# Analytical Method and Analysis of Cold-Joint Interface
## 3 Analytical Method
### Curve of Interface without Connectors  
1. During the analysis of the literature describing the experimental results of interfaces without connectors, it was noticed that, from the start of the shear loading to the maximum interface shear strength (\(\tau_{max}\)), interface behaviour was more rigid than that for interfaces with shear connectors. The experimental curves displayed a linear stress-slip relationship from the start of the loading to the maximum shear stress. This means that point \(\tau_{A}\) does not exist. It can be assumed that the interface starts cracking and fails at the same time, which is at point \(\tau_{max}\). The elimination of the shear point \(\tau_{A}\) results in stage 1 starting from the start of the loading and ending at point \(\tau_{max}\) (Equation (12)). The calculation of shear stress \(\tau_{max}\) is similar to that for concrete interfaces with connectors; however, coefficient \(\theta\) is used instead of coefficient \(\kappa\) (Equation (13)). The physical meaning of \(\theta\) is that the larger the interface area \(A_{c}\) and the higher the concrete's strength, the larger the value of \(\theta\). This value has to meet the condition of \(\theta\geq 1\). The higher the value of \(\theta\), the further \(\tau_{max}\) is from the next shear stress point \(\tau_{C}\). Shear stress \(\tau_{max}\) can be determined by: \[\tau_{MAX}=\theta\cdot\tau_{fr}\] (12) Coefficient \(\theta\) can be determined by: \[\left\{ \begin{matrix}\theta=1.71f_{c}A_{c}+2.95\\ \theta\geq 1.0\end{matrix} \right.\] (13) The calculation of slip \(s_{max}\) is quite similar to that for an interface with connectors; however, in order to determine the slip value for a concrete interface with no connectors, all equation components that describe connector properties, such as \(f_{y}\) and \(\rho\), have to be equal to 0 (Equation (14)). This leads to the equation having only one component, which describes how the slip is influenced by concrete's compressive strength \(f_{c}\). Slip \(s_{max}\) is determined by: \[s_{MAX}=106\cdot f_{c}^{-2.08}\] (14)
2. Stage 2 of the behaviour of the interface without connectors ends with shear stress \(\tau_{C}\). At this stage, interface behaviour is only governed by friction stress. That is why bowel action should be eliminated (Equation (15)). Shear stress \(\tau_{C}\) is determined by: \[\tau_{C}=\tau_{fr}\] (15) All factors that cause friction stress related to connectors should be eliminated (Equation (16)). Friction stress \(\tau_{fr}\) can be determined by: \[\tau_{fr}=\mu(\sigma_{n}+0.005f_{c})\] (16) when calculating slip \(s_{C}\), the connector area \(A_{s}\) must be taken to be equal to 0. By doing so, the equation is only left with a part describing how the slip is influenced by concrete's compressive strength \(f_{c}\). Slip \(s_{C}\) is determined by: \[s_{C}=2.3\cdot f_{c}^{-0.28}\] (17)
3. At stage 3, the interface is slipping until the final point \(\tau_{D}\). Here, bowel action also has to be eliminated (Equation (18)). Shear stress \(\tau_{D}\) is determined by: \[\tau_{D}=0.5\tau_{fr}\] (18) Interface slips to the final point \(s_{D}\). Slip \(s_{D}\) can be taken as: \[s_{D}=6\text{ mm}\] (19)The authors of this paper also developed shear stress functions at different interface behaviour stages, with shear slip \(s\) being the unknown variable. Functions for interface with connectors (Equations (20)-(23)) were constructed using interface shear stress and slippage equations.