A method for constructing a universal potential sliding surface model for two-dimensional slopes in polar angle mode

Abstract

The present invention provides a method for constructing a potential sliding surface model universal for two-dimensional slopes under a polar angle mode, and belongs to the field of engineering disaster prevention and mitigation. The method introduces end boundary condition constraints and a polar angle increasing mechanism, and generates an unlimited potential sliding surface curve. The method provided by the present invention, on the one hand, describes the generation process of the potential sliding surface from the lower sliding out point to the upper sliding out point using a function of the sliding surface polar angle, and constructs the sliding surface polar angle function in an unrestricted sliding surface form using a polynomial method, thereby ensuring the universality of the potential sliding surface generation. On the other hand, the end boundary constraint conditions and the polar angle increasing mechanism are introduced to ensure the effectiveness and rationality of the potential sliding surface generation. In addition, combined with a discrete method, the potential sliding surface is easily generated. Therefore, the present invention can effectively improve the accuracy of slope stability assessment, has the advantages of being simple and easy to implement, high in precision, and wide in application, and can provide a strong scientific basis for reliable early warning of slope instability and collapse.

Description

Technical Field

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

[0002] Landslide disasters, as a major geological disaster, have a serious impact on human production and life. In order to protect people's lives and property and avoid or reduce the losses caused by landslides, it is necessary to accurately and reliably evaluate slope stability and predict the most dangerous potential sliding surface. For slope stability evaluation and the most dangerous sliding surface prediction, choosing a reasonable sliding surface model is the key to ensuring the accuracy and reliability of the calculation results. A reasonable sliding surface model can more accurately simulate the actual damage of the slope during the instability process, thereby obtaining more effective slope safety factor calculation results and slope collapse range, providing strong technical support for landslide disaster warning and slope engineering construction.

[0003] In slope stability analysis, existing sliding surface models are mainly divided into simple specific sliding surface models and complex random sliding surface models. For simple specific sliding surface models, the sliding surface is mainly simplified into a certain type of regular curve, such as a straight line, circular arc, logarithmic spiral, etc. In the calculation, the straight line model simulates the translation failure of the slope by giving two boundary points, and the circular arc and logarithmic spiral simulate the rotation failure of the slope by determining the center (or polar coordinate origin) and controlling the radius (or polar diameter) rotation. The simple specific sliding surface model requires few parameters for calculation, is simple to implement and easy to construct, and is therefore widely used in current engineering calculations. For complex random sliding surface models, existing methods usually simulate a series of discrete points through a certain connection method. Specifically, random parameters are given in the calculation, and then discrete points are obtained according to a polynomial function or parametric equation, and the sliding surface morphology is determined by a certain connection method (straight line or spline curve, etc.). The complex random sliding surface model can theoretically obtain a more accurate and most dangerous sliding surface morphology, and can also be better applied to the stability analysis of various complex slopes.

[0004] Although the above methods have their own advantages, they also have certain limitations. For example, the simple specific sliding surface model limits the sliding surface morphology, and the potential sliding surface constructed is not universal, which makes it difficult to accurately and effectively capture the most dangerous potential sliding surface of the slope in the slope with complex geological conditions. The complex random sliding surface model needs to set more parameters to simulate any sliding surface morphology, which makes the calculation efficiency low. At the same time, the sliding surface generated by random means often needs to introduce a large number of constraints to ensure the rationality of its morphology, making its construction more cumbersome. In addition, the existing sliding surface model does not incorporate the physical constraints of the end, which makes the constructed potential most dangerous sliding surface lack rationality and reliability.

[0005] With the increasing complexity and refinement of slope engineering design and construction theory, the need to accurately identify the most dangerous potential sliding surface of the slope in a simple and easy way, and then scientifically evaluate the stability of the slope has become more urgent. However, the sliding surface models currently used often cannot meet such requirements. Therefore, building a simpler and more effective sliding surface model is an important issue to be solved in the field of slope engineering. The key lies in simulating the widest possible range of sliding surface types with the simplest implementation method possible, while ensuring the rationality of the simulated sliding surface morphology.

[0006] Therefore, a new method for constructing a universal potential sliding surface model for two-dimensional slopes is needed in this field. Summary of the invention

[0007] The present invention provides a new method for constructing a potential sliding surface model universal for two-dimensional slopes. The present invention uses the inclination angle in the horizontal direction of the line connecting any point on the potential sliding surface and the sliding out point as the polar angle, adopts the sliding surface polar angle rotation method to describe the potential sliding surface generation process, and applies the polynomial method to construct the sliding surface polar angle function. At the same time, the end boundary conditions and the polar angle increment mechanism are introduced to ensure the effectiveness and rationality of the potential sliding surface generation. The present invention proposes an efficient and practical sliding surface construction technology solution, which effectively improves the efficiency and accuracy of identifying the most potentially dangerous sliding surface in the process of slope stability assessment, and can widely cover the common sliding surface forms in engineering practice, and has strong adaptability and high-precision analysis characteristics. In addition, the technology is simple to operate and easy to implement, providing strong support for the safe design and assessment of slope engineering.

[0008] The present invention firstly provides a method for constructing a universal potential sliding surface model of a two-dimensional slope. The method introduces end boundary condition constraints and a polar angle increasing mechanism to obtain the universal potential sliding surface model of the two-dimensional slope.

[0009] In a specific embodiment, a non-limited potential slip surface curve is generated in the method.

[0010] In a specific implementation, the sliding surface polar angle of any point P on the potential sliding surface of the two-dimensional slope is θ and the coordinates of point P are (x, y), and the sliding surface polar angle is the inclination angle of the line between point P and the downward sliding outward point A in the horizontal direction; in the method, a cubic polynomial and a sum of high-order error remainders are used to construct a function of the sliding surface polar angle θ, and a polar angle increasing mechanism is adopted, and the first-order derivative of the sliding surface polar angle function is applied to be greater than 0 to achieve it, and a discrete generation method is adopted to equally separate the potential sliding surface into n segments;

[0011] The function of the sliding surface polar angle θ is shown in formula (1):

[0012]

[0013] And for any point P on the potential sliding surface, its coordinate calculation formula is shown in formula (2):

[0014] y=y A +(xx A )tanθ(2)

[0015] In the formula, λ 1 , 2 , 3 and m are potential sliding surface shape parameters, where λ 1 , 2 and m are both free variables; the coordinates of the sliding out point A and the sliding out point B (x A ,y A ) and (x B ,y B ); α AB is the horizontal inclination of the line connecting the sliding out point A and the upper sliding out point B; α A It is the horizontal inclination angle of the tangent line of the potential sliding surface at the sliding exit point A.

[0016] In a specific embodiment, the method comprises the following steps:

[0017] S1: Given the potential sliding surface shape parameter λ 1 , 2 and m;

[0018] S2: Determine the coordinates of the downward sliding point A and the upward sliding point B (x A ,y A ) and (x B ,y B ) and its corresponding slope surface tangent direction in the horizontal direction A and β B ;

[0019] S3: Obtain the internal friction angle of the potential sliding surface at the sliding out point A and the upper sliding out point B and

[0020] S4: Calculate the inclination angle α of the most unfavorable shear direction in the horizontal direction at the lower slide-out point A and the upper slide-out point B A and α B ; That is α A is the horizontal inclination angle of the potential sliding surface tangent direction at the sliding exit point A, α B is the horizontal inclination angle of the potential sliding surface tangent direction at the upper sliding point B, where: and

[0021] S5: Using the coordinates of the downward sliding point A and the upward sliding point B (xA ,y A ) and (x B ,y B ), calculate the horizontal inclination angle α of the line connecting the lower slide-out point A and the upper slide-out point B AB , the straight-line distance between the downward sliding point A and the upward sliding point B l AB And the horizontal distance d between the downward sliding point A and the upward sliding point B AB ;

[0022] S6: Divide the potential sliding surface into n equal segments and calculate the horizontal length d of each segment. AB / n;

[0023] S7: Use equation (4) to solve the polar angle differential dθ B ;

[0024]

[0025] In the formula, dx B is the horizontal increment of the sliding surface micro-segment at the upper sliding point B, where dx B =d AB / n;l AB The distance between the downward sliding point A and the upward sliding point B;

[0026] S8: Use equation (5) to solve the potential sliding surface shape parameter λ 3 ; and formula (5) is obtained by combining formula (4) with formula (1) to calculate the polar angle differential at the upper slide-out point B;

[0027]

[0028] S9: Use equation (6) to calculate the coordinates (x i ,y i )(1≤i≤n);

[0029]

[0030] S10: Determine whether the discrete point i on the potential discrete sliding surface satisfies equation (3).

[0031]

[0032] If the discrete point i on the potential discrete sliding surface does not satisfy formula (3), it is considered that the shape of the constructed potential sliding surface is unreasonable; otherwise, if it is a potential sliding surface of reasonable shape, that is, the n-1 discrete points i except point A and point B all meet formula (3), then the result is output, that is, the potential sliding surface model is obtained.

[0033] In the present invention, step S1 and step S2 can be performed in any order, step S3 is performed after step S2, step S4 is performed after step S1 and step S2, step S5 is performed after step S2, step S6 is performed after step S5, step S7 is performed after step S1, step S2, step S5 and step S6, step S8 is performed after steps S1 to S7, step S9 is performed after steps S1 to S8, and step S10 is performed after steps S1 to S9.

[0034] In a specific implementation, in step S1, λ 1 and λ 2 The value range of is -10 to 10, and the value range of m is 1 to 10.

[0035] The present invention also provides an application of the potential sliding surface model obtained by the construction method as described above to capture the most potentially dangerous sliding surface of the slope when it is unstable and damaged.

[0036] The advantages of the present invention are that: in the case of non-limited sliding surface shape, the present invention introduces the end boundary constraint condition and the polar angle increment mechanism, which ensures the rationality of the construction of the sliding surface model and makes it more in line with the actual situation. At the same time, in addition to the upper and lower sliding points of the potential sliding surface end, the sliding surface shape parameters only include 4 parameters, namely, λ 1 , 2 , 3 and m, where λ 1 , 2 The three parameters m and m are free variables, which means that the generation of potential sliding surface in the present invention is easy to control; in addition, the potential sliding surface can be generated discretely in the present invention; the implementation method of the method of the present invention is simple, with high precision and a wide range of applications, which can effectively improve the accuracy of slope stability assessment and provide a strong scientific basis for reliable early warning of slope instability and landslide. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is a schematic diagram of the two-dimensional slope and its potential sliding surface of the present invention.

[0038] Figure 2 It is a schematic diagram of the principal stress action surface at the end of the universal potential sliding surface of the two-dimensional slope of the present invention; among them, Figure m is a schematic diagram of the principal stress action surface at the lower sliding out point A, and Figure n is a schematic diagram of the principal stress action surface at the upper sliding out point B.

[0039] Figure 3 It is a schematic diagram of the Mohr stress circle and the shear strength curve of the rock and soil mass of the stress state at the end of the universal potential sliding surface of the two-dimensional slope of the present invention; wherein, Figure m is a schematic diagram of the Mohr stress circle and the shear strength curve at the lower sliding out point A, and Figure n is a schematic diagram of the Mohr stress circle and the shear strength curve at the upper sliding out point B.

[0040] Figure 4 It is a schematic diagram of the horizontal inclination angle of the sliding surface tangent direction at the end of the universal potential sliding surface of the two-dimensional slope of the present invention; among them, Figure m is a schematic diagram of the horizontal inclination angle of the sliding surface tangent direction at the lower sliding out point A, and Figure n is a schematic diagram of the horizontal inclination angle of the sliding surface tangent direction at the upper sliding out point B.

[0041] Figure 5 A schematic diagram of constructing a general potential sliding surface model for a two-dimensional slope generated by the end boundary condition constraints and the polar angle increasing mechanism of the present invention.

[0042] Figure 6 It is a schematic diagram of the horizontal inclination angle constraint condition of the sliding surface tangent at the sliding-out point on the end of the universal potential sliding surface of the two-dimensional slope of the present invention.

[0043] Figure 7 It is a schematic diagram of the discrete generation method of the universal potential sliding surface of a two-dimensional slope according to the present invention.

[0044] Figure 8 The present invention is a flowchart of the steps for constructing a universal potential sliding surface model of a two-dimensional slope.

[0045] In the figure, 1. Two-dimensional slope; 2. Potential sliding surface; 3. Slope surface; 4. End of potential sliding surface; 5. Small principal stress action surface; 6. Large principal stress action surface; 7. Tangent direction of sliding surface; 8. Moore stress circle; 9. Shear strength curve; 10. Most unfavorable shear direction; 11. Tangent direction of slope surface; 12. Horizontal direction; 13. Any point on the potential sliding surface; 14. Sliding surface polar angle; 15. Micro-segmentation of sliding surface at the upper sliding point; 16. Discrete points on the potential sliding surface. DETAILED DESCRIPTION

[0046] The present invention designs a method for constructing a general potential sliding surface model of a two-dimensional slope under the constraint of end boundary conditions and the mechanism of increasing polar angle. The method does not limit the form of the sliding surface, and describes the generation process of the potential sliding surface from the downward sliding exit point to the upward sliding exit point by using the function of the sliding surface polar angle, wherein the sliding surface polar angle is the inclination angle of the line connecting any point on the potential sliding surface and the downward sliding exit point in the horizontal direction. Then, a polynomial method is used to construct the sliding surface polar angle function. At the same time, the end boundary conditions and the mechanism of increasing polar angle are introduced to ensure the effectiveness and rationality of the generation of the potential sliding surface. Based on this, the discrete method is combined to realize the simple generation of the potential sliding surface. For the end boundary conditions of the potential sliding surface, it is based on the potential sliding surface tangent direction representing the most unfavorable shear direction when the rock and soil body fails in shear. The known external boundary information of the end of the potential sliding surface is used, and the relationship between the Moore stress circle at the end of the potential sliding surface and the shear strength curve of the rock and soil body is combined to obtain the calculation formula of the horizontal inclination angle of the sliding surface tangent direction at the end of the potential sliding surface, so as to form the boundary condition constraint of the end of the potential sliding surface. The potential sliding surface polar angle increasing mechanism is realized by applying the first-order derivative of the sliding surface polar angle function being greater than 0.

[0047] In general, the present invention relates to a method for constructing a universal potential sliding surface model for slope stability analysis, which adopts a sliding surface polar angle function in the form of a non-restricted sliding surface to ensure the universality of potential sliding surface generation, and introduces end boundary constraints and a polar angle incremental generation mechanism to ensure the effectiveness and rationality of potential sliding surface generation, thereby achieving a balance of multiple advantages such as universality, accuracy, applicability and ease of use in the generation of potential sliding surfaces in complex actual slope engineering.

[0048] The specific model building method and implementation process of the present invention are as follows:

[0049] (1) Slope instability is caused by shear failure of rock mass and the formation of a connected sliding surface. On the one hand, the shear failure of rock mass is controlled by the strength criterion. Here, the general expression of the shear strength criterion of rock mass is τ f =f(σ), where τ f is the shear strength of the rock mass, σ is the normal stress acting on the failure surface of the rock mass, on the other hand, the shear failure of the rock mass usually occurs along its most unfavorable shear direction, therefore, the tangent direction of the potential sliding surface of the slope represents the most unfavorable shear direction when the rock mass fails in shear.

[0050] (2) Figure 1 As shown in the figure, in the two-dimensional slope, the ends of the potential sliding surface (i.e., the lower sliding point A and the upper sliding point B) are located both on the potential sliding surface 2 and on the slope surface 3. Since the slope surface is the external boundary, the stress information on it is known. Therefore, the stress state at the end of the potential sliding surface will be constrained by the known boundary conditions and affect the most unfavorable shear direction of the rock and soil at the end, thereby restricting the shape of the potential sliding surface.

[0051] (3) Figure 2 As shown in the figure, for the sliding out point A at the end of the potential sliding surface of the two-dimensional slope, it is usually in a compressive state. If there is no external load on the slope surface here, the slope surface here is the surface of small principal stress, and the surface perpendicular to the slope surface is the surface of large principal stress at this location. For the sliding out point B at the end of the potential sliding surface of the two-dimensional slope, it is usually in a tensile state. If there is no external load on the slope surface here, the slope surface here is the surface of large principal stress, and the surface perpendicular to the slope surface is the surface of small principal stress at this location.

[0052] (4) Figure 3 As shown, let σ 1_A and σ 3_A The major and minor principal stresses at the sliding out point A are respectively established, and the rectangular coordinate system of normal stress and shear stress is established. The Mohr stress circle and the shear strength curve of the rock and soil body reflecting the stress state at the sliding out point A are drawn. Among them, σ 3_A =0, at the same time, let σ 1_B and σ 3_Bare the major and minor principal stresses at the upper sliding point B, respectively. The Moore stress circle and the shear strength curve of the rock and soil mass reflecting the stress state at the upper sliding point B are also drawn. Among them, σ 1_B =0.

[0053] (5) Figure 3 As shown, in the Mohr stress circle at the sliding out point A, the center point is O A , the intersection of the Mohr stress circle and the normal stress axis is point E A and point F A , and point E A and point F A The corresponding normal stress values ​​are the minor principal stress σ at the sliding out point A. 3_A and major principal stress σ 1_A In addition, O A E A represents the direction of the minor principal stress action surface, O A F A Represents the direction of the major principal stress action surface, O A T A The point O is the center of the circle A The perpendicular line to the shear strength curve of rock and soil is A T A The point O is the center of the circle A The shortest distance to the shear strength curve of the rock mass, therefore, O A T A It also reflects the most unfavorable shear direction when shear failure occurs in the rock mass at the downward sliding point A.

[0054] (6) Figure 3 As shown, in the Mohr stress circle at the upper sliding point B, the center point is O B , the intersection of the Mohr stress circle and the normal stress axis is point E B and point F B , and point E B and point F B The corresponding normal stress values ​​are the minor principal stress σ at the upper sliding point B. 3_B and major principal stress σ 1_B In addition, O B E B represents the direction of the minor principal stress action surface, O B F B Represents the direction of the major principal stress action surface, O B T B The point O is the center of the circle B The perpendicular line to the shear strength curve of rock and soil is B T B The point O is the center of the circle B The shortest distance to the shear strength curve of the rock mass, therefore, O B TB It also reflects the most unfavorable shear direction when shear failure occurs in the rock mass at the upper sliding point B.

[0055] (7) Figure 3 As shown, through point T A Draw the tangent line of the shear strength curve of the rock and soil mass. The horizontal inclination angle of this tangent line is It is the internal friction angle of the sliding surface at the point A where the slide goes down. At the same time, the angle passing through point T B Draw the tangent line of the shear strength curve of the rock and soil mass. The horizontal inclination angle of this tangent line is It is the internal friction angle of the sliding surface at the upper sliding point B. If the linear strength criterion is used, and are all constant values. If the nonlinear strength criterion is adopted, and are functions related to the sliding surface stress at the lower sliding point A and the upper sliding point B, respectively, and need to be solved using an iterative cycle strategy.

[0056] (8) Figure 3 As shown, based on O A T A With point T A The tangent line of the shear strength curve of the rock and soil mass is perpendicular, then O A T A With O A E A The angle between It reflects the relationship between the most unfavorable shear direction and the direction of the small principal stress surface when the rock mass at the sliding out point A fails in shear. Since the Moore stress circle is twice the angle in the actual model, the angle between the most unfavorable shear direction and the direction of the small principal stress surface when the rock mass at the sliding out point A fails in shear in the actual model (see the right triangle in the upper left corner of Figure m) is And the most unfavorable shear direction is below the direction of the small principal stress action surface.

[0057] (9) Figure 3 As shown, based on O B T B With point T B The tangent line of the shear strength curve of the rock and soil mass is perpendicular, then O B T B With O B E B The angle between It reflects the relationship between the most unfavorable shear direction and the direction of the small principal stress surface when the rock mass at the upper sliding point B fails in shear. Since the Moore stress circle is twice the angle in the actual model, the angle between the most unfavorable shear direction and the direction of the small principal stress surface when the rock mass at the upper sliding point B fails in shear in the actual model (see the right triangle in the upper left corner of Figure n) is And the most unfavorable shear direction is on the right side of the direction of the small principal stress action surface.

[0058] (10) Figure 4 As shown in the figure, since the slope surface at the sliding exit point A is the surface of the small principal stress, the tangent direction of the slope surface at the sliding exit point A represents the direction of the small principal stress surface. Let the inclination angle of the tangent direction of the slope surface at the sliding exit point A in the horizontal direction be β A Then, according to the actual model, the angle between the most unfavorable shear direction and the direction of the small principal stress action surface when the rock mass at the sliding point A fails in shear is At the same time, the tangent direction of the potential sliding surface of the slope represents the most unfavorable shear direction when the rock mass fails in shear, and the inclination angle of the tangent direction of the sliding surface at the sliding exit point A in the horizontal direction is α A , then we can get the inclination angle of the tangent line of the sliding surface at the sliding point A in the horizontal direction When the tangent direction of the sliding surface at the sliding exit point A is above the horizontal direction, this angle is positive, otherwise it is negative.

[0059] (11) Figure 4 As shown in the figure, since the slope surface at the upper sliding point B is the surface of the major principal stress, the tangent direction of the slope surface at the upper sliding point B represents the direction of the major principal stress surface. Let the inclination angle of the tangent direction of the slope surface at the upper sliding point B in the horizontal direction be β B , then the horizontal inclination angle of the direction of the minor principal stress action surface at the upper sliding point B is (π / 2+β B ), then, according to the angle between the most unfavorable shear direction and the direction of the small principal stress action surface when the rock mass at the upper sliding point B in the actual model fails in shear, At the same time, the tangent direction of the potential sliding surface of the slope represents the most unfavorable shear direction when the rock mass fails, and the inclination angle of the tangent direction of the sliding surface at the upper sliding point B in the horizontal direction is α B , then we can get the inclination angle of the tangent line of the sliding surface at the sliding point B in the horizontal direction When the tangent direction of the slope surface at the upper sliding point B is above the horizontal direction, β B is a positive value, otherwise it is a negative value.

[0060] (12) Figure 5As shown in the figure, for the potential sliding surface of the two-dimensional slope, since the principal stress action surface direction of the rock and soil in the slope is continuously deflected from the lower sliding point A to the upper sliding point B, it means that the inclination angle of the tangent direction of the sliding surface of the two-dimensional slope in the horizontal direction will also change from α to α from bottom to top. A Continuously change to α B In order to describe this continuous change process, the horizontal inclination angle of the line connecting any point on the potential sliding surface and the sliding exit point is taken as the sliding surface polar angle, and the positive direction of the sliding surface polar angle is counterclockwise. Then, the tangent direction of the two-dimensional slope sliding surface is converted from continuous deflection from bottom to top into the sliding surface polar angle from α A Continuously change to α AB , where α AB It is the horizontal inclination angle of the line connecting the lower slide-out point A and the upper slide-out point B.

[0061] (13) Figure 5 As shown in the figure, the xy-axis rectangular coordinate system is established with the foot of the slope as the origin. The coordinates of the sliding out point A and the sliding out point B on the sliding surface are (x A ,y A ) and (x B ,y B ), take any point P on the potential sliding surface of the two-dimensional slope, its corresponding sliding surface polar angle is θ and its corresponding coordinates are (x, y). When point P changes continuously from the downward sliding point A to the upward sliding point B, the sliding surface polar angle θ will also change from α A Continuously change to α AB , that is, the sliding surface polar angle θ is a continuity function of the corresponding sliding surface point coordinates, and the sliding surface polar angles at the lower sliding out point A and the upper sliding out point B are α A and α AB Based on the Taylor series expansion definition, any continuous function can be regressed by using a finite-order polynomial and the sum of high-order error residuals. The higher the polynomial order, the higher the degree of regression approximation, but the more corresponding parameters. In order to ensure the effectiveness and simplicity of sliding surface generation, the sum of cubic polynomial and high-order error residuals is used to construct the function of the sliding surface polar angle θ. The specific calculation formula is:

[0062]

[0063] In the formula, λ 1 , 2 , 3 and m are the potential sliding surface shape parameters.

[0064] For any point P on the potential sliding surface, its coordinate calculation formula is:

[0065] y=y A +(xx A )tanθ(2)

[0066] In order to ensure the rationality of the generation of potential sliding surface and avoid the overlapping phenomenon of sliding surface, the polar angle increasing mechanism should be adopted, that is, the first-order derivative of the sliding surface polar angle function is greater than 0. Therefore, the shape parameters of the potential sliding surface should satisfy the following relationship:

[0067]

[0068] (14) Figure 6 As shown, when the sliding surface polar angle θ changes from α A Continuously change to α AB When the horizontal inclination constraint of the sliding surface tangent at the sliding exit point A at the end of the potential sliding surface is satisfied, in order to satisfy the horizontal inclination constraint of the sliding surface tangent at the upper sliding exit point B at the end of the potential sliding surface, the upper polar angle differential dθ of the sliding surface at the upper sliding exit point B is established. B The inclination angle α with the tangent line of the sliding surface at the upper sliding point B B For the differential segment of the sliding surface at the upper sliding point B, it can form a differential triangle with the lower sliding point A. Then, based on the trigonometric relationship of the differential triangle, the polar angle differential dθ can be derived. B Inclination angle α with the sliding surface tangent B The specific relationship between them is:

[0069]

[0070] In the formula, dx B is the micro-segmented horizontal increment of the sliding surface at the upper sliding point B, l AB is the distance between the downward sliding point A and the upward sliding point B.

[0071] Using the principle of mathematical differentiation, the polar angle differential dθ can also be obtained based on formula (1): B Then, the calculation formula of the polar angle differential at the upper sliding point B is combined with the formula (4) and the formula (1). Thus, the sliding surface shape parameter λ that satisfies the horizontal inclination constraint condition of the sliding surface tangent at the upper sliding point B at the end of the potential sliding surface is established. 3 The calculation formula is:

[0072]

[0073] (15) Figure 7 As shown in the figure, in order to facilitate the construction of the potential sliding surface, the discrete generation method is adopted to discretize the potential sliding surface into n segments, and the sliding surface will be composed of n+1 points. Except that the starting point is the sliding out point A and the ending point is the upper sliding out point B, the other points are controlled by the sliding surface shape parameters. If the sliding surface is divided into n equal parts according to the horizontal distance between the lower sliding out point A and the upper sliding out point B, the horizontal length of each equal segment is d AB / n, in addition, the horizontal increment dx of the sliding surface at the upper sliding point B B It can also be approximately equal to dAB / n, where d AB is the horizontal distance between the lower sliding-out point A and the upper sliding-out point B.

[0074] For a discrete point i on the potential sliding surface, let its coordinates be (x i ,y i ), then the specific calculation formula for the coordinates of point i is:

[0075]

[0076] (16) Figure 8 As shown in the figure, the steps of constructing the general potential sliding surface model of two-dimensional slope under the constraint of end boundary conditions and the mechanism of increasing polar angle are as follows: ① Given the potential sliding surface shape parameter λ 1 , 2 and m; ② Determine the coordinates of the downward sliding point A and the upward sliding point B (x A ,y A ) and (x B ,y B ) and its corresponding slope surface tangent direction in the horizontal direction A and β B ; ③ Obtain the internal friction angle of the potential sliding surface at the downward sliding point A and the upward sliding point B and ④ Calculate the inclination angle α of the most unfavorable shear direction in the horizontal direction at the lower sliding point A and the upper sliding point B A and α B ⑤Use the coordinates of the downward sliding point A and the upward sliding point B (x A ,y A ) and (x B ,y B ), calculate the horizontal inclination angle α of the line connecting the lower slide-out point A and the upper slide-out point B AB , the straight-line distance between the downward sliding point A and the upward sliding point B l AB And the horizontal distance d between the downward sliding point A and the upward sliding point B AB ⑥ Divide the potential sliding surface into n equal segments and calculate the horizontal length d of each segment AB / n; ⑦Use equation (4) to solve the polar angle differential dθ B ⑧Use formula (5) to solve the potential sliding surface shape parameter λ 3 ; ⑨Use formula (6) to calculate the coordinates (x i ,y i )(1≤i≤n); ⑩ Determine whether the discrete point i on the potential discrete sliding surface satisfies equation (3). If not, the constructed potential sliding surface is considered to have an unreasonable shape. Otherwise, it is a potential sliding surface of reasonable shape, and the result is output.

[0077] The characteristics of the above-mentioned universal potential sliding surface of the present invention are: the tangent direction of the potential sliding surface of the slope represents the most unfavorable shear direction when shear failure occurs in the rock and soil body, and its generation is constrained by the end boundary conditions and follows the polar angle increasing mechanism.

[0078] The above-mentioned end boundary condition constraint of the present invention is characterized in that: based on the feature that the end of the potential sliding surface is located both on the potential sliding surface and on the slope surface, the slope surface is used as a known external boundary to determine the large and small principal stress action surfaces and their directions at the end of the potential sliding surface. Then, the relationship between the Moore stress circle at the end of the potential sliding surface and the shear strength curve of the rock and soil mass is applied to obtain the most unfavorable shear direction and the angle between it and the small principal stress action surface when shear failure occurs in the rock and soil mass at the end of the potential sliding surface, and then the calculation formula for the horizontal inclination angle of the sliding surface tangent at the end of the potential sliding surface is derived, thereby forming the end boundary condition constraint.

[0079] The characteristics of the generation of the above-mentioned polar angle increasing mechanism of the present invention are: the universal potential sliding surface polar angle of the two-dimensional slope is the inclination angle of the line connecting any point on the potential sliding surface and the sliding exit point in the horizontal direction. When the potential sliding surface is generated from the sliding exit point to the sliding exit point, the horizontal inclination angle of the slope sliding surface tangent will also change continuously from bottom to top. This continuous change can be reflected by the sliding surface polar angle function. In order to ensure the effectiveness and simplicity of sliding surface generation, the sliding surface polar angle function is constructed by the sum of cubic polynomials and high-order error residues. At the same time, in order to ensure the rationality of potential sliding surface generation and avoid the overlapping phenomenon of sliding surface, the polar angle increasing mechanism is adopted, and the first-order derivative of the sliding surface polar angle function is applied to be greater than 0 to achieve it. In order to facilitate the construction of potential sliding surface, a discrete generation method is adopted, and the potential sliding surface is equally dispersed into n segments.

[0080] The advantages of the present invention are as follows: in the case of unrestricted sliding surface shape, the end boundary conditions and the polar angle increasing mechanism are introduced to ensure the rationality of the sliding surface model construction and make it more in line with the actual situation. At the same time, in addition to the positions of the upper and lower sliding points at the end of the potential sliding surface, the sliding surface shape parameters only include 4, namely, λ 1 , 2 , 3 and m, where λ 1 , 2 The three parameters m and m are free variables, which means that the generation of potential sliding surface is easy to control. In addition, the potential sliding surface can be generated discretely. The implementation method of the present invention is simple, with high precision and a wide range of applications. It can effectively improve the accuracy of slope stability assessment and provide a strong scientific basis for reliable early warning of slope instability and landslide.

[0081] Example

[0082] A kind of Figures 1 to 8The construction method of the two-dimensional slope general sliding surface model under the end boundary condition constraint and polar angle increment mechanism is shown in the figure. In a highway construction project, a mountain slope with complex geological conditions, low rock and soil strength and great safety hazards is passed through. The slope is 21m high, the average slope angle is 40°, and the slope soil weight is 18kN / m 3 . In order to ensure the safe and smooth progress of highway construction, it is of primary importance to accurately and reliably capture the most dangerous sliding surface range of the slope when it fails and to efficiently and accurately evaluate the slope stability. Therefore, the two-dimensional slope universal potential sliding surface model construction method invented in this paper was introduced and applied, which successfully and effectively guided the definition of the potential range of slope collapse and the implementation of support and reinforcement measures. The specific operations are as follows:

[0083] (1) According to the requirements of the Code for Geotechnical Engineering Investigation (GB50021-2001), a field survey was conducted on the highway slope to obtain the two-dimensional slope surface shape and stratigraphic data and the corresponding rock and soil strength parameters in the potential sliding area. The shear failure of the slope rock and soil obeys the linear MC strength criterion, and the soil strength parameters are c = 23 kPa and

[0084] (2) The two-dimensional slope universal potential sliding surface model of the present invention is selected, and then, based on the possible range of the potential sliding body, the end position of the sliding surface is estimated, thereby determining the horizontal coordinates x of the lower and upper sliding points of the potential sliding surface. A and x B The value range is: when the xy axis coordinate system is established with the slope foot point as the origin, the horizontal coordinate x of the sliding point on the potential sliding surface is A The value range is -5m to 5m. The horizontal coordinate x of the sliding point on the potential sliding surface is B The value range is 25m to 35m. Furthermore, within the range of the horizontal coordinates of the lower and upper sliding points of the potential sliding surface, the vertical coordinate value of the sliding point on the corresponding potential sliding surface is obtained based on the measured slope surface shape. A =0m(x A <0m) and y A =x A tan40°(x A ≧0m), and the vertical coordinate of the sliding point on the corresponding potential sliding surface is y B = 21m. In addition, the horizontal inclination angle of the tangent line of the slope at the sliding point under the potential sliding surface is β A =0°(x A <0m) and β A =40°(x A ≧0m), and the inclination angle of the tangent line of the slope surface at the sliding point on the potential sliding surface in the horizontal direction is β B =0°;

[0085] (3) Take the potential sliding surface shape parameter λ 1 and λ 2 The value range is -10 to 10 and the value range of the potential sliding surface shape parameter m is 1 to 10, and the number of potential sliding surface segmentation n is taken as 100. Then, within the estimated range of the sliding surface end position and the value range of the potential sliding surface shape parameter, a series of potential sliding surfaces of the slope are formed for slope stability analysis.

[0086] (4) Combine the limit equilibrium method and embed the mathematical optimization algorithm to realize the slope stability analysis. In this process, when any set of sliding surface shape parameters λ is taken 1 , 2 and m and the horizontal coordinates x of the sliding points below and above the sliding surface A and x B After that, a reasonable potential sliding surface can be generated according to the end boundary condition constraints and the polar angle increasing mechanism. Then, the limit equilibrium method is used to solve the safety factor of the lower slope corresponding to the potential sliding surface. Then, with the assistance of the mathematical optimization algorithm, the minimum safety factor of the slope is used as the optimization target to search for the most dangerous sliding surface of the slope. In this way, the purpose of slope stability analysis is completed, the minimum safety factor of the slope is output, and the result of the most dangerous sliding surface of the slope is given. Among them, the horizontal coordinates of the lower and upper sliding points of the most dangerous sliding surface of the slope are x A =0m and x B =31.6m and the most dangerous sliding surface shape parameter is λ 1 =-1.329, λ 2 =1.983,λ 3 = -0.460 and m = 8.266;

[0087] (5) Based on the slope stability analysis results, the slope stability is evaluated in accordance with the stability evaluation standard of the Technical Code for Building Slope Engineering (GB 50330-2013). At the same time, the potential instability range of the slope is clarified to provide a scientific basis for subsequent safe construction and reliable reinforcement of the slope.

[0088] When constructing the potential sliding surface model of the present invention, the potential sliding surface shape parameter λ is first given in step S1. 1 , 2 and m, all of which are specific data, are used to obtain the potential sliding surface model of the present invention. Then, when the potential sliding surface model of the present invention is used to predict the potential sliding surface of a two-dimensional slope, the sliding surface shape parameter λ introduced into the model is 1 , 2and m are both a data range. After automatic calculation by the computer, multiple different (a series of) potential sliding surfaces are obtained. Then, some means and methods (such as the above-mentioned limit equilibrium method and mathematical optimization algorithm) are used to determine which of these potential sliding surfaces is closest to the actual most dangerous sliding surface, thereby providing strong technical support for landslide disaster warning and slope engineering construction.

[0089] In general, in the existing technical methods, simple specific sliding surface models often limit the sliding surface form and have poor versatility. Under complex conditions, it is difficult to accurately reflect the actual shape and range of slope collapse. The complex random sliding surface model does not limit the sliding surface form, but the generation of smooth curved sliding surfaces involves many sliding surface shape parameters and is difficult to achieve. At the same time, the end boundary constraints and the polar angle increment generation mechanism are not introduced, and the rationality and effectiveness of the generated sliding surface cannot be ensured. Therefore, the existing methods have not solved the contradiction between the versatility, accuracy, applicability and ease of use of potential sliding surface generation, making it difficult for the existing slope stability theory analysis to meet the needs of actual engineering.

[0090] However, the present invention provides a new method for constructing a universal potential sliding surface model for two-dimensional slopes. On the one hand, the generation process of the potential sliding surface from the downward sliding exit point to the upward sliding exit point is described by a function of the sliding surface polar angle, and a polynomial method is used to construct the sliding surface polar angle function without limiting the sliding surface form, thereby ensuring the universality of the potential sliding surface generation. On the other hand, end boundary constraints and a polar angle increment mechanism are introduced to ensure the effectiveness and rationality of the potential sliding surface generation. In addition, the potential sliding surface is easily generated by combining a discrete method. Therefore, the present invention can effectively improve the accuracy of slope stability assessment, has the advantages of being simple and easy to operate, high in precision, and having a wide range of applications, and can provide a strong scientific basis for reliable early warning of slope instability and landslides.

[0091] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for constructing a universal potential sliding surface model for two-dimensional slopes in a polar angle mode, characterized in that: The construction method introduces the end boundary condition constraint and the polar angle increasing mechanism to obtain the potential sliding surface model universal for the two-dimensional slope, and generates an unrestricted potential sliding surface curve in the method; in the method, the sliding surface polar angle of any point P on the potential sliding surface of the two-dimensional slope is θ and the coordinates of point P are (x, y), and the sliding surface polar angle is the inclination angle of the line between point P and the downward sliding out point A in the horizontal direction; in the method, the sum of a cubic polynomial and a high-order error remainder is used to construct a function of the sliding surface polar angle θ, and the polar angle increasing mechanism is adopted, and the first-order derivative of the sliding surface polar angle function is applied to be greater than 0 to achieve it, and a discrete generation method is adopted to equally separate the potential sliding surface into n segments; The function of the sliding surface polar angle θ is shown in formula (1): And for any point P on the potential sliding surface, its coordinate calculation formula is shown in formula (2): y=y A +(x-x A )tanθ(2) Where λ1, λ2, λ3 and m are potential sliding surface shape parameters, among which λ1, λ2 and m are free variables; the coordinates of the sliding out point A and the sliding out point B (x A ,y A ) and (x B ,y B ); α AB is the horizontal inclination of the line connecting the sliding out point A and the upper sliding out point B; α A is the horizontal inclination of the tangent line of the potential sliding surface at the sliding exit point A; And the method comprises the following steps, S1: Given potential sliding surface shape parameters λ1, λ2 and m; S2: Determine the coordinates of the downward sliding out point A and the upward sliding out point B (x A ,y A ) and (x B ,y B ) and its corresponding slope surface tangent direction in the horizontal direction A and β B ; S3: Obtain the internal friction angle of the potential sliding surface at the sliding out point A and the upper sliding out point B and S4: Calculate the inclination angle α of the most unfavorable shear direction in the horizontal direction at the lower slide-out point A and the upper slide-out point B A and α B ; That is α A is the horizontal inclination angle of the potential sliding surface tangent direction at the sliding exit point A, α B is the horizontal inclination angle of the potential sliding surface tangent direction at the upper sliding point B, where: and S5: Using the coordinates of the downward sliding point A and the upward sliding point B (x A ,y A ) and (x B ,y B ), calculate the horizontal inclination angle α of the line connecting the lower slide-out point A and the upper slide-out point B AB , the straight-line distance between the downward sliding point A and the upward sliding point B, l AB And the horizontal distance d between the downward sliding point A and the upward sliding point B AB ; S6: Divide the potential sliding surface into n equal segments and calculate the horizontal length d of each segment. AB / n; S7: Use equation (4) to solve the polar angle differential dθ B ; In the formula, dx B is the micro-segmented horizontal increment of the sliding surface at the upper sliding point B, where dx B =d AB / n;l AB is the distance between the downward sliding point A and the upward sliding point B; S8: Use equation (5) to solve the potential sliding surface shape parameter λ3; and equation (5) is obtained by combining equation (4) with equation (1) to calculate the polar angle differential at the upper sliding point B; S9: Use equation (6) to calculate the coordinates (x i ,y i )(1≤i≤n); S10: Determine whether the discrete point i on the potential discrete sliding surface satisfies equation (3). If the discrete point i on the potential discrete sliding surface does not satisfy formula (3), it is considered that the shape of the constructed potential sliding surface is unreasonable; otherwise, if it is a potential sliding surface of reasonable shape, that is, the n-1 discrete points i except point A and point B all meet formula (3), then the result is output, that is, the potential sliding surface model is obtained.

2. The method according to claim 1, characterized in that In the step S1, the value ranges of λ1 and λ2 are both data in the range of -10 to 10, and the value range of m is both data in the range of 1 to 10.

3. An application of a potential sliding surface model obtained by the construction method as claimed in claim 1 or 2 to capture the most potentially dangerous sliding surface of a slope when it fails due to instability.

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