Plastic Zone Model of Surrounding Rock of Circular Tunnel Considering Rock Mass Self-Weight and Its Construction Method

Abstract

The present invention discloses a circular tunnel surrounding rock plastic zone model considering rock mass self-weight and its construction method, and establishes a theoretical model of the circular tunnel surrounding rock plastic zone that simultaneously considers rock mass self-weight, intermediate principal stress and the interaction between surrounding rock and support structure. Through parameter sensitivity analysis, the influence laws of initial in-situ stress, rock mass shear modulus, rock mass self-weight, rock cohesion, support structure stiffness and intermediate principal stress coefficient on the calculation results of the surrounding rock plastic zone are studied. It is found that with the increase of initial in-situ stress, rock mass shear modulus and rock unit weight, the surrounding rock plastic zone gradually increases, while with the increase of rock cohesion, support structure stiffness and intermediate principal stress coefficient, the surrounding rock plastic zone gradually decreases, but the increase or decrease amplitude is different.

Description

Technical Field

[0001] The present invention belongs to the technical field of surrounding rock stability control in tunnel engineering, and particularly relates to a plastic zone model of circular tunnel surrounding rock considering rock mass self-weight and a construction method thereof. Background Art

[0002] Tunnel excavation will inevitably generate a certain range of plastic zones in the surrounding rock, and the size of the surrounding rock plastic zone will directly affect the safety and economy of tunnel construction, thus attracting the attention of many scholars. They have conducted in-depth research on it by using model tests, field tests, theoretical analysis, numerical simulation and other methods. However, due to the stronger universality of the theoretical method and the convenience of popularizing and applying the obtained results, Kastner earlier proposed a theoretical model for calculating the plastic zone of circular tunnel surrounding rock under hydrostatic pressure, which has been widely recognized and become the basis for the theoretical research on the plastic zone of tunnel surrounding rock. However, this method follows the following basic assumptions, that is: not considering the rock mass self-weight, the rock is an isotropic homogeneous body and obeys the Mohr-Coulomb (M-C) strength criterion and the ideal elastic-plastic model, and regarding the support force as a fixed value. With the gradual deepening of the research, many scholars believe that the above assumptions do not conform to the actual situation. In terms of rock strength criteria and constitutive models, many scholars believe that the M-C criterion does not consider the influence of the intermediate principal stress, and the ideal elastic-plastic model cannot reflect the dilatancy and softening characteristics of the rock. Therefore, many theoretical models of the plastic zone of surrounding rock based on other strength criteria such as the Drucker-Prager (D-P) criterion, the unified strength theory and the rock dilatancy-softening model have been proposed. At the same time, in terms of the value of the support force, Hou Gongyu et al. believe that the support force is not a fixed value, but is closely related to the surrounding rock deformation. Liu et al. have also conducted relatively in-depth research on this.

[0003] However, although many improvements have been made to the Kastner theory, there are few reports on the improvement of the first assumption of this method, that is, not considering the deadweight of the rock mass. On the one hand, when the tunnel is buried deep, ignoring the deadweight of the rock mass will not cause large errors in the calculation results. On the other hand, when the deadweight of the rock mass is considered, the problem cannot be simplified to an axisymmetric problem, so it is difficult to solve the problem theoretically, or even there is no theoretical solution. However, relevant studies have shown that when the tunnel is buried shallow, the probability and severity of roof collapse are much higher than the corresponding bottom drum and sidewalls, which means that the damage range and degree of the tunnel top are much higher than the bottom plate and the two sides, that is, the plastic zone of the surrounding rock is no longer circularly symmetrical. Therefore, how to establish a theoretical model of the plastic zone of the tunnel surrounding rock considering the deadweight of the rock mass is an important issue that needs to be solved urgently. Some scholars have also carried out useful explorations on this. For example, Zhi Xizhe studied the calculation method of polar coordinate stress components under constant body force conditions earlier, and gave the expressions of three stress components in plane polar coordinates under constant body force conditions, but he did not further explore the calculation method of the plastic zone of the tunnel surrounding rock under constant body force conditions. Song Haoran et al. obtained the analytical solution of the stress field of the surrounding rock of a shallow buried tunnel under the condition of ground load and surrounding rock self-weight based on the theory of complex variables. They believed that when the tunnel is shallow, the surrounding rock is mainly subjected to a composite failure of tension and shear, while when the tunnel is deep, the main failure mode of the surrounding rock is shear failure of the side wall. Zhou Wei derived the calculation formula of the plastic zone at the arch crown, arch waist and arch bottom of the tunnel under the condition of rock mass self-weight based on the unified strength theory. They believed that when the rock mass self-weight is considered, the size of the plastic zone at the three locations is the arch crown, arch waist and arch bottom. This study provides a good research idea for the calculation of the plastic zone of the tunnel surrounding rock under the condition of rock mass self-weight, but it assumes that the tunnel support force is constant, and fails to consider the change of support force caused by the interaction between surrounding rock and support structure.

[0004] In conclusion, the current research status shows that for shallow buried tunnels, the influence of rock mass deadweight on the plastic zone of tunnel surrounding rock cannot be ignored and should be considered. However, there is little theoretical research in this area. Therefore, based on the previous research, this paper proposes a theoretical model of the plastic zone of circular tunnel surrounding rock considering the deadweight of rock mass based on the ideal elastic-plastic theory, and at the same time considers the influence of intermediate principal stress and the interaction between surrounding rock and supporting structure on the plastic zone of surrounding rock. Summary of the invention

[0005] To solve the above problems, the present invention proposes a plastic zone model of surrounding rock of a circular tunnel considering the self-weight of rock mass and a construction method thereof. First, the unified strength theory considering the intermediate principal stress is introduced. Secondly, based on the ideal elastoplastic theory, the theoretical models of the plastic zones of the surrounding rock at three positions, namely the crown, the waist and the invert of the circular tunnel, are established. Then, by introducing the theory of interaction between the surrounding rock and the support structure, the above theoretical models are corrected. Finally, a theoretical model of the plastic zone of the surrounding rock of the circular tunnel considering the self-weight of rock mass, the intermediate principal stress and the interaction between the surrounding rock and the support structure is proposed. Finally, the influence laws of parameters such as the initial in-situ stress, the shear modulus of rock mass, the unit weight, the cohesion of rock, the stiffness of the support structure and the intermediate principal stress coefficient on the plastic zone of the surrounding rock are studied by using parameter sensitivity analysis.

[0006] The plastic zone model of the surrounding rock of a circular tunnel considering the self-weight of rock mass in the present invention is a plastic zone model of the surrounding rock of a tunnel considering the interaction between the surrounding rock and the support structure. The model corresponds to 4 points on the surrounding rock of the circular tunnel at three positions, namely the crown, the waist and the invert of the tunnel. After obtaining the plastic zone radii of the above 4 points, a smooth curve is used to connect them to obtain a closed curve, and this curve is the boundary line between the elastic and plastic zones of the surrounding rock, and the area between this curve and the tunnel contour line is the range of the plastic zone of the surrounding rock. The calculation formula for the plastic zone radius is

[0007]

[0008] Equation (12) is an implicit equation about r p and cannot be directly solved, and an iterative method needs to be used for solving; where r p is the plastic zone radius of the surrounding rock, r 0 is the tunnel radius, k s is the tensile and compressive stiffness of the support structure, γ is the unit weight of the rock mass, p 0 is the initial in-situ stress, G is the shear modulus of the rock, c and are the cohesion of the rock and the internal friction angle respectively, and b (0 ≤ b ≤ 1) is the intermediate principal stress coefficient.

[0009] The construction method of the plastic zone model of the surrounding rock of a circular tunnel considering the self-weight of rock mass in the present invention includes the following steps:

[0010] S1. Establish the unified strength theory considering the intermediate principal stress;

[0011] S2. Based on the ideal elastoplastic theory, establish the mechanical model of the circular tunnel considering the self-weight of the rock mass;

[0012] S3. Introduce the theory of interaction between the surrounding rock and the support structure to correct the mechanical model of the circular tunnel, and obtain the plastic zone model of the surrounding rock of the tunnel considering the self-weight of the rock mass, the intermediate principal stress and the interaction between the surrounding rock and the support structure.

[0013] The unified strength theory formula of S1 considering the intermediate principal stress is

[0014]

[0015] In the formula, c and are the cohesion and internal friction angle of the rock respectively, b(0≤b≤1) is the intermediate principal stress coefficient, σ θ is the circumferential stress of the rock, and σ r is the radial stress of the rock.

[0016] The mechanical model of the circular tunnel considering the self-weight of the rock mass in S2 is the model at the crown, waist and invert of the tunnel. The mechanical model of the circular tunnel considering the self-weight of the rock mass includes the surrounding rock stress model and the surrounding rock displacement model.

[0017] The radial stress and the radius of the plastic zone of the surrounding rock on the elastic-plastic interface in the surrounding rock stress model are respectively

[0018] ① Crown:

[0019]

[0020] ② Waist:

[0021]

[0022] ③ Invert:

[0023]

[0024] In the formula, only formula (5b) is an explicit equation and can be directly solved, while formulas (5a) and (5c) are implicit equations and cannot be directly solved. Therefore, an iterative method can be used to first find r p , and then find p r ; p r is the radial stress, r p is the radius of the plastic zone of the surrounding rock, p s is the tunnel support force, γ is the unit weight of the rock, p 0 is the initial in-situ stress, and r 0 is the tunnel radius.

[0025] The surrounding rock displacement model is

[0026]

[0027] In the formula, u r0 is the displacement of the tunnel inner wall, γ is the unit weight of the rock, p 0 is the initial in-situ stress, G is the shear modulus of the rock, r p is the radius of the plastic zone of the surrounding rock, r 0 is the tunnel radius, c and are the cohesion and internal friction angle of the rock respectively, and b (0≤b≤1) is the intermediate principal stress coefficient.

[0028] The p s is calculated by the formula

[0029] p s = k s u r0 (11)

[0030] In the formula, p s is the tunnel support force, k s is the tensile and compressive stiffness of the support structure, and u r0 is the displacement of the tunnel inner wall.

[0031] The beneficial effect of the present invention is that the present invention establishes a theoretical model of the plastic zone of the surrounding rock of a circular tunnel that simultaneously considers the self-weight of the rock mass, the intermediate principal stress, and the interaction between the surrounding rock and the support structure. Through parametric sensitivity analysis, the influence laws of the initial in-situ stress, rock mass shear modulus, rock mass self-weight, rock cohesion, support structure stiffness, and intermediate principal stress coefficient on the calculation results of the plastic zone of the surrounding rock are studied. It is found that with the increase of the initial in-situ stress, rock mass shear modulus, and rock mass unit weight, the plastic zone of the surrounding rock gradually increases, while with the increase of the rock cohesion, support structure stiffness, and intermediate principal stress coefficient, the plastic zone of the surrounding rock gradually decreases, but the increase or decrease amplitude is different. Description of the Drawings

[0032] Figure 1 is the force analysis diagram of the plastic zone of the surrounding rock of the circular tunnel and the micro-elements at different positions of the present invention.

[0033] Figure 2 is the calculation result diagram of the plastic zone of the surrounding rock of the present invention.

[0034] Figure 3 is the influence diagram of the initial in-situ stress on the calculation results of the plastic zone of the surrounding rock of the present invention.

[0035] Figure 4 is the influence diagram of the rock mass shear modulus on the calculation results of the plastic zone of the surrounding rock of the present invention.

[0036] Figure 5 is the influence diagram of the rock mass unit weight on the calculation results of the plastic zone of the surrounding rock of the present invention.

[0037] Figure 6 is the influence diagram of the cohesion on the calculation results of the plastic zone of the surrounding rock of the present invention.

[0038] Figure 7 is the influence diagram of the support structure stiffness on the calculation results of the plastic zone of the surrounding rock of the present invention.

[0039] Figure 8 It is the influence diagram of the intermediate principal stress coefficient of the present invention on the calculation result of the plastic zone of surrounding rock. Specific implementation manners

[0040] The embodiments of the present invention will be described in detail below. The examples of the embodiments are shown in the drawings. The embodiments described below by referring to the drawings are exemplary and are intended to explain the present invention, and should not be construed as a limitation to the present invention.

[0041] The construction method of the plastic zone model of the surrounding rock of a circular tunnel considering the self-weight of rock mass of the present invention includes the following steps:

[0042] S1. Establish a unified strength theory considering the intermediate principal stress;

[0043] When taking σ 2 =(σ 1 +σ 3 ) / 2, the unified strength theory can be written in a form similar to the M-C criterion, that is (represented in polar coordinates):

[0044]

[0045] In the formula: c and are the cohesion and internal friction angle of rock respectively, b(0≤b≤1) is the intermediate principal stress coefficient, σ θ is the circumferential stress of rock, σ r is the radial stress of rock.

[0046] S2. Establish a mechanical model of a circular tunnel considering the self-weight of rock mass based on the ideal elastoplastic theory;

[0047] S201. Stress of surrounding rock

[0048] Since the dimension in the tunnel length direction is much larger than the dimension in its cross-section direction, it can be regarded as a plane strain problem. Therefore, a certain cross-section can be taken for research. As Figure 1 shown is the cross-section calculation model of a circular tunnel under the action of uniformly distributed initial ground stress p 0 . Since the tunnel cross-section is circular, polar coordinates are used for solution here, and the direction perpendicular to the cross-section is set as the z direction. Thus, σ θ , σ r , σ z , ε θ , ε r , ε z are the circumferential, radial and axial stress and strain components of the tunnel calculation model respectively, and satisfy σ θ >σ z >σ r .

[0049] When considering the self-weight of the rock mass, the plastic zone of the surrounding rock will no longer be a circular area symmetric about the tunnel center, but will be an asymmetric area with a larger top range, a smaller bottom range, and a middle waist as shown in Figure 1 the figure.

[0050] The force conditions of the micro-elements at the tunnel crown, arch waist, and arch bottom are as shown in Figure 1 the figure. When considering the self-weight of the rock mass, due to symmetry, the tangential stress at these three positions is zero, while the tangential stress at the remaining positions is no longer zero. Therefore, this problem is not a completely axisymmetric problem. For the convenience of research, only the micro-elements at these three positions will be taken as examples for research below.

[0051] The equilibrium equations of the micro-elements at the above three positions of the tunnel crown, arch waist, and arch bottom are as follows:

[0052]

[0053] In the formula: σ r , σ θ are the radial and circumferential normal stresses respectively, γ is the unit weight of the rock mass, and r is the distance from a certain point A to be determined to the tunnel center.

[0054] Below, the physical quantities of the elastic zone and plastic zone of the tunnel surrounding rock are represented by the subscripts "e" and "p" respectively. If the unified strength theory is adopted as the rock strength criterion. Then, by combining equations (1) and (2), the stress component expressions of the plastic zone of the tunnel surrounding rock at the above three positions can be obtained as (r 0 ≤r≤r p , where r p is the radius of the plastic zone, and r 0 is the radius of the tunnel).

[0055] ① Crown:

[0056]

[0057] ② Arch waist:

[0058]

[0059] ③ Arch bottom:

[0060]

[0061] In the formula: p s is the tunnel support force, which acts uniformly on the inner wall of the tunnel.

[0062] If the radial stress at the elastic-plastic interface is assumed to be p r , similarly, when considering body force, the stress components of the elastic zone at the above three positions can be obtained respectively as (r≥r p ):

[0063] ① Crown:

[0064] ② Springing:

[0065] ③ Invert:

[0066] Since the stress at the elastic - plastic interface is continuous, i.e., σ re = σ rp , σ θe = σ θp , substituting r = r p into Eqs. (3) - (4) and solving the equations simultaneously, the radial stress p r on the elastic - plastic interface and the radius r p of the surrounding rock plastic zone at the crown, springing and invert of the tunnel can be obtained respectively as follows:

[0067] ① Crown:

[0068]

[0069] ② Springing:

[0070]

[0071] ③ Invert:

[0072]

[0073] It can be seen that only Eq. (5b) is an explicit equation and can be solved directly, while Eqs. (5a) and (5c) are implicit equations and cannot be solved directly. Therefore, an iterative method can be used to first find r p , and then find p r .

[0074] S202. Surrounding Rock Displacement

[0075] Similarly, only the surrounding rock displacements at the crown, springing and invert of the tunnel are studied here. According to the elastic theory, when the radial stress p r on the elastic - plastic interface is obtained, the stresses in the elastic zones of the surrounding rock at the above - mentioned three positions can be obtained as:

[0076]

[0077] Yu et al. believe that before the tunnel excavation, there are initial in - situ stresses in the rock mass. Tunnel excavation causes stress redistribution on the one hand and leads to convergence displacement of the surrounding rock on the other hand. Therefore, the convergence displacement of the surrounding rock is caused by the stress increment due to tunnel excavation. Thus, from Eq. (6) and the elastic constitutive relationship, the surrounding rock displacements at the above - mentioned three positions can be obtained as:

[0078]

[0079] where: E and v are the elastic modulus and Poisson's ratio of the rock, respectively.

[0080] Since the plastic deformation of the rock mainly involves shape change, it is assumed here that the volume of the rock in the plastic zone remains unchanged. Combining with the geometric equation, we have:

[0081]

[0082] Meanwhile, according to the displacement continuity condition at the interface between the elastic and plastic zones of the surrounding rock, the displacement of the surrounding rock in the plastic zone can be obtained as (here, the direction towards the tunnel center is taken as positive):

[0083]

[0084] where: G is the shear modulus of the rock.

[0085] It can be seen that the displacement expressions in the elastic and plastic zones are the same. Substituting r = r 0 and p in Equation (5) r into Equation (9), the displacement u of the tunnel inner wall at the above three positions can be obtained as r0 follows:

[0086]

[0087] S3. Introduce the theory of the interaction between the surrounding rock and the support structure to correct the mechanical model of the circular tunnel, and obtain the plastic zone model of the tunnel surrounding rock considering the self-weight of the rock mass, the intermediate principal stress, and the interaction between the surrounding rock and the support structure.

[0088] At present, the commonly used tunnel support structures in engineering mainly include reinforced concrete linings, steel arches, bolts, and their combinations. Therefore, compared with the rock, their stiffness is generally large, so it can be considered that they only produce elastic deformation. At the same time, in order to achieve good support effects, it is generally required that the support structure should be supported in a timely manner and closely fit with the surrounding rock, and there is no relative sliding between the two. Therefore, it can be considered that the radial convergence displacement of the surrounding rock is equal to the elongation of the support structure. From this, we can obtain:

[0089] p s = k s u r0 (11)

[0090] where: k s is the tensile and compressive stiffness of the support structure.

[0091] Then, substituting Equations (10) and (11) into the second equation of Equation (5), the calculation formula for the radius of the plastic zone of the surrounding rock can be obtained as:

[0092]

[0093] It can be seen that Equation (12) is an implicit equation with respect to r p and cannot be directly solved. Therefore, an iterative method needs to be used for solving.

[0094] It should be noted that the present invention only proposes a calculation method for the plastic zone of the surrounding rock at three positions such as the crown, the waist and the invert of the tunnel. For other positions, since it is not an axisymmetric problem, there is usually no theoretical solution. The present invention intends to adopt an approximate method for processing, that is, corresponding to the four points on the circular tunnel surrounding rock at the above three positions. When the plastic zone radii of the above four points are obtained, a closed curve is obtained by connecting them with a smooth curve. Then this curve is the boundary between the elastic and plastic zones of the surrounding rock, and the area between this curve and the tunnel contour line is the range of the plastic zone of the surrounding rock.

[0095] Example analysis

[0096] 1. Calculation model

[0097] Take the circular tunnel as shown in Figure 1 as the calculation model, and the calculation parameters are shown in Table 1. Next, the plastic zone of the tunnel surrounding rock is calculated using the theory in this paper.

[0098] Table 1 Calculation parameter table

[0099] Tab.1The calculation parameters

[0100]

[0101] According to the above parameters, the plastic zone of the surrounding rock is as shown in Figure 2As shown in the figure, it can be seen that when the deadweight of the rock mass is considered, the plastic zone of the tunnel surrounding rock will no longer be a circle with the tunnel center point O as the center, but will present a shape with a large arch, a small arch bottom, and a central arch waist, which fully illustrates the influence of the deadweight of the rock mass on the plastic zone of the tunnel surrounding rock. For this example, when the deadweight of the rock mass is not considered, the radius of the plastic zone of the surrounding rock is the radius of the plastic zone of the tunnel waist, that is, 4.86m, while when the deadweight of the rock mass is considered, the plastic zone radii of the arch and arch bottom are 5.74m and 4.07m respectively, and the increase or decrease is 18.11% and 16.26% respectively. This is because when the deadweight of the rock mass is considered, the gravity direction of the rock mass on the arch is toward the center of the tunnel, which is a favorable factor for inducing roof collapse and damage to the rock mass, and is therefore beneficial to the formation of the plastic zone of the surrounding rock. Correspondingly, the radius of the plastic zone at the arch is larger. The gravity direction of the rock mass at the arch bottom is away from the center of the tunnel, so it has a restraining effect on the bottom drum damage at the arch bottom, which is not conducive to the formation of the plastic zone of the surrounding rock. Correspondingly, the radius of the plastic zone at the arch bottom is also small. The deadweight of the rock mass at the arch waist is perpendicular to the radius of the tunnel, so it has no effect on the range of the plastic zone. This can also be seen from the second equation of equation (12). In short, it can be considered that when the deadweight of the rock mass is considered, the range of the plastic zone of the surrounding rock is no longer a circular area with the center of the tunnel as the center, but a non-circular area with a large upper part, a small bottom and a central waist. At the same time, it should be noted that in this example, p 0 = 100kPa. According to the γ value in Table 1, when only the self-weight stress is considered, the thickness of the overburden is only 5m, and the radius of the tunnel is 3m. Therefore, it can be considered that the tunnel is shallow or even ultra-shallow. Figure 2 The calculation results of the surrounding rock plastic zone shown in the figure are obtained when the tunnel is shallowly buried or even ultra-shallowly buried. As the tunnel depth increases, the directional characteristics of the tunnel surrounding rock plastic zone will gradually weaken. This can be seen from the following calculation for p 0 In conclusion, it can be considered that it is very necessary to consider the influence of rock mass deadweight on the plastic zone of tunnel surrounding rock when the tunnel is buried at a shallow depth.

[0102] 2. Parameter sensitivity analysis

[0103] From the above calculation examples, we can see that the deadweight of the rock mass has an important influence on the range of the plastic zone of the surrounding rock at different locations of the tunnel. However, the influence of the above calculation parameters on the calculation results of the plastic zone of the surrounding rock needs further study. 0 , rock mass shear modulus G, gravity γ, rock shear strength (here taking cohesion c as an example), support structure stiffness k s , intermediate principal stress coefficient b, etc. on the calculation results of the plastic zone of surrounding rock.

[0104] (1) Initial geostress p 0 The influence of

[0105] Take p 0 as 100 kPa, 1000 kPa, 10000 kPa and 100000 kPa respectively, with the remaining parameters unchanged. The variation laws of the plastic zones of the surrounding rock at three positions, namely the crown, the waist and the invert of the tunnel, are as follows Figure 3 . It can be seen that: ① As p 0 increases, the range of the plastic zone of the surrounding rock gradually increases, but the increasing trend gradually slows down and finally tends to a certain value. Taking the plastic zone at the crown as an example, when p 0 gradually increases from 100 kPa to 1000 kPa, 10000 kPa and 100000 kPa, its plastic zone increases from 5.74 m to 6.80 m, 6.89 m and 6.9 m respectively, that is, the increase amplitude gradually decreases. At the same time, it shows that the tunnel excavation only affects the rock mass within a certain range, that is, the stress concentration caused by the tunnel excavation is local, so the influence range on the surrounding rock is also local and will not increase infinitely. ② As p 0 increases, the overall characteristic that the plastic zone range is the largest at the top, in the middle at the waist and the smallest at the bottom gradually tends to the plastic zone ranges at all positions around the tunnel being equal, that is, the distribution of the plastic zone becomes more uniform. This is because when p 0 is relatively small, the self-weight of the rock mass within the plastic zone accounts for a relatively large proportion in the total stress field, while when p 0 increases to a certain extent, the plastic zone range no longer increases significantly, so the self-weight of the rock mass in the plastic zone will not increase significantly either. At this time, the proportion of the self-weight of the rock mass in the plastic zone in the total stress field gradually decreases. Therefore, when p 0 is large enough, the influence of the self-weight of the rock mass within the plastic zone on the calculation result of the plastic zone of the surrounding rock can be ignored, so the plastic zone range gradually approaches a circle. This is also the reason why the self-weight of the rock mass is usually not considered when calculating the plastic zone of the surrounding rock of a deep-buried tunnel

[0106] (2) Influence law of the shear modulus G of the rock mass

[0107] Take G as 40 MPa, 400 MPa, 4000 MPa and 40000 MPa respectively, with the remaining parameters unchanged. The variation laws of the plastic zones of the surrounding rock at three positions, namely the crown, the waist and the invert of the tunnel, are as follows Figure 4It can be seen that: ① As G increases, the range of the surrounding rock plastic zone gradually increases, but the increasing trend gradually slows down and finally tends to a certain value. Taking the surrounding rock at the crown as an example, when G gradually increases from 40 MPa to 400 MPa, 4000 MPa, and 40000 MPa, the plastic zone increases from 5.74 m to 5.89 m, 5.91 m, and 5.91 m respectively, that is, the increase amplitude gradually decreases and gradually tends to a certain value. ② As G increases, the overall characteristic of the plastic zone range being large at the top, medium at the waist, and small at the bottom gradually tends to the plastic zone ranges in all directions being approximately equal, that is, the plastic zone distribution becomes more uniform. This is because when G is small, the deformation of the rock mass is large, and correspondingly the deformation of the support structure will also increase, resulting in an increase in the support force. Therefore, under the condition that the shear strength of the rock mass remains unchanged, when the support force increases, the surrounding rock plastic zone will decrease accordingly. As G increases, the surrounding rock displacement decreases, the deformation of the support structure decreases accordingly, and the support force also decreases, so the plastic zone will increase. ③ From the perspective of the increase amplitude of the surrounding rock plastic zone at different positions, as G increases, the increasing speed of the surrounding rock plastic zone range at the crown is much smaller than that at the bottom. This is because as G increases, the deformation of the surrounding rock at the crown decreases rapidly and gradually converges to the deformation at other positions, so the surrounding rock plastic zone ranges at different positions are also more uniform.

[0108] (3) Influence law of rock unit weight γ

[0109] Take γ as 18 kN / m 3 , 19 kN / m 3 , 20 kN / m 3 and 21 kN / m 3 , with other parameters unchanged, the variation laws of the surrounding rock plastic zones at three positions, namely the crown, the waist, and the invert of the tunnel, are as follows Figure 5 . It can be seen that: ① As γ increases, the variation laws of the surrounding rock plastic zones at different positions are inconsistent. Among them, the plastic zone at the crown gradually increases, the plastic zone at the waist remains unchanged, and the plastic zone at the invert gradually decreases. This is because the gravity of the rock mass at the crown is towards the center of the tunnel. Therefore, as the self-weight of the rock mass increases, the rock mass at the crown is more likely to be damaged due to self-weight, so the range of its plastic zone increases. From Equation (12), it can be seen that the surrounding rock plastic zone at the waist is independent of the rock unit weight, while the plastic zone at the invert decreases with the increase of the rock self-weight. This is because the gravity of the rock mass at the invert deviates from the center of the tunnel, which plays a role in curbing adverse deformations such as the uplift of the tunnel bottom. Therefore, it is less likely to form a plastic zone under the action of the gradually increasing self-weight. ② As γ increases, the increase amplitudes of the plastic zones at the crown and the invert change little. Taking the surrounding rock at the crown as an example, when γ increases from 18 kN / m 3 to 19 kN / m 3 , 20 kN / m 3and 21 kN / m 3 When it is, the plastic zones increase from 5.56 m to 5.62 m, 5.74 m and 5.78 m respectively, that is, the increase range is not significant, indicating that the influence of rock mass self-weight on the plastic of surrounding rock is approximately linear. ③ As γ increases, the characteristic of the plastic zone range with the largest at the top, moderate in the waist and smallest at the bottom gradually evolves into a feature with a larger top, a smaller bottom and an unchanged waist, that is, the plastic zone range gradually migrates from the bottom to the top.

[0110] (4) Influence law of rock mass shear strength

[0111] The rock mass shear strength parameters include cohesion c and internal friction angle The influence laws of the two are basically the same. Therefore, the former is taken as an example for research here. Take c as 10 kPa, 20 kPa, 30 kPa and 40 kPa respectively, and keep the other parameters unchanged. The variation laws of the plastic zones of the surrounding rock at three positions, namely the tunnel crown, the arch waist and the arch bottom, are as follows Figure 6 . It can be seen that: ① As c increases, the range of the plastic zone of the surrounding rock gradually decreases, but the decreasing trend gradually slows down and finally tends to a certain value. Taking the surrounding rock at the crown as an example, when c increases from 10 kPa to 20 kPa, 30 kPa and 40 kPa respectively, its plastic zone decreases from 5.74 m to 4.32 m, 3.76 m and 3.43 m respectively, that is, the decreasing range gradually decreases. On the one hand, because as c increases, the rock mass shear strength increases and the ability to resist shear failure improves, so the range of the plastic zone gradually decreases. On the other hand, it can be seen from Equation (12) that the plastic zone radius and c do not show a simple linear relationship, so as c increases, the plastic zone radius does not decrease linearly. ② From the influence of c on the plastic zones at different positions of the tunnel, as c increases, although the ranges of the plastic zones at the crown, the arch waist and the arch bottom all decrease, their decreasing amplitudes are still slightly different. For example, when c increases from 10 kPa to 40 kPa, the decreasing amplitudes of the plastic zones at the three positions are 40.24%, 30.65% and 26.29% respectively, that is, the decreasing amplitude of the plastic zone radius at the crown is the largest. This is because the crown is more likely to yield and fail under the combined action of rock mass in-situ stress and self-weight, so when the rock mass shear strength increases, the decreasing amplitude of its plastic zone is the largest.

[0112] (5) Support structure stiffness k s Influence law

[0113] Take k s as 1.5 MPa / m, 3 MPa / m, 6 MPa / m and 9 MPa / m respectively, and keep the other parameters unchanged. The variation laws of the plastic zones of the surrounding rock at three positions, namely the tunnel crown, the arch waist and the arch bottom, are as follows Figure 7 . It can be seen that: ① As k sWith the increase of s , the range of the plastic zone of the surrounding rock gradually decreases, but the decreasing amplitude varies at different positions. The decreasing amplitude at the crown changes little, while the decreasing amplitude at the waist and the invert gradually decreases. Taking the surrounding rock at the invert as an example, when k s increases from 1.5 MPa to 3 MPa, 6 MPa and 9 MPa respectively, the radius of its plastic zone decreases from 4.07 m to 3.57 m, 3.09 m and 3 m respectively, that is, its decreasing amplitude gradually decreases. This is because with the increase of k s , the supporting force provided by the supporting structure also increases accordingly. Therefore, it is not easy for the surrounding rock to produce large deformation, and thus the radius of its plastic zone also decreases. ② With the increase of k

[0114] (6) Influence law of the intermediate principal stress coefficient b

[0115] Take b as 0, 0.33, 0.67 and 1 respectively, and keep the other parameters unchanged. The variation law of the radius of the plastic zone of the surrounding rock at three positions, namely the crown, the waist and the invert of the tunnel, is as shown in Figure 8 . It can be seen that: ① With the increase of b, the radius of the plastic zone of the surrounding rock gradually decreases, but its decreasing trend gradually slows down and finally tends to a certain value. Taking the surrounding rock at the crown as an example, when b increases from 0 to 0.33, 0.67 and 1 respectively, its plastic zone decreases from 6.5 m to 5.74 m, 5.29 m and 5.05 m respectively, that is, its decreasing amplitude gradually decreases. On the one hand, this shows that when considering the intermediate principal stress, the plastic zone of the surrounding rock gradually decreases, that is, the intermediate principal stress has a certain influence on the stability of the surrounding rock. On the other hand, it also shows that the influence of the intermediate principal stress on the plastic zone of the surrounding rock has a certain limit. ② With the increase of b, the difference in the range of the plastic zone at each position will gradually decrease. This is because with the increase of b, the influence weight of the intermediate principal stress on the plastic zone of the tunnel surrounding rock gradually increases. That is, when b = 0 and 1, the unified strength theory corresponds to the M-C strength theory that completely does not consider the intermediate principal stress and the twin-shear strength theory that completely considers the intermediate principal stress respectively. Therefore, the surrounding rock gradually evolves from a two-way stress state to a three-way stress state, and thus the failure conditions of the surrounding rock in all directions are more uniform.

[0116] Although the above embodiments have been shown and described, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Any changes, modifications, substitutions and variations made by those of ordinary skill in the art to the above embodiments are within the protection scope of the present invention.

Claims

1. A plastic zone model of the surrounding rock of a circular tunnel considering the deadweight of the rock mass, characterized in that: The model is a tunnel surrounding rock plastic zone model that takes into account the interaction between surrounding rock and supporting structure. The model corresponds the three parts of the tunnel vault, arch waist and arch bottom to four points on the circular tunnel surrounding rock. After the plastic zone radius of the above four points is obtained, they are connected by a smooth curve to obtain a closed curve. The curve is the boundary between the elastic and plastic zones of the surrounding rock, and the area between the curve and the tunnel contour is the range of the surrounding rock plastic zone. The calculation formula of the plastic zone radius is: ① Vault: （12a） ② Arch waist: (12b) ③ Arch bottom: （12c） Formulas (12a), (12b), and (12c) are about r p The implicit equation cannot be solved directly and needs to be solved by iterative method; in, r p is the radius of the plastic zone of the surrounding rock, r 0 is the tunnel radius, k s is the tensile and compressive stiffness of the supporting structure, γ is the rock mass, p 0 is the initial ground stress, G is the rock shear modulus, , , c and φ are rock cohesion and internal friction angle, b (0≤b≤1) is the intermediate principal stress coefficient.

2. A method for constructing a plastic zone model of a circular tunnel surrounding rock taking into account the deadweight of the rock mass according to claim 1, characterized in that: The following steps are involved: S1. Establish a unified strength theory considering intermediate principal stresses; S2. Based on the ideal elastic-plastic theory, a circular tunnel mechanical model considering the deadweight of the rock mass is established; S3. The theory of surrounding rock-support structure interaction is introduced to modify the mechanical model of circular tunnel, and a tunnel surrounding rock plastic zone model is obtained which takes into account the deadweight of rock mass, intermediate principal stress and the interaction between surrounding rock and support structure.

3. The method for constructing a plastic zone model of a circular tunnel surrounding rock considering the deadweight of the rock mass according to claim 2, characterized in that: The unified strength theory formula of S1 considering the intermediate principal stress is: （1） In the formula, , , c and φ are rock cohesion and internal friction angle, b (0≤b≤1) is the intermediate principal stress coefficient, σ θ is the hoop stress of rock, σ r is the radial stress of rock.

4. The method for constructing a plastic zone model of a circular tunnel surrounding rock considering the deadweight of the rock mass according to claim 2, characterized in that: The circular tunnel mechanical model of S2 considering the deadweight of the rock mass is a model of the tunnel vault, the haunch and the bottom of the vault. The circular tunnel mechanical model considering the deadweight of the rock mass includes a surrounding rock stress model and a surrounding rock displacement model.

5. The method for constructing a plastic zone model of a circular tunnel surrounding rock considering the deadweight of the rock mass according to claim 4, characterized in that: The radial stress on the elastic-plastic interface and the radius of the surrounding rock plastic zone in the surrounding rock stress model are respectively, ① Vault: (5a) ② Arch waist: (5b) ③ Arch bottom: (5c) Among them, only equation (5b) is an explicit equation and can be solved directly, while equations (5a) and (5c) are both implicit equations and cannot be solved directly. Therefore, an iterative method can be used to first solve r p , then find p r ; p r is the radial stress, r p is the radius of the plastic zone of the surrounding rock, p s is the tunnel support force, γ is the rock mass, p 0 is the initial ground stress, r 0 is the tunnel radius.

6. The method for constructing a plastic zone model of a circular tunnel surrounding rock considering the deadweight of the rock mass according to claim 4, characterized in that: The surrounding rock displacement model is ① Vault: (10a) ② Arch waist: (10b) ③ Arch bottom: (10c) In the formula, u r0 is the tunnel wall displacement, γ is the rock mass, p 0 is the initial ground stress, G is the rock shear modulus, r p is the radius of the plastic zone of the surrounding rock, r 0 is the tunnel radius, , , c and φ are rock cohesion and internal friction angle, b (0≤b≤1) is the intermediate principal stress coefficient.

7. The method for constructing a plastic zone model of a circular tunnel surrounding rock considering the deadweight of the rock mass according to claim 5, characterized in that: Said p s The calculation formula is (11) In the formula, p s is the tunnel support force, k s is the tensile and compressive stiffness of the supporting structure, u r0 is the displacement of the tunnel wall.

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