# Effect of Bond-Slip on Dynamic Response of FRP-Confined RC Columns with Non-Linear Damping
## 3 The Unit Energy Dissipation
### Establishment of the Unit Energy Dissipation Formula  
For the random combination of the four parameters in Table 2, 81 different calculation schemes can be derived by OpenSees for circular column and square column, respectively. Using function drawing software Origin, nonlinear integration of the hysteretic curve under each condition above is carried out. By calculating the area enclosed by the time-delay loop with different horizontal displacement amplitude and dividing by the volume of the component, the energy dissipation per unit volume of the FRP-C RC columns with different parameter combinations can be obtained.  
Therefore, establishing a calculation formula of the energy dissipation per unit volume is a five-element (the maximum horizontal displacement amplitude, concrete strength, reinforcement ratio, FRP volume, axial compression ratio) nonlinear regression problem. A method of fixing other parameters and examining a single parameter is adopted to analyze  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Parameters**} \\ \hline Cross section type &
\begin{tabular}{c} Concrete strength \\ \(f_{xx}^{a}\)/MPa \\ \end{tabular} & Reinforcement ratio \(\rho\)/\% & FRP volume \(\gamma\)/\% & Axial compression ratio/\(n\) \\ \hline Circular column & 30, 40, 50 & 2.67 ((\(\Phi\) = 10 mm)), 5.23 (\(\Phi\) = 12 mm), & 0.45 (1 layer), 0.91 & 0.1, 0.2, 0.3 \\  & 8.64 (\(\Phi\) = 18 mm) & (2 layer), 1.36 (3 layer) & 0.1, 0.2, 0.3 \\ \hline Square column & 30, 40, 50 & 1.56 (\(\Phi\) = 14 mm), 2.58 (\(\Phi\) = 18 mm), & 0.32 (1 layer), 0.96 & 0.1, 0.2, 0.3 \\  & 3.85 (\(\Phi\) = 22 mm) & (3 layer), 1.61 (5 layer) & 0.1, 0.2, 0.3 \\ \hline \hline \end{tabular} Note: \(\Phi\) is the diameter of the longitudinal reinforcement.  
\end{table}
Table 2: Case study parameter matrix of FRP-C RC material energy dissipation analysis.  
Figure 5: Section shape and loading form of FRP-C RC columns.  
its influence on the whole curve one by one. The Statistical Program for Social Sciences (SPSS) is used to conduct a regression analysis for each single parameter and material constant until achieve a unified expression by combining these formulas.  
Due to the limit of the length of the paper, the effects of the concrete strength and reinforcement ratio on the unit energy dissipation of FRP-C RC circular columns and square columns are only plotted in Figure 6a-d.  
Figure 6: _Cont._  
After the above calculation and analysis, the formulas of the energy dissipation per unit volume of FRP-C RC circular columns and square columns considering bond-slippage can be obtained, as shown in the following Equations (17) and (18).  
For circular columns:  
\[\Delta U(S,\rho,\gamma,n,f^{\prime}_{eu})=a^{*}\times f(a+b\times S+c\times S^{ 2}+d\times S^{3})^{b^{*}}\times(1+\rho)^{c^{*}}\times(1+\gamma)^{d^{*}}\times( 1+f^{\prime}_{eu})^{c^{*}}\times(1+f^{*}\times n+g^{*}\times n^{2}) \tag{17}\]  
where, \(a^{*}=1.095\), \(a=9\),531,170.569, \(b=58\),665.047, \(c=-117\),075.512, \(d=1280.522\), \(b^{*}=1.146\), \(c^{*}=18.073\), \(d^{*}=33.528\), \(e^{*}=0.359\), \(f^{*}=40.162\), \(g^{*}=-100.427\), the R-squared value of the regression results is 0.942.  
For square columns:  
\[\Delta U(S,\rho,\gamma,n,f^{\prime}_{eu})=a^{*}\times f(a+b\times S+c\times S ^{2}+d\times S^{3}+e\times S^{4})^{b^{*}}\times(1+\rho)^{c^{*}}\times(1+\gamma )^{d^{*}}\times(1+f^{\prime}_{eu})^{c^{*}}\times(1+f^{*}\times n+g^{*}\times n ^{2}) \tag{18}\]  
where, \(a^{*}=2.818\), \(a=646\),248.526, \(b=-853010.979\), \(c=204\),865.377, \(d=1895.954\), \(e=0.814\), \(b^{*}=1.526\), \(c^{*}=35.627\), \(d^{*}=2.121\), \(e^{*}=0.014\), \(f^{*}=-0.008\), \(g^{*}=-0.752\), the R-squared value of the regression results is 0.996.