# Numerical Modeling of Non-Uniformly Reinforced Carbon Concrete Lightweight Ceiling Elements
## 3 Smeared Model of a Cross Section
### Calibration of a Smeared Cross Section Model Using a Tensile Test  
To identify the damage functions with the same effective stress-strain behavior assumed in all material points of a cross section, the data from the tensile test following the RILEM recommendation [29] was used as input. The cross sections with similar fabric layups as those used in the tested girder were tested in two series. The specimens were reinforced with a single layer of fabrics, one with 3300 tex and one with 800 tex. Figure 8 shows the stress-strain response obtained in four tests for each cross section as gray dashed lines, and the average response curves as black solid lines. Given the cross-sectional thickness \(d^{\text{test}}\) and area of the textile fabrics per unit width \(a_{\text{f}}\), the reinforcement ratio \(\rho^{\text{test}}\) reads  
\[\rho^{\text{test}}=\frac{a_{\text{f}}}{d^{\text{test}}}. \tag{1}\]  
The fabric area per unit length and the corresponding reinforcement ratios for the tested specimens reinforced with both types of fabrics are listed in Table 1.  
Figure 7: Microplane model within a single layer of a shell cross section.  
The parameters of the microplane damage model include the [PERSON]'s modulus (\(E_{\rm m}=28\) GPa) and [PERSON]'s ratio (0.2) and the function \(\phi(e)=1-\omega(e)\) defining the diminishing integrity evolution for an increasing normal microplane strain. The integrity functions obtained using the incremental calibration procedure are plotted in Figure 9a,b for the two tested cross sections. As observed for the cross section with 800 tex fabrics in Figure 9a, the integrity does not drop to zero but remains at a constant level corresponding to the final elastic stage of the strain-hardening behavior at which no further matrix cracks appear. In case of the 3300 tex specimens, the shape of the damage function shown in Figure 9b reveals an increasing integrity in the range of strains \(0.001<\varepsilon<0.002\). This effect is owing to the delayed activation of filaments within the yarn cross section, usually referred to as slack, of stretched filaments in this range of strains.  
\begin{table}
\begin{tabular}{l l l l l} \hline fabric fineness & [tex] & **800** & **3300** & 3300 + 800 \\ \hline fabric area & \(a_{\rm f}\) & [m\({}^{2}\)/m] & **0.616 \(\cdot\) 10\({}^{-4}\)** & **1.713 \(\cdot\) 10\({}^{-4}\)** & 2.329 \(\cdot\) 10\({}^{-4}\) \\ \hline specimen thickness & \(d\) & [m] & **0.01** & **0.01** & 0.01 \\ \hline reinforcement ratio & \(\rho=a_{\rm f}/\bar{d}\) & [-] & \(0.616\cdot 10^{-2}\) & \(1.713\cdot 10^{-2}\) & 2.329 \(\cdot\) 10\({}^{-2}\) \\ \hline hybrid section ratio & \(\eta\) (Equation (9)) & [-] & 0.264 & 0.735 & 1.000 \\ \hline final composite stiffness & \(E_{\rm cf}\) & [GPa] & 1.48 & 3.22 & 4.70 \\ \hline effective fabric stiffness & \(E_{\rm f}\) & [GPa] & 250 & 188 & 202 \\ \hline \end{tabular}
\end{table}
Table 1: Cross section characteristics; parameters of performed tests in boldface font, derived values in normal font.  
Figure 8: Tensile response of the cross sections with (**a**) 800 tex, (**b**) 3300+tex reinforcement and (**c**) of the hybrid cross section 3300+800 tex.