## 2 Basic Assumptions and General Analytical Formulation  
The study considers a straight uniform slender transversely cracked beam with a rectangular constant cross-section resting on a deformable (elastic) soil. Since the bending of a beam with a transverse crack in the vertical plane is analyzed, the Opening Mode (Mode I) is considered. This mode is reflected in the different slopes to the left and right of the crack. Therefore, the discrete crack is introduced into the mathematical model through [PERSON]'s approach [1]. Although each transverse crack is geometrically defined by the location \(L_{1}\) and depth \(d\), it is modeled by a rotational spring in the considered solution, as shown in Figure 1.  
[PERSON] et al. defined the genuine spring stiffness \(K_{r}\) as:  
\[K_{r}=\frac{EI_{z}}{h\cdot 6\cdot(1-\
u^{2})\cdot F(\delta^{\prime})}. \tag{1}\]  
Figure 1: Mathematical model of a beam with crack on the two-parametric soil.  
This original spring stiffness definition takes into account the non-cracked bending stiffness of the beam \(El_{z}\), the height of the non-cracked part of the cross-section \(h\), [PERSON]'s ratio \(\
u\), the relative crack depth \(\delta^{\prime}=d/h\), and the compliance function \(F(\delta^{\prime})\).  
\[F(\delta^{\prime})=1.98{\delta^{\prime}}^{2}-3.277{\delta^{\prime}}^{3}+14.43{ \delta^{\prime}}^{4}-31.26{\delta^{\prime}}^{5}+63.56{\delta^{\prime}}^{6}-103.36{\delta^{\prime}}^{7}+147.52{\delta^{\prime}}^{8}-127.69{\delta^{\prime}}^{ 9}+61.50{\delta^{\prime}}^{10} \tag{2}\]  
In addition to [PERSON], several other authors have provided their definitions, which implement the same general definition form of \(K_{r}\) but with different compliance functions [9; 10; 11; 12].  
The derivations represent the soil (defined as isotropic, homogeneous, and linear material) as a two-parameter elastic medium. The governing equilibrium differential equation for the non-cracked beams takes the form [8]:  
\[El_{z}\cdot\frac{d^{4}v(x)}{dx^{4}}-k_{G}\cdot\frac{d^{2}v(x)}{dx^{2}}+k\cdot v (x)=q(x) \tag{3}\]  
where \(v(x)\)--transverse displacement function; \(k\)--coefficient of proportionality derived from the genuine [PERSON]'s soil coefficient; \(k_{G}\)--the second soil parameter; \(q(x)\)--continuous transverse load. When searching for the analytical solution, two auxiliary parameters (\(\lambda_{1}\) and \(\lambda_{2}\)) are implemented, which can be altogether presented by:  
\[\lambda_{i}=\sqrt{\frac{\sqrt{4\cdot EI_{z}\cdot k}-(-1)^{i}\cdot k_{G}}{4 \cdot EI_{z}}}i=1,2. \tag{4}\]  
If the value of \(k_{G}\) equals zero, both expressions reduce into a single parameter, which is already known from the genuine Winkler model solution. However, when solving the governing differential equation of a two-parametric soil, there is an important difference compared to the equivalent Winkler soil differential equation where a single analytical solution is obtained regardless of the combination of the beam and soil coefficients. Namely, for some combinations of two-parametric soil mechanical parameters, the second parameter becomes an imaginary value. Consequently, there are two different forms of analytical solutions (\(v_{GDE}\)) of Equation (3) [13]. For the case of \({k_{G}}^{2}<4\cdot EI_{z}\cdot k\), the solution of Equation (3) obtains the following form:  
\[v_{GDE}(x)=\mathrm{e}^{\lambda_{1}\cdot x}\cdot(a_{1}\cdot\cos(\lambda_{2} \cdot x)+a_{2}\cdot\sin(\lambda_{2}\cdot x))+\mathrm{e}^{\lambda_{1}\cdot x} \cdot(a_{3}\cdot\cos(\lambda_{2}\cdot x)+a_{4}\cdot\sin(\lambda_{2}\cdot x)) \tag{5}\]  
while for the case of \({k_{G}}^{2}\geq 4\cdot EI_{z}\cdot k\), the solution is:  
\[v_{GDE}(x)=a_{1}\cdot\mathrm{e}^{\lambda_{1}\cdot x}+a_{2}\cdot\mathrm{e}^{ \lambda_{2}\cdot x}+a_{3}\cdot\mathrm{e}^{-\lambda_{1}\cdot x}+a_{4}\cdot \mathrm{e}^{-\lambda_{2}\cdot x} \tag{6}\]  
where \(a_{i}\) (\(i=1,\,2,\,3,\,4\)) are integration constants.