Unified elastoplastic solution for the stress and displacement of tunnel lining and surrounding rock in cold areas considering heterogeneous characteristics

The lining and surrounding rock around tunnels constructed in cold areas exhibit nonuniform material properties due to the existence of a temperature field. This study considered the effects of these properties on the integrity of tunnel structures. By establishing an elastoplastic mechanical model, analytical solutions to the stress and displacement under five different elastoplastic states were derived and compared based on distinct yield criteria. The findings showed that with increasing relative radius, the displacement in the lining elastic zone initially decreased before increasing, whereas the shift in the plastic zone continued to increase. The displacement in the elastic zone of the frozen surrounding rock intensified with increasing relative radius, whereas the shift in the plastic zone experienced a gradual decline. The displacement of the inner wall of the lining was always greater than that of the outer wall, and this phenomenon occurred only after the frozen surrounding rock exhibited a plastic zone. The maximum displacements of the liner in its elastically limited and plastically limited states were 1.39, 1.77, 2.28, and 2.37 mm and 15.93, 25.51, 44.28, and 48.58 mm based on the Drucker–Prager (DP), Mohr–Coulomb (MC), Tresca, and double-shear strength criteria, respectively; the maximum limit displacements of the frozen surrounding rock were 12.74, 20.41, 35.43, and 38.87 mm and 85.32, 103.38, 569.23, and 680.43 mm, respectively. With increasing relative radius, the radial stresses within both the lining and the frozen surrounding rock intensified; and the tangential stress in the elastic zone of the lining decreased whereas the opposite change rule was observed in the plastic zone. The tangential stresses in the frozen surrounding rock and lining exhibited the same variation trend. Based on calculations with four distinct strength criteria, the elastic and plastic ultimate bearing capacities of the lining were 1.81, 2.31, 2.95, and 3.07 MPa, and 3.31, 4.84, 7.48, and 8.05 MPa, while those of the frozen surrounding rock were 8.52, 13.24, 22.17, and 24.18 MPa, and 16.76, 32.46, 74.15, and 85.64 MPa. In addition, with the expansion of the plastic zone, the phenomenon of a sudden change in the tangential stress at location r₂ became progressively attenuated. The study findings can provide some theoretical guidance for the design and construction of tunnels in cold areas.