Rock mass self-weight-considered circular tunnel surrounding rock plastic zone radius calculation method

Abstract

The invention discloses a round tunnel surrounding rock plastic zone radius calculation method considering rock mass self weight. A round tunnel surrounding rock plastic zone theoretical model considering rock mass self weight, middle principal stress and surrounding rock-support structure interaction at the same time is established. The rule of influence of initial crustal stress, rock mass shear modulus, rock mass dead weight, rock cohesive force, support structure rigidity and intermediate principal stress coefficient on the calculation result of the plastic zone of the surrounding rock is analyzed and researched through parameter sensitivity. As the initial crustal stress, the rock mass shear modulus and the rock mass weight are increased, the plastic zone of the surrounding rock is gradually increased, and as the rock mass cohesive force, the supporting structure rigidity and the middle principal stress coefficient are increased, the plastic zone of the surrounding rock is gradually decreased, but the increasing or decreasing amplitude of the plastic zone of the surrounding rock is different.

Description

Technical Field

[0001] The invention belongs to the technical field of tunnel engineering surrounding rock stability control, and particularly relates to a method for calculating the radius of the plastic zone of a circular tunnel surrounding rock taking into account the deadweight of the rock mass. Background Art

[0002] Tunnel excavation inevitably creates a certain plastic zone in the surrounding rock. The size of this zone directly affects the safety and economic efficiency of tunnel construction, attracting the attention of numerous researchers. This research has been extensively studied using model and field tests, theoretical analysis, and numerical simulation. However, because theoretical methods offer greater universality and the results obtained are easily generalizable and applicable, Kastner early on proposed a theoretical model for calculating the plastic zone of circular tunnel surrounding rock under hydrostatic pressure. This model has been widely recognized and has become the foundation for theoretical research on the plastic zone of tunnel surrounding rock. However, this method relies on the following basic assumptions: the rock mass's deadweight is not considered; the rock is isotropic and homogeneous, obeying the Mohr-Coulomb (MC) strength criterion and the ideal elastic-plastic model; and the support force is considered constant. However, as research progresses, many researchers believe that these assumptions are not fully consistent with actual conditions. For example, regarding rock strength criteria and constitutive models, many scholars believe that the MC criterion fails to consider the influence of intermediate principal stresses, and that the ideal elastic-plastic model cannot reflect the dilatancy and softening properties of rock. Consequently, numerous theoretical models for the surrounding rock plastic zone have been proposed based on alternative strength criteria, such as the Drucker-Prager (DP) criterion, unified strength theory, and the rock dilatancy-softening model. Furthermore, regarding the value of support force, Hou Gongyu et al. believe that support force is not a constant but is closely related to surrounding rock deformation. Liu et al. have also conducted in-depth research on this topic.

[0003] However, although many improvements have been made to Kastner's theory, there are few studies on the first assumption of the method, i.e. the self-weight of the rock mass is not considered. On the one hand, when the tunnel is deeply buried, the error caused by ignoring the self-weight of the rock mass is not large. On the other hand, when the self-weight of the rock mass is considered, the problem cannot be simplified as an axisymmetric problem, and thus it is difficult to obtain a theoretical solution. However, relevant studies have shown that when the tunnel is shallowly buried, the probability and severity of roof collapse are much higher than those of floor heave and rib spalling, which indicates that the damage range and degree of the tunnel roof are much higher than those of the floor and the two sides, i.e. the plastic zone of the surrounding rock is no longer circularly symmetric. Therefore, how to establish a theoretical model of the plastic zone of the surrounding rock of a tunnel considering the self-weight of the rock mass is an important issue to be solved. Some scholars have also made beneficial explorations in this regard. For example, Zhi-zhe ZHOU studied the calculation method of the stress components in polar coordinates under the condition of constant body force at an early stage, and gave the expressions of the three stress components in polar coordinates under the condition of constant body force. However, he did not further explore the calculation method of the plastic zone of the surrounding rock of a tunnel under the condition of constant body force. Based on the theory of complex variable functions, Hao-ran SONG et al. obtained an analytical solution of the stress field of the surrounding rock of a shallow tunnel considering the ground load and the self-weight of the surrounding rock, and believed that when the tunnel is shallowly buried, the main failure mode of the surrounding rock is the combined failure of tension and shear, while when the tunnel is deeply buried, the main failure mode of the surrounding rock is the shear failure of the sidewalls. Based on the unified strength theory, Wei ZHOU derived the calculation formulas of the plastic zone at the crown, haunch and invert of a tunnel considering the self-weight of the rock mass, and believed that when the self-weight of the rock mass is considered, the sizes of the plastic zones at the three positions are in the order of crown, haunch and invert. This study provides a good research idea for the calculation of the plastic zone of the surrounding rock of a tunnel considering the self-weight of the rock mass, but it assumes that the support force of the tunnel is constant, and fails to consider the change of the support force caused by the interaction between the surrounding rock and the support structure.

[0004] In summary, for shallow tunnels, the influence of the self-weight of the rock mass on the plastic zone of the surrounding rock cannot be ignored and should be considered. Therefore, based on the previous studies, a theoretical model of the plastic zone of the surrounding rock of a circular tunnel considering the self-weight of the rock mass is proposed based on the ideal elastic-plastic theory, and the influence of the intermediate principal stress and the interaction between the surrounding rock and the support structure on the plastic zone of the surrounding rock is also considered. SUMMARY

[0005] To address the above issues, this paper proposes a method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel, taking into account the deadweight of the rock mass. First, a unified strength theory considering intermediate principal stresses is introduced. Next, a theoretical model for the plastic zone of the surrounding rock at three locations—the crown, haunch, and base of a circular tunnel—is established based on the ideal elastic-plastic theory. This theoretical model is then modified by introducing the theory of surrounding rock-support structure interaction. Finally, a theoretical model for the plastic zone of the surrounding rock of a circular tunnel is proposed that simultaneously considers the deadweight of the rock mass, intermediate principal stresses, and the interaction between the surrounding rock and the support structure. Finally, a parameter sensitivity analysis is used to investigate the influence of parameters such as initial in situ stress, rock mass shear modulus, rock density, rock cohesion, support structure stiffness, and intermediate principal stress coefficient on the surrounding rock plastic zone.

[0006] The circular tunnel surrounding rock plastic zone model of the present invention takes into account the deadweight of the rock mass. The model is a tunnel surrounding rock plastic zone model that takes into account the interaction between the surrounding rock and the supporting structure. The model corresponds to four points on the circular tunnel surrounding rock at the three positions of the tunnel vault, the arch waist and the arch bottom. After obtaining the plastic zone radius of the above four points, 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 line is the range of the surrounding rock plastic zone. The calculation formula of the plastic zone radius is:

[0007]

[0008] Formula (12) is about r p The implicit equation cannot be solved directly and needs to be solved by iterative method; among them, r p is the radius of the plastic zone of the surrounding rock, r0 is the radius of the tunnel, k s is the tensile and compressive stiffness of the support structure, γ is the weight of the rock mass, p0 is the initial ground stress, G is the rock shear modulus, c and are rock cohesion and internal friction angle respectively, and b (0≤b≤1) is the intermediate principal stress coefficient.

[0009] The method for constructing a plastic zone model of a circular tunnel surrounding rock taking into account the deadweight of the rock mass of the present invention comprises the following steps:

[0010] S1. Establish a unified strength theory considering intermediate principal stresses;

[0011] S2. Establish a circular tunnel mechanical model based on the ideal elastic-plastic theory taking into account the rock mass self-weight;

[0012] S3. The theory of surrounding rock-support structure interaction is introduced to modify the circular tunnel mechanical model, and a tunnel surrounding rock plastic zone model is obtained that takes into account the rock mass deadweight, intermediate principal stress, and surrounding rock-support structure interaction.

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

[0014]

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

[0016] The S2 round tunnel mechanics model considering the rock self weight is a model at the tunnel vault, haunch and arch bottom, and the round tunnel mechanics model considering the rock self weight includes a surrounding rock stress model and a surrounding rock displacement model.

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

[0018] ① Vault:

[0019]

[0020] ② Haunch:

[0021]

[0022] ③ Arch bottom:

[0023]

[0024] wherein, only formula (5b) is an explicit equation and can be directly solved, while formula (5a) and (5c) are implicit equations and cannot be directly solved, and thus an iteration method can be used to first solve r p , and then solve p r ; p r is the radial stress, r p is the surrounding rock plastic zone radius, p s is the tunnel support force, and γ is the rock body specific weight, p0 is the initial ground stress, and r0 is the tunnel radius.

[0025] The surrounding rock displacement model is

[0026]

[0027] wherein, u r0 is the tunnel inner wall displacement, γ is the rock body specific weight, p0 is the initial ground stress, G is the rock shear modulus, r p is the surrounding rock plastic zone radius, r0 is the tunnel radius, c and are the rock cohesion and internal friction angle respectively, and b (0≤b≤1) is the intermediate principal stress coefficient.

[0028] The p s The calculation formula is

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

[0030] Where 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.

[0031] The beneficial effect of the present invention is that it establishes a theoretical model of the plastic zone of the surrounding rock of a circular tunnel that simultaneously considers the deadweight of the rock mass, the intermediate principal stress, and the interaction between the surrounding rock and the support structure. The influence of initial geostress, rock mass shear modulus, rock mass deadweight, rock cohesion, support structure stiffness, and intermediate principal stress coefficient on the calculation results of the surrounding rock plastic zone was studied through parameter sensitivity analysis. It was found that with the increase of initial geostress, rock mass shear modulus, and rock mass weight, the surrounding rock plastic zone gradually increases, while with the increase of rock mass cohesion, support structure stiffness, and intermediate principal stress coefficient, the surrounding rock plastic zone gradually decreases, but the increase or decrease amplitude is different. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 This is a force analysis diagram of the plastic zone of the surrounding rock of a circular tunnel and micro-element bodies at different positions according to the present invention.

[0033] Figure 2 This is a diagram showing the calculation results of the surrounding rock plastic zone according to the present invention.

[0034] Figure 3 This is a diagram showing the influence of the initial ground stress on the calculation results of the plastic zone of the surrounding rock according to the present invention.

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

[0036] Figure 5 This is a diagram showing the influence of rock mass weight on the calculation results of the plastic zone of the surrounding rock.

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

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

[0039] Figure 8 This is a diagram showing the influence of the intermediate principal stress coefficient on the calculation results of the plastic zone of the surrounding rock. DETAILED DESCRIPTION

[0040] The embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to be used to explain the present invention, but should not be understood as limiting the present invention.

[0041] The method for constructing a plastic zone model of a circular tunnel surrounding rock taking into account the deadweight of the rock mass of the present invention comprises the following steps:

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

[0043] When σ2 = (σ1 + σ3) / 2, the unified strength theory can be written in a form similar to the MC criterion, that is (expressed in polar coordinates):

[0044]

[0045] Where: c and 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.

[0046] S2. Establish a circular tunnel mechanical model based on the ideal elastic-plastic theory taking into account the rock mass self-weight;

[0047] S201. Surrounding rock stress

[0048] Since the tunnel's length dimension is much larger than its cross-sectional dimension, it can be considered as a plane strain problem, so a certain cross section can be selected for study. Figure 1 The figure shows the calculation model of a circular tunnel cross section subjected to the far-field uniformly distributed initial ground stress p0. Since the tunnel cross section is circular, polar coordinates are used for the solution, and the direction perpendicular to the cross section is set as the z direction, so 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, and satisfy σ θ >σ z >σ r .

[0049] When the deadweight of the rock mass is taken into account, the plastic zone of the surrounding rock will no longer be a circular area symmetrical with the center of the tunnel, but will appear as follows: Figure 1 Asymmetrical area shown with a larger top range, smaller bottom range, and a centered waist.

[0050] The stress conditions of the micro-element at the tunnel vault, waist and bottom are as follows: Figure 1 As shown in the figure, when the deadweight of the rock mass is considered, the tangential stresses at the three locations mentioned above are all zero due to symmetry, while the tangential stresses at the remaining locations are no longer zero. Therefore, the problem is not completely axisymmetric. Therefore, for the sake of convenience, the following study only uses the microelement at the three locations mentioned above as an example.

[0051] The equilibrium equations of the microelement at the three locations of the tunnel vault, haunch and base are:

[0052]

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

[0054] The subscripts "e" and "p" are used below to represent the physical quantities of the elastic zone and plastic zone of the tunnel surrounding rock. If the unified strength theory is used as the rock strength criterion, then the stress component expressions of the tunnel surrounding rock plastic zone at the above three locations can be obtained by combining equations (1) and (2): (r0≤r≤r p , where r p is the radius of the plastic zone, and r0 is the radius of the tunnel.

[0055] ① Vault:

[0056]

[0057] ② Arched waist:

[0058]

[0059] ③ Arch bottom:

[0060]

[0061] Where: p s It is the tunnel support force, acting evenly on the inner wall of the tunnel.

[0062] If the radial stress on the elastic-plastic interface is assumed to be p r Similarly, when considering physical force, the stress components of the elastic zone at the above three locations are (r≥r p ):

[0063] ① Vault:

[0064] ② Arched waist:

[0065] ③ Arch bottom:

[0066] Since the stress at the elastic-plastic interface is continuous, that is, σ re =σ rp , σ θe =σ θp , so r=r p Substituting into equations (3) and (4), and solving them separately, we can obtain the radial stress p on the elastic-plastic interface at the three locations of the tunnel vault, arch waist and arch bottom: r and the radius of the surrounding rock plastic zone r p They are:

[0067] ① Vault:

[0068]

[0069] ② Arched waist:

[0070]

[0071] ③ Arch bottom:

[0072]

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

[0074] S202.Surrounding rock displacement

[0075] Here we only study the displacement of the surrounding rock at the tunnel vault, waist and bottom. According to the elastic theory, when the radial stress p on the elastic-plastic interface is obtained, r After that, the elastic zone stress of the surrounding rock at the above three locations can be obtained as follows:

[0076]

[0077] Yu et al. believed that before tunnel excavation, there is initial ground stress in the rock mass. Tunnel excavation causes stress redistribution on the one hand and convergence displacement of the surrounding rock on the other. Therefore, the convergence displacement of the surrounding rock is caused by the stress increment caused by tunnel excavation. Therefore, according to Equation (6) and the elastic constitutive relationship, the displacement of the surrounding rock at the above three locations can be obtained as:

[0078]

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

[0080] Since the main change in shape occurs when the rock undergoes plastic deformation, it is assumed here that the volume of the rock in the plastic zone remains unchanged. Combined with the geometric equation, we have:

[0081]

[0082] At the same time, according to the displacement continuity condition on 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 follows (the direction toward the tunnel center is considered positive):

[0083]

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

[0085] It can be seen that the displacement expressions of the elastic and plastic zones are the same. r Substituting into formula (9), we can get the tunnel inner wall displacement u at the above three locations: r0 for:

[0086]

[0087] S3. The theory of surrounding rock-support structure interaction is introduced to modify the circular tunnel mechanical model, and a tunnel surrounding rock plastic zone model is obtained that takes into account the rock mass deadweight, intermediate principal stress, and surrounding rock-support structure interaction.

[0088] The tunnel support structures commonly used in current engineering projects mainly include reinforced concrete lining, steel arches, anchors, and their combinations. Therefore, their stiffness is generally greater than that of the rock, so it can be assumed that they only produce elastic deformation. At the same time, in order to achieve a good support effect, it is generally required that the support structure should be in a timely manner and fit closely with the surrounding rock, without relative sliding between the two. Therefore, the radial convergence displacement of the surrounding rock can be considered to be equal to the elongation of the support structure, which can be obtained:

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

[0090] Where: k s It is the tensile and compressive stiffness of the supporting structure.

[0091] Then, by 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 follows:

[0092]

[0093] It can be seen that formula (12) is about r p The implicit equation cannot be solved directly, so an iterative method is needed to solve it.

[0094] It should be noted that this invention only proposes a calculation method for the plastic zone of the surrounding rock at three locations: the tunnel vault, haunch, and base. For other locations, there is usually no theoretical solution because they are not axisymmetric. This invention proposes an approximate method to handle this problem. That is, the three locations are corresponding to four points on the circular tunnel surrounding rock. After calculating the plastic zone radius of these four points, they are connected by a smooth curve to obtain a closed curve. 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 is the range of the surrounding rock plastic zone.

[0095] Case Analysis

[0096] 1. Computational Model

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

[0098] Table 1 The calculation parameters

[0099]

[0100] According to the above parameters, the plastic zone of surrounding rock can be obtained as follows Figure 2It can be seen that the plastic zone of surrounding rock is no longer a circular region with the tunnel center as the center, but a non-circular region with the upper part larger, the bottom smaller and the waist in the middle. This fully illustrates the influence of rock mass self-weight on the plastic zone of surrounding rock. For this example, the plastic zone radius of surrounding rock when not considering the rock mass self-weight is the plastic zone radius of the tunnel waist, i.e. 4.86 m, while when considering the rock mass self-weight, the plastic zone radius of the vault and the arch bottom is 5.74 m and 4.07 m respectively, with an increase or decrease of 18.11% and 16.26% respectively. This is because when considering the rock mass self-weight, the gravity direction of the vault rock mass is towards the tunnel center, which is a favorable factor for inducing roof fall failure of rock mass, thus being favorable for the formation of the plastic zone of surrounding rock, and accordingly the plastic zone radius at the vault is larger. While the gravity direction of the arch bottom rock mass is away from the tunnel center, thus being unfavorable for the formation of the plastic zone of surrounding rock, and accordingly the plastic zone radius at the arch bottom is smaller. While the rock mass self-weight at the waist is perpendicular to the tunnel radius, thus having no influence on the plastic zone range, which can also be seen from the second formula of formula (12). In summary, it can be considered that when considering the rock mass self-weight, the plastic zone range of surrounding rock is no longer a circular region with the tunnel center as the center, but a non-circular region with the upper part larger, the bottom smaller and the waist in the middle. It should be noted that in this example, p0=100 kPa, and according to the value of γ in table 1, the overlying rock thickness is only 5 m when only considering the self-weight stress, and the tunnel radius is 3 m, thus it can be considered that the tunnel is shallow or even super shallow. That is, the plastic zone calculation results shown are obtained under the condition that the tunnel is shallow or even super shallow, and with the increase of the tunnel depth, the directional characteristics of the plastic zone of surrounding rock will gradually weaken, which can be seen from the following parameter sensitivity analysis of p0. In summary, it can be considered that when the tunnel depth is shallow, it is very necessary to consider the influence of rock mass self-weight on the plastic zone of surrounding rock. Figure 2 The plastic zone calculation results shown are obtained under the condition that the tunnel is shallow or even super shallow, and with the increase of the tunnel depth, the directional characteristics of the plastic zone of surrounding rock will gradually weaken, which can be seen from the following parameter sensitivity analysis of p0. In summary, it can be considered that when the tunnel depth is shallow, it is very necessary to consider the influence of rock mass self-weight on the plastic zone of surrounding rock.

[0101] 2. Parameter sensitivity analysis

[0102] From the above example, it can be seen that the rock mass self-weight has an important influence on the plastic zone range of surrounding rock at different positions of the tunnel, while how the above calculation parameters influence the plastic zone calculation results needs further study. Therefore, the following parameter sensitivity analysis is used to study the influence law of the initial ground stress p0, the rock mass shear modulus G, the specific weight γ, the rock shear strength (here taking the cohesion c as an example), the support structure stiffness k s , the intermediate principal stress coefficient b, etc. on the plastic zone calculation results of surrounding rock.

[0103] (1) Influence law of initial ground stress p0

[0104] Taking p0 as 100kPa, 1000kPa, 10000kPa and 100000kPa respectively, and keeping other parameters unchanged, the variation law of the plastic zone of the surrounding rock at the three parts of the tunnel arch crown, arch waist and arch bottom is as follows: Figure 3 It can be seen that: ① With the increase of p0, the range of the plastic zone of the surrounding rock gradually increases, but its increasing trend gradually slows down and eventually tends to a certain value. Taking the plastic zone at the arch as an example, when p0 gradually increases from 100kPa to 1000kPa, 10000kPa and 100000kPa, its plastic zone increases from 5.74m to 6.80m, 6.89m and 6.9m respectively, that is, its increase rate gradually decreases. At the same time, it shows that tunnel excavation only affects a certain range of rock mass, that is, the stress concentration caused by tunnel excavation is localized, and therefore the range of influence on the surrounding rock is also localized, and will not increase indefinitely. ② As p0 increases, the plastic zone range changes from being large at the top, middle at the waist, and smallest at the bottom to becoming uniformly distributed throughout the tunnel perimeter. This means the plastic zone distribution becomes more uniform. This is because when p0 is small, the deadweight of the rock mass within the plastic zone accounts for a large proportion of the total stress field. However, when p0 increases to a certain level, the plastic zone range no longer increases significantly, and therefore the deadweight of the rock mass within the plastic zone also does not increase significantly. At this point, the proportion of the deadweight of the rock mass within the plastic zone in the total stress field gradually decreases. Therefore, when p0 is sufficiently large, the influence of the deadweight of the rock mass within the plastic zone on the calculation results of the surrounding rock plastic zone is negligible, and the plastic zone range gradually approaches a circular shape. This is also the reason why the deadweight of the rock mass is usually not considered when calculating the plastic zone of the surrounding rock of deep tunnels.

[0105] (2) Influence of rock mass shear modulus G

[0106] Taking G as 40MPa, 400MPa, 4000MPa and 40000MPa respectively, and keeping other parameters unchanged, the variation law of the plastic zone of the surrounding rock at the three parts of the tunnel arch crown, arch waist and arch bottom is as follows: Figure 4It can be seen that: ① With the increase of G, the plastic zone range of surrounding rock is gradually increased, but its increasing trend is gradually slowed down and finally tends to a certain value. Taking the surrounding rock at the arch top as an example, when G gradually increases from 40 MPa to 400 MPa, 4000 MPa and 40000 MPa, its plastic zone increases from 5.74 m to 5.89 m, 5.91 m and 5.91 m respectively, that is, its increasing amplitude is gradually reduced and gradually tends to a certain value. ② With the increase of G, the overall characteristics of the plastic zone range from large at the top, moderate at the waist and minimum at the bottom gradually tend to approximately equal in each direction, that is, the plastic zone distribution is more uniform, because when G is small, the rock mass deformation is large, and the deformation of the supporting structure will also increase accordingly, which will lead to an increase in the supporting force, so under the condition that the shear strength of the rock mass is unchanged, when the supporting force increases, the plastic zone of the surrounding rock will decrease accordingly. With the increase of G, the displacement of the surrounding rock decreases, and the deformation of the supporting structure decreases accordingly, and the supporting force also decreases, so the plastic zone will increase. ③ From the increase of the plastic zone of the surrounding rock at different positions, with the increase of G, the increase speed of the plastic zone range of the surrounding rock at the arch top is much smaller than that at the bottom, because with the increase of G, the deformation of the surrounding rock at the arch top decreases rapidly and gradually tends to the deformation at other positions, so the plastic zone range of the surrounding rock at different positions is more uniform.

[0107] (3) Influence law of rock mass unit weight γ

[0108] Take γ as 18 kN / m 3 , 19 kN / m 3 , 20 kN / m 3 and 21 kN / m 3 , and the rest of the parameters remain unchanged, the change law of the plastic zone of the surrounding rock at the three positions of the tunnel arch top, arch waist and arch bottom is as follows Figure 5 . It can be seen that: ① With the increase of γ, the change law of the plastic zone of the surrounding rock at different positions is not consistent, among which the arch top gradually increases, the arch waist remains unchanged, and the arch bottom gradually decreases, because the gravity of the arch top rock mass is towards the center of the tunnel, so with the increase of the self-weight of the rock mass, the rock mass at the arch top is more likely to be damaged due to the action of gravity, so its plastic zone range increases. According to formula (12), the plastic zone of the surrounding rock at the arch waist is not related to the unit weight of the rock mass, and the plastic zone at the arch bottom decreases with the increase of the self-weight of the rock mass, because the gravity of the arch bottom rock mass deviates from the center of the tunnel, which plays a role in restraining the uplift and other adverse deformations at the bottom of the tunnel, so it is not easy to form a plastic zone under the action of the gradually increasing self-weight. ② With the increase of γ, the increase amplitude of the plastic zone at the arch top and the arch bottom changes little, taking the surrounding rock at the arch top as an example, when γ increases from 18 kN / m 3 to 19 kN / m 3 , 20 kN / m 3and 21kN / m 3 When γ increases, the plastic zone increases from 5.56m to 5.62m, 5.74m, and 5.78m, respectively. This indicates that the increase in the plastic zone is not significant, indicating that the effect of the rock mass's deadweight on the plasticity of the surrounding rock is approximately linear. ③ With the increase of γ, the range of the plastic zone gradually evolves from being large at the top, in the middle of the waist, and smallest at the bottom to being larger at the top, smaller at the bottom, and with an unchanged waist. In other words, the range of the plastic zone gradually shifts from bottom to top.

[0109] (4) Influence of rock mass shear strength

[0110] Rock mass shear strength parameters include cohesion c and internal friction angle The influence of the two is basically the same. Therefore, the former is taken as an example for research. Taking c as 10kPa, 20kPa, 30kPa and 40kPa respectively, and keeping the other parameters unchanged, the change law of the surrounding rock plastic zone at the three parts of the tunnel arch, arch waist and arch bottom is as follows Figure 6 . It can be seen that: ① With the increase of c, the range of the plastic zone of the surrounding rock gradually decreases, but its decreasing trend gradually slows down and eventually tends to a certain value. Taking the surrounding rock at the arch as an example, when c increases from 10kPa to 20kPa, 30kPa and 40kPa respectively, its plastic zone decreases from 5.74m to 4.32m, 3.76m and 3.43m respectively, that is, its decrease gradually decreases. On the one hand, this is because with the increase of c, the shear strength of the rock mass 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 formula (12) that the radius of the plastic zone is not a simple linear relationship with c, so with the increase of c, the radius of the plastic zone does not decrease linearly. ② Regarding the effect of c on the plastic zone at different locations in the tunnel, while the plastic zone ranges at the crown, haunch, and base all decrease with increasing c, the extent of these decreases varies slightly. For example, when c increases from 10 kPa to 40 kPa, the plastic zones at these three locations decrease by 40.24%, 30.65%, and 26.29%, respectively. The crown exhibits the largest decrease in plastic zone radius. This is because the crown is more susceptible to yield failure under the combined effects of rock mass in-situ stress and deadweight. Therefore, when the rock mass shear strength increases, its plastic zone decreases the most.

[0111] (5) Support structure stiffness k s The influence of

[0112] Take k s The plastic zone changes of the surrounding rock at the three locations of the tunnel vault, arch waist and arch bottom are as follows: Figure 7 It can be seen that: ① As k sAs k increases, the plastic zone of the surrounding rock gradually decreases, but the reduction range varies in different locations. The reduction range at the arch crown does not change much, while the reduction range at the arch waist and arch bottom gradually decreases. Taking the surrounding rock at the arch bottom as an example, when k s When the pressure increases from 1.5MPa to 3MPa, 6MPa and 9MPa respectively, the radius of the plastic zone decreases from 4.07m to 3.57m, 3.09m and 3m respectively, that is, the reduction rate gradually decreases. This is because as k s As k increases, the support force provided by the support structure also increases, so the surrounding rock will not easily produce large deformation, and the radius of its plastic zone will also decrease. s As the stiffness increases, the plastic zones at different locations exhibit a characteristic pattern of being larger at the top, smaller at the bottom, and centered at the waist. The top plastic zone resembles a common collapse arch found in tunnels. This suggests that increasing the stiffness of the support structure can reduce the plastic zone of the surrounding rock and improve its stability, but this will correspondingly increase support costs. Therefore, a comprehensive balance between safety and efficiency should be considered.

[0113] (6) Influence of the intermediate principal stress coefficient b

[0114] Taking b as 0, 0.33, 0.67 and 1 respectively, and keeping other parameters unchanged, the variation law of the plastic zone radius of the surrounding rock at the three parts of the tunnel arch crown, arch waist and arch bottom is as follows: 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 eventually tends to a certain value. Taking the surrounding rock at the arch as an example, when b increases from 0 to 0.33, 0.67 and 1 respectively, its plastic zone decreases from 6.5m to 5.74m, 5.29m and 5.05m respectively, that is, its decrease is gradually decreasing. On the one hand, this shows that when the intermediate principal stress is considered, the plastic zone of the surrounding rock is gradually decreasing, 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 is limited. ② As b increases, the difference in the range of plastic zones in various parts will gradually decrease. This is because as b increases, 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 MC strength theory that completely ignores the intermediate principal stress and the double shear strength theory that completely considers the intermediate principal stress, respectively. Therefore, the surrounding rock gradually evolves from a biaxial stress state to a triaxial stress state, and the damage of the surrounding rock in all directions also tends to be more uniform.

[0115] Although the above embodiments have been shown and described, it is understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. Changes, modifications, substitutions and variations of the above embodiments by those skilled in the art are all within the scope of protection of the present invention.

Claims

1. A method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel taking into account the deadweight of the rock mass, characterized in that: The following steps are involved: S1. Calculate the stress components in the plastic zone of the tunnel surrounding rock at the arch crown, arch haunch, and arch base; S2. Calculate the stress components in the elastic zone at the crown, haunch, and base of the arch; S3. Calculate the radial stress on the elastic-plastic interface and the radius of the plastic zone of the surrounding rock at the arch crown, arch waist, and arch base; S4. Calculate the stresses in the elastic zone of the surrounding rock at the arch crown, arch haunch, and arch base; S5. Calculate the displacement of the surrounding rock in the elastic zone at the arch crown, arch haunch, and arch base; S6. Calculate the displacement of surrounding rock in the plastic zone; S7. Calculate the tunnel wall displacements at the arch crown, arch haunch, and arch base. S8. The rock support structure interaction theory is introduced to correct the radius of the plastic zone of the circular tunnel surrounding rock, and the corrected calculation formula of the plastic zone radius of the surrounding rock is obtained.

2. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 1 is characterized in that: The radius of the surrounding rock plastic zone in S3 is ① Vault: ② Arched waist: ③ Arch bottom: Among them, only equation (5b) is an explicit equation and can be solved directly, while equations (5a) and (5c) are implicit equations and cannot be solved directly. Therefore, an iterative method can be used to first calculate 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 weight of rock mass, p0 is the initial ground stress, and r0 is the tunnel radius.

3. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 2, characterized in that: The radius of the plastic zone of the surrounding rock in S3 does not take into account the interaction between the surrounding rock support structure. When the interaction theory of the surrounding rock support structure is introduced to correct the radius of the plastic zone of the circular tunnel, the calculation formula of the radius of the plastic zone of the surrounding rock is: p s =k s u r0 (11) Where: k s is the tensile and compressive stiffness of the supporting structure, u r0 is the displacement of the tunnel wall.

4. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 1, characterized in that: The stress in the elastic zone of the surrounding rock in S4 is Where, σ r , σ θ are radial and hoop normal stresses respectively, r is the distance from a certain point A to be determined to the center of the tunnel, p r is the radial stress on the elastic-plastic interface, p0 is the initial ground stress, r p is the radius of the plastic zone.

5. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 1, characterized in that: The calculation formula for the displacement of the surrounding rock in the plastic zone of S6 is: Where E and v are the rock elastic modulus and Poisson's ratio respectively, G is the rock shear modulus, and p is r is the radial stress on the elastic-plastic interface, p0 is the initial ground stress, r is the distance from a certain point A to be determined to the center of the tunnel, r0≤r≤r p , where r p is the radius of the plastic zone, and r0 is the radius of the tunnel.

6. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 5, characterized in that: The tunnel inner wall displacements at the arch crown, arch waist and arch bottom in S7 are: ① Vault: ② Arched waist: ③ Arch bottom: Where u r0 is the tunnel wall displacement, γ is the rock mass, p0 is the initial ground stress, G is the rock shear modulus, r p is the radius of the plastic zone of the surrounding rock, r0 is the radius of the tunnel, c and are rock cohesion and internal friction angle respectively, and b (0≤b≤1) is the intermediate principal stress coefficient.

7. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 1, characterized in that: The corrected calculation formula for the radius of the surrounding rock plastic zone is: ① Vault: ② Arched waist: ③ Arch bottom: Formulas (12a), (12b), and (12c) are about r p The implicit equation cannot be solved directly and needs to be solved by iterative method; among them, r p is the radius of the plastic zone of the surrounding rock, r0 is the radius of the tunnel, k s is the tensile and compressive stiffness of the support structure, γ is the weight of the rock mass, p0 is the initial ground stress, G is the rock shear modulus, c and are rock cohesion and internal friction angle respectively, and b (0≤b≤1) is the intermediate principal stress coefficient.

8. The method for calculating the radius of the plastic zone of the surrounding rock of a circular tunnel considering the deadweight of the rock mass according to claim 7, characterized in that: The revised formula for calculating the radius of the plastic zone of the surrounding rock is a formula for calculating the radius of the plastic zone of four points on the circular tunnel surrounding rock corresponding to the tunnel vault, arch waist and arch bottom. After obtaining the radius of the plastic zone of these four points, they are connected by a smooth curve to obtain a closed curve. This curve is the boundary between the elastic and plastic zones of the surrounding rock, and the area between the curve and the tunnel contour line is the range of the plastic zone of the surrounding rock.

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