# Seismic Fragility Analysis of Existing RC Frame Structures Strengthened with the External Self-Centering Substructure

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Information of RC Frame Structures

#### 2.1. Frame KJ1

^{2}. The length was 25.5 m in the east–west direction, and the width was 18.0 m in the north–south direction. The total height of the frame KJ1 was 14.8 m, and the storey heights from the ground floor to the top floor were 4.11 m, 3.30 m, 3.30 m and 4.09 m, respectively. The RC frame structure was located in a seismic precautionary intensity of 8 degrees. The site class was Class II. The design earthquake group was Group II. Additionally, the characteristic period value was 0.4 s.

#### 2.2. Frame KJ2

## 3. Establishment and Verification of Finite Element Models

#### 3.1. Finite Element Modelling Methods

#### 3.2. One-Story One-Span Frame Structures

#### 3.2.1. Test Overview

#### 3.2.2. Model Establishment of One-Story One-Span Frame Structures

#### 3.2.3. Model Verification of One-Story One-Span Frame Structures

#### 3.3. RC Frame Structures

#### 3.3.1. Model Establishment of RC Frame Structures

#### 3.3.2. Verification of RC Frame Structures

## 4. Earthquake Ground Motion Input

## 5. Incremental Dynamic Analysis of RC Frame Structures

_{PG}) was selected as the intensity measure (IM), and the maximum inter-story drift (θ

_{max}) was selected as the damage measure (DM). The peak ground acceleration underwent a total of 34 amplitude modulations during the IDA. The amplitude modulation increments of 0.1~1.0 g were 0.1 g; the amplitude modulation increments of 1.2~5.0 g were 0.2 g; and the amplitude modulation increments of 5.5~7.0 g were 0.5 g. In order to reduce the computational complexity and reflect various performance points on the IDA curves based on different seismic demand parameters, the ultimate state of IDA analysis was the point where the structure reached dynamic instability, the nonlinear time history analysis did not converge, the maximum inter-story drift reached 10%, or the slope reached the initial slope of 20% in the IDA curve. Additionally, the point with the smallest value was selected as the ultimate state of collapse.

#### 5.1. Distribution Diagram of Maximum Inter-Story Drifts

_{max}). Under different performance states, the reference value C

^{Δ}of the maximum inter-story drift is exhibited in Table 5.

#### 5.2. Distribution Diagram of Residual Inter-Story Drifts

_{R}) and gives the reference value C

^{Δ}for θ

_{R}in different performance states. In this paper, three performance points were selected for analysis, as shown in Table 6. Among them, DS1 corresponded to the minor repair state; DS2 corresponded to the major repair state; and DS3 corresponded to the collapse state.

#### 5.3. Incremental Dynamic Analysis

_{PG}was small, the initial stage of the IDA curve showed approximately linear growth, indicating that the structures were in an elastic state. As a

_{PG}increased, the slope of the IDA curve gradually decreased, indicating that the structures were in an elastic–plastic state.

_{max}, and the median and logarithmic standard deviation of θ

_{max}were calculated. Thus, fractile curves of 16%, 50%, and 84% were obtained. It could be seen that (1) the shape of IDA curves was different. The curves had characteristics of “excessive softening”, “excessive hardening”, and “fluctuation”. The IDA curve could comprehensively reflect the possible seismic responses of frame structures under different intensity levels. (2) Under the same earthquake direction, the difference between the mean values of θ

_{max}was significant for frames KJ1 and KJ2. The IDA curve of frame KJ2 showed overall data points moving to the left, which indicated that the seismic performance of the existing frame structure could be significantly improved by the enlarging cross-section method and the external self-centering substructure.

## 6. Seismic Fragility Analysis of RC Frame Structures

#### 6.1. Probabilistic Seismic Demand Models

_{PG}follow an exponential relationship [31], as shown in Equation (1):

_{d}of the logarithmic normal distribution function for the seismic demand capacity is presented in Equation (3):

_{i}is the result of the time history analysis (i = 1, 2, …, N).

_{d}was approximately 0.2, and the fitting effect was good.

#### 6.2. Seismic Fragility Curve of Frame Structures

_{c}is the logarithmic standard deviation of the seismic capacity of the frame structures, and the uncertainty was not considered.

_{PG}is 0.2 g, the exceedance probability of the IO state for frame KJ2 increases from 86.9% to 100.0%. When a

_{PG}is 0.4 g, the exceedance probability of the IO state increases from 0.9% to 96.9%. When a

_{PG}is 0.6 g, the exceedance probability of the IO state increases from 0.0% to 30.4%; the exceedance probability of the LS state increases from 19.8% to 69.6%; and the exceedance probability of the CP state decreases from 79.9% to 0.0%. When a

_{PG}is 1.0 g, the exceedance probability of the LS state increases from 0.0% to 77.4%; the exceedance probability of the CP state decreases from 55.6% to 22.6%; and the exceedance probability of the collapse state decreases from 44.4% to 0.0%. When a

_{PG}is 2.0 g, the exceedance probability of the CP state increases from 0.0% to 82.8%, and the exceedance probability of the collapse state decreases from 100.0% to 17.2%. It is evident that the exceedance probability of the frame structures is significantly reduced, and the safety margin is greatly increased.

## 7. Conclusions

- (1)
- The proposed modeling method was reasonable, and the error between the experiment results and finite element simulation results was small;
- (2)
- Under the same peak ground acceleration, the maximum inter-story drift of frame KJ1 decreased with the increase in storeys. However, the maximum inter-story drift of frame KJ2 moved upwards, and was significantly reduced;
- (3)
- When the peak ground acceleration was small, the IDA curves showed a linear increase. As the peak ground acceleration increased, the slope of the IDA curves gradually decreased;
- (4)
- Based on the results of IDA, the probabilistic seismic demand model curve was fitted, and the fitting effect was good;
- (5)
- When the same peak ground acceleration was applied to the frame structures, the exceedance probability of frame KJ2 was significantly lower than that of frame KJ1 at various performance levels.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Structural plan layout of frame KJ1. (

**a**) Plan layout of the first to the third storey; (

**b**) Plan layout of the fourth storey; (

**c**) Cross-sectional dimensions of beams and columns.

**Figure 3.**Load plan layout of frame KJ1. (

**a**) First storey; (

**b**) Second storey; (

**c**) Third storey; (

**d**) Fourth storey.

**Figure 4.**Structural plan layout of frame KJ2. (

**a**) Plan layout of the first to the third storey; (

**b**) Plan layout of the fourth storey; (

**c**) Cross-sectional dimensions of beams and columns.

**Figure 5.**Load plan layout of frame KJ2. (

**a**) First to second storey; (

**b**) Third storey; (

**c**) Fourth storey.

**Figure 7.**Schematic diagram and cross-section reinforcement of one-story one-span frame structures. (

**a**) K1-1; (

**b**) K1-2; (

**c**) K2-1; (

**d**) K2-2.

**Figure 9.**Finite element models of one-story one-span frame structures. (

**a**) K1-1 (

**b**) K1-2 (

**c**) K2-1 (

**d**) K2-2.

**Figure 10.**Comparisons of test and simulation results for one-story one-span frame structures. (

**a**) K1-1; (

**b**) K1-2; (

**c**) K2-1; (

**d**) K2-2.

**Figure 13.**Distribution diagram of maximum inter-story drifts. (

**a**) KJ1-x; (

**b**) KJ2-x; (

**c**) KJ1-y; (

**d**) KJ2-y.

**Figure 14.**Distribution diagram of residual inter-story drifts. (

**a**) KJ1-x; (

**b**) KJ2-x; (

**c**) KJ1-y; (

**d**) KJ2-y.

**Figure 15.**IDA curve diagram of the No. 1 earthquake ground motion record. (

**a**) x-direction earthquakes (

**b**) y-direction earthquakes.

**Figure 17.**Regression curve of probabilistic seismic demand models. (

**a**) KJ1-x; (

**b**) KJ2-x; (

**c**) KJ1-y; (

**d**) KJ2-y.

Concrete Strength Grade | f_{cu}/MPa | E_{c}/(×10^{4} N/mm^{2}) |
---|---|---|

C20 | 25.0 | 2.80 |

C40 | 39.3 | 3.25 |

Steel Type | f_{y}/MPa | E_{s}/(×10^{5} N/mm^{2}) |
---|---|---|

A6 | 293 | 2.05 |

A8 | 298 | 2.05 |

C12 | 497 | 2.01 |

C14 | 443 | 2.01 |

C18 | 459 | 2.00 |

A^{s}12.5 | 1680 | 1.95 |

Frame Number | Calculation Software | T1/s | T2/s | T3/s |
---|---|---|---|---|

KJ1 | YJK | 1.159 | 1.143 | 1.015 |

OpenSees | 0.966 | 0.954 | 0.869 | |

KJ2 | YJK | 0.511 | 0.443 | 0.382 |

OpenSees | 0.498 | 0.487 | 0.389 |

No. | Database Code | Year | Magnitude | PGA/Gal | Duration Time/s | Time Interval/s |
---|---|---|---|---|---|---|

1 | NGA_no_40_A-SON033 | 1968 | 6.63 | 40.252 | 39.995 | 0.005 |

2 | NGA_no_366_H-VC6090 | 1983 | 6.36 | 74.745 | 39.990 | 0.010 |

3 | NGA_no_1010_5082A-325 | 1994 | 6.69 | 247.560 | 55.325 | 0.005 |

4 | NGA_no_1184_CHY010-W | 1999 | 7.62 | 222.011 | 131.996 | 0.004 |

5 | NGA_no_2676_TTN024-V | 1999 | 6.20 | 4.225 | 34.995 | 0.005 |

6 | NGA_no_2721_CHY057-N | 1999 | 6.20 | 24.109 | 49.995 | 0.005 |

7 | NGA_no_2787_HWA039-V | 1999 | 6.20 | 15.959 | 48.995 | 0.005 |

8 | NGA_no_2914_TTN018-N | 1999 | 6.20 | 10.057 | 42.995 | 0.005 |

9 | NGA_no_3160_TCU014-N | 1999 | 6.20 | 14.362 | 61.990 | 0.005 |

10 | NGA_no_3291_CHY061-N | 1999 | 6.30 | 27.555 | 50.990 | 0.005 |

11 | NGA_no_3454_TCU046-N | 1999 | 6.30 | 25.643 | 47.945 | 0.005 |

12 | NGA_no_3485_TCU095-V | 1999 | 6.30 | 16.870 | 66.995 | 0.005 |

13 | 000538ZA | 1992 | 5.48 | 18.031 | 33.140 | 0.010 |

14 | 001967YA | 1985 | 5.50 | 8.349 | 13.080 | 0.010 |

15 | 006968ZA | 1999 | 6.20 | 3.267 | 33.780 | 0.010 |

16 | AKT016908222_M | 2017 | - | 95.404 | 95.000 | 0.020 |

17 | AKT0130807240026NS | 2008 | 6.80 | 23.963 | 174.000 | 0.010 |

18 | AKTH01618222_M | 2019 | - | 44.269 | 204.000 | 0.020 |

19 | AKTH02311144_M | 2011 | - | 70.392 | 300.000 | 0.020 |

20 | SZO0190908110507EW | 2009 | 6.50 | 96.006 | 138.000 | 0.010 |

Performance States | Immediate Occupancy (IO) | Life Safe (LS) | Collapse Prevention (CP) |
---|---|---|---|

θ_{max} | 1/100 | 1/50 | 1/25 |

Performance States | DS1 | DS2 | DS3 |
---|---|---|---|

θ_{R} | 1/500 | 1/200 | 1/100 |

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**MDPI and ACS Style**

Liu, W.; Zhang, J.; Liu, H.; Wang, F.; Liu, J.; Han, M.
Seismic Fragility Analysis of Existing RC Frame Structures Strengthened with the External Self-Centering Substructure. *Buildings* **2023**, *13*, 2117.
https://doi.org/10.3390/buildings13082117

**AMA Style**

Liu W, Zhang J, Liu H, Wang F, Liu J, Han M.
Seismic Fragility Analysis of Existing RC Frame Structures Strengthened with the External Self-Centering Substructure. *Buildings*. 2023; 13(8):2117.
https://doi.org/10.3390/buildings13082117

**Chicago/Turabian Style**

Liu, Weiheng, Jianwei Zhang, Hang Liu, Fei Wang, Juan Liu, and Mingjie Han.
2023. "Seismic Fragility Analysis of Existing RC Frame Structures Strengthened with the External Self-Centering Substructure" *Buildings* 13, no. 8: 2117.
https://doi.org/10.3390/buildings13082117