## 4 Validation of the Calculation Method
### Passive Pile Analysis Based on Staged Descriptions of Additional Loads  
A passive pile test in Ningbo [6] was selected to judge the correctness of the method and program in this paper further. Table 2 describes the soil layer distribution and calculation parameters of the test site. The site test pile is a cast-in-place bored pile with a length of 45 m, a diameter of 1.0 m, and a pile spacing of 3.0 m. The stiffness of this reinforced concrete pile is \(EI=1.65\times 10^{6}\)\(\mathrm{kN}\cdot\mathrm{m}^{2}\), and the surcharge strength is 70.0 kPa. In this case the load is formed by filling the soil in a rectangular area (surcharge length (\(A\) = 25.0 m), \begin{table}
\begin{tabular}{c c} \hline \hline
**Computational Model** & \(p\)**-\(y\) Curve Expression** \\ \hline Zhang Model [17] & \(p=\left\{\begin{array}{ll}0.5p_{u}(y/y_{50})^{1/3}&(y\leq 8y_{50})\\ \left[F_{s}+(1-F_{s})x/x_{r}\right]p_{u}&(y>8y_{50},z\leq z_{rs})\\ p_{u}&(y>8y_{50},z>z_{rs})\end{array}\right.\) \\ Wang Model [18] & \(p=\left\{\begin{array}{ll}\dfrac{y/y_{50}}{a+by/y_{50}}p_{u} \left(y\leq\beta y_{50}\right)\\ p_{u}&(y>\beta y_{50})\end{array}\right.\) \\ Matlock model [19] & \(p=\left\{\begin{array}{ll}0.5p_{u}(y/y_{50})^{1/3}\left(y\leq 8y_{50}\right)\\ p_{u}&(y>8y_{50})\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(p\)–\(y\) curve expression. Figure 6: Test and calculation results of different \(p\)–\(y\) curves. (**a**) Horizontal displacement of the pile top; (**b**) maximum bending moment of the pile shaft. surcharge width (\(B\) = 10.0 m)). Finally, the surcharge distances (\(e\) in Figure 1b) are 15 m (working condition 1) and 10 m (working condition 2). The test results for the pile displacement and bending moment under the two working conditions were compared with our calculation results, as shown in Figures 7 and 8. In the figures, the positive displacement indicates that the pile deviates away from the surcharge load, and the positive bending moment means that the pile is strained on the side of the surcharge load. According to the measured results, the maximum horizontal displacement occurs at the top of the pile under both the working conditions, and the pile displacement gradually decreases along the depth direction. If the horizontal displacement of the pile is greater than or equal to 0.2 mm, the horizontal displacement of the pile in condition 1 and condition 2 occurs in the range of about 22 m (2.2\(B\)) and 28 m (2.8\(B\)) below the ground, respectively. This is obviously greater than the pile displacement depth 9.4 m calculated by elastic analysis when the pile is in an active pile condition (calculated by the \"\(m\)\" method, at \(\alpha z=4.0\)[15]), and it increases as the distance between the pile and surcharge load decreases. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Soil Layer** & **Soil Thickness (m)** & \begin{tabular}{c} **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} & \begin{tabular}{c} **Modulus of** \\ **Compression (MPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Cohesion** \\ **(kPa)** \\ \end{tabular} &
\begin{tabular}{c} **Internal Friction** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} \\ \hline Miscellaneous fill & 4.8 & 1800 & - & 5 & 11.5 \\ Mud & 8.3 & 1642 & 2.09 & 10.5 & 10.2 \\ Silty day with silty soil & 4.8 & 1877 & 4.31 & 12.2 & 12.8 \\ Muddy silty clay & 12.6 & 1733 & 3.24 & 11.9 & 11.2 \\ Silty clay & 12.7 & 1742 & 4.63 & 15.3 & 12.3 \\ Silty clay & 1.9 & 1774 & 3.77 & 24.1 & 11.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation parameters of the soil layers. Figure 7: The calculated and measured values for working condition 1 (\(e\) = 15 m). (**a**) Pile displacement; (**b**) pile-bending moment. From the pile top to pile bottom, the pile-bending moment changes from a negative moment of load side compression to a positive moment of load side tension under both conditions.