CN116226961A - Bayesian network-based large-scale steel bridge deck fatigue crack space uneven distribution evaluation method - Google Patents
Bayesian network-based large-scale steel bridge deck fatigue crack space uneven distribution evaluation method Download PDFInfo
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Abstract
The invention provides a Bayesian network-based large-scale steel bridge deck fatigue crack space uneven distribution evaluation method, which comprises the steps of dividing a bridge deck into finite cells through regular grid division, wherein each cell is a research unit, and defining the state of the unit based on structural construction factors, load environment factors and crack states; establishing a Bayesian network to describe probability dependency relationships among the variables according to causal relationships among the unit state variables; and estimating Bayesian network parameters by using a expectation maximization algorithm, and performing sensitivity analysis between variables. According to the invention, network parameter learning can be performed by utilizing crack detection data of the sections of the orthotropic steel deck slab, and then the number of cracks of the undetected sections of the deck slab is estimated, so that the rapid assessment of the number of fatigue cracks of the whole deck slab is realized, and the time cost and economic cost of large-span bridge crack detection are greatly reduced.
Description
Technical Field
The invention relates to the technical field of safety assessment of large bridge structures, in particular to a Bayesian network-based assessment method for fatigue crack spatial uneven distribution of a large steel bridge deck.
Background
Failure of large bridge structures can cause serious economic loss, environmental damage and social impact. Among them, the orthotropic steel deck system is an important component of large bridges. Because of the direct bearing of the load of the vehicle and various long-term environmental effects, fatigue cracking inevitably occurs in the service period, and the initiation and development of the fatigue cracking of the whole bridge deck slab of the bridge can lead to the degradation of the bearing performance. In order to ensure the operation safety of the bridge structure in the service period, the accurate detection of fatigue cracks on the steel bridge deck is important, and the method is a foundation for improving the reliability of the whole bridge structure, preventing bridge failure and reducing the operation and maintenance cost of the bridge. Therefore, the grasping of the spatial uneven distribution of the fatigue cracks of the steel bridge deck slab has important significance for guaranteeing the service safety of the orthotropic steel bridge deck slab.
For the steel box girder structure adopting the orthotropic steel bridge deck, the number and the development speed of fatigue crack initiation can be increased rapidly along with the increase of the service time. At present, bridge management units begin to pay attention to the fatigue condition of orthotropic plates, perform manual inspection on a monthly basis to count the number, position, width, length and other information of fatigue cracks of the main beam, and adopt maintenance measures mainly based on welding for remediation. The related detection results show that over 8000 various fatigue cracks are found on the orthotropic bridge deck of a large-scale steel box girder cable-stayed bridge in an accumulated way, and typical fatigue vulnerable parts of the bridge deck are mainly divided into six types: a) Welding seams between the U-shaped stiffening ribs and the top plate; b) Butt welding seams of the U-shaped stiffening ribs; c) Welding seams between the U-shaped stiffening ribs and the transverse partition plates; d) Welding seams between the cable brackets and the U-shaped stiffening ribs; e) Welding seams between the top plate and the diaphragm plate; f) Horizontal welding seams at arc-shaped passing welding holes of the diaphragm plates. The arc-shaped overselding holes of the diaphragm plates are cut at high temperature, so that the edges of the holes are subjected to larger residual stress and initial defects after cooling, and researches show that the residual stress can even reach the yield limit of the material. In addition, there is a stress concentration effect at the center of the over-welded hole, and therefore, fatigue cracks are extremely liable to develop at the structure, and the fatigue cracks often appear to propagate approximately horizontally. Except the arc-shaped openings, the other fatigue vulnerable parts are all splicing welding seams of the steel plates. As shown in fig. 1, typical fatigue cracks found in practical detection are remarkable in that c-type cracks and f-type cracks are close in distance, so that when cracks appear at one part, stress at the other part is redistributed, and initiation and development of the cracks are restrained, and therefore, the c-type cracks and the f-type cracks do not appear at the same time.
In the fatigue crack detection of a steel box girder of an actual large-scale bridge, the spatial distribution of cracks on an orthotropic steel bridge deck is found to have non-uniformity, namely, a plurality of crack frequent areas and a plurality of crack sparse areas exist. At present, research on fatigue performance of an orthotropic steel bridge deck often focuses on a single fatigue vulnerable part, as shown in fig. 1, local details of occurrence of each fatigue crack are researched, and residual fatigue life, crack growth rate, crack initiation mechanism and the like are not researched, but the spatial distribution of fatigue cracks on the orthotropic steel bridge deck is not researched. The main factors influencing the fatigue crack space distribution comprise structural construction factors and load environment factors, and the sensitivity of different influencing factors to crack initiation probability is different. In order to solve the problems, the invention provides a Bayesian network-based evaluation method for the spatial uneven distribution of fatigue cracks of a large-sized steel bridge deck.
Disclosure of Invention
In order to solve the problems, the bridge deck is divided into limited cells through regular grid division, each cell is a research unit, and the cell states are defined based on structural construction factors, load environment factors and crack states; establishing a Bayesian network to describe probability dependency relationships among the variables according to causal relationships among the unit state variables; and estimating Bayesian network parameters by using a expectation maximization algorithm, and performing sensitivity analysis between variables.
The invention provides a Bayesian network-based evaluation method for the spatial uneven distribution of fatigue cracks of a large-sized steel bridge deck, which comprises the following specific schemes:
a Bayesian network-based large-scale steel bridge deck fatigue crack spatial uneven distribution evaluation method comprises the following steps:
step 1: dividing the bridge deck into finite cells through regular grid division, wherein each cell is a research unit, and defining a unit state based on structural construction factors, load environment factors and crack states;
step 2: establishing a Bayesian network according to the unit state variables to describe probability dependency relationships among the variables;
step 3: and estimating Bayesian network parameters by using a expectation maximization method, performing sensitivity analysis among variables, and evaluating the spatial uneven distribution of fatigue cracks of the large-scale steel bridge deck.
Further, the step 1 specifically includes:
defining bridge deck research units, discretizing the bridge deck into limited cells, wherein each cell is used as a research unit, and the more the number of the divided units is, the more samples are, so that the parameter learning of a network model is facilitated;
defining unit state variables, defining the unit states according to the crack states on the research units, the unit load environments and the space positions due to the fact that the crack initiation probability of different units is different, so that the fatigue crack distribution problem is converted into the crack initiation probability problem on the units,
defining cell state variables includes the following four variable types:
(1) Taking into account spatial position variables
The units are different in space positions on the bridge deck, so that the units are influenced by vehicle loads and are divided into forward-direction positions and transverse-direction positions, and as vehicles running at high speed usually do not change lanes frequently, the forward-direction positions of the research units are identical and the load effects of the research units with the same structure are basically consistent, and therefore the influence of the forward-direction positions of the research units is not considered;
the load of the vehicles is different for the units with different transverse bridge directions, the fatigue crack of the bridge deck is mainly concentrated in the heavy road and the middle road, the transverse position of the unit i is described by two variables, X 1,i Indicating the lane in which the unit is located, X 2,i Representing a specific location of the unit within the lane;
(2) Taking into account unit construction variables
Comprising a transverseBaffle thickness variable X 3,i ＝T dia Variable of unit length X 4,i ＝L e Variable X of top plate thickness 5,i ＝T dec Five values are available for the diaphragm thickness variable, three values are available for the unit length variable, and three values are available for the top plate thickness variable;
(3) Taking into account temperature interval variables
The temperature interval in which the unit is located is described by a variable: x is X 6,i ＝Tem dec ，Tem dec Coding the temperature zone of the unit, tem dec =1, 2 indicates that the cell is in the low temperature region and the high temperature region, respectively;
(4) Taking into account crack state variables
The cell crack states are described using four variables for four classes of typical cracks: x is X 7,i ＝I 1 ，X 8,i ＝I 2 ，X 9,i ＝I 3 ，X 10,i ＝I 4, in the formula I1 Indicating type I crack status coding on a cell, I 1 =1, 2, respectively, indicates unexplosive and type i cracks on the cell; i 2 Representing type II crack status codes on a cell, I 2 =1, 2, respectively, indicates unexplosive and type ii cracks on the cell; i 3 Representing type III crack status encodings on a cell, I 3 =1, 2, respectively, indicates unexplosive and type iii cracks on the cell; i 4 Representing type IV crack status codes on a cell, I 4 =1, 2, respectively indicating that no crack of type iv was initiated and that a crack of type iv was initiated on the cell;
(5) Complete state X of a cell i Represented as a vector of the ten state variables above: x is X i ＝{X j,i ；j＝1,2,…,10}。
Further, the division of cells follows the following two principles:
1. the residual state variables of the units are kept consistent except for the difference in the construction to be researched and the load environment;
2. each unit contains the various types of fatigue vulnerable sites studied, and the numbers remain consistent.
Further, the step 2 specifically includes:
step 2.1: performing causal relation analysis between unit state variables, and designing causal relation judgment criteria between the unit state variables: when the state of the variable X is changed to influence the reliability of the variable Y by researchers, and the state of the variable Y is changed but cannot influence the reliability of the variable X by the researchers, the variable X is called as the reason of the variable Y;
in unit construction variable X 5 And crack state variable X 9 When the thickness of the top plate is increased, the fatigue resistance of the welding seam between the top plate and the U rib is increased, so that the occurrence probability of III type cracks is reduced; when a III type crack is manually added, the judgment of the thickness of the top plate is not influenced, and the structural variable X 5 Is a crack state variable X 9 Reasons for (2);
based on causal Markov assumption, the unit space position variable, the component variable and the temperature interval variable are all reasons of the unit crack state variable;
step 2.2: the hidden node is introduced to simplify network parameters, a father node separation method is adopted, partial father nodes are combined in the middle node introduction process, the network independent parameters are reduced after the middle node introduction process is realized, and the I-type crack is provided with a Bayesian network model of the middle node;
step 2.3: establishing a unit Bayesian network, introducing a unit Bayesian network of intermediate nodes, wherein node X i I=11, 12, …,16 is an intermediate node having two discrete states, the values of the position variable, the construction variable and the temperature interval variable being known for a certain determination unit, i.e. node X i I=1, 2, …,6 is state observable, and the conditional probability p (X j |X 1 ,X 2 ,X 3 ,X 4 ,X 5 ,X 6 )j＝7,8,9,10；
The bayesian network is denoted bn= (G, θ), where G is directed acyclic; θ is a network parameter, and for discrete state variables, a conditional probability representation is used; each node in the Bayesian network corresponds to a random variable in the data set, the directed edges between the nodes represent probability dependency relationships among the variables, and the directed edges in the Bayesian network are established based on the causal relationshipsRepresenting causal relation among variables, wherein a starting node of a directed edge represents a factor and is a father node; the pointing node represents a fruit and is a child node; based on d-separation criteria and chain law of bayesian network, a set of random variables x= { X 1 ,X 2 ,…,X n The joint probability distribution of } is decomposed into Markov conditional equations:
wherein pa (X i ) Is node X i Is a parent node set of (a); p (X) i |pa(X i ) A) component node X i I.e. bayesian network parameters.
Further, the step 3 specifically includes:
step 3.1: the Bayesian network for dividing the sample into complete and missing samples comprises 16 variables based on whether the state of the variables is known in its entirety, i.e. one complete data sample is represented as a 16-dimensional vector d i = (1,1,1,3,3,1,2,1,1,2,? Representing the unknown state of the intermediate node, the meaning of the 16-dimensional state vector is: is positioned in lane 1 (X) 1 Position No. 1 (X) =1) 2 Unit of =1) with diaphragm thickness of 8mm (X 3 =1), length 3000mm (X 4 =3), top plate thickness 16mm (X 5 =3), located in a low temperature region (X 6 =1), sprouting of type i (X 7 =2) and type iv cracks (X 10 ＝2)；
Step 3.2: let o i Representing data sample d i Non-observable part of (h) i Is an observable part, d i ＝o i ∪h i The method comprises the steps of carrying out a first treatment on the surface of the Let o= { O 1 ,o 2 ,…,o N The data is observable in the training set, and H= { H 1 ,h 2 ,…,h N The training set d=o u H; the parameter learning containing the deficiency value data adopts an expected maximization algorithm;
step 3.3: for the complete dataset d= { D 1 ,d 2 ,…d m Use ofThe maximum likelihood estimation obtains an accurate estimation result, and a log likelihood function based on the condition independence and the chain rule is represented by the following formula:
wherein ,
representing variable X j At sample d i State of (a); pa (pa) i (X j ) Representing variable X j Is at sample d i State of (a); n represents the number of cell states; n represents the number of samples, and the parameter maximizing the log likelihood function is used as the optimal estimation
Step 3.4: for a missing sample, the probability of missing data p (d) cannot be directly calculated at a given parameter i θ), a desired maximization algorithm is adopted, and the method mainly comprises the following two steps of iteration: given the current network parameter θ t And observing the data set O, calculating a conditional expectation Q (θ|θ t )＝E[logp(D|θ)|θ t ,O]Find the maximization Q (θ|θ) t ) As initial value of the next step, i.e
Further, the sensitivity analysis process includes:
the conditional probabilities of the nodes cooperatively change when the conditional probabilities of the nodes are changed
Other conditional probability of the node +.>
And also changes at the same time to meet the requirement that the probability sum is one:
wherein ,
representing variable X i A j-th state value of (2); pi represents the variable X i A combined state of the parent nodes;
when the probability parameters change cooperatively, the prior probability lambda=p (X 1 =1), η is a certain state posterior probability η=p (X 7 =1), η is expressed as a linear function of λ:
η＝αλ+β (4)
wherein α and β are constants related to λ and η, referred to as the sensitivity of α to λ of η, and further referred to as α η-λ ；
Given a disturbance Δλ to λ, η will also change accordingly, and the change is denoted Δη, and the sensitivity is expressed by the following equation:
the beneficial effects are that:
the invention overcomes the defect that the traditional technology only pays attention to a single fatigue vulnerable part, and can obtain the spatial distribution of fatigue cracks on the orthotropic steel bridge deck;
the position variable, the construction variable and the temperature interval variable of the unit are considered in the Bayesian network model established by the invention, so that the model captures main factors influencing the space uneven distribution of fatigue cracks;
the intermediate nodes are introduced to summarize the Bayesian network part nodes, so that the number of the undetermined parameters of the Bayesian network is obviously reduced, and the problem of excessive network parameters caused by directly connecting the root nodes and the leaf nodes of the Bayesian network is solved;
the influence degree of the unit position variable, the construction variable and the temperature variable on the initiation probability of various fatigue cracks can be revealed through Bayesian network sensitivity analysis;
according to the invention, network parameter learning can be performed by utilizing crack detection data of the sections of the orthotropic steel deck slab, and then the number of cracks of the undetected sections of the deck slab is estimated, so that the rapid assessment of the number of fatigue cracks of the whole deck slab is realized, and the time cost and economic cost of large-span bridge crack detection are greatly reduced.
Drawings
FIG. 1 is a schematic diagram of an exemplary fatigue crack in a steel box girder detected in situ;
FIG. 2 is a flowchart of a method for evaluating the spatial non-uniform distribution of fatigue cracks of a large-scale steel bridge deck based on a Bayesian network;
FIG. 3 is a schematic diagram of orthotropic steel deck slab unit division;
FIG. 4 is a schematic diagram of cell space position variables;
FIG. 5 is a schematic diagram of a Bayesian network model of type I cracks with intermediate nodes;
fig. 6 is a schematic diagram of a unit bayesian network including intermediate nodes.
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.
Referring to fig. 1 to 6, the invention provides a bayesian network-based assessment method for fatigue crack spatial non-uniform distribution of a large-scale steel bridge deck, which comprises the following steps:
step 1: dividing the bridge deck into finite cells through regular grid division, wherein each cell is a research unit, and defining a unit state based on structural construction factors, load environment factors and crack states;
the step 1 specifically comprises the following steps:
step 1.1: bridge deck research units are defined, bridge decks are discretized into limited cells, each cell is used as one research unit, and the more the number of divided units is, the more samples are, so that parameter learning of a network model is facilitated.
The division of cells follows the following two principles: firstly, the state variables of the units are kept consistent except for the difference in the construction and load environments to be researched; secondly, each unit contains various fatigue vulnerable parts which are researched, and the quantity is kept consistent.
(2) The regular meshing is shown in fig. 3. The dividing boundaries of the bridge deck along the forward bridge direction and the transverse bridge direction are respectively the centers of the adjacent transverse partition plates and the U-shaped stiffening ribs.
Step 1.2: cell state variables are defined, which result in spatially non-uniform distribution of cracks due to differences in crack initiation probability for different cells. Therefore, the cell state is defined according to the crack state on the research cell, the cell load environment and the space position, so that the fatigue crack distribution problem is converted into the crack initiation probability problem on the cell.
In step 1.2, defining the cell state variables requires consideration of the following four variable types:
(1) Taking into account spatial position variables
The units are different in space positions on the bridge deck, so that the units are also different in influence of vehicle load and are divided into a forward bridge position and a transverse bridge position. Since vehicles traveling at high speeds generally do not change lanes frequently, the loads applied to the research units having the same forward direction and the same structure are substantially uniform, and thus the influence of the forward direction of the research units on the forward direction of the axles is not considered.
The vehicle loading applied to the units at different lateral positions is also different due to the different traffic volumes of the different lanes and the different types of main vehicles passing through the different lanes. In practical tests, it was found that deck slab fatigue cracks were mainly concentrated in heavy and intermediate lanes. The typical lane section is shown in fig. 4, then the lateral position of the unit i is described by two variables, X 1,i Indicating the lane in which the unit is located,X 2,i indicating the specific location of the unit within the lane.
(2) Taking into account unit construction variables
Comprising diaphragm thickness variation X 3,i ＝T dia Variable of unit length X 4,i ＝L e Variable X of top plate thickness 5,i ＝T dec . The range of values of each variable is shown in Table 1, five values of the diaphragm thickness variable, three values of the unit length variable and three values of the top plate thickness variable are obtained.
Table 1 unit construction variable value table
(3) Taking into account temperature interval variables
The temperature interval in which the unit is located is described by a variable: x is X 6,i ＝Tem dec ，Tem dec Coding the temperature zone of the unit, tem dec =1, 2 indicates that the cell is in the low temperature region and the high temperature region, respectively.
(4) Taking into account crack state variables
The real bridge detection result shows that the number of V-shaped cracks is small, and the V-shaped cracks have no meaning of parameter learning, so that the V-shaped cracks are not considered in the Bayesian network. For the type of fatigue crack that occurs in fig. 4, four variables are used to describe the cell crack state for four types of typical cracks: x is X 7,i ＝I 1 ，X 8,i ＝I 2 ，X 9,i ＝I 3 ，X 10,i ＝I 4, in the formula I1 Indicating type I crack status coding on a cell, I 1 =1, 2, respectively, indicates unexplosive and type i cracks on the cell; i 2 Representing type II crack status codes on a cell, I 2 =1, 2, respectively, indicates unexplosive and type ii cracks on the cell; i 3 Representing type III crack status encodings on a cell, I 3 =1, 2, respectively, indicates unexplosive and type iii cracks on the cell; i 4 Representing type IV crack status codes on a cell, I 4 =1, 2 indicates unexplosive and burst type iv on the unit, respectivelyAnd (5) lines.
(5) Finally, the complete state X of a cell i Represented as a vector of the ten state variables above: x is X i ＝{X j,i ；j＝1,2,…,10}。
Step 2: establishing a Bayesian network to describe probability dependency relationships among the variables according to causal relationships among the unit state variables;
the step 2 specifically comprises the following steps:
step 2.1: causal relationship analysis between cell state variables
(1) Designing causal relationship judgment criteria between unit state variables: if the state of the variable X is changed, which affects the reliability of the variable Y by researchers, but the state of the variable Y is changed, which cannot affect the reliability of the variable X by researchers, the variable X is called as the reason of the variable Y.
In unit construction variable X 5 And crack state variable X 9 For example, when the thickness of the top plate is increased, fatigue resistance of the weld between the top plate and the U rib is increased, thereby resulting in a decrease in the occurrence probability of III type cracks; and when a III type crack is manually added, the judgment of the thickness of the top plate is not influenced. Thus, the construction variable X 5 Is a crack state variable X 9 For reasons of (2).
(2) Based on the causal Markov assumption, the unit space position variable, the component variable and the temperature interval variable are all the reasons of the unit crack state variable, and the specific causal relationship is shown in Table 2.
TABLE 2 Unit State variable causal relationship Table
Step 2.2: the hidden node is introduced to simplify network parameters; and a father node separation method is adopted, partial father nodes are combined after the intermediate nodes are introduced, the independent parameters of the network are reduced after the intermediate nodes are introduced, and a Bayesian network model with I-type cracks and intermediate nodes is shown in figure 5.
The conditional probability calculation formula is
p(X 7 |X 1 ,X 2 ,X 5 ,X 6 )＝p(X 11 |X 1 ,X 2 )p(X 7 |X 11 ,X 5 ,X 6 ) (1)
Step 2.3: the main procedure for building a unit bayesian network is as follows. A schematic diagram of a unit bayesian network incorporating intermediate nodes is shown in fig. 6, in which node X i I=11, 12, …,16 is an intermediate node, having two discrete states.
For a certain determination unit, the values of its position variable, construction variable and temperature interval variable are known, i.e. node X i I=1, 2, …,6 is state observable. Calculation of conditional probability p (X) by Bayesian network forward reasoning j |X 1 ,X 2 ,X 3 ,X 4 ,X 5 ,X 6 )j＝7,8,9,10。
The bayesian network is denoted bn= (G, θ), where G is directed acyclic; θ is a network parameter, and for discrete state variables, a conditional probability representation is used; each node in the Bayesian network corresponds to a random variable in the data set, the directed edges between the nodes represent probability dependency relationships among the variables, the directed edges in the Bayesian network established based on the causal relationships represent causal relationships among the variables, and the initial node of the directed edges represents a 'factor' and is a father node; the pointing node represents a fruit and is a child node; based on d-separation criteria and chain law of bayesian network, a set of random variables x= { X 1 ,X 2 ,…,X n The joint probability distribution of } is decomposed into Markov conditional equations:
wherein pa (X i ) Is node X i Is a parent node set of (a); p (X) i |pa(X i ) A) component node X i I.e. bayesian network parameters.
Step 3: estimating Bayesian network parameters by using an expectation maximization algorithm, and performing sensitivity analysis between variables;
in step 3, the main steps of estimating bayesian network parameters using a expectation maximization algorithm include:
(1) The samples are divided into complete samples and missing samples according to whether the variable states are all known. The bayesian network shown in fig. 6 contains 16 variables in total, i.e. one complete data sample is represented as a 16-dimensional vector d i = (1,1,1,3,3,1,2,1,1,2,? Representing the unknown state of the intermediate node. The meaning of the 16-dimensional state vector is as follows: is positioned in lane 1 (X) 1 Position No. 1 (X) =1) 2 Unit of =1) with diaphragm thickness of 8mm (X 3 =1), length 3000mm (X 4 =3), top plate thickness 16mm (X 5 =3), located in a low temperature region (X 6 =1), sprouting of type i (X 7 =2) and type iv cracks (X 10 ＝2)。
(2) Let o i Representing data sample d i Non-observable part of (h) i Is an observable part, d i ＝o i ∪h i . Let o= { O 1 ,o 2 ,…,o N The data is observable in the training set, and H= { H 1 ,h 2 ,…,h N And the training set d=o u H. Parameter learning with missing data employs a expectation maximization algorithm.
(3) For the complete dataset d= { D 1 ,d 2 ,…d m Obtaining a more accurate estimation result by adopting maximum likelihood estimation, wherein a log likelihood function based on the condition independence and a chain rule is as follows:
in the formula ,
representing variable X j At sample d i State of (a); pa (pa) i (X j ) Representing variable X j Is at sample d i State of (a); n represents the number of cell statesThe method comprises the steps of carrying out a first treatment on the surface of the N represents the number of samples. Maximizing parameters of log likelihood functions as optimal estimates
(4) For a missing sample, the probability of missing data p (d) cannot be directly calculated at a given parameter i θ) and therefore employs a expectation-maximization algorithm, consisting essentially of two iterations:
first, given the current network parameter θ t And observing the data set O, calculating a conditional expectation Q (θ|θ t )＝E[logp(D|θ)|θ t ,O]. Then find the maximization Q (θ|θ) t ) As initial value of the next step, i.e
In step 3, the sensitivity analysis process includes:
the conditional probabilities of the nodes cooperatively change when the conditional probabilities of the nodes are changed
Other conditional probability of the node +.>
And also changes at the same time to meet the requirement that the probability sum is one:
in the formula ,
representing variable X i A j-th state value of (2); pi represents the variable X i A combined state of the parent nodes.
When the probability parameters change cooperatively, the prior probability lambda=p (X 1 =1), η is a state posterior of a network leaf nodeProbability η=p (X 7 =1), η is expressed as a linear function of λ:
η＝αλ+β (4)
wherein α and β are constants related to λ and η, referred to as the sensitivity of η to λ, and further referred to as α η-λ 。
Let lambda be disturbance delta lambda, eta will change correspondingly, and the change is marked as delta eta, the sensitivity calculation formula is
According to the technical scheme, the invention provides a Bayesian network-based evaluation method for the fatigue crack spatial uneven distribution of a large-scale steel bridge deck, which has the following effects:
(1) The defect that the traditional technology only focuses on a single fatigue vulnerable part is overcome, and the spatial distribution of fatigue cracks on an orthotropic steel bridge deck can be obtained;
(2) The built Bayesian network model considers the position variable, the construction variable and the temperature interval variable of the unit, so that the model grabs the main factors influencing the fatigue crack space uneven distribution;
(3) The intermediate nodes are introduced to summarize the Bayesian network part nodes, so that the number of the Bayesian network undetermined parameters is obviously reduced, and the problem of excessive network parameters caused by directly connecting the Bayesian network root nodes with the leaf nodes is solved;
(4) The influence degree of the unit position variable, the construction variable and the temperature variable on the initiation probability of various fatigue cracks can be revealed through Bayesian network sensitivity analysis;
(5) According to the invention, network parameter learning can be performed by utilizing crack detection data of the sections of the orthotropic steel deck slab, and then the number of cracks of the undetected sections of the deck slab is estimated, so that the rapid assessment of the number of fatigue cracks of the whole deck slab is realized, and the time cost and economic cost of large-span bridge crack detection are greatly reduced.
The invention provides a Bayesian network-based evaluation method for the spatial uneven distribution of fatigue cracks of a large-scale steel bridge deck, which is described in detail above, and the invention applies specific examples to illustrate the principles and the implementation modes of the invention, and the description of the examples is only used for helping to understand the method and the core ideas of the invention; meanwhile, as those skilled in the art will have variations in the specific embodiments and application scope in accordance with the ideas of the present invention, the present description should not be construed as limiting the present invention in view of the above.
Claims (6)
1. A Bayesian network-based large-scale steel bridge deck fatigue crack spatial uneven distribution evaluation method is characterized by comprising the following steps:
step 1: dividing the bridge deck into finite cells through regular grid division, wherein each cell is a research unit, and defining a unit state based on structural construction factors, load environment factors and crack states;
step 2: establishing a Bayesian network according to the unit state variables to describe probability dependency relationships among the variables;
step 3: and estimating Bayesian network parameters by using a expectation maximization method, performing sensitivity analysis among variables, and evaluating the spatial uneven distribution of fatigue cracks of the large-scale steel bridge deck.
2. The method for evaluating the spatial non-uniform distribution of fatigue cracks of a large-scale steel bridge deck slab based on a bayesian network according to claim 1, wherein the step 1 is specifically as follows:
defining bridge deck research units, discretizing the bridge deck into limited cells, wherein each cell is used as a research unit, and the more the number of the divided units is, the more samples are, so that the parameter learning of a network model is facilitated;
defining unit state variables, defining the unit states according to the crack states on the research units, the unit load environments and the space positions due to the fact that the crack initiation probability of different units is different, so that the fatigue crack distribution problem is converted into the crack initiation probability problem on the units,
defining cell state variables includes the following four variable types:
(1) Taking into account spatial position variables
The units are different in space positions on the bridge deck, so that the units are influenced by vehicle loads and are divided into forward-direction positions and transverse-direction positions, and as vehicles running at high speed usually do not change lanes frequently, the forward-direction positions of the research units are identical and the load effects of the research units with the same structure are basically consistent, and therefore the influence of the forward-direction positions of the research units is not considered;
the load of the vehicles is different for the units with different transverse bridge directions, the fatigue crack of the bridge deck is mainly concentrated in the heavy road and the middle road, the transverse position of the unit i is described by two variables, X 1,i Indicating the lane in which the unit is located, X 2,i Representing a specific location of the unit within the lane;
(2) Taking into account unit construction variables
Comprising diaphragm thickness variation X 3,i ＝T dia Variable of unit length X 4,i ＝L e Variable X of top plate thickness 5,i ＝T dec Five values are available for the diaphragm thickness variable, three values are available for the unit length variable, and three values are available for the top plate thickness variable;
(3) Taking into account temperature interval variables
The temperature interval in which the unit is located is described by a variable: x is X 6,i ＝Tem dec ，Tem dec Coding the temperature zone of the unit, tem dec =1, 2 indicates that the cell is in the low temperature region and the high temperature region, respectively;
(4) Taking into account crack state variables
The cell crack states are described using four variables for four classes of typical cracks: x is X 7,i ＝I 1 ，X 8,i ＝I 2 ，X 9,i ＝I 3 ，X 10,i ＝I 4, in the formula I1 Indicating type I crack status coding on a cell, I 1 =1, 2, respectively, indicates unexplosive and type i cracks on the cell; i 2 Representing type II cracks on a cellState encoding, I 2 =1, 2, respectively, indicates unexplosive and type ii cracks on the cell; i 3 Representing type III crack status encodings on a cell, I 3 =1, 2, respectively, indicates unexplosive and type iii cracks on the cell; i 4 Representing type IV crack status codes on a cell, I 4 =1, 2, respectively indicating that no crack of type iv was initiated and that a crack of type iv was initiated on the cell;
(5) Complete state X of a cell i Represented as a vector of the ten state variables above: x is X i ＝{X j,i ；j＝1,2,…,10}。
3. The bayesian network-based assessment method for the spatial non-uniform distribution of fatigue cracks of a large-scale steel bridge deck according to claim 2, wherein the division of units follows the following two principles:
1. the residual state variables of the units are kept consistent except for the difference in the construction to be researched and the load environment;
2. each unit contains the various types of fatigue vulnerable sites studied, and the numbers remain consistent.
4. The method for evaluating the spatial non-uniform distribution of fatigue cracks of the large-scale steel bridge deck slab based on the Bayesian network according to claim 3, wherein the step 2 is specifically:
step 2.1: performing causal relation analysis between unit state variables, and designing causal relation judgment criteria between the unit state variables: when the state of the variable X is changed to influence the reliability of the variable Y by researchers, and the state of the variable Y is changed but cannot influence the reliability of the variable X by the researchers, the variable X is called as the reason of the variable Y;
in unit construction variable X 5 And crack state variable X 9 When the thickness of the top plate is increased, the fatigue resistance of the welding seam between the top plate and the U rib is increased, so that the occurrence probability of III type cracks is reduced; when a III type crack is manually added, the judgment of the thickness of the top plate is not influenced, and the structural variable X 5 Is a crack state variable X 9 Reasons for (2);
based on causal Markov assumption, the unit space position variable, the component variable and the temperature interval variable are all reasons of the unit crack state variable;
step 2.2: the hidden node is introduced to simplify network parameters, a father node separation method is adopted, partial father nodes are combined in the middle node introduction process, the network independent parameters are reduced after the middle node introduction process is realized, and the I-type crack is provided with a Bayesian network model of the middle node;
step 2.3: establishing a unit Bayesian network, introducing a unit Bayesian network of intermediate nodes, wherein node X i I=11, 12, …,16 is an intermediate node having two discrete states, the values of the position variable, the construction variable and the temperature interval variable being known for a certain determination unit, i.e. node X i I=1, 2, …,6 is state observable, and the conditional probability p (X j |X 1 ,X 2 ,X 3 ,X 4 ,X 5 ,X 6 )j＝7,8,9,10；
The bayesian network is denoted bn= (G, θ), where G is directed acyclic; θ is a network parameter, and for discrete state variables, a conditional probability representation is used; each node in the Bayesian network corresponds to a random variable in the data set, the directed edges between the nodes represent probability dependency relationships among the variables, the directed edges in the Bayesian network established based on the causal relationships represent causal relationships among the variables, and the initial node of the directed edges represents a 'factor' and is a father node; the pointing node represents a fruit and is a child node; based on d-separation criteria and chain law of bayesian network, a set of random variables x= { X 1 ,X 2 ,…,X n The joint probability distribution of } is decomposed into Markov conditional equations:
wherein pa (X i ) Is node X i Is a parent node set of (a); p (X) i |pa(X i ) A) component node X i Is provided with a conditional probability table of (a),i.e. bayesian network parameters.
5. The method for evaluating the spatial non-uniform distribution of fatigue cracks of a large-scale steel bridge deck based on a bayesian network according to claim 4, wherein the step 3 is specifically as follows:
step 3.1: the Bayesian network for dividing the sample into complete and missing samples comprises 16 variables based on whether the state of the variables is known in its entirety, i.e. one complete data sample is represented as a 16-dimensional vector d i = (1,1,1,3,3,1,2,1,1,2,? Representing the unknown state of the intermediate node, the meaning of the 16-dimensional state vector is: is positioned in lane 1 (X) 1 Position No. 1 (X) =1) 2 Unit of =1) with diaphragm thickness of 8mm (X 3 =1), length 3000mm (X 4 =3), top plate thickness 16mm (X 5 =3), located in a low temperature region (X 6 =1), sprouting of type i (X 7 =2) and type iv cracks (X 10 ＝2)；
Step 3.2: let o i Representing data sample d i Non-observable part of (h) i Is an observable part, d i ＝o i ∪h i The method comprises the steps of carrying out a first treatment on the surface of the Let o= { O 1 ,o 2 ,…,o N The data is observable in the training set, and H= { H 1 ,h 2 ,…,h N The training set d=o u H; the parameter learning containing the deficiency value data adopts an expected maximization algorithm;
step 3.3: for the complete dataset d= { D 1 ,d 2 ,…d m Obtaining an accurate estimation result by adopting maximum likelihood estimation, wherein a log likelihood function based on conditional independence and a chain rule is represented by the following formula:
wherein ,
representing variable X j At sample d i State of (a); pa (pa) i (X j ) Representing variable X j Is at sample d i State of (a); n represents the number of cell states; n represents the number of samples, and the parameter maximizing the log likelihood function is used as the optimal estimation
Step 3.4: for a missing sample, the probability of missing data p (d) cannot be directly calculated at a given parameter i θ), a desired maximization algorithm is adopted, and the method mainly comprises the following two steps of iteration: given the current network parameter θ t And observing the data set O, calculating a conditional expectation Q (θ|θ t )＝E[logp(D|θ)|θ t ,O]Find the maximization Q (θ|θ) t ) As initial value of the next step, i.e
6. The method for evaluating the spatial non-uniform distribution of fatigue cracks of a large-scale steel bridge deck based on a Bayesian network according to claim 5,
the sensitivity analysis process includes:
the conditional probabilities of the nodes cooperatively change when the conditional probabilities of the nodes are changed
Other conditional probability of the node +.>
And also changes at the same time to meet the requirement that the probability sum is one:
wherein ,
representing variable X i A j-th state value of (2); pi represents the variable X i A combined state of the parent nodes;
when the probability parameters change cooperatively, the prior probability lambda=p (X 1 =1), η is a certain state posterior probability η=p (X 7 =1), η is expressed as a linear function of λ:
η＝αλ+β (4)
wherein α and β are constants related to λ and η, referred to as the sensitivity of α to λ of η, and further referred to as α η-λ ；
Given a disturbance Δλ to λ, η will also change accordingly, and the change is denoted Δη, and the sensitivity is expressed by the following equation:
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