# Influence of the Plan Structural Symmetry on the Non-Linear Seismic Response of Framed Reinforced Concrete Buildings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

- Interstory drifts;
- Roof rotations;
- Demand–capacity ratios of structural elements.

#### 2.1. Description of the Archetypes

#### Geometry of the Archetypes

#### 2.2. Analysis Type

- Nonlinear static analysis with incremental pushover;
- Nonlinear dynamic time-history analysis using artificial accelerograms;
- Nonlinear dynamic time-history analysis using recorded events.

#### 2.2.1. Static Non-Linear Analysis

#### 2.2.2. Time-History Non-Linear Dynamic Analysis Using Artificial Accelerograms

#### 2.2.3. Time-History Non-Linear Dynamic Analysis Using Seismic Records

## 3. Results and Discussion

#### 3.1. Results of Linear Analysis

#### 3.2. Results from Non-Linear Analysis

#### 3.2.1. Interstory Drift Control

#### 3.2.2. Acceptance Criteria Based on the Transient Interstory Drift

#### 3.2.3. Rotation Control

#### 3.2.4. Damage Control

## 4. Conclusions

- Influence of Asymmetry on Dynamic Response: The linear analysis indicated a significant impact of asymmetry on the dynamic response of the buildings, a factor that becomes increasingly critical as the slope of the terrain, and hence the irregularity of the buildings in both plan and elevation, increases. This underscores the need for considering a higher number of vibration modes to accurately capture the seismic behavior of structures located on sloped terrains.
- Nonlinear Ductility and Capacity: The nonlinear analyses, particularly with incremental pushover, demonstrated that the archetypes exhibited adequate ductility in both the X and Y directions. This finding is pivotal, as it suggests that despite the geometric complexities introduced by slope-induced asymmetry, the designed archetypes can still achieve the desirable level of seismic performance.
- Transient Interstory Drift and Damage Control: The dynamic nonlinear analysis, essential for closely approximating the seismic response of buildings, highlights the significance of transient interstory drift as a critical measure of seismic damage. This study confirmed that the maximum allowable drift values, as dictated by ASCE 7, were not exceeded, ensuring that all the dynamic time-history analyses results can be considered reliable for performance evaluation.
- Acceptance Criteria and Structural Design Suitability: The application of ASCE 7 acceptance criteria based on transient interstory drift offers a robust framework for evaluating the seismic suitability of structural designs. Our findings reveal that despite the potential increases in drift due to architectural asymmetry, the structural integrity remains within safe limits, affirming the effectiveness of conventional design practices in mitigating seismic risks.
- Rotations Control and Irregularity Effects: Contrary to expectations, archetypes with higher degrees of irregularity (45°) did not exhibit the highest rotations, challenging the preconceived notions about the relationship between structural asymmetry and torsional responses. This suggests that the presence of structural supports and the method of displacement restriction play a significant role in mitigating potential damage, especially in highly irregular structures.
- Comprehensive Damage Analysis: The extensive damage control analysis, considering both demand versus capacity ratios and specific architectural features like short columns and reduced stiffness on upper stories, provides a nuanced understanding of seismic vulnerability. This approach allows for a detailed examination of how structural and non-structural elements contribute to the overall seismic resilience of buildings on sloped terrains.
- Incorporation into Current Standards: There is a pressing need for current seismic design standards to include provisions that directly address the design of buildings located on slopes, ensuring that their dynamic response and the irregularity caused by asymmetry are considered in conventional design procedures. This inclusion would enhance the seismic safety and performance of such structures, acknowledging the unique challenges posed by sloped terrains.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Elevation views of the archetypes: (

**a**) 0° archetype, (

**b**) 15° archetype, (

**c**) 30° archetype, and (

**d**) 45° archetype. In this figure, A, B, C and D are structural axis.

**Figure 3.**Response spectra of the matched records, with the SB soil elastic design spectrum and the mean spectrum.

**Figure 5.**Capacity curves of the analyzed archetypes: (

**a**) 0° Archetype, (

**b**) 15° Archetype, (

**c**) 30° Archetype, (

**d**) 45° Archetype. In this figure, the light green dot represents the performance point for the limit state of the operational level (1-A), while the dark green, orange, and red dots represent the limit states of immediate occupancy (1-B), life safety (3-C), and collapse prevention (5-D), respectively.

**Figure 6.**Results of the pushover analyses: (

**a**) ductility and (

**b**) overstrength. In this figure, the blue bars represent the drifts of the Basic Archetype, the light blue bars represent the drifts of the 15° Archetype, the green bars represent the drifts of the 30° Archetype, and the yellow bars represent the drifts of the 45° Archetype.

**Figure 7.**Maximum interstory drift in X direction of columns (

**a**) C1, (

**b**) C5, (

**c**) C16, and (

**d**) C20. In this figure, each color represents the interstory drifts calculated using each pair of seismic records.

**Figure 8.**Maximum interstory drift in Y direction of columns (

**a**) C1, (

**b**) C5, (

**c**) C16, and (

**d**) C20. In this figure, each color represents the interstory drifts calculated using each pair of seismic records.

**Figure 14.**Elements that exceed the limit for the demand-capacity relationship in the archetypes (

**a**) 0°, (

**b**) 15°, (

**c**) 30°, and (

**d**) 45° subjected to the action of synthetic accelerograms for the Immediate Occupancy limit state (1-B).

Parameter | Concrete G30 |
---|---|

Compression strength (MPa) | 38.00 |

Strength lower bound (MPa) | 30.00 |

Tension strength (MPa) | 2.90 |

Modulus of elasticity (MPa) | 28,973.00 |

Specific weight (kN/m^{3}) | 24.00 |

Parameter | Value |
---|---|

Modulus of elasticity $E$ (MPa) | 200,000.00 |

Yield strength ${F}_{y}$ (MPa) | 490.00 |

Strain hardening parameter (dimensionless) | 0.0050 |

Transition curve initial shape parameter (dimensionless) | 20.00 |

Transition curve shape calibrating coefficient A1 (dimensionless) | 18.50 |

Transition curve shape calibrating coefficient A2 (dimensionless) | 0.15 |

Transition curve shape calibrating coefficient A3 (dimensionless) | 0.00 |

Transition curve shape calibrating coefficient A4 (dimensionless) | 1.00 |

Fracture/buckling strain (dimensionless) | 1.00 |

Specific weight (kN/m^{3}) | 78.00 |

Columns | Beams X and Y | ||||||
---|---|---|---|---|---|---|---|

Archetype | 1-2 | 3-4 | 5-6-7 | 8-9 | 1-2-3 | 4-5-6 | 7-8-9 |

Basic | 70 × 70 | 60 × 60 | 50 × 50 | 40 × 40 | 30 × 70 | 30 × 60 | 30 × 50 |

15° | 70 × 70 | 60 × 60 | 50 × 50 | 40 × 40 | 30 × 70 | 30 × 60 | 30 × 50 |

30° | 70 × 70 | 60 × 60 | 50 × 50 | 40 × 40 | 30 × 70 | 30 × 60 | 30 × 50 |

45° | 70 × 70 | 60 × 60 | 50 × 50 | 40 × 40 | 30 × 70 | 30 × 60 | 30 × 50 |

**Table 4.**Definition of performance limit states with their corresponding associated exceedance probabilities.

EarthquakeHazard Level | Exceedance Probability | Target Building Performance Level | |||

1-A | 1-B | 3-C | 5-D | ||

50% in 50 years | a | b | c | d | |

20% in 50 years | e | f | g | h | |

5% in 50 years | i | j | k | l | |

2% in 50 years | m | n | o | p |

Earthquake | Date | $\mathbf{Magnitude}\left({\mathit{M}}_{\mathit{w}}\right)$ | Station | Epicentral Distance (km) | Component | PGA (cm/sec^{2}) |
---|---|---|---|---|---|---|

Maule | 27-02-2010 | 8.8 | Angol | 209 | E–W | 684 |

N–S | 916 | |||||

Maule | 27-02-2010 | 8.8 | Concepción San Pedro | 109 | 97 | 598 |

7 | 667 | |||||

Maule | 27-02-2010 | 8.8 | Constitución | 70 | E–W | 530 |

N–S | 618 | |||||

Maule | 27-02-2010 | 8.8 | Llolleo | 274 | E–W | 324 |

N–S | 549 | |||||

Maule | 27-02-2010 | 8.8 | Santiago Maipú | 69 | E–W | 481 |

N–S | 549 | |||||

Coquimbo | 16-09-2015 | 8.3 | El pedregal | 92 | 90 | 677 |

360 | 561 | |||||

Coquimbo | 16-09-2015 | 8.3 | Tololo | 175 | 90 | 234 |

360 | 338 | |||||

Coquimbo | 16-09-2015 | 8.3 | San Esteban | 168 | 90 | 268 |

360 | 182 | |||||

Puerto Quellón | 25-12-2016 | 7.6 | Loncomilla | 136 | 90 | 136 |

360 | 148 | |||||

Puerto Quellón | 25-12-2016 | 7.6 | Hotel Espejo de Luna | 75 | 90 | 371 |

360 | 350 | |||||

Valparaíso | 24-07-2015 | 6.9 | Torpederas | 39 | 90 | 889 |

360 | 731 |

**Table 6.**Normalized displacements of the performance points calculated for the different archetypes and limit states.

Limit State | Basic | 15° | 30° | 45° | ||||
---|---|---|---|---|---|---|---|---|

X (%) | Y (%) | X (%) | Y (%) | X (%) | Y (%) | X (%) | Y (%) | |

Operational Level (1-A) | 0.428 | 0.449 | 0.370 | 0.391 | 0.362 | 0.375 | 0.261 | 0.265 |

Immediate Occupancy (1-B) | 0.571 | 0.599 | 0.493 | 0.522 | 0.483 | 0.499 | 0.354 | 0.356 |

Life Safety (3-C) | 0.713 | 0.749 | 0.617 | 0.652 | 0.603 | 0.624 | 0.451 | 0.450 |

Collapse Prevention (5-D) | 0.856 | 0.899 | 0.740 | 0.783 | 0.724 | 0.749 | 0.553 | 0.546 |

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**MDPI and ACS Style**

Vielma-Quintero, J.C.; Diaz-Segura, E.G.; Vielma, J.C.
Influence of the Plan Structural Symmetry on the Non-Linear Seismic Response of Framed Reinforced Concrete Buildings. *Symmetry* **2024**, *16*, 370.
https://doi.org/10.3390/sym16030370

**AMA Style**

Vielma-Quintero JC, Diaz-Segura EG, Vielma JC.
Influence of the Plan Structural Symmetry on the Non-Linear Seismic Response of Framed Reinforced Concrete Buildings. *Symmetry*. 2024; 16(3):370.
https://doi.org/10.3390/sym16030370

**Chicago/Turabian Style**

Vielma-Quintero, Juan Carlos, Edgar Giovanny Diaz-Segura, and Juan Carlos Vielma.
2024. "Influence of the Plan Structural Symmetry on the Non-Linear Seismic Response of Framed Reinforced Concrete Buildings" *Symmetry* 16, no. 3: 370.
https://doi.org/10.3390/sym16030370