## 2 Similarity Theory and the Model's Test Parameters
### Loading and Data Collection  
According to _the Code for the Design of the Subgrade of Railways_ (TB 10001-2016), the track is a high-speed railway ballast track, and the load on the top surface of the subgrade should meet those requirements. According to _the Code for the Design of Retaining Structures of Railway Subgrades_ (TB 10025-2019), the displacement of the pile's top should be less than 1/100 of the length of the pile cantilever and should not exceed 10 cm. The cantilever end  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{2}{c}{**Grading Test**} & \multicolumn{2}{c}{**Compaction Test**} & \multicolumn{2}{c}{**Shear Test**} \\ \cline{2-7}  & **C\({}_{\textbf{u}}\)** & **C\({}_{\textbf{c}}\)** & **W\({}_{\textbf{o}}\)/\%** & **\(\rho_{\textbf{o}}\)/(g/cm\({}^{3}\))** & **c/kPa** & **\(\varphi^{c}\)** \\ \hline Subgrade filling & 49 & 0.3 & 5.7 & 2.05 & 13.82 & 35.4 \\ Filling soil & — & — & 16.4 & 1.81 & 10.1 & 35.3 \\ Pebble soil & 23.5 & 1.15 & 8.5 & 2.03 & 5.2 & 36.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the filling material.  
Figure 6: The large shear test’s equipment.  
lengths of the existing and new piles in this test were 0.56 and 0.92 m, respectively. Thus, the displacements were to be less than 5.6 and 9.2 mm, respectively.  
(1) Model test loading scheme  
The ZK load of the new line was 36.8 kPa, the self-weight load on the track was 17.3 kPa, the total load was 54.1 kPa (P), and the width was 0.425 m. When only the existing sheet pile wall was loaded, the freight line load was preloaded with 68.4 kN/m\({}^{2}\). The freight line load was removed and the soil was backfilled after the new piles were installed. The line was then loaded in two steps to a design load of 54.1 kN/m\({}^{2}\). In accordance with the conversion relationship, the top load was applied to the subgrade at 1.5 P\(\rightarrow\)2.0 P\(\rightarrow\)2.5 P\(\rightarrow\) until the pile top displacement was exceeded (with an increment of 0.5 P). The pile deformation, load, and pile strain were measured at 5, 15, 30, 45, and 60 min after each load level was applied, and subsequently at 30 min intervals. The deformation of the pile top did not exceed 0.1 mm per hour, and the pile deformation could be considered stable when the above-mentioned deformation occurred twice in succession. When the deformation rate of the pile top stabilized, the next load level was applied, as shown in Figure 7.  
(2) Test data collection and processing  
The strain values were obtained from the static strain test and analysis system. The internal force and the earth pressure behind the composite sheet pile wall were calculated as follows.  
1 Calculation of pile bending moment  
The pile curvature \(\varphi(z)\) at relatively small strains can be obtained using the elastic foundation beam theory [19]:  
\[\varphi(z)=\frac{\Delta\varepsilon}{a} \tag{1}\]  
where \(z\) is the distance from the pile top to the strain gauge (unit: m). \(\Delta\varepsilon\) is the strain difference between the back of the pile and the surface of the pile at depth \(z\) derived from the strain gauges. \(a\) is the height of the pile section, which was 0.25 m for the test pile.  
The relationship between the pile curvature and the bending moment gives the bending moment of the model pile at different sections:  
\[M(z)=EI\cdot\varphi(z) \tag{2}\]  
where \(EI\) is the flexural stiffness of the test pile, kN\(\cdot\)m\({}^{2}\).  
Figure 7: The model test: (**a**) horizontal view (**b**) side view.  
Earth pressure calculation  
The force \(F\) acting on the pile is obtained from the load sensor installed between the retaining plate and the pile. If the earth pressure on each layer is uniform, it can be converted to \(p\)[7]:  
\[p=\frac{2F}{L\cdot a} \tag{3}\]  
where \(L\) is the span of the retaining plate (unit: m). \(a\) is the width of the retaining plate (unit: m).