## Appendix A Solutions of the Governing Equation
### Flexible Joint Model  
For the flexible joint model (\(m=1\)), the solutions for the upper sub-beam (\(s=1\)) are given for two cases (a) and (b) based on [15]. The two cases represent the solutions of the differential equation for real material and geometry parameters. The two cases depend on the roots \(R\) of the characteristic equation that takes the form:  
\[\texttt{a}_{5}\texttt{x}^{6}+\texttt{a}_{4}\texttt{x}^{4}+\texttt{a}_{3} \texttt{x}^{2}+\texttt{a}_{2}=0\] (A12)  
with:  
\[\texttt{a}_{5}=\frac{1}{K_{1}K_{2}},\quad\texttt{a}_{4}=-\frac{1} {K_{1}}\left(\xi\frac{h^{(1)}}{2}+\eta\right)-\frac{1}{K_{2}}\left(l^{(1)}+l^{(2 )}\right),\] (A13) \[\texttt{K}_{2}=\frac{1}{C_{2}^{(1)}+C_{2}^{(2)}},\quad\texttt{K}_ {1}=\frac{1}{C_{1}^{(1)}+C_{1}^{(2)}}\]  
Case (a): \(\pm R_{1}\), \(\pm R_{2}\), and \(\pm R_{3}\)  
Then, the axial and shear force and the moment can be expressed as:  
\[\mathcal{N}^{(31)}=\sum_{i=1}^{3}c_{i}^{(3)}e^{-R_{i}x}+\mathcal{N }^{(11)},\] (A14) \[\mathcal{Q}^{(31)}=\sum_{i=1}^{3}c_{i}^{(3)}E_{Qi}e^{-R_{i}x}+ \mathcal{Q}^{(11)},\] \[\mathcal{M}^{(31)}=\sum_{i=1}^{3}c_{i}^{(3)}E_{Mi}e^{-R_{i}x}+ \mathcal{M}^{(11)}\]  
with  
\[E_{Qi} = R_{i}\bigg{(}\frac{R_{i}^{2}}{\xi K_{2}}-\frac{\eta}{\xi}-\frac{h^{( 1)}}{2}\bigg{)},\ \ E_{Mi}=-\frac{R_{i}^{2}}{\xi K_{2}}+\frac{\eta}{\xi}\] (A15)  
and \(c_{i}^{(3)}\) (\(i=1,2,..,6\)) are coefficients estimated using the boundary and continuity conditions. At \(x=0^{+}\), one can now estimate the crack-tip forces as:  
\[\mathcal{N}^{(1^{\prime})}|_{x^{\prime}=\texttt{a}^{-}}=\mathcal{ N}^{(31)}|_{x=0^{+}},\] (A16) \[\mathcal{Q}^{(1^{\prime})}|_{x^{\prime}=\texttt{a}^{-}}=\mathcal{ Q}^{(31)}|_{x=0^{+}},\] \[\mathcal{M}^{(1^{\prime})}|_{x^{\prime}=\texttt{a}^{-}}=\mathcal{ M}^{(31)}|_{x=0^{+}}\]  
with \(c_{i}^{(3)}\) given by:  
\[ \begin{bmatrix}c_{1}^{(3)}\\ c_{2}^{(3)}\\ c_{3}^{(3)}\end{bmatrix} = \begin{bmatrix}c_{11}^{(3)}&c_{12}^{(3)}&c_{13}^{(3)} \\ c_{21}^{(3)}&c_{22}^{(3)}&c_{23}^{(3)}\\ c_{31}^{(3)}&c_{32}^{(3)}&c_{33}^{(3)}\end{bmatrix}\begin{bmatrix}N\\ M\\ Q\end{bmatrix} =\frac{1}{Y} \begin{bmatrix}E_{M3}E_{Q2}-E_{M2}E_{Q3}&E_{Q3}-E_{Q2 }&E_{M2}-E_{M3}\\ E_{M1}E_{Q3}-E_{M3}E_{Q1}&E_{Q1}-E_{Q3}&E_{M3}-E_{M1}\\ E_{M2}E_{Q1}-E_{M1}E_{Q2}&E_{Q2}-E_{Q1}&E_{M1}-E_{M2}\end{bmatrix}\begin{bmatrix}N \\ M\\ Q\end{bmatrix} \] (A17)  
and  
\[Y=E_{M2}E_{Q1}-E_{M3}E_{Q1}-E_{M1}E_{Q2}+E_{M3}E_{Q2}+E_{M1}E_{Q3}-E_{M2}E_{Q3}\] (A18)  
Case (b): \(\pm R_{1}\) and \(\pm R_{3}\)\(\pm\)i\(R_{3}\), with \(\mathrm{i}=\sqrt{-1}\)The solutions of the axial and shear forces and the moment are:  
\[ \begin{split}\mathcal{N}^{(31)}&=c_{1}^{(3)}e^{-R_{1}x}+ e^{-R_{2}x}\big{(}c_{2}^{(3)}cos(R_{3}x)+c_{3}^{(3)}sin(R_{3}x)\big{)}+\mathcal{N}^{(11)} \\ \mathcal{Q}^{(31)}&=c_{1}^{(3)}E_{\Omega 1}e^{-R_{1}x}+ e^{-R_{2}x}\big{[}c_{2}^{(3)}\big{(}E_{\Omega 2}cos(R_{3}x)+E_{\Omega 3}sin(R_{3}x)\big{)}\\ &\qquad+c_{3}^{(3)}\big{(}E_{\Omega 3}cos(R_{3}x)+E_{\Omega 2}sin(R_{3}x) \big{)}\big{]}+\mathcal{Q}^{(11)}\\ \mathcal{M}^{(31)}&=c_{1}^{(3)}E_{\Lambda 1}e^{-R_{1}x}+ e^{-R_{2}x}\big{[}c_{2}^{(3)}\big{(}E_{\Lambda 2}cos(R_{3}x)+E_{\Lambda 3}sin(R_{3}x) \big{)}\\ &\qquad+c_{3}^{(3)}\big{(}E_{\Lambda 3}cos(R_{3}x)+E_{\Lambda 3}sin(R_{3}x) \big{)}\big{]}+\mathcal{M}^{(11)}\end{split}  \tag{19}\]  
with  
\[ \begin{split}& E_{M1}=-\frac{R_{1}^{2}}{\xi K_{2}}+\frac{\eta}{ \xi},\ E_{M2}=-\frac{R_{2}^{2}-R_{3}^{2}}{\xi K_{2}}+\frac{\eta}{\xi},\ E_{M3}= \frac{2R_{2}R_{3}}{\xi K_{2}},\ \ E_{\Omega 1}=-R_{1}\bigg{(}E_{M1}+\frac{h^{(1)}}{2} \bigg{)},\\ & E_{Q2}=-R_{2}\bigg{(}E_{M2}+\frac{h^{(1)}}{2}\bigg{)}+E_{M3}R_ {3},\ \ E_{Q3}=-R_{3}\bigg{(}E_{M3}+\frac{h^{(1)}}{2}\bigg{)}+E_{M2}R_{3} \end{split}  \tag{20}\]  
At \(x=0^{+}\), the coefficients \(c_{i}^{(3)}\) are calculated using:  
\[ \begin{bmatrix}c_{1}^{(3)}\\ c_{2}^{(3)}\\ c_{3}^{(3)}\end{bmatrix} = \begin{bmatrix}c_{11}^{(3)}&c_{12}^{(3)}&c_{13}^{(3)} \\ c_{21}^{(3)}&c_{22}^{(3)}&c_{23}^{(3)}\\ c_{31}^{(3)}&c_{32}^{(3)}&c_{33}^{(3)}\end{bmatrix}\begin{bmatrix}N\\ Q\end{bmatrix} =\frac{1}{Y}