Although the Mexican engineering community has been interested for more than three decades in hysteretic passive energy dissipation devices (HEDDs), today applications are fewer than 30 including RC and steel buildings (Tena-Colunga, 2007). Applications in other countries worldwide are not much higher, taking aside leading countries in applications, such as Japan. The relatively small number of applications is directly related to the absence of global design parameters and detailed guidelines in official building codes of Mexico and other countries. The absence of such recommendations are due to the lack of enough studies available in the literature (Mexican and worldwide) focused on defining global design parameters that could be easily inserted in a traditional seismic building code format such as those found in Mexican Codes (for example, Tena-Colunga et al., 2009) or US recommendations (for example, ASCE 7, 2010).

Most previous studies oriented to define global seismic design parameters considered single degree of freedom systems (SDOFs) (for example, Wu and Hanson, 1987) or SDOFs tuned with simple multi-degree of freedom systems (for example, Ciampi et al., 1995; Ramírez et al., 2001; Ruiz and Badillo, 2001; Vargas and Bruneau, 2009). In several studies, equivalent design procedures for structures with HEDD using a supplemental viscous damping approach were validated (for example, Wu and Hanson, 1987; Scholl, 1993; Foti et al., 1998; Ramírez et al., 2001; Benedetti et al., 2014) and/or displacement-based design procedures (for example, Ramírez et al., 2001; Chen et al., 2007; Symans et al., 2008).

There are only few studies available where global design parameters suitable to be implemented in building codes have been assessed (Vargas and Bruneau, 2009; Tena-Colunga and Nangullasmú-Hernández, 2015). In order to ensure an elastic behavior for the frame, Vargas and Bruneau (2009) associated the design base shear with: (a) the elastic stiffness ratio (α) between the bracing-HEDD and the steel moment frame and (b) the peak ductility capacity (μ_d) for the hysteretic energy dissipation device. Vargas and Bruneau (2009) also assessed the overstrength factor (Ω) related to the yielding of the HEDDs while the frame remains elastic. Tena-Colunga and Nangullasmú-Hernández (2015) conducted a parametric study devoted to the evaluation, using pushover analyses, of the seismic behavior of low to medium rise regular reinforced concrete intermediate moment-resisting frames (RC-IMRFs) with hysteretic energy dissipation devices (HEDDs) mounted on chevron steel bracing. The main purpose of the described study was to assess global seismic design parameters that could be easily inserted in the seismic design philosophy of Mexican codes. Different elastic stiffness ratios between the moment frame system and the whole structure (α) and between the hysteretic device and the supporting braces (β) were considered, among other relevant structural parameters, as described in following sections.

The research reported in this paper is directly related to the one reported by Tena-Colunga and Nangullasmú-Hernández, as it is a parallel study for a different structural system and material: special moment-resisting steel frames (SMRSFs). The purpose of the study was to assess and validate the range for global design parameters and stiffness balances under study for SMRSFs, which do not necessarily have to coincide with those for a RC-IMRFs. However, the global code-oriented design strategy is general, and therefore, the design strategy and methods coincide with those reported in Tena-Colunga and Nangullasmú-Hernández (2015).

Hysteretic energy dissipation devices are considered mounted into SMRSFs using chevron bracing. The following design hypothesis were done: (a) SMRSFs were designed to carry gravitational loads plus their share of seismic lateral loads and respond essentially in the elastic range, but designed according to the capacity-design approach for ductile frames of Mexican codes (NTCEM-04, 2004), which are similar to those specified for special moment frames in standards such as AISC 341 (2010), (b) the supporting chevron system should remain essentially elastic under seismic loading also and, (c) HEDDs are designed under seismic loading to behave inelastically up their maximum local displacement ductility capacity μ. Therefore, the structural system under lateral loading is composed by the SMRSFs and the chevron-bracing-HEDDs system. SMRSFs should be able to carry the gravitational loads after a strong earthquake (remaining essentially elastic) and have additional strength and deformation reserve capacity, whereas HEDDs should respond in the inelastic range of response as structural fuses and the supporting chevron bracing should remain essentially elastic.

For the parametric study, buildings from 5 to 25 stories were considered with a typical plan layout depicted in Figure 1. The floor system is composed of steel decks. In order to balance the lateral stiffness and strength of the floor system in the main orthogonal directions, a chessboard distribution of supporting beams and steel sheets was proposed (Figure 1). HEDDs are mounted on chevron bracing in perimeter frames, as shown in Figure 1.

Concurrent bracing at building corners were considered. The reasons behind it are the following: (a) it is commonly used in buildings in Mexico City and worldwide (Figure 2), because architects prefer to use central bays of perimeter frames for the building’s access and, (b) this configuration is the worst-case design scenario for this system from a seismic design perspective, because it favors a high concentration of axial loads in columns, especially due to biaxial dynamic effects. Hence, if the concentration of axial loads in corner columns is limited within reasonable limits while using the proposed design procedure, then, the design strategy should be much more efficient in other bracing configurations.

The angle of the chevron bracing with respect to a horizontal plane (θ) was studied because in a previous parametric study (Tena-Colunga, 2002) it was found that the efficiency of HEDD on chevron devices depend on this parameter. Two angles were considered, θ = 40° (story height h = 336 cm) and θ = 45° (h = 400 cm), taking into account typical story heights for regular buildings in Mexico.

Therefore, building models had 5, 10, 15, 20, and 25 stories. Cross sections of beams (W sections), columns (square box sections), and braces (square box sections) were varied along the height (that is, different sections were used for beams, columns, braces, and HEDDs through the height of the studied frames) as schematically depicted with different colors in Figure 3. As a consequence of this design strategy, higher sections were used at the lower levels and relatively smaller sections were used at the top levels due to variation of story shears. It can be noticed also in Figure 3 that, in order to minimize the potential formation of intermediate soft stories (stiffness and/or strength), cross sections of beams and columns are changed at different stories with respect to cross sections of braces. The designed cross sections are reported in detail elsewhere (Hernández, 2015).

Four structural parameters (three stiffness-related) were considered with the intention of assessing their extent of application under the general design hypothesis previously mentioned.

The first stiffness parameter is α, defined as the ratio between the global average elastic lateral stiffness for the frame (K_frame, as evaluated as base shear over top-story displacement) with respect to the global average lateral stiffness of the whole frame-bracing-HEDD system (K_total): (1)α=KframeKtotal

Three values of α were selected: α = 0.25, where the SMRSFs are more flexible than the bracing-HEDD system, α = 0.50, where the SMRSFs and the bracing-HEDDs system have the same elastic lateral stiffness and, α = 0.75, where the SMRSFs are stiffer than the bracing-HEDD system.

The second stiffness parameter is β, the ratio between the elastic stiffness for the HEDD (K_ELD, Figure 4) with respect to the elastic lateral stiffness of the supporting chevron braces (K_diag): (2)β=KELDKdiag

Three values of β were chosen: β = 0.75, β = 0.50, and β = 0.25. When β < 1.0, the bracing system is stiffer than the HEDD, a desirable requirement for most available HEDDs (for example, ADAS, TADAS, etc.) supported in chevron bracing.

The third stiffness parameter is the post- to pre-yielding stiffness ratios for the HEDD (K₂/K_ELD, Figure 4). Bilinear ratios for elastic-perfectly plastic devices (K₂/K_ELD = 0.0) and hardening devices (K₂/K_ELD = 0.03 and 0.05) were studied. Most structural engineers frequently model HEDD as bilinear, elastic-perfectly plastic. However, it has been observed from most reported experimental research studies that most HEDD develop a secondary stiffness different from 0 (for example, Whittaker et al., 1989; Tsai et. al, 1993). Important differences in peak structural responses may be obtained considering a secondary stiffness different from 0 with respect to an elastic-perfectly plastic idealization for the HEDD for large ductility demands, as suggested in previous parametric studies (Tena-Colunga, 2002; Tena-Colunga and Nangullasmú-Hernández, 2015).

The angle of inclination of the chevron bracing with respect to a horizontal plane (θ) was also assessed for the following reason. In a good design under the structural fuse design concept, the non-linear action should takes place only in the energy dissipator (Figure 4), whereas the bracing system should behave elastically. Therefore, if the “equivalent diagonal” concept is used for the bracing-HEDD system, under a chevron bracing mounting (Figure 2), one can easily demonstrate that the equivalent instantaneous effective stiffness of each brace in the non-linear range (K_eqnl) is (Tena-Colunga, 2002): (3)Keqnl=Kbrace[KELD+K2(μ−1)]KELD+K2(μ−1)+2μKbracecos2θ where K_brace is the elastic stiffness of the bracing element, K_ELD is the elastic stiffness of the HEDD (Figure 4), K₂ is the postyielding stiffness of the HEDD (Figure 4) and μ is the instantaneous ductility demand of the HEDD (Figure 4). It is clear from the observation of Eq. 3 that the angle of the chevron bracing has an impact on the equivalent instantaneous non-linear effective stiffness and, for a given ductility demand μ for the HEDD, the equivalent non-linear effective stiffness gets smaller as θ is smaller. A smaller effective non-linear stiffness leads to smaller axial forces in the brace-HEDD system and therefore, to smaller axial and shear forces transmitted to beams and columns and hence, leading in theory to smaller demands. That was the purpose of studying θ, to discern if smaller yielding of beams and columns can be obtained for a smaller θ for a given V/W ratio.

In the conducted parametric study that is reported in detail elsewhere (Hernández, 2015), for each height and θ angle, there are 27 different models or combinations of parameters α, β, and K₂/K_ELD (Figure 5). Thus, 135 different models were designed for each θ angle. Then, 270 different models were strictly designed according to code recommendations and analyzed to carry out this parametric study.

All models were designed under the structural fuse concept using a procedure adapted to a building seismic code format using the initial lateral elastic stiffnesses of the resisting elements. From a code-oriented design perspective, the effective design spectral acceleration, which can also be understood as a V/W ratio, should be obtained using the procedures outlined by the code (this is, from the design spectrum, Figure 6). However, as it is shown later when discussing non-linear dynamic analyses, an effective design base shear V/W = 0.10 is obtained for the design spectrum for the lake-bed zone of Mexico City when the highest ductility (Q = 4) and overstrength (R) factors of Mexico’s Federal District Code (NTCS-04, 2004) are considered, as the resulting reduced design spectrum is flat for almost all periods (0 < T < 2.8 s, Figure 14). As a consequence of this fact, all models of this parametric study were finally designed for a yielding design base shear of 10% of the total weight (V/W = 0.10), which provides a common frame of comparison. The target (objective) maximum local displacement ductility (μ) for all hysteretic energy dissipation devices was assumed to be 10 (μ = 10), independently of their location along the building height.

The code-oriented design procedure is schematically synthesized in Figure 6 and reported in detail elsewhere (Hernández, 2015). The proposed method is an adaptation for SMRSFs of the design procedure reported for RC-IMRFs in Tena-Colunga and Nangullasmú-Hernández (2015).

It can be observed that for the proposed design method, a preliminary decision is done on the elastic lateral stiffness ratio α, this is, a proportion of the global elastic lateral frame stiffness with respect to the global lateral stiffness of structure (step 3, Figure 6). Then, the design base shear that the moment frame should resist behaving essentially elastic in absence of the bracing-hysteretic device system is computed (step 4, Figure 6) and distributed along the height (step 5, Figure 6). Then, a preliminary design for the frame sections considering vertical loads plus their share of lateral loads in absence of the bracing and the structural fuses could be done (steps 6 and 7, Figure 6). It is worth noting that the proposed α ratio must be checked at all stories (storey shear over story displacement), where cross sections for columns, beams, and braces are changed. The frame must satisfy all serviceability state limits under gravitational loads also.

Once frame sections have been pre-designed, the lateral stiffness for the moment frame (K_frame) could be estimated using any accepted method already available in the literature (step 8, Figure 6). Then, the stiffness of an equivalent global axial element (nK_eq) is assessed (step 9, Figure 6), where n is the number of braces required to support the HEDDs. The stiffness K_eq considers the elastic stiffness for each brace (K_brace) and the effective stiffness of the HEDD at the objective maximum local displacement ductility μ, K_EFD (Figure 4). A decision should be made about the design parameters for the HEDDs, mainly β, μ, and η = K₂/K_ELD (step 10, Figure 6). The needed elastic stiffnesses for the bracing and the HEDDs are defined based upon these hypotheses (step 11, Figure 6). The bracing is then pre-dimensioned taking into account stiffness (step 12, Figure 6) and strength criteria. An elastic behavior for the braces must be warrant when the HEDDs would develop their maximum displacement ductility μ, designing the braces also to have a reasonable safety factor for buckling (Hernández, 2015; Tena-Colunga and Nangullasmú-Hernández, 2015). Once the braces are pre-designed, the required lateral stiffness of the HEDDs is assessed (step 13, Figure 6) and the devices predesigned (step 14, Figure 6).

Once the preliminary design for all members has been done, an ad hoc elastic analytical model for the whole building is built (step 15, Figure 6), using the equivalent secant stiffness K_EFD at the target ductility μ for the HEDD (Figure 4), that should be analyzed again under gravitational loads and the lateral load profile corresponding to the design base-shear (step 16, Figure 6). 3D elastic models were built in ETABS (CSI-2005, 2005) for this purpose. All elements should be revised again for strength and deformation in the described order (step 17, Figure 6) corresponding to a capacity-design approach to warrant a design according to the structural fuse concept: HEDDs first, then braces (weakest HEDDs—strong brace), then beams, columns, and panel zone (weak beam—strong column—strongest joint). The entire structure should also be reviewed to comply with lateral deformations limit states according to code recommendations or other performance objectives. A final check is also needed to evaluate how close are the obtained values of α (particularly at each story where changes of cross sections were proposed) and β (for each HEDDs-bracing) with those initially assumed in the design. If the design procedure is successful, only HEDDs would respond in the non-linear range, having the beams of frames as reserve source of energy dissipation if expected demands are exceeded.

Non-linear static analyses (pushover) were conducted for each designed model under study using Drain-2DX (Prakash et al., 1992). All elements (columns, beams, braces, and HEDDs) were modeled to monitor the possibility of developing a non-linear behavior, as described in detail elsewhere (Hernández, 2015). Beams and columns (A50 steel) were modeled as beam-column elements with lumped plasticity at element ends and assuming and elastic-perfectly plastic behavior. Beams were modeled as beams type (type 1) and columns as steel columns type (type 2) according to Drain-2DX manuals. Yield interaction surfaces were computed for each cross section using basic plasticity principles for steel elements. Braces (A36 steel) were modeled as truss elements with yielding in tension and elastic bucking in compression. Buckling load capacities were assessed according to NTCEM-04 (2004). HEDDs (A36 steel) were modeled using equivalent beam-column elements connected to the chevron braces and beams, according to a previously validated procedure which is outlined in detail elsewhere (Tena-Colunga, 1997). P-Δ effects were considered in the analyses. For simplicity, lateral load distribution profiles based upon the first mode of vibration were used in the pushover analysis. This was done to have a general framework of comparison, taken into account that: (a) building height ranges from 5 to 25 stories, (b) the modal mass associated to the fundamental mode is higher than 60% for most buildings, and (c) for similar buildings, it was demonstrated (Godínez and Tena, 2014) that modal pushover analysis is not a suitable method to determine average global design parameters for the structural system of interest, which strongly depends on the first mode of vibration. In addition, for this system and for the purpose of assessing global design parameters only, higher mode effects were found to have a reduced impact in assessing peak lateral drifts even for the upper stories, where larger differences are expected when comparing the results obtained with pushover analyses based upon the fundamental mode with those obtained with modal pushover analyses as presented in the literature (i.e., Chopra and Goel, 2002). Peak effective lateral forces were not importantly modified either when considering higher modes. Of course, higher mode effects are very important in the non-linear dynamic response of multi-story and very tall buildings, something which it is out of the scope of the present study.

The principal results obtained from pushover analyses were: (a) normalized story and global lateral shear vs drift curves (V/W vs Δ) and, (b) yielding mapping related to the load step where the collapse mechanism occurred. The following information was processed from the story and global shear vs drift curves: (a) overstrength factors (Ω), (b) ductility reduction factors (Q), (c) apparent peak story and global ductility capacities, (d) equivalent story drift at yielding (Δ_y), (e) peak story drifts (Δ_max).

Once all inelastic demand mappings and their relation with the ductility demand curves for the HEDDs for all considered models were carefully studied, the information obtained from non-linear static analyses was used to assess global design parameters, as briefly discussed in following sections.

In order to help illustrate that the proposed design procedure is reasonable for strong ground shaking, non-linear dynamic analyses were conducted for the models subjected to artificial records for a postulated M_s = 8.1 subduction earthquake similar to the September 19, 1985, Michoacán earthquake in the lake-bed region of Mexico City (soft soils). Artificial records were obtained for stations installed after the 1985 earthquake where recorded ground motions are usually stronger than the well-known SCT station.

One of these stations is Station 56, located at the Rome District, where many midrise buildings collapsed during the 1985 Earthquake. The artificial S56-EW record and its corresponding response spectra for 2 and 5% equivalent viscous damping is depicted in Figure 13. In the non-linear dynamic analyses, an equivalent viscous damping ratio ζ = 2% was used for all models, as values of ζ ranging from 2 to 4% have been measured experimentally in existing steel buildings. The site period for S56 is T_s = 2.2 s, where it has a peak pseudo-acceleration close to 1.5 g for ζ = 5%, which in fact it would surpass in 25% the peak pseudo-acceleration 1.2 g considered in the elastic design spectrum (Q = 1) for that region in Mexico’s Federal District Code (NTCS-04, 2004), as depicted in Figure 13B. However, if one considers the assumed viscous damping ratio ζ = 2% for the steel structure, the peak pseudo-acceleration is S_a = 2.32 g, which surpass in 93% the design acceleration for the elastic design spectrum (Q = 1), as also depicted in Figure 13B. Therefore, this artificial record is a good test for the 25-story models, which computed fundamental periods ranged from T = 2.14s to T = 2.33 s and, according to NTCS-04, these models should be designed with an effective design base shear V/W = 0.10 for a site with T_s = 2.2 s if Q = 4 and R = 2 are assumed in the design process (Figure 13B).

For that reason, the following 25-story models were considered in the simulations with S56-EW record when β = 0.75, K₂/K_ELD = 0.03 and θ = 45°: (a) α = 0.25, (b) α = 0.50, and (c) α = 0.75. As expected, because of their corresponding inelastic demand mappings obtained from pushover analyses for a target ductility μ = 10 in the HEDDs (Figures 9G–I, respectively), the most demanded model under the action of S56-EW record was the model where α = 0.75, which results would be discussed as follows. The peak inelastic demand mapping obtained from the non-linear dynamic simulation under the action of the S56-EW record for the model where α = β = 0.75 are depicted in Figure 14 and compared with the one obtained from pushover analyses. It is observed from Figure 14A that most of the inelastic action is developed at the HEDDs, which work efficiently, as it can also be observed in the normalized hysteresis curves for each HEDDs depicted in Figure 15. Despite the fact that the response spectrum for the artificial S56-EW record for ζ = 2% considerably surpasses the elastic design spectrum (Figure 13B), the non-linear action in beams is moderate in stories 6–17 (Figure 14A). Therefore, the design procedure was found adequate, as after the capacity of the HEDDs is surpassed by the extraordinary shaking, the next elements that work non-linearly are the beams, as assumed in the proposed capacity-design method. In fact, it can be observed from the pushover results that for this model, beam yielding was expected at intermediate levels (Figure 14B). Also, although magnitudes of peak inelastic demands are different, one can observed that the topological locations of peak demands for HEDDs and beams from the pushover analyses (Figure 14B) reasonably coincide with those obtained from the non-linear dynamic analysis under the action of S56-EW record (Figure 14A). Thus, this fact confirms that for well-designed low-rise and midrise regular buildings with HEDDs under lateral loading, there is a characteristic mechanics of inelastic deformation that mostly depends on the first mode of vibration and less on higher modes and the characteristics of the ground motion. Therefore, if capacity-design methods are used to define clearly the desired yielding sequence of structural elements for an ultimate or collapse mechanism, then pushover analyses based upon the first mode of vibration are reasonable enough to assess global design parameters (ductility capacity, overstrength, drifts, etc.) from a code-oriented design viewpoint for low-rise and mid-rise regular buildings with HEDDs as structural fuses.

Finally, it is worth noting that peak story dynamic drifts were near 1% (stories 11–14, not shown) and despite that, the model (Figure 14A) and particularly the HEDDs (Figure 15) dissipate an important amount of energy in a stable way.

Many observations can be made from the extensive and detailed parametric study that was conducted and reported to obtain global parameters for a code-oriented design of buildings structured with SMRSFs with HEDDs. Because of space constraints, only few additional comments from those already available in previous sections are made. From the results of the study one can conclude that an important global ductility capacity for the whole system (Q in terms of Mexican codes) could be achieved for SMRSFs with HEDDs as structural fuses without experiencing important inelastic response in the frame elements (beams and columns) for the whole range of parameters under study: (a) 0.25 ≤ α ≤ 0.75, (b) 0.25 ≤ β ≤ 0.75, and (c) μ ≤ 10.

The following general conclusions can also be drawn for SMRSFs with HEDDs as structural fuses when designed with a code-oriented procedure for a yielding base shear of 10% of the total weight (V/W = 0.10):

For a given value of α, inelastic demands at HEDDs tend to increase as β decreases. In fact, as the number of stories increases, some beams start to yield as β decreases.

Non-linear dynamic analyses were helpful to illustrate that the proposed code-oriented capacity-design procedure is reasonable for strong ground shaking typical of soft soils of Mexico City and related to current Mexico’s Federal District design spectra for such sites. Therefore, if capacity-design methods are used to define clearly the desired yielding sequence of structural elements for an ultimate or collapse mechanism, then pushover analyses based upon the first mode of vibration are reasonable enough to assess global design parameters (ductility capacity, overstrength, drift limits, etc.) from a code-oriented design viewpoint for low-rise and mid-rise regular buildings with HEDDs as structural fuses.

Despite the fact that the parametric study was broad and comprehensive, further studies are required from a code-oriented perspective, for example: (a) additional non-linear dynamic analyses for completely code-designed buildings with SMRSFs with HEDDs using the proposed global design parameters in this study when subjected to a comprehensive set of acceleration records typical of soft and firm soils and related to the corresponding soil site and design spectra, (b) a broader range for the design base shear (V/W) and its impact on the studied stiffness ratios and global design parameters, (c) other sections for columns (for example, W sections, which are commonly used in many countries, for example, the United States of America), (d) other structural detailing (non-ductile steel frames) and their impact to the global response, and (e) the impact of the flexibility of the moment connection (partially restrained moment connections).

HH-R engineered the structural models and carried out the numerical non-linear analyses. AT-C organized the research group, defined the design procedure, and supervised the design of models and numerical non-linear analyses. All authors read and approved the final manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.