# Unified Simplified Capacity Model for Beam-Column Joints into RC Moment Resisting Frame
## 3 Analytical Model
### Equilibrium Condition of the RC Joint
In particular, F9 and F10 are internal forces to be fixed, while Pc and Pb are the external axial forces applied to the upper column and beam, respectively, which are usually known parameters. For the X-shaped RC joint, a single load direction of the seismic action can be analyzed. A load reversal provides a symmetrical stress state. The beam column joint is clearly symmetric if the joint is exactly at \(\mathrm{L_{b}}/2\) and \(\mathrm{L_{c}}/2\). These geometrical characteristics are generally available in the experimental setup and in new RC joints. However, this does not mean that steel reinforcements have the same cross-section or full area to have an equal resultant. In particular, RC joints show a symmetry in terms of forces (internal and external) but not necessarily in terms of geometrical characteristics of reinforcement bars. The number of non-linear independent equations is five, in five unknowns (C, F1, F2, F3, and F4), due to the symmetry. \[F_{i}=C_{i}+S_{i} \tag{1}\]  
All internal forces can be evaluated as functions of the column shear, \(\mathrm{V_{c}}\), by solving the following system of equations, both for positive and negative column shear applied on T or X-shaped RC joints:  
\[F_{1}+F_{4}(1-i)+F_{6}i-C\cdot sin\vartheta-V_{  
In the previous equations, \(\mathrm{f_{c}}\) represents the compressive strength of concrete, while \(\alpha\) is the aspect ratio \(\mathrm{L_{c}}/\mathrm{L_{b}}\). Furthermore, in the previous equations, column shear, beam shear, and bending moments on beam and column (V\({}_{c}\), V\({}_{b}\), M\({}_{c}\) and M\({}_{b}\) respectively), also satisfy the following conditions, representing additional equations:  
\[\left(V_{c}-\frac{V_{b}}{(2-i)\alpha}\right)=0 \tag{11}\]  
\[\left(V_{c}-\frac{2M_{c}}{L_{c}-H_{b}}\right)=0 \tag{12}\]  
\[\left(V_{c}-\frac{(2-i)M_{b}}{(L_{b}-H_{c})\alpha}\right)=0 \tag{13}\]  
\[\left(V_{b}-2\alpha\cdot V_{c}\right)i=0 \tag{14}\]  
The i-index allows the assessment of the structural behavior of any shape of the RC joint (T or X-shaped). Some internal forces, as shown in Figure 2, must be permutated when the shape shifts from T to X. Therefore, a generalized system of equations can be written in order to assess the behavior of any RC joints, by using the following math assumptions, and noting that identity 0 = 0 means that the equation is redundant for the purpose of solving the problem:  
\[i= \begin{cases}0\text{ for X-shaped RC joints}\\ 1\text{ for T-shaped RC joints}\end{cases}  \tag{15}\]  
The symbols shown in the previous equations are reported in the Figures 1 and 2, as well as in the notation list (Table 1). The width, B, of the RC joint can be identical to the width of the beams or column, \(\mathrm{B_{b}}\) and \(\mathrm{B_{c}}\), respectively, or it can be different. Once the previous system of equations is solved, each internal force can be correlated to column shear, V\({}_{c}\). This correlation allows for the estimation of the maximum value of the column shear, due to the first failure mode (an internal force reaches its threshold value). The ultimate condition, in fact, depends on the ultimate value of internal forces F\({}_{i}\). In particular, for each ultimate value of internal forces F\({}_{i}\), the correlated column shear, V\({}_{c}\), can be evaluated according to the failure modes outlined in Table 1. The actual capacity of the RC-joint is given by the minimum value among all column shear forces.