## 4 Verification Numerical Examples and Discussion
### First Example
these discrete values were implemented with newly derived polynomial interpolation functions to obtain transverse displacement values between the nodes, as shown in Equations (8) and (9):  
\[v_{1,1}(x)=-4.4899\cdot 10^{-3}+1.8797\cdot 10^{-3}\cdot x-1.6973\cdot 10^{-4 }\cdot x^{2}+9.2677\cdot 10^{-6}\cdot x^{3}\qquad 0\leq x\leq 2\ \text{m}\] \[v_{2,1}(x)=-3.2780\cdot 10^{-3}+1.2242\cdot 10^{-3}\cdot x-1.3259 \cdot 10^{-4}\cdot x^{2}+3.0761\cdot 10^{-6}\cdot x^{3}\qquad 2\ \text{m}\leq x\leq 6\ \text{m}\] \[v_{1,2}(x)=-1.7714\cdot 10^{-3}+4.3860\cdot 10^{-4}\cdot x+3.7231 \cdot 10^{-6}\cdot x^{2}-4.7954\cdot 10^{-6}\cdot x^{3}\qquad 6\ \text{m}\leq x\leq 10\ \text{m}\] \[v_{2,2}(x)=1.2968\cdot 10^{-2}-2.5558\cdot 10^{-3}\cdot x+2.318 0\cdot 10^{-4}\cdot x^{2}-1.2398\cdot 10^{-5}\cdot x^{3}\qquad 10\ \text{m}\leq x\leq 12\ \text{m}.\]  
The results for transverse displacements are shown in Figure 3, where the horizontal axis represents the distance from the left end of the beam. The vertical axis shows the transverse displacement, and the red and black lines show the solutions of GDEs and the FEM model, respectively. It is clear from Figure 3a that both models exhibit decent agreement at the end nodes of the beam, but the discrepancies are more obvious at some other points. However, it should be noted that the implemented interpolation functions were polynomial interpolation functions, Equations (10) and (11), which do not include the genuine Winkler soil parameter. Nevertheless, mesh refinements had an obvious positive impact on the results as the differences between approximate solutions and the GDEs solutions effectively decreased. In Figure 3b, the results for transverse displacements are presented for the FEM computational model consisting of four cb2 psFE elements of equal lengths (where for the non-cracked elements with \(\delta^{\prime}\) = 0, the value of parameter \(\psi\) was taken as 0). The progress over the original two cb2 psFE model is clearly evident as the discrepancies already almost vanished. Furthermore, a comparison of displacements at five locations (points A, B, and C, which are located at the left end, mid-span, and right end, respectively, as well as both crack locations (C1 and C2)) for different finite element meshes is given in Table 1. While the discrete displacements of points A, B, and C of all meshes were obtained directly from the model system of linear equations, the displacement values at the crack locations (\(v_{\text{C1}}\) and \(v_{\text{C2}}\)) were generally obtained by interpolation (denoted by * in the table). The only exceptions are meshes where the element node and the crack coincided in the model. Figure 3: Comparison of transverse displacements for two (**a**) and four (**b**) FE meshes, respectively. Obviously, the comparison of displacement values in the table leads to the conclusion that the convergence of the results is evident. The values demonstrate that the directly obtained nodal values are generally slightly more accurate than the interpolated values between the nodes. However, it should be noted that the first soil parameter (i.e., \(k\)) is included solely in the calculation of the nodal displacement (through the newly presented stiffness matrices), while it is not included in the interpolation functions. The discrete nodal displacements and rotations additionally allow for the matching nodal values of bending moments as well as shear forces to be calculated directly through the vector \(\{Q\}\). By further implementations, these discrete values allow for bending moment functions to be calculated. Thus, Figure 4 shows the values of bending moment functions on the vertical axis obtained from the four implemented approaches, while the horizontal axis represents the distance from the left beam end. The red line shows the benchmark solutions of GDEs, while the black line shows the values obtained by DEDC i.e., through the second derivatives of transverse displacement functions.