Abstract
Hermite models are verified to describe analytically the statistics of nonlinear responses based on the re-sponse mean, standard deviation, skewness and kur-tosis. Cothparion are made bétween the numerical fittings and data obtainéd fröm a nonlinear moor-ing simulation programme obtained for rioored 250 FCdwt FPSO. Good agreerents ä.re obtained own to the lowest levels of probability in thé simulations. This inestigation is an intermediate step in the de velopment of a complete method to calculate the statistics of nonlinear mooring loads loads in arbi-trarv wave spectra. Since the Hetmite models turn out to give ver accurate results, the main attention should be addressed to thé calculation of the statis-tical response moments in nonlinear conditions.
i
Introduction
To calculate fatigue damage or extreme loads in the mooring system of an FPSO, dynamic responses are usually defined as linear or slightly non-linear pro-cesses. lt is known however,tha.t the mooring loads
can show a signißcant nonlinear behaviour, both
from the low frequency motions of the vessel á.s well is from ihe high fréquecy chain dynamics. This
aftcts hoi h the distribution of extremes used in ul tniiate si iength problems and of the rainflow
distri-hittion which is required to perform fatigue
assess-vcor
rchtf
Mekeweg 2 2E23 CD Oft 0th - o1 i'81$
L
Fatigue assessment in FPSO mooring design
using moment based Hermite Approximation
R.H.M.Huij smans (Maritime Research Institute Netherlands) L.J.M. Adegeest (Det Norske Ventas, Høvik)
February 1998
ments. First validation of our numerical approach for realistic type of vessels is based on results from a study performed for the Royal Netherlands Navy for
an Mfrigate type of vessel. The attention in that project was focused on the global hull girder loads in the nonlinear regime The experiments comprised amongst others, fre running tests in head waves and following waves. The free-running tests were per-formed in regular and irregular waves, In this study significant nonlinear hull girder loads were observed under severe relative motion conditions either in the
aft or at the bow. Sag/hog ratio's higher than 2 were observed both in head seas and in following seas.(partly reported in Adegeest (1995)
Thé present article inVestigates the applicability of momentbased Hetnite models to describe analvt-icallv the marginal and extreme probability distri-butions and crossing rates of. a nonlinear response. These models use the first N statistical moments of the response (mean, standard deviation, skewness, kurtosis, etc.). The analytical results are compared with numerical simulation results and are found to cömpare well.
This investigation is an intermediate step in the de-velopment of a complete method to calculate the statistics of nonlinear hill girder loads in arbitrar' wave spectra. Since thè Hermite models turn out to give very accurate results, the development of higher ordér Vôlterra approach to calculate directly the statistical moments in arbitrary wave spectra in the frequency dotnaip is supported after which a
very efficient procedure to perform nonllnea tatis-tical analysis can be developed. However the knOwl-edge of the second order transfer function (QTF) of the drift forces or low frequency motions is essential
for the estimation of the distribution of the turret loads. From model tests or numerical simulations the underlying statistics is by no mans easy to be retrieved. The numerical simulations that are used in this article are based on the work of Wichers and Huijsmans (1992) and Dercksen et al (1994). Their numerical simulatiOns compare favourably with the results of model tests.
Recently Standberg (1997) discussed a procedure to obtain the underlying statistical properties using
crossbispctral techniques and a "preguessed"
linear response deconvolution for the estimation of the QTF of the wave drift forces. However for moor-ing systeths where highly nonlinear restormoor-ing forces and dynamic chain effects play a prominent role, such procedure may prove to be very rude and inac-curate. The suggested procethire of standsberg for applying simplified deconvolution techniques may i hen be used in an iterative manner.
2
Short-term statistical analysis
of nonlinear responses
A significant nonlinear behaviour of the loads on i tirrets in FPSO's has been observed in. the model tc'sts. Irregular wave tests showedconsiderable skew
atid kurtosis in the probability density functions of
tlic' t urret loads.
lliis severely nonlinear behaviour has an important itii pact on the strategy that has to be followed in a
statistical analysis of turret lOads, both of the pre-(hei ion of extreme loads and of fatigue load distri-hut ions.
\ r('view is given of different methods to perform si ¿il istica.l analyses of nonlinear responses, in
partic-ilat those which are applicable to analyze
nonlin-r t tirret loads. Subjects of interest are firstly the.
2
marginal response probability density function and secondly the extreme value and range distributions using crossing rate statistics.
2.1
Closed
form
nonlinear
statistical
models
Analytical derivations of nonlinear statistical
quan-tities such as probability densities and crossing statistics are based on the. knowledge of the linear and nonlinear frequency response functions.
Un-der certain conditions, it is possible to Un-derive closed formulations describing the statistical properties of nonlinear processes that can be described using such a set of frequency response functions.
Kac and Siegert (1947) presented the inathemati-cal fundamentals to inathemati-calculate the probability den-sity function of the secord order problem, followed by Bedrosian and Rice (1971) nd Neal (1974). Use was made of a second order Volterra modelling of
the system's response.
For practical applications, the main attention should
be focused on the distribution of the peak
val-ues and of the distribution of response ranges.
Based on the same principles
as applied in the
second order theories, referred to above, Longuet-Higgins (1963), Longuet-Longuet-Higgins (1964) and Vinje and Skjordal (1975) both showed that for slightly quadratic processes, the distribution of the extreme values can be found using the joint probability density function of the variable itself and of its time derivative. The resulting expressions are presented in the form of perturbatión series expansions, called Edgeworth or Charlier series. Jensen and Feder-sen (1981) applied this method to predict extreme hull girder loads in ships. The fatigue damage in-flicted on a construction by non-Gaussian wave-induced sttesses Was analyzed by Jensen (1991). The results of these practical applications were
ob-tained on the assumption of, again, a weakly second order behaviour arid a narrow band response
The complication with the measured turret load responses is the significant contribution to the to-tal response of not only second order but also of
third order components. This phenomenon has bêen observed from model tests exemplifying the effects of chain dynamics on the low frequency motions (Huijsmans and Wichers (1992)) The higher than second order behaviour of the responses is in
con-flict vith the basic assumption made in the
an-alytical approaches above i.e. the occurrence of small second order non linearities. In another study (Adegeest (1995)) showed that neglecting the third order contribution, it was not possb1e to make ac-curate predictions of the nonlinear hull girder loads for flared hull shapes
DaIzell (1984) reported theoretical derivatiOns in-cluding comparisons with simulations foi the proba bility dénity fünctions of third order systems. The theory was based on an extension of Vinje and Skjordalîs method (1975). In a similar way use vas made of a third. order functIonal polynomial modelling or Volterra modelling in which the linear and nonlinear frequency response functions were as-suined to be available. Including third order nonlin-earities required the derivation of the joint densty of t he response and two derivatives instead of one. Yo legant way was found to approximate the max-hua and minima of the responses. In conditions with strong nonlinearities, sometimes negative probabil-irv densIties vere found which are obviously
unreal-istic results. For relatively weak nonlinerities, how-ever. good predictions could be made of the lower
uuid upper tails of the distributions.
2.2
Approximate moment-based
statisti-cal models
:\s ong as closed formulations for the statistical characteristics of third order responses do not give "ahst Ic and reliable answers in all the conditions. i Ir pId)babilistic quantities can be computed using
ai approximate statistical method. These
approxi-ita t r odds are based on the knowledge that that
the response. statistics are reasonably well described
by a limited number of spectral or statistical
re-sponse moments.
Wirsching and Light (1980) published a method to predict the fatigue damage under wi4e band
stresses. Based on spectral moments in combina-tion with the applicacombina-tion of the rain flow counting method, the fatigue damage was predicted.
Non-linear responses generally result in wider response spectra because of sub- and super-harmonic contri-butions. This by nonlinearities increased spectral broadness is accounted for by the larger higher or-der spectral moments, which are very sensitive es-pecially to super-harmonic components. \Virsching and Light presuted expressiors which only depeid on the spectral moments m0, m2 and m4.
Using Hermite moment formulations as initially pre-sented by Winterstein Ç1988), it is possible to cal-culate approximate probability density functions, crossing rates and extreme values based solely on knowledge of the statistical response moments. The basic idea of a Hermit e moment node1 is that a non-linear response y(t) is matched to a Gaussian pro-cess U(t) by an appropriate function g according to y(t) = g[U(t)J. The transforration function g is taken as a N-terms I-Termite series. The shàpe of the function is controlled by the statistical moments of the response. Winterstéin showed that uing.the first four moments. good results could be obtained for the probability density functions, spectral densi-ties, extremes and ranges for various nonlinear
pro-cesses.
2.3
Proposed method
Since the proper working of the proposed Volterra modelling was validated for hull girder loads in head waves (Adegeest 1994b). it's worthwhile to investi-gate the pp1hcability of the afòremetttiored approx-irnate ethóds to analyze the statistical propetties of the nonlinear hull girder load responses.
re-quired statistical moments in arbitrary conditions, i.e. the mean, variance, skewness and kurtosis in dif-ferent wave spectra. The Volterra modelling makes it possible to calculate these quantities directly in the frequency domain.
in summary, the following procedure is proposed to calculate the short-term statistics of nonlinear hull girder loads in arbitrary wave spectra:
Calculation of the set of first, second and third order frequency response functions using a non-linear time-domain program or from model test identification techniques Standsberg (1997). Definition of a realistic wave scatter diagram,
pec ra and operational profile.
Calculation of statistical and/or spectral mo-inents according to the ã.pproxinate higher
or-(1er Volterra modelling.
-I. Calculation of the marginal distributions,
cross-ing rates and other statistics uscross-ing
moment-based Hermite models.
Prior to fully developing the outlined method, par-ticulariv the development of the direct method to compute the statistical moments in the frequency domain, t he applicability of the Hermite models is
verified.
3
Momeit-based Hermite Models
lit lit eral lire, \Vinterstein's approach based on Her-unte polynomials is widely used and seems to have t he most consistePt theoretical background. In this sN:l ion a brief review is given of the theory on Her-.mmijte imiodels as described by \Vinterstein in. a
num-ber of papers (1987, 1988). The starting point of the derivatiofl is the assumption that tie marginal
dis-I 111)111 iùii of virtually any nonlinear response can.be miat :hed to a Gaussian process by applying an ap-l)ToPriite monotone function g. Polynomial approx-nieit tamis Io!J are defined based on a limited number
4
of statistical moments of the response. A distinction had to be made between softening responses, which are characterized by wider tails than the Gaussian distribution i.e. kurtosis a4 larger than 3, and hard-ening responses, which have narrower tails i.e. kur tosis a smaller than. 3.
3.1
Matching Hermite models with
sta-tistical moments
Given N response moments of a slightly nonlinear response y(t), the transformation to a standardized Gaussian repbnse U(t) is taken as an N-term Her-mite series (Winterstein (1987)):
= p0(t) = g(U(t))
= + cnHeni(U(t))]
(1)
where p and o are respectively the mean and Stan-dard deviation of y(t). After substitution of the first thtee Hermite polynomials by their equivalents in terms of powers of U, respectively Heo(U) = i
He1(U) = U, He2(U) = U2 - i and He3(U)
U3 - 3U, one gets
o=K[U+c3(U2-1)+c4(U3-3U)+...j
(2)The time variable t as been omitted for brevity. The shape of the staidardized distribution i
con-trolled by the coefficients c,1. the scaling factor ensures that has unit variance. It was derived by \Vintersteîn that for 1V = 4. the approximate analyt-ical solutiön for these coefficients can be expressed in terms of the central response moments, defined as
E[(t)1 where for n = .3 the skewness a3 and n 4 the kurtosis a4 are found. The results for c3 and c4 are found by matching the following equations
24 a3 6 O4 - 3 c3 = 6c3c4 c4 2c32 + 9c42
The scaling parameter i is given by
i
K
The first order fitting is foi.ind after neglecting all second order contributions in equation 3 to 5, re-sulting in i añd a4 - -3 24 (6) a3 (7) 6
After ignoring the term 2c in equation 4, an ap-proximate so1utio for c3 and. c4 was found, the so-called second order Wiïtersteiri model:
4 + 2./i + 1.5(û4 - 3)
Matching the Hermite series coefficients with the exact, numerically computed result, slightly
dii-frent
coefficients were found, by Mansòur aridJensen (1.994):
Torhaug (1996) reports a third set of coefficients which were derived by \Vinterstein t al in an un published report (1994): [1 + 1.25(a4 .3)]l/3 - i. C4 = -- 10
L-
c14-3 1.43a-., i-0 0.8 (1 2) a3fi -
0.0151a31 + 0.3a32-
6 L1+0.2(a4)
(13)This set of coefficients results -in distributions which ate very similar to Mansour's and Jensen's fit.
3.2
Crossing rates arid probability
densi-ties
The mean rate-at which the stationary resporse
yo(t) crosses level Y from below is defined as vy(y).
If the Hermite series is monotone, the crossing
statistics are defined -by
vy(y)=vyo(o)±voexP(_)
(14)where u0 is defined as max[vy(g); oc < y < which is a measure of the rate of response cycles or the 'average' response frequency. The normalized rate Liy(y)/vo approximates both the failure rate per response cycle (for extremes) and the probability thä.t a response peak exceeds level y in narrow-band
responses (for fatigue).
The tuargirial distribution of the nonlinear response i given by
py(y)o' = pv0(o) =
dg0
du
(1.5)
where (u) is the standard normal probability deny sitv function
(u)çcxP()
(16)v/1 +15(a - 1
(4 = 30 (11) 5.8 + 2v"l ± 1.5(a4 - 3)l'liese coefficieiits vere shown tó give better results
iii i lic tails of the ditribution and vill be used from
no\ on. /i + 1.5(c14 - 3) - i C4 18 a3 C3 = C4 C3
Since He,(u) nHe...1(u), it follows that du/do = I/(d?ja/du) equals
N -1
[K(1
+ >J(n -
1)cnllen.2(u))](7)
This equation should be larger than, O to satisfy the condition of monotony, or, if.N = 4:
The results for the crossing statistics and margin.l distribution depend explicitly on u. This requires that for a specified response level o the
standard-ized response g(u) has tò be inverted to tnd u(0) = g' (go).
Comparison
with
numerical
simulations
The ca1cu1ted irregular wave results of the moored FP.SO are used to verify the last step in the proposed method, i.e. the calculatiòn of probability densities and crossing rates using approximate moment-based Herinite modéls.
The degree of fitting of the different Hérmite models is verified by substituting the measured statistical niotnents into the expressions for the coefficients c3 and c4. Three different Hermite modelings defined in the previous chapter are considered:
The second order fitting defined by the coeffi-deitts given in equation S and 9.
The set of coefficients as derived by Mansour and Jensen, given in equatiOn 10 and 11. The First order coefficients derived by Winter-stein et al, given in equation 6 and 7.
Results are presented for the turret loads in head
seas. lite Qbtained statistical moments per condi-I jutis a r' list ed itt table i
6 Heading: Wave height: Peak period: 14kN] o [kÑ] Skewness Kurtosis
4.1
MarginaI probability densities
The resulting marginal distributions are plotted on linear scale see. figures 1, 2. It is clearly shown that the nonsyrnmetryin the probability densities is ac-curately predicted especially using Mansour's and Jensen's fit or Winterstein's second order fit. Down to the lowest measured probability levels the densi-ties are accurately approximated especially for the survival Wave con4ition as shown in figure 1.
Slightly less accurate results are obtainéd using the original first order fit, in particular in the tails of
the distribution.
5
Rainflow damage simulations
For sufficiently narrow bafided stress responses 'it is common to relate the growth in fatigue damageD(t) to ranges of the stress responsé y(t). Particularily Miner's rule assumes that a stress range R causes an incremental fatigue damage 5D = cR". The con-stants c and n depend on material and construction details.For narrow-band Gaussian processes, the expected damage is equal to
E[!.D1Lirtear = E[CR''Linear
= (19)
Head seas Head seas JI. 16.0 m H5 = 5.6 m T0=13.Os
T0=8.Os
-2350'
-537 3040 680 1.027 0.496 3.929 3.392Table 1: Measured statistical moments of the turret < 3c(1 - 3c4) (18) load on .a moored FPSO
du
using the. Hermite model formulation, a correction factor -y can be. derived which accounts för band-width and nonlinear effects such that the expected. damage for the nonlinear systems becomes
'1 D'1
-
.i..1CLt. jNonlineer= 7E[CR]
Linear (20) with (Winterstein (1987)): -y= (V1_E2k)fl[1+n(n_1){
c4(1 - E2)± 2E2c +((1 _2e2)22±5)±9E4)
]] (21) in which z is the response band width. Thecorrec-i correc-ion facto fór a narrow band process satisfying the first order \Vinterstein model follows after substitu-tion of z 0, 1 and for c4 and c3 the first order
results ¿.s given in equations 6 and 7, hence
-y = i + n(n 1)c4 (22) Normally, n is between 2 and 5. Since we only want to illustrate the effects of nnlinèar contrIbutions in
fatigue calculations, we consider n to be constant and ecival 3 and c equal 1. Using this values, the correction factor for gamma becomes
-y=1+6c4=1+
- 3
(23)The estimation of the factor obtained from the rainflow calculations are displayed in the next table
2.
6
Concluding remarks and
rec-ommendations
..\ wet hod was presented to perform statistical anal-ys('s of nonlinear hull girder loads. The method is
E[D]
NonlinearTable 2: Measured and approximated correction fac-tors fatigue life
based on the calculation of the first four statisti-cal moments of the nonlinear response in combina-tion with an approximate moment based statistical model to calculate the probability densities, crossing rates.
The mean, standard deviation, skewness and kurto-sis obtained from simulations are used as input to three different approximate moment-based Hermite
models. The resulting probability densities have been compared with the sirnuated equialents \'er good cörtiparisotis were found for the urret loads on the FPSO. Also the effect of nofl-linearity on the fatigue life was considered A.s vas shon for an ar tificial SN curve, the effect of nonlinearit on the fatiguge life was demonstrated to be quite signifi= cant.
The approximate higher order Volterra. series expan-sion is a good candidate for the. further development
of the iderttification technique as put forward by Standsberg (1997)
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Technol-Heading: - Head séas Héad seas
Wave height: 16.0 m 5.6 m Peak period: 13.0 s 8;0 S Skivness 1.027 - O.496 Ku rtosis 3.929 3.392 approx. eq.(21) 1.232 1.082 y rainfiow 1.32 1.003
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-3
1(.tctt
ri-haI,Iit iit. otwrr Iöids t ¿, ni
itisrc min fr.,rn SiniuIatiOnS
Henrut .\ppro. 2nJ crdrWiecs'in
t Appras.. i rst otder Winerstein
M&czist,ur .ter.scit
Figure I: Probability density funçtions of the turret load for an moored FPSO in head waves of 16.0 rn
sign. waVe height
Tr.-hthditv rstr rttirr icdc S n ni
j' ',d
-n5,.
-
t tiz,,.,rzj'i Sir.ut,tic,nt tr?n; r- .-\pprQs.. 2 nJ .rdr W irt- n
- - - t ..,i r t ri rd er \ in,.r.t I' - - - ?¼1çri 5t,LII .Ii:ri-cn
'igure 2: Probabilit density functions of the turret load for ari moored FPSO in head waves of .6 in sign. wave height
