# Numerical Study on Scale Effect of Repetitive Plate-Loading Test
## 2 Theoretical Background
### Previous Study Cases on the Scale Effect of the Loading Plate  
[PERSON] et al. [2] proposed a method of estimating the elastic modulus from the bearing-capacity factor derived from the plate-loading test based on the existing elastic theory and empirical correlations. In particular, they explained that the difference between the subgrade reaction and elastic moduli is caused by the nonlinear behavior of the plate-loading test in the region generally treated as havingelastic behavior and that the measured properties approach the actual properties as the size of the loading plate increases. They also identified the deformation-level dependence, load duration, and loading-width dependence as the three major factors that cause such nonlinearity. Among them, the loading-width dependence refers to the characteristic in which the subgrade reaction modulus decreases as the loading width increases in the same subgrade. This result suggested the need for the scale effect. Further, the subgrade reaction and the elastic moduli have a suggested estimated ratio of 2:1, while the subgrade strain and elastic moduli have that of 1:1.  
[PERSON] et al. [6] suggested the need for a loading plate that is smaller than 300 mm for the laboratory repetitive plate-loading test. For this, they examined the previous study cases and compared the repetitive plate-loading test results of the 50- and 300-mm loading plates through a field test. They found that the test results using the two loading plates were similar (correlation coefficient: 0.987), but there was an average difference of 6%. Therefore, they mentioned that caution must be exercised when the results using the 50-mm loading plate are used. The results had a limited number of data because of the limitations in the field test, and the 50-mm loading plate could have been more affected by the particle sizes of the embankment roadbed because they were irregular in the field.  
[PERSON] and [PERSON] [10] mentioned that the scale effect applied to the plate-loading test can be applied to pure sand or clay and that such an application suffers from problems for mixed sand and clay in the field. As such, they proposed an estimation equation for the reasonable application of the scale effect according to the mixing ratios of sand and clay when the bearing capacity and settlement amount of the actual foundation are estimated. Their proposal suggested that different scale effects must be applied depending on the subgrade condition.  
[PERSON] et al. [14] conducted numerical analyses and model tests on band and circular bases and drew the following conclusions. In dry sand, when the base size increased from 0.1 m to 10 m, the bearing capacity increased from 0.62 MPa to 12.3 MPa in the band base and from 0.29 MPa to 6.73 MPa in the circular base. When the base increased by approximately 10 times, the bearing-capacity factor decreased by 55%. Similar results were obtained in the centrifugal model test.  
To use unbonded granulated material mixed with the rubber particles of scrap tires as raw material for roads and railways, [PERSON] et al. [15] performed laboratory and field tests to assess the bearing capacity of the mixture. In the laboratory, a repetitive plate-loading test was performed using a 75-mm loading plate, while in the field test, a repetitive plate-loading test was performed using a 300-mm loading plate. A linear regression analysis was performed for the stiffness values measured in the laboratory and field tests. The results of the two tests had a similar correlation. However, the number of samples used for statistical analysis was small (four data points), and the subjects evaluated were the same. This study has limitations as it was conducted under different conditions for the laboratory and field tests.