FATIGUE STRENGTH ASSESSMENT OF SHIP STRUCTURES. Y. Garbatov 1

Transcription

1 XIV Portuguese Conference on Fracture (214) FATIGUE STRENGTH ASSESSMENT OF SHIP STRUCTURES Y. Garbatov 1 Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal ABSTRACT Fatigue strength assessment of complex double hull oil tanker structures based on different local structural finite element approaches is performed here. Wave-induced vertical and horizontal bending moments and pressure are considered in the analysis. Stress analyses are performed based on the local hotspot and notch stress approaches. A linear elastic finite element analysis is used to determine the stress distribution around the welded details and to estimate structural stresses of all critical locations. Fatigue damage is estimated by employing the Palmgren-Miner approach. The time-dependent hotspot stresses affected by corrosion deterioration are defined by making use of a nonlinear regression analysis. The statistical properties of fatigue damage and service life of hotspots are estimated taking into account random nonuniform time-dependent corrosion wastage. Fatigue reliability, during the service life, is modelled as a series system of correlated components. KEYWORDS: Fatigue, ship, case studies, structural stress approach, notch spot. 1. INTRODUCTION The process of ship structural design goes from the primary structure (mid-ship section) to the detailed design of substructures and components such as plates and welded joints. The design of primary load-carrying structures is mainly governed by fatigue and ultimate strength. Fatigue analysis and ultimate collapse are based on different principles. Nowadays, there is a trend to consider explicitly the effect of degradations such as corrosion and fatigue. Fatigue damage is a strength degradation phenomenon that can also increase with corrosion degradation. Fatigue is one of the most complicated problems in engineering, especially for the structural components subjected to stochastic loading. Both the environmental loads and the corresponding stresses in a structural component vary with time and can be modelled as stochastic processes. To design a marine structure with respect to fatigue damage, the hot spot stress approach is one of the most practical methods and it is usually combined with detailed finite element analysis [1-3]. It has to be pointed out that the calculated local stress around the structural singularities depends very much on the structural idealization, the element types used and the mesh density. Some application of this approach can be found in [4-6]. Engineering systems such as ship structures are designed to ensure an economical operation throughout the anticipated service life in compliance with given requirements and acceptance criteria. Deterioration processes such as fatigue crack growth and corrosion degradation are always present to some degree and depending on the adapted design in terms of degradation, allowance and protective measures; the deterioration processes may reduce the performance of the system beyond what is acceptable. In order to ensure that the given acceptance criteria are fulfilled throughout the service life of the engineering systems it is necessary to control the development of deterioration and, if required, to apply corrective maintenance measures. Normally, both fatigue and corrosion will be present and their combined effect needs to be considered in that the decreased net section due to corrosion will increase the stress levels, which in turn increase the rate of crack growth. This effect has been considered in [7], showing that, depending on the repair policy adopted, one of the two phenomena would be the dominating one. The reliability of the ship hull under the effect of both fatigue and corrosion, where the local loading of the side shell is modelled, was considered for the case of tankers in [8]. Failure is considered possible in components subjected to the combined loading as well as the global hull section. The previous approach was also applied to the ship hull structures accounting for a cost optimization in [9]. Ships are subjected to periodic major inspections and repair action and the assessment of the reliability as a function of time needs to consider this reality [1]. 1 Corresponding author address: yordan.garbatov@ist.utl.pt 3

2 XIV Portuguese Conference on Fracture (214) The present work deals with fatigue analysis of complex double hull oil tanker welded structures. It begins with an abstract followed by an introduction. Section 2 reviews structural stress approaches. Section 3 presents fatigue loads. Section 4 and 5 and 6 deal with the case studies. Section 7 presents the results of the fatigue damage assessment and Section 8 analyses reliability. Section 9 presents a discussion and some conclusions. the region to reflect the stress gradient near the weld toe. Hexahedral elements with element size 1x1x1 mm are used in this approach. 2. STRUCTURAL STRESS APPROACHES For ship structures, the fatigue analysis is usually developed accordingly to procedures with a predefined long term stress range distribution and assuming a design life corresponding to the service life, typically for 25 years and the fail-safe principle is applied. The fatigue damage assessment, based on the S-N curves, employing hot spot and notch stress approaches, are generally accepted and the Palmgren-Miner linear damage summation is applied. Differences remain on the definition of wave loads, the S-N curves and the calculation of hotspot and notch spot stresses [11]. The simplified procedures of the Classification Societies [12] are sufficient for a fatigue screening of the details of a ship structure for pointing out the potentially fatigue critical ones but they cannot be realistically applied to the design of new types of structures (catamarans, fast ships etc.). For a realistic fatigue life assessment these procedures need to be further calibrated for the different types of ships and the database of ship structural details should be completed and integrated with the results of both tests and theoretical investigations [13]. Two approaches are applied to analyse the stress concentrations and to evaluate the fatigue life here, including the structural hot spot stress and effective notch stress approach. The hot spot stresses are based on the surface stress extrapolation by using the reference points at.5d and 1.5d from the weld toe, as is proposed in the guidelines [14] (Figure 1), where d is the plate thickness. The models are analysed with three-dimensional coarser mesh. There is only one layer of hexahedral elements throughout the thickness of plates with the element size d xd xd. Poutiainen [15] recommends that the analyst must take care with respect to the stress component that is being used. Stress components may be selected as perpendicular to the weld toe or in the principal stress direction. In this analysis, the principal stress component (σ 1 ) is chosen. The Xiao-Yamada approach [16] is based on the stress found at 1 mm below the surface in the direction corresponding to the expected crack path. The finer mesh with element size of 1 mm or less must be used in Figure 1 Hot spot stress approach The effective notch stress approach is using the fictitious effective notches rounded reference radius r ref = 1 mm, for assessing both the welded toes and welded roots (Figure 2). In this analysis, the maximum principal stresses are used at the middle cross-section of the weld toe path and weld root path. Figure 2 Effective notch stress approach radius 1 mm To define the effective notch stress, the element sizes are recommended to be no more than 1/6 of the radius, in the case of linear elements, and 1/4 of the radius in the case of the higher order elements. These sizes have to be observed in the curved parts as well as in the beginning of the straight part of the notch surfaces in both directions, tangential and normal to the surface, respectively. 3. WAVE INDUCED LOADING The ocean surface can be represented as a superposition of a large number of regular waves having different heights, lengths, direction and random phase differences. Such consideration allows the ocean surface to be described mathematically and also allows the use of statistical methods to predict the loads in a ship's life. Assuming stationarity over a short period of time 1~3 hours, the sea elevation can be described as a stationary, relatively narrow-banded, Gaussian random process, where the distribution of wave energy over different frequencies is expressed by a wave spectrum. It is common practice to assume that the sea states are 4

3 North Atlantic Scatter Diagram S(w),[ m*m*s] XIV Portuguese Conference on Fracture (214) described by a single peaked spectrum, which is well modelled by ISSC parametric of the Pierson and Moskowitz [17] form. A more precise formulation would use the Pierson Moskowitz spectrum only for developed sea states and to adopt the JONSWAP [18] spectrum for developing seas and also a double peaked spectrum for mixed seas [19] as can be seen in Figure w, [rad/s] Pirson-Moskow itz JONSWAP Tw o-peak Figure 3 Pirson-Moskowitz, JONSWAP and Two-peak spectra, H = 6 m s The general practice in establishing the ship design for wave-induced loading has been to adopt the North Atlantic [2] wave climate (see Figure 4) as the reference situation based on the argument that it has the most extreme sea states of all ocean going areas [21]. From the worldwide mission profile of the ship, the relative time period within each Mardsen zone [22] is estimated and the frequency of occurrence of different sea conditions is found as the weighted average of the available wave statistics in the different zones Tz, sec Figure 4: The North Atlantic wave scatter diagram Hs, m If the ship response to wave excitation is linear, the total response in a seaway is described by a superposition of the responses to all regular wave components that constitute the irregular sea, which can be performed in a frequency domain analysis. Given the linearity, the response is described by a stationary and ergodic but not necessarily the narrow-banded Gaussian process. Linear strip theory is the established method of predicting wave-induced load effects despite the emerging availability of computer codes based on the discretization of the hull on panels and on the application of three dimensional diffraction theory [23]. However, since for fatigue it is the stress range that is important, wave induced loads can still be calculated by linear theories in that the stress range shows a very small degree of non-linearity. The transfer function modelling is the response due to a sinusoidal wave with unit amplitude of different frequencies is usually obtained from calculations based on the theory of ship motion in potential flow with linearized free surface conditions. The transfer function is however valid for a specified ship velocity, wave heading angle and loading condition. The loading conditions are typically represented by two discrete cases of full load and ballast load, while a more detailed discretization of the parameters speed and heading angles is required. In the evaluation of the dynamic stress levels of the structural joint of consideration, the total dynamic stress components need to be considered. The stress component included in the present analysis is as a result of wave-induced vertical and horizontal hull girder bending stresses and the local pressure stresses. An adequate approximation for the long-term distribution of wave induced stress range can be described by the Weibull distribution [21]. 4. CASE STUDY 1 Fatigue cracks may initiate at hotspot points of the weld toe and grow through the thickness of the plate such as the connection between the stiffener attached to a transverse web frame (web-stiffener) and the flange of a longitudinal stiffener at the side shell. Typical connections of longitudinal to transverse structural elements at double side of an oil tanker are considered (see Figure 5). In the majority of cases, there are two major hotspot points, one is at the toe of the web-stiffener (HSA), and the second is at the flange of the stiffener close to the cut-out (HSB), as can be seen from Figure 6. To define the local stress distribution at the intersection between the longitudinal stiffeners and the transverse web frame, a finite element analysis based on global and local structural finite element models employing the sub model techniques is performed here. The hotspot and an effective notch stress, approaches are applied to investigating the fatigue damage of the welded joints at 5

4 XIV Portuguese Conference on Fracture (214) the two hotspot positions. Shell finite elements with 4 and 8 nodes are used in the finite element analysis of the cargo - tank and local models respectively. The solid finite element with 2 nodes is applied in solid sub models for the notch stress approaches by employing the commercial software ANSYS [24]. as the combination of them. Several levels of finite element analyses are used by employing the sub modelling techniques to achieve a better accuracy. All the finite element models were created based on the same thickness scantlings and material properties. The modulus of elasticity and the Poisson ratio are 26 GPa and.3 respectively. Sub-modelling techniques are used here as a refined analysis of the sub region, which contains singularities or geometry discontinuities, based on global analysis results. For the local region one can create a sub-model with fine mesh while keeping the global model with a coarse mesh. In this way, the finite element analysis may be controlled step by step, ensuring maximum accuracy and less computing time. The global finite element model represents the global stress distribution of the primary members in the hull. The cargo tank model is generated to cover ½ +1+ ½ cargo tank length in the amid-ship region, which is sufficient to estimate the behaviour of the oil tanker structure (see Figure 7 and Figure 8). Figure 5 - Midship section of the oil tanker. L The simplified fatigue load cases for the oil tanker are expected to be 42.5% of its lifetime in each of the full and ballast load condition. The remaining time of 15% is assumed to be spent in harbour water. The fatigue loads are divided into two levels: global hull girder and local dynamic pressure loads. The global loads are composed of vertical and horizontal waveinduced bending moment. The local dynamic pressure loads are to be considered as external and internal pressures acting on the hull and tank boundaries of the ship in the two loaded conditions, respectively. web frame flat bar HSB shell plating HSA Figure 6 - Hotspot A and B locations flange plate The load cases at 1-4 probability level of the long-term Weibull distribution is applied to the finite element global and local models for the estimation of the fatigue stress response analysis of the hull ship structure. For each loading condition, the global ( g ) and local ( l ) dynamic stress range components are considered as well Figure 7 - Global finite element model. In order to estimate stresses in an accurate manner for the fatigue assessment, the longitudinal stiffeners are included in this model (see Figure 9). Four-nod shell elements (SHELL63) are used for modelling the structural details in the global model with a mesh size of 9 mm, smaller than spacing between longitudinal stiffeners. The global model is analysed using the load cases and boundary conditions as given in [25]. The full breadth cargo hold model is subjected to lateral pressures and it is connected by vertical springs using the COMBIN14 element in the vertical direction at the intersection of the transverse bulkheads with both side shells and the longitudinal bulkheads. 6

5 XIV Portuguese Conference on Fracture (214) The local web frame sub model is modelled at the same location with respect to the global origin co-ordinate at the mid-ship region of the oil tanker. The cut-boundary conditions interpolation is performed on the model. The high order 8-nodes finite elements (SHELL93) are used for the local finite element model with an element size of 25 mm creating a finer mesh than the one of the global model. As shown in Figure 9 and Figure 1, three levels of solid sub-models were necessary for transferring nodal displacements as boundary conditions from the solid model to local sub-models. The solid finite elements can provide a better stress distribution because they may model precisely the shapes of the weld connections. Solid model 1 st sub-model HS2 HS1 Displacements interpolated from shell-to-solid model Figure 9 - Solid FE model and 1 st sub-model. 2 nd sub-model 1/ elements 3 rd sub-model 1/ elements Figure 8 Intermidiate finite element model (shell plating is not shown). Figure 9 shows the local shell model for the finite element sub-model analysis on the critical position on the side shell. It is extended to 2 transverse web frame spacing and 4 longitudinal stiffener spacing. The stiffeners are attached to the web frame and longitudinal stiffeners by the length equals to spacing between longitudinal stiffeners on the side and the inner side shell of the double hull. In addition, the shell model included cut-out at the intersections in the web - frame. The load cases and cut-boundary conditions are applied in the same way. The SHELL93 finite elements are used for analysing with an element size of 1 mm. Finite element solid sub-models were created in order to calculate the stress distribution and deformations in a more accurate manner for fatigue damage assessment of the structural details. Finite element analysis accuracy of welded joints based on the effective notch stress approach depends very much on the element types and mesh density [26, 27]. Shell-to-solid sub-modelling techniques were used for achieving accuracy in the computation of the local stress at the critical hotspots in front weld toes of flat bar attached to flange plate (see Figure 9). r = 1 mm 45 o gap of.1 mm Figure 1-2 nd and 3 rd sub-models for hotspots 1 and 2. Three dimensional solid finite elements (SOLID95) are used for the finite element solid models. The SOLID95 finite element is a higher order element with 2 nodes, three degrees of freedom per node and translation in x, y and z directions. This type of finite element may tolerate irregular shapes without loss of accuracy. It is used to comply with the recommendation presented in [14, 28]. The SOLID95 finite element has been also applied as for examples in [26, 29, 3] for studying welded structural details. For this study, the sub-models are analysed by using the tetrahedral-shaped solid elements. The fictitious effective notch radius of 1 mm is included around the welded toes as recommended in [14, 28]. Relatively fine meshes are set up with an element size of.2 mm along the circumference of effective notch in the third sub-models. 7

6 XIV Portuguese Conference on Fracture (214) 5. CASE STUDY 2 The web-frame welded structure, analysed here, is composed of a ship side shell plate, a longitudinal stiffener, five equally spaced transverse frames with stiffener and welded lugs. The structural geometry is defined as 12,8 x 8 x 16 mm, five frames of 13 x 8 x 12 mm, which is welded on the ship side sell about longitudinal direction, the lugs are about 18 x 149 x12 mm, which are attached to the longitudinal stiffener and the frame. The longitudinal stiffener, which is made of HP 32 x 14, is 12,8 mm long. The stiffeners on the frames are made of HP 26 x 12 (see Figure 11). concentration caused by welding defects are not taken into account. Figure 12 shows some high stress concentration areas. Figure 12 - Hotspots in welded joints Figure 11 - Web-frame structural section The stresses and their distributions are influenced by the mesh density and element properties, which requires following the guidelines in [31] in predefining of element type, mesh size as well as the stress evaluation at the extrapolation points. The complexity of the structural geometry and the existence of many hot spots, defined by welding toes, require a three dimensional finite element model. The finite elements used in the present model are 2-node quadratic solid finite elements, SOLID95. A 3-D detailed finite element model of the middle part (three web-frame joints) of the entire welded structure with five web-frame joints is defined accounting for the periodic symmetry of the web-frame structure. The finite element mesh is generated in a such way that a relatively coarse one is used in the zones away from the welded joints and a fine mesh of element size doxd o is generated around the welded joints studied, where represents the thickness of the local plate. The finite element analysis performed here, applies the linear static analysis capability of Ansys [32]. The aim of the finite element analysis is to determine the structural stress distributions around the welded joints and to locate the hot spots in the structure. Based on the results of structural analysis, the structural hotspot stresses are defined. The weld joint is assumed to be polished and a fully penetrated by butt welds. The stress d o The hotspot is a weld toe position in a welded structure where a fatigue crack may initiate due to the combined effect of structural stress fluctuation and weld geometry. The aim of the hotspot stress analysis is to evaluate the principal stress distribution around the welded structural details. The structural stress at a hotspot is defined as HS. The structural stress includes all stress raising effects excluding the stress due to the local weld profile itself. The stresses are determined on the structural surface at the point of the vicinity of the welded joints as may be seen from Figure 13. For the fatigue assessment, the hotspot stress, HS is determined at the hot spot of the welded joint, where the maximum principal stresses are analysed. The hotspot stress, HS is the maximum principal stress at the weld toe calculated at reference points and extrapolated at the hotspot in consideration [11, 33]. The hotspot stress extrapolation procedures may be based on a surface stress extrapolation, with the assumption that the hotspot stress varies linearly through the plate thickness and the effect of the weld notch is localized within a distance close to the weld, where the locations of the reference points are expressed as a function of plate thickness. First, the stresses at the reference points are determined and secondly, the hotspot stresses, HS are calculated using a linear extrapolation equation [34] HS 1.5.5d.5 1.5d, where.5d is the principal stress of reference point, which is located at a distance.5d away from the weld toe (hotspot) and 1.5d is the principal stress at a distance of 1.5d. The positions of all hotspots, analysed in the web-frame joint, are marked as black dots and shown in Figure 13. 8

7 XIV Portuguese Conference on Fracture (214) represent the details of all corrosion mechanisms that may be developed with the main trend by fitting the field data. The large number of parameters that can affect corrosion demonstrates the difficulty of developing a model of corrosion wastage that explicitly considers them. Therefore, the estimation of the corrosion depth of the present stage needs to have an empirical component and to be very much based on the historical data collected for a certain type of ship. Figure 13 - Hotspots of web-frame welded structures 6. CASE STUDY 3 Marine structures operate in complex environments. Water properties such as salinity, temperature, oxygen content, ph level and chemical composition may vary according to location and water depth. Time spent in ballast or cargo, tank washing and inerting, corrosion protection effectiveness and component location and orientation have a significative effect on the corrosion phenomena. Some types of corrosive attack on metals may be defined as general corrosion, galvanic cells, under-deposit corrosion, top-of-line corrosion, weld attack, erosion corrosion, corrosion fatigue, pitting corrosion, microbiological corrosion and stress corrosion cracking. Three fundamental approaches can be applied to corrosion deterioration modelling. The conventional approach is just to consider that corrosion grows linearly with time, but this is a very crude model. The second one can be based on the results of experiments in specific conditions, which suggest laws of the growth of corrosion as a function of specific parameters. The corrosion model can be developed by considering all those laws derived from experiments in specific conditions as is being pursued in [35]. This approach involves one difficulty in generalizing results from coupons at coastal corrosion stations to full-scale conditions. The other difficulty is related to the general lack of data on the environmental conditions, which affect corrosion in full-scale. The third approach is to consider that a model should provide the trend that is derived from the dominating mechanism and then it should be fit to the field data. The parameters of the dominant mechanism are not derived from experimental work but are fitted to full-scale data. The fitting to fullscale data compensates for the potential errors that the omission of less important corrosion mechanisms may cause. Although the model adopted in [36-38] does not Guedes Soares and Garbatov [39] developed a model for the nonlinear time-dependent function of general corrosion wastage. This time-dependent model separates corrosion degradation into several phases. In the first one, there is in fact no corrosion because the protection of the metal surface works properly. The second phase is initiated when the corrosion protection is damaged and corresponds really to the start of corrosion, which decreases the thickness of the plate. The third phase corresponds to slowly growing corrosion and the last one, corresponds to a stop in the corrosion process when the corrosion rate becomes zero. Figure 14 illustrates the time dependent model of corrosion degradation. dt O c O t B A Figure 14 Thickness of corrosion wastage, [39, 4] The model can be described by the solution of a differential equation of the corrosion wastage: d d t d t d (1) where d is the long-term thickness of the corrosion wastage, at time t, and dt is the thickness of the corrosion wastage dt is the corrosion rate. The solution of Eqn (1) can have the general form leading to: d C t 9

8 Standard Deviation of Corrosion Depth, mm Corrosion Depth, mm XIV Portuguese Conference on Fracture (214) d t t C t d 1 e, t c, t c (2) StDev d t aln t b (4) where a and b are regression parameters. where d is the long- corrosion wastage, dt is the corrosion wastage of time t, C is the time without corrosion which corresponds to the start of the failure of the corrosion protection coating (when there is one), and t is the transition time duration, which may be calculated as: Deck Plates - Ballast Tanks d =1.85 mm t = years c =1.541 years d t (3) tg Two sets of corrosion data, deck plates of ballast and cargo tanks of tankers described in [41-43] were analysed in [44]. The frequency scatter diagram of corrosion wastage of deck plates of ballast tanks is shown in Figure 15 and the mean value of corrosion depth of deck plates of ballast tanks in Figure Time, years Figure 16 Corrosion depth of deck plates of ballast tanks, mean value Figure 15 Frequency scatter diagram of corrosion wastage The parameters of the regression analysis of the corrosion depth, as a function of time, were determined under the assumption that the function given in Eqn (2) is fitted to the real measurement data. It should be noted that the long-term corrosion wastage for deck plates of ballast tanks is 1.85 mm, the time without d,ballast corrosion is 1.54 years and the transition period is years. Another important statistical descriptor of the data set is the standard deviation, which is given in Eqn (4), for each yearly subset of data. The standard deviation, which is modelled as a function of time, is fit to a logarithmic function (see Figure 17): Figure 17 Corrosion depth of deck plates of ballast tanks, standard deviation The corroded structure in finite the element models generated here are modelled by random thicknesses that results in the random vertical position of the coordinates of the corroded surface for equally spaced reference points positioned along the x and y direction of the structures, as shown in Figure 18. These reference points are defined by Monte Carlo simulations as being the node thickness of the finite element used for the finite element analysis later on. The corroded plate thickness, dt c ij, at any reference point with coordinates x, y for the corroded surface, is defined by the random thickness of the intact plate 1

9 XIV Portuguese Conference on Fracture (214) surface, d ij affected by the random reduction resulting from the corrosion depth, dt ij [45, 46] as: d c d d (5) where d are the matrixes of the corroded, intact and the corrosion depths. This convention is used to describe the thickness of the non-linear corroded plate resulting in randomly distributed thicknesses for randomly defined reference nodes at a specific time based on Eqn (5) and applying the corrosion degradation levels as defined by Eqn (2). The vertical random coordinates of the corroded and intact surfaces and corrosion depths are described as a lognormal distribution. The intact surface coordinates and corresponding corrosion depths are considered as not correlated. However, this assumption is not essential for the model just presented here and if the correlation between different corroded locations is known, it may be easily incorporated. defined as taking the same position with the surface nodes of the finite element model of the web frame structure. The random corrosion wastage is generated to model the original thickness by updating the vertical coordinate of each corrosion-dent in the finite element model. Therefore, 6 samples of finite element models of the corroded web frame structure with random non-uniform corrosion wastage were generated at different points of the service life, for the 15 th, 17 th, 2 th, 22 th, 25 th, 28 th and 3 th year, respectively. The web frame structure at any year of analysis includes 36,78 wastage data points/measurements? of corrosionpitting from the whole finite element model of the web frame, which leads to 21,646,8 pits modelled for each time studied. A total of 4,2 samples of finite element models was generated to cover the whole anticipated service life range including 151,527,6 pits.. Because of the presence of the random non-uniform corrosion wastage in the web frame, the structural surfaces are not flat and corrosion decreases the actual area of the plate cross section. The degradation will become more severe with time. The local finite element model of the web frame structure accounting for the plate thickness reduction caused by random distributed non-uniform corrosion wastage, an example of that is shown in Figure 19. Figure 18 Corroded plate at the 15th year The mean value and the standard deviation of the averaged corrosion depth are considered as the ones on the deck plate of ballast tanks of a tanker ship in the present study here. There is a large amount of uncertainties in the corrosion process, the occurrence of corrosion, their spatial distribution on a structure, and the time-dependent growth and interactions in-service are all random phenomena, which results in the random non-uniform thickness in a corroded structure. In order to realistically investigate the effect of random non-uniform corrosion wastage on the structural stress distribution and its scatter, the Monte Carlo simulation technique is employed for generating the samples of the random wastage at any random corroded point of the surfaces of the corroded structure. These corrosion dents are Figure 19 FEM of web frame at the 3 th year Based on the results of structural stress analysis, the structural hotspot stresses are defined. Some highlight stress concentration areas in a welded joint are shown in Figure 2. The non-uniform corrosion wastage is considered here to be a random variable and as a result of that stresses in the structure are also random. So the local structural stress of each hotspot should be considered as a random variable following a certain distribution. Based on the generated Monte Carlo samples of the finite element model, the structural stresses are calculated for each hotspot in a web frame at a different service time. The 11

10 Standard Deviation Hot Spot Stress, MPa XIV Portuguese Conference on Fracture (214) statistical properties of each stochastic hotspot stress, the mean value and standard deviation of stresses were then estimated. structural stress at the hotspot 4 is shown in Figure Observed Estimated Time, year Figure 2 Hotspots in welded joints Figure 21 Hotspot stress at No.4 vs. time, mean value Therefore, by using the nonlinear regression analysis, the structural stresses at any hotspot is fitted with a timedependent exponential function as: Observed Estimated initial, t C Mean t t C initial 1 exp t C T t, where (4) t is the structural stress of the hotspot in consideration at time t, accounting for the effect of random non-uniform corrosion wastage, which is the mean value of a hotspot stress at time t, initial is the initial structural stress of the hotspot, without accounting the effect of corrosion wastage, there is no corrosion when t, C, is the long term increase of the hotspot stress, Tt, is the transition time, C is the time without corrosion, which takes the same value as the one in the corrosion model adopted here. The regression curves for the mean value of the structural stress at the hotspot 4 is shown in Error! Reference source not found.. It is also assumed that the stress range of structural stress of a hotspot of time t is 2 t. The standard deviation of hotspot stress at any hotspot location, which is also a nonlinear function of time, is fitted by the following equation: t C StDev () t aln( t C) b t C where a and b are regression parameters. (5) The regression curve for the standard deviation of the Time, year Figure 22 Hotspot stress at No.4 vs. time, standard deviation 7. FATIGUE DAMAGE ASSESSMENT The fatigue damage of ship structural details during the service life of 25 years is calculated based on the Palmgren-Miner [47] approach. The design S-N curve is chosen according to [14, 48] as: log N log K mlog o (6) where N is predicted the number of cycles to failure for a given reference stress range, m is the negative inverse slope of the S-N curve, K is the constant depending on material property and log K is the intercept of log N -axis of the S-N curve. In this study, the effective notch stress approach is 12

11 Fatigue Damage Damage Damage XIV Portuguese Conference on Fracture (214) applied according to the IIW recommendation [14],where the single FAT225 S-N curve is used for the effective notch stress approach and for plate thickness 5 mm with the curve slope m = 3 and log K = and for the hotspot approach m = 3 and log K = of the S-N curve for welded joints in air/cathodic protected environment [25]. For marine structures, the probability density function of the long-term stress range is represented by the twoparameter Weibull distribution, and the fatigue damage ratio based on Palmgren-Miner approach may be calculated as [49]: m d m/ K Lnn vt m D 1 (7) where the long-term average response zero-crossing frequency is taken as =.1 Hz, T d is the ship design life, is the Gamma function, and is the Weibull stress range shape. The long-term stress range,, over the time may be calculated as [25]: f f (8) e m where f e =.8 is the environmental wave climate correction factor and the effect of mean stress correction factor fm.9 [25]. The shakedown effect is not considered here. 7.1 Fatigue damage - Case Study 1 Fatigue damage and fatigue life for the two hotspot points A and B are shown in Figure 23. The index 1 is for fatigue damage considering that the SCFs are taken from the list of standard details, SCF n, where n is the nominal stress range, [25], index 2 is for fatigue damage estimated based on the SCF defined using the hotspot stress approach with a coarse mesh and index 3 is for a fine mesh. The index 4 is for fatigue damage calculated based on the effective notch approach. It can be seen that the total fatigue damage estimated from the hotspot stress approach is smaller than the one obtained from the effective notch stress approach. It is also found from the effective notch stress approach that the fatigue damage is bigger at point B. It should be noted, as observed in Figure 23, that for the hotspot stress approach, the fatigue damage, which are estimated by the coarse and fine mesh in the finite element analysis, compared with the fatigue damage based on the standard list of details [25] are different. At the hotspot point A, the fatigue damage obtained from the coarse and fine mesh is smaller than the one considering that the SCFs are taken from the list of standard details of DnV [25]. Moreover, the results of the damage are nearly equal between the coarse and fine mesh. At the hotspot point B, the fatigue damage of the fine mesh is higher than the two remaining cases. It is also found that the fatigue damage at point B is smaller than at point A A1 B1 A2 B2 A3 B3 A4 B4 Hotspot Figure 23 - Fatigue damage (case study 1) 7.1 Fatigue damage - Case Study 2 The fatigue life of each individual hotspot of the case study 2 is shown in Figure 24. As may be seen, the maximum fatigue damage is observed at the hotspot 4, which equals to This hotspot contains a sharp corner where may start growing a fatigue crack when the structure is subjected to fatigue loads, thus reducing or eliminating the crack initiation stage of the fatigue life Hot-spot Figure 24 - Fatigue damage (case study 2) at the 3th year at the 25th year at the 2th year at the 15th year at the 1.54th year The principal ways to reduce the probability of crack initiation is by reducing the stress concentration factor of the weld, removing the crack-like defects at the weld toe or by removing the harmful tensile welding residual stresses or introducing compressive stresses. The first two methods involve altering the local geometry and are identified as weld geometry modification methods and the third one as residual stress methods. 13

12 Fatigue Damage XIV Portuguese Conference on Fracture (214) 7.1 Fatigue damage - Case Study 3 Because corrosion deterioration becomes more and more severe as a function of time, the structural stresses at each hotspot increases as a function of time. So the structural stress time history at the hotspots may be divided into several stress stages during the whole service life as a function of the progress of corrosion wastage. The comparison of fatigue damage accounting for corrosion behaviour between different hotspots at different service times is shown in Figure 25, in which the increment of cumulative fatigue damage is nonlinear and it is increasing with time. As may be seen, from Figure 25, the hotspots 3, 4, 5, 6, 9, 12, 15 and 16 are the most critical fatigue details that govern the fatigue life of the web frame welded structure. The maximum fatigue damage is observed at the hotspot Hot-spot at the 3th year at the 25th year at the 2th year at the 15th year at the 1.54th year Figure 25 Fatigue damage accounting for corrosion deterioration An efficient probabilistic fatigue life assessment depends on the S-N curve model ad it is also expressed as follows. T m K Ln n m m m vo B 1 (9) The reliability calculation can be performed using the total uncertainties on fatigue stress estimation represented by the random variable K, B and. The mean value and the coefficient of variation of the parameter K are taken as K 1 and COV K.15 respectively. In Eqn 9, B represents the uncertainty in fatigue stress estimation, which is modelled by a Normal distributed random variable. As the stress calculation has several steps, each of which with its own uncertainty, the stochastic variable B can be split into four components including B modelling the uncertainty in the load L calculation, B modelling the uncertainty in the nominal stress calculation, B H modelling the uncertainty in the hotspot concentration factor calculation and B Q modelling the uncertainty in the weld quality estimation and in the misalignment. The stochastic models used for the present reliability calculations are based on the extensive study reported in [5] as described in the following. The mean value and the coefficient of variation of the random variable B are taken as B 1.1 and COV B.25 respectively. The first two moments of fatigue life are linearized about K, B and and are expressed by [7]: 8. FATIGUE RELIABILITY ANALYSIS The reliability of structures can be defined as the likelihood of maintaining its ability to fulfil its design purpose for some time period. The fatigue reliability assessment conducted in the present study deals with the structural failure hypothesis due to fatigue damage and corrosion deterioration during the service life of the web frame. The corrosion degradation can influence the ship structural reliability due to improper surveillance or due to strength degradation induced by plate thickness reduction caused by corrosion wastage. Fatigue damage of the case study 3 is analysed here. The reliability is modelled as a series system, assuming that the structure will fail due to fatigue strength degradation accounting for the corrosion wastage and correlation between them. Mean T K Lnn m m m vo B N N N Var T K B K B m N Ln n K m m m v o B 1 m N K Ln n m B B m m vo 1 m1 (1) (11) (12) (13) 14

13 XIV Portuguese Conference on Fracture (214) m1 m m m vb o 1 m N K Ln n (14) Fatigue damage of the weld detail is considered to be a random variable and the confidence limits of fatigue damage at the confidence level of 95% may be calculated by the following equations: D D 95_ U 95_L m m 95_ U 95_ U m K95_ L Ln( n ) B N m 1 m m 95_L 95_L m K95_U Ln( n ) B N m 1 (15) (16) where the subscript 95_U represents the upper 95% confidence limit of the mean of a random and 95_ L is the lower 95% confidence limit for the mean of a random variable. The hotspot 3, 4, 5, 6, 9, 12, 15 and 16 are the most critical fatigue details in the studied stiffened frame. The system reliability of the web frame, containing those eight critical hotspots, is defined as a series system. The occurrence of fatigue failure at any hotspot (component) for a series system of components will result in the failure of the web frame (entire system). Reliability index, t, t, T hotspot location may be defined as: t of any individual 1 1 R t (17) 1 where is the inverse of the standard normal distribution function and Rt is the structural reliability. Based on the First Order Second Moment, the reliability index may be expressed as: t where t t T d (18) t and t are the mean value and standard deviation of the estimated fatigue life and design life. T d is the If the fatigue life of each individual critical hotspot is considered as independent, the reliability of the series system, composed of independent components, may be calculated as: R ( t) R ( t) R ( t) R ( t) R ( t) R ( t) S R ( t) R ( t) R ( t) (19) If those critical hotspots in a web frame are considered to be correlated, the failure of the entire system composed of k components, E is a union of all possible component failures, which can be expressed as follows: E E1 E2... Ek (2) where E i represents a failure occurrence at any component. The failure probability can be computed either by use of Morgan s law: P E P E E E k i1 1 2 i P two components P E P three components k 1 1 PE E E 1 2 k k (21) or by the use of bounding techniques as the narrow bounds on the probability of failure of a series system proposed by Ditlevsen [51]: P E P E P E P E i2 j k i1 1 max i ij, (22) k k i max ij (23) P E P E P E i1 i2 ji where the joint probabilities of failure, PE ij can be expressed [52] as: 1 2 2,, 2,, P E x x dx dx (24) ij i j ij i j ij i j i j ij i j 2 xi, xj, ij exp ij ij where x x 2 x x (25) and i i T are the reliability j j T indices corresponding to the i th and j th component failure respectively, is the correlation coefficient between ij the i th and j th component failure and and are the probability density function and cumulative distribution function respectively of the two-dimension standard normal distribution Eqn 23 represents the narrow bounds for the system probability of failure, and they still regard the joint 15

14 Reliability Reliability(N ) XIV Portuguese Conference on Fracture (214) failure modes. Ditlevsen [51] also proposed a method for the bounding joint probability of failure, PE ij as: max P, P P E P P (26) where P P A B A B ij A B j ij i 1 ij i 2 i ij j 1 ij j 2 Upper Bound,ρ=.1 Lower Bound,ρ=.1 Lower Bound,ρ=.5 Upper Bound,ρ=.5 (27) (28) Lower Bound,ρ=.75 Upper Bound,ρ=.75 Lower Bound,ρ= Upper Bound,ρ= No Correlation, with Corrosion No Correlation, without Corrosion Fully correlation, with Corrosion Fully correlation, without Corrosion Coating Life Corrosion Degradation Time, N, years years Figure 26 Reliability of web frame as a function of time Ditlevsen bounds demonstrated to be more practical and usable than the exact expressions in a related study [5, 53]. Four different correlations of.1,.5,.75 and between the reliability of the hotspots are considered here. Using Eqn 23 for estimating the joint probability of failure, the fatigue reliability of the corroded web frame is modelled as a series of components, as a function of time. In the time interval up to the 1.45 th year, the resulting fatigue reliability, accounting for corrosion behaviour, is the same as for the one without corrosion, since there is no corrosion occurrence before the 1.45 th year, where the corrosion protection system of metal structure surface is effective. The fatigue reliability decreases significantly with time after the 1.54 th year. The failure of the protection system at a random point of time will result in corrosion wastage and will accelerate the progressive deterioration of the steel structure. Figure 26 presents the comparison of the fatigue reliability assessment of the web frame, operating during a 3-year service life, accounting for the correlation between the reliability of different hotspots. 9. CONCLUSIONS Fatigue damage assessment of complex ship hull structures subjected to fatigue loading has been performed based on the three dimensional linear finite element analysis and SN approach. Fatigue assessment of a side-longitudinal stiffener considering the two most probable crack initiation locations in a double hull oil tanker based on the three stress approaches has been performed here (case study 1). The analyses were performed for two basic loading conditions, full and ballast, accounting for the expected operation time in each of the considered conditions. The stress distributions at the studied locations of the weld toe were analysed. The finite element analysis of the effective notch stress approach showed that the stresses at the hotspot B are always highest in all the loading cases. The local hot spot stress approach was employed and compared with the effective notch stress approach with respect to the estimated fatigue damage. In this study, significant differences between the local stress approaches were found with respect to the calculated fatigue damage. The fatigue damage, based on the effective notch stress approach, is higher than the hotspot stress approach and the one considering that the SCFs are taken from the list of standard details. The comparison of fatigue damage assessment of each hotspot of welded joint detail in the case study 2 showed that the fatigue damage of the hot spot No.4 has the maximum and exceeds 1. during the service life of the structure resulting in the fact that the fatigue life is less than the design life. This reveals that the welded joint between longitudinal stiffener and lug (hot spot No.4) cannot provide adequate fatigue strength and a fatigue crack initiation is most probable to be expected. The present work analysed the effect of random nonuniform distributed corrosion wastage on fatigue reliability of a complex stiffened frame. A total of 151,527,6 pits were simulated in the threedimensional finite element model of the stiffener web frame, accounting for random non-uniform corrosion wastage at different service times by using the Monte Carlo simulation technique. The time-dependent nonlinear corrosion wastage technique, adopted here, has been demonstrated to be a practical tool to simulate corroded structures. Based on nonlinear regression analysis, the timedependent function for each hotspot stress that weakens the capacity of the stiffener web frame was fitted. The probabilistic characteristics of fatigue damage and life at each hotspot in the corroded welded structure, during the anticipated service life, were quantified based on the S-N approach and the first-order reliability method. 16