# Fatigue Reliability Assessment for Orthotropic Steel Decks: Considering Multicrack Coupling Effects

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{eq}is the equivalent stress amplitude under varying amplitude load; Y is a geometric function considering the crack shape of the member.

_{1}and N

_{2}, then the integral of Equation (1) can be obtained as follows:

_{1}to α

_{2}[26], which is defined as follows:

_{2}− N

_{1}) stress cycle can be defined as:

_{N}is the crack size of the in-service structure after N stress cycles, which can be redefined as the crack evolving from the initial size of α

_{0}(the N

_{0}stress cycle) to the α

_{2}size (the N stress cycle). Once α

_{N}exceeds the critical crack size α

_{C}, a failure problem can be considered to have occurred.

_{2}, α

_{1}) is monotonically increasing with crack size, the limit state function of Equation (6) can be redefined as:

_{c}, α

_{0}) is the fatigue damage accumulation function from the initial crack size to the critical crack size, namely, the critical threshold of the limit state equation. Ψ(α

_{N}, α

_{0}) is the damage accumulation function from the initial crack size α

_{0}through N stress cycles to α

_{N}, namely, the load-effect part of the limit state function.

_{0}is equivalent to a single-crack depth α

_{e}after the extended coupling effect, and the coupling equivalent to a single crack continues to expand to the critical depth α

_{c}. According to the recommendation of the IIW (International Institute of Welding) [27], when the crack propagation depth reaches half of the thickness of the roof plate, the component is considered to have failed. During this process, Y changes with the crack size; the expression on the right of Equation (8) is processed with piecewise integral.

_{d}is the number of daily cycles of stress; n is the service life of the bridge. Y

_{0}and Y

_{e}are the boundary correction factors for the reference stress intensity factors, respectively [28].

_{e}is the crack depth of the collinear double crack considering the coupling effect equivalent to a single crack, which is calculated using the ABAQUS-FRANC3D interactive technique; T is the plate thickness; and R

_{s}is the collinear double-crack spacing ratio.

_{y}are further added to the limit state function.

## 3. Structural Reliability Analysis Based on iHL-RF Method

_{u}is the reliability index corresponding to the failure probability; Φ(·) is the standard normal cumulative distribution function. In the standard normal space, the reliability index β

_{u}can be calculated as:

**u*** is a design point in standard normal space. In a geometric sense, β

_{u}is the point corresponding to the minimum Euclidean distance from the limit state plane to the origin in standard space. The solution of design point u* involves a constrained optimization solution problem, which is defined as

**x**one by one and can be calculated from variable

**x**by equal probability transformation method to obtain

**u**.

**α**

_{u}are calculated as:

_{1}and x

_{2}are standard normal random variables. The iHL-RF method is used to solve the example, and the reliability of the example is 2.873. The iterative route in the process of solving the iHL-RF method is shown in Figure 1.

## 4. Structural Reliability Analysis Method Based on AK-MCS Method

#### 4.1. Kriging Modeling

^{2}, whose covariance function is defined as follows:

**x**

_{i}and

**x**

_{j}; this paper adopts the form of the Gaussian function. θ is a parameter variable of $1\times n$. The superscript k represents the k-th component of sample

**x**

_{i}.

**X**and response set

**G**, the regression coefficient

**β**and variance σ

^{2}are estimated as follows:

**F**is the regression coefficient matrix of the training sample;

**R**is the regression coefficient matrix of the training sample: $\mathit{R}={\left[{R}_{ij}\right]}_{m\times m}$, ${R}_{ij}=R(\mathit{\theta},{\mathit{x}}_{i},{\mathit{x}}_{j})$.

**x**is calculated as follows:

#### 4.2. AK-MCS Method

**x**) and the input random variable

**x**. The direct MCS method requires a large sample sampling of input random variables and a calculation of output response values of each group of samples in turn. The whole calculation process of the MCS method is lengthy in duration, leading the proxy model to not be adopted into the field of structural reliability analysis.

_{1}and x

_{2}are random variables subject to a standard normal distribution, respectively. AK-MCS is used to solve the reliability problem of the above series system, and the reliability index is 2.845. The point selection process of the

**U**function is shown in Figure 2. The selected sample points are uniformly distributed near the limit state surface. Due to the

**U**function, we can balance the searchability of the region near the limit state surface and the global region and effectively prevent the agglomeration phenomenon of the selected sample points while ensuring the selection of points near the limit state surface.

#### 4.3. AK-MCS Calculation Process

**S**

_{c}is generated. According to the distribution of input variables, a candidate sample set

**S**

_{c}with size

**N**

_{c}is obtained with sampling.

**X**and the corresponding response value

**G**are generated. To construct an initial Kriging model, it is necessary to obtain N initial samples $\mathit{X}={\left[{\mathit{x}}_{1},{\mathit{x}}_{2},\dots ,{\mathit{x}}_{N}\right]}^{T}$ first and then calculate its true performance function response value $\mathit{G}={\left[g({\mathit{x}}_{1}),g({\mathit{x}}_{2}),\dots g({\mathit{x}}_{N})\right]}^{T}$. The size of the initial sample set

**X**defines $N=\mathrm{max}\left(n+2,12\right)$.

**X**and its response value

**G**, the Kriging model is constructed based on the DACE toolbox.

**S**

_{c}is calculated, and its failure probability is calculated as follows:

**x**

_{b}. Based on the Kriging method, the prediction response values and prediction variance of all samples in candidate sample set

**S**

_{c}are estimated, respectively. Then, function $U$ values in all candidate samples are calculated according to Equation (35), and the sample corresponding to the minimum value is selected as the best update point

**x**

_{b}.

**x**

_{b}) of the best update point

**x**

_{b}and add it to the sample set

**X**and its response value

**G**to reconstruct the Kriging model and return to step (4).

## 5. Simulation Method of Crack Propagation Based on ABAQUS-FRANC3D Interactive Technique

^{5}MPa, and the Poisson ratio was 0.3. There were two transverse partitions and two U-ribs. The model length and width were 3200 mm and 1400 mm, respectively. The thicknesses of the top plate and U-rib were 16 mm and 8 mm, respectively. The top plate U-rib upper mouth width was 300 mm, the lower mouth width 170 mm, the height 280 mm, and the transverse partition thickness was 10 mm. The welding seam with 80% was used to connect the U-rib and the top plate of the steel bridge panel. The assembly clearance parameter g between the top plate and the U-rib was 0.5 mm. The finite element model adopted two-point loading; the loading area was 200 mm × 200 mm. The three-way displacement of all nodes at the bottom of the transverse partition of the steel bridge panel was constrained.

_{e}, α

_{0}and R

_{s}. In this study, a polynomial response surface is used to construct the relationship between the three based on the test data. The overall trend of the test data points and the response surface is consistent.

## 6. Reliability Calculation

_{eq}and the number of daily stress cycles N

_{d}of different lanes are corrected accordingly; namely, different variable distribution parameters are adopted.

_{y}is 0%, 1%, 2% and 3%, respectively. Figure 9, Figure 10, Figure 11 and Figure 12 shows the analysis result. Figure 9, Figure 10, Figure 11 and Figure 12 shows that both the iHL-RF method and AK-MCS method can efficiently solve the fatigue reliability problem of steel bridge decks, and the errors are within the acceptable range of engineering. To further compare the computational efficiency of the iHL-RF method and the AK-MCS method, the reliability results of the single crack and multicrack under different working conditions are shown in Table 3 and Table 4. The AK-MCS method has a great reduction in the number of function calls compared with the iHL-RF method in both single-crack and double-crack cases, and the estimated relative error is less than 2% in all cases.

## 7. Conclusions

_{eq}and the number of daily stress cycles N

_{d}of the passing lane are smaller. The fatigue reliability of the passing lane on the steel bridge deck is higher under the same working conditions (annual traffic increase and service time).

_{c}, α

_{0}) in the fatigue reliability function is relatively small, which leads to an increase in the probability of fatigue failure of steel bridge decks. When the design life reaches 100 years and the annual traffic growth amount α

_{y}= 3%, the multicrack fatigue reliability of the steel bridge deck driving lane is lower than 2.

_{y}has a crucial influence on the fatigue reliability of steel bridge decks. For the driving lanes, the reliability difference between annual traffic growth α

_{y}= 1% and α

_{y}= 3% after 100 years of service is 1.55 (single crack) and 1.53 (double crack). For the passing lane, the difference in reliability between annual traffic growth α

_{y}= 1% and α

_{y}= 3% after 240 years of service is 2.59 (single crack) and 2.46 (double crack).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

α | fatigue crack size | β_{u} | reliability index corresponding to the failure probability |

N | number loading cycles | Φ(·) | standard normal cumulative distribution function |

C | fatigue growth correlation coefficient | u* | design point in standard normal space |

m | fatigue growth correlation coefficient | u | random variable in a standard normal space |

ΔK | amplitude of stress intensity factor | k | number of iterations |

Seq | equivalent stress amplitude under varying amplitude load | d | search direction |

Y | geometric function considering the crack shape of the member | λ | search step |

α_{N} | crack size of the in-service structure after N stress cycles | ▽g(u) | gradient vector of the function |

ψ(α_{c}, α_{0}) | fatigue damage accumulation function from the initial crack size to the critical crack size | m(·) | value function |

ψ(α_{N}, α_{0}) | damage accumulation function from the initial crack size α0 through N stress cycles to α_{N} | c | penalty parameter |

N_{d} | number of daily cycles of stress | f(x) | polynomial function variable |

n | service life of the bridge | β | regression coefficient vector |

Y_{0} | boundary correction factors for the reference stress intensity factor | ξ(x) | random process |

Y_{e} | boundary correction factors for the reference stress intensity factor | R(θ, x_{i}, x_{j}) | correlation |

α_{e} | crack depth of collinear double crack considering the coupling effect equivalent to a single crack | θ | parameter variable |

T | plate thickness | F | regression coefficient matrix of the training sample |

R_{s} | collinear double-crack spacing ratio | R | regression coefficient matrix of the training sample |

R_{s} | double-crack spacing ratio | U(x) | consistent with the sign of the actual function |

α_{0} | initial crack depth | $\widehat{g}(\mathit{x})$ | related to the low confidence bounding function |

e | wheel track transverse distribution coefficient |

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**Figure 8.**Fatigue reliability analysis based on MCS (driving lane). (

**a**) Single crack. (

**b**) Double crack.

**Figure 9.**Fatigue reliability analysis of single crack based on iHL-RF (driving lane). (

**a**) Reliability calculation results. (

**b**) Estimation error.

**Figure 10.**Fatigue reliability analysis of double cracks based on iHL-RF (driving lane). (

**a**) Reliability calculation results. (

**b**) Estimation error.

**Figure 11.**Fatigue reliability analysis of single crack based on AK-MCS (driving lane). (

**a**) Reliability calculation results. (

**b**) Estimation error.

**Figure 12.**Fatigue reliability analysis of double cracks based on AK-MCS (driving lane). (

**a**) Reliability calculation results. (

**b**) Estimation error.

**Figure 13.**Fatigue reliability analysis based on MCS (passing lane). (

**a**) Single crack. (

**b**) Double cracks.

**Figure 14.**Single-crack fatigue reliability analysis based on iHL-RF (passing lane). (

**a**) Single crack. (

**b**) Double cracks.

**Figure 15.**Double-crack fatigue reliability analysis based on iHL-RF (passing lane). (

**a**) Reliability calculation results. (

**b**) Estimation error.

**Figure 16.**Single-crack fatigue reliability analysis based on AK-MCS (passing lane). (

**a**) Reliability calculation results, (

**b**) Estimation error.

**Figure 17.**Double-crack fatigue reliability analysis based on AK-MCS (passing lane). (

**a**) Reliability calculation results, (

**b**) Estimation error.

Variable | Distribution | Mean | Variation Coefficient |
---|---|---|---|

R_{s} | Uniform [29] | 0.1 | 3 |

α_{0} | lognormal [30] | 0.5 | 0.2 |

C | lognormal [31] | 5.21 | 0.6 |

e | lognormal [32] | 0.78 | 0.1 |

S_{eq} | normal (driving lane) [33] normal (passing lane) [33] | 17.67 16.87 | 0.44 1.53 |

N_{d} | normal (driving lane) [34] normal (passing lane) [34] | 5685 1140 | 480 52 |

${\mathit{\alpha}}_{0}$/mm | ${\mathit{R}}_{\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{e}}$/mm | ${\mathit{\alpha}}_{0}$/mm | ${\mathit{R}}_{\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{e}}$/mm |
---|---|---|---|---|---|

1 | 0.5 | 1.2948 | 3 | 0.33 | 3.348 |

1 | 1 | 1.6284 | 3 | 0.5 | 3.5868 |

1 | 1.5 | 1.9169 | 3 | 0.66 | 3.8187 |

1 | 2 | 2.2819 | 3 | 0.83 | 4.174 |

1 | 2.5 | 2.6597 | 3 | 1 | 4.41 |

1 | 3 | 3.06 | 4 | 0.25 | 4.271 |

2 | 0.5 | 2.437 | 4 | 0.375 | 4.4913 |

2 | 1.5 | 2.682 | 4 | 0.5 | 4.681 |

2 | 1 | 3 | 4 | 0.625 | 4.8997 |

2 | 1.25 | 3.364 | 4 | 0.75 | 4.8 |

2 | 1.5 | 3.79 |

Conditions | MCS | iHL-RF | AK-MCS | ||
---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | |

$n=40$, ${\alpha}_{y}=0\%$ | 5.42 | 5.45 | 328 | 5.42 | 121 |

$n=40$, ${\alpha}_{y}=1\%$ | 5.24 | 5.14 | 306 | 5.27 | 156 |

$n=40$, ${\alpha}_{y}=2\%$ | 4.90 | 4.88 | 302 | 4.90 | 188 |

$n=40$, ${\alpha}_{y}=3\%$ | 4.66 | 4.66 | 286 | 4.66 | 156 |

$n=60$, ${\alpha}_{y}=0\%$ | 4.76 | 4.75 | 300 | 4.76 | 212 |

$n=60$, ${\alpha}_{y}=1\%$ | 4.32 | 4.31 | 278 | 4.32 | 247 |

$n=60$, ${\alpha}_{y}=2\%$ | 3.96 | 3.96 | 274 | 3.96 | 259 |

$n=60$, ${\alpha}_{y}=3\%$ | 3.67 | 3.67 | 272 | 3.68 | 129 |

$n=80$, ${\alpha}_{y}=0\%$ | 4.27 | 4.26 | 278 | 4.27 | 206 |

$n=80$, ${\alpha}_{y}=1\%$ | 3.70 | 3.69 | 272 | 3.70 | 186 |

$n=80$, ${\alpha}_{y}=2\%$ | 3.27 | 3.27 | 262 | 3.27 | 143 |

$n=80$, ${\alpha}_{y}=3\%$ | 2.93 | 2.92 | 252 | 2.93 | 157 |

$n=100$, ${\alpha}_{y}=0\%$ | 3.88 | 3.88 | 274 | 3.88 | 125 |

$n=100$, ${\alpha}_{y}=1\%$ | 3.19 | 3.19 | 252 | 3.19 | 143 |

$n=100$, ${\alpha}_{y}=2\%$ | 2.71 | 2.70 | 252 | 2.71 | 162 |

$n=100$, ${\alpha}_{y}=3\%$ | 2.33 | 2.32 | 254 | 2.33 | 179 |

Conditions | MCS | iHL-RF | AK-MCS | ||
---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | |

$n=40$, ${\alpha}_{y}=0\%$ | 4.84 | 4.83 | 346 | 4.83 | 104 |

$n=40$, ${\alpha}_{y}=1\%$ | 4.55 | 4.52 | 324 | 4.54 | 122 |

$n=40$, ${\alpha}_{y}=2\%$ | 4.29 | 4.27 | 324 | 4.29 | 127 |

$n=40$, ${\alpha}_{y}=3\%$ | 4.07 | 4.05 | 310 | 4.07 | 135 |

$n=60$, ${\alpha}_{y}=0\%$ | 4.16 | 4.15 | 302 | 4.17 | 153 |

$n=60$, ${\alpha}_{y}=1\%$ | 3.72 | 3.71 | 294 | 3.73 | 125 |

$n=60$, ${\alpha}_{y}=2\%$ | 3.38 | 3.36 | 286 | 3.38 | 139 |

$n=60$, ${\alpha}_{y}=3\%$ | 3.09 | 3.08 | 276 | 3.10 | 154 |

$n=80$, ${\alpha}_{y}=0\%$ | 3.67 | 3.66 | 282 | 3.68 | 129 |

$n=80$, ${\alpha}_{y}=1\%$ | 3.12 | 3.10 | 276 | 3.12 | 153 |

$n=80$, ${\alpha}_{y}=2\%$ | 2.70 | 2.69 | 276 | 2.70 | 171 |

$n=80$, ${\alpha}_{y}=3\%$ | 2.36 | 2.35 | 270 | 2.36 | 185 |

$n=100$, ${\alpha}_{y}=0\%$ | 3.30 | 3.29 | 286 | 3.30 | 146 |

$n=100$, ${\alpha}_{y}=1\%$ | 2.63 | 2.61 | 274 | 2.62 | 169 |

$n=100$, ${\alpha}_{y}=2\%$ | 2.15 | 2.13 | 266 | 2.14 | 193 |

$n=100$, ${\alpha}_{y}=3\%$ | 1.77 | 1.76 | 260 | 1.77 | 112 |

Conditions | MCS | iHL-RF | AK-MCS | ||
---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | |

$n=180$, ${\alpha}_{y}=0\%$ | 5.49 | 5.62 | 346 | 5.49 | 156 |

$n=180$, ${\alpha}_{y}=1\%$ | 4.58 | 4.56 | 286 | 4.58 | 162 |

$n=180$, ${\alpha}_{y}=2\%$ | 3.94 | 3.93 | 274 | 3.93 | 161 |

$n=180$, ${\alpha}_{y}=3\%$ | 3.48 | 3.47 | 270 | 3.48 | 138 |

$n=200$, ${\alpha}_{y}=0\%$ | 5.37 | 5.44 | 342 | 5.37 | 164 |

$n=200$, ${\alpha}_{y}=1\%$ | 4.33 | 4.30 | 278 | 4.33 | 197 |

$n=200$, ${\alpha}_{y}=2\%$ | 3.65 | 3.64 | 272 | 3.65 | 136 |

$n=200$, ${\alpha}_{y}=3\%$ | 3.18 | 3.17 | 252 | 3.17 | 147 |

$n=220$, ${\alpha}_{y}=0\%$ | 5.29 | 5.29 | 326 | 5.30 | 172 |

$n=220$, ${\alpha}_{y}=1\%$ | 4.08 | 4.07 | 274 | 4.07 | 189 |

$n=220$, ${\alpha}_{y}=2\%$ | 3.39 | 3.38 | 252 | 3.39 | 142 |

$n=220$, ${\alpha}_{y}=3\%$ | 2.90 | 2.89 | 252 | 2.90 | 172 |

$n=240$, ${\alpha}_{y}=0\%$ | 5.24 | 5.14 | 322 | 5.24 | 139 |

$n=240$, ${\alpha}_{y}=1\%$ | 3.86 | 3.85 | 274 | 3.86 | 160 |

$n=240$, ${\alpha}_{y}=2\%$ | 3.15 | 3.13 | 252 | 3.14 | 148 |

$n=240$, ${\alpha}_{y}=3\%$ | 2.65 | 2.64 | 252 | 2.65 | 176 |

Conditions | MCS | iHL-RF | AK-MCS | ||
---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$ | |

$n=180$, ${\alpha}_{y}=0\%$ | 5.10 | 5.02 | 350 | 5.10 | 176 |

$n=180$, ${\alpha}_{y}=1\%$ | 4.00 | 3.99 | 310 | 3.99 | 169 |

$n=180$, ${\alpha}_{y}=2\%$ | 3.38 | 3.36 | 286 | 3.37 | 152 |

$n=180$, ${\alpha}_{y}=3\%$ | 2.93 | 2.91 | 278 | 2.93 | 164 |

$n=200$, ${\alpha}_{y}=0\%$ | 4.88 | 4.85 | 348 | 4.87 | 171 |

$n=200$, ${\alpha}_{y}=1\%$ | 3.75 | 3.73 | 304 | 3.75 | 184 |

$n=200$, ${\alpha}_{y}=2\%$ | 3.10 | 3.08 | 276 | 3.10 | 163 |

$n=200$, ${\alpha}_{y}=3\%$ | 2.63 | 2.62 | 274 | 2.64 | 180 |

$n=220$, ${\alpha}_{y}=0\%$ | 4.73 | 4.69 | 346 | 4.71 | 178 |

$n=220$, ${\alpha}_{y}=1\%$ | 3.52 | 3.50 | 284 | 3.52 | 141 |

$n=220$, ${\alpha}_{y}=2\%$ | 2.84 | 2.82 | 278 | 2.84 | 172 |

$n=220$, ${\alpha}_{y}=3\%$ | 2.36 | 2.35 | 272 | 2.36 | 193 |

$n=240$, ${\alpha}_{y}=0\%$ | 4.59 | 4.55 | 328 | 4.59 | 145 |

$n=240$, ${\alpha}_{y}=1\%$ | 3.30 | 3.28 | 286 | 3.30 | 147 |

$n=240$, ${\alpha}_{y}=2\%$ | 2.60 | 2.62 | 258 | 2.60 | 185 |

$n=240$, ${\alpha}_{y}=3\%$ | 2.12 | 2.10 | 266 | 2.12 | 117 |

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## Share and Cite

**MDPI and ACS Style**

Liu, J.; Liu, Y.; Wang, G.; Lu, N.; Cui, J.; Wang, H.
Fatigue Reliability Assessment for Orthotropic Steel Decks: Considering Multicrack Coupling Effects. *Metals* **2024**, *14*, 272.
https://doi.org/10.3390/met14030272

**AMA Style**

Liu J, Liu Y, Wang G, Lu N, Cui J, Wang H.
Fatigue Reliability Assessment for Orthotropic Steel Decks: Considering Multicrack Coupling Effects. *Metals*. 2024; 14(3):272.
https://doi.org/10.3390/met14030272

**Chicago/Turabian Style**

Liu, Jing, Yang Liu, Guodong Wang, Naiwei Lu, Jian Cui, and Honghao Wang.
2024. "Fatigue Reliability Assessment for Orthotropic Steel Decks: Considering Multicrack Coupling Effects" *Metals* 14, no. 3: 272.
https://doi.org/10.3390/met14030272