## 3 Numerical Study
### Numerical Modelling  
To carry out nonlinear time history (NTH) analyses, nonlinear numerical models of the prototypes were developed using Inelastic Dynamic Analysis of Reinforced Concrete IDARC-2D [27]. All frames in the \(X\) direction were considered for each model. The nonlinearities in the RC elements were concentrated on plastic hinges located at both ends of the beams and columns. To take into account the extension of the plastic hinge length as the level of plastic deformation increases, the spread of plasticity formulation implemented in IDARC-2D was adopted. The hysteretic moment rotation loops followed by the plastic hinges build on a non-symmetric moment curvature backbone of the cross section. This moment curvature backbone was determined with a fibre model analysis based on the nonlinear behaviour of the materials and the cross-sectional properties. The nonlinear properties of materials adopted for numerical modelling are as follows: concrete: compressive strength 20.6 MPa, initial Young's modulus 21.5 GPa, strain at maximum strength 0.2%, stress at tension cracking 2.47 MPa, ultimate strain in compression 0.95% and slope of falling branch \(-\)2.2 GPa; steel: yield strength 274 MPa, ultimate strength 385 MPa, modulus of elasticity 206 GPa, modulus of strain hardening 3.33 GPa and strain at start of hardening 0.3%; and infill walls: prism strength of masonry 2.3 MPa, initial Young's modulus 2.1 GPa, cracking stress of masonry 0.12 MPa, strain corresponding to prism strength 0.2%, shear strength of masonry bed joints 0.28 MPa and coefficient of friction of frame-infill interface 0.3. The response under cyclic loadings (i.e., the hysteretic rule) is governed by four parameters that account for the effects of stiffness degradation (HC), strength degradation (HBE, HBD) and pinching (HS). The values HC = 10, HBE = 0.01, HBD = 0.45 and HS = 0.25 were adopted for beams and HC = 10, HBE = 0.01, HBD = 0.45 and HS = 0.25 for columns. For the masonry infill walls, two compression struts linking correlative floors diagonally were defined using the lateral yield force and lateral stiffness determined in FEMA-356 [4]. Their nonlinear behaviour of the struts was modelled by the Bouc-Wen model [28] using the default values recommended by IDARC-2D. Finally, a tube-in-tube (TTD) metallic damper was considered in this study [29]. The TTD device exhibits stable quasi-rectangular loops under cyclic deformations. Its hysteretic behaviour was also represented with a Bouc-Wen model, whose parameters were calibrated against experimental results. The dampers were designed by applying the energy balance concept explained in Section 2.2, as follows. First, the values of \(Q_{yi}\), \(\phi_{yi}\) and \(\mu_{i}=\beta Q_{yi}/\rho_{yi}\) were determined from a pushover analysis of the bare frame with infills; the fundamental period \(T_{1}\) was calculated with an eigenvalue analysis, and the ratio \(c/a^{2}\) was estimated using Equation (8). Second, the dampers were designed for the design earthquake associated with a return period of 475 years and soft-soil conditions prescribed by the current seismic code of the Dominican Republic [25]. The values of \(S_{V}\) (=\(V_{D}\)) obtained with this code are \(S_{V}\) = 1.08 m/s for prototype N3, \(V\)\(S_{V}\) = 0.97 m/s for prototype N6 and \(S_{V}\) = 0.92 m/s for prototype N9. The yield horizontal displacement of the dampers was made equal to 0.15\(\phi_{y1}\) so that the dampers start dissipating energy far before the onset of plastic deformations in the frame. The required horizontal strength of the dampers expressed in terms of \({}_{s}\alpha_{1}\) was determined by solving \({}_{s}\alpha_{1}\) in the energy balance equation of the structure, Equation (17), with \(\mu_{max1}=\beta Q_{y1}/Mg\) and \(\eta=26\). The value adopted for \(\eta\) corresponds to a low level of damage on the dampers, and it is far below the ultimate energy dissipation capacity of the type of damper (TTD damper) used in this study [27].