Numerical analysis of hard rock tunnel excavated by double shield TBM based on CWFS model

3.2.1. Longitudinal stresses of surrounding rock

For the intrinsic excavation condition, the stress value of the vault (1^#) is equal to that of the invert (3^#) due to symmetry of excavation boundary. Therefore, only the longitudinal stress value of the vault (1^#) and sidewall (2^#) are shown in Fig. 7.

The stress paths of the vault are shown in Fig. 7(a). In the face of the heading, ${\sigma }_1$ and ${\sigma }_2$ increase rapidly to about 1.7 times and 1.5 times of their initial values, respectively, as well as ${\sigma }_3$ drops to approximate three fourths of its initial value, which due to effect of excavation. Behind the work face (excavated part), ${\sigma }_1$ decreases sharply at first, then increases quickly, and rises to about 1.5 times of its initial value eventually. Meanwhile, ${\sigma }_2$ drops to approximate half of its initial value quickly, and ${\sigma }_3$ decreases at first, then increases and finally stabilizes at approximate 0. The above phenomena are stress concentration caused by the stress rotation [17]. In front of the work face (the unexcavated part), ${\sigma }_1$ and ${\sigma }_2$ decrease to their initial values, respectively at the face 15 m away from the work face. Meanwhile, ${\sigma }_3$ decreases slightly at first, then increases obviously, and restores to its initial value at the face 8 m away from the work face.

Fig. 7The stress paths of surrounding rock along the longitudinal direction of the tunnel: a) along point 1#, b) along point 2#

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b)

The stress paths of sidewall are illustrated in Fig. 7(b). In the face of the heading, ${\sigma }_1$ increases rapidly to about double its initial value. Meanwhile, ${\sigma }_2$ and ${\sigma }_3$ become about four fifths and six fifths of their initial values, respectively. Behind the work face, ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_3$ decrease to 25 MPa, 4.5 MPa and 0 MPa, respectively. In front of the work face, ${\sigma }_1$ and ${\sigma }_3$ decrease to their initial values, respectively, and ${\sigma }_2$ rises to its initial value slowly.

In conclusion, the stress concentration appears in the vault and invert due to excavation in intrinsic condition, and the stress concentration value is about 76 MPa, while the sidewall experiences excavation unloading.

3.2.2. Circumferential stresses and radial stresses

The circumferential stresses and radial stresses are mainly influenced by in situ stresses and excavation effects in intrinsic condition. In this paper, the stresses in two key cross sections are analyzed in detail, where key cross sections are the work face and the face 13 m away from the work face (Corresponding to I and Ⅳ positions in Fig. 6(b)). In order to simplify the illustration, only the stress values of the vault (1^#) and the sidewall (2^#) are shown in Fig. 8.

As shown in Fig. 8(a), the circumferential stress of point 1^# reaches a maximum value of about 80.0 MPa near the tunnel wall, and then decreases to its initial value gradually with the depth increase of surrounding rock. The radial stress of point 1^#, however, has a minimum value of approximate 21.0 MPa near the tunnel wall, and then increases to its initial value slowly. In addition, near the tunnel wall, the circumferential stress of point 2^# reaches its maximum value of about 32.4 MPa, but the radial stress of point 2^# has its minimum value of about 35.3 MPa. With the increase depth of surrounding rock, they restore gradually to their initial values, respectively.

The stresses in the cross section 13 m away from the work face are shown in Fig. 8(b). The circumferential stresses of points 1^# and 2^# have similar tendency to increase first and then decrease, but the circumferential stress of point 1^# varied more than that of point 2^# with the increase of the depth of the surrounding rock. However, the radial stresses rise to their initial values gradually with the increase depth of surrounding rock. Moreover, radial stress of point 1^# restores to its initial value earlier than point 2^#.

Fig. 8Circumferential stresses and radial stresses of internal surrounding rock points in intrinsic condition: a) the cross section of work face and, b) the cross section 13 m away from the work face

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b)

3.2.3. Plastic zone distribution

The plastic zone can clearly indicate the stress state (whether it is yielding or not) of the surrounding rock, which reflects its mechanical response after stress redistribution due to excavation.

As shown in Fig. 9, the failure zones by the positions Ⅰ and Ⅱ are mainly concentrated in the tunnel circumference. The v-shape failure zone is shown in Ⅲ to occur at the vault and the invert, which is a form of brittle failure caused by excavation unloading under high in situ stress [18, 19]. The vault and invert underwent tensile plastic yield but became shear plastic yield after stress adjustment. According to the previous study on stress paths by true triaxial tests on granite [20, 21], samples failed in shear when the intermediate principal stress ${\sigma }_2$ was less than 20 MPa, but failed in slabbing when ${\sigma }_2$ was equal to and greater than 20 MPa. The stress paths simulated in those tests are similar to the vault stress paths as shown in Fig. 7(a) in this paper, which further proves 1^# point fails in spalling. In addition, shear failure occurs at the sidewall.

By the positions Ⅳ and Ⅴ, the angle and depth of v-shape failure zone increase due to the distance from the work face and the unloading time increase, which means the development of brittle failure. Moreover, the failure zones of vault and invert distribute symmetrically. In addition, there are tension-p failure zones around tunnel hole, which is due to the Poisson’s effect that lateral extension strain with a relative larger ${\sigma }_2$ may larger than a critical value (${\epsilon }_c$). This will leads to tension failure prior to shear failure for the hard rocks [13]. It is note that the depth of the v-shape of the vault is larger than that of the invert due to the influence of dilatancy deformation in the actual excavation process. However, it is hard to consider for the analysis software using continuous elements [7].

It can be seen from Fig. 9 that excavation will not immediately lead to obvious spalling failure. The spalling failure occurs at a certain distance away from the excavation face (the cross section III). This is because the stress concentration of rock mass near the excavation face has not increased to a critical stress value of the occurrence of spalling failure. However, spalling failure occurs at the cross section III. The stress concentration value of cross section III accounts for about 45 % of the unaxial compressive strength (UCS) of the rock mass, which can be taken as the critical stress value of occurrence of spalling failure. Therefore, it is considered that when the stress concentration value reaches about 45 % of the compressive strength, spalling failure occurs in surrounding rock. The result is located in the range of the reported results for spalling occurrence, which is about 40 %-60 % UCS for hard rock [9, 22].

Fig. 9The plastic zone distributions within surrounding rock on profile Ⅰ-Ⅴ (intrinsic condition)

3.3.1. Longitudinal stresses of surrounding rock

In DS-TBM excavation condition, the stress of vault is different from that of invert because of nonuniform gap between the shields and surrounding rock. Therefore, Fig. 10(a), (b) and (c) illustrate the longitudinal stress magnitudes along 1^#, 2^# and 3^#, respectively. It can be seen from Fig. 7 and Fig. 10 that the stress paths of TBM model on the work face and in front of the work face are similar to those of the intrinsic condition. Therefore, only the stress in the areas behind the work face is analyzed in detail.

As shown in Fig. 10(a), the stress concentration of surrounding rock takes place in the area before soften backfill (less than 13 m away from work face) as the shields don’t contact with the surrounding rock, and the stress concentration value is about 64 MPa. In the meanwhile, ${\sigma }_2$ and ${\sigma }_3$ decrease to their relative stabilized values, respectively. After backfill and lining, the level of stress concentration decreases rapidly with increasingly harden grouting material. Therefore, ${\sigma }_1$ decreases to about 35 MPa, and ${\sigma }_2$ decreases to about 16 MPa. In addition, ${\sigma }_3$ has a very weak unloading process. As shown in Fig. 10(b), the 2^# point stress paths of complete TBM model are similar to those of intrinsic condition because the stress of sidewall is mainly affected by the in situ stress.

Furthermore, in the area before soften backfill, the stress paths of invert (Fig. 10(c)) are more complex than those of vault (Fig. 10(a)) due to the contact between the shields and the rock mass at the tunnel bottom. Eventually ${\sigma }_1$ decreases to 25 MPa and ${\sigma }_2$ increases to 24 MPa due to supporting effect. Meanwhile, ${\sigma }_3$ remains about 1 MPa after a number of small level unloading and loading processes.

Notably, the number of unloading processes at different monitor points is different. A brief description of the obvious unloading processes of each monitoring point and its causes are listed as follows:

• When the tunnel walls are no longer supported, the entire tunnel boundary undergoes the first unloading process.

• The invert experiences the second unloading process at the end of rear shield (11 m away from the work face) due to the contact between the rear shield and the invert.

• Since the backfill is injected in the gap between the lining and the rock mass, the vault and invert experience the final unloading process.

In conclusion, the vault, sidewall and invert experience two, one and three unloading processes respectively.

Fig. 10Longitudinal stresses of surrounding rock in complete TBM model: a) along point 1#, b) along point 2#, c) along point 3#

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b)

c)

3.3.2. Circumferential stresses and radial stresses

In simulation process of complete TBM excavation, differences exist in the stress paths between the vault and the invert because of interaction between shields and surrounding rock. Circumferential stresses and radial stresses of three cross sections (corresponding to Fig. 6(b) Ⅰ, Ⅲ and Ⅳ positions) are analyzed. Moreover, the stress of sidewall is hardly influenced by excavation method because the initial stress dominated by horizontal geostress, so the stresses of vault and invert are mainly analyzed.

It can be seen from Fig. 8(a) and Fig. 11(a) that the stress paths of surrounding rock in complete TBM model are similar to those in the intrinsic condition. However, there is a small difference in radial stresses between of 1^# and 3^# as the contact between the cutter-head and the surrounding rock behind work face.

In Fig. 11(b), the stress paths of 1^# point are different from those of 3^# point due to the interaction between the rear shield and the surrounding rock, as well as the conicity between the front and rear shields. Moreover, the maximum circumferential stress value of 1^# and 3^# point become higher, and those peak positions are also located deeper surrounding rock. In addition, the cross section before soften backfill is shown in Fig. 11(c). The difference in stress paths between 1^# and 3^# point still exists due to nonuniform gap caused by the eccentricity of the shield. The peak circumferential stress value of 1^# and 3^# point become higher, on account of longer stress relaxation time.

Fig. 11Circumferential stresses and radial stresses of internal surrounding rock points in complete TBM model: a) the cross section of work face, b) the cross section at front end of rear shield, c) the cross section before soften backfill

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b)

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3.3.3. Plastic zone distribution

Analyzing the plastic zone of surrounding rock under TBM tunneling can help to estimate the pressure on the shields. Fig. 12 shows failure zones in cross sections in steady state when interaction of TBM is activated. It can be seen from Fig. 9 and Fig. 12 that the failure zones of TBM model by the positions Ⅰ and Ⅱ are similar to those of the intrinsic condition. The v-shape failure zone in complete model also occurs in Ⅲ cross section. It can be seen that plastic zones of invert are less than those of vault, as shields and their self-weight produce a support at the invert, which closely relates to the shield eccentricity. This is consistent with the in situ observations [23, 24].

Fig. 12The plastic zone distributions within surrounding rock on profile Ⅰ-Ⅴ (complete TBM model)

In addition, some zones exit the plastic state (the zones of shear-p or tension-p) in IV position due to the stress adjustment caused by interaction between grouting and surrounding rock. In the cross section V, as increasingly grouting material strength, the surrounding rock has been stabilized by the stress adjustment.

Brittle surrounding rock in high geostress mainly fails in spalling or slabbing, and forms v-shape failure zone eventually. It can be seen from Fig. 9 and Fig. 12 that the excavation influence area of the intrinsic condition is smaller than that of the complete TBM model. This is because the high geostress conditions have the negative effects on breaking intact rock mass, but the positive impacts on TBM excavation, which are similar to the phenomena observed by Boniface A. et al. [25, 26].