# Seismic Risk of Weak First-Story RC Structures with Inerter Dampers Subjected to Narrow-Band Seismic Excitations

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## Abstract

**:**

## 1. Introduction

## 2. Designed Buildings Considered

#### 2.1. Dynamic Analysis of the Buildings

#### 2.2. Modeling of the Inerter Damper

## 3. Seismic Hazard and Ground Motion Selection

#### 3.1. Seismic Hazard at Soft Sites

#### 3.2. Set of Earthquake Ground Motions Considered

^{th}, 2017 Mw7.1 and September 19

^{th}, 1985 M8.2 earthquakes recorded at stations CH84 and SCT, respectively. Figure 8 presents the SA pseudo-acceleration (upper-left corner), SV pseudo-velocity (upper-right corner), and SD displacement (lower-left corner) seismic response spectra from synthetic ground motions for sites CH84 (pink lines) and SCT (blue lines). It can be observed that these ground motions preserve the main characteristics of the seed records; for instance, the frequency content and intensities take into account the variation of soil dominant period ${T}_{s}$ due to the drying process that occurs in the lacustrine zone of the city.

#### 3.3. Criteria for Scaling Ground Motion

## 4. Earthquake-Induced Response of Buildings with Inerter Dampers

#### 4.1. Seismic Response of Sdof Systems for Synthetic Ground Motion

#### 4.2. Influence of Seismic Intensities Associated to Limit State of Collapse

#### 4.3. Influence of Seismic Intensities Associated to Limit State of Damage Limitation

## 5. Fragility Functions

#### 5.1. Probability of Failure for Maximum Peak Story Drifts

#### 5.2. Results of Probability of Failure for Maximum and Residual Displacements

## 6. Earthquake-Induced Risk Assessment

#### 6.1. Maximum Peak Story Drift

#### 6.2. Maximum Peak Story Drift for Buildings Q = 6 Versus Q = 4

## 7. Conclusions

- 1.
- The results of fragility functions indicate that the ground motions with high frequency (e.g., site CH84, ${T}_{s}~1.35\text{}\mathrm{s}$) yield larger peak drifts for structures with inerter dampers compared to the ground motions with low frequency (e.g., site SCT, ${T}_{s}~1.9\text{}\mathrm{s}$). 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.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Seismic Response for Buildings of Q = 4 Subjected to a Set Earthquakes Whose Seismic Intensities Are Associated to Limit State of Collapse

**Figure A1.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story buildings of Q = 4 with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of collapse (i.e., ${\lambda}_{y}=4\xb7{10}^{-3}$ annual exceedance of SA, T_R = 250 years) at site CH84. (

**Left**) median peak story drift demands; (

**middle**) median normalized peak floor acceleration demands with respect to the ground; and (

**right**) median normalized peak shear demands with the building weight.

**Figure A2.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story buildings of Q = 4 with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of collapse (i.e., ${\lambda}_{y}=4\xb7{10}^{-3}$ annual exceedance of SA, T_R = 250 years) at site SCT. (

**Left**) median peak story drift demands; (

**middle**) median normalized peak floor acceleration demands with respect to the ground; and (

**right**) median normalized peak shear demands with the building weight.

## Appendix B. Seismic Response for Buildings of Q = 4 Subjected to a Set Earthquakes Whose Seismic Intensities Are Associated to Limit State of Damage Limitation

**Figure A3.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story buildings of Q = 4 with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of damage limitation (i.e., ${\lambda}_{y}={10}^{-1}$ annual exceedance of SA, T_R = 10 years) at site CH84. (

**Left**) median peak story drift demands; (

**middle**) median normalized peak floor acceleration demands with respect to the ground; and (

**right**) median normalized peak shear demands with the building weight.

**Figure A4.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story buildings of Q = 4 with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of damage limitation (i.e., ${\lambda}_{y}={10}^{-1}$ annual exceedance of SA, T_R = 10 years) at site SCT. (

**Left**) median peak story drift demands; (

**middle**) median normalized peak floor acceleration demands with respect to the ground; and (

**right**) median normalized peak shear demands with the building weight.

## Appendix C. Fragility Function for Maximum Peak Story Drift for Buildings of Q = 4

**Figure A5.**Fragility functions for maximum peak story drift for 4- (

**top**), 6- (

**middle**), and 8-story buildings of Q = 4 located at sites CH84 (

**left**) and SCT (

**right**) considering inerter dampers with σ = 0.7 (dotted lines) and without them (solid lines) at their ground level.

## Appendix D. Risk Curves for Maximum Peak Story Drift for Buildings of Q = 4

**Figure A6.**Risk curves for maximum peak story drift for 4- (

**top**), 6- (

**middle**), and 8-story buildings of Q = 4 located at sites CH84 (

**left**) and SCT (

**right**) considering inerter dampers and without them at their ground level 4.

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**Figure 1.**Examples of upper soft-story failures in Mexico City caused by the 2017 Mw7.1 Morelos–Puebla seismic event. (

**a**) 241 Rebsamen St., Narvarte neighborhood; (

**b**) 37 Patricio Sanz St., Del Valle Norte neighborhood; and (

**c**) 47 Gallinas St., Lomas Estrella neighborhood.

**Figure 4.**Elevation view of the weak first-story buildings provided with inerter dampers at their ground level with 4 stories (

**top**), 6 stories (

**middle**), and 8 stories (

**bottom**). Infill masonry walls with and without openings correspond to exterior and interior walls, respectively.

**Figure 5.**Schematic representation of an inerter damper with the main parameters, after [15].

**Figure 8.**(

**a**) Acceleration ($SA$), (

**b**) velocity ($SV$), and (

**c**) displacement ($SD$) response spectra of the set of input ground motions considered for the risk analyses in Mexico City for two sites: CH84 (pink lines) and SCT (blue lines). Thick lines are the median response values.

**Figure 9.**Response spectra of a simple bilinear SDOF system (gray lines) and bilinear SDOF system equipped with an inerter damper with $\sigma =1$ on a stiff frame (pink lines) when excited by a set of input ground motions considered for the risk analyses in Mexico City for two sites: (

**a**) CH84 and (

**b**) SCT considering ${Q}_{1}/{m}_{1}g=0.1$, $\alpha =0.5$, $\beta =0.5$, $\gamma =0.5$, $n=10$, and ${\xi}_{1}=0.05$. Thick lines are the median response values.

**Figure 10.**Seismic response of 4- (top), 6- (middle), and 8-story (bottom) buildings of $Q=6$ with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of collapse (i.e., ${\lambda}_{y}=4\xb7{10}^{-3}$ annual exceedance of SA, ${T}_{R}$ = 250 years) at site CH84. Left: median peak story drift demands; middle: median normalized peak floor acceleration demands with respect to the ground; and right: median normalized peak shear demands with the building weight ${W}_{T}$.

**Figure 11.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story (

**bottom**) buildings of $Q=6$ with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of collapse (i.e., ${\lambda}_{y}=4\xb7{10}^{-3}$ annual exceedance of SA, ${T}_{R}$ = 250 years) at site SCT. Left: median peak story drift demands; middle: median normalized peak floor acceleration demands with respect to the ground; and right: median normalized peak shear demands with the building weight.

**Figure 12.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story (

**bottom**) buildings of $Q=6$ with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of damage limitation (i.e., ${\lambda}_{y}={10}^{-1}$ annual exceedance of SA, ${T}_{R}$ = 10 years) at site CH84. Left: median peak story drift demands; middle: median normalized peak floor acceleration demands with respect to the ground; and right: median normalized peak shear demands with the building weight.

**Figure 13.**Seismic response of 4- (

**top**), 6- (

**middle**), and 8-story (

**bottom**) buildings of $Q=6$ with and without inerter dampers subjected to a set of earthquakes whose seismic intensities are associated to limit state of damage limitation (i.e., ${\lambda}_{y}={10}^{-1}$ annual exceedance of SA, ${T}_{R}$ = 10 years) at site SCT. Left: median peak story drift demands; middle: median normalized peak floor acceleration demands with respect to the ground; and right: median normalized peak shear demands with the building weight.

**Figure 14.**Fragility functions for maximum peak story drift for 4- (

**top**), 6- (

**middle**), and 8-story (

**bottom**) buildings of $Q=6$ located at sites CH84 (left) and SCT (right) considering inerter dampers with $\sigma =0.7$ (dotted lines) and without them (solid lines) at their ground level.

**Figure 15.**Risk curves for maximum peak story drift for 4- (

**top**), 6- (

**middle**), and 8-story (

**bottom**) buildings of $Q=6$ located at sites CH84 (

**left**) and SCT (

**right**) with and without inerter dampers at their ground level.

**Figure 16.**Risk curves for maximum peak story drift for 4- (

**top**), 6- (

**middle**), and 8-story (

**bottom**) buildings of $Q=6$ (blue lines) and $Q=4$ (pink lines) located at sites CH84 (

**left**) and SCT (

**right**) with and without inerter dampers at their ground level.

**Table 1.**Total weights and periods of fundamental mode for each building designed with the NTCS-1976 [25].

Model | Total Weight $\mathit{W}$ (kN) | Period ${\mathit{T}}_{1}\text{}\left(\mathbf{s}\right)$ | Participating Mass Percentage (%) First, Second, and Third Mode |
---|---|---|---|

E4Q6 | 5928 | 1.02 | 89.4, 8.8, 1.1 |

E6Q6 | 9186 | 1.04 | 87.3, 9.8, 2.0 |

E8Q6 | 13,574 | 0.98 | 83.9, 11.6, 2.5 |

E4Q4 | 5972 | 0.87 | 89.3, 9.1, 1.2 |

E6Q4 | 9330 | 0.97 | 87.2, 10, 1.9 |

E8Q4 | 13,771 | 0.92 | 83.8, 10.8, 2.5 |

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## Share and Cite

**MDPI and ACS Style**

Jaimes, M.A.; Niño, M.; Franco, I.; Trejo, S.; Godínez, F.A.; García-Soto, A.D. Seismic Risk of Weak First-Story RC Structures with Inerter Dampers Subjected to Narrow-Band Seismic Excitations. *Buildings* **2023**, *13*, 929.
https://doi.org/10.3390/buildings13040929

**AMA Style**

Jaimes MA, Niño M, Franco I, Trejo S, Godínez FA, García-Soto AD. Seismic Risk of Weak First-Story RC Structures with Inerter Dampers Subjected to Narrow-Band Seismic Excitations. *Buildings*. 2023; 13(4):929.
https://doi.org/10.3390/buildings13040929

**Chicago/Turabian Style**

Jaimes, Miguel A., Mauro Niño, Isaac Franco, Salatiel Trejo, Francisco A. Godínez, and Adrián D. García-Soto. 2023. "Seismic Risk of Weak First-Story RC Structures with Inerter Dampers Subjected to Narrow-Band Seismic Excitations" *Buildings* 13, no. 4: 929.
https://doi.org/10.3390/buildings13040929