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Extreme and fatigue response of offshore platforms due to three-dimensional stochastic wave fields
Extreme and fatigue response of offshore platforms due to three-dimensional stochastic wave fields Ragnar Sigbj6rnsson*
Engineering Research Institute, University of Iceland, R ey kjavzTq Iceland (Received September 1980)
This paper deals with stochastic aspects of wave induced vibrations of fixed offshore platforms with view towards probabilistic design. This includes stochastic treatment of ocean waves, wave loading and structural response processes, emphasizing predictions of extremes and fatigue. The load effects of short crested wave fields are exemplified.
Introduction The rapidly expanding off industry offshore requires structures for operations in deep waters and hostile environments. Under these conditions the structures are exposed to the action of random natural forces including wind, waves, current and even strong motion earthquakes. To achieve optimal structural safety and serviceability it is of fundamental importance to predict these loads and their load effects realistically. Such predictions call for rational and reliable methods of analys~s. 1-7 The purpose of this paper is to discuss some stochastic aspects of wave induced vibrations of fixed offshore structures with a view towards probabilistic design. The basic principles are, however, general and can be applied to other types of structures. 8 O c e a n w a v e s as a s t o c h a s t i c p r o c e s s
Long-term model The most characteristic feature of ocean waves is randomness in time and space. Hence, the most suitable mathematical description of their behaviour and effects is furnished by the theory of stochastic processes. Heuristically it seems necessary to apply non-stationary processes to be able to account for their intrinsic long-term variability. The development of a non-stationary model requires, strictly speaking, a finite ensemble of sample functions describing the same time sequence. Such an ensemble is unfortunately generally not obtainable in the case of geophysical processes. This difficulty may, however, be overcome by recognizing that the dominating long-term fluctuations in many cases, for instance in the North Atlantic * Previously at SINTEF, the Foundation of Scientific and Industrial Research at the Norwegian Institute of Technology, Trondheim. 0141-0296/81/040210-06/$02.00 © 1981 IPC Business Press
Ocean and the North Sea, are primarily governed by movements of whole weather systems with a characteristic period which is much longer than the average wave period in question. A suitable engineering model may then be obtained from a special class of non-stationary processes characterized by evolutionary spectral densities. 9 The evolutionary spectral density is a time dependent spectral density changing continuously with time. The advantage of this model is, that it is possible to estimate the spectral properties of the wave process from a single sample function by time-averaging over short segments provided that the spectral density evolves sufficiently slowly with time. This approach yields, of course, time dependent spectral moments and hence time dependent significant wave height, expected zero-crossing wave period (or an alternative statistical measure of wave periods) and average wave direction. The most common standard wave spectral densities can be specified completely in terms of these parameters. 1° The essentials of the long-term model of ocean waves are therefore the joint statistical description of these parameters. The structural response is sensitive to the spectral composition of ocean waves. This is especially the case for fixed offshore structures subjected to high frequency wave action. It is therefore important to establish an upper boundary of the evolutionary wave spectral density. Intuitively, the existence of such a boundary seems plausible due to limited fetch length and breaking of individual waves. This boundary is given by the envelope of wave spectra put forward by BretschneiderA 1 Together with proper standard wave spectral densities the envelope of wave spectra also yields a lower boundary of the expected zero-crossing wave period Tz and an upper boundary of the significant wave height/t s. This is of great importance for
Eng. Struct., 1981, Vol. 3, October
219
Response of offshore p/atforms: R. Sigbjornsson fixed offshore platforms as sea states described by small Tz and large/1s normally induce relatively large structural response. These boundaries should be accounted for properly in the estimation of the joint probability density of/1 s and ;rz-
Here, n is the normal to the area element dA pointing positive outwards. This equation may be rewritten as follows:
q(P)(t) = q(O(t) -- ( m ( h ) ( 6 0 ) e it~t dZ~~(60)
Short-term model The aforementioned evolutionary wave model may be approximated by a stationary (ergodic) process within a sufficiently short time interval (Priestly, 9 p. 212). For these short-term sea states the notion of generalized harmonic analysis of zero mean ergodic processes applies. This leaves the following expression for the wave sea surface:
~(X, t) = ;
e i(t°t-r'x) d(iZ~(K, 60))
(1)
Here, x = {x, y } 7"denotes the spatial coordinates; K is the two-dimensional wave number vector, t is the time; 60 is the circular frequency; and Z n is the spectral process associated with 77. Further discussion of this model is given elsewhere.12,13
Wave loading processes Following the notion of the preceding section the wave loading may be approximated by a stationary (ergodic) stochastic process within a sufficiently short interval in time and space. In the case of large volume structures this stochastic process may be derived by applying the linearized probabilistic potential theory. This approach assumes the existence of a stochastic velocity potential qb(x, t) which satisfies the Laplace equation with appropriate linearized boundary conditions. By applying the principle of superposition this potential is assumed given as the sum of the velocity potentials of, respectively, the incident waves and the waves diffracted by a restrained structure and the waves generated by the various modes of structural vibrations. 14 It follows that each of these potentials must satisfy the Laplace equation. The velocity potential of incident waves can be expressed as follows applying a Cartesian coordinate system placed on the sea floor with the z-axis pointing upwards:
-- f c(h)((.O)e it°t dZ~: (60)
(4)
where qU) is the inertia force due to incident and diffracted waves; 13 In (h) and c(h) are the hydrodynamic mass and the
radiational damping due to wave generation by the vibrating structures 14 and Z~ and Zr" are the spectral processes associated with the acceleration and the velocity of the structure. The potential theory is not adequate for slender bodies due to viscous effects like drag forces and lift forces. As these forces are nonlinear, a reliable prediction is difficult. It seems unlikely that lift forces should yield a significant contribution to estimates of the total loading on fixed offshore platforms. This is, however, not necessarily the case for local load effects. They are therefore neglected in the present study. The drag forces per unit length are commonly assumed given by the following expression:
q(a)(t) = }pCDDIft - i l(t~ - i)
(5)
Here, Co is the drag force coefficient; ti is the water particle velocity; and i is the structural velocity. This equation relates the drag forces to the relative velocity, which seems reasonable as long as the structural member in question is sufficiently slender in the hydrodynamic sense. On the other hand, if the structural member is not slender, the drag forces are of minor importance and equation (5) may therefore still apply with reasonable accuracy. Due to the fact that equation (5) is an empirical approximation rather than a physical law, it is important in the analysis to apply a Co value derived from measurements using an estimator consistent with the method of analysis. Thus, if the stochastic linearization method is used in the analysis, it should be preferred in the estimation of Co (see Borgman I for detailed discussion).
oo
• (W)(x, t) = f
Response processes
g cosh{Kz} ei(tot_r.x ) dZn(/~, 60) 60 cosh{rd}
-=
(2)
Here,g is the acceleration of gravity; and d is the water depth. The water particle velocity and acceleration in the undisturbed stochastic wave field are readily obtained from equation (2) by differentiation.12 The remaining stochastic velocity potentials can be expressed formaUy applying equations similar to equation (2). In this case it is generally required to use numerical methods to calculate the transfer functions, is When all the velocity potentials have been determined, the stochastic wave loading process may be derived by applying the Bernoulli equation. This yields, disregarding second-order terms: 0cb f (3) q(P) = - P 3t n dA A
220
Eng. Struct., 1981, Vol. 3, October
Analysis It is assumed that the structure can be modelled as an assembly of finite (beam) elements applying boundary spring-dashpot elements to idealize the soft-foundation system. These spring-dashpot elements will generally be both nonlinear and time dependent. 16 This implies that the linearized equation of motion may be expressed as follows by introducing the Dirac delta function in the time space for frequency independent system parameters:
M(t - r) J:(r) dr +
C(t - r) i(r) dr
+ f K(t - r) r(r) dr = Q(t)
(6)
Response of offshore platforms: R. Sigbj6rnsson Here, r is the nodal response process; M is the virtual mass of the system including the hydrodynamic mass; C is the virtual damping including hydrodynamic effects; K is the stiffness matrix; and Q(t) is the wave loading process, which strictly speaking is a function of k (see equation (5)). This Coupling is assumed to have been removed in the following by applying the stochastic linearization method including Gaussianization of the excitation as well as the response. Other nonlinearities are treated in the same way. Hence, a drag damping proportional to the rms relative velocity is introduced into the C matrix, i.e. ~pCD(E[(u- f)2] 8fir) 1/2. In the solution of this equation the notion of the preceding sections is carried a step further by assuming the evolution of the sea state is slow enough to neglect the transient structural response. Then, the long-term probability density of the response can be expressed as:
pRr)=f f
(7)
Here, p~s~zg denotes the long-term joint probability density of significant wave height and expected zerocrossing wave period and average wave propagation; Pr IPsl"z~ is the conditional probability density of the response derived applying the statistics of the response process as approximated by: oo
r(t) = f ei¢°t dZr(w )
(8)
--oo
where Z r is the spectral process o f r possessing orthegonal increments. It follows that the response process has zero mean and spectral density given as: Srr(¢o) = H(w)
SQQ(co) H T*(w)
(9)
where: //(co) = [K(co)
-
co:M(w) + icoC(co)l-I
(10)
is the virtual frequency response function of the system; and S@Q is the spectral density of the wave excitation. By this approach the system given in equation (6) is in the general case approximated by a hierarchy of linear systems due to the fact that H(¢o) is a function of sea state severity. For each of these linear systems the response is approximated by an equivalent Gaussian process. Hence, the response statistics are completely defined in terms of equations (7), (9) and (10). It is worth pointing out that this direct frequency response method does not put any restrictions on the damping matrix, and the frequency dependent system matrices cause no extra problems. Further, it is generally most convenient to use complex arithmetic in the evaluation of//(co) (further discussion may be found elsewhere 17). The spectral density of wave excitation may be expressed as follows in terms of the one-dimensional wave spectral density:
So0(w ) = F(co) Snn(¢o)
In some cases it is desirable to carry out synthesis of the response process, i.e. to generate a time-space sample function from given response statistics. In the case of stationary multivariate Gaussian response processes this may be done effectively by Monte Carlo simulation, for instance combined with (a) direct time space filtering, 18 (b) spectral factorization combined with harmonic decomposition, 19 and (c) autoregressive moving-average time series model. 2° The factorization of the response spectral densities combined with harmonic decomposition yields:
rm(t) = Re 2 ~
m
N
E
~
1=1
Brnl(Wk)
k=l
× exp {i(eot + q~tk)}
(12)
Here, A~ is the frequency resolution;N = max(eo)/Aeo; ¢ is a random phase angle uniformly distributed between 0 and 2n;and:
fPrlFs~zg(rlh,t,v)PFs~zg(h,t,v)
HsTzO x dv dt dh
Synthesis
(11)
where F(¢o) is a Hermitian matrix, the so-called hydrodynamic transfer function, which in the general case depends on the sea state severity. Further discussion of this equation is given by Sigbj6rnsson) 2,13
B(6o) BT*(~) =Srr(¢O)
(13)
where B is a triangular matrix which may be determined by a Cholesky factorization. This procedure can be made very effective by applying the fast Fourier transformation algorithm. However, it is a drawback that the series will be periodic with the period 21r/.A¢o. Therefore, a limitation on the length of the time series is set by a finite computer memory. It is also worth pointing out that this method yields no random variation ~n the sample spectral densities. These problems may be avoided at least partly by applying a related method, 21 which, however, will be more time consuming. An attractive alternative to this method as well as the direct time-space filtering is in some cases furnished by the autogressive moving-average time series. In this case a limitation in computer memory is generally no problem and the periodicity of the sample function follows the periodicity of the pseudo-random number generator. The estimation of the model parameters may, however, be rather tedious. The method is therefore most tractable to generate long sample functions.
On the effects of short-crestedness of waves To exemplify the effects of the short-crestedness of the wave field on the structural response, a gravity platform of typical design has been selected. A sketch of the platform is shown in Figure 1 indicating the finite element model applied. The foundation-soil system is modelled by frequency dependent spring-dashpot elements using the elastic half space model. It can be argued that this model yields somewhat too high radiational damping in the case of layered soil. Therefore, the material (hysteretic) damping in the soil is neglected. The wave loading on the towers is obtained by applying the strip approach and neglecting interaction effects. The importance of mutual interaction can be evaluated by applying results given by Chakrabarti. za The drag forces are found to be almost insignificant in the whole frequency range in question. The wave forces on the caisson are based on potential theory and lumped as moment and shear force at mudline. More detailed discussion on the modelling including numerical results may be found in Sigbj6rnsson et al) 2,13,17 Figure 2 displays the statistics of bending moment about they-axis at the top of tower A. The wave field is specified in terms of the parameterized JONSWAP wave
Eng. Struct., 1981, Vol. 3, October
221
Response of offshore platforms: R. Sigbjornsson distribution for maxima. The joint distribution of significant wave height and expected zero-crossing period is derived applying wave data from the weather ship Famita. 24 The effects of short-crestedness are especially pronounced for the probability density. The strange shape of the peak of the probability density should also be noted. This is due to the resonance effects, which by increasing the response virtually move a part of the density to higher response levels. This study indicates that the effects of short-crestedness are generally most significant for small and moderate response levels.
,j" I
I
.•
i
:
f ~ ~.4-t
Applications in probabilistic design Extremes
a Figure I
10 -
b (a), sketch o f platform. (b), finite element idealization
~12 -
8t17~
o_
I~
Expected
The basic problem in structural design concerns first excursion failures and requires traditionally information concernin the characteristic extreme response and in modern approaches even the extreme value distribution. 25 An estimate of the 'most probable' largest maximum during the time interval T may be obtained from the longterm distribution of maxima as: Pr [r > ~] = (vT) -~
/
z e r o - crossing
(14)
Here, v denotes the expected zero-crossing frequency given as follows: ,
"=2-gf f J4 E-p] p°se= (h't'v)dvdt
£
1D
Ha r s o
gt~ o 5 L
O-
O-
0
4
I I I 8 12 14 Expected zero- crossing wave p e r i o d , ( s )
applying the notion of the slowly evolving response. The reliability function given in Figure 4 is based on this approach and derived from the results given in Figure 3. It is seen that the short-crestedness of the wave field influences the extreme response only moderately, which is in accordance with earlier findings, s An extreme value distribution for stationary processes put forward by Ditlevsen 26 is:
Figure 2
Bending m o m e n t at top o f t o w e r A as function of expected zero-crossing wave period. Significant wave height equal to 3 . 2 5 m; ( - A - ) based on idealized long crested waves; ( - o - ) based on short crested waves described by spreading function (2/1r) cos 2 O
spectral density 23 and the cos n 0 spreading function. The figure shows both results for idealized long crested waves (n -- o0) and short crested waves (n = 2) with the average direction of wave propagation in the direction of the x-axis. The statistical response quantities (a definition is given in Sigbj6rnsson et al. 17, p. 270) are expressed as functions of the expected zero-crossing wave period for a given significant wave height (defined as 4(E [I'~2]1/2). It is seen that the reduction in the rms moment due to short-crestedness is greatest close to resonance (fundamental natural frequency is 1.31 rad/s) but decreases with increasing low frequency quasi-static response, while the expected zero-(up) crossing period of the moment shows the opposite effect. The rms bandwidth of the moment shows only a small increase due to short-crestedness. The increase is, however, barely significant in an intermediate range with approximately equal contributions of resonance and quasi-static response. Figure 3 shows the long-term distribution of maxima of moment at the top of tower A induced by a unidirectional evolutionary sea state. This distribution is based on the Gaussianization of the response process applying the Rice
222
Eng. Struct., 1981, Vol. 3, October
(15)
F(~, T) ~- F(~, O) e -v~ T/F(~, O)
(1 6)
where v~ denotes the expected number of upcrossings per unit time at level ~. The validity of this distribution has been tested in an extensive simulation study by applying a variety of rather broad banded processes including nonGaussian processes. 26 Heuristically, it seems possible to generalize equation (14) to the semi-stationary processes discussed by Priestly. 9 Then, v~ is given by an expression similar to equation (15). 5
~
4
08
3
06
2
04
I
-2
10
a
/j~
0
,.~
,
,
,
-2 0 2 4 6 Moment (10 MNm) Figure 3 Long-term distribution of maxima o f moment at top of tower A. (a), p r o b a b i l i t y densities; (b), p r o b a b i l i t y distribution; (-z~-) based on idealized long crested waves; ( - o - ) based on short crested waves described by spreading f u n c t i o n (2/7r) cos20
8
Response of offshore platforms: R. Sigbj6rnsson
ooo
/
Then a simple estimate of the fatigue life of the structure is given as:
/
noo
//
f =
a
(19)
/_[l)or(Or) om do r /0
~i 6 c o
i
(3
I
I
0
I
1 Moment
I
2 (IOaMNm)
I
I
3
I
4
Figure 4 Long-term probability distribution of maxima of moment at top of tower A: (-A-), based on idealized long crested waves; (-o-), based on short crested waves described by spreading function (2H) cos20
Fatigue The continuous wave action subjects the structures to an extremely large number of oscillations. It is therefore necessary to consider fatigue in the structural design. A variety of theories has been proposed to account for fatigue under complex stressing, but none has apparently received universal acceptance. In the present study the concept of the maximum principal stress as a fatigue governing parameter is applied. At least for in-phase stresses this concept seems to give reasonable results. 27 Unfortunately, most structural elements in an offshore structure virtually experience multiaxial out-of-phase stochastic stressing. To facilitate a proper counting of stress cycles a generation of sample functions of stresses is desirable. The first step is the computation of the spectral densities of hot spot stresses, which generally requires refined f'mite element analysis. Secondly, time series of hot spot stresses are simulated using equations (12) and (13). Finally, the number of stress cycles is counted using the so-called NLR method (see Appendix), 2s which is a rain-flow type procedure. The long-term distribution of fatigue stress cycles per unit time may then be estimated by applying the concept of slowly varying evolutionary processes. This yields: 29 /g°r(Or)=
f
This estimate may serve as a useful measure in design decision. Figure 5 shows the long-term distribution of stress cycles and maxima in a given 'hot spot' at the deck/tower connection of the example platform subjected to unidirectional long-crested waves described by the parameterized JONSWAP wave spectrum. 29 Fortunately, in this case it is found that the principal axis corresponding to maxima of principal stresses are almost fixed in space. The distribution of partial damage is obtained applying the standards of the Norwegian Petroleum Directorate (Curve No. D). This yields, applying the Palmgren-Miner rule, an estimate of the fatigue life equal to four years. In the cases of shortcrested waves this estimate is roughly twice as high. The distribution of stress derived in this way may be applied directly in a more refined probabilistic analysis 3° or even in fracture mechanics approaches.
Final remarks A consistent stochastic analysis of waves and wave load effects is presented. The potential of the analysis is demonstrated by numerical examples. Uncertainties in the model of the wave field are discussed emphasizing the effects of short-crestedness on the structural response. It is found that the response induced by light and moderate sea states may be quite inaccurate, implying a serious drawback for the fatigue analysis. The accuracy of the extreme response appears on the other hand to be satisfactory, al This may, however, not be so surprising bearing in mind that the wave climate on the Norwegian Continental Shelf is governed by very unstable wind fields concerning homogeneity, direction and fetch. Therefore, more research is needed before wave induced response can be predicted with the desired accuracy. Acknowledgements The results reported here were obtained partly under a project sponsored by the Royal Norwegian Council for Scientific and Industrial Research (NTNF).
f flS°rllTs~z~(°rlh't'V)PlTs~zo(h't'v)
gs Tz ~ x dh dt dv
_E6
(17)
where, #arlITs¢zOrefers to the conditional distribution of fatigue stress cycles induced by a given short-term sea state. The most common estimates of fatigue life are based on the Palmgren-Miner hypothesis by assuming linear accumulation of damage. Then the accumulated damage per unit time may be expressed formally as follows: o~
= a-, f Pot(Or)o2 dot
= = =
u 5 E~ 2_~ o
1 I ~MQXImQ
4 u
I '
I i
Stress
cycles
E ~
b o ~
/Partial
2
r-]
E
(18) h
(l) 1 ~. i.. ~
~ -
r-~/' r-~ I I
, WIl i
I I
I
-2Q
0
20
40
I
,, i Ir -k~ t - ,! I I
I I
O
Here, the S-N curves are assumed specified in terms of a and m. According to the Palmgren-Miner hypothesis failure will occur when the accumulated damage is equal to unity.
damage
oI
io~ -' - w - i I ~--] ~-t I I i ~ I
'
:
~ : ', ~-I
60 80 100 Stress level, ( MPQ )
Figure 5 Long-term distribution of
r''l
!--'_ 120
140
160
stress cycles and maxima and
partial damage
Eng. S t r u c t . , 1981, V o l . 3, O c t o b e r
223
Response of offshore platforms: R. Sigbjornsson The co-operation o f Senior Structural Engineers S.-E. Jensen and E. T. Moe at A. S. Bergens Mekaniske Verksteder on fatigue problems is greatly acknowledged. Further, the author is indebted to Assistant Prof. K. Syvertsen and Research Assistant N. Spid6e at the Norwegian Institute o f Technology for assistance in preparing some of the numerical results reported here.
References 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15
16 17
18 19 20 21 22 23
24
224
Borgman, L'. E. 'Statistical models for ocean waves and wave forces',Advances in Hydroscience 1972, 8,139. Academic Press Nath, J. H. and Harleman, D. F. R. 'Dynamics of fixed towers in deep-water random waves', J. Waterways Harbors Div., ASCE 1969, 95 (WW4), 373 Forster, E. T. 'Model for nonlinear dynamics of offshore towers',J. Eng. Mech. Div., ASCE 1970, 86 (EM1), 41 Malhotra, A. K. and Penzien, J. 'Nondeterministic analysis of offshore structures', J. Eng. Mech. Div., ASCE 1970, 86 (EM6) 895 Berge, B. and Penzien, P. Offshore Technology Conference 1974, Houston Eatock Taylor, R. 'Structural dynamics of offshore platforms', Offshore Struct. Conf., London 1975, 125-132 Moan, T. and Sigbj6rnsson, R. 'Stochastic sea load effect analysis for probabilistic design of fixed offshore structures', 2nd Int. Conf. Struct. Safety Reliability, Munich, 1977, 227-246 Langen, 1. and Sigbj6rnsson, R. 'Stochastic dynamics of floating bridges',Eng. Struct. 1980, 2, 123 Priestly, M. B., 'Evolutionary spectra and non-stationary processes',J. Roy. Stat. Soc., 1965, 27, B204 Price, W. G. and Bishop, R. E. D. 'Probabilistic theory of ship dynamics', Chapman and Hall, London, 1974 Bretschneider, C. L. 'The envelope wave spectrum', Third Int. Conf. Port Ocean Eng. under Arctic Conditions, POAC-75, Fairbanks, USA, 1975 Sigbj6rnsson, R. 'Stochastic theory of wave loading processes',Eng. Struct. 1979, 1,58 Sigbj6rnsson, R. and Smith, E. K. 'Wave induced vibrations of gravity platforms: a stochastic theory', Appl. Math. Modelling, 1980, 4,155 Faltinsen, O. and Michelsen, F. C. 'Motion of large structures in waves at zero Froude number', lnt. Symp. dyn. marine vehicles struct, in waves, London 1974 Garrison, C. J. 'Hydrodynamic loading of large offshore structures: three-dimensional source distribution methods', in 'Numerical methods in offshore engineering' (ed. Zienkiewicz, O. C. et al.), John Wiley, Chichester, UK, 1978 Sigbj6rnsson, R. and Spidsfe, N. 'Long term response of gravity platforms with load dependent soil behaviour', Norwegian Maritime Research, 1980, 8 (3), 20 Sigbj6rnsson, R. et al. 'Dynamic response of framed and gravity structures', in 'Numerical methods in offshore engineering' (ed. Zienkiewicz, O. C. et al. ), John Wiley, Chiehester, UK, 1978 Borgman, L. E. 'Ocean wave simulation for engineering design', J. Waterways Harbors Div., ASCE 1969, 95 (WW4), 556 Shinozuka, M. 'Monte Carlo solution of structural dynamics', Cornput. Struct. 1972, 2,855 Gersch, W. and Luo, S. 'Discrete time series synthesis of randomly excited structural response', J. Acoust. Soc. Amer. 1972,51,402 Shinozuka, M. 'Simulation of multivariate and multi-dimensional random processes', J. Acoust. Soc. Amer. 1971,49 (1), 357 Chakrabarti, S. K. 'Wave forces on multiple vertical cylinders', J. Waterway, Port, Coastal Ocean Div., ASCE 1978, 104, WW2, 147 Houmb, O. G. and Overvik, T. 'Parameterization of wave spectra and long term joint distribution of wave height and period', First Int. Conf. Behaviour Offshore Struct., BOSS-76, Trondheim 1976 Sigbj6rnsson, R. et al. 'Estimation of the joint distribution of significant wave height and average wave period', SINTEF Report STF71 A76041, Trondheim 1976
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25 26 27 28 29
30 31
Flint, A. R. and Baker, M. J. 'Safety approach for structures subjected to stochastic loads', Int. Syrup. Integrity Offshore Struct., Glasgow 1978 Ditlevsen, O. 'Extremes and first passage times' (doctoral dissertation), Copenhagen 1971 Toor, P. M. 'A unified engineering approach to the prediction of multiaxial fatigue fracture of aircraft structures', Fract. Mech. 1975, 7,731 De Jonge, J. B. 'The monitoring of fatigue loads', NLR Report MP 700101, 1970 Sigbj6rnsson, R. and Syvertsen, K. 'Fatigue analysis of offshore structures', in 'Safety of structures under dynamic loading' (ed. Holand et al.), Tapir Publishers, Trondheim 1978 Wirshing, P. H. and Haugen, E. B. 'Probabilistic design for random fatigue loads', J. Eng. Mech. Div., ASCE 1973, 99 (EM6), 1165 Spids6e, N. and Sigbj6rnsson, R. 'On the reliability of standard wave spectra in structural response analysis', Eng. Struct. 1980, 2,123
Appendix
Counting of fatigue stress cycles The counting o f fatigue stress cycles is based on series consisting of extremes o f the maximum principal stresses o~ derived for each short-term sea state. The counting procedure applied is the so-called N L R method which is a rainflow type approach operating in two phases. In the first phase all intermediate stress cycles are detected. Starting with m = 1, four successive extremes are considered. If the following conditions are fulfilled: (ol(m + 3 ) > ~ o l ( m ) ) A ( o l ( m + 1) <~ o l ( m + 3 ) ) A (ol(m + 2) ~> o,(m))
( o l ( m + 3) < o , ( m ) ) A ( o l ( m + 1) ) o,(rn + 3))
(A1)
A (o,(m + 2) ~< cq (m)) a count is made o f one fatigue stress cycle defined by the amplitude: Or = ½loi(m + 2) - o l ( m +1)1
(A2)
and the mean value: Oa
=
~ l o l ( m + 2) + ol(m + 1)1
(A3)
and the extremes ol(m + 1) and o2(m + 2) are deleted from the series. The procedure is then repeated for the next four extremes ol(m), ol(m + 3), o l ( m + 4), ol(m + 5). If the condition (A1) is not met, the value of m is simply increased by one and the procedure is then repeated. When the whole series has been analysed in this way, a residue series remains which contains no intermediate stress cycles and consists simply of a sequence of cycles with increasing amplitudes followed by a sequence o f cycles with decreasing amplitudes, etc. In the second phase, the residue series is analysed by applying a simple range count procedure. That is, a count is made o f successive half stress cycles defined by the following amplitude and mean value: or = ~lo,(n + 1 ) - o , ( n ) l
(A4)
= 2tla,(n +1) + o,(n)[
(AS)
Oa
These counts are then added to the results of the first phase. The counting results are finally presented in one dimensional form for each or in terms of the number (distribution) o f fatigue stress cycles P(or) per unit time, the average mean stress and the root mean square mean stress. This should be seen in the light of the fact that the codified fatigue damage is primarily defined by the stress range and to a lesser extent b y the mean stress.
Engineering Research Institute, University of Iceland, R ey kjavzTq Iceland (Received September 1980)
This paper deals with stochastic aspects of wave induced vibrations of fixed offshore platforms with view towards probabilistic design. This includes stochastic treatment of ocean waves, wave loading and structural response processes, emphasizing predictions of extremes and fatigue. The load effects of short crested wave fields are exemplified.
Introduction The rapidly expanding off industry offshore requires structures for operations in deep waters and hostile environments. Under these conditions the structures are exposed to the action of random natural forces including wind, waves, current and even strong motion earthquakes. To achieve optimal structural safety and serviceability it is of fundamental importance to predict these loads and their load effects realistically. Such predictions call for rational and reliable methods of analys~s. 1-7 The purpose of this paper is to discuss some stochastic aspects of wave induced vibrations of fixed offshore structures with a view towards probabilistic design. The basic principles are, however, general and can be applied to other types of structures. 8 O c e a n w a v e s as a s t o c h a s t i c p r o c e s s
Long-term model The most characteristic feature of ocean waves is randomness in time and space. Hence, the most suitable mathematical description of their behaviour and effects is furnished by the theory of stochastic processes. Heuristically it seems necessary to apply non-stationary processes to be able to account for their intrinsic long-term variability. The development of a non-stationary model requires, strictly speaking, a finite ensemble of sample functions describing the same time sequence. Such an ensemble is unfortunately generally not obtainable in the case of geophysical processes. This difficulty may, however, be overcome by recognizing that the dominating long-term fluctuations in many cases, for instance in the North Atlantic * Previously at SINTEF, the Foundation of Scientific and Industrial Research at the Norwegian Institute of Technology, Trondheim. 0141-0296/81/040210-06/$02.00 © 1981 IPC Business Press
Ocean and the North Sea, are primarily governed by movements of whole weather systems with a characteristic period which is much longer than the average wave period in question. A suitable engineering model may then be obtained from a special class of non-stationary processes characterized by evolutionary spectral densities. 9 The evolutionary spectral density is a time dependent spectral density changing continuously with time. The advantage of this model is, that it is possible to estimate the spectral properties of the wave process from a single sample function by time-averaging over short segments provided that the spectral density evolves sufficiently slowly with time. This approach yields, of course, time dependent spectral moments and hence time dependent significant wave height, expected zero-crossing wave period (or an alternative statistical measure of wave periods) and average wave direction. The most common standard wave spectral densities can be specified completely in terms of these parameters. 1° The essentials of the long-term model of ocean waves are therefore the joint statistical description of these parameters. The structural response is sensitive to the spectral composition of ocean waves. This is especially the case for fixed offshore structures subjected to high frequency wave action. It is therefore important to establish an upper boundary of the evolutionary wave spectral density. Intuitively, the existence of such a boundary seems plausible due to limited fetch length and breaking of individual waves. This boundary is given by the envelope of wave spectra put forward by BretschneiderA 1 Together with proper standard wave spectral densities the envelope of wave spectra also yields a lower boundary of the expected zero-crossing wave period Tz and an upper boundary of the significant wave height/t s. This is of great importance for
Eng. Struct., 1981, Vol. 3, October
219
Response of offshore p/atforms: R. Sigbjornsson fixed offshore platforms as sea states described by small Tz and large/1s normally induce relatively large structural response. These boundaries should be accounted for properly in the estimation of the joint probability density of/1 s and ;rz-
Here, n is the normal to the area element dA pointing positive outwards. This equation may be rewritten as follows:
q(P)(t) = q(O(t) -- ( m ( h ) ( 6 0 ) e it~t dZ~~(60)
Short-term model The aforementioned evolutionary wave model may be approximated by a stationary (ergodic) process within a sufficiently short time interval (Priestly, 9 p. 212). For these short-term sea states the notion of generalized harmonic analysis of zero mean ergodic processes applies. This leaves the following expression for the wave sea surface:
~(X, t) = ;
e i(t°t-r'x) d(iZ~(K, 60))
(1)
Here, x = {x, y } 7"denotes the spatial coordinates; K is the two-dimensional wave number vector, t is the time; 60 is the circular frequency; and Z n is the spectral process associated with 77. Further discussion of this model is given elsewhere.12,13
Wave loading processes Following the notion of the preceding section the wave loading may be approximated by a stationary (ergodic) stochastic process within a sufficiently short interval in time and space. In the case of large volume structures this stochastic process may be derived by applying the linearized probabilistic potential theory. This approach assumes the existence of a stochastic velocity potential qb(x, t) which satisfies the Laplace equation with appropriate linearized boundary conditions. By applying the principle of superposition this potential is assumed given as the sum of the velocity potentials of, respectively, the incident waves and the waves diffracted by a restrained structure and the waves generated by the various modes of structural vibrations. 14 It follows that each of these potentials must satisfy the Laplace equation. The velocity potential of incident waves can be expressed as follows applying a Cartesian coordinate system placed on the sea floor with the z-axis pointing upwards:
-- f c(h)((.O)e it°t dZ~: (60)
(4)
where qU) is the inertia force due to incident and diffracted waves; 13 In (h) and c(h) are the hydrodynamic mass and the
radiational damping due to wave generation by the vibrating structures 14 and Z~ and Zr" are the spectral processes associated with the acceleration and the velocity of the structure. The potential theory is not adequate for slender bodies due to viscous effects like drag forces and lift forces. As these forces are nonlinear, a reliable prediction is difficult. It seems unlikely that lift forces should yield a significant contribution to estimates of the total loading on fixed offshore platforms. This is, however, not necessarily the case for local load effects. They are therefore neglected in the present study. The drag forces per unit length are commonly assumed given by the following expression:
q(a)(t) = }pCDDIft - i l(t~ - i)
(5)
Here, Co is the drag force coefficient; ti is the water particle velocity; and i is the structural velocity. This equation relates the drag forces to the relative velocity, which seems reasonable as long as the structural member in question is sufficiently slender in the hydrodynamic sense. On the other hand, if the structural member is not slender, the drag forces are of minor importance and equation (5) may therefore still apply with reasonable accuracy. Due to the fact that equation (5) is an empirical approximation rather than a physical law, it is important in the analysis to apply a Co value derived from measurements using an estimator consistent with the method of analysis. Thus, if the stochastic linearization method is used in the analysis, it should be preferred in the estimation of Co (see Borgman I for detailed discussion).
oo
• (W)(x, t) = f
Response processes
g cosh{Kz} ei(tot_r.x ) dZn(/~, 60) 60 cosh{rd}
-=
(2)
Here,g is the acceleration of gravity; and d is the water depth. The water particle velocity and acceleration in the undisturbed stochastic wave field are readily obtained from equation (2) by differentiation.12 The remaining stochastic velocity potentials can be expressed formaUy applying equations similar to equation (2). In this case it is generally required to use numerical methods to calculate the transfer functions, is When all the velocity potentials have been determined, the stochastic wave loading process may be derived by applying the Bernoulli equation. This yields, disregarding second-order terms: 0cb f (3) q(P) = - P 3t n dA A
220
Eng. Struct., 1981, Vol. 3, October
Analysis It is assumed that the structure can be modelled as an assembly of finite (beam) elements applying boundary spring-dashpot elements to idealize the soft-foundation system. These spring-dashpot elements will generally be both nonlinear and time dependent. 16 This implies that the linearized equation of motion may be expressed as follows by introducing the Dirac delta function in the time space for frequency independent system parameters:
M(t - r) J:(r) dr +
C(t - r) i(r) dr
+ f K(t - r) r(r) dr = Q(t)
(6)
Response of offshore platforms: R. Sigbj6rnsson Here, r is the nodal response process; M is the virtual mass of the system including the hydrodynamic mass; C is the virtual damping including hydrodynamic effects; K is the stiffness matrix; and Q(t) is the wave loading process, which strictly speaking is a function of k (see equation (5)). This Coupling is assumed to have been removed in the following by applying the stochastic linearization method including Gaussianization of the excitation as well as the response. Other nonlinearities are treated in the same way. Hence, a drag damping proportional to the rms relative velocity is introduced into the C matrix, i.e. ~pCD(E[(u- f)2] 8fir) 1/2. In the solution of this equation the notion of the preceding sections is carried a step further by assuming the evolution of the sea state is slow enough to neglect the transient structural response. Then, the long-term probability density of the response can be expressed as:
pRr)=f f
(7)
Here, p~s~zg denotes the long-term joint probability density of significant wave height and expected zerocrossing wave period and average wave propagation; Pr IPsl"z~ is the conditional probability density of the response derived applying the statistics of the response process as approximated by: oo
r(t) = f ei¢°t dZr(w )
(8)
--oo
where Z r is the spectral process o f r possessing orthegonal increments. It follows that the response process has zero mean and spectral density given as: Srr(¢o) = H(w)
SQQ(co) H T*(w)
(9)
where: //(co) = [K(co)
-
co:M(w) + icoC(co)l-I
(10)
is the virtual frequency response function of the system; and S@Q is the spectral density of the wave excitation. By this approach the system given in equation (6) is in the general case approximated by a hierarchy of linear systems due to the fact that H(¢o) is a function of sea state severity. For each of these linear systems the response is approximated by an equivalent Gaussian process. Hence, the response statistics are completely defined in terms of equations (7), (9) and (10). It is worth pointing out that this direct frequency response method does not put any restrictions on the damping matrix, and the frequency dependent system matrices cause no extra problems. Further, it is generally most convenient to use complex arithmetic in the evaluation of//(co) (further discussion may be found elsewhere 17). The spectral density of wave excitation may be expressed as follows in terms of the one-dimensional wave spectral density:
So0(w ) = F(co) Snn(¢o)
In some cases it is desirable to carry out synthesis of the response process, i.e. to generate a time-space sample function from given response statistics. In the case of stationary multivariate Gaussian response processes this may be done effectively by Monte Carlo simulation, for instance combined with (a) direct time space filtering, 18 (b) spectral factorization combined with harmonic decomposition, 19 and (c) autoregressive moving-average time series model. 2° The factorization of the response spectral densities combined with harmonic decomposition yields:
rm(t) = Re 2 ~
m
N
E
~
1=1
Brnl(Wk)
k=l
× exp {i(eot + q~tk)}
(12)
Here, A~ is the frequency resolution;N = max(eo)/Aeo; ¢ is a random phase angle uniformly distributed between 0 and 2n;and:
fPrlFs~zg(rlh,t,v)PFs~zg(h,t,v)
HsTzO x dv dt dh
Synthesis
(11)
where F(¢o) is a Hermitian matrix, the so-called hydrodynamic transfer function, which in the general case depends on the sea state severity. Further discussion of this equation is given by Sigbj6rnsson) 2,13
B(6o) BT*(~) =Srr(¢O)
(13)
where B is a triangular matrix which may be determined by a Cholesky factorization. This procedure can be made very effective by applying the fast Fourier transformation algorithm. However, it is a drawback that the series will be periodic with the period 21r/.A¢o. Therefore, a limitation on the length of the time series is set by a finite computer memory. It is also worth pointing out that this method yields no random variation ~n the sample spectral densities. These problems may be avoided at least partly by applying a related method, 21 which, however, will be more time consuming. An attractive alternative to this method as well as the direct time-space filtering is in some cases furnished by the autogressive moving-average time series. In this case a limitation in computer memory is generally no problem and the periodicity of the sample function follows the periodicity of the pseudo-random number generator. The estimation of the model parameters may, however, be rather tedious. The method is therefore most tractable to generate long sample functions.
On the effects of short-crestedness of waves To exemplify the effects of the short-crestedness of the wave field on the structural response, a gravity platform of typical design has been selected. A sketch of the platform is shown in Figure 1 indicating the finite element model applied. The foundation-soil system is modelled by frequency dependent spring-dashpot elements using the elastic half space model. It can be argued that this model yields somewhat too high radiational damping in the case of layered soil. Therefore, the material (hysteretic) damping in the soil is neglected. The wave loading on the towers is obtained by applying the strip approach and neglecting interaction effects. The importance of mutual interaction can be evaluated by applying results given by Chakrabarti. za The drag forces are found to be almost insignificant in the whole frequency range in question. The wave forces on the caisson are based on potential theory and lumped as moment and shear force at mudline. More detailed discussion on the modelling including numerical results may be found in Sigbj6rnsson et al) 2,13,17 Figure 2 displays the statistics of bending moment about they-axis at the top of tower A. The wave field is specified in terms of the parameterized JONSWAP wave
Eng. Struct., 1981, Vol. 3, October
221
Response of offshore platforms: R. Sigbjornsson distribution for maxima. The joint distribution of significant wave height and expected zero-crossing period is derived applying wave data from the weather ship Famita. 24 The effects of short-crestedness are especially pronounced for the probability density. The strange shape of the peak of the probability density should also be noted. This is due to the resonance effects, which by increasing the response virtually move a part of the density to higher response levels. This study indicates that the effects of short-crestedness are generally most significant for small and moderate response levels.
,j" I
I
.•
i
:
f ~ ~.4-t
Applications in probabilistic design Extremes
a Figure I
10 -
b (a), sketch o f platform. (b), finite element idealization
~12 -
8t17~
o_
I~
Expected
The basic problem in structural design concerns first excursion failures and requires traditionally information concernin the characteristic extreme response and in modern approaches even the extreme value distribution. 25 An estimate of the 'most probable' largest maximum during the time interval T may be obtained from the longterm distribution of maxima as: Pr [r > ~] = (vT) -~
/
z e r o - crossing
(14)
Here, v denotes the expected zero-crossing frequency given as follows: ,
"=2-gf f J4 E-p] p°se= (h't'v)dvdt
£
1D
Ha r s o
gt~ o 5 L
O-
O-
0
4
I I I 8 12 14 Expected zero- crossing wave p e r i o d , ( s )
applying the notion of the slowly evolving response. The reliability function given in Figure 4 is based on this approach and derived from the results given in Figure 3. It is seen that the short-crestedness of the wave field influences the extreme response only moderately, which is in accordance with earlier findings, s An extreme value distribution for stationary processes put forward by Ditlevsen 26 is:
Figure 2
Bending m o m e n t at top o f t o w e r A as function of expected zero-crossing wave period. Significant wave height equal to 3 . 2 5 m; ( - A - ) based on idealized long crested waves; ( - o - ) based on short crested waves described by spreading function (2/1r) cos 2 O
spectral density 23 and the cos n 0 spreading function. The figure shows both results for idealized long crested waves (n -- o0) and short crested waves (n = 2) with the average direction of wave propagation in the direction of the x-axis. The statistical response quantities (a definition is given in Sigbj6rnsson et al. 17, p. 270) are expressed as functions of the expected zero-crossing wave period for a given significant wave height (defined as 4(E [I'~2]1/2). It is seen that the reduction in the rms moment due to short-crestedness is greatest close to resonance (fundamental natural frequency is 1.31 rad/s) but decreases with increasing low frequency quasi-static response, while the expected zero-(up) crossing period of the moment shows the opposite effect. The rms bandwidth of the moment shows only a small increase due to short-crestedness. The increase is, however, barely significant in an intermediate range with approximately equal contributions of resonance and quasi-static response. Figure 3 shows the long-term distribution of maxima of moment at the top of tower A induced by a unidirectional evolutionary sea state. This distribution is based on the Gaussianization of the response process applying the Rice
222
Eng. Struct., 1981, Vol. 3, October
(15)
F(~, T) ~- F(~, O) e -v~ T/F(~, O)
(1 6)
where v~ denotes the expected number of upcrossings per unit time at level ~. The validity of this distribution has been tested in an extensive simulation study by applying a variety of rather broad banded processes including nonGaussian processes. 26 Heuristically, it seems possible to generalize equation (14) to the semi-stationary processes discussed by Priestly. 9 Then, v~ is given by an expression similar to equation (15). 5
~
4
08
3
06
2
04
I
-2
10
a
/j~
0
,.~
,
,
,
-2 0 2 4 6 Moment (10 MNm) Figure 3 Long-term distribution of maxima o f moment at top of tower A. (a), p r o b a b i l i t y densities; (b), p r o b a b i l i t y distribution; (-z~-) based on idealized long crested waves; ( - o - ) based on short crested waves described by spreading f u n c t i o n (2/7r) cos20
8
Response of offshore platforms: R. Sigbj6rnsson
ooo
/
Then a simple estimate of the fatigue life of the structure is given as:
/
noo
//
f =
a
(19)
/_[l)or(Or) om do r /0
~i 6 c o
i
(3
I
I
0
I
1 Moment
I
2 (IOaMNm)
I
I
3
I
4
Figure 4 Long-term probability distribution of maxima of moment at top of tower A: (-A-), based on idealized long crested waves; (-o-), based on short crested waves described by spreading function (2H) cos20
Fatigue The continuous wave action subjects the structures to an extremely large number of oscillations. It is therefore necessary to consider fatigue in the structural design. A variety of theories has been proposed to account for fatigue under complex stressing, but none has apparently received universal acceptance. In the present study the concept of the maximum principal stress as a fatigue governing parameter is applied. At least for in-phase stresses this concept seems to give reasonable results. 27 Unfortunately, most structural elements in an offshore structure virtually experience multiaxial out-of-phase stochastic stressing. To facilitate a proper counting of stress cycles a generation of sample functions of stresses is desirable. The first step is the computation of the spectral densities of hot spot stresses, which generally requires refined f'mite element analysis. Secondly, time series of hot spot stresses are simulated using equations (12) and (13). Finally, the number of stress cycles is counted using the so-called NLR method (see Appendix), 2s which is a rain-flow type procedure. The long-term distribution of fatigue stress cycles per unit time may then be estimated by applying the concept of slowly varying evolutionary processes. This yields: 29 /g°r(Or)=
f
This estimate may serve as a useful measure in design decision. Figure 5 shows the long-term distribution of stress cycles and maxima in a given 'hot spot' at the deck/tower connection of the example platform subjected to unidirectional long-crested waves described by the parameterized JONSWAP wave spectrum. 29 Fortunately, in this case it is found that the principal axis corresponding to maxima of principal stresses are almost fixed in space. The distribution of partial damage is obtained applying the standards of the Norwegian Petroleum Directorate (Curve No. D). This yields, applying the Palmgren-Miner rule, an estimate of the fatigue life equal to four years. In the cases of shortcrested waves this estimate is roughly twice as high. The distribution of stress derived in this way may be applied directly in a more refined probabilistic analysis 3° or even in fracture mechanics approaches.
Final remarks A consistent stochastic analysis of waves and wave load effects is presented. The potential of the analysis is demonstrated by numerical examples. Uncertainties in the model of the wave field are discussed emphasizing the effects of short-crestedness on the structural response. It is found that the response induced by light and moderate sea states may be quite inaccurate, implying a serious drawback for the fatigue analysis. The accuracy of the extreme response appears on the other hand to be satisfactory, al This may, however, not be so surprising bearing in mind that the wave climate on the Norwegian Continental Shelf is governed by very unstable wind fields concerning homogeneity, direction and fetch. Therefore, more research is needed before wave induced response can be predicted with the desired accuracy. Acknowledgements The results reported here were obtained partly under a project sponsored by the Royal Norwegian Council for Scientific and Industrial Research (NTNF).
f flS°rllTs~z~(°rlh't'V)PlTs~zo(h't'v)
gs Tz ~ x dh dt dv
_E6
(17)
where, #arlITs¢zOrefers to the conditional distribution of fatigue stress cycles induced by a given short-term sea state. The most common estimates of fatigue life are based on the Palmgren-Miner hypothesis by assuming linear accumulation of damage. Then the accumulated damage per unit time may be expressed formally as follows: o~
= a-, f Pot(Or)o2 dot
= = =
u 5 E~ 2_~ o
1 I ~MQXImQ
4 u
I '
I i
Stress
cycles
E ~
b o ~
/Partial
2
r-]
E
(18) h
(l) 1 ~. i.. ~
~ -
r-~/' r-~ I I
, WIl i
I I
I
-2Q
0
20
40
I
,, i Ir -k~ t - ,! I I
I I
O
Here, the S-N curves are assumed specified in terms of a and m. According to the Palmgren-Miner hypothesis failure will occur when the accumulated damage is equal to unity.
damage
oI
io~ -' - w - i I ~--] ~-t I I i ~ I
'
:
~ : ', ~-I
60 80 100 Stress level, ( MPQ )
Figure 5 Long-term distribution of
r''l
!--'_ 120
140
160
stress cycles and maxima and
partial damage
Eng. S t r u c t . , 1981, V o l . 3, O c t o b e r
223
Response of offshore platforms: R. Sigbjornsson The co-operation o f Senior Structural Engineers S.-E. Jensen and E. T. Moe at A. S. Bergens Mekaniske Verksteder on fatigue problems is greatly acknowledged. Further, the author is indebted to Assistant Prof. K. Syvertsen and Research Assistant N. Spid6e at the Norwegian Institute o f Technology for assistance in preparing some of the numerical results reported here.
References 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15
16 17
18 19 20 21 22 23
24
224
Borgman, L'. E. 'Statistical models for ocean waves and wave forces',Advances in Hydroscience 1972, 8,139. Academic Press Nath, J. H. and Harleman, D. F. R. 'Dynamics of fixed towers in deep-water random waves', J. Waterways Harbors Div., ASCE 1969, 95 (WW4), 373 Forster, E. T. 'Model for nonlinear dynamics of offshore towers',J. Eng. Mech. Div., ASCE 1970, 86 (EM1), 41 Malhotra, A. K. and Penzien, J. 'Nondeterministic analysis of offshore structures', J. Eng. Mech. Div., ASCE 1970, 86 (EM6) 895 Berge, B. and Penzien, P. Offshore Technology Conference 1974, Houston Eatock Taylor, R. 'Structural dynamics of offshore platforms', Offshore Struct. Conf., London 1975, 125-132 Moan, T. and Sigbj6rnsson, R. 'Stochastic sea load effect analysis for probabilistic design of fixed offshore structures', 2nd Int. Conf. Struct. Safety Reliability, Munich, 1977, 227-246 Langen, 1. and Sigbj6rnsson, R. 'Stochastic dynamics of floating bridges',Eng. Struct. 1980, 2, 123 Priestly, M. B., 'Evolutionary spectra and non-stationary processes',J. Roy. Stat. Soc., 1965, 27, B204 Price, W. G. and Bishop, R. E. D. 'Probabilistic theory of ship dynamics', Chapman and Hall, London, 1974 Bretschneider, C. L. 'The envelope wave spectrum', Third Int. Conf. Port Ocean Eng. under Arctic Conditions, POAC-75, Fairbanks, USA, 1975 Sigbj6rnsson, R. 'Stochastic theory of wave loading processes',Eng. Struct. 1979, 1,58 Sigbj6rnsson, R. and Smith, E. K. 'Wave induced vibrations of gravity platforms: a stochastic theory', Appl. Math. Modelling, 1980, 4,155 Faltinsen, O. and Michelsen, F. C. 'Motion of large structures in waves at zero Froude number', lnt. Symp. dyn. marine vehicles struct, in waves, London 1974 Garrison, C. J. 'Hydrodynamic loading of large offshore structures: three-dimensional source distribution methods', in 'Numerical methods in offshore engineering' (ed. Zienkiewicz, O. C. et al.), John Wiley, Chichester, UK, 1978 Sigbj6rnsson, R. and Spidsfe, N. 'Long term response of gravity platforms with load dependent soil behaviour', Norwegian Maritime Research, 1980, 8 (3), 20 Sigbj6rnsson, R. et al. 'Dynamic response of framed and gravity structures', in 'Numerical methods in offshore engineering' (ed. Zienkiewicz, O. C. et al. ), John Wiley, Chiehester, UK, 1978 Borgman, L. E. 'Ocean wave simulation for engineering design', J. Waterways Harbors Div., ASCE 1969, 95 (WW4), 556 Shinozuka, M. 'Monte Carlo solution of structural dynamics', Cornput. Struct. 1972, 2,855 Gersch, W. and Luo, S. 'Discrete time series synthesis of randomly excited structural response', J. Acoust. Soc. Amer. 1972,51,402 Shinozuka, M. 'Simulation of multivariate and multi-dimensional random processes', J. Acoust. Soc. Amer. 1971,49 (1), 357 Chakrabarti, S. K. 'Wave forces on multiple vertical cylinders', J. Waterway, Port, Coastal Ocean Div., ASCE 1978, 104, WW2, 147 Houmb, O. G. and Overvik, T. 'Parameterization of wave spectra and long term joint distribution of wave height and period', First Int. Conf. Behaviour Offshore Struct., BOSS-76, Trondheim 1976 Sigbj6rnsson, R. et al. 'Estimation of the joint distribution of significant wave height and average wave period', SINTEF Report STF71 A76041, Trondheim 1976
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25 26 27 28 29
30 31
Flint, A. R. and Baker, M. J. 'Safety approach for structures subjected to stochastic loads', Int. Syrup. Integrity Offshore Struct., Glasgow 1978 Ditlevsen, O. 'Extremes and first passage times' (doctoral dissertation), Copenhagen 1971 Toor, P. M. 'A unified engineering approach to the prediction of multiaxial fatigue fracture of aircraft structures', Fract. Mech. 1975, 7,731 De Jonge, J. B. 'The monitoring of fatigue loads', NLR Report MP 700101, 1970 Sigbj6rnsson, R. and Syvertsen, K. 'Fatigue analysis of offshore structures', in 'Safety of structures under dynamic loading' (ed. Holand et al.), Tapir Publishers, Trondheim 1978 Wirshing, P. H. and Haugen, E. B. 'Probabilistic design for random fatigue loads', J. Eng. Mech. Div., ASCE 1973, 99 (EM6), 1165 Spids6e, N. and Sigbj6rnsson, R. 'On the reliability of standard wave spectra in structural response analysis', Eng. Struct. 1980, 2,123
Appendix
Counting of fatigue stress cycles The counting o f fatigue stress cycles is based on series consisting of extremes o f the maximum principal stresses o~ derived for each short-term sea state. The counting procedure applied is the so-called N L R method which is a rainflow type approach operating in two phases. In the first phase all intermediate stress cycles are detected. Starting with m = 1, four successive extremes are considered. If the following conditions are fulfilled: (ol(m + 3 ) > ~ o l ( m ) ) A ( o l ( m + 1) <~ o l ( m + 3 ) ) A (ol(m + 2) ~> o,(m))
( o l ( m + 3) < o , ( m ) ) A ( o l ( m + 1) ) o,(rn + 3))
(A1)
A (o,(m + 2) ~< cq (m)) a count is made o f one fatigue stress cycle defined by the amplitude: Or = ½loi(m + 2) - o l ( m +1)1
(A2)
and the mean value: Oa
=
~ l o l ( m + 2) + ol(m + 1)1
(A3)
and the extremes ol(m + 1) and o2(m + 2) are deleted from the series. The procedure is then repeated for the next four extremes ol(m), ol(m + 3), o l ( m + 4), ol(m + 5). If the condition (A1) is not met, the value of m is simply increased by one and the procedure is then repeated. When the whole series has been analysed in this way, a residue series remains which contains no intermediate stress cycles and consists simply of a sequence of cycles with increasing amplitudes followed by a sequence o f cycles with decreasing amplitudes, etc. In the second phase, the residue series is analysed by applying a simple range count procedure. That is, a count is made o f successive half stress cycles defined by the following amplitude and mean value: or = ~lo,(n + 1 ) - o , ( n ) l
(A4)
= 2tla,(n +1) + o,(n)[
(AS)
Oa
These counts are then added to the results of the first phase. The counting results are finally presented in one dimensional form for each or in terms of the number (distribution) o f fatigue stress cycles P(or) per unit time, the average mean stress and the root mean square mean stress. This should be seen in the light of the fact that the codified fatigue damage is primarily defined by the stress range and to a lesser extent b y the mean stress.