# Fragility Curves of Existing RC Buildings Accounting for Bidirectional Ground Motion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Case Study

- the lack or the total absence of capacity design provisions and details;
- realized with the philosophy of strong-beams and weak-columns;
- the presence of vertical loads resisting frames in only one of the main directions of the building;
- the lack of confinement effects in the panel joint regions;

## 3. Ground Motion Selection and Scaling

_{1}/PGA

_{2}ratio varies between 0.5 and 2.0, as shown in Table 1. Figure 2 shows the horizontal geometric mean (geo-mean) of elastic acceleration response spectra for a 5% soil damping ratio (grey lines) and mean of the 30 response spectra (red line). Each record pair was applied twice to the structure by rotating 90°. Therefore, the largest spectral acceleration was applied first in the X direction and then in the Y direction.

## 4. Finite Element Modelling

_{y}, Θ

_{y}) and at the failure point, i.e., the ultimate values, (M

_{u}, Θ

_{u}). The main points of the backbone curves of the plastic hinges were calculated as recommended in [2] for existing buildings, with the bending moment in the columns calculated by taking into account the axial compression forces due to the presence of gravitational loading.

_{c}= 24 MPa and E

_{c}= 30.94 GPa (respective compression stress strength and Young modulus) for concrete, whereas the values f

_{y}=375 MPa and E

_{s}= 210 GPa (respective yielding stress and Young modulus) have been assumed for the steel of the reinforcement bars. At the ground level, the columns were fully clamped, neglecting the soil-structure interaction. Masses corresponding to structural dead loads, non-structural dead loads and live loads were considered as equivalent distributed masses on the beams.

- the ultimate rotation for a column;
- the ultimate rotation for a beam;
- the displacement capacity in one of the shear sliding hinges;
- an interstorey drift ratio equal to 5%.

## 5. Structural Analysis Results

_{1X}= 0.40 s and T

_{1Y}= 0.75 s, respectively, in the X and Y directions.

_{1}): reporting the trend of 50% (solid red line), 16%, and 84% percentiles (dashed red lines) for drift distribution and the results of the single time-history (blue circles).

## 6. Damage States and Fragility Curves

_{i}[46,47].

_{y}, significant damage state corresponds to ¾ of the rotational capacity θ

_{u}and collapse limit condition rotation corresponds to the attainment of the column rotational capacity θ

_{u}. Then, three-chord rotation limit values according to the geometrical and mechanical characteristics of the vertical seismic-resistant structural elements can be calculated. Table 2 indicates the calculated values for the case study and their association to the damage state of EMS-98. Therefore, in order to use the thresholds adopted by Borzi et al. in 2008, the IDR is assumed to be comparable to the required columns’ chord rotation neglecting the contribution of joints and beams.

## 7. Conclusions

- The non-linear three-dimensional model together with the bi-directional ground motion allowed for highlighting a different seismic behavior of the structure in the two main directions (X and Y), revealing a higher vulnerability in the Y direction with respect to X. The RC bare frame in the Y direction reaches collapse at 30% in 50 years of HLs, whereas in the X direction, at the same HLs, the first cracking condition in some external joints is achieved.
- The non-linear behaviours attributed to structural elements account for shear and flexural behavior of beams and columns, and the moment-rotation relationship attributed to the joint panel allowed us to underline the different activation sequences in the two main directions of the building. In particular, it is possible to identify a structural behavior governed by the bending failure of beams and columns in the Y direction and a behavior controlled by the shear failure of the joint in the Y direction.
- The damage thresholds are defined following two criteria: the first one proposed by HAZUS-MH MR5 (2009), with given values of IDR depending on the different classes of RC frame buildings; the second one [46] is a local criterion in which the thresholds are defined on the basis of the ultimate and yielding rotation of columns. The adopted criteria significantly affect the fragility curves shape and the parameters values of the lognormal distribution adopted to fit the numerical points. In particular, for the damage state DS2–DS3 and DS4, the HAZUS criterion is more conservative, leading to a higher vulnerability characterized by lower median values of the parameter distribution. Contrarily, for DS5, the local criterion is more restrictive.
- Consistent with the numerical results, the fragility curves show a higher vulnerability for the Y direction of the building, with lognormal distribution median values lower than in X direction for both damage thresholds criteria. Nevertheless, with the local criterion, the slope of the functions is more marked, showing that the overcoming of the damage state occurs nearly always for the various time-history analyses and the different HLs.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 2.**Main characteristics of the adopted accelerograms. (

**a**) Elastic acceleration response spectra; (

**b**) PGA

_{1}/PGA

_{2}Vs. R (distance to fault).

**Figure 4.**Non-linear behavior adopted in the FE models. (

**a**) Zero-length elements at the extremities of beams and columns; (

**b**) Plastic hinges at the extremities of beams and columns; (

**c**) Zero-length elements simulating the non-linear behavior of the joint panels.

**Figure 5.**Median values of IDR for the time-history analyses at (

**a**) 63%, (

**b**) 30% and (

**c**) 10% in 50 years of HLs (solid line X direction, dashed line Y direction).

**Figure 6.**Peak interstorey drift ratio (IDR) vs Sa (T

_{1}). (

**a**) Case I with IDR measured in X direction; (

**b**) Case II with IDR measured in Y direction. The green triangles indicate the results of the first time-history of Table 1.

**Figure 7.**Deformability contribution of joint panel, upper column and lower column to the total rotation on an external node of the first floor during the first time-history analyses of the intensity level at 30% in 50 years HL. (

**a**) Time-step corresponding to the attainment of the maximum interstorey drift ratio (instant 22.23 s of the time history); (

**b**) Time-step corresponding to the attainment of the maximum rotation at the investigated joint panel (instant 27.35 s of the time history); (

**c**) Example of rotation recorded for the joint panel, upper column and lower column; (

**d**) Histograms diagram showing the distribution of the percentage values of the joint panel contribution during the time-history analyses.

**Figure 8.**Activation sequence of the different non-linear behaviours attributed to the different zero-length elements.

**Figure 9.**Fragility curves with damage thresholds defined according to HAZUS (

**a**,

**b**) and CR criterion (

**c**,

**d**) for X (

**a**,

**c**) and Y (

**b**,

**d**) directions.

**Figure 10.**Median exp(μ) of fragility curves for X and Y direction and for the adopted IDR damage state thresholds.

**Table 1.**Time history records adopted for the analysis (* Moment magnitude; ** Closest distance to fault rupture; *** Fault mechanism, where R: reverse; SS: strike-slip; RO: reverse-oblique).

No. | Event Name | Station | M * | R ** [km] | Mech. *** | PGA_{1} [g] | PGA_{2} [g] |
---|---|---|---|---|---|---|---|

1 | San Fernando, 1971 | LA-Hollywood Stor FF | 6.61 | 22.77 | R | 0.225 | 0.163 |

2 | Imperial Valley-06, 1979 | Parachute Test Site | 6.53 | 12.69 | SS | 0.113 | 0.206 |

3 | Superstition Hills-02, 1987 | Brawley Airport | 6.54 | 17.03 | SS | 0.131 | 0.111 |

4 | Superstition Hills-02, 1987 | Poe Road (temp) | 6.54 | 11.16 | SS | 0.475 | 0.286 |

5 | Spitak_Armenia, 1988 | Gukasian | 6.77 | 23.99 | RO | 0.200 | 0.174 |

6 | Loma Prieta, 1989 | Coyote Lake Dam-SW | 6.93 | 19.97 | RO | 0.132 | 0.280 |

7 | Loma Prieta, 1989 | Fremont—Emerson Court | 6.93 | 39.66 | RO | 0.192 | 0.099 |

8 | Landers, 1992 | Mission Creek Fault | 7.28 | 26.96 | SS | 0.097 | 0.132 |

9 | Northridge-01, 1994 | LA—Pico & Sentous | 6.69 | 27.82 | R | 0.103 | 0.186 |

10 | Northridge-01, 1994 | LA—S. Vermont Ave | 6.69 | 27.89 | R | 0.137 | 0.068 |

11 | Northridge-01, 1994 | LA—Temple & Hope | 6.69 | 28.82 | R | 0.124 | 0.165 |

12 | Kobe_Japan, 1995 | Abeno | 6.90 | 24.85 | SS | 0.149 | 0.231 |

13 | Denali_Alaska, 2002 | Carlo (temp) | 7.90 | 49.94 | SS | 0.081 | 0.084 |

14 | San Simeon_CA, 2003 | Cambria-Hwy1 Caltrans Bridge | 6.52 | 6.97 | R | 0.179 | 0.126 |

15 | Niigata_Japan, 2004 | FKS028 | 6.63 | 30.11 | R | 0.135 | 0.170 |

16 | Niigata_Japan, 2004 | NIG023 | 6.63 | 25.33 | R | 0.405 | 0.248 |

17 | Chuetsu-oki_Japan, 2007 | Nadachiku Joetsu City | 6.80 | 35.79 | R | 0.119 | 0.155 |

18 | Chuetsu-oki_Japan, 2007 | Tokamachi Chitosecho | 6.80 | 25.35 | R | 0.201 | 0.251 |

19 | Chuetsu-oki_Japan, 2007 | Kawaguchi | 6.80 | 23.63 | R | 0.147 | 0.147 |

20 | Chuetsu-oki_Japan, 2007 | NIG022 | 6.80 | 37.79 | R | 0.155 | 0.126 |

21 | Iwate_Japan, 2008 | IWT010 | 6.90 | 16.26 | R | 0.226 | 0.289 |

22 | Iwate_Japan, 2008 | Kami_ Miyagi Miyazaki City | 6.90 | 25.15 | R | 0.117 | 0.156 |

23 | Iwate_Japan, 2008 | Iwadeyama | 6.90 | 20.77 | R | 0.269 | 0.354 |

24 | Iwate_Japan, 2008 | Oomagari Hanazono-cho_Daisen | 6.90 | 46.32 | R | 0.093 | 0.127 |

25 | Iwate_Japan, 2008 | Mizusawaku Interior O ganecho | 6.90 | 7.82 | R | 0.361 | 0.257 |

26 | Darfield_New Zealand, 2010 | DFHS | 7.00 | 11.86 | SS | 0.275 | 0.333 |

27 | Darfield_New Zealand, 2010 | DORC | 7.00 | 29.96 | SS | 0.070 | 0.084 |

28 | Darfield_New Zealand, 2010 | OXZ | 7.00 | 30.63 | SS | 0.119 | 0.105 |

29 | Darfield_New Zealand, 2010 | RKAC | 7.00 | 13.37 | SS | 0.167 | 0.191 |

30 | Cucapah_Mexico, 2010 | El Centro Array #4 | 7.20 | 35.08 | SS | 0.238 | 0.310 |

Reference | DS1 | DS2 | DS3–DS4 | DS5 |
---|---|---|---|---|

HAZUS MH-MR5 TM (2009) | 0.4% | 0.6% | 1.6% | 4.0% |

Chord Rotation CR [49] | - | 1.1% (θ_{y}) | 2.5% (3/4 θ_{u}) | 3.3% (θ_{u}) |

**Table 3.**Parameters of lognormal fragility curves for X and Y direction and for the adopted IDR damage states thresholds.

HAZUS | CR Criterion | |||||||
---|---|---|---|---|---|---|---|---|

Damage State | Drift X | Drift Y | Drift X | Drift Y | ||||

Exp(μ) [g] | β | Exp(μ) [g] | β | Exp(μ) [g] | β | Exp(μ) [g] | β | |

DS1 | 0.078 | 0.213 | 0.064 | 0.207 | - | - | - | - |

DS2 | 0.103 | 0.213 | 0.096 | 0.207 | 0.138 | 0.196 | 0.109 | 0.067 |

DS3–DS4 | 0.160 | 0.338 | 0.105 | 0.207 | 0.197 | 0.306 | 0.109 | 0.067 |

DS5 | 0.270 | 0.338 | 0.153 | 0.207 | 0.238 | 0.368 | 0.133 | 0.054 |

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**MDPI and ACS Style**

Zucconi, M.; Bovo, M.; Ferracuti, B.
Fragility Curves of Existing RC Buildings Accounting for Bidirectional Ground Motion. *Buildings* **2022**, *12*, 872.
https://doi.org/10.3390/buildings12070872

**AMA Style**

Zucconi M, Bovo M, Ferracuti B.
Fragility Curves of Existing RC Buildings Accounting for Bidirectional Ground Motion. *Buildings*. 2022; 12(7):872.
https://doi.org/10.3390/buildings12070872

**Chicago/Turabian Style**

Zucconi, Maria, Marco Bovo, and Barbara Ferracuti.
2022. "Fragility Curves of Existing RC Buildings Accounting for Bidirectional Ground Motion" *Buildings* 12, no. 7: 872.
https://doi.org/10.3390/buildings12070872