CN104899380B - A kind of Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation - Google Patents
A kind of Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation Download PDFInfo
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Abstract
The invention provides a kind of Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation, including：Step 1, the joint probability density function of uncertain parameters is constructed；Step 2, side slope failure probability is obtained using Monte-carlo Simulation Method, and obtains failure sample；Step 3, a variety of sensitivity analysis schemes are designed, and construct the joint probability density function of uncertain parameters under each sensitivity analysis scheme respectively；Step 4, the side slope failure probability under each sensitivity analysis scheme is obtained；Step 5, according to the side slope failure probability of each sensitivity analysis scheme, variation tendency of the side slope failure probability with the statistical nature of uncertain parameters is obtained.The present invention is applied widely, calculating process is simple, computational efficiency is high, and can effectively disclose the horizontal response pattern, side slope risk control, design optimization etc. between the statistical nature of uncertain parameters of reliability of slope has certain directive significance.
Description
Technical field
The present invention relates to a kind of slope parameter Sensitivity Analysis, more particularly to a kind of side based on Monte Carlo simulation
Slope ultimate carrying capacity Sensitivity Analysis.
Background technology
Side slope sensitivity analysis based on Reliability Theory can effectively characterize the uncertainty and its statistics of input parameter
The influence of feature (average, variance etc.) Slope Stability reliability, side slope design, reinforcing etc. has great importance, closely
Received significant attention over year.Monte-carlo Simulation Method is as a kind of Method of Stochastic for solving reliability, because of the letter of its concept
Single, strong applicability is used widely in Geotechnical Engineering reliability field.But there is uncertainty in Monte-carlo Simulation Method
The indefinite shortcoming of the mechanism of transmission, response pattern of the failure probability with the uncertain change of input parameter can not be directly obtained.
Therefore, when carrying out the sensitivity analysis of uncertain parameters using Monte Carlo simulation, it usually needs repeat to simulate, calculate defeated
Enter failure probability corresponding to the series of discrete point of stochastic variable statistical nature, then between analytic statistics feature and failure probability
Changing rule.In engineering practice, failure probability corresponding to general Side slope designing project it is smaller (such as less than 0.001), and
For small probability problem, Monte-carlo Simulation Method computational efficiency is relatively low, repeats simulation needs and takes a substantial amount of time and calculate
Machine resource, required calculating time and resource increase with the increase of the complexity of Analysis of Slope Stability model.
Wang[1]Etc. a kind of Sensitivity Analysis based on Monte Carlo simulation is proposed, this method can effectively determine
Amount analysis uncertain parameters are for condition corresponding to the Relative Contribution and the different uncertain parameters values of calculating of failure probability
Failure probability, but the response pattern between failure probability and the statistical nature of uncertain parameters is not studied.Then
Wang[2]A kind of Sensitivity Analysis of combination bayesian theory is proposed, this method need not re-execute Monte Carlo mould
Intend, it is possible to calculate single uncertain parameters statistical nature change after side slope failure probability, obtain failure probability with not
Response pattern between the statistical nature of deterministic parameter.But this method needs to make frequency histogram, calculating process is more
Complexity, and can not solve the problems, such as the sensitivity analysis of multiple statistics of variable features while change.
It is related to following bibliography in text：
[1]Wang Y,Cao Z J,Au S K.Efficient Monte Carlo Simulation of
parameter sensitivity in probabilistic slope stability analysis[J].Computers
And Geotechnics, 2010,37 (7-8):1015-1022.
[2]Wang Y.Uncertain parameter sensitivity in Monte Carlo Simulation
by sample reassembling[J].Computers and Geotechnics,2012,46:39-47.
[3]Kiureghian A D,Liu P L.Structural reliability under incomplete
probability information[J].Journal of Engineering Mechanics,1986,112:86-114.
The content of the invention
In view of the deficienciess of the prior art, the present invention based on Monte Carlo simulation propose a kind of calculating process simplicity,
The high Reliability of Slope Stability Sensitivity Analysis of computational efficiency.
In order to solve the above technical problems, the present invention adopts the following technical scheme that：
Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation, including step：
Step 1, the statistical nature of uncertain parameters is determined, and constructs the joint probability density function of uncertain parameters
f(x)；
Step 2, based on joint probability density function f (x), side slope failure probability is obtained using Monte-carlo Simulation Method
Pf, and obtain failure sample；
Step 3, a variety of sensitivity analysis schemes are designed, according to statistical nature corresponding to sensitivity analysis scheme, difference structure
Make the joint probability density function f of uncertain parameters under each sensitivity analysis schemek(x)；
Described sensitivity analysis scheme obtains with the following method：The statistical nature determined based on step 1, consider not true
Determine the actual change scope of parameter, change the statistical characteristics of one or more uncertain parameters, keep other uncertainties
The statistical nature probability distribution of parameter is constant, that is, obtains a kind of sensitivity analysis scheme；
Step 4, the side slope failure probability under each sensitivity analysis scheme is obtained, this step further comprises：
4.1 according to joint probability density function fk(x) and failure sample, obtain failure sample weight index f(xj) and fk(xj) be respectively the weight index of j-th of failure sample, uncertain parameters joint probability density function and
Joint probability density function under k-th of sensitivity analysis scheme；
4.2 obtain the side slope failure probability of sensitivity analysis scheme according to weight index For kth
The side slope failure probability of individual sensitivity analysis scheme, nsFor the sample number that fails, N is to be produced in step 2 Monte-carlo Simulation Method
Random vector sample number；
Step 5, according to the side slope failure probability of each sensitivity analysis scheme, obtain side slope failure probability and join with uncertainty
The variation tendency of number statistical nature, so as to identify the crucial uncertain parameters for influenceing Reliability of Slope Stability.
Step 1 further comprises sub-step：
1.1 according to the test datas of uncertain parameters, or in existing literature uncertain parameters statistical nature value,
Determine uncertain parameters statistical nature and each uncertain parameters between coefficient correlation；
1.2 is uncertain according to the coefficient correlation between the statistical nature of uncertain parameters and each uncertain parameters, construction
The joint probability density function f (x) of property parameter.
Sub-step 1.2 is specially：
(1) if uncertain parameters are all separate, the joint probability density functions of uncertain parameters for it is each not
The product of deterministic parameter probability density function；
(2) if uncertain parameters are all related, based on Copula theories or the uncertain ginseng of Nataf converter techniques construction
Several joint probability density functions；
(3) if uncertain parameters part is independent, partly related, first, obtained using method in step (1) separate
Uncertain parameters joint probability density function, be designated asWherein, q is separate uncertain parameters
Number；Then, the joint probability density function f (x of related uncertain parameter are obtained using method in step (2)q+1,…,xn)；
The then joint probability density function of uncertain parameters
Step 2 further comprises sub-step：
2.1 produce the random vector sample for submitting to joint probability density function f (x)；
2.2 pairs of random vector samples carry out Analysis of Slope Stability, whether judge side slope corresponding to each random vector sample
Failure；
2.3 statistics failure sample numbers, obtain side slope failure probabilityWherein, nsFor the sample number that fails, N is random
Vectorial sample number；Described failure sample random vector sample i.e. corresponding to failure side slope；
2.4 export and preserve failure sample.
Compared with prior art, the present invention has advantages below and beneficial effect：
1st, it is applied widely：
The present invention is based on Monte Carlo simulation, clear concept, is easily understood.In addition, the present invention can efficiently solve list
The Reliability of Slope Stability sensitivity analysis problem of one uncertain parameters and a variety of uncertain parameters, while also can be preferably
It is implicit complex slope ultimate carrying capacity sensitivity analysis problem suitable for power function expression formula, it is applied widely.
2nd, calculating process is simple, and computational efficiency is high：
Traditional Monte-carlo Simulation Method needs to repeat when calculating the side slope failure probability corresponding to sensitivity analysis scheme
Monte Carlo simulation is performed, computational efficiency is relatively low, for the ultimate carrying capacity sensitivity analysis of complex slope, covers special
It is long the time required to side slope deterministic parsing in the simulation of Carlow, repeat Monte Carlo simulation and carry out the sensitivity analysis calculating time
It is long, high is required to computing resource.The present invention makes full use of the calculating of uncertain parameters original probability distribution Monte Carlo simulation to believe
Breath, during sensitivity analysis scheme failure probability is solved, the weight index of failure sample need to be only calculated, avoids Meng Teka
Repeating for Lip river simulation, simplifies calculating process, improves computational efficiency, is a kind of simple efficient Reliability of Slope Stability
Sensitivity Analysis.For the ultimate carrying capacity sensitivity analysis of above-mentioned complex slope, the present invention is imitated to calculating
The raising of rate seems especially prominent, is advantageous to the application for promoting Monte-carlo Simulation Method in reliability of slope engineering.
3rd, uncertainty propagation mechanism is disclosed, important reference frame is provided for In Slope Engineering Design：
The present invention can effectively disclose the horizontal response relation between the statistical nature of uncertain parameters of reliability of slope,
So as to specify that, probabilistic mechanism of transmission, side slope risk control, design optimization etc. has during Monte Carlo simulation
Certain directive significance.
To sum up, the present invention can simplify calculating process, improve computational efficiency, disclose uncertainty during Monte Carlo simulation
The mechanism of transmission, to promoting Geotechnical Engineering reliability and Risk Theory in engineering practice using there is important practical value.
Brief description of the drawings
Fig. 1 is the idiographic flow schematic diagram of the present invention；
Fig. 2 is the reliability of slope analysis process figure based on Monte Carlo simulation；
Fig. 3 is the profile of side slope in embodiment；
Fig. 4 is the probability distribution graph of the uncertain parameters of different sensitiveness schemes in embodiment, with hard paste thickness
TcrExemplified by；
Fig. 5~7 are sensitivity analysis result in embodiment.
Embodiment
The present invention is further illustrated below in conjunction with the drawings and specific embodiments.
The idiographic flow of the present invention is shown in Fig. 1, comprises the following steps that：
Step 1, the statistical nature of uncertain parameters is determined, and constructs the joint probability density function of uncertain parameters
The joint probability density function f (x) of the former distribution of f (x), i.e. uncertain parameters, X be characterize uncertain parameters it is random to
Amount.
In reliability analysis of geotechnical engineering, Soil Parameters can be divided into two class parameters：Deterministic parameter and uncertain parameters,
Uncertain parameters include soil body cohesive strength, internal friction angle, soil body unit weight, soil thickness etc..For uncertain parameters, it is not
Deterministic quantization can be divided into two kinds of situations according to test data is whether there is：One, if having scene or laboratory test knot in Slope Design
Fruit, statistical method can be taken to quantify it uncertain according to the test data of uncertain parameters.For example join for uncertainty
For number soil body severe, multiple measuring point samples are taken at the scene, according to the soil body severe of each measuring point sample, are determined using statistical method
Its statistical nature, such as average, variance etc., then quantify the uncertainty of soil body severe according to the statistical nature of acquisition.Two, if
The not testing data on uncertain parameters, can be according to the statistical nature value of existing literature uncertain parameters, to institute
The statistical nature of corresponding uncertain parameters is reasonably set in research side slope.
This step further comprises following sub-step：
1.1 according to the test datas of uncertain parameters, or the statistical nature value of existing literature uncertain parameters, really
Determine uncertain parameters statistical nature and each uncertain parameters between correlation coefficient ρ.
1.2 is true according to the coefficient correlation between the statistical nature of uncertain parameters and each uncertain parameters, construction
The joint probability density function f (x) of qualitative parameter.
Joint probability density function f (x) construction can be divided into three kinds of situations：(1) uncertain parameters are all separate
In the case of joint probability density function construction；(2) joint probability density function under uncertain parameters whole correlation circumstance
Construction；(3) construction of joint probability density function in the case that uncertain parameters part is independent, part is related.
For separate uncertain parameters, n ties up the joint probability density function f (x) of uncertain parameters
For the product of the probability density function of each uncertain parameters, therefore f (x) can be expressed as：
In formula (1)：f(xi) it is i-th of uncertain parameters XiProbability density function, wherein xiFor XiValue, n is
Uncertain parameters number, Π represent even multiplication.
For the uncertain parameters of correlation, the construction of joint probability density function can use Copula it is theoretical,
Nataf converter techniques etc..Using the joint of the related uncertain parameters of more common Nataf converter techniques construction in this specific implementation
Probability density function.Now with two-dimensional correlation uncertain parameters X1And X2Exemplified by, introduce the general principles of Nataf converter techniques.
By the uncertain parameters X of correlation1And X2Probability density function be designated as f (x respectively1) and f (x2), wherein, x1With
x2X is represented respectively1And X2Value, its Cumulative Distribution Function is designated as F respectively1(x1) and F2(x2), uncertain parameters X1And X2Between
Pearson correlation coefficient be ρ.To uncertain parameters X1And X2Carry out following equiprobability conversion：
In formula (2)：Φ(yi) it is standard normal variable YiCumulative Distribution Function, Φ-1() tires out for standard normal variable
The inverse function of cloth function phi of scoring (), YiFor XiCorresponding standardized normal distribution variable after equiprobability conversion, xiAnd yiRepresent Xi
And YiValue.
According to Nataf converter techniques, uncertain parameters X can be derived1And X2Joint probability density function f (x) be：
In formula (3)：ρ0For variable Y1And Y2Pearson correlation coefficient；φ () is the probability density of standardized normal distribution
Function；φ2() is two-dimentional standardized normal distribution joint probability density function, and its expression formula is：
Defined according to coefficient correlation, ρ and ρ0With relation：
In formula (5)：μ1And μ2Respectively uncertain parameters X1And X2Average；σ1And σ2Respectively uncertain parameters X1
And X2Standard deviation.
Variable Y1And Y2Correlation coefficient ρ0Can by formula (5) iterative, can also according to Der Kiureghian and
Liu[3]The empirical equation for different probability statistical nature provided solves, and can also be asked according to Gauss-Hermite integration
Solution.But generally, ρ0It is more or less the same with ρ, ρ can be directly taken in approximate solution0=ρ.
It is pointed out that the usual Normal Distribution of Rock And Soil uncertain parameters.If uncertain parameters X1And X2Clothes
From normal distribution, now according to formula (3) and (4), uncertain parameters X1And X2Joint probability density function f (x) can express
For：
For the uncertain parameters that part is independent, part is related, independent uncertainty is obtained according to formula (1)
The joint probability density function of parameter, is designated asPhase can be obtained according to above-mentioned Nataf transform methods or Copula theories
Joint probability density function f (the x of the uncertain parameter of passq+1,…,xn), therefore it is uncertain under the independent sector correlation circumstance of part
The joint probability density function of parameterWherein, f (xi) represent independent uncertain ginseng
Number Xi, (i=1 ... probability density function q), Xq+1、Xq+2、…XnRepresent related uncertain parameters.
Step 2, the joint probability density function f (x) built according to step 1, side is calculated using Monte-carlo Simulation Method
Slope failure probability Pf, and export and preserve failure sample xj, j=1,2 ..., ns。
This step realizes that idiographic flow is shown in Fig. 2, mainly including following sub-step using traditional Monte-carlo Simulation Method：
The 2.1 joint probability density function f (x) built according to step 1, generation submit to joint probability density function f (x)
Random vector sample xi, i=1,2 ... N, i represent random vector sample number.
2.2 structure Analysis of Slope Stability models, calculate random vector sample xiCorresponding side slope power function value g (xi)
=FS (xi) -1, and according to power function value g (xi) judge whether side slope fails.
Currently used Analysis of Slope Stability method has limit equilibrium method and FInite Element, and wherein limit equilibrium method includes auspicious
Allusion quotation slices method, simplification form, Morgenstern-Price method etc..Business software for slope stability mainly has
GeoStudio (limit equilibrium method), ANSYS (FInite Element), ABAQUS (FInite Element) etc..The present invention is applicable any side
Analysis of Stability of Front Slope method.
For specific slope project, according to the Method for Slope Stability Analysis to be used, with corresponding business software
Establish Analysis of Slope Stability model.Then, random vector x is substituted intoi, calculate its corresponding side slope power function value g (xi)=FS
(xi)-1,As side slope power function value g (xi)>0, i.e. Side Slope Safety Coefficient FS (xi) be less than 1 when, represent side slope be in stablizes shape
State；As g (xiDuring)=0, represent that side slope is in critical condition；As g (xi)<When 0, slope instability is represented, i.e. side slope fails.
Circulation performs sub-step 2.1 and sub-step 2.2, until i=N.
2.3 statistics failure number of samples ns, calculate side slope failure probability
Meet g (xi)<0 random vector sample xiFor the sample that fails.Statistics failure sample size is calculated as ns, fail sample
It is designated as xj, j=1,2 ..., ns.According to Monte-carlo Simulation Method principle, side slope failure probability PfIt can be expressed as：
In formula (7),Represent side slope failure probability PfValuation, I [g (x)] is indicator function, as g (x)<When 0, I [g
(x) 1] is taken；As g (x) >=0, I [g (x)] is taken as 0.
2.4 export and preserve failure sample xj, j=1,2 ..., ns。
Step 3, the uncertain parameters statistical nature determined based on step 1, by increasing and reducing uncertain parameters
Statistical nature obtains a series of sensitivity analysis schemes, according to the statistics of uncertain parameters corresponding to each sensitivity analysis scheme
Latent structure joint probability density function.
For the uncertain parameters studied, by changing its statistical characteristics, ensuring it without departing from accordingly not true
Under the premise of the actual change scope of qualitative parameter, M kind sensitivity analysis schemes are set；According to corresponding to each sensitivity analysis scheme
Statistical nature, corresponding uncertain parameters joint probability density function f is constructed respectivelyk(x), k=1,2 ... M, M are sensitivity
Property analytical plan quantity, k represent sensitivity analysis Protocol Numbers, fk(x) represent that k-th of sensitivity analysis scheme be not corresponding true
The joint probability density function of qualitative parameter.
, it is necessary to uncertain by calculating input when analyzing uncertain parameters sensitiveness using Monte-carlo Simulation Method
The side slope failure probability of the series of discrete point of parametric statistics feature (such as average, variance), to probe into side slope failure probability and not
Response pattern between the statistical nature of deterministic parameter.
, can be by increasing and reducing the mean μ or mark of cohesive strength when such as carrying out sensitivity analysis to soil body adhesive aggregation force parameter
Accurate poor σ, the different distributions of M kind cohesive strengths are set as sensitivity analysis scheme.Then, according under each sensitivity analysis scheme
Statistical nature, construct the uncertain parameters joint probability density function f under each sensitivity analysis schemek(x).Joint probability is close
Spend function fk(x) specific configuration method is the same as sub-step 1.2.
It is to be herein pointed out the present invention can carry out sensitivity analysis simultaneously to multiple uncertain parameters, such as may be used
To set the sensitivity analysis scheme of mean μ that is a series of while changing multiple uncertain parameters and standard deviation sigma.
Step 4, the joint probability density function f obtained using step 3k(x) the failure sample x obtained with step 2j, j=
1,2,...,ns, solve the side slope failure probability P under each sensitivity analysis schemef k, according to side slope failure probability Pf kWith uncertain
Property parametric statistics feature changing rule sensitivity analysis is carried out to the statistical nature and the coefficient of variation of uncertain parameters.
According to side slope failure probability general formula, the side slope failure probability P under k-th of sensitivity analysis schemef kIt can represent
For：
In formula (8)：For in joint probability density function fk(x) the mathematic expectaion operator under.
With reference to the joint probability density function f (x) of the former distribution of uncertain parameters, i.e., the joint probability that step 1 constructs is close
Function f (x) is spent, the side slope failure probability P under k-th of sensitivity analysis schemef kIt can be further represented as：
In formula (9), ωkIt is as follows for the customized weight index of the present invention, its calculation expression：
Pass through weight index ωkCan be the side slope failure probability P under the conditions of solution sensitivity analysis schemef kIt is converted into and asks
Solve I [g (x)] ωkMathematic expectaion under f (x) distributions.Therefore, can directly utilize step 2.1 in Monte Carlo simulation with
This x of press proofi, i=1,2, N, calculate side slope failure probability Pf k.Therefore it is based on N caused by joint probability density function f (x)
Group random sample solves the side slope failure probability P of k-th of sensitivity analysis schemef kIt is represented by：
In view of indicator function I [g (x)] value feature, only fail sample xj, j=1,2, nsCan be to failure
Probability Pf kContribution is produced, formula (11) can be further expressed as：
Pf kEstimateVarianceIt is represented by：
The coefficient of variationIt is represented by：
The coefficient of variationThe accuracy of result of calculation can be effectively characterized, is calculated for the inventive method
As a result reliability provides foundation.
It can be seen from formula (12) when solving the side slope failure probability of sensitivity analysis scheme, the present invention only needs to count
Calculate the weight index caused by former distribution at failure sample and sum, without re-starting Monte Carlo simulation, calculated
Journey is relatively simple, computational efficiency is high.
It should be noted that joint probability density function used in failure probability is calculated in this step as in failure sample range
Joint probability density function, be different from the joint probability density function in all sample ranges in step 1 to 3.
The specific sub-step of this step is as follows：
The specific sub-step of this step is as follows：
4.1 according to failure sample joint probability density function f (xj) and fk(xj), obtain j-th of failure sample xjPower
Weight index f(xj) and fk(xj) be respectively j-th of failure sample weight index, uncertain parameters connection
Close the joint probability density function under probability density function and k-th of sensitivity analysis scheme.
4.2 according to weight indexCalculate the side slope failure probability of sensitivity analysis scheme
Step 5, according to the side slope failure probability of sensitivity analysis scheme, side slope failure probability is drawn with uncertain parameters
The trend curve of statistical nature change, identification influence the crucial uncertain parameters of Reliability of Slope Stability.
Embodiment 1
First, Engineering Notice
James gulf dykes and dams are located at a hydraulic engineering in Quebec, CAN area, the long 50km of dykes and dams, and one of which is set
Meter scheme section is as shown in figure 3, there is a 56m Wide Worktable the high 12m of dykes and dams, centre, and side slope slope angle is 18.4 °, and slope ratio is 3:1.
The following soil layer of dam body is followed successively by hard paste, marine clay, lake caly and drift sheet from top to bottom.
Bibliography [1], the present embodiment consider 6 stochastic variables, i.e., uncertain parameters to be studied, respectively hard
Clay thickness Tcr, marine clay undrained shear strength SuM, lake caly undrained shear strength SuL, the dam foundation to drift sheet
Distance DTill, build a dam the angle of friction φ to banketFillWith severe γFill.The equal Normal Distribution of uncertain parameters and mutually solely
Vertical, corresponding statistical nature is shown in Table 1.Other deterministic parameters are as follows：The thickness of marine clay is 8m, and hard paste not draining resists
It is 41kPa to cut intensity, and hard paste, marine clay, the severe of lake caly are respectively 19kN/m3、19kN/m3、20.5kN/m3。
In addition, lake caly thickness T is understood by geometrical relationshipL=DTill-Tcr-8。
Calculated to simplify, document [1] increases most dangerous sliding surface two limitations：(1) most dangerous sliding surface and drift sheet
Tangent and mistake point (x, y), wherein, x=4.9m, y=36m；(2) most dangerous sliding surface center of circle abscissa x=85.9m.This place
Reason method causes distance D of the position of most dangerous sliding surface only with the dam foundation to drift sheetTillIt is relevant.
The statistical nature of the uncertain parameters of table 1
2nd, flow is embodied
Step 1, according to the statistical nature of each uncertain parameters in table 1, the joint of the former distribution of construction uncertain parameters
Probability density function f (φFill,γFill,Tcr,SuM,SuL,DTill)。
Due to each uncertain parameters Normal Distribution in the present embodiment and independently of each other, therefore joint probability density letter
Number f (φFill,γFill,Tcr,SuM,SuL,DTill) can be expressed as：
f(φFill,γFill,Tcr,SuM,SuL,DTill)=f (φFill)f(γFill)f(Tcr)f(SuM)f(SuL)f(DTill) (15)
Wherein：
In formula, μ1、μ2、μ3、μ4、μ5、μ6And σ1、σ2、σ3、σ4、σ5、σ6Respectively parameter phiFill、γFill、Tcr、SuM、SuL、
DTillAverage and standard deviation, its concrete numerical value is shown in Table 1.
Step 2, according to the joint probability density function f (φ of the former distribution of uncertain parametersFill,γFill,Tcr,SuM,SuL,
DTill), utilize the side slope failure probability P of Monte Carlo Analogue Method acquisition James gulf dykes and damsf, and export and preserve failure sample
xj, j=1,2 ..., ns。
2.1 submit to joint probability density function f (φ using the generation of VBA programsFill,γFill,Tcr,SuM,SuL,DTill)
Random vector sample xi, i=1,2 ... N, i represent random vector sample number, N=2 × 105。
2.2 structure Analysis of Slope Stability models, calculate random vector sample xiCorresponding side slope power function value g (xi)
=FS (xi) -1, and according to side slope power function value g (xi) judge whether James gulf dykes and dams fail.
First, using simplification form, James gulf stability of earth dams analysis model is built using EXCEL lists.So
Afterwards, brought into using random vector sample as input parameter in the stability analysis model on James's dike gulf dam.Finally, according to stable
Property analysis model calculate random vector variable xiCorresponding Side Slope Safety Coefficient FS (xi) and side slope power function value g (xi)。
Circulation performs sub-step 2.1 and sub-step 2.2, until i=N.
2.3 statistics failure sample number ns, share failure sample ns=452.James gulf dykes and dams are calculated according to formula (7)
Side slope failure probability Pf=452/200000=2.26 × 10-3。
2.4 will failure sample xjIt is stored in EXCEL lists, is used for sensitivity analysis.
(200,000 times) about 10 hours times of calculating of Monte Carlo Analogue Method based on the former distribution of uncertain parameters above, institute
Allocation of computer is used as internal memory 4GB, CPU Intel Core i3 and dominant frequency 3.3GHz.Therefore, it is if special using traditional illiteracy
The sensitivity analysis problem of Carlow analogue approach James gulf dykes and dams, for every kind of sensitivity analysis scheme, in identical mould
The calculating time required under plan number may each be about 10 hours.As can be seen here, side is solved using traditional Monte Carlo Analogue Method
Slope ultimate carrying capacity sensitivity analysis problem, computational efficiency are very low.
Step 3, the statistical characteristics of uncertain parameters is studied by changing, a series of sensitivity analysis sides are set
Case.According to statistical nature corresponding to each sensitivity analysis scheme, the connection of uncertain parameters under each sensitivity analysis scheme is constructed
Close probability density function fk(φFill,γFill,Tcr,SuM,SuL,DTill), fk(φFill,γFill,Tcr,SuM,SuL,DTill) represent
Uncertain parameters joint probability density function under kth kind sensitivity analysis scheme.
This step is by changing the average, variance or other statistical natures of certain uncertain parameters, to obtain a variety of sensitivities
Property analytical plan, and by analyzing the side slope failure probability of each sensitivity analysis scheme, obtain the statistics of the uncertain parameters
The influence degree of feature Slope Stability reliability, so as to obtain influenceing the crucial uncertain parameters of Reliability of Slope Stability.
Because uncertain parameters are more in the present embodiment, hard paste thickness T is only chosen hereincr, drift sheet thickness
DTillWith the undrained shear strength S of lake calyuL, study Tcr、DTill、SuLChange in Mean to James gulf embankment slope lose
Imitate the influence of probability.As shown in table 2, by changing uncertain parameters T in the present embodimentcr、DTill、SuLAverage, altogether set
18 kinds of sensitivity analysis schemes：In scheme 1~6, hard paste thickness TcrAverage be set to u3-3σ3、u3-2σ3、u3-σ3、
u3+σ3、u3+2σ3、u3+3σ3, u3、σ3Respectively TcrThe average and standard deviation of original distribution, its probability distribution curve are shown in Fig. 4；Other
The statistical characteristics of uncertain parameters keeps the former statistical characteristics being distributed.Similar, respectively to parameter SuLAnd DTillDo
Same processing, scheme 7~12 and scheme 13~18 can be obtained.
The joint probability density function f of each sensitivity analysis schemek(φFill,γFill,Tcr,SuM,SuL,DTill) construction
It is similar with analysis of cases step 1, it will not be repeated here.
The sensitivity analysis scheme of table 2
Step 4, according to the joint probability density function f tried to achievek(φFill,γFill,Tcr,SuM,SuL,DTill) and step 2
The failure sample x of middle outputj, j=1,2, ns, solve the side slope failure probability P of each sensitivity analysis schemef k。
According to formula (10), weight index ω corresponding to k-th of sensitivity analysis schemekIt is represented by：
Such as when k takes 1, weight index ω1It can be expressed as
The weight index value ω of failure samplej kBring formula (12) into, then can calculate corresponding to each sensitivity analysis scheme
Side slope failure probability.
In this respect it is to be noted that the calculating of weight index can be with reference to probability density function (such as normal state built in EXCEL
Distribution probability density function NORM.DIST), calculate very easy.Meanwhile solving sensitiveness it can be seen from step 3~4
During analytical plan, due to taking full advantage of the failure sample of former probability distribution Monte Carlo simulation, failure is directly utilized
The weight index of sample solves failure probability, and without re-executing Monte Carlo simulation, calculating process is very simple efficient, calculates
It is only 5.4 × 10 the time required to side slope failure probability corresponding to a kind of sensitivity analysis scheme-3s。
Step 5, according to the side slope failure probability of sensitivity analysis scheme, system of the failure probability with uncertain parameters is drawn
The trend curve of changing features is counted, identification influences the crucial uncertain parameters of James gulf dykes and dams ultimate carrying capacity.
Fig. 5~7 are respectively James gulf embankment slope failure probability with uncertain parameters Tcr、SuLAnd DTillChange in Mean
Trend curve.As shown in figure 5, with hard paste thickness TcrAverage is stepped up to 5.44m, failure probability from 8 from 2.56m
×10-3It is slowly declined to 1 × 10-3Left and right.As shown in fig. 6, work as lake caly cohesive strength SuLAverage from 12.27kN/m2Increase
50.13kN/m2When, James gulf dykes and dams failure probability is then from 10-1Left and right drops sharply to 10-7, illustrate lake caly cohesive strength
SuLThe increase of average James gulf dykes and dams level of reliability can be caused to drastically reduce.As shown in fig. 7, with drift sheet thickness
DTillAverage is stepped up 21.5m from 15.5m, and the failure probabilities of James gulf dykes and dams is from 10-5Increase to 10-2Left and right, it is seen that Zhan
The failure probability of this gulf dykes and dams of nurse can be with uncertain parameters drift sheet thickness DTillThe increase of average and increase.
It can be drawn by above-mentioned analysis, among the present embodiment studies three parameters, the failure of James gulf dykes and dams
Probability is for drift sheet thickness DTill, lake caly cohesive strength SuLIt is more sensitive, wherein, lake caly cohesive strength SuLThe change of average
James gulf dykes and dams ultimate carrying capacity is influenceed maximum.Therefore in engineering, it should take corresponding engineering technology to obtain more close
In lake caly cohesive strength SuLInformation, so as to more accurately assess side slope security.
Above-described embodiment shows, the Reliability of Slope Stability sensitivity analysis side of Monte Carlo simulation proposed by the invention
Method, the failure probability of sensitivity analysis scheme can be efficiently solved, it is horizontal with not knowing ginseng effectively to disclose reliability of slope
Response relation between several statistical natures, and the crucial uncertain parameters for influenceing Reliability of Slope Stability are identified, it is side
Slope risk control, stabilization and reinforcement, optimization design etc. provide reference frame.
Claims (3)
1. the Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation, it is characterized in that, including step：
Step 1, the statistical nature of uncertain parameters is determined, and constructs the joint probability density function f of uncertain parameters
(x)；
Step 2, based on joint probability density function f (x), side slope failure probability P is obtained using Monte-carlo Simulation Methodf, and obtain
Must be failed sample；
Step 3, a variety of sensitivity analysis schemes are designed, according to statistical nature corresponding to sensitivity analysis scheme, construction is each respectively
The joint probability density function f of uncertain parameters under sensitivity analysis schemek(x)；
Described sensitivity analysis scheme obtains with the following method：The statistical nature determined based on step 1, consider uncertain ginseng
Several actual change scopes, change the statistical characteristics of one or more uncertain parameters, keep other uncertain parameters
Statistical nature probability distribution it is constant, that is, obtain a kind of sensitivity analysis scheme；
Step 4, the side slope failure probability under each sensitivity analysis scheme is obtained, this step further comprises：
4.1 according to joint probability density function fk(x) and failure sample, obtain failure sample weight index f(xj) and fk(xj) be respectively j-th of failure sample weight index, uncertain parameters joint probability density function and
Joint probability density function under k sensitivity analysis scheme；
4.2 obtain the side slope failure probability of sensitivity analysis scheme according to weight index It is sensitive for k-th
The side slope failure probability of property analytical plan, nsFor the sample number that fails, N is caused random in step 2 Monte-carlo Simulation Method
Vectorial sample number；
Step 5, according to the side slope failure probability of each sensitivity analysis scheme, obtain side slope failure probability and united with uncertain parameters
The variation tendency of feature is counted, so as to identify the crucial uncertain parameters for influenceing Reliability of Slope Stability.
2. the Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation as claimed in claim 1, its feature
It is：
Step 1 further comprises sub-step：
1.1, according to the test datas of uncertain parameters, determine the statistical nature of uncertain parameters and each uncertain ginseng
The coefficient correlation of several；
1.2, according to the coefficient correlation between the statistical nature of uncertain parameters and each uncertain parameters, construct uncertain ginseng
Several joint probability density function f (x).
3. the Reliability of Slope Stability Sensitivity Analysis based on Monte Carlo simulation as claimed in claim 2, its feature
It is：Sub-step 1.2 is specially：
(1) if uncertain parameters are all separate, the joint probability density function of uncertain parameters is each uncertain
The product of property parameter probability density function；
(2) if uncertain parameters are all related, uncertain parameters are constructed based on Copula theories or Nataf converter techniques
Joint probability density function；
(3) if uncertain parameters part is independent, partly related, first, obtained independently of each other not using method in step (1)
The joint probability density function of deterministic parameter, is designated asWherein, q is separate uncertain parameters number；
Then, the joint probability density function f (x of related uncertain parameter are obtained using method in step (2)q+1,...,xn)；Then
The joint probability density function of uncertain parameters
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