# Effects of Dissipative Systems on the Seismic Behavior of Irregular Buildings—Two Case Studies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling and Analysis of Existing Buildings

#### 2.1. Verifications of the Structural Elements

_{m}), which is equal to 1.5 for concrete and 1.15 for steel [14].

_{Ed}is the acting moment and M

_{Rd}the resisting (yield) moment.

_{c,Rd}is calculated for the axial stress levels present in the column due to the seismic combinations of actions.

_{Rd}, which for cyclic loads, such as seismic loads, is calculated using the following equation [15]. The equation considers not only the three contributions due to axial force, concrete, and steel, but also the interaction with the flexural rotation of the element as function of the plastic ductility demand, ${\mu}_{\mathsf{\Delta},pl}$, Equation (3):

_{el}is equal to 1.15, h is the depth of cross-section, x is the compression zone depth, N is the compressive axial force, L

_{V}is the shear span obtained by dividing the acting moment by the acting shear, ${A}_{c}$ is the cross-section area, ${f}_{c}$ is the concrete compressive strength, ρ

_{tot}is the total longitudinal reinforcement ratio, and ${V}_{W}$ is the contribution of transverse reinforcement to shear strength. V

_{W}is calculated as ${V}_{W}={\rho}_{sx}{b}_{w}z{f}_{yw}$, where ${\rho}_{sx}$ is the transverse reinforcement ratio, z is the the internal level arm, ${f}_{yw}$ is the stirrup yield strength. ${\mu}_{\mathsf{\Delta},pl}$ is the plastic part of ductility demand equal to ${\mu}_{\mathsf{\Delta},pl}=\text{}{\mu}_{\mathsf{\Delta}}-1$, where ${\mu}_{\mathsf{\Delta}}$ is calculated as the maximum chord rotation at the considered seismic action, ${\theta}_{m}$, normalized to the chord rotation at yielding, ${\theta}_{y}$.

_{i}are the distances of the longitudinal bars held by tie rods or stirrups present on the perimeter.

_{N}, representing the number of years in which the structure is to be used for its purposes, is assumed to be 50 years, which is typical for housing. The coefficient of use, C

_{U}, related to the importance of the building, is assumed equal to 1, which is the value normally adopted for private buildings. The reference time period for seismic action, V

_{R}, is obtained from C

_{U}·V

_{N}, hence it is equal to 50 years. The topographic category is T1 (flat surface), and the soil is B-type (coarse-grained soil).

_{R}, the Italian Building Code [14] establishes four reference seismic levels, namely, Frequent Design Earthquake (FDE, probability of 81%), Serviceability Design Earthquake (SDE, probability of 50%), Basic Design Earthquake (BDE, probability of 10%) and Maximum Considered Earthquake (MCE, probability of 5%).

#### 2.2. Modeling of the Case Studies

_{inf}is the strut length and $\lambda $ is a dimensionless parameter that takes into account the effect of relative stiffness of the masonry panel to the frame, given by Equation (9):

_{inf}the height of the masonry wall, and E

_{f}and I

_{col}the elastic modulus and the moment of inertia of the column, respectively. Angle θ is calculated as Equation (10):

_{n}associated to the n

_{th}mode remain constant throughout the analysis, even when the structural stiffness changes. Thus, to define the modal damping it is sufficient to specify the damping ratios ξ

_{n}for each mode.

_{n}= 0.05 has been assumed, while for the retrofitted structures ξ

_{n}= 0.02 has been assumed.

## 3. First Case Study—Building A

#### 3.1. Geometrical and Structural Characteristics of the Building

^{2}. Secondary beams, oriented parallel to the Y direction, are present only in the exterior sides of the building and have a section of 33 × 23.5 cm

^{2}. All under-roof floors have beams parallel to the X direction, with a section of 80 × 15 cm

^{2}, and beams parallel to the y direction with a section of 20 × 15 cm

^{2}. The longitudinal reinforcement of all beams is made by Φ16. The transverse reinforcement for beams in the X direction consists of Φ8 every 7 cm close to the ends and every 25 cm in the middle, while beams parallel to the Y direction are reinforced by Φ6 every 25 cm.

^{2}on habitable stories, 2.35 kN/m

^{2}on the under-roof story, and 2.70 kN/m

^{2}for the roof. The weight of the infill perimeter walls is considered assuming a linearly distributed load of 9.97 kN/m acting on the beams below. Live loads of 2 kN/m

^{2}are applied on all habitable stories, as well as 4 kN/m

^{2}on balconies and staircases, 0.5 kN/m

^{2}on under-roof floors, and 1.2 kN/m

^{2}on the horizontal projection of the roof, due to the snow.

#### 3.2. Results of the Modal and FNA Analyses

#### 3.3. Retrofit Intervention with BRADs and Effects

#### 3.4. Building Torsional Stiffness

_{tor}around the Z-axis is applied to the center of mass of a diaphragm, while all other diaphragms are embedded. The rotation of the center of mass, γ, due to this torsional moment, is obtain from the FE model, and the torsional stiffness, k

_{tor}, is calculated as Equation (12):

_{tor}is calculated in the same way on the FE model with dissipative devices. Relative results are reported in Table 2.

_{s}

^{2}and l

_{s}

^{2}, where r

_{s}

^{2}= k

_{tor}/k

_{f}is the ratio between k

_{tor}and the flexural stiffness, k

_{f}, of each floor, and l

_{s}

^{2}, in cases of a rectangular plan, is equal to (L

^{2}+ B

^{2})/12, with L and B the dimensions of the floor in the plan. For building A, considering the floor coinciding with a diaphragm, l

_{s}

^{2}= 61.9 m

^{2}for all diaphragms. Since l

_{s}

^{2}is constant, to obtain the minimum ratio r

_{s}

^{2}/l

_{s}

^{2}, the minimum value of r

_{s}has to be considered; hence, the maximum k

_{f}has to be calculated. To do this, a unitary point load, F, is applied, along one of the building’s principal directions, to the center of rigidity of the considered diaphragm, while the other diaphragms are embedded. This results in a translational displacement, d, of the floor without torsional effect. The flexural stiffness, is then obtained from Equation (13):

_{s}

^{2}/l

_{s}

^{2}are higher than 1.0 for all levels. This is the limit above which a building can be considered not torsionally deformable according to [14]. For this reason, from the point of view of torsional deformability, the behavior of building A before retrofitting cannot be regarded as satisfactory.

_{s}

^{2}/l

_{s}

^{2}increases by about 11%, and on the other two levels it increases by more than 6%.

## 4. Second Case Study—Building B

#### 4.1. Geometrical and Structural Characteristics of the Building

^{2}. The building can be divided into three main blocks plus two staircases, separated from each other by the seismic joints that are indicated by dotted lines in Figure 13. The two lateral blocks have no infill walls on the ground floor, while in the central block there are masonry infill walls between the perimeter ground floor columns, with a continuous strip window at the top.

^{2}on the ground floor, and 30 × 30 cm

^{2}on the upper floors. The longitudinal reinforcement of the column is made by 4Φ16, one for each corner, for all the levels. In the two lateral blocks, the bearing structure is characterized by a single RC wall in the Y direction and, in the X direction, by three RC frames in the right-side block, and four RC frames in the left side one. As for the central block, columns have a square section of 35 × 35 cm

^{2}on the ground floor, and 30 × 30 cm

^{2}on the upper floors. The longitudinal reinforcement of columns C1, C3, C5 and C7 in Figure 13 on the ground and first floor consists of two layers, each made by 3Φ20 parallel to y direction. On the second floor, the longitudinal reinforcement of the same columns consists of 6Φ16, distributed in two layers, each layer made of three bars, parallel to y directions; on the other floors, the longitudinal bars are reduced to 4Φ16, one bar for each corner. The reinforcement of C2 and C8 on the ground floor consists of 8Φ20, three parallel to each side; in the same orientation, there are 8Φ16 on the first floor. On the second floor, longitudinal reinforcement consists of a total of 6Φ16, divided into two layers of three bars parallel to the longer side of the section. On the other floors, reinforcement consists of 4Φ16, one for each corner. The longitudinal reinforcement of columns C4 and C6 on the ground floor comprises 4Φ20, one for each corner; this is reduced to 4Φ16 on the other floors. The transverse reinforcement of the columns consists of Φ6 every 17.5 cm.

^{2}, while perimeter beams have a section of 60 × 23.5 cm

^{2}. There are no secondary beams in Y directions. On the under-roof floor, which is not practicable, beams have section of 40 × 15.5 cm

^{2}. Frame elements of the central and the right block have the same dimensions as those of the left block. Floors are made of precast RC joists, oriented in the Y direction, having a depth of 20.5 cm and an interaxle spacing of 25 cm, completed with clay bricks and a cast-in-place concrete slab 3-cm thick.

^{2}on the story floors and 2.35 kN/m

^{2}on the under-roof floor. Infill walls are considered assuming a linearly distributed load of 7.15 kN/m acting on the beams below. Live loads are equal to 2 kN/m

^{2}on all the story floors, 4 kN/m

^{2}on balconies and staircases, 0.5 kN/m

^{2}on the under-roof floor, and 1.2 kN/m

^{2}on the horizontal projection of the roof due to the snow.

#### 4.2. Results of the Modal and FNA Analyses

#### 4.3. Retrofit Intervention with Viscous Fluid Dampers and Effects

^{0.15}are used. As shown in the previous section, the verifications required by [14] are not satisfied due to both the characteristics of the structural members and the torsional behavior of the building lateral block. To regularize this behavior and reduce the seismic vulnerability, the viscous fluid dampers of Jarret type BC5A-105 are used as a retrofit solution. Pairs of dissipative devices are mounted at the tops of concentric V-shaped steel braces. The braces are connected to the concrete frame by metal anchors. The disposition of Jarret dampers, shown in Figure 18, is the result of a comprehensive design process, whose goal is the best performance behavior of the dissipative system.

#### 4.4. Building Torsional Stiffness

_{tor}, of all floors in the as-built condition and after retrofit is calculated (Table 5).

_{tor}reported in Table 5 show that, after retrofit, the torsional stiffness of the floors increases to values that range from about 185% to 208% of the as-built stiffness. In this way, the torsional behavior of the lateral block exhibits great improvement after the retrofit.

_{s}

^{2}= 27.8 m

^{2}. The values of the ratio r

_{s}

^{2}/l

_{s}

^{2}are calculated as explained in Section 3.4., and the obtained results are shown in Table 5. In the as-built configuration, the r

_{s}

^{2}/l

_{s}

^{2}ratio is about 0.04 for all floors. This is a value lower than 1.0, hence the building is considered torsionally deformable according to [14]. After retrofit, the ratio increases to the value of 0.12 for the first floor and 0.13 for the others, with variations ranging from about +183% to about +217%, respectively. The final values of the ratio are not sufficient to consider the building not torsionally deformable. Nevertheless, the increase is substantial and demonstrates an improvement in the building torsional behavior. Thus, it is possible to say that the use of dissipative devices is beneficial not only to absorb part of the input energy and reduce the force acting on bearing structures, but also to improve the building behavior by reducing negative effects due to its irregularities.

## 5. Conclusions

- 1.
- The verifications performed according to the Italian Building Code prescriptions for existing buildings show that many structural members of irregular buildings designed without anti-seismic criteria are not satisfied. This is mainly due to insufficient transverse reinforcement of the members and column deformations caused by the additional displacements that arise from the building torsional behavior.
- 2.
- To increase the torsional stiffness of the building, the steel braces where the dissipative devices are mounted should be placed, if at all possible, in the perimeter frames.
- 3.
- Since the shear acting on the building due to seismic action decreases along the height, the use of devices that become progressively smaller and less stiff along the height optimizes the performance of the dissipative system.
- 4.
- The dissipative systems designed for the two case studies adhere as much as possible to the previous principles and:
- Are able to absorb more than 50% of the seismic input energy for building A, and more than 80% of that for building B, thus reducing the forces acting on the existing building structural members;
- Increase the building torsional stiffness by about 7% for building A, which already had a good initial value of this parameter, and by more than 200% for building B, thus reducing the columns displacements, and consequently, their stresses;
- Increase the ratio r
_{s}^{2}/l_{s}^{2}, which is a measure of the building torsional deformability, by about 7% for building A and about 200% for building B, which was initially very torsionally deformable.

- 5.
- Thanks to the introduction of the dissipative systems, the reduction in the acting shear is 64% for the columns of building A, and 65% for the columns of building B.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Infill wall: (

**a**) schematization and (

**b**) static scheme to determine the equivalent stiffness of the infill walls.

**Figure 10.**Cyclic response of different Buckling-Restrained Axial Damper (BRAD) types under an accelerogram of the Maximum Considered Earthquake (MCE): (

**a**) 34/30-b (

**b**) 27/30-b (

**c**) 21/30-b (

**d**) 14/30-b.

Vibration Mode | Period | Modal Participating Mass Ratios | ||
---|---|---|---|---|

(s) | Ux | Uy | Rz | |

1 | 0.67 | 0.00002 | 0.001 | 0.0017 |

8 | 0.59 | 0.0005 | 0.10 | 0.62 |

10 | 0.46 | 0.02 | 0.60 | 0.14 |

12 | 0.28 | 0.71 | 0.02 | 0.0007 |

**Table 2.**Comparison of torsional stiffness and deformability of building A before and after retrofitting.

Level | Before Retrofitting | After Retrofitting | k_{tor} Variation | r_{s}^{2}/l_{s}^{2} Variation | ||||||
---|---|---|---|---|---|---|---|---|---|---|

k_{tor}(GN·m/rad) | k_{f}(GN/m) | r_{s}^{2}(m ^{2}/rad) | r_{s}^{2}/l_{s}^{2} | k_{tor}(GN·m/rad) | k_{f}(GN/m) | r_{s}^{2}(m ^{2}/rad) | r_{s}^{2}/l_{s}^{2} | |||

1 | 90.91 | 5.30 | 17.15 | 0.28 | 101.32 | 53.22 | 19.04 | 0.31 | +11.5% | +11.0% |

2 | 100.00 | 5.74 | 17.41 | 0.28 | 106.78 | 57.70 | 18.51 | 0.30 | +6.8% | +6.3% |

3 | 66.67 | 4.33 | 15.41 | 0.25 | 71.43 | 43.33 | 16.49 | 0.27 | +7.1% | +7.0% |

Vibration Mode | Period | Modal Participating Mass Ratios | ||
---|---|---|---|---|

(s) | Ux | Uy | Rz | |

1 | 1.13 | 0.79 | 0.0 | 0.0001 |

2 | 0.39 | 0.0 | 0.79 | 0.0 |

4 | 0.27 | 0.0 | 0.0 | 0.74 |

Vibration Mode | Period | Modal Participating Mass Ratios | ||
---|---|---|---|---|

(s) | Ux | Uy | Rz | |

1 | 1.82 | 0.0 | 0.32 | 0.50 |

2 | 0.84 | 0.02 | 0.0 | 0.0 |

7 | 0.28 | 0.0 | 0.41 | 0.28 |

**Table 5.**Comparison of torsional stiffness and deformability of building B before and after retrofitting.

Level | Before Retrofitting | After Retrofitting | k_{tor} Variation | r_{s}^{2}/l_{s}^{2} Variation | ||||||
---|---|---|---|---|---|---|---|---|---|---|

k_{tor}(GN·m/rad) | k_{f}(GN/m) | r_{s}^{2}(m ^{2}/rad) | r_{s}^{2}/l_{s}^{2} | k_{tor}(GN·m/rad) | k_{f}(GN/m) | r_{s}^{2}(m ^{2}/rad) | r_{s}^{2}/l_{s}^{2} | |||

1 | 10.64 | 9.33 | 1.14 | 0.041 | 30.30 | 9.38 | 3.23 | 0.116 | +184.8% | +183.3% |

2 | 9.35 | 7.84 | 1.19 | 0.043 | 28.68 | 7.87 | 3.63 | 0.131 | +206.7% | +205.0% |

3 | 9.43 | 7.84 | 1.20 | 0.043 | 28.57 | 7.87 | 3.63 | 0.131 | +203.0% | +202.5% |

4 | 9.35 | 7.58 | 1.23 | 0.044 | 28.52 | 7.60 | 3.76 | 0.135 | +205.0% | +205.7% |

5 | 9.26 | 7.97 | 1.16 | 0.042 | 28.54 | 7.77 | 3.68 | 0.132 | +208.2% | +217.2% |

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**MDPI and ACS Style**

Miani, M.; Di Marco, C.; Frappa, G.; Pauletta, M.
Effects of Dissipative Systems on the Seismic Behavior of Irregular Buildings—Two Case Studies. *Buildings* **2020**, *10*, 202.
https://doi.org/10.3390/buildings10110202

**AMA Style**

Miani M, Di Marco C, Frappa G, Pauletta M.
Effects of Dissipative Systems on the Seismic Behavior of Irregular Buildings—Two Case Studies. *Buildings*. 2020; 10(11):202.
https://doi.org/10.3390/buildings10110202

**Chicago/Turabian Style**

Miani, Marco, Caterina Di Marco, Giada Frappa, and Margherita Pauletta.
2020. "Effects of Dissipative Systems on the Seismic Behavior of Irregular Buildings—Two Case Studies" *Buildings* 10, no. 11: 202.
https://doi.org/10.3390/buildings10110202