## Appendix A Vectors and Matrices of Equation (27)  
\[\mathbf{K}_{1}=\frac{1}{l^{4}}\begin{bmatrix}A_{1}&A_{2}&2&&&&0\\ A_{3}&5&-4&1&&&\\ 1&-4&6&-4&1&&&\\ &1&-4&6&-4&1&&&\\ &&\vdots&\vdots&\vdots&\vdots&\vdots&&&\\ &&&1&-4&6&-4&1\\ &&&1&-4&6&-4&1\\ &&&&1&-4&5&B_{3}\\ 0&&&&2&B_{2}&B_{1}\end{bmatrix}_{(n+1)\times(n+1)} \tag{11}\]  
where, \(A_{1}=\frac{k_{0}^{2}D_{\mathrm{e}}^{2}l^{4}}{(\kappa GA)_{\mathrm{eq}}^{2}}- \frac{(k_{1}-k_{-1})D_{\mathrm{e}}l^{2}}{(\kappa GA)_{\mathrm{eq}}}+2\), \(A_{2}=-\frac{2k_{0}D_{\mathrm{e}}l^{2}}{(\kappa GA)_{\mathrm{eq}}}-4\), \(A_{3}=\frac{k_{0}D_{\mathrm{e}}l^{2}}{(\kappa GA)_{\mathrm{eq}}}-2\), \(B_{1}=\frac{k_{0}^{2}D_{\mathrm{e}}^{2}l^{4}}{(\kappa GA)_{\mathrm{eq}}}-\frac {(k_{-1}-k_{-1})D_{\mathrm{e}}l^{2}}{(\kappa GA)_{\mathrm{eq}}}+2\), \(B_{2}=-\frac{2k_{0}D_{\mathrm{e}}l^{2}}{(\kappa GA)_{\mathrm{eq}}}-4\), \(B_{3}=\frac{k_{0}D_{\mathrm{e}}l^{2}}{(\kappa GA)_{\mathrm{eq}}}-2\).  
\[\mathbf{K}_{2}=\frac{k_{i}D_{\mathrm{e}}}{(\kappa GA)_{\mathrm{eq}}l^{2}} \begin{bmatrix}C_{1}&&&&0\\ 1&-2&1&&&&\\ &1&-2&1&&&\\ &\vdots&\vdots&\vdots&\vdots&\vdots&\\ &&&1&-2&1&\\ &&&&1&-2&1\\ 0&&&&1&-2&1\\ 0&&&&&&&C_{2}\end{bmatrix}_{(n+1)\times(n+1)} \tag{12}\]where, \(C_{1}=\frac{k_{0}D_{\text{e}}l^{2}}{(\kappa GA)_{\text{eq}}},C_{2}=\frac{k_{1}D_{ \text{e}}l^{2}}{(\kappa GA)_{\text{eq}}}\).  
\[\mathbf{K}_{3}=\frac{D_{\text{e}}}{(EI)_{\text{eq}}}\begin{bmatrix}k_{0}&&&&0\\ k_{1}&&&&\\ &k_{2}&&&&\\ &&\ddots&&\\ &&&k_{i}&&&\\ &&&&\ddots&&\\ &&&&k_{n-2}&&\\ &&&&k_{n-1}&\\ 0&&&&&&&k_{n}\end{bmatrix}_{(n+1)\times(n+1)} \tag{10}\]  
\[\mathbf{K}_{4}=\frac{(k_{i+1}-k_{i-1})D_{\text{e}}}{2l^{2}(\kappa GA)_{\text{ eq}}}\begin{bmatrix}C_{3}&2&&&&0\\ -1&0&1&&\\ &-1&0&1&&\\ &&\ddots&&\\ &&&-1&0&1&\\ 0&&&&-2&C_{4}\end{bmatrix}_{(n+1)\times(n+1)} \tag{11}\]  
where, \(C_{3}=-\frac{k_{0}D_{\text{e}}l^{2}}{(\kappa GA)_{\text{eq}}}-2\), \(C_{4}=\frac{k_{n}D_{\text{e}}l^{2}}{(\kappa GA)_{\text{eq}}}+2\).  
\[\mathbf{K}_{5}=\frac{k_{i+1}-2k_{i}+k_{i-1}}{l^{2}}\frac{D_{\text{e}}}{(\kappa GA )_{\text{eq}}}\begin{bmatrix}1&&&&0\\ &1&&&&\\ &&\ddots&&\\ &&&1&\\ &&&1&\\ 0&&&&1\end{bmatrix}_{(n+1)\times(n+1)} \tag{12}\]  
\[\mathbf{f}_{1}=\frac{D_{\text{e}}}{(EI)_{\text{eq}}}\{k_{0}U_{0}\quad k_{1}U_{ 1}\quad\cdots\quad k_{n-1}U_{n-1}\quad k_{n}U_{n}\}_{{}_{(n+1)}}^{T} \tag{13}\]  
\[\mathbf{f}_{2}=\frac{D_{\text{e}}}{(\kappa GA)_{\text{eq}}l^{2}}\begin{B} \begin{B}\begin{B}(k_{1}-2k_{0}+k_{-1})U_{0}\\ (k_{2}-2k_{1}+k_{0})U_{1}\\ \vdots\\ (k_{n}-2k_{n-1}+k_{n-2})U_{n-1}\\ (k_{n+1}-2k_{n}+k_{n-1})U_{n}\end{B}_{1\times(n+1)}\end{B}_{1\times(n+1)} \tag{14}\]  
\[\mathbf{f}_{3}=\frac{D_{\text{e}}}{(\kappa GA)_{\text{eq}}l^{2}}\begin{B} \begin{B}\begin{B}(U_{1}-2U_{0}+U_{-1})k_{0}\\ (U_{2}-2U_{1}+U_{0})k_{1}\\ \vdots\\ (U_{n}-2U_{n-1}+U_{n-2})k_{n-1}\\ (U_{n+1}-2U_{n}+U_{n-1})k_{n}\end{B}_{1\times(n+1)}\end{B}_{1\times(n+1)} \tag{15}\]\[\mathbf{f}_{4}=\frac{D_{\mathbf{e}}}{2l^{2}(\kappa GA)_{\mathrm{eq}}}\left\{ \begin{array}{c}(k_{1}-k_{-1})(U_{1}-U_{-1})\\ (k_{2}-k_{0})(U_{2}-U_{0})\\ \vdots\\ (k_{n}-k_{n-2})(U_{n}-U_{n-2})\\ (k_{n+1}-k_{n-1})(U_{n+1}-U_{n-1})\end{array}\right\}_{1\times(n+1)} \tag{120}\]  
\[\mathbf{f}_{5}=\left\{D_{1}\quad D_{2}\quad 0\quad\cdots\quad 0\quad D_{3}\quad D_{4} \right\}^{T}_{(n+1)} \tag{121}\]  
where, \(D_{1}=\frac{2k_{0}U_{0}D_{\mathbf{e}}}{(\kappa GA)_{\mathrm{eq}}l^{2}}+\frac{ k_{0}D_{\mathbf{e}}(U_{1}-U_{-1})}{(\kappa GA)_{\mathrm{eq}}l^{2}}+\frac{U_{0}D_{ \mathbf{e}}(k_{1}-k_{-1})}{(\kappa GA)_{\mathrm{eq}}l^{2}}-\frac{(k_{1}-k_{-1} )k_{0}U_{0}D_{\mathbf{e}}^{2}}{2(\kappa GA)_{\mathrm{eq}}l^{2}}\), \(D_{4}=\frac{2k_{n}U_{0}D_{\mathbf{e}}}{(\kappa GA)_{\mathrm{eq}}l^{2}}+\frac{ k_{n}D_{\mathbf{e}}(U_{n-1}-U_{n+1})}{(\kappa GA)_{\mathrm{eq}}l^{2}}+\frac{(k_{n}-k_{n +1})k_{n}U_{0}D_{\mathbf{e}}^{2}}{2(\kappa GA)_{\mathrm{eq}}l^{2}}\), \(D_{2}=-\frac{k_{0}U_{0}D_{\mathbf{e}}}{(\kappa GA)_{\mathrm{eq}}l^{2}}\), \(D_{3}=-\frac{k_{n}U_{0}D_{\mathbf{e}}}{(\kappa GA)_{\mathrm{eq}}l^{2}}\).