Probabilistic seismic response analysis of tunnel linings considering coupled rock mass property and earthquake excitation uncertainties

Abstract

Tunnel lining seismic performance is significantly influenced by the spatial variability of geological parameters and the uncertainty of earthquake excitation factors, which are conventionally treated in isolation. This study proposes a novel probabilistic framework that integrates random field theory with an enhanced Clough-Penzien spectrum to concurrently model both uncertainty sources. The approach offers a more realistic and integrated assessment of seismic risk for tunnels under complex geological and loading conditions. The case analysis of a railway project reveals that considering both spatial variability of rock mass and uncertainty in seismic excitation leads to significant increases in internal forces and their variability, with mean values rising up to 278.9% and coefficients of variation (COV) up to 262.8%, compared to single-factor random analyses. The non-normal distribution of responses under seismic uncertainty, combined with the broader dispersion from rock variability, necessitates integrating both random factors for reliable seismic performance assessment of tunnels. Parametric studies demonstrate spectral parameters, including initial circular frequency (ω₀), equivalent damping ratio (ξ₀), and peak acceleration (a_max), significantly influence results: increasing ω₀ and ξ₀ markedly reduces both the mean and COV of lining mechanical response-by up to 83.5% and 82.5%, respectively-potentially underestimating failure risk and underscoring the need to adopt lower-bound values in design for enhanced safety. Meanwhile, a_max positively correlates with mean structural response, while variability in internal forces follows distinct trajectories; moreover, the interaction between rock spatial variability and seismic uncertainty raises failure probabilities by 3%-38%, emphasizing the necessity of integrating both randomness sources, especially in high-intensity seismic regions.

Keywords

Tunnel lining structure / Spatial variability / Random earthquake excitation / Random field / Clough-Penzien power spectrum / Random dynamic response

Data availability

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

CRediT authorship contribution statement

Xiancheng Mei: Writing – review & editing, Writing –origin al draft, Visualization, Funding acquisition. Jiajun Wu: Writing – review & editing, Writing – original draft, Methodology. Baiyi Li: Supervision, Funding acquisition. Zhen Cui: Supervision, Funding acquisition. Chong Yu: Supervision. Qian Sheng: Supervision. Jian Chen: Supervision.

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 Key R&D Programs for Young Scientists (2023YFB2390400), the National Natural Science Foundation of China (Grant Nos. U21A20159, 52079133, and 12441510), the Hubei Provincial Natural Science Foundation of China (2024AFB041), State Key Laboratory of Intelligent Con-struction and Healthy Operation and Maintenance of Deep Underground Engineering (SDGZK2412), and Seed Fund for Basic Research and Original Innovation of Young Talents of State Key Laboratory of Geomechanics and Geotechnical Engineering Safety (SKLGGES-ZZJJ2503). The authors want to thank all the members who gave us lots of help and cooperation.

References

[1] Alibeikloo, M., Khabbaz, H., & Fatahi, B. (2022). Random field reliability analysis for time-dependent behaviour of soft soils considering spatial variability of elastic visco-plastic parameters. Reliability Engineering & System Safety, 219, 108254.

[2] Cacciola, P., & Deodatis, G. (2011). A method for generating fully non-stationary and spectrum-compatible ground motion vector processes. Soil Dynamics and Earthquake Engineering, 31(3), 351-360.

[3] Cao, Z., & Wang, Y. (2014). Bayesian model comparison and selection of spatial correlation functions for soil parameters. Structural Safety, 49, 10-17.

[4] Chen, F. Y., Wang, L., & Zhang, W. G. (2019). Reliability assessment on stability of tunnelling perpendicularly beneath an existing tunnel considering spatial variabilities of rock mass properties. Tunnelling and Underground Space Technology, 88, 276-289.

[5] Chen, P., Geng, P., Chen, J., & Gu, W. (2023). The seismic damage mechanism of Daliang tunnel by fault dislocation during the 2022 Menyuan Ms6.9 earthquake based on unidirectional velocity pulse input. Engineering Failure Analysis, 145, 107047.

[6] Chen, X., & Li, J. (2025). Stochastic nonlinear dynamic analysis and system reliability evaluation of RC structures involving spatial variation under stochastic ground motions. Structural Safety, 114, 102581.

[7] China Railway Second Design Institute Engineering Group Co., Ltd. ( 2017). TB 10003—2016: Code for Design of Railway Tunnel. Beijing: China Railway Publishing House Co., LTD. (in Chinese).

[8] Ching, J., & Phoon, K. K. (2013a). Probability distribution for mobilised shear strengths of spatially variable soils under uniform stress states. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 7(3), 209-224.

[9] Ching, J., & Phoon, K. K. (2013b). Effect of element sizes in random field finite element simulations of soil shear strength. Computers and Structures, 126(15), 120-134.

[10] Ching, J., Hu, Y. G., Yang, Z. Y., Shiau, J. Q., Chen, J. C., & Li, Y. S. (2011). Reliability-based design for allowable bearing capacity of footings on rock masses by considering angle of distortion. International Journal of Rock Mechanics and Mining Sciences, 48(5), 728-740.

[11] Cui, L., Yang, W. Y., Sheng, Q., Zheng, J. J., & Ali, N. (2024). Deformation behaviour of strain-softening rock mass in tunnels considering deterioration model of elastic modulus. Geomechanics and Geophysics for Geo-Energy and Geo-Resources, 10(1), 171.

[12] Deodatis, G. (1996). Non-stationary stochastic vector processes: Seismic ground motion applications. Probabilistic Engineering Mechanics, 11(3), 149-167.

[13] Ding, J. Y., Zhou, J. F., & Cai, W. (2023). An efficient variable selection-based Kriging model method for the reliability analysis of slopes with spatially variable soils. Reliability Engineering and System Safety, 235, 109234.

[14] Dong, X., Tan, X., Lu, Z., Zhang, P., Zha, F., & Xu, L. (2024). Reliability analysis of two-pile group in spatially variable soil considering differential settlement. Ocean Engineering, 307, 118182.

[15] Fenton, G. A., & Griffiths, D. V. (2002). Probabilistic foundation settlement on spatially random soil. Journal of Geotechnical and Geoenvironmental Engineering, 128(5), 381-390.

[16] Fenton, G. A., Griffiths, D. V., & Williams, M. B. (2005). Reliability of traditional retaining wall design. Géotechnique, 55(1), 55-62.

[17] Guo, X., Du, D., & Dias, D. (2019). Reliability analysis of tunnel lining considering soil spatial variability. Engineering Structures, 196, 109332.

[18] Housner, G. W. (1947). Characteristics of strong-motion earthquakes. Bulletin of the Seismological Society of America, 37(1), 19-31.

[19] Huang, G., & Chen, X. (2009). Wavelets-based estimation of multivariate evolutionary spectra and its application to nonstationary downburst winds. Engineering Structures, 31(4), 976-989.

[20] Huang, H. W., Xiao, L., Zhang, D. M., & Zhang, J. (2017). Influence of spatial variability of soil Young’s modulus on tunnel convergence in soft soils. Engineering Geology, 228, 357-370.

[21] Institute of Geophysics, China Earthquake Administration, et al. (2015). GB 18306—2015: Seismic Ground Motion Parameters Zonation of China. Beijing: China Planning Press. (in Chinese).

[22] Jerez, D. J., Chwala, M., Jensen, H. A., & Beer, M. (2024). Optimal borehole placement for the design of rectangular shallow foundation systems under undrained soil conditions: A stochastic framework. Reliability Engineering & System Safety, 242, 109771.

[23] Ji, J., Wang, C. W., Gao, Y. F., & Zhang, L. M. (2021). Probabilistic investigation of the seismic displacement of earth slopes under stochastic ground motion: A rotational sliding block analysis. Canadian Geotechnical Journal, 58(7), 952-968.

[24] Jiang, S., Li, D., Phoon, K. K., Cao, Z., & Zhou, C. (2015). Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation. Journal of Geotechnical and Geoenvironmental Engineering, 141(2), 04014096.

[25] Johari, A., & Rahmati, H. (2019). System reliability analysis of slopes based on the method of slices using sequential compounding method. Computers and Geotechnics, 114, 103116.

[26] Johari, A., & Sabzi, A. (2017). Reliability analysis of foundation settlement by stochastic response surface and random finite element method. Scientia Iranica Transactions A: Civil Engineering, 24(6), 2741-2751.

[27] Johari, A., Sabzi, A., & Gholaminejad, A. (2019). Reliability analysis of differential settlement of strip footings by stochastic response surface method. Iranian Journal of Science and Technology-transactions of Civil Engineering, 43(1), 37-48.

[28] Johari, A., & Fooladi, H. (2022). Simulation of the conditional models of borehole’s characteristics for slope reliability assessment. Transportation Geotechnics, 35, 100778.

[29] Johari, A., & Talebi, A. (2021). Stochastic analysis of piled-raft foundations using the random finite-element method. International Journal of Geomechanics, 21(4), 04021020.

[30] Khosravi-Hajivand, A., & Johari, A. (2024). Unsaturated soil nailing wall system reliability analysis using random finite element. Computers and Geotechnics, 173, 106554.

[31] Kroetz, H. M., Do, N., Dias, D., & Beck, A. T. (2018). Reliability of tunnel lining design using the hyperstatic reaction method. Tunnelling and Underground Space Technology, 77, 59-67.

[32] Kuhlemeyer, R. L., & Lysmer, J. (1973). Finite element method accuracy for wave propagation problems. Journal of the Geotechnical Engineering Division, 99(5), 421-427.

[33] Lee, J., & Fenves, G. L. (1998). Plastic-damage model for cyclic loading of concrete structures. Journal of Engineering Mechanics, 124(8), 892-900.

[34] Li, J., Tian, Y., & Cassidy, M. J. (2015). Failure mechanism and bearing capacity of footings buried at various depths in spatially random soil. Journal of Geotechnical and Geoenvironmental Engineering, 141(2), 04014099.

[35] Li, J. D., Yang, T. H., Liu, F. Y., Zhao, Y., Liu, H. L., Deng, W. X., Gao, Y., & Li, H. B. (2024). Modeling spatial variability of mechanical parameters of layered rock masses and its application in slope optimization at the open-pit mine. International Journal of Rock Mechanics and Mining Sciences, 181, 105859.

[36] Luo, Z., Li, Y., Zhou, S., & Di, H. (2018). Effects of vertical spatial variability on supported excavations in sands considering multiple geotechnical and structural failure modes. Computers and Geotechnics, 95, 16-29.

[37] Luo, C. R., & Peng, Y. B. (2024). Stochastic simulation of earthquake ground motions based on improved finite-fault model. Soil Dynamics and Earthquake Engineering, 176, 108669.

[38] Marano, G. C., Trentadue, F., Morrone, E., & Amara, L. (2008). Sensitivity analysis of optimum stochastic nonstationary response spectra under uncertain soil parameters. Soil Dynamics and Earthquake Engineering, 28(12), 1078-1093.

[39] Mei, X. C., Sheng, Q., Cui, Z., Zhang, M. C., & Dias, D. (2023). Experimental investigation on the mechanical and damping properties of rubber-sand-concrete prepared with recycled waste tires for aseismic isolation layer. Soil Dynamics and Earthquake Engineering, 165, 107718.

[40] Ministry of Railways of the People’s Republic of China (2006). GB 50111—2006: Code for seismic design of railway engineering. Beijing: China Planning Press (in Chinese).

[41] Ministry of Housing and Urban-Rural Development of the People’s Republic of China (2010). GB 50010—2010: Code for Design of Concrete Structures. Beijing: China Planning Press (in Chinese).

[42] Napoli, M. L., Barbero, M., & Scavia, C. (2021). Tunneling in heterogeneous rock masses with a block-in-matrix fabric. International Journal of Rock Mechanics and Mining Sciences, 138, 104655.

[43] Pang, R., Xu, B., Zou, D., & Kong, X. (2018a). Stochastic seismic performance assessment of high CFRDs based on generalized probability density evolution method. Computers and Geotechnics, 97, 233-245.

[44] Pang, R., Xu, B., Kong, X. J., Zhou, Y., & Zou, D. G. (2018b). Seismic performance evaluation of high CFRD slopes subjected to near-fault ground motions based on generalized probability density evolution method. Engineering Geology, 246, 391-401.

[45] Pang, R., Xu, B., Zhou, Y., & Song, L. F. (2021). Seismic time-history response and system reliability analysis of slopes considering uncertainty of multi-parameters and earthquake excitations. Computers and Geotechnics, 136, 104245.

[46] Phoon, K. K., Quek, S. T., & An, P. (2003). Identification of statistically homogeneous soil layers using modified Bartlett statistics. Journal of Geotechnical and Geoenvironmental Engineering, 129(7), 649-659.

[47] Renani, H. R., Martin, C. D., Varona, P., & Lorig, L. (2019). Stability analysis of slopes with spatially variable strength properties. Rock Mechanics and Rock Engineering, 52(10), 3791-3808.

[48] Singh, D. K., Mandal, A., Karumanchi, S. R., Murmu, A., & Sivakumar, N. (2018). Seismic behaviour of damaged tunnel during aftershock. Engineering Failure Analysis, 93, 44-54.

[49] Sun, B., Wang, P., Deng, M., Fang, H., Xu, J., Zhang, S., & Wang, C. (2024). Seismic performance assessment of hydraulic tunnels considering oblique incoming nonstationary stochastic SV waves based on the generalized PDEM. Tunnelling and Underground Space Technology, 143, 105481.

[50] Vanmarcke, E. (1983). Random fields:Analysis and synthesis. Cambridge: MIT Press.

[51] Wang, L., Cui, G., Zhang, C., Zhao, Y., Ma, J., & Min, B. (2024a). Failure characteristics and seismic behavior of steel basalt hybrid fiber reinforced concrete lining for the tunnel in strong earthquake areas. Engineering Failure Analysis, 162, 108357.

[52] Wang, T. T., Kwok, O. L. A., & Jeng, F. S. (2021). Seismic response of tunnels revealed in two decades following the 1999 Chi-Chi earthquake (Mw 7.6) in Taiwan: A review. Engineering Geology, 287, 106090.

[53] Wang, Y. B., He, J. J., & Shu, S. (2022). Seismic responses of rectangular tunnels in liquefiable soil considering spatial variability of soil properties. Soil Dynamics and Earthquake Engineering, 162, 107498.

[54] Wang, Z. K., Song, Z. Q., Li, C., & Liu, Y. H. (2024b). Random dynamic response analysis of an asphalt concrete core wall dam on deep overburden with double random factors. Computers and Geotechnics, 171, 106364.

[55] Wu, Y., Wang, J., Cheng, J., & Yang, S. (2024). Dimension-reduction spectral representation of soil spatial variability and its application in the efficient reliability analysis of seismic response in tunnels. Reliability Engineering & System Safety, 248, 110175.

[56] Wu, Y., Zhou, X., Gao, Y., Zhang, L., & Yang, J. (2019). Effect of soil variability on bearing capacity accounting for non-stationary characteristics of undrained shear strength. Computers and Geotechnics, 110, 199-210.

[57] Xu, B., Lu, Y. Z., & Pang, R. (2024). Seismic safety assessment of dam slopes considering rockfill softening characteristics, shear strength uncertainties, and stochastic ground motion. Structures, 62, 106180.

[58] Xu, B., Wang, G., Peng, R., & Zhou, Y. (2025). Stochastic simulation method for ground motions considering parameters uncertainty of evolutionary power spectral density model for structural reliability assessment. Mechanical Systems and Signal Processing, 237, 112941.

[59] Yao, C., He, C., Wang, T., Chen, C., Geng, P., Dong, W., Yuan, F., & Xu, G. (2024). Damages of highway tunnels during 2022 luding earthquake (Mw = 6.6). Soil Dynamics and Earthquake Engineering, 177, 108357.

[60] Zhang, H., Luo, F., Yang, S., & Wu, Y. (2023). Probabilistic analysis of crown settlement in high-speed railway tunnel constructed by sequential excavation method considering soil spatial variability. Tunnelling and Underground Space Technology, 140, 105342.

[61] Zhang, L., Fredlund, M., Fredlund, D. G., Lu, H., & Wilson, G. W. (2015). Comparison of 2-D and 3-D slope stability analyses for unsaturated soil slopes. Engineering Geology, 193, 374-383.

[62] Zhang, S., Wang, Y., Gao, Q., Ma, X., Zhou, H., & Wang, Z. (2024). Probabilistic analysis of ground settlement induced by tunnel excavation in multilayered soil considering spatial variability. Computers and Geotechnics, 165, 105951.

[63] Zhang, W., Nagger, M., Ni, P., Zhao, M., & Du, X. (2025). Nonlinear seismic response analysis of underground structures considering spatial variability of soil parameters. Tunnelling and Underground Space Technology, 159, 106445.