# Analytical Method and Analysis of Cold-Joint Interface
## 3 Analytical Method
### Curve of Concrete Interface with Connectors  
1. At stage 1, shear at the concrete interface is only transferred by cohesive bonds between concrete layers. The authors of experimental shear tests noted that shear stress \(\tau_{A}\) (Equation (1)) is the point at which the first cracks develop at the interface: \[\tau_{A}=0.5f_{ctm}\] (1) where \(f_{ctm}\) is the tensile strength of concrete. The measurement of slip \(s_{A}\) is a difficult task, as shear stress \(\tau_{A}\) occurs at particularly low slippage. Model Code 2010 [5] suggests that this occurs at values lower than 0.05 mm. After analysing different experimental interface shear tests, it was determined that the value of \(s_{A}\) should not be lower than 0.01 mm. Ultimately, the authors of this research propose that the value of \(s_{A}\) chosen should be in the interval of 0.01 mm-0.05 mm (Equation (2)): \[s_{A}=a=0.01\div 0.05\;\text{mm}\] (2)
2. At the start of stage 2, cohesive forces at the interface begin to decrease and the other shear transfer component, which consists of the connectors, joins the shear resistance. However, the analysis of the experimental results indicates that, at this stage, cohesion is still the main contributor to shear transfer. When this analytical model was built, it was assumed that, at the end of stage 2, the highest shear stress of \(\tau_{max}\) (Equation (3)) was reached at the concrete interface. The analysis of experimental tests led to the conclusion that this value depended on coefficient \(\kappa\) (Equation (4)), which is influenced by the ratio of \(\frac{f_{\gamma}p}{f_{c}}\). \(f_{c}\) is the compressive strength of the concrete, \(f_{y}\) is the strength of shear connectors and \(\rho\) is the ratio of connectors which cross the interface. When the ratio of \(\frac{f_{\gamma}p}{f_{c}}\) is equal to an approximate value of 0.18, the highest shear stress \(\tau_{max}\) is equal to the following shear stress point \(\tau_{C}\). If the ratio is lower than 0.18, \(\tau_{max}\) is higher than \(\tau_{C}\). Additionally, the value of \(\kappa\) must meet the condition of \(\kappa\geq 1\). The physical meaning of this ratio is that the higher the concrete strength, the more that \(\tau_{max}\) is influenced by cohesion bonds. The lower the concrete strength, the more it is assumed that cohesion had already been lost and that \(\tau_{max}\) had been reached only by the shear resistance of connectors. Shear stress \(\tau_{max}\) can be determined by: \[\tau_{MAX}=\kappa\Big{(}\tau_{fr}+0.5\tau_{dw}\Big{)}\] (3) where \(\tau_{fr}\) is the interface shear stress due to friction and \(\tau_{dw}\) is the interface shear stress due to dowel action. Coefficient \(\kappa\) determines the influence of cohesion for the highest interface shear capacity: \[ \begin{cases}\kappa=0.35(\frac{f_{\gamma}p}{f_{c}})^{-0.62}\\ \kappa\geq 1.0\end{cases} \] (4) Shear slip \(s_{max}\) (Equation (5)) is composed of two components. The first component defines the magnitude of the slip based on the value of concrete strength \(f_{c}\). The greater the value, the lower the slippage. The second component defines the influence of connectors on the value of the slip. It was noticed that the lower the ratio of \(\frac{f_{\gamma}p}{f_{c}}\), the stiffer the shear stress-slip behaviour and the smaller the slip \(s_{max}\). On the contrary, the higher the ratio, the more ductile the behaviour and the larger the slip. Shear slip can be determined by: \[s_{MAX}=106\cdot{f_{c}}^{-2.08}+18\cdot\left(\frac{f_{y}\rho}{f_{c}}\right)^{ 2.08}\] (5)At stage 3, it is assumed that all cohesive bonds between the concrete layers are lost and that the friction and dowel action are the only shear mechanisms that transfer shear stress along the interface. This is indicated by the shear stress \(\tau_{C}\) (Equation (6)). The equation for this stress was acquired from Model Code 2010 [5] and modified with reference to the experimental results to suit the behaviour of this type of interface.