# Development of Analytical Model for Bonding of CFRP Rod in Concrete Subjected to Cyclic Loads
## 6 Analytical Approach
### Jacobi Matrix  
In the Finite Element Method (FEM), the Jacobian matrix of an element connects the quantities expressed in natural coordinate space to their counterparts in real space.  
\[\therefore\mathrm{J}=\begin{bmatrix}\frac{\mathrm{d}\mathrm{x}}{ \mathrm{d}\xi}&\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\xi}&\frac{\mathrm{d} \mathrm{z}}{\mathrm{d}\xi}\\ \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\eta}&\frac{\mathrm{d}\mathrm{y}}{ \mathrm{d}\eta}&\frac{\mathrm{d}\mathrm{z}}{\mathrm{d}\eta}\\ \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\zeta}&\frac{\mathrm{d}\mathrm{y}}{ \mathrm{d}\zeta}&\frac{\mathrm{d}\mathrm{z}}{\mathrm{d}\zeta}\\ \end{bmatrix}[\mathrm{J}]=\begin{bmatrix}\frac{\mathrm{d}\mathrm{x}_{1}}{ \mathrm{d}\xi}&\frac{\mathrm{d}\mathrm{x}_{2}}{\mathrm{d}\xi}&\frac{\mathrm{d} \mathrm{x}_{3}}{\mathrm{d}\xi}&\frac{\mathrm{d}\mathrm{x}_{4}}{\mathrm{d}\xi} \\ \frac{\mathrm{d}\mathrm{x}_{1}}{\mathrm{d}\eta}&\frac{\mathrm{d}\mathrm{x}_{2}}{ \mathrm{d}\eta}&\frac{\mathrm{d}\mathrm{x}_{3}}{\mathrm{d}\eta}&\frac{\mathrm{ d}\mathrm{x}_{3}}{\mathrm{d}\eta}&\frac{\mathrm{d}\mathrm{x}_{3}}{\mathrm{d}\eta}\\ \frac{\mathrm{d}\mathrm{x}_{1}}{\mathrm{d}\zeta}&\frac{\mathrm{d}\mathrm{x}_{2}}{ \mathrm{d}\zeta}&\frac{\mathrm{d}\mathrm{x}_{3}}{\mathrm{d}\zeta}&\frac{\mathrm{ d}\mathrm{x}_{2}}{\mathrm{d}\zeta}\\ \end{bmatrix}\times\begin{bmatrix}\mathrm{x}_{1}&\mathrm{y}_{1}&\mathrm{z}_{1} \\ \mathrm{x}_{2}&\mathrm{y}_{2}&\mathrm{z}_{2}\\ \mathrm{x}_{3}&\mathrm{y}_{3}&\mathrm{z}_{3}\\ \vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots\\ \chi_{22}&\mathrm{y}_{22}&\mathrm{z}_{22}\end{bmatrix} \tag{8}\]  
The derivatives with respect to global coordinates are computed using the inverse of the Jacobian matrix from local coordinates.  
\[\begin{split}\begin{cases}\frac{\mathrm{d}\mathrm{N}_{1}}{ \mathrm{d}\mathrm{y}\zeta}\\ \frac{\mathrm{d}\mathrm{N}_{2}}{\mathrm{d}\eta}\\ \frac{\mathrm{d}\mathrm{N}_{3}}{\mathrm{d}\zeta}\end{cases}=\mathrm{J}^{-1} \begin{cases}\frac{\mathrm{d}\mathrm{N}_{1}}{\mathrm{d}\xi}\\ \frac{\mathrm{d}\mathrm{N}_{2}}{\mathrm{d}\eta}\\ \frac{\mathrm{d}\mathrm{N}_{3}}{\mathrm{d}\zeta}\end{cases}|\mathrm{J}|= \frac{\mathrm{A}}{4}=\frac{700\,*\,400}{4}=28\,*\,10^{5}\,\mathrm{mm}^{2}[ \mathrm{thickness}=700\,\,\mathrm{mm}]\end{split} \tag{9}\]  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Shape Function** & **Derivation to \(\xi\)** & **Derivation to \(\eta\)** \\ \hline \(\mathrm{N}_{1}=-1/4\,\xi(1-\xi)(1-\eta)\) & \(\mathrm{d}\mathrm{N}_{1}/\mathrm{d}\xi=-1/4\,(1-2\xi)(1-\eta)\) & \(\mathrm{d}\mathrm{N}_{1}/\mathrm{d}\eta=1/4\,\xi(1-\xi)\) & (1) \\ \(\mathrm{N}_{2}=1/2\,(1-\xi^{2})(1-\eta)\) & \(\mathrm{d}\mathrm{N}_{2}/\mathrm{d}\xi=-\The CFRP rod is modelled as a two-node bar element. Subsequently, the constitutive model was incorporated for the two-node bar element within the 12 \(\times\) 12 [K\({}_{\text{System}}\)] to ensure compatibility with the concrete element.  
\[\text{K}_{\text{CFRP}}=\begin{bmatrix}\frac{\text{EA}}{\text{L}}&-\frac{\text{ EA}}{\text{L}}\\ \frac{\text{EA}}{\text{L}}&\frac{\text{EA}}{\text{L}}\end{bmatrix}_{\begin{array} []{c}\text{E}=155,000\text{ Gpa}\\ \text{I}=73.9\text{ mm}^{2}\end{array}} \tag{10}\]  
The CFRP rod stiffness matrix (K\({}_{\text{CFRP}}\)) was assembled into the stiffness matrix of the 2D model, which was defined as K\({}_{\text{SYSTEM}}\). The assembly was performed based on the corresponding nodes, aligning the CFRP's nodal positions with those of the concrete element [31].  
\[\text{K}_{\text{SYSTEM}}=\begin{bmatrix}0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&\frac{\text{EA}}{\text{L}}&0&0&0&0&0&\frac{-\text{EA}}{\text{L}}&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&\frac{-\text{EA}}{\text{L}}&0&0&0&0&0&\frac{\text{EA}}{\text{L}}&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ \end{bmatrix} \tag{11}\]