## 4 Discussion  
Fibre reinforced composite is essentially a type of multi-scale material. Uncertainty consideration from different scales would provide more comprehensive understanding on accurate uncertainty characterization. Previous work has highlighted possible existence of significant correlation betweenply mechanical properties [19; 22]. This correlation would affect reliability of composite structures considering structure failure defined by structure deformation, fracture or buckling [20]. It is known that natural frequency of composite structures not only depends on ply elastic properties but also laminate density. Therefore, as a new observation on the statistical correlation associated to ply density, it would be interesting to see how much the correlation contributes to the statistics of structure natural frequency. Here, a further comparison is conducted on deriving the statistics of structure natural frequency by considering two scenarios: (1) considering only correlation between ply mechanical properties (\(E_{11}\), \(E_{22}\), \(G_{12}\), \(\
u_{12}\)); (2) considering correlation between ply mechanical properties and also density (\(E_{11}\), \(E_{22}\), \(G_{12}\), \(\
u_{12}\), \(\rho\)). The derived standard deviations of the first order natural frequency of laminate 1 (see Table 3) and (90)\({}_{8}\) laminate cylindrical shell are shown in Table 5. It is interesting to observe that neglecting correlation associated with ply density results into 30-50% overestimation of  
Figure 5: (a) Geometrical dimension and mesh of laminate cylindrical shell; (b) first order vibration model of (\(\pm 45\))\({}_{4}\) laminate.  
Figure 6: Standard deviation of first order natural frequency of (\(\pm 0\))\({}_{4}\) laminate cylindrical shell derived by considering different uncertainty.  
the standard deviation, while neglecting all correlation between meso-scale uncertainty results into around 10% overestimation.  
Overestimation of standard deviation by 30-50% would lead to very large error in corresponding cumulative probability estimation. Therefore, it is important to account for the statistical correlation associated with ply density to obtain accurate stochastic free vibration behaviour of laminated structures.  
A recent study [25] reports that larger standard deviation of natural frequency was obtained for the case of'micro-scale uncertainty' than that of the 'independent meso-scale uncertainty', which seems to contradict with our observations in Table 4 and Figure 5. Actually, the comparison in [25] was conducted by assuming identical coefficient of variation (CoV) for both meso-scale and micro-scale uncertainties, for example the CoV of all of meso-scale and micro-scale random variables were assumed to be 5%. However, Tables 1 and 2 tell that micro-scale uncertainty with CoV at 5% actually results in CoV of meso-scale uncertainty at around 10%. The uncertainty amplification from micro-scale to meso-scale was also addressed in [18]. The uncertainty amplification explains why larger standard deviation was obtained for the case of'micro-scale uncertainty' in [25]. Meanwhile, statistical correlation between meso-scale uncertainty was not considered and discussed in [25].