# A Probabilistic Method for Estimating the Percentage of Corrosion Depth on the Inner Bottom Plates of Aging Bulk Carriers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Collecting Methodology

#### 2.2. A Brief Description of the Input Data Set

#### 2.3. The Proposed Problem and Related Methodology

## 3. Results

#### 3.1. Appropriate Statistical Analysis Related to Measurements of Inner Bottom Plates

- The choice of an adequate model, i.e., the choice of optimal theoretical distributions,
- The determination of the optimal values for the parameters that characterize theoretical distributions,
- The determination of the significance level and the quality of the distributions fitted.

_{E}(x), and ${F}_{LP}\left(x\right)$) denote the probability density functions of the best fitted Generalized Pareto, Error, and Log-Pearson 3 distributions (the cumulative distribution functions) for $\frac{{c}_{1}}{\overline{{d}_{0}}}$, respectively.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Paik, J.F.; Brennan, F.; Carlsen, C.A.; Daley, C.; Garbatov, Y.; Ivanov, L.; Rizzo, C.; Simonsen, B.C.; Yamamoto, N.; Zhuang, H.Z. Report of Committee V.6 Condition Assessment of Aging Ships. In Proceedings of the 16th International Ship and Offshore Structures Congress, Southampton, UK, 20–25 August 2006. [Google Scholar]
- Roberts, S.E.; Marlow, P.B. Casualties in dry bulk shipping (1963–1996). Mar. Policy
**2002**, 26, 437–450. [Google Scholar] [CrossRef] - Bulk carrier casualty report, IMO, MSC 83/INF.6, 3 July 2007. Available online: http://www.safedor.org/resources/MSC_83-INF-8.pdf (accessed on 16 June 2020).
- Paik, J.K.; Kim, S.K.; Lee, S.K. A probabilistic corrosion rate estimation model for longitudinal strength members of bulk carriers. Ocean Eng.
**1998**, 25, 837–860. [Google Scholar] [CrossRef] - Guedes Soares, C.; Garbatov, Y.; Zayed, A. Effect of environmental factors on steel plate corrosion under marine immersion conditions. Corros. Eng. Sci. Technol.
**2011**, 46, 524–541. [Google Scholar] [CrossRef] - Paik, J.K.; Thayamballi, A.K.; Park, Y.I.; Hwang, J.S. A time-dependent corrosion wastage model for seawater ballast tank structures of ships. Corros. Sci.
**2004**, 46, 471–486. [Google Scholar] [CrossRef] - Yamamoto, N.; Ikagaki, K. A Study on the Degradation of Coating and Corrosion on Ship’s Hull Based on the Probabilistic Approach. J. Offshore Mech. Arct. Eng.
**1998**, 120, 121–128. [Google Scholar] [CrossRef] - Guo, J.; Wang, G.; Ivanov, L.; Perakis, A.N. Time-varying ultimate strength of aging tanker deck plate considering corrosion effect. Mar. Struct.
**2008**, 21, 402–419. [Google Scholar] [CrossRef] - Garbatov, Y.; Guedes Soares, C.; Wang, G. Non-linear time dependent corrosion wastage of deck plates of ballast and cargo tanks of tankers. In Proceedings of the 22nd International Conference on Offshore Mechanics and Arctic Engineering, OMAE 2005–67579, Halkidiki, Greece, 12–17 June 2005. [Google Scholar]
- Jurišić, P.; Parunov, J.; Garbatov, Y. Aging effects on Ship Structural integrity. Brodogradnja/Shipbuilding
**2017**, 68, 15–28. [Google Scholar] [CrossRef] [Green Version] - Jurišić, P.; Parunov, J.; Garbatov, Y. Comparative analysis based on two nonlinear corrosion models commonly used for prediction of structural degradation of oil tankers. Trans. FAMENA
**2014**, 38, 21–30. [Google Scholar] - Paik, J.K.; Lee, J.M.; Park, Y.I.; Hwang, J.S.; Kim, C.W. Time–variant ultimate longitudinal strength of corroded bulk carriers. Mar. Struct.
**2003**, 16, 567–600. [Google Scholar] [CrossRef] - Paik, J.K.; Thayamballi, A.K.; Park, Y.I.; Hwang, J.S. A time-dependent corrosion wastage model for bulk carrier structures. Int. J. Marit. Eng.
**2003**, 145 Pt A2, 61–87. [Google Scholar] - Ivošević, Š.; Meštrović, R.; Kovač, N. Probabilistic estimates of corrosion rate of fuel tank structures of aging bulk carriers. Int. J. Naval Arch. Ocean Eng.
**2019**, 11, 165–177. [Google Scholar] [CrossRef] - Ivošević, Š. Analysis of Ships Hull Structural Degradation. Ph.D. Thesis, University of Montenegro, Maritime Faculty Kotor, Kotor, Montenegro, 2012. [Google Scholar]
- Ivošević, Š.; Meštrović, R.; Kovač, N. An approach to the probabilistic corrosion rate estimation model for inner bottom plates of bulk carriers. Brodogradnja/Shipbuilding
**2017**, 68, 57–70. [Google Scholar] [CrossRef] [Green Version] - Ivošević, Š.; Meštrović, R.; Kovač, N. A comparison of some multi-parameter distributions related to estimation of corrosion rate of aging bulk carriers. In Proceedings of the 7th International Conference on Marine Structures, Dubrovnik, Croatia, 6–8 May 2019; CRC Press: Boca Raton, FL, USA, 2019; pp. 403–410. [Google Scholar]
- Qin, S.; Cui, W. Effect of corrosion models on the time-dependent reliability of steel plated elements. Mar. Struct.
**2003**, 16, 15–34. [Google Scholar] [CrossRef] - Paik, J.K.; Thayamballi, A.K. Ultimate strength of aging ships. J. Eng. Marit. Environ.
**2002**, 1, 57–77. [Google Scholar] - Guedes Soares, C.; Garbatov, Y. Reliability of maintained ship hulls subjected to corrosion. J. Ship Res.
**1996**, 40, 235–243. [Google Scholar] - Guedes Soares, C.; Garbatov, Y. Reliability of maintained ship hull girders subjected to corrosion and fatigue. Struct. Saf.
**1998**, 20, 201–219. [Google Scholar] [CrossRef] - Guedes Soares, C.; Garbatov, Y. Reliability of plate elements subjected to compressive loads and accounting for corrosion and repair. In Structural Safety and Reliability; Shiraishi, N., Shinozuka, M., Wen, Y.K., Eds.; Balkema: Rotterdam, The Netherlands, 1998; Volume 3, pp. 2013–2020. [Google Scholar]
- Guedes Soares, C.; Garbatov, Y. Reliability of corrosion protected and maintained ship hulls subjected to corrosion and fatigue. J. Ship Res.
**1998**, 43, 65–78. [Google Scholar] - Guedes Soares, C.; Garbatov, Y. Reliability of maintained ship hulls subjected to corrosion and fatigue under combined loading. J. Constr. Steel Res.
**1999**, 52, 93–115. [Google Scholar] [CrossRef] - Katalinić, M.; Parunov, J. Uncertainties of Estimating Extreme SignificantWave Height for Engineering Applications Depending on the Approach and Fitting Technique—Adriatic Sea Case Study. J. Mar. Sci. Eng.
**2019**, 8, 259. [Google Scholar] [CrossRef] [Green Version] - Hays, W.L. Statistics for the Social Sciences, 2nd ed.; Rinehart & Winston: New York, NY, USA, 1973. [Google Scholar]
- Stephens, M.A. Introduction to Kolmogorov (1033) on the empirical determination of a distribution. In Breakthroughs in Statistics; Springer: New York, NY, USA, 1992. [Google Scholar]
- De Smith, M.J. Statistical Analysis Handbook a Comprehensive Handbook of Statistical Concepts, Techniques and Software Tools; The Winchelsea Press: London, UK, 2018. [Google Scholar]

**Figure 2.**The data of 570 thickness measurements of inner bottom plates obtained through 38 ship surveys.

**Figure 3.**The empirical PDF and the PDF of the three best fitted three-parameter distributions of $\frac{{c}_{1}}{\overline{{d}_{0}}}$ for IBP with ${\mathrm{T}}_{\mathrm{c}\mathrm{l}}=4$ years.

**Figure 4.**The empirical CDF and the CDF of the three best fitted three-parameter distributions of $\frac{{c}_{1}}{\overline{{\mathrm{d}}_{0}}}$ for IBP with ${\mathrm{T}}_{\mathrm{c}\mathrm{l}}=4$ years.

**Figure 5.**The probability difference graph of the three best fitted three-parameter distributions of $\frac{{c}_{1}}{\overline{{d}_{0}}}$ for IBP with ${\mathrm{T}}_{\mathrm{c}\mathrm{l}}=4$ years.

**Figure 6.**Probability-probability (P-P) plots of the three best fitted three-parameter distributions of $\frac{{c}_{1}}{\overline{{d}_{0}}}$ for IBP with ${\mathrm{T}}_{\mathrm{c}\mathrm{l}}=4$ years.

**Figure 7.**Estimates of $\frac{\mathrm{d}\left(\mathrm{t}\right)}{\overline{{\mathrm{d}}_{0}}}$ for IBP as a function of t.

The Age of Ships (Years) | The Number of Ship Surveys | The Number of Tanks | The Number of Measured Points | The Number of Sections | The Average Values of Plate Thickness Reduction Caused by Corrosion (%) |
---|---|---|---|---|---|

0–5 | 4 | 9 | 230 | 45 | 0.5 |

5–10 | 4 | 10 | 266 | 55 | 2.8 |

10–15 | 7 | 19 | 500 | 100 | 9.8 |

15–20 | 13 | 43 | 998 | 220 | 11.7 |

20–25 | 10 | 29 | 816 | 150 | 17.7 |

SUM: | 38 | 110 | 2810 | 570 |

Statistic | Value | Percentile | Value |
---|---|---|---|

Sample Size | 465 | Min | 0.00129 |

Range | 0.02041 | 5% | 0.00197 |

Mean | 0.00788 | 10% | 0.00284 |

Variance | 1.5617 × 10^{-5} | 25% (Q1) | 0.00448 |

Standard Deviation | 0.00395 | 50% (Median) | 0.00746 |

Coefficient of Variation | 0.5017 | 75% (Q3) | 0.01111 |

Standard Error | 1.8326 × 10^{-4} | 90% | 0.01336 |

Skewness | 0.22245 | 95% | 0.01424 |

Excess Kurtosis | −0.83477 | Max | 0.0217 |

**Table 3.**Fitted three-parameter distributions of $\frac{{\mathrm{c}}_{1}}{\overline{{\mathrm{d}}_{0}}}$ for IBP where ${\mathrm{T}}_{\mathrm{cl}}=4$ years.

Distribution | Parameters |
---|---|

Burr | $k=4.2273E+13\alpha =0.93196\beta =3.4010x{10}^{12}$ |

Dagum | $k=0.08817\alpha =15.238\beta =0.01399$ |

Erlang | $m=7\beta =0.0016\gamma =-0.00262$ |

Error | $k=4.1549\sigma =0.00395\mu =0.00788$ |

Fatigue Life | $\alpha =0.21019\beta =0.01867\gamma =-0.0112$ |

Frechet | $\alpha =1.0402E+8\beta =3.4816x{10}^{5}\gamma =-3.4816x{10}^{5}$ |

Gamma | $\alpha =6.5552\beta =0.0016\gamma =-0.00262$ |

Gen. Extreme Value | $k=-0.1979\sigma =0.00381\mu =0.00631$ |

Gen. Gamma | $k=0.94793\alpha =3.6606\beta =0.00198$ |

Gen. Logistic | $k=0.04888\sigma =0.00226\mu =0.00769$ |

Gen. Pareto | $k=-0.81361\sigma =0.01158\mu =0.00149$ |

Inv. Gaussian | $\lambda =0.45761\mu =0.0195\gamma =-0.01163$ |

Log-Logistic | $\alpha =7.6305\beta =0.01799\gamma =-0.01047$ |

Log-Pearson3 | $\alpha =6.1211\beta =-0.2504\gamma =-3.4722$ |

Lognormal | $\sigma =0.18192\mu =-3.8403\gamma =-0.01397$ |

Pearson 5 | $\alpha =57.607\beta =1.6874\gamma =-0.02194$ |

Pearson 6 | $\alpha 1=3.4686\alpha 2=6330.4\beta =14.369$ |

Pert | $m=0.00571a=7.0282x{10}^{-4}b=0.0232$ |

Power Function | $\alpha =0.57172a=0.00129b=0.0217$ |

Triangular | $m=0.00129a=0.00128b=0.02175$ |

Weibull | $\alpha =1.9729\beta =0.00845\gamma =3.8091x{10}^{-4}$ |

**Table 4.**The three best fitted three-parameter distributions of $\frac{{c}_{1}}{\overline{{d}_{0}}}$ for IBP with ${\mathrm{T}}_{\mathrm{cl}}=4$ years.

Distribution | Parameters |
---|---|

Gen. Pareto | k = −0.8 σ = 0.01 μ = 0.00149 |

Error | k = 7.0 σ = 0.08 μ = 0.16 |

Log-Pearson 3 | α = 5.7 β = −0.26 γ = −0.54 |

**Table 5.**The empirical values and the values of PDF for the three best fitted three-parameter distributions for $\frac{{c}_{1}}{\overline{{d}_{0}}}$ with respect to IBP.

$\mathbf{IBP}\text{}\mathbf{with}\text{}{\mathit{T}}_{\mathit{c}\mathit{l}}\mathbf{=}\mathbf{4}\mathbf{Years}$ | |||||
---|---|---|---|---|---|

Lower Bound | Upper Bound | Empirical PDF of $\frac{{\mathit{c}}_{\mathbf{1}}}{\overline{{\mathit{d}}_{\mathbf{0}}}}$ | f_{GP}(x) | f_{E}(x) | f_{LP}(x) |

0 | 0.00227 | 0.073 | 0.06681 | 0.08237 | 0.05704 |

0.00227 | 0.00454 | 0.187 | 0.18958 | 0.15123 | 0.18019 |

0.00454 | 0.00681 | 0.191 | 0.18092 | 0.18039 | 0.21897 |

0.00681 | 0.009079 | 0.155 | 0.17050 | 0.18290 | 0.19497 |

0.009079 | 0.011349 | 0.157 | 0.15752 | 0.17981 | 0.14648 |

0.011349 | 0.013619 | 0.161 | 0.13914 | 0.14778 | 0.09680 |

0.013619 | 0.015889 | 0.067 | 0.09552 | 0.06668 | 0.05684 |

0.015889 | 0.018159 | 0.006 | 0.00001 | 0.00870 | 0.02941 |

0.018159 | 0.020429 | 0.000 | 0 | 0.00014 | 0.01306 |

0.020429 | 0.022699 | 0.002 | 0 | 0 | 0.00474 |

**Table 6.**The empirical values and the values of cumulative density functions (CDF) for the three best fitted distributions for $\frac{{c}_{1}}{\overline{{d}_{0}}}$ with respect to IBP.

$\mathbf{IBP}\text{}{\mathit{T}}_{\mathit{c}\mathit{l}}\mathbf{=}\mathbf{4}\mathbf{Years}$ | |||||
---|---|---|---|---|---|

Lower Bound | Upper Bound | Empirical CDF of $\frac{{\mathit{c}}_{\mathbf{1}}}{\overline{{\mathit{d}}_{\mathbf{0}}}}$ | F_{GP}(x) | F_{E}(x) | F_{LP}(x) |

0 | 0.00227 | 0.073 | 0.06681 | 0.08237 | 0.05704 |

0.00227 | 0.00454 | 0.260 | 0.25639 | 0.23360 | 0.23723 |

0.00454 | 0.00681 | 0.451 | 0.43731 | 0.41399 | 0.45620 |

0.00681 | 0.009079 | 0.606 | 0.60781 | 0.59689 | 0.65117 |

0.009079 | 0.011349 | 0.763 | 0.76533 | 0.77670 | 0.79765 |

0.011349 | 0.013619 | 0.924 | 0.90447 | 0.92448 | 0.89445 |

0.013619 | 0.015889 | 0.991 | 0.99999 | 0.99116 | 0.95129 |

0.015889 | 0.018159 | 0.997 | 1 | 0.99986 | 0.98070 |

0.018159 | 0.020429 | 0.999 | 1 | 1 | 0.99376 |

0.020429 | 0.022699 | 0.999 | 1 | 1 | 0.99850 |

Distribution | Statistic | p-Value |
---|---|---|

Gen. Pareto | 0.03075 | 0.75958 |

Error | 0.05153 | 0.16339 |

Log-Pearson 3 | 0.05805 | 0.08375 |

Years | $\frac{\mathit{d}\mathbf{\left(}\mathit{t}\mathbf{\right)}}{\overline{{\mathbf{d}}_{\mathbf{0}}}}$ |
---|---|

5 | 0.0216986 |

10 | 0.0399617 |

15 | 0.104289 |

20 | 0.116896 |

25 | 0.166935 |

**Table 9.**Estimated values of $p\left(t\right)=\frac{d\left(t\right)}{\overline{{d}_{0}}}$ for IBP with ${T}_{cl}>4$ years (expressed as a percentage) in accordance with Equation (16).

T_{cl} | p(t) | T_{cl} | p(t) | T_{cl} | p(t) | T_{cl} | p(t) |
---|---|---|---|---|---|---|---|

5 | 0.704556 | 10 | 4.22734 | 15 | 7.75012 | 20 | 11.2729 |

6 | 1.40911 | 11 | 4.93189 | 16 | 8.45467 | 21 | 11.9775 |

7 | 2.11367 | 12 | 5.63645 | 17 | 9.15923 | 22 | 12.682 |

8 | 2.81822 | 13 | 6.341 | 18 | 9.86378 | 23 | 13.3866 |

9 | 3.52278 | 14 | 7.04556 | 19 | 10.5683 | 24 | 14.0911 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ivošević, Š.; Meštrović, R.; Kovač, N.
A Probabilistic Method for Estimating the Percentage of Corrosion Depth on the Inner Bottom Plates of Aging Bulk Carriers. *J. Mar. Sci. Eng.* **2020**, *8*, 442.
https://doi.org/10.3390/jmse8060442

**AMA Style**

Ivošević Š, Meštrović R, Kovač N.
A Probabilistic Method for Estimating the Percentage of Corrosion Depth on the Inner Bottom Plates of Aging Bulk Carriers. *Journal of Marine Science and Engineering*. 2020; 8(6):442.
https://doi.org/10.3390/jmse8060442

**Chicago/Turabian Style**

Ivošević, Špiro, Romeo Meštrović, and Nataša Kovač.
2020. "A Probabilistic Method for Estimating the Percentage of Corrosion Depth on the Inner Bottom Plates of Aging Bulk Carriers" *Journal of Marine Science and Engineering* 8, no. 6: 442.
https://doi.org/10.3390/jmse8060442