CN112528517B - Steel box girder fatigue reliability analysis method based on two-stage convergence criterion - Google Patents

Steel box girder fatigue reliability analysis method based on two-stage convergence criterion Download PDF

Info

Publication number
CN112528517B
CN112528517B CN202011548489.8A CN202011548489A CN112528517B CN 112528517 B CN112528517 B CN 112528517B CN 202011548489 A CN202011548489 A CN 202011548489A CN 112528517 B CN112528517 B CN 112528517B
Authority
CN
China
Prior art keywords
failure
function
probability
failure probability
estimation
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN202011548489.8A
Other languages
Chinese (zh)
Other versions
CN112528517A (en
Inventor
李惠
徐阳
陈家辉
陈智成
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Harbin Institute of Technology
Original Assignee
Harbin Institute of Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Harbin Institute of Technology filed Critical Harbin Institute of Technology
Priority to CN202011548489.8A priority Critical patent/CN112528517B/en
Publication of CN112528517A publication Critical patent/CN112528517A/en
Application granted granted Critical
Publication of CN112528517B publication Critical patent/CN112528517B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/15—Correlation function computation including computation of convolution operations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/04—Ageing analysis or optimisation against ageing

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Mathematical Physics (AREA)
  • General Engineering & Computer Science (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Data Mining & Analysis (AREA)
  • Computational Mathematics (AREA)
  • Algebra (AREA)
  • Software Systems (AREA)
  • Databases & Information Systems (AREA)
  • Computing Systems (AREA)
  • Computer Hardware Design (AREA)
  • Evolutionary Computation (AREA)
  • Geometry (AREA)
  • Testing Of Devices, Machine Parts, Or Other Structures Thereof (AREA)
  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)

Abstract

The invention discloses a steel box girder fatigue reliability analysis method based on a two-stage convergence criterion, and belongs to the technical field of bridge fatigue reliability analysis. Initializing an internal Kriging agent model, continuously increasing sample points near a failure surface by adopting a U function in an active learning mode for training, and obtaining an estimation error of failure probability; step two, constructing a subset simulation external framework by defining a middle failure event, converting a minimum failure probability into a series of larger conditional failure probabilities, and training a kriging agent model in the subsets of different levels until convergence; and step three, repeating the steps, continuously defining a new subset, and training the kriging agent model until a two-stage convergence criterion is met. The method can ensure the estimation precision of the failure probability, improve the efficiency and is insensitive to the numerical value of the failure probability, thereby verifying the accuracy and the efficiency of the method.

Description

Steel box girder fatigue reliability analysis method based on two-stage convergence criterion
Technical Field
The invention relates to a steel box girder fatigue reliability analysis method based on a two-stage convergence criterion, and belongs to the technical field of bridge fatigue reliability analysis.
Background
The main girder structure of the long-span bridge generally adopts a flat steel box girder form, is composed of main components such as orthotropic steel bridge deck boards, transverse partition boards, longitudinal partition boards and the like, and has the advantages of light weight, high strength, convenience in installation and construction and the like. Along with the continuous increase of the traffic flow and the single axle weight, the fatigue cracking phenomenon of the steel box girder is increasingly serious due to the coupling effect of complex factors such as welding residual stress, material internal defects, reciprocating vehicle load and the like, the fatigue crack is continuously expanded to cause local failure or fracture, the safety and the durability of the long-span bridge are seriously influenced, and even extreme catastrophic accidents can be caused.
Because the fatigue cracking of the steel box girder can obviously cause the reduction of the structural reliability, the research on the high-efficiency analysis method of the fatigue reliability of the steel box girder is an important content of the safety evaluation of the steel box girder. At present, a sub-model method is mainly adopted to determine the boundary condition of a local model by analyzing an overall model, or a substructure method is adopted to condense the local model into a super-unit to construct the overall model, and finally multi-scale finite element modeling is realized, wherein the multi-scale finite element modeling comprises the steps of carrying out fatigue failure modeling and experimental research on a steel bridge model, evaluating the fatigue performance of the steel bridge based on monitoring data such as stress, vehicle load and the like. The method for evaluating the fatigue life of the steel bridge based on finite element and fracture mechanics highly depends on the accurate value of model parameters, and the related parameters are influenced by load, environment and material properties and often have larger variability. In consideration of uncertainty of the fatigue degradation process of the steel box girder, domestic and foreign scholars develop a series of methods for evaluating the fatigue reliability of the steel bridge deck under the random vehicle load effect and research the influence rule of factors such as operation load and construction detail design on the fatigue reliability of the steel bridge deck.
At present, a first moment and a second moment based method is generally adopted for analyzing the fatigue reliability of the steel bridge deck, Taylor series expansion is carried out on an explicit nonlinear limit state function near a design point, and high-order terms of the explicit nonlinear limit state function are ignored, however, the estimation accuracy of the method under the high-dimensional nonlinear condition is poor. A monte carlo simulation method was subsequently developed to estimate failure probability by generating random samples, providing a reliability analysis method that does not require explicit extreme state functions, but reliability assessment for small probability fatigue failure events of actual steel box beam structures is inefficient because monte carlo simulation requires a large number of samples to guarantee accuracy. For the defects of the Monte Carlo simulation method, researchers later propose a reliability analysis method based on variance reduction, which comprises importance sampling, subset simulation and the like, and random samples are generated according to an auxiliary probability density function, so that a sampling center is moved to a design point, the sampling efficiency is improved, and fewer samples are needed under the condition of ensuring the same variation coefficient. Such algorithms are very efficient for estimation of small probability events, are robust to the dimensionality of the input variables, but require a great deal of effort in determining failure domains and selecting appropriate secondary probability density functions. The subset simulation method converts the small failure probability into a product of a series of larger conditional failure probabilities by defining an intermediate conditional failure event, and can estimate the larger conditional failure probability by using fewer samples, so that the method has obvious efficiency under the condition of ensuring the same precision. However, for large engineering and infrastructure structures, the reliability analysis method based on variance reduction requires huge consumption of computational resources, which is unacceptable in practical engineering, since its extreme state function is usually highly non-linear and implicit, has no direct explicit expression, and requires finite element analysis.
To solve this problem, researchers have subsequently proposed a series of reliability analysis methods based on proxy models, which replace the process of finite element analysis by establishing implicit input-output relationships, thereby obtaining sample points at a lower time cost. The currently widely adopted proxy models include a response surface model, polynomial chaotic expansion, a support vector machine, a neural network, a kriging model and the like. The kriging model can simultaneously obtain an unbiased estimation of an untested sample and an estimated variance based on a gaussian process, and the precision and efficiency of the model depend on a Design of experiment (DoE). While conventional random sampling or latin hypercube sampling are quite inefficient for constructing a DoE because they require the entire state space to be completely filled with random variables. In fact, the actual sampling process should be focused on the region near the extreme state surface, i.e. the boundary between the security domain and the failure domain is concerned. In recent years, researchers have developed a sampling method based on an active learning strategy, and new samples around the extreme state plane are added to the DoE in an iterative sampling manner, so that a good balance between accuracy and efficiency is achieved.
Reliability analysis methods based on kriging models can generally be divided into three main categories: (1) researching a new learning function, and continuously selecting new samples to enrich the DoE, wherein a U function is the most widely adopted form; (2) optimizing the construction strategy of a sample base and reducing the sample capacity, wherein the variance reduction method belongs to the large category; (3) new convergence criteria for the active learning process are explored. Currently, a suitable threshold is generally selected to determine whether the iteration is terminated, for example, by determining whether the value of the U function exceeds 2. In this case, the probability of obtaining a correct estimate per sample needs to exceed 0.977 (following a standard normal distribution calculation according to the U function), and this condition is usually too strict for each local sample, because it is true that the accuracy of the corresponding failure probability estimate can be guaranteed as long as the global error of the symbols of the extreme state function can be controlled.
Disclosure of Invention
The invention aims to provide a method for analyzing the fatigue reliability of a steel box girder based on a two-stage convergence criterion so as to solve the problems in the prior art.
The steel box girder fatigue reliability analysis method based on the two-stage convergence criterion comprises the following steps:
initializing an internal Kriging agent model, continuously increasing sample points near a failure surface by adopting a U function in an active learning mode for training, and obtaining an estimation error of failure probability;
step two, constructing a subset simulation external framework by defining a middle failure event, converting a minimum failure probability into a series of larger conditional failure probabilities, and training a kriging agent model in the subsets of different levels until convergence;
and step three, repeating the steps, continuously defining a new subset, and training the kriging agent model until a two-stage convergence criterion is met.
Further, in step one, specifically, in kriging theory, the exact extreme state function is modeled as a random process.
Linear regression of global mean and random modeling of local bias, expressed as:
G(x)＝f(x)Tβ+Z(x) (1)
in the formula, f (x)TIs a set of known basis function vectors, β is a regression parameter vector, and z (x) is a stationary gaussian random process whose covariance function is established based on a gaussian correlation function and can be calculated according to the following formula:
cov(Z(xi)，Z(xj))＝σ2R(xi，xj) (2)
in the formula, σ2For random process variance, R (x)i，xj) Is a Gaussian correlation function based on least squares regression algorithm, and has regression parameter beta in formula (1) and process variance sigma in formula (2)2Estimating from the sample, and obtaining the optimal linear unbiased estimation
Figure BDA0002856402000000031
And its variance
Figure BDA0002856402000000032
According to kriging theory, the estimation of any point in the random variable state space follows a normal distribution:
Figure BDA0002856402000000033
the sample-based failure probability estimate may be calculated by:
Figure BDA0002856402000000034
wherein q isX(x) A joint probability density function for a random variable X; x is a vector consisting of N-dimensional random variables, and is converted into mutually independent normal distribution random variables through Naphv transformation in advance in the structural reliability analysis; i isF(x) To binarize the indicative function, defined by:
Figure BDA0002856402000000041
wherein, b is a failure threshold value,
Figure BDA0002856402000000042
is a surface in a limit state,
considering the variance of the kriging model, the estimation value of the indicative function in the above formula also has randomness, and therefore, the estimation value of the failure probability should be adjusted to be:
Figure BDA0002856402000000043
in the formula (I), the compound is shown in the specification,
Figure BDA0002856402000000044
to account for the failure probability estimation of the kriging model error,
Figure BDA0002856402000000045
is a sample point xiProbability of failure, Padj(xi) In order to be an adjustment function of the indicative function,
based on the kriging mean and the estimated variance,
Figure BDA0002856402000000046
calculated from the following formula:
Figure BDA0002856402000000047
where Φ is the cumulative distribution function of the standard normal distribution, U (x)i) Is defined as:
Figure BDA0002856402000000048
thus, a smaller value of the U function indicates that the sample point is closer to the extreme state surface or its estimated variance
Figure BDA0002856402000000049
And the estimation error of the sample indicative function is increased by the two methods, the U function is used as a learning function in the active learning process, and the selection of the new training sample is performed according to the following formula:
Figure BDA00028564020000000410
wherein, S represents a candidate sample pool,
to this end, the adjustment function is calculated as:
Figure BDA00028564020000000411
further, in step two, specifically, the samples of the subset simulation first layer are directly generated by the monte carlo simulation, so the samples are independent from each other, whereas the samples of the other layers are generated by the markov chain monte carlo simulation, and there is a correlation between the first samples of different markov chains, so the samples of the other layers are correlated, and as a simplification condition, it is assumed that the sample-based adjustment functions are independent, that is:
cov(Padj(xi)，Padj(xj))＝0，i≠j (11)
to study the impact of kriging model randomness on failure probability estimation, the expectation of defining the tuning function is:
Eadj＝E[Padj(x)]＝∫Padj(x)fX(x)dx (12)
wherein f isX(x) As a function of the joint probability density of the random variable X, i.e. f for the first layer of the subset simulationX(x)＝qX(x) For the ith layer f of the subset simulationX(x)＝qX(x|Fl-1) According to the central limit theorem, assume EadjThe approximation follows a normal distribution, which is based on a sample estimate of:
Figure BDA0002856402000000051
the corresponding variance estimate is:
Figure BDA0002856402000000052
defining the model error rate as:
Figure BDA0002856402000000053
wherein the content of the first and second substances,
Figure BDA0002856402000000054
is a scalar quantity, because EadjThe approximation follows a normal distribution, so η also follows a normal distribution, i.e.
Figure BDA0002856402000000055
Converting a smaller failure probability into a product of a series of larger conditional failure probabilities based on subset simulations, given failure event F, defining intermediate failure events { F }1，F2，…，FLIs Fl＝{G(x)≤bl1, 2, …, L, wherein b1＞…＞bl＞…＞bLWith 0 being the failure threshold value,
the final failure probability is calculated by:
Figure BDA0002856402000000056
wherein, P1＝P{F1Is the failure probability of the first failure event, Pl＝P{Fl|Fl-1(L2, 3, …, L) is the conditional failure probability, a simulated estimate is made based on markov chain monte carlo,
estimating failure probability
Figure BDA0002856402000000061
The coefficient of variation calculation formula is:
Figure BDA0002856402000000062
wherein n islNumber of samples for direct Monte Carlo simulation (L ═ 1) or Markov chain Monte Carlo simulation (L ═ 2 … L)，γlTo account for non-negative scalars of sample correlation, γlSmaller indicates weaker correlation between samples, and thus gamma1＝0，
Final failure probability estimation
Figure BDA0002856402000000063
The coefficient of variation of (A) is:
Figure BDA0002856402000000064
further, in step three, specifically, on the one hand, the conditional probability PlThe estimation error of (L ═ 1, 2, …, L) should be controlled, defining the estimated error rate of the L-th layer failure probability as ηlObeying a normal distribution according to equation (16), the first stage constraint being |. etal|＜α1The confidence of (2) is not lower than 99.7%:
Figure BDA0002856402000000065
Figure BDA0002856402000000066
wherein the content of the first and second substances,
Figure BDA0002856402000000067
and
Figure BDA0002856402000000068
in order to control the boundaries of the process,
Figure BDA0002856402000000069
as an estimate of the failure probability of the l-th failure event, a1A preset threshold value greater than zero,
Figure BDA00028564020000000610
for the estimate of the adjustment function expectation on the l-th subset, var represents the variance computation functionEquation (20) is the first stage convergence criterion,
on the other hand, since the final failure probability is the product of a series of conditional failure probabilities, the calculation formula for obtaining the final failure probability is as follows:
Figure BDA00028564020000000611
wherein the content of the first and second substances,
Figure BDA00028564020000000612
to account for the estimation of the ultimate failure probability of the model error,
Figure BDA00028564020000000613
for the estimate of the adjustment function expectation on the l-th subset, due to alpha1Is set to a very small value, here 0.03, so that the higher order terms in equation (21) are ignored,
the model cumulative error is defined as:
Figure BDA0002856402000000071
and ηlUniformity xilObey a normal distribution, i.e.:
Figure BDA0002856402000000072
to control model accumulation error, guarantee xil∈[-α2，α2]The confidence of the second stage convergence criterion is not lower than 99.7 percent, namely the second stage convergence criterion provided by the invention is as follows:
Figure BDA0002856402000000073
Figure BDA0002856402000000074
wherein alpha is2For a preset threshold value greater than zero, 0.05 is assumed here.
The main advantages of the invention are as follows:
(1) the method is based on the internal Kriging model, and can be used for carrying out error analysis on the estimated value of the failure probability.
(2) The method is based on the subset simulation external framework, and can realize the reliability evaluation of the small failure probability event under the condition of less calculation amount.
(3) The method is based on the two-stage convergence criterion, can simultaneously realize the hierarchical control and the global control of the failure probability estimation errors of the subsets of different levels, and can improve the reliability analysis efficiency on the basis of keeping the failure probability estimation precision.
(4) The whole modeling process is data-driven, sample points close to failure surfaces of different levels are continuously selected according to an active learning strategy, and sampling efficiency is remarkably improved.
(5) The method improves the automation and intelligence degree and analysis efficiency of the evaluation of the fatigue cracking reliability of the steel box girder orthotropic steel bridge panel of the large-span bridge, and provides technical support for the autonomous intelligent evaluation of the service state of the large-span cable-stayed bridge.
Drawings
FIG. 1 is a flow chart of one embodiment of a steel box girder fatigue reliability analysis method based on a two-stage convergence criterion of the present invention;
FIG. 2 is a schematic structural view of an orthotropic steel deck plate comprising 15 fatigue cracks;
FIG. 3 is a three-dimensional finite element model of a fatigue crack orthotropic steel deck slab structure;
FIG. 4 is a graph of the true response of the new sample points in different subsets versus the middle extreme state plane.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The first embodiment of the invention: the embodiment is a steel box girder fatigue reliability analysis method based on a two-stage convergence criterion, as shown in fig. 1, a subset simulation outer frame is adopted to solve the reliability evaluation problem of an extremely-small failure probability event, an active learning method is adopted to select sample points in each subset in an active learning mode to train an internal Kriging model, and the two-stage convergence criterion is designed to control the layering and global estimation errors in the subset simulation process. The method specifically comprises the following steps:
the method comprises the following steps: initializing an internal Kriging agent model, continuously increasing sample points near a failure surface by adopting a U function in an active learning mode for training, and obtaining an estimation error of failure probability;
step two: constructing a subset simulation external framework by defining a middle failure event, converting a minimum failure probability into a series of larger conditional failure probabilities, and training a kriging agent model in the subsets of different levels until convergence;
step three: and repeating the steps, continuously defining new subsets, and training the kriging agent model until a two-stage convergence criterion is met.
Embodiment two of the present invention:
the first step specifically comprises the following steps:
the kriging model as a common proxy model can approximate the complex input and output function relationship of the proxy so as to replace the extreme state function. In kriging theory, the exact extreme state function is usually modeled as a random process, consisting of two parts: the linear regression of the global mean and the stochastic modeling of the local bias can be expressed as:
G(x)＝f(x)Tβ+Z(x) (1)
in the formula, f (x)TIs a set of known basis function vectors, beta is a regression parameter vector, Z (x) is a stationary Gaussian random process whose covariance function is based onThe gaussian correlation function is established and can be calculated according to the following formula:
cov(Z(xi)，Z(xj))＝σ2R(xi，xj) (2)
in the formula, σ2For random process variance, R (x)i，xj) Is a gaussian correlation function. Based on least squares regression algorithm, regression parameter beta in formula (1) and process variance sigma in formula (2)2Can be estimated from the samples and then an optimal linear unbiased estimate can be obtained
Figure BDA0002856402000000091
And its variance
Figure BDA0002856402000000092
According to kriging theory, the estimation of any point in the random variable state space follows a normal distribution:
Figure BDA0002856402000000093
the sample-based failure probability estimate may be calculated by:
Figure BDA0002856402000000094
wherein q isX(x) A joint probability density function for a random variable X; x is a vector consisting of N-dimensional random variables, and is usually converted into mutually independent normal distribution random variables in advance through the Naphv transformation in the structural reliability analysis; i isF(x) To binarize the indicative function, defined by:
Figure BDA0002856402000000095
wherein, b is a failure threshold value,
Figure BDA0002856402000000096
is a limit state surface.
The indicative function estimates in the above equation will also be random, taking into account the kriging model variance. Therefore, the estimated value of the failure probability should be adjusted to:
Figure BDA0002856402000000097
in the formula (I), the compound is shown in the specification,
Figure BDA0002856402000000098
to account for the failure probability estimation of the kriging model error,
Figure BDA0002856402000000099
is a sample point xiProbability of failure, Padj(xi) An adjustment function that is an indicative function.
Based on the kriging mean and the estimated variance,
Figure BDA00028564020000000910
can be calculated from
Figure BDA0002856402000000101
Where Φ is the cumulative distribution function of the standard normal distribution, U (x)i) Is defined as
Figure BDA0002856402000000102
Thus, a smaller value of the U-function may indicate that the sample point is closer to the extreme state plane or its estimated variance
Figure BDA0002856402000000103
Larger, both increase the estimation error of the sample indicative function. The U-function is used as a learning function in the active learning process, and the selection of the new training sample is performed according to the following formula:
Figure BDA0002856402000000104
where S represents a candidate pool.
To this end, an adjustment function of
Figure BDA0002856402000000105
The third embodiment of the invention:
the second step specifically comprises:
the samples of the subset simulation first layer are generated directly from the monte carlo simulation, and thus the samples are independent of each other. However, the samples of other layers are generated by markov chain monte carlo simulation, and the first sample of different markov chains has correlation, so the samples of other layers are correlated. As a simplification condition, the invention assumes that the sample-based adjustment functions are independent, i.e.
cov(Padj(xi)，Padj(xj))＝0，i≠j (11)
To study the effect of the randomness of the kriging model on the failure probability estimation, the expectation of the tuning function is defined as
Eadj＝E[Padj(x)]＝∫Padj(x)fX(x)dx (12)
Wherein f isX(x) As a function of the joint probability density of the random variable X, i.e. f for the first layer of the subset simulationX(x)＝qX(x) For the ith layer f of the subset simulationX(x)＝qX(x|Fl-1). According to the central limit theorem, assume EadjApproximately obey a normal distribution based on an estimate of the sample as
Figure BDA0002856402000000111
The corresponding variance estimate is:
Figure BDA0002856402000000112
defining the model error rate as:
Figure BDA0002856402000000113
wherein the content of the first and second substances,
Figure BDA0002856402000000114
is a scalar quantity. Due to EadjThe approximation follows a normal distribution, so η also follows a normal distribution, i.e.
Figure BDA0002856402000000115
A smaller probability of failure may be converted to a product of a series of larger conditional probabilities of failure based on subset simulations. Given failure event F, define intermediate failure events { F1，F2，…，FLIs Fl＝{G(x)≤b l1, 2, …, L, wherein b1＞…＞bl＞…＞b L0 is the failure threshold.
The final failure probability can be calculated by:
Figure BDA0002856402000000116
wherein, P1＝P{F1Is the failure probability of the first failure event, Pl＝P{Fl|Fl-1The conditional failure probability (L ═ 2, 3, …, L) can be estimated based on markov chain monte carlo simulations.
Estimating failure probability
Figure BDA0002856402000000117
The coefficient of variation calculation formula is:
Figure BDA0002856402000000118
wherein n islNumber of samples, γ, for direct monte carlo simulation (L ═ 1) or markov chain monte carlo simulation (L ═ 2 … L)lTo account for non-negative scalars of sample correlation, γlSmaller indicates weaker correlation between samples, and thus gamma1＝0。
Final failure probability estimation
Figure BDA0002856402000000119
Has a coefficient of variation of
Figure BDA00028564020000001110
The fourth embodiment of the invention:
the third step specifically comprises:
on the one hand, conditional probability PlThe estimation error of (L ═ 1, 2, …, L) should be controlled. Defining an estimated error rate of the l-th layer failure probability as etalA normal distribution is followed according to equation (16). The constraint condition of the first stage is |. etal|＜α1The confidence of (2) is not lower than 99.7%:
Figure BDA0002856402000000121
Figure BDA0002856402000000122
wherein the content of the first and second substances,
Figure BDA0002856402000000123
and
Figure BDA0002856402000000124
in order to control the boundaries of the process,
Figure BDA0002856402000000125
as an estimate of the failure probability of the l-th failure event, a1A preset threshold value greater than zero,
Figure BDA0002856402000000126
for the desired estimate of the adjustment function on the/th subset, var represents the variance calculation function. Equation (20) is the first stage convergence criterion proposed by the present invention.
On the other hand, since the final failure probability is the product of a series of conditional failure probabilities, the calculation formula for obtaining the final failure probability is
Figure BDA0002856402000000127
Wherein the content of the first and second substances,
Figure BDA0002856402000000128
to account for the estimation of the ultimate failure probability of the model error,
Figure BDA0002856402000000129
the desired estimate of the function is adjusted for the ith subset. Due to alpha1Can be set to a very small value (here, 0.03) and thus the higher order terms in equation (21) can be ignored.
The model cumulative error is defined as:
Figure BDA00028564020000001210
and ηlUniformity xilSubject to a normal distribution, i.e.
Figure BDA00028564020000001211
To control model accumulation error, guarantee xil∈[-α2，α2]Degree of confidence ofNot less than 99.7%, which is the second stage convergence criterion proposed by the present invention:
Figure BDA0002856402000000131
Figure BDA0002856402000000132
wherein alpha is2For a preset threshold value greater than zero, 0.05 is assumed here.
Example five of the present invention:
the difference between this embodiment and one of the first to fourth embodiments is:
the embodiment is a concrete implementation of the steel box girder fatigue reliability analysis method based on the two-stage convergence criterion on an orthotropic steel bridge deck with 15 fatigue cracks.
FIG. 2 is a schematic structural view of an orthotropic steel deck plate containing fatigue cracks, wherein C represents a crack, and the numbers 1-15 correspond to 15 different cracks. The structure consists of 3U ribs and 2 diaphragms.
Based on ANSYS finite element modeling software, a three-dimensional finite element model of the fatigue cracking orthotropic steel bridge deck plate structure is established, and is used for calculating the response of the structure under the action of wheel load as shown in figure 3. The finite element model is mainly built on the basis of shell units, fatigue cracks are simulated on the basis of replication nodes to realize crack surface separation, fixed constraint is applied to the bottom of the model, and other boundaries are symmetric constraint. The wheel load acts on the center of the bridge deck, and the acting area is 200mm multiplied by 600 mm. The thickness of the steel plate, the elastic modulus of steel, the load amplitude of the wheel and the length of fifteen fatigue cracks are taken as random variables, and the total is twenty dimensions. The probability distribution and parameters of random variables of the crack structure are shown in table 1.
Figure BDA0002856402000000133
Figure BDA0002856402000000141
TABLE 1 probability distribution and parameters of random variables for crack structures
Establishing extreme state functions based on mid-span vertical displacement, i.e.
Figure BDA0002856402000000142
Wherein X is a random variable vector and comprises 5 structural parameter random variables and 15 crack length random variables, and delta (X) is midspan vertical displacement under the action of load and delta0And gamma is a threshold value for the mid-span vertical displacement when the fatigue cracks are not considered and the mean value of the structural parameter random variable is measured.
In order to verify the robustness of the method of the present invention to the parameters, four cases are considered, i.e. γ ═ 1.5, 1.6, 1.7, 1.8, and the calculation results are shown in table 2, wherein,
Figure BDA0002856402000000143
for the estimated probability value of failure,
Figure BDA0002856402000000144
to estimate the coefficient of variation of the failure probability, beta is the estimated reliability index, NcallThe number of calls to the finite element model. The direct monte carlo simulation method is not applicable in this case due to the high cost of finite element calculations.
In addition, the method provided by the invention is compared with the traditional Active Learning Kriging-Monte Carlo method (AK-MCS), and the result shows that the calculation result of the failure probability is close to that of the traditional Active Learning Kriging-Monte Carlo Simulation method, but the method provided by the invention needs obviously less NcallTherefore, the method can not only ensure the precision, but also improve the calculation efficiency. In addition, when the failure probability is from 1.312 × 10-2Down to 3.919X 10-5When N is presentcallThe increase from 33.2 to 40.5 is only to show the inventionThe efficiency of the method provided by the invention has the characteristic of insensitivity to failure probability.
Figure BDA0002856402000000145
Figure BDA0002856402000000151
TABLE 2 comparison of reliability calculations
FIG. 4 is a graph of the true response of the new sample points in different subsets versus the middle extreme state plane. The results show that the final failure surface is achieved through the subset simulation of five levels in the embodiment, and the selected samples in the subsets of the levels are close to the middle limit state surface, thereby further proving the effectiveness of the method in selecting the sample points.

Claims (3)

1. The steel box girder fatigue reliability analysis method based on the two-stage convergence criterion is characterized by comprising the following steps of:
initializing an internal Kriging agent model, continuously increasing sample points near a failure surface by adopting a U function in an active learning mode for training, and obtaining an estimation error of failure probability;
step two, constructing a subset simulation external framework by defining a middle failure event, converting a minimum failure probability into a series of larger conditional failure probabilities, and training a kriging agent model in the subsets of different levels until convergence;
step three, repeating the steps, continuously defining new subsets, training the kriging agent model until meeting the two-stage convergence criterion,
in step one, specifically, in kriging theory, the exact extreme state function is modeled as a random process,
linear regression of global mean and random modeling of local bias, expressed as:
G(x)＝f(x)Tβ+Z(x) (1)
in the formula, f (x)TIs a set of known basis function vectors, β is a regression parameter vector, and z (x) is a stationary gaussian random process whose covariance function is established based on a gaussian correlation function and can be calculated according to the following formula:
cov(Z(xi)，Z(xj))＝σ2R(xi，xj) (2)
in the formula, σ2For random process variance, R (x)i，xj) Is a Gaussian correlation function based on least squares regression algorithm, and has regression parameter beta in formula (1) and process variance sigma in formula (2)2Estimating from the sample, and obtaining the optimal linear unbiased estimation
Figure FDA0003256266190000011
And its variance
Figure FDA0003256266190000012
According to kriging theory, the estimation of any point in the random variable state space follows a normal distribution:
Figure FDA0003256266190000013
the sample-based failure probability estimate may be calculated by:
Figure FDA0003256266190000014
wherein q isX(x) A joint probability density function for a random variable X; x is a vector consisting of N-dimensional random variables, and is converted into mutually independent normal distribution random variables through Naphv transformation in advance in the structural reliability analysis; i isF(x) To binarize the indicative function, defined by:
Figure FDA0003256266190000021
wherein, b is a failure threshold value,
Figure FDA0003256266190000022
is a surface in a limit state,
considering the variance of the kriging model, the estimation value of the indicative function in equation (5) will also have randomness, and therefore, the estimation value of the failure probability should be adjusted to:
Figure FDA0003256266190000023
in the formula (I), the compound is shown in the specification,
Figure FDA0003256266190000024
to account for the failure probability estimation of the kriging model error,
Figure FDA0003256266190000025
is a sample point xiProbability of failure, Padj(xi) In order to be an adjustment function of the indicative function,
based on the kriging mean and the estimated variance,
Figure FDA0003256266190000026
calculated from the following formula:
Figure FDA0003256266190000027
where Φ is the cumulative distribution function of the standard normal distribution, U (x)i) Is defined as:
Figure FDA0003256266190000028
thus, a smaller value of the U function indicates that the sample point is closer to the extreme state surface or its estimatorDifference (D)
Figure FDA0003256266190000029
And the estimation error of the sample indicative function is increased by the two methods, the U function is used as a learning function in the active learning process, and the selection of the new training sample is performed according to the following formula:
Figure FDA00032562661900000210
wherein, S represents a candidate sample pool,
to this end, the adjustment function is calculated as:
Figure FDA00032562661900000211
2. the method for analyzing the fatigue reliability of the steel box girder based on the two-stage convergence criterion as claimed in claim 1, wherein in the second step, in particular, the samples of the subset simulation first layer are directly generated by the monte carlo simulation, so that the samples are independent from each other, whereas the samples of the other layers are generated by the markov chain monte carlo simulation, and there is a correlation between the first samples of different markov chains, so that the samples of the other layers are correlated, and as a simplification condition, the sample-based adjustment function is independent, that is:
cov(Padj(xi)，Padj(xj))＝0，i≠j (11)
to study the impact of kriging model randomness on failure probability estimation, the expectation of defining the tuning function is:
Eadj＝E[Padj(x)]＝∫Padj(x)fX(x)dx (12)
wherein f isX(x) As a function of the joint probability density of the random variable X, i.e. f for the first layer of the subset simulationX(x)＝qX(x) For the ith layer f of the subset simulationX(x)＝qx(x|Fl-1) According to the central limit theorem, EadjThe approximation follows a normal distribution, which is based on a sample estimate of:
Figure FDA0003256266190000031
the corresponding variance estimate is:
Figure FDA0003256266190000032
defining the model error rate as:
Figure FDA0003256266190000033
wherein the content of the first and second substances,
Figure FDA0003256266190000034
is a scalar quantity, because EadjThe approximation follows a normal distribution, so η also follows a normal distribution, i.e.
Figure FDA0003256266190000035
Converting a smaller failure probability into a product of a series of larger conditional failure probabilities based on subset simulations, given failure event F, defining intermediate failure events { F }1，F2，…，FLIs Fl＝{G(x)≤bl1, 2, …, L, wherein b1＞…＞bl＞…＞bLWith 0 being the failure threshold value,
the final failure probability is calculated by:
Figure FDA0003256266190000036
wherein, P1＝P{F1Is the failure probability of the first failure event, Pl＝P{Fl|Fl-1Where L is 2, 3, …, L, PlPerforming simulation estimation based on Markov chain Monte Carlo for conditional failure probability,
estimating failure probability
Figure FDA00032562661900000411
The coefficient of variation calculation formula is:
Figure FDA0003256266190000041
wherein, when l is 1, nlNumber of samples for direct monte carlo simulation, or when L ═ 2.. L, nlNumber of samples, gamma, for Markov chain Monte Carlo simulationlTo account for non-negative scalars of sample correlation, γlSmaller indicates weaker correlation between samples, and thus gamma1＝0，
Final failure probability estimation
Figure FDA0003256266190000042
The coefficient of variation of (A) is:
Figure FDA0003256266190000043
3. the method for analyzing the fatigue reliability of the steel box girder based on the two-stage convergence criterion as claimed in claim 1, wherein in step three, in particular, in one aspect, when L ═ 1, 2, …, L, the conditional probability P islShould be controlled, defining an estimated error rate of the l-th layer failure probability as etalObeying a normal distribution according to equation (16), the first stage constraint being |. etal|＜α1The confidence of (2) is not lower than 99.7%:
Figure FDA0003256266190000044
Figure FDA0003256266190000045
wherein the content of the first and second substances,
Figure FDA0003256266190000046
and
Figure FDA0003256266190000047
in order to control the boundaries of the process,
Figure FDA0003256266190000048
estimate of the probability of failure, α, for the 1 st failure event1A preset threshold value greater than zero,
Figure FDA0003256266190000049
for the desired estimate of the adjustment function on the ith subset, var represents the variance calculation function, equation (20) is the first stage convergence criterion,
on the other hand, since the final failure probability is the product of a series of conditional failure probabilities, the calculation formula for obtaining the final failure probability is as follows:
Figure FDA00032562661900000410
wherein the content of the first and second substances,
Figure FDA0003256266190000051
to account for the estimation of the ultimate failure probability of the model error,
Figure FDA0003256266190000052
for the estimate of the adjustment function expectation on the l-th subset, due to alpha1Is set to a very small value, here 0.03, so that the higher order terms in equation (21) are ignored,
the model cumulative error is defined as:
Figure FDA0003256266190000053
and ηlUniformity xilObey a normal distribution, i.e.:
Figure FDA0003256266190000054
to control model accumulation error, guarantee xil∈[-α2，α2]The confidence of the second stage convergence criterion is not lower than 99.7 percent, namely:
Figure FDA0003256266190000055
Figure FDA0003256266190000056
wherein alpha is2For a preset threshold value greater than zero, 0.05 is assumed here.
CN202011548489.8A 2020-12-24 2020-12-24 Steel box girder fatigue reliability analysis method based on two-stage convergence criterion Active CN112528517B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202011548489.8A CN112528517B (en) 2020-12-24 2020-12-24 Steel box girder fatigue reliability analysis method based on two-stage convergence criterion

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202011548489.8A CN112528517B (en) 2020-12-24 2020-12-24 Steel box girder fatigue reliability analysis method based on two-stage convergence criterion

Publications (2)

Publication Number Publication Date
CN112528517A CN112528517A (en) 2021-03-19
CN112528517B true CN112528517B (en) 2021-10-22

Family

ID=74976188

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202011548489.8A Active CN112528517B (en) 2020-12-24 2020-12-24 Steel box girder fatigue reliability analysis method based on two-stage convergence criterion

Country Status (1)

Country Link
CN (1) CN112528517B (en)

Families Citing this family (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113705045B (en) * 2021-08-20 2024-04-12 上海交通大学 Agent model-based friction reliability analysis method for rotor-stator subsystem
CN114861489B (en) * 2022-04-18 2023-09-19 河海大学 rPCK proxy model aided structure dynamic parameter identification method
CN117454668B (en) * 2023-12-18 2024-03-19 中国人民解放军国防科技大学 Method, device, equipment and medium for predicting failure probability of parts

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109255144A (en) * 2018-07-09 2019-01-22 南京航空航天大学 A kind of undercarriage braking member light-weight design method
CN111783209A (en) * 2020-07-03 2020-10-16 吉林大学 Self-adaptive structure reliability analysis method combining learning function and kriging model

Family Cites Families (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108304679A (en) * 2018-03-07 2018-07-20 西北工业大学 A kind of adaptive reliability analysis method
CN109165425B (en) * 2018-08-03 2022-04-12 湖南大学 Gear contact fatigue reliability analysis method
CN110532723B (en) * 2019-09-06 2021-06-22 北京航空航天大学 EGRA-based turbine disk multi-failure-mode reliability optimization method
CN111339488B (en) * 2020-03-24 2023-06-13 成都理工大学 Slope system failure probability calculation method based on Kriging

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109255144A (en) * 2018-07-09 2019-01-22 南京航空航天大学 A kind of undercarriage braking member light-weight design method
CN111783209A (en) * 2020-07-03 2020-10-16 吉林大学 Self-adaptive structure reliability analysis method combining learning function and kriging model

Also Published As

Publication number Publication date
CN112528517A (en) 2021-03-19

Similar Documents

Publication Publication Date Title
CN112528517B (en) Steel box girder fatigue reliability analysis method based on two-stage convergence criterion
Srikanth et al. Deterioration models for prediction of remaining useful life of timber and concrete bridges: A review
CN111950180B (en) Reliability analysis method and system for cable bearing bridge structure system
Yi et al. A novel fidelity selection strategy-guided multifidelity kriging algorithm for structural reliability analysis
Morse et al. A multi-fidelity boundary element method for structural reliability analysis with higher-order sensitivities
CN111444649B (en) Slope system reliability analysis method based on intensity reduction method
CN115982141A (en) Characteristic optimization method for time series data prediction
CN116127818A (en) Bridge seismic vulnerability assessment method based on influence line correction model
Lin et al. Dynamic finite element model updating of prestressed concrete continuous box-girder bridge
Nikolaidis et al. Neural networks and response surface polynomials for design of vehicle joints
Lima et al. Hyperparameter-optimized multi-fidelity deep neural network model associated with subset simulation for structural reliability analysis
Potrzeszcz-Sut Reliability analysis of shell truss structure by hybrid Monte Carlo method
CN116629080B (en) Method for predicting rolling of steel pipe concrete superposed member impact displacement time course chart
CN113706328A (en) Intelligent manufacturing capability maturity evaluation method based on FASSA-BP algorithm
Cheng Serviceability reliability analysis of prestressed concrete bridges
Inzunza-Aragón et al. Use of artificial neural networks and response surface methodology for evaluating the reliability index of steel wind towers
CN111582634A (en) Multi-factor safety grading method and system for underground large-space construction
Segura et al. Expected seismic performance of gravity dams using machine learning techniques
CN115659750A (en) Physical structure system numerical modeling parameter identification method based on SMA-aSCE
Xu et al. Assessing the capacity reliability of ageing water distribution systems
CN109376957A (en) A kind of prediction technique of thermal power plant's load
Lu et al. Prediction of the static load test results of bridges based on the dynamic load test and the Kriging model
CN115114819B (en) Failure surface simulation method of planar woven panel honeycomb sandwich structure based on adaboost algorithm
Khorramian et al. Efficient Representation of Random Fields for Training the Kriging Predictor in Adaptive Kriging Reliability Assessment of Civil Structures
Mei et al. Model updating for bridge structures based on the Kriging meta-model enhanced with DE algorithm and analytic hierarchy process

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant