## 3 Derivation of a New Finite Element
### Derivation of Matrices of the Cracked Beam on the Two-Parametric Medium  
The entire deformation energy \(U\) can be expressed as a sum of the contributions from the beam (from both elastic parts as well as the spring) and the soil:  
\[ \begin{array}{l}U=\frac{1}{2}\cdot\int\limits_{x=0}^{L_{1}}EI_{z}\cdot \Big{(}\frac{\partial^{2}v_{1}}{\partial x^{2}}\Big{)}^{2}\cdot dx+\frac{1}{2} \cdot\int\limits_{x=L_{1}}^{L}EI_{z}\cdot\Big{(}\frac{\partial^{2}v_{2}}{ \partial x^{2}}\Big{)}^{2}\cdot dx+\frac{1}{2}\cdot K_{r}\cdot\bigg{(}\frac{ \partial v_{1}}{\partial x}\Big{|}_{L_{1}}-\frac{\partial v_{2}}{\partial x} \Big{|}_{L_{1}}\bigg{)}^{2}+\frac{1}{2}\cdot\int\limits_{x=0}^{L_{1}}k\cdot v _{1}{}^{2}\cdot dx\\ +\frac{1}{2}\cdot\int\limits_{x=L_{1}}^{L}k\cdot v_{2}{}^{2}\cdot dx+\frac{1}{ 2}\cdot\int\limits_{x=0}^{L_{1}}k_{G}\cdot\bigg{(}\frac{\partial v_{1}}{ \partial x}\bigg{)}^{2}\cdot dx+\frac{1}{2}\cdot\int\limits_{x=L_{1}}^{L}k_{G} \cdot\bigg{(}\frac{\partial v_{2}}{\partial x}\bigg{)}^{2}\cdot dx\\ \end{array}  \tag{20}\]  
where the first two terms belong to the strain energy in the beam elastic non-cracked parts, the third term corresponds to the contribution of the crack through the rotational spring energy, the fourth and the fifth term represent the strain energy of the Winkler soil, and the last two terms stand for the second soil parameter contribution, respectively. Since the interpolation functions are known, transverse displacements from Equations (8) and (9) can be implemented in Equation (20). Furthermore, the complete 4 \(\times\) 4 stiffness matrix \([k]\) of a cracked beam finite-element on two-parametric soil can be thus expressed as:  
\[ \begin{array}{l}[k]=\int\limits_{x=0}^{L_{1}}EI_{z}\cdot\big{\{}N_{1}^{ \prime\prime}\big{\}}^{T}\cdot\big{\{}N_{1}^{\prime\prime}\big{\}}\cdot dx+ \int\limits_{x=L_{1}}^{L}EI_{z}\cdot\big{\{}N_{2}^{\prime\prime}\big{\}}^{T} \cdot\big{\{}N_{2}^{\prime\prime}\big{\}}\cdot dx+K_{r}\cdot\big{(}\big{\{}N_{ 1}^{\prime}(L_{1})\big{\}}-\big{\{}N_{2}^{\prime}(L_{1})\big{\}}\big{)}^{T} \cdot\big{(}\big{\{}N_{1}^{\prime}(L_{1})\big{\}}-\big{\{}N_{2}^{\prime}(L_{1}) \big{\}}\big{)}\\ +k\cdot\int\limits_{x=0}^{L_{1}}\{N_{1}\}^{T}\cdot\{N_{1}\}\cdot dx+k\cdot\int \limits_{x=L_{1}}^{L}\{N_{2}\}^{T}\cdot\{N_{2}\}\cdot dx+\underbrace{k_{G}\cdot \int\limits_{x=0}^{L_{1}}\{N_{1}^{\prime}\}^{T}\cdot\{N_{1}^{\prime}\}\cdot dx +k_{G}\cdot\int\limits_{x=L_{1}}^{L}\{N_{2}^{\prime}\}^{T}\cdot\{N_{2}^{ \prime}\}\cdot dx}_{[k_{2}]}\\ =[k_{b}]+[k_{s1}]+[k_{s2}]\end{array}  \tag{21}\]  
where \(\{N_{1}\}\) and \(\{N_{2}\}\) represent the interpolation function vectors of the first and the second elastic part, respectively. Thus, Equation (21) clearly shows the separate contributions of all the beam and soil parameters of the stiffness matrix. Every element of each separate contribution of \(k_{b,j,i}\), \(k_{s1,j,i}\) and \(k_{s2,ij}\) (\(i,j=1,2,3,4\)) of stiffness matrix \([k]\) is presented in a closed form expression as:  
\[ \begin{array}{l}k_{b,ij}=4 EI_{z}\psi(A_{i,3}+3L_{1}A_{i4})\big{(}A_{j,3}+3L_{ 1}A_{j4})+\\ \frac{EI_{z}}{L_{1}^{2}L_{3}^{4}}\sum\limits_{m=3}^{4}\sum\limits_{n=3}^{4} \frac{(m-1)(m-2)(n-1)(n-2)}{(m+n-5)}\Big{(}L^{k+l}L_{1}^{5}B_{i,m}B_{j,n}+L^{5 }L_{1}^{k+l}\big{(}A_{i,m}A_{j,n}-B_{i,m}B_{j,n}\big{)}\Big{)}\\ \\ k_{s1,ij}=\frac{k}{LL_{1}}\sum\limits_{m=1}^{4}\sum\limits_{n=1}^{4} \frac{\big{(}L^{m+n}L_{1}B_{i,m}B_{j,n}+LL_{1}\frac{m+n}{m}(A_{i,m}A_{j,n}-B_{ i,m}B_{j,n})\big{)}}{(m+n-1)}\\ \\ k_{s2,ij}=\frac{k_{G}}{L_{1}^{3}L_{3}^{3}}\sum\limits_{k=2}^{4} \sum\limits_{l=2}^{4}\frac{\big{(}-1)(n-1)}{(m+n-3)}\big{(}L^{m+n}L_{1}^{3}B_{ i,m}B_{j,n}+L^{3}L_{1}^{m+n}\big{(}A_{i,m}A_{j,n}-B_{i,m}B_{j,n}\big{)}\big{)}.\end{array}  \tag{22}\]Each element \(k_{ij}\) (\(I\), \(j\) = 1, 2, 3, 4) of the entire stiffness matrix can be further straightforwardly presented as a sum of all three separate contributions:  
\[k_{ij}=k_{b,ij}+k_{s1,ij}+k_{s2,ij}. \tag{25}\]  
Moreover, Equations (22)-(25) may be generally presented in a more convenient form with a single expression:  
\[