Multidimensional seismic fragility analysis of subway station structures using the adaptive bandwidth kernel density estimation and Copula function

Abstract

Structural damages during an earthquake are typically controlled by seismic demands, which are represented by the combination of amplitude of ground motion and cyclic load effects. Since traditional methods normally assume the lognormal distributions of seismic demands and resistance parameters, uncertainties are inevitably induced in the seismic fragility analysis. In this paper, the Copula function and adaptive bandwidth kernel density estimation method (ABKDE) are used to establish a novel multidimensional seismic fragility analysis framework. Based on the results of incremental dynamic analysis for subway station structures, ABKDE is adopted to establish single-parameter seismic fragility curves for both the maximum inter-story drift ratio (MIDR) and cumulated dissipated hysteretic energy (CDHE), respectively. Subsequently, the Copula function is used to formulate a bivariate seismic fragility function considering the correlations among seismic demand measures and establish the corresponding fragility curves. Finally, comparative analyses are conducted to evaluate seismic fragility curves using Copula-based dual and single-parameter damage models as well as the traditional damage models. It is found that the seismic fragility analysis method using the Copula function has the ability to gain a comprehensive consideration of the MIDR and CDHE during the damage process of subway station structures. Moreover, this newly developed seismic fragility analysis framework can capture the influence of the correlation between deformation and energy under various peak ground accelerations on structural damage. Thus, this framework can provide a scientific basis for predicting structural damage in subway stations subjected to varying intensities of ground motion while considering multiple damage indicators.

Keywords

Multidimensional seismic fragility / Subway station structure / Adaptive bandwidth kernel density estimation / Gaussian kernel function / Copula function

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

CRediT authorship contribution statement

Chunyi Cui: Writing - review & editing, Supervision, Resources, Project administration, Funding acquisition. Jingtong Zhao: Writing - original draft, Methodology, Conceptualization. Minze Xu: Validation, Supervision. Chengshun Xu: Validation, Methodology. Hailong Liu: Validation. Kunpeng Wang: Data curation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The work is supported by the National Natural Science Foundation of China (Grant Nos. 52178315, and 51578100), the Fundamental Research Funds for the Central Universities (Grant No. 3132023504), the Dalian Science and Technology Innovation Fund (Grant No. 2022JJ12GX031), and the Project of Shenyang Key Laboratory of Safety Evaluation and Disaster Prevention of Engineering Structures (Grant No. S230184).

References

[1] Argyroudis, S. A., & Pitilakis, K. D. (2012). Seismic fragility curves of shallow tunnels in alluvial deposits. Soil Dynamics and Earthquake Engineering, 35, 1-12.

[2] Bu, X. B., Ledesma, A., & López-Almansa, F. (2022). Novel seismic design solution for underground structures. Case study of a 2 -story 3bay subway station. Soil Dynamics and Earthquake Engineering, 153, 107087.

[3] Cao, X. Y., Feng, D. C., & Beer, M. (2023). A KDE-based nonparametric cloud approach for efficient seismic fragility estimation of structures under non-stationary excitation. Mechanical Systems and Signal Processing, 205, 110873.

[4] Chen, J. Y., Wen, R. Z., Yu, P. Q., & He, W. (2009). Numerical analysis of seismic response of a shallow-buried subway station structure in soft soil. World Earthquake Engineering, 25(2), 46-53 (in Chinese).

[5] Cimellaro, G. P., & Reinhorn, A. M. (2010). Multidimensional performance limit state for hazard fragility functions. Journal of Engineering Mechanics, 137(1), 47-60.

[6] Cornell, C. A., Jalayer, F., Hamburger, R. O., & Foutch, D. A. (2002). Probabilistic basis for 2000 SAC federal emergency management agency steel moment frame guidelines. Journal of Structural Engineering, 128(4), 526-533.

[7] Cui, C. Y., Xin, Y., Xu, C. S., Zhang, P., Liang, Z. M., & Liu, H. L. (2024). Analytical solution to horizontal vibrations of large-diameter piles in layered pasternak soils. International Journal of Geomechanics, 24(5), 04024058.

[8] Dong, R., Jing, L. P., Li, Y. Q., Yin, Z. Y., Wang, G., & Xu, K. P. (2020). Seismic deformation mode transformation of rectangular underground structure caused by component failure. Tunnelling and Underground Space Technology, 98, 103298.

[9] Du, X. L., Jiang, J. W., Naggar, M. H. E., Xu, C. S., & Xu, Z. G. (2021). Interstory drift ratio associated with performance objectives for shallow-buried multistory and span subway stations in inhomogeneous soil profiles. Earthquake Engineering Structural Dynamics, 50(2), 655-672.

[10] Elgamal, A., Yang, Z. H., Parra, E., & Ragheb, A. (2003). Modeling of cyclic mobility in saturated cohesionless soils. International Journal of Plasticity, 19(6), 883-905.

[11] Fan, X. Y., Liu, B., Luo, J., Pan, K., Han, S. Y., & Zhou, Z. L. (2023). Comparison of earthquake-induced shallow landslide susceptibility assessment based on two-category LR and KDE-MLR. Scientific Reports, 13(1), 1-14.

[12] He, H. X., Cheng, S. T., & Liao, L. C. (2022). Performance evaluation and maintenance optimization strategy of seismic damaged frame structure considering residual displacement. Journal of Vibration Engineering, 35 (1), 23-33 (in Chinese).

[13] Huang, G., Qiu, W. G., & Zhang, J. R. (2017). Modelling seismic fragility of a rock mountain tunnel based on support vector machine. Soil Dynamics and Earthquake Engineering, 102, 160-171.

[14] Huang, Z. K., Pitilakis, K., Tsinidis, G., Argyroudis, S., & Zhang, D. M. (2020). Seismic vulnerability of circular tunnels in soft soil deposits: The case of Shanghai metropolitan system. Tunnelling and Underground Space Technology, 98, 103341.

[15] Huh, J., Tran, Q. H., Haldar, A., Park, I., & Ahn, J. H. (2017). Seismic vulnerability assessment of a shallow two-story underground RC box structure. Applied Sciences, 7(7), 735.

[16] Iida, H., Hiroto, T., Yoshida, N., & Iwafuji, M. (1996). Damage to Daikai subway station. Soils and Foundations, 36, 283-300.

[17] Jara, J. M., Galván, A., Jara, M., & Olmos, B. (2013). Procedure for determining the seismic vulnerability of an irregular isolated bridge. Structure and Infrastructure Engineering, 9(6), 516-528.

[18] Kontoe, S., Zdravkovic, L., Potts, D. M., & Menkiti, C. O. (2008). Case study on seismic tunnel response. Canadian Geotechnical Journal, 45 (12), 1743-1764.

[19] Li, J. B., Cui, C. Y., Xiao, Z. S., Wang, B. L., & Xu, C. S. (2024). Reliability and sensitivity analyses of monopile supported offshore wind turbines based on probability density evolution method with prescreening of controlling parameters. Ocean Engineering, 310, 118746.

[20] Liang, Z. M., Cui, C. Y., Xu, C. S., Zhang, P., & Wang, K. P. (2023). A close-formed solution for the horizontal vibration of a pipe pile in saturated soils considering the radial heterogeneity effect. Computers and Geotechnics, 158, 105379.

[21] Liu, T., Chen, Z. Y., Yuan, Y., & Shao, X. Y. (2017). Fragility analysis of a subway station structure by incremental dynamic analysis. Advances in Structural Engineering, 20(7), 1111-1124.

[22] Liu, X. X., Wu, Z. Y., & Liang, F. (2016). Multidimensional performance limit state for probabilistic seismic demand analysis. Bulletin of Earthquake Engineering, 14(12), 3389-3408.

[23] Lu, D. C., Ma, C., Du, X. L., & Wang, X. (2019). A new method for the evaluation of the ultimate seismic capacity of rectangular underground structures. Soil Dynamics and Earthquake Engineering, 126, 105776.

[24] Mai, C., Konakli, K., & Sudret, B. (2017). Seismic fragility curves for structures using non-parametric representations. Frontiers of Structural and Civil Engineering, 11(2), 169-186.

[25] Nguyen, D. D., Park, D., Shamsher, S., Nguyen, V. Q., & Lee, T. H. (2019). Seismic vulnerability assessment of rectangular cut-and-cover subway tunnels. Tunnelling and Underground Space Technology, 86, 247-261.

[26] Niu, D. T., & Ren, L. J. (1996). A modified seismic damage model with double variables for reinforced concrete structures. Earthquake Engineering and Engineering Vibration, 16(4), 44-54 (in Chinese).

[27] Nourzadeh, D., Mortazavi, P., Ghalandarzadeh, A., Takada, S., Najma, A., & Rahimi, S. (2020). Numerical, experimental and fragility analysis of urban lifelines under seismic wave propagation: Study on gas distribution pipelines in the greater Tehran area. Tunnelling and Underground Space Technology, 106, 103607.

[28] Opabola, E. A., & Elwood, K. J. (2020). Incorporating failure mode uncertainty into probabilistic assessment of RC components. Structural Safety, 87, 101997.

[29] Plisetskaya, E., & Ivanov, S. (2021). An empirical analysis of KDE-based generative models on small datasets. Procedia Computer Science, 193, 442-452.

[30] Rossetto, T., & Elnashai, A. (2003). Derivation of vulnerability functions for European-type RC structures based on observational data. Engineering Structures, 25(10), 1241-1263.

[31] Provost, S. B., & Zang, Y. S. (2024). Nonparametric copula density estimation methodologies. Mathematics, 12(3), 398.

[32] Shen, Y. M., Zhang, D. M., Wang, R. L., Li, J. P., & Huang, Z. K. (2023). SBD-K-medoids-based long-term settlement analysis of shield tunnel. Transportation Geotechnics, 42, 101053.

[33] Shinozuka, M., Feng, M. Q., Lee, J. H., & Naganuma, T. (2000). Statistical analysis of fragility curves. Journal of Engineering Mechanics, 126(12), 1224-1231.

[34] Sreelakshmi, N. (2018). An introduction to copula-based bivariate reliability concepts. Communications in Statistics - Theory and Methods, 47(4), 996-1012.

[35] Wang, B. L., Cui, C. Y., Xu, C. S., Meng, K., Li, J. B., & Xu, L. N. (2024). A novel analytical solution for horizontal vibration of partially embedded offshore piles considering the distribution effect of wave loads. Ocean Engineering, 307, 118179.

[36] Wang, Q. A., Wu, Z. Y., & Jia, Z. P. (2013). Multi-dimensional fragility analysis of bridge system under earthquake. Engineering Mechanics, 30 (10), 192-198 (in Chinese).

[37] Wang, W. L., Wang, T. T., Su, J. J., Lin, C. H., Seng, C. R., & Huang, T. H. (2001). Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi Earthquake. Tunnelling and Underground Space Technology, 16(3), 133-150.

[38] Wang, Z. Z., Gao, B., Jiang, Y. J., & Yuan, S. (2009). Investigation and assessment on mountain tunnels and geotechnical damage after the Wenchuan earthquake. Science in China Series E: Technological Sciences, 52(2), 546-558.

[39] Wu, Z. Y., Cai, Z. M., Zhang, K., & Wang, Q. A. (2018). Multidimensional fragility analysis of structures based on cumulative plastic strain. Journal of Basic Science and Engineering, 26(6), 1259-1270.

[40] Xu, M. Z., Cui, C. Y., Xu, C. S., Zhang, P., & Zhao, J. T. (2023). Seismic risk analysis of subway station structures combining the epistemic uncertainties from both seismic hazard and numerical simulation. Journal of Earthquake Engineering, 28(5), 1474-1494.

[41] Xu, M. Z., Cui, C. Y., Zhao, J. T., Xu, C. S., Zhang, P., & Su, J. (2024). Fuzzy seismic fragility analysis of underground structures considering multiple failure criteria. Tunnelling and Underground Space Technology, 145, 105614.

[42] Xu, Q., Zheng, S. S., Han, Y. Z., Cheng, Y., & Tian, J. (2014). Steel frame seismic vulnerability based on a global structural damage index. Journal of Vibration and Shock, 33(11), 78-82 (in Chinese).

[43] Xu, Z. G., Du, X. L., Xu, C. S., Hao, H., Bi, K. M., & Jiang, J. W. (2019). Numerical research on seismic response characteristics of shallow buried rectangular underground structure. Soil Dynamics and Earthquake Engineering, 116, 242-252.

[44] Yang, J., Yin, Z. Y., Liu, X. F., & Gao, F. P. (2020). Numerical analysis for the role of soil properties to the load transfer in clay foundation due to the traffic load of the metro tunnel. Transportation Geotechnics, 23, 100336.

[45] Zakeri, B., Padgett, J. E., & Ghodrati Amiri, G. (2015). Fragility assessment for seismically retrofitted skewed reinforced concrete box girder bridges. Journal of Performance of Constructed Facilities, 29(2), 04014043.

[46] Zhai, P. P., Zhao, P., Lu, Y., Ye, C. Y., & Xiong, F. (2019). Seismic fragility analysis of buildings based on double-parameter damage models considering soil-structure interaction. Advances in Materials Science and Engineering, 2019(1), 4592847.

[47] Zhang, C. M., Zhao, M., Zhong, Z. L., & Du, X. L. (2022). Optimum intensity measures for probabilistic seismic demand model of subway stations with different burial depths. Soil Dynamics and Earthquake Engineering, 154, 107138.

[48] Zhang, X. P., Jiang, Y. J., & Sugimoto, S. (2018). Seismic damage assessment of mountain tunnel: a case study on the Tawarayama tunnel due to the 2016 Kumamoto Earthquake. Tunnelling and Underground Space Technology, 71, 138-148.

[49] Zhu, T., Wang, R., & Zhang, J. M. (2021). Effect of nearby ground structures on the seismic response of underground structures in saturated sand. Soil Dynamics and Earthquake Engineering, 146, 106756.

[50] Zhuang, H. Y., Yang, J., Chen, S., Fu, J. S., & Chen, G. X. (2019). Seismic performance of underground subway station structure considering connection modes and diaphragm wall. Soil Dynamics and Earthquake Engineering, 127, 105842.

[51] Zhong, Z. L., Shen, Y. Y., Zhao, M., Li, L. Y., Du, X. L., & Hao, H. (2020). Seismic fragility assessment of the Daikai subway station in layered soil. Soil Dynamics and Earthquake Engineering, 132, 106044.

[52] Zougab, N., Adjabi, S., & Kokonendji, C. C. (2014). Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation. Computational Statistics & Data Analysis, 75, 28-38.