Abstract
An algorithm for calculating the effective ratio of the transverse bending rigidity is established based on the segment longitudinal joint bending stiffness. With the knowledge of this effective ratio, the bending rigidity of a modified uniform rigidity ring is fully defined. To verify this developed algorithm, the effective ratios and convergence deformations of the modified uniform rigidity rings obtained with different methods are compared. Moreover, the responses of the modified uniform rigidity ring model under loading obtained from this algorithm are compared to those obtained with the existing generally accepted beam-spring model. The results show that although the bending moments obtained from these two models are different, the axial forces, horizontal convergent deformations, and vertical convergent deformations are quite consistent with each other. The modified uniform rigidity ring model built on the developed effective ratio algorithm is applicable for the analysis of the tunnel convergence deformation and the interaction between the tunnel structure and the ground during operation. This modified uniform rigidity ring model is simpler and easier to use than the beam-spring model; thus, the significance of the developed algorithm for the effective ratio of the transverse bending rigidity is demonstrated.
Highlights
Many segment ring models, such as the uniform rigidity ring model [1], modified uniform rigidity ring model [2], multi-hinge ring model [3,4], beam-spring model [5,6,7], beam-discontinuous joint model [8], and shell-spring model [9], are used for the transverse analysis of the internal force and deformation of shield-driven tunnels
An algorithm for calculating the effective ratio of the transverse bending rigidity was established, and the responses of the modified uniform rigidity ring model under loading obtained from this algorithm were compared to those obtained with the existing beam-spring model
The following conclusions are drawn: (1) Based on the assumption of the equivalence for transverse bending rigidity between the installed segment ring and the modified uniform rigidity ring, an algorithm to determine the effective ratio of the transverse bending rigidity was obtained
Summary
Many segment ring models, such as the uniform rigidity ring model [1], modified uniform rigidity ring model [2], multi-hinge ring model [3,4], beam-spring model [5,6,7], beam-discontinuous joint model [8], and shell-spring model [9], are used for the transverse analysis of the internal force and deformation of shield-driven tunnels. The correlation between the longitudinal joint bending stiffness of an installed segment ring and the effective ratio of the transverse bending rigidity of a modified uniform rigidity ring is investigated. Without considering the rotation angles of all joints, the relative rotation angle for the end sections of the straight beam spread from the installed segment ring may be expressed as Equation (3), when subjected to the pure bending moment M: LM θ1 = EI (3). Under the pure bending moment M, the relative rotation angle for the end sections of the straight beam spread from the installed segment ring can be calculated by Equation (6): θ2. Under the pure bending moment, the relative rotation angle of the end sections of the straight beam spread from the modified uniform rigidity ring can be expressed as Equation (7):. According to the equivalent condition of the transverse bending rigidity, Equation (8) can be derived and simplified into Equation (9), which is denoted as the equivalent equation of the transverse bending rigidity:
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