Commentary on the Guide for the Fatigue Assessment of Offshore Structures (April 2003)
COMMENTARY ON THE GUIDE FOR THE
FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES (APRIL 2003)
JANUARY 2004 (Updated July 2014 – see next page)
American Bureau of Shipping Incorporated by Act of Legislature of the State of New York 1862
Copyright 2004 American Bureau of Shipping ABS Plaza 16855 Northchase Drive Houston, TX 77060 USA
Updates July 2014 consolidation includes: February 2013 version plus Corrigenda/Editorials February 2013 consolidation includes: January 2004 version plus Notice No. 1 April 2010 consolidation includes: June 2007 version plus Corrigenda/Editorials June 2007 consolidation includes: June 2007 – Corrigenda/Editorials
Foreword
Foreword This Commentary provides background, including source and additional technical details, for the ABS Guide for the Fatigue Assessment of Offshore Structures, April 2003, which is referred to herein as “the Guide”. The criteria contained in the Guide are necessarily brief in order to give clear descriptions of the fatigue assessment process. This Commentary allows the presentation of supplementary information to better explain the basis and intent of the criteria that are used in the fatigue assessment process. It should be understood that the Commentary is applicable only to the indicated version of the Guide. The order of presentation of the material in this Commentary generally follows that of the Guide. The major topics of the Sections in both the Guide and Commentary are the same, but the detailed contents of the individual Subsections and Paragraphs will not typically correspond between the Guide and the Commentary. In case of a conflict between anything presented herein and the ABS Rules or the Guide, precedence is given to the Rules or the Guide. This Commentary shall not be considered as being more authoritative than the Guide to which it refers. ABS welcomes comments and suggestions for improvement of this Commentary. Comments or suggestions can be sent electronically to
[email protected].
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
iii
Table of Contents
COMMENTARY ON THE GUIDE FOR THE
FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES (APRIL 2003) CONTENTS SECTION 1
SECTION 2
SECTION 3
iv
Introduction ............................................................................................ 1 1 General Comments ............................................................................. 1 2 Basic Terminology .............................................................................. 1 3 The Deterministic Method and the Palmgren-Miner Rule to Define Fatigue Damage ................................................................................. 1 4 Application of the Palmgren-Miner (PM) Rule ....................................2 5 Safety Checking with Respect to Fatigue ........................................... 3 TABLE 1
Deterministic Stress Spectra ..................................................... 2
TABLE 2
Tubular Joints: Statistics on Damage at Failure, (δ Lognormal Distribution Assumed) ......................................... 2
TABLE 3
Plated Joints: Statistics on Damage at Failure, (δ Lognormal Distribution Assumed) ......................................... 3
Fatigue Strength Based on S-N Curves – General Concepts ............. 4 1 Preliminary Comments........................................................................ 4 2 Statistical Analysis of S-N Data .......................................................... 5 3 The Design Curve ............................................................................... 5 4 The Endurance Range ........................................................................ 6 5 Stress Concentration Factors – Tubular Intersections ....................... 7 TABLE 1
Details of the Basic “In-Air” S-N Curves....................................6
FIGURE 1
An Example of S-N Fatigue Data Showing the Least Squares Line and the Design Line [HSE(1995)] ....................... 4
FIGURE 2
The Design S-N Curve for the ABS-(A) Class D Joint .............. 7
FIGURE 3
Weld Toe Extrapolation Points for a Tubular Joint ................... 8
S-N Curves .............................................................................................. 9 1 Introduction ......................................................................................... 9 2 A Digest of the S-N Curves Used for the Structural Details of Offshore Structures ............................................................................. 9 3 General Comparison ......................................................................... 10
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
4
Tubular Intersection Connections ..................................................... 11 4.1
Without Weld Profile Control ......................................................... 11
4.2
With Weld Improvement ................................................................ 12
5
Plated Connections ........................................................................... 13
6
Discussion of the Thickness Effect ................................................... 14
7
6.1
Introduction.................................................................................... 14
6.2
Fatigue Test Data on Plated Joints ............................................... 15
6.3
Design F-Curves with Thickness Adjustment ................................ 15
6.4
Thickness Adjustments to Test Data and Their Regressed S-N Curves .................................................................................... 15
6.5
Discussion ..................................................................................... 16
6.6
Postscript....................................................................................... 16
Effects of Corrosion on Fatigue Strength.......................................... 28 7.1
Preliminary Remarks ..................................................................... 28
7.2
A Summary of the Results ............................................................. 28
7.3
The Summaries ............................................................................. 28
TABLE 1
Coverage of the Two Main Sources of S-N Curves Used for Offshore Structures ............................................................ 11
TABLE 2
AWS-HSE/DEn Curves for Similar Detail Classes ................. 13
TABLE 3
Parameters of Plate Thickness Adjustment for Plated Joints ....................................................................................... 14
TABLE 4
Parameters of Plate Thickness Adjustment for Tubular Joints ....................................................................................... 15
TABLE 5
Parameters of F-curves .......................................................... 15
TABLE 6
Details of Basic Design S-N Curves HSE(1995) .................... 29
TABLE 7
Life Reduction Factors to be Applied to the Lower Cycle Segment of the Design S-N HSE Curves ............................... 29
TABLE 8
Life Reduction Factors to be Applied to the Lower Segment of the Design S-N DNV Curves............................................... 30
FIGURE 1
API, DEn, and ABS S-N design Curves for Tubular Joints; Effective Cathodic Protection; No Profile Control Specified ................................................................................. 12
FIGURE 2
F-Curves with Thickness Adjustment and Test Data; 16 mm Plate ............................................................................ 17
FIGURE 3
F-Curves with Thickness Adjustment and Test Data; 20 mm Plate ............................................................................ 17
FIGURE 4
F-Curves with Thickness Adjustment and Test Data; 22 mm Plate ............................................................................ 18
FIGURE 5
F-Curves with Thickness Adjustment and Test Data; 25 mm Plate ............................................................................ 18
FIGURE 6
F-Curves with Thickness Adjustment and Test Data; 26 mm Plate ............................................................................ 19
FIGURE 7
F-Curves with Thickness Adjustment and Test Data; 38 mm Plate ............................................................................ 19
FIGURE 8
F-Curves with Thickness Adjustment and Test Data; 40 mm Plate ............................................................................ 20
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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FIGURE 9
F-Curves with Thickness Adjustment and Test Data; 50 mm Plate ............................................................................ 20
FIGURE 10
F-Curves with Thickness Adjustment and Test Data; 52 mm Plate ............................................................................ 21
FIGURE 11
F-Curves with Thickness Adjustment and Test Data; 70 mm Plate ............................................................................ 21
FIGURE 12
F-Curves with Thickness Adjustment and Test Data; 75 mm Plate ............................................................................ 22
FIGURE 13
F-Curves with Thickness Adjustment and Test Data; 78 mm Plate ............................................................................ 22
FIGURE 14
F-Curves with Thickness Adjustment and Test Data; 80 mm Plate ............................................................................ 23
FIGURE 15
F-Curves with Thickness Adjustment and Test Data; 100 mm Plate .......................................................................... 23
FIGURE 16
F-Curves with Thickness Adjustment and Test Data; 103 mm Plate .......................................................................... 24
FIGURE 17
F-Curves with Thickness Adjustment and Test Data; 150 mm Plate .......................................................................... 24
FIGURE 18
F-Curves with Thickness Adjustment and Test Data; 160 mm Plate .......................................................................... 25
FIGURE 19
F-Curves with Thickness Adjustment and Test Data; 200 mm Plate .......................................................................... 25
FIGURE 20
Test data with DEn(1990) Thickness Adjustment and their Regressed S-N Curves (All Thicknesses) .............................. 26
FIGURE 21
Test Data with HSE(1995) Thickness Adjustment and their Regressed S-N Curves (All Thicknesses) .............................. 26
FIGURE 22
Test Data with DNV(2000) Thickness Adjustment and their Regressed S-N Curves (All Thicknesses) .............................. 27
FIGURE 23
Regressed S-N Curves and Design F-curves ......................... 27
SECTION 4
Fatigue Design Factors ........................................................................ 31 1 Preliminary Remarks ......................................................................... 31 2 The Safety Check Expression........................................................... 31 3 Summaries of FDFs Specified by Others ......................................... 32
SECTION 5
The Simplified Fatigue Assessment Method ..................................... 34 1 Introduction ....................................................................................... 34 2 The Weibull Distribution for Long Term Stress Ranges ................... 34
3
4
vi
2.1
Definition of the Weibull Distribution .............................................. 34
2.2
A Modified Form of the Weibull Distribution for Offshore Structural Analysis ......................................................................... 35
Typical Values of the Weibull Shape Parameter γ for Stress ........... 35 3.1
Experience with Offshore Structures ............................................. 35
3.2
Experience with Ships ................................................................... 36
Fatigue Damage: General.................................................................36 4.1
Preliminary Remarks ..................................................................... 36
4.2
General Expression for Fatigue Damage ....................................... 36
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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5.1
Expression for Damage at Life, NR ................................................ 37
5.2
Miner’s Stress ................................................................................ 38
5.3
The Damage Expression for Weibull Distribution of Stress Ranges .......................................................................................... 38
6
Fatigue Damage for Bilinear S-N Curve ........................................... 38
7
Safety Check Using Allowable Stress Range ................................... 40
8
The Simplified Method for Which Stress is a Function of Wave Height ................................................................................................ 40
9
SECTION 6
Fatigue Damage for Single Segment S-N Curve .............................. 37
8.1
The Weibull Model for Stress Range; Stress as a Function of Wave Height .................................................................................. 40
8.2
The Weibull Model for Stress Range; Stress as a Function of Wave Height; Considering Two Wave Climates ............................ 41
The Weibull Distribution; Statistical Considerations ......................... 42 9.1
Preliminary Remarks ..................................................................... 42
9.2
Estimating the Parameters from Long-Term Data; Method of Moment Estimators ....................................................................... 42
9.3
Estimating the Parameters from Long-Term Data; Probability Plotting .......................................................................................... 42
9.4
Another Representation of the Weibull Distribution Function ........ 45
9.5
Fitting the Weibull to Deterministic Spectra ................................... 46
9.6
Fitting the Weibull Distribution to the Spectral Method .................. 47
TABLE 1
Data Analysis for Weibull Plot ................................................. 43
TABLE 2
Deterministic Spectra .............................................................. 46
FIGURE 1
A Short Term Realization of a Long-Term Stress Record ...... 34
FIGURE 2
Probability Density Function of s ............................................. 36
FIGURE 3
Characteristic S-N curve ......................................................... 37
FIGURE 4
Bilinear Characteristic S-N curve ............................................ 39
FIGURE 5
Weibull Probability Plot ........................................................... 44
FIGURE 6
Long Term Distribution of Fatigue Stress as a Function of the Weibull Shape Parameter ................................................. 45
FIGURE 7
Long-Term Stress Range Distribution of Large Tankers, Bulk Carriers, and Dry Cargo Vessels Compared with the Weibull .............................................................................. 46
FIGURE 8
Probability Density Function of Stress Ranges of the i-th Sea State ................................................................................ 47
The Spectral Based Fatigue Assessment Method ............................ 49 1 Preliminary Comments...................................................................... 49 2 Basic Assumptions............................................................................ 49 3 The Rayleigh Distribution for Short Term Stress Ranges................. 50 4 Spectral Analysis; More Detail .......................................................... 51 5 Wave Data ........................................................................................ 51 6 Additional Detail on Fatigue Stress Analysis; Global Performance Analysis ............................................................................................. 52
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7
8
SECTION 7
SECTION 8
viii
7.1
General Considerations ................................................................. 53
7.2
The Stress Process in Each Cell ................................................... 53
Fatigue Damage Expression for Wide Band Stress ......................... 54 8.1
Preliminary Comments .................................................................. 54
8.2
Definitions ...................................................................................... 55
8.3
The Equivalent Narrow Band Process ........................................... 56
8.4
The Rainflow Method ..................................................................... 56
8.5
A Closed Form Expression for Wide Band Damage ...................... 57
9
The Damage Calculation for Single Segment S-N Curve ................. 58
10
The Damage Calculation for Bi-Linear S-N Curve ............................ 59
TABLE 1
A Sample Wave Scatter Diagram ........................................... 52
FIGURE 1
Fatigue Assessments by Spectral Analysis Method ............... 50
FIGURE 2
Realizations of a Narrow Band and Wide Band Process (Both Have the Same RMS and Rate of Zero Crossings) ...... 55
FIGURE 3
Segment of Stress Process to Demonstrate Rainflow Method .................................................................................... 56
Deterministic Method of Fatigue Assessment ................................... 61 1 General ............................................................................................. 61 2 Application to a Self-Elevating Unit ................................................... 61 TABLE 1
Deterministic Stress Spectra ................................................... 61
TABLE 2
Wave and Other Parameters to be Used in the Fatigue Assessment............................................................................. 62
Fracture Mechanics Fatigue Model ..................................................... 63 1 Introduction ....................................................................................... 63 2 Crack Growth Model (Fatigue Strength) ........................................... 63
3
SECTION 9
The Safety Check Process ............................................................... 53
2.1
Stress Intensity Factor Range ....................................................... 63
2.2
The Paris Law ................................................................................ 63
2.3
Determination of the Paris Parameters, C and m ........................... 64
Life Prediction ................................................................................... 65 3.1
Relationship Between Cycles and Crack Depth ............................ 65
3.2
Determination of Initial Crack Size, ai ............................................ 65
3.3
Determination of the Failure (Critical) Crack Length, ac. ................ 66
TABLE 1
Paris Parameters for Structural Steel .....................................65
FIGURE 1
A Model of Crack Propagation Rate versus Stress Intensity Factor Range .......................................................................... 64
References ............................................................................................ 67
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section 1: Introduction
1
SECTION
1
Introduction
General Comments For over a half century, ABS has been involved in the development of fatigue technology, starting in 1946 with the formation of the Ship Structure Committee (SSC) for the specific goal of addressing avoidance of serious fracture in ships. The SSC, with strong financial support from ABS, has executed several fatigue research projects. Over the years, ABS has also provided support to numerous joint industry/agency fatigue projects in addition to independent investigators for their own in-house projects. The current state of the art in fatigue technology represents worldwide contributions of a large numbers of investigators from government agencies, professional organizations, classification societies, universities and private industry, most notably petroleum companies. ABS has synthesized this body of knowledge to provide fatigue design criteria for marine structures. This document provides a review of the most relevant literature, describes how ABS criteria were established and compares ABS criteria with those of other organizations. Because welded joints are subject to a variety of flaws, it is generally expected that fatigue cracks will start first at the joints. Therefore, the focus of this document will be on the joints, but the general principles and some of the fatigue strength data will apply to the base material.
2
Basic Terminology NT (or T)
=
Design life; the intended service life of the structure in cycles (or time)
Nf (or Tf)
=
Calculated fatigue life; the computed life in cycles (or time) of the structure using the design S-N curve
D
=
fatigue damage at the design life of the structure
∆
=
maximum allowable fatigue damage at the design life of the structure
FDF
=
fatigue design factor; FDF ≥ 1.0
The FDF accounts for:
3
i)
Uncertainty in the fatigue life estimation process
ii)
Consequences of failure (i.e., criticality)
iii)
Difficulty of inspection
The Deterministic Method and the Palmgren-Miner Rule to Define Fatigue Damage Fatigue assessment in the Guide relies on the characteristic S-N curve to define fatigue strength under constant amplitude stress and a linear damage accumulation rule (Palmgren-Miner) to define fatigue strength under variable amplitude stress. Fatigue stress is a random process. Stress ranges in the long-term process form a sequence of dependent random variables, Si; i = 1, NT. For purposes of fatigue analysis and design, it is assumed that Si are mutually independent. The set of Si can be decomposed and discretized into J blocks of constant amplitude stress, as illustrated in Section 1, Table 1.
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
1
Section
1
Introduction
TABLE 1 Deterministic Stress Spectra Stress Range Si
Number of Cycles ni
S1
n1
S2
n2
S3
. .
n3
SJ-1
nJ-1
SJ
nJ
Applying the Palmgren-Miner linear cumulative damage hypothesis to the block loading of Section 1, Table 1, cumulative fatigue damage, D, is defined as: D=
J
ni
∑N i =1
................................................................................................................................. (1.1)
i
where Ni is the number of cycles to failure at stress range Si, as determined by the appropriate S-N curve. Failure is then said to occur if: D > 1.0......................................................................................................................................... (1.2)
4
Application of the Palmgren-Miner (PM) Rule The PM rule is a simple algorithm for predicting an extremely complex phenomenon (i.e., fatigue under random stress processes). Results of tests, however, have suggested that the PM rule is a reasonable engineering tool for predicting fatigue in welded joints subjected to random loading. Statistical summaries of random fatigue tests have been reported by the UK Health and Safety Executive [HSE(1995)]. Let δ be a random variable denoting damage at failure and let δi denote damage at failure in a test of the i-th specimen in a sample of size, n. δi will depend on how the constant amplitude S-N curve is defined (e.g., as a median (best fit) curve through the center of the data or a design curve on the safe side (lower) of the data). The sample mean and standard deviation of δ can be computed from the random sample (δi ; i = 1, n). An empirical distribution can be fitted as well. A limited number of tests on tubular joints is available. In HSE(1995), a lognormal distribution is assumed for δ. Statistics computed from the data presented are summarized in Section 1, Table 2. It is noted that the scatter is quite broad, and it is likely that the wide distribution is largely a result of the inherent scatter in fatigue data and not the suitability of the PM algorithm. For reference purposes, the probability of δ being less than the reference curve is also presented in Section 1, Table 2.
TABLE 2 Tubular Joints: Statistics on Damage at Failure, (δ Lognormal Distribution Assumed) Best fit curve Design curve
2
~ Median, δ
COV, Cδ
1.41 4.42
0.98 0.98
Percent less than S-N curve 34 3.5
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
1
Introduction
For plated joints *, there is a relatively large database. Again, a lognormal distribution for δ is assumed, and the statistics are presented in Section 1, Table 3.
TABLE 3 Plated Joints: Statistics on Damage at Failure, (δ Lognormal Distribution Assumed) Best fit curve Design curve
5
~ Median, δ
COV, Cδ
Percent less than S-N curve
1.38 4.44
0.70 0.70
33 1.5
Safety Checking with Respect to Fatigue The safety check expression can be based on damage or life. While the damage approach is featured in the Guide, either approach below can be used. Damage The design is considered to be safe if: D ≤ ∆ .......................................................................................................................................... (1.3) where
∆ = 1.0/FDF ............................................................................................................................... (1.4) Life The design is considered to be safe if: Nf ≥ NT ∙ FDF.............................................................................................................................. (1.5)
* Note:
In the Guide, to conform to practice, the two general categories of structural details are referred to as “tubular” (really meaning “tubular intersection”) details and “non-tubular” details. In the context of the HSE (1995), the “non-tubular” details are referred to as “plate” or “plate type” details. The “plate” terminology will be used in this Commentary.
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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Section 2: Fatigue Strength Based on S-N Curves – General Concepts
2
SECTION
1
Fatigue Strength Based on S-N Curves – General Concepts
Preliminary Comments This Section introduces general concepts related to the S-N curve-based method of fatigue assessment. The next Section contains detailed information regarding S-N curves. For the stress-based approach to fatigue, the S-N curve defines fatigue strength. An example of S-N data and a design curve are shown in Section 2, Figure 1. Each point represents the cycles to failure N of a specimen subjected to constant range stress S. Log(N) is plotted versus Log(S). Section 2, Figure 1 presents the results of fatigue tests on tubular joints where failure is defined as first through wall cracking.
FIGURE 1 An Example of S-N Fatigue Data Showing the Least Squares Line and the Design Line [HSE(1995)] 1000
Hot Spot Stress Range, S (MPa)
Least Squares Line
++ + + + ++ +++ ++ +++ +++ ++ + Design +++++++ + + + ++++++ + + Line ++ +
100
+
++++ + + + + ++ +
+
Best Fit S-N Line Through 16 mm Data Design Line for 16 mm Data Experimental Data for 16 mm Thick Tubular Joints
10 10 000
100 000
1 000 000
107
108
Fatigue Endurance, N (Cycles)
A design curve is defined on the safe (lower) side of the data. Note that an implicit fatigue design factor is thereby introduced. For purposes of safety checking, the design S-N curve defines fatigue strength, but one should keep in mind that there is a large statistical scatter in fatigue data (relative to other structural design factors) with cycles-to-failure data often spanning more than two orders of magnitude.
4
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
2
2
Fatigue Strength Based on S-N Curves – General Concepts
Statistical Analysis of S-N Data The design curve is established as follows: First, it is noted that when S-N data are plotted in a log-log space, the data tend to plot as a straight line, as suggested in Section 2, Figure 1. A linear model can be employed, the form of which is: log(N) = log(A) – m log(S)........................................................................................................... (2.1) Base 10 logarithms are generally used. A and m are empirical constants to be determined from the data. A is called the fatigue strength coefficient and m is called the fatigue strength exponent. The parameter m is the negative reciprocal slope of the S-N curve, but for convenience, it is often referred to simply as the “slope”. Another component of the model is the standard deviation of N given S, denoted as σ(N|S), or simply, σ. This parameter describes the scatter in life. To estimate A, m and σ, the least squares method can be employed, thus providing parameters (A and m) to define the median S-N curve (i.e., a curve that passes through the center of the data). Note that S is the independent variable and N is the dependent variable. It is assumed that log(N) has a normal distribution, which means that N will have a lognormal distribution. For many welded joint fatigue data, the parameter m is approximately equal to 3.0. Therefore, for convenience and consistency, a fixed value of m = 3 is assumed and least squares analysis is then employed to estimate A and σ. Let A′ and σ′ denote the estimates. For the sample data of Section 2, Figure 1: m=3 log(A′) = 12.942
σ′ = 0.233 The coefficient of variation (standard deviation/mean) of cycle life N is required for a reliability analysis. The form for the COV is: CN = 10 (σ
2
/ 0.434 )
− 1 ................................................................................................................ (2.2)
For the example: CN = 0.58, or 58%
3
The Design Curve The design S-N curve is defined as the median curve minus two standard deviations on a log basis. Thus, the basic S-N curves are of the form: log(N) = log(A) − m log(S) where log(A) =
log(A1) − 2σ
N
=
predicted number of cycles to failure under stress range S
A1
=
constant relating to the mean S-N curve
σ
=
standard deviation of log N
m
=
inverse slope of the S-N curve
The relevant values of these terms are shown in the table below for the ABS “In-Air” S-N curves for platetype (non-tubular) details. The “in-air” S-N curves have a change of inverse slope from m to m + 2 at N = 107 cycles.
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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Section
2
Fatigue Strength Based on S-N Curves – General Concepts
TABLE 1 Details of the Basic “In-Air” S-N Curves A1
Standard Deviation
Class B
A1 2.343 × 1015
log10 15.3697
loge 35.3900
m 4.0
log10 0.1821
loge 0.4194
A 1.01 × 1015
C
1.082 × 1014
14.0342
32.3153
3.5
0.2041
0.4700
4.23 × 1013
D
3.988 × 1012
12.6007
29.0144
3.0
0.2095
0.4824
1.52 × 1012
E
3.289 × 10
12
12.5169
28.8216
3.0
0.2509
0.5777
1.04 × 1012
F
1.726 × 1012
12.2370
28.1770
3.0
0.2183
0.5027
0.63 × 1012
F2
1.231 × 1012
12.0900
27.8387
3.0
0.2279
0.5248
0.43 × 1012
G
0.566 × 1012
11.7525
27.0614
3.0
0.1793
0.4129
0.25 × 1012
W
0.368 × 1012
11.5662
26.6324
3.0
0.1846
0.4251
0.16 × 1012
If cycles to failure were lognormally distributed, then a specimen selected at random would have a probability of 2.3% of falling below the design curve. There may be confusion over this probability compared to those mentioned previously in Section 1, Tables 2 and 3. Different random variables are being referred to. In Section 1, Tables 2 and 3, the random variable is delta, the damage at failure. The statistics for delta are computed for both the best-fit curve and the design curve. Note that the fatigue test results are based on random stresses. The title of the column in the tables labeled, “Percent less than S-N curve” could have been alternatively labeled, “Percent of specimens that had lives below the S-N curve”. The basic S-N curves are established from constant amplitude tests. Assuming a lognormal distribution for life, the design curve is that curve below which 2.3% of the specimens are expected to fall. So, random fatigue test results are being compared to constant amplitude test results. It would not necessarily be expected that the results would be the same, but it is gratifying to see that the results are so close.
4
The Endurance Range Test data are much more limited in the range beyond 107 cycles. It appears that there may be an endurance limit near this point (i.e., a stress below which fatigue life would be infinite). However, a more prudent extrapolation of the S-N curve into the high cycle range involves a change in slope. For in-air structure, the slope (actually the negative reciprocal slope) beyond 107 cycles is: r = m + 2 ..................................................................................................................................... (2.3) While defined by engineering judgment, this form seems to have performed well for an extended period of time. This algorithm is used by DEn(1990) and others, but ISO(2000) specifies the knee of the curve at 108 cycles.
6
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
2
Fatigue Strength Based on S-N Curves – General Concepts
As an example, consider the ABS-(A) class D curve.
FIGURE 2 The Design S-N Curve for the ABS-(A) Class D Joint
Stress Range, S (MPa)
1000
100
Cycles to Failure, N 10 4 10
10
5
10
6
10
7
10
8
For plate joints that are cathodically protected, HSE(1995) specifies the knee at 106 cycles. For joints exposed to free corrosion, most organizations do not specify an endurance limit (i.e., the S-N curve is extrapolated into the high cycle range without a change in slope).
5
Stress Concentration Factors – Tubular Intersections A major theme of the presentation in Section 2 of the Guide is that the fatigue assessment should employ applicable stress concentration factors (SCFs) and the appropriate S-N curve. For a tubular joint, the S-N curves recommended by DEn(1990)/HSE(1995) and API RP2A are meant to be used with SCFs obtained for the hot-spot locations at the weld toe. The SCF equations referenced in the Guide’s Appendix 2 are meant to have precedence. However, allowance is made (Guide Paragraph 3/5.5) to also use, as appropriate, the parametric equations referenced in the API RP2A when it is permitted to use the API’s tubular joint S-N curves (e.g., structure sited on the U.S. Outer Continental Shelf, subject to US Minerals Management Service Regulation). Where conditions are such that the recommended parametric SCF equations cannot be applied confidently, then the SCFs can be obtained experimentally or numerically via finite element analysis. In either case, it is necessary to have a stress extrapolation procedure to weld toe locations that is compatible with the S-N curve. This is directly analogous to the extrapolation procedure for non-tubular details given in the Guide. The DEn provided guidance, as shown in Section 2, Figure 3, on the specific locations where the stresses should be obtained for extrapolation to the hot-spot locations at the weld toe.
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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Section
2
Fatigue Strength Based on S-N Curves – General Concepts
FIGURE 3 Weld Toe Extrapolation Points for a Tubular Joint a = 0.2(rt)0.5, but not smaller than 4 mm. r t Line 1.
Line 2. Brace.
A1 0.65(rt)0.5 Line 3.
A3
A2
B1 a B3
B2
0.65(rt)0.5
a
a 0.4(rtRT)0.25 5°
T
R
a B4 A4
Line 4. Chord.
8
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section 3: S-N Curves
SECTION
1
3
S-N Curves
Introduction In the offshore industry, fatigue assessment and design are based primarily on S-N curves to define strength. These curves define the integrity of both plate-type details and tubular welded joints under oscillatory loading. ABS has performed a comprehensive review of fatigue test results and fatigue strength models employed for steel structural details for the purpose of defining the ABS requirements. For the sources of design S-N curves, documents from three organizations, API (2000), AWS(2002), DEn (1990)/HSE(1995), are commonly cited by designers and analysts in the offshore industry. Agencies and organizations that provide structural design criteria for welded joints use these S-N curves and variations thereof. In order to gain a perspective on current practice, a digest of the S-N curves cited in various design criteria documents is provided in Subsection 3/2 below. The approach used in the ABS Guide for the classification of details, the S-N curves and adjustments made to the curves, may be referred to as a “hybrid” of the DEn(1990) and HSE(1995) criteria. The ABS Guide criteria uses:
2
•
The classification of details and basic S-N curves from the DEn(1990), which is almost identical to that found in HSE(1995) for plate-type details [a comparative description of DEn(1990) and HSE(1995) is given below in Subsection 3/2ii)].
•
For plate-type details, the thickness adjustment applies when t > 22 mm using tref = 22 mm and exponent of 0.25, and for tubular intersection details, the thickness adjustment applies when t > 22 mm using tref = 32 mm and exponent of 0.25.
•
The HSE(1995) “Environmental Reduction Factors” (ERFs), which is akin to “Corrosiveness” in the ABS Guide are for plate type details: 2.5 where effective Cathodic Protection (CP) is provided and 3.0 for Free Corrosion (FC) conditions, and for tubular intersection details, the ERFs are 2.0 for CP and 3.0 for FC conditions.
A Digest of the S-N Curves Used for the Structural Details of Offshore Structures i)
DEn (1990), Gurney (1979); A suite of eight curves for plated joints. Change in slope at 1E7 cycles, used successfully for many years by DEn and other criteria based on DEn
ii)
HSE(1995). Citations and comparisons to HSE and DEn criteria are difficult. The version of the fatigue criteria contained in the DEn “Guidance Notes” that was issued in 1990 was labeled the 4th Edition. It is referred to here as “DEn(1990).” Following DEn practice, changes to an edition were issued as “amendments” to that edition. Revision of the fatigue criteria in the 4th Edition was planned for publication in the 3rd amendment of the DEn “Guidance Notes” in 1995. At the same time, the DEn was undergoing organizational change, and the HSE became its successor organization. The document planned for release was relabeled, and is referred to here as “HSE(1995).” There were changes in the details of the criteria presentation between what had been planned as the 3rd amendment of the “Guidance Notes”, 4th Edition in 1995 and the superseding HSE(1995) document. However, immediately after the HSE(1995) fatigue criteria were issued, it was withdrawn along with all of the other DEn “Guidance Notes”.
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Section
3
S-N Curves
For information, the essential features of the HSE(1995) fatigue criteria compared to DEn(1990) are as follows.
3
The guidance provided on the classification of structural details and the assigned S-N curve to each class remained the same (see Appendix 1 of the Guide showing the classifications using the sketches of the various structural details and loading). Changes included in HSE(1995) were added guidance related to tubular member details and a change in the “W” S-N curve.
Also, in the detail classification guidance (for plate type details), it was planned to replace mention of the individual (8) S-N categories with one S-N curve, the “P” curve that was equivalent to the “D” curve in DEn(1990). Then, the detail classes would be related to the “P” curve by a ‘classification factor.’
The basic S-N curve for tubular intersection details was revised. In DEn(1990), the T curve is close to the “D”. The revised HSE(1995) T′ curve (in air) is higher than the 1990 T curve. However, the application of Environmental Reduction Factors (EFRs) and a revised thickness adjustment might produce significant reductions from the basic case.
In the DEn(1990), no reduction to an (in air) S-N curve is called for when effective Cathodic Protection is present. Based on additional testing, it was deemed necessary to include in HSE(1995) penalties for the Cathodic Protection (CP) case and to increase the penalties for the Free Corrosion case. For plate type details, the penalty factors are 2.5 and 3.0 for (CP) and (FC), respectively. For tubular intersection details, the respective penalty factors were 2.0 and 3.0. (The specific details of how these are applied are discussed in Subsection 3/7.)
Another planned, significant change between HSE(1995) and DEn(1990) concerns the adjustment to the S-N curves for thickness. The limiting thickness (above which adjustments are to be made), and the exponent and reference thickness in the adjustment equation were all affected.
iii)
ABS (2001) Rules for Building and Classing Steel Vessels. Since the original introduction in 1994, the criteria for fatigue strength in these Rules employ the DEn (1990) curves.
iv)
Eurocode 3 (1992). Uses a suite of 14 curves, with initial segments having slopes of 3.0. Beyond 5E6 cycles, the slopes are 5.0 for the curves up to 1E8 cycles, beyond which the curves are flat (endurance limit).
v)
IIW (1996). In general application, a suite of 14 S-N curves is presented. Each has an endurance limit at 5E6 cycles, after which the curve is flat. For marine application to be used together with Palmgren-Miner summation, another suite of 14 S-N curves that basically matches the Eurocode 3 curves is recommended: Beyond 5E6 cycles the curve has a slope of 5 and the curve has a cut-off limit at 1E8. The concept of a FAT class defines the joint detail.
vi)
DNV (2000); RP-C203 for offshore structures. Uses a suite of 14 curves [as in iv) and v)] that also incorporate the HSE(1995) curves. This reference also has S-N curves that reflect FC and CP conditions. It also has a curve for tubular joints, in-air and for CP and FC conditions in seawater.
vii)
ISO/CD 19902 (2000). The ISO draft standard appears to be based on DEn(1990), but the basic 2-segment S-N curves have a change of slope at 1E8 cycles, which is not the same as DEn(1990). S-N curves are also provided for tubular intersection details and cast steel tubular joints.
viii)
API (2001 a & b). RP2A (both WSD and LRFD) has S-N curves for tubular intersection joints. Defines X and X’ curves for joints with and without weld profile control, respectively. Cites ANSI/AWS D1.1- for plate joints.
ix)
API RP2T(1997). Cites RP2A for definition of S-N curves.
General Comparison Section 3, Table 1 summarizes the characteristics of the S-N design curves of DEn(1990)/HSE(1995) and API/AWS relative to environment, cathodic protection, and weld improvement.
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ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
3
S-N Curves
TABLE 1 Coverage of the Two Main Sources of S-N Curves Used for Offshore Structures Detail Type Tubular Intersection
Non-Tubular (Plate) Notes:
1&2
Corrosion Condition
API (2000) Notes 1&4
AWS D1.1
DEn(1990) HSE(1995) Notes 2&3
In-Air
√
-
√
Cathodic Protection
√
-
√
Free Corrosion in the Sea Water
√
-
√
In-Air
-
√
√
Cathodic Protection
-
Note 5
√
Free Corrosion in the Sea Water
-
-
√
Fatigue life enhancement via “Weld Improvement” techniques is explicitly permitted: --in API RP2A by weld profiling --in DEn/HSE by weld toe grinding
3
DEn/HSE is the basis of the ABS criteria
4
API RP 2A treats corrosion differently from the other codes. API RP 2A uses one curve with different endurance limits to represent the three corrosion cases (in-air, in seawater with free corrosion, and in seawater with cathodic protection). DEn/HSE use three curves to represent the three cases.
5
While AWS does not address modification of S-N curves for CP, API RP2A specifies an endurance limit at 2 × 108 cycles for plate type details.
4
Tubular Intersection Connections
4.1
Without Weld Profile Control A summary of the API and HSE(1995) having no weld profile control is presented as follows. API RP 2A(2000) uses the X′ curve for the following three corrosion cases with various endurance limits: •
In the air, endurance limit = 2 × 107 cycles
•
Cathodic protection, endurance limit = 2 × 108 cycles
•
Free corrosion in sea water, no endurance limit
HSE(1995) defines a T’ curve and its derivatives for the three corrosion cases: •
In-air,
•
Cathodic protection, (CP)
•
Free corrosion in sea water, (FC)
The ABS Guide specifies a T curve and recognizes three corrosion cases: •
In-air, (A)
•
Cathodic protection, (CP)
•
Free corrosion in sea water, (FC)
Section 3, Figure 1 presents the S-N curves for the CP case for tubular joints for: HSE (1995) T’ with CP, API RP2A the X’ curve, and the ABS T (CP) curve, as provided in the ABS Guide. The latter is based on the use of the DEn(1990) T curve, which is adjusted as recommended in HSE (1995). ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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Section
3
S-N Curves
FIGURE 1 API, DEn, and ABS S-N design Curves for Tubular Joints; Effective Cathodic Protection; No Profile Control Specified
4.2
With Weld Improvement (1 February 2013) A summary of the API and HSE/DEn S-N curves for joints of tubular members having weld improvement is presented in the following. API RP 2A(2000) uses the X curve for the following three corrosion cases with various endurance limits: •
In-air, endurance limit = 107 cycles
•
Cathodic protection, endurance limit = 2 × 108 cycles
•
Free corrosion in seawater, no endurance limit.
The crediting of weld profile control (i.e., concave weld profile) and other fatigue strength enhancements are not mentioned in the Guide for use with the ABS S-N curves. The main reason for this is to discourage (however, not ban) the use of such a credit in design. In this way, the credit will be available if needed in the future [say, if design changes occur after structural fabrication begins and even later in the structure’s life should reconditioning or reuse be considered]. Out of necessity and in a limited, particular circumstance, the Guide (in its Appendix 3) allows the use of the API X curve, which requires weld profile control and NDE. Grinding is preferably to be carried out by rotary burr and to extend below the plate surface in order to remove toe defects and the ground area is to have effective corrosion protection. The treatment is to produce a smooth concave profile at the weld toe with the depth of the depression penetrating into the plate surface to at least 0.5 mm below the bottom of any visible undercut. The depth of groove produced is to be kept to a minimum, and, in general, kept to a maximum of 1 mm. In no circumstances is the grinding depth to exceed 2 mm or 7% of the plate gross thickness, whichever is smaller. Grinding has to extend to areas well outside the highest stress region. The finished shape of a weld surface treated by ultrasonic/hammer peening is to be smooth and all traces of the weld toe are to be removed. Peening depth below the original surface is to be maintained at least 0.2 mm. Maximum depth is generally not to exceed 0.5 mm. Provided these recommendations are followed, when using the ABS S-N curves, a credit of 2 on fatigue life may be permitted when suitable toe grinding or ultrasonic/hammer peening are provided. Credit for an alternative life enhancement measure may be granted based on the submission of a well-documented, project-specific investigation that substantiates the claimed benefit of the technique to be used.
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Section
5
3
S-N Curves
Plated Connections For plated connections, API RP2A cites the ANSI/AWS D1.1-92 [AWS(1992)] S-N design curves. The S-N curves of the newer AWS(2002) document are essentially the same as AWS(1992). The AWS and DEn (1990) curves are compared below. Both references use sketches to help the designer in the selection of a detail’s classification. The comparison is not exact. Observations that contrast the two main reference sources are: i)
DEn has eight classes or categories of joint types. AWS has six.
ii)
DEn is more discriminating in the number of joint types or details.
iii)
There are differences in the definition of the detail category.
iv)
DEn employs a thickness adjustment (see Subsection 3/6). There is no thickness adjustment in the AWS criteria.
v)
Except for free corrosion in seawater, AWS specifies a stress “endurance limit” in the high cycle range. DEn changes to a shallower slope.
vi)
Overall, there is no direct correspondence of categories, but there are a few that are similar. These are summarized in Section 3, Table 2.
TABLE 2 AWS-HSE/DEn Curves for Similar Detail Classes Detail Class Base or parent material Full penetration butt welds, groove welds
ANSI/AWS(1992)
DEn(1990)
A
B
B
C
C (L < 50 mm) D (50 < L < 100) E (L > 100)
F (L < 150 mm) F2 (L > 150 mm)
Parent material of cruciform T-joints
C
F
Load carrying fillet welds transverse to the direction of stress (parent material)
E
F (d > 10 mm) G (d < 10)
Load carrying fillet welds transverse to the direction of stress (weld material)
F
W
Parent material at the end of butt welded attachments
For conditions of effective cathodic protection (CP): i)
API specifies a stress endurance limit on the AWS curves at 2 × 108 cycles.
ii)
The DEn CP curves have a break at 106 cycles. The slope to the left is m; to the right, it is m + 2). The DEn curves are lowered from the in-air curves by a factor of 2.5 on life, again maintaining the break point at 106 cycles.
For conditions of free corrosion, both curves have no endurance limit or slope change in the high cycle range (i.e., the low cycle curve with a slope of 3.0 is continued into the high cycle range). In addition, the DEn curves are lowered by a factor of 3.0 on life.
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Section
3
S-N Curves
6
Discussion of the Thickness Effect
6.1
Introduction The ABS-recommended thickness adjustment (size effect) is based on studies of fatigue test data as well as models used by others. A summary of this study is presented below. The basic S-N design curve has the functional form: log10 N = log10 A – mlog10 S ......................................................................................................... (3.1) where N is cycles to failure, S is stress range, and A and m are respectively, the fatigue strength coefficient and exponent. The size effect in fatigue in which larger sections tend to be weaker is manifest in welded joint fatigue by a thickness adjustment. In API, HSE/DEn and other codes, the effect of plate thickness is addressed by a similar adjustment formula: t Sf = S tR
−q
Sf = S
t > t0.................................................................................................................. (3.2) t ≤ t0.................................................................................................................. (3.3)
where Sf
=
allowable stress range,
S
=
allowable stress range from the nominal S-N design curve,
q, tR =
parameters (tR is the “reference” thickness),
t0
=
thickness above which adjustments should be made,
t
=
actual thickness.
A “thickness adjusted” S-N curve can be constructed when t > t0. t log10 (N) = log10 (A) – m log S tR
+q
.................................................................................... (3.4)
The parameters q and tR are determined empirically. For plated joints, Section 3, Table 3 summarizes these parameter values from the references: DEn (1990), HSE (1995) and DNV (2000). (Size effect is not considered in ANSI/AWS D1.1.)
TABLE 3 Parameters of Plate Thickness Adjustment for Plated Joints Parameters
DEn (1990)
HSE (1995)
DNV (2000)
q
0.25
0.30
0.0 – 0.25 depending on detail classification; 0.25 for F-curve
tR
22 mm
16 mm
25 mm
These values do not depend upon the environment (i.e., they are the same for the in-air, cathodic protection and free corrosion curves). The objective of this section is to compare the three parameter sets with the test data on plated joints that were used in reviewing the thickness effect by HSE (1995) and to recommend the algorithm to be used by ABS in the Guide. 14
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Section
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S-N Curves
For reference, the tubular joint parameters are also given in Section 3, Table 4.
TABLE 4 Parameters of Plate Thickness Adjustment for Tubular Joints Parameters
API (2000, 1993)
HSE (1995)
DNV (2000)
q
0.25
0.30
0.25 for SCF < 10.0 0.30 for SCF > 10.0
tR
25 mm
16 mm
32 mm
6.2
Fatigue Test Data on Plated Joints An analysis was undertaken of data from tests on as-welded T-butt and cruciform joints that belong to the F classification [HSE(1995)]. The specimens varied in thickness from 16 mm to 200 mm. There are a total of 146 specimens in which 125 specimens have equal main plate and attachment thickness. Stress ranges in the tests varied from 56 MPa to 341 MPa and only four specimens had a fatigue life exceeding 107 cycles.
6.3
Design F-Curves with Thickness Adjustment The parameters of the basic F-curves used in the three codes are shown in Section 3, Table 5. The F-curves in DEn (1990) and HSE (1995) are identical, but with different thickness adjustment formulae. The DNV (2000) F-curve is slightly less conservative than the other two.
TABLE 5 Parameters of F-curves N < 107
N > 107
Codes
log10 (A)
m
log10 (A)
m
DEn (1990)
11.801
3
15.001
5
HSE (1995)
11.801
3
15.001
5
DNV (2000)
11.855
3
15.091
5
The design F-curves with thickness adjustment (Equation 3.4) are plotted in Section 3, Figures 2 through 19. In ascending order, each curve has a different thickness. The test data for each thickness are plotted. The HSE (1995) F-curve of 16 mm thickness (i.e., without thickness adjustment) is also plotted in figures where it is appropriate for reference. These series of figures demonstrate the general detrimental effect of increasing plate thickness. There exist relatively large safety margins between the test data and design curves, with the HSE (1995) curve having the largest gap.
6.4
Thickness Adjustments to Test Data and Their Regressed S-N Curves For a different viewpoint, the adjustment of Equation 3.2 is applied to the data and then compared to the basic curves (without the thickness adjustment). In this analysis, only data for specimens with equal main plate and attachment thicknesses were included because HSE used the same strategy in their study on thickness effect. Data with fatigue lives longer than 107 cycles were also excluded due to the small sample size (i.e., insufficient data to regress the curve segment for N > 107). With the adjusted data, “quasi-design” S-N curves were produced. These curves were constructed by taking the least squares line and shifting it two standard deviations (on a log basis) to the left. The adjusted data, (the quasi-design S-N curves,) and the basic F-curves, without thickness adjustments, are plotted together for comparison. The results for DEn (1990), HSE (1995) and DNV (2000) are shown in Section 3, Figures 20 through 22, respectively. The comparison across the codes is demonstrated in Section 3, Figure 23. The conclusion stated previously is justified. There are relatively large safety margins between the regressed S-N curves and design curves, with HSE (1995) curve having the largest margin.
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Section
6.5
3
S-N Curves
Discussion In reviewing the commentary document [HSE (1992)] that supports the HSE Fatigue Criteria [HSE (1995)], it is found that with the thickness adjustment of HSE (1995), all test data locate above the P-curve [i.e., D-curve in DEn (1990)], while the test specimens were as-welded T-butt and cruciform joints that belong to F-curve of joint classification. This gap indicates that HSE (1995) thickness adjustment formula is too conservative. Perhaps, in recognition of the possible excessive conservatism for particular details, a clause is included in HSE (1995) so that alternative adjustments may be used if they are supported by results from experiments or from fracture mechanics analyses. A statement that the basic 16 mm P-curve is equivalent to the 22 mm D-curve in DEn (1990) is found in a commentary paper on the HSE (1995) [Stacey and Sharp (1995)]. Therefore, one may ask why it is necessary to make a thickness adjustment to joints with a 22 mm thickness. In a commentary paper of DNV RP-C203 [Lotsberg and Larsen (2001)], a similar study was conducted and a conclusion is that use of the F-curve for this detail with reference thickness 16 mm is conservative.
6.6
16
Postscript Due to the discrepancy between the thickness adjustment formulae, there is a question as to how the thickness adjustment formula of HSE (1995) was derived. It is speculated by the authors of this Commentary that the algorithm was obtained by borrowing the form for tubular joints, or by using a curve other than the F-curve as the target curve for regression analysis, or perhaps using some other procedure. The origin of the algorithm is not documented in HSE (1992). Thus, the procedure used to derive the thickness adjustment formula of HSE (1995), particularly the choice of 16 mm as basic thickness, is not clear.
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Section
3
S-N Curves
FIGURE 2 F-Curves with Thickness Adjustment and Test Data; 16 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 3 F-Curves with Thickness Adjustment and Test Data; 20 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
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Section
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S-N Curves
FIGURE 4 F-Curves with Thickness Adjustment and Test Data; 22 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 5 F-Curves with Thickness Adjustment and Test Data; 25 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995)-16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
18
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S-N Curves
FIGURE 6 F-Curves with Thickness Adjustment and Test Data; 26 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 7 F-Curves with Thickness Adjustment and Test Data; 38 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
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Section
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S-N Curves
FIGURE 8 F-Curves with Thickness Adjustment and Test Data; 40 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 9 F-Curves with Thickness Adjustment and Test Data; 50 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
20
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
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S-N Curves
FIGURE 10 F-Curves with Thickness Adjustment and Test Data; 52 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 11 F-Curves with Thickness Adjustment and Test Data; 70 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
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Section
3
S-N Curves
FIGURE 12 F-Curves with Thickness Adjustment and Test Data; 75 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 13 F-Curves with Thickness Adjustment and Test Data; 78 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
22
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S-N Curves
FIGURE 14 F-Curves with Thickness Adjustment and Test Data; 80 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 15 F-Curves with Thickness Adjustment and Test Data; 100 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000) Test Data
Stress Range (MPa)
HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
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Section
3
S-N Curves
FIGURE 16 F-Curves with Thickness Adjustment and Test Data; 103 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data HSE(1995) 16mm
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 17 F-Curves with Thickness Adjustment and Test Data; 150 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data HSE(1995) 16mm 100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
24
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Section
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S-N Curves
FIGURE 18 F-Curves with Thickness Adjustment and Test Data; 160 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data HSE(1995) 16mm 100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 19 F-Curves with Thickness Adjustment and Test Data; 200 mm Plate
1000
DEn(1990) HSE(1995) DNV(2000)
Stress Range (MPa)
Test Data HSE(1995) 16mm 100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
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Section
3
S-N Curves
FIGURE 20 Test data with DEn(1990) Thickness Adjustment and their Regressed S-N Curves (All Thicknesses)
1000
DEn F-Curve without Thickness Correction Test Data with Thickness Correction
Stress Range (MPa)
Regressed S-N Curve
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 21 Test Data with HSE(1995) Thickness Adjustment and their Regressed S-N Curves (All Thicknesses)
1000
HSE F-Curve without Thickness Correction Test Data with Thickness Correction
Stress Range (MPa)
Regressed S-N curve
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
26
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Section
3
S-N Curves
FIGURE 22 Test Data with DNV(2000) Thickness Adjustment and their Regressed S-N Curves (All Thicknesses)
1000
DNV F-Curve without Thickness Correction Test Data with Thickness Correction
Stress Range (MPa)
Regressed S-N Curve
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
FIGURE 23 Regressed S-N Curves and Design F-curves
1000
Stress Range (MPa)
DEn F-Curve without Thickness Correction HSE F-Curve without Thickness Correction DNV F-Curve without Thickness Correction Regressed S-N Curve with HSE Thickness Correction Regressed S-N Curve with DEn Thickness Correction Regressed S-N Curve with DNV Thickness Correction
100
10 1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
1.00E+08
1.00E+09
N
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Section
3
S-N Curves
7
Effects of Corrosion on Fatigue Strength
7.1
Preliminary Remarks ABS recommendations for considering the effects of corrosion on fatigue strength are based on a review of corrosion effects published in specifications, guidance and recommended practice documents relating to marine structures. A digest of the corrosion requirements relative to fatigue is presented for each of several documents in 3/7.3, below. There is no particular significance to the ordering of the documents presented.
7.2
A Summary of the Results A review of the requirements suggests only that fatigue strength is reduced in the presence of free corrosion. One approach is providing separate S-N curves for in-air and free corrosion conditions. Another is to specify a reduction factor on in-air life when operating in a corrosive environment. It is generally thought that effective cathodic protection restores fatigue strength to in-air values. However, both HSE and DNV specify a reduction of the in-air curves for CP joints exposed to seawater. Moreover, for DNV ship requirements, factors are provided for reduction of in-air S-N curves for those cases where cathodic protection has become ineffective later in life. Some documents provide no adjustments for corrosive environments. ABS archives contain results of corrosion studies on marine structures. These results suggest: (1) it is very difficult to characterize corrosion in a general, useful engineering context, and (2) there is enormous statistical variability in corrosion rates.
7.3
The Summaries API RP2T [API(1997)] No specific reference to corrosion requirements. API RP2A [API(2000, 1993)] i) For all non-tubular members, refer to ANSI/AWS D1.1-92 (Table 10.2, Figure 10.6). No endurance limit should be considered for those members exposed to corrosion. For submerged members where cathodic protection is present, the endurance limit is set at 2 × 108 cycles.
ii)
The S-N curves are the X and X′ curves. These curves assume effective cathodic protection. For splash zone, free corrosion or excessive corrosion conditions, no endurance limit should be considered.
Fatigue Design of Welded Joints and Components [IIW (1996)] The basic fatigue requirements presented assume corrosion protection. If there is unprotected exposure, the fatigue class should be reduced. The fatigue limit may also be reduced considerably. Offshore Installations: Guide on Design, Construction, and Certification, [HSE (1995)]
This document defines basic design curves for plates (P curve) and for tubular joints (T′ curve). A classification factor is applied to the P curve to account for different joint types. There are three sets of the basic curves: (1) in-air, (2) seawater with corrosion protection, and (3) free corrosion. (3) is lower than (2) and (2) is lower than (1). The S-N curves are defined in Section 3, Table 6.
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Section
3
S-N Curves
TABLE 6 Details of Basic Design S-N Curves HSE(1995) Class P P
Environment Air
Log10A 12.182 15.637*
m 3 5**
SQ (N/mm2) 53
NQ (cycles) 107
P P P
Seawater (CP) Seawater (CP) Seawater (FC)
11.784 15.637* 11.705
3 5** 3
84
1.026 × 106
T′
Air
12.476
3
67
107
95
1.745 × 106
T′
16.127*
5**
T′
Seawater (CP)
12.175
3
T′
Seawater (CP)
16.127*
5**
T′
Seawater (FC)
12.000
3
*
Fatigue strength coefficient (C; see Section 5, Figure 4) beyond NQ
**
Fatigue strength exponent (r; see Section 5, Figure 4) beyond NQ
The parameters of Section 3, Table 6 can be translated into reduction factors to be applied to life in the lower life segment of the in-air S-N curves. These factors are defined in Section 3, Table 7.
TABLE 7 Life Reduction Factors to be Applied to the Lower Cycle Segment of the Design S-N HSE Curves Tubular Joints
Plated Joints
Cathodic Protected
2.0
2.5
Free Corrosion
3.0
3.0
ISO CD 19902, International Standards Organization [ISO/CD 19902 (2000)] This is a draft document.
Basic in-air S-N curves are defined for tubular joints, cast joints and other joints. Joints with cathodic protection. The basic in-air curves apply for N greater than 106 cycles. If significant damage may occur with N less than 106 cycles, a factor of 2 reduction on life is recommended. Free corrosion. A reduction factor of 3 on life is required. There is to be no slope change at 108 cycles. Note:
The editing panel found these statements confusing, so they have requested a re-write.
RP-C203, Fatigue Strength Analysis of Offshore Structures, Det norske Veritas [DNV (2000)] There are 14 S-N curves, each representing a joint classification. These S-N curves are specified separately for: (1) in-air, (2) seawater with cathodic protection, and (3) seawater with free corrosion.
In-air. The S-N curves have a break at 107 cycles with a slope of m = 3 in the low cycle range and m = 5 in the high cycle range. Cathodic protection. The S-N curves in the low cycle range are reduced by the factor of 2.5 on life for both tubular and plated joints. The curves have a break at 106 cycles. Free Corrosion. The S-N curves in the low cycle range are reduced by the factor of 3.0 on life for both tubular and plated joints (see Section 3, Table 8). There is no break in the curves (i.e., m = 3) for all values of S. ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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Section
3
S-N Curves
TABLE 8 Life Reduction Factors to be Applied to the Lower Segment of the Design S-N DNV Curves Tubular Joints
Plated Joints
Cathodic Protected
2.5
2.5
Free Corrosion
3.0
3.0
Eurocode 3 Design of Steel Structures, BSI Standards, 1992 [Eurocode 3, (1992)] No specific reference to corrosion. Fatigue Assessment of Ship Structures, Classification Notes No. 30.7, Det norske Veritas, [DNV (1998)] A factor is specified for reduction of in-air curves for those cases where cathodic protection is effective for only a fraction of the life. BS 7608 Fatigue Design and Assessment of Steel Structures, British Standards Institute [BS 7608 (1993)] For unprotected joints exposed to seawater, a factor of safety on life of 2 is required. For steels having a yield strength in excess of 400 MPa, this penalty may not be adequate. ABS Design Curves; Guide on the Fatigue Assessment of Offshore Structures The ABS in-air curves for both plated and tubular members are those given in DEn(1990). The basis for this choice is: (1) the history of successful practice, (2) worldwide acceptance, and (3) relatively conservative performance in the high cycle range.
The API (2000) curves are permitted as an alternative for application in the Gulf of Mexico based on the history of successful practice and their mandated use by U.S. Regulatory Bodies. Adjustment for thickness (see Equations 3.2 and 3.3) For plated details: q = 0.25; tR = 22 mm For tubular details: q = 0.25; tR = 32 mm; This applies for thicknesses greater than 22 mm. The following adjustments to the in-air curves for corrosion were subsequently recommended by the HSE(1995), these were adopted by ABS. Tubular Details •
With CP. A penalty factor of 2.0 on life applied to the low cycle segment of the in-air S-N curve and no penalty on life applied to the high cycle segment of the in-air S-N curve.
•
Free corrosion. A penalty factor of 3.0 on life applied to the low cycle segment of the in-air S-N curve and continuation of the obtained curve to the high cycle range.
Plated Details •
With CP. A penalty factor of 2.5 on life applied to the low cycle segment of the in-air S-N curve and no penalty on life applied to the high cycle segment of the in-air S-N curve.
•
Free corrosion. A penalty factor of 3.0 on life applied to the low cycle segment of the in-air S-N curve and extrapolation of the obtained curve to the high cycle range.
The following adjustments to the in-air curves for corrosion are recommended for the API X and X′ curves. Tubular joints
30
•
CP; endurance limit at 2 × 108 cycles.
•
FC; no endurance limit. ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section 4: Fatigue Design Factors
4
SECTION
1
Fatigue Design Factors
Preliminary Remarks The purpose of a fatigue design factor is to account for uncertainties in the fatigue assessment and design process. The process includes operations of estimating dynamic response and stresses under environmental conditions. The uncertainties include the following: •
Statistical models used to describe the sea states
•
Prediction of the wave-induced loads from sea state data
•
Computation of nominal element loads given the wave-induced loads
•
Computation of fatigue stresses at the hot spot from nominal member forces
•
Application of Miner’s rule
•
Fatigue strength as seen in the scatter in test data, where a typical coefficient of variation on life is approximately 50-60%.
•
Environmental effects on fatigue strength (e.g., corrosion)
•
Size effects on fatigue strength
•
Manufacturing, assembly and installation operations
In addition to uncertainties, the fatigue design factor should also account for: •
Ease of in-service inspection of a detail
•
Consequences of failure (criticality) of a detail
While reliability methods promise the most rational way of managing uncertainty, the concept of a factor of safety on life [referred herein as a fatigue design factor (FDF)], maintains universal acceptance.
2
The Safety Check Expression The safety check expression can be based on damage or life. While the damage approach is featured in the Guide, either approach below can be used and are exactly equivalent. Refer to Subsection 1/2 for terminology. Subsection 1/5 is repeated here for reference. Damage. The design is considered to be safe if: D ≤ ∆ .......................................................................................................................................... (1.3) where
∆=
1.0 ................................................................................................................................... (1.4) FDF
Life. The design is considered to be safe if: Nf ≥ (FDF) (NT) .......................................................................................................................... (1.5) Fatigue design factors specified in relevant documents are summarized in this Section. ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
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Section
3
4
Fatigue Design Factors
4
Summaries of FDFs Specified by Others The following is a summary of factors of safety on life that have been extracted from documents relevant to marine structural fatigue. The safety factors by themselves do not tell the whole story and may not address all of the issues raised above. However, it is instructive and helpful in the development of the Guide to review those factors that have been published in relevant documents. It should be noted that safety factors associated with free corrosion and CP are not included in these factors and should be applied separately. API RP2T [API (1997)] “General structure. In general, it is recommended that the design fatigue life of each structural element of the platform be at least three times the intended service life of the platform.”
“Tendons. … high uncertainties exist … The component fatigue life factor of ten is considered a reasonable blanket requirement.” API RP2A [API (2000, 1993)] “In general, the design fatigue life of each joint and member should be at least twice the intended service life of the structure (i.e., FDF = 2.0).” Fatigue Design of Welded Joints and Components, [IIW (1996)] For fatigue verification, it has to be shown that the total accumulated damage is less than 0.5 (i.e., FDF = 2.0). ABS Rules for Building and Classing Steel Vessels, Part 5, The American Bureau of Shipping [ABS (2001)] No safety factor specified (i.e., an implied factor of safety on life of 1.0). However, since computed stress is based on “net” scantlings, the nominal FDF is greater than 1.0. Offshore Installations: Guidance on Design, Construction and Certification, UK Department of Energy [DEn (1990)] No specific value given. “In defining the factor of safety on life, account should be taken of the accessibility of the joint and the proposed degree of inspection as well as the consequences of failure.” ISO CD 19902, International Standards Organization [ISO CD 19902 (2000)] In lieu of more detailed fatigue assessment, the FDF can be taken from the following table: Failure Critical
Inspectable
Uninspectable
No
2.0
5.0
Yes
5.0
10.0
RP-C203 Fatigue Strength Analysis of Offshore Structures, Det norske Veritas [DNV (2000)] “Design fatigue factor from OS-C101, Section 6, Fatigue Limit States”
Design Fatigue Factor (DFF) (Table A1 of DNV-OS-C101 “Design of Offshore Steel Structures, General (LRFD Method)”, Section 6) The following DFFs are valid for units with low consequence of failure and where it can be demonstrated that the structure satisfies the requirement for the damaged condition according to the Accidental Limit State (ALS) with failure in the actual joint as the defined damage. DFF
32
Structural element
1
Internal structure, accessible and not welded directly to the submerged part.
1
External structure, accessible for regular inspection and repair in dry and clean conditions.
2
Internal structure, accessible and welded directly to the submerged part.
2
External structure, not accessible for regular inspection and repair in dry and clean conditions.
3
Non-accessible areas, areas not planned to be accessible for inspection and repair during operation. ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
4
Fatigue Design Factors
4
Eurocode 3 Design of Steel Structures, BSI Standards [Eurocode 3, (1992)] This document lists safety factors on stress. These are converted to FDF in the following table. Fail Safe (a) Components
Non Fail Safe (b) Components
Periodic inspection and maintenance (accessible joint)
1.00
1.95
Periodic inspection and maintenance (poor accessibility)
1.52
2.46
Inspection and access
Notes: (a)
local failure of one component does not result in failure of the structure
(b)
local failure of one component leads rapidly to failure of the structure
Fatigue Assessment of Ship Structures, Classification Notes No. 30.7, Det norske Veritas [DNV (1998)] “Accepted usage factor is defined as 1.0” (FDF = 1.0) BS 7608 Fatigue Design and Assessment of Steel Structures, British Standards Institute [BS 7608 (1993)] The standard basic S-N curves are based on a mean minus two standard deviations.... Thus, an additional factor on life (i.e., the use of S-N curves based on the mean minus more that two standard deviations) should be considered for cases of inadequate structural redundancy.
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Section 5: The Simplified Fatigue Assessment Method
SECTION
1
5
The Simplified Fatigue Assessment Method
Introduction The simplified fatigue assessment method employs the Weibull distribution to model the long-term distribution of sea states. In fact, other distributions could be used, but the Weibull is standard practice in the marine industry. In this Section, the Weibull distribution is defined and described. Expressions for fatigue damage at the design life NT of the structure are derived. Also, the allowable stress range approach to safety checking is derived. Statistical considerations associated with the Weibull distribution are provided in Subsection 5/9.
2
The Weibull Distribution for Long Term Stress Ranges
2.1
Definition of the Weibull Distribution A segment of a long-term stress record at a fatigue sensitive point is shown in Section 5, Figure 1.
FIGURE 1 A Short Term Realization of a Long-Term Stress Record S
i
S
Stress, S(t)
i+1
time, t
The stress range, Si, for the i-th trough and peak is defined. Stress ranges, Si, i = 1, n, form a sequence of n dependent random variables. In the linear damage accumulation model, this dependency is ignored. Thus, it will be assumed that Si, i = 1, n is a random sample of independent and identically distributed random variables. Let S be a random variable denoting a single stress range in a long term stress history. Assume that S has a two-parameter Weibull distribution. The distribution function is:
s r Fs(s) = P(S ≤ s) = 1 – exp − δ
s > 0 ......................................................................... (5.1)
where γ and δ are the Weibull shape and scale parameters, respectively. The shape parameter is predetermined from a detailed stress spectrum analysis or by using historical, empirical data (see Subsections 5/3 and 5/9). 34
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The Simplified Fatigue Assessment Method
The parameters in terms of the mean and standard deviation of S are: S S
1.08
S 1 1
.................................................................................... (5.2)
where S and S are the mean and standard deviation of S respectively. The expression for is approximate, but for engineering purposes, very close to the exact. (x) is the gamma function defined as: (x) =
0
t x 1e t dt ..................................................................................................................... (5.3)
The gamma function is widely available in mathematical analysis programs (e.g., MatLab) and also in some programmable calculators.
2.2
A Modified Form of the Weibull Distribution for Offshore Structural Analysis The magnitude of stresses is defined by . However, for design and safety check purposes, it is convenient to represent in terms of the long term stress spectra as described in the following. Define a reference life, NR. This could be a time over which records are available (e.g., three years). It could also be chosen as the design life NT. Define a reference stress range SR which characterizes the largest stress anticipated during NR. The probability statement defines SR: P( S S R )
1 ........................................................................................................................ (5.4) NR
SR is the value that the fatigue stress range S exceeds on the average once every NR cycles. From the definition of the distribution function, FS(SR) = P(S ≤ SR), it follows from Equations 5.1 and 5.4 that:
SR
ln N R 1 /
.......................................................................................................................... (5.5)
The parameter, , is a measure of the amplitude of S(t) and will be independent of the length of time NR considered. In the special case where NR is taken as the design life NT, the corresponding stress range ST is defined by Equation 5.4. The fatigue stress range S will exceed ST on the average once every NT cycles. Thus, ST can be interpreted as a maximum stress applied during the design life and may be used for static design. ST is sometimes called the “once-in-a-lifetime” stress. Using the Weibull formulation, as described in Subsection 5/6, the fatigue strength can be formulated in terms of maximum allowable stress and a safety check expression based on stress can be derived.
3
Typical Values of the Weibull Shape Parameter for Stress
3.1
Experience with Offshore Structures The Weibull shape parameter for stress, S typically varies between 0.7 to 1.4, depending on the dominant period of the structural response and the considered wave environments. For example, Gulf of Mexico fixed platforms experience = 0.7, whereas the same platforms in the North Sea would have > 1, maybe as high as 1.4 if the platform is slender and experiences significant dynamics. API RP2A separates the longterm distribution into components, hurricane waves and operational sea state. = 1.0 is used for both. In general, in the absence of data, one might choose a suitable value based on experience from the fatigue analysis of similar structures.
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Section
3.2
5
The Simplified Fatigue Assessment Method
Experience with Ships Ships operating mainly in the Northern or Southern oceans will generally have a Weibull shape parameter, γ > 1, maybe as high as 1.3 or a little more. However, there are cases where γ < 1 (see Section 5, Figure 7). Part 5C of the ABS Rules for Building and Classing Steel Vessels provides an empirical algorithm for the shape parameter (“long term stress distribution parameter”) that depends upon the length of the ship and also the location of the detail in the ship.
4
Fatigue Damage: General
4.1
Preliminary Remarks This Subsection provides detail on the fatigue damage expressions that are used in the Guide. Considered are both the single segment and the bi-linear S-N curves which are used to describe fatigue strength in design criteria documents.
4.2
General Expression for Fatigue Damage Assume that a statistical model (e.g., Weibull) has been fit to stress range data. Let S be a random variable denoting stress range. The probability density function of S is shown in Section 5, Figure 2.
FIGURE 2 Probability Density Function of s
f (s) S
f(s ) i
∆s
s
i
Stress, s
An expression for damage is derived in the following. Consider a small increment of s of width ∆s at stress level si, as shown in the figure. The number of cycles to failure at stress level si as obtained from the S-N curve defining fatigue strength is: Ni = N(si). ..................................................................................................................................... (5.6) The number of cycles of applied stress at level Si is: ni = NR [fs(si)∆s] ........................................................................................................................... (5.7)
36
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The Simplified Fatigue Assessment Method
where NR is any reference life. The term in brackets is the fraction of the total cycles associated with si. Substituting Equation 5.7 into Equation. 1.1 (the basic damage expression of the PM rule defined in Subsection 1/3), it follows that the damage at reference life NR is:
DR =
∑
N R f s ( s i )∆s ................................................................................................................ (5.8) N (si )
where the summation is taken over the whole sample space of s. In the limit as ∆s → 0, the summation becomes an integral. ∞
DR = N R
∫ 0
f s ( s )ds .................................................................................................................... (5.9) N (s)
Equation 5.9 is the damage at life NR. This expression will be used in the following to derive closed form expressions for damage.
5
Fatigue Damage for Single Segment S-N Curve
5.1
Expression for Damage at Life, NR Assume that the single segment S-N curve, as shown in Section 5, Figure 3, defines the fatigue strength.
FIGURE 3 Characteristic S-N curve
Log(S)
NS
m
=A
1 m
Log(N)
The analytical form of the S-N curve is: N(s) = As-m ................................................................................................................................ (5.10) Damage becomes, from Equation 5.9:
DR =
NR A
∞
∫s
m
f ( s )ds .............................................................................................................. (5.11)
0
But the integral is by definition the expected value of Sm. ∞
∫
E ( S m ) = s m f ( s )ds ................................................................................................................ (5.12) 0
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Section
5
The Simplified Fatigue Assessment Method
Thus the expression for damage at any arbitrary life NR can be written:
D=
5.2
N R E (S m ) ......................................................................................................................... (5.13) A
Miner’s Stress In the special case where s is constant amplitude: E(Sm) = Sm ................................................................................................................................. (5.14) Given E(Sm) from a random stress process, Equation 5.14 is used to define se as: Se = [E(Sm)]1/m ........................................................................................................................... (5.15) Se can be thought of as an equivalent constant amplitude stress which produces the same fatigue damage as the random stress process. Se is called Miner’s stress. Using Se, damage at any life NR for the single segment S-N curve can be written as: DR =
5.3
N R S em ............................................................................................................................ (5.16) A
The Damage Expression for Weibull Distribution of Stress Ranges The expression for damage is derived from Equations 5.11 and 5.12. It is necessary to determine the probability density function for the Weibull distribution. The probability density function is fs(s) = dFs/ds, where Fs is defined in Equation 5.1. γ s f s ( s ) = δ δ
r −1
s r exp − δ
s > 0 ....................................................................... (5.17)
Upon substituting Equation 5.17 into Equation 5.12, it follows that: m E ( s m ) = δ m Γ + 1 ............................................................................................................... (5.18) γ where Γ(.) is the gamma function defined in Equation 5.3. The general expression for damage at life NR is obtained by substituting Equation 5.18 into Equation 5.13. Damage at the design life NT is obtained by letting NR = NT. D=
NT m m δ Γ + 1 ............................................................................................................... (5.19) A γ
The fatigue safety check is:
D≤
6
1.0 ................................................................................................................................. (5.20) FDF
Fatigue Damage for Bilinear S-N Curve The bilinear S-N curve, as shown in Section 5, Figure 4, is specified in the Guide for in-air and cathodic protection. Fatigue damage for the case of Weibull-distributed stress and the bilinear S-N curve is addressed in this Subsection.
38
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Section
5
The Simplified Fatigue Assessment Method
FIGURE 4 Bilinear Characteristic S-N curve
Log(S)
NS
m
=A
1
r
NS = C
m SQ
1 r
Log (N)
NQ
Fatigue strength is given as: N(s) = As–m
for s > sQ
N(s) = Cs–r
for s < sQ .......................................................................................... (5.21)
The basic damage equation is given by Equation 5.9. For the bilinear case, Equation 5.9, damage at reference life NR, becomes:
DR =
NR C
∫
sQ
0
s r f s ( s )ds +
NR A
∫
∞
sQ
s m f s ( s )ds ........................................................................... (5.22)
Upon integration of Equation 5.22 and after some reduction, the expression for damage at the design life NT, for the two-segment case of Section 5, Figure 4 is derived [Wirsching and Chen (1988)]. D=
NT δ m A
m N δr r Γ + 1, z + T Γo + 1, z ........................................................................ (5.23) γ γ C
Γ(z,a) and Γo(z,a) are incomplete gamma functions (integrals z to ∞ and 0 to z, respectively). For reference, these functions are available in MatLab. Γ( z , a ) =
∫
Γo ( z , a) = a= b=
m
γ r
γ
∞
t a −1 e −t dt ........................................................................................................... (5.24)
z
∫
z O
t a −1 e −t dt .......................................................................................................... (5.25)
+ 1 ................................................................................................................................. (5.26) + 1 .................................................................................................................................. (5.27)
sQ z = δ
γ
................................................................................................................................ (5.28)
The safety check expression of Equation 5.20 is applied.
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39
Section
7
5
The Simplified Fatigue Assessment Method
Safety Check Using Allowable Stress Range A safety check expression can also be developed in terms of the once-in-a-lifetime stress range, SR. First, consider the single segment S-N curve. Let D = (1/FDF). From Equation 5.5, δ can be expressed in terms of SR. Substituting D and δ into Equation 5.19 and solving for SR, it follows that: m/γ A(ln N R ) S R′ = m (FDF )N T Γ + 1 γ
1/ m
................................................................................................. (5.29)
This is actually the maximum allowable stress range that would be permitted once on the average every NR cycles, and as such would be interpreted as fatigue strength. The prime is included to distinguish it from stress. The fatigue stress is SR and is the computed fatigue stress that will be exceeded once on the average every NR cycles. The safety check expression is: SR ≤ S′R ...................................................................................................................................... (5.30) In practice, it is most likely that NR will be taken as the design life NT so that stress and strength (SR and S′R) will refer to NT. If S′R has been computed, the strength (maximum allowable stress range) S′S at any other life NS can be computed using Equation 5.5. ln N S S S′ = S R′ ln N R
1/ γ
.................................................................................................................. (5.31)
Now consider the bilinear S-N curve (Section 5, Figure 4 and Equation 5.23). Let D = (1/FDF). From Equation 5.5, δ can be expressed in terms of SR. Substituting D and δ into Equation 5.23 and solving for SR, it follows that:
m/γ ( ln N R ) S R′ = (FDF )N Γ m + 1, z / A + δ r − m Γ r + 1, z / C T o γ γ
1/ m
................................................ (5.32)
Again, the prime superscript is used to denote strength. The strength modification of Equation 5.31 will apply. The safety check expression of Equation 5.30 would apply.
8
The Simplified Method for Which Stress is a Function of Wave Height
8.1
The Weibull Model for Stress Range; Stress as a Function of Wave Height Another refinement of the simplified method is a model in which: (a) the long term distribution of wave heights H is known (or assumed), (b) H is assumed to have a Weibull distribution, and (c) the fatigue stress range S is related to H by the power law. Such a model is presented in API RP2A [API(2000, 1993)] and described in the following. This model is not presented in the Guide. It appears here only for reference. API RP2A [API(2000, 1993)] defines a damage model based on Weibull wave heights. This model was based on an earlier work by Nolte and Hansford (1976). Consider the random instantaneous ocean wave elevation process as a sequence of waves. Let H be a random variable denoting the height (i.e., the vertical distance from the trough to the following peak) of a single wave.
40
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Section
5
The Simplified Fatigue Assessment Method
It is assumed that stress range S is proportional to the wave height H that produces that stress. S = ChHg ................................................................................................................................... (5.33) where Ch and g are empirical constants. If H is Weibull, then S will also be Weibull (from basic probability theory). To compute Miner’s stress: Sem = E(Sm) = E(Chm Hgm) = ChmE(Hgm) .................................................................................. (5.34) From Equations 5.5 and 5.18, it follows that:
(
)
E H gm = H Rmg (ln N R )
− gm / γ
gm Γ + 1 ................................................................................. (5.35) γ
where γ is the Weibull shape parameter of H. HR is a wave height that is exceeded on the average once every NR cycles (analogous to SR of Equation 5.5). Damage can be computed by substituting Equations 5.34 and 5.35 into Equation 5.16. D=
8.2
NR m C h E ( H gm ) ................................................................................................................ (5.36) A
The Weibull Model for Stress Range; Stress as a Function of Wave Height; Considering Two Wave Climates Assume that there are two distinct wave climates in a region, one characterized as the usual wind-generated seas and the other the exceptional cyclonic storm seas of a hurricane (e.g., Gulf of Mexico). Both are assumed to have Weibull distributions. The parameters are: Ho1
=
design wave height for the period of wave climate 1
Ho2
=
design wave height for the period of wave climate 2
N1
=
number of wave cycles for the period of wave climate 1
N2
=
number of wave cycles for the period of wave climate 2
γ1
=
Weibull shape parameter for wave heights for wave climate 1
γ2
=
Weibull shape parameter for wave heights for wave climate 2
Λ1
=
as needed, a “calibration factor” for wave climate 1
Λ2
=
as needed, a “calibration factor” for wave climate 2
Damage at life n is equal to the sum of the damage in each wave climate. D=
NR (Yo + Y1 ) ..................................................................................................................... (5.37) A
where NR = N1 + N2 ............................................................................................................................. (5.38)
Y1 =
N 2 mg − gm / γ 1 gm Γ + 1 ..................................................................................... (5.39) H o1 (ln N 1 ) NR γ1
Y2 =
N 2 mg − gm / γ 2 gm Γ + 1 ................................................................................... (5.40) H o 2 (ln N 2 ) NR γ2
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41
Section
5
The Simplified Fatigue Assessment Method
9
The Weibull Distribution; Statistical Considerations
9.1
Preliminary Remarks The Weibull distribution is commonly used to describe long term wave height and long term stress range processes. Statistical considerations of the Weibull distribution are presented below for reference. Specifically, parameter estimation and probability plotting are described. Also, the operations of fitting the Weibull distribution to deterministic and spectral models are presented.
9.2
Estimating the Parameters from Long-Term Data; Method of Moment Estimators Given a long-term measured history, the stress ranges Si, i = 1, n are recorded. It is assumed that the observations in this random sample are independent and identically distributed (i.e., all dependencies are ignored). It must be emphasized that this history be representative of the entire service life. The parameters can be estimated using the method of moments. Compute the sample mean and standard deviation. S=
1 n
n
∑S i =1
1 sS = n − 1
i
.............................................................................................................................. (5.41) n
∑ (S i =1
i
2
)
−S
1/ 2
..................................................................................................... (5.42)
S and sS are estimates for µs and σS. From Equation 5.2, the method of moment estimators for γ and δ are: sS S
γˆ = δˆ =
−1.08
S 1 Γ + 1 ˆ γ
............................................................................................................................ (5.43) ............................................................................................................................ (5.44)
While the more complicated maximum likelihood estimators are generally considered by mathematicians to be of higher quality, extensive and unpublished Monte Carlo simulation by Wirsching has demonstrated that the performance of the two estimator types to be essentially identical. Example. An illustration of a Weibull statistical analysis for which the sample size, n = 25. The random sample:
S = (12.5377 9.8050 8.2736 12.3759 11.0321 14.6142 21.6679 9.4326 17.2867 4.3187 24.5273 5.7944 5.5171 17.1173 13.2568 3.7245 0.8759 17.8508 4.7251 6.1859 11.7465 5.9809 8.6302 2.3726 26.4654) Using Equations 5.41 and 5.42, the sample mean and standard deviation are, respectively, S = 11.04, sS = 6.83. Substituting these values into Equations 5.43 and 5.44, the method of moment estimates for γ and δ are:
γˆ = 1.68 9.3
δˆ = 12.36
Estimating the Parameters from Long-Term Data; Probability Plotting A probability plot involves a transformation that allows the distribution function to be plotted as a straight line. Consider the distribution function of Equation 5.1. Taking the log of both sides twice results in: ln[–ln(1 – FS)] = γ ln s – γ ln δ ................................................................................................. (5.45) Let Y = ln[–ln(1 – FS)] X = ln s ...................................................................................................................................... (5.46)
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Section
5
The Simplified Fatigue Assessment Method
The result is a linear relationship in which γ is the slope of the plotted line and –γ ln δ is the y-intercept. Given the random sample, the distribution function is estimated in the following manner: i)
The sample is ordered, smallest to largest. S(i) denotes the i-th smallest value.
ii)
The estimate of the distribution function corresponding to S(i) is:
Fi =
i − 0.5 .............................................................................................................................. (5.47) n
Fi and Si are translated into Yi and Xi using Equation 5.46. The points are plotted on rectangular paper. If the data follow a linear trend, it is suggested that the Weibull distribution is a reasonable model. This decision is purely subjective. A straight line can be fitted through the points. The least squares estimators of γ and δ are then: γ~ = slope of Y-X line .............................................................................................................. (5.48) ~
δ = exp −
y - intercept ......................................................................................................... (5.49) γ~
Example: Consider the random sample as given above. The following table provides the details of the preparation of the data for plotting. Xi and Yi are computed using Equation 5.46.
TABLE 1 Data Analysis for Weibull Plot i
S(i)
Fi = (i – 0.5)/n
Yi
Xi
1
0.88
0.02
−3.90
−0.13
2
2.37
0.06
−2.78
0.86
3
3.72
0.10
−2.25
1.31
4
4.32
0.14
−1.89
1.46
5
4.73
0.18
−1.62
1.55
6
5.52
0.22
−1.39
1.71
7
5.79
0.26
−1.20
1.76
8
5.98
0.30
−1.03
1.79
9
6.19
0.34
−0.88
1.82
10
8.27
0.38
−0.74
2.11
11
8.63
0.42
−0.61
2.16
12
9.43
0.46
−0.48
2.24
13
9.81
0.50
−0.37
2.28
14
11.03
0.54
−0.25
2.40
15
11.75
0.58
−0.14
2.46
16
12.38
0.62
−0.03
2.52
17
12.54
0.66
0.08
2.53
18
13.26
0.70
0.19
2.58
19
14.61
0.74
0.30
2.68
20
17.12
0.78
0.41
2.84
21
17.29
0.82
0.54
2.85
22
17.85
0.86
0.68
2.88
23
21.67
0.90
0.83
3.08
24
24.53
0.94
1.03
3.20
25
26.47
0.98
1.36
3.28
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43
Section
5
The Simplified Fatigue Assessment Method
Y versus X is shown in Section 5, Figure 5.
FIGURE 5 Weibull Probability Plot 2
Y = ln(-ln(1-F )) S
1 0 -1 -2
y = -4.01 + 1.59x
-3
X = ln(S) -4 -0.5
0
0.5
1
1.5
2
2.5
3
3.5
As shown in the figure, the slope is 1.59 and the y-intercept is −4.01. Thus, the least squares estimators from Equations 5.48 and 5.49 are: γ~ = 1.59
~
4.01 = 12.45 1.59
δ = exp
Some observations:
44
i)
The data plot is nearly a straight line on Weibull paper. This suggests that the Weibull is a reasonable model for the random variable, S. This does not mean that nature actually chose the Weibull.
ii)
The method of moment estimators and least squares estimators are in relatively good agreement. The difference is due to the randomness in the process.
iii)
For large sample sizes, the maximum likelihood estimators are minimum variance unbiased estimators. However, method of moment estimators have been found to be insignificantly different from maximum likelihood.
iv)
When the data do not plot as a straight line, the Weibull model might be a poor choice and the analyst must exercise engineering judgment regarding the possible choice of another distribution. Another indication that the Weibull model might be inappropriate would be poor agreement between method of moment and least squares estimators.
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Section
9.4
5
The Simplified Fatigue Assessment Method
Another Representation of the Weibull Distribution Function For fatigue purposes, the long-term Weibull distribution of fatigue stresses is often described graphically as shown in Section 5, Figure 6.
FIGURE 6 Long Term Distribution of Fatigue Stress as a Function of the Weibull Shape Parameter 0.8 1.4 0.6
S/So = 1.0
0.4 = 0.7 0.2 Number of Exceedances 0 1
10
100
1000
10
4
10
5
10
6
10
7
10
8
Section 5, Figure 6 is another graphical representation of the distribution function. Let NT be the total number of stress cycles in the design life. Let n be the number of values of the stress range during the service life that exceed the value, s (i.e., the number of exceedances of S by stress level, s). Then, for any value of S selected at random, the probability of exceeding the value s is simply equal to n/NT. P(S > s) =
number of values of S which exceed s n = = 1 – P(S s) = 1 – FS(s) ....... (5.50) total number of stress ranges in design life NT
By combining Equations 5.1, 5.4 and 5.50, the following equation, a restatement of the distribution function, is obtained. S exp ST
n ln N T ..................................................................................................... (5.51) NT
where n is the number of exceedances of the fatigue stress of stress level S. Section 5, Figure 6 is a plot of (S/ST) versus n for the case where NT = 108 and for three values of . An example of the use of this plot for marine structures is shown in Section 5, Figure 7.
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45
Section
5
The Simplified Fatigue Assessment Method
FIGURE 7 Long-Term Stress Range Distribution of Large Tankers, Bulk Carriers, and Dry Cargo Vessels Compared with the Weibull
Bending Stress - Range (S/So)
W.H. Munse "Fatigue Characterization of Fabricated Ship Details for Design"..... SSC-318 (1983)
12
102
102
103 10422 105 1066 Number of Exceedances, N Cycles
107
108 = NT
A crude method for estimating the Weibull shape parameter, , is to plot observed stress data on the figure as shown and then choose that value of which provides the curve matching the experimental data. The method of moments estimator for is recommended.
9.5
Fitting the Weibull to Deterministic Spectra A deterministic spectra defined over the service life (Section 5, Table 2) provides the stress information for computing fatigue damage and executing a safety check. However, a Weibull distribution can be fitted to this deterministic spectra. One reason for doing this is to develop information on the Weibull shape parameter working towards characteristic values for various structure types.
TABLE 2 Deterministic Spectra Stress Range
Number of Cycles
s1
n1
si
ni
sk
Note that: (1) N S S=
n
i
nk
, and (2) there are ni values of si. Thus, the random sample is generated as:
(n1 values of s1… plus n2 values of s2… plus ni values of si… plus nk values of sk.) ......(5.52)
The method of moment estimators, as described above in 5/9.2, are used to estimate and . 46
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Section
9.6
5
The Simplified Fatigue Assessment Method
Fitting the Weibull Distribution to the Spectral Method While the spectral method provides definition of fatigue stresses for analysis, the Weibull parameters can be extracted from the spectra. As in the case of the deterministic method, a random sample is constructed. In the spectral method, the long-term stress process is discretized into J short term stationary sea states. Consider the i-th sea state. The stress ranges in this sea state will have a density function (e.g., as shown in Section 5, Figure 8).
FIGURE 8 Probability Density Function of Stress Ranges of the i-th Sea State
f (s) i
f i (S m )
s
S
Stress, s
If the stress process is assumed to be Gaussian and narrow band, then S will have a Rayleigh distribution. However, this is not a necessary assumption to execute this process. To construct the random sample of stress ranges, divide the sample space into M “small” increments. One such increment of width, s, is shown in the figure. The continuous distribution of S is transformed to a discrete distribution of M constant amplitude stresses. One of those stresses, sm, is shown at the center of the interval. The following definitions lead to the number of cycles of Sm. TS
=
service life
i
=
% of time in the i-th sea state
Ti
=
i
TS
=
time in the i-th sea state
i
=
center frequency of the stress process in the i-th sea state
Ni
=
i
Ti
=
number of cycles in the i-th sea state
nm
=
[fi(sm) s]
Ni
=
number of cycles of sm
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47
Section
5
The Simplified Fatigue Assessment Method
The contribution to the random sample is nm cycles of sm. Then,
48
Repeat the above for all of the intervals of the i-th sea state
Repeat for all sea states
Pool all of the values of S and use the method of moments
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Section 6: The Spectral Based Fatigue Assessment Method
SECTION
1
6
The Spectral Based Fatigue Assessment Method
Preliminary Comments The spectral fatigue analysis is a systematic approach to develop a rational, first-principle approach whereby the dynamic loads and the wave environments are explicitly considered. The essence of the spectral fatigue analysis method is to model the long-term random sea state process as several short-term stationary Gaussian processes, each defined by the wave spectral density function. The fraction of time that each short-term process acts is specified. A frequency response function relating the wave spectral density and the spectral density of fatigue critical components, and including the structural dynamics, is developed. The fatigue stress spectral density is computed. If the process can be assumed to be narrow band (a conservative assumption), the stress ranges will have a Rayleigh distribution and a closed form expression for fatigue damage contributed by each of the short-term processes can be used. A summary of the process of fatigue assessment by the spectral fatigue analysis method is provided in Section 6, Figure 1.
2
Basic Assumptions Basic assumptions in spectral fatigue analysis are: i)
The global performance analysis and the associated structural analysis are assumed to be linear. Hence, scaling and superposition of stress range transfer functions due to unit wave height are valid.
ii)
Non-linearities due to non-linear motions and wave loadings are treated by the equivalent linearization.
iii)
Structural dynamic amplification, transient loads and effects such as springing and ringing are insignificant and, hence, use of quasi-static finite element analysis is valid.
iv)
Short-term stress processes are assumed to be stationary Gaussian.
v)
Short-term stress processes are also assumed to be narrow band so that the stress ranges have a Rayleigh distribution. (If a decision is made that the stress process must be treated as wide band, a correction to the narrow band assumption is provided.)
vi)
The linear damage accumulation rule (Palmgren-Miner) applies.
A fundamental limitation of the spectral method is the assumption of linearity (i.e., items i) and ii) above). Thus, the spectral method may not be appropriate in those cases where non-linearities are considered to be important (e.g., TLP tendons).
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49
Section
6
The Spectral-based Fatigue Assessment Method
FIGURE 1 Fatigue Assessments by Spectral Analysis Method DEFINE OPERATIONAL PROFILE
DEFINE THE LONGTERM DESIGN WAVE ENVIRONMENT
DEFINE THE FATIGUE CRITICAL STRUCTURAL DETAILS
PERFORM FREQUENCY DOMAIN ANALYSES OF SHORT-TERM FATIGUE STRESS PROCESSES
direction seakeeping analysis for ships global finite element analysis local finite element analysis frequency response functions for fatigue stresses spectral density functions for fatigue stresses
DERIVE SHORT-TERM DISTRIBUTIONS OF STRESS RANGE
CALCULATE AND SUM SHORT-TERM FATIGUE DAMAGES
SAFETY CHECK EXPRRESSION FOR FATIGUE
3
The Rayleigh Distribution for Short Term Stress Ranges The long term (non-stationary) sea state is modeled by several short-term stationary sea states. It is assumed that the short term processes are Gaussian and narrow band. Thus, the distribution of the peaks is Rayleigh [Wirsching, Paez, and Ortiz (1995)]. The Rayleigh distribution is a special form of the Weibull distribution having parameters:
γ = 2.0
δ = 2 2σ ....................................................................................................... (6.1)
Thus, it follows from Equation 5.19 that damage at the any reference life NR is: DR =
(
NR 2 2σ A
)
m
m Γ + 1 ..................................................................................................... (6.2) 2
This expression forms the basis of the damage expression for the spectral method.
50
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Section
4
6
The Spectral-based Fatigue Assessment Method
Spectral Analysis; More Detail Under the previously stated assumptions, a more detailed description of the spectral analysis method is provided in the following:
5
i)
Characterization of Sea Environment. The sea environment is represented by the number of occurrences of various sea states, each defined by a set of spectra. A two-parameter (significant wave height HS, zero up-crossing period, TZ) wave-scatter diagram shall be used to characterize the sea states (See Section 6, Table 1 for example). All sea state spectra are defined by the wave spectrum relationship such as the Pierson-Moskowitz relationship. Wave direction probability is included in the sea environmental characterization.
ii)
Global Performance Analysis. Waves of appropriate frequencies, heights and directions are selected. The response and the loading of the structure are computed for each wave condition.
iii)
Structural Analysis. A global structural analysis is performed to determine the applied loading or displacement for the critical structural details (load transfer function per unit wave amplitude as a function of frequency). The local structural analysis is performed to determine the stress transfer function (per unit wave amplitude as a function of wave frequency) at each location of interest in the structural detail.
iv)
Stress Concentration Factor (SCF). The geometric SCF is accounted for in the fatigue assessment. The SCF can be determined either by parametric equation or by fine mesh FEA.
v)
Hotspot Stress Transfer Function. As needed the stress transfer function is multiplied by the SCF to determine the hotspot stress transfer function.
vi)
Long-term Stress Range. Based on the wave spectrum, wave scatter diagram and hotspot stress response per unit wave amplitude, the long-term stress range is determined. This is done by multiplying the ordinate of the wave amplitude spectrum for each sea state by the ordinate squared of the hotspot stress transfer function to determine the stress spectrum. The stress range distribution is assumed to follow a Rayleigh distribution. The long-term stress range is then defined through a short-term Rayleigh distribution within each sea state for different wave directions.
vii)
S-N Classification. For each critical location considered in the analysis, S-N curves are assigned based on the structural geometry, applied loading and, when applicable, welding procedure and quality.
viii)
Fatigue Life. Estimation of fatigue damage (or life) using the Palmgren-Miner Rule and subsequent comparison with design requirements.
Wave Data The basis of the spectral fatigue approach lies in the wave environment data that are contained within the “wave scatter” diagram that is to be used. Clearly, physically measured wave data is the most reliable form. In lieu of the physical measurement data, visually observed wave data and some well compiled “authoritative” sources (e.g., ABSWAVE, BMT Global Wave Statistics, Walden Data, etc.) can be used. The wave data should be available in “scatter diagram” form which consists of M number, of “cells” that contain the probability of occurrence of specific “sea states”. Any cell will effectively contain three data items, namely i)
The significant wave height, Hs, (typically in meters),
ii)
The characteristic wave period, TZ, (in seconds) and
iii)
The fraction of the total time that the wave condition exists, pi.
Wave data are characterized as a wave scatter diagram, an example of which is shown in Section 6, Table 1.
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51
Section
6
The Spectral-based Fatigue Assessment Method
TABLE 1 A Sample Wave Scatter Diagram Wave Period (sec)
Wave Height (m)
5s
7s
9s
11s
13s
15s
Sum Over All Periods
17s
0.0-0.75
20.91
11.79
4.57
2.24
0.47
0.06
0.00
0.60
40.64
0.75-1.75
72.78
131.08
63.08
17.26
2.39
0.33
0.11
0.77
287.80
1.75-2.75
21.24
126.41
118.31
30.24
3.68
0.47
0.09
0.56
301.00
2.75-3.75
3.28
49.60
92.69
32.99
5.46
0.68
0.12
0.27
185.09
3.75-4.75
0.53
16.19
44.36
22.28
4.79
1.14
0.08
0.29
89.66
4.75-5.75
0.12
4.34
17.30
12.89
3.13
0.56
0.13
0.04
38.51
5.75-6.75
0.07
2.90
9.90
8.86
3.03
0.59
0.08
0.03
25.46
6.75-7.75
0.03
1.39
4.47
5.22
1.93
0.38
0.04
0.04
13.50
1.09
2.55
3.92
1.98
0.50
0.03
0.02
10.09
0.54
1.36
2.26
1.54
0.68
0.20
0.04
6.62
0.01
0.10
0.11
0.10
0.05
0.02
0.00
10.75-11.75
0.00
0.03
0.08
0.17
0.06
0.00
11.75-12.75
0.05
0.00
0.14
0.22
0.06
0.01
12.75-13.75
0.02
0.07
0.09
0.03
0.02
0.06
0.02
0.00
0.01
0.11
0.01
0.01
0.02
0.01
0.01
0.08
138.59
29.05
5.63
0.92
2.69
1000.00
7.75-8.75 8.75-9.75 9.75-10.75
0.01
13.75-14.75 14.75-15.75 Sum Over All Heights
0.02 118.97
345.43
358.72
0.40 0.34 0.48
0.01
0.22
The long-term sea state is discretized as a set of M short-term sea states as shown. Each sea state, which is assumed to be a stationary Gaussian process, contributes to the total fatigue damage. The numbers in each cell (divided by 1000) define the joint probability of occurrence of Hs and Tz, denoted as pi. The fraction of the total time each cell is acting is defined by pi, i = 1, M. For a more refined analysis, wave directionality can be accounted for. Each of the M cells can be subdivided into discretized wave direction. The probability distribution of wave direction needs to be defined. Fatigue damage is computed for each of the M cells and summed to compute total damage.
6
Additional Detail on Fatigue Stress Analysis; Global Performance Analysis In order to translate wave data into fatigue stresses, it is necessary to construct a frequency response function, and to do this, it is necessary to perform a motion response analysis. This analysis should be based on sixdegree-of-freedom global motion models with full simulation of the mass properties of the operating platform or ship. For a TLP, the global performance model may be “uncoupled” from the tendons and risers, meaning that the responses of these components are not included in the model, although the static stiffness must be included. The analysis should be conducted in the frequency domain based on panel model diffraction loads or strip theory to determine the hydrodynamic loads, stiffness matrix at the quasi-static offset position in the environment, and full spectral definition of the sea states. A suitable range of wave frequency, number of frequency points and wave headings shall be used in the analysis. Ideally, for ship type structure, the following are suggested:
52
i)
Frequency range:
0.20 to 1.80 rad/sec
ii)
Frequency Increment:
0.05 rad/sec
iii)
Wave headings:
0 to 360 degrees in 15 degree increments
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Section
6
The Spectral-based Fatigue Assessment Method
The global motion analyses will serve as the basis for load development in FEA, discussed later. The actual interface from the global analysis to the structural analysis consists of six loading components for each analyzed wave period and direction: the real and imaginary applied unit amplitude wave diffraction and radiation loads, the associated inertial loads and the other cyclic loading (e.g., TLP tendon dynamic reactions). The successful interface of these load components, as discussed above, is dependent on a consistent geometry and mass model between the motion and structural analyses and consistent generation of the loading components in the motion analysis. Consistent modeling can be obtained by interfacing the model geometry directly from the motion analysis wherever possible. Consistent mass can be obtained by interfacing with the same weight control database for both the motions and structural analyses, when available.
7
The Safety Check Process
7.1
General Considerations As a general reference to random processes for the following discussions, see Wirsching, Paez, and Ortiz (1995).
7.2
The Stress Process in Each Cell Fatigue stresses are to be established by the following process: i)
Determine the stress transfer function, Hσ(ω|θ), for a particular location of interest in the structural detail for a particular load condition. This can be done in a direct manner, whereby, the structural analysis is carried out at every frequency and heading for which the spectral analysis is intended to be carried out and the resulting stresses are used to generate the stress transfer function explicitly.
ii)
For each of the M cells or short-term sea states, generate a stress spectra, Sσ(ω|Hs, Tz, θ), by linearly scaling the wave spectral density, Sη(ω|Hs, Tz), in the following manner: Sσ(ω|Hs, Tz, θ) = | Hσ(ω|θ)|2 · Sη(ω|Hs, Tz) .................................................................................. (6.3)
iii)
For each of the M cells or short-term sea states, calculate the spectral moments. The n-th spectral moment, μn, is calculated as follows: ∞
μn = ω n S σ (ω H S , TZ , θ)dω ....................................................................................................... (6.4)
∫ 0
Most fatigue damage is associated with low or moderate seas. Therefore, confused short-crested sea conditions should be considered. Confused short-crested seas result in a kinetic energy spread which is modelled using the cosine-squared approach, (2/π)cos2θ. Generally, cosine-squared spreading is assumed from +90 to –90 degrees on either side of the selected wave heading. Applying the wave spreading function, the spectral moment may be modified as follows: μn =
∞ θ' = θ + 90
2 2 ' n cos θ ω S σ (ω H S , TZ , θ)dω ........................................................................... (6.5) π θ = θ −90
∫ ∑ 0
Often, the correction for short crested seas is ignored, leading to conservative damage estimates. iv)
Using the spectral moments, compute: (1) the RMS σi, (equal to the standard deviation for a zero mean process), (2) rate of zero crossings (equivalent frequency) foi, and, if the decision is made to include a wide band correction, (3) rate of peaks fpi, and (4) the spectral width parameter for each of the M cells. Let µni be the n-th spectral moment of the i-th cell. General reference for the following discussion is Wirsching, Paez, and Ortiz (1995).
RMS
σ i = µ 2i (6.6) ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
53
Section
6
The Spectral-based Fatigue Assessment Method
Rate of Zero Crossings (equivalent frequency) The rate of zero crossings (in Hertz) for the i-th cell is:
f oi
1 2
2i ...........................................................................................................................(6.7) oi
The expression for foi is based on a stress spectral density that is a function of (rad/sec) per Equation 6.4. If the spectral density function is in terms of f (Hertz), then the factor (1/2) should be missing. See Subsection 6/8. If the decision is made to also apply the wide band correction factor, the following must be computed. Ignoring this factor produces conservative results. Rate of Peaks The rate of peaks (number per second) for the i-th cell is:
f pi
4i ...................................................................................................................................(6.8) 2i
The expression for fpi is based on a stress spectral density that is a function of (rad/sec) per Equation 6.4. If the spectral density function is in terms of f (Hertz), then the factor (1/2) should be missing. See Subsection 6/8.
Spectral Width Parameter The spectral width parameter is an index of the spectrum (i.e., the stress spectral density function). For the i-th sea state:
i 1 i2
0 < i < 1.0 .......................................................................................................(6.9)
where i is the irregularity factor defined as:
i
f oi f pi
0 < i < 1.0 ...........................................................................................................(6.10)
A process can be assumed to be narrow band if i will be “close to” 1.0 (say 0.95) and i close to zero. i becomes smaller as the process becomes more wide band. The spectral shape will differ with each sea state and, thus, i must be defined for each sea state.
8
Fatigue Damage Expression for Wide Band Stress
8.1
Preliminary Comments As a general reference, consider Wirsching, Paez, and Ortiz (1995). The expression for damage used by ABS is based on the assumption of a narrow band stress process in which the stress ranges have a Rayleigh distribution. However, the stress process will, in general, be wide band. A rainflow correction factor, an index of the wide band spectrum, is applied to the narrow band equation for damage to form a damage expression for wide band stress. For the i-th sea state, the rainflow correction factor is denoted as i and the frequency of loading is taken to be foi. Realizations of a narrow band and a wide band process are shown in Section 6, Figure 2. Each has the same RMS and rate of zero crossings.
54
ABS COMMENTARY ON THE GUIDE FOR THE FATIGUE ASSESSMENT OF OFFSHORE STRUCTURES . 2004
Section
6
The Spectral-based Fatigue Assessment Method
FIGURE 2 Realizations of a Narrow Band and Wide Band Process (Both Have the Same RMS and Rate of Zero Crossings) Stress Narrow band process
S
One cycle
Wide band process
time
For the narrow band process, stress cycles are easily identified and the application of Miner’s rule is straightforward. While it is not immediately obvious how to identify individual stress cycles in a wide band process, the rainflow algorithm provides a formal method for stress cycle counting. 6/8.4 describes the rainflow method and presents a closed form expression for computing damage based on a correction factor applied to the equivalent narrow band process.
8.2
Definitions The spectral density function of the stress process, S(t), is given as WS(f) where frequency, f, is in Hertz. An index of the frequency spectrum of the stress process, S(t), is the irregularity factor:
α=
fo fp
0 < α
