# Application of Finite Layer Method in Pavement Structural Analysis
## 2 Numerical Solution of EasyFEM
### FE Solution of Displacements  
The pavement is assumed to be supported at side edges (\(x=0\), \(x=a\), \(z=0\) and \(z=c\)) in a manner preventing all displacements in the vertical planes but permitting \"unrestricted\" movement in the \(x\)- and \(z\)-directions. The shape functions are re-written as the conventional 1D shape functions multiplied by the Fourier series, in which \(x\) ranges between zero and \(a\); \(z\) ranges between zero and \(c\). If the displacement functions with three components \(u\), \(v\) and \(w\) are formulated in the following form, the boundary condition will meet this above-mentioned requirement:  
\[ \begin{array}{c}d=\left\{\begin{array}{c}u\\ v\\ w\end{array}\right\}=\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}\left[\begin{array} []{ccc}cos\frac{m\pi x}{a}sin\frac{n\pi z}{c}&0&0\\ 0&sin\frac{m\pi x}{a}sin\frac{n\pi z}{c}&0\\ 0&0&sin\frac{m\pi x}{a}cos\frac{n\pi z}{c}\end{array}\right]\left\{\begin{array} []{c}u^{mn}\\ v^{mn}\\ w^{mn}\end{array}\right\}\\ \hskip 14.226378 pt\text{with }N^{mn}=\left[\begin{array}{ccc}cos\frac{m\pi x }{a}sin\frac{n\pi z}{c}&0&0\\ 0&sin\frac{m\pi x}{a}sin\frac{n\pi z}{c}&0\\ 0&0&sin\frac{m\pi x}{a}cos\frac{n\pi z}{c}\end{array}\right],\\ \hskip 14.226378 ptd^{mn}=\left\{\begin{array}{c}u^{mn}\\ v^{mn}\\ w^{mn}\end{array}\right\}=\left[N_{k}(y)\right]\!\left\{d^{mn}_{k}\right\}.\end{array}  \tag{2}\]  
\(N^{mn}\) is the matrix for the Fourier series expansion in the two horizontal directions, \(d^{mn}\) is the displacement vector without the Fourier series expansion, \([N_{k}(y)]\) is the conventional 1D shape function applied in the FEM model, and \(\left\{d^{mn}_{k}\right\}\) represents the corresponding nodal displacements. Similarly, the loading function for the pavement analysis can be formulated as:  
\[ \begin{array}{c}f=\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}p(y)sin\frac{m \pi x}{a}sin\frac{n\pi z}{c}=\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}\{p\}^ {mn},\\ p(y)=\sum_{s=1}^{S}\!\left(\frac{2P_{s}}{m\pi}\right)\!\left[cos\frac{m\pi}{a} X_{s1}-cos\frac{m\pi}{a}X_{s2}\right]\sum_{t=1}^{T}\!\left(\frac{2P_{s}}{n\pi} \right)\!\left[cos\frac{n\pi}{c}Z_{t1}-cos\frac{n\pi}{c}Z_{t2}\right]\!,\end{array}  \tag{3}\]  
where \(m\) and \(n\) are the terms of Fourier series in the \(x\) and \(z\) directions, respectively. In this study, both \(M\) and \(N\) are adopted as 100, which has been previously proved to be sufficiently accurate to represent the load variation [16; 17]. The product of \(P_{s}\) and \(P_{t}\) represents the tire load pressure, when \(m=n\); \(X_{s1}\) and \(Z_{t1}\) are the \(x\) and \(z\) coordinates, where the tire load area starts, respectively; \(X_{s2}\) and \(Z_{t2}\) are the \(x\) and \(z\) coordinates where the tire load area ends, respectively. Figure 2: The natural coordinates for 1D quadratic elements. At the nodes the strains are determined through displacements:  
\[\varepsilon=\left\{ \begin{array}{l}\varepsilon_{x}\\ \varepsilon_{y}\\ \varepsilon_{z}\\ \gamma_{xy}\\ \gamma_{yz}\\ \gamma_{zx}\end{array} \right\}=\left[ \begin{array}{cccc}\frac{\partial}{ \partial x}&0&0\\ 0&\frac{\partial}{\partial y}&0\\ 0&0&\frac{\partial}{\partial z}\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}&0\\ 0&\frac{\partial}{\partial z}&\frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}&0&\frac{\partial}{\partial x}\end{array} \right]d =\left(L_{1}\frac{\partial}{\partial x}+L_{2}\frac{\partial}{\partial y}+L_{3} \frac{\partial}{\partial z}\right)\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}N^ {mn}\cdot d^{mn} \tag{4}\] \[=\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}E^{mn}d^{mn}=\sum \limits_{m=1}^{M}\sum\limits_{n=1}^{N}B^{mn}\cdot\left\{d_{k}^{mn}\right\},\]  
where L\({}_{1}\), L\({}_{2}\), L\({}_{3}\) are three different matrices of size 6 \(\times\) 3 whose elements are either 0 or 1,  
\[E^{mn}=L_{1}\frac{\partial N^{mn}}{\partial x}+L_{2}N^{mn}\frac{\partial}{ \partial y}+L_{3}\frac{\partial N^{mn}}{\partial z}, \tag{5}\]  
and matrix  
\[B^{mn}=