# SHM-Based Probabilistic Fatigue Life Prediction for Bridges Based on FE Model Updating

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Probabilistic Fatigue Life Prediction Using FE Model Updating Based on SHM Data

#### 2.1. Limit-State Function Formulations for Fatigue Failure

^{0}is the initial crack length, N is the total number of loading applications at frequency ν

_{0}and T is the time duration. If it is assumed that a crack failure occurs when the crack length exceeds the critical crack length a

^{c}, the time required for crack growth from a

^{0}to a

^{c}, T

_{0}, can be expressed as follows:

^{0}and ΔS

_{0}are the initial crack length and stress range, respectively. The limit-state function for the failure of a member within a given time interval [0, T

_{s}] can be expressed as follows:

**X**denotes the vector of random variables. In structural reliability, g(

**X**) ≤ 0 typically indicates the occurrence of a failure event.

^{1}

_{up}, a recursive formulation of the time duration from that moment to the crack failure can be developed as follows. Consider an auxiliary “damage” function:

^{1}and ΔS

_{1}denote the crack length and stress, respectively, at the moment that the FE model is updated. Equations (8) and (9) represent the crack growth before and after the FE model is updated, respectively. Summing Equations (8) and (9), one obtains the following:

_{1},

_{0}/ΔS

_{1}incorporates the effect of the stress change obtained by the FE model updating. Similarly, if another FE model updating occurs at T

^{2}

_{up}, the time required for the crack failure after the second updating is derived as follows:

_{j}denote the number of FE model updates and the time duration from the last SHM to the crack failure, respectively. The fatigue failure within a given time interval [0, T

_{s}] is then described as follows:

#### 2.2. Component and System Reliability Analysis

**x**), and the event of interest (often called the “failure”) is expressed by g(

**x**) ≤ 0, where

**x**is a column vector of n random variables (i.e.,

**x**= [x

_{1}, x

_{2}, …, x

_{n}]

^{T}) representing the uncertainties in the given problem. The probability of the event P

_{f}is expressed as follows:

**(**

_{x}**x**) is the joint probability density function (PDF) of

**x**. By transforming the space of the random variables into the standard normal space, the probability P

_{f}can be expressed as follows:

**u**) = g(

**T**

^{−1}(

**u**)) is the transformed limit-state function in the standard normal space, φ

_{n}(·) denotes the n-th order standard normal PDF,

**u**is the column vector of n standard normal variables and

**T**is the one-to-one mapping transformation matrix that satisfies

**u**=

**T**(

**x**).

_{f}in Equation (15)) can be approximately calculated by use of the linearized function of G(

**u**) at point

**u***, which is defined by the following constrained optimization problem:

^{2}norm. As an example of the first-order approximation concept of the FORM, Figure 1 shows the approximated limit-state function in a two-dimensional space.

**u*** has the highest probability among all of the nodes in the failure domain G(

**u**) ≤ 0. In this sense,

**u*** is an optimal point and is commonly called the design point or most probable point (MPP).

**u**) = [∂G/∂u

_{1}, …, ∂G/∂u

_{n}] denotes the gradient vector,

**α**= −∇G(

**u***)/||∇G(

**u***)|| is the normalized negative gradient vector at MPP and β = −

**αu*** is the reliability index. After FORM analysis, in this study, SORM is employed to achieve more accurate results than those from FORM. Der Kiureghian [20] provided more details about FORM and SORM.

#### 2.3. Finite Element Model Updating Based on Structural Health Monitoring Data

_{i}is the i-th natural frequency, φ

_{ji}denotes the j-th component of the i-th normalized mode shape ϕ

_{i}, w

_{i}and ε are the weighting factor for the i-th mode and the admissible error bound for the mode shape, respectively, and the superscripts “m” and “c” indicate data from the measurement and calculation of the FE model under updating, respectively. The details of the FE model updating procedure can be found in the following section and in Yi et al. [13,14].

## 3. Numerical Example

#### 3.1. Example Bridge: Samseung Bridge

#### 3.2. FE Model Updating

#### 3.3. Statistical Parameters

#### 3.4. Random Variables and Deterministic Parameters

^{0}are considered to be random variables with mean values of 2.18 × 10

^{−13}mm/cycle/(MPa·mm)

^{m}and 0.1 mm, respectively, and the coefficients of variation (COVs) are 0.2 and 0.1, respectively. The parameter m in the Paris equation can also be considered as a random variable. However, a preliminary analysis showed that the consideration gave a negligible impact to the result of life prediction for a large amount of additional time costs, so only C is considered as a random variable in this example. The uncertainties of the stresses are also introduced using a load scale factor S, whose mean and COV are assumed to be 1.0 and 0.1, respectively. It is assumed that the initial crack length a

^{0}follows an exponential distribution, whereas the other parameters follow a lognormal distribution. The statistical properties of the random variables in this numerical example are summarized in Table 3.

^{0}) of the five girders (correlation coefficient: 0.6). The correlation coefficients in these cases are not known, so they were initially assumed to be 0.6, which indicates that Girders 1–5 were manufactured by the same manufacturer and that their material properties are thus highly correlated. In addition, a parametric study with various correlation coefficients was performed to investigate the effects of these correlations on the fatigue life.

_{th}): 30 mm; the critical crack length (a

^{c}): 30 mm; and the time of the SHM test (T

^{1}

_{up}): four years. The average daily truck traffic (ADTT) was assumed to be 5388/day, based on actual passing truck data provided by the Korea Expressway Corporation. For the geometry function Y(a) in Equation (2), the following function from Wang et al. [34] for I-beams is introduced:

#### 3.5. Analysis Results

^{−4}) with a service life of 75 years for steel and prestressed concrete components. The fatigue lives of Girders 1–5 and the bridge system were estimated using the target reliability index (i.e., the black lines in Figure 6), as listed in Table 4. With the updated FE model, the fatigue lives of the girders and bridge system were estimated to be much greater, and all of them meet the AASHTO requirement, with fatigue lives longer than 75 years.

^{7}samples, and the results obtained with the proposed method match those obtained from the MCS, except in the range of relatively large reliability index values, where accurate results cannot be expected, even with MCS using 10

^{7}samples in nature.

^{0}of the five girders were assumed to be 0.6, which accounts for the high dependency due to the same manufacturer assumption. To investigate the effect of the correlation coefficient, the fatigue life of the bridge system was evaluated using a range of correlation coefficient values, as a parametric study. As Table 5 shows, the fatigue life increases by 2–4 years as the correlation coefficient increases, which means that the effect of the correlation coefficient is not very significant. This is because, although the failure event of the bridge system is assumed to occur if any of the five girders fails, the system failure event is actually dominated by the failure events of Girders 2 and 4, as shown in Figure 7 and Table 4.

## 4. Conclusions

## Acknowledgment

## Author Contributions

## Conflicts of Interest

## References

- Byers, W.G.; Marley, M.J.; Mohammadi, J.; Nielsen, R.J.; Sarkani, S. Fatigue reliability reassessment applications: State-of-the-art paper. J. Struct. Eng.
**1997**, 123, 277–285. [Google Scholar] [CrossRef] - Karamchandani, A.; Dalane, J.I.; Bjerager, P. Systems reliability approach to fatigue of structures. Struct. Eng.
**1992**, 118, 684–700. [Google Scholar] [CrossRef] - Millwater, H.R.; Wieland, D.H. Probabilistic sensitivity-based ranking of damage tolerance analysis elements. J. Aircr.
**2010**, 47, 161–171. [Google Scholar] [CrossRef] - Imam, B.M.; Righiniotis, T.D.; Chryssanthopoulos, M.K. Probabilistic fatigue evaluation of riveted railway bridges. J. Bridge Eng.
**2008**, 13, 237–244. [Google Scholar] [CrossRef] - Park, Y.-S.; Han, S.-Y.; Suh, B.-C. Fatigue reliability analysis of steel bridge welding member by fracture mechanics method. Struct. Eng. Mech.
**2005**, 19, 347–359. [Google Scholar] [CrossRef] - Zhao, Z.; Haldar, A.; Breen, F.L. Fatigue-reliability evaluation of steel bridges. J. Struct. Eng.
**1994**, 120, 1608–1623. [Google Scholar] [CrossRef] - Madsen, H.O. Probabilistic and Deterministic Models for Predicting Damage Accumulation Due to Time Varying Loading. In DIALOG 5-82; Danish Engineering Academy: Lyngby, Denmark, 1983. [Google Scholar]
- Lukić, M.; Cremona, C. Probabilistic assessment of welded joints versus fatigue and fracture. J. Struct. Eng.
**2001**, 127, 211–218. [Google Scholar] [CrossRef] - Kwon, K.; Frangopol, D.M. Bridge fatigue reliability assessment using probability density functions of equivalent stress range based on field monitoring data. Int. J. Fatigue
**2010**, 32, 1221–1232. [Google Scholar] [CrossRef] - Ni, Y.Q.; Ye, X.W.; Ko, J.M. Monitoring-based fatigue reliability assessment of steel bridges: Analytical model and application. J. Struct. Eng.
**2010**, 132, 1563–1573. [Google Scholar] [CrossRef] - Zhao, Z.; Haldar, A. Bridge fatigue damage evaluation and updating using non-destructive inspections. Eng. Fract. Mech.
**1996**, 53, 775–788. [Google Scholar] [CrossRef] - Nagayama, T.; Sim, S.H.; Miyamori, Y.; Spencer, B.F. Issues in structural health monitoring employing smart sensors. Smart Struct. Syst.
**2007**, 3, 299–320. [Google Scholar] [CrossRef] - Yi, J.-H.; Cho, S.; Koo, K.-Y.; Yun, C.-B.; Kim, J.-T.; Lee, C.-G.; Lee, W.-T. Structural performance evaluation of a steel-plate girder bridge using ambient acceleration measurements. Smart Struct. Syst.
**2007**, 3, 281–298. [Google Scholar] [CrossRef] - Yi, J.-H.; Kim, D.; Go, S.; Kim, J.-T.; Park, J.-H.; Feng, M.Q.; Kang, K.-S. Application of structural health monitoring system for reliable seismic performance of infrastructures. Adv. Struct. Eng.
**2012**, 15, 955–967. [Google Scholar] - Miner, M.A. Cumulative damage in fatigue. J. Appl. Mech.
**1945**, 12, 159–164. [Google Scholar] - Paris, P.C.; Erdogan, F. An effective approximation to evaluate multinormal integrals. Struct. Saf.
**1963**, 20, 51–67. [Google Scholar] - Lee, Y.-J.; Song, J. Risk analysis of fatigue-induced sequential failures by branch-and-bound method employing system reliability bounds. J. Eng. Mech.
**2011**, 137, 807–821. [Google Scholar] [CrossRef] - Newman, J.C.; Raju, I.S. An empirical stress intensity factor equation for the surface crack. Eng. Fract. Mech.
**1981**, 15, 185–192. [Google Scholar] [CrossRef] - Melchers, R.E. Structural Reliability: Analysis and Prediction; John Wiley & Sons: New York, NY, USA, 1999. [Google Scholar]
- Der Kiureghian, A. Chapter 14. First- and Second-Order Reliability Methods. In Engineering Design Reliability Handbook; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Song, J.; Der Kiureghian, A. Bounds on system reliability by linear programming. J. Eng. Mech.
**2003**, 129, 627–636. [Google Scholar] [CrossRef] - Song, J.; Kang, W.-H. System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Struct. Saf.
**2009**, 31, 148–156. [Google Scholar] [CrossRef] - Hohenbichler, M.; Rackwitz, R. First-order concepts in system reliability. Struct. Saf.
**1983**, 1, 177–188. [Google Scholar] [CrossRef] - Genz, A. Numerical computation of multivariate normal probabilities. J. Comput. Graph. Stat.
**1992**, 1, 141–149. [Google Scholar] - Lee, Y.-J.; Song, J. Finite-element-based system reliability analysis of fatigue-induced sequential failures. Reliab. Eng. Syst. Saf.
**2012**, 108, 131–141. [Google Scholar] [CrossRef] - Nelder, J.A.; Mead, R. A simplex method for function minimization. Comput. J.
**1965**, 7, 308–313. [Google Scholar] [CrossRef] - Yi, J.H.; Yun, C.B. Comparative study on modal identification methods using output-only information. Struct. Eng. Mech.
**2004**, 17, 445–466. [Google Scholar] [CrossRef] - Ministry of Construction and Transportation. Korea Highway Bridge Design Specifications; Korea Society of Civil Engineers: Seoul, Korea, 2005. [Google Scholar]
- Moan, T.; Song, R. Implications of inspection updating on system fatigue reliability of offshore structures. J. Offshore Mech. Arct. Eng.
**2000**, 122, 173–180. [Google Scholar] [CrossRef] - Riahi, H.; Bressolette, P.; Chateauneuf, A.; Bouraoui, C.; Fathallah, R. Reliability analysis and inspection updating by stochastic response surface of fatigue cracks in mixed mode. Eng. Struct.
**2011**, 33, 3392–3401. [Google Scholar] [CrossRef] - Zheng, R.; Ellingwood, B.R. Stochastic fatigue crack growth in steel structures subject to random loading. Struct. Saf.
**1998**, 20, 303–323. [Google Scholar] [CrossRef] - Borrego, L.P.; Ferreira, J.M.; Costa, J.M. Fatigue crack growth and crack closure in an AlMgSi alloy. Fatigue Fract. Eng. Mater. Struct.
**2001**, 24, 255–266. [Google Scholar] [CrossRef] - Yarema, S.Y. Correlation of the parameters of the Paris equation and the cyclic crack resistance characteristics of materials. Strength Mater.
**1980**, 13, 1090–1098. [Google Scholar] [CrossRef] - Wang, C.S.; Chen, A.R.; Chen, W.Z.; Xu, Y. Application of probabilistic fracture mechanics in evaluation of existing riveted bridges. Bridge Struct. Assess. Des. Constr.
**2006**, 2, 223–232. [Google Scholar] [CrossRef]

**Figure 7.**Reliability indices of girders and the bridge system using the proposed method and Monte Carlo simulation (MCS).

**Table 1.**Updating parameters of Samseung Bridge (modified from [13]).

Members | Updating Parameters | Count | |
---|---|---|---|

First Step | Second Step | ||

Support | Rotational Spring Constant | 1 | 2 |

Concrete Slab | Young’s Modulus | 1 | 1 |

Main Girder | Second Moment of Inertia | 5 | 5 |

Torsional Coefficient | 0 | 5 | |

Floor Beam | Second Moment of Inertia | 1 | 9 |

Torsional Coefficient | 1 | 9 | |

Total | 9 | 31 |

Stress (MPa) | Girder 1 | Girder 2 | Girder 3 | Girder 4 | Girder 5 |
---|---|---|---|---|---|

Initial FE model | 18.24 | 20.77 | 20.03 | 20.77 | 18.24 |

Updated FE model | 15.95 | 17.52 | 17.11 | 17.52 | 15.83 |

Random Variables (RVs) | Mean | COV | Distribution Type | Number of RVs |
---|---|---|---|---|

Paris law parameter © | 2.18 × 10^{−13} (mm/cycle/(MPa·mm)^{m}) | 0.2 | Lognormal | 5 |

Initial crack length (a^{0}) | 0.1 (mm) | 1.0 | Exponential | 5 |

Live load scale factor (S) | 1 | 0.1 | Lognormal | 1 |

Fatigue Life (Years) | Girder 1 | Girder 2 | Girder 3 | Girder 4 | Girder 5 | System |
---|---|---|---|---|---|---|

Initial FE model | 125.6 | 81.2 | 95 | 81.2 | 125.6 | 74.3 |

Updated FE model | 170 | 119.4 | 133.6 | 118.3 | 175 | 108 |

Bridge System Fatigue Life (Years) | ρ = 0.0 | ρ = 0.2 | ρ = 0.4 | ρ = 0.6 | ρ = 0.8 |
---|---|---|---|---|---|

Initial FE model | 73 | 73.3 | 73.7 | 74.3 | 75.7 |

Updated FE model | 106 | 106.5 | 107 | 108 | 110 |

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Lee, Y.-J.; Cho, S.
SHM-Based Probabilistic Fatigue Life Prediction for Bridges Based on FE Model Updating. *Sensors* **2016**, *16*, 317.
https://doi.org/10.3390/s16030317

**AMA Style**

Lee Y-J, Cho S.
SHM-Based Probabilistic Fatigue Life Prediction for Bridges Based on FE Model Updating. *Sensors*. 2016; 16(3):317.
https://doi.org/10.3390/s16030317

**Chicago/Turabian Style**

Lee, Young-Joo, and Soojin Cho.
2016. "SHM-Based Probabilistic Fatigue Life Prediction for Bridges Based on FE Model Updating" *Sensors* 16, no. 3: 317.
https://doi.org/10.3390/s16030317