PSI - Issue 44

Sourav Das et al. / Procedia Structural Integrity 44 (2023) 1680–1687 Das and Tesfamariam/ Structural Integrity Procedia 00 (2022) 000–000

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where θ ll and θ ul denote the lower and upper bounds of θ , respectively. Fig. 5 shows the PDF of Y EEV and response surface of the probability of failure corresponding to design variables, θ . The optimal value is taken where the failure probability is the minimum. The optimal locations of the outriggers from the ground level are 21m and 57m, respectively. The other obtained optimal design parameters are [ F 0 , μ s , x s,max ] = [0.27, 22.15, 0.18], respectively.

Fig 5. (a) PDF of equivalent extreme event and (b) Probability of failure surface corresponding to design parameters

5. Conclusions The numerical investigations presented in this study are focused on reliability analysis of stochastic systems using the probability density evolution method (PDEM). The PDEM is computationally expensive for high-fidelity models as it requires a large number of sample points. To reduce the computational burden, stochastic spectral embedding is used as a surrogate model. The major observations from this study are listed below: • PDEM enables to estimate the joint probability density function of the equivalent-extreme event of an MDOF system by solving GDEEs efficiently. • When compared to Monte Carlo simulation, surrogate assisted PDEM can accurately estimate the probability of failure while using fewer samples of design variables. This study clearly indicates the efficiency of the proposed surrogate model based PDEM for the time-independent reliability analysis of the stochastic system. This method can be extended to time-varying system such as deteriorating system, which the authors wish to address in their future works. Acknowledgements This study was funded by Natural Sciences and Engineering Research Council of Canada under the Discovery Grant programs (RGPIN-2019- 05584). References Au, S.K., Beck, J.L., 2001. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 16(4), 263–277. Bourinet, J.M., Deheeger, F., Lemaire, M., 2011. Assessing small failure probabilities by combined subset simulation and support vector machines. Structural Safety 33(6), 343–353. Chen, J., Yang, J., Li, J., 2016. A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters. Structural Safety 59, 20–31. Das, S., Tesfamariam, S., 2022. Multiobjective design optimization of multi-outrigger tall-timber building: Using SMA-based damper and Lagrangian model. Journal of Building Engineering 51, 104358. Li, J., Chen, J.B., Fan, W.l., 2007. The equivalent extreme-value event and evaluation of the structural system reliability. Structural Safety 29(2), 112–131. Li, J., Chen, J., 2008. The principle of preservation of probability and the generalized density evolution equation. Structural Safety 30(1), 65–77. Marelli, S., Wagner, P.R., Lataniotis, C., Sudret, B., 2021. Stochastic spectral embedding. International Journal for Uncertainty Quantification 11(2), 25-47. NBCC (National Building Code of Canada). (2015). National Building Code of Canada 2015. National Research Council of Canada, Ottawa.