1. Introduction
Recent earthquakes have exposed the high vulnerability of several ductile moment-resisting frames due to their soft story located either at ground level or several stories above ground [
1,
2,
3,
4]. According to [
1], the contributing factors to soft-story failures are low lateral stiffness and the strength of one or more resisting frames within a given story level, the ground-motion amplitude and duration, as well as large plastic deformations at column joints that induce lateral instability due to
p-
δ effects. This collapse mechanism has been a problem for a long time not only in Mexico City [
1,
5] but also in other seismic regions worldwide, e.g., [
6,
7]. One example of the high seismic vulnerability of soft-story framed structures was displayed in the Michoacan 1985 Mw8.1 earthquake, where approximately 37% of the collapsed buildings (78 out of 210) in Mexico City involved the failure of one or more soft stories. Similarly, 44 multi-story buildings collapsed due to the 2017 Mw7.1 Morelos–Puebla earthquake, 40 of which were built before 1985; a total of 25 out of the 40 presented a soft-story mechanism (
Figure 1) according to Galvis et al. [
8].
Therefore, it is essential to undertake actions to control excessive deformation demands in buildings that are prone to this failure mechanism. There are different strategies to improve the structural safety and resilience by implementing innovative seismic protection devices in new and existing structures, e.g., [
9]. These technologies are based on passive control, isolation, and damping systems, which could be used to avoid damage in structural and non-structural elements. The most popular damping devices are those dependent on displacement and velocity; they have advantages but also limitations. However, a third category that is not as common is based on inertial dampers, in which the response is controlled by adding inertia to the structure as investigated here. Therefore, it is considered that the present study increases the understanding of the advantages and disadvantages of inerter dampers with respect to other seismic protection devices as described in the body and conclusions of this work. The inerter damper (ID) is a control mechanical device that has gained popularity to passively suppress and control vibrations in structures [
10,
11,
12,
13,
14,
15,
16,
17]. An ID is a mechanical response modification device whose resisting force is proportional to the relative accelerations between their terminal nodes [
18]. An ID relies on the inertance property
with units of mass [
18,
19] that can scale up independently of the weight of the device. This inertance property can be achieved, for instance, using supplemental flywheels that are engaged in motion through rack-and-pinion or ball-screw systems [
10,
11,
20]. The whole mechanical arrangement is called the inerter damper. If a yielding single-degree-of-freedom (SDOF) system with strength
and yield force
is provided with an ID supported on a stiff Chevron frame (
Figure 2), the governing equation is [
17]:
In Equations (1) and (2),
is the ground acceleration,
is the nominal damping ratio,
is the nominal frequency with mass
and pre-yielding stiffness
, and
is the ratio of pre-yielding to post-yielding stiffnesses ratio.
is a parameter, which introduces the hysteretic behavior in the columns as given by the so-called Bouc–Wen model, where
,
, and
are the parameters defining the hysteretic loop [
21,
22], and
is the yield displacement. Finally,
is the inertance ratio or apparent mass ratio [
18]. This coefficient
can assume any desired value with sufficient size and number of supplemental flywheels engaged in motion through rack-and-pinion or ball-screw systems [
10,
11,
20].
Given the effectiveness of IDs to control the seismic displacements, we study the seismic risk of weak first-story RC structures with IDs at their ground level to control lateral displacements. The study uses narrow-band ground motions, as those recorded at several sites located in Mexico City. Narrow-band ground motions in Mexico City are known to generate a significant amount of energy content within a narrow range of structural periods [
23]. When structural systems are subjected to narrow-band ground motions with a long duration, their capacity to dissipate energy is decreased as the structural period tends to zero [
23]. The amount of energy input to the structural systems due to narrow-band ground motions is substantial as compared to the energy introduced by wide-band ground motions [
23]. For the scope of this study, a set of three regular RC buildings with masonry infill walls above the first story was used [
24]. These buildings were designed according to the NTCS-1976 [
25]. Before carrying out a retrofitting or rehabilitation, a preliminary and adequate seismic assessment is a fundamental phase in order to retrofit existent reinforced concrete structures with a suitable rehabilitation strategy [
26]. Therefore, herein, we hypothesize that the performance of weak first-story RC structures can be improved, and their likelihood of experiencing a soft-story collapse during narrow-band seismic excitations can be significantly reduced by using IDs at their ground level especially for large ground motions, i.e., seismic intensities associated to limit state of incipient collapse. This study includes records of large earthquakes (1985 and 2017), since these events caused important damage to several low-to-medium rise weak first-story structures located on the Mexico City basin. As it is known, a real building has many degrees of freedom, which somewhat contributes to its irregularity, e.g., [
27]. In the present study, the lateral–torsional and the inertial-damper effect coupling is neglected; this could lead to an overestimation of the effectiveness of the inerter. Therefore, a future study investigating the influence of location and direction of the inerter dampers for irregular buildings subjected to horizontal seismic excitations is recommended.
2. Designed Buildings Considered
Regular structures (symmetrical) in plan and elevation are adopted from Ruiz-García and Cárdenas [
24] for this study. These buildings are considered first weak-floor structures, not only because no infill walls are provided at the first floor (leading to a much smaller rigidity in that level), but also because they are adopted as representative of typical structures with this failure mechanism observed during a field-reconnaissance report after the 2017 Mw7.1 Morelos–Puebla earthquake [
24].
They are assumed as office buildings with a height equal to 4.5 m from the ground to the first floor, and an interstory height equal to 3 m for the other levels. They have a squared plan view with three bays in each direction.
The plan view indicated in
Figure 3 is common to the three considered buildings, except that they have a different number of stories: 4, 6, and 8 stories as shown in
Figure 4. This figure also shows the exterior infill masonry walls with openings and the interior infill masonry walls with no openings; no walls are provided on the first floor. For the openings, a simple configuration is considered just to distinguish between façade and interior walls [
24]. Extensive detailing of the reinforced concrete, column, and beam locations; sections and reinforcement steel arrangements; as well as other technical information are thoroughly described and can be inspected by interested readers in [
24] and other references [
28,
29]. The Mexico City 1976 version of the Technical Norms for Seismic Design [
25] is employed in the design using a so-called Seismic Behavior Factor
with values of 4 and 6 (i.e., a factor related to demand reduction by ductility that is high and very high for the selected values, respectively). More details are available in the study conducted by Ruiz-García and Cárdenas [
24], who compared the seismic performance of these buildings versus the performance of their counterparts, but as if they were designed with the updated version of the code, which expressively includes design regulations for first weak-floor buildings. It is noteworthy to mention that Ruiz-García and Cárdenas [
24] found, in general terms, that the new regulations successfully improve the seismic behavior and are effective for the seismic demands expected in Mexico City for the ground motions considered. Nevertheless, it is also concluded that the updated regulations delay the formation of the weak-story mechanism, and that the peak drift (possibly damage) could be migrated to the upper levels. To draw a parallel between the results presented for the overall seismic behavior of the buildings designed with the old and updated Mexico City seismic provisions by Ruiz-García and Cárdenas (in terms of normalized shear versus roof drifts) and simple material behavior, it is as if the first weak-floor structures designed as per the 1976 regulations would behave as confined or unconfined concrete (in compression), and as if those designed as per the updated regulations would behave as reinforcement steel of different grades (in tension).
In the present study, we rather considered the buildings as representative of existent structures designed as per the 1976 regulations, which could be retrofitted with inerter dampers to improve their seismic behavior as urged by Ruiz-García and Cárdenas [
24] and other authors regarding this type of existent buildings.
2.1. Dynamic Analysis of the Buildings
Nonlinear dynamic analyses were performed using OpenSees [
30] modelling on only half of each building due to their symmetry in the plan (see the rectangle depicted with the dashed lines in
Figure 3). Hence, one exterior frame and one interior frame are considered linked through rigid frame elements (i.e., two-dimensional centerline models) implying that each floor has the same lateral deformation (i.e., a rigid diaphragm assumption) as can be observed in
Figure 4, where it can also be observed that the IDs are placed only in the exterior frames at the depicted locations. The reinforced concrete elements of the model (columns and beams) were modelled with the fiber section, since the most promising models for the nonlinear analysis of reinforced concrete members are, to date, flexibility-based fiber elements. In order to consider the stress–strain behavior of the concrete element under cyclic loads,
Concrete02 material was implemented, which allows for the construction of a uniaxial concrete material object with tensile strength and linear tension softening behavior specified by the Kent–Park stress–strain relationship [
31]. Similarly, the Giuffre–Menegotto–Pinto steel material object with isotropic strain hardening model was used to model the reinforcement steel fibers, which is identified in OpenSees as
Steel02 material [
32]. Total weights
, first-mode periods
, and percentage of the effective mass of the required modes to accomplish the minimum 90% of the total mass required in the standards employed are listed in
Table 1 for each analyzed building. All buildings have a fundamental period
near unity as shown in
Table 1. In
Table 1, each model designation is similar to that in Ruiz-García and Cárdenas [
24], i.e., the label EXQY (column 1 in
Table 1) denotes the number of floors for the X and
factor for Y (e.g., E4Q6 corresponds to a 4-story building designed for a
factor equal to 6). As explained before, this factor is linked to the ductility behavior of the buildings and varies for different versions of the Mexico City seismic provisions; it depends on different aspects. In this study, it is only pointed out that different
factors lead to different designed structural elements. The interested reader is referred to Ruiz-García and Cárdenas [
24] and references therein [
28,
29] for more details.
As it is observed from the previous table, the fundamental mode participates with more than 80% of the response; however, higher modes are also considered in the computation of the lateral response of the structure. As can be observed from
Table 1, the fundamental mode controls the response as each one of the higher modes contributes with less than ~10% in most of the cases. It is expected that these characteristics do not vary when the inerter dampers are allocated in the structures, as they are located in the first floor; besides, the mass of the damper is allocated over the chevron frame, which is supported directly by the foundation structure. Therefore, the mass and the stiffness of the chevron frame will not change either the vibration periods or the modal shapes of the structure, and the same applies for the effective mass associated to each mode. Although issues related to the stored energy in the inerter damper and subsequent transfer forces to the supporting system are ignored in the present study, they could be investigated in future research.
2.2. Modeling of the Inerter Damper
Figure 5 shows a schematic representation of an inerter with the main parameters used to establish the numerical model as proposed by Málaga-Chuquitaype et al. [
15]. In
Figure 5, two nodes connected through a rigid link with the length denoted as
are observed. A rotational inertia or angular mass denoted as
is assigned to the rotational degree of freedom (Node A). This is done so that the relative lateral displacement between the two nodes (A and B) translates into a rotation in Node A. Node A corresponds to the pivot of the flywheel. On the other hand, Node B represents the pinion gear. Depending on the desired value of the inertance property obtained as
,
and
can be varied. The so-called apparent mass ratio
can be determined for the considered building as
, where
is the mass of story
i, and
is the mass of the whole structure (total mass). For the scope of this study, every building has
and
is either equal to
or
. Such values were calculated considering the placement of the inerter dampers.
4. Earthquake-Induced Response of Buildings with Inerter Dampers
To evaluate the influence of ground motions (e.g., intensity and frequency content) on the seismic response of weak first-story RC structures with IDs at their ground level due to narrow-band earthquakes in Mexico City, the earthquake-induced response demands were inspected. First, a SDOF-yielding system with inerter dampers is subjected to narrow-band seismic motions, and its response is compared to an analog system without inerter dampers. Second, the earthquake-induced response demands for weak first-story reinforced concrete (RC) structures for two limit states were examined: (1) seismic intensities associated with the limit state of incipient collapse and (2) seismic intensities associated with the limit state of damage limitation. The former limit state (i.e., incipient collapse) is associated with rare extreme events capable of generating high seismic demands, for which collapse should be prevented to avoid major structural failures and fatalities, although significant damage and residual deformations could occur and may lead to operation interruption and important retrofitting tasks. The latter limit state (i.e., damage limitation) is associate with moderate seismic demands that may occur several times during the life service of the structure, for which only minor damage should occur so that the structure operation and occupancy is not interrupted.
Despite that the studied structures were designed according to the Mexico City 1976 version of the Technical Norms for Seismic Design [
25], the current accepted limit states are prescribed in the 2017 edition of the Technical Norms for Seismic Design (NTCS-2017) for Mexico City, which considers the spectral intensities of the limit state of collapse to be associated with a return period of 250 years and the intensities of the limit state of damage limitation to be associated with a return period of 10 years.
This situation obeys to the fact that, currently in Mexico City, there is an important number of structures built in the 1980s; however, due to the scarce economic sources available, it is not feasible to demolish and rebuild such structures. Instead, retrofitting strategies need to be proposed.
4.1. Seismic Response of Sdof Systems for Synthetic Ground Motion
Figure 9 presents the maximum seismic inelastic displacement of a bilinear (
) SDOF system with inerter dampers as described by Equations 1 to 2 with
(pink solid lines), which is compared against the response of the same simple bilinear SDOF system but with
(gray solid lines). The systems are subjected to the referred synthetic ground motions (station CH84,
Figure 8a; station SCT,
Figure 8b). The studies on yielding concrete structures to identify parameter values of the Bouc–Wen model (e.g., [
50,
51] Kunnath et al. 1997; Goda et al. 2009) are the basis for adopting 0.05, 0.95, and 2 for
,
and
, respectively. Moreover, a pre-yielding stiffness equal to
is stipulated as well as a post-yielding stiffness
and
as strength. Accordingly, assuming
= 0.05, the yield displacement
can be determined as [
17]
. The other parameter to fully define the structural properties of the SDOF system is the normalized strength
.
Figure 9 shows the efficiency of the SDOF system with inerter dampers on a stiff frame in reducing the maximum inelastic displacements (solid pink lines for the SDOF system with inerter dampers, both sites). Therefore, introducing inerter damper systems seems to be a very effective measure to reduce peak inelastic responses when they are subjected to narrow-band ground-motions.
4.2. Influence of Seismic Intensities Associated to Limit State of Collapse
Figure 10 shows a comparison of the height-wise distribution of median peak story drift (left), normalized peak floor acceleration with respect to the ground (middle), and normalized peak shear demands with the building weight (right) for the 4-story building (top), 6-story building (middle), and 8-story building (bottom) with Q = 6 under seismic intensities associated to the limit state of collapse at site CH84 (
= 1.4 s), i.e., the drifts (relative displacement between two consecutive stories) should be smaller than the limiting 1.5% drift (NTCS-2017) to be code-compliant. In
Figure 10, blue lines show responses for the original buildings without inerters (i.e., O), and orange and yellow lines show the responses for the buildings retrofitted with inerters (i.e., ID) for
σ equal to 0.7 and 0.5, respectively. Under these intensities associated to the limit state of collapse, the buildings are considered to have a non-linear behavior. In
Figure 10 (left), it can clearly be seen that the drift peak demands are concentrated in the first story, which is typical of a weak first-story mechanism. Notice that the largest drifts are concentrated in the first story for the case without inerters that occurs in the 4-story building (the lowest building); this behavior, also reported by Ruiz-García and Cárdenas [
24], was partially attributed to the contribution of the higher modes, which may delay the formation of the weak first-story mechanism in the tallest building [
24]. Nevertheless, it is noteworthy that the addition of IDs does not eliminate the weak story. It rather controls the problem induced by the soft-story mechanism from a dynamic standpoint provided that an adequate apparent mass ratio is selected. However, providing an ID device with a larger apparent mass ratio
(e.g., values larger than 1 for the cases under study) leads to damage in the upper stories as indicated below. Therefore, care should be exercised to adequately select IDs to control lateral displacements.
Figure 10 shows an evident decrease in the median peak story drift as well as in the normalized peak floor acceleration when the buildings are provided with inerter dampers at their ground level (solid yellow and orange lines) versus the original case (solid blue lines). Likewise, similar or reduced median normalized peak shears can be appreciated (depending on the value of
) for buildings with inerter dampers with respect to the original buildings. Therefore, introducing inerter dampers can be an effective measure to reduce peak responses concentrated in the first story when buildings are subjected to narrow-band ground motions. However, inerter dampers placed at their ground level could cause drifts to migrate to upper stories because of the backstay effect introduced by the inerter [
52,
53]. This situation is due to the high
values of the inerter dampers (e.g.,
); this can be especially observed in the 6- and 8-story buildings. As mentioned before, the placement of IDs as a seismic displacement-control strategy should be carefully reviewed on a case-by-case basis so that proper parameters are selected to avoid self-defeating results. This migration of larger drifts to upper levels is also reported by Ruiz-García and Cárdenas [
24] when the buildings are designed as per the regulations for first weak story in updated seismic provisions (rather than by retrofitting them with inerters as in this study).
On the other hand,
Figure 11 presents a similar comparison as in
Figure 10 but for the SCT site (
= 1.4 s). It is possible to observe that the original buildings continue to exceed the drift limit value than that observed at the CH84 site (i.e., 1.5%) indicated in the current regulation. However, they present a better behavior (e.g., an evident decrease in the median peak story drift, normalized peak floor acceleration, and normalized peak shear) when the buildings are provided with inerter dampers at their ground level. Therefore, in this case, introducing inerter damper systems is a very effective measure to reduce peak responses along the height of the buildings when they are subjected to narrow-band ground motions associated to limit state of collapse and are designed with Q = 6.
Appendix A contains additional figures for the buildings with
. They exhibit similar trends as those showed in
Figure 10 and
Figure 11 and are not discussed for brevity.
4.3. Influence of Seismic Intensities Associated to Limit State of Damage Limitation
Similarly depicted as in
Figure 10 and
Figure 11,
Figure 12 and
Figure 13 show a comparison of the height-wise distribution of median peak story drift (left), normalized peak floor acceleration with respect to the ground (middle), and normalized peak shear demands with the building weight (right) for the 4-story building (top), 6-story building (middle), and 8-story building (bottom) with Q=6 but under seismic intensities associated with the limit state of damage limitation at CH84 (
= 1.4 s,
Figure 12) and SCT (
= 1.4 s,
Figure 13), i.e., the drifts should be smaller than the limiting values of 0.4% drift when elements not being part of the structural system are not attached to the building (NTCS-2017). Under these intensities associated with the limit state of damage limitation, the buildings are considered to have a linear behavior. For buildings located at CH84 (
Figure 12), it is possible to see a decrease in the median peak story drift (except in the 8-story building) as well as in the normalized peak floor acceleration and peak shear demands when the buildings are provided with inerter dampers at their ground level (solid yellow and orange lines) versus the original case (solid blue lines). Contrarily, for buildings located at SCT (
Figure 13), an increase is seen in the median peak story drift and peak shear demands when the buildings are provided with inerter dampers at their ground level (solid yellow and orange lines) versus the original case (solid blue lines). Therefore, for site SCT and intensities associated with the limit state of damage limitation, the inerter dampers do not offer any tangible benefit compared to the original case; it could even be self-defeating. A possible explanation is that, since for this limit state (damage limitation) structures at site SCT are subjected to low intensities and behave linearly, it may cause a delay in the ID deacceleration leading to a faster inerter movement than that of the structure. Subsequently, the ID pushes the structure so that it moves, while in turn the ID and its support must withstand a larger load, generating larger drifts and shear demands at the first floor. This may not occur when the structure exhibits a non-linear behavior as slightly observed for the 8-story building (see
Figure 11). Therefore, under high seismic intensities (non-linear behavior), the use of IDs could be beneficial, but this may be reversed under low seismic intensities (linear behavior). In any case, since the frequency content and other aspects may play a role, the design should be carefully reviewed for each case as noted before.
Appendix B contains additional figures for the buildings with
. They exhibit similar trends as those showed in
Figure 12 and
Figure 13 and are not discussed here for brevity.
5. Fragility Functions
For a given engineering demand parameter (), its fragility function is defined as the probability that EDP exceeds the demand threshold conditioned on a ground-motion intensity , i.e., . For the scope of the present work, the is considered to be the peak story drift denoted as .
The well-known incremental dynamic analysis (IDA) method [
54] is used herein to obtain the fragility functions. For this aim, the ground motions were incrementally scaled at constant values of the pseudo-accelerations
, where
is the fundamental period of the weak first-story RC structure. Then, the probabilities of an
exceeding a test value
were obtained simply by dividing the number of simulations for which
by the total number of simulations.
For the development of the fragility functions, scaling of the ground motions to SA ( = 1 s) ranging from 0.01 to 1.4 g and 0.01 to 0.7 g for sites CH84 and SCT, respectively, was carried out.
5.1. Probability of Failure for Maximum Peak Story Drifts
The probability of failure for the maximum peak story drift demands was evaluated for peak story drifts ranging from 0.001 to 0.04, which accounts for very small or negligible drift demands to very large ones that could trigger secondary effects such as pounding against adjacent structures or lateral instability due to p- effects among others.
Let
represent a structure demand of interest. To separate zero and non-zero
(maximum drift demand), the conditional probability of the demand given a ground-motion intensity
is represented by
where
is the Dirac delta function;
denotes the level deemed null (or practically null) for a structure; all demands that meet
are concentrated in
corresponding to a mass probability denoted as
; and
denotes the probability density function for non-zero displacements. Equation 3 can be used to derive
(i.e., the exceedance probability for a
threshold given an intensity measure value).
5.2. Results of Probability of Failure for Maximum and Residual Displacements
By carrying out incremental dynamic analyses, fragility functions for peak story drifts are computed for each of the structure configurations described in
Section 2.
An illustrative example of the fragility functions of maximum peak story drift conditioned on three ground-motion intensities,
and
, is presented in
Figure 14 for the 4-story building (top), 6-story building (middle), and 8-story building (bottom) with Q = 6. Fragility functions are presented in
Figure 14 for sites at CH84 (left) as well as for SCT (right). This example illustrates that the ground motion’s frequency content has a huge effect on the peak story drift fragility (right) for structures considering inerter dampers. For instance, it is apparent that the probability that
exceeds a value
as a function of the intensity
is larger for site CH84 (i.e., high-frequency ground motions of approximately 0.71 Hz;
Figure 14 left) than for site SCT (i.e., low-frequency ground motions of approximately 0.52 Hz;
Figure 14 right). Finally, it is observed that, when structures include inerter dampers at their ground level, the probabilities of exceeding certain peak story drift are smaller than for a structure without inerter dampers for large ground-motion intensities (e.g.,
g) at both sites. However, for moderate ground-motion intensities (e.g.,
g), this trend can be reversed mainly for site SCT where the inerter dampers do not offer any benefit compared to the original case.
Appendix C contains additional figures for the buildings with
. They exhibit similar trends as those showed in
Figure 14 and are not discussed for brevity.
7. Conclusions
This study evaluates the reliability of weak first-story reinforced concrete (RC) structures with inerter dampers at their ground level in terms of their maximum story drift due to narrow-band seismic excitations. A set of low-to-medium height buildings designed in accordance with the 1976 old edition of the Technical Norms for Seismic Design (NTC-1976) of Mexico City was used to inspect the feasibility of retrofitting old existing structures with inerter dampers. Fragility functions were determined by means of incremental dynamic analyses (IDA) and a dataset of narrow-band ground motions in Mexico City consistent with the current seismic regulations. In addition, the damage rate function for bare structures and structures equipped with inerter dampers was computed and compared.
The seismic response of weak first-story RC structures is computed considering narrow-band seismic excitations recorded during the large 1985 and 2017 Mexico earthquakes. These seismic events caused significant damage to structures in Mexico City mostly in the lakebed zone. Furthermore, the site effects for places located on soft soils are investigated thoroughly because, for these narrow-band seismic excitations, the frequency contents can differ importantly, which is very relevant in relation to the seismic demand of the structures with inerters. Low-frequency seismic excitations (i.e., predominant period Ts~1.9 s) result in larger seismic demands in relation to those from the high-frequency ground motions (i.e., Ts~1.35s) according to the findings in this study. It should be noted that different results could be obtained for structures with different dynamic characteristics (e.g., buildings with larger structural periods). This could be investigated in future research.
From the cases analyzed, the following findings can be highlighted:
- 1.
The results of fragility functions indicate that the ground motions with high frequency (e.g., site CH84, ) yield larger peak drifts for structures with inerter dampers compared to the ground motions with low frequency (e.g., site SCT, ). Likewise, it is observed that, when structures include inerter dampers at their ground level, the probabilities of exceeding certain peak story drift are less than those in structures without inerter dampers for large ground-motion intensities at both sites. On the contrary, for moderate ground-motion intensities, this trend can be reversed, which is particularly evident for site SCT where the inerter dampers do not offer benefits compared to the original case.
- 2.
The results show that, for the maximum peak story drift, the reliabilities of structures with inerter dampers at their ground level are in general higher or the risks are lower for buildings under seismic intensities associated with limit state of collapse (i.e., very high seismic demands), especially for low-height buildings. Improvements in order of half of the original response were observed. However, for buildings under intensities associated with the limit state of damage limitation (i.e., relatively common seismic demands during the service life of the building), the reliability of structures with inerter dampers could be less than those of structures without IDs.
- 3.
Therefore, it is concluded that inerter dampers are an effective retrofitting alternative for improving the seismic behavior of weak first-story buildings that undergo inelastic behavior (very large seismic intensities associated with the incipient collapse limit state); however, this is not the case (and it actually could be self-defeating) for controlling lateral demands for buildings that behave linearly (under moderate seismic intensities associated with the limit state of damage limitation). This applies to the cases studied in the present research. Further investigation is recommended for other structures and ground motions.
- 4.
It is noteworthy that providing inerter dampers to the building does not eliminate the weak first story, but it controls the problem induced by the weak first-story mechanism from a dynamic point of view provided that adequate parameters of the inerter dampers are selected. If an ID device with larger apparent mass ratio is selected (e.g., >1 for the studied cases), damage in the upper stories could be expected. Therefore, care should be taken to adequately select IDs to control lateral displacements.
Finally, it is considered that future investigations should be aimed at better exploring control systems for seismic protection, e.g., clutched inerters, to improve the response of structures under intensities associated to the limit state of damage limitation, and a cost- benefit analysis should be performed.