Abstract
This study presents a theoretical analysis of the deformation induced in an existing curved subway tunnel by a new shield tunnel crossing diagonally beneath it. A refined two-stage method is developed to address this engineering problem. In the first stage, the additional stress on the existing tunnel is calculated using Mindlin’s solution. In the second stage, the existing tunnel is modeled as an Euler–Bernoulli beam on a Pasternak foundation, explicitly incorporating the effects of tunnel curvature and a stress reduction factor for the grout-reinforced zone. The proposed method is validated against monitoring data from a case study of the Zhengzhou Metro, showing good agreement. A systematic parametric analysis investigates the influence of key factors: the clearance and intersection angle between tunnels, the curvature radius of the existing tunnel, the length of the grouted section, and Poisson’s ratio of the grouted soil. Results demonstrate that the crossing angle and grouting length are the most significant parameters affecting deformation, whereas the existing tunnel’s curvature and the grout’s Poisson’s ratio have a negligible impact.
Data availability
All data or models that support the findings of this study are available from the corresponding author upon reasonable request.
References
Mair, R. J., Taylor, R. N. & Bracegirdle, A. Subsurface settlement profiles above tunnels in clays. Geotechnique 43 (2), 315–320 (1993).
Peck, R. B. Deep excavations and tunnelling in soft ground. In: Proceedings of 7th International Conference of Soil Mechanics and Foundation Engineering, State of the Art, pp. 225–290. (1969).
Vorster, T. E., Klar, A., Soga, K. & Mair, R. J. Estimating the effects of tunneling on existing pipelines. J. Geotech. Geoenviron. Eng. 131(11), 1399–1410 (2005).
Bai, X. F. & Wang, M. S. A two-stage method for analyzing the effects of twin tunnel excavation on adjacent tunnels. China Civ. Eng. J. 49(10), 123–128 (2016).
Liu, Z., Xue, J., Ye, J. & Qian, J. A simplified two-stage method to estimate the settlement and bending moment of upper tunnel considering the interaction of undercrossing twin tunnels. Transp. Geotech. 29, 100558 (2021).
Zhang, Z., Zhang, M. X. & Wang, W. D. Two-stage method for analyzing effects on adjacent metro tunnels due to foundation pit excavation. Rock Soil Mech. 32(7), 2085–2092 (2011).
Mindlin, R. D. Force at a point in the interior of a semi-infinite solid. Physics 7 (5), 195–202 (1936).
Sagaseta, C. Analysis of undrained soil deformation due to ground loss. Geotechnique 37 (3), 301–320 (1987).
Loganathan, N. & Poulos, H. G. Analytical prediction for tunneling-induced ground movements in clays. J. Geotech. Geoenviron. Eng. 124(9), 846–856 (1998).
Liu, W., Dai, X., Sun, K., Ai, G. & Lei, T. Calculation method of longitudinal deformation of metro shield tunnel overpassing existing line at short distance. Rock. Soil. Mech. 43, 831–842 (2022).
Liang, R., Xia, T., Hong, Y. & Yu, F. Effects of above-crossing tunnelling on the existing shield tunnels. Tunn. Undergr. Space Technol. 58, 159–176 (2016).
Zhang, Z. & Huang, M. Geotechnical influence on existing subway tunnels induced by multiline tunneling in Shanghai soft soil. Comput. Geotech. 56, 121–132 (2014).
Feng, G. et al. Study of tunnel-soil interaction induced by tunneling underlying. Eng. Mech. 40 (5), 59–68 (2023).
Xu, H., Huang, H. & Zhang, D. Longitudinal deflection of existing shield tunnels due to deep excavation. Chin. J. Geotech. Eng. 34 (7), 1241–1249 (2012).
Zhang, D. M., Zong, X. & Huang, H. W. Longitudinal deformation of existing tunnel due to underlying shield tunneling. Rock. Soil. Mech. 35 (9), 2659–2666 (2014).
Ke, W. H. et al. Research on upper pipeline-soil interaction induced by shield tunnelling. Rock. Soil. Mech. 41 (1), 221–228 (2020).
Liu, T., Jiang, X., Luo, W. & Yan, J. Settlement of an existing tunnel induced by crossing shield tunneling involving residual jacking force. Symmetry 14 (7), 1462 (2022).
Ding, Z., Zhang, M. B., Zhang, X. & Wei, X. J. Theoretical analysis on the deformation of existing tunnel caused by under-crossing of large-diameter slurry shield considering construction factors. Tunn. Undergr. Space Technol. 133, 104913 (2023).
Lai, H. X., Zhou, Z. L., Tan, Z. S. & Zonglin Li, Zhao, J. P. Development of open TBM tunnelling performance prediction model based on Hydropower Classification (HC) system: A case study in Xinjiang. Rock. Mech. Bull. 100264, 2773–2304 (2025).
Zhao, J. P., Tan, Z. & Liu, X. Large Deformation Characteristics and Mechanisms of Deep-Buried Foliated Basalt Tunnel: A Case Study of the Haba Snow Mountain Tunnel. Rock. Mech. Rock. Eng. 58, 3523–3544 (2025).
Zhao, J. P., Fu, W., Yuan, S., Liu, X. & Ma, Q. A Method for Testing Rock Mass Elastic Modulus Based on Anchor Pullout Tests and Its Optimization. Int. J. Numer. Anal. Meth. Geomech. 49, 15: 3531–3547 (2025).
Tanahashi, H. Formulas for an infinitely long Bernoulli-Euler beam on the pasternak model. Soils Found. 44 (5), 109–101 (2004).
Shiba, Y., Kawashima, K., Obinata, N. & Kano, T. Evaluation procedure for seismic stress developed in shield tunnels based on seismic deformation method. Doboku Gakkai Ronbunshu. 1989 (404), 385–394 (1989).
Attewell, P. B., Yeates, J. & Selby, A. R. Soil movements induced by tunnelling and their effects on pipelines and structures. (1986).
Acknowledgements
We would like to acknowledge gratefully that the work presented in this paper is supported by Henan Provincial Natural Science Foundation General Program (252300421855) and the Key R&D Program of the Science and Technology Department of Henan Province, China (Grant No. 231111322100) and Henan Provincial Central Guidance Fund for Local Science and Technology Development Projects (Z20231811149). And this paper was completed through the collective efforts of all authors, to whom we extend our deepest gratitude.
Funding
We would like to acknowledge gratefully that the work presented in this paper is supported by the General Program of Natural Science Foundation of Henan Province, China (252300421855), the Key R&D Program of the Science and Technology Department of Henan Province, China (Grant No. 231111322100), and Henan Provincial Central Guidance Fund for Local Science and Technology Development Projects (Z20231811149).
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**Yonghui Li: ** Writing - review & editing, Supervision, Funding acquisition. **Yucheng Zhao: ** Writing - original draft, Visualization, Investigation, Data curation. **Gang Shi: ** Writing - original draft, review & editing, Methodology. **Xiaoming Wang: ** Writing - original draft, Data curation. **Shuai Wu: ** Writing - original draft, Data curation. **Wenmin Yao: ** Writing - original draft, Data curation.
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Li, Y., Zhao, Y., Shi, G. et al. Two-stage-method-based calculation and analysis of the deformation of the existing subway tunnel caused by the diagonal crossing of the new tunnel. Sci Rep (2026). https://doi.org/10.1038/s41598-026-40967-9
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DOI: https://doi.org/10.1038/s41598-026-40967-9
