# Numerical Study on Scale Effect of Repetitive Plate-Loading Test
## 2 Theoretical Background  
### Plate Loading and Repetitive Plate-Loading Tests  
The plate-loading test is a representative field-testing method that determines the bearing-capacity factor considering the deformation characteristics of the subgrade. The average vertical stress and settlement amount are measured by applying a load to the surface of the roadbed after completion of the construction using a circular rigid plate with a thickness of 22 mm or more. The subgrade reaction modulus can be obtained using Equation (1).  
\[k=\frac{\sigma}{s} \tag{1}\]  
where \(k\) is the subgrade reaction modulus (MN/m\({}^{3}\)), \(s\) is the significant settlement amount (asphal concrete: 2.5 mm; cement concrete: 1.25 mm), and \(\sigma\) is the vertical stress corresponding to the significant settlement amount (kPa).  
This subgrade reaction modulus represents the subgrade reaction of the foundation, i.e., the gradient in the relationship between the resistance of the subgrade and displacement per unit area. It is the solution of the boundary-value problem, instead of a property of the subgrade, and is affected by the size of the loading plate and loading conditions [2]. For this scale effect, many experimental studies have been conducted on the relationship between the width of the foundation and settlement amount. When the stress-settlement curve is assumed to be linear, the scale effect of the subgrade reaction modulus can be expressed using its reciprocal [7]. Equation (2) is the empirical formula proposed using the results of the loading test conducted by [PERSON] and [PERSON] [8], which is most widely used. The scale effect proposed by other studies is summarized by Equation (3).  
\[\begin{array}{ll}\frac{S}{S_{o}}=\frac{B}{B_{o}}&(\textit{Cohesive soil})\\ \frac{S}{S_{o}}=\left(\frac{2B}{B+B_{o}}\right)^{2}&(\textit{Sandy soil})\end{array} \tag{2}\]  
where \(S\) is the settlement amount of the foundation with width \(B\), and \(S_{o}\) is the settlement amount measured using the loading plate with width \(B_{o}\).  
\[\frac{k}{k_{o}}=\left(\frac{B}{B_{o}}\right)^{n} \tag{3}\]  
where:  
\(n=-0.2\) to \(-0.4\) (loose-medium dense sand, Bond [9])  
\(n=-0.4\) to \(-0.5\) (dense sand, Bond [9])  
\(n=-0.495\) (pure sand, [PERSON] and Kim [10])  
\(n=-0.965\) (pure clay, [PERSON] and Kim [10])  
\(n=-0.750\) (Korean Standard Association [11])  
The repetitive plate-loading test is commonly used in Europe, especially in Germany. The average vertical stress and settlement amount are measured by applying loading, unloading, and reloading on a circular rigid plate. The subgrade strain modulus (\(E_{v}\)) can be obtained using the measured stress-settlement curve and Equation (4). The strain modulus of the first loading curve is referred to as the primary strain modulus (\(E_{v1}\)), and that of the second loading curve is referred to as the secondary strain modulus (\(E_{v2}\)). Quality control of the subgrade is performed using the secondary strain modulus, and this value represents the strain modulus of the roadbed [12].  
\[E_{v}=1.5\cdot r\frac{\Delta\sigma}{\Delta s} \tag{4}\]  
where \(r\) is the radius of the loading plate (mm), \(\Delta s\) is the difference in the settlement amount between the points with 30% and 70% of the maximum stress, and \(\Delta\sigma\) is the difference in the stress.  
Equation (4) is a theoretical equation that derives the vertical displacement due to the load applied to the rigid circular plate placed on a homogeneous, isotropic, elastic, and semi-infinite mass based on the elastic theory [13]. It does not consider the scale effect of the loading plate because the radius of the loading plate is considered a variable. However, because it is based on the elastic theory derived under the assumption of homogeneous subgrade, caution must be exercised when it is actually applied to the field subgrade in which sand and clay with various particle sizes are mixed. Examining the use of the scale effect in lieu of the radius of the loading plate is also necessary [6].