Deformation-based longitudinal equivalent stiffness beam model for shield tunnel and its application in seismic deformation method

In the longitudinal seismic deformation method for shield tunnels, one of the most commonly used is the longitudinal equivalent stiffness beam model (LES) for simulating the mechanical behavior of the lining. In this model, axial deformation and bending deformation are independent, so the equivalent stiffness is a constant value. However, the actual situation is that axial deformation and bending deformation occur simultaneously, which is not considered in LES. At present, we are not clear about the effect on the calculation results when axial deformation and bending deformation occur simultaneously. Therefore, in this paper, we improve the traditional LES by taking the relative deformation as a load and considering the coordinated deformation of axial and bending degrees of freedom. This improved model is called DNLES, and its neutral axis equations are an explicit expression. Then, we propose an iterative algorithm to solve the calculation model of the DNLES-based longitudinal seismic deformation method. Through a calculation example, we find that the internal forces based on LES are notably underestimated than those of DNLES in the compression bending zone, while are overestimated in the tension bending zone. When considering the combined effect, the maximum bending moment reached 13.7 times that of the LES model, and the axial pressure and tension were about 1.14 and 0.96 times, respectively. Further analysis reveals the coordinated deformation process in the axial and bending directions of the shield tunnel, which leads to a consequent change in equivalent stiffness. This explains why, in the longitudinal seismic deformation method, the traditional LES may result in unreasonable calculation results.