A method for constructing a universal three-dimensional potential sliding surface model of slopes under rotation mode

Abstract

The present invention belongs to the field of slope engineering, and specifically relates to a method for constructing a general three-dimensional slope potential sliding surface model under a rotational mode, including constructing the potential sliding surface by making the potential sliding surface a rotational shear failure surface and utilizing the physical and mechanical relationship between the shear failure direction and the motion direction of the potential sliding surface. The solution of the potential sliding surface is then converted into the determination of the shear failure plane at any point on the potential sliding surface, and then an angle function characterizing the position of the shear failure plane under a Taylor series expansion is established. On this basis, discrete technology and recursive calculation methods are applied, and a potential sliding surface range limitation and a curved surface concave mechanism are introduced to achieve effective generation of the potential sliding surface. The present invention has the advantages of being simple and reliable, having a wide range of applications, strong versatility, high accuracy, and being fast and convenient. The potential sliding surface is generated simply, reasonably, and effectively, thereby providing a prerequisite for reliable slope stability analysis and providing powerful guidance for the implementation of slope engineering reinforcement measures under complex conditions.

Description

Technical Field

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

[0002] Among existing models of potential sliding surfaces for slopes, the two-dimensional (2D) model remains the mainstream. For this model, the slope is simplified as an infinite half-space volume, and the potential sliding surface is represented by a curve. Furthermore, based on various analytical approaches and methods, various curve morphologies are constructed to simulate slope slip on a two-dimensional plane, thereby enabling the construction of 2D potential sliding surface models. These include arcs, straight lines, logarithmic spirals, combined curves, and arbitrary curves defined by functions. Since slopes in practical engineering projects are typically three-dimensional, 2D potential sliding surface models have certain limitations in their application. Therefore, developing 3D potential sliding surface models is crucial for reliable slope stability analysis. Existing 3D potential sliding surface models primarily rely on specific surfaces, such as cylinders, ellipsoids, and logarithmic spirals. Within this limited surface form, these models manipulate the sliding surface morphology through varying surface geometric parameters to construct a regular potential sliding surface. In addition, there is also the construction of special-shaped ellipsoids by giving different combinations of the major semi-axis, median semi-axis and minor semi-axis of the ellipsoid to simulate the potential sliding surface of a complex three-dimensional slope, and the use of two-dimensional arbitrary generatrix and directrix and the mutual movement and spatial expansion between the generatrix and directrix to obtain the potential arbitrary sliding surface of a three-dimensional slope.

[0003] Existing 2D slope potential sliding surface models rely on the simplified assumption of an infinite half-space volume, making the potential sliding surface construction very simple. However, the resulting slope stability analysis results are relatively conservative, which is not conducive to project cost control. Furthermore, they fail to demonstrate the 3D failure characteristics of actual slopes, making it difficult to provide more accurate guidance for slope engineering. The simple regular surfaces used in existing 3D sliding surface models can reflect 3D slope failure characteristics to a certain extent, but are constrained by the limited surface form and are not conducive to the arbitrary display of the 3D slope potential sliding surface morphology. Complex irregular surfaces, which account for the morphological differences between different parts of the potential sliding surface, are still constrained by a set of assumed surface or curve morphologies, making it difficult to further demonstrate the spatial variation trends of the 3D slope potential sliding surface under complex conditions.

[0004] As slope engineering projects advance into challenging mountainous areas, accurately and efficiently identifying the most dangerous potential sliding surfaces within slopes and providing scientific and detailed construction guidance requires the construction of a comprehensive and universal three-dimensional potential sliding surface model. However, current methods for constructing sliding surface models have limitations and are unable to fully meet the requirements for accurate and reliable analysis in slope engineering. Therefore, a new and universal method for constructing a three-dimensional potential sliding surface model for slopes is needed. Summary of the Invention

[0005] The present invention provides a method for constructing a universal three-dimensional slope potential sliding surface model under a rotational mode, the method comprising: setting the potential sliding surface as a rotational shear failure surface, and utilizing the physical and mechanical relationship between the shear failure direction and the motion direction of the potential sliding surface to construct the potential sliding surface, and then converting the solution of the potential sliding surface into the determination of the shear failure plane of any point on the potential sliding surface, and then establishing an angle function characterizing the position of the shear failure plane under a Taylor series expansion. On this basis, discrete technology and recursive calculation methods are applied, and a potential sliding surface range limitation and a curved surface concave mechanism are introduced to achieve effective generation of the potential sliding surface; and the method comprises the following steps: step S1, establishing a three-dimensional slope with any point as the origin x.y.z Axis coordinate system; Step S2, give the sliding point on the lower side of the potential sliding surface A exist x 、 y and z Coordinates on the axis ( x A , y A , z A )as well as AB Connected xyz Projection on a plane x Angle between axes ψ , where point B is the potential sliding surface upper sliding point; step S3, xyz Axis coordinate system around z Axis rotation angle ψ , so that x' Axis and AB Connected xyz The projections on the plane are parallel, thus establishing x'y'z Axis coordinate system, at the same time, use formula (1) to solve the point A exist x' and y' Coordinates on the axis ( , );

[0006] (1)

[0007] Step S4: Set the maximum width of the left and right potential sliding surfaces w left and w right and the maximum length of the potential sliding surface L Step S5: Develop the three-dimensional slope potential sliding surface model parameters, i.e., point A Sliding surface extreme diameter r A 、 OAConnection x' Angle between axes i A , variable factors k and sliding surface shape control parameters l 0~ l 9, among which, point O Step S6, using the potential sliding surface rotation center point and the boundary discrete point coordinate solving step, solve the potential sliding surface rotation center point O and boundary discrete points P 0j and P i0 exist x' 、 y' and z The coordinates on the axis, where i = 0 ~ n and j = – m 1~ m 2, n is the number of discrete directional lines of the potential sliding surface, m 1 and m 2 are the number of discrete column lines on the left and right potential sliding surfaces respectively; Step S7, using the coordinates of discrete points on the potential sliding surface to recursively solve the steps, calculate the discrete points on the potential sliding surface P ij exist x' 、 y' and z The coordinates on the axis and its shear failure plane along x' Axis and y' The inclination angle in the direction of the axis; Step S8, the feasibility judgment step of the potential sliding surface is controlled by the number of intersection points, and the feasibility of the constructed potential sliding surface is analyzed. If it is not feasible, the shape control parameters of the potential sliding surface need to be re-selected; Step S9, the rationality judgment step of the potential sliding surface is controlled by the concave mechanism, and the rationality of the constructed potential sliding surface is analyzed. If it is not rational, the shape control parameters of the potential sliding surface need to be re-selected; Step S10, based on the discrete points on the potential sliding surface P ij and the corresponding slope points G ij , using these two points z The size of the axis is used to determine the valid and invalid discrete points, that is, when z ij ≤ s ij The corresponding discrete points are valid discrete points, otherwise they are invalid discrete points. z ij for point P ij of z Axis coordinates, sij for point G ij of z Axis coordinates; Step S11, when the discrete point P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points P ij When there are 3 or more valid discrete points in the grid, determine the discrete grid ij is a valid grid; step S12, outputting the valid discrete points and valid grid of the potential sliding surface.

[0008] In a specific embodiment, in step S6, according to the sliding point on the lower side of the potential sliding surface A , first calculate the potential sliding surface rotation center point O The coordinates of the sliding surface are then calculated using the shear failure plane of the sliding surface at the discrete points. x' and y' The inclination angle in the axial direction is obtained to obtain the coordinates of the discrete points on the 0th column line and the 0th row line; and the steps for solving the coordinates of the potential sliding surface rotation center point and the boundary discrete points specifically include the following steps: Step S6-1: Use formula (25) to solve the point O exist x' 、 y' and z Coordinates on the axis ( , , );

[0009] (25)

[0010] Step S6-2: Based on the potential sliding surface of the three-dimensional slope, [ , ] within the range y ′ axis coordinates are divided into m 1 copy, , ] within the range y ′ axis coordinates are divided into m 2 parts, [ , ] within the range x ′ axis is divided into n Then, the potential sliding surface is discretized within the scope of the potential sliding surface, and ( n +1) ×( m 1+ m 2+1) discrete points on the potential sliding surface, and then use equations (17) and (18) to calculate the discrete points Pij exist x' and y' The coordinates on the axis, where i = 0 ~ n and j = – m 1~ m 2; among them, is the left boundary point of the potential slip surface y' Minimum axis coordinate, is the right boundary point of the potential sliding surface y' The maximum value of the axis coordinate, The upper sliding point of the potential sliding surface x' Maximum value of axis coordinate;

[0011] (17)

[0012] (18)

[0013] in, and Discrete points P ij of x' Axis and y' axis coordinates;

[0014] Step S6-3: Make i = 1; Step S6-4: Use equations (20) to (23) to solve the discrete points P (i-1)0 Angle oh (i-1)0 , sliding surface extreme diameter r (i-1)0 , sliding surface polar angle or (i-1)0 The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i-1)0 ;

[0015] (20)

[0016] (twenty one)

[0017] (twenty two)

[0018] (twenty three)

[0019] in, oh ij for point P ij Angle, rij for point P ij Sliding surface extreme diameter, or ij for point P ij The polar angle of the sliding surface, x is the dilatancy angle between the sliding surface movement direction and the shear failure direction;

[0020] Step S6-5: Use equation (26) to solve the discrete points P i0 of z axis coordinates;

[0021] (26)

[0022] Step S6-6: If i ≥ n , then end the loop, otherwise, let i = i + 1, and return to step S6-4; step S6-7: let j =–1; Step S6-8: Use equations (20) to (22) and (24) to solve the discrete points P 0(j+1) Angle oh 0(j+1) , sliding surface extreme diameter r 0(j+1) , sliding surface polar angle or 0(j+1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_0(j+1) ;

[0023] (twenty four)

[0024] Step S6-9: Use equation (27) to solve the discrete points P 0j of z Axis coordinates z 0j ;

[0025] (27)

[0026] Step S6-10: If i ≤ – m 1, then end the loop, otherwise, let j = j – 1, and return to step S6-8; step S6-11: let j = 1; Step S6-12: Use equations (20) to (22) and (24) to solve the discrete pointsP 0(j-1) Angle oh 0(j-1) , sliding surface extreme diameter r 0(j-1) , sliding surface polar angle or 0(j-1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_0(j-1) ; Step S6-13: Use formula (27) to solve the discrete points P 0j of z Axis coordinates z 0j ; Step S6-14: If i ≥ m 2, then end the loop, otherwise, let j = j + 1, and return to step S6-12; Step S6-15: Output the calculation result, that is, the potential sliding surface rotation center point O exist x' 、 y' and z Coordinates on the axis and discrete points on the potential slip surface boundary P 0j and P i0 exist z Coordinates on the axis.

[0027] In a specific embodiment, in step S7, after the coordinates of the potential sliding surface rotation center point and the boundary discrete points are determined, the coordinates of the discrete points on the left and right potential sliding surfaces are further recursively obtained. z Axis coordinates, and then obtain the spatial position and shape characteristics of discrete points on the potential sliding surface; and the recursive solution step of the coordinates of discrete points on the potential sliding surface specifically includes the following steps: Step S7-1: Get i = 0 and j = 0; Step S7-2: For the potential sliding surface on the left, j <0, use equations (20) to (24) to solve the discrete points P (i+1)j Angle oh (i+1)j , sliding surface extreme diameter r (i+1)j , sliding surface polar angle or (i+1)j The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i+1)j and discrete points P i(j-1) Angle ohi(j-1) , sliding surface extreme diameter r i(j-1) , sliding surface polar angle or i(j-1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_i(j-1) ; Step S7-3: For the potential sliding surface on the right side, that is j ≥0, it is necessary to use formula (20) to formula (24) to calculate the discrete points respectively P (i+1)j Angle oh (i+1)j , sliding surface extreme diameter r (i+1)j , sliding surface polar angle or (i+1)j The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i+1)j and discrete points P i(j+1) Angle oh i(j+1) , sliding surface extreme diameter r i(j+1) , sliding surface polar angle or i(j+1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_i(j+1) ; Step S7-4: For the potential sliding surface on the left, that is j <0, use formula (19) to calculate the discrete points P (i+1)(j-1) exist z Coordinates on the axis z (i+1)(j-1) ;

[0028] (19)

[0029] Step S7-5: For the potential sliding surface on the right side, j ≥ 0, use formula (19) to calculate the discrete points P (i+1)(j+1) exist z Coordinates on the axis z (i+1)(j+1) ; Step S7-6: For the potential sliding surface on the left, that is j <0, determine whether it is satisfied j ≤ – m 1+1, if not satisfied, then let j = j –1 and return to step S7-2; Step S7-7: For the potential sliding surface on the right, that is, j ≥ 0, determine whether it is satisfiedj ≥ m 2–1, if not satisfied, then let j = j +1 and return to step S7-3; Step S7-8: Determine whether i ≥ n – 1, if not satisfied, then let i = i +1 and return to step S7-2; Step S7-9: Output the coordinates of the discrete points on the potential sliding surface and the shear failure plane of the sliding surface x' Axis and y' The inclination angle of the axis.

[0030] In a specific embodiment, in step S8, the specific implementation step of determining whether the potential slip surface is feasible, that is, the step of determining the feasibility of the potential slip surface under the control of the number of intersection points, specifically includes the following steps: Step S8-1: Let i The number of intersection points between the broken line segments composed of discrete points on the potential sliding surface and the slope surface is Row ( i ), and take i = 1; Step S8-2: Get Row ( i ) = 0 and j = – m 1+1; Step S8-3: Obtain discrete points on the potential sliding surface P i(j-1) and P ij of z Axis coordinates z i(j-1) and z ij and the corresponding slope points G i(j-1) and G ij of z Axis coordinates s i(j-1) and s ij ; Step S8-4: Determine whether z i(j-1) ≤ s i(j-1) and z ij > s ij or z i(j-1) > s i(j-1) and z ij ≤ s ij, if satisfied, then let Row ( i ) = Row ( i ) + 1, if not satisfied, go to the next step; Step S8-5: Determine whether it is satisfied j ≥ m 2. If not satisfied, then j = j + 1, and return to step S8-3; Step S8-6: If Row ( i ) = 2, it indicates that the potential sliding surface shape of this row meets the intersection control condition, otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible; Step S8-7: Determine whether it meets i ≥ n If it is not satisfied, then i = i + 1, and return to step S8-2; step S8-8: let j The number of intersections between the broken line segments formed by the discrete points on the potential sliding surface and the slope surface is Col ( j ), and take j = – m 1+1; Step S8-9: Take Col ( j ) = 0 and i = 1; Step S8-10: 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 corresponding slope points G (i-1)j and G ij of z Axis coordinates s (i-1)j and s ij ; Step S8-11: Determine whether z (i-1)j ≤ s (i-1)j and z ij > s ij or z (i-1)j > s (i-1)j and z ij ≤s ij , if satisfied, then let Col ( j ) = Col ( j ) + 1, if not satisfied, go to the next step; Step S8-12: Determine whether it is satisfied i ≥ n If it is not satisfied, then i = i + 1, and return to step S8-10; step S8-13: If Col ( i ) = 2, it indicates that the potential sliding surface shape of this column meets the intersection control condition, otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible; Step S8-14: Determine whether it meets j ≥ m 2. If not satisfied, then j = j + 1, and return to step S8-9; if satisfied, it indicates that the potential slip surface is feasible.

[0031] In a specific embodiment, in step S9, the rationality of the shape of the potential sliding surface is determined by whether the potential sliding surface satisfies the first and second conditions, wherein the first condition is the first condition. i Discrete points on the potential sliding surface P ij Corresponding edge y' Horizontal inclination of tangent line in the axis direction α y'_ij is an increasing function, 0 ≤ i ≤ n , condition 2 is j List discrete points on the potential sliding surface P ij Corresponding edge x' Horizontal inclination angle of tangent line in the axis direction tan α x'_ij is an increasing function, – m 1≤ j ≤ m 2, and the rationality judgment step of the potential sliding surface under the concave mechanism control specifically includes the following steps: Step S9-1: Let i = 0 and j = – m 1; Step S9-2: Determine discrete points on the potential sliding surface P ij and P i(j+1) Failure plane along y' Horizontal inclination of the axis α y'_ij and αy'_i(j+1) and discrete points on the potential sliding surface P ij and P (i+1)j Failure plane along x' Horizontal inclination of the axis α x'_ij and α x'_(i+1)j ; Step S9-3: Determine whether α y'_ij ≤ α y'_i(j+1) and α x'_ij ≤ α x'_(i+1)j If it is satisfied, it means that the sliding surface at the discrete point meets the concave characteristic mechanism, otherwise, it indicates that the potential sliding surface is unreasonable; Step S9-4: Determine whether it meets i ≥ n If it is not satisfied, then i = i + 1, and return to step S9-2; Step S9-5: Determine whether j ≥ m 2. If not satisfied, then j = j + 1, and return to step S9-2; if satisfied, it indicates that the potential slip surface is reasonable.

[0032] The present invention has the advantages of simplicity, reliability, wide applicability, strong versatility, high accuracy, and speed and convenience. The constructed three-dimensional slope potential sliding surface model follows the principle of rotational shear slip. At the same time, under the strict physical and mechanical mechanisms of potential sliding surface generation, not only the sliding surface shape and range are controllable but also the complex actual sliding surface morphology can be displayed. 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 landslide. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 This is a schematic diagram of the potential sliding surface of the three-dimensional slope of the present invention.

[0034] Figure 2 The potential sliding surface division of the left and right sides of the three-dimensional slope of the present invention and x'y'z Schematic diagram of establishing the axis coordinate system.

[0035] Figure 3 It is a schematic diagram of the scope of the potential sliding surface of the three-dimensional slope of the present invention.

[0036] Figure 4 This is a schematic diagram of the movement direction and shear failure direction at any point on the potential sliding surface of the three-dimensional slope of the present invention.

[0037] Figure 5 It is a schematic diagram of the motion plane and shear failure plane of the sliding surface at any point on the potential sliding surface of the three-dimensional slope of the present invention.

[0038] Figure 6 The shear failure plane of the sliding surface at any point on the potential sliding surface of the three-dimensional slope of the present invention is x' Axis and y' Schematic diagram of axial inclination.

[0039] Figure 7 It is a schematic diagram of the differential relationship of coordinate increments at any point on the potential sliding surface of a three-dimensional slope according to the present invention.

[0040] Figure 8 This is a schematic diagram of the discrete potential sliding surface of a three-dimensional slope according to the present invention.

[0041] Figure 9 This is a schematic diagram of the rotation center point and boundary discrete points of the potential sliding surface of the three-dimensional slope of the present invention.

[0042] Figure 10 This is a flow chart of the steps for solving the coordinates of the rotation center point and boundary discrete points of the potential sliding surface of a three-dimensional slope in the present invention.

[0043] Figure 11 This is a flow chart of the steps for recursively solving the coordinates of discrete points on the potential sliding surface of a three-dimensional slope in the present invention.

[0044] Figure 12 This is a schematic diagram of the intersection of the potential sliding surface and the slope surface of the three-dimensional slope of the present invention.

[0045] Figure 13 This is a flow chart of the steps for determining the feasibility of a potential sliding surface under the control of the number of intersection points of the present invention.

[0046] Figure 14 This is a flow chart of the steps for determining the rationality of a potential sliding surface under the control of the concave mechanism of the present invention.

[0047] Figure 15 This is a schematic diagram of the potential sliding surface discrete points and grid validity judgment of the three-dimensional slope in the present invention.

[0048] Figure 16 This is a flow chart of the steps for constructing a universal three-dimensional slope potential sliding surface model in the rotation mode of the present invention.

[0049] In the figure: 1. 3D slope, 2. Potential sliding surface, 3. Rotation center, 4. Any point on the potential sliding surface P , 5. Movement direction, 6. Shear failure direction, 7. Slope surface, 8. Sliding boundary line, 9. Sliding boundary line xyz Projection on the plane, 10, bottom slide point A , 11. Upper side slide out pointB , 12, AB Connected xyz Projection on the plane, 13, AB The vertical plane of the connecting line, 14, the potential sliding surface on the left, 15, the potential sliding surface on the right, 16, the passing point O of x'y' Plane, 17, Plane 1, 18, Plane 2, 19, Sliding surface polar diameter, 20, Sliding surface polar angle, 21, Point P The motion plane of the sliding surface, 22, point P Shear failure plane of the sliding surface, 23, x'z plane, 24, y'z Plane, 25, No. i Line 26, j Column-oriented line, 27, discrete points on the potential sliding surface, 28, the 0th row-oriented line, 29, the 0th column-oriented line, 30, the 0th i The vertical plane of the line, 31, through the j Vertical plane of the strip line, 32. Valid discrete points, 33. Invalid discrete points, 34. Valid discrete grid, 35. Invalid discrete grid. DETAILED DESCRIPTION

[0050] The present invention constructs the potential sliding surface by making the potential sliding surface a rotational shear failure surface and utilizing the physical and mechanical relationship between the shear failure direction and the movement direction of the potential sliding surface. Then, the solution of the potential sliding surface is converted into the determination of the shear failure plane at any point on the potential sliding surface, and then an angular function characterizing the position of the shear failure plane under the Taylor series expansion is established. On this basis, discrete technology and recursive calculation methods are applied, and the potential sliding surface range limitation and curved surface concave mechanism are introduced to achieve the simple, reasonable and effective generation of the potential sliding surface, thereby providing a prerequisite for reliable slope stability analysis and providing strong guidance for the implementation of slope engineering reinforcement measures under complex conditions.

[0051] Specifically, this method assumes the potential sliding surface is a shear failure surface, with the rotation center as the reference point. The potential sliding surface is then constructed using the physical and mechanical relationship between the shear failure direction and the direction of motion of the potential sliding surface. The rotation center and the slip points below and above the potential sliding surface are located on the same vertical plane. Furthermore, the solution to the potential sliding surface is converted into the determination of the shear failure plane at any point on the potential sliding surface, and the coordinate increments of adjacent points on the potential sliding surface are associated with the horizontal inclination angle of the shear failure plane of the potential sliding surface along the corresponding coordinate axis. The shear failure plane of any point on the potential sliding surface is constrained by the shear failure direction at that point, which in turn depends on the plane passing through the rotation center. Accordingly, this plane is controlled by the position of its intersection with the horizontal plane and can be represented by an angle. At the same time, this angle corresponds one-to-one with the coordinates of the points on the potential sliding surface. Furthermore, a Taylor series expansion method is used to establish its functional expression. Based on this, the range of the potential sliding surface of the three-dimensional slope is delineated, and within this range, the potential sliding surface is discretized using equally divided row and column lines, and then the sliding point under the potential sliding surface is solved. A The coordinates of discrete points on the row and column lines are then calculated using the coordinates of adjacent points, recursively solving for the coordinates of other discrete points on the potential sliding surface using a mean calculation method. Irrational and infeasible potential sliding surfaces are eliminated, subject to the potential sliding surface range and concave surface mechanism. Furthermore, discrete points on the potential sliding surface are obtained using discretization techniques, and valid discrete points and discrete grids on the potential sliding surface are identified accordingly, making them suitable for slope stability analysis using the limit equilibrium method.

[0052] The specific model construction method and implementation process of the present invention are as follows:

[0053] like Figure 1 As shown in Figure 1, the three-dimensional slope collapse is caused by the slope rock and soil reaching the shear failure state and forming a connected potential sliding surface. At the same time, the three-dimensional slope collapse is a rotational failure, that is, the three-dimensional slope collapse is the potential sliding body rotating around the center point. O The phenomenon of sliding along the potential sliding surface and becoming unstable toward the free surface. P The direction of movement is usually perpendicular to the direction of the line connecting the rotation center to the point. Further, the potential sliding surface is the three-dimensional slope shear failure surface, so the point on the potential sliding surface is P The tangent direction of is the shear failure direction of the point. According to the correlation and non-correlation flow rules, it can be known that the point on the potential slip surface P There is a correlation between the movement direction and the shear failure direction.

[0054] like Figure 1 As shown in the figure, in the three-dimensional slope, any point is taken as the origin and the vertical direction is z Axis direction, with the longitudinal direction of the slope on the horizontal plane as xAxis direction, and the horizontal slope direction y Axis direction, establish x.y.z Axis coordinate system. For the potential sliding surface of the three-dimensional slope, the intersection line with the slope surface is the sliding boundary line. For the sliding boundary line, let it be xyz The lowest boundary point projected on the plane is the sliding point below the potential sliding surface of the three-dimensional slope A ,point A exist x.y.z The coordinates in the axis coordinate system are ( x A , y A , z A ).

[0055] like Figure 2 As shown, for the sliding boundary line, let it be xyz The uppermost boundary point projected on the plane is the upper sliding point of the potential sliding surface of the three-dimensional slope B .further, AB The vertical plane where the connecting line is located is the main sliding plane of the potential sliding body of the three-dimensional slope, and the rotation center of the potential sliding body is O Located in AB In the vertical plane of the connecting line. AB Connection, which is xyz Projection on a plane x The angle between the axes is ψ , the angle is positive in the clockwise direction and negative in the counterclockwise direction, thus, ψ The value range is (– π / 2, π / 2). Then, based on the AB The vertical plane of the connecting line divides the three-dimensional slope potential sliding surface into two parts: the left potential sliding surface and the right potential sliding surface. x.y.z Axis coordinate system around z Axis rotation angle ψ , so that x' Axis and AB Connected xyz The projections on the plane are parallel, and thus the x'y'z Axis coordinate system. Here, let the rotation center point O exist x'y'z The coordinates in the axis coordinate system are ( , , ), and according to the spatial geometric relationship between the rotating coordinate systems, we can get the point A exist x'y' The coordinates in the axis coordinate system are:

[0056] (1)

[0057] Where, and The lower slide-out points A of x' Axis and y' Axis coordinates.

[0058] like Figure 3 As shown in the figure, the scope of the potential sliding surface of the three-dimensional slope is delineated. For the potential sliding surface on the left, its maximum width is w left , for the potential sliding surface on the right, its maximum width is w right Correspondingly, the left boundary point of the potential slip surface y' The minimum axis coordinate is , the right boundary point of the potential slip surface y' The maximum value of the axis coordinate is In addition, for the entire potential slip surface, its maximum length is L Correspondingly, the potential slip surface upper slip point x' The maximum value of the axis coordinate is , thus, all points on the potential sliding surface of the three-dimensional slope x' and y' The axis coordinate range is [ , ]and[ , ].

[0059] like Figure 4 As shown, let any point on the potential sliding surface P exist x'y'z The coordinates in the axis coordinate system are ( , , z P ), and ordered OP The vertical plane of is plane 1, and point P In the past O of x'y' The projection point on the plane is point E . Plane 2 is a passing point P The shear failure direction of the sliding surface is OP The plane in the direction of the line, and plane 2 and the point O of x'y' Intersect at point F At the same time, click F Located at the passing point E and parallel to x' In addition, the point P The sliding surface movement direction is also located in plane 2. According to the correlation and non-correlation flow laws, there is a dilatancy angle between the sliding surface movement direction and the shear failure direction. x , and the dilatancy angle x Internal friction angle with sliding surface f There is a functional relationship. Here, let x = arctan( k tan f ),in, k is a variable factor and reflects the sliding surface friction angle f and dilatancy angle x A dimensionless parameter of the relationship between them.

[0060] In plane 2, let ∠ FOP for point P Polar angle of the slip surface or , OP The distance between the lines is the point P Sliding surface extreme diameter r If you click P The slip surface increases the polar angle differential along the direction of motion d , then the corresponding sliding surface diameter also increases by a differential amount dr . Further, based on the potential sliding surface point P The direction of movement is perpendicular to the center of rotation O Solstice P The direction of the connecting line, and according to the spatial geometric relationship between the polar angle and the polar diameter differential of the sliding surface, we can get:

[0061] (2)

[0062] like Figure 5 As shown, let OF and x' The angle between the axes is oh ,and oh is the clockwise angle, and its value range is [0, 2 π ]. Further, based on point P The motion plane of the sliding surface not only passes through the point P The sliding surface is in the direction of motion and is perpendicular to plane 2, thus determining point P At the same time, based on the point P The shear failure plane of the sliding surface not only passes through the point P The shear failure direction of the sliding surface at is also perpendicular to plane 2, so the point P The shear failure plane of the sliding surface at . P The shear failure direction of the sliding surface at point O of x'y' The intersection point of the planes is a point F' , for point P The direction of movement of the sliding surface at point O of x'y' The intersection point of the planes is a pointF'' In addition, according to the point P The angle relationship between the sliding surface movement direction and the shear failure direction and the sliding surface movement direction OP The lines are perpendicular, so ∠ OPF' = 90° – x .

[0063] like Figure 6 As shown, point P The shear failure plane of the sliding surface and the passing point P of x'z The intersection line between the planes is PC , PC Connection x' The angle between the axes (i.e. the point P The shear failure plane of the sliding surface is along x' The inclination angle in the axial direction is α x' Similarly, click P The shear failure plane of the sliding surface and the passing point P of y'z The intersection line between the planes is PD , PD Connection y' The angle between the axes (i.e. the point P The shear failure plane of the sliding surface is along y' The inclination angle in the axial direction is α y' .

[0064] exist x'y'z In the axis coordinate system, OP The direction vector is , parallel to OF The unit vector of the direction is , thus, we can get the plane OPF (i.e., the normal vector of plane 2) n OPF respectively x' axis, y' Axis and z The components in the axial direction are:

[0065] (3)

[0066] (4)

[0067] (5)

[0068] Where, n OPF_x' , n OPF_ y' and n OPF_zThey are n OPF exist x' axis, y' Axis and z Component in the direction of the axis.

[0069] In addition, using PF' The geometric relationship of the triangle can be deduced PF' Direction vector n PF′ respectively x' axis, y' Axis and z The components in the axial direction are:

[0070] (6)

[0071] (7)

[0072] (8)

[0073] Where, n PF'_x' n PF'_ y' and n PF'_z They are n PF′ exist x' axis, y' Axis and z Component in the direction of the axis.

[0074] Thus, using the plane OPF The normal vector n OPF and PF' Direction vector n PF′ , good points P The normal vector of the shear failure plane of the sliding surface is n P respectively x' axis, y' Axis and z The components in the axial direction are:

[0075] (9)

[0076] (10)

[0077] (11)

[0078] Where, n P_x' , nP_y' and n P_z They are n P exist x' 、 y' and z Component in the direction of the axis.

[0079] like Figure 6 As shown, the midpoints of formula (9) to formula (11) P Sliding surface extreme diameter r and the sliding surface polar angle or Available points O with dot P Coordinate relationships and trigonometric OPF The geometric relationship is derived:

[0080] (12)

[0081] (13)

[0082] Therefore, using the point P The normal vector of the shear failure plane of the sliding surface at n P , you can determine the point P The shear failure plane equation of the sliding surface at point P The shear failure plane of the sliding surface is along x' Axis tilt α x' for:

[0083] (14)

[0084] Similarly, use the P The normal vector of the shear failure plane of the sliding surface at n P , get point P The shear failure plane of the sliding surface is along y' Axis tilt α y' for:

[0085] (15)

[0086] like Figure 7 As shown, according to the point P The shear failure plane of the sliding surface is along x' Axis tilt α x' It can be seen that if P Potential slip surface x' Increase the differential amount in the axial direction dx' , then accordinglyz Axis coordinates increase differential increment dz x' ,and dz x' = dx' tan α x' , similarly, according to the point P The shear failure plane of the sliding surface is along y' Axis tilt α y' It can be seen that if P Potential slip surface y' Increase the amount of trace you' , then accordingly z Axis coordinates increase differential increment dz y' ,and dz y' = you' tan α y' .

[0087] For any point on the potential sliding surface of a three-dimensional slope P , the plane 2 where the sliding surface movement direction and shear failure direction are located will be determined by the angle oh When the angle oh In [0, 2 π ]When the value changes within the range, click P The spatial position of plane 2 changes accordingly. At the same time, the point P The direction of the shear failure plane of the sliding surface will also change synchronously, and then the angle oh The change of will affect the shape of the potential sliding surface. In addition, the midpoint of the potential sliding surface of the three-dimensional slope P Coordinates and angles oh One-to-one correspondence, that is, angle oh with dot P There is a functional relationship between the coordinates of . Therefore, within the range of potential sliding surface, the Taylor series expansion method is applied to establish the angle oh Functional form, so as to realize the continuous and arbitrary construction of universal potential sliding surface. Thus, the angle oh The functional form is:

[0088] (16)

[0089] Where, and are any points on the potential sliding surface of the three-dimensional slope P of x' Axis and y' axis coordinates; l 0. l 1. l 2. l 3. l 4. l 5. l 6. l 7. l 8 and l 9 is the sliding surface shape control parameter, where l 0~ l The value range of 9 is –1 to 1.

[0090] It should be noted that the angle oh The value range is [0, 2 π ], when the angle calculated by formula (16) oh More than 2 π Or when it is less than 0, it can be corrected to [0, 2 π ] range, and the potential sliding surface generated thereby is also continuous and reasonable.

[0091] like Figure 8 As shown, AB Connected x ' y ′ plane as the reference, [ , ] within the range y ′ axis coordinates are divided equally and drawn m 1 column line, numbered from left to right – m 1~ 0. At the same time, [ , ] within the range y ′ axis coordinates are divided equally and drawn m 2+1 column lines, numbered from left to right from 0 to m 2. In addition, [ , ] within the range x ′ axis coordinates are divided equally and drawn n +1 row line, numbered from bottom to top from 0 to n Then, within the scope of potential slip surface, construct n ×( m 1+ m 2) discrete grids and ( n +1)×( m 1+ m 2+1) discrete points. i The row line and the j The discrete point on the potential sliding surface corresponding to the intersection of the strip-line is the point P ij (0≦ i ≦n and- m 1≦ j ≦ m 2) Point P ij exist x' and y' The coordinates on the axis are:

[0092] (17)

[0093] (18)

[0094] Where, and Discrete points P ij of x' Axis and y' Axis coordinates.

[0095] For discrete points P ij of z Axis coordinates, which can be used as points P (i+1)j (Potential slip surface on the left) or point P (i-1)j The shear failure plane of the sliding surface at (potential sliding surface on the right) is along x' The inclination angle in the axial direction, x' When the axis coordinate increases the differential z The axis coordinates are obtained by increasing the corresponding differential increment, or by using the point P i(j-1) The shear failure plane of the sliding surface is along y' The inclination angle in the axial direction, y' When the axis coordinate increases the differential z The axis coordinates are obtained by increasing the corresponding differential increment. P (i+1)j (Potential slip surface on the left) or point P (i-1)j (Potential sliding surface on the right) and point P i(j-1) The two points P ij of z Axis coordinates are affected, and the mean calculation method is used to solve the point P ij of z Axis coordinates. Therefore, the point P ij of z Axis coordinates z ij The specific calculation formula is:

[0096] (19)

[0097] As for the point P ij Angle oh ij , sliding surface extreme diameter r ij Polar angle with sliding surface or ij , can be respectively replaced in formula (16), formula (12) and formula (13) 、 and z P use 、 and z ij The solution can be obtained by replacing it. The specific calculation formula is:

[0098] (20)

[0099] (twenty one)

[0100] (twenty two)

[0101] As for the point P ij The shear failure plane of the sliding surface is along x' and y' Axis tilt α x'_ij and α y'_ij , can be respectively replaced in formula (14) and formula (15) oh 、 or 、 r 、 、 and z P use oh ij 、 or ij 、 r ij 、 、 and z ij The solution can be obtained by replacing it. The specific calculation formula is:

[0102] (twenty three)

[0103] (twenty four)

[0104] like Figure 9 As shown, the rotation center point of the potential sliding surface of the three-dimensional slopeO In terms of AB On the vertical plane where the connecting line is located. When the sliding point on the lower side of the given potential sliding surface A Sliding surface extreme diameter r A as well as OA Connection x' Angle between axes i A When, based on the known point A exist x'y'z The coordinates in the axis coordinate system can be solved to get the point O The coordinates of ( , , ), the specific calculation formula is:

[0105] (25)

[0106] In order to facilitate the recursive solution of the coordinates of discrete points on the potential sliding surface, first, based on the known sliding points on the lower side of the potential sliding surface A , using the shear failure plane of the sliding surface at the discrete point along x' The inclination angle in the axial direction is used to obtain the coordinates of the discrete points on the 0th column line. Similarly, the shear failure plane of the sliding surface at the discrete point is used to calculate the shear failure plane along the sliding surface. y' The inclination angle in the axial direction is obtained, and the coordinates of the discrete points on the 0th row line are obtained. Then, the 0th row line and the 0th column line are used as the boundaries of the left and right potential slip surfaces, and the unknown discrete points (left potential slip surface) are recursively solved from the two adjacent known discrete points on the right and below, or the unknown discrete points (right potential slip surface) are recursively solved from the two adjacent known discrete points on the left and below.

[0107] For the discrete point on the 0th column line, z The axis coordinate calculation formula is:

[0108] (26)

[0109] For the discrete points on the 0th row line, z The axis coordinate calculation formula is:

[0110] (27)

[0111] like Figure 10 As shown, according to the sliding point on the lower side of the potential sliding surface A , we can first calculate the potential sliding surface rotation center point O Then, the shear failure plane of the sliding surface at the discrete point is used to calculate the coordinates of x' and y'The inclination angle in the direction of the axis is used to obtain the coordinates of the discrete points on the 0th column line and the 0th row line. The specific process (called the step of solving the coordinates of the potential sliding surface rotation center point and the boundary discrete point) is as follows: ① Use formula (25) to solve the point O exist x' 、 y' and z Coordinates on the axis ( , , );② Based on the scope of potential sliding surface of three-dimensional slope, [ , ] within the range y ′ axis coordinates are divided into m 1 copy, , ] within the range y ′ axis coordinates are divided into m 2 parts, [ , ] within the range x ′ axis is divided into n Then, the potential sliding surface is discretized within the range of the potential sliding surface, and ( n +1)×( m 1+ m 2+1) discrete points on the potential sliding surface, and then, using equations (17) and (18) to calculate the discrete points P ij exist x' and y' The coordinates on the axis, where i = 0 ~ n and j = – m 1~ m 2; ③ Order i =1; ④ Use equations (20) to (23) to solve the discrete points P (i-1)0 Angle oh (i-1)0 , sliding surface extreme diameter r (i-1)0 , sliding surface polar angle or (i-1)0 The shear failure plane of the sliding surface is along y' Axis tilt α x'_(i-1)0 ⑤ Use formula (26) to solve the discrete points P i0 of z Axis coordinates z i0 ⑥ If i ≥ n , then end the loop, otherwise, leti = i + 1, and return to step ④; ⑦ j = –1;⑧Use equations (20) to (22) and (24) to solve the discrete points P 0(j+1) Angle oh 0(j+1) , sliding surface extreme diameter r 0(j+1) , sliding surface polar angle or 0(j+1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_0(j+1) ; ⑨ Use formula (27) to solve the discrete points P 0j of z Axis coordinates z 0j ⑩ If i ≤ – m 1, then end the loop, otherwise, let j = j – 1, and return to step ⑧; make j = 1; Using equations (20) to (22) and (24) to solve the discrete points P 0(j-1) Angle oh 0(j-1) , sliding surface extreme diameter r 0(j-1) , sliding surface polar angle or 0(j-1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_0(j-1) ; Use formula (27) to solve the discrete points P 0j of z Axis coordinates z 0j ; like i ≥ m 2, then end the loop, otherwise, let j = j + 1 and return to step ; Output calculation results, i.e. the potential sliding surface rotation center point O exist x' 、 y' and z Coordinates on the axis and discrete points on the potential slip surface boundary P 0j andP i0 exist z Coordinates on the axis.

[0112] like Figure 11 As shown in the figure, after the coordinates of the potential sliding surface rotation center point and the boundary discrete points are determined, the coordinates of the discrete points on the left and right potential sliding surfaces can be further recursively obtained. z Axis coordinates, and then, obtain the spatial position and shape characteristics of discrete points on the potential sliding surface. The specific steps (called the recursive solution steps for the coordinates of discrete points on the potential sliding surface) are as follows: ① Get i = 0 and j = 0; ② For the left potential slip surface ( j <0), use equations (20) to (24) to solve the discrete points P (i+1)j Angle oh (i+1)j , sliding surface extreme diameter r (i+1)j , sliding surface polar angle or (i+1)j The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i+1)j and discrete points P i(j-1) Angle oh i(j-1) , sliding surface extreme diameter r i(j-1) , sliding surface polar angle or i(j-1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_i(j-1) ; ③ For the potential sliding surface on the right ( j ≥ 0), it is necessary to use formula (20) to formula (24) to calculate the discrete points respectively P (i+1)j Angle oh (i+1)j , sliding surface extreme diameter r (i+1)j , sliding surface polar angle or (i+1)j The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i+1)j and discrete points P i(j+1) Angle oh i(j+1) , sliding surface extreme diameter r i(j+1) , sliding surface polar angle or i(j+1) The shear failure plane of the sliding surface is alongy' Axis tilt α y'_i(j+1) ④ For the potential sliding surface on the left ( j <0), use formula (19) to calculate the discrete points P (i+1)(j-1) exist z Coordinates on the axis z (i+1)(j-1) ⑤ For the potential sliding surface on the right ( j ≥ 0), use formula (19) to calculate the discrete points P (i+1)(j+1) exist z Coordinates on the axis z (i+1)(j+1) ⑥ For the potential sliding surface on the left ( j <0), determine whether j ≤ – m 1+1, if not satisfied, then let j = j –1 and return to step ②; ⑦ For the potential sliding surface on the right ( j ≥ 0), to determine whether j ≥ m 2–1, if not satisfied, then let j = j +1 and return to step ③;⑧ Determine whether it satisfies i ≥ n – 1, if not satisfied, then let i = i +1 and return to step ②; ⑨ Output the coordinates of the discrete points on the potential sliding surface and the shear failure plane of the sliding surface x' Axis and y' The inclination angle of the axis.

[0113] like Figure 12 As shown in the figure, due to the complexity of the three-dimensional slope surface, when the potential sliding surface parameters are not properly selected, there may be more than two intersections between the potential sliding surface and the slope surface in the vertical plane of the row or column line within the scope of the three-dimensional slope potential sliding surface (condition 1), no intersection (condition 2), and only one intersection (condition 3). These conditions indicate that the constructed potential sliding surface is not feasible. Therefore, in order to ensure the feasibility of the potential sliding surface construction, let the discrete points P ij The corresponding point on the three-dimensional slope surface is point G ij , and based on the potential sliding surface point P ij Points on the slope G ij of z The relationship between the axis coordinates determines whether the potential sliding surface intersects the slope surface.i Row (0 ≤ i ≤ n ) Discrete points on the potential sliding surface P ij (– m 1≤ j ≤ m 2) The number of intersections between the broken line segments and the slope surface formed by sequential connection and the number of intersections between the broken line segments and the slope surface j List(- m 1≤ j ≤ m 2) Discrete points on the potential sliding surface P ij (0 ≤ i ≤ n ) is used to determine whether the potential sliding surface is feasible by the number of intersection points between the broken line segments formed by sequentially connecting the two segments and the slope surface.

[0114] like Figure 13 As shown in the figure, the specific implementation steps for judging whether the potential sliding surface is feasible (called the feasibility judgment steps of the potential sliding surface under the control of the number of intersection points) are as follows: ① Let i The number of intersection points between the broken line segments composed of discrete points on the potential sliding surface and the slope surface is Row ( i ), and take i = 1;② Take Row ( i ) = 0 and j = – m 1+1; ③ Obtain discrete points on the potential sliding surface P i(j-1) and P ij of z Axis coordinates z i(j-1) and z ij and the corresponding slope points G i(j-1) and G ij of z Axis coordinates s i(j-1) and s ij ④ Determine whether z i(j-1) ≤ s i(j-1) and z ij > s ij or z i(j-1) > s i(j-1) andz ij ≤ s ij , if satisfied, then let Row ( i ) = Row ( i ) + 1, if not satisfied, go to the next step; ⑤ Determine whether it is satisfied j ≥ m 2. If not satisfied, then j = j + 1, and return to step ③; ⑥ If Row ( i ) = 2, it means that the potential sliding surface shape of this row meets the intersection control condition, otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible; ⑦ Determine whether it meets i ≥ n If it is not satisfied, then i = i + 1, and return to step ②; ⑧ Let j The number of intersections between the broken line segments formed by the discrete points on the potential sliding surface and the slope surface is Col ( j ), and take j = – m 1+1; ⑨ Take Col ( j ) = 0 and i = 1; ⑩ Get 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 corresponding slope points G (i-1)j and G ij of z Axis coordinates s (i-1)j and s ij ; Determine whether it is satisfied z (i-1)j ≤ s (i-1)j and z ij > s ij or z (i-1)j > s (i-1)j andz ij ≤ s ij , if satisfied, then let Col ( j ) = Col ( j ) + 1, if not satisfied, go to the next step; Determine whether it is satisfied i ≥ n If it is not satisfied, then i = i + 1, and return to step ⑩; like Col ( i ) = 2, it indicates that the potential sliding surface shape of this column meets the intersection control condition; otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible; Determine whether it is satisfied j ≥ m 2. If not satisfied, then j = j + 1, and return to step ⑨; if satisfied, it indicates that the potential sliding surface is feasible.

[0115] like Figure 14 As shown in the figure, based on the slope rotation failure mechanism, it can be seen that the reasonable three-dimensional slope potential sliding surface should be a concave surface. Therefore, the rationality of the potential sliding surface shape is judged by whether the potential sliding surface meets conditions one and two. Among them, condition one is the first condition. i Row (0 ≤ i ≤ n ) Discrete points on the potential sliding surface P ij Corresponding edge y' Horizontal inclination of tangent line in the axis direction α y'_ij is an increasing function, condition 2 is j List(- m 1≤ j ≤ m 2) Discrete points on the potential sliding surface P ij Corresponding edge x' Horizontal inclination angle of tangent line in the axis direction tan α x'_ij is an increasing function. The specific implementation steps (called the rationality judgment steps of potential sliding surface under concave mechanism control) are as follows: ① Let i = 0 and j = – m 1;② Determine discrete points on the potential sliding surface P ij and P i(j+1) Failure plane alongy' Horizontal inclination of the axis α y'_ij and α y'_i(j+1) and discrete points on the potential sliding surface P ij and P (i+1)j Failure plane along x' Horizontal inclination of the axis α x'_ij and α x'_(i+1)j ; ③ Determine whether it is satisfied α y'_ij ≤ α y'_i(j+1) and α x'_ij ≤ α x'_(i+1)j , if satisfied, it means that the sliding surface at the discrete point satisfies the concave characteristic mechanism, otherwise, it indicates that the potential sliding surface is unreasonable; ④ Determine whether it satisfies i ≥ n If it is not satisfied, then i = i + 1, and return to step ②; ⑤ Determine whether it is satisfied j ≥ m 2. If not satisfied, then j = j + 1, and return to step ②; if satisfied, it indicates that the potential slip surface is reasonable.

[0116] like Figure 15 As shown in the figure, in the three-dimensional slope potential sliding surface discrete generation mode, only the discrete points on the slope surface and below the slope surface are valid discrete points of the potential sliding surface. In order to facilitate the combination with the limit equilibrium calculation method to carry out slope stability analysis, it is necessary to judge the validity of the discrete points. Here, based on the discrete points on the potential sliding surface P ij and the corresponding slope points G ij , be on point P ij with dot G ij of z Axis coordinates satisfy z ij > s ij When the discrete points on the potential sliding surface are determined P ij If it exceeds the slope, it is defined as an invalid discrete point, otherwise it is a valid discrete point. Further, according to the validity of the discrete point, the validity of the discrete grid divided by the corresponding row and column lines is determined, that is, for the discrete grid ij, which consists of 4 discrete points, namely discrete points P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points 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.

[0117] like Figure 16 As shown in the figure, the steps for constructing a general 3D slope potential sliding surface model in the rotation mode are as follows: ① In the 3D slope, establish a x.y.z Axis coordinate system; ② Given the sliding point on the lower side of the potential sliding surface A exist x 、 y and z Coordinates on the axis ( x A , y A , z A )as well as AB Connected xyz Projection on a plane x Angle between axes ψ ③ will xyz Axis coordinate system around z Axis rotation angle ψ , so that x' Axis and AB Connected xyz The projections on the plane are parallel, thus establishing x'y'z Axis coordinate system, at the same time, use formula (1) to solve the point A exist x' and y' Coordinates on the axis ( , );④ Set the maximum width of the potential sliding surface on the left and right sides w left and w right and the maximum length of the potential sliding surface L ⑤ Develop the parameters of the 3D slope potential sliding surface model, i.e. A Sliding surface extreme diameter r A 、 OA Connection x' Angle between axes i A , variable factors k and sliding surface shape control parameters l 0~ l 9;⑥ Use the coordinates of the potential sliding surface rotation center point and the boundary discrete point solution steps to solve the potential sliding surface rotation center point O and boundary discrete points P 0j and P i0 exist x' 、 y' and z The coordinates on the axis, where i = 0 ~ n and j = – m 1~ m 2; ⑦ Use the recursive solution steps of the coordinates of discrete points on the potential sliding surface to calculate the discrete points on the potential sliding surface P ij exist x' 、 y' and z The coordinates on the axis and its shear failure plane along x' Axis and y' 8. Using the number of intersection points to control the feasibility of the potential sliding surface, analyze the feasibility of the constructed potential sliding surface. If it is not feasible, reselect the shape control parameters of the potential sliding surface. 9. Using the concave mechanism to control the rationality of the potential sliding surface, analyze the rationality of the constructed potential sliding surface. If it is not rational, reselect the shape control parameters of the potential sliding surface. 10. Based on the discrete points on the potential sliding surface, the potential sliding surface is judged to be reasonable. P ij and the corresponding slope points G ij , using these two points z The size of the axis is used to determine the valid and invalid discrete points, that is, when z ij ≤ s ij When , the corresponding discrete point is a valid discrete point, otherwise it is an invalid discrete point; When discrete points P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points P ij When there are 3 or more valid discrete points in the grid, determine the discrete grid ij is a valid grid; Output the effective discrete points and effective mesh of the potential sliding surface.

[0118] Example

[0119] A kind of Figure 1~Figure 16The method for constructing a general three-dimensional slope potential sliding surface model under the rotation mode is shown. The slope height of a certain mountain slope is about 20 m, the average slope angle is 45°, and the soil weight is 19 kN / m 3 The shear failure of the slope rock and soil obeys the linear MC strength criterion, and the soil strength parameters are: cohesion c = 20 kPa and internal friction angle f = 27°. Heavy rainfall caused significant cracking and deformation in the houses on the rear edge of this mountain slope. To prevent further slope collapse, urgent reinforcement and protection measures were required. Accurately and reliably determining the most potentially dangerous sliding surface of this slope in the event of instability and failure was a top priority. Therefore, the potential sliding surface construction method invented in this paper was introduced and applied. The specific steps are as follows:

[0120] Established with a point in space as the origin x.y.z Then, based on the existing survey data, the slip point on the lower side of the potential slip surface of the slope is estimated. A exist x 、 y and z The coordinates on the axis are (0 m, 0 m, 0 m), AB Connection direction xyz Projection on the plane and x Angle between axes ψ = 0°, the maximum width of the potential sliding surface on the left and right sides is w left = 20 m and w right = 20 m and the maximum length of the potential sliding surface is L = 30 m.

[0121] Setting the 3D potential sliding surface morphology control parameters l 0~ l The value range of 9 is –1~1, variable factor k The value range is 0~3, point A Sliding surface extreme diameter r A The value range of is [0 m, 100 m] and OA Connection x' Angle between axes i A The value range is [0, π / 2], and take the maximum width of the left and right potential sliding surfaces as the number of equal parts m 1 = 40 and m 2 = 40 and x' The coordinate range is divided into n= 60, and then, within the scope of potential sliding surface and the range of potential sliding surface shape parameters, a series of potential sliding surfaces of the slope are formed for use in slope stability analysis;

[0122] Combined with the limit equilibrium method and embedded in the mathematical optimization algorithm, the slope stability analysis is realized. In this process, when any set of sliding surface shape parameters is taken l 0~ l 9. k 、 r A and i A After that, a reasonable and feasible potential sliding surface can be generated according to the steps of constructing the general three-dimensional slope potential sliding surface model under the rotation mode. 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 value of the slope safety factor is used as the optimization target, and the most dangerous potential sliding surface of the slope is searched. In this way, the purpose of slope stability analysis is completed, the minimum slope safety factor and the most dangerous potential sliding surface result of the slope are output, among which the minimum slope safety factor is F s_min = 1.166 and the parameters of the most potentially dangerous slip surface are: l 0 = 0.00, l 1 = –0.05, l 2 = 0.10, l 3 = –0.18, l 4 = –0.16, l 5= 0.08, l 6 = –0.20, l 7= 0.23, l 8= 0.33, l 9= –0.25, k = 0.47, r A = 48.9m and i A= 75.3°, and the most dangerous sliding surface is x The maximum slip range in the axial direction is 0.0 m to 24.2 m. y The maximum slip range in the axial direction is –19.8m to 19.3m;

[0123] According to the evaluation standard of Technical Specification for Building Slope Engineering (GB 50330-2013), the safety level of this slope project is classified as Level 2. Based on the slope stability analysis results and the slope stability evaluation standard in the specification, the minimum safety factor of this slope is F s_min = 1.166 between the specification value (1.05 ≤ Fs_min <1.25), its stability status is judged to be basically stable, and regular monitoring is required in subsequent projects to focus on preventing the potential risk of slope collapse.

[0124] In general, the present invention realizes the simple, reasonable and effective generation of potential sliding surfaces, thereby providing a prerequisite for reliable slope stability analysis and providing powerful guidance for the implementation of slope engineering reinforcement measures under complex conditions.

[0125] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for constructing a universal three-dimensional potential sliding surface model of a slope in a rotational mode, characterized in that: The method comprises constructing the potential sliding surface by setting the potential sliding surface as a rotational shear failure surface and utilizing the physical and mechanical relationship between the shear failure direction and the motion direction of the potential sliding surface. The solution of the potential sliding surface is then converted into the determination of the shear failure plane at any point on the potential sliding surface. An angular function characterizing the position of the shear failure plane under a Taylor series expansion is then established. On this basis, discrete technology and a recursive calculation method are applied, and a potential sliding surface range limitation and a surface concave mechanism are introduced to achieve effective generation of the potential sliding surface. The method comprises the following steps: Step S1: In the three-dimensional slope, establish a xyz Axis coordinate system; Step S2: Given the potential sliding surface bottom side sliding point A exist x 、 y and z Coordinates on the axis ( x A , y A , z A )as well as AB Connected xy Projection on a plane x Angle between axes ψ , where point B It is the sliding exit point on the upper side of the potential sliding surface; Step S3: xy Axis coordinate system around z Axis rotation angle ψ , so that x' Axis and AB Connected xy The projections on the plane are parallel, and thus the x'y'z Axis coordinate system, at the same time, use formula (1) to solve the point A exist x' and y' Coordinates on the axis ( , ); (1) Step S4: Set the maximum width of the left and right potential sliding surfaces w left and w right and the maximum length of the potential sliding surface L ; Step S5: Develop the parameters of the three-dimensional slope potential sliding surface model, i.e., point A Sliding surface extreme diameter r A 、 OA Connection x' Angle between axes θ A , variable factors κ and sliding surface shape control parameters λ 0~ λ 9, among which, point O is the potential sliding surface rotation center point; Step S6: Using the coordinates of the potential sliding surface rotation center point and the boundary discrete point solution step, the potential sliding surface rotation center point is solved. O and boundary discrete points P 0j and P i0 exist x' 、 y' and z The coordinates on the axis, where i = 0 ~ n and j = – m 1 ~ m 2, n is the number of discrete directional lines of the potential sliding surface, m 1 and m 2 is the number of discrete columnar lines on the left and right potential sliding surfaces respectively; Step S7: Calculate the discrete points on the potential sliding surface using the recursive solution step. P ij exist x' 、 y' and z The coordinates on the axis and its shear failure plane along x' Axis and y' The inclination angle of the axis; Step S8: using the number of intersection points to control the feasibility of the potential sliding surface, analyzing the feasibility of the constructed potential sliding surface. If it is not feasible, it is necessary to reselect the potential sliding surface shape control parameters; Step S9: using the concave mechanism to control the rationality of the potential sliding surface, analyzing the rationality of the constructed potential sliding surface. If it is unreasonable, it is necessary to reselect the potential sliding surface shape control parameters; Step S10: Based on the discrete points on the potential sliding surface P ij and the corresponding slope points G ij , using these two points z The size of the axis is used to determine the valid and invalid discrete points, that is, when z ij ≤ s ij The corresponding discrete points are valid discrete points, otherwise they are invalid discrete points. z ij for point P ij of z Axis coordinates, s ij for point G ij of z axis coordinates; Step S11: When the discrete points P (i-1)(j-1) , discrete points P (i-1)j , discrete points P i(j-1) and discrete points P ij When there are 3 or more valid discrete points in the grid, determine the discrete grid ij is a valid grid; Step S12: Output the effective discrete points and effective grids of the potential sliding surface.

2. The method for constructing a universal three-dimensional potential sliding surface model of a slope in a rotational mode according to claim 1, characterized in that: In step S6, according to the sliding point on the lower side of the potential sliding surface A , first calculate the potential sliding surface rotation center point O The coordinates of the sliding surface are then calculated using the shear failure plane of the sliding surface at the discrete points. x' and y' The inclination angle in the axial direction is obtained to obtain the coordinates of the discrete points on the 0th column line and the 0th row line; and the steps of solving the coordinates of the potential sliding surface rotation center point and the boundary discrete points specifically include the following steps: Step S6-1: Use equation (25) to solve the point O exist x' 、 y' and z Coordinates on the axis ( , , ); (25) Step S6-2: Based on the potential sliding surface of the three-dimensional slope, [ , ] within the range y ′ axis coordinates are divided into m 1 copy, , ] within the range y ′ axis coordinates are divided into m 2 parts, [ , ] within the range x ′ axis is divided into n Then, the potential sliding surface is discretized within the scope of the potential sliding surface, and ( n +1)×( m 1+ m 2+1) discrete points on the potential sliding surface, and then use equations (17) and (18) to calculate the discrete points P ij exist x' and y' The coordinates on the axis, where i = 0 ~ n and j =– m 1 ~ m 2; among them, is the left boundary point of the potential slip surface y' Minimum axis coordinate, is the right boundary point of the potential sliding surface y' The maximum value of the axis coordinate, The upper sliding point of the potential sliding surface x' Maximum value of axis coordinate; (17) (18) in, and Discrete points P ij of x' Axis and y' axis coordinates; Step S6-3: Make i = 1; Step S6-4: Use equations (20) to (23) to solve the discrete points P (i-1)0 Angle ω (i-1)0 , sliding surface extreme diameter r (i-1)0 , sliding surface polar angle η (i-1)0 The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i-1)0 ; (20) (21) (22) (23) in, ω ij for point P ij Angle, r ij for point P ij Sliding surface extreme diameter, η ij for point P ij The polar angle of the sliding surface, ξ is the dilatancy angle between the sliding surface movement direction and the shear failure direction; Step S6-5: Use equation (26) to solve the discrete points P i0 of z axis coordinates; (26) Step S6-6: If i ≥ n , then end the loop, otherwise, let i = i + 1, and return to step S6-4; Step S6-7: Make j = –1; Step S6-8: Use equations (20) to (22) and (24) to solve the discrete points P 0(j+1) Angle ω 0(j+1) , sliding surface extreme diameter r 0(j+1) , sliding surface polar angle η 0(j+1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_0(j+1) ; (24) Step S6-9: Use equation (27) to solve the discrete points P 0j of z Axis coordinates z 0j ; (27) Step S6-10: If i ≤ – m 1, then end the loop, otherwise, let j = j – 1, and return to step S6-8; Step S6-11: Make j = 1; Step S6-12: Use equations (20) to (22) and (24) to solve the discrete points P 0(j-1) Angle ω 0(j-1) , sliding surface extreme diameter r 0(j-1) , sliding surface polar angle η 0(j-1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_0(j-1) ; Step S6-13: Use equation (27) to solve the discrete points P 0j of z Axis coordinates z 0j ; Step S6-14: If i ≥ m 2, then end the loop, otherwise, let j = j + 1, and return to step S6-12; Step S6-15: Output the calculation result, i.e. the potential sliding surface rotation center point O exist x' 、 y' and z Coordinates on the axis and discrete points on the potential slip surface boundary P 0j and P i0 exist z Coordinates on the axis.

3. The method for constructing a universal three-dimensional potential sliding surface model of a slope in a rotational mode according to claim 2, characterized in that: In step S7, after the coordinates of the potential sliding surface rotation center point and the boundary discrete points are determined, the coordinates of the discrete points on the left and right potential sliding surfaces are further recursively obtained. z Axis coordinates, and then obtain the spatial position and shape characteristics of the discrete points on the potential sliding surface; and the recursive solution steps for the coordinates of the discrete points on the potential sliding surface specifically include the following steps: Step S7-1: Get i = 0 and j = 0; Step S7-2: For the potential sliding surface on the left, j < 0, use equations (20) to (24) to solve the discrete points P (i+1)j Angle ω (i+1)j , sliding surface extreme diameter r (i+1)j , sliding surface polar angle η (i+1)j The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i+1)j and discrete points P i(j-1) Angle ω i(j-1) , sliding surface extreme diameter r i(j-1) , sliding surface polar angle η i(j-1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_i(j-1) ; Step S7-3: For the potential sliding surface on the right side, j ≥ 0, it is necessary to use formula (20) to formula (24) to calculate the discrete points respectively P (i+1)j Angle ω (i+1)j , sliding surface extreme diameter r (i+1)j , sliding surface polar angle η (i+1)j The shear failure plane of the sliding surface is along x' Axis tilt α x'_(i+1)j and discrete points P i(j+1) Angle ω i(j+1) , sliding surface extreme diameter r i(j+1) , sliding surface polar angle η i(j+1) The shear failure plane of the sliding surface is along y' Axis tilt α y'_i(j+1) ; Step S7-4: For the potential sliding surface on the left, j < 0, use formula (19) to calculate the discrete points P (i+1)(j-1) exist z Coordinates on the axis z (i+1)(j-1) ; (19) Step S7-5: For the potential sliding surface on the right side, j ≥ 0, use formula (19) to calculate the discrete points P (i+1)(j+1) exist z Coordinates on the axis z (i+1)(j+1) ; Step S7-6: For the potential sliding surface on the left, j < 0, determine whether j ≤ – m 1+1, if not satisfied, then let j = j –1 and return to step S7-2; Step S7-7: For the potential sliding surface on the right side, j ≥ 0, determine whether it is satisfied j ≥ m 2–1, if not satisfied, then let j = j +1 and return to step S7-3; Step S7-8: Determine whether i ≥ n – 1, if not satisfied, then let i = i +1 and return to step S7-2; Step S7-9: Output the coordinates of discrete points on the potential sliding surface and the shear failure plane of the sliding surface x' Axis and y' The inclination angle of the axis.

4. The method for constructing a universal three-dimensional potential sliding surface model of a slope in a rotational mode according to claim 3, characterized in that: In step S8, the specific implementation steps of determining whether the potential slip surface is feasible, that is, the feasibility determination step of the potential slip surface under the control of the number of intersection points, specifically include the following steps: Step S8-1: Make i The number of intersection points between the broken line segments composed of discrete points on the potential sliding surface and the slope surface is Row ( i ), and take i = 1; Step S8-2: Get Row ( i ) = 0 and j = – m 1+1; Step S8-3: Obtaining discrete points on the potential sliding surface P i(j-1) and P ij of z Axis coordinates z i(j-1) and z ij and the corresponding slope points G i(j-1) and G ij of z Axis coordinates s i(j-1) and s ij ; Step S8-4: Determine whether z i(j-1) ≤ s i(j-1) and z ij > s ij or z i(j-1) > s i(j-1) and z ij ≤ s ij , if satisfied, then let Row ( i ) = Row ( i ) + 1, if not satisfied, go to the next step; Step S8-5: Determine whether j ≥ m 2. If not satisfied, then j = j + 1, and return to step S8-3; Step S8-6: If Row ( i ) = 2, it indicates that the potential sliding surface shape of this row meets the intersection control condition; otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible; Step S8-7: Determine whether i ≥ n If it is not satisfied, then i = i + 1, and return to step S8-2; Step S8-8: Make j The number of intersections between the broken line segments formed by the discrete points on the potential sliding surface and the slope surface is Col ( j ), and take j = – m 1+1; Step S8-9: Get Col ( j ) = 0 and i = 1; Step S8-10: 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 corresponding slope points G (i-1)j and G ij of z Axis coordinates s (i-1)j and s ij ; Step S8-11: Determine whether z (i-1)j ≤ s (i-1)j and z ij > s ij or z (i-1)j > s (i-1)j and z ij ≤ s ij , if satisfied, then let Col ( j ) = Col ( j ) + 1, if not satisfied, go to the next step; Step S8-12: Determine whether i ≥ n If it is not satisfied, then i = i + 1, and return to step S8-10; Step S8-13: If Col ( i ) = 2, it indicates that the potential sliding surface shape of this column meets the intersection control condition; otherwise, the selected potential sliding surface shape control parameters make the constructed potential sliding surface infeasible; Step S8-14: Determine whether j ≥ m 2. If not satisfied, then j = j + 1, and return to step S8-9; if satisfied, it indicates that the potential slip surface is feasible.

5. The method for constructing a universal three-dimensional potential sliding surface model of a slope in a rotational mode according to claim 4, characterized in that: In step S9, the rationality of the potential sliding surface shape is judged by whether the potential sliding surface satisfies conditions 1 and 2, wherein condition 1 is i Discrete points on the potential sliding surface P ij Corresponding edge y' Horizontal inclination of tangent line in the axis direction α y'_ij is an increasing function, 0 ≤ i ≤ n , condition 2 is j List discrete points on the potential sliding surface P ij Corresponding edge x' Horizontal inclination angle of tangent line in the axis direction tan α x'_ij is an increasing function, – m 1 ≤ j ≤ m 2. The rationality judgment step of the potential slip surface under the control of the concave mechanism specifically includes the following steps: Step S9-1: Make i = 0 and j = – m 1; Step S9-2: Determine discrete points on the potential slip surface P ij and P i(j+1) Failure plane along y' Horizontal inclination of the axis α y'_ij and α y'_i(j+1) and discrete points on the potential sliding surface P ij and P (i+1)j Failure plane along x' Horizontal inclination of the axis α x'_ij and α x'_(i+1)j ; Step S9-3: Determine whether α y'_ij ≤ α y'_i(j+1) and α x'_ij ≤ α x'_(i+1)j , if it is satisfied, it means that the sliding surface at the discrete point satisfies the concave characteristic mechanism, otherwise, it indicates that the potential sliding surface is unreasonable; Step S9-4: Determine whether i ≥ n If it is not satisfied, then i = i + 1, and return to step S9-2; Step S9-5: Determine whether j ≥ m 2. If not satisfied, then j = j + 1, and return to step S9-2; if satisfied, it indicates that the potential slip surface is reasonable.

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