# 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 equations shown in Table 4 are generally used for low-stress levels and enable either tangential slip, or full bonding between the lining and ground. The calculations employ the full bonding assumption between the lining structure and surrounding ground. Figure 6: Shaft stress conditions with depth. The seismic loads (Load Case 4), as detailed in the Ontario Building Code assuming a Class B site, were considered separately for the shaft. The maximum deflection from the seismic loads was assessed as below:  
\[\gamma=\frac{V_{s}}{C_{\text{se}}} \tag{8}\]  
where \(V_{s}\) is peak wave velocity and \(C_{\text{se}}\) is shear wave velocity. Based on the Ontario Building Code, the shear wave velocity ranges from 760 to 1500 m/s, and peak wave velocity is 0.8 m/s. The induced radial displacement from the seismic loads is calculated as below:  
\[\Delta D_{e}=2\gamma(1-\gamma)D \tag{9}\]  
The mobilised maximum bending moment from the seismic loads can be calculated as below and added to the uniform load in Cases 1 and 2:  
\[\text{M}=\frac{\frac{\Delta D_{e}}{2}\text{EI}}{\text{R}^{2}}(\text{Morgan}) \tag{10}\]  
The lining capacity envelope can be determined by plotting an interaction diagram, and the interaction diagram is plotted by calculating the bending moment and hoop force values. This gives a parabolic curve, which defines the limit of the concrete lining capacity. Three possible cases can be defined after the determination of design loads in the interaction diagram. If the load case coincides with the interaction diagram curve, it represents the limit state. In addition, values plotted inside the capacity curve indicated that the lining was structurally within the section capacity, whereas values plotted outside of the capacity curve indicated that it is estimated as insufficient to this criterion and considered unsafe [55]. Figure 7 illustrates the capacity curves for RC and SFRC lining, calculated from the spreadsheet, to check the structural capacity of designed linings. It was apparent that RC demonstrated relatively high hoop thrust and bending moment capacities. SFRC had an axial load corresponding to RC, it is estimated that the compressive strength of SFRC is not affected by the fibres. In addition, SFRC showed a bending moment of up to about 380 kNm/m, even without reinforcement steel bars, it is estimated that the steel fibres improved the cracking resistance and residual strength of the concrete. The interaction diagram also depicts the final design force/moment combinations for the four cases considered; these combinations were based on the deepest soft ground and bedrock sections. The maximum lateral loads for the shaft in soil layers and shale bedrock were 2772 and 3979 kN, respectively. The factored maximum and minimum design hoop loads derived from the Duddeck and Erdmann equations were 2738.3 and  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Full Bond** & **Tangential Slip** \\ \hline Maximum bending moment & \(\sigma_{v}(1-\text{K}_{o})\text{R}^{2}\frac{1}{4+\frac{1-\gamma_{o}}{(1+ \gamma)(2-\gamma)}}\frac{\text{E}_{\text{e}}\text{R}^{2}}\) & \(\sigma_{v}(1-\text{K}_{o})\text{R}^{2}\frac{1}{\frac{1-\gamma_{o}}{(1-\gamma)( 3-\gamma)}}\frac{\text{E}_{\text{e}}\text{R}^{2}}\) \\ \hline Average hoop thrust & \(\sigma_{v}(1+\text{K}_{o})\text{R}\frac{1}{2+\frac{1}{1+\gamma}\frac{\text{E} _{\text{e}}\text{R}^{2}}}\) & \\ \hline Variable hoop thrust & \(\sigma_{v}(1+\text{K}_{o})\text{R}\frac{1}{2+\frac{1}{1+\gamma}\frac{\text{E} _{\text{e}}\text{R}^{2}}}\) & \(\sigma_{v}(1-\text{K}_{o})\text{R}\frac{1}{\frac{1-\gamma_{o}}{(1-\gamma)( 3-\gamma)}}\frac{\text{E}_{\text{e}}\text{R}^{2}}{\frac{1-\gamma_{o}}{(1+ \gamma)(3-\gamma)}}\frac{\text{E}_{\text{e}}\text{R}^{2}}{\frac{1-\gamma_{o}}{(1 +\gamma)(3-\gamma)}}\frac{\text{E}  
2455.7 kN/m, respectively, and the maximum bending moment was 157.2 kNm/m.