# Methodology for the Concept Design of Locally Reinforced Composites
## 2 Approach and Models
### Mechanical Model
However, from a technological and economic point of view, these small clusters and incremental changes in material properties are not feasible for FRP components, considering economic and manufacturing aspects [4]. Similarly, the clustering and homogenization of too large areas is mechanically unreasonable due to the high deviation from the mechanically optimal orientation of the orthotropic material [5; 6]. Thus, to satisfy both the economic and the mechanical domains, the most suitable number of clusters with a constant angle must be determined. Therefore, elements with similar properties regarding the location and local orientation of the principal stresses are clustered within the reinforced area using the cluster algorithm k-means [29] (Step 5, Figures 3 and 5). The entire reinforcement area is thus divided into clusters. Each of the principal stress directions is considered individually, since each principal stress direction results in different reinforcement areas. For each cluster, a homogenized angle and a deviation factor are calculated (Step 6 Figure 5). The homogenized angle \(\overline{\Phi_{i}}\) is the mean angle of the principal stress angles \(\varphi_{i}\) in the respective cluster, weighted by their local stress. For the calculation of the homoge  
Figure 5: Clustering of reinforcement area into multiple areas with similar angles of principal stresses using deviation factor \(c_{d,\max}\). (**a**) Shows the resulting principal stress directions for \(\mu=60\%\). In (**b,c**) the calculated cluster angles are shown for \(c_{d,\max}=8^{\circ}\) and \(20^{\circ}\). For \(c_{d,\max}=8^{\circ}\) three clusters are required, for \(c_{d,\max}=20^{\circ}\) only one cluster is required. nized angle, higher stressed elements are also associated with a higher weight, due to the higher priority and greater demand for reinforcements (Step 6, Figure 5). The deviation factor \(\mathrm{c_{d,max}}\) is the maximum allowed deviation from the mechanically ideal angle and is consequently a measure of mechanical quality for the respective area. It is calculated by the stress-weighted standard deviation of the local element principal stress angles \(\varphi_{\mathrm{i}}\) (Step 7, Figure 5). If the deviation factor exceeds the predefined value \(\mathrm{c_{d,max}}\), the number of clusters is iteratively increased until the desired value is achieved. The higher the value of \(\mathrm{c_{d,max}}\) chosen, the fewer clusters are required to satisfy this constraint (see Figures 2 and 5). However, the deviation from the mechanically optimal solution increases respectively [6]. The deviation factor and stress threshold are the main parameters of the model that control the trade-off between a mechanically optimal structure and economic efficiency. The parameter dependencies and interactions are examined in more detail in Section 3, as part of the analysis with different parts and load cases. In the last step, the base-laminate and the reinforcement concepts of the 1 st and 2 nd principal stress directions are combined additively (step 8). It should be noted that the reinforcements of the principal stress directions 1 and 2 overlap in some areas. However, this behavior strongly depends on the specific dimensions, loads, and boundary conditions. The final results of this mechanical model are simplified concepts for the design of the base-laminate and reinforcements (clusters, dimensions, homogenized angles, etc.) as a function of the input parameters: stress threshold \(\upmu\) and the maximum deviation factor \(\mathrm{c_{d,max}}\). The presented approach primarily focuses on an increase in stiffness. The material thicknesses must be defined manually for each specific application. In the mechanical model, the mean stress is calculated for the base-laminate and each cluster within the reinforcement area, which can be used for an appropriate initial estimation of thickness values. To satisfy specific strength requirements, the layer thicknesses can then be estimated iteratively or by evolutionary algorithms [21,30].