# Design Study of Steel Fibre Reinforced Concrete Shaft Lining for Swelling Ground in Toronto, Canada
## 5 Numerical Analysis  
### Analysis Method and Input Data  
The numerical analysis was performed to investigate whether the recommended installation timing of the shaft lining is appropriate under the swelling behaviour of Georgian Bay Formation bedrock [40]. In addition, the long-term stability of the shaft lining installed 90 days after excavation was assessed. The Fast Lagrangian Analysis of Continua (FLAC) 2D, two-dimensional continuum explicit finite element analysis code developed by Iasca consulting group Inc., was used. This software has been widely used for geotechnical analysis application in the fields of civil mining engineering [56; 57; 58]. A user-defined time-dependent deformation (TDD) constitutive model developed by the author, specifically for Georgian Bay shale, was used with a creep process in FLAC 2D to simulate the swelling behaviour of the bedrocks [44]. This constitutive model, based on the Mohr-Coulomb elastic/perfectly plastic material model, is derived from observations in the laboratory experiments that the swelling strains in the principle swelling directions of a shaley rock specimen increase linearly with the logarithm of time, and the swelling strains are reduced in both the parallel and perpendicular directions by the application of stress on the rock specimen [59; 60; 61; 62; 63]. The geotechnical parameters required for numerical analysis in the FLAC model were derived from geological reports, lab test results, and historical project data from around the proposed Toronto region. It is known that the shaley bedrock in the Southern Ontario region in Toronto shows high horizontal in-situ stresses. The maximum and minimum  
Figure 7: Thrust—Moment interaction diagram for the shaft. vertical stresses at the bottom of the excavation (70 m-deep) determined by the weight of overburden and the maximum and minimum horizontal stresses were 15.0 and 5.0 MPa, respectively, based on the geological baseline report [40]. The estimated parameters applied in the numerical analysis are summarised in Table 5.  
First, the initial ground and circular shape of the shaft boundary were modelled using the mesh (green cross lines) generation technique to divide a component into a large number of finite elements, as depicted in Figure 8. For computational efficiency, fine mesh was used around the shaft, which is the region of interest, while a coarser mesh was used other regions. The model boundaries were set to at least four times the excavation dimensions of the shaft wall to minimise the boundary effect. The maximum and minimum in-situ horizontal field stresses were initially applied to the ground. Joints and dilation of rock were not considered in this analysis, and surcharge loads also were not considered because the vertical stress includes both the soil and rock overburden stress depending on the depth. Adequate drainage during the construction process was assumed to relieve any hydrostatic pressures within the rock joints. The effect of the long-term groundwater pressure was included in both the in-situ stress condition and the modelling of the shaft excavation. After the initialisation of the model, the entire shaft was excavated in one stage, and surrounding rock mass was modelled left to freely deform elastically with swelling behaviour for 90 days. After the relaxation of the ground, the shaft lining was installed. The shaft linings were subjected to the stresses because of the swelling behaviour and field horizontal stresses. The concrete lining is considered as a linear-elastic material. The mobilised lining forces were analysed for 100-years to assess the long-term TDD-related pressure on the shaft walls. The shaft wall was designed for a 200 kPa TDD loading, and a Mohr-Coulomb elastic/perfectly plastic failure criterion was applied to the intact rock mass.