# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
## 3 Deterministic Static Analysis of FGM Timoshenko Nanobeam
### Formulas of SFPM  
Consider a FGM Timoshenko nanobeam with length \(L\), the beam can be divided by several points in \(x\)-axial direction, as shown in Figure 3. We assume the number of the points are \(n+1\); then, the position of the points can be described as  
\[x_{i}=x_{0}+i\Delta l\ \ \ \ \ (i=0,1,2,\ldots,n) \tag{21}\]  
where \(\Delta l=l/n\) stands for the distance between two adjacent two points, and  
\[0=x_{0}<x_{1}<x_{2}<\ldots<x_{n-1}<x_{n}=l \tag{22}\]  
Figure 3: Diagram of finite point discretization for FGM nanobeam.  
In this work, the cubic B-spline function is adopted to the discrete displacement field, due to its superiority in approximation. The detail expression of the cubic B-spline \(\varphi_{3}(x)\) can be written as follows  
\[\varphi_{3}(x)=\frac{1}{6}\begin{cases}(x+2)^{3}&x\in[-2,-1]\\ (x+2)^{3}-4(x+1)^{3}&x\in[-1,0]\\ (2-x)^{3}-4(1-x)^{3}&x\in[0,1]\\ (2-x)^{3}&x\in[1,2]\\ 0&x\
otin[-2,2]\end{cases} \tag{23}\]  
A function \(F(x)\) can be expressed as a group of spline basic functions, i.e.,  
\[F(x)=\sum_{i=-1}^{n+1}\phi_{i}a_{i}=[\phi]\{a\} \tag{24}\]  
where \([\phi]\) and \(\{a\}\) denote the vector of spline basic function and coefficient, respectively. The basic function is constructed by cubic B-spline function (see Figure 4), which can be written as  
\[\begin{array}{ll}\phi_{i}(x)&=\frac{10}{3}\varphi_{3}\Big{(}\frac{x-x_{i}}{ l}-i\Big{)}-\frac{4}{3}\varphi_{3}\Big{(}\frac{x-x_{i}}{l}-i+\frac{1}{2}\Big{)}- \frac{4}{3}\varphi_{3}\Big{(}\frac{x-x_{i}}{l}-i-\frac{1}{2}\Big{)}\\ &+\frac{1}{6}\varphi_{3}\Big{(}\frac{x-x_{i}}{l}-i+1\Big{)}+\frac{1}{6}\varphi _{3}\Big{(}\frac{x-x_{i}}{l}-i-1\Big{)}\end{array} \tag{25}\]  
where \(\phi_{i}\) satisfies \(\phi_{i}(x_{j})=\delta_{ij}\), and \(\delta_{ij}\) is the Kronecker delta.  
Similarly, the displacement can also be described based on the spline function, as follows  
\[\begin{array}{ll}u_{0}=\sum\limits_{i=-1}^{n+1}\phi_{i}U_{i}\\ w_{0}=\sum\limits_{i=-1}^{n+1}\phi_{i}W_{i}\\ \varphi_{0}=\sum\limits_{i=-1}^{n+1}\phi_{i}\phi_{i}\end{array} \tag{26}\]  
It is more convenient to express via a matrix; to do so, the displacement field can be written as:  
\[U=[u_{0},w_{0},\varphi]^{\mathrm{T}}=[N]\{C\} \tag{27}\]  
where \([N]\) and \(\{C\}\) are the shape function and the nodal displacement, respectively. \([N]\) can be written as  
\[[N]=\begin{bmatrix}[\phi]&[0]&[0]\\ [0]&[\phi]&[0]\\ [0]&[0]&[\phi]\end{bmatrix} \tag{28}\]  
Figure 4: Basic function constructed by cubic B-spline function.  
where \([\phi]\) is the spline basic function (SBF), which is constructed by the cubic B-spline function. To guarantee that the boundary condition of the Timoshenko beam is satisfied at both of the ends, the SBF is determined by the following conditions:  
\[\begin{array}{l}\phi_{i}=0\;(i\
eq-1)\\ \phi_{i}=0\;(i\
eq n+1)\\ \phi^{\prime}_{i}=0\;(i\
eq-1,0)\\ \phi^{\prime}_{i}=0\;(i\
eq n,n+1)\end{array} \tag{29}\]  
Submitting the Equations (26)-(28) to the Equation (10), the equilibrium equations of SFPM can be derived as follows:  
\[[G]\{C\}=\{P\} \tag{30}\]  
where \([G]\) is stiffness matrix, \(\{C\}\) is nodal displacement vector, and \(\{P\}\) is the load vector.