# Development of Stochastic Fatigue Model of Reinforcement for Reliability of Concrete Structures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Test Data

#### 2.2. Statistical Analysis of Fatigue Data of Steel Reinforcing Bars

#### 2.3. Bootstrap Method

#### 2.4. Bayesian Inference with Markov Chain Monte Carlo Implementation

- Select initial parameter vector
- Iterate as follows for $k=1,2,3,\dots $
- a.
- Create a new trial position ${\theta}^{*}={\theta}^{k-1}+\Delta \theta ,$ where $\Delta \theta $ is randomly sampled from the jumping distribution $q\left(\Delta \theta \right).$
- b.
- Create the Metropolis ratio.

- 3.
- Accept a new sample if:

## 3. Results of Uncertainty Modelling

## 4. Case Study: Crêt De l’Anneau Viaduct

#### 4.1. Limit State Equation

- t indicates time 0 < t < T
_{L}in years, - ${T}_{L}$ is the service life time of the structure,
- ${R}_{D}$ is modelling the ratio of design parameters, here the section modulus of the deck slab,
- $\mathsf{\Delta}{s}_{i}$ is the stress range for the ith load bin.

#### 4.2. Reliability Analysis

#### 4.3. Reliability Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**S-N curve for rebar data [1].

**Table 1.**Data [1].

Data Number (Index) | Number of Cycles to Failure | Stress Range [MPa] | Run-Out |
---|---|---|---|

1 | 7,875,829 | 337 | 1 |

2 | 4,485,923 | 335 | 1 |

3 | 9,182,542 | 391 | 1 |

4 | 3,981,071 | 385 | 1 |

5 | 347,328 | 396 | 0 |

6 | 589,346 | 403 | 0 |

7 | 441,005 | 405 | 0 |

8 | 371,852 | 408 | 0 |

9 | 341,454 | 408 | 0 |

10 | 238,658 | 405 | 0 |

11 | 255,509 | 408 | 0 |

12 | 255,509 | 420 | 0 |

13 | 273,550 | 430 | 0 |

14 | 215,443 | 430 | 0 |

15 | 411,921 | 439 | 0 |

16 | 398,107 | 419 | 0 |

17 | 411,921 | 424 | 0 |

18 | 255,509 | 467 | 0 |

19 | 184,784 | 488 | 0 |

20 | 161,215 | 488 | 0 |

21 | 161,215 | 494 | 0 |

22 | 131,376 | 503 | 0 |

23 | 114,619 | 505 | 0 |

24 | 129,154 | 506 | 0 |

25 | 158,489 | 507 | 0 |

26 | 140652 | 536 | 0 |

27 | 105,250 | 536 | 0 |

28 | 80,113 | 561 | 0 |

29 | 53,201 | 572 | 0 |

30 | 48,026 | 572 | 0 |

31 | 50,547 | 572 | 0 |

Parameter | Mean by MLM | Mean by Bayesian Approach | Standard Deviation by MLM | Standard Deviation by Bayesian Approach | Distribution | Remark |
---|---|---|---|---|---|---|

$\epsilon $ | 0 | 0 | --- | --- | Normal | Error term |

${\sigma}_{\epsilon}$ | 0.39 | 0.21 | 0.06 | 0.04 | Normal | Standard deviation of error term |

$\mathrm{log}k$ | 18.77 | 18.72 | 0.07 | 0.05 | Normal | Location parameter in Wöhler curve |

$m$ | Fixed to 5 | 5.03 | --- | 0.02 | Fixed/Deterministic | Slope of Wöhler curve |

${\rho}_{\mathrm{log}k,{\sigma}_{\epsilon}}$ | 0.06 | 0.03 | Deterministic | Correlation coefficient between location and standard deviation of error |

Parameter | Distribution | Mean | Standard Deviation | Remark |
---|---|---|---|---|

$\Delta $ | Lognormal | 1 | 0.30 | Model uncertainty related to PM Rule ^{1} |

${X}_{w}$ | Lognormal | 1 | 0.05 | Uncertainty in strain measurements |

${X}_{n}$ | Lognormal | 1 | 0.01 | Uncertainty in number of vehicles |

$logk$ | Normal | 18.77 | 0.07 | Location parameter in Wöhler curve |

$m$ | Fixed | 5 | --- | Slope of Wöhler curve fixed to 5 ^{2} |

$\u03f5$ | Normal | 0 | ${\sigma}_{\epsilon}$ | Error term taken from Table 2 |

${\sigma}_{\epsilon}$ | Normal | 0.39/0.21 ^{3} | 0.06/0.004 ^{3} | Standard deviation of error term taken from Table 2 |

${\rho}_{logk,\sigma \epsilon}$ | Deterministic | 0.06/0.003 ^{3} | --- | Correlation coefficient between location and standard deviation of error taken from Table 2 |

^{1}model uncertainty obtained by fitting lognormal distribution to test data in [45];

^{2}slope of Wöhler curve fixed to 5 as $\mathrm{log}k$ and m are highly correlated with correlation coefficient equal to 0.9997;

^{3}two values are used for analysis first one from MLM approach, while the second one is from Bayesian approach.

$\mathbf{CoV}\mathbf{of}\mathbf{log}\mathit{k}$ | Annual Reliability Index at 120 Years |
---|---|

0.39 (MLM) | 3.90 |

0.20 (Bayesian) | 4.25 |

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**MDPI and ACS Style**

Rastayesh, S.; Mankar, A.; Dalsgaard Sørensen, J.; Bahrebar, S.
Development of Stochastic Fatigue Model of Reinforcement for Reliability of Concrete Structures. *Appl. Sci.* **2020**, *10*, 604.
https://doi.org/10.3390/app10020604

**AMA Style**

Rastayesh S, Mankar A, Dalsgaard Sørensen J, Bahrebar S.
Development of Stochastic Fatigue Model of Reinforcement for Reliability of Concrete Structures. *Applied Sciences*. 2020; 10(2):604.
https://doi.org/10.3390/app10020604

**Chicago/Turabian Style**

Rastayesh, Sima, Amol Mankar, John Dalsgaard Sørensen, and Sajjad Bahrebar.
2020. "Development of Stochastic Fatigue Model of Reinforcement for Reliability of Concrete Structures" *Applied Sciences* 10, no. 2: 604.
https://doi.org/10.3390/app10020604