# Experimental and Theoretical Study of One-Way Cyclic Lateral Responses of Piles in Sloping Ground
## Appendix A Finite Beam Element Method  
Figure 11 illustrates a pile element with the length \(l_{i}\), flexural rigidity \(EI\), and axial rigidity \(EA\), and the element ends are noted as the node \(i\) and node \(j\). Six element end forces (\(\overline{F}_{Ni}\), \(\overline{F}_{Qi}\), \(\overline{M}_{i}\), \(\overline{F}_{Nj}\), \(\overline{F}_{Qi}\), and \(\overline{M}_{j}\)) and six element-end displacements (\(\overline{u}_{i}\), \(\overline{v}_{i}\), \(\overline{\varphi}_{i}\), \(\overline{u}_{j}\), \(\overline{v}_{j}\), and \(\overline{\varphi}_{j}\)) can be written in the vector form:  
\[\overline{F}^{\mathbf{e}}=\begin{bmatrix}\overline{F}_{Ni}&\overline{F}_{Qi}& \overline{M}_{i}&\overline{F}_{Nj}&\overline{F}_{Qi}&\overline{M}_{j}\end{bmatrix} ^{\mathsf{T}} \tag{12}\]  
\[\overline{\delta}^{\mathbf{e}}=\begin{bmatrix}\overline{u}_{i}&\overline{v}_{i}& \overline{\varphi}_{i}&\overline{u}_{j}&\overline{v}_{j}&\overline{\varphi}_{ j}\end{bmatrix}^{\mathsf{T}} \tag{13}\]  
The relationship between these two vectors can be obtained by the principle of virtual work, Equation (10).  
\[\overline{F}^{\mathbf{e}}=\mathbf{k}\overline{\delta}^{\mathbf{e}} \tag{14}\]  
where \(\mathbf{k}\) is the element stiffness matrix, \(\mathbf{k}=\mathbf{k_{0}}+\mathbf{k_{P}}+\mathbf{k_{I}}+\mathbf{k_{G}}\), \(\mathbf{k_{0}}\) is the basis element stiffness matrix in Equation (12), \(\mathbf{k_{P}}\), \(\mathbf{k_{I}}\), and \(\mathbf{k_{G}}\) are the modified element stiffness matrixes considering the second-order effect, the strengthening effect of surrounding soils on the pile stiffness, and the shear resistance induced by the Pasternak model, expressed as Equations (A5)-(A7), respectively.  
\[\mathbf{k_{0}}=\begin{bmatrix}\frac{EA}{l_{i}}&0&0&-\frac{EA}{l_{i}}&0&0\\ 0&\frac{12 EI}{l_{i}^{3}}&\frac{6 EI}{l_{i}^{2}}&0&-\frac{12 EI}{l_{i}^{3}}&\frac{6 EI }{l_{i}^{2}}\\ 0&\frac{6 EI}{l_{i}^{2}}&\frac{4 EI}{l_{i}}&0&-\frac{6 EI}{l_{i}^{2}}&\frac{2 EI}{l _{i}}\\ -\frac{EA}{l_{i}}&0&0&\frac{EA}{l_{i}}&0&0\\ 0&-\frac{12 EI}{l_{i}^{3}}&-\frac{6 EI}{l_{i}^{2}}&0&\frac{12 EI}{l_{i}^{3}}&- \frac{6 EI}{l_{i}^{2}}\\ 0&\frac{6 EI}{l_{i}^{2}}&\frac{2 EI}{l_{i}}&0&-\frac{6 EI}{l_{i}^{2}}&\frac{4 EI}{ l_{i}}\end{bmatrix}\] (A4)  
\[\mathbf{k_{P}}=\begin{bmatrix}0&0&0&0&0&0\\ 0&\frac{p_{ij}}{l_{i}}&0&0&-\frac{p_{ij}}{l_{i}}&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&-\frac{p_{ij}}{l_{i}}&0&0&\frac{p_{ij}}{l_{i}}&0\\ 0&0&0&0&0\end{bmatrix}\] (A5)  
\[\mathbf{k_{I}}=m(h_{i}+\frac{l_{i}}{2})b_{0}\begin{bmatrix}0&0&0&0&0\\ 0&\frac{13}{32}l_{i}&\frac{11}{10}l_{i}&0&\frac{9}{70}l_{i}&-\frac{13}{25}l_{ i}^{2}\\ 0&\frac{11}{10}l_{i}&\frac{1}{105}l_{i}^{3}&0&-\frac{19}{10}l_{i}^{2}&-\frac{1 }{10}l_{i}^{3}\\ 0&0&0&0&0&0\\ 0&\frac{9}{70}l_{i}&-\frac{19}{10}l_{i}^{2}&0&\frac{13}{35}l_{i}&-\frac{11}{ 10}l_{i}^{2}\\ 0&-\frac{13}{42}l_{i}^{2}&-\frac{1}{140}l_{i}^{3}&0&-\frac{1}{140}l_{i}^{3}& \frac{1}{105}l_{i}^{3}\end{bmatrix}\] (A6)  
\[\mathbf{k_{G}}=G_{p}b_{0}\begin{bmatrix}0&0&0&0\\ 0&\frac{1}{5l_{i}}&\frac{2}{15l_{i}}&0&-\frac{1}{10}&-\frac{l_{i}}{30}\\ 0&0&0&0&0&0\\ 0&-\frac{6}{5l_{i}}&-\frac{1}{10}&0&\frac{6}{5l_{i}}&-\frac{1}{10}\\ 0&\frac{1}{10}&-\frac{l_{i}}{30}&0&-\frac{1}{10}&\frac{2}{15l_{i}}\end{bmatrix}\] (A7)  
where \(p_{ij}\) is the axial force at the middle of an element; \(h_{i}\) represents the depth of an element top; \(G_{p}\) is the shear modulus of soils.  
Figure 1: Element end forces and displacements.  
Furthermore, the element stiffness matrix \(k\) can be written in symmetry matrix form, Equation (A8) for simplification, with the double-subscript notation employed to represent the force and displacement, respectively. Hence, the global stiffness matrix of piles can be given by assembling the element stiffness matrix as shown in Figure A2, the dimensions of which are \((3N+3)\times(3N+3)\), while the element numbers are \(N\).