A bridge structure reliability evaluation method based on a Gaussian mixture model

Abstract

The application relates to a bridge structure reliability evaluation method based on a Gaussian mixture model and belongs to the technical field of bridge structure reliability analysis. The method obtains random input variables of a bridge structure, an original joint probability density function and a limit state function, and establishes a failure probability integral expression; a parameterized important sampling proposal distribution is constructed by using a Gaussian mixture model, and an intermediate threshold sequence is constructed by using a cross-entropy criterion, so that the proposal distribution is gradually pushed to the vicinity of a failure domain of the bridge structure; the external sample weight of each round of training samples is calculated, and an EM algorithm with the external weight is used to update the mixed weight, the mean vector and the covariance matrix of the Gaussian mixture model; in the final important sampling stage, the learned proposal distribution and the original distribution are combined to form a defensive mixed proposal distribution, the failure probability is estimated by using a likelihood ratio correction, and reliability indexes, a standard error, a relative variation coefficient, an effective sample number and a risk level are output.

Description

Technical Field

[0001] This invention belongs to the field of bridge structure reliability analysis technology, specifically relating to a bridge structure reliability assessment method based on a Gaussian mixture model. Background Technology

[0002] During service, bridge structures are affected by factors such as vehicle loads, wind loads, temperature effects, material dispersion, geometric errors, component degradation, and uncertainties in boundary conditions, resulting in significant randomness in their structural response and safety margin. Bridge reliability assessment typically requires estimating the probability of structural failure under random input conditions and further converting it into reliability indices to provide a basis for inspection, reinforcement, load limiting, and operation and maintenance.

[0003] Existing structural reliability methods include first-order second-order reliability methods, direct Monte Carlo simulation, surrogate model methods, and variance reduction methods such as importance sampling and subset simulation. First-order second-order reliability methods rely on local linearization or quadratic approximation near the design point, and are easily affected by the geometry of the limit state surface in high-dimensional, strongly nonlinear, multi-failure-mode, or implicit finite element model scenarios. Although direct Monte Carlo simulation has universality, it requires a large number of samples when the failure probability is very small, which is difficult to meet the computational efficiency requirements of complex bridge models.

[0004] Surrogate modeling methods can reduce the number of calls to high-fidelity structural models. Methods such as Kriging, radial basis functions, neural networks, and support vector machines have been used in reliability analysis, but their accuracy depends on the sample layout near the limit state surface and surrogate error control. When a bridge structure has multiple failure regions or potential design points, insufficient local fitting may lead to misjudgment of the failure domain.

[0005] Importance sampling concentrates samples in regions that contribute significantly to the failure probability by constructing a proposal distribution, and uses likelihood ratios to ensure that the estimation is unbiased or nearly unbiased. The theoretically optimal proposal distribution is the conditional distribution of the original distribution over the failure domain, but this distribution depends on unknown failure probabilities and is difficult to obtain directly. Existing techniques include constructing proposal distributions around design points, optimizing proposal distributions using cross-entropy, or combining Gaussian mixture models with reliability analysis; for example, existing bridge multi-design-point reliability analysis techniques typically require first screening or determining multiple design points, and then conducting importance sampling around these design points; some studies also combine cross-entropy importance sampling with surrogate models and Gaussian mixture models to handle structural reliability problems.

[0006] The above-mentioned scheme still has the following shortcomings: First, relying on a single design point or a single Gaussian proposal distribution makes it difficult to simultaneously cover the multimodal failure domain formed by the coupling of multiple components and multiple failure modes in bridges; second, if the proposal distribution is trained directly in the real failure domain, there are insufficient failure samples under low failure probability conditions, and parameter training is prone to instability; third, the learned proposal distribution may be over-concentrated in local areas, resulting in insufficient tail coverage of the original distribution, which in turn leads to likelihood ratio weight degradation, variance inflation, and fluctuations in estimation results. Therefore, a reliability assessment method that can adaptively learn for complex bridge failure domains, cover multiple modes, and has tail robustness is needed. Summary of the Invention

[0007] In view of this, the purpose of this invention is to provide a bridge structure reliability assessment method based on a Gaussian mixture model. This method transforms the estimation of bridge structure failure probability into an important sampling problem. Using a Gaussian mixture model as a trainable family of proposal distributions, it gradually approximates the optimal failure condition distribution through the cross-entropy principle and intermediate threshold sequences, and updates the mixture distribution parameters using an EM algorithm with external sample weights. To improve tail coverage, this invention constructs a defensive mixture proposal distribution by combining the learned Gaussian mixture proposal distribution with the original distribution in the final estimation stage, thereby suppressing likelihood ratio weight explosion and improving estimation stability.

[0008] To achieve the above objectives, the present invention provides the following technical solution: A bridge structural reliability assessment method based on a Gaussian mixture model includes the following steps: S1. Obtain the uncertainty input random vector of the bridge structure. Original joint probability density function and the limit state function of bridge structure and with As a structural failure condition; S2, Construction includes Gaussian mixture proposal distribution of Gaussian components. and initialize its mixture weights, mean vector and covariance matrix; S3. In the adaptive training round, the current Gaussian mixture proposal distribution is used as the reference distribution to generate training samples. The preset bridge structure response model is called to calculate the limit state value of each training sample, and an intermediate threshold that gradually approaches the failure boundary is constructed based on the limit state value. S4. Calculate the external sample weights of the training samples based on the intermediate threshold, the original joint probability density function, and the current reference distribution density, and normalize the external sample weights. S5. Update the mixture weights, mean vector and covariance matrix of the Gaussian mixture proposal distribution using the EM algorithm with external sample weights until the intermediate threshold reaches the preset termination condition. S6. Combine the trained Gaussian mixture proposal distribution with the original joint probability density according to a preset ratio to form a defensive mixture proposal distribution. S7. Independent importance sampling is performed based on the defensive hybrid proposal distribution. The failure probability of the bridge structure is calculated using the likelihood ratio weight, and the reliability index of the bridge structure is obtained based on the failure probability.

[0009] As a further preferred embodiment of the present invention, the intermediate threshold in S3 is determined based on a preset quantile of the limit state values ​​of the training samples, and when the bridge failure condition is... When the intermediate threshold is reached, it progresses from the safety domain to 0 according to the training rounds, and adaptive training stops when the intermediate threshold is less than or equal to 0 or when the change in consecutive rounds is less than the preset tolerance.

[0010] As a further preferred embodiment of the present invention, in S5, in the EM algorithm with external sample weights, the sample weights are first calculated. Belongs to the Posterior responsibility of each Gaussian component Then based on , , and Update the parameters of the Gaussian mixture model, where For the first The mixing weights of Gaussian components, For the first The mean vector of Gaussian components, For the first The covariance matrix of Gaussian components, It is a multivariate Gaussian probability density function.

[0011] As a further preferred embodiment of the present invention, the defensive hybrid proposal distribution described in S6 is represented as follows: ,in A defense mixing ratio greater than 0 and less than 1, such that This is to reduce the risk of important sampling weight degradation and weight explosion.

[0012] As a further preferred embodiment of the present invention, the bridge structure failure probability in S7 is calculated according to... Calculation, where The final sample is drawn independently from the defensive mixed proposal distribution. This represents the final sample size; reliability indicators are calculated as follows: =- ( ) calculate, where It is the inverse function of the standard normal distribution function.

[0013] As a further preferred embodiment of the present invention, each Gaussian component of the Gaussian mixture proposal distribution described in S2 is pre-mapped to a specific component of the bridge structure, and the initial mixing weight of each Gaussian component is set according to the preset contribution ratio of each component in the bridge system; in the EM algorithm update process with external sample weights in S5, the update of the mean vector of each Gaussian component is constrained by the mechanical transmission path between bridge components. The mechanical transmission path constraint is as follows: when the limit state value corresponding to the i-th component is less than the intermediate threshold of the current round, the load transmission ratio between the i-th component and the j-th component is used as the constraint coefficient, and the mean vector of the Gaussian component corresponding to the j-th component is shifted in the direction of decreasing limit state function value. The load transmission ratio is predetermined based on the mechanical analysis of the bridge structure; after completing the independent importance sampling based on the defensive mixture proposal distribution in S7 and obtaining the overall reliability index, the component reliability index corresponding to each Gaussian component is weighted and combined according to the final mixing weight obtained after training of each Gaussian component, and the system reliability index of the bridge structure is output.

[0014] As a further preferred embodiment of the present invention, during the adaptive training process from S3 to S5, the changes in the mean vector and the changes in the mixing weight of each Gaussian component of the Gaussian mixture proposal distribution are recorded after each training round to form the parameter evolution trajectory of each Gaussian component. Based on the parameter evolution trajectory, potential damage modes of the bridge structure are identified: when the mean vector of a certain Gaussian component continuously shifts in the same direction in multiple consecutive training rounds and the cumulative shift exceeds a preset damage judgment threshold, the failure mode corresponding to that Gaussian component is determined to be the dominant failure mode; when the mean vectors of two Gaussian components approach each other in multiple consecutive training rounds and the final distance is less than a preset merging judgment threshold, a coupling relationship is determined between the failure modes corresponding to the two Gaussian components; in S6, the defensive mixing ratio of the defensive mixing proposal distribution is dynamically set according to the identified dominant failure mode and coupling relationship: when a dominant failure mode is identified, the mixing ratio of the learned Gaussian mixture proposal distribution is increased to enhance the sampling focus on that failure mode; when a coupling relationship is identified between failure modes, the mixing ratio of the Gaussian mixture proposal distribution is decreased to increase the coverage of the original distribution over the transition region of the failure domain.

[0015] As a further preferred embodiment of the present invention, the uncertain input random vector in S1 is divided into three levels according to the spatial scale of the bridge structure: material scale parameters, component scale parameters, and system scale parameters. In S2, each Gaussian component of the Gaussian mixture proposal distribution corresponds to one of the three scale levels, and the covariance matrix of the Gaussian components corresponding to different scale levels adopts different structured forms: the covariance matrix of the Gaussian components corresponding to the material scale parameters adopts a diagonal form, the covariance matrix of the Gaussian components corresponding to the component scale parameters adopts a block diagonal form, and the covariance matrix of the Gaussian components corresponding to the system scale parameters adopts a complete matrix form. In S5, the EM algorithm with external sample weights alternately updates the parameters between different scale levels: first, the Gaussian components corresponding to the system scale parameters are fixed, and the Gaussian components corresponding to the material scale parameters and component scale parameters are updated; then, the Gaussian components corresponding to the material scale parameters and component scale parameters are fixed, and the Gaussian components corresponding to the system scale parameters are updated; the above alternating update process is repeated until the Gaussian component parameters of each scale level converge.

[0016] As a further preferred embodiment of the present invention, in the adaptive training process from S3 to S5, each training round is assigned a physical timescale corresponding to the actual service time of the bridge, and the rate of change of the mean vector of each Gaussian component with the training round is recorded as the degradation rate of the component corresponding to that Gaussian component under the physical timescale; when the mean vector of a certain Gaussian component continuously shifts in the same direction in multiple consecutive training rounds, the ratio of the cumulative shift of the mean vector to the degradation rate is used as the estimated remaining life of the component corresponding to that Gaussian component; when the eigenvalue of the covariance matrix of a certain Gaussian component continuously increases in multiple consecutive training rounds and the increase ratio exceeds a preset uncertainty inflation threshold, it is determined that the damage of the component corresponding to that Gaussian component is accumulating at an accelerated rate, and the estimated remaining life is corrected by the rate of increase of the eigenvalue of the covariance matrix; the bridge structural reliability index output in S7 includes the probability distribution information of the remaining life of each component at the current service time.

[0017] As a further preferred embodiment of the present invention, in S1, each random variable in the uncertain input random vector is pre-divided into several component groups according to the spatial connection relationship of the bridge components, and the mechanical transmission path between any two component groups is pre-determined by the finite element model of the bridge structure; in S2, the Gaussian mixture proposal distribution, in addition to the mixture weights, mean vector, and covariance matrix, also includes an inter-component coupling matrix. Each element of the coupling matrix represents the mechanical influence coefficient between the component groups corresponding to the two Gaussian components. The initial value of the mechanical influence coefficient is determined by the load transmission ratio between the corresponding component groups in the finite element model of the bridge structure; in S5, the EM algorithm with external sample weights updates the mixture weights, mean vector, and covariance matrix of each Gaussian component in each update, and also updates the value of the coupling matrix synchronously; in S7, the failure probability of the bridge structure is the joint failure probability obtained by propagating the failure probabilities corresponding to each Gaussian component through the updated coupling matrix. The joint failure probability includes the contribution of the mechanical coupling effect between components to the failure of the system.

[0018] The beneficial effects of this invention are as follows: (1) It can describe multiple failure regions simultaneously through multiple Gaussian components, and is suitable for multi-modal failure domains formed by multiple components, multiple load effects or multiple failure modes in bridge structures.

[0019] (2) The training objective is advanced step by step through the intermediate threshold sequence to avoid the problem of insufficient failure samples when training directly in the real low failure probability region.

[0020] (3) EM update with external sample weights unifies cross-entropy target, likelihood ratio correction and Gaussian mixture parameter training. The parameter update has clear calculation steps, which is convenient for engineering implementation.

[0021] (4) Defensive hybrid proposal distribution retains the global coverage capability of the original distribution and can reduce the risk of weight degradation, weight explosion and variance inflation.

[0022] (5) It can output evaluation quantities such as failure probability, reliability index, relative coefficient of variation and effective sample number, providing quantitative basis for bridge operation and maintenance decisions. Attached Figure Description

[0023] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 This is a flowchart of the bridge structure reliability assessment based on GMM-IS in an embodiment of the present invention.

[0024] Figure 2 This is a block diagram of the bridge structure reliability assessment system module in an embodiment of the present invention. Detailed Implementation

[0025] The specific embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the following embodiments are used to illustrate the technical solutions of the present invention, and are not intended to limit the scope of protection of the present invention.

[0026] Example 1 like Figures 1-2 As shown, this invention discloses a bridge structure reliability assessment method based on a Gaussian mixture model.

[0027] (I) Stochastic Models and Failure Probability Expressions for Bridge Structures Assume the uncertainty of the bridge structure is input by a random vector. Its original joint probability density function is Random vector This may include vehicle load amplitude, vehicle lateral distribution coefficient, concrete strength, steel reinforcement or steel strand strength, cross-sectional geometry, elastic modulus, temperature gradient, support stiffness, corrosion or fatigue degradation parameters, etc.

[0028] Define the limit state function of bridge structure ,when The time is a safe state, when The current state is a failure state. For multiple failure modes, it can be set as follows: ,in These correspond to modes such as load-bearing capacity, deformation limit, fatigue damage, local component failure, or system failure. The structural failure probability is expressed as: in For indicator functions. Direct Monte Carlo simulation is performed by... Sampling estimation ,exist Small failure probabilities require a large number of samples. This invention improves the efficiency of small failure probability estimation by constructing an adaptive importance sampling proposal distribution.

[0029] (ii) Gaussian Mixture Proposed Distribution This invention uses a Gaussian mixture model to represent the distribution of important sampling proposals, enabling the proposal distribution to simultaneously cover multiple failure regions. The Gaussian mixture proposal distribution is represented as follows: in The number of Gaussian components. and ; For the first The mean vector of Gaussian components; For the first The covariance matrix of Gaussian components; It is a multivariate Gaussian density function; = These are the parameters to be trained.

[0030] (III) Cross-entropy adaptive threshold training Ideally, the important sampling proposal distribution should approximate the conditional distribution of the original distribution over the failure domain. Since the true failure probability is unknown, this invention employs the cross-entropy criterion to transform the proposal distribution construction into a weighted maximum likelihood training problem of a parameterized distribution, and gradually approximates the failure domain through an intermediate threshold sequence.

[0031] In the During the training round, the current Gaussian mixture proposal distribution is used as the reference distribution. From the current reference distribution... Generate training samples {x_i} calls the bridge finite element model, analytical response model, or surrogate response model to calculate... .according to The preset quantile determines the intermediate threshold. When the failure condition is hour, Progressing gradually from the safety side towards 0; if If the threshold value is ≤0 or the threshold change is less than the preset tolerance in consecutive rounds, then adaptive training will stop.

[0032] For the For each training sample, calculate the unnormalized weights of the external samples: in For the first training samples, For the first Intermediate threshold of the wheel For indicator functions, For the first The reference distribution density used to generate training samples is used in rounds; and the normalized weights are further calculated. The aforementioned weights unify the original distribution, the current reference distribution, and intermediate events into the same training objective, enabling high-weighted samples to have a greater impact on subsequent parameter updates.

[0033] (iv) EM parameter update with external weights Since the log-likelihood of a Gaussian mixture model includes a mixture summation term, direct optimization is quite difficult. This invention introduces latent variables representing sample component affiliations and uses the EM algorithm with external sample weights to update the Gaussian mixture parameters.

[0034] In step E, the samples are calculated. Belongs to the Posterior responsibility of each Gaussian component: In the M-step, normalized external weights are incorporated. Calculate the first Effective sample weights of each component And update the parameters: To avoid singularity or excessive shrinkage of the covariance matrix, this invention employs covariance regularization: in The regularization coefficient is . This is the identity matrix. Optionally, when the effective sample weights of a certain Gaussian component are lower than a preset threshold, the component is reinitialized, merged, or deleted to further improve training stability.

[0035] (v) Defensive Hybrid Proposal Distribution and Final Estimation In the final critical sampling stage, if only the trained Gaussian mixture proposal distribution is used, the original distribution tail coverage may be insufficient due to excessive concentration of samples in local failure regions. Therefore, this invention constructs a defensive mixture proposal distribution: because Therefore, the likelihood is higher than This property can suppress extreme weights and improve the stability of small failure probability estimation.

[0036] from The final sample is generated independently in the middle. }, call the bridge structural response model to calculate The failure probability is estimated using the following formula: Further calculation of reliability indicators ,in It is the inverse function of the standard normal distribution function. It can also be determined based on the final sample variables. Calculate the standard error, relative coefficient of variation, and effective sample size (ESS) to evaluate the accuracy of the estimation and the degree of weight balance.

[0037] (vi) Algorithm Flow One embodiment of the present invention can be performed according to the following steps: S101, Establish a random input model for the bridge structure and determine the original density. Sampler and limit state function .

[0038] S102, Set the number of Gaussian mixture components Number of training samples per round Final sample size Elite ratio Defense Mixing Ratio Regularization coefficient And the maximum number of iterations, and initialize the Gaussian mixture parameters. .

[0039] S103, from the current reference distribution Generate training samples and calculate the bridge structural response and limit state value corresponding to each training sample.

[0040] S104, Construct an intermediate threshold based on the quantiles of the limit state values. Calculate and normalize the weights of the external samples.

[0041] S105 uses the EM algorithm with external weights to update the Gaussian mixture proposal distribution parameters and performs covariance regularization.

[0042] S106, Determine If the failure boundary is not reached, the updated distribution will be used as the reference distribution for the next round of training; if it is reached, the final estimation will proceed.

[0043] S107, Constructing a defensive hybrid proposal distribution Independent importance sampling is performed to output failure probability, reliability index, standard error, relative coefficient of variation, ESS and risk level.

[0044] (vii) Parameter setting example In engineering implementation, Gaussian mixture components The number of training samples per round can be 3 to 8. The cost of calling up the bridge model can be set from 500 to 5000; elite ratio A value of 0.05 to 0.30 is acceptable; Defense mixing ratio It can be taken as 0.01 to 0.20; covariance regularization coefficient The values ​​can range from 10^-8 to 10^-4. These values ​​are merely examples and do not constitute a limitation on the scope of the claims.

[0045] For non-Gaussian random inputs, the original density can be used directly in the physical variable space. The input variables can be trained with the sampler; alternatively, the input variables can be transformed to the standard normal space first, and then a Gaussian mixture proposal distribution can be constructed in the transformed space, with the corresponding transformation relationship included in the weight calculation.

[0046] (viii) Bridge Examples Taking the reliability assessment of a bridge structure as an example, vehicle load factor, material strength, cross-sectional dimensions, bearing stiffness, temperature effect, and degradation parameters can be used as random inputs; the bending moment, shear force, displacement, or stress response of key sections can be calculated through a finite element model; the ultimate limit state function of bearing capacity, the serviceability limit state function, and the fatigue damage limit state function can be established respectively, and the minimum value of multiple limit state functions can be taken as the ultimate limit state function of the system.

[0047] During adaptive training, Gaussian mixture components can cluster near different components or different failure modes, thus forming a joint coverage of multiple failure regions. Finally, a defensive mixture proposal distribution is used for independent sampling, outputting the bridge system failure probability, reliability index, and estimation accuracy evaluation. If the relative coefficient of variation exceeds a preset threshold, the final sample size can be increased or the proposal distribution can be retrained; if the reliability index is lower than the operation and maintenance threshold, suggestions for inspection, reinforcement, or load limiting can be generated.

[0048] Numerical examples in the appendix demonstrate that this type of GMM-IS scheme is applicable to bimodal nonlinear failure domains, Waarts multi-design point functions, non-Gaussian input distributions, and high-dimensional linear reliability problems, indicating that the present invention can maintain good estimation stability under multimodal, nonlinear, non-Gaussian, and medium-to-high-dimensional random space conditions.

[0049] (ix) Reliability Assessment System This invention also discloses a bridge structure reliability assessment system based on a Gaussian mixture model, comprising: a random input modeling module for acquiring random input variables, the original joint probability density function, and the limit state function of the bridge structure; an intermediate threshold construction module for generating intermediate thresholds that progressively approximate the failure domain based on the limit state values ​​of training samples; a weighted EM training module for updating the Gaussian mixture proposal distribution parameters according to the weights of external samples; a defensive mixture proposal module for mixing the trained Gaussian mixture proposal distribution with the original distribution; and a reliability output module for performing final important sampling and outputting the failure probability, reliability index, and accuracy evaluation index.

[0050] Example 2 Based on Example 1, this embodiment takes into account the characteristics of a bridge structure composed of multiple components. It maps each Gaussian component of the proposed Gaussian mixture distribution to each component of the bridge, and introduces mechanical transmission path constraints between components during the parameter update process, ultimately outputting the system reliability index.

[0051] Step 1: Component-Gaussian component mapping and initial weight setting Suppose the bridge structure consists of N key components, and let the number of Gaussian mixture components be K = N. The k-th Gaussian component is pre-mapped to the k-th component. The initial mixing weight of each Gaussian component is set according to the preset contribution ratio of each component in the bridge system: in The preset contribution ratio for the k-th component can be determined based on the component's proportion of the load-bearing capacity in the system, its stiffness contribution, or an importance coefficient based on structural mechanics analysis, satisfying the following conditions: .

[0052] Mean vector of each Gaussian component Initialize to the nominal value (such as design value or mean) of the random input parameter of the corresponding component, covariance matrix Initialize the initial variance estimate of the random input parameters of the corresponding component.

[0053] Step 2: Mean Vector Update under Mechanical Transfer Path Constraints In the S5 EM algorithm update process with external sample weights, the mean vector update of each Gaussian component is not only based on the standard EM formula, but also constrained by the mechanical transmission path between bridge components.

[0054] For the i-th component, when its corresponding limit state value Less than the median threshold of the current round When (i.e., the component is close to or has entered a failure state), define the load transfer ratio between the component and the j-th component. . Based on the pre-determined structural mechanics analysis of the bridge, for example, the proportion of load transferred from the i-th component to the j-th component when the i-th component experiences load capacity degradation can be calculated using a finite element model, satisfying the following conditions: .

[0055] In step M, the mean vector update of the Gaussian component corresponding to the j-th component is subject to a constraint offset: in: The maximum likelihood estimate of the mean vector of the j-th Gaussian component given by the standard EM formula; The extreme state value in the current round is less than the intermediate threshold. A collection of components; Let be the offset direction vector of the mean vector of the j-th component when the i-th component fails, as determined by mechanical analysis. This direction causes the limit state function value of the j-th component to decrease (i.e., shift towards the failure direction). The constraint coefficient (0 < η ≤ 1) is used to control the strength of the constraint offset.

[0056] The offset direction vector The specific values ​​are predetermined based on the structural mechanics analysis of the bridge: when the failure of the i-th component leads to load redistribution, the direction of the increase in the internal force on the j-th component is... The direction, its magnitude and the load transfer ratio It is related to the degree of deviation of the limit state value of the failed component.

[0057] Step 3: Calculation of System Reliability Indicators After completing the independent importance sampling based on the defensive hybrid proposal distribution in S7 and obtaining the reliability indices of each component, the final hybrid weights are obtained by training on each Gaussian component. The reliability indices of each component are weighted and combined: in The reliability index is the one corresponding to the k-th component (estimated from the failure region covered by the Gaussian component corresponding to that component). It serves as a system reliability indicator for bridge structures.

[0058] Reliability indicators of each component The following method can be used to obtain the final sample based on the defensive hybrid proposal distribution: calculate the limit state value of each component for each sample, and statistically analyze the failure probability of each component. Then by Calculated.

[0059] By mapping Gaussian components to bridge components one by one, the training process of the proposal distribution has a clear physical meaning; the mechanical transmission path constraint ensures that when a component is close to failure, the proposal distribution of other components with load transmission relationship with it can be adjusted synchronously in the direction of failure, reflecting the mechanical coupling characteristics of the bridge structure as a whole system; the final output system reliability index integrates the failure probability contribution of each component, and more accurately reflects the overall safety status of the bridge.

[0060] Example 3 Based on Example 1, this embodiment records the parameter evolution trajectory of each Gaussian component during adaptive training, thereby identifying potential damage modes of the bridge structure and dynamically adjusting the mixing ratio of the defensive hybrid proposal distribution.

[0061] Step 1: Recording the parameter evolution trajectory During the adaptive training process from S3 to S5, the changes in the mean vector and the changes in the mixture weights of each Gaussian component of the Gaussian mixture proposal distribution are recorded after each training round t.

[0062] The change in the mean vector of the k-th Gaussian component in round t is defined as: The change in mixed weights is defined as: This forms the parameter evolution trajectory of the k-th Gaussian component: Where T is the total number of rounds of adaptive training.

[0063] Step 2: Dominant Failure Mode Identification Potential damage modes of bridge structures are identified based on parameter evolution trajectories. When the mean vector of a certain Gaussian component continuously shifts in the same direction over M consecutive training rounds, and the cumulative shift exceeds a preset damage assessment threshold... When the failure mode corresponding to the Gaussian component is determined to be the dominant failure mode, the failure mode is determined to be the dominant failure mode.

[0064] The offset accumulation is defined as: in For the Euclidean norm. When And each When the directional angle is less than a preset angle threshold (e.g., 30°), the failure mode corresponding to the kth Gaussian component is determined to be the dominant failure mode.

[0065] Step 3: Failure Mode Coupling Relationship Identification When the mean vectors of two Gaussian components approach each other in multiple consecutive training rounds and the final distance is less than a preset merging threshold. When the failure modes corresponding to the two Gaussian components are determined to be coupled, it is determined that there is a coupling relationship between them.

[0066] The distance between the p-th and q-th Gaussian components in round t is defined as: If in M ​​consecutive rounds Monotonically decreasing, and eventually If so, it is determined that there is a coupling relationship between the p-th and q-th failure modes.

[0067] Step 4: Dynamically set the defense mixing ratio The defensive hybrid proportion of the defensive hybrid proposal distribution in S6 Dynamic settings are configured based on the identified dominant failure modes and coupling relationships.

[0068] The defensive hybrid proposal distribution is expressed as: in The Gaussian mixture proposal distribution obtained during training, This is the original joint probability density function.

[0069] The rules for dynamic setting are as follows: (1) When R dominant failure modes are identified, increase the proportion of the Gaussian mixture proposal distribution obtained from the learning: in The baseline defense mixing ratio is (e.g., 0.10). This is the attenuation coefficient (e.g., 0.3). This represents the number of dominant failure modes. This setting ensures that the more significant the dominant failure mode, the more focused the proposed distribution will be on the failure region corresponding to that mode.

[0070] (2) When coupling relationships are identified between failure modes, reduce the proportion of the Gaussian mixture proposal distribution in the mixture: Where C is the number of identified coupling pairs. This is the coupling enhancement factor (e.g., 0.05). The upper limit is set to 0.5 to prevent the proposal distribution from being overly dispersed.

[0071] When both dominant failure modes and coupling relationships exist simultaneously, both rules should be applied. A larger value is used to ensure that the proposed distribution focuses on the dominant failure mode while retaining coverage of the transition region of the failure domain.

[0072] By recording the parameter evolution trajectory, dynamic identification of potential damage modes of bridge structures is achieved without the need for additional damage detection equipment; the identification of dominant failure modes helps to concentrate sampling resources on the most critical risk areas; the identification of coupling relationships avoids excessive localization of the proposal distribution when multiple failure modes influence each other; the dynamic setting of the defense ratio enables the proposal distribution to adaptively adjust according to the structural damage state, taking into account both sampling efficiency and estimation robustness.

[0073] Example 4 Based on Example 1, this embodiment addresses the characteristic that the uncertainty parameters of bridge structures span multiple spatial scales by layering the random input vector by scale, adopting different covariance matrix structures for the Gaussian components corresponding to each scale, and implementing an alternating update strategy in EM updates.

[0074] Step 1: Multi-scale partitioning of random input vectors Input the uncertainty in S1 into a random vector Based on the spatial scale of the bridge structure, it is divided into three levels: Material dimensional parameters This includes material constitutive parameters such as concrete strength, steel yield strength, elastic modulus, and Poisson's ratio. Component dimensional parameters This includes component-level parameters such as cross-sectional dimensions, reinforcement ratio, prestress loss, component stiffness, and local defects. System Scale Parameters This includes system-level parameters such as vehicle load, wind load, temperature effect, support settlement, and overall geometric alignment.

[0075] The parameter vectors at three scales satisfy Their respective dimensions are , , Total dimensions .

[0076] Step 2: Structured Form of the Covariance Matrix Each Gaussian component of the Gaussian mixture proposal distribution corresponds to one of the three scale levels. Let the k-th Gaussian component correspond to the scale level. ( = 1,2,3), and their covariance matrices adopt different structured forms: (1) Material dimensional parameters ( = 1) The corresponding Gaussian components, the covariance matrix is ​​in diagonal form: This form assumes that the parameters at the material scale are independent of each other, and the number of parameters is... This is beneficial for stable estimation under limited samples.

[0077] (2) Component dimensional parameters ( = 2) The corresponding Gaussian components, the covariance matrix is ​​in block diagonal form: Each block The covariance matrix corresponds to the parameters (such as cross-sectional width, height, reinforcement ratio, etc.) within the same component. The covariance between different components is set to zero, reflecting the assumption that the parameters within the component are related but the components are independent.

[0078] (3) System scale parameters ( = 3) The covariance matrix of the corresponding Gaussian components is in full matrix form: The complete matrix form allows for arbitrary correlations between parameters at the system scale (such as vehicle load and temperature effects) to capture the complete correlation structure of system-level uncertainties.

[0079] Step 3: Alternating Update EM Algorithm In the EM algorithm with external sample weights in S5, parameters are updated alternately between different scale levels. Let the parameters for the current round be... .

[0080] Sub-step 3.1: Fix the Gaussian components corresponding to the system scale parameters (i.e., and (Keep it unchanged), update the Gaussian components corresponding to the material scale parameters and component scale parameters.

[0081] In step E, the posterior responsibility of each sample belonging to each Gaussian component is calculated. (The formula is the same as described above). In the M-step, only the scale is considered. = 1 and = 2 corresponds to the following Gaussian component update parameters: in For the i-th sample, the corresponding scale The parameter subvector, Let be the external sample weights of the i-th sample.

[0082] For material scale ( = 1), Force after update Off-diagonal elements are set to zero to preserve diagonal form. For component dimensions ( = 2), after the update, force the covariance between different component blocks to be set to zero in order to maintain the diagonal form of the blocks.

[0083] Sub-step 3.2: Fix the Gaussian components corresponding to the material dimensional parameters and component dimensional parameters (i.e., , , , (Keep it unchanged), update the Gaussian components corresponding to the system scale parameters.

[0084] Only for scale The Gaussian components corresponding to = 3 are updated according to the standard EM formula. and (Complete matrix form, without structural constraints).

[0085] Sub-step 3.3: Repeat sub-steps 3.1 and 3.2 until the Gaussian component parameters at each scale level converge. The convergence criterion is: after two consecutive alternating updates, the relative change of all parameters is less than the preset tolerance. (e.g., 10^-4).

[0086] Multi-scale partitioning allows for the separate handling of uncertain parameters with different physical properties, avoiding scale confusion issues in unified modeling. Different scales employ different covariance matrix structures, achieving a reasonable balance between model flexibility and parameter estimation stability. Material-scale parameters, which are highly independent, are reduced by using a diagonal form. Component-scale parameters, which are correlated with the component itself, are partitioned diagonally. System-scale parameters, which have complex correlation structures, are represented by a complete matrix. The alternating update strategy reduces the difficulty of simultaneous optimization in high-dimensional parameter spaces and improves the convergence stability of the EM algorithm.

[0087] Example 5 Based on Example 1, this embodiment introduces a physical time scale during the adaptive training process. It estimates the component degradation rate and remaining lifetime by recording the evolution rate of the mean vector of each Gaussian component, and corrects the lifetime estimate by utilizing the changes in the eigenvalues ​​of the covariance matrix.

[0088] Step 1: Correlation between training rounds and physical time scale During the adaptive training process from S3 to S5, each training round t is assigned a physical timescale corresponding to the actual service time of the bridge. . It can be determined in the following way: in The current service life of the bridge. The physical time step corresponds to each training round (e.g., one year per round). The physical time step can be flexibly set according to the bridge's degradation rate and the frequency of monitoring data updates.

[0089] Step 2: Degradation rate recording The rate of change of the mean vector of each Gaussian component with each training epoch is recorded as the degradation rate of the corresponding component on the physical time scale. The degradation rate of the k-th Gaussian component in epoch t is defined as: The physical meaning of degradation rate is: the offset of the mean vector per unit physical time, which reflects the evolution trend of the corresponding random parameters of the component during service.

[0090] To eliminate the impact of single-round fluctuations, a moving average degradation rate is used: Where M is the width of the sliding window (e.g., 5 slides).

[0091] Step 3: Preliminary estimate of remaining lifetime When the mean vector of a certain Gaussian component continuously shifts in the same direction in multiple consecutive training rounds, the ratio of the cumulative shift of the mean vector to the degradation rate is used as the estimated remaining lifetime of the component corresponding to that Gaussian component.

[0092] Let the mean vector of the k-th Gaussian component be derived from the first... Wheel starts along direction The cumulative offset up to the current round T is: The average degradation rate in this direction is: The preliminary estimate of the remaining life of this component is: in It is the magnitude of the critical mean vector corresponding to the k-th component (i.e., the magnitude of the mean vector when the component reaches its limit state). This represents the magnitude of the current mean vector.

[0093] Step 4: Lifetime Correction Based on Covariance Eigenvalues When the eigenvalues ​​of the covariance matrix of a certain Gaussian component continue to increase in multiple consecutive training rounds, it indicates that the uncertainty of the component parameters is expanding, reflecting the physical process of accelerated damage accumulation.

[0094] Let the covariance matrix of the k-th Gaussian component be... The j-th feature is Define the eigenvalue increment ratio: when ( When the preset uncertainty expansion threshold (e.g., 0.20) is reached, it is determined that the damage to the component is accumulating at an accelerated rate.

[0095] Define the average rate of increase of the eigenvalues ​​of the covariance matrix: in Let be the parameter dimension corresponding to the k-th Gaussian component.

[0096] The remaining lifetime estimate is corrected by the rate of increase of the eigenvalues ​​of the covariance matrix: in This is a correction factor (greater than 0), used to control the degree to which uncertainty expansion reduces the remaining lifetime. When Time (without accelerated degradation) ;when At that time, the remaining lifespan decreases exponentially.

[0097] Step 5: Output reliable metrics containing remaining lifetime information Bridge structural reliability indicators output in S7 It includes the probability distribution of the remaining lifetime of each component at the current service moment. Specifically, for each component, the output is a triplet: in: This is the estimated remaining life (in years) for the k-th component. The standard deviation of the remaining lifetime estimate is obtained from the uncertainty propagation of the degradation rate estimate; For the k-th component within the preset service life The probability of no failure occurring is determined by... Approximate estimation.

[0098] By linking the adaptive training process with physical time scales, the training process of Gaussian mixture models, which was originally just a numerical optimization tool, is endowed with the physical meaning of time evolution; the recording of degradation rates provides a data-driven basis for the long-term performance prediction of bridges; changes in covariance eigenvalues ​​serve as an indicator of accelerated damage accumulation, enabling dynamic correction of traditional remaining life estimates; and the output reliable index containing the probability distribution of remaining life provides richer information for bridge operation and maintenance decisions, supporting the formulation of risk-based preventive maintenance strategies.

[0099] Example 6 Based on Example 1, this embodiment groups random input variables according to the spatial connection relationship of components, introduces an inter-component coupling matrix in the Gaussian mixture proposal distribution, and updates the coupling matrix synchronously through the EM algorithm, finally outputting the joint failure probability considering the mechanical coupling effect between components.

[0100] Step 1: Component group division and determination of mechanical transfer path Input the uncertainty in S1 into a random vector The random variables in the model are divided into G component groups according to the spatial connection relationship of the bridge components. Each component group contains the relevant random parameters of one or more adjacent components. The mechanical transmission path between any two component groups is predetermined by the finite element model of the bridge structure.

[0101] Assume there is a mechanical transfer path between component group a and component group b, and define the load transfer ratio. The proportion of the load borne by component group a that is transferred to component group b satisfies the following condition. .

[0102] Step 2: Introduction and initialization of the coupling matrix Gaussian mixture proposals are distributed in the mixture weights Mean vector Covariance Matrix In addition, it includes a G × G component coupling matrix. Each element of the coupling matrix This represents the mechanical influence coefficient between the component groups corresponding to the a-th Gaussian component and the b-th Gaussian component.

[0103] The initial value of the mechanical influence coefficient is determined by the load transfer ratio between corresponding component groups in the finite element model of the bridge structure: That is, the off-diagonal elements of the initial coupling matrix are equal to the load transfer ratio, and the diagonal elements are zero (the influence of the component group on itself is already reflected in its own Gaussian components).

[0104] Step 3: Synchronous Update of Coupling Matrix In each round of the EM algorithm with external sample weights in S5, in addition to updating the mixture weights, mean vectors and covariance matrices of each Gaussian component, the values ​​of the coupling matrix are also updated simultaneously.

[0105] The coupling matrix is ​​updated based on the parameter changes of the corresponding Gaussian components of each component group: in The coupling update rate coefficient between the a-th and b-th component groups can be predetermined based on the mechanical transmission path length and stiffness ratio between the two component groups.

[0106] The updated coupling matrix elements are: And perform row normalization to ensure : in Keep it at zero.

[0107] Step 4: Calculation of Joint Failure Probability In S7, the failure probability of the bridge structure is not a simple summation of the failure probabilities of each component, but a joint failure probability obtained by propagating the failure probabilities corresponding to each Gaussian component through the updated coupling matrix.

[0108] First, based on the final samples drawn from the defensive hybrid proposal distribution, the marginal failure probability corresponding to each component group is estimated. ( ).

[0109] Secondly, construct the coupling failure propagation vector. : in To account for the equivalent failure probability of the k-th component group after the failure of other component groups is transmitted through mechanical coupling.

[0110] Ultimately, the combined failure probability of the bridge structure is: This formula combines the equivalent failure probabilities of each component group through a product, reflecting the serial logic of system failure: the failure of any component group leads to the failure of the entire system.

[0111] Joint failure probability This includes the contribution of the mechanical coupling effect between components to the system failure: when the coupling matrix... When the off-diagonal elements are large (i.e., load transfer between components is significant), the failure of a component will be transmitted through... The propagation term increases the equivalent failure probability of other component groups, thereby increasing the joint failure probability of the system.

[0112] Step 5: Sensitivity analysis of coupling effect The sensitivity of each component group to the combined failure probability of the system can be further calculated: Sensitivity It reflects the degree of influence of the change in the marginal failure probability of the kth component group on the joint failure probability of the system, and can be used to identify key component groups that play a controlling role in the safety of the system.

[0113] The division of components into groups provides the possibility of dimensionality reduction for large-scale bridge structures, avoiding the computational burden of modeling each component individually; the introduction of the coupling matrix enables the Gaussian mixture model to not only describe the randomness of each component group itself, but also characterize the mechanical coupling relationship between component groups; the synchronous update of the coupling matrix allows the coupling effect to be adaptively adjusted during the training process, reflecting the dynamic changes of the mechanical transmission path during the structural damage evolution; the joint failure probability is calculated through coupling propagation, which is more scientific and reasonable than simply adding failure probabilities or taking the maximum or minimum value, and can accurately reflect the actual contribution of mechanical coupling between components to the failure of the system; sensitivity analysis provides a quantitative basis for prioritizing bridge reinforcement.

[0114] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made to it in form and detail without departing from the scope defined by the claims of the present invention.

Claims

1. A bridge structural reliability assessment method based on a Gaussian mixture model, characterized in that, Includes the following steps: S1. Obtain the uncertainty input random vector of the bridge structure. Original joint probability density function and the limit state function of bridge structure and with As a structural failure condition; S2, Construction includes Gaussian mixture proposal distribution of Gaussian components. , and initialize its mixture weights, mean vector and covariance matrix; S3. In the adaptive training round, the current Gaussian mixture proposal distribution is used as the reference distribution to generate training samples. The preset bridge structure response model is called to calculate the limit state value of each training sample, and an intermediate threshold that gradually approaches the failure boundary is constructed based on the limit state value. S4. Calculate the external sample weights of the training samples based on the intermediate threshold, the original joint probability density function, and the current reference distribution density, and normalize the external sample weights. S5. Update the mixture weights, mean vector and covariance matrix of the Gaussian mixture proposal distribution using the EM algorithm with external sample weights until the intermediate threshold reaches the preset termination condition. S6. Combine the trained Gaussian mixture proposal distribution with the original joint probability density according to a preset ratio to form a defensive mixture proposal distribution. S7. Independent importance sampling is performed based on the defensive hybrid proposal distribution. The failure probability of the bridge structure is calculated using the likelihood ratio weight, and the reliability index of the bridge structure is obtained based on the failure probability.

2. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: The intermediate threshold mentioned in S3 is determined based on a preset quantile of the limit state values ​​of the training samples, and when the bridge failure condition is... When the intermediate threshold is reached, it progresses from the safety domain to 0 according to the training rounds, and adaptive training stops when the intermediate threshold is less than or equal to 0 or when the change in consecutive rounds is less than the preset tolerance.

3. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: In the EM algorithm with external sample weights in S5, the sample weights are first calculated. Belongs to the Posterior responsibility of each Gaussian component Based on , , and Update the parameters of the Gaussian mixture model, where For the first The mixing weights of Gaussian components, For the first The mean vector of Gaussian components, For the first The covariance matrix of Gaussian components, It is a multivariate Gaussian probability density function.

4. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: The defensive hybrid proposal distribution described in S6 is represented as follows: ,in A defense mixing ratio greater than 0 and less than 1, such that This is to reduce the risk of important sampling weight degradation and weight explosion.

5. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 4, characterized in that: Bridge structure failure probability in S7 is calculated as follows Calculation, where The final sample is drawn independently from the defensive mixed proposal distribution. This represents the final sample size; reliability indicators are calculated as follows: =- ( ) calculate, where It is the inverse function of the standard normal distribution function.

6. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: In S2, each Gaussian component of the Gaussian mixture proposal distribution is pre-mapped to a specific component of the bridge structure. The initial mixing weight of each Gaussian component is set according to the preset contribution ratio of each component in the bridge system. In S5, during the update process of the EM algorithm with external sample weights, the update of the mean vector of each Gaussian component is constrained by the mechanical transfer path between bridge components. The mechanical transfer path constraint is as follows: when the limit state value corresponding to the i-th component is less than the intermediate threshold of the current round, the load transfer ratio between the i-th component and the j-th component is used as the constraint coefficient, and the mean vector of the Gaussian component corresponding to the j-th component is shifted in the direction of decreasing limit state function value. The load transfer ratio is predetermined based on the mechanical analysis of the bridge structure. After completing the independent importance sampling based on the defensive mixture proposal distribution in S7 and obtaining the overall reliability index, the component reliability index corresponding to each Gaussian component is weighted and combined according to the final mixing weight obtained after training, and the system reliability index of the bridge structure is output.

7. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: During the adaptive training process from S3 to S5, the changes in the mean vector and mixing weight of each Gaussian component of the Gaussian mixture proposal distribution are recorded after each training round, forming the parameter evolution trajectory of each Gaussian component. Based on the parameter evolution trajectory, potential damage modes of the bridge structure are identified: when the mean vector of a certain Gaussian component continuously shifts in the same direction in multiple consecutive training rounds and the cumulative shift exceeds a preset damage judgment threshold, the failure mode corresponding to that Gaussian component is determined to be the dominant failure mode; when the mean vectors of two Gaussian components approach each other in multiple consecutive training rounds and the final distance is less than a preset merging judgment threshold, a coupling relationship is determined between the failure modes corresponding to the two Gaussian components; in S6, the defensive mixing ratio of the defensive mixing proposal distribution is dynamically set based on the identified dominant failure mode and coupling relationship: when a dominant failure mode is identified, the mixing ratio of the learned Gaussian mixture proposal distribution is increased to enhance the sampling focus on that failure mode; when a coupling relationship is identified between failure modes, the mixing ratio of the Gaussian mixture proposal distribution is decreased to increase the coverage of the original distribution over the transition region of the failure domain.

8. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: In S1, the uncertain input random vector is divided into three levels according to the spatial scale of the bridge structure: material scale parameters, component scale parameters, and system scale parameters. In S2, each Gaussian component of the Gaussian mixture proposal distribution corresponds to one of the three scale levels, and the covariance matrix of the Gaussian component corresponding to different scale levels adopts different structured forms: the covariance matrix of the Gaussian component corresponding to the material scale parameter adopts a diagonal form, the covariance matrix of the Gaussian component corresponding to the component scale parameter adopts a block diagonal form, and the covariance matrix of the Gaussian component corresponding to the system scale parameter adopts a complete matrix form. In S5, the EM algorithm with external sample weights alternately updates the parameters between different scale levels: first, fix the Gaussian component corresponding to the system scale parameter, and update the Gaussian components corresponding to the material scale parameter and the component scale parameter; then fix the Gaussian components corresponding to the material scale parameter and the component scale parameter, and update the Gaussian component corresponding to the system scale parameter; repeat the above alternating update process until the Gaussian component parameters of each scale level converge.

9. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: During the adaptive training process from S3 to S5, each training round is assigned a physical timescale corresponding to the actual service time of the bridge. The rate of change of the mean vector of each Gaussian component with each training round is recorded as the degradation rate of the component corresponding to that Gaussian component under the physical timescale. When the mean vector of a certain Gaussian component continuously shifts in the same direction in multiple consecutive training rounds, the ratio of the cumulative shift of the mean vector to the degradation rate is used as the estimated remaining life of the component corresponding to that Gaussian component. When the eigenvalue of the covariance matrix of a certain Gaussian component continuously increases in multiple consecutive training rounds and the increase rate exceeds a preset uncertainty inflation threshold, it is determined that the damage of the component corresponding to that Gaussian component is accumulating at an accelerated rate, and the estimated remaining life is corrected by the rate of increase of the eigenvalue of the covariance matrix. The bridge structural reliability index output in S7 includes the probability distribution information of the remaining life of each component at the current service time.

10. The bridge structure reliability assessment method based on Gaussian mixture model according to claim 1, characterized in that: In S1, the random variables in the uncertain input random vector are pre-divided into several component groups according to the spatial connection relationship of bridge components. The mechanical transmission path between any two component groups is pre-determined by the finite element model of the bridge structure. In S2, the Gaussian mixture proposal is distributed in addition to the mixture weights, mean vector, and covariance matrix, and also includes an inter-component coupling matrix. Each element of the coupling matrix represents the mechanical influence coefficient between the component groups corresponding to the two Gaussian components. The initial value of the mechanical influence coefficient is determined by the load transmission ratio between the corresponding component groups in the finite element model of the bridge structure. In S5, the EM algorithm with external sample weights updates the mixture weights, mean vector, and covariance matrix of each Gaussian component in each update, and also updates the value of the coupling matrix synchronously. In S7, the failure probability of the bridge structure is the joint failure probability obtained by propagating the failure probabilities corresponding to each Gaussian component through the updated coupling matrix. The joint failure probability includes the contribution of the mechanical coupling effect between components to the failure of the system.

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