## 2 Simple Hysteresis Loop Model  
The Simple Hysteresis Loop Model (SHLM) is an empirical model created as a result of analyzing stress-strain plots. It utilizes the simple dependence of plastic strain ablation [42] during repeated loading performed in constant stress conditions. The constant-stress loading is more suitable to describe what might be imposed on the soil subbase in comparison to the constant-strain loading  
Figure 3: Schema of dissipated energy and elastic strain energy in one cycle.  
Figure 2: Soil respond to repeating loading in strain-controlled test (**a**) and stress-controlled test (**b**).  
conditions. To describe the plastic strain ablation phenomena, we established the abation parameter \(A_{x}\). The abation parameter can be defined as the quotient of plastic strain \(\varepsilon_{a}^{p}\) in \(i\) cycle to the strain achieved at maximal deviator stress \(\varepsilon_{a}^{T}\) in \(i\) cycle in Equation (13):  
\[A_{x}(i)=\frac{\varepsilon_{a}^{p,i}}{\varepsilon_{a}^{T,i}}=\frac{\varepsilon_ {a}^{2}-\varepsilon_{a}^{1}}{\varepsilon_{a}^{max}-\varepsilon_{a}^{1}}\ [-], \tag{13}\]  
where \(\varepsilon_{a}^{1}\) is a strain at the beginning of the cycle, \(\varepsilon_{a}^{2}\) is a strain at the end of the loading-unloading procedure, and \(\varepsilon_{a}^{max}\) is a strain at maximal stress during one cycle. Figure 4 presents the dependencies mentioned above. The ablation parameter \(A_{x}\) will have a value between 0 and 1 when the abation parameter is equal to zero, meaning that the strain which occurred during one cycle was fully recovered, and the material behaved as elastic. When the \(A_{x}\) is equal to 1, the strain during the cyclic loading was fully plastic in such stress conditions.  
The strain dependences might be extended with the plastic strain normalization parameter. This dependence is a quotient of the plastic strain in the first cycle to the plastic strain in \(i^{th}\) cycle so, the plastic strain normalization parameter \(P_{x}\) describes the development of plastic strain increment during the next phases of repeated loading as in Equation (14):  
\[P_{x}(i)=\frac{\varepsilon_{a}^{p,1}}{\varepsilon_{a}^{p,i}}[-], \tag{14}\]  
The \(P_{x}\) parameter utilizes the proportion of plastic strain in the first cycle \(\varepsilon_{a}^{p,1}\) to its growth in the further cycles \(\varepsilon_{a}^{p,i}\). We applied a similar solution to maximal strain in one cycle. The total strain normalization parameter \(T_{x}\) deals with the proportion of maximal strains as in Equation (15):  
\[T_{x}(i)=\frac{\varepsilon_{a}^{T,1}}{\varepsilon_{a}^{T,i}}[-], \tag{15}\]  
The plastic and total strain normalization parameters which describe the distribution of strains in constant stress conditions are supplemented by a ratio of initial cycle as in Equation \(O_{x}\) (16):  
\[O_{x}(i)=\frac{\varepsilon_{a}^{p,1}}{\varepsilon_{a}^{T,1}}[-], \tag{16}\]  
Figure 4: Schema for abation parameter \(A_{x}\) parameters calculation.  
The SHLM parameters describe only the soil response to cyclic loading in constant stress loading conditions. Such simplification is favorable from an engineering point of view. When the soil subbase is loaded by repeating excitation, the strain which will develop as its result will be the sum of strains that occurred in certain loading conditions. When long-term cyclic loading is analyzed, the set of loading cycles can be further simplified to the uniform constant stress cyclic loading with amplitude of loading equal to the highest stress value, which occurred during the service phase.  
By such an assumption, we omit the stress component from calculations in this model, and the strain relationships are employed for the deformation calculation of the soil. We designed and conducted the program of cyclic triaxial tests to derive the mathematical relationships between the stiffness and stress conditions and the strain characteristics represented by SHLM parameters.