Slope stability analysis method and system based on first-order reliability

Abstract

The invention discloses a slope stability analysis method and system based on first-order reliability, and belongs to the technical field of slope reliability analysis of water conservancy and hydropower engineering. The method comprises the following steps: constructing a performance function of a target slope according to pre-acquired parameter information of the target slope; determining random variables in the performance function and probability distribution information of the random variables; converting the random variable into an independent standard normal random variable based on the probability distribution information of the random variable; and based on the independent standard normal random variable, calculating a reliable index of the target slope, and analyzing according to the reliable index to obtain the stability condition of the target slope. The method breaks through the bottleneck of a traditional first-order reliability method in stability and iteration speed, provides solid technical support for slope risk assessment and design in water conservancy and hydropower engineering, and has wide application value in water conservancy and hydropower engineering practice.

Description

Technical Field

[0001] The invention belongs to the technical field of slope reliability analysis of water conservancy and hydropower engineering, and particularly relates to a slope stability analysis method and system based on first-order reliability. Background Art

[0002] Landslide disasters, as a common natural disaster, pose a serious threat to human life and property safety. Their suddenness and destructiveness make slope stability analysis and risk assessment particularly important in water conservancy and hydropower projects. In recent decades, with the continuous advancement of engineering technology and the deepening of scientific research, it has become a common and scientific practice to regard rock and soil parameters as random variables and use probabilistic methods to evaluate the stability of natural or engineered soil slopes. This method can better reflect the inherent uncertainty of rock and soil parameters and provide a more reliable basis for the design and construction of slope projects. The uncertainty of rock and soil parameters is a major problem in slope stability analysis, but thanks to the continuous efforts of scientific researchers, this uncertainty problem has been well handled. However, despite such progress, researchers still face the challenge of developing effective probabilistic slope stability assessment procedures in long-term practice. This is mainly because the assessment of slope stability involves many complex factors, such as geological conditions, climatic conditions, human activities, etc. The combined effect of these factors makes the assessment of slope stability extremely complicated.

[0003] Among the numerous slope stability analysis methods, the Monte Carlo simulation method plays an important role in slope reliability analysis because it can fully consider the uncertainty of parameters and give probabilistic evaluation results. It is regarded as the basic method of slope reliability analysis. However, the Monte Carlo simulation method requires a huge amount of calculation and high calculation cost when simulating the minimum failure probability in slope design. This may be one of the reasons why it is well-known in geotechnical engineering academic research but relatively less used in actual engineering applications. In contrast, the first-order reliability method has become a common tool for probabilistic slope design due to its advantages such as high computational efficiency and easy implementation. The first-order reliability method provides useful information for reliability-based slope design by finding design points and calculating reliability indicators, which helps engineers better understand and grasp the stability of slopes.

[0004] The essence of the first-order reliability method is to find the design point closest to the limit state curve in the independent standard normal space, and this process involves iterative optimization. The traditional HLRF method has relatively low storage and computational requirements during the iteration process, and can quickly converge to the optimal solution in most cases. However, for highly complex nonlinear limit state functions, the HLRF method may converge slowly or even diverge, causing the design point to fall into periodic and chaotic solutions. These challenges severely limit the scope of application of the first-order reliability method in slope design, construction, and maintenance. Summary of the invention

[0005] The purpose of the present invention is to overcome the deficiencies in the prior art and provide a slope stability analysis method and system based on first-order reliability, which breaks through the bottleneck of the traditional first-order reliability method in terms of stability and iteration speed, and provides a solid technical support for slope risk assessment and design in water conservancy and hydropower projects.

[0006] The present invention provides the following technical solutions:

[0007] In a first aspect, a slope stability analysis method based on first-order reliability is provided, comprising: constructing a functional function of a target slope based on parameter information of the target slope acquired in advance; determining random variables in the functional function and probability distribution information of the random variables; based on the probability distribution information of the random variables, converting the random variables into independent standard normal random variables; based on the independent standard normal random variables, calculating a reliability index of the target slope, and obtaining the stability status of the target slope based on the reliability index analysis.

[0008] As an optional technical solution of the present invention, the functional function of the target slope It is expressed as:

[0009] ;

[0010] Among them, FS represents the slope safety factor including shear strength S and sliding load F, which is expressed as:

[0011] ;

[0012] Where c represents the effective cohesion of the soil on the potential sliding surface, represents the effective internal friction angle of the soil on the potential sliding surface, H represents the thickness of the soil layer above the potential sliding surface, represents the target slope inclination, Indicates the total weight of soil.

[0013] As an optional technical solution of the present invention, the random variable x in the functional function includes the effective cohesion c and the effective internal friction angle φ, which is expressed as:

[0014] ;

[0015] Where T represents the transpose of the vector, 1 and 2 represent the random variable numbers respectively;

[0016] The probability distribution information of the random variable x includes the cumulative distribution function And the correlation coefficient matrix R.

[0017] As an optional technical solution of the present invention, the random variable is converted into an independent standard normal random variable based on the probability distribution information of the random variable, including:

[0018] Convert a random variable x to a related standard normal random variable y, expressed as:

[0019] ;

[0020] The conversion process is expressed as:

[0021] ;

[0022] in, represents the inverse function of the standard normal distribution function, represents the cumulative distribution function;

[0023] Convert the dependent standard normal random variable y to an independent standard normal random variable u, expressed as:

[0024] ;

[0025] The conversion process is expressed as:

[0026] ;

[0027] in, Represents the inverse of the lower triangular matrix of the correlation matrix R after Cholesky decomposition.

[0028] As an optional technical solution of the present invention, the initial calculation parameters need to be set before calculating the reliability index of the target slope; the initial calculation parameters include: the allowable reliability index error , allowable function value error , allowable design point error , iterative design point , iterative function value , the gradient at the iterative design point , iterate the BFGS matrix and iterative reliability indicators ;

[0029] Among them, the iteration reliability index It is expressed as:

[0030] ;

[0031] Where k represents the number of iterations, and k = 1 in the initial state.

[0032] As an optional technical solution of the present invention, before calculating the reliability index of the target slope, it is necessary to calculate the design point according to the initial calculation parameters, including:

[0033] Calculate iteration direction , expressed as:

[0034] ;

[0035] Calculate updated iterative design points , expressed as:

[0036] ;

[0037] in, Indicates the iterative step size calculated by the Armijo criterion;

[0038] Calculate the updated iterative function value , updated iterative design point The gradient at and updated iterative reliability indicators ,

[0039] Among them, the updated iterative reliability index It is expressed as:

[0040] ;

[0041] Compute Lagrange multipliers , expressed as:

[0042] ;

[0043] Compute the updated iterative BFGS matrix , expressed as:

[0044] ;

[0045] in, Represents the iteration increment, expressed as:

[0046] ;

[0047] Represents the iterative increment of the gradient of the Lagrangian function, expressed as:

[0048] ;

[0049] in, represents the iterative increment of the uncorrected Lagrangian function gradient, expressed as:

[0050] ;

[0051] represents the correction factor, expressed as:

[0052] ;

[0053] Determine iterative design points based on convergence criteria Whether it is used as a design point.

[0054] As an optional technical solution of the present invention, the convergence criterion includes:

[0055] ;

[0056] ;

[0057] ;

[0058] If convergence criteria 1, 2 and 3 are satisfied at the same time, let the iterative design point As a design point ; If convergence criteria 1, 2 and 3 are not met at the same time, set the number of iterations k = k+1 and perform the next iterative calculation.

[0059] As an optional technical solution of the present invention, the reliability index It is expressed as:

[0060] ;

[0061] in, Indicates the design point.

[0062] As an optional technical solution of the present invention, the step of obtaining the stability of the target slope according to the reliable index analysis includes:

[0063] If reliable indicators , it means that the target slope is extremely unstable;

[0064] If reliable indicators , it means that the target slope is relatively unstable;

[0065] If reliable indicators , it means that the target slope is potentially unstable;

[0066] If reliable indicators , it means that the target slope is below the average stability level;

[0067] If reliable indicators , it means that the target slope is higher than the average stability level;

[0068] If reliable indicators , it means that the target slope is relatively stable;

[0069] If reliable indicators , it means that the target slope is extremely stable.

[0070] In a second aspect, a slope stability analysis system based on first-order reliability is provided, comprising: a construction module for constructing a functional function of a target slope according to pre-acquired parameter information of the target slope;

[0071] An information determination module, used to determine the random variables in the functional function and the probability distribution information of the random variables;

[0072] A conversion module, used for converting the random variable into an independent standard normal random variable based on the probability distribution information of the random variable;

[0073] The calculation and analysis module is used to calculate the reliability index of the target slope based on the independent standard normal random variable, and obtain the stability status of the target slope according to the reliability index analysis.

[0074] Compared with the prior art, the present invention has the following beneficial effects:

[0075] The slope stability analysis method based on first-order reliability provided by the present invention calculates the reliability index through the optimized first-order reliability method, breaking through the bottleneck of the traditional first-order reliability method in stability and iteration speed, and providing solid technical support for slope risk assessment and design in water conservancy and hydropower projects. At the same time, it will deepen the understanding of the mechanism of the influence of rock and soil parameter uncertainty on slope stability, provide engineers with a more accurate and reliable slope stability assessment scheme, and thus promote the safe and efficient construction and operation of water conservancy and hydropower projects. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 is a flow chart of a slope stability analysis method based on first-order reliability in an embodiment of the present invention;

[0077] Figure 2 is a schematic diagram of target slope stability analysis in an embodiment of the present invention;

[0078] Figure 3 is a reliability indicator iteration diagram in an embodiment of the present invention;

[0079] Figure 4It is a functional function value iteration diagram in an embodiment of the present invention;

[0080] Figure 5 is a design point iteration diagram in an embodiment of the present invention;

[0081] Figure 6 It is a design point iteration cloud diagram in an embodiment of the present invention. DETAILED DESCRIPTION

[0082] The present invention will be further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present invention.

[0083] Example 1

[0084] This embodiment provides a slope stability analysis method based on first-order reliability, which optimizes the first-order reliability method by using the Armijo criterion and BFGS technology, breaking through the bottleneck of the traditional first-order reliability method in terms of stability and iteration speed. Figure 1 As shown, the analysis method includes the following steps:

[0085] Step 1: Construct the functional function of the target slope based on the pre-acquired parameter information of the target slope.

[0086] Specifically, the target slope function is It is expressed as:

[0087] ;

[0088] Among them, FS represents the slope safety factor including shear strength S and sliding load F, which is expressed as:

[0089] ;

[0090] Where c represents the effective cohesion of the soil on the potential sliding surface, represents the effective internal friction angle of the soil on the potential sliding surface, H represents the thickness of the soil layer above the potential sliding surface, represents the target slope inclination, Indicates the total weight of the soil.

[0091] like Figure 2 As shown in the figure, it is a schematic diagram of the target slope stability analysis, and the parameter information of the target slope can be obtained in the figure. In this embodiment, the thickness of the soil layer above the potential sliding surface is H = 4 m, the slope inclination angle is α = 25°, and the total weight of the soil is γ = 17 kN / m 3 .

[0092] Step 2: Determine the random variables in the functional function and the probability distribution information of the random variables.

[0093] Specifically, the random variable x in the functional function includes the effective cohesion c and the effective internal friction angle φ, which can be expressed as:

[0094] ;

[0095] Where T represents the transpose of the vector, and 1 and 2 represent the random variable numbers.

[0096] The probability distribution information of the random variable x includes the cumulative distribution function And the correlation coefficient matrix R.

[0097] In this embodiment, the statistical information of the effective cohesion c and the effective internal friction angle φ is shown in Table 1, and the cumulative distribution function Uniquely determined by statistical information, the correlation coefficient matrix R is expressed as:

[0098] .

[0099] Table 1 Statistical information of effective cohesion c and effective internal friction angle φ

[0100] Geotechnical parameters unit symbol Distribution Type Mean Standard Deviation Effective cohesion c kPa x1 Lognormal 12 4.8 Effective internal friction angle φ ° x2 Normal 30 6

[0101] Step 3: Based on the probability distribution information of the random variable, convert the random variable into an independent standard normal random variable. Specifically including:

[0102] 3.1. Convert the random variable x into the related standard normal random variable y, expressed as:

[0103] ;

[0104] The conversion process is expressed as:

[0105] ;

[0106] in, represents the inverse function of the standard normal distribution function, represents the cumulative distribution function.

[0107] 3.2. Convert the correlated standard normal random variable y into an independent standard normal random variable u, expressed as:

[0108] ;

[0109] The conversion process is expressed as:

[0110] ;

[0111] in, Represents the inverse of the lower triangular matrix of the correlation matrix R after Cholesky decomposition.

[0112] In this embodiment, the expression of the lower triangular matrix A after the correlation coefficient matrix R is decomposed by Cholesky is as follows:

[0113] .

[0114] Step 4: Based on the independent standard normal random variables, the reliability index of the target slope is calculated, and the stability of the target slope is obtained according to the reliability index analysis.

[0115] Before calculating the reliability index of the target slope, initial calculation parameters need to be set;.

[0116] Specifically, the initial calculation parameters include: allowable reliability index error , allowable function value error , allowable design point error , iterative design point , iterative function value , the gradient at the iterative design point , iterate the BFGS matrix , iterative reliability index .

[0117] Among them, the iteration reliability index It is expressed as:

[0118] ;

[0119] Where k represents the number of iterations. In the initial state, k = 1 and the maximum number of iterations is N.

[0120] In this embodiment, the reliability index error is allowed , allowable function value error , allowable design point error , iterative design point , iterative function value , the gradient at the iterative design point , iterative reliability index , iterate the BFGS matrix It is expressed as:

[0121] .

[0122] The use of BFGS technology in this step can improve the iteration efficiency and enhance the robustness of the method. The BFGS technology is used to achieve approximate calculation of the second-order curvature information of the limit state function, so that this method can quickly converge to a stable design point when dealing with highly complex nonlinear limit state functions, and significantly improve the calculation efficiency.

[0123] Before calculating the reliability index of the target slope, it is necessary to calculate the design point based on the initial calculation parameters. Specifically, it includes:

[0124] 4.1. Calculation Iteration Direction , expressed as:

[0125] .

[0126] 4.2. Calculate the updated iterative design point , expressed as:

[0127] ;

[0128] in, Represents the iteration step size calculated by the Armijo criterion. The Armijo criterion can optimize the iteration step size and ensure that each iteration step is carried out in the direction of the objective function value decreasing, thereby effectively avoiding the instability problem in the iteration process.

[0129] 4.3. Calculate the updated iterative function value , updated iterative design point The gradient at and updated iterative reliability indicators ,

[0130] Among them, the updated iterative reliability index It is expressed as:

[0131] .

[0132] 4.4. Calculation of Lagrange multipliers , expressed as:

[0133] .

[0134] 4.5. Calculate the updated iterative BFGS matrix , expressed as:

[0135] ;

[0136] in, Represents the iteration increment, expressed as:

[0137] .

[0138] Represents the iterative increment of the gradient of the Lagrangian function, expressed as:

[0139] ;

[0140] in, represents the iterative increment of the uncorrected Lagrangian function gradient, expressed as:

[0141] ;

[0142] represents the correction factor, expressed as:

[0143] .

[0144] 4.6. Determine the iterative design point based on the convergence criterion Whether it is used as a design point.

[0145] Convergence criteria include:

[0146] ;

[0147] ;

[0148] ;

[0149] If convergence criteria 1, 2 and 3 are satisfied at the same time, let the iterative design point As a design point If convergence criteria 1, 2, and 3 are not satisfied at the same time, set the number of iterations k = k+1, perform the next iteration, and return to step 4.1.

[0150] In this embodiment, the reliability index iteration diagram is shown in Figure 3 , the function value iteration diagram is shown in Figure 4 , the design point iteration diagram is shown in Figure 5 , design point iteration cloud diagram see Figure 6 , design point .

[0151] 4.7. Calculate the reliability index based on the design point. The reliability index It is expressed as:

[0152] ;

[0153] in, Indicates the design point.

[0154] If reliable indicators , it means that the target slope is extremely unstable. , it means that the target slope is relatively unstable. , it indicates that the target slope is potentially unstable; if the reliability index , it means that the target slope is below the average stability level. , it means that the target slope is above the average stability level. , it means that the target slope is relatively stable. , it means that the target slope is extremely stable.

[0155] In this example, the reliability index β = 2.864, and the target slope is above the average stability level.

[0156] Figure 3-6 It is clearly shown that this method achieves accurate convergence of reliability indicators, performance function values ​​and design points after only 6 iterations, which greatly improves the computational efficiency.

[0157] Example 2

[0158] This embodiment provides a slope stability analysis system based on first-order reliability, including:

[0159] The construction module is used to construct the functional function of the target slope according to the parameter information of the target slope obtained in advance.

[0160] The information determination module is used to determine the random variables in the functional function and the probability distribution information of the random variables.

[0161] The conversion module is used to convert the random variable into an independent standard normal random variable based on the probability distribution information of the random variable.

[0162] The calculation and analysis module is used to calculate the reliability index of the target slope based on the independent standard normal random variable, and obtain the stability status of the target slope according to the reliability index analysis.

[0163] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.

[0164] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0165] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0166] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0167] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the technical principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A slope stability analysis method based on first-order reliability, characterized in that: include: Constructing a functional function of the target slope according to the parameter information of the target slope obtained in advance; Determining random variables in the functional function and probability distribution information of the random variables; Based on the probability distribution information of the random variable, the random variable is converted into an independent standard normal random variable; Based on the independent standard normal random variables, the reliability index of the target slope is calculated, and the stability of the target slope is obtained according to the reliability index analysis.

2. The slope stability analysis method based on first-order reliability according to claim 1 is characterized in that: The target slope function It is expressed as: ； Among them, FS represents the slope safety factor including shear strength S and sliding load F, which is expressed as: ； Where c represents the effective cohesion of the soil on the potential sliding surface, represents the effective internal friction angle of the soil on the potential sliding surface, H represents the thickness of the soil layer above the potential sliding surface, represents the target slope inclination, Indicates the total weight of soil.

3. The slope stability analysis method based on first-order reliability according to claim 2 is characterized in that: The random variable x in the performance function includes the effective cohesion c and the effective internal friction angle φ, which can be expressed as: ； Where T represents the transpose of the vector, 1 and 2 represent the random variable numbers respectively; The probability distribution information of the random variable x includes the cumulative distribution function And the correlation coefficient matrix R.

4. The slope stability analysis method based on first-order reliability according to claim 1 is characterized in that: The method of converting the random variable into an independent standard normal random variable based on the probability distribution information of the random variable includes: Convert a random variable x to a related standard normal random variable y, expressed as: ； The conversion process is expressed as: ； in, represents the inverse function of the standard normal distribution function, represents the cumulative distribution function; Convert the dependent standard normal random variable y to an independent standard normal random variable u, expressed as: ； The conversion process is expressed as: ； in, Represents the inverse of the lower triangular matrix of the correlation matrix R after Cholesky decomposition.

5. The slope stability analysis method based on first-order reliability according to claim 1 is characterized in that: Before calculating the reliability index of the target slope, initial calculation parameters need to be set; The initial calculation parameters include: allowable reliability index error , allowable function value error , allowable design point error , iterative design point , iterative function value , the gradient at the iterative design point , iterate the BFGS matrix and iterative reliability indicators ; Among them, the iteration reliability index It is expressed as: ； Where k represents the number of iterations, and k = 1 in the initial state.

6. The slope stability analysis method based on first-order reliability according to claim 5 is characterized in that: Before calculating the reliability index of the target slope, it is necessary to calculate the design point based on the initial calculation parameters, including: Calculate iteration direction , expressed as: ； Calculate updated iterative design points , expressed as: ； in, Indicates the iterative step size calculated by the Armijo criterion; Calculate the updated iterative function value , updated iterative design point The gradient at and updated iterative reliability indicators , Among them, the updated iterative reliability index It is expressed as: ； Compute Lagrange multipliers , expressed as: ； Compute the updated iterative BFGS matrix , expressed as: ； in, Represents the iteration increment, expressed as: ； Represents the iterative increment of the gradient of the Lagrangian function, expressed as: ； in, represents the iterative increment of the uncorrected Lagrangian function gradient, expressed as: ； represents the correction factor, expressed as: ； Determine iterative design points based on convergence criteria Whether it is used as a design point.

7. The slope stability analysis method based on first-order reliability according to claim 6 is characterized in that: The convergence criteria include: ； ； ； If convergence criteria 1, 2 and 3 are satisfied at the same time, let the iterative design point As a design point ; If convergence criteria 1, 2 and 3 are not met at the same time, set the number of iterations k = k+1 and perform the next iterative calculation.

8. The slope stability analysis method based on first-order reliability according to claim 1 is characterized in that: The reliability index It is expressed as: ； in, Indicates the design point.

9. The slope stability analysis method based on first-order reliability according to claim 1 is characterized in that: The stability of the target slope is obtained by analyzing the reliability index, including: If reliable indicators , it means that the target slope is extremely unstable; If reliable indicators , it means that the target slope is relatively unstable; If reliable indicators , it indicates that the target slope is potentially unstable; If reliable indicators , it means that the target slope is below the average stability level; If reliable indicators , it means that the target slope is higher than the average stability level; If reliable indicators , it means that the target slope is relatively stable; If reliable indicators , it means that the target slope is extremely stable.

10. A slope stability analysis system based on first-order reliability, characterized in that: include: A construction module, used for constructing a functional function of the target slope according to the parameter information of the target slope acquired in advance; An information determination module, used to determine the random variables in the functional function and the probability distribution information of the random variables; A conversion module, used for converting the random variable into an independent standard normal random variable based on the probability distribution information of the random variable; The calculation and analysis module is used to calculate the reliability index of the target slope based on the independent standard normal random variable, and obtain the stability status of the target slope according to the reliability index analysis.

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