## 2 Methods and Analysis
 \tag{8}\]  
While considering the obtained expressions (7) and (8) on the vertical axis z (\(x=0\); \(y=0\)), their brief forms can be expressed as follows:  
\[\sigma_{\text{z}}=\frac{2q}{\pi}\left[\text{arctg}\frac{ab}{z\sqrt{a^{2}+b^{2 }+z^{2}}}+\frac{abz(a^{2}+b^{2}+2z^{2})}{(a^{2}+z^{2})(b^{2}+z^{2})\sqrt{a^{2}+ b^{2}+z^{2}}}\right] \tag{9}\]  
\[\sigma_{\text{m}}=\frac{4q(1+\
u)}{3\pi}\left[\text{arctg}\left(\frac{ab}{z \sqrt{a^{2}+b^{2}+z^{2}}}\right)\right] \tag{10}\]  
Under the generalized Hooke's Law, the total strain \(\varepsilon_{\text{z}}\) of a soil element can be considered through two components, volumetric strain \(\varepsilon_{\text{v}}\) and shear strain \(\varepsilon_{\gamma}\), which are dependent on vertical stress, mean stress, and the stiffness of the soil. \[\varepsilon_{\text{z}}=\varepsilon_{\gamma}+\varepsilon_{\text{v}}=\frac{ \sigma_{z}-\sigma_{\text{m}}}{2G}+\frac{\sigma_{\text{m}}}{K} \tag{11}\]  
The obtained expressions above reveal that, for the linear elastic soil medium, the shear strain and volumetric strain of soil elements at any point in Cartesian coordinates (x, y, z) can be determined using expressions of \(\sigma_{\text{z}}\) and \(\sigma_{\text{m}}\) along with the stiffness parameters. In the particular case of a rigid rectangular footing, in order to evaluate its settlement, the expression (11) requires an integration with respect to z (from 0 to H, where H is the depth of the compressed thickness of the soil massif under the footing). Therefore, the total settlement is presented as follows:  
\[S=S_{\gamma}+S_{\text{v}}=\int\limits_{0}^{H}\varepsilon_{\gamma}d\text{z}+ \int\limits_{0}^{H}\varepsilon_{\text{v}}d\text{z}=\int\limits_{0}^{H}\frac{ \sigma_{z}-\sigma_{\text{m}}}{2G}d\text{z}+\int\limits_{0}^{H}\frac{\sigma_{ \text{m}}}{K}d\text{z} \tag{12}\]  
in which \(S_{\gamma}\) and \(S_{\text{v}}\) are the values of settlement due to shear strain and volumetric strain, respectively. For the purpose of convenience, the authors propose an analytical method of total settlement assessment using expressions of \(\sigma_{\text{z}}\) and \(\sigma_{\text{m}}\) on the axis that passes through the center (\(0,0,0\)) of a rigid rectangular footing. The expressions of \(S_{\gamma}\) and \(S_{\text{v}}\) are illustrated as:  
\[S_{\gamma}=\int\limits_{0}^{H}\frac{\sigma_{z}-\sigma_{\text{m}}}{2G}d\text{z }=\int\limits_{0}^{H}\left[\frac{q(1-2\
u)}{3\pi G}\text{arctg}\frac{ab}{z\sqrt {a^{2}+b^{2}+z^{2}}}+\frac{q}{\pi G}\frac{abz\big{(}a^{2}+b^{2}+2z^{2}\big{)}} {(a^{2}+z^{2})(b^{2}+z^{2})\sqrt{a^{2}+b^{2}+z^{2}}}\right]\text{d}\text{z} \tag{13}\]  
Figure 3: Stress in an elastic half-space caused by a uniformly distributed load on the surface. \[S_{\rm v}=\int\limits_{0}^{H}\frac{\sigma_{\rm m}}{K}dz=\frac{4q(1+\
u)}{3\pi K} \int\limits_{0}^{H}\text{arctg}\bigg{(}\frac{ab}{z\sqrt{a^{2}+b^{2}+z^{2}}} \bigg{)}dz \tag{14}\]  
The analyzed expression of settlement \(S_{\rm Y}\) due to shear strain consists of three components \(A\)\(B\)\(C\) as follows:  
\[S_{\rm Y}=A+B+C \tag{15}\]  
in which  
\[A=\frac{q(1-2\
u)}{3\pi G}h\text{ arctg}\frac{ab}{h\sqrt{a^{2}+b^{2}+h^{2}}} \tag{16}\]  
\[ \begin{array}{l}B=\frac{q(1-2\
u)}{3\pi G}\bigg{\{}H_{1}\ln\left|\frac{8H_{1 }^{3}-2\left(a^{2}+b^{2}\right)H_{1}+ab\sqrt{a^{2}+b^{2}+h^{2}}}{-8H_{1}^{3}+ 2\left(a^{2}+b^{2}\right)H_{1}+ab\sqrt{a^{2}+b^{2}+h^{2}}}\right|\\ \left.+H_{2}\ln\left|\frac{8H_{2}^{3}-2\left(a^{2}+b^{2}\right)H_{2}+ab\sqrt{ a^{2}+b^{2}+h^{2}}}{-8H_{2}^{3}+2\left(a^{2}+b^{2}\right)H_{2}+ab\sqrt{a^{2}+b^{2}+h^{2}} }\right|\\ \left.-H_{1}\ln\left|\frac{8H_{1}^{3}-2\left(a^{2}+b^{2}\right)H_{1}+ab\sqrt{ a^{2}+b^{2}}}{-8H_{1}^{3}+2\left(a^{2}+b^{2}\right)H_{1}+ab\sqrt{a^{2}+b^{2}}} \right|\\ \left.-H_{2}\ln\left|\frac{8H_{1}^{3}-2\left(a^{2}+b^{2}\right)H_{2}+ab\sqrt{ a^{2}+b^{2}}}{-8H_{2}^{3}+2\left(a^{2}+b^{2}\right)H_{2}+ab\sqrt{a^{2}+b^{2}}} \right|\right\}\bigg{\}}\end{array}  \tag{17}\]  
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