# A Unified Solution for Surrounding Rock of Roadway Considering Seepage, Dilatancy, Strain-Softening and Intermediate Principal Stress

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Unified Strength Theory

_{3}, σ

_{2}and σ

_{1}represent minor, intermediate and major principal stresses, respectively; b represents weighted coefficient accounting for influencing of σ

_{2}which ranges from zero to one. In the present paper, positive value is for compressive stress.

## 3. Mechanical Model for Surrounding Rock of Roadway

#### 3.1. Distribution of Pore Water Pressure

_{0}, and it is imposed by a hydrostatic stress of σ

_{0}before excavation. If the support pressure p

_{i}is not great enough, two plastic zones, namely broken zone with radius being R

_{b}and plastic softening zone with radius being R

_{p}, would be commonly formed around the roadway, while the surrounding rock with radius greater than R

_{p}is still under elastic state.

_{0}is existed throughout surrounding rock (Figure 1). After roadway excavation, internal water pressure is decreased to zero. As is well known, the subsurface water will flow due to pore water pressure difference between inner and outer domain. According to above assumptions, the water seepage would only occur in radial direction, which results in the axisymmetric redistribution of pore water pressure. To obtain analytical solution of water pressure distribution, a boundary on which water pressure is unaltered should be determined. Related studies [2] showed that when a point is located on boundary with radius of R

_{0}≥ 30 r

_{0}, the alternation of pore water pressure p

_{w}and stress can be negligible compared to infinite condition. Thus, the mechanical model is established with a maximum radial dimension of R

_{0}= 30r

_{0}(shown in Figure 1). In this study, neglecting the buoyancy part of seepage body force and compressibility of water, the governing equation for steady water seepage can be obtained [11]:

_{w}(r)|

_{r}

_{=r0}= 0 and p

_{w}(r)|

_{r}

_{=R0}= p

_{0}, the distribution of water pressure around roadway surrounding rock is given:

_{0}/ln(r

_{0}/R

_{0}).

#### 3.2. Elastoplastic Solution of Mechanical Model

#### 3.2.1. Basic Equations under Polar Coordinate System

_{θ}and σ

_{r}, respectively. Under the assumption of (1), the strain in axial direction of roadway is zero, and then stress σ

_{z}in axial direction can be expressed by σ

_{z}= a(σ

_{r}+ σ

_{θ})/2, in which a is a constant. If the surrounding rock is in elastic zone, then a = 2v in which v represents Poisson’s ratio with value interval of [−1, 0.5]; while if the surrounding rock is in the two plastic zones, then a = 1 [13]. Hence, the relations among minor, intermediate and major principal stresses under polar coordinate system are σ

_{3}= σ

_{r}≤ σ

_{2}= σ

_{z}≤ σ

_{1}= σ

_{θ}, and then intermediate principal stresses in surrounding rock satisfy the relation of σ

_{2}≥ (σ

_{1}+ σ

_{3})/2 − (σ

_{1}− σ

_{3})sin φ/2. Then unified strength theory of Equation (1) can be rewritten as an alternative form:

_{θ}and ε

_{r}denote circumferential strain and radial strain, respectively; and u denotes displacement in radial direction.

#### 3.2.2. Analytical Solutions of Displacement and Stress in Elastic Zone

_{p}, the surrounding rock is on the verge of yielding, which satisfies yield condition of Equation (4); and the boundary condition of radial stress at r = R

_{0}is σ

_{0}+ p

_{0}, namely

_{0}have been accomplished before excavation of roadway, so the actual strain components are given by [29,30]:

^{e}* is solved by substituting Equation (12) into left-hand side equation of Equation (6):

#### 3.2.3. Analytical Solutions of Displacement and Stress in Plastic Softening Zone

_{1}(α

_{1}≥ 1) denotes dilatancy coefficient, which is slope between increments of radial and circumferential plastic strains shown in Figure 3. When α

_{1}= 1, it represents a condition of surrounding rock without volumetric dilatancy.

_{2}(α

_{2}≥ 1) denotes dilatancy coefficient, which is slope between increments of radial and circumferential plastic strains in broken zone as illustrated in Figure 3, and it can be calculated similarly to Equation (19). When α

_{2}= 1, it means that no volumetric dilatancy occurs in the broken zone.

_{p}.

^{p}* = u

^{e}* at r = R

_{p}(see Equation (13)), we obtain the displacement u

^{p}* and circumferential strain ${\epsilon}_{\theta}^{p}$ through solving ordinary differential Equations (24) and (6):

^{p}denotes compressive strength of surrounding rock as illustrated in Figure 4.

^{p}. The compressive strength of surrounding rock, n

^{p}, is given by

_{p}, and substituting Equation (30) into Equation (5), the stress components in plastic softening zone are expressed as

#### 3.2.4. Analytical Solutions for Displacement and Stress in Broken Zone

_{r}

_{=r0}= p

_{i}, and substituting Equation (30) into Equation (5), stress components in broken zone are then given as:

#### 3.2.5. Radii of Plastic Zones

^{p}= n* at r = R

_{b}and Equation (29), we obtain

_{b}, the radius of softening zone is be given by

#### 3.3. Validation of Unified Analytical Solution

_{0}= 0, the water seepage influence is not regarded; ${n}^{\ast}=n$, the strain softening of surrounding rock is ignored; ${\alpha}_{1}={\alpha}_{2}=1$, the rock dilatancy is not considered; b = 0, the unified strength theory is simplified as Mohr–Coulomb criterion. For the analytic solution of plastic zone radius (i.e., strain-softening zone radius or broken zone radius in present study, as the two radii are the same under such condition), substituting the above parameters into Equation (42), and with combination of Equations (33), (11), and (4), the plastic zone radius is given:

## 4. Example Analysis

_{0}is 2 m with uniform original stress σ

_{0}being 15 MPa. Table 1 lists the calculation parameters for mechanical model of surrounding rock of roadway.

#### 4.1. Influence of Water Seepage

_{0}is increased from 1 MPa to 3 MPa, peak circumferential stress increases by 18% from 28.7 MPa to 33.8 MPa. Figure 5b,c shows variations of radii of plastic zones and of roadway surface displacements under different water pressure. As can be seen, radii of broken and plastic softening zones and surface displacement with water seepage considered are greater than that of without water seepage considered. The reason is that effective stress in surrounding rock would increase with seepage effect considered [2,6]; moreover, radii of two plastic zones and surface displacement of roadway all present an increase trend as water pressure increases with water seepage considered and without water seepage considered although their tendencies are different. The former increases exponentially with water pressure, while the latter increases linearly with water pressure. For instance, with p

_{0}increasing from 1 MPa to 3 MPa, plastic softening zone radius increases by 22% from 3.2 m to 3.9 m, surface displacement of roadway increases by 80% from 0.05 m to 0.09 m. In short, the obtained results indicate that water seepage plays a significant role in radii of plastic zones and surface displacement of roadway, and its influence should be carefully considered for the stability of roadway under groundwater occurrence environment.

#### 4.2. Influence of Strain Softening

_{b}but less influence on radius of strain-softening zone as illustrated in Figure 6b. With the increase in M, R

_{b}would gradually expand with expansion velocity decreasing. For instance, when softening modulus M is increased from 1000 MPa to 2000 MPa, radius of broken zone increases by 21% from 2.4 m to 2.9 m; while softening modulus M is increased from 2000 MPa to 3000 MPa, radius of broken zone increases by 7% from 2.9 m to 3.1 m. Ratio of R

_{p}and R

_{b}decreases in a decay form with softening modulus increasing, which further indicates that softening modulus has a greater influence on R

_{b}than R

_{p}. The variation curve between softening modulus and surface displacement of roadway. As seen from Figure 6c, when softening modulus M is increased from 1000 MPa to 3000 MPa, surface displacement of roadway only increases from 51.8 mm to 62.3 mm with its slope between softening modulus, and surface displacement of roadway is decreasing gradually, which showed that softening modulus M would affect surface displacement of roadway to an extent. In short, the obtained results indicate that strain-softening characteristics of surrounding rock should be analyzed to accurately predict broken zone radius and surface displacement of roadway; for bolting support roadway, the length of rock bolt is usually determined by the radius of plastic zones [1], so the length of rock bolt should be increased with softening modulus increasing.

#### 4.3. Influence of Dilatancy

_{1}and α

_{2}in this paper (see Figure 3). As described in theoretical analysis of Section 3, dilatancy coefficient α

_{2}has no influence on radii of broken and softening zones. Noting that when rock dilatancy coefficient is 3.4, it is corresponding to associated flow rule (i.e., internal friction angle is equal to dilatancy angle); while rock dilatancy coefficient is set to 1, it is corresponding to the condition without dilatancy considered (i.e., dilatancy angle is set to zero).

_{1}, the greater of radius of broken zone, which agrees well with the related literature [18]. The broken zone radius increases linearly with dilatancy coefficient α

_{1}increasing. Dilatancy coefficients play an important role in surface displacements of roadway (Figure 7c). As can be seen, surface displacements of roadway increase linearly with dilatancy coefficient α

_{1}. When dilatancy coefficient α

_{1}is increased from 1 to 3.4, surface displacement of roadway increases by 32.4% from 50.0 mm to 66.2 mm under condition of α

_{2}= 1; while surface displacement of roadway increases by 37.8% from 52.6 mm to 72.5 mm under condition of α

_{2}= 1.6, the increment of surface displacement of roadway is about 7.52 mm with dilatancy coefficient increasing by one. Relevant studies showed that the general value of dilatancy coefficient is 1.5 [22]. Hence, the radius of broken zone and surface displacement of roadway is underestimated with dilatancy not considered, while it is overestimated using the associated flow rule of rock.

#### 4.4. Influence of Intermediate Principal Stress

#### 4.5. Influence of Residual Cohesion

## 5. Conclusions

- (1)
- With water pressure increasing, peak circumferential stress, radii of two plastic zones and surface displacement of roadway with water seepage considered would increase, and their magnitudes are all greater than their corresponding conditions with water seepage not considered. Particularly, radii of plastic zones and surface displacement of roadway increases exponentially with water pressure increasing, water seepage influence should be carefully considered to ensure roadway stability under groundwater environment.
- (2)
- With softening modulus increasing, the magnitude of peak circumferential stress is kept unchanged, and location of peak circumferential stress would slightly shift to the deeper surrounding rock. Softening modulus has a greater influence on broken zone radius than that of strain-softening zone radius, and it also affects surface displacement of roadway to an extent. To accurately predict broken zone radius and surface displacement of roadway, strain-softening characteristics of surrounding rock should be analyzed; and length of rock bolt should be increased with softening modulus increasing for bolting support roadway.
- (3)
- Rock dilatancy coefficient has little effect on magnitude of peak circumferential stress and plastic softening zone radius, while radius of broken zone and surface displacement of roadway increase linearly with dilatancy coefficient α1 increasing. For actual engineering, surface displacement of roadway would be underestimated with rock dilatancy not considered, while they are overestimated if associated flow rule is adopted, indicating that flow rule of rock should be reasonably chosen for calculating roadway surface displacement and broken zone radius of roadway.
- (4)
- With weighted coefficient increasing, stress components in plastic zones at the same distance from roadway center would increase; while the distance of peak circumferential stress location from roadway center, radii of broken and plastic softening zones, and surface displacement of roadway are reduced. Self-bearing capacity enhancement of surrounding rock resulting from effect of intermediate principal stress should be considered. The traditional Mohr–Coulomb criterion is more conservative. Rock bolt length and grouting range can be decreased after considering effect of inter-mediate principal stress, which is helpful for reasonable measure selection of surrounding rock.
- (5)
- With residual cohesion increasing, peak circumferential stress remains unchanged, stress components in plastic zones at the same distance from roadway center would increase, and the distance of plastic–elastic interface from roadway center decreases, which implicates that grouting measure can be adopted to improve roadway stability effectively.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Influence of water seepage on (

**a**) stress distributions, (

**b**) radii of plastic zones, (

**c**) surface displacements of roadway.

**Figure 6.**Influence of strain softening on (

**a**) stress distributions, (

**b**) radii of plastic zones, (

**c**) surface displacements of roadway.

**Figure 7.**Influence of dilatancy on (

**a**) stress distributions, (

**b**) radii of plastic zones, (

**c**) surface displacements of roadway.

**Figure 8.**Influence of intermediate principal stress on (

**a**) stress distributions, (

**b**) radii of plastic zones, (

**c**) surface displacements of roadway.

Parameters | Values | Parameters | Values |
---|---|---|---|

Poisson’s ratio v | 0.25 | Dilatancy coefficient in plastic softening zone α_{1} | 2 |

Elastic modulus E/MPa | 2000 | Dilatancy coefficient in broken zone α_{2} | 1.5 |

Residual Cohesion c*/MPa | 1.0 | Softening modulus M/MPa | 2000 |

Original Cohesion c/MPa | 3.0 | Internal friction angle φ/° | 30 |

Weighted coefficient b | 0.5 | Initial pore water pressure p_{0}/MPa | 2 |

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## Share and Cite

**MDPI and ACS Style**

Yuan, Z.; Zhao, J.; Li, S.; Jiang, Z.; Huang, F.
A Unified Solution for Surrounding Rock of Roadway Considering Seepage, Dilatancy, Strain-Softening and Intermediate Principal Stress. *Sustainability* **2022**, *14*, 8099.
https://doi.org/10.3390/su14138099

**AMA Style**

Yuan Z, Zhao J, Li S, Jiang Z, Huang F.
A Unified Solution for Surrounding Rock of Roadway Considering Seepage, Dilatancy, Strain-Softening and Intermediate Principal Stress. *Sustainability*. 2022; 14(13):8099.
https://doi.org/10.3390/su14138099

**Chicago/Turabian Style**

Yuan, Zhigang, Jintao Zhao, Shuqing Li, Zehua Jiang, and Fei Huang.
2022. "A Unified Solution for Surrounding Rock of Roadway Considering Seepage, Dilatancy, Strain-Softening and Intermediate Principal Stress" *Sustainability* 14, no. 13: 8099.
https://doi.org/10.3390/su14138099