Hierarchical Dynamic Bayesian Network-Based Fatigue Crack Propagation Modeling Considering Initial Defects
Abstract
:1. Introduction
2. Hierarchical Dynamic Bayesian Network for Fatigue Crack Propagation (FCP)
2.1. Fracture Mechanics-Based Multi-Source Uncertainty Analysis
2.2. Hierarchical Dynamic Bayesian Network for FCP of OSD System
2.3. Refined Exact Inference Algorithm for Dynamic Bayesian Network Updating
3. Implementation Details
3.1. Stochastic Traffic Flow Model
3.2. Multi-Scale Finite Element Analysis
3.3. Parameter Setups in Hierarchical DBN for FCP Analysis
4. Results and Discussion
4.1. Demonstration by Numerical Simulations
- (1)
- (2)
- For each combination of , FCP analysis is performed in a continuous space based on Equation (6). The comparison of the probability distribution of crack length at t = 0 and t = 300 is shown in Figure 11, which indicates a significant increase in crack length.
- (3)
- Then, crack lengths under each combination at time steps t = 300:10:390 are predicted and transferred into the discrete states based on the discretization scheme shown in Table 1.
- (4)
- The simulated states of different cracks at time steps t = 300:10:390 are utilized to update the prior probability distribution of parent-node hyperparameters and the EICS using the refined exact inference algorithm introduced in Section 2.3.
4.2. Demonstration by Actual Bridge Inspections
4.3. Prediction of Component-Level Fatigue Reliability
5. Conclusions
- (1)
- Fracture mechanics-based uncertainty analysis is performed to determine uncertainty sources in the Paris crack propagation model, material property, and observation data of crack length. The uncertainty of monitoring data of vehicle loads is ignored because of the high accuracy of the WIM system. Actual applications also demonstrate that the vehicle-load-related uncertainty can be ignored.
- (2)
- A hierarchical DBN model is introduced to construct mutual dependences and correlations among various uncertainties for fatigue crack propagation in different components. A refined exact inference algorithm is adopted to estimate the posterior distribution of EICS and update the DBN model in a forward–backward–forward manner. In addition, the component-level reliability index can be updated.
- (3)
- A stochastic traffic model is established using the WIM monitoring data and multi-scale finite element analysis via substructure techniques. Numerical case studies and actual bridge inspections are performed to validate the effectiveness of the established hierarchical DBN model for FCP analysis.
- (4)
- The results show that the established DBN model is robust to random observation errors and that the current and future states of the concerned fatigue crack can be updated by the observation data of other cracks based on the hierarchical DBN model, indicating that the proposed method constructs the dependencies and correlations among different fatigue cracks well.
- (5)
- Numerical experiments demonstrate that using more temporal–spatial inspection data leads to a much more precise estimation of the probability distribution of the initial crack size. It also demonstrates that considering the observation uncertainty of crack length enables the accurate estimation of the probability distribution for EICS. The predictability of the proposed method on fatigue crack propagation forecast is also verified.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Random Variable | Number of States | Interval Boundaries |
---|---|---|
Vehicle Type | Truck Silhouette | Axle Span (m) |
---|---|---|
A | | |
B | | |
C | | |
D | | |
E | | |
F | | |
G | | |
Vehicle Type | Mean Value of Axle Weight Ratio | |||||
---|---|---|---|---|---|---|
Axle 1 | Axle 2 | Axle 3 | Axle 4 | Axle 5 | Axle 6 | |
A | 0.392 | 0.608 | - | - | - | - |
B | 0.264 | 0.266 | 0.470 | - | - | - |
C | 0.185 | 0.207 | 0.291 | 0.317 | - | - |
D | 0.154 | 0.274 | 0.199 | 0.184 | 0.190 | - |
E | 0.109 | 0.138 | 0.204 | 0.193 | 0.178 | 0.177 |
G | 0.392 | 0.608 | - | - | - | - |
Variable | Description | Distribution Type | Mean Value | Standard Deviation |
---|---|---|---|---|
EICS | Lognormal | 0.15 | 0.1 | |
Deterministic | 9.282 × 108 | - | ||
Material-related parameter | Normal | −27.76 | 0.23 | |
Multiplier for crack geometric factor | Normal | 1 | 0.1 | |
Hyperparameters of parent-nodes | Normal | 0 | 1 | |
Standard deviation of observation error | Deterministic | 5 | - |
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Xu, Y.; Zhu, B.; Zhang, Z.; Chen, J. Hierarchical Dynamic Bayesian Network-Based Fatigue Crack Propagation Modeling Considering Initial Defects. Sensors 2022, 22, 6777. https://doi.org/10.3390/s22186777
Xu Y, Zhu B, Zhang Z, Chen J. Hierarchical Dynamic Bayesian Network-Based Fatigue Crack Propagation Modeling Considering Initial Defects. Sensors. 2022; 22(18):6777. https://doi.org/10.3390/s22186777
Chicago/Turabian StyleXu, Yang, Bin Zhu, Zheng Zhang, and Jiahui Chen. 2022. "Hierarchical Dynamic Bayesian Network-Based Fatigue Crack Propagation Modeling Considering Initial Defects" Sensors 22, no. 18: 6777. https://doi.org/10.3390/s22186777