The calculation of the factor of safety (FOS) is an important means of slope evaluation. This paper proposed an improved double strength reduction method (DRM) to analyze the safety of layered slopes. The physical properties of different soil layers of the slopes are different, so the single coefficient strength reduction method (SRM) is not enough to reflect the actual critical state of the slopes. Considering that the water content of the soil in the natural state is the main factor for the strength of the soil, the attenuation law of shear strength of clayey soil changing with water content is fitted. This paper also establishes the functional relationship between different reduction coefficients. Then, a USDFLD subroutine is programmed using the secondary development function of finite element software. Controlling the relationship between field variables and calculation time realizes double strength reduction applicable to the layered slope. Finally, by comparing the calculation results of different examples, it is proved that the stress and displacement distribution of the critical slope state obtained by the improved method is more realistic, and the calculated safety factor is more reliable. The newly proposed method considers the difference of intensity attenuation between different soil layers under natural conditions and avoids the disadvantage of the strength reduction method with uniform parameters, which provides a new idea and method for stability analysis of layered and complex slopes.

The strength reduction method (SRM) is widely used in slope engineering because of its simple principle and convenient application, which can deal with complex geological conditions. In slope stability analysis, the SRM can be used to search the potential sliding surface and the corresponding safety factor for the slope through specific criteria without assuming the location of the sliding surface in advance. The basic idea of the SRM is to make the slope just in the critical failure state through continuous trial calculation by reducing the strength parameters of rock and soil. At this state, the corresponding reduction coefficient is the minimum safety factor for the slope. Zienkiewicz et al. [

In the general SRM calculation, the cohesion

The above studies have enriched and developed the theoretical system of the SRM. However, these studies have been conducted for slopes with homogeneous single soil layers, and fewer studies have been conducted for non-homogeneous layered slopes. The applicability of the SRM in homogeneous slopes has been confirmed, but non-homogeneous slopes are more common in practical engineering. Griffiths et al. [

In this paper, an improved double-strength reduction method for calculating non-homogeneous layered soil slopes is proposed. First, the attenuation law of shear strength for typical clayey soil with water content change is studied. Moreover, the functional relationship of its shear strength change with the change in water content is fitted. Then the connection of the reduction coefficients between different soil layers was established by mathematical derivation. The subroutine is written using the secondary development function of ABAQUS finite element software to realize the reduction of several strength parameters of layered slopes with increased calculation time. Then, two different methods are proposed to calculate the comprehensive safety factor for layered slopes. Finally, the reasonableness of this method to calculate the safety coefficient of the heterogeneous slope is verified calculation examples.

The DRM and the SRM have the same theoretical basis, which are belong to the category of strength reduction methods. The definition of the factor of safety for both methods is based on the strength reserve. When the shear strength parameters

where

The factor of safety is defined as the ratio of soil anti-sliding force to sliding force, such as

Rewrite

The shear strength of soil is an essential index of its mechanical properties, which is mainly affected by soil type, structure, and moisture content. In engineering practice, factors such as rainfall, drought, groundwater seepage will change the water content greatly. The variation of water content is the main factor affecting the attenuation of shear strength parameters.

In this paper, soil samples of different layers of typical soil slope were collected. The water content was measured, and the shear strength parameters under different water content conditions were obtained through laboratory direct shear tests. Three groups of test data from three soil layers were obtained, and the test results are shown in

The above three groups of data are fitted by exponential regression, which is suitable for the research. The fitting accuracy

In the reduction process of the DRM, the initial strength parameter of soil is reduced to the strength parameter value of critical instability, and multiple groups of values can be obtained to meet the conditions of slope instability. However, the state described by these parameters only reflects one possible state of slope failure, not necessarily the actual state of slope failure. Considering the natural deterioration law of shear strength parameters of slope soil in the present research, the combination of shear strength parameters obtained after reduction can more accurately describe the stress balance state of the slope and obtain the instability state of the slope. In this paper, the corresponding function relationship between each reduction coefficient is established through the influence law of water content on the shear strength of the soil.

The following exponential function is used to describe the variation trend of soil shear strength with water:

where

Let

where

It can be inferred from

During the process of strength reduction, the attenuation rates of each strength parameter for each soil layer are different. When the critical instability state is finally reached, the attenuation degree of each parameter is also different. According to the above process, the functional relationship between the reduction factors can be further applied to complex soil slopes with three or more layers, which satisfies the following equation:

At present, there is no unified standard for the definition of the comprehensive factor of safety by the DRM. Moreover, there is no conclusion on obtaining a reasonable comprehensive safety factor. In this paper, two methods are proposed to calculate the comprehensive safety factor for a layered slope; one is the mean method, and the other is the attenuation weight method.

The double-strength reduction method is used to calculate the comprehensive safety coefficient of the homogeneous slope; the average value is a standard method [

where

The weight analysis method is a common idea in the study of soil mechanics. For example, the influence of weight is considered in the study of permeability and foundation modulus for layered soil. When the slope is from the initial state to the critical instability state, the attenuation degree of the shear strength parameters of each soil layer is different. Therefore, for layered soil slopes, the attenuation weight of shear strength parameters is used to define the comprehensive safety factor. For slopes with two soil layers, the comprehensive safety factor is defined as follows.

For the shear strength parameter of any soil layer, take the value of its initial state and slope instability state to calculate its reduction degree.

where

and

The comprehensive factor of safety obtained from the attenuation weight, that is

For layered soil slope, its comprehensive factor of safety can be obtained from the following formula:

where

The above comprehensive safety factor is weighted by the attenuation degree of soil shear strength, and considering the soil moisture content, the reduction relationship in the DRM is determined, which provides a theoretical basis and technical support for the stability analysis of layered soil.

The traditional SRM adopts the dichotomy method to obtain the safety factor for the slope, which requires multiple trial calculations and computes inefficiently. In the process of SRM, the field variable is set as the safety factor and then the internal friction angle changes

where

A reduction factor is taken to satisfy the above equation, and the relationship between the remaining reduction factor and this factor is

When the traditional limit equilibrium method is used to calculate the safety factor for slope stability, it is assumed that rock and soil are ideal rigid plastic materials, which is different from the fact. The slope instability is mainly dominated by shear failure. The Mohr-Coulomb strength criterion is considered the shear failure criterion, which can describe the shear strength characteristics of rock and soil under tensile stress or compressive stress. Mohr-Coulomb strength criterion is adopted in this paper. The yield surface of the Mohr-Coulomb strength criterion is an irregular hexagonal cone with cusps and corners in the principal stress space. The shape is shown in

The criterion selection of slope instability affects the factor of safety. There are three methods to judge the failure of slope instability by finite element strength reduction method: (I) Based on the non-convergence of numerical calculation; (II) Based on a feature point at the top or middle of the slope and the sudden change of the displacement for the feature point; (III) Based on the penetration of plastic zone from slope toe to slope top.

For the above three criteria, the calculation non-convergence result is too large because the calculation has not stopped immediately when the slope is unstable and destroyed, and the setting of the calculation time step and the selection of the unit type will also affect the convergence result of the slope. The calculation results of the two criteria of feature point displacement mutation and plastic zone penetration are close. In this paper, the displacement mutation of the characteristic point is used as the basis for judging slope instability.

An example of a two-layer soil slope model is established. The improved DRM proposed in this study is compared with the traditional SRM to analyze the applicability of the methods. Model calculations are shown in

Soil number | Modulus of elasticity ( |
Poisson’s ratio | Bulk density ( |
Cohesion ( |
Friction angle ( |
---|---|---|---|---|---|

1 | 1E5 | 0.30 | 19.39 | 48.10 | 14.60 |

2 | 1E5 | 0.35 | 19.22 | 44.90 | 13.80 |

The sliding surface in slope instability is shown in

Strength parameters | ||||
---|---|---|---|---|

Initial value | 13.8 | 44.9 | 14.6 | 48.1 |

Reduced parameter values | 12.01 | 33.73 | 13.26 | 40.96 |

Degree of attenuation | 12.97% | 24.90% | 9.20% | 14.84% |

The results of the traditional SRM and improved DRM are compared in the displacement cloud chart, as shown in

Furthermore, the critical state stress cloud diagrams of the two methods are shown in

Value | 1.154 | 1.331 | 1.105 | 1.174 |

Different methods | Safety factor | Error calculation formula | Percentage error |
---|---|---|---|

① Limit equilibrium method | 1.188 | – | – |

② SRM | 1.233 | (② − ①)/① | 3.7% |

③ Calculated by average value | 1.191 | (③ − ①)/① | 0.3% |

④ Calculated by weight | 1.222 | (④ − ①)/① | 2.8% |

This section takes an example of a slope calculation test of the Australian Computer Application Association (ACADS). The limit equilibrium method must presuppose a sliding surface and make some ideal assumptions on the soil. Then, an equilibrium equation is established to obtain the numerical solution of the safety factor. The DRM and SRM are used to compare with the limit equilibrium method. The calculation diagram of the slope is shown in

Soil number | Modulus of elasticity ( |
Poisson’s ratio | Bulk density ( |
Cohesion ( |
Friction angle ( |
---|---|---|---|---|---|

1 | 1E5 | 0.35 | 19.22 | 44.90 | 13.80 |

2 | 1E5 | 0.30 | 19.39 | 48.10 | 14.60 |

3 | 1E5 | 0.35 | 18.84 | 29.40 | 13.60 |

When the DRM is applied, from the initial to the instability state of the slope, each parameter is reduced asynchronously, and its attenuation rate is different. The attenuation rate of the bottom soil layer is the fastest. In slope instability, the plastic zone is often the first to form at the foot of the slope and then develops to the top of the slope until it penetrates. Therefore, the process illustrated by this method conforms to the characteristics of slope progressive instability.

As shown in

Strength parameters | ||||||
---|---|---|---|---|---|---|

Initial value | 13.8 | 44.9 | 14.6 | 48.1 | 13.6 | 29.4 |

Reduced parameter values | 8.48 | 16.57 | 10.41 | 27.48 | 8.45 | 9.64 |

Degree of attenuation | 38.6% | 63.1% | 28.7% | 42.9% | 37.9% | 67.2% |

For layered slopes, the stress distribution is related to the geometric shape and physical properties of each part. The final stress field obtained by using the same reduction coefficient is different from the actual situation. The improved DRM considers the difference in attenuation between different soil layer parameters to obtain a more accurate stress field. As shown in

As shown in

Different methods | Safety factor | Error calculation formula | Percentage error |
---|---|---|---|

① Limit equilibrium method | 2.088 | – | – |

② SRM | 2.080 | (② − ①)/① | 0.3% |

③ Calculated by average value | 2.034 | (③ − ①)/① | 2.6% |

④ Calculated by weight value | 2.210 | (④ − ①)/① | 5.8% |

A two-layer and a complex three-layer slope example are analyzed. After comparison of the modified DRM with the traditional SRM, it is proved that the improved DRM proposed in this study can be applied to simulate the attenuation difference of various soils of layered slope in the natural state. Moreover, its displacement and stress in the critical state align with the actual slope failure situation. The limit equilibrium method and the factor of safety obtained by the proposed method are compared, and the rationality of the two DRM methods proposed in this paper for calculating the comprehensive safety factor for the layered slope is verified.

In this study, a modified DRM is proposed to make it applicable to the slopes of layered slopes, and the applicability of this method is verified by different calculations, and the following conclusions are obtained:

The shear strength parameters of cohesive soils of different soil layers at different water contents were analyzed. Moreover, the exponential function was considered a better expression for the decay law of shear strength parameters with increasing water content.

The USDFLD subroutine for Abaqus software is programed, which is suitable for the improved double-intensity discounting method proposed in this paper. All values in the intensity discounting calculation process can be non-linearly discounted with the increase in the calculation time, which improves the accuracy and efficiency of the calculation for slope stability analysis.

The functional relationship between the reduction coefficients of different soil layers is established. The shear strength parameters of each soil layer in progressive instability are decayed to different degrees. Furthermore, the stress and displacement clouds of the critical state obtained through this way are more consistent with the actual natural state. By calculation examples, calculation results by this method are proven correct and reasonable. It provides a reference for the analysis of layered slope stability in practical engineering.

The average and weight methods are proposed, which is suitable for multiple reduction factors to find the integrated safety factor. The coefficient of safety calculated by the average method is closer to that by the limit equilibrium method. The safety coefficient calculated by the attenuation weight method has its advantages but still exists a reasonable error range. Meanwhile, the method considers the weights of different soil layer strength parameters to influence slope instability for calculation, and its physical meaning is clear.

The authors are grateful for the support by the National Natural Science Foundation of China, Qinglan Project of Jiangsu Province, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering.

This research was funded by the National Natural Science Foundation of China (51709194), Qinglan Project of Jiangsu University, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering.

Conceptualization: F.S. and B.L.; methodology: F.S. and B.L.; software: Y.Z.; validation: Y.Z. and K.W.; formal analysis: Y.Z.; investigation: K.W.; data curation: Y.Z.; writing original draft preparation: F.S. and Y.Z.; writing review and editing: B.L.; visualization: Y.Z.; supervision, F.S.; funding acquisition, F.S. All authors have read and agreed to the published version of the manuscript.

The data supporting the conclusions of this article are included within the article. Any queries regarding these data may be directed to the corresponding author.

The authors declare that they have no conflicts of interest to report regarding the present study.