# A Novel Strutless Double-Row Structure for Deep Excavation: Working Mechanism Study and Analysis
## 2 Model Test and Numerical Model Introduction
### Retaining Structures  
In the context of model testing for retaining structures in excavation projects, the precise fabrication of model retaining structures is crucial for the success of the experiment. The model material must meet a series of strict standards: uniformity, isotropy, stability of performance, excellent linear elastic characteristics, and the ability to be easily processed and shaped. As retaining structures should maintain elastic behavior under normal operating conditions, the focus is on replicating the elastic response of real retaining structures using a material that can simulate this behavior. Polypropylene is selected for the model retaining structures due to its good elastic properties, which can simulate the response of actual structures, and its lightweight nature, which simplifies the operational procedures of the model test. The elastic modulus of polypropylene, as show in Figure 5, determined using a universal testing machine, is 1.5 GPa. The diameter of the model front-row pile (FRP) is set at 50 mm, with a length of 1000 mm, and the thickness and height of the model back-row retaining wall are both 30 mm and 1000 mm, respectively. Assuming pile elasticity during the test, the bending moment of the pile is calculated using the following formula:  
\[M=\frac{E_{p}I\Delta\epsilon}{b} \tag{1}\]  
\[I=\frac{\pi d^{4}}{64} \tag{2}\]  
\(M\) represents the bending moment of acting on the section; \(E_{p}\) is the elastic modulus of the pile material; \(I\) is the moment of inertia; \(\Delta\epsilon\) is the strain difference measured by two strain gauges in the same section; \(b\) is the horizontal spacing between two strain gauges of the same pile; and \(d\) is the diameter of the pile. The flexural stiffness (\(E_{p}\cdot I\)) of the pile is 460.18 N\(\cdot\)m\({}^{2}\), and the strain value of the strain gauge is 54.34 \(\mu\) when the section is subjected to a unit bending moment (1 N\(\cdot\)m). Based on the similitude relationship in Table 1, the prototype retaining piles (made of C30 reinforced concrete) were approximately 1.0 m in diameter, values that are frequently used in practical engineering. Other geometric and physical parameters of the model and prototype excavations are presented in Table 3.  
Figure 5: Model pile. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Prototype** & **Model** \\ \hline Excavation depth (m) & 10 & 0.5 \\ Structure embedment length (m) & 10 & 0.5 \\ Pile spacing (m) & 1.6 & 0.08 \\ Pile diameter (m) & 1 & 0.05 \\ Back-row wall width (m) & 0.6 & 0.03 \\ _EI_ (N\(\cdot\)m) & 460 & 1.472 \(\times\) 10\({}^{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the piles and excavations in the prototype and model. This experiment assumes that the crossbeam (Figure 6) is completely rigid and that the crossbeam is constructed from a 3 mm thick rectangular iron plate to ensure a rigid connection with the model structure. Welded connectors beneath the steel plate are attached to the front row-piles and back-row walls (piles), creating a structural connection between them, thus simulating the interconnection used in actual engineering practices (as shown in Figure 7). Furthermore, to prevent sand leakage through the gaps between the inner row of piles, a geotextile is installed behind this row. This geotextile serves the purpose of mimicking the sand leakage prevention provided by shotcrete with a net in real-world engineering scenarios. Figure 6: Sketch of crossbeam. (a) SPW; (b) DRPW; (c) ARPW. Figure 7: Layout of retaining structure. The layouts of the retaining structures are shown in Figure 7. To impartially assess the supporting capabilities of ARPW, control groups including SPW and DRPW were established. The excavation depth and pile embedment length were both standardized to 500 mm, with the excavation process replicating that of ARPW structures. Simultaneously, the stress deformation and alterations in the piles were compared between DRPW and ARPW structures by varying the row spacing.