# On the Modeling and Analysis of Brittle Failure in Existing R/C Structures Due to Seismic Loads
## 3 Example Application of the Proposed Numerical Simulation Procedures
### Numerical Simulation and Analysis of the SPEAR Building  
In order to evaluate the seismic response of the SPEAR test building, the structural system was simulated as a 3D-FE model comprising linear and zero-length elements as described in Section 2, using the OpenSees platform [41]. At first, the flexural response of the columns was modeled using two equal-length nonlinear Beam-Column element objects with a longitudinal fiber section discretization along the column clear length \(H_{cl}\) (see Figure 1). The reinforcing steel material properties were represented by the uniaxial, bilinear stress-strain relationship with kinematic hardening (Steel01 Material) with an initial elastic modulus equal to 200 GPa and a strain-hardening ratio of 3.3% and 5.3% in the cases of the \(\mathcal{O}\)12 mm and the \(\mathcal{O}\)20 mm bars, respectively. The inelastic concrete behavior was modeled using the uniaxial Kent-Scott-Park stress-strain envelope with degrading unloading/reloading hysteresis loops and no tensile strength (Concrete01 Material). Specifically, the concrete had an initial tangent modulus of \(2\cdot f_{cm}/\epsilon_{cy}\) (\(\epsilon_{cy}=0.0022\)) and an ultimate strength and deformation capacity of \(f_{cu}=0.2\cdot f_{cm}\) and \(\epsilon_{cu}=0.0035\) according to [PERSON], [PERSON], and [PERSON] [45; 46]. Details of the fiber discretization of the column cross-sections are depicted in Figure 2.  
Next, the modeling of the flexural response of beams was similar to that of columns, using the nonlinear Beam-Column element objects with longitudinal fiber section discretization. The cross-section of beam elements was a T-shape for all beams with effective width equal to \(b_{w}+2\cdot d\) (\(b_{w}\): beam web width; \(d\): beam static height), except for beams [B1, B2, B5, B6, B7, and B8] which had an L-shape cross-section with effective width equal to \(b_{w}+d\). Beam finite elements spanning areas within the beam-column joints were modeled as rigid zones (see Figure 1). Finally, the brittle response mechanisms of column failure (shear failure of the column web, of the anchorage, and lap-splice failure of column longitudinal reinforcements) were all simulated using Zero-Length element objects in the locations shown in Figure 1. Mechanical properties of the Zero-Length elements were defined according to the discussion in Sections 2.2-2.4, further considering column shear strengths corresponding to the different  
Figure 7: Time-histories of the two components of ground acceleration used in the testing of the SPEAR building at ELSA-JRC. response mechanisms as calculated from Equations (A1)-(A6). The normalized average axial load \(\
u\) in the columns that were used in Equations (A1)-(A6) was calculated from a static analysis of the FE building model under the self-weight of the structural elements and the additional gravity loads applied to the SPEAR building during the tests [42; 43; 44]. It is noted that the true axial load is variable, fluctuating about an average value due to overturning effects in the ground motion, with maximum values recorded in the columns that are furthest from the center of mass. Table 1 presents the normalized average axial load and the shear strengths of the SPEAR building columns corresponding to the mechanisms considered in the RSA system [30; 31; 32], whereas Table 2 presents the parameters defining the multilinear \(M\)-\(\theta\) relationships used for simulating the responses of the zero-length elements parallel to the two principal plan directions. Regarding the building mass needed for conducting a dynamic analysis, the numerical simulations used lumped masses defined at the centroid of each structural element (i.e., the slabs, the clear length of columns and beams, and the beam-column joints). Lumped masses were also added at individual nodes which were inserted at locations on the building floors where water tanks were placed during the tests for simulating the operational loads.