# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 4 Numerical Model in OpenSees  
Fully 3D numerical models of the case-study structures were formulated using the OpenSees software program. The _forceBeamColumn_ element object, in the form of the _beamWithHinges_ element [57], was implemented for both beam and column members. A linear elastic material behavior was assigned to the internal sub-element of the _forceBeamColumn_ object, while a fiber-based cross-section discretization was associated with the two external sub-elements. The length of these two sub-elements where inelastic behavior can be triggered was taken as \(L_{pl}=0.22\ L_{v}\) according to references [58; 75], where \(L_{v}\) is the shear length. The cyclic deterioration of the stiffness and strength of the structural members was disregarded as the bidirectional NLDAs were performed considering artificial ground motions characterized by a total duration of 25 s, as recommended in [51].  
A two-point Gauss-Radau integration scheme for each sub-element was used in the element state determination, for a total of six integration points across the whole element object [71]. The cross-section of each external sub-element was discretized into 24 fibers, specifically eight fibers for each flange and eight fibers for the web [77; 78; 79; 80]. The uniaxial Giuffre-Menegotto-Pinto constitutive law [74], corresponding to the _Steel02_ material model with isotropic strain hardening [81], was assigned to each fiber. The yield strength \(f_{y}\), the modulus of elasticity \(E_{s}\) and the strain-hardening ratio \(b\) were taken equal to 355 MPa, 210,000 MPa and 0.01, respectively; the model parameters that control the transition from the elastic to the plastic branch were assumed as \(R_{0}=18\), \(C_{R1}=0.925\) and \(C_{R2}=0.15\), consistently with reference [57].  
The floor slabs were modeled as rigid diaphragms and the masses of the beams, columns and slabs were concentrated at the center of mass of each floor. The seismic masses at each floor were calculated according to code recommendations [51; 76], taking a combination factor of 1.0 for the permanent loads and of 0.3 for the live loads as prescribed for use category B. The dead and live loads were calculated according to the tributary area concept and uniformly distributed on the beams along the X-direction (Figure 1). P-Delta effects were included in the analysis. The columns had fixed supports at the ground level, simulating rigid foundations. The structural damping of the frame was modeled as a function of the tangent stiffness matrix only, considering 3% on the first mode in line with references [82; 83].  
As in reference [55], the three MRFs were upgraded considering the seismic loads defined in the IBC [51] for life-safety limit state (SLV), in the municipality of Lamezia Terme (latitude 38.57\({}^{\circ}\), longitude 16.18\({}^{\circ}\)), for functional class \(c_{u}=2\), nominal life \(V_{n}=100\), PGA = 4.47 m/s\({}^{2}\), soil type C and topographic factor T\({}_{1}\). The seismic retrofit of the structures is performed by means of steel braces equipped with steel hysteretic devices and arranged in a chevron (or \"reversed V\") brace configuration. Such a configuration was adopted because it provides a direct relationship between the axial deflection of the damper and the structural inter-story drift [55; 84; 85]; nevertheless, alternative configurations can be implemented in the model as well. The braces are modeled as truss elements [57] associated with an _uniaxialMaterial_ model with elastic behavior. The damper, placed at the intersection of the two braces and connected to the midsection of the beam, is modeled by a _zeroLength_ element object [57] associated with an _uniaxialMaterial_ model with elastic-perfectly plastic behavior [56; 57], as commonly assumed in literature for hysteretic dampers [2; 9; 11; 12; 47; 55].