CN113449429A - Slope stability evaluation and correction method based on local average - Google Patents
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Abstract
The invention discloses a slope stability evaluation and correction method based on local average, which comprises the following steps: A. setting initial soil characteristic parameters to carry out Monte Carlo simulation, generating random samples, judging whether the slope is invalid or not through each group of samples, counting the number of invalid samples, and calculating the probability estimation value of the original slope invalid
B. Statistical failure sample x under original Monte Carlo simulationjJ 1, 2.... am; C. calculating the original and corrected combined probability density function according to the original and corrected soil characteristic parameters to obtain the weight index omegajJ 1, 2.... am; D. calculating the corrected failure probability and the estimated value of the variation coefficient thereof. The method solves the problems that the Monte Carlo simulation method which is a widely applied reliability analysis method needs to rebuild the model after considering the fluctuation range or other soil characteristic parameters, and particularly for the slope with small probability of failure, a large amount of calculation consumes huge time, energy and computer resources.
Description
Technical Field
The invention relates to a slope stability evaluation and correction method based on local average, and belongs to the technical field of geotechnical engineering.
Background
The landslide problem caused by slope instability is a geological disaster with extremely strong destructive power, however, many uncertain factors exist in slope engineering, such as uncertainty of characteristic parameters of rock and soil mass in fig. 2, uncertainty of a calculation model and the like. The traditional slope stability analysis ignores the uncertainties, the safety degree of the slope cannot be truly reflected, and the slope stability analysis based on the reliability theory can consider the factors. Slope stability analysis methods based on the reliability theory are increasingly researched in recent years, and the design specifications of the slope of the hydraulic and hydroelectric engineering clearly suggest that the reliability analysis should be carried out when the grade 1 slope is conditioned.
For the problem of stability and reliability of the side slope, statistical parameters of soil features, such as mean, variance, distribution type and the like, are determined by a mathematical statistical method according to prior information such as field or indoor test data in preliminary construction. And obtaining an estimated value of the slope failure probability according to the prior information, wherein the estimated value is only a preliminary estimated value of the slope failure probability. With the deep progress of slope engineering construction, engineers can obtain new statistical data, the data is beneficial to deepening the understanding of the engineers on soil characteristic uncertainty, then the originally estimated soil characteristic statistical parameters are corrected, the correction of the parameters can further cause the reevaluation of the slope reliability, and particularly the problem of slope reliability updating caused by the uncertainty correction of the fluctuation range is solved. Some existing reliability analysis methods include a monte carlo simulation, a first order moment method, a probability moment point estimation method, a response surface and the like, wherein the monte carlo simulation method is a widely used analysis method. However, after the fluctuation range or other soil characteristic parameters are considered to be updated, the model is required to be reconstructed, and especially for the slope with small probability of failure, huge time, energy and computer resources are consumed by a large amount of calculation.
Disclosure of Invention
The invention aims to provide a slope stability evaluation and correction method based on local average. The method solves the problems that the Monte Carlo simulation method which is a widely applied reliability analysis method needs to rebuild the model after considering the fluctuation range or other soil characteristic parameters, and particularly for the slope with small probability of failure, a large amount of calculation consumes huge time, energy and computer resources.
The technical scheme of the invention is as follows: a slope stability evaluation and correction method based on local average comprises the following steps:
A. setting initial soil characteristic parameters to carry out Monte Carlo simulation, generating random samples, judging whether the slope is invalid or not through each group of samples, counting the number of invalid samples, and calculating the probability estimation value of the original slope invalid
B. Statistical failure sample x under original Monte Carlo simulationj,j＝1，2，......，M；
C. Calculating the original and corrected combined probability density function according to the original and corrected soil characteristic parameters to obtain the weight index omegaj,j＝1，2，......，M；
D. And calculating the corrected failure probability and the estimated value of the variation coefficient thereof.
In the slope stability evaluation and correction method based on local average, in the step C, it is assumed that m layers of soil exist in the slope, m soil variables correspond to the m soil variables, and the m soil variables are all subject to normal distribution, that is, the method is characterized in that
Dividing each layer of soil variable into n by local average1,n2,···nmAnd in the interval part, if the soil body variables of each layer are not related, the joint probability density function is as follows:
in the formula, xiN generated by local averaging of i-th layer soil variableiA vector representation of a random variable; mu.siN for i-th layer soil variable after local averagingiVector representation of individual variables to mean, i.e.
BiN for i-th layer soil variable after local averagingiOf a variable
The covariance matrix of (1) is similar to that when calculating the soil slope of one layer,
σij＝σi[Γ(Δij|δ)]0.5 i＝1,2,···m；j＝1,2,···ni，
in the formula, σijThe standard deviation corresponding to the jth variable after local averaging of the ith layer soil body variable is obtained; deltaijDividing the j-th interval of the corresponding sliding arc for the soil body variable in the ith layer of soil body of the side slope through local averaging; delta is the fluctuation range;
the correlation coefficient of the local average variable corresponding to the interval between the j section and the k section of the sliding arc in the i-th layer soil body is as follows:
in the formula,. DELTA.ij,ikFor the ith section delta on the sliding arc in the ith layer soil bodyiAnd j segment ΔjThe distance between the curves on the sliding arc.
In the slope stability evaluation and correction method based on local average, in the step C, according to the following formula:
defining:
when the value of i is equal to j,
the covariance matrix of the ith layer of soil slope can be obtained as follows:
thus, the joint probability density function value of the failed samples at the original fluctuation range δ can be found.
In the above slope stability evaluation and correction method based on local average, in step C, similarly under the multi-layer soil slope with the new fluctuation range δ', the joint probability density function is:
in the formula, xiN generated by local averaging of i-th layer soil variableiThe vector representation of random variable sample only needs to use original failure sample x(i)；μiN is the i-th layer soil variable which is locally averaged under a new fluctuation range deltaiThe individual variables correspond to the vector representation of the mean, i.e. are still:
B′in for i-th layer soil body variable under new fluctuation range delta' through local averagingiA variable quantity
The covariance matrix of (a) may, similarly,
σ′ij＝σ′i[Γ(Δij|δ′)]0.5 i＝1,2,…m；j＝1,2,…ni，
wherein sigmaij'is the standard deviation corresponding to the jth variable after local averaging of the ith layer soil body variable in the new fluctuation range delta'; deltaijDividing the j-th interval of the corresponding sliding arc for the soil body variable in the ith soil body of the original lower side slope through local averaging; delta is the fluctuation range;
under the new fluctuation range delta', the correlation coefficient of the local average variable corresponding to the interval between the j section and the k section of the sliding arc in the i-th layer soil body is as follows:
wherein Δij,ikThe section I delta on the sliding arc in the layer I soil body under the original divisioniAnd ith segment deltajThe distance between the curves on the sliding arc; then:
when the value of i is equal to j,
the covariance matrix of the ith layer of soil slope can be obtained as follows:
the joint probability density function value of the failed samples under the new fluctuation range can be obtained.
In the slope stability evaluation and correction method based on local average, in the step D, the soil characteristic parameter estimation value is estimated based on new dataMaking a correction of the corrected failure probability
Expressed as:
combining with weighing theory, utilizing the corrected combined probability density function f (x) of the initial distribution of the soil characteristic parameters, and the corrected slope failure probability
Can be expressed as:
in the formula, omega is a weight index; thus, under the original joint probability density function f (x), the corrected failure probability estimation value
Can be expressed as:
random samples generated by original monte carlo simulation; omegajAnd j is 1,2 …, and M is a weight index of a failure sample generated by the original Monte Carlo simulation.
In the slope stability evaluation and correction method based on local average, in step D, the failure probability estimation value under the new fluctuation range δ' is obtained by combining the original joint probability density function value and according to the high-efficiency slope reliability correction formula combined with weighing theory:
wherein N is the Monte Carlo simulation times; n issThe number of failed samples.
The invention has the beneficial effects that: compared with the existing Monte Carlo simulation method, the method has the advantages that the spatial variability of the soil body parameters is described by considering the fluctuation range, the point variability and the spatial variability are connected, and the failure samples which consume a large amount of time for calculation in the original simulation can be used in calculating the corrected failure probability by deducing a formula, so that the time required by the corrected large amount of simulation calculation is avoided. Therefore, the method can effectively solve the problem of correcting the multilayer slope reliability under the consideration of the spatial variability of rock-soil body parameters; the method is irrelevant to the function when in correction calculation, so the method can be well suitable for the slope reliability correction problem of the implicit expression function. The variation coefficient of the failure probability can quantitatively represent the accuracy of the calculation result of the failure probability, and the variation coefficient of the failure probability after slope correction is smaller than that of the original calculation result, so the calculation precision of the method is superior to that of a direct Monte Carlo simulation method. When the failure sample space can cover the whole failure area, the corrected failure probability is more accurate. For example, for the correction condition that the parameter variation coefficient is changed from big to small in practical engineering, the method has high calculation efficiency and calculation precision, and is obviously superior to a Monte Carlo simulation method. The slope stability correction method does not need to execute Monte Carlo simulation again in the correction process, has high calculation efficiency, can shorten the calculation time to one percent of the calculation time of the primary slope, has clear concept and simple calculation, and is easy to master by engineers.
Drawings
FIG. 1 is a schematic diagram of a random field of characteristic parameters of a slope rock-soil mass;
FIG. 2 is a flow chart of the computing concept of the present invention;
FIG. 3 is a cross-sectional view of a soil slope;
FIG. 4 is an error analysis diagram.
Detailed Description
The invention is further illustrated by the following figures and examples, which are not to be construed as limiting the invention.
The embodiment of the invention comprises the following steps: a slope stability evaluation and correction method based on local average is shown in figure 1 and comprises the following steps:
A. setting initial soil characteristic parameters to carry out Monte Carlo simulation, generating random samples, judging whether the slope is invalid or not through each group of samples, counting the number of invalid samples, and calculating the probability estimation value of the original slope invalid
B. Statistical failure sample x under original Monte Carlo simulationj,j＝1，2，......，M；
C. Calculating the original and corrected combined probability density function according to the original and corrected soil characteristic parameters to obtain the weight index omegaj,j＝1，2，......，M；
D. And calculating the corrected failure probability and the estimated value of the variation coefficient thereof.
In the step C, assuming that m layers of soil bodies exist in the side slope, m soil body variables are correspondingly arranged and are all subjected to normal distribution, namely
Dividing each layer of soil variable into n by local average1,n2,···nmAnd in the interval part, if the soil body variables of each layer are not related, the joint probability density function is as follows:
in the formula, xiN generated by local averaging of i-th layer soil variableiA vector representation of a random variable; mu.siN for i-th layer soil variable after local averagingiVector representation of individual variables to mean, i.e.
BiN for i-th layer soil variable after local averagingiOf a variable
The covariance matrix of (1) is similar to that when calculating the soil slope of one layer,
σij＝σi[Γ(Δij|δ)]0.5 i＝1,2,···m；j＝1,2,···ni，
in the formula, σijThe standard deviation corresponding to the jth variable after local averaging of the ith layer soil body variable is obtained; deltaijDividing the j-th interval of the corresponding sliding arc for the soil body variable in the ith layer of soil body of the side slope through local averaging; δ is the fluctuation range.
The correlation coefficient of the local average variable corresponding to the interval between the j section and the k section of the sliding arc in the i-th layer soil body is as follows:
in the formula,. DELTA.ij,ikFor the ith section delta on the sliding arc in the ith layer soil bodyiAnd j segment ΔjThe distance between the curves on the sliding arc.
In step C, according to the following formula:
defining:
when the value of i is equal to j,
the covariance matrix of the ith layer of soil slope can be obtained as follows:
thus, the joint probability density function value of the failed samples at the original fluctuation range δ can be found.
In step C, similarly under the multi-layer soil slope of the new fluctuation range δ', the joint probability density function is:
in the formula, xiN generated by local averaging of i-th layer soil variableiThe vector representation of random variable sample only needs to use original failure sample x(i)；μiN is the i-th layer soil variable which is locally averaged under a new fluctuation range deltaiThe individual variables correspond to the vector representation of the mean, i.e. are still:
B′in for i-th layer soil body variable under new fluctuation range delta' through local averagingiA variable quantity
The covariance matrix of (a) may, similarly,
σ′ij＝σ′i[Γ(Δij|δ′)]0.5 i＝1,2,…m；j＝1,2,…ni，
wherein sigmaij'is the standard deviation corresponding to the jth variable after local averaging of the ith layer soil body variable in the new fluctuation range delta'; deltaijDividing the j-th interval of the corresponding sliding arc for the soil body variable in the ith soil body of the original lower side slope through local averaging; δ is the fluctuation range.
Under the new fluctuation range delta', the correlation coefficient of the local average variable corresponding to the interval between the j section and the k section of the sliding arc in the i-th layer soil body is as follows:
wherein Δij,ikThe section I delta on the sliding arc in the layer I soil body under the original divisioniAnd ith segment deltajThe distance between the curves on the sliding arc. Then:
when the value of i is equal to j,
the covariance matrix of the ith layer of soil slope can be obtained as follows:
the joint probability density function value of the failed samples under the new fluctuation range can be obtained.
In step D, the soil body characteristic parameter estimated value is corrected based on the new data, and the corrected failure probability is
Expressed as:
combining with weighing theory, utilizing the corrected combined probability density function f (x) of the initial distribution of the soil characteristic parameters, and the corrected slope failure probability
Can be expressed as:
where ω is a weight index. Thus, under the original joint probability density function f (x), the corrected failure probability estimation value
Can be expressed as:
In the step D, combining the original joint probability density function value and according to a high-efficiency slope reliability correction formula combined with a weighing theory, the failure probability estimation value under the new fluctuation range delta' can be obtained as follows:
Specific application examples are as follows:
the calculation example is a simplified soil slope with the height H of 10m and the slope angle psi of 26 degrees, and soil body parameter characteristic values such as the volume weight gamma, the effective internal friction angle phi ', the effective cohesion force c' and the like are shown in table 1. Assuming that the plane sliding fails, the sliding plane inclination angle θ becomes 20 °. The functional function of the soil slope stability analysis can be expressed as:
TABLE 1 statistical parameter Table of original random variables
After deep exploration and correction, the parameter change condition is shown in table 2.
TABLE 2 statistical parameter Table of modified random variables
Based on the established analytical model, by 1 × 108Sub Monte Carlo simulation, and calculating to obtain the failure probability of 5.033 × 10-3The failure probability after correction is calculated and obtained according to the method and is 7.67 multiplied by 10-4. The calculation of the initial Monte Carlo simulation in the EXCEL needs about 2h, the time needed after the correction is less than 2s, which shows that the method of the invention has high calculation efficiency, and if the correction is still carried out by repeatedly executing the Monte Carlo simulation after the correction, a large amount of calculation resources and time are needed, therefore. As shown in fig. 4, the Monte Carlo simulation and the comparative analysis of the method (as shown in fig. 4) by correcting the variation coefficient of the volume weight can show that the method has higher accuracy.
Claims (6)
1. A slope stability evaluation and correction method based on local average is characterized by comprising the following steps: the method comprises the following steps:
A. setting initial soil characteristic parameters to carry out Monte Carlo simulation, generating random samples, judging whether the slope is invalid or not through each group of samples, counting the number of invalid samples, and calculating the probability estimation value of the original slope invalid
B. Statistical failure sample x under original Monte Carlo simulationj,j＝1，2，......，M；
C. Calculating the original and corrected combined probability density function according to the original and corrected soil characteristic parameters to obtain the weight index omegaj,j＝1，2，......，M；
D. And calculating the corrected failure probability and the estimated value of the variation coefficient thereof.
2. The slope stability evaluation and correction method based on local average according to claim 1, characterized in that: in the step C, it is assumed that m layers of soil bodies exist in the side slope, m soil body variables correspond to the side slope, and the side slope are subjected to normal distribution, namely
Dividing each layer of soil variable into n by local average1,n2,···nmAnd in the interval part, if the soil body variables of each layer are not related, the joint probability density function is as follows:
in the formula, xiN generated by local averaging of i-th layer soil variableiA vector representation of a random variable; mu.siN for i-th layer soil variable after local averagingiVector representation of individual variables to mean, i.e.
BiN for i-th layer soil variable after local averagingiOf a variable
The covariance matrix of (1) is similar to that when calculating the soil slope of one layer,
σij＝σi[Γ(Δij|δ)]0.5i＝1,2,···m；j＝1,2,···ni，
in the formula, σijThe standard deviation corresponding to the jth variable after local averaging of the ith layer soil body variable is obtained; deltaijDividing the j-th interval of the corresponding sliding arc for the soil body variable in the ith layer of soil body of the side slope through local averaging; delta is the fluctuation range;
the correlation coefficient of the local average variable corresponding to the interval between the j section and the k section of the sliding arc in the i-th layer soil body is as follows:
in the formula,. DELTA.ij,ikFor the ith section delta on the sliding arc in the ith layer soil bodyiAnd j segment ΔjThe distance between the curves on the sliding arc.
3. The slope stability evaluation and correction method based on local average according to claim 2, characterized in that: in the step C, according to the following formula:
defining:
when the value of i is equal to j,
the covariance matrix of the ith layer of soil slope can be obtained as follows:
thus, the joint probability density function value of the failed samples at the original fluctuation range δ can be found.
4. The slope stability evaluation and correction method based on local average according to claim 3, characterized in that: in the step C, similarly under the multilayer soil slope of the new fluctuation range δ', the joint probability density function is:
in the formula, xiN generated by local averaging of i-th layer soil variableiThe vector representation of random variable sample only needs to use original failure sample x(i)；μiN is the i-th layer soil variable which is locally averaged under a new fluctuation range deltaiThe individual variables correspond to the vector representation of the mean, i.e. are still:
μi'＝(μi1,μi2,...μini)′＝(μi,μi,...μi)′,
B′in for i-th layer soil body variable under new fluctuation range delta' through local averagingiA variable quantity
The covariance matrix of (a) may, similarly,
σ′ij＝σ′i[Γ(Δij|δ′)]0.5 i＝1,2,…m；j＝1,2,…ni，
wherein sigmaij'is the standard deviation corresponding to the jth variable after local averaging of the ith layer soil body variable in the new fluctuation range delta'; deltaijDividing the j-th interval of the corresponding sliding arc for the soil body variable in the ith soil body of the original lower side slope through local averaging; delta is the fluctuation range;
under the new fluctuation range delta', the correlation coefficient of the local average variable corresponding to the interval between the j section and the k section of the sliding arc in the i-th layer soil body is as follows:
wherein Δij,ikThe section I delta on the sliding arc in the layer I soil body under the original divisioniAnd ith segment deltajThe distance between the curves on the sliding arc; then:
when the value of i is equal to j,
the covariance matrix of the ith layer of soil slope can be obtained as follows:
the joint probability density function value of the failed samples under the new fluctuation range can be obtained.
5. The slope stability evaluation and correction method based on local average according to claim 1, characterized in that: in the step D, the soil body characteristic parameter estimated value is corrected based on the new data, and the corrected failure probability is
Expressed as:
combining with weighing theory, utilizing the corrected combined probability density function f (x) of the initial distribution of the soil characteristic parameters, and the corrected slope failure probability
Can be expressed as:
in the formula, omega is a weight index; thus, under the original joint probability density function f (x), the correction is madeLater failure probability estimate
Can be expressed as:
6. The slope stability evaluation and correction method based on local average according to claim 5, characterized in that: in the step D, combining the original joint probability density function value and according to a high-efficiency slope reliability correction formula combined with a weighing theory, the failure probability estimation value under a new fluctuation range delta' can be obtained as follows:
wherein N is the Monte Carlo simulation times; n issThe number of failed samples.
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