# Comparative Assessment of Shear Demand for RC Beam-Column Joints under Earthquake Loading
## 3 Numerical Study
### Validation  
The structural analyses were validated against both experimental and numerical results derived from the literature. Results are briefly discussed, in the following, assuming the global behavior of the structure as a target, e.g., inter-story drifts, capacity curves, displacement time histories.  
The influence of beam-column joints modeling was recognized by modal analysis. In particular, rigid offsets gave shorter periods for the first three modes as can be inferred from Table 3. Furthermore, the first three modes (Figure 12a) amount to almost 80% of the participating mass in both x and y directions. Clearly, the third mode involved torsional behavior, although only partly confirmed by previous numerical investigation. Nevertheless, MRSA produced comparable results with respect to [PERSON] [34] for drift values as shown in Figure 12b.  
Results of NLSA are shown in Figure 13. For the sake of validation, two different load patterns were adopted, i.e., uniform distribution of forces and modal; the latter resulting in reduced lateral load bearing capacity. Plastic hinges formed in the columns. Most of the beams remained in the elastic regime. This evidence agreed with what was presented by [PERSON] [34].  
Figure 14 shows the resulting displacement history (roof) in the y direction obtained via NLTH. The comparison with experimental outcomes shows that agreement was found for the oscillation period, the differences in peak values being in line with those obtained, for instance, by  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Method** & **Seismic Demand** & **Notes** & **Ref. (a)** & **Bi-Dir. (b)** \\ \hline MRSA & Design Response Spectrum with q = 3.45 & & & & & \\ according to Figure 11. & & & & & & \\ Capacity Spectrum with 100 mm target & & & & & & \\ displacement in both directions. & & & & & & \\ Hereg-Novi Accelerogram (PGA = 0.25 g) & & & & & & \\ according to Figure 11 & & & & & & \\ \hline \hline \end{tabular}  
Notes. (a) References used to compare the numerical results. (b) Method used to account for bi-directional seismic demand.  
\end{table}
Table 2: Details for structural analyses.  
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Author/Model** & **Ref.** & \(\mathbf{T_{1}}\) & \(\mathbf{M_{X}}\) & \(\mathbf{M_{Y}}\) & \(\mathbf{T_{2}}\) & \(\mathbf{M_{X}}\) & \(\mathbf{M_{Y}}\) & \(\mathbf{T_{3}}\) & \(\mathbf{M_{X}}\) & \(\mathbf{M_{Y}}\) \\  & **(s)** & **(\%)** & **(\%)** & **(\%)** & **(s)** & **(\%)** & **(\%)** & **(s)** & **(\%)** & **(\%)** \\ \hline [PERSON] & [27] & 0.62 & 71.80 & 5.80 & 0.54 & 12.40 & 60.50 & 0.43 & 12.40 & 60.50 \\ [PERSON] & [21] & 0.64 & - & - & 0.54 & - & - & 0.42 & - & - \\ [PERSON] & [36] & 0.57 & - & - & 0.48 & - & - & 0.39 & - & - \\ [PERSON] & [34] & 0.80 & 69.00 & 4.80 & 0.69 & 15.60 & 47.80 & 0.58 & 2.70 & 30.30 \\ [PERSON] (Experimental) & [19] & 0.84 & - & - & 0.78 & - & - & 0.67 & - & - \\ RIGID & This paper & 0.85 & 55.00 & 0.00 & 0.47 & 2.18 & 56.00 & 0.35 & 31.00 & 27.00 \\ [PERSON] & This paper & 1.24 & 54.00 & 0.00 & 0.64 & 2.10 & 56.00 & 0.50 & 31.00 & 27.00 \\ \hline \hline \end{tabular}  
Notes. \(M_{X}\) and \(M_{Y}\) are the participating masses in X and Y direction, respectively.  
\end{table}
Table 3: Validation of Modal Analysis results.  
Figure 12: Validation of MRSA applied to the SPEAR frame: (**a**) Modal analysis results [Note. The first three relevant modes amount 86% and 82% of participating masses in X and Y directions, respectively]; (**b**) MRSA drifts results.  
Figure 13: Validation of NLSA applied to the SPEAR frame: (**a,b**) capacity curves in y and x directions, respectively; (**c,d**) plastic hinges rotation at 10 cm roof displacement in y and x–directions, respectively. The Reader is referred to the color version of this figure.