# Comparative Assessment of Shear Demand for RC Beam-Column Joints under Earthquake Loading
## 4 Results and Discussion  
### Shear Demand at a Joint  
The shear demand at a beam-column joint is not conventionally given as an output by commercial FEM software, thus post-processing is needed. Figure 15 shows the key aspects of the implemented post-processing. At the joint labeled \"i\" each converging beam delivers a bending moment, represented as arrows in the x-y plane. The horizontal shear demand (\(V_{jlx}\) and \(V_{jhy}\)) is given in each flexural plane as the summation of \(T\)-forces contribution coming from left and right side of the column neglecting the column's shear. The \(T\) forces represent the resultant of stresses at the column-face cross section under the applied bending moment. For the sake of simplicity, a fixed value of the internal lever arm is assumed equal to 0.83 times the effective depth of the beam. Different assumptions should not lead to significant changes for the final shear demand [48].  
Figure 14: Validation of NLTH applied to the SPEAR frame, y–displacement at the roof.  
Figure 15: Post-process procedure to evaluate shear demand at beam–column joint “i”. (Note “a”, “b”, “c”, “d” labels define the converging beams at the node).  
Considering a practical assessment scenario, the designer might be interested in envelope values. In this regard, the following additional assumptions either on modal combination or bi-directional seismic demand were made for the different analysis methods:  
* (MRSA) Maximum values of shear demand at a joint are evaluated, considering the 30% rule for the bi-directional combination. Modal contributions were summed using the complete-quadratic combination (CQC);
* (NLSA) Pushover curves were calculated using the modal force distribution for the first three modes. Specifically, target displacements were assumed in the x-direction for the mode \"1\" and in the y-direction for the mode \"2\". Mode \"3\" being torsional, both the \(\pm\)x and \(\pm\)y directions of target displacements were considered. Subsequently, idealization of the pushover curve into a bi-linear curve is made. Shear demand at a joint is extracted for each idealized pushover curve, at the target displacement. The final shear demand is then computed, summing the modal contribution using CQC. This procedure is compliant with what was proposed by [PERSON] [46], named modal-pushover-analysis (MPA);
* (NLTH) Extreme values of shear demand at a joint are extracted from time histories. Acceleration of time histories apply simultaneously in \(\pm\)x and \(\pm\)y directions. Envelope of the results coming from different direction signs is made.  
To summarize, the shear demand at each flexural plane of a beam-column joint is computed as a post-process starting from values of beams' bending moments converging at one node. Two major hypotheses were made: (i) the internal lever arm at the column face cross section of the beam is assumed to be constant; and (ii) the contribution to the horizontal equilibrium of the column's shear is neglected. The latter assumption has already been discussed in Section 2.1, defining the joint shear demand for a planar joint.  
Different strategies with respect to the one proposed in this study might facilitate the output of shear demand. Usually they imply additional dofs for the numerical model. As an example, explicit joint modeling can be pursued via single or multi-springs [5]. Besides, the introduction of a stiff \"shear hinge\" was recently proposed by [PERSON] [33]. It is worth mentioning, though, that the introduction of a large stiffness, compared with the flexural stiffness of the beam, remains dubious in the light of numerical errors (the reader is refereed to paragraph 8.2.5 of [49]).