A method for constructing a segmented general potential slip surface model

Abstract

The present application relates to a kind of construction method of segmented general potential sliding surface model, belong to traffic side slope engineering field.The present application is by considering the difference of stress state of each part of side slope and leading to the difference of shear failure mechanism and shear failure characteristics of different parts of rock-soil mass, using three segmented general potential sliding surface to simulate the complex failure mode of side slope, and make the segmented general potential sliding surface constructed contain the possible type of existing sliding surface, can ensure the rationality of sliding surface shape under the requirement of concave characteristics of sliding surface curve and the consistency of movement direction at the junction point of adjacent segment sliding surface, and can be controlled by several easily determined parameters to control the shape of segmented general potential sliding surface, and be generated in discrete mode, to realize the general consideration of multiple advantages such as universality, precision and ease of use of potential sliding surface determination in actual side slope engineering under complex conditions, further improve the accuracy of side slope stability evaluation.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of traffic slope engineering, and particularly relates to a construction method of a segmented general potential sliding surface model. BACKGROUND

[0002] In the development of slope stability evaluation analysis, selecting a proper sliding surface model is the basis for improving the evaluation accuracy and reliability. A reasonable sliding surface model can more truly reflect the actual failure state of the slope in the instability process, and can provide a solid scientific basis for landslide disaster warning, and can provide sufficient technical support for effective design and safe construction of slope engineering.

[0003] In two-dimensional slope stability analysis, existing sliding surface models are mainly divided into simple specific sliding surface models and complex random sliding surface models. For simple specific sliding surface models, the sliding surface is simplified as a certain type of regular curve, such as a straight line, a circular arc, a logarithmic spiral curve, etc. In simple specific sliding surface models, a straight line sliding surface is simulated by giving two boundary points to simulate the translational sliding failure of the slope, and a circular arc and a logarithmic spiral curve sliding surface is simulated by giving a center (or a polar coordinate origin) and controlling a radius (or a polar radius) to simulate the rotational sliding failure of the slope. These types of sliding surface models have few calculation requirements, simple methods and are easy to construct, and thus are widely used in current slope engineering stability analysis. For complex random sliding surface models, a given random parameter is usually used, and then a sliding surface discrete point is obtained according to a certain polynomial function or parameter equation, and then the sliding surface discrete points are sequentially connected in a certain connection manner (such as a polyline or a spline curve) to form a sliding surface. In theory, complex random sliding surface models can obtain more accurate most dangerous sliding surface shapes, and can be better applied to various complex slope stability analysis.

[0004] The above methods have advantages, but also have certain limitations. For example, simple specific sliding surface models limit the sliding surface shape, and on the same sliding surface, it is difficult to reflect the differences in stress states of various parts of the slope, resulting in different sliding trends in different parts, so that the sliding surface model may not accurately and effectively capture the most dangerous sliding surface for slopes with complex geological conditions. Complex random sliding surface models need to set more parameters to simulate arbitrary sliding surface shapes. At the same time, the sliding surface obtained by random means cannot guarantee the rationality of its shape, and often needs to introduce a large number of constraint conditions, making the construction process more complicated and not conducive to efficient slope stability analysis.

[0005] With the increasing complexity and refinement of slope engineering design and construction technology, the demand for accurately identifying the most dangerous sliding surface of the slope in a simple and easy way and then scientifically assessing the stability of the slope becomes increasingly urgent. However, the current commonly used sliding surface model often fails to meet such requirements, therefore, constructing a more simple and accurate sliding surface model is an important issue to be solved in the field of slope engineering.

[0006] Chinese invention patent CN202210520129.X discloses a non-circular arc slope sliding surface search method based on improved wolf pack algorithm, including the following steps: establishing a numerical model, extracting the basic information of the numerical model; classifying the unit cells according to the unit material number, and drawing the slope image in layers; setting the area of the slope surface within a certain range of the slope top and slope foot in the slope image as the possible activity range of the alpha wolf and the beta wolf; the alpha wolf, the beta wolf and the explorer wolf walk with a fixed step length, together forming a wolf pack combination, and determining the stress information of each point on the path; calculating the sliding force and the anti-sliding force between the adjacent two wolves according to the stress information, defining the fitness of the wolf pack as the reciprocal of the safety factor, and taking the spline curve with the largest fitness value as the optimized spline curve. The non-circular arc slope sliding surface search method based on improved wolf pack algorithm provided by the invention can make up for the shortcomings of the circular arc sliding surface assumption not matching the actual situation, quickly and accurately determine the position of the slope sliding surface and calculate the safety factor. However, the invention patent equally divides the slope sliding surface into three segments, which to some extent excludes part of the reasonable sliding surface. At the same time, the vertical position of the sliding surface equal division control point is not reasonably limited, allowing it to take any value between the given minimum point and maximum point, which leads to the generation of a large number of unreasonable or invalid sliding surfaces. In addition, the sliding surface segment is assumed to be a cubic spline curve, and the sliding surface segment should remain smooth and continuous at the equal division control point. Although this approach makes the sliding surface type more diversified, it also makes the calculation more complex, and the cubic spline curve cannot guarantee that the sliding surface is a concave curve.

[0007] Therefore, there is a need in the art for a new method for constructing a segmented general potential sliding surface model. SUMMARY

[0008] To this end, the application designs a construction method of a segmented general potential sliding surface model. The method considers that the stress state of each part of the slope is different, which leads to different shear failure mechanisms and shear failure characteristics of the rock-soil mass at different parts, adopts a three-segment general potential sliding surface to simulate the complex failure mode of the slope, further applies a polar coordinate function to describe the shape of the general potential sliding surface, then takes the polar coordinate origin as the rotation center point of the slope sliding body, and establishes the polar coordinate equation of the general potential sliding surface by using the mutual relationship between the shear failure direction and the movement direction of the rock-soil mass at the sliding surface. The segmented general potential sliding surface constructed not only contains the possible types of the existing sliding surface, but also can ensure the rationality of the sliding surface shape under the requirements of the concave characteristics of the sliding surface curve and the consistency of the movement direction at the connection points of adjacent segments of the sliding surface. In addition, the shape of the segmented general potential sliding surface can be controlled by several easily determined parameters, and is generated in a discrete manner, which is beneficial to the automatic acquisition of the segmented general potential sliding surface under complex conditions, and is also convenient for the intelligent search of the most dangerous sliding surface combined with the limit equilibrium method of slope stability. The application has the characteristics of wide application range, good universality, high precision, simple implementation, strong operability and the like, can reliably improve the accuracy of slope stability evaluation, and provides strong support for scientifically carrying out slope instability and collapse warning.

[0009] Therefore, the application provides a construction method of a segmented general potential sliding surface model, which adopts a three-segment general potential sliding surface to simulate the complex failure mode of the slope, and controls the shape of the segmented general potential sliding surface by several easily determined parameters and generates the segmented general potential sliding surface in a discrete manner. The method comprises the following steps:

[0010] S1, the x-axis coordinates of the given points Z3 and Z0, the sliding surface connection point position calculation variables κ1-κ4, and the sliding surface shape control calculation variables λ1 i and λ2 i (1≤i≤3) and the starting polar radius r 0_1 of the first sliding surface segment; wherein r is the polar radius of the sliding surface.

[0011] Where Z3 is the termination point of the potential third sliding surface segment, and Z0 is the starting point of the potential first sliding surface segment; Z1 is the termination point of the potential first sliding surface segment, and Z2 is the termination point of the potential second sliding surface segment; κ1 is a variable for calculating the horizontal position of the sliding surface connection point, with a value range of [0, 1]. When κ1 = 0, point Z1 coincides with point Z3, and when κ1 = 1, point Z1 coincides with point Z0; κ2 is a variable for calculating the horizontal position of the sliding surface connection point, with a value range of [0, 1]. When κ2 = 0, point Z2 coincides with point Z3, and when κ2 = 1, point Z2 coincides with point Z3. When κ = 1, point Z2 coincides with point Z1; κ3 is the variable for calculating the vertical position of the sliding surface connection point, and its value range is (0, 1). When κ3 = 0, the horizontal inclination angle of the line connecting points Z2 and Z3 is -45°, with the counterclockwise direction being positive. When κ3 = 1, point Z2 is located on the line connecting points Z3 and Z0; κ4 is the variable for calculating the vertical position of the sliding surface connection point, and its value range is (0, 1). When κ4 = 0, point Z1 is on the extension line connecting points Z3 and Z2. When κ4 = 1, point Z1 is located on the line connecting points Z3 and Z0.

[0012] S2. Apply the slope equation to calculate the y-axis coordinates of points Z3 and Z0;

[0013] S3. Using equations (13) to (16), calculate the x and y coordinates of points Z1 and Z2.

[0014]

[0015] In the formula, and Let x and y be the x and y coordinates of point Z1, respectively. and Let x and y be the x and y coordinates of point Z2, respectively. and Let Z0 and Z3 be the x-coordinates of points Z0 and Z3, respectively.

[0016] S4. Solve for the initial polar angle θ of the first sliding surface segment. 0_1 The termination polar angle θ of the first sliding segment 1_1 and the terminal polar diameter r of the first sliding section 1_1 Where θ is the polar angle of the sliding surface;

[0017] S5. Solve for the initial polar radius r of the second or third sliding surface segment. 0_i The terminal radius r of the second or third sliding section 1_i The termination polar angle θ of the second or third sliding segment 1_i (2≤i≤3);

[0018] S6. Using equations (9) and (10), calculate the polar coordinate origin O of the sliding surface segment i. i The coordinates of i in the xy-axis coordinate system, where 1≤i≤3;

[0019]

[0020] wherein, and are the x and y axis coordinates of the polar coordinate origin O i , 1≤i≤3, and i=1, 2 and 3 correspond to point O1, point O2 and point O3 respectively;

[0021] S7, discretize the sliding surface segment i into n i (1≤i≤3) equal parts using the polar angle equal increment mode, and apply formula (5) to determine whether the calculation variables λ 1_i and λ 2_i can guarantee the concave feature of the curve of each sliding surface segment in the segmented general potential sliding surface; if not, end the calculation and need to re-define the calculation variables λ i and λ i ; if yes, use formula (11) and formula (12) to calculate the x and y axis coordinates x j_i and y j_i (0≤j≤n i and 1≤i≤3) of the jth point on the sliding surface segment i;

[0022]

[0023] In formula (5), dθ j_i is the sliding surface polar angle increment of the sliding surface segment i at the sliding surface polar angle θ j_i ;

[0024]

[0025] In formula (11) and formula (12), when j=0, x j_i and y j_i are the x and y axis coordinates of point Z i-1 , i.e. and When j=n i , x j_i and y j_i are the x and y axis coordinates of point Z i , i.e. and

[0026] S8, output the discrete calculation result of the segmented general potential sliding surface.

[0027] In a specific embodiment, the following method is used to solve the starting polar angle θ 0_1 of the first sliding surface segment, the ending polar angle θ 1_1 of the first sliding surface segment and the ending polar radius r 1_1 of the first sliding surface segment in step S4.;

[0028] S4-1: Let θ 0_1 and θ 1_1 Calculate the variables for the iterative loop and take θ. 0_1 and θ 1_1 The initial values ​​are θ 0_1 (0) and θ 1_1 (0) And in the first calculation, θ 0_1 (0) =0 and θ 1_1 (0) =0;

[0029] S4-2: Substitute θ 0_1 (0) and θ 1_1 (0) Used to solve r 1_1 ,in,

[0030] In the formula, η is the internal friction angle of the rock and soil at the slip surface. i Let i be the angle of inclination of the line connecting the starting point to the ending point of the sliding surface segment in the horizontal direction, and i = 1, 2 and 3 correspond to the first sliding surface segment, the second sliding surface segment and the third sliding surface segment, respectively;

[0031] S4-3: Using the coordinates of points Z0 and Z1, calculate the length of the line connecting points Z0 and Z1. in,

[0032]

[0033] S4-4: Using equation (6), calculate the new θ 0_1 ;

[0034]

[0035] In the formula, when i is 1, r 1_i This is the terminal radius of the first sliding section;

[0036] S4-5: Using equation (8), calculate the new θ 1_1 ;

[0037]

[0038] Where i is 1, θ 1_i θ is the terminal polar angle of the first sliding segment. 0_i This is the initial polar angle of the first sliding segment;

[0039] S4-6: If |θ 0_1 –θ0_1 (0) |≤ε1and |θ 1_1 -θ 1_1 (0) |≤ε1, where ε1=0.001°, then the calculated θ 0_1 , θ 1_1 and r 1_1 are the final results; otherwise, let θ 0_1 (0) = θ 0_1 and θ 1_1 (0) = θ 1_1 , and repeat steps S4-2 to S4-5.

[0040] In one specific embodiment, the following method is used to solve the initial polar radius r 0_i , the terminal polar radius r 1_i and the terminal polar angle θ 1_i of the second or third sliding surface segment in step S5, where 2≤i≤3; and where the piecewise universal potential sliding surface is used to keep the motion direction consistency at the junction point, i.e. θ 0_2 = θ 1_1 and θ 0_3 = θ 1_2 ;

[0041] where θ 1_1 , θ 1_2 and θ 1_3 are the terminal polar angles of the first, second and third sliding surface segments, r 1_1 , r 1_2 and r 1_3 are the terminal polar radii of the first, second and third sliding surface segments, θ 0_1 , θ 0_2 and θ 0_3 are the initial polar angles of the first, second and third sliding surface segments, r 0_1 , r 0_2 and r 0_3 are the initial polar radii of the first, second and third sliding surface segments.

[0042] S5-1, let r 0_i and r 1_i be the iteration loop calculation variables, and take the initial values of r 0_i and r 1_i as r 0_i (0) and r 1_i (0) , and when calculating for the first time, r 0_i(0) = r 1_i-1 and r 1_i (0) = r 1_i–1 where r 1_i–1 is the polar radius of the previous slide surface segment;

[0043] S5-2, using the coordinates of point Z i-1 and point Z i , calculate the length of the line connecting point Z i–1 and point Z i where,

[0044]

[0045] S5-3, substitute r 0_i (0) and r 1_i (0) , calculate θ 1_i using equation (8);

[0046] S5-4, substitute θ 0_i and θ 1_i to solve for r 1_i ,

[0047] where

[0048] S5-5, using equation (7), calculate the new r 0_i ;

[0049]

[0050] S5-6, if |r 0_i -r 0_i (0) |≤ε2 and |r 1_i -r 1_i (0) |≤ε2, where ε2=0.001m, then the calculated r 0_i , r 1_i and θ 1_i are the final results; otherwise, let r 0_i (0) = r 0_i and r 1_i (0) = r 1_i , and repeat steps S5-3 to S5-5.

[0051] ​Compared with the prior art invention patent CN202210520129.X, the present application adopts a non-equal division type to segment the sliding surface, and the sliding surface segment is a composite curve composed of general logarithmic spiral and straight line, which contains the straight line type and circular arc type sliding surface commonly used in existing theories and confirmed by actual engineering; at the same time, the present application applies the consistency of the motion direction of adjacent sliding surface segments at the control point of the sliding surface segment to simply realize the smooth continuity at the control point of the sliding surface segment; in addition, the present application limits the shape parameters of the sliding surface and the vertical position of the control point of the sliding surface segment under the concave feature of the sliding surface, so that the method has multiple advantages such as universality, precision, ease of use and the like.

[0052] In summary, the present application provides a segmented universal potential sliding surface model construction method with wide application range, good universality, high precision, simple implementation and strong operability, which effectively improves the ease and accuracy of searching the most dangerous sliding surface in the process of slope stability evaluation under complex conditions, and can widely cover the common sliding surface forms in engineering practice, and can reflect the differences in stress state of each part of the slope, resulting in different shear failure mechanisms and shear failure characteristics of the rock-soil mass at different parts, so that it has strong adaptive ability and intelligent characteristics. At the same time, the present application can reliably improve the accuracy of slope stability evaluation, and provide strong support for scientific development of slope instability and collapse warning. BRIEF DESCRIPTION OF DRAWINGS

[0053] Figure 1 It is a two-dimensional slope segmented universal potential sliding surface diagram of the present application.

[0054] Figure 2 It is a two-dimensional slope segmented universal potential sliding surface model diagram under the polar coordinate system of the present application.

[0055] Figure 3 It is a two-dimensional slope segmented universal potential sliding surface shape parameter diagram of the present application.

[0056] Figure 4 It is a two-dimensional slope segmented universal potential sliding surface polar coordinate equation and each sliding surface segment junction point feature diagram of the present application.

[0057] Figure 5 It is a two-dimensional slope segmented universal potential sliding surface space geometric relationship diagram of the present application.

[0058] Figure 6 It is a two-dimensional slope segmented universal potential sliding surface parameter iterative loop calculation step 1 flowchart of the present application.

[0059] Figure 7 It is a two-dimensional slope segmented universal potential sliding surface parameter iterative loop calculation step 2 flowchart of the present application.

[0060] Figure 8 This is a schematic diagram of the discrete generation of a two-dimensional slope segmented universal potential slip surface according to the present invention.

[0061] Figure 9 This is a schematic diagram illustrating the range of horizontal and vertical position variations of the connection point of the two-dimensional slope segmented universal potential slip surface according to the present invention.

[0062] Figure 10 This is a flowchart illustrating the steps involved in constructing a two-dimensional slope segmented universal potential slip surface model according to the present invention.

[0063] In the figure: 1. Two-dimensional slope, 2. Segmented general potential sliding surface, 3. Upper potential sliding surface, 4. Middle potential sliding surface, 5. Lower potential sliding surface, 6. Polar diameter of sliding surface, 7. Polar angle of sliding surface, 8. Polar coordinate origin of sliding surface segment, 9. First sliding surface segment, 10. Second sliding surface segment, 11. Third sliding surface segment, 12. Starting point of sliding surface segment, 13. Ending point of sliding surface segment, 14. Direction of rock and soil movement at sliding surface, 15. Tangential direction of sliding surface, 16. Shear dilatation angle, 17. Connection point, 18. Discrete point. Detailed Implementation

[0064] The specific model construction method and implementation process of this invention are as follows:

[0065] (1) As Figure 1 As shown, in a two-dimensional slope, slope instability is caused by the slope's soil and rock reaching a shear failure state and forming a continuous sliding surface. Generally, the stress state varies in different parts of the slope, leading to differences in the shear failure mechanism and shear failure characteristics of the soil and rock at different locations. Consequently, the slope exhibits different sliding trends and forms different sliding surface morphologies in different areas. For example, the upper part of the slope typically experiences tensile-shear failure, where the soil and rock failure manifests as tensile cracking and may show obvious fissures or faulting, resulting in a steeply dipping sliding surface. The middle and lower parts of the slope are often subjected to compression-shear failure, with the shear exit at the lower part being the most intense. The slip surface formed by the failure of the rock and soil mass at this point is often relatively gentle and approximately curved. It is precisely because there is a significant difference in slip surface morphology between the steep slip surface at the upper part of the slope and the gentle slip surface at the shear exit of the slope that a transition section is needed between the two to connect them, thereby ensuring the continuity of the slope slip surface stress and slip surface morphology changes. For this reason, a segmented approach is adopted to construct a general potential slip surface model. The segmented general potential slip surface includes three segments: the upper segment, the middle segment, and the lower segment potential slip surface.

[0066] (2) Figure 2As shown, slope instability is mainly caused by rotational or translational motion. When slope failure is rotational sliding, the potential sliding surface of a two-dimensional slope can mostly be represented by a circular arc or a logarithmic spiral curve. When slope failure is translational sliding, the potential sliding surface of a two-dimensional slope can be represented by a straight line. However, the sliding surface formed by either rotational or translational motion can be described using polar coordinate functions. We select any point as the origin of polar coordinates, and let r be the polar radius of the sliding surface and θ be the polar angle of the sliding surface. The polar radius of the sliding surface is the distance from a point on the sliding surface to the origin of polar coordinates, and the polar angle of the sliding surface is the horizontal inclination angle of the line connecting a point on the sliding surface and the origin of polar coordinates, with clockwise values ​​being positive. For segmented general potential sliding surfaces, since the shape characteristics and variation patterns of each segment of the sliding surface are different, the origin of polar coordinates and the polar coordinate function of each segment will also differ. Therefore, we let point O be the origin of polar coordinates for each segment of the sliding surface. i (1≤i≤3, and i=1, 2 and 3 correspond to the polar coordinate origins O1, O2 and O3 of the potential sliding surfaces of the upper, middle and lower segments, respectively), and let the polar coordinate function of the sliding surface segment be r=f i (θ)(1≤i≤3, and i=1,2 and3 correspond to the polar coordinate functions f1(θ), f2(θ) and f3(θ) of the potential sliding surfaces of the upper, middle and lower segments, respectively).

[0067] (3) Figure 3 As shown, for ease of description, in the segmented universal potential sliding surface, let the upper, middle, and lower segments of the potential sliding surface correspond to the first sliding surface segment (i.e., sliding surface segment 1), the second sliding surface segment (i.e., sliding surface segment 2), and the third sliding surface segment (i.e., sliding surface segment 3), respectively. For the first sliding surface segment, the starting point of the sliding surface segment is Z0, and the ending point of the sliding surface segment is Z1. O1Z0 is the starting polar radius r of the sliding surface segment. 0_1 O1Z1 is the terminal radius r of the sliding segment. 1_1 The horizontal inclination angle of O1Z0 is the initial polar angle θ of the sliding surface segment. 0_1 The horizontal inclination angle of O1Z1 is the terminal polar angle θ of the sliding segment. 1_1 For the second sliding segment, the starting point of the sliding segment is Z1, and the ending point of the sliding segment is Z2. O2Z1 is the initial polar radius r of the sliding segment. 0_2 O2Z2 is the terminal radius r of the sliding segment. 1_2 The horizontal inclination angle of O2Z1 is the initial polar angle θ of the sliding segment. 0_2 The horizontal inclination angle of O2Z2 is the terminal polar angle θ of the sliding segment. 1_2 For the third sliding segment, the starting point of the sliding segment is Z2, and the ending point of the sliding segment is Z3. O3Z2 is the initial polar radius r of the sliding segment. 0_3 O3Z3 is the terminal radius r of the sliding segment. 1_3 The horizontal inclination angle of O3Z2 is the initial polar angle θ of the sliding surface segment. 0_3The horizontal inclination angle of O3Z3 is the terminal polar angle θ of the sliding segment. 1_3 Among them, points Z1 and Z2 are the connection points of adjacent sliding surface segments in the segmented universal potential sliding surface.

[0068] (4) Figure 4 As shown, when using polar coordinate functions to describe the potential sliding surface shape, the origin of the polar coordinates can be set as the rotation center of the slope sliding body. Then, the perpendicular direction of the polar radius r of the sliding surface is the direction of motion of the soil and rock mass at the sliding surface, and the tangent direction of the sliding surface is the direction of shear failure of the soil and rock mass at the sliding surface. According to the relevant and unrelated flow rules, there is a dilatation angle ξ between the direction of motion of the soil and rock mass at the sliding surface and the direction of shear failure. Furthermore, the dilatation angle ξ is related to the internal friction angle of the soil and rock mass at the sliding surface. There is a certain correlation. To facilitate calculation and analysis, let Where k is a dimensionless control parameter for the shape of the potential slip surface curve. Based on this, the general potential slip surface equation in polar coordinates is:

[0069]

[0070] In the formula, θ0 is the initial polar angle of the sliding segment; r0 is the initial polar diameter of the sliding segment.

[0071] In equation (1), when the constructed potential sliding surface is a straight line, based on the spatial geometric relationship of the sliding surface, we should have ξ=θ+η–π / 2. Where η is the horizontal inclination angle of the line connecting the starting point to the ending point of the sliding surface segment, and η is positive in the counterclockwise direction; when the constructed potential sliding surface is a circular arc, based on the spatial geometry of the sliding surface, ξ = 0, and k = 0; when the constructed potential sliding surface is a general logarithmic spiral curve, based on the spatial geometry of the sliding surface, k should be an arbitrary variable and k > 0. Therefore, based on the above analysis and considering the universality of the sliding surface, the formula for calculating the dimensionless control parameter k of the potential sliding surface curve shape is:

[0072]

[0073] In the formula, λ1 and λ2 are the variables for controlling the shape of the sliding surface, and λ1≥0 and λ2≥0.

[0074] Furthermore, to ensure the concave characteristic of the sliding surface curve, the tangential horizontal inclination angle of the sliding surface should decrease as the polar angle of the sliding surface increases. Therefore, based on the spatial geometric relationship between the incremental polar angle dθ and the incremental dilatation angle dξ, when the incremental dilatation angle is less than the corresponding incremental polar angle, the tangential horizontal inclination angle of the sliding surface will decrease as the polar angle increases, thus ensuring the concave characteristic of the sliding surface curve. In this case, the calculated variables λ1 and λ2 in parameter k should satisfy the following relationship:

[0075]

[0076] Based on this, using equations (1) and (2), the polar coordinate function of the sliding surface segment i (1≤i≤3, and i=1, 2 and 3 correspond to the upper, middle and lower sliding surfaces respectively) in the segmented general potential sliding surface model can be obtained as follows:

[0077]

[0078] In the formula, r j_i Let P be any point on the sliding surface segment i. i The corresponding sliding radius, θ j_i Let P be any point on the sliding surface segment i. i The corresponding polar angle of the sliding surface, when θ j_i =θ 0_i At that time, r j_i =r 0_i When θ j_i =θ 1_i At that time, r j_i =r 1_i ;λ 1_i and λ 2_i Let λ be the variable controlling the shape of the sliding surface segment i, and let λ be the variable used in the calculation. 1_i ≥0 and λ 2_i ≥0; η i Z is the starting point of the sliding surface segment i. i–1 The termination point Z of the sliding surface segment i i The angle of inclination of the line in the horizontal direction; Let P be any point on the sliding surface segment i. i The internal friction angle of the soil and rock, if the linear strength criterion is adopted, then If a constant value is used, then... The function related to the normal stress on the sliding surface at this point needs to be determined using an iterative solution strategy.

[0079] Meanwhile, to ensure the continuity of the sliding body's motion along the segmented universal potential sliding surface—that is, the segmented universal potential sliding surfaces not only connect to each other but also maintain the same direction of motion at the connection points (i.e., points Z1 and Z2)—θ must be satisfied. 0_2 =θ 1_1 and θ 0_3 =θ 1_2 Furthermore, in order to ensure the concave curve characteristics of each sliding surface segment in the segmented general potential sliding surface, the calculated variable λ in each sliding surface segment can be obtained using equation (3). 1_i and λ 2_i The following relationship must be satisfied:

[0080]

[0081] In the formula, dθ j_i For the sliding segment i at the polar angle θ j_iThe increase in the polar angle of the slip surface at that location.

[0082] (5) Figure 5 As shown, a rectangular coordinate system with the toe of the slope as the origin is established, and point Z... i The coordinates of (0≤i≤3, and i=0,1,2 and3 correspond to points Z0, Z1, Z2 and Z3 respectively) are Then, using triangle O i Z i-1 Z i (1≤i≤3, and i=1, 2 and 3 correspond to triangles O1Z0Z1, O2Z1Z2 and O3Z2Z3 respectively) spatial geometric relationship, when the initial polar radius r of the sliding surface segment i 0_i Given the initial polar angle θ of the sliding surface segment i, it can be established. 0_i The implicit computation formula is:

[0083]

[0084] In the formula, r 1_i Let be the terminal polar radius of the sliding segment i, and For point Z i-1 With point Z i The length of the connecting line, and

[0085] At the same time, using triangle O i Z i–1 Z i Spatial geometric relationship, when the initial polar angle θ of the sliding surface segment i 0_i Given the initial polar radius r of the sliding segment i, it is possible to establish the initial polar radius r of the sliding segment i. 0_i The formula for calculation is:

[0086]

[0087] Furthermore, using triangle O i Z i–1 Z i Based on the spatial geometric relationship, the initial polar angle θ of the sliding surface segment i can be established. 0_i and termination polar angle θ 1_i The relationships between them are as follows:

[0088]

[0089] For the first slip segment, it is usually based on the known initial polar radius r of the slip segment. 0_1 The initial polar angle θ of the sliding surface segment is then solved using equations (6) and (8). 0_1 The terminal polar angle θ of the sliding segment 1_1 and the terminal radius r of the sliding section 1_1For the second and third sliding surface segments, since the segmented universal potential sliding surfaces maintain a consistent direction of motion at the connection point, the polar angle of the sliding surface satisfies θ. 0_2 =θ 1_1 and θ 0_3 =θ 1_2 Furthermore, here the initial polar angle θ of the known slip surface segment is typically... 0_2 (Second sliding segment) or the initial polar angle θ of the sliding segment 0_3 (Third sliding section) The initial polar radius r of the sliding section is solved using equations (7) and (8). 0_2 The terminal radius r of the sliding section 1_2 The polar angle θ at the end of the segment of the sliding surface 1_2 (Second sliding segment) or the initial polar diameter r of the sliding segment 0_3 The terminal radius r of the sliding section 1_3 The terminal polar angle θ of the sliding segment 1_3 (Third sliding section)

[0090] To obtain the sliding parameters of sliding segment i (i.e., the initial polar angle θ of the sliding segment) 0_i The terminal polar angle θ of the sliding segment 1_i The initial polar diameter r of the sliding section 1_i The terminal radius r of the sliding section 1_i After that, triangle O can also be used. i Z i–1 Z i Given the spatial geometric relationships (1≤i≤3), calculate the polar coordinate origin O of the sliding surface segment i. i The coordinates in the xy-axis coordinate system are calculated using the following formula:

[0091]

[0092] In the formula, and The origin O of polar coordinates i The x and y coordinates of (1≤i≤3, and i=1, 2 and 3 correspond to points O1, O2 and O3 respectively).

[0093] (6) Figure 6 As shown, in the first sliding segment, when the initial polar radius r of the sliding segment is known... 0_1 When the initial polar angle θ of the sliding surface segment can be solved by combining equations (6) and (8) and using an iterative calculation strategy. 0_1 The terminal polar angle θ of the sliding segment 1_1 and the terminal radius r of the sliding section 1_1 The specific calculation steps (referred to as step 1 of the piecewise general potential sliding surface parameter iterative loop calculation) are as follows: ① Let θ 0_1 and θ 1_1 Calculate the variables for the iterative loop and take θ.0_1 and θ 1_1 The initial values ​​are θ 0_1 (0) and θ 1_1 (0) And in the first calculation, θ 0_1 (0) =0 and θ 1_1 (0) =0; ② Substitute θ 0_1 (0) and θ 1_1 (0) Used to solve r 1_1 ,in, ③ Using the coordinates of points Z0 and Z1, calculate the length of the line connecting points Z0 and Z1. in, ④ Using equation (6), calculate the new θ 0_1 ⑤ Using equation (8), calculate the new θ 1_1 ⑥ If |θ 0_1 –θ 0_1 (0) |≤ε1 and |θ 1_1 –θ 1_1 (0) If |≤ε1 (where ε1 = 0.001° is taken here), then the calculated θ 0_1 θ 1_1 and r 1_1 For the final result, otherwise, let θ 0_1 (0) =θ 0_1 and θ 1_1 (0) =θ 1_1 And repeat steps ② to ⑤.

[0094] (7) Figure 7 As shown, in the second or third sliding surface segment, the segmented universal potential sliding surface is used to maintain the consistency of the motion direction at the connection point, with θ 0_2 =θ 1_1 and θ 0_3 =θ 1_2 Furthermore, after the termination polar angle of the previous sliding segment is solved, the starting polar angle of the next sliding segment can be obtained. Based on this, the starting polar radius r of the sliding segment is solved by combining equations (7) and (8) and using an iterative calculation strategy. 0_i The terminal radius r of the sliding section 1_i The terminal polar angle θ of the sliding segment 1_i (2≤i≤3), the specific calculation steps (referred to as step 2 of the piecewise general potential sliding surface parameter iterative loop calculation) are as follows: ① Let r 0_i and r 1_i Calculate the variable for the iterative loop and take r.0_i and r 1_i 0_i (0) and r 1_i (0) , and the first time calculation, r 0_i (0) = r 1_i-1 and r 1_i (0) = r 1_i–1 , wherein r 1_i–1 is the previous slide surface segment of the slide surface end polar radius; ② using the coordinates of point Z i-1 and point Z i , calculate the length of the line connecting point Z i–1 and point Z i wherein, ③ substitute r 0_i (0) and r 1_i (0) , calculate θ 1_i using equation (8); ④ substitute θ 0_i and θ 1_i for solving r 1_i , wherein ⑤ using equation (7), calculate the new r 0_i ; ⑥ if |r 0_i – r 0_i (0) |≤ε2and |r 1_i – r 1_i (0) |≤ε2 (here, take ε2=0.001 m), then the calculated r 0_i , r 1_i and θ 1_i are the final results, otherwise, let r 0_i (0) = r 0_i and r 1_i (0) = r 1_i , and repeat steps ③-⑤.

[0095] (8) As shown in Figure 8 , in order to facilitate calculation, the polar coordinate system under the segmented type general potential sliding surface function is converted into a rectangular coordinate system under the segmented type general potential sliding surface function, at the same time, a discrete way is used to construct the segmented type general potential sliding surface, here, the polar angle is used. Equidistance mode divides ∠Z i–1 O i Z i into n i (1≤i≤3) equal parts, and then, the sliding surface segment i can also be divided into n i ​​Divide into equal parts, at which point the smooth surface segment i is composed of n i It consists of +1 discrete points. For the j-th point on the i-th sliding surface segment, its corresponding polar angle is θ. j_i Polar angle θ of the sliding surface j_i The increment of the polar angle of the slip surface is dθ j_i And the radius of the sliding surface is r j_i (0≤j≤n i ),and dθ j_i =(θ 1_i –θ 0_i ) / n i and r j_i Using equation (4) for calculation, the x and y coordinates of the j-th point on the sliding surface segment i are obtained as follows:

[0096]

[0097] In equations (11) and (12), when j = 0, x j_i and y j_i For point Z i-1 The x and y coordinates, i.e. and When j = n i At that time, x j_i and y j_i For point Z i The x and y coordinates, i.e. and

[0098] (9) such as Figure 9 As shown, in a segmented general potential sliding surface, the connection points Z1 and Z2 of adjacent sliding surfaces should be located between points Z0 and Z3, and point Z2 should also be located between points Z1 and Z3. Therefore, the relationship between point Z1 and points Z0 and Z3 with respect to the x-axis coordinate can be established as follows:

[0099]

[0100] In the formula, κ1 is the variable for calculating the horizontal position of the sliding surface connection point, and its value range is [0, 1]. When κ1 = 0, point Z1 coincides with point Z3, and when κ1 = 1, point Z1 coincides with point Z0.

[0101] Furthermore, the relationship between point Z2 and points Z0 and Z3 with respect to their x-axis coordinates is established as follows:

[0102]

[0103] In the formula, κ2 is the variable for calculating the horizontal position of the sliding surface connection point, and its value range is [0, 1]. When κ2 = 0, point Z2 coincides with point Z3, and when κ2 = 1, point Z2 coincides with point Z1.

[0104] Further, in order to ensure the rationality of the sliding surface, the potential sliding surface should be a concave curve, which means that the slope of the tangent line of the sliding surface keeps increasing in the xy coordinate system, and any point on the potential sliding surface cannot be beyond the line connecting the lower sliding-out point (i.e. point Z3) and the upper sliding-out point (i.e. point Z0) of the potential sliding surface, and at the same time, the horizontal inclination of the tangent line of the potential sliding surface at the lower sliding-out point (point Z3) cannot be less than -45° (counterclockwise direction is positive). Thus, the relationship between point Z2 and points Z0 and Z3 with respect to the y-axis coordinate can be obtained as follows:

[0105]

[0106] In the formula, κ3 is a calculation variable of the vertical position of the sliding surface junction point, and its value range is (0, 1). When κ3 = 0, the horizontal inclination of the line connecting points Z2 and Z3 is -45° (counterclockwise direction is positive), and when κ3 = 1, point Z2 is located on the line connecting points Z3 and Z0.

[0107] In addition, the relationship between point Z1 and points Z0 and Z3 with respect to the y-axis coordinate can be established as follows:

[0108]

[0109] In the formula, κ4 is a calculation variable of the vertical position of the sliding surface junction point, and its value range is (0, 1). When κ4 = 0, point Z1 is on the extension line of the line connecting points Z3 and Z2, and when κ4 = 1, point Z1 is located on the line connecting points Z3 and Z0.

[0110] Thus, by applying the calculation variables κ1, κ, κ3 and κ4, the coordinates of the junction points of adjacent segments of the segmented general potential sliding surface can be solved from the coordinates of the lower and upper sliding-out points of the potential sliding surface. For the lower and upper sliding-out points (i.e. points Z3 and Z0), they are both on the sliding surface and on the slope surface. Therefore, when the x-axis coordinates of points Z3 and Z0 are given, their corresponding y-axis coordinates can be obtained by using the slope surface equation. Further, for the segmented general potential sliding surface, its shape and position are determined by the x-axis coordinates of points Z3 and Z0, the sliding surface junction point position calculation variables κ1-κ4, and the sliding surface shape control calculation variables λ1_ i and λ2_ i (1≤i≤3) of the sliding surface segment i. At the same time, if the x-axis coordinates of points Z3 and Z0, the sliding surface junction point position calculation variables κ1-κ4, and the sliding surface shape control calculation variables λ1_ i and λ2_ iWhen the setting is within a reasonable range and is arbitrarily valued, a series of potential sliding surfaces can be obtained. In this series of potential sliding surfaces, if the limit equilibrium method of slope stability is used to solve the slope safety factor corresponding to the potential sliding surface, the most dangerous sliding surface corresponding to the minimum slope safety factor can be obtained by searching for the minimum value of the slope safety factor as the optimization objective, which is the most likely sliding surface of slope instability.

[0111] (10) As shown in Figure 10 , the segmented general potential sliding surface model construction and implementation process is as follows: ① Given the x-axis coordinates of points Z3 and Z0, sliding surface connection point position calculation variables κ1-κ4, and sliding surface shape control calculation variables λ1 i and λ2 i (1≤i≤3) and the starting polar radius r 0_1 of the first sliding surface segment; ② Apply the slope surface equation to calculate the y-axis coordinates of points Z3 and Z0; ③ Use equations (13)-(16) to calculate the x and y-axis coordinates of points Z1 and Z2; ④ Use the segmented general potential sliding surface parameter iterative loop calculation step 1 to solve the starting polar angle θ 0_1 , the ending polar angle θ 1_1 , and the ending polar radius r 1_1 of the first sliding surface segment; ⑤ Use the segmented general potential sliding surface parameter iterative loop calculation step 2 to solve the starting polar radius r 0_i , the ending polar radius r 1_i , and the ending polar angle θ 1_i (2≤i≤3) of the second or third sliding surface segment; ⑥ Use equations (9) and (10) to calculate the coordinates of the polar coordinate origin O i (1≤i≤3) in the xy-axis coordinate system; ⑦ Discretize the sliding surface segment i into n i (1≤i≤3) equal parts using the polar angle equal increment mode, and apply equation (5) to determine whether the calculation variables λ 1_i and λ 2_i can guarantee the concave characteristics of the curves of each sliding surface segment in the segmented general potential sliding surface. If it cannot be guaranteed, the calculation is ended and the calculation variables λ1 i and λ2 i need to be redefined. If it can be guaranteed, use equations (11) and (12) to calculate the x and y-axis coordinates x j_i and y j_i (0≤j≤n i and 1≤i≤3) of the jth point on the sliding surface segment i; ⑧ Output the discrete calculation results of the segmented general potential sliding surface.

[0112] The general potential sliding surface has the characteristics that: a polar coordinate function is used to describe the shape of the general potential sliding surface, then the polar coordinate origin is taken as the rotation center point of the slope sliding body, the mutual relationship between the shear failure direction (i.e. the tangent direction of the sliding surface) and the movement direction (i.e. the vertical direction of the polar radius of the sliding surface) of the rock and soil at the sliding surface is used to establish the polar coordinate equation of the general potential sliding surface, the shape of the general potential sliding surface is controlled by two calculation variables, and is related to the internal friction angle of the rock and soil at the sliding surface, the general potential sliding surface can be degraded into a straight line sliding surface when the slope failure is translational sliding, can be degraded into a circular arc or logarithmic spiral curve sliding surface when the slope failure is rotational sliding, and the concave feature of the sliding surface curve is ensured.

[0113] The segmented general potential sliding surface has the characteristics that: the stress state of each part of the slope is different, which leads to the different shear failure mechanism and shear failure characteristics of the rock and soil at different parts, a three-segment general potential sliding surface is used to simulate the complex failure mode of the slope, at the same time, different polar coordinate origins and polar coordinate functions are used to describe the shape and position of the three-segment potential sliding surface, and the segmented general potential sliding surface is required to not only be connected with each other but also keep the movement direction consistency at the connection point of adjacent segments.

[0114] The segmented general potential sliding surface can effectively describe the different shear failure mechanism and shear failure characteristics of the rock and soil at different parts of the slope due to the difference of the stress state of each part of the slope, at the same time, the segmented general potential sliding surface not only contains the possible types of the existing sliding surface, but also makes the constructed sliding surface shape rationalized under the condition of ensuring the concave feature of the sliding surface curve, in addition, the shape of the segmented general potential sliding surface is simply controlled by several easily determined parameters and is generated in a discrete manner, which is beneficial to the automatic acquisition of the segmented general potential sliding surface under complex conditions and is convenient for the intelligent search of the most dangerous sliding surface combined with the limit equilibrium method of slope stability, so that the application has the characteristics of wide application range, good universality, high precision, simple implementation, strong operability and the like, can reliably improve the accuracy of slope stability evaluation, and provides strong support for scientifically carrying out slope instability and collapse warning.

[0115] Embodiment

[0116] A kind of as Figures 1-10The segment type general potential sliding surface model construction method is shown. A mountain slope with steep terrain and complex geological conditions, the preliminary survey results show that the mountain slope is 23.8 m high, the average slope is 50°, and there is a certain possibility of sliding. In order to ensure the safety of the infrastructure on the mountain slope, it is necessary to accurately obtain the most dangerous sliding surface shape and position of the slope in the instability and destruction, and to efficiently and reliably realize the slope stability evaluation, at the same time, to guide the definition of the potential range of the slope collapse and the implementation of the supporting protection work. Therefore, the segment type general potential sliding surface model construction method and implementation process are introduced and applied, and the specific operation steps are as follows:

[0117] (1) According to the "Geotechnical Engineering Investigation Specification" (GB50021-2001), the field survey test is carried out on the mountain slope, the two-dimensional slope surface shape and stratum data in the potential sliding area of the slope are determined, and the corresponding rock-soil body strength parameters are determined, wherein the rock-soil body is considered to obey the linear M-C strength criterion, and the rock-soil body obtained by the survey test is γ = 19 kN / m 3 , c = 25 kPa and the internal friction angle

[0118] (2) The two-dimensional slope segment type general potential sliding surface model in the application is selected, the starting point Z0 of the potential first sliding surface segment and the ending point Z3 of the potential third sliding surface segment are estimated and given, and the range of the horizontal coordinates and is given, wherein when the slope foot point is taken as the origin to establish the xy axis coordinate system, the range is -5 m to 5 m, the range is 20 m to 30 m;

[0119] (3) The range of the sliding surface connection point position calculation variables κ1, κ2, κ3 and κ4 is 0 to 1, the shape control parameters λ 1_i and λ 2_i of the sliding surface segment i (1 ≤ i ≤ 3) are given, the value range of the starting polar radius r 1_0 of the first sliding surface segment is 30 m to 150 m, the sliding surface segment numbers n1, n2 and n3 of the first sliding surface segment, the second sliding surface segment and the third sliding surface segment are respectively taken as 30, 50 and 30, and then a series of potential sliding surfaces of the slope are formed in the given parameter value range, and the slope stability analysis is carried out based on the potential sliding surfaces;

[0120] (4) The limit equilibrium method is combined, and the mathematical optimization algorithm is embedded, so as to realize the slope stability analysis, and in the process, when the horizontal coordinates and The sliding surface connection point position calculation variables κ1, κ2, κ3 and κ4, the shape control parameters λ of the sliding surface segment i (1≤i≤3) 1_i and λ 2_i , and the starting polar radius r of the first sliding surface segment 1_0 After that, a reasonable potential sliding surface can be generated according to the segmented general potential sliding surface model construction step, then, the safety factor of the slope under the corresponding potential sliding surface is solved by using the limit equilibrium method, and then the search for the most dangerous potential sliding surface of the slope is carried out with the minimum value of the safety factor of the slope as the optimization goal with the aid of the mathematical optimization algorithm, thereby completing the slope stability analysis and outputting the minimum slope safety factor and the most dangerous potential sliding surface result, wherein the minimum slope safety factor is 1.04, and the horizontal coordinates of the starting point Z0 of the first sliding surface segment and the ending point Z3 of the third sliding surface segment of the most dangerous potential sliding surface are and The sliding surface connection point position calculation variables are κ1=0.98, κ2=0.56, κ3=0.63 and κ4=0.84, the shape control parameters of the sliding surface segment i (1≤i≤3) are λ 1_1 =0.02, λ 2_1 =0.26, λ 1_2 =0.27, λ 2_2 =0.14, λ 1_3 =0.16 and λ 2_3 =0.71 respectively, and the starting polar radius r 1_0 =36.40m of the first sliding surface segment.

[0121] (5) According to the stability evaluation standard in the Technical Code for Building Slope Engineering (GB 50330-2013), the required safety factor of the slope (first-grade slope) should reach 1.30, and the calculation result shows that the slope stability result is less than the value, so the slope needs to be reinforced and treated, and anchor rods and retaining walls are used for support within the range of the most dangerous sliding surface.

[0122] Therefore, the application provides a construction method of a segmented general potential sliding surface model, which considers the differences in stress states of various parts of the slope, which lead to different shear failure mechanisms and shear failure characteristics of the rock-soil mass at different parts, adopts a three-segment general potential sliding surface to simulate the complex failure shape of the slope, and makes the constructed segmented general potential sliding surface contain possible types of existing sliding surfaces, ensure the rationality of the shape of the sliding surface under the requirement of the concave features of the sliding surface curve and the consistency of the movement direction at the connection points of adjacent segments, and be controlled by several easily determined parameters to control the shape of the segmented general potential sliding surface and be generated in a discrete manner, so as to realize the generalization, accuracy and ease of use of the determination of the potential sliding surface in actual slope engineering under complex conditions, and further improve the accuracy of the slope stability evaluation.

[0123] It is to be understood that the above description is intended to be illustrative and not restrictive. Many other embodiments will be apparent to those of skill in the art upon reading and understanding the above description. The scope of the application should, therefore, be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled.

Claims

1. A method of constructing a piecewise universal potential slip surface model, characterized by, The three-section general potential sliding surface is used to simulate the complex failure mode of the slope, and is simple and convenient for several parameters to control the shape of the section general potential sliding surface and is generated in a discrete manner, and the method comprises the following steps: S1, x-axis coordinates of given points Z3 and Z0, slide surface connection point position calculation variables κ1-κ4, and slide surface shape control calculation variable λ1 of slide surface section i i and λ2 i (1≤i≤3) and the start polar radius r of the first slide surface section 0_1 ; wherein r is the slide surface polar radius Wherein, Z3 is the end point of the third potential sliding surface section, Z0 is the starting point of the first potential sliding surface section; and Z1 is the end point of the first potential sliding surface section, Z2 is the end point of the second potential sliding surface section; κ1 is a horizontal position calculation variable of the sliding surface connection point, and the value range is [0, 1], when κ1 = 0, the point Z1 coincides with the point Z3, when κ1 = 1, the point Z1 coincides with the point Z0; κ2 is a horizontal position calculation variable of the sliding surface connection point, and the value range is [0, 1], when κ2 = 0, the point Z2 coincides with the point Z3, when κ2 = 1, the point Z2 coincides with the point Z1; κ3 is a vertical position calculation variable of the sliding surface connection point, and the value range is (0, 1), when κ3 = 0, the horizontal inclination angle of the line connecting the point Z2 and Z3 is -45°, and the clockwise direction is positive, when κ3 = 1, the point Z2 is located on the line connecting the point Z3 and Z0; κ4 is a vertical position calculation variable of the sliding surface connection point, and the value range is (0, 1), when κ4 = 0, the point Z1 is on the extension line of the line connecting the point Z3 and Z2, when κ4 = 1, the point Z1 is located on the line connecting the point Z3 and Z0; S2, the slope surface equation is applied to calculate the y-axis coordinates of the points Z3 and Z0; S3, the x and y axis coordinates of the points Z1 and Z2 are calculated by using formula (13) to formula (16); wherein and are the x and y axis coordinates of point Z1 respectively, and are the x and y axis coordinates of point Z2 respectively, and are the x axis coordinates of point Z0 and point Z3 respectively; S4, solving the starting polar angle θ of the first sliding surface segment 0_1 , the ending polar angle θ of the first sliding surface segment 1_1 , and the ending polar radius r of the first sliding surface segment 1_1 ; wherein θ is the polar angle of the sliding surface; S5, solving the starting polar radius r of the second sliding surface segment or the third sliding surface segment 0_i , the ending polar radius r of the second sliding surface segment or the third sliding surface segment 1_i , and the ending polar angle θ of the second sliding surface segment or the third sliding surface segment 1_i (2≤i≤3); S6. Using equations (9) and (10), calculate the polar coordinate origin O of the sliding surface segment i. i The coordinates of i in the xy-axis coordinate system, where 1≤i≤3; wherein and are the x and y coordinates of the polar coordinate origin O i respectively, 1≤i≤3, and i = 1, 2 and 3 correspond to point O1, point O2 and point O3 respectively; S7, discretize the sliding surface segment i into n i (1≤i≤3) equal parts, and apply formula (5) to determine the variable λ 1_i and λ 2_i whether the concave feature of the curve of each sliding surface segment in the segmented universal potential sliding surface can be guaranteed; if not, end the calculation and need to re-define the calculation variable λ1 i and λ2 i ; if yes, use formula (11) and formula (12) to calculate the x and y axis coordinates x j_i and y j_i of the jth point on the sliding surface segment i (0≤j≤n i and 1≤i≤3); In formula (5), dθ j_i is the slide polar angle increment of the slide section i at the slide polar angle θ j_i at the slide polar angle θ In formula (11) and formula (12), when j = 0, x j_i and y j_i are the x and y axis coordinates of the point Z i-1 , i.e. and When j = n i , x j_i and y j_i are the x and y axis coordinates of the point Z i , i.e. and S8, the discrete calculation results of the section general potential sliding surface are output.

2. The method of claim 1, wherein, The starting polar angle θ of the first sliding surface section is solved in step S4 by using the following method 0_1 The ending polar angle θ of the first sliding surface section 1_1 And the ending polar radius r of the first sliding surface section 1_1 ; S4-1: Let θ 0_1 and θ 1_1 be iteration loop variables, and take the initial values of θ 0_1 and θ 1_1 to be θ 0_1 (0) and θ 1_1 (0) , and for the first calculation, θ 0_1 (0) = 0 and θ 1_1 (0) = 0; S4-2: Substitute θ 0_1 (0) and θ 1_1 (0) for solving r 1_1 where, wherein is the internal friction angle of the rock-soil mass at the sliding surface, η i is the inclination angle of the line connecting the starting point of the sliding surface segment to the ending point of the sliding surface segment in the horizontal direction, and i = 1, 2, and 3 correspond to the first sliding surface segment, the second sliding surface segment, and the third sliding surface segment, respectively; S4-3: Calculate the length of the line connecting point Z0 and point Z1 using the coordinates of point Z0 and point Z1 wherein, S4-4: Calculate a new θ using Equation (6) 0_1 ; wherein, when i is 1, r 1_i is the terminal polar radius of the first sliding surface section; S4-5: Calculate a new θ using Equation (8) 1_1 ; wherein, when i is 1, θ 1_i is the end polar angle of the first sliding surface section, θ 0_i is the start polar angle of the first sliding surface section; S4-6: If |θ 0_1 - θ 0_1 (0) |≤ ε1 and |θ 1_1 - θ 1_1 (0) |≤ ε1, where ε1=0.001° is taken, then the calculated θ 0_1 , θ 1_1 and r 1_1 are the final results; otherwise, let θ 0_1 (0) = θ 0_1 and θ 1_1 (0) = θ 1_1 , and repeat steps S4-2 to S4-5.

3. The method of claim 2, wherein, The starting polar radius r of the second sliding surface section or the third sliding surface section is solved in step S5 by using the following method 0_i The ending polar radius r of the second sliding surface section or the third sliding surface section 1_i And the ending polar angle θ of the second sliding surface section or the third sliding surface section 1_i Wherein 2≤i≤3; and wherein the consistency of the movement direction is maintained at the junction point by using the piecewise general potential sliding surface 0_2 = θ 1_1 And θ 0_3 = θ 1_2 Solving wherein θ 1_1 , θ 1_2 and θ 1_3 are the terminal polar angles of the first, second and third slide surface sections, respectively, r 1_1 , r 1_2 and r 1_3 are the terminal polar radii of the first, second and third slide surface sections, respectively, θ 0_1 , θ 0_2 and θ 0_3 are the initial polar angles of the first, second and third slide surface sections, respectively, r 0_1 , r 0_2 and r 0_3 are the initial polar radii of the first, second and third slide surface sections, respectively. S5-1, let r 0_i and r 1_i be the initial values of r 0_i and r 1_i respectively, and r 0_i (0) and r 1_i (0) for the first calculation, r 0_i (0) = r 1_i-1 and r 1_i (0) = r 1_i–1 , where r 1_i–1 is the polar radius of the end of the previous slide surface segment. S5-2, point Z i-1 and the coordinates of point Z i S5-2, point Z i–1 and the coordinates of point Z i the length of the line connecting point Z wherein, S5-3, substitute r 0_i (0) and r 1_i (0) , calculate θ 1_i ; S5-4, substitute θ 0_i and θ 1_i for solving r 1_i , wherein S5-5, using equation (7), calculate new r 0_i ; S5-6, if |r 0_i -r 0_i (0) |r 1_i -r 1_i (0) |r 0_i , r 1_i and θ 1_i are the final results; otherwise, let r 0_i (0) = r 0_i and r 1_i (0) = r 1_i and repeat steps S5-3 through S5-5.

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