Mathematical Problems in Engineering

Volume 2018, Article ID 8718274, 14 pages

https://doi.org/10.1155/2018/8718274

## Analysis of Soil-Compacting Effect Caused by Shield Tunneling Using Three-Dimensional Elastoplastic Solution of Cylindrical Cavity Expansion

^{1}Department of Civil Engineering, Shanghai University, 99 Shangda Road, Shanghai 200444, China^{2}Department of Civil Engineering, Shanghai University, 99 Shangda Road, Shanghai 200072, China^{3}Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong^{4}Department of Engineering, College of Engineering Mathematics and Physical Sciences, University of Exeter, Exeter, Devon EX4 4QF, UK

Correspondence should be addressed to Mengxi Zhang; nc.ude.uhs.i@gnahzxm

Received 16 September 2017; Revised 10 March 2018; Accepted 1 April 2018; Published 27 May 2018

Academic Editor: Edoardo Artioli

Copyright © 2018 Mengxi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Soil squeezing effect and formation disturbance caused by tunnel excavation can be simulated by cylindrical cavity expansion due to the comparability between tunneling and cavity expansion. Although most of the existing theoretical derivation is based on simple constitutive model of soil foundation, not only the relation between principal stress components was simplified in the solution process, but also the stress history, initial stress anisotropy, and stress-induced anisotropy of structural soil were neglected. The mechanical characteristics of soil are closely related to its stress history, so there is a gap between the above research and the actual engineering conditions. A three-dimensional elastoplastic solution of cylindrical cavity expansion is obtained based on the theory of critical state soil mechanics and engineering characteristics of shield tunneling. In order to fully consider the influence of initial anisotropy and induced anisotropy on the mechanical behavior of soils, the soil elastoplastic constitutive relation of cavity expansion is described in the course of -based modified Cam-clay (-MCC) model after soil yielding. An equation with equal number of variables is obtained under the elastic-plastic boundary condition based on the Lagrange multiplier method. By solving the extreme value of the original function, the analytical solution of radial, tangential, and vertical effective stresses distribution around the circular tunnel excavation is obtained. In addition, changes of elastic deformation area and plastic deformation area for soil during the shield excavation have been analyzed. Calculation results are compared with the numerical solutions which usually consider isotropic soil behavior as the basic assumption. In this paper, a constitutive model which is more consistent with the actual mechanical behavior of the soil and the construction process of the shield tunnel is considered. Therefore, the numerical solutions are more realistic and suitable for the shield excavation analysis and can provide theoretical guidance required for design of shield tunneling.

#### 1. Introduction

Theoretical analysis of cylindrical cavity expansion theory in geotechnical engineering has been widely used in pile sinking, shield tunneling, static cone penetration test (CPT), pressuremeter test (PMT), and mixing pile construction disturbance problems [1–6]. Hill (1950) first proposed the spherical cavity expansion method and derived a general solution of stresses and displacements in Tresca material [7]. Vesic (1972) obtained the classical solutions considering the plastic zone volume change of ideal elastic-plastic expansion problem based on Mohr-Coulomb model [8]. Cao et al. (2001) presented closed form solutions for both spherical and cylindrical cavities by assuming the ultimate deviator stress distributions in the plastic zone and taking the in situ stress conditions into consideration [9]. Alonso et al. (2003) obtained the self-similar solution for the circular tunnel in strain-softening rock masses [10]. Wheeler et al. (2003) and Nakano et al. (2005) adjusted yield surface equation on the basis of -based modified Cam-clay, considering consolidation caused by induced anisotropy and its evolution law in the loading process [11, 12]. Yin and Hicher (2008) identified parameters of modified Cam-clay model with viscosity of soil from the cavity expansion [13]. The problem of cavity expansion and cavity contraction has attracted much attention in geotechnical problems with application to the bearing capacity of deep foundations, interpretation of pressuremeter tests, breakout resistance of anchors, pile driving, wellbore instability, underground excavation, and blasting fracturing by explosives.

However, based on the above-mentioned researches, the conventional cylinder expansion theory assumes that initial stress is isotropic (static earth pressure coefficient ). Due to the sedimentation history, mineralogical characteristics, and other several factors, the initial stress of soil layer is usually anisotropic. For example, the vertical stress and horizontal stress of the cylinder bore are not the same in the horizontal tunneling construction of the underground pipe laying, tunnel engineering. As a result, the traditional theory of cylindrical expansion generally used in practical engineering fails in explaining some practical phenomenon. Also, the traditional cylindrical expansion theory assumes that the cylindrical boundary condition is controlled by displacement, which presents around the cylinder in the form of symmetrical distribution as a function of distance. The traditional control theory can be applied in geotechnical problems analysis such as cylindrical pile and static cone penetration test. But in the practical engineering of shield tunnel, boundary condition of the cylinder expansion is controlled by stress and displacement of the cylinder, which was unable to satisfy the condition of axial symmetry. Hence, it may not be reasonable to explain such geotechnical problems considering traditional theory of cylindrical expansion.

In this paper, a three-dimensional elastoplastic solution of cylindrical cavity expansion is obtained based on the theory of critical state soil mechanics and engineering characteristics of shield tunneling to calculate the undrained cylindrical cavity expansion considered anisotropic clay of . This research work not only contributes to the theoretical values in field of geotechnical engineering but also has an important practical engineering significance.

#### 2. Definition of Soil-Compacting Effect and Mechanical Model

##### 2.1. Definition of Soil-Compacting Effect

Shield tunneling is a typical three-dimensional problem. Soil deformation is closely related to the relative position of the shield machine. In actual shield construction, earth pressure and support pressure ahead of excavation face are not completely balanced [15]. When support pressure is greater than passive earth pressure, the soil is extruded and then soil-compacting effect presents. And the free surface may collapse if support pressure is less than active earth pressure. This paper mainly aims to study the soil-compacting effect produced when the support pressure is larger than passive earth pressure. Soil-compacting effect of shield tunneling is shown in Figure 1.