An analytical formulation involving residual force is proposed to predict the displacement of an existing structure by simplifying the tunnel as an infinite Euler–Bernoulli beam resting on a two-parameter Pasternak foundation. The feasibility is confirmed by two actual measurements at sites in the published literature. Parametric studies—including consideration of jacking force alone, jacking force and ground loss, and jacking force and equivalent bending stiffness—are carried out to study the influence on the deformation of the existing tunnel. The results show that the residual jacking force can decrease the settlement of the existing tunnel, but the significance of its effect varies under different engineering conditions. With the increase in ground loss, the beneficial effect of jacking force in preventing further aggravation of tunnel settlement gradually becomes obvious, and the jacking force can reduce the deformation more effectively when the tunnel has lower bending stiffness. As a result, it is recommended that some effective measures should be adopted to maintain sufficient residual jacking force in the segments, so as to prolong the life of the tunnel.

In this paper, a semi-analytical method called null space method is proposed to realize fast processing of bending deflection for Euler–Bernoulli beam among various types of boundary constraints. The null space method employs a basis function such as trigonometric function with unknown weight coefficients to approximate the field function. The unknown weight coefficients can be independently solved by the boundary constraint matrix, thus this method can easily keep away from the high demanding of field function to match the given boundary constraint and fastly switch among various types of boundary constraints. The effect of basis function on the final results are first confirmed as compared with the analytical solutions, and the null space method is then applied to predict the settlement of the existing tunnel due to new shield tunneling. The calculated results by null space method gives a good prediction of FEM and in-situ test results.

Existing tunnels are often subject to longitudinal bending when undercrossing tunnelling. It is of significance to more accurately evaluate the longitudinal equivalent bending stiffness (LEBS) of the existing tunnel within the influential zone. A new analytical method is proposed for the LEBS of tunnel segmental lining joints with consideration of incorporating combined action of residual jacking force and bending moment. The solution can degenerate into a special case with no residual jacking force, which agrees well with other classical solutions and validates the model and solutions. Sensitivity analyses are carried out for the bending moment, tunnel geometry, tensile stiffness of bolts and concrete grade on the LEBS, and effective ratio of the LEBS considering residual jacking force. The LEBS and the effective ratio of LEBS increase nonlinearly as an S-curve with the residual jacking force and decrease with an increasing bending moment. The results show that the LEBS of the shield tunnel is variable stiffness, which exhibits a significant nonlinearity. The maximum increment of the LEBS reaches 80.3% as the ring width increases from 1 m to 2 m, and the LEBS of the shield tunnel increases by approximately 1.3 × 10⁷ kN·m² for every 4-bolt added. The influential order on the LEBS of shield tunnels is the tunnel diameter > lining thickness > bolts diameter > ring width > the number of bolts > elastic modulus of bolts. When the effective ratio of LEBS is more than 0.85, it does not change with the ring width, lining thickness, tensile stiffness of bolts, and concrete grade. The response characteristics of the tunnel parameters on the LEBS, considering the residual jacking force, could provide a theoretical basis for the design and deformation control of shield tunnels when undercrossing tunnelling.