An Analytical Solution for Deep Circular Tunnels in Rock Based on GZZ Criterion
Authors: Chen Haohua , Zhu Hehua , Zhang Lianyang 
Published: August 2024

ABSTRACT:
Tunnels in rock are often used for transportation, mining, and storage. It is important to perform accurate analysis and design so as to ensure the safety of tunnels during construction and operation. Currently, most of the analysis and design methods of tunnels in rock are based on the Hoek-Brown criterion, which ignores the effect of the intermediate principal stress (i.e., the three-dimensional (3D) strength) and might underestimate the rock strength. In this regard, this paper presents a new analytical solution for deep circular tunnels in rock with consideration of disturbed zone and 3D strength. The rock is assumed to be elastic–brittle– plastic and governed by a 3D Hoek–Brown yield criterion, the newly modified generalized Zhang-Zhu (GZZ) criterion. Based on the equilibrium equation, constitutive law, and large-strain theory, the governing equations for the stresses and radial displacement around the tunnel were derived and solved by using the programming language MATLAB. The proposed solution was validated by using it to analyze a tunnel and comparing the results with those from numerical analysis using commercial software FLAC3D. Finally, extensive parametric studies were performed on tunnels in both poor-quality and good-quality rock masses with a special focus on stresses and radial displacement of the rock mass. The proposed analytical solution can be used for the preliminary analysis and design of tunnels in different rock masses with the consideration of 3D strength.

1. INTRODUCTION
Tunnels in rock are often used for transportation, mining, and storage. It is important to perform accurate analysis and design so as to ensure the safety of tunnels during construction and operation. Cavity contraction theory has been used to analyze tunnels by regarding the tunnel as a cylindrical cavity (Mair and Taylor 1993, Wang 1996, Carranza-Torres and Fairhurst 1999, Yu 1999). Early studies on cavity contraction were developed mostly using the linear Mohr-Coulomb criterion (Florence and Scher 1978; Kennedy and Lindberg 1978). To consider the non-linearity of rock, stresses and displacements around a tunnel were analyzed based on the original Hoek-Brown criterion with some simplifying assumptions (Brown et al. 1983; Wang 1996). Later, Carranza-Torres and Fairhurst (1999) presented an analytical solution to the cavity contraction problem in the original Hoek-Brown criterion-based rock mass without additional assumptions, but the elastic-perfectly-plastic behavior of rock mass was considered. Park and Kim (2006) provided a solution to tunnel in an elastic-brittle-plastic rock mass, but the Mohr-Coulomb criterion rather than the Hoek-Brown criterion was adopted as the potential function in their solution. Since both the Mohr-Coulomb and Hoek-Brown criteria are two-dimensional (2D) strength criteria with respect to the major and minor principal stresses, the intermediate principal stress was ignored in the abovementioned solutions. Wang et al. (2012) proposed a more general solution to the cavity contraction problem using the Mohr-Coulomb criterion by considering the out-of-plane stress, and the plastic strain in the out-of-plane direction was simply assumed to be zero. More recently, to consider the large strain around a tunnel, solutions based on the Mohr-Coulomb criterion were presented to analyze excavated tunnels (Park 2014; Zhang et al. 2019).