## 3 Formulation and Solution of Governing Equations
### The Solution to the Equations for the Surrounding Soil and Core Soil of the Pile
\begin{array}{ccc}-E_{p}I_{p}\bigg{[}\chi_{1}\alpha_{1}+\sum\limits_{n=1}^{ \infty}\gamma_{11}^{\prime}g_{n}+\sum\limits_{n=1}^{\infty}\gamma_{21}^{\prime}g _{n}\bigg{]}&\kappa_{2}\chi_{2}-E_{p}I_{p}\bigg{[}\sum\limits_{n=1}^{\infty} \gamma_{12}^{\prime}g_{n}+\sum\limits_{n=1}^{\infty}\gamma_{22}^{\prime}g_{n} \bigg{]}\\ 0&\chi_{2}-\alpha_{1}\\ \cos(\alpha_{1}L)&\sin(\alpha_{1}L)\\ \chi_{1}\sin(\alpha_{1}L)+\sum\limits_{n=1}^{\infty}\gamma_{11}^{\prime}gh(g_{ n}L)+\sum\limits_{n=1}^{\infty}\gamma_{12}^{\prime}gh(g_{n}L)&\chi_{2}\cos( \alpha_{1}L)+\sum\limits_{n=1}^{\infty}\gamma_{12}^{\prime}g_{n}(g_{n}L)+\sum \limits_{n=1}^{\infty}\gamma_{22}^{\prime}gh(g_{n}L)\\ -E_{p}I_{p}\bigg{[}\chi_{2}\alpha_{2}+\sum\limits_{n=1}^{\infty}\gamma_{13}^{ \prime}g_{n}+\sum\limits_{n=1}^{\infty}\gamma_{23}^{\prime}g_{n}\bigg{]}&\kappa _{2}\chi_{4}-E_{p}I_{p}\bigg{[}\sum\limits_{n=1}^{\infty}\gamma_{14}^{\prime}g_ {n}+\sum\limits_{n=1}^{\infty}\gamma_{24}^{\prime}g_{n}\bigg{]}\\ 0&\chi_{4}-\alpha_{2}\\ \par ch(g_{2}L)&\begin{array}{c}sh(g_{2}L)\\ \end{array}\\ \chi_{3}sh(g_{2}L)\\ \end{array} \right)\left( \begin{array}{c}K_{1}\\ K_{2}\\ K_{3}\\ K_{4}\\ \end{array} \right)=\left( \begin{array}{c}\left(\frac{\kappa_{2}}{\kappa_{1}} -1\right)\overline{M}_{c}\\ \overline{p}\\ \overline{A}G_{p}\overline{p}\\ 0\\ \end{array} \right) \tag{39}\]  
When \(\overline{M}(0,s)\leq\overline{M}_{c}(0,s)\), let \(\overline{M}_{c}=0\), replacing the first row of the coefficient matrix on the left-hand side of the above system of equations with:  
\[ \begin{array}{c}-E_{p}I_{p}\bigg{[}\chi_{1}\alpha_{1}+\sum\limits_{n=1}^{ \infty}\gamma_{11}^{\prime}g_{n}+\sum\limits_{n=1}^{\infty}\gamma_{21}^{ \prime}g_{n}\bigg{]}&\kappa_{1}\chi_{2}-E_{p}I_{p}\bigg{[}\sum\limits_{n=1}^{ \infty}\gamma_{12}^{\prime}g_{n}+\sum\limits_{n=1}^{\infty}\gamma_{22}^{ \prime}g_{n}\bigg{]}\\ -E_{p}I_{p}\bigg{[}\chi_{3}\alpha_{2}+\sum\limits_{n=1}^{\infty}\gamma_{13}^{ \prime}g_{n}+\sum\limits_{n=1}^{\infty}\gamma_{23}^{\prime}g_{n}\bigg{]}&\kappa _{1}\chi_{4}-E_{p}I_{p}\bigg{[}\sum\limits_{n=1}^{\infty}\gamma_{14}^{\prime}g _{n}+\sum\limits_{n=1}^{\infty}\gamma_{24}^{\prime}g_{n}\bigg{]}\\ \end{array} \]  
The above system of linear equations is solved to determine the coefficients \(K_{1}\), \(K_{2}\), \(K_{3}\), and \(K_{4}\) under the flexible constraint conditions at the pile head. Introduce two parameters: the pile head displacement ratio \(\chi\) and the constraint degree \(\lambda\)[22]. \[\chi=\overline{u}(0,s)/\overline{u}_{free}(0,s)\times 100,\;\lambda=\overline{M}(0,s )/\overline{M}_{fixed}(0,s)\times 100 \tag{40}\]  
The parameter \(\overline{u}_{free}\) is obtained by substituting the coefficients derived from the system of Equation (36) into Equation (32). The boundary condition (15) is defined as \(\theta|_{z=0}=0\), which involves replacing the first row of the coefficient matrix in Equation (36) with \(\left( \begin{array}{ccc}0&\chi_{2}&0&\chi_{4}\\ \end{array} \right)\). Solving the modified system yields the undetermined coefficients \(K_{1}\), \(K_{2}\), \(K_{3}\), and \(K_{4}\) under fixed-head conditions. These coefficients are, subsequently, substituted into Equation (34) to calculate \(\overline{M}_{fixed}\). The load profiles for triangular, half-sine, and rectangular horizontal impact forces are described as follows:  
\[P(t)=\frac{2P_{\max}t}{t_{0}}H(t_{0}/2-t)+2P_{\max}(1-t/t_{0})[H(t-t_{0}/2)-H( t-t_{0})] \tag{41}\]  
\[P(t)=P_{\max}\sin(\pi t/t_{0})H(t_{0}-t) \tag{42}\]  
\[P(t)=P_{\max}H(t_{0}-t) \tag{43}\]  
The resulting expressions are obtained after applying the Laplace transform:  
\[\overline{P}(s)=\frac{2P_{\max}\Big{(}1-e^{-t_{0}s/2}\Big{)}^{2}}{t_{0}s^{2}} \tag{44}\]  
\[\overline{P}(s)=\frac{P_{\max}rt_{0}\big{(}1+e^{-t_{0}s}\big{)}}{\pi^{2}+t_{0}^ {2}s^{2}} \tag{45}\]  
\[\overline{P}(s)=\frac{P_{\max}\big{(}1-e^{-t_{0}s}\big{)}}{s} \tag{46}\]  
\(H(. )\) is the Heaviside step function. The solutions obtained above are expressed in the Laplace transform domain.