A method for calculating generalized limit residual thrust of three-dimensional stability evaluation of slope

Abstract

The application develops a generalized limit residual thrust calculation method for three-dimensional stability evaluation of slope, and belongs to the field of geotechnical engineering safety analysis. The application effectively solves the inherent limitation problem that the traditional two-dimensional limit equilibrium method cannot truly reflect the three-dimensional space effect and lateral constraint effect of slope instability, and also solves the problem that the existing three-dimensional method leads to non-unique calculation results and insufficient precision due to too many assumptions of inter-strip forces. By establishing a micro-unit body sliding surface control differential equation under a symmetric mode and fusing a fuzzy programming search strategy, the most dangerous three-dimensional sliding surface form is automatically identified, and a unique minimum safety factor solution is given. The method significantly improves the objectivity, accuracy and reliability of the slope stability evaluation, and provides a scientific basis and evaluation theoretical method for the safety analysis and disaster prevention decision of slope engineering under complex load.

Description

Technical Field

[0001] This invention relates to the field of slope stability analysis in geotechnical engineering, specifically to a method for calculating the generalized limit residual thrust for three-dimensional slope stability evaluation. Background Technology

[0002] In the construction of highways, railways, water conservancy and hydropower projects, landslide mitigation, and open-pit mines, slope stability assessment is a core aspect of ensuring project safety. Traditional stability analysis methods mainly rely on the two-dimensional limit equilibrium method, which simplifies the complex spatial problem into a plane strain problem. While this method is computationally convenient, it cannot truly reflect the three-dimensional spatial failure and lateral constraint effects of slope instability.

[0003] Although the three-dimensional limit equilibrium method is more consistent with reality in theory, its application has long been limited by two major problems: first, the spatial shape of the three-dimensional sliding surface is difficult to determine accurately in advance; second, when discretizing the sliding body into elements for force analysis, the complex interaction forces between elements, especially the lateral shear forces, make the mechanical model statically indeterminate, making the solution difficult and not unique. Some existing three-dimensional methods usually require making many assumptions about inter-strip forces to simplify the calculation, which undoubtedly affects the accuracy and reliability of the results.

[0004] Therefore, developing an analytical method that can reasonably determine the three-dimensional slip surface, effectively handle the inter-strip force problem, and ultimately output a deterministic safety factor solution is of great theoretical and engineering significance for improving the accuracy of slope stability evaluation. Summary of the Invention

[0005] To address the difficulties in determining the slip surface and the non-uniqueness of solutions encountered in existing three-dimensional limit equilibrium methods, this invention aims to provide a unique solution method for three-dimensional slope stability calculation. This method constructs a generalized limit residual thrust differential equation under symmetric failure modes and combines iterative search using fuzzy programming theory. Under comprehensive consideration of multiple loads such as self-weight, groundwater, earthquakes, and blasting vibrations, it can automatically find the most dangerous three-dimensional slip surface and its corresponding minimum safety factor (i.e., the unique optimal solution).

[0006] To achieve the above objectives, this invention provides a method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation, comprising: S1: Define the coordinate system and symmetric slip surface equation based on the three-dimensional spatial morphology of the slope; S2: Based on the coordinate system and the equation of the symmetric sliding surface, the potential sliding body is discretized into multiple vertical prism elements, and the residual thrust transfer relationship between the elements is established. S3: Based on the residual thrust transmission relationship, establish the force balance equation based on the limit equilibrium principle and derive the control differential equation of the residual thrust change, and then construct the functional expression of the safety factor. By using the variational method, the problem of finding the most dangerous slip surface is transformed into solving the Euler-Lagrange equation. S4: Based on the Euler-Lagrange equations, a numerical strategy of inverse connection and fuzzy programming is adopted to iteratively search for the minimum safety factor and the corresponding most dangerous slip surface that satisfy the mechanical boundary conditions, and to complete the calculation of the ultimate residual thrust.

[0007] Preferably, S1 includes: Establish a right-handed coordinate system, with the principal sliding direction as... Axis, vertically upward Axis and transverse are Axis; When the unit is very small, set the potential sliding body about If the plane is symmetric, then the equation of the smooth surface is a curved surface: And it satisfies the symmetry relation: Preferably, S2 includes: along The axis divides the sliding body into sections of length . of A vertical prism unit cell, when At that time, according to the generalized residual thrust transmission relationship, the first Residual thrust at unit exit end Satisfy the following formula: In the formula: For safety reasons, For the density of the sliding force, It is the anti-slip density.

[0008] Preferably, S3 includes: When the number of elements approaches infinity, the residual thrust function The governing differential equations are satisfied along the main sliding direction: According to the mechanical boundary closure condition, the following should be satisfied: By integrating the control differential equation, the functional expression for the safety factor is obtained: Then, by using the variational method, the functional is minimized, thus obtaining the most dangerous slip surface function. The Euler-Lagrange equations that must be satisfied.

[0009] Preferably, S4 includes: Starting from the exit area at the toe of the slope, given the exit chamfer angle, candidate slip surfaces are generated by inverse numerical integration of the Euler-Lagrange equations, and the remaining thrust is calculated by simultaneously solving the control differential equations. A fuzzy programming model is constructed with the remaining thrust in the exit zone at the toe of the slope approaching zero as a fuzzy constraint and the minimum safety factor as a fuzzy objective. The safety factor is dynamically adjusted based on the gradient of the closure error and the change in the safety factor until the closure convergence and extreme value stability are satisfied. Then the minimum safety factor and the most dangerous slip surface can be obtained.

[0010] Preferred, sliding force density It includes the weight of the sliding body, the inertial external force components such as earthquakes or explosions, and the water pressure gradient or seepage force components along the sliding direction. Anti-slip density It is calculated based on the Mohr-Coulomb strength criterion and by calculating the effective normal stress through three-dimensional contact normal projection. If the rock mass is damaged or deteriorated, the density of the equivalent anti-slip force is obtained by introducing a strength reduction factor based on experimental results to reduce the cohesion and friction coefficient respectively. and in the governing differential equation Alternative .

[0011] Preferably, fuzzy constraints are obtained by constructing membership functions and setting closure errors: , in, For the inlet end of the slip surface (the top of the slope) at Position coordinates on the axis, P(x) export The remaining thrust at the toe of the slope is expressed as an exponential membership function. In the formula, This is the sensitivity coefficient; it should satisfy P(x) upon iteration convergence. export ).

[0012] Compared with existing calculation methods, the innovative achievements of this invention are as follows: This invention effectively overcomes the inherent limitations of traditional two-dimensional limit equilibrium methods, which cannot accurately reflect the three-dimensional spatial effects and lateral constraints of slope instability. It also solves the problems of inaccurate calculation results and insufficient accuracy caused by excessive assumptions about inter-strip forces in existing three-dimensional methods. By establishing the governing differential equations under a symmetric mode and integrating a fuzzy programming inverse search approach, it solves the problem of automatically identifying the most dangerous three-dimensional slip surface morphology while simultaneously determining the unique minimum safety factor. This method can significantly improve the objectivity, accuracy, and reliability of slope stability evaluation, providing a more scientific theoretical approach for slope engineering safety evaluation and disaster prevention decision-making under complex loads. Without changing the original governing equation structure and solution process, this invention, based on experimental results, introduces material damage effects by reducing strength parameters, achieving an integrated coupled calculation method for rock mass damage and stability analysis. This makes the calculation model applicable to stability calculations under seismic or blasting vibration disturbance conditions. Attached Figure Description

[0013] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0014] Figure 1 This is a schematic diagram of the potential sliding body coordinate system and element division in an embodiment of the present invention; Figure 2 These are the unit blocks and state points in this embodiment of the invention; Figure 3 This is a force distribution diagram of the unit block in an embodiment of the present invention. Detailed Implementation

[0015] The present invention will be further described below with reference to the accompanying drawings and embodiments. It should be understood that the following embodiments are only used to explain the technical solutions of the present invention and are not intended to limit the scope of protection of the present invention.

[0016] Example 1: This embodiment provides a generalized ultimate residual thrust calculation method for three-dimensional slope stability evaluation, and its steps are as follows.

[0017] S1. Establish the coordinate system and the equation of the symmetric sliding surface.

[0018] like Figure 1 As shown, a right-handed coordinate system is established, with the principal sliding direction as... The axis, vertically upward is The axis is horizontal. Axis. Assume the potential sliding body is about Planar symmetry, the smooth surface morphology is expressed by the surface function expression (1). (1) Equation (1) satisfies Equation (2): (2) When the unit volume is infinitely small, assuming the sliding surface is symmetric, this applies only to the half-domain. The calculation is performed, and then the global result is recovered through symmetry.

[0019] To describe the geometric change of the sliding surface along the principal sliding direction, a slope metric for the sliding surface in the principal sliding direction is defined. When force transmission analysis is performed with the main sliding direction as the controlling direction, the sliding surface inclination angle can be described by the equivalent relationship of equation (3).

[0020] (3) S2. Discretize the potential sliding body and establish the remaining thrust transfer relationship.

[0021] Discretize the potential three-dimensional sliding body along the principal sliding direction as follows: A length of Tiny vertical prism unit cells, Figure 2 The distribution of unit blocks and their state points is shown. When... The time-slip surface can be approximated as a smooth surface by the broken line of the element's bottom surface. Based on the generalized residual thrust method, the first... Residual thrust at unit exit end The result is the sum of the thrust of the previous unit and the contribution of the driving force and resistance of this unit, which satisfies equation (4).

[0022] (4) In the formula: For safety factor; The density of the sliding force; It is the anti-slip density.

[0023] To ensure consistency in the three-dimensional force projection, the three-dimensional geometric quantities of the sliding surface are first given. Let: (5) The unit normal vector of the smooth surface is taken as: (6) The area element of the smooth surface is: (7) In the given At this location, the width of the horizontal half-domain is denoted as The integration range is .

[0024] (1) Sliding force density The sliding force density is obtained by projecting the external load onto the sliding direction and integrating it laterally. Define the unit tangential vector along the principal sliding direction: (8) set up For density, The vector of gravitational acceleration. This refers to the equivalent acceleration field caused by earthquakes or explosions. Let the equivalent thickness under unit horizontal projection be such that the volume element can be written as The sliding force density per unit principal sliding length can then be expressed as: (9) In the formula: This is the additional distributed load density term along the sliding direction, used to characterize the slope load along the sliding direction component, seepage force along the sliding direction component, etc.

[0025] (2) Anti-slip density The anti-slip force density is based on the Mohr-Coulomb strength criterion. The effective normal stress at the slip surface is obtained by three-dimensional contact normal projection. Let the pore water pressure be... Then the normal stress can be written as: (10) The effective normal stress is: (11) The local shear strength of the sliding surface is: (12) in For cohesion, Let be the internal friction angle. Integrating over the transverse half-domain and incorporating the area element yields the anti-slip force density: (13) (3) Equivalent anti-skid force density under damage reduction When considering material damage caused by seismic disturbances, blasting, or weathering, a strength reduction factor is introduced. and ,satisfy , and define: (14) (15) Thus, the anti-slip force density after damage reduction is obtained: (16) In the calculation under damage conditions, with Alternative The rest of the derivation and solution process remains unchanged.

[0026] S3. Establish the governing differential equation and construct the safety factor functional, and derive the variational conditions.

[0027] based on Figure 3 The force analysis of the element shown is used to establish the force balance equation between elements based on the principle of limit equilibrium, and the governing differential equation for the change of residual thrust is derived as follows.

[0028] The thrust transmission relationship from S2: (17) Divide both sides And order The governing differential equations under the continuous limit are obtained as follows: (18) If damage reduction is taken into account, then Replace with : (19) Using the mechanical boundary closure condition, the residual thrust at the inlet and outlet of the sliding surface is zero: (20) For the governing differential equation by Points to Combining the boundary conditions, we can obtain: (twenty one) Thus, the functional expression for the safety factor is obtained: (twenty two) When considering damage reduction, the corresponding situation is: (twenty three) The most dangerous slip surface is equivalent to the slip surface that minimizes the functional of the safety factor: (twenty four) The minimization problem can be transformed into solving the Euler-Lagrange equations using the variational method. These Euler-Lagrange equations are slip surface functions. (Under symmetric simplification, it can be written as) (a necessary condition for ).

[0029] S4: Inverse connection and fuzzy programming iterative search for minimum safety factor and most dangerous slip surface This embodiment employs a numerical strategy of "reverse connection + fuzzy programming". Let the exit end of the sliding surface be... The entrance end is Given the elevation of the exit point Chamfer at the exit ,Pick: (25) Given a trial safety factor Below, the Euler-Lagrange equations are self-contained. Towards Inverse integration generates candidate smooth surfaces The remaining thrust is calculated by simultaneously solving the control differential equations during the inverse integration process. The discrete implementation can be written as: (26) in For the integration step size, , or (27) Integrate backwards to the entry point The remaining thrust at the foot of the slope was then obtained: P(x export )=P(x a (28) in For the inlet end of the slip surface (the top of the slope) at Position coordinates on the axis. Define the closure error. ΔP=P(x export (29) Physical closure requirements Meanwhile, the goal is to make Minimize it as much as possible. Construct the closure condition as a fuzzy constraint and use an exponential membership function: (30) In the formula This is the sensitivity coefficient.

[0030] During the iteration process, the optimization variables are dynamically adjusted based on the changing trends of the closure error and the safety factor. Let the... The variable for the next iteration is , A unified symbol-driven update approach can be adopted: (31) (32) in and This is the step size parameter, which can be gradually decreased with iterations to ensure stability.

[0031] The iteration ends and the result is output when the double convergence criterion is met: (1) Physical closure (33) (2) Extreme value stability (34) Output the minimum safety factor after the iteration is complete. With the most dangerous slippery surface And recover the global sliding surface under the symmetry assumption. This allows for the completion of a three-dimensional stability evaluation of the slope.

[0032] Example 2 This invention proposes a generalized ultimate residual thrust calculation system for three-dimensional slope stability evaluation, used to implement the method described in Embodiment 1. The system includes a definition module, a construction module, a solution module, and a calculation module. The definition module is used to establish the coordinate system and the equation of the symmetric slip surface; the construction module is used to discretize the potential slip body into vertical prism elements and establish the thrust transfer relationship; the solution module is used to establish the governing differential equation and the safety factor functional and form the Euler-Lagrange equation; the calculation module is used to perform inverse connection and fuzzy programming iteration, outputting the minimum safety factor and the most dangerous slip surface. When considering damage reduction, the system introduces a reduction coefficient in the anti-slip force density calculation stage and replaces r(x) with rd(x), realizing the integrated coupling of damage effect and stability evaluation.

[0033] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation, characterized by the following steps: include: S1: Define the coordinate system and symmetric slip surface equation based on the three-dimensional spatial morphology of the slope; S2: Based on the coordinate system and the equation of the symmetric sliding surface, the potential sliding body is discretized into multiple vertical prism elements, and the residual thrust transfer relationship between the elements is established. S3: Based on the residual thrust transmission relationship, establish the force balance equation based on the limit equilibrium principle and derive the control differential equation of the residual thrust change, and then construct the functional expression of the safety factor. By using the variational method, the problem of finding the most dangerous slip surface is transformed into solving the Euler-Lagrange equation. S4: Based on the Euler-Lagrange equations, a numerical strategy of inverse connection and fuzzy programming is adopted to iteratively search for the minimum safety factor and the corresponding most dangerous slip surface that satisfy the mechanical boundary conditions, and to complete the calculation of the ultimate residual thrust.

2. The method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation according to claim 1, characterized in that, S1 includes: Establish a right-handed coordinate system, with the principal sliding direction as... Axis, vertically upward Axis and transverse are Axis; setting the potential sliding body about If the plane is symmetric, then the smooth surface has the following equation: And it satisfies the symmetry relation: 。 3. The method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation according to claim 1, characterized in that, S2 includes: along The axis divides the sliding body into sections of length . of A vertical prism unit cell, when At that time, according to the generalized residual thrust transmission relationship, the first Residual thrust at unit exit end Satisfy the following formula: In the formula: For safety reasons, For the density of the sliding force, It is the anti-slip density.

4. The method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation according to claim 1, characterized in that, S3 includes: When the number of elements approaches infinity, the residual thrust function The governing differential equations are satisfied along the main sliding direction: According to the mechanical boundary closure condition, the following should be satisfied: By integrating the control differential equation, the functional expression for the safety factor is obtained: Then, by using the variational method, the functional is minimized, thus obtaining the most dangerous slip surface function. The Euler-Lagrange equations that must be satisfied.

5. The method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation according to claim 1, characterized in that, S4 includes: Starting from the exit area at the toe of the slope, given the exit chamfer angle, candidate slip surfaces are generated by inverse numerical integration of the Euler-Lagrange equations, and the remaining thrust is calculated by simultaneously solving the control differential equations. A fuzzy programming model is constructed with the remaining thrust in the exit zone at the toe of the slope approaching zero as a fuzzy constraint and the minimum safety factor as a fuzzy objective. The safety factor is dynamically adjusted based on the gradient of the closure error and the change of the safety factor until the closure convergence and extreme value stability are satisfied, and then the minimum safety factor and the most dangerous slip surface are obtained.

6. The method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation according to claim 4, characterized in that, Sliding force density It includes the sliding body's own weight component, the inertial external force component, and the water pressure gradient or seepage force component along the sliding direction; Anti-slip density It is based on the Mohr-Coulomb strength criterion and is obtained by calculating the effective normal stress through three-dimensional contact normal projection. If the rock mass is damaged or deteriorated, the density of the equivalent anti-slip force can be obtained by introducing a strength reduction factor based on experimental results to reduce the cohesion and friction coefficient respectively. and in the governing differential equation Alternative .

7. The method for calculating the generalized ultimate residual thrust for three-dimensional slope stability evaluation according to claim 5, characterized in that, Fuzzy constraints are obtained by constructing membership functions and setting closure errors: ， in, For the smooth inlet end at Position coordinates on the axis, P(x) export The remaining thrust at the toe of the slope is expressed as an exponential membership function. In the formula, This is the sensitivity coefficient; it should satisfy P(x) upon iteration convergence. export ).

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