2.1 Establishment of mechanical model

The excavation of the cavern causes stress redistribution and tangential stress increase. The roof loosens and breaks as the limit strength is exceeded. For shallow-buried large-span caverns, the thin overlying strata and large cross-section span, the roof arises fracture zone after excavation. When the boundary of the fracture zone does not reach the ground level, the rock mass between caverns roof and the ground level partially collapses, ie., arched failure [31, 32], as shown in Figure 1(a). When the boundary of the fracture zone reaches the ground level, the rock mass between the cavern roof and the ground level will overall collapse, ie., collapsed failure [33, 34], as shown in Figure 1(b).

Figure 1全屏查看

Roof failure mechanical modes of shallow-buried large-span caverns: (a) Arched failure mode; (b) Collapsed failure mode

Take the circular arch shallow-buried large-span caverns commonly used on site as an example. The distance from the top of the cavern to the ground level is h_d. The span of the cavern is 2l_t. The height of the straight wall on both sides is h_w. The expression equation of roof arch is g(x). The corresponding center angle is α, and the circle center is point O.

The non-caving part of the roof is modelled as an ideal rigid-plastic material, and the caving part is modelled as a rigid material, which settles downward along the y-axis with the velocity . Assuming that the collapsed rock is symmetrical, the corresponding fracture curve equation is f(x). The top width of the collapsed rock is 2l_g (l_g is 0 in arched failure mode). The bottom width is 2l_c, and the height is h_c. The distance from the top of the collapsed rock to the vault is h_y, and the minimum distance between caved rock and straight wall is h_m. Point M and point N are the intersection points between the boundary line of the collapsed rock and the ground level in collapsed failure mode. Point L is the intersection point between MO and the circular arch roof. The angle between MO and NO in collapsed arched failure modes is β.

The stress is modelled as the distributed stress q_r acting on the collapsed rock and perpendicular to the collapsed rock. The corresponding principal stress at caving rock surface is σ_n, the angle from the x-axis is φ, and the shear stress is τ_n. The bolt is evenly arranged according to the angle of the cavern roof, the prestress of the bolt is P, and the angle between the adjacent bolts is θ.

2.2 Upper limit method of bolting support for shallow-buried large-span caverns

The upper limit method uses the principle of virtual work to construct the allowable field of velocity boundary condition and deformation coordination condition, so that the power of internal force and external force are equal, to determine the ultimate load required for rock mass failure.

The collapsed rock is symmetrically distributed. Take the collapsed rock in the range of [0, l_c] as the research object to analyze the failure mechanism of the roof. The failure surface of the roof is regarded as the layer with thickness ω. Under the condition of considering dead weight, bolt prestress, and surrounding rock stress, the principle of equal internal and external power can be expressed as:

(1)

where is the power of internal force of caving rock mass; is the power produced by falling body weight of rock; is the power by surrounding rock stress; is the power by water pressure; is the power by bolt prestress.

2.2.1 Failure analysis of roof based on arched failure mode

(1) The power of internal force to do work

According to the plastic potential theory [35], assume that the plastic potential surface of rock mass coincides with the yield surface. The energy dissipation rate per unit volume of caving rock surface is [36]:

(2)

where and are the rate of plastic normal strain and shear strain of caving rock surface, respectively; σ_c and σ_t are the compressive strength and tensile strength, respectively; A and B are the empirical parameters.

Assume that the arch curve total length of the collapsed roof fracture surface in the range of [0, l_c] is s. Integrating Eq. (2) in the range of [0, s], the power of internal forces to do work generated at the thin deformation layer of the fracture surface is shown as follows:

(3)

where is satisfied.

(2) The power of external force to do work

When the roof is broken, the dead weight, the surrounding rock stress and the bolt prestress will accelerate or restrain the failure and do work.

1) Work power by dead weight of collapsed rock

Assuming that the weight density of rock is γ, the power of rock dead weight accelerating roof rupture can be expressed as:

(4)

where the curve equation of the vault is .

2) Work power by surrounding rock stress

The surface force of caving rock is complicated, and the surrounding rock stress distributed along the fracture curve is calculated by definite integral. The power can be expressed as:

(5)

3) Work power by water pressure

The water pressure distributed along the fracture curve is calculated by definite integral, and the power can be expressed as:

(6)

4) Work power of bolt prestress

According to Figure 1(a), the number of bolts arranged in the surrounding rock collapse range

[0, l_c] can be expressed as follows:

(7)

where ψ is the central angle corresponding to the collapsed rock, and its expression is . When n is a decimal number calculated by Eq. (7), n takes the integer part and the decimal part is deleted. The power of bolt prestress can be expressed as follows:

(8)

where κn is a coefficient related to the number of bolts.

If n is even, κn can be expressed as follows:

(9)

If n is odd, κn can be expressed as follows:

(10)

(3) Design of bolt support parameters

The key design parameters of the bolt include bolt length, prestress, spacing, etc. To quantify the design parameters, this section explores the main support parameters of bolts using theoretical calculation methods, and designs the prestress and length of bolt.

1) Design of bolt prestress

Substitute the calculation results of the power of internal forces to do work , work power of dead weight of collapsed rock , work power of surrounding rock stress , work power of water pressyre , work power of bolt prestress into Eq. (1):

(11)

The prestress P of single bolt can be expressed as follows:

(12)

where,

(13)

The prestress required for a single bolt is expressed by Eq. (12), which is a universal function of f (x). When Eq. (12) takes the maximum value, the solution of this equation can be obtained. So, Euler’s equation can solve it:

(14)

Bring the relevant parameters into the equation:

(15)

The following equation can be obtained after sorting out,

(16)

By integrating Eq. (16), the fracture curve equation can be obtained:

(17)

where c₁ and c₂ are undetermined constants. The failure curve f(x) is symmetrically distributed along the y-axis because of the symmetry of the collapsed rock. Therefore, c₁=0 should be satisfied. And the relationship is as follows:

(18)

Equrations (17) and (18) can be solved simultaneously:

(19)

The fracture curve equation can be obtained:

(20)

Combining Eq. (20) with Eq. (12), the prestress required for a single bolt can be expressed as follows:

(21)

2) Design of bolt support length

To ensure that the shallow-buried large-span caverns roof rock bolt has a good control effect, the bolts are anchored outside the damage range. The height h_y from the vault to the top of the collapsed rock under arched failure mode can be calculated by Eq. (20):

(22)

The bolt support length l_b should be designed as:

(23)

2.2.2 Failure analysis of roof based on collapse failure mode

(1) The power of internal force to do work

According to the above derivation results, in the range of [0, l_c], the internal force work power at the fracture surface can be expressed as:

(24)

(2) The power of external force to do work

1) Work power by dead weight of collapsed rock

The power of the dead weight can be obtained:

(25)

According to geometric relationship of collapse failure model, the curve equation of the roof can be obtained as .

2) Work power of surrounding rock stress

The caving surface stress distributed in the [l_g, l_c] range is integrated, and the work power of surrounding rock stress can be expressed as:

(26)

3) Work power of water pressure

The water pressure distributed in the [l_g, l_c] range is integrated, and the work power of water pressure can be expressed as:

(27)

4) Work power of bolt prestress

Because the boundary curve of the caving body is in contact with the ground plane, the bolt cannot be anchored in the stable rock stratum in this area. Therefore, this section does not consider the support force of the bolt in this area, and the number of bolts n required to be arranged in the roof [l_g, l_c] range is:

(28)

When n is a decimal number calculated, n takes the integer part and the decimal part is deleted. The power of bolt prestress can be obtained:

(29)

κn can be expressed as follows:

(30)

(3) Design of bolt support parameters

1) Design of bolt prestress

Substitute the calculation results of the power of internal forces to do work , work power of dead weight , work power of surrounding rock stress , work power of water pressure , and work power of bolt prestress into Eq. (1):

(31)

After arranging Eq. (31), the expression of the prestress P of single bolt can be obtained:

(32)

In the equation, φ[x, f(x), f '(x)] is the same as that of Eq. (13), so the fracture surface equation can be expressed Eq. (20). Solving Eq. (20) and Eq. (32) simultaneously, the prestress P of single bolt can be obtained:

(33)

In the equation,

.

2) Design of bolt support length

Because the bolts cannot play a supporting effect in the range of [0, l_g], the bolt length should be greater than the length of ML in the collapsed failure mode. The expression of the length of ML can be expressed as follows:

(34)

According to the collapse failure model, the expression of β can be expressed as follows:

(35)

According to Eq. (34), the bolt support length l_b is:

(36)

2.3 Bolt support design under different failure modes

Based on the above theoretical derivation, the reasonable design value of bolt support parameters for shallow-buried large-span caverns can be carried out. The most conservative calculation takes into account the most unfavorable situation of roof failure, and the roof of the caverns collapses in the whole range, that is, l_c=l_t.

(1) Failure mode discrimination

By substituting l_c=l_t into Eq. (22), the discriminant of surrounding rock failure mode can be expressed as follows:

(37)

When h_y<h_d, the boundary of the collapsed rock does not reach the ground level, which belongs to arched failure. On the contrary, the boundary of the collapsed rock reaches the ground level, which belongs to collapsed failure.

(2) Design of bolt support

By substituting l_c=l_t into the analytical solutions of bolt prestress and support length, the design equations of bolt support parameters can be obtained.

Arched failure mode: Roof collapse height does not reach the ground line after excavation. The reasonable design value of single bolt prestress in the failure model can be obtained:

(38)

The bolt support design length l_b can be expressed as follows:

(39)

Collapsed failure mode: Roof collapse height reaches the ground line after excavation. The reasonable design value of single bolt prestress in the failure model can be obtained:

(40)

In the equation,

.

The bolt support length l_b can be expressed as follows:

(41)

The expression of β is:

(42)

3.1 Influence mechanism of bolt support length

According to the calculation equation of bolt support length in Section 2.3, the relationship curves of the weight density of rock γ, the compressive strength σ_c, the cavern width 2l_t on the bolt support length are drawn (see Figure 2). The sensitivity coefficient of bolt support length is compared (see Figure 3).

Figure 2全屏查看

Influence mechanism of bolt support length

Figure 3全屏查看

Comparison of sensitivity coefficients of bolt support length: (a) The proportion of each parameter; (b) The sensitivity coefficient of each parameter

1) The length of bolt support is negatively correlated with the compressive strength, positively correlated with the weight density of rock, and exponentially varied with the width of cavern.

2) The sensitivity coefficients of the compressive strength, the weight density of rock and the cavern width are 1.68, 1.61 and 14.86 respectively. Compared with other influencing factors, the width of the cavern has a greater influence on the length of the bolt support. Under the condition the design requirements of the safety and function for the cavern, the design value of the cavern width should be reduced as far as possible.

3.2 Influence mechanism of bolt prestress

According to the calculation equation of bolt prestress in Section 2.3, the relationship curves between the bolt prestress and the compressive strength σ_c, the weight density of rock γ, the tensile strength σ_t, the surrounding rock stress q, the cavern width 2l_t are drawn, as shown in Figure 4. Comparison of sensitivity coefficients of bolt prestress is shown in Figure 5.

Figure 4全屏查看

Influence mechanism of bolt prestress: (a) Results for different compressive strengths; (b) Results for different tensile strengths; (c) Results for different weights densities; (d) Results for different surrounding rock stress; (e) Results for different cavern widths

Figure 5全屏查看

Comparison of sensitivity coefficients of bolt prestress: (a) The proportion of each parameter; (b) The sensitivity coefficient of each parameter

1) Under the same influence factors, the prestress of single bolt is positively correlated with the angle between bolts. Increasing the bolt density can reduce the prestress of single bolt.

2) The prestress of single bolt is negatively correlated with the tensile strength, which has great influence on it. It is also negatively correlated with compressive strength and has little influence on it. The better the mechanical properties and integrity of rock, the lower the prestress of single bolt.

3) The prestress of single bolt is positively correlated with the surrounding rock stress, which has great influence on it. It is also positively correlated with the weight density of rock and has little influence on it. Complex environment and large density of rock will lead to the increase of single bolt prestress.

4) The prestress of single bolt and the cavern width show an exponential growth trend. The bolt prestress required for a cavern with a width of 28 m is 162.5% higher than that of a cavern with a width of 12 m. The results show that the large-span caverns need high prestress bolt for support.

5) Taking the bolt arrangement scheme with an angle of 6° between adjacent bolts as an example, the influence mechanism of bolt prestress is analyzed by sensitivity coefficient. The sensitivity coefficients of weight density of rock, compressive strength, cavern width, tensile strength, surrounding rock stress are 0.30, 0.51, 1.22, 52.40, and 147.62, respectively. Analysis of sensitivity coefficient shows that the surrounding rock stress and the tensile strength have significant influence on the bolt prestress.

Under the conditions of large surrounding rock stress and poor rock properties, the roof is prone to collapsed failure. Therefore, high prestress bolt should be used to support the caverns to provide high strength support force and ensure the safety and stability of caverns.

4.1 Mechanical properties of high prestress bolt

The material comparison tests between CREA bolt and HRB335 bolt are carried out by static tensile test. The size parameter of bolt is Φ18 mm×1200 mm. The test device and bolts are shown in Figure 6. The strength-elongation curves of the CREA bolt body and HRB335 bolt body are shown in Figure 7. The elongation of each section for the bolts is counted after the test, as shown in Figure 8. The statistical results of the strength and elongation of the two types for bolts are shown in Figures 9 and 10.

Figure 6全屏查看

The test device and bolts

Figure 7全屏查看

Strength-elongation curves

Figure 8全屏查看

Segment elongation lengths

Figure 9全屏查看

Yield strength and elongation

Figure 10全屏查看

Breaking strength and elongation

1) Comparison of mechanical properties

The yield strength values of HRB335 bolt and CREA bolt are 388.0 MPa and 812.6 MPa respectively; the breaking strength values of HRB335 bolt and CREA bolt are 576.0 MPa and 907.8 MPa, respectively. Compared with the former, the yield strength and breaking strength of the latter material are increased by 109.4% and 57.6%, respectively. At the same time, CREA bolt experienced two stages of elasticity and constant resistance during the tensile process. The tensile curve of static test shows that CREA bolt has high strength bearing capacity and ideal elastic-plastic properties. In the support control of shallow-buried large-span caverns, CREA bolt can be used to achieve high prestress active supporting for surrounding rock.

2) Comparison of deformation ability

The yield elongation and breaking elongation of HRB335 bolt are 1.5% and 15.1% respectively, and those of CREA bolt are 2.5% and 26.9% respectively. Compared with the former, the yield elongation and breaking elongation of the latter are increased by 66.7% and 78.1%, respectively. The average deformation of each section of HRB335 bolt is 13.0 mm, and the variance is 12.9 mm², while those of CREA bolt are 20.5 mm and 5.4 mm². Compared with the former, the average deformation of the latter increased by 57.7% and the variance decreased by 58.1%. The experimental data show that CREA bolt has significant advantages in the total deformation and uniform deformation of the bolt body, and the diameter of the bolt fracture has little change.

3) Temperature comparison of each section

A thermal infrared imaging system is used to record the temperature variation of bolt during tension. The temperature distribution of each section for the two types bolts are shown in Figures 11 and 12.

Figure 11全屏查看

Temperature distribution of each section for CREA bolt

Figure 12全屏查看

Temperature distribution of each section for HRB335 bolt

The fracture temperature, average temperature, and variance of HRB335 bolt are 42.33 ℃, 31.58 ℃ and 10.88 ℃², respectively, and those of CREA bolt are 35.78 ℃, 33.37 ℃ and 1.68 ℃², respectively. Compared with the former, the fracture temperature of the latter is 6.55 ℃ lower and the average temperature is 1.79 ℃ higher. The temperature variance of each section for HRB335 bolt is 6.48 times that of CREA bolt. Thermal infrared system data show that CREA bolt has the mechanical characteristics of uniform stress and uniform deformation.

4.2 High prestress bolt calculation model

While the carven is excavated, the high prestress can restore the normal stress on the free surface to the maximum extent in the shortest time. The stress environment of the near surface surrounding rock degraded by excavation is improved to resist the deformation of surrounding rock. At the same time, it plays an important role in improving the self-bearing capacity of surrounding rock [37-39]. Therefore, the higher the prestress of the bolt, the better the control effect of surrounding rock.

The calculation model of CREA bolt is obtained according to its mechanical properties, as shown in Figure 13. CREA bolt experiences two stages of elasticity and constant resistance during the tensile process, and its mechanical properties behave as quasi-ideal elastoplastic. Compared with HRB335 bolt, CREA bolt has a higher safety margin even if the prestress increases to 60%-70% of the yield load.

Figure 13全屏查看

Calculation models of CREA bolt and HRB335 bolt

5.1 High prestress bolt support design method

The design method of high prestress bolt for shallow-buried large-span caverns is shown in Figure 14. Failure mode discrimination, design of bolt support length and bolt prestress are as follows:

Figure 14全屏查看

High prestress bolt design method for shallow-buried large-span caverns

1) Failure mode discrimination: The distance h_y from the roof to the top of collapsed rock is derived by bringing the relevant calculation parameters into Eq. (37), and the damage type of the roof is obtained by comparing the values of h_y and h_d from the roof to the ground line.

2) Design of bolt support length: Combined with different failure modes and Eq. (39) or

Eq. (41), the bolt length l_b is designed, ensureing that the length of the bolt exceeds the fracture line to play the anchoring role.

3) Design of bolt prestress: The bolt spacing is designed according to the requirements of the specifications for bolt-shotcrete support [40].

Eq. (38) or Eq. (40) is used to calculate the prestress P required for a single bolt corresponding to different design schemes, and to ensure that the prestress P of the bolt is less than 70% of its yield load f_y.

5.2 Field support design and application

According to the site parameters of Qingdao metro station in China, combined with the above design basis, the roof support of shallow-buried large-span cavern is designed with high prestress bolt.

1) Failure mode discrimination: According to Eq. (37), the height h_y from the roof to the top of the collapsed rock is 2.3 m, and the falling body fails to meet the ground line. The roof at this site belongs to the arched failure mode.

2) Design of bolt support length: To prevent the bolt from anchoring in the loose rock formation, and ensure that it plays the anchoring role, the length of the bolt should exceed the fracture curve. According to Eq. (39), the minimum bolt support length can be calculated to be 2.3 m; therefore, the bolt support length is designed to be 4.0 m.

3) Design of bolt prestress: The high prestress bolt spacing is designed according to the requirements of the specifications for bolt-shotcrete support. Combined with the bolt prestress design Eq. (38), the prestress of single bolt under different bolt spacing can be calculated. In Table 1, the prestress value of a single bolt corresponding to different bolt spacing is calculated.

According to the mechanical properties of CREA bolt in Section 4.1, the yield load of CREA bolt is 206.8 kN. The prestress that can be applied is 70% of the yield load; therefore, the prestress of CREA bolt is 144.76 kN. The prestress values of single bolt in scheme I and scheme II are lower than the maximum prestress that can be applied to high prestress bolt, which meets the requirements of support design. At the same time, considering that the larger the spacing between bolts, the less the number of bolts required, and the higher the economy of the support scheme. Scheme II is selected as the optimal support scheme for field application, and the bolt prestress is designed to be 120 kN.

Bolt installation steps: 1) The construction of bolt boreholes is carried out on the surrounding rock. 2) According to the design requirements, the design length of the resin anchoring agent is filled. 3) The whole length of the bolt is inserted into the borehole and the anchoring agent is stirred to play the anchoring role. 4) The bolt is tensioned and locked with lock while the prestress design value is reached. 5) The bolt installation is completed. The surrounding rock support scheme and its effect are shown in Figure 15.

Figure 15全屏查看

Field surrounding rock support effect

The displacement of roof and axial force of bolt are systematically monitored regularly, and the control effect is analyzed, as shown in Figure 16.

Figure 16全屏查看

Monitoring results of roof surrounding rock settlement and axial force of bolt

1) According to the different falling speed, the roof settlement can be roughly divided into three processes: fast, slow, and stable. In the initial stage AB of support, due to the excavation of the cavern, the roof deformation is faster, the stress of the

bolt changes greatly, and the settlement rate is 0.44 mm/d. In the middle stage BC of support, the settling velocity is slow down, and the average settling velocity is 0.16 mm/d. In the later stage CD of support, there is little change in roof displacement, and the final settlement of the roof is 6.8 mm. The above field monitoring data show that the high prestress support design method can effectively control the settlement of shallow-buried large-span cavern.

2) When the roof settlement is stable, the maximum force of high prestress bolt is 136.8 kN. Compared with the high prestress bolt yield load of 206.8 kN, its maximum axial force is 66.2% of the yield load. The high prestress bolt still has a safety margin load of 70.0 kN. The results of cavern roof control effect and bolt axial force monitoring verify the correctness of the design method of high prestressed support with high prestress bolt.