# Analytical Method and Analysis of Cold-Joint Interface
## 3 Analytical Method
### Curve of Concrete Interface with Connectors
It is presumed that, at the start of the stage (\(\tau_{C}\)), friction is a predominant shear mechanism and dowel action is a secondary mechanism. This is indicated by coefficients of 1 and 0.5 before the friction and dowel action components, respectively. Shear stress \(\tau_{C}\) can be determined by:  
\[\tau_{C}=\tau_{fr}+0.5\tau_{dw} \tag{6}\]  
Friction stress is composed of three friction stress values that are multiplied by the friction coefficient \(\mu\) (Equation (7)). \(\sigma_{n}\) is the value of external compressive stress that is imposed perpendicular to the interface plane. \(f_{y}\rho\) determines the friction stress value due to connector clamping force and \(0.005f_{c}\) defines the minimal value of friction stress. This equation shows how much friction stress remains when there is no external compression (\(\sigma_{n}\)) and no connectors crossing the interface. Friction stress \(\tau_{fr}\) can be determined by:  
\[\tau_{fr}=\mu\big{(}\sigma_{n}+f_{y}\rho+0.005f_{c}\big{)} \tag{7}\]  
The magnitude of dowel action stress depends on the strength of concrete and connector steel and the connector ratio (Equation (8)). The more the interface is crossed by connectors and the stronger the materials that are used, the more dowel action partakes in shear resistance. Dowel action stress \(\tau_{dw}\) can be determined by:  
\[\tau_{dw}=\rho\sqrt{f_{c}f_{y}} \tag{8}\]  
Stage 3 ends with a slip \(s_{C}\) (Equation (9)). Shear slip is composed of two components. Just like the previously defined shear slip (\(s_{max}\)), the value of \(s_{C}\) decreases with increasing concrete strength \(f_{c}\). The second component determines how much the slippage is suppressed by the presence of connectors. The larger the total connectors' cross-sectional area \(A_{s}\), the smaller the total value of \(s_{C}\). Slip \(s_{C}\) can be determined:  
\[s_{C}=2.3\cdot f_{c}^{-0.28}-0.00003\cdot A_{s}{}^{1.4} \tag{9}\]  
At stage 4, concrete interface behaviour is governed by friction resistance and dowel action. At the end of stage (\(\tau_{D}\)) (Equation (10)), dowel action becomes a predominant shear mechanism and friction becomes a secondary mechanism. This is indicated by the coefficients of 0.5 and 1 before the friction and dowel action components, respectively. Connectors yield later at this stage. Shear stress \(\tau_{D}\) can be determined by:  
\[\tau_{D}=0.5\tau_{fr}+\tau_{dw} \tag{10}\]  
The final component of this shear stress-slippage relationship curve is the slip \(s_{D}\) (Equation (11)). The value of this component was chosen to be 6 mm. The analysis of experimental results led to the conclusion that 6 mm was an optimal value to end the estimation of concrete interface behaviour. On average, the value of 6 mm was the point at which many experimental tests were stopped. Additionally, the authors of this paper decided that concrete interface subjected to slip larger than 6 mm could be considered severely broken and no further evaluation would be necessary. Slip \(s_{D}\) can be taken as:  
\[s_{D}=6\;\text{mm} \tag{11}\]