# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
## 4 Uncertain Analysis of FGM Timoshenko Nanobeam with Defects
### Hybrid Method of Polynomial Chaos Expansion and Dimension-Wise Analysis (PCE-WD)
#### 4.3.2 Formulas of Dimension-Wise (DW) Analysis
The zero-roots can be obtained by  
\[\frac{\text{d}T_{j}^{(i)}(\zeta_{i})}{\text{d}\zeta_{i}}=0 \tag{58}\]  
The solutions are denoted as \(\zeta_{j,i}^{(l)}(l=1,2,\ldots,n_{L}-1)\) and translated into \([-1,\ 1]\), according to the definition domain of Legendre polynomials, as follows:  
\[\zeta_{j,i}^{(l)}= \begin{cases}1&\text{if}\ \text{Im}\big{(}\zeta_{j,i}^{(l)} \big{)}\
eq 0\\ -1&\text{if}\ \text{Im}\big{(}\zeta_{j,i}^{(l)}\big{)}\
eq 0\mathbf{\xi}\big{|} \text{Re}\big{(}\zeta_{j,i}^{(l)}\big{)}\Bigg{|}>1&l=1,2,\ldots,n_{L}-1\\ \zeta_{j,i}^{(l)}&\text{if}\ \text{Im}\big{(}\zeta_{j,i}^{(l)}\big{)}\
eq 0\mathbf{\xi} \big{|}\text{Re}\big{(}\zeta_{j,i}^{(l)}\big{)}\Bigg{|}<1\end{cases}  \tag{59}\]  
where \(\text{Im}(\bullet)\) and \(\text{Re}(\bullet)\) are the imaginary and real parts, respectively. Thus, the \(\text{min}/\text{max}\) points of \(w_{j}(\beta(\zeta_{i}))\) are expressed as \(\zeta_{j,i}^{\text{min}}\) and \(\zeta_{j,i}^{\text{max}}\), which can be obtained by  
\[T_{j}^{(i)}\big{(}\zeta_{j,i}^{\text{sym}}\big{)}=\text{sym}_{\zeta_{j}\in \zeta_{j,i}^{\text{ext}}}T_{j}^{(i)}\big{(}\zeta_{j}\big{)},\ \text{sym}\triangleq\text{min},\ \text{max} \tag{60}\]  
where  
\[\xi_{j,i}^{\text{ext}}=\left[-1,1,\zeta_{j,i}^{(l)},\ldots,\zeta_{j,i}^{(n_{L }-1)}\right] \tag{61}\]  
Therefore, the \(\text{min}/\text{max}\) points of the \(j\)-th component in \(\mathbf{w}(\beta)\) can be assembled by traversing \(i\) from \(1\) to \(n\) parallelly, i.e.,  
\[\beta_{j}^{\text{sym}}=\left[\beta_{j,1}^{\text{sym}},\beta_{j,2}^{\text{sym }},\ldots,\beta_{j,n}^{\text{sym}}\right],\ j=1,2,\ldots,2N;\ \text{sym}\triangleq\text{min},\ \text{max} \tag{62}\]  
Then, \(\mathbf{w}\big{(}\beta_{j}^{\text{sym}}\big{)}\) can be determined by the PCE, i.e.,  
\[\mathbf{w}_{j\text{-out}}^{\text{sym}}=\left[ \begin{array}{c}\mathbf{C}_{\text{1 col },\ j\text{-out}}^{\text{sym}}\\ \mathbf{C}_{\text{-out}}^{\text{sym}}\circ\mathbf{C}_{\text{-out}}^{\text{sym}}(\mathbf{ \Phi}(\mathbf{\xi})\circ\mathbf{\Phi}(\mathbf{\xi}))-\mathbf{C}_{\text{1 col},\ j\text{-out}}^{ \text{sym}}\circ\mathbf{C}_{\text{1 col},\ j\text{-out}}^{\text{sym}}\end{array}  \right],\ \text{sym}\triangleq\text{min},\ \text{max} \tag{63}\]  
To obtain the bounds of \(\mathbf{w}(\beta)\), the \(\text{min}/\text{max}\) matrixes can be assembled by traversing \(j\) from \(1\) to \(2\)\(N\),  
\[\mathbf{w}_{\text{mat}}^{\text{sym}}=\left[w_{\text{1-out}}^{\text{sym}},w_{2 \text{-out}}^{\text{sym}},\ldots,w_{2N\text{-out}}^{\text{sym}}\right] \tag{64}\]  
Then, the bounds of \(\mathbf{w}(\mathbf{\beta})\) can be obtained as follows  
\[w=\left[\underline{w},\overline{w}\right]=\left[\text{diag}\Big{(}\mathbf{w}_{ \text{mat}}^{\text{min}}\Big{)},\text{diag}(\mathbf{w}_{\text{mat}}^{\text{max}} )\right] \tag{65}\]  
Combining the PCE and DW methods, the hybrid uncertainty propagation analysis method, namely PCE-DW, is established. Its execution process is shown in Figure 7.