A multi-design point reliability analysis method for bridge structures

Abstract

The present invention relates to the field of reliability analysis of engineering structures, and in particular to a multi-design point reliability analysis method for bridge structures, the method comprising determining the original structural function and the positions and numbers of design points according to the bridge structure analysis; selecting a design point, and using an additional function to shield the remaining design points to determine the shielded structural function; based on an improved cross entropy method, performing MCMC pre-sampling on the shielded structural function function and determining the distribution parameters of the important sampling function; performing important sampling on the shielded structural function function to calculate the structural failure probability corresponding to the currently selected design point; and obtaining the failure probability of the entire structure by superimposing the structural failure probabilities of all design points. The scheme of the present invention realizes an efficient and robust solution to the reliability problem of multi-design point structures, which is beneficial to reliability optimization, structural design and structural evaluation of large bridge structures.

Description

Technical Field

[0001] The invention relates to the field of engineering structure reliability analysis, and in particular to a multi-design point reliability analysis method for a bridge structure. Background Art

[0002] Modern large-scale bridges and other engineering structures are all highly indeterminate structural systems with complex structural components. The use of probability-based structural reliability theory to analyze failure probability is one of the core issues in current research on structural safety performance. However, due to the complexity of large-scale bridge engineering structures, their structural performance functions may have multiple design points, resulting in the following two difficult problems: (1) The failure domains near multiple design points have an important impact on their overall failure probability; (2) Structural failure is a problem of small failure probability (rare event simulation), and its probability estimation accuracy and computational efficiency are both low. Using traditional first-order and second-order reliability calculation methods to deal with such problems will cause large errors, and it is difficult to properly consider the contribution of multiple failure domains.

[0003] To address this problem, different scholars currently use methods based on numerical sampling to analyze and discuss the failure probability estimation of structures under multiple design points. Bjerager proposed a uniform directional sampling method suitable for failure probability estimation of structural systems earlier. This method can effectively consider the probability distribution characteristics of the limit state equations of multiple design points by sampling the entire normal space without focus. However, due to the lack of emphasis in sampling, its estimation variance is relatively large. In order to reduce the sampling variance and further expand the scope of application, Au SK proposed a famous subset sampling method based on the Markov Chain Monte Carlo (MCMC) conditional sampling technology. This method has a strong ability to span multiple design point regions by using multiple Markov chain splitting conditional sampling, but it may lead to large errors in some cases where the failure surface is extremely discontinuous. Corresponding to the subset sampling method, the important sampling method based on cross entropy (CE) is a new sampling method developed in recent years. This method can adaptively obtain the optimization parameters of the important sampling density function by using cross entropy maximization as a parameter optimization method. Compared with other methods, it has extremely excellent sampling variance reduction capabilities. In theory, this method can converge the important sampling density function to the global optimum through cross entropy, and has the ability to solve the reliability problem of multiple design points of the structure. However, in practice, it is found that the multi-design point cross entropy iteration often converges to the local optimum, which will cause large errors. At present, this method has obvious defects in applicability.

[0004] In summary, the existing analysis methods for the reliability of multiple design points of modern large bridge structures still have problems of poor stability and low efficiency, and cannot accurately quantify the impact of each failure mode (corresponding to different design points) on the overall failure probability of the structure, which is extremely unfavorable for in-depth understanding of the evolution of structural failure paths and identification of weak parts in reliability optimization, structural design and structural evaluation. Therefore, this field urgently needs a method for analyzing the reliability of multiple design points of bridge structures to achieve efficient and robust calculation of the overall failure probability of bridge structures, so as to facilitate subsequent reliability optimization, structural design and structural evaluation of large bridge structures. Summary of the invention

[0005] The object of the present invention is to provide a multi-design point reliability analysis method for a bridge structure, thereby solving the above-mentioned problems existing in the prior art.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A multi-design point reliability analysis method for a bridge structure, the method comprising the following steps:

[0008] Determine the original structural function based on the structural parameters, functional parameters and loads of the bridge structure;

[0009] Determining the design point position and the number of design points according to the original structural function, and constructing corresponding additional functions for different design points;

[0010] Selecting a design point, and using the additional function to shield the remaining design points, and determining the structure function function after shielding;

[0011] Based on the improved cross entropy method, MCMC pre-sampling is performed on the masked structural function function and the parameter value of the important sampling probability density function is determined according to the MCMC pre-sampled samples;

[0012] Using the importance sampling probability density function to perform importance sampling on the shielded structural function function to determine the structural failure probability corresponding to the currently selected design point;

[0013] After obtaining the structural failure probabilities corresponding to all design points, the failure probability of the entire structure is obtained by superposition and combination.

[0014] Preferably, the additional function is:

[0015]

[0016] Among them, r i is the shielding surface radius, s i is the proportionality coefficient, x i *is the design point to be shielded; i and i The values ​​of are determined by the following formula:

[0017] r i =γβ form,i

[0018]

[0019] Among them, β form,i For x i * The corresponding first-order reliability index is, is the original structure function G(x i * ), γ and δ are empirical parameters.

[0020] Preferably, the shielded structure function is:

[0021] G j (x)=G(x)+ΣB i (x)

[0022] Among them, G j (x) is the shielded structure function function of the currently selected design point, G(x) is the original structure function function, B i (x) is the shielded design point x i * The corresponding additional functions.

[0023] Preferably, when the additional function B i (x) in||xx i * ||=r i When B i The first and second derivatives of (x) are:

[0024] B i ′(x|||xx i * ||=r i )=0

[0025] B i ″(x|||xx i * ||=r i )=8s i r i 2 >0.

[0026] Preferably, the MCMC pre-sampling of the shielded structure function function specifically comprises the following steps:

[0027] Step 1: Randomly generate an n-dimensional initial variable x1=[x1 (1) ,…,x1 (i) ,…,x1 (n) ]∈F;

[0028] Step 2: The i-th dimension variable x obtained by the k-th round of sampling k (i) Construct normal proposal distribution φ i (·; x k (i) ), draw samples x from the normal proposal distribution i ′;

[0029] Step 3: Calculate the transfer kernel:

[0030]

[0031] Step 4: Randomly draw a sample U from a uniform distribution [0,1] i , and order

[0032]

[0033] Step 5: Repeat steps 2 to 4 until n dimensions are completed Sampling, determine the k+1th round of generated sample x k+1 for:

[0034]

[0035] Step 6: Repeat steps 2 to 5 until the required number of samples are drawn.

[0036] Preferably, the parameter values ​​of the importance sampling probability density function include a mean and a covariance matrix;

[0037] The mean is determined based on the MCMC pre-sampled samples, and the mean is calculated as:

[0038]

[0039] in, is the mean, x k is the sample pre-sampled for the MCMC.

[0040] Preferably, the covariance matrix is ​​determined based on the MCMC pre-sampled samples and the mean, specifically:

[0041] A new coordinate system is constructed with the direction indicated by the mean as the coordinate axis, and the variance in the direction indicated by the mean is calculated based on the MCMC pre-sampled samples, and the covariance matrix in the original coordinate system is obtained using the transformation matrix; the covariance matrix is ​​calculated as:

[0042]

[0043]

[0044] in, is the variance in the direction of the mean, is the covariance matrix.

[0045] Preferably, the determination of the structural failure probability corresponding to the currently selected design point is specifically:

[0046]

[0047] Among them, f(x) is the original probability density function, h(x) is the importance sampling probability density function, E h (·) is the expectation corresponding to h(x); I f (·) is an indicative function, which is 1 when x is in the failure domain F and 0 in other cases.

[0048] The beneficial effects of the present invention are:

[0049] The present invention proposes a multi-design point reliability analysis method for bridge structures, which includes two parts: local shielding of the structural limit state surface and improved cross entropy important sampling. The method transforms the original multi-design point structural function function into multiple single design point structural function functions by local shielding of the bridge structure limit state surface, and calculates the failure probability by using the improved cross entropy method suitable for solving single peak or single design point reliability. The two are organically combined to form a good complement, realizing an efficient and robust solution to the multi-design point reliability problem of bridge structures, which is beneficial to reliability optimization, structural design and structural evaluation of large bridge structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 It is a schematic flow chart of a multi-design point reliability analysis method for a bridge structure in an embodiment of the present invention;

[0051] Figure 2 It is a schematic diagram of the general force of the structure in the embodiment of the present invention;

[0052] Figure 3 is a schematic diagram of a limit state surface of a structure with multiple design points in an embodiment of the present invention;

[0053] Figure 4 is the limit state surface x2 of the multi-design point structure in the embodiment of the present invention * Schematic diagram of regional local shielding;

[0054] Figure 5 is the limit state surface x1 of the multi-design point structure in the embodiment of the present invention *Schematic diagram of regional local shielding;

[0055] Figure 6 is the optimal important sampling probability density function h in the embodiment of the present invention * (x) Sample distribution diagram;

[0056] Figure 7 is a schematic diagram of MCMC pre-sampling in an embodiment of the present invention;

[0057] Figure 8 is a schematic diagram of a key direction coordinate system in an embodiment of the present invention;

[0058] Fig. 9 is the shielding design point x2 in the test example of the present invention * The ultimate limit state surface of the structure after

[0059] Fig.10 is the shielding design point x1 in the test example of the present invention * The ultimate limit state surface of the structure after

[0060] Fig.11 It is a graph showing the variation of the coefficient of variation of the structural failure probability calculated by the method provided by the present invention in the test example of the present invention versus the sampling number. DETAILED DESCRIPTION

[0061] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings. It should be understood that the specific implementation methods described herein are only used to explain the present invention and are not used to limit the present invention.

[0062] The present invention proposes a multi-design point reliability analysis method for bridge structures by combining local shielding of limit state surfaces and an improved cross-entropy important sampling method. Figure 1 As shown, the method comprises the following steps:

[0063] S101, determining an original structural functional function G(x) based on structural parameters, functional parameters and borne loads of the bridge structure.

[0064] S102, determining the design point position and the number of design points according to the original structural function G(x), and constructing corresponding additional functions B for different design points i (x).

[0065] S103, select a design point j, and use the additional function B i (x) Shield the remaining design points and determine the shielded structure function G j (x); where j and i are design point labels to distinguish each design point.

[0066] S104, based on the improved cross entropy method, the shielded structure function function G j (x) performing MCMC pre-sampling and determining parameter values ​​of the importance sampling probability density function h(x) based on the MCMC pre-sampling samples.

[0067] S105, using the importance sampling probability density function h(x) to analyze the masked structure function function G j (x) Perform importance sampling to determine the structural failure probability P corresponding to the currently selected design point fj =P(G j (x)<0).

[0068] S106, repeat S103 to S105, change the selected design point (i.e. j = j + 1), until all design points are sampled, and the failure probability P corresponding to each design point is calculated. fj Perform superposition combination (accumulation method can be used) to calculate the failure probability P of the entire bridge structure. f =ΣP fj ; Based on the failure probability P of the entire bridge structure f By combining the information of each design point, the failure behavior characteristics can be analyzed and the reliability of the bridge structure can be evaluated.

[0069] In this embodiment, the basic goal of bridge structure design is to make the designed bridge structure meet the requirements of safety, applicability and durability, that is, to make the structure have sufficient reliability. The probability measurement of structural reliability is called structural reliability. In other words, the reliability of a structure refers to the probability that the structure completes its predetermined function within a specified time and under specified conditions. The sign that the structure completes various functions can be measured by the corresponding limit state. When the structure as a whole or a part exceeds a certain state, the structure cannot meet a certain functional requirement specified in the design. This specific state is called the limit state of the structure. Therefore, the limit state of the structure is the dividing line that distinguishes whether the structure works reliably or unreliably.

[0070] In structural reliability analysis, the limit state of the structure is defined by describing the functional function of the structure. Let X1, X2, ..., X n are n random variables that affect the structure function, then the following random function Z=G(X1,...,X n ) is called the structural performance function, and the random variables can be the resistance of the structure, the external loads it bears, etc.

[0071] When Z>0, the structure has the specified function, that is, it is in a reliable state;

[0072] When Z < 0, the structure loses its specified function, that is, it is in a failed state;

[0073] When Z = 0, the structure is in a critical state, also called a limit state;

[0074] And Z=G(X1,...,X n )=0 is the limit state equation of the structure. The probability that the structural function is less than 0 is called the failure probability of the structure, which is expressed as P f express.

[0075] The original structural function function G(x) in the present invention is a structural function function constructed based on the structural parameters of the bridge structure to be analyzed (such as the geometric dimensions of the components, the physical parameters of the materials, etc.), the functional parameters and the random variables involved in the load. The original structural function function is obtained by analyzing the stress of the bridge structure. Figure 2 As shown in the figure, a structure is subjected to a certain external load. Based on the structural force analysis, the load effect of the external load on the structure can be obtained as S(x i ,x j ), and the resistance of the structure itself is calculated to be R(x k ). According to the definition of structure-function function, the original structure-function function is G(x)=R(x k )-S(x i ,x j ). For other structural forms, G(x) can also be obtained by following the above method.

[0076] In this embodiment, the corresponding structural limit state equation G(x)=0 can be obtained according to the original structural function G(x), and its geometric form in the variable space is the structural limit state surface. The shortest distance from the limit state surface (global or local) to the origin of the coordinate corresponds to the key point on the limit state surface (global or local) - the most probable failure point (MPP), which is usually called the design point.

[0077] Due to its complexity, large bridge structures often have multiple design points. By performing a first-order reliability analysis on its limit state equation G(x)=0, the specific location and number of design points can be obtained. Studies have shown that in the reliability problem of bridge structures with multiple design points, the location distribution of adjacent design points is not very close. At the same time, from the probability characteristics of structural reliability, it can be seen that its failure probability is mainly contributed by the probability density of the area near the design point. For other locations in the space far away from this area, its contribution to the failure probability is extremely small. Therefore, in this embodiment, by controlling and intervening in the morphology of the structural limit state surface and improving its functional function form, other redundant design point areas are pushed as far as possible into the failure domain, eliminating the contribution of the failure probability of the corresponding area (i.e., shielding redundant design points).

[0078] The above operation can transform the reliability problem of multi-design point structure into multiple single design point problems for analysis, thereby avoiding the influence of multi-design point failure contribution and having little influence on the failure probability of unshielded areas. Figure 3 As shown, for such a surface, when analyzing the design point x1 * When the failure probability corresponding to the area is , in order to avoid the influence of other design points, the area near other design points can be partially shielded, such as Figure 4 The mathematical expression of the shielded surface can be obtained by adding an additional function B to the original structural function G(x): i (x) Implementation:

[0079] G j (x)=G(x)+ΣB i (x) (1)

[0080] In formula (1), G j (x) can be regarded as the shielding structure performance function for a specific design point or failure mode, B i (x) for each shielded design point x i * The corresponding additional functions.

[0081] Thus, the original multiple design point problem can be eliminated and simplified to a single design point x1 * Problem. For the design point x2 * The analysis can also be done in the same way as above. Figure 5 shown.

[0082] Since the additional function B i The selection of (x) directly determines the quality of shielding, so its determination is crucial. i The basic principle of (x) is to push the shielded design point area as far as possible to the failure domain, so that the local morphology of the limit state surface changes, eliminating the contribution of the failure probability of the corresponding area, and thus making its probability as small as possible. At the same time, the additional function surface should maintain a smooth transition near the intersection with the original surface to avoid sharp points, so that the shielded structure function G j (x) remains continuous to facilitate subsequent sampling analysis. In this embodiment, the additional function B i (x) is:

[0083]

[0084] In formula (2), r i is the shielding surface radius, s i is the proportionality coefficient, x i * is the design point to be shielded. iand i The values ​​can be determined by the following formula:

[0085] r i =γβ form,i (3)

[0086]

[0087] In formula (3) and (4), β form,i For x i * The corresponding first-order reliability index is, is the original structure function G(x i * ), γ and δ are empirical parameters that can be set according to the specific implementation scenario. In this embodiment, the value of the empirical parameter δ is preferably 0.75. As for the empirical parameter γ, it can be seen from formula (3) that it actually directly determines the shielding radius r i If the value is too small, the function cannot guarantee the integrity of the shielding. If the value is too large, other key areas may be affected. Therefore, on the basis of ensuring the reliability of the shielding, the value of γ should be as small as possible. According to research and analysis, the value of the empirical parameter γ is preferably 1.1, at which time the shielding surface can cover the required area more completely and its related interference is small.

[0088] In this embodiment, for the additional function B i (x) Boundary conditions (smooth transition between the shielding surface and the original surface), which can be obtained by B i (x) The first and second order derivatives are verified, when || xx i * ||=r i When B i (x) The first and second order derivatives are:

[0089] B i ′(x|||xx i * ||=r i )=0 (5)

[0090] B i ″(x|||xx i * ||=r i )=8s i r i 2 >0 (6)

[0091] At this time B i (x) The first-order derivative is 0, the second-order derivative is greater than 0, and the reconstructed structural function G j(x) No additional slope is caused here and it has a strong upward concavity, and the surface meets the transition roundness requirements.

[0092] In this embodiment, the importance sampling method is used to calculate the failure probability P of the structure. fj ,This method has extremely excellent rare event simulation capability and sampling efficiency.

[0093] Based on the importance sampling method, the structural failure probability P corresponding to the current design point j is fj The sampling principle is:

[0094]

[0095] In formula (7), f(x) is the original probability density function of the random variable in the masked structural function, h(x) is the importance sampling probability density function (its parameters include the mean and covariance matrix ), E h (·) is the expectation corresponding to h(x) (obtained through importance sampling), I f (·) is an indicative function, which is 1 when x is in the failure domain F and 0 in other cases.

[0096] In the above calculation process, the parameter value of the important sampling probability density function h(x) has an important influence on the accuracy and stability of the calculation. Therefore, the cross entropy theory is used to optimize the h(x) parameter. According to the cross entropy theory, the Kullback-Leibler (KL) divergence is used to measure the "distance" between two probability density functions h(x) and f′(x) on f′(x):

[0097]

[0098] In formula (8), f′(x) is the conditional probability density function defined only on the failure domain F, f′(x)=I f (x∈F)·f(x) / P fj , the meaning of each function is the same as in formula (7). This "distance" actually reflects the similarity between the two, and it is always non-negative. When h(x) = f'(x), D(f',h) = 0. In order to find the optimal important sampling probability density function h(x), according to the KL divergence property, it should be as small as possible, so it is necessary to solve the corresponding parameters for the minimum value of D(f',h). According to the definition of formula (8), D(f',h) can be further transformed into:

[0099] D(f',h)=∫f'(x)log(f'(x))dx-∫f'(x)log(h(x))dx (9)

[0100] In formula (9), the first term on the right side is a constant that is independent of h(x), and the second term is the cross entropy between f′(x) and h(x). Therefore, the problem of minimizing D(f′,h) to find the optimal parameter for h(x) can be simplified to the problem of optimizing the parameters to maximize its cross entropy:

[0101]

[0102] Among them, v * is the optimal parameter value of the important sampling probability density function h(x; v). This equation is generally called the cross entropy problem. Since equation (10) cannot usually be obtained analytically, it can be optimized using a discrete sampling method:

[0103]

[0104] Currently h(x; v) can be in the form of a Gaussian function. At this time, the optimization result of formula (11) has the following analytical form:

[0105]

[0106]

[0107] In formula (12) and (13), and are the calculation results (usually estimated values) of the mean and covariance matrix of the Gaussian distribution h(x; v); w k =w k ′ / Σ q w q ′,w k ′=I f (G j (x k )<0)·f(x k ) / h(x k ; v), x k is a sample drawn from a Gaussian distribution h(x;v).

[0108] From the above formula, we can see that due to x k It needs to be extracted from h(x; v), so Equations (12) and (13) do not actually give an explicit expression of the parameters. To solve this problem and overcome the difficulty of simulating rare events, the current conventional practice is to use a sequential sampling method to gradually approximate and introduce an important sampling process to adaptively update the parameters. However, it has two inherent disadvantages: first, parameter update generally requires multiple rounds of pre-sampling, which is inefficient; second, for high-dimensional problems, its likelihood ratio w k ′ will degenerate and lead to the curse of dimensionality.

[0109] In this embodiment, the cross entropy method is improved by combining Markov Chain Monte Carlo (MCMC) and important direction one-dimensional projection variance estimation technology (i.e., improved cross entropy method). According to the cross entropy theory, the quality of the structural sample has a great influence on the parameter And the failure probability P fj The volatility of the estimated variance has a significant impact. According to the definition of KL divergence, the theoretically optimal important sampling probability density function is:

[0110] h * (x) = f′(x) = I f (x∈F)·f(x) / P fj (14)

[0111] Formula (14) is essentially a conditional distribution defined only on the failure domain F (see Figure 6 ), and its parameter P fj is unknown, so it cannot be used directly for analysis. However, for high-quality sampling and parameter estimation, it is well applicable and has significant advantages: for the conditional distribution of formula (14), since it is the theoretically optimal h * (x), so the quality of the corresponding sample generation is extremely high, and its application in estimating distribution parameters is expected to achieve better results. It should be noted that f′(x) is a function defined only on the failure domain F. Unlike f(x), it cannot calculate the function value and can only extract samples that conform to its distribution.

[0112] For h * (x) can be used to generate samples using MCMC sampling technology. This method can solve the conditional sampling problem well by constructing an MC transfer kernel and combining the "acceptance-rejection" sampling technology. For the original probability density function f(x)=∏ in Gaussian space i φ i (x (i) ), in order to further improve the acceptance rate of MCMC sampling and prevent jump "blocking", the improved Metropolis-Hastings algorithm is used for pre-sampling. The purpose of pre-sampling is to generate failure domain samples x∈F through formula (14), which are then used to estimate the parameter value of the important sampling probability density function h(x; v) and

[0113] The corresponding pre-sampling steps are as follows:

[0114] Step 1: Randomly generate an n-dimensional initial variable x1=[x1 (1) ,…,x1 (i) ,…,x1 (n) ]∈F (x1 is also called random seed);

[0115] Step 2: The i-th dimension variable x obtained by the k-th round of sampling k (i) Construct a normal proposal distribution φ i (·; x k (i) ), draw a sample x from this distribution i ′;

[0116] Step 3: Calculate the transfer kernel:

[0117]

[0118] Step 4: Randomly draw a sample U from a uniform distribution [0,1] i , and order

[0119]

[0120] Step 5: Repeat steps 2 to 4, i = i + 1, until n dimensions are completed Sampling, determine the k+1th round of generated sample x k+1 for:

[0121]

[0122] Step 6: Repeat steps 2 to 5, k=k+1, until the required number of samples are extracted.

[0123] The above is the step of MCMC pre-sampling to generate parameter estimation samples. In order to further improve the sampling quality and reduce the sample correlation, it is preferred to use more than 10 Markov chains to jointly generate all samples, such as Figure 7 As shown. Applying the above sample, we can get the Gaussian distribution parameters and According to formula (14) and the likelihood ratio w k ′ definition, we can get w at this time k ′ degenerates to a constant, and equations (12) and (13) are simplified to:

[0124]

[0125]

[0126] For the above formula, the mean The calculation effect is better, but for the covariance matrix Due to the existence of sampling fluctuations, equation (16) will introduce a lot of random "noise" and may obliterate real information, which is the main reason for the dimensionality curse. To overcome the above problem, this embodiment further improves the above calculation method by combining the important direction one-dimensional projection variance estimation technology.

[0127] It has been found that in n-dimensional random space, the covariance matrix Only a very small number of key elements have a significant impact on the efficiency and stability of the important sampling method. Accurately clarifying the corresponding variance or covariance can effectively ensure the efficiency and stability of the sampling. Therefore, the above problems can be solved by seeking key spatial directions (or elements) and accurately estimating their projection variances. For all directions in the random space, the key directions should have the following characteristics: the projection variance of the sample in the key direction should be significantly different from that in other directions to a large extent, so that the projection of the sample on it brings more effective information without being obliterated by "noise". Based on the above principles and combined with the characteristics of the structural limit state surface, it can be known that the mean The direction pointed to is a good choice that meets its requirements. Based on this information, the covariance matrix Improve the calculation.

[0128] For ease of description, the direction Construct a new coordinate system Y for the coordinate axis Y1, such as Figure 8 As shown. Each coordinate axis Y i The transformation relationship with the original coordinate system X is shown as follows:

[0129] Y=α T X (17)

[0130] Where α is the transformation matrix, T is the transpose symbol, and α is defined as:

[0131] α=[α1,…,α i ,…,α n ] (18)

[0132] α is an orthogonal matrix, and its α1 represents the direction from the origin of space to the mean point The unit vector of .

[0133] From the above, we can see that for the new coordinate system Y, the covariance matrix can also be constructed and calculated under this frame: And obtain the covariance matrix in the original coordinate system X through the transformation matrix α It is then used for subsequent failure probability sampling analysis. Therefore, for the covariance matrix C Y The sample projection in the α1 direction, its variance v1 can be calculated through the samples as:

[0134]

[0135] As for other directions, in view of the research findings, using a suitable n-dimensional independent Gaussian distribution as the important sampling density function does not affect the sampling efficiency and accuracy, but its covariance matrix should be as close to the unit matrix as possible (the variance in other directions can be set to 1), otherwise the algorithm stability cannot be guaranteed. Based on the above information and making full use of the results of formula (19), C Y Can be constructed as:

[0136]

[0137] In the formula, I n is the n-dimensional identity matrix.

[0138] Further considering the transformation relationship of formula (17), the inverse transformation between X and Y can be obtained as follows:

[0139] X=αY (21)

[0140] To find the covariance matrix of X, we have

[0141] DX=C σ =αDYα T =αC Y α T (twenty two)

[0142] Further simplify the covariance matrix Finally available

[0143]

[0144] This is the key direction The covariance matrix obtained by projection As can be seen from formula (23), the number of parameter estimates in the variance matrix is ​​reduced from n(n+1) / 2 to 1, which greatly reduces the computational workload. At the same time, while fully retaining the probability information of key directions, it significantly reduces the impact of random noise, thereby effectively alleviating the dimensionality curse problem and improving the stability of important sampling.

[0145] After obtaining the structural importance sampling probability density function h(x) based on the improved cross entropy method, the structural failure probability P can be obtained by combining the importance sampling method of formula (7). fj Calculate and superimpose the combination to finally obtain the failure probability P of the entire structure f The corresponding calculation results can be used to analyze the characteristics of structural failure behavior and evaluate the reliability of bridge structures.

[0146] By adopting the above technical solution disclosed in the present invention, the following beneficial effects are obtained:

[0147] The present invention proposes a multi-design point reliability analysis method for bridge structures, which includes two parts: local shielding of the structural limit state surface and improved cross entropy important sampling. The method transforms the original multi-design point structural function function into multiple single design point structural function functions by local shielding of the bridge structure limit state surface, and solves the reliability by an improved cross entropy method suitable for solving single peak or single design point reliability. The two are organically combined to form a good complement, realizing an efficient and robust solution to the multi-design point reliability problem of bridge structures, which is beneficial to reliability optimization, structural design and structural evaluation of large bridge structures.

[0148] In order to further illustrate the scheme of the present invention, the failure probability of a bridge structure is calculated in combination with a specific test example, and the calculation results are compared and analyzed with other traditional algorithms to verify the correctness and corresponding performance of the reliability analysis method proposed in the present invention.

[0149] Test example

[0150] By analyzing the stress of a bridge structure, we can find that its limit state equation is in the form of a parabola: G(x)=5-x2-0.5×(x1-0.1) 2 =0, where the random variables x1 and x2 of the structure both obey the standard normal distribution N(0,1) and are independent of each other.

[0151] By performing first-order reliability analysis on the above limit state equation, it can be obtained that there are two design points, namely x1 * (-2.7409,0.9648) and x2 * (2.9159,1.0355), the corresponding first-order reliability indexes of the structure are β form1 =2.9057 and β form2 =3.0943.

[0152] Based on the above information, additional functions are constructed to locally shield each design point area, and the limit state surface and design point distribution after shielding can be obtained as follows: Fig. 9 and Fig.10 The reliability analysis method of the present invention and other methods are used to calculate the failure probability of the structure, and the results are shown in Table 1:

[0153] Table 1

[0154]

[0155] Note: The coefficient of variation of each calculation method was obtained by repeating the calculation 100 times.

[0156] Comparing the calculation results in Table 1, it can be found that since the test case has two design points, its limit state surface is extremely complex. Therefore, the use of traditional important sampling and line sampling methods will lead to excessive errors, and the results can only reflect x1 * The failure probability contribution of the nearby area is difficult to effectively improve even if the number of sampling times is increased. This problem is mainly caused by the fact that the algorithm principle focuses on the direction of a local single design point. The local shielded cross entropy method proposed in the present invention can effectively solve the above problem because it traverses the contribution of the failure domain of each design point. And the relatively simplified second-order reliability approximation method (SORM) has higher calculation accuracy. The calculation result is only 0.5% error compared to the standard Monte Carlo method (Monte Carlo simulation, MCS), indicating that the structural multi-design point reliability analysis method provided by the present invention has excellent applicability and calculation accuracy.

[0157] Further analysis of the calculation efficiency of the method of the present invention can yield the failure probability P f The variation law of the coefficient of variation of the calculated results with the number of sampling times is as follows: Fig.11 As shown. Fig.11 It can be seen that the method provided by the present invention has a good variance reduction capability. When the sampling exceeds 2000 times, the coefficient of variation is below 2.5%. f This result shows that the multi-design point reliability analysis method for bridge structures provided by the present invention has excellent convergence performance and analysis efficiency. When solving the multi-design point reliability problem of complex bridges, its sampling workload can be greatly reduced, which is more conducive to the subsequent reliability optimization, structural design and structural evaluation of large bridge structures.

[0158] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be considered as the scope of protection of the present invention.

Claims

1. A multi-design point reliability analysis method for a bridge structure, characterized in that: The method comprises the following steps: Determine the original structural function based on the structural parameters, functional parameters and loads of the bridge structure; Determining the design point position and the number of design points according to the original structural function, and constructing corresponding additional functions for different design points; Selecting a design point, and using the additional function to shield the remaining design points, and determining the structure function function after shielding; Based on the improved cross entropy method, MCMC pre-sampling is performed on the masked structural function function and the parameter value of the important sampling probability density function is determined according to the MCMC pre-sampled samples; Using the importance sampling probability density function to perform importance sampling on the shielded structural function function to determine the structural failure probability corresponding to the currently selected design point; After obtaining the structural failure probabilities corresponding to all design points, the failure probability of the entire structure is obtained by superposition and combination; The additional function is: Among them, r i is the shielding surface radius, s i is the proportionality coefficient, x i * is the design point to be shielded; i and i The values ​​of are determined by the following formula: r i =γβ form,i Among them, β form,i For x i * The corresponding first-order reliability index is, is the original structure function G(x i * ), γ and δ are empirical parameters; The MCMC pre-sampling of the shielded structure function function specifically comprises the following steps: Step 1: Randomly generate an n-dimensional initial variable x1=[x1 (1) ,…,x1 (i) ,…,x1 (n) ]∈F; Step 2: The i-th dimension variable x obtained by the k-th round of sampling k (i) Construct a normal proposal distribution φ i (·; x k (i) ), draw samples x from the normal proposal distribution i ′; Step 3: Calculate the transfer kernel: Step 4: Randomly draw a sample U from a uniform distribution [0,1] i , and order Step 5: Repeat steps 2 to 4 until n dimensions are completed Sampling, determine the k+1th round of generated sample x k+1 for: Step 6: Repeat steps 2 to 5 until the required number of samples are drawn; The parameter "F" represents the failure domain of the structure; The parameter values ​​of the importance sampling probability density function include mean and covariance matrix; The mean is determined based on the MCMC pre-sampled samples, and the mean is calculated as: in, is the mean, x k is the sample pre-sampled by the MCMC; The covariance matrix is ​​determined based on the MCMC pre-sampled samples and the mean, specifically: A new coordinate system is constructed with the direction indicated by the mean as the coordinate axis, and the variance in the direction indicated by the mean is calculated based on the MCMC pre-sampled samples, and the covariance matrix in the original coordinate system is obtained using the transformation matrix; the covariance matrix is ​​calculated as: in, is the variance in the direction of the mean, is the covariance matrix; α is the transformation matrix that transforms the sample from the original coordinate system to the new coordinate system; α T is the transpose of matrix α, T is the transpose symbol, that is α1 is the direction from the origin of the space to the mean point The unit vector of α1 T is the transpose of column vector α1; I n is the n-dimensional identity matrix.

2. The multi-design point reliability analysis method for bridge structure according to claim 1 is characterized in that: The shielded structure function is: G j (x)=G(x)+ΣB i (x) Among them, G j (x) is the shielded structure function function of the currently selected design point, G(x) is the original structure function function, B i (x) is the shielded design point The corresponding additional functions.

3. The multi-design point reliability analysis method for bridge structure according to claim 2 is characterized in that: When the additional function B i (x) When B i The first and second derivatives of (x) are:

4. The multi-design point reliability analysis method for bridge structure according to claim 3 is characterized in that: The determination of the structural failure probability corresponding to the currently selected design point is specifically: Among them, G j (x) means the structure function after shielding; f(x) is the original probability density function, h(x) is the importance sampling probability density function, E h (·) is the expectation corresponding to h(x); I f (·) is an indicative function, which is 1 when x is in the failure domain F and 0 in other cases.

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