# Comparison of Strengthening Solutions with Optimized Passive Energy Dissipation Systems in Symmetric Buildings

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Description of Benchmark Investigated Buildings

## 3. Building’s Modeling

## 4. Optimal Design of Passive Energy Dissipation Systems

#### 4.1. Tuned Mass Damper

#### 4.2. Viscous Dampers (VDs)

#### 4.3. Friction Dampers (FDs)

- Force independent of the apparent contact surface
- Force proportional to the total normal force acting through the interface
- Force independent of speed even with a slip at low speed

## 5. Results and Discussion

#### 5.1. Displacement at the Top of the Structures

#### 5.2. Base Shear Load

#### 5.3. Interstory Drift

_{s,i}, divided by story height, h

_{i}. The relationship between the interstory drift index and the global drift index δ

_{t}/h

_{t}depends on the extent of inelasticity in the structure, the type of plastic hinge mechanism, and the importance of higher mode effects. This comparison validates the general conclusion of this study that is presented in the Section 6.

#### 5.4. Hysteretic Loops

#### 5.5. General Remarks

- It is well known that the structural response reduction increases as the mass of TMD increases, but this mass has a limit in practice, due to geometrical and economical constraints. That is the reason why the mass ratio is not considered as an important value to optimize, and therefore, it is hard to achieve high reduction values practically. The results show that TMD systems are not effective for low and mid-rise buildings, because both the displacement and base shear values are barely affected, unlike high-rise building values. In fact, TMD are motion-based systems that demonstrate how their effectiveness is very limited for rigid buildings. As for the high-rise building, even though damped case with PTMD provides less reduction compared to the two other damped cases, it is considered acceptable and more suitable for this kind of structure.
- Structural strengthening with viscous damper systems is defined by the desired additional damping fixed in the preliminary design. From the results obtained, it has been observed that the structural response with the viscus dampers decreases well, showing better performance in terms of the displacement and base shear. In addition, viscous dampers are velocity-dependent systems, where its effectiveness increases with high velocities, usually for flexible buildings. Even though these systems are considered effective for the three studied buildings, they are considered more suitable for mid-rise buildings.
- Friction dampers’ incorporation into the structures reduces considerably the building’s response after optimizing dampers slip forces, their numbers, and locations under all earthquakes and types of buildings considered. It can be seen from the results obtained that the friction dampers are effective for both rigid and flexible buildings.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The bidirectional recorded accelerograms, (

**b**) response spectra relative to the selected accelerograms compared to the EC8 elastic response spectrum Type 1 with peak ground acceleration equal to $0.3\mathrm{g}$, ground type B, and 5% damping.

**Figure 6.**Modeling configuration of (

**a**) TMD for low-rise and mid-rise buildings, (

**b**) PTMD for high-rise building.

**Figure 10.**(

**a**) Force-Displacement hysteresis loop of a friction damper, (

**b**) optimal slip force effect on structural response.

**Figure 12.**Horizontal displacement at the top of (

**a**,

**b**) the low-rise building, (

**c**,

**d**) the mid-rise building, and (

**e**,

**f**) the high-rise building.

**Figure 13.**Base shear load for (

**a**,

**b**) the low-rise building, (

**c**,

**d**) the mid-rise building and (

**e**,

**f**) the high-rise building.

**Figure 14.**Maximum interstory drift for (

**a**) low-rise, (

**b**) mid-rise, and (

**c**) high-rise building in the longitudinal direction.

**Figure 15.**Maximum interstory drift for (

**a**) low-rise, (

**b**) mid-rise, and (

**c**) high-rise building in the transversal direction.

Building Type | Low-Rise | Mid-Rise | High-Rise |
---|---|---|---|

Number of stories | G + 03 | G + 08 | G + 15 |

Story height | 3.5 m | ||

Total height including roof level | 14 m | 31.5 m | 56 m |

Columns cross section | 30 × 50 cm | 30 × 80 cm | 30 × 80 cm |

Beams cross section | 30 × 30 cm | ||

Walls thickness | 30 cm | ||

Slab thickness | 20 cm |

Expressions | ||
---|---|---|

Optimal tuning parameters of TMD given in [38] | $\nu =\frac{1}{1+\overline{m}}\xb7\left[1-\xi \xb7\sqrt{\frac{\overline{m}}{1+\overline{m}}}\right]$ | (2) |

${\xi}_{d}=\frac{\xi}{1+\overline{m}}+\sqrt{\frac{\overline{m}}{1+\overline{m}}}$ | (3) | |

Optimized absorber parameter | ${k}_{d}={\omega}_{d}{}^{2}\xb7{m}_{d}={\nu}^{2}\xb7{\omega}^{2}\xb7\overline{m}\xb7m$ | (4) |

${c}_{d}=2\xb7{\xi}_{d}\xb7{\omega}_{d}\xb7{m}_{d}=2\xb7{\xi}_{d}\xb7\upsilon \xb7\omega \xb7\overline{m}\xb7m$ | (5) |

Building Type | Structure | TMD | |||
---|---|---|---|---|---|

Parameter | Value | Parameter | Values | ||

Low-rise | $T$ [s] | 0.216 | $\overline{m}=0.5\%$ | $\nu =0.9915$ | ${\xi}_{d}=12.03\%$ |

$m$ [t] | 1572.29 | $\overline{m}=1\%$ | $\nu =0.9852$ | ${\xi}_{d}=14.90\%$ | |

Mid-rise | $T$ [s] | 0.985 | $\overline{m}=1\%$ | $\nu =0.9852$ | ${\xi}_{d}=$14.90% |

$m$ [t] | 4626.92 | $\overline{m}=3\%$ | $\nu =0.9626$ | ${\xi}_{d}=21.92\%$ | |

High-rise | $T$ [s] | 2.202 | $\overline{m}=2\%$ | $\nu =0.9735$ | ${\xi}_{d}=18.90\%$ |

$m$ [t] | 8560.39 | $\overline{m}=3\%$ | $\nu =0.9626$ | ${\xi}_{d}=21.92\%$ |

Building Type | $\overline{\mathit{m}}$ | m_{d}[t] | k_{d}[kN/m] | c_{d}[kN·s/m] |
---|---|---|---|---|

Low-rise | 0.50% | 7.86 | 6561.47 | 54.63 |

1% | 15.72 | 12,955.58 | 134.49 | |

Mid-rise | 1% | 46.27 | 1829.10 | 86.70 |

3% | 138.8 | 5238.61 | 373.86 | |

High-rise | 2% | 171.2 | 1321.15 | 179.82 |

3% | 256.8 | 1937.40 | 309.24 |

**Table 5.**Responses of the three investigated buildings for the two different mass ratio of TMD with margin from undamped case.

Building Type | Case | Direction | Fundamental Period | Top Roof Displacement | Base Shear | |||
---|---|---|---|---|---|---|---|---|

Value [s] | Margin | Value [cm] | Margin | Value [kN] | Margin | |||

Low-rise | Undamped | Longitudinal | 0.156 | 0.8556 | 9536.64 | |||

Transversal | 0.216 | 0.4248 | 2772.68 | |||||

Damped with $\overline{m}=0.5\%$ | Longitudinal | 0.158 | −1.28% | 0.8243 | 3.66% | 9530.28 | 0.07% | |

Transversal | 0.218 | −0.93% | 0.4183 | 1.53% | 2751.67 | 0.76% | ||

Damped with $\overline{m}=1\%$ | Longitudinal | 0.157 | −0.64% | 0.8237 | 3.73% | 9287.71 | 2.61% | |

Transversal | 0.216 | 0.00% | 0.4164 | 1.98% | 2693.44 | 2.86% | ||

Mid-rise | Undamped | Longitudinal | 0.697 | 13.2068 | 24,870.8 | |||

Transversal | 0.985 | 5.4141 | 4652.36 | |||||

Damped with $\overline{m}=1\%$ | Longitudinal | 0.71 | −1.87% | 12.6213 | 4.43% | 23,726.16 | 4.60% | |

Transversal | 0.997 | −1.22% | 5.3094 | 1.93% | 4567.02 | 1.83% | ||

Damped with $\overline{m}=3\%$ | Longitudinal | 0.733 | −5.16% | 12.5036 | 5.32% | 22,507.32 | 9.50% | |

Transversal | 1.031 | −4.67% | 4.7931 | 11.47% | 4559.55 | 1.99% | ||

High-rise | Undamped | Longitudinal | 1.983 | 38.377 | 19,306.7 | |||

Transversal | 2.202 | 12.5677 | 5222.92 | |||||

Damped with $\overline{m}=2\%$ | Longitudinal | 2.058 | −3.78% | 35.9845 | 6.23% | 18,077.8 | 6.37% | |

Transversal | 2.284 | −3.72% | 10.2574 | 18.38% | 5112.19 | 2.12% | ||

Damped with $\overline{m}=3\%$ | Longitudinal | 2.091 | −5.45% | 36.3981 | 5.16% | 17,476.06 | 9.48% | |

Transversal | 2.322 | −5.45% | 10.1373 | 19.34% | 5331.14 | −2.07% |

Building Type | $\overline{\mathit{m}}$ | ${\mathit{\omega}}_{\mathit{d}}[\mathbf{rad}/\mathbf{s}]$ | $\mathit{L}$ [m] |
---|---|---|---|

Low-rise | 0.50% | 28.893 | 0.012 |

Mid-rise | 3.00% | 6.143 | 0.259 |

High-rise | 3.00% | 2.747 | 1.300 |

Case | Longitudinal | Transversal | ||||
---|---|---|---|---|---|---|

Undamped | With TMD | With PTMD | Undamped | With TMD | With PTMD | |

Fundamental period [s] | 1.983 | 2.091 | 1.801 | 2.202 | 2.322 | 2.67 |

Top roof displacement [cm] | 38.377 | 36.39 | 25.01 | 12.567 | 10.14 | 9.23 |

Base shear [kN] | 19,306.7 | 17,476.06 | 16,974.11 | 5222.9 | 5331.14 | 4386.71 |

**Table 8.**Effective damping and calculated damping coefficient for the three investigated buildings (α = 0.3).

Building Type | Direction | Fundamental Period [s] | Structural Rigidity Description | Suggested Velocity [m/s] | $\mathbf{Suggestedeffective}\mathbf{Damping}{\mathit{\xi}}_{\mathit{e}\mathit{f}\mathit{f}}$ | $\sum}{\mathit{C}}_{\mathit{j}$ [kN·(s/m)] |
---|---|---|---|---|---|---|

Low-rise | Longitudinal | 0.156 | Rigid | 0.127 | 30% | 71,537.07 |

Transversal | 0.216 | 0.127 | 30% | 59,405.58 | ||

Mid-rise | Longitudinal | 0.697 | Semi-rigid | 0.254 | 35% | 56,981.16 |

Transversal | 0.985 | 0.254 | 35% | 39,325.16 | ||

High-rise | Longitudinal | 1.983 | Flexible | 0.381 | 40% | 12,658.71 |

Transversal | 2.202 | 0.381 | 40% | 12,095.19 |

Building Type | Direction | Fundamental Period [s] | Top Roof Displacement [cm] | Base Shear [kN] | |||
---|---|---|---|---|---|---|---|

Altern. 1 | Altern. 2 | Altern. 1 | Altern. 2 | Altern. 1 | Altern. 2 | ||

Low-rise | Longitudinal | 0.112 | 0.112 | 0.167 | 0.176 | 1973.2 | 1905.6 |

Transversal | 0.148 | 0.147 | 0.218 | 0.345 | 2308.6 | 2070.0 | |

Mid-rise | Longitudinal | 0.387 | 0.387 | 3.121 | 3.763 | 1888.1 | 7567.0 |

Transversal | 0.775 | 0.775 | 2.997 | 1.760 | 4991.5 | 83.3 | |

High-rise | Longitudinal | 0.543 | 0.543 | 5.812 | 6.821 | 10,828 | 322.5 |

Transversal | 0.989 | 0.989 | 4.604 | 4.803 | 178.6 | 14,737.6 |

Building Type | Direction | Fundamental Period [s] | Top Roof Displacement [cm] | Base Shear [kN] | |||
---|---|---|---|---|---|---|---|

Altern. 1 | Altern. 2 | Altern. 1 | Altern. 2 | Altern. 1 | Altern. 2 | ||

Low-rise | Longitudinal | 0.158 | 0.158 | 0.228 | 0.290 | 6971.2 | 7296.0 |

Transversal | 0.218 | 0.218 | 0.126 | 0.057 | 1525.1 | 2000.0 | |

Mid-rise | Longitudinal | 0.707 | 0.708 | 1.682 | 1.775 | 14,030.2 | 14,639.4 |

Transversal | 0.998 | 0.999 | 1.124 | 0.298 | 3100.0 | 3500.0 | |

High-rise | Longitudinal | 2.016 | 2.016 | 6.915 | 7.787 | 12,373.0 | 13,950.1 |

Transversal | 2.239 | 2.239 | 4.349 | 2.672 | 3494.58 | 4000.0 |

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**MDPI and ACS Style**

Mrad, C.; Titirla, M.D.; Larbi, W.
Comparison of Strengthening Solutions with Optimized Passive Energy Dissipation Systems in Symmetric Buildings. *Appl. Sci.* **2021**, *11*, 10103.
https://doi.org/10.3390/app112110103

**AMA Style**

Mrad C, Titirla MD, Larbi W.
Comparison of Strengthening Solutions with Optimized Passive Energy Dissipation Systems in Symmetric Buildings. *Applied Sciences*. 2021; 11(21):10103.
https://doi.org/10.3390/app112110103

**Chicago/Turabian Style**

Mrad, Charbel, Magdalini D. Titirla, and Walid Larbi.
2021. "Comparison of Strengthening Solutions with Optimized Passive Energy Dissipation Systems in Symmetric Buildings" *Applied Sciences* 11, no. 21: 10103.
https://doi.org/10.3390/app112110103