Three-dimensional numerical simulation of rock deformation in bolt-supported tunnels: A homogenization approach Samir Maghous ⇑ , Denise Bernaud, Eduardo Couto Department of Civil Engineering, Federal University of Rio Grande do Sul, Porto Alegre-RS, Brazil article info Article history: Received 7 October 2011 Received in revised form 27 February 2012 Accepted 2 April 2012 Available online 28 April 2012 Keywords: Tunnel Bolted rock Bolt/rock interface Finite element method Homogenization Embedded model abstract Despite fully grouted bolts are nowadays widely used for tunnel support, bolting design is still based in many cases on empirical or semi-empirical considerations. This paper describes a three-dimensional the- oretical and numerical model for the behavior of tunnels reinforced by bolts. From a theoretical view- point, the elastoplastic constitutive equations for the reinforced rock are derived in the framework of homogenization method. Emphasis is given to the formulation of the homogenized strength criterion with account for the bolt/rock interface properties. The anisotropic constitutive equations are then imple- mented in a 3D finite element computer code in which the processes of excavation, installation of bolts and lining placement are simulated by means of the ‘‘activation/deactivation’’ technique. Due to the multi-potential nature of the plastic flow, a specific iterative algorithm for plastic integration is proposed. The finite element model is applied to the analysis of the Kielder experimental tunnel for which in situ measured data are available. The accuracy of the numerical predictions based on homogenization method is also assessed by comparison with the results derived from implementation of the so-called embedded model. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The idea of installing metallic bolts within the ground for tunnel support has originally emerged in 1913 with the request of a technical patent to Germany authorities (Kovari, 2003). However, their effective use began only in the 1940s as roof support in the context of the American mining industry. From 1970s up today rock bolting has experienced a rapid increase and nowadays it is widely applied in both civil and mine tunnel projects. The success of such reinforcing system can be explained by its ease to carry out since it does not require heavy equipment for installation and by its relative flexibility in the sense that bolt density can be punctually adapted to account for local geotechnical properties. Furthermore, the use of bolts is an essential component of the New Austrian Tunneling Method (NATM) or the so-called ‘convergence-confinement’ approach for tunnel construction (Panet, 1995) and, for this reason, application of these approaches has accompanied the expansion of this reinforcement technique for tunnels excavated in soft rocks. Schematically rock bolts fall within the general category of rock reinforcement systems that act as structural inclusion installed within the ground with the objective to improve the behavior of the ground (Windsor and Thompson, 1993). Mechanically speak- ing, the mechanisms governing the interaction prevailing between the rock bolts and the surrounding rock mass are complex and still not well understood. Hence, the modeling of such interaction is still a work in progress, and a comprehensive mechanical descrip- tion is needed for reliable predictions regarding the performance of the rock bolting. The reader may refer for instance to the recent pa- per by Bobet and Einstein (2010) for a review on the technique of rock bolting. The geocomposite defined by association of rock bolts and the rock mass is strongly heterogeneous. Due to the large number of rock bolts involved in such a reinforcement technique, the imple- mentation of direct numerical finite element analyses based on the explicit description by anchor element embedded within the rock mass, would require the use of sophisticated computational capabilities. Indeed, an accurate 3D geometrical discretization of the reinforced zones should involve rock mass finite elements whose typical size is close to the bolt diameter. This results in tedious mesh generation procedures with a high number of de- grees of freedom. These difficulties have given rise to alternatives approaches to handle this kind of problem. On one hand, the homogenization approach is a theoretical-based alternative which offers an appropriate framework in the field of rock bolting. On the other hand, the so-called ‘‘embedded model’’ appears as a quite efficient numerical direct method to simulate bolt-supported tun- nels in the case of perfect bonding assumed for the interface bolts/ surrounding rock. In terms of numerical implementation of both methods, the rock bolts are not treated as individual structural 0886-7798/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tust.2012.04.008 ⇑ Corresponding author. Tel.: +55 51 33 08 35 88; fax: +55 51 33 08 39 99. E-mail address: samir.maghous@ufrgs.br (S. Maghous). Tunnelling and Underground Space Technology 31 (2012) 68–79 Contents lists available at SciVerse ScienceDirect Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust