# A Simple Three-Dimensional Failure Criterion for Jointed Rock Masses under True Triaxial Compression

Authors: Yaohui Gao; Chunsheng Zhang; Zhaofeng Wang; Jun Chen
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The joint configuration and the intermediate principal stress have a significant influence on the strength of rock masses in underground engineering. A simple three-dimensional failure criterion is developed in this study to predict the true triaxial strength of jointed rock masses. The proposed failure criterion in the deviatoric and meridian planes adopts the elliptic and hyperbolic forms to approximate the Willam–Warnke and Mohr–Coulomb failure criterion, respectively. The four parameters in the proposed failure criterion have close relationships with the cohesion and the internal friction angle and can be linked with the joint inclination angle using a cosine function. Two suits of true triaxial strength data are collected to validate the correctness of the proposed failure criterion. Compared with other failure criteria, the proposed failure criterion is more reasonable and acceptable to describe the strength of jointed rock masses.


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## Body

## 1. Introduction

Rock mass strength is an extremely important parameter in predicting the stability of geoengineering such as rock slopes, dam foundation, and deeply buried tunnels [1]. Due to the existing joints, rock mass strength is anisotropic [2–8]. In addition, true triaxial stress state (σ1 > σ2 > σ3) is more universal with the increasing depth of engineering. The influence of σ2 on the compressive strength of jointed rock masses has been investigated extensively in the experimental tests [9–14]. Therefore, establishing a strength criterion considering both the joint and σ2 dependency is essential for better designing the layout and construction of underground engineering.A large number of research papers are concentrated on rock strength. Among others, the Mohr–Coulomb failure criterion [15] and the Hoek–Brown failure criterion [16] are most widely used because of their simplicities. However, these two failure criteria ignore the σ2 effect. Based on the true triaxial strength data, the authors in [17–21] have proposed the three-dimensional failure criteria which can describe the variation in the rock strength with increasing σ2 very well. There are also some popular true triaxial failure criteria [22–24]. However, these failure criteria are applicable for intact rock and do not consider the joint effect.To develop a failure criterion considering the joint effect under true triaxial compression, Tiwari and Rao [14] introduced the joint effect into the generalized von Mises theory [20]. In addition, enormous research studies [25–27] have extended the Hoek–Brown failure criterion into the three-dimensional form to predict rock mass strength. Moreover, Singh and Singh and Zhang et al. [28, 29] established a modified Mohr–Coulomb criterion for the polyaxial strength of rock masses, and Rafiai [30] proposed an empirical criterion for rock mass strength under multiaxial state on the basis of a comprehensive experimental database. Based on the research in [7, 10, 30], the modified nonlinear criteria are presented to determine the strength of rock masses.Most of the abovementioned failure criteria can only consider the effect of the joint orσ2 on the prediction of rock mass strength. In addition, some aforementioned failure surfaces in three-dimensional principal stress space are not absolutely continuous, which could present difficulties in numerical calculations. To overcome these problems, the true triaxial failure criterion proposed in this paper adopts the modified Willam–Warnke yield criterion and hyperbolic function in the deviatoric and meridian planes, respectively. Moreover, the adopted hyperbolic function takes the Mohr–Coulomb criterion as the asymptote in the meridian plane. To valid the correctness of the developed failure criterion, it is employed to fit two suits of true triaxial strength data of rock masses.


## 2. A Failure Criterion for Jointed Rock Masses

Within this paper, tensile stress is considered positive and compressive stress is negative. A general form of a failure criterion suggested in [31] can be expressed in a quadratic function:(1)Fτoct,σoct,θσ=ασoct2+βσoct+γ+σ+¯2=0,where σ+¯=τoct/gθσ; θσ is the Lode angle (θσ=1/3cos−133J3/2J23/2), where J2 (J2=1/6σ1−σ22σ2−σ32+σ3−σ12) and J3 (J3=σ1−σoctσ2−σoctσ3−σoct) are the second and third invariants of the deviatoric stress tensor, respectively; τoct and σoct are the octahedral shear and normal stresses (τoct=1/3σ1−σ22+σ2−σ32+σ3−σ12 and σoct=σ1+σ2+σ3/3); and α, β, and γ are material constants. First, the dependency of τoct on θσ in the deviatoric plane perpendicular to the hydrostatic axis is determined; namely, gθσ is obtained. Then, the influence of σoct on σ+¯ can be investigated in the meridian plane.


### 2.1. True Triaxial Failure Criterion in the Deviatoric Plane

The failure curve in the deviatoric plane displays the following significant characteristics of rock failure: (i) continuous, smooth, and convex; (ii) a closed curve or vertex; and (iii) symmetric with respect to three principal stress axes. For this purpose, an elliptical form developed in [32] is adopted here to express the relationship between gθσ and θσ:(2)gθσ=R+2K−12Rcosθσ+5K2−4K2Rcosθσ+1−2K2,where R=21−K2cosθσ; K is the ratio (K=τoct,ae/τoct,ac) between the strength (τoct,ae) for axisymmetric extension and the strength (τoct,ac) for axisymmetric compression; 0° ≤ θσ ≤ 60° and θσ  = 0° (gθσ=K) for axisymmetric extension; and θσ  = 60°gθσ=1 for axisymmetric compression.It is notable thatgθσ in (2) unconditionally satisfies the abovementioned properties of a failure criterion in the deviatoric plane for the range 0.5 ≤ K ≤ 1. Therefore, taking θσ as the independent variable, gθσ is only a function of K. Figure 1 shows that the shapes of the failure curves in the deviatoric plane changes with K. More specifically, as K varies from 0.5 to 1, failure surfaces gradually change from a triangle to a circle. It can be seen that the Willam–Warnke failure criterion is more appropriate than the Mohr–Coulomb failure criterion, and the Willam–Warnke criterion can change into the Mises criterion when K = 1. Here, to present the joint effect on the failure function in the deviatoric plane, the parameter K depends on both the joint inclination angle β and σoct, namely, K=Kσoct,β. If K is constant at different σoct, gθσ is only a function of β. Thus, the joint inclination angle β can describe the influence of the joint on the failure surfaces in the deviatoric plane.Figure 1
The Willam–Warnke failure criterion in the deviatoric plane with different K values.


### 2.2. True Triaxial Failure Criterion in the Meridian Plane

As pointed out by equation (1), once gθσ is determined using equation (2), a true triaxial failure criterion can be obtained if the relationship between σ+¯ and σoct is known. Here, a hyperbolic function (as shown in Figure 2) is adopted because of its simplicity and continuity to present the failure criterion in the meridian plane. Thus, equation (1) can be defined by(3)Fτoct,σoct,θσ=σoct−da2−σ+¯b2−1=0,where a, b, and d are material constants. Figure 3 exhibits a typical failure surface of the proposed criterion in the principal stress space.Figure 2
The hyperbolic function to approximate the Mohr–Coulomb failure criterion for axisymmetric compression in the meridian plane.Figure 3
The typical failure surface in the principal stress space.


### 2.3. Determination of the Parameters in the Failure Criterion

Generally, equation (3) has four parameters: K, a, b, and d. As mentioned above, K can be determined by the strength ratio under axisymmetric extension and compression at different β and σoct and be given by(4)K=τoct,aeσoct,βτoct,acσoct,β.To determine the values of the other parameters, the hyperbolic function in equation (3) approximates the Mohr–Coulomb failure criterion for axisymmetric compression (gθσ=60°=1) in the meridian plane (as shown in Figure 2). The Mohr–Coulomb failure criterion at θσ=60° can be expressed by(5)Fτoct,σoct,θσ=σ+¯+22sinφ3−sinφσoct−22ccosφ3−sinφ=0,where c and φ are the cohesion force and internal friction angle. Taking the Mohr–Coulomb line as the hyperbolic asymptote, the parameters a, b, and d are then obtained as follows:(6)ba=22sinφ3−sinφ,d=ccotφ.The parametera can be fitted by a series of simple experimental tests under axisymmetric compression. As the parameter a tends to be 0, equation (3) is getting closer to the Mohr–Coulomb failure surfaces in the meridian plane. In addition, gθσMC in equation (5) is given by(7)gθσMC=3−sinφ23cosθσ−π/6−2sinφsinθσ−π/6.Thus, the strength ratioKMC under axisymmetric extension and compression can be calculated using the Mohr–Coulomb failure criterion.(8)KMC=gθσ=0°MC=3−sinφ3+sinφ.If equation (4) is simply substituted by equation (8), four unknown parameters (K, a, b, and d) in equation (3) become three (a, c, and φ) using equations (6) and (8). It is noted that the joint inclination angle β affects the unknown parameters (aβ, cβ, and φβ).


## 2.1. True Triaxial Failure Criterion in the Deviatoric Plane

The failure curve in the deviatoric plane displays the following significant characteristics of rock failure: (i) continuous, smooth, and convex; (ii) a closed curve or vertex; and (iii) symmetric with respect to three principal stress axes. For this purpose, an elliptical form developed in [32] is adopted here to express the relationship between gθσ and θσ:(2)gθσ=R+2K−12Rcosθσ+5K2−4K2Rcosθσ+1−2K2,where R=21−K2cosθσ; K is the ratio (K=τoct,ae/τoct,ac) between the strength (τoct,ae) for axisymmetric extension and the strength (τoct,ac) for axisymmetric compression; 0° ≤ θσ ≤ 60° and θσ  = 0° (gθσ=K) for axisymmetric extension; and θσ  = 60°gθσ=1 for axisymmetric compression.It is notable thatgθσ in (2) unconditionally satisfies the abovementioned properties of a failure criterion in the deviatoric plane for the range 0.5 ≤ K ≤ 1. Therefore, taking θσ as the independent variable, gθσ is only a function of K. Figure 1 shows that the shapes of the failure curves in the deviatoric plane changes with K. More specifically, as K varies from 0.5 to 1, failure surfaces gradually change from a triangle to a circle. It can be seen that the Willam–Warnke failure criterion is more appropriate than the Mohr–Coulomb failure criterion, and the Willam–Warnke criterion can change into the Mises criterion when K = 1. Here, to present the joint effect on the failure function in the deviatoric plane, the parameter K depends on both the joint inclination angle β and σoct, namely, K=Kσoct,β. If K is constant at different σoct, gθσ is only a function of β. Thus, the joint inclination angle β can describe the influence of the joint on the failure surfaces in the deviatoric plane.Figure 1
The Willam–Warnke failure criterion in the deviatoric plane with different K values.


## 2.2. True Triaxial Failure Criterion in the Meridian Plane

As pointed out by equation (1), once gθσ is determined using equation (2), a true triaxial failure criterion can be obtained if the relationship between σ+¯ and σoct is known. Here, a hyperbolic function (as shown in Figure 2) is adopted because of its simplicity and continuity to present the failure criterion in the meridian plane. Thus, equation (1) can be defined by(3)Fτoct,σoct,θσ=σoct−da2−σ+¯b2−1=0,where a, b, and d are material constants. Figure 3 exhibits a typical failure surface of the proposed criterion in the principal stress space.Figure 2
The hyperbolic function to approximate the Mohr–Coulomb failure criterion for axisymmetric compression in the meridian plane.Figure 3
The typical failure surface in the principal stress space.


## 2.3. Determination of the Parameters in the Failure Criterion

Generally, equation (3) has four parameters: K, a, b, and d. As mentioned above, K can be determined by the strength ratio under axisymmetric extension and compression at different β and σoct and be given by(4)K=τoct,aeσoct,βτoct,acσoct,β.To determine the values of the other parameters, the hyperbolic function in equation (3) approximates the Mohr–Coulomb failure criterion for axisymmetric compression (gθσ=60°=1) in the meridian plane (as shown in Figure 2). The Mohr–Coulomb failure criterion at θσ=60° can be expressed by(5)Fτoct,σoct,θσ=σ+¯+22sinφ3−sinφσoct−22ccosφ3−sinφ=0,where c and φ are the cohesion force and internal friction angle. Taking the Mohr–Coulomb line as the hyperbolic asymptote, the parameters a, b, and d are then obtained as follows:(6)ba=22sinφ3−sinφ,d=ccotφ.The parametera can be fitted by a series of simple experimental tests under axisymmetric compression. As the parameter a tends to be 0, equation (3) is getting closer to the Mohr–Coulomb failure surfaces in the meridian plane. In addition, gθσMC in equation (5) is given by(7)gθσMC=3−sinφ23cosθσ−π/6−2sinφsinθσ−π/6.Thus, the strength ratioKMC under axisymmetric extension and compression can be calculated using the Mohr–Coulomb failure criterion.(8)KMC=gθσ=0°MC=3−sinφ3+sinφ.If equation (4) is simply substituted by equation (8), four unknown parameters (K, a, b, and d) in equation (3) become three (a, c, and φ) using equations (6) and (8). It is noted that the joint inclination angle β affects the unknown parameters (aβ, cβ, and φβ).


## 3. Validation of the Proposed Failure Criterion

The validity of the proposed failure criterion in this study is discussed using two following examples. The prediction accuracy is also analyzed and compared with other strength criteria.


### 3.1. Example 1

The true triaxial strength data for the first example are derived from the experimental results of [9]. This test was conducted on jointed marble which contains a natural joint plane with a rectangular prismatic size (50 × 50 × 100 mm3). Table 1 lists the derived test data.Table 1
True triaxial compression strength of experimental and predicted results of jointed marble (data from [9]; here, compressive stress is considered positive).
σ3 (MPa)σ2 (MPa)σ1exp (MPa), test data at joint inclination angle β (°)σ1cal (MPa), calculated data at joint inclination angle β (°)020406080900204060809000158.59100.7973.7677.57112.21141.07146.23110.1681.8184.31118.00144.631060279.94296.43272.40240.12242.96258.62276.72254.21220.28202.30224.88240.923060332.25361.55330.96305.12338.63357.16354.39339.66319.75307.68300.75304.063090409.76387.18363.12337.69311.01297.24395.24384.82363.43337.62330.11331.2630120377.51366.23350.79333.50316.97309.78427.12421.00397.62357.13350.11349.71K______0.71960.72210.70110.66510.70840.7272a______22.6423.385.5812.2123.0725.38b______17.9819.344.8410.3717.1517.24d______40.9630.2717.5420.2638.9755.32The proposed failure criterion is used to model these selected strength data, and the calculated results are also listed in Table1. Figure 4 presents the comparisons of the predicted strength σ1cal with the experimental data σ1exp. The reference line (σ1cal = σ1exp) is also demonstrated in Figure 4 to exhibit the calculation accuracy. All of the data points are concentrated around the reference line, which indicates that the predictions by the proposed failure criterion are generally consistent with the experimental data.Figure 4
The comparison results of the calculated and experimental strength of jointed marble for different joint inclination angles: (a–f)β = 0°, 20°, 40°, 60°, 80°, and 90°.
(a)(b)(c)(d)(e)(f)The linear Mogi–Coulomb failure criterion [17] was adopted in [9] to fit the test data and is defined by(9)τoct=223ccosφ+223sinφσm,2,σm,2=σ1+σ32,where σm,2 is the effective mean stress. Here, the fitting results by these two failure criteria are compared in Figure 5, and the relationship between σ1cal and σ1exp can be expressed by(10)σ1cal=1.0049σ1expR2=0.9042,σ1cal=1.0059σ1expR2=0.9189,where equation (10) refers to the fitting results by the linear Mogi–Coulomb and proposed failure criteria, respectively. Table 2 also shows the percentage error in predicting strength data of jointed marble. Obviously, the proposed failure criterion has higher accuracy than the Mogi–Coulomb criterion. Therefore, the proposed failure criterion is reasonable and applicable to predict the true triaxial strength of jointed rock masses.Figure 5
The fitting results of all the experimental strength of jointed marble: (a) the linear Mogi–Coulomb failure criterion and (b) the proposed failure criterion.
(a)(b)Table 2
Percentage error in predicting strength data of jointed marble.
σ3 (MPa)σ2 (MPa)Percentage error in predicting strength data at joint inclination angleβ (°)02040608090007.799.3010.918.695.162.5210601.1514.2419.1315.757.446.8430606.666.053.390.8411.1914.8730903.540.610.080.026.1411.453012013.1414.9513.357.0910.4612.89Average of this paper6.469.039.376.488.089.71Average of the Mogi–Coulomb criterion [17]4.8810.4311.808.529.1711.57


### 3.2. Example 2

The strength data for example 2 are collected from the test results of [14]. This series of true triaxial tests were carried out on rock mass models with different joint inclination angles, which is made using similar material and contains three joint sets. Table 3 summarizes the strength data of both experimental and calculated results.Table 3
True triaxial compression strength of experimental and predicted results of rock mass models (data from [14]; here, compressive stress is considered positive).
σ3 (MPa)σ2 (MPa)σ1exp (MPa), test data at joint inclination angle β (°)1cal (MPa), calculated data at joint inclination angle β (°)02040608090020406080900.310.316.814.232.921.844.917.286.814.352.961.875.087.040.310.597.785.015.322.856.768.687.425.053.852.275.717.660.310.957.656.057.343.677.698.347.875.784.892.646.398.110.311.229.327.417.964.248.277.338.096.265.622.866.848.350.311.629.328.688.547.538.549.188.336.906.673.097.448.590.780.789.767.236.263.547.739.549.766.996.183.477.3910.050.781.2210.69.019.384.998.5910.6510.688.087.594.108.3610.990.782.2411.4510.611.917.6610.8611.9711.6710.0310.525.0810.1312.001.221.2212.519.329.134.939.3613.1312.529.459.184.969.5512.87K______0.61090.74750.84330.83090.76660.6085a______0.0011910.0013930.0012690.0011990.0014740.002107b______0.0010730.0011930.0011850.0007530.0011790.001917d______0.92520.56770.14520.34200.91180.9330Figure6 shows the comparison results of the calculated and experimental strength of jointed blocky mass. The data points except the joint inclination 60° are almost around the reference line. Thus, Figure 7 gives the fitting results of all the data and the data except the joint inclination 60°, and the relationship between σ1cal and σ1exp can be given by(11)σ1cal=0.9247σ1expR2=0.8738,σ1cal=0.9400σ1expR2=0.9907,where equation (11) refers to the fitting results using all data and all data except the joint inclination 60°, respectively. In addition, two similar relationships suggested in [10, 28] are defined as follows:(12)σ1cal=1.0150σ1expR2=0.9383,σ1cal=1.0706σ1expR2=0.8268.Figure 6
The comparison results of the calculated and experimental strength of jointed blocky mass for different joint inclination angles: (a–f)β = 0°, 20°, 40°, 60°, 80°, and 90°.
(a)(b)(c)(d)(e)(f)Figure 7
The fitting results of all the experimental strength of jointed blocky mass: (a) all the data and (b) all the data exceptβ = 60°.
(a)(b)Table3 also exhibits the percentage error in predicting strength data of rock mass models using these three strength criteria. Generally, compared with the modified Mohr–Coulomb failure criterion [28] and the empirical strength criterion [10], the proposed failure criterion in this study has higher correlation coefficient and smaller average percentage error, as shown in Table 4, which indicates that the proposed failure criterion is acceptable.Table 4
Percentage error in predicting strength data of rock mass models.
σ3 (MPa)σ2 (MPa)Percentage error in predicting strength data at joint inclination angleβ (°)020406080900.310.310.062.901.521.773.523.350.310.594.620.7727.5520.3315.5611.790.310.952.834.4233.4328.0116.942.720.311.2213.1515.5029.4132.6317.3213.890.311.6210.5920.4921.9558.9212.896.400.780.780.013.351.352.014.415.350.781.220.7510.3519.0717.792.683.190.782.241.885.4211.6833.636.730.251.221.220.051.440.560.682.001.97Average of this paper3.777.1816.2821.759.125.43Average of [14]8.7011.3015.3049.6013.3011.80Average of [10]17.808.3615.6029.707.7012.50


## 3.1. Example 1

The true triaxial strength data for the first example are derived from the experimental results of [9]. This test was conducted on jointed marble which contains a natural joint plane with a rectangular prismatic size (50 × 50 × 100 mm3). Table 1 lists the derived test data.Table 1
True triaxial compression strength of experimental and predicted results of jointed marble (data from [9]; here, compressive stress is considered positive).
σ3 (MPa)σ2 (MPa)σ1exp (MPa), test data at joint inclination angle β (°)σ1cal (MPa), calculated data at joint inclination angle β (°)020406080900204060809000158.59100.7973.7677.57112.21141.07146.23110.1681.8184.31118.00144.631060279.94296.43272.40240.12242.96258.62276.72254.21220.28202.30224.88240.923060332.25361.55330.96305.12338.63357.16354.39339.66319.75307.68300.75304.063090409.76387.18363.12337.69311.01297.24395.24384.82363.43337.62330.11331.2630120377.51366.23350.79333.50316.97309.78427.12421.00397.62357.13350.11349.71K______0.71960.72210.70110.66510.70840.7272a______22.6423.385.5812.2123.0725.38b______17.9819.344.8410.3717.1517.24d______40.9630.2717.5420.2638.9755.32The proposed failure criterion is used to model these selected strength data, and the calculated results are also listed in Table1. Figure 4 presents the comparisons of the predicted strength σ1cal with the experimental data σ1exp. The reference line (σ1cal = σ1exp) is also demonstrated in Figure 4 to exhibit the calculation accuracy. All of the data points are concentrated around the reference line, which indicates that the predictions by the proposed failure criterion are generally consistent with the experimental data.Figure 4
The comparison results of the calculated and experimental strength of jointed marble for different joint inclination angles: (a–f)β = 0°, 20°, 40°, 60°, 80°, and 90°.
(a)(b)(c)(d)(e)(f)The linear Mogi–Coulomb failure criterion [17] was adopted in [9] to fit the test data and is defined by(9)τoct=223ccosφ+223sinφσm,2,σm,2=σ1+σ32,where σm,2 is the effective mean stress. Here, the fitting results by these two failure criteria are compared in Figure 5, and the relationship between σ1cal and σ1exp can be expressed by(10)σ1cal=1.0049σ1expR2=0.9042,σ1cal=1.0059σ1expR2=0.9189,where equation (10) refers to the fitting results by the linear Mogi–Coulomb and proposed failure criteria, respectively. Table 2 also shows the percentage error in predicting strength data of jointed marble. Obviously, the proposed failure criterion has higher accuracy than the Mogi–Coulomb criterion. Therefore, the proposed failure criterion is reasonable and applicable to predict the true triaxial strength of jointed rock masses.Figure 5
The fitting results of all the experimental strength of jointed marble: (a) the linear Mogi–Coulomb failure criterion and (b) the proposed failure criterion.
(a)(b)Table 2
Percentage error in predicting strength data of jointed marble.
σ3 (MPa)σ2 (MPa)Percentage error in predicting strength data at joint inclination angleβ (°)02040608090007.799.3010.918.695.162.5210601.1514.2419.1315.757.446.8430606.666.053.390.8411.1914.8730903.540.610.080.026.1411.453012013.1414.9513.357.0910.4612.89Average of this paper6.469.039.376.488.089.71Average of the Mogi–Coulomb criterion [17]4.8810.4311.808.529.1711.57


## 3.2. Example 2

The strength data for example 2 are collected from the test results of [14]. This series of true triaxial tests were carried out on rock mass models with different joint inclination angles, which is made using similar material and contains three joint sets. Table 3 summarizes the strength data of both experimental and calculated results.Table 3
True triaxial compression strength of experimental and predicted results of rock mass models (data from [14]; here, compressive stress is considered positive).
σ3 (MPa)σ2 (MPa)σ1exp (MPa), test data at joint inclination angle β (°)1cal (MPa), calculated data at joint inclination angle β (°)02040608090020406080900.310.316.814.232.921.844.917.286.814.352.961.875.087.040.310.597.785.015.322.856.768.687.425.053.852.275.717.660.310.957.656.057.343.677.698.347.875.784.892.646.398.110.311.229.327.417.964.248.277.338.096.265.622.866.848.350.311.629.328.688.547.538.549.188.336.906.673.097.448.590.780.789.767.236.263.547.739.549.766.996.183.477.3910.050.781.2210.69.019.384.998.5910.6510.688.087.594.108.3610.990.782.2411.4510.611.917.6610.8611.9711.6710.0310.525.0810.1312.001.221.2212.519.329.134.939.3613.1312.529.459.184.969.5512.87K______0.61090.74750.84330.83090.76660.6085a______0.0011910.0013930.0012690.0011990.0014740.002107b______0.0010730.0011930.0011850.0007530.0011790.001917d______0.92520.56770.14520.34200.91180.9330Figure6 shows the comparison results of the calculated and experimental strength of jointed blocky mass. The data points except the joint inclination 60° are almost around the reference line. Thus, Figure 7 gives the fitting results of all the data and the data except the joint inclination 60°, and the relationship between σ1cal and σ1exp can be given by(11)σ1cal=0.9247σ1expR2=0.8738,σ1cal=0.9400σ1expR2=0.9907,where equation (11) refers to the fitting results using all data and all data except the joint inclination 60°, respectively. In addition, two similar relationships suggested in [10, 28] are defined as follows:(12)σ1cal=1.0150σ1expR2=0.9383,σ1cal=1.0706σ1expR2=0.8268.Figure 6
The comparison results of the calculated and experimental strength of jointed blocky mass for different joint inclination angles: (a–f)β = 0°, 20°, 40°, 60°, 80°, and 90°.
(a)(b)(c)(d)(e)(f)Figure 7
The fitting results of all the experimental strength of jointed blocky mass: (a) all the data and (b) all the data exceptβ = 60°.
(a)(b)Table3 also exhibits the percentage error in predicting strength data of rock mass models using these three strength criteria. Generally, compared with the modified Mohr–Coulomb failure criterion [28] and the empirical strength criterion [10], the proposed failure criterion in this study has higher correlation coefficient and smaller average percentage error, as shown in Table 4, which indicates that the proposed failure criterion is acceptable.Table 4
Percentage error in predicting strength data of rock mass models.
σ3 (MPa)σ2 (MPa)Percentage error in predicting strength data at joint inclination angleβ (°)020406080900.310.310.062.901.521.773.523.350.310.594.620.7727.5520.3315.5611.790.310.952.834.4233.4328.0116.942.720.311.2213.1515.5029.4132.6317.3213.890.311.6210.5920.4921.9558.9212.896.400.780.780.013.351.352.014.415.350.781.220.7510.3519.0717.792.683.190.782.241.885.4211.6833.636.730.251.221.220.051.440.560.682.001.97Average of this paper3.777.1816.2821.759.125.43Average of [14]8.7011.3015.3049.6013.3011.80Average of [10]17.808.3615.6029.707.7012.50


## 4. Conclusions

To investigate the influence of the joint orientation andσ2 on the strength characteristics of jointed rock masses, a three-dimensional failure criterion is developed and is validated by two examples of true triaxial test results. Some important conclusions can be drawn as follows:(1)
The proposed failure criterion in the deviatoric plane adopts the elliptic form based on the Willam–Warnke failure criterion. The joint effect is considered by only one parameter, strength ratioK.(2)
The proposed failure criterion in the meridian plane uses the hyperbolic function to approximate the Mohr–Coulomb failure criterion so that the parameters can be linked with the cohesion and the internal friction angle.(3)
The proposed failure criterion in the presented study can achieve small percentage error and high correlation coefficient to predict the true triaxial strength data in two examples.



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*Source: 9939144-2021-05-27.xml*