# Unified Simplified Capacity Model for Beam-Column Joints into RC Moment Resisting Frame
## 3 Analytical Model
### Failure Modes
Once the effective length, L\({}_{\text{eb}}\), is estimated, the bond capacity of concrete can be compared to the bond demand. The bond capacity is estimated assuming a uniform bond stress, \(\tau\). For T-shaped RC joints under negative shear, the bond force is F1 + S4 in the upper layer of bars, while it becomes F7 + S6, at the bottom side. When the shear becomes negative, the bond force is F4 + S1 and F6 + S7, respectively, at the upper and lower side. For X-shaped RC joints, only the Equation (24) must be considered, due to the symmetry. For this reason, it is important to split the resultant compressive force into two contributions, due to the steel, S\({}_{\text{i}}\), and concrete, C\({}_{\text{i}}\). Figure 2a,b show the contribution due to the internal force under positive column shear (Figure 1b). Under a generic column shear (\(\Gamma\) or X-shaped RC joint), there are four steel contributions that can be calculated, for instance, starting from the following elastic equilibrium Equations (21)-(24). The neutral axis, \(y\), can be evaluated, for instance, in pure flexure, equating the first order moment to zero [77]. \[L_{eb}=\frac{h_{b}^{*}}{tan\theta} \tag{20}\]  
\[\left(S4-F_{7}\frac{A_{s}^{\prime}\big{(}2y-H_{b}+h_{b}^{*}\big{)}}{A_{s} \big{(}H_{b}-h_{b}^{*}-2y\big{)}}\right)i=0 \tag{21}\]\[\left(S6-F_{1}\frac{A_{s}\big{(}2y-H_{b}+h_{b}^{*}\big{)}}{A_{s}^{\prime}\big{(}H_{ b}+h_{b}^{*}-2y\big{)}}\right)i=0 \tag{22}\]  
\[\left(S7-F_{4}\frac{A_{s}\big{(}2y-H_{b}+h_{b}^{*}\big{)}}{A_{s}^{\prime}\big{(}H _{b}+h_{b}^{*}-2y\big{)}}\right)i=0 \tag{23}\]  
\[S1-\left(F_{6}\frac{A_{s}}{A_{s}^{\prime}}i+F_{4}\frac{A_{s}^{\prime}}{A_{s}} (1-i)\right)\frac{\big{(}2y-H_{b}+h_{b}^{*}\big{)}}{\big{(}H_{b}+h_{b}^{*}-2y \big{)}}=0 \tag{24}\]  
In previous equations, the cross-section of reinforcement in compression, \(A_{s}^{\prime}\), and tension, \(A_{s}\), is introduced. The column shear, V\({}_{c}\), due to the maximum bond force, provides the ultimate column shears, V\({}_{c8}\) to V\({}_{c10}\). This value depends on the bond quality conditions. The crushing of concrete strut yields to another failure mode. The maximum column shear, V\({}_{c11}\), is related to this failure mode. The maximum value of the compressed concrete force, C, (please note that it is different from C\({}_{i}\)) is governed by the compression strength (of concrete), f\({}_{c}\):  
\[F_{max,j}=\sum_{i=1}^{n_{j}}A_{k,i}\text{:}f_{yk} \tag{16}\]0  
\(\theta\) is the inclination angle of the compressed concrete force, C (Figure 3). This value is potentially different from the diagonal crack angle \(\theta\) (Figure 3). Concrete contact force, C, inclination can be assumed to be different from the diagonal crack (i.e., \(\vartheta\
eq\theta\)) if shear friction is assumed along cracks. Hence, shear friction, measured as deviation from diagonal (i.e., \(|\vartheta-\theta|\)), should be limited to a threshold capacity value. Compressed strength achievement yields to the corresponding column shear ultimate value of V\({}_{c11}\). The previously discussed failure modes are related to the RC joint only. However, a check on the flexural capacity of both beam, M\({}_{b}\), and column, M\({}_{c}\), is necessary. It can be easily performed by using the classical capacity model of RC cross-sections. The top and bottom columns present different flexural capacities due to different axial loads. The maximum column shear related to the collapse of beam or columns can be easily expressed with Equations (11)-(14).