# Numerical Modeling of Non-Uniformly Reinforced Carbon Concrete Lightweight Ceiling Elements
## 4 Resolved Model of a Cross Section
### Mixture Rule for Hybrid Fabric Reinforcement  
In the girder member described in Section 2, four regions were reinforced with both types of fabrics i.e., 3300 + 800 tex. However, no tensile test that could be used for calibration of such a cross section has been conducted. Thus, to derive the stress-strain curve of such a cross section, a mixture rule must be used to extract the theoretical tensile behavior from the separately tested 3300 tex and 800 text cross sections. Consider a hybrid cross section combining several types of fabrics \(j=1\dots m\) with a cross-sectional area per unit length \(a_{\mathrm{f}}^{(j)}\). For each of these fabrics, a composite tensile test has been conducted  
Figure 11: Stress–strain curves of the matrix (black solid lines), reinforcement (dashed lines) and composite (grey solid lines) for cross sections with (**a**) 800 tex and (**b**) 3300 tex reinforcements. delivering the stress-strain curves \(\sigma_{\rm c}^{(j)}\). To combine the behavior of the fabrics in a hybrid cross section, the composite stress of each test \(j\) must first be decomposed using the procedure described in Section 4.1 into the fractions associated with fabrics and matrix \((\sigma_{t}^{(j)},\sigma_{\rm m}^{(j)})\). Then, the fabrics stresses \(\sigma_{t}^{(j)}\) can be mixed into a single hybrid reinforcement layer using the area fractions of each reinforcement type in the total reinforcement area \(\bar{a}_{\rm f}=\sum_{j}^{m}a_{\rm f}^{(j)}\) with the weight factors  
\[\eta^{(j)}=\frac{a_{\rm f}^{(j)}}{\bar{a}_{\rm f}}. \tag{5}\]  
The effective fabric stress within the hybrid fabric cross section is then given as  
\[\bar{\sigma}_{\rm f}=\sum_{i}^{m}\eta^{(j)}\,\sigma_{\rm f}^{(j)}. \tag{6}\]  
The matrix stress is obtained by averaging the contributions determined in the individual tests  
\[\bar{\sigma}_{\rm m}=\frac{1}{m}\sum_{i}^{m}\sigma_{\rm m}^{(j)}. \tag{7}\]  
Then, the composite stress in a cross section of thickness \(\bar{d}\) reinforced with a hybrid fabric reads  
\[\bar{\sigma}_{\rm c}=\left(1-\bar{p}\right)\,\bar{\sigma}_{\rm m}+\bar{p}\, \bar{\sigma}_{\rm f} \tag{8}\]  
with the reinforcement ratio  
\[\bar{p}=\bar{a}_{\rm f}/\bar{d}.\]  
In reality, the usage of hybrid fabrics can introduce additional damage effects that are not reflected by this simple scaling of one-dimensional springs. In particular, a finer grid structure of overlapping fabrics reduces the contact area between the lower and upper concrete layers and can thus lead to surface delamination in the fabric plane at a low level of loading. However, in the case of the reinforcement ratio applied in the studied girder, the validity of the scaling can be assumed. To relate the mixture rule to the studied girder, let us rewrite Equations (5)-(8) considering the case of hybrid reinforcement layer consisting of \(3300+800\) tex fabrics. Using cross-sectional characteristics of the test specimens summarized in boldface font in Table 1, the weighting factors of the fabric mixture rule read  
\[\eta^{800}=\frac{a_{\rm f}^{800}}{\bar{a}_{\rm f}},\ \ \eta^{3300}=\frac{a_{\rm f}^{3300}}{ \bar{a}_{\rm f}},\ \ \bar{a}_{\rm f}=a_{\rm f}^{800}+a_{\rm f}^{3300}. \tag{9}\]  
The fractions of fabric stress and of the matrix stress within a unit cross section are expressed as  
\[\sigma_{\rm f}^{3300+800} =\eta^{800}\,\sigma_{\rm f}^{800}+\eta^{3300}\,\sigma_{\rm f}^{330 0}, \tag{10}\] \[\sigma_{\rm m}^{3300+800} =\frac{1}{2}(\sigma_{\rm m}^{3300}+\sigma_{\rm m}^{800}) \tag{11}\]  
and the corresponding stress-strain curve of the composite cross section reads  
\[\bar{\sigma}_{\rm c}^{3300+800} =\left(1-\rho^{3300+800}\right)\,\sigma_{\rm m}^{3300+800} \tag{12}\] \[\eta^{(j)}=\frac{a_{\rm f}^{(j)}}{\bar{a}_{\rm f}}. \tag{5}\]0  
The result of this mixture rule is plotted in Figure 12 both as the composite stress fraction ascribed to the fabrics \(\sigma_{\rm cf}\) (a) and the composite stress \(\sigma_{\rm c}\) including also the matrix stress fraction \(\sigma_{\rm cm}\) (b).