First order reliability method based on copula theory in geotechnical engineering

Abstract

The application discloses a first-order reliability method based on a Copula theory in geotechnical engineering, and comprises the following steps: analyzing a geotechnical structure to determine a function function Z of the geotechnical structure and related random variables and independent random variables of the function function Z; determining an edge probability distribution function of the related random variables and a Copula function under a calculation working condition according to engineering experience and measured data; determining an edge probability distribution function of the independent random variables under the calculation working condition according to the engineering experience and the measured data; and calculating a reliability index, a design value and a failure probability of the geotechnical structure according to the edge probability distribution function of the related random variables, the Copula function and the edge probability distribution function of the independent random variables. The application can directly implement the first-order reliability method based on the Copula theory in the original space of the random variables familiar to geotechnical practitioners, and does not involve conversion of additional random variables, so that the complexity of reliability calculation is greatly reduced.

Description

Technical Field

[0001] This invention belongs to the technical field of reliability analysis in geotechnical engineering, specifically relating to a first-order reliability method based on Copula theory in geotechnical engineering. Background Technology

[0002] Complex correlations often exist between soil and rock mass parameters in geotechnical engineering. For example, there is a positive correlation between horizontal wind loads and vertical superstructure loads, as well as a positive correlation between the maximum and residual inter-story drift ratios of multi-story buildings during earthquakes. Similarly, there are negative correlations between shear strength parameters cohesion and internal friction angle, and between the fitting parameters of the pile load-settlement curve and the soil-water characteristic curve. Establishing a reasonable probabilistic model of soil and rock mass parameters is crucial for reliability analysis and risk assessment in geotechnical engineering. Unfortunately, field test or experimental data in geotechnical engineering is often scarce, making it impossible to determine the complete binary probability distribution of soil and rock mass parameters, i.e., lacking complete probabilistic information. Therefore, geotechnical practitioners must conduct reliability analysis based on incomplete probabilistic information, relying only on the marginal probability distributions of soil and rock mass parameters and the correlation coefficients between the two parameters.

[0003] For a long time, the Nataf transform has been an effective tool for geotechnical professionals to model the joint probability distribution of soil and rock mass parameters using incomplete probabilistic information. However, multiple studies have pointed out that the Nataf transform assumes a Gaussian correlation structure between random variables, therefore, the reliability analysis results based on this transform are only one type of reliability analysis result obtained under incomplete probabilistic information. Recently, Copula theory has emerged as an effective method for simulating non-Gaussian correlation structures between random variables. From the perspective of Copula theory, the Nataf transform is equivalent to using a Gaussian Copula function to represent the correlation structure. Therefore, integrating Copula theory into reliability analysis is a significant advancement in the past application of the Nataf transform, especially in geotechnical engineering designs where the Nataf transform may lead to more dangerous situations.

[0004] Currently, many researchers combine Copula theory with Monte Carlo simulation to solve complex geotechnical engineering reliability analysis problems. These studies confirm that the choice of Copula function has a significant impact on geotechnical engineering reliability analysis. The essential reason for the differences in reliability analysis results among different Copula functions is the difference in the number of samples within the failure domain. Although this method is conceptually simple and easy to implement, it is less efficient for implicit function functions involved in numerical simulation and is more suitable for theoretical research. Therefore, approximate reliability analysis methods based on Taylor expansion of function functions, such as first-order reliability methods, are more suitable for practical engineering applications. First-order reliability methods typically obtain satisfactory accuracy reliability analysis results with lower computational costs. A few studies have combined Copula theory with first-order reliability methods and performed reliability analysis in independent standard normal spaces. However, this method requires geotechnical practitioners to use complex mathematical formulas to transform random variables from the original space into independent standard normal spaces for reliability analysis. This poses an invisible obstacle for geotechnical practitioners to apply first-order reliability methods based on Copula theory in geotechnical engineering. Therefore, there is an urgent need to develop a first-order reliability method based on Copula theory that is more suitable for geotechnical engineering, so that geotechnical practitioners can easily apply it even if they do not have professional knowledge of probability theory and statistics. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing technologies by providing a first-order reliability method based on Copula theory in geotechnical engineering. This method directly implements the first-order reliability method based on Copula theory in the original space of random variables familiar to geotechnical practitioners, without involving additional transformations of random variables, thus greatly reducing the complexity of reliability calculation. This allows geotechnical practitioners to easily apply the method even if they do not have professional knowledge of probability theory and statistics.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0007] A first-order reliability method based on Copula theory in geotechnical engineering includes the following steps:

[0008] The rock and soil structure is analyzed to determine the function Z of the rock and soil structure and the related random variables and independent random variables of the function Z;

[0009] Based on engineering experience and measured data, determine the marginal probability distribution function and Copula function of the relevant random variables under the calculation conditions;

[0010] Determine the marginal probability distribution function of the independent random variable under the calculation conditions based on engineering experience and measured data;

[0011] The reliability index, design value, and failure probability of geotechnical structures are calculated based on the marginal probability distribution functions and Copula functions of relevant random variables and the marginal probability distribution functions of independent random variables.

[0012] Furthermore, the method for analyzing the soil and rock structure and determining its function Z and related and independent random variables is as follows: First, a simplified analysis is performed on a specific soil and rock structure to determine its model. Then, a stability analysis is conducted on the model to determine the function Z. Finally, the soil and rock mass parameters that require consideration of uncertainty in the function are divided into related random variables x. C x C ={(x 11 ,x 12 (x) 21 ,x 22 )···(x i1 ,x i2 )···(x P1 ,x P2 )}, where x i1 and x i2 Let x be the i-th pair of related random variables, where i is the index of the related random variable, i = 1, 2, ..., P, and P be the total number of pairs of related random variables, and x be the independent random variable. I x I ={x1 x2···x j ···x Q}, where x j Let be the j-th independent random variable, where j is the index of the independent random variable, j = 1, 2, ..., Q, and Q is the total number of independent random variables.

[0013] Furthermore, the method for determining the function Z of the geotechnical structure is as follows: perform stability analysis on the model of the geotechnical structure, determine the resistance term R and load term Q on the potential sliding surface based on the self-weight, equivalent seismic force, hydrostatic pressure and anchoring force of the rock slope, and then determine the function Z based on the resistance term R and load term Q.

[0014] Furthermore, the resistance term R is the resultant force parallel to the potential sliding surface and upward, and its expression is:

[0015] R = cA + N′tanφ

[0016] In the formula, c is the cohesion of the rock, φ is the internal friction angle of the rock, A is the potential sliding surface area per unit width, and N′ is the normal pressure on the potential sliding surface.

[0017] Furthermore, the load term Q is the resultant force parallel to the potential sliding surface and downwards, and its expression is:

[0018] Q = W(sinψ) p +αcosψ p )+V w cosψ p -Tsinε

[0019] In the formula, W is the weight of the slider, and V w For the resultant force of water pressure on the crack, ψ p Let θ be the potential sliding surface inclination angle, T be the anchoring force, and ε be the angle between the anchoring force and the normal to the potential sliding surface.

[0020] Furthermore, based on the resistance term R and the load term Q, the expression for the function Z is determined as follows:

[0021]

[0022] Furthermore, the method for determining the marginal probability distribution function and Copula function of the relevant random variables is as follows: First, based on engineering experience and measured data, determine the marginal probability distribution function F of the i-th pair of relevant random variables under the calculation condition. i1 (x i1 ) and F i2 (x i2 );

[0023] Then, based on engineering experience and measured data, the Kendall rank correlation coefficient τ of the i-th pair of related random variables under the calculation condition is determined. i ;

[0024] Finally, based on engineering experience and measured data, the Copula function C for the i-th pair of related random variables under the computational condition was determined. i (w i1 ,w i2 ;θ i ), where w i1 For a standard homogenized random variable x i1 w i1 =F i1 (x i1 ), w i2 For a standard homogenized random variable x i2 w i2 =F i2 (x i2 ), θ i For Copula function C i (w i1 ,w i2 ;θ i The Copula parameter of ) can be determined from the Kendall rank correlation coefficient τ. i The inverse solution is used to obtain the result, and the calculation formula is as follows:

[0025]

[0026] Furthermore, the Copula function includes any one of the Gaussian Copula function, Plackett Copula function, Frank Copula function, and No. 16 Copula function.

[0027] Furthermore, the method for calculating the reliability index, design value, and failure probability of a geotechnical structure is as follows: First, based on the Copula function C... i (w i1 ,w i2 ;θ i )Calculation conditions Copula function h i (w i1 ,w i2 ;θ i Its expression is:

[0028]

[0029] Then, based on the conditional Copula function, the marginal probability distribution functions of the relevant random variables, and the marginal probability distribution functions of the independent random variables, constrained nonlinear programming is used to calculate the reliability index β of the geotechnical structure and the random variable x. i1 Design value x i1 * Random variable x i2 Design value x i2 * and random variable x j Design value x j * ;

[0030] Finally, the failure probability P of the geotechnical structure is calculated based on the reliability index β of the geotechnical structure. f Its expression is:

[0031] P f =Φ(-β)

[0032] In the formula, Φ(·) is the cumulative distribution function of the standard normal distribution.

[0033] Furthermore, the formula for calculating the reliability index β is:

[0034]

[0035] In the formula, Φ -1 (·) represents the inverse cumulative distribution function of the standard normal distribution, F i1 (x i1 * ) and F i2 (x i2* (x) is the design value i1 * and design value x i2 * Marginal probability distribution function value and F j (x j * (x) is the design value j * The marginal probability distribution function value at that location.

[0036] Compared with existing technologies, the advantages of this invention are as follows: In existing technologies, most methods apply Nataf transform for geotechnical engineering reliability analysis. The assumption of Gaussian correlation structure contained therein may lead to the reliability analysis results being biased towards danger. Therefore, reliability analysis based on Copula theory is an important advancement over the past application of Nataf transform. However, existing first-order reliability methods based on Copula theory are not practical because they require geotechnical practitioners to use complex mathematical formulas to transform random variables from the original space into an independent standard normal space. In view of the problems of existing technologies, this invention aims to develop a first-order reliability method based on Copula theory that is more suitable for geotechnical engineering. This method directly implements the first-order reliability method based on Copula theory in the original space of random variables familiar to geotechnical practitioners, without involving additional transformation of random variables, greatly reducing the complexity of reliability calculation, so that geotechnical practitioners can easily apply it even if they do not have professional knowledge of probability theory and statistics. At the same time, the calculation results of this invention directly give the design values ​​of geotechnical parameters, which can automatically reflect the degree of variability, correlation structure and marginal probability distribution of random variables in the reliability analysis under the current calculation conditions. Attached Figure Description

[0037] Figure 1 A flowchart of a first-order reliability method based on Copula theory in geotechnical engineering provided in this embodiment of the invention;

[0038] Figure 2 This is a simplified schematic diagram of the stability analysis model of the Sau Mau Ping rock slope in Hong Kong, as described in this embodiment of the invention. Detailed Implementation

[0039] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0040] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0041] The present invention will be further described below with reference to specific embodiments, but these are not intended to limit the scope of the invention.

[0042] This invention discloses a first-order reliability method based on Copula theory in geotechnical engineering, such as... Figure 1 As shown, it includes the following steps:

[0043] The rock and soil structure is analyzed to determine the function Z of the rock and soil structure and the related random variables and independent random variables of the function Z.

[0044] In this step, a simplified analysis of a certain rock and soil structure is first performed to determine the model of the rock and soil structure.

[0045] In this embodiment of the invention, a simplified analysis is performed on the rocky slope of Sau Mau Ping in Hong Kong. Figure 2 A simplified schematic diagram of the stability analysis model for the Sau Mau Ping rock slope in Hong Kong.

[0046] Then, a stability analysis is performed on the model of the soil and rock structure to determine the function Z of the soil and rock structure.

[0047] In this embodiment of the invention, the goal of stability analysis is to investigate whether the Sau Mau Ping rock slope in Hong Kong will slide along the potential sliding surface through stress analysis. The method for determining the function Z of the rock and soil structure is as follows: a stability analysis is performed on the model of the rock and soil structure, and the resistance term R and load term Q on the potential sliding surface are determined based on the self-weight, equivalent seismic force, hydrostatic pressure, and anchoring force acting on the rock slope. Then, the function Z is determined based on the resistance term R and the load term Q.

[0048] The resistance term R on the potential sliding surface of the Sau Mau Ping rock slope in Hong Kong is the resultant force (in kN / m) parallel to the potential sliding surface and pointing upwards. Its expression is:

[0049] R = cA + N′tanφ

[0050] In the formula, c is the cohesion of the rock (in kPa), φ is the internal friction angle of the rock (in °), and A = (Hz) / sinψ p Potential sliding surface area per unit width (in m²) 2 / m), N′=W(cosψ p -αsinψ p )-U w -V w sinψ p +Tcosε is the normal force on the potential sliding surface (in kN / m), W = 0.5γrock H 2 {[1-(z / H) 2 ]cotψ p -cotψ f} represents the weight of the slider (in kN / m), U w =0.5γ w rzA is the resultant force of water pressure on the potential sliding surface (in kN / m), V w =0.5γ w r 2 z 2 The resultant force of water pressure on the crack (in kN / m), r = z w / z is the water filling depth coefficient of the tensile fracture (unitless), and z is the area of ​​the tensile fracture per unit width (unit: m). 2 / m), z w The area of ​​the water-filled tension crack per unit width (in m²) 2 / m), α is the horizontal seismic acceleration coefficient (unitless), H=60m is the slope height, ψ f =50° is the overall slope inclination angle, ψ p =35° is the potential sliding surface inclination angle, γ rock =26kN / m 3 For the specific weight of the rock, γ w =10kN / m 3 Let T = 3000 kN / m be the specific weight of water, T = 3000 kN / m be the anchoring force, and ε = 55° be the angle between the anchoring force and the normal to the sliding surface.

[0051] The load term Q on the potential sliding surface of the Sau Mau Ping rock slope in Hong Kong is the resultant force (in kN / m) parallel to the potential sliding surface and pointing downwards. Its expression is:

[0052] Q = W(sinψ) p +αcosψ p )+V w cosψ p -T sinε.

[0053] Based on the resistance term R and the load term Q, the expression for the function Z of the rock slope in Sau Mau Ping, Hong Kong is as follows:

[0054]

[0055] Finally, the soil and rock parameters that need to be considered for uncertainty in the function are divided into relevant random variables x. C x C ={(x 11 ,x 12 (x) 21 ,x 22 )···(x i1,x i2 )···(x P1 ,x P2 )}, where x i1 and x i2 Let x be the i-th pair of related random variables, where i is the index of the related random variable, i = 1, 2, ..., P, and P be the total number of pairs of related random variables, and x be the independent random variable. I x I ={x1 x2···x j ···x Q}, where x j Let be the j-th independent random variable, where j is the index of the independent random variable, j = 1, 2, ..., Q, and Q is the total number of independent random variables.

[0056] In this embodiment of the invention, there are a total of 2 pairs of related random variables, that is, the total number of pairs of related random variables P = 2. The first pair of related random variables are cohesion c and internal friction angle φ, which are denoted as random variables x and x, respectively. 11 and random variable x 12 The second pair of relevant random variables are the area z of the tension crack per unit width and the water-filling depth coefficient r of the tension crack, denoted as random variables x and x, respectively. 21 and random variable x 22 There is a total of 1 independent random variable, that is, the total number of independent random variables Q = 1. The first independent random variable is the horizontal seismic acceleration coefficient α, denoted as random variable x1.

[0057] Based on engineering experience and measured data, determine the marginal probability distribution function and Copula function of the relevant random variables under the calculation conditions.

[0058] In this step, the marginal probability distribution function F of the i-th pair of related random variables under the calculation condition is first determined based on engineering experience and measured data. i1 (x i1 ) and F i2 (x i2 ).

[0059] In this embodiment of the invention, the edge probability distribution information of cohesion c, internal friction angle φ, tensile crack area per unit width z, and tensile crack water filling depth coefficient r is shown in Table 1. This information can be directly used to calculate the edge probability distribution function F. i1 (x i1 ) and F i2 (x i2 In the table, the symbol "-" indicates that the corresponding parameter does not need to be applied.

[0060] Table 1 Marginal probability distribution information of relevant random variables

[0061]

[0062] Then, based on engineering experience and measured data, the Kendall rank correlation coefficient τ of the i-th pair of related random variables under the calculation condition is determined. i .

[0063] In this embodiment of the invention, the Kendall rank correlation coefficient τ1 for cohesion c and internal friction angle φ is -0.5, and the Kendall rank correlation coefficient τ2 for tensile crack area z per unit width and tensile crack water filling depth coefficient r is -0.5.

[0064] Finally, based on engineering experience and measured data, the Copula function C for the i-th pair of related random variables under the computational condition was determined. i (w i1 ,w i2 ;θ i ), where w i1 For a standard homogenized random variable x i1 w i1 =F i1 (x i1 ), w i2 For a standard homogenized random variable x i2 w i2 =F i2 (x i2 ), and θ i For Copula function C i (w i1 ,w i2 ;θ i The Copula parameter of ) can be determined from the Kendall rank correlation coefficient τ. i The inverse solution is used to obtain the result, and the calculation formula is as follows:

[0065]

[0066] The Copula function includes any one of the Gaussian Copula, Plackett Copula, Frank Copula, and No. 16 Copula functions, and its specific expression is as follows:

[0067] The expression for the Gaussian Copula function is:

[0068] C i (w i1 ,w i2 ;θ i )=Φ2(Φ -1 (w i1 ),Φ -1 (w i2 );θ i )

[0069] In the formula, Φ2(·,·;θ i Φ is the cumulative distribution function of a two-dimensional standard normal distribution. -1 (·) is the inverse cumulative distribution function of the standard normal distribution;

[0070] The expression for the Plackett Copula function is:

[0071]

[0072] The expression for the Frank Copula function is:

[0073]

[0074] The expression for function No. 16 Copula is:

[0075]

[0076] In this embodiment of the invention, Table 2 summarizes eight different combinations of Copula functions to represent different calculation conditions for the reliability analysis of the Sau Mau Ping rock slope in Hong Kong, which is used to explore the impact of the selection of Copula functions on the reliability analysis results of the Sau Mau Ping rock slope in Hong Kong.

[0077] Table 2. Copula functions and parameters under 8 calculation conditions

[0078]

[0079]

[0080] The marginal probability distribution function of the independent random variable under the calculation conditions is determined based on engineering experience and measured data.

[0081] In this step, the method for determining the marginal probability distribution function of the independent random variable is as follows: Based on engineering experience and measured data, determine the marginal probability distribution function F of the j-th independent random variable under the calculation condition. j (x j ).

[0082] In this embodiment of the invention, the marginal probability distribution information of the horizontal seismic acceleration coefficient α is shown in Table 3. This information can be directly used to calculate the marginal probability distribution function F. j (x j In the table, the symbol "-" indicates that the corresponding parameter does not need to be applied.

[0083] Table 3 Marginal probability distribution information of independent random variables

[0084]

[0085] The reliability index, design value, and failure probability of geotechnical structures are calculated based on the marginal probability distribution functions and Copula functions of relevant random variables and the marginal probability distribution functions of independent random variables.

[0086] In this step, firstly, according to the Copula function C... i (w i1 ,w i2 ;θ i )Calculation conditions Copula function h i (w i1 ,w i2 ;θ i Its expression is:

[0087]

[0088] Then, based on the conditional Copula function, the marginal probability distribution functions of the relevant random variables, and the marginal probability distribution functions of the independent random variables, constrained nonlinear programming is used to calculate the reliability index β of the geotechnical structure and the random variable x. i1 Design value x i1 * Random variable x i2 Design value x i2 * and random variable x j Design value x j * The formula for calculating the reliability index β is:

[0089]

[0090] In the formula, Φ -1 (·) represents the inverse cumulative distribution function of the standard normal distribution, F i1 (x i1 * ) and F i2 (x i2 * (x) is the design value i1 * and design value x i2 * Marginal probability distribution function value and F j (x j * (x) is the design value j * The marginal probability distribution function value at that location.

[0091] Finally, the failure probability P of the geotechnical structure is calculated based on the reliability index β of the geotechnical structure. f Its expression is:

[0092] P f =Φ(-β)

[0093] In the formula, Φ(·) is the cumulative distribution function of the standard normal distribution.

[0094] In this embodiment of the invention, the proposed technology was used to calculate the reliability index, design value and failure probability of the Sau Mau Ping rock slope in Hong Kong under eight different calculation conditions. The calculation results are shown in Table 4.

[0095] Table 4 Reliability Indicators, Design Values, and Failure Probability of the Sau Mau Ping Rock Slope in Hong Kong

[0096]

[0097] The calculation results in Table 4 show that the choice of the Copula function representing the correlation structure among the relevant random variables has a significant impact on the reliability analysis results of the Sau Mau Ping rock slope in Hong Kong. Case 1 represents the conventional scheme for handling incomplete probability information in geotechnical engineering design, in which the Gaussian Copula function is used entirely for reliability analysis. Compared with other cases, the maximum ratio of failure probabilities is close to 103 times. In other words, using Nataf transform to handle incomplete probability information may seriously underestimate the failure probability of the geotechnical structure, thus making the geotechnical engineering design results somewhat risky. The proposed technique provides a more conservative option for geotechnical engineering design with incomplete probability information, and can be easily applied by geotechnical practitioners even without expertise in probability theory and statistics. Furthermore, the reliability analysis results verify the importance of rationally selecting the correlation structure among the relevant random variables for geotechnical engineering reliability analysis. The technique of this invention can easily adapt to various complex correlation structures, and the calculation results also directly provide design values ​​for geotechnical parameters. Therefore, the results of the embodiments of this invention verify the beneficial effects of this invention, indicating its ease of practical application and its innovativeness.

[0098] The above are merely preferred embodiments of the present invention and are not intended to limit the implementation methods and protection scope of the present invention. Those skilled in the art should recognize that any equivalent substitutions and obvious changes made based on the content of this specification should be included within the protection scope of the present invention.

Claims

1. A method for first order reliability in geotechnical engineering based on Copula theory, characterized by, Includes the following steps: Analyze the soil and rock structure to determine its function. Z and function Z Related random variables and independent random variables; Based on engineering experience and measured data, determine the marginal probability distribution function and Copula function of the relevant random variables under the calculation conditions; Determine the marginal probability distribution function of the independent random variable under the calculation conditions based on engineering experience and measured data; Calculate the reliability index, design value, and failure probability of geotechnical structures based on the marginal probability distribution functions and Copula functions of relevant random variables and the marginal probability distribution functions of independent random variables; Among them, the functional functions of rock and soil structures Z The determination method is as follows: a stability analysis is performed on the model of the rock and soil structure, and the resistance terms on the potential sliding surface are determined based on the self-weight, equivalent seismic force, hydrostatic pressure, and anchoring force of the rock slope. R and load items Q Then, based on the resistance item R and load items Q Determine the function Z ; Resistance item R The resultant force is parallel to the potential sliding surface and upwards, and its expression is: ； In the formula, c For the cohesion of rocks, The internal friction angle of the rock. A The potential sliding surface area per unit width and N ′ represents the normal force on the potential sliding surface; Load Item Q The resultant force is parallel to the potential sliding surface and downwards, and its expression is: ； In the formula, W For the weight of the slider, V w The resultant force of water pressure on the crack, ψ p For the potential sliding surface inclination angle, T For anchoring force and ε The angle between the anchoring force and the normal to the potential sliding surface; According to the resistance term R and load items Q Determine the function Z The expression is: 。 2. The first-order reliability method based on Copula theory in geotechnical engineering according to claim 1, characterized in that, Analyze the soil and rock structure to determine its function. Z and function Z The method for determining related and independent random variables is as follows: First, a simplified analysis of a certain soil and rock structure is performed to determine the model of the soil and rock structure. Then, a stability analysis is performed on the model of the soil and rock structure to determine the function of the soil and rock structure. Z Finally, the geotechnical parameters that need to be considered for uncertainty in the functional group are divided into relevant random variables. x C , x C = {( x 11 , x 12 ) ( x 21 , x 22 ) ··· ( x i1 , x i2 )··· ( x P1 , x P2 )},in x i1 and x i2 For the first i For relevant random variables, i For the relevant random variable, i = 1,2, ..., P , P The total logarithm of the correlated random variables and the logarithm of the independent random variables. x I , x I = { x 1 x 2 ··· x j ··· x Q },in x j For the first j 1 independent random variable j For the index of the independent random variable, j = 1, 2, ..., Q , Q The total number of independent random variables.

3. The first-order reliability method based on Copula theory in geotechnical engineering according to claim 1, characterized in that, The method for determining the marginal probability distribution function and Copula function of the relevant random variables is as follows: First, determine the marginal probability distribution function and Copula function under the calculation condition based on engineering experience and measured data. i Marginal probability distribution function of relevant random variables F i1 ( x i1 )and F i2 ( x i2 ); Then, based on engineering experience and measured data, the calculation condition is determined. i Kendall rank correlation coefficient for relevant random variables τ i ; Finally, based on engineering experience and measured data, the calculation condition was determined. i Copula function for relevant random variables C i ( w i1 , w i2 ; θ i ),in, w i1 For standard homogenized random variables x i1 , w i1 = F i1 ( x i1 ), w i2 For standard homogenized random variables x i2 , w i2 = F i2 ( x i2 ), θ i Copula function C i ( w i1 , w i2 ; θ i The Copula parameter of ) is based on the Kendall rank correlation coefficient. τ i The inverse solution is used to obtain the result, and the calculation formula is as follows: 。 4. The first-order reliability method based on Copula theory in geotechnical engineering according to claim 3, characterized in that, Copula functions include any one of the GaussianCopula, PlackettCopula, FrankCopula, and No.16Copula functions.

5. The first-order reliability method based on Copula theory in geotechnical engineering according to claim 1, characterized in that, The method for calculating the reliability index, design value, and failure probability of geotechnical structures is as follows: First, based on the Copula function... C i ( w i1 , w i2 ; θ i )Calculation conditions Copula function h i ( w i1 , w i2 ; θ i Its expression is: ； Then, based on the conditional Copula function, the marginal probability distribution functions of the relevant random variables, and the marginal probability distribution functions of the independent random variables, a constrained nonlinear programming approach is used to calculate the reliability index of the geotechnical structure. β ,random variable x i1 Design value x i1 * ,random variable x i2 Design value x i2 * and random variables x j Design value x j * ; Finally, based on the reliability indicators of the geotechnical structure β Calculate the failure probability of geotechnical structures P f Its expression is: ； In the formula, Φ (·) is the cumulative distribution function of the standard normal distribution.

6. The first-order reliability method based on Copula theory in geotechnical engineering according to claim 5, characterized in that, Reliability indicators β The calculation formula is: ； In the formula, Φ -1 (·) represents the inverse cumulative distribution function of the standard normal distribution. F i1 ( x i1 * )and F i2 ( x i2 * () is the design value x i1 * and design value x i2 * Marginal probability distribution function value; F j ( x j * () is the design value x j * The marginal probability distribution function value at that location.

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