# 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  
This section describes the methodology used to analyse the possibility of substituting SFRC for reinforcement bar in concrete solutions. The RILEM TC 162-TDF [43], which is based on the conventional design of standard RC, was used to generate a thrust-moment interaction curve. The bending tests on beams with three-points or four-points are generally used to evaluate the post-cracking behaviour of SFRC for precast segmental linings [46]. According to the RC design reports, the original design for the shaft included a 400 mm-thick RC lining with 20 mm and 15 mm-diameter radial and vertical reinforcements respectively, both spaced at 300 mm. The analysis was based on a rigid method [47], in which both materials acting in the ultimate state of resistance and ultimate capacity were compared. To evaluate the performance of the original RC design, its mechanical properties of RC under flexural and tension stress were assessed, and an equivalent structure using SFRC was proposed. The maximum flexural capacity of the traditional flat cross-section was calculated using the following formula (ACI 318-05) for simplified flexural capacity for a flexural member:  
\[M_{res}=0.9\big{(}f_{y}*A_{S}*d\big{)} \tag{1}\]  
where \(M_{res}\) is the flexural resistance for cross-section with reinforcement bar, \(f_{y}\) is the ultimate tensile strength of the steel, \(A_{s}\) is the steel area of reinforcement bar, and \(d\) is the depth of flexural reinforcement bar position. The performance of SFRC depends on the types of fibre, volume fraction, aspect ratio [48]; thus, in order to evaluate the performance of SFRC, two types of steel fibre were used. The volume fraction of steel fibres was chosen based on the literature review of previous research, and the dimensions and properties of the fibre are shown in Table 1[41; 49; 50; 51]. Figure 2: Proposed tunnel profile on simplified stratigraphy. Rectangular cross-section flexural response in SFRC can be calculated using  
\[M_{r}=R_{eq}\frac{bh^{2}}{6} \tag{2}\]  
where \(M_{r}\) is the resistant flexural moment and \(R_{eq}\) is the equivalent resistance for fibre-RC obtained by the flexural test based on ASTM C1609, assuming the cross-section to be uniform and homogeneously reinforced, \(b\) and \(h\) are the base and thickness of cross-section, respectively. To determine the equivalent resistance of the SFRC, the flexural test results for codes ASTM C1609 and UNI 11039 were considered, taking into account a first crack moment of 3.8-4.5 MPa, with a minimum ductility of 35-120% by a dosage of 0, 20 (0.25%), 30 (0.38 %), 40 (0.51%) kg/m\({}^{3}\) of Wirand(r)/MasterFiber(r) FF1 and FF3 fibres. The concrete specimens were made based on cast-in-place concrete specification section 03300 with compressive strength class of 35 MPa, and four-point bending tests were carried out with a closed-loop hydraulic testing machine by using crack mouth opening displacement (CMOD) as a control parameter. A linear variable differential transformer (LVDT) was used to measure the crack tip opening displacement (CTOD), and the test setup and geometry are shown in Figure 3.  
Figure 4 illustrated the nominal stress-crack tip opening displacement (CTOD) curves of SFRC for several types of fibre and dosage rates based on ASTM C1609/UNI 11039. Overall, plain concrete and SFRC for all fibre types demonstrated a similar trend with linear elastic behaviour up to the first-peak, and plain concrete and SFRC shows different behaviour after first-peak. Plain concrete showed brittle failure after the first-peak, while SFRC demonstrated ductile behaviour due to the bridging effect of fibres. The nominal stresses increased with increasing the volume of steel fibres for all cases, and the residual stresses also increased after the first-peak. The FF3 type of steel fibre showed significantly higher nominal stress than the FF1 type.