# Unified Simplified Capacity Model for Beam-Column Joints into RC Moment Resisting Frame
## 3 Analytical Model  
The present study focuses on the behavior assessment of RC joints in moment resisting frames. In these structures, perimetric and internal joints can be identified in the frame. The internal joints benefit from confinement of the beams coming to the sides of the vertical elements. In fact, according to the modern building codes [1; 2; 12], these RC joints could not be verified due to this beneficial effect. The perimetric RC joints are more critical, because the confinement provided by the converging RC members is lower. The proposed analytical model can be used to assess both X-shaped and T-shaped beam column RC joints. The analytical model is based on knowledge of few key parameters.  
Experimental programs [27; 32; 35] allowed for the assessment of the main parameters that govern the ultimate behavior of RC joints. The experimental joint database [79; 80; 81] indicates the typical failure modes of RC joints. The experimental programs refer to a significant number of RC joints with different geometries, assessing the (usually cyclic) load levels pushing the joint to failure. The failure mode can be due to several mechanisms strongly influenced by the geometrical and mechanical characteristics of RC joints:  
* shear failure modes generally with or without the yielding of longitudinal reinforcement bars of beams;
* concrete cracking is generally linked to bond or yielding failure mode of longitudinal reinforcement bars of beams.  
The effects of different mechanisms are also clear on the global force displacement capacity curves, where several mechanisms differently alter the stiffness. Starting from the existing, available literature [70; 79], key parameters have been identified to develop the proposed simplified model. The main parameters that govern the behavior of the RC joint can be identified as:  
* material properties;
* geometry of the RC joint panel;
* type of RC joint (X-shaped or T-shaped);
* reinforcement bars in the RC joint panel;
* axial load acting on the RC joint panel;* bond condition of reinforcement;
* geometrical and mechanical properties of strengthening systems;
* shear and flexural capacity of the converging members (beams and columns), possibly coupled with strengthening systems.  
The analytical model represents an improvement on the model proposed by [PERSON] [82]. The proposed model provides a mathematical formulation applicable in the cases of X-shaped and T-shaped RC joints. For X-shaped RC joints, the symmetry allows for simplification of the numerical formulation. In particular, the direction of the seismic action does not influence the analytical formulations. For T-shaped RC joints, the mathematical formulation assumes different forms as the direction of the seismic action changes. This is due to the missing symmetry, which causes the capacity of T-shaped RC joints to depend on the direction of the beam and column shear. Starting from the parameters previously discussed, the shear applied to columns (namely, column shear) was considered to be the main parameter in a frame analysis. The proposed model was applied to several cases, and experimentally tested and related failure modes have been detected for each of them. The experimental data [79] refer to about 500 X-shaped and T-shaped RC joints. According to the experimental setup for each RC joint, some parameters have changed to assess the relevant changes in terms of failure modes and capacity.