# Comparative Assessment of Shear Demand for RC Beam-Column Joints under Earthquake Loading
## 2 Fundamentals of RC Beam-Column Joints  
### Joint Shear Demand  
Considering an RC seismic primary frame such as the one shown in the Figure 1, a beam-column joint panel is defined as the zone where both the beam and column converge. A remarkable definition of the joint shear demand (\(V_{j}\)) is due to [PERSON] and [PERSON] [8], who assumed the joint panel as a continuous part of the column (the same considerations apply if it is considered as a part of the beam). When the diagram of bending moments are characterized by a seismic prevalent condition, a sign inversion is needed, passing from the top face to the bottom face of the joint panel. In this regard, \(V_{j}\) can be defined as follows, by considering it as the gradient of the bending moment within the beam depth:  
\[V_{jh}=\frac{V_{c}(H-h_{b})}{h_{b}}, \tag{1}\]  
where \(V_{c}\) is the column shear, \(H\) is the distance between the column's contraflexure points, \(h_{b}\) is the beam depth. As a result, by considering a common value of 3 m for \(H\) and 0.30 for \(h_{b}\), the ratio \(V_{jh}/V_{c}\) amounts to 9, roughly suggesting that the shear demand in the beam-column joint has a different order of magnitude with respect to shear force either in the column or in the beam. Different results have been suggested by [PERSON] [8], who claimed the occurrence of a reduced gradient. In fact, \(V_{jh}/V_{c}\) was supposed to vary from four to six. Similarly, [PERSON] [6] suggested \(V_{jh}/V_{c}\) ranging from three to five for interior joints and half of those values for the exterior.  
Figure 1: Beam–column joint definition for a planar moment-resisting frame under gravity and lateral forces. (Notes. The diagram of bending moment (M) are obtained from a simplified 2D analysis carried out on SPEAR frame).  
The conventional definition of \(V_{jh}\), widely adopted in design codes, is shown in Figure 2a. An interior beam-column joint is considered when the hogging moment acts on the right beam. After cutting with a horizontal plane, the horizontal free-body equilibrium requires:  
\[V_{jh}=T_{R}+C_{L}-V_{c}, \tag{2}\]  
where \(T_{R}\) is the tensile force of the top reinforcement on the right part, \(C_{L}\) is the result of compression stresses on the left part. This study assumes neglecting \(V_{c}\) in Equation (2). Such an assumption will always result in being conservative as long as the seismic condition is dominant for the bending moment diagram.  
A similar definition can apply, considering the case of a bi-dimensional joint as the one shown in Figure 2b. In this case, Equation (2) should apply both in x-z and y-z planes.  
Frequently, shear stresses are used instead of shear forces. The definition is given as follows:  
\[\tau_{j}=\frac{V_{jh}}{A_{j}}, \tag{3}\]  
where \(A_{j}\) is the horizontal projection of the joint area.  
Figure 2: Horizontal joint shear (\(V_{jh}\)) definition: (**a**) Case of a 2D interior joint; (**b**) 3D corner joint.  
According to ACI318 [9], the forces \(T_{R}\) and \(C_{L}\), in Equation (2), should be calculated either from: (i) the maximum moments transferred between the beam and column as determined from factored-load analysis for beam-column joints with continuous beams in the direction of joint shear considered; or (ii) from the beam nominal moments strength possibly with overstrength. The latter is intended to account for: (a) the actual yield stress of a typical reinforcing bar being commonly 10 to 25% higher than the nominal value; and (b) the reinforcing bars strain hardening at member displacements only slightly larger than the yield rotation. EC8 [10] and ACI352 [11], specifying an overstrength factor (\(\gamma_{Rd}\)) equal to 1.20 and 1.25, respectively. As an example, by defining \(A_{s}\) as the area of the beam reinforcement anchored within the joint and \(f_{y}\) the yielding stress, the yielding force (\(T_{y}\)) can be written as follows:  
\[T_{y}=A_{s}f_{y}\gamma_{Rd}. \tag{4}\]