The stability of shallow circular tunnels in soil considering variations in cohesion with depth Nima Khezri a , Hisham Mohamad a,⇑ , Mohsen HajiHassani a , Behzad Fatahi b a UTM Construction Research Centre, Faculty of Civil Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia b School of Civil and Environmental Engineering, University of Technology, Sydney, Australia article info Article history: Received 6 March 2014 Received in revised form 28 February 2015 Accepted 13 April 2015 Keywords: Tunnel face pressure Upper bound solution Tunnel face stability abstract This paper presents an upper bound investigation of the three dimensional stability of a tunnel face in a deposit of soil whose strength varies with depth. The upper bound theorem of limit analysis incorporat- ing the linear variation of the soil cohesion with depth was used to calculate the pressure at the tunnel face of a closed face excavation. For an open face excavation, the factor of safety against the tunnel face instability was calculated using the strength reduction technique and the upper bound theorem. The results, in terms of the minimum required face pressure, were then compared with other solutions avail- able from the literature for verification, and the numerical results in the form of dimensionless design charts are also presented. In addition, a comparative study between the simplified approaches adopting a singular soil cohesion parameter representing the whole layer instead of considering its actual variation with depth is presented. It was concluded that adopting the mean soil cohesion that does not vary with depth would lead to a conservative design, that is, a higher minimum face pressure being required during construction and a lower factor of safety against face instability. However, adopting the local cohesion obtained from the tunnel face may result in underestimating the required face pressure and may lead to an unsafe design. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction The tunnel face stability and ground surface settlement due to underground tunnelling are two important criteria to be consid- ered in the design of shallow tunnels. In mechanised tunnelling, the imposing tunnel face pressure must be sufficient to maintain its stability (prevent the face from collapsing) during boring but not excessively high, as this may cause a surface blow-out. Numerous researchers have investigated the problem of tunnel face stability by laboratory and experimental methods (Broms and Bennermark, 1967; Kimura and Mair, 1981; Chambon and Corte, 1994; Takano et al., 2006). Based on the laboratory tests and field observation for tunnelling in purely cohesive soils, Broms and Bennermark (1967) introduced the stability ratio N as follows: N ¼ r s þ cH  r T c u ð1Þ where r s is the surcharge acting on the ground surface, c is the soil unit weight, H is the tunnel depth, r T is the tunnel face pressure, and c u is the undrained shear strength of the soil. Eq. (1) shows that the higher the ratio N is, the less stable it is, for example, Broms and Bennermark (1967) suggested that where N < 6, stability is main- tained. Kimura and Mair (1981) conducted centrifuge tests and derived stability charts for clays. Their test results suggested a wider range for the stability factor N of between 5 and 10, depend- ing on the ratio of the depth to the tunnel diameter (H/D) and the unlined length of the tunnel. Chambon and Corte (1994) conducted model centrifuge tests to study the stability of a tunnel face in sand. Their research aimed to determine the minimum face pressure to be applied to the face of the tunnel to optimise the cost of excavation as well as to prevent ground surface heave. They investigated the effects of the geometry and material parameters on the failure mechanism and the shape of the failure surface. Chambon and Corte (1994) concluded that the shape of the failure surface is like a chimney in the longitudinal direction where the magnitude of the limiting pressure is related to the unsupported length of the tunnel. Their results showed that the depth of a tunnel had an insignificant effect on limiting pressure; it was the diameter of the tunnel that was the influencing parameter. Takano et al. (2006) conducted some model tests using an X-ray computed tomography scanner to obtain a three dimensional (3D) visualisa- tion of the failure zone. Their results showed that the failure surface was like a logarithmic-spiral curve in a longitudinal direction with http://dx.doi.org/10.1016/j.tust.2015.04.014 0886-7798/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +60 75533109; fax: +60 75566157. E-mail address: mhisham@utm.my (H. Mohamad). Tunnelling and Underground Space Technology 49 (2015) 230–240 Contents lists available at ScienceDirect Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust