# Integrated Plug High-Strength Geocell Reinforcement in Foundation Design for Square Footing
## 2 Methodology and Analytical Equation
### Enhancement of Ultimate Bearing Capacity by the IPGC
#### 2.2.3 The Relationship between Hoop Stress and Cell Wall Tearing Strength of the IPGC  
The tearing force (\(F\)) of the cell wall can be calculated as follows:  
\[F=\sigma_{t}th \tag{9}\]where \(t\) and \(\sigma_{t}\) are the thickness and the linear elastic tensile stress of the cell wall, respectively, and \(h\) is the height of the cell wall. Due to the high stiffness and low strain of the cell wall, the stress distribution is considered uniform during loading. Consequently, the cell is assumed to be circular in theoretical calculations for computational simplicity. The stress of a single cell under an overburden loading can be calculated by the theory of thin-walled rings in the linear elastic range, as depicted in Figure 7. Although this approach may introduce some discrepancies between theoretical values and experimental results, it aids in establishing a foundational understanding of IPGC performance. This understanding can serve as a basis for more extensive analyses in subsequent investigations. Considering the symmetry, the tearing force of the cell wall (\(F\)) can be calculated as follows:  
\[2F=\int_{0}^{\pi}\sigma_{r}h\frac{d_{0}}{2}(\sin\theta)d\theta=\sigma_{r}hd_{0} \tag{10}\]  
where \(\sigma_{r}\) is the lateral confinement stress of the cell induced by the movement of soil particles, and \(d_{0}\) is the diameter of the maximum cell size. Considering the circle shape of a cell, the following equation can also be obtained:  
\[\sigma_{r}=\sigma_{x}=\sigma_{y} \tag{11}\]  
From Equations (10) and (11), the following equation can be obtained:  
\[\sigma_{x}=\sigma_{y}=\sigma_{r}=\frac{2F}{hd_{0}} \tag{12}\]  
Three tensile tests were conducted to measure the strength of the cell wall and plug-in joints, as depicted in Figure 8. It is worth noting that the samples were not filled with soil during the test. The results indicated minimal variation among the three sets of tests, demonstrating good consistency. The peak tensile strength standard deviation for the tested samples was 2.6 MPa. Figure 8a displays the photographs and stress-strain curve of the cell wall, while Figure 8b illustrates the photographs and stress-strain curve of geocell joints after tension tests. Figure 8a shows that the stress-strain relationship of the cell wall remains linear within the 10% strain range. Meanwhile, from Figure 8b, when the stress is 32.5 MPa (see point A in Figure 8b), the cell wall at the pin plug-in joint begins to tear significantly (see comparison from the dotted line box to the solid line box in Figure 8b). In the model experiments, it was observed that the bearing capacity of the foundation could be significantly diminished, resulting in rapid failure when tearing occurs along the integrated plug-in joint of the cell wall laterally [14]. Therefore, the overburden loading corresponding to the tearing strength of the cutting cell wall at the plug-in joint of the IPGC can represent the cell wall's tearing force (\(F\)). According to Equations (5), (7) and (12), the bearing capacity increase by the cell directly below the foundation of IPGC (\(\Delta p\)) can be expressed as follows:  
\[\Delta p=\frac{2F}{hd_{0}K_{0}\alpha} \tag{13}\]  
Figure 7: Stress distribution in a cell. Confinement from adjacent cells provides additional resistance against the loaded cell through passive resistance, while the lateral expansion of the infill restricted by high hoop strength should also be considered:  
\[\sum_{i=1}^{n}\Delta p_{i}=\sum_{i=1}^{n}\frac{2F}{hd_{0}K_{0}\alpha} \tag{14}\]  
where \(n\) is the number of cells, and \(i\) is the \(i\)th cell. The possibility of the superposition of the load-bearing capacity of geocells and the soil contained in them is supported by several mechanisms: the joint action between the basement structure and internal soil [12], the reinforcement effect of the soil [14], their mutual interaction [16], and existing research findings [7; 20; 21; 22].