Hull Girder Ultimate Strength Assessment Considering Local Corrosion

The residual strength capacity of ship hull with full corrosion appearance in every structural member has been considered in a large number of research works; however, the influence of local corrosion on the ultimate strength and cross-section properties has not been taken into account and analyzed. Hence, this study aims to assess the effect of corrosion appearance in the flange section and web section on the ultimate vertical bending moment and several cross-section properties of a bulk carrier. To perform this task, a probabilistic corrosion rate estimation model and the common structural rule model are introduced and employed. The incremental-iterative method given by the International Association of Classification Societies-Common Structural Rules (IACS-CSR) is applied to determine the ultimate vertical bending moment, neutral axis position at the limit state, and other properties of the cross-section. The calculation results and discussions relative to the effect of corrosion on ship hull are presented.


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.

Article Highlights
• The total cross-sectional area (CSA) of all longitudinal stiffeners in a panel is always smaller than this value of the attached plating in the same panel. • For bulk carriers, the CSA of side panel is the biggest, but the influence of the CSA reduction on ultimate bending moment (UBM) degradation is the smallest. • Deck panel has the most effect on the UBM of bulk carriers.
(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 , 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  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 Figure 3 Corrosion addition margins of a bulk carrier structure specified by the CSR Figure 4 Probabilistic corrosion rate estimation model Torsional buckling Web local buckling of flanged profiles σ CR3 ¼ Ф Web local buckling of flat bars Shortened stiffened pate element Plate buckling Hard corner element Elasto-plastic collapse σ = ФR eHA 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 Notes: The ending letter "P" is denoted for plating, "W" is denoted for the web, and "F" is denoted for the flange Figure 5 Corrosion rate of member location groups of bulk carrier  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 incrementaliterative method combined with two corrosion models is applied. The results and discussions are also presented.

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  & 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 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 Δχ and go to step 4.
Following the aforementioned process, the UBM, M u , is identified as the peak point of the M-χ curve, which results from the bending moment components of all incremental steps expressed by where: σ ui is the stress of element, A i is the net cross-sectional area of the element, Z i is the vertical distance from the baseline to the horizontal neutral axis of the element, and Z NA_cur is the current position of the neutral axis from the baseline.

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).

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).
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): where C 1 is the annualized corrosion rate which follows the Weibull distribution, C 2 is the coefficient and assumed to be 1.0, t is the age of the ship, and T 0 is the coating life. 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.  Table 3.
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.
The corrosion wastage is applied with three assumptions of corrosion appearance to desire a full assessment of the effect   Table 4 present the detailed information of the thickness degradation of the attached plating (t p ), thickness of web (t w ), and thickness of flange (t f ). The "gr" is denoted for gross, and "net" is defined as the net scantling of the thickness. 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 (M u ) and neutral axis position at the limit state (g u ) 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.

Case of Corrosion Appearance in the Bottom Panel
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%. 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 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

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. 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). (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) 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) 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) 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) On the contrary, UHBM degradation is smaller than USBM oneforbothcasesofthecorrodeddeckandbottompanels.The appearanceofcorrosionatthedeckpanelisconsideredtobethe mosthazardousontheresidualstrengthoftheselectedship.
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