## 1 Introduction  
Shield tunnelling is widely used in metro tunnels because of its high safety, efficiency, and small disturbance to surroundings. Shield tunnels are made of lining rings assembled with prefabricated concrete segments bolted at the radial and longitudinal joints. The presence of longitudinal joints between the lining rings significantly affects the LEBS of the shield tunnels [1; 2]. The bending stiffness of a uniform beam having the same stress-deformation relation as that of the segmental tunnel is called the longitudinal equivalent bending stiffness (LEBS), which differs from the bending stiffness without segmental joints. The LEBS is of significance in the shield tunnels, which potentially causes some quality problems, such as excessive tunnel deformation, cracking, and leakage on the tunnel linings and the separation between the track bed and tunnel linings [3; 4; 5]. Therefore, it is of great engineering significance to more accurately evaluate the LEBS of the existing tunnel when undercrossing tunnelling. During shield tunnelling, the segmental linings are assembled ring-by-ring to provide the jacking reaction force for driving the tunnel boring machine (TBM). After shield construction completion, a certain amount of longitudinal jacking force is trapped in the segmental linings balanced by the friction resistance along the lining outer surface, which is called residual jacking force here below. The residual jacking force may have a major influence on the LEBS, such as controlling tunnel deformation and preventing water leakage between the lining rings [6; 7; 8]. Field tests have demonstrated that the residual jacking force between lining rings varies with time and generally undergoes four evolution stages: cyclical and violent fluctuations, dynamic stability, gradual decay, and stabilization [9]. Even for shield tunnels in soft soils, the long-term residual jacking force can be more than 30% of the initial peak jacking force imposed during tunnelling [9]. For tunnelling in rocks, [PERSON] [10] found that the long-term residual jacking force during the tunnel operation was approximately 70% of the initial peak jacking force. However, a theoretical study conducted by [PERSON] et al. [11] indicated that the residual jacking force would decay gradually to 20.4%-44% of the initial peak jacking force, depending on the creep coefficient of lining concrete and the deformation modulus of the surrounding ground. Previous research has confirmed that the residual jacking force exists during shield tunnelling and tunnel operation, including field tests and analytical methods. This means the residual jacking force of the shield tunnel is an objective structural internal force and shall be considered when studying the LEBS of the shield tunnel. [PERSON] et al. [12; 13] established the solution of the LEBS for a shield tunnel without residual jacking force based on the joints between the lining, and they concluded that the joints between the lining should be considered when studying the LEBS of a shield tunnel. [PERSON] et al. [14; 15; 16] carried out model tests and theoretical calculations on the LEBS of shield tunnels with no account of longitudinal pressure, and the LEBS of shield tunnels was found to be of significance on the tunnel-longitudinal deformation. [PERSON] et al. [17; 18] proposed a solution for the LEBS of a shield tunnel under the action of bending moment and discussed the influence of tunnel radius, ring width, and the number of bolts on the LEBS of a shield tunnel. Based on the numerical calculation method for longitudinal equivalent bending stiffness of a shield tunnel, [PERSON] [19] found that segment ring width and the number of longitudinal bolts influence the longitudinal bending stiffness largely. However, the studies mentioned above failed to consider the effects of the residual jacking force on the LEBS of the shield tunnel. [PERSON] [20] carried out model tests of the LEBS for shield tunnels and indicated that the effective ratio of LEBS increased with longitudinal axial force.