A limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability

Abstract

The present invention relates to a limit equilibrium analysis method for three-dimensional slope stability, and belongs to the field of slope engineering. The present invention combines the advantages of existing limit equilibrium methods, ensuring the rationality of the assumption of the direction of the inter-strip force increment and the clarity of the physical meaning, while at the same time adaptively adjusting the direction of the inter-strip force increment by satisfying the mechanical equilibrium conditions of the three-dimensional sliding body, and converting the mechanical equilibrium conditions required for the solution from the column mechanical equilibrium conditions to the overall mechanical equilibrium conditions of the three-dimensional sliding body, avoiding the dependence on the column division, simplifying the derivation process, and also solving the problems of low calculation accuracy, solution convergence difficulties, and deviation of analysis results from reality caused by unreasonable inter-strip force assumptions and single calculation algorithms in existing three-dimensional slope stability analysis limit equilibrium methods under complex conditions. The present invention provides a simple, reliable, efficient, and widely applicable calculation approach and implementation process for the limit equilibrium analysis of three-dimensional slope stability under complex conditions.

Description

Technical Field

[0001] The invention belongs to the field of traffic slope engineering, and in particular relates to a limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability. Background Art

[0002] my country has many mountainous areas and is prone to widespread geological disasters, with slope landslides being the most frequent. Slope landslides pose a serious threat to people's lives and property, and severely hinder national economic and social development. To prevent and mitigate the losses caused by slope landslides, slope stability analysis is essential. The most commonly used approach in slope stability research is to transform the actual slope into a plane strain model and then employ a two-dimensional limit equilibrium calculation method to conduct slope stability analysis. However, the two-dimensional limit equilibrium calculation method cannot account for factors such as three-dimensional spatial effects, resulting in conservative calculation results and unnecessary waste in slope management projects. Therefore, conducting three-dimensional slope stability analysis is a reasonable and effective approach to reducing costs and increasing efficiency in slope engineering projects.

[0003] Existing three-dimensional limit equilibrium calculation methods primarily employ two models: the inter-strip force assumption model and the sliding surface stress assumption model. The inter-strip force assumption model discretizes the sliding mass into multiple vertical bars and makes appropriate assumptions about the location and direction of the inter-strip forces. The mechanical equilibrium conditions satisfied by the sliding mass and the inter-strip force boundary conditions are then used to derive a slope safety factor formula, thereby completing the three-dimensional slope stability analysis. Most of the inter-strip force assumption models employed in existing methods extend the inter-strip force assumptions used in two-dimensional limit equilibrium calculation methods. Examples include the simplified three-dimensional Janbu method, which assumes the inter-strip force location is 1 / 3 to 1 / 2 of the bar height, the three-dimensional unbalanced thrust method, and the three-dimensional Morgenstern-Price method, which assumes the inter-strip force direction follows a specific distribution pattern. While these methods inherit the advantages of the two-dimensional limit equilibrium method, such as simple formulas and the elimination of the need to solve systems of equations, they also retain or even amplify the drawbacks of the inter-strip force assumption model. For example, the partial inter-strip force assumption model ignores certain inter-strip forces, preventing the sliding body from satisfying all mechanical equilibrium conditions and resulting in low computational accuracy. Furthermore, the partial inter-strip force assumption model is unrealistic and theoretically insufficient, leading to serious convergence issues in the calculation method and even deviations from actual results. Regarding the sliding surface stress assumption model, this method assumes the distribution pattern of normal stress on the sliding surface. Subsequently, the shear stress distribution pattern on the sliding surface is constructed based on the shear strength failure criterion. Furthermore, the slope safety factor is calculated using the mechanical equilibrium conditions satisfied by the sliding body. This method eliminates the need for assumptions about the position or direction of inter-strip forces, while simultaneously obtaining a rigorous solution that satisfies all mechanical equilibrium conditions and offers high computational accuracy. However, the physical meaning of the sliding surface stress distribution model assumed by this method is unclear, making it difficult to apply to slope stability analysis in complex situations. Furthermore, the slope safety factor calculation formulas derived by this method are often systems of multi-variable equations, which are difficult to solve, and the existence and uniqueness of the safety factor solution cannot be guaranteed. In summary, the existing three-dimensional limit equilibrium calculation method still has some shortcomings and is difficult to cope with actual slope engineering problems under complex conditions. It is urgent to establish a three-dimensional limit equilibrium calculation method that is practical, simple in theory, concise in calculation, and easy for engineering personnel to master.

[0004] Chinese invention patent application CN201910680998.7 discloses a three-dimensional slope stability analysis method for reservoir areas under water-level fluctuation conditions, comprising the following steps: S1. Selecting a specific landslide to be analyzed, determining its sliding surface, and discretizing the landslide into columns; S2. Providing basic assumptions for the stability analysis; S3. Writing the functional relationship of the basic assumptions in step S2 into incremental form, introducing a correction coefficient c, and assigning initial values to each parameter; S4. Solving the safety factor and the introduced correction coefficient and parameters based on the three-force equilibrium equation, the three-torque equilibrium equation, and the boundary conditions; S5. Comparing the safety factor solved in step S4 with the introduced correction coefficient and parameters to see if they meet the requirements. If not, an iterative method is used to solve them until they meet the requirements. This invention effectively improves the efficiency of safety factor calculation and the reliability of analysis. However, this method introduces a large number of unknown variables, making the calculation process more complicated. In addition, the number of known equations is far less than the number of unknown variables. Therefore, the unknown variables cannot be solved one by one, and reasonable values can only be selected based on the conditions of the inequality. Therefore, when the calculation results are unreasonable, further manual adjustments are required, which reduces the ease of use and practicality of the method. At the same time, the invented method fails to incorporate nonlinear strength criteria.

[0005] Chinese invention patent application CN202111332366.5 discloses a method for predicting the three-dimensional stability of vertically uniformly striped rock slopes. This method involves selecting a specific slope, discretizing the three-dimensional slope, and establishing an equilibrium equation for the strip column force; establishing an expression for the strip column shear force based on the Moore-Coulomb criterion; combining these equations, eliminating the shear force, and finally obtaining an expression for the horizontal seismic acceleration coefficient of the entire slope; and further discussing the direction of the inter-strip force. This method, along with several other steps, yields a relationship between the horizontal seismic acceleration coefficient and the safety factor, thereby determining the three-dimensional slope stability coefficient in the absence of seismic loads. The advantage of this invention is that it takes into account the magnitude and direction of the inter-strip force, facilitating calculations and improving prediction accuracy. However, the inventive method does not meet all the boundary conditions of the inter-column force in the calculation method of the discrete inter-column force, the rigor of its calculation method is relatively weak, and the constructed limit equilibrium equation does not directly solve the slope safety factor, which makes the calculation process still relatively complicated; at the same time, the invention limits the direction of the shear force of the discrete column sliding surface, and this direction is unrelated to the main sliding direction of the slope, which is obviously deviated from the actual situation; in addition, the inventive method only combines the linear strength criterion, and it is difficult to solve the three-dimensional slope stability assessment problem under the nonlinear strength criterion.

[0006] Therefore, there is still a need in this field to provide a limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability. Summary of the Invention

[0007] Among existing three-dimensional limit equilibrium calculation methods, the one based on the inter-strip force assumption model is difficult to effectively perform three-dimensional slope stability analysis under complex conditions. This is mainly manifested in two aspects: first, some inter-strip force assumptions ignore certain inter-strip forces, making it impossible for the sliding body to meet all mechanical equilibrium conditions, resulting in low calculation accuracy. Second, some inter-strip force assumptions are not realistic and theoretically rigorous, resulting in serious convergence problems in the calculation method and even deviations from actual results. Furthermore, the calculation method based on the sliding surface stress assumption model has unclear physical meaning in its sliding surface stress distribution pattern, making it difficult to apply to slope stability analysis under complex conditions. Furthermore, the derived slope safety factor calculation formulas are mostly systems of single-variable multi-order equations, which are difficult to solve and the existence and uniqueness of the solution cannot be guaranteed.

[0008] To this end, the present invention designs a limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability analysis. This method uses the coordinate range of the three-dimensional sliding body in the longitudinal and transverse directions of the slope as a reference, and uses equally spaced column and row planes to divide the three-dimensional sliding body into multiple vertically bounded bars. A mechanical analysis of the bars is then performed, and the shear stress reciprocity theorem is applied to ignore the transverse shear force between bars within the bars. A functional relationship between the inter-strip normal force and the vertical shear force increment is established. Based on this, combined with the shear strength criterion of the rock mass, the overall mechanical equilibrium equation of the three-dimensional sliding body is constructed. Boundary conditions for the inter-strip normal force increment and the inter-strip vertical shear force increment are introduced to achieve a rigorous solution for the three-dimensional slope stability analysis. To ensure the reliability and convergence of the solution process, an optimization objective function is constructed to calculate the easily divergent variables. An iterative loop solution strategy is proposed, embedded in the optimization objective function calculation genetic algorithm and implementation process, to obtain the slope stability analysis results. The present invention has the advantages of wide application range, simple theory, rigorous process, easy implementation, strong scalability, high solution efficiency, reliable calculation, strict solution, and easy mastery by engineering personnel. It provides a reasonable and effective implementation approach for the limit equilibrium analysis of three-dimensional slope stability under complex conditions.

[0009] Therefore, the present invention provides a limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability, which includes the following steps:

[0010] Step 1: Given the three-dimensional slope surface and sliding surface geometric parameters and the physical and mechanical parameters of the rock and soil, then establish the xyz axis rectangular coordinate system and determine the three-dimensional sliding body range [x min , x max ] and [y min ,y max ];

[0011] Step 2: Given the row plane spacing dx and the column plane spacing dy, the three-dimensional sliding body is divided into n×m bars with vertical boundaries, and the total number of valid bars M is determined based on the range of the three-dimensional sliding body;

[0012] Step 3: Calculate the x, y, and z coordinates of the center point on the sliding surface of the column ij. i,j 、y i,j and z i,j , where 1≤i≤n and 1≤j≤m; calculate the vertical height h of column ij i,j , the horizontal inclination angle α of the tangent direction of the sliding surface of the column ij in the x direction x_(i,j) , the horizontal inclination angle α of the tangent direction of the sliding surface of the column ij in the y direction y_(i,j) , inter-stripe force correction function f x_(i,j) and f y_(i,j) , and the normal force N on the sliding surface of column ij i,j Direction cosines n x_(i,j) 、n y_(i,j) and n z_(i,j) ;

[0013] Step 4: Given the three-dimensional slope seismic load and slope surface load, determine the known force acting on the column ij, that is, the gravity W acting on the centroid of the column ij i,j , horizontal earthquake force k H W i,j and vertical seismic force k V W i,j , and the horizontal x-axis load PX acting on the slope of column ij i,j , horizontal y-axis load PY i,j and vertical load PZ i,j ;

[0014] Step 5: Combine the iterative loop solution strategy and use the optimization objective function to calculate the genetic algorithm to solve the slope safety factor F s , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment correction between strips, and proportional coefficients λ1 and λ2 of the vertical shear force increment correction between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j , and then complete the three-dimensional slope stability analysis.

[0015] In a specific embodiment, in step 1, the physical and mechanical parameters of the rock and soil mass include the average weight of the soil mass and the cohesion c of the rock and soil mass on the sliding surface of the column ij. i,j and internal friction angle If the slope failure obeys the linear shear strength criterion, then c i,j and is a constant value. If the slope instability obeys the nonlinear shear strength criterion, then c i,j and is the normal force N on the sliding surface of column ij i,j Related functions need to be calculated using an iterative loop solution strategy.

[0016] In a specific implementation, in step 2, the 50≤n≤200, and the 50≤m≤200.

[0017] In a specific embodiment, the step 5 specifically includes the following steps:

[0018] Step 5A, setting the value range of the proportional coefficients λ1 and λ2;

[0019] Step 5B: Based on the genetic algorithm, the value ranges of the proportional coefficients λ1 and λ2 are encoded, and a population of N individuals is randomly generated, that is, a population of N value combinations of the proportional coefficients λ1 and λ2 is randomly generated;

[0020] Step 5C: Calculate the slope safety factor F using an iterative loop solution strategy. s , angle ρ and the constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i, j;

[0021] Step 5D: Use formula (32) to calculate the objective function f value under different combinations of λ1 and λ2, and convert it into fitness value;

[0022] f=min{[λ1-g(λ2)] 2 +[λ2-h(λ1)] 2} (32)

[0023] Where g(λ2) is a function about λ2; h(λ1) is a function about λ1;

[0024] Step 5E: Determine whether the number of iterations of the genetic algorithm has reached the genetic generation G. If so, stop searching; otherwise, perform selection, crossover, and mutation operations to generate a new population, and repeat steps 5B to 5D.

[0025] Step 5F: Output the proportional coefficients λ1 and λ2 corresponding to the minimum objective function f value, and give the slope safety factor Fs , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and the force N on the strip ij i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j The calculation results of .

[0026] In a specific embodiment, the values of λ1 and λ2 in step 5A range from 0 to 1, N in step 5B ranges from 40 to 60, and the value of the genetic generation G in step 5E ranges from 50 to 200.

[0027] In a specific embodiment, step 5C specifically includes the following steps:

[0028] Step 5C-1: Assume the slope safety factor F s The initial values of the constant variables ΔF1 and ΔF2 for the correction of the vertical shear force increment between strips are F s (0) , ρ (0) , ΔF1 (0) and ΔF2 (0) , and when calculating for the first time, take F s (0) =1,ρ (0) =0, ΔF1 (0) =0 and ΔF2 (0) =0;

[0029] Step 5C-2: Calculate the shear force S on the sliding surface of column ij using equations (10) to (12) i,j Direction cosines τ x_(i,j) , τ y_(i,j) and τ z_(i,j) Then, use equations (22) to (24) to calculate the parameters Φ i,j 、R i,j and T i,j ;

[0030]

[0031] Step 5C-3: Calculate the slope safety factor F using formula (26) s ;

[0032]

[0033] Step 5C-4: Calculate the inter-bar normal force increment dE on the front and rear sides of the bar ij using equations (21) and (1) respectively: i,j and vertical shear force increment dX i,j ;

[0034]

[0035] dX i,j =ΔF1+λ1f x_(i,j) dE i,j (1)

[0036] Step 5C-5: Calculate the normal force N on the sliding surface of column ij using equations (19) and (20) respectively: i,j and shear force S i,j ;

[0037]

[0038]

[0039] Step 5C-6: Calculate the angle ρ between the vertical plane where the main sliding direction of the three-dimensional sliding body is located and the xz plane using equation (27);

[0040]

[0041] Step 5C-7: Calculate the inter-bar normal force increment dL on the left and right sides of the bar ij using equations (25) and (2) respectively. i,j and vertical shear force increment dV i,j ;

[0042]

[0043] dV i,j =ΔF2+λ2f y_(i,j) dL i,j (2)

[0044] Step 5C-8, using equations (28) and (29), respectively calculate the constant variables ΔF1 and ΔF2 for the correction of the inter-strip vertical shear force increment;

[0045]

[0046] Step 5C-9, if |F s (0) -F s |≤ε1、|ρ (0) -ρ|≤ε2、|ΔF1 (0) -ΔF1|≤ε3 and |ΔF2 (0) -ΔF2|≤ε4, where ε1=0.001, ε2=0.001° and ε3=ε4=0.001kPa, then the iteration cycle ends; otherwise, let F s (0) =F s , ρ (0) =ρ, ΔF1 (0)=ΔF1 and ΔF2 (0) =ΔF2, repeat steps 5C-2 to 5C-8;

[0047] Step 5C-10: Output slope safety factor F s , angle ρ and the constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j .

[0048] Compared to patent application CN201910680998.7, the present invention introduces fewer unknown variables, which fully match the number of known equations. Furthermore, by combining an iterative loop solution strategy with a genetic algorithm, the present invention can independently achieve efficient and reasonable calculation of slope safety factors and solve the problem of slope stability analysis under nonlinear strength criteria. Compared to patent application CN202111332366.5, the present invention effectively solves all the problems encountered in that invention. In addition, for example, Chinese invention patent CN201310036626.3 discloses a three-dimensional slope stability prediction method, Chinese invention patent CN201310036780.0 discloses a three-dimensional slope stability prediction method under seismic load, and Chinese invention patent CN201610405311.5 discloses a three-dimensional slope stability prediction method under rainfall conditions; compared with these existing patents, the present invention not only establishes a more reasonable and efficient method for calculating the inter-strip force of discrete columns, but also can strictly meet all inter-strip force boundary conditions; at the same time, under the embedded iterative loop solution strategy, combined with the genetic algorithm, a simple and fast solution of the slope safety factor is achieved, and the reliability and validity of the calculation results are ensured; and in the process of three-dimensional slope stability analysis, the present invention associates the shear force direction of the discrete column sliding surface with the main sliding direction of the slope, and can incorporate nonlinear strength criteria, which is more conducive to solving slope stability problems under more complex conditions.

[0049] The advantages of the present invention are that it combines the advantages of the existing limit equilibrium method for three-dimensional slope stability analysis, constructs a reasonable inter-strip force increment function, and realizes the simplification and lightweight of the calculation formula derivation process. At the same time, it proposes an optimization objective function for calculating easily divergent variables, and establishes an iterative loop solution strategy embedded in the optimization objective function calculation genetic algorithm, thereby overcoming the shortcomings of the existing three-dimensional slope stability analysis limit equilibrium method under complex conditions, such as low calculation accuracy, difficulty in solution convergence, and deviation of analysis results from reality due to unreasonable inter-strip force assumptions and single calculation algorithms. The present invention has the advantages of a wide range of applications, simple theory, rigorous process, easy implementation, strong scalability, high solution efficiency, reliable calculation, strict solution, and easy mastery by engineering personnel. It provides a set of practical theoretical calculation methods and implementation processes for three-dimensional slope stability analysis under complex conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 It is a schematic diagram of the division of three-dimensional slope and three-dimensional sliding body columns of the present invention.

[0051] Figure 2 Schematic diagram of force analysis of column ij in the three-dimensional sliding body of the present invention.

[0052] Figure 3 This is a schematic diagram of the mechanical calculation of the column ij in the limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability analysis of the present invention.

[0053] Figure 4 This is a schematic diagram of the mechanical equilibrium conditions in the limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability analysis of the present invention.

[0054] Figure 5 This is a flow chart of the genetic algorithm for optimizing the objective function calculation in the limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability analysis of the present invention.

[0055] Figure 6 This is a flowchart of the iterative cycle solution in the limit equilibrium inter-strip force increment analysis method of the present invention for three-dimensional slope stability analysis.

[0056] Figure 7 This is a flow chart for implementing the limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability analysis of the present invention.

[0057] In the figure: 1. three-dimensional slope, 2. longitudinal slope, 3. transverse slope, 4. three-dimensional sliding body, 5. column, 6. slope surface, 7. slope sliding surface, 8. left side of column ij, 9. right side of column ij, 10. front side of column ij, 11. rear side of column ij, 12. slope surface of column ij, 13. sliding surface of column ij, 14. main sliding direction of three-dimensional sliding body, 15. vertical plane where the main sliding direction of three-dimensional sliding body is located, 16. xz plane, 17. projection of the sliding boundary line of three-dimensional sliding body on xy plane, 18. sliding boundary line of three-dimensional sliding body, 19. valid column, 20. invalid column. DETAILED DESCRIPTION

[0058] The implementation approach and implementation process of the present invention are as follows:

[0059] (1) Figure 1 As shown, in the three-dimensional slope, an xyz-axis rectangular coordinate system is established, where the x-axis direction is the longitudinal direction of the slope, the y-axis direction is the transverse direction of the slope, and the z-axis direction is the vertical direction. For the three-dimensional sliding body, its range in the x-axis direction is [x min , x max ] and the range in the y-axis direction is [y min ,y max ], then, a series of column planes parallel to the xz plane and row planes parallel to the yz plane are used to divide the three-dimensional sliding body into multiple bars with vertical boundaries, the spacing of the column planes is dy, which divides the three-dimensional sliding body into m columns in the horizontal direction, and the spacing of the row planes is dx, which divides the three-dimensional sliding body into n rows in the vertical direction.

[0060] (2) Figure 2 As shown, taking the i-th row and j-th column bar (name it bar ij, 1≤i≤n and 1≤j≤m) as an example, p i–1,j–1 、p i,j–1 、p i–1,j and p i,j is the point of column ij on the slope surface, s i–1,j–1 、s i,j–1 、s i–1,j and s i,j is the point of column ij on the slope sliding surface, p i–1,j–1 s i–1,j–1 s i,j–1 p i,j–1 is the left side of the bar ij, p i–1,j s i–1,j s i,j p i,j is the right side of column ij, p i–1,j–1 s i–1,j–1 s i–1,j p i–1,j is the front side of column ij, p i,j–1s i,j–1 s i,j p i,j is the rear side of column ij, p i–1,j– 1p i,j–1 p i–1,j p i,j is the slope of column ij, s i–1,j–1 s i,j–1 s i–1,j s i,j The sliding surface of column ij is known to have gravity W acting on column ij. i,j , horizontal earthquake force k H W i,j and vertical seismic force k V W i,j , the action point is at the centroid of column ij, and the horizontal x-axis load PX acts on the slope of column ij i,j , horizontal y-axis load PY i,j and vertical load PZ i,j , the points of action are all at the center of the slope surface of column ij. The unknown forces acting on the sliding surface of column ij are normal forces N i,j and shear force S i,j , the points of action are all at the center of the sliding surface of the column ij. The unknown forces acting on the left side of the column ij are the inter-strip normal force L i,j-1 , inter-strip vertical shear force V i,j-1 and the transverse shear force Q between strips i,j-1 The unknown forces acting on the right side of the column ij are the inter-column normal force L i,j , vertical shear force between strips V i,j and the transverse shear force Q between strips i,j The unknown forces acting on the front side of the bar ij are the normal forces between bars E i-1,j , vertical shear force between strips X i-1,j and the transverse shear force between strips H i-1,j The unknown forces acting on the rear side of the column ij are the inter-column normal force E i,j , vertical shear force between strips X i,j and the transverse shear force between strips H i,j .

[0061] (3) Figure 3 As shown, considering that the scales of the bar ij in the longitudinal and transverse directions are all differential, based on the shear stress reciprocity theorem, it can be seen that the transverse shear forces between the bars on each side of the bar ij not only appear in pairs but also cancel each other out. Therefore, the transverse shear forces between the bars on each side of the bar ij can be ignored. i-1,j 、H i,j , Q i,j-1 and Q i,jAt the same time, this approach helps to ensure the moment balance of the forces acting on the column ij around the z-axis. Furthermore, the normal force increment between the bars on the front and rear sides of the column ij is dE i,j =E i,j -E i-1,j , the vertical shear force increment between the front and rear side strips is dX i,j =X i,j -X i-1,j , the normal force increment between the left and right sides is dL i,j =L i,j -L i,j-1 , and the vertical shear force increment between the left and right sides is dV i,j =V i,j -V i,j-1 , when the directions of the resultant forces of the front and rear side inter-strip force increments and the left and right side inter-strip force increments are all parallel to the tangent direction of the sliding surface of column ij, the slope sliding force caused by the inter-strip force is the largest, resulting in the slope sliding surface being in the most unfavorable stress state. In fact, in order to meet all the static equilibrium conditions of column ij, the direction of the resultant forces of the inter-strip force increments is not necessarily parallel to the tangent direction of the sliding surface. Therefore, there will be a certain functional relationship between the inter-strip force increments. For this reason, it is assumed that the functional relationship between the inter-strip force increments is:

[0062] dX i,j =ΔF1+λ1f x_(i,j) dE i,j (1)

[0063] dV i,j =ΔF2+λ2f y_(i,j) dL i,j (2)

[0064] Where ΔF1 and ΔF2 are constant variables corresponding to the incremental correction of the vertical shear force between the strips, which aims to ensure that the inter-strip force strictly meets the boundary conditions; f x_(i,j) and f y_(i,j) is the inter-strip force correction function that needs to be satisfied when the inter-strip force increment direction is parallel to the sliding surface of the column ij, which is used to describe the influence of the geometric morphology characteristics of the sliding surface of the column ij on the inter-strip vertical shear force, and f x_(i,j) =tanα x_(i,j) and f y_(i,j) =tanα y_(i,j) , where α x_(i,j) is the horizontal inclination angle of the sliding surface of column ij in the x direction, α y_(i,j) is the horizontal inclination angle of the sliding surface of column ij in the y direction; λ1 and λ2 are the proportional coefficients for the correction of the vertical shear force increment between the corresponding bars.

[0065] Slope instability is caused by the sliding of a three-dimensional sliding body along a main sliding direction. As a result, the shear force S on the sliding surface of the column ij is i,jLet the vertical plane in or parallel to the main sliding direction of the three-dimensional sliding body be the included angle between the vertical plane in which the main sliding direction of the three-dimensional sliding body is located and the xz plane be ρ, and ρ is positive in the clockwise direction, otherwise it is negative.

[0066] When the slope instability is shear failure, the shear force S on the sliding surface of the column ij is i,j With normal force N i,j The following relationship is satisfied:

[0067]

[0068] Where c i,j and are the cohesion and internal friction angle of the rock mass on the sliding surface of column ij, respectively. If the slope instability obeys the linear shear strength criterion, then c i,j and is a constant value. If the slope instability obeys the nonlinear shear strength criterion, then c i,j and is the normal force N on the sliding surface of column ij i,j Related functions need to be calculated using an iterative loop solution strategy; F s is the slope safety factor.

[0069] (4) Figure 4 As shown, the mechanical equilibrium equations of all forces acting on the column ij along the x, y and z axes are:

[0070] n x_(i,j) N i,j +τ x_(i,j) S i,j -dE i,j -k H W i,j +PX i,j =0 (4)

[0071] n y_(i,j) N i,j +τ y_(i,j) S i,j -dL i,j +PY i,j =0 (5)

[0072] n z_(i,j) N i,j +τ z_(i,j) S i,j -dV i,j -dX i,j -(1-k V )W i,j -PZ i,j =0 (6)

[0073] Where nx_(i,j) 、n y_(i,j) and n z_(i,j) is the normal force N on the sliding surface of column ij i,j direction cosines of τ x_(i,j) , τ y_(i,j) and τ z_(i,j) is the shear force S on the sliding surface of column ij i,j The direction cosines of the shear force S on the sliding surface of the column ij i,j Let the vertical plane in or parallel to the main sliding direction of the three-dimensional sliding body be the included angle between the vertical plane in which the main sliding direction of the three-dimensional sliding body is located and the xz plane be ρ, then τ y_(i,j) / τ x_(i,j) =tanρ.

[0074] In equations (4) to (6), the normal force N on the sliding surface of the column ij is i,j Direction cosines n x_(i,j) 、n y_(i,j) and n z_(i,j) The calculation formulas are:

[0075]

[0076] In equations (4) to (6), the shear force S on the sliding surface of the column ij is i,j Direction cosines τ x_(i,j) , τ y_(i,j) and τ z_(i,j) The calculation formulas are:

[0077]

[0078] Furthermore, the moment equilibrium equations of all forces acting on the three-dimensional sliding body around the x-axis and y-axis are:

[0079]

[0080] Where x i,j 、y i,j and z i,j are the x-, y-, and z-axis coordinates of the center point on the sliding surface of column ij; h i,j is the vertical height of column ij.

[0081] It should be noted that although the three-dimensional sliding body is divided into m columns in the horizontal direction and n rows in the vertical direction, not all the columns consisting of m columns and n rows are within the range of the sliding body. For the column ij, the x-axis and y-axis coordinates of the center point on the sliding surface are x and y, respectively. i,j and y i,j , and x i,j =x min +(i–0.5)×(x max –xmin ) / n and y i,j =y min +(j–0.5)×(y max –y min ) / m. If this point is within the projection of the three-dimensional sliding body's sliding boundary line onto the xy plane, then the bin ij is a valid bin; otherwise, it is an invalid bin. For invalid bins, all forces acting on them can be set to zero. The three-dimensional sliding body's sliding boundary line is the boundary line where the slope's sliding surface intersects the slope's surface.

[0082] As for the increments of normal force between strips and vertical shear force between strips, when boundary conditions are met, their calculation formulas are:

[0083]

[0084] (5) Combining equations (4) to (6) and combining equations (1) and (2), we can obtain the normal force N on the sliding surface of the column ij: i,j and shear force S i,j The calculation formulas are:

[0085]

[0086] Furthermore, substituting equations (19) and (20) into equation (3), we can obtain the inter-bar normal force increment dE on the front and rear sides of the column ij: i,j The calculation formula is:

[0087]

[0088] In formula (21), the parameter Φ i,j 、R i,j and T i,j The calculation formulas are:

[0089]

[0090]

[0091] Then, substituting Equation (11) into Equation (5), we can obtain the normal force increment dL between the bars on the left and right sides of the bar ij: i,j The calculation formula is:

[0092]

[0093] Then, by substituting Equation (21) into Equation (15), the slope safety factor F can be established. s The calculation formula is:

[0094]

[0095] At the same time, by substituting formula (25) into formula (16), the calculation formula of angle ρ can be established as follows:

[0096]

[0097] For the constant variable ΔF1 of the inter-strip vertical shear force increment correction, equation (1) can be substituted into equation (17) to establish its calculation formula:

[0098]

[0099] Where M is the total number of valid bars.

[0100] For the constant variable ΔF2 of the vertical shear force increment correction between strips, substitute equation (2) into equation (18) to establish its calculation formula:

[0101]

[0102] For the proportional coefficient λ1 of the vertical shear force increment correction between strips, substitute equations (5) and (6) into equation (13), and combine equations (1) and (2) to establish its calculation formula:

[0103]

[0104] For the proportional coefficient λ2 of the vertical shear force increment correction between strips, substitute equations (4) and (6) into equation (14), and combine equations (1) and (2) to establish its calculation formula:

[0105]

[0106] According to the above formula, N can be solved by using formula (19) to formula (21), formula (1), formula (25), formula (2) and formula (26) to formula (31) i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j 、dV i,j 、F s , ρ, ΔF1, ΔF2, λ1 and λ2.

[0107] (6) Equations (26) to (31) give the slope safety factor F s, angle ρ, constant variables ΔF1 and ΔF2 of the inter-strip vertical shear force increment correction, and proportional coefficients λ1 and λ2 of the inter-strip vertical shear force increment correction are obtained. However, Equations (26) to (31) are all implicit calculation formulas and need to be calculated using an iterative loop solution strategy. At the same time, considering that there is a positive correlation between the calculation of the proportional coefficients λ1 and λ2, relying solely on the iterative loop solution strategy cannot guarantee the convergence of its calculation. Therefore, Equations (30) and (31) used for the calculation of the proportional coefficients λ1 and λ2 are converted into the following optimization objective function:

[0108] f=min{[λ1-g(λ2)] 2 +[λ2-h(λ1)] 2} (32)

[0109] Where g(λ2) is the function on the right side of equation (30); h(λ1) is the function on the right side of equation (31).

[0110] In formula (32), the value range of λ1 and λ2 is usually 0 to 1. When λ1 and λ2 are within this range, if formula (32) can be made close to 0, then the corresponding λ1 and λ2 results are the approximate solutions of formulas (30) and (31).

[0111] like Figure 5 As shown in Figure 2, the specific implementation steps of the genetic algorithm for optimizing the objective function calculation are as follows:

[0112] ① Given the value range of the proportional coefficients λ1 and λ2, here, let the value range of λ1 and λ2 be 0~1;

[0113] ② Based on the genetic algorithm, encode the value ranges of the proportional coefficients λ1 and λ2 and randomly generate a population of N individuals (i.e., N combinations of the value of the proportional coefficients λ1 and λ2). It is recommended to take N = 50;

[0114] ③ Using the iterative cycle solution strategy, calculate the slope safety factor F s , angle ρ and the constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j ;

[0115] ④ Use formula (32) to calculate the objective function f value under different combinations of λ1 and λ2, and convert it into fitness value;

[0116] ⑤ Determine whether the number of iterations of the genetic algorithm has reached the genetic generation G (G = 100 is recommended). If so, stop searching. Otherwise, perform selection, crossover, and mutation operations to generate a new population and repeat steps ② to ④.

[0117] ⑥ Output the proportional coefficients λ1 and λ2 corresponding to the minimum objective function f value, and give the slope safety factor F s , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and the force N on the strip ij i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j The calculation results of .

[0118] (7) Figure 6 As shown, the slope safety factor F s The iterative loop solution strategy and specific implementation steps of the constant variables ΔF1 and ΔF2 for the correction of the vertical shear force increment between strips are as follows:

[0119] ① Assuming the slope safety factor F s The initial values of the constant variables ΔF1 and ΔF2 for the correction of the vertical shear force increment between strips are F s (0) , ρ (0) , ΔF1 (0) and ΔF2 (0) , and when calculating for the first time, take F s (0) =1,ρ (0) =0, ΔF1 (0) =0 and ΔF2 (0) =0;

[0120] ② Use equations (10) to (12) to calculate the direction cosine τ x_(i,j) , τ y_(i,j) and τ z_(i,j) Then, use equations (22) to (24) to calculate the parameters Φ i,j 、R i,j and T i,j ;

[0121] ③ Calculate the slope safety factor F using formula (26) s ;

[0122] ④ Use equations (21) and (1) to calculate the inter-strip normal force increment dE on the front and rear sides of the column ij respectively i,j and vertical shear force increment dX i,j ;

[0123] ⑤ Use equations (19) and (20) to calculate the normal force N on the sliding surface of column ij i,j and shear force S i,j ;

[0124] ⑥ Calculate the angle ρ between the vertical plane where the main sliding direction of the three-dimensional sliding body is located and the xz plane using formula (27);

[0125] ⑦Use equation (25) and equation (2) to calculate the normal force increment dL between the bars on the left and right sides of the bar ij respectively: i,j and vertical shear force increment dV i,j ;

[0126] ⑧Use equations (28) and (29) to calculate the constant variables ΔF1 and ΔF2 for the correction of the vertical shear force increment between strips respectively;

[0127] ⑨If |F s (0) -F s |≤ε1、|ρ (0) -ρ|≤ε2、|ΔF1 (0) -ΔF1|≤ε3 and |ΔF2 (0) -ΔF2|≤ε4(ε1=0.001, ε2=0.001° and ε3=ε4=0.001kPa), then the iteration cycle ends, otherwise, let F s (0) =F s , ρ (0) =ρ, ΔF1 (0) =ΔF1 and ΔF2 (0) =ΔF2, repeat steps ② to ⑧;

[0128] ⑩Output slope safety factor F s , angle ρ and the constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j .

[0129] (8) Figure 7 As shown in Figure 2, the implementation process of the limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability analysis is as follows:

[0130] ① Given the three-dimensional slope surface and sliding surface geometric parameters and the physical and mechanical parameters of the rock and soil, then establish the xyz axis rectangular coordinate system and determine the three-dimensional sliding body range [x min , x max ] and [y min ,y max];

[0131] ② Given the row plane spacing dx and column plane spacing dy, the 3D sliding body is divided into n×m bars with vertical boundaries, and the total number of valid bars M is determined based on the range of the 3D sliding body;

[0132] ③ Calculate the x, y and z axis coordinates of the center point on the sliding surface of the column ij (1≤i≤n and 1≤j≤m) i,j 、y i,j and z i,j , vertical height h i,j , the horizontal inclination angle α of the tangent direction of the sliding surface of the column ij in the x direction x_(i,j) , the horizontal inclination angle α of the tangent direction of the sliding surface of the column ij in the y direction y_(i,j) , inter-stripe force correction function f x_(i,j) and f y_(i,j) , and the normal force N on the sliding surface of column ij i,j Direction cosines n x_(i,j) 、n y_(i,j) and n z_(i,j) ;

[0133] ④ Given the three-dimensional slope seismic load and slope surface load, the known force acting on the column ij is determined, that is, the gravity W acting on the centroid of the column ij. i,j , horizontal earthquake force k H W i,j and vertical seismic force k V W i,j , and the horizontal x-axis load PX acting on the slope of column ij i,j , horizontal y-axis load PY i,j and vertical load PZ i,j ;

[0134] ⑤ Combined with the iterative loop solution strategy, the specific implementation steps of the genetic algorithm are calculated using the optimization objective function to solve the slope safety factor F s , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment correction between strips, and proportional coefficients λ1 and λ2 of the vertical shear force increment correction between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j , and then, complete the three-dimensional slope stability analysis.

[0135] The characteristic of the inter-strip force increment functional relationship of the present invention is that it combines the advantages of the existing limit equilibrium method, ensuring the rationality of the assumption of the inter-strip force increment direction and the clarity of the physical meaning, while adaptively adjusting the inter-strip force increment direction by satisfying the mechanical equilibrium conditions of the three-dimensional sliding body. At the same time, the mechanical equilibrium conditions required for the solution are converted from the strip-column mechanical equilibrium conditions to the overall mechanical equilibrium conditions of the three-dimensional sliding body, avoiding the dependence on the strip-column division, simplifying the derivation process, and facilitating fast, effective and reliable calculation. In addition, it also makes up for the deficiency of the existing inter-strip force assumption model that it is difficult to consider the boundary conditions of the inter-strip shear force ends.

[0136] Example

[0137] A kind of Figures 1 to 7 The three-dimensional slope stability analysis method for limit equilibrium inter-strip force increment is shown in Figure 2. A highway subgrade slope with a height of approximately 12.2 m and a slope angle of approximately 26.57° was analyzed using the three-dimensional slope stability analysis method for limit equilibrium inter-strip force increment to investigate the stability of this slope and the rationality of its design.

[0138] The specific operations are as follows:

[0139] (1) In accordance with the requirements of the Code for Investigation of Geotechnical Engineering (GB50021-2001), field investigation and testing of the slope soil were carried out to determine the physical and mechanical parameters of the rock and soil. It was determined that the rock and soil obeyed the Mohr-Coulomb shear strength criterion when the slope slipped. The specific test results were: the average soil density was 19.2 kN / m 3 The cohesion of the rock and soil on the sliding surface is 29.3 kPa and the internal friction angle is 20° (which indicates that the cohesion of the rock and soil on the sliding surface of the subsequent column ij is c i,j =29.3kPa and internal friction angle );

[0140] (2) The xyz-axis rectangular coordinate system is established with the slope foot point as the origin, where the x-axis is the longitudinal direction of the slope, the y-axis is the transverse direction of the slope, and the z-axis represents the vertical direction. According to the geological survey and slope displacement monitoring results, the potential sliding surface of the slope is approximately an ellipsoid, that is, the three-dimensional sliding body is approximately an ellipsoid, and the symmetry plane of the ellipsoid is parallel to the xz plane. The equation of the potential sliding surface of the slope is:

[0141]

[0142] (3) Based on the potential sliding surface equation of the slope and the slope surface morphology, determine the lateral range of the three-dimensional sliding body [x min , x max ] and the vertical range [y min ,y max] are [-10.000m, 28.487m] and [-58.159m, 58.159m] respectively, thus setting the row plane spacing dx to 0.385m and the column plane spacing dy to 1.163m, and then dividing the three-dimensional sliding body into n×m bars with vertical boundaries, where n=100, m=100 and the total number of valid bars M=7626;

[0143] (4) Obtain the z-axis coordinate (i.e., z) of the center point on the sliding surface of the column ij (1≤i≤n and 1≤j≤m) i,j , where the x- and y-axis coordinates of the center point on the sliding surface of column ij are x i,j =x min +(i–0.5)×(x max –x min ) / n and y i,j =y min +(j–0.5)×(y max –y min ) / m), vertical height h i,j , the horizontal inclination angle α of the sliding surface of column ij in the x direction x_(i,j) and the horizontal inclination angle α in the y direction y_(i,j) , inter-strip force correction function f x_(i,j) and f y_(i,j) and the normal force N on the sliding surface of column ij i,j Direction cosines n x_(i,j) 、n y_(i,j) and n z_(i,j) ;

[0144] (5) Calculate the gravity W acting on the centroid of column ij i,j At the same time, geological data show that this slope is not in a seismic activity zone and the seismic effect can be ignored. At the same time, there is no slope load on the slope. Therefore, the horizontal seismic force k on the slope is H W i,j and vertical seismic force k V W i,j are all 0, and the horizontal x-axis load PX acting on the slope of column ij i,j , horizontal y-axis load PY i,j and vertical load PZ i,j are also 0;

[0145] (6) Combined with the iterative loop solution strategy, the specific implementation steps of the genetic algorithm are calculated using the optimization objective function to solve the slope safety factor F s , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment correction between strips, and proportional coefficients λ1 and λ2 of the vertical shear force increment correction between strips, and then the force N on the strip ij is given i,j 、Si,j 、dE i,j 、dX i,j 、dL i,j and dV i,j , where the slope safety factor F s =2.235, angle ρ = 0.000°, constant variables ΔF1 = 5.578 kPa and ΔF2 = 0.191 kPa for the correction of the vertical shear force increment between strips, and proportional coefficients λ1 = 0.693 and λ2 = 0.199 for the correction of the vertical shear force increment between strips;

[0146] (7) According to the stability evaluation standard of the Technical Code for Construction Slope Engineering (GB 50330-2013), combined with the slope safety factor calculated by the three-dimensional slope stability analysis limit equilibrium inter-strip force increment analysis method, it is determined that this slope is stable and meets the first-level safety grade assessment standard, which shows that the design of this roadbed slope is reasonable.

[0147] In general, the present invention relates to a limit equilibrium analysis method for three-dimensional slope stability, which belongs to the field of slope engineering. The present invention combines the advantages of existing limit equilibrium methods, ensuring the rationality of the assumption of the direction of the inter-strip force increment and the clarity of the physical meaning, while adaptively adjusting the direction of the inter-strip force increment by satisfying the mechanical equilibrium conditions of the three-dimensional sliding body, and converting the mechanical equilibrium conditions required for the solution from the column mechanical equilibrium conditions to the overall mechanical equilibrium conditions of the three-dimensional sliding body, avoiding the dependence on the column division and simplifying the derivation process. In addition, it can also be used to solve the problems of low calculation accuracy, solution convergence difficulties and deviation of analysis results from reality caused by unreasonable inter-strip force assumptions and single calculation algorithms in existing three-dimensional slope stability analysis limit equilibrium methods under complex conditions. The present invention provides a simple, reliable, efficient and widely applicable calculation approach and implementation process for the limit equilibrium analysis of three-dimensional slope stability under complex conditions.

[0148] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability, characterized by: The three-dimensional slope stability analysis method includes the following steps: Step 1: Given the three-dimensional slope surface and sliding surface geometric parameters and the physical and mechanical parameters of the rock and soil, then establish the xyz axis rectangular coordinate system and determine the three-dimensional sliding body range [x min , x max ] and [y min ,y max ]; Step 2: Given the row plane spacing dx and the column plane spacing dy, the three-dimensional sliding body is divided into n×m bars with vertical boundaries, and the total number of valid bars M is determined based on the range of the three-dimensional sliding body; Step 3: Calculate the x, y, and z coordinates of the center point on the sliding surface of the column ij. i,j 、y i,j and z i,j , where 1≤i≤n and 1≤j≤m; calculate the vertical height h of bin ij i,j , the horizontal inclination angle α of the tangent direction of the sliding surface of the column ij in the x direction x_(i,j) , the horizontal inclination angle α of the tangent direction of the sliding surface of the column ij in the y direction y_(i,j) , inter-stripe force correction function f x_(i,j) and f y_(i,j) , and the normal force N on the sliding surface of column ij i,j Direction cosines n x_(i,j) 、n y_(i,j) and n z_(i,j) ; Step 4: Given the three-dimensional slope seismic load and slope surface load, determine the known force acting on the column ij, that is, the gravity W acting on the centroid of the column ij i,j , horizontal seismic force k H W i,j and vertical seismic force k V W i,j , and the horizontal x-axis load PX acting on the slope of column ij i,j , horizontal y-axis load PY i,j and vertical load PZ i,j ; Step 5: Combine the iterative loop solution strategy and use the optimization objective function to calculate the genetic algorithm to solve the slope safety factor F s , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment correction between strips, and proportional coefficients λ1 and λ2 of the vertical shear force increment correction between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j , and then complete the three-dimensional slope stability analysis.

2. The limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability according to claim 1 is characterized in that: In step 1, the physical and mechanical parameters of the rock and soil mass include the average weight of the soil mass and the cohesion c of the rock and soil mass on the sliding surface of the column ij. i,j and internal friction angle If the slope failure obeys the linear shear strength criterion, then c i,j and is a constant value. If the slope instability obeys the nonlinear shear strength criterion, then c i,j and is the normal force N on the sliding surface of column ij i,j Related functions need to be calculated using an iterative loop solution strategy.

3. The limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability according to claim 1 is characterized in that: In step 2, the 50≤n≤200, and the 50≤m≤200.

4. The limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability according to any one of claims 1 to 3, characterized in that: The step 5 specifically includes the following steps: Step 5A, setting the value range of the proportional coefficients λ1 and λ2; Step 5B: Based on the genetic algorithm, the value ranges of the proportional coefficients λ1 and λ2 are encoded, and a population of N individuals is randomly generated, that is, a population of N value combinations of the proportional coefficients λ1 and λ2 is randomly generated; Step 5C: Calculate the slope safety factor F using an iterative loop solution strategy. s , angle ρ and the constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i ,j; Step 5D: Use formula (32) to calculate the objective function f value under different combinations of λ1 and λ2, and convert it into fitness value; f=min{[λ1-g(λ2)] 2 +[λ2-h(λ1)] 2 } (32) Where g(λ2) is a function about λ2; h(λ1) is a function about λ1; Step 5E: Determine whether the number of iterations of the genetic algorithm has reached the genetic generation G. If so, stop searching; otherwise, perform selection, crossover, and mutation operations to generate a new population, and repeat steps 5B to 5D. Step 5F: Output the proportional coefficients λ1 and λ2 corresponding to the minimum objective function f value, and give the slope safety factor F s , angle ρ, constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and the force N on the strip ij i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j The calculation results of .

5. The limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability according to claim 4 is characterized in that: In step 5A, the values of λ1 and λ2 range from 0 to 1, in step 5B, N=40 to 60, and in step 5E, the value of the genetic generation G ranges from 50 to 200.

6. The limit equilibrium inter-strip force increment analysis method for three-dimensional slope stability according to claim 5 is characterized in that: Step 5C specifically includes the following steps: Step 5C-1: Assume the slope safety factor F s The initial values of the constant variables ΔF1 and ΔF2 for the correction of the vertical shear force increment between strips are F s (0) , ρ (0) , ΔF1 (0) and ΔF2 (0) , and when calculating for the first time, take F s (0) =1,ρ (0) =0, ΔF1 (0) =0 and ΔF2 (0) =0; Step 5C-2: Calculate the shear force S on the sliding surface of column ij using equations (10) to (12) i,j Direction cosines τ x_(i ,j),τ y_(i,j) and τ z_(i,j) Then, use equations (22) to (24) to calculate the parameters Φ i,j 、R i,j and T i,j ; Step 5C-3: Calculate the slope safety factor F using formula (26) s ; Step 5C-4: Calculate the inter-bar normal force increment dE on the front and rear sides of the bar ij using equations (21) and (1) respectively: i,j and vertical shear force increment dX i,j ; dX i,j =ΔF1+λ1f x_(i,j) dE i,j (1) Step 5C-5: Calculate the normal force N on the sliding surface of column ij using equations (19) and (20) respectively: i,j and shear force S i,j ; Step 5C-6: Calculate the angle ρ between the vertical plane where the main sliding direction of the three-dimensional sliding body is located and the xz plane using equation (27); Step 5C-7: Calculate the inter-bar normal force increment dL on the left and right sides of the bar ij using equations (25) and (2) respectively. i,j and vertical shear force increment dV i,j ; dV i,j =ΔF2+λ2f y_(i,j) dL i,j (2) Step 5C-8, using equations (28) and (29), respectively calculate the constant variables ΔF1 and ΔF2 for the correction of the inter-strip vertical shear force increment; Step 5C-9, if |F s (0) -F s |≤ε1、|ρ (0) -ρ|≤ε2、|ΔF1 (0) -ΔF1|≤ε3 and |ΔF2 (0) -ΔF2|≤ε4, where ε1=0.001, ε2=0.001° and ε3=ε4=0.001kPa, then the iteration cycle ends; otherwise, let F s (0) =F s , ρ (0) =ρ, ΔF1 (0) =ΔF1 and ΔF2 (0) =ΔF2, repeat steps 5C-2 to 5C-8; Step 5C-10: Output slope safety factor F s , angle ρ and the constant variables ΔF1 and ΔF2 of the vertical shear force increment between strips, and then the force N on the strip ij is given i,j 、S i,j 、dE i,j 、dX i,j 、dL i,j and dV i,j .

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