Construction method of sectional type general potential sliding surface model

Abstract

The invention relates to a construction method of a sectional type universal potential sliding surface model, and belongs to the field of traffic slope engineering. According to the method, different shear failure mechanisms and different shear failure characteristics of rock-soil bodies at different parts due to different stress states of all parts of the slope are considered, a three-section type universal potential sliding surface is adopted to simulate the complex failure form of the slope, and the constructed sectional type universal potential sliding surface comprises the possible type of the existing sliding surface and the possible type of the existing sliding surface; the reasonability of the shape of the sliding surface can be ensured under the condition that the concave characteristic of the curve of the sliding surface and the consistency of the motion directions of the connecting points of the adjacent sections of sliding surfaces are required, and the shape of the sectional type universal potential sliding surface can be controlled by a plurality of easily determined parameters simply and conveniently and is generated in a discrete mode; therefore, multiple advantages of universality, accuracy, usability and the like of potential sliding surface determination in actual slope engineering under complex conditions are taken into consideration, and the accuracy of slope stability evaluation is further improved.

Description

Technical Field

[0001] The invention belongs to the field of traffic slope engineering, and in particular relates to a method for constructing a segmented universal potential sliding surface model. Background Art

[0002] In the slope stability assessment and analysis, the selection of an appropriate sliding surface model is the basis for improving the assessment accuracy and reliability. A reasonable sliding surface model can more realistically reflect the actual damage state of the slope during the instability process, and can provide a solid scientific basis for landslide disaster warning, and can provide sufficient technical support for the effective design and safe construction of slope engineering.

[0003] In the two-dimensional slope stability analysis, the existing sliding surface models are mainly divided into simple specific sliding surface models and complex random sliding surface models. For the simple specific sliding surface model, the sliding surface is mainly simplified into a certain type of regular curve, such as a straight line, an arc, a logarithmic spiral curve, etc. In the simple specific sliding surface model, the straight line sliding surface simulates the slope translation sliding failure by giving two boundary points, and the arc and logarithmic spiral curve sliding surfaces simulate the slope rotation sliding failure by giving a circle center (or polar coordinate origin) and controlling the radius (or polar diameter). These types of sliding surface models require few calculation parameters, are simple in method and easy to construct, and therefore are widely used in the current slope engineering stability analysis. For the complex random sliding surface model, the given random parameters are usually used, and then the sliding surface discrete points are obtained according to a polynomial function or parametric equation. Then, the sliding surface discrete points are connected in sequence in a certain connection method (such as a broken line or a spline curve) to form a sliding surface. In theory, the complex random sliding surface model can obtain a more accurate and most dangerous sliding surface morphology, and can also be better applied to various types of complex slope stability analysis.

[0004] Although the above methods have their own advantages, they also have certain limitations. For example, the simple specific sliding surface model limits the sliding surface morphology. On the same sliding surface, it is difficult to reflect the differences in stress states of different parts of the slope, which leads to different sliding trends in different parts. As a result, the sliding surface model may not be able to accurately and effectively capture the most dangerous sliding surface for slopes with complex geological conditions. The complex random sliding surface model requires setting more parameters to simulate any sliding surface morphology. At the same time, the sliding surface obtained by random means cannot guarantee the rationality of its morphology, and often requires the introduction of a large number of constraints, making its construction process more cumbersome and not conducive to the efficient implementation of slope stability analysis.

[0005] With the increasing complexity and refinement of slope engineering design and construction technology, the need to accurately identify the most dangerous sliding surface of the slope in a simple and easy way and then scientifically evaluate the stability of the slope has become more urgent. However, the sliding surface models commonly used at present often cannot meet such requirements. Therefore, constructing a simpler and more accurate sliding surface model is an important issue that needs to be solved in the field of slope engineering.

[0006] Chinese invention patent CN202210520129.X discloses a non-circular slope sliding surface search method based on an improved wolf pack algorithm, including the following steps: establish a numerical model and extract the basic information of the numerical model; classify the cells according to the unit material number and draw the slope image in layers; set the slope surface area within a certain range of the top and foot of the slope image as the possible activity range of the head wolf and the female wolf; the head wolf, the female wolf, and the scout wolf walk with a fixed step length to form a wolf pack combination and determine the stress information of each point on the path; calculate the sliding force and anti-sliding force between two adjacent wolves based on the stress information, define the fitness of the wolf pack as the inverse of the safety factor, and use the spline curve with the largest fitness value as the optimal spline curve. The invention provides a non-circular slope sliding surface search method based on an improved wolf pack algorithm, which can make up for the shortcomings of the circular sliding surface assumption that does not match the actual situation, quickly and accurately determine the position of the slope sliding surface and calculate the safety factor. However, this invention patent divides the slope sliding surface into three equal sections. This practice excludes some reasonable sliding surfaces to a certain extent. At the same time, the vertical position of the sliding surface equal division control point is not reasonably limited, and it is allowed to take any value between the given lowest point and the highest point, which leads to the generation of a large number of unreasonable or invalid sliding surfaces. In addition, its sliding surface segment is assumed to be a cubic spline curve, and it is required that the sliding surface segment should remain smooth and continuous at the equal division control points. Although this approach will make the sliding surface type more diverse, the calculation method is also more complicated, and the cubic spline curve cannot guarantee that the sliding surface has the characteristics of a concave curve.

[0007] Therefore, a new method for constructing a segmented universal potential sliding surface model is needed in the art. Summary of the invention

[0008] To this end, the present invention designs a method for constructing a segmented universal potential sliding surface model. The method considers the differences in stress states of various parts of the slope, resulting in different shear failure mechanisms and shear failure characteristics of the rock and soil at different parts. A three-segment universal potential sliding surface is used to simulate the complex failure morphology of the slope. Further, a polar coordinate function is used to describe the shape of the universal potential sliding surface. Then, the polar coordinate origin is used as the rotation center of the slope sliding body, and the relationship between the shear failure direction and the movement direction of the rock and soil at the sliding surface is used to establish the polar coordinate equation of the universal potential sliding surface. The constructed segmented universal potential sliding surface not only includes the possible types of existing sliding surfaces, but also ensures 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 the adjacent segments of the sliding surface. In addition, the shape of the segmented universal potential sliding surface can be simply controlled by several easy-to-determine parameters, and is generated in a discrete manner, which is conducive to the automatic acquisition of the segmented universal potential sliding surface under complex conditions, and is also convenient for combining with the limit equilibrium method of slope stability to carry out intelligent search for the most dangerous sliding surface. The present invention has the characteristics of wide application range, good versatility, high precision, simple implementation, strong operability, etc. It can reliably improve the accuracy of slope stability assessment and provide strong support for the scientific implementation of slope instability and landslide early warning.

[0009] Therefore, the present invention provides a method for constructing a segmented universal potential sliding surface model, which uses a three-segment universal potential sliding surface to simulate the complex failure form of the slope, and is simple to control the shape of the segmented universal potential sliding surface with a few easily determined parameters and generate it 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 variable λ1_ i and λ2_ i (1≤i≤3) and the starting pole diameter r of the first sliding surface segment 0_1 ; Where r is the sliding surface extreme diameter;

[0011] Among them, Z3 is the end point of the potential third sliding surface segment, and Z0 is the starting point of the potential first sliding surface segment; Z1 is the end point of the potential first sliding surface segment, and Z2 is the end point of the potential second sliding surface segment; κ1 is the sliding surface connection point horizontal position calculation variable, 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; κ2 is the sliding surface connection point horizontal position calculation variable, and its value range is [0, 1]. When κ2=0, point Z2 coincides with point Z3, and when κ2=1, point Z1 coincides with point Z0. =1, point Z2 coincides with point Z1; κ3 is the calculation variable of 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°, and the counterclockwise direction is positive. When κ3=1, point Z2 is located on the line connecting points Z3 and Z0; κ4 is the calculation variable of 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 of the line connecting points Z3 and Z2. When κ4=1, point Z1 is located on the line connecting points Z3 and Z0.

[0012] S2, using the slope surface equation, calculate the y-axis coordinates of points Z3 and Z0;

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

[0014]

[0015] In the formula, and are the x and y axis coordinates of point Z1, and are the x and y axis coordinates of point Z2, and are the x-axis coordinates of point Z0 and point Z3 respectively;

[0016] S4. Solve for the starting polar angle θ of the first sliding surface segment 0_1 , the terminal polar angle θ of the first sliding surface segment 1_1 and the terminal diameter r of the first sliding surface segment 1_1 ; where θ is the polar angle of the sliding surface;

[0017] S5. Calculate the starting diameter r of the second or third sliding surface segment 0_i , the terminal diameter r of the second sliding surface segment or the third sliding surface segment 1_i The termination polar angle θ of the second sliding surface segment or the third sliding surface 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 Coordinates in the xy axis coordinate system, where 1≤i≤3;

[0019]

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

[0021] S7, using the polar angle equal increment mode to discretize the sliding surface segment i into n i (1≤i≤3) and use formula (5) to determine the calculated variable λ 1_i and λ 2_i Can the concave characteristics of each sliding surface segment curve in the piecewise universal potential sliding surface be guaranteed? If not, the calculation is terminated and the calculation variable λ1_ needs to be re-given. i and λ2_ i ; If it can be guaranteed, then use equations (11) and (12) to calculate the x and y axis coordinates x of the jth point on the sliding surface segment i j_i and j_i (0≤j≤n i and 1≤i≤3);

[0022]

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

[0024]

[0025] In equations (11) and (12), when j = 0, x j_i and j_i Point Z i-1 The x and y axis coordinates, that is and When j = n i When x j_i and j_i Point Z i The x and y axis coordinates, that is and

[0026] S8. Output the discrete calculation results of the piecewise general potential sliding surface.

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

[0028] S4-1: Let θ 0_1 and θ 1_1 Calculate variables for the iteration loop and take θ 0_1 and θ 1_1 The initial values ​​are θ 0_1 (0) and θ 1_1 (0) , and when the first calculation is performed, θ 0_1 (0) =0 and θ 1_1 (0) =0;

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

[0030] In the formula, is the friction angle of rock and soil 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;

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

[0032]

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

[0034]

[0035] In the formula, when the value of i is 1, r 1_i is the terminal pole diameter of the first sliding surface segment;

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

[0037]

[0038] When the value of i is 1, θ 1_i is the terminal polar angle of the first sliding surface segment, θ 0_i is the starting polar angle of the first sliding surface segment;

[0039] S4-6: If |θ 0_1 –θ0_1 (0) |≤ε1 and |θ 1_1 –θ 1_1 (0) |≤ε1, here ε1=0.001°, then the calculated θ 0_1 ,θ 1_1 and r 1_1 is the final result; 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 a specific embodiment, the following method is used in step S5 to solve the starting diameter r of the second sliding surface segment or the third sliding surface segment: 0_i , the terminal diameter r of the second sliding surface segment or the third sliding surface segment 1_i The termination polar angle θ of the second sliding surface segment or the third sliding surface segment 1_i , where 2≤i≤3; and where the segmented universal potential sliding surface is used to maintain the consistency of the motion direction at the connection point under θ 0_2 =θ 1_1 and θ 0_3 =θ 1_2 Solution;

[0041] Among them, θ 1_1 ,θ 1_2 and θ 1_3 are the terminal polar angles of the first, second and third sliding surface segments, respectively, r 1_1 、r 1_2 and r 1_3 are the terminal pole diameters of the first sliding surface segment, the second sliding surface segment and the third sliding surface segment, θ 0_1 ,θ 0_2 and θ 0_3 are the starting polar angles of the first, second and third sliding surface segments, respectively, r 0_1 、r 0_2 and r 0_3 are the starting pole diameters of the first sliding surface segment, the second sliding surface segment and the third sliding surface segment respectively;

[0042] S5-1, let r 0_i and r 1_i Calculate variables for the iteration loop and take r 0_i and r 1_i The initial values ​​are 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 end pole diameter of the sliding surface of the previous sliding surface segment;

[0043] S5-2, use point Z i-1 and point Z i The coordinates of point Z i–1 and point Z i Length of the connection in,

[0044]

[0045] S5-3, substitute r 0_i (0) and r 1_i (0) , use formula (8) to calculate θ 1_i ;

[0046] S5-4, Substitute θ 0_i and θ 1_i To solve r 1_i ,

[0047] in

[0048] S5-5. Using formula (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, here we take ε2=0.001m, then the calculated r 0_i 、r 1_i and θ 1_i is the final result; 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 invention patent CN202210520129.X in the prior art, the present invention adopts a non-equally divided type to segment the sliding surface, and the sliding surface segment is a composite curve constructed by combining a general logarithmic spiral and a straight line, which includes the straight and arc sliding surfaces commonly used in existing theories and confirmed by actual engineering; at the same time, the present invention uses the consistency of the movement directions of adjacent sliding surface segments at the sliding surface segment control points to simply achieve smooth continuity at the sliding surface segment control points; in addition, the present invention limits the sliding surface shape parameters and the vertical position of the sliding surface segment control points under the concave characteristics of the sliding surface, so that the method described in the present invention has multiple advantages such as versatility, accuracy and ease of use.

[0052] In general, the present invention provides a method for constructing a segmented universal potential sliding surface model with a wide range of applications, good versatility, high accuracy, simple implementation, and strong operability, which effectively improves the feasibility and accuracy of searching for the most dangerous sliding surface in the process of slope stability assessment under complex conditions, and can widely cover the common sliding surface forms in engineering practice, and can reflect the differences in stress states of various parts of the slope, resulting in different shear failure mechanisms and shear failure characteristics of rock and soil bodies at different parts, so that it has strong adaptive capabilities and intelligent features. At the same time, the present invention can reliably improve the accuracy of slope stability assessment and provide strong support for the scientific development of slope instability and landslide warning. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0055] Figure 3 It is a schematic diagram of the shape parameters of the two-dimensional segmented universal potential sliding surface of the slope according to the present invention.

[0056] Figure 4 It is a schematic diagram of the characteristic of the segmented universal potential sliding surface polar coordinate equation of the two-dimensional slope and the connection points of each sliding surface segment of the present invention.

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

[0058] Figure 6 This is a flow chart of step 1 of iterative cycle calculation of two-dimensional slope segmented universal potential sliding surface parameters of the present invention.

[0059] Figure 7 This is a flow chart of step 2 of iterative cycle calculation of segmented universal potential sliding surface parameters for two-dimensional slopes of the present invention.

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

[0061] Fig. 9 It is a schematic diagram of the horizontal and vertical position variation range of the segmented universal potential sliding surface connection point of the two-dimensional slope of the present invention.

[0062] Fig.10 The present invention is a flowchart of the steps of constructing a two-dimensional segmented universal potential sliding surface model of slope.

[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, sliding surface polar diameter, 7, sliding surface polar angle, 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, movement direction of rock and soil at sliding surface, 15, sliding surface tangent direction, 16, shear dilation angle, 17, connection point, 18, discrete point. DETAILED DESCRIPTION

[0064] The specific model building method and implementation process of the present invention are as follows:

[0065] (1) Figure 1 As shown in the figure, in a two-dimensional slope, the instability of the slope is caused by the fact that the rock and soil of the slope reaches a shear failure state and forms a continuous sliding surface. Generally speaking, the stress state of each part of the slope is different, which leads to differences in the shear failure mechanism and shear failure characteristics of the rock and soil at different parts. As a result, the slope will show different sliding trends in different areas and form different sliding surface forms. For example, the upper part of the slope is usually tensile shear failure, where the rock and soil failure is manifested as tensile cracking and may have obvious cracks or faults. The sliding surface morphology is steep, and the slope is The middle and lower parts of the slope are often subjected to compression and shear failure, and the shear outlet at the lower part is the strongest. The sliding surface formed by the destruction of the rock and soil here is often relatively gentle and approximates an arc. It is precisely because there is a large difference in the sliding surface morphology between the steep sliding surface at the upper part of the slope and the gentle sliding surface at the shear outlet of the slope that a transition section is needed between the two to connect them, so as to ensure the continuity of the slope sliding surface stress and sliding surface morphology changes. For this reason, a segmented method is used to construct a general potential sliding surface model, in which the segmented general potential sliding surface includes three sections, namely the upper, middle and lower potential sliding surfaces.

[0066] (2) Figure 2As shown in the figure, the instability of the slope is mainly caused by rotation or translation movement. When the slope failure is rotation sliding, the potential sliding surface of the two-dimensional slope can mostly be an arc or a logarithmic spiral curve. When the slope failure is translation sliding, the potential sliding surface of the two-dimensional slope can be a straight line. However, whether it is a sliding surface formed by rotational motion or translational motion, it can be described by a polar coordinate function. Select an arbitrary point as the polar coordinate origin, and let r be the polar diameter of the sliding surface and θ be the polar angle of the sliding surface, where the polar diameter of the sliding surface is the distance from the point on the sliding surface to the polar coordinate origin, and the polar angle of the sliding surface is the inclination angle of the line connecting the point on the sliding surface and the polar coordinate origin in the horizontal direction, and the value is taken in the clockwise direction. For the segmented universal potential sliding surface, due to the different shape characteristics and change laws of each segment of the sliding surface, the polar coordinate origin and polar coordinate function of each segment of the potential sliding surface will also be different. Therefore, let the polar coordinate origin of the sliding surface segment be point O i (1≤i≤3, and i=1, 2 and 3 correspond to the polar coordinate origins O1, O2 and O3 of the upper, middle and lower potential sliding surfaces respectively), and let the polar coordinate function of the sliding surface segment be r=f i (θ) (1≤i≤3, and i=1, 2 and 3 correspond to the polar coordinate functions f1(θ), f2(θ) and f3(θ) of the potential sliding surfaces of the upper, middle and lower sections respectively).

[0067] (3) Figure 3 As shown, for the convenience of description, in the segmented universal potential sliding surface, the upper, middle and lower potential sliding surfaces are respectively corresponding 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). For the first sliding surface segment, the starting point of the sliding surface segment is Z0, the ending point of the sliding surface segment is Z1, and O1Z0 is the starting polar diameter r of the sliding surface segment. 0_1 , O1Z1 is the terminal diameter r of the sliding surface segment 1_1 , the inclination angle of O1Z0 in the horizontal direction is the starting polar angle θ of the sliding surface segment 0_1 , the inclination angle of O1Z1 in the horizontal direction is the terminal polar angle θ of the sliding surface segment 1_1 For the second sliding surface segment, the starting point of the sliding surface segment is Z1, the ending point of the sliding surface segment is Z2, and O2Z1 is the starting pole diameter r of the sliding surface segment. 0_2 , O2Z2 is the terminal diameter r of the sliding surface segment 1_2 , the inclination angle of O2Z1 in the horizontal direction is the starting polar angle θ of the sliding surface segment 0_2 , the inclination angle of O2Z2 in the horizontal direction is the terminal polar angle θ of the sliding surface segment 1_2 For the third sliding surface segment, the starting point of the sliding surface segment is Z2, the ending point of the sliding surface segment is Z3, and O3Z2 is the starting pole diameter r of the sliding surface segment. 0_3 , O3Z3 is the terminal diameter r of the sliding surface segment 1_3 , the inclination angle of O3Z2 in the horizontal direction is the starting polar angle θ of the sliding surface segment 0_3, the inclination angle of O3Z3 in the horizontal direction is the terminal polar angle θ of the sliding surface segment 1_3 , where point Z1 and point Z2 are the connection points of adjacent sliding surface segments in the segmented universal potential sliding surface.

[0068] (4) Figure 4 As shown in the figure, when the polar coordinate function is used to describe the shape of the potential sliding surface, the origin of the polar coordinates can be set as the rotation center of the slope sliding body, then the vertical direction of the sliding surface polar diameter r is the movement direction of the rock and soil at the sliding surface, and the tangent direction of the sliding surface is the shear failure direction of the rock and soil at the sliding surface. According to the relevant and non-relevant flow laws, it can be known that there is a dilatancy angle ξ between the movement direction of the rock and soil at the sliding surface and the shear failure direction, and the dilatancy angle ξ is related to the friction angle of the rock and soil at the sliding surface. There is a certain correlation. In order to facilitate calculation and analysis, Among them, k is the dimensionless control parameter of the potential slip surface curve shape. Based on this, the general potential slip surface equation in the polar coordinate system is:

[0069]

[0070] Where θ0 is the starting polar angle of the sliding surface segment; r0 is the starting polar diameter of the sliding surface segment.

[0071] In formula (1), when the constructed potential sliding surface is a straight line, based on the spatial geometric relationship of the sliding surface, ξ = θ + η – π / 2, then Among them, η is the horizontal inclination of the line connecting the starting point of the sliding surface segment to the end point of the sliding surface segment, and η is positive in the counterclockwise direction; when the potential sliding surface constructed is an arc, based on the spatial geometric relationship of the sliding surface, ξ=0, and k=0 at this time; when the potential sliding surface constructed is a general logarithmic spiral curve, based on the spatial geometric relationship of the sliding surface, k should be an arbitrary variable and k>0. Therefore, according to the above analysis and taking into account the universality of the sliding surface, the calculation formula of the dimensionless control parameter k of the potential sliding surface curve shape is:

[0072]

[0073] Where λ1 and λ2 are sliding surface shape control calculation variables, and λ1≥0 and λ2≥0.

[0074] Furthermore, in order to ensure the concave characteristics of the sliding surface curve, the sliding surface tangent horizontal inclination angle should decrease as the sliding surface polar angle increases. Therefore, according to the spatial geometric relationship between the sliding surface polar angle increment dθ and the dilatancy angle increment dξ, when the dilatancy angle increment on the sliding surface is less than the corresponding sliding surface polar angle increment, the sliding surface tangent horizontal inclination angle will decrease as the sliding surface polar angle increases, thereby ensuring the concave characteristics of the sliding surface curve. At this time, the calculation variables λ1 and λ2 in the 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) of the segmented general potential sliding surface model can be obtained as follows:

[0077]

[0078] In the formula, r j_i is any point P on the sliding surface segment i i The corresponding sliding surface polar diameter, θ j_i is any point P on the sliding surface segment i i The corresponding sliding surface polar angle is j_i =θ 0_i When j_i =r 0_i , when θ j_i =θ 1_i When j_i =r 1_i ; 1_i and λ 2_i is the sliding surface shape control calculation variable of sliding surface segment i, and λ 1_i ≥0 and λ 2_i ≥0;η i is the starting point Z of the sliding surface segment i i–1 The end point Z of the sliding surface segment i i The inclination of the connecting line in the horizontal direction; is any point P on the sliding surface segment i i If the linear strength criterion is used, then is a constant value. If the nonlinear strength criterion is adopted, then It is a function related to the normal stress of the sliding surface at that point and needs to be determined by an iterative solution strategy.

[0079] At the same time, in order to ensure the continuity of the sliding body along the segmented universal potential sliding surface, that is, the segmented universal potential sliding surfaces are not only connected to each other, but also the adjacent segment sliding surfaces should maintain the consistency of movement direction at the connection points (i.e., points Z1 and Z2), it is necessary to satisfy θ 0_2 =θ 1_1 and θ 0_3 =θ 1_2 In addition, in order to ensure the concave characteristics of each sliding surface segment curve in the segmented universal potential sliding surface, the calculation variable λ in each sliding surface segment can be obtained by using formula (3): 1_i and λ 2_i The following relationship must be satisfied:

[0080]

[0081] Where dθ j_i is the sliding surface polar angle θ of the sliding surface segment i j_iThe increment of the sliding surface polar angle at .

[0082] (5) Figure 5 As shown in the figure, the xy-axis rectangular coordinate system is established with the slope foot point as the origin, and point Z i The coordinates of (0≤i≤3, and i=0, 1, 2 and 3 correspond to point Z0, point Z1, point Z2 and point Z3 respectively) are Then, using triangle O i Z i-1 Z i (1≤i≤3, and i=1, 2 and 3 correspond to triangle O1Z0Z1, triangle O2Z1Z2 and triangle O3Z2Z3 respectively), when the starting polar radius r of the sliding surface segment i 0_i When known, the starting polar angle θ of the sliding surface segment i can be established 0_i The implicit calculation formula is:

[0083]

[0084] In the formula, r 1_i is the terminal pole diameter of the sliding surface segment i, and Point Z i-1 With point Z i The length of the line, and

[0085] At the same time, using triangle O i Z i–1 Z i The spatial geometric relationship of the sliding surface segment i is 0_i When known, the starting radius r of the sliding surface segment i can be established 0_i The calculation formula is:

[0086]

[0087] Further, using triangle O i Z i–1 Z i The spatial geometric relationship of can establish the starting polar angle θ of the sliding surface segment i. 0_i and the end polar angle θ 1_i The relationship between them is:

[0088]

[0089] For the first sliding surface segment, it is usually at the starting pole radius r of the known sliding surface segment. 0_1 Next, we use equations (6) and (8) to solve the starting polar angle θ of the sliding surface segment: 0_1 , the end polar angle θ of the sliding surface segment 1_1 and the terminal diameter r of the sliding surface segment 1_1For the second and third sliding surface segments, the segmented universal potential sliding surface maintains the consistency of motion direction at the connection point, so that the sliding surface polar angle satisfies θ 0_2 =θ 1_1 and θ 0_3 =θ 1_2 , and thus, here usually at the starting polar angle θ of the known sliding surface segment 0_2 (Second sliding surface segment) or the starting polar angle θ of the sliding surface segment 0_3 (The third sliding surface segment) Use equations (7) and (8) to solve the starting radius r of the sliding surface segment 0_2 , the terminal diameter r of the sliding surface segment 1_2 and the segment termination polar angle θ of the sliding surface 1_2 (Second sliding surface segment) or the starting diameter r of the sliding surface segment 0_3 , the terminal diameter r of the sliding surface segment 1_3 and the end polar angle θ of the sliding surface segment 1_3 (Third sliding surface segment).

[0090] When obtaining the sliding surface parameters of sliding surface segment i (i.e. the starting polar angle θ of the sliding surface segment) 0_i , the end polar angle θ of the sliding surface segment 1_i , the starting diameter r of the sliding surface segment 1_i and the terminal diameter r of the sliding surface segment 1_i ) can also be used to i Z i–1 Z i (1≤i≤3) to calculate the polar coordinate origin O of the sliding surface segment i i The coordinates in the xy axis coordinate system are calculated as follows:

[0091]

[0092] In the formula, and They are 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) x and y axis coordinates.

[0093] (6) Figure 6 As shown, in the first sliding surface segment, when the starting diameter r of the sliding surface segment is known 0_1 When , we can combine equation (6) and equation (8) and use the iterative cycle calculation strategy to solve the starting polar angle θ of the sliding surface segment. 0_1 , the end polar angle θ of the sliding surface segment 1_1 and the terminal diameter r of the sliding surface segment 1_1 The specific calculation steps (called the iterative cycle calculation step 1 of the piecewise general potential sliding surface parameter) are as follows: ① Let θ 0_1 and θ 1_1 Calculate variables for the iteration loop and take θ0_1 and θ 1_1 The initial values ​​are θ 0_1 (0) and θ 1_1 (0) , and when the first calculation is performed, θ 0_1 (0) =0 and θ 1_1 (0) =0;②Substitute into θ 0_1 (0) and θ 1_1 (0) To solve r 1_1 ,in, ③ Using the coordinates of point Z0 and point Z1, calculate the length of the line connecting point Z0 and point Z1 in, ④ Using formula (6), calculate the new θ 0_1 ⑤ Using formula (8), calculate the new θ 1_1 ;⑥If |θ 0_1 –θ 0_1 (0) |≤ε1 and |θ 1_1 –θ 1_1 (0) |≤ε1 (here ε1=0.001°), then the calculated θ 0_1 ,θ 1_1 and r 1_1 is 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 sliding surface segment or the third sliding surface segment, the segmented universal potential sliding surface is used to maintain the consistency of the movement direction at the connection point. 0_2 =θ 1_1 and θ 0_3 =θ 1_2 , and then, after the ending polar angle of the previous sliding surface segment is solved, the starting polar angle of the next sliding surface segment can be obtained. On this basis, by combining equations (7) and (8) and using an iterative loop calculation strategy, the starting polar radius r of the sliding surface segment is solved. 0_i , the terminal diameter r of the sliding surface segment 1_i and the end polar angle θ of the sliding surface segment 1_i (2≤i≤3), the specific calculation steps (called the iterative cycle calculation step 2 of the piecewise general potential sliding surface parameter) are as follows: ① Let r 0_i and r 1_i Calculate variables for the iteration loop and take r0_i and r 1_i The initial values ​​are 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 end pole diameter of the sliding surface of the previous sliding surface segment; ② Use point Z i-1 and point Z i The coordinates of point Z i–1 and point Z i Length of the connection in, ③Substitute r 0_i (0) and r 1_i (0) , use formula (8) to calculate θ 1_i ; ④Substitute into θ 0_i and θ 1_i To solve r 1_i ,in ⑤ Using formula (7), calculate the new r 0_i ⑥If | r 0_i –r 0_i (0) |≤ε2 and |r 1_i –r 1_i (0) |≤ε2 (here ε2=0.001 m), then the calculated r 0_i 、r 1_i and θ 1_i is the final result, otherwise, let r 0_i (0) =r 0_i and r 1_i (0) =r 1_i , and repeat steps ③ to ⑤.

[0095] (8) Figure 8 As shown in the figure, for the convenience of calculation, the piecewise universal potential sliding surface function in the polar coordinate system is transformed into the piecewise universal potential sliding surface function in the rectangular coordinate system. At the same time, the piecewise universal potential sliding surface is constructed in a discrete manner. Here, the polar angle equal increment mode is used to convert the ∠Z in the sliding surface segment i i–1 O i Z i Divide into n i (1≤i≤3) equal parts, and then the sliding surface segment i can also be divided into n iequal parts, at this time the sliding surface segment i is n i +1 discrete points, for the jth point on the sliding surface segment i, the corresponding sliding surface polar angle is θ j_i 、Sliding surface polar angle θ j_i The increment of the sliding surface polar angle is dθ j_i And the sliding surface diameter is r j_i (0≤j≤n i ),and dθ j_i =(θ 1_i –θ 0_i ) / n i and r j_i Formula (4) is used for calculation, and thus, the x-axis and y-axis coordinates of the j-th point on the sliding surface segment i are calculated as follows:

[0096]

[0097] In equations (11) and (12), when j = 0, x j_i and j_i Point Z i-1 The x and y axis coordinates, that is and When j = n i When x j_i and j_i Point Z i The x and y axis coordinates, that is and

[0098] (9) Fig. 9 As shown in the figure, in the segmented universal potential slip surface, the connecting points Z1 and Z2 of the adjacent segment slip surfaces should be between points Z0 and Z3, and point Z2 should also be between points Z1 and Z3. Then, the relationship between point Z1 and points Z0 and Z3 about the x-axis coordinate can be established as follows:

[0099]

[0100] Where κ1 is the calculation variable of 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] In addition, the relationship between point Z2 and points Z0 and Z3 about the x-axis coordinate is established as:

[0102]

[0103] Where κ2 is the calculation variable of 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] Furthermore, in order to ensure the rationality of the sliding surface generation, the potential sliding surface should be a concave curve, which means that the slope of the tangent line of the sliding surface in the xy axis coordinate system remains increasing, and any point on the potential sliding surface cannot exceed the line connecting the lower sliding point (i.e. point Z3) and the upper sliding point (i.e. point Z0) on the potential sliding surface. At the same time, the horizontal inclination angle of the sliding surface tangent line at the lower sliding point (point Z3) of the potential sliding surface cannot be less than -45° (positive in the counterclockwise direction). Therefore, the relationship between point Z2 and points Z0 and Z3 on the y axis coordinate is:

[0105]

[0106] Where κ3 is the calculation variable of 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° (counterclockwise is positive). 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 about the y-axis coordinate is established as:

[0108]

[0109] Wherein, κ4 is the calculation variable of 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 of the line connecting points Z3 and Z2. When κ4=1, point Z1 is on the line connecting points Z3 and Z0.

[0110] Therefore, by applying the calculation variables κ1, κ, κ3 and κ4, the coordinates of the connection points of the adjacent sections of the potential sliding surface in the piecewise universal potential sliding surface can be solved from the coordinates between the lower and upper sliding points of the potential sliding surface. For the lower and upper sliding points of the potential sliding surface (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 using the slope surface equation. Furthermore, for the piecewise universal potential sliding surface, its shape and position are determined by the x-axis coordinates of points Z3 and Z0, the sliding surface connection point position calculation variables κ1~κ4, and the sliding surface shape control calculation variable λ1_ i and λ2_ i At the same time, if the x-axis coordinates of points Z3 and Z0, the sliding surface connection point position calculation variables κ1~κ4 and the sliding surface shape control calculation variable λ1_ i and λ2_ iWhen the value is set within a reasonable range and taken arbitrarily, a series of potential sliding surfaces can be obtained. In this series of potential sliding surfaces, if the slope stability limit equilibrium method is combined 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 slope safety factor as the optimization target, which is the sliding surface where the slope instability is most likely to occur.

[0111] (10) Fig.10 As shown in the figure, the construction and implementation process of the segmented general potential sliding surface model is as follows: ① Given the x-axis coordinates of points Z3 and Z0, the sliding surface connection point position calculation variables κ1~κ4 and the sliding surface shape control calculation variable λ1_ i and λ2_ i (1≤i≤3) and the starting pole diameter r of the first sliding surface segment 0_1 ; ② Apply the slope surface equation to calculate the y-axis coordinates of points Z3 and Z0; ③ Use equations (13) to (16) to calculate the x-axis and y-axis coordinates of points Z1 and Z2; ④ Use the segmented universal potential sliding surface parameter to iterate and loop through step 1 to solve for the starting polar angle θ of the first sliding surface segment 0_1 、Termination polar angle θ 1_1 and the terminal diameter r 1_1 ⑤ Use the segmented universal potential sliding surface parameter to iteratively calculate step 2 to solve the starting polar diameter r of the second sliding surface segment or the third sliding surface segment. 0_i , terminal diameter r 1_i and the end polar angle θ 1_i (2≤i≤3); ⑥ Using equations (9) and (10), calculate the polar coordinate origin O of the sliding surface segment i i (1≤i≤3) is the coordinate in the xy axis coordinate system; ⑦ Use the polar angle equal increment mode to discretize the sliding surface segment i into n i (1≤i≤3) and use formula (5) to determine the calculated variable λ 1_i and λ 2_i Can the concave characteristics of each sliding surface segment curve in the piecewise universal potential sliding surface be guaranteed? If not, the calculation is terminated and the calculation variable λ1_ needs to be re-given. i and λ2_ i , if it can be guaranteed, then use equations (11) and (12) to calculate the x and y axis coordinates x of the jth point on the sliding surface segment i j_i and j_i (0≤j≤n i and 1≤i≤3);⑧ output the discrete calculation results of the piecewise general potential sliding surface.

[0112] The characteristics of the above-mentioned universal potential sliding surface of the present invention are: using polar coordinate function to describe the shape of the universal potential sliding surface, then taking the polar coordinate origin as the rotation center point of the slope sliding body, and utilizing the relationship between the shear failure direction of the rock and soil at the sliding surface (i.e. the tangent direction of the sliding surface) and the movement direction (i.e. the perpendicular direction of the sliding surface polar diameter), a universal potential sliding surface polar coordinate equation is established. The shape of this universal 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. It can degenerate into a straight sliding surface when the slope is destroyed as translational sliding, or it can degenerate into a circular arc or logarithmic spiral curve sliding surface when the slope is destroyed as rotational sliding. In addition, the concave characteristics of the sliding surface curve are also guaranteed.

[0113] The characteristics of the above-mentioned segmented universal potential sliding surface of the present invention are: taking into account the differences in stress states of various parts of the slope, resulting in different shear failure mechanisms and shear failure characteristics of the rock and soil at different parts, a three-segment universal potential sliding surface is used to simulate the complex failure form of the slope. At the same time, different polar coordinate origins and polar coordinate functions are used to describe the shapes and positions of the three segments of potential sliding surfaces, and it is required that the segmented universal potential sliding surfaces are not only connected to each other, but also that the adjacent segments of the sliding surfaces should maintain consistency in the movement direction at the connection points.

[0114] The advantages of the present invention are as follows: the segmented universal potential sliding surface can effectively describe the differences in stress states of various parts of the slope, which leads to different shear failure mechanisms and shear damage characteristics of rock and soil at different parts. At the same time, the segmented universal potential sliding surface not only includes possible types of existing sliding surfaces, but also can rationalize the shape of the constructed sliding surface while ensuring the concave characteristics of the sliding surface curve. In addition, the shape of the segmented universal potential sliding surface is simply controlled by several easy-to-determine parameters and is generated in a discrete manner, which is conducive to the automatic acquisition of the segmented universal potential sliding surface under complex conditions, and is also convenient for combining with the limit equilibrium method of slope stability to carry out intelligent search for the most dangerous sliding surface. Therefore, the present invention has the characteristics of wide application range, good versatility, high accuracy, simple implementation, strong operability, etc., can reliably improve the accuracy of slope stability assessment, and provide strong support for the scientific development of slope instability and landslide warning.

[0115] Example

[0116] A kind of Figures 1 to 10The method for constructing a segmented universal potential sliding surface model is shown. A mountain slope with steep terrain and complex geological conditions. Preliminary survey results show that the mountain slope is 23.8m high and has an average slope of 50°, and there is a certain possibility of sliding. In order to ensure the safety of infrastructure construction on the mountain slope, it is necessary to accurately obtain the most dangerous sliding surface shape and position of the slope when it becomes unstable and damaged, and thereby efficiently and reliably implement slope stability assessment. At the same time, it is used to guide the definition of the potential range of slope collapse and the implementation of support protection work. Therefore, the segmented universal potential sliding surface model construction method and implementation process invented in this article are introduced and applied, and the specific operating steps are as follows:

[0117] (1) According to the Code for Geotechnical Engineering Investigation (GB50021-2001), a field survey was conducted on the mountain slope to determine the two-dimensional slope surface shape and stratigraphic data and the corresponding rock and soil strength parameters in the potential sliding area of ​​the slope. It was considered that the slope rock and soil obeyed the linear MC strength criterion, and the rock and soil weight obtained from the survey was γ = 19 kN / m 3 , cohesion c = 25 kPa and internal friction angle

[0118] (2) The two-dimensional slope segmented universal potential sliding surface model of the present invention is selected, and the horizontal coordinates of 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 according to the possible range of the sliding body. and When the xy axis coordinate system is established with the slope foot point as the origin, The range is -5m~5m, The range is 20m to 30m;

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

[0120] (4) Combined with the limit equilibrium method and embedded in the mathematical optimization algorithm, the slope stability analysis is realized. In this process, when 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 are randomly selected, and The calculation variables κ1, κ2, κ3 and κ4 of the sliding surface connection point position and the morphological control parameter λ of the sliding surface segment i (1≤i≤3) 1_i and λ 2_i and the starting diameter r of the first sliding surface segment 1_0 After that, a reasonable potential sliding surface can be generated according to the construction steps of the segmented general potential sliding surface model. Then, the limit equilibrium method is used to solve the slope safety factor under the corresponding potential sliding surface. Then, with the assistance of the mathematical optimization algorithm, the minimum slope safety factor is used as the optimization target to search for the most dangerous sliding surface of the slope. In this way, the slope stability analysis is completed, and the minimum slope safety factor and the most dangerous sliding surface results are output. Among them, the minimum slope safety factor is 1.04. As for the most dangerous sliding surface, 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 are respectively and The calculation variables of the sliding surface connection point position are κ1=0.98, κ2=0.56, κ3=0.63 and κ4=0.84, and the morphological 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, the starting diameter r of the first sliding surface segment 1_0 =36.40m;

[0121] (5) According to the stability evaluation standard of the Technical Code for Slope Engineering of Buildings (GB 50330-2013), the required slope safety factor of this slope (first-level slope) should reach 1.30. The calculation results show that the slope stability result is less than this value. Therefore, the slope needs to be reinforced and supported by anchor rods and retaining walls within the most dangerous sliding surface range.

[0122] Therefore, the present invention provides a method for constructing a segmented universal potential sliding surface model, which takes into account the differences in stress states of various parts of the slope, resulting in different shear failure mechanisms and shear failure characteristics of rock and soil at different parts, and adopts a three-segment universal potential sliding surface to simulate the complex failure forms of the slope. The constructed segmented universal potential sliding surface not only includes possible types of existing sliding surfaces, but also ensures the rationality of the sliding surface shape under the requirements of concave characteristics of the sliding surface curve and consistency of movement directions at the connection points of adjacent segments of the sliding surface. Moreover, it can be simplified to several easily determined parameters to control the shape of the segmented universal potential sliding surface and generate it in a discrete manner, thereby achieving a balance of multiple advantages such as universality, accuracy and ease of use in determining the potential sliding surface in actual slope engineering under complex conditions, and further improving the accuracy of slope stability assessment.

[0123] 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 protection scope of the present invention.

Claims

1. A method for constructing a segmented universal potential sliding surface model, characterized in that: A three-segment universal potential sliding surface is used to simulate the complex failure form of the slope, and the shape of the segmented universal potential sliding surface is controlled by a few easily determined parameters and generated in a discrete manner. The method comprises the following steps: 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 variable λ1_ i and λ2_ i (1≤i≤3) and the starting pole diameter r of the first sliding surface segment 0_1 ; Where r is the sliding surface extreme diameter; Among them, Z3 is the end point of the potential third sliding surface segment, and Z0 is the starting point of the potential first sliding surface segment; Z1 is the end point of the potential first sliding surface segment, and Z2 is the end point of the potential second sliding surface segment; κ1 is the sliding surface connection point horizontal position calculation variable, 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; κ2 is the sliding surface connection point horizontal position calculation variable, and its value range is [0, 1]. When κ2=0, point Z2 coincides with point Z3, and when κ2=1, point Z1 coincides with point Z0. =1, point Z2 coincides with point Z1; κ3 is the calculation variable of 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°, and the counterclockwise direction is positive. When κ3=1, point Z2 is located on the line connecting points Z3 and Z0; κ4 is the calculation variable of 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 of the line connecting points Z3 and Z2. When κ4=1, point Z1 is located on the line connecting points Z3 and Z0. S2, using the slope surface equation, calculate the y-axis coordinates of points Z3 and Z0; S3. Using equations (13) to (16), calculate the x-axis and y-axis coordinates of point Z1 and point Z2; In the formula, and are the x and y axis coordinates of point Z1, and are the x and y axis coordinates of point Z2, and are the x-axis coordinates of point Z0 and point Z3 respectively; S4. Solve for the starting polar angle θ of the first sliding surface segment 0_1 , the terminal polar angle θ of the first sliding surface segment 1_1 and the terminal diameter r of the first sliding surface segment 1_1 ; where θ is the polar angle of the sliding surface; S5. Calculate the starting radius r of the second or third sliding surface segment 0_i , the terminal diameter r of the second sliding surface segment or the third sliding surface segment 1_i The termination 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 Coordinates in the xy axis coordinate system, where 1≤i≤3; In the formula, and They are the polar coordinate origin O i The x- and y-axis coordinates of , 1≤i≤3, and i=1, 2, and 3 correspond to point O1, point O2, and point O3 respectively; S7, using the polar angle equal increment mode to discretize the sliding surface segment i into n i (1≤i≤3) and use formula (5) to determine the calculated variable λ 1_i and λ 2_i Can the concave characteristics of each sliding surface segment curve in the piecewise universal potential sliding surface be guaranteed? If not, the calculation is terminated and the calculation variable λ1_ needs to be re-given. i and λ2_ i ; If it can be guaranteed, then use equations (11) and (12) to calculate the x and y axis coordinates x of the jth point on the sliding surface segment i j_i and j_i (0≤j≤n i and 1≤i≤3); In formula (5), dθ j_i is the sliding surface polar angle θ of the sliding surface segment i j_i The increment of the sliding surface polar angle at ; In equations (11) and (12), when j = 0, x j_i and j_i Point Z i-1 The x and y axis coordinates, that is and When j = n i When x j_i and j_i Point Z i The x and y axis coordinates, that is and S8. Output the discrete calculation results of the piecewise general potential sliding surface.

2. The method for constructing a segmented universal potential sliding surface model according to claim 1, characterized in that: In step S4, the starting polar angle θ of the first sliding surface segment is solved by the following method: 0_1 , the terminal polar angle θ of the first sliding surface segment 1_1 and the terminal diameter r of the first sliding surface segment 1_1 ; S4-1: Let θ 0_1 and θ 1_1 Calculate variables for the iteration loop and take θ 0_1 and θ 1_1 The initial values ​​are θ 0_1 (0) and θ 1_1 (0) , and when the first calculation is performed, θ 0_1 (0) =0 and θ 1_1 (0) =0; S4-2: Substitute θ 0_1 (0) and θ 1_1 (0) To solve r 1_1 ,in, In the formula, is the friction angle of rock and soil 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: Using the coordinates of point Z0 and point Z1, calculate the length of the line connecting point Z0 and point Z1 in, S4-4: Using equation (6), calculate the new θ 0_1 ; In the formula, when the value of i is 1, r 1_i is the terminal pole diameter of the first sliding surface segment; S4-5: Using formula (8), calculate the new θ 1_1 ; When the value of i is 1, θ 1_i is the terminal polar angle of the first sliding surface segment, θ 0_i is the starting polar angle of the first sliding surface segment; S4-6: If |θ 0_1 –θ 0_1 (0) |≤ε1 and |θ 1_1 –θ 1_1 (0) |≤ε1, here ε1=0.001°, then the calculated θ 0_1 ,θ 1_1 and r 1_1 is the final result; 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 for constructing a segmented universal potential sliding surface model according to claim 2, characterized in that: In step S5, the starting diameter r of the second sliding surface segment or the third sliding surface segment is solved by the following method: 0_i , the terminal diameter r of the second sliding surface segment or the third sliding surface segment 1_i The termination polar angle θ of the second sliding surface segment or the third sliding surface segment 1_i , where 2≤i≤3; and where the segmented universal potential sliding surface is used to maintain the consistency of the motion direction at the connection point under θ 0_2 =θ 1_1 and θ 0_3 =θ 1_2 Solution; Among them, θ 1_1 ,θ 1_2 and θ 1_3 are the terminal polar angles of the first sliding surface segment, the second sliding surface segment and the third sliding surface segment, r 1_1 、r 1_2 and r 1_3 are the terminal pole diameters of the first sliding surface segment, the second sliding surface segment and the third sliding surface segment, θ 0_1 ,θ 0_2 and θ 0_3 are the starting polar angles of the first, second and third sliding surface segments, respectively, r 0_1 、r 0_2 and r 0_3 are the starting pole diameters of the first sliding surface segment, the second sliding surface segment and the third sliding surface segment respectively; S5-1, let r 0_i and r 1_i Calculate variables for the iteration loop and take r 0_i and r 1_i The initial values ​​are 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 end pole diameter of the sliding surface of the previous sliding surface segment; S5-2, use point Z i-1 and point Z i The coordinates of point Z i–1 and point Z i Length of the connection in, S5-3, substitute r 0_i (0) and r 1_i (0) , use formula (8) to calculate θ 1_i ; S5-4, Substitute θ 0_i and θ 1_i To solve r 1_i , in S5-5. Using formula (7), calculate the new r 0_i ; S5-6, if | r 0_i –r 0_i (0) |≤ε2 and |r 1_i –r 1_i (0) |≤ε2, here we take ε2=0.001m, then the calculated r 0_i 、r 1_i and θ 1_i is the final result; 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.

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