# 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
#### 2.2.2 Zero-Length Section Element  
As illustrated by [PERSON] and [PERSON] ([PERSON] and [PERSON], 2017), the bond-slip model of reinforcement in zero-length element is represented by a stress-slip relationship, which can be described by  
Figure 1: Fiber-based modeling of strain penetration effects.  
an elastic straight line for the elastic stage and a curved section for the post-yield region. The schematic diagram is shown in Figure 2.  
The stress-slip curve relations are described as follows:  
\[\sigma=K\text{s for }0\leq s\leq s_{y} \tag{1}\]  
\[\sigma=\widetilde{\sigma}\times\left(f_{u}-f_{y}\right)+f_{y}\text{ for }s_{y}\leq s\leq s_{u} \tag{2}\]  
\[\widetilde{\sigma}=\frac{\frac{\widetilde{s}}{\mu-\widetilde{s}}}{\sqrt[k_{e} \sqrt[k_{e}]{\left(\frac{1}{\mu\times b}\right)^{R_{e}}}+\left(\frac{ \widetilde{s}}{\mu-\widetilde{s}}\right)^{R_{e}}} \tag{3}\]  
\[\widetilde{s}=\frac{s-s_{y}}{s_{y}} \tag{4}\]  
\[\gamma=\frac{s_{u}-s_{y}}{s_{y}} \tag{5}\]  
where, \(k\) is the slope in the elastic region; \(\widetilde{\sigma}\) and \(\widetilde{s}\) are the normalized bar stress and slip, respectively; \(f_{y}\) and \(f_{u}\) are the yield and ultimate strength of the steel reinforcing bar; \(\gamma\) is the ductility coefficient; \(b\) is the stiffness (\(K\)) reduction factor; \(s_{y}\) and \(s_{u}\) are the loaded-end slips when bar stresses are \(f_{y}\) and \(f_{u}\), respectively; \(\mu=(s_{u}-s_{y})/\!s_{y}\) is the ductility coefficient; and the value of factor \(R_{c}\) is taken as 1.01.  
By simulating the pull-out test data of the steel, [PERSON] and [PERSON] [33] obtained a regression empirical formula of \(s_{y}\).  
\[s_{y}=2.54\times\left[\frac{d_{b}}{8437}\times\frac{f_{y}}{\sqrt{f_{c}^{\prime} }}\times(2\alpha+1)\right]^{\frac{1}{2}}+0.34 \tag{6}\]  
where \(\alpha\) is the parameter for the local bond-slip relation; \(d_{b}\) is the bar diameter; and \(f_{c}^{\prime}\) is the concrete compressive strength.  
The limited test information available in the literature [33] indicated that \(30s_{y}\leq s_{u}\leq 40s_{y}\) and \(0.3\leq b\leq 0.5\) would be appropriate. In this study, \(s_{u}=35s_{y}\), \(\alpha\) and \(b\) are taken as 0.4 and 0.5 separately. The hysteresis curve schematic diagram of the bond-slip model is shown in Figure 3.  
Figure 2: The bar stress versus loaded-end slip relationship for the zero-length element.  
A reloading path of this model is defined in the following Equations (7)-(10):  
\[\sigma=\sigma^{*}maxrl\text{ or }\sigma=\sigma^{*}minl \tag{7}\]  
\[\sigma^{*}=\frac{\frac{s^{*}}{s_{uy}-s^{*}}}{\left[\left(\frac{1}{s_{uy}}\right) ^{R_{c}}+\left(\frac{s^{*}}{s_{uy}-s^{*}}\right)^{R_{c}}\right]^{\frac{1}{R_{c }}}} \tag{8}\]  
\[s^{*}=\frac{s-rsvg}{s^{\prime}_{y}} \tag{9}\]  
\[s_{uy}=\frac{maxrl-rsvg}{s^{\prime}_{y}}\text{ or }\frac{minrl-rsvg}{s^{\prime}_{y}} \tag{10}\]  
where, \(\sigma^{*}\) is the bar stress ratio; \(s^{*}\) is the slip ratio; \(s_{uy}\) is the stress limit ratio; \(s_{y^{\prime}}\) is the slip recovered elastically as determined by the return stress divided by the initial slope (\(K\)); the variable (maxrs, maxrl) denotes the maximum bar stress and the corresponding slips; and the variable (minrs, minrl) denotes the minimum bar stress and the corresponding slips; the variable (\(rsvg\), 0) is the intersection between the straight unloading/reloading path and the \(s\)-axis; and \(R_{c}\) is the coefficient defining the shape of the reloading curve. \(R_{c}\) is taken as 1.0.  
The Kent-Scott-Park model proposed by [PERSON] et al. (2016) is still used for concrete in the zero-length section element, and the corresponding hysteretic criterion can be referring to the Concrete 01 model in the OpenSees.