# Design Study of Steel Fibre Reinforced Concrete Shaft Lining for Swelling Ground in Toronto, Canada
## 4 Structural Design of Lining
### Design of Shaft Using SFRC  
The capacity of the lining can be determined by plotting an interaction diagram by calculating the bending moment and hoop force values. The lining design was based on a shaft diameter of 8 m with a 400 mm-thick lining of 35 MPa-grade concrete. The RC and SFRC lining capacities were derived based on the Canadian concrete design code CSA A23.3, using the RILEM method [43] for the SFRC lining. Four different methods with uniform and non-uniform loading cases were adopted to calculate the applied load on the shaft lining. The loading conditions from the soil and groundwater are depicted in Figure 6. The standard penetration test (SPT) is a widely used subsurface exploration geotechnical in-situ test method to measure the bearing capacity of the ground, and the total blow count number required to penetrate the 30 cm-depth becomes the standard penetration resistance (N-value). The modulus of the ground was calculated as 4 N for silts, silty sands, and a slightly cohesive mixture of soil with an N-value of 30 taken from Figure 2, which indicated that the lowest N-value of the interpreted stratigraphy was in the range of 30-40. The maximum lateral pressure exerted on the shaft in the soil layers was calculated from the soil densities and friction angles presented in the geotechnical baseline report with K  
Figure 5: Final proposed SFRC solution for shaft lining. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **RC** & **SFRC** \\ \cline{2-3}  & **Reinforcement Bar** & **FF3** \\  & **(20 mm@300)** & \((40\) kg/m\({}^{3})\) \\ \hline Cross-section width (mm) & \(1000\) & \(1000\) \\ Thickness of cross-section (mm) & \(400\) & \(400\) \\ Tensile strength of reinforcement bar (MPa) & \(414\) & - \\ Area of reinforcement bar (mm\({}^{2}\)) & \(1047.20\) & - \\ Reduction factor (-) & \(0.90\) & - \\ \hline Reduction factor for tensile strength by CNR-204 & - & \(0.45\) \\ Reduction factor for members in tension & - & \(1.30\) \\ Concrete first crack strength (MPa) & - & \(4.70\) \\ Equivalent resistance by flexural test ASTM C1609 (\%) & - & \(102\) \\ Equivalent resistance by flexural test ASTM C1609 (MPa) & - & \(4.79\) \\ \hline Cross-section tensile force (kN) & \(390.2\) & \(663.8\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of tension resistance of conventional RC and SFRC. calculated as \(1-\sin\varnothing\) (Load Case 1). The load factor was 1.4 for the ultimate limit state (National Annex to BS EN 1990:2002) and Load Case 1 was calculated using Equation (6):  
\[N=\left(P_{gw}+P_{soil}\right)\times r\times L.F. \tag{6}\]  
where \(N\) is axial force, \(P_{gw}\) is groundwater pressure, \(P_{soil}\) is soil pressure, \(r\) is radius of tunnel, and L.F. is load factor. The load from a residual 200 kPa rock swelling pressure was applied to the lining below the shale bedrock level (Load Case 2). The total load from the rock swelling pressure in addition to the water pressure at the deepest shaft level was plotted against the capacity in the interaction diagram and calculated using Equation (7):  
\[N=\left(P_{gw}+P_{steel}\right)\times r\times L.F. \tag{7}\]  
where \(N\) is axial force, \(P_{gw}\) is groundwater pressure, \(P_{steel}\) is rock swelling pressure, \(r\) is radius of tunnel, and L.F. is load factor. A non-uniform loading case (Load Case 3) was considered as an extreme case and was based on the assumption that the full soil and groundwater pressures were present on only one side of the shaft, with 80% ground loading and 100% water pressure on the other side. The bending moment of the lining has been assessed by the Duddeck and Erdmann equation; it is based on the closed-form solutions to estimate the ground-structure interaction for circular tunnels [54]. This model was assumed to be in a continuous elastic environment under plane strain conditions; therefore, it does not consider any volume lost or relaxation of the ground.