# Effect of Residual Stresses on the Elastoplastic Behavior of Welded Steel Plates

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Implementation of Residual Stresses due to Welding

#### 2.1.1. Theoretical Model of Residual Stresses for Perfect Plates

#### 2.1.2. Implementation of Residual Stresses by FE Modelling

^{−5}K

^{−1}resulting in a minimum difference of temperatures $\Delta T$ of 208 °C.

#### 2.1.3. Implementation of Procedure in FE Modelling

#### 2.2. Parameters Affecting the Strength of Unstiffened Plates

#### 2.2.1. Plate Slenderness

#### 2.2.2. Geometric Imperfections

#### 2.2.3. Boundary Conditions

- Unrestrained; all lateral edges and tops, respectively long and short edges in a rectangular plate, are supported perpendicularly to the plate’s plane, but in-plane movements along the lateral edges are allowed and rotations are free.
- Restrained; all lateral edges and tops are supported perpendicularly to the plate’s plane, but in-plane movements along the edges are not allowed, called as fixed condition, and rotations are free.
- Constrained; all lateral edges and tops are supported perpendicularly to the plate plane, in-plane movements of the lateral edges are allowed but kept straight and rotations are free.
- Clamped; displacements and rotations are null in all edges.

## 3. Results

#### 3.1. Stocky Plates

_{i}, is 2 mm according to Equation (9). Three different levels of imperfections are considered: 2, 5 and 10 mm, which corresponds to w

_{i}/t ratio of 0.1, 0.25 and 0.5, respectively. Two modes of imperfections are analyzed: the fundamental, m = 1, and the critical one, m = 3. The effect of residual stresses for each plate is evaluated for three levels of width of the tensile strip at the edges: η = 0, 2 and 3 to which corresponds a normalized residual stress level of 0.0, 0.15 and 0.25.

_{r}/σ

_{o}to 2 by an almost straight cut. The ultimate stress passes to occur at normalized ultimate shortening of 2.

_{i}= 2 mm) has an ultimate strength close to the values found in plates with fundamental mode of imperfections.

#### 3.2. Intermediate Plates

_{i,}is 2.74 mm according to Equation (9). Three different levels of imperfections are considered: 2, 3 and 5 mm, which correspond to w

_{i}/t ratios of 0.13, 0.2 and 0.33, respectively. The structural analysis concentrates on the critical mode, m = 3. The effect of residual stresses for each plate is evaluated for three levels of the width of the tensile strip at the edges: η = 0, 2 and 3 to which corresponds a normalized residual stress level of 0, 0.11 and 0.18.

#### 3.3. Slender Plates

_{i,}is 4.11 mm according to Equation (9) corresponding to a w

_{i}/t ratio of 0.411. Three levels of imperfections, w

_{i}= 2, 5 and 6 mm and residual stresses of 0, 0.07 and 0.15 σ

_{y}are considered. The LSC’s of this group of plates are shown in Figure 10. Table 3 presents the main results of the ultimate strength and the corresponding normalized shortening for the different cases.

#### 3.4. Very Slender Plates

_{i,}is 5.14 mm according to Equation (9) and a w

_{i}/t ratio of 0.643. Three levels of imperfections, w

_{i}= 2, 5 and 6 mm and residual stresses of 0, 0.06 and 0.12 σ

_{y}are considered. The LSC’s of this group of plates are shown in Figure 12.

## 4. Discussion

- Effect of slenderness on ultimate strength and correlation with imperfections and residual stresses;
- Dependency of structural tangent modulus from initial conditions of plate, i.e., geometry, imperfections and residual stresses;
- Effect of residual stresses in the LSC’s.

#### 4.1. Effect of Slenderness on Ultimate Strength and Correlation with Imperfections and Residual Stresses

#### 4.2. Dependency of Structural Tangent Modulus from Initial Conditions

_{i}in both cases, as seen by comparison of yellow and red curves in both graphics. The presence of residual stresses represents a further reduction on initial effectiveness measured as E

_{t}/E. The total reduction can be very high, representing a great softening of the structural element in elastic range.

#### 4.3. Effect of Residual Stresses in the LSC’s

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Idealized residual stress model for an unstiffened perfect plate welded in the edges (

**left**) and results by FE thermal analysis (

**right**).

**Figure 3.**Example of state of stresses of a plate after HAZ heating (

**left**), HAZ cooling (

**middle**) and at collapse (

**right**).

**Figure 5.**Boundary conditions on a rectangular plate loaded longitudinally (constrained conditions).

**Figure 6.**Load shortening curves for stocky plates with different initial imperfections in amplitude (i = 2, 5 and 10 mm) and mode (m = 1, 3), and several levels of residual stresses (r = 0, 0.15 and 0.25%).

**Figure 7.**Variation of ultimate strength and corresponding normalized strain for plates with t = 20 mm, m = 3 and no residual stresses.

**Figure 8.**Load shortening curves for intermediate plates (β = 1.352) with different initial imperfections in fundamental and critical mode and residual stresses.

**Figure 9.**Variation of ultimate strength and corresponding normalized strain for plates with t = 15 mm, m = 3 and no residual stresses.

**Figure 10.**Load shortening curves for intermediate plates (β = 2.03) with different initial imperfections in critical mode and residual stresses.

**Figure 11.**Variation of ultimate strength and corresponding normalized strain for plates with t = 10 mm, m = 3 and no residual stresses.

**Figure 12.**Load shortening curves for very slender plates with different amplitude of initial imperfections in critical mode and residual stresses.

**Figure 13.**Variation of ultimate strength and corresponding normalized strain for plates with t = 8 mm, m = 3 and residual stresses with η = 0, 4.

**Figure 14.**Comparison of results for constraint plates considering residual stress and initial imperfections and standard formulations.

Identification | Mode i | w_{i} (mm) | w_{i}/t | η | Ultimate Stress | Ultimate Strain | Residual Stress % |
---|---|---|---|---|---|---|---|

t20m1i2r0 | 1 | 2 | 0.10 | 0 | 1.001 | 1.00 | 0.00 |

t20m1i2r25 | 1 | 2 | 0.10 | 3 | 1.002 | 2.00 | 0.25 |

t20m1i10r0 | 1 | 10 | 0.50 | 0 | 0.991 | 1.46 | 0.00 |

t20m1i10r15 | 1 | 10 | 0.50 | 2 | 0.989 | 2.36 | 0.15 |

t20m1i10r25 | 1 | 10 | 0.50 | 3 | 0.990 | 2.37 | 0.25 |

t20m3i2r0 | 3 | 2 | 0.10 | 0 | 0.982 | 0.98 | 0.00 |

t20m3i2r15 | 3 | 2 | 0.10 | 2 | 0.954 | 1.94 | 0.15 |

t20m3i2r25 | 3 | 2 | 0.10 | 3 | 0.952 | 1.97 | 0.25 |

t20m3i5r0 | 3 | 5 | 0.25 | 0 | 0.904 | 1.31 | 0.00 |

t20m3i5r15 | 3 | 5 | 0.25 | 2 | 0.861 | 1.92 | 0.15 |

t20m3i5r25 | 3 | 5 | 0.25 | 3 | 0.851 | 1.99 | 0.25 |

t20m3i10r0 | 3 | 10 | 0.50 | 0 | 0.789 | 1.53 | 0.00 |

t20m3i10r15 | 3 | 10 | 0.50 | 2 | 0.772 | 1.94 | 0.15 |

t20m3i10r15 | 3 | 10 | 0.50 | 3 | 0.765 | 2.03 | 0.25 |

Identification | Mode i | w_{i} (mm) | w_{i}/t | η | Ultimate Stress | Ultimate Strain | Residual Stress % |
---|---|---|---|---|---|---|---|

t15m1i3r0 | 1 | 3 | 0.20 | 0 | 1.000 | 1.402 | 0.00 |

t15m1i3r11 | 1 | 3 | 0.20 | 2 | 0.999 | 2.114 | 0.11 |

t15m3i2r0 | 3 | 2 | 0.13 | 0 | 0.919 | 1.113 | 0.00 |

t15m3i2r18 | 3 | 2 | 0.13 | 3 | 0.837 | 1.231 | 0.18 |

t15m3i3r0 | 3 | 3 | 0.20 | 0 | 0.874 | 1.129 | 0.00 |

t15m3i3r11 | 3 | 3 | 0.20 | 2 | 0.801 | 1.449 | 0.11 |

t15m3i3r18 | 3 | 3 | 0.20 | 3 | 0.782 | 1.742 | 0.18 |

t15m3i5r0 | 3 | 5 | 0.33 | 0 | 0.802 | 1.276 | 0.00 |

t15m3i5r11 | 3 | 5 | 0.33 | 2 | 0.755 | 1.660 | 0.11 |

t15m3i5r25 | 3 | 5 | 0.33 | 4 | 0.737 | 1.969 | 0.25 |

Identification | Mode i | w_{i} (mm) | w_{i}/t | η | Ultimate Stress | Ultimate Strain | Residual Stress % |
---|---|---|---|---|---|---|---|

t10m3i2r0 | 3 | 2 | 0.2 | 0 | 0.724 | 1.052 | 0.00 |

t10m3i2r07 | 3 | 2 | 0.2 | 2 | 0.670 | 1.105 | 0.07 |

t10m3i2r15 | 3 | 2 | 0.2 | 4 | 0.639 | 1.787 | 0.15 |

t10m3i5r0 | 3 | 5 | 0.5 | 0 | 0.648 | 1.215 | 0.00 |

t10m3i5r07 | 3 | 5 | 0.5 | 2 | 0.626 | 1.641 | 0.07 |

t10m3i5r15 | 3 | 5 | 0.5 | 4 | 0.615 | 1.748 | 0.15 |

t10m3i6r0 | 3 | 6 | 0.6 | 0 | 0.629 | 1.183 | 0.00 |

t10m3i6r07 | 3 | 6 | 0.6 | 2 | 0.615 | 1.635 | 0.07 |

t10m3i6r15 | 3 | 6 | 0.6 | 4 | 0.606 | 1.733 | 0.15 |

t10m3i8r0 | 3 | 8 | 0.8 | 0 | 0.598 | 1.432 | 0.00 |

t10m3i8r15 | 3 | 8 | 0.8 | 4 | 0.588 | 1.756 | 0.15 |

Identification | Mode i | w_{i} (mm) | w_{i}/t | η | Ultimate Stress | Ultimate Strain | Residual Stress % |
---|---|---|---|---|---|---|---|

t8m3i2r0 | 3 | 2 | 0.25 | 0 | 0.627 | 1.063 | 0.00 |

t8m3i2r07 | 3 | 2 | 0.25 | 2 | 0.593 | 1.505 | 0.06 |

t8m3i2r15 | 3 | 2 | 0.25 | 4 | 0.582 | 1.654 | 0.12 |

t8m3i5r0 | 3 | 5 | 0.63 | 0 | 0.575 | 1.289 | 0.00 |

t8m3i5r07 | 3 | 5 | 0.63 | 2 | 0.569 | 1.544 | 0.06 |

t8m3i5r15 | 3 | 5 | 0.63 | 4 | 0.563 | 1.668 | 0.12 |

t8m3i6r0 | 3 | 8 | 1.00 | 0 | 0.542 | 1.633 | 0.00 |

t8m3i6r07 | 3 | 8 | 1.00 | 4 | 0.542 | 1.792 | 0.12 |

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**MDPI and ACS Style**

Gordo, J.M.
Effect of Residual Stresses on the Elastoplastic Behavior of Welded Steel Plates. *J. Mar. Sci. Eng.* **2020**, *8*, 702.
https://doi.org/10.3390/jmse8090702

**AMA Style**

Gordo JM.
Effect of Residual Stresses on the Elastoplastic Behavior of Welded Steel Plates. *Journal of Marine Science and Engineering*. 2020; 8(9):702.
https://doi.org/10.3390/jmse8090702

**Chicago/Turabian Style**

Gordo, José Manuel.
2020. "Effect of Residual Stresses on the Elastoplastic Behavior of Welded Steel Plates" *Journal of Marine Science and Engineering* 8, no. 9: 702.
https://doi.org/10.3390/jmse8090702