Applied Ocean
Research
E L S E V I E R Applied Ocean Research 27 (2005) 173-185 ^ = = ^ ^ = ^ ^ = www.elsevier.com/locate/apor
Fatigue damage assessments and the influence of wave directionality
J . H . V u g t s *
Faculty of Civil Engineering wid Geosciences, Delft University of Technology, Van Hogenhoucklaan 56A, 2596 TE The Hague, Tlie Netherlands
Accepted 10 November 2005 Available online 7 February 2006
Abstract
A fatigue assessment is a complex and time-consuming process with an uncertain outcome. The result is sensitive to comparatively small differences in the structure (globally or locally) or the loading environment. The predominant cause of fatigue damage for marine structures is usually wave action. It is then often asked what the beneficial effect of wave directionality is. There are two sources of wave directionahty, i.e. firstly, the variable direction of uni-dhectional seas, and secondly, wave spreading around a given mean direction. These two effects need to be distinguished and considered separately. The influence of wave directionality is not easily predictable and its quantification normally requires extensive and detailed analysis by numerical calculations. However, under two general restrictions the influence can be rather easily determined, i.e. for a rotationally symmetric strucmre (a vertical cir-cular cylinder) and for wave conditions that are of the same severity in afl directions. The paper investigates this special problem and presents quantitative results for vaiious cases.
© 2005 Elsevier Ltd. All rights reserved.
Keywords: Offshore engineering; Wave duectionality; Fatigue damage assessment; Reduction factor
1. Introduction
Structures in a marine environment are all subject to potential fatigue problems due to wave action. Examples of structures in a marine environment are ships, offshore structures (both floating and fixed), and more recently also support structures of offshore wind turbines. The predominant cause of stress variations for such structures is normally wave action. This often also applies to structures supporting offshore wind turbines, although the variable rotor thnast caused by wind action clearly also is a significant source of variable stresses in the support structure.
Fatigue damage assessment is a complex and uncertain process. Its prediction at the design stage is difficult and time consuming, while the outcome is usually far from accurate. Marine structures are normally welded thi-oughout. As welded structures always contain imperfections (flaws) at the welds which can act as fatigue initiators, fatigue in welded structures is only associated with crack propagation. A fatigue assessment of a marine structure is neaiiy always performed as follows. The stress variations that are expected to occur during its
* Tel.: - f 3 1 70 324 8385; fax: + 3 1 70 324 8355.
E-mail address: j.h.vugts@offshore.tudelft.nl.
0141-1187/$ - see front matter © 2005 Elsevier Ltd. A l l rights reserved. doi:10.1016/j.apor.2005.11.003
lifetime are determined and counted in accordance with some counting method; this information represents the input, the 'loading side' of the fatigue assessment. The capacity of the material t o endure repeated stress cycling is given in the f o m of empirical SN-cmve& (i.e. stress range S versus allowable number of cycles N). The SN-curve represents the 'resistance side' of the fatigue assessment. The damage done by each stress range magnitude is assumed to be the number of occuiTences n divided by the allowable number of cycles N from the SN-cuxve. The elementary damages n/N for each stress range magnitude are next summed to obtain an estimate of the total damage during the structure's service life; this is the output of the fatigue assessment. The procedure is described in more detail in Section 2.
Fatigue always relates to very local damage in a structure. Two points a small distance from each other on the same structural component can experience markedly different stress variations and can exhibit coirespondingly large differences in fatigue damage. These differences are the result of differences in overall structural configuration or i n local detail of construction; these can both cause significant differences in
'fatigue loading' for the environment in which the stnacture is placed. Additionally, the resistance of the local material can also differ appreciably with local detail of construction. Consequentiy, fatigue assessments are very sensitive to comparatively minor differences i n the stracture (globally or locally) as well as in the loading environment. This sensitivity
174 J.H. Vugts /Applied Ocean Research 27(2005) 173-185
Symbols
CAT multiplication factor on Cs multiplication factor on
c„ multiplication factor on all numbers of occuiTences Csk multiplication factor on all stress ranges 5^ Cj = constant in the mathematical formulation of the
5A^-curve branch for which S>Sr
C j = NX'' constant in the mathematical formulation of the j'A^-curve branch for which 5'<iS'r
D calculated fatigue damage done within duration T Dd design value of calculated fatigue damage D D(d) wave directional spreading function
Z ) A ( M ) fatigue damage at point A on the cylinder for uni-directional waves in direction
DA total fatigue damage at point A on the cylinder: (a) for uni-directional waves of equal severity in all
directions and probabiUties of o c c u i T e n c e p{fj,i) per
mean wave direction;
(b) for spread waves of equal severity in all directions, with probabilities of occurrence p(dj\iXi) for each spreading sector and p(i.ii) for each mean direction sector
DYi{4i) fatigue damage at point B on the cyUnder at an
angle (j) from point A on the x-axis, for uni-directional waves of equal severity in all directions and probabilities of occurrence p{(pj)
f(l.i) probability density function of mean wave directions
( , c o u n t e r s for mean wave directions, ! = 1, 2, ..., / a n d j = l , 2 ,
k counter for stress range classes, fe = 1, 2, K kj counter k for which Sii=Sr
Lf calculated fatigue life (usually expressed in years) slope of that part of the SN-cnvve for which >S'>5'r ( m , > 0 )
1)12 slope of that part of the SN-cmve. for which S <
5",-{»l2>0)
n number of stress range cycles S occurring within duration T
Uk number of stress range cycles in discrete class occurring within duration T
" A , < : > '^B,k number of stress range cycles in discrete class S^^k
occurring at point A (SB,k occurring at point B , respectively) within duration T for uni-directional waves
N the allowable number of stress range cycles S from the 5'A^-curve
Nk the allowable number of stress range cycles in discrete class Sf^ from the i'A'-curve
Nr absciss of a reference point on the 5A^-curve, or of the intersection point of the two branches of the SN-curve with different slopes
p(Ati). p((Pj) probability of occurrence that the mean wave
direction jx is within the fth sector (or mean direction rp is within the ;th sector, respectively) p(di) probability of o c c u i T e n c e that the spread waves are
within the fth spreading sector for mean direction
M = 0
pi6j\iii) conditional probability of occurrence that the spread waves ai'e within the j t h spreading sector, given that the mean wave direction is within the fth sector
R D A , I ( M ) reduction factor for fatigue damage at point A on the cylinder for uni-directional waves in duection IX =^0, compared with maximum fatigue damage at point A for uni-directional waves in direction p = 0
R D A , 2 reduction factor for fatigue damage at point A on the cylinder for uni-duectional waves of equal severity i n all directions and probabilities of occurrence p(^,) per direction, compared with fatigue damage at point A for all waves occuning in direction p = 0
R D A , 3 ( M ; ) reduction factor for fatigue damage at point A on the cylinder for spread waves around mean direction Hi, compared with fatigue damage at point A for uni-directional waves in direction jXi R D A . 4 total reduction factor for fatigue damage at point A
on the cylinder for spread waves around mean direction /.t,-, with probabilities of occurrence pifij) per mean direction, compared with fatigue damage at point A for uni-directional waves along the x-axis
(11 = 0)
R D B , A ( 0 ) reduction factor for fatigue damage at point B on the cylinder for uni-directional waves of equal severity in all directions and probabilities of occurrence piiXj) per direction, compared with fatigue damage at point A
sit) random stress history S stress range magnitude
Sk /rth discrete class of stress range magnitudes, arranged in increasing order
SA.k, ^B.k kih discrete class of stress range magnitudes at point A (point B , respectively) on the cylinder,
a i T a n g e d in increasing order
Sy ordinate of a reference point on the 5'A'^-curve, or of the intersection point of the two branches of the SN-curve with different slopes
S(io) wave frequency spectrum (uni-directional wave spectrum)
Siio, 9) duectional wave spectrum t time
T duration, the period of time considered (usually in years)
Tj period of time that waves are within the rth wave direction sector
YPMS safety factor on calculated fatigue damage for checking the Palmgren-Miner sum
J.H. Vugts / Applied Ocean Research 27 (2005) 173-185 175
6 wave spreading angle around the mean wave direction fi
df, lower limit of the /th spreading sector
du upper Umit of the (th spreading sector
JJ. mean wave direction of uni-directional waves
will be illustrated in Section 2 for any sort of stress variation, whatever the cause.
It is of course not the wave environment itself that causes fatigue, but the stress variations that are due to the wave environment. A n accurate determination of stress variations usually requires detailed finite element analyses, i.e. extensive and complicated numerical calculations. Consequently, the influence of various factors on a fatigue assessment is usually not easily predictable. However, under two general restrictions overall trends can relatively easily be determined. These restrictions are, firstly, that a simple and rotationally symmetric configuration is considered (i.e. a vertical circular cylinder) and, secondly, that the severity of the wave conditions (no cuirent) is independent of their direction (i.e. the severity is the same for all directions). In this paper, we w i l l investigate such trends for the influence of wave directionaUty on fatigue damage.
The wave environment consists of sea states, which are random processes that can be described by a Gaussian random wave model with zero mean. The random wave model is fiilly characterised by a wave spectrum. The model may be visualized as the summation of a large (theoretically an infinite) number of wavelets, where each wavelet is a periodic wave with its own frequency, amplitude and direction; each wavelet has arbitrary phases relative to the other wavelets. When all wavelets are assumed to have the same direction we have a so-called uni-directional sea with infinitely long crests; such a sea state is fully defined by the wave frequency spectrum and a mean wave direction. When the individual wavelets also have different directions we have a so-called directional sea in which the lengths of the wave crests are finite. There are thus two different directional effects of waves, which should be cleariy distinguished. The first directional effect refers to the variable mean direction of uni-directional sea states; this influence is addressed in Section 3. The second directional effect refers to wave spreading around a given mean direction; this influence is discussed in Section 4. Finally, Section 5 combines these two influences.
2. General background to fatigue assessments 2.1. Calculation of fatigue damage
Let s(t) a random stress history occurring at a particular point P of a structure during a period 0 < f < r . To determine the fatigue damage done by s(t) the stress history is characterized by the pairs iS,n), where 5 is the stress range magnitude and n is the number of occurrences of S within the period T. For a randomly varying stress history s{t) the definition of a stress range s and the counting method to determine the associated number of cycles n
lij /'th mean wave direction for point A on the cylinder
(Pj Jth mean wave direction for point B on the cylinder 0 angle of point B on the cylinder with respect to
point A
w wave frequency (rad/s)
are not immediately obvious. Several definitions of stress ranges and counting methods have been proposed, such as peak counting, zero-crossing range counting, rainflow counting, etc., but these are not relevant to the purpose of this investigation and w i l l not be discussed here.
For practical purposes the stress ranges S within s{t) are distributed over a finite number of discrete (usually equally spaced) stress range classes of increasing magnitude that are numbered from k= 1 to K. Thus, Sj, is the hXh stress range class and nj, is the number of times during the period T that stress ranges fall into class 5^. With this procedure the sequence in which the stress ranges occur is lost; the pans {Syi^) can only reflect a purely statistical representation of the stress history.
The pairs {Syi^) for l<k<K form a set of values that in conjunction with a given 57V-curve fully determine the fatigue damage done for welded structures. The paii's (5^.,n^) represent the statistical aspects of the fatigue loading at point P. As already noted, the definition of the stress ranges and their counting method make no difference for the present discussion. The actual number of stress ranges within class Sj, during the period T is According to the SN-c\xxv&, the allowable number of stress range cycles for class is 7V^. The partial damage due to Sj, is then assumed to be n^lN^ and the total damage done is assumed to be given by the Palmgren-Miner rale of the linear accumulation of damage
^ = ^ 2 . 1 )
The SV-curve specifies the relationship between the stress range S and the allowable number of cycles N untfl failure due to progressive fatigue cracking occurs. Like the definition of the stress ranges and the counting method, the definition of what exactly constitutes 'failure' can also vary, but this is again iiTelevant for the present discussion. The 5'A^-curve used for design includes a suitable safety factor on the mean curve through the experimental data points. The 5A^-curve is normally presented as a bi-linear curve on a double logarithmic basis; see Fig. 1. I f {Sj,N,) is the intersection point of the two branches of the SN-curve, the curve can be formulated as (see Appendix A)
N=N, = (NX" )^"" = C2^"" (S < S,), ^^-^^
176 J.H. Vugts / Applied Ocean Researcli 27 (2005) 173-185
Fig. 1. A typical SW-curve.
Applying Eq. (2.2) t o t h e discrete stress range classes Sj, and s u b s t i t u t i n g the a s s o c i a t e d i n t o Eq. (2.1), while n o t i n g t h a t t h e 5'^ v a l u e s are a i T a n g e d in classes of i n c r e a s i n g m a g n i t u d e f o r l<k<K, t h e r e s u l t i s found to b e
(2.3)
where Ic, is the counter of the stress range class for which Fatigue failure is assumed to occur when D reaches a value of 1.0. The calculated damage D is usually multiplied by a safety factor to obtain the design value D j , so that
= Y P M S Ö < 1 . 0 (2.4)
Common values of the safety factor YPMS on the Palmgren-Miner sum range from 1.0 to 10. The stress range pairs (SkAk) relate to a duration T of the stress history, where T is usually expressed in years. Consequently, the fatigue life Lf that corresponds with 0^=1.0 is
T
h = j r (years) i->i
2.2. Discussion of the fatigue damage equation
(2.5)
To enhance understanding of the results of fatigue damage calculations it is useful to briefly discuss the characteristics of the fatigue damage Eq. (2.3). This is most easily done for a constant slope 5"A'-curve for which m\ = m2. Both terms of Eq. (2.3) are of the same form so that this does not imply loss of generality of the observations. However, it should be noted that the contribution to the total fatigue damage from each branch depends on all variables involved in the calculation, being:
(a) the absolute magnitudes and numbers of the stress range pairs {Sk,nu)\
(b) the absciss and ordinate of the intersection point {S,,Nr) of the two branches of the SN-cmN&;
(c) the two slopes and 77)2.
The influence of variations in these variables is further comphcated by the fact that the ^T^^curve is bilinear on a l o g -log basis, which does not make the effect of changes very transparent. For real data statements on influences and sensitivities can therefore only be made by performing appropriate numerical calculations. Before presenting some typical results of such calculations under the two restrictions discussed in the introduction we w i l l first discuss some general relationships that can be derived from the global nature of Eq. (2.3).
For a single slope ^A^-curve the point (5r,A'r) cannot be related to the intersection point of the two branches and should simply be r e f e i T e d to as a reference point. Fatigue damage is then governed by
D = 1 E"kSl (2.6)
k=i
I f Nr of the reference point increases but S,. and 77? are kept the same, the SN-curve moves to the right, see Fig. 2. For a given stress range history this means that the pairs (5^:,77.j.) are evaluated against a less severe SN-curve (higher allowable A'^^ for the same 5^,); i.e. less damage and a longer fatigue life. For decreasing N, the 5'A''-curve moves to the left, is more severe and fatigue damage is coixespondingly increased. I f Nr_b = c^Nr^a the ratio of the two damage sums is easily quantified
CN
(2.7) Therefore, i f N, moves one log-cycle to the right (c/v= 10) or to the left {c{^=0.l) the damage is reduced or increased, respectively, by a factor of 10.
I f Sr increases but A^i- and 777 are kept the same, the SN-curve moves upwards, see again Fig. 2. A n SN-curve that either moves up or to the right achieves the same result. Consequentiy, an increase in S, or A'^ both represents a less severe 5A^-curve (higher allowable N^ for the same 5"^), or less damage and a longer fatigue life. For decreasing S, or N, the reverse is true. However, i f we describe the change in S,
J.H. Vugts / Applied Ocean Reseai ch 27 (2005) 173-185 111 10,000 UI E E m m a) 1,000 • Sr,b 100 -Sr,a 10 Nr,a Nr,b 1.E+12 1.E+02 1.E+04 1.E+06 1 .E+08 1.E-H0
n u m b e r of c y c l e s to failure N
Fig, 2. Changes of a single slope OT-curve due to increases in S, or A'r of the reference point
tlie same factor analogously as Sr^h = CsSj,a the influence of the factor Cs on the
fatigue damage is much greater because is raised to the power m A , (2.8) D„ k=\ E «*,«(5A.)" (2.10)
I f for the same reference point {Sr,Nr), all stress ranges are increased or decreased by a particular factor (as could, for example, be due to a change of the stress concentration factor) the influence on the fatigue damage is similar but opposite to a change of the reference stress range S,. When Sk,b = CskSk,a the influence on the damage sum is
E nkiS.^t)" D ^
" E ' h i S k j '
k=l
= icskr (2.9)
If, on the other hand, the numbers of cycles iip. are changed by a factor c„ {nk,b = c,fli.,a) the damage is simply changed by
This latter situation would, for example, occur when the duration T is changed. In such a situation, the fatigue life will obviously not be affected, because when 7'è = c„r„ and Db = c„D„ the fatigue life Lf = TbIDb = TJD„ remains the same.
An increase in the slope m of the SN-cuive through the same reference point (S^Nr) affects the damage calculation thi-ough the summation of the terms (Sk/Sj" over all k. For Sk>Sr, each term is larger than 1.0 because ?n is positive; the coiTcsponding damage contributions are hence increased. However, for 5^ < bl-each term is smaller than 1.0 and the coiTcsponding damage contributions are reduced. The influence of m can also be explained in another way. When m increases the SN-cmve rotates counter-clockwise around the reference point (Fig. 3). To the right of (5r,iVr) the .SA^-curve moves upwards and the associated damage due to this part of the curve is reduced.
1,000 UI UI I 100 Sr 10 Increasing m increasing m increasing m Nr 1
1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 n u m b e r of c y c l e s to failure N
1.E+12
178 J.H. Vugts / Applied Ocean Research 27 (2005) 173-185 700 600 500 400 300 200 100
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 cumulative n u m b e r of o c c u r r e n c e s
Fig. 4. Tliree examples of a long-term stress range distribution for a service life of 20 years, representative of light, moderate and severe fatigue loading situations.
a.
m
E
w •.•1
However, to the left of (Sy,N,) the ^A^-curve moves downwards and the damage is hence increased. The total net effect depends entirely on the severity of the stress range pairs {Syi/,) compared to the reference point (SM). Please note that this discussion relates to a single slope iS'A^-curve and has therefore nothing to do with damage contributions from the two branches of a bi-linear .SA^-curve.
To illustrate the influence of m, some numerical calculations are presented for thi'ee different stress range distributions that ai'e considered to be representative for a service life with an assumed duration of 20 years. The probability distributions of the stress ranges are taken as straight lines on semi-logarithmic plots, where S;. is plotted vertically on a linear scale and the cumulative values of iik are plotted horizontally on a logarithmic scale; see Fig. 4. The maximum stress range occurring once during the 20 years service life is assumed to be, respectively, 100 MPa (representing a light fatigue loading situation); 345 MPa (with single sided stress amplitude variations up to half the yield strength of a normal quality construction steel; this represents moderate fatigue loading situations); and 690 MPa (with single sided stress amplitude variations up to the yield
strength of a normal quality construction steel, representing severe fatigue loading situations). The total number of cycles i n a 20-year period is assumed to be 10^ for each of the three cases. The slope of the SN-cmws is arbitrarily varied from ;n.= 1 to 9; common values for 5'A'^-curves for welded steel structures are 1111=3 and in2 = S. However, an 5'A^-curye cannot only be defined by its slope; it also needs a point to fix i t in an absolute position. For the present illustration, this point is chosen to be the point (5=67.1 MPa, A'^^10''), which corresponds with an often used intersection point of the two branches of a realistic 5A^-curve for marine structures. A l l 5'A^-curves for the different slopes hence pass through this point, see Fig. 5. It should be noted that these ^A^-curves are only used for the purpose of illustrating the influence of slope on fatigue damage calculations and should not be interpreted as realistic curves. The results of the calculations are summarized in Table 1.
For the fight fatigue loading regime with a maximum 5"^.= 100 MPa the fatigue damage decreases (fatigue life increases) continuously when m increases. For the severe fatigue loading regime with a maximum 5^.=690 MPa, the reverse is true: fatigue damage increases and fatigue life
1,000 cs Q. S 100 (fl Sr=67.1 111 C 1^ 10 ^ • " " ^ ^ m = 5 m = 9 \ ^ n n = 3 , m = 1 Nr=10'
\
1.E+02 1.E+04 1.E+06 1.E+08 1,E+10 n u m b e r of c y c l e s to failure N
1.E+12
J.H. Vugts/Applied Ocean Research 27 (2005) 173-185 179
Table 1
The influence of the slope of the SA'-curve for the three long-teim stress range distributions of Fig. 4 and the five SN-cmves of Fi". 5
Max. St = 100MPa
Total damage sum Fatigue life (years)
Fatigue life (;;!)/fatigue life (;» p.c. damage for St > 5 r p.c. damage for S^KS,
••i)
Max. Si:=345MPa
Total damage sum Fatigue Ufe (years)
Fatigue hfe (Hi)/fatigue life (HI = 3) p.c. damage for St>S,
p.c. damage for S,;<Sr
Max. 5^=690 MPa
Total damage sum Fatigue life (yeai's)
Fatigue life (;)0/fatigue life (;H = 3) p.c. damage for Si,>S^
p.c. damage for S^KS^
Slope of the SN-cmve
0 8.133X10" 24.65 0.04 0.01 99.99 2.799 7.146 0.47 12.62 87.38 5.598 3.573 1.87 47.71 52.29 3.297X10" 628.3 1.00 0.16 99.84 1.307 15.30 1.00 51.69 48.31 10.46 1.913 1.00 89.93 10.07 m = 5 4.168X10" 4799 7.64 1.54 98.46 2,037 9.818 0.64 84.53 15.47 65.19 0.3068 0.16 99.09 0.91 1.145X10" 17,470 27.81 7.13 92.87 6.662 3.002 0.20 96.96 3.04 852.7 0.02345 0.01 99.95 0.05 5.370X10" 37,240 59.27 79.84 20.16 37.18 0.5379 0.04 99.60 0.40 19 040 0.001051 0.0005 100.00 0.00
decreases consistently with increasing m. For the intermediate, moderate fatigue loading regime with a maximum S/,= 345 MPa, the trend reverses from a decrease into an increase in fatigue damage for a value of around m = 3 or 4 . This difference in behaviour is associated with which part of the stress ranges (Sk>S,. or Sk<Sj) contributes most to the fatigue damage incurred. For each of the three cases and each value of m, the percentage contributions are also shown in Table 1.
3. Fatigue damage due to uni-directional waves in different mean directions
3.1. Waves with a single mean wave direction
The influence of mean wave direction on the stress history j ( 0 at a point of an arbitrary structure is normally rather complex, will vary from case to case and can only be established by numerical calculations for each particular case. However, for the elementary case of wave loading on a vertical circular cylinder and wave conditions that are the same i n all directions the influence can (within reason) be determined analytically. In these circumstances the fatigue damage is due to wave induced variable bending stresses i n the cylinder.
Let us consider that point A in an arbitrary cross-section of the cyUnder (Fig. 6 ) is the point of interest. Let further the stresses occurring at point A for wave direction (1 = 0 be indicated by JA(AM = 0 ) . The stress history occurring at another point B for a wave direction 0 < ^ < 90° is identical to the stress history occuning at point A for / i = 0, i.e.
sji(t,/x)=SA{t,ii = 0) ( 3 . 1 ) For a wave direction 0 < ^ < 9 0 ° the relationship between
the bending stresses occurring at points A and B is clearly
( 3 . 2 )
This results in the following stress history at point A for
0 < M < 9 0 °
SAif> M) = SAit, M = 0)COS fL
We are only interested i n the stress ranges from the f u l l stress history J A ( ^ M ) and we w i l l indicate the stress range class S^. occuning at A for wave direction ^ by 5A,A(M)- The stress range classes for a direction 0 < yti < 9 0 ° are of course related to the stress range classes for jtt = 0 by the same relationship SA.f.-W = Sam(/J- = O)cos IX
Stress ranges are by definition always positive. For symmetric wave directions (±ix) and for diametrically opposite wave directions ( ^ ± 1 8 0 ° ) , the bending stresses are equal to those for 0 < ^ < 9 0 ° . Therefore, the generafisation of the above relationship for all fx is
SA,k(P) = • 5 A , i ( M = O)|cos fi\ ( 3 . 3 )
180 J.H. Vugts/Applied Ocean Researcli 27 (2005) 173-185
Thus, for fi^O all stress range classes are a factor |cos n\ smaller than for /( = 0. For corresponding classes with reduced magnitudes the number of cycles of ^A.^^ are not affected by the wave direction, i.e. the numbers nA,k are unchanged
Table 2
( 3 . 4 )
Substituting the pairs iSA,k(p)^iiA,kil^)) from Eqs. ( 3 . 3 ) and ( 3 . 4 ) into Eq. ( 2 . 3 ) we obtain for the fatigue damage due to bending of the cylinder
1 ^
DaW = 7 ^ Icos Ml"" ^ « A , . . ( M = 0){5A,,(M = 0)}'"' ^ 1 k=k,
1
+ — I c o s M I ' " ^ ^ n A , i . ( M = 0){5A,,(M = 0)}"
k=\
( 3 . 5 )
The reduction factor R D A , I ( M ) fof the fatigue damage at point A and uni-directional waves in direction yU^^O, compared with the maximum fatigue damage at point A when the waves occur in the direction p.=0, is consequently
R D A , I ( M )
D A ( M = 0 )
( 3 . 6 )
The outcome of this ratio depends on the spread of the stress range classes ^ A . ^ over the two branches of the ^A^-curve and cannot be evaluated in a general manner. However, the results can be bracketed by considering the SN-curve to have a single slope for all 5A,*; the position of the reference point of the 5A^-curve is iiTelevant when this ratio of damage sums is considered. For welded steel structures, the two common slopes are mi = 3 and ;«2 = 5. As jcos / i | < 1 . 0 , the fatigue reduction is greatest (the reduction factor is lowest) for the higher exponent. Therefore, the reduction factor R D A , i ( j U ) for fatigue damage is bracketed by
jcos j t i p < R D A , I ( M ; ' " 1 , ' " 2 ) ^ Icos ( 3 . 7 )
The value of R D A . I within this range for a bi-linear SN-cmve depends entirely on the stress range pairs {SA,hnA,k) in relation to the intersection point of the two branches, as was discussed in Section 2. Please note that the influence of the slope of the j'A'-curve with regard to wave direction differs from the influence of the slope for stress ranges in Section 2.2. In the present case, the slope operates as an exponent on the external factor |cos fi\ in front of the fatigue sum, whereas in Section 2.2 it operated as an exponent on the stress ranges ^ A , * behind the fatigue sum.
Numerical results for Eq. ( 3 . 7 ) are summarized in Table 2 for 0 < I M I < 9 0 ° . For 9 0 ° < I M I < 1 8 0 ° the reduction factor is
R D A , I ( I M - 1 8 0 1 ) = R D A , I ( I M + 1801) = R D A , I ( I M I ) .
The results in Table 2 can, of course, also be interpreted as the fatigue damage D B ( ( ^ = 0 ) at a point B located at an angle
\(p\ from the x-axis, when the uni-directional waves approach
the cylinder all the time along the x-axis.
Reduction factor for fatij ;ue damage at point A for uni-directional waves in direction p. and 0 < < 9 0°, compared with maximum fatigue damage at point A for uni-directional waves in direction ; i = 0
p (deg) R D A , I ( M ) ' " = 5 RDA,i(/i)m = 3 0 1.000 1.000 15 0.841 0.901 30 0.487 0.650 45 0.177 0.354 60 0.031 0.125 75 0.001 0.017 90 0.000 0.000
3.2. Waves distributed over several mean directions The wave climate is assumed to be the same for all mean directions. However, the waves now approach the vertical cylinder from several instead of one mean direction during the service life. The fatigue damage for this situation is a weighted summation of the results of the previous section, where the weighting factors are the probabilities of occurrence of the various directions.
We w i l l look at the same situation that point A on the x-axis is the point of interest (see Fig. 6). A l l wave directions are measured relative to the x-axis. Let the envrronment be diseretized into I sectors of (uni-duectional) wave fields with mean directions jti,-and i = 1, 2 , . . . , ƒ . The widths of the sectors may be mutually the same or may differ from sector to sector; the width of the rth sector is indicated by Ajtt,-. The probability of occuiTence /?(ytt,) that the wave direction falls within the /th sector is
/(M)dM (3.
w h e r e / ( M ) is the probabifity density function of the mean wave directions p. Obviously, the sum of these probabifities is unity
1.0 ( 3 . 9 )
1=1
The probabifity pipi) represents the fraction T; of the total time r t h a t the wave du'ection is within the /th sector, i.e. Tj=p{jx,)'V. The number of o c c u i T e n c e s per sector is hence also reduced by the factor piiJ,)- Thus, the numbers of cycles are, in analogy with
E q . ( 3 . 4 )
«A,<:(M;) = P(M/)"A,A-(M = 0) ( 3 . 1 0 )
The stress ranges are unaffected by the distribution of mean wave directions and remain the same as in Section 3 . 1 . Thus, in analogy withEq. (3.3)
ShAp-i) = SA,k(f^ = O)|cos Hi\ ( 3 . 1 1 )
Substituting Eqs. (3.10) and ( 3 . 1 1 ) into Eq. (2.3) gives for the partial fatigue damage per mean wave duection
J.H. Vugts/Applied Ocean Research 27 (2005) 173-185 181 1 ^' Ö A ( M / ) = 7 ^ i ' ( M ; ) | c O S V « . A , i . ( M = 0 ) { V t ( M = 0 ) } " " •-1
fee
1 + — ; 7 ( M , ) | c o s Mil'"^ E " A , A ( M = 0 ) { 5 A , , ( M = 0 ) } " ( 3 . 1 2 )This is equal to Eq. (3.5) for uni-duectional waves with a single mean wave du-ection, multiplied by the weighting factor pifi,) per direction, as was to be expected.
The total damage at point A is the sum of the partial damages for each direction yU,-, or
Ö A = £ Ö A ( M , ) ( 3 . 1 3 )
1=1
Consequently, the reduction factor R D A , 2 for the fatigue
damage at point A for uni-directional waves of equal severity in all directions and probabilities of occuiTence pipi) per direction, compared with the fatigue damage at point A for all waves occuning in direction ;tt = 0, is
R D A , 2 =
Z ) A according to equation (3.13)
Ö A ( M = 0 ) according to equation ( 3 . 5 ) ( 3 . 1 4 )
As before, this reduction factor cannot be evaluated in general, but it can again be bracketed by considering the SN-curve to have a constant slope equal to each of the two gradients. Thus, for welded steel structures with ; 7 J i = 3 and wi2 = 5 the reduction factor is bounded by
j][p(M/)|cos < RDA.2(mi,m2)< Ê[p(M,)lcos
1=1 1=1
( 3 . 1 5 )
By way of example. Table 3 gives results for Eq. (3.15) for a discretization of wave fields into uniform sectors of 30° and an assumed distribution of the probabilities p(,p,) as shown i n the table. The co-ordinate system is aligned such that / = 1
Table 3
Reduction factor for fatigue damage at point A for uni-duectional waves of equal severity in all directions and probabilities o f occurrence p{p() per direction, compared with fatigue damage at point A for all waves occurring in direction p = 0
p, (deg) p(pi) in % p(^,) |cos (pi)\^ p(pd Icos (M,)P
1 0 23 0.230 0.230 2 30 17 0.083 0.110 3 60 12 0.004 0.015 4 90 7 0.000 0.000 5 120 6 0.002 0.008 6 150 5 0.024 0.032 7 180 3 0.030 0.030 8 210 2 0.010 0.013 9 240 2 0.001 0.003 10 270 3 0.000 0.000 11 300 8 0.003 0.010 12 330 12 0.058 0.078 Total 100 ^ = 0 . 4 4 5 = R D A , 2 = 0.529 = R D A . 2
coiTcsponds with the predominant mean wave direction, so that the probability p(jJi) is by definition the highest.
3.3. Fatigue damage at another point B for waves distributed over several mean directions
For another point B with an angle 0 from the x-axis (cf> being measured positively in the same sense as p.) the formulation of the problem is entirely the same as for point A, provided that the x-y co-ordinate system is rotated over 0. The x-y system for point A then transforms into the u-v system for point B, as shown in Fig. 7. The relationship between the angles in the two systems is (pj = Hj —(j), where for convenience we have introduced the counter for point B, to distinguish it from the counter ; for point A. The fatigue damage at point B is then, fully analogous with Eqs. ( 3 . 1 2 ) and (3.13)
1
DsiVj) = ^pi(Pj)\cos CPP
y^n^^cp =
0 ) { 5 B . , ( ^ = 0 ) } ' " ' 1D B ( 0 ) = £ Z ) B ( < P , )
( 3 . 1 6 )
(3.17)
7 = 1
Note that the damage at point B is now a function of the angle 0 over which the co-ordinate system is rotated. To compare the fatigue damage at point B with the fatigue damage at point A we need to compare the results of Eqs. ( 3 . 1 7 ) and
( 3 . 1 3 ) , which are the integrals of Eqs. ( 3 . 1 6 ) and ( 3 . 1 2 ) ,
respectively. The environment is unchanged and the prob-ability of occuiTence of each wave direction sector is invariant to a rotation of the co-ordinate system. Therefore,
pirpj) = p(,Xi) ( 3 . 1 8 )
mean wave direction
Fig. 7. Relationship between points B and A f o r uni-dbrectional waves i n mean direction Pi.
182 J.H. Vugts / Applied Ocean Research 27 (2005) 173-185
However, as the counters / and j are not the same they are associated with different angles and hence different multipli-cation factors Icos (pj\"' or |cos /Lt,|'". In contrast, for the case that all waves travel in the direction 0 = 0 for point B and all waves travel in the direction (1 = 0 for point A , each of the two damage sums are the same
K K Y^n^Micp = 0){5B,,(<P = 0)}"" = X ] n A , , ( M = 0 ) { 5 A , , ( M = 0)}""
(3.19)
Table 4
Reduction factor for fatigue damage at point B for uni-directional waves of equal severity in all directions and probabilities of occuirence p{pj) per direction, compaied witb fatigue damage at point A
Rotations, <j> RDB,A(<^) for /» = 5 R D B , A ( 0 ) for m = 3 0 and 180° 1.000 1.000 30 and 210° 0.886 0.914 60 and 240° 0.662 0.716 90 and 270° 0.558 0.618 120 and 300° 0.640 0.691 150 and 330° 0.842 0.881 k=l K Y,it^A<p = 0 ) { 5 B . , ( < P = 0)}"'^ = 5 ] « A . . ( M = 0 ) { 5 A , , ( M = 0)}"'^ k=\
Therefore, for a single slope 5A^-curve the ratio of the total fatigue damage at point B to the total fatigue damage at point
A is R D B , A ( 0 ) ^ Y.^p{(pj)\cOS (pj\ DBW _ ; = i
E[p(M/)|cosAi,|
/=i (3.20)Due to the thi-ee-dimensionality of the wave envii-onment in terms of the probability of occurrence per direction (not in terms of severity!), the value of the numerator depends on the rotation <f> of the co-ordinate system; the value of the denominator is a constant reference value.
For a wave chmate of equal severity in all directions, the ratio of the damage at point B to the damage at point A is thus bracketed as shown in Eq. (3.21)
points B that are located at either side of point A , that is at rotations + 0 and —0, are no longer the same.
For the purpose of illustration. Table 4 gives numerical results for various points B; the distribution of the probabilities of occurrence is the same as shown in Table 3.
4. Fatigue damage due to spread waves around one mean direction
A sea state with spread waves around one mean wave direction (for which ^ti = 0 wiU be chosen) can be described by a directional wave spectrum Si(jj,d). The directional wave spectram is usually expressed in the following form by assuming that the influences of the frequency parameter oj and the directional parameter 6 can be split into two separate functions S(oj, d) = Did)Sioj) and (4.1) y ^ i É[p(M,)lcos M , f ] 1=1 < R D B , A ( 0 ) ^ E[pi(Pj)\cos (pj\ Ö B ( 0 ) ^ 7 = 1 E[p(M,)|cosM,f] i=l (3.21)
The absolute value of a cosine-function is periodic with intervals of 180°, hence
[cos (pj\ = Icos i(pj±m°)\ (3.22)
which means that the numerator of Eq. (3.20) is the same for aU rotations for which
180° (3.23)
The physical meaning of this relationship is the observation that the fatigue damage is the same at points B that are diametrically opposite to one another. However, despite the geometrical symmetry of the cylinder the fatigue damage for
-I-7T/2
D(d)dÖ = 1.0 (4.2)
-7r/2
where
S((JL)) the wave frequency spectrum
D{d) the wave directional spreading function.
For the spreading function D(6), several formulations have been proposed. One simple formulation that is often applied in practical applications and will also be used to illustrate the influence of wave spreading in this paper is
Did) -cos^d ( - 9 0 ° < ö < + 9 0 ° ) (4.3)
Recognizing that within a discretized directional sector AÖ; the spread waves are uni-directional, the case of spread waves around one mean wave direction is fully analogous with the case described in Section 3,2, where the spreading angle should take the place of the mean direction jij. The probability of o c c u i T e n c e p(Oi) that waves propagate within the sector
A(9,-J.H. Vugts / Applied Ocean Research 27(2005) 173-185 183
Table 5
Reduction factor for fatigue damage at point A for spread waves according to Eq. (4.3) around mean direction p = G, compared with fatigue damage at point A for uni-directional waves in direction ^L = 0
I Wave direction sector 6j (deg) Lower limit of ;th sector ^8 (deg) Upper limit of ;th sector da (deg)
pid,) in % p ( 0 , ) Icos 0,1= p(e,) Icos ö , p
1 0 - 1 5 - f 15 32.58 0.326 0.326 2 30 -1-15 +45 24.62 0.120 0.160 3 60 -b45 + 75 8.71 0.003 0.011 4 90 - f 7 5 + 90 ( < + 105) 0.38 0.000 0.000 120
-
- 0 0 0 6 150 _ _ 0 0 0 7 180 _ - 0 0 0 8 210 ( - 1 5 0 ) - - 0 0 0 9 240 ( - 1 2 0 ) - - 0 0 0 10 270 ( - 9 0 ) - 9 0 ( > - 1 0 5 ) - 7 5 0.38 0.000 0.000 11 300 ( - 6 0 ) - 7 5 - 4 5 8.71 0.003 0.011 12 330 ( - 3 0 ) - 4 5 - 1 5 24.62 0.120 0.160 Total 100.00 E = 0.572 = R D A , 3 X ; = 0 . 6 6 8 = R D ,for -9Qi°<e< + 9 0 ° is now equal to
Did)dd TL cos^
e de
-4 A 9/ TT[ ^ i n 2 ö + i ö ] i
(4.4)The fatigue damage evaluation in Section 3.2 remains entirely valid, with the results given by Eqs. (3.12)-(3.15). We only need to replace fij by Of, the mean wave direction for the spread waves is M = JUI = 0. For uniform sectors of 30°, as in Section 3.2, this produces the results shown in Table 5, which can be directly compared with Table 3.
5. Fatigue damage due to spread waves around several mean directions
The case of spread waves around several mean directions is also analogous with Section 3.2, but its foiTnulation is a bit more comphcated. The physical situation is illustrated in Fig. 8. For a mean wave direction /tt and a spreading angle 6 around p the total angle a of a particular group of wavelets is
a=ix + 6 (5.1) In discrete form this becomes
« , V = Mi + dj (5.2)
where j assumes the values j = l , 2, J for each i and summation over j should take place before / is changed.
The sti-ess ranges 5A,ic(a,j) are i n f u l l analogy with Eq. (3.11)
SA.kic'u) = SAAP = 0 = 0)|cos a. (5.3)
Similarly, the associated number of cycles 7JA,k(o;;\;) is in f u l l analogy with Eq. (3.10)
"A,A-(«,v) = P ( M , - ) P ( Ö / I M , ) « A , A . ( M = 0 = 0) (5.4)
where pijx^) is the probability that the mean wave direction is within sector A/u,- as before (see Eq. (3.8)), w h i l e p ( d j \ n ^ is the conditional probability that the spread waves are within the sector Mj for — 9 0 ° < i 9 < + 9 0 ° around the mean direction fij.
Substitution of Eqs. (5.3) and (5.4) into Eq. (2.3) gives the partial damage DAiciij) due to waves within the directional sector Aöy around the mean direction /tt,-. The result is fully analogous with Eq. (3.12); for a single slope 57\^-curve 0^(011 j) is thus equal to Ö A ( « V - ) = ~ E " A . * ( ' ^ V O { ^ A . A ( « / J ) } " C K = p(fii)p{dj\iJi)\cos a j j 1 V -X - ^ 2 ^ « A , A ( A i (5.5) S = 0){SA,kifi = 0 = 0)}" = p(Ai,)Möyl/ti,)|cos a , / " D A ( M = 0 = 0)
In words, the fatigue damage at point A due to spread waves within the directional sector Mj around the mean direction jti,- is equal to the damage at point A due to uni-directional waves along the x-axis, multiplied by the weighting factor \p(pi) p{dj\ixi) Icos «,^1"'].
direction of wavelets
mean wave direction
Fig. 8. Wave duection of wavelets with a spreading angle around mean wave duection p j.
184 J.H. Vugts/Applied Ocean Research 27 (2005) 173-185
The damage associated with mean direction M,- is obtained by summing o v e r j
= P(f^i) J ] p ( ö > , ) | c o s aJ"DA(li = 0 = 0) (5.6)
Further elaboration of the factor |cos a,j|'" is complicated and impractical as a result of the absolute value and the power m( = 3 or = 5 ) involved. Therefore, numerical evaluation of Eqs. (5.8) and (5.9) is the best way forward. A n illustrative example is shown in Table 6, where forp(/t,) the same values as in Table 3 and forp(ö,lM,) the same directional distribution as in Section 4 have been adopted; the associated probabilities are given in Table 5.
while for the total damage we should also subsequently sum Rg^^jt^ conclusions over
/ ƒ
DA = E ^ A ( M ; ) = ^
(=1 1=1 X D A ( M = ^ = 0)
The reduction factor R D A , 3 ( M / ) for fatigue damage at point
A due to spreading around a given mean direction jUj, compared with fatigue damage at point A due to uni-directional waves i n the same direction /Xj is obtained from Eq. (5.6) as
R D A , 3 ( M , )
Ö A ( M , ) D A ( M = i^i, e = 0)
J
^p{6j\iXi)\cos aij\
Similarly, the total reduction factor R D A , 4 for fatigue damage at point A for spread waves around mean direction p,;, with probabilities of occurrence pifid per mean direction, compared with fatigue damage at point A for uni-directional waves along the x-axis (fi = 0) is obtained from Eq. (5.7) as
For any type of stress variation, whatever the cause, the sensitivity of fatigue damage calculations has been formulated (5.7) jjjg parameters on the 'loading side', i.e. the magnitudes
(Sk) and the number of cycles (;i^.) of the stress ranges experienced, as well as for the parameters on the 'resistance side', i.e. the ordinate (S,) and the absciss (Nr) of the intersection point and the slopes (nii and 1112) of the two branches of the .W-curve. The impact of is by far the greatest. The influence of a variation of the slope of the SN-cmve with respect to the base value in=3 has further been quantified for thi'ee typical long-term stress range distributions during a stracture's lifefime. _ .
For the bending stress variations in a vertical circular (5.8) cyUnder due to wave action in wave conditions that are the same in all directions, the influence of wave directionality was next detennined for tlnee different cases:
R D A , 4 = D, D A C M = 9 = 0 )
E
1 = 1 P(P'i)^p(9j\pi)\cos a,_1. uni-directional seas with varying mean wave directions; 2. spread seas around one mean wave direction;
3. spread seas and different mean wave directions.
The reduction in fatigue damage for case 1 and an assumed probability distribution of mean wave directions is typically around 50%. For case 2 and a cosine squared directional spreading function, the reduction in fatigue damage is typically (5.9) around 40%, while for the combination of these two wave
conditions in case 3 the reduction is of the order of 60%.
Table 6
The reduction factors RDA,3(At;) for fatigue damage at point A for spread waves around mean direction /t„ compared with fatigue damage at point A for uni-directional waves in direction ^t,, and the total reduction factor R D A . 4 for fatigue damage at point A for spread waves ai'ound mean direction p,- with probabilities of occuirence p(pi) per mean direction, compared with fatigue damage at point A for uni-directional waves along the .v-axis (/n = 0 )
(deg) pipd in % K D A , 3 ( / 1 / ) = 5 p(p,) R D A , 3 ( M / ) = 5 R D A . 3 ( M , ) = 3 p(pd R D A , 3 ( / < / ) ni = 3 1 0 2 3 0 . 5 7 1 1 0 . 1 3 1 4 0 . 6 6 7 4 0 . 1 5 3 5 2 3 0 1 7 0 . 4 5 5 3 0 . 0 7 7 4 0 . 5 4 6 1 0 . 0 9 2 8 3 6 0 1 2 0 . 2 2 3 6 0 . 0 2 6 8 0 . 3 0 3 6 0 . 0 3 6 4 4 9 0 7 0 . 1 0 7 8 0 . 0 0 7 5 0 . 1 8 2 3 0 . 0 1 2 8 5 1 2 0 6 0 . 2 2 3 6 0 . 0 1 3 4 0 . 3 0 3 6 0 . 0 1 8 2 6 1 5 0 5 0 . 4 5 5 3 0 . 0 2 2 8 0 . 5 4 6 1 0 . 0 2 7 3 7 1 8 0 3 0 . 5 7 1 1 0 . 0 1 7 1 0 . 6 6 7 4 0 . 0 2 0 0 8 2 1 0 2 0 . 4 5 5 3 0 . 0 0 9 1 0 . 5 4 6 1 0 . 0 1 0 9 9 2 4 0 2 0 . 2 2 3 6 0 . 0 0 4 5 0 . 3 0 3 6 0 . 0 0 6 1 1 0 2 7 0 3 0 . 1 0 7 8 0 . 0 0 3 2 0 . 1 8 2 3 0 . 0 0 5 5 1 1 3 0 0 8 0 . 2 2 3 6 0 . 0 1 7 9 0 . 3 0 3 6 0 . 0 2 4 3 1 2 3 3 0 1 2 0 . 4 5 5 3 0 . 0 5 4 6 0 . 5 4 6 1 0 . 0 6 5 5 Total 1 0 0 J ] = 0 . 3 8 5 8 = R D A , 4 ^ = 0 . 4 7 3 4 = R D A , 4
J.H. Vugts / Applied Ocean Researcli 27 (2005) 173-185 1 8 5 1.E+12 (11 £1 I 1.E+02 1 ^ - \ 1 = 10 ' 100 1.000 stress range x - S (MPa)
>-Fig. A l . Alternative way of plotting an SN-cuive, with the usual convention of the independent variable S along the horizontal axis and the dependent vaiiable A' along the vertical axis.
The reductions that have been found are reliable indications for the special case of a simple, rotationally symmetric structure and an idealized wave environment. However, it should be noted that for a real structure and real data regarding the wave environment the influence of wave directionally could only be determined by pertinent and detailed calculations for the specific case concerned.
Appendix A. Derivation of tlie equation for the SN-curve The normal way in which SN-cmves are presented is shown in Fig. 1, i.e. with the axis for the stress range vertical and the axis for the allowable number of cycles horizontal, both on logarithmic scales. However, it should be appreciated that the stress range S is the independent and the allowable number of cycles N the dependent parameter. In the conventional representation of a straight line graph with the independent (x-) axis horizontal and the dependent (}>-) axis vertical the graph would hence look like Fig. (Al). The mathematical formulation of a straight line passing through the point (AV, }'r) and sloping downwards with a slope m>0 for increasing x is
y - J r = -m{x — x,) (Al)
Applying this formulation to the two branches of the SN-cmve, these can be formulated as follows
log A' - log N, = -/)J2(log S - log S,) for S<S,
log A' - log N, = -in 1 (log 5 - log S,) for S>S, (A2)
These two expressions can be combined into one general expression
log N - log N, = -7«(log S - log S,)
with m = m i > 0 for S>S, (A3) m = ;«2 > 0 for S<S,.
which can be rewritten as
log N = log N, — Hj(log S-logSj) = log
4
Therefore 'S' N =N, CS^'" with c=N,s; +111 (A4) (A5) For the two branches of the ^Af-curve separately we thus obtainN =N,\ CiS-"" with CI=NX" for S>S, - H I ,
N=N, — =C2S-""- with C2=NX" foi- S<S,
(A6) which is the form of Eq. (2.2).
As both n!i > 0 and «72 > 0, it should be noted that Eqs. (A6) result in
N<N, for S>S,
N>N, for S<S,
