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Many experimental data have illustrated that the strength envelops for soils are not linear. Nevertheless, the linear Mohr–Coulomb (MC) strength parameters are widely applied for the conventional method, software codes, and engineering standards in the slope design practice. Hence, this paper developed the 3D limit analysis for the stability of soil slopes with the nonlinear strength criterion. Based on a numerical optimization procedure written in Matlab software codes, the equivalent MC parameters (the equivalent friction angle and the equivalent cohesion) from the nonlinear strength envelopes were derived with respect to the least upper-bound solutions. Further investigations were made to assess the influences of nonlinear strength parameters and slope geometries on the equivalent MC parameters. The presented results indicate that the equivalent MC parameters are closely related to the nonlinear strength parameters. As the inclination angle increases, the equivalent friction angle becomes bigger, but the equivalent cohesion becomes smaller. Besides, 3D effects on the equivalent MC parameters were found to be slight. The presented approach for the determination of MC strength parameters is analytical and rigorous, and the approximate MC strength parameters in the provided design tables can be alternative references for practical use.

Strength criterion is critical for all types of materials in the area of slope stability analysis. The strengths of soils and rocks are universally presented by the linear Mohr–Coulomb (MC) failure envelope, which represents the shear strength by two MC strength parameters: the friction angle and the cohesion. The MC strength parameters have been widely applied in the conventional limit equilibrium methods for the calculations of slope safety factors. Besides, the computer codes and engineering standards for slope design are commonly on the base of the MC strength criterion. However, according to the experimental data, many studies have illustrated that the strength envelops for soils and rocks are not linear [

To solve this problem, many attempts have been made to derive the equivalent MC parameters from nonlinear strength criteria. Hoek and his partners [

For slopes in soils, a number of researchers have used a Power-Law (PL) type of nonlinear strength envelope for the evaluation of slope safety (e.g., Charles & Soares [

Since the nonlinear PL failure criterion was firstly proposed by Zhang & Chen [_{0},_{0,} and

PL strength envelope and tangent line.

To implement the use of nonlinear strength criteria for slope engineering, the tangential method was originally proposed by Drescher & Christopoulos [

As illustrated in Figure

For the PL strength criterion, the gradient of the tangent line at some point T can be derived from the deviation of the expression of (_{e} can be expressed as follows:

To establish the 3D limit analysis method, Michalowski & Drescher [

As presented by Gao et al. [_{0} and

Modified 3D failure mechanisms: (a) face failure mechanism and (b) base failure mechanism [

Modified 3D failure mechanisms with the insertosome: (a) face failure mechanism and (b) base failure mechanism [

On the basis of the above 3D failure mechanisms, the energy-balance equation can be established by equating the soil weight work rate

For 3D face failure mechanism (Figure

For 3D base failure mechanism (Figure

On the base of the energy-balance equation, the upper-bound solutions (i.e., the critical height

As presented in the 3D limit analysis method, the equivalent friction angle

Selecting two 3D slopes (

Effect of

Effect of

For gentle slopes with

Nevertheless, for steep slopes (Figure

Figures

Effect of

Effect of

To explore the effects of the slope angle

Effect of

It is obvious that the equivalent friction angle

Figure

Effect of

Based on the above results and analyses, it can be concluded that the nonlinear strength parameters and the slope inclination have significant influences on the equivalent MC parameters (

The equivalent friction angle

| | m | ||||||
---|---|---|---|---|---|---|---|---|

1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 2.5 | ||

0.5 | 20 | 14.30 | 10.57 | 8.31 | 6.78 | 5.69 | 4.87 | 3.98 |

30 | 17.52 | 13.15 | 10.50 | 8.72 | 7.44 | 6.49 | 5.43 | |

40 | 18.96 | 14.63 | 11.86 | 9.95 | 8.56 | 7.50 | 6.32 | |

50 | 19.78 | 15.60 | 12.82 | 10.85 | 9.39 | 8.27 | 7.01 | |

60 | 20.33 | 16.30 | 13.53 | 11.54 | 10.05 | 8.89 | 7.57 | |

| ||||||||

1.0 | 20 | 15.41 | 12.07 | 9.76 | 8.03 | 6.75 | 5.79 | 4.72 |

30 | 22.99 | 17.44 | 13.84 | 11.40 | 9.66 | 8.37 | 6.95 | |

40 | 28.73 | 21.55 | 17.08 | 14.08 | 11.95 | 10.36 | 8.62 | |

50 | 32.06 | 24.49 | 19.61 | 16.27 | 13.86 | 12.05 | 10.06 | |

60 | 34.01 | 26.63 | 21.61 | 18.07 | 15.48 | 13.52 | 11.34 | |

| ||||||||

2.0 | 20 | 15.45 | 12.35 | 10.13 | 8.47 | 7.19 | 6.19 | 5.08 |

30 | 23.54 | 18.77 | 15.28 | 12.74 | 10.86 | 9.43 | 7.85 | |

40 | 32.06 | 25.32 | 20.36 | 16.85 | 14.29 | 12.37 | 10.26 | |

50 | 40.63 | 31.64 | 25.26 | 20.81 | 17.60 | 15.21 | 12.59 | |

60 | 47.52 | 37.19 | 29.79 | 24.56 | 20.78 | 17.96 | 14.87 | |

| ||||||||

3.0 | 20 | 15.46 | 12.38 | 10.20 | 8.55 | 7.28 | 6.29 | 5.17 |

30 | 23.55 | 18.91 | 15.52 | 13.03 | 11.17 | 9.73 | 8.12 | |

40 | 32.17 | 25.88 | 21.11 | 17.61 | 15.00 | 13.02 | 10.82 | |

50 | 41.45 | 33.36 | 27.00 | 22.36 | 18.95 | 16.38 | 13.57 | |

60 | 51.13 | 41.09 | 33.12 | 27.30 | 23.05 | 19.87 | 16.40 | |

| ||||||||

5.0 | 20 | 15.46 | 12.40 | 10.23 | 8.59 | 7.33 | 6.35 | 5.23 |

30 | 23.55 | 18.94 | 15.62 | 13.18 | 11.34 | 9.91 | 8.30 | |

40 | 32.18 | 26.05 | 21.45 | 18.02 | 15.43 | 13.44 | 11.21 | |

50 | 41.54 | 33.97 | 27.90 | 23.30 | 19.85 | 17.22 | 14.30 | |

60 | 51.86 | 43.03 | 35.25 | 29.25 | 24.77 | 21.38 | 17.66 |

The equivalent cohesive

| | m | ||||||
---|---|---|---|---|---|---|---|---|

1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 2.5 | ||

0.5 | 20 | 2.46 | 1.82 | 1.62 | 1.52 | 1.45 | 1.41 | 1.35 |

30 | 1.30 | 1.29 | 1.27 | 1.24 | 1.22 | 1.20 | 1.18 | |

40 | 1.12 | 1.15 | 1.15 | 1.14 | 1.13 | 1.12 | 1.11 | |

50 | 1.07 | 1.09 | 1.09 | 1.09 | 1.09 | 1.08 | 1.08 | |

60 | 1.04 | 1.06 | 1.06 | 1.06 | 1.06 | 1.06 | 1.05 | |

| ||||||||

1.0 | 20 | 42.32 | 6.05 | 3.39 | 2.61 | 2.23 | 2.01 | 1.80 |

30 | 5.30 | 2.54 | 2.02 | 1.78 | 1.64 | 1.54 | 1.45 | |

40 | 1.90 | 1.65 | 1.53 | 1.45 | 1.39 | 1.35 | 1.30 | |

50 | 1.32 | 1.34 | 1.31 | 1.29 | 1.26 | 1.24 | 1.21 | |

60 | 1.15 | 1.19 | 1.20 | 1.19 | 1.18 | 1.17 | 1.15 | |

| ||||||||

2.0 | 20 | 1329.29 | 31.19 | 9.68 | 5.55 | 4.03 | 3.26 | 2.64 |

30 | 136.66 | 10.52 | 4.87 | 3.37 | 2.70 | 2.33 | 2.01 | |

40 | 22.55 | 4.76 | 3.02 | 2.41 | 2.09 | 1.89 | 1.71 | |

50 | 5.04 | 2.65 | 2.14 | 1.89 | 1.73 | 1.63 | 1.52 | |

60 | 1.93 | 1.77 | 1.66 | 1.58 | 1.51 | 1.45 | 1.38 | |

| ||||||||

3.0 | 20 | 10061.7 | 85.18 | 18.72 | 9.03 | 5.91 | 4.47 | 3.39 |

30 | 1033.92 | 28.09 | 9.13 | 5.32 | 3.87 | 3.13 | 2.53 | |

40 | 165.56 | 11.87 | 5.35 | 3.64 | 2.89 | 2.47 | 2.11 | |

50 | 30.59 | 5.68 | 3.46 | 2.69 | 2.30 | 2.06 | 1.83 | |

60 | 5.95 | 3.00 | 2.40 | 2.10 | 1.90 | 1.77 | 1.63 | |

| ||||||||

5.0 | 20 | 129396 | 304.20 | 43.49 | 16.95 | 9.74 | 6.76 | 4.71 |

30 | 13296.2 | 99.87 | 21.00 | 9.83 | 6.27 | 4.66 | 3.47 | |

40 | 2123.08 | 41.33 | 11.97 | 6.56 | 4.58 | 3.61 | 2.84 | |

50 | 383.73 | 18.62 | 7.34 | 4.65 | 3.53 | 2.93 | 2.42 | |

60 | 62.77 | 8.37 | 4.61 | 3.40 | 2.80 | 2.44 | 2.11 |

To verify the accuracy of the presented method and the applicability of the given approximate MC parameters, this section provides two examples of uniform dry soil slopes in plain-strain conditions. Since the limit analysis method focuses on the critical state of slope failure, the safety factors for slopes are assumed as

For the slope in example 1, the geometry parameters are given as_{0} = 0.98 kPa,_{0} = 0.33 kPa, and^{3}.

Based on the presented method for this slope with nonlinear parameters, the minimum safety factor is calculated as 1.64. This slope problem has been analyzed by Eid [

For slope design, the approximate MC parameters for this slope example can be obtained from Tables

The problem considered in this example adopts the test data reported by Baker [_{0} = 0.06 kPa,_{0} = 0.02 kPa, and^{3}.

For such a problem, the limit analysis method presented in this study yielded the safety factor of 1.14, which is a little larger than the result of_{0}/_{0} = 0.06/0.02 = 3.0, and

On the base of 3D failure mechanisms for soil slopes with the MC strength criterion, this paper employed the tangential method to develop the upper-bound limit analysis of slope stability with the nonlinear PL strength criterion. A numerical optimization procedure written in a computer code of Matlab software was applied to calculate the upper-bound solutions of slope stability. The equivalent MC strength parameters from the PL strength envelope were then derived with respect to the least upper-bound solutions. Effects of nonlinear strength parameters and slope geometries on the equivalent MC parameters have been well studied, and design chats of approximate MC strength parameters have been provided for various soil slopes. From this study, the main conclusions can be made as follows:

The equivalent MC strength parameters

As the inclination angle

Although 3D effect has significant influences on the safety of soil slopes, 3D effect on the equivalent MC strength parameters seems to be slight.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This study was financially supported by National Natural Science Foundation of China (Grant Nos. 51708310 and 51809160), Shandong Provincial Natural Science Foundation, China (Grant Nos. ZR2017BEE066 and ZR201702160366), a Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J17KB049), and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant No. 2017RCJJ004).