## 3 Results
### \(G_{0}\) Mathematical Modeling  
The presented test results showed that the sandy silt in this study would respond to dynamic loadings in a manner that is well established in the literature. The mathematical modeling of dynamic soil response relies on the estimation of \(G_{0}\) value. In this study, we found that the soil's initial shear modulus was dependent on soil suction. Another factor that impacted the soil's initial shear modulus was the pressure in the triaxial cell \(\sigma_{3}\). In the literature, we can find numerous equations for the  
Figure 11: Results of the bender element test in the unsaturated triaxial apparatus: post-compaction suction in terms of the saturation ratio (**a**); maximum shear modulus in different suction conditions (**b**). calculation of the \(G_{0}\) modulus [12; 13]. The most popular ones for the saturated soil in tests with uniform stress (\(\sigma^{\prime}_{\rm m}=\sigma^{\prime}{}_{3}\)) are Equations (16) and (17):  
\[G_{0}=625{\cdot}(p_{a}\sigma^{\prime}{}_{3})^{0.5}/\Big{(}0.3+0.7e^{2}\Big{)}, \tag{16}\]  
\[G_{0}=22{\cdot}K_{2,\ max}{\cdot}(p_{a}\sigma^{\prime}{}_{3})^{0.5}, \tag{17}\]  
where \(p_{a}\) is atmospheric pressure (\(p_{a}\) = 100 kPa), \(e\) is the void ratio, and \(K_{2,max}\) is the constant determined by the void ratio (\(K_{2,\ max}=-44.66\cdot\ln(e)+28.79\)). The maximum shear modulus is calculated in both cases in kPa. The equations that account for the unsaturated state are more complex than the equations for the saturated state. The equations have to consider the saturation ratio, as well as the soil suction; therefore, the equations are often limited to one or two types of soil. Examples of such equations are as follows [46; 47], Equations (18) and (19):  
\[\frac{G_{0}}{G_{r,\ sat}}=A{\cdot}f(e)\Bigg{[}\frac{(p+S_{r}{}^{\kappa}s)}{p_ {a}}\Bigg{]}^{n}, \tag{18}\]  
\[G_{0(unsat)}=G_{0(sat)}\Bigg{[}1+\zeta\Bigg{(}\frac{s}{p_{a}/101.3}\Bigg{)} \Big{(}S_{r}{}^{\xi}\Big{)}\Bigg{]}, \tag{19}\]  
where \(p\) is net pressure, \(\zeta\) and \(\xi\) are constants of Equation (19), which relies on the coefficient of uniformity and soil classification, and \(\kappa\) and \(n\) (stiffness coefficient \(n\) = 0.2 for silty sand, \(n\) = 0.6 for sandy silt in this study) are constants of Equation (18) where \(\kappa\) is a parameter related to the plasticity index (for silty sand soil \(\kappa\) = 1.75, \(\kappa\) = 5.0 for sandy silt in this study). The \(f(e)\) parameter is a void ratio function that can be an exponential function (\(f(e)\) = e\({}^{-\chi}\) where \(\chi\) is the exponent parameter, equal to 1.7 for silty sand and sandy silt [48]). The \(G_{r,sat}\) parameter is the reference shear modulus, which for silty sand was equal to 7 MPa and 13 MPa for sandy silt in this study. The \(A\) parameter is associated with the soil structure, which was further analyzed, and the correlation with the degree of saturation was found as follows (20):  
\[A(S_{r})=a\bigg{(}\frac{S_{max}}{S_{r}}-1\bigg{)}^{b}, \tag{20}\]  
where \(S_{max}\) is the maximum saturation ratio that can be obtained from compaction tests and \(a\), \(b\) are constants (\(a\) = 4.55 and \(b\) = 0.25 for silty sand and \(a\) = 4.0 and \(b\) = 0.15 for sandy silt in this study). Additionally, we analyzed the resonant column and bender element tests, and we found that the relationship between the maximum shear modulus and stress state and the void ratio function can be described as (21):  
\[G_{0}=G_{0,u0}+150{\cdot}f(e){\cdot}(\sigma_{3}+s), \tag{21}\]  
where \(G_{0,u0}\) is the shear modulus value when \(S_{r}\) = 0 and \(\sigma_{3}\) = 0 kPa. For Equations (16)-(19) and (21), we evaluated the quality of \(G_{0}\) prognosis with the use of ex-post methods. In order to do so, we used the test results as a reference. In Table 3, the data concerning the value of \(G_{0}\) are presented. We also conducted the ex-post analysis to answer the question of how well the models performed if they were used before the tests that evaluated the models' reliability.