# Distortion-Induced Fatigue Reassessment of a Welded Bridge Detail Based on Structural Stress Methods

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## Abstract

**:**

## 1. Introduction

## 2. Structural Stress Methods

#### 2.1. Hot-Spot Stress Method

_{w}and 1 t

_{w}from the weld toe, where t

_{w}corresponds to the plate thickness/plate of the element subjected to loading (Figure 1). In this context, according to Saini et al. [38], the hot-spot stress method considers the effect of increased stress due to geometric discontinuity, but disregards the stress located in the notch of the weld of the structural element under analysis. The hot-spot stress, also known as geometric stress, does not consider the non-linear peak resulting from the discontinuity existing in the limit of the weld bead of the structural element (refer to Figure 1). Surface extrapolated (Figure 1a) or through thickness linearized structural stresses may be computed (Figure 1b).

^{6}cycles is described by Equation (1), where Δσ

_{hs}is the variation of the hot-spot stress amplitude (in MPa), N is the number of cycles to failure, m is the slope of the curve S–N, and C is the design value of the fatigue strength of the joint.

_{w}, located at a distance of 0.4 t

_{w}and 1.0 t

_{w}from the edge of the weld bead, Equation (2) should be applied:

_{w}, the evaluations of nodal stresses are performed at three reference points located at distances of 0.4 t

_{w}, 0.9 t

_{w,}and 1.4 t

_{w}from the edge of the weld bead, according to Equation (3). This method is recommended for cases of a marked increase in the bending component of the structural stress. Like the previous method (linear extrapolation), the quadratic extrapolation method in fine meshes was adopted for analysis in the present work.

#### 2.2. The Master S-N Curve

_{m}], can be obtained by dividing the linear forces at each node (unit: N/m) by the plate thickness (unit: m), t, (or length of “integration” for partial through-thickness failure). The matrix of bending stresses, [σ

_{b}], can be obtained by dividing the linear moments at each node (unit: Nm/m) by the bending elastic modulus per unit of length (unit: m

^{3}/m) of a rectangular section of height t:

#### 2.2.1. The Master S-N Curve (Mode I)

_{S}), taking into account both plate thickness and bending ratio effect, was proposed by Dong et al. [21] under the dominated Mode I loading condition (Equation (10)):

_{s}is the equilibrium equivalent structural stress range; m is the inverse slope of Paris–Law with a two-stage crack growth model, usually taken as 3.6; t* is a dimensionless plate thickness ratio about a unit thickness (t* = t/t

_{ref}, with t

_{ref}= 1 mm), thus handling the equivalent stress parameter with stress units (MPa); and the integral I(r

_{b}) is a dimensionless function of the bending ratio, r

_{b}, computed according to Equation (11):

_{b})

^{1/m}(Equations (12) and (13) were provided in SI units in [43]:

_{S}, versus cycles to failure, N, for various types of joints, with different thicknesses, dimensions, and loading modes. The analysis demonstrated the capability to collapse several fatigue tests of welded components into a narrow band. The derived curve was then designated (according to Equation (14)) as the master S-N curve:

#### 2.2.2. The Master S-N Curve Method for Multiaxial Fatigue Analysis (Mixed-Mode I + III)

_{e}[44], parameter in the following form, Equation (15):

_{S}is the equivalent in-plane shear structural stress parameter, accounting for Mode III contribution and defined as follows:

_{s}is computed according to Equation (19); m

_{τ}taken as equal to 5 is the inverse slope of the Paris–Law curve in a log–log scale with a two-stage crack growth model for specimens under pure Mode III loading; and I(r

_{τ})

^{1/m}

_{τ}is a dimensionless 4th order polynomial taking into account the in-plane shear bending ratio effect, r

_{τ}, both proposed by Hong and Forte [44] as equal to:

_{τ})

^{1/m}

_{τ}can be found in [44]. In this case, transverse shear (Mode II) is usually negligible for most applications in welded structures and hence neglected in the above formula. Proceeding in this form, the same fatigue strength parameters for the master S-N curve derived for Mode I, C, and h (Table 1), can be used for multiaxial purposes under Modes I + III with the employment of the effective equivalent structural stress parameter in the y-axis combined computed on the basis of the von Mises interaction criterion.

_{z}, are usually neglected because they tend to be lower than the other structural stress components for welded structures.

_{S}, defined as the sum of in-plane membrane shear stress ranges, Δτ

_{m}, and in-plane bending shear stress ranges Δτ

_{b}, following Equation (19):

## 3. Description of the Fatigue Test Setup

#### 3.1. Description of the Numerical Models

#### 3.2. Numerical Model Calibration Methodology

_{1}to θ

_{4}) which were not explicitly given in the NCHRP fatigue program report [47]. The process is based on a black-box numerical optimization algorithm available in MATLAB Optimization Toolbox (Figure 10), referred to as PatternSearch, which is suitable for dealing with problems without known derivatives (Audet and Kokkolaras, ref. [48]). The algorithm is put to work together with ANSYS Mechanical in a structured framework, aiming at minimizing the objective function defined by Equation (20) consisted of the sum of the squared differences between the observed and computed modified hot-spot stress ranges (MHS) which should be below a predefined tolerance of 1 MPa. In addition, the algorithms were designed to be able to apply a post-processor, developed by the authors in reference [6], for mesh insensitive or equilibrium equivalent structural stresses computation during fatigue induced distortion of welded joints.

_{1}, θ

_{2}, θ

_{3}, and θ

_{4}) with values limited by the vectors lb and ub, standing for lower bound and upper bound, respectively, which were adopted between 0 and 45 degrees due to geometric constraints. These values are passed to “fobj.m”, which is responsible for erasing the contents and writing it to file “bar_angles.txt”. The set of angles in question is also appended to “angle_iterations.txt” for register purposes. Then, an instance of ANSYS Mechanical is launched with the macro “Macro_Pair_N.inp”, which may have different configurations for each pair of girders being calibrated (Figure 11). This macro is basically responsible for: (i) resuming a shell global FE model of the test setup without the bracings; (ii) reading the proposed angles and automatically model the bracings; (iii) performing static linear analysis in the global and local sub-models; and (iv) compute modified hot-spot stresses and apply Equation (20). The objective function value is then returned to the PatternSearch algorithm in MATLAB, which will evaluate the need to propose a new set of angles (θ′

_{1}, θ′

_{2}, θ′

_{3}, and θ′

_{4}) and restart the analysis if convergence is not verified. N is the number of the tested pair of beams. For each iteration step along the optimization process, the process described by Figure 10 and Figure 11 takes nearly 30 s. The number of iterations for each calibrated girder will be described in the next section.

## 4. Results and Discussion

_{w}and 1.0 t

_{w}of the weld toes. The optimization was performed separately for each girder taking advantage of the symmetry, which was carried out by pairs.

_{e}, of a local web gap sub-model obtained with the developed post-processor tool [6] is shown in Figure 12a. In general, the overall pattern of the maximum values for the structural stress are in agreement with the expected fatigue crack growth behaviour (Figure 12b). The reassessment of the S-N data of transverse stiffeners web gap details in terms of hot-spot stress and Equation SS ranges was performed by using the values shown in Table 3.

#### 4.1. Validation of the S-N Hot-Spot Curve Approach for the Distortion-Induced Fatigue Program

_{HS}by 1.05, to cover their unfavourable influence on the fatigue life up to a margin of 5%. The reason is that the FE models are idealized geometries, and did not explicitly include the misalignments in solid modelling, which is not practical.

#### 4.2. Validation of the Master S-N Curve Approach for the Distortion-Induced Fatigue Program

^{2}= 0.831 and SD = 0.23, R

^{2}= 0.856, respectively, are of the same magnitude, and the differences are negligible, on the order of 0.04 for SD, and 0.025 for R

^{2}. However, the possibility of using a unique S-N master curve for any details makes the usage of the equilibrium-equivalent structural stress method very attractive for bridge welded joints.

## 5. Conclusions and Future Prospects

- Based on the real system, it was modelled numerically in order to perform the computational analysis, and then a methodology for the calibration was proposed. In this way, it resulted that the local and global stresses were successfully determined, reaching a difference of less than 1%.
- Both methods (hot-spot stress and master S-N curve) were successfully applied. With the hot-spot stress method, the points were distributed above FAT 90 and with respect to the S-N master curve method, the points collapsed within the narrow band (curves interval). Nevertheless, the master curve can be more attractive, taking into account the need for one S-N curve usage, even for structural details.
- This study highlights the advantages of using the master curve and the structural equilibrium equivalent mesh insensitive structural stresses in the fatigue design of welded bridge details, even in cases of complex load transfer mechanisms. Moreover, the availability of a dedicated post-processor made the analysis quite effective.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Structural stress components at the weld toe (adapted from [45]).

**Figure 4.**Flowchart of multiaxial fatigue analysis (mixed-mode I + III) using the master S-N curve method.

**Figure 5.**Global model of the structural system of beams; fatigue test of details sensitive to distortion-induced fatigue (units: millimetres).

**Figure 6.**Fatigue test of distortion-induced fatigue sensitive detail: (

**a**,

**c**) girders and (

**b**,

**d**) sections dimensions (units: millimetres).

**Figure 7.**Fatigue test of distortion-induced fatigue sensitive detail: section (units: millimetres).

**Figure 8.**The developed global numerical model of the fatigue test of the NCHRP [47] program for fatigue strength assessment of web gap details: (

**a**) global view, (

**b**) longitudinal view, and (

**c**) side view.

**Figure 9.**(

**a**) Local numerical models of the web gap detail: 76 mm web gap and (

**b**) placement of strain gauges to measure the modified hot-spot stresses [47].

**Figure 10.**Flowchart for the calibration procedure of web gaps test setup for each girder pair: MATLAB/ANSYS framework.

**Figure 12.**(

**a**) Contour-plot of the Equation SS found after the calibration process for a girder specimen series and (

**b**) crack pattern around the stiffener termination [49].

**Figure 13.**Surface extrapolated structural stresses computation with reference points: (

**a**) linear extrapolation compatible with experimental measurements; (

**b**) quadratic extrapolation at weld corner looking for critical hot-spot stresses.

**Figure 14.**Modified hot-spot S-N curve for transverse web stiffener (refer to Figure 12a).

**Figure 15.**Structural hot-spot stress results for welded bridge details (refer to Figure 12b).

**Table 1.**Master S-N curve parameters according to ASME BPVC Sec VIII Div 2 [43].

Statistical Basis | C | h |
---|---|---|

Mean | 19,930.2 | −0.3195 |

+2 SD | 28,626.5 | |

−2 SD | 13,875.8 | |

+3 SD | 34,308.1 | |

−3 SD | 11,577.9 |

Steel Bracings Slopes (Decimal Degrees °) | No. of Iterations | ||||
---|---|---|---|---|---|

Girder | Left Girders | Right Girders | |||

θ_{1} | θ_{2} | θ_{3} | θ_{4} | ||

G2 | 3.21 | 10.97 | 10.65 | 12.83 | 53 |

G5 | 8.25 | 12.82 | 12.10 | 5.72 | 46 |

G6 | 4.58 | 12.20 | 7.76 | 13.95 | 51 |

G8 | 4.43 | 12.40 | 6.88 | 14.51 | 51 |

G10 | 6.25 | 12.09 | 12.03 | 7.82 | 61 |

G11 | 4.69 | 11.44 | 4.50 ^{1} | 4.50 ^{1} | 28 |

G12 | 4.50 ^{1} | 4.50 ^{1} | 0.72 | 6.78 | 39 |

^{1}Defined as fixed slopes for these optimization simulations.

Girder | Applied FE Loads, Pmax (kN) | Web Gap Length ^{2} (mm) | In-Plane Stress Range ^{1}, Δσ (MPa)Δσ = σ _{MAX} × (1 − R) | Out-of-Plane Hot-Spot Stress Range ^{2}, Δσ_{HS} (MPa) | Hot-Spot (Inclined) (MPa) Computed Quadratic (Figure 12b) | EESS Range, ΔS_{e} (MPa) | |||
---|---|---|---|---|---|---|---|---|---|

Δσ_{HS} = σ_{HSMAX} × (1 − R) | Modes I + III | ||||||||

Linear Rule (0.4 t_{w}, 1.0 t_{w})(Figure 12a) | ΔS_{e} = SS × (1 − R) | ||||||||

Measured | Computed | Measured ^{3} | Computed | Error | Computed | ||||

G2 | 122.04 | 45.97 | 41.36 | 41.27 | 61.36 | 61.78 | 0.68% | 109.57 | 153.64 |

G5 | 122.04 | 49.02 | 41.36 | 41.25 | 60.67 | 61.28 | 1.01% | 65.01 | 87.75 |

G6 | 122.04 | 49.53 | 41.36 | 41.22 | 132.38 | 132.00 | 0.29% | 165.29 | 223.95 |

G8 | 122.04 | 51.05 | 41.36 | 41.20 | 160.65 | 160.15 | 0.31% | 188.38 | 255.72 |

G10 | 229.62 | 51.31 | 82.74 | 83.54 | 102.04 | 101.43 | 0.60% | 104.19 | 151.13 |

G11 | 229.62 | 53.59 | 82.74 | 83.10 | 170.30 | 169.61 | 0.41% | 225.28 | 329.39 |

G12 | 229.62 | 48.26 | 82.74 | 82.43 | 186.85 | 186.52 | 0.18% | 233.66 | 337.72 |

^{1}Measured in the bottom flange at the midspan.

^{2}Measured from the toes of stiffener-to-web and flange-to-web welds.

^{3}At the time the test was carried out, the stresses were measured using linear extrapolation with nonstandard positions for the strain gauges.

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**MDPI and ACS Style**

Quissanga, V.; Alencar, G.; de Jesus, A.; Calçada, R.; da Silva, J.G.S.
Distortion-Induced Fatigue Reassessment of a Welded Bridge Detail Based on Structural Stress Methods. *Metals* **2021**, *11*, 1952.
https://doi.org/10.3390/met11121952

**AMA Style**

Quissanga V, Alencar G, de Jesus A, Calçada R, da Silva JGS.
Distortion-Induced Fatigue Reassessment of a Welded Bridge Detail Based on Structural Stress Methods. *Metals*. 2021; 11(12):1952.
https://doi.org/10.3390/met11121952

**Chicago/Turabian Style**

Quissanga, Vencislau, Guilherme Alencar, Abílio de Jesus, Rui Calçada, and José Guilherme S. da Silva.
2021. "Distortion-Induced Fatigue Reassessment of a Welded Bridge Detail Based on Structural Stress Methods" *Metals* 11, no. 12: 1952.
https://doi.org/10.3390/met11121952