# Design Study of Steel Fibre Reinforced Concrete Shaft Lining for Swelling Ground in Toronto, Canada
## 4 Structural Design of Lining
### Theoretical Approaches for the Alternative Design Using SFRC
In particular, the FF3 type showed a strain hardening characteristic with ultimate strength, and continued a softening behaviour after the hardening behaviour at a dosage rate larger than 0.38%. Therefore, the FF3 type of fibre with 40 kg/m\({}^{3}\) (0.51%) was chosen as alternative reinforcement material for the concrete section, and the calculated moment resistances for RC and SFRC based on the tests are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameters** & \multicolumn{2}{c}{**Type of Steel Fibre**} \\ \cline{3-4}  & & **FF1** & **FF3** \\ \hline \multirow{3}{*}{Nominal dimensions} & Diameter (mm) & 1.00 & 0.75 \\  & Length (mm) & 50 & 50 \\  & Aspect ratio (length/diameter) (\(-\)) & 50 & 67 \\ \hline \multirow{2}{*}{Mechanical properties} & Tensile strength of the wire (MPa) & \textgreater{}1100 & \textgreater{}1100 \\  & Ultimate strain (\%) & \textless{}4 & \textless{}4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions and mechanical properties of Fibres. Figure 3: Four-point bending test: (**a**) Example of the test setup; (**b**) specimen geometry. In order to determine the tensile capacity of the critical cross-section for the original design with reinforcement bar, the tensile strength of concrete was not considered based on the CSA A23.3 and literature reviews [52; 53]; it can be assumed that  
\[T=0.9*\big{(}f_{y}*A_{S}\big{)} \tag{3}\]  
where \(T\) is the tensile force, \(f_{y}\) is the steel ultimate tensile strength, and \(A_{s}\) is the steel area of reinforcement bar. CNR-204 was used to evaluate the SFRC solution in tension. This code proposes the tensile behaviour of SFRC by dividing it into a linear-elastic model, represented by a stress-deformation perspective and a plastic-rigid model, represented by a stress-crack perspective. For the linear-elastic model, the behaviour of strain softening and strain hardening is presented separately. The main characteristic of this code is not the use of residual flexural tensile strength used by the RILEM code, but the use of equivalent flexural tensile strength. It is required to be calculated from a uniaxial tensile test based on a four-point bending test or UNI 11039/11188. The linear elastic model identifies two reference values, SFRC service residual tensile strength (\(f_{\text{FIs}}\)) and SFRC ultimate residual tensile strength (\(f_{\text{FFlu}}\)), and can be defined through equivalent values of flexural strength, using the following equations:  
\[f_{\text{FIs}}=0.45f_{\text{e}q1} \tag{4}\]  
\[f_{\text{Ftu}}=k\bigg{[}f_{\text{FIs}}-\frac{w_{u}}{w_{l2}}\big{(}f_{\text{FIs }}-0.5f_{\text{e}q2}+0.2f_{\text{e}q1}\big{)}\bigg{]}\geq 0 \tag{5}\]  
where \(f_{\text{e}q1}\) and \(f_{\text{e}q2}\) are the post-cracking equivalent strengths, which are useful for the serviceability limit states (SLS) and ultimate limit states (ULS), respectively. The coefficient \(k\) is assumed to be a cross-section, fully subjected to tensile stresses. The parameter \(w_{u}\) is the ultimate crack opening value and \(w_{l2}\) is the mean value of the crack opening at the endpoints of the interval. The calculated tensile forces capacity for RC and SFRC are summarised in Table 3.  
Figure 4: Example of test curves of steel fibre reinforced concrete (SFRC) for dosage rates of 0–40 kg/m\({}^{3}\): (**a**) FF1 type; (**b**) FF3 type.