## 4 Pile-Soil Shaking Table Model Preparation
### Calculation and Analysis of the NSF of the Pile  
A suspended pile model was used in the test, where the model pile was subjected to tensile forces from the model box crossbeam and frictional forces from the soil around the pile. In a section of the pile unit, the sum of the pile unit sidewall frictional forces is equal to the pile body axial force difference, as shown in Figure 13. The pile body strain data from each measurement point (the strain-time history for pile depths of 0.1 m, 0.2 m, 0.3 m, 0.4 m, and 0.5 m under 1 Hz, 0.10 g sine wave loading are shown in Figure 14a-f, respectively) are used to calculate the pile body shaft force, and then the NSF can be found according to the force balance relationship [27], as shown in Equation (1). Where E is the pile modulus of elasticity, A is the pile cross-sectional area, and \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the strain values at the upper and lower measurement points of the pile unit. Negative values are negative skin friction, indicating that the soil is pulling down on the pile, and positive values are positive skin friction, indicating that the soil is supporting the pile. \[\mathbf{f}=\frac{\mathbf{EA}(\varepsilon_{2}-\varepsilon_{1})}{\mathbf{h\pi d}} \tag{1}\]  
Subplots a, b, and c in Figure 15 show the distribution of NSF for a sine wave with an input frequency of 1 Hz and peak accelerations of 0.1 g, 0.2 g, and 0.3 g, respectively. The maximum NSF of the pile occurs at pile depths of 0.22 m, 0.25 m, and 0.15 m, and the maximum values of the pile NSF are approximately 12.1 kPa, 43.6 kPa, and 75.1 kPa, with the locations of the neutral points occurring at pile depths of 0.35 m, 0.37 m, and 0.43 m, respectively. The maximum NSF of the pile body occurs at the upper part of the pile body, with an increase of 260.3% and 72.2% from 0.1 g to 0.2 g and from 0.2 g to 0.3 g under a 1 Hz sine wave, with the neutral point occurring between 0.35 m and 0.43 m, i.e., the middle and lower parts of the pile body. Subplots a, b, and c in Figure 16 show the distribution of NSF for a sine wave with an input frequency of 3 Hz and peak accelerations of 0.1 g, 0.2 g, and 0.3 g, respectively. The maximum NSF of the pile occurs at pile depths of 0.23 m, 0.20 m, and 0.20 m, and the maximum values of the pile NSF are approximately 45 kPa, 458 kPa, and 1015 kPa, with the locations of the neutral points occurring at pile depths of 0.38 m, 0.28 m, and 0.33 m, respectively. The maximum NSF of the pile occurs at the upper part of the pile, with an increase of 917.7% and 121.6% from 0.1 g to 0.2 g and from 0.2 g to 0.3 g under a 3 Hz sine wave, with the neutral point occurring between 0.28 m and 0.38 m, i.e., the lower and middle parts of the pile. Figure 13: Schematic diagram of the calculation of the NSF of the pile body. Subplots a, b, and c in Figure 17 show the distribution of NSF for a sine wave with an input frequency of 5 Hz and peak accelerations of 0.1 g, 0.2 g, and 0.3 g, respectively. The maximum NSF occurs at pile depths of 0.12 m, 0.28 m, and 0.29 m, with maximum values of pile NSF of approximately 350 kPa, 604 kPa, and 1502 kPa, and the locations of the neutral points occur at pile depths of 0.29 m, 0.38 m, and 0.40 m, respectively. The maximum NSF of the pile occurs in the middle of the pile, with an increase of 72.6% and 148.7% from 0.1 g to 0.2 g and from 0.2 g to 0.3 g at 5 Hz sine waves, with the neutral point occurring between 0.29 m and 0.40 m, i.e., the middle and lower parts of the pile. Figure 14: Strain-time history of different pile depths under 1 Hz, 0.10 g sine wave loading. (**a**) Pile-soil contact surface; (**b**) 0.1 m; (**c**) 0.2 m; (**d**) 0.3 m; (**e**) 0.4 m; (**f**) 0.5 m.  
Figure 15: Distribution of the NSF of a pile under 1 Hz with different PGA sine waves.