## 3 Derivation of a New Finite Element
### Derivation of Interpolation Functions of the Cracked Beam on the Two-Parametric Medium  
As in many similar derivations, and primarily in order to universally cover both possible cases related to the parameter \(\lambda_{2}\) by a unique finite element solution, the complete cubic polynomials are implemented as interpolation functions. Due to the presence of the crack, which divides the beam into two elastic parts, separate interpolation functions are required for the parts to the left and the right. Since the finite element has four standard degrees of freedom, this consequently requires eight constants to be determined. They are determined from four boundary as well as four continuity conditions (the equality of displacements, the condition for the discrete slope increase, the equality of bending moments, and the equality of shear forces) at the crack location where the influence of the crack is introduced as a slope discontinuity. Although the general description of the continuity conditions is identical to the similar Winkler soil, there is an important computational difference since the second soil parameter \(k_{G}\) is now directly introduced into the derivation of interpolation functions through the condition of shear forces equilibrium, i.e.,  
\[EI_{2}\cdot v_{1}{}^{\prime\prime\prime}(L_{1})-k_{G}\cdot v_{1}{}^{\prime}(L_ {1})=EI_{2}\cdot v_{2}{}^{\prime\prime\prime}(L_{1})-k_{G}\cdot v_{2}{}^{ \prime}(L_{1}) \tag{7}\]  
Thus, the following functions of transverse displacements are obtained for the sections to the left and right of the crack, as shown in Equations (8) and (9):  
\[v_{1}(x)=Y_{1}\cdot N_{1,1}(x)+\Phi_{1}\cdot N_{1,2}(x)+Y_{2}\cdot N_{1,3}(x)+ \Phi_{2}\cdot N_{1,4}(x)\qquad 0\leq x\leq L_{1} \tag{8}\]  
\[v_{2}(x)=Y_{1}\cdot N_{2,1}(x)+\Phi_{1}\cdot N_{2,2}(x)+Y_{2}\cdot N_{2,3}(x)+ \Phi_{2}\cdot N_{2,4}(x)\qquad L_{1}\leq x\leq L \tag{9}\]  
with the following interpolation functions \(N_{1,i}(x)\) and \(N_{2,i}(x)\) (\(i=1,\,2,\,3,\,4\)), as shown in Equations (10) and (11):  
\[N_{1,i}(x)=A_{i,1}+A_{i,2}\cdot x+A_{i,3}\cdot x^{2}+A_{i,4}\cdot x^{3} \tag{10}\]  
\[N_{2,i}(x)=B_{i,1}+B_{i,2}\cdot x+B_{i,3}\cdot x^{2}+B_{i,4}\cdot x^{3}. \tag{11}\]  
The coefficients \(A_{i,j}\) (\(i,j=1,\,2,\,3,\,4\)) from interpolation functions of the first elastic part can be expressed by implementing the Dirac \(\delta\) function as:  
\[A_{i,1}=\delta_{i1} \tag{12}\]  
\[A_{i,2}=\delta_{i2} \tag{13}\]  
\[A_{i,3}=\frac{3\big{(}L^{2}+\alpha_{2}L_{1}\psi\big{)}(\delta_{i1}+L\delta_{i2 }-\delta_{i3})-\big{(}L^{3}+\alpha_{6}(L-L_{1})L_{1}\psi\big{)}(\delta_{i2}- \delta_{i4})}{L(L^{3}+2\psi(L^{2}\beta_{2}-2 LL_{1}\beta_{3}+L_{1}{}^{2}\beta_ {6}))} \tag{14}\]  
\[A_{i,4}=\frac{(2L+\psi\alpha_{2})\delta_{i1}+\big{(}3L^{2}+\psi(6L_{1}+\alpha_ {0}(2L+L_{1}))\big{)}\delta_{i2}-(6L+\psi\alpha_{6})\delta_{i3}+\big{(}3L^{2} +\psi\alpha_{6}(L-L_{1})\big{)}\delta_{i4}}{3L(L^{3}+2\psi(L^{2}\beta_{2}-2 LL_{1}\beta_{3}+L_{1}{}^{2}\beta_{6}))} \tag{15}\]  
with the three following abbreviations:  
\[\psi=\frac{EI_{2}}{K_{r}} \tag{16}\]  
\[\alpha_{i}=i+\frac{k_{G}(L-L_{1})^{2}}{EI_{2}} \tag{17}\]  
\[\beta_{i}=i+\frac{k_{G}L_{1}{}^{2}}{EI_{2}}. \tag{18}\]Considering all four conditions of continuity at the crack location, the coefficients of the interpolation functions of the second part can be written by the coefficients of the first part as follows:  
\[B_{i,j}=A_{ij}+\psi\cdot(A_{i3}+3\cdot L_{1}\cdot A_{i4})\Bigg{(}2\delta_{j1}-2L _{1}\delta_{j1}+\frac{k_{G}(-1)^{k}L_{1}^{4-j}}{EI_{z}\big{(}\delta_{j2}+\delta_ {j3}+3(\delta_{j1}+\delta_{j4})\big{)}}\Bigg{)}. \tag{19}\]  
It should be noted that for the non-cracked case (\(\delta^{\prime}=0\)), these functions reduce into standard polynomial interpolation functions for the non-cracked case.