# Investigation of the Fatigue Stress of Orthotropic Steel Decks Based on an Arch Bridge with the Application of the Arlequin Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Bridge

_{y}and σ

_{u}denote the yield strength and ultimate strength, respectively; and A represents the elongation ratio. Its top plates adopt the orthotropic steel deck form with closed U-ribs, fabricated by cold bending of 8 mm steel plates. All the U-ribs pass through the 30 mm diaphragm set every 4 m. The bottom plates of the main girder are composed of orthotropic plates with L-shaped stiffeners, which are made of 200-type flat-bulb steel. In the carriageway area, the thickness of the top plates and bottom plates is 18 mm and 14 mm, respectively. Details of the trough-to-deck weld joint are shown in Figure 1d.

## 3. Arlequin Method

_{1}and Ω

_{2}—and the overlapping area of these two regions is defined as coupling region S. According to analysis requirements, different mesh sizes and element types can be assigned in Ω

_{1}and Ω

_{2}. The primary purpose of the Arlequin method is to establish transition elements in coupling region S to conduct global/local analysis, with the usage of the energy partition function and reliable coupling operators between Ω

_{1}and Ω

_{2}. The following is a brief explanation of the stiffness matrix of the transition elements.

_{i}denotes the displacement tensor of region Ω

_{i}.

**λ**is the Lagrange multiplier within the coupling area, and Γ

_{u}, f, ε(u) and σ(u) represent the displacement constraints, load tensor, strain tensor and stress tensor of boundary ∂Ω

_{i}, respectively. Then, the possible displacement fields in each region can be expressed as:

^{1}is the scalar product defined in Sobolev space. Assuming ${W}_{\mathrm{h}i}\subset {W}_{i}$ and ${W}_{\mathrm{h}\lambda}\subset {W}_{\lambda}$ denote the FE discrete spacing of regions Ω

_{i}and S

_{λ}, respectively, the Arlequin algorithm for this model can be interpreted as the solution to the following saddle point equation:

_{i}, and the weight function α

_{i}and β

_{i}of strain energy and work satisfy the following relationship:

**W**

_{hi}and

**W**

_{hλ}, $\widehat{{u}_{i}}$ and $\widehat{\lambda}$ as the coordinates of

**u**

_{hi}and

**λ**

_{h}along these primary functions, a variational solution for Equation (4) can be obtained:

## 4. Finite Element Models

#### 4.1. Global Model

#### 4.2. Local Model

## 5. Fatigue Stress Analysis of Orthotropic Steel Deck

_{l}and σ

_{g}are the stress range derived from the local model and the global model, respectively. According to the degree of stress concentration derived from the stress nephogram in subsequent simulations, A, B, C and D nodes are selected as the most vulnerable sites for stress comparison, as shown in Figure 7. The stress nephogram is depicted in the next section. For vulnerable points B–D, the stress range is the nominal stress by direct calculation; for point A located at the root of the weld toe, the stress range is the hot spot stress obtained by the linear extrapolation formula recommended by the International Institute of Welding (IIW) [38]:

_{0.4t}and σ

_{1.0t}are the nodal stresses at reference points 0.4 times and 1.0 times the plate thickness away from the weld toe, respectively, as illustrated in Figure 7.

#### 5.1. Fatigue Stresses of the Local and Global Models

#### 5.2. Fatigue Stresses Provoked by Other Vehicle Loads

_{1}are calculated based on the stress of the MC model and the local model, respectively. As can be observed, compared with single-vehicle loading, fatigue stress aroused by traffic loads increases about 10%. For some vulnerable sites, the stress change rate can be up to 14%. This indicates that the practice of single-fatigue-vehicle loading stipulated in the design code can roughly reflect real traffic conditions. From traffic flow case 1 to case 4, the outright load applied on the bridge is constant, but the concentration of vehicle distribution around the refined solid region is gradually weakened. Correspondingly, the stress change rate δ of each column in Table 2 has a gradual decline from top to bottom. This means that the fatigue performance of the bridge deck is not only related to the total number of vehicles, but its scattered status is also a key factor. The stress change rate δ

_{1}reflects the effects of the three sub-systems on fatigue stress calculations; its impact is about 30–40% for each traffic condition. As far as the arch bridge analyzed in this paper, the local approach is not suitable for fatigue evaluation because of the huge stress difference. The practice of obtaining fatigue stress through local analysis is not available for all bridge structures. This may help to explain the premature cracks observed in bridges under service.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the arch bridge: (

**a**) deck cross-section at the center span; (

**b**) plan view; (

**c**) elevation view; (

**d**) structural details of orthotropic deck (unit: mm).

**Figure 4.**Schematic diagram of the local model (unit: mm): (

**a**) full model; (

**b**) half model; (

**c**) mesh details.

**Figure 6.**Convergence verification: (

**a**) influence of different solid area; (

**b**) validation model (unit: mm).

**Figure 8.**Load case for the global model: (

**a**) load location on the deck panel; (

**b**) fatigue vehicle load [35].

E (MPa) | σ_{y} (MPa) | σ_{u} (MPa) | A (%) |
---|---|---|---|

198,221 | 351.10 | 508.57 | 40.60 |

Model | Position A | Position B | Position C | Position D | ||||
---|---|---|---|---|---|---|---|---|

Stress Range | δ | Stress Range | δ | Stress Range | δ | Stress Range | δ | |

Local Model | 23.0 | - | 51.2 | - | 44.4 | - | 16.0 | - |

MC | 27.8 | 20.87% | 62.3 | 21.68% | 55.7 | 25.45% | 24.3 | 51.87% |

MS | 28.3 | 23.04% | 62.9 | 22.85% | 56.6 | 27.48% | 25.4 | 58.75% |

QC | 27.5 | 19.57% | 61.7 | 20.51% | 55.2 | 24.32% | 23.2 | 45.00% |

QS | 28.4 | 23.48% | 62.2 | 21.48% | 56.3 | 26.80% | 24.8 | 55.00% |

Traffic Flow Case | Position A | Position B | ||||

Stress Range | δ | δ_{1} | Stress Range | δ | δ_{1} | |

Local Model | 23.0 | - | - | 51.2 | - | - |

MC | 27.8 | - | - | 62.3 | - | - |

Case 1 | 30.3 | 8.99% | 31.74% | 68.9 | 10.59% | 34.57% |

Case 2 | 30.2 | 8.63% | 31.30% | 68.6 | 10.11% | 33.98% |

Case 3 | 29.6 | 6.47% | 28.70% | 65.7 | 5.46% | 28.32% |

Case 4 | 29.5 | 6.12% | 28.26% | 64.9 | 4.17% | 26.76% |

Traffic Flow Case | Position C | Position D | ||||

Stress Range | δ | δ_{1} | Stress Range | δ | δ_{1} | |

Local Model | 44.4 | - | - | 16.0 | - | - |

MC | 55.7 | - | - | 24.3 | - | - |

Case 1 | 63.3 | 13.64% | 42.57% | 26.4 | 8.64% | 65.00% |

Case 2 | 62.7 | 12.57% | 41.22% | 25.9 | 6.58% | 61.88% |

Case 3 | 59.3 | 6.46% | 33.56% | 23.5 | −3.29% | 46.88% |

Case 4 | 58.1 | 4.31% | 30.86% | 24.1 | 0.82% | 50.63% |

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**MDPI and ACS Style**

Cheng, C.; Xie, X.; Yu, W.
Investigation of the Fatigue Stress of Orthotropic Steel Decks Based on an Arch Bridge with the Application of the Arlequin Method. *Materials* **2021**, *14*, 7653.
https://doi.org/10.3390/ma14247653

**AMA Style**

Cheng C, Xie X, Yu W.
Investigation of the Fatigue Stress of Orthotropic Steel Decks Based on an Arch Bridge with the Application of the Arlequin Method. *Materials*. 2021; 14(24):7653.
https://doi.org/10.3390/ma14247653

**Chicago/Turabian Style**

Cheng, Cheng, Xu Xie, and Wentao Yu.
2021. "Investigation of the Fatigue Stress of Orthotropic Steel Decks Based on an Arch Bridge with the Application of the Arlequin Method" *Materials* 14, no. 24: 7653.
https://doi.org/10.3390/ma14247653