Introduction

With developments in deep underground engineering, the application of rectangular-section tunnels has become increasingly common. Because of the complex three-dimensional (3D) stress environment of a deep rock mass, the sectional shape of a rectangular tunnel with different height-to-width ratios (H/Ws) causes different degrees of disaster. A reasonable sectional shape is helpful for improving the stability of a tunnel-surrounding-rock structure and reducing the damage degree of surrounding rock (Cristescu and Paraschiv 1995; Ren et al. 2005). In addition, many on-site engineering examples show that under the condition of deep high stress, the rectangular tunnel experiences an obvious spalling (slabbing) failure, which can bring enormous risks to production safety (Jiang et al. 2017; Hoek and Brown 2018; Hoek et al. 1995; Exadaktylos and Tsoutrelis 1995; Cai and Kaiser 2018; Gong et al. 2019a). For example, Gong et al. (2019a) carried out a study on spalling failure of a rectangular tunnel with a single H/W, which indicated that a deep rectangular tunnel was prone to stress-induced failure under 3D high-stress conditions, creating an approximately parallel maximum shear stress on the sidewall and resulting in obvious spalling failure. Therefore, the selection of a reasonable sectional shape, which reduces the occurrence of other rock disasters such as spalling in deep hard-rock tunnels and caverns, has attracted widespread attention from researchers in rock mechanics (Carter 1992; Martin 1993, Martin 1997; Hoek 1965; He et al. 2010, 2012; Zhu et al. 2014; He et al. 2014; Zhou et al. 2016; Du et al. 2016; Li et al. 2018a, b; Gong et al. 2018a, b, 2019a, b; Su et al. 2019).

In the past decades, researchers have carried out a large number of laboratory tests to understand the mechanism of spalling failure of the tunnel sectional shape more fully (Tuncay and Hasancebi 2009; Liang et al. 2016; Du et al. 2019; Gong et al. 2012; Zhao and He 2016; Li et al. 2018a). For example, Du et al. (2019) found that the fracture mode of rectangular and square prism specimens was more likely to change from shear fracture to spalling fracture when the shape ratio was reduced from 3 to 0.5. Li et al. (2018a) conducted a true triaxial unloading test on granite cubic specimens with different H/Ws (between 2.0 and 0.5). It was found that as the H/W of the specimen decreased, a spalling failure was more likely to occur in the case where the intermediate principal stress was low. These experiments were conducted to analyse the influence of different aspect ratios on the spalling of the sidewall (from the perspective of rock-material failure), but the influence of the spatial structure of the section in an actual tunnel on the spalling failure was not considered.

In fact, because of the deep 3D high stress, the damage to the surrounding rocks is affected by the tunnel space structure. Usually, spalling is only a local damage after the tunnel excavation has ended (Li et al. 2012; Xu et al. 2016). For example, after the excavation of Jinping II Hydropower Station, obvious local spalling occurred on the sidewall of the tunnel (Hou et al. 2013; Xu et al. 2016). Therefore, researchers simulated the influence of a spatial structure of a tunnel section on the spalling using rock materials with holes (Carter 1992; Martin 1993, 1997; Hoek 1965; Zhou et al. 2016). Martin (1993, 1997) performed an indoor uniaxial compression test on a circular hole in the Lac du Bonnet granite with a diameter of 5 to 103 mm and observed the spalling failure on the sidewall of the hole. It was also pointed out that when the diameter of the circular hole was larger than approximately 75 mm, the initial stress of the spalling failure was approximately equal to the unconfined compressive strength of granite. Carter (1992) carried out uniaxial compression tests on physical models of nine different hole sizes (diameters between 3.2 and 62 mm) and analysed the effect of size and stress gradient on the spalling of the surrounding rocks. Hoek (1965) and Martin (1993) performed a biaxial compression test on a circular open rock specimen; a failure zone formed by the spalling failure was observed on the sidewall of the hole, and tensile cracks occurred due to spalling failure. Zhou et al. (2016) studied the spalling-failure characteristics of circular holes with different bore diameters and straight-wall arch holes with different sizes under a 3D high stress using a true triaxial compression test, then analysed the influence of the radius of curvature of the excavated section on the spalling failure. Martin (1997) drilled a series of vertical boreholes with different diameters of 75, 150, 300, 600 and 1250 mm from the floor of room 405 in the underground research laboratory to analyse and determine the strength-scale effect of different diameters in the field. These studies mainly focused on the spalling test of a sectional shape of a circular tunnel with different diameters, whereas studies on the spalling test of a sectional shape of a rectangular tunnel with different H/Ws are few. In addition, these experimental studies demonstrated that the failure characteristics of spalling are affected by the cross-sectional shape of the tunnel. Therefore, studying the sidewall spalling of rectangular tunnels with different H/Ws under 3D conditions is necessary.

In the present study, five types of rectangular section shapes with different H/Ws (1/2, 3/4, 1/1, 4/3 and 2/1) are designed, and a series of true triaxial tests are carried out on cubic granite specimens with rectangular holes to study the influence of different H/Ws of rectangular holes on the spalling. In the entire test process, a wireless microcamera is used to monitor and record the whole process of sidewall damage in real time, which can correspond to the 3D stress in real time. By analysing and summarising the spalling-failure process and the rectangular holes with different H/Ws, the influence and mechanism of different H/Ws of the rectangular-hole section on the spalling were obtained.

Experimental design

Rock specimen

To study the process and mechanism of spalling failure in deep rectangular tunnels with different H/Ws, the specimens were selected from granite rock with good integrity and texture. The microstructures of the granite specimens were observed using optical microscopy, and their mineral compositions were analysed (Fig. 1a). They had a primary mineral composition of quartz (~ 49%; grain size 0.1–0.5 mm), plagioclase (~ 43%; grain size 0.1 × 0.4–0.4 × 1.3 mm), biotite (~ 4%; grain size of 0.04 × 0.1–0.4 × 0.8 mm), amphibole (~ 3%; grain size of 0.2–0.8 mm), and apatite and opaque minerals (~ 1%; grain size of 0.05–0.2 mm). The uniaxial compressive stress-strain curve of the granite specimens are shown in Fig. 2a, and the basic physical and mechanical properties of the granite are obtained. The uniaxial compressive strength, average density and wave velocity of the granite specimens were 261.55 MPa, 2.80 g/cm3 and 5419.00 m/s, respectively. The uniaxial compression failure mode of the granite specimen is shown in Fig. 2b, which shows a sudden brittle failure, and the specimen has a fragment ejection phenomenon, which has a typical rockburst impact tendency (Gong et al. 2019c).

Fig. 1
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The granite specimens: a microstructures of granite specimens (Q is quartz, Pl is potassium feldspar, Bt is biotite, Hbl is amphibole) (Gong et al. 2019a); b specimens shapes

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Fig. 2
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Results of uniaxial compression tests on cylindrical specimens (Gong et al. 2019c): a typical stress-strain curves, and b typical failure characteristics

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According to existing research results (Gong et al. 2019a), the granite was processed into cubic blocks of 100 × 100 × 100 mm3, and rectangular holes were cut through the cube blocks. To simulate the failure process of rectangular holes with different H/Ws under the 3D high-stress state, the rectangular holes were designed according to the list in Table 1. The prepared specimens are shown in Fig. 1b.

Table 1 Specification of rectangular holes
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Test equipment

The test was carried using a TRW-3000 rock true triaxial electro-hydraulic servo mutagenesis test machine with a vertical loading capacity of 3000 KN and a horizontal loading capacity of 2000 KN, as shown in Figs. 3a and b. In addition, to observe the process of real-time damage of the rectangular holes under 3D loading conditions, a video surveillance system was installed, as shown in Figs. 3c and d. For the detailed parameters of the experimental equipment, the readers are referred to Gong et al. (2018a).

Fig. 3
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Test equipment: a trw-3000 true triaxial equipment; b 3D loading device; c video monitoring work diagram; d wireless micro camera

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Three-dimensional stress condition

The failure process of the 3D initial stress environment of the deep rectangular tunnel at the depth of 1000 m was simulated. The calculation method of the 3D initial stress is described in the literature (Gong et al. 2018a), and the calculation results are listed in Table 2. In the test, the rectangular tunnel was arranged along the direction of the maximum initial horizontal direction, and the influence of the surrounding rock stress adjustment on the tunnel sidewall damage was simulated by increasing the Z-direction stress to the maximum principal stress during the test. Therefore, the 3D initial stress of the rectangular hole in the simulation test was σx = 51.0 MPa, σy = 34.0 MPa and σz = 27.0 MPa. The 3D initial stress loading mode is shown in Fig. 4.

Table 2 Initial in-situ stress at 1000 m depth
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Fig. 4
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3D initial stress loading diagram

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Three-dimensional stress loading path

The 3D stress loading path of the specimens in the test is shown in Fig. 5. The specific loading method is described as follows. (I) Initial stress loading stage: the three directions of the specimen were loaded at the same loading rate (0.1 MPa/s) using the load-displacement control method up to the 3D initial stress level. (II) Stress adjustment stage: the Y-direction stress was kept unchanged, and the X-direction was then adjusted to the displacement-load control mode. The displacement was kept unchanged (simplified as plane strain), and finally increase the Z-direction stress with the original loading rate (0.1 MPa/s). (III) Z-direction grading loading stage: to effectively observe the spalling on the sidewall of the specimen, when an obvious local damage of the sidewall of the specimen was observed, the Z-direction stress is adjusted to be graded and kept for a certain period of time. (IV) Unloading stage: when no obvious damage on the sidewall occurred, to prevent overall instability of the specimen, the Z-direction stress loading was stopped and maintain loading for a certain period of time, and finally stop loading in the three directions. The unloading process in this study is described as follows. First, σz was unloaded to σx. Then, they were unloaded together to σy. Finally, they were all unloaded to zero.

Fig. 5
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3D stress path loading schematic

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Experimental results and discussion

Test curve

The 3D stress path curve of rectangular holes with different H/Ws after the end of the entire test is shown in Fig. 6. Figure 6 shows that after all the specimens were loaded to the initial stress level, the X-direction control mode was adjusted from the loading to displacement control, and the X-direction deformation was kept unchanged (i.e. simulating a plane strain problem). With the increase in σz, σx fluctuated and slowly increased with the increase in σz, which was influenced by the Poisson effect and accuracy of the true triaxial-test equipment.

Fig. 6
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The actual 3D stress loading diagram of rectangular hole: a HW-20-40; b HW-30-40; c HW-40-40 (Gong et al. 2019a); d HW-40-30; e HW-40-20

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Failure mode of the sidewall

The overall and local failures of the left and right sidewalls in the rectangular holes after the test are shown in Fig. 7, which show that under 3D high-stress conditions, cracks were generated on the specimen surfaces, but all specimens remained intact. Significant spalling failure occurred on the sidewalls of all specimens, and the spalling degree was different, because of the different H/Ws of the rectangular holes in the specimens. Some of the sidewalls of the specimens formed obvious thin rock slabs, whereas others exhibited obvious failure zones after the cracks penetrated. Figures 7a–d show that significant spalling failure occurred on the sidewalls, forming thin rock slabs that were approximately parallel to the sidewall. Some thin rock slabs were partially opened due to the cracks that did not completely penetrate, and they were not completely separated from the sidewalls (Figs. 7a and d). Additionally, some thin rock slabs accumulated on the sidewall or fell off to the bottom of the holes, because cracks penetrated through the sidewall (Figs. 7a–c). However, because of the H/Ws of the specimens, no obvious thin rock slabs were generated on the sidewall, and failure zones after the crack penetration were observed only at the shoulder angle of the sidewall (Fig. 7e). Moreover, HW-40-20 exhibited shear cracks that spread from the shoulder angle to the surface of the specimen, especially at the right lower shoulder angle, which indicated that the specimen is prone to overall shear failure (Fig. 7e).

Fig. 7
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The failure of the whole specimen and the left and right sidewalls after the test: a HW-20-40; b HW-30-40; c HW-40-40; d HW-40-30; e HW-40-20

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To compare the spalling phenomena in deep tunnels, Fig. 8 shows the typical spalling in the surrounding rock of a deep engineering site. Figures 8a–d show obvious spalling failure of the tunnel sidewall due to the redistribution of the surrounding rock stress after excavation of the deep tunnel. Compared with those shown in Figs. 7 and 8, the failure modes of the sidewalls in the rectangular holes in the test were consistent with those of the in situ rectangular tunnels, demonstrating typical spalling characteristics. This result indicates that the spalling simulation test in this study is reliable for simulating the spalling-failure process of rectangular holes under a 3D stress.

Fig. 8
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Examples of on-site spalling: a mine tunnel sidewall spalling failure (Hoek and Brown 2018); b under high stress right sidewall spalling failure (Hoek et al. 1995); c in the transport sidewall spalling (Stacey 1981); d sidewall spalling failure (Jiang et al. 2017)

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Analysis of spalling-failure process

To understand the specific process of spalling of the sidewall in the rectangular hole in detail, the entire test process was recorded using a wireless miniature camera, and the typical damage image of the sidewall of the hole was recorded by video to illustrate the spalling-failure process, as shown in Fig. 9. Obvious spalling failure occurred on the sidewalls in the rectangular holes, and some thin rock slabs fell off. Owing to the different H/Ws of each specimen, the degree of spalling on the sidewalls was also different. However, the failure modes of all specimens were the same, which indicated that the typical tensile failure and spalling-failure processes on the sidewall were the same. Therefore, by considering the spalling-failure process of the HW-20-40 specimen as an example, the spalling-failure process was analysed in detail as follows (Fig. 9a): When loaded at the 3D initial-stress level, the sidewall of the specimen was not damaged (Fig. 9a1). When σz was increased to 52.1 MPa, microcracks were generated at the upper and lower shoulders of the sidewall (Fig. 9a2). With the increase in σz, the crack propagated along the lower shoulder angle on the left sidewall and upper shoulder angle on the right sidewall, accompanied by ejection of fine particles during the crack propagation (Fig. 9a3). Next, the crack continuously propagated along the axis at the shoulder angles on both sidewalls (Figs. 9a4–a6). When σz = 144.0 MPa, the cracks partially penetrated the shoulder angles of the left and right sidewalls, resulting in a large quantity of fine particles falling off (Fig. 9a7). As σz increased to 148.0 MPa, thin rock slabs parallel to the maximum vertical principal stress were formed on the right sidewall and opened, which was an obvious indication of a spalling failure (Fig. 9a8). When σz = 152.3 MPa, the initial thin rock slab on the right sidewall finally fell off to the bottom of the hole, and cracks continued to propagate and partially penetrated, resulting in a new thin rock slab on the right sidewall. Meanwhile, the thin rock slab formed on the left sidewall remained in an open state (Fig. 9a9). When σz increased to 153.8 MPa, the thin rock slabs on both sidewalls continued to open, and the thin rock slabs on the right sidewalls were more obviously separated (Fig. 9a10). At 154.0 MPa, the thin rock slabs on both sidewalls fell off, and the new thin rock slabs on the right sidewall were separated (Fig. 9a11). Finally, when σz = 156.0 MPa, the right thin rock slab completely fell off, and no obvious damage on both sidewalls was observed until the end of the test (Fig. 9a12).

Fig. 9
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Screenshots of specimen spalling process: a HW-20-40; b HW-30-40; c HW-40-40; d HW-40-30; e HW-40-20

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Influence of different H/Ws on spalling failure

The vertical stress (σz) of each specimen at the initial failure is shown in Fig. 10, which shows that when the widths of the specimen were the same, σz required for the initial failure of the specimen was not much different as the height increased. Meanwhile, when the height was consistent, σz required for the initial failure of specimens significantly decreased with the decrease in the width. These results indicated that initial failure of the specimens is more likely to occur with the increase in H/W. Therefore, to analyse the influence of different H/Ws on the spalling-failure characteristics and stability of hard-rock rectangular tunnels, detailed analysis and discussion were carried out.

Fig. 10
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Initial vertical stress magnitude in spalling failure

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Figure 11 shows a comparison of the different heights of rectangular tunnels for the same width. When σz = 150.0 MPa, the sidewalls of HW-20-40 were destroyed. In particular, the thin rock slab on the right sidewall was opened (Fig. 11a1). Meanwhile, those of HW-30-40 only exhibited fine particles falling off at the lower shoulder angle on the left sidewall, whereas the whole specimen remained intact (Fig. 11b1). However, both sidewalls of the HW-40-40 exhibited obvious thin rock-slab opening (see Fig. 11c1). With the increase in σz to 156.0 MPa, obvious thin rock-slab shedding phenomena appeared on both sidewalls of HW-20-40 (Fig. 11a2). The thin rock slab on the right sidewall of HW-30-40 fell off, and the left sidewall remained intact (Fig. 11b2). The damage degree of the HW-40-40 sidewall was obviously more serious than that of the other two specimens, the new thin rock slabs on the left sidewall continued to fall off, and the failure zone became larger (Fig. 11c2). By combining Figs. 11 and 9a–c, a comparison of the rectangular-hole specimens at three different heights showed that the degree of damage of the rectangular holes from high to low followed the specimen order HW-40-40, HW-20-40, and HW-30-40. The spalling-failure degree did not decrease with the increase in the height.

Fig. 11
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Contrast chart of spalling failure degree of section shape with same width and different height under the same vertical stress: a HW-20-40; b HW-30-40; c HW-40-40

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Figure 12 shows the comparison of different widths of rectangular tunnels at the same height. When σz = 134.0 MPa, the thin rock slabs on the right sidewalls of HW-40-40 obviously opened (Fig. 12a1). Cracks penetrated the upper left shoulder and lower right shoulder angles of HW-40-30, and bulges appeared on both sidewalls (Fig. 12b1). Cracks penetrated the upper left shoulder and lower right shoulder angles of HW-40-20, but both sidewalls were not damaged (Fig. 12c1). With the increase in σz to 140.0 MPa, the thin rock slab on the right sidewall of HW-40-40 further opened, whereas the crack penetration on the left sidewall led to opening of the thin rock slab (Fig. 12a2). In contrast, both sidewalls of the HW-40-30 further bulged, and the damage at the shoulder angles on both sidewalls was more obvious (Fig. 12b2). Conversely, at the shoulder angles on both sidewalls of the HW-40-20, the crack continued to propagate and penetrate in which fine particles fell off but the sidewall remained intact (Fig. 12c2). By combining Fig. 12 and Figs. 9a–c, comparison of the rectangular-hole specimens at three different widths revealed that the degree of damage of the rectangular holes from high to low follow the following order: HW-40-40, HW-40-30 and HW-40-20. The spalling failure degree decreased with the decrease in the width.

Fig. 12
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Contrast chart of spalling failure degree of section shape with same height and different width under the same vertical stress: a HW-40-40; b HW-40-30; c HW-40-20

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Figures 13 and 14 show comparisons of different H/Ws for the same rectangular-section area. Figure 13 shows that under the same vertical stress, when the H/Ws were 1/2 (Fig. 13a1) and 2/1 (Fig. 13 b1), the stress concentration at the shoulder angle on both sidewalls was very obvious, which led to the continuous propagation and penetration of cracks at the shoulder angle and formation of an obvious failure zone. However, when the H/W was 1/2, obvious damage was observed on the sidewall of the specimen, which released part of the energy (Fig. 13a2). In contrast, when H/W was 2/1, the specimen was damaged only at the shoulder angle, but the sidewall remained intact (Fig. 13b2). Further, cracks that propagated from the shoulder angle to the surface in HW-40-20 were observed, as shown in Fig. 7e, and obvious shear cracks were formed, which presented a greater risk of overall shear failure of the specimen. In summary, the spalling failure degree of the 2/1 H/W was lower than that of the 1/2 H/W, which could effectively restrain the development of spalling. Figure 14 shows that when σz = 154.0 MPa, thin rock-slab openings appeared on the sidewalls of both specimens (Figs. 14a1 and b1), but the thin rock-slab openings were more obvious in HW-30-40. When σz was increased to 156.0 MPa, the thin rock slabs formed on the right sidewall of HW-30-40 were detached, and new thin rock slabs were opened (Fig. 14a2). In contrast, cracks on the sidewalls of HW-40-30 continued to develop and penetrated, and no obvious phenomenon of thin rock-slab opening was observed. Both sidewalls of the specimens were basically intact (Fig. 14b2). Obviously, the spalling-failure degree of the 4/3 H/W was lower than that of the 3/4 H/W, which could effectively restrain the development of spalling. In summary, we can conclude that when the rectangular cross-sectional areas are the same, the arrangement in which the height is more than the width is adopted, which can improve the stability of the surrounding rocks of a tunnel and effectively reduce the degree of spalling.

Fig. 13
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The same cross-section area under the same vertical stress: a HW-20-40; b HW-40-20

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Fig. 14
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The same cross-section area under the same vertical stress: a HW-30-40; b HW-40-30

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Conclusions

In this study, simulation tests of the spalling failure of a granite rectangular hole with different H/Ws were carried out, to determine the influence of the H/W of specimens on spalling failure. During the entire test process, the spalling failure in the rectangular holes was successfully reproduced by adjusting the vertical principal stress to the maximum principal stress. The test results were analysed in detail, and the following conclusions were drawn.

  1. (1)

    Spalling process of the sidewall: a crack was first created at the shoulder angle of the sidewall in the rectangular hole and propagated along the axis. After the crack spread to a certain extent, spalling occurred in a local area on the sidewall. With the increase in stress, the crack continued to develop and penetrate, and the spalling failure gradually developed from the local area to the entire sidewall along the axial direction. The thin rock slabs continuously fell off.

  2. (2)

    Influence of the different H/Ws in the rectangular holes on the spalling failure. (a) When the rectangular-hole widths were the same, the spalling-failure degree did not decrease with the increase in the height. (b) When the heights of the rectangular holes were the same, the spalling failure degree decreased with the decrease in the width. (c) When the cross-sectional areas of the rectangular hole were the same, the arrangement in which the height was more than the width could effectively reduce the degree of spalling failure. (d) The larger the H/W of the rectangular hole, the more likely the stress concentrated at the shoulder angle of the sidewall, resulting in obvious crack propagation and penetration at the shoulder angle, which is prone to overall shear failure. (e) When the H/W of the rectangular hole was 1/1, the degree of spalling failure was the highest, which presented the greatest risk.

The aforementioned conclusions provide a theoretical basis for the selection and safety management of the sectional size of deeply buried rectangular tunnels. For safety management of rectangular tunnels with different H/Ws, crack monitoring at the shoulder angle of the sidewall should be emphasised, and supporting measures such as providing bolts on the sidewalls should be applied to improve the stability of the surrounding rock. As far as possible, the arrangement in which the height is more than the width can effectively reduce the degree of spalling failure.