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1. INTRODUCTION
Ecuador is a country located in the Pacific Ring of Fire. For this reason, earthquakes constitute one of the leading causes of structural destruction. They generate damage and, in some cases, collapse, as evidenced in the past, as evidenced by the Manabí earthquake in 2016 (Vanegas, 2018). In addition to the subduction of the Nazca and South American plates, there is a system of local faults that can generate significant earthquakes at shallow depths (Parra, 2016). This is Quito’s seismic scenario, where earthquakes generated by local faults can cause significant damage to structures and their occupants.
Some structural damage is tolerated when designing against earthquakes. The methodology of the Ecuadorian Construction Stan- dards, in its chapter on seismic resistant design, states that structures should have non-damage behavior from frequent earthquakes. These should maintain the integrity of the systems and aim to avoid the collapse of said structures for earthquakes of greater intensity (MIDUVI, ).
To create resilient cities, the quandary of damage accepted in the regulations and tolerated by society must be addressed. The global trend emphasizes continuous functionality, especially for essential structures or those critical to the community. In conventional de- signs, the aim is to minimize damage. According to Soong and Spencer (2002), it is necessary to find energy dissipation, base isolation, and active control techniques to prevent extensive damage to the structure and lower repair costs. Energy dissipation devices such as TADAS (Triangular plate added damping and stiffness), SLB (Shear Link Bozzo), or Restricted Buckling Bars (RBB) are intended to resist the largest portion of seismic forces. These also concentrate damage onto these devices to protect structural elements such as beams or columns. In addition, these systems make it possible to rigidize structures and increase damping; consequently, the response in displacements and accelerations can be reduced.
Another aspect in which the advantage of using energy dissipators is evident is that the seismicity in Ecuador is not yet fully known, and the uncertainty in seismic hazard studies is great. It is, therefore, possible to have a higher seismic energy than expected by the designers. In short, more significant damage could be manifested in conventional structures if the intensity exceeds the design inten- sity. However, by including energy dissipators, excess energy can be resisted through damage to the dissipators, thus safeguarding the integrity of the structures and protecting their occupants.
This paper presents the design of 4-, 8- and 12-story steel and reinforced concrete portal frames with and without energy dissipaters, considering TADAS, SLB, and RBB devices. Non-linear behavior under static and dynamic loads is studied, structural performance is evaluated, and fragility curves are determined. Damage states corresponding to mild, moderate, severe, and complete are considered. In addition, the design of the structures is verified by considering their structural performance.
2.STRUCTURE DESIGN
Four-, eight-, and twelve-story portal frames with three spans are analyzed and designed. The TADAS and SLB dissipaters are placed on steel diagonals; these elements are located in the central span, similar to the RBB case, see Figure .
The total number of structures analyzed is 24.
The analysis and design of the structures is carried out applying national regulatory guidelines (MIDUVI, , , ), and their corresponding international standards (American Concrete Institute, ; American Institute of Steel Construction, , , ; American Society of Civil Engineers, ). The loads are determined by the chapter on Non- seismic Loads (MIDUVI,
). Dead load for slabs is considered at 500 
and 380 for the roof. Furthermore, office occupancy is assumed at a live load of 250 
and 100
for the roof. To define the design earthquake, the spectrum for the city of Quito is used considering type D soil and a seismic response reduction factor of 6. The reactive seismic load is assumed at 100% of the dead load and 15% of the live load (Cagua et al., 2022).
In structures with TADAS dissipators, 20% damping is assumed; for structures with SLB, 15%, and other structures are modeled with 5%. For damping greater than 5%, the design spectrum is divided by value B, which is the function of the damping factor of the structure with dissipators 
, as indicated in Equation ().This approach to modifying the design spectrum is proposed in the methodology utilized by Aguiar et al. (2017).
The structures are analyzed and designed using “CEINCI LAB Computing System functions” (Aguiar & Cagua, 2022), developed by said authors using the rigidity method. For the design, the linear properties of the structure are considered. However, when analyzing the dissipator, non-linear properties are operated through a bilinear model, and secant stiffness is taken into consideration. This is determined with an iterative analysis (Aguiar et al., ).
The models with TADAS and SLB dissipators contemplate these as a short vertical element (Aguiar et al., 2019). For the seismic case, the degrees of freedom consider a rigid floor, and therefore the horizontal degree of freedom is unique in each level; in addition, the dissipators are considered axially rigid in the vertical sense. For the static analysis, 3 degrees of freedom per node are used due to being two-dimensional models.
The sections of the structural elements, beams, columns, diagonals, and steel core of the RBB, taken into account in the models, are presented in Table .
The reinforcement of reinforced concrete beams and columns meets the regulatory requirements, and 20mm and 25mm rods are used.
TADAS dissipators with 6 triangular plates are considered. These plates have a geometry of 17cm in base and height, and a thickness of 2.5cm is also assumed. The SLBs used have the properties of an SLB3_25_2 device as indicated by the manufacturer (Bozzo, 2022).
In all cases, compliance with a maximum inelastic drift of 2% is verified, which is the permissible limit of the MIDUVI (). An equivalent spectral and static modal analysis is performed. Figure shows the result of the seismic analysis for a 4-story steel structure without including dissipators (Cagua et al., 2021).
In steel structures, it is verified that the axial capacity of the columns is greater than the demand before the effects envelope, including the seismic over resistance factor, as indicated in ASCE 7. It is also shown that the demand relationship versus axial bending and shear capacity is less than 1 for all structural elements. These results can be corroborated graphically in the generated programs, as indicated in Figure .
In reinforced concrete frames, the analysis and design processes are similar. Still, the reinforcement of the elements is taken into consideration for verification. This reinforcement is evaluated at the beginning, middle, and end of each component. The demand vs. capacity relationship in the beam is considered for upper and lower steel due to the moment envelope. An interaction diagram is usedfor columns. In all cases, “Strong Column and Weak Beam” is checked to verify the connection.
Designs with similar inelastic drift responses are performed to simulate comparable steel and reinforced concrete structures. For example, in systems without the inclusion of dissipators, there are drifts of 2%. In the case of a dissipator, these drifts are close to 1%.
3.SEISMIC HAZARD
A seismic hazard analysis is performed for the site of interest, where the structures will be implemented, using the CRISIS program (Ordaz et al., ). The ground motion prediction models are incorporated into the calculation using a logic tree composed by Akkar et al. (2014), Chiou and Youngs (2014), and Zhao et al. (2006), all with a weight equal to 33%. The source zones of Beauval et al. (2018) are considered.
From the disaggregation of a seismic hazard, it is determined that for a design earthquake (EQ),i.e., with Tr = 475 years, the earth- quakes contributing the most danger correspond to events with magnitudes of 6.25 on faults with focal distances of up to 40 km. In this case, eleven accelerograms are chosen from the “PEER Ground Motion Database” (), corresponding to this seismic scenario, as shown in Table .
Earthquake scaling is performed using the weighted factors procedure. Amplitude scaling aims to reduce the error, defined as the sum of the squares of the difference between the spectral acceleration of the scaled component and the target spectral acceleration in a rangeof periods. This range for the analysis structures corresponds to 0.2T to 1.50T .
4.NON-LINEAR BEHAVIOR AND SEISMIC PERFORMANCE EVALUATION
Non-linear analyses are carried out in OpenSees (Zhu et al., 2018), ForceBeamColumn models are considered for columns and beams, “Two Node Link” type models are used for the dissipators. Truss models are used for the diagonals. “Steel02” and concrete, confined and unconfined, are used with “Concrete02”. These models are shown in Figure .
The capacity curve of the structures is determined by non-linear static analysis. Figure shows the investigation results. It can be seen that the curves for both steel structures (SS) and reinforced concrete frames (RC) are similar when considering the same typology (floor number and type of dissipator).
Objective target displacement δ t for an intensity level defined by Sa is estimated by using ASCE 7 methodology, where coefficients C 0,
C 1, and C 2 are used to modify the response of a one-degree-of-freedom system (considered with an effective fundamental T e period); this is summarized in Equation ().
On the other hand, effective yield displacement δ y e f f is estimated using Quantification of Building Seismic Performance Factors (FEMA-695) (), expressed in Equation (). This displacement corresponds to the beginning of the non-linear behavior of the structure, i.e., the first plastic hinge is formed.
For this equation, Γ1 is the elastic modal participation factor of the first vibration mode. T 1 is the period of the fundamental vibration mode; V max is the maximum shear of the capacity curve; W is the total reactive load.
For the non-linear dynamic analyses, simplified models of one degree of freedom are used, with a response based on a bilinear model. These simulations are calibrated with Pushover capacity curves of the structures, and the damping assumed in each type of analysis. The models do not consider strain hardening (the post-yield line has zero slopes) or the effect of crack closure. They do not contem- plate deterioration of rigidity and resistance, either. However, they are applicable because the structures are two-dimensional frames with a response controlled by the first vibration mode.
The performance limit states are defined by the displacement on the roof. The fully occupational limit (O) is considered the effective yield displacement. The collapse limit (C) to the ultimate displacement, for functional (F) 30% is assumed, and for life safety (LS) 60%. In collapse prevention (CP), 80% of inelastic displacement ∆ P is measured from yield displacement (SEAOC, ). Figure shows a capacity curve of the structure, identifying the performance limits and the maximum displacements in the roof for the non-linear dynamic response of each earthquake (EQ) scaled to TR = 475 years.
The performances of the structures are presented in Table , evaluated with the ASCE 7 methodology for the design and maximum earthquake considered. The performances evaluated with the responses of the non-linear dynamic analyzes estimate a level of damage lower than those presented in Table by the ASCE 7 methodology. Dynamic incremental analyses are carried out to obtain the fragility curves for each structural typology (see Figure ). Four levels of damage are defined, correlated to the described performance levels, corresponding to mild (occupational), moderate (functional), severe (life safety), and complete (collapse prevention). A compendium of the individual results for each structure and of all the analyses is presented in Cagua (2022).
5. CONCLUSIONS
Structures designed without energy dissipators present drifts close to 2%. However, by including dissipators, these decrease to almost 1%, which means an improvement in the expected structural response. In addition, the stress on the portal frame elements, beams, or columns is substantially reduced. Consequently, the probable damages from a seismic event will be concentrated on the energy dissipators.
The Pushover curve shows that, at the first moment of analysis, the elastic stiffness of the structures is practically the same for each type of structure regardless of the kind of dissipator. However, an advantage is observed in the inelastic range when this device is included.
The capacity of the structures, in terms of shear force, when including BRB, is higher than with TADAS or SLB for this particular case. Although all the dissipators increase the rigidity of the system, this increase is higher in the case of the BRB. In cases with TADAS or SLB dissipators, the increase in damping is its main advantage. The global drifts of structures, the displacement in ceiling concerning its total height, is close to 0.5% for the effective creep displacement and 2.5% for the ultimate removal. Although in reinforced concrete structures of 8 and 12 floors, this decreases to 2%.
The ASCE 7 accelerogram scaling methodology, applied over a range of periods, using weighted averages, allows maintaining the variability of the seismic hazard in the analysis for the types of structures based on the number of floors. This approach facilitates generation of seismic scenarios consistent with the analysis site, in this case corresponding to the city of Quito.
The fragility curves of the structures show that the probability of reaching and exceeding a specific damage state decreases when including energy dissipators. In addition, for 12- story facilities, lower levels of intensity are required to achieve the same probability of damage as for 8- or 4-story structures.
In the fragility curves for slight damage, it can be seen that the damage probabilities are similar regardless of the type of dissipators because in this range of intensities, the structure responds elastically, and the dissipators do not yet contribute predominantly. On the other hand, there are differences for higher damage levels depending on the dissipator type. In addition, a more significant reduction of damage probabilities is observed when dissipators are included compared to only the portal frame structure for severe or complete damage states. This is explained because the inclusion of the design in the inelastic range is more significant, and the additional damping, stiffness, and resistance of the dissipators contribute significantly to the structure.
Choosing one dissipation system over another implies a precise analysis of each project. This requires studying other variables in finances, construction, and device availability, among other aspects not considered in this article. Therefore, it is impossible to infer that one energy dissipation system is better than another because they are different structures. However, it can be concluded that the performance of the networks improve significantly by including these devices. For this reason, implementing energy dissipators in formats is recommended to contribute to the facilities’ resilience.
