# The Numerical Analysis of Textile Reinforced Concrete Shells: Basic Principles
## 6 Trial-Based Calculation  
To validate the material models provided above and the selected FEM procedures and to create a basis for the further development of the workflow, an arch-like shell was selected to perform a bending test according to the reference project [44], Figure 12. The total length of the shell is 2700 mm and the width is 600 mm. The shell provided has articulated linear supports from both ends in longitudinal direction of the shell. The clear span between the supports is 2500 mm. The transversal profile follows a chain line with a rise in the shell in the middle point of 100 mm. Due to manufacturing constraints, the thickness of the shell was set to be 40 mm. Prior to the experiments, the TRC shell was simulated; in the first step, it was simulated only with one single main layer of the solidian GRIDQ85-CCE-21-E5 completely over the whole element. The loading was selected according to EN 1991-1-1 [45] for residential use, comprising a 2.0 kN/m2 area load and a 3.0 kN point load. In the ultimate limit state, the textile stresses in the tension zone were higher than the design value for the textile. Therefore, it was decided to place two 200 mm wide additional textile stripes symmetrically in the tensile zone of the shell (Figure 12). The textile layers were arranged analogously to the uniaxial tensile samples with 10 mm spacing between the layers and a 15 mm concrete cover. For the validation of the numerical model, laboratory experiments with simulated TRC shells were performed. According to the technical constraints of the lab, the initial area load was converted into four equal line loads (see Figure 13a), resulting in a six-point bending test. The load was introduced in a displacement-controlled manner with a velocity of 0.05 mm/s until failure. In total, two shells were tested. Figure 13 gives the results for the limit stress state that corresponds to 12.1 kN/m for each line load or a total of 31 kN of an accumulated load. Figure 13a,b show the stress state of the composite, where \(\sigma_{i}^{+}\) stands for the top and \(\sigma_{i}^{-}\) stands for the bottom side of the shell. As may be seen from the legends, the compression zone lies between the top and bottom sides of the shell. Consequently, the TRC shell possesses significant bending action that is in contradiction with the accepted calculation hypothesis. The maximum stress of the concrete matrix reaches a magnitude of -32.7 MPa. That magnitude corresponds to only 28.5% of the experimentally determined concrete strength of \(f_{cm}=114.8\) MPa. Thus, it is expected that a reinforcement failure will be obtained during the experiment. Additionally, this result indicates that there is still a relatively large potential for material optimization that may reduce the compression zone and increase the utilization ratio of the concrete under compression. The reinforcement diagram in Figure 13c shows that in the middle section of the shell, the magnitude of \(\widetilde{\epsilon}_{i}>0\), and consequently, the reinforcement is considered to be activated. In the critical tensile zone of the shell, two layers of reinforcement reach the limit of \(f_{tm,tex,Q85}=3750.0\) MPa. The textile tensile strength magnitude was directly calculated from the mean value of composite strength from the conducted uniaxial tensile experiments (Figures 6 and 7). It can be calculated with a modified Equation (7) as follows:  
\[f_{tm,tex,Q85}=\frac{\widetilde{\sigma}_{cu}}{V_{f}} \tag{26}\]  
where \(V_{f}=0.4\%\) and \(\widetilde{\sigma}_{cu}=15\) MPa (composite's ultimate strength). In the bending tests, we detected midpoint deflections between 121.9 and 128.6 mm and reached failure loads between 33.0 and 35.7 kN, resulting in averages of 125.3 mm and 34.4 kN. The failure load according to the textile failure estimation procedure described above, derived at levels of 27.8 kN and 29.7 kN for the material models M2 and M3, respectively. Figure 12: General view of the TRC shell for the trial-based calculation. Graphic: [PERSON].