analyses. Results of smart 3D analysis have been shown for different soil models. Mohr-Coulumb Model gives too wide and too shallow settlement troughs, but the Hardening Soil Model can improve a lot, at least for overconsolidated soils. Results for structural forces in tunnel linings from ‘step-by-step’ simulation have been shown. Both bending moments and normal forces appear to have a zigzagging pattern within one lining segment. It has been shown, that relatively fine FE meshes have to be used in order to compute exact maximums and minimums of the observed zigzagging. Even though these results are believed to be highly realistic the expense of such a full 3D analysis is difficult to justify at least for straight ahead tunnelling. Therefore for the computation of structural forces in tunnel linings it is advocated to use a 2D analysis. Appropriate unloading factors may come from the proposed “Smart 3D Analysis”.
REFERENCES Bonnier, P.G., Möller, S.C. & Vermeer, P.A. 2002. Bending Moments and Normal Forces in Tunnel linings. Paper presented at the 5th European Conference of Numerical Methods in Geotechnical Engineering, Paris, France. Brinkgreve, R.B.J. & Vermeer, P.A. 2002. Plaxis 3D Manual. Delft: A.A. Balkema. Cai, F. & Ugai, K. 2003. Reinforcing mechanism of anchors in slopes: a numerical comparison of results of LEM and FEM. Numerical and Analytical Methods in Geomechanics (27): 549 -564. Möller, S.C., Vermeer, P.A. & Bonnier, P.G. 2003. A fast 3D tunnel analysis. Paper presented at the Second MIT Conference on Computational Fluid and Solid Mechanics, Boston, USA. Panet, M. & Guenot, A. 1982. Analysis of convergence behind the face of tunnel. Institution of Mining and Metallurgy, Proc. Tunneling 82. Vermeer, P.A., Bonnier, P.G. & Möller, S.C. 2002. On A Smart Use of 3D-FEM in Tunneling. Paper presented at the 8th International Symposium on Numerical Models in Geomechanics, Rom, Italy. Vermeer, P.A., Ruse, N. & Marcher, T. 2002. Tunnel heading stability in drained ground. Felsbau: 8 – 18. Zienkiewicz, O.C., Humpheson, C. & Lewis, R.W. 1975. Assosciated and nonassociated visco-plasticity in soil mechanics. Géotechnique (25) : 671 - 689.
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