# A New Method to Evaluate the Post-Earthquake Performance and Safety of Reinforced Concrete Structural Frame Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reinforced Concrete Frames under Consideration

#### 2.1. General Description

- (a)
- 1.35G + 1.50Q
- (b)
- 1.00G + ψQ + 1.00E
- (c)
- 1.00G + ψQ – 1.00E

_{RC}and ΣM

_{RB}are the sums of the design values for the moments corresponds to the columns and beams framing the joint, respectively. This demand is waived at the top floor of the frames of Group A.

_{ef}, were assumed, i.e., for columns I

_{ef}= 90%·I

_{g}and for beams I

_{ef}= 50%·I

_{g}where I

_{g}is the gross section moment of inertia [23].

#### 2.2. Nonlinear Modeling

^{T}is the tangent stiffness matrix and ${\ddot{u}}_{g}\left(t\right)$ denotes the ground acceleration. The solution of the dynamic equilibrium equation has been provided by the Ruaumoko program [20]. Beam and column elements were both modeled as nonlinear frame elements with lumped plasticity by defining plastic hinges at both ends. For the beams, axial forces were assumed to be practically zero, since all floor slabs are considered to be rigid in their plan to simulate the diaphragm action. Typical input data for strength that are essential for Ruaumoko [20] are the axial force -bending moment interaction relations for vertical members (columns) and maximum and minimum strength values associated with bending moments for horizontal members (beams). In this research study, a home-made design and analysis program was developed for each reinforced concrete section using the model of fibers. The cyclic behavior of all members was evaluated using the well-known Takeda hysteresis model which is available in Ruaumoko [20] and it is shown in Figure 1. The phenomenological parameters of Takeda’s model are mainly influenced by the end resistances of columns and beams, which are evidently dissimilar for the corresponding structures of Group B. It should be mentioned that due to inadequate seismic design provisions for Group B, all these structures appear to have stiffness and strength degradation after yielding. One can consult the Ruaumoko user manual by Carr [20] for an in-depth description of degradation factors and their effect on the shape of the hysteresis model.

_{ph}for each member is equal to half of its section’s height, H:

_{ph}= 0.5·H.

#### 2.3. Description of Structures and their Reinforcement Amount and Arrangement

## 3. Seismic Input

## 4. Results and Discussion

_{1}–a

_{3}are appropriate parameters which have been determined numerically to have the best fit for Equation (4). These parameters appear in Table 2, where the correlation coefficient R

^{2}is also provided.

_{max,Eq.4}= 149 mm since this value is very close to that from dynamic inelastic analysis, i.e., u

_{max,analysis}= 148 mm. Thus, the relative error is r.e. = 100%|148 − 149|/148 = 0.68%.

_{max,Eq.4}= 61 mm since this value is very close to that from dynamic inelastic analysis, i.e., u

_{max,analysis}= 60 mm. Thus, the relative error in this case is r.e. = 100%|61 − 60|/61 = 1.64%.

_{max}/H) with the residual roof drift (u

_{res}/H) and therefore can be used to assess the global performance level of a reinforced concrete frame system.

## 5. Conclusions

- The maximum seismic deformation of reinforced concrete structures can be successfully assessed using the pertinent residual deformation. The new method proposed herein can be applied for both regular and irregular (with setbacks) reinforced concrete buildings. Additionally, this new approach can be used for both well-designed and under-designed structures.
- The correlation coefficient between the ‘exact’ results from dynamic inelastic analyses and those produced by the proposed empirical relation is equal to 95.9% for the case of well-design reinforced concrete structures, and 95.1% for the set of under-designed structures. Therefore, the proposed empirical relation appears to be reliable for a direct and rapid evaluation of the maximum response.
- The effectiveness and the accuracy of the proposed method are additionally confirmed using two verification examples. It is found that the maximum displacements evaluated by the proposed approach are very close to those computed by dynamic inelastic analysis. Thus, the relative error between these methods for the first verification example is 0.68%, and for the second example is equal to 1.64%.
- In order to apply this method in 3-D structures, an extension of the proposed methodology is required in order to also consider deformation due to vertically (torsional) rotations. Furthermore, more analyses are required for reinforced concrete systems with shear walls.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The modified Takeda model (adopted from reference [20]).

**Figure 10.**Evaluation of maximum displacements for well-designed frames—Results from 224 dynamic inelastic analyses.

**Figure 11.**Evaluation of maximum displacements for poor-designed frames—Results from 224 dynamic inelastic analyses.

**Figure 15.**Time history of top displacement of Frame B1 under the Loma Prieta (18/10/1989) earthquake.

**Table 1.**Strong ground motions examined in this study (downloaded from reference [25]).

Date. | Event | Direction | Station | PGA (g) |
---|---|---|---|---|

20/9/1999 | Chi-Chi, Taiwan | N034 | TCU046 | 0.133 |

8/7/1986 | N. Palm Springs | NS | 12206 Silent Valley | 0.139 |

28/6/1992 | Landers | EW | 21081 Amboy | 0.146 |

2/5/1983 | Coalinga | EW | 36227 Parkfield | 0.147 |

25/4/1992 | Cape Mendocino | NS | 89509 Eureka | 0.154 |

9/2/1971 | San Fernando | N069 | 127 Lake Hughes #9 | 0.157 |

4/10/1987 | Whittier Narrows | NS | 24399 Mt Wilson - CIT Station | 0.158 |

17/8/1999 | Kocaeli, Turkey | EW | Atakoy | 0.164 |

17/1/1994 | Northridge | N005 | 90017 LA - Wonderland Ave | 0.172 |

20/9/1999 | Chi-Chi, Taiwan | NS | TAP103 | 0.177 |

7/6/1975 | Northern Calif | N150 | 1249 Cape Mendocino, Petrolia | 0.179 |

1/10/1987 | Whittier Narrows | NS | 24399 Mt Wilson - CIT Station | 0.186 |

15/10/1979 | Imperial Valley | N015 | 6622 Compuertas | 0.186 |

9/2/1971 | San Fernando | EW | 135 LA – Hollywood | 0.210 |

24/4/1984 | Morgan Hill | NS | 57382 Gilroy Array #4 | 0.224 |

17/8/1999 | Kocaeli, Turkey | EW | Gebze | 0.244 |

18/10/1989 | Loma Prieta | NS | 1028 Hollister City Hall | 0.247 |

17/1/1994 | Northridge | NS | 90019 San Gabriel - E. Gr. Ave. | 0.256 |

24/11/1987 | Superstition Hills(B) | NS | 01335 El Centro Imp. Co. Cent | 0.258 |

15/10/1979 | Imperial Valley | N012 | 6621 Chihuahua | 0.270 |

27/1/1980 | Livermore | EW | 57187 San Ramon | 0.301 |

20/9/1999 | Chi-Chi, Taiwan | EW | NST | 0.309 |

24/4/1984 | Morgan Hill | EW | 57382 Gilroy Array #4 | 0.348 |

26/4/1981 | Westmorland | NS | 5169 Westmorland Fire Sta | 0.368 |

17/1/1994 | Northridge | EW | 90057 Canyon Country | 0.410 |

18/10/1989 | Loma Prieta | NS | 47379 Gilroy Array #1 | 0.473 |

17/1/1994 | Northridge | NS | 90057 Canyon Country | 0.482 |

12/11/1999 | Duzce, Turkey | NS | Bolu | 0.728 |

a_{1} | a_{2} | a_{3} | R^{2} | |
---|---|---|---|---|

Well-designed frames A1–A4 | 0.002124 | −0.000322 | 1.1623 | 0.959 |

Poor-designed frames B1–B4 | 0.001982 | −0.000283 | 1.1498 | 0.951 |

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**MDPI and ACS Style**

Konstandakopoulou, F.; Hatzigeorgiou, G.; Evangelinos, K.; Tsalis, T.; Nikolaou, I.
A New Method to Evaluate the Post-Earthquake Performance and Safety of Reinforced Concrete Structural Frame Systems. *Infrastructures* **2020**, *5*, 16.
https://doi.org/10.3390/infrastructures5020016

**AMA Style**

Konstandakopoulou F, Hatzigeorgiou G, Evangelinos K, Tsalis T, Nikolaou I.
A New Method to Evaluate the Post-Earthquake Performance and Safety of Reinforced Concrete Structural Frame Systems. *Infrastructures*. 2020; 5(2):16.
https://doi.org/10.3390/infrastructures5020016

**Chicago/Turabian Style**

Konstandakopoulou, Foteini, George Hatzigeorgiou, Konstantinos Evangelinos, Thomas Tsalis, and Ioannis Nikolaou.
2020. "A New Method to Evaluate the Post-Earthquake Performance and Safety of Reinforced Concrete Structural Frame Systems" *Infrastructures* 5, no. 2: 16.
https://doi.org/10.3390/infrastructures5020016