# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 2 Modeling Approaches for Steel Structures in OpenSees  
Non-linear analyses of steel structures fall into two main categories: distributed plasticity and concentrated (or lumped) plasticity analyses [58]. In the lumped plasticity approach [55; 58; 59; 60; 61; 62; 63; 64], the beams and columns are modeled as elastic elements with plasticity concentrated in zero-length rotational springs at either ends. The zero-length rotational spring elements (implemented e.g., as _zeroLength_ element object in the OpenSees framework [56; 57]) generally include the modified Ibarra-Medina-Krawinkler (IMK) material model [65], which estimates the hysteretic behavior of beam-column connections capturing the strength degradation occurring both during monotonic loading after the strength cap is reached, and during consecutive cycles even though the strength cap has not been reached [64; 54]. However, for MRFs, this model has been shown to have convergence issues in case of dynamic loading [58], due to high stiffness that must be assigned to the rotational springs in order to preserve the elastic stiffness of the beam. Moreover, the structural response is significantly affected by the selection of the damping model, and the Rayleigh damping model leads to unreliable results [66]. In contrast, the damping model has been demonstrated in several studies [67; 68; 69; 70] to have less influence when the structural elements are formulated according to distributed plasticity models, such as e.g., fiber-based cross-section discretization [66]. According to the fiber-based cross-section discretization model, anelastic behavior can spread across the whole length of each element [60; 67] and plastic hinges are activated at the most stressed sections when the yield limit is exceeded [58]. Another feature of this approach is that the influence of the axial force on the column moment-rotation relationship is directly taken into account [21], differently from the lumped plasticity model that requires each time to adjust the axial load-moment interaction based on the current axial load and moment gradient along the member [71]. Conversely, the lumped plasticity is more versatile than the distributed plasticity for simulating the cyclic degradation of strength and stiffness (low-cycle fatigue) [21], which has been demonstrated to significantly affect the seismic response of steel structures [58; 72], especially in the event of long earthquakes [68]. In fact, mainly in the case of the I-shaped cross-sections, which are commonly used for the MRF beams, section deterioration begins in the sections subjected to the maximum moment, due to the triggering of flange local buckling in the compressed zone [73]. Since distributed plasticity models do not allow to incorporate explicitly the cyclic degradation, [PERSON] and [PERSON] [58] proposed a refined beam formulation, which is able to reproduce the strength and stiffness degradation of the I-shaped cross-sections and to replicate the failure mechanism caused by low-cycle fatigue. In particular, they formulated a fiber-based damage accumulation model by introducing a fatigue material wrapped around the uniaxial steel material used for the non-linear fiber section model assigned to the plastic hinges of the _beamWithHinges_ element object [56; 57]. In the _beamWithHinges_ object, which can be used both for the beams and columns, the structural member is discretized into three sub-elements: a linear elastic material behavior is assigned to the internal sub-element while non-linear material behavior is associated with the two external sub-elements, whose length is assigned by the user. Different integration methods can be used with the _beamWithHinges_ element object, varying with the number and location of the integration points [58].