# 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  
Aiming to determine the bounds of the expectation and variance of the response for the nanobeam, the dimension-wise approach is adopted to integrate with the PCE. The idea of the WD is that separating the variables from a high-dimensional problem and converting the original problem to that combining univariate function, i.e.,  
\[f(x_{1},x_{2},\ldots,x_{n})=f_{0}+\sum_{i=1}^{n}f_{i}(x_{i}) \tag{49}\]  
where \(f_{0}\) is the zeroth-order effect, and \(f_{i}(x_{i})\) provides the first-order effect associated with \(x_{i}\), acting independently upon the output. \(f_{i}(x_{i})\) is a univariate function, which can be considered as the dimension-wise cut along the \(i\)-th input dimensionality. It is noted that the hierarchical expansion of the response function in Equation (49) is just to interpret the idea of DW, rather than utilized as the surrogate model. To identify the min/max point, the Legendre polynomials are adopted, which are orthogonal to each other in the interval \([-1,\ 1]\). The interval parameters should be standardized with the following rule  
\[\zeta=\frac{\beta-\beta^{c}}{\beta^{r}} \tag{50}\]  
In order to be convenient, the first two statistical moments are denoted as  
\[\mathbf{w}(\mathbf{\beta})=\left[ \begin{array}{c}\mu(\mathbf{\beta})\\ \sigma^{2}(\mathbf{\beta})\end{array} \right] \tag{51}\]  
Furthermore, the \(i\)-th dimensional cut of \(\mathbf{w}(\mathbf{\beta})\) can be expressed as  
\[\mathbf{w}(\mathbf{\beta}(\zeta_{i}))=\left[ \begin{array}{c}\mathcal{C}_{\text{ local}}(\mathbf{\beta}(\zeta_{i}))\\ \mathcal{C}(\mathbf{\beta}(\zeta_{i}))\circ\mathcal{C}(\mathbf{\beta}(\zeta_{i}))(\bm {\Phi}(\mathbf{\xi})\circ\mathbf{\Phi}(\mathbf{\xi}))-\mathcal{C}_{\text{local}}(\mathbf{\beta }(\zeta_{i}))\circ\mathcal{C}_{\text{local}}(\mathbf{\beta}(\zeta_{i}))\end{array} \right] \tag{52}\]  
where \(\mathbf{\beta}(\zeta_{i})\) is defined as  
\[\mathbf{\beta}(\zeta_{i})=\left[\beta_{1}^{c},\ldots\beta_{i-1}^{c},\beta_{i}^{c }+\beta_{i}^{r}\zeta_{i},\beta_{i+1}^{c},\ldots,\beta_{n}^{c}\right]^{\text{T} }\!\!,\ \zeta_{i}\in[-1,\ 1] \tag{53}\]  
To approximate the synthesis vector, \(\mathbf{w}(\mathbf{\beta}(\zeta_{i}))\) can be expressed as  
\[\mathbf{w}(\mathbf{\beta}(\zeta_{i}))=\mathbf{T}^{(i)}(\zeta_{i})=\left[T_{1}^{(i)}(\zeta _{i}),T_{2}^{(i)}(\zeta_{i}),\ldots,T_{2N}^{(i)}(\zeta_{i})\right]=\mathbf{D}^{(i) }\mathbf{L}(\zeta_{i}) \tag{54}\]  
where \(T_{1}^{(i)}(\zeta_{i})\) denotes the approximate polynomial of the \(i\)-th dimension-wise cut of the \(j\)-the component in \(\mathbf{w}(\mathbf{\beta})\); \(\mathbf{D}^{(i)}\) is the coefficient matrix; \(\mathbf{L}(\zeta_{i})\) is a vector composed of the first \(n_{L}\)-order Legendre polynomials, i.e.,  
\[\mathbf{D}^{(i)}=\left[ \begin{array}{cccc}d_{1,0}^{(i)}&d_{1,1}^{(i)}&\cdots&d_ {1,n_{L}}^{(i)}\\ d_{2,0}^{(i)}&d_{2,1}^{(i)}&\cdots&d_{2,n_{L}}^{(i)}\\ \vdots&\vdots&\ddots&\vdots\\ d_{2N,0}^{(i)}&d_{2N,1}^{(i)}&\cdots&d_{2N,n_{L}}^{(i)}\end{array} \right] \tag{55}\]  
\[\mathbf{L}(\zeta_{i})=\left[L_{0}(\zeta_{i}),L_{1}(\zeta_{i}),\ldots,L_{n_{L}}( \zeta_{i})\right]^{\text{T}}\!\!. \tag{56}\]  
The elements of \(\mathbf{D}^{(i)}\) can be obtained in the following way  
\[d_{j,k}^{(i)}=(2k+1)\sum_{l=1}^{s}\frac{L_{k}\big{(}\zeta_{l}^{\text{int}}\big{)} w_{j}\big{(}\mathbf{\beta}\big{(}\zeta_{l}^{\text{int}}\big{)}\big{)}}{\left(1-\big{(} \zeta_{l}^{\text{int}}\big{)}^{2}\right)\big{[}L^{\prime}\big{(}\zeta_{l}^{ \text{int}}\big{)}\big{]}^{2}} \tag{57}\]It is the Gauss integral formula, in which \(\beta_{l}^{\text{int}}\) is the \(l\)-th integral point; \(w_{j}\big{(}\beta_{l}^{\text{int}}\big{)}\) is the \(j\)-th component of \(\mathbf{w}(\mathbf{\beta})\); \(\zeta_{i}^{\text{int}}(l=1,2,\ldots,s)\) is the zero-roots of the \(s\)-order Legendre polynomial.