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Reliability analysis of fatigue life of the connectors—the US Mobile Offshore Base
Marine Structures 15 (2002) 233–250
Reliability analysis of fatigue life of the connectorsFthe US Mobile Offshore Base D.V. Ramsamooja,*, T.A. Shugarb a
Department of Civil Engineering, California State University, P.O. Box 6870, Fullerton, CA 92834-6870, USA b Naval Facilities Engineering Service Center, Port Hueneme, CA, USA
Received 1 May 2001; received in revised form 25 August 2001; accepted 3 October 2001
Abstract Reliability-based design analysis of the fatigue life of the connectors of the five sections of the 2-km long US Mobile Offshore Base (MOB) is demonstrated. A performance function is defined in terms of the nominal stress range, inherent defect or starter crack, and appropriate material properties, which are considered random variables. The reliability analysis is performed for a sea state 1–8 (SS1–8) random loading having a Gumbel distribution. Where possible, uncertainty data for random variables are obtained from published data relating to the fatigue of metal and metal alloys. Otherwise, judgmental coefficients of variation are prescribed for purposes of demonstration. The fatigue life is assumed to follow the Weibull distribution. The reliability function is defined in terms of the mean life and the total uncertainty in the fatigue life. Preliminary reliability calculations suggest that current design stress levels be reduced to meet the current fatigue life target reliability level for the MOB connectors. An illustrative design is demonstrated and the metal selected for a fatigue design of the connectors for 10 million cycles with a reliability of 0.99 at a nominal stress of 203 MPa is HY-130 steel. r 2002 Published by Elsevier Science Ltd.
*Corresponding author. Tel.: +1-714-278-3968; fax: +1-714-278-3916. E-mail addresses: [email protected] (D.V. Ramsamooj), [email protected] (T.A. Shugar). 0951-8339/02/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 0 9 5 1 - 8 3 3 9 ( 0 1 ) 0 0 0 2 8 - 4
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Nomenclature a0 ; b0 af c E L N N; Nf R R R% S T Z dg Dg KI ; KII ; KIII DKI KImax DKth; 0 DKth sZ OZ mZ
Semi-minor and semi-major axes of the semi-elliptical starter crack Values of a at failure Crack length in the connector cylindrical tube Young’s modulus Reliability function Cyclic strain hardening function Number of cycles and fatigue life Yield strength Minimum stress/maximum stress Radius of cylindrical connector tube Nominal stress Thickness of cylindrical connector tube Performance function Scatter of the data about the model prediction line Inaccuracy of the fatigue model Modes I, II and III stress intensity factors (SIFs) Increment of the SIF in Mode I Maximum value of KI Threshold SIF at R ¼ 0 Threshold SIF at any value of R Standard deviation of Z Total uncertainty in Z Mean value of Z
1. Introduction 1.1. Brief description of the US Mobile Offshore Base (Brown and Root concept) The Mobile Offshore Base (MOB) program is advancing the design of five platforms with fundamentally different degrees of module connectivity [1]. The design by Brown and Root [2] comprises five semi-submersible units connected by eight connectors at the module interfaces. This design concept is presented in Fig. 1. The objective of this paper is to perform a reliability-based design analysis of the fatigue life of the Brown and Root [2] connectors between the modules, using fracture mechanics. 1.2. Rigid connector conceptual design The Brown and Root Rigid MOB connection system design is based on Tension Leg Platform (TLP) connection systems that are currently in operation. Each connector consists of male and female halves, made of 6.1 m diameter, 152 mm wall
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Fig. 1. Rigidly connected semi-submersible modules of the US Mobile Offshore Base.
thickness tubing. Each half pivots on a 3.66 m diameter tube, and can move or slide on the pivot tube. Both sliding and rotating are accomplished with hydraulic cylinders. This provides assistance in alignment for connection and latching purposes. Figs. 2(a) and (b) show the proposed ABB Vetco Grey latching system. The conceptual design deploys eight rigid connectors. The first four are connected at module joining. The second two are required to be connected before encountering sea state 6, and the final two must be connected before encountering sea state 8. The Buckled Strut elastomeric members provide limited flexibility in the Brown and Root connection system. They are located in an outer circle around the connector well. Their purpose is threefold. Firstly, they provide for initial shock attenuation upon impact of the modules during mating. Secondly, they are compressed upon latching such that they are pre-loaded in the connected mode. This means that no movement is encountered until the pre-load is reached, then limited flexing can occur in pitch rotation. Thirdly, upon disconnecting, there is stored energy present to push the modules apart. Pull-in cylinders will be provided to make the final pull-in, compress the buckled struts, and engage the latches. Once a rigid connector is latched, some type of shear mechanism will be required to eliminate movement at the module interface. The details are given in Brown and Root [2]. The ABB Vetco Gray latching interface conceptual design is similar to those used on TLP connectors. Forty-eight actuation wedges around the 6.1 m diameter circumference were required to reach the initial conceptual design load of 100,000 tonnes per connector. This load is the design load or the maximum operating load based on a maximum yield of 448 MPa with a safety factor of 1.5 based on yield, and is analyzed in detail by Finite Element Analysis. Fig. 3 shows an enlarged view of the connector, and Fig. 4 shows the Von Mises stresses for an axial load of 100,000 tonnes determined by ANSYS finite element program.
2. Fracture mechanics reliability-based analysis An empirical method, using the UK’s DOE S–N curves [3] cast within a reliability framework [4,5] handles fatigue design at the structural component level for MOB. Ang et al. [5] developed a technical procedure for a reliability-based approach to fatigue analysis and applied it to the stress-range data measured on an existing LNG tanker and to a newly designed double hull tanker with ABS’s DLA rating. In both
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Fig. 2. ABB Vetco Grey latching system for connector.
cases the level of reliabilty was only 0.85. Other equivalent reliability models are also based on the same physics of fatigue [6]. Fatigue life estimates vary greatly according to the values of the COVs assumed. Estimates of COVs are given in Mansour et al. [7]. Other methods of reliability design, including the direct-reliabilty-based design and the load and resistance design (LRFD) are discussed by Assakkaf et al. [8]. However, MOB connectors are a key technology issue by virtue of their importance, uniqueness and high estimated cost of approximately $80 million per
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Fig. 3. Enlarged view of the rigid connector.
module interface, for one particular concept. The ABS Guide [9] currently specifies a lifetime target reliability of 0.9999 for fatigue design of connectors. Further, the MOB Connector Working Group has estimated that a reasonable design life for the connectors is 5–10 years. Moreover, there are no SN-curve data in existence for large monolithic connector concepts, which currently involve plate thickness of 150 mm or greater. Accordingly, fundamental approaches to MOB connector design are therefore warranted. Thus a fracture mechanics approach to fatigue design has been recommended by the MOB Standards and Criteria Working Group and prescribed by the ABS Guide. Ramsamooj [10] presented a newly developed linear elastic fracture mechanics method (LEFM) for fatigue design. In the present paper, this method is cast within a reliability-based design procedure.
3. Fatigue crack propagation An approximate analytical model [10–12] for the rate of crack propagation under constant amplitude cyclic loading is expressed by da 5 1 ¼ 2 ðDKI DKth Þ2 dN 3p ð1 þ nÞER 1 ðKImax =KIc Þ2
ð1Þ
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Fig. 4. Von Mises stresses computed by ANSYS for the connector.
in which E is Young’s modulus, R is the yield strength, KIc is the fracture toughness, KImax is the applied stress intensity factor range (Mode I), which measures the intensity of the stress field in the vicinity of the crack, n is the cyclic strain hardening coefficient, and DKth is threshold SIF below which the crack growth is smaller than 1010 m/cycle, according to the ASTM definition. Eq. (1) is for laboratory air environment. Corrosion fatigue is not considered in this paper. The threshold variation with the stress ratio R (minimum/maximum stress) can be derived from the ASTM definition, as follows. From Eq. (1) dc 1 ¼ CðDK DKth Þ2 ; ð2Þ dN 1 ðKmax =KIc Þ2 where C ¼ 5=ð3p2 ð1 þ nÞERÞ: For R ¼ 0; KImax ¼ DKth;0 : When DK ¼ DKth ; the net SIF range is DK DKth ; the FCGRE0, or dc 1 ¼ 0 ¼ CðDK DKth;0 Þ2 : ð3Þ dN 1 ðDKth;0 =KIc Þ2 When Kmax ¼ DKth;0 ; but R a0; DK ¼ DKth;0 ð1 RÞ: Therefore dc 1 ¼ 0ECðDKth;0 ð1 RÞ DKth Þ2 ; dN 1 ðDKth;0 =KIc Þ2 DKth ¼ DKth;0 ð1 RÞ:
ð4Þ ð5Þ
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This expression for the threshold variation with R shows good agreement with the experimental data [12], except for large positive or negative values of R: In particular for R ¼ 0:9; the experimental data is about 10% smaller. This is probably due to the residual tension caused by compression part of the load. The fatigue life is then Z af 1 da ð6Þ Nf ¼ da=dN a0 in which a0 is the size of the starter crack, which is usually the size of the largest defect, depending on its location, af ¼ KIc2 =pS 2 ; is the crack length at failure and S is the applied stress. The integral needs to be evaluated numerically. We have compared the model predictions for 37 sets of data, including steels, steel alloys, aluminum and aluminum alloys, titanium, nickel and plastics [10–12]. Despite the fact the fatigue crack propagation rates of the above metals and plastics at the same stress intensity factor extend over a range of 1000, there was good agreement in almost all cases between the model predictions and the experimental data. We know of no other model having the overall visual resemblance fidelity exhibited in the numerous graphs presented in our papers. Some of the advantages of the analytical model presented herein are: (a) It is formulated in terms of the material properties E; R; KIc and DKth;0 : There are no empirical fitting constants. It captures the physical process of crack growth as discovered by numerous experimental crack propagation measurements. (b) There is no need for the several regions of crack growth. It treats FCGR as a continuous process. (c) There is no scaling factor. The fatigue behavior of any size structure can be predicted. According to Paris and Hermann [14] ‘‘unifying theories and models remain badly needed as a substitute for zillions of data points required by all of the potential applications if we do not develop unifying theories’’. (d) It provides a means of quantitative evaluation of alternative materials and for making trade-offs to optimize the design process for engineering efficiency and business economy.
4. Performance function Z A performance function Z is defined as Z ¼ Nf ¼ gðS; a0 ; KIc ; DKth ; R; EÞ;
ð7Þ
where S is the stress level. The stress ratio is fully accounted for DKth : Let mZ and sZ be the mean and standard deviation of Z: Then from a first order approximation, the mean value of Z is mZ ¼ gðmS ; ma0 ; mKIc ; mKth ; mR ; mE Þ:
ð8Þ
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and the standard deviation of Z is given by s2Z ¼
6 X
c2i s2Xi þ
1
6 X 6 X 1
ð9Þ
rij ci cj sXi sXj ;
1
where c1 ¼
qNf ; qS
c5 ¼
qNf Nf ¼ ; qR R
c2 ¼
qNf ; qa0 c6 ¼
c3 ¼
qNf ; qKIc
c4 ¼
qNf qKth
qNf Nf ¼ ; qE E
ð10Þ
where sX 1 ¼ sS ; sX 2 ¼ sa0 ; similar terms exist for KIc ; Kth ; R; and E: The term rij is the correlation between the ith and jth parameters. Since there is no correlation between the parameters, rij ¼ 0: The total uncertainty in the fatigue life is expressed by [4] O2Z ¼ O2g þ
1 2 2 ðc s þ c22 s2a0 þ c3 s2KIc þ c4 s2Kth þ c25 s2R þ c26 s2E Þ; m2Z 1 S
ð11Þ
where O2g ¼ d2g þ D2g ; dg is the (average) scatter of the mean fatigue life data about the model prediction line for the test specimens, and Dg is the inaccuracy of the fatigue model in the predicted mean life associated with the imperfection in the fatigue theory, including the effects of impact. Impact forces may result from mooring of ocean-going vessels alongside the MOB. Eq. (11) uses a first order approximation or a small variance theory, which may be applicable for COVs of 0.15–0.30. Since the COV of the inherent flaw or starter crack is estimated to be about 0.5, there may be a significant error. In such cases, Ang [4] recommends a second order approximation. However, noting that qNf sa ¼ ðNf Þa0 ðNf Þa0 þsa 0 qa0 0
ð12Þ
is the change in the fatigue life due to a change of one COV in a0 ; and that Eq. (6) has to be differentiated numerically anyway to get terms such as qNf =qa0 ; the total uncertainty may be expressed as O2Z ¼ O2g þ
1 ððDNfS Þ2 þ ðDNfa0 Þ2 þ ðDNfKIc Þ2 þ ðDNfR Þ2 þ ðDNfE Þ2 Þ2 ; m2Z
ð13Þ
where DNfa0 is the change in the fatigue life due to a change of one COV in a0 ; or DNfa0 ¼ ðNf Þa0 ðNf Þa0 þsa : 0
ð14Þ
These terms are obtained numerically from the computer program called MOBYDICK.
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5. Reliability function Pertaining to the reliability analysis of the MOB connectors, the American Bureau of Shipping (ABS) [9] states ‘‘A direct reliability method is preferred. Unnecessary simplifications to the limit state or the basic variables, such as the lognormal assumption for basic variables as a substitute for rigorous analysis of the data, should be avoided. In many situations, however, selective assumptions can significantly reduce the computations without compromising the accuracy of the analysis. For example, Ang [4] and Ang et al. [5] assumed the fatigue life could be represented by a Weibull distribution. Unlike a normal distribution, a Weibull distribution has a monotonically increasing hazard function, which is physically consistent with the cumulative nature of fatigue damage’’. The lognormal distribution may fit some data better, but the hazard function decreases at a later life, which is contrary to the nature of fatigue damage. The probability of no fatigue failure or the reliabilty function is [4] ! )O1:08 Z Nf 1:08 LðNf Þ ¼ exp ; G 1 þ OZ Nf (
ð15Þ
where N% f =the fatigue life predicted from Eq. (1), herein after referred to as ‘‘the mean fatigue life’’ using the mean values of each parameter, and Nf is the fatigue life for a specified reliability, herein after referred to as ‘‘the reduced fatigue life’’, GðxÞ is the gamma function, OZ =the total uncertainty in the fatigue life, assuming that the minimum performance level is zero. Thus the fatigue life for a specific reliability is dependent on the mean fatigue life and the total uncertainty.
6. Uncertainty in fatigue life variables The reliability parameters consisting of the means and standard deviations are discussed below. Due to the huge monetary investment that a MOB implies, the cost of developing data beyond usual standards should be evaluated in reliability-based design. The uncertainty associated with each random variable is discussed below. 6.1. Wave load-induced stress range Wave loads have significant uncertainty. They are developed from random environmental force input data, and computed by imperfect hydrodynamic computer programs. The resulting uncertainty in the nominal stress range can be substantial. Sea state data in this study were based on standard conditions for the North Sea and were taken from Schutz and Pook [15]. They are reproduced in Table 1. The relative occurrence of sea states within small intervals of significant wave height H1=3 was for use in calculating fatigue loads on fixed and moored platforms. Although
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Table 1 Approximation of sea state data (from Schutz and Pook [15]) Sea state
H1=3 a (m)
Range (m)
Wave period (s)
Time (%)
0 1 2 3 4 5 6 7 8 9 10 11
1.75 2.55 3.4 4.15 4.80 5.40 6.15 6.90 7.80 8.80 10.35 13.60
0.00–1.95 1.95–2.85 2.85–3.80 3.80–4.45 4.50–5.10 5.10–5.75 5.75–6.55 6.55–7.35 7.35–8.30 8.30–9.40 9.40–12.55 12.55–
5.9 6.6 6.9 7.3 7.7 8.0 8.4 8.7 9.1 9.5 10.2 11.5
38.5 28.5 17.5 7.18 3.40 2.16 1.31 0.678 0.334 0.156 0.0797 0.0043
a
Average height of the highest 1/3 of all the waves in a particular sea state.
more applicable to fixed platforms than to floaters, it is assumed that wave loads are linearly proportional to wave heights. The MOB modules are to be disconnected in SS7 or SS8, freeing the connectors from SS9 and greater load-induced stresses. The maximum nominal stresses are correlated with the average H1=3 value for the highest applicable sea-state (SS8) or 7.84 m. The H1=3 waves of SS8 are generally more damaging than the lower sea-state waves. However, for high cycle fatigue, a relatively few very high waves are not as critical as for fracture. During the connected mode operation, the MOB will have an automatic wave-measuring device, and the connectors will be fully instrumented with strain gages and other sensors. Preliminary comparisons of predictions to experimental data for assessment of the linearized hydrodynamic computer programs for MOB application have been completed (Bechtel National, Inc. [16]). The results suggest a COV from 0.1 to 0.2 for prediction of motion response of the five connected MOB single base units in regular waves. However, physical model testing that includes hydroelastic behavior is needed to assess this uncertainty accurately. Uncertainty in the load-induced stress is largely obviated if it is assumed that conditions leading to crack growth in the MOB connectors will be continuously monitored, and the crack growth is controlled through manual intervention. Accordingly, a rather low COV of 0.05 was selected for load-induced stress range. 6.2. Starter crack Knowledge of the maximum crack sizes is critical to the success of LEFM applications in both design and maintenance of MOB connectors. Recent improvement in non-destructive evaluation (NDE) methods, such as magnetic and eddy-current methods, allows detection of cracks as small as 0.1 mm, with confidence. This relates to precision of measurement and thus to scatter or
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Table 2 Reliability parameters for HY-130 steel
a
Starter crack (mm) Hot-spot axial. stress (MPa) Fracture toughness (MPam0.5) Yield strength, MPa Young’s modulus, MPa Fatigue life model: Scatter dg Uncertainty Dg a
Mean
SD
COV
0.1 517 120 1007 207,000
0.05 25.8 26.4 30.2 4140
0.50 0.05 0.12 0.03 0.02 0.10 0.015
Larger of the value in the Table 2 and that in Eq. (16).
variability in the starter crack size, and that in turn relates to the total uncertainty in starter crack size. It is assumed that the initial crack size distribution is determined practically from the measurement by NDT during production, and that all crack sizes exceeding a0 will be repaired (Table 2). pffiffiffi Barsom and McNicol [17] applied the non-dimensional parameter DKI =R r; where r is the minimum radius of the starter crack, to normalize the fatigue-crackinitiation threshold behavior of various steels subject to stress ratios between 1.0 and 0.5. Good correlation was found with the experimental data. Most cracks initiate from surface defects, which are usually semi-elliptical with a shape factor (minor/major semi-axes) of 1/2 to 1/4. The Metals handbook recommends that fatigue-crack initiation for steels in Mode I, occurs when pffiffiffiffi DKI ¼ 10 R: r
ð16Þ
The crack size is considered as a normal distribution variable with a mean a0 =2 and a COV of 0.5 [18]. Therefore the mean value of a0 is given by ma0 ¼
DKI pffiffiffiffi: 20 R
ð17Þ
However, Barsom and Rolfe [19] found experimentally that for polished steel and steel alloy surfaces, the minimum size of the fatigue-crack-initiation was 0.2 mm (Table 3). 6.3. Fracture toughness Fracture toughness is a random material variable. Fracture mechanics-based failure analysis [20] of experimental data over a period of 12 years (1981–1993) show a COV of 0.09–0.12 in the fracture toughness values of low to high strength steel and aluminum. A value of 0.12 is taken for the COV.
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Table 3 Mechanical properties of the metals and metal alloys Metal/metal alloy
E (GPa)
R (MPa)
KIc (MPam1/2)
R=Kth (MPam1/2)
Test/Thick (mm)
Orient
Steel Alloys A533B-1 A517 Gr. F A36 8630 HY-130 12Ni–5Cr–3Mo
207 207 207 207 207 207 [13]
483 758 248 985 1007 1241
220 151 151 99 104 98
0.1/8.0 0/7.0 0/11.0 0/6.6 0.1/4.10 0/2.45
CT/5 TR
RW
CT/8 CT/50
Note: The references from which the above material properties were obtained are given in Ref. [10].
6.4. Threshold SIF As the value of the SIF approaches DKth ; the fatigue life approaches infinity asymptotically. This discontinuity causes numerical problems in the usual manner of handling the reliability analysis of the fatigue life. Consequently, the COV is not handled in the usual manner. The uncertainty is accounted for by reducing the value of DKth by the COV, which in this case is 0.2. This procedure essentially treats DKth deterministically, so that COV is in effect zero. 6.5. Yield strength A COV of 0.03 is assumed for good laboratory testing. 6.6. Young’s modulus A COV of 0.02 is recommended by ABS Guide [9]. 6.7. Fatigue life model The average scatter dg of the fatigue data in 33 cases of published crack propagation data is approximately 0.1, and the inaccuracy of the fatigue model Dg was approximately 0.05 [10].
7. Crack growth rates under random loading The MOB connectors are to be designed for a typical sea state spectrum loading up to SS8. Fatigue cracks grow by blunting and resharpening of the crack tip. When a sequence of much smaller loads follows a tensile overload, there is a tendency for the fatigue crack growth rate (FCGR) to be retarded. This is caused by the large compressive stresses left in the wake of the crack by the overload, so that the tensile stresses by the smaller succeeding loads are too small to reopen the crack. However,
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when smaller compressive loads follow a compressive overload, there is a tendency for the FCGR to be accelerated. This is caused by residual tensile stresses caused by the compressive overload. Thus retardation and acceleration are more significant the greater the overload. There is also the effect of rest periods. Garg [21] found that an overload followed by a rest period tends to accelerate the rate of crack growth because of the relaxation on of the compressive stresses in the wake of the crack. Newman [22] concluded from the ASTM round-robin tests, that in most of the spectra, the effects of retardation and acceleration nearly cancel each other. The MOB connectors are to be disconnected at SS8, so that stress level will be relatively small and the number of cycles large. The model automatically considers the magnitude of the stresses generated by each wave height. Because of the large number of cycles of relatively small stresses that the MOB connectors must withstand, the effects of the sequence of the random loading are assumed to be negligible. Sea state has the appearance of a random process. Schutz and Pook [15] examined the sea state data from four sources to see if the distribution of the H1=3 waves was Gaussian, but the fit was not very satisfactory. The distribution was, however, accurately represented by a Gumbel distribution given by ð18Þ PðH1=3 Þ ¼ 1 exp expð1:9H1=3 Þ=1:06 ; where H1=3 =the average height of the highest 1/3 of all waves in a particular sea state, PðHÞ is the probability of exceedance of H: Typical data is shown in Table 1. The average of the H1=3 waves for SS8 generates the maximum stress on the connector given in the finite element analysis (FEA) by Brown and Root [2]. It is assumed that the stresses generated on the connectors are proportional to the height of the waves. The rate of crack propagation under random loading is obtained by numerical integration of Eq. (1) in the computer program MOBYDICK. For greater accuracy, about 20 stress levels should be used for the sea state spectrum. 8. Input data for fatigue crack initiation and propagation The input data consist of: (a) The material properties are the nominal stress S; yield strength R; fracture toughness KIc ; threshold SIF DKth and starter crack, a0 ; together with the mean and standard deviation of each variable. The cyclic strain hardening exponent n is obtained by correlation with the yield strength [19]. In particular, the fracture toughness should be known as a function of the thickness and service temperatures. (b) The stresses and the critical combination of stresses in the connector by finite element or other methods for SS8 are needed. The crack is assumed to grow from a semi-elliptical defect in the outer surface of the 20-ft diameter steel tube at the edge of the hole loaded by the shear pin (Brown and Root design [2]) as shown in Fig. 5. The shape factor of the semi-elliptical equal to the ratio of the minor and major semi-axes is taken as 1/2. The direction of crack propagation is along the meridian perpendicular to the axial force.
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Fig. 5. Crack Pattern in the monolithic steel connector.
(c) The wave heights for the design sea-state spectrum in digital form. A minimum of 20 ranges is needed. (d) Minimum/maximum stress ratio R : (e) The stress intensity factors for Modes I–III by finite elements or by other means for the dead, live, impact, and environmental loads. 9. Illustrative reliability design The design procedure is embodied in a computer program called MOBYDICK. It features the crack growth law given by Eqs. (1) and (5), and the reliabilty analysis described above. The maximum Von Mises stress for an axial load of 100,000 tonnes is obtained from Fig. 4 as 517 MPa [2]. The stresses generated by the vertical and horizontal shear forces were not obtained in the Brown and Root draft report [2]. However, the total combined stress is estimated to 552 MPa, This includes the effect of stress concentration, so that the nominal stress approximately half of the maximum combined stress [19]. The following loads, with the Von Mises stress for the axial load in parentheses, are used in the illustrative reliability design. 9.1. Connector loads and stresses Axial Vertical Lateral Maximum combined load Maximum combined stress (est.) Stress ratio R
100,000 tonnes (517 MPa) 36,000 tonnes 12,000 tonnes 106,900 tonnes 552 MPa 0.9
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9.2. Crack initiation The fatigue crack initiation criterion is given in Eq. (16). MOBYDICK finds that the initiation for HY-130 steel occurs in SS7. 9.3. Crack propagation The starter crack is assumed to be semi-elliptical, at the outer surface of the pinloaded hole of the rigid connector just outside the reinforcing. Newman and Raju [23] give the stress intensity factors for the semi-elliptical crack. The crack then grows in accordance with Eq. (1) until breaks through to the inner surface. Henceforth the K-values for a crack in a cylindrical shell are given by Erdogan and Kibler [24] as KI ¼ ðAe þ Ab ÞKp
at the outer surface of the cylindrical shell
ð19Þ
KI ¼ ðAe Ab ÞKp
at the inner surface of the cylindrical shell;
ð20Þ
where Ae and Ab are the coefficients that are functions of l defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12ð1 n2 Þ pffiffiffiffiffiffiffiffi l¼ cKp ; % RT
ð21Þ
where c is the half-crack length, R% is the radius of the cylindrical shell, and T is the wall thickness. Numerical values of Ae ; Ab and Kp are given in [24].
10. Results of the reliability calculation MOBYDICK computed the mean fatigue life and the useful fatigue life as a function of the reliability for HY-130 steel, A533B-1 steel, 517 Gr. F steel, 12Ni– 5Cr–3Mo, and Ti–6Al–4V. The highest mean fatigue life in air (without stress corrosion) is obtained with 517 Gr. F (T1) steel and the lowest by 12Ni–5Cr–3Mo. However, the useful fatigue life for a reliability of 0.99 is highest for HY-130 steel (Table 4). A comparison of the mean fatigue lives of the five metal and metal alloys for a nominal stress of 276 MPa and those for a reliability of 0.99 as computed by
Table 4 Predicted mean fatigue life and useful fatigue life for a reliability of 0.99 at a nominal stress of 276 MPa, for several steels and metal alloys Metal or metal alloy
Nf Eq. (1) 106 cycles
Nf (L ¼ 0:99) 106 cycles
A533B-1 A517 Gr. F HY-130 12Ni–5Cr–3Mo Ti–6Al–4V
4.44 12.72 3.06 1.94 4.93
0.889 1.336 1.607 1.021 0.862
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Table 5 Sensitivity of the predicted mean fatigue life to one COV for each parameter at a nominal stress of 276 MPa, and total uncertainty Metal/metal alloy
Design parameters
Total uncertainty
S A533B-1 517Gr. F HY-130 12Ni–5Cr–3Mo Ti–6Al–4V
0.317 0.440 0.10 0.10 0.374
0.22 0.305 0.10 0.11 0.203
0.082 0.015 0 0.03 0.005
R
E
0.03 0.03 0.03 0.03 0.03
0.02 0.02 0.02 0.02 0.02
0.411 0.549 0.183 0.173 0.442
Fig. 6. Predicted useful fatigue life versus the nominal stress.
MOBYDICK is given on Table 5. The sensitivity of the mean fatigue life, which is defined as the change in the fatigue life for one COV of each of the design parameters, together with the total uncertainty is shown in Table 5. The most sensitive parameter is the starter crack as expected. For a reliability of 0.99 with a maximum nominal stress of 276 MPa, the useful fatigue life for HY-130 steel connector is only about 1.61 million cycles. To obtain a fatigue life of 10 million cycles at a reliability of 0.99, the stress must be reduced significantly. The results for the useful fatigue life with a reliability of 0.99 for HY130 steel are presented graphically in Fig. 6, showing that 10 million cycles is reached at nominal stress of 203 MPa. Since this is the highest stress level at which 10 million cycles of wave loading can be obtained for the 5 metals investigated, HY-130 is the steel selected for the fatigue design of the connectors.
11. Conclusions A fracture mechanics reliability-based design procedure has been demonstrated for large monolithic steel components typical of inter-module connectors systems.
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The reliability analysis of the connectors for the US Mobile Offshore Base shows that the connectors can be designed for 10 million cycles of wave loading with a reliability of 0.99 at a nominal stress of 135 MPa. Assuming that the design life for connectors is 5–10 years, structural reliability calculations suggest that the design stress level for the Brown and Root MOB connector concepts should be reduced to 135 MPa to achieve current life-time target reliability.
Acknowledgements The authors wish to thank the Office of Naval Research for sponsoring this project as part of the MOB Science & Technology Program.
References [1] Remmers G, Taylor R, Paulo P, Brackett R. Mobile offshore base: a seabasing option (keynote address), Third International Workshop on Very Large Floating Structures, Honolulu, September 1999. [2] Brown and Root. Mobile offshore base connector system. Final Report No. CM-0031, Houston, June 1999. [3] UK, DOE, Offshore. Installations: guidance on design, construction and certification. 4th ed. App. A21.2.13, 1990. [4] Ang AHS. A comprehensive basis for reliability analysis, design, Japan–USA Joint Seminar on Reliability Approach in Structural Engineering, Marzuen, Co. Ltd., Tokyo, 1974. p. 29–47. [5] Ang AHS, Cheung MC, Shugar TA, Fernie J. Reliability-based fatigue analysis and design of floating structures. Third International Workshop on Very Large Floating Structures, Honolulu, 22–24, September 1999. [6] Mansour AE, Wirsching PH, White G, Ayyub BM. Probability-based ship design: implementation of design guidelines. SSC 392, NTIS, Washington, DC, 1995, 200p. [7] Mansour AE et al. Assessment of reliability of ship structures. Third International Workshop on Very Large Floating Structures, Honolulu, 22–24, September 1999. [8] Assakkaf IA, Ayyub BM. Reliability-based design for ship structures. Third Workshop on Very large Floating Structures, Honolulu, 1999. [9] ABS Americas. MOB classification guide and commentary. ABS, Americas, Houston, January 2000. [10] Ramsamooj DV. Fracture mechanics-based fatigue design of the US Mobile Offshore Base connectors. Costa Mesa, CA: MCA Engineers, Inc., 1999. [11] Ramsamooj DV, Shugar TA. Prediction of fracture-based fatigue life of the connectors of the US Mobile Offshore Base connectors, September 1999. [12] Ramsamooj DV, Shugar TA. Prediction of fracture–based fatigue life of connectors of the US Mobile Offshore Base. Mar Struct 2001;14:197–214. [13] Taylor D. Fatigue thresholds, Butterworths, London, 1989, 218p. [14] Paris PC, Hermann L. Twenty years of reflection on questions involving fatigue crack growth. In: Blacklund J, Beevers CJ, editors. First Symposium on Fatigue Thresholds, Stockholm, Vol. 1, Warley, UK: EMAS Publications, 1982. p. 3–32. [15] Schutz W, Pook LP. WASH (Wave Action Standard History)Fa standardized stress time history. In: Noordhoek C, de Back J, editors. Steel in Marine Structures, Amsterdam: Elsevier Science Publishers, 1987. [16] Bechtel, National Inc. Mobile Offshore Base design tools and procedures. Validation Summary Report, Contract No. N47408-93-D-7001, San Francisco, 1999.
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[17] Barsom JM, McNicol RC. Effect of stress concentration of fatigue-crack initiation in HY-130 steel. ASTMSTP 559, 1974. [18] Jiao G, Moan T. Reliability-based fatigue and fracture design cruteria for welded offshore structures. Eng Fract Mech 1992;41(2):271–82. [19] Barsom JM, Rolfe ST. Fracture and fatigue control in structures. New Jersey: Prentice-Hall, 1987. [20] Rosenfield AR, Marshall CW. Fracture-mechanics based failure analysis. Eng Fract Mech 1991;45(3):333–8. [21] Garg SBL. Influence of mechanical properties, overload on fatigue crack retardation. In: Goel VS, editor. Fatigue Life Analysis and Prediction, American Society of Metals, 1986, 400p. [22] Newman JC. A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. Methods and models for predicting fatigue crack growth under random loading, ASTM STP 748, 1981. p. 53–84. [23] Newman JC, Raju IS. An empirical stress intensity factor equation for the surface crack. Eng Fract Mech 1981;11(1–2):185–97. [24] Erdogan F, Kibler JJ. Cylindrical and spherical shells with cracks. Int J Fract Mech 1969;5(3):229–36.
Reliability analysis of fatigue life of the connectorsFthe US Mobile Offshore Base D.V. Ramsamooja,*, T.A. Shugarb a
Department of Civil Engineering, California State University, P.O. Box 6870, Fullerton, CA 92834-6870, USA b Naval Facilities Engineering Service Center, Port Hueneme, CA, USA
Received 1 May 2001; received in revised form 25 August 2001; accepted 3 October 2001
Abstract Reliability-based design analysis of the fatigue life of the connectors of the five sections of the 2-km long US Mobile Offshore Base (MOB) is demonstrated. A performance function is defined in terms of the nominal stress range, inherent defect or starter crack, and appropriate material properties, which are considered random variables. The reliability analysis is performed for a sea state 1–8 (SS1–8) random loading having a Gumbel distribution. Where possible, uncertainty data for random variables are obtained from published data relating to the fatigue of metal and metal alloys. Otherwise, judgmental coefficients of variation are prescribed for purposes of demonstration. The fatigue life is assumed to follow the Weibull distribution. The reliability function is defined in terms of the mean life and the total uncertainty in the fatigue life. Preliminary reliability calculations suggest that current design stress levels be reduced to meet the current fatigue life target reliability level for the MOB connectors. An illustrative design is demonstrated and the metal selected for a fatigue design of the connectors for 10 million cycles with a reliability of 0.99 at a nominal stress of 203 MPa is HY-130 steel. r 2002 Published by Elsevier Science Ltd.
*Corresponding author. Tel.: +1-714-278-3968; fax: +1-714-278-3916. E-mail addresses: [email protected] (D.V. Ramsamooj), [email protected] (T.A. Shugar). 0951-8339/02/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 0 9 5 1 - 8 3 3 9 ( 0 1 ) 0 0 0 2 8 - 4
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Nomenclature a0 ; b0 af c E L N N; Nf R R R% S T Z dg Dg KI ; KII ; KIII DKI KImax DKth; 0 DKth sZ OZ mZ
Semi-minor and semi-major axes of the semi-elliptical starter crack Values of a at failure Crack length in the connector cylindrical tube Young’s modulus Reliability function Cyclic strain hardening function Number of cycles and fatigue life Yield strength Minimum stress/maximum stress Radius of cylindrical connector tube Nominal stress Thickness of cylindrical connector tube Performance function Scatter of the data about the model prediction line Inaccuracy of the fatigue model Modes I, II and III stress intensity factors (SIFs) Increment of the SIF in Mode I Maximum value of KI Threshold SIF at R ¼ 0 Threshold SIF at any value of R Standard deviation of Z Total uncertainty in Z Mean value of Z
1. Introduction 1.1. Brief description of the US Mobile Offshore Base (Brown and Root concept) The Mobile Offshore Base (MOB) program is advancing the design of five platforms with fundamentally different degrees of module connectivity [1]. The design by Brown and Root [2] comprises five semi-submersible units connected by eight connectors at the module interfaces. This design concept is presented in Fig. 1. The objective of this paper is to perform a reliability-based design analysis of the fatigue life of the Brown and Root [2] connectors between the modules, using fracture mechanics. 1.2. Rigid connector conceptual design The Brown and Root Rigid MOB connection system design is based on Tension Leg Platform (TLP) connection systems that are currently in operation. Each connector consists of male and female halves, made of 6.1 m diameter, 152 mm wall
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Fig. 1. Rigidly connected semi-submersible modules of the US Mobile Offshore Base.
thickness tubing. Each half pivots on a 3.66 m diameter tube, and can move or slide on the pivot tube. Both sliding and rotating are accomplished with hydraulic cylinders. This provides assistance in alignment for connection and latching purposes. Figs. 2(a) and (b) show the proposed ABB Vetco Grey latching system. The conceptual design deploys eight rigid connectors. The first four are connected at module joining. The second two are required to be connected before encountering sea state 6, and the final two must be connected before encountering sea state 8. The Buckled Strut elastomeric members provide limited flexibility in the Brown and Root connection system. They are located in an outer circle around the connector well. Their purpose is threefold. Firstly, they provide for initial shock attenuation upon impact of the modules during mating. Secondly, they are compressed upon latching such that they are pre-loaded in the connected mode. This means that no movement is encountered until the pre-load is reached, then limited flexing can occur in pitch rotation. Thirdly, upon disconnecting, there is stored energy present to push the modules apart. Pull-in cylinders will be provided to make the final pull-in, compress the buckled struts, and engage the latches. Once a rigid connector is latched, some type of shear mechanism will be required to eliminate movement at the module interface. The details are given in Brown and Root [2]. The ABB Vetco Gray latching interface conceptual design is similar to those used on TLP connectors. Forty-eight actuation wedges around the 6.1 m diameter circumference were required to reach the initial conceptual design load of 100,000 tonnes per connector. This load is the design load or the maximum operating load based on a maximum yield of 448 MPa with a safety factor of 1.5 based on yield, and is analyzed in detail by Finite Element Analysis. Fig. 3 shows an enlarged view of the connector, and Fig. 4 shows the Von Mises stresses for an axial load of 100,000 tonnes determined by ANSYS finite element program.
2. Fracture mechanics reliability-based analysis An empirical method, using the UK’s DOE S–N curves [3] cast within a reliability framework [4,5] handles fatigue design at the structural component level for MOB. Ang et al. [5] developed a technical procedure for a reliability-based approach to fatigue analysis and applied it to the stress-range data measured on an existing LNG tanker and to a newly designed double hull tanker with ABS’s DLA rating. In both
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Fig. 2. ABB Vetco Grey latching system for connector.
cases the level of reliabilty was only 0.85. Other equivalent reliability models are also based on the same physics of fatigue [6]. Fatigue life estimates vary greatly according to the values of the COVs assumed. Estimates of COVs are given in Mansour et al. [7]. Other methods of reliability design, including the direct-reliabilty-based design and the load and resistance design (LRFD) are discussed by Assakkaf et al. [8]. However, MOB connectors are a key technology issue by virtue of their importance, uniqueness and high estimated cost of approximately $80 million per
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Fig. 3. Enlarged view of the rigid connector.
module interface, for one particular concept. The ABS Guide [9] currently specifies a lifetime target reliability of 0.9999 for fatigue design of connectors. Further, the MOB Connector Working Group has estimated that a reasonable design life for the connectors is 5–10 years. Moreover, there are no SN-curve data in existence for large monolithic connector concepts, which currently involve plate thickness of 150 mm or greater. Accordingly, fundamental approaches to MOB connector design are therefore warranted. Thus a fracture mechanics approach to fatigue design has been recommended by the MOB Standards and Criteria Working Group and prescribed by the ABS Guide. Ramsamooj [10] presented a newly developed linear elastic fracture mechanics method (LEFM) for fatigue design. In the present paper, this method is cast within a reliability-based design procedure.
3. Fatigue crack propagation An approximate analytical model [10–12] for the rate of crack propagation under constant amplitude cyclic loading is expressed by da 5 1 ¼ 2 ðDKI DKth Þ2 dN 3p ð1 þ nÞER 1 ðKImax =KIc Þ2
ð1Þ
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Fig. 4. Von Mises stresses computed by ANSYS for the connector.
in which E is Young’s modulus, R is the yield strength, KIc is the fracture toughness, KImax is the applied stress intensity factor range (Mode I), which measures the intensity of the stress field in the vicinity of the crack, n is the cyclic strain hardening coefficient, and DKth is threshold SIF below which the crack growth is smaller than 1010 m/cycle, according to the ASTM definition. Eq. (1) is for laboratory air environment. Corrosion fatigue is not considered in this paper. The threshold variation with the stress ratio R (minimum/maximum stress) can be derived from the ASTM definition, as follows. From Eq. (1) dc 1 ¼ CðDK DKth Þ2 ; ð2Þ dN 1 ðKmax =KIc Þ2 where C ¼ 5=ð3p2 ð1 þ nÞERÞ: For R ¼ 0; KImax ¼ DKth;0 : When DK ¼ DKth ; the net SIF range is DK DKth ; the FCGRE0, or dc 1 ¼ 0 ¼ CðDK DKth;0 Þ2 : ð3Þ dN 1 ðDKth;0 =KIc Þ2 When Kmax ¼ DKth;0 ; but R a0; DK ¼ DKth;0 ð1 RÞ: Therefore dc 1 ¼ 0ECðDKth;0 ð1 RÞ DKth Þ2 ; dN 1 ðDKth;0 =KIc Þ2 DKth ¼ DKth;0 ð1 RÞ:
ð4Þ ð5Þ
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This expression for the threshold variation with R shows good agreement with the experimental data [12], except for large positive or negative values of R: In particular for R ¼ 0:9; the experimental data is about 10% smaller. This is probably due to the residual tension caused by compression part of the load. The fatigue life is then Z af 1 da ð6Þ Nf ¼ da=dN a0 in which a0 is the size of the starter crack, which is usually the size of the largest defect, depending on its location, af ¼ KIc2 =pS 2 ; is the crack length at failure and S is the applied stress. The integral needs to be evaluated numerically. We have compared the model predictions for 37 sets of data, including steels, steel alloys, aluminum and aluminum alloys, titanium, nickel and plastics [10–12]. Despite the fact the fatigue crack propagation rates of the above metals and plastics at the same stress intensity factor extend over a range of 1000, there was good agreement in almost all cases between the model predictions and the experimental data. We know of no other model having the overall visual resemblance fidelity exhibited in the numerous graphs presented in our papers. Some of the advantages of the analytical model presented herein are: (a) It is formulated in terms of the material properties E; R; KIc and DKth;0 : There are no empirical fitting constants. It captures the physical process of crack growth as discovered by numerous experimental crack propagation measurements. (b) There is no need for the several regions of crack growth. It treats FCGR as a continuous process. (c) There is no scaling factor. The fatigue behavior of any size structure can be predicted. According to Paris and Hermann [14] ‘‘unifying theories and models remain badly needed as a substitute for zillions of data points required by all of the potential applications if we do not develop unifying theories’’. (d) It provides a means of quantitative evaluation of alternative materials and for making trade-offs to optimize the design process for engineering efficiency and business economy.
4. Performance function Z A performance function Z is defined as Z ¼ Nf ¼ gðS; a0 ; KIc ; DKth ; R; EÞ;
ð7Þ
where S is the stress level. The stress ratio is fully accounted for DKth : Let mZ and sZ be the mean and standard deviation of Z: Then from a first order approximation, the mean value of Z is mZ ¼ gðmS ; ma0 ; mKIc ; mKth ; mR ; mE Þ:
ð8Þ
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and the standard deviation of Z is given by s2Z ¼
6 X
c2i s2Xi þ
1
6 X 6 X 1
ð9Þ
rij ci cj sXi sXj ;
1
where c1 ¼
qNf ; qS
c5 ¼
qNf Nf ¼ ; qR R
c2 ¼
qNf ; qa0 c6 ¼
c3 ¼
qNf ; qKIc
c4 ¼
qNf qKth
qNf Nf ¼ ; qE E
ð10Þ
where sX 1 ¼ sS ; sX 2 ¼ sa0 ; similar terms exist for KIc ; Kth ; R; and E: The term rij is the correlation between the ith and jth parameters. Since there is no correlation between the parameters, rij ¼ 0: The total uncertainty in the fatigue life is expressed by [4] O2Z ¼ O2g þ
1 2 2 ðc s þ c22 s2a0 þ c3 s2KIc þ c4 s2Kth þ c25 s2R þ c26 s2E Þ; m2Z 1 S
ð11Þ
where O2g ¼ d2g þ D2g ; dg is the (average) scatter of the mean fatigue life data about the model prediction line for the test specimens, and Dg is the inaccuracy of the fatigue model in the predicted mean life associated with the imperfection in the fatigue theory, including the effects of impact. Impact forces may result from mooring of ocean-going vessels alongside the MOB. Eq. (11) uses a first order approximation or a small variance theory, which may be applicable for COVs of 0.15–0.30. Since the COV of the inherent flaw or starter crack is estimated to be about 0.5, there may be a significant error. In such cases, Ang [4] recommends a second order approximation. However, noting that qNf sa ¼ ðNf Þa0 ðNf Þa0 þsa 0 qa0 0
ð12Þ
is the change in the fatigue life due to a change of one COV in a0 ; and that Eq. (6) has to be differentiated numerically anyway to get terms such as qNf =qa0 ; the total uncertainty may be expressed as O2Z ¼ O2g þ
1 ððDNfS Þ2 þ ðDNfa0 Þ2 þ ðDNfKIc Þ2 þ ðDNfR Þ2 þ ðDNfE Þ2 Þ2 ; m2Z
ð13Þ
where DNfa0 is the change in the fatigue life due to a change of one COV in a0 ; or DNfa0 ¼ ðNf Þa0 ðNf Þa0 þsa : 0
ð14Þ
These terms are obtained numerically from the computer program called MOBYDICK.
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5. Reliability function Pertaining to the reliability analysis of the MOB connectors, the American Bureau of Shipping (ABS) [9] states ‘‘A direct reliability method is preferred. Unnecessary simplifications to the limit state or the basic variables, such as the lognormal assumption for basic variables as a substitute for rigorous analysis of the data, should be avoided. In many situations, however, selective assumptions can significantly reduce the computations without compromising the accuracy of the analysis. For example, Ang [4] and Ang et al. [5] assumed the fatigue life could be represented by a Weibull distribution. Unlike a normal distribution, a Weibull distribution has a monotonically increasing hazard function, which is physically consistent with the cumulative nature of fatigue damage’’. The lognormal distribution may fit some data better, but the hazard function decreases at a later life, which is contrary to the nature of fatigue damage. The probability of no fatigue failure or the reliabilty function is [4] ! )O1:08 Z Nf 1:08 LðNf Þ ¼ exp ; G 1 þ OZ Nf (
ð15Þ
where N% f =the fatigue life predicted from Eq. (1), herein after referred to as ‘‘the mean fatigue life’’ using the mean values of each parameter, and Nf is the fatigue life for a specified reliability, herein after referred to as ‘‘the reduced fatigue life’’, GðxÞ is the gamma function, OZ =the total uncertainty in the fatigue life, assuming that the minimum performance level is zero. Thus the fatigue life for a specific reliability is dependent on the mean fatigue life and the total uncertainty.
6. Uncertainty in fatigue life variables The reliability parameters consisting of the means and standard deviations are discussed below. Due to the huge monetary investment that a MOB implies, the cost of developing data beyond usual standards should be evaluated in reliability-based design. The uncertainty associated with each random variable is discussed below. 6.1. Wave load-induced stress range Wave loads have significant uncertainty. They are developed from random environmental force input data, and computed by imperfect hydrodynamic computer programs. The resulting uncertainty in the nominal stress range can be substantial. Sea state data in this study were based on standard conditions for the North Sea and were taken from Schutz and Pook [15]. They are reproduced in Table 1. The relative occurrence of sea states within small intervals of significant wave height H1=3 was for use in calculating fatigue loads on fixed and moored platforms. Although
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Table 1 Approximation of sea state data (from Schutz and Pook [15]) Sea state
H1=3 a (m)
Range (m)
Wave period (s)
Time (%)
0 1 2 3 4 5 6 7 8 9 10 11
1.75 2.55 3.4 4.15 4.80 5.40 6.15 6.90 7.80 8.80 10.35 13.60
0.00–1.95 1.95–2.85 2.85–3.80 3.80–4.45 4.50–5.10 5.10–5.75 5.75–6.55 6.55–7.35 7.35–8.30 8.30–9.40 9.40–12.55 12.55–
5.9 6.6 6.9 7.3 7.7 8.0 8.4 8.7 9.1 9.5 10.2 11.5
38.5 28.5 17.5 7.18 3.40 2.16 1.31 0.678 0.334 0.156 0.0797 0.0043
a
Average height of the highest 1/3 of all the waves in a particular sea state.
more applicable to fixed platforms than to floaters, it is assumed that wave loads are linearly proportional to wave heights. The MOB modules are to be disconnected in SS7 or SS8, freeing the connectors from SS9 and greater load-induced stresses. The maximum nominal stresses are correlated with the average H1=3 value for the highest applicable sea-state (SS8) or 7.84 m. The H1=3 waves of SS8 are generally more damaging than the lower sea-state waves. However, for high cycle fatigue, a relatively few very high waves are not as critical as for fracture. During the connected mode operation, the MOB will have an automatic wave-measuring device, and the connectors will be fully instrumented with strain gages and other sensors. Preliminary comparisons of predictions to experimental data for assessment of the linearized hydrodynamic computer programs for MOB application have been completed (Bechtel National, Inc. [16]). The results suggest a COV from 0.1 to 0.2 for prediction of motion response of the five connected MOB single base units in regular waves. However, physical model testing that includes hydroelastic behavior is needed to assess this uncertainty accurately. Uncertainty in the load-induced stress is largely obviated if it is assumed that conditions leading to crack growth in the MOB connectors will be continuously monitored, and the crack growth is controlled through manual intervention. Accordingly, a rather low COV of 0.05 was selected for load-induced stress range. 6.2. Starter crack Knowledge of the maximum crack sizes is critical to the success of LEFM applications in both design and maintenance of MOB connectors. Recent improvement in non-destructive evaluation (NDE) methods, such as magnetic and eddy-current methods, allows detection of cracks as small as 0.1 mm, with confidence. This relates to precision of measurement and thus to scatter or
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Table 2 Reliability parameters for HY-130 steel
a
Starter crack (mm) Hot-spot axial. stress (MPa) Fracture toughness (MPam0.5) Yield strength, MPa Young’s modulus, MPa Fatigue life model: Scatter dg Uncertainty Dg a
Mean
SD
COV
0.1 517 120 1007 207,000
0.05 25.8 26.4 30.2 4140
0.50 0.05 0.12 0.03 0.02 0.10 0.015
Larger of the value in the Table 2 and that in Eq. (16).
variability in the starter crack size, and that in turn relates to the total uncertainty in starter crack size. It is assumed that the initial crack size distribution is determined practically from the measurement by NDT during production, and that all crack sizes exceeding a0 will be repaired (Table 2). pffiffiffi Barsom and McNicol [17] applied the non-dimensional parameter DKI =R r; where r is the minimum radius of the starter crack, to normalize the fatigue-crackinitiation threshold behavior of various steels subject to stress ratios between 1.0 and 0.5. Good correlation was found with the experimental data. Most cracks initiate from surface defects, which are usually semi-elliptical with a shape factor (minor/major semi-axes) of 1/2 to 1/4. The Metals handbook recommends that fatigue-crack initiation for steels in Mode I, occurs when pffiffiffiffi DKI ¼ 10 R: r
ð16Þ
The crack size is considered as a normal distribution variable with a mean a0 =2 and a COV of 0.5 [18]. Therefore the mean value of a0 is given by ma0 ¼
DKI pffiffiffiffi: 20 R
ð17Þ
However, Barsom and Rolfe [19] found experimentally that for polished steel and steel alloy surfaces, the minimum size of the fatigue-crack-initiation was 0.2 mm (Table 3). 6.3. Fracture toughness Fracture toughness is a random material variable. Fracture mechanics-based failure analysis [20] of experimental data over a period of 12 years (1981–1993) show a COV of 0.09–0.12 in the fracture toughness values of low to high strength steel and aluminum. A value of 0.12 is taken for the COV.
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Table 3 Mechanical properties of the metals and metal alloys Metal/metal alloy
E (GPa)
R (MPa)
KIc (MPam1/2)
R=Kth (MPam1/2)
Test/Thick (mm)
Orient
Steel Alloys A533B-1 A517 Gr. F A36 8630 HY-130 12Ni–5Cr–3Mo
207 207 207 207 207 207 [13]
483 758 248 985 1007 1241
220 151 151 99 104 98
0.1/8.0 0/7.0 0/11.0 0/6.6 0.1/4.10 0/2.45
CT/5 TR
RW
CT/8 CT/50
Note: The references from which the above material properties were obtained are given in Ref. [10].
6.4. Threshold SIF As the value of the SIF approaches DKth ; the fatigue life approaches infinity asymptotically. This discontinuity causes numerical problems in the usual manner of handling the reliability analysis of the fatigue life. Consequently, the COV is not handled in the usual manner. The uncertainty is accounted for by reducing the value of DKth by the COV, which in this case is 0.2. This procedure essentially treats DKth deterministically, so that COV is in effect zero. 6.5. Yield strength A COV of 0.03 is assumed for good laboratory testing. 6.6. Young’s modulus A COV of 0.02 is recommended by ABS Guide [9]. 6.7. Fatigue life model The average scatter dg of the fatigue data in 33 cases of published crack propagation data is approximately 0.1, and the inaccuracy of the fatigue model Dg was approximately 0.05 [10].
7. Crack growth rates under random loading The MOB connectors are to be designed for a typical sea state spectrum loading up to SS8. Fatigue cracks grow by blunting and resharpening of the crack tip. When a sequence of much smaller loads follows a tensile overload, there is a tendency for the fatigue crack growth rate (FCGR) to be retarded. This is caused by the large compressive stresses left in the wake of the crack by the overload, so that the tensile stresses by the smaller succeeding loads are too small to reopen the crack. However,
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when smaller compressive loads follow a compressive overload, there is a tendency for the FCGR to be accelerated. This is caused by residual tensile stresses caused by the compressive overload. Thus retardation and acceleration are more significant the greater the overload. There is also the effect of rest periods. Garg [21] found that an overload followed by a rest period tends to accelerate the rate of crack growth because of the relaxation on of the compressive stresses in the wake of the crack. Newman [22] concluded from the ASTM round-robin tests, that in most of the spectra, the effects of retardation and acceleration nearly cancel each other. The MOB connectors are to be disconnected at SS8, so that stress level will be relatively small and the number of cycles large. The model automatically considers the magnitude of the stresses generated by each wave height. Because of the large number of cycles of relatively small stresses that the MOB connectors must withstand, the effects of the sequence of the random loading are assumed to be negligible. Sea state has the appearance of a random process. Schutz and Pook [15] examined the sea state data from four sources to see if the distribution of the H1=3 waves was Gaussian, but the fit was not very satisfactory. The distribution was, however, accurately represented by a Gumbel distribution given by ð18Þ PðH1=3 Þ ¼ 1 exp expð1:9H1=3 Þ=1:06 ; where H1=3 =the average height of the highest 1/3 of all waves in a particular sea state, PðHÞ is the probability of exceedance of H: Typical data is shown in Table 1. The average of the H1=3 waves for SS8 generates the maximum stress on the connector given in the finite element analysis (FEA) by Brown and Root [2]. It is assumed that the stresses generated on the connectors are proportional to the height of the waves. The rate of crack propagation under random loading is obtained by numerical integration of Eq. (1) in the computer program MOBYDICK. For greater accuracy, about 20 stress levels should be used for the sea state spectrum. 8. Input data for fatigue crack initiation and propagation The input data consist of: (a) The material properties are the nominal stress S; yield strength R; fracture toughness KIc ; threshold SIF DKth and starter crack, a0 ; together with the mean and standard deviation of each variable. The cyclic strain hardening exponent n is obtained by correlation with the yield strength [19]. In particular, the fracture toughness should be known as a function of the thickness and service temperatures. (b) The stresses and the critical combination of stresses in the connector by finite element or other methods for SS8 are needed. The crack is assumed to grow from a semi-elliptical defect in the outer surface of the 20-ft diameter steel tube at the edge of the hole loaded by the shear pin (Brown and Root design [2]) as shown in Fig. 5. The shape factor of the semi-elliptical equal to the ratio of the minor and major semi-axes is taken as 1/2. The direction of crack propagation is along the meridian perpendicular to the axial force.
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Fig. 5. Crack Pattern in the monolithic steel connector.
(c) The wave heights for the design sea-state spectrum in digital form. A minimum of 20 ranges is needed. (d) Minimum/maximum stress ratio R : (e) The stress intensity factors for Modes I–III by finite elements or by other means for the dead, live, impact, and environmental loads. 9. Illustrative reliability design The design procedure is embodied in a computer program called MOBYDICK. It features the crack growth law given by Eqs. (1) and (5), and the reliabilty analysis described above. The maximum Von Mises stress for an axial load of 100,000 tonnes is obtained from Fig. 4 as 517 MPa [2]. The stresses generated by the vertical and horizontal shear forces were not obtained in the Brown and Root draft report [2]. However, the total combined stress is estimated to 552 MPa, This includes the effect of stress concentration, so that the nominal stress approximately half of the maximum combined stress [19]. The following loads, with the Von Mises stress for the axial load in parentheses, are used in the illustrative reliability design. 9.1. Connector loads and stresses Axial Vertical Lateral Maximum combined load Maximum combined stress (est.) Stress ratio R
100,000 tonnes (517 MPa) 36,000 tonnes 12,000 tonnes 106,900 tonnes 552 MPa 0.9
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9.2. Crack initiation The fatigue crack initiation criterion is given in Eq. (16). MOBYDICK finds that the initiation for HY-130 steel occurs in SS7. 9.3. Crack propagation The starter crack is assumed to be semi-elliptical, at the outer surface of the pinloaded hole of the rigid connector just outside the reinforcing. Newman and Raju [23] give the stress intensity factors for the semi-elliptical crack. The crack then grows in accordance with Eq. (1) until breaks through to the inner surface. Henceforth the K-values for a crack in a cylindrical shell are given by Erdogan and Kibler [24] as KI ¼ ðAe þ Ab ÞKp
at the outer surface of the cylindrical shell
ð19Þ
KI ¼ ðAe Ab ÞKp
at the inner surface of the cylindrical shell;
ð20Þ
where Ae and Ab are the coefficients that are functions of l defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12ð1 n2 Þ pffiffiffiffiffiffiffiffi l¼ cKp ; % RT
ð21Þ
where c is the half-crack length, R% is the radius of the cylindrical shell, and T is the wall thickness. Numerical values of Ae ; Ab and Kp are given in [24].
10. Results of the reliability calculation MOBYDICK computed the mean fatigue life and the useful fatigue life as a function of the reliability for HY-130 steel, A533B-1 steel, 517 Gr. F steel, 12Ni– 5Cr–3Mo, and Ti–6Al–4V. The highest mean fatigue life in air (without stress corrosion) is obtained with 517 Gr. F (T1) steel and the lowest by 12Ni–5Cr–3Mo. However, the useful fatigue life for a reliability of 0.99 is highest for HY-130 steel (Table 4). A comparison of the mean fatigue lives of the five metal and metal alloys for a nominal stress of 276 MPa and those for a reliability of 0.99 as computed by
Table 4 Predicted mean fatigue life and useful fatigue life for a reliability of 0.99 at a nominal stress of 276 MPa, for several steels and metal alloys Metal or metal alloy
Nf Eq. (1) 106 cycles
Nf (L ¼ 0:99) 106 cycles
A533B-1 A517 Gr. F HY-130 12Ni–5Cr–3Mo Ti–6Al–4V
4.44 12.72 3.06 1.94 4.93
0.889 1.336 1.607 1.021 0.862
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Table 5 Sensitivity of the predicted mean fatigue life to one COV for each parameter at a nominal stress of 276 MPa, and total uncertainty Metal/metal alloy
Design parameters
Total uncertainty
S A533B-1 517Gr. F HY-130 12Ni–5Cr–3Mo Ti–6Al–4V
0.317 0.440 0.10 0.10 0.374
0.22 0.305 0.10 0.11 0.203
0.082 0.015 0 0.03 0.005
R
E
0.03 0.03 0.03 0.03 0.03
0.02 0.02 0.02 0.02 0.02
0.411 0.549 0.183 0.173 0.442
Fig. 6. Predicted useful fatigue life versus the nominal stress.
MOBYDICK is given on Table 5. The sensitivity of the mean fatigue life, which is defined as the change in the fatigue life for one COV of each of the design parameters, together with the total uncertainty is shown in Table 5. The most sensitive parameter is the starter crack as expected. For a reliability of 0.99 with a maximum nominal stress of 276 MPa, the useful fatigue life for HY-130 steel connector is only about 1.61 million cycles. To obtain a fatigue life of 10 million cycles at a reliability of 0.99, the stress must be reduced significantly. The results for the useful fatigue life with a reliability of 0.99 for HY130 steel are presented graphically in Fig. 6, showing that 10 million cycles is reached at nominal stress of 203 MPa. Since this is the highest stress level at which 10 million cycles of wave loading can be obtained for the 5 metals investigated, HY-130 is the steel selected for the fatigue design of the connectors.
11. Conclusions A fracture mechanics reliability-based design procedure has been demonstrated for large monolithic steel components typical of inter-module connectors systems.
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The reliability analysis of the connectors for the US Mobile Offshore Base shows that the connectors can be designed for 10 million cycles of wave loading with a reliability of 0.99 at a nominal stress of 135 MPa. Assuming that the design life for connectors is 5–10 years, structural reliability calculations suggest that the design stress level for the Brown and Root MOB connector concepts should be reduced to 135 MPa to achieve current life-time target reliability.
Acknowledgements The authors wish to thank the Office of Naval Research for sponsoring this project as part of the MOB Science & Technology Program.
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