Reliability Calculation Method for Landslide Systems Based on Multivariate Normal Distribution Function

Abstract

This invention discloses a method for calculating the reliability of landslide systems based on the multivariate normal distribution (MVN) function. The multivariate normal cumulative distribution function in the system reliability is transformed into an integral form of multiple bivariate normal probability products, effectively solving the problem of difficulty in solving high-dimensional integrals. First, the function functions and covariances of each failure mode of the landslide are calculated using the rigid body limit equilibrium method and the multiple response surface method. Then, the numerical integral form of the MVN function under bivariate conditions is determined, and it is reordered using univariate methods to select a suitable numerical integration order. Finally, the system reliability of the landslide is solved by combining the covariance matrix. This method is computationally simple, has high accuracy, and can effectively calculate the reliability of landslide systems with multiple potential failure surfaces.

Description

Technical Field

[0001] This invention belongs to the field of landslide assessment technology, specifically relating to a method for calculating the reliability of landslide systems based on a multivariate normal distribution function. Background Technology

[0002] The system reliability of landslide stability is a probabilistic measure of landslide reliability. Due to the heterogeneity of soil and rock materials, landslide bodies often have a large number of potential sliding surfaces. The failure of any sliding surface will lead to the failure of the landslide system, which is a typical reliability problem of a series system.

[0003] System reliability is a typical high-dimensional numerical integral calculation problem. The fundamental challenge lies in solving multidimensional numerical integrals, where the integration dimension depends on the number of sliding surfaces. Due to the increased dimensionality, the integral calculation of the multivariate cumulative distribution function becomes complex, even difficult to solve. Currently, there are no good direct methods for calculating system reliability; most methods rely on point estimation or interval estimation, which are computationally inefficient and lack sufficient accuracy. Summary of the Invention

[0004] The purpose of this invention is to address the above-mentioned problems by transforming the multivariate normal cumulative distribution function in system reliability into an integral form of multiple bivariate normal probability products, which can be directly solved and calculated. This effectively solves the problem of the difficulty in solving high-dimensional integrals and improves the accuracy of the calculation results by reordering variables.

[0005] To solve the above-mentioned technical problems, the technical solution adopted in this invention is a landslide system reliability calculation method based on a multivariate normal distribution function, which includes the following steps:

[0006] S1: Analyze and determine the failure modes of the landslide system;

[0007] S2: Select the function to determine each failure mode of the landslide system;

[0008] S3: Calculate the correlation of each failure mode to obtain the covariance matrix;

[0009] S4: The numerical integral expression of the multivariate normal distribution function under the bivariate condition is derived;

[0010] S5: Sort the integration variables and select the appropriate order of numerical integration;

[0011] S6: The reliability of the landslide system is calculated based on the covariance matrix and the binary numerical integral expression of the sorted multivariate normal distribution function.

[0012] Preferably, the failure mode of the landslide system in step S1 is any sliding surface that causes the slope to lose stability.

[0013] Preferably, the function of the failure mode in step S2 is:

[0014] Z = F⁻¹ (1)

[0015] In the formula: Z is the function of the sliding surface, and F is the safety factor of the sliding surface.

[0016] Preferably, the response surface function in step S2 is:

[0017]

[0018] In the formula, g(x) represents the function, and a and b are... i i = 1, 2, ..., n, C i i = 1, 2, ..., n, d ij i = 1, 2, ..., n, j = 1, 2, ..., n represent undetermined coefficients; X i i = 1, 2, ..., n, X j j = 1, 2, ..., n represents different random variables;

[0019] Experimental points can be located along the coordinate axes at the mean point μ. x Nearby selection, where along coordinate x i The experimental points of the axis have coordinates Where f > 0 is an arbitrary factor.

[0020] Initial iteration point

[0021] In the formula, X represents a random variable. The vector formed; i = 1, 2…n represents the initial iteration points of the random variable; the average value is taken in the initial calculation, and the coefficients a and b of the linear equation system are calculated according to equation (2). i c i .

[0022] Preferably, the covariance between the failure mode function functions in step S3 is calculated as follows:

[0023] For nonnormal variables, equivalent normalization is performed using the following formula:

[0024]

[0025]

[0026] In the formula Represents the mean and standard deviation of the equivalent normalized variable; This is a verification point; The cumulative distribution function is... Φ is the probability density function; -1() denotes the inverse function of the cumulative distribution function; Represents the density function;

[0027] Function Z L mean and standard deviation Calculate according to the following formulas:

[0028]

[0029]

[0030] Represents the mean of a variable; Indicates the standard deviation of the variable;

[0031] Define variable X i The sensitivity coefficient is:

[0032]

[0033] In the formula Representing point x * The angle between the point and the origin O in the standardized space;

[0034] Equation (7) represents the normal form hyperplane equation in the space of standard normal random variables Y, where the normal is p on the limit state surface. * The direction cosine of the line connecting to the origin O in the standard normalized space is: p * Represents x in space X * The point corresponding to the limit state surface;

[0035]

[0036] In the formula C jk This represents the covariance between the function functions of the smooth surfaces j and k; Let represent the direction cosines of the function of the smooth surface j and k, respectively; Let represent the standard deviations of the function functions of the smooth surfaces j and k, respectively;

[0037] The steps for calculating the covariance between two different sliding surfaces are as follows:

[0038] (1) Assume the initial verification point x * Generally, x is taken. * =μ x ;

[0039] (2) Equivalent normalization: Calculate using equations (3) and (4). and

[0040] (3) Calculate the standard deviation of different functional functions using equation (6).

[0041] (4) Calculate the sliding surfaces using equation (7).

[0042] (5) Calculate the covariance between different sliding surfaces using equation (8). At this time, the correlation coefficient between different sliding surfaces is obtained.

[0043] (6) Obtain the covariance matrix Σ according to step (5).

[0044] Preferably, the step S4 of determining the numerical integral form of the multivariate normal distribution function under bivariate conditions is as follows:

[0045]

[0046] Q2(x;μ,Σ)=(x-μ) T ∑ -1 (x-μ) (10)

[0047] In the formula, f() represents the density function; μ represents the mean of the variables;

[0048] In the formula:

[0049] Where X = (X1, ..., X) n ) T Let μ be an n-dimensional random variable. i , σ ii They represent variables X = (X1, ..., X...). n ) T The mean and variance, σ ij For variable X i X j The covariance between them, 1≤i <j≤n;

[0050] Consider the general integral form of the multivariate normal distribution function as follows: ∫ A f(x)g(x)dx;

[0051] In the formula, x = (x1, x2, ... x n ) t ∈R k , A∈R k ,dx=dx n dx n-1 …dx1, f(x) is the density function of the multivariate normal distribution function, g(x) is an expectation function with a defined application. To simplify the calculation, we take g(x) = 1. Therefore, the integration region A is a hyperrectangle.

[0052] Where: [a,b],-∞≤a i ≤bi , i = 1, ... k.

[0053] The corresponding integral form of the multivariate normal distribution function is:

[0054]

[0055] Perform the Choliski decomposition on matrix Σ, ∑=CC t Where C is a lower triangular matrix, then x t ∑ -1 x = x t C -t C -1 Let x be the variable X, and let x = Cy be the expression. x t ∑ -1 x = y t y; Since C is a lower triangular structure, the function Φ n The integration region (a, b; ∑) becomes the following form:

[0056] a1 / c 11 ≤y1≤b1 / c 11 , (a2-c 21 y1) / x 22 ≤y2≤(b2-c 21 y1) / c 22 .

[0057]

[0058] make

[0059] function Φ n It can be written in the following form:

[0060]

[0061] The key to the univariate integral method for the multivariate normal distribution function lies in replacing y in equation (12). i The value of the one-dimensional truncation expectation E is defined as follows:

[0062]

[0063] In the formula, x represents the integral variable, and the symbols φ and Φ represent the probability density function and cumulative distribution function of the normal distribution, respectively.

[0064]

[0065]

[0066] make j = 1, ..., i-1 replace a′ i (y1,…,y i-1 ), b′ i (y1,…,y i-1 ) in y i The value is obtained. Finally, the approximate value for the univariate numerical integration problem of the multivariate normal distribution function is obtained as follows:

[0067]

[0068] The above formula yields the univariate numerical integral form of the multivariate normal distribution function. The derivation of the bivariate numerical integral form of the multivariate normal distribution function is given below.

[0069] Perform LDL on the covariance matrix ∑ t Decomposition: If n is even, the matrices L and D take the following forms:

[0070]

[0071] Where D i ,L i,j Let L be a 2x2 matrix and O2 be a 2x2 zero matrix. If n is odd, then matrix L will have an extra row, and the last column of matrix D will be d. nn ;

[0072] Divide the covariance matrix ∑ into blocks as follows:

[0073] Where ∑ 1.1 Let D1 be a second-order matrix, and ∑ 1.1 , R represents the matrix Σ and ∑ 1.1 Corresponding sub-blocks;

[0074] right Perform the same decomposition process;

[0075] Through the above decomposition process, L and D are obtained. By transforming x = Ly, we obtain dx = |L|dy, where a ≤ Ly ≤ b.

[0076] Define (α,β) j =(ag,bg) j in Let y 2k =(y 2k-1 ,y 2k ) t The bivariate numerical integral form of the multivariate normal distribution function is obtained as follows:

[0077]

[0078] make d ii This represents the element in the i-th row and i-th column of matrix D;

[0079] Therefore, the above formula can be written as:

[0080]

[0081] In the formula z i , i = 1, ..., n represent variables; Ω 12 ,…Ω 2k-1,2k Representing ρ1, ..., ρ k The extended matrix;

[0082]

[0083] ρ k This represents the k-th simplified variable corresponding to the D matrix;

[0084] Define the integral form of the outermost bivariate normal distribution (BVN) function as p1=Φ2((a′1,a′2),(b′1,b′2); Ω 12 To approximate the inner integral, we need to obtain the values ​​of z1 and z2. According to the univariate numerical integration algorithm using the multivariate normal distribution function, z1 and z2 are the expected values ​​of the variables in the truncated bivariate distribution.

[0085] In the formula, p1 represents the numerical integral of the bivariate normal distribution function; Φ2() represents the bivariate normal distribution function;

[0086] make In a i When the value is -∞, the expected values ​​u1 and u2 of the truncated bivariate normal distribution function are as follows:

[0087]

[0088] In the formula, ρ1 represents the first simplified variable corresponding to the D matrix; q1 represents the replacement variable of ρ1; u and v are both mean variables;

[0089] The u1 in the above formula is calculated by the following formula:

[0090]

[0091] The formula for calculating u2 is the same as that for calculating u1, except that a′1 and a′2 are interchanged and b′1 and b′2 are interchanged.

[0092] Through formula To approximate the integral of the second outer bivariate normal distribution function, where, For a′ i b′ iThe values ​​of z1 and z2 are replaced by u1 and u2, and then let... ρ2 represents the second simplified variable corresponding to matrix D; q2 represents the substitution variable for ρ2;

[0093] Therefore, we obtain

[0094]

[0095] And calculated by formula (20), the calculated u3, u4 values ​​replace the corresponding z3, z4 values, thus obtaining the calculation formula for step i.

[0096]

[0097] In the formula: For a′ i b′ i The values ​​of z1,…z 2i-2 Given the expected values ​​u1,…u 2i-2 replace;

[0098] After the k-th step of calculation, the formula for calculating the bivariate numerical integral of the multivariate normal distribution function is obtained:

[0099]

[0100] Preferably, step S5, which sorts the integration variables and selects a suitable numerical integration order, is as follows:

[0101] By changing the priority of variables, the variables are sorted so that the innermost integration variable has a smaller expected value; the outermost integration variable is selected by choosing variable i.

[0102]

[0103] Interchange the first and i-th variables in the first row and column of ∑, and then calculate the first column of C in the Cholesky decomposition of ∑. make

[0104]

[0105] The upper and lower limits of integration, the rows and columns of ∑, and c 12 With c i1 Perform the interchange, then calculate the second column of C in the ∑ Cholisky decomposition as follows:

[0106]

[0107] make

[0108] In step j, the integral variable for step j is determined by selecting variable i, as shown in the following equation:

[0109]

[0110] Interchange the upper and lower limits of integration, the rows and columns of ∑, and the i-th and j-th rows of C. Then calculate the j-th column of C.

[0111] make

[0112] In the formula Indicates a j Alternative variables; b j Alternative variables; μ j μ m These represent the expected values ​​calculated according to equation (15);

[0113] The entire variable sorting process ends when j = n;

[0114] Preferably, the reliability of the landslide system is calculated using the bivariate numerical integral expression of the covariance matrix and the sorted multivariate normal distribution function in step S6 as follows:

[0115] Let E be the event that occurs in the i-th failure mode. i An event that does not occur is represented as System failure events are represented by E, and system reliability events are represented by...

[0116]

[0117] The reliability probability of the system is calculated from equation (26).

[0118]

[0119] In the formula: P r , This represents the system's reliability probability.

[0120] The beneficial effects of this invention are:

[0121] 1. Strong applicability: This invention takes into account the correlation between failure modes. The correlation between failure modes has a certain impact on the reliability index of the system. If the correlation between failure modes is not considered, the reliability of the slope system will be underestimated.

[0122] 2. High accuracy: This invention directly solves high-dimensional numerical integrals, which is more accurate than the traditional interval estimation method to a certain extent. The calculation accuracy is further improved by sorting variables.

[0123] 3. High efficiency: The algorithm of this invention effectively avoids the complex high-dimensional numerical integration of the multivariate cumulative distribution function. It can be directly calculated after the basic parameters are given, which greatly improves the computational efficiency. Attached Figure Description

[0124] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0125] Figure 1 This is a flowchart illustrating an embodiment of the present invention.

[0126] Figure 2 This is a calculation example diagram of the present invention. Detailed Implementation

[0127] like Figure 2 The specific implementation of this invention will be illustrated using a classic example. This invention is a method for calculating the reliability of a landslide system based on a multivariate normal distribution function, and its flowchart is shown below. Figure 1 As shown.

[0128] The computational examples selected in this invention are as follows: Figure 2 As shown in Table 1, the soil parameters are as follows.

[0129] Table 1 Soil Parameters

[0130]

[0131]

[0132] For other dimensions, see Figure 2 .

[0133] The landslide system reliability calculation method in this embodiment includes the following steps:

[0134] S1: Analyze and determine the failure mode of the landslide system: assume the failure mode is an arbitrary sliding surface.

[0135] S2: The steps for selecting and determining the function for each failure mode of the landslide system are as follows:

[0136] The function function of the failure mode is Z = F⁻¹. The function function of the failure mode is calculated using the response surface methodology, and the response surface function is set as follows:

[0137]

[0138] By using data from different experimental points, simultaneous equations are solved to obtain the correction coefficients a and b. i C i .

[0139] The correction coefficients for the function of different sliding surfaces are shown in Table 2.

[0140] Table 2 Correction Coefficients for Functional Functions

[0141] coefficient Sliding surface 1 Sliding surface 2 Sliding surface 3 a -0.99248 -0.95641 -1.4408 <![CDATA[b1]]> 0.049366 0.004943 0.003185 <![CDATA[b2]]> 0 0.017859 0.009424 <![CDATA[b3]]> 0 0 0.015832 <![CDATA[C1]]> -0.0001 -1.2E-6 -2.4E-6 <![CDATA[C2]]> 0 2.97E-5 -4.5E-5 <![CDATA[C3]]> 0 0 -3.2E-5

[0142] S3: Calculate the correlation of each failure mode to obtain the covariance matrix. The specific process is as follows:

[0143] (1) Assume the initial verification point x * Generally, x is taken. * =μ x ;

[0144] (2) Equivalent normalization: Calculate using equations (3) and (4). and

[0145] (3) Calculate the standard deviation of different functional functions using equation (6).

[0146] (4) Calculate the different sliding surfaces using equation (7).

[0147] (5) Calculate the covariance between different sliding surfaces using equation (8). At this time, the correlation coefficient between different sliding surfaces is obtained.

[0148] (6) Obtain the covariance matrix ∑ according to step (4).

[0149] The calculations in steps (3) and (4) As shown in Table 3.

[0150] Table 3 Standard deviation, mean, sensitivity coefficient, and reliability of different sliding surfaces

[0151]

[0152]

[0153] The calculated covariance matrix Σ and correlation coefficients are as follows:

[0154]

[0155] Correlation coefficient: ρ 12 =0.5305, ρ 13 =0.3184, ρ 13 =0.3774.

[0156] S4: The numerical integral expression of the multivariate normal distribution function under the bivariate condition is derived.

[0157] S5: Sort the integration variables and select the appropriate numerical integration order. The calculation steps are as follows:

[0158] To improve computational accuracy and efficiency, it is necessary to sort the integration variables and minimize the external integral by replacing the variables.

[0159] First, perform the Choliski decomposition on the matrix ∑, then ∑ = C0. t To minimize the outer integral, the expected value of the outer integral must be minimized. The expected value E is calculated using equation (13); to simplify the calculation process, the outermost integration variable is selected by choosing variable i.

[0160] First, we choose the first-level integration variable, when 1 ≤ i ≤ n.

[0161]

[0162] Then, the integration limits corresponding to the i-th integration variable are interchanged with the integration limits corresponding to the first variable; the i-th row and the first column of ∑ are interchanged, and the i-th column and the first column are interchanged.

[0163] Then let

[0164] Obtain the first column of matrix C;

[0165] make Obtain the expected value of the outermost integral

[0166]

[0167] Choose the second level of integration variables, where 2≤i≤n.

[0168]

[0169] Swap the rows and columns of ∑ corresponding to the integration limits of the second and i-th variables; change the c in the second column... 12 and c i2 exchange.

[0170] make

[0171] We obtain the second column of matrix C; at this point...

[0172]

[0173] And so on, when j≤i≤n,

[0174]

[0175] After performing the interchange steps described above, the integration limit corresponding to the j-th integration variable is now...

[0176]

[0177]

[0178] expect

[0179] When j = n, the entire variable sorting process ends;

[0180] S6: The reliability of the landslide system is calculated based on the covariance matrix and the bivariate numerical integral expression of the multivariate normal distribution function as follows:

[0181] Let E be the event that occurs in the i-th failure mode. i An event that does not occur is represented as System failure events are represented by E, and system reliability events are represented by...

[0182]

[0183] P r This represents the system's reliability probability.

[0184] The scope of the calculation is as follows

[0185]

[0186] b = [+∞,+∞,+∞] T The calculated reliability probability of the system is P. r =0.3871.

[0187] Reliability probability P r The relationship between P and the system reliability β is P r =Φ(β), and the system reliability β = 0.6507 is obtained.

[0188] The above description only discloses specific embodiments of the present invention, but the specific protection scope of the present invention is not limited thereto. Any variations or modifications that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the protection scope of the present invention.

Claims

1. A method for calculating the reliability of a landslide system based on a multivariate normal distribution function, characterized in that, Includes the following steps: S1: Analyze and determine the failure modes of the landslide system; S2: Select the function to determine each failure mode of the landslide system; S3: Calculate the correlation of each failure mode to obtain the covariance matrix; S4: The numerical integral expression of the multivariate normal distribution function under the bivariate condition is derived; S5: Sort the integration variables and select an appropriate order for numerical integration; S6: The reliability of the landslide system is calculated using the system reliability calculation formula based on the bivariate numerical integral expression of the covariance matrix and the sorted multivariate normal distribution function. In step S2, the function of the failure mode of the landslide system is: ； (1) In the formula Z For the function of the sliding surface, F The safety factor for the sliding surface; The response surface methodology approximates the function as an explicit expression of a random variable. Based on this methodology, the response surface function can be defined as: ； (2) In the formula Indicates a function. , , , They represent the undetermined coefficients; , Representing different random variables; Using an incomplete quadratic polynomial that ignores cross-product terms , ；(3) Experimental points can be located along the coordinate axes at the mean point. Nearby selection, where along coordinates The experimental points of the axis have coordinates ,in f These are the fitting coefficients. Let the initial iteration point be: ； In the formula X Represents random variables The vector formed; Represents the initial iteration point of the random variable; The initial calculation is taken as the average value, and the coefficients of the linear equation system are calculated according to equation (3). , , ; In step S3, the correlation between each failure mode is calculated to obtain the covariance matrix. The formula for calculating the covariance between the function functions of the failure modes is as follows: For nonnormal variables, use equation (4) to perform equivalent normalization: ；(4) ； (5) In the formula Represents the mean and standard deviation of the equivalent normalized variable; This is a verification point; The cumulative distribution function is... It is the probability density function; It represents the inverse function of the cumulative distribution function; Represents the density function; At point Expand the function using a Taylor series and take the first term: ； (6) In the formula and Indicates a function; In random variables Space, Equations For passing point The tangent plane to the limit state surface at that point; Using the properties of linear combinations of independent normal random variables, mean and standard deviation They are respectively: ；(7) ；(8) Represents the mean of a variable; Indicates the standard deviation of the variable; Define variables Sensitivity coefficient as follows: ； (9) In the formula Point The angle between the point and the origin O in the standardized space; Equation (9) represents the expression for a standard normal random variable The equation of a hyperplane in space is expressed in normal form, where the normal is the line on the limit state surface. The direction cosine of the line connecting a point to the origin O in the standard normalized space is . , Represents the space in X The point corresponding to the limit state surface; The formula for calculating the covariance between different slip surfaces is as follows: ； (10) In the formula Indicates smooth surface j , k Covariance between functional functions; They represent the smooth surfaces. j , k The direction cosine of the function; , They represent the smooth surfaces. j , k The standard deviation of the function; The covariance calculation process between two different sliding surfaces is as follows: 1) Select the initial verification point It is acceptable ; 2) Equivalent normalization, using equations (4) and (5) to calculate and ; 3) Calculate the standard deviation of different functional functions using equation (8). ; 4) Calculate the values ​​of different sliding surfaces using equation (9). ; 5) Calculate different sliding surfaces using equation (10) j , k Covariance between them; 6) Obtain the covariance matrix according to step 5). .

2. The landslide system reliability calculation method according to claim 1, characterized in that, In step S1, the failure mode of the landslide system is any sliding surface that causes the slope to lose stability.

3. The landslide system reliability calculation method according to claim 2, characterized in that, In step S4, the derivation of the numerical integral form of the multivariate normal distribution function under the bivariate condition is as follows: ；(11) make (12) In the formula f ( ) represents the density function; Represents the mean of a variable; ， ； in for n 3D random variable, ; Representing variables respectively The mean and variance of For variables and Covariance between ; Consider the integral form of the general multivariate normal distribution function as follows: ； In the formula ; A Indicates the integration region; ， Let be the probability density function of the multivariate normal distribution. Let be the expected function, and to simplify the calculation, take . Therefore, the integration region A A hyperrectangle can be represented as , in , ; The integral form of the corresponding multivariate normal distribution function is as follows: ； (13) matrix Perform Jolisky decomposition. ,in C It is a lower triangular matrix; then ； variables Using relational expressions Replacement, therefore there is , ; because C It has a lower triangular structure, and the function... The integration region becomes the following form: ； ； ； make: , Then the function It can be represented in the following form: ； (14) The key to calculating the univariate numerical integral of the multivariate normal distribution function is the substitution in equation (14). The value; Define one-dimensional truncation expectation value : ； (15) In the formula t Represents the integral variable; Let represent the probability density function and cumulative distribution function of the normal distribution, respectively; ；(16) ； (17) make , replace , middle ,get , Finally, the approximate value for the univariate numerical integration problem of the multivariate normal distribution function is obtained as follows: ； (18) In the formula Represents the cumulative distribution function; Equation (18) yields the univariate numerical integral form of the multivariate normal distribution function. The derivation of the bivariate numerical integral form of the multivariate normal distribution function is given below. The covariance matrix conduct LDL T Decomposition, if n If it is even, then ； in Represents a lower triangular matrix; Represents a diagonal matrix; It is a second-order matrix. It is a second-order zero matrix; if n If the number is odd, then the matrix There is an extra row in the matrix. The last column is ; The covariance matrix Blocking: ， ； in It is a second-order matrix. , ; Represents the relationship between the Σ matrix and Corresponding sub-blocks; right Perform the same decomposition process; The matrix was obtained through the above decomposition process. By transformation ,get , ; ; definition ,in ,make The bivariate numerical integral form of the multivariate normal distribution function is obtained as follows: ； (19) make , express D Matrix number i Line number i Column elements; then ； (20) In the formula Represents variables; They represent The extended matrix; ， ， ； Indicates correspondence D The first of the matrix k A simplified variable; Define the integral form of the outermost bivariate normal distribution function as follows: ； In the formula Represents the numerical integral of a bivariate normal distribution function; Represents the bivariate normal distribution function; To approximate the inner integral, we need to obtain... The value, The expected value of the variable in a truncated bivariate distribution; make ,exist In the case of a truncated bivariate normal distribution function, the expected value is... as follows: ； (21) In the formula Indicates correspondence D The first simplified variable of the matrix; express Alternative variables; , Mean variable; In the above formula The result is obtained from equation (22): ；(22) With calculation Similarly, calculations yielded ; Through formula To approximate the integral of the second outer bivariate normal distribution function, where, , for , The value, Depend on replace; Again , Indicates correspondence D The second simplified variable of the matrix; express Alternative variables; Therefore, we obtain ； And it is calculated from equation (22), the calculated result is Replace the corresponding value The value; obtain the first Step calculation formula ， In the formula , for , The value, From expected value replace; After the first k After the first step of calculation, the formula for calculating the bivariate numerical integral of the integral function of the multivariate normal distribution function is obtained: ；(23)。 4. The landslide system reliability calculation method according to claim 3, characterized in that: In step S5, the steps for sorting the integration variables and selecting a suitable numerical integration order are as follows: By changing the priority of variables and ordering them, the innermost integration variable can have a smaller expected value; this can be achieved by selecting variables. To select the outermost integration variable; ； (24) Set the upper and lower limits of the integral. The first variable in the middle row and column and the first variable in the middle row and column Swap the variables, then calculate. In the decomposition of Jolisky C The first column is ， ， make , , , In the formula express Substitute variables; express Alternative variables; ； (25) The upper and lower limits of the integral and Bank and column and and Perform the interchange, then calculate. In the decomposition of Jolisky C The second column is , ,make , , ; And so on. Step, the first The integral variable of the step is selected by choosing the variable. To decide; ； (26) Set the upper and lower limits of the points, Bank and column and C The Middle i Line and number j Swap the rows, then calculate. C The j-th column is ； ， ， ； In the formula express Alternative variables; express Alternative variables; This represents the expected value calculated according to equation (15); when j = n The entire sorting process is now complete.

5. The landslide system reliability calculation method according to claim 4, characterized in that, In step S6, the system reliability probability The calculation is as follows: Let the first i The events that occur in each failure mode are represented as follows: An event that does not occur is represented as System failure events are represented as Reliable events in the system are represented as ；（27） The reliability probability of the system is calculated from equation (28). ；（28） In the formula This represents the system's reliability probability.

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