Impact of small-angle shield undercrossing on existing tunnels

A three-dimensional finite element model was developed to investigate the effects of small-angle tunnels on existing tunnels in terms of surface settlement, arch bottom settlement, and ellipticalization of existing tunnel linings. The model simulated two small-angle tunnels crossing two existing tunnels. The maximum surface subsidence was observed at the recently excavated tunnel centers. The presence of existing tunnels reduced surface subsidence caused by ring-by-ring excavation. Surface settlement followed a “W” curve pattern, with two local minima gradually getting close as the tunnel was excavated. The maximum value of the arch bottom settlement of the existing tunnel gradually approached the intersection of the existing tunnel and the newly excavated tunnel. When the absolute value of the rate of change of the arch bottom settlement was high, the tunnel produced a large inter-ring misalignment. The line connecting the centers of the existing tunnel and recently excavated tunnel was parallel to the major axis direction of the ring after the deformation of the lining on the existing tunnel. Therefore, a small-angle tunnel passing through an existing tunnel may result in misalignment and torsion of the existing tunnel, requiring careful monitoring of the inter-ring bolt force to ensure safety.


Introduction
Following their continuous expansion in China, the complexity of urban underground networks in China has increased.Consequently, assessing the ramifications of new tunnel crossings on existing tunnels has become crucial.To determine the impact of new tunnels on existing tunnels, Cheng [1] studied the impact of vertical and parallel tunnel crossing on pipelines; Zhang [2] used ANASYS modeling to analyze the impact of new tunnels with different spacings vertically passing through the existing tunnels; Li [3] used three-dimensional simulation to study the deformation of the existing tunnels when new tunnels are vertically passing through the existing tunnels; Gan [4] proposed a simplified formula for calculating the longitudinal deformation of existing tunnels when a two-lane tunnel passes vertically underneath the existing tunnel; Hua [5] used three-dimensional numerical simulation to study the displacement of an existing tunnel when a new tunnel is vertically upwards and downwards passing through an existing tunnel, and determined that the maximum displacement is located where the existing shield tunnel and the new shield tunnel intersect; Kong [6] used three-dimensional numerical simulation to study the displacement of new tunnel vertically passing through two-lane subway tunnel, and investigated the law of surface settlement.
Numerous studies have examined the vertical crossing of new tunnels through existing tunnels; however, the increasing complexity of underground transportation networks have availed additional examples of new tunnels crossing existing tunnels at a small angle; research on this topic is becoming increasingly important.Ma [7] illustrated the settlement law of existing tunnel when it passes through the existing tunnel at small angle; Yin [8] established a simplified formula for the nonlinear deformation of pipeline when the tunnel passes through the pipeline vertically or at a small angle; Zhang [9] analyzed the vertical displacement, horizontal displacement, and stress evolution law during the process of passing through the existing tunnel at a small angle of the new tunnel; Hu [10] analyzed that when the small-angle tunnel passes through the double-line double-arch tunnel, the deformation of the doubleline double-arch tunnel is mainly vertical settlement, accompanied by torsional deformation.When a small-angle tunnel passes through an existing tunnel, the existing tunnel produces settlement as well as ellipticalization in different directions.However, the aforementioned scholars have only studied the tunnel settlement and surface settlement of existing tunnels in the case of small-angle tunnels crossing existing tunnels.There has been no research on the elliptical deformation of existing tunnels and the degree of ellipticalization, and the characteristics of existing tunnels under the combined effect of ellipticalization deformation, and tunnel settlement have not been studied.In this study, a threedimensional finite element model was established to study the surface settlement, settlement of the existing tunnel, and degree of ellipticalization of the existing tunnel in the case of a small-angle tunnel crossing the existing tunnel, and to analyze the risk potential of the existing tunnel under the combined effect of elliptical deformation and tunnel settlement.

Overview of supporting projects
The model is centered on a specific section within a designated area of a subway line Phase II project, focusing on the impact of two small-angle tunnels crossing two existing tunnels.The two existing tunnels are the left line (EL) and right line (ER), similar to the green axial tunnel shown in figure 1(a).The newly excavated tunnels are represented by the left line (RL) and right line (RR), as shown by the red axial tunnel in figure 1(a).The construction of this section involves the employment of earth pressure balance shield tunneling.The outer diameter of the shield tunnels for both existing and recently excavated tunnels is 6.48 m, with a grouting layer thickness of 0.14 m.Concrete pipe segments have an outer diameter of 6.30 m, and the wall thickness of these segments is 0.35 m.The standard ring width of the lining is 1.5 m, and is assembled with staggered joints.The top view at the node of the zone is shown in figure 1(a), with an angle of 20° between EL and RL and an angle of 15° between ER and RR.A surface settlement analysis was performed on the four profiles depicted in the figure .A side view of a node in this zone is shown in figure 1(b).The surface elevation of the soil layer is +5 m.There are 8 soil layers from the surface to the tunnel arch, including plain fill, silty clay, gravel sand, gravel cohesive soil, fully weathered granite, strongly weathered granite, moderately weathered granite, and slightly weathered granite.The burial depth of the existing tunnel is 4.02 m, whereas that of the newly constructed tunnel is 28.52 m.The upper part of the existing tunnel was located in silty clay, whereas the lower part was situated in gravelly sand.The newly constructed tunnel is situated entirely in slightly weathered granite, as shown in figure 1(b).

Model overview
The three-dimensional finite element model was 150 m long, 100 m wide, and 55 m deep.Four tunnels were established in the model, as shown in figure 2: the left line (EL) and right line (ER) of the existing tunnel, and the left line (RL) and right line (RR) of the newly excavated tunnel.In this model, the soil, grouting layer, and lining were represented using three-dimensional solid elements to reflect the actual dimensions.The modeling of the shield machine was simplified based on its external structure by utilizing three-dimensional shell elements in accordance with its true dimensions.To improve the accuracy of the model, a seven-ring shell unit was used to simulate the shield machine.The tunnel segments progress vertically along the tunnel axis.The excavation time for the two rings is set as the grouting-hardening time.The three-dimensional model incorporates normal displacement constraints except for the upper surface, which was treated as a free boundary.The contact between the grout layer and tunnel lining, and between the grout layer and the soil were set to bind.

Parameter values
The soil was modeled using the Mohr-Coulomb [11] model, and soil parameters were extracted from the geological survey report, as outlined in table 1.
The modulus of the segment lining was decreased by 75% to incorporate the influence of the segment joints in the 3D model.In addition, the weight of the shield shell was computed based on the overall mass of the shield tunnel.The grouting layer was modeled using semisolid slurry parameters.The material parameters of the model are listed in table 2.
The support force on the tunnel surface increased by 20 KPa in addition to the static soil pressure, and the support force gradient along the direction of gravity corresponded to the weight of the soil.The grouting pressure was increased by 10 KPa on the basis of 0.6 times the static soil pressure at the top of the excavation section, with the grouting pressure gradient along the gravity direction representing the grouting weight.Meshing was performed after setting the parameters.Because the new tunnel was a small-angle tunnel, the transition layer between the existing tunnel and the new tunnel was meshed with a tetrahedral mesh, and the remaining parts were meshed with a hexahedral mesh.

Simulation of crossing process
The model selected 66 rings in the newly excavated tunnel as the key simulation objects, and the edge ring of the tunnel was simplified using a trapezoidal tube sheet.In the construction sequence of the newly excavated tunnel, the right line is constructed first, and the left line is constructed after the right line has been fully constructed.The construction process simulation is as follows.1) Existing tunnels are constructed.2) Soil corresponding to the analysis step is excavated, with activation of the shield shell unit and excavation support force ahead of the palm.3) after pushing the top seven rings, the tail of the shield machine emerges.The tailpipe segments of the shield machine are then activated, and grouting pressure is synchronously initiated.4) Following the activation of the grouting pressure for two cycles, the grouting layer hardens, activating the solid unit of the grouting layer, and subsequently deactivating the grouting pressure.

Green field
The effect of ring-by-ring tunnel excavation on surface settlement was observed by building a threedimensional finite element model with a green field.Surface subsidence is illustrated in figure 3(a), where ring-by-ring excavation can cause surface subsidence.Maximum surface subsidence was observed at the recently excavated tunnel center, reaching a value of 0.42 mm.As the two recently excavated tunnels approached each other, the surface settlement between them gradually increased by 0.079 mm.

Existing tunnels
The effect of ring-by-ring tunnel excavation on surface settlement was observed by building a threedimensional finite element model containing an existing tunnel.The surface subsidence is presented in figure 3(b), where the presence of existing tunnels reduces the surface subsidence caused by ring-byring excavation.The maximum surface subsidence is observed at the recently excavated tunnels center, reaching a value of 0.38 mm, which is 9.5% lower than that of the greenfield situation.As the two recently excavated tunnels approach each other, the surface settlement between the tunnels gradually increases by 0.057 mm, which is 27.8% lower than that of the greenfield situation.

Comparison
In the case of the green field and existing tunnels, the surface settlement from sections 1 to 4 was extracted and plotted in figure 4, where the effect of the existing tunnel on the surface settlement is shown.Analyzing the surface settlement for different scenarios from section 1 to 4 in figure 4. The surface settlement for all four sections in the green field was greater than that of the case of existence of existing tunnels.In the case of green field and existence of existing tunnels, the surface settlement is "W" shaped owing to the excavation of both tunnels.The local minimum values of "W" gradually converge with each other from sections 1 to 4, and the x-axis coordinates of the local minimum points converge from -40 m and 40 m in sections 1 to -30 m and 22 m in section 4. The local maximum values of the settlement differences from section 1 to 4 occurred at the center of the excavated tunnels and in the middle of the two excavated tunnels.

Bottom settlement of existing tunnels
When a new tunnel passes under an existing tunnel, the existing tunnel generates bottom settlement.The change in bottom settlement along the existing tunnel results in a misalignment between the rings of the existing tunnel.Inter-ring misalignment increases the shear stresses on the bolts between the tubes in an existing tunnel, thereby creating a hazardous situation for the existing tunnel.By extracting the bottom settlements of the existing tunnels in all the analysis steps, the maximum bottom settlements and maximum changes in the bottom settlements were analyzed.
Figure 5(a) shows the arch bottom settlement from excavation to Section 1 to the ER of excavation to Section 4 and the absolute values of the rate of change of the arch bottom settlement at excavation to section four.Figure 5(b) shows the arch bottom settlement of the EL from excavation to section 1 to excavation to section 4, and the absolute values of the rate of change of the arch bottom settlement from excavation to section four.Data analysis reveals that the bottom settlement of the ER increases owing to the RR crossing, and the RL crossing produces a small effect; the bottom settlement of the EL increases owing to the RL crossing, and the RR crossing produces a small effect.As shown in figure 5(a), the maximum value of the arch bottom settlement of ER when excavating to section 4 is 0.5 mm, at 40 m from the initial end of ER.As shown in figure 5(b), the maximum value of the arch bottom settlement of EL when excavating to section 4 is 0.42 mm, at 66 m from the initial end of EL.From excavation to section 1, to excavation to section 4, the location of the maximum arch bottom settlement of ER gradually approached the intersection of the ER and RR, and the location of the maximum arch bottom settlement of EL gradually approached the intersection of EL and RL.Comparing figure 5(a) and figure 5(b) excavation to the same section of the bottom of the arch settlement, the larger the angle between the newly excavated and existing tunnels, the smaller the scope of influence on the bottom of the existing tunnel settlement during the excavation process.Data analysis revealed that when the distance between the center of the recently excavated tunnel crossing ring and the center of the existing tunnel was < 20 m, the crossing of the recently excavated tunnel had a significant impact on the settlement at the bottom of the existing tunnel.

Ellipticity of existing tunnels
By extracting the coordinates and displacements of the existing tunnels in the x, y, and z directions in all the analysis steps, the deformation of the existing tunnel lining during the tunneling process was calculated to determine the effect of tunnel excavation on the ellipticity of the existing tunnels (figure 6 and figure 7).
A section of the existing tunnel was selected to draw the initial shape of the lining; the shape after deformation is shown in figure 6 and figure 7, and the boring position of the newly excavated tunnel is shown in figure 5 and figure 6.According to the cross-sectional diagram, the line connecting the center of the ER to the center of the RR is parallel to the major axis direction of the ring after the deformation of the lining on the ER.Similarly, the line connecting the centers of the EL and RL becomes parallel to the major axis direction after the deformation of the lining on the EL.The data were extracted, and the ellipticity of the existing tunnel and distance between the new tunnel and the center of the existing tunnel were calculated and plotted in figure 8.The ellipticity of the tunnel section lining after deformation is the value of the long axis divided by the short axis after the deformation is magnified 50,000 times.As shown in figure 8, when the angle between the recently excavated tunnel and the existing tunnel is constant, the impact of the tunnel crossing on the ellipticity of the existing tunnel diminishes as the distance between the center of the corresponding ring of the recently excavated tunnel and the center of the corresponding ring of the existing tunnel increases.
The angle between the EL and RL was 20°, and the angle between the ER and RR was 15°.The ellipticalization formed by EL is greater when the distance from the center of the circle is 24.5 m in both cases.Consequently, with a fixed distance between the centers of the rings, a smaller angle between the recently excavated and existing tunnels results in reduced impact of tunnel excavation on the ellipticity of the existing tunnel.When small-angle tunnels pass through existing tunnels, the existing tunnels are not only misaligned between rings, but also have different directions along the long axes owing to the different ellipticalizations of the different sections.Therefore, the adjacent rings of the existing tunnel are not only subjected to vertical shear, but also to torsional shear owing to ellipticalization in different directions.Compared with the newly excavated tunnels crossing the existing tunnels vertically, the small-angle tunnel crossing the existing tunnels will further complicate the shear force on the neighboring inter-ring bolts, causing potential safety hazards; therefore, deformation and stress monitoring on the inter-ring bolts are crucial.

Conclusions
This study presents a three-dimensional finite element model of two small-angle tunnels passing through two existing tunnels.Through the collection and organization of data, the influence of existing tunnels on the surface settlement after small-angle tunnel crossing, and the impact of small-angle tunnel crossing on the bottom settlement of existing tunnels, and the impact of small-angle tunnel crossing on the ovality change of the existing tunnel lining were examined.The following conclusions were drawn: 1) Maximum surface subsidence was observed at the recently excavated tunnel center.The presence of existing tunnels reduced the surface subsidence caused by ring-by-ring excavation, and the maximum settlement was reduced by 27.8%.Owing to the excavation of the two tunnels, the surface settlement is characterized by a "W" curve, where the two local minima gradually converge with the excavation of the tunnels.
2) The location of the maximum value of the arch bottom settlement of the existing tunnel gradually approaches the intersection of the existing tunnel and the newly excavated tunnel.When the absolute value of the rate of change of the arch bottom settlement is a local maximum, the tunnel will produce a large inter-ring misalignment, which will produce a large safety hazard.The line connecting the centers of the existing and recently excavated tunnels was parallel to the major axis direction of the ring after the deformation of the lining on the existing tunnel.The degree of ellipticalization of the established tunnels decreased with increasing distance from the center of the circle and decreased with decreasing pinch angle.
As a result, a small-angle tunnel crossing of an existing tunnel produces not only inter-ring misalignments in the existing tunnel, but also inter-ring torsions owing to ellipticalization in different directions.Compared with vertical crossings, the shear stress on the bolts between the rings of existing tunnels is more complicated and needs to be monitored with an emphasis on deformation and stress to ensure the safety of the structure.

Figure 1 .
Schematic of the node situation: (a) top view and (b) side view; (unit: m).

Figure 2 .
Finite element model: (a) tunnel location schematic and (b) status of grid demarcation.

Figure 3 .
Figure 3. Schematic of surface settlement: (a) green field and (b) existing tunnels.

Figure 5 .
Tunnel bottom settlement and rate of change of sedimentation; (a) RR crossing ER and (b) RL crossing EL.As in figure 5(a), the absolute local maximum values of the sedimentation change rate are 0.89×10-5 and 1.04×10-5 at 14 m and 80 m from the initial end of ER, respectively.As in figure 5(b), the local maximum values of absolute values of sedimentation change rate are 1.38×10-5 and 1.48×10-5 at 44 m and 86 m from the initial end of ER, respectively.The absolute value of the rate of change of the bottom of the arch settlement first increased to the first local maximum value point, then decreased to 0, which was the location of the maximum value of the bottom of the arch settlement, increased to the second local maximum value point, and finally gradually decreased.The absolute value of the rate of change of the settlement at the bottom of the arch for the local maximum value indicates that the tunnel produces a large inter-ring misalignment, resulting in greater safety risks.The construction of the project should focus on monitoring and protecting extreme-value points.

Figure 8 .
Ellipticity and center distance of existing tunnels: (a) right line and (b) left view.

Table 1 .
Values of physical and mechanical parameters of strata.