A Construction Method for a Generalized Potential Slip Surface Model of a 3D Slope in Polar Angle Mode

Abstract

The present invention belongs to the field of slope engineering, and specifically relates to a method for constructing a general potential sliding surface model of a three-dimensional slope in the polar angle mode. This method generates a potential sliding surface by controlling the polar angle of the sliding surface, defines the range of the potential sliding surface using four boundary points on the lower side, upper side, left side, and right side of the sliding surface, establishes stress constraint conditions that the polar angles of the sliding surface 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 method of the present invention solves the problem of limiting the sliding surface type of the three-dimensional slope potential sliding surface model, ensures the feasibility and reasonableness of the sliding surface shape and range, strictly satisfies the physical and mechanical mechanisms for generating the potential sliding surface, and facilitates the 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 application range, strong generality, high accuracy, etc., and provides a strong scientific basis for reliable early warning and prevention of slope instability and collapse.

Description

Technical Field

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

[0002] In order to carry out landslide treatment and prevention work efficiently and reliably, it is necessary to accurately carry out 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 ensuring the effective evaluation of slope stability and the accurate positioning of the most dangerous sliding surface.

[0003] In the existing slope stability analysis, the construction of the two-dimensional slope potential sliding surface model is still the mainstream. In the two-dimensional plane, the slope is regarded as a semi-infinite body in space, and the potential sliding surface of the slope can be represented by a certain type of curve. For example, specific types of curves such as straight lines, circular arcs, and logarithmic spirals are often used, as well as arbitrary random curves applicable to complex conditions. In the two-dimensional plane, the potential sliding surface model simplifies the actual sliding surface morphology to a great extent, and the two-dimensional most dangerous sliding surface often exceeds 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 on the conservative side, which leads to unnecessary waste in the design and reinforcement costs of engineering slopes. For the existing three-dimensional slope potential sliding surface models, like the two-dimensional potential sliding surface models, there are specific rotational type sliding surfaces such as spherical surfaces, ellipsoidal surfaces, and logarithmic spiral surfaces. In addition, there are also arbitrary-shaped sliding surfaces obtained by translating the generatrix based on the sliding direction and the directrix perpendicular to the sliding direction under complex conditions.

[0004] However, there are still certain limitations in the construction methods of the above-mentioned sliding surface models. 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, they are all constructed based on the 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 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, making the slope stability analysis results tend to be conservative and not conducive to the saving of engineering costs. In the existing three-dimensional sliding surface models, although the three-dimensional space effect is considered to a certain extent, the specific sliding surface models limit the type of sliding surface and cannot show the complex morphology of the actual sliding surface. At the same time, although the arbitrary-shaped sliding surface model improves the diversity of the three-dimensional sliding surface model, the method of generating by the generatrix and the directrix is still difficult to truly achieve the arbitrariness of the sliding surface morphology. In addition, the existing models cannot make the constructed potential sliding surface strictly meet the physical geometry and mechanical failure characteristics, and thus it is difficult to capture the more accurate most dangerous potential sliding surface under complex conditions.

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

[0006] With the continuous expansion of engineering slopes into dangerous mountainous areas, accurately and efficiently identifying the most dangerous potential sliding surface of the slope and providing scientific and detailed construction guidance based on this have become urgent requirements to ensure the safety and smooth progress of the project. However, the existing slope smooth surface calculation models are generally restricted by reasons such as unreasonable physical and mechanical models and cannot provide reliable guidance for slope engineering.

[0007] Therefore, there is a need in the art for a new method for constructing a general potential smooth surface model for three-dimensional slopes. Summary of the Invention

[0008] To this end, the present invention generates a potential smooth surface by controlling the polar angle of the smooth surface, without limiting the smooth surface type, ensuring the arbitrariness of the potential smooth surface. At the same time, the range of the potential smooth surface is defined by four boundary points on the lower side, upper side, left side, and right side, and the polar angle of the smooth surface 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 smooth surface strictly satisfies the physical geometry and mechanical failure characteristics. In addition, the potential smooth surface range limitation and the surface concave mechanism are incorporated to ensure the feasibility and rationality of the potential smooth surface model. Subsequently, discrete technology is applied to obtain discrete points on the potential smooth surface, and correspondingly, the effective discrete points and discrete grids on the potential smooth surface are identified, which is conducive to combining with the limit equilibrium method for carrying out the stability analysis of three-dimensional slopes under complex conditions. The present invention has the advantages of being simple and easy to implement, wide application range, strong generality, high accuracy, etc., and provides a strong scientific basis for reliably warning and preventing slope instability and collapse.

[0009] Specifically, the present invention provides a method for constructing a general potential slip surface model of a three-dimensional slope in the polar angle mode. The method generates a potential slip surface by using the slip surface polar angle control method, defines the range of the potential slip surface by using four boundary points on the lower side, upper side, left side, and right side of the slip surface, and then uses the position extreme value characteristics of the lower side boundary point and the upper side boundary point of the slip surface. Based on the sliding trends of the rock and soil masses on both sides at these two boundary points, the approximate occurrence planes of shear failure of the rock and soil masses at the lower side boundary point and the upper side boundary point are determined. Then, according to the fact that the tangent direction of the potential slip surface is the most unfavorable shear direction of the rock and soil mass, when the acting force on the slope surface is known, combined with the shear strength criterion of the rock and soil mass, the stress constraint conditions that the slip surface polar angle at the lower side boundary point and the upper side boundary point should satisfy are established; and the slip surface polar angle also matches the position constraint conditions at the above four boundary points; and the slip surface polar angle is related to the slip surface coordinates. An approximate function of the slip surface polar angle and the slip surface coordinates is constructed by using the Taylor series expansion method. In addition to satisfying the stress and position constraint conditions at the potential slip surface boundary points, the slip surface polar angle also conforms to the potential slip surface range limitation and the surface concave mechanism, so as to ensure that the constructed potential slip surface is feasible and reasonable; and the discrete technology is applied to obtain the discrete points on the potential slip surface, and then the effective discrete points and discrete grids on the potential slip surface are identified and output, so as to construct a general potential slip surface model of a three-dimensional slope in the polar angle mode.

[0010] In a specific implementation manner, the method includes the following steps: Step S1: In a three-dimensional slope, establish a xyz axis space coordinate system with any point as the coordinate origin; Step S2: Given the lower side boundary point A , upper side boundary point B , left side boundary point C and right side boundary point D and their coordinates, that is, ( 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 points and upper boundary points of the potential slip surface according to the occurrence of the three-dimensional slope surface A and B The horizontal dip angle of the tangent line of the slope surface in the xz and yz plane, that is β A_x and β A_y , β B_x and β B_y ; Step S4: Determine the control parameters of the potential slip surface shape λ 0,[[]] λ 1,[[]] λ 2,[[]] λ 4,[[]] λ 7,[[]] λ 8,[[]] ξ 1,[[]] ξ 2,[[]] η 1 and η 2; Step S5: According to the tangent dip angles of the slope surface at the known potential slip surface boundary points β A and β B and the soil strength parameters φ A and φ B , use the boundary stress constraint conditions to solve the horizontal dip angles α A and α B of the tangent line of the potential slip surface in the approximate plane of shear failure of the rock and soil mass at the boundary points, and determine the horizontal dip angles of the tangent plane of the potential slip surface at the boundary points in the xz and yz plane, that is α A_x , α A_y , α B_x and α B_y ; Step S6: Use Equation (16) to solve the control parameters of the potential slip surface shape λ 3,[[]] λ 5,[[]] λ 6,[[]] λ 9,[[]] λ 10 , λ 11 and λ 12 ;

[0011] (16)

[0012] Wherein, 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 all coefficients of the system of equations; b 1 to b 7 are the constant terms of the system of equations; Step S7: Using Equation (24), Equation (25) and Equation (26), calculate the P ij 's x 、 y and z axis coordinates, that is x ij 、 y ij and z ij , 0 ≤ i ≤ n and 0 ≤ j ≤ m ;

[0013] (24)

[0014] (25)

[0015] (26)

[0016] Step S8: Analyze the feasibility of the constructed potential slip surface by using the discriminant step of the potential slip surface feasibility under the control of the intersection points of the potential slip surface and the slope surface; if it is not feasible, the control parameters of the potential slip surface shape need to be reselected; Step S9: Analyze the rationality of the constructed potential slip surface by using the discriminant step of the potential slip surface rationality under the concave mechanism; if it is not reasonable, the control parameters of the potential slip surface shape need to be reselected; Step S10: Compare the P ij of z axial coordinates z ij of the discrete points on the potential slip surface and the z axial coordinates s ij of the corresponding slope points. If z ij ≤ s ij , then the discrete point P ij is an effective discrete point, 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S11: Determine the effectiveness of the discrete grid by using the effectiveness of the discrete points on the potential slip surface. If among the discrete points P (i-1)(j-1) , discrete point P (i-1)j , discrete point P i(j-1) and discrete point P ij there are 3 or more effective discrete points, 1 ≤ i ≤ n and 1 ≤ j ≤ m , then the discrete grid ij is an effective grid; Step S12: Output the effective discrete points and effective grids of the potential slip surface.

[0017] In a specific embodiment, in the said Step S4, λ 0, λ 1, λ 2, λ 4, λ 7 and λ 8 have a value range of -10 to 10, ξ 1, η 1, ξ 2 and η 2 have a value range of 1 to 10.

[0018] In a specific embodiment, in the said Step S8, in order to eliminate the infeasible potential slip surface, according to the j column, the discrete points of the potential slip surfaceP ij The feasibility of the potential slip surface is determined by the number of intersection points between the broken line segments connected in sequence and the slope surface, where 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S8 specifically includes the following steps: Step S8-1: Let the number of intersection points between the broken line segment formed by the discrete points of the j th column of potential slip surfaces and the slope surface be J d ( j ), and take j = 1; Step S8-2: Take J d ( j ) = 0 and i = 1; Step S8-3: Obtain the P (i-1)j -axis coordinates P ij and z of the discrete points z (i-1)j and z ij on the potential slip surface, as well as the z -axis coordinates s (i-1)j and s ij of the slope surface at the corresponding positions; 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 not satisfied, let i = i + 1, and return to Step S8-3; Step S8-7: If J d (j ) = 2, it indicates that the shape of the potential slip surface meets the intersection control condition; otherwise, the selected potential slip surface shape control parameters make the constructed potential slip surface infeasible, and the potential slip surface shape control parameters need to be reselected; Step S8-8: Determine whether it meets j < m , if it does not meet, then let j = j + 1, and return to Step S8-2.

[0019] In a specific implementation manner, in the step S9, the pole angle of the slip surface should conform to the concave mechanism of the potential slip surface curve, and the rationality of the potential slip surface is determined by whether the following Condition 1 and Condition 2 are met. Condition 1 is that the broken line segment formed by connecting the discrete points i of the potential slip surface in the P ij row in sequence shows an increasing trend in the slip surface curvature. Condition 2 is that the broken line segment formed by connecting the discrete points j of the potential slip surface in the P ij column in sequence shows an increasing trend in the slip surface curvature; 0 ≤ i ≤ n and 0 ≤ j ≤ m ; The step S9 specifically includes the following steps: Step S9-1: Let i = 0 and j = 0; Step S9-2: Obtain the coordinates ([[]]END]] P ij , P (i+1)j and P i(j+1) ) of the discrete points x ij , y ij , z ij )、([[]]END]] x (i+1)j , y (i+1)j , z (i+1)j ) and ([[]]END]] x i(j+1) , y i(j+1) , z i(j+1) ) of the potential slip surface; Step S9-3: Obtain the horizontal inclination angle P ij of the tangential plane of the slip surface at the discrete point xz on the α x_(i,j) plane, where

[0020]

[0021] Step S9-4: Obtain discrete points P ij The horizontal dip angle of the tangential plane of the slip surface at the point yz on the plane α y_(i,j) , where

[0022]

[0023] Step S9-5: Determine whether the horizontal dip angle of the slip surface tangent satisfies:

[0024] and ;

[0025] If not satisfied, it indicates that the selected potential slip surface shape control parameters make the constructed potential slip surface unreasonable, and the potential slip surface shape control parameters need to be reselected; Step S9-6: Determine whether it satisfies i < n ; if not satisfied, then let i = i + 1, and return to Step S9-2; Step S9-7: Determine whether it satisfies j < m - 1 ; if not satisfied, then let j = j + 1, and return to Step S9-2.

[0026] Compared with the prior method for constructing a general potential slip surface model in the two-dimensional case, the present invention solves the problem of constructing a three-dimensional general potential slip surface under complex conditions. In addition to ensuring the feasibility and reasonableness of the slip surface shape and range, it can also strictly meet the physical and mechanical mechanisms for generating the potential slip surface, and can be combined with the three-dimensional slope stability calculation method to effectively evaluate the three-dimensional slope stability state.

[0027] The advantages of the present invention are as follows: it is simple and easy to implement, has a wide application range, strong generality, high accuracy, etc. The constructed three-dimensional slope potential slip surface model not only ensures the feasibility and reasonableness of the slip surface shape and range but also strictly meets the physical and mechanical mechanisms for generating the potential slip surface under non-limited slip surface types. In addition, by applying discrete technology, it is beneficial to embed the potential slip surface model into the existing limit equilibrium method to carry out three-dimensional slope stability analysis under complex conditions, thus providing a strong scientific basis for reliably warning and preventing slope instability and collapse. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 is a schematic diagram of the three-dimensional slope potential slip surface and potential slip surface boundary points of the present invention.

[0029] Figure 2 is the lower boundary point of the three-dimensional slope potential slip surface of the present inventionA Schematic diagram of the horizontal inclination angle of the slope surface at A , where (a) is the lower boundary point xz Schematic diagram of the three-dimensional spatial distribution of the yz plane and the A plane, (b) is the lower boundary point xz Schematic diagram of the horizontal inclination angle of the slope surface on the A plane, (c) is the lower boundary point yz Schematic diagram of the horizontal inclination angle of the slope surface on the A plane, (d) is the schematic diagram of the boundary line of the two-side micro-segment sliding perimeter at A , (e) is the lower boundary point

[0030] Figure 3 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 angle of the tangential plane of the potential sliding surface at A , where (a) is the lower boundary point A Schematic diagram of the three-dimensional spatial distribution of the two-dimensional slice 1 at A , (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 at

[0031] Figure 4 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 angle of the slope surface at B , where (a) is the upper boundary point xz Schematic diagram of the three-dimensional spatial distribution of the yz plane and the B plane, (b) is the upper boundary point xz Schematic diagram of the horizontal inclination angle of the slope surface on the B plane, (c) is the upper boundary point yz Schematic diagram of the horizontal inclination angle of the slope surface on the B plane, (d) is the schematic diagram of the boundary line of the two-side micro-segment sliding perimeter at B , (e) is the upper boundary point

[0032] Figure 5 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 angle of the tangential plane of the potential sliding surface at B , where (a) is the upper boundary point B Schematic diagram of the three-dimensional spatial distribution of the two-dimensional slice 2 atSchematic diagram of the most unfavorable shear direction of the upslope surface at the two-dimensional slice 2, (c) is the upper boundary point B Schematic diagram of the geometric relationship of the tangential dip angle of the potential slip surface at, (d) is the upper boundary point B Schematic diagram of the Mohr stress circle of the stress state and the shear strength curve of the rock and soil mass at.

[0033] Figure 6 It is a schematic diagram of an arbitrary point on the potential slip surface of the three-dimensional slope of the present invention.

[0034] Figure 7 It is the lower boundary point of the potential slip 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 at xz plane and yz Schematic diagram of the three-dimensional space distribution of the plane, (b) is the lower boundary point A at xz Schematic diagram of the inclination angle constraint of the potential slip surface on the plane, (c) is the lower boundary point A at yz Schematic diagram of the inclination angle constraint of the potential slip surface on the plane.

[0035] Figure 8 It is the upper boundary point of the potential slip 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 at xz Schematic diagram of the three-dimensional space of the plane distribution and the inclination angle constraint of the slip surface, (b) is the upper boundary point B at xz Schematic diagram of the inclination angle constraint of the potential slip surface on the plane, (c) is the upper boundary point B at yz Schematic diagram of the three-dimensional space of the plane distribution and the inclination angle constraint of the slip surface, (d) is the upper boundary point B at yz Schematic diagram of the inclination angle constraint of the potential slip surface on the plane.

[0036] Figure 9 It is a schematic diagram of the introduction of the coordinate matching condition at the boundary point of the potential slip surface of the three-dimensional slope of the present invention, where (a) is the point B and the point C and the point D Schematic diagram of the three-dimensional space of the polar angle and polar radius of the potential slip surface at, (b) is the left boundary point C Schematic diagram of the spatial relationship between the polar angle and polar radius of the potential slip surface at, (c) is the upper boundary point B Schematic diagram of the spatial relationship between the polar angle and polar radius of the potential slip surface at, (d) is the right boundary point D Schematic diagram of the spatial relationship between the polar angle and polar radius of the potential slip surface at.

[0037] Figure 10 It is a schematic diagram for determining the intersection points of the potential slip surface and the slope surface of the three-dimensional slope in the present invention. Among them, (a) is a schematic diagram for determining the discrete points of the three-dimensional potential slip surface and the corresponding slope points, and (b) is the j schematic diagram for determining the intersection points of the discrete points in the

[0038] Figure 11 It is a flowchart of the feasibility discrimination steps of the potential slip surface under the control of the intersection points of the potential slip surface and the slope surface of the three-dimensional slope in the present invention.

[0039] Figure 12 It is a flowchart of the rationality discrimination steps of the potential slip surface under the concave mechanism of the three-dimensional slope in the present invention.

[0040] Figure 13 It is a schematic diagram of the effectiveness of the discrete points and grids of the potential slip surface of the three-dimensional slope in the present invention.

[0041] Figure 14 It is a flowchart of the steps for constructing a general potential slip surface model of a three-dimensional slope in the polar angle mode in the present invention.

[0042] In the figure: 1. Three-dimensional slope, 2. Potential slip surface, 3. Slope surface, 4. Slip perimeter boundary, 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 at point xz , 10. yz , 11. A The direction of the minor principal stress acting surface at point A , 13. Mohr stress circle, 14. Shear strength curve, 15. Point A The most unfavorable shear direction when the rock and soil mass fails in shear at point B , 17. The tangential plane of the potential slip surface at point B , 18. The plane where the slope surface is located at point B The direction of the major principal stress acting surface at point B , 20. The tangential plane of the potential slip surface at point P , 22. Polar angle of the slip surface, 23. Horizontal line, 24. Vertical line, 25. Discrete points on the potential slip surface P ij , 26. Intersection point, 27. Effective discrete point, 28. Invalid discrete point, 29. Effective discrete grid, 30. Invalid discrete grid. Detailed implementation method

[0043] The specific model construction method and implementation process of the present invention are as follows:

[0044] Slope instability is caused by the shear failure of rock and soil masses and the formation of a connected slip surface. Whether the shear of rock and soil masses fails is controlled by the shear strength criterion of rock and soil masses. Therefore, let the general expression of the shear strength criterion of rock and soil masses be τ f = f ( σ ), where τ f is the shear strength of the rock and soil mass, σ is the normal stress acting on the failure surface of the rock and soil mass. At the same time, the shear failure of the rock and soil mass indicates that the rock and soil mass will slide along its most unfavorable shear direction. That is to say, the shear failure direction of the rock and soil mass is the most unfavorable shear direction of the rock and soil mass, and it is also the tangent direction of the potential slip surface.

[0045] As Figure 1 shown, in a three-dimensional slope, taking any point as the origin of the coordinate system, and taking the vertical direction as the z axis direction, taking the longitudinal direction of the slope on the horizontal plane as the x axis direction, and taking the transverse direction of the slope on the horizontal plane as the y axis direction. Furthermore, establish a xyz axis space coordinate system. For the potential slip surface of the three-dimensional slope, the intersection line with the slope surface is the slip perimeter boundary. The coordinate range of the slip perimeter boundary on the x axis is x min , x max and its coordinate range on the y axis is y min , y max . At the same time, the slip perimeter boundary point corresponding to x min is point A , the slip perimeter boundary point corresponding to x max is point B , the slip perimeter boundary point corresponding to y min is point C , the slip perimeter boundary point corresponding to y max is point D . Further, name point A , point B , point C and point D as the lower boundary point, upper boundary point, left boundary point and right boundary point of the potential slip surface respectively.

[0046] As shown Figure 2 in the figure, for the lower boundary point of the potential sliding surface A on the slope surface, its horizontal inclination angle on the xz plane is β A_x and its horizontal inclination angle on the yz plane is β A_y . And when the horizontal inclination angle is above the horizontal direction, it is positive, otherwise it is negative. Subsequently, move the origin of the xyz axis coordinate system to point A and rotate the x axis coordinate system around the yz axis by an angle β A_y to obtain the local coordinate system . Among them, the direction of the axis in the local coordinate systemis the tangential direction of the slope surface at point on the A plane. Based on this, then rotate the yz axis coordinate system around the axis by an angle β A to obtain the local coordinate system . Among them, β A = arctan(tan β A_x cos β A_y ) and the direction of the axis in the local coordinate system is the tangential direction of the slope surface at point A on the plane. Thus, the plane is the plane where the slope surface at point A is located, and the plane is the vertical plane of the slope surface at point A . At the same time, take a micro-segment sliding peripheral boundary line on both sides of point A . It can be approximately considered that the micro-segment sliding peripheral boundary line is located in the A plane like point . According to the spatial geometric relationship between the coordinate axis translation and rotation, it can be known that the x axis coordinate of the sliding peripheral boundary line and its axis coordinate have the same property, that is, point A is the peak point of the micro-segment sliding peripheral boundary line on both sides in the x axis direction, and it is also the peak point of the micro-segment sliding peripheral boundary line on both sides in the axis direction. Further, when the three-dimensional sliding body slides longitudinally along the slope, pointA The sliding trends of the rock and soil masses along the boundary lines of the two side micro-segments are opposite along the axis direction. Therefore, based on the continuity of the movement trends, there will inevitably be no sliding trend along the A axis direction at point . This means that the shear failure of the rock and soil mass at point A will approximately occur in the (or ) plane.

[0047] As shown in Figure 3 , using the plane A passing through point (or ) as the cutting plane to cut the three-dimensional slope, a two-dimensional slice of the three-dimensional slope is obtained, which is the plane where the shear failure of the rock and soil mass at point A approximately occurs. Name this two-dimensional slice as two-dimensional slice 1 and analyze two-dimensional slice 1. Among them, point A is both on the slope surface and on the potential sliding surface. Since the slope surface is the external boundary, when clarifying its position information, the force conditions on the slope surface at point A are known. Furthermore, boundary stress constraint conditions are formed at this point, so that the stress state of the potential sliding surface at point A is controlled. Thus, it affects the most unfavorable shear direction of the rock and soil mass here and further restricts the shape of the potential sliding surface.

[0048] In two-dimensional slice 1, the rock and soil mass at point A is usually in a compressed state. If there is no external load on the slope surface at this point, the slope surface at point A is the plane of the minor principal stress at point A , and the tangent direction of the slope surface at point A is the direction of the plane of the minor principal stress at point A . Here, let σ 1_A and σ 3_A be the major and minor principal stresses of the rock and soil mass at point A respectively, and σ 3_A = 0. Subsequently, in the coordinate system of normal stress and shear stress, draw the Mohr stress circle reflecting the stress state at point A and the shear strength curve of the rock and soil mass. Among them, the center point of the Mohr stress circle is O A , the intersection points of the Mohr stress circle and the normal stress axis are point E A and point F A , and point E A and pointF A correspond to the minor principal stress A at the point σ 3_A and the major principal stress σ 1_A , O A E A represents the direction of the plane on which the minor principal stress acts, O A T A is the perpendicular line to the shear strength curve of the rock and soil mass and passes through the center point O A . Since O A T A is the shortest distance from the center point O A to the shear strength curve of the rock and soil mass, therefore, O A T A represents the most unfavorable shear direction when the rock and soil mass at the point A undergoes shear failure. If a tangent line to the shear strength curve of the rock and soil mass is drawn through the point T A , and the horizontal inclination angle of this tangent line direction is φ A , then ∠ T A O A E A = ( π / 2 – φ A ), where φ A is the internal friction angle of the slip surface at the point A , and ∠ T A O A E A represents the angle between the most unfavorable shear direction and the direction of the plane on which the minor principal stress acts when the rock and soil mass at the point A undergoes shear failure. According to the double angle relationship between the Mohr stress circle and the actual model, the angle between the most unfavorable shear direction and the direction of the plane on which the minor principal stress acts at the point A in the actual model when the rock and soil mass undergoes shear failure is ( π / 4 – φ A / 2), and the most unfavorable shear direction is below the direction of the minor principal stress plane. Further, the most unfavorable shear direction is the tangent direction of the potential slip surface, and the horizontal inclination angle of the tangent line of the slope surface at point A is β A , then the horizontal inclination angle of the tangent line of the potential slip surface at point A can be obtained as α A = β A + φ A / 2 – π / 4, which is also located within the two-dimensional slice 1. In addition, the tangential plane of the potential slip surface at point A is the plane passing through the axis and the tangent line of the potential slip surface at point A within the two-dimensional slice 1. Then, using the spatial geometric relationship, the horizontal inclination angle of the tangential plane of the potential slip surface at point A on the xz plane is α A_x = arctan(tan α A / cos β A_y ) and its horizontal inclination angle on the yz plane is α A_y = β A_y .

[0049] As Figure 4 shown, for the slope surface at the upper boundary point B of the potential slip surface, its horizontal inclination angle on the xz plane is β B_x and its horizontal inclination angle on the yz plane is β B_y , and this horizontal inclination angle is positive when it is above the horizontal direction and negative otherwise. Subsequently, move the origin of the xyz axis coordinate system to point B , and rotate the x axis coordinate system around the yz axis by an angle β B_y to obtain the local coordinate system , where the direction of the axis in the local coordinate system is the tangential direction of the slope surface at point B on the yz plane. Based on this, then rotate the axis around the Axis coordinate system rotation angle β B , and thus obtain the local coordinate system , where β B = arctan(tan β B_x cos β B_y ) and the local coordinate system in the direction of the axis is the tangent of the slope surface at the point B on the plane of the plane, thus, the plane is the plane where the slope surface at the point B is located, the plane is the vertical plane of the slope surface at the point B . At the same time, take a micro-segment sliding perimeter boundary on both sides of the point B , and it can be approximately considered that this micro-segment sliding perimeter boundary is located in the plane of B like the point . According to the spatial geometric relationship between the translation and rotation of the coordinate axes, it can be known that the x axis coordinate of the sliding perimeter boundary is the same as its axis coordinate, that is, the point B is the peak point of the micro-segment sliding perimeter boundaries on both sides of it in the x axis direction, then it is also the peak point of the micro-segment sliding perimeter boundaries on both sides of it in the axis direction. Further, when the three-dimensional sliding body slides longitudinally along the slope, the sliding trend of the rock and soil mass on the micro-segment sliding perimeter boundaries on both sides of the point B along the axis is opposite. Thus, based on the continuity of the movement trend, there will inevitably be no sliding trend along the B axis at the point . This means that the shear failure of the rock and soil mass at the point B will approximately occur in the plane of (or ).

[0050] As Figure 5 shown, use the plane B passing through the point as the cutting plane to cut the three-dimensional slope, and obtain a two-dimensional slice of the three-dimensional slope, which is the plane where the shear failure of the rock and soil mass at the point B approximately occurs. Name this two-dimensional slice as two-dimensional slice 2 and analyze two-dimensional slice 2. Among them, the point B is both on the slope surface and on the potential sliding surface. Since the slope surface is the external boundary, when clarifying its position information, the point BThe acting force conditions on the slope surface at a certain location are known. Furthermore, boundary stress constraint conditions are formed at this location, such that the stress state of the potential slip surface at point B is controlled. Thus, it affects the most unfavorable shear direction of the rock and soil mass here and further restricts the shape of the potential slip surface.

[0051] In the two-dimensional slice 2, the rock and soil mass at point B is usually in a tensile state. If there is no external load on the slope surface at this location, then the slope surface at point B is the major principal stress acting surface at point B , and the tangent direction of the slope surface at point B is the direction of the major principal stress acting surface at point B . Here, let σ 1_B and σ 3_B be the major and minor principal stresses of the rock and soil mass at point B respectively, and σ 1_B = 0. Subsequently, in the coordinate system of normal stress and shear stress, a Mohr stress circle reflecting the stress state at point B and the shear strength curve of the rock and soil mass are plotted. Among them, the center point of the Mohr stress circle is O B , the intersection points of the Mohr stress circle and the normal stress axis are point E B and point F B , and point E B and point F B correspond to the minor principal stress B σ 3_B and the major principal stress σ 1_B at point respectively. O B F B represents the direction of the major principal stress acting surface. O B T B is the perpendicular line from the center point O B to the shear strength curve of the rock and soil mass. Since O B T B is the shortest distance from the center point O B to the shear strength curve of the rock and soil mass, therefore, O B TB Representative point B The most unfavorable shear direction when the rock and soil mass fails in shear at this point. If passing through point T B Make a tangent to the shear strength curve of the rock and soil mass, and the horizontal inclination angle of this tangent direction is φ B , then ∠ T B O B F B = ( π / 2 + φ B ), where φ B is the internal friction angle of the slip surface at point B , ∠ T B O B F B represents the angle between the most unfavorable shear direction and the direction of the major principal stress plane when the rock and soil mass fails in shear at point B . According to the double-angle relationship between the Mohr stress circle and the actual model, the angle between the most unfavorable shear direction and the direction of the major principal stress plane when the rock and soil mass fails in shear at point B in the actual model can be obtained as ( π / 4 + φ B / 2), and the most unfavorable shear direction is below the direction of the major principal stress plane. Further, the most unfavorable shear direction is the tangent direction of the potential slip surface, and the horizontal inclination angle of the tangent of the slope surface at point B is β B , then the horizontal inclination angle of the tangent of the potential slip surface at point B can be obtained as α B = β B + φ B / 2 + π / 4, and it is also within the two-dimensional slice 2. In addition, the tangential plane of the potential slip surface at point B is the plane passing through the axis and the tangent of the potential slip surface at point B within the two-dimensional slice 2. Then, using the spatial geometric relationship, the horizontal inclination angle of the tangential plane of the potential slip surface at point B on the xz plane is α B_x = arctan(tan αB / cos β B_y ) and its horizontal inclination angle on the yz plane is α B_y = β B_y .

[0052] As Figure 6 shown, in a three-dimensional slope, any point P is randomly taken on the potential slip surface, and the inclination angle of the line connecting it with point A on the xy plane is θ P . This inclination angle is named the slip surface polar angle. As for the slip surface polar angle θ P , its magnitude is associated with the P and x axis coordinates of point y , that is, it changes with the P and x axis coordinates of point y . This means that the slip surface polar angle θ P is a function of the slip surface coordinates. According to the definition of the Taylor series, any continuous function can be approximately described by the sum of a polynomial and a higher-order error remainder. Further, considering the reliability and ease of use of slip surface generation, the sum of a cubic polynomial and a higher-order error remainder is used to construct the functional relationship between the slip surface polar angle θ P and the slip surface coordinates. The specific functional formula is:

[0053] (1)

[0054] In the formula, α A is the horizontal inclination angle of the tangent direction of the potential slip surface at point A in two-dimensional slice 1; λ 0, λ 1, λ 2, λ 3, λ 4, λ 5, λ 6, λ 7, λ 8, λ 9, λ 10 , λ 11 , λ 12 , ξ 1, η 1, ξ 2 andη 2 is the control parameter of the potential slip surface shape, λ 0, λ 1, λ 2, λ 4, λ 7 and λ 8 have a value range of -10 to 10, λ 3 , λ 5 , λ 6 , λ 9 , λ 10 , λ 11 and λ 12 can be solved based on the stress and position constraint conditions at the boundary points of the potential slip surface, ξ 1, η 1, ξ 2 and η 2 have a value range of 1 to 10; x P and y P are the P and x coordinates of any point y on the potential slip surface along the x A and x B are the A coordinates of the lower boundary point B and the upper boundary point x of the potential slip surface along the y C and y D are the C coordinates of the left boundary point D and the right boundary point y of the potential slip surface along the

[0055] Meanwhile, according to the spatial geometric relationship, the calculation formula for the P coordinate of any point z on the potential slip surface along the

[0056] (2)

[0057] In the formula, z P is the P coordinate of any point z on the potential slip surface along the y A and z A are the A coordinates of the lower boundary pointy and z axis coordinates.

[0058] In addition, when the P axis coordinate remains unchanged at a point, if y the x axis coordinate increases by a differential dx quantity, then a differential P quantity of the slip surface pole angle will be generated at any point dθ P_x on the potential slip surface. Further, applying Equation (1), the calculation formula for dθ P_x can be obtained as:

[0059] (3)

[0060] Similarly, when the P axis coordinate remains unchanged at a point, if x the y axis coordinate increases by a differential dy quantity, then a differential P quantity of the slip surface pole angle will be generated at any point dθ P_y on the potential slip surface. Further, applying Equation (1), the calculation formula for dθ P_y can be obtained as:

[0061] (4)

[0062] As Figure 7 shown, based on the stress constraint condition at the lower boundary point A of the potential slip surface, it can be known that the horizontal inclination angle of the tangential plane of the slip surface at the point A on the xz plane is α A_x ( α A_x = arctan(tan α A / cos β A_y )), and its horizontal inclination angle on the yz plane is α A_y ( α A_y = β A_y ). Therefore, if a differential segment A of the slip surface is selected on the xz plane passing through the point AA' , then the horizontal inclination angle of the differential segment AA' of the slip surface on the xz plane is αA_x , where the point A' compared with the point A has a differential increment in the x axis coordinate dx . If a differential segment of the slip surface is selected on the A plane passing through the point yz , then the horizontal inclination angle of the differential segment AA'' of the slip surface on the AA'' plane is yz α A_y , where the point A'' compared with the point A has a differential increment in the y axis coordinate dy . Further, by combining the slip surface polar angle function formula (i.e., Equation (1)), we can obtain:

[0063] (5)

[0064] (6)

[0065] Using Equation (5) and Equation (6), and substituting α A_x = arctan(tan α A / cos β A_y ) and α A_y = β A_y , we can obtain the stress constraint condition that the slip surface polar angle at the lower boundary point A should satisfy:

[0066] (7)

[0067] (8)

[0068] As Figure 8 shown, based on the stress constraint condition at the upper boundary point B of the potential slip surface, it can be known that the horizontal inclination angle of the tangential plane of the potential slip surface at the point B on the xz plane is α B_x ( α B_x = arctan(tan α B / cos β B_y )) and its horizontal inclination angle on the yz plane is α B_y (​α B_y = β B_y ) For this purpose, on the plane passing through point B of xz , a micro-segment of the sliding surface B'B is selected, and the partial differential component of the sliding surface polar angle in the B'B axis direction in the micro-segment of the sliding surface x is dθ B_x . Among them, the B' coordinate of point B compared with that of point x in the dx axis increases by a differential component of - B'B . Further, according to the geometric definition of the partial differential component of the sliding surface polar angle of the potential sliding surface micro-segment x in the dθ B_x axis direction, the functional relationship between the partial differential component of the sliding surface polar angle α B_x and the horizontal dip angle

[0069] (9)

[0070] In the formula, y B and z B are the B coordinates of the upper boundary point of the potential sliding surface y and z axis respectively.

[0071] Similarly, on the plane passing through point B of yz , a micro-segment of the sliding surface BB'' is selected, and the partial differential component of the sliding surface polar angle in the BB'' axis direction in the micro-segment of the sliding surface y is dθ B_y . Among them, the B'' coordinate of point B compared with that of point y in the dy axis increases by a differential component of BB'' . Further, according to the geometric definition of the partial differential component of the sliding surface polar angle of the sliding surface micro-segment y in the d θ B_y axis direction, the functional relationship between the partial differential component of the sliding surface polar angle α B_y and the horizontal dip angle

[0072] (10)

[0073] Then, by using Equations (3) and (4), the partial differential component of the slip surface pole angle at point B can also be obtained as dθ B_x and dθ B_y . On this basis, by combining Equations (9) and (10), the stress constraint condition that the slip surface pole angle at the upper boundary point B should satisfy is:

[0074] (11)

[0075] (12)

[0076] As Figure 9 shown, when the upper boundary point B of the given potential slip surface (corresponding to the slip surface pole angle θ B ), the left boundary point C (corresponding to the slip surface pole angle θ C ), and the right boundary point D (corresponding to the slip surface pole angle θ D ) are known points, the slip surface pole angle function also needs to satisfy the matching of the coordinates of the upper, left, and right boundary points of the potential slip surface, that is, the boundary position constraint condition. Furthermore, by combining Equation (1) and using the correlation relationship between the spatial point coordinates and the slip surface pole angle, the position constraint conditions that the slip surface pole angles at the upper boundary point B , the left boundary point C , and the right boundary point D should match are:

[0077] (13)

[0078] (14)

[0079] (15)

[0080] By using the stress constraint conditions that the slip surface pole angles at the lower and upper boundary points should satisfy and the position constraint conditions that the slip surface pole angles at the four boundary points should match, namely, Equations (7), (8), (11), (12), (13), (14), and (15), the shape control parameters of the potential slip surface λ 3, λ 5, λ 6, λ 9, λ 10 , λ 11 and λ12 The non - homogeneous system of equations is as follows:

[0081] (16)

[0082] Wherein, 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 all coefficients of the system of equations; b 1 to b 7 are the constant terms of the system of equations.

[0083] In equation (16), a 11 ~ a 17 and b The calculation formulas of 1 are respectively:

[0084] (17)

[0085] In equation (16), a 21 ~ a 27 and b The calculation formulas of 2 are respectively:

[0086] (18)

[0087] In equation (16), a 31 ~ a 37 and b The calculation formulas of 3 are respectively:

[0088] (19)

[0089] In formula (16), a 41 ~ a 47 and b the calculation formulas of 4 are respectively:

[0090] (20)

[0091] In formula (16), a 51 ~ a 57 and b the calculation formulas of 5 are respectively:

[0092] (21)

[0093] In formula (16), a 61 ~ a 67 and b the calculation formulas of 6 are respectively:

[0094] (22)

[0095] In formula (16), a 71 ~ a 77 and b the calculation formulas of 7 are respectively:

[0096] (23)

[0097] If the potential slip 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 are given, then by using formula (16) and combining with matrix calculation methods, the potential slip surface shape control parameters λ 3, λ 5, λ 6, λ 9, λ 10 , λ 11 and λ 12 can be solved.

[0098] For exampleFigure 10 As shown, the discrete technology is adopted to construct the potential slip surface. Furthermore, at the lower boundary points A and the upper boundary points B of x the x A and x B range), the x axis is equally divided into n parts, and n +1 horizontal lines are drawn accordingly. At the left boundary points C and the right boundary points D of y the y C and y D range), the y axis is equally divided into m parts, and m +1 vertical lines are drawn accordingly. For the discrete points P ij (0 ≤ i ≤ n and 0 ≤ j ≤ m ), their projections on the xy plane correspond to the intersection points of the i th horizontal line and the j th vertical line. Furthermore, the P ij of the discrete points on the potential slip surface x and y axis coordinates are respectively:

[0099] (24)

[0100] (25)

[0101] In the formula, x ij and y ij are respectively the P ij of the discrete points on the potential slip surface x and y axis coordinates;

[0102] Furthermore, combining the functional relationship between the slip surface polar angle and the slip surface coordinates (i.e., formula (2)), and respectively substituting x P and y P adopt xij and y ij are substituted to obtain the discrete points of the potential slip surface P ij of z The axial coordinate is:

[0103] (26)

[0104] In the formula, z ij is the discrete point on the potential slip surface P ij of z axial coordinate.

[0105] In addition, based on the discrete grid divided by the longitudinal and transverse lines, during the calculation of the potential slip surface x the differential of the axial coordinate dx and y the differential of the axial coordinate dy are respectively:

[0106] (27)

[0107] (28)

[0108] As Figure 10 and 11 shown, considering the complexity of the three-dimensional slope surface, when the slip surface shape parameters are not properly selected, there may be cases where the discrete points P ij on the potential slip surface all exceed the slope surface, or all do not exceed the slope surface, or some exceed the slope surface but the point A point B point C and point D one of them is not a boundary point on one side of the potential slip surface of the three-dimensional slope. These situations all indicate that the constructed potential slip surface is not feasible. Therefore, the slip surface polar angle should be subject to the potential slip surface range limiting mechanism. To eliminate the infeasible potential slip surface, it is possible to determine the feasibility of the potential slip surface according to the number of intersection points between the broken line segment formed by connecting the discrete points j in the j th column (0 ≤ m ) of the potential slip surface and the slope surface. The specific operation steps (referred to as the potential slip surface feasibility discrimination steps under the control of the intersection points between the potential slip surface and the slope surface) are as follows: ① Let the number of intersection points between the broken line segment composed of the discrete points of the potential slip surface in the P ij th column (0 ≤ i ≤ n ) and the slope surface be j d J ​( j ), and take j = 1; ② Take J d ( j ) = 0 and i = 1; ③ Obtain the P (i-1)j and P ij of the z axis coordinates z (i-1)j and z ij and the z axis coordinates s (i-1)j and s ij at the corresponding positions on the slope surface; ④ Determine whether it satisfies 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 Let J d ( j ) = J d ( j ) + 1; ⑥ Determine whether it satisfies i < n , if not satisfied, let i = i + 1, and return to step ③; ⑦ If J d ( j ) = 2, it indicates that the shape of the potential slip surface satisfies the intersection control condition, otherwise, the selected potential slip surface shape control parameters make the constructed potential slip surface infeasible, and the potential slip surface shape control parameters need to be reselected; ⑧ Determine whether it satisfies j < m , if not satisfied, let j = j + 1, and return to step ②.

[0109] Such as Figure 12As 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 ij The 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 not satisfied, it indicates that the selected potential slip surface shape control parameters make the constructed potential slip surface unreasonable, and the potential slip surface shape control parameters need to be reselected; ⑥ Determine whether it satisfies i < n , if not satisfied, then let i = i + 1, and return to step ②; ⑦ Determine whether it satisfies j < m - 1 , if not satisfied, then let j = j + 1, and return to step ②.

[0110] As Figure 13 shown, in the three-dimensional potential slip surface discrete generation mode, using the vertical and horizontal grid lines, ([[]] n + 1) × ([[]] m + 1) discrete points can be obtained. Among them, only the discrete points located within the slip perimeter boundary are the discrete points on the potential slip surface. To facilitate the combination with the limit equilibrium calculation method for slope stability analysis, the discrete points located within the slip perimeter boundary are defined as effective discrete points, and the discrete points located outside the slip perimeter boundary are defined as invalid discrete points, and the effectiveness of the discrete grids divided by the corresponding vertical and horizontal lines is determined accordingly. That is, for the discrete grid ij , it consists of 4 discrete points, namely point P (i-1)(j-1) , point P (i-1)j , point P i(j-1) and point P ij . If 3 or more discrete points are effective discrete points, then the discrete grid ij is an effective grid; otherwise, it is an invalid grid.

[0111] As Figure 14 shown, the construction steps of the three-dimensional slope general potential slip surface model in the polar angle mode are as follows: ① In the three-dimensional slope, establish a xyz axis space coordinate system with any point as the coordinate origin; ② Given the lower boundary point A , upper boundary point B , left boundary point C and right boundary point D of the potential slip surface and their coordinates, that is, ([[]] x A , y A , z A ), ([[]] x B , y B , zB ), ( x C , y C , z C ), and ( x D , y D , z D ); ③ According to the occurrence of the three-dimensional slope surface, obtain the lower boundary points and upper boundary points of the potential slip surface A of the slope surface at the B horizontal dip angle of the tangent line of the slope surface in the xz and yz planes, that is, β A_x and β A_y , β B_x and β B_y ; ④ Determine the control parameters of the potential slip surface shape λ 0, λ 1, λ 2, λ 4, λ 7, λ 8, ξ 1, ξ 2, η 1 and η 2; ⑤ According to the tangent dip angles of the slope surface at the known potential slip surface boundary points ( β A and β B ) and the soil strength parameters ( φ A and φ B ), use the boundary stress constraint conditions to solve the horizontal dip angles α A and α B of the tangent line of the potential slip surface in the approximate shear failure plane at the boundary points, and determine the horizontal dip angles of the tangent plane of the potential slip surface at the boundary points in the xz and yz planes, that is, α A_x , α A_y , α B_x and α B_y ; ⑥ Use equation (16) to solve the control parameters of the potential slip surface shape λ 3,λ 5. λ 6. λ 9. λ 10 , λ 11 and λ 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 ), the discrete grid ij is a valid grid; Output the effective discrete points and effective grids on the potential slip surface.

[0112] The characteristics of the above-mentioned three-dimensional slope general potential slip surface of the present invention are as follows: The potential slip surface is generated by using the slip surface polar angle control method, without limiting the slip surface type, ensuring the arbitrariness of the potential slip surface. At the same time, the range of the potential slip surface is defined by using four boundary points on the lower side, upper side, left side and right side, and the slip surface polar angle satisfies the stress constraint conditions at the upper and lower side boundary points and the position constraint conditions at the four boundary points, so that the constructed potential slip surface strictly satisfies the physical geometry and mechanical failure characteristics. In addition, the potential slip surface range limitation and the surface concave mechanism are incorporated to ensure the feasibility and rationality of the potential slip surface model. Subsequently, the discrete points on the potential slip surface are obtained by applying the discrete technology, and accordingly, the effective discrete points and discrete grids on the potential slip surface are identified, which is conducive to combining with the limit equilibrium method for carrying out the three-dimensional slope stability analysis under complex conditions.

[0113] The characteristics of the above-mentioned slip surface polar angle of the present invention are as follows: The slip surface polar angle is related to the slip surface coordinates, and an approximate function of the slip surface polar angle and the slip surface coordinates is constructed by using the Taylor series expansion method. At the same time, in order to ensure the simplicity and diversity of the potential slip surface model, the slip surface polar angle approximate function is expressed as the sum of a cubic polynomial and a high-order error remainder in the Taylor series expansion. It can also make the potential slip surface shape control parameters contained in the slip surface polar angle approximate function more than the boundary constraint conditions, thereby ensuring the arbitrariness of the generation of the potential slip surface.

[0114] The characteristics of the above boundary constraint conditions of the present invention are as follows: The boundary constraint conditions are the constraint conditions at the four boundary points on the lower side, upper side, left side, and right side of the potential slip surface. This includes both boundary stress constraint conditions and boundary position constraint conditions. Among them, for the boundary stress constraint conditions, by using the position extreme value characteristics of the upper and lower side boundary points and based on the sliding trends of the rock and soil masses on both sides of the boundary points, the approximate occurrence planes of shear failure of the rock and soil masses at the upper and lower side boundary points are determined. On this basis, by applying the property that the tangent direction of the potential slip surface is the most unfavorable shear direction of the rock and soil mass, when the acting force on the slope surface is known, combined with the shear strength criterion of the rock and soil mass, the most unfavorable shear direction of the rock and soil mass at the upper and lower side boundary points and its tangential plane of the potential slip surface are obtained. Further, according to the spatial geometric relationship, the horizontal inclination angle of the tangential plane of the potential slip surface at the upper and lower side boundary points is solved. In addition, by selecting a micro-segment of the potential slip surface for analysis, the mathematical relationship between the horizontal inclination angle of the tangential plane of the potential slip surface at the upper and lower side boundary points and the partial differential component of the pole angle of the slip surface is derived. Thus, the boundary stress conditions are used to constrain the potential slip surface shape control parameters in the approximate function of the pole angle of the slip surface. As for the boundary position constraint conditions, by using the known coordinates of the four boundary points, the pole angle of the slip surface at the boundary points is calculated. Furthermore, the boundary position conditions are used to constrain the potential slip surface shape control parameters in the approximate function of the pole angle of the slip surface. Embodiment

[0115] A method for constructing a general three-dimensional slope potential slip surface model in a pole angle mode as Figures 1 - 14 shown. For a mountain slope along a certain highway, the slope height is about 10 m, the average slope angle is 45°, and the unit weight of the slope soil is 19 kN / m 3 , the shear of the soil follows the linear M-C strength criterion, and the soil strength parameters are c = 20 kPa and φ = 15°. Under the action of heavy rainfall, obvious cracks and deformations appeared in the houses at the rear edge of the mountain body of this mountain slope. In order to ensure traffic safety and prevent further landslides of the slope, it is necessary to reinforce and protect this slope. Among them, accurately and reliably capturing the potential most dangerous sliding surface of this slope during instability failure becomes the primary task. Thus, the general three-dimensional slope potential slip surface construction method proposed by the present invention is introduced and applied, and the specific operations are as follows:

[0116] Taking a certain point in the area where the slope is located as the origin to establish xyz a spatial coordinate system, where z the z-axis direction is the vertical direction, taking x the x-axis direction as the longitudinal direction of the slope on the horizontal plane, y the y-axis direction as the transverse direction of the slope on the horizontal plane. Then, based on the existing survey data, the coordinates of the lower side boundary point A of the potential slip surface of the slope are obtained as (0 m, 0 m, 0 m) and the inclination angle βA_x and β A_y are respectively β A_x = 45° and β A_y = 0°, the coordinates of the upper side boundary point of the potential slip surface B are (15 m, 0 m, 10 m) and the dip angle of the tangent line of the slip surface β B_x and β B_y are respectively β B_x = 0° and β B_y = 0°, the coordinates of the left side boundary point of the potential slip surface C are marked as (5 m, -5 m, 5 m) and the coordinates of the left side boundary point of the potential slip surface D are (5 m, 5 m, 5 m).

[0117] The shape control parameters of the proposed potential slip surface λ 0, λ 1, λ 2, λ 4, λ 7 and λ 8 all range from -10 to 10, and the shape control parameters of the potential slip surface ξ 1, ξ 2, η 1 and η 2 all range from 1 to 10, and in the z-axis coordinate range of the lower side boundary point A and the upper side boundary point B of the potential slip surface, the z-axis is equally divided into x n segments, and x n n = 100, and in the y-axis coordinate range of the left side boundary point n and the right side boundary point C of the potential slip surface, the y-axis is equally divided into D m segments, and y m y = 100. Furthermore, within the value range of the potential slip surface shape control parameters, a series of potential slip surfaces of the slope can be formed for slope stability analysis; m m m = 100, and then, within the value range of the potential slip surface shape control parameters, a series of potential slip surfaces of the slope can be formed for slope stability analysis;

[0118] Combined with the three-dimensional slope stability limit equilibrium method and embedded with a mathematical optimization algorithm, the slope stability analysis is realized. In this process, when any set of slip surface shape parameters λ 0, λ 1, λ 2, λ 4,λ 7、 λ 8、 ξ 1、 ξ 2、 η 1 and η 2, a reasonable potential slip surface can be generated according to the construction steps of the three-dimensional slope general potential slip surface model in the polar angle mode. Subsequently, using the limit equilibrium method, the slope safety factor corresponding to the potential slip surface is solved. Then, with the assistance of the mathematical optimization algorithm, taking the minimum value of the slope safety factor as the optimization goal, the search for the potentially most dangerous slip surface of the slope is carried out. Thus, the purpose of slope stability analysis is completed, and the minimum slope safety factor is output and the result of the potentially most dangerous slip surface of the slope is given. Among them, the minimum slope safety factor is F s = 1.31, and the shape control parameters of the potentially most dangerous slip surface are respectively: λ 0 = 3.32, λ 1 = -1.25, λ 2 = -0.59, λ 3 = 1.75, λ 4 = 1.78, λ 5 = -5.05, λ 6 = 3.94, λ 7 = -8.40, λ 8 = 7.64, λ 9 = -2.55, λ 10 = 2.94, λ 11 = 1.93, λ 12 = 0.64, ξ 1 = 1.41, ξ 2 = 2.57, η 1 = 2.07 and η 2 = 6.69. Based on the results of slope stability analysis, according to the stability evaluation criteria of the Technical Code for Building Slope Engineering (GB 50330-2013), the slope stability state is evaluated. At the same time, the potentially unstable range of the slope is clarified, providing a scientific basis for subsequent safe construction and reliable reinforcement of the slope.

[0119] The present invention adopts a smooth surface pole angle control method to generate a potential sliding surface, uses four boundary points on the lower side, upper side, left side and right side of the sliding surface to define the range of the potential sliding surface, establishes stress constraint conditions that the pole angles of the sliding surface 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 in the three-dimensional slope potential sliding surface model, ensures the feasibility and reasonableness of the shape and range of the sliding surface, strictly meets the physical and mechanical mechanisms for generating the potential sliding surface, and facilitates the stability analysis of the three-dimensional slope under complex conditions. The present invention has the advantages of being simple and easy to implement, wide application range, strong generality, high accuracy, etc., and provides a strong scientific basis for reliable early warning and prevention of slope instability and landslide.

[0120] It is easy for those skilled in the art to understand that the above description is only a preferred embodiment of the present invention and is not used to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A construction method for a general potential sliding surface model of a three-dimensional slope in the polar angle mode, characterized in that The method adopts a smooth surface polar angle control method to generate a potential sliding surface, and uses four boundary points on 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 side boundary point and the upper side boundary point of the sliding surface, and the sliding trends of the rock and soil masses on both sides at these two boundary points, the approximate occurrence planes of shear failure of the rock and soil masses at the lower side boundary point and the upper side boundary point are determined. According to the fact that the tangent direction of the potential sliding surface is the most unfavorable shear direction of the rock and soil mass, when the acting force on the slope surface is known, combined with the shear strength criterion of the rock and soil mass, the stress constraint conditions that the polar angles of the sliding surface at the lower side boundary point and the upper side boundary point should satisfy are established. 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 related to the sliding surface coordinates, and an approximate function between the polar angle of the sliding surface and the sliding surface coordinates is constructed by using the Taylor series expansion method. In addition to satisfying the stress and position constraint conditions at the boundary points of the potential sliding surface, the polar angle of the sliding surface also conforms 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 effective 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 the three-dimensional slope in the polar angle mode. The generation process of the discrete points on the potential sliding surface includes the following steps: Step S1: In a three-dimensional slope, establish a xyz axis spatial coordinate system with any point as the origin of the coordinate system; Step S2: Specify the lower boundary points of the potential slip surface A , the upper boundary points B , the left boundary points C and the right boundary points D and their 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 points and upper boundary points of the potential slip surface according to the occurrence of the three-dimensional slope surface A and the upper boundary points B The horizontal dip angles of the tangents of the slope surface on xz and yz planes, that is β A_x and β A_y , β B_x and β B_y ; Step S4: Determine the control parameters for the potential slip surface shape 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 tangent inclination angle of the slope surface at the known potential slip surface boundary points β A and β B as well as the soil strength parameters φ A and φ B , using the boundary stress constraint conditions, solve for the horizontal inclination angles α A and α B of the tangent of the potential slip surface in the approximate plane of shear failure of the rock and soil mass at the boundary points, and thereby determine the horizontal inclination angles of the tangents of the potential slip surface tangential plane at the boundary points in the xz and yz planes, namely α A_x , α A_y , α B_x and α B_y ; Step S6: Solve the potential slip surface shape control parameters using Equation (16). 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 all coefficients of the system of equations; b 1 ~ b 7 are the constant terms of the system of equations; Step S7: Calculate the P ij - x -, y - z axis coordinates of the discrete points on the potential slip surface using Equations (24), (25) and (26), i.e., x ij -, y ij - z ij , where 0 ≤ i ≤ n and 0 ≤ j ≤ m ; n refers to the number of equal divisions of the x axis, and m refers to the number of equal divisions of the y axis; (24) (25) (26)。 2. The construction method of the three-dimensional slope general potential slip surface model in the polar angle mode according to claim 1, characterized in that The step of conforming to the potential sliding surface range limitation and the surface concave mechanism and identifying and outputting the effective discrete points and discrete grids on the potential sliding surface includes the following steps: Step S8: Analyze the feasibility of the constructed potential sliding surface by using the feasibility discrimination step of the potential sliding surface controlled by the intersection point of the potential sliding surface and the slope surface; if it is not feasible, the shape control parameters of the potential sliding surface need to be reselected; Step S9: Analyze the rationality of the constructed potential sliding surface by using the rationality discrimination step of the potential sliding surface under the concave mechanism; if it is not reasonable, the shape control parameters of the potential sliding surface need to be reselected; Step S10: Compare the discrete points on the potential slip surface P ij of the z axial coordinates z ij and the z axial coordinates s ij of the corresponding slope points. If z ij ≤ s ij , then the discrete point P ij is a valid discrete point, where 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S11: Determine the validity of the discrete grid by using the validity of the discrete points on the potential slip surface. If among the discrete points P (i-1)(j-1) , discrete point P (i-1)j , discrete point P i(j-1) and discrete point P ij there are 3 or more 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 construction method of the three-dimensional slope general potential sliding surface model in the polar angle mode according to claim 1, characterized in that, In the said step S4, λ 0, λ 1, λ 2, λ 4, λ 7 and λ 8 have a value range of -10 to 10, ξ 1, η 1, ξ 2 and η 2 have a value range of 1 to 10.

4. The construction method of the three-dimensional slope general potential slip surface model in the polar angle mode according to claim 2, characterized in that In the said step S8, in order to eliminate infeasible potential slip surfaces, according to the j number of intersection points between the broken line formed by successively connecting the discrete points of the potential slip surface in the P ij column and the slope surface, the feasibility of the potential slip surface is determined, where 0 ≤ i ≤ n and 0 ≤ j ≤ m ; Step S8 specifically includes the following steps: Step S8-1: Let the number of intersection points between the polyline composed of the discrete points of the potential slip surface in the j th column and the slope surface be J d ( j ), and take j = 1; Step S8-2: Take J d ( j ) = 0 and i = 1; Step S8-3: Obtain discrete points on the potential slip surface P (i-1)j and P ij of the z axial coordinates z (i-1)j and z ij as well as the z axial coordinates s (i-1)j and s ij ; Step S8-4: Determine whether it satisfies 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 the condition i < n is satisfied. If not, let i = i + 1, and return to Step S8-3; Step S8-7: If J d ( j ) = 2, it indicates that the shape of the potential slip surface meets the intersection control condition; otherwise, the selected potential slip surface shape control parameters make the constructed potential slip surface infeasible, and the potential slip surface shape control parameters need to be reselected; Step S8-8: Determine whether the condition j < m is satisfied. If not, set j = j + 1, and return to Step S8-2.

5. The construction method of the three-dimensional slope general potential sliding surface model in the polar angle mode according to claim 2, characterized in that In the said step S9, the sliding surface pole angle should conform to the concave mechanism of the potential sliding surface curve, and the rationality of the potential sliding surface is determined by whether the following Condition 1 and Condition 2 are satisfied. Condition 1 is that the broken line segment formed by sequentially connecting the discrete points of the potential sliding surface in the i row shows an increasing trend in the sliding surface curvature, and Condition 2 is that the broken line segment formed by sequentially connecting the discrete points of the potential sliding surface in the P ij column shows an increasing trend in the sliding surface curvature; 0 ≤ j ≤ P ij and 0 ≤ i ≤ n and 0 ≤ j ≤ m ; The said step S9 specifically includes the following steps: Step S9-1: Let 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: Obtain discrete points P ij The horizontal dip angle of the tangential plane of the slip surface at xz on the plane α x_(i,j) , where Step S9-4: Obtain discrete points P ij The horizontal dip angle of the tangential plane of the slip surface at yz on the plane α y_(i,j) , where Step S9-5: Determine whether the horizontal inclination angle of the tangent of the sliding surface satisfies: and ; If it does not satisfy, it indicates that the selected shape control parameters of the potential sliding surface make the constructed potential sliding surface unreasonable, and the shape control parameters of the potential sliding surface need to be reselected; Step S9-6: Determine whether the following is satisfied i < n ; if not, then let i = i + 1, and return to Step S9-2; Step S9-7: Determine whether it satisfies j < m-1 ; if not, then set j = j + 1, and return to Step S9-2.

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