# Model Test Study on the Anisotropic Characteristics of Columnar Jointed Rock Mass

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structural Characteristics of the CJRM

_{2}β

_{3}

^{3−2}strata, with strikes trending N30°–50°E [18]. The statistical results demonstrate that the quadrangular, pentagonal and hexagonal columns account for 32.1, 46.7 and 17.6% of the total data, respectively [1]. The diameters of columns are mainly in the range of 13–25 cm, and the aspect ratios are in the range of 2–5 [17].

_{4}and l

_{6}are the side lengths of the regular quadrilateral and hexagon, respectively. l

_{5}is the length of the other four sides in the pentagon, excluding the bottom side. The angle between the parallel directions in the top and bottom sides of the pentagon is 50°. Because of the geological structure, the influence of the column deflection needs to be considered when studying the mechanical properties of CJRM. As shown in Figure 2, α is the dip direction of columnar joints in the horizontal plane. β is the dip angle of columnar joints in the vertical plane. In the following model tests, the effect of dip direction on the anisotropic behavior of the three CJRM models was investigated.

## 3. Model Tests

#### 3.1. Similar Material and Model Size

_{σ}, C

_{γ}, C

_{L}, C

_{δ}, C

_{ε}, C

_{E}, C

_{f}, C

_{φ}, C

_{μ}, C

_{c}, C

_{σc}and C

_{σt}are the similarity constants of stress, bulk density, geometry, displacement, strain, deformation modulus, friction coefficient, internal friction angle, Poisson’s ratio, cohesion, compressive strength and tensile strength, respectively.

_{g}:m

_{s}:m

_{w}= 3:1:2.4 as the rock-like material, where m

_{g}, m

_{s}and m

_{w}represented the qualities of high-strength gypsum, fine sand and water, respectively. The brand of gypsum was “White Magnolia”, and the particle size of fine sand was less than 1 mm. To avoid errors caused by the brands of raw materials and working conditions, the uniaxial compression tests, Brazilian splitting tests and direct shear tests were performed on the rock-like material with the same mixing ratio, and the mechanical parameters were obtained. As shown in Table 1, the physical and mechanical parameters of rock-like material and intact basalt are listed. By combining the geometric structure of the Baihetan CJRM and the similarity principle, the similarity constants of geometry, density, stress, deformation modulus and dimensionless parameters were determined as 6, 2.5, 15, 15 and 1, respectively.

_{4}= 2 cm, l

_{5}= 1.7 cm and l

_{6}= 1.4 cm, respectively. The cement slurry with a mixing ratio of m

_{c}:m

_{w}= 1:0.4 was selected to simulate the joint surface and act as a binder between the columns, where m

_{c}and m

_{w}represent the qualities of cement and water, respectively [16]. Through laboratory tests, the cohesion and friction angle of the joint filler were determined as 1.23 MPa and 32.0°, respectively.

#### 3.2. Manufacturing Process of Specimens

_{g}:m

_{s}:m

_{w}= 3:1:2.4 (Figure 4a), and then the mixture was slowly poured into the molds to obtain the columnar bars with different cross-sectional shapes (Figure 4b). There were six sets of molds, each of which consisted of two symmetrical acrylic plates with grooves. Before casting, engine oil was daubed on each groove to help the columns release from the molds, and six sets of molds were neatly assembled together and fastened using a fixed clamp to prevent slurry loss.

_{c}:m

_{w}= 1:0.4 was used to glue the columns with the same cross-sectional shape together (Figure 4e) and then allowed to cure at 22 ± 2 °C for 20 days.

#### 3.3. Testing Equipment and Procedure

## 4. Test Results

#### 4.1. Deformation and Strength Behavior

_{cj}was calculated by the following equation [15]:

_{cj}

_{(0.2)}and ε

_{cj}

_{(0.2)}are the axial stress and strain corresponding to 0.2 times the peak strength σ

_{cj}, respectively. σ

_{cj}

_{(0.8)}and ε

_{cj}

_{(0.8)}are the axial stress and strain corresponding to 0.8 times the peak strength σ

_{cj}, respectively. The elastic modulus E

_{cj}and uniaxial compression strength (UCS) σ

_{cj}of the CJRM specimens are shown in Table 2.

#### 4.2. Failure Modes and Mechanisms

- Mode I:
- Splitting failure through joint planes. This mode occurs in the 4P- and 5P-CJRM specimens with the dip directions of 0, 15, 75 and 90° and in all the 6P-CJRM specimens. As observed from Figure 9a, the cracks initiate at the column material and propagate in the vertical direction. Eventually, failure surfaces passing through the joint planes appear on the specimen. Therefore, no sliding along the joint plane is observed, and the damage of the specimen is tensile failure.
- Mode II:
- Sliding failure along persistent joint plane. This mode occurs in the 4P-CJRM specimens with the dip directions of 30, 45 and 60° and in the 5P-CJRM specimen with the dip direction of 30°. The cracks initiate at the persistent joint plane and propagate along the persistent joint (Figure 9b). Eventually, sliding failure occurs along the persistent joint plane because the shear stress acting on the joint plane is greater than its shear strength.
- Mode III:
- Mixed failure along joint plane. This mode occurs in the 5P-CJRM specimens with the dip directions of 45 and 60°. The cracks initiate at the interlocking joints on the top of the pentagon, and appear at the joints on the sides of the pentagon under the further load (Figure 9c). Eventually, the cracks at the two locations connect and form a failure surface. The initial cracks are caused by tension because the angle between the joint and the load direction is small. The further load makes the columns slide along the sides of the pentagon, and the lateral deformation of the specimen accelerates the propagation of the cracks at the interlocking joints. Therefore, the failure mode of the specimen is mixed failure along the joint plane.

## 5. Discussion

#### 5.1. Anisotropic Degrees in Horizontal Plane

_{c}is introduced. R

_{c}is the ratio of the maximum value to the minimum value in a variation curve [30]:

_{cs}and R

_{cd}are the strength and deformation anisotropic ratios, respectively. σ

_{cjmax}and σ

_{cjmin}are the maximum and minimum UCS values in the variation curve, respectively. E

_{cjmax}and E

_{cjmin}are the maximum and minimum elastic modulus values in the variation curve, respectively. The larger the R

_{cs}or R

_{cd}value, the more significant the strength or deformation anisotropy of the rock.

#### 5.2. Anisotropic Characteristics of the Three CJRM Models

#### 5.3. Theoretical Prediction of Strength and Deformation

_{c}is the UCS of jointed rock mass when the joint angle is θ. θ

_{m}is the joint angle value corresponding to the minimum value of UCS. A and B are the constant terms that can be determined using the test results at θ = 0°, θ

_{m}and θ = θ

_{m}, 90°.

_{cr}and E

_{cr}are the normalized UCS and normalized elastic modulus, respectively. σ

_{ci}and E

_{ci}are the UCS and elastic modulus of intact rock-like material, respectively. Then, the empirical equations for estimating the normalized UCS and normalized elastic modulus of the CJRM can be rewritten as:

## 6. Conclusions

- (1)
- The differences between the strength and deformation behaviors of the three CJRM models were mainly caused by the structural features of these three models. The curves of 4P-CJRM were symmetrical, while the curves of 6P-CJRM were approximately straight. For 6P-CJRM, the variation curves combined the characteristics of the above two models.
- (2)
- Three typical failure modes of the CJRM specimens with different dip directions α were summarized, including the splitting failure through joint planes, the sliding failure along a persistent joint plane and the mixed failure along a joint plane.
- (3)
- The anisotropy ratio was introduced to classify the anisotropic degrees of the three CJRM models. The anisotropic degrees of the 4P-, 5P- and 6P-CJRM models in the horizontal plane were medium, low and low, respectively.
- (4)
- The anisotropic characteristics of the three CJRM models were described by placing the test results in the polar coordinates. The curves of the 4P- and 5P-CJRM models in the horizontal plane were all an axisymmetric diagram resembling a gyro, which indicated their orthotropy. The curves of the 6P-CJRM were approximately circular, which revealed its quasi-transverse isotropy.
- (5)
- A simple empirical expression was adopted to estimate the strength and deformation of the CJRM, and the derived equations were used in the Baihetan CJRM. The calculated values were all within the ranges of the existing research results, which indicates that the derived empirical equations are valuable for related engineering applications.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Columnar jointed rock mass (CJRM) at Baihetan Hydropower Station in China: (

**a**) CJRM in the vertical plane; (

**b**) CJRM in the horizontal plane.

**Figure 2.**Structural characteristics of the three CJRM models: (

**a**) 4P-CJRM model; (

**b**) 6P-CJRM model; (

**c**) 5P-CJRM model.

**Figure 3.**CJRM specimens with different dip directions α: (

**a**) 4P-CJRM specimens; (

**b**) 5P-CJRM specimens; (

**c**) 6P-CJRM specimens.

**Figure 4.**Manufacturing process of the artificial CJRM specimens: (

**a**) Rock-like material with a mixing ratio of m

_{g}:m

_{s}:m

_{w}= 3:1:2.4; (

**b**) Acrylic molds for making the columns with different cross-sectional shapes; (

**c**) Removing the column bars from the molds; (

**d**) Checking the size of the three types of columns; (

**e**) Gluing the columns with the same cross-sectional shape using the cement slurry to form a model block; (

**f**) Standard cubic CJRM specimens.

**Figure 6.**Stress-strain curves of the CJRM specimens: (

**a**) 4P-CJRM specimens; (

**b**) 5P-CJRM specimens; (

**c**) 6P-CJRM specimens.

**Figure 7.**Variations in elastic modulus and uniaxial compression strength (UCS) with dip direction α: (

**a**) Elastic modulus anisotropy curves of the three CJRM models; (

**b**) UCS anisotropy curves of the three CJRM models.

**Figure 8.**Final failure appearances of three types of artificial CJRM specimens with different dip directions: (

**a**) 4P-CJRM specimens; (

**b**) 5P-CJRM specimens; (

**c**) 6P-CJRM specimens.

**Figure 9.**Typical failure processes of the CJRM specimens: (

**a**) Failure process and mechanism of Mode I; (

**b**) Failure process and mechanism of Mode II; (

**c**) Failure process and mechanism of Mode III.

**Figure 10.**Elastic modulus and UCS of the three CJRM models in the polar coordinate system: (

**a**) Elastic modulus E

_{cj}; (

**b**) UCS σ

_{cj}.

**Figure 11.**Theoretical curves of the normalized mechanical parameters: (

**a**) Normalized UCS σ

_{cr}; (

**b**) Normalized elastic modulus E

_{cr}.

Material | Density (g/cm ^{3}) | Compressive Strength (MPa) | Elastic Modulus (GPa) | Poisson’s Ratio | Cohesion (MPa) | Friction Angle (°) |
---|---|---|---|---|---|---|

Intact basalt ^{1} | 2.83–2.93 | 47.7–255.0 | 30.0–86.6 | 0.17–0.26 | 10.0–13.0 | 45–50 |

Rock-like material | 1.17 | 6.58 | 1.24 | 0.19 | 1.37 | 51.3 |

^{1}The physical and mechanical parameters of intact basalt are taken from Jin [17].

Shape of Columns | Dip Direction (°) | Elastic Modulus (GPa) | UCS (MPa) |
---|---|---|---|

Quadrangular prism | 0 | 0.429 | 3.193 |

15 | 0.311 | 2.052 | |

30 | 0.258 | 1.427 | |

45 | 0.214 | 1.396 | |

60 | 0.258 | 1.427 | |

75 | 0.311 | 2.052 | |

90 | 0.429 | 3.193 | |

Pentagonal prism | 0 | 0.282 | 2.376 |

15 | 0.267 | 1.973 | |

30 | 0.229 | 1.495 | |

45 | 0.196 | 1.267 | |

60 | 0.237 | 1.384 | |

75 | 0.262 | 1.962 | |

90 | 0.275 | 2.210 | |

Hexagonal prism | 0 | 0.255 | 2.092 |

15 | 0.240 | 1.937 | |

30 | 0.243 | 1.892 | |

45 | 0.242 | 1.933 | |

60 | 0.255 | 2.092 | |

75 | 0.240 | 1.937 | |

90 | 0.243 | 1.892 |

Normalized UCS σ_{cr} | Normalized Elastic Modulus E_{cr} |
---|---|

σ = 0.388 − 0.155cos [2(45° − α)] (0° ≤ α < 45°)_{cr}σ = 0.370 − 0.137cos[2 (45° − α)] (45° ≤ α ≤ 90°)_{cr} | E = 0.260 − 0.084cos[2(45° − α)] (0° ≤ α < 45°)_{cr}E = 0.256 − 0.080cos[2(45° − α)] (45° ≤ α ≤ 90°)_{cr} |

Result | Normalized UCS σ_{cr} | Normalized Elastic Modulus E_{cr} |
---|---|---|

Calculated result | 0.24 | 0.181 |

Existing research result ^{2} | 0.1430.536 | 0.157–0.552 |

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**MDPI and ACS Style**

Zhu, Z.; Que, X.; Niu, Z.; Lu, W.
Model Test Study on the Anisotropic Characteristics of Columnar Jointed Rock Mass. *Symmetry* **2020**, *12*, 1528.
https://doi.org/10.3390/sym12091528

**AMA Style**

Zhu Z, Que X, Niu Z, Lu W.
Model Test Study on the Anisotropic Characteristics of Columnar Jointed Rock Mass. *Symmetry*. 2020; 12(9):1528.
https://doi.org/10.3390/sym12091528

**Chicago/Turabian Style**

Zhu, Zhende, Xiangcheng Que, Zihao Niu, and Wenbin Lu.
2020. "Model Test Study on the Anisotropic Characteristics of Columnar Jointed Rock Mass" *Symmetry* 12, no. 9: 1528.
https://doi.org/10.3390/sym12091528