CN116561875B - Bridge network vulnerability analysis method considering bridge seismic response correlation - Google Patents

Bridge network vulnerability analysis method considering bridge seismic response correlation Download PDF

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CN116561875B
CN116561875B CN202310827938.XA CN202310827938A CN116561875B CN 116561875 B CN116561875 B CN 116561875B CN 202310827938 A CN202310827938 A CN 202310827938A CN 116561875 B CN116561875 B CN 116561875B
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road section
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maximum drift
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CN116561875A (en
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钟剑
周思恩
史正骁
吴俊骁
陈政帆
朱运涛
史龙飞
徐伟
郑香林
吴乔飞
毛永恒
蔡耀鑫
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Hefei University of Technology
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Abstract

The invention relates to the technical field of traffic network vulnerability analysis, in particular to a bridge network vulnerability analysis method considering bridge seismic response correlation, which comprises the following steps: calculating the maximum drift ratio of bridge piers of each bridge; calculating the correlation between any two bridges in the same road section through a correlation formula; establishing the association between the correlation of the bridge and the maximum drift ratio of the corresponding bridge piers, and selecting a preset number of bridge seismic displacement responses through the association; comparing the maximum drift ratio of the bridge piers of each bridge with the limit value of the maximum drift ratio of the bridge piers, and calculating the vulnerability of each road section according to the comparison result; judging the magnitude relation between the vulnerability of each road section in the bridge network and the set value, and calculating the vulnerability of the bridge network according to the judging result; the method can accurately calculate the seismic vulnerability of the bridge network with correlation among the bridges.

Description

Bridge network vulnerability analysis method considering bridge seismic response correlation
Technical Field
The invention relates to the technical field of traffic network vulnerability analysis, in particular to a bridge network vulnerability analysis method considering bridge seismic response correlation.
Background
The bridge is used as a ring in the traffic network and can play a role in connecting roads. However, bridges are more easily damaged under the action of an earthquake than roads, and the post-earthquake state of the bridge directly affects the traffic capacity of the traffic network. Therefore, it is necessary to evaluate the post-earthquake state of the bridge and further determine the actual traffic capacity of the traffic network.
When evaluating the post-earthquake state of a bridge, the vulnerability of the bridge is generally used as a measurement index. The traditional vulnerability calculation method needs to carry out fine modeling on the bridges, and for bridges with a small number, the accuracy of the method is high. However, in a practical traffic network, the number of bridges is huge, the constructed bridge network is complex, and there are a large number of similar bridges in the complex network, which results in that a large cost is spent and accuracy is lowered by adopting the method.
The reason that a large number of similar bridges exist is that the design and construction methods are similar, the reference specifications are the same, so that the bridges have quite similar structural characteristics in aspects of types, bridge spans, seismic reduction and isolation measures and the like, and correlation is established between the bridges. The method is affected by regions, similar traffic flows exist on each road section or bridge on the bridge network, and the same extreme events are experienced under similar environmental conditions, so that the inter-bridge compatibility is more obvious when an earthquake happens. The traditional vulnerability calculation method does not consider the correlation between bridges, which is one of the important reasons for low calculation accuracy.
Disclosure of Invention
In order to avoid and overcome the technical problems in the prior art, the invention provides a bridge network vulnerability analysis method considering the correlation of bridge seismic response. The method can accurately calculate the seismic vulnerability of the bridge network with correlation among the bridges.
In order to achieve the above purpose, the present invention provides the following technical solutions:
a bridge network vulnerability analysis method considering bridge seismic response correlation comprises the following steps:
s1, calculating the maximum drift ratio of bridge piers of each bridge;
s2, calculating the correlation between any two bridges in the same road section through a correlation formula;
s3, establishing association between the correlation of the bridge and the corresponding maximum drift ratio of the bridge piers, and selecting a preset number of maximum drift ratios of the bridge piers through the association; comparing the maximum drift ratio of the bridge piers of each bridge with the limit value of the maximum drift ratio of the bridge piers, and calculating the vulnerability of each road section according to the comparison result;
and S4, judging the magnitude relation between the vulnerability of each road section in the bridge network and the set value, and calculating the vulnerability of the bridge network according to the judging result.
As still further aspects of the invention: the specific steps of step S1 are as follows:
s11, firstly, acquiring a calculation formula of the bridge seismic displacement response, wherein the calculation formula is as follows:
wherein, μrepresenting the median value of the seismic displacement response of the bridge;Trepresenting the fundamental period of the bridge;PGVrepresenting peak ground speed;a，b，cthe parameters are parameters of a bridge seismic displacement response calculation formula;
s12, the basic period of each bridge is shortenedT、Peak ground speedPGVCorresponding median value of bridge seismic displacement responseμIntroducing into a pier maximum drift ratio calculation formula, and solving parameters by regression analysisa，b，cThe specific value of (3);
s13, median value of earthquake displacement response through bridgeμCalculating the maximum drift ratio of bridge piersM，MThe calculation formula of (2) is as follows:
wherein, Mrepresenting the maximum drift ratio of the bridge pier;Hthe height of the bridge pier is set;
s14, acquiring the basic period of each bridge in the step S12T、Bridge pier heightHPeak ground speed of the bridge locationPGVAnd leading the values into a bridge pier maximum drift ratio calculation formula to calculate so as to obtain the bridge pier maximum drift ratio of each bridge.
As still further aspects of the invention: the specific steps of the step S2 are as follows:
s21, acquiring the basic period of each bridge in the same road sectionTRespectively comparing the basic periods of any two bridges in the road section, and generating a basic period ratio;
s22, calculating covariance between seismic displacement responses of any two bridges in the road section according to a covariance calculation principle;
s23, substituting the calculated basic cycle ratio and the corresponding covariance into a correlation calculation formula, fitting the calculation formula, and solving the specific numerical value of each parameter in the calculation formula through regression analysis;
s24, substituting the calculated basic period ratios in the road section into a correlation calculation formula determined by parameters to deduce the corresponding correlation between the bridges.
As still further aspects of the invention: the specific steps of step S3 are as follows:
s31, selecting a road section, and calculating the ground speed of each bridge in the road section at each peak valuePGVMaximum drift ratio of bridge pierM；
S32, selectingN L Group same peak ground speedPGVMaximum drift ratio of bridge pier of each lower bridgeM；
S33, selecting the maximum drift ratio of each pierMLimit value of maximum drift ratio of bridge pierM max Comparing the sizes;
s34, counting maximum drift ratio of bridge pierMIs larger than the limit value of the maximum drift ratio of the bridge pierM max The number of groups of (a)n L Will beN L Andn L introducing a road section vulnerability probability function representing vulnerability of the road section to obtain the ground speed of the road section at the peak valuePGVThe vulnerable probability of the lower road section;
s35, calculating the ground speed of each road section in the bridge network at each peak value by using the road section vulnerable probability functionPGVThe probability of vulnerability of the lower road section.
As still further aspects of the invention: the specific steps of step S4 are as follows:
s41, intersecting and connecting all road sections through the end points of the road sections, and forming corresponding paths, wherein all paths are combined to form a bridge network;
s42, randomly selecting each road section from the set rangeN W Group random numberP W As an observation value, the link vulnerability probability of the link is comparedP L And observed valueP W Is of a size of (2);
s43, if the vulnerable probability of the road section exists in the pathP L Greater than the corresponding observed valueP W The path does not exist, otherwise it exists;
s44, statisticsN W Group number of group random numbers such that no path exists between the start point and the end point of bridge networkn W And willN W Andn W and (5) leading the probability of the vulnerability of the bridge network into a calculation formula of the probability of the vulnerability of the bridge network to calculate the probability of the vulnerability of the bridge network.
As still further aspects of the invention: the basic cycle ratio is calculated as follows:
wherein, R k,i,j represent the firstkIn road segmentsiBridge seat and the firstjThe basic cycle ratio of the seat bridge;T k,i represent the firstkIn road segmentsiThe basic period of the bridge seat;T k,j represent the firstkIn road segmentsjBasic cycle of the bridge.
As still further aspects of the invention: the correlation calculation formula is expressed as follows:
wherein, Cov[i,j]represent the firstiSeat bridgeBeam and the firstjCovariance between base bridge fundamental periods, the covariance characterizing the firstiBridge seat and the firstjCorrelation between seat bridges;eandfare parameters of the correlation calculation formula.
As still further aspects of the invention: the calculation formula of the road section vulnerable probability function is as follows:
wherein, P L the link vulnerability probability of the link is represented.
As still further aspects of the invention: the calculation formula of the probability of vulnerability of the bridge network is as follows:
wherein, P W representing the probability of vulnerability of the bridge network.
As still further aspects of the invention: the setting range in step S42 is [0,1].
Compared with the prior art, the invention has the beneficial effects that:
1. the method and the device can accurately calculate the vulnerability of the bridge network with correlation among the bridges.
2. The method and the device can calculate the earthquake displacement response of the bridge pier more quickly and obtain the correlation between the bridge displacement responses more quickly.
Drawings
FIG. 1 is a schematic diagram of the operation flow structure of the present invention.
FIG. 2 is a scatter plot of median values of seismic displacement responses of 20 bridges in accordance with the present invention.
FIG. 3 is a spatial curved surface diagram of median composition of seismic displacement responses of 20 bridges in the invention.
FIG. 4 is a correlation data distribution diagram of 20 bridges in the present invention.
Fig. 5 is a network diagram of a bridge formed by 20 bridges according to the present invention.
Fig. 6 is a network diagram of a bridge formed by 20 bridges after earthquake in the invention.
Fig. 7 is a graph of vulnerability of the bridge network from point a to point E in the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Because bridges are more easily damaged than roads in an earthquake, the invention assumes that the traffic capacity of the roads remains intact when studying the earthquake vulnerability of road sections and traffic networks, i.e. does not consider the influence of the damage probability of the roads. The vulnerability of a road segment may be obtained from bridges on the road segment. Likewise, the vulnerability of the bridge network may be obtained from a combination of vulnerabilities of road segments. Generally expressed as:
（1）
（2）
wherein, P road section Representing the vulnerable probability of the road section;P mbridge-shaped structure Represent the firstmThe probability of the easy damage of the seat bridge,Mrepresenting the total number of bridges in the road section;P network system Representing the probability of vulnerability of the bridge network;P nroad segment- Represent the firstnThe probability of vulnerability of the road segment,Nrepresenting the total number of road segments in the bridge network.
In order to achieve the above object, the present invention thus employs the operation steps as shown in fig. 1:
s1, calculating the maximum drift ratio of bridge piers of each bridge.
Median seismic displacement response of bridge based on assumptions of Probabilistic Seismic Demand Model (PSDM)μAnd earthquake intensity indexIMThe logarithmic linear relation exists, and the logarithmic linear relation is a bridge seismic displacement response calculation formula, and the calculation formula is as follows:
or->（3）
Wherein, a，cthe parameters are parameters of a bridge seismic displacement response calculation formula.
By peak ground speedPGVRepresenting the intensity of earthquake motionIM. The bridge seismic displacement response calculation formula becomes the following form:
（4）
and carrying out nonlinear time-course analysis on bridges with 20 different basic periods to obtain the median value of the seismic displacement response. Research finds that the median value of earthquake displacement response and the fundamental period of bridgeTSince there is a quantitative relationship, the formula (4) can be rewritten as:
（5）
wherein, band calculating parameters of a formula for the bridge seismic displacement response.
The data obtained by nonlinear time-course analysis of the bridges with 20 different basic periods are shown in fig. 2, and the data in fig. 2 are subjected to statistical regression analysis according to a formula (5), and the result is shown in fig. 3. In combination with fig. 2 and 3, givenPGVUnder the condition of (1), obtaining the quantitative relationship between the median value of the seismic displacement response of the bridge and the fundamental period of the bridge, and determining the parametersaTaking the weight of the mixture to be 0.36,btaking the sample of 0.82 of the total weight of the mixture,ctaking 1.21, equation (5) can be rewritten as:
（6）
maximum drift ratio of bridge pierMPDMedian value of seismic displacement response to bridgeμThe relation of (2) is:MPD=μ/Husing charactersMAlternate representationMPD. The bridge pier height is also added to the formula (6)HEquation (6) translates to equation (7), equation (7) being as follows:
（7）
according to formula (7), at the basic period of the known bridgeTAnd pier heightHOn the premise of the method, the median value of the earthquake displacement response of the bridge can be rapidly obtained. The parameters of the bridge sections for 20 different fundamental periods are shown in table 1.
Table 1 bridge parameters
S2, calculating the correlation between any two bridges in the same road section through a correlation formula.
The covariance reflects the cooperative relation between the two variables, and when the covariance is positive, and the larger the covariance is, the larger the degree of the same direction of the two variables is; the two variables change inversely when the covariance is negative. The invention uses covariance to describe the correlation of bridge seismic displacement response. In PSDM, it is generally assumed that the seismic displacement response and seismic capacity of the bridge are log-normal distribution-compliant. Thus, when considering the correlation of the bridge seismic displacement response, the natural logarithm of the bridge seismic displacement response is taken first, and then the covariance is calculated. The 20 models were analyzed for 400 sets of covariance and the data obtained are shown in fig. 4. The following quantization model is given to determine the relationship between covariance and fundamental period ratio.
（8）
Wherein, R k,i,j represent the firstkIn road segmentsiBridge seat and the firstjThe basic cycle ratio of the seat bridge;T k,i represent the firstkIn road segmentsiThe basic period of the bridge seat;T k,j represent the firstkIn road segmentsjBasic cycle of the bridge.
（9）
Wherein, Cov[i,j]represent the firstiBridge seat and the firstjCovariance between base bridge fundamental periods, the covariance characterizing the firstiBridge seat and the firstjCorrelation between seat bridges;eandfare parameters of the correlation calculation formula.
Substituting the data into equation (9) can be done as follows from the data scatter and fit curve represented in fig. 4: parameters (parameters)eTaking 0.58, parametersfTaking-0.50, then equation (9) is updated to equation (10), equation (10) being expressed as follows:
（10）
through the formula (10), the correlation between bridge seismic displacement responses can be obtained quickly when the basic period of the bridge is known.
S3, establishing association between the correlation of the bridge and the corresponding maximum drift ratio of the bridge piers, and selecting the maximum drift ratio of the bridge piers with a preset number through the association; and comparing the maximum drift ratio of the bridge piers with the limit value of the maximum drift ratio of the bridge piers, and calculating the vulnerability of each road section according to the comparison result.
When calculating the vulnerability of the road sections containing the bridge, the correlation between the bridge seismic displacement responses needs to be considered.
The invention acquires a bridge structure response sample considering correlation by means of a multi-element normal sampling function in a Python platform.
The Monte Carlo method is used for simulating the vulnerability of the road section by considering the completely uncorrelated unconnected probability calculation principle with reference to a series system.
The detailed steps for calculating the vulnerability of the road section are as follows:
1. for a certain road section, selecting one road sectionPGV。
2. Obtaining the maximum drift ratio of the bridge pier in the step S1MAnd that obtained in step S2Cov[i,j]As a parameter of the multiple normal sampling function, by selectingN L Group same peak ground speedVMaximum drift ratio of bridge pierM. In practice, a bridge is at a peak ground speedPGVMaximum drift ratio of bridge pierMIs a value, which can be recorded as a numberM 1 . However, if the earthquake happens again under the same condition, the bridge pier maximum drift ratio of the bridgeMIs marked asM 2 。M 1 And (3) withM 2 Are generally not equal, the same happensN L Group same peak ground speedPGVDuring the earthquake, the bridge will correspondingly generateN L Personal (S)M. Among the seismic vulnerability analysis systems we consider thisN L Personal (S)MObeys normal distribution, and because the invention considers a plurality of bridges, each bridgeN L Personal (S)MFor example, 20 bridges have 20×N L Personal (S)MIs subject to a multivariate normal distribution of mean and covariance.
Sequentially sorting according to the measured time to obtain 20 bridges, wherein each bridge is respectively measured into 10000 groups, namelyN L Each group comprises the maximum drift ratio of the corresponding bridge piers generated by each bridgeM。
3. Judging whether the selected data sets have the maximum drift ratio of the bridge pierMIs larger than the limit value of the maximum drift ratio of the bridge pierM max If so, the bridge damage is indicated, and the variable is countedcount 1 The number of the two-way valve is added with one,count 1 is 0; if not, count variablecount 1 Is unchanged. Up toN L The group data were all analyzed at this timecount 1 =n L RepresentingN L Among the group data aren L Maximum drift ratio of bridge piersMIs larger than the limit value of the maximum drift ratio of the bridge pierM max Is the case in (a).count 1 /N L The vulnerable probability of the road section is expressed as follows by adopting a formula (11):
（11）
wherein, P L the link vulnerability probability of the link is represented.
4. Modification ofPGVRepeating the steps to obtain the vulnerable probability of the road section under each peak ground speed, wherein the corresponding discrete data point is #PGV，P L ) A smooth line is used to connect the discrete data points together to form a vulnerability curve.
And S4, judging the magnitude relation between the vulnerability of each road section in the bridge network and the set value, and calculating the vulnerability of the bridge network according to the judging result.
In the connectivity analysis of a bridge network, it is generally necessary to use adjacency matrices to represent connection relationships between nodes in the bridge network by means of graph theory. The road sections are connected by the end points of the road sections in a crossing way, and form corresponding paths, and each path is combined to form a bridge network. The nodes are the end points of each road section and are also junction points.
In the theory of diagrams, the representation of a diagram is generallyG= (Y,E)，YIs a set of points that are to be selected,Y= {y 1 ,y 2 , …,y p ,…,y q ,…,y m },Eis an edge set of the two-dimensional object,E= {e 1 ,e 2 , …,e n for adjacent matrixA m m× =[a pq ]The representation is made of a combination of a first and a second color,a pq is referred to as formula (12):
（12）
the detailed steps for simulating the vulnerability of the bridge network are as follows:
1. for a certain road section, selecting one road sectionPGV。
2. The vulnerable probability of each road section obtained in the step S3 is calculatedP L As a known value, at [0,1]Obtaining random value for each road sectionP W As an observation. Comparison ofP L And (3) withP W Is of a size of (a) and (b). If it isP W Less thanP L Representing the road segment damage, the road segment corresponding in the adjacency matrixa pq Marked as "0", otherwiseP W Not less thanP L Indicating that the road section works normally, the road section corresponding to the adjacent matrixa pq And is designated as "1".
3. After the states of all road sections are determined, the adjacency matrix is determined, and whether paths exist from the starting point to the destination point can be judged by analyzing the adjacency matrix according to the graph theory principle.
Randomly selecting each road section from the set rangeN W Group random numberP W As an observation value, the link vulnerability probability of the link is comparedP L And observed valueP W Is of a size of (a) and (b).
If there is vulnerable probability of road section in the pathP L Greater than the corresponding observed valueP W Then the path does not exist, the variable is countedcount 2 The number of the two-way valve is added with one,count 2 is 0; if not, count variablecount 2 Is unchanged.
StatisticsN W Group number of group random numbers such that no path exists between the start point and the end point of bridge networkn W At this timecount 2 =n W ，count 2 /N W The probability of easy damage of the bridge network is obtained,N W =10000. expressed using equation (13), equation (13) is expressed as follows:
（13）
wherein, P W representing the probability of vulnerability of the bridge network.
4. Modification ofPGVRepeating the steps to obtain the probability of easy damage of the bridge network under each peak ground speed, wherein the corresponding discrete data points are @PGV，P W ) A smooth line is used to connect the discrete data points together to form a vulnerability curve.
Examples:
the example object is a bridge network composed of 5 nodes and 7 road sections, the bridge network is shown in FIG. 5, 20 bridges in step S1 are adopted, and the bridge pier height of each bridgeHAnd basic periodTAs shown in table 2.
Table 2 bridge parameter table
The calculation flow is as follows:
1. according to the basic period of the bridge in Table 2TObtaining the maximum drift ratio of the bridge piers of each bridge by the bridge pier height H and the formula (7)MPDThe results are shown in Table 3:
TABLE 3 expression of maximum drift ratio of pier
According to research, bridge pier maximum drift ratio limit valueM max The values of (2) are shown in Table 4:
TABLE 4 maximum drift ratio limit for piers
2. According to the basic period ratio of the bridgeT r As can be obtained from equation (10), the covariances are combined to form a covariance matrix, the covariance matrix generated by 20 bridges is 20×20, and the covariance matrix is a symmetric matrix, i.e. each column or each row respectively represents data of one bridge. The first column of data for the covariance matrix is listed as shown in table 5.
3. For a givenPGVFor example, initiallyPGV=0.03 m/s. Sampling the samples considering the correlation of the seismic displacement response by using a plurality of normal sampling functions in a Python platform, and designatingN L =100000, the sample obtained is oneN L X 20 data matrix, thiColumn data represents the firstiBridge seatN L And response samples.
4. The damaged state, such as the medium damaged state, of a road segment, such as road segment AB, is analyzed. As shown in fig. 5, the section AB in the embodiment includes the bridge 1 and the bridge 2.
When (when)N L When=1, the bridge pier maximum drift ratio of two bridgesMPDIn a data matrix, usingDr[1,1]、Dr[1,2]And (3) representing. The maximum drift ratio of bridge piers of two bridgesMPDLimit value of maximum drift ratio of bridge pier in slight damage stateMPD max Comparison is performed as long as there isMPD>MPD max The road section is considered to be in a slight damage state, and the variable is countedcount 1 Adding one; if not, continuing to judge the second group of data.
When (when)N L When=2, the data matrix is formedDr[2,1]、Dr[2,2]The corresponding data are respectively compared with the limit value of the maximum drift ratio of the bridge pierMPD max Comparison is performed as long as there isMPD>MPD max Then it is recognized thatTo achieve a slight injury state on the road section, the variables are countedcount 1 Adding one; if not, continuing to judge other groups of data.
When (when)N L When=3, 4 …, the above procedure is repeated untilN L =100000。N L After all of the group samples have been analyzed,count 1 /N L the road section vulnerability probability in the slight damage state of the road section AB is the road section vulnerability probability. Changing limit value of maximum drift ratio of bridge pierMPD max The link vulnerability probability of the link AB in the other three damaged states can be obtained. And the road section vulnerability probability calculation process of other road sections is the same as that of the road section AB.
TABLE 5 covariance matrix portion data
5. Modification ofPGVFor example, the nextPGV=0.06 m/s, which can be obtained by repeating the above stepsPGVRoad segment vulnerability probability of all road segments below. In the present embodimentPGVThe values are from 0.03m/s to 1.50m/s, 50 groups are total, and partial data of the medium probability of occurrence of all road sections are shown in the following table 6.
Table 6 moderate probability of vulnerability
Each is to bePGVThe vulnerable probability of 7 road sections forms a matrixU 1×7 For example:
U 1×7 (PGV=0.30m/s)=[ 0.20278, 0.23943, 0.34123, 0.41875, 0.39404, 0.46120, 0.46119 ]；
U 1×7 (PGV=0.60m/s)=[ 0.70286, 0.73475, 0.84276, 0.89045, 0.86740, 0.90796, 0.89630]；
U 1×7 (PGV=0.90m/s)=[ 0.90655, 0.92385, 0.96950, 0.97995, 0.97310, 0.98578, 0.98148]。
5. from the graph theory knowledge, the adjacency matrix of the bridge network in the embodiment is written directly, which is a 5×5 matrix:
（14）
for matrix A 5×5 The values of (2) are explained as follows: rows 1 through 5 represent A, C, D, E, F five nodes, and columns 1 through 5 represent these five nodes, respectively; a value of 0 at row 1 and column 3 indicates that there is no direct link between nodes A and C in the bridge network of the embodiment, and a value of 2 at row 3 and column 3a BC Indicating that there is a link between node B and node C when no earthquake has occurred, the state parameter is used here because the damaged state of the link BC after the earthquake is unknowna BC Instead, the value of the state parameter refers to formula (12).
The matrix is determined as followsaIs the value of (1):
to be used forPGVFor example, apply np.random.rand function in Python to generate random numbers between 0 and 1 for 7 road segments, and form a random number matrix:
P W 1×7 (PGV=0.30m/s)=[ 0.03153, 0.45179, 0.90813, 0.03868, 0.78933, 0.39648, 0.26016 ]. Comparing the known matrix using a random number matrix:
U 1×7 (PGV=0.30m/s)=[ 0.20278,0.23943, 0.34123, 0.41875, 0.39404, 0.46120, 0.46119 ]。
judging according to the description in the formula (11) and the step S4aIs the value of (1):
0.03153＜0.20278，a AB =0；
0.45179＞0.23943，a AD =1；
the remaining results were as follows:
a BC =1；a BE =0；a CD =1；a CE =0；a DE =0. The adjacency matrix in equation (14) is rewritten as:
（15）
after the adjacent matrix is updated, the corresponding bridge network is also updated synchronously, and a bridge network diagram corresponding to the adjacent matrix expressed by the formula (15) is shown in fig. 6.
By using the networkx function library in Python, it is possible to obtain whether an undamaged path exists between two designated points, for example, in this example, a is used as a starting point and E is used as an end point. Calculating that no path exists between A and E according to the above adjacency matrix, counting variablescount 2 And adding one.
Regenerating random number matrixU 1×7 Repeating the above operation untilN W The analysis of the fractions was complete. 50 groups were analyzedPGVThe process is carried outPGV，P W ) And the scattered points are connected smoothly, so that the vulnerability curve of the bridge network can be obtained. Other lesion status analysis steps are as above. The vulnerability curves of the four types of damage states of the final example are shown in fig. 7 below.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.

Claims (6)

1. A bridge network vulnerability analysis method considering bridge seismic response correlation is characterized by comprising the following steps:
s1, calculating the maximum drift ratio of bridge piers of each bridge;
s2, calculating the correlation between any two bridges in the same road section through a correlation formula;
s3, establishing association between the correlation of the bridge and the corresponding maximum drift ratio of the bridge piers, and selecting a preset number of maximum drift ratios of the bridge piers through the association; comparing the maximum drift ratio of the bridge piers of each bridge with the limit value of the maximum drift ratio of the bridge piers, and calculating the vulnerability of each road section according to the comparison result;
s4, judging the magnitude relation between the vulnerability of each road section in the bridge network and the set value, and calculating the vulnerability of the bridge network according to the judging result;
the specific steps of step S1 are as follows:
s11, firstly, acquiring a calculation formula of the bridge seismic displacement response, wherein the calculation formula is as follows:
wherein, μrepresenting the median value of the seismic displacement response of the bridge;Trepresenting the fundamental period of the bridge;PGVrepresenting peak ground speed;a，b，cthe parameters are parameters of a bridge seismic displacement response calculation formula;
s12, the basic period of each bridge is shortenedT、Peak ground speedPGVCorresponding median value of bridge seismic displacement responseμIntroducing into a pier maximum drift ratio calculation formula, and solving parameters by regression analysisa，b，cThe specific value of (3);
s13, median value of earthquake displacement response through bridgeμCalculating the maximum drift ratio of bridge piersM，MThe calculation formula of (2) is as follows:
wherein, Mrepresenting the maximum drift ratio of the bridge pier;Hthe height of the bridge pier is set;
s14, acquiring the basic period of each bridge in the step S12T、Bridge pier heightHPeak ground speed of the bridge locationPGVThe values are led into a bridge pier maximum drift ratio calculation formula to be calculated, so that the bridge pier maximum drift ratio of each bridge is obtained;
the specific steps of the step S2 are as follows:
s21, acquiring the basic period of each bridge in the same road sectionTRespectively comparing the basic periods of any two bridges in the road section, and generating a basic period ratio;
s22, calculating covariance between seismic displacement responses of any two bridges in the road section according to a covariance calculation principle;
s23, substituting the calculated basic cycle ratio and the corresponding covariance into a correlation calculation formula, fitting the calculation formula, and solving the specific numerical value of each parameter in the calculation formula through regression analysis;
s24, substituting each basic period ratio calculated in the road section into a correlation calculation formula determined by parameters to deduce the corresponding correlation between bridges;
the specific steps of step S3 are as follows:
s31, selecting a road section, and calculating the ground speed of each bridge in the road section at each peak valuePGVMaximum drift ratio of bridge pierM；
S32, selectingN L Group same peak ground speedPGVMaximum drift ratio of bridge pier of each lower bridgeM；
S33, selecting the maximum drift ratio of each pierMLimit value of maximum drift ratio of bridge pierM max Comparing the sizes;
s34, counting maximum drift ratio of bridge pierMIs larger than the limit value of the maximum drift ratio of the bridge pierM max The number of groups of (a)n L Will beN L Andn L introducing a road section vulnerability probability function representing vulnerability of the road section to obtain the ground speed of the road section at the peak valuePGVThe vulnerable probability of the lower road section;
s35, calculating the ground speed of each road section in the bridge network at each peak value by using the road section vulnerable probability functionPGVThe vulnerable probability of the lower road section;
the specific steps of step S4 are as follows:
s41, intersecting and connecting all road sections through the end points of the road sections, and forming corresponding paths, wherein all paths are combined to form a bridge network;
s42, randomly selecting each road section from the set rangeN W Group random numberP W As an observation value, the link vulnerability probability of the link is comparedP L And observed valueP W Is of a size of (2);
s43, if the vulnerable probability of the road section exists in the pathP L Greater than the corresponding observed valueP W The path does not exist, otherwise it exists;
s44, statisticsN W Group number of group random numbers such that no path exists between the start point and the end point of bridge networkn W And willN W Andn W and (5) leading the probability of the vulnerability of the bridge network into a calculation formula of the probability of the vulnerability of the bridge network to calculate the probability of the vulnerability of the bridge network.
2. The bridge network vulnerability analysis method considering bridge seismic response correlation according to claim 1, wherein the basic cycle ratio is calculated according to the following formula:
wherein, R k,i,j represent the firstkIn road segmentsiBridge seat and the firstjThe basic cycle ratio of the seat bridge;T k,i represent the firstkIn road segmentsiThe basic period of the bridge seat;T k,j represent the firstkIn road segmentsjBasic cycle of the bridge.
3. The bridge network vulnerability analysis method considering bridge seismic response correlation according to claim 2, wherein the correlation calculation formula is expressed as follows:
wherein, Cov[i, j]represent the firstiBridge seat and the firstjCovariance between base bridge fundamental periods, the covariance characterizing the firstiBridge seat and the firstjCorrelation between seat bridges;eandfare parameters of the correlation calculation formula.
4. The bridge network vulnerability analysis method considering bridge seismic response correlation as claimed in claim 3, wherein the calculation formula of the road section vulnerability probability function is as follows:
wherein, P L the link vulnerability probability of the link is represented.
5. The bridge network vulnerability analysis method considering bridge seismic response correlation as claimed in claim 4, wherein the bridge network vulnerability probability calculation formula is as follows:
wherein, P W representing the probability of vulnerability of the bridge network.
6. The bridge network vulnerability analysis method considering bridge seismic response correlation as claimed in claim 5, wherein the setting range in step S42 is [0,1].
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