Slope stability analysis is a core issue in geotechnical engineering. This paper proposes a method of upper bound limit stability analysis for a slope with multiparameter coordinated variation based on the comprehensive consideration of the nonuniform distribution of slope soil parameters. This method starts from the perspective of energy balance, establishes a slope failure mechanism which meets velocity separation requirements, deduces its calculation formula for external force power and internal energy dissipation power, develops a cycle program for the most dangerous slip surface searching and stability coefficient calculation through computer programming technology, and finally forms a calculation method of upper bound limit stability analysis for the soil slope with nonuniform multiparameter distribution. At the same time, this method takes a dump slope in an open-pit mine as the engineering background, considers the nonuniformity of density, cohesion, and internal friction angle of the slope soil under subsidence, applies upper bound limit analysis to analyze the slope stability, and evaluates the accuracy of analysis results by using the residual thrust method. The results show that upper bound limit analysis has highly accuracy in stability coefficient calculation; compared with the residual thrust method, the stability coefficient calculation result by upper bound limit analysis is a strict upper bound solution, and the calculation error is easy to be estimated and eliminated. Simultaneously, the most dangerous slip surface obtained by upper bound limit analysis can fully satisfy the velocity separation requirement and has a greater engineering reference value.
Slope stability analysis is a core issue in geotechnical engineering [
In the recent years, with the rapid development of plastic mechanics, the limit analysis method has become an important measure in slope stability analysis [
Based on the above analysis, this paper creatively proposes an upper bound limit stability analysis for a soil slope with coordinated variation of multiparameter. The method discretizes the slope failure mechanism, and each discrete block can fully meet the associated flow rules. At the same time, the method takes the whole slip surface as the research object without establishing a complicated velocity field equation for the internal slip surface and has the advantages of simple calculation process, accurate calculation result, strong engineering applicability, and the like, thus providing a new idea for heterogeneous slope stability.
Soil is a kind of a natural product, and so its mechanical parameter distribution is extremely complicated. Among them, the soil mechanical parameters of accumulation slopes are approximately linearly distributed along the depth below the slope due to natural sedimentation [
Mechanical parameters varying along the depth below the slope.
According to the distribution of slope soil mechanical parameters, the slope failure mechanism is dispersed into the
Slope failure mechanism.
According to the uniqueness of velocity for each point of the failure mechanism, it can be determined that the discrete block has the same angular velocity and center of rotation. Obviously, each discrete block can be approximated as a homogeneous block soil when
Upper bound limit analysis analyzes slope stability from the perspective of energy balance can effectively avoid inconsistent analysis results caused by different mapping methods. When the slope is in the limit equilibrium state, external force power should be equal to internal energy dissipation power in the slope failure area. In natural conditions, external force power is provided by gravity power only [
There are two assumptions about slope soil when calculating the internal energy dissipation power: the slip mass is regarded as a rigid body; the slip surface is regarded as a velocity intermittent surface [
Internal energy dissipation power of the
Until now, external force power and internal energy dissipation power of the slope failure area can be obtained. In order to establish an energy balance equation, constraint conditions should be determined according to the slope failure law. First of all, each discrete point of the slope failure mechanism should have consistent coordinates, and the constraint condition can be expressed as
Secondly, the failure mechanism intersects the slope surface at the slope toe, and this constraint condition can be expressed as
At last, the failure mechanism should be located inside the slope surface except the point at the slope toe, and this constraint condition can be expressed as
The slope energy balance equation can be finally established with the energy calculation method and equation constraint conditions.
When the slope is in the limit equilibrium state, external force power should be equal to internal energy dissipation power in the slope failure area. A gradual transition to the limit equilibrium state through repeated strength reduction is needed for the slope under the nonlimit equilibrium state [
According to the definition of the stability coefficient, when the strength reduction coefficient
Calculation process of upper bound limit analysis.
In Figure
This paper takes a dump slope in an open-pit mine as the engineering background. The dump slope is composed of loose sandstone, the slope height is 150 meters, and the slope surface angle is 20°. Due to soil sedimentation, density, cohesion, and internal friction angle of soil at the top of the slope are, respectively, 18 kN/m3, 20 kPa, and 21°. At the bottom of the slope, density, cohesion, and internal friction angle are, respectively, 24 kN/m3, 65 kPa, and 18°. In the depth below the slope, the variable quantity of density, cohesion, and internal friction angle is, respectively, 0.04 kN/m3, 0.3 kPa, and 0.02° for per unit length. The slope does not have large cracks and slips, and the slope has been stable since formation from the perspective of geological characteristics and monitoring data and slope state analysis. The slope shape and mechanical parameter change law are shown in Figure
Slope shape and mechanical parameter change law.
According to the calculation process of upper bound limit analysis above, the slope stability with nonuniform distribution of each parameter can be obtained. The relationship between the number of discrete blocks and the slope stability coefficient is shown in Figure
Variation law of the slope stability coefficient.
It can be shown from Figure
Calculation results of the slope stability coefficient with upper bound limit analysis and the residual thrust method.
Calculation results of stability coefficient | |||
---|---|---|---|
Upper bound limit analysis | Residual thrust method | Error rate (%) | |
Linear distribution of density only | 1.306 | 1.256 | 3.98 |
Linear distribution of cohesion only | 1.41 | 1.365 | 3.30 |
Linear distribution of internal friction angle only | 1.435 | 1.393 | 3.02 |
Density and cohesion changed together | 1.454 | 1.41 | 3.12 |
Density and internal friction angle changed together | 1.312 | 1.265 | 3.72 |
Cohesion and internal friction angle changed together | 1.343 | 1.298 | 3.47 |
Density, cohesion, and internal friction angle changed together | 1.355 | 1.313 | 3.20 |
Most dangerous slip surface shape.
Upper bound limit analysis analyzes slope stability from the perspective of energy balance, while the limit equilibrium method analyzes slope stability through a static equilibrium equation. Two methods are quite different in the analysis perspective, but the results should be highly consistent. Therefore, this paper selects the residual thrust method to evaluate the accuracy of results of upper bound limit analysis. Residual thrust method assumes that the thrust direction of the current block to the next block at the interface is parallel to the bottom slip surface of the block, and the values based on zero resultant force in the two directions of the parallel bottom slip surface and the vertical bottom slip surface and the zero residual thrust of the leading edge strip are iteratively obtained [
From the perspective of soil parameter influence on the slope stability coefficient, compared with a homogeneous slope (the slope is a homogeneous slope when the number of discrete block is 1), linear distribution of density and cohesion will improve the slope stability, but linear distribution of internal friction angle will reduce the slope stability. Among these three parameters, linear distribution of the internal friction angle has the most significant influence on the slope stability; therefore, the reason for reduction in accumulation slope stability is closely related to the linear distribution of the internal friction angle. In engineering practice, keeping relatively a stable internal friction angle is important for accumulation slope stability.
From the perspective of calculation results of the stability coefficient, since the residual thrust method assumes that the thrust direction of the current block to the next block at the interface is parallel to the bottom slip surface of the block, shear force at the interface may exceed shear strength of the slope when the bottom slip surface of the block is steep, and it may cause stability coefficient calculation results to be critical. To the opposite, shear force at the interface may be far less than shear strength of the slope when the bottom slip surface of the block is flat, and it can fully meet reasonable conditions but may cause stability coefficient calculation results to be too conservative. That is to say, the stability coefficient obtained by the residual thrust method is not an upper bound limit solution or a lower limit solution in strict sense; calculation errors are difficult to be estimated and eliminated although the calculation results have certain accuracy, while upper bound limit analysis takes the velocity field satisfying the velocity boundary condition and the volume invariable condition as the admissible function; the obtained functional value is an upper bound limit value, which results in the greater value shown in the calculation results of the stability coefficient by upper bound limit analysis, and calculation errors are easily estimated and eliminated. The calculation result error rate of two methods is less than 5%, which meet the requirements of engineering practice and fully verify the accuracy of the calculation results. At the same time, the calculation results show that the slope is stable, which is highly consistent with engineering analysis.
From the perspective of the most dangerous slip surface, the starting point and the end point of the most dangerous slip surface obtained by two methods are basically the same, and the slip surface shape is similar. However, the most dangerous slip surface shape obtained by the residual thrust method is a shear arc slip surface, and the most dangerous slip surface shape obtained by upper bound limit analysis is a combined logarithmic spiral slip surface. According to the basic theory of plastic mechanics, the most dangerous slip surface obtained by upper bound limit analysis can fully meet velocity separation requirement. Therefore, the most dangerous slip surface obtained by upper bound limit analysis has a greater engineering reference significance.
Main conclusions of this paper are as follows: A slope discrete failure mechanism that meets the velocity separation requirements is established based on nonuniformity analysis of slope soil. A calculation method of external force power and internal energy dissipation rate is proposed, and calculation process of slope stability upper bound limit analysis method is compiled. The engineering case is used to calculate the stability of the slope with nonuniform multiparameter distribution by upper bound limit analysis, and the residual thrust method is used to verify the accuracy of the calculation results.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This paper was supported by the Key Research Project for Higher Education in Henan Province (20B560009) and the Scientific and Technological Project in Henan Province (202102310567).