Uncertainty quantification of fatigue S-N curves with sparse data using hierarchical Bayesian data augmentation

Jie Chen, Siying Liu, Wei Zhang, Yongming Liu

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


A novel statistical uncertainty quantification (UQ) method for fatigue S-N curves with sparse data is proposed in this paper. Sparse data observation is very common in fatigue testing due to various reasons, such as time and budget constraints, availability of testing materials and resources. A brief review of existing UQ methods for fatigue properties with sparse data is given. Following this, a new method, called Hierarchical Bayesian data augmentation (HBDA) is proposed to integrate the hierarchical Bayesian modeling (HBM) and Bayesian data augmentation (BDA) to deal with sparse data problem specifically for fatigue S-N curves. The key idea is to use: (1) HBM for analyzing the variability of S-N curves both within one stress level and across stress levels; (2) BDA to build up a large-size sample of fatigue life data based on the observed sparse samples. Four strategies to estimate the probabilistic S-N curves with sparse data are proposed: (1) hierarchical Bayesian modeling (HBM) only, (2) Bayesian data augmentation (BDA) only, (3) posterior information from HBM used as prior information for BDA (HBM + BDA), and (4) augmented data from BDA used by HBM (BDA + HBM). The strategy (3) and (4) are named HBDA hereafter. Next, the four strategies are validated and compared using aluminum alloy data and laminate panel data from open literature. Convergence study and confidence estimation is performed, and it is shown that the HBDA methods (i.e., HBM + BDA or BDA + HBM) have better performance compared with the classical method and HBM/BDA alone. The performance gain is especially significant when the number of available data samples is small. Finally, the proposed methodology is applied to a practical engineering problem for fatigue property quantification of the demolished Pearl Harbor Memorial Bridge, where only limited samples are available for testing. Conclusions and future work are drawn based on the proposed study.

Original languageEnglish (US)
Article number105511
JournalInternational Journal of Fatigue
StatePublished - May 2020


  • Bayesian data augmentation
  • Fatigue probabilistic S-N curve
  • Sparse data
  • hierarchical Bayesian model

ASJC Scopus subject areas

  • Modeling and Simulation
  • Materials Science(all)
  • Mechanics of Materials
  • Mechanical Engineering
  • Industrial and Manufacturing Engineering


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