# The Influence of Interlock Loss between Rebar and Concrete on Bond Performance of RC Member
## 4 FE Analysis for Reproduction on Beam Loading Experiment
### FE Analytical System \"COM3\" and Its Constitutive Laws
#### 4.2.1 Model of Concrete and Rebar  
An elasto-plastic and continuum fracture model [28; 33] was applied for the concrete compression. Compressive stiffness was modeled by multiple springs and cumulative plastic damage due to micro crack or local crush was expressed by decrease of springs, namely stiffness reduction. Under this concept, the constitutive law of concrete in compression was given by Equation (1)  
\[{\sigma_{c}}^{\prime}=K_{c}E_{0}\big{(}\epsilon^{\prime}-{\epsilon_{p}}^{ \prime}\big{)} \tag{1}\]  
where \({\sigma_{c}}^{\prime}\) is normalized compressive stress, \(E_{0}\) is initial tangent elastic modulus, \(\epsilon^{\prime}\) is compressive strain, \({\epsilon_{p}}^{\prime}\) is plastic compressive strain, and \(K_{c}\) is failure parameter. \(K_{c}\) means the residual ratio of springs keeping the load bearing function, namely stiffness ratio to sound elastic modulus. \(K_{c}\) and \({\epsilon_{p}}^{\prime}\), which are residual strain in an actual situation, are given Equations (2) and (3) in normal strength concrete under one axial stress field, respectively. \[K_{c}=\exp\big{[}-0.73\big{(}\epsilon_{max}^{\prime}\big{)}\big{\{}1-\exp\big{(} -1.25\epsilon_{max}^{\prime}\big{)}\big{\}}\big{]} \tag{2}\]  
\[\epsilon_{p}^{\prime}=\epsilon_{max}^{\prime}-(20/7)\ \big{\{}1-\ \exp\big{(}-0.35\epsilon_{max}^{\prime}\big{)}\big{\}} \tag{3}\]  
where \(\epsilon_{max}^{\prime}\) is maximum compressive strain in stress hysteresis. This compression model can be expanded to the three-dimensional constitutive law by equivalent stress and strain composed with average stress and deviatoric stress in a solid element. The tension model of concrete based on averaged stress and averaged strain relationship [34]. This model had good agreement with the smeared crack model by [PERSON] and [PERSON] [35], and tension softening behavior was separately considered by RC zone which was affected by bond between rebar and concrete and plain concrete zone based on the zoning method [30]. In the RC zone, concrete has resistance against tensile stress even after cracking because rebar stress can be transferred by bond known as tension stiffening. Concrete tensile model after crack with tension stiffening can be given by Equation (4). \[\sigma_{c}=f_{t}(\epsilon_{tu}/\epsilon_{t})^{c} \tag{4}\]  
where \(\sigma_{c}\) is averaged tensile stress, \(\epsilon_{tu}\) cracking strain, \(\epsilon_{t}\) averaged tensile strain, and \(c\) is stiffening parameter which represents the bond condition. Based on the previous research,  
Figure 8: Constitutive law of concrete and joint element model. \(\varepsilon_{tu}\) and c were set to 0.0002 and 0.4 for the RC zone, respectively [26]. In plain concrete, tension softening behavior should be determined by fracture energy and reference length on which the average stress-strain relationship was defined. Stiffening parameter of plain concrete can be given by Equation (5) [26]. \[\int\ \sigma_{c}\ d\varepsilon_{t}=G_{f}/l_{r} \tag{5}\]  
where \(G_{f}\) is fracture energy of concrete and \(l_{r}\) is reference length of finite element. Reference length was equal to the element dimension. Parameter \(c\), which reproduced tension softening behavior, was determined on the reference length, thus average stress-strain relationship was dependent on the element size. Shear model of cracked concrete was based on the contact density function model [36]. Equation (6) shows the relationship between shear stress \(\tau\) and shear strain \(\gamma\) by shear stiffness \(G\). \[\tau=G\gamma \tag{6}\]  
Shear stiffness \(G\) of cracked concrete is described as shown in Equations (7) and (8). \[1/G=1/G_{st}+1/G_{c} \tag{7}\]  
\[G_{st}=\frac{\tau_{st}}{\gamma}=f_{st}\frac{\beta^{2}}{1+\beta^{2}},\ \beta=\frac{\gamma}{\varepsilon_{t}} \tag{8}\]  
where \(G_{st}\) is intrinsic shear stiffness, \(G_{c}\) is shear stiffness of uncracked concrete, \(\varepsilon_{t}\) is tensile strain normal to crack surface, and \(f_{st}\) is intrinsic shear strength.