# Numerical Study on Scale Effect of Repetitive Plate-Loading Test
## 4 Analysis of the Results
### Loading-Plate Scale Effect of the Repetitive Plate-Loading Test under Field-Test Roadbed Condition  
Because the subgrade reaction modulus of the plate-loading test is the solution to the boundary-value problem in which the loading plate contacts the subgrade, the original properties of the subgrade were approached, and the modulus decreased when the size of the loading plate increased. Therefore, the subgrade reaction modulus was used considering the effect of the loading plate size, as expressed in Equation (3). Because the repetitive plate-loading test is also the solution of the boundary-value problem in the same manner, the scale effect must be considered and corrected when the original properties of the subgrade were approached when the loading-plate size increased. The calculated value of the strain modulus also varied depending on the loading-plate size in the same subgrade.  
Figure 4: Regression analysis comparison between the field-test and numerical analysis results: (**a**) field-test results ([PERSON] et al. [6]); (**b**) numerical analysis results.  
Under the condition of the field-test embankment roadbed, we confirmed that the results of the 50-mm loading plate were approximately 19% lower than those of the 300-mm loading plate. Therefore, further research was conducted using 150-mm, 400-mm, 600-mm, and 762-mm loading plates under the same subgrade condition, and the results were compared with those of the 762-mm loading plate, which had the largest diameter among the employed loading plates, as listed in Table 5. The analysis results showed that the calculated strain-modulus value decreased as the loading-plate diameter decreased under the same subgrade condition. The ratios of the result of the 762-mm loading plate to each of the loading plates were 1.44, 1.22, 1.17, 1.09, and 1.08 for the 50-mm, 150-mm, 300-mm, 400-mm, and 600-mm loading plates, respectively. The deviation was close to zero under all subgrade conditions with different elastic moduli.  
Figure 5 shows the stress-settlement curves of the 762-mm, 600-mm, 300-mm, and 50-mm loading plates in the subgrade with an elastic modulus of 20 MPa. Figure 5 shows that the nonlinearity of the stress-settlement curve increased as the loading-plate diameter decreased. In other words, nonlinear behavior was observed as the loading-plate size decreased in the subgrade area with the same elastic behavior. Because the strain-modulus estimation equation of the repetitive plate-loading test is a theoretical equation derived based on the elastic theory, the error increased as the nonlinear behavior increased, which was determined to cause the decrease in the calculated strain-modulus value as the loading-plate size decreased. Apparently, therefore, the scale effect must be considered, as expressed in Equation (7), when a loading plate with a diameter of 762 mm or less is used.  
\[E_{v2,762 mm}=1.08\cdot E_{v2,600 mm}=1.09\cdot E_{v2,400 mm}=1.17\cdot E_{v2,300 mm}=1.22\cdot E_{v2,150 mm}=1.44\cdot E_{v2,50 mm} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Strain Modulus, \(E_{v2}\) (MPa)**} \\ \cline{2-6} \multirow{2}{*}{\(E\) (MPa)} & \multicolumn{2}{c}{**50-mm**} & \multicolumn{2}{c}{**150-mm**} & \multicolumn{2}{c}{**300-mm**} \\  & \multicolumn{2}{c}{**Loading Plate**} & \multicolumn{2}{c}{**Loading Plate**} & \multicolumn{2}{c}{**Loading Plate**} & \\ \hline
20 & 12.55 & 1.45 & 14.79 & 1.23 & 15.59 & 1.16 \\
40 & 25.3 & 1.44 & 29.74 & 1.22 & 31.09 & 1.17 \\
60 & 37.93