A method for calculating seismic vulnerability of underground structures based on multidimensional performance indicators

Abstract

The present invention discloses a method for calculating the seismic vulnerability of underground structures based on multi-dimensional performance indicators, including: establishing a finite element model of the underground structure of a single-layer double-span subway station; obtaining a kernel density function model; establishing a joint distribution function of the inter-layer displacement angle and the cumulative hysteresis energy dissipation about the Copula function; establishing a seismic vulnerability model about the inter-layer displacement angle and the cumulative hysteresis energy dissipation based on the Copula function, so as to obtain the failure probability of the underground structure, and then evaluate the seismic vulnerability of the underground structure. The present invention can fully consider the cumulative destructive effect of the underground structure on the structure under the action of the peak acceleration of the seismic motion, and on the basis of the dual-parameter damage model of deformation and energy, fully consider the influence of the two damage indicators of the inter-layer displacement angle and the cumulative hysteresis energy dissipation on the seismic vulnerability, and has a strong engineering application value and significance.

Description

Technical Field

[0001] The present invention relates to the technical field of underground structure damage assessment, and in particular to a method for calculating the seismic vulnerability of underground structures based on multi-dimensional performance indicators. Background Art

[0002] Most existing studies on the seismic vulnerability of underground structures use a single seismic demand parameter to characterize the failure probability of subway stations under seismic motion. However, due to the complexity and randomness of seismic motion, the structural seismic demand is not only affected by the seismic amplitude characteristics, but also by the spectrum characteristics and duration characteristics. The use of a single seismic parameter often results in a large discreteness in the structural seismic demand analysis, which in turn leads to inaccurate and incomplete calculated vulnerability. Under earthquake action, the response of the structure is essentially a problem of composite random vibration, that is, a problem of a structure with random parameters being subjected to random excitation, so it is difficult to effectively evaluate the seismic performance of the structure. Summary of the invention

[0003] The present invention provides a method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators to overcome the above technical problems.

[0004] In order to achieve the above object, the technical solution of the present invention is:

[0005] A method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators comprises the following steps:

[0006] S1: Establish an underground structure finite element model of a single-story, double-span subway station to obtain a sample set of seismic demand;

[0007] S2: obtaining a kernel density function model according to the earthquake demand sample set;

[0008] S3: According to the kernel density function model, a kernel density function is established regarding the inter-story displacement angle and the cumulative hysteresis energy dissipation to obtain the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy dissipation;

[0009] S4: establishing a joint distribution function of the inter-story displacement angle and the cumulative hysteresis energy dissipation with respect to the Copula function according to the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy dissipation;

[0010] S5: Based on the seismic demand sample set, a failure probability model for inter-story displacement angle and a failure probability model for accumulated hysteresis energy dissipation are established;

[0011] S6: According to the joint distribution function of the inter-story displacement angle and the cumulative hysteresis energy dissipation related to the Copula function, and the failure probability model related to the inter-story displacement angle and the failure probability model related to the cumulative hysteresis energy dissipation, a seismic vulnerability model of the inter-story displacement angle and the cumulative hysteresis energy dissipation based on the Copula function is established to obtain the failure probability of the underground structure, and then the seismic vulnerability of the underground structure is evaluated.

[0012] Furthermore, in S1, the earthquake demand sample set obtained is as follows:

[0013] X={(dm1,im1),(dm2,im2),...,(dm i ,im i ),...,(dm n ,im n )} (1)

[0014] Where: X is the earthquake demand sample set; dm i is the i-th earthquake demand sample, im i is the i-th ground motion intensity sample; (dm i ,im i ) is the seismic demand under the i-th seismic motion intensity; n is the total number of seismic demand samples; i is the number of the seismic demand sample.

[0015] Furthermore, in S2, the method for obtaining the kernel density function model is as follows:

[0016] Construct the probability density function of the kernel density estimate:

[0017]

[0018] Where: f h (dm) is the probability density function of the kernel density estimate; h is the window width; K(·) is the kernel function, and dm is the seismic demand of any earthquake intensity;

[0019] in,

[0020]

[0021] The kernel density function model of earthquake demand is obtained as follows:

[0022]

[0023] Furthermore, the optimal value of the window width in the kernel density function model is calculated as follows:

[0024] MISE(f h )=E{∫[f h (dm)-f(dm)] 2d(dm)} (5)

[0025] Where: MISE(f h ) is a function of the window width h; E{·} is the average integrated square error of the seismic demand; f h (dm) is the probability density function of kernel density estimation; f(dm) is the empirical probability density function estimated based on the data set;

[0026] MISE(h)=AMISE(h)+p(1 / (nh)+h 4 ) (6)

[0027] Where: AMISE(h) is the asymptotic MISE(h); MISE(h) is the integrated square error function with respect to the window width h; p(·) is a high-order infinitesimal with respect to the window width h;

[0028]

[0029] in:

[0030] m=∫[f(dm)] 2 d(dm) (8)

[0031] m2=∫(dm) 2 f(dm)d(dm) (9)

[0032] According to formula (7), the optimal window width is calculated as follows:

[0033]

[0034] Where: h AMISE is the optimal window width; m is the first-order origin moment of the sample set, m2 is the second-order origin moment of the sample set, and f (2) (dm) is the second-order derivative of the probability density function of the kernel density estimate.

[0035] Furthermore, in S3, a kernel density function of the inter-story displacement angle and the accumulated hysteresis energy consumption is established as follows:

[0036]

[0037]

[0038] Where: f(IDR) is the kernel density function of the interlayer displacement angle; IDR is the interlayer displacement angle; idr i is the inter-layer displacement angle of the underground structure of the ith ground motion intensity sample; f(CHE) is the kernel density function of the accumulated hysteretic energy dissipation; CHE is the accumulated hysteretic energy dissipation; che i is the cumulative hysteretic energy dissipation of the underground structure of the i-th earthquake intensity sample.

[0039] Furthermore, in S4, a joint distribution function of the inter-story displacement angle and the cumulative hysteresis energy dissipation of the Copula function is established as follows:

[0040] F(IDR,CHE)=C[F(IDR),F(CHE)] (13)

[0041] but

[0042] C(IDR,CHE)=F(F (-1) (IDR),F (-1) (CHE)) (14)

[0043] Where: C[F(IDR),F(CHE)] is the Copula function of the inter-story displacement angle and the cumulative hysteretic energy dissipation; F(IDR,CHE) is the binary joint distribution function of the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteretic energy dissipation for the incremental dynamic analysis of underground structures; F(IDR) is the marginal distribution function of the inter-story displacement angle; F(CHE) is the marginal distribution function of the cumulative hysteretic energy dissipation.

[0044] Furthermore, in S5, the method of establishing the failure probability model for the inter-story displacement angle and the failure probability model for the accumulated hysteresis energy dissipation is as follows;

[0045] First: The vulnerability model of underground structure with a single demand parameter is established as follows:

[0046]

[0047] in:

[0048] ln(DM)=aln(IM)+b (16)

[0049]

[0050] Where: P f (im) is the failure probability of underground structures under earthquake action, i.e., seismic vulnerability; IM is the seismic intensity when the underground structure reaches a certain damage state, EDP is the seismic demand of the underground structure; DM is the mean seismic demand of the underground structure, and DC is the quantitative index limit of the underground structure under different states; β d is the logarithmic standard deviation of seismic demand, β c is the logarithmic standard deviation of the structural limit state; a is the regression slope; b is the regression intercept;

[0051] Secondly, the failure probability model of the inter-story displacement angle is established as follows:

[0052]

[0053] Where: P f (IDR) is the failure probability of underground structure under earthquake with respect to inter-layer displacement angle; Φ[·] is the cumulative function of normal distribution;

[0054] in,

[0055] ln(IDR)=x+yln(IM) (19)

[0056] Where: x is the intercept of the regression between interlayer displacement angle and earthquake intensity; y is the slope of the regression between interlayer displacement angle and earthquake intensity;

[0057] The failure probability model for cumulative hysteresis energy dissipation is established as follows:

[0058]

[0059] Where: P f (CHE) is the failure probability of underground structures under earthquake action due to accumulated hysteretic energy dissipation;

[0060] in,

[0061] ln(CHE)=z+wln(IM) (21)

[0062] Where: z is the intercept of the regression between the cumulative hysteresis energy dissipation and the earthquake intensity; w is the slope of the regression between the cumulative hysteresis energy dissipation and the earthquake intensity.

[0063] Furthermore, in S6, a seismic vulnerability model based on the Copula function regarding the inter-story displacement angle and the accumulated hysteretic energy dissipation is established as follows:

[0064] First, the Copula function of the inter-story displacement angle and the cumulative hysteresis energy failure probability is established:

[0065] C[P f (IDR),P f (CHE);IM] = P(P (-1) (IDR),P (-1) (CHE)) (22)

[0066] Where: P (-1) (IDR) is the inverse function of the failure probability function of the interlayer displacement angle of the underground structure; P (-1) (CHE) is the inverse function of the cumulative hysteresis energy failure probability function of underground structures; C[P f (IDR),P f (CHE); IM] is the Copula function of the inter-story displacement angle and the cumulative hysteresis energy failure probability;

[0067] Secondly, the seismic vulnerability model of underground structures regarding interlayer displacement angle and accumulated hysteresis energy dissipation is established as follows:

[0068] P[IDR,CHE;IM]=C(IDR,CHE;IM) (23)

[0069] Where: P[IDR,CHE;IM] is the failure probability of the inter-layer displacement angle and the accumulated hysteretic energy dissipation of the underground structure under the action of earthquake motion; C(IDR,CHE;IM) is the Copula function of the inter-layer displacement angle and the accumulated hysteretic energy dissipation under the action of earthquake motion;

[0070] Finally, the seismic vulnerability model based on the Copula function regarding the inter-story displacement angle and the accumulated hysteretic energy dissipation is obtained as follows:

[0071] P fs =P f (IDR)+P f (CHE)-C[(P f (IDR),P f (CHE); IM)] (24)

[0072] Where: P fs is the failure probability of the underground structure.

[0073] Beneficial effects: The method for calculating the seismic vulnerability of underground structures based on multi-dimensional performance indicators of the present invention solves the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy dissipation of the subway station structure by establishing a kernel density function model. Because it does not require assumptions about the overall distribution, it can effectively reduce the influence of knowledge uncertainty caused by the log-normal distribution assumption and the discreteness of seismic demand as the ground shaking intensity increases, making the obtained marginal distribution function more practical. From the perspective of the correlation between the inter-story displacement angle and the cumulative hysteresis energy dissipation, the Copula function is used so that the function does not change when the variable changes monotonically, so that the Copula function can easily handle the nonlinear correlation between the two variables of the inter-story displacement angle and the cumulative hysteresis energy dissipation. Thereby, a more accurate seismic vulnerability function of the underground structure can be obtained.

[0074] The present invention fully considers the cumulative destructive effect of sustained earthquake motion on underground structures under the action of peak earthquake acceleration. On the basis of the dual-parameter damage model of deformation and energy, it fully considers the influence of two damage indicators, namely, inter-story displacement angle and cumulative hysteresis energy dissipation, on seismic vulnerability, and has strong engineering application value and significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0076] Figure 1a A schematic cross-sectional view of a subway station structure in an embodiment of the present invention;

[0077] Figure 1b A schematic diagram of the reinforcement arrangement of the middle column of a large-scale railway station in an embodiment of the present invention;

[0078] Figure 2 A finite element model diagram of an embodiment of the present invention;

[0079] Figure 3 is an acceleration time history curve diagram of input ground motion in an embodiment of the present invention;

[0080] Figure 4a It is a diagram of the empirical distribution function of the seismic demand for inter-storey displacement angle of a subway station in an embodiment of the present invention;

[0081] Figure 4b It is a diagram of empirical distribution function of cumulative hysteretic energy dissipation seismic demand of subway stations in an embodiment of the present invention;

[0082] Figure 5 is a distribution function diagram of a binary Copula function in an embodiment of the present invention;

[0083] Figure 6 It is a seismic vulnerability diagram of underground structure based on multivariate seismic demand parameters of Copula function in an embodiment of the present invention;

[0084] Figure 7 The figure is a flow chart of the method for calculating seismic vulnerability of underground structures according to the present invention. DETAILED DESCRIPTION

[0085] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0086] This embodiment provides a method for calculating the seismic vulnerability of underground structures based on multi-dimensional performance indicators. Figure 7 As shown,

[0087] S1: Establish an underground structure finite element model of a single-story, double-span subway station to obtain a sample set of seismic demand;

[0088] This embodiment takes a single-layer double-span subway station as the research object, and uses finite element software (such as: OpenSees open source program platform) to establish a finite element calculation model. Specifically, 10 to 20 earthquake motion records are selected according to the construction site conditions of the single-layer double-span subway station, and based on the finite element model, nonlinear time history analysis (existing technology) is performed according to the earthquake motion records to obtain a sample set of earthquake demand for underground structures.

[0089] Specifically, the research methods for data sample distribution can be divided into the following two categories: parameter estimation and non-parametric estimation. Parameter estimation requires making assumptions about the distribution of data samples in advance based on experience, while non-parametric methods do not need to make assumptions about the overall distribution, and the marginal distribution function obtained is more in line with reality. Kernel density estimation is a type of non-parametric estimation, which constructs a continuous probability density function through the linear addition of discrete sample points to obtain a smooth sample distribution.

[0090] The obtained earthquake demand sample set is as follows:

[0091] X={(dm1,im1),(dm2,im2),...,(dm i ,im i ),...,(dm n ,im n )} (1)

[0092] Where: X is the earthquake demand sample set; dm i is the i-th earthquake demand sample, im i is the i-th ground motion intensity sample; (dm i ,im i ) is the seismic demand under the i-th seismic motion intensity; n is the total number of seismic demand samples; i is the number of the seismic demand sample.

[0093] S2: According to the seismic demand sample set, a method for obtaining a kernel density function model of seismic demand dm corresponding to any seismic motion intensity im is as follows:

[0094] Construct the probability density function of the kernel density estimate:

[0095]

[0096] Where: f h (dm) is the probability density function of the kernel density estimate; h is the window width; K(·) is the kernel function, and dm is the seismic demand of any earthquake intensity;

[0097] In order to ensure the rationality of the kernel density function estimation, the kernel function K(·) is required to satisfy:

[0098]

[0099] Specifically, commonly used kernel functions include: linear kernel function, polynomial kernel function, radial basis kernel function and Gaussian kernel function. Among them, Gaussian kernel function is smoother and has more convenient mathematical properties, so Gaussian kernel function is selected as the kernel function of seismic demand kernel density estimation.

[0100] The Gaussian kernel density function model of earthquake demand is obtained as follows:

[0101]

[0102] The smoothness of the kernel function is determined by the window width h. The optimal value of the window width in the kernel density function model is calculated as follows:

[0103] MISE(f h )=E{∫[f h (dm)-f(dm)] 2 d(dm)} (5)

[0104] Where: MISE(f h ) is a function of the window width h; E{·} is the average integrated square error of the seismic demand; f h (dm) is the probability density function of kernel density estimation; f(dm) is the empirical probability density function estimated based on the data set;

[0105] MISE(h)=AMISE(h)+o(1 / (nh)+h 4 ) (6)

[0106] Where: AMISE(h) is the asymptotic MISE(h); MISE(h) is the integrated square error function with respect to the window width h; o(·) is a high-order infinitesimal with respect to the window width h;

[0107]

[0108] in:

[0109] m=∫[f(dm)] 2 d(dm)(8)

[0110] m2=∫(dm) 2 f(dm)d(dm)(9)

[0111] According to formula (7), the optimal window width is calculated as follows:

[0112]

[0113] Where: h AMISE is the optimal window width; m is the first-order origin moment of the sample set, m2 is the second-order origin moment of the sample set, and f (2) (dm) is the second-order derivative of the probability density function of the kernel density estimate.

[0114] S3: According to the kernel density function model about dm, a kernel density function about inter-story displacement angle (IDR) and cumulative hysteresis energy (CHE) is established to obtain the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy;

[0115] Preferably, the kernel density function about the inter-story displacement angle (IDR) and the accumulated hysteresis energy (CHE) is established as follows:

[0116]

[0117]

[0118] Where: f(IDR) is the kernel density function of the interlayer displacement angle; IDR is the interlayer displacement angle; idr i is the inter-layer displacement angle of the underground structure of the ith ground motion intensity sample; f(CHE) is the kernel density function of the accumulated hysteretic energy dissipation; CHE is the accumulated hysteretic energy dissipation; che i is the accumulated hysteresis energy dissipation of the underground structure of the i-th ground motion intensity sample;

[0119] Specifically, by establishing the kernel density function of the inter-story displacement angle (IDR) and the cumulative hysteretic energy (CHE), the correlation between the inter-story displacement angle and the cumulative hysteretic energy can be described without being restricted by the form of their distribution function, and the influence of knowledge uncertainty caused by the assumption of log-normal distribution can be avoided.

[0120] According to the f(IDR), by integrating it, the marginal distribution function F(IDR) of the inter-story displacement angle can be obtained; similarly, according to the f(CHE), by integrating it, the marginal distribution function F(CHE) of the inter-story displacement angle can be obtained;

[0121] Specifically, in this embodiment, the marginal distribution function F(IDR) of the inter-story displacement angle can be connected with the marginal distribution function F(CHE) of the cumulative hysteretic energy loss according to the Copula function. F(IDR) and F(CHE) are the cumulative distribution functions of f(IDR) and f(CHE), respectively, and then the joint distribution function F(IDR, CHE) of the inter-story displacement angle and the cumulative hysteretic energy loss is obtained.

[0122] S4: establishing a joint distribution function of the inter-story displacement angle and the cumulative hysteresis energy dissipation with respect to the Copula function according to the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy dissipation;

[0123] Preferably, according to Sklar's theorem, if F(IDR,CHE) is the binary joint distribution function of the marginal distribution function F(IDR) of the inter-story displacement angle and the marginal distribution function F(CHE) of the cumulative hysteresis energy dissipation for the incremental dynamic analysis of underground structures, then there exists a Copula function that satisfies:

[0124] F(IDR,CHE)=C[F(IDR),F(CHE)] (13)

[0125] Where: C[F(IDR), F(CHE)] is the Copula function of the inter-story displacement angle and the cumulative hysteretic energy dissipation; F(IDR, CHE) is the binary joint distribution function of the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteretic energy dissipation for the incremental dynamic analysis of underground structures; F(IDR) is the marginal distribution function of the inter-story displacement angle; F(CHE) is the marginal distribution function of the cumulative hysteretic energy dissipation;

[0126] Specifically, according to the properties of the Copula function, if the marginal distribution functions F(IDR) and F(CHE) of the interlayer displacement angle and the accumulated hysteretic energy dissipation of the underground structure are continuous functions, then C[F(IDR), F(CHE)] is uniquely determined; conversely, if F(IDR) and F(CHE) are univariate distribution functions and C[F(IDR), F(CHE)] is the corresponding Copula function, then F(IDR, CHE) determined by the above formula is a multivariate Copula function with marginal distribution functions F(IDR) and F(CHE), and the marginal distribution function is estimated using the kernel density function.

[0127] According to the inverse function F of F(IDR), F(CHE) (-1) (IDR),F (-1) (CHE), then the Copula function of interlayer displacement angle and cumulative hysteresis energy dissipation is established, that is, the Copula function model of underground structure is as follows:

[0128] C(IDR,CHE)=F(F (-1) (IDR),F (-1) (CHE)) (14)

[0129] According to the inverse function and Copula function of the marginal distribution of the inter-story displacement angle and the cumulative hysteretic energy dissipation, the corresponding Copula function can be obtained to describe the correlation between the inter-story displacement angle and the cumulative hysteretic energy dissipation; the correlation between the inter-story displacement angle and the cumulative hysteretic energy dissipation can be separated from their marginal distribution function through the Copula function, thereby reducing the difficulty of establishing a multivariate probability model.

[0130] S5: Establish a fragility model of the Copula function of the inter-story displacement angle and the accumulated hysteresis energy dissipation, that is, establish a multivariate Copula fragility model; that is, establish a failure probability model of the inter-story displacement angle and a failure probability model of the accumulated hysteresis energy dissipation; the details are as follows:

[0131] The vulnerability model of underground structure with a single demand parameter is established as follows:

[0132]

[0133] in:

[0134] ln(DM)=aln(IM)+b (16)

[0135]

[0136] Where: P f (im) is the failure probability of underground structures under earthquake action, i.e., seismic vulnerability; IM is the seismic intensity when the underground structure reaches a certain damage state, EDP is the seismic demand of the underground structure; DM is the mean seismic demand of the underground structure, and DC is the quantitative index limit of the underground structure under different states; β d is the logarithmic standard deviation of seismic demand, β c is the logarithmic standard deviation of the structural limit state; a is the regression slope; b is the regression intercept;

[0137] The vulnerability model of inter-story displacement angle and accumulated hysteresis energy dissipation is established as follows:

[0138]

[0139] Where: P f (IDR) is the failure probability of underground structure under earthquake with respect to inter-layer displacement angle; Φ[·] is the cumulative function of normal distribution;

[0140] in,

[0141] ln(IDR)=x+yln(IM) (19)

[0142] Where: x is the intercept of the regression between interlayer displacement angle and earthquake intensity; y is the slope of the regression between interlayer displacement angle and earthquake intensity;

[0143]

[0144] Where: P f (CHE) is the failure probability of underground structures under earthquake action due to accumulated hysteretic energy dissipation;

[0145] in,

[0146] ln(CHE)=z+wln(IM) (21)

[0147] Where: z is the intercept of the regression between the cumulative hysteresis energy dissipation and the earthquake intensity; w is the slope of the regression between the cumulative hysteresis energy dissipation and the earthquake intensity.

[0148] S6: According to the joint distribution function of the inter-story displacement angle and the cumulative hysteretic energy dissipation about the Copula function, and the failure probability model about the inter-story displacement angle and the failure probability model about the cumulative hysteretic energy dissipation, a seismic vulnerability model about the inter-story displacement angle and the cumulative hysteretic energy dissipation based on the Copula function is established to obtain the failure probability of the underground structure, and then the seismic vulnerability of the underground structure is evaluated.

[0149] Preferably, a seismic vulnerability model based on the Copula function regarding the inter-story displacement angle and the accumulated hysteretic energy dissipation is established as follows:

[0150] First, the Copula function of the inter-story displacement angle and the cumulative hysteresis energy failure probability is established:

[0151] C[(P f (IDR),P f (CHE);IM] = P(P (-1) (IDR),P (-1) (CHE)) (22)

[0152] Where: P (-1) (IDR) is the inverse function of the failure probability function of the interlayer displacement angle of the underground structure; P (-1) (CHE) is the inverse function of the cumulative hysteresis energy failure probability function of underground structures; C[P f (IDR),P f (CHE); IM] is the Copula function of the inter-story displacement angle and the cumulative hysteresis energy failure probability;

[0153] Secondly, according to the properties of the Copula function, a seismic vulnerability model of multivariate seismic demand parameters of underground structures is established, that is, a seismic vulnerability model of interlayer displacement angle and accumulated hysteresis energy dissipation of underground structures is established as follows:

[0154] P[IDR,CHE;IM]=C(IDR,CHE;IM) (23)

[0155] Where: P[IDR,CHE;IM] is the failure probability of the inter-layer displacement angle and the accumulated hysteretic energy dissipation of the underground structure under the action of earthquake motion; C(IDR,CHE;IM) is the Copula function of the inter-layer displacement angle and the accumulated hysteretic energy dissipation under the action of earthquake motion.

[0156] Finally, according to the correlation between the seismic demand parameters, they can be regarded as a series effect, that is, if the failure probability is greater than 1 under any seismic demand parameter, it is regarded as a failure of the entire structure, and then the seismic vulnerability model of multivariate seismic demand parameters based on the Copula function is obtained, that is, the seismic vulnerability model of inter-story displacement angle and accumulated hysteretic energy dissipation based on the Copula function is as follows:

[0157] P fs =P f (IDR)+P f (CHE)-C[(P f (IDR),P f (CHE); IM)] (24)

[0158] Where: P fs is the failure probability of the underground structure.

[0159] It can be seen from the above formula that when solving the multidimensional vulnerability, the one-dimensional vulnerability of the structure and the correlation between them can be considered separately, which can make the solution of the seismic vulnerability of the underground structure simpler.

[0160] Case Description

[0161] The present invention is described in detail in conjunction with the following specific examples. This example is implemented based on the technical method of the present invention, and provides a detailed implementation method and a specific operation process, but the protection scope of the present invention is not limited to the following examples.

[0162] Taking a single-layer double-span saturated sand field subway station as an example, this subway station is a 17m×7.17m rectangular frame with a burial depth of 5m, a top and bottom plate reinforcement ratio of 1.0%, a side wall reinforcement ratio of 0.8%, a middle column cross-section size of 0.4m×1m, and a reinforcement ratio of 6.0%. Its cross section and middle column reinforcement diagram are shown in Figure 1. The physical parameters of the saturated site soil where the station is located are shown in Table 1.

[0163] Table 1 Soil parameters

[0164]

[0165] (1) Establishing a finite element model of underground structures and conducting nonlinear time-history analysis

[0166] The finite element model of the subway station in a saturated sand field was established based on the open source program platform of OpenSees. The calculation size of the overall model is 170m×30m. The top of the soil model is free, and the bottom is a vertical fixed constraint. The base consistent excitation is used in OpenSees to input the ground motion; the binding boundary is set on both sides of the model through equalDOF. Similarly, the contact surface of the soil of the subway station is also bound.

[0167] The subway station is a reinforced concrete structure, which is simulated by a fiber-section-based beam-column unit. The concrete grade is C30, and the Scott-Kent-Park constitutive model Concrete02 is used to consider the influence of the constraint of the transverse stirrups in the concrete. The steel bars use the uniaxial isotropic reinforced Giuffre-Menegotto-Pinto constitutive model Steel02. The saturated sand field soil is simulated by a plane strain four-node unit quadUP with full fluid-solid coupling; the constitutive model of the soil body uses the multi-yield surface elastoplastic material PressureDependMultiYield that is sensitive to response under general load conditions. The finite element model is as follows Figure 2 shown.

[0168] The ground peak acceleration PGA is used as the intensity index of the earthquake motion, and the inter-layer displacement angle and the accumulated hysteresis energy dissipation are selected as the structural seismic demand parameters. According to the "Code for Seismic Design of Building Structures": The limit value of the elastic inter-layer displacement angle is [θ e ], the elastic-plastic interlayer displacement angle limit is [θ p ],in:

[0169]

[0170]

[0171] The damage index of cumulative hysteresis energy dissipation can be expressed by the following formula:

[0172]

[0173] In the formula, E DE is the structural hysteresis energy demand; E CE The ability of a structure to dissipate energy.

[0174] The relationship between the seismic damage level and performance index of underground structures is shown in Table 2, and the correspondence between the seismic damage level and its quantitative index is shown in Table 3.

[0175] Table 2 Relationship between seismic damage level and performance index of underground structures

[0176]

[0177] Table 3 Damage level, performance and quantitative indicators of underground structures

[0178]

[0179] 20 earthquake records were selected from the PEER strong earthquake record database, and their peak values ​​were adjusted to 0.05g to 1.0g, totaling 200 earthquake records (10 records at each PGA level). The acceleration time history curve of the input earthquake is as follows: Figure 3 As shown, a nonlinear time history analysis was performed on the finite element model of a saturated sand field and underground structure.

[0180] (2) Establishing vulnerability based on multivariate Copula function

[0181] According to the results of incremental dynamic analysis, the probability distribution of inter-story displacement angle and cumulative hysteresis energy dissipation seismic demand is estimated by non-parametric kernel density estimation method, and its empirical distribution function is shown in Figure 4. According to formulas (1) and (8), the binary Gaussian Copula joint distribution function is solved, and the binary Copula function with inter-story displacement angle and cumulative hysteresis energy dissipation as marginal distribution functions is obtained as follows: Figure 5 As shown. Substituting the obtained Copula function and the marginal distribution function of the inter-story displacement angle and the cumulative hysteretic energy dissipation into formula (9), the seismic vulnerability of the subway station structure based on the binary Copula function is obtained, as follows: Figure 6 As shown in the figure, when the ground motion peak acceleration is small, the damage degree of the subway station structure develops rapidly, and the exceedance probability of each damage index is large. As the PGA increases, the structural damage rate slows down. The correlation dual-parameter damage model better considers the influence of inter-story displacement angle and cumulative hysteresis energy dissipation in the damage process of the subway station structure, as well as the influence of the correlation between deformation and energy on structural damage under different ground motion peak accelerations. The use of kernel density function to estimate the marginal distribution function can effectively reduce the influence of knowledge uncertainty caused by the log-normal distribution assumption and the discreteness of seismic demand with increasing earthquake intensity, making the final fragility function more practical.

[0182] The present invention provides a method for calculating the seismic vulnerability of underground structures based on multidimensional performance indicators. Starting from the perspective of solving the relationship between the seismic intensity index and the damage index, the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy consumption of the subway station structure are solved by establishing a kernel density function model. Because it does not require assumptions about the overall distribution, it can effectively reduce the influence of knowledge uncertainty caused by the log-normal distribution assumption and the discreteness of the seismic demand as the seismic intensity increases, making the obtained marginal distribution function more practical. Starting from the perspective of the correlation between the inter-story displacement angle and the cumulative hysteresis energy consumption, the Copula function is used so that the function does not change when the variable changes monotonically, so that the Copula function can conveniently handle the nonlinear correlation between the two variables of the inter-story displacement angle and the cumulative hysteresis energy consumption. Moreover, as the connection function of the marginal distribution function, the Copula function form is not restricted by the marginal distribution, so that a more accurate seismic vulnerability function of the underground structure can be obtained.

[0183] The present invention fully considers the cumulative destructive effect of the underground structure under the peak acceleration of the earthquake. The joint probability density function is connected with the edge function through the Copula function, and is not affected by the form of the edge distribution function. On the basis of the double parameter damage model of deformation and energy, the influence of the two damage indicators of interlayer displacement angle and cumulative hysteresis energy dissipation on seismic vulnerability is fully considered, which has strong engineering application value and significance.

[0184] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators, characterized in that: The steps include: S1: Establish an underground structure finite element model of a single-story, double-span subway station to obtain a sample set of seismic demand; S2: obtaining a kernel density function model according to the earthquake demand sample set; S3: According to the kernel density function model, a kernel density function is established regarding the inter-story displacement angle and the cumulative hysteresis energy dissipation to obtain the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy dissipation; S4: establishing a joint distribution function of the inter-story displacement angle and the cumulative hysteresis energy dissipation with respect to the Copula function according to the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteresis energy dissipation; S5: Based on the seismic demand sample set, a failure probability model for inter-story displacement angle and a failure probability model for accumulated hysteresis energy dissipation are established; S6: According to the joint distribution function of the inter-story displacement angle and the cumulative hysteretic energy dissipation about the Copula function, and the failure probability model about the inter-story displacement angle and the failure probability model about the cumulative hysteretic energy dissipation, a seismic vulnerability model about the inter-story displacement angle and the cumulative hysteretic energy dissipation based on the Copula function is established to obtain the failure probability of the underground structure, and then the seismic vulnerability of the underground structure is evaluated.

2. A method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 1, characterized in that: In S1, the obtained earthquake demand sample set is as follows: X={(dm1,im1),(dm2,im2),...,(dm i ,im i ),...,(dm n ,im n )} (1) Where: X is the earthquake demand sample set; dm i is the i-th earthquake demand sample, im i is the i-th ground motion intensity sample; (dm i ,im i ) is the seismic demand under the i-th ground motion intensity; n is the total number of earthquake demand samples; i is the number of the earthquake demand sample.

3. The method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 1, characterized in that: In S2, the method for obtaining the kernel density function model is as follows: Construct the probability density function of the kernel density estimate: Where: f h (dm) is the probability density function of the kernel density estimate; h is the window width; K(·) is the kernel function, and dm is the seismic demand of any earthquake intensity; in, The kernel density function model of earthquake demand is obtained as follows:

4. The method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 3, characterized in that: The optimal value of the window width in the kernel density function model is calculated as follows: MISE(f h )=E{∫[f h (dm)-f(dm)] 2 d(dm)} (5) Where: MISE(f h ) is a function of the window width h; E{·} is the average integrated square error of the seismic demand; f h (dm) is the probability density function of kernel density estimation; f(dm) is the empirical probability density function estimated based on the data set; MISE(h)=AMISE(h)+o(1 / (nh)+h 4 ) (6) Where: AMISE(h) is the asymptotic MISE(h); MISE(h) is the integrated square error function with respect to the window width h; o(·) is a high-order infinitesimal with respect to the window width h; in: m=∫[f(dm)] 2 d(dm) (8) m2=∫(dm) 2 f(dm)d(dm) (9) According to formula (7), the optimal window width is calculated as follows: Where: h AMISE is the optimal window width; m is the first-order origin moment of the sample set, m2 is the second-order origin moment of the sample set, and f (2) (dm) is the second-order derivative of the probability density function of the kernel density estimate.

5. The method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 3, characterized in that: In S3, the kernel density function of the inter-story displacement angle and the accumulated hysteresis energy consumption is established as follows: Where: f(IDR) is the kernel density function of the interlayer displacement angle; IDR is the interlayer displacement angle; idr i is the inter-story displacement angle of the underground structure of the i-th ground motion intensity sample; f(CHE) is the kernel density function of the cumulative hysteresis energy dissipation; CHE is the cumulative hysteresis energy dissipation; che i is the cumulative hysteretic energy dissipation of the underground structure of the i-th earthquake intensity sample.

6. A method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 5, characterized in that: In S4, the joint distribution function of the inter-story displacement angle and the cumulative hysteresis energy dissipation about the Copula function is established as follows: F(IDR,CHE)=C[F(IDR),F(CHE)] (13) but C(IDR,CHE)=F(F (-1) (IDR),F (-1) (WHAT)) (14) Where: C[F(IDR), F(CHE)] is the Copula function of the inter-story displacement angle and the cumulative hysteretic energy dissipation; F(IDR, CHE) is the binary joint distribution function of the marginal distribution function of the inter-story displacement angle and the marginal distribution function of the cumulative hysteretic energy dissipation for the incremental dynamic analysis of underground structures; F(IDR) is the marginal distribution function of the inter-story displacement angle; F(CHE) is the marginal distribution function of the cumulative hysteretic energy dissipation.

7. A method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 6, characterized in that: In said S5, the method of establishing the failure probability model about the inter-story displacement angle and the failure probability model about the accumulated hysteresis energy dissipation is as follows; First: The vulnerability model of underground structure with a single demand parameter is established as follows: in: ln(DM)=aln(IM)+b (16) Where: P f (im) is the failure probability of underground structures under earthquake action, i.e., seismic vulnerability; IM is the seismic intensity when the underground structure reaches a certain damage state, EDP is the seismic demand of the underground structure; DM is the mean seismic demand of the underground structure, and DC is the quantitative index limit of the underground structure under different states; β d is the logarithmic standard deviation of seismic demand, β c is the logarithmic standard deviation of the structural limit state; a is the regression slope; b is the regression intercept; Secondly, the failure probability model of the inter-story displacement angle is established as follows: Where: P f (IDR) is the failure probability of underground structure under earthquake with respect to inter-layer displacement angle; Φ[·] is the cumulative function of normal distribution; in, ln(IDR)=x+yln(IM) (19) Where: x is the intercept of the regression between interlayer displacement angle and earthquake intensity; y is the slope of the regression between interlayer displacement angle and earthquake intensity; The failure probability model for cumulative hysteresis energy dissipation is established as follows: Where: P f (CHE) is the failure probability of underground structures under earthquake action due to accumulated hysteretic energy dissipation; in, ln(CHE)=z+wln(IM) (21) Where: z is the intercept of the regression between the cumulative hysteresis energy dissipation and the earthquake intensity; w is the slope of the regression between the cumulative hysteresis energy dissipation and the earthquake intensity.

8. The method for calculating seismic vulnerability of underground structures based on multi-dimensional performance indicators according to claim 7, characterized in that: In S6, a seismic vulnerability model based on the Copula function regarding the inter-story displacement angle and the accumulated hysteretic energy dissipation is established as follows: First, the Copula function of the inter-story displacement angle and the cumulative hysteresis energy failure probability is established: C[P f (IDR),P f (WHAT);IM]=P(P (-1) (IDR),P (-1) (WHAT)) (22) Where: P (-1) (IDR) is the inverse function of the failure probability function of the interlayer displacement angle of the underground structure; P (-1) (CHE) is the inverse function of the cumulative hysteresis energy failure probability function of underground structures; C[P f (IDR), P f (CHE); IM] is the Copula function of the inter-story displacement angle and the cumulative hysteresis energy failure probability; Secondly, the seismic vulnerability model of underground structures regarding interlayer displacement angle and accumulated hysteresis energy dissipation is established as follows: P[IDR,CHE;IM]=C(IDR,CHE;IM) (23) Where: P[IDR, CHE; IM] is the failure probability of the inter-layer displacement angle and the accumulated hysteretic energy dissipation of the underground structure under the action of earthquake motion; C(IDR, CHE; IM) is the Copula function of the inter-layer displacement angle and the accumulated hysteretic energy dissipation under the action of earthquake motion; Finally, the seismic vulnerability model based on the Copula function regarding the inter-story displacement angle and the accumulated hysteretic energy dissipation is obtained as follows: P fs =P f (IDR)+P f (WHAT)-C[(P f (IDR),P f (WHAT);IM)] (24) Where: P fs is the failure probability of the underground structure.

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