# Unified Simplified Capacity Model for Beam-Column Joints into RC Moment Resisting Frame
## 3 Analytical Model
### Equilibrium Condition of the RC Joint  
The mathematical formulation is based on internal equilibrium conditions, and it allows for the assessment of the failure mode under ultimate conditions. The direction of the seismic action alters the internal equilibrium conditions when the RC joint is characterized by a T-shape. In this paper, a unified mathematical formulation will be discussed. It allows the application of the proposed approach to RC joints of any shape (X and T type). According to modern guidelines [1; 2; 12], internal RC joints can also be analyzed by using a plane model once the main joint panel has been both mechanically and geometrically characterized. The equilibrium conditions can be written for the T-shaped and X-shaped RC joint, once the geometrical characteristics and the internal forces are known. Figure 1a shows the case of T-shaped RC joints subjected to positive or negative column shear, respectively. The parameter \(\alpha\) represents the ratio between the length of the column and beams, called \(\mathrm{L_{c}}\) and \(\mathrm{L_{b}}\), respectively. Conversely, the internal equilibrium condition is shown in Figure 2a,b as the direction of the seismic action changes. The main difference is clear in Figure 1b due to the symmetry of the internal stresses for the X-shaped RC joint, drawn with different colors in order to clarify the relevant longitudinal reinforcements depicted in the corresponding cross-sections, as well. Figure 2a,b show the two different directions of the column shear, \(\mathrm{V_{c}}\), due to the asymmetry of the T-shaped RC joint. In particular, the internal stresses acting in the RC joint change with the direction of the column shear. This effect does not occur for the X-shaped RC joint, as shown in Figure 2c. The depth of RC joint can be also different from the depth of the beam and column. Figure 1: Characteristics and external forces acting on the RC joints: (**a**) positive direction of the column shear acting on the T-shaped RC joint, (**b**) negative direction of the column shear acting on the T-shaped RC joint, and (**c**) positive direction of the column shear acting on the X-shaped RC joint (negative is symmetrical). The compressive force \(\rm F_{i}\) is obtained by adding the contribution of the concrete \(\rm C_{i}\) and the reinforcement \(\rm S_{i}\) in compression, as shown in Figure 3.  
Figure 2: Internal equilibrium conditions for the RC joints: (**a**) positive direction of the column shear acting on the T-shaped RC joint, (**b**) negative direction of the column shear acting on the T-shaped RC joint, and (**c**) positive direction of the column shear acting on the X-shaped RC joint (negative is symmetrical). The behavior of the RC joint panel is assessed by means of a discretized approach in cracked condition. In particular, the RC joint panel is modeled as four portions (i.e., rigid bodies) obtained by intersecting the two diagonals of the RC joint panel, each with an inclination \(\theta\). Under a generic load, the RC joint is subjected to several internal forces (Figure 2). The internal forces are derived from the resultants of the internal stress acting on the longitudinal bars, stirrups, and compressed concrete. The contributions provided by the bars and concrete around compressed bars have been supposed to be applied at bars location. The contribution provided by the compressed concrete diagonal strut in the joint is modeled as a resultant force, C, acting on the boundary of each portion of the RC-joint panel. It has been supposed to act equally on each half of the diagonal of the RC joint panel. The inclination \(\vartheta\) of this resultant force depends on the internal stress distribution. However, it can be assumed to act normally to the crack (hence \(\vartheta=\theta\)), and it is centered with respect to the diagonal of the RC joint. Both for the T-shaped and the X-shaped RC joint, twelve equilibrium equations can be written to assess the ultimate behavior and the occurred failure mode.