An Innovative Elastoplastic Analysis for Soft Surrounding Rock considering Supporting Opportunity Based on Drucker-Prager Strength Criterion

State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China Key Laboratory of Deep Coal Resource Mining, Ministry of Education of China, School of Mines, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China Taiyuan Coal Gas Group Transportation and Marketing Branch, Taiyuan 030021, Shanxi, China Institute of Mining Engineering and Geology, Xinjiang Institute of Engineering, Urumqi 830091, China School of Science, Yangzhou Polytechnic Institute, Yangzhou 225127, China


Introduction
Coal has long been the main energy source in China [1,2]. In view of the exhaustion of shallow resources in recent years, the resource exploitation and space development gradually have been transferred to the deep earth [3,4]. e complex mechanical environment of "three heights and one disturbance" in deep coal and rock body leads to many problems and challenges in deep mining [5,6]. ey lead to the strength degradation of rock mass, which affects the stability of underground engineering [7]. e development and establishment of new theories and methods are the theoretical basis of surrounding rock control in deep underground engineering, which is of great significance to guide the surrounding rock control technology and engineering practice of kilometer deep roadway. e stress and plastic zone distribution state of surrounding rock of deep roadway is an important basis for evaluating the stability of surrounding rock and the reliability of quantitative support design. In the past, many researchers [8][9][10][11][12][13] have done research on circular tunnel problems in an elastoplastic, elasto-brittle-plastic, and strain-softening way. Generally, the Mohr-Coulomb criteria, Hoek-Brown criteria, and generalized Hoek-Brown criteria are employed as associated and nonassociated potential flow laws. On this basis, the surrounding rock stress and deformation distribution is analyzed by ideal elastoplastic and elasto-brittle-plastic mechanical models. However, engineering practices show that, applying to the different strength criteria, the stress state and plastic zone of the surrounding rock are different. erefore, choosing the appropriate strength criterion can make the mechanical state of surrounding rock closer to engineering practice. What is more, the elastic-plastic solution of surrounding rock under supporting condition could be obtained, and the deformation of surrounding rock may be too large in the process of soft rock roadway excavation, which affects the safety of chamber construction seriously.
At present, the supporting theory of surrounding rock after excavation has become mature, and many scholars have done a lot of research in this respect [14][15][16][17][18][19][20]. Although the elastic-plastic solution of surrounding rock under supporting condition is obtained, the deformation of surrounding rock may be too large in the process of soft rock chamber excavation, which affects the safety of roadway construction seriously. So, the supporting time should be considered, and it should be considered earlier than before. In summary, it is urgent to develop the theoretical study of supporting time in the process of roadway excavation. Due to the traditional elastic-plastic analysis of the roadway based on two-dimensional plane strain, considering the "space effect" can analyze the stress, displacement, and other analytical changes caused by threedimensional excavation of the roadway. Both theory and practice have shown that, after roadway excavation, most of the rock mass is still in the state of three directions. In addition to the maximum and minimum principal stress, the magnitude of intermediate principal stress also has an important effect on the deformation and failure of roadway surrounding rock.
Based on the Drucker-Prager (D-P) strength criterion and noncorrelation flow rule, this paper analyzes the fact that brittle fracture zone exists obviously in deep roadway. e closed analytical solutions of stress field, displacement field, and plastic zone radius of roadway are derived. e influence of different parameters on the stress, deformation, and plastic zone radius of surrounding rock is studied. e results can provide important theoretical basis for the stability evaluation and support design of surrounding rock.

Theoretical Model
In the process of excavation, the excavation surface of roadway is within a certain range, and the development of elastoplastic deformation and stress redistribution of surrounding rock are restricted by its own constraints, which leads to the fact that the deformation of surrounding rock cannot be fully released. Stress redistribution cannot be completed quickly, called excavation "space effect ." Considering the "space effect" of excavation, the timely support of roadway has theoretical basis. With the advance of excavation facing forward, the virtual support force is gradually released with the secondary deformation of surrounding rock, while the pretightening force of the actual support of roadway increases gradually, and the stability of surrounding rock is maintained.
According to the failure characteristics of the surrounding rock of the roadway, the zoning of the surrounding rock of the circular roadway is shown in Figure 1 and the following assumptions are made: (1) e roadway is considered to be infinite long, and the rock mass is isotropic approximately and homogeneous continuous medium; (2) e section of roadway is circular, and the radius of the roadway is R 0 , and the radii of residual plastic zone, softening plastic zone, and elastic zone are R r , R s and R e ; (3) Before excavation, the surrounding rock of roadway is in the hydrostatic pressure field; that is, the original rock stress is σ 0 , MPa. And supporting lining structure provides supporting resistance p i . σ r , σ θ are defined by the radial stress and the circumferential stress of the roadway, respectively, and σ θ > σ r are satisfied. ε θ , ε r are defined by the radial and circumferential strain of the surrounding rock, respectively. In the following upper corner marks, "e", "s" and "r" denote the amount of elastic zone, plastic softening zone, and plastic residual zone, respectively.

Drucker-Prager Strength Criterion
At present, the Drucker-Prager strength criterion has been applied to lots of finite element numerical modeling software; meanwhile, Drucker-Prager criterion is a linear expression of the generalized Mises criterion. e effects of the intermediate principal stress and the hydrostatic pressure on the yield characteristics of rock materials are considered. Hence, the Drucker-Prager criterion is adopted in this paper, and its form is given in the following equation [21]: where I 1 is the first stress invariant, and J 2 is the second stress deviation invariant.
Regarding the values of β and k f , it is usually to associate the D-P criterion with the M-C criterion, that is, to think that the D-P criterion is a smooth approximate treatment method for the M-C criterion to remove angular points (also singular points) on the π plane, because the D-P criterion is a circle on the π plane. e characteristics of tensile resistance, different compressive strength, and the effect of the intermediate principal stress on the material need to be considered since the coal and rock mass are a strength differential (SD) material [22][23][24][25][26][27] with different compressive properties, and the elastic core zone is in a triaxial stress state.
If σ 1 , σ 2 and σ 3 are assumed to be maximum principal stress, intermediate principal stress, and minimum principal stress, and then the specific expressions of I 1 and J 2 are, respectively, To simplify the selection of the parameters in the Drucker-Prager criterion, the converted relationship between β, k f , the internal frictional angle (φ), and the cohesion (c) in the Mohr-Coulomb criterion with the plane strain assumption is provided in equation (3).
In practical engineering, the intermediate principal stress coefficient m is often introduced to indicate the relationship between the three principal stresses: where 0 ≤ m ≤ 1, and the greater the m, the greater the effect of the intermediate principal stress, and the smaller the effect.
Equation (4) is simplified, then substituting in equations (2a) and (2b), where λ m � ������������ � (m 2 − m + 1)/3. By substituting equations (5a), (5b) into equation (1), the expression of D-P criterion including to the parameters m, λ m , β, k f is derived: 3.1. Initial Bearing Capacity. Elastic-plastic problem of surrounding rock of roadway can be regarded as plane strain problem. When the support resistance and original rock stress meet the p i < σ 0 , it is generally considered that the stress state of surrounding rock satisfies the following relationship: According to the finite element simulation results, it is pointed out that, due to the influence of "space effect" on the working surface of roadway, the released load acting on the roadway excavation section will not immediately reach the initial in situ stress state but has a time course. e release load varies with time, where R 0 is the radius of circular roadway, m; V is the average speed of roadway excavation, m/d; t is the starting time of the moment of excavation from section. e default value in this paper t � 1 d, so different roadway velocities represent different distances between section and excavation surface. e expression of "virtual support resistance" of "space effect" roadway is At this point, the bearing capacity p c of the initial support can be expressed as

Plastic Flow Equation (Definition of Dilatancy Coefficient).
Considering the dilatation of rock mass in plastic softening zone and plastic residual zone, the relationship between dilatancy coefficient and strain is shown in Figure 2. e dilatation of rock mass in plastic softening zone and plastic residual zone is considered.
where η 1 is the dilatancy coefficient of plastic softening zone, and its value is greater than 1. e plastic deformation of rock is nonlinear and generally satisfies the noncorrelated flow rule, which can be determined by the plastic potential function (φ). e yield function (f ) and φ have the same expression form. e internal friction angle (φ) in f can be transformed into the Elastic zone Plastic softening zone Plastic residual zone shear dilatancy angle ψ. e expression of plastic potential function is where A i,ψ is the rock material parameter. According to the plastic potential theory, where dε p ij , σ ij are the plastic strain increment and stress tensor, respectively; dλ is a constant related to the plastic potential function.
According to the nonassociated flow rule of dilatation of rock mass in the plastic residual zone, we have where Δε r r , Δε r θ are the radial and circumferential strain increments in the plastic residual zone, respectively; η 2 is the dilatancy coefficient in the plastic residual zone, and its range is 1.3-1.5 generally.

Elastoplastic Analysis of Surrounding
Rock before Supporting Roadways

Basic Equation.
According to the elastoplastic theory, the surrounding rock satisfies the equilibrium equation: Geometric equation: where u is the radial displacement of the surrounding rock. Constitutive equations satisfying the plane strain are shown:

Stresses and Displacement in Elastic
Zone. When considering the "space effect," the stresses of elastic zone are Radial displacement is where μ is Poisson's ratio of rock mass.

Stress and Displacement in Plastic Softening
Zone. e total strains in plastic softening zone consist of elastic strain and plastic strain. e strains in plastic softening zone are e compatible equation of displacement in plastic softening zone can be obtained from eqs. (20), (16), and (14), and the displacement at the boundary condition r � R s can be obtained continuously.
where G � (2(1 + u)(p 0 − p c )/E)(r 0 /R e ) 2 (R s /r) 1+η 1 , ]. e radial stresses are derived by connecting eq. (15) when r � R s , . φ is the internal friction angle of rock mass. σ c , σ m c , and σ r c are the yield strength, peak strength, and residual strength of rock mass, respectively.  Advances in Civil Engineering

Stress and Displacement in Plastic Residual
Zone. e strains in plastic residual zone are e displacement solution process is consistent with the plastic softening zone, and the differential equation can be obtained: where e compatible equation of displacement in this zone can be obtained from eqs. (25), (15), and (13), and the displacement at the boundary condition r � R r can be obtained continuously.
e radial stresses can be obtained by combining eq (15) when r � R r ,

Plastic Softening Zone and Plastic Residual Zone.
At the boundary of the plastic softening zone and the plastic residual zone (i.e., r � R r ), the strength parameter of plastic softening zone softens to the residual strength value. So the ratio of radius of plastic residual zone to radius of plastic softening zone is At the boundary of the elastic zone and the plastic softening zone (i.e., r � R s ), the radius of the plastic softening zone can be obtained: e radius of the plastic residual zone can be obtained by combining eqs. (29) and. (30):

Coupling Effect between Supporting Structures and Surrounding Rock
Assuming that the bolts and liners of the supporting structure are contacting with the surrounding rock closely, the coupling between the bolt and the surrounding rock is considered as a composite bearing body, and the time difference between bolts and liners is 0 approximately. In addition, considering the timeliness of concrete strength hardening, rockbolts and liners bear the load of surrounding rock together.

Equivalent Material Parameters under Coupling Effect of Surrounding Rock and Rockbolts.
Because of the supporting force between rockbolts and surrounding rock acting on the surrounding rock in the form of equivalent volume force, the stresses and the displacement of plastic softening zone change nonlinearly. When the rockbolt anchoring acts on these two sections, it is more complicated to solve the stresses and displacement of surrounding rock. According to reference [29], using homogenization method, bolts are uniformly arranged. Bolts and surrounding rock are regarded as equivalent materials, and its elastic modulus expression is as follows where E b is the elastic modulus of bolt; r b is the diameter of bolt; E is the elastic modulus of surrounding rock; f 1 is the row spacing of bolt, f 2 is the hoop spacing of bolt. When the plastic zone is formed after the excavation of the surrounding rock, the internal friction angle changes to be smaller than that of the elastic zone. When the bolt is applied in the plastic zone, the internal friction angle will be close to the elastic zone. According to reference [30], the equivalent material cohesive force of bolt pretightening force should be considered, and the equivalent material cohesive force is where c i is the cohesion of each zone, and the cohesion of the elastic zone, the plastic softening zone, and the plastic residual zone is c 0 , c s and c r , respectively; c j is the cohesion force of the rockbolt andc j � σ s πr 2 b /4 . In this equation, σ s is the yield strength of the rockbolt; c t is the additional cohesion force formed by the pretightening force of rockbolt and c t � (F 0 /f 1 f 2 )cos(45 ∘ − φ)tan φ. In this equation, F 0 is the pretightening force of rockbolt.
Advances in Civil Engineering e stresses, displacement, and the zoning range of surrounding rock after rockbolt supporting can be obtained by substituting the equivalent material parameters E * and c * into the above equations.

Displacement Analysis of Concrete Shotcrete Supporting.
e supporting resistance is provided by passive compression after spraying, but the strength required for the bearing capacity of the spraying layer needs a time course. e strength is mainly related to the elastic modulus.
According to reference [31], the elastic modulus can be obtained: where E 0 is the ultimate elastic modulus of concrete shotcrete layer; t 1 is the time of concrete shotcrete layer achieving the design strength.
When the rockbolts and surrounding rock are regarded as composite bearing body, the load of composite bearing body is assumed by lining supporting resistance and virtual supporting force. e radial displacement of the plastic residual zone is expressed as follows: For a given time increment ∆t, the radial displacement increment of the plastic residual zone is e concrete spray layer is equivalent to the curved beam with stiffness K c as the incremental constitutive equation of the roadway. e equation is where , μ 0 is Poisson's ratio of the concrete sprayed layer, and a is the inner diameter of the concrete sprayed layer.
According to eq. (37), we can obtain

Example Analysis
e basic parameters of circular roadway design are as follows: radius of roadway r 0 � 3 m, elastic modulus E � 2.0 GPa, Poisson's ratio μ � 0.32, crustal stress p 0 � 20 MPa, initial cohesion c 0 � 0.8 MPa, peak hardening cohesion c m � 1.6 MPa, residual cohesion c r � 0.4 MPa, internal friction angle � 30°, and initial support force p i � 0 MPa. e rockbolts cover the plastic zone, and the specific parameters are as follows: modulus of elasticity E b � 210 GPa, yield strength σ b � 345 MPa, pretightening force F � 60 kN, r b � 20 mm, row spacing, and hoop spacing f 1 � f 2 � 800 mm. e lining parameters are as follows: elastic modulus E 0 � 2.8 × 10 4 MPa, r 0 − a � 0.25 m. When considering the supporting time, the excavation progress is V � 5 m/d, and the excavation section follows the principle of excavation to support. Before and after supporting, the roadway area is shown in Table 1 [13,25,28].
As shown in Table 1, the thickness of plastic residual zone is 1.87 m under high crustal stress and only 0.96 m under the supporting. e supporting can reduce the range of plastic zone effectively. When considering the "space effect," the range of plastic zones is further reduced. In addition, the more nearer the distance between the excavation face and the supporting surface, the more obvious the "space effect." Figure 3, the larger the distance between rows, the smaller the circumferential stress and radial stress, and the peak value of circumferential stress moves away from the center of the roadway, which is not conducive to the safety of the roadway. e radial stress has an inflection point at the radius of the plastic softening zone, developing faster in the plastic softening zone. In the plastic softening zone, the circumferential stress increases sharply, while the elastic zone tends to be stable. Although the reduction of row spacing between rockbolts can increase the stress of surrounding rock and strengthen the bearing capacity of plastic zone, the increasement is not obvious. Figure 4 shows that the displacement of surrounding rock increases with the distance increasement of spacingrow of rockbolts. e displacement of plastic zone is mainly affected by the distance of spacing-row of rockbolts. But the displacement of elastic zone is almost negligible. When the distance between spacing-row of rockbolts is 1.2 m, the displacement of surrounding rock is compared with that of the spacing-row of rockbolts, which is 0.4 m. e displacement of the latter is 35.6% less than that of the former. e influence of spacing-row of rockbolts is mainly reflecting in the displacement of surrounding rock.

Effect of "Space Effect" and Support Time on Displacement of Surrounding Rock in Roadway.
e distance between the excavation face and the supporting surface is defined as x and x � Vt 0 . So, x is related to the "space effect." e influence of "space effect" on the displacement of surrounding rock is shown in Figure 5. e smaller the value x is, the greater the virtual supporting resistance is, and the shorter the distance between the supporting surface and the excavation surface is. So, the smaller the radial displacement of the tunnel wall is, the more favorable the stability of the roadway is. e displacement of the surrounding rock at x � 15 m is compared with the displacement without considering the "space effect" and the supporting time, and the displacement of the former is 95.9% of that of the latter when the displacement of surrounding rock is equal to r � 3 m. erefore, when the displacement of surrounding rock is        greater than 15 m, the influence of the "space effect" on the displacement of surrounding rock can be neglected.
In the process of soft rock roadway excavation, a layer of concrete is sprayed on the excavated part to restrain the deformation of surrounding rock, and then the bolt is applied. e influence of supporting time on surrounding rock displacement is shown in Figure 6. Before the supporting is applied, a section of displacement has been produced on the wall of the roadway, which can be recorded as u 0 . e displacement after the supporting is applied, which can be recorded as u 1 . When the driving speed is constant, x is correlated with t 0 positively. e bigger the t 0 , the bigger the u 0 , and the smaller the u 1 . But the increase of u 0 is far greater than the reduction of u 1 and the total deformation of the tunnel wall is increasing. is is a result of the larger displacement of the roadway wall before supporting, and the limited displacement of the surrounding rock is restrained by supporting. For example, when t 0 is 6 d, u 1 is only 47.4% of u 0 . On the contrary, when t 0 is 1 d, u 1 is only 28.8% of u 0 . e total displacement is only 60.7% when t 0 is 6 d, and the supporting effect is obvious.

Conclusions
In this paper, based on D-P strength criterion and the noncorrelation flow rule, an equivalent circular roadway solution method is proposed. e stress and deformation of surrounding rock of underground roadways are obtained by the elastoplastic theory. en, the failure mechanism of roadway with soft rock is analyzed. Based on the D-P strength criterion, considering the softening and dilatation of surrounding rock, the surrounding rock of the roadway is divided into three zones, and the "space effect" is introduced. e analytical expressions of stress, displacement, and partition range of surrounding rock without rockbolt and lining supporting are obtained. In the process of supporting, the rockbolt is evenly arranged in the roadway, and the composite bearing body is composed of the rockbolt and the surrounding rock. e equivalent material parameters are calculated reasonably, and the elastoplastic solution of surrounding rock under the supporting condition is obtained. erefore, considering the aging characteristics of the concrete shotcrete layer, the theoretical calculation is closer to the field engineering practice. Under the support condition, the plastic softening zone can be effectively reduced, and the plastic residual zone can be mainly reduced. Considering the "space effect" and the supporting time, the zones can be calculated more accurately. By analyzing the influencing factors of the stress and displacement of the surrounding rock, it is concluded that the spacing-row between rockbolts has little effect on the surrounding rock stress, while the influence on the displacement of surrounding rock is mainly reflected in the plastic residual zone.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.