# Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation
## 3 Solution of Multilayer Pavement under Pavement-Unevenness Excitation
### Interlayer Transfer Relationship Based on the DVP Method
\begin{bmatrix}\mathbf{\alpha}_{j}\end{bmatrix} \begin{bmatrix}\mathbf{U}(z_{j+})\\ \mathbf{T}(z_{j-})\end{bmatrix}  \tag{45}\]  
where the matrix of [a] is  
\[[\mathbf{\alpha}]= \begin{bmatrix}\alpha_{z}&0\\ 0&\alpha_{r}\end{bmatrix}  \tag{46}\]  
That is, when the interface \(z_{j}\) is a discontinuous interface, the transfer relation between layer \(j-1\) and layer \(j\) can be divided into two parts. The transfer relation of the first part is transferred from layer \(j-1\) to the upper interface of the interface of layer \(j\), and the coordinate of the interface is \(z=z_{j-}\). Since the transfer relation of this part does not involve the discontinuous interface, the following relation can be obtained:  
\[ \begin{bmatrix}\mathbf{U}(z_{j-1})\\ \mathbf{T}(z_{j-})\end{bmatrix} = \begin{bmatrix}\mathbf{S}^{j}_{11}&\mathbf{S}^{j}_{12}\\ \mathbf{S}^{j}_{21}&\mathbf{S}^{j}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j-})\\ \mathbf{T}(z_{j-1})\end{bmatrix}  \tag{47}\]  
In the second part, stress and displacement are transferred on both sides of the non-completely continuous interface \(z_{j}\), and the transfer relationship can be obtained as follows:  
\[ \begin{bmatrix}\mathbf{U}(z_{j-})\\ \mathbf{T}(z_{j+})\end{bmatrix} = \begin{bmatrix}\mathbf{I}&\mathbf{\alpha}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j+})\\ \mathbf{T}(z_{j-})\end{bmatrix}  \tag{48}\]  
Combining the above two formulas can be obtained as follows:  
\[ \begin{bmatrix}\mathbf{U}(z_{j-1})\\ \mathbf{T}(z_{j+})\end{bmatrix} = \begin{bmatrix}\mathbf{S}^{j}_{11 new}&\mathbf{S}^{j}_{12 new }\\ \mathbf{S}^{j}_{21 new}&\mathbf{S}^{j}_{22 new}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{j+}) \\ \mathbf{T}(z_{j-1})\end{bmatrix}  \tag{49}\]  
The new transfer matrix coefficients are as follows:  
\[ \begin{split}\mathbf{S}^{j}_{11 new}&=\mathbf{S}^{j}_{11}+\mathbf{S}^{j}_{11} \mathbf{\alpha}\begin{bmatrix}I-\mathbf{S}^{j}_{21}\mathbf{\alpha}\end{bmatrix}^{-1}\mathbf{S} ^{j}_{21}\\ \mathbf{S}^{j}_{12 new}&=\mathbf{S}^{j}_{12}+\mathbf{S}^{j}_{11}\mathbf{\alpha} \begin{bmatrix}I-\mathbf{S}^{j}_{21}\mathbf{\alpha}\end{bmatrix}^{-1}\mathbf{S}^{j}_{21}\\ \mathbf{S}^{j}_{21 new}&=\begin{bmatrix}I-\mathbf{S}^{j}_{21}\mathbf{\alpha} \end{bmatrix}^{-1}\mathbf{S}^{j}_{21}\\ \mathbf{S}^{j}_{22 new}&=\begin{bmatrix}I-\mathbf{S}^{j}_{21}\mathbf{\alpha} \end{bmatrix}^{-1}\mathbf{S}^{j}_{22}\end{split}  \tag{50}\]  
Then, by using the known boundary conditions, the undetermined vector coefficients corresponding to any position inside the layered body are obtained. The coefficient propagates from the surface to the last layer of the interface where \(z=z_{n}\); we have  
\[ \begin{bmatrix}\mathbf{U}(z_{0})\\ \mathbf{T}(z_{n})\end{bmatrix} = \begin{bmatrix}\mathbf{S}^{j}_{11}&\mathbf{S}^{j}_{12}\\ \mathbf{S}^{j}_{21}&\mathbf{S}^{j}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{U}(z_{n})\\ \mathbf{T}(z_{o})\end{bmatrix}  \tag{51}\]  
The boundary conditions of the surface layer and the expression of the corresponding expansion coefficient in the conversion domain were introduced in detail before, and the underlying boundary conditions will be introduced in detail below. For the underlying boundary conditions, there are two cases, as follows:  
Using the boundary condition Equation (29), the vector coefficients in the surface can be expressed as  
\[T(z_{0})= \begin{bmatrix}\frac{\sqrt{2\pi}qRJ_{1}(\lambda r)}{\lambda},0\end{bmatrix} ^ {T} \tag{52}\]When the lowest layer is a rigid base, the displacement of the bottom is 0, then the boundary condition at the bottom is \(U(z_{n})=0\), and the solution at any field point can be obtained by combining the Formula (51) as follows:  
\[ \begin{bmatrix}\mathbf{U}(z_{0})\\ \mathbf{T}(z_{n})\end{bmatrix} = \begin{bmatrix}\mathbf{S}_{11}^{1:n}&\mathbf{S}_{12}^ {1:n}\\ \mathbf{S}_{21}^{1:n}&\mathbf{S}_{22}^{1:n}\end{bmatrix}\begin{bmatrix}\mathbf{0} \\ \mathbf{T}(z_{0})\end{bmatrix}