A method for seismic vulnerability analysis of underground structures

Abstract

The application discloses a method for seismic vulnerability analysis of underground structures, comprising the following steps: obtaining an initial sample set of seismic demand of underground structures and an initial sample set of limit state threshold; obtaining a sample set of parameters for describing statistical uncertainty of seismic vulnerability of underground structures; obtaining an edge probability density function of the parameters for describing statistical uncertainty of seismic vulnerability; obtaining a joint probability density function of the parameters for describing statistical uncertainty of seismic vulnerability; obtaining a mean value of seismic failure probability and a standard deviation of seismic failure probability; and obtaining a failure probability interval of underground structures, and evaluating seismic performance of the underground structures. The statistical uncertainty of sample data of seismic demand and sample data of limit state threshold is quantified, and the seismic vulnerability of underground structures is expressed as an interval value with certain dispersion degree, so that evaluators can more reasonably consider the variability of seismic vulnerability of underground structures, and the seismic risk evaluation result of the underground structures is more accurate.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of seismic risk assessment of underground structures, and in particular to a seismic vulnerability analysis method for underground structures. BACKGROUND

[0002] At present, the seismic vulnerability assessment methods for underground structures mainly include: ① seismic vulnerability assessment based on experimental data; ② seismic vulnerability assessment based on historical earthquake damage data; and ③ seismic vulnerability assessment based on numerical simulation. The common feature of the above three methods is that the seismic vulnerability curve of the underground structure is constructed by using limited seismic demand samples (even small samples), which leads to statistical uncertainty of the seismic demand samples, so that the calculated values of the mean m D|IM and the logarithmic standard deviation β d of the seismic demand in the probabilistic seismic demand analysis are the estimated values of the true values.

[0003] Similarly, as an indispensable part of seismic vulnerability analysis, the probabilistic seismic capacity analysis of underground structures also has the problem of statistical uncertainty similar to that of the probabilistic seismic demand analysis. This leads to statistical uncertainty of the limit state threshold sample of the underground structure, so that the mean m C and the logarithmic standard deviation β c of the limit state threshold in the probabilistic seismic capacity analysis are the estimated values of the true values.

[0004] Considering that the calculation of the seismic vulnerability of the underground structure inevitably causes cognitive uncertainty of the vulnerability and the difficulty of improving the accuracy of the seismic vulnerability analysis of the underground structure by increasing the number of seismic demand samples and the number of limit state threshold samples, how to quantify the cognitive uncertainty of the seismic vulnerability of the underground structure caused by limited seismic demand sample data and limited limit state threshold sample data becomes a problem to be solved. SUMMARY

[0005] The present application provides a seismic vulnerability analysis method for underground structures to overcome the above technical problems.

[0006] To achieve the above purpose, the technical solution of the present application is:

[0007] A seismic vulnerability analysis method for underground structures, comprising the following steps:

[0008] S1: obtaining a seismic demand initial sample set and a limit state threshold initial sample set of the underground structure;

[0009] S2: Based on the initial sample set of earthquake demand and the initial sample set of limit state threshold, obtain a sample set of parameters describing the statistical uncertainty of seismic vulnerability of underground structures;

[0010] The parameters describing the statistical uncertainty of the seismic vulnerability of the underground structure include parameters describing the statistical uncertainty of seismic demand and parameters describing the statistical uncertainty of the limit state threshold.

[0011] The parameters describing the statistical uncertainty of earthquake demand include fitting parameters A and B, and the logarithmic standard deviation β of earthquake demand. d ;

[0012] The parameters describing the statistical uncertainty of the limit state threshold include the mean m of the limit state threshold. C The logarithmic standard deviation β of the limit state threshold C ;

[0013] S3: Based on the sample set of parameters describing the statistical uncertainty of seismic vulnerability, obtain the marginal probability density function of the parameters describing the statistical uncertainty of seismic vulnerability;

[0014] S4: Based on the marginal probability density function of the parameters describing the statistical uncertainty of seismic vulnerability of the underground structure, obtain the joint probability density function of the parameters describing the statistical uncertainty of seismic vulnerability;

[0015] S5: Based on the joint probability density function of the parameters describing the statistical uncertainty of seismic vulnerability, obtain the mean and standard deviation of the seismic failure probability.

[0016] S6: Based on the mean and standard deviation of the earthquake failure probability, obtain the failure probability range of the underground structure, thereby evaluating the seismic performance of the underground structure.

[0017] Furthermore, in S1, the initial sample set of earthquake demand is obtained as follows:

[0018] X={(im1,edp1),(im2,edp2),...,(im n edp n )} (1)

[0019] In the formula: im n For the nth ground motion intensity sample, edp n For the nth earthquake demand sample; (im n edp n Let X be the earthquake demand sample under the nth earthquake intensity sample; X is the initial earthquake demand sample set; and n is the number of samples in the earthquake demand sample set.

[0020] The limit state threshold initial sample set is as follows:

[0021] η i = (η i1 , η i2 ,..., η im )i = 1, 2, 3, 4 (2)

[0022] In the formula, η i represents a set of limit state thresholds under the i-th limit state; η im represents the m-th limit state threshold sample under the i-th limit state; m represents the total number of data in the limit state threshold set under the i-th limit state; i is the number of the limit state of the seismic damage level of the underground structure.

[0023] Further, in S2, the method for obtaining the sample set of parameters describing the statistical uncertainty of seismic vulnerability is as follows:

[0024] S21: Establish a seismic vulnerability model as follows:

[0025]

[0026] wherein,

[0027] ln(m D|IM ) = Aln(IM) + B (4)

[0028]

[0029]

[0030]

[0031] In the formula, P f (im) is the failure probability of the underground structure under the action of an earthquake, i.e., seismic vulnerability; IM is the intensity of ground motion; EDP is the seismic demand of the underground structure; m D|IM is the mean value of the seismic demand of the structure; β d is the logarithmic standard deviation of the seismic demand; edp k is the k-th seismic demand sample, k is the number of the seismic demand sample; C is the seismic capacity of the underground structure; m C is the mean value of the limit state threshold; β c is the logarithmic standard deviation of the limit state threshold; η h is the h-th limit state threshold sample;

[0032] S22: Sample the seismic demand initial sample set N B times to obtain N B seismic demand sub-sample data sets;

[0033] The seismic demand sub-sample data set is:

[0034]

[0035] wherein, is the seismic demand sub-sample data set of the jth sampling; j is the number of sampling times; is the seismic demand sub-sample data set of the jth sampling; j is the number of sampling times; is the seismic demand sample under the nth seismic intensity sample in the seismic demand sub-sample data set;

[0036] S23: Obtain the sample set of the parameters describing the statistical uncertainty of the seismic demand as follows:

[0037] Obtain the sample set of the fitting parameter A and the fitting parameter B as follows:

[0038] According to the formula (4) and the seismic demand sub-sample data set, obtain the sample set of the fitting parameter A and the fitting parameter B as follows:

[0039]

[0040]

[0041] In the formula: A * is the sample set of the fitting parameter A; is the sample set of the fitting parameter A; * is the N B th sample of the fitting parameter A in the sample set A * is the sample set of the fitting parameter B; is the N * th sample of the fitting parameter B in the sample set B B

[0042] Obtain the sample set of the seismic demand logarithmic standard deviation β d as follows:

[0043] According to the formula (5) and the seismic demand sub-sample data set, obtain the sample set of the seismic demand logarithmic standard deviation as follows:

[0044]

[0045] In the formula: β d * is the sample set of the seismic demand logarithmic standard deviation; is the N d th sample of the seismic demand logarithmic standard deviation in the sample set β * B

[0046] S24: Sample the initial sample set of the limit state threshold N B times to obtain N​​B a limit state threshold value sub-sample data set;

[0047] The limit state threshold value sub-sample data set is:

[0048]

[0049] wherein, is the limit state threshold value sub-sample data set of the jth sampling; is the mth data in

[0050] The sample set of parameters describing the statistical uncertainty of the limit state threshold value is obtained as follows in S25:

[0051] The sample set of the mean value m C of the limit state threshold value is obtained as follows:

[0052] According to formula (6) and the limit state threshold value sub-sample data set, we have:

[0053]

[0054] wherein: m c * is the sample set of the mean value m C of the limit state threshold value; is the sample of the mean value of the N c * th limit state threshold value in B

[0055] According to formula (7) and the limit state threshold value sub-sample data set, we have:

[0056]

[0057] wherein: β c * is the sample set of the logarithmic standard deviation β C of the limit state threshold value; is the sample of the logarithmic standard deviation of the N c * th limit state threshold value in B

[0058] Further, in S3, the marginal probability density function describing the parameters of the statistical uncertainty of the seismic vulnerability is obtained as follows:

[0059]

[0060] wherein: θ is a set of parameters describing the statistical uncertainty of the seismic vulnerability, θ = (A, B, β d , m​​​c ,β c );b t (t = 0, 1, 2, 3, 4) represents Lagrange multipliers; t represents the power of the parameter describing the statistical uncertainty of seismic vulnerability;

[0061] The fitting parameter A, the fitting parameter B, the logarithmic standard deviation of seismic demand β d , the mean value of limit state threshold m C , the logarithmic standard deviation of limit state threshold β C are substituted into formula (15) respectively, that is, the marginal probability density function f(A) of fitting parameter A, the marginal probability density function f(B) of fitting parameter B, the marginal probability density function f(β d ) of the logarithmic standard deviation of seismic demand β d , the marginal probability density function f(m C ) of the mean value of limit state threshold m c , and the marginal probability density function f(β C ) of the logarithmic standard deviation of limit state threshold β c can be obtained respectively.

[0062] Further, in the S4, the joint probability density function f J (θ) describing the statistical uncertainty of the parameters of seismic vulnerability is obtained as follows:

[0063]

[0064] In the formula, D1[·] is the density function of the Copula function of the correlation between the parameters A, B, β d describing the statistical uncertainty of seismic demand; D2[·] is the density function of the Copula function of the correlation between the parameters m c , β c describing the statistical uncertainty of limit state threshold; F(A), F(B), F(β d ), F(m c ), and F(β c ) are the marginal cumulative distribution functions of A, B, β d , m c , and β c respectively.

[0065] Further, the mean value of the seismic failure probability is calculated as follows:

[0066]

[0067] The standard deviation of the seismic failure probability is calculated as follows:

[0068]

[0069] Ω θ Ω f (im; θ) represents the seismic vulnerability expression.

[0070] Further, the underground structure failure probability interval is obtained as follows:

[0071]

[0072] α is a discrete degree coefficient considered; represents the lower limit value of the seismic failure probability interval of the underground structure; represents the upper limit value of the seismic failure probability interval of the underground structure.

[0073] Beneficial effects: the seismic vulnerability analysis method for the underground structure provided by the present application proposes a seismic vulnerability analysis method considering the statistical uncertainty of the seismic demand and the limit state threshold value of the underground structure. By quantifying the statistical uncertainty of the seismic demand sample data and the limit state threshold value sample data, and representing the seismic vulnerability of the underground structure as an interval value with a certain discrete degree, the evaluator can more reasonably consider the variability of the seismic vulnerability of the underground structure caused by the statistical uncertainty of the seismic demand and the limit state threshold value of the underground structure, and the seismic risk assessment result of the underground structure is more accurate. BRIEF DESCRIPTION OF DRAWINGS

[0074] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or the prior art description will be briefly introduced. Obviously, the drawings in the following description are some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0075] Figure 1 is the flow chart of the underground structure vulnerability analysis of the present application;

[0076] Figure 2a is the schematic diagram of the cross section of the large opening station structure in the embodiment of the present application;

[0077] Figure 2b is the schematic diagram of the middle column reinforcement of the large opening station in the embodiment of the present application;

[0078] Figure 3 is the ground motion response spectrum curve at the bedrock in the embodiment of the present application;

[0079] Figure 4 is the finite element model diagram of the embodiment of the present application;

[0080] Figure 5A schematic diagram of initial sample data of seismic demand and fitting results in an embodiment of the present application;

[0081] Figure 6a A schematic diagram of influence of mean value of failure probability on seismic vulnerability of a subway station in an embodiment of the present application;

[0082] Figure 6b A schematic diagram of influence of standard deviation of failure probability on seismic vulnerability of a subway station in an embodiment of the present application;

[0083] Figure 7 A schematic diagram of seismic vulnerability interval of a subway station considering statistical uncertainty in an embodiment of the present application. DETAILED DESCRIPTION

[0084] To make the objectives, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described below in a clear and complete manner with reference to the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0085] The present embodiment provides a seismic vulnerability analysis method for underground structures, as shown in Figure 1 , comprising the following steps:

[0086] S1: obtaining an initial sample set of seismic demand and an initial sample set of limit state threshold of the underground structure;

[0087] Specifically, the present embodiment considers the randomness of the geometric size, material strength and ground motion of the underground structure, and obtains the initial sample set of seismic demand of the underground structure under different seismic intensity by means of shaking table experiment, historical earthquake damage data investigation or numerical simulation, etc. as follows:

[0088] X={(im1,edp1),(im2,edp2),...,(im n ,edp n )} (1)

[0089] In the formula, im n is the n th seismic intensity sample, edp n is the n th seismic demand sample, (im n , edp n ) is the seismic demand sample under the n th seismic intensity sample, X is the initial sample set of seismic demand, and n is the number of samples in the seismic demand sample set.

[0090] Considering the randomness of the form, geometric size, material strength and other factors of the underground structure system, the limit state threshold of the underground structure is calibrated based on the methods such as seismic damage investigation, physical test, numerical simulation, qualitative analysis and expert judgment, and the initial sample set of the limit state threshold of the underground structure is obtained as follows:

[0091] η i =(η i1 ,η i2 ,...,η im )i=1,2,3,4 (2)

[0092] In the formula, η i represents the set of limit state thresholds under the i th limit state; η im represents the m th limit state threshold sample under the i th limit state; m represents the total number of data in the set of limit state thresholds under the i th limit state; i is the number of the limit state of the seismic damage level of the underground structure, wherein i = 1 represents the limit state of slight damage; i = 2 represents the limit state of moderate damage; i = 3 represents the limit state of severe damage; i = 4 represents the limit state of complete damage.

[0093] Specifically, in the embodiment, the definition methods of the four limit states of slight damage limit state, moderate damage limit state, severe damage limit state and complete damage limit state are existing technologies, and will not be described in detail here.

[0094] S2: obtaining a sample set of parameters for describing statistical uncertainty of seismic vulnerability of the underground structure;

[0095] The parameters for describing statistical uncertainty of seismic vulnerability of the underground structure include parameters for describing statistical uncertainty of seismic demand and parameters for describing statistical uncertainty of limit state threshold;

[0096] The parameters for describing statistical uncertainty of seismic demand include fitting parameters A and fitting parameters B and the logarithmic standard deviation β d of seismic demand;

[0097] The parameters for describing statistical uncertainty of limit state threshold include the mean m C of limit state threshold and the logarithmic standard deviation β C of limit state threshold;

[0098] Specifically, due to the finiteness of the initial sample set of seismic demand and the initial sample set of limit state threshold of the underground structure, the parameters A, B, β d , m c , β c in the traditional seismic vulnerability analysis will have certain variability, and therefore, these parameters need to be analyzed.

[0099] The method for obtaining the sample set of parameters describing the statistical uncertainty of seismic vulnerability in S2 is as follows:

[0100] S21: Establish a seismic vulnerability model as follows:

[0101]

[0102] wherein,

[0103] ln(m D|IM )=Aln(IM)+B (4)

[0104]

[0105]

[0106]

[0107] In the formula, P f (im) is the failure probability of the underground structure under the action of an earthquake, that is, the seismic vulnerability. IM is the ground motion intensity, and EDP is the seismic demand of the underground structure; m D|IM is the mean value of the seismic demand of the structure, β d is the logarithmic standard deviation of the seismic demand, which can be obtained by probabilistic seismic demand analysis; edp k is the kth seismic demand sample, k is the number of seismic demand samples; C is the seismic capacity of the underground structure, m C is the mean value of the limit state threshold, β c is the logarithmic standard deviation of the limit state threshold, η h is the hth limit state threshold sample;

[0108] wherein, the fitting parameter A is the coefficient of the first-order term in the logarithmic linear regression formula (4) in the probabilistic seismic demand analysis; wherein, the fitting parameter B is the constant term coefficient in the logarithmic linear regression formula (4) in the probabilistic seismic demand analysis;

[0109] S22: Perform N B times of Bootstrap sampling on the initial sample set of seismic demands to obtain N B Bootstrap sampling seismic demand sub-sample data sets with the same sample capacity as the original sample capacity;

[0110] The Bootstrap sampling seismic demand sub-sample data set is:

[0111]

[0112] wherein, Bootstrap sampling seismic demand sub-sample dataset for the jth sampling; j is the number of sampling times; Bootstrap sampling seismic demand sub-sample dataset for the jth sampling; j is the number of sampling times; Bootstrap sampling seismic demand sub-sample dataset for the jth sampling; j is the number of sampling times; B Bootstrap sampling seismic demand sub-sample dataset for the jth sampling; j is the number of sampling times; B Bootstrap sampling seismic demand sub-sample dataset for the jth sampling; j is the number of sampling times; 5 .

[0113] S23: Obtain a sample set of parameters describing the statistical uncertainty of the seismic demand of the underground structure as follows:

[0114] Obtain a sample set of fitting parameters A and fitting parameters B as follows:

[0115] According to formula (4) and Bootstrap sampling seismic demand sub-sample dataset, the probability seismic demand analysis is performed on each Bootstrap sampling seismic demand sub-sample dataset, and a sample set of fitting parameters A and fitting parameters B is obtained as follows:

[0116]

[0117]

[0118] In the formula: A * is a sample set of fitting parameters A; is the N B th fitting parameter A in A * ; B * is a sample set of fitting parameters B; is the N B th fitting parameter B in B * ;

[0119] Obtain a sample set of seismic demand log standard deviation β d as follows:

[0120] According to formula (5) and Bootstrap sampling seismic demand sub-sample dataset, the probability seismic demand analysis is performed on each Bootstrap sampling seismic demand sub-sample dataset, and a sample set of seismic demand log standard deviation is obtained as follows:

[0121]

[0122] In the formula: β d * is a sample set of seismic demand log standard deviation; is the N B th seismic demand log standard deviation in β d * ;

[0123] S24: N times Bootstrap sampling is performed on the initial limit state threshold sample set to obtain N Bootstrap sampling limit state threshold subsample data sets with the same sample size as the original sample set; B B S25: A sample set of parameters describing the statistical uncertainty of the limit state threshold of the underground structure is obtained as follows:

[0124] The Bootstrap sampling limit state threshold subsample data set is:

[0125]

[0126] wherein, is the Bootstrap sampling limit state threshold subsample data set for the jth sampling; is the mth data in

[0127] S25: A sample set of parameters describing the statistical uncertainty of the limit state threshold of the underground structure is obtained as follows:

[0128] A sample set of the mean value m C of the limit state threshold is obtained as follows:

[0129] According to formula (6) and the Bootstrap sampling limit state threshold subsample data set, we have:

[0130]

[0131] wherein: m c * is the sample set of the mean value m C of the limit state threshold; is the N c * th sample of the mean value of the limit state threshold in m B

[0132] According to formula (7) and the Bootstrap sampling limit state threshold subsample data set, we have:

[0133]

[0134] wherein: β c * is the sample set of the logarithmic standard deviation β C of the limit state threshold; is the N c * th sample of the logarithmic standard deviation of the limit state threshold in β B

[0135] ​​​​S3: obtaining a sample set of parameters describing statistical uncertainty of seismic vulnerability of the underground structure according to the description of the underground structure; obtaining an edge probability density function of parameters describing statistical uncertainty of seismic vulnerability of the underground structure;

[0136] Specifically, the edge probability density functions of the parameters describing statistical uncertainty of seismic vulnerability of the underground structure are obtained as follows: calculating the first four order origin moments of θ=(A, B, β d ,m c ,β c ), wherein θ is a set of parameters describing statistical uncertainty of seismic vulnerability, and fitting the edge probability density functions of the parameters describing statistical uncertainty of seismic vulnerability according to the maximum entropy principle, which are in the following form:

[0137]

[0138] In the formula, θ is a set of parameters describing statistical uncertainty of seismic vulnerability; b t (t=0, 1, 2, 3, 4) represents a Lagrange multiplier, which is a to-be-determined parameter, and the optimal solution of the b t value of each statistical uncertainty parameter is obtained by using a nonlinear programming algorithm in combination with the first four order origin moments of the set of statistical uncertainty parameters θ; t represents the power of the parameter describing statistical uncertainty of seismic vulnerability.

[0139] Based on formula (15), the fitting parameter A, the fitting parameter B, the logarithmic standard deviation β d of seismic demand, the mean value m C of the limit state threshold, and the logarithmic standard deviation β C of the limit state threshold are substituted into formula (15) respectively, that is, the edge probability density function f(A) of the fitting parameter A, the edge probability density function f(B) of the fitting parameter B, the edge probability density function f(β d ) of the logarithmic standard deviation β d of seismic demand, the edge probability density function f(m C ) of the mean value m c of the limit state threshold, and the edge probability density function f(β C ) of the logarithmic standard deviation β c of the limit state threshold can be obtained respectively.

[0140] S4: obtaining a joint probability density function of parameters describing statistical uncertainty of seismic vulnerability according to the edge probability density functions of the parameters describing statistical uncertainty of seismic vulnerability of the underground structure;

[0141] Specifically, the embodiment further considers the correlation between the parameters describing statistical uncertainty of seismic vulnerability, that is, the parameters A, B, βd the correlation between the parameters m c , β c , and the correlation between the parameters and the parameters m J . The Copula function with the optimal correlation between the parameters describing the statistical uncertainty of the seismic demand is selected based on the AIC criterion (the minimum information criterion), and the joint probability density function f J (θ) of the parameters describing the statistical uncertainty of the seismic demand is further constructed based on the Copula theory. Therefore, the joint probability density function f d (θ) of the parameters describing the statistical uncertainty of the seismic demand based on the Copula theory can be expressed as:

[0142]

[0143] where D1[·] is the density function of the Copula function describing the correlation between the parameters A, B, β c , and D2[·] is the density function of the Copula function describing the correlation between the parameters m c , β d ; F(A), F(B), F(β c ), F(m c ), and F(β d ) are the marginal cumulative distribution functions of A, B, β c , m c , and β

[0144] S5: According to the joint probability density function of the parameters describing the statistical uncertainty of the seismic demand, the mean of the seismic failure probability and the standard deviation of the seismic failure probability are obtained to quantify the cognitive uncertainty of the seismic vulnerability of the underground structure. Specifically, the cognitive uncertainty in the embodiment refers to the uncertainty caused by insufficient data;

[0145] The mean of the seismic failure probability is calculated as follows:

[0146]

[0147] The standard deviation of the seismic failure probability is calculated as follows:

[0148]

[0149] where Ω θ is the definition domain of θ; P f (im; θ) represents the probability that the underground structure fails under the earthquake action with the intensity im when the parameters are θ = (A, B, βd ,m c ,β c ) as the seismic fragility expression after considering the statistical uncertainty parameters, expressed as

[0150] S6: obtaining the underground structure failure probability interval according to the mean value of the seismic failure probability and the standard deviation of the seismic failure probability, so as to evaluate the seismic performance of the underground structure.

[0151] The failure probability of the underground structure is expressed as an interval value with a certain degree of dispersion (as shown in formula (19)) to replace the single fragility curve in the traditional fragility analysis. On this basis, the failure probability interval of the underground structure under different seismic intensity (such as frequent earthquake, basic earthquake, rare earthquake, and extremely rare earthquake) is calculated, so as to evaluate the seismic performance of the underground structure under different seismic intensity.

[0152]

[0153] In the formula, α is the dispersion coefficient considered; represents the lower limit value of the seismic failure probability interval of the underground structure; represents the upper limit value of the seismic failure probability interval of the underground structure.

[0154] Case description

[0155] The present application is described in detail in combination with the following specific embodiments. The embodiments are implemented on the premise of the technical method of the present application, and detailed implementation modes and specific operation processes are given, but the protection scope of the present application is not limited to the following embodiments.

[0156] Taking the Daikai subway station in Japan as an example, the structure has a buried depth of 4.8 m, a width of 17 m, a height of 7.17 m, a middle column spacing of 3.5 m, a middle column section of 0.4 m x 1 m, and a reinforcement ratio of 6.0%. The structure cross section and middle column reinforcement diagram are shown in FIGS. 1 and 2. Figure 2a and Figure 2b The site where the station is located is mainly composed of Holocene sand, Pleistocene clay, and Pleistocene sand. The equivalent shear wave velocity of the site is 191.9 m / s, and the soil physical parameters are shown in Table 1.

[0157] Table 1 Soil physical properties

[0158]

[0159] Considering the influence of the essential uncertainty of the strength parameters of the subway station structure on the seismic response and seismic capacity of the structure, the density ρ of the structure, the peak compressive strength σ cp,core of the core area concrete, the peak compressive strain ε cp,core of the core area concrete, and the ultimate compressive strength σcu,core , ultimate compressive strain of core concrete ε cu,core , peak compressive strength of protective concrete σ cp,cover , peak compressive strain of protective concrete ε cp,cover , ultimate compressive strain of protective concrete ε cu,cover , tensile strength of concrete E t , yield strength of steel f y , initial stiffness of steel E s as random variables. The related statistical characteristics and probability distributions are shown in Table 2 and Table 3.

[0160] Table 2 Probability distributions of random variables

[0161]

[0162] Table 3 Correlation coefficient matrix of random variables

[0163]

[0164]

[0165] According to the equivalent shear wave velocity of the site and considering the site conditions of the large open subway station and the actual suffered ground motion, 100 seismic waves are selected from the PEER library, with the magnitude range of 5.5-8.0, the epicentral distance range of 10-30 km, V s30 range of 180-360 m / s, and PGA range of 0.05g-0.9g. Further, the selected ground surface seismic waves are inverted by the EERA program (Equivalent-Linear Earthquake Site Response Analyses program) to obtain the seismic waves at the bedrock, and the acceleration response spectrum of the inverted seismic waves is shown in Figure 3

[0166] To further reflect the influence of the statistical uncertainty of the seismic demand of underground structures, 100 sets of subway station-soil interaction finite element models are established based on the Latin hypercube sampling technique and the OpenSees open source program platform according to Table 2 and Table 3, and are randomly matched with 100 sets of seismic waves, and nonlinear dynamic time history analysis is performed to obtain 100 sets of seismic demand data of the subway station structure. The finite element model is shown in Figure 4 , and the initial sample data and fitting results with the inter-story drift angle as the index are shown in Figure 5

[0167] ​​To further reflect the influence of statistical uncertainty of the limit state threshold of underground structure, 20 groups are selected from the above 100 groups of models to carry out random underground structure pushover analysis. On the basis of the obtained pushover curve, the limit state threshold is calibrated by using a four-fold line model of bending moment-rotation angle considering P-Δ effect, so as to obtain 20 groups of limit state threshold samples, as shown in Table 4.

[0168] Table 4 Limit state threshold samples of subway station

[0169]

[0170]

[0171] The above obtained 100 groups of seismic demand data and 20 groups of limit state threshold data are taken as original samples, and the statistical uncertainty of seismic demand and limit state threshold of underground structure is quantified according to the process described in the application, so as to obtain the marginal probability distribution of the statistical uncertainty parameters. Further, the AIC values of Gaussian Copula function, t Copula function, Gumbel Copula function, Clayton Copula function and Frank Copula function are calculated and compared, and it is confirmed that the Gaussian Copula function is the optimal function for describing the correlation of the statistical uncertainty parameters, and thus the joint probability density function f J (θ) of the statistical uncertainty parameters is finally obtained. On the basis of the above, the mean value and standard deviation of the seismic vulnerability of underground structure considering statistical uncertainty are calculated. Figure 6a It can be seen from the above that there is a certain difference between the seismic failure probability of subway station based on limited seismic demand and limited limit state threshold data and the failure probability considering statistical uncertainty. In the state of slight damage, severe damage and complete damage, the mean value of failure probability is less than the failure probability based on limited seismic demand and limited limit state threshold data, and the difference between the two is most obvious in the state of severe damage, indicating that ignoring statistical uncertainty will overestimate the seismic vulnerability of subway station in the corresponding damage state; in the state of moderate damage, the mean value of failure probability is slightly greater than the failure probability based on limited seismic demand and limit state threshold data, indicating that ignoring statistical uncertainty will underestimate the seismic vulnerability of subway station. It can be seen from the above that with the increase of ground motion intensity, the standard deviation of the failure probability of subway station presents a trend of first increasing and then decreasing, indicating that under the action of ground motion of different intensity, the variability of the failure probability of subway station is different. The above results further show that when evaluating the seismic vulnerability of underground structure, the vulnerability cognitive uncertainty caused by the statistical uncertainty of seismic demand and limit state threshold cannot be ignored. Figure 6b

[0172] ​In view of the difficulty of accurately solving the probabilistic seismic demand model and the probabilistic seismic capacity model of the underground structure, the case further takes the discrete degree coefficient α=0.5 to calculate the seismic vulnerability interval value of the subway station in the range of one deviation around the mean value of the failure probability, such as Figure 7 As shown in the figure, the evaluator can quantify the statistical uncertainty of the seismic demand and the threshold of the limit state of the subway station, and reasonably consider the variability of the seismic vulnerability of the subway station structure through the vulnerability interval determined by the upper and lower bounds of the failure probability of the subway station, so as to replace the single vulnerability curve in the traditional vulnerability analysis, so as to make effective evaluation on the seismic performance of the subway station.

[0173] The seismic vulnerability analysis method for the underground structure of the present application proposes a seismic vulnerability analysis method considering the statistical uncertainty of the seismic demand and the threshold of the limit state of the underground structure. By quantifying the statistical uncertainty of the seismic demand sample data and the threshold of the limit state sample data, and expressing the seismic vulnerability of the underground structure as an interval value with a certain degree of dispersion, the single vulnerability curve in the traditional vulnerability analysis is replaced, so that the evaluator can more reasonably consider the variability of the seismic vulnerability of the underground structure caused by the statistical uncertainty of the seismic demand and the threshold of the limit state, and the seismic risk assessment result of the underground structure is more accurate.

[0174] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application, but not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement to part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A method for seismic vulnerability analysis of underground structures, characterized in that, Includes the following steps: S1: Obtain the initial sample set of seismic demand and the initial sample set of limit state threshold for underground structures; S2: Based on the initial sample set of earthquake demand and the initial sample set of limit state threshold, obtain a sample set of parameters describing the statistical uncertainty of seismic vulnerability of underground structures; The parameters describing the statistical uncertainty of the seismic vulnerability of the underground structure include parameters describing the statistical uncertainty of seismic demand and parameters describing the statistical uncertainty of the limit state threshold. The parameters describing the statistical uncertainty of earthquake demand include fitting parameters A and B, and the logarithmic standard deviation β of earthquake demand. d ; The parameters describing the statistical uncertainty of the limit state threshold include the mean m of the limit state threshold. C The logarithmic standard deviation β of the limit state threshold C ; S3: Based on the sample set of parameters describing the statistical uncertainty of seismic vulnerability, obtain the marginal probability density function of the parameters describing the statistical uncertainty of seismic vulnerability; S4: Based on the marginal probability density function of the parameters describing the statistical uncertainty of seismic vulnerability of the underground structure, obtain the joint probability density function of the parameters describing the statistical uncertainty of seismic vulnerability; S5: Based on the joint probability density function of the parameters describing the statistical uncertainty of seismic vulnerability, obtain the mean and standard deviation of the seismic failure probability. S6: Based on the mean and standard deviation of the earthquake failure probability, obtain the failure probability range of the underground structure, thereby evaluating the seismic performance of the underground structure.

2. The seismic vulnerability analysis method for underground structures according to claim 1, characterized in that, In step S1, the initial sample set of earthquake demand is obtained as follows: X={(im1,edp1),(im2,edp2),...,(im n ,edp n )} (1) In the formula: im n For the nth ground motion intensity sample, edp n For the nth earthquake demand sample; (im n edp n ) represents the earthquake demand sample under the nth earthquake intensity sample; X represents the initial earthquake demand sample set; n represents the number of samples in the earthquake demand sample set; The initial sample set for the limit state threshold is as follows: or i =(the i1 ,or i2 ,...,or im )i=1,2,3,4 (2) In the formula: η i This represents the set of limit state thresholds under the i-th limit state; η im This represents the threshold sample of the m-th limit state under the i-th limit state. m represents the total number of data in the limit state threshold set under the i-th limit state; i is the limit state number of the seismic damage level of the underground structure.

3. The seismic vulnerability analysis method for underground structures according to claim 2, characterized in that, In step S2, the method for obtaining the sample set of parameters describing the statistical uncertainty of seismic vulnerability is as follows: S21: The seismic vulnerability model is established as follows: in, ln(m D|IM )=Aln(IM)+B (4) In the formula, P f (im) represents the failure probability of the underground structure under seismic loading, i.e., seismic vulnerability; IM represents the seismic intensity; EDP represents the seismic demand of the underground structure; m D|IM β is the mean of structural seismic demand. d The standard deviation of earthquake demand; edp k Let k be the k-th earthquake demand sample, where k is the sample number; C represents the seismic resistance capacity of the underground structure, and m C β is the mean of the limit state threshold. c η is the logarithmic standard deviation of the limit state threshold. h This is the h-th limit state threshold sample; S22: Perform N operations on the initial sample set of earthquake demand. B The second sampling yields N. B A dataset of earthquake demand subsamples; The earthquake demand subsample dataset is as follows: in, This represents the earthquake demand subsample dataset from the j-th sampling; j is the sampling number. for Earthquake demand sample under the nth earthquake intensity sample; S23: The sample set of parameters describing the statistical uncertainty of earthquake demand is obtained as follows: The sample sets for fitting parameters A and B are obtained as follows; Based on formula (4) and the earthquake demand subsample dataset, the sample sets for fitting parameters A and B are as follows: In the formula: A * The sample set for fitting parameter A; For A * The Nth B A sample of fitted parameter A; B * The sample set for fitting parameter B; For B * The Nth B A sample of fitted parameter B; Obtain the logarithm standard deviation β of earthquake demand d The sample set is as follows: Based on formula (5) and the earthquake demand subsample dataset, the sample set of the log standard deviation of earthquake demand is as follows: Where: β d * The sample set is the logarithmic standard deviation of earthquake demand; For β d * The Nth B A sample of the logarithmic standard deviation of earthquake demand; S24: Perform N operations on the initial sample set of the limit state threshold. B The second sampling yields N. B A subset of extreme state threshold datasets; The threshold subsample dataset for the extreme state is: in, The limit state threshold subsample dataset is the sampled data of the j-th sampling. for The m-th data in; S25: The following is a sample set of parameters describing the statistical uncertainty of the limit state threshold; Obtain the mean m of the limit state threshold C The sample set is as follows: Based on formula (6) and the limit state threshold subsample dataset, we get: Where: m c * The mean m of the limit state threshold C The sample set; For m c * The Nth B A sample of the mean values ​​of each limit state threshold; Based on formula (7) and the limit state threshold subsample dataset, we get: Where: β c * β is the logarithmic standard deviation of the limit state threshold. C The sample set; For β c * The Nth B A sample of the logarithmic standard deviation of the limit state thresholds.

4. The seismic vulnerability analysis method for underground structures according to claim 3, characterized in that, In step S3, the marginal probability density function describing the statistical uncertainty of seismic vulnerability is obtained as follows: In the formula: θ is the set of parameters describing the statistical uncertainty of seismic vulnerability, θ=(A,B,β) d ,m c ,β c );b t (t=0,1,2,3,4) represents the Lagrange multiplier; t represents the power of the parameter describing the statistical uncertainty of seismic vulnerability; The fitting parameters A, B, and the logarithmic standard deviation β of the seismic demand are used. d The mean m of the limit state threshold C The logarithmic standard deviation β of the limit state threshold C Substituting these values ​​into formula (15), we can obtain the marginal probability density function f(A) of the fitting parameter A, the marginal probability density function f(B) of the fitting parameter B, and the logarithmic standard deviation β of the seismic demand, respectively. d Marginal probability density function f(β) d The mean m of the limit state threshold C Marginal probability density function f(m) c ), the logarithmic standard deviation β of the limit state threshold C Marginal probability density function f(β) c ).

5. The seismic vulnerability analysis method for underground structures according to claim 4, characterized in that, In S4, the joint probability density function f of the parameters describing the statistical uncertainty of seismic vulnerability is... J (θ) is obtained as follows: In the formula, D1[·] represents the parameters A, B, and β that describe the statistical uncertainty of earthquake demand. d The density function of the Copula function of the correlation between them; D2[·] is the parameter m describing the statistical uncertainty of the limit state threshold. c β c The density functions of the Copula function relating the two factors; F(A), F(B), F(β) d ), F(m c ) and F(β) c ) are A, B, and β respectively. d m c and β c The marginal cumulative distribution function.

6. The seismic vulnerability analysis method for underground structures according to claim 5, characterized in that, The mean value of the earthquake failure probability is calculated as follows: The standard deviation of the earthquake failure probability is calculated as follows: In the formula, Ω θ P is the domain of θ; f (im;θ) represents the expression for seismic vulnerability.

7. The seismic vulnerability analysis method for underground structures according to claim 6, characterized in that, The failure probability range of the underground structure is obtained as follows: In the formula, α is the coefficient of dispersion to be considered; This represents the lower limit of the probability interval for seismic failure of underground structures. This represents the upper limit of the range of probability intervals for seismic failure of underground structures.

Images