# 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
#### 3.2.1 Elastic Stage
\tag{9}\]  
In the formula, \(\varphi\) denotes the through steel bar section's corner (rad); \(S\) denotes the through steel bar section's shear force (N); \(M\) denotes the through steel bar section's moment of inertia (N-mm); \(F_{s}\) denotes the horizontal shear force (N). The integral constants can be obtained by combining (7)-(9):  
\[\left\{ \begin{array}{l}C_{1}=Cd\Big{(}e^{-2\beta l}+\cos 2\beta l-\sin 2 \beta l+2\Big{)}\\ C_{2}=Cd\Big{(}\cos 2\beta l+\sin 2\beta l-e^{-2\beta l}\Big{)}\\ C_{3}=Cd\Big{(}\cos 2\beta l+\sin 2\beta l+2\Big{)}\\ C_{4}=Cd\Big{(}e^{2\beta l}+\sin 2\beta l-\cos 2\beta l\Big{)}\end{array} \right. \tag{10}\]  
where \(Cd=\frac{F_{s}}{8 EI\beta^{3}(\sin 2\beta l+\sinh 2\beta l)}\). Because the penetrating steel bar and concrete work together and the deformation is coordinated, the slip of the penetrating steel bar can represent the overall slip of the PBL shear key. Therefore, for a single shear key, the maximum slip is the opening position of the joint plate: that is, the position of \(x=0\); thus, the shear stiffness of the free section of the PBL shear key is as follows:  
\[K_{z}=\frac{F_{s}}{y_{\text{max}}}=\frac{F_{s}}{y_{x=0}}=\frac{4 EI\beta^{3}( \sin 2\beta l+\sinh 2\beta l)}{\cos 2\beta l+\cosh 2\beta l+2} \tag{11}\]The calculation process of the middle section is similar to that of the free section, and the calculation diagram is shown in Figure 6. For the meso-level element, the difference in macroscopic boundary conditions will not make the force type of the element different; thus, under the elastic condition, the flexure-depth differential equation obtained is also consistent with Equation (6). The boundary conditions are changed as follows:  
\[\left\{ \begin{array}{l}\phi_{x=0}=0\\ S_{x=0}=F_{ms}\\ M_{x=ln}=0\\ S_{x=ln}=F_{ms}\end{array} \right. \tag{12}\]  
The integral constants can be obtained by combining three expressions: (7), (8) and (12):  
\[\left\{ \begin{array}{l}C_{m1}=-Cd_{m}\left(e^{\beta l_{m}}\cos\beta l_{m}+e^ {\beta l_{m}}\sin\beta l_{m}+1\right)\\ C_{m2}=Cd_{m}\left(e^{\beta l_{m}}\cos\beta l_{m}-e^{\beta l_{m}}\sin\beta l_{ m}+1\right)\\ C_{m3}=Cd_{m}\cdot e^{\beta l_{m}}\left(e^{\beta l_{m}}+\cos\beta l_{m}-\sin \beta l_{m}\right)\\ C_{m4}=Cd_{m}\cdot e^{\beta l_{m}}\left(e^{\beta l_{m}}+\cos\beta l_{m}+\sin \beta l_{m}\right)\end{array} \right. \tag{13}\]  
Among them, \(Cd_{m}=\frac{F_{m}}{4E\beta^{3}}\left(e^{\beta l_{m}}+e^{\beta l_{m}}\cos\beta l _{m}+1\right)\); \(C_{m1}\), \(C_{m2}\), \(C_{m3}\), and \(C_{m4}\) denote integral constants of the middle section; \(l_{m}\) denotes the calculated length of the middle section of the through the steel bar (mm). In the same manner, the shear stiffness of the PBL shear key in the middle section is as follows:  
\[K_{m}=\frac{F_{sm}}{y_{mm\max}}=\frac{F_{sm}}{y_{mx=0}}=\frac{4 EI\beta^{3}(\cos \beta l_{m}+\cosh\beta l_{m})}{\sin\beta l_{m}-\sinh\beta l_{m}} \tag{14}\]  
The shear stiffness of the elastic stage of a single PBL shear key in the elastic stage joint is as follows:  
\[K_{\text{PBL}}=2K_{z}+2K_{m}=\frac{8 EI\beta^{3}(\sin 2\beta l+\sinh 2\beta l)}{ \cos 2\beta l+\cosh 2\beta l+2}+\frac{8 EI\beta^{3}(\cos\beta l_{m}+\cosh\beta l _{m})}{\sin\beta l_{m}-\sinh\beta l_{m}}\]  
It was observed that the shear stiffness is only related to the displacement of the bracing end under the action of the unit shear force. Moreover, considering the displacement continuity of the penetrating steel bar, for the section of the penetrating steel bar that belongs to two sections at the center of the opening of the joint plate, the displacement of the bracing end of the same section is equal; that is, the displacement of the free section of the PBL shear key is equal to that of the middle section at the opening position of the joint plate. Because the penetrating steel bar is all embedded in concrete and combined with the concept of a semi-infinite long pile in foundation engineering and its judgment standard,  
Figure 6: Calculation diagram of the middle section.