## 3 Results and Discussion  
Calculation of Total Strain in Soil Layers Taking into Account the Non-Linear Deformability and Their Various Mechanical Parameters  
Throughout this article, it has been emphasized that when estimating the settlement of a rectangular footing, [PERSON]'s system of physical equations prominently manifests its special feature of analyzing the total deformations as shear and volumetric strains, taking into consideration the varying values of soil stiffness parameters, which depends on the stress-state of the soil elements. The equation for total strain is briefly displayed as follows:  
\[\varepsilon_{\mathrm{z}}=\varepsilon_{\mathrm{\gamma}}+\varepsilon_{\mathrm{ \varrho}}=\frac{\sigma_{\mathrm{z}}-\sigma_{\mathrm{m}}}{G\big{(}\sigma_{ \mathrm{m}},\tau_{\mathrm{i}}/\tau_{\mathrm{i}}^{*}\big{)}}+\frac{\sigma_{ \mathrm{m}}}{K(\sigma_{\mathrm{m}})} \tag{30}\]  
Putting Equations (24) and (29) into the procured expression (30) above, we get:  
\[\varepsilon_{\mathrm{z}}=\frac{\sigma_{\mathrm{z}}-\sigma_{\mathrm{m}}}{2G^{ \mathrm{\varrho}}\Big{(}1-\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}^{*}} \Big{)}}+\varepsilon^{*}\big{(}1-e^{-\alpha\sigma_{\mathrm{m}}}\big{)} \tag{31}\]  
in which the definitions of \(G^{\mathrm{\varrho}}\), \(\tau_{\mathrm{i}}^{*}\), \(\varepsilon^{*}\), \(\sigma_{\mathrm{z}}\), and \(\sigma_{\mathrm{m}}\) are detailed above. According to [16], [PERSON]'s circle reveals that the maximum shear stress in soil elements takes a value that is equal to half of the difference between the maximum principal stress \(\sigma_{1}\) and \(\sigma_{3}\). On the axis that passes through the center (0, 0, 0) of a rigid rectangular footing, the vertical stress \(\sigma_{z}\) and the two components \(\sigma_{x}\) and \(\sigma_{y}\) act as the principal stress \(\sigma_{1}\), \(\sigma_{2}\)  
Figure 6: Color shadings of \(\sigma_{\mathrm{m}}\) (**a**), \(\sigma_{\mathrm{z}}\) (**b**), and \(\sigma_{\mathrm{z}}-\sigma_{\mathrm{m}}\) (**c**) (at a = 0.75 m; b = 1.5 m; q = 1000 kPa) and the calculation scheme for calculating shear and volumetric strain based on [PERSON]’s system of physical equations (**d**). and \(\sigma_{3}\), respectively. Therefore, the value of shear stress that occurs in a cubic soil element \(\tau_{i}\) can be accepted as:  
\[\tau_{i}=\frac{\sigma_{1}-\sigma_{3}}{2}=\frac{\sigma_{z}-\sigma_{x}}{2} \tag{32}\]  
where \(\sigma_{x}\) is calculated by integrating \(\sigma_{\rm p.x}\), which corresponds to the case of point load with respect to \(\eta\) (from \(-\)a to a) and to \(\xi\) (from \(-\)b to b). \[\sigma_{x}=\int\limits_{-a}^{a}\int\limits_{-b}^{b}\frac{q}{2}\left[\frac{3(x -\eta)^{2}z}{{R_{1}}^{5}}-(1-2\
u)\left(\frac{(x-\eta)^{2}-(y-\xi)^{2}}{\left( (x-\eta)^{2}+(y-\xi)^{2}\right)R_{1}(z+R_{1})}+\frac{(y-\xi)^{2}z}{\left((x- \eta)^{2}+(y-\xi)^{2}\right)R_{1}}^{3}\right)\right]d\eta d\xi \tag{33}\]  
\[R_{1}=\sqrt{\left(x-\eta\right)^{2}+(y-\xi)^{2}+z^{2}}\]  
With a rigid square footing, thanks to its symmetrical shape, the stress-strain state (SSS) of the soil subgrade under the footing can be considered as the SSS of a soil sample in a symmetric triaxial test, in which \(\sigma_{3}\) and \(\sigma_{2}\) are the same. Therefore, the equation \(\tau_{i}\) can be written in the following form (see also [17]):  
\[\tau_{i}=\frac{\sigma_{z}-\sigma_{x}}{2}=\frac{\sigma_{z}-\frac{3\sigma_{\rm m }-\sigma_{z}}{2}}{2}=\frac{3}{2}(\sigma_{z}-\sigma_{\rm m}) \tag{34}\]  
where \(\sigma_{z}\) and \(\sigma_{\rm m}\) are calculated by Equations (9) and (10). From analysis of Equation (31), it can be concluded that the component deformations grow according to the loading increment. The curve of the shear strain \(\varepsilon_{\gamma}\) takes a parabolic form and has a tendency of infinity, while the progress of the volumetric strain \(\varepsilon_{v}\) tends to be dampened.