# Plate Size Effects in Gravelly Soil Based on In Situ Plate Load Tests and Finite Element Analysis
## 4 Finite Element Analysis
### Results and Analysis  
#### 4.2.1 Average Contact Pressure-Settlement Curves  
As shown in Figure 8a, the variation pattern of the average contact pressure-settlement curve (\(p\)-\(s\) curves) changed with the increasing diameter of the test plate. Specifically, as the test plate diameter increased, the \(p\)-\(s\) curve exhibited more pronounced nonlinear characteristics. Additionally, at the same contact stress, larger diameter test plates resulted in greater displacements. As shown in Figure 8b, despite the variation in test plate diameters, the average contact stress-settlement gradient (\(\Delta s/\Delta p\)) followed a consistent pattern with the average contact stress (\(p\)). This can be effectively described using a three-stage linear model. Figure 8: The relationship between average contact stress and settlement. (**a**) Average contact stress versus settlement. (**b**) Average contact stress–settlement gradient versus contact stress. In the first stage, \(\Delta s/\Delta p\) was constant, indicating that its slope was zero as the load increased. This suggests that the \(p\)-\(s\) curve remained in the elastic phase. In the second stage, as the load increased, the \(\Delta s/\Delta p\) began to rise, with its rate of increase becoming a non-zero constant. This indicates that the \(p\)-\(s\) curve entered the yield phase. In the third stage, as the load continued to increase, the rate of increase of the \(\Delta s/\Delta p\) rose sharply, suggesting that the soil reached the failure stage. Table 3 presents the elastic and yield limit loads derived from the average contact stress-settlement gradient curves. For test plates of different diameters, the load at the elastic limit state remained constant at 1 MPa. In contrast, the yield limit load progressively increased with the test plate size and eventually leveled off at 4.6 MPa. Notably, the yield limit load did not follow a linear relationship with the increasing size of the loading plate, aligning with findings from load tests on pebble layer foundations in literature [23]. To clearly illustrate the shear failure behavior of the soil, the relative shear stress [24] was introduced, and it is defined as:  
\[\tau_{rel}=\frac{\tau}{\tau_{max}} \tag{2}\]  
\(\tau_{max}\) represents the maximum shear stress when the Mohr stress circle is tangent to the Coulomb failure envelope. \(\tau\) represents the shear stress in the soil. When \(\tau_{rel}\) equals 1, it signifies that the Coulomb failure envelope has been reached. When \(\tau_{rel}\) equals 0, it indicates that the soil remains entirely undisturbed. Figure 9 illustrates the shear failure condition of the soil beneath test plates with diameters of 2 m and 16 m at an average contact stress of 1 MPa, as determined by the relative shear stress \(\tau_{rel}\). The figure shows significant differences in the shear failure behavior of the soil under the two test plate sizes. According to Table 3, the 2 m test plate experienced a settlement of 2.42 mm at an average pressure of 1 MPa, with a complete shear band from beneath it. In contrast, under the same load, the 16 m diameter test plate underwent a settlement of 19.89 mm, with only localized shear bands forming near the plate's edge. This phenomenon suggests that under the same average contact stress conditions, the soil beneath the smaller test plate reached its failure point within a limited deformation range, resulting in the formation of a continuous shear plane. However, this does not imply that this level of deformation approached or reached the soil's actual ultimate deformation capacity. Therefore, the ultimate bearing capacity determined from small test plate tests may satisfy safety standards based on strength failure, but it may fall short of the deformation safety requirements in real-world engineering applications.