Stress–strain assessment of tunnel lining with regard to physical and structural nonlinearity

The article describes an approach to stress–strain assessment of precast tunnel lining with regard to physical and structural nonlinearity. The analysis involved three types of precast lining used in the Saint-Petersburg Metro: reinforced concrete segmental lining with a diameter of 9.8 m, cast-iron segmental lining with a diameter of 5.5 m and reinforced concrete block lining 5-BNL-2 with a diameter of 5.62 m. Using the finite element method, numerical stress–strain modeling of the lining segments was performed with regard to the contact interaction between them. The calculation involved nonlinear models of deformation of structural materials: model of plastic deformation of gray cast iron and elastoplastic model of deformation of concrete with accumulation of defects. The dependences of the achieved value of a bending moment on the included angle and the graphic charts of a rotational stiffness and axial loads are obtained. It is found that the influence of the axial load on the rotational stiffness greatly differs in different types of lining: from 15% in the cast-iron lining to a three-time increase in the reinforced concrete lining. The asymmetry is revealed in the behavior of joints at the positive and negative included angles, conditioned by a configuration of bolting. The research findings can be used for optimizing structural designs of precast lining for transportation tunnels. 

Karasev M. A., Tay Tien Nguyen, Marinchenko M. A. Stress–strain assessment of tunnel lining with regard to physical and structural nonlinearity. MIAB. Mining Inf. Anal. Bull. 2026;(10):53-70. [In Russ]. DOI: 10.25018/0236_1493_2026_10_0_53.

M.A. Karasev¹, Dr. Sci. (Eng.), Professor, e-mail: karasevma@gmail.com, ORCID ID: 0000-0001-8939-0807, 
Tay Tien Nguyen, Cand. Sci. (Eng.), Lecturer, Hanoi University of Mining and Geology, Hanoi, Vietnam, e-mail: taitien12@gmail.com, ORCID ID: 0000-0002-5246-9252, 
M.A. Marinchenko¹, Graduate Student, e-mail: mikhailmarincenko@gmail.com,
¹ Empress Catherine II Saint-Petersburg Mining University, 199106, Saint-Petersburg, Russia.

M.A. Marinchenko, e-mail: mikhailmarincenko@gmail.com.

1. Jusoh S., Mohamad H., Marto A., Kasim F. Assessment on segment joint to improve soil-tunnel interaction. Proceedings of the MATEC Web of Conferences. 2018, vol. 250, article 01005. DOI: 10.1051/matecconf/201825001005.

2. Yang F., Liu G., Liu W., Liu K., Wang Y.- qiao, Yao H. Modeling the segmental lining behavior using an analytical segmental joint model. Buildings. 2024, vol. 14, no. 11, article 3542. DOI: 10.3311/PPci.37309.

3. Protosenya A., Ivanov A. Data-driven prediction of stress–strain fields around interacting mining excavations in jointed rock: A comparative study of surrogate models. Preprints. 2025, 2025120880. DOI: 10.20944/preprints202512.0880.v1.

4. Li X., Yang S., Dai R., Ai Z. Analytical solutions for the internal forces of lining structure in shallow double-arched tunnel subject to unsymmetrical loads. Journal of Rock Mechanics and Geotechnical Engineering. 2020, vol. 12, no. 6, pp. 1214—1225. DOI: 10.1088/1755-1315/570/5/052029.

5. Shi L., Wang D., Zhang Y., Han F. An analytical solution for internal forces of shallow circular low-to-vacuum tunnel linings in soft soils. Scientific Reports. 2024, vol. 15, article 10234. DOI: 10.1038/s41598-024-83062-7.

6. Yang F., Cao S., Li Q. An analytical model for the rotational behavior of concrete segmental joints with gaskets. Journal of Bridge Engineering. 2019, vol. 24, no. 5, article 04019030. DOI: 10.1177/1369433219852566.

7. Zhang j., Mang H., Liu X., Yuan Y. On a nonlinear hybrid method for multiscale analysis of a bearing-capacity test of a real-scale segmental tunnel ring. International Journal for Numerical and Analytical Methods in Geomechanics. 2019, vol. 43, no. 5, pp. 1120—1139. DOI: 10.1002/nag.2894.

8. Xiaodi Z., Wang J., Chen Q., Lim Z. Analytical method for segmental tunnel linings reinforced by secondary lining considering interfacial slippage and detachment. Journal of Engineering Mechanics. 2021, vol. 147, no. 6, article 04021027. DOI: 10.1061/(ASCE)GM.1943-5622.0002025.

9. Han X., Xia Y., Wang X., Chai L. Complex variable solutions for forces and displacements of circular lined tunnels. Computers and Geotechnics. 2018, vol. 101, pp. 45—57. DOI: 10.1155/2018/2037845.

10. Amjad R., Zhang Y., Liu X. An analytical solution for shear bearing capacity of circumferential joints of precast concrete segmental tunnel linings considering dowel action. Canadian Journal of Civil Engineering. 2024, vol. 51, no. 8, pp. 625—637. DOI: 10.1002/suco.202400250.

11. Tang J., He M., Bian H., Qiao Y. Analytical solution for longitudinal seismic responses of circular tunnel crossing fault zone. Tunnelling and Underground Space Technology. 2024, vol. 150, article 105831. DOI: 10.1002/nag.3841.

12. Moreno-Martínez J. Y., Galván A., Peña F., Carpio F. Practical model proposed for the structural analysis of segmental tunnels. Materials. 2020, vol. 13, no. 24, article 5749. DOI: 10.3390/app10238514.

13. Lakirouhani A. Evaluation of analytical solutions and two-dimensional models in estimating the internal forces of tunnel lining against seismic loading, compared with three-dimensional analysis. Journal of Earthquake Engineering. 2022, vol. 26, no. 6, pp. 3182—3204. DOI: 10.1007/S12517-022-10385-Y.

14. Jiang X., Luo W., Zhu B., Yan J., Liu T. Evaluation of longitudinal equivalent bending stiffness of shield tunnel with residual jacking force. Underground Space. 2023, vol. 12, pp. 1—15. DOI: 10.1016/j.undsp.2023.03.002.

15. Rossi M., Mroueh H., Bian H., Cormey F. Synthesis of numerical methods for the design of segmental tunnel lining. MATEC Web of Conferences. EDP Sciences. 2019, vol. 295, article 03008. DOI: 10.1051/matecconf/201929503008.

16. Zhang W., De Corte W., Liu X., Taerwe L. Influence of rotational stiffness modeling on the joint behavior of quasi-rectangular shield tunnel linings. Applied Sciences. 2020, vol. 10, no. 23, article 8396. DOI: 10.3390/app10238396.

17. Wu L., Li M., Bai H., Yun Y., Li H., Feng Y. An iterative calculation method for internal forces and deformation of curved tunnel lining. KSCE Journal of Civil Engineering. 2024, vol. 28, pp. 1522—1533. DOI: 10.1007/s12205-024-0302-6.

18. Topchiy D., Yurgaytis A., Sayadova J., Shinkarenko A. Basic prerequisites for technical inspection of castiron lining elements in the construction of Moscow Metro facilities. E3S Web of Conferences. 2019, vol. 164, no. 1, article 02027. DOI: 10.1051/e3sconf/202016402027.

19. Li W., Li P., Xie X., Cao Y. A novel back-analysis approach for the external loads on shield tunnel lining in service based on monitored deformation. Advances in Civil Engineering. 2023, vol. 2023, article 5571682. DOI: 10.1155/2023/8128701.

20. Protosenya A., Kumov V. Geomechanical substantiation of soil stability during tunnel construction by shield tunneling complexes in layered massifs. Eng. 2026, vol. 7, no. 3, article 108. DOI: 10.3390/eng7030108.

21. Majdi A., Ajamzadeh H., Nadimi S. Investigation of moment‑rotation relation in different joint types and evaluation of their effects on segmental tunnel lining. Arabian Journal of Geosciences. 2016, vol. 9, no. 7, article 512. DOI: 10.1007/s12517-016-2538-z.

22. Petraroi D., Mark P. Variable, full-scale tester for tunnel linings. Structural Concrete. 2021, vol. 22, no. 6, pp. 3353—3367. DOI: 10.1002/suco.202000806.

23. Yu Y., Wang J., Hu X. Damage evolution and failure mechanism of segmental tunnel lining. Civil Engineering Journal. 2025, vol. 11, no. 8. DOI: 10.28991/CEJ-2025-011-08-03.

24. Zhang Y., Ai S., Yang X., Zhou L. Mechanical properties of segmental joints strengthened by assembled reinforcement structures for shield tunnel linings. Journal of Zhejiang University – Science  A. 2024, vol. 25, pp. 1037—1050. DOI: 10.1631/jzus.A2300455.

25. Protosenya A., Vilner M. Assessment of excavation intersections' stability in jointed rock masses using the discontinuum approach. The Mining-Geological-Petroleum Engineering Bulletin. 2022, vol. 37, no. 2, pp. 137—147. DOI: 10.17794/rgn.2022.2.12.

26. Belyakov N., Smirnova O., Alekseev A., Tan H. Numerical simulation of the mechanical behavior of fiber-reinforced cement composites subjected dynamic loading. Applied Sciences. 2021, vol. 11, no. 3, article 1112. DOI: 10.3390/app11031112.

27. Volokhov E. M., Kozhukharova V. K., Zelentsov S. N., Mukminova D. Z., Isaev A. A. Assessment of foundation-base interaction effects by numerical modeling methods under undermining conditions of the Mariinsky Theatre building in St. Petersburg. Journal of Mining Institute. 2025, vol. 276, issue 1, pp. 16—29. [In Russ].

28. Shi Q., Liu Y., Wang Z., Peng Z. Failure analysis of high-strength bolts in spherical joint assembly under in-plane quasi-static loading. Journal of Failure Analysis and Prevention. 2023. DOI: 10.1007/s11668-023-01656-1.

29. Iovlev G.. Katerov A., Andreeva A., Ageeva A. Concrete damage plasticity model application to predict stress-strain behavior of waterproof strata in deep rock salt deposits. Preprints 2026, 2026030207. DOI: 10.20944/preprints202603.0207.v1.

30. Lubliner J., Oliver J., Oller S., Onate E. Concrete damaged plasticity model: everything you need to know. Journal of Engineering Mechanics. 1989, vol. 113, no. 10, pp. 1548—1565. DOI: 10.1061/(ASCE)0733-9399(1989)113:10(1548).

31. Trushko V. L., Klimov V. Y., Baeva E. K., Ozhigin A. Y. Geomechanical substantiation of the technology of constructing modular pile foundations of technological platforms in permafrost rocks. Geotechnics. 2025, vol. 5, article 79. DOI: 10.3390/geotechnics5040079.

32. Protosenya A. G., Alekseev A. V., Verbilo P. E. Prediction of the stress-strain state and stability of the front of tunnel face at the intersection of disturbed zones of the soil mass. Journal of Mining Institute. 2022, vol. 254, pp. 252—260. [In Russ]. DOI: 10.31897/PMI.2022.26.

33. Karasev M. A., Nguyen T. T. Method for predicting the stress state of the lining of underground structures of quasi-rectangular and arched shapes. Journal of Mining Institute. 2022, vol. 257, pp. 807—821. [In Russ]. DOI: 10.31897/PMI.2022.17.

34. Miyachi T., Fukuda T. Numerical simulation and simplified analytical model for the longitudinal joint bending stiffness of a tunnel considering axial force. Structural Concrete. 2021, vol. 22, no. 6, pp. 3368—3384. DOI: 10.1002/suco.202000796.

35. Wang Y., Yang F., Liu G., Liu W., Liu K.- qiao, Yao H. Modeling the segmental lining behavior using an analytical segmental joint model. Periodica Polytechnica Civil Engineering. 2025, vol. 69, no. 1, pp. 280—292. DOI: 10.3311/PPci.37309.

36. Liu Y., Huang B., Yuan M., Xu Y., Wang W. Analytical solution to the problem of lining void in deeply buried circular tunnel. Mechanics of Solids. 2024, vol. 59, pp. 873—884. DOI: 10.1134/S0025654423602537. 

37. Demenkov P. A., Romanova E. L. Regularities of plastic deformation zone formation around unsupported shafts in tectonically disturbed massive rock. Geosciences. 2025, vol. 15, no. 1, article 23. DOI: 10.3390/geosciences15010023.

38. Belikov A. A., Belyakov N. A. Method for ensuring geomechanical safety during under-mining of waterproof strata. Sustainable Development of Mountain Territories. 2025, vol. 17, no. 1, pp. 115—125. [In Russ]. DOI: 10.21177/1998-4502-2025-17-1-115-125.

39. Trushko O. V., Trushko V. L., Demenkov P. A. Arrangement of multistory underground parking garages in complex engineering and geological environment. International Journal of Mathematical, Engineering and Management Sciences. 2020, vol. 5, no. 5, pp. 897—912. DOI: 10.33889/IJMEMS.2020.5.5.069.