Time-domain fatigue assessment of ship side-shell structures

International Journal of Fatigue 55 (2013) 276–290

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Time-domain fatigue assessment of ship side-shell structures Zhiyuan Li a,⇑, Jonas W. Ringsberg a, Gaute Storhaug b a b

Chalmers University of Technology, Department of Shipping and Marine Technology, SE-412 96 Gothenburg, Sweden Det Norske Veritas AS, Veritasveien 1, 1322 Høvik, Norway

a r t i c l e

i n f o

Article history: Received 13 December 2012 Received in revised form 21 June 2013 Accepted 8 July 2013 Available online 16 July 2013 Keywords: Fatigue Full-scale measurement Nonlinear time-domain approach Ship side-shell structure Stress concentration factor

a b s t r a c t Loads acting on ship side-shell structures are complex and vary randomly over time. The current study proposes a direct calculation procedure for the fatigue assessment of ship side-shell structures. The calculation procedure is characterised by nonlinear time-domain hydrodynamic simulations followed by finite element (FE) analyses. Sensitivity and feasibility analyses of the proposed time-domain procedure were carried out, and the calculated fatigue damages were compared with full-scale measurements made on a container vessel. Fatigue life analyses were carried out by both the spectral method and the timedomain approach. In addition, two approaches for local stress analysis are presented and discussed: an engineering-based definition of the stress concentration factor (SCF) and a proposed local stress factor (LSF) that utilises stress ranges extracted from the stress history. The results from the fatigue analysis using the LSF indicated a shorter fatigue life than the results obtained using the SCF. This difference is observed because the LSF accounts for the effects of wave-induced loads under ship operation conditions in a more realistic manner. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Fatigue damages in ship side-shell structures have for decades been detected and frequently repaired on ships. Schulte-Strathaus and Bea [1] studied different tanker designs and critical locations for fatigue damage. They found that fatigue cracks often occurred in connections between side-shell longitudinal and transverse bulkheads or the web frames. A similar investigation by ClassNK [2] of their very large crude carriers (VLCCs) indicated that fatigue cracks in side-shell structures represented approximately 35% of all cracks found in this type of vessel. This problem has recently been brought to the attention of container vessels, largely as a consequence of the ever-expanding container ship fleet. From 2003 to 2006, a growing number of fatigue cracks were observed in sideshell structures of approximately 10-year-old Panamax container ships (see Müller [3]). A vessel can suffer from fatigue cracks early in its service life for several reasons. One possible explanation is poor welding quality. However, it is also relevant and necessary to examine the current ship fatigue design methodology with respect to the representation of wave-induced loads acting on ship side-shell structures and the effects from trade and routing. Loads acting on ship side-shell structures are complex. In addition to the global and local girder loads, the side-shell is also subjected to external sea pressure. For example, at the area close to the still waterline, or the splash zone, the intermittency of sea pressure ⇑ Corresponding author. E-mail address: [email protected] (Z. Li). 0142-1123/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2013.07.007

introduces fatigue loads on the side-shell structures. Due to the relative motion between the ship and wave, the pressure load is particularly nonlinear with respect to wave height. Structural details in the splash zone are fully submerged for some regular waves, whereas they are either above or below the water surface for other waves. This example of nonlinearity in the splash zone, between wave height and pressure load, cannot be addressed accurately via a traditional frequency-domain analysis. Since the 1990s, researchers have tried to account for this nonlinear pressure load using linear strip theory. Some studies note the importance of load combination effects and demonstrate that the nonlinear pressure load contributes significantly to the sideshell fatigue damage. Friss Hansen and Winterstein [4] propose a realistic wave model that considers nonlinearities due to the instantaneous position of the hull relative to the waves. The stress caused by the wave pressure is combined with the stress from hull bending calculated via Euler–Bernoulli beam theory. Folsø [5] incorporates the nonlinear wave pressure load via a modification of the pressure response amplitude operator (RAO). The total fatigue damage due to hull bending and local sea pressure is calculated based on the modified RAOs of both hull bending and local sea pressure. Instead of solving the nonlinear sea pressure load in the frequency domain, Berstad [6] models the sea pressure in the time domain, which is consistent with linear strip theory. A local finite-element (FE) model is built to undertake the sea pressure, and the hull bending loads are applied to a beam model. Thus, the total fatigue stress is the summation of the stresses caused by the local sea pressure and global bending. These studies have

Z. Li et al. / International Journal of Fatigue 55 (2013) 276–290

succeeded in extending linear strip theory to account for the nonlinear sea pressure load. However, the wave loads and ship motions are derived from linear strip theory and are thus not as accurate as those calculated using the panel method. This is more important for certain wave lengths and locations. 1.1. Structure response analysis due to wave-induced loads Euler–Bernoulli beam theory (hereafter referred to as beam theory) is often used for ship structure stress response analysis; however, it has some limitations. Under beam theory, the ship hull is considered a girder, and the stress is calculated using such load types as the vertical bending moment, horizontal bending moment, torsion moment, shear forces, and axial force. Local loads can also be considered using beam theory, but the combination of these loads and how they should be incorporated into the ship structure response analysis is often complicated. For ships with an open deck like container ships, the U-shaped cross-section makes the hull structure sensitive to wave-induced torsion; thus, warping stresses occur. These stresses may contribute to the total longitudinal stress and must be included in an accurate prediction of fatigue lives (see Li and Ringsberg [7] for a detailed discussion). Compared to the stresses caused by bending-induced loads, the distribution of the warping (normal) stresses is more complicated and strongly depends on the cross-section geometry and longitudinal position of the detail along the hull of the ship. Thus, it is not an easy task to express the combined stress from bending and torsion loads with simplified formulas through the use of beam theory. In light of this difficulty, FE analysis (FEA) is preferred over beam theory when deriving the structure response. Nonlinear effects in wave loads and ship motions must be properly addressed in ship fatigue assessment. These nonlinearities arise from the global wave loads and ship motions, which are due to the non-vertical hull form in the fore and aft part and characterised by the asymmetry between sagging and hogging moments. Thus, this geometric nonlinearity should be distinguished from the nonlinear pressure loads applied to the side-shell structures. Nevertheless, in practice, a fatigue analysis of ship structures is often treated as a linear problem and assessed using the spectral method [8–10]. The linear relationship is assumed between the wave loads and the responses of the ship. The linear assumption allows the solution of the hydrodynamic problem in the frequency domain, so it is termed the spectral method. The spectral method is widely used in the fatigue assessment of marine structures; this method assumes that the major contribution to the accumulation of fatigue damage originates from small and moderate waves (see DNV [8]). However, a spectral methodbased fatigue assessment can be questioned if more severe sea states occur in the wave environment of the ship. Mao [11] finds that more moderate waves were encountered for a container vessel on the North Atlantic route than in the standard wave scatter diagram. A similar observation has been made by Sternsson and Björkenstam [12], who find that more waves between 4 and 6 m were encountered by a car carrier during winter crossings compared to the tabulated results in Hogben et al. [13]. These observations are important because the nonlinear effects in the hydrodynamic loads and the corresponding fatigue damage of the hull structure become significant with the increase in wave height. According to Jensen [14], during moderate sea states, nonlinear effects will result in an increase in fatigue damage by 50–100% in the hull plating of a container ship. Similarly, Gu and Moan [15] demonstrate that nonlinear wave loads result in a 10–100% increase in fatigue damage compared to linear wave loads. However, all of these studies focussed on the global bending in deck. How the side-shell fatigue is affected by the nonlinear wave loads must be investigated further.

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1.2. The objective with the current study The current study presents a direct calculation procedure useful for the fatigue assessment of ship side-shell structures. The calculation procedure is characterised by a nonlinear time-domain hydrodynamic analysis followed by (hotspot) stress calculations using global and/or local FE models. The main objective is to establish a time-domain procedure for fatigue assessment of ship sideshell structural details under complex loadings. Due considerations are given to nonlinear wave loads, and variation in local stress response, as well as to the reliability and feasibility of the time-domain procedure. The main advantage of using this procedure is that the complex wave-induced load combination acting on the side-shell structure is realistically considered, which is believed to yield improved fatigue assessment of ship side-shell structures. The outline of the study is as follows. Section 2 presents a case study vessel that has been used in the numerical analyses throughout the investigation. This Panamax container vessel has been studied by the authors in previous studies (see, for example, references [7,16–18]). Section 3 presents a time-domain-based analysis procedure that is used to carry out fatigue damage assessments. The emphasis of this section is on the hydrodynamic analysis and how the analysis should be carried out to ensure a realistic modelling and representation of wave loads, which is crucial for the correct ship-motion behaviour and for the loads calculated and applied in the following structure response analysis. In Section 4, short-term fatigue analyses based on real-life sea states are studied and compared with full-scale measurement. The reliability and sensitivity of time-domain methods are discussed. Long-term fatigue analysis is studied using the proposed time-domain approach in Section 5, and fatigue assessment is also carried out with the traditional spectral method for comparison. In Section 6, methods to derive local stresses are discussed, and a new approach to derive a local stress useful for fatigue analysis is proposed. Finally, the conclusions from the study are presented in Section 7. 2. The case study container vessel A Panamax container vessel sailing on the North Atlantic trade between Europe and Canada is used as a case study vessel (see Fig. 1 and Table 1 for the main particulars of the vessel). The North Atlantic trade is considered one of the harshest sea environments in the world. During winter seasons, one to three low pressures typically occur for every crossing, and 15–25% of the significant wave heights encountered exceed 5 m (see Mao [11]). The vessel was built to the DNV class. The vessel operates with a slightly lower draft than similar-sized container vessels. Its cross-section has a conventional structural design that is made of HT32-grade steel, except for above the upper deck, which is constructed with HT36-grade steel.

Fig. 1. Picture of the Panamax container vessel.

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angle because the measurement of this property was deemed reliable as long as only one-directional sea is under study.

Table 1 Main particulars of the Panamax container ship. Length overall, LOA (m) Length between perpendiculars, LPP (m) Breadth, B (m) Depth, D (m) Draft design, T (m)

294.0 281.0 32.26 21.50 10.78

Deadweight at design, dwt (tonnes) Service speed at design draft (kn) Block coefficient, CB (–) Horsepower M.C.R. at 104.7 RPM (kW) Max TEU

47,754 23.0 0.69 37,275 4400

This vessel is instrumented with an onboard measurement system from a leading hull monitoring supplier that comprises strain sensors in several locations of the ship and other devices that measure ship motions, encountered waves, and ship speed. The strain sensors have been mounted on both the port and starboard sides. Fig. 2 shows the location of one of the strain sensors, the side-shell longitudinal (SL), on one side amidships. This strain sensor is located in the No. 5 wing water ballast tank between frames 72 and 73, which are 133.5 and 136.75 m forward of the aft perpendicular, respectively (see Storhaug and Moe [17] for details). This tank is a heeling tank, so the internal pressure may introduce uncertainties. The centreline of the longitudinal stiffener is 3.8 m below the full-loaded waterline. In this study, all fatigue assessments as well as stress concentration calculations are carried out for the side-shell longitudinal at this specific location on both the port and starboard sides. The strain sensors record the longitudinal strain responses at a sampling rate of 42 Hz. Elastic strain is assumed and re-calculated for stresses using Hooke’s law. The position of the sensor is based on the lowest estimated fatigue life of the stiffeners in the side-shell according to the simplified fatigue check in NAUTICUS (Newbuilding) notation [19]. The onboard measurement records are composed of components from wave frequency loads and high-frequency loads. However, in this study, the emphasis is on wave frequency loads, and the high-frequency loads are excluded by a low-pass filter at 0.3 Hz, which is consistent with the hydrodynamic model. The encountered waves are measured by an onboard directional wave radar. Previous investigations have shown that the wave radar tend to overestimates the wave heights (see, for instance, [9]). Thus, the significant wave height, Hs, the wave period, Tp, and the mean wave direction were retrieved from hindcast data from the ECMWF ERA-interim database. The hindcast data yield a 1.5  1.5° grid resolution with four daily observations. A detailed description of the accuracy, re-analysis model, and input observation is presented in Dee et al. [20]. The only property measured by the onboard wave radar that was used in the current study was the wave encounter

(a)

3. Analysis procedure A numerical analysis procedure suitable for time-domain-based direct calculation of fatigue damage in ship structures is presented in Fig. 3. The numerical analysis procedure is considered particularly useful for side-shell fatigue assessment, but the methodology can also be used for the analysis of structural details at other locations. Similar detailed descriptions of the parts of the analysis procedure have been presented by the authors in [7], among others. Thus, in the current section, only a brief overview is presented with the emphasis on hydrodynamic load analysis followed by the structure response analysis. In the flowchart in Fig. 3, a square-shaped symbol indicates that the DNV SESAM software [21] has been used; a rhombus symbolises input to the analysis, such as material data/factors or results from prior analyses; a square with rounded corners denotes that an in-house code has been used in the calculations. The dashed areas indicate two options for approximation or calculation of the local stress, either by a conventional stress concentration factor (SCF) or a local stress factor (LSF), which are presented in Section 6. The DNV software SESAM is used for hydrodynamic and structural analyses, whereas in-house codes are used for stress evaluation and fatigue assessment. For each representative sea state, a timedomain hydrodynamic analysis is carried out followed by a global linear FEA. The global FEA is used for screening for fatigue critical locations in the hull. Nominal stresses can be extracted from the global FEA in the locations of interest. By multiplying the nominal stress with a SCF, the hotspot stress can be calculated at the corresponding structural detail. A SCF can be assumed based on tabulated values from classification societies or other references. Alternatively, the stress concentration relationship can be obtained by a comparison of the hotspot stress from a local FEA with the nominal stress from a global FEA also according to class procedures. The latter approach is preferred for fatigue-critical locations where a SCF is unavailable or susceptible. In such locations, local FE models are created to investigate the local stress response in more detail. The hotspot stress can then be obtained from the local FEA through sub-modelling, which transfers the global loadings to the local FE model (see Sections 3.2 and 6 for details). Short-term fatigue damage is calculated by rainflow counting of the fatigue stress history in combination with hotspot stress S–N curves for the structural detail under consideration. Long-term fatigue damage is calculated by summation of the fatigue damages from individual sea states, multiplied by the encountered probabilities extracted from wave scatter diagrams for the ship route of

(b)

SL

Fig. 2. (a) Location of the side-shell longitudinal (SL) strain sensor on one side and (b) a geometric model of the current SL geometry.

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279

Fig. 3. Flowchart of analysis procedure for the direct calculation of fatigue damage.

interest. Details regarding the fatigue assessment are presented in Sections 4 and 5. 3.1. Hydrodynamic load analysis Hydrodynamic loads acting on a ship structure can be computed using strip theory or the panel method. According to strip theory, the ship hull is treated as a series of strips that represent the cross-sections along the hull. Many authors have reported that the results predicted by strip theory are in good agreement with experiments in the vertical plane but less satisfactory in the lateral plane (see, for example, Beck et al. [22]). In the panel method, the hull form is modelled by a number of panels. The wave loads are represented by water pressure distributed on these panels along the entire hull of the ship. In contrast to strip theory, the panel method considers the three-dimensional (3D) effects. Thus, the panel method could provide more realistic and accurate results with respect to ship motions and section loads in the lateral plane. In the current investigation, the 3D hydrodynamic Rankine panel method based code WASIM (see DNV [23]) is used, and the calculations are carried out in the time domain. Waves can come from arbitrary directions and ship motions are calculated in all six degrees of freedom. Forward speed is included, and the incident waves are modelled here with irregular short-crested (cosine square) waves. The Pierson–Moskowitz (PM) spectrum is used to describe the sea state conditions (see DNV [24] for the detailed expression of the PM spectrum). In WASIM, wave loads can be computed either by linear or nonlinear algorithms. For the linear solution, the Froude-Krylov and restoring hydrostatic forces are determined by integrating the pressure up to the mean wetted surface. For the splash zone, the sea pressure above the mean wetted surface is always zero, whereas negative pressure exists below the mean wetted surface, which leads to an unrealistic stress distribution in the splash zone. In contrast, in the nonlinear solution, the Froude-Krylov and

restoring hydrostatic forces are integrated on the instantaneously wetted hull. Thus, the nonlinear load caused by sea pressure near the mean free surface is more realistically modelled. Through calculating pressure according to the instantaneously wetted hull, the geometric nonlinearity due to hull shape is also considered, which yields more accurate values of the wave-induced hogging/sagging moments. However, this nonlinear effect is still a type of simplification because the radiation/diffraction parts are linearised and solved on the mean free surface and mean wetted surface. Time-varying sea loads on each panel of the wetted hull are calculated separately and then transferred to the SESAM software structure analysis module SESTRA, a linear FE solver. These loads are applied on the FE model of the vessel and enable the full analysis of structure response and stress calculation via FEA from wave and inertia loads. 3.2. Stress response analysis The objective of the stress response analysis is to calculate the stress history for various sea states, thereafter used in fatigue assessment (see Fig. 3). Note that despite the demanding computational efforts using the FE method compared to a simplified method, such as engineering beam theory, the FEA of a ship structure is considered more reliable and accurate, particularly for the calculation of local stresses and load combination effects when loads are directly transferred from hydrodynamic analysis. The structure response analyses of the case study container vessel is divided into steps of analyses using FE models of different sizes and element resolution depending on the purpose of the FEA. Fig. 4 presents an example of an FE model library of the part of the global FE model indicated by the ellipse in the figure. The global model is the largest-sized FE model that has the coarsest element resolution of all models and has the top position in a hierarchy of models. Local structure elements, such as brackets, are not represented in this model, and stiffeners are modelled

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Global model

Local model 1

Local model 2

Local model 3

Hotspot model

Fig. 4. Example of FE models of the case study container vessel: global, local (1–3), and hotspot models.

simplistically using representative beam elements. When the dimensions of the FE model become smaller (left to right in the figure, local models 1–3), the element size becomes smaller and the mesh density increases; more structural details are modelled and beam elements are replaced by shell elements, resulting in improved accuracy in stress calculation. The finite element size is typically on the order of the plate thickness in the hotspot FE model, whereas an element in the side-shell structure is approximately the frame distance in the global model. The wave-induced loads and inertia loads from WASIM are transferred directly to the local FE models by means of a submodelling technique available in SESAM. The submodelling technique ensures that the boundary conditions of a local FE match with the corresponding nodes in an FE model higher up in the hierarchy of models. Moreover, the local sea pressure is obtained from the hydrodynamic analysis, and potential local cargo loads and internal liquid pressure (disregarded in the current investigation) can also be included.

4. Short-term fatigue damage calculation During local elastic or elastic shakedown material responses, fatigue assessment of ship structures is typically carried out using the S–N approach. Stress cycles from stress histories can be obtained from various approaches, with the rainflow counting approach considered the most accurate; thus, the rainflow counting approach is used in the current study. The short-term fatigue damage was calculated using the Palmgren–Miner cumulative damage rule in combination with the S–N curves for the material and structural details of interest:

D¼

X N i Sm i

i

a

ð1Þ

In Eq. (1), Ni is the number of stress cycles at stress range Si, which is calculated using rain-flow counting. The outer side-shell longitudinal is assessed with respect to fatigue using the hotspot stress S–N curve (for welded joints) with the material parameters a = 1012.164 and m = 3 adapted from DNV [6], which corresponds to FAT90. In the current investigation, calculation of the short-term fatigue damage, D, was constrained to a 20-min sea state. The calculated short-term fatigue damages were compared with results from the onboard measurements, which are presented in more detail in Section 4.1. In the following subsections, important issues that affect the results from a time-domain fatigue assessment were investigated, including the influence from time history

length (see Section 4.2), the sensitivity from the time step length in the analysis (see Section 4.3), and the influence of wave loads calculated from linear and nonlinear algorithms (see Section 4.4). 4.1. Comparison of fatigue damages - measurements and numerical calculations Three actual sea states from a winter voyage are selected for comparison representing high-, median-, and low-wave environments, respectively. The main particulars of the sea states and operational profiles are presented in Table 2; the significant wave height, Hs, and the wave peak period, Tp, were calibrated by the hindcast data. Both the measured main wave heading angle (HDG) from the onboard wave radar and the ship speed, U, are assumed to be unchanged within a 1-h period. The HDGs are illustrated in Fig. 5. The measured ship speeds are lower than the service speed of 11.8 m/s (23 knots). The reason for this lower speed is that voluntary or involuntary speed reduction occurs due to harsh sea state conditions. Fig. 6 presents the measured global nominal longitudinal stresses, rx, in the side-shell structure longitudinals on the port (SLP) and starboard (SLS) sides for sea states 1–3. Table 3 presents the fatigue damages based on results from full-scale measurements and from the current numerical simulations and fatigue damage calculations. For each sea state, the 1-h measured record was divided into three adjacent 20-min intervals. The mean value (mean) is the average value of the sea state interval fatigue damages, and the standard deviation (std) is the scatter in fatigue damage between the three intervals that form a sea state. The results from the measurements exhibit some variations. One reason for these variations may be that the assumption of stationary conditions for the 1-h sea state may have been violated. Another possible cause is that between the adjacent 20-min intervals, the course of the ship to the wave direction could have been changed a bit, but this course change has not been modelled in the numerical simulations. These observations underline the difficulties and challenges in the use and interpretation of the results from full-scale measurements with variability in wave environments and ship operation conditions. Table 3 presents a comparison of accumulated fatigue damages from full-scale measurements and numerical calculations based on the input presented in Table 2. The SLP and SLS locations in the case study vessel are presented in Section 2. The results are based on the analysis of nominal longitudinal stresses for a 20-min sea state. The nominal longitudinal stress history in the numerical calculations has been calculated using the global FE model. The

Z. Li et al. / International Journal of Fatigue 55 (2013) 276–290 Table 2 Main particulars of the studied sea states. Sea state

Date and time

Hs (m)

Tp (s)

1

2008-12-17, 08:00– 09:00 2008-12-17, 17:00– 18:00 2008-12-16, 21:00– 22:00

6.1

13.7

8.5

130

4.7

12

9.7

140

3.5

10

10.2

160

2 3

U (m/ s)

HDG (degrees)

Fig. 5. Definition of HDGs used in the current investigation.

results in Table 3 illustrate that the side-shell structure on the starboard side accumulates more fatigue damage compared to its counterpart on the port side. This difference is also captured by the numerical calculations. The agreement in results between the measurements and numerical calculations is not fully satisfactory. One possible reason may be that the internal pressure was not modelled at the measured locations. Thus, further investigation of the internal pressure loading is necessary. 4.2. Influence of time history length In time-domain analyses, the time history length is an important factor that impacts both the accuracy of the results and the applicability of the time-domain procedure. Ship structures are subject to random sea loads that must be analysed statistically. To ensure the reliability of the statistical analysis, the time history should comprise at least 100 pairs of peaks and troughs, which corresponds to 20–30 min of data. Nevertheless, very long records should be avoided as well because a stationary sea state typically lasts no more than several hours [25]. In practice, it is customary to assume a sea state between 20 min and 3 h for time-domain analyses. Fig. 7 presents fatigue damages calculated using various lengths of the time record from 1200 s to 10,800 s. To compare the influence of the time record length on fatigue assessment results, the accumulated fatigue damage was normalised for 20-min intervals, D20min,norm. The diagram presents D20min,norm in the two locations SLP and SLS of the case study container vessel for the sea states 1–3. The calculated fatigue damages are observed to be, in principle, convergent when the time records are longer than 20 min. In conclusion, a time history of 20 min is sufficient for a single realisation for time-domain-based fatigue assessments. However, it needs to be pointed out that more realisations are necessary for investigation of uncertainties associated with numerical analysis.

281

Fig. 3, the time steps in both the hydrodynamic simulation and FE analysis must be considered. The time step used in a hydrodynamic simulation is important with respect to numerical stability and is determined based on the speed of the ship and the panel size in the hydrodynamic model. In the current study, the ship has a service speed of 23 knots, and there are approximately 100 panels in the longitudinal direction of the ship. Thus, a 0.10-s time step was used, which is generally considered sufficient for the nonlinear hydrodynamic simulations; more discussion is referred to DNV [19]. The time step required to ensure numerical accuracy in the linear FE analysis can be set similarly to the time step used in the hydrodynamic simulation. However, a larger time step is often used in the FE analysis to reduce the computational effort. In the current study, dynamic effects are disregarded, and the ship structure responses are taken as rigid body responses; thus, the FE analysis is considered quasi-static. These assumptions allow a considerably larger time step in the FE analysis compared to the time step used in the hydrodynamic simulation. The question is then how large this time step can be without losing accuracy in the results. If the time step used is too large, the number of peak loads will be too low because some peaks are excluded. This exclusion will directly affect the stress history in the same manner, and the fatigue damage based on the stress history will be underestimated. For this reason, a time-step sensitivity analysis was carried out. The time step used in the FE analysis was varied, and the accumulated fatigue damage during 20 min was calculated at SLS and SLP, respectively. Fig. 8 presents the results for sea state 1, where the FE analysis time steps were selected as 0.1, 0.25, 0.50, 0.75, and 1.00 s. The results indicate that the accumulated fatigue damage decreases with an increasing time step. The deviations in fatigue damage between the smallest and largest time steps are approximately 22.6% and 11.0% for SLP and SLS, respectively. Considering the balance between engineering accuracy and computational effort, the time steps used in FE analyses were set to 0.5 s, which is utilised in the fatigue calculations presented in this paper unless stated otherwise. 4.4. Influence of nonlinearity in wave loads and ship motions This section presents the assumption and selection of a linear or nonlinear wave load algorithm in WASIM and how this selection influences the fatigue life. The nonlinear results are presented in Table 3 as the calculated fatigue damages at SLP and SLS under different sea states. Linear WASIM simulations were carried out for the sea states and under the same operation conditions as those stated in Table 2. The results in Fig. 9 indicate that there is some variation between the linear and nonlinear WASIM simulations. However, the difference is less than expected. Compared to nonlinear WASIM, the linear algorithm predicts higher values of fatigue damage in SLS but a lower amount of fatigue damage in SLP. Nevertheless, linear WASIM integrates pressure only up to the still water line and should thus be considered an approximation of the nonlinear solution. For these global FE analyses, the calculated global nominal stresses seem to be dominated by global girder forces. In other words, the difference in local pressure between linear and nonlinear solvers is relatively small and likely not captured adequately in the global nominal stresses.

4.3. Influence of time step length

5. Long-term fatigue damage assessment

Another essential influencing factor in time-domain analysis is the length of the time step. With the approach presented in

A long-term fatigue assessment of fatigue damage accumulation can be carried out using the spectral method in the frequency

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Fig. 6. Measured nominal longitudinal stresses for sea states 1–3 in the SLP and SLS locations.

Table 3 Accumulated fatigue damage (unit: 1  106) during 20 min from measurements and numerical calculations. SLS: outer side-shell longitudinal on starboard side. SLP: outer side-shell longitudinal on port side. Std: standard deviation. See the text for details. Sea state

Measurements SLS

1 2 3

Numerical calculations SLP

Mean

Std

Mean

Std

2.122 1.237 0.372

0.072 0.022 0.020

0.389 0.220 0.067

0.033 0.021 0.007

SLS

SLP

3.597 2.044 0.743

1.241 0.629 0.250

domain or using the time-domain method. The advantage of using the former method is that it requires significantly less computational effort than the latter. Therefore, the spectral method is often used for long-term fatigue assessments. On the other hand, in the long-term fatigue assessment of side-shell structures, the time-domain method is preferable because the pressure loads are more realistically computed. Thus, in the current study, both the spectral method and the time-domain method are investigated. Linear and nonlinear WASIM are used for hydrodynamic simulations for the spectral method and time-domain method, respectively. In both approaches, a general SCF of 2 and the S–N data presented in Section 4 are used.

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Fig. 7. Time record length sensitivity analysis using five different lengths of time record. D20min,norm is the accumulated fatigue damage normalised for 20-min intervals. The results are presented in SLP and SLS for sea states 1–3.

Fig. 9. Comparison of accumulated fatigue damage over 20 min using linear and nonlinear WASIM: (a) port (SLP) and (b) starboard (SLS) side.

Fig. 8. Time step sensitivity analysis using four different time steps in the FE analysis. The diagram presents the accumulated fatigue damage during 20 min in two locations of the case study container vessel for sea state 1.

together different levels of exceedance probabilities. These parameters are average values of the entire stress range distribution calculated by fitting the Weibull function to all distribution points by a least-squares technique.

5.1. Results using the spectral method In the spectral method, the ship response is assumed to be proportional to the wave load, which can be described by a superposition of the response of all regular wave components that comprise an irregular sea. The long-term distribution of wave loads may be estimated from wave statistics, where the scatter diagram is assumed to be equal for all wave directions. Based on the assumption of linearity, the ship response spectrum can be derived directly from the wave spectrum using unit wave heights. The short-term stress range can be assumed to be Rayleigh distributed within each short-term condition. The stress range distribution for a given sea state i and heading direction j can be expressed by Eq. (2), where m0 is the zero-order spectral moment:

 F Drij ðrÞ ¼ 1  exp 

r2 8m0ij

 ð2Þ

The long-term stress range is assumed to follow a Weibull distribution according to Eq. (3), where h is the shape factor and q is the scale factor. The Weibull parameters are calculated by fitting

F Dr ðrÞ ¼ 1  exp 



r2 q

h ! ð3Þ

The hydrodynamic analysis is carried out using linear WASIM with 29 wave components. As recommended in DNV [8], the wave direction spacing can be assumed to be evenly distributed, and a constant ship speed can be used for the spectral analysis. Here, the wave headings are set to 15° with an equal probability of occurrence. A constant speed of 8 m/s is assumed, which corresponds to two thirds of the service speed. This representative speed should be considered a simplification, and it varies among classification guidelines. For instance, ABS [26] suggests a speed of three fourths of the service speed instead. A full stochastic fatigue assessment is performed in the frequency-domain assuming a wave environment of the North Atlantic (see IACS [27] for the wave scatter diagram). The target service life of the case study vessel is 20 years with 85% time at sea. The commercial code STOFAT [28] is employed, and the fatigue damage is calculated based on the nominal stress derived from global FE analyses, resulting a fatigue life of 7.1 years. The spectral method

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used in this investigation excludes the splash zone pressure reduction. This is based on the fact the side-shell structures are located relatively far below the water line and the splash zone effect can thus be assumed small. Nevertheless, neglecting the splash zone reduction may introduce risk of overestimated fatigue damage. However, for the location and structural detail of interest, the missing global nonlinear effects tend to override the splash zone effect. In view of this consideration, it is unclear whether the spectral approach used here yields conservative fatigue results. 5.2. Results using the time-domain method In a long-term fatigue assessment using the time-domain method, the accumulated fatigue damage is calculated as a summation of partial fatigue damages, Di, multiplied by their respective probabilities of occurrence, pi for all sea states encountered during the ship’s lifetime:

Dtotal ¼

X p i Di

ð4Þ

i

A representative wave scatter diagram for the case study container vessel and its route was designed. This wave scatter diagram is presented in Table 4 and is based on 20 encountered sea states (called ‘‘block’’ in Table 4) and the corresponding probabilities of occurrence, pi, and the ship speed. These sea states are based on the standard wave scatter diagram for the North Atlantic [27] in which sea states have 1 m intervals for significant wave height and 1 s intervals in wave period (Tz). In the current study, the standard wave scatter diagram is modified to represent five significant wave heights with four wave periods each. Significant wave heights up to 9.5 m are incorporated. Larger significant wave heights were excluded because current vessels have access to general weather advice services that assist in avoiding large/extreme storms. Additionally, according to Storhaug et al. [18], more than 80% of the wave-induced fatigue damage is caused by waves between 3.5 and 9.5 m, though for particular vessel it may differ slightly. Thus, the chosen sea states are considered adequate. The simplified scatter diagram in Table 4 was necessary to maintain a reasonable computational effort. The ship speed for each encountered wave height is presented in Table 4. The speed reduction is related to the encountered wave height because of added resistance in higher waves and also because speed reduction is often made voluntarily by the captain,

Table 4 Simplified wave scatter diagram used in the long-term fatigue assessment in the time domain. Block

Hs (m)

Tp (s)

pi (%)

U (m/s)

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20

1.5 1.5 1.5 1.5 3.5 3.5 3.5 3.5 5.5 5.5 5.5 5.5 7.5 7.5 7.5 7.5 9.5 9.5 9.5 9.5

10.6 13.4 16.3 19.1 10.6 13.4 16.3 19.1 10.6 13.4 16.3 19.1 10.6 13.4 16.3 19.1 10.6 13.4 16.3 19.1

30.36 9.36 0.69 0.08 12.69 20.02 4.42 0.39 2.01 9.35 3.29 0.65 0.26 2.39 2.12 0.50 0.08 0.46 0.66 0.25

11.8 11.8 11.8 11.8 9.8 9.8 9.8 9.8 8.2 8.2 8.2 8.2 6.2 6.2 6.2 6.2 4.1 4.1 4.1 4.1

who is responsible for comfort and cargo safety. In the current investigation, the reduced speeds are determined based on onboard observations of the case study vessel and extrapolated linearly for higher wave heights when the measurements are not available. The HDG of the ship in relation to the wave encounter direction is another important factor that significantly influences fatigue damage. To achieve an accurate fatigue estimate, fatigue damage must be calculated for each wave heading. In this study, the variation in HDG is treated by using eight representative heading angles, as illustrated in Fig. 5. For side-shell structures, much more fatigue damage is expected for side longitudinals located on the windward side than on the leeward side. The oblique waves from the windward side are defined by a positive HDG. Accordingly, the leeward oblique waves are defined by negative HDG values (see Table 5). The HDGs are assumed to have variable occurrence probabilities, i.e., head seas and following seas are encountered with a probability of 20%, whereas the oblique seas and beam seasare encountered with a probability of 10%, as presented in Table 5. This uneven occurrence probability of HDGs is because the target vessel operates in the North Atlantic, in which head seas and following seas are more common according to observations [29]). The occurrence probabilities in Table 5 apply for all of the representative sea states in Table 4. Table 6 presents 20-min partial damages under various wave headings for the representative sea states. All individual partial damages are calculated according to the specific ship speed as presented in Table 4. For each sea state, partial fatigue damages for all heading angles are summed and multiplied by the occurrence probability of this specific sea state (pi in Table 4). The summed and weighted partial damages are listed in the ‘‘Sum. HDGs’’ column in Table 6. A summation of this column yields the resultant 20-min accumulated fatigue damage D20min for all representative sea states. The total accumulated damage during for a target service life of 20 years, D20years, is then obtained from Eq. (5), assuming 85% of the life time at sea. For the side longitudinals, D20years = 2.99, corresponding to a fatigue life of 6.7 years.

D20years ¼ D20 min  ð20  365  24  3  0:85Þ

ð5Þ

For the current case study, the calculated fatigue life using the time-domain procedure is similar to the results from the spectral approach (7.1 years) (see Section 5.1). However, the time-domain procedure considers more factors in the simulations, such as the nonlinear effects in loads, speed reduction and variation in encountered wave heading. Thus, this time-domain-based procedure should be considered more realistic than the spectral method. It is noticeable that both method predict fatigue lives significantly shorter than the target service life, which may imply that the current fatigue design practice for side-shell structures should be looked over. Partial damages for the four sea states of the same wave height are summed together to investigate the contribution of individual representative wave heights. The summed damages for each significant wave height are given in the rightmost column ‘‘Sum. for Hs’’. Using the current representative sea states, most damages are caused by moderate to severe sea states (Hs is between 3.5 and 7.5 m), as illustrated in Fig. 10. This observation is in contradiction with the statement in DNV [8] that ship fatigue damages are mainly due to low and moderate sea states. Thus, further investigation of this topic is required.

6. Calculation of local stress In a ship structure with good-quality welds without defects and details without misalignments, fatigue cracks can still occur in

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Z. Li et al. / International Journal of Fatigue 55 (2013) 276–290 Table 5 Definition and occurrence probability (pb) of the HDGs.

HDG pb

Bow sea leeward

Beam sea leeward

Quarter sea leeward

Following sea

Quarter sea windward

Beam sea windward

Bow sea windward

Head sea

135° 0.1

90° 0.1

45° 0.1

0° 0.2

45° 0.1

90° 0.1

135° 0.1

180° 0.2

Table 6 20-min partial damages (unit: 1  106) for various wave headings and sea states. Hs (m)

HDG 135°

HDG 90°

HDG 45°

HDG 0°

HDG 45°

HDG 90°

HDG 135°

HDG 180°

Sum. HDGs

Sum. for Hs

1.5

Block #1 #2 #3 #4

0.0116 0.0018 0.0001 0.0000

0.0112 0.0035 0.0001 0.0000

0.0117 0.0036 0.0001 0.0000

0.0534 0.0165 0.0014 0.0002

0.0313 0.0096 0.0004 0.0000

0.0395 0.0122 0.0003 0.0000

0.0267 0.0048 0.0002 0.0000

0.0201 0.0062 0.0002 0.0000

0.2055 0.0581 0.0028 0.0003

0.2668

3.5

#5 #6 #7 #8

0.0472 0.0542 0.0071 0.0003

0.0428 0.0676 0.0051 0.0003

0.0199 0.0315 0.0053 0.0004

0.0540 0.0852 0.0119 0.0014

0.0982 0.1549 0.0199 0.0009

0.1507 0.2378 0.0158 0.0009

0.1358 0.1154 0.0151 0.0008

0.1007 0.1588 0.0185 0.0009

0.6493 0.9054 0.0986 0.0059

1.6591

5.5

#9 #10 #11 #12

0.0301 0.0873 0.0174 0.0021

0.0210 0.0978 0.0155 0.0022

0.0086 0.0400 0.0085 0.0011

0.0183 0.0854 0.0442 0.0045

0.0503 0.2343 0.0499 0.0065

0.0777 0.3617 0.0449 0.0061

0.0798 0.2142 0.0376 0.0045

0.0536 0.2497 0.0482 0.0058

0.3395 1.3703 0.2662 0.0328

2.0087

7.5

#13 #14 #15 #16

0.0099 0.0663 0.0588 0.0057

0.0078 0.0722 0.0433 0.0038

0.0040 0.0370 0.0267 0.0023

0.0086 0.0792 0.0666 0.0092

0.0194 0.1795 0.1528 0.0135

0.0235 0.2169 0.1416 0.0097

0.0223 0.1283 0.1138 0.0090

0.0171 0.1581 0.1351 0.0111

0.1126 0.9375 0.7387 0.0642

1.8530

9.5

#17 #18 #19 #20

0.0061 0.0255 0.0279 0.0045

0.0066 0.0386 0.0178 0.0038

0.0027 0.0157 0.0126 0.0023

0.0056 0.0326 0.0789 0.0096

0.0144 0.0843 0.0499 0.0093

0.0202 0.1178 0.0477 0.0097

0.0131 0.0572 0.0551 0.0086

0.0106 0.0619 0.0506 0.0089

0.0793 0.4334 0.3405 0.0568

0.9100

uncertainty in the value of the SCF accounts for 28% of the total uncertainty in a fatigue life calculation. In this section, the methodology for deriving the local fatigue stress is studied in depth. Section 6.1 presents the hotspot stress analysis in engineering practice, followed by an SCF analysis procedure for ship structure details that is proposed by DNV [31]. This analysis procedure and its possible limitations are discussed in relation to the resulting uncertainty in fatigue life predictions. A new approach for local stress analysis is proposed in Section 6.2. The purpose of this approach is to avoid the limitations of traditional SCF analysis procedures and thereby enable more realistic fatigue analysis predictions. Section 6.3 presents examples of hotspot stress analyses and a convergence analysis of a LSF in the proposed approach. In Section 6.4, an example of fatigue analysis using LSFs is presented.

Fig. 10. Relative contribution of different significant wave heights to fatigue damage.

positions where local stresses are high. Local structure details in the deck, side-shell, and bottom are typical locations with high local stresses, and the design for persistence against metal fatigue is a challenge. An accurate computation of hotspot stresses via a detailed FEA in all locations of interest in a ship structure requires intensive computation resources and is not feasible in practice. Instead, it is more convenient to use a global FE model for the calculation of nominal stresses in the structure. Then, the nominal stress is multiplied with a SCF of specific detail to obtain the (local) hotspot stress. Consequently, the value of the SCF is critical for an accurate fatigue assessment because a small variation in the SCF leads to a large variation in the calculated fatigue life. Determining the SCF involves large uncertainties. According to Sharp [30],

6.1. Hotspot stress analysis: one definition of SCF According to Pilkey [32], a stress concentration factor is defined as the ratio between the maximum local stress, rmax, and the nominal stress, rnom, as K = rmax/rnom. A maximum local stress can be derived analytically, numerically, or experimentally. The hotspot stress methodology is often used in ship fatigue analysis, and thus, the maximum local stress is often represented by the hotspot stress that is computed numerically using FE analysis. This is consistent with the S–N curve used. Numerous design codes, handbooks, and other literature on the subject provide SCFs in the form of graphs, look-up tables, and formulae for structures with typical geometrical configurations, such as holes, notches, and shoulder fillets. The comprehensive book by Pilkey [32] presents the SCFs for the most commonly used structural details. However, few of these structural details are ship-like details. Tabulated SCF values for ship-like details are provided by

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classification societies. These SCFs are normally based on simplistic loading conditions such as uniaxial loading conditions. However, structure details in ship structures are seldom loaded in one direction only, and thus, the definition and usage of SCF must be based on a more appropriate stress measure as a result of combined loadings. The resultant local maximum stress in a structural member is a function of the type, amplitude, and phase of all loads acting on it. The loading situation can be complex because the various loads acting on a specific structural member can be in or out of phase and proportional or non-proportional to each other. For a ship side-shell structural detail, the following categories of loads should be accounted for in the calculation of the SCF, considering how they relate to each other and to the change in time:  Hull girder wave loads such as the global vertical bending moment, horizontal bending moment, torsion moment, axial load, and shear force.  Secondary girder loads, i.e., double-hull bending from sea pressure or cargo loads.  Local plating loads, i.e., water pressure from ballast water and sea pressure. 6.1.1. Hotspot stress analysis – a formula based on engineering practice One of the most commonly used methods for calculating the hotspot stress in a structure has evolved as a result of good engineering design philosophy and experience from practice. The hotspot stress is calculated by the superposition of hotspot stresses from axial and bending loading conditions. Look-up tables for the most common geometrical configurations exist where contributing factors for the axial load, Kaxial, and the bending load, Kbending, for a structure detail under consideration can be obtained. The hotspot stress, rhs, or the local maximum stress, is then derived by summation of the maximum axial stress and maximum bending stress, as denoted in Eq. (6). These stresses are often calculated from design rules, i.e., there is often no need for detailed numerical models, such as a FE model. This methodology is used by, among others, DNV [8] to obtain the local hotspot stress for a fatigue analysis of various geometries in ship side-shell structures.

rmax ¼ rmaxaxial þ rmaxbending ¼ K axial  rnomaxial þ K bending  rnombending

ð6Þ

An actual loading situation of a ship side-shell structure is not only a combination of bending and axial loads. There may also be other critical details that are more affected by shear stress, such as slots, lugs, and frame openings. Despite its simplicity and practical use in simplistic engineering estimations, the local hotspot stress may be defined and calculated using a more accurate loading acting on ship side-shell structures. Thus, the definition of the hotspot stress in Eq. (6) will not be investigated further in the current study. Instead, two other approaches are compared (see Sections 6.1.2 and 6.2). 6.1.2. Hotspot stress analysis – calculation of SCF according to DNV Calculation of the local stress response in a structural detail, as a result of complex loading, dictates the use of FE analysis. The objective here is to define and calculate the hotspot stress so that a SCF can be obtained. In DNV [31], a procedure is proposed for SCF derivation using two FE models with different levels of mesh resolutions, where empirical loads are applied on both FE models. The nominal stress is estimated by the node stresses from a seminominal-mesh FE model with a typical element size of 50  50 mm, whereas the hotspot stress is extracted from a fine-mesh FE model, where the element size is t  t, with t being

the plate thickness of the structure detail. Fig. 11 presents an example of these two FE models for a part of the outer side-shell structure of the case study container vessel. It must be pointed out that both FE models are invented by the authors based on the scantling. The geometrical configuration of the actual structural details in the case study vessel differs from those of the FE models. The purpose of using these FE models is mainly to demonstrate the methodology for deriving local stresses for fatigue analysis. Three empirical load cases (LCs) representing the combined loads of pressure, shear and axial loads are used in the DNV procedure. For all load cases, each individual load case is assumed to introduce a stress of certain amplitude at locations away from the hotspots. The maximum membrane principal stress is extracted from the semi-nominal model, and the surface principal stress (upper or lower element surface) is extracted from the fine-mesh model. The SCF is defined as the ratio between these two stresses according to Eq. (7). Two important issues should be noted. First, the sum of the absolute principal stresses from pressure, shear, and axial loads is to be used for both models, and second, according to the procedure, the local stress values should be extracted at a distance t/2 from the true hotspot location. To compensate for this, a correction factor K = 1.12 should be used (see DNV [31] for details).

PLC3 LC1 jrtt j SCF ¼ PLC3 K LC1 jr5050 j

ð7Þ

Compared to the hotspot stress formulation in Eq. (6), the current approach more accurately considers the actual structure and loading configuration, i.e., the error originating from simplification in geometry remains at a low level and more realistic loads are applied. However, the computational effort is greater with respect to modelling, analysis time, and the processing of results. The definition of the SCF in Eq. (7) can be developed further due to its restriction in terms of the empirical loads used in the stress response analysis. 6.2. Local stress response and fatigue analysis using loads from hydrodynamic analysis It is proposed in the current study that a time-domain-based calculation procedure should be used to calculate the local stress response. To allow for a reliable and realistic fatigue analysis prediction, the time-domain-based calculation should be carried out using the local stress records obtained from the direct calculation of local hydrodynamic loads followed by local FE analyses (see Fig. 3 for the procedure). The nonlinear sea pressure load, which has not been fully accounted for in the global numerical analyses (hydrodynamic and FE analyses), is then considered in the resultant values of the LSF. This approach is motivated by the fact that local stress responses are a result of a combination of all loads acting on the structure and their variations with time. The hotspot stress is extracted from (principal) stress records using the fine-mesh (t  t) FE model. The nominal (longitudinal principal) stress is obtained from the full-ship global FE model (see Fig. 4). The advantage of this method is that more realistic and representative stresses for true loading conditions are used in the analysis, represented by local and global records that are obtained from the hotspot and global FE models, respectively. The idea behind this approach is that local fatigue assessment can be related to fatigue damages on the local and global levels and be based on realistic loadings. The stress history should be used instead of simplified stress magnitudes from empirical or single load cases. An equivalent stress range is then defined to represent all of the stress cycles within a certain time period of the

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287

Fig. 11. FE models of a part of the outer side-shell structure: (a) semi-nominal mesh: 50  50-mm mesh and (b) fine mesh: t  t mesh, where t is the plate thickness.

practicable and useful for engineers. Finally, the LSF is based on fatigue damage analysis, which is closely related to the rainflow cycle counting technique, making the proposed methodology particularly useful for the time-domain procedure.

(a)

6.3. Computation of SCF and LSF values in four hotspots in an actual structural detail

Hotspot 1a

Hotspot 2a

(b) Hotspot 1b Hotspot 2b

Fig. 12. Four hotspots where the LSFs and SCFs are compared: (a) outer and (b) inner side-shell.

stress history (see Eq. (8)). Here, Dreq is the equivalent stress range, m is a material constant that is typically set at put to 3 for steel, Drk denotes individual stress cycles identified by the rainflow counting method, and Nk is the total number of stress cycles for stress range k. In Eq. (9), a LSF is introduced that is used in following sections for, among others, convergence analyses related to time length of the stress record and selection of time steps in the numerical analyses (subscripts hs = hotspot and nom = nominal).

Dreq ¼

X

m

ðDrk Þ =Nk

!1=m 

ð8Þ

k

LSF ¼ Dreqhs =Dreqnom

ð9Þ

Several advantages are obtained with the proposed concept compared to the previous approaches. More realistic and representative stresses for true local loading conditions are obtained that can be used in the fatigue life predictions. The nominal stress is obtained from the global FE analysis, which makes this approach

The SCF and LSF values were computed using a structural detail in the case study container vessel (see Fig. 12 for the geometry). In the figure, two FE models are presented having the same geometry. The difference between the models is their location on the outer and inner side-shell structures, respectively. Thus, only the former is subjected to local pressure from wave loads. Both models are subjected to the same global wave-induced loads that give rise to, for example, horizontal bending, vertical bending, and torsion loading. The FE meshes are comprised of eight-node shell elements, and the element resolutions and design of the FE mesh follow the recommendations of DNV [31]. The four hotspots at which the stress is calculated are shown in Fig. 12. Hotspots 1a and 1b are located where the stiffener is welded to the web, whereas Hotspots 2a and 2b are located in the notch where the stress concentration is high and cracks are likely to occur. Welds are not modelled in the current FE models and, thus, only the influence on local stress from the geometry can be considered in Hotspots 1a and 1b. To account for the additional stresses from the weld stress raiser effect, a hotspot stress S–N curve in DNV [8] is used in the fatigue damage calculations. Fig. 13 presents an example for demonstration using a longitudinal stiffener along the outer side-shell of the case study container vessel. The Figure presents a 1200-s stress history signal of the calculated nominal stress as well as the hotspot stress. The nominal stress is extracted from the stiffener in the global FE model. And the hotspot stress is derived from the hotspot FE model, which is defined as the maximum principal stress at a point on a structural detail where the stiffener is connected to the transverse web (see Hotspot 1a in Fig. 12a). Furthermore, the nominal global stress suggests that the stress is compressive because the side-shell structures are located below the neutral axis. However, the local response indicates that the principal stress is high and tensile. Hence, fatigue assessment must be carried out for the hotspots. Table 7 lists the correlation between the hotspot stresses at the selected locations and their corresponding nominal stresses under the sea states listed in Table 2. The correlation between the hotspot stresses and nominal stresses is strong in general. Stronger correlations are found for the inner shell, represented by Hotspots 1b and 2b, compared to the outer shell because the local sea pressure contributes to the hotspot stresses in the outer shell, whereas the hotspot stresses in the inner shell originate mainly from global girder loadings. The different approaches referred to as the LSF and SCF are now discussed and compared. In Section 6.3.1, the influence of

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in a fatigue assessment. However, the LSF and SCF values cannot be compared in magnitude because their definitions are very different. A comparison between these values can only be conducted through fatigue analysis. The LSFs in Hotspots 2a and 2b are less than one, which means that the local stress ranges in these locations are less than the nominal stresses derived from the global FE analyses and are thus not fatigue critical. However, it is unreasonable to assume that structural configurations similar to Hotspots 2a and 2b in other locations of the ship hull are all not fatigue critical because, as discussed earlier, a LSF is affected by not only the geometry but also the loadings.

Fig. 13. Example of calculated stress histories in a longitudinal stiffener in the outer side-shell: nominal stress and hotspot stress (see the text for details).

Table 7 Correlation between the hotspot stresses and corresponding nominal stresses. Sea state

Hotspot 1a

Hotspot 2a

Hotspot 1b

Hotspot 2b

1 2 3

0.870 0.904 0.855

0.867 0.900 0.843

0.997 0.998 0.997

0.982 0.980 0.982

geometry and loads is discussed. Convergence analysis of the LSF with respect to length of the stress record is presented and discussed in Section 6.3.2, followed by a discussion of the influence of the sea state conditions in Section 6.3.3. 6.3.1. Influence of geometry and loads The SCF presented in Section 6.1.2 and the LSF presented in Section 6.2 are calculated for the four hotspots shown in Fig. 12. For the SCF, the reference stress was extracted from the semi-nominal FE model, as illustrated in Fig. 11. For the LSF, a 1200-s stress history was simulated from a winter voyage of the case study container vessel). This voyage is referred to as sea state 2 in Section 5 of the fatigue analysis. The results of the SCF calculations are presented in Table 8. The SCF is calculated based on empirical loads, i.e., the stress history signal has not been used. Detailed calculation procedure is referred to DNV [31]. The SCF does not distinguish between the difference in loads acting on the outer and inner side-shell structures. Moreover, Table 8 also clarifies how the hotspot stress should be calculated, i.e., which FE model should be used to obtain the nominal global stress. In Table 8, the largest LSF value occurs in Hotspot 1a for the current sea state conditions and loading scenario. The LSFs are generally larger for the outer side-shell compared with the inner side-shell, implying that the applied sea pressure contributes to the hotspot stress. This observation highlights the importance of considering the sea pressure load acting on the side-shell structure

6.3.2. Influence of time record length – a convergence analysis A numerical analysis in the time domain is inevitably influenced by the length of the time record. Considering the random nature of wave loads, the stress responses are expected to be random as well. The numerical results are expected to be more accurate for a longer time record. However, the simulation of a long time record requires excessive computational effort. A convergence analysis with respect to the value of LSF as a function of time record length was carried out. The objective was to ensure that the LSF values presented in the previous section had been calculated using a sufficiently long time record of the stress history signal. The LSFs were calculated using stress history lengths from 300 s to 1200 s with a 300-s interval. The wave environment and operational conditions were the same as those presented in Section 6.3.1. Fig. 14 presents the results from a convergence analysis in Hotspot 1a. The results suggest that a time record length of at least 600 s should be used. 6.3.3. Influence of sea state conditions The influence of the wave-load amplitude was studied by altering the severity of encountered sea states while keeping the wave heading constant at 135°. Representative sea states (#1, 6, 10, 14, and 19) and ship speeds in Table 4 were selected, representing five significant wave heights for comparison: 1.5, 3.5, 5.5, 7.5, and 9.5 m. Because the results presented in Section 6.3.1 exhibit the largest local stress response in Hotspot 1a, the comparison focuses on this location. Fig. 15 presents the results for the LSF; SCF is not presented because according to its definition, its value is independent of the load amplitude. The results in Fig. 15 demonstrate that the calculated LSF values vary with wave loads. There is not a linear relationship between LSF and Hs. However, LSF tend to increase with increasing Hs. In reality, wave loads that act on a ship side-shell structure are random. These loads vary due to variation in ship speed, significant wave height, wave period, and wave encounter direction. Consequently, the LSF values may vary, as well. The significant wave height is considered most influential in the LSF calculations. However, the influence of other factors may also be significant and deserve further investigation. 6.4. Example of a fatigue life prediction using a LSF Hotspot 1a is the most fatigue-critical location of the four hotspots in Fig. 12. A comparison of the time-domain-based long-term

Table 8 A summary of the LSF and SCF values in the four hotspot locations. Outer side-shell

Inner side-shell

Factor

Hotspot 1a

Hotspot 2a

Hotspot 1b

Hotspot 2b

FE model used to calculate nominal stress

SCF (–) LSF ()

2.27 3.26

1.05 0.97

2.27 1.65

1.05 0.87

Semi-nominal FE model, 50  50 mesh; see Fig. 5 Global FE model; see Fig. 4

Z. Li et al. / International Journal of Fatigue 55 (2013) 276–290

Fig. 14. LSF versus time record length.

Fig. 15. Values of the LSF in Hotspot 1a for sea states with different Hs.

fatigue life prediction in this hotspot was carried out using a nominal longitudinal stress history calculated using the global FE model (see Section 5.2 for the definition of sea states). The local stress response needed in the fatigue analysis was calculated using (see Fig. 3): (i) a constant value of the empirical SCF = 2, which is typical value often used in this type of analysis (see, e.g., Storhaug and Moe [17]); and (ii) the LSFs presented in Fig. 15 were matched with the corresponding sea states. The SCF-based analysis resulted in a fatigue life of 6.7 years (see Section 5.2), whereas the LSF-based analysis resulted in a fatigue life of 1.67 years. Both fatigue lives, and particularly the latter, are quite short. However, as mentioned in Section 6.1.2, the hotspot FE models used here are not identical as the actual structural details in the case study vessel. Consequently, the actual LSF is most likely of another value and the resultant fatigue life accordingly differs than the above calculated ones. 7. Conclusions A direct calculation procedure for the fatigue assessment of ship side-shell structures was presented that is characterised by nonlinear time-domain hydrodynamic simulations followed by linear FE analyses. A Panamax container vessel sailing on the North Atlantic trade between Europe and Canada was used as a case study vessel.

289

A comparison between short-term fatigue analyses from full-scale measurements and numerical calculations was presented and discussed. The conclusions from the study can be summarised as follows. The proposed time-domain-based procedure accounts for the nonlinearity in wave loads and ship speed reduction as well as the variation in occurrence probability of encountered waves. Based on the investigations and the results presented in the current study, the proposed procedure is considered useful and appropriate for fatigue assessments of ship side-shell structures. Further work should include fatigue analysis using the proposed procedure in more locations in the hull. A comparison between linear and nonlinear WASIM simulations was presented. The results indicated that there are slight variations between the simulations. Furthermore, the influence of the time step length in the linear FE analysis on the calculation results is rather insignificant. Sensitivity analysis of the influence on stress history (time record) length indicated that a 20-min time record is sufficient for a single short-term fatigue damage calculation using the proposed procedure for wave induced loads. Long-term fatigue analysis was carried out in a fatigue-critical hotspot location in the outer shell of the ship side-shell structure. The results from the spectral method and proposed time-domain procedure were compared. The former method resulted in a fatigue life of 7.1 years, whereas the latter resulted in a fatigue life of 6.7 years for the studied case. The time-domain procedure considers more factors in the simulations, such as the nonlinear effects in loads, speed reduction and variation in the encountered wave heading. Thus, the time-domain-based procedure is considered more realistic than the commonly used spectral method. The computation of local stresses in ship side-shell structural details is a challenging task. Accordingly, a realistic stress concentration relationship between the global stress and local stress under complex loading plays a vital role in the process of fatigue analysis. In this investigation, a definition of a factor for local stress used in fatigue damage evaluation, the LSF, was proposed and methods to derive local stresses using the LSF and conventional SCF were discussed. Essential issues when using the LSF were investigated for ship side-shell structural details. The proposed LSF is believed to provide a more realistic description of the fatigue stress.

Acknowledgments The authors wish to acknowledge the support from the Swedish Governmental Agency for Innovation Systems (VINNOVA) and the Lighthouse Maritime Competence Centre (www.lighthouse.nu) in Sweden. The authors would like to thank Wengang Mao, Ph.D., for providing the hindcast data and for fruitful discussions.

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