## 5 Analytical Modelling
### Implementation of the Stochastic Cracking Model  
To apply the stochastic cracking model to the set of TRC composites and compare the obtained analytical responses with the experimental results along the warp direction, some assumptions regarding the base materials and the efficiency of the composites were made. As regards the failure strength \(f_{ctm}\) of the two matrices, different approaches were considered, following the observations of Section 2.1: (i) average values of the \(f_{ct}\) tensile strengths, computed with Equation (1); (ii) average tensile strengths empirically derived, according to the MC2010, from the average compressive strengths; and (iii) data provided by the premixed mortar manufacturer (the obtained values are collected in Table 9). Analyses \"A\" (see Section 5.3) considered the average strengths calculated from the three-point bending flexural strengths (\(f_{ctm}=6.31\) MPa for M1 and \(f_{ctm}=3.10\) MPa for M2), while Analyses \"B\" instead assumed \(f_{ctm}=4.59\) MPa for M1 and \(f_{ctm}=4.47\) MPa for M2; the former value is the one obtained from the average compressive strength, while the latter is the one provided by the thixotropic mortar manufacturer. Regarding M2, the selection of the highest tensile strength was consistent with a greater compaction of the mortar (as resulting in the composites, produced by means of a hand lay-up technique) and the stabilising effect offered by polyacrylonitrile-based microfibres originally introduced in the pre-mixed powder to minimise shrinkage effects. For both analyses, the matrix elastic modulus \(E_{m}\) and the Weibull shape coefficient m were assumed equal to 42.9 GPa and 11.58 for the M1-based composites and 28.0 GPa and 7.15 for the M2-based ones, respectively (please note that the m Weibull parameter associated to the tensile strength distribution was hence assumed equal to the experimental flexural one). The efficiency of the cement-based matrix, as defined in the original theory, was set at \(\eta_{m}=1\). Regarding the textiles, an elastic modulus \(E_{f}\) of 70 GPa was assumed, as suggested by the glass filaments manufacturer. The fabric volume fraction \(V_{f}\) was calculated as the ratio between \(A_{f}\) (Table 5) and the average cross section of the samples belonging to each set (see Table 7), while \(V_{m}\) was equal to 1-\(V_{f}\). The reinforcement efficiency \(\eta_{f}\) was taken equal to \(EF_{f}\), where \(EF_{f}\) is the fabric efficiency factor introduced in Section 2.2 (see Equation (2) and Table 4). This latter assumption followed the hypothesis that the filaments that effectively contribute to the resisting mechanism (\(A_{f}^{*}=A_{f}\cdot EF_{f}\)) work in parallel and each of them is capable of reaching the maximum tensile material strength (\(\sigma_{fit}=2000\) MPa). The hypothesis seems to be confirmed by Figure 15, showing a reworking of the plain fabric tensile responses of Figure 4 carried out by calculating the nominal fabric stress \(\sigma_{f}^{*}\) as the ratio between the load \(P_{f}\) and the effective fabric area \(A_{f}^{*}\). As one should note, the scatter of the seven average curves in terms of stiffness was limited, proving that a fabric efficiency factor merely estimated on the failure loads (see Equation (2)) may be sufficiently representative of the global textile behaviour. The value of the average saturated crack spacing \(X\) was obtained experimentally, dividing the LVDT gauge length (the nominal value of 200 mm was assumed) by the number of cracks detected at the end of each tensile test (see Figure 10). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Approach** & **Matrix M1** & **Matrix M2** \\ \hline (i) from bending tests & 6.31 MPa & 3.10 MPa \\ (ii) from compressive tests & 4.59 MPa & 3.46 MPa \\ (iii) manufacturer data & n.a. & 4.47 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Average matrix failure strengths \(f_{ctm}\).