# The Numerical Analysis of Textile Reinforced Concrete Shells: Basic Principles
## 5 Textile Failure Criteria Post-Processing  
To start with Step III of the workflow, the resulting values have to be exported to stand-alone Python-based scripts. For further processing, fields strain values \(\varepsilon_{i}^{+}\big{(}\varphi_{j}\big{)}\) and \(\varepsilon_{i}^{-}\big{(}\varphi_{j}\big{)}\) for the positive and negative sides of the shell elements, respectively, should be exported for each FEM node point \(i\) for two principal stress axis directions \(j\). Thus, the FEM nodes form a set of vectors \(\mathbf{x}\in X\) that defines a space \(X\). Figure 10 shows the strain components derived from the FEM calculation. The selected smeared material model approach possesses a significant limitation: it is impossible to correctly model the position of the reinforcement and, as a result, the inner lever arm in the case of bending. This may lead to an overestimation of stiffness in the case of significant bending [19]. It is assumed that the textile reinforcement is arranged symmetrically in the shell cross-section. Based on the central position of the reinforcement, the strain level \(\widetilde{\varepsilon}_{i}\) of each FEM point from \(X\) for the textile reinforcement layers and corresponding in-plane tensile forces \(n_{Et}\big{(}\varphi_{j}\big{)}\) can be estimated as follows:  
\[\widetilde{\varepsilon}_{i}=0.5\cdot\sum\big{(}\varepsilon_{i}^{+}\big{(} \varphi_{j}\big{)},\varepsilon_{i}^{-}\big{(}\varphi_{j}\big{)}\big{)};\;n_{Et,i}\big{(}\varphi_{j}\big{)}=\widetilde{\varepsilon}_{i}\cdot E_{f}\cdot A_{ tex}\text{ for }\widetilde{\varepsilon}_{i}\geq 0 \tag{16}\]  
where \(A_{tex}\) (mm\({}^{2}\)/m) is the cross-section area of the textile reinforcement from Table 3. Such an approach gives insight into the resulting stress contained within the textile reinforcement  
Figure 10: The shell strain and stress state. Graphic: [PERSON]. Figure 9: Reconstructed uniaxial tests in RFEM using various material models. Graphic: [PERSON]. for further comparative analysis. The resulting concrete in-plane force may be estimated according to Equation (23):  
\[\sigma_{i,Ec}(\varphi_{j})=-\int_{-h_{c}/2}^{+h_{c}/2}\varepsilon_{i}(\varphi_{j },h)\cdot E_{m}dh;\ n_{i,Ec}(\varphi_{j})=\sigma_{i,Ec}(\varphi_{j})\cdot h_{c} \text{ for }\varepsilon_{i}\leq 0 \tag{17}\]  
where \(h_{c}\) is the height of the TRC shell. After the assessment of the principal stress field, another question arises: how can we estimate the limit of the structure properly if axial forces as well as bending moments act simultaneously? A solution is the use of the \(n\)-\(m\) interaction diagram in a similar way as for steel-reinforced concrete [41]. In recent research on TRC structures, a simplified and normalized \(\eta_{n}\)-\(\eta_{m}\) interaction diagram was proposed [41; 42; 43]. This can be constructed using experimentally derived strength characteristics for uniaxial tension \(n_{RI}\) (kN/m), uniaxial compression \(n_{Rc}\) (kN/m), and pure bending \(m_{R}\) (kNm/m), assuming a linear interpolation between them (Figure 11). The normalized stress resultants can be calculated based on the characteristic strength values:  
\[\eta_{mt}=\frac{n_{E}}{n_{RI}};\quad\eta_{nc}=-\frac{n_{E}}{n_{Rc}};\quad\eta_{ m}=\frac{|m_{E}|}{m_{R}} \tag{18}\]  
where \(0\leq\eta\leq 1.0\). In the general case of combined loading, the utilization ratio \(\eta_{nm}\) can be calculated according to [41] as follows:  
\[\eta_{mn}=max(\eta_{nt},\eta_{nc})+\eta_{m} \tag{19}\]  
The simplified envelope diagram shows the following characteristics [41]:  
* In the tension-bending range, linear interpolation gives a relatively good representation of real behaviour. * In the compression-bending range, the ultimate strength is underestimated by the linear interpolation, staying on the safe side. In the first stages of the project, for simplification reasons, only in-plane actions were considered according to the general aim of the project to proceed with shell-like structures and membrane action.