# Safety Concept for Textile-Reinforced Concrete Structures with Bending Load
## 2 Boundary Conditions
### Design Model  
For the verification of the ultimate limit state, the bending load capacity \(M_{\rm Rd}\) should, at least, be equal to the applied design moment \(M_{\rm Ed}\) (Equation (1)). The calculation of \(M_{\rm Ed}\) for the specified static system (Figure 1) is carried out according to Equation (2). \[M_{Ed}\leq M_{Rd} \tag{1}\]  
\[M_{Ed}=\gamma_{G}\frac{g_{k}\,l^{2}}{8}+\gamma_{G}\frac{q_{k}\,l^{2}}{8} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter 1** & **Description** & **DT 2** & **Unit** & **\(\mu_{\rm X}\)3** & \(\sigma_{\rm X}\)4** \\ \hline \(d_{\rm I}\) & effective depth & ND & (mm) & 15 & 2 \\ \hline \(d_{\rm II}\) & effective depth & ND & (mm) & 60 & 3 \\ \hline \(h_{\rm I}\) & height & det. & (mm) & 30 & – \\ \hline \(h_{\rm II}\) & height & det. & (mm) & 80 & – \\ \hline \(B\) & width & det. & (mm) & 1000 & – \\ \hline \(L\) & span & det. & (mm) & 1000 & – \\ \hline \(\rho_{\rm I}\) & reinforcement ratio & varies & (–) & – & – \\ \hline \({}^{1}\) Basic variable; \({}^{2}\) distribution type: ND = normally distributed, det. = deterministic value; \({}^{3}\) expected value of the basic variable \(X\); \({}^{4}\) standard deviation of the basic variable \(X\). & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of the geometric dimensions—statistical description of the basic variables [9]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Description** & **DT 1** & **Unit** & **Char. Value** & \(\mu_{\rm X}\) & \(\sigma_{\rm X}\) & \(v_{\rm X}\) \\ \hline \(G\) & dead loads [19] & ND & (N/mm2) & \(g_{\rm X}\) & \(g_{\rm X}\) & \(0.1\,g_{\rm X}\) & \(0.1\) \\ \hline \(Q\) & live loads [19] & Ext. I & (N/mm2) & \(q_{\rm X}\) & \(0.7\,q_{\rm X}\) & \(0.14\,q_{\rm X}\) & \(0.2\) \\ \hline \(\eta_{\rm I}\) & load ratio \(g/q\) & det. & (–) & – & \(0.5\) & – & – \\ \hline \(\eta_{2}\) & load ratio \(g/q\) & det. & (–) & – & \(1.0\) & – & – \\ \hline \multicolumn{6}{c}{\({}^{1}\) Distribution type: Ext I = generalized extreme value distribution type-I (Gumbel distribution).} \\ \end{tabular}
\end{table} 
Table 2: Overview of the loads—statistical description of the applied loads [9]. The values for the partial safety factors \(\gamma_{\text{G}}\) and \(\gamma_{\text{Q}}\) are taken from EN 1990 [21] and are summarized in Table 3.  
The bending load capacity \(M_{\text{Rd}}\) is obtained by applying a linear stress-strain curve for the concrete with Equation (3) for a textile failure and with Equation (4) for a concrete failure. The type of failure can be verified with Equation (5). \[M_{Rd}=F_{td}\;z=A_{t}\;f_{td}\cdot\left(d-\frac{\frac{\alpha_{cd}\;A_{t}}{b} \left[-1+\sqrt{1+\frac{2b\;d}{\alpha_{cd}\;A_{t}}}\right]}{3}\right) \tag{3}\]  
\[M_{Rd}=F_{cd}\;z=f_{cd}\cdot\frac{\alpha_{cd}\;A_{t}\left[-1+\sqrt{1+\frac{2b\; d}{\alpha_{cd}\;A_{t}}}\right]}{2}\left(d-\frac{\frac{\alpha_{cd}\;A_{t}}{b} \left[-1+\sqrt{1+\frac{2b\;d}{\alpha_{cd}\;A_{t}}}\right]}{3}\right) \tag{4}\]  
\[\sigma_{c}=\frac{F_{td}\;2}{b\;x}=\frac{A_{t}\;f_{td}\;2}{\alpha_{cd}\;A_{t} \left[-1+\sqrt{1+\frac{2b\;d}{\alpha_{cd}\;A_{t}}}\right]}\leq f_{cd} \tag{5}\]  
Both equations depend on the relation \(\alpha_{\text{Ed}}\) between the elastic modulus of the concrete \(E_{\text{cd}}\) and the textile reinforcement \(E_{\text{tm}}\). Figures 2 and 3 illustrate both elastic modulus and Equation (6) expresses the relation between them. \[\alpha_{cd}=\frac{E_{tm}}{E_{cd}} \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Unit** & **Value** \\ \hline \(\gamma_{\text{G}}\) & dead load & (–) & 1.35 \\ \hline \(\gamma_{\text{Q}}\) & variable load & (–) & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overview of the safety elements for the impact side. Figure 2: \(\sigma-\varepsilon\) diagram of the reinforcement for the design. Furthermore, the closed solution for a rectangular stress-strain curve for the concrete is also taken into account for the assessment of the bending load capacity \(M_{\text{Rd}}\).