# Simplified Plastic Hinge Model for Reinforced Concrete Beam-Column Joints with Eccentric Beams
## 3 Joint Shear Strength Degradation Model
### Lateral Load-Drift Ratio Relationship  
Figure 4a shows the envelop curve of the simplified plastic hinge model [37] for a joint, and EY, EU, and EF in the figure indicate the yield, ultimate, and failure points, respectively. In the model, the lateral load \(P_{y}\) at EY corresponds to the nominal moment strength of the beam section at the joint interface. The lateral drift ratio \(\delta_{y}\) at EY includes the elastic deformations of beams and columns framing into the joint (affected by effective flexural stiffness \(EI\)) and elastic joint shear deformation of the joint panel (affected by the rotation angle \(\theta_{yy}\)[22]).  
For interior beam-column joints,  
\[P_{y}=\frac{L}{2H}\bigg{(}\frac{M^{+}+M^{-}}{L_{s}}\bigg{)} \tag{5}\]  
\[\delta_{y}=\frac{2P_{y}H_{s}^{3}}{3E_{c}I_{c}}\frac{1}{H}+\frac{2P_{y}L_{s}^{3} }{3E_{c}I_{b}}\frac{H}{L^{2}}+\frac{2\theta_{yy}L_{s}}{L} \tag{6}\]  
\[\theta_{yy}=8.4\times 10^{-4}\big{(}f^{\prime}_{\ c}\big{)}^{0.44}\frac{(BI) ^{0.48}}{(JI)^{0.16}}\bigg{(}1-\frac{h_{c}}{2L}-\frac{h_{b}}{2H}\bigg{)}\frac{L }{2L_{s}} \tag{7}\]  
For exterior beam-column joints,  
\[P_{y}=\frac{L}{2H}\bigg{(}\frac{M^{-}}{L_{s}}\bigg{)} \tag{8}\]  
Figure 4: Load–displacement relationship of plastic hinge model: (**a**) lateral load–drift relationship of beam–column joint; (**b**) moment–rotation relationship of plastic hinge.  
Figure 3: Plastic hinge model for beam–column joints: (**a**) interior beam–column joint; (**b**) exterior beam–column joint.  
\[\delta_{y}=\frac{2P_{y}H_{s}^{3}}{3E_{c}I_{c}}\frac{1}{H}+\frac{4P_{y}L_{s}^{3}}{3E _{c}I_{b}}\frac{H}{L^{2}}+\frac{2\theta_{yy}L_{s}}{L} \tag{9}\]  
\[\theta_{yy}=5.2\times 10^{-4}\big{(}f^{\prime}_{\ c}\big{)}^{0.44}\frac{(BI)^{0.4 8}}{(JI)^{0.16}}\bigg{(}1-\frac{h_{c}}{2L}-\frac{h_{b}}{2H}\bigg{)}\frac{L}{2L_ {s}} \tag{10}\]  
\[BI=\frac{(A_{s}+A^{\prime}_{s})f_{y}}{b_{b}h_{b}f^{\prime}_{\ c}} \tag{11}\]  
\[JI=\max\bigg{[}0.0128,\frac{A_{h}f_{yh}}{b_{c}h_{b}f^{\prime}_{\ c}}\bigg{]} \tag{12}\]  
where \(L\) is the beam length; \(H\) is the column height; \(H_{s}\) is the column shear span; \(L_{s}\) is the beam shear span; \(h_{b}\) is the beam height; \(M^{+}\) and \(M^{-}\) are the positive and negative moments of the beam section, respectively; \(A_{s}\) and \(A_{s}^{\prime}\) are the areas of tension and compression bars in the beam section, respectively; \(A_{h}\) and \(f_{yh}\) are the area and yield strength of joint hoop bars parallel to beam flexural bars, respectively; and \(f_{c}^{\prime}\) is the concrete compressive strength.  
According to ASCE/SEI 41-17 [16], the effective flexural stiffness of beams (\(E_{c}I_{b}\)) and columns (\(E_{c}I_{c}\)) are defined as follows:  
\[E_{c}I_{b}=0.3E_{c}I_{gb} \tag{13}\]  
\[E_{c}I_{c}=0.3E_{c}I_{gc}\leq\bigg{(}0.2+\frac{N_{c}}{b_{c}I_{c}f^{\prime}_{\ c}} \bigg{)}E_{c}I_{gc}\leq 0.7E_{c}I_{gc} \tag{14}\]  
where \(E_{c}\) is the elastic modulus of concrete (= 4700\(\sqrt{f_{c}}\)); \(I_{gb}\) and \(I_{gc}\) are the second-order moments of inertia for gross section of beams and columns, respectively; and \(N_{c}\) is the axial force in columns.  
The lateral drift ratios \(\delta_{u}\) and \(\delta_{f}\) at EU and EF are defined as follows:  
\[\delta_{u}=\delta_{y}+a_{p}\frac{2L_{s}}{L} \tag{15}\]  
\[\delta_{f}=\delta_{y}+b_{p}\frac{2L_{s}}{L} \tag{16}\]  
where \(a_{p}\) (= \(a_{j}+a_{\theta}\)) and \(b_{p}\) (= \(b_{j}+b_{\theta}\)) are the plastic hinge rotation angles of a rotational spring element at EU and EF (Figure 4b), which represent the sum of the joint plastic shear angle and beam plastic rotation. The definitions of \(a_{j}\), \(a_{f}\), \(b_{j}\), and \(b_{f}\) are given in the following section.