CN114239348A - Bridge seismic reliability analysis method - Google Patents
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Abstract
The invention relates to the technical field of bridge seismic resistance, in particular to a bridge seismic reliability analysis method, which comprises the following steps: s1, determining N bridge random parameters, generating B seismic waves, acquiring M Gaussian interpolation points of each bridge random parameter, and establishing A bridge structure finite element model analysis samples; s2, carrying out bridge certainty dynamic response analysis to obtain a key component response extreme value fractional order moment; and S3, obtaining the failure probability of the bridge system under the earthquake action. According to the bridge anti-seismic reliability analysis method, the bridge structural member response extreme value distribution and the tiny failure probability under the action of the earthquake can be analyzed, the correlation characteristics among different member earthquake demands are determined, the complex bridge system anti-seismic reliability theoretical method based on the dimension reduction principle is established, the random performance evolution rule of the bridge system under the action of the earthquake is disclosed, and meanwhile, the calculated amount in the bridge anti-seismic reliability analysis process is effectively reduced.
Description
Technical Field
The invention relates to the technical field of bridge seismic resistance, in particular to a bridge seismic reliability analysis method.
Background
The bridge structure is used as a traffic hub crossing rivers, valleys or other cutting obstacles, is widely applied to railway and highway main lines, and plays a significant role in the development of national economy and society. China is located between the Pacific earthquake zone and the Asia-Europe earthquake zone, earthquakes occur frequently, and earthquake disasters are one of the most main natural disasters faced by China. With the improvement of the requirement on the service performance of the bridge structure, the safety risk caused by strong dynamic action such as earthquake disaster and the like is obviously enhanced. For actual seismic waves, the peak value, the time duration and the frequency spectrum of the seismic waves have strong randomness and non-stationarity. On the other hand, due to construction errors, manufacturing defects and limited statistical samples, significant uncertainties exist for the parameters of the structure (defined as random parameters of the bridge structure), and significant uncertainties also exist for the parameters of the material (defined as random parameters of the bridge material). Under the influence of dual uncertainties of structural and non-stationary seismic waves, significant randomness must also exist in the seismic response of the structure. How to efficiently and accurately evaluate the response extreme value distribution and the tiny failure probability of the bridge structure under the action of strong earthquake becomes a core problem of the evaluation of the earthquake-resistant reliability of the bridge structure.
Considering the influence of dual random parameters of the structure and the non-stationary seismic waves, a probability analysis method is adopted to research and evaluate the seismic performance of the bridge structure. In the probability analysis method, structural earthquake-resistant reliability analysis is a main probability method for evaluating that the structural earthquake-resistant requirement does not exceed the corresponding limit bearing capacity. In addition, for a large-span bridge structure with multiple components and multiple failure modes, the random response analysis of the large-span bridge structure is always a great challenge for scholars and engineers. Meanwhile, the large-span bridge is used as a high-order statically indeterminate structure, and the failure of a certain member does not mean the failure of the whole structure, so that the seismic reliability of a bridge system considering the relevant characteristics of the member is always a hotspot and a difficulty in research. The existing solving method is mainly limited by the high-dimensional calculated quantity of the structure and the characteristics of related components, and a new effective seismic reliability analysis method of the bridge structure system without losing precision is urgently needed to be provided.
Disclosure of Invention
The invention aims to solve the technical problem that the reliability theoretical method of the existing bridge system is insufficient, provides a bridge anti-seismic reliability analysis method, can analyze the response extreme value distribution and the tiny failure probability of bridge structural members under the action of an earthquake, determines the relevant characteristics among different member earthquake requirements, establishes a complex bridge system anti-seismic reliability theoretical method of a dimension reduction principle, reveals the random performance evolution rule of the bridge system under the action of the earthquake, and reduces the calculated amount in the analysis process of the bridge anti-seismic reliability.
In order to achieve the purpose, the invention adopts the technical scheme that:
a bridge seismic reliability analysis method comprises the following steps:
s1, determining N bridge random parameters, generating B seismic waves representing bridge randomness, acquiring M Gaussian interpolation points of each bridge random parameter, and establishing A bridge structure finite element model analysis samples according to the M Gaussian interpolation points, wherein A is M.N;
s2, combining B seismic waves with A bridge structure finite element model analysis samples to perform A-B times of bridge deterministic dynamic response analysis to obtain a key component response extreme value fractional order moment;
and S3, obtaining the failure probability and the reliability index of each key component under the earthquake action based on the response extreme value fractional order moment of the key component, and further obtaining the failure probability of the bridge system.
According to the bridge anti-seismic reliability analysis method, M Gaussian interpolation points of random parameters of each bridge are obtained, on the basis, the fractional order moment of the response extreme value of a key component is obtained, the failure probability of a bridge system is further obtained, the distribution of the response extreme value and the tiny failure probability of a bridge structural component under the action of an earthquake can be analyzed, the correlation characteristics among different component earthquake demands are determined, the complex bridge system anti-seismic reliability theoretical method of a dimension reduction principle is established, the random performance evolution rule of the bridge system under the action of the earthquake is revealed, the distribution of the random parameters of the bridge is reduced to the M Gaussian interpolation points, and the calculated amount in the analysis process of the bridge anti-seismic reliability is further effectively reduced.
Preferably, step S1 includes the following steps;
s11, determining N bridge random parameters and the distribution form of each bridge random parameter, and randomly generating B seismic waves by considering the seismic wave phase;
s12, selecting a corresponding Gaussian polynomial according to a random parameter distribution form of the bridge, and determining Gaussian weight and an integral point;
s13, generating interpolation points by the bridge random parameters according to the integral points, wherein M interpolation points are correspondingly generated by each bridge random parameter;
s14, establishing a bridge three-dimensional finite element model according to design data of bridge structures and materials, and generating bridge finite element model analysis samples under M Gaussian interpolation points of random parameters of each bridge, wherein A bridge structure finite element model analysis samples are total.
Preferably, before step S14, there is a step of three-dimensional finite element model verification, specifically:
a1: taking a median value for each bridge structure random parameter;
a2: respectively establishing a bridge three-dimensional finite element model by utilizing at least two nonlinear finite element analysis software, respectively inputting a median value into all the bridge three-dimensional finite element models for modal analysis, and comparing results obtained by the at least two nonlinear finite element analysis software to verify the correctness of the bridge three-dimensional finite element models;
then, in step S14, the median is replaced with gaussian interpolation point values, and a bridge structure finite element model analysis samples are further established.
In the above scheme, modal analysis of each bridge three-dimensional finite element model is only required to be performed once, namely, modal analysis is performed on the bridge finite element model with the parameters taking the median value, and the first to tenth order modal vibration modes can be obtained.
Preferably, the bridge random parameters comprise bridge structure random parameters and bridge material random parameters, the bridge structure random parameters comprise concrete volume weight random parameters, member integral size random parameters and damping ratio random parameters, the materials comprise concrete elastic modulus random parameters, concrete compressive strength random parameters and reinforcing steel bar yield strength random parameters,
the step S1 of determining N bridge random parameters specifically includes: summarizing and summarizing the randomness of the existing bridge structure parameters and the randomness of the material parameters, obtaining a bridge random parameter database, obtaining N bridge random parameters based on the bridge random parameter database, and determining the distribution form of each bridge random parameter, wherein the N bridge random parameters comprise N1 bridge structure random parameters and N2 bridge material random parameters.
Preferably, the seismic waves include longitudinal artificial seismic waves, transverse artificial seismic waves and vertical artificial seismic waves,
in step S1, generating B seismic waves representing the randomness of the bridge is specifically:
and determining a power spectrum model by combining actual field soil layer parameters of a bridge site, converting the seismic wave frequency domain power spectrum into time domain seismic waves according to a multi-dimensional multi-point non-stationary seismic wave triangular series method seismic wave synthesis theory and considering seismic wave phase randomness, and synthesizing B pieces of artificial seismic waves based on the time domain seismic waves.
Preferably, step S2 specifically includes the following steps:
s21: combining the B seismic waves with the A bridge structure finite element model analysis samples, performing nonlinear dynamic analysis, and obtaining a bridge structure key component response extreme value and a key component response extreme value arbitrary order moment;
s22: and obtaining the fractional order moment of the response extreme value of the key component based on the random order moment of the response extreme value of the key component, wherein the key component comprises a pier, a bridge tower, a support, a main beam and a stay cable.
Preferably, the step S3 specifically includes the following steps:
s31: respectively determining damage indexes of each key component according to a deformation damage criterion or a strength damage criterion, and obtaining a probability density function f of the key component by combining the response extreme value fractional order moment of the key component according to the maximum entropy principleY(y)；
S32: integrating the probability density function f of the key componentY(y) indexes of damage to respective key members and respectiveCombining the failure states of the key components to obtain the failure probability of each key component under the action of the earthquake;
s33: and combining the failure probabilities of the key components by using a conditional marginal probability density Product (PCM) method and considering the relevant characteristics among the failure modes of the key components to obtain the failure probability of the bridge system.
Preferably, in step S31, a dual-loop optimization function is obtained according to the maximum entropy principle, and the fraction order α and the lagrangian multiplier λ obtained by optimization are substituted into the estimated probability density function of the critical component response extremum by finding the fraction order α and the lagrangian multiplier λ in the minimum optimization function of the dual-loop optimization function
Obtaining the probability density function f of the key componentY(y)。
Preferably, in step S32, the probability density function is substituted into the reliable probability function of the component according to the failure index of the component, and the probability that each key component first surpasses the damage index threshold is calculated, so as to obtain the failure probability of each main component of the bridge.
Specifically, the reliability probability is 1 — failure probability.
Preferably, step S33 specifically includes the following steps,
s331. probability density function f based on key componentY(y) determining the correlation between failure modes of all main components, and calculating the correlation coefficient between every two components;
s332, assembling the failure probability of a single key component according to a serial mode, a parallel mode and a series-parallel mode by using a conditional marginal probability density Product (PCM) method, and converting the failure probability of the component into the failure probability of the system.
In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:
1. according to the bridge anti-seismic reliability analysis method, M Gaussian interpolation points of random parameters of each bridge are obtained, on the basis, the fractional order moment of the response extreme value of a key component is obtained, the failure probability of a bridge system is further obtained, the distribution of the response extreme value and the tiny failure probability of a bridge structural component under the action of an earthquake can be analyzed, the correlation characteristics among different component earthquake demands are determined, the complex bridge system anti-seismic reliability theoretical method of a dimension reduction principle is established, the random performance evolution rule of the bridge system under the action of the earthquake is revealed, the distribution of the random parameters of the bridge is reduced to the M Gaussian interpolation points, and the calculated amount in the analysis process of the bridge anti-seismic reliability is further effectively reduced.
2. The invention provides a bridge earthquake-proof reliability analysis method based on a dimension reduction principle and a conditional marginal probability density product, which is used for calculating the earthquake reliability of a bridge structure by using original design parameters of the bridge structure and earthquake waves which are possibly generated, so that earthquake risk evaluation of a single component and an integral structure of the bridge under the action of an earthquake is realized.
Drawings
Fig. 1 is a schematic flow chart of a method for analyzing seismic reliability of a bridge in embodiment 2 of the present invention.
FIG. 2 is a schematic diagram (longitudinal direction) of the probability density function curve of the earthquake lower support displacement in the embodiment 2 of the invention.
FIG. 2-1 is a schematic diagram (lateral direction) of the probability density function curve of seismic sub-mount displacement in example 2 of the present invention.
FIG. 3 is a schematic diagram (longitudinal direction) of the failure probability of each main component under the action of 30 seismic waves in embodiment 2 of the present invention.
Fig. 3-1 is a schematic diagram (transverse direction) of the failure probability of each main component under the action of 30 seismic waves in embodiment 2 of the invention.
FIG. 4 is a graph (longitudinal direction) of correlation coefficients of main components under the action of 30 seismic waves in embodiment 2 of the present invention.
FIG. 4-1 is a schematic diagram (transverse direction) of correlation coefficient curves of 30 main members under the action of seismic waves in embodiment 2 of the present invention.
FIG. 5 is a schematic diagram of the failure probability of a bridge system under the action of 30 seismic waves (longitudinal direction) in embodiment 2 of the present invention.
Fig. 5-1 is a schematic diagram (transverse direction) of the failure probability of the bridge system under the action of 30 seismic waves in embodiment 2 of the invention.
FIG. 6 is a schematic diagram of the upper and lower bounds of the system failure probability under 30 seismic waves in embodiment 2 of the present invention (longitudinal direction).
FIG. 6-1 is a schematic diagram (transverse direction) of the upper and lower bounds of the system failure probability under 0 seismic waves in embodiment 2 of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1
The method for analyzing the seismic reliability of the bridge comprises the following steps:
s1, determining N bridge random parameters, generating B seismic waves representing bridge randomness, acquiring M Gaussian interpolation points of each bridge random parameter, and establishing A bridge structure finite element model analysis samples according to the M Gaussian interpolation points, wherein A is M.N;
s2, combining B seismic waves with A bridge structure finite element model analysis samples to perform A-B times of bridge deterministic dynamic response analysis to obtain a key component response extreme value fractional order moment;
and S3, obtaining the failure probability and the reliability index of each key component under the earthquake action based on the response extreme value fractional order moment of the key component, and further obtaining the failure probability of the bridge system.
According to the bridge anti-seismic reliability analysis method, M Gaussian interpolation points of random parameters of each bridge are obtained, on the basis, the fractional order moment of the response extreme value of a key component is obtained, the failure probability of a bridge system is further obtained, the distribution of the response extreme value and the tiny failure probability of a bridge structural component under the action of an earthquake can be analyzed, the correlation characteristics among different component earthquake demands are determined, the complex bridge system anti-seismic reliability theoretical method of a dimension reduction principle is established, the random performance evolution rule of the bridge system under the action of the earthquake is revealed, the distribution of the random parameters of the bridge is reduced to the M Gaussian interpolation points, and the calculated amount in the analysis process of the bridge anti-seismic reliability is further effectively reduced.
Specifically, step S1 includes the following steps;
s11, determining N bridge random parameters and the distribution form of each bridge random parameter, and randomly generating B seismic waves by considering the seismic wave phase;
s12, selecting a corresponding Gaussian polynomial according to a random parameter distribution form of the bridge, and determining Gaussian weight and an integral point;
s13, generating interpolation points by the bridge random parameters according to the integral points, wherein M interpolation points are correspondingly generated by each bridge random parameter;
s14, establishing a bridge three-dimensional finite element model according to design data of bridge structures and materials, and generating bridge finite element model analysis samples under M Gaussian interpolation points of random parameters of each bridge, wherein A bridge structure finite element model analysis samples are total.
Specifically, before step S14, there is a step of three-dimensional finite element model verification, specifically:
a1: taking a median value for each bridge structure random parameter;
a2: respectively establishing a bridge three-dimensional finite element model by utilizing at least two nonlinear finite element analysis software, respectively inputting a median value into all the bridge three-dimensional finite element models for modal analysis, and comparing results obtained by the at least two nonlinear finite element analysis software to verify the correctness of the bridge three-dimensional finite element models;
then, in step S14, the median is replaced with gaussian interpolation point values, and a bridge structure finite element model analysis samples are further established.
In the above scheme, modal analysis of each bridge three-dimensional finite element model is only required to be performed once, namely, modal analysis is performed on the bridge finite element model with the parameters taking the median value, and the first to tenth order modal vibration modes can be obtained.
Specifically, the bridge random parameters comprise bridge structure random parameters and bridge material random parameters, the bridge structure random parameters comprise concrete volume weight random parameters, member integral size random parameters and damping ratio random parameters, the materials comprise concrete elastic modulus random parameters, concrete compressive strength random parameters and reinforcing steel bar yield strength random parameters,
the step S1 of determining N bridge random parameters specifically includes: summarizing and summarizing the randomness of the existing bridge structure parameters and the randomness of the material parameters, obtaining a bridge random parameter database, obtaining N bridge random parameters based on the bridge random parameter database, and determining the distribution form of each bridge random parameter, wherein the N bridge random parameters comprise N1 bridge structure random parameters and N2 bridge material random parameters.
In particular, the seismic waves comprise longitudinal artificial seismic waves, transverse artificial seismic waves and vertical artificial seismic waves,
in step S1, generating B seismic waves representing the randomness of the bridge is specifically:
and determining a power spectrum model by combining actual field soil layer parameters of a bridge site, converting the seismic wave frequency domain power spectrum into time domain seismic waves according to a multi-dimensional multi-point non-stationary seismic wave triangular series method seismic wave synthesis theory and considering seismic wave phase randomness, and synthesizing B pieces of artificial seismic waves based on the time domain seismic waves.
Specifically, step S2 specifically includes the following steps:
s21: combining the B seismic waves with the A bridge structure finite element model analysis samples, performing nonlinear dynamic analysis, and obtaining a bridge structure key component response extreme value and a key component response extreme value arbitrary order moment;
s22: and obtaining the fractional order moment of the response extreme value of the key component based on the random order moment of the response extreme value of the key component, wherein the key component comprises a pier, a bridge tower, a support, a main beam and a stay cable.
Specifically, the step S3 specifically includes the following steps:
s31: according to a deformation failure criterion or a strength failure criterion, determining damage indexes of each key component respectively, and according to a maximum entropy principle, combining the response extreme value fractional order moment of the key component to obtain the probability of the key componentDensity function fY(y)；
S32: integrating the probability density function f of the key componentY(y) combining the damage indexes of the key components and the failure states of the key components to obtain the failure probability of the key components under the action of the earthquake;
s33: and combining the failure probabilities of the key components by using a conditional marginal probability density Product (PCM) method and considering the relevant characteristics among the failure modes of the key components to obtain the failure probability of the bridge system.
Specifically, in step S31, a dual-cycle optimization function is obtained according to the maximum entropy principle, and the fraction order α and the lagrangian multiplier λ obtained by optimization are substituted into the estimated probability density function of the critical component response extremum by finding the fraction order α and the lagrangian multiplier λ in the minimum optimization function of the dual-cycle optimization function
Obtaining the probability density function f of the key componentY(y)。
Specifically, in step S32, a probability density function is substituted into a member reliability probability function according to the failure index of the member, the probability that each key member first transcends to damage the index threshold is calculated, and the failure probability of each main member of the bridge is further obtained, where the reliability probability is 1-failure probability.
Specifically, step S33 specifically includes the following steps,
s331. probability density function f based on key componentY(y) determining the correlation between failure modes of all main components, and calculating the correlation coefficient between every two components;
s332, assembling the failure probability of a single key component according to a serial mode, a parallel mode and a series-parallel mode by using a conditional marginal probability density Product (PCM) method, and converting the failure probability of the component into the failure probability of the system.
The beneficial effects of this embodiment: a bridge anti-seismic reliability analysis method is characterized in that M Gaussian interpolation points of each bridge random parameter are obtained, a key component response extreme value fractional order moment is obtained on the basis of the M Gaussian interpolation points, failure probability of a bridge system is further obtained, bridge structure component response extreme value distribution and micro failure probability under the action of an earthquake can be analyzed, correlation characteristics among different component earthquake requirements are determined, a complex bridge system anti-seismic reliability theoretical method of a dimensionality reduction principle is established, a random performance evolution rule of the bridge system under the action of the earthquake is revealed, meanwhile, the bridge random parameter distribution is reduced to the M Gaussian interpolation points, and further, calculated amount in the bridge anti-seismic reliability analysis process is effectively reduced.
Example 2
A bridge earthquake-resistant reliability analysis method mainly comprises the following steps:
(1) the uncertainty (namely randomness) of the bridge structure and the uncertainty (namely randomness) of the materials are summarized through a large amount of reference documents. The structural uncertainty mainly comprises uncertainty related to parameters such as concrete volume weight, macroscopic size of a member, damping ratio and the like. The method mainly considers the uncertainty of parameters such as the elastic modulus of concrete, the compressive strength of concrete, the yield strength of reinforcing steel bars and the like, and determines N1 random parameters N of the bridge structure and N2 random parameters N of the bridge material, wherein N1 is generally 8-12, and N2 is generally 6-8, in the scheme, the bridge structure is the structure after the bridge is built, namely the characteristics of the geometric position, the member size, the damping ratio and the like after the bridge is built are considered, and the concrete is used when bridge members (including main beams, piers, bridge towers and the like) are built;
(2) determining a power spectrum function by combining detailed soil layer parameters of an actual field of a bridge site, converting a seismic wave frequency domain power spectrum into time domain seismic waves at random based on the power spectrum function and the seismic wave phase according to a multi-dimensional multi-point non-stationary seismic wave trigonometric series seismic wave synthesis theory and considering the random seismic wave phase, and forming 30 longitudinal artificial seismic waves, transverse artificial seismic waves and vertical artificial seismic waves based on the time domain seismic waves and 30 artificial seismic waves (each artificial seismic wave has a longitudinal artificial seismic wave component, a transverse artificial seismic wave component and a vertical artificial seismic wave component) to represent the uncertainty of the seismic waves;
(3) taking a median value for each bridge random parameter to form a median value set, establishing a bridge structure three-dimensional finite element model based on nonlinear finite element analysis software, inputting the median value to a corresponding position of the finite element model for modal analysis according to the action of different random parameters, comparing the modal analysis with the natural vibration period of the bridge and the bridge modal vibration mode obtained by other universal finite element software models, verifying the correctness of the established finite element model, and when the deviation of the result is less than five percent (including five percent), indicating that the finite element model is correct;
(4) selecting a Gaussian polynomial, determining Gaussian weights and Gaussian integral points, determining 5 Gaussian interpolation points of random parameters of each bridge based on a dimensionality reduction principle, replacing median values of the parameters in the previous model with Gaussian interpolation point sample values, and further establishing 5N bridge structure finite element model analysis samples in total, wherein the number of the interpolation points is odd, generally five are selected, and if seven are selected, the calculation workload is large, but the accuracy is poor. (ii) a
(5) Taking 30 artificial seismic waves obtained by randomly considering the soil layer characteristics and the phases of the bridge site in the step (2) as input, combining the input with the finite element model sample established in the step (4), carrying out a large amount of nonlinear dynamic analysis, obtaining a bridge structure key component response extreme value and any order moment of the bridge structure key component response extreme value, and obtaining a key component response extreme value fractional order moment from the bridge structure key component response extreme value any order moment;
(6) according to the deformation damage criterion or the strength damage criterion, respectively determining damage indexes of the bridge pier, the bridge tower, the support, the main beam and the stay cable and giving the damage indexes in a parallel table;
(7) according to the maximum entropy principle, combining the fractional order moment of the critical component response extreme value obtained in the step (5), and solving an estimated probability density function of the bridge critical component response extreme value based on mathematical calculation software
(8) The estimated probability density function of the response extremum of the key component obtained by the solution in the step (7)
Combining the damage index of the key component determined in the step (6) and the failure state of the key construction to further obtain the failure probability and reliability index of each key component under the action of the earthquake;
(9) and combining the failure probabilities of the bridge members by using a conditional marginal probability density Product (PCM) method and considering the relevant characteristics among the failure modes of the members so as to obtain the failure probability and the reliability index of the bridge system.
The invention aims to provide a bridge earthquake-resistant reliability analysis method based on a dimension reduction principle and a conditional marginal probability density product, which utilizes original design parameters of a bridge structure and earthquake waves possibly generated to calculate the earthquake reliability of the bridge structure and realizes earthquake risk evaluation of a single member and an integral structure of the bridge under the action of an earthquake, and the details are as follows.
On the basis, in a further preferable mode, in the step (2), according to a Tajimi-Kanai power spectrum model, a seismic wave frequency domain power spectrum S is determinedg(ω), the formula is as follows:
wherein, | HPI is a correction function for suppressing low frequency seismic wave components, S0(omega) is a spectral intensity factor, and can be obtained by looking up a table corresponding to different site conditions, wherein omega is angular frequency, and omega isfThe circular frequency of the low and high pass filters; xifDamping ratio parameter, omega, for low and high pass filtersgIs the prominent angular frequency, xi, of the field soilgThe damping ratio of the field soil is shown as gamma, and the self-spectral density of the basement rock acceleration (white noise) is shown as gamma.
On the basis of the above, it is further preferable that, in step (5), the fractional order moment of the key member is calculated
The formula of (1) is as follows:
in the above formula theta [ ·]To expect operator, Yq(. h) is the q-order moment of the response,
representing a k-order Gaussian weight, Xi kRepresenting a Gaussian interpolation point of order k, k being 5, ujJ is 1, …, and N is the average value of 1 to N bridge random parameters;
on the basis of the above, in a further preferred manner, in step (7), a two-cycle optimization function can be obtained according to the maximum entropy principle, and α and λ in the two-cycle optimization function are optimized by finding the minimum value of the two-cycle optimization function, where the two-cycle optimization function is as follows:
in the formula, lambda is Lagrange multiplier, alpha is fraction order, and y is component response;
using the Langerian constraint equation, an estimated probability density function of the bridge member can be obtained
The formula is as follows:
on the basis of the above, in a further preferable mode, in the step (8), according to the first transcending probability, a calculation formula of the member reliability probability R is as follows;
wherein Pr represents a reliable probability operator; t is seismic wave duration; zmIs a structural limit state under the action of an earthquake;
probability density function, Z, being the bridge structure responseextAnd (t) is an extreme value of the bridge structure under the action of the earthquake.
In step (9), for the parallel system, the conditional marginal probability density Product (PCM) is calculated as:
wherein,
is a reliable indicator of each failure mode of the parallel system, X1～XnRespectively represent failure modes of main components of the bridge structure,
for a linear correlation coefficient matrix between the individual failure modes, phin(. cndot.) is a cumulative distribution function of an n-dimensional standard normal distribution.
On the basis of the above, in a further preferred mode, in step (9), the calculation formula of the failure probability of the conditional marginal probability density Product (PCM) for the parallel-serial system is:
in the formula, XkThe k-th failure mode of the main member of the bridge structure, wherein
Is calculated as shown in equation (6).
On the basis of the above, in a further preferred mode, in step (9), the correlation of the component failure mode is first replaced by the correlation between the probability density functions of the responses of the key components of each component, and specifically calculated as follows:
in the formula, Cov (X)i，Xj) Random parameter X for representing bridge structure layer and material layer bridgei，XjThe covariance of (a) of (b),
respectively represent the random parameters X of the bridgei，XjRoot mean square value of variance.
And (3) based on the failure probability of each main component and the related structure and the related coefficient between every two components obtained by the formula (8), finally, respectively obtaining the reliability index and the failure probability of the bridge system of the series-connection system, the parallel-connection system and the series-parallel system by using a PCM method, and verifying by adopting a most common first-order boundary method for calculating the system failure probability, thereby proving the validity of the seismic reliability of the structural system solved in the aspect.
Specifically, as shown in fig. 1, the method mainly includes the following steps:
(F1) selecting N bridge random parameters related to bridge structure and material uncertainty according to the existing research and prior knowledge
Determining the random parameter distribution form of each bridge, and giving main statistical parameters of mean value and standard deviation;
(F2) considering the field condition of the actual bridge site of the bridge, selecting a Tajimi-Kanai power spectrum model, converting a power spectrum from a frequency domain to a time domain according to a trigonometric series non-stationary seismic wave synthesis theory and considering phase randomness, and synthesizing 30 pieces of longitudinal artificial seismic waves, transverse artificial seismic waves and vertical artificial seismic waves to represent the uncertainty of the seismic waves;
(F3) according to the distribution form of the random parameters of the bridge determined in the step (F1), selecting a corresponding Gaussian polynomial such as Gauss-Hermite polynomial, determining the weight and the integral point of the Gaussian, and determining the random parameters of the bridge
Generating an interpolation point X from the integration pointsi1, 2., 5, and generating 5 interpolation points corresponding to each bridge random parameter;
(F4) establishing an equivalent three-dimensional finite element model by using an open source finite element platform OpenSees according to the basic design data of a real bridge, generating bridge finite element model analysis samples under 5 Gaussian interpolation points of random parameters of each bridge, combining 30 artificially synthesized seismic waves, and performing 30 x 5 x N times of nonlinear time-course dynamic calculation based on nonlinear finite element analysis software, thereby obtaining a structural response extreme value theta of a key componentext；
(F5) The response extreme value theta obtained in the step (4) is processedextSubstituting the formula (2) and combining with the weight coefficient corresponding to the interpolation integral point to obtain the fractional order moment of the random parameter response of the single bridge
The statistical moment of the response extreme value is any order moment;
(F6) searching for the optimal parameters of the formula (3) by using a MATLAB simplex search method, and substituting the optimized alpha and lambda into the formula (4) to obtain a probability density function f of the componentY(y) representative pedestal response extremum probability density curves are shown in fig. 2 and 2-1;
(F7) according to the failure indexes of the components, the obtained probability density function is substituted into formula (5), the probability that each component exceeds the damage index threshold value for the first time is calculated, and then the failure probability of each main component of the bridge is obtained, as shown in fig. 3 and 3-1, wherein the damage indexes generally comprise indexes such as slight, general and serious failures, the failure indexes generally correspond to the serious failures, and the failure index is one of the damage indexes;
(F8) determining the correlation between failure modes of each main component by adopting the probability density function of the seismic response of the main components of the bridge obtained in the step (F6) and an equation (8), and calculating the correlation coefficient between every two components, as shown in figures 4 and 4-1;
(F9) assembling the individual component failure probabilities (equation (6), equation (7)) in series, parallel, and series-parallel modes according to the PCM method, and converting the component failure probabilities into system failure probabilities, as shown in fig. 5 and 5-1;
(10) and comparing the failure probability of the system with the upper and lower boundaries of a first-order boundary method, and determining the reliability of solving the failure probability of the system with the complex structure by adopting a down-conversion principle and a PCM method as shown in figures 6 and 6-1.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (10)
1. A bridge earthquake-resistant reliability analysis method is characterized by comprising the following steps:
s1, determining N bridge random parameters, generating B seismic waves representing bridge randomness, acquiring M Gaussian interpolation points of each bridge random parameter, and establishing A bridge structure finite element model analysis samples according to the M Gaussian interpolation points, wherein A is M.N;
s2, combining B seismic waves with A bridge structure finite element model analysis samples to perform A-B times of bridge deterministic dynamic response analysis to obtain a key component response extreme value fractional order moment;
and S3, obtaining the failure probability and the reliability index of each key component under the earthquake action based on the response extreme value fractional order moment of the key component, and further obtaining the failure probability of the bridge system.
2. The method for analyzing earthquake resistance reliability of a bridge according to claim 1, wherein the step S1 includes the following steps;
s11, determining N bridge random parameters and the distribution form of each bridge random parameter, and randomly generating B seismic waves by considering the seismic wave phase;
s12, selecting a corresponding Gaussian polynomial according to a random parameter distribution form of the bridge, and determining Gaussian weight and an integral point;
s13, generating interpolation points by the bridge random parameters according to the integral points, wherein M interpolation points are correspondingly generated by each bridge random parameter;
s14, establishing a bridge three-dimensional finite element model according to design data of bridge structures and materials, and generating bridge finite element model analysis samples under M Gaussian interpolation points of random parameters of each bridge, wherein A bridge structure finite element model analysis samples are total.
3. The method for analyzing earthquake resistance reliability of a bridge according to claim 2,
before step S14, there is a step of three-dimensional finite element model verification, specifically:
a1: taking a median value for each bridge structure random parameter;
a2: respectively establishing a bridge three-dimensional finite element model by utilizing at least two nonlinear finite element analysis software, respectively inputting a median value into all the bridge three-dimensional finite element models for modal analysis, and comparing results obtained by the at least two nonlinear finite element analysis software to verify the correctness of the bridge three-dimensional finite element models;
then, in step S14, the median is replaced with gaussian interpolation point values, and a bridge structure finite element model analysis samples are further established.
4. The method for analyzing earthquake resistance reliability of a bridge according to claim 3,
the bridge random parameters comprise bridge structure random parameters and bridge material random parameters, the bridge structure random parameters comprise concrete volume weight random parameters, member integral size random parameters and damping ratio random parameters, the materials comprise concrete elastic modulus random parameters, concrete compressive strength random parameters and reinforcing steel bar yield strength random parameters,
the step S1 of determining N bridge random parameters specifically includes: summarizing and summarizing the randomness of the existing bridge structure parameters and the randomness of the material parameters, obtaining a bridge random parameter database, obtaining N bridge random parameters based on the bridge random parameter database, and determining the distribution form of each bridge random parameter, wherein the N bridge random parameters comprise N1 bridge structure random parameters and N2 bridge material random parameters.
5. The method for analyzing earthquake resistance reliability of a bridge according to claim 4,
in step S1, generating B seismic waves representing the randomness of the bridge is specifically:
and determining a power spectrum model by combining actual field soil layer parameters of a bridge site, converting the seismic wave frequency domain power spectrum into time domain seismic waves according to a multi-dimensional multi-point non-stationary seismic wave triangular series method seismic wave synthesis theory and considering seismic wave phase randomness, and synthesizing B pieces of artificial seismic waves based on the time domain seismic waves.
6. The method for analyzing the seismic reliability of the bridge according to any one of claims 1 to 5, wherein the step S2 specifically comprises the following steps:
s21: combining the B seismic waves with the A bridge structure finite element model analysis samples, performing nonlinear dynamic analysis, and obtaining a bridge structure key component response extreme value and a key component response extreme value arbitrary order moment;
s22: and obtaining the fractional order moment of the response extreme value of the key component based on the random order moment of the response extreme value of the key component, wherein the key component comprises a pier, a bridge tower, a support, a main beam and a stay cable.
7. The method for analyzing seismic reliability of a bridge according to any one of claims 1 to 5, wherein the step S3 specifically comprises the following steps:
s31: respectively determining damage indexes of each key component according to a deformation damage criterion or a strength damage criterion, and obtaining a probability density function f of the key component by combining the response extreme value fractional order moment of the key component according to the maximum entropy principleY(y)；
S32: integrating the probability density function f of the key componentY(y) combining the damage indexes of the key components and the failure states of the key components to obtain the failure probability of the key components under the action of the earthquake;
s33: and combining the failure probabilities of the key components by using a conditional marginal probability density product method and considering the relevant characteristics among the failure modes of the key components to obtain the failure probability of the bridge system.
8. The method for analyzing earthquake resistance reliability of a bridge according to claim 7,
in step S31, a dual-loop optimization function is obtained according to the maximum entropy principle, and the fraction order α and the lagrangian multiplier λ obtained by optimization are substituted into the estimated probability density function of the critical component response extremum by finding the fraction order α and the lagrangian multiplier λ in the minimum optimization function of the dual-loop optimization function
Obtaining the probability density function f of the key componentY(y)。
9. The method for analyzing earthquake resistance reliability of a bridge according to claim 8,
in step S32, the probability density function is substituted into the member reliability probability function according to the failure index of the member, and the probability that each key member first surpasses the damage index threshold is calculated, thereby obtaining the failure probability of each main member of the bridge.
10. The method for analyzing earthquake resistance reliability of a bridge according to claim 8,
step S33 specifically includes the following steps,
s331. probability density function f based on key componentY(y) determining the correlation between failure modes of all main components, and calculating the correlation coefficient between every two components;
s332, assembling the failure probability of a single key component according to a serial mode, a parallel mode and a series-parallel mode by using a conditional marginal probability density Product (PCM) method, and converting the failure probability of the component into the failure probability of the system.
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