## 2 Research on Flexural Strengthening with Textile Reinforced Concrete
### Calculation Model  
#### 3.3.1 Iteration Process  
Similar to RC, the design model is based on the equilibrium of internal and external forces. As is well known, this equilibrium cannot be solved in a closed way, one or more iterations are necessary. The iteration process is described e.g., in References [43; 46; 47]. In this paper, the basic formulas will be given for a cross section with forces, stresses, and strains according to Figure 4 (compare e.g., [46; 47]). The definitions of the symbols are summarized in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \multicolumn{2}{c}{**Indices**} \\ E & external \\ c & concrete \\ s & steel \\ t & textile \\ d & design value \\ l & tensile stress area \\ 2 & compressive stress area \\ 0 & preload condition \\ \multicolumn{2}{c}{**Forces and Moments**} \\ F & force \\ N & normal force \\ M & bending moment \\ \multicolumn{2}{c}{**Geometrical Values**} \\ h & height \\ d & effective depth \\ b & width \\ A & reinforcement area \\ a & distance of compression force to top \\ layer \\ x & neutral axis depth \\ z & inner lever arm \\ k\({}_{\text{a}}\) & coefficient for distance a \\ \(\alpha_{\text{R}}\) & block coefficient \\ \multicolumn{2}{c}{**Stresses and Strains**} \\ \(\sigma\) & stress \\ \(\epsilon\) & strain \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of symbols. Figure 4: Principle of strengthened reinforced concrete (RC) cross section with outer and inner forces, strains and stresses. The maximum resistance of the cross section is reached when at least one of the three materials reaches its ultimate capacity. The goal is to fully utilize the textile reinforcement. This is therefore the first limiting case: The textile reinforcement achieves its ultimate strain (\(\varepsilon_{\rm tu}\)) or ultimate strength. The second possibility is a concrete failure in the compression zone (\(\varepsilon_{\rm c}=\varepsilon_{\rm cu2}\)). Last but not least, the steel reinforcement can fail as well, when the steel strain reaches \(\varepsilon_{\rm su}\), but this scenario would require an extremely large pre-deformation of the existing component prior to the strengthening measure (see \"consideration of a preload\", Section 3.3.3), and would apply only to exceptional cases. All other types of failure, e.g., debonding or anchorage failure, are to be excluded by separate proofs or by complying with design regulations analogous to steel reinforced concrete construction. The first equation for the iteration is the equilibrium for horizontal forces. Irrespective of the graphic representation in Figure 4, compressive forces are introduced in equilibrium conditions under consideration of the sign (negative!). \[\rm N_{\rm Ed}=F_{s1d}+F_{td}+F_{s2d}+F_{cd} \tag{1}\]  
In Equation (2), the moment's equilibrium is formed in the centre of gravity of the textile reinforcement. \[\rm M_{\rm Ed}-N_{\rm Ed}\cdot z_{t1}=-F_{s2d}\cdot z_{t82}-F_{cd}\cdot z_{t}- F_{s1d}\cdot z_{t81} \tag{2}\]  
In the Equations (1) and (2), there are four unknown forces (\(F_{s1d}\), \(F_{td}\), \(F_{s2d}\), \(F_{cd}\)) and one unknown geometrical value \(z_{t}\). Overall, there are two equations but five unknown variables. To solve this mathematical problem, more equations are necessary. For this reason, the strains will be brought into relation, according to the Bernoulli hypothesis. \[\varepsilon_{s1}=\ \varepsilon_{c2}+(\varepsilon_{t}+\varepsilon_{t0}- \varepsilon_{c2})\cdot\frac{d_{s}}{d_{t}} \tag{3}\]  
\[\varepsilon_{s2}=\ \varepsilon_{c2}+(\varepsilon_{s1}-\varepsilon_{c2})\cdot \frac{d_{2}}{d_{s}} \tag{4}\]  
\[\varepsilon_{t0}=\ \varepsilon_{s10}+(\varepsilon_{s10}-\varepsilon_{c20}) \cdot\frac{z_{t81}}{d_{s}} \tag{5}\]  
For iteration, a first strain distribution must be estimated. Then, the position of the neutral axis \(x\), including all coefficients such as \(k_{a}\) and \(\alpha_{r}\), and the compression force \(F_{cd}\) can be calculated.