# Methodology for the Concept Design of Locally Reinforced Composites
## 3 Demonstration of the Approach
### Analysis of the Deviation Factor  
To analyze the effect of the deviation factor \(\mathrm{c_{d,max}}\), a single layer was created by using a shell FE model and execution of the mechanical model, as presented in Section 2.1. Anisotropic material properties of a unidirectional layer (UD) (see Table 2) were then applied for each cluster, whereby the material orientations were provided by the mechanical model.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline E\({}_{11}\) (GPa) & 138 \\ E\({}_{22}\) (GPa) & 9.5 \\ G\({}_{12}\) (GPa) & 5.2 \\ v\({}_{12}\) & 0.28 \\ \(\mathrm{\rho}\) (\(10^{3}\) kg/m\({}^{3}\)) & 1.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of each unidirectional carbon/epoxy composite ply.  
Figure 8: As a demonstration example, a simple quadratic shell with a center hole and a shear load case is examined. The deformed geometry is illustrated by the dotted line.  
The stress threshold was set to a constant value of \(\mu=100\%\) and, hence, the entire part, except the excluded areas near to the constraints, is covered by reinforcements. The maximum principal stress was considered for each of the elements, and no distinction was made between 1 st and 2 nd principal stress. The relative displacement in respect to the maximum displacement, as well as the number of clusters as a function of the deviation factor, were examined. Figure 9 shows the results of the investigations.  
For the demonstrator, a significant dependence of the relative displacement and the number of clusters from \(c_{d,\max}\) can be identified. For small deviation factors, e.g., \(c_{d,\max}=1^{\circ}\), the relative displacement can be reduced by approx. 70%. Concurrently, the number of clusters increases as the angular tolerance for each individual cluster factor decreases. The number of clusters is small for high deviation factors, since the angular tolerance is wider and more elements can be summarized in one cluster. It should be noted that a high deviation factor, at about 45\({}^{\circ}\) or more, usually results in a single cluster (see Figure 10 \(c_{d,\max}=45^{\circ}\)) and a homogenized angle for the entire part, which is effectively equivalent to a single UD-layer.  
Both the displacement and the number of clusters change progressively with decreasing \(c_{d,\max}\). This behavior is caused by the fact that a maximum limit value is defined with  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Cost & 564.94 \(\xi\) \\ Max. displacement & 1.6 mm \\ Weight & 13.597 g \\ \hline \end{tabular}
\end{table}
Table 2: Cost, max. displacement, and weight of the reference laminate.  
Figure 10: Resulting clusters for deviation factors of \(c_{d,\max}=15^{\circ}\), \(20^{\circ}\), and \(45^{\circ}\).  
Figure 9: Total number of clusters that are required to satisfy the deviation factor \(c_{d,\max}\) (black) and the corresponding relative displacement (red) in relation to the max. displacement.  
c\({}_{\text{d,max}}\) for all clusters (refer Figure 5), although the actual deviation in several clusters is often significantly lower. Thus, many clusters typically achieve a lower deviation prior to that point. For values of c\({}_{\text{d,max}}\) up to approx. 10\({}^{\circ}\), the effect of reducing the displacement is most significant. Below 10\({}^{\circ}\), the displacement is only slightly reduced, while the number of clusters increases excessively. Economic factors must be considered in this context, as higher costs are to be expected for a large number of small clusters [4] (see Section 3.2).