J. Cent. South Univ. Technol. (2011) 18: 2143-2149

DOI: 10.1007/s11771-011-0955-4

Macro-mechanical properties of columnar jointed basaltic rock masses

DI Sheng-jie(狄圣杰)¹, XU Wei-ya(徐卫亚)¹, NING Yu(宁宇)², WANG Wei(王伟)¹, WU Guan-ye(吴关叶)³

1. Institute of Geotechnical Engineering, Hohai University, Nanjing 210098, China;

2. Kunming Investigation and Design Institute, China Hydropower Engineering Group Co., Kunming 650051, China;

3. East China Investigation and Design Institute,China Hydropower Engineering Group Co., Hangzhou 310014, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: From the geological structure of the columnar jointed rock mass, a visual model was established in software AUTOCAD by programming based on the algorithm of the Voronoi diagram. Furthermore, a program to convert the AUTOCAD model into 3DEC (3-dimensional distinct element code) model was developed, and a numerical model was established in 3DEC. Moreover, the results of triaxial compression tests of columnar jointed rock masses were simulated numerically. The REV (representative element volume) scale was studied, and the result shows that the REV size is 3 m × 3 m. The proposed approach, the established model and the numerical simulation were applied to study the macro-mechanical properties and the equivalent strength parameters of the columnar jointed rock mass. The numerical simulation results are in good accordance with the in-situ test results.

Key words: columnar joint; Voronoi diagram; three-dimensional distinct element code; representative element volume; mechanical parameters

1 Introduction

A columnar jointed basaltic rock mass is a kind of volcanic lava geomaterial that is discontinuous, non-uniform, and exhibits anisotropy. Columnar jointed basaltic rock mass has a wide distribution and is often encountered during civil and architectural engineering projects as well as in hydraulic and hydro-power engineering projects. It possesses certain geological structure characteristics and plays a role in engineering. There are many irregular polygons in the cross-section of a columnar jointed basaltic rock mass, which is not the same as common rock masses. Due to the dominant joint plane, a columnar rock mass has strong anisotropy and complicated mechanical behavior. The study of the macroscopic mechanical properties of the columnar jointed basaltic rock masses is of particular interest to the civil, architectural, hydraulic and hydropower engineering fields.

Over the last few decades, comprehensive references on columnar jointed basaltic rock masses have been covered in several review papers. The Basalt Waste Isolation Project (BWIP) was carried out in USA in 1980s, and the geometrical characteristics and engineering properties of columnar jointed basaltic rock masses were summarized and analyzed. HART and CUNDALL [1] simulated and analyzed the columnar jointed rock mass of BWIP using regular hexagonal prism columnar jointed rock mass. SCHULTZ [2] evaluated the engineering quality and estimated the mechanical parameters of this kind of rock mass. JUSTO et al [3] analyzed the sedimentation of a 40-story building on Tenerife Island that is built on a foundation of columnar jointed rock masses. Previous researches have failed to consider the irregular shape of the cross-section of the columnar jointed rock masses. The planned engineering of the Baihetan Hydropower Project in China has attracted the attention of many scholars. The columnar jointed rock masses at the abutment and the foundation of dam show significant anisotropy, which has become one of the key technical issues for the Baihetan Hydropower Station. A special research fund was set up, and field test research and geological engineering investigation were carried out [4-5]. Furthermore, the geological origin, the anisotropic mechanical behavior and the constitutive model of columnar jointed rock masses were extensively studied at the dam site of the Baihetan Hydropower Project [6]. Because a columnar jointed basaltic rock mass is a discontinuous medium, the deformation and strength behaviors of this geomaterial are quite different from those of continuous media [7]. LU et al [8] analyzed the strength anisotropic characteristic of rock masses under uniaxial compression situation. Numerical methods can be used to simulate the compression test of rock masses at any scale and can also make up for the deficiencies of current large-scale rock mass tests. The three- dimensional discrete element method is a classical example of a powerful discontinuum modeling approach for simulating the behavior of jointed rock masses. This method regards the medium as an assembly of rigid or deformable blocks interacting through discontinuities, and the discontinuities are treated as boundary interactions between blocks [9]. The discrete element method is much more flexible than any other method. It can not only simulate the triaxial test of rock masses that possesses complex structure, but also consider the interaction of several groups of joints and directly reflect the deformation, yielding, and failure of jointed rock masses.

In this work, we focus on the modeling method and the anisotropic equivalent strength of columnar jointed rock masses based on a numerical method. Taking the columnar jointed basaltic rock mass at the Baihetan Hydropower Station as an example, the approach of estimating the anisotropic mechanical parameters of irregular jointed rock masses is proposed. The numerical models are established based on the Voronoi diagram, and compression tests are simulated based on three-dimensional distinct element code (3DEC) to study the mechanical parameters and anisotropy of rock masses.

2 Models and computational methods

The Voronoi diagram is a data structure describing space division that is widely used in data handling and in the management of the computer software field [10]. Voronoi structures are widespread in nature (Fig.1), and they are widely used in materials science research. Various mechanical properties and the macro and micro constitutive relations of polycrystalline aggregates based on the three-dimensional Voronoi models were studied [11-12]. The elastic and plastic properties of the anisotropic model were investigated based on the Voronoi diagram [13].

Assume that p₁ and p₂ are two points on a plane and L is the perpendicular bisector of the line p₁p₂. The plane is divided into two parts, L_L and L_R, through the line p₁p₂. The Euclidean distance between points p_l and p_i (i=1, 2) is d(p_l, p₁)l, p₂), and d(p_l, p_i). Thus, the points in L_L are closer to point p_i than the other points on the plane, the collection of which is defined as V(p_i). We use H(p₁, p₂) to represent the half plane L_L, and L_R=H(p₂, p₁); thus, V(p₁)=H(p₁, p₂) and V(p₂)=H(p₂, p₁). The representation is shown in Fig.2.

Fig.1 Voronoi structures in nature: (a) Plant cells; (b) Basaltic rock

Fig.2 Sketch of V(p₁) and V(p₂)

Assume that there is a group of points S (S={p₁, p₂, …, p_n}) involving n points on the plane such that V(p_i)=∩_i=jH(p_i, p_j); thus V(p_i) represents the intersection of n-1 half planes when the point is closer than any other points to the trace of p_i. V(p_i) is a convex polygon field, which is called the Voronoi field relative to p_i. It can make Voronoi polygons for each point in S. Using the equation V(p_i)=∩_i=jH(p_i, p_j), we can obtain the Voronoi polygons of p_i and draw the Voronoi graphics of S point-by-point [14]. The Voronoi models are established in AUTOCAD software based on the geometric features of the rock mass structure. The typical cross-section of the columnar jointed rock mass is more closely related to the structures observed in nature, which is illustrated in Fig.3.

 

Fig.3 Enlarged sketch of irregular columnar jointed rock mass (a) and Voronoi model established (b)
 

The two-dimensional Voronoi models are created with the format of DXF in AUTOCAD by programming. Furthermore, an import and export program to transfer data between AUTOCAD and 3DEC is developed. This program allows two-dimensional Voronoi models to be imported into 3DEC and can stretch the models in the direction of different angles into 3DEC. Three- dimensional numerical models of the columnar jointed rock mass are established in 3DEC. Random polygon meshes of the model with the average edge of 20 cm are established according to Voronoi algorithm. The maximum columnar diameter is 27 cm and the minimum is 15 cm at the dam site of the Baihetan Hydropower Station according to the statistical analysis of the columnar jointed rock mass; thus, the geometrical properties of the numerical model are close to the actual properties. Considering the inner cracks and blind joints in the horizontal direction of the rock mass, the persistence of the rock mass is set to be 50% in 3DEC, meaning that be 50% of the blocks will be split on average. The structure of the model is shown in Fig.4. Figure 5 presents the models of different dip angles.

Fig.4 Blind joints in columnar jointed rock mass and joint structures of model

The failure envelope for the Mohr-Coulomb model in 3DEC consists of a Mohr-Coulomb criterion with a tension cutoff. There is a non-associated flow rule for shear failure, and no flow rule for tension failure is considered in this model. The basic joint constitutive model incorporated in 3DEC is the generalization of the Coulomb friction law. This law works in a similar fashion for both sub-contacts between rigid blocks and sub-contacts between deformable blocks. Both shear and tensile failure are considered, and the joint dilation is included. Equations (1)-(4) are the formulas describing the Mohr-Coulomb yield criterion in 3DEC [9]:

                             (1)

                                       (2)

                                      (3)

                                      (4)

Fig.5 Models (3 m × 3 m) with different dip angles: (a) 15°;  (b) 30°; (c) 45°; (d) 60°

The relationship of the three principal stresses is σ₁≤σ₂≤σ₃, where φ is the friction angle, c is the cohesion, σ ^t is the tensile strength, ψ is the dilation angle, f_t and f  ^s are the yield functions, and N is the friction index. The potential function, g^s, used to define the shear plastic flow, corresponds to a non-associated law. If shear failure takes place, the stress point is placed on the curve f  ^s=0 using a flow rule derived from the potential function g^s. If the tensile failure is declared, the new stress point is simply reset to conform to f  ^t=0; no flow rule is used in this case.

The detailed steps of the numerical compression test using 3DEC are as follows:

1) Build 3D models with different dip angles in 3DEC, and then, after meshing the models, determine the parameters of the corresponding materials. The intact rock is modeled using a block constitutive model, and the joints are modeled using an elasto-plastic constitutive model.

2) Given a certain confining pressure σ_x and σ_z, continue to increase the axial compression σ_y step by step until an obvious plastic deformation appears. Note the axial stress σ_y and strain ε_y of the specimen, and draw the curves for the corresponding axial stress and strain (σ_y－ε_y). We can get the yield stress from the curves of each specimen with different dip angles.

3) Given another confining pressure, repeat steps 1) and 2) to obtain a series of Mohr’s strength envelopes. Draw the envelope of these Mohr-circles, and then linearly extrapolate the envelope until it intersects with the σ-axis and the τ-axis. The intersections correspond to the equivalent strength parameters φ_e and c_e of the rock mass.

The method for numerical simulation of uniaxial compression is shown in Fig.6. As shown in Fig.7, the envelope of the jointed rock mass strength is a linear function. Therefore, the equivalent strength parameters of the jointed rock mass can be obtained using Eq.(5):

                         (5)

where i+1 and i represent two independent triaxial tests with different confining pressures,  and σ_x are the maximum normal stress and minimum normal stress, respectively, and φ_e and c_e are the equivalent friction angle and cohesion, respectively.

3 REV scale of columnar jointed rock mass

The equivalent parameters of a complex rock mass have an obvious size effect. Generally, the parameters of a complex rock mass will decrease with an increase in the rock mass scale. However, when the scale increases to a certain critical value, the parameters of the rock mass will become nearly constant. This scale value is referred to as the representative element volume (REV) scale. The REV scale determination is the basis of calculating equivalent parameters of rock masses. Theoretically, when the model scale reaches the REV scale, the parameters derived from experimental or numerical methods can not only be equivalent to the parameters of the macroscopic rock mass but also actually reflect the mechanical characteristics of macroscopic rock mass. ZHOU et al [15] proposed a general theory to confirm the scale of REV, a numerical simulation approach that can describe the actual characteristics of the structure of the rock masses, and a numerical estimation that can reflect the discontinuity, anisotropy, size effect, and environment of the rock mass. JING et al [16-17] simulated random fractures in rock masses as part of the DECOVALEX III and BENCHPAR projects. They performed several numerical tests using UDEC and studied the size of the REV and the equivalent mechanical parameters.

Fig.6 Representation of numerical compression simulations with load along coordinate axis

Fig.7 Mohr’s diagram and strength envelope line

In this work, random specimens of different scales were selected for numerical triaxial compression testing from the whole models. The selection of specimens is shown in Fig.8. Five actual sizes were used from 0.5 m ×0.5 m to 4.0 m × 4.0 m (0.5 m × 0.5 m, 1.0 m × 1.0 m, 2.0 m × 2.0 m, 3.0 m × 3.0 m and 4.0 m × 4.0 m). Four selected dip angles were used for each size of four groups (0°, 50°, 70° and 90°), and six random specimens of each size were analyzed. Thus, there were five different rock sizes, six random specimens of each size and four dip angles to be chosen, analyzed and simulated.

Fig.8 Specimens with different sizes
 

The numerical triaxial compression tests of 120 conditions (6×4×5=120) were completed. The objective of the selection was to reduce the amount of computing required.

The experimental specimens with four different dip angles were representative. When the dip angle was 0° or 90°, the failure stress of the specimen was mainly the result of the intact rocks. The failure stress of rock masses was the result of both the rocks and the joints when the dip angle was 50° or 70°; this behavior represents the compound failure mode.

The physico-mechanical parameters of the rocks and joints for class-III rock masses at the dam site of the Baihetan Hydropower Station are listed in Table 1 and Table 2, which were determined through experiments in the laboratory.

The scatter plots in Fig.9 are the failure stress values of randomly selected specimens. When the size of specimens is greater than or equal to 3 m × 3 m, the failure stress of the four different dip angles tends to be stable. Furthermore, the calculated equivalent deformation parameters of columnar jointed rock masses also tend to be stable when the size reaches 3 m × 3 m. Therefore, the REV scale of a columnar jointed rock mass is 3 m × 3 m.
 

Table 1 Mechanical parameters of intact rock

Table 2 Mechanical parameters of joint

Fig.9 Scatter plots of numerical experiment results for specimens with different dip angles: (a) 0°; (b) 50°; (c) 70°; (d) 90°
 

4 Estimation of equivalent strength parameters

The dip angle of columnar basaltic jointed rock masses is about 70°-80° in the dam region of the Baihetan Hydropower Station, and the average is 75°, as shown in Fig.10.

Fig.10 Dip angle of columnar basalt in Baihetan, approximately 75° (β=15°, α=75°), relative to material’s principal axis (1-3-axis) and global coordinates (X-Z-axis)

Taking the size effect into account, the size of the model selected is 3 m × 3 m, and the class-III columnar jointed rock masses are used as an example to analyze the stress-strain relationship at different dip angles and under different confining pressures.

In succession, the equivalent strength parameters (friction angle and cohesion) of the rock mass are obtained to validate the model and the approach. The confining pressures are from 0 MPa to 25 MPa at a regular interval of 5 MPa, and the dip angles are from 0° to 90° at a regular interval of 15°. The anisotropy of the columnar jointed rock mass strength under various dip angles and confining pressures can be obtained and analyzed from the curves.

From the analysis of the stress-strain curves, when the columnar dip angle is 90° or 0°, the failure stress of the rock mass reaches a maximum. This result indicates that the maximum compressive strength occurs when the load is applied perpendicular to the principle axis of the column and that the strength is mainly under the control of the rocks. When the dip angle is about 60°-75°, the failure stress of the rock mass reaches a minimum, and the compressive strength is mainly under the control of the joints. The columnar jointed rock mass obviously has anisotropic mechanical behaviors based on the curves, as shown in Fig.11.

Fig.11 Failure stress curves with different confining pressures and dip angles

The equivalent strength parameters are obtained under the Mohr-Coulomb criterion with different dip angles, and the results are given in Table 3. The failure stress of each numerical test and the equivalent mechanical parameters are listed in Table 3. The calculated equivalent cohesion is 1.145 MPa, and the friction angle is 44.441° for class-III columnar jointed rock masses. By comparing the numerical simulation results with the in-situ test results, it is found that the results of the numerical simulation are similar to real data. The calculated parameters are within the given range, as listed in Table 4. The numerical test results indicate that the Voronoi model and the analytical method are reasonable tools for the analysis of columnar jointed rock masses.
 

Table 3 Failure stress and equivalent strength parameters with different dip angles and different confining pressures

Table 4 Comparison of calculated and in-situ test results

5 Conclusions

1) According to the Voronoi diagram method, a random model of the columnar jointed rock mass is established, and a procedure for using 3DEC is proposed. The Voronoi graphics are produced in AUTOCAD. The program can import points and lines into 3DEC from AUTOCAD and can stretch the rock mass model at different angles.

2) Random specimens are selected from the whole model, and they are used to conduct a series of numerical tests. It is found that the REV size is 3 m × 3 m. The method for choosing random specimens of different angles is reasonable and representative.

3) The method presented can be applied directly to the simulation of specimens of large dimension that cannot be studied in the laboratory. The failure stress is obtained under different dip angles and different confining pressures, and it is clear that the failure stress shows strong anisotropy from the strength curves. The equivalent friction angle and cohesion are calculated and agree with the experimental data. The results indicate that the proposed approach can provide a solid basis for practical engineering design and that this approach can be used as technical guidance for situations involving other similar materials.

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(Edited by YANG Bing)

Foundation item: Projects(50911130366, 50979030) supported by the National Natural Science Foundation of China

Received date: 2010-11-29; Accepted date: 2011-03-07

Corresponding author: XU Wei-ya, Professor, PhD; Tel: +86-25-83787379; E-mail: wyxu@hhu.edu.cn