Numerical investigation of the effects of tunnelling on existing tunnels Open Access

As more tunnels are constructed in urban areas to facilitate infrastructure systems, the subsurface environment becomes more congested. The situation where a new tunnel (NT) is constructed in the vicinity of an existing tunnel (ET) is therefore becoming increasingly frequent and the consequent interaction has to be controlled – the ETs need to remain operative during and after completion of the NT and, consequently, limits on allowable deformations should be specified (Kimmance et al., 1996).

Although field measurements of interactions between tunnels are generally rare, when available they offer a valuable database for future design and an important reference against which numerical models can be validated (Cooper et al., 2002). The available field measurements of such problems generally identify three patterns of tunnel displacement and deformation: settlement, rotation and distortion. Table 1 summarises tunnel–tunnel interactions, published from various case studies in London, in which field monitoring data were provided. In the case study reported by Cooper et al. (2002), the following observations were made based on the intensive monitoring performed.

• (a)

  The settlements of the existing Piccadilly line increased with the successive construction of the three new Heathrow Express tunnels and, in the long term, the asymmetry of the ground surface settlement troughs was attributed to the sequence of construction of the NTs.

• (b)

  The cross-section of the existing Piccadilly line tunnel, just above each of the NTs, rotated towards the excavation face as it approached and then, with further advancement of the face below and beyond it, it rotated in the opposite sense. The cross-section ultimately experienced a residual rotation that was asymmetrical around the NTs’ alignments, a fact attributed to the relative skew angle between the NT and the ETs.

• (c)

  The cross-section of the existing Piccadilly line tunnel was distorted to an ‘egg-shaped’ form with its elongated axis following the face of the NT excavation.

Numerical analyses have been used to obtain insight into tunnel–tunnel interaction problems. The first tunnel–tunnel interaction problems that were investigated numerically considered tunnels with parallel axes. This general geometry can be investigated with plane-strain 2D analysis. Addenbrooke (1996) looked into different relative positions of tunnels with parallel axes in London Clay, focusing on the influence of spacing, rest period between constructions and the construction sequence. For a side-by-side tunnel geometry, there was found to be an interaction between the two tunnels when they were spaced less than seven tunnel diameters apart. The first excavated tunnel ‘squatted’ due to the construction of the second and the lining of the second tunnel was generally stressed less than that of the first tunnel. An increase in rest time (i.e. the time between construction of each tunnel) masked the effect of the second tunnel on the first tunnel's lining while the opposite was noted for the effect of the first tunnel on the second tunnel's lining. For a piggy-back tunnel geometry, the critical vertical spacing between the tunnels was found to be four tunnel diameters. The influence of the second tunnel on the internal stresses of the first tunnel's lining was more pronounced when the second tunnel was located below the first, with the influence becoming more pronounced with a decrease in rest time.

Ng et al. (2004) investigated the interaction of two side-by-side tunnels, 20 m below the ground surface in London Clay, constructed with the new Austrian tunnelling method (now more commonly referred to as the sprayed concrete lining (SCL) method) using a three-dimensional (3D) step-by-step approach (Katzenbach & Breth, 1981). The unsupported length (before constructing the lining) was 5 m and the distance between the leading and lagging tunnels was varied parametrically. Ng et al. established the influence of the distance of the tunnels’ faces on the surface settlement troughs. The authors also showed that the smaller this distance was, the more uniformly the loads were shared between the two tunnels and, the larger this distance, the greater the bending moment that developed within the leading tunnel.

In the region of Sydney, Liu et al. (2008) investigated the effects of tunnelling on an existing adjacent tunnel that was located 15 m below ground level with the NT having horizontal, vertical or staggered parallel alignments. A step-by-step approach was adopted for the excavation of both tunnels with an unsupported length of 4 m. It was concluded that the shotcrete lining of the ET was significantly affected when the NT's face passed below it and was less affected as the NT progressed further away from it. Their results indicated that the effect on the existing shotcrete tunnel lining differed for the various relative geometries considered.

As an extension to that work, Liu et al. (2009) investigated the effect of perpendicularly crossing tunnels, also in the Sydney region, with a clearance of 3·5 m. They observed that the circumferential bending moments in the existing shotcrete lining were affected first in the leading side, then at the invert and the crown and finally at the far side as the new excavation progressed towards, under and beyond the ET. The interaction effect was restricted to a close area where the two tunnels crossed.

The literature review revealed that extensive numerical research has not been performed for the case where ETs and NTs do not have parallel axes. The impetus for this research was the recent construction of the Crossrail tunnels beneath the existing London Underground (LU) Central line tunnels in central London. Analyses were performed by simulating the actual stratigraphy of the site and were based on real LU Central line and Crossrail tunnel dimensions and the clearance between them. However, since this work was focused on a more ‘general’ tunnel interaction case, a single tunnel being constructed perpendicularly beneath an ET was modelled. The use of numerical analyses allows the investigation of tunnel interaction problems with regard to hoop forces, bending moment changes and deformation characteristics of the ET as excavation of the NT progresses.

As already mentioned, the impetus for this paper was the crossing of the new Crossrail tunnels beneath the existing LU Central Line tunnels, the former passing the latter at a 40° skew angle at the edge of Hyde Park. The Central line tunnel axes are at 24 m below ground level and the Crossrail tunnels run below them with a relative crown to invert clearance that varies from 4·2 m to 4·9 m. The Crossrail tunnels were constructed using earth pressure balance machines (EPBMs) within which conventional bolted precast concrete segmental lining rings were erected. Each ring is 1·6 m long with internal and external diameters of D_in = 6·2 m and D_ex = 6·8 m, respectively.

The soil profile adopted for the finite-element (FE) analysis is presented schematically in Fig. 1 and is based on the deepest of the boreholes sunk to install instrumentation at greenfield site HP6 (Wan et al., 2017). The adopted profile comprised a 6 m thick layer of superficial deposits (made ground, alluvium and Terrace Gravels) overlying units B2, A3 and A2 of London Clay (King, 1981) of thickness 30 m, 12·5 m and 11·4 m, respectively. As shown in Fig. 1, the London Clay is underlain by the Lambeth Group, which was divided into two layers representing the upper more clayey and less permeable units and the lower more granular and more permeable units.

Several analyses were performed, each one involving various stages spread over several increments. The excavation and construction of the ET was achieved using a volume-loss control procedure (Potts & Zdravković, 2001). The target volume loss assumed for construction of the ET was 1·6% (a reasonable estimate for open-face shield tunnelling in London Clay), with construction completed in less than 100 h. It should be mentioned that, for the original Central line tunnel, each ring was erected in less than 20 min (Dalrymple-Hay & Jenkins, 1900). A 100 year period of consolidation followed. The length of the NT considered in the FE analyses was 120 m and the rate of excavation/construction was modelled as 100 m/week assuming an excavation length of 2 m (discussed further in the section headed ‘Boundary conditions’).

In the primary analysis discussed in this paper, the NT was excavated assuming the face pressure of the EPBM to be 200 kPa (which was the mean face pressure measured by eight face pressure transducers inside the EPBM plenum chamber in the case of the westbound Crossrail tunnel in the vicinity of the intersection with the Central line tunnels). Volume losses were not an input in the 3D FE analyses presented herein. The volume loss obtained from numerically modelling the construction of the NT was an output of the analyses and depended on the way that the modelling was performed (see the section ‘Boundary conditions’). Generally, 3D FE tunnelling analyses produce volume losses that exceed those measured in the field as a result of a too-large unsupported excavation length. However, the results of such analyses can be scaled to a desired volume loss, as discussed by Franzius & Potts (2005).

Two series of parametric analyses were also performed in order to investigate

• (a)

  varying the face pressure of the EPBM

• (b)

  varying the longitudinal axial and bending stiffness of the ET's lining (using shell elements).

A 3D FE model was constructed in order to investigate the tunnel–tunnel interaction problem in an accurate and computationally efficient way. Regarding the general geometry of the model, the following were considered.

• (a)

  One ET of diameter D_ET = 3·8 m and one NT of diameter D_NT = 6·8 m were simulated with their axes crossing perpendicularly to take advantage of symmetry.

• (b)

  The clearance between the invert of the ET and the crown of the NT was assumed to be 5 m.

• (c)

  Each ring of the NT was modelled to be 2·0 m wide. The 2 m excavation length (L_exc) used was small enough not to result in an unrealistically high volume loss but somewhat longer than typical concrete rings used in recent projects (e.g. the Crossrail rings were 1·6 m long).

The 3D FE mesh used in the analyses is shown in Fig. 2. It consists of 21 528 20-noded hexahedral isoparametric solid elements for modelling the soil and 1031 eight-noded shell elements (Schroeder, 2003) for modelling the two linings, 719 of which simulated the NT's lining. The axis of the ET runs along the x-direction (from 0 m to −110 m) and the axis of the NT runs along the z-direction (from 120 m to 0 m).

The FE mesh dimensions were selected carefully in view of their potential influence when analysing boundary value problems involving tunnelling, as highlighted by Franzius & Potts (2005). The length of the NT to be excavated was 120 m (i.e. L_tun = 120 m) and the vertical boundary of the mesh perpendicular to its axis was placed at a distance of 50 m (L_soil = 50 m, around 8D_NT) in front of the final position of the face (see Fig. 2). Both L_tun and L_soil were proven to be of sufficient length since ‘steady-state’ conditions were achieved for the final position of the NT's face (described when the results are presented later in the paper). The ET was located halfway along the excavated length of the NT.

Each node of the solid elements had three degrees of freedom, one for each component of displacement (u, v and w) and the eight corner nodes had an additional degree of freedom for the pore water pressure. The shell elements had the same three displacement degrees of freedom and, in addition, three rotational degrees of freedom.

The analyses performed involved coupled consolidation for all solution increments. All the analyses were performed with the Imperial College FE Program (ICFEP) in 3D and 2×2×2 integration was used. An accelerated modified Newton–Raphson technique with an error-controlled substepping stress point algorithm was used as the solver for the non-linear FE equations (Potts & Zdravković, 1999). Each analysis took between 15 and 20 days to run.

A pre-yield non-linear elastic model (model J4, based on Jardine et al. (1986)) coupled with Mohr–Coulomb yield and plastic potential surfaces was used for all the soil layers apart from the superficial deposits, which were modelled as linear elastic–perfectly plastic with Mohr–Coulomb yield and plastic potential surfaces. The pre-yield model J4 has been described and calibrated by various researchers (e.g. Addenbrooke et al., 1997). The parameters for model J4 and the Mohr–Coulomb model for all the soil layers are listed in Tables 2 and 3, respectively. During consolidation between the two tunnelling events, the angle of dilation, ν, of the soil layers was set to zero to prevent the Mohr–Coulomb model predicting unrealistic dilation in parts of the mesh where the soil yielded. This assumption concerning the angle of dilation is believed not to affect the results since the soil's behaviour due to tunnelling is primarily affected by the soil's stiffness and not its strength.

Anisotropic permeability profiles, with permeability values reducing with depth, were assumed, as shown in Fig. 3(a). These are consistent with the under-drained pore water pressure profile measured at the Hyde Park site close to the intersection of the Crossrail and Central line tunnels (see Fig. 3(b)) and were used in the FE analysis. The K₀ profile used in the FE analysis is shown in Fig. 3(c) (K₀ being the coefficient of earth pressure at rest).

The tunnel linings were represented by elastic isotropic shell elements (Schroeder, 2003). Each ring of the ET's lining was formed of 26 elements while the half perimeter of the new lining was formed of 12 shell elements. The joints between segments and rings were not modelled. The lining of the ET was modelled assuming the properties of grey cast iron (unit weight γ = 69·16 kN/m³, Young's modulus E = 100 000 MPa and Poisson's ratio of μ = 0·26) while the shell thickness was set as t = 0·0781 m to match the second moment of area per unit metre of the original segment. The lining of the NT was modelled with γ = 30 kN/m³, E = 40 000 MPa, μ = 0·15 and t = 0·30 m – properties that simulate segments used for the Crossrail tunnels (bolted precast concrete segments grade of C50/60 concrete with steel fibre dosage of 30 kg/m³). The shear correction factor for both tunnel linings was k = 0·8. Finally, for the analysis in which the longitudinal stiffness of the ET's lining was reduced, as a means of simulating the joints between successive rings, anisotropic elastic shell elements were used to model the lining with their longitudinal axial and bending stiffnesses being 1% of the circumferential stiffness.

Throughout each analysis, movements in all the three directions (x, y and z) were restricted on the bottom boundary of the 3D mesh. All the lateral boundaries were prevented from moving in a direction normal to the boundary while the remaining components were not restricted. The top boundary of the mesh was free to move.

The ET excavation was modelled with the volume-loss control method (Potts & Zdravković, 2001), implying plane-strain conditions. The excavation/construction of the NT followed a step-by-step approach (Katzenbach & Breth, 1981), also modelling the face pressure of the EPBM used (Fig. 4). In each increment of the excavation/construction of the NT, soil elements in front of the previous face position were excavated and, simultaneously, pressure was applied to the NT's face, located at L_exc in front of the previous tunnel face. The shell elements representing the lining were also constructed during this increment but their stiffness was only activated at the end of the increment, allowing soil movement into the tunnel. When the tunnel was advanced again, the next set of soil elements was excavated and consequently the active boundary of the mesh changed and the face pressure applied on the previous boundary disappeared with the excavated elements. This procedure was repeated for all of the tunnel excavation steps.

As the analyses employed coupled consolidation, it was necessary to define hydraulic boundary conditions. No change in pore water pressure was applied throughout the analyses, either at the top of the London Clay or at the bottom of the Upper Lambeth Group, leaving their interface with the non-consolidating elements (of the superficial deposits and Lower Lambeth Group) free to drain (or take in water). During excavation of the ET and the subsequent 100 year consolidation period, the two lateral boundaries (normal to the z-direction) were free to drain (i.e. no change in pore water pressure), these being far away from the tunnel. The remaining two vertical boundaries (normal to the x-direction) were impermeable as they were planes of symmetry for that stage of the analysis. During excavation of the NT, all the vertical boundaries were considered impermeable apart from the one that was far away from its axis (normal to the x-direction), which was left free to drain. Around the perimeter of the ET, zero pore water pressure was prescribed, allowing water to flow into the tunnel (simulating a permeable lining). No flow from the ET to the surrounding soil was encountered as its excavation procedure did not generate tensile pore water pressures in the surrounding soil. Finally, the completed NT was assumed to be fully impermeable. It should be noted that the pore water pressure profile prior to the NT excavation was not the same as that assumed in the beginning of the analysis (see Fig. 3(a)): it varied spatially and depended on the permeability of the soil, the permeability assumed for the ET and the initial profile itself.

Figure 5 shows three monitoring areas in the plane transverse to the ET (hence along the length of the NT). Throughout the paper these areas are consistently referred to as behind, at and in front of the ET's axis.

As the NT progressed in the analysis, transverse surface settlements developed. These are shown in Fig. 6(a) for the section located directly above the axis of the ET. A ‘steady state’ was reached for the last 20 m of the NT's advancement in front of the ET, where negligible additional settlements occurred (less than 0·5 mm). This indicated that the results could be considered reliable and were not significantly influenced by boundary effects.

The results of the analysis also indicate that the shape of the surface settlement trough did not change, either for various positions of the NT's excavation face in relation to a certain monitoring section or between different ‘monitoring sections’. The normalised settlement troughs relating to the NT's final face position (i.e. 120 m from the boundary of the mesh where tunnelling started) as determined for different monitoring sections are presented in Fig. 6(b): they are practically identical. This implies that, in the transverse direction (to the NT's axis), the presence of the ET did not affect the width/shape of the surface settlement troughs.

The evolution of volume loss with the advancing NT's excavation face for a number of monitoring sections is presented in Fig. 7. It can be seen that the rate of increase in volume loss diminished considerably over the final 20 m of the NT's advance for each of the monitoring sections, apart from that at +30 m from the ET (where settlements were still developing). This further illustrates that a satisfactory ‘steady state’ was reached for most of the mesh, especially in the close vicinity of the ET. The arrows above each curve in Fig. 7 mark the positions at which the tunnel face reached the corresponding monitoring section. The variation of volume loss along these points (illustrated by the thick grey dotted line) indicates that the rate at which the surface settlements developed was affected by the presence of the ET, despite the fact that the ultimate volume loss at the end of the NT excavation was constant for all monitoring sections. It seems that the earlier construction of the ET caused the soil adjacent to it to soften, as evident from the larger volume losses behind the ET. The effect of softening the ground around the ET was offset to a degree in its close vicinity by the intrinsic stiffness of the tunnel itself (as evident from the small decrease in volume loss in the region close to its axis).

Longitudinal surface settlement profiles along a monitoring line located vertically above the NT's axis for different NT excavation face positions (every 10 m) are shown in Fig. 8. The arrows mark the position of the face on each curve. The profiles for the first 50 m of the NT excavation (grey curves) (i.e. behind the ET's axis) have the expected shape of a cumulative distribution curve (Attewell & Woodman, 1982). The figure shows that, as excavation of the NT progresses, passing below and in front of the ET's axis, the profiles distort from the form of a cumulative distribution and exhibit a localised maximum settlement behind the ET's axis. The position of maximum settlement moves towards the ET's axis, reaching a maximum value of about 14·5 mm, 15–20 m behind it, where it becomes stable. As the NT excavation advances further, the surface settlements decrease locally around the ET because of the greater stiffness of the lining compared with that of the surrounding soil. Therefore, in the vicinity of the NT, there is a combined effect of soil relaxation (due to excavation/construction of the ET) and stiffening of the ground (due to the presence of the ET) as the NT excavation progresses.

Circumferential hoop forces, bending moment distributions and the respective interaction diagram around the tunnel lining for various stages of the NT excavation are presented in Fig. 9.

The distribution of hoop forces shown in Fig. 9(a) is almost uniform, fluctuating around an average of about −370 kN/m prior to the beginning of excavation of the NT (the sign convention usually adopted in structural engineering is used, where compressive forces are negative). The distribution remains almost constant until the NT's face reaches 10 m behind the ET's axis, at which point it starts becoming less uniform. The distribution reaches its most non-uniform state when the NT's face is just below the axis of the ET. At this stage, the hoop forces are up to 20% more compressive in the first and third quadrants and up to 30% less compressive in the second and the fourth quadrants (see inset diagram and defined zones in Fig. 9(a)) compared with the situation prior to the beginning of the NT excavation. With further tunnel advance the hoop forces equilibrate to a new distribution and remain practically unchanged for the last 30 m of the NT excavation. At this stage, the hoop forces are up to 10% more compressive around the crown and up to 10% more tensile everywhere else compared with the situation prior to the NT excavation.

The circumferential bending moments, which are practically zero prior to the beginning of the NT excavation, follow a similar trend to that of the hoop forces (see Fig. 9(b)). As the NT approaches the ET's axis, the arcs of the sections of the lining in the first and third quadrants, respectively (looking in the negative x-direction) start experiencing tension at the extrados. The remaining part of the lining experiences tension at the intrados. The change in the bending moments (relative to values prior to the new excavation) becomes a maximum (±12 kNm/m) when the new excavation face is below the axis of the ET. Generally, an anti-clockwise rotation of the lining sections experiencing tension at the extrados is evident with advancement of the NT. At the final stage of the NT excavation the differences in bending moments (compared with prior to excavation) are ±4 kNm/m.

The combined effect of hoop forces/bending moments acting on different positions of the ring (for any NT face position) are well within the failure envelope of the analysed section of the ET (see Fig. 9(c)). This indicates that, despite the sizeable changes in hoop forces and bending moments, no structural failure (e.g. cracking of the lining) is predicted due to excavation of the NT either in the temporary condition or after the NT construction has progressed far from the intersection area.

The overall deformed shapes of the ET as the NT excavation progresses are shown in Fig. 10, along with superimposed contours of displacement (representing a combination of all three components).

The ET's lining starts experiencing displacements and deformations due to the new excavation when the excavation face is about 40 m behind its axis (Fig. 10(a)). With further advancement of the NT, the magnitudes of both increase (Figs 10(b)–10(f)). The closer a section of the ET is to the NT's axis, the greater and more rapidly it deforms, while there is negligible influence at distances greater than 30–40 m from the NT. Once the NT's face is at a greater distance from the ET (about 40 m in front of it) its overall deformed shape does not change, indicating that the influence zone is no longer affecting the NT (Figs 10(g) and 10(h)).

As far as the magnitudes of displacements are concerned, the vertical component (in the y-direction) is the greatest, with a maximum magnitude of around 23 mm. Displacements in the two horizontal directions are an order of magnitude smaller (longitudinally up to around 2 mm and transversely up to around 7 mm).

The following two major modes of deformations can be noted from Fig. 10 and these are discussed in the following subsections.

• (a)

  Elliptical deformation of the ET's cross-section, forming either ‘squatting’ or ‘egg-shaped’ profiles, and rotation as the NT advances (Standing & Selman (2001) observed this mode of deformation from field measurements).

• (b)

  Longitudinal bending of the lining caused by its vertical settlement.

In the initial few metres of the NT advancement (Fig. 10(a)) a lengthening of the horizontal axis and shortening of the vertical axis of the ET occurs. Viewing along the ET (negative x-direction), there is a resulting ‘squatting’ form that rotates anti-clockwise as the NT advances towards the ET's axis, with lengthening along a chord connecting the left-hand haunch to the right-hand shoulder of the ring directly above the NT and shortening in the orthogonal direction. As the NT advances further, in front of the ET, the major axis of the latter's elliptical section has rotated so that it is essentially vertical, thus becoming ‘egg-shaped’ (Fig. 10(e)). In all cases, deformations of the ET's cross-section diminish with distance from the NT.

Figure 11 shows the cross-sectional deformations of sections directly above the NT's axis and at distances of 10 m and 20 m in the form of changes in span of several chords. In practice, changes in such chords might be measured using a tape extensometer from within the ET. The changes in chords/diameters obtained from the FE analysis show that their length remains unaffected until the NT's face is about 30 m behind the axis of the ET. From that tunnel face position onwards, the following observations can be made from the results of the numerical analysis.

• (a)

  Diameter BD increases in length as the NT approaches the ET, reaching a maximum when its face is directly below the axis of the ET. It subsequently decreases as the NT progresses.

• (b)

  The behaviour of diameter AC (orthogonal to diameter BD) follows an almost mirror image mode to that of BD.

• (c)

  Chord BC, which connects the two haunches of the lining, increases in length as the NT approaches the ET, reaching a maximum when it is 5 m behind it. The span then reduces, reaching a minimum when the excavation face is 5 m in front of the axis of the ET, and then increases again before becoming stable with further tunnel face advancement.

• (d)

  Changes in the lengths of the other chords (AD, AB and CD) were found to be negligible (less than 1 mm for any NT position)  and so are not plotted in Fig. 11.

For tunnel advancement beyond about 40 m in front of the ET, there are negligible further changes in the length of the chords/diameters and the deformed shape of the section remains (see Figs 11, 10(g) and 10(h)).

As expected, the largest span changes are observed in the section just above the NT's axis (Fig. 11(a)) and their magnitude decreases with increasing distance of the section from the NT (Figs 11(b) and 11(c)) (note that different scales are used for the changes shown in Figs 11(a)–11(c)).

Figure 11 clearly shows that rapid changes in span lengths of the ET occur when the excavation face of the NT is in the close vicinity of the ET. Most significant changes in span length occur over approximately 20 m of the NT's excavation advancement (from −15 m to +5 m). Given that modern tunnel boring machines (in soil conditions like London Clay) can readily achieve advancement rates greater than 15 m/day, Fig. 11 suggests that when transient distortions are of interest and need to be measured in tunnel–tunnel interaction problems, remote automated measurement systems should be used. Monitoring such distortions using manual field measurements during ‘engineering hours’ is likely to result in key responses being missed.

The results shown in Fig. 11 were compared with field measurements taken when the first Crossrail tunnel was excavated beneath the Central line tunnels at Hyde Park, reported by Yu (2014). The results showed an overall satisfactory agreement in terms of the magnitude and the manner of changes in the chord lengths, despite the simplified basic scenario analysed in the current work.

Results from the numerical analysis regarding the development of longitudinal strains along the ET crown and invert due to excavation of the NT are summarised in Figs 12(a) and 12(b). The influence of the NT excavation first becomes evident at both crown and invert level when the face of the NT is about 10 m behind the intersection of the two alignments. The magnitude of strains develops appreciably as the NT's face advances from 10 m behind to 10 m in front of the ET's axis, after which they remain essentially constant.

The form of the longitudinal strains developing in the crown (Fig. 12(a)) is the same as would be expected in the ground transverse to a NT in greenfield conditions, with maximum compression strains directly above the tunnel axis, reducing laterally and becoming zero at a certain offset (the points of inflection when considering tunnelling in greenfield conditions). Tensile strains develop beyond these offsets (c.f. again points of inflection), reaching a maximum before diminishing towards zero with increased distance. In the analysis, the offset distance from the axis of the NT at which longitudinal strains in the ET change sense is about 20 m (Fig. 12(a)). As the NT advances, the magnitudes of the compressive and tensile strains increase. Ultimately, after the strain evolution stabilises, the maximum compressive and tensile longitudinal strains in the crown of the ET are 240 με and 60 με, located directly above and 32 m from the NT's axis, respectively, with the offset where strains switch from compression to tension being 17·1 m (2·6D or 0·5z₀ where z₀ is the depth to NT axis) from the axis. The mode of strains along the invert (Fig. 12(b)) is slightly more complex, with final maximum compressive and tensile strains of 120 με and 85 με, respectively.

FE predictions of longitudinal strains along the crown and the invert of an ET could be used in practice when assessing tunnel–tunnel interaction problems provided that the results are scaled to the design value of volume loss. This methodology was proposed by Franzius & Potts (2005) in an analysis of tunnelling-induced transverse deformation of a building.

It is generally thought that the instruments used to measure the pressure in the plenum chamber of an EPBM measure lower pressures than those applied to the soil in front of the cutter-head. In order to investigate this further in relation to its effect on the ET's response, additional analyses were run in which the face pressure applied in the model was set to 0 kPa and 500 kPa (a value of 200 kPa was used in the analyses described so far; see Fig. 9 for the distributions of absolute values of hoop force and bending moment with this face pressure). An increase in face pressure is thought to contribute to reducing the rate of development and the magnitude of ground surface settlements. However, in numerical analysis, the unsupported length (L_exc) seems to be a much more influential factor. The grout pressure behind the rings was not modelled in this study and, as such, only the effect of face pressure is discussed here.

The hoop forces developed in the ET for the three NT excavation face pressures are presented in Figs 13(a) and 13(b) (comparisons are made between cases with no face pressure and face pressures of either 200 kPa or 500 kPa). Generally, the hoop forces are within 5% all around the lining for any position of the NT's excavation face irrespective of the face pressure, apart from the position of 10 m behind the crossing. This result indicates that the stabilised hoop force distributions are not greatly affected by the EPBM face pressure. However, when the face of the new excavation is 10 m behind the crossing, an increase in the face pressure results in generally more compressive hoop forces, with maximum increases of 8% and 18% for face pressures of 200 kPa and 500 kPa, respectively.

Equally, as shown in Figs 13(c) and 13(d), the circumferential bending moments are mostly affected by excavation of the NT when its face is 10 m behind the ET's axis, with a maximum monitored difference of 5 kNm/m. Comparing Figs 13 and 9, it can be concluded from the numerical analyses that an increase in EPBM face pressure is beneficial for the circumferential bending moments of the ET for any position of the EPBM face because a reduction in their magnitude at any point around the cross-section is predicted.

These results suggest that there are benefits to increasing the EPBM face pressure in cases where the combined hoop forces/bending moments within an existing lining are likely to become adverse because of the construction of a NT beneath the ET.

The effect of not applying and applying 500 kPa face pressure on the deformation of the various cross-sections of the ET's lining is shown in Fig. 14. The results are presented in the same way as Fig. 11 (where the face pressure was 200 kPa). An increase in face pressure results in a delayed response of the ET with noticeable changes in span not starting until the face is within about 20 m (compared with 30 m when the modelled face pressure was 200 kPa). This was found to be the case for all the sections considered (Figs 14(a)–14(c)), regardless of their distance from the NT's axis. As the NT advances further in front of the ET, the increased face pressure seems to cause the spans to change length more rapidly. Consequently, the final stable lengths are different, although these effects diminish with increasing distance of the cross-section from the NT's axis.

The ET's lining was modelled as a sequence of successive rings of bolted grey cast-iron segments. In all the analyses presented so far, a rigid connection between successive rings was inherently assumed. This implies that the tunnel was actually simulated as a long continuous tube within the ground. In practical terms, this could be interpreted as a situation where the circumferential bolts connecting two successive rings are tightened very tightly. However, this is not always the case and successive rings can have their bolted connections loose with circumferential bolts not being tightened. As a way of numerically modelling the effect of an extreme case (where successive segments are practically unbolted), anisotropic shell elements were used to simulate the ET with their longitudinal axial and rotational stiffness reduced to be 1% of their circumferential values for the case of an applied face pressure of 200 kPa.

Figure 15 shows the evolution of longitudinal strains along the crown and the invert of the ET when successive rings were modelled as practically unbolted. The shapes of the predicted strains along the crown and invert are generally similar to those in which the ET was modelled as a continuous tube (Fig. 12) but, in the ‘loose ring’ case, the magnitudes of longitudinal strains are about six to ten times larger. This difference in the magnitude of strains for the two cases shows that for the ‘loose ring’ larger axial displacements are anticipated and consequently the stresses in the longitudinal direction are likely to be reduced.

A series of 3D FE analyses was conducted to investigate the interaction and influence of a NT crossing perpendicularly beneath an ET. The motivation for the analyses presented here was the new Crossrail tunnels excavated beneath the existing Central line tunnels in central London. The results provide insight into the tunnel interaction problem with regard to changes in hoop forces, bending moments, cross-sectional deformations and longitudinal strains of the ET. The effects of the face pressure of the EPBM' and the longitudinal stiffness of the ET were also explored.

The predicted ground surface movements show that the presence of an ET affects the rate at which settlements develop for transverse sections near to it. The ground surface settlements also indicate a combination of soil disturbance from the first (existing) tunnel construction and stiffening of the ground due to the presence of the ET.

Regarding the structural capacity of the ET, the most critical combination of circumferential hoop force and bending moment was identified as the point when the NT's excavation face is directly below the ET's axis.

For the geometry investigated in this study, the ET experienced deformations when the NT approached within 20 m of its axis. These deformations developed with further advance of the face of the NT and stabilised once it had reached a distance 20 m beyond the ET's axis. Various cross-sections of the ET deformed into an elliptical shape as the NT's face advanced. The major and minor axes of the ellipse rotated as the NT advanced, with the major axis following the NT's excavation face. It was found that the most significant changes occurred over a very short distance (when the NT was in the very close vicinity of the ET), suggesting that remote in-tunnel monitoring (rather than manual measurements made during engineering hours) would be necessary to observe the complete tunnel response. Longitudinally, the most significant strains were predicted along the crown of the ET.

The results presented in this paper are for a certain volume loss. However, the results can be scaled to an appropriate volume loss in order to be used for a particular scenario, as discussed by Franzius & Potts (2005).

The influence of the face pressure of the EPBM was also studied parametrically. An increased face pressure was found to be generally beneficial for the ET, reducing the combined effects of lining hoop forces and bending moments. Changing the longitudinal stiffness of the ET can be used as a means of modelling unbolted lining rings.

The authors acknowledge the Engineering and Physical Sciences Research Council (EPSRC research grant EP/G063486/1), Crossrail and Morgan Sindall, who were major sponsors of the research project conducted at Imperial College. Many thanks are due to other members of the Imperial College research team, in particular Professor John Burland, Dr Jessica Yu and Dr Katerina Tsiampousi.