The Longitudinal Deformation Profile of a Rock Tunnel: An Elastic Analysis

Yonghong Wang,Guohui Zhang,Wen Du,Yang Song,Rossana Dimitri

doi:10.1155/2021/6684035

Abstract

The longitudinal deformation profile (LDP) is the profile of wall displacement versus the distance from the tunnel face. To develop LDP equations, numerical methods and in situ experiments have been used to obtain the deformation of a tunnel in three-dimensional space. However, extant approaches are inadequate in terms of explaining the mechanical relation between the wall displacement and the conditions of a tunnel (e.g., properties of rock). In this paper, an analytical approach is proposed to develop a new LDP equation. First, on the basis of the axisymmetric elastic model of a tunnel, a closed-form solution of wall displacement is derived. Then, a new LDP equation is presented according to the solution developed above; the coefficient β, defined as the ratio of the effective range of the “face effect” to the radius of the tunnel, is proposed for the first time. Finally, a case study is proposed to validate the practicability of this equation.

Highlights

The wall displacements are usually normalized with respect to the maximum wall displacement at the far end of the tunnel in order to eliminate the effects of elastic properties (E and μ) and tunnel radius (r)
E mathematical difficulties associated with calculating the wall displacements in three-dimensional space have motivated researchers to the utilization of numerical methods and in situ experiments
Where u∗ u/umax, in which u is the wall displacement at a specified x and umax is the ultimate radial displacement far away from the tunnel face, and x∗ x/r0, in which x is the distance from the face and r0 is the radius of the tunnel

Summary

Model Conditions

In equations (3)–(5), the LDPs ahead of the face (i.e., x < 0) and the LDPs behind the face (i.e., x > 0) are expressed separately. In equations (1), (2), (6), and (8), only the LDPs behind the face are proposed. E LDP ahead of the face reflects the wall deformation that has already been released before any excavation. It is not necessary to determine the exact LDP equation ahead of the face because only the LDP behind the face is considered in the calculation of actual wall displacement when the supports are installed [9, 10]. Erefore, this paper focuses on wall displacement behind the face, and the model for the rock behind the face is presented as follows. Expressed by a nonlinear monotonic increasing function, for instance, the sine function

Solutions to the Proposed Model

Case Study

Stress Equations Solved by the Stress Function