Intake tower anti-seismic reliability maximum entropy calculation method based on dynamic integral boundary

Abstract

The invention discloses an intake tower anti-seismic reliability maximum entropy calculation method based on a dynamic integral boundary. The method comprises the following steps: firstly, determining a plurality of anti-seismic performance levels of an intake tower and corresponding tower top displacement thresholds; taking the seismic oscillation peak acceleration as a random variable, and generating sample points by using a layered Latin hypercube sampling method; adopting a C-P power spectrum model to generate a random seismic wave sample; obtaining a structure response value through finite element dynamic analysis, and constructing a performance function, thereby calculating first four-order statistical moments of the performance function; dynamically determining an integral boundary for each performance level; and by taking the statistical moment as a constraint, fitting a probability density function of the performance function by applying a maximum entropy method in a dynamic boundary, and carrying out integral calculation on the probability density function to obtain a failure probability and a reliable index. Thus, the problem that the fixed integral boundary and the performance level of a traditional maximum entropy method are disjointed is solved, the calculation efficiency is guaranteed, and meanwhile an accurate quantification means is provided for evaluating the anti-seismic reliability of the water inlet tower.

Description

Technical Field

[0001] This application relates to the field of seismic reliability analysis technology for water intake towers, and in particular to a method for calculating the maximum entropy of seismic reliability of water intake towers based on dynamic integral boundary. Background Technology

[0002] Intake towers are common and important structures in hydraulic engineering projects, mainly built in reservoirs, dams, or water diversion projects. Their structural safety is crucial to the overall project, but their tall, isolated, and thin-walled characteristics make them susceptible to damage or even collapse under strong earthquakes. Since strong earthquakes are random, accurately simulating these random vibrations and efficiently assessing the seismic reliability of intake towers has become a core issue in ensuring their safety.

[0003] Accurate simulation of seismic ground motion is fundamental to the seismic reliability analysis of water intake towers, and the power spectral density function is a core tool for describing the frequency characteristics of random seismic ground motion. In existing technologies, the stationary white noise power spectrum model pioneered seismic ground motion simulation. Subsequently, related research proposed filtering the stationary white noise process through a single-degree-of-freedom system to obtain the absolute acceleration power spectrum (i.e., the Kanai-Tajimi power spectrum model). However, this model has significant drawbacks: it overestimates the low-frequency components of seismic ground motion, fails to accurately reflect the actual frequency characteristics of bedrock ground motion, and is difficult to calculate the finite variances of surface displacement, velocity, and acceleration.

[0004] To compensate for the shortcomings of the Kanai-Tajimi power spectrum, existing technologies include various improved models, all based on modifications and optimizations of the Kanai-Tajimi power spectrum. Among them, the Clough-Penzien (CP) power spectral density model is widely accepted due to its clear physical meaning and superior low-frequency component handling capabilities. This model assumes that seismic waves must pass through two soil layers before reaching the structural foundation, with each soil layer considered as a filter. As the seismic wave passes through the filter sequentially, its low-frequency components are gradually corrected. Compared to traditional models, this model better reflects the actual propagation patterns of seismic motions and has become one of the preferred models for generating random ground motions.

[0005] In the field of structural stochastic dynamic reliability analysis, classical Monte Carlo simulation (MCS) is the traditional benchmark method. However, it requires tens of thousands of finite element iterations, resulting in extremely low efficiency and making it difficult to meet the analytical needs of high-dimensional nonlinear systems. To improve computational efficiency, researchers have proposed the method of moments for structural reliability. This method does not require iteration or differentiation, and the calculation process is simple. However, it suffers from large computational errors, making it difficult to meet the requirements of high-precision analysis.

[0006] The maximum entropy method, based on information entropy theory, is gradually becoming an important technical direction for structural reliability calculation. This method constructs a probability density function (PDF) and a cumulative distribution function (CDF) to maximize entropy under given statistical moment constraints. It requires only a small number of sample moments to accurately fit the failure probability, significantly improving computational efficiency. Related research has successfully achieved seismic reliability assessment of transmission towers with varying leg lengths using the sample moment-maximum entropy method, with an error of less than 3%, verifying the feasibility of this method in engineering structures. Subsequent research further demonstrates its robustness in the analysis of strongly nonlinear systems. Existing techniques, by introducing a refined moment estimation model, achieve quantitative characterization of nonlinear correlations, verifying the accuracy of the maximum entropy method in handling complex nonlinear problems.

[0007] However, applying the traditional maximum entropy method to the seismic reliability assessment of water intake towers reveals key technical limitations: First, the fixed integration boundary is disconnected from the classification of seismic performance levels of the water intake tower. The seismic performance of the water intake tower is divided into clearly defined performance levels based on the tower top displacement threshold (such as basically intact, slightly damaged, severely damaged, collapsed, etc.). The fixed boundary cannot adapt to the performance index distribution characteristics of different failure modes, resulting in a decrease in the fitting accuracy of the PDF in the critical threshold range. Second, insufficient tail accuracy. For low-probability extreme events such as collapse, the fixed boundary easily truncates the response tail data, causing deviations in the failure probability calculation. These limitations make it difficult for the traditional maximum entropy method to simultaneously consider both the "fitting accuracy of the main damage range" and the "calculation accuracy of the extreme failure tail," failing to meet the high-precision seismic reliability assessment requirements of water intake towers as critical structures.

[0008] Currently, although significant progress has been made in random ground motion simulation technology and structural reliability calculation methods, there are still core technology gaps in the engineering application of seismic reliability analysis for water intake towers. On the one hand, some existing ground motion simulation models suffer from problems such as unreasonable low-frequency processing and unclear physical meaning, while the engineering application of high-quality models such as CP spectrum still requires parameter matching optimization based on specific structural characteristics. On the other hand, among existing reliability calculation methods, the MCS method is too inefficient, the traditional maximum entropy method has poor boundary adaptability, and other methods such as response surface methodology are difficult to balance computational accuracy and efficiency. Summary of the Invention

[0009] This application provides a method for calculating the maximum entropy of the seismic reliability of water intake towers based on dynamic integral boundaries. This method solves the problems of disconnect between fixed integral boundaries and performance levels, and insufficient accuracy of tail calculations in the prior art, and provides technical support for the seismic design and safety assessment of water intake towers.

[0010] To achieve the above objectives, the technical solution of this invention is as follows:

[0011] In a first aspect, embodiments of the present invention provide a method for calculating the maximum entropy of the seismic reliability of a water intake tower based on a dynamic integral boundary, comprising: determining multiple seismic performance levels of the water intake tower structure and determining the tower top displacement threshold corresponding to each seismic performance level; taking the peak ground acceleration as a random variable and, based on its probability distribution, extracting a set of sample points from the random variable using a stratified Latin hypercube sampling method; for each sample point, generating a random seismic wave sample corresponding to the sample point using a Clough-Penzien power spectrum model; inputting the random seismic wave sample into the finite element model of the water intake tower structure for dynamic response analysis, and obtaining the tower top of the water intake tower corresponding to each sample point. The maximum horizontal relative displacement is used as the structural response value. For each seismic performance level, a function is constructed based on the tower top displacement threshold and structural response value corresponding to the seismic performance level. Based on the function value calculated for each sample point, at least the first four origin moments of the function value set are calculated as statistical moments. The integration boundary used for the maximum entropy method is dynamically determined based on the statistical characteristics of the tower top displacement threshold and structural response value corresponding to the current seismic performance level. Based on the integration boundary, the probability density function of the function is fitted within the integration boundary using the maximum entropy method. The fitted probability density function is integrated to calculate the failure probability and reliability index under the current seismic performance level.

[0012] In some possible implementations, seismic performance levels include basically intact, slightly damaged, moderately damaged, severely damaged, and collapsed / destroyed; among them, the tower top displacement threshold corresponding to basically intact is 0~0.02m, the tower top displacement threshold corresponding to slightly damaged is 0.02~0.2m, the tower top displacement threshold corresponding to moderately damaged is 0.2~0.3m, the tower top displacement threshold corresponding to severely damaged is 0.3~0.8m, and the tower top displacement threshold corresponding to collapsed / destroyed is greater than 0.8m.

[0013] In some possible implementations, a set of sample points is drawn from a random variable using a stratified Latin hypercube sampling method, including: determining the global range of values ​​for the random variable as [ , ], divide it into There are three equal-volume first-level subregions, and the interval length of each first-level subregion is expressed as follows:

[0014] , =1, 2, ..., ;

[0015] The variable range within each first-level subregion is further divided into... There are two equal-volume second-level hierarchical intervals, and the length of each second-level hierarchical interval is expressed as:

[0016] , =1, 2, ..., M;

[0017] In the The first-level sub-region Sample points are randomly selected from each of the two-level stratified intervals. The formula for the sample points is as follows:

[0018] ;

[0019] in, ~(0,1) is a uniform random number within the interval (0,1), used to randomly select sample points within the second-level stratified interval; For the first The first-level sub-region, the first Random samples generated in two secondary strata.

[0020] In some possible implementations, the power spectral density of the Clough-Penzien power spectral model is expressed as:

[0021] ;

[0022] in, and These are the optimal circular frequency and damping ratio of the first filter layer, respectively; and These are the superior circular frequency and damping ratio of the second filter layer, respectively; Represents angular frequency. Spectral density;

[0023] stochastic parameter vectors of the Clough-Penzien power spectrum model for:

[0024] .

[0025] In some possible implementations, the function expression is as follows:

[0026] ;

[0027] in, It is the first The tower top displacement threshold corresponding to each seismic performance level It is the structural response value; when When <0, it indicates that the water intake tower structure is in the first stage. The failure states corresponding to each seismic performance level.

[0028] In some possible implementations, the origin moments of the set of function values ​​are represented as:

[0029] ;

[0030] ;

[0031] ;

[0032] in, for First-order origin moment, Let Z be the mean of the function Z. For the sample size, Let Z be the standard deviation.

[0033] In some possible implementations, the statistical characteristics of the structural response values ​​include the average value of the structural response values. and standard deviation The dynamic determination of the integral boundary is based on the type calculation of the seismic performance level, and is expressed as:

[0034] When the current seismic performance level is severely damaged, the main body of the response distribution is covered, and the mean ±3 is used. With threshold correction, the integral boundary is represented as:

[0035] ;

[0036] in, The threshold for tower top displacement that severely disrupts the leveling system. An extension factor greater than 1;

[0037] When the current seismic performance level is at collapse failure, focus on the tail response and use a threshold +2. Extending this, the boundary of the integral is expressed as:

[0038] ;

[0039] in, The threshold displacement of the tower top at the level of collapse and destruction. , The integral boundary of the structural response value. , This represents the integral boundary of the function.

[0040] Among some possible implementations, the maximum entropy method is used to fit the probability density function of the function function within the integral boundary, including:

[0041] Will Substituting the first-order origin moment into the Lagrange multiplier method optimization model, dynamic adaptation between the integral boundary and the seismic performance threshold of the intake tower is achieved; the objective function of the Lagrange multiplier method optimization model is:

[0042] ;

[0043] in, For entropy, , It is a Lagrange multiplier;

[0044] The probability density function of the function is expressed as:

[0045] ;

[0046] Based on the probability density function expression, the failure probability and reliability index are calculated according to the following formula:

[0047] ;

[0048] in, This represents the probability of failure. As a reliable indicator, This is the cumulative distribution function of the standard normal distribution.

[0049] One or more technical solutions provided in the embodiments of the present invention have at least the following technical effects or advantages:

[0050] In this embodiment of the invention, based on the displacement thresholds of different seismic performance levels of the intake tower, the integral boundary is dynamically determined by combining the statistical characteristics of the structural response, establishing a correlation mechanism between performance level, displacement threshold, and integral boundary to avoid fitting bias caused by the disconnect between fixed boundaries and failure modes. A hierarchical Latin hypercube sampling method is used to extract random variable samples, avoiding the local clustering problem of traditional sampling and achieving a more uniform spatial distribution with a small number of samples, providing a high-quality data foundation for statistical moment calculation. Simultaneously, a Clough-Penzien power spectrum model is used to generate random seismic waves, and low-frequency components are corrected through two layers of soil filters, conforming to the actual physical mechanism of seismic wave propagation, improving the realism of the seismic motion input, and ensuring the accuracy of structural dynamic response analysis. Furthermore, by leveraging the efficiency of hierarchical Latin hypercube sampling, the optimization of the integral range by dynamic boundaries, and the efficient use of a small number of statistical moments by the maximum entropy method, the number of calculation iterations is significantly reduced, solving the problem of low simulation efficiency in traditional methods and significantly shortening the seismic reliability assessment cycle. While improving efficiency, it maintains high computational accuracy, outperforming traditional fixed-boundary maximum entropy methods and response surface methods. It resolves the contradiction of insufficient accuracy or low efficiency in traditional methods, and meets the practical needs of seismic design and safety assessment of water intake towers. Attached Figure Description

[0051] To more clearly illustrate the embodiments of the present invention, the accompanying drawings used in the embodiments of the present invention will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0052] Figure 1 A schematic flowchart of an embodiment of the method for calculating the maximum entropy of seismic reliability of a water intake tower based on a dynamic integral boundary, provided by an embodiment of the present invention;

[0053] Figure 2 This is a schematic diagram of the tower top displacement IDA curve in an embodiment of the present invention;

[0054] Figure 3 This is a schematic diagram of seismic wave samples in an embodiment of the present invention;

[0055] Figure 4 This is a schematic diagram of the seismic wave power spectrum in an embodiment of the present invention;

[0056] Figure 5 This is a schematic diagram comparing the two-dimensional distribution of samples in an embodiment of the present invention;

[0057] Figure 6 This is a schematic diagram of the three-dimensional finite element model of the water intake tower in an embodiment of the present invention;

[0058] Figure 7 This is a schematic diagram of the finite element mesh generation of the water intake tower body in an embodiment of the present invention;

[0059] Figure 8 This is a schematic diagram of the x-direction of the maximum horizontal relative displacement at the top of the tower in an embodiment of the present invention;

[0060] Figure 9 This is a schematic diagram of the extreme value of the maximum horizontal relative displacement at the top of the tower in the y-direction in an embodiment of the present invention;

[0061] Figure 10 This is a schematic diagram of the structural response along the x-direction corresponding to the seismic wave sample in an embodiment of the present invention;

[0062] Figure 11 This is a schematic diagram of the structural response along the y-direction corresponding to the seismic wave sample in an embodiment of the present invention;

[0063] Figure 12 This is a PDF schematic diagram of the water intake tower structure under severe damage in an embodiment of the present invention;

[0064] Figure 13 This is a PDF schematic diagram of the collapsed and damaged water intake tower structure in an embodiment of the present invention;

[0065] Figure 14A comparison chart of CDF calculated by the method of this invention and empirical CDF under severe damage to the intake tower structure;

[0066] Figure 15 A comparison chart of CDF calculated by the method of this invention and empirical CDF under the condition of water tower structure collapse and destruction. Detailed Implementation

[0067] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0068] In the relevant descriptions of this embodiment, the terms "including," "containing," and "possessing" are all open terms and are generally understood to include but not be limited to; the term "at least one" is generally understood to mean one or more, where "multiple" refers to two or more; the term "at least one of the following" or similar expressions refer to any combination of these items, including any combination of single or multiple items, for example, "at least one of a, b, or c", or "at least one of a, b, and c", which can all mean: a, b, c, ab (i.e., a and b), ac, bc, or abc, where a, b, and c can be single or multiple; the symbol "A / B" is used to describe the selection relationship of associated objects, generally indicating an "or" relationship.

[0069] In the following description of the embodiments, the terminology used in the embodiments of this application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. The singular forms "a" and "the" as used in the embodiments of this application and the appended claims are also intended to include the plural forms, unless the context clearly indicates otherwise.

[0070] Those skilled in the art should understand that, in the following description of the embodiments of this application, the sequence of numbers does not imply the order of execution. Some or all steps may be executed in parallel or sequentially. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of this application.

[0071] Those skilled in the art will understand that the numerical ranges in the embodiments of this application should be understood to specifically disclose each intermediate value between the upper and lower limits of the range. Any stated value or intermediate value within a stated range, as well as any other stated value or each smaller range between intermediate values ​​within a range, are also included within this invention. The upper and lower limits of these smaller ranges may be independently included or excluded from the range.

[0072] Unless otherwise stated, the technical / scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains. While this application describes only preferred methods and materials, any methods and materials similar or equivalent to those described herein may be used in the implementation or testing of this application. All references to this specification are incorporated by way of citation to disclose and describe the methods and / or materials associated with those references. In the event of any conflict with any incorporated reference, the content of this specification shall prevail.

[0073] To illustrate the technical solution of the present invention, specific embodiments are described below.

[0074] Figure 1 A schematic flowchart illustrating an embodiment of the maximum entropy calculation method for seismic reliability of water intake towers based on dynamic integral boundary conditions provided by this invention is shown below. Figure 1 As shown, the above method may include:

[0075] S101, determine multiple seismic performance levels of the water intake tower structure, and determine the tower top displacement threshold corresponding to each seismic performance level;

[0076] S102, taking peak ground acceleration as a random variable, and based on its probability distribution, a set of sample points are drawn from the random variable using the stratified Latin hypercube sampling method;

[0077] S103, For each sample point, the Clough-Penzien power spectrum model is used to generate random seismic wave samples corresponding to the sample point;

[0078] S104. Input random seismic wave samples into the finite element model of the water intake tower structure for dynamic response analysis, and obtain the maximum horizontal relative displacement of the top of the water intake tower corresponding to each sample point as the structural response value.

[0079] S105, for each seismic performance level, construct a function based on the tower top displacement threshold and structural response value corresponding to the seismic performance level, and calculate at least the first four original moments of the function value set as statistical moments based on the function function value calculated for each sample point;

[0080] S106, Based on the statistical characteristics of the tower top displacement threshold and structural response value corresponding to the current seismic performance level, dynamically determine the integral boundary used for the maximum entropy method calculation;

[0081] S107, based on the integral boundary, the probability density function of the function is fitted within the integral boundary using the maximum entropy method;

[0082] S108, integrate the fitted probability density function to calculate the failure probability and reliability index under the current seismic performance level.

[0083] It should be noted that the Kanai-Tajimi spectrum assumes that bedrock earthquake acceleration is white noise, meaning it cannot accurately reflect the frequency characteristics of bedrock ground motion. Furthermore, it overestimates the low-frequency components of ground motion and cannot calculate the finite variances of surface displacement, velocity, and acceleration. To address this issue, the Kanai-Tajimi power spectrum model can be modified by cascading a single-degree-of-freedom filter. This results in the Clough-Penzien power spectrum model, which effectively filters out extremely low-frequency excitations. By cascading an additional soil filter into the Kanai-Tajimi power spectrum model and considering the process of seismic waves traveling from bedrock through two soil layers to the structure, the corresponding power spectral density expression is obtained:

[0084] ;

[0085] in, and These are the optimal circular frequency and damping ratio of the first filter layer, respectively; and These are the superior circular frequency and damping ratio of the second filter layer, respectively; Represents angular frequency. Spectral density;

[0086] For example, the high-frequency cutoff frequency and the low-frequency cutoff frequency can be 15 rad / s and 0.3 rad / s respectively, and the values ​​of the above parameters can be shown in Table 1:

[0087] Table 1

[0088]

[0089] At this point, the random parameter vector of the Clough-Penzien power spectral model for:

[0090] .

[0091] Among them, the above 5 parameters, The value can range from 0.1 to 0.15 times. , Can be taken and Same value.

[0092] In this embodiment of the invention, the state variable model for structural reliability analysis can be the state variables of the function g(X), expressed as:

[0093] ;

[0094] in, It is structural resistance. It is the structural response value, in this invention, here It is the displacement threshold of the top of the water intake tower. It is the maximum horizontal relative displacement of the tower top obtained from finite element analysis.

[0095] In some embodiments, by combining the structural characteristics of the tall and thin-walled water intake tower, five seismic performance levels can be divided into basically intact, slightly damaged, moderately damaged, severely damaged, and collapsed. For the structural damage state corresponding to each seismic performance level, the maximum horizontal relative displacement of the tower top is clearly defined as the core judgment indicator, and the tower top displacement threshold corresponding to each level is formulated to provide a clear judgment standard for subsequent failure mode identification and reliability calculation.

[0096] Specifically, in this embodiment of the invention, incremental dynamic analysis (IDA) curves can be used to quantify the performance limit state of the intake tower structure. Using seismic waves with different phase characteristics generated by the CP spectrum, the ground acceleration is gradually increased from 0.1g to 1.0g with an amplification step size of 0.1g. The IDA curve of the maximum horizontal relative displacement at the top of the intake tower structure is obtained through analysis. Based on other evaluation systems for hydraulic concrete structures, according to... Figure 2 Four performance level limits based on displacement response were defined: LS1=0.02m, LS2=0.2m, LS3=0.3m, and LS4=0.8m, as shown in Table 2.

[0097] Table 2

[0098]

[0099] Based on the earthquake damage levels and performance standards shown in Table 2, and combined with the formulas of the state variable model above, taking the severe damage and collapse levels as examples, the state variable expressions under the severe damage and collapse levels can be expressed as follows:

[0100] ;

[0101] Among them, state variables When this occurs, it indicates severe damage or collapse of the intake tower structure, meaning the intake tower structure is at its most critical stage. The failure states corresponding to each seismic performance level.

[0102] In some embodiments, a set of sample points is drawn from the random variable using a stratified Latin hypercube sampling method, including:

[0103] The global range of values ​​for a random variable is determined as [ , ], divide it into There are three equal-volume first-level subregions, and the interval length of each first-level subregion is expressed as follows:

[0104] , =1, 2, ..., ;

[0105] The variable range within each first-level subregion is further divided into... There are two equal-volume second-level hierarchical intervals, and the length of each second-level hierarchical interval is expressed as:

[0106] , =1, 2, ..., M;

[0107] In the The first-level sub-region Sample points are randomly selected from each of the two-level stratified intervals. The formula for the sample points is as follows:

[0108] ;

[0109] in, ~(0,1) is a uniform random number within the interval (0,1), used to randomly select sample points within the second-level stratified interval; For the first The first-level sub-region, the first Random samples generated in two secondary strata.

[0110] It should be noted that the estimation of statistical moments of the state variables of the intake tower structure based on stratified Latin hypercube sampling (SLHS) actually involves generating random sample points of the intake tower structure using an efficient stratified Latin hypercube sampling method, obtaining the corresponding random function values, and then using statistical methods to obtain the statistical moments of the state variable Z of the intake tower structure. The SLHS method adds a two-layer sampling logic of global sub-region stratification to the traditional single-dimensional stratification of Latin hypercube sampling (LHS): first, the entire parameter space of the random variable is divided into several equal-volume sub-regions, and then LHS sampling is performed independently within each sub-region. That is, each random variable within each sub-region is further stratified, and only one sample point is extracted from each layer. This "block-first, then sample" strategy can avoid sample clustering in local space through global sub-region division and ensure sample uniformity in one dimension through LHS within sub-regions. Therefore, it requires fewer simulations to generate random samples with better spatial filling and more uniform distribution, providing a more accurate sample basis for subsequent calculation of the statistical moments of the state variable Z of the intake tower structure.

[0111] In some embodiments, the random variable peak ground acceleration (PGA) and its probability distribution parameters are shown in Table 3:

[0112] Table 3

[0113]

[0114] For example, based on Tables 1 to 3 and the SLHS method, 200 sets of 20s random seismic wave samples were generated using the CP power spectrum. The generated 0.4g (section 31) seismic wave sample is shown below. Figure 3 As shown, see Figure 3 As shown, the generated seismic waves fully demonstrate the randomness of ground motion. The power spectra of the 0.4g seismic wave sample in the X and Y directions are as follows. Figure 4 As shown, the energy is mainly concentrated around 5Hz, and the fitting effect is good, which verifies the rationality of generating seismic waves using the CP spectrum.

[0115] In some embodiments, to verify the uniformity of the SLHS method, 500 sets of pseudo-random sequences and SLHS numbers are generated within a 1cm × 1cm area, and the two-dimensional distribution diagram is shown below. Figure 5 As shown. Among them, Figure 5 In the diagram, 'a' represents a pseudo-random sequence, and 'b' represents an SLHS sequence. Pseudo-random sequences are prone to local clustering, while the SLHS distribution is more uniform than that of pseudo-random sequences. It is divided into 25 sub-layers, with 20 sets of data in each sub-layer.

[0116] After obtaining random sample points, the function values ​​corresponding to each sample point are calculated using the finite element method. Through statistical analysis of the function set, the origin moments of Z are obtained as follows:

[0117] ;

[0118] ;

[0119] ;

[0120] in, for First-order origin moment, Let O be the order of the moment at the origin. Let Z be the mean of the function Z. For the sample size, Let Z be the standard deviation.

[0121] In this embodiment of the invention, based on the maximum entropy method, an improved strategy for dynamic correction of integral boundaries is proposed to solve the problem of mismatch between fixed boundaries and the differentiated failure modes of hydraulic structures, and finally form a reliability calculation process suitable for water intake towers.

[0122] Specifically, the basic principle of the maximum entropy method includes: for the function in the seismic reliability analysis of water intake towers... Its failure probability and reliability index are:

[0123] ;

[0124] in, Let Z be the probability density function of the function Z.

[0125] Based on the statistical moments of the function Z obtained above, let the objective function be the maximum value of the entropy of the function Z, and simultaneously... First-order origin moment As constrained, the optimization model is established as follows:

[0126] ;

[0127] ;

[0128] Here, H represents entropy.

[0129] Using the Lagrange multiplier method, the constrained optimization problem of the above optimization model formula is transformed into an unconstrained optimization problem, which is then solved using Newton's method, as follows:

[0130] ;

[0131] in, It is a Lagrange multiplier.

[0132] The probability density function of the function Z is:

[0133] ;

[0134] In traditional methods, the integral domain is bounded by a fixed limit. However, in the seismic analysis of water intake towers, the displacement thresholds for different failure modes differ significantly, and a fixed limit can easily lead to fitting bias.

[0135] To address the limitations of traditional methods, and considering the physical characteristics of the seismic performance level of water intake towers, this invention proposes a dynamic correction method for integral boundaries based on performance thresholds and statistical characteristics. The specific implementation includes:

[0136] The dynamic correction uses the displacement threshold of the intake tower's seismic performance level as the base point, and the function is: , It is the first The tower top displacement thresholds corresponding to each seismic performance level are established. Based on the statistical characteristics of the state variables of the combined function, a correlation criterion between performance level, displacement threshold, and integral boundary is established. Taking the seismic performance level, which includes severe damage and collapse, as an example, the correlation criterion for the integral boundary is shown in Table 4.

[0137] Table 4

[0138]

[0139] In some embodiments, the association criteria satisfy two conditions: first, it covers the core displacement range of the target failure mode; second, it includes the statistical extreme values ​​of the response distribution to avoid truncating the tail data and affecting the accuracy of low-probability failure calculations.

[0140] In some embodiments, the statistical characteristics of the structural response values ​​include the average value of the structural response values. and standard deviation The dynamic determination of the integral boundary is based on the type calculation of the seismic performance level, and is expressed as:

[0141] When the current seismic performance level is severely damaged, the main body of the response distribution is covered, and the mean ±3 is used. With threshold correction, the integral boundary is represented as:

[0142] ;

[0143] in, The threshold for tower top displacement that severely disrupts the leveling system. An extension factor greater than 1;

[0144] When the current seismic performance level is at collapse failure, focus on the tail response and use a threshold +2. Extending this, the boundary of the integral is expressed as:

[0145] ;

[0146] in, The threshold displacement of the tower top at the level of collapse and destruction. , The integral boundary of the structural response value. , This represents the integral boundary of the function.

[0147] Referring to the basic principle of the maximum entropy method described above, in this embodiment of the invention, the maximum entropy method is used to fit the probability density function of the function function within the integral boundary, including:

[0148] Will Substituting the first-order origin moment into the Lagrange multiplier method optimization model, dynamic adaptation between the integral boundary and the seismic performance threshold of the intake tower is achieved; the objective function of the Lagrange multiplier method optimization model is:

[0149] ;

[0150] in, For entropy, , It is a Lagrange multiplier;

[0151] The probability density function of the function is expressed as:

[0152] ;

[0153] Based on the probability density function expression, the failure probability and reliability index are calculated according to the following formula:

[0154] ;

[0155] in, This represents the probability of failure. As a reliable indicator, This is the cumulative distribution function of the standard normal distribution.

[0156] In traditional methods, the integral domain is bounded by a fixed limit. However, in the seismic analysis of water intake towers, the displacement thresholds for different failure modes differ significantly, and a fixed limit can easily lead to fitting bias.

[0157] In this embodiment of the invention, based on the displacement thresholds of different seismic performance levels of the intake tower, the integral boundary is dynamically determined by combining the statistical characteristics of the structural response, establishing a correlation mechanism between performance level, displacement threshold, and integral boundary to avoid fitting bias caused by the disconnect between fixed boundaries and failure modes. A hierarchical Latin hypercube sampling method is used to extract random variable samples, avoiding the local clustering problem of traditional sampling and achieving a more uniform spatial distribution with a small number of samples, providing a high-quality data foundation for statistical moment calculation. Simultaneously, a Clough-Penzien power spectrum model is used to generate random seismic waves, and low-frequency components are corrected through two layers of soil filters, conforming to the actual physical mechanism of seismic wave propagation, improving the realism of the seismic motion input, and ensuring the accuracy of structural dynamic response analysis. Furthermore, by leveraging the efficiency of hierarchical Latin hypercube sampling, the optimization of the integral range by dynamic boundaries, and the efficient use of a small number of statistical moments by the maximum entropy method, the number of calculation iterations is significantly reduced, solving the problem of low simulation efficiency in traditional methods and significantly shortening the seismic reliability assessment cycle. While improving efficiency, it maintains high computational accuracy, outperforming traditional fixed-boundary maximum entropy methods and response surface methods. It resolves the contradiction of insufficient accuracy or low efficiency in traditional methods, and meets the practical needs of seismic design and safety assessment of water intake towers.

[0158] The process of steps S104 to S109 described above will be specifically explained below with a specific embodiment.

[0159] The site is classified as a Class II site. The intake tower is 62.5m high, with a normal water level of 1875m. The tower top elevation is 1877.5m, and the tower bottom elevation is 1815m. The water diversion direction of the structure is along the river at an angle of 60°. The concrete backfill at the back of the tower is up to 1850.8m, and the concrete backfill on both sides of the tower body is up to 1835m. The outer profile cross-section dimensions are 8m × 13m (length × width). The material parameters are listed in Table 5.

[0160] Table 5

[0161]

[0162] This invention utilizes the commercial software HyperMesh and ABAQUS to implement three-dimensional finite element modeling of the intake tower structure and its foundation. (See attached image.) Figure 6 , Figure 7 The three-dimensional finite element model uses a rectangular coordinate system, where the positive X-axis points to the right perpendicular to the water flow direction, the positive Y-axis points downstream, and the positive Z-axis points to the top of the tower. The foundation depth is taken as 1.5 times the tower height, while the upstream, downstream, and left and right side boundaries extend to twice the tower height. The intake tower body, backfill, and foundation are all discretized using hexahedral solid elements. During the analysis, the influence of dynamic water pressure on the inner and outer walls of the intake tower is considered, introduced through equivalent added mass. Calculations are performed according to the "Code for Seismic Design of Hydraulic Structures in Hydropower Projects," and the calculated mass is applied layer by layer to the inner and outer nodes of the tower body according to the tower height to simulate the dynamic water pressure exerted on the tower body by the water inside and outside. For boundary conditions, to better simulate the influence of truncated boundaries, the front, rear, left, and right side boundaries of the foundation are set as normal constraints, while the bottom boundary is fully constrained. Seismic motion input uses a massless, uniform foundation input.

[0163] This invention uses an 8-degree rare earthquake as an example, employing 200 sets of random earthquakes generated by the CP power spectrum model as ground motion inputs to perform finite element dynamic response analysis on the intake tower structure. The extreme values ​​of the maximum relative horizontal displacement of the intake tower top in the x and y directions corresponding to the 200 representative samples are shown below. Figure 8 and Figure 9 As shown. Among them, Figure 8 In the x-direction, Figure 9 In the y-direction. From Figure 8 and Figure 9 It can be seen that the maximum displacement in both directions exhibits obvious random fluctuations with the changes in the ground motion samples. The maximum relative displacements of the structure in the x and y directions are approximately 0.3m and 0.33m, respectively, with the displacement fluctuation in the y direction being larger. This is closely related to the direction of water flow acting on the structure. In seismic design, the stress analysis and structural measures in the direction of water flow should be strengthened.

[0164] The time history curve of the displacement at the top of the water intake tower under the action of a 0.4g seismic wave sample is as follows: Figures 10 to 11 As shown. , where, Figure 10 In the x-direction, Figure 11 In the y-direction. See also Figures 10 to 11 As shown, the structural dynamic response in the x-direction is strongest at approximately the 7th second, while the structural dynamic response in the y-direction is strongest at approximately the 12th second. This is because the input random ground motion reaches its peak at the 7th second in the x-direction and the 12th second in the y-direction, respectively. Furthermore, by analyzing the changes in the maximum horizontal relative displacement curve at the top of the tower and the structural response time history curve corresponding to the 0.4g seismic wave, both clearly demonstrate that the generated ground motion exhibits significant randomness.

[0165] Based on Table 3 above, 200 random sample points are generated. The first four original moments of the structural function Z of the water tower are combined with the PDF of the structure. Then, the CDF is obtained by fitting the PDF with the improved maximum entropy method according to the present invention. The CDF is then obtained by integrating the corresponding PDF. Figures 12 to 13 PDFs under severe damage and collapse of the intake tower structure are presented and compared with the PDFs obtained by kernel density estimation (KDE). The fitting errors are shown in Table 6.

[0166] Table 6

[0167]

[0168] As can be seen from the PDF comparison chart, the improved maximum entropy method with dynamic boundary correction has smaller errors compared to the maximum entropy method with fixed boundaries and KDE. Under severe damage, the errors with KDE are 1.8% and 5.2%, respectively. The dynamic boundary correction method and KDE have a high degree of fit in the 0.3-0.8m range, while the fixed boundary method has a relatively wider fit. It differs from the former two in the peak and tail areas, demonstrating the advantage of the dynamic boundary in fitting the main distribution. In collapse damage, the errors with KDE are 2.3% and 9.8%, respectively. The dynamic boundary correction method has the same decay trend as KDE in the tail area, while the fixed boundary method has no data at 1.0m, which shows that the dynamic boundary optimizes the fitting of low-probability events in the tail area. Figures 14 to 15 The CDF calculated by the method of this embodiment of the invention is compared with the empirical CDF. The two curves under the two performance levels are quite similar. The reliability is 99.5% under the severe damage performance level and 99.9998% under the collapse and destruction performance level. The seismic performance meets the actual requirements of the project.

[0169] In addition, to verify the accuracy and efficiency of the method of the present invention, the reliability of the response surface method was also calculated, and the MCS calculation result was used as the standard solution to compare and verify the three methods. The calculation results are shown in Table 6.

[0170] Table 6 shows that the response surface methodology (RSM) calculates the failure probability of the water intake tower under severe structural damage as 4.2 × 10⁻³ and the reliability index as 2.634. The maximum entropy method (fixed boundary) calculates the failure probability and reliability index as 4.12 × 10⁻³ and 2.642, respectively. The improved maximum entropy method (dynamic boundary correction) calculates the failure probability as 4.13 × 10⁻³ and the reliability index as 2.64. Compared to the MCS's failure probability of 4.15 × 10⁻³ and reliability index of 2.6389, the relative errors of the reliability indices of the three methods are significant. The failure probabilities of the three methods under collapse damage were 0.19%, 0.12%, and 0.07%, respectively. The failure probability and reliability index of the response surface methodology (RSM) under collapse damage were 2 × 10⁻⁶ and 4.417, respectively. The failure probability and reliability index of the maximum entropy method (fixed boundary) were 1.9 × 10⁻⁶ and 4.622, respectively. The failure probability and reliability index of the improved maximum entropy method (dynamic boundary correction) were 1.92 × 10⁻⁶ and 4.58, respectively. The failure probability and reliability index of the MCS were 1.94 × 10⁻⁶ and 4.52, respectively. The relative errors of the three methods were 2.28%, 2.26%, and 1.33%, respectively. This indicates that the method of this invention can achieve good accuracy at both performance levels. Furthermore, the MCS method requires 10⁶ calculations, while the method of this invention only requires 200 calculations to achieve results close to those of the MCS method. The calculation time is only 0.02% of that of the MCS method, similar to the calculation efficiency of the RSM, significantly reducing the required calculation time.

[0171] In this embodiment of the invention, addressing the reliability issue of water intake towers under seismic loading, 200 random ground motion acceleration time histories are generated using the CP power spectrum model. An improved maximum entropy method combined with SLHS is then employed to conduct a reliability analysis of the water intake tower structure. Taking a water intake tower structure in a certain region as the engineering object, the following conclusions are drawn:

[0172] 1) A three-dimensional finite element model of the intake tower structure was established, and the performance levels of the intake tower structure were divided based on displacement response. The function functions of the intake tower under different failure performance standards were given, and an improved maximum entropy method based on dynamic correction of integral boundaries of performance levels was proposed. By associating the physical threshold of the seismic performance level of the intake tower, the problem of the disconnect between fixed boundaries and failure modes was solved, and the fitting accuracy of the probability density function in the critical interval was significantly improved.

[0173] 2) Reliability analysis revealed that the relative error between the reliability index calculation results under severe damage and the MCS method was 0.07%, and the relative error between the reliability index calculation results under collapse and damage and the MCS method was 1.33%, while the number of calculations was only 0.02% of that of the MCS method, and the results were close to those of the response surface methodology. This verifies that the method of the present invention has high accuracy and computational efficiency, and provides a quantitative method for seismic reliability analysis of water intake tower structures.

[0174] 3) This invention currently only conducts reliability analysis based on the tower top displacement performance index and single seismic load. Future research will expand to reliability analysis based on multiple performance indices and multiple random variables, further broadening its application scenarios.

[0175] The various embodiments in this specification are described in a progressive manner. For the same or similar parts between the various embodiments, please refer to each other. Each embodiment focuses on describing the differences from other embodiments.

[0176] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit this application. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of this application.

Claims

1. A dynamic integral boundary-based water intake tower seismic reliability maximum entropy calculation method, characterized in that, The application relates to a method for calculating the failure probability and reliability index of a water intake tower structure. The method comprises the following steps: determining a plurality of seismic performance levels of the water intake tower structure, and determining a tower top displacement threshold value corresponding to each of the seismic performance levels; taking the peak ground acceleration as a random variable, and extracting a group of sample points from the random variable based on the probability distribution of the random variable by using a stratified Latin hypercube sampling method; for each of the sample points, generating a random seismic wave sample corresponding to the sample point by using a Clough-Penzien power spectrum model; inputting the random seismic wave sample into a finite element model of the water intake tower structure to perform dynamic response analysis, and obtaining the maximum relative horizontal displacement of the tower top of the water intake tower corresponding to each of the sample points as a structural response value; for each of the seismic performance levels, constructing a performance function based on the tower top displacement threshold value and the structural response value corresponding to the seismic performance level, and calculating at least the first four order origin moments of the performance function value set as statistical moments according to the performance function value calculated for each of the sample points; dynamically determining the integral boundary for the maximum entropy method calculation based on the tower top displacement threshold value and the statistical characteristics of the structural response value corresponding to the current seismic performance level; fitting the probability density function of the performance function within the integral boundary by using the maximum entropy method based on the integral boundary; 2. The method of claim 1, wherein, integrating the fitted probability density function to obtain the failure probability and the reliability index under the current seismic performance level.

3. The method of claim 2, wherein, The seismic performance levels include basic integrity, slight damage, moderate damage, severe damage and collapse destruction; wherein the tower top displacement threshold value corresponding to the basic integrity is 0-0.02m, the tower top displacement threshold value corresponding to the slight damage is 0.02-0.2m, the tower top displacement threshold value corresponding to the moderate damage is 0.2-0.3m, the tower top displacement threshold value corresponding to the severe damage is 0.3-0.8m, and the tower top displacement threshold value corresponding to the collapse destruction is greater than 0.8m. determining a global value range of the random variable as , ] and dividing it into equal-volume first-level sub-regions, and an interval length of each of the first-level sub-regions is represented as: ， =1，2，…， ； further dividing the variable range in each of the primary sub-regions into a number of secondary stratified intervals of equal volume, each of the secondary stratified intervals having a length represented by: ， =1，2，…，M； In the first second hierarchical interval of the first sample point in the first primary sub-region, and the formula of the sample point is: ； wherein, (0, 1) is a uniform random number in the interval (0, 1) for randomly selecting a sample point in the two-level hierarchical interval; is the first first-level sub-region, the first random sample generated in the second-level hierarchical interval.

4. The method of claim 3, wherein, The stratified Latin hypercube sampling method is used to extract a group of sample points from the random variable, and the method comprises the following steps: ； wherein, and are the dominant circular frequency and damping ratio of the first filter layer, respectively; and are the dominant circular frequency and damping ratio of the second filter layer, respectively; denotes the circular frequency, is the spectral degree. a random parameter vector of the clough-penzien power spectral model is: 。 5. The method of claim 4, wherein, The power spectral density of the Clough-Penzien power spectrum model is expressed as: ； in, It is the first The tower top displacement threshold corresponding to each seismic performance level It is the structural response value; when When <0, it indicates that the water intake tower structure is in the first stage. The failure states corresponding to each seismic performance level.

6. The method of claim 5, wherein, The expression of the performance function is: ； ； ； wherein is the origin of the scale, is the mean of the function Z, is the number of samples, is the standard deviation of Z.

7. The method of claim 6, wherein, the statistical characteristics of the structural response values include a mean value of the structural response values and a standard deviation the dynamic determination of the integral boundary is based on the type of the seismic performance level, expressed as: At the current performance level of severe damage, the integral bounds are expressed as mean ± 3 threshold correction, the integral bounds are expressed as: ； wherein, a threshold for top-of-tower displacement at a severe damage level, an extension factor greater than 1; At the current performance level of collapse, focusing on the tail response, the threshold +2 is extended, the integral boundary is expressed as: ； wherein is a collapse damage level tower top displacement threshold, , is an integral boundary for the structural response value, , is an integral boundary for the performance function.

8. The method of claim 6, wherein, The origin moments of the performance function value set are expressed as: The method comprises the following steps: The Lagrange multiplier method optimization model is obtained by substituting the step origin moment into the Lagrange multiplier method optimization model, and dynamic adaptation of the integral boundary and the water inlet tower seismic performance threshold is realized; and the objective function of the Lagrange multiplier method optimization model is: ； wherein is the entropy, , is the Lagrange multiplier; The maximum entropy method is used to fit the probability density function of the performance function within the integral boundary, and the method comprises the following steps: ； The expression of the probability density function of the performance function is: Based on the expression of the probability density function, the failure probability and the reliability index are calculated according to the following formula: ； wherein, is the failure probability, is the reliability index, is the standard normal cumulative distribution function.

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