# 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
#### 3.2.3 Strengthening Stage  
When the PBL shear key is in the strengthening section, the concrete at the opening position of the joint plate reaches the ultimate compressive strain, and it is destroyed and collapses; only the penetrating steel bar slips and bears the shear force.  
The process of the shear and final yield of the steel bar is accompanied by the extension of the concrete failure area along its length direction; thus, the interaction between them is more complex at this stage.  
In order to not significantly increase the workload of calculations and analyses, it is assumed that the PBL shear key carries all horizontal loads via the steel bar in the entire reinforced section regardless of the bearing capacity of concrete in gradual failure processes, and the calculation diagram shown in Figure 8 is adopted. When the steel bar is separately sheared and yielded, according to the four-strength theory of material mechanics, the following is obtained:  
\[\left[\tau_{j}\right]=0.58f_{y} \tag{21}\]In the formula, \(f_{y}\) denotes the steel joint plate's tensile strength design value (N/mm\({}^{2}\)); \(\left[\tau_{J}\right]\) denotes the steel shear strength design value (N/mm\({}^{2}\)). The shear force acting on the penetrating steel bar at this time (kN) is as follows:  
\[\Delta F_{s3}=A_{g}\big{[}\tau_{J}\big{]} \tag{22}\]  
where \(A_{g}\) denotes the area of the through steel bar's reinforcement (mm\({}^{2}\)), which, in this case, is \(\pi d_{\text{PBL}}{}^{2}/4\). Then, the slip of the penetrating steel bar at the opening position of the joint plate is  
\[\Delta u_{03}=\frac{k_{s}\Delta F_{s3}l}{GA_{g}} \tag{23}\]  
\[G=\frac{E}{2(1+\
u)} \tag{24}\]  
In the formula, \(f_{y}\) denotes the steel joint plate's tensile strength design value (N/mm\({}^{2}\)); \(\left[\tau_{J}\right]\) denotes steel the shear strength design value (N/mm\({}^{2}\)); \(k_{s}\) denotes the shear stress distribution's non-uniform correction coefficient, with a circular section value of 10/9; \(\
u\) denotes the Poisson's ratio of the penetrating steel bar; \(G\) denotes the shear modulus of the penetrating steel bar (N/mm\({}^{2}\)). On the load curve of the shear key of the free segment, point \((u_{cu}+\Delta u_{03},F_{s2}+\Delta F_{s3})\) is the end point of the strengthening stage. For the PBL shear key's free segment of the strengthened section, the shear stiffness is as follows:  
\[K_{zs}=\frac{\Delta F_{s3}}{\Delta u_{03}} \tag{25}\]  
The overall shear stiffness of a single reinforced PBL shear key section is as follows:  
\[K_{sq}=4K_{zq} \tag{26}\]  
In summary, the slip behavior of PBL shear keys is mainly related to the diameter (\(d_{\text{PBL}}\)) and calculated length (\(l_{j}\)) of the through steel bar, as well as the diameter (\(D\)) and thickness (\(t\)) of the opening in the node plate. The calculation formula for the load (\(F\))-slip (\(u\)) curve of the PBL key inside the joint of the steel-concrete composite truss is as follows:  
\[F=\begin{cases}2(K_{z}+K_{m})\cdot u,u\leq u_{0}&\text{Elastic stage}\\ au^{b}+c,u_{0}<u<u_{cu}&\text{Plastic stage}\\ K_{sq}u,u\geq u_{cu}&\text{Strengthing stage}\end{cases}\]