# Fragility Assessment of RC Bridges Exposed to Seismic Loads and Corrosion over Time

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fragility Estimation

## 3. Corrosion Assessment

#### 3.1. Corrosion Initiation Time

#### 3.2. Corrosion Evolution

#### 3.3. Cracking Initiation Time

## 4. Cumulative Damage Estimation

Algorithm 1 The pseudocode of cumulative damage |

1: Begin 2: $n$ bridges with uncertain properties are generated 3: Realizations of seismic occurrences associated with each bridge model are generated 4: Time thresholds of interest, $m$, associated with corrosion are estimated 5: Different time stages, $T$, are selected 6: Initialize counters $i=1$, $k=1$ and ${t}_{0}=0$ 7: while $k\le n$ 8: while $i\le m$ 9: $t={t}_{0}+\mathsf{\Delta}{t}_{i+1}$ 10: while $t\le T$ 11: if $i=1$ 12: The $i-\mathrm{th}$ and $\left(i+1\right)-\mathrm{th}$ intensities are associated with the $k-\mathrm{th}$ structural model 13: Two seismic records are associated with the $i-\mathrm{th}$ and $\left(i+1\right)-\mathrm{th}$ intensities 14: Each record is modified by a factor ${\psi}_{e}={i}_{sim}/{i}_{T}$ that relates the intensity and the value of spectral acceleration at the fundamental period of the $k-\mathrm{th}$ system 15: $D{i}_{corr}{}_{\left|y,t\right.}$ of the $k-\mathrm{th}$ system is calculated 16: A random ground motion, ${S}_{ki}$, is modified by a factor, $\beta m$, that matches $D{i}_{corr}{}_{\left|y,t\right.}$ 17: else 18: A random seismic record, ${r}_{i}$, is associated with the $\left(i+2\right)-\mathrm{th}$ simulated intensity and is scaled by the factor ${\psi}_{e}$ 19: The system is subjected to a seismic signal composed of the seismic record, ${S}_{ki}$, and the seismic record, $ri$ 20: $D{i}_{corr}{}_{\left|y,t\right.}$ of the $k-\mathrm{th}$ system is calculated 21: A ground motion, ${S}_{k\left(i+1\right)}$, is selected randomly, and it is modified by a factor, $\beta m$, that matches $D{i}_{corr}{}_{\left|y,t\right.}$ 22: A reduction of the cross-sectional area of the reinforcement steel is performed 23: The ground motion ${S}_{k\left(i+1\right)}$ at the stage $t$ is scaled up until the structure fails 24: add one to the intensities counter 25: add one to the simulated bridges counter 26: end |

## 5. Illustrative Example

#### 5.1. Uncertainties for RC Bridges

#### 5.2. Nonlinear Modelling

#### 5.3. Waiting Times and Intensities

^{2}associated with an exceedance rate equal to ${v}_{0}=$0.05081; ${y}_{max}$ represents the last intensity of the SHC; $r$ and $\epsilon $ fitted the SHC. It is assumed that seismic occurrences follow a Poisson-type process so that the waiting times between events are distributed exponentially [63]. After some mathematical steps about CDF, the time occurrence of seismic loads is ${T}_{i}=-\left|\mathrm{ln}\left(u\right)/{v}_{0}\right|$ where $u$ is estimated based on a uniform distribution [13].

#### 5.4. Seismic Loadings

#### 5.5. Structural Demand over Time

^{2}[65] and 0.523 and 0.622 [53], respectively. For stages less than ${T}_{corr}$, the reinforcement cross-sectional area is intact. On the other hand, the area of the reinforcement bars is reduced for stages greater than ${T}_{corr}$ (Equation (4)). The statistical parameters shown in Table 4 and Table 5, which are associated with the $i-\mathrm{th}$ structural model, are used to simulate both corrosion initiation and cracking times. The mean value of corrosion initiation time, ${\widehat{T}}_{corr}$, is equal to 45 years, and the mean cracking time is equal to 57 years. The 75 years threshold refers to the life span of bridges based on the AASHTO code [66]. Time thresholds of 100, 125 and 150 years are considered to observe the evolution of the corrosion deterioration. The structural demand is obtained based on the cumulative damage process described previously. One hundred realizations of waiting times and seismic intensities associated with one hundred structural models with uncertain properties are considered. The structural demand is estimated by means of nonlinear, step-by-step, dynamic analysis. Figure 7 shows an example of the global response of the system after 50 years, expressed in terms of global drift versus base shear, considering both cases: seismic sequences (S) and seismic sequences plus the effect of corrosion (S + C). At 50 years, the corrosion process has begun, and only the diameter of the steel reinforcement is reduced. It is noticed that the structural response under the effect of S + C presents both a greater reduction in stiffness and an increase in global drift compared with the case in which the system is subjected only to seismic loads S. In the case of S + C, corrosion affects the moment–curvature relationship because it is estimated considering the reduced diameter at time $t$, $H\left(t\right)$. Then, there is a reduction in both the yield moment and ultimate moment, which explains the differences between S versus S + C on the structural response. On the other hand, the parameters that contribute to the response of the system are earthquakes with a high magnitude whose dominant period of their response spectra is close to the dominant period of the structure. In the case of corrosion, its initiation time is reduced with high values of both critical ion concentration, ${C}_{cr}$, and temperature, $\varphi $. Cracking time is reduced in the case of either a high corrosion rate, ${i}_{corr}$, or low steel density, ${\rho}_{steel}$.

#### 5.6. Fragility Curves over Time

## 6. Research Significance

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Bridge structure located near to Mexican Pacific coast: (

**a**) longitudinal view and (

**b**) structural element with deterioration.

**Figure 7.**Global structural response due to seismic loads (S) and due to seismic loads plus corrosion (S + C).

**Figure 10.**Median demand for stages of 0, 50, 75, 100, 125 and 150 years due to seismic loads (S) and due to seismic loads plus corrosion (S + C).

Material | Nominal Resistance (MPa) | Distribution | Mean (MPa) | C.V. | Reference |
---|---|---|---|---|---|

Concrete | 27.60 | Normal | 34.22 | 0.15 | [54] |

31.00 | Normal | 37.21 | 0.14 | ||

41.40 | Normal | 47.61 | 0.125 | ||

Steel reinforcement | 412 | Normal | 448.85 | 0.0369 | [55] |

**Table 2.**Geometric uncertainties of structural elements [55].

Distribution | Bias Factor | C.V. | |
---|---|---|---|

Slab element | Normal | +7.62 × 10^{−4} | 6.60 × 10^{−3} |

Beam height | Normal | −5.334 × 10^{−3} | 6.35 × 10^{−3} |

Beam width | Normal | +2.54 × 10^{−3} | 3.81 × 10^{−3} |

Column dimension | Normal | +1.524 × 10^{−3} | 6.35 × 10^{−3} |

Cover | Normal | +8.128 × 10^{−3} | 4.318 × 10^{−3} |

**Table 3.**Uncertainties for structural and nonstructural elements [56].

Distribution | Bias Factor | C.V. | |
---|---|---|---|

Factory items | Normal | 1.03 | 0.08 |

Site elements | Normal | 1.05 | 0.10 |

Asphalt | Normal | 0.075 * | 0.25 |

Nonstructural elements | Normal | 1.03–1.05 | 0.08–0.01 |

Parameter | Distribution | Mean | Standard Deviation | Reference |
---|---|---|---|---|

$\mathrm{Cover},{d}_{0}$ (m) | Normal | 8.128 × 10^{−3} | 4.318 × 10^{−2} | [55] |

$\mathrm{Chloride}\mathrm{concentration}\mathrm{in}\mathrm{the}\mathrm{exposed}\mathrm{zone},{C}_{0}$ (%) | Normal | 10.918 × 10^{−2} | 6.56 × 10^{−2} | [57] |

$\mathrm{Initial}\mathrm{chloride}\mathrm{concentration},{C}_{i}$ (%) | Deterministic | 0.00 | - | [25] |

$\mathrm{Critical}\mathrm{concentration}\mathrm{of}\mathrm{chloride}\mathrm{ions},{C}_{cr}$ (%) | Uniform | 2.5 × 10^{−2} | 3.75 × 10^{−2} | [57] |

$\mathrm{Temperature}$ (°C) | Normal | 27.92 | 1.47 | [58] |

Parameter | Distribution | Mean | Standard Deviation | Reference |
---|---|---|---|---|

$\mathrm{Rust}\mathrm{density},{\rho}_{rust}$ (ton/m^{3}) | Normal | 3.60 | 0.36 | [26] |

$\mathrm{Pore}\mathrm{cement}\mathrm{size},{t}_{pore}$ (mm) | Lognormal | 12.5 | 2.54 | [25] |

$\mathrm{Diameter}\mathrm{of}\mathrm{rebar}\mathrm{for}\mathrm{cap}\mathrm{beams},{\eta}_{0beam}$ (m) | Normal | 2.5 × 10^{−2} | $\pm $4 | [59] |

$\mathrm{Diameter}\mathrm{of}\mathrm{rebar}\mathrm{for}\mathrm{cap}\mathrm{beams},{\eta}_{0col}$ (m) | Normal | 3.2 × 10^{−2} | $\pm $4 | [59] |

$\mathrm{Steel}\mathrm{density},{\rho}_{steel}$ (ton/m^{3}) | Normal | 8.00 | 0.80 | [26] |

$\mathrm{Poisson}\mathrm{ratio},{\nu}_{c}$ | Deterministic | 0.25 | - | - |

y/g | Seismic Loads (S) | Seismic Loads Plus Corrosion (S + C) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 Years | 50 Years | 75 Years | 100 Years | 125 Years | 150 Years | 0 Years | 50 Years | 75 Years | 100 Years | 125 Years | 150 Years | |

0.10 | 6.9 × 10^{−3} | 1.6 × 10^{−2} | 2.6 × 10^{−2} | 3.7 × 10^{−2} | 4.6 × 10^{−2} | 5.9 × 10^{−2} | 1.3 × 10^{−2} | 2.8 × 10^{−2} | 4.6 × 10^{−2} | 7.2 × 10^{−2} | 9.8 × 10^{−2} | 1.2 × 10^{−1} |

0.20 | 1.2 × 10^{−2} | 2.1 × 10^{−2} | 3.3 × 10^{−2} | 4.7 × 10^{−2} | 5.7 × 10^{−2} | 6.9 × 10^{−2} | 2.8 × 10^{−2} | 5.1 × 10^{−2} | 7.14 × 10^{−2} | 9.1 × 10^{−2} | 1.3 × 10^{−1} | 1.4 × 10^{−1} |

0.30 | 1.8 × 10^{−2} | 3.4 × 10^{−2} | 4.7 × 10^{−2} | 6.1 × 10^{−2} | 7.7 × 10^{−2} | 9.8 × 10^{−2} | 3.8 × 10^{−2} | 7.7 × 10^{−2} | 1.1 × 10^{−1} | 1.4 × 10^{−1} | 1.6 × 10^{−1} | 1.9 × 10^{−1} |

0.40 | 2.3 × 10^{−2} | 3.8 × 10^{−2} | 6.0 × 10^{−2} | 8.8 × 10^{−2} | 1.10 × 10^{−1} | 1.27 × 10^{−1} | 7.1 × 10^{−2} | 1.0 × 10^{−1} | 1.3 × 10^{−1} | 1.5 × 10^{−1} | 1.7 × 10^{−1} | 1.9 × 10^{−1} |

0.50 | 3.3 × 10^{−2} | 5.9 × 10^{−}2 | 9.3 × 10^{−2} | 1.2 × 10^{−1} | 1.41 × 10^{−1} | 1.69 × 10^{−1} | 9.28 × 10^{−2} | 1.22 × 10^{−1} | 1.48 × 10^{−1} | 1.75 × 10^{−1} | 1.96 × 10^{−1} | 2.17 × 10^{−1} |

0.60 | 8.3 × 10^{−2} | 1.0 × 10^{−1} | 1.2 × 10^{−1} | 1.4 × 10^{−1} | 1.61 × 10^{−1} | 1.84 × 10^{−1} | 1.22 × 10^{−1} | 1.61 × 10^{−1} | 1.87 × 10^{−1} | 2.14 × 10^{−1} | 2.44 × 10^{−1} | 2.92 × 10^{−1} |

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**MDPI and ACS Style**

Herrera, D.; Tolentino, D.
Fragility Assessment of RC Bridges Exposed to Seismic Loads and Corrosion over Time. *Materials* **2023**, *16*, 1100.
https://doi.org/10.3390/ma16031100

**AMA Style**

Herrera D, Tolentino D.
Fragility Assessment of RC Bridges Exposed to Seismic Loads and Corrosion over Time. *Materials*. 2023; 16(3):1100.
https://doi.org/10.3390/ma16031100

**Chicago/Turabian Style**

Herrera, Daniel, and Dante Tolentino.
2023. "Fragility Assessment of RC Bridges Exposed to Seismic Loads and Corrosion over Time" *Materials* 16, no. 3: 1100.
https://doi.org/10.3390/ma16031100