# Fragility Analysis of RC Frame Structures Subjected to Obliquely Incident Seismic Waves

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## Abstract

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## 1. Introduction

## 2. Model for Analysis

#### 2.1. Structural Model

#### 2.2. Foundation Soil Model

#### 2.3. Soil-Structure Interface Modeling

#### 2.4. Soil and Structural Damping

#### 2.5. Selection of Input Seismic Motions

## 3. Seismic Oblique Incidence on the Infinite Element Boundary

#### 3.1. Seismic Oblique Incidence in a Free Field

#### 3.2. The Equivalent Loads on the Infinite Element Boundary

## 4. Dynamic Response of Multi-Story RC Frame Structures Subjected to Obliquely Incident Seismic Waves

#### 4.1. The Influence of Oblique Incidence of SV Waves on the Column

#### 4.2. The Influence of Obliquely Incident SV-Waves on the Displacement of Each Story

## 5. Fragility Analysis of Multi-Story RC Frame Structures under Obliquely Incident Seismic Waves

#### 5.1. The Method of Fragility Analysis

#### 5.2. The Influence of Oblique Incidence of SV Waves on the Seismic Fragility of Structures

## 6. Conclusions

- For homogenous foundation soil, the internal force distribution within the structure is significantly asymmetrical, i.e., shear forces on the columns close to the seismic source are much larger than those far from the seismic source.
- Among the 4 specific different incident angles investigated in the case of homogenious foundation soil, the maximum inter-story displacement is smallest when the incident angle is 20° and largest when the incident angle is 30°. Moreover, this rule also applies to the fragility of structures. At each structural damage level, the exceedance probability is smallest when the incident angle is 20° and largest when 30°.
- For silty valley foundation, there is no obvious asymmetry of internal force distribution since the reflection and refraction of seismic waves at the terrain interface attenuate the influence of obliquely incident seismic waves. However, the valley terrain does magnify the dynamic response of the superstructure. Moreover, the fragility of structures built on a silty valley foundation soil is strongly influenced by the incident angles, particularly at the moderate damage level.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Soil-structure systems used in the simulation (unit: m): (

**a**) Homogeneous foundation soil model. (

**b**) Silty valley foundation soil model.

**Figure 4.**Details of the 18 selected earthquake motions: (

**a**) Spectral accelerations. (

**b**) (${M}_{s},R)$ pairs.

**Figure 7.**Comparison of the analytical and simulated shear stress-time series at the central point of the ground surface.

**Figure 8.**The distribution of maximum inter-story displacement along the height under the obliquely incident Coyote wave: (

**a**) Homogenous foundation soil case. (

**b**) Silty valley foundation soil case.

**Figure 9.**The seismic fragility curves of homogeneous foundation soil model at different seismic damage levels: (

**a**) Slight demage (

**b**) Moderate damage (

**c**) Severe damage (

**d**) Collapse.

**Figure 10.**The seismic fragility curves of silty valley model at different seismic damage levels: (

**a**) Slight demage (

**b**) Moderate damage (

**c**) Severe damage (

**d**) Collapse.

**Figure 11.**Differences of exceedance probability between the normal incidence and the oblique incidences at different seismic damage levels in the case of homogenous foundation soil: (

**a**) Slight demage (

**b**) Moderate damage (

**c**) Severe damage (

**d**) Collapse.

**Figure 12.**Differences of exceedance probability between the normal incidence and the oblique incidences at different seismic damage levels in the case of silty valley foundation soil: (

**a**) Slight demage (

**b**) Moderate damage (

**c**) Severe damage (

**d**) Collapse.

${\mathit{E}}_{\mathit{e}}$ (MPa) | $\mathit{\nu}$ | ${\mathit{f}}_{\mathit{c},\mathit{r}}$ (MPa) | ${\mathit{\u03f5}}_{\mathit{c},\mathit{r}}$ | ${\mathit{f}}_{\mathit{t},\mathit{r}}$ (MPa) | ${\mathit{\u03f5}}_{\mathit{t},\mathit{r}}$ |
---|---|---|---|---|---|

2.2 × 10${}^{-4}$ | 0.2 | 3.0 | 1.36 × 10${}^{-4}$ | 39.6 | 3.15 × 10${}^{-3}$ |

Soil Layer | $\mathit{\rho}$ (kg/m${}^{3}$) | E (MPa) | $\mathit{\nu}$ | c (kPa) | $\mathit{\varphi}$ (°) |
---|---|---|---|---|---|

Mud | 1820 | 149 | 0.49 | 5 | 20 |

Silt clay | 2000 | 329 | 0.49 | 8 | 25 |

**Table 3.**Peak shear force on each column in different foundation soil under the Coyote wave (unit: kN).

Foundation Soil | Incident Angle | Column A | Column B | Column C | Column D | ||||
---|---|---|---|---|---|---|---|---|---|

Top | Bottom | Top | Bottom | Top | Bottom | Top | Bottom | ||

Homogenous | $\alpha =0\xb0$ | 1.110 | 6.204 | 4.879 | 7.562 | 4.875 | 6.418 | 1.084 | 4.921 |

$\alpha =10\xb0$ | 1.005 | 6.038 | 3.380 | 7.208 | 3.603 | 6.107 | 0.546 | 4.498 | |

$\alpha =20\xb0$ | 0.569 | 4.175 | 3.152 | 4.216 | 3.749 | 3.950 | 0.826 | 3.327 | |

$\alpha =30\xb0$ | 0.884 | 7.796 | 3.560 | 7.756 | 3.572 | 5.605 | 1.083 | 3.522 | |

Silty valley | $\alpha =0\xb0$ | 1.542 | 6.875 | 6.088 | 7.494 | 6.017 | 6.847 | 1.497 | 6.384 |

$\alpha =10\xb0$ | 1.301 | 7.753 | 4.887 | 8.171 | 4.929 | 6.311 | 1.009 | 5.426 | |

$\alpha =20\xb0$ | 1.077 | 5.196 | 4.153 | 6.117 | 3.635 | 6.123 | 1.100 | 4.984 | |

$\alpha =30\xb0$ | 1.381 | 7.632 | 4.479 | 7.641 | 4.118 | 8.103 | 1.417 | 7.133 |

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**MDPI and ACS Style**

Huang, B.; Guo, J.; Liao, K.; Zhao, Y.
Fragility Analysis of RC Frame Structures Subjected to Obliquely Incident Seismic Waves. *Sustainability* **2021**, *13*, 1108.
https://doi.org/10.3390/su13031108

**AMA Style**

Huang B, Guo J, Liao K, Zhao Y.
Fragility Analysis of RC Frame Structures Subjected to Obliquely Incident Seismic Waves. *Sustainability*. 2021; 13(3):1108.
https://doi.org/10.3390/su13031108

**Chicago/Turabian Style**

Huang, Bo, Jiachen Guo, Kailong Liao, and Yu Zhao.
2021. "Fragility Analysis of RC Frame Structures Subjected to Obliquely Incident Seismic Waves" *Sustainability* 13, no. 3: 1108.
https://doi.org/10.3390/su13031108