# Effect of Bond-Slip on Dynamic Response of FRP-Confined RC Columns with Non-Linear Damping
## 3 The Unit Energy Dissipation  
### Definition  
To study the dynamic response of FRP-C RC columns based on a nonlinear damping theory, this paper introduces a damping calculating method with the so-called loss factor proposed by [PERSON] [41], given by  
\[\eta=\frac{W}{\pi\int_{V}\frac{c^{2}}{E}dV}=\frac{W}{\pi k^{\prime}{u_{0}}^{2}} \tag{11}\]  
\[W=\int_{V}\Delta U(\sigma)dV=\pi c\omega{u_{0}}^{2} \tag{12}\]  
where, \(\eta\) is the loss factor, \(W\) is the energy dissipation per cycle for the total volume (\(V\)) of structure member, \(f^{\prime}_{cu}\) is the stress amplitude, \(\Delta U\) is the unit energy dissipation, \(c\) is the damping coefficient, \(u_{0}\) is the displacement amplitude, \(\omega\) is the exciting frequency, and \(k^{\prime}\) is the stiffness of the system.  
[PERSON] [41] found that the damping energy dissipation of materials in the hysteresis process is mainly related to the maximum stress amplitude. Moreover, the relationship between the unit energy dissipation (\(\Delta U\)) and maximum stress amplitude (\(\sigma_{\max}\)) is obtained as follows:  
\[\Delta U(\sigma)=J\sigma^{m} \tag{13}\]  
where, \(J\) and \(m\) are constants determined according to experiments.  
[PERSON] and [PERSON] [19] proposed a five-element nonlinear regression formula of the unit energy dissipation of FRP-C RC columns, in the form of the following equation:  
\[\Delta U(\sigma,\rho,\gamma,n,f^{\prime}_{cu})=a\times\sigma^{b}\times(1+\rho )^{c}\times(1+\gamma)^{d}\times(1+f^{\prime}_{cu})^{e}\times(1+f\times n+g \times n^{2}) \tag{14}\]  
where, \(\rho\) is the longitudinal reinforcement ratio, \(\rho=A_{s}/A_{column}\), with \(A_{s}\) is the cross-sectional area of longitudinal reinforcement and \(A_{column}\) is the effective cross-sectional area of the column; \(\gamma\) is the FRP volume ratio, \(\gamma=4t_{f}/D\), with \(t_{f}\) is the thickness of FRP one-layer and \(D\) is the diameter of a cylinder or the length of a square column; \(n\) is the axial compression ratio; and \(f^{\prime}_{cu}\) is the strength of concrete.  
In the process of calculating hysteretic energy dissipation of materials, a specific command (Recorder Element-file ele1 sec1 Stress-Strain. Out-time-ele1 section1 fiber $y $z < $mat ID > Stress-Strain) provided by the OpenSees can obtain the stress-strain relationship. However, under the action of the horizontal reciprocating forces, the theoretical maximum stress amplitude (\(\sigma_{\max}\)) appears at the bottom of the column, and the integral point cannot be defined on the zero-length section element. As a result, the above command in the OpenSees does not capture the stress-strain relationship on a zero-length element. Therefore, we propose a method to calculate the energy dissipation by using the maximum horizontal displacement amplitude (\(S_{\max}\)) at the top of the column to replace the maximum stress amplitude (\(\sigma_{\max}\)), with the form of the following equations:  
\[\sigma_{\max}=f(S_{\max})=a+b\times S_{\max}+c\times{S_{\max}}^{2}+\cdots\cdots+ z\times{S_{\max}}^{n+1}+\theta(S_{\max}) \tag{15}\]  
\[\Delta U(S_{\max},\rho,\gamma,n,f^{\prime}_{cu})=a^{*}\times f(S_{\max})^{b^{*} }\times(1+\rho)^{c^{*}}\times(1+\gamma)^{d^{*}}\times(1+f^{\prime}_{cu})^{e^{*} }\times(1+f^{*}\times n+g^{*}\times n^{2}) \tag{16}\]where \(a,b,c\ldots,a^{*},b^{*},\ldots\ldots\) are undetermined constants.