# Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation
## 3 Governing Equations and Their Solutions
### Solutions of the Equations  
For the steady-state lateral vibration of the pile-soil system, the dynamic equilibrium equation of the helical pile can be directly solved from the complex frequency domain, and the solutions of the equation can be written as  
\[u(z,t)=U(z)\cdot e^{i\omega t}\varphi(z,t)=\psi(z)\cdot e^{i\omega t}, \tag{8}\]  
where \(U(z)\) and \(\psi(z)\) denote the amplitudes of the LDD and rotation angle of the helical pile, respectively. i denotes the imaginary unit. Figure 3: Schematic of pile–soil interaction model. For the convenience of solving, the following equations are set:  
\[W_{\rm p}=E_{\rm p}^{\prime}I_{\rm p}^{\prime},J_{\rm p}=k^{\prime}A_{\rm p}G_{ \rm p},k_{\rm s}=k_{x}-m_{\rm p}\omega^{2}+c_{x}\omega\rm i, \tag{9}\]  
Substituting Equations (8) and (9) into Equation (7) yields  
\[W_{\rm p}\frac{d^{4}U(z)}{dz^{4}}-\frac{k_{\rm s}\cdot W_{\rm p}}{J_{\rm p}} \cdot\frac{d^{2}U(z)}{dz^{2}}+k_{\rm s}\cdot U(z)=0, \tag{10}\]  
\[\psi(z)=\frac{W_{\rm p}}{J_{\rm p}}\cdot\frac{d^{3}U(z)}{dz^{3}}+(1-\frac{k_{ \rm s}\cdot W_{\rm p}}{J_{\rm p}^{2}})\cdot\frac{dU(z)}{dz}, \tag{11}\]  
The solution of Equation (10) can be derived as  
\[U(z)=e^{az}(A_{1}\cos\beta z+B_{1}\sin\beta z)+e^{-az}(C_{1}\cos\beta z+D_{1} \sin\beta z), \tag{12}\]  
where \(A_{1}\), \(B_{1}\), \(C_{1}\), and \(D_{1}\) are constants determined by the boundary conditions. The coefficients \(\alpha\) and \(\beta\) satisfy the following equations:  
\[\alpha=\sqrt{\sqrt{\frac{k_{\rm s}}{4W_{\rm p}}}+\frac{k_{\rm s}}{4f_{\rm p}}}, \tag{13}\]  
\[\beta=\sqrt{\sqrt{\frac{k_{\rm s}}{4W_{\rm p}}}-\frac{k_{\rm s}}{4f_{\rm p}}}, \tag{14}\]  
Substituting Equation (12) into Equation (11) gives  
\[\psi(z)=e^{az}(A_{2}\cos\beta z+B_{2}\sin\beta z)+e^{-az}(C_{2}\cos\beta z+D_{ 2}\sin\beta z), \tag{15}\]  
According to the Material Mechanics, the BM and SF of the helical pile can be obtained as  
\[M(z) =-W_{\rm p}\frac{\mathrm{d}\psi(z)}{dz} \tag{16}\] Figure 3:0  
Figure 3:1 Figure 3:2  
Combined with, \(B_{1}\), \(C_{1}\), and \(D_{1}\), the undetermined coefficients in Equations (15)-(17) can be obtained as  
\[\left\{ \begin{array}{l}A_{2}=A_{1}\cdot t_{5}+B_{1}\cdot t_{6}\\ B_{2}=-A_{1}\cdot t_{6}+B_{1}\cdot t_{5}\\ C_{2}=-C_{1}\cdot t_{5}+D_{1}\cdot t_{6}\\ D_{2}=-C_{1}\cdot t_{6}-D_{1}\cdot t_{5}\end{array} \right., \tag{18}\]  
\[\left\{ \begin{array}{l}A_{3}=A_{1}\cdot t_{7}+B_{1}\cdot t_{8}\\ B_{3}=-A_{1}\cdot t_{8}+B_{1}\cdot t_{7}\\ C_{3}=C_{1}\cdot t_{7}-D_{1}\cdot t_{8}\\ D_{3}=C_{1}\cdot t_{8}+D_{1}\cdot t_{7}\end{array} \right., \tag{19}\]  
\[\left\{ \begin{array}{l}A_{4}=A_{1}\cdot t_{9}+B_{1}\cdot t_{0}\\ B_{4}=-A_{1}\cdot t_{0}+B_{1}\cdot t_{9}\\ C_{4}=-C_{1}\cdot t_{9}+D_{1}\cdot t_{0}\\ D_{4}=-C_{1}\cdot t_{0}-D_{1}\cdot t_{9}\end{array} \right., \tag{20}\]  
Since Equations (15)-(17) are derived from Equation (12), the undetermined coefficients satisfy certain equation relationships, which can be expressed as follows:\[t_{1}=e^{\alpha z}\cdot\cos\beta z, \tag{21}\]  
\[t_{2}=e^{\alpha z}\cdot\sin\beta z, \tag{22}\]  
\[t_{3}=e^{-\alpha z}\cdot\cos\beta z, \tag{23}\]  
\[t_{4}=e^{-\alpha z}\cdot\sin\beta z, \tag{24}\]  
\[t_{5}=\alpha(1-\frac{k_{s}\cdot W_{p}}{J_{p}^{2}})+\frac{W_{p}}{J_{p}}(\alpha ^{3}-3\alpha\beta^{2}), \tag{25}\]  
\[t_{6}=\beta(1-\frac{k_{s}\cdot W_{p}}{J_{p}^{2}})+\frac{W_{p}}{J_{p}}(-\beta^{ 3}+3\alpha^{2}\beta), \tag{26}\]  
\[t_{7}=\alpha^{2}-\beta^{2}-\frac{k_{s}}{J_{p}}, \tag{27}\]  
\[t_{8}=2\alpha\beta, \tag{28}\]  
\[t_{9}=\alpha\frac{k_{s}}{J_{p}}-\alpha^{3}+3\alpha\beta^{2}, \tag{29}\]  
\[t_{0}=\beta\frac{k_{s}}{J_{p}}+\beta^{3}-3\alpha^{2}\beta, \tag{30}\]  
When the rotation angle of the pile top is constrained and the pile bottom is fixed, the boundary conditions can be obtained as  
\[\left. \begin{array}{ccc}\psi(z)\right|_{z=0}=0;Q(z)\right|_{z=0}=Q_{0}\\ U(z)\right|_{z=L}=0;\psi(z)\right|_{z=L}=0\end{array} , \tag{31}\]  
Utilizing Equations (21)-(30), the boundary conditions can be rewritten as  
\[\left[