Deformation-based longitudinal equivalent stiffness beam model for shield tunnel and its application in seismic deformation method

Abstract

In the longitudinal seismic deformation method for shield tunnels, one of the most commonly used is the longitudinal equivalent stiffness beam model (LES) for simulating the mechanical behavior of the lining. In this model, axial deformation and bending deformation are independent, so the equivalent stiffness is a constant value. However, the actual situation is that axial deformation and bending deformation occur simultaneously, which is not considered in LES. At present, we are not clear about the effect on the calculation results when axial deformation and bending deformation occur simultaneously. Therefore, in this paper, we improve the traditional LES by taking the relative deformation as a load and considering the coordinated deformation of axial and bending degrees of freedom. This improved model is called DNLES, and its neutral axis equations are an explicit expression. Then, we propose an iterative algorithm to solve the calculation model of the DNLES-based longitudinal seismic deformation method. Through a calculation example, we find that the internal forces based on LES are notably underestimated than those of DNLES in the compression bending zone, while are overestimated in the tension bending zone. When considering the combined effect, the maximum bending moment reached 13.7 times that of the LES model, and the axial pressure and tension were about 1.14 and 0.96 times, respectively. Further analysis reveals the coordinated deformation process in the axial and bending directions of the shield tunnel, which leads to a consequent change in equivalent stiffness. This explains why, in the longitudinal seismic deformation method, the traditional LES may result in unreasonable calculation results.

Keywords

Shield tunnel / Longitudinal seismic deformation method / Longitudinal equivalent stiffness beam / Combined axial and bending deformation

CRediT authorship contribution statement

Pingliang Chen: Conceptualization, Methodology, Validation, Visualization, Writing - original draft, Writing - review & editing. Ping Geng: Conceptualization, Funding acquisition, Project administration, Supervision, Writing - review & editing. Junbo Chen: Data curation, Software, Visualization. Qi Yang: Data curation, Software, Visualization.

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 authors gratefully acknowledge the National Natural Science Foundation of China (Grant Nos. 52130808 and 51878566) and National Key R & D Program of China (Key Projects for International Science and Technology Innovation Cooperation between Governments, Grant No. 2022YFE0104300).

References

[1] Geng, P., Chen, P. L., Zhang, J., He, C., & Yan, Q. X. (2017). Nonlinear longitudinal equivalent bending stiffness of shield tunnel under the combined effect of axial force and bending moment. Chinese Journal of Rock Mechanics and Engineering, 36, 2522-2534 (in Chinese).

[2] Geng, P., Mei, S. Y., Zhang, J., Chen, P. L., Zhang, Y. Y., & Yan, Q. X. (2019). Study on seismic performance of shield tunnels under combined effect of axial force and bending moment in the longitudinal direction. Tunnelling and Underground Space Technology, 91, 103004.

[3] Guo, X., Geng, P., Wang, Q., Chen, C., Tang, R., & Yang, Q. (2021). Full-scale test on seismic performance of circumferential joint of shield-driven tunnel. Soil Dynamics and Earthquake Engineering, 151, 106957.

[4] He, C., & Koizumi, A. (1999). Shaking tests and dynamic behavior analyses in longitudinal direction of shield tunnel under transverse seismic force. Doboku Gakkai Ronbunshu, 1999(624), 145-164.

[5] Huang, D., Jiang, H., Xu, C., Li, X., & Zhan, T. (2023). Analytical algorithm of longitudinal bending stiffness of shield tunnel considering longitudinal residual jacking force. Tunnelling and Underground Space Technology, 137, 105146.

[6] Kiyomiya, O. (1995). Earthquake-resistant design features of immersed tunnels in Japan. Tunnelling and Underground Space Technology, 10(4), 463-475.

[7] Kuribayashi, E., Iwasaki, T., & Kawashima, K. (1974). Dynamic behaviour of a subsurface tubular structure. Bulletin of the NEWZEALAND Society for Earthquake Engineering, 7(4), 200-209.

[8] Li, X. J., Zhou, X. Z., Hong, B. C., & Zhu, H. H. (2019). Experimental and analytical study on longitudinal bending behavior of shield tunnel subjected to longitudinal axial forces. Tunnelling and Underground Space Technology, 86, 128-137.

[9] Li, X. Y., Liu, G. B., Yang, X., & Han, C. L. (2014). Deformation and stress of tunnel structures based on modified longitudinal equivalent continuous model. Chinese Journal of Geotechnical Engineering, 36(4), 662-670 (in Chinese).

[10] Liang, R. Z., Wang, K. C., Huang, L., Sun, L. W., Li, Z. C., Zhang, L., & Wu, X. J. (2022). Analytical solution for longitudinal equivalent bending stiffness of quasi-rectangular shield tunnels. Chinese Journal of Geotechnical Engineering, 44(2), 212-223 (in Chinese).

[11] Liu, D., Tian, C., Wang, F., Hu, Q., & Zuo, J. (2021). Longitudinal structural deformation mechanism of shield tunnel linings considering shearing dislocation of circumferential joints. Computers and Geotechnics, 139, 104384.

[12] Liu, J., Wang, D., & Bao, X. (2021). Longitudinal integral response deformation method for the seismic analysis of a tunnel structure. Earthquake Engineering and Engineering Vibration, 20(4), 887-904.

[13] Mikio, T., Takahiro, I., & Senzai, S. (1996). The damages and the destructive peculiarities of tunnels in urban area caused by the HYOFOKEN-HANBU earthquake. In Proceedings of the 1st JSCE Colloquium on Seismic Isolation and Seismic Response Control (pp.163-170).

[14] Ministry of Housing and Urban-Rural Development of the People’s Republic of China. (2014) Code for seismic design of urban rail transit structures, GB 50909—2014.

[15] China Planning Press Okamoto, S., Tamura, C., Kato, K., & Hamada, M. (1973). Behaviors of submerged tunnels during earthquakes. Proceedings 5th World Conference on Earthquake Engineering, 1, 544-553. (in Chinese).

[16] Owen, G.N., & Scholl, R.E. (1981). Earthquake engineering of large underground structures.

[17] Qian, Q., He, C., & Yan, Q. (2009). Analysis of dynamic response characteristics of tunnel engineering and seismic damage of tunnels in Wenchuan earthquake. Analysis and Research on Earthquake Damage of Wenchuan Earthquake, 608-618 (in Chinese).

[18] Shao, R., & Lei, Y. (2013). Longitudinal anti-seismic analysis of shield tunnel based on response deformation method. China Civil Engineering Journal, 46(S2), 260-265 (in Chinese).

[19] Shiba, Y., Kawashima, K., Obinata, N., & Kano, T. (1988). An evaluation method of longitudinal stiffness of shield tunnel linings for application to seismic response analyses. Journal of JSCE, 319-327.

[20] Shiba, Y., Kawashima, K., Obinata, N., & Kano, T. (1989). Evaluation procedure for seismic stress developed in shield tunnels based on seismic deformation method. Journal of JSCE, 385-394.

[21] Tsinidis, G., De Silva, F., Anastasopoulos, I., Bilotta, E., Bobet, A., Hashash, Y. M. A., He, C., Kampas, G., Knappett, J., Madabhushi, G., Nikitas, N., Pitilakis, K., Silvestri, F., Viggiani, G., & Fuentes, R. (2020). Seismic behaviour of tunnels: From experiments to analysis. Tunnelling and Underground Space Technology, 99, 103334.

[22] Wang, Z., Shi, C., Gong, C., Lei, M., Liu, J., & Cao, C. (2022). An enhanced analytical model for predicting the nonlinear longitudinal equivalent bending stiffness of shield tunnels incorporating combined NM actions. Tunnelling and Underground Space Technology (pp.126). Tunnelling and Underground Space Technology.

[23] Ye, F., He, C., Zhu, H. H., & Sun, H. D. (2011). Longitudinal equivalent rigidity analysis of shield tunnel considering transverse characteristics. Chinese Journal of Geotechnical Engineering, 33, 1870-1876 (in Chinese).

[24] Yu, H., Yuan, Y., Zhang, Z., Zhang, Q., & Wang, J. (2011). Application of response displacement method on seismic design of a complex underground structure. Chinese Journal of Underground Space and Engineering, 7(5), 857-862 (in Chinese).

[25] Zang, X.L. (2003). Research on longitudinal deformation of soft soil shield tunnel. [Master’s thesis, Tongji University]

[26] Zhang, W. J., Xu, X., Li, X. H., Shen, Y. D., & Zhang, M. X. (2009). Research on generalized longitudinal equivalent continuous model of shield tunnels. Chinese Journal of Rock Mechanics and Engineering, 28, 3938-3944 (in Chinese).