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Fatigue crack propagation prediction for marine structures based on a spectral method
Ocean Engineering 163 (2018) 706–717
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Fatigue crack propagation prediction for marine structures based on a spectral method
T
Yongkuang Zhanga, Xiaoping Huanga,∗, Fang Wangb a b
Collaborative Innovation Center for Advanced Ship and Deep-sea Exploration, Shanghai Jiao Tong University, Shanghai, 200240, China Shanghai Engineering Research Center of Hadal Science and Technology, College of Marine Science, Shanghai Ocean University, Shanghai, 201306, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Fatigue crack propagation Marine structure Spectrum analysis Short term distribution Stress intensity factor Improved Euler method
A method based on spectrum analysis is proposed for predicting fatigue crack propagation (FCP) of a crack in marine structures. The stress intensity factors (SIF) for a crack can be different under different load cases, such as loading conditions, heading angles and wave frequencies, even at the same level of nominal or hotspot stress. Therefore, the load spectrum obtained from the stress transfer function and wave spectrum may be failing to calculate the FCP accurately. In current study, SIF transfer functions were evaluated through detailed structural analysis for different crack sizes, and the short term SIF distribution obtained from the SIF transfer function. FCP life could then be calculated from numerous short term SIF distribution and the improved Paris formula. A submodel technique was integrated, facilitating SIF re-analysis due to crack growth, and an Improved Euler method was adopted to reduce the computational steps. FCP of a semi elliptical surface crack at the weld toe between hatch coaming and forward bulkhead of a container ship superstructure was used to demonstrate the application of the proposed spectral method. This paper offers a way of combining the spectral analysis and fracture mechanics in fatigue crack growth calculation for marine structures.
1. Introduction Ships and marine structures are subjected to fluctuating loading at sea, which could lead to inducing fatigue crack growth. Generally, ships are designed with a fatigue life of 25 years using conventional high cycle fatigue principles, i.e., the S-N method (Det Norsk Veritas, 2010). However, large uncertainties are often disregarded in ship structural fatigue analysis, such as real wave environments, corrosion, weld defects, etc. (Cui, 2003; Fricke et al., 2002). These factors contribute to fatigue cracks being initiated much earlier than expected in many vessels, which challenge ship safety and reliability. The high expenses of repairs and lost time mean that it is not practical or possible to repair every minor to moderate crack at once (Mao, 2014). Hence, forecasting fatigue crack growth and identifying specific cracks that are critical to structural integrity are important issues (Okawa et al., 2007). Therefore, tools and methods based on fracture mechanics are required to predict fatigue crack propagation (FCP) lifetime and crack growing path rationally and accurately. Among S-N methods, spectral based fatigue analysis provides a more accurate method to consider some factors, with a clear and reasonable calculation process (American Bureau Of Shipping, 2016). However, FCP using spectral based methods for marine structures is rarely applied. Some
∗
Corresponding author. E-mail address: [email protected] (X. Huang).
https://doi.org/10.1016/j.oceaneng.2018.06.032 Received 8 April 2017; Received in revised form 5 June 2018; Accepted 9 June 2018 0029-8018/ © 2018 Published by Elsevier Ltd.
scholars (Jang et al., 2010; Mao, 2014; Yan et al., 2016a; 2016b) have derived the load spectrum, consisting of stress ranges, from the stress transfer function and wave spectrum, with the stress intensity factor (SIF) then calculated by empirical formula or a finite element model (FEM) to simulate the FCP. This is referred to as the stress spectral method. Unfortunately, there are two drawbacks when using load spectra obtained from stress transfer functions in FCP. (1) It is not easy to find the relationship between the SIF and the stress when the crack in the complex stress field. One needs to choose the appropriate stress (normally the maximum stress on the top surfaces), and then calculates the SIFs by empirical formula or FEA. However, many critical spots of marine structure are in complex stress states and may subject to tension, bending, shear, and non-uniform distributed boundary stresses synchronously. Thus, it can be difficult to determine how SIF can be calculated accurately from stress for the actual structure. (2) The boundary conditions of actual structures are very complicated, and a single stress value cannot represent the true loading conditions. That is, the SIF for a crack can be different under different load cases, such as loading conditions and heading angles etc., even for the same value of nominal or hotspot stress. Thus, it may be difficult to determine a harmonious relationship between stress and
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Nomenclature
f λ 0 λ2 da/ dN C, m
ω θ U
Angular frequency (rad/s) Heading angles Ship forward speed used in seakeeping analysis and spectral analysis, equal to 2/3 times design speed Hs Significant wave height Tz Wave period Hsif (ω|θ) SIF transfer function Wave energy spectrum S (ω|Hs , Tz ) Ssif (ω|θ,Hs , Tz ) SIF energy spectrum N-th order SIF energy spectral moment λn K SIF amplitude induced by waves ΔK SIF range induced by waves ΔKn Nominal SIF range induced by waves Kstatic SIF induced by still water bending moment ΔK eff Effective SIF range Equivalent SIF of mixed fracture modes K eq Kmax , Kmin Maximum and minimum SIF value Kres SIF induced by residual stress ς(K ) Probability density function of SIF amplitude induced by waves Probability density function of nominal SIF range ς ΔK (ΔKn ) induced by waves σx Rayleigh distribution scale parameter
β , β1 R MR ΔKth ΔKth′ dai E (·) ΔNi a, c a0 , c0 Δatol af
Average zero-up crossing frequency Zeroth, second order SIF energy spectral moment Fatigue crack propagation rate Coefficient and exponent in crack growth relationship Exponential parameters in Huang's model Stress ratio Correction factor for the effect of stress ratio Threshold of SIF range corresponding to stress ratio R=0 Value of ΔK when MR ΔK ≥ ΔKth Crack increment in a sea state i Expected value The number of load cycles in sea state i Depth and half length of the semi-elliptical crack The initial crack size Accuracy controlling parameter Critical crack depth
Subscripts
i I, II, III a, c
Sea state i Opening, sliding and tearing fracture mode Deepest point, surface ends point of a semi-elliptical crack
surface crack in a rectangular plate and a semi elliptical surface crack at the weld joint toe of a detail between hatch coaming and superstructure of a container ship in section 4 and section 5.
SIF for a crack in complex structural detail. This study proposes an approach to predict fatigue crack propagation incorporating spectral analysis. Section 2 presents a method to generate short term SIF distributions for real structural details. Combining the short-term SIF distributions and improved Paris formula to calculate FCP directly is described in section 3. Validation for the proposed spectral methods are demonstrated using a semi-elliptical
2. Spectral analysis for marine structures Linear elastic fracture mechanics (LEFM) is often used to predict FCP (Stephens et al., 2000). The empirical Paris Law has become a
Fig. 1. Proposed spectral analysis procedure for fatigue crack propagation in marine structures. 707
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2.2. Stress intensity factor transfer functions
widely recognized approach to estimate the FCP, based on the SIF range, rather than stress, and FCP can be obtained directly if the short term SIF distribution can be derived. Generally, fatigue analysis of marine structures involves the local structure. Fracture mechanics sub-model is employed to simulate FCP of actual structures, and the displacement boundary conditions of the sub-model should be obtained from the global structural response analysis. Fig. 1 presents a brief description of the proposed spectral based FCP. The Pre-processor mainly contains global hydrodynamic and structural response analysis and extract displacement boundary conditions for sub-model analysis. Then, according to the encounter waves and load state, continuously apply the corresponding boundary conditions to the fracture sub-model, calculate the SIFs of cracks with different sizes. Crack propagation can be simulated through update the sub-model step-by-step.
After the global structural stress analysis, dangerous regions with cracks are selected as the critical regions or hot spots. To balance the demands of computational efficiency and precision, element types of the whole ship FE-model are mainly two-dimensional shell element and one-dimensional beam element, and the element size is usually very coarse. To calculate SIFs for three-dimensional (3D) surface cracks, it is necessary to refine the shell elements in the vicinity of the hot spot, create solid elements with surface cracks and build multi-point constraints (MPCs) (ANSYS, 2013) between the shell and solid elements. Fig. 2 is an example of a sub-model in a ship structure. Boundary displacements obtained from the global structural analysis and all relevant local loads should be applied to the sub-models. The SIF of the crack is then calculated using FEM. The sub-model regions should be large enough to consider the load shedding induced by cracks growth and structural redundancy (Okawa et al., 2007). It needs to rebuild the sub fracture model when the crack size is large enough, and increasing the area of submodel, until the SIF value gradually converging. FEA has been widely used in SIF calculation of a 3-D crack in structural detail. Newman and Raju (1981) evaluated the SIF of a semi elliptical surface crack in a rectangular plate under uniform tensile and bending load. Bowness and Lee (1996, 2000) numerically calculated the SIFs of surface cracks in weld toe of T joints and proposed corresponding empirical formulations. Some scholars (Bowness and Lee, 1995; Chiew et al., 2001; Lee et al., 2005) proposed methods to model 3-D doubly curved semi-elliptical crack at weld toe in tubular joints. Some specialized software, such as Franc3d, Zencrack, FEACrack, etc.
2.1. Global structural response Fatigue damage in ships is mainly caused by wave loads acting on ship structures. For more reliable fatigue assessment, a direct calculation method is often used to evaluate the wave loads under different sailing conditions and loadings (American Bureau Of Shipping, 2016). The wave loads can be computed by hydrodynamic software, and structural analysis provides the structural stress/strain response by either simple hull girder theory or FEM. In general, the response is calculated by linear analysis in the frequency domain, as described by Lewis (1988).
Fig. 2. Finite element model of a structural detail with a semi-elliptic surface crack. Multi-point constraints (MPCs) were used to connect the shell and solid elements. 708
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2.4. Short term stress intensity factor distribution
provide non-planar crack growth prediction via automated adaptive meshing techniques. Dolbow and Belytschko (1999), Fries and Belytschko (2010) proposed a finite element method for crack growth without remeshing, i.e. extended finite element method (XFEM). These studies have greatly facilitated SIF calculation and simulation of FCP. The current study mainly focuses on a spectral method and how to model a fracture FE-model is not the focus of this article. The authors take semi-elliptical surface cracks at butt weld toe as an example, to illustrate the proposed spectral method. Referring to BSI, 2013, where the flaws often treated as an equivalent elliptical crack, two parameters were used to describe propagation of the surface crack, the crack depth, a, and the crack half-length, c. The current study followed common practice (Branco et al., 2015), singular element (dedicated to SIF calculation, collapsed from a 20 nodes hexahedral element) and SOLID95 element (20 nodes hexahedral element) was adopted when creating the solid model using commercial software ANSYS (2013), shown in Fig. 2. The radius of the first ring element at the crack tip, r = 0.0375a when a > 15 mm to ensure accurate results. This method has been validated by our research group (Yan et al., 2016a; 2016b). In the linear spectral analysis, global structural analysis is performed for a series of unit amplitude harmonic waves, ω , and various heading angles, θ . The SIF transfer function Hsif (ω|θ) is then obtained for a given crack size.
In a stationary sea state, it is generally assumed that the waves are zero-mean narrow-banded random Gaussian process (American Bureau Of Shipping, 2016). Thus, for linear systems, fluctuating stress/SIF induced by wave loads are also a zero-mean narrow-banded Gaussian process. According to the theory of random process, SIF amplitude induced by waves, K, will follow the Rayleigh distribution. The corresponding probability density function is:
ς(K ) =
ς ΔK (ΔK ) =
λ 0 and = 1/2π λ2 / λ 0 ,
(5)
3. Fatigue crack propagation calculation 3.1. Fatigue crack growth rate model Kujawski et al.(2001; 2004) proposed a two-parameter driving force model to account for R-ratio. The crack growth rate is expressed as:
da = C·(ΔK ∗)m dN
(6)
ΔK ∗ = (Kmax )α ·(ΔK+)1 − α = MΔK
(7)
ΔK+
Where α is the correlation parameter, and = ΔK for R ≥ 0 and ΔK+ = Kmax for R < 0 . This model is based on the premise that for a negative R-ratio, the negative part of ΔK does not contribute to the crack growth. However, it has been mentioned that the model was better for positive R-ratios than for negative ones from the comparison with the experimental data (Huang and Moan, 2007; Noroozi et al., 2007). Huang and Moan (2007; 2009) further studied the effect of R-ratio from an extensive analysis using experimental data from the literature, and proposed an improved FCP rate model. In the current study, Huang's model was selected in the spectral FCP, the basic expression is:
(1)
where S (ω|Hs , Tz ) is the wave energy spectrum. The nth order SIF energy spectral moment at the ship forward speed, U, is similar to the stress spectral moment (American Bureau Of Shipping, 2016), and can be expressed as ∞
0
(4)
where λ 0 and λ2 are the zeroth and second order moment of the SIF response spectrum.
Generally, the relationship between wave loads and induced stress is linear in spectral-based fatigue. For a given crack size, there is also a linear relationship between SIF and stress in LEFM, i.e., there is a linear relationship between SIF and wave loads, which is a precondition for applying spectral analysis. For the fatigue assessment of ships, the response under real wave environments is often divided into a series of stationary process, known as sea states. A sea state can be described by the classic wave energy spectrum, S (ω|Hs , Tz ) , for example, Pierson-Moskowitz (P-M) or JONSWAP (Lewis, 1988), which can be expressed as a function of significant wave height, Hs, and wave period, Tz. Thus, for a given crack size, the SIF energy spectrum under arbitrary sea states can be computed by
∫
ΔKn ΔK 2 exp ⎛⎜− 2 ⎟⎞, 4σx2 ⎝ 8σx ⎠
where σx , and the average zero-up crossing frequency, f can be expressed as
2.3. Spectral analysis
λn =
(3)
Theoretically, compression force does NOT have SIF values, in the fatigue crack growth model, we employed equivalent SIF to consider the effect of the mean stress and the residual stress is considered as part of the mean stress. The SIF range, ΔK , equals to two times of SIF amplitude and coresponding probability density function, can be expressed as:
σx =
Ssif (ω|θ,Hs , Tz ) = Hsif (ω|θ) 2 *S (ω|Hs , Tz ),
K K2 exp ⎛⎜− 2 ⎟⎞. 2 σx ⎝ 2σx ⎠
ω − ω2Ucosθ / g n Ssif (ω|θ,Hs , Tz ) dω. (2)
This study used a wave scatter diagram to describe the expected sea states encountered. The North Atlantic scatter diagram was selected, and a sea state was assumed to last two hours and to be described by the P-M spectrum. Suppose that each short-sea condition lasts two hours and determine the number of cycles in two hours based on the average zero-crossing rate. Generate load block for each short-sea condition which consists of a number of random stress ranges that following the Rayleigh distribution. Determine the probability of occurrence for each load block based on the wave scatter diagram. The total number of cycles is determined by the average zero-crossing rate. Various short-sea conditions appear randomly in accordance with a certain probability constitute the long-term fatigue load spectrum.
da = C [(ΔKE )m − (ΔKth)m], dN
(8)
ΔKE = MR ΔK,
(9)
and
(1 − R)−β1 (−5 < R < 0) ⎧ ⎪ MR = , (1 − R)−β (0 ≤ R < 0.5) ⎨ ⎪ (1.05 − 1.4R + 0.6R2)−β (0.5 ≤ R < 1) ⎩
(10)
Kmin + Kres −0.5ΔKn + Kstatic + Kres = , Kmax + Kres 0.5ΔKn +Kstatic + Kres
(11)
R=
where N is the number of cycles, ΔKE is the equivalent SIF range, C and m are the Paris parameters corresponding to stress ratio R = 0, ΔKth is the threshold of SIF range corresponding to stress ratio R = 0, β and β1 709
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Substituting equations (13) and (8) into (12), the expected crack increments in a sea state can be expressed as
are shape exponents depending on the material property and environment, MR is the correction factor for the effect of stress ratio, Kmin is the minimum SIF, Kmax is the maximum SIF, Kres is the SIF induced by residual stress, and Kstatic is the SIF induced by static load. Since Huang model is valid for R > − 5, Kstatic and Kres are set to zero for safety consideration when mean stress induced by still water bending, σstatic and residual stress are negative. In this study, owing to absence of specified material parameters, the Paris law parameters for steel recommended by International Institute of Welding (IIW) (2016) were adopted as C= 1.65 × 10−11, m = 3, and ΔKth = 5.4 MPa m for da/dN in m/cycle, and ΔK in MPa m . For demonstration, the exponential parameters of steel in Huang's model were chosen as β = 0.5, β1 = 0.8 (Huang and Moan, 2007).
E(dai ) = C
⎭
0
ΔTi 2π
λ2i , λ 0i
+∞
x mMR (x )mςiΔK (x ) dx − ΔKthm
⎝ ΔKth′
∫ ςiΔK (x ) dx ⎞⎟.
ΔKth ′
⎠
If the FEM mesh and ς ΔK (ΔK ) are updated for every stationary sea state, although the result would be more accurate, too much time would be required to perform the FCP simulation. Therefore, the forward Euler method has been widely used in FCP simulation (He et al., 2014; Mao, 2014; Sumi et al., 2005). When the crack increment is < Δatol , e.g. 1 mm, the FEM mesh is not updated, to save calculation time. Unfortunately, there is a major drawback associated with the forward Euler method, in that it requires choosing a small step size to maintain stability and accuracy of the solution. Thus, an improved Euler method is presented in the current study, the main procedure is shown in Fig. 3. 4. Validation of the proposed spectral based method
(12)
The crack growth analysis of a semi-elliptical surface crack in a structural detail was performed by using the proposed procedure to check the accuracy of the proposed spectral method. It was assumed that the structural detail had the same stress state, i.e., uniform tension, as shown in Fig. 4. Thus, there was a consistent relationship between stress and SIF. A widely recognized empirical formula proposed by Newman and Raju (1981) was used calculate SIF. We also computed it numerically using ANSYS, which could be used for more complex
with
ΔNi = ΔTi fi =
⎜
3.3. Crack growth within a computational step
∞
da ⎫ (x )*ςiΔK (x ) dx , ∫ dN ⎬
+∞
⎛∫
Where ΔKth′ is the value of ΔKn when MR ΔKn ≥ ΔKth (Since xMR (x ) is strictly monotonic increasing, there must exists a unique x that meets xMR (x ) = ΔKth , the unique value x namely is ΔKth′ ). It allows to compute E(dai ) easily through equation (14) using numeral integral.
In general, FCP under the one sea state is vanishingly small, and thousands of sea states may be needed for a small crack to grow to a given length, e.g. 1 mm. Therefore, during the ith sea state, the SIF transfer function can be treated as constant. Thus, the SIF range probability density function is also constant. In a sea state, since the SIF da range, X, is a random variable, the crack growth rate, dN (X ) , is a function of random variables, the expected crack increment, E (dai ) can be expressed as
⎧ ⎨ ⎩
λ2i λ 0i
(14)
3.2. Closed form expression for fatigue crack propagation during one sea state
E (dai ) = CΔNi
ΔTi 2π
(13)
where ΔNi is the number of load cycles in sea state i , ΔTi is the duration time in one sea state, fi is the average zero crossing frequency.
Fig. 3. Flow chart of numerical procedure for the improved Euler method, where a, c are the depth and length of the elliptical crack, respectively; a0, c0 are the initial crack size; Δatol is the accuracy controlling parameter, and for good precision, Δatol should be 1–2 mm. 710
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Fig. 4. Plate with surface crack, initial crack size a0 = 5 mm, c0 = 20 mm.
Fig. 5. Stress transfer function.
Fig. 6. Stress intensity factor (SIF) transfer functions evaluated at (a) the deepest points, and (b) the surface points of the semi elliptical crack.
structural details. The flat plate size is 600 × 300 × 65 mm; with initial crack size a0 = 5 mm, c0 = 20 mm; and critical crack depth, af = 42 mm. There is no residual/mean stress in the structure, and only head waves were encountered. Fig. 5 shows the nominal stress transfer functions. First, the short-term distributions of stress ranges were obtained from the stress transfer function in combination with the wave scatter diagram. FCP life was then calculated based on empirical SIF formula and FCP model discussed in Section 3.1, using the cycle-by-cycle (CBC) counting method (Cui et al., 2011), a time-consuming but accurate method to calculate FCP. Alternatively, the SIF transfer function was derived from ANSYS, and we also calculated it using the Newman-Raju empirical formula. Fig. 6 shows that the results are in close agreement. The short term ΔK distributions were obtained following Sections 2.3 and 2.4. Then, FCP life was calculated following Section 3. Calculation accuracy may differ due to the different crack growth calculation steps, Δatol , and the crack growth curves for different step sizes and stress spectral method using CBC are shown in Fig. 7. The FCP life obtained from SIF spectral method closely agrees with stress spectral method when the stress state (loading mode) of critical spot are the same or similar among different load cases. The Euler and improved Euler methods both provide good precision with small step size. However, the improved Euler method shows better consistency with the CBC. When the step size was set to 1–2 mm, sufficient accuracy could be
obtained through the improved Euler method. 5. Example application of the proposed method to a container ship detail To demonstrate the proposed spectral methods, a joint between the hatch coaming and front bulkhead of superstructure in a large container ship was selected as a real structure example. Fig. 8 shows the sketch of hot spot location. Fig. 9 shows the detail structural geometry of the hot spot. The hatch coaming plate and bulkhead plate were welded together. And the fillet weld was assumed to have a throat thickness of 10 mm, and the weld toe radius ρ = 1 mm. The hatch coaming was constructed from E40 steel with yield stress σY = 390 MPa, and a semielliptical surface crack at the weld toe found in an inspection, with initial crack size a0 = 2 mm, c0 = 8 mm. The proposed spectral based FCP method was used to compute the sailing hours until the crack depth grows to af = 0.6T , for safety consideration (Yan et al., 2016a; 2016b). For a flaw lying in a plane parallel to the welding direction, residual stress σres should be considered. Referring to BSI, 2013, residual stress was assumed to be 20% of σY in the structures subject to post-weld heattreated. Then, Kres could be calculated through weight function method (Wu and Carlsson, 1991). The mean stress induced by still water bending, σstatic approximately equal to 39 MPa tensile stress for the full load case. 711
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Fig. 9. Detail structure of hot spot.
5.1. Global structural stress response Hydrodynamic software WALCS developed by the China Classification Society (2013) was used to compute the wave loads on the target marine structure, based on 3D frequency domain linear hydrodynamics principles. Parameters for the hydrodynamic analysis were:
• wave frequency = 0.1–1.8 rad/s, step = 0.1 rad/s, 18 frequencies; • ship speed, U = 2/3 times design speed, the design speed is 25 knots; • heading angle = 0–180°, step = 30° (7 directions, given the symmetry of the ship hull; heading angle 180° represents the head sea, and a heading angle 0° denotes the following sea).
Only the full load condition was considered in this study. For each wave with determined heading angle and frequency, unit wave amplitude was used to generate the real and imaginary hydrodynamic wave pressures on the panels. After FE-analysis, the stress and displacement responses of the target structure for different wave frequencies and directions were obtained. Fig. 10 shows the global structure response of the container ship induced by waves load.
Fig. 7. Comparison of fatigue crack growth with different methods. (a) the Euler method with different crack size interval and the cycle by cycle method, (b) the improved Euler method with different crack size interval and the cycle by cycle method.
5.2. SIFs for different load cases The structural detail shown in Fig. 11 was a sub-model of the whole ship FEM model, and it was modeled following procedure described in Section 2.2. Three types of crack opening modes were considered, i.e., opening (Mode I), sliding (Mode II), and tearing (Mode III). Owing to the imaginary and real part of wave pressure are loaded separately, the 2 2 + Kimaginary SIF amplitude becomes K = Kreal . The SIF transfer function at different heading angle shown in Fig. 12 and the data in Table 1 and Table 2 show that the Mode I constitutes the main component of the SIF for both surface end point and deepest point. More specifically, compared with surface end point, the effect of Mode II and Mode III is smaller in the deepest point. In head and following waves, SIFs of cracks have smaller share of Mode II than that in oblique and beam waves. This can be caused by shearing force induced by the torsion and horizontal bending. The equivalent SIF (Bowness and Lee, 1996) is
K eq =
KI2 + KII2 +
2 KIII , 1−ν
(15)
where ν is Poisson's ratio. The average proportions of KI , KII and KIII to K eq in a heading angle are summarized in Table 1 and Table 2. The share of Mode II and Mode III will be less for larger crack size, and more for smaller crack size. But for simulate the crack growth direction, it can be done by rebuilt the crack increment with an angle determined by an adopted criteria, such
Fig. 8. Schematic diagram of target structure.
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Fig. 10. The representative global structure response induced by waves load (target spot is in the red circle). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 11. FEM model of the detail with a surface crack in ANSYS (a sub-model of the ship).
inconsistent relationship. The equivalent SIF value at unit HSS can be obtained through SIF transfer functions divided by corresponding HSS transfer functions, as shown in Fig. 14. It is difficult to define a harmonious relationship between hot spot stress and SIF that consider all the load cases. More specifically, an approximate consistent correspondence can be founded in heading waves and following waves, i.e. 180° and 0°, the stress status of the detail is near uniform tension longitudinally, as shown in Fig. 15, while in the oblique waves, the ship structure is under horizontal bending and torsion. The stress status of local structure mainly is under combined tension and bending, as shown in Fig. 15. The degree of bending (DoB) (Ahmadi et al., 2015) in hatch coaming plate varies with waves frequency, as shown in Fig. 16, and, there is a similar trend between Figs. 14 and 16. It can be the specific reason for the inconsistent relationship. The results represent
as the maximum circumferential stress criterion in the submode when the K_II or K_III is big enough. This issue will be discussed in the future. 5.3. Inconsistent relationship between SIF and stress in different load cases Since the nominal stress is not always possible to define in real structures, hot spot stress (HSS) is widely used in the S-N curve spectral method. The HSS is adopted to illustrate the inconsistent relationship. And it was generated by the recommended extrapolation approach (American Bureau Of Shipping, 2016). The HSS amplitude is 2 2 σh = σreal + σimaginary . Fig. 13 shows the HSS transfer functions of the local structure. For a relative large crack size, i.e. a = 10, a/c = 0.3, the K eq is basically equal to KI (see Table 2). K eq is used to illustrate the 713
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Fig. 12. SIF transfer function (a = 5, a/c = 0.3). (a) the deepest points, and (b) the surface points of the semi elliptical crack. 714
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Table 1 Average proportions of KI , KII and KIII to K eq in a heading angle (a = 5 mm, a/ c = 0.3).
KIa KIIa KIIIa KIc KIIc KIIIc
180
150
120
90
60
30
0
99.5% 7.3% 5.6% 97.0% 5.5% 19.5%
96.6% 13.8% 14.3% 93.0% 19.6% 22.5%
93.8% 14.0% 18.4% 92.8% 23.1% 22.6%
96.7% 13.4% 13.9% 92.5% 21.7% 24.4%
98.0% 12.1% 11.4% 94.6% 15.9% 22.1%
98.9% 11.5% 7.2% 95.4% 12.6% 22.1%
98.2% 7.2% 10.9% 96.3% 5.0% 19.1%
Table 2 Average proportions of KI , KII and KIII to K eq in a heading angle (a = 10 mm, a/ c = 0.3).
KIa KIIa KIIIa KIc KIIc KIIIc
180
150
120
90
60
30
0
99.2% 0.9% 6.9% 97.2% 7.4% 17.1%
99.0% 5.7% 12.8% 95.6% 18.0% 20.6%
97.5% 5.7% 12.1% 93.9% 19.8% 20.2%
98.4% 3.7% 11.2% 94.3% 18.6% 21.5%
99.2% 3.2% 8.4% 96.1% 12.4% 19.4%
99.7% 2.7% 9.0% 96.7% 13.6% 19.5%
98.1% 1.4% 11.3% 96.4% 7.2% 16.8%
Fig. 14. Equivalent SIF value at unit HSS. It is difficult to determine a harmonious relationship between stress and SIF. (a) the deepest points, (b) the surface points of the semi elliptical crack.
eliminate the effect of uncertainty of pseudo-random stress signal. Comparison of crack increment computed by the two methods are presented in Table 3. Apparently, different relationships between stress signal and SIF have quite large impact on FCP in CBC method. According to Fig. 14, when using the stress spectral method, it could be difficult to determine an exact relationship that take into account of all the heading angles and frequencies, while the proposed method would not have the problem. It is noticed that the relationships almost are consistent in heading angle 180 and 0, and the corresponding FCP results using CBC and integral are nearly equal. It confirms that SIF spectral method closely agrees with stress spectral method when the loading mode of critical spot are the same or similar among different heading angles and frequencies.
Fig. 13. Hot spot stress transfer functions.
only one structural detail. However, for other details, it is foreseeable that there is an inconsistent relationship between stress and SIF if the loading situation of local structure changes considerably among different load case, otherwise, the relationship would be consistent.
5.4. Comparison of crack increment in a short sea state For comparison, we take the crack increment in a sea state when the crack depth, a = 10 mm, crack aspect ratio, a/c = 0.3, for example. Since the crack increment in a short sea state is very small, both the SIF transfer functions in proposed method and the relationship between stress signal and SIF in stress spectral method remain constant. The significant wave height of the short sea state is 8 m and the wave period is 11 s. In the stress spectral method, we calculate FCP through CBC method using the simulated stress signal based on the calculated stress spectra. The relationship between stress signal and SIF, i.e. SIF value at unit HSS, as shown in Fig. 14, is not consistent in different heading angles and frequencies. We chose relationship in a frequency that the HSS transfer function is at its maximum to calculate FCP. The CBC method was repeated 100 times, and the average crack increment was used to
5.5. Fatigue crack propagation Following the improved Euler method in Section 3, the FCP of the structural detail was simulated step by step with step size Δatol = 1 mm . A program code was developed to update the FEM mesh automatically and calculate the fatigue life. The probability of heading angle is assumed to follow a uniform distribution. Fig. 17 shows the FCP curves, using K eq and KI , the expected fatigue life is 35244 h (approximately 35244/365/24 = 4.02 years) and 37418 sailing hours respectively. For simulating FCP by FEA, additional calculation of KII and 715
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Fig. 15. Stress distribution of intact local structure (half of the model in Fig. 9).
Fig. 16. The DoB in hatch coaming of local structure under different load cases.
Fig. 17. Fatigue crack propagation curves with uniform distribution of heading angle.
Table 3 Comparison of crack increment computed by two methods in a short sea state. Heading angle
180 150 120 90 60 30 0
crack increment (half length, mm)
crack increment (depth, mm)
CBC (1e5)
Integral (1e5)
Relative value (%)
CBC (1e5)
Integral (1e5)
Relative value (%)
4.61 1.55 5.22 0.23 3.15 3.44 1.40
4.62 2.24 4.55 0.17 2.72 3.61 1.4
−0.24 −31.00 14.81 35.35 15.69 −4.60 0.21
4.53 1.59 4.22 0.17 2.59 3.10 1.38
4.54 2.11 3.76 0.14 2.31 3.22 1.38
−0.17 −24.58 12.28 17.96 12.19 −3.58 −0.11
changes of the surface crack using K eq . Since the SIFs were evaluated according to the actual boundary conditions, this method automatically considers the impact of stress distribution through the plate thickness. 6. Conclusions A compatible relationship may not exist between SIF and nominal or hotspot stress for a specific crack size at different loading conditions, heading angles, and wave frequencies. Therefore, a method to calculate the FCP from multiple short-term SIF distributions which derived from the SIF transfer functions and wave spectrum has been proposed. Some conclusions can be drawn. . The proposed method needs to calculate the SIF transfer function for crack at different sizes which requires too much computation cost for reanalysis and updating the FEM model. Therefore, an improved Euler method was proposed to reduce the computation steps and improve precision.
KIII almost does not increase the computational cost, and neglecting the influence of KII and KIII would lead to no conservative results. Therefore, K eq is recommended in the present demo. Fig. 18 shows shape 716
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Fig. 18. Geometric change of the surface crack using equivalent SIF.
A case study of a real structural detail with an initial semi-elliptical surface crack was used to demonstrate the application of the proposed procedure. The results showed that the expected service time for the ship was approximately 4 years sailing before the crack depth reached 0.6 T. The proposed spectral based method which uses short term SIF distribution instead of short term stress distribution to predict the FCP of a detail under random fatigue loading is a new trial and still in its early stage. Compared with the stress spectral method, SIF spectral method is still complicated in computation of SIFs. Further work should be focused on overcoming this drawback and developing a much faster and easier use of numerical computation method.
Cui, W., Wang, F., Huang, X., 2011. A unified fatigue life prediction method for marine structures. Mar. Struct. 24, 153–181. Det Norsk Veritas, 2010. Fatigue Assessment of Ship Structures. Classification Notes No. 30.7. Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46, 131–150. Fricke, W., Cui, W., Kierkegaard, H., Kihl, D., Koval, M., Mikkola, T., Parmentier, G., Toyosada, M., Yoon, J.H., 2002. Comparative fatigue strength assessment of a structural detail in a containership using various approaches of classification societies. Mar. Struct. 15, 1–13. Fries, T., Belytschko, T., 2010. The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Meth. Eng. 84, 253–304. He, W., Liu, J., Xie, D., 2014. Numerical study on fatigue crack growth at a web-stiffener of ship structural details by an objected-oriented approach in conjunction with ABAQUS. Mar. Struct. 35, 45–69. Huang, X., Moan, T., 2007. Improved modeling of the effect of -ratio on crack growth rate. Int. J. Fatig. 29, 591–602. Huang, X., Moan, T., Cui, W., 2009. A unique crack growth rate curve method for fatigue life prediction of steel structures. Ships Offshore Struct. 4, 165–173. IIW, 2016. Recommendations for Fatigue Design of Welded Joints and Components. Jang, B.S., Ito, H., Kim, K.S., Suh, Y.S., Jeon, H.T., Ha, Y.S., 2010. A study of fatigue crack propagation at a web stiffener on a longitudinal stiffener. J. Mar. Sci. Technol. 15, 176–189. Kujawski, D., 2001. A new (Δ K + K max ) 0.5 driving force parameter for crack growth in aluminum alloys. Int. J. Fatig. 23, 733–740. Lee, C.K., Tjhen, L.S., Ping, C.S., Yongbo, S., 2005. Numerical models verification of cracked tubular T, Y and K-joints under combined loads. Eng. Fract. Mech. 72, 983–1009. Lewis, E.V., 1988. Principles of Naval Architecture Second Revision. Jersey: SNAME. Mao, W., 2014. Development of a spectral method and a statistical wave model for crack propagation prediction in ship structures. J. Ship Res. 58, 479–499. Newman Jr., J.C, Raju, I.S., 1981. An empirical stress-intensity factor equation for the surface crack. Eng. Fract. Mech. 15, 185–192. Noroozi, A.H., Glinka, G., Lambert, S., 2007. A study of the stress ratio effects on fatigue crack growth using the unified two-parameter fatigue crack growth driving force. Int. J. Fatig. 29, 1616–1633. Okawa, T., Sumi, Y., Mohri, M., 2007. Simulation-based fatigue crack management of ship structural details applied to longitudinal and transverse connections. Mar. Struct. 19, 217–240. Stephens, R.I., Fatemi, A., Stephens, R.R., Fuchs, H.O., 2000. Metal Fatigue in Engineering, second ed. (Engineering Computations). Sumi, Y., Mohri, M., Kawamura, Y., 2005. Computational prediction of fatigue crack paths in ship structural details. Fatigue Fract. Eng. Mater. Struct 28, 107–115. Wu, X.-R., Carlsson, J., 1991. Weight Functions and Stress Intensity Factor Solutions. Pergamon. Yan, X., Huang, X., Huang, Y., Cui, W., 2016a. Prediction of fatigue crack growth in a ship detail under wave-induced loading. Ocean Eng. 113, 246–254. Yan, X.S., Huang, X.P., Cui, W.C., 2016b. An engineering method to predict fatigue crack propagation life for marine structures. J. Ship Mech. 20, 323–334.
Acknowledgement This study was supported by the State Key Program of National Natural Science of China (Grant No. 51439004) and the National Key Basic Research Special Foundation of China (2013CB036105). References Ahmadi, H., Lotfollahi-Yaghin, M.A., Asoodeh, S., 2015. Degree of bending (DoB) in tubular K-joints of offshore structures subjected to in-plane bending (IPB) loads: study of geometrical effects and parametric formulation. Ocean Eng. 102, 105–116. American Bureau Of Shipping, 2016. Guide for Spectral-based Fatigue Analysis for Vessels. ANSYS, 2013. ANSYS Mechanical APDL Theory Reference. ANSYS Inc. Bowness, D., Lee, M.M.K., 1995. The development of an accurate model for the fatigue assessment of doubly curved cracks in tubular joints. Int. J. Fract 73, 129–147. Bowness, D., Lee, M.M.K., 1996. Stress intensity factor solutions for semi-elliptical weldtoe cracks in t-butt geometries. Fatigue Fract. Eng. Mater. Struct 19, 787–797. Bowness, D., Lee, M.M.K., 2000. Weld toe magnification factors for semi-elliptical cracks in T–butt joints — comparison with existing solutions. Int. J. Fatig. 22, 389–396. Branco, R., Antunes, F.V., Costa, J.D., 2015. A review on 3D-FE adaptive remeshing techniques for crack growth modelling. Eng. Fract. Mech. 141, 170–195. BSI, 2013. BS 7910, Guide to Methods for Assessing the Acceptability of Flaws in Metallic Structures. BSI Standards Publication. Chiew, S.P., Lie, S.T., Lee, C.K., Huang, Z.W., 2001. Stress Intensity Factors for a Surface Crack in a Tubular T-joint, vol. 78. Int. J. Press, pp. 677–685 Vessel. Pip. China Classification Society, 2013. WALCS Users Manual. Cui, W.C., 2003. A feasible study of fatigue life prediction for marine structures based on crack propagation analysis. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ 217, 11–23.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Fatigue crack propagation prediction for marine structures based on a spectral method
T
Yongkuang Zhanga, Xiaoping Huanga,∗, Fang Wangb a b
Collaborative Innovation Center for Advanced Ship and Deep-sea Exploration, Shanghai Jiao Tong University, Shanghai, 200240, China Shanghai Engineering Research Center of Hadal Science and Technology, College of Marine Science, Shanghai Ocean University, Shanghai, 201306, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Fatigue crack propagation Marine structure Spectrum analysis Short term distribution Stress intensity factor Improved Euler method
A method based on spectrum analysis is proposed for predicting fatigue crack propagation (FCP) of a crack in marine structures. The stress intensity factors (SIF) for a crack can be different under different load cases, such as loading conditions, heading angles and wave frequencies, even at the same level of nominal or hotspot stress. Therefore, the load spectrum obtained from the stress transfer function and wave spectrum may be failing to calculate the FCP accurately. In current study, SIF transfer functions were evaluated through detailed structural analysis for different crack sizes, and the short term SIF distribution obtained from the SIF transfer function. FCP life could then be calculated from numerous short term SIF distribution and the improved Paris formula. A submodel technique was integrated, facilitating SIF re-analysis due to crack growth, and an Improved Euler method was adopted to reduce the computational steps. FCP of a semi elliptical surface crack at the weld toe between hatch coaming and forward bulkhead of a container ship superstructure was used to demonstrate the application of the proposed spectral method. This paper offers a way of combining the spectral analysis and fracture mechanics in fatigue crack growth calculation for marine structures.
1. Introduction Ships and marine structures are subjected to fluctuating loading at sea, which could lead to inducing fatigue crack growth. Generally, ships are designed with a fatigue life of 25 years using conventional high cycle fatigue principles, i.e., the S-N method (Det Norsk Veritas, 2010). However, large uncertainties are often disregarded in ship structural fatigue analysis, such as real wave environments, corrosion, weld defects, etc. (Cui, 2003; Fricke et al., 2002). These factors contribute to fatigue cracks being initiated much earlier than expected in many vessels, which challenge ship safety and reliability. The high expenses of repairs and lost time mean that it is not practical or possible to repair every minor to moderate crack at once (Mao, 2014). Hence, forecasting fatigue crack growth and identifying specific cracks that are critical to structural integrity are important issues (Okawa et al., 2007). Therefore, tools and methods based on fracture mechanics are required to predict fatigue crack propagation (FCP) lifetime and crack growing path rationally and accurately. Among S-N methods, spectral based fatigue analysis provides a more accurate method to consider some factors, with a clear and reasonable calculation process (American Bureau Of Shipping, 2016). However, FCP using spectral based methods for marine structures is rarely applied. Some
∗
Corresponding author. E-mail address: [email protected] (X. Huang).
https://doi.org/10.1016/j.oceaneng.2018.06.032 Received 8 April 2017; Received in revised form 5 June 2018; Accepted 9 June 2018 0029-8018/ © 2018 Published by Elsevier Ltd.
scholars (Jang et al., 2010; Mao, 2014; Yan et al., 2016a; 2016b) have derived the load spectrum, consisting of stress ranges, from the stress transfer function and wave spectrum, with the stress intensity factor (SIF) then calculated by empirical formula or a finite element model (FEM) to simulate the FCP. This is referred to as the stress spectral method. Unfortunately, there are two drawbacks when using load spectra obtained from stress transfer functions in FCP. (1) It is not easy to find the relationship between the SIF and the stress when the crack in the complex stress field. One needs to choose the appropriate stress (normally the maximum stress on the top surfaces), and then calculates the SIFs by empirical formula or FEA. However, many critical spots of marine structure are in complex stress states and may subject to tension, bending, shear, and non-uniform distributed boundary stresses synchronously. Thus, it can be difficult to determine how SIF can be calculated accurately from stress for the actual structure. (2) The boundary conditions of actual structures are very complicated, and a single stress value cannot represent the true loading conditions. That is, the SIF for a crack can be different under different load cases, such as loading conditions and heading angles etc., even for the same value of nominal or hotspot stress. Thus, it may be difficult to determine a harmonious relationship between stress and
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Nomenclature
f λ 0 λ2 da/ dN C, m
ω θ U
Angular frequency (rad/s) Heading angles Ship forward speed used in seakeeping analysis and spectral analysis, equal to 2/3 times design speed Hs Significant wave height Tz Wave period Hsif (ω|θ) SIF transfer function Wave energy spectrum S (ω|Hs , Tz ) Ssif (ω|θ,Hs , Tz ) SIF energy spectrum N-th order SIF energy spectral moment λn K SIF amplitude induced by waves ΔK SIF range induced by waves ΔKn Nominal SIF range induced by waves Kstatic SIF induced by still water bending moment ΔK eff Effective SIF range Equivalent SIF of mixed fracture modes K eq Kmax , Kmin Maximum and minimum SIF value Kres SIF induced by residual stress ς(K ) Probability density function of SIF amplitude induced by waves Probability density function of nominal SIF range ς ΔK (ΔKn ) induced by waves σx Rayleigh distribution scale parameter
β , β1 R MR ΔKth ΔKth′ dai E (·) ΔNi a, c a0 , c0 Δatol af
Average zero-up crossing frequency Zeroth, second order SIF energy spectral moment Fatigue crack propagation rate Coefficient and exponent in crack growth relationship Exponential parameters in Huang's model Stress ratio Correction factor for the effect of stress ratio Threshold of SIF range corresponding to stress ratio R=0 Value of ΔK when MR ΔK ≥ ΔKth Crack increment in a sea state i Expected value The number of load cycles in sea state i Depth and half length of the semi-elliptical crack The initial crack size Accuracy controlling parameter Critical crack depth
Subscripts
i I, II, III a, c
Sea state i Opening, sliding and tearing fracture mode Deepest point, surface ends point of a semi-elliptical crack
surface crack in a rectangular plate and a semi elliptical surface crack at the weld joint toe of a detail between hatch coaming and superstructure of a container ship in section 4 and section 5.
SIF for a crack in complex structural detail. This study proposes an approach to predict fatigue crack propagation incorporating spectral analysis. Section 2 presents a method to generate short term SIF distributions for real structural details. Combining the short-term SIF distributions and improved Paris formula to calculate FCP directly is described in section 3. Validation for the proposed spectral methods are demonstrated using a semi-elliptical
2. Spectral analysis for marine structures Linear elastic fracture mechanics (LEFM) is often used to predict FCP (Stephens et al., 2000). The empirical Paris Law has become a
Fig. 1. Proposed spectral analysis procedure for fatigue crack propagation in marine structures. 707
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2.2. Stress intensity factor transfer functions
widely recognized approach to estimate the FCP, based on the SIF range, rather than stress, and FCP can be obtained directly if the short term SIF distribution can be derived. Generally, fatigue analysis of marine structures involves the local structure. Fracture mechanics sub-model is employed to simulate FCP of actual structures, and the displacement boundary conditions of the sub-model should be obtained from the global structural response analysis. Fig. 1 presents a brief description of the proposed spectral based FCP. The Pre-processor mainly contains global hydrodynamic and structural response analysis and extract displacement boundary conditions for sub-model analysis. Then, according to the encounter waves and load state, continuously apply the corresponding boundary conditions to the fracture sub-model, calculate the SIFs of cracks with different sizes. Crack propagation can be simulated through update the sub-model step-by-step.
After the global structural stress analysis, dangerous regions with cracks are selected as the critical regions or hot spots. To balance the demands of computational efficiency and precision, element types of the whole ship FE-model are mainly two-dimensional shell element and one-dimensional beam element, and the element size is usually very coarse. To calculate SIFs for three-dimensional (3D) surface cracks, it is necessary to refine the shell elements in the vicinity of the hot spot, create solid elements with surface cracks and build multi-point constraints (MPCs) (ANSYS, 2013) between the shell and solid elements. Fig. 2 is an example of a sub-model in a ship structure. Boundary displacements obtained from the global structural analysis and all relevant local loads should be applied to the sub-models. The SIF of the crack is then calculated using FEM. The sub-model regions should be large enough to consider the load shedding induced by cracks growth and structural redundancy (Okawa et al., 2007). It needs to rebuild the sub fracture model when the crack size is large enough, and increasing the area of submodel, until the SIF value gradually converging. FEA has been widely used in SIF calculation of a 3-D crack in structural detail. Newman and Raju (1981) evaluated the SIF of a semi elliptical surface crack in a rectangular plate under uniform tensile and bending load. Bowness and Lee (1996, 2000) numerically calculated the SIFs of surface cracks in weld toe of T joints and proposed corresponding empirical formulations. Some scholars (Bowness and Lee, 1995; Chiew et al., 2001; Lee et al., 2005) proposed methods to model 3-D doubly curved semi-elliptical crack at weld toe in tubular joints. Some specialized software, such as Franc3d, Zencrack, FEACrack, etc.
2.1. Global structural response Fatigue damage in ships is mainly caused by wave loads acting on ship structures. For more reliable fatigue assessment, a direct calculation method is often used to evaluate the wave loads under different sailing conditions and loadings (American Bureau Of Shipping, 2016). The wave loads can be computed by hydrodynamic software, and structural analysis provides the structural stress/strain response by either simple hull girder theory or FEM. In general, the response is calculated by linear analysis in the frequency domain, as described by Lewis (1988).
Fig. 2. Finite element model of a structural detail with a semi-elliptic surface crack. Multi-point constraints (MPCs) were used to connect the shell and solid elements. 708
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2.4. Short term stress intensity factor distribution
provide non-planar crack growth prediction via automated adaptive meshing techniques. Dolbow and Belytschko (1999), Fries and Belytschko (2010) proposed a finite element method for crack growth without remeshing, i.e. extended finite element method (XFEM). These studies have greatly facilitated SIF calculation and simulation of FCP. The current study mainly focuses on a spectral method and how to model a fracture FE-model is not the focus of this article. The authors take semi-elliptical surface cracks at butt weld toe as an example, to illustrate the proposed spectral method. Referring to BSI, 2013, where the flaws often treated as an equivalent elliptical crack, two parameters were used to describe propagation of the surface crack, the crack depth, a, and the crack half-length, c. The current study followed common practice (Branco et al., 2015), singular element (dedicated to SIF calculation, collapsed from a 20 nodes hexahedral element) and SOLID95 element (20 nodes hexahedral element) was adopted when creating the solid model using commercial software ANSYS (2013), shown in Fig. 2. The radius of the first ring element at the crack tip, r = 0.0375a when a > 15 mm to ensure accurate results. This method has been validated by our research group (Yan et al., 2016a; 2016b). In the linear spectral analysis, global structural analysis is performed for a series of unit amplitude harmonic waves, ω , and various heading angles, θ . The SIF transfer function Hsif (ω|θ) is then obtained for a given crack size.
In a stationary sea state, it is generally assumed that the waves are zero-mean narrow-banded random Gaussian process (American Bureau Of Shipping, 2016). Thus, for linear systems, fluctuating stress/SIF induced by wave loads are also a zero-mean narrow-banded Gaussian process. According to the theory of random process, SIF amplitude induced by waves, K, will follow the Rayleigh distribution. The corresponding probability density function is:
ς(K ) =
ς ΔK (ΔK ) =
λ 0 and = 1/2π λ2 / λ 0 ,
(5)
3. Fatigue crack propagation calculation 3.1. Fatigue crack growth rate model Kujawski et al.(2001; 2004) proposed a two-parameter driving force model to account for R-ratio. The crack growth rate is expressed as:
da = C·(ΔK ∗)m dN
(6)
ΔK ∗ = (Kmax )α ·(ΔK+)1 − α = MΔK
(7)
ΔK+
Where α is the correlation parameter, and = ΔK for R ≥ 0 and ΔK+ = Kmax for R < 0 . This model is based on the premise that for a negative R-ratio, the negative part of ΔK does not contribute to the crack growth. However, it has been mentioned that the model was better for positive R-ratios than for negative ones from the comparison with the experimental data (Huang and Moan, 2007; Noroozi et al., 2007). Huang and Moan (2007; 2009) further studied the effect of R-ratio from an extensive analysis using experimental data from the literature, and proposed an improved FCP rate model. In the current study, Huang's model was selected in the spectral FCP, the basic expression is:
(1)
where S (ω|Hs , Tz ) is the wave energy spectrum. The nth order SIF energy spectral moment at the ship forward speed, U, is similar to the stress spectral moment (American Bureau Of Shipping, 2016), and can be expressed as ∞
0
(4)
where λ 0 and λ2 are the zeroth and second order moment of the SIF response spectrum.
Generally, the relationship between wave loads and induced stress is linear in spectral-based fatigue. For a given crack size, there is also a linear relationship between SIF and stress in LEFM, i.e., there is a linear relationship between SIF and wave loads, which is a precondition for applying spectral analysis. For the fatigue assessment of ships, the response under real wave environments is often divided into a series of stationary process, known as sea states. A sea state can be described by the classic wave energy spectrum, S (ω|Hs , Tz ) , for example, Pierson-Moskowitz (P-M) or JONSWAP (Lewis, 1988), which can be expressed as a function of significant wave height, Hs, and wave period, Tz. Thus, for a given crack size, the SIF energy spectrum under arbitrary sea states can be computed by
∫
ΔKn ΔK 2 exp ⎛⎜− 2 ⎟⎞, 4σx2 ⎝ 8σx ⎠
where σx , and the average zero-up crossing frequency, f can be expressed as
2.3. Spectral analysis
λn =
(3)
Theoretically, compression force does NOT have SIF values, in the fatigue crack growth model, we employed equivalent SIF to consider the effect of the mean stress and the residual stress is considered as part of the mean stress. The SIF range, ΔK , equals to two times of SIF amplitude and coresponding probability density function, can be expressed as:
σx =
Ssif (ω|θ,Hs , Tz ) = Hsif (ω|θ) 2 *S (ω|Hs , Tz ),
K K2 exp ⎛⎜− 2 ⎟⎞. 2 σx ⎝ 2σx ⎠
ω − ω2Ucosθ / g n Ssif (ω|θ,Hs , Tz ) dω. (2)
This study used a wave scatter diagram to describe the expected sea states encountered. The North Atlantic scatter diagram was selected, and a sea state was assumed to last two hours and to be described by the P-M spectrum. Suppose that each short-sea condition lasts two hours and determine the number of cycles in two hours based on the average zero-crossing rate. Generate load block for each short-sea condition which consists of a number of random stress ranges that following the Rayleigh distribution. Determine the probability of occurrence for each load block based on the wave scatter diagram. The total number of cycles is determined by the average zero-crossing rate. Various short-sea conditions appear randomly in accordance with a certain probability constitute the long-term fatigue load spectrum.
da = C [(ΔKE )m − (ΔKth)m], dN
(8)
ΔKE = MR ΔK,
(9)
and
(1 − R)−β1 (−5 < R < 0) ⎧ ⎪ MR = , (1 − R)−β (0 ≤ R < 0.5) ⎨ ⎪ (1.05 − 1.4R + 0.6R2)−β (0.5 ≤ R < 1) ⎩
(10)
Kmin + Kres −0.5ΔKn + Kstatic + Kres = , Kmax + Kres 0.5ΔKn +Kstatic + Kres
(11)
R=
where N is the number of cycles, ΔKE is the equivalent SIF range, C and m are the Paris parameters corresponding to stress ratio R = 0, ΔKth is the threshold of SIF range corresponding to stress ratio R = 0, β and β1 709
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Substituting equations (13) and (8) into (12), the expected crack increments in a sea state can be expressed as
are shape exponents depending on the material property and environment, MR is the correction factor for the effect of stress ratio, Kmin is the minimum SIF, Kmax is the maximum SIF, Kres is the SIF induced by residual stress, and Kstatic is the SIF induced by static load. Since Huang model is valid for R > − 5, Kstatic and Kres are set to zero for safety consideration when mean stress induced by still water bending, σstatic and residual stress are negative. In this study, owing to absence of specified material parameters, the Paris law parameters for steel recommended by International Institute of Welding (IIW) (2016) were adopted as C= 1.65 × 10−11, m = 3, and ΔKth = 5.4 MPa m for da/dN in m/cycle, and ΔK in MPa m . For demonstration, the exponential parameters of steel in Huang's model were chosen as β = 0.5, β1 = 0.8 (Huang and Moan, 2007).
E(dai ) = C
⎭
0
ΔTi 2π
λ2i , λ 0i
+∞
x mMR (x )mςiΔK (x ) dx − ΔKthm
⎝ ΔKth′
∫ ςiΔK (x ) dx ⎞⎟.
ΔKth ′
⎠
If the FEM mesh and ς ΔK (ΔK ) are updated for every stationary sea state, although the result would be more accurate, too much time would be required to perform the FCP simulation. Therefore, the forward Euler method has been widely used in FCP simulation (He et al., 2014; Mao, 2014; Sumi et al., 2005). When the crack increment is < Δatol , e.g. 1 mm, the FEM mesh is not updated, to save calculation time. Unfortunately, there is a major drawback associated with the forward Euler method, in that it requires choosing a small step size to maintain stability and accuracy of the solution. Thus, an improved Euler method is presented in the current study, the main procedure is shown in Fig. 3. 4. Validation of the proposed spectral based method
(12)
The crack growth analysis of a semi-elliptical surface crack in a structural detail was performed by using the proposed procedure to check the accuracy of the proposed spectral method. It was assumed that the structural detail had the same stress state, i.e., uniform tension, as shown in Fig. 4. Thus, there was a consistent relationship between stress and SIF. A widely recognized empirical formula proposed by Newman and Raju (1981) was used calculate SIF. We also computed it numerically using ANSYS, which could be used for more complex
with
ΔNi = ΔTi fi =
⎜
3.3. Crack growth within a computational step
∞
da ⎫ (x )*ςiΔK (x ) dx , ∫ dN ⎬
+∞
⎛∫
Where ΔKth′ is the value of ΔKn when MR ΔKn ≥ ΔKth (Since xMR (x ) is strictly monotonic increasing, there must exists a unique x that meets xMR (x ) = ΔKth , the unique value x namely is ΔKth′ ). It allows to compute E(dai ) easily through equation (14) using numeral integral.
In general, FCP under the one sea state is vanishingly small, and thousands of sea states may be needed for a small crack to grow to a given length, e.g. 1 mm. Therefore, during the ith sea state, the SIF transfer function can be treated as constant. Thus, the SIF range probability density function is also constant. In a sea state, since the SIF da range, X, is a random variable, the crack growth rate, dN (X ) , is a function of random variables, the expected crack increment, E (dai ) can be expressed as
⎧ ⎨ ⎩
λ2i λ 0i
(14)
3.2. Closed form expression for fatigue crack propagation during one sea state
E (dai ) = CΔNi
ΔTi 2π
(13)
where ΔNi is the number of load cycles in sea state i , ΔTi is the duration time in one sea state, fi is the average zero crossing frequency.
Fig. 3. Flow chart of numerical procedure for the improved Euler method, where a, c are the depth and length of the elliptical crack, respectively; a0, c0 are the initial crack size; Δatol is the accuracy controlling parameter, and for good precision, Δatol should be 1–2 mm. 710
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Fig. 4. Plate with surface crack, initial crack size a0 = 5 mm, c0 = 20 mm.
Fig. 5. Stress transfer function.
Fig. 6. Stress intensity factor (SIF) transfer functions evaluated at (a) the deepest points, and (b) the surface points of the semi elliptical crack.
structural details. The flat plate size is 600 × 300 × 65 mm; with initial crack size a0 = 5 mm, c0 = 20 mm; and critical crack depth, af = 42 mm. There is no residual/mean stress in the structure, and only head waves were encountered. Fig. 5 shows the nominal stress transfer functions. First, the short-term distributions of stress ranges were obtained from the stress transfer function in combination with the wave scatter diagram. FCP life was then calculated based on empirical SIF formula and FCP model discussed in Section 3.1, using the cycle-by-cycle (CBC) counting method (Cui et al., 2011), a time-consuming but accurate method to calculate FCP. Alternatively, the SIF transfer function was derived from ANSYS, and we also calculated it using the Newman-Raju empirical formula. Fig. 6 shows that the results are in close agreement. The short term ΔK distributions were obtained following Sections 2.3 and 2.4. Then, FCP life was calculated following Section 3. Calculation accuracy may differ due to the different crack growth calculation steps, Δatol , and the crack growth curves for different step sizes and stress spectral method using CBC are shown in Fig. 7. The FCP life obtained from SIF spectral method closely agrees with stress spectral method when the stress state (loading mode) of critical spot are the same or similar among different load cases. The Euler and improved Euler methods both provide good precision with small step size. However, the improved Euler method shows better consistency with the CBC. When the step size was set to 1–2 mm, sufficient accuracy could be
obtained through the improved Euler method. 5. Example application of the proposed method to a container ship detail To demonstrate the proposed spectral methods, a joint between the hatch coaming and front bulkhead of superstructure in a large container ship was selected as a real structure example. Fig. 8 shows the sketch of hot spot location. Fig. 9 shows the detail structural geometry of the hot spot. The hatch coaming plate and bulkhead plate were welded together. And the fillet weld was assumed to have a throat thickness of 10 mm, and the weld toe radius ρ = 1 mm. The hatch coaming was constructed from E40 steel with yield stress σY = 390 MPa, and a semielliptical surface crack at the weld toe found in an inspection, with initial crack size a0 = 2 mm, c0 = 8 mm. The proposed spectral based FCP method was used to compute the sailing hours until the crack depth grows to af = 0.6T , for safety consideration (Yan et al., 2016a; 2016b). For a flaw lying in a plane parallel to the welding direction, residual stress σres should be considered. Referring to BSI, 2013, residual stress was assumed to be 20% of σY in the structures subject to post-weld heattreated. Then, Kres could be calculated through weight function method (Wu and Carlsson, 1991). The mean stress induced by still water bending, σstatic approximately equal to 39 MPa tensile stress for the full load case. 711
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Fig. 9. Detail structure of hot spot.
5.1. Global structural stress response Hydrodynamic software WALCS developed by the China Classification Society (2013) was used to compute the wave loads on the target marine structure, based on 3D frequency domain linear hydrodynamics principles. Parameters for the hydrodynamic analysis were:
• wave frequency = 0.1–1.8 rad/s, step = 0.1 rad/s, 18 frequencies; • ship speed, U = 2/3 times design speed, the design speed is 25 knots; • heading angle = 0–180°, step = 30° (7 directions, given the symmetry of the ship hull; heading angle 180° represents the head sea, and a heading angle 0° denotes the following sea).
Only the full load condition was considered in this study. For each wave with determined heading angle and frequency, unit wave amplitude was used to generate the real and imaginary hydrodynamic wave pressures on the panels. After FE-analysis, the stress and displacement responses of the target structure for different wave frequencies and directions were obtained. Fig. 10 shows the global structure response of the container ship induced by waves load.
Fig. 7. Comparison of fatigue crack growth with different methods. (a) the Euler method with different crack size interval and the cycle by cycle method, (b) the improved Euler method with different crack size interval and the cycle by cycle method.
5.2. SIFs for different load cases The structural detail shown in Fig. 11 was a sub-model of the whole ship FEM model, and it was modeled following procedure described in Section 2.2. Three types of crack opening modes were considered, i.e., opening (Mode I), sliding (Mode II), and tearing (Mode III). Owing to the imaginary and real part of wave pressure are loaded separately, the 2 2 + Kimaginary SIF amplitude becomes K = Kreal . The SIF transfer function at different heading angle shown in Fig. 12 and the data in Table 1 and Table 2 show that the Mode I constitutes the main component of the SIF for both surface end point and deepest point. More specifically, compared with surface end point, the effect of Mode II and Mode III is smaller in the deepest point. In head and following waves, SIFs of cracks have smaller share of Mode II than that in oblique and beam waves. This can be caused by shearing force induced by the torsion and horizontal bending. The equivalent SIF (Bowness and Lee, 1996) is
K eq =
KI2 + KII2 +
2 KIII , 1−ν
(15)
where ν is Poisson's ratio. The average proportions of KI , KII and KIII to K eq in a heading angle are summarized in Table 1 and Table 2. The share of Mode II and Mode III will be less for larger crack size, and more for smaller crack size. But for simulate the crack growth direction, it can be done by rebuilt the crack increment with an angle determined by an adopted criteria, such
Fig. 8. Schematic diagram of target structure.
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Fig. 10. The representative global structure response induced by waves load (target spot is in the red circle). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 11. FEM model of the detail with a surface crack in ANSYS (a sub-model of the ship).
inconsistent relationship. The equivalent SIF value at unit HSS can be obtained through SIF transfer functions divided by corresponding HSS transfer functions, as shown in Fig. 14. It is difficult to define a harmonious relationship between hot spot stress and SIF that consider all the load cases. More specifically, an approximate consistent correspondence can be founded in heading waves and following waves, i.e. 180° and 0°, the stress status of the detail is near uniform tension longitudinally, as shown in Fig. 15, while in the oblique waves, the ship structure is under horizontal bending and torsion. The stress status of local structure mainly is under combined tension and bending, as shown in Fig. 15. The degree of bending (DoB) (Ahmadi et al., 2015) in hatch coaming plate varies with waves frequency, as shown in Fig. 16, and, there is a similar trend between Figs. 14 and 16. It can be the specific reason for the inconsistent relationship. The results represent
as the maximum circumferential stress criterion in the submode when the K_II or K_III is big enough. This issue will be discussed in the future. 5.3. Inconsistent relationship between SIF and stress in different load cases Since the nominal stress is not always possible to define in real structures, hot spot stress (HSS) is widely used in the S-N curve spectral method. The HSS is adopted to illustrate the inconsistent relationship. And it was generated by the recommended extrapolation approach (American Bureau Of Shipping, 2016). The HSS amplitude is 2 2 σh = σreal + σimaginary . Fig. 13 shows the HSS transfer functions of the local structure. For a relative large crack size, i.e. a = 10, a/c = 0.3, the K eq is basically equal to KI (see Table 2). K eq is used to illustrate the 713
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Fig. 12. SIF transfer function (a = 5, a/c = 0.3). (a) the deepest points, and (b) the surface points of the semi elliptical crack. 714
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Table 1 Average proportions of KI , KII and KIII to K eq in a heading angle (a = 5 mm, a/ c = 0.3).
KIa KIIa KIIIa KIc KIIc KIIIc
180
150
120
90
60
30
0
99.5% 7.3% 5.6% 97.0% 5.5% 19.5%
96.6% 13.8% 14.3% 93.0% 19.6% 22.5%
93.8% 14.0% 18.4% 92.8% 23.1% 22.6%
96.7% 13.4% 13.9% 92.5% 21.7% 24.4%
98.0% 12.1% 11.4% 94.6% 15.9% 22.1%
98.9% 11.5% 7.2% 95.4% 12.6% 22.1%
98.2% 7.2% 10.9% 96.3% 5.0% 19.1%
Table 2 Average proportions of KI , KII and KIII to K eq in a heading angle (a = 10 mm, a/ c = 0.3).
KIa KIIa KIIIa KIc KIIc KIIIc
180
150
120
90
60
30
0
99.2% 0.9% 6.9% 97.2% 7.4% 17.1%
99.0% 5.7% 12.8% 95.6% 18.0% 20.6%
97.5% 5.7% 12.1% 93.9% 19.8% 20.2%
98.4% 3.7% 11.2% 94.3% 18.6% 21.5%
99.2% 3.2% 8.4% 96.1% 12.4% 19.4%
99.7% 2.7% 9.0% 96.7% 13.6% 19.5%
98.1% 1.4% 11.3% 96.4% 7.2% 16.8%
Fig. 14. Equivalent SIF value at unit HSS. It is difficult to determine a harmonious relationship between stress and SIF. (a) the deepest points, (b) the surface points of the semi elliptical crack.
eliminate the effect of uncertainty of pseudo-random stress signal. Comparison of crack increment computed by the two methods are presented in Table 3. Apparently, different relationships between stress signal and SIF have quite large impact on FCP in CBC method. According to Fig. 14, when using the stress spectral method, it could be difficult to determine an exact relationship that take into account of all the heading angles and frequencies, while the proposed method would not have the problem. It is noticed that the relationships almost are consistent in heading angle 180 and 0, and the corresponding FCP results using CBC and integral are nearly equal. It confirms that SIF spectral method closely agrees with stress spectral method when the loading mode of critical spot are the same or similar among different heading angles and frequencies.
Fig. 13. Hot spot stress transfer functions.
only one structural detail. However, for other details, it is foreseeable that there is an inconsistent relationship between stress and SIF if the loading situation of local structure changes considerably among different load case, otherwise, the relationship would be consistent.
5.4. Comparison of crack increment in a short sea state For comparison, we take the crack increment in a sea state when the crack depth, a = 10 mm, crack aspect ratio, a/c = 0.3, for example. Since the crack increment in a short sea state is very small, both the SIF transfer functions in proposed method and the relationship between stress signal and SIF in stress spectral method remain constant. The significant wave height of the short sea state is 8 m and the wave period is 11 s. In the stress spectral method, we calculate FCP through CBC method using the simulated stress signal based on the calculated stress spectra. The relationship between stress signal and SIF, i.e. SIF value at unit HSS, as shown in Fig. 14, is not consistent in different heading angles and frequencies. We chose relationship in a frequency that the HSS transfer function is at its maximum to calculate FCP. The CBC method was repeated 100 times, and the average crack increment was used to
5.5. Fatigue crack propagation Following the improved Euler method in Section 3, the FCP of the structural detail was simulated step by step with step size Δatol = 1 mm . A program code was developed to update the FEM mesh automatically and calculate the fatigue life. The probability of heading angle is assumed to follow a uniform distribution. Fig. 17 shows the FCP curves, using K eq and KI , the expected fatigue life is 35244 h (approximately 35244/365/24 = 4.02 years) and 37418 sailing hours respectively. For simulating FCP by FEA, additional calculation of KII and 715
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Fig. 15. Stress distribution of intact local structure (half of the model in Fig. 9).
Fig. 16. The DoB in hatch coaming of local structure under different load cases.
Fig. 17. Fatigue crack propagation curves with uniform distribution of heading angle.
Table 3 Comparison of crack increment computed by two methods in a short sea state. Heading angle
180 150 120 90 60 30 0
crack increment (half length, mm)
crack increment (depth, mm)
CBC (1e5)
Integral (1e5)
Relative value (%)
CBC (1e5)
Integral (1e5)
Relative value (%)
4.61 1.55 5.22 0.23 3.15 3.44 1.40
4.62 2.24 4.55 0.17 2.72 3.61 1.4
−0.24 −31.00 14.81 35.35 15.69 −4.60 0.21
4.53 1.59 4.22 0.17 2.59 3.10 1.38
4.54 2.11 3.76 0.14 2.31 3.22 1.38
−0.17 −24.58 12.28 17.96 12.19 −3.58 −0.11
changes of the surface crack using K eq . Since the SIFs were evaluated according to the actual boundary conditions, this method automatically considers the impact of stress distribution through the plate thickness. 6. Conclusions A compatible relationship may not exist between SIF and nominal or hotspot stress for a specific crack size at different loading conditions, heading angles, and wave frequencies. Therefore, a method to calculate the FCP from multiple short-term SIF distributions which derived from the SIF transfer functions and wave spectrum has been proposed. Some conclusions can be drawn. . The proposed method needs to calculate the SIF transfer function for crack at different sizes which requires too much computation cost for reanalysis and updating the FEM model. Therefore, an improved Euler method was proposed to reduce the computation steps and improve precision.
KIII almost does not increase the computational cost, and neglecting the influence of KII and KIII would lead to no conservative results. Therefore, K eq is recommended in the present demo. Fig. 18 shows shape 716
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Fig. 18. Geometric change of the surface crack using equivalent SIF.
A case study of a real structural detail with an initial semi-elliptical surface crack was used to demonstrate the application of the proposed procedure. The results showed that the expected service time for the ship was approximately 4 years sailing before the crack depth reached 0.6 T. The proposed spectral based method which uses short term SIF distribution instead of short term stress distribution to predict the FCP of a detail under random fatigue loading is a new trial and still in its early stage. Compared with the stress spectral method, SIF spectral method is still complicated in computation of SIFs. Further work should be focused on overcoming this drawback and developing a much faster and easier use of numerical computation method.
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