## 4 Verification Numerical Examples and Discussion
### First Example  
In the first study, the rectangular cross-section height was taken as \(h=0.8\) m (with \(K_{r}=2.76494\cdot 10^{8}\) Nm obtained from Equation (1)). This value led to the case where \({k_{G}}^{2}<4\cdot El_{z}\cdot k\), and both coefficients \(\lambda_{1}\) and \(\lambda_{2}\) were positive and real. Therefore, the solutions obtained from the coupled system of GDEs were acquired the form of Equation (5):  
\(v_{GDE,1}(x)=\mathrm{e}^{0.4233\cdot x}\cdot\left(7.4811\cdot 10^{-7}\cdot \cos(0.3837\cdot x)\right.\left.+2.7792\cdot 10^{-4}\cdot\sin(0.3837\cdot x) \right)\)  
\(+\mathrm{e}^{-0.4233\cdot x}\cdot\left(-4.2377\cdot 10^{-3}\cdot\cos(0.3837 \cdot x)\right.\left.-1.3896\cdot 10^{-4}\cdot\sin(0.3837\cdot x) \right)-3.3333\cdot 10^{-4}\) \(0\leq x\leq 2\) m  
\(v_{GDE,2}(x)=\mathrm{e}^{0.4233\cdot x}\cdot\left(8.8944\cdot 10^{-6}\cdot \cos(0.3837\cdot x)\right.\left.+1.5382\cdot 10^{-5}\cdot\sin(0.3837\cdot x) \right)\)  
\(+\mathrm{e}^{-0.4233\cdot x}\cdot\left(-2.8128\cdot 10^{-3}\cdot\cos(0.3837 \cdot x)\right.\left.-2.3464\cdot 10^{-4}\cdot\sin(0.3837\cdot x) \right)-3.3333\cdot 10^{-4}\) \(2\) m \(\leq x\leq 6\) m  
\(v_{GDE,3}(x)=\mathrm{e}^{0.4233\cdot x}\cdot\left(4.2036\cdot 10^{-6}\cdot \cos(0.3837\cdot x)+2.1772\cdot 10^{-5}\cdot\sin(0.3837\cdot x)\right)\)  
\(+\mathrm{e}^{-0.4233\cdot x}\cdot\left(-3.7523\cdot 10^{-3}\cdot\cos(0.3837 \cdot x)+6.2555\cdot 10^{-4}\cdot\sin(0.3837\cdot x)\right)-5.3333\cdot 10^{-4}\) \(6\) m \(\leq x\leq 10\) m  
\(v_{GDE,4}(x)=\mathrm{e}^{0.4233\cdot x}\cdot\left(4.6530\cdot 10^{-6}\cdot \cos(0.3837\cdot x)+3.2713\cdot 10^{-5}\cdot\sin(0.3837\cdot x)\right)\)  
\(+\mathrm{e}^{-0.4233\cdot x}\cdot\left(-5.5265\cdot 10^{-2}\cdot\cos(0.3837 \cdot x)+7.83935\cdot 10^{-3}\cdot\sin(0.3837\cdot x)\right)-5.3333\cdot 10^{-4}\) \(10\) m \(\leq x\leq 12\) m  
In the initial FE model, the applied two-parametric soil stiffness matrices and load vectors obtained from Equations (25) and (29) of the implemented two cb2 psFE of equal lengths were:  
\[[k_{b}]=\left[ \begin{array}{cccc}0.094107&0.070474&-0.094107&0.044168\\ 0.070474&0.517836&-0.070474&-0.094990\\ -0.094107&-0.070474&0.094107&-0.044168\\ 0.044168&-0.094990&-0.044168&0.360000\\ \end{array} \right]\cdot 10^{8}\]  
\[[k_{2}]=\left[ \begin{array}{cccc}0.094107&0.070474&-0.094107&0.044168\\ 0.070474&0.517836&-0.070474&-0.094990\\ -0.094107&-0.070474&0.094107&-0.044168\\ 0.044168&-0.094990&-0.044168&0.360000\\ \end{array} \right]\cdot 10^{8}\]  
\[[k_{1}]=\left[ \begin{array}{cccc}2.196411&2.719004&0.106533&0.356000\\ 2.719004&6.053453&-0.151340&0.939812\\ 0.106533&-0.151340&2.090523&-2.606000\\ 0.356000&0.939812&-2.606000&6.596190\\ \end{array} \right]\cdot 10^{8}\]  
\[[k_{2}]=\left[ \begin{array}{cccc}2.090523&2.606000&0.106533&0.151340\\ 2.606000&6.596190&-0.356000&0.939812\\ 0.106533&-0.356000&2.196411&-2.719004\\ 0.151340&0.939812&-2.719004&6.053453\\ \end{array} \right]\cdot 10^{8}\]  
where all matrix components of the complete stiffness matrix \([k_{1}]\) obtained from Equations (22)-(24) are also shown for the first finite element. Afterwards, element matrices, as well as load vectors, were assembled and three discrete nodal displacement values, as well as rotations, were calculated next. Subsequently,  
Figure 2: Cracked beam resting on elastic soil.