# Unified Simplified Capacity Model for Beam-Column Joints into RC Moment Resisting Frame
## 3 Analytical Model
### Failure Modes  
Each failure mode is attained when the stress related to the involved element reaches its capacity. The weaker mechanism governs the structural capacity of the RC-joint, and three failure modes can be considered:  
* A first failure mode is due to the conventional ultimate condition of reinforcement bars in tension: the yielding of longitudinal (V\({}_{c5}\)) or vertical reinforcement bars (V\({}_{c7}\)); \begin{table}
\begin{tabular}{c c} \hline \hline
**Column Shear** & **Failure Mode** \\ \hline V\({}_{c1}\) & Flexural capacity of beam (FSB) \\ V\({}_{c2}\) & Flexural capacity of column (FSC) \\ V\({}_{c3}\) & Shear capacity of beam (SSB) \\ V\({}_{c4}\) & Shear capacity of column (SSC) \\ V\({}_{c5}\) & Yielding of horizontal bottom (YBB) or top bars (YTB) in beams \\ V\({}_{c6}\) & Yielding of vertical bars on the top column (YTC) \\ V\({}_{c7}\) & Yielding of vertical bars on the bottom column (YBC) \\ V\({}_{c8}\) & Maximum bond capacity exceedance (BC1) \\ V\({}_{c9}\) & Average bond capacity exceedance (BC2) \\ V\({}_{c10}\) & Minimum bond capacity exceedance (BC3) \\ V\({}_{c11}\) & Crushing capacity of concrete strut (SC) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ultimate column shear, V\({}_{c}\), due to each linked failure mode. The debonding of longitudinal reinforcements associated with V\({}_{\text{c8}}\), V\({}_{\text{c9}}\), and V\({}_{\text{C10}}\), related to maximum, average, and minimum bond capacity;  
Failure of concrete strut associated to column shear value V\({}_{\text{c11}}\). The breakage or yielding of steel bars can be considered to be an ultimate condition for the cracked joint. In a cross-section with different number of bars, nj, each with cross-section area, A\({}_{\text{k,i}}\), and characteristic ultimate stress due to yielding or ultimate stress, fyk, the axial strength of longitudinal reinforcement can be easily calculated as:  
\[F_{max,j}=\sum_{i=1}^{n_{j}}A_{k,i}\text{:}f_{yk} \tag{16}\]  
The maximum value, Fmax,j, is related to the vertical or horizontal reinforcement layer. When this maximum value is reached in one of the reinforcement layers, the associated column shear, V\({}_{\text{c}}\), corresponds to the ultimate value, V\({}_{\text{c5}}\), for horizontal bars, and V\({}_{\text{c6}}\) and V\({}_{\text{c7}}\) involve the vertical steel bars of the top or bottom columns, respectively. The different values for top and bottom columns are due to the different axial load acting on the columns. The failure mode, due to the de-bonding mechanism, is affected by a strong sensibility to the characteristics of steel bars. Different capacity models can be considered according to the existing literature. For this study, the model proposed in the model Code 2010 [86] has been chosen. It considered three values (\(\tau_{\text{min}}\), \(\tau_{\text{av}}\), \(\tau_{\text{max}}\)) of the bond strength, \(\tau\), as a function of the compressive strength of concrete, f\({}_{\text{c}}\), both expressed in MPa in dimensional Equations (17)-(19), and the characteristics of steel bars. In such equations, the numerical coefficients, \(k_{min},\ k_{av}\), and \(k_{max}\), are in MPa\({}^{1/2}\). The minimum value is related to the smooth bars, typical of existing RC buildings built before the 1960s. \[\tau_{min}=k_{min}\sqrt{f_{c}}=0.3\sqrt{f_{c}} \tag{17}\]  
\[\tau_{av}=k_{av}\sqrt{f_{c}}=1.25\sqrt{f_{c}} \tag{18}\]  
\[\tau_{max}=k_{max}\sqrt{f_{c}}=2.5\sqrt{f_{c}} \tag{19}\]  
After cracking, the RC joint is ideally discretized by means of four rigid portions. The maximum bond force is given by the lateral surface of bars multiplied by the strength, \(\tau\). Diagonal cracks cut the joint in rigid portions, providing the length, L\({}_{\text{eb}}\), where bond strength contribution develops (Equation (20)). For the T-shaped RC joint, the bond force changes with the shear (positive or negative direction) and with the location of bars (upper and bottom layer).