# Study on the Anisotropy of Strength Properties of Columnar Jointed Rock Masses Using a Geometric Model Reconstruction Method Based on a Single-Random Movement Voronoi Diagram of Uniform Seed Points

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## Abstract

**:**

## 1. Introduction

## 2. Geometric Model Reconstruction Method for CJRM

#### 2.1. Engineering Background of the Baihetan Hydropower Station and Feasible Geological Statistical Parameters

_{2}β

_{3}–P

_{2}β

_{4}(a transitional interval between the Lower Permian P2β3 and Upper Permian P2β4) rock stratum [23], the CJRM horizontal cross-sectional shapes are composed of irregular polygons, primarily including quadrilateral, pentagonal, and hexagonal polygons, as shown in Figure 2b. Additionally, there are transverse joints with a large penetration rate in the vertical cross-section, as shown in Figure 2c.

_{i}, and the total area of the visible region, A, can be acquired. For the description of the geometric characteristics of the irregular polygons of the CJRM horizontal cross-section, the polygon density, Q, of the horizontal cross-section and the irregular factor, I

_{r}, of the cross-sectional polygons are adopted [25,26]. The horizontal cross-sectional polygon density, Q, can be calculated by dividing the number of irregular polygons in the exploration region by the area of the exploration region, A. To describe the irregular factor, I

_{r}, of the horizontal cross-sectional polygons, the variation coefficient of the polygon area is adopted, that is, the ratio of the standard deviation of each polygon area within the exploration region to the average polygon area, as shown in the following Equation (1):

_{i}is the area of the polygon, i.

#### 2.2. Single-Random Movement with a Range Constraint Voronoi Diagram of Uniform Seed Points

_{r}, a method to generate single-random movement with a range constraint Voronoi diagram of uniform seed points is proposed. The method consists of the following four procedures (taking a target reconstruction 2D model with plane size of 1 m × 1 m, polygon density, Q, of 10 m

^{−2}, and a factor, I

_{r}, of 35%) for example:

_{c}, is calculated by Equation (2):

_{e}is the area of the expanded region, and the expansion multiple is set as 2 (A

_{e}= 2L × 2H) for the 3D geometry rotation in the subsequent procedure.

_{i}(x

_{i},y

_{i}), as shown in Figure 4a.

_{r}, of the Voronoi diagram is 0, as shown in Figure 4b.

_{ri}, of the new generated Voronoi diagram. Calculate the error range, E

_{i}, between the irregular factor, I

_{ri}, and the target irregular factor, I

_{r}. If the error range does not meet the requirement (set by specific engineering conditions), perform a new random-single movement according to procedure (b) and repeat procedure (c) until the error range meets the requirement, as shown in Figure 4f. Before performing a new random-single movement, compare the irregular factor, I

_{ri}, with the target irregular factor, I

_{r}; if I

_{ri}< I

_{r}, then i = i + 1, if I

_{ri}≥ I

_{r}, then the defined range factor η remains unchanged in the subsequent iterations.

#### 2.3. CJRM Geometric Model Reconstruction Method

_{r}. The CJRM geometric model needs to be generated on this 2D plane, which is divided into the following three procedures:

_{i}polygons randomly in each 2D diagram, and N

_{i}is calculated by Equation (6):

_{s}is the number of the selected polygons in each 2D diagram, N

_{c}is the number of polygons in the 2D diagram, and P is the transverse joint penetration rate.

_{r}, dip angle, Da, strike angle, Sa, transverse joint spacing, S, and transverse joint penetration rate, P, can be generated to better simulate the CJRM in numerical research and physical model tests. The entire procedure of the CJRM geometric model reconstruction method is shown in Figure 6.

#### 2.4. Application of the CJRM Geometric Model Reconstruction Method

## 3. CJRM Numerical Simulation

#### 3.1. Numerical Modeling of CJRM

#### 3.2. Validation of the Numerical Modeling

## 4. Calculation Results and Discussion

#### 4.1. The Effect of Strike Angle on CJRM Strength Behavior with Different Irregular Factors

^{−3}mm/step. The mechanical properties of the numerical model are listed in Table 1 and Table 2. The geometric parameters of the numerical model are listed in Table 5.

_{r}= 0, representing the numerical model as a regular hexagonal CJRM model, the curve of the uniaxial compressive strengths versus the strike angle curve is W-shaped. The strength has a maximum of 31.06 MPa when the strike angel is 0°, and the strength has a minimum of 28.10 MPa when the strike angle is 30°. As the irregular factor increases, the uniaxial compressive strength with different strike angles decreases, and the decrease is relatively uniform. The curve of the strength versus strike angle gradually changes from W-shaped to flat. When the irregularity factor is 40%, the curve of uniaxial compressive strength versus the strike angle shows an approximate straight line. For the numerical CJRM model, the maximum uniaxial compressive strength is 17.25 MPa when the strike angle is 0°, while the minimum uniaxial compressive strength is 16.59 MPa when the strike angle is 90°.

_{c}proposed by Singh et al. [31] was introduced. R

_{c}is defined as the ratio of the maximum strength to the minimum strength versus the dip angle or strike angle under the same stress conditions. The specific expression is:

_{max}and σ

_{min}are the maximum and minimum values of failure strength under the same stress conditions, respectively. With the increase in the anisotropy ratio R

_{c}, the jointed rock mass shows more obvious anisotropy, and the degree of anisotropy are divided into isotropy (1 < R

_{c}≤ 1.1), low degree anisotropy (1.1 < R

_{c}≤ 2), medium degree anisotropy (2 < R

_{c}≤ 4), and high degree anisotropy (R

_{c}> 4) based on the anisotropy classification method of rock mass proposed by Rammamurthy [32].

_{c-Sa}with different irregular factors. With the increase in the irregular factor, R

_{c-Sa}gradually decreases from 1.1057 to 1.0395, and changes from low degree anisotropy to isotropy, indicating that the effect of the strike angle on the strength behavior of the CJRM weakens. The basalt CJRM encountered during project construction has an irregular factor ranging from 30% to 40% and a strike angle anisotropy below 1.1, which is approximately isotropic. Therefore, the effect of strike angle on strength anisotropy of basalt CJRM can be considered negligible.

#### 4.2. The Effect of Dip Angle on CJRM Strength Behavior with Different Irregular Factors

_{r}= 0, the numerical model is a regular hexagonal CJRM model with prevalent physical samples; with the increase in the dip angle, the compressive strength initially decreases to the minimum value of 18.77 MPa when Da = 60°, and then increases to the maximum value of 81.42 Mpa when Da = 90°, showing a U-shaped curve that is consistent with current physical model test results. With the increase in the irregular factor, the uniaxial compressive strength with different dip angles decreases, and the decreases are all from sharp to steady. When Da = 60°, the effect of the irregular factor on strength behavior has the strongest impact with a decrease of 59.4% from I

_{r}= 0 to I

_{r}= 40%. The weakest impact of irregular factor occurs when Da = 90°, with a decrease of 23.17% from I

_{r}= 0 to I

_{r}= 40%. According to the relevant test analysis, the uniaxial compressive strength is mainly affected by the strength of the column when the dip angle is 90°, and the influence of the irregular coefficient is insignificant.

_{c-Da}with different irregular factors. As the irregular factor increases, R

_{c-Da}increases continuously from 4.3381 to 6.7953 at a high degree, showing strong anisotropy, which indicates that the effect of the dip angle on the strength behavior is enhanced. Therefore, in the engineering design and construction of CJRM, great attention should be paid to the effect of the dip angle on strength anisotropy.

#### 4.3. Three-Dimensional Strength Anisotropy of CJRM

_{r}= 0, the three-dimensional strength anisotropy in the three-dimensional space shows a gyroscopic shape, and the horizontal plane figure changing with the strike angle and the vertical plane figure changing with the dip angle shows an approximate hexagonal shape and gyroscopic shape, respectively. With the increase in the irregular factor, the three-dimensional shape remains a gyroscopic shape but continues to shrink. The approximate hexagonal shape in the horizontal plane changes into a smaller approximate circular shape and tends towards isotropic. The gyroscopic shape in the vertical plane has a reduced overall shape with a larger reduction on both sides, showing increasing anisotropy. Therefore, the irregular factor increases the strength anisotropy of the dip angle and decreases the strength anisotropy of the strike angle. The dip angle has a great effect on the strength anisotropy of CJRM.

## 5. Conclusions

- (1)
- A single-random movement with a range constraint Voronoi diagram of uniform seed points is proposed for simulating the irregular horizontal section of the CJRM. Based on this, a geometric model reconstruction method is developed using feasible geological statistical parameters, including horizontal polygon density, Q, irregular factor, I
_{r}, dip angle, Da, strike angle, Sa, transverse joint spacing, S, and transverse joint penetration rate, P. With this method, a geometric modeling approach that is more practical and relatively simple, with better integration and actual engineering provided for the production of similar material models and numerical simulation modeling. In addition, the method also has exploitable value for other types of sedimentary rocks, such as conglomerates, sandstones, metamorphic rocks, such as shales, and intrusive rocks, such as granites. - (2)
- In the numerical simulation of numerical CJRM models with different strike angles and irregular factors under uniaxial compression, when I
_{r}= 0, the curve of the uniaxial compressive strengths versus the strike angle was W-shaped, the maximum strength and minimum strength with a small gap are at the strike angles of 0°and 30°, respectively. As the irregular factor increases, the uniaxial compressive strength with different strike angles decreases uniformly, and the strength anisotropy of the strike angle decreases from 1.1057 to 1.0395, indicating that the numerical CJRM models change from low degree anisotropy to isotropy. Therefore, the effect of the strike angle on the strength anisotropy of basalt CJRM can be considered negligible. - (3)
- In the numerical simulation of numerical CJRM models with different dip angles and irregular factors under uniaxial compression, when I
_{r}= 0, the curve of the uniaxial compressive strengths versus the dip angle shows a U-shape, with the maximum strength and minimum strength at the dip angles of 90°and 60°, respectively. As the irregular factor increases, the uniaxial compressive strength with different dip angles decreases, and the decreases are all from sharp to steady. The effect of the irregular factor on strength behavior has the strongest and weakest impact on the dip angles at 60° and 90°, respectively. The strength anisotropy of the dip angle increases from 4.3381 to 6.7953, showing an increasing strong anisotropy at a high degree, indicating that the effect of the dip angle on the strength behavior is enhanced. Therefore, in the engineering design and construction of the CJRM, great attention should be paid to the effect of the dip angle on strength anisotropy.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Baihetan Hydropower Station. (

**a**) Location of the hydropower station. (

**b**) Layout of the project.

**Figure 2.**Geological environment of the Baihetan dam foundation. (

**a**) Typical engineering geological section of the dam foundation. (

**b**) Vertical cross-section of the CJRM (parallel to the column axis direction). (

**c**) Horizontal cross-section of the CJRM (perpendicular to the column axis direction).

**Figure 3.**Voronoi diagram. (

**a**) The Voronoi diagram generated by random seed points. (

**b**) The Voronoi polygon with adjacent seed points.

**Figure 4.**Single-random movement with range constraint Voronoi diagram of uniform seed points. (

**a**) Expanded region (2 m × 2 m) with uniformly distributed seed points. (

**b**) Voronoi diagram generated by uniform distributed seed points (I

_{r}= 0). (

**c**) Single-random movement with range constraint diagram. (

**d**) Single-random moved seed points. (

**e**) Voronoi diagram generated by single-random moved seed points. (

**f**) Voronoi diagram with I

_{r}= 35%. (

**g**) Existence and elimination of relative short edges. (

**h**) Voronoi diagram with target polygon density and irregular factor without relative short edges.

**Figure 5.**CJRM geometric model reconstruction method. (

**a**) Generation of transverse joint. (

**b**) Generation of the coordinate axis. (

**c**) Rotation of the cube. (

**d**) The final CJRM geometric model with six geological statistical parameters.

**Figure 7.**CJRM geometric model reconstruction method. (

**a**) Special mold for sample preparation. (

**b**) Generation of the real project geometric model.

**Figure 9.**Comparison of the physical model test and numerical simulation under uniaxial compression.

**Figure 10.**Comparison of the uniaxial compressive strength between experimental data [15] and numerical results.

**Figure 11.**Variation in uniaxial compression strength of the numerical CJRM model with different strike angles and irregular factors. (

**a**) Varying strike angles. (

**b**) Varying irregular factors.

**Figure 13.**Variation in the uniaxial compression strength of the numerical CJRM model with different dip angles and irregular factors. (

**a**) Varying dip angles. (

**b**) Varying irregular factors.

**Figure 15.**Visualization of three-dimensional anisotropy characteristics in the CJRM with varied irregular factors using Polar Coordinate System Transformation. (

**a**) I

_{r}= 0; (

**b**) I

_{r}= 10%; (

**c**) I

_{r}= 20%; (

**d**) I

_{r}= 30%; (

**e**) I

_{r}= 40%.

Bulk Density (kg/m ^{3}) | Elastic Modulus (GPa) | Friction Angle (°) | Poisson’s Ratio | Tensile Strength (MPa) | Cohesion (MPa) |
---|---|---|---|---|---|

2800 | 60 | 56.13 | 0.2 | 5 | 12.4 |

Cohesion (MPa) | Normal Stiffness (GPa/m) | Shear Stiffness (GPa/m) | Friction Angle (°) | Tensile Strength (MPa) |
---|---|---|---|---|

0.6 | 100 | 50 | 36 | 0 |

Bulk Density (kg/m ^{3}) | Elastic Modulus (GPa) | Friction Angle (°) | Poisson’s Ratio | Tensile Strength (MPa) | Cohesion (MPa) |
---|---|---|---|---|---|

1170 | 1.24 | 51.3 | 0.19 | 0.84 | 1.37 |

Cohesion (MPa) | Normal Stiffness (GPa/m) | Shear Stiffness (GPa/m) | Friction Angle (°) | Tensile Strength (MPa) |
---|---|---|---|---|

0.04 | 9.21 | 4.72 | 32 | 0.04 |

**Table 5.**Geometric parameters of the numerical models with different strike angles and irregular factors.

Size (m) | Polygon Density (m^{−2}) | Irregular Factor | Dip Angle (°) | Strike Angle (°) | Transverse Joint Spacing (m) | Transverse Joint Penetration |
---|---|---|---|---|---|---|

3 × 3 × 3 | 20.15 | 0 10% 20% 30% 40% | 0 | 0 | 0 | 0 |

15 | ||||||

30 | ||||||

45 | ||||||

60 | ||||||

75 | ||||||

90 |

**Table 6.**Geometric parameters of the numerical modells with different dip angles and irregular factors.

Size (m) | Polygon Density (m^{−2}) | Irregular Factor | Dip Angle (°) | Strike Angle (°) | Transverse Joint Spacing (m) | Transverse Joint Penetration |
---|---|---|---|---|---|---|

3 × 3 × 3 | 20.15 | 0 10% 20% 30% 40% | 0 | 0 | 0 | 0 |

15 | ||||||

30 | ||||||

45 | ||||||

60 | ||||||

75 | ||||||

90 |

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## Share and Cite

**MDPI and ACS Style**

Zhu, Z.; Wang, L.; Zhu, S.; Wu, J.
Study on the Anisotropy of Strength Properties of Columnar Jointed Rock Masses Using a Geometric Model Reconstruction Method Based on a Single-Random Movement Voronoi Diagram of Uniform Seed Points. *Symmetry* **2023**, *15*, 944.
https://doi.org/10.3390/sym15040944

**AMA Style**

Zhu Z, Wang L, Zhu S, Wu J.
Study on the Anisotropy of Strength Properties of Columnar Jointed Rock Masses Using a Geometric Model Reconstruction Method Based on a Single-Random Movement Voronoi Diagram of Uniform Seed Points. *Symmetry*. 2023; 15(4):944.
https://doi.org/10.3390/sym15040944

**Chicago/Turabian Style**

Zhu, Zhende, Luxiang Wang, Shu Zhu, and Junyu Wu.
2023. "Study on the Anisotropy of Strength Properties of Columnar Jointed Rock Masses Using a Geometric Model Reconstruction Method Based on a Single-Random Movement Voronoi Diagram of Uniform Seed Points" *Symmetry* 15, no. 4: 944.
https://doi.org/10.3390/sym15040944