The 1995-1999 Madrid Metro Extension included 37km of new lines and 36 stations, all under heavily populated areas. Some 5,400 monitoring devices were installed to record ground movements during and after excavation, as well as measure the stresses and strains on the lining rings. A 3D numerical model simulating the excavation process was developed that takes into account the main processes affecting soil movements. Successful estimation of these movements before the passage of the shield was made through the use of this model (Medina 1999, 2000), allowing further study of the behaviour of the Madrid soils upon completion of the Project.

However, insufficient care was taken when modelling the lining behaviour. It was intended to behave as a continuum cylindrical shield. This is believed not to have important repercussions on soil movement predictions but significant discrepancies may occur if an accurate response of the rings is required. In this work an improved simulation of the lining rings behaviour has been carried out. Thrust forces from the hydraulic jacks, left out in the former work, have also been taken into account. A simpler 2D model has also been developed to compare the results with those from the 3D approach and decide on the necessity of this complex model to undertake the analysed problem. Field data from total pressure cells, gathered during excavation, will be employed to qualify and validate the developed lining models.

Characteristics of materials

According to present theories, the Madrid soils have historically experienced an overconsolidation process. The past maximum elevation of ground surface is believed to be 725m, today it varies between 600m-700m. The coefficient earth pressure at rest, K0, in Madrid soils give values in some cases higher than 1, and as high as 1.6. In this work, K0 values are obtained from the Alpan expression (1967): K0 = K0NC.OCRff (1). Where K0NC is the coefficient of earth pressure at rest for normally consolidated soils; OCR is the Over Consolidation Ratio; and ff is a parameter depending on the soil.

Two constitutive models have been analysed; the ‘linear elastic model’ and the ‘Mohr-Coulomb elasto-plastic model’. The main properties corresponding to the Madrid soils required by the models have been obtained from thousands of soil tests taken during the recent extension.

Design and description of the lining

During the last Metro extension a precast reinforced concrete segmental lining was used, the universal ring consists of six segments plus a keystone, which varies in position for each ring. By turning the segments in the ring, with respect to the preceding one, its geometrical design allows one to obtain the desired alignments in plan and elevation with one kind of ring. Using steel bolts the segments belonging to the same ring are restrained and each ring is linked to the next along the tunnel axis. Links between segments in the same ring cannot be in line with those of the preceding ring: ^ joints should be used instead of + joints (not allowed in the last extension) to reduce seepage through the lining.

During analysis of the mechanical behaviour of the segments the tunnel soil pressures were considered alongside other actions such as those due to manipulation, storage, stress concentration on the corners and near the bolts, jack thrusts etc.

Model description

Numerical simulation of EPB excavation used the FLAC3D finite difference code (Itasca 1997) to account for tunnel face deformations and the shield effect. The excavation sequence was reproduced, including simulated over-excavation (void between ground and shield), the gap (tailpiece void between soil and liner), soil chamber pressures (top, centre and bottom), EPB parameters and the grouting process through the pipes in the shield tail. The lining was modelled using a continuum cylindricall shell focusing on ground movements, not effects on lining. In the present work a segmental lining is considered, with beam elements simulating steel bolts. Interface elements are used to reproduce the contact between the lining segments. During shield advancement, the hydraulic jacks shoving loads (80MN) are taken by the segments in the longitudinal direction. The model used takes into account these longitudinal thrusts.

The EPB frustum conic shape was modelled and the large strain option adopted. For each excavation step the following sequence is applied:

  • Removal of face elements;

  • Interface generation on the new soil surfaces, to avoid penetration of soil into the EPB mesh and allow contact forces to be applied between them where and when they contact each other;

  • Shield displacement for a length equal to the removed soil cylinder length;

  • Grout pressure applied to tunnel surfaces. To reduce the volume loss, VS (ratio of the volume of the surface settlement trough per meter run to the excavated area) and the surface settlement, the soil and lining gap soil and lining is filled with grout. The authors believe a hydrostatic pressure distribution is believed to be the most realistic approach to model grout effect on the ground;

  • Generation of grouted elements. Grouting material is designed to harden 12 hours after applied.

  • Generation of lining ring elements. The linear elastic model echoes the behaviour of both the lining and the grouted ements;

  • Application of the hydraulic jack thrusts (80MN load) on the front face of the new lining ring.

  • Soil chamber pressure application on the tunnel face. Madrid EPB’s have normaly worked with 60 to 100kPa pressure at the upper cell in the soil chamber, and as high as 270 to 300kPa at the bottom cells;

  • Finally, the weight of the back up is applied on the corresponding lining rings.
  • Technical parameters of the EPB’s, lining rings (LR) and injection grout (IJ) are as follows:

  • EPB shield o.d. = 9.33m, length = 10.8m, thickness = 80mm, conicity = 0.01m/m, weight = 9750kN; back-up length = 115m, weight 5050kN.

  • LR: i.d . = 8.43m, thickness = 0.32m, length = 1.5m. Concrete: fck = 40MPa, elastic modulus: Ec = 28GPa. Steel: fyk = 510MPa.

  • IJ: Pressure at the shield tail, end of grouting pipes: 200-600kPa. Young’s modulus (hardened)= 2.57GPa. Poisson’s ratio (hardened) = 0.286.
  • Ground pressures acting on the lining are obtained from unbalanced forces at soil grid points after deleting the lining and grout elements.

    2D Model

    Close to the shield tail, the grout still hasn’t hardened and soil pressures can’t be sustained by the lining. Grout hardening happens 20-30m away from the tunnel face (depending on EPB speed) so a 2D model can be used when studying the lining behaviour. As the analysed ring is far from the EPB, the longitudinal hydraulic jack thrusts have not been considered in this model.

    For soil movement prediction, 3D analysis should be used, as final displacements depend on the whole excavation process and the tunnel heading. However, lining behaviour only depends on stress and strain states of the soil when the lining starts working. The same mesh used for the 3D model has been adopted, but with only one element in thickness. Construction consists of the following:

  • Excavation phase: removal of elements corresponding to excavation, interface generation on new soil surfaces created and EPB generation;

  • Non-hardened grout phase: grout pressure

    application on tunnel surfaces, with a hydrostatic distribution in depth;

  • Lining phase: the generation of both grouted and lining ring elements, with the corresponding bolts and interfaces between the segments.
  • Technical parameters of the EPB’s, lining rings and injection are the same used in the 3D analysis. Soil pressures acting on the lining are obtained as in the 3D model.


    Three sections on the Extension have been analysed (figure 1). Geotechnical profiles of these are represented in Figure 2. Results are presented in three sections: final ground pressures on the lining, far away from the shield tail, where plane strain conditions may be supposed; Ground pressures evolution, from the installation of the cells until the plane strain state is achieved; Analysis of the stress state of the lining at the end of excavation.

    Figures 3, 4 and 5 show predictions of final soil pressures acting on the continuum and segmented linings obtained from 2D and 3D models relating to Sections I, II and III. Numerical results are compared with data from total pressure cells.

    Results from the four simulations are very close, although ‘peaks’ on the pressure distributions corresponding to the segment links are observed when using the non-continuum 2D mode. No significant peaks were noticed in the pressure distribution from the non-continuum 3D model, probably due to the links existing between successive rings along the tunnel axis. These links provide additional stiffness to the radial joints (those between segments of the same ring). Few differences in time consumption are observed between continuum and segmented approaches. However computation times vary widely when comparing 2D (4min) and 3D (24hrs) simulations when using a standard Pentium II PC.

    Figure 4 shows a good agreement between predicted and measured pressures in Section II in spite of a significant discrepancy on the cell located at 191°. Results from all predictive models are similar as in the previous section. Peaks in the pressure distribution are only detected in the results from the segmented 2D model. The peaks coincide to the links between the segments in the same ring. Results for Section III are represented in figure 5. Data from three pressure cells have correctly been reproduced though an important error (27%), is seen in the cell at 157°. Again, the different approaches used for soil pressure prediction give very similar results, in shape and magnitude, as in the rest of the studied sections. The main difference between them is the ‘corners’ or ‘peaks’ in the pressure distribution corresponding to the segmented 2D model.

    From the 3D model soil pressure evolution can be obtained where the pressure cells are placed. Numerical predictions and monitoring data relating to Sections I, II and III are seen in figures 6, 7 and 8, respectively. Although available field data extends more than 100 days, it stabilises long before that. To obtain results from the 3D model within a reasonable time, simulations cannot represent more than 12-14 day periods. Thus comparisons between predicted pressure evolutions and monitoring data have been made a few days after the shield passage.

    Figure 6 shows soil pressure evolution corresponding to five pressure cells in Section I. An acceptable estimation of real data has been obtained by numerical simulation. The greatest discrepancy occurs in the cell at 170° seen in figure 3. Results in Section II are presented in figure 7. Data from five pressure cells have again been analysed. Predictions from the 3D model disagree with field data within the represented period of time. However, final pressures are acceptably reproduced. Finally, the soil pressure evolutions in Section III are reproduced in figure 8. Four total pressure cells have been studied. A reasonable prediction has been made by the numerical model.

    Stress state of the lining

    The state of stresses throughout the lining due to soil loads and hydraulic jack pressures should be analysed to check whether they can withstand the segments. The distribution of minimum and maximum principal stresses on the lining corresponding to Section I were obtained from the continuum 3D model. Very similar results were achieved from a segmented 3D model, probably due to the influence of grout on the lining’s loads. The grout filling the gap between soil and lining is stiff enough to smooth the links effects between the segments on the distribution of soil pressures. The bolts between segments and consecutive rings should also play a role in this behaviour, making the segmented model work almost as a continuum one.

    To avoid the effect of the boundary condition (y = 0) on results, the part of the mesh between y = 0 and y = 6 was deleted from the 3D continuum model.

    The minimum principal stresses (compression) acting on the lining are 10,000kPa, far below the compression strength of the concrete: fck = 40,000kPa. Interestingly, these minimum stresses are very similar to the pressures applied by the hydraulic jacks, in the order of 9100kPa. Thus, thrust forces become the most important action on the rings. Stresses due to bending from ground loads are not important if compared with those caused by the former. All mechanical analysis of lining rejecting the jack thrusts are far from being realistic. The maximum principal stresses (tensile) continuum 3D model for Section I shows that most of the lining is under compression stresses and the maximum tensile stresses are over 50kPa, very far from the tensile strength of the concrete, that may be obtained from the following expression:

    fct, k=0.21³√⌊²=2.46N/mm²=2460kPa

    Where fck is the compression strength of the concrete, expressed in N/mm².


    Results from the proposed numerical prediction models have been compared with soil pressures acting on the lining rings. Both final values and pressure evolutions in time have been analysed. Agreement is generally good although some pressure cells show discrepancies.

    The grout hardening happens away from the tunnel face so that the lining starts working when the plane strain conditions have almost been achieved. Thus similar ground pressure distributions have been obtained when using the 2D or 3D models. However, large computation time differences have been noticed. Nor has a relevant difference been noticed between the predictions from the continuum and segmental lining simulation approaches, and the computation times are quite similar (in both 2D and 3D cases). Some ‘peaks’ on the pressure distributions are observed when using the non-continuum 2D model (segmented lining). These peaks coincide in position to the links between the segments. No significant peaks are detected in the pressure distributions from the non-continuum 3D model, probably due to the links existing between successive rings along the tunnel axis. These links provide an additional stiffness to the radial joints.

    The similarity between pressure distributions from all the proposed models should also be caused by the strong influence of the grout on the loads that reach the lining. The grout filling the gap between the soil and the lining is stiff enough to smooth the effect of the links between the segments on the distribution of soil pressures. The presence of bolts between both segments and consecutive rings should also play an important role in this behaviour, making the segmented model work almost as a continuum one.

    Study of the development of the ground pressures on the lining needs a 3D approach. If only final pressures acting on the ring were required a two-dimensional model would have given a reliable response.

    Finally, the state of stresses throughout the tunnel lining due to the soil loads and the hydraulic jack pressures have been analysed to check whether they can be withstood by the concrete of the segments. The minimum principal stresses are very similar to the pressures applied by the hydraulic jacks, thus, thrust forces become the most important action on the rings. Stresses due to bending from the ground loads are not important if compared with those caused by the former.

    All mechanical analyses of linings rejecting the jack thrusts are far from realistic and accurate. Compression and tensile stresses within the lining, under these load conditions, have been shown to be far from the strength characteristics of the concrete forming the lining.

    Related Files
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