# Comparative Assessment of Shear Demand for RC Beam-Column Joints under Earthquake Loading
## 3 Numerical Study
### Structural Modeling and Seismic Demand  
A three dimensional frame structural model was developed using SAP2000 [37]. Details are given in Figure 8. Beams and columns were modeled using a lumped-plasticity [38] approach. Specifically, elastic 3D frame elements (two nodes, 12 dofs) were used in conjunction with rotational non-linear springs (plastic hinges) assigned at the same location (zero \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Author** & **Ele (a)** & \multicolumn{3}{l}{**Type of Analysis**} & & \\  & **MRSA** & **NLSA** & **NLTH** & **Scope of Investigation** & **Refs.** \\ \hline [PERSON] & F & ✗ & ✓ & ✓ & Validation of non-linear static procedures. & [22] \\ [PERSON] & F & ✗ & ✓ & ✓ & Extension of Capacity Spectrum Method [23] to non-symmetric case. & [24] \\ [PERSON] & F & ✗ & ✗ & ✓ & Validation of GC [25] time integration algorithms. & [26] \\ [PERSON] & F & ✗ & ✓ & ✓ & Assessment of experimental results. & [27] \\ [PERSON] & F & ✗ & ✓ & ✗ & Validation of a probabilistic seismic performance & [28] \\ [PERSON] & F & ✓ & ✗ & ✓ & Definition of torsional amplification to be applied to N2 method [29]. & [30; 31] \\ [PERSON] & F & ✓ & ✗ & ✓ & Validation of chord rotation demand from MRSA. & [17] \\ [PERSON] & F & ✗ & ✓ & ✓ & Assessment of experimental results. & [32] \\ [PERSON] & F & ✗ & ✗ & ✓ & Validation of lumped non-linear hinges representing brittle failures. & [33] \\ [PERSON] & B & ✗ & ✗ & ✓ & Assessment of experimental results. & [21] \\ [PERSON] & F & ✓ & ✓ & ✗ & Comparison of seismic conforming variants of SPEAR frame. & [34] \\ [PERSON] & F & ✗ & ✓ & ✓ & Assessment of experimental results. Influence of modelling assumptions. & [35; 36] \\ \hline \hline \end{tabular} Notes. (a) Type of elements: [F] frame; [B] Bricks or shells.  
\end{table} 
Table 1: Literature survey of numerical studies addressing SPEAR frame. length) of the extreme nodes. The element size was kept equal to one half of the inter-storey height for the columns and half-bay for the beams. Columns were clamped at the base. Floor masses were lumped at the centers of mass. Diaphragm kinematic constraint was assigned at each floor. The plastic hinges' constitutive law (moment vs rotation) was defined for each beam and columns, considering different axial load levels for the latter. In particular, a preliminary sectional analysis was carried out using Response2000 [39]. The yielding moment was evaluated from the moment-curvature diagrams (examples are given in Figure 9). Plastic rotation capacities were attributed empirically as shown in Figure 10, complying with the FEMA356 [40], similarly with what was presented by [PERSON] [35]. Takeda [41] model was used for the hysteretic response. Slab contribution to the flexural capacity of the beam was neglected because it would activate only for large ductility demand [42], which was not reached at the imposed seismic level. Furthermore, a weak column condition is recognized even using the beam's nominal dimensions, as can be inferred from Figure 9. Indeed, despite the case of the strong column (cross-section 25 \(\times\) 75 cm), beam bending capacity (nominal cross section 25 \(\times\) 50 cm) results in being larger with respect to the columns having a 25 \(\times\) 25 cm cross-section. Figure 8: Modeling features. Figure 9: Moment–curvature diagrams for the cross sections of beams and columns of SPEAR frame. Results were obtained using the software Response 2000 [39]. Beam-column joints were modeled as rigid. The half depth of both the column and beam was set as a rigid length. A discussion about differences in adopting center-line modeling, which assumes beams and columns that are entirely flexible, can be generally found in [43]. Moreover, [PERSON] [35] studied the influence for SPEAR frame's joint. In essence, higher shear demand is a consequence of rigid offsets. Center-line modeling may be used as a simple way to account for both the reduction of stiffness and strength due to additional deformation in the joint regions [44].