## 5 Finite Element Analysis and Shear Capacity Checking Calculation
### Finite Element Modelling  
Leveraging the capabilities of the finite element analysis software, namely ABAQUS 2020, we established four separate models of newly constructed concrete beam-column joints and a single model of a joint created in situ; these models were constructed in accordance with the test sizes. The three-dimensional solid element responsible for concrete, C3D8R (linear reduced integration element), was utilized. The simulation and analysis of the concrete material was performed using the Concrete Damage Plasticity model (CDP model). During the elastic stress phase of concrete, the mechanical aspects of the material were directly illustrated with the initial elastic modulus through the CDP model. However, upon the concrete reaching the material damage phase, the computational expression of the elastic modulus is modified as follows:  
\[E=(1-d)E_{0} \tag{5}\]  
where \(E\) signifies the elastic modulus; \(E_{0}\) represents the initial elastic modulus; and \(d\) signifies the plastic damage factor, which varies from 0 to 1. Here, 0 implies there is no  
Figure 10: Strength degradation curve. material damage to the concrete, and 1 denotes complete damage, signifying no remaining strength. The expression for the uniaxial stress-strain curve of concrete under tensile and compressive conditions is as follows:  
\[\sigma_{t}=(1-d_{t})E_{0}\Big{(}\varepsilon_{t}-\varepsilon_{t}^{pl}\Big{)} \tag{6}\]  
\[\sigma_{c}=(1-d_{c})E_{0}\Big{(}\varepsilon_{c}-\varepsilon_{c}^{pl}\Big{)}\  
In the model, the three-dimensional, two-node linear truss element T3D2, is utilized for reinforcement. The constitutive model for a steel bar, incorporated within, is a modified update of the double-fold steel bar constitutive model developed by Professor [PERSON] of Shenzhen University. This model, as displayed in Figure 13, accounts for the bond-slip effect between the steel bar and concrete and considers the influence exerted by concrete materials on the steel bar. It is, therefore, an appropriate model to apply for the hysteresis analysis of the structure. Unloading stiffness calculation formula:  
\[E_{\text{sr}}= \begin{cases}\begin{array}{cc}E_{s},&\beta<1\\ (1.05-0.05\beta)\,E_{s}\,,1\leq\beta\leq 4\\ 0.85&E_{s},&\beta>4\end{array}\end{cases}  \tag{8}\]  
Load path formula:  
\[\sigma=\gamma\Big{(}\vec{\varepsilon}^{2}-\vec{\varepsilon}^{2}\Big{)}+(1- \alpha)\sigma_{m}\vec{\varepsilon}+\alpha\sigma_{m} \tag{9}\]  
where \(\gamma=E_{sh}(\varepsilon_{m}-\varepsilon_{L})-(1-\alpha)\sigma_{m};\vec{ \varepsilon}=(\varepsilon-\varepsilon_{L})/(\varepsilon_{m}-\varepsilon_{L})\) ; \(\varepsilon_{L}\) is the strain corresponding to point \(L\) or point \(N\) in Figure 13; and \(\alpha\) is the influence coefficient of hysteresis energy dissipation. Finally, it is essential to take into account the elastic modulus, Poisson's ratio, yield strength, ultimate strength, and the associated parameters of reinforced concrete. These should be in accordance with the norms and measured values. Their specific values are exhibited in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Steel Type**} & **Diameter** & **Elasticity Modulus** & **Yield Strength** & **Inelastic** \\  & **(mm)** & **(N/m2)** & **(MPa)** & **Strain** \\ \hline \multirow{2}{*}{HRB400} & \multirow{2}{*}{6} & \multirow{2}{*}{2.0 \(\times\) 10\({}^{11}\)} & 429.32 & 0 \\ \cline{3-5}  & & & 540.24 & 0.002 \\ \hline \multirow{2}{*}{HRB400} & \multirow{2}{*}{10} & \multirow{2}{*}{2.0 \(\times\) 10\({}^{11}\)} & 443.25 & 0 \\ \cline{3-5}  & & & 625.16 & 0.002 \\ \hline \multirow{2}{*}{HRB400} & \multirow{2}{*}{12} & \multirow{2}{*}{2.0 \(\times\) 10\({}^{11}\)} & 450.13 & 0 \\ \cline{3-5}  & & & 631.02 & 0.0021 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Reinforcement of constitutive data. Figure 13: Modified constitutive model of steel bar. In the FE model, three-dimensional solid elements are utilized for components such as the beam, column, and steel pipe.