Underground station earthquake vulnerability analysis method based on probability damage weight

Abstract

The invention relates to the technical field of underground structure damage evaluation, in particular to an underground station earthquake vulnerability analysis method based on probability damage weight, which comprises the steps of collecting geological exploration reports and underground structure materials, generating random earthquake oscillation and determining corresponding assigned probability, establishing random earthquake oscillation with the same model input intensity, and adjusting the earthquake oscillation intensity. Carrying out incremental dynamic analysis; the method comprises the steps of solving a generalized probability density equation, calculating a cumulative probability density function of an extreme value of a seismic performance index of each structural component, deriving a vulnerability curve of each structural component, calculating a damage weight coefficient of each structural component, and deriving a vulnerability curve of the whole station structure. It is not needed to assume that the seismic oscillation intensity and the structural damage degree obey specific probability distribution in advance, and meanwhile contribution of damage of different structural members to overall structural damage can be reflected.

Description

Technical Field

[0001] This invention relates to the field of underground structure damage assessment technology, and in particular to a method for seismic vulnerability analysis of underground railway stations based on probability damage weights. Background Technology

[0002] Seismic vulnerability refers to the conditional probability that a structure will reach or exceed a certain limit state under a specific seismic intensity. Traditional vulnerability analysis methods typically assume that seismic intensity indices and structural damage indices follow a log-normal distribution. However, seismic intensity indices and structural damage indices do not perfectly conform to the log-normal distribution assumption, especially at higher seismic intensities where the deviation becomes more significant. Furthermore, such methods usually use only a limited number of seismic motion records as input, failing to fully consider the inherent randomness of seismic motions and structural responses. Therefore, traditional parametric vulnerability methods are insufficient to accurately characterize the stochastic characteristics of seismic motions and the resulting stochastic dynamic responses of structures.

[0003] As a crucial underground structure for urban transportation, subway stations have varying load-bearing capacities among their structural components, and their seismic damage contributes differently to the overall structural system damage. Most current studies treat damage to the central column as the sole failure criterion, neglecting damage to other components (such as the roof slab and sidewalls). Summary of the Invention

[0004] The purpose of this application is to provide a seismic vulnerability analysis method for underground railway stations based on probabilistic damage weights, aiming to solve the problems in the prior art.

[0005] This application provides a method for seismic vulnerability analysis of underground railway stations based on probabilistic damage weights. The method includes the following steps:

[0006] (1) Collect geological survey reports and underground structure materials;

[0007] (2) By sampling in probability space, a set of representative points with probability weights are obtained, and a random ground motion model is used to synthesize the random bedrock ground motion corresponding to each representative point;

[0008] (3) Adjust the amplitude of all bedrock ground motion time histories generated in step (2) to the same target intensity and apply them as input loads to the numerical model established in step (1). Perform deterministic nonlinear time history analysis on each ground motion record to obtain the ground motion acceleration time history curve.

[0009] (4) Adjust the ground motion intensity, repeat step (3), perform incremental dynamic analysis, and obtain the key response time history of structures with different ground motion intensities;

[0010] (5) Substitute the structural critical response time histories obtained in steps (3) and (4) into the generalized probability density evolution equation, and apply the equivalent extreme event theory to calculate the cumulative distribution of the extreme values ​​of the structural seismic performance index under different ground motion intensities.

[0011] (6) Derive the vulnerability curves of each subway station structure based on the cumulative probability density function obtained in step (5) and determine the damage probability of different seismic intensities.

[0012] The calculation of the improved seismic vulnerability function based on GPDEM is expressed as follows:

[0013]

[0014] L is the performance level limit, R is the seismic response of the structural component under IM, and R max For different damage states of structural components, the exceedance probability of structural damage under each seismic intensity IM is the cumulative probability density function of each IM, i.e., the definite value of the limit state in the CDF curve. R│IM To determine the CDF value of PI under IM conditions, p R│IM It is a PDF that determines the extreme values ​​of the structural component response under IM conditions;

[0015] (7) Calculate the damage weight coefficient for each structural component based on the damage probability calculated in step (6);

[0016] Determine the damage contribution of each structural component to the overall structure, i.e., the damage weighting coefficient κ:

[0017]

[0018] Where κ is the damage weighting coefficient of structural member i when the damage state is Ri under a given seismic intensity IM, p i Let represent the damage probability of structural component i when the damage state is Ri, and n represent the total number of components;

[0019] (8) Based on the damage weight coefficient calculated in step (7) and the damage probability calculated in step (6), calculate the overall vulnerability curve of the entire subway station structure;

[0020] Based on the failure probability and damage weight coefficient of each structural component under each damage state, the seismic vulnerability curve of the underground structure is obtained:

[0021]

[0022] In the formula, P i Let κ represent the exceedance probability of structural member i under different damage states under earthquake intensity IM. i, where represents the damage weighting coefficient for structural member i under different damage states under earthquake intensity IM.

[0023] Furthermore, the derivation process of the seismic vulnerability function is as follows: seismic vulnerability represents the probability that the structural demand parameters exceed a threshold.

[0024]

[0025] Where P[DM│IM] is the probability of damage to structural components exceeding a certain performance level under seismic intensity IM, R is the seismic response of structural components under IM, and L is the limit value of the performance level.

[0026] Furthermore, the specific operation process of step (2) is to solve the time history curve of the ground motion acceleration:

[0027] (1)

[0028] Among them, F0(X) g1，。。。， X gn， ω) is the random Fourier spectral function, X g1，。。。， X gn The random variables inside are the base amplitude, frequency, and damping ratio, where ω is the angular frequency and i represents the imaginary number. unit, and These are the equivalent damping ratio and fundamental circular frequency of the engineering site, respectively, and the random ground motion at the bedrock is a white noise process.

[0029] By performing an inverse Fourier transform on formula (1), the corresponding acceleration time history generated by the stochastic ground motion model based on physical mechanisms is obtained. It can be represented as:

[0030] (2)

[0031] Where t is time and e is the natural constant.

[0032] Further, the specific operation steps of step (5) are as follows: the seismic response Y(t) of the structure:

[0033] (3)

[0034] In the formula, t is time, and H is... y It is the seismic response function, where Θ represents an independent random variable;

[0035] Based on the principle of probability conservation, the probability density function of the structural seismic response Y can be solved:

[0036] (4)

[0037] In the formula, This represents the joint probability density function of a stochastic dynamical system. This is the partial derivative of the physical quantity of interest with respect to time when Θ=θ.

[0038] By summing all the discrete numerical solutions above, we can obtain the instantaneous probability density function of Y at any time:

[0039] (5)

[0040] Furthermore, the specific solution process is as follows:

[0041] (1) Select a typical representative point θ in the probability space x And determine the corresponding probability of assignment;

[0042] (2) Based on the selected typical representative point θ x A large number of random samples are generated, followed by extensive deterministic dynamical system analysis to obtain the stochastic dynamic response of the structure and the partial derivatives of the physical quantities of interest with respect to time. , where j = 1, 2, ..., n;

[0043] (3) For each representative point θ x The corresponding partial derivative of the structural response Y(t) with respect to time Introducing the generalized probability density evolution formula (4), the numerical solution P is obtained based on the initial and boundary conditions. YΘ (y,θ,t);

[0044] (4) Use formula (5) to process all discrete solutions P obtained in step 3. YΘ Integrating (y, θ, t), we finally obtain the numerical solution P of the generalized probability density evolution equation. Y (y,t).

[0045] Furthermore, the derivation process of the improved seismic vulnerability function based on GPDEM is as follows: seismic vulnerability represents the probability that the structural demand parameters exceed a threshold, as shown in equation (6):

[0046] (6)

[0047] Wherein, P[DM│IM] is the probability of damage to structural components exceeding a certain performance level under seismic intensity IM, R is the seismic response of structural components under IM, and L is the limit value of the performance level;

[0048] For seismic vulnerability analysis of engineering structures, the extreme value of the structural response is used as the performance index (Rmax), as shown in equation (7):

[0049] (7)

[0050] Therefore, equation (6) can also be equivalently expressed as:

[0051] (8)

[0052] In the formula, Rmax is the threshold for different damage states of structural components, and the exceedance probability of structural damage under each seismic intensity IM is the determined value of the limit state in the cumulative probability density function curve of each IM.

[0053] The beneficial effects of this invention are:

[0054] (1) Traditional vulnerability methods require assumptions about the distribution of ground motion intensity and damage indicators, such as log-normal distribution. This method uses probability density evolution, which does not require pre-setting the distribution form and preserves the true distribution characteristics of the data.

[0055] (2) Traditional vulnerability methods rely on a limited number of deterministic ground motion records and cannot take into account the random characteristics of ground motion. This method can reasonably characterize the uncertainty of ground motion and avoid the evaluation error caused by ground motion selection bias.

[0056] (3) Traditional vulnerability methods for double-span subway station structures generally only consider the damage to the central column structure of the subway and ignore the damage to other structural components. This method considers the impact of all structural components under different damage states on the overall structural vulnerability.

[0057] (4) The Monte Carlo method requires a large number of samples and has high computational cost. This method uses the GF difference selection method to optimize the distribution of points in the probability space, which effectively improves the computational efficiency (by about 33%). Attached Figure Description

[0058] Figure 1 A flowchart of the analysis method provided in the embodiments of this application.

[0059] Figure 2 This is a schematic diagram of random bedrock ground motion provided in an embodiment of this application.

[0060] Figure 3 This is the cumulative probability density curve of the sidewall provided in the embodiments of this application.

[0061] Figure 4 This is the cumulative probability density curve of the central column provided in the embodiments of this application.

[0062] Figure 5 This is the cumulative probability density curve of the top plate provided in the embodiments of this application.

[0063] Figure 6The seismic vulnerability curves of each component of a typical double-span underground station structure provided in the embodiments of this application. Figure 7 The damage weighting coefficients for each structural component under minor damage conditions are provided in the embodiments of this application. Figure 8 The damage weighting coefficients for each structural component under moderate damage conditions are provided in the embodiments of this application. Figure 9 This refers to the damage weighting coefficients for each structural component under severe damage conditions provided in the embodiments of this application. Figure 10 The seismic vulnerability curve of a typical double-span underground station structure provided in the embodiments of this application. Detailed Implementation

[0064] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0065] like Figure 1 This paper presents a method for seismic vulnerability analysis of underground railway stations based on probability damage weights.

[0066] First, for the selected research object, the geometric dimensions of the underground station were determined, and relevant geological survey reports and material mechanical parameters of concrete and steel reinforcement were collected. Based on this, this invention uses ABAQUS finite element software to establish a two-dimensional double-span subway station numerical model considering the dynamic interaction between the soil and the station structure.

[0067] Step (2): Through probability space sampling, a set of representative points with probability weights are obtained. Then, using a stochastic ground motion model, stochastic bedrock ground motions corresponding to each representative point are synthesized, such as... Figure 2 As shown;

[0068] Using a physical mechanism-based stochastic ground motion model, the ground motion acceleration time history curve is generated in the frequency domain using the following formula. :

[0069] (1)

[0070] Among them, F0(X) g1，。。。， X gn， ω) is the random Fourier spectral function, X g1，。。。， X gnThe random variables within are the base amplitude, frequency, and damping ratio. ω is the angular frequency, and i represents the imaginary number. The unit. and These are the equivalent damping ratio and the fundamental circular frequency of the engineering site, respectively. Random ground motions in the bedrock are usually assumed to be white noise processes.

[0071] By performing an inverse Fourier transform on formula (1), the corresponding acceleration time history generated by the stochastic ground motion model based on physical mechanisms is obtained. It can be represented as:

[0072] (2)

[0073] Where t is time and e is the natural constant.

[0074] This invention targets a subway station structure, with the input ground motion being bedrock ground motion; therefore, the site type is Class I. Based on this Class I site type, the basic random variable (X) in the stochastic ground motion model is determined. g1 , ..., X gn The mean and coefficient of variation of the random variables in the random ground motion model were determined by using the generalized F-bias point selection method to divide the probability space and select representative points of the basic random variables, thus generating the corresponding ground motion acceleration time history curves. Based on a literature review, the mean and coefficient of variation of the random variables for site type I in the random ground motion model are shown in Table 1.

[0075] Table 1. Mean and coefficient of variation of random variables in the random ground motion model

[0076] mean coefficient of variation Base amplitude 0.22 0.50 frequency 18 0.40 Damping ratio 0.65 0.30

[0077] Step (3): Adjust the amplitude of all bedrock ground motion time histories generated in step (2) to the same target intensity and apply them as input loads to the numerical model established in step (1). Subsequently, perform deterministic nonlinear time history analysis on each ground motion record to obtain the key response time histories of the structure;

[0078] Step (4): Adjust the seismic intensity, repeat step (3), and perform incremental dynamic analysis to obtain the key response time history of structures with different seismic intensities;

[0079] Step (5): Substitute the structural critical response time histories obtained in steps (3) and (4) into the generalized probability density evolution equation, and apply the equivalent extreme event theory to calculate the cumulative distribution function of the extreme values ​​of the structural seismic performance index under different seismic motion intensities, such as... Figure 3 As shown;

[0080] The seismic response Y(t) of the structure can be calculated using formula (3):

[0081] (3)

[0082] In the formula, t is time, and H is... y It is the seismic response function, where Θ represents an independent random variable. The probability density function of the random ground motion modulated to the same intensity in step (3) of this invention is denoted as P. Θ (θ).

[0083] The above-mentioned stochastic system satisfies the probability conservation condition. Based on the description of random events according to the principle of probability conservation, the generalized probability density evolution equation of the nonlinear system can be obtained, and thus the probability density function of the structural seismic response Y can be solved:

[0084] (4)

[0085] In the formula, This represents the joint probability density function of a stochastic dynamical system. This is the partial derivative of the physical quantity of interest with respect to time when Θ=θ. By summing all the discrete numerical solutions above, the instantaneous probability density function (PDF) of Y at any given time can be calculated using the following formula:

[0086] (5)

[0087] Equation (4) is solved using the General probability density evolution method (GPDEM) proposed by Li Jie and Chen Jianbing. The specific solution process is as follows:

[0088] (1) Partitioning of the probability space. A typical representative point θ is selected in the probability space. x And determine the corresponding probability of assignment.

[0089] (2) Generation of random samples and calculation of dynamic response. Based on the selected typical representative point θ x A large number of random samples are generated, followed by extensive deterministic dynamical system analysis to obtain the stochastic dynamic response of the structure and the partial derivatives of the physical quantities of interest with respect to time. , where j=1,2,…,n.

[0090] (3) Solving the probability density evolution equation. For each representative point θ... x The corresponding partial derivative of the structural response Y(t) with respect to time The generalized probability density evolution formula (4) is introduced, and the numerical solution P is obtained based on the initial conditions and boundary conditions. YΘ(y,θ,t).

[0091] (4) Summing over all discrete points yields the final solution. Formula (5) is used to sum all discrete solutions P obtained in step 3. YΘ Integrating (y, θ, t), we finally obtain the numerical solution P of the generalized probability density evolution equation. Y (y,t).

[0092] Step (6): Derive the vulnerability curves of each subway station structure based on the cumulative probability density function obtained in step (5), such as... Figure 4 As shown, the damage probability for different seismic intensities is determined.

[0093] Seismic vulnerability represents the probability that a structure's demand parameters exceed a threshold (or capacity), as shown in Equation (6):

[0094] (6)

[0095] Where P[DM│IM] is the probability of damage to structural components exceeding a certain performance level under seismic intensity IM. R is the seismic response of the structural component under IM, and L is the limit value of the performance level. As mentioned earlier, traditional vulnerability analysis assumes a log-normal distribution of structural demand and capacity, which may lead to inaccurate vulnerability functions. Therefore, this paper improves the calculation of seismic vulnerability functions based on GPDEM, which can effectively consider the complete non-stationarity of ground motion.

[0096] For seismic vulnerability analysis of engineering structures, the extreme values ​​of the structural response are usually used as performance indicators (R0). max As shown in equation (7):

[0097] (7)

[0098] Therefore, equation (6) can also be equivalently expressed as:

[0099] (8)

[0100] In the formula, R max The threshold values ​​represent the structural components under different damage states. The exceedance probability of structural damage under each seismic intensity IM is the definite value of the limit state in the cumulative probability density function (CDF) curve of each IM. Therefore, equation (8) can also be expressed as:

[0101] (9)

[0102] Among them, CDF R│IM To determine the CDF value of PI under IM conditions; pR│IM It is a PDF that determines the extreme values ​​of the structural component response under IM conditions.

[0103] Through literature review, R values ​​under different damage states of the station were obtained. max As shown in Table 2.

[0104] Table 2 Classification of damage state thresholds for different structural components

[0105] member Central column side walls roof <![CDATA[Maximum tilt angle θ max > <![CDATA[Maximum tilt angle θ max > <![CDATA[Maximum settlement S max (mm)]]> No damage state R0 <![CDATA[θ max <0.08%]]> <![CDATA[θ max <0.046%]]> <![CDATA[S max <0.1526]]> Minor injury status R1 <![CDATA[0.08%≤θ max <0.37%]]> <![CDATA[0.046%≤θ max <0.23%]]> <![CDATA[0.1526≤S max <0.3297]]> Moderate injury state R2 <![CDATA[0.37%≤θ max <0.84%]]> <![CDATA[0.23%≤θ max <0.48%]]> <![CDATA[0.3297≤S max <8.398]]> Severe injury status R3 <![CDATA[0.84%≤θ max ]]> <![CDATA[0.48%≤θ max ]]> <![CDATA[8.398≤S max ]]>

[0106] Step (7): Calculate the damage weight coefficient for each structural component based on the damage probability calculated in Step 6.

[0107] To discuss the seismic vulnerability of the overall structure, we should first determine the damage contribution of each structural member to the overall structure, i.e., the damage weighting coefficient κ, which is defined by equation (10):

[0108] (10)

[0109] Where κ is the damage weight coefficient of structural member i when the damage state is Ri under a given seismic intensity IM. i Let represent the damage probability of structural member i under damage state Ri, and n represent the total number of members. The damage weighting coefficients for different damage states are as follows: Figure 5 As shown, some ground motion intensities do not reach the level of moderate or severe damage, therefore no damage weighting coefficient is applied.

[0110] Step (8): Based on the damage weighting coefficient calculated in Step 7 and the damage probability calculated in Step 6, calculate the overall vulnerability curve of the entire subway station structure as follows: Figure 6 As shown.

[0111] Based on the failure probability and damage weight coefficient of each structural component under each damage state, the seismic vulnerability curve of the underground structure can be obtained from equations (6) to (10) as equation (11):

[0112] (11)

[0113] In the formula, P i Let κ represent the exceedance probability of structural member i under different damage states under earthquake intensity IM. i , where represents the damage weighting coefficient for structural member i under different damage states under earthquake intensity IM.

[0114] The above embodiments are not intended to limit the present invention. Unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; a mechanical connection or an electrical connection; a direct connection or an indirect connection through an intermediate medium; or a connection within two components. Those skilled in the art can understand the specific meaning of the above terms in this application based on the specific circumstances. The present invention is not limited to the above examples. Changes, modifications, additions, or substitutions made by those skilled in the art within the scope of the technical solutions of the present invention are also within the protection scope of the present invention. Furthermore, the technical features involved in the different embodiments of the present application described above can be combined with each other as long as they do not conflict with each other.

[0115] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

Claims

1. A method for seismic vulnerability analysis of underground railway stations based on probabilistic damage weighting, characterized in that, The steps of the method include: Step (1): Collect geological survey reports and underground structure materials; Step (2): By sampling in probability space, a set of representative points with probability weights are obtained. Using a random ground motion model, random bedrock ground motions corresponding to each representative point are synthesized. Step (3): Adjust the amplitude of all bedrock ground motion time histories generated in step (2) to the same target intensity, and apply them as input loads to the numerical model established in step (1). Perform deterministic nonlinear time history analysis on each ground motion record to obtain the ground motion acceleration time history curve. Step (4): Adjust the seismic intensity, repeat step (3), perform incremental dynamic analysis, and obtain the key response time history of structures with different seismic intensities; Step (5): Substitute the structural critical response time histories obtained in Step (3) and Step (4) into the generalized probability density evolution equation, and apply the equivalent extreme event theory to calculate the cumulative distribution of the extreme values ​​of the structural seismic performance index under different ground motion intensities. Step (6): Derive the vulnerability curves of each subway station structure based on the cumulative probability density function obtained in step (5) and determine the damage probability of different seismic intensities. The calculation of the improved seismic vulnerability function based on GPDEM is expressed as follows: , In the above formula, L is the limit value of the performance level, R is the seismic response of the structural component under IM, and R max For different damage states of structural components, the exceedance probability of structural damage under each seismic intensity IM is the cumulative probability density function of each IM, i.e., the definite value of the limit state in the CDF curve. R│IM To determine the CDF value of PI under IM conditions, p R│IM It is a PDF that determines the extreme values ​​of the structural component response under IM conditions; Step (7): Calculate the damage weight coefficient for each structural component based on the damage probability calculated in step (6); determine the damage contribution of each structural component to the overall structure, i.e., the damage weight coefficient κ: ， Where κ is the damage weighting coefficient of structural member i when the damage state is Ri under a given seismic intensity IM, p i Let represent the damage probability of structural component i when the damage state is Ri, and n represent the total number of components; Step (8): Based on the damage weight coefficient calculated in step (7) and the damage probability calculated in step (6), calculate the overall vulnerability curve of the entire subway station structure; based on the failure probability and damage weight coefficient of each structural component under each damage state, obtain the seismic vulnerability curve of the underground structure: ， In the above formula, P i Let κ represent the exceedance probability of structural member i under different damage states under earthquake intensity IM. i , where represents the damage weighting coefficient of structural member i under different damage states under earthquake intensity IM.

2. The seismic vulnerability analysis method for underground railway stations based on probabilistic damage weights according to claim 1, characterized in that, The derivation of the seismic vulnerability function is as follows: seismic vulnerability represents the probability that the structural demand parameters exceed a threshold. , Where P[DM│IM] is the probability of damage to structural components exceeding a certain performance level under seismic intensity IM, R is the seismic response of structural components under IM, and L is the limit value of the performance level.

3. The seismic vulnerability analysis method for underground railway stations based on probabilistic damage weights according to claim 1, characterized in that, The specific operation process of step (2) is to solve the time history curve of ground motion acceleration: (1)， Among them, F0(X) g1，。。。， X gn， ω) is the random Fourier spectral function, X g1，。。。， X gn The random variables inside are the base amplitude, frequency, and damping ratio, where ω is the angular frequency and i represents the imaginary number. unit, and These are the equivalent damping ratio and fundamental circular frequency of the engineering site, respectively, and the random ground motion at the bedrock is a white noise process. By performing an inverse Fourier transform on formula (1), the corresponding acceleration time history generated by the stochastic ground motion model based on physical mechanisms is obtained. It can be represented as: (2)， Where t is time and e is the natural constant.

4. The seismic vulnerability analysis method for underground railway stations based on probabilistic damage weights according to claim 1, characterized in that, The specific operation steps of step (5) are as follows: the seismic response Y(t) of the structure: (3)， In the formula, t is time, and H is... y It is the seismic response function, where Θ represents an independent random variable; Based on the principle of probability conservation, the probability density function of the structural seismic response Y can be solved: (4) In the formula, This represents the joint probability density function of a stochastic dynamical system. This is the partial derivative of the physical quantity of interest with respect to time when Θ=θ. By summing all the discrete numerical solutions above, we can obtain the instantaneous probability density function of Y at any time: (5)。 5. The seismic vulnerability analysis method for underground railway stations based on probabilistic damage weights according to claim 4, characterized in that, The specific solution process is as follows: (1) Select a typical representative point θ in the probability space x And determine the corresponding probability of assignment; (2) Based on the selected typical representative point θ x A large number of random samples are generated, followed by extensive deterministic dynamical system analysis to obtain the stochastic dynamic response of the structure and the partial derivatives of the physical quantities of interest with respect to time. , where j = 1, 2, ..., n; (3) For each representative point θ x The corresponding partial derivative of the structural response Y(t) with respect to time Introducing the generalized probability density evolution formula (4), the numerical solution P is obtained based on the initial and boundary conditions. YΘ (y,θ,t); (4) Use formula (5) to process all discrete solutions P obtained in step 3. YΘ Integrating (y, θ, t), we finally obtain the numerical solution P of the generalized probability density evolution equation. Y (y,t).

6. The seismic vulnerability analysis method for underground railway stations based on probabilistic damage weights according to claim 1, characterized in that, The derivation process of the improved seismic vulnerability function based on GPDEM is as follows: seismic vulnerability represents the probability that the structural demand parameters exceed a threshold, as shown in equation (6): (6) Wherein, P[DM│IM] is the probability of damage to structural components exceeding a certain performance level under seismic intensity IM, R is the seismic response of structural components under IM, and L is the limit value of the performance level; For seismic vulnerability analysis of engineering structures, the extreme value of the structural response is used as the performance index (Rmax), as shown in equation (7): (7) Therefore, equation (6) can also be equivalently expressed as: (8) In the formula, Rmax is the threshold for different damage states of structural components, and the exceedance probability of structural damage under each seismic intensity IM is the determined value of the limit state in the cumulative probability density function curve of each IM.

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