A correction method for slope stability evaluation based on local average

Abstract

The present invention discloses a slope stability evaluation correction method based on local average, comprising the following steps: A. setting initial soil characteristic parameters for Monte Carlo simulation, generating random samples, and judging whether the slope is invalid through each group of samples, counting the number of invalid samples, and calculating the original slope failure probability estimation value #imgabs0#B. counting the invalid samples x under the original Monte Carlo simulation j ,j=1,2,......,M;C. Calculate the original and revised joint probability density functions according to the original and revised soil characteristic parameters to obtain the weight index ω j ,j=1,2,……,M;D. Calculate the estimated value of the corrected failure probability and its coefficient of variation. This overcomes the problem that the existing widely used reliability analysis method, the Monte Carlo simulation method, needs to rebuild the model after considering the fluctuation range or other soil characteristic parameters, especially for slopes with low probability of failure, and its large amount of calculation will consume huge time, energy and computer resources.

Description

Technical Field

[0001] The invention relates to a slope stability evaluation correction method based on local average, belonging to the technical field of geotechnical engineering. Background Art

[0002] Landslides caused by slope instability are a highly destructive geological disaster. However, there are many uncertain factors in slope engineering, such as Figure 2 The uncertainty of the characteristic parameters of the rock and soil mass, the uncertainty of the calculation model, etc. The traditional slope stability analysis ignores these uncertainties and cannot truly reflect the safety of the slope. The slope stability analysis based on reliability theory can take the above factors into consideration. The slope stability analysis method based on reliability theory has been increasingly studied in recent years. The "Code for Slope Design of Water Conservancy and Hydropower Engineering" clearly recommends that reliability analysis should be carried out for Class 1 slopes when conditions permit.

[0003] For the reliability problem of slope stability, mathematical statistical methods are usually used to determine the statistical parameters of soil characteristics, such as mean, variance and distribution type, based on prior information such as field or indoor test data during preliminary construction. The estimated value of the slope failure probability is obtained based on the prior information. This estimated value is only a preliminary estimate of the failure probability of the slope. As the slope engineering construction progresses, engineers will obtain new statistical data, which will help to deepen their understanding of the uncertainty of soil characteristics. Then, the original estimated statistical parameters of soil characteristics will be corrected. The correction of the parameters will further lead to the re-evaluation of the slope reliability, especially the update of the slope reliability caused by the correction of the uncertainty of the fluctuation range. Some of the current reliability analysis methods include Monte Carlo simulation, first-order second moment method, probability moment point estimation method, response surface, etc. Among them, Monte Carlo simulation is a widely used analysis method. However, after considering the fluctuation range or other soil characteristic parameters, it is often necessary to rebuild the model. Especially for slopes with low probability of failure, a large amount of calculations often consumes a lot of time, energy and computer resources. Summary of the invention

[0004] The purpose of the present invention is to provide a slope stability evaluation correction method based on local average, which overcomes the problem that the existing widely used reliability analysis method, the Monte Carlo simulation method, needs to rebuild the model after considering the fluctuation range or other soil characteristic parameters, especially for slopes with low probability of failure, and its large amount of calculation will consume huge time, energy and computer resources.

[0005] The technical solution of the present invention is a slope stability evaluation correction method based on local average, comprising the following steps:

[0006] A. Set the initial soil characteristic parameters for Monte Carlo simulation, generate random samples, and use each group of samples to determine whether the slope has failed, count the number of failed samples, and calculate the estimated failure probability of the original slope

[0007] B. Statistics of failure samples x under the original Monte Carlo simulation j ,j=1,2,......,M;

[0008] C. Calculate the original and revised joint probability density functions based on the original and revised soil characteristic parameters to obtain the weight index ω j ,j=1,2,......,M;

[0009] D. Calculate an estimate of the corrected failure probability and its coefficient of variation.

[0010] In the above-mentioned slope stability evaluation correction method based on local average, in step C, it is assumed that there are m layers of soil in the slope, corresponding to m soil variables, and all of them obey the normal distribution, that is, The soil variables of each layer are divided into n1, n2, ... n by local averaging. m Assuming that the variables of each layer of soil are unrelated, the joint probability density function is:

[0011]

[0012] In the formula, x i is the n generated by local averaging of the soil variables in the i-th layer i A vector representation of random variables; μ i is the local averaged n-th layer soil variable i The vector representation of the mean value of each variable is Β i is the local averaged n-th layer soil variable i variables The covariance matrix of is similar to that of a layer of soil slope.

[0013] σ ij =σ i [Γ(Δ ij |δ)] 0.5 i=1,2,·…·m; j=1,2,…n i ,

[0014] In the formula, σ ij is the standard deviation of the jth variable of the i-th layer soil mass variable after local averaging; Δ ij is the jth interval of the sliding arc corresponding to the soil variable in the i-th layer of the slope divided by local averaging; δ is the fluctuation range;

[0015] The correlation coefficient of the local average variable between the jth and kth intervals of the sliding arc in the i-th layer of soil is:

[0016]

[0017] In the formula, Δ ij,ik is the i-th segment Δ on the sliding arc in the i-th layer of soil i and the jth segment Δ j The curve distance between them on the sliding arc.

[0018] In the above-mentioned slope stability evaluation correction method based on local average, in step C, according to the following formula:

[0019]

[0020] definition:

[0021]

[0022] When i=j, The covariance matrix of the i-th layer of soil slope can be obtained as:

[0023]

[0024] Therefore, the joint probability density function value of the failure samples under the original fluctuation range δ can be obtained.

[0025] In the aforementioned slope stability evaluation correction method based on local average, in step C, similarly, under the multi-layer soil slope with a new fluctuation range δ', its joint probability density function is:

[0026]

[0027] In the formula, x i is still the n generated by local averaging of the i-th layer soil variables i The vector representation of random variable samples only needs to use the original failure sample x (i) ;μ i ' is the local average of the soil variables of the i-th layer under the new fluctuation range δ' i The vector representation of the mean value of the variables is still:

[0028]

[0029] Β′ i is the local averaged n of the soil variables in the i-th layer under the new fluctuation range δ' i variables The covariance matrix of , similarly,

[0030] σ′ij =σ′ i [Γ(Δ ij |δ′)] 0.5 i=1,2,…m;j=1,2,…n i ,

[0031] where σ ij ' is the standard deviation of the jth variable of the i-th layer soil mass variable after local averaging under the new fluctuation range δ'; Δ ij Still under the original division, the soil variables in the i-th layer of the slope are divided into the j-th interval of the corresponding sliding arc by local averaging; δ is the fluctuation range;

[0032] The correlation coefficient of the local average variable between the jth and kth intervals of the sliding arc in the i-th layer of soil under the new fluctuation range δ' is:

[0033]

[0034] Where Δ ij,ik Still under the original division, the i-th segment Δ on the sliding arc in the i-th layer of soil i and the i-th segment Δ j The curve distance between them on the sliding arc; then:

[0035]

[0036] When i=j, The covariance matrix of the i-th layer of soil slope can be obtained as:

[0037]

[0038] The joint probability density function value of failure samples under the new fluctuation range can be obtained.

[0039] In the above-mentioned slope stability evaluation correction method based on local average, in step D, the estimated value of soil characteristic parameters is corrected based on the new data, and the corrected failure probability is It is expressed as:

[0040]

[0041] Combined with the Weighting theory, the modified joint probability density function f(x) of the initial distribution of soil characteristic parameters is used to calculate the modified slope failure probability. It can be expressed as:

[0042]

[0043] Where ω is the weight index; thus, under the original joint probability density function f(x), the revised failure probability estimate is It can be expressed as:

[0044]

[0045] Random samples generated by the original Monte Carlo simulation; ω j ,j=1,2…,M is the weight index of the failure samples generated by the original Monte Carlo simulation.

[0046] In the aforementioned slope stability evaluation correction method based on local average, in step D, combined with the original joint probability density function value, according to the efficient slope reliability correction formula combined with the Weighting theory, the estimated failure probability value under the new fluctuation range δ' can be obtained as:

[0047]

[0048] Where N is the number of Monte Carlo simulations; n s is the number of failed samples.

[0049] Beneficial effects of the present invention: Compared with the existing Monte Carlo simulation method, the present invention describes the spatial variability of soil parameters by considering the fluctuation range, links the "point" variability with the "space" variability, and through the derivation formula, it can be seen that the failure samples that take a lot of time to calculate in the original simulation are used in calculating the corrected failure probability, avoiding the time required for a large number of simulation calculations after the correction. Therefore, the present invention can effectively solve the problem of multi-layer slope reliability correction under the consideration of the spatial variability of rock and soil parameters; because the present invention is independent of the functional function during the correction calculation, it can also be well applied to the slope reliability correction problem of implicitly expressing the functional function. The coefficient of variation of the failure probability can quantitatively characterize the accuracy of the failure probability calculation result, and the coefficient of variation of the failure probability after the slope correction is less than the coefficient of variation of the original calculation result, so the calculation accuracy of the present invention is better than the direct Monte Carlo simulation method. When the failure sample space can cover the entire failure area, the corrected failure probability is more accurate. For example, for the correction situation where the coefficient of variation of the parameter changes from large to small, which is often encountered in actual engineering, the calculation efficiency and calculation accuracy of the present invention are very high, which is significantly better than the Monte Carlo simulation method. The slope stability correction method does not need to re-execute Monte Carlo simulation during the correction process, and has very high calculation efficiency. The calculation time can be shortened to a few hundredths of the original slope calculation time. The concept is clear, the calculation is simple, and it is easy for engineers to master. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Attached Figure 1 It is a schematic diagram of the random field of characteristic parameters of slope rock and soil;

[0051] Attached Figure 2 This is a flow chart of the calculation idea of ​​the present invention;

[0052] Attached Figure 3 It is the cross section of soil slope;

[0053] Attached Figure 4 This is the error analysis diagram. DETAILED DESCRIPTION

[0054] The present invention is further described below in conjunction with the accompanying drawings and embodiments, but they are not intended to limit the present invention.

[0055] Embodiment of the present invention: A slope stability evaluation correction method based on local average, as shown in the attached Figure 1 As shown, the following steps are included:

[0056] A. Set the initial soil characteristic parameters for Monte Carlo simulation, generate random samples, and use each group of samples to determine whether the slope has failed, count the number of failed samples, and calculate the estimated failure probability of the original slope

[0057] B. Statistics of failure samples x under the original Monte Carlo simulation j ,j=1,2,......,M;

[0058] C. Calculate the original and revised joint probability density functions based on the original and revised soil characteristic parameters to obtain the weight index ω j ,j=1,2,......,M;

[0059] D. Calculate an estimate of the corrected failure probability and its coefficient of variation.

[0060] In step C, it is assumed that there are m layers of soil in the slope, corresponding to m soil variables, and all of them obey the normal distribution, that is, The soil variables of each layer are divided into n1, n2, ... n by local averaging. m Assuming that the variables of each layer of soil are unrelated, the joint probability density function is:

[0061]

[0062] In the formula, x i is the n generated by local averaging of the soil variables in the i-th layer i A vector representation of random variables; μ i is the local averaged n-th layer soil variable i The vector representation of the mean value of each variable is Β i is the local averaged n-th layer soil variable i variables The covariance matrix of is similar to that of a layer of soil slope.

[0063] σij =σ i [Γ(Δ ij |δ)] 0.5 i=1,2,…m;j=1,2,…n i ,

[0064] In the formula, σ ij is the standard deviation of the jth variable of the i-th layer soil mass variable after local averaging; Δ ij is the jth interval of the sliding arc corresponding to the soil variable in the i-th layer of the slope divided by local averaging; δ is the fluctuation range.

[0065] The correlation coefficient of the local average variable between the jth and kth intervals of the sliding arc in the i-th layer of soil is:

[0066]

[0067] In the formula, Δ ij,ik is the i-th segment Δ on the sliding arc in the i-th layer of soil i and the jth segment Δ j The curve distance between them on the sliding arc.

[0068] In step C, according to the following formula:

[0069]

[0070] definition:

[0071]

[0072] When i=j, The covariance matrix of the i-th layer of soil slope can be obtained as:

[0073]

[0074] Therefore, the joint probability density function value of the failure samples under the original fluctuation range δ can be obtained.

[0075] In step C, similarly, under the multi-layer soil slope with a new fluctuation range δ', its joint probability density function is:

[0076]

[0077] In the formula, x i is still the n generated by local averaging of the i-th layer soil variables i The vector representation of random variable samples only needs to use the original failure sample x (i) ;μ i ' is the local average of the soil variables of the i-th layer under the new fluctuation range δ' i The vector representation of the mean value of the variables is still:

[0078]

[0079] Β′ i is the local averaged n of the soil variables in the i-th layer under the new fluctuation range δ' i variables The covariance matrix of , similarly,

[0080]

[0081] where σ ij ' is the standard deviation of the jth variable of the i-th layer soil mass variable after local averaging under the new fluctuation range δ'; Δ ij The jth interval of the sliding arc is divided by local averaging of the soil variables in the i-th layer of the slope under the original division; δ is the fluctuation range.

[0082] The correlation coefficient of the local average variable between the jth and kth intervals of the sliding arc in the i-th layer of soil under the new fluctuation range δ' is:

[0083]

[0084] Where Δ ij,ik Still under the original division, the i-th segment Δ on the sliding arc in the i-th layer of soil i and the i-th segment Δ j The curve distance between them on the sliding arc. Then:

[0085]

[0086] When i=j, The covariance matrix of the i-th layer of soil slope can be obtained as:

[0087]

[0088] The joint probability density function value of failure samples under the new fluctuation range can be obtained.

[0089] In step D, the estimated values ​​of soil characteristic parameters are corrected based on the new data, and the corrected failure probability is It is expressed as:

[0090]

[0091] Combined with the Weighting theory, the modified joint probability density function f(x) of the initial distribution of soil characteristic parameters is used to calculate the modified slope failure probability. It can be expressed as:

[0092]

[0093] Where ω is the weight index. In this way, under the original joint probability density function f(x), the revised failure probability estimate is It can be expressed as:

[0094] Where x i ,i=1,2…,N is the random sample generated by the original Monte Carlo simulation; ω j ,j=1,2…,M is the weight index of the failure samples generated by the original Monte Carlo simulation.

[0095] In step D, combined with the original joint probability density function value, according to the efficient slope reliability correction formula combined with the Weighting theory, the estimated failure probability under the new fluctuation range δ' can be obtained as:

[0096]

[0097] , where N is the number of Monte Carlo simulations; n s is the number of failed samples.

[0098] Specific application examples:

[0099] This example is a simplified soil slope with a height of H = 10m and a slope angle of ψ = 26°. The characteristic values ​​of soil parameters such as bulk density γ, effective internal friction angle φ', and effective cohesion c' are shown in Table 1. Assuming plane sliding failure, the sliding surface inclination angle θ = 20°. The function function of soil slope stability analysis can be expressed as:

[0100] When g≤0, the slope is unstable. The cross section of the homogeneous slope is as follows Figure 3 shown.

[0101] Table 1 Statistical parameters of original random variables

[0102]

[0103] After in-depth exploration and correction, the parameter changes are shown in Table 2.

[0104] Table 2 Statistical parameters of corrected random variables

[0105]

[0106] Based on the established analytical model, 1×10 8 The Monte Carlo simulation calculated the failure probability to be 5.033×10 -3 According to the method, the corrected failure probability is 7.67×10 -4It takes about 2 hours to complete the initial Monte Carlo simulation in EXCEL, and less than 2 seconds after correction, which shows that the method of the present invention has high computational efficiency. If the correction analysis is still performed by repeating the Monte Carlo simulation after correction, a lot of computing resources and time are required. Figure 4 As shown, the Monte Carlo simulation was performed by correcting the coefficient of variation of the bulk density and comparing it with the method described above (as shown in Figure 4 This indicates that the method has high accuracy.

Claims

1. A slope stability evaluation correction method based on local average, characterized in that: The steps include: A. Set the initial soil characteristic parameters for Monte Carlo simulation, generate random samples, and use each group of samples to determine whether the slope has failed, count the number of failed samples, and calculate the estimated failure probability of the original slope B. Count the M failure samples under the original Monte Carlo simulation; C. Calculate the original and revised joint probability density functions of the failure samples according to the original and revised soil characteristic parameters to obtain the weight indexes ω1, ω2, ..., ω of the M failure samples M ; D. Calculate the estimated value of the corrected failure probability and its coefficient of variation; In step C, it is assumed that there are m layers of soil in the slope, corresponding to m soil variables, and all of them obey the normal distribution, that is, The soil variables of each layer are divided into n1, n2, ..., n by local averaging. m Assuming that the variables of each layer of soil are unrelated, the joint probability density function is: In the formula, x i is the corresponding n generated by local averaging of the variables of the i-th layer of soil in the m-layer soil i A vector representation of random variables; μ i is the local averaged n of the variables of the corresponding i-th layer of the m-layer soil mass i The vector representation of the mean of the random variables; i is the local averaged n of the variables of the corresponding i-th layer of the m-layer soil mass i The covariance matrix of the random variables, s ij =s i [C(D ij |d)] 0.5 i=1,2,…,m;j=1,2,…,n i , In the formula, σ ij is the standard deviation of the jth random variable of the soil variable of the i-th soil layer in the m-layer soil mass of the slope divided by local averaging; Δ ij is the jth interval of the sliding arc divided by the local average of the soil variables of the i-th layer of the slope; δ is the fluctuation range; The correlation coefficient of the local average variable between the jth and kth intervals of the sliding arc in the i-th layer of soil is:

2. The slope stability evaluation correction method based on local average according to claim 1 is characterized in that: In step C, according to the following formula: definition: When j = k, The covariance matrix of the i-th layer of soil is obtained as: Therefore, the joint probability density function value of the failure samples under the original fluctuation range δ is obtained.

3. The slope stability evaluation correction method based on local average according to claim 2 is characterized in that: In step C, under the multi-layer soil with a new fluctuation range δ′, the joint probability density function is: In the formula, x i is still the soil variable of the i-th layer generated by local averaging. i The vector representation of random variable samples only needs to use the original failure samples; μ i ′ is the local averaged variables of the corresponding layers of soil in the m-layer soil under the new fluctuation range δ′, n1, n2, …, n m The vector representation of the mean of the random variables; i ′ is the local average of the soil variables of the corresponding layers in the m-layer soil mass n i The covariance matrix of the random variables, where σ ij ′ is the standard deviation of the jth random variable of the soil variable of the i-th layer of soil under the new fluctuation range δ′ after local averaging; Δ ij The soil variables of the i-th layer of the slope under the original division are divided into the j-th interval of the sliding arc by local averaging; the correlation coefficient of the j-th and k-th intervals of the sliding arc in the i-th layer of soil under the new fluctuation range δ′ corresponding to the local average variables is: but: When j = k, The covariance matrix of the i-th layer of soil is obtained as: Obtain the joint probability density function value of failure samples under the new fluctuation range.

4. The slope stability evaluation correction method based on local average according to claim 3 is characterized in that: In step D, the estimated value of the soil characteristic parameter is corrected based on the new data, and the corrected failure probability It is expressed as: Combined with the Weighting theory, the modified joint probability density function f(x) of the initial distribution of soil characteristic parameters is used to calculate the modified slope failure probability. It is expressed as: Where ω is the weight index; thus, under the original joint probability density function f(x), the revised failure probability estimate is It is expressed as: Where x L is the random sample generated by the original Monte Carlo simulation, L = 1, 2, ..., N; ω J is the weight index of the M failure samples generated by the original Monte Carlo simulation, J = 1, 2, ..., M.

5. The slope stability evaluation correction method based on local average according to claim 4 is characterized in that: In step D, combined with the original joint probability density function value, according to the efficient slope reliability correction formula combined with the Weighting theory, the estimated failure probability value under the new fluctuation range δ′ is obtained as: 。

Images