# Unified Simplified Capacity Model for Beam-Column Joints into RC Moment Resisting Frame
## 3 Analytical Model
### Equilibrium Condition of the RC Joint
For the X-shaped RC joints, the equilibrium equations can be simplified, exploiting the symmetry. In particular, the number of independent equations is reduced to five, but the unknown terms are at least eight. Some of them could be fixed according to typical experimentally detected behavior [79] or directly assigned in the design phase. However, in order not to fix a great number of unknown terms a priori (assumed at their maximum capacity or supported by experimental results), different solutions will be discussed to assess the failure modes as the input parameters change. Furthermore, the proposed model allows the inclusion of the RC converging members failure mode (beams and column) into a unified RC joint model. For this reason, a distinct check on the flexural (or shear) failure for beams and columns was added. These assumptions allow the merging of the concrete compression, C\({}_{i}\), and the steel compression, S\({}_{i}\), in a compression resultant only, acting at both beam and column ends (Figure 2). This simplification provides a negligible approximation on the equilibrium of beams and columns [83]. Furthermore, the proposed model allows the assessment of the potential failure mode due to the RC joint or the beam or the column. The capacity of the RC joint is the smallest of the capacities related to their failure modes. Finally, the evolution of all other unknown terms can be assessed as the column shear increases up to the ultimate condition. In fact, the column shear is not an unknown parameter, but is the main input parameter. The evolution of other internal forces (unknown terms) depends on the column shear, V\({}_{\rm c}\). It represents a key aspect, due to the compatibility with the modern basic capacity design principles [1,2,12]. In particular, the joint capacity, in terms of shear force, is provided for each failure mode. The equilibrium configuration was fixed by setting the geometrical parameters and the internal stresses, as shown in Figure 2, for the T-shaped (Figure 2a,b) and the X-shaped (Figure 2c) RC joint. For the T-shaped RC joint, under positive column shear, V\({}_{\rm c}\) (Figure 2a), the tensile resulting forces of longitudinal bars are named F1, F2, F7, and F8, while the compressive forces are F3, F4, F5, and F6. These internal forces cross the RC joint panel, as clarified in Figure 1a. The resultant forces provided by the stirrups and/or eventual strengthening  
Figure 3: Analysis of compressive forces, F\({}_{i}\), components in concrete, C\({}_{i}\), reinforcement, S\({}_{i}\), and compressive forces, C, in joint core (inclination \(\theta\) for cracked portion and \(\vartheta\) for C). systems on the RC joint are named F9 and F10. Finally, C represents the compression resultant force of concrete acting on the RC joint panel. Several strengthening systems can be coupled with the existing reinforcement to provide an additional capacity [84,85]. In particular, F9 is the resultant force provided by steel stirrups or externally bonded reinforcement (EBR), placed horizontally. Conversely, F10 is the resultant force provided by EBR, placed vertically. These additional EBRs can be made of several classical or innovative materials. Therefore, the proposed model allows the assessment of the increases in capacity due to the strengthening systems. When the column shear becomes negative (Figure 2b), the compressed resultants are named F1, F2, F7, and F8, while those in traction are named F3, F4, F5, and F6. F9, F10, and C have been previously defined. The internal forces can be evaluated by solving a system of equations according to three degrees of freedom for each of the discretized portions (four rigid portions) of the RC joints. In particular, equilibrium equations on vertical and horizontal translations, as well as rotation, must be used. The equilibrium equations provide twelve equations, where only nine are independent. It is possible to reduce the unknown internal forces to nine once the internal forces F9, F10, Pb, and Pc are known.