Method for efficiently calculating reliability of general-shaped sliding surface soil slope system

Abstract

The invention relates to the technical field related to geotechnical engineering, in particular to an efficient calculation method for reliability of a general-shaped sliding surface soil slope system, and the method comprises the steps: 1, employing a critical horizontal acceleration coefficient Kc to replace a safety coefficient Fs to evaluate the stability of a slope; 2, a performance function adopts the difference between Kc and a known seismic coefficient Kc0; step 3, reliability calculation; according to the method, a Rosenblue sampling point method is used for setting parameter combinations of random variables, a GCSF method is used for searching a critical sliding surface in any shape under each combination condition, a two-dimensional slope reliability calculation method based on strict limit equilibrium is used for calculating a reliability index of the critical sliding surface, a PNET method is used for identifying a representative sliding surface, and the reliability of a slope system is calculated in combination with an SCM method.

Description

Technical Field

[0001] The invention relates to the technical field related to geotechnical engineering, and in particular to a method for efficiently calculating the reliability of a sliding surface soil slope system of a general shape. Background Art

[0002] Slope stability evaluation is one of the important topics in the field of geotechnical engineering. Slope stability reliability analysis can quantitatively consider the uncertainties in geotechnical engineering and is considered to be a useful supplement to deterministic methods. Most slopes contain multiple soil layers, and the soil layer parameters have spatial variability. These factors lead to the possibility that the slope may fail along multiple different sliding surfaces. Sliding along any sliding surface will cause slope failure. It can be seen that the slope is a series structural system with multiple failure modes. Slope stability reliability analysis is essentially a slope system reliability problem.

[0003] The Monte Carlo simulation method is recognized as the most direct and effective method for calculating the reliability of the slope system. However, there are a large number of potential sliding surfaces in the slope. The MCS method requires a large amount of deterministic analysis for each possible sliding surface, which has a high computational cost. Based on the MCS method, some scholars have proposed some methods that can effectively improve the efficiency of slope system reliability calculation. According to the research ideas, these methods can be roughly divided into two categories: one is to identify several representative sliding surfaces that have an important impact on the reliability of the slope system, and then calculate the failure probability of the slope system. For example, the multiple response surface method based on representative sliding surfaces was used to perform system reliability analysis on multiple slopes. The other is to use efficient methods that can directly reduce the number of sampling times, such as the generalized subset simulation (GSS) method, the WUS probability density weight method, and the adaptive Monte Carlo simulation (AMCS) method.

[0004] However, the above literature generally assumes that the potential sliding surface is arc-shaped, and the functional function adopts the safety factor expression. Studies have shown that the shape of the sliding surface has a significant impact on the calculation results of the slope system reliability. For simple clay slopes, the arc-shaped sliding surface is more realistic. When the slope soil layer distribution is more complex or there are weak interlayers, the real sliding surface is usually non-arc-shaped. In addition, the non-convex, non-smooth, and nonlinear implicit function characteristics of the safety factor determine the difficulty of solving it. For this reason, the present invention proposes an efficient calculation method for the reliability of soil slope systems with general sliding surfaces to solve the above problems. Summary of the invention

[0005] The purpose of the present invention is to provide an efficient calculation method for the reliability of a sliding surface soil slope system of a general shape, so as to solve the problems raised in the above-mentioned background technology.

[0006] To achieve the above object, the present invention provides the following technical solution: an efficient calculation method for the reliability of a soil slope system with a general shape sliding surface, the efficient calculation method for the reliability of a soil slope system with a general shape sliding surface comprises:

[0007] Step 1: Use the critical horizontal acceleration coefficient K c Instead of safety factor F s To evaluate the slope stability, the two-dimensional slope has a sliding surface of general shape. Assuming that a static horizontal seismic force KW (expressed by the product of the horizontal acceleration coefficient K and the deadweight W) is applied to the sliding body, when the sliding body reaches the limit equilibrium state, that is, F s =1, the value of the horizontal acceleration coefficient K is K c ;

[0008] Step 2: The function uses K c With the known seismic coefficient K c0 The difference between , as shown in formula (1), is used to calculate the slope reliability;

[0009] Z=K c -K c0 (1)

[0010] Where K c is the critical horizontal acceleration coefficient, K c 0 is the known seismic coefficient. When there is no seismic load on the slope, K c 0=0;

[0011] Step 3: Reliability calculation, assuming that the random variables affecting slope stability are {x:X1,X2,…,X n}, the failure probability of the slope can be expressed as:

[0012]

[0013] Where f x (x) is the probability density function;

[0014] The reliability index β can be calculated by the inverse function of the standard normal distribution function in formula (3);

[0015] β=Φ -1 (1-P f )(3)

[0016] Where Φ -1 (x) is the inverse of the standard normal distribution function.

[0017] Preferably, when the slope sliding surface is a representative sliding surface of a general shape, in this case there are a large number of possible sliding surfaces on the slope, and the shape of the sliding surface is usually not circular. Only a few sliding surfaces have a significant impact on the calculation results of the slope system reliability, which are called representative sliding surfaces. On the basis of the representative sliding surfaces, the time consumed in calculating the slope system reliability is less. According to the general shape of the sliding surface, the coupled Rosenblueth sampling point method, the global critical sliding field (GCSF) method and the probabilistic network evaluation technology (PNET) are used to identify the representative sliding surface.

[0018] Preferably, the coupled Rosenblueth sampling point method is specifically:

[0019] When the random variable x is known i When the mean and standard deviation of (i=1,2,…,n) are calculated, two sampling points with a standard deviation distance from the mean can be symmetrically selected, that is:

[0020]

[0021] The two random variables are x1 and x2, and a rectangular coordinate system is established with them as coordinate axes. The four sampling points set by the Rosenblueth method correspond to the four corner points of the rectangle, as shown in the following figure. Figure 2 As shown;

[0022] When the number of random variables is 3, the 8 sampling points set by the Rosenblueth method are the 8 corner points of the cuboid in the three-dimensional rectangular coordinate system, as shown in the following figure. Figure 3 As shown;

[0023] In order to ensure the high accuracy of slope system reliability calculation results, the Rosenblueth sampling point method is used to set 2 n A random variable parameter combination of soil.

[0024] Preferably, when the slope sliding surface is a non-homogeneous slope with large differences in soil parameters, its sliding surface is generally not circular, and the reliability of the slope system is analyzed by using the global critical sliding field (GCSF) method. The critical sliding field method is based on the principle of maximum thrust between strips, that is, for any point in the slope, there is only one optimal path to maximize its thrust;

[0025] This method discretizes the slope uniformly into multiple state points, such as Figure 4 As shown in the figure, taking the maximum thrust as the target, the dangerous sliding direction of each state point is obtained. The dangerous sliding directions of all state points constitute the dangerous sliding field of the slope. Starting from the exit state point of the slip surface with the largest residual thrust, the critical slip surface (CSF) is obtained by reverse tracking.

[0026] The global critical slip field (GCSF) consists of the critical slip surfaces of all exit state points of the slip surface, such as Figure 5 As shown;

[0027] The identification method of representative sliding surface is as follows: according to the Rosenblueth sampling point method, set 2 n parameter combinations, 2 can be searched by critical sliding field method n The representative sliding surfaces are identified by PNET method based on the reliability index of each sliding surface and the correlation coefficient between sliding surfaces.

[0028] Preferably, the correlation analysis between the sliding surfaces is:

[0029] The correlation coefficient is introduced to measure the correlation between failure modes. For the kth and lth failure modes, the approximate correlation coefficient can be calculated by formula (5);

[0030]

[0031] Where: kl is the correlation coefficient between the kth and lth failure modes; Z k and Z l are the functional functions of the kth and lth failure modes respectively, and m is the number of random variables. k and Z l For a random variable x j The partial derivative of is:

[0032]

[0033] Where Z k +、Z k -、Z l + and Z l - is the performance function value obtained by adding or subtracting one standard deviation from the mean of the random variable corresponding to the kth and lth failure modes, respectively, and is calculated by formula (1).

[0034] Preferably, the representative sliding surface identification method is to use Rosenblueth sampling point method and GCSF method to identify the representative sliding surface, and the identification steps include:

[0035] ①The reliability index of 2n critical sliding surfaces found by GCSF method was calculated by SS method;

[0036] ② Find the critical slip surface with the smallest reliability index as the first representative slip surface;

[0037] ③ The correlation coefficient between the representative sliding surface and other remaining critical sliding surfaces was calculated;

[0038] ④ Eliminate the sliding surface whose correlation coefficient ρ with the representative sliding surface is greater than the threshold ρ0;

[0039] ⑤ Find the critical sliding surface with the smallest reliability index among the remaining sliding surfaces as the new representative sliding surface;

[0040] ⑥ Repeat steps ③ to ⑤ until no new representative sliding surface is found.

[0041] Preferably, the slope system reliability calculation method of the representative sliding surface is a sequential combination method (SCM), and the sequential combination method (SCM) is specifically:

[0042] Assume first that there are N slopes r There are N representative sliding surfaces, which can be regarded as r The SCM is a series system consisting of two components, S1 and S2, coupled into an equivalent component S 1or2 , and then the equivalent element S 1or2 The next step is to couple with element S3 until all elements are coupled into a final equivalent element S 1or2or…orNr ,like Figure 6 As shown, according to the above idea, the failure probability of the slope system can be expressed by equation (7);

[0043]

[0044] Due to the one-to-one correspondence between reliability index and failure probability, the equivalent component S 1or2 The reliability index of can be calculated using equation (8);

[0045] β 1or2 =-Φ -1 [P f(1or2) ]=-Φ -1 [Φ(-β1)+Φ(-β2)-Φ2(-β1,-β2;ρ 1,2 )] (8)

[0046] where Φ2(-β1,-β2;ρ 1,2 ) is the two-dimensional standard normal cumulative distribution function; ρ 1,2 is the correlation coefficient between S1 and S2, β1, β2 and P f1 ,P f2 are their reliability index and failure probability respectively, Φ2(-β1,-β2; ρ 1,2 ) can be reduced in dimension by conditional probability, and then approximated by normal distribution function;

[0047]

[0048] Where: is the conditional reliability index of component 1 under the condition of component 2 failure, and the parameters can be calculated by formula (10):

[0049]

[0050] B2=A2(-β2+A2)(10c)

[0051] Where φ(-β2) is the standard normal probability density function;

[0052] Substituting equations (9)-(10) into equation (8), we can obtain:

[0053] β 1or2 =-Φ -1 [Φ(-β1)+Φ(-β2)-Φ(-β 12 )Φ(-β2)](11)

[0054] Similarly, when S1or2 is coupled with Si, under the condition of Si failure, the conditional reliability index can be expressed by formula (12-13):

[0055] β (1or2)i =-Φ -1 [Φ(-β 1|i )+Φ(-β 2|i )-Φ2(-β 1|i ,-β 2|i ρ 1,2|i )] (12)

[0056]

[0057] Where:

[0058]

[0059] B i =A i (-β i +A i ),(i=3,4,L,N r )(14e)

[0060] By solving equations (12) and (13) simultaneously and restricting them to the range [-1,1], we can obtain ρ (1or2),i The value of

[0061] In view of the simplicity and practicability of Rosenblueth method, the accuracy of global critical sliding field method in searching for critical sliding surface of arbitrary shape, the simplicity and effectiveness of sequential combination method, and the high efficiency of strict limit equilibrium method based on sliding surface normal stress correction mode in calculating slope reliability, an efficient calculation method for slope system reliability is proposed.

[0062] This method uses the Rosenblueth sampling point method to construct 2 n The GCSF method searches for the sliding surfaces under these parameter combinations to find the critical sliding surface with the smallest safety factor in each combination, and then uses the PNET method to identify the representative sliding surface, and then uses the SCM method to calculate the reliability index of the slope system.

[0063] In order to compare the influence of sliding surface shape on the calculation results, circular and non-circular sliding surfaces were considered respectively, the reliability of the slope system was calculated, and the circular sliding surface was searched by Geo-studio software.

[0064] Preferably, in the efficient calculation method of the slope system reliability, according to the mean and standard deviation of the undrained shear strength, four combinations of random variable parameters c1 and c2 are generated according to formulas (4a) to (4b), and for each combination, 1000 circular arc sliding surfaces are searched using Geo-studio software, as shown in the attached figure. Fig. 9 -Attached Fig.12 As shown;

[0065] And find the critical sliding surface with the smallest safety factor in each group of sliding surfaces, named sliding surfaces 1-4 (hereinafter referred to as S1-S4), where S1 and S4 coincide, such as Fig.13 As shown;

[0066] And by using the efficient calculation method of slope reliability based on strict limit equilibrium, the reliability indexes of the four critical sliding surfaces are obtained: β1=β4=2.9889, β2=3.1044, β3=2.7822;

[0067] Among them, S3 has the smallest reliability index. As the first representative sliding surface, the correlation coefficients between the three sliding surfaces calculated by formula (5) are: ρ 12 =1.000,ρ 13 =0.348,ρ 23 =0.331. 13 and ρ 23 All of them are less than ρ0 (ρ0=0.9), no sliding surface is excluded, and S1 with the smallest reliability index among the remaining sliding surfaces is selected as the second representative sliding surface. 12 is greater than ρ0, so S2 can be excluded. So far, all representative slip surfaces have been found, such as Fig.10 shown.

[0068] Since the reliability index of S3 is smaller than that of S1, S3 and S1 can be named as components 1 and 2 respectively, and the reliability index β of the composite component can be calculated using equation (11): (1or2)=2.6485. Since there are only two representative sliding surfaces in this example, the slope system reliability index is β sys =2.6485, and according to the standard normal distribution function, the probability of system failure can be calculated as P fsys =0.0040.

[0069] Preferably, when the sliding surface is a non-circular sliding surface, the soil layer parameters are averaged, and considering that the sliding surface is a non-circular arc shape, the safety factor of the most dangerous sliding surface searched by the GCSF method is 1.810, which is less than 1.863 of the Morgenstern-Price method in the Geo-studio software, and also less than the minimum safety factor of 2.003 in the case of a circular sliding surface;

[0070] It can be seen that the GCSF method can search for non-circular sliding surfaces with smaller safety factors;

[0071] Considering non-circular sliding surfaces, the global critical sliding field method is also used to search for 50 sliding surfaces of arbitrary shapes for each random variable parameter combination. The positions of dangerous sliding surfaces searched for under the four parameter combinations are as follows: Figure 15-18 As shown;

[0072] The critical sliding surface with the smallest safety factor among the four groups of dangerous sliding surfaces is named S1-S4, as shown in the figure Figure 15-18 As shown, S1 and sliding surface S4 still coincide with each other, and the reliability index values ​​of these four sliding surfaces are: β1=β4=2.3199, β2=2.2451, β3=2.6203.

[0073] S2 has the smallest reliability index and is the first representative sliding surface. The correlation coefficients between S2 and sliding surfaces S1 and S3 are ρ 12 =0.991,ρ 23 =0.177, according to ρ 12 >ρ0, S1 can be excluded, and the representative sliding surfaces are S2 and S3, such as Fig. 20 As shown;

[0074] According to the size relationship of the reliability index, S2 and S3 are named as component 1 and component 2 respectively. After calculation, the system failure probability P fsys =0.0166, which is about 4 times the result of considering the circular sliding surface. It can be seen that the non-circular sliding surface searched by the GCSF method is more dangerous than the circular sliding surface, and the probability of failure of the slope system is also greater;

[0075] In order to compare the sliding surface positions more intuitively, the circular and non-circular representative sliding surfaces obtained by the proposed method and the sliding surfaces in the literature are plotted on Fig.21 middle.

[0076] Compared with the prior art, the present invention has the following beneficial effects:

[0077] 1. This method uses the Rosenblueth sampling point method to set the parameter combination of random variables, uses the GCSF method to search for the critical sliding surface of any shape in each combination, calculates the reliability index of the critical sliding surface based on the two-dimensional slope reliability calculation method of strict limit equilibrium, uses the PNET method to identify the representative sliding surface, and combines the SCM method to calculate the reliability of the slope system;

[0078] 2. This method only needs a small amount of calculation to quickly identify the main representative sliding surface of the slope, and conveniently and effectively calculate the reliability index and failure probability of the slope system. Compared with the MCS method, this method has higher calculation accuracy and efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is the horizontal earthquake force diagram acting on the sliding body;

[0080] Figure 2 The weight graph of Rosenblueth sampling points and random variables when there are two random variables;

[0081] Figure 3 The weight diagram of Rosenblueth sampling points and random variables when there are three random variables;

[0082] Figure 4 is a distribution diagram of discrete state points;

[0083] Figure 5 is the global critical slip field diagram;

[0084] Figure 6 It is a schematic diagram of the sequential combination method of the present invention;

[0085] Figure 7 This is a flow chart of the efficient calculation method for slope system reliability of the present invention;

[0086] Figure 8 It is the cross-section diagram of the two-layer slope in Example 1;

[0087] Fig. 9 Schematic diagram of arc sliding surface Figure 1 ;

[0088] Fig.10 Schematic diagram of arc sliding surface Figure 2 ;

[0089] Fig.11 Schematic diagram of arc sliding surface Figure 3 ;

[0090] Fig.12 Schematic diagram of arc sliding surface Figure 4 ;

[0091] Fig.13 are the four arc-shaped critical sliding surfaces in Example 1;

[0092] Fig.14 These are the two arc-shaped representative sliding surfaces of Example 1;

[0093] Fig.15 Schematic diagram of non-circular sliding surface Figure 1 ;

[0094] Fig.16 Schematic diagram of non-circular sliding surface Figure 2 ;

[0095] Fig.17 Schematic diagram of non-circular sliding surface Figure 3 ;

[0096] Fig.18 Schematic diagram of non-circular sliding surface Figure 4 ;

[0097] Fig.19 are the four non-circular critical sliding surfaces in Example 1;

[0098] Fig. 20 These are the two non-circular representative sliding surfaces of Example 1;

[0099] Fig.21 It is a comparison diagram of the representative sliding surface positions in Example 1;

[0100] Fig. 22 This is the sliding surface diagram obtained by Geostudio software in Example 1. DETAILED DESCRIPTION

[0101] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0102] See also Figure 1-22 The present invention provides the following preferred embodiments:

[0103] Embodiment 1

[0104] The efficient calculation method for reliability of soil slope system with general sliding surface includes:

[0105] Step 1: Use the critical horizontal acceleration coefficient K c Instead of safety factor F sTo evaluate the slope stability, the two-dimensional slope has a sliding surface of general shape. Assuming that a static horizontal seismic force KW (expressed by the product of the horizontal acceleration coefficient K and the deadweight W) is applied to the sliding body, when the sliding body reaches the limit equilibrium state, that is, F s =1, the value of the horizontal acceleration coefficient K is K c ;

[0106] Step 2: The function uses K c With the known seismic coefficient K c0 The difference between , as shown in formula (1), is used to calculate the slope reliability;

[0107] Z=K c -K c0 (1)

[0108] Where K c is the critical horizontal acceleration coefficient, K c 0 is the known seismic coefficient. When there is no seismic load on the slope, K c 0=0;

[0109] Step 3: Reliability calculation, assuming that the random variables affecting slope stability are {x:X1,X2,…,X n}, the failure probability of the slope can be expressed as:

[0110]

[0111] Where f x (x) is the probability density function;

[0112] The reliability index β can be calculated by the inverse function of the standard normal distribution function in formula (3);

[0113] β=Φ -1 (1-P f ) (3)

[0114] Where Φ -1 (x) is the inverse of the standard normal distribution function.

[0115] When the slope sliding surface is a representative sliding surface of general shape, there are a large number of possible sliding surfaces on the slope, and the shape of the sliding surface is usually not circular. Only a few sliding surfaces have a significant impact on the calculation results of the slope system reliability, which are called representative sliding surfaces. On the basis of the representative sliding surface, the time consumed in calculating the slope system reliability is less. According to the general shape of the sliding surface, the coupled Rosenblueth sampling point method, the global critical sliding field (GCSF) method and the probabilistic network evaluation technology (PNET) are used to identify the representative sliding surface.

[0116] The coupled Rosenblueth sampling point method is specifically:

[0117] When the random variable x is known i When the mean and standard deviation of (i=1,2,…,n) are calculated, two sampling points with a standard deviation distance from the mean can be symmetrically selected, that is:

[0118]

[0119] The two random variables are x1 and x2, and a rectangular coordinate system is established with them as coordinate axes. The four sampling points set by the Rosenblueth method correspond to the four corner points of the rectangle, as shown in the following figure. Figure 2 As shown;

[0120] When the number of random variables is 3, the 8 sampling points set by the Rosenblueth method are the 8 corner points of the cuboid in the three-dimensional rectangular coordinate system, as shown in the following figure. Figure 3 As shown;

[0121] In order to ensure the high accuracy of slope system reliability calculation results, the Rosenblueth sampling point method is used to set 2 n A random variable parameter combination of soil.

[0122] When the sliding surface of the slope is a non-homogeneous slope with large differences in soil parameters, its sliding surface is generally not circular. The global critical sliding field (GCSF) method is used to analyze the reliability of the slope system. The critical sliding field method is based on the principle of maximum thrust between strips, that is, for any point in the slope, there is only one optimal path to maximize its thrust.

[0123] This method discretizes the slope uniformly into multiple state points, such as Figure 4 As shown in the figure, taking the maximum thrust as the target, the dangerous sliding direction of each state point is obtained. The dangerous sliding directions of all state points constitute the dangerous sliding field of the slope. Starting from the exit state point of the slip surface with the largest residual thrust, the critical slip surface (CSF) is obtained by reverse tracking.

[0124] The global critical slip field (GCSF) consists of the critical slip surfaces of all exit state points of the slip surface, such as Figure 5 As shown;

[0125] The GCSF method does not require the assumption of an initial sliding surface and can quickly and accurately determine the critical sliding surface of any shape, and the calculation results are stable and unique.

[0126] The identification method of representative sliding surface is as follows: according to the Rosenblueth sampling point method, set 2 n parameter combinations, 2 can be searched by critical sliding field method nThe same sedimentation history and other factors make the soil parameters have a certain correlation. Therefore, there may be a certain correlation between the sliding surfaces. Based on the reliability index of each sliding surface and the correlation coefficient between sliding surfaces, the PNET method is used to identify the representative sliding surface.

[0127] Correlation analysis between sliding surfaces:

[0128] The correlation coefficient is introduced to measure the correlation between failure modes. For the kth and lth failure modes, the approximate correlation coefficient can be calculated by formula (5);

[0129]

[0130] Where: kl is the correlation coefficient between the kth and lth failure modes; Z k and Z l are the functional functions of the kth and lth failure modes respectively, and m is the number of random variables. k and Z l For a random variable x j The partial derivative of is:

[0131]

[0132] Where Z k +、Z k -、Z l + and Z l - is the performance function value obtained by adding or subtracting one standard deviation from the mean of the random variable corresponding to the kth and lth failure modes, respectively, and is calculated by formula (1).

[0133] The representative sliding surface identification method is to use Rosenblueth sampling point method and GCSF method to identify the representative sliding surface, and the identification steps include:

[0134] ①The reliability index of 2n critical sliding surfaces found by GCSF method was calculated by SS method;

[0135] ② Find the critical slip surface with the smallest reliability index as the first representative slip surface;

[0136] ③ The correlation coefficient between the representative sliding surface and other remaining critical sliding surfaces was calculated;

[0137] ④ Eliminate the sliding surface whose correlation coefficient ρ with the representative sliding surface is greater than the threshold ρ0;

[0138] ⑤ Find the critical sliding surface with the smallest reliability index among the remaining sliding surfaces as the new representative sliding surface;

[0139] ⑥ Repeat steps ③ to ⑤ until no new representative sliding surface is found;

[0140] At present, there is no unified conclusion on the selection of correlation coefficient threshold, and most of them are 0.8 or 0.9 [4, 6, 22]. Here, it is assumed that the correlation coefficient threshold is 0.9.

[0141] The slope system reliability calculation method of the representative sliding surface is the sequential combination method (SCM), which is as follows:

[0142] Assume first that there are N slopes r There are N representative sliding surfaces, which can be regarded as r The SCM is a series system consisting of two components, S1 and S2, coupled into an equivalent component S 1or2 , and then the equivalent element S 1or2 The next step is to couple with element S3 until all elements are coupled into a final equivalent element S 1or2or…orNr ,like Figure 6 As shown, according to the above idea, the failure probability of the slope system can be expressed by equation (7);

[0143]

[0144] Due to the one-to-one correspondence between reliability index and failure probability, the equivalent component S 1or2 The reliability index of can be calculated using equation (8);

[0145] β 1or2 =-Φ -1 [P f(1or2) ]=-Φ -1 [Φ(-β1)+Φ(-β2)-Φ2(-β1,-β2;ρ 1,2 )] (8)

[0146] where Φ2(-β1,-β2;ρ 1,2 ) is the two-dimensional standard normal cumulative distribution function; ρ 1,2 is the correlation coefficient between S1 and S2, β1, β2 and P f1 ,P f2 are their reliability index and failure probability respectively, Φ2(-β1,-β2; ρ 1,2 ) can be reduced in dimension by conditional probability, and then approximated by normal distribution function;

[0147]

[0148] Where: is the conditional reliability index of component 1 under the condition of component 2 failure, and the parameters can be calculated by formula (10):

[0149]

[0150] B2=A2(-β2+A2)(10c)

[0151] Where φ(-β2) is the standard normal probability density function;

[0152] Substituting equations (9)-(10) into equation (8), we can obtain:

[0153] β 1or2 =-Φ -1 [Φ(-β1)+Φ(-β2)-Φ(-β 12 )Φ(-β2)](11)

[0154] Similarly, when S1or2 is coupled with Si, under the condition of Si failure, the conditional reliability index can be expressed by formula (12-13):

[0155] β (1or2)i =-Φ -1 [Φ(-β 1|i )+Φ(-β 2|i )-Φ2(-β 1|i ,-β 2|i ρ 1,2|i )] (12)

[0156]

[0157] Where:

[0158]

[0159]

[0160] B i =A i (-β i +A i ),(i=3,4,L,N r )(14e)

[0161] By solving equations (12) and (13) simultaneously and restricting them to the range [-1,1], we can obtain ρ (1or2),i The value of

[0162] In view of the simplicity and practicability of Rosenblueth method, the accuracy of global critical sliding field method in searching for critical sliding surface of arbitrary shape, the simplicity and effectiveness of sequential combination method, and the high efficiency of strict limit equilibrium method based on sliding surface normal stress correction mode in calculating slope reliability, an efficient calculation method for slope system reliability is proposed.

[0163] This method uses the Rosenblueth sampling point method to construct 2 nThe GCSF method searches for the sliding surfaces under these parameter combinations to find the critical sliding surface with the smallest safety factor in each combination, and then uses the PNET method to identify the representative sliding surface, and then uses the SCM method to calculate the reliability index of the slope system.

[0164] In order to compare the influence of sliding surface shape on the calculation results, circular and non-circular sliding surfaces were considered respectively, the reliability of the slope system was calculated, and the circular sliding surface was searched by Geo-studio software.

[0165] Table 1 Soil layer parameters of Example 1

[0166]

[0167] In the efficient calculation method of slope system reliability, according to the mean and standard deviation of undrained shear strength, four combinations of random variable parameters c1 and c2 are generated according to formulas (4a) to (4b), and for each combination, 1000 circular arc sliding surfaces are searched using Geo-studio software, as shown in the attached figure. Fig. 9 -Attached Fig.12 As shown;

[0168] And find the critical sliding surface with the smallest safety factor in each group of sliding surfaces, named sliding surfaces 1-4 (hereinafter referred to as S1-S4), where S1 and S4 coincide, such as Fig.13 As shown;

[0169] And by using the efficient calculation method of slope reliability based on strict limit equilibrium, the reliability indexes of the four critical sliding surfaces are obtained: β1=β4=2.9889, β2=3.1044, β3=2.7822;

[0170] Among them, S3 has the smallest reliability index. As the first representative sliding surface, the correlation coefficients between the three sliding surfaces calculated by formula (5) are: ρ 12 =1.000,ρ 13 =0.348,ρ 23 =0.331. 13 and ρ 23 All of them are less than ρ0 (ρ0=0.9), no sliding surface is excluded, and S1 with the smallest reliability index among the remaining sliding surfaces is selected as the second representative sliding surface. 12 is greater than ρ0, so S2 can be excluded. So far, all representative slip surfaces have been found, such as Fig.10 shown.

[0171] Since the reliability index of S3 is smaller than that of S1, S3 and S1 can be named as components 1 and 2 respectively, and the reliability index β of the composite component can be calculated using equation (11): (1or2)=2.6485. Since there are only two representative sliding surfaces in this example, the slope system reliability index is β sys =2.6485, and according to the standard normal distribution function, the probability of system failure can be calculated as P fsys =0.0040.

[0172] The calculation results of this paper and the above scholars are listed in Table 2

[0173] Table 2 System reliability calculation results of different methods in Example 1

[0174]

[0175]

[0176] When the sliding surface is non-circular, the soil layer parameters are averaged. Considering that the sliding surface is non-circular, the safety factor of the most dangerous sliding surface searched by the GCSF method is 1.810, which is smaller than 1.863 of the Morgenstern-Price method in Geo-studio software, and also smaller than the minimum safety factor of 2.003 in the case of circular sliding surface.

[0177] It can be seen that the GCSF method can search for non-circular sliding surfaces with smaller safety factors;

[0178] Considering non-circular sliding surfaces, the global critical sliding field method is also used to search for 50 sliding surfaces of arbitrary shapes for each random variable parameter combination. The positions of dangerous sliding surfaces searched for under the four parameter combinations are as follows: Figure 15-18 As shown;

[0179] The critical sliding surface with the smallest safety factor among the four groups of dangerous sliding surfaces is named S1-S4, as shown in the figure Figure 15-18 As shown, S1 and sliding surface S4 still coincide with each other, and the reliability index values ​​of these four sliding surfaces are: β1=β4=2.3199, β2=2.2451, β3=2.6203.

[0180] S2 has the smallest reliability index and is the first representative sliding surface. The correlation coefficients between S2 and sliding surfaces S1 and S3 are ρ 12 =0.991,ρ 23 =0.177, according to ρ 12 >ρ0, S1 can be excluded, and the representative sliding surfaces are S2 and S3, such as Fig. 20 As shown;

[0181] According to the size relationship of the reliability index, S2 and S3 are named as component 1 and component 2 respectively. After calculation, the system failure probability P fsys=0.0166, which is about 4 times the result of considering the circular sliding surface. It can be seen that the non-circular sliding surface searched by the GCSF method is more dangerous than the circular sliding surface, and the probability of failure of the slope system is also greater;

[0182] In order to compare the sliding surface positions more intuitively, the circular and non-circular representative sliding surfaces obtained by the proposed method and the sliding surfaces in the literature are plotted on Fig.21 In, by Fig.21 It can be seen that the position of the non-circular sliding surface obtained by the proposed method is relatively close to the circular sliding surface in this study and the references, with only a certain difference at the entrance of the sliding surface. The main reason for the difference is that the GCSF method takes into account the non-tensile properties of geotechnical materials and sets tension cracks where tension occurs;

[0183] In order to verify the accuracy of the calculation results of this method in the case of non-circular sliding surfaces, the Geo-studio software was used to calculate the system reliability of Example 1, and a total of 275 sliding surfaces were obtained, such as Fig. 22 As shown in the figure, the reliability of the slope system under 25,000 samples was calculated using the MCS method, and the system failure probability P fsys =0.0170, reliability index β sys =2.1114, the errors of the proposed method and the MCS method are -2.35% and 0.73% respectively, which shows that the slope system reliability calculation method based on the GCSF method is effective;

[0184] On the same computer, the actual calculation time of this method is 184 seconds, while the calculation time of Geo-studio software using MCS method is 3498 seconds, which is about 19 times that of this method. Therefore, this method is a slope system reliability calculation method that can achieve both high precision and high efficiency.

[0185] Although the above describes the illustrative specific implementation methods of the present application so that technicians in this technical field can understand the present application, the present application is not limited to the scope of the specific implementation methods. For ordinary technicians in this technical field, as long as various changes are within the spirit and scope of the present application defined and determined by the attached claims, all application creations using the concept of the present application are protected.

Claims

1. An efficient calculation method for system reliability of soil slope with general sliding surface, characterized by: The high-efficiency calculation method for the reliability of the shape sliding surface soil slope system includes: Step 1: Use the critical horizontal acceleration coefficient K c Instead of safety factor F s To evaluate the slope stability, the two-dimensional slope has a sliding surface of general shape. Assuming that a static horizontal seismic force KW (expressed by the product of the horizontal acceleration coefficient K and the deadweight W) is applied to the sliding body, when the sliding body reaches the limit equilibrium state, that is, F s =1, the value of the horizontal acceleration coefficient K is K c ; Step 2: The function uses K c With the known seismic coefficient K c0 The difference between , as shown in formula (1), is used to calculate the slope reliability; Z=K c -K c0 (1) Where K c is the critical horizontal acceleration coefficient, K c 0 is the known seismic coefficient. When there is no seismic load on the slope, K c 0=0; Step 3: Reliability calculation, assuming that the random variables affecting slope stability are {x:X1,X2,…,X n }, the failure probability of the slope can be expressed as: Where f x (x) is the probability density function; The reliability index β can be calculated by the inverse function of the standard normal distribution function in formula (3); β=Φ -1 (1-P f ) (3) Where Φ -1 (x) is the inverse of the standard normal distribution function.

2. The high-efficiency calculation method for system reliability of sliding surface soil slope of general shape according to claim 1 is characterized by: When the slope sliding surface is a representative sliding surface of a general shape, there are a large number of possible sliding surfaces in the slope, and the shape of the sliding surface is usually not circular. Only a few sliding surfaces have a significant impact on the calculation results of the slope system reliability, which are called representative sliding surfaces. On the basis of the representative sliding surface, the time consumed in calculating the slope system reliability is less. According to the general shape of the sliding surface, the coupled Rosenblueth sampling point method, the global critical sliding field (GCSF) method and the probabilistic network evaluation technology (PNET) are used to identify the representative sliding surface.

3. The high-efficiency calculation method for system reliability of sliding surface soil slope of general shape according to claim 2 is characterized by: The coupled Rosenblueth sampling point method is specifically: When the random variable x is known i When the mean and standard deviation of (i=1,2,…,n) are calculated, two sampling points with a standard deviation distance from the mean can be symmetrically selected, that is: The two random variables are x1 and x2, and a rectangular coordinate system is established with them as coordinate axes. The four sampling points set by the Rosenblueth method correspond to the four corner points of the rectangle, as shown in Figure 2; When the number of random variables is 3, the 8 sampling points set by the Rosenblueth method are the 8 corner points of the cuboid in the three-dimensional rectangular coordinate system, as shown in Figure 3; In order to ensure the high accuracy of slope system reliability calculation results, the Rosenblueth sampling point method is used to set 2 n A random variable parameter combination of soil.

4. The high-efficiency calculation method for system reliability of sliding surface soil slope of general shape according to claim 3 is characterized by: When the slope sliding surface is a non-homogeneous slope with large differences in soil parameters, its sliding surface is generally not circular, so the reliability of the slope system is analyzed using the global critical sliding field (GCSF) method. The critical sliding field method is based on the principle of maximum thrust between strips, that is, for any point in the slope, there is only one optimal path to maximize its thrust; This method discretizes the slope uniformly into multiple state points, as shown in Figure 4. Taking the maximum thrust as the goal, the dangerous sliding direction of each state point is obtained. The dangerous sliding directions of all state points constitute the dangerous sliding field of the slope. Starting from the exit state point of the slip surface with the largest residual thrust, the critical slip surface (CSF) is obtained by reverse tracking. The global critical slip field (GCSF) consists of the critical slip surfaces of all exit state points of the slip surface, as shown in Figure 5; The identification method of representative sliding surface is as follows: according to the Rosenblueth sampling point method, set 2 n parameter combinations, 2 can be searched by critical sliding field method n The representative sliding surfaces are identified by PNET method based on the reliability index of each sliding surface and the correlation coefficient between sliding surfaces.

5. The high-efficiency calculation method for system reliability of sliding surface soil slope of general shape according to claim 4 is characterized by: Correlation analysis between the sliding surfaces: The correlation coefficient is introduced to measure the correlation between failure modes. For the kth and lth failure modes, the approximate correlation coefficient can be calculated by formula (5); Where: kl is the correlation coefficient between the kth and lth failure modes; Z k and Z l are the functional functions of the kth and lth failure modes respectively, m is the number of random variables, Z k and Z l For a random variable x j The partial derivative of is: Where Z k +, Z k -、Z l + and Z l - is the performance function value obtained by adding or subtracting one standard deviation from the mean of the random variable corresponding to the kth and lth failure modes, respectively, and is calculated by formula (1).

6. The efficient calculation method for system reliability of sliding surface soil slope of general shape according to claim 5 is characterized by: The representative sliding surface identification method is to use Rosenblueth sampling point method and GCSF method to identify the representative sliding surface, and the identification steps include: ①The reliability index of 2n critical sliding surfaces found by GCSF method was calculated by SS method; ② Find the critical slip surface with the smallest reliability index as the first representative slip surface; ③ The correlation coefficient between the representative sliding surface and other remaining critical sliding surfaces was calculated; ④ Eliminate the sliding surface whose correlation coefficient ρ with the representative sliding surface is greater than the threshold ρ0; ⑤ Find the critical sliding surface with the smallest reliability index among the remaining sliding surfaces as the new representative sliding surface; ⑥ Repeat steps ③ to ⑤ until no new representative sliding surface is found.

7. The efficient calculation method for system reliability of sliding surface soil slope of general shape according to claim 6 is characterized by: The slope system reliability calculation method of the representative sliding surface is the sequential combination method (SCM), and the sequential combination method (SCM) is specifically: Assume first that there are N slopes r There are N representative sliding surfaces, which can be regarded as r The SCM is a series system consisting of two components, S1 and S2, coupled into an equivalent component S 1or2 , and then the equivalent element S 1or2 The next step is to couple with element S3 until all elements are coupled into a final equivalent element S 1or2or…orNr , as shown in Figure 6, according to the above idea, the failure probability of the slope system can be expressed by equation (7); Due to the one-to-one correspondence between reliability index and failure probability, the equivalent component S 1or2 The reliability index of can be calculated using equation (8); b 1or2 =-Φ -1 [P f(1or2) ]=-Φ -1 [Φ(-β1)+Φ(-β2)-Φ2(-β1,-β2;ρ 1,2 )] (8) where Φ2(-β1,-β2;ρ 1,2 ) is the two-dimensional standard normal cumulative distribution function; ρ 1,2 is the correlation coefficient between S1 and S2, β1, β2 and P f1 ,P f2 are their reliability index and failure probability respectively, Φ2(-β1,-β2; ρ 1,2 ) can be reduced in dimension by conditional probability, and then approximated by normal distribution function; Where: is the conditional reliability index of component 1 under the condition of component 2 failure, and the parameters can be calculated by formula (10): B2=A2(-β2+A2) (10c) Where φ(-β2) is the standard normal probability density function; Substituting equations (9)-(10) into equation (8), we can obtain: b 1or2 =-Φ -1 [Φ(-β1)+Φ(-β2)-Φ(-β 12 )Φ(-β2)] (11) Similarly, when S1or2 is coupled with Si, under the condition of Si failure, the conditional reliability index can be expressed by formula (12-13): b (1or2)i =-Φ -1 [Φ(-β 1|i )+Φ(-β 2|i )-Φ2(-β 1|i ,-b 2|i ;r 1,2|i )] (12) Where: B i =A i (-b i +A i ),(i=3,4,L,N r ) (14e) By solving equations (12) and (13) simultaneously and restricting them to the range [-1,1], we can obtain ρ (1or2),i The value of In view of the simplicity and practicability of Rosenblueth method, the accuracy of global critical sliding field method in searching for critical sliding surface of arbitrary shape, the simplicity and effectiveness of sequential combination method, and the high efficiency of strict limit equilibrium method based on sliding surface normal stress correction mode in calculating slope reliability, an efficient calculation method for slope system reliability is proposed. This method uses the Rosenblueth sampling point method to construct 2 n The GCSF method searches for the sliding surfaces under these parameter combinations to find the critical sliding surface with the smallest safety factor in each combination, and then uses the PNET method to identify the representative sliding surface, and then uses the SCM method to calculate the reliability index of the slope system. In order to compare the influence of sliding surface shape on the calculation results, circular and non-circular sliding surfaces were considered respectively, the reliability of the slope system was calculated, and the circular sliding surface was searched by Geo-studio software.

8. The efficient calculation method for system reliability of a sliding soil slope of a general shape according to claim 7 is characterized by: In the efficient calculation method of the slope system reliability, according to the mean and standard deviation of the undrained shear strength, four combinations of random variable parameters c1 and c2 are generated according to formulas (4a) to (4b), and for each combination, 1000 circular arc sliding surfaces are searched using Geo-studio software, as shown in Figures 9 to 12; And find the critical sliding surface with the smallest safety factor in each group of sliding surfaces, named sliding surfaces 1-4 (hereinafter referred to as S1-S4), where S1 coincides with S4, as shown in Figure 13; And by using the efficient calculation method of slope reliability based on strict limit equilibrium, the reliability indexes of the four critical sliding surfaces are obtained: β1=β4=2.9889, β2=3.1044, β3=2.7822; Among them, S3 has the smallest reliability index. As the first representative sliding surface, the correlation coefficients between the three sliding surfaces calculated by formula (5) are: ρ 12 =1.000,ρ 13 =0.348,ρ 23 =0.

331. 13 and ρ 23 All of them are less than ρ0 (ρ0=0.9), no sliding surface is excluded, and S1 with the smallest reliability index among the remaining sliding surfaces is selected as the second representative sliding surface. 12 is greater than ρ0, so S2 can be excluded. So far, all representative slip surfaces have been found, as shown in Fig.

10. Since the reliability index of S3 is smaller than that of S1, S3 and S1 can be named as components 1 and 2 respectively, and the reliability index β of the composite component can be calculated using equation (11): (1or2) =2.6485. Since there are only two representative sliding surfaces in this example, the slope system reliability index is β sys =2.6485, and according to the standard normal distribution function, the probability of system failure can be calculated as P fsys =0.0040.

9. The efficient calculation method for system reliability of sliding surface soil slope of general shape according to claim 8 is characterized by: When the sliding surface is a non-circular sliding surface, the soil layer parameters are averaged. Considering that the sliding surface is a non-circular arc shape, the safety factor of the most dangerous sliding surface searched by the GCSF method is 1.810, which is less than 1.863 of the Morgenstern-Price method in the Geo-studio software, and also less than the minimum safety factor of 2.003 in the case of a circular sliding surface; It can be seen that the GCSF method can search for non-circular sliding surfaces with smaller safety factors; Considering non-circular sliding surfaces, the global critical sliding field method is also used to search for 50 sliding surfaces of arbitrary shapes for each random variable parameter combination. The positions of dangerous sliding surfaces searched under the four parameter combinations are shown in Figures 15 to 18 respectively; The critical sliding surfaces with the smallest safety factors among the four groups of dangerous sliding surfaces are named S1-S4, as shown in Figures 15-18, where S1 and sliding surface S4 still coincide with each other, and the reliability index values ​​of these four sliding surfaces are: β1=β4=2.3199, β2=2.2451, β3=2.6203. S2 has the smallest reliability index and is the first representative sliding surface. The correlation coefficients between S2 and sliding surfaces S1 and S3 are ρ 12 =0.991,ρ 23 =0.177, according to ρ 12 >ρ0, S1 can be excluded, and the representative sliding surfaces are S2 and S3, as shown in Figure 20; According to the size relationship of the reliability index, S2 and S3 are named as component 1 and component 2 respectively. After calculation, the system failure probability P fsys =0.0166, which is about 4 times the result of considering the circular sliding surface. It can be seen that the non-circular sliding surface searched by the GCSF method is more dangerous than the circular sliding surface, and the probability of failure of the slope system is also greater; In order to compare the sliding surface positions more intuitively, the circular and non-circular representative sliding surfaces obtained by the proposed method and the sliding surfaces in the literature are plotted in Figure 21.

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