# Study on Load-Slip Curve of a PBL Shear Key at a Steel-Concrete Composite Joint
## 4 Formula Verification  
An ABAQUS finite element model was established based on the PBL shear key design parameters from reference [19], as shown in Figure 9. Concrete was simulated using solid elements with a plastic damage model composed of relationships. The steel plates and  
Figure 8: Reinforcement force diagram. penetrating steel bars were also simulated using solid elements, and a bi-linear constitutive model was used for both. The connection relationship between each component is set as follows: The penetrating steel bars and steel plates were embedded in concrete, and the circular holes in the steel plate and the penetrating steel bars were bound by constraints. The comparison between the finite element results and the calculated results is shown in Figure 10, and it is easy to see that the calculated results agree well with experimental values. The design parameters of PBL shear keys in reference [9] are selected to verify the formula. The experimental values of specimens SB60-22-1 and SB45-12-3 in reference [9] are compared with the calculated results in Figure 11a,b, and their correlation is shown in Table 1. It can be observed that the results of the formula are in good agreement with the results of the finite element calculation and test. In Figure 10, the experimental values in the elastic stage are slightly higher than the calculated results due to the assumption that the bond between steel and concrete is not considered in the calculation formula. In the actual experiment, the bond effect exists, which results in improved load-carrying capacity in the elastic stage compared to the calculated values. However, the error is small and within an acceptable range. \begin{table}
\begin{tabular}{c c} \hline \hline
**Finite Element/Specimen** & **Related Coefficient (r)** \\ \hline Finite element [19] & 0.9983 \\ Specim SB60-22-1 [9] & 0.9411 \\ Specim SB45-12-3 [9] & 0.9612 \\ Specim PBC-75 [20] & 0.9907 \\ Specim PBC-200 [20] & 0.9356 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The correlation between the calculated results and finite element results and experimental values. Figure 10: The results are compared with finite element results. Figure 9: Finite element model meshing. The design parameters of the PBL shear key in reference [20] are selected to verify the formula. With the exception of different spacings between the two joints, the other parameters of the two specimens--PBC-75 and 200--are the same, and the relevant parameters are included in the formula. The comparison between the test values and the calculated results is shown in Figure 12, and the correlation is shown in Table 1. It can be observed that the test values of PBC-75 and PBC-200 are in good agreement with the calculation results of the formula. The experimental values in the strengthening stage are slightly lower than the calculated values because the calculation formula only considered the main horizontal force for convenience, while the actual failure process is subject to more complex forces, but the error is not significant. According to the results of the formula, it can be observed that the slope of the curve of the two specimens in the elastic section is slightly different, which increases slightly with the increase in plate spacing, but it does not change much; moreover, the load--slip curve after the elastic section has no obvious differences. The difference between the two specimens lies in a single variable, that is, the lengths of the free section of the steel bar, which are 190 mm and 130 mm. This shows that, within a reasonable range, the length distribution of the penetrating steel bar between the two joints is more uniform, which will improve the initial stiffness of the PBL key. The diameter of the penetrating steel bar is 20 mm, and the aspect ratios of the two specimens are all greater than 6. Because the two types of aspect ratios are all within the range of the semi-infinite pile's length, the influence is not significant.