# Study on Load-Slip Curve of a PBL Shear Key at a Steel-Concrete Composite Joint
## 3 Formula for the Load-Slip Curve of the PBL Shear Key
### Analytical Formula of Load-Slip Curve
#### 3.2.2 Plastic Stage  
Considering the continuity of the curve, the endpoint of the elastic stage (the OA section in Figure 3) is the starting point of the plastic stage (the AB section in Figure 3). The calculation formula of the elastic stage of PBL shear key is used to study the starting point of the plastic stage. The calculation diagram is shown in Figure 7a.  
Because the internal deformation of the PBL shear key is coordinated, the displacement of the steel bar in the opening position is equal to the deformation of concrete (\(u_{0}\)); thus, for this position, there is \(y_{x=0}=u_{0}\). With reference to the calculation method of the plastic deformation of soil with respect to the composite foundation problem [17; 18], within the unit length of the shear key in the free section, the width of concrete subjected to compressive stress is larger than the diameter of penetrating steel bar, and the width can be taken as \(\kappa\cdot d_{\mathrm{PBL}}\).  
\[u_{0}=\kappa\varepsilon_{0}d_{\mathrm{PBL}} \tag{16}\]  
Figure 7: Strain diagram of concrete in the plastic stage.  
In the formula, \(\kappa\) denotes the equivalent width coefficient of the compressive stress of concrete, which is 2.5 here. The horizontal shear force \(F_{s_{1}}\) at the opening position of the joint plate in the PBL shear key is calculated by the combination of Formulas (16) and (7).  
\[F_{s_{1}}=\frac{10\epsilon_{0}d_{\text{PBL}}EI\beta^{3}(\sin 2\beta l+\sinh 2 \beta l)}{\cos 2\beta l+\cosh 2\beta l+2} \tag{17}\]  
In the load-slip curve of the PBL shear key, point (\(u_{0}\), \(F_{s_{1}}\)) is the end point of the elastic stage and the starting point of the plastic stage. Because the stress condition of the plastic stage is very complex, if the result of the elastic stage method is too large, in order to simplify the calculation and facilitate the application, the power function in the shape of \(y=ax^{b}+c\) is selected as the curve formula of the plastic stage. With reference to the fitting coefficient of the plastic stage of the PBL key [12], the values are as follows:  
\[\left\{\begin{array}{l}a=384.536-16.528t+56.903f_{t}\left(D^{2}-d^{2}\right) -2.165f_{yk}d^{2}\\ b=0.345-0.002t-0.006f_{t}\left(D^{2}-d^{2}\right)+0.001f_{yk}d^{2}\end{array}\right. \tag{18}\]  
Among them, \(t\) denotes the thickness of the joint plate; \(f_{t}\) denotes the standard value of the shear strength of tenon concrete; \(f_{yk}\) denotes the standard value of tensile strength of the penetrating steel bar; \(D\) and \(d\) denote the diameter of the opening and the diameter of penetrating steel bar, respectively. The end point of the elastic section is regarded as the starting point of the plastic section, and the \(c\) value can be determined according to reference point (\(u_{0}\), \(F_{s_{1}}\)); thus, the load-displacement curve formula of the plastic section can be obtained.  
As shown in Figure 7b, for the plastic endpoint of the PBL shear key in the free section, the concrete at the position of \(x=0\) reaches the ultimate compressive strain. Due to the coordination of the deformation within the PBL shear key, the deflection of the steel bar through the opening position is equal to the deformation of the concrete; thus, we have the following.  
\[u_{cu}=2.5\epsilon_{cu}d_{\text{PBL}} \tag{19}\]  
In the formula, \(u_{cu}\) denotes the ultimate deformation of concrete (mm). The combined formulas of (18) and (19) then solves shear force \(F_{s_{2}}\):  
\[F_{s_{2}}={au_{cu}}^{b}+c \tag{20}\]  
Point (\(u_{cu}\), \(F_{s_{2}}\)) obtained by the solution is the plastic end point coordinate of a single PBL shear key on its load-slip curve.