# Effect of Bond-Slip on Dynamic Response of FRP-Confined RC Columns with Non-Linear Damping
## 2 Numerical Model of FRP-C RC Columns Considering Bond-Slip Effect
### Material Constitutive Models and Loading-Unloading Criteria
#### 2.2.3 Verification of the Proposed Model of FRP-C RC Columns Considering Bond-Slip  
After determining the material constitutive relations and corresponding hysteretic criterions, the nonlinear beam and column element based on the flexibility method and zero-length element considering bond-slip of steel bars are available for FRP-C RC columns modeling. Firstly, we adopt a load control to apply axial load on the top of the column while using a displacement control scheme for cyclic load at the horizontal position, and each load increment step carries out a cyclic iteration until convergence. The iterative algorithm of Newton-Raphson is adopted to control convergence. Here, the simulation results of FRP-C RC square and circular columns are separately compared with the experimental results provided by [PERSON] et al. (2018) and [PERSON] (2010), as shown in Figure 4.  
Figure 3: Hysteretic model schematic diagram for the bar stress vs. loaded-end slip relationship. Figure 4: _Cont._  
It can be seen from Figure 4 that almost all the simulated hysteresis curves show a more significant pinching effect. Whether the square columns [18] or circular columns [40], the simulated hysteresis curves are in good agreement with the experimental hysteretic curves, especially for the square specimens coated with a layer of FRP, such as AR-1 and BR-1. In addition, the reliability of the proposed model can be proved from the perspective of energy dissipation by comparing the areas of hysteresis loops under different lateral drift rates. Here, we still take AR-1 and BR-1 square specimens [18] and the tall circular column specimen [40] as examples, the energy dissipation versus the lateral drift rate (\(\mu^{\prime}=\frac{\Delta}{L}\), where \(\Delta\) represents the average of the forward and reverse peak point displacements of a certain cycle, \(L\) indicates the vertical height of the specimen loading point to the upper surface of the bottom beam) curves of the specimens are evaluated, and the specific results are shown in Table 1.  
For AR-1 and BR-1 square specimens, the calculated values of the proposed model in this paper fit better to the test results. The relative errors of the simulation and test results are less than 10%, except for the case of the AR-1 specimen with a lateral drift rate at \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Specimen** & **Compare the Item** & \multicolumn{4}{c}{**Value of Calculated Results**} \\ \hline \multirow{4}{*}{AR-1 [18]} & Lateral drift rate & 0.002 & 0.005 & 0.009 & 0.011 & 0.013 \\ \cline{2-6}  & Tests data & 5.16 & 31.36 & 76.85 & 118.66 & 157.52 \\  & Numerical results & 5.03 & 27.58 & 70.63 & 108.28 & 147.83 \\  & Relative error (\%) & 2.52 & 12.05 & 8.09 & 8.75 & 6.15 \\ \hline \multirow{4}{*}{BR-1 [18]} & Lateral drift rate & 0.002 & 0.005 & 0.009 & 0.014 & 0.018 \\ \cline{2-6}  & Tests data & 3.30 & 30.65 & 83.28 & 191.90 & 288.12 \\ \cline{2-6}  & Numerical results & 3.15 & 28.37 & 77.32 & 185.89 & 267.95 \\ \cline{2-6}  & Relative error (\%) & 4.55 & 7.44 & 7.16 & 3.13 & 7.00 \\ \hline \multirow{4}{*}{Tall circular column [40]} & Lateral drift rate & 0.002 & 0.0033 & 0.046 & 0.006 & 0.073 \\ \cline{2-6}  & Tests data & 70.83 & 445.38 & 896.75 & 1379.16 & 1882.67 \\ \cline{1-1} \cline{2-6}  & Numerical results & 65.1 & 422.35 & 810.92 & 1301.75 & 1743.83 \\ \cline{1-1} \cline{2-6}  & Relative error (\%) & 8.09 & 5.17 & 9.57 & 5.61 & 7.37 \\ \hline \hline \end{tabular} Note: The value unit of test data and numerical results in Table 1 are kN-mm.  
\end{table} 
Table 1: Comparison of tests data and numerical results for energy dissipation of fiber reinforced polymer-confined reinforced concrete (FRP-C RC) columns. Figure 4: The results of test in literatures [18] and [40] versus numerical results for hysteretic Scheme 1. (**a**) AR-1, (**b**) Scheme 3. (**c**) Specim BR-1, and (**d**) Tall Circular Column specimen. 0.005, the relative error is 12.05%.