# Resilient Modulus Characterization of Compacted Cohesive Subgrade Soil
## 3 Results and Discussion
### Analytical Model for \(M_{r}\) Calculation  
Calculations on the average resilient modulus value (\(M_{r\,avg}\)) and analysis of effective stress paths (see Figure 4a-c) led to the estimation of the function describing the change of resilient response of tested soils as a function of the maximal deviator stress in the actual mean effective stress conditions. The stress paths 'critical state line for both tests in radial stress \(\sigma\)'\({}_{3}\) conditions was employed for the model. The inclination \(M\) of the critical state line was calculated based on Equation (5):  
\[M=\frac{q_{max}}{p\prime}, \tag{5}\]  
in which \(q_{max}\) is stated as the maximal deviator stress in the actual mean effective stress conditions. The \(M\) parameter is equal to 2.4. For characterization of how distant the actual maximal deviator stress is from the maximal deviator stress in the critical state, the \(T\) parameter is introduced. The maximal deviator stress \(q_{max}(p^{\prime})\) is a value of deviator stress which must occur to reach the critical state. Equation (6) presents the abovementioned parameter:  
\[q_{max}\big{(}p^{\prime}\big{)}=M\cdot p\prime, \tag{6}\]  
The \(T\) parameter, therefore, is simply the difference between \(q_{max}(p^{\prime})\) and \(q_{max}\) from Equation (7):  
\[T=q_{max}\big{(}p^{\prime}\big{)}-q_{max} \tag{7}\]  
The \(T\) parameter change is presented in Figure 10. The figure plots the value of the resilient modulus and \(T\) parameter in 3D space. Note that the resilient modulus is almost independent from the number of cycles. Additionally, the \(M_{r}\) value corresponds to the value of the \(T\) parameter. This phenomenon is very similar for tests in the effective confining pressure \(\sigma\)'\({}_{3}\) equal to 45 kPa, 90 kPa, and 135 kPa. This fact strongly suggests that the change of \(M_{r}\) can be modeled in a reasonable manner through the use of an equation involving the \(T\) parameter and number of cycles \(N\). Based on the test results, the selection of parameters to formulate the resilient modulus equation should be conducted carefully. On one hand, the formula must have its limitations, and the results derived from their use must be carefully interpreted. On the other hand, the use of the resilient modulus formula which have parameters significantly different for each test condition results in different \(M_{r}\) values. According to previously-presented test results, the level of resilient modulus in cohesive soil subjected to cyclic loading can be estimated through the combination of the number of repeated loading and \(T\) parameter. The linear formula based on this two parameters and constants which characterize the cyclic loading conditions is presented in Equation (8):  
\[M_{r}=k_{1}+\left(-\frac{1}{k_{2}E_{max}}\cdot N\right)+\left(\chi\cdot\Delta u _{1}\right)^{k_{3}}\cdot T \tag{8}\]  
The \(\chi\) parameter is termed amplitude ratio (MPa) and can be calculated based on Equation (9):  
\[\chi=\frac{q_{m}}{q_{a}}=\frac{1+R}{1-R} \tag{9}\]  
The \(\Delta u_{1}\) is the excess pore water pressure in the first cycle in MPa. The \(k_{1}\), \(k_{2}\), andThe resilient modulus was calculated from Equation (8). This is confronted by the Uzan-Witczak model. The results of the Uzan-Witczak resilient modulus \(M_{r}\) was calculated based on Equation (2). The result show that the resilient modulus calculated based on Equation (8) (constants \(k_{1}\) = 43, \(k_{2}\) = 10, \(k_{3}\) = 0.2) better fits the obtained data. The Uzan-Witczak model parameters were fitted for this study in each test. The \(k_{1}\), \(k_{2}\), and \(k_{3}\) parameters were equal to 0.19, 0.91, and \(-\)0.45, respectively, for \(\sigma^{{}^{\prime}}_{3}\) equal to 45 kPa, 0.15, 0.91, and \(-\)0.45 for \(\sigma^{{}^{\prime}}_{3}\) equal to 90 kPa, 0.4, 0.1 and \(-\)0.31 for \(\sigma^{{}^{\prime}}_{3}\) equal to 135 kPa. The results of resilient modulus calculation also shows that if we consider a 10% error level for \(M_{r}\) estimation, in almost all cases the resilient modulus is in this range.