# A Strain Fitting Strategy to Eliminate the Impact of Measuring Points Failure in Longitudinal Bending Moment Identification

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## Abstract

**:**

## 1. Introduction

- (1)
- A four-point bending experiment based on a box girder was conducted to realize the loading identification based on the measured signals.
- (2)
- The correlation between the measurement point position and identification accuracy is explored in the experiment scenario to investigate the impact of measuring point failure on longitudinal bending moment identification.
- (3)
- Based on the investigation of correlation, 17 failure conditions were designed to assess the effectiveness of the fitting method based on XGboost, and the method’s fitting capabilities were thoroughly evaluated.
- (4)
- To clarify the sources of fitting errors and further explore the applicability of fitting method, the connections between the fitting values and the training set were comprehensively studied under typical load cases using finite element analysis. Then, the minimum training set required for fitting was determined for the simplification of the training set.
- (5)
- Recommendations were provided for the implementation of the strain fitting method for longitudinal bending moment identification.

## 2. Experiment on Longitudinal Bending Moment Identification

#### 2.1. Experimental Details

#### 2.1.1. Design of the Box Girder

#### 2.1.2. Experiment Setup

#### 2.2. Longitudinal Bending Moment Identification Method

#### 2.3. Gradient Descent Method for Load Identification

## 3. XGboost Fitting of the Strains at Measuring Points

^{4}or the number of iterations exceeds 1000, the iteration stops. And the maximum depth of the tree is 10.

## 4. Investigations into Strain Fitting Method Based on the XGboost Method

#### 4.1. Correlation between Measuring Point Position and Identification Accuracy

#### 4.1.1. Correlation between Measuring Points at Different Locations and Identification Accuracy

#### 4.1.2. Correlation between Measuring Points at Different Structures and Identification Accuracy

#### 4.2. Strain Fitting Based on Experiment Results

- (a)
- Remove failed points.
- (b)

#### 4.2.1. Strain Fitting of Failed Points on Different Positions

#### Strain Fitting at Different Locations

#### Strain Fitting at Different Structures

#### 4.2.2. Strain Fitting on Different Sections

- (a)
- It makes the identification results more stable.
- (b)
- It shows a better improvement capability in cases of high-correlation measurement point failure and small-scale measurement point layouts.
- (c)
- Using strain fitting can estimate the approximate strain of failed points, which is more conducive to structural safety assessment.

#### 4.3. Strain Fitting Based on Numerical Analysis

#### 4.3.1. Finite Element Model

#### 4.3.2. Convergence Study on the Mesh Size of Finite Elements

^{5}N is applied to the box girder, the $MSE$ increases with the increase of the mesh size. When the mesh size is less than 40 mm, the $MSE$ grows slowly. Therefore, the mesh size of 20 mm is adopted for numerical analysis. The number of elements for the model is 11,511.

#### 4.3.3. Analysis of the Sources of Fitting Errors

#### 4.3.4. Impact of Lateral Bending

#### 4.3.5. Impact of Torsion-Lateral Bending

#### 4.4. Investigation of Fitting Schemes

## 5. Discussion

#### 5.1. Strain Fitting Model Based on XGboost in the Case of Failed Points

#### 5.2. Application Prospects of the XGboost Method for the Treatment of Failed Points

## 6. Conclusions

- (1)
- The failure of highly correlated measuring points has a significant impact on the accuracy of identifying the total longitudinal bending moment. This result indicates that the points on the deck stiffener have the highest correlation, followed by the deck plate, the side plate, and the side stiffener.
- (2)
- Based on the experimental results, the XGboost fitting method can effectively improve the identification accuracy of the longitudinal bending moment in the case of the failure of measuring points. Compared to common methods of removing failed points, the XGboost method makes the identification results more stable and shows a better improvement capability in the case of high-correlation measurement point failure. In this study, after using the strain fitting method, the relative error decreased to less than 6%, and that was less than 5% in most conditions. By contrast, the relative error of some conditions still reaches 10% by removing failed points.
- (3)
- There is a fitting error between the fitted strain and the measured strain due to the measurement error. However, the trend of the fitted values is almost consistent with the actual values. Therefore, this method is able to estimate strain to an approximate value.
- (4)
- Based on the FEM analysis, the XGboost fitting method is effective for complex load conditions. After further numerical investigation, it is suggested as the optimal fitting scheme which adopts the failed section and its adjacent section as the training set.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Katsoudas, A.S.; Silionis, N.E.; Anyfantis, K.N. Structural Health Monitoring for Corrosion Induced Thickness Loss in Marine Plates Subjected to Random Loads. Ocean Eng.
**2023**, 273, 114037. [Google Scholar] [CrossRef] - Rahgozar, R.; Bitaraf, M. A Summary Evaluation of Output-Only Damage-Sensitive Features for Structural Health Monitoring of Offshore Platforms Subjected to Ambient Loads. Ocean Eng.
**2022**, 266, 112892. [Google Scholar] [CrossRef] - Gordo, J.; Guedes Soares, C.; Faulkner, D. Approximate Assessment of the Ultimate Longitudinal Strength of the Hull Girder. J. Ship Res.
**1996**, 40, 60–69. [Google Scholar] [CrossRef] - Tanaka, Y.; Ando, T.; Anai, Y.; Yao, T.; Fujikubo, M.; Iijima, K. Longitudinal Strength of Container Ships Under Combined Torsional and Bending Moments. In Proceedings of the Nineteenth International Offshore and Polar Engineering Conference, Osaka, Japan, 21–26 July 2009; OnePetro: Richardson, TX, USA, 2009. [Google Scholar]
- American Bureau of Shipping. Guide for Hull Condition Monitoring System 2020. Available online: https://ww2.eagle.org/content/dam/eagle/rules-and-guides/current/conventional_ocean_service/73_Hull_Condition_Monitoring_2016/hull-condition-monitoring-guide-july20.pdf (accessed on 23 February 2023).
- China Classification Society. Rules for Intelligent Ship. Available online: https://www.ccs.org.cn/ccswz/specialDetail?id=201900001000009739 (accessed on 23 February 2023).
- Nippon Kaiji Kyokai. Rules for Hull Monitoring Systems. Available online: https://www.classnk.or.jp/account/zh/Rules_Guidance/ssl/tech_rules.aspx (accessed on 24 February 2023).
- Li, M.; Boulougouris, E.; Lazakis, I.; Theotokatos, G. Wave-Induced Vertical Bending Moment Estimation by Onboard Tiltmeters Units on Container Ship. In Proceedings of the International Conference on Maritime Safety and Operations, Glasgow, UK, 13–14 October 2016; Available online: https://strathprints.strath.ac.uk/58370/1/Li_etal_MSO2016_Wave_induced_vertical_bending_moment_estimation_by_onboard_tiltmeters.pdf (accessed on 24 February 2023).
- Tatsumi, A.; Iijima, K.; Fujikubo, M. Estimation of Still-Water Bending Moment of Ship Hull Girder Using Beam Finite Element Model and Ensemble Kalman Filter. In Proceedings of the ASME 2022 41st International Conference on Ocean, Offshore and Arctic Engineering, Hamburg, Germany, 5–10 June 2022; American Society of Mechanical Engineers Digital Collection: Washington, DC, USA, 2022. [Google Scholar]
- Cusano, G.; La Marca, L.S. Evaluation and Forecasting of Elapsed Fatigue Life of Ship Structures by Analyzing Data from Full Scale Ship Structural Monitoring. J. Shipp. Ocean Eng.
**2015**, 5, 59–74. [Google Scholar] [CrossRef] - Moreira, L.; Soares, C.G. Neural Network Model for Estimation of Hull Bending Moment and Shear Force of Ships in Waves. Ocean Eng.
**2020**, 206, 107347. [Google Scholar] [CrossRef] - Yu, H.; Ha, M.; Choi, J.; Tai, J.S. Design and Implementation of a Comprehensive Full-Scale Measurement System for a Large Container Carrier. In Proceedings of the Design & Operation of Container Ships conference, London, UK, 22–23 November 2006; RINA: Genoa, Italy, 2006; pp. 51–60. [Google Scholar]
- Liu, D. Fatigue Assessment Methodfor Hull Structure Based on Strength Monitoring System. Master’s Thesis, Harbin Engineering University, Harbin, China, 2018. [Google Scholar]
- Li, J. Research on Load Inversion Method and Model Experiment of Ultra Large Ships. Master’s Thesis, Harbin Engineering University, Harbin, China, 2021. [Google Scholar]
- Li, J.; Dong, W.; Liu, N.; Feng, G.; Ren, H. Theoretical and Experimental Study on Load Inversion of Ship Hull Girder. Shipengineering
**2022**, 44, 41–46. [Google Scholar] [CrossRef] - Liu, Y.; Xing, Y.; Shen, Z. Identification of Cross-section Loads Based on Measured Strain of Missile Body. Acta Amamentaii
**2016**, 37, 332–337. [Google Scholar] - Wang, J.; Chen, X.; Duan, Q.; Ji, S. Eliminating the Influence of Measuring Point Failure in Ice Load Identification of Polar Ship Structures. Ocean Eng.
**2022**, 261, 112082. [Google Scholar] [CrossRef] - Shrive, F.M.; Stuart, H.; Quan, H.; Ghali, W.A. Dealing with Missing Data in a Multi-Question Depression Scale: A Comparison of Imputation Methods. BMC Med. Res. Methodol.
**2006**, 6, 57. [Google Scholar] [CrossRef] [PubMed] - Troyanskaya, O.; Cantor, M.; Sherlock, G.; Brown, P.; Hastie, T.; Tibshirani, R.; Botstein, D.; Altman, R.B. Missing Value Estimation Methods for DNA Microarrays. Bioinformatics
**2001**, 17, 520–525. [Google Scholar] [CrossRef] [PubMed] - Stekhoven, D.J.; Buhlmann, P. MissForest-Non-Parametric Missing Value Imputation for Mixed-Type Data. Bioinformatics
**2012**, 28, 112–118. [Google Scholar] [CrossRef] - Chen, T.; Guestrin, C. XGBoost: A Scalable Tree Boosting System. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016; pp. 785–794. [Google Scholar]
- Kankanamge, K.D.; Witharanage, Y.R.; Withanage, C.S.; Hansini, M.; Lakmal, D.; Thayasivam, U. Taxi Trip Travel Time Prediction with Isolated XGBoost Regression. In Proceedings of the 2019 Moratuwa Engineering Research Conference (MERCon), Moratuwa, Sri Lanka, 3–5 July 2019; pp. 54–59. [Google Scholar]
- Li, W.; Yin, Y.; Quan, X.; Zhang, H. Gene Expression Value Prediction Based on XGBoost Algorithm. Front. Genet.
**2019**, 10, 1077. [Google Scholar] [CrossRef] [PubMed] - Saad-Eldeen, S.; Garbatov, Y.; Soares, C.G. Experimental Assessment of the Ultimate Strength of a Box Girder Subjected to Severe Corrosion. Mar. Struct.
**2011**, 24, 338–357. [Google Scholar] [CrossRef] - Xu, W.; Iijima, K.; Wada, R.; Fujikubo, M. Experimental Evaluation of the Post-Ultimate Strength Behavior of a Ship’s Hull Girder in Waves. J. Mar. Sci. Appl.
**2012**, 11, 34–43. [Google Scholar] [CrossRef] - Saad-Eldeen, S.; Garbatov, Y.; Guedes Soares, C. Effect of Corrosion Severity on the Ultimate Strength of a Steel Box Girder. Eng. Struct.
**2013**, 49, 560–571. [Google Scholar] [CrossRef] - Akhras, G.; Gibson, S.; Yang, S.; Morchat, R. Ultimate Strength of a Box Girder Simulating the Hull of a Ship. Can. J. Civ. Eng.
**1998**, 25, 829–843. [Google Scholar] [CrossRef] - Hughes, O. Ship Structural Design: A Rationally-Based, Computer-Aided Optimization Approach; Wiley-Interscience: New York, NY, USA, 1983. [Google Scholar]
- Hauser, R. Line Search Methods for Unconstrained Optimisation, Lecture 8; Oxford University Computing Laboratory: Oxford, UK, 2007. [Google Scholar]
- Kong, S.; Cui, H.; Tian, Y.; Ji, S. Identification of Ice Loads on Shell Structure of Ice-Going Vessel with Green Kernel and Regularization Method. Mar. Struct.
**2020**, 74, 102820. [Google Scholar] [CrossRef] - Ma, H.; Xiong, Q.; Wang, D. Experimental and Numerical Study on the Ultimate Strength of Stiffened Plates Subjected to Combined Biaxial Compression and Lateral Loads. Ocean Eng.
**2021**, 228, 108928. [Google Scholar] [CrossRef] - Santhanam, R.; Uzir, N.; Raman, S.; Banerjee, S. Experimenting XGBoost Algorithm for Prediction and Classification of Different Datasets. In Proceedings of the National Conference on Recent Innovations in Software Engineering and Computer Technologies (NCRISECT), Pittsburgh, PA, USA, 23–24 March 2017. [Google Scholar]
- Zhang, X.; Yan, C.; Gao, C.; Malin, B.A.; Chen, Y. Predicting Missing Values in Medical Data Via XGBoost Regression. J. Healthc. Inform. Res.
**2020**, 4, 383–394. [Google Scholar] [CrossRef] - Choi, J.; Kang, Y. Two-Plane Hull Girder Stress Monitoring System for Container Ship. J. Ship Ocean Technol.
**2004**, 8, 17–25. [Google Scholar] - Park, J.; Kang, Y.-T. Strain Decomposition Method in Hull Stress Monitoring System for Container Ship. J. Ship Ocean Technol.
**2003**, 7, 56–65. [Google Scholar] - Yu, H.; Choi, J.; Park, G.; Han, S.; Tai, S.; Ha, M. Full Scale Measurement of a Large Container Carrier on the Far East—Europe Route. In Proceedings of the SNAME Maritime Convention, Houston, TX, USA, 15–17 October 2008; OnePetro: Richardson, TX, USA, 2008. [Google Scholar]
- Suominen, M.; Kujala, P.; Romanoff, J.; Remes, H. Influence of Load Length on Short-Term Ice Load Statistics in Full-Scale. Mar. Struct.
**2017**, 52, 153–172. [Google Scholar] [CrossRef] - Lee, T.; Lee, J.; Kim, H.; Rim, C. Field Measurement of Local Ice Pressures on the ARAON in the Beaufort Sea. Int. J. Nav. Archit. Ocean Eng.
**2014**, 6, 788–799. [Google Scholar] [CrossRef] - Kefal, A.; Oterkus, E. Displacement and Stress Monitoring of a Chemical Tanker Based on Inverse Finite Element Method. Ocean Eng.
**2016**, 112, 33–46. [Google Scholar] [CrossRef] - Kefal, A.; Oterkus, E. Displacement and Stress Monitoring of a Panamax Containership Using Inverse Finite Element Method. Ocean Eng.
**2016**, 119, 16–29. [Google Scholar] [CrossRef]

**Figure 4.**Layout of strain gauges on the box girder. (

**a**) Layout of strain gauges on the deck. (

**b**) Layout of strain gauges on the left side. (

**c**) Layout of strain gauges on the right side.

**Figure 5.**Bending moment identification of different sections. (

**a**) Distribution of identified sections. (

**b**) 5t. (

**c**) 10t. (

**d**) 15t. (

**e**) 20t. (

**f**) 25t. (

**g**) 30t.

**Figure 8.**Failure conditions at different locations. (

**a**) One failed point (deck). (

**b**) One failed point (side). (

**c**) Two failed points (side). (

**d**) Two failed points (deck). (

**e**) Three failed points (deck: one, side: two). (

**f**) Three failed points (deck: two, side: one). (

**g**) Four failed points (deck: two, side: two).

**Figure 9.**Bending moment identification of point failures at different locations. (

**a**) 5t. (

**b**) 10t. (

**c**) 15t. (

**d**) 20t. (

**e**) 25t. (

**f**) 30t.

**Figure 10.**Failure conditions at different structures. (

**a**) Two failed points on the deck (plate). (

**b**) Two failed points on the deck (plate, stiffener). (

**c**) Two failed points on the deck (stiffener). (

**d**) Two failed points on the side (plate). (

**e**) Two failed points on the side (plate, stiffener). (

**f**) Two failed points on the side (stiffener).

**Figure 11.**Bending moment identification of point failure at different structures. (

**a**) 5t. (

**b**) 10t. (

**c**) 15t. (

**d**) 20t. (

**e**) 25t. (

**f**) 30t.

**Figure 12.**Bending moment identification of strain fitting at different locations. (

**a**) Condition a. (

**b**) Condition b. (

**c**) Condition c. (

**d**) Condition d. (

**e**) Condition e. (

**f**) Condition f. (

**g**) Condition g.

**Figure 13.**Strain fitting at different locations. (

**a**) Condition a. (

**b**) Condition b. (

**c**) Condition c. (

**d**) Condition d. (

**e**) Condition e. (

**f**) Condition f. (

**g**) Condition g.

**Figure 14.**Bending moment identification of strain fitting at different structures. (

**a**) Condition a. (

**b**) Condition b. (

**c**) Condition c. (

**d**) Condition d. (

**e**) Condition e. (

**f**) Condition f.

**Figure 15.**Strain fitting at different structures. (

**a**) Condition a. (

**b**) Condition b. (

**c**) Condition c. (

**d**) Condition d. (

**e**) Condition e. (

**f**) Condition f.

**Figure 16.**Conditions of lack of points. (

**a**) Two points deleted on the deck stiffener. (

**b**) Two points deleted on the side plate. (

**c**) Four points deleted (deck: two; side: two). (

**d**) Four points deleted (deck: two; side: two).

**Figure 17.**Longitudinal bending moment of strain fitting under different layouts of points. (

**a**) Condition a. (

**b**) Condition b. (

**c**) Condition c. (

**d**) Condition d.

**Figure 22.**Longitudinal strain under different conditions. (

**a**) Lateral bending. (

**b**) Torsion-lateral bending.

Component (mm) | Dimension (Extension Part) | Dimension (Middle Part) | |
---|---|---|---|

Entire model | Length | 1250 | 1450 |

Breadth | 600 | 600 | |

Depth | 450 | 450 | |

Deck | Plate thickness | 6 | 3 |

Longitudinal stiffener | FB50 × 6 | FB30 × 3 | |

Transverse stiffener | FB50 × 6 | FB80 × 3 | |

Bottom | Plate thickness | 6 | 4 |

Longitudinal stiffener | FB50 × 6 | FB40 × 4 | |

Transverse stiffener | FB80 × 3 | FB80 × 3 | |

Side | Plate thickness | 6 | 3 |

Longitudinal stiffener | FB50 × 6 | FB40 × 3 | |

Transverse stiffener | FB50 × 6 | FB80 × 3 | |

Bulkhead | Both ends | 6 | 6 |

Middle | 6 | 6 |

Condition No. | Longitudinal Bending Moment (N × mm) |
---|---|

1 | 1.000 × 10^{8} |

2 | 5.000 × 10^{7} |

3 | 2.500 × 10^{7} |

Condition No. | Identified Moment (N × mm) | True Moment (N × mm) |
---|---|---|

1 | 9.994 × 10^{8} | 1.000 × 10^{8} |

2 | 4.997 × 10^{7} | 5.000 × 10^{7} |

3 | 2.499 × 10^{7} | 2.500 × 10^{7} |

Condition No. | Lateral Bending Moment (N × mm) | Longitudinal Bending Moment (N × mm) |
---|---|---|

1 | 2.500 × 10^{7} | 1.000 × 10^{8} |

2 | 5.000 × 10^{7} | 5.000 × 10^{7} |

3 | 1.000 × 10^{8} | 2.500 × 10^{7} |

Condition No. | Identified Moment (N × mm) | True Moment (N × mm) |
---|---|---|

1 | 1.000 × 10^{8} | 1.000 × 10^{8} |

2 | 5.000 × 10^{7} | 5.000 × 10^{7} |

3 | 2.498 × 10^{7} | 2.500 × 10^{7} |

Condition No. | Lateral Bending Moment (N × mm) | Longitudinal Bending Moment (N × mm) | Torque (N × mm) |
---|---|---|---|

1 | 2.5 × 10^{7} | 1 × 10^{8} | 5 × 10^{7} |

2 | 5 × 10^{7} | 5 × 10^{7} | 5 × 10^{7} |

3 | 1 × 10^{8} | 2.5 × 10^{7} | 5 × 10^{7} |

Condition No. | Identified Moment (N × mm) | True Moment (N × mm) |
---|---|---|

1 | 1.000 × 10^{8} | 1.000 × 10^{8} |

2 | 5.000 × 10^{7} | 5.000 × 10^{7} |

3 | 2.499 × 10^{7} | 2.500 × 10^{7} |

Scheme No. | Training Set |
---|---|

1 | Failed Section |

2 | Failed Section + Adjacent Section (Two Sections) |

3 | Three Sections |

4 | Four Sections |

5 | All Sections (Whole model) |

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## Share and Cite

**MDPI and ACS Style**

Xu, G.; Gan, J.; Li, J.; Liu, H.; Wu, W.
A Strain Fitting Strategy to Eliminate the Impact of Measuring Points Failure in Longitudinal Bending Moment Identification. *J. Mar. Sci. Eng.* **2023**, *11*, 2282.
https://doi.org/10.3390/jmse11122282

**AMA Style**

Xu G, Gan J, Li J, Liu H, Wu W.
A Strain Fitting Strategy to Eliminate the Impact of Measuring Points Failure in Longitudinal Bending Moment Identification. *Journal of Marine Science and Engineering*. 2023; 11(12):2282.
https://doi.org/10.3390/jmse11122282

**Chicago/Turabian Style**

Xu, Gengdu, Jin Gan, Jun Li, Huabing Liu, and Weiguo Wu.
2023. "A Strain Fitting Strategy to Eliminate the Impact of Measuring Points Failure in Longitudinal Bending Moment Identification" *Journal of Marine Science and Engineering* 11, no. 12: 2282.
https://doi.org/10.3390/jmse11122282