1 Introduction

The aging ship structures always comprise several damages such as corrosion, fatigue cracks, and local dents. Among these factors, corrosion is the most important and serious (Kim et al. 2012a). The corrosion wastage might appear in every zone of ship structures, but several locations in the hold are more susceptible to corrosion than others such as the inner bottom, lower and upper sloping plates, hold frame, side shell plate, deck plate, shear strake, and deck longitudinal (Paik et al. 1998). The corrosion appearance in the marine structures is caused by a large number of factors such as protection systems, type of cargo, temperature, humidity, oxygen content, pH level, and chemical composition (Paik et al. 2003c). Due to the complex working environment of marine structures, many modes of corrosion can appear in the metal’s surface such as general corrosion, pitting corrosion, grooving corrosion, crevice corrosion, galvanic corrosion, erosion-corrosion, cavitation, stress corrosion, and corrosion fatigue (Guedes Soares and Garbatov 1997).

Facts indicate that the pitting corrosion and general corrosion are the most popular in the marine structures. To the pitting corrosion, several parameters such as pit depth, pit diameter, shape of pit, location of pit, and degree of pit (DOP) are employed to assess the effect of pitting corrosion on the structure behaviors (Paik et al. 2003b; Guedes Soares et al. 2009). Meanwhile, the annual corrosion rate is often applied to evaluate the effect of general corrosion on the structure responses. Concerning this, the existing research works investigated and published the annual corrosion rate for different types of ships. Specifically, Paik et al. (1998) surveyed 7503 corrosion data of 44 existing bulk carriers and suggested the annual corrosion rate for 34 different location groups of bulk carriers. Paik et al. (2003a, d) measured corrosion data for 230 aging single-hull tankers with 33 820 measurements for 34 different member categories including 14 groups of plate parts, 11 groups of stiffener webs, and 9 groups of stiffener flanges. This study also gave the corrosion rate for 34 different member groups of double-hull tankers or floating production storage and offloading (FPSOs). The annual corrosion rate of oil tankers and FPSOs was also presented in research works of Guedes Soares and Garbatov (1999), Akpan et al. (2002), and Kim et al. (2014b). Guedes Soares and his partners did an investigation of the effect of environmental factors on corrosion of ship structures in the marine atmosphere. A new corrosion model was proposed in predicting corrosion degradation in short-term and long-term ship lifetime (Guedes Soares et al. 2009). The real parameters of corrosion are used to evaluate the existing corrosion wastage models (Garbatov et al. 2007; Garbatov and Guedes Soares 2008). The corrosion addition was suggested to apply to container ships as the common corrosion addition rule to determine the ultimate longitudinal strength (Kim et al. 2012a). According to the IACS-CSR guidelines, the corrosion addition is applied to both bulk carriers and oil tankers for the design problem or strength assessment (IACS 2017). Zayed et al. (2018) developed a new corrosion model to predict corrosion degradation of ship plates on both sides in different ship spaces. This model was built based on the assessment of the local environmental conditions including chemical and physical characteristics. A Suezmax oil tanker, an Aframax oil tanker, and double-hull tankers are selected in studies of Guedes Soares and Parunov (2008), Parunov and Guedes Soares (2008), and Gaspar et al. (2011), respectively. These studies mentioned the new common structural rules given by IACS which affect the hull girder ultimate strength of tankers.

The effect of corrosion wastage on the ultimate strength of hull structural plates was covered in several studies. To be more specific, several parameters of pitting corrosion including the location of pit, pit depth, and one-sided or two-sided corrosion were employed to assess the ultimate strength of plate under uniaxial compression (Ok et al. 2007; Jiang and Guedes Soares 2013). The biaxial load and other factors such as volume loss, plate slenderness, aspect ratio, distribution, and pit depth were applied to evaluate the residual strength of the hull structural plate (Huang et al. 2010). Meanwhile, Paik (2005) assessed the ultimate strength of the dented steel plate under the edge shear loads. This study considered the parameters namely pit shape, pith depth, pit diameter, location of pit, different plate thickness, and aspect ratio, and the nonlinear finite element method was used to analyze the dented plates. Another type of load is the tensile force which was employed to analyze the strength capacity of corroded wide steel plates. The effective thickness, initial thickness, and standard deviation were also applied (Kaita et al. 2011). According to the study by Zhang et al. (2016), the combination of loadings was applied. With regard to this, biaxial load combined with shear load was employed, and other factors related to corrosion wastage were also used to assess the variations of the ultimate strength of hull structural plates.

Regarding the stiffened plates, Dunbar et al. (2004) conducted an assessment of the initial buckling, ultimate collapse, and post-ultimate responses of a corroded stiffened panel. This study used local corrosion and initial imperfections (initial distortions and residual stresses) to assess the strength of a stiffened panel. Wang et al. (2015) applied grooving corrosion and other factors such as initial imperfections (initial deflections and residual stresses) and corrosion parameters (grooving depth and volume loss) to assess the ultimate strength of a stiffened panel. The grooving corrosion is assumed to appear in the attached plating and the bottom of the stiffener web. The corroded stiffened plates with irregular surfaces with different corrosion patterns, amount of corrosion loss, and roughness of surface were employed to assess the elastic buckling strength (Rahbar-Ranji 2015) while pitting corrosion was used to evaluate the ultimate strength of stiffened plate (Jing et al. 2017; Rahbar-Ranji et al. 2015).

The corrosion wastage was employed to assess the ship hull girder responses by several authors groups. Paik et al. (2001) developed a closed-form ultimate strength formulation to determine the UBM and neutral axis position at the limit state. This formulation was applied to evaluate the effect of some factors such as initial imperfections (initial distortions and residual stresses) and potential structural damages (corrosion, collision, and grounding) on the ultimate vertical bending moment of a bulk carrier. Meanwhile, Paik et al. (2003d) applied two annual corrosion rate models namely average and severe corrosion rate to analyze the influence of them on the ultimate longitudinal bending moment and section modulus of a bulk carrier. The ultimate longitudinal strength and section modulus of a single-hull VLCC under the thickness degradation due to corrosion wastage were also analyzed. The 15% diminution of section modulus and the linear relationship of the ultimate longitudinal strength were found (Ikeda et al. 2001). The residual strength of a container ship with corrosion and fatigue cracks was investigated. The results showed that uniform corrosion has the biggest reduction in hogging ultimate strength, while the sagging ultimate strength has the smallest diminution (Gao et al. 2012). Kim et al. performed a series of assessments of the effect of corrosion wastage on the ultimate strength responses of four types of oil tankers namely VLCC, Suezmax, Aframax, and Panamax (Kim et al. 2012b, 2014a, b; Guedes Soares and Garbatov 1999). The average general corrosion thickness degradation model and real corrosion model are expropriated to assess the influence of corrosion on the ultimate bending moment of two box girders (Saad-Eldeen et al. 2013). The ultimate strength reduction and cross-section properties of a double-hull VLCC were assessed. Two corrosion rate models with three assumptions of coating lifetime were employed. The study showed that all the reductions follow a linear relationship except a neutral axis position (Vu Van and Yang 2017). Zayed et al. (2013) conducted an assessment of lifetime reliability of the ship hull structures consideration corrosion degradation, inspection activity, and ship loading. The ultimate vertical bending moment capacity of the ship hull is employed in a reliability problem and an algorithm was developed to assess the structural reliability of ship hull.

The above-mentioned research works have limitations in assessing the effect of corrosion wastage on the ultimate strength of the unstiffened plates, stiffened plates, or hull girders. In terms of ship hull girder, the corrosion wastage is assumed that entire cross-section structures are corrosive with different corrosion levels. Therefore, this study aims to assess the UBM and cross-section properties under the effect of corrosion appearance in stiffeners, attached plating, or whole panel. The incremental-iterative method combined with two corrosion models is applied. The results and discussions are also presented.

2 Applied Methods and Parameters

2.1 Incremental-Iterative Method

The IACS-CSR developed the incremental-iterative method for determining the bending moment of ship hull girder which considers the critical failure modes of all main longitudinal structural elements belonging to each transverse section and has some assumptions as follows (IACS 2017):

  • The ultimate strength is calculated based on the hull transverse sections between the two adjacent transverse webs.

  • The hull girder transverse section remains plane during each curvature increment.

  • The hull material displays an elasto-plastic behavior.

The hull girder transverse section is categorized into a set of elements, which act independently from one another. In determining the UBM, the following main steps are followed concurrently as in the flow chart shown in Figure 1, which results in the M-χ curve.

  • Step 1: Divide the transverse section of the hull into several stiffened plate elements.

  • Step 2: Define the stress-strain relationships among all elements.

  • Step 3: Initialize neutral axis and curvature χ1 for the first incremental step.

  • Step 4: Calculate the corresponding stress σi and corresponding strain εi for each element.

  • Step 5: Identify the neutral axis ZNA_cur at each incremental step.

  • Step 6: Identify the corresponding moment through calculation.

  • Step 7: Compare the moment in the current incremental step with that in the previous one. The process is terminated and the peak value is defined if the slope in the M-χ relationship is less than a negative fixed value. Otherwise, increase the curvature by the amount of ∆χ and go to step 4.

Figure 1
figure 1

Flow chart of the procedure for the evaluation of the M-χ curve

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Following the aforementioned process, the UBM, Mu, is identified as the peak point of the M-χ curve, which results from the bending moment components of all incremental steps expressed by

$$ {M}_u=\sum {\sigma}_{ui}{A}_{i-n50}\left({Z}_i-{Z}_{\mathrm{NA}\_\mathrm{cur}}\right) $$
(1)

where: σui is the stress of element, Ai is the net cross-sectional area of the element, Zi is the vertical distance from the baseline to the horizontal neutral axis of the element, and ZNA_cur is the current position of the neutral axis from the baseline.

2.2 Ultimate Stress of the Structural Elements

By the IACS-CSR guidelines, the transverse cross-section is divided into three types of elements namely hard corner element, stiffened plate element, and stiffener element, as shown in Figure 2. In identifying the weakest inter-frame failure mode, all relevant failure modes for individual structural elements are considered. Table 1 presents the corresponding failure modes and stress-strain relationships (IACS 2017).

Figure 2
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Element definition

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Table 1 Modes of failure of individual structural elements
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2.3 Corrosion Model

To take into account the corrosion wastage, two corrosion models are employed such as the CSR corrosion addition model and probabilistic corrosion rate estimation model. The former is given by IACS-CSR, and the latter is suggested by Paik et al.

The corrosion addition is required in IACS-CSR guidelines for the ship design to achieve a 25-year design life. The corrosion addition margin is different for every thickness member such as the attached plating, flange thickness, and web thickness and different structural location member groups such as the deck, side, and bottom. The net scantlings in the CSR design are taken into account as the reference scantlings in the current study. Figure 3 presents the corrosion addition margins of different location member groups of a bulk carrier (IACS 2017).

Figure 3
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Corrosion addition margins of a bulk carrier structure specified by the CSR

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Paik et al. proposed a probabilistic corrosion model which was developed based on the measured data obtained from corroded oil tankers. Hence, the authors consider this model as the most suitable among other options in assessing the effect of corrosion on the responses of oil tanker structures. As indicated in Figure 4, the corrosion model is divided into two phases, one related to the coating life and the other concerned with the progress of corrosion, but without transition duration. The corrosion rate r(t) and corrosion wear d(t) are defined respectively as (Paik et al. 2003a):

$$ r(t)={C}_1 $$
(2)
$$ d(t)={C}_1{\left(t-{T}_0\right)}^{C_2} $$
(3)

where C1 is the annualized corrosion rate which follows the Weibull distribution, C2 is the coefficient and assumed to be 1.0, t is the age of the ship, and T0 is the coating life.

Figure 4
figure 4

Probabilistic corrosion rate estimation model

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Figure 5 shows the annual corrosion rate of individual structural member groups of a bulk carrier. The corrosion rate is the average level of corrosion wastage with the assumed coating lifetime of 7.5 years. Meanwhile, the identifications of member location groups are declared in Table 2.

Figure 5
figure 5

Corrosion rate of member location groups of bulk carrier

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Table 2 Identification of member location groups of bulk carrier
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3 Demonstrative Example

A bulk carrier is applied to assess the effect of local corrosion wastage on the vertical bending moment capacity and other properties of the cross-section. The main principles of the selected ship are presented in Table 3.

Table 3 Principal dimensions of selected bulk carrier
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To determine the influence of the local corrosion wastage, three assumptions of member structural groups are defined. The first group is the deck panel which includes deck plating, deck longitudinal stiffeners, and deck girders. The second group contains the longitudinal members of the side shell including side plating, side longitudinal stiffeners, and side girders. The last one is the bottom panel including bottom plating, bottom longitudinal stiffeners, and bottom girders. These mentioned categories are shown in Figure 6.

Figure 6
figure 6

Member location groups of the bottom, side, and deck panel of bulk carrier

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The corrosion wastage is applied with three assumptions of corrosion appearance to desire a full assessment of the effect of local corrosion on the behaviors of structures. Firstly, the corrosion is assumed that only the attached plating is corrosive. Then, all the thickness members of the longitudinal stiffeners are reduced due to corrosion wastage. Finally, the entire panel is assumed as a corroded panel.

Figures 7 and 8 combined with Table 4 present the detailed information of the thickness degradation of the attached plating (tp), thickness of web (tw), and thickness of flange (tf). The “gr” is denoted for gross, and “net” is defined as the net scantling of the thickness.

Figure 7
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Nomenclature of the plate-stiffener combination element dimensions. a Flat bar. b L profile

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Figure 8
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Schematic of a corroded plate-stiffener combination element. a Flat bar. b L profile

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Table 4 Thickness degradation due to corrosion wastage
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4 Results and Discussions

4.1 Case of Corrosion Appearance in the Bottom Panel

Table 5 and Figure 9a, b present the calculation results of UBMs and several cross-section properties under the corrosion appearance in the bottom panel of the selected bulk carrier. The information is classified into six cases of corrosion wastage and three structural member groups such as attached plating, stiffeners, and entire panel. The variations of UBM (Mu) and neutral axis position at the limit state (gu) combined with cross-section properties such as total cross-sectional area (A), moment of inertia (I), section modulus at deck and bottom (SM), and neutral axis of cross-section (NA) are illustrated.

Table 5 UBM and cross-section properties in case of the corroded bottom panel
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Figure 9
figure 9

Case of corrosion appearance in the bottom panel. a Variations of UBM under the corroded bottom panel. b Variations of CSA under the corroded bottom panel

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The appearance of corrosion at the bottom panel causes the reduction of cross-sectional area (CSA) of both attached plating and stiffeners by 0.8% (case PL5) for the former and 0.5% (case ST5) for the later. The effect of corroded plating is much more significant than that of corroded stiffeners on the UBM reduction. For instance, the ultimate sagging bending moment (USBM) reductions are 3.3% for the case of PL5 and 1.0% for the case of ST5. When the whole bottom panel is corroded, the USBM reduction is considerable (4.3% for case PA5) while the ultimate hogging bending moment (UHBM) reduction is slight (about 0.7% for case PA5). This degradation of UBM meets the biggest value in the case of PA6 as nearly 12%.

4.2 Case of Corrosion Appearance in the Side Panel

Table 6 and Figure 10a, b show the information related to the effect of corrosion appearance in the side panel on the UBMs and other properties. It can be seen clearly from the table that the CSA reduction of attached plating, stiffeners, and entire side panel are the similar trends in comparison with the case of the corroded bottom. The corrosion appeared in the deck panel affects clearly on the USBM while the side panel under corrosion influences apparently on the UHBM. Specifically, the corrosion diminishes the UHBM by 2.3% for case PA5 and 7.8% for case PA6; meanwhile, these values of USMBs reduction are 1.8% and 6.2%, respectively.

Table 6 UBM and cross-section properties in case of the corroded side panel
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Figure 10
figure 10

Case of corrosion appearance in side panel. a Variations of UBM under the corroded side panel. b Variations of CSA under the corroded side panel

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4.3 Case of Corrosion Appearance in the Deck Panel

Table 7 and Figure 11a, b give the information of UBMs, neutral axis position at the limit state, and other properties of the cross-section. The CSA reduction of the panel due to corrosion is small. It is about 1.1% for case PA5, but the UBM reduction is the biggest in comparison with other locations of corrosion appearance. To be more specific, the UHBM and USBM are reduced by 4.0% and 4.9% for case PA5, respectively.

Table 7 UBM and cross-section properties in case of the corroded deck panel
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Figure 11
figure 11

Case of corrosion appearance in deck panel. a Variations of UBM under corroded deck panel. b Variations of CSA under corroded deck panel

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4.4 Comparison of UBM and CSA of Panel

The UBMs for hogging and sagging conditions are summarized and shown in Figure 12a. The UBM is obtained for different cases of corrosion appearance locations such as deck, side, and bottom panel. Meanwhile, the variations of CSA of deck, side, and bottom panel under corrosion rate or corrosion addition are drawn as in Figure 12b.

Figure 12
figure 12

Ultimate bending moment and cross-sectional area comparisons. a Comparison of the USBM and UHBM. b Comparison of the CSA

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It can be seen from Figure 12b that the corrosion gives the biggest influence on the CSA reduction of the side panel and this reduction reaches 4.3% under the corrosion addition specified by the CSR. Although the CSA reduction of deck panel under corrosion is small, its effect on the UBMs and other cross-section properties is the most serious, especially USBM with the reductions are 4.9% (case PA5) and 10.6% (case PA6).

5 Concluding Remarks

Several concluding remarks can be drawn as the following list:

  1. (1)

    The analyses of the ultimate vertical bending moment and cross-section properties of a bulk carrier are performed and compared using the corrosion rate estimation model and CSR corrosion addition.

  2. (2)

    The total CSA of all longitudinal stiffeners in a panel is smaller than that of the attached plating of the same panel. The longer is the service year, the more obvious discrepancy is between these areas. The diminution of CSA of the deck panel is the smallest, while that of the side panel is the biggest.

  3. (3)

    The effect of corroded stiffeners on the UBM is smaller than the effect of corroded plating on the UBM. This is true for all of the defined panels such as deck, side, and bottom.

  4. (4)

    Under the thickness degradation due to corrosion wastage at the side panel, the CSA reduction is biggest, while UBM reduction is smallest in comparison with cases of the corroded deck and corroded bottom. This damaged structure also brings the diminution of UHBM which is bigger than USBM reduction.

  5. (5)

    On the contrary, UHBM degradation is smaller than USBM one for both cases of the corroded deck and bottom panels. The appearance of corrosion at the deck panel is considered to be the most hazardous on the residual strength of the selected ship.