# Numerical Prediction of Welding Distortion Considering Gravity Force on General Ship Grillage Structure by Elastic Finite Element Method Using Inherent Strain

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Elastic FEM Using Inherent Strain

#### 2.1. Basic Concept of Inherent Strain and Inherent Displacment

#### 2.2. Caculation of Inherent Deforatmion and Inherent Strain

_{2}arc butt-welding of a plain plate (200 mm × 200 mm × 10 mm) using high-strength steel (HT50), Equations (3)–(9) were derived [12] to calculate the inherent displacement. In these equations, the amount of inherent deformation is determined by the amount of heat input ${Q}^{*}$ of CO

_{2}arc butt-welding. In [13], it was validated that the relationship between heat input ${Q}^{*}$ and net heat input ${Q}_{net}$ in the welding process is decided by the thickness h of the welded steel plate, and it can be expressed as ${Q}^{*}={Q}_{net}/{h}^{2}$. In [14], it was concluded that the longitudinal shrinkage is generally evaluated by the contraction force ${F}_{T}$, and; the relationship between the contraction force ${F}_{T}$ and the net heat input ${Q}_{net}$ is derived as Equation (9). Table 1 and Table 2 present, respectively, the conditions of welding and the mechanical properties of HT50 steel.

- Transverse Shrinkage$$S={C}_{t}\left(L\right){S}_{0}$$$${S}_{0}=\{\begin{array}{c}1.16\times {10}^{-3}{Q}_{net}/h\left({Q}^{*}\le 6.27\right)\\ h\{1.44\times {10}^{-4}[{\left({Q}^{*}\right)}^{2}-{Q}^{*}]+2.5\times {10}^{-3}\}(6.27{Q}^{*}\le 20)\\ 2.85\times {10}^{-3}{Q}_{net}/h(20{Q}^{*})\end{array}$$$${C}_{t}\left(L\right)=[4ta{n}^{-1}\left(L/200\right)+\left(L/100\right)\times \mathrm{log}\left(1+40000/{L}^{2}\right)]/3.74$$
- Angular Deformation$$\theta ={C}_{a}\left(L\right){\theta}_{0}$$$${\theta}_{0}=\{\begin{array}{c}1.44\times {10}^{-3}{Q}^{*}\left({Q}^{*}\le 6.27\right)\\ 1.06\times {10}^{-1}{Q}^{*}/\{{({Q}^{*}-6.16)}^{2}+73.6\}\left(6.27{Q}^{*}\right)\end{array}$$$${C}_{a}\left(L\right)=[8ta{n}^{-1}\left(L/120\right)+\left(1+v\right)\left(L/60\right)\times \mathrm{log}\left(1+14400/{L}^{2}\right)]/8.84$$
- Longitudinal Shrinkage (Contraction Force)$${F}_{T}=0.2{Q}_{net}$$
- ${\theta}_{0}$ is the angular deformation at a welding length of 200 mm;
- ${C}_{t}\left(L\right)$ is the welding length compensation coefficient for lateral shrinkage;
- ${C}_{a}\left(L\right)$ is the welding length compensation coefficient for angular deformation;
- ${F}_{t}$ is the vertical contraction force;
- $L$ is the welding length [mm];
- $v$ is Poisson’s ratio;
- ${Q}_{net}$ is the net heat input [J/mm];
- $h$ is the plate thickness [mm];
- ${Q}^{*}$ is ${Q}_{net}/{h}^{2}$ [J/mm
^{3}];

^{3}];

#### 2.3. Interface Element Method

#### 2.4. Multipoint Constraint Function

## 3. Analysis Model

_{2}arc butt-welding is used herein as Table 1. All the cross-section upper and bottom points were originally installed MPC to assume tack welding and outline the complete structure before being in a full welding operation. A total of 29 welding lines are used, categorized in three groups, as follows: longitudinal (1–5), transverse (6–9), and vertical (10–29). The 29 welding lines with their numbers are schematically drawn in Figure 6.

## 4. Boundary Condition of the Rails Considering the Gravity Force

## 5. Effect of the Gravity Force under the Rail Boundary Condition on Welding Sequence

#### 5.1. Effect of Each Welding Line on the Bottom Plate

#### 5.2. Welding Sequence

#### 5.3. Result and Discussion of Effect of Gravity Force Under the Rail Boundary Condition on Welding Sequence

## 6. Effect of the Change in Gravity Direction on the Numerical Prediction of Welding Displacements

#### Results and Discussion of Effect of Change in Gravity Direction on the Numerical Prediction of Welding Displacements

## 7. Conclusions

- (1)
- Although the bottom plate and rails touch each other under the gravity force while processing the welding sequence under the rail boundary condition, these considerably constraint the structure, and therefore, significantly mitigate the welding displacement without additional clamps for the restriction of its movement. In other words, the numerical prediction of welding displacements without precisely reflecting the real work environment would lead to enormous errors in heavy industries.
- (2)
- In the rail boundary condition under the effect of the gravity force, the optimal welding sequence is to weld first vertically for improving the stiffness of the structure and then horizontally. It is preferable to begin the horizontal welding lines, which generate a direct heat effect on the bottom plate, as late as possible. Moreover, welding the transverse lines before the longitudinal lines is preferred for minimizing welding displacements. The conclusion is the same as that in the previous study of D. Woo et al. (2019), which was validated according to the result of the simple boundary condition.
- (3)
- The change in direction of the gravity force according to the design plan has significant effects on the change in the distribution of welding displacements. Without consideration of these effects, the prediction of the additional production cost for the revision work could involve a substantial error. Thus, in the numerical prediction of welding displacements in the welding process, consideration of the change in direction of the gravity force with respect the structure is technically essential.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 14.**Z-axis displacement distribution along line T of Sequence B-10 under two different boundary conditions.

**Figure 15.**Z-axis displacement distribution along line L of Sequence B-10 under two different boundary conditions.

Current [A] | Voltage [V] | Travel Speed [mm/s] | Heat Efficiency | Net Heat [J/mm^{2}] |
---|---|---|---|---|

230 | 23 | 5 | 0.77 | 500 |

Density [kg/m^{3}] | Young’s Modulus [MPa] | Specific Heat [J/kg/°C] | Yield Stress [MPa] | Poisson’s Ratio |
---|---|---|---|---|

7720 | 2.0 × 10^{5} | 659.4 | 440 | 0.3 |

Sequence | Welding Sequence Preference | Welding Sequence | ||||
---|---|---|---|---|---|---|

Category A | A-1 | Horizontal | H | Vertical | H | 6→9→7→8→3→2→4→5→1→20→19→24→23→15→16→14→ 17→22→25→18→21→27→28→11→12→26→29→10→13 |

A-2 | H | L | 6→9→7→8→3→2→4→5→1→10→13→26→29→11→12→27→ 28→18→21→22→25→14→17→15→16→23→24→19→20 | |||

A-3 | L | H | 1→5→4→2→3→7→8→6→9→20→19→24→23→15→16→14→ 17→22→25→18→21→27→28→11→12→26→29→10→13 | |||

A-4 | L | L | 1→5→4→2→3→7→8→6→9→10→13→26→29→11→12→27→ 28→18→21→22→25→14→17→15→16→23→24→19→20 | |||

A-5 | Vertical | H | Horizontal | H | 20→19→24→23→15→16→14→17→22→25→18→21→27→28→11→12→26→29→10→13→6→9→7→8→3→2→4→5→1 | |

A-6 | H | L | 20→19→24→23→15→16→14→17→22→25→18→21→27→28→11→12→26→29→10→13→1→5→4→2→3→7→8→6→9 | |||

A-7 | L | H | 10→13→26→29→11→12→27→28→18→21→22→25→14→17→15→16→23→24→19→20→6→9→7→8→3→2→4→5→1 | |||

A-8 | L | L | 10→13→26→29→11→12→27→28→18→21→22→25→14→17→15→16→23→24→19→20→1→5→4→2→3→7→8→6→9 |

Sequence | Welding Sequence Preferences | Welding Sequence | ||||||
---|---|---|---|---|---|---|---|---|

Category B | B-1 | Vertical | H | Longitudinal | H | Transverse | H | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→3→2→4→5→1→6→9→7→8 |

B-2 | H | H | L | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→3→2→4→5→1→7→8→6→9 | ||||

B-3 | H | L | H | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→1→5→4→2→3→6→9→7→8 | ||||

B-4 | H | L | L | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→1→5→4→2→3→7→8→6→9 | ||||

B-5 | L | H | H | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→3→2→4→5→1→6→9→7→8 | ||||

B-6 | L | H | L | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→3→2→4→5→1→7→8→6→9 | ||||

B-7 | L | L | H | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→1→5→4→2→3→6→9→7→8 | ||||

B-8 | L | L | L | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→1→5→4→2→3→7→8→6→9 | ||||

B-9 | Vertical | H | Transverse | H | Longitudinal | H | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→6→9→7→8→3→2→4→5→1 | |

B-10 | H | H | L | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→6→9→7→8→1→5→4→2→3 | ||||

B-11 | H | L | H | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→7→8→6→9→3→2→4→5→1 | ||||

B-12 | H | L | L | 13→10→26→29→12→11→27→28→18→21→25→14→22→17→16→23→15→24→19→20→7→8→6→9→1→5→4→2→3 | ||||

B-13 | L | H | H | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→6→9→7→8→3→2→4→5→1 | ||||

B-14 | L | H | L | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→6→9→7→8→1→5→4→2→3 | ||||

B-15 | L | L | H | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→7→8→6→9→3→2→4→5→1 | ||||

B-16 | L | L | L | 19→20→24→15→23→16→17→22→14→25→18→21→28→27→11→12→29→26→10→13→7→8→6→9→3→2→4→5→1 |

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## Share and Cite

**MDPI and ACS Style**

Woo, D.; Kitamura, M.
Numerical Prediction of Welding Distortion Considering Gravity Force on General Ship Grillage Structure by Elastic Finite Element Method Using Inherent Strain. *J. Mar. Sci. Eng.* **2020**, *8*, 454.
https://doi.org/10.3390/jmse8060454

**AMA Style**

Woo D, Kitamura M.
Numerical Prediction of Welding Distortion Considering Gravity Force on General Ship Grillage Structure by Elastic Finite Element Method Using Inherent Strain. *Journal of Marine Science and Engineering*. 2020; 8(6):454.
https://doi.org/10.3390/jmse8060454

**Chicago/Turabian Style**

Woo, Donghan, and Mitsuru Kitamura.
2020. "Numerical Prediction of Welding Distortion Considering Gravity Force on General Ship Grillage Structure by Elastic Finite Element Method Using Inherent Strain" *Journal of Marine Science and Engineering* 8, no. 6: 454.
https://doi.org/10.3390/jmse8060454