# Numerical Study on Scale Effect of Repetitive Plate-Loading Test
## 4 Analysis of the Results
### Examination through Comparison with the Scale Effect of the Existing Plate-Loading Test  
For the examination and reliability verification of the scale-effect results of the repetitive plate-loading test obtained through the numerical analysis in this study, the results were compared with the scale-effect results of the existing plate-loading test. Thus, the scale effect of the repetitive plate-loading test proposed in this study was converted into a form similar to Equation (3), which was the scale effect of the plate-loading test as expressed by Equations (12) and (13). In Equation (12), \(E_{k}\) is the ratio of the amount of stress change to that of the settlement change and had a form similar to the subgrade reaction modulus of the plate-loading test, i.e., k. Therefore, if the two moduli are assumed to be the same, an equation on the scale effect of the repetitive plate-loading test is derived with the same form as Equation (3), as expressed in Equation (13). Here, B, which is the diameter of the foundation in the plate-loading test, was changed to the \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{3}{c}{**Subgrade Condition**} & & \\ \cline{2-5}
**Loading-Plate Diameter** & \multicolumn{1}{c}{**Gravel and**} & \multicolumn{1}{c}{**Sand**} & \multicolumn{1}{c}{**Sandy Soil**} & \multicolumn{1}{c}{**Cohesive Soil**} & \multicolumn{1}{c}{**Field-Test**} \\  & \multicolumn{1}{c}{**Fine Gravel**} & \multicolumn{1}{c}{**Sand**} & \multicolumn{1}{c}{**Sandy Soil**} & \multicolumn{1}{c}{**Cohesive Soil**} & \multicolumn{1}{c}{**Roadbed**} \\ \hline
50-mm loading plate & 2.23 & 1.74 & 1.41 & 1.38 & 1.44 \\
150-mm loading plate & 1.67 & 1.45 & 1.20 & 1.24 & 1.22 \\
300-mm loading plate & 1.28 & 1.24 & 1.13 & 1.18 & 1.17 \\
400-mm loading plate & 1.16 & 1.16 & 1.09 & 1.12 & 1.09 \\
600-mm loading plate & 1.09 & 1.10 & 1.06 & 1.08 & 1.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Scale effect of repetitive plate-loading test according to the subgrade condition. Figure 6: Regression analysis of the scale-effect results according to the subgrade condition: (**a**) gravel and fine gravel; (**b**) sand; (**c**) sandy soil; (**d**) cohesive soil. reference loading plate. Thus, in this study, the 762-mm loading plate selected as the reference loading plate was applied. \[\frac{E_{v}}{E_{v0}}=\frac{1.5\frac{B}{2}.\frac{\Delta\alpha}{\Delta s}}{1.5\frac {B_{v}}{2}.\frac{\Delta\alpha}{\Delta s}}=\frac{B}{B_{v}}.\frac{E_{k}}{E_{k0}} \tag{12}\]  
\[\frac{E_{v}}{E_{v0}}=\frac{B}{B_{o}}\left(\frac{B}{B_{o}}\right)^{n}=\left( \frac{B}{B_{o}}\right)^{n+1}=\left(\frac{B}{B_{o}}\right)^{a}\qquad\quad(k=E_ {k}) \tag{13}\]  
where B is the diameter of the reference loading plate (mm), B\({}_{0}\) is the diameter of the used loading plate (mm), \(E_{k}\) is the ratio of the amount of stress change to that of the settlement change at points with 30% and 70% of the maximum stress (\(\Delta\sigma\)/\(\Delta\)s), N is the scale-effect calculation factor of the plate-loading test, a is the scale-effect calculation factor of the repetitive plate-loading test (\(n+1\)), and \(k\) is the subgrade reaction modulus of the plate-loading test (\(\sigma\)/s). To determine the scale-effect calculation factor of the repetitive plate-loading test, i.e., a, the loading-plate diameter, which is the x axis shown in Figure 6, was normalized to the diameter of the reference loading plate (762 mm), as shown in Figure 7. Regression equations were derived through regression analysis, and the value of the x-intercept was close to one under all subgrade conditions. Therefore, when this value was ignored, the scale-effect calculation factor of the strain modulus, namely, a, was found to be 0.3 for the gravel and fine gravel, 0.2 for sand, 0.12 for sandy soil, and 0.11 for cohesive soil.