Construction method of three-dimensional side slope universal potential sliding surface model in polar angle mode

Abstract

The invention belongs to the field of slope engineering, and particularly relates to a construction method of a three-dimensional slope universal potential sliding surface model in a polar angle mode, which adopts a sliding surface polar angle control mode to generate a potential sliding surface, and utilizes four boundary points of the lower side, the upper side, the left side and the right side of the sliding surface to define a potential sliding surface range. And stress constraint conditions which should be met by the polar angles of the sliding surfaces at the boundary points of the lower side and the upper side are established, and a discrete technology is applied, so that the potential sliding surface model is embedded into the existing limit equilibrium method. The sliding surface model constructed by the method solves the problem of limiting the sliding surface type of a three-dimensional side slope potential sliding surface model, ensures that the shape and range of the sliding surface are feasible and reasonable, strictly meets the physical and mechanical mechanisms of potential sliding surface generation, and facilitates the development of stability analysis of a three-dimensional side slope under complex conditions. The method has the advantages of being simple and easy to implement, wide in application range, high in universality, high in precision and the like, and a powerful scientific basis is provided for reliable early warning and prevention of slope instability and collapse.

Description

Technical Field

[0001] The invention belongs to the field of slope engineering, and in particular relates to a method for constructing a universal potential sliding surface model of a three-dimensional slope in a polar angle mode. Background Art

[0002] In order to efficiently and reliably handle and prevent landslides, it is necessary to accurately conduct slope stability evaluation and obtain the most dangerous sliding surface of the slope. Among them, a feasible and reasonable potential sliding surface model is the key to ensure effective slope stability evaluation and accurate positioning of the most dangerous sliding surface.

[0003] In the existing slope stability analysis, the construction of two-dimensional slope potential sliding surface model is still the mainstream. In the two-dimensional plane, the slope is regarded as an infinite body in half space, and the potential sliding surface of the slope can be represented by a certain type of curve, such as a specific type of curve commonly used such as a straight line, a circular arc and a logarithmic spiral, as well as any random curve suitable for complex conditions. In the two-dimensional plane, the potential sliding surface model greatly simplifies the actual sliding surface morphology, and the two-dimensional most dangerous sliding surface is often larger than the actual failure range of the three-dimensional slope without considering the boundary effect. In addition, the slope stability analysis results in the two-dimensional case are also conservative, which leads to unnecessary waste in the cost of engineering slope design and reinforcement. For the existing three-dimensional slope potential sliding surface model, like the two-dimensional potential sliding surface model, there are specific rotational sliding surfaces such as spheres, ellipsoids and logarithmic spirals. In addition, there are also sliding surfaces of arbitrary shapes obtained by translating the generatrix based on the sliding direction and the directrix perpendicular to the sliding direction under complex conditions.

[0004] However, the above-mentioned sliding surface models still have certain limitations in the construction method. For example, in the existing two-dimensional sliding surface models, whether it is a specific type of sliding surface model or a complex random sliding surface model, it is constructed based on a simplified two-dimensional plane. On the one hand, the two-dimensional sliding surface cannot show the three-dimensional failure characteristics of the actual slope, and it is difficult to accurately guide the project. On the other hand, in the slope stability analysis, the two-dimensional sliding surface ignores the three-dimensional space effect, which makes the slope stability analysis results often conservative, which is not conducive to the saving of engineering costs. In the existing three-dimensional sliding surface model, although the three-dimensional space effect is considered to a certain extent, the specific sliding surface model limits the sliding surface type and cannot show the complex shape of the actual sliding surface. At the same time, although the arbitrary shape sliding surface model improves the diversity of the three-dimensional sliding surface model, the method of generating by the generatrix and the directrix control is still difficult to truly achieve the arbitrary shape of the sliding surface. In addition, the existing model cannot make the constructed potential sliding surface strictly meet the physical geometry and mechanical failure characteristics, and thus, it is difficult to capture the most accurate and dangerous potential sliding surface under complex conditions.

[0005] The inventors of this application have previously provided some methods for constructing sliding surface models, such as invention patent CN119129054A, which provides a method for constructing a potential sliding surface model for a two-dimensional slope in a polar angle mode, and invention patent application CN119249738A. A method for constructing a two-dimensional slope potential sliding surface model under a principal stress deflection mode is provided, and invention patent application CN119513994A provides a method for constructing a segmented universal potential sliding surface model. These methods are all methods for constructing a universal potential sliding surface model in a two-dimensional case. However, the two-dimensional universal potential sliding surface fails to consider the spatial effect of the three-dimensional slope, which results in that the results obtained by using the two-dimensional universal potential sliding surface to analyze the slope stability are often conservative and difficult to reflect the actual slope stability state. At the same time, the three-dimensional universal potential sliding surface not only adds a spatial dimension on the basis of the two-dimensional universal potential sliding surface, but also changes the intersection boundary of the three-dimensional potential sliding surface and the slope surface from two points in the two-dimensional case to a spatial curve, making the feasibility of the three-dimensional universal potential sliding surface (such as the three-dimensional universal potential sliding surface should satisfy the shear failure mechanism in addition to being an arbitrary surface in space) and the boundary controllability (such as whether the three-dimensional universal potential sliding surface intersects with the slope surface) extremely complicated, and it is also impossible to apply the two-dimensional universal potential sliding surface model construction method to expand it to three-dimensional space to construct a three-dimensional universal potential sliding surface.

[0006] As engineering slopes continue to expand into dangerous mountainous areas, accurately and efficiently identifying the most dangerous potential sliding surfaces of slopes and providing scientific and detailed construction guidance based on this has become an urgent requirement to ensure the safety and smooth progress of the project. However, the existing slope sliding surface calculation models are generally subject to unreasonable physical and mechanical models and are unable to provide reliable guidance for slope engineering.

[0007] Therefore, a new method for constructing a universal potential sliding surface model of three-dimensional slope is needed in this field. Summary of the invention

[0008] To this end, the present invention adopts the sliding surface polar angle control method to generate the potential sliding surface, without limiting the sliding surface type, ensuring the arbitrariness of the potential sliding surface, at the same time, the potential sliding surface range is defined by using the four boundary points of the lower side, upper side, left side and right side, and the sliding surface polar angle satisfies the stress constraint conditions at the upper and lower boundary points and the position constraint conditions at the four boundary points, so that the constructed potential sliding surface strictly meets the physical geometry and mechanical failure characteristics, in addition, the potential sliding surface range limitation and the curved surface concave mechanism are included to ensure the feasibility and rationality of the potential sliding surface model, and then the discrete technology is applied to obtain the discrete points on the potential sliding surface, and correspondingly, the effective discrete points and discrete grids on the potential sliding surface are identified, which is conducive to combining with the limit equilibrium method for carrying out three-dimensional slope stability analysis under complex conditions. The present invention has the advantages of being simple and easy to implement, wide application range, strong versatility, high accuracy, etc., and provides a strong scientific basis for reliable early warning and prevention of slope instability and collapse.

[0009] Specifically, the present invention provides a method for constructing a universal potential sliding surface model of a three-dimensional slope under a polar angle mode. The method generates a potential sliding surface by controlling the sliding surface polar angle, and defines the range of the potential sliding surface by using four boundary points at the lower, upper, left and right sides of the sliding surface. Then, based on the position extreme value characteristics of the lower boundary point and the upper boundary point of the sliding surface, based on the sliding trend of the rock and soil bodies on both sides at the two boundary points, the approximate occurrence plane of shear failure of the rock and soil bodies at the lower boundary point and the upper boundary point is determined. Then, according to the tangent direction of the potential sliding surface as the most unfavorable shear direction of the rock and soil body, when the slope surface force is known, combined with the shear strength criterion of the rock and soil body, the potential sliding surface polar angle control method is used to generate a potential sliding surface, and the potential sliding surface polar angle control method is used to generate a potential sliding surface. The polar angle of the sliding surface at the boundary point should satisfy the stress constraint conditions; and the polar angle of the sliding surface also matches the position constraint conditions at the above four boundary points; the polar angle of the sliding surface is associated with the sliding surface coordinates, and the approximate function of the sliding surface polar angle and the sliding surface coordinates is constructed using the Taylor series expansion method. In addition to satisfying the stress and position constraints at the boundary points of the potential sliding surface, the polar angle of the sliding surface is also subject to the potential sliding surface range limitation and the concave surface mechanism, so as to ensure that the constructed potential sliding surface is feasible and reasonable; and the discrete technology is applied to obtain the discrete points on the potential sliding surface, and then the valid discrete points and discrete grids on the potential sliding surface are identified and output, so as to construct a general potential sliding surface model of three-dimensional slope under the polar angle mode.

[0010] In a specific embodiment, the method comprises the following steps: Step S1: In a three-dimensional slope, establish a coordinate system with any point as the origin. x.y.z Axis space coordinate system; Step S2: Given the lower boundary point of the potential sliding surface A , upper boundary point B , left boundary point C and the right boundary point D and its coordinates, namely ( x A , y A , z A )、( x B , y B , z B )、( x C , y C , z C )and( x D , y D , z D); Step S3: According to the three-dimensional slope surface, obtain the lower boundary point of the potential sliding surface A and upper boundary point B Slope surface xz and yz The horizontal inclination angle of the tangent line on the plane is β A_x and β A_y , β B_x and β B_y ; Step S4: Formulate potential sliding surface shape control parameters l 0. l 1. l 2. l 4. l 7. l 8. x 1. x 2. or 1 and or 2; Step S5: Based on the known slope tangent inclination angle at the potential sliding surface boundary point β A and β B And soil strength parameters f A and f B , using the boundary stress constraint condition, solve the horizontal inclination angle of the potential sliding surface tangent in the plane where the shear failure of the rock mass occurs at the boundary point α A and α B , and use this to determine the potential sliding surface tangent plane at the boundary point xz and yz The horizontal inclination angle of the tangent line on the plane is α A_x , α A_y , α B_x and α B_y Step S6: Using equation (16), solve the potential sliding surface shape control parameters l 3. l 5. l 6. l 9. l 10 , l 11 and l 12 ; (16) In the formula,a 11 ~ a 17 , a 21 ~ a 27 , a 31 ~ a 37 , a 41 ~ a 47 , a 51 ~ a 57 , a 61 ~ a 67 and a 71 ~ a 77 are coefficients of the equation system; b 1~ b 7 is the constant term of the equation system; Step S7: Using equations (24), (25) and (26), calculate the discrete points on the potential sliding surface respectively. P ij of x , y and z Axis coordinates, i.e. x ij , y ij and z ij , 0 ≤ i ≤ n and 0 ≤ j ≤ m ; (twenty four) (25) (26) Step S8: Analyze the feasibility of the constructed potential sliding surface by using the potential sliding surface feasibility judgment step under the control of the intersection of the potential sliding surface and the slope surface; if it is not feasible, it is necessary to re-select the potential sliding surface shape control parameters; Step S9: Analyze the rationality of the constructed potential sliding surface by using the potential sliding surface rationality judgment step under the concave mechanism; if it is not rational, it is necessary to re-select the potential sliding surface shape control parameters; Step S10: Compare the discrete points on the potential sliding surface P ij of z Axis coordinates z ij and the corresponding slope pointz Axis coordinates s ij ,like z ij ≤ s ij , then the discrete points P ij is a valid discrete point, 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S11: Determine the validity of the discrete grid using the validity of the discrete points on the potential sliding surface. P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points P ij There are 3 or more points in the table that are valid discrete points, 1 ≤ i ≤ n and 1 ≤ j ≤ m , then the discrete grid ij is a valid grid; Step S12: output the valid discrete points and valid grid of the potential sliding surface.

[0011] In a specific implementation, in step S4, l 0. l 1. l 2. l 4. l 7 and l The value range of 8 is -10 ~10, x 1. or 1. x 2 and or The value range of 2 is 1 ~ 10.

[0012] In a specific implementation manner, in step S8, in order to eliminate infeasible potential sliding surfaces, according to j Potential sliding surface discrete points in the column P ij The feasibility of the potential sliding surface is determined by the number of intersections between the broken line segments and the slope surface, 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S8 specifically includes the following steps: Step S8-1: Order j The number of intersection points between the broken line segments composed of discrete points on the potential sliding surface and the slope surface is J d ( j ), and takej = 1; Step S8-2: Get J d ( j ) = 0 and i = 1; Step S8-3: Obtain discrete points on the potential sliding surface P (i-1)j and P ij of z Axis coordinates z (i-1)j and z ij And the slope surface at the corresponding position z Axis coordinates s (i-1)j and s ij ; Step S8-4: Determine whether z (i-1)j ≤ s (i-1)j and z ij > s ij or z (i-1)j > s i-1j and z ij ≤ s ij ; If not satisfied, jump to step S8-6; Step S8-5: Let J d ( j ) = J d ( j ) + 1; Step S8-6: Determine whether i < n If it is not satisfied, then i = i + 1, and return to step S8-3; Step S8-7: If J d ( j ) = 2, it indicates that the potential sliding surface shape satisfies the intersection control condition; otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible, and the potential sliding surface shape control parameters need to be reselected; Step S8-8: Determine whether it satisfies j < m If it is not satisfied, then j = j + 1, and return to step S8-2.

[0013] In a specific implementation, in step S9, the sliding surface polar angle should be subject to the potential sliding surface concave mechanism, and the rationality of the potential sliding surface is determined by whether the following conditions 1 and 2 are met. Condition 1 is i Potential slip surface discrete points in the row P ij The curvature of the sliding surface of the broken line segments formed by connecting them in sequence is increasing. The second condition is j Potential sliding surface discrete points in the column P ij The curvature of the sliding surface formed by the broken line segments connected in sequence is increasing; 0 ≤ i ≤ n and 0 ≤ j ≤ m ; The step S9 specifically includes the following steps: Step S9-1: i = 0 and j = 0; Step S9-2: Obtain discrete points on the potential sliding surface P ij , P (i+1)j and P i(j+1) The coordinates of ( x ij , y ij , z ij )、( x (i+1)j , y (i+1)j , z (i+1)j )and( x i(j+1) , y i(j+1) , z i(j+1) ); Step S9-3: Obtaining discrete points P ij The tangent plane of the sliding surface is xz Horizontal inclination on a plane α x_(i,j) ,in,

[0014] Step S9-4: Obtaining discrete points P ij The tangent plane of the sliding surface is yz Horizontal inclination on a plane α y_(i,j) ,in,

[0015] Step S9-5: Determine whether the horizontal inclination angle of the sliding surface tangent satisfies: and ; If not, it indicates that the selected potential sliding surface shape control parameters make the constructed potential sliding surface unreasonable, and the potential sliding surface shape control parameters need to be reselected; Step S9-6: Determine whether i < n If not satisfied, then i = i + 1, and return to step S9-2; Step S9-7: Determine whether j <m-1 If not satisfied, then j = j + 1, and return to step S9-2.

[0016] Compared with the previous method of constructing a general potential sliding surface model in two-dimensional conditions, the present invention solves the problem of constructing a three-dimensional general potential sliding surface under complex conditions. In addition to ensuring the feasibility and rationality of the sliding surface shape and range, it can also strictly meet the physical and mechanical mechanisms of potential sliding surface generation, and can be combined with the three-dimensional slope stability calculation method to effectively judge the three-dimensional slope stability state.

[0017] The advantages of the present invention are: it is simple and easy to operate, has a wide range of applications, is highly versatile, and has high accuracy. The constructed three-dimensional slope potential sliding surface model not only ensures the feasibility and rationality of the sliding surface shape and range under non-restricted sliding surface types, but also strictly meets the physical and mechanical mechanisms of potential sliding surface generation. In addition, the application of discrete technology is beneficial to embedding the potential sliding surface model into the existing limit equilibrium method to carry out three-dimensional slope stability analysis under complex conditions, thereby providing a strong scientific basis for reliable early warning and prevention of slope instability and landslides. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It is a schematic diagram of the potential sliding surface and boundary points of the three-dimensional slope of the present invention.

[0019] Figure 2 It is the lower boundary point of the potential sliding surface of the three-dimensional slope of the present invention. A Schematic diagram of the horizontal inclination of the slope surface, where (a) is the lower boundary point A Department xz Plane and yz Schematic diagram of the three-dimensional space distribution of the plane, (b) is the lower boundary point A Department xz Schematic diagram of the horizontal inclination of the slope surface on the plane, (c) is the lower boundary point A Department yz Schematic diagram of the horizontal inclination of the slope surface on the plane, (d) is the lower boundary point ASchematic diagram of the micro-segment sliding boundary line on both sides, (e) is the lower boundary point A Schematic diagram of the geometric relationship of the horizontal inclination angle of the slope surface.

[0020] Figure 3 It is the lower boundary point of the potential sliding surface of the three-dimensional slope of the present invention. A Schematic diagram of the horizontal inclination of the tangential plane of the potential sliding surface, where (a) is the lower boundary point A Schematic diagram of the three-dimensional spatial distribution of the two-dimensional slice 1, (b) is the lower boundary point A Schematic diagram of the most unfavorable shear direction of the sliding surface on the two-dimensional slice 1, (c) is the lower boundary point A Schematic diagram of the geometric relationship of the tangential inclination angle of the potential sliding surface, (d) is the lower boundary point A Schematic diagram of Mohr stress circle and shear strength curve of rock and soil at stress state.

[0021] Figure 4 It is the upper boundary point of the potential sliding surface of the three-dimensional slope of the present invention. B Schematic diagram of the horizontal inclination of the slope surface at the location where (a) is the upper boundary point B Department xz Plane and yz Schematic diagram of the three-dimensional spatial distribution of the plane, (b) is the upper boundary point B Department xz Schematic diagram of the horizontal inclination of the slope surface on the plane, (c) is the upper boundary point B Department yz Schematic diagram of the horizontal inclination of the slope surface on the plane, (d) is the upper boundary point B Schematic diagram of the micro-segment sliding boundary line on both sides, (e) is the upper boundary point B Schematic diagram of the geometric relationship of the horizontal inclination angle of the slope surface.

[0022] Figure 5 It is the upper boundary point of the potential sliding surface of the three-dimensional slope of the present invention. B Schematic diagram of the horizontal inclination of the tangent plane of the potential sliding surface, where (a) is the upper boundary point B Schematic diagram of the three-dimensional spatial distribution of the two-dimensional slice 2, (b) is the upper boundary point B Schematic diagram of the most unfavorable shear direction of the sliding surface on the two-dimensional slice 2, (c) is the upper boundary point B Schematic diagram of the geometric relationship of the tangential inclination angle of the potential sliding surface at (d) is the upper boundary point B Schematic diagram of Mohr stress circle and shear strength curve of rock and soil at stress state.

[0023] Figure 6 It is a schematic diagram of any point on the potential sliding surface of the three-dimensional slope of the present invention.

[0024] Figure 7 It is the lower boundary point of the potential sliding surface of the three-dimensional slope of the present invention.A Schematic diagram of the introduction of stress constraint conditions at , where (a) is the lower boundary point A Department xz Plane and yz Schematic diagram of the three-dimensional space distribution of the plane, (b) is the lower boundary point A Department xz Schematic diagram of potential sliding surface inclination constraint on a plane, (c) is the lower boundary point A Department yz Schematic diagram of the potential sliding surface inclination constraint on a plane.

[0025] Figure 8 It is the upper boundary point of the potential sliding surface of the three-dimensional slope of the present invention. B Schematic diagram of the introduction of stress constraint conditions at , where (a) is the upper boundary point B Department xz Schematic diagram of plane distribution and sliding surface inclination constraint in three-dimensional space, (b) is the upper boundary point B Department xz Schematic diagram of potential sliding surface inclination constraint on a plane, (c) is the upper boundary point B Department yz Schematic diagram of plane distribution and sliding surface inclination constraint in three-dimensional space, (d) is the upper boundary point B Department yz Schematic diagram of the potential sliding surface inclination constraint on a plane.

[0026] Figure 9 Schematic diagram for introducing coordinate matching conditions at the boundary points of the potential sliding surface of the three-dimensional slope of the present invention, where (a) is the point B and Point C And point D Schematic diagram of the potential sliding surface polar angle and polar diameter in three dimensions, (b) is the left boundary point C Schematic diagram of the spatial relationship between the potential sliding surface polar angle and polar diameter, (c) is the upper boundary point B Schematic diagram of the spatial relationship between the potential sliding surface polar angle and polar diameter, (d) is the right boundary point D Schematic diagram of the spatial relationship between the polar angle and polar diameter of the potential sliding surface.

[0027] Figure 10 Schematic diagram of the present invention for discretizing the potential sliding surface of a three-dimensional slope and determining the intersection points between the potential sliding surface and the slope surface, wherein (a) is a schematic diagram for determining the discrete points of the three-dimensional potential sliding surface and the corresponding slope surface points, and (b) is a schematic diagram for determining the potential sliding surface. j Schematic diagram of determining the intersection points of a column of discrete points and the slope surface.

[0028] Figure 11 It is a flow chart of the feasibility determination steps of a potential sliding surface under the control of the intersection point of a three-dimensional potential sliding surface of a slope and a slope surface according to the present invention.

[0029] Figure 12It is a flow chart of the steps for judging the rationality of potential sliding surface under the concave mechanism of three-dimensional slope of the present invention.

[0030] Figure 13 It is a schematic diagram of discrete points and grid validity of potential sliding surface of three-dimensional slope of the present invention.

[0031] Figure 14 It is a flow chart of the steps for constructing a universal potential sliding surface model of a three-dimensional slope in the polar angle mode of the present invention.

[0032] In the figure: 1. 3D slope, 2. Potential sliding surface, 3. Slope surface, 4. Sliding boundary line, 5. Lower boundary point A , 6. Upper boundary point B , 7. Left boundary point C , 8. Right boundary point D , 9. Point A The plane where the slope surface is located, 10, xz Plane, 11. yz Plane, 12, Point A The direction of the minor principal stress surface, 13, Mohr stress circle, 14, shear strength curve, 15, point A The most unfavorable shear direction when shear failure occurs in the rock mass is 16. A The potential sliding surface tangent plane at 17, point B The plane where the slope surface is located, 18, point B The direction of the major principal stress action surface, 19, point B The most unfavorable shear direction when shear failure occurs in the rock mass is 20, point B At the potential sliding surface tangent plane, 21, any point on the potential sliding surface P , 22, sliding surface polar angle, 23, transverse line, 24, longitudinal line, 25, discrete points on the potential sliding surface P ij , 26. Intersection points, 27. Valid discrete points, 28. Invalid discrete points, 29. Valid discrete grids, 30. Invalid discrete grids. DETAILED DESCRIPTION

[0033] The specific model building method and implementation process of the present invention are as follows: Slope instability is caused by shear failure of rock mass and the formation of connected sliding surfaces. Whether the rock mass shear fails is controlled by the rock mass shear strength criterion. Therefore, the general expression of the rock mass shear strength criterion is: t f = f ( s ),in, t f is the shear strength of rock mass, sIt is the normal stress acting on the failure surface of the rock mass. At the same time, the shear failure of the rock mass indicates that the rock mass will slide along its most unfavorable shear direction. That is to say, the shear failure direction of the rock mass is the most unfavorable shear direction of the rock mass, which is also the tangent direction of the potential sliding surface.

[0034] like Figure 1 As shown in the figure, in a three-dimensional slope, any point is taken as the origin of the coordinate system, and the vertical direction is z Axis direction, with the longitudinal direction of the slope on the horizontal plane as x Axis direction, and the horizontal slope transverse direction y Axis direction, thus, establishing x.y.z Axis space coordinate system, for the three-dimensional potential sliding surface of the slope, its intersection line with the slope surface is the sliding boundary line. x The coordinate range on the axis is [ x min , x max ] and its y The coordinate range on the axis is [ y min , y max ], at the same time, corresponding to x min The sliding boundary point is point A , corresponding to x max The sliding boundary point is point B , corresponding to y min The sliding boundary point is point C , corresponding to y max The sliding boundary point is point D , further, point A ,point B ,point C and Point D They are named as the lower boundary point, upper boundary point, left boundary point and right boundary point of the potential slip surface respectively.

[0035] like Figure 2 As shown, for the lower boundary point of the potential sliding surface A At the slope surface, xz The horizontal inclination angle on the plane is β A_x and in yz The horizontal inclination angle on the plane is β A_y , and the horizontal inclination angle is positive when it is above the horizontal direction, and negative otherwise. x.y.z The origin of the axis coordinate system moves to point A , and aroundx Axis yz Axis coordinate system rotation angle β A_y , to obtain the local coordinate system , where the local coordinate system middle The direction of the axis is the point A The slope surface is yz The tangent on the plane, based on this, then around Axis Axis coordinate system rotation angle β A , to obtain the local coordinate system ,in, β A = arctan(tan β A_x cos β A_y ) and the local coordinate system middle The direction of the axis is the point A The slope surface is The tangent direction on the plane, thus, the plane For point A The plane where the slope surface is located, the plane For point A At the same time, at point A Taking the micro-segment sliding boundary line on both sides, it can be approximately considered that the micro-segment sliding boundary line and point A Also located in In the plane, according to the spatial geometric relationship between the translation and rotation of the coordinate axis, it can be known that the sliding boundary line x Axis coordinates and The axis coordinates retain the same properties, i.e. the point A The micro-segment sliding boundary lines on both sides are x The peak point in the axial direction is also the micro-segment sliding boundary line on both sides. The peak point in the axial direction. Furthermore, when the three-dimensional sliding body slides longitudinally along the slope, the point A The rock and soil along the boundary line of the micro-segment sliding on both sides The sliding trend in the axial direction is relative, so based on the continuity of the movement trend, it is bound to be at point A Nowhere The sliding trend in the axis direction means that the point A Shear failure of the soil mass will occur approximately at (or ) plane.

[0036] like Figure 3 As shown, the pointA Plane (or ) is the section plane that cuts the three-dimensional slope to obtain a two-dimensional slice of the three-dimensional slope, which is the point A The plane where the shear failure of the rock mass occurs approximately is named as 2D slice 1, and 2D slice 1 is analyzed. A It is located both on the slope surface and on the potential sliding surface. Since the slope surface is the external boundary, when its location information is clear, the point A The force on the slope surface at is known, and then the boundary stress constraint condition is formed at this point, so that the point A The stress state of the potential sliding surface is controlled, thereby affecting the most unfavorable shear direction of the rock and soil mass here and further restricting the shape of the potential sliding surface.

[0037] In 2D slice 1, point A The rock mass at point 1 is usually in a compressive state. If there is no external load on the slope surface at this point, A The slope surface is a point A The minor principal stress action surface is A The tangent direction of the slope surface is point A In the direction of the minor principal stress action surface, s 1_A and s 3_A Points A The major and minor principal stresses of the rock mass at s 3_A = 0. Then, in the coordinate system of normal stress and shear stress, the reflection point is plotted A The Mohr stress circle of the stress state and the shear strength curve of the rock and soil mass, where the center point of the Mohr stress circle is O A , the intersection of the Mohr stress circle and the normal stress axis is point E A and Point F A , and click E A and Point F A Corresponding to the points A Minor principal stress at s 3_A and major principal stress s 1_A , O A E A represents the direction of the minor principal stress action surface, O A T Ais a vertical line of the shear strength curve of the rock mass and passes through the center point O A .because O A T A Through the center point O A The shortest distance to the shear strength curve of the rock mass, therefore, O A T A Representative Points A The most unfavorable shear direction when shear failure occurs in the rock mass. T A Draw a tangent to the shear strength curve of the rock and soil mass, and the horizontal inclination angle of this tangent direction is f A , then ∠ T A O A E A = ( π / 2 – f A ),in, f A For point A Internal friction angle of sliding surface, ∠ T A O A E A Representative Points A The angle between the most unfavorable shear direction and the direction of the minor principal stress action surface when shear failure occurs in the rock mass. According to the double angle relationship between the Moore stress circle and the actual model, the midpoint of the actual model can be obtained. A When shear failure occurs in the soil mass, the angle between the most unfavorable shear direction and the direction of the minor principal stress action surface is ( π / 4 – f A / 2), and the most unfavorable shear direction is below the direction of the minor principal stress action surface. Furthermore, the most unfavorable shear direction is the tangent direction of the potential sliding surface, and the point A The horizontal inclination angle of the slope tangent direction is β A , then you can get points A The horizontal inclination angle of the tangent line of the potential sliding surface at α A = β A + f A / 2 – π / 4, which is also located in the two-dimensional slice 1. In addition, point A The tangent plane of the potential sliding surface is Axis and points within 2D slice 1 A Then, using the spatial geometric relationship, we can get the point A The tangent plane of the potential sliding surface is xz The horizontal inclination angle on the plane is α A_x = arctan(tan α A / cos β A_y ) and its yz The horizontal inclination angle on the plane is α A_y = β A_y .

[0038] like Figure 4 As shown, for the upper boundary point of the potential sliding surface B At the slope surface, xz The horizontal inclination angle on the plane is β B_x and in yz The horizontal inclination angle on the plane is β B_y , and the horizontal inclination angle is positive when it is above the horizontal direction, and negative otherwise. x.y.z The origin of the axis coordinate system moves to point B , and around x Axis yz Axis coordinate system rotation angle β B_y , to obtain the local coordinate system , where the local coordinate system middle The direction of the axis is the point B The slope surface is yz The tangent on the plane, based on this, then around Axis Axis coordinate system rotation angle β B , to obtain the local coordinate system ,in, β B = arctan(tan β B_x cos β B_y ) and the local coordinate system middle The direction of the axis is the point B The slope surface is The tangent direction on the plane, thus, the plane For point B The plane where the slope surface is located, the plane For point B At the same time, at point B Taking the micro-segment sliding boundary line on both sides, it can be approximately considered that the micro-segment sliding boundary line and point B Also located in In the plane, according to the spatial geometric relationship between the translation and rotation of the coordinate axis, it can be known that the sliding boundary line x Axis coordinates and The axis coordinates retain the same properties, i.e. the point B The micro-segment sliding boundary lines on both sides are x The peak point in the axial direction is also the micro-segment sliding boundary line on both sides. The peak point in the axial direction. Furthermore, when the three-dimensional sliding body slides longitudinally along the slope, the point B The rock and soil along the boundary line of the micro-segment sliding on both sides The sliding trend in the axial direction is relative, so based on the continuity of the movement trend, it is bound to be at point B Nowhere The sliding trend in the axis direction means that the point B Shear failure of the soil mass will occur approximately at (or ) plane.

[0039] like Figure 5 As shown, the point B Plane The 3D slope is cut by the section plane to obtain a 2D slice of the 3D slope, which is the point B The plane where the shear failure of the rock mass occurs approximately is named as 2D slice 2, and 2D slice 2 is analyzed. B It is located both on the slope surface and on the potential sliding surface. Since the slope surface is the external boundary, when its location information is clear, the point B The force on the slope surface at is known, and then the boundary stress constraint condition is formed at this point, so that the point B The stress state of the potential sliding surface is controlled, thereby affecting the most unfavorable shear direction of the rock and soil mass here and further restricting the shape of the potential sliding surface.

[0040] In 2D slice 2, point B The rock mass at point 1 is usually in a tensile state. If there is no external load on the slope surface, then point 2 B The slope surface is a point B The principal stress action surface is at B The tangent direction of the slope surface is point BIn the direction of the major principal stress action surface. Here, let s 1_B and s 3_B Points B The major and minor principal stresses of the rock mass at s 1_B = 0. Then, in the coordinate system of normal stress and shear stress, the reflection point is plotted B The Mohr stress circle of the stress state and the shear strength curve of the rock and soil mass, where the center point of the Mohr stress circle is O B , the intersection of the Mohr stress circle and the normal stress axis is point E B and Point F B , and click E B and Point F B Corresponding to the points B Minor principal stress at s 3_B and major principal stress s 1_B , O B F B represents the direction of the major principal stress action surface, O B T B Through the center point O B The perpendicular line to the shear strength curve of the rock and soil. O B T B Through the center point O B The shortest distance to the shear strength curve of the rock mass, therefore, O B T B Representative Points B The most unfavorable shear direction when shear failure occurs in the rock mass. T B Draw a tangent to the shear strength curve of the rock and soil mass, and the horizontal inclination angle of this tangent direction is f B , then ∠ T B O B F B = ( π / 2 + f B ),in, f B For point B Internal friction angle of sliding surface, ∠ T B O B F B Representative Points B The angle between the most unfavorable shear direction and the direction of the major principal stress action surface when shear failure occurs in the rock mass. According to the double angle relationship between the Moore stress circle and the actual model, the midpoint of the actual model can be obtained. B When shear failure occurs in the soil mass, the angle between the most unfavorable shear direction and the direction of the major principal stress action surface is ( π / 4 + f B / 2), and the most unfavorable shear direction is below the direction of the large principal stress action surface. Furthermore, the most unfavorable shear direction is the tangent direction of the potential sliding surface, and the point B The horizontal inclination angle of the slope tangent direction is β B , then you can get points B The horizontal inclination angle of the tangent line of the potential sliding surface at α B = β B + f B / 2 + π / 4, which is also located in the two-dimensional slice 2. In addition, point B The tangent plane of the potential sliding surface is Axis and points inside 2D slice 2 B Then, using the spatial geometric relationship, we can get the point B The tangent plane of the potential sliding surface is xz The horizontal inclination angle on the plane is α B_x = arctan(tan α B / cos β B_y ) and its yz The horizontal inclination angle on the plane is α B_y = β B_y .

[0041] like Figure 6 As shown in the figure, in a three-dimensional slope, any point on the potential sliding surface P , and its A Connect at xyz The inclination angle on the plane is i P, the inclination angle is named the sliding surface polar angle. i P , whose size is similar to that of the point P of x and y The axis coordinates are associated, that is, P of x and y The axis coordinate changes, which means that the sliding surface polar angle i P is a function of the sliding surface coordinates. According to the Taylor series definition, any continuous function can be approximately described by the sum of a polynomial and a high-order error residual. Furthermore, considering the reliability and ease of sliding surface generation, the sum of a cubic polynomial and a high-order error residual is used to construct the sliding surface polar angle. i P The functional relationship with the sliding surface coordinates is as follows: (1) In the formula, α A is the interior point of 2D slice 1 A The horizontal inclination of the tangent line of the potential sliding surface; l 0. l 1. l 2. l 3. l 4. l 5. l 6. l 7. l 8. l 9. l 10 , l 11 , l 12 , x 1. or 1. x 2 and or 2 is the potential sliding surface shape control parameter, l 0. l 1. l 2. l 4. l 7 and l The value range of 8 is -10 ~ 10, l 3 、l 5 、l 6 、l 9 、l 10 、l 11 and l 12 It can be solved based on the stress and position constraints at the boundary points of the potential sliding surface. x 1. or 1. x 2 and or The value range of 2 is 1 to 10; x P and y P are any points on the potential sliding surface. P of x and y Axis coordinates; x A and x B are the lower boundary points of the potential sliding surface A and upper boundary point B of x Axis coordinates; y C and y D are the left boundary points of the potential sliding surface C and the right boundary point D of y Axis coordinates.

[0042] At the same time, according to the spatial geometric relationship, any point on the potential sliding surface can be obtained P of z The axis coordinate calculation formula is: (2) In the formula, z P is any point on the potential sliding surface P of z Axis coordinates; y A and z A are the lower boundary points of the potential sliding surface A of y and z Axis coordinates.

[0043] In addition, when P Department y When the axis coordinates remain unchanged, if x Axis coordinates add differential dx , then any point on the potential sliding surface P The sliding surface polar angle differential component will be generated at dth P_x . Further, applying formula (1), we can get dth P_x The calculation formula is: (3) Similarly, when P Department xWhen the axis coordinates remain unchanged, if y Axis coordinates add differential day , then any point on the potential sliding surface P The sliding surface polar angle differential component will be generated at dth P_y . Further, applying formula (1), we can get dth P_y The calculation formula is: (4) like Figure 7 As shown, based on the lower boundary point of the potential sliding surface A The stress constraint condition at point A The tangent plane of the sliding surface is xz The horizontal inclination angle on the plane is α A_x ( α A_x = arctan(tan α A / cos β A_y )) and its yz The horizontal inclination angle on the plane is α A_y ( α A_y = β A_y ). Therefore, if you A of xz Select the sliding surface micro-segment on the plane AA' , then the sliding surface micro-segmentation AA' exist xz The horizontal inclination angle on the plane is α A_x , where point A' Comparison point A of x Axis coordinates add differential dx If you pass the point A of yz Select the sliding surface micro-segment on the plane AA'' , then the sliding surface micro-segmentation AA'' exist yz The horizontal inclination angle on the plane is α A_y , where point A'' Comparison point A of y Axis coordinates add differential day . Further, combined with the sliding surface polar angle function (i.e., formula (1)), we can obtain: (5) (6) Using equations (5) and (6), and substituting α A_x = arctan(tan α A / cos β A_y )and α A_y = β A_y , we can get the lower boundary point A The stress constraint condition that the sliding surface polar angle should satisfy is: (7) (8) like Figure 8 As shown, based on the upper boundary point of the potential sliding surface B The stress constraint condition at point B The tangent plane of the potential sliding surface is xz The horizontal inclination angle on the plane is α B_x ( α B_x = arctan(tan α B / cos β B_y )) and its yz The horizontal inclination angle on the plane is α B_y ( α B_y = β B_y ). To this end, B of xz Select the sliding surface micro-segment on the plane B'B , so that the sliding surface is micro-segmented B'B The polar angle of the middle sliding surface is x The partial differential component in the axial direction is dth B_x , where point B' Comparison point B of x Axis coordinates add differentials - dx Further, according to the potential sliding surface micro-segmentation B'B exist x The geometric definition of the differential component of the sliding surface polar angle in the axial direction can be derived as the differential component of the sliding surface polar angle dth B_x Angle with horizontal α B_x The functional relationship is: (9) In the formula,y B and z B are the upper boundary points of the potential sliding surface. B of y and z Axis coordinates.

[0044] Similarly, in the past B of yz Select the sliding surface micro-segment on the plane BB'' , so that the sliding surface is micro-segmented BB'' The polar angle of the middle sliding surface is y The partial differential component in the axial direction is dth B_y , where point B'' Comparison point B of y Axis coordinates add differential day . Further, according to the sliding surface micro-segmentation BB'' exist y The geometric definition of the differential component of the sliding surface polar angle in the axial direction can be derived as the differential component of the sliding surface polar angle d i B_y Angle with horizontal α B_y The functional relationship is: (10) Then, using equations (3) and (4), we can also get the point B The partial component of the polar angle of the slip surface dth B_x and dth B_y On this basis, combining equations (9) and (10), we can get the upper boundary point B The stress constraint condition that the sliding surface polar angle should satisfy is: (11) (12) like Figure 9 As shown, when the upper boundary point of the potential sliding surface B (The corresponding sliding surface polar angle is i B ), left boundary point C (The corresponding sliding surface polar angle is i C ) and the right boundary point D (The corresponding sliding surface polar angle is i D) is a known point, the sliding surface polar angle function must also satisfy the matching of the coordinates of the upper, left and right boundary points of the potential sliding surface, that is, the boundary position constraint condition. Then, combined with formula (1), and using the relationship between the coordinates of the spatial point and the sliding surface polar angle, the upper boundary point can be obtained. B , left boundary point C and the right boundary point D The position constraint condition where the sliding surface polar angle should match is: (13) (14) (15) The potential sliding surface shape control parameters can be established by using the stress constraints that the sliding surface polar angles at the lower and upper boundary points should satisfy and the position constraints that the sliding surface polar angles at the four boundary points should match, namely, equations (7), (8), (11), (12), (13), (14) and (15). l 3. l 5. l 6. l 9. l 10 , l 11 and l 12 The non-homogeneous system of equations is: (16) In the formula, a 11 ~ a 17 , a 21 ~ a 27 , a 31 ~ a 37 , a 41 ~ a 47 , a 51 ~ a 57 , a 61 ~ a 67 and a 71 ~ a 77 are coefficients of the equation system; b 1~ b 7 is the constant term of the equation system.

[0045] In formula (16), a 11 ~ a 17 and b The calculation formulas for 1 are: (17) In formula (16), a 21 ~ a 27 and b The calculation formulas for 2 are: (18) In formula (16), a 31 ~ a 37 and b The calculation formulas for 3 are: (19) In formula (16), a 41 ~ a 47 and b The calculation formulas for 4 are: (20) In formula (16), a 51 ~ a 57 and b The calculation formulas for 5 are: (twenty one) In formula (16), a 61 ~ a 67 and b The calculation formulas for 6 are: (twenty two) In formula (16), a 71 ~ a 77 and b The calculation formulas for 7 are: (twenty three) If the potential sliding surface shape control parameter is given l 0. l 1. l 2. l 4. l 7. l 8. x 1. x 2. or 1 and or 2, then the potential sliding surface shape control parameters can be solved by using formula (16) combined with the matrix calculation method: l 3. l 5. l 6. l 9. l 10 , l 11 and l 12 .

[0046] like Figure 10 As shown in the figure, the potential sliding surface is constructed by discrete technology, and then, at the boundary point below the potential sliding surface A and upper boundary point B of x Axis coordinate range (i.e. x A and x B range) will x The axis is divided into n and draw it based on n +1 lateral line at the left boundary point of the potential slip surface C and the right boundary point D of y Axis coordinate range (i.e. y C and y D range) will y The axis is divided into m and draw it based on m +1 vertical line for discrete points on the potential slip surface P ij (0 ≤ i ≤ n and 0 ≤ j ≤ m ), which is in xyz The projection on the plane corresponds to i The horizontal line and j The intersection points of the longitudinal lines, and then the potential sliding surface discrete points can be obtained P ij of x and y The axis coordinates are: (twenty four) (25) In the formula, xij and y ij are discrete points on the potential sliding surface. P ij of x and y Axis coordinates; Furthermore, the functional relationship between the sliding surface polar angle and the sliding surface coordinates (i.e., formula (2)) is combined, and x P and y P use x ij and y ij Substituting, we can get the potential sliding surface discrete points P ij of z The axis coordinates are: (26) In the formula, z ij Discrete points on the potential sliding surface P ij of z Axis coordinates.

[0047] In addition, the discrete grids divided by the longitudinal and transverse lines make the potential sliding surface calculation process x Axis coordinate differential dx and y Axis coordinate differential day They are: (27) (28) like Figure 10 and 11 As shown in the figure, considering the complexity of the three-dimensional slope surface, when the sliding surface shape parameters are not properly selected, there may be discrete points on the potential sliding surface. P ij All exceed the slope surface, or none exceed the slope surface, or partly exceed the slope surface but the point A ,point B ,point C and Point D In the case where one of the points is not a boundary point on one side of the potential sliding surface of the three-dimensional slope, these cases all indicate that the constructed potential sliding surface is not feasible. Therefore, the sliding surface polar angle should be subject to the potential sliding surface range limitation mechanism. In order to eliminate the infeasible potential sliding surface, the following can be used: j Column (0 ≤ j ≤ m ) in the potential sliding surface discrete points P ij (0 ≤i ≤ n ) are connected in sequence to form a broken line segment and the number of intersection points of the slope surface to determine the feasibility of the potential sliding surface. The specific operation steps (called the feasibility determination steps of the potential sliding surface under the control of the intersection points of the potential sliding surface and the slope surface) are as follows: ① Let j The number of intersections between the broken line segments composed of discrete points on the potential sliding surface and the slope surface is J d ( j ), and take j = 1; ② Take J d ( j ) = 0 and i = 1; ③ Obtain discrete points on the potential sliding surface P (i-1)j and P ij of z Axis coordinates z (i-1)j and z ij And the slope surface at the corresponding position z Axis coordinates s (i-1)j and s ij ④ Determine whether z (i-1)j ≤ s (i-1)j and z ij > s ij or z (i-1)j > s i-1j and z ij ≤ s ij , if not satisfied, jump to step ⑥; ⑤ J d ( j ) = J d ( j ) + 1; ⑥ Determine whether it is satisfied i < n If it is not satisfied, then i = i + 1, and return to step ③; ⑦ If J d ( j) = 2, it indicates that the potential sliding surface shape meets the intersection control condition. Otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible, and the potential sliding surface shape control parameters need to be reselected;⑧ Determine whether it meets j < m If it is not satisfied, then j = j + 1, and return to step ②.

[0048] like Figure 12 As shown in Figure 1, according to the slope failure mechanism, the reasonable potential sliding surface should be a concave surface in space. Therefore, the sliding surface polar angle should also obey the concave mechanism of the potential sliding surface. To this end, the rationality of the potential sliding surface is determined by whether conditions one and two are met. Condition one is the first condition. i Row (0 ≤ i ≤ n ) in the potential sliding surface discrete points P ij (0 ≤ j ≤ m ) are connected in sequence to form a broken line segment with increasing curvature on the sliding surface. The second condition is j Column (0 ≤ j ≤ m ) in the potential sliding surface discrete points P ij (0 ≤ i ≤ n ) are connected in sequence to form a broken line segment with increasing curvature on the sliding surface. The specific operation steps (called the rationality judgment steps of the potential sliding surface under the concave mechanism) are as follows: ① Let i = 0 and j = 0; ② Obtain discrete points on the potential sliding surface P ij , P (i+1)j and P i(j+1) coordinate( x ij , y ij , z ij )、( x (i+1)j , y (i+1)j , z (i+1)j )and( x i(j+1) , y i(j+1) , z i(j+1) );③Find discrete points P ijThe tangent plane of the sliding surface is xz Horizontal inclination on a plane α x_(i,j) ,in, ; ④ Obtain discrete points P ij The tangent plane of the sliding surface is yz Horizontal inclination on a plane α y_(i,j) ,in, ⑤ Determine whether the horizontal inclination of the sliding surface tangent satisfies and If it is not satisfied, it means that the selected potential sliding surface shape control parameters make the constructed potential sliding surface unreasonable, and the potential sliding surface shape control parameters need to be reselected; ⑥ Determine whether it satisfies i < n If it is not satisfied, then i = i + 1, and return to step ②; ⑦ Determine whether it is satisfied j <m-1 If it is not satisfied, then j = j + 1, and return to step ②.

[0049] like Figure 13 As shown in the figure, in the three-dimensional potential sliding surface discrete generation mode, the vertical and horizontal grid lines can be used to obtain ( n + 1)×( m + 1) discrete points, among which only the discrete points within the sliding boundary are discrete points on the potential sliding surface. In order to facilitate the combination with the limit equilibrium calculation method to carry out slope stability, the discrete points within the sliding boundary are defined as valid discrete points, and the discrete points outside the sliding boundary are defined as invalid discrete points. The validity of the discrete grids divided by the corresponding longitudinal and transverse lines is determined accordingly. That is, for the discrete grids ij , which consists of 4 discrete points, namely P (i-1)(j-1) ,point P (i-1)j ,point P i(j-1) and Point P ij If there are 3 or more discrete points as valid discrete points, the discrete grid ij is a valid grid, otherwise it is an invalid grid.

[0050] like Figure 14 As shown in the figure, the steps of constructing the universal potential sliding surface model of three-dimensional slope in polar angle mode are as follows: ① In the three-dimensional slope, take any point as the origin of the coordinate system to establish x.y.z Axis space coordinate system; ② Given the lower boundary point of the potential sliding surface A , upper boundary point B, left boundary point C and the right boundary point D and its coordinates, namely ( x A , y A , z A )、( x B , y B , z B )、( x C , y C , z C )and( x D , y D , z D ); ③According to the three-dimensional slope surface, obtain the lower boundary point of the potential sliding surface A and upper boundary point B Slope surface xz and yz The horizontal inclination angle of the tangent line on the plane is β A_x and β A_y , β B_x and β B_y ④ Formulate potential sliding surface shape control parameters l 0. l 1. l 2. l 4. l 7. l 8. x 1. x 2. or 1 and or 2; ⑤ According to the known slope tangent inclination angle at the potential sliding surface boundary point ( β A and β B ) and soil strength parameters ( f A and f B ), using the boundary stress constraint condition, solve the horizontal inclination angle of the potential sliding surface tangent in the plane where the shear failure of the rock mass occurs at the boundary point α A and α B, and use this to determine the potential sliding surface tangent plane at the boundary point xz and yz The horizontal inclination angle of the tangent line on the plane is α A_x , α A_y , α B_x and α B_y ⑥ Using equation (16), solve the potential sliding surface shape control parameters l 3. l 5. l 6. l 9. l 10 , l 11 and l 12 ⑦ Using equations (24), (25) and (26), the discrete points on the potential sliding surface are calculated respectively. P ij of x , y and z Axis coordinates, i.e. x ij , y ij and z ij (0 ≤ i ≤ n and 0 ≤ j ≤ m ⑧ Analyze the feasibility of the constructed potential sliding surface by using the potential sliding surface feasibility judgment step under the control of the intersection of the potential sliding surface and the slope surface. If it is not feasible, it is necessary to re-select the potential sliding surface shape control parameters; ⑨ Analyze the rationality of the constructed potential sliding surface by using the potential sliding surface rationality judgment step under the concave mechanism. If it is not reasonable, it is necessary to re-select the potential sliding surface shape control parameters; ⑩ Compare the discrete points on the potential sliding surface P ij of z Axis coordinates z ij and the corresponding slope point z Axis coordinates s ij ,like z ij ≤ s ij , then the discrete points P ij is a valid discrete point (0 ≤ i ≤ n and 0 ≤ j ≤ m ); The validity of the discrete points on the potential sliding surface is used to determine the validity of the discrete grid. P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points P ij There are 3 or more points in the i ≤ n and 1 ≤ j ≤ m ), then the discrete grid ij is a valid grid; Output the effective discrete points and effective mesh of the potential sliding surface.

[0051] The characteristics of the above-mentioned universal potential sliding surface of the three-dimensional slope of the present invention are: the potential sliding surface is generated by the sliding surface polar angle control method, and there is no need to limit the sliding surface type, thereby ensuring the arbitrariness of the potential sliding surface. At the same time, the four boundary points of the lower side, upper side, left side and right side are used to define the range of the potential sliding surface, and the sliding surface polar angle satisfies the stress constraint conditions at the upper and lower boundary points and the position constraint conditions at the four boundary points, so that the constructed potential sliding surface strictly satisfies the physical geometry and mechanical failure characteristics. In addition, the potential sliding surface range limitation and the concave surface mechanism are incorporated to ensure the feasibility and rationality of the potential sliding surface model. Subsequently, discrete technology is applied to obtain discrete points on the potential sliding surface, and correspondingly, effective discrete points and discrete grids on the potential sliding surface are identified, which is conducive to combining with the limit equilibrium method for conducting three-dimensional slope stability analysis under complex conditions.

[0052] The characteristics of the sliding surface polar angle of the present invention are: the sliding surface polar angle is associated with the sliding surface coordinates, and the Taylor series expansion method is used to construct an approximate function of the sliding surface polar angle and the sliding surface coordinates. At the same time, in order to ensure the simplicity, feasibility and diversity of the potential sliding surface model, the sliding surface polar angle approximate function is expressed by the sum of the cubic polynomial and the high-order error remainder in the Taylor series expansion, which can also make the potential sliding surface shape control parameters contained in the sliding surface polar angle approximate function more than the boundary constraints, thereby ensuring the arbitrariness of the potential sliding surface generation.

[0053] The characteristics of the above-mentioned boundary constraint conditions of the present invention are: the boundary constraint conditions are constraint conditions at four boundary points: the lower side, the upper side, the left side and the right side of the potential sliding surface, which include both boundary stress constraint conditions and boundary position constraint conditions, wherein the boundary stress constraint conditions utilize the position extreme value characteristics of the upper and lower boundary points, and based on the sliding trend of the rock and soil bodies on both sides of the boundary points, determine the approximate occurrence plane of shear failure of the rock and soil bodies at the upper and lower boundary points, on this basis, apply the property that the tangent direction of the potential sliding surface is the most unfavorable shear direction of the rock and soil body, and when the slope surface force is known, combine the shear strength criterion of the rock and soil body to obtain the most unfavorable shear direction of the rock and soil bodies at the upper and lower boundary points. The shear direction and the tangent plane of the potential sliding surface are taken into consideration. Furthermore, according to the spatial geometric relationship, the horizontal inclination angles of the tangent planes of the potential sliding surface at the upper and lower boundary points are solved. In addition, the potential sliding surface micro-segments are selected for analysis, and the mathematical relationship between the horizontal inclination angles of the tangent planes of the potential sliding surface at the upper and lower boundary points and the partial differential components of the sliding surface polar angle are derived. Thus, the boundary stress conditions are realized to constrain the potential sliding surface shape control parameters in the sliding surface polar angle approximation function. As for the boundary position constraint conditions, the known coordinates of the four boundary points are used to calculate the sliding surface polar angles at the boundary points. Then, the boundary position conditions are realized to constrain the potential sliding surface shape control parameters in the sliding surface polar angle approximation function. Example

[0054] A kind of Figure 1~Figure 14 A method for constructing a universal potential sliding surface model of a three-dimensional slope in a polar angle mode is shown in FIG. The slope height of a mountain slope along a highway is about 10 m, the average slope angle is 45°, and the slope soil weight is 19 kN / m 3 , the soil shear obeys the linear MC strength criterion, and the soil strength parameter is c = 20 kPa and f = 15°. Under the action of heavy rainfall, the houses on the rear edge of the mountain slope showed obvious cracks and deformation. In order to ensure traffic safety and prevent further slope collapse, the slope needs to be reinforced and protected. Among them, accurately and reliably capturing the most potentially dangerous sliding surface of the slope when it is unstable and damaged becomes the primary task. Therefore, the three-dimensional slope universal potential sliding surface construction method proposed in the present invention is introduced and applied, and the specific operation is as follows: Take a point in the slope area as the origin to establish x.y.z Space coordinate system, where z The axis direction is the vertical direction. x The axis direction is the longitudinal direction of the slope on the horizontal plane. y The axis direction is the horizontal direction of the slope. Then, based on the existing survey data, the boundary points on the lower side of the potential sliding surface of the slope are obtained. A The coordinates at (0 m, 0 m, 0 m) and the inclination angle of the sliding surface tangent β A_x andβ A_y They are β A_x = 45° and β A_y = 0°, upper boundary point of potential sliding surface B The coordinates of the location are (15 m, 0 m, 10 m) and the inclination angle of the sliding surface tangent β B_x and β B_y They are β B_x = 0° and β B_y = 0°, left boundary point of potential slip surface C The points marked as (5 m, -5 m, 5 m) and the left boundary point of the potential sliding surface D The coordinates are (5 m, 5 m, 5 m).

[0055] Propose potential sliding surface shape control parameters l 0. l 1. l 2. l 4. l 7 and l The value range of 8 is -10 ~ 10, as well as the potential sliding surface shape control parameter x 1. x 2. or 1 and or The value range of 2 is 1~10, and it is located at the lower boundary point of the potential sliding surface. A and upper boundary point B of x The axis coordinate range will be x The axis is divided into n , and n = 100, at the left boundary point of the potential slip surface C and the right boundary point D of y The axis coordinate range will be y The axis is divided into m , and m = 100, and then, within the value range of the potential sliding surface shape control parameter, a series of potential sliding surfaces of the slope can be formed for slope stability analysis; The three-dimensional slope stability limit equilibrium method is combined with a mathematical optimization algorithm to achieve slope stability analysis. In this process, when any set of sliding surface shape parameters is taken, l 0. l 1. l 2. l 4. l 7. l 8. x 1. x 2. or 1 and or 2, a reasonable potential sliding surface can be generated according to the steps of constructing the universal potential sliding surface model of the three-dimensional slope in the polar angle mode. Then, the limit equilibrium method is used to solve the safety factor of the slope under the corresponding potential sliding surface. Then, with the assistance of the mathematical optimization algorithm, the minimum value of the slope safety factor is taken as the optimization target, and the potential most dangerous sliding surface of the slope is searched. Thus, the purpose of slope stability analysis is completed, the minimum slope safety factor is output, and the result of the potential most dangerous sliding surface of the slope is given. Among them, the minimum slope safety factor is F s =1.31, the most potentially dangerous sliding surface shape control parameters are: l 0 = 3.32, l 1 = -1.25, l 2 = -0.59, l 3 = 1.75, l 4=1.78, l 5 = -5.05, l 6 = 3.94, l 7 = -8.40, l 8 = 7.64, l 9 = -2.55, l 10 = 2.94, l 11 = 1.93, l 12 =0.64, x 1 = 1.41, x 2 = 2.57, or 1 = 2.07 and or 2= ​​6.69. Based on the slope stability analysis results, according to the stability evaluation standard of the Technical Code for Building Slope Engineering (GB 50330-2013), the slope stability state is evaluated. At the same time, the potential instability range of the slope is clarified to provide a scientific basis for subsequent safe construction and reliable reinforcement of the slope.

[0056] The present invention uses the sliding surface polar angle control method to generate a potential sliding surface, and uses four boundary points of the lower side, upper side, left side and right side of the sliding surface to define the range of the potential sliding surface, and establishes stress constraints that the sliding surface polar angles at the lower and upper boundary points should satisfy, and applies discrete technology to embed the potential sliding surface model into the existing limit equilibrium method. The sliding surface model constructed by the present invention solves the problem of limiting the sliding surface type of the potential sliding surface model of the three-dimensional slope, and ensures the feasibility and rationality of the sliding surface shape and range, and strictly meets the physical and mechanical mechanisms of the potential sliding surface generation, which is convenient for carrying out stability analysis of three-dimensional slopes under complex conditions. The present invention has the advantages of being simple and easy to implement, having a wide range of applications, strong versatility, and high precision, and provides a strong scientific basis for reliable early warning and prevention of slope instability and landslides.

[0057] 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 universal potential sliding surface model of a three-dimensional slope in a polar angle mode, characterized in that: The method uses the sliding surface polar angle control method to generate a potential sliding surface, and uses four boundary points at the lower side, upper side, left side and right side of the sliding surface to define the range of the potential sliding surface. Then, based on the position extreme value characteristics of the lower boundary point and the upper boundary point of the sliding surface and the sliding trend of the rock and soil bodies on both sides at the two boundary points, the approximate occurrence plane of shear failure of the rock and soil bodies at the lower boundary point and the upper boundary point is determined. Then, according to the tangent direction of the potential sliding surface as the most unfavorable shear direction of the rock and soil body, when the slope surface force is known, combined with the shear strength criterion of the rock and soil body, the stress constraint conditions that the sliding surface polar angles at the lower boundary point and the upper boundary point should satisfy are established. parts; and the sliding surface polar angle also matches the position constraints at the above four boundary points; the sliding surface polar angle is associated with the sliding surface coordinates, and the approximate function of the sliding surface polar angle and the sliding surface coordinates is constructed using the Taylor series expansion method. In addition to satisfying the stress and position constraints at the potential sliding surface boundary points, the sliding surface polar angle is also subject to the potential sliding surface range limitation and the surface concave mechanism, so as to ensure that the constructed potential sliding surface is feasible and reasonable; and the discrete technology is applied to obtain the discrete points on the potential sliding surface, and then the valid discrete points and discrete grids on the potential sliding surface are identified and output, so as to construct a general potential sliding surface model of three-dimensional slope under the polar angle mode.

2. The method for constructing a three-dimensional slope universal potential sliding surface model in polar angle mode according to claim 1, characterized in that: The method comprises the following steps: Step S1: In the three-dimensional slope, take any point as the origin of the coordinate system to establish xyz Axis space coordinate system; Step S2: Given the lower boundary point of the potential sliding surface A , upper boundary point B , left boundary point C and the right boundary point D and its coordinates, namely ( x A , y A , z A )、( x B , y B , z B )、( x C , y C , z C )and( x D , y D , z D ); Step S3: Obtain the lower boundary point of the potential sliding surface according to the three-dimensional slope surface orientation A and upper boundary point B Slope surface xz and yz The horizontal inclination angle of the tangent line on the plane is β A_x and β A_y , β B_x and β B_y ; Step S4: Determine potential sliding surface shape control parameters λ 0. λ 1. λ 2. λ 4. λ 7. λ 8. ξ 1. ξ 2. η 1 and η 2; Step S5: Based on the known slope tangent inclination angle at the potential sliding surface boundary point β A and β B And soil strength parameters φ A and φ B , using the boundary stress constraint condition, solve the horizontal inclination angle of the potential sliding surface tangent in the plane where the shear failure of the rock mass occurs at the boundary point α A and α B , and use this to determine the potential sliding surface tangent plane at the boundary point xz and yz The horizontal inclination angle of the tangent line on the plane is α A_x , α A_y , α B_x and α B_y ; Step S6: Using equation (16), solve the potential sliding surface shape control parameter λ 3. λ 5. λ 6. λ 9. λ 10 , λ 11 and λ 12 ; (16) In the formula, a 11 ~ a 17 , a 21 ~ a 27 , a 31 ~ a 37 , a 41 ~ a 47 , a 51 ~ a 57 , a 61 ~ a 67 and a 71 ~ a 77 are coefficients of the equation system; b 1 ~ b 7 is the constant term of the equation system; Step S7: Using equations (24), (25) and (26), the discrete points on the potential sliding surface are calculated respectively: P ij of x , y and z Axis coordinates, i.e. x ij , y ij and z ij , 0 ≤ i ≤ n and 0 ≤ j ≤ m ; (24) (25) (26) Step S8: Analyze the feasibility of the constructed potential sliding surface by using the potential sliding surface feasibility judgment step under the control of the intersection point of the potential sliding surface and the slope surface; if it is not feasible, it is necessary to reselect the potential sliding surface shape control parameters; Step S9: Analyze the rationality of the constructed potential sliding surface by using the rationality judgment step of the potential sliding surface under the concave mechanism; if it is unreasonable, it is necessary to reselect the potential sliding surface shape control parameters; Step S10: Comparing discrete points on the potential sliding surface P ij of z Axis coordinates z ij and the corresponding slope point z Axis coordinates s ij ,like z ij ≤ s ij , then the discrete points P ij is a valid discrete point, 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S11: Determine the validity of the discrete grid using the validity of the discrete points on the potential sliding surface. P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points P ij There are 3 or more points in the table that are valid discrete points, 1 ≤ i ≤ n and 1≤ j ≤ m , then the discrete grid ij is a valid grid; Step S12: Output the effective discrete points and effective grids of the potential sliding surface.

3. The method for constructing a three-dimensional slope universal potential sliding surface model in polar angle mode according to claim 2, characterized in that: In the step S4, λ 0. λ 1. λ 2. λ 4. λ 7 and λ The value range of 8 is -10 ~ 10, ξ 1. η 1. ξ 2 and η The value range of 2 is 1 ~ 10.

4. The method for constructing a three-dimensional slope universal potential sliding surface model in polar angle mode according to claim 2, characterized in that: In step S8, in order to eliminate the infeasible potential sliding surface, according to j Potential sliding surface discrete points in the column P ij The feasibility of the potential sliding surface is determined by the number of intersections between the broken line segments and the slope surface, 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S8 specifically includes the following steps: Step S8-1: j The number of intersection points between the broken line segments composed of discrete points on the potential sliding surface and the slope surface is J d ( j ), and take j = 1; Step S8-2: Get J d ( j ) = 0 and i = 1; Step S8-3: Obtaining discrete points on the potential sliding surface P (i-1)j and P ij of z Axis coordinates z (i-1)j and z ij And the slope surface at the corresponding position z Axis coordinates s (i-1)j and s ij ; Step S8-4: Determine whether z (i-1)j ≤ s (i-1)j and z ij > s ij or z (i-1)j > s i-1j and z ij ≤ s ij ; If not satisfied, jump to step S8-6; Step S8-5: J d ( j ) = J d ( j ) + 1; Step S8-6: Determine whether i < n If it is not satisfied, then i = i + 1, and return to step S8-3; Step S8-7: If J d ( j ) = 2, it indicates that the potential sliding surface shape satisfies the intersection control condition; otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible, and the potential sliding surface shape control parameters need to be reselected; Step S8-8: Determine whether j < m If it is not satisfied, then j = j + 1, and return to step S8-2.

5. The method for constructing a three-dimensional slope universal potential sliding surface model in polar angle mode according to claim 2, characterized in that: In step S9, the sliding surface polar angle should be subject to the potential sliding surface concave mechanism, and the rationality of the potential sliding surface is determined by whether the following conditions 1 and 2 are met. Condition 1 is i Potential slip surface discrete points in the row P ij The curvature of the sliding surface of the broken line segments formed by connecting them in sequence is increasing. The second condition is j Potential sliding surface discrete points in the column P ij The curvature of the sliding surface formed by the broken line segments connected in sequence is increasing; 0 ≤ i ≤ n and 0 ≤ j ≤ m ; The step S9 specifically comprises the following steps: Step S9-1: i = 0 and j = 0; Step S9-2: Obtaining discrete points on the potential sliding surface P ij , P (i+1)j and P i(j+1) The coordinates of ( x ij , y ij , z ij )、( x (i+1)j , y (i+1)j , z (i+1)j )and( x i(j+1) , y i(j+1) , z i(j+1) ); Step S9-3: Obtaining discrete points P ij The tangent plane of the sliding surface is xz Horizontal inclination on a plane α x_(i,j) ,in, Step S9-4: Obtaining discrete points P ij The tangent plane of the sliding surface is yz Horizontal inclination on a plane α y_(i,j) ,in, Step S9-5: Determine whether the horizontal inclination angle of the sliding surface tangent satisfies: and ; If it is not satisfied, it indicates that the selected potential sliding surface shape control parameters make the constructed potential sliding surface unreasonable, and the potential sliding surface shape control parameters need to be reselected; Step S9-6: Determine whether i < n If not satisfied, then i = i + 1, and return to step S9-2; Step S9-7: Determine whether j < m-1 If not satisfied, then j = j + 1, and return to step S9-2.

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