# The Structural Reliability of the Usumacinta Bridge Using InSAR Time Series of Semi-Static Displacements

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

_{R}) of the bridge were extracted, respectively. Based on the results of the experimental part of the paper, the displacements indicated maximum and minimum values of reliability index of 8.1 and 3.4, respectively. Within this context, the mean and standard deviation obtained were 5.9 and 1.4, respectively. On the other hand, the monthly velocities showed a maximum probability of risk of 2.61%, minimum value of 1.5 × 10

^{−5}%, mean of 0.4%, and standard deviation of 0.8%. Hence, the above-documented results indicate that the Usumacinta Bridge did not suffer any damage during its overloading condition period.

## 1. Introduction

## 2. Case Study

## 3. Methodology

#### 3.1. PS InSAR

#### 3.2. Reliability Index

_{R}). The β value represents a factor indicating the level of structural safety, and the P

_{R}considers the proportion of the calculated discrepancies of the mean resistance and the influence of force, with respect to the combination of their standard deviation [29]. Based on the theory of probability, one way to compute the β value is by considering its corresponding P

_{R}, which expresses the likelihood that the bridge under consideration presents displacements larger than the threshold limits. Within this frame of reference, the P

_{R}can be calculated using the following equation, considering the threshold limits a and b, respectively [30]:

_{R}. Conversely, P

_{S}, generally known as the probability of security, represents the probability of obtaining displacements within the thresholds; in other words, P

_{S}is the complement of P

_{R}.

_{x}(x)dx is the best-fitted PDF of displacements, and which is selected based on the Chi-squared test of several analyzed distributions [30]. Considering this context, in this paper, the best-fitted PDF was selected between the following eleven PDFs: (1) t-student, (2) logistics, (3) log-logistics, (4) stable, (5) Weibull, (6) generalized extreme value, (7) extreme value, (8) gamma, (9) log-normal, (10) normal, and (11) t-location scale. These PDFs have been validated in other investigations for similar purposes [32].

_{R}and can be solved by the next equation [30]:

^{−1}is the inverse Cumulative Distribution Function (CDF), related to the PDF of the data (displacements) under consideration.

## 4. Controlled Testing Using CR

^{2}); Clutter is the background noise (−12); and SCR is equal to 30 dB. The result indicates that the CR must generate an RCS equal to or greater than 40.4 dBm

^{2}. The value of the RCS of the CR depends on its geometry and size; in this case, a square tetrahedron was used due to its high value of RCS and easy construction. The maximum value of RCS for such a figure is given by Equation (9) [37], where d is the length of the sides.

^{2}, a CR with sides of approximately 1 m is required. The CR built can be seen in Figure 5, before it was installed and oriented.

## 5. Field Testing on the Usumacinta Bridge

#### Structural Reliability of the Usumacinta Bridge

_{R}value of 1.64% is one of the highest values of all points in the structure.

_{R}is presented in Table 7, where the minimum value was 1.5 × 10

^{−5}% at point number 8, the maximum value was 2.61% at point number 4, the mean probability of risk was 0.4%, and the standard deviation was 0.8%. In general, the resulting probabilities of overpassing the threshold limits are very small. Thus, the safety of the Usumacinta Bridge can be declared to be adequate.

## 6. Discussion

_{R}, the Usumacinta Bridge did not suffer considerable damages from the demands of the heavy vehicles that were overloading it for 5 months. In addition, no changes were found concerning the tendency of the InSAR time series during the period analyzed. In this manner, the capabilities of the proposed approach for defining the structural reliability using semi-static displacements from InSAR were demonstrated in a real case study. Considering the competencies of InSAR for determining accurate displacement in the vertical component with the acquisition of information once per two weeks, the proposed probabilistic approach, which can extract structural risk associated with the bridge performance, and using specific limit states according to the main characteristics of bridge structures, it is possible to improve the administration of the structures during the maintenance and rehabilitation process.

## 7. Conclusions

- The displacements of the Usumacinta Bridge achieved a maximum reliability index value (β) of 8.1 and a minimum one of 3.4. The mean value of β was 5.9, and the standard deviation was 1.4. On the other hand, the calculated monthly velocities presented a maximum probability of risk (P
_{R}) of 2.61%. The minimum value was 1.5 × 10^{−5}%, the mean 0.4%, and the standard deviation 0.8%. Based on the results, it can be established that the areas of the bridge analyzed did not present damages produced by the heavy vehicles overloading the main structure. - InSAR is a useful technology to determine the semi-static displacements of bridges and estimate their structural reliability. Therefore, a support decision system can be developed to improve the quality of the road infrastructure with the methodology presented in this manuscript.
- Due to the Sentinel-1 image resolution, a few zones of the bridge were analyzed, which represents a general idea of the actual reliability of the Usumacinta Bridge. An ideal assessment would be a study in detail of the bridge considering the relevant structural elements. This can be accomplished using commercial images and corner reflectors.
- The proposed probabilistic assessment can be improved by using specific limit states for each structure instead of employing a general one. In addition, more PDFs can be integrated into the methodology to obtain the structural risk.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Vehicles that affected the Usumacinta Bridge: (

**a**) eighteen-wheeler vehicle of total length of 30 m; (

**b**) eighteen-wheeler vehicle of a total length of 68 m.

Image Mode | Azimuthal Resolution (m) | Slant Range Resolution (m) | Ground Range Resolution (m) | Ground Range Resolution Area (m^{2}) | Clutter (dB) |
---|---|---|---|---|---|

Interferometric wide swath | 20.0 | 5.0 | 8.7 | 174.3 | −12 |

Number of the Test | Ground Truth (mm) | InSAR Displacements (mm) | Differences (mm) |
---|---|---|---|

1 | 0 | 1 | 1 |

2 | 0 | 1.6 | 1.6 |

3 | 0 | 1 | 1 |

4 | 7.9 | 10 | 2.1 |

5 | 7.9 | 8.4 | 0.5 |

6 | 15.8 | 14.4 | −1.4 |

Number of the Test | Ground Truth (mm) | InSAR Displacements (mm) | Differences (mm) |
---|---|---|---|

1 | 0 | 0.2 | 0.2 |

2 | 0 | 0.2 | 0.2 |

3 | 0 | 1.2 | 1.2 |

4 | 7.9 | 9.1 | 1.2 |

5 | 7.9 | 8.2 | 0.3 |

6 | 15.8 | 14.6 | −1.2 |

Number of the Point | Velocity (mm/year) | Cumulative Displacements (mm) |
---|---|---|

1 | −3.5 ± 1.4 | −7 |

2 | −0.5 ± 0.35 | −1 |

3 | −0.66 ± 0.45 | −1.3 |

4 | −0.8 ± 0.59 | −1.6 |

5 | 0.37 ± 0.59 | 0.8 |

6 | 0.45 ± 0.59 | 0.9 |

7 | −0.2 ± 0.34 | −0.4 |

8 | −0.45 ± 0.57 | −0.9 |

9 | −0.19 ± 0.37 | −0.4 |

10 | 0.51 ± 0.4 | 1 |

11 | 0.37 ± 0.28 | 0.8 |

12 | 1.68 ± 0.8 | 3.4 |

13 | −0.83 ± 0.56 | −1.6 |

14 | 1.22 ± 0.9 | 2.4 |

15 | −0.29 ± 0.39 | −0.6 |

16 | −0.23 ± 0.49 | −0.4 |

Number of the Point | Mean Velocity (mm/month) | Standard Deviation (mm) |
---|---|---|

1 | −0.37 | 2.45 |

2 | 0.71 | 3.23 |

3 | −1.59 | 4.13 |

4 | −0.38 | 3.39 |

5 | 0.1 | 2.29 |

6 | −0.003 | 3.34 |

7 | 0.06 | 2.12 |

8 | 0.14 | 1.44 |

9 | 0.7 | 3.56 |

10 | 0.77 | 5.03 |

11 | 0.57 | 3.53 |

12 | 0.26 | 2.06 |

13 | −0.23 | 1.45 |

14 | −0.25 | 2.47 |

15 | 0.59 | 3.31 |

16 | 0.64 | 2.92 |

Point | Best-Fitted PDF | β |
---|---|---|

1 | Extreme Value | 8.1 |

2 | Extreme Value | 6.6 |

3 | Weibull | 6.1 |

4 | Generalized Extreme Value | 6.7 |

5 | Extreme Value | 7.5 |

6 | Extreme Value | 6.0 |

7 | Weibull | 7.6 |

8 | Stable | 5.2 |

9 | Extreme Value | 4.9 |

10 | Stable | 3.4 |

11 | Extreme Value | 6.5 |

12 | Weibull | 7.2 |

13 | T Location Scale | 5.9 |

14 | Stable | 3.6 |

15 | Extreme Value | 5.9 |

16 | Weibull | 3.7 |

Point | Best-Fitted PDF | P_{R} |
---|---|---|

1 | Logistic | 0.013 |

2 | Logistic | 0.17 |

3 | t student | 1.64 |

4 | Stable | 2.61 |

5 | Logistic | 0.006 |

6 | Fatigue life | 0.012 |

7 | Fisk | 0.036 |

8 | Logistic | 1.5 × 10^{−5} |

9 | Stable | 1.78 |

10 | Generalized Extreme Value | 8.4 × 10^{−5} |

11 | Logistic | 0.22 |

12 | Logistic | 0.002 |

13 | Stable | 0.001 |

14 | Logistic | 0.001 |

15 | Generalized Extreme Value | 0.03 |

16 | t student | 0.35 |

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## Share and Cite

**MDPI and ACS Style**

Guzman-Acevedo, G.M.; Quintana-Rodriguez, J.A.; Gaxiola-Camacho, J.R.; Vazquez-Becerra, G.E.; Torres-Moreno, V.; Monjardin-Quevedo, J.G.
The Structural Reliability of the Usumacinta Bridge Using InSAR Time Series of Semi-Static Displacements. *Infrastructures* **2023**, *8*, 173.
https://doi.org/10.3390/infrastructures8120173

**AMA Style**

Guzman-Acevedo GM, Quintana-Rodriguez JA, Gaxiola-Camacho JR, Vazquez-Becerra GE, Torres-Moreno V, Monjardin-Quevedo JG.
The Structural Reliability of the Usumacinta Bridge Using InSAR Time Series of Semi-Static Displacements. *Infrastructures*. 2023; 8(12):173.
https://doi.org/10.3390/infrastructures8120173

**Chicago/Turabian Style**

Guzman-Acevedo, German Michel, Juan A. Quintana-Rodriguez, Jose Ramon Gaxiola-Camacho, Guadalupe Esteban Vazquez-Becerra, Vanessa Torres-Moreno, and Jesus Guadalupe Monjardin-Quevedo.
2023. "The Structural Reliability of the Usumacinta Bridge Using InSAR Time Series of Semi-Static Displacements" *Infrastructures* 8, no. 12: 173.
https://doi.org/10.3390/infrastructures8120173