# The Numerical Analysis of Textile Reinforced Concrete Shells: Basic Principles
## 6 Trial-Based Calculation
Figure 14 reflects the midspan deflections derived from tested shells and deflection results obtained from numerical analysis using the first linear model, M1, as a reference and the two different models described above, M2 and M3, with variations provided by input data from trilinear and ACK material models. As expected, the linear model M1 gives good agreement only in the uncracked state. In general, all models show deficits in State II. Furthermore, there is a clear influence of the input data developed according to the trilinear and ACK material models. The ACK approach (M2) gives slightly less stiffness behaviour than the M3 material model, which matches the experimental data at the end of test a bit better. In general, the overestimation of the stiffness may be a cumulative result of various reasons. Nevertheless, one of the key results is that the smeared model does not give a correct bending load response at the first stages of loading history due to weak behaviour for non-membrane actions, regardless of the utilized material model. In more details such effects were studied in [19]. Accordingly, all the evaluated layers that the cross-section of the shell can be subdivided in over the height follow the same strain-hardening curve. The stress profile was calculated from the strain level distribution. As a result, all the layers of the composite material contribute to the bending stiffness with the corresponding lever arm, even from the areas where the concrete is cracked. Consequently, it is not possible to reflect the effect of the position of the textile reinforcement in the cross-section; in turn, it is impossible to represent the effect of the reinforcement lever arm. Figure 13: Composite stress state field of the shell according to an accumulated load of 31.0 kN; (**a**) shell’s top side \(\sigma_{i}^{+}\); (**b**) bottom side of the shell \(\sigma_{i}^{-}\); (**c**) textile reinforcement stress field. Graphic: [PERSON]. The discussion and consequent hypothesis regarding a possible reason for the overestimation of the structural stiffness of the tested shell-like structures need to be strengthened. For this purpose, experiments carried out earlier at our institute in another project were used as alternative shell structures (Figure 15); more details arme described in [19; 44]. The main shell thickness was 1 cm with minor differences in the middle area of the shell element. The neutral line was located approx. 50 mm from the shell's vertex [44]. Therefore, it may be indicated that the top and bottom parts of the shell possess mainly membrane forces, in contrast to the previously presented shells with a neutral line inside the element's thickness. For comparison purposes, the alternative shell was modelled in RFEM software (Ver. 5.19) using the M2 and M3 material models. The stress-strain relations for the models are based on uniaxial tests on the used textiles [19] according to ACK methodology. The key points of the stress-strain diagrams are provided in Table 6.  
Figure 14: Force–midspan deflection relationships determined in the lab tests and in simulations by using M1–M3 material models. Graphic: [PERSON]. Figure 15: Alternative TRC shell structure. Graphic: [PERSON] based on [19; 45]. In Figure 16, in addition to test values, force-midspan deflection curves from calculations using M2 and M3 models in RFEM are shown, as well as curves derived with a smeared approach using the ABAQUS (Ver. 6.10) software package for simplified model with 10 mm shell thick and exact model according to the design [19]. It can be seen from the figure that the models M2 and M3 give good agreement with the reference smeared approach calculation and the experimental results. This agreement supports the hypothesis that the smeared TRC calculation approach can be used in RFEM for structures with predominantly membrane actions. It is also important to note that model M3 with [PERSON] hypothesis provides good agreement, even though the TRC material behaves in a strain-hardening manner.