Residual life assessment of structures under random loading based on a one-dimensional convolutional neural network

A model is proposed for assessing the residual life of structural elements under random vibration loading, based on a one-dimensional convolutional neural network (1D-CNN) that takes the stress-process maxima distribution as input and outputs the relative durability. A training set of 12,000 realizations of stationary and non-stationary random processes – including processes with shock impacts and real operational records – was generated; the reference durability values were obtained by the rainflow method with linear damage summation, and the hyperparameters were tuned by Bayesian optimization (minimum error RMSE = 0.0403). Experimental verification on MA-15 light-alloy specimens tested on an electrodynamic shaker at three loading levels yielded a prediction error of 6 to 25 %, increasing at high levels owing to low-cycle fatigue. Taking the scatter of the experimental data into account, the model is considered verified.

1. A comparative review of time- and frequency-domain methods for fatigue damage assessment / M. Muñiz-Calvente [et al.] // International Journal of Fatigue. – 2022. – Vol. 163. – Art. 107069. – DOI 10.1016/j.ijfatigue.2022.107069.

2. Dirlik, T. Dirlik and Tovo-Benasciutti spectral methods in vibration fatigue: a review with a historical perspective / T. Dirlik, D. Benasciutti // Metals. – 2021. – Vol. 11, No. 9. – Art. 1333. – DOI 10.3390/met11091333.

3. Miner, M. A. Cumulative damage in fatigue / M. A. Miner // Journal of Applied Mechanics. – 1945. – Vol. 12, No. 3. – P. A159–A164. – DOI 10.1115/1.4009458.

4. Vibration fatigue by spectral methods: from structural dynamics to fatigue damage – theory and experiments / J. Slavic, M. Boltezar, M. Mrsnik [et al.]. – Amsterdam : Elsevier, 2021. – 300 p. – DOI 10.1016/C2019-0-04580-3.

5. Park, J. B. Fatigue damage model comparison with formulated tri-modal spectrum loadings under stationary Gaussian random processes / J. B. Park, C. Y. Song // Ocean Engineering. – 2015. – Vol. 105. – P. 72–82. – DOI 10.1016/j.oceaneng.2015.05.039.

6. Braccesi, C. Random fatigue. A new frequency domain criterion for the damage evaluation of mechanical components / C. Braccesi, F. Cianetti, L. Tomassini // International Journal of Fatigue. – 2015. – Vol. 70. – P. 417–427. – DOI 10.1016/j.ijfatigue.2014.07.005.

7. Trapp, A. Fatigue assessment of non-stationary random loading in the frequency domain by a quasi-stationary Gaussian approximation / A. Trapp, P. Wolfsteiner // International Journal of Fatigue. – 2021. – Vol. 148. – Art. 106214. – DOI 10.1016/j.ijfatigue.2021.106214.

8. Zorman, A. Short-time fatigue-life estimation for non-stationary processes considering structural dynamics / A. Zorman, J. Slavič, M. Boltežar // International Journal of Fatigue. – 2021. – Vol. 147. – Art. 106178. – DOI 10.1016/j.ijfatigue.2021.106178.

9. Wolfsteiner, P. Fatigue assessment of non-stationary random vibrations by using decomposition in Gaussian portions / P. Wolfsteiner // International Journal of Mechanical Sciences. – 2017. – Vol. 127. – P. 10–22. – DOI 10.1016/j.ijmecsci.2016.05.024.

10. Wolfsteiner, P. Fatigue life due to non-Gaussian excitation – an analysis of the fatigue damage spectrum using higher order spectra / P. Wolfsteiner, A. Trapp // International Journal of Fatigue. – 2019. – Vol. 127. – P. 203–216. – DOI 10.1016/j.ijfatigue.2019.06.005.

11. Non-stationarity and non-Gaussianity in vibration fatigue / J. Slavič [et al.] // Conference Proceedings of the Society for Experimental Mechanics Series. – 2020. – Vol. 97. – P. 73–76. – DOI 10.1007/978-3-030-12676-6_7.

12. Assessment of fatigue damage induced by non-Gaussian bimodal processes with emphasis on spectral methods / S. Gao [et al.] // Ocean Engineering. – 2021. – Vol. 220. – Art. 108489. – DOI 10.1016/j.oceaneng.2020.108489.

13. Marques, J. M. E. Variance of the fatigue damage in non-Gaussian stochastic processes with narrow-band power spectrum / J. M. E. Marques, D. Benasciutti // Structural Safety. – 2021. – Vol. 93. – Art. 102131. – DOI 10.1016/j.strusafe.2021.102131.

14. Marques, J. M. E. More on variance of fatigue damage in non-Gaussian random loadings – effect of skewness and kurtosis / J. M. E. Marques, D. Benasciutti // Procedia Structural Integrity. – 2020. – Vol. 25. – P. 101–111. – DOI 10.1016/j.prostr.2020.04.014.

15. Marques, J. M. E. Variability of the fatigue damage due to the randomness of a stationary vibration load / J. M. E. Marques, D. Benasciutti, R. Tovo // International Journal of Fatigue. – 2020. – Vol. 141. – Art. 105891. – DOI 10.1016/j.ijfatigue.2020.105891.

16. Benasciutti, D. The use of fractional order statistics for estimating nonparametric confidence intervals for quantiles of the fatigue damage computed in service random loadings / D. Benasciutti, J. M. E. Marques // IOP Conference Series: Materials Science and Engineering. – 2023. – Vol. 1275, No. 1. – Art. 012020. – DOI 10.1088/1757-899X/1275/1/012020.

17. Orlando, A. Structural response and fatigue assessment of a small vertical axis wind turbine under stationary and non-stationary excitation / A. Orlando, L. Pagnini, M. P. Repetto // Renewable Energy. – 2021. – Vol. 170. – P. 251–266. – DOI 10.1016/j.renene.2021.01.123.

18. Niu, Q. An empirical mode decomposition-based frequency-domain approach for the fatigue analysis of nonstationary processes / Q. Niu, S. Yang, X. Li // Fatigue & Fracture of Engineering Materials & Structures. – 2018. – Vol. 41, No. 9. – P. 1980–1996. – DOI 10.1111/ffe.12836.

19. A novel approach for stress cycle analysis based on empirical mode decomposition / R. Li [et al.] // MFPT 2018: Intelligent Technologies and Equipment for Human Performance Monitoring. – 2018. – P. 4–12.

20. An improved VMD-based denoising method for time domain load signal combining wavelet with singular spectrum analysis / J. Fu [et al.] // Mathematical Problems in Engineering. – 2020. – Vol. 2020. – Art. 1485937. – DOI 10.1155/2020/1485937.

21. Soman, R. Semi-automated methodology for damage assessment of a scaled wind turbine tripod using enhanced empirical mode decomposition and statistical analysis / R. Soman // International Journal of Fatigue. – 2020. – Vol. 134. – Art. 105475. – DOI 10.1016/j.ijfatigue.2020.105475.

22. Median ensemble empirical mode decomposition / X. Lang [et al.] // Signal Processing. – 2020. – Vol. 176. – Art. 107686. – DOI 10.1016/j.sigpro.2020.107686.

23. Method for assessing the durability of structures under stationary and non-stationary random loading using variational mode decomposition (VMD) / A. V. Erpalov, K. A. Khoroshevskii, E. A. Rumyanceva, I. V. Gadolina // Industrial Laboratory. Diagnostics of Materials. – 2024. – Vol. 90, No. 9. – P. 63–74. – DOI 10.26896/1028-6861-2024-90-9-63-74.

24. Niesłony, A. Mean stress effect correction in frequency-domain methods for fatigue life assessment / A. Niesłony, M. Böhm // Procedia Engineering. – 2015. – Vol. 101. – P. 347–354. – DOI 10.1016/j.proeng.2015.02.042.

25. Niesłony, A. Frequency-domain fatigue life estimation with mean stress correction / A. Niesłony, M. Böhm // International Journal of Fatigue. – 2016. – Vol. 91. – P. 373–381. – DOI 10.1016/j.ijfatigue.2016.02.031.

26. A unified mean stress correction model for fatigue thresholds prediction of metals / Y. Liu [et al.] // Engineering Fracture Mechanics. – 2020. – Vol. 223. – Art. 106787. – DOI 10.1016/j.engfracmech.2019.106787.

27. Oh, G. Effective stress and fatigue life prediction with mean stress correction models on a ferritic stainless steel sheet / G. Oh // International Journal of Fatigue. – 2022. – Vol. 157. – Art. 106707. – DOI 10.1016/j.ijfatigue.2021.106707.

28. The choice of the sampling frequency and optimal method of signal digital processing in the problems considering a random loading process for assessing durability / I. V. Gadolina, N. G. Lisachenko, Yu. A. Svirskiy, D. A. Dubin // Industrial Laboratory. Diagnostics of Materials. – 2019. – Vol. 85, No. 7. – P. 64–72. – DOI 10.26896/1028-6861-2019-85-7-64-72.

29. Gadolina, I. V. Varied approaches to loading assessment in fatigue studies / I. V. Gadolina, N. A. Makhutov, A. V. Erpalov // International Journal of Fatigue. – 2021. – Vol. 144. – Art. 106035. – DOI 10.1016/j.ijfatigue.2020.106035.

30. Khoroshevsky, K. A. A computational and experimental study of the influence of structural damping on a structure's durability under random loading / K. A. Khoroshevsky, A. V. Erpalov, I. V. Gadolina // Bulletin of the South Ural State University. Series "Mathematics, Mechanics, Physics". – 2023. – Vol. 15, No. 4. – P. 47–57. – DOI 10.14529/mmph230406.

31. 1D convolutional neural networks and applications: a survey / S. Kiranyaz, O. Avci, O. Abdeljaber [et al.] // Mechanical Systems and Signal Processing. – 2021. – Vol. 151. – Art. 107398. – DOI 10.1016/j.ymssp.2020.107398.

32. Chen, J. Fatigue modeling using neural networks: a comprehensive review / J. Chen, Y. Liu // Fatigue & Fracture of Engineering Materials & Structures. – 2022. – Vol. 45, No. 4. – P. 945–979. – DOI 10.1111/ffe.13640.

33. Li, X. Remaining useful life estimation in prognostics using deep convolution neural networks / X. Li, Q. Ding, J.-Q. Sun // Reliability Engineering & System Safety. – 2018. – Vol. 172. – P. 1–11. – DOI 10.1016/j.ress.2017.11.021.

34. A novel method of multiaxial fatigue life prediction based on deep learning / J. Yang, G. Kang, Y. Liu, Q. Kan // International Journal of Fatigue. – 2021. – Vol. 151. – Art. 106356. – DOI 10.1016/j.ijfatigue.2021.106356.

35. Machine learning-based predictions of fatigue life and fatigue limit for steels / L. He, Z. Wang, H. Akebono, A. Sugeta // Journal of Materials Science & Technology. – 2021. – Vol. 90. – P. 9–19.

36. Jimenez-Martinez, M. Fatigue damage effect approach by artificial neural network / M. Jimenez-Martinez, M. Alfaro-Ponce // International Journal of Fatigue. – 2019. – Vol. 124. – P. 42–47.

37. Zhang, X.-C. A deep learning based life prediction method for components under creep, fatigue and creep-fatigue conditions / X.-C. Zhang, J.-G. Gong, F.-Z. Xuan // International Journal of Fatigue. – 2021. – Vol. 148. – Art. 106236. – DOI 10.1016/j.ijfatigue.2021.106236.

38. Modelling fatigue life prediction of additively manufactured Ti-6Al-4V samples using machine learning approach / J. Horňas, J. Běhal, P. Homola [et al.] // International Journal of Fatigue. – 2023. – Vol. 169. – Art. 107483. – DOI 10.1016/j.ijfatigue.2022.107483.

39. Graphical feature construction-based deep learning model for fatigue life prediction of AM alloys / H. Wu, A. Wang, Z. Gan, L. Gan // Materials. – 2024. – Vol. 18, No. 1. – Art. 11. – DOI 10.3390/ma18010011.

40. Multiaxial fatigue life prediction for various metallic materials based on the hybrid CNN-LSTM neural network / F. Heng, J. Gao, R. Xu [et al.] // Fatigue & Fracture of Engineering Materials & Structures. – 2023. – Vol. 46, No. 5. – P. 1979–1996. – DOI 10.1111/ffe.13977.

41. A physics-informed neural network approach to fatigue life prediction using small quantity of samples / D. Chen, Y. Li, K. Liu, Y. Li // International Journal of Fatigue. – 2023. – Vol. 166. – Art. 107270. – DOI 10.1016/j.ijfatigue.2022.107270.

42. An image recognition based multiaxial low-cycle fatigue life prediction method with CNN model / X. Sun, T. Zhou, K. Song, X. Chen // International Journal of Fatigue. – 2023. – Vol. 167. – Art. 107324. – DOI 10.1016/j.ijfatigue.2022.107324.

43. Remaining useful life prediction via a deep adaptive transformer framework enhanced by graph attention network / P. Liang, Y. Li, B. Wang, X. Yuan, L. Zhang // International Journal of Fatigue. – 2023. – Vol. 174. – Art. 107722.

44. Remaining useful life prediction of aero-engine enabled by fusing knowledge and deep learning models / Y. Li, Y. Chen, Z. Hu, H. Zhang // Reliability Engineering & System Safety. – 2023. – Vol. 229. – Art. 108869.

45. Sun, H. A novel artificial neural network model for wide-band random fatigue life prediction / H. Sun, Y. Qiu, J. Li // International Journal of Fatigue. – 2022. – Vol. 157. – Art. 106701. – DOI 10.1016/j.ijfatigue.2021.106701.

46. Novel models for fatigue life prediction under wideband random loads based on machine learning / H. Sun, Y. Qiu, J. Li [et al.] // Fatigue & Fracture of Engineering Materials & Structures. – 2024. – Vol. 47, No. 9. – P. 3342–3360. – DOI 10.1111/ffe.14371.

47. Fatigue life prediction based on modified narrow-band method under broadband random vibration loading / S.-D. Wu, D.-G. Shang, P.-C. Liu [et al.] // International Journal of Fatigue. – 2022. – Vol. 159. – Art. 106832. – DOI 10.1016/j.ijfatigue.2022.106832.

48. A new frequency-domain method for fatigue prediction of offshore structures under wideband random loadings / B. Wu, W. Lu, X. Li [at al.] // Ocean Engineering. – 2023. – Vol. 281. – Art. 114812. – DOI 10.1016/j.oceaneng.2023.114812.

49. Random vibration fatigue behavior of directionally solidified superalloy: experiments and evaluation of life prediction methods / H. Lu, J. Wang, Y. Lian [et al.] // International Journal of Fatigue. – 2023. – Vol. 175. – Art. 107746. – DOI 10.1016/j.ijfatigue.2023.107746.

50. Zhang, H. Data-based deep learning for random vibration fatigue life prediction of car seat frame / H. Zhang, S. Wang // Nonlinear Dynamics. – 2025. – Vol. 113, No. 5. – P. 4121–4145. – DOI 10.1007/s11071-024-09972-3.

51. Alyunov, D. Yu. Recurrent neural network for controlling the spectrum width of a non-stationary random signal / D. Yu. Alyunov // Bulletin of the Chuvash University. – 2023. – No. 2. – P. 5–17. – DOI 10.47026/1810-1909-2023-2-5-17.

52. Fedotov, M. V. Predictive analytics of the tech-nical condition of diesel locomotive systems using neural network predictive models / M. V. Fedotov, V. V. Grachev // Bulletin of Scientific Research Results. – 2021. – No. 3. – P. 102–114. – DOI 10.20295/2223-9987-2021-3-102-114.

53. Lomakina, L. S. Diagnostics and forecasting of technical and technological object states based on ensemble machine learning technologies / L. S. Lomakina, A. N. Dvitovskaya, K. A. Korelin // Information and Mathematical Technologies in Sci-ence and Management. – 2025. – No. 4 (40). – P. 26–37. – DOI 10.25729/ESI.2025.40.4.003.