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Wave loading effect in offshore structural reliability
Structural Safety, 7 (1990) 1-10 Elsevier
1
WAVE LOADING EFFECT IN OFFSHORE STRUCTURAL RELIABILITY H.Y. Chan and R.E. Melchers Department of Civil Engineering and Surveying, The University of Newcastle, IV.S. W. 2308 (Australia)
(Received November 20, 1987; accepted in revised form November 10, 1988)
Key words: load modelling; offshore structures; reliability; risk; structural engineering; waves.
ABSTRACT
The reliabifity analysis of multi-membered structural systems such as steel tubular offshore platforms (jacket-type) has proved to be a challenging task. One matter of particular importance is the modelling of applied wave loading. For the structural analyses used in reliability estimation, one single wave loading system only is most commonly considered. Real wave trains, however, reach the various structural components at different times. This may have an effect on the sequence of member failure and hence on structural failure. This can, in turn, affect the calculated probability of failure for the structure as a whole. The importance of this possibility is investigated in the paper for a reasonably realistic but idealized offshore structure.
1. I N T R O D U C T I O N A n offshore structure may fail in a variety of ways. One of these is collapse under the action of an extreme seastate, another is through fatigue failure under all seastates, yet a third is failure under dynamic wave action. However m a n y other modes of failure are possible and need to be considered in practical design. A number of methods to predict the reliability of structures under predefined failure modes have been developed elsewhere [1-3]. Nevertheless complex structures such as jacket platforms still present a challenge for reliability analysis because the possible n u m b e r of failure modes, considering the combination of all m e m b e r failures, is often extremely large. In addition, careful attention must be given to correctly modelling the loading applied to the structure. Much previous work has considered the one-time application of an extreme wave load based on the probabilistically worst seastate. Such an approach is reasonable for structures which fail 0167-4730/90/$03.50
© 1990 Elsevier Science Publishers B.V.
when one member fails ("elastic" structures, or "weakest-link" structures) but not for redundant structures for which more than one member must fail for the structure to fail. For these, it is by no means obvious that there is a single location of the extreme wave which will produce the worst loading for individual members in the structure even under extreme wave conditions. Such conditions imply high waves of long period. Guenard [4] assumed that all members of the structure experience their maximum stress at the same time, which would occur when the crest of the wave passed through the structure. Murotsu et al. [5], on the other hand, concluded from a numerical study that the sum of the mean values of the horizontal wave forces acting on all the members is a maximum when the distance between wave crest and jacket structure centreline is approximately (wavelength/40). It was recognized that this location did not necessarily produce the maximum force on all the members simultaneously. In principle, as a wave moves through a jacket-type structure, the forces on the members and hence the member internal actions change. This means that the most "highly stressed" member is likely to be a different member for different locations of the wave peak relative to the structure. In general, with uncertainty in member strength also considered this means that the member most likely to fail next will also depend on wave location. It follows, that the sequence of member failure calculated for a stationary, extreme wave is not necessarily related to the sequence of member failure under progressive extreme wave action. The effect of this phenomenon on the calculated probability of failure of the structure as a whole is discussed herein. A brief review of the modelling of forces acting on circular cylindrical members due to wave loads is presented. This is followed in Section 3 by a description, in terms of probability calculations, of the effect of a transient wave through the structure, and how conventional structural systems reliability calculation techniques may be used. An example application to a reasonably simple two-dimensional structure is given in Section 4.
2. WAVE LOAD MODEL Jacket platforms consist of a collection of cylindrical members and connecting nodes. These are the principal elements encountered by sea waves. Stemming from earlier work on single, vertical piles, Morison's equation
Q=
(kau I u I q - k i u ) dz
(1)
can be used to compute the horizontal force Q acting on cylindrical piles between vertical locations z = (a, b). Here u describes the water particle velocity and ti its acceleration. The coefficients k d and k i are empirical and relate to drag forces and inertia forces respectively. The wave particle velocity and acceleration terms may be obtained from an appropriate wave theory. For simplicity of discussion, linear wave theory will be assumed [6]: H 7/(x, t) = ~ - c o s ( k x - wt)
(2)
u(t) = wH cosh k ( z + d) c o s ( w t - kx) 2 sinh kd
(3a)
i~(t) - w2H2 cOShsinhk(Zkd + d) sin(wt - kx)
(3b)
where ,/ is the water surface elevation, u is the horizontal water particle velocity, ~ is the horizontal water particle acceleration, x is the horizontal distance, z is the location relative to water surface, d is the water depth, H is the wave height, k is the wave n u m b e r = 2~r/L, L is the wavelength, w is the wave angular frequency = 2~r/T, T is the wave period, and t is time. Substituting into eqn. (1) produces Q = Qd c o s ( k x - wt)[cos(kx- wt)[+Qi s i n ( k x - wt)
(4)
where Qd and Qi are the m a x i m u m drag and inertia forces respectively, defines as:
' - - - - r cos" s ds
sinhZkd -fo 2
Qi = ½CmPD2-T--~H
1 (acosh2ks ds sinh~kd Jo
(6)
where Cd and C m are the well-known Morison drag and mass coefficients, D is the pile diameter, and p is the water density. By introducing the phase angle 0 = kx - wt, expression (4) for the force profile as a function of 0 becomes:
(7)
Q(o) = Qd cos 0 [cos 0 [ + Qi sin 0 The m a x i m u m total force can now be obtained by setting OQ/OO = 0, from which
f sin-l(Qi/2Qd), 0° = ~ ~r/2,
Qi ~< 2Qd Qi > 2Qd
The phase angle 00 is a measure of the distance between the location of the m a x i m u m wave force and to the peak of the wave. They are clearly not coincident, in general (See Fig. 1). Substituting 00 into eqn. (7) produces an expression for the m a x i m u m wave force. Using this expression it is possible to derive the probability density function f M ( m ) for the m a x i m u m wave force, given the probability density function for the wave height H and assuming all other parameters deterministic. Such a procedure has been described by Borgman [7]. It is also possible, with considerably more integration, to incorporate the probability density functions for Cd, C m and T through Qd and Qi (e.g., [8]). M o n t e Carlo methods may need to be employed. Q
O
Fig. 1. Wave load profile.
Rather than determining the complete probability density function (p.d.f.) for the loading Q as a function of O, an alternative is to determine only the first two moments (mean and variance).
,~(o) = f f f f o ( o ) / ( H ,
v, ca, Cm)dH dT dC~ dCm
(8)
.~(ot = f f f f o 2 I o l / I H ,
T, ca, c~t dH dT dCa dCm-(~,~t ~
(9)
where f( ) is the joint p.d.f, of H, T, Ca and Cm, independent of 0. Integrations (8) and (9) are generally still too difficult to carry out even using Monte Carlo methods. A further alternative is to work entirely with a Second Moment approximation of each variable: mean(Q)
= I£Q-~
O(~g, I-tr, ~cd, ~c,,)
0Q 0~yQj variance(Q) = o ~ = ~ ~ ~ oy,%pii J
(10) (11)
where Q( ) is given by eqn (4), y is the vector (H, T, Ca, Cm), .~ is the vector of means (~H, t~T, t~cd, I~Cm), and Pij is the correlation coefficient between i and j. The correlation coefficient for forces acting at piles 1 and 2 is given by
E( Q1Q2) - I~Q'I~Q2 Peioj =
(12)
oeoe2
with
E(Q, Q2)--ffffo,o2f(H, T, Ca, era)dH aT dCa dCm
(13)
The choice of the joint probability density function f( ) for use in eqns. (8), (9) and (13) depends on the type of problem being considered. In a reliability analysis concerned with structural collapse due to extreme conditions (rather than fatigue or dynamic action), maximum wave seastates (i.e., fully developed seastates) only are of interest. It follows that the p.d.f, for the wave height H may be obtained as a first approximation from the p.d.f, of individual wave heights using the well-known extreme value argument:
FHma~(h ) = [ FH( h )] tl
(14)
where F#( ) is the cumulative distribution function of wave heights, given by the Rayleigh distribution [6], n is the number of waves during a sea-state and FHma.(h) represents the cumulative distribution function for the maximum wave height during the sea-state. The period T is not generally independent of H or Hm~,. Weigel [9] proposed that for extreme sea-states the period can be approximated by a deterministic expression: T(nmax) = T ( n s ) ~-- 1 5 . 6 ( n s / g ) 1/2
(15)
where H s is the "significant" wave height = mean wave height of the highest third of all the waves occurring during a storm. However, for narrow-banded sea-states, as would be usual for storm conditions, Longuet-Higgins [10] indicated that for large wave heights, the wave period is much less variable than for smaller wave heights. Since the forces on a jacket structure are likely to be dominated by drag rather than by inertia effects and since the drag force is more sensitive
to change in wave height than to changes in wave period, it is reasonable to use eqn. (15) as an approximation [5]. Further, for a narrow-banded seastate it is reasonable to treat the period T as a constant, T = 1/fo, where f0 is the "central" frequency. Because Morison's equation is not a close descriptor of wave forces on cylinders, the scatter in the values of Cd and C m obtained in experiments is quite large. Nevertheless, independent Normal and Lognormal distributions are most commonly r e c o m m e n d e d (e.g., [6]). Once the statistical properties of the various parameters have been fixed, the Second M o m e n t statistical description of the wave force acting on a cylindrical m e m b e r can be obtained from eqns. (8), (9) and (12).
3. SYSTEMS RELIABILITY CALCULATION As noted, the magnitude of loads acting on the structure and hence the load effects within structural members change as a wave passes through the structure. In determining whether the structure as a whole survives, various combinations of members and their associated load effects need to be assessed. For a probabilistic calculation the "worst" loading case need not correspond to the most likely structural failure model. The uncertainty of m e m b e r resistance and the frequency of member failure events need also to be considered. At least one complete cycle of load patterns caused by the motion of a wave through the structure needs to be studied. Assuming for simplicity that the total load cycle may be represented by n discrete load patterns, the probability of m e m b e r failure (for the j t h member) will be
P,j= P [ E I U E 2 U - . . UE.]
(16)
where E i is the event "failure under load pattern i ". Since all load patterns are the result of the same wave they are perfectly correlated (assuming the wave does not change shape in passing through the structure). The m e m b e r failure probability, therefore, may be written as:
Pfj
=
max ( P [ Ei]}
(17)
1
The question of application and removal of each load pattern may now be ignored. The evaluation of terms such as eqn. (17) requires knowledge of the statistical properties of the m e m b e r strength as well as the loading acting on the member. For simplicity, the loading which the wave exerts on the structure will be assumed to be applied at the nodes only (see Fig. 2). The relative distribution of nodal forces makes up the applied load pattern. This will be assumed to be independent of the wave height. (The magnitude of the forces is not, of course, independent of wave height.) For a given sequence (A, B, C) of members to cause structural failure, the failure probability is given by:
pf=P[ANBAC]
(18)
where event A corresponds to failure of m e m b e r A and correspondingly for B and C. In principle, all possible combinations of load sequence applications need to be considered to evaluate eqn. (18). This is extremely tedious but can be done using a slight modification of existing techniques for structural systems reliability analysis [2,3]. The technique used in the present study is the Truncated Enumeration (T.E.) method described in detail elsewhere [11]. It is based on using an elastic structural analysis as an artifice to generate all failure modes for the
structure. It does this by gradually incrementing one or other of the applied loads; however, the load so used has no direct implications for the totality of failure modes generated and hence no direct influence on the calculated system reliability. This important aspect is often misunderstood. As considered in some detail in Chan and Melchers [12], the fact that the loading used for incrementation in the T.E. method is only an artifice, means that, because also of the dependent nature of all the load sequences, it is permissible to consider event A in eqn. (18) as that due to cycling the complete load sequence through the structure and selecting the critical sequence as in eqn. (17). Once this is done, the next member may be similarly examined. The process of considering a load sequence in this way would not be valid if complete dependence did not exist between them. A more concise argument is given in the Appendix. The complementary situation, with mutually independent load sequences has been considered by Wen and Chen [13].
4. E X A M P L E
A two-dimensional truss modified from the special truss considered by Bjerager [14] will be used for analysis (see Fig. 2). Member behaviour will be assumed to be elastic-plastic and the same for all members [11]. For the present, the mean wave height will be assumed 12.5 m, the wave period fixed at 15 s and the wave length 300 m. This corresponds to realistic storm conditions. Table 1 gives the numerical data for the truss members with Ri being the mean strength of the ith member, whose area is Ai and diameter is D~. For purposes of illustration, the coefficient of variation of the m e m b e r strengths will be assumed to be 0.15 for all members, and the coefficient 20.Om i~ Q7
~i Q~
M.s.L.
•/"
~-,
'o7
_
I
!~
39.1m
Fig. 2. Example offshore structure.
°
TABLE 1 Numerical data of truss tower Member i
Ri (kN)
Area, A~ (mE)
Diameter, (m)
1-6 7 8 9 1o- 11 12-13 14-15
51890.0 84400.0 18535.0 33600.0 6675.0 11535.0 18535.0
0.324 0.053 0.116 0.210 0.042 0.074 0.116
2.5 1.0 1.5 2.0 0.9 1.2 1.5
Di
of correlation between member strength 0.5. The drag and mass coefficients will be assumed to be Ca = 1.3 and C m = 2.0. The deck load Q7 acting vertically on the tower has a mean of /~07 = 22 x 103 k N with a coefficient of variation VQ7 = 0.1. This load will be assumed to be completely independent of the wave loading. Using the above data and those of Table 1, eqn. (10) together with eqns. (5)-(7) yielded a total mean horizontal load on one leg of /-to = 428.11 cos 0 Icos 0 1+ 5 4 8 . 5 4 sin 0
(19)
which has a maximum mean value of 603.82 k N at phase angle 0 = 36 o. This corresponds to the location of the maximum total wave force for the whole structure. (Here 0 is referenced to joint no. 1.) For the wave forces, a coefficient of variation VQ = 0.4 will be assumed in the absence of sufficient data with which to calculate it using the theory herein. To replicate the wave force profile with loading only at the nodes, the following mean nodal forces were used for each leg: QI=Q2=
15.56kN
03 = Q4 = 243.54 k N Q5 = Q6 = 242.41 kN The results from the reliability analysis are given in Tables 2 and 3. The "failure path" refers to the sequence of member failure, using the m e m b e r notation given in Fig. 2. It will be seen that the failure modes which dominate the system reliability index are closely similar, at least for the first few modes, and that the reliability indices are also similar. It also follows that the bounds on system reliability are similar. As found earlier [12], the use of only one load profile (i.e., at 0 = 36 °) overestimates the system reliability index (i.e., underestimates the system probability of failure). However, for the realistic loading magnitude of the present example, the error is not large, being about 1%. As seen from Table 3, this observation also appears to be valid for a range of the deck loads Q7The effect of shorter wavelength conditions on the system reliability index for a complete wave cycle was not detectable. Table 4 shows the b o u n d s on the reliability indices obtained for a range of wavelengths. Short wavelength waves cause a greater force on the leading leg compared with the trailing leg. However, at any one level these forces are additive and, as shown in Table 1, the horizontal member connecting pairs of nodes is not involved in critical failure paths. The critical
8 TABLE 2 D o m i n a n t failure modes a n d reliability index fl = - ~
l(pf) 0 = 00 = 36 o (where m a x i m u m horizontal load occurred)
C o m p l e t e wave cycle Failure path
Reliability index
Failure path
Bs 2-10 1-11 4-12 3 13 4-13 3-12
4.0762 4.0711 4.0814 4.2969 4.3340 4.3348
Bounds
3.8150 3.8101
Reliability Index
Bs 2-10 1-11 4-12 4-13 3-12 3-13
4.0762 4.0828 4.2814 4.3340 4.3404 4.3908 3.8308 3.8274
Note: Q7 = deck load = 22,000 kN, wavelength = 300 m, wave period = 15 s.
TABLE 3 Effect of deck load o n b o u n d s on system reliability index fls Deck load
40,000 kN 30,000 kN 22,000 kN
Complete wave cycle
0 = 36 °
Lowerbound
Upperbound
Lowerbound
Upperbound
1.4504 2.7218 3.8150
1.3460 2.6969 3.8101
1.4562 2.7357 3.8308
1.3662 2.7202 3.8274
Note: Wavelength = 300 m.
members are, realistically, the "verticals" and, to a lesser extent, the diagonals. In a more detailed modelling of the way loads are applied to the structure, allowing, for example, for lateral loading along the horizontals, it would be expected that wavelength will have some effect on the results for reliability index.
TABLE 4 Effect of wavelength on b o u n d s on system reliability index fls Wavelength
Bounds on Bs
(m) 350 300 250 200 150 100 Note: Q2 = 22,000 kN.
3.8150 3.8150 3.8149 3.8150 3.8145 3.8167
3.8102 3.8101 3.8101 3.8102 3.8101 3.8123
5. C O N C L U S I O N T w o d i f f e r e n t t e c h n i q u e s w e r e c o m p a r e d herein for the a s s e s s m e n t o f the reliability of s i m p l i f i e d m u l t i - m e m b e r e d j a c k e t - t y p e o f f s h o r e s t r u c t u r e s u n d e r the a c t i o n o f a fully d e v e l o p e d , e x t r e m e seastate. It was a s s u m e d t h a t for such l o a d i n g c o n d i t i o n s the w a v e p e r i o d is essentially constant. It w a s f o u n d t h a t the c o n s i d e r a b l y m o r e i n v o l v e d a n d t i m e - c o n s u m i n g c a l c u l a t i o n of the s y s t e m reliability i n d e x c o r r e s p o n d i n g to the c o m p l e t e l o a d i n g cycle led to a slightly l o w e r e s t i m a t e o f the s y s t e m reliability i n d e x o b t a i n e d f r o m a n e s t i m a t e b a s e d o n a n a r b i t r a r i l y selected, b u t realistically e x t r e m e p o s i t i o n for the w a v e p r o d u c i n g the m o s t severe t o t a l w a v e lateral force c o n d i t i o n . It is c o n c l u d e d t h a t for m o s t p r a c t i c a l p u r p o s e s the s i m p l e r m e t h o d is a d e q u a t e a n d t h a t cycling o f the c o m p l e t e w a v e t h r o u g h a j a c k e t - t y p e o f f s h o r e s t r u c t u r e is n o t necessary.
REFERENCES 1 P. Thoft-Christensen and M.J. Baker, Structural Reliability Theory and Its Applications, Springer-Verlag, Berlin,
1982. 2 0 . Ditlevsen and P. Bjerager, Methods of structural systems reliability, Structural Safety, 3 (1986) 195-229. 3 R.E. Melchers, Structural Reliability Analysis and Prediction, Ellis Horwood/J. Wiley, Chichester, UK, 1987. 4 Y.F. Guenard, Application of System Reliability Analysis to Offshore Structures, Report No. 71, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA, 1984. 5 Y. Murotsu, M. Kishi, H. Okada, Y. Ikeda and S. Matsuzaki, Probabilistic collapse analysis of offshore structures, in: J.S. Chung et al. (Eds.), Proc. 4th Int. Offshore Mechanics and Arctic Engineering Symposium, ASME, Vol. 1, 1985, pp. 250-258. 6 T. Sarpkaya and M. Isaacson, Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 1982. 7 L.E. Borgman, Wave forces on piling for narrow-band spectra, J. Waterways Harbors Div., ASCE, 91(WW3) (1965) 65-90. 8 A.M. Nafday and J. Wang, Probabilistic model of wave forces on cylindrical piles, J. Waterway Port Ocean Eng., ASCE, 109(2) (May 1983). 9 R.L. Wiegel, Oceanographical Engineering, Prentice-Hall, Englewood Cliffs, N J, 1964. 10 M.S. Longuet-Higgins, On the Joint Distribution of the Periods and Amplitudes of Sea Waves, J. Geophys. Res., 80(18) (1975) 2688-2694. 11 R.E. Melchers and L.K. Tang, Dominant failure modes in stochastic structural systems, Structural Safety, 2 (1984) 127-142. 12 H.Y. Chan and R.E. Melchers Reliability of complex structures under wave loads, in: N.C. Lind (Ed.), Proc. 5th Int. Conf. Applications of Statistics and Probability in Soil and Structural Engineering, Vancouver, Canada, 1987, pp. 166-173. 13 Y.K. Wen and H.C. Chen, System reliability under multiple hazards, Struct. Res. Ser. No. 526, University of Illinois, Urbana, IL, 1986. 14 P. Bjerager, Reliability Analysis of Structural Systems, Department of Structural Engineering, Technical University of Denmark, Lyngby, Denmark, 1984.
APPENDIX C o n s i d e r a s i m p l e structure, with piles at l o c a t i o n s 1 a n d 2. L e t A r e p r e s e n t the e v e n t " f a i l u r e of m e m b e r A " a n d let A 1 r e p r e s e n t the e v e n t " f a i l u r e of m e m b e r A u n d e r the a p p l i e d l o a d at l o c a t i o n 1". L e t a n a l o g o u s n o t a t i o n a p p l y for m e m b e r s B a n d C for l o c a t i o n 2.
10
Member A will fail if it fails under the internal action (load effect) resulting in it from force being applied at 1 or at 2. Let these events be denoted A 1 and A 2 respectively, then
P ( A ) = P ( A 1 UA2)
(A.1)
However, since the actions result from the same wave, (which is deterministic in time, for simplicity), it follows that 2
P(A~ U AE) = m a x P ( A i )
(1.2)
i~l
and similarly for members B and C. The mode failure probability for the member failure sequence A, B, C then becomes Pf =
P ( A C~B ~ C)
= e[(A, u A )n (B, u B )n (C, uC ) = P[( AI f-) B, n CI) U ( A1N B, f') C2) U ( AI A B2 (3 C,) u (A, n B2 n C2)u (12 n B, n C , ) u (A2 n B, n C2) U ( A 2 n B 2 N C,) U (A 2 n B 2 n C2) ]
= m a x P [ A , NBjC3Ck], i,j,k
i, j , k = l , 2
(A.3) (A.4)
This shows that the probability of failure associated with the failure mode (A, B, C) is the maximum of all such probabilities for all possible load sequences. Expression (A.4) is significantly easier to evaluate than (A.3). As a result of eqn. (A.2) it may be seen that it is sufficient to run through all load cases in the load sequence to determine the next most likely m e m b e r to fail and so determine the load case under which this occurs. For the member failure sequence (A, B, C), this procedure fixes the indices in (A.4). That it is permissible to consider all load cases in the load sequence at each stage of the Truncated Enumeration analysis has been considered in a different m a n n e r in Chan and Melchers [12].
1
WAVE LOADING EFFECT IN OFFSHORE STRUCTURAL RELIABILITY H.Y. Chan and R.E. Melchers Department of Civil Engineering and Surveying, The University of Newcastle, IV.S. W. 2308 (Australia)
(Received November 20, 1987; accepted in revised form November 10, 1988)
Key words: load modelling; offshore structures; reliability; risk; structural engineering; waves.
ABSTRACT
The reliabifity analysis of multi-membered structural systems such as steel tubular offshore platforms (jacket-type) has proved to be a challenging task. One matter of particular importance is the modelling of applied wave loading. For the structural analyses used in reliability estimation, one single wave loading system only is most commonly considered. Real wave trains, however, reach the various structural components at different times. This may have an effect on the sequence of member failure and hence on structural failure. This can, in turn, affect the calculated probability of failure for the structure as a whole. The importance of this possibility is investigated in the paper for a reasonably realistic but idealized offshore structure.
1. I N T R O D U C T I O N A n offshore structure may fail in a variety of ways. One of these is collapse under the action of an extreme seastate, another is through fatigue failure under all seastates, yet a third is failure under dynamic wave action. However m a n y other modes of failure are possible and need to be considered in practical design. A number of methods to predict the reliability of structures under predefined failure modes have been developed elsewhere [1-3]. Nevertheless complex structures such as jacket platforms still present a challenge for reliability analysis because the possible n u m b e r of failure modes, considering the combination of all m e m b e r failures, is often extremely large. In addition, careful attention must be given to correctly modelling the loading applied to the structure. Much previous work has considered the one-time application of an extreme wave load based on the probabilistically worst seastate. Such an approach is reasonable for structures which fail 0167-4730/90/$03.50
© 1990 Elsevier Science Publishers B.V.
when one member fails ("elastic" structures, or "weakest-link" structures) but not for redundant structures for which more than one member must fail for the structure to fail. For these, it is by no means obvious that there is a single location of the extreme wave which will produce the worst loading for individual members in the structure even under extreme wave conditions. Such conditions imply high waves of long period. Guenard [4] assumed that all members of the structure experience their maximum stress at the same time, which would occur when the crest of the wave passed through the structure. Murotsu et al. [5], on the other hand, concluded from a numerical study that the sum of the mean values of the horizontal wave forces acting on all the members is a maximum when the distance between wave crest and jacket structure centreline is approximately (wavelength/40). It was recognized that this location did not necessarily produce the maximum force on all the members simultaneously. In principle, as a wave moves through a jacket-type structure, the forces on the members and hence the member internal actions change. This means that the most "highly stressed" member is likely to be a different member for different locations of the wave peak relative to the structure. In general, with uncertainty in member strength also considered this means that the member most likely to fail next will also depend on wave location. It follows, that the sequence of member failure calculated for a stationary, extreme wave is not necessarily related to the sequence of member failure under progressive extreme wave action. The effect of this phenomenon on the calculated probability of failure of the structure as a whole is discussed herein. A brief review of the modelling of forces acting on circular cylindrical members due to wave loads is presented. This is followed in Section 3 by a description, in terms of probability calculations, of the effect of a transient wave through the structure, and how conventional structural systems reliability calculation techniques may be used. An example application to a reasonably simple two-dimensional structure is given in Section 4.
2. WAVE LOAD MODEL Jacket platforms consist of a collection of cylindrical members and connecting nodes. These are the principal elements encountered by sea waves. Stemming from earlier work on single, vertical piles, Morison's equation
Q=
(kau I u I q - k i u ) dz
(1)
can be used to compute the horizontal force Q acting on cylindrical piles between vertical locations z = (a, b). Here u describes the water particle velocity and ti its acceleration. The coefficients k d and k i are empirical and relate to drag forces and inertia forces respectively. The wave particle velocity and acceleration terms may be obtained from an appropriate wave theory. For simplicity of discussion, linear wave theory will be assumed [6]: H 7/(x, t) = ~ - c o s ( k x - wt)
(2)
u(t) = wH cosh k ( z + d) c o s ( w t - kx) 2 sinh kd
(3a)
i~(t) - w2H2 cOShsinhk(Zkd + d) sin(wt - kx)
(3b)
where ,/ is the water surface elevation, u is the horizontal water particle velocity, ~ is the horizontal water particle acceleration, x is the horizontal distance, z is the location relative to water surface, d is the water depth, H is the wave height, k is the wave n u m b e r = 2~r/L, L is the wavelength, w is the wave angular frequency = 2~r/T, T is the wave period, and t is time. Substituting into eqn. (1) produces Q = Qd c o s ( k x - wt)[cos(kx- wt)[+Qi s i n ( k x - wt)
(4)
where Qd and Qi are the m a x i m u m drag and inertia forces respectively, defines as:
' - - - - r cos" s ds
sinhZkd -fo 2
Qi = ½CmPD2-T--~H
1 (acosh2ks ds sinh~kd Jo
(6)
where Cd and C m are the well-known Morison drag and mass coefficients, D is the pile diameter, and p is the water density. By introducing the phase angle 0 = kx - wt, expression (4) for the force profile as a function of 0 becomes:
(7)
Q(o) = Qd cos 0 [cos 0 [ + Qi sin 0 The m a x i m u m total force can now be obtained by setting OQ/OO = 0, from which
f sin-l(Qi/2Qd), 0° = ~ ~r/2,
Qi ~< 2Qd Qi > 2Qd
The phase angle 00 is a measure of the distance between the location of the m a x i m u m wave force and to the peak of the wave. They are clearly not coincident, in general (See Fig. 1). Substituting 00 into eqn. (7) produces an expression for the m a x i m u m wave force. Using this expression it is possible to derive the probability density function f M ( m ) for the m a x i m u m wave force, given the probability density function for the wave height H and assuming all other parameters deterministic. Such a procedure has been described by Borgman [7]. It is also possible, with considerably more integration, to incorporate the probability density functions for Cd, C m and T through Qd and Qi (e.g., [8]). M o n t e Carlo methods may need to be employed. Q
O
Fig. 1. Wave load profile.
Rather than determining the complete probability density function (p.d.f.) for the loading Q as a function of O, an alternative is to determine only the first two moments (mean and variance).
,~(o) = f f f f o ( o ) / ( H ,
v, ca, Cm)dH dT dC~ dCm
(8)
.~(ot = f f f f o 2 I o l / I H ,
T, ca, c~t dH dT dCa dCm-(~,~t ~
(9)
where f( ) is the joint p.d.f, of H, T, Ca and Cm, independent of 0. Integrations (8) and (9) are generally still too difficult to carry out even using Monte Carlo methods. A further alternative is to work entirely with a Second Moment approximation of each variable: mean(Q)
= I£Q-~
O(~g, I-tr, ~cd, ~c,,)
0Q 0~yQj variance(Q) = o ~ = ~ ~ ~ oy,%pii J
(10) (11)
where Q( ) is given by eqn (4), y is the vector (H, T, Ca, Cm), .~ is the vector of means (~H, t~T, t~cd, I~Cm), and Pij is the correlation coefficient between i and j. The correlation coefficient for forces acting at piles 1 and 2 is given by
E( Q1Q2) - I~Q'I~Q2 Peioj =
(12)
oeoe2
with
E(Q, Q2)--ffffo,o2f(H, T, Ca, era)dH aT dCa dCm
(13)
The choice of the joint probability density function f( ) for use in eqns. (8), (9) and (13) depends on the type of problem being considered. In a reliability analysis concerned with structural collapse due to extreme conditions (rather than fatigue or dynamic action), maximum wave seastates (i.e., fully developed seastates) only are of interest. It follows that the p.d.f, for the wave height H may be obtained as a first approximation from the p.d.f, of individual wave heights using the well-known extreme value argument:
FHma~(h ) = [ FH( h )] tl
(14)
where F#( ) is the cumulative distribution function of wave heights, given by the Rayleigh distribution [6], n is the number of waves during a sea-state and FHma.(h) represents the cumulative distribution function for the maximum wave height during the sea-state. The period T is not generally independent of H or Hm~,. Weigel [9] proposed that for extreme sea-states the period can be approximated by a deterministic expression: T(nmax) = T ( n s ) ~-- 1 5 . 6 ( n s / g ) 1/2
(15)
where H s is the "significant" wave height = mean wave height of the highest third of all the waves occurring during a storm. However, for narrow-banded sea-states, as would be usual for storm conditions, Longuet-Higgins [10] indicated that for large wave heights, the wave period is much less variable than for smaller wave heights. Since the forces on a jacket structure are likely to be dominated by drag rather than by inertia effects and since the drag force is more sensitive
to change in wave height than to changes in wave period, it is reasonable to use eqn. (15) as an approximation [5]. Further, for a narrow-banded seastate it is reasonable to treat the period T as a constant, T = 1/fo, where f0 is the "central" frequency. Because Morison's equation is not a close descriptor of wave forces on cylinders, the scatter in the values of Cd and C m obtained in experiments is quite large. Nevertheless, independent Normal and Lognormal distributions are most commonly r e c o m m e n d e d (e.g., [6]). Once the statistical properties of the various parameters have been fixed, the Second M o m e n t statistical description of the wave force acting on a cylindrical m e m b e r can be obtained from eqns. (8), (9) and (12).
3. SYSTEMS RELIABILITY CALCULATION As noted, the magnitude of loads acting on the structure and hence the load effects within structural members change as a wave passes through the structure. In determining whether the structure as a whole survives, various combinations of members and their associated load effects need to be assessed. For a probabilistic calculation the "worst" loading case need not correspond to the most likely structural failure model. The uncertainty of m e m b e r resistance and the frequency of member failure events need also to be considered. At least one complete cycle of load patterns caused by the motion of a wave through the structure needs to be studied. Assuming for simplicity that the total load cycle may be represented by n discrete load patterns, the probability of m e m b e r failure (for the j t h member) will be
P,j= P [ E I U E 2 U - . . UE.]
(16)
where E i is the event "failure under load pattern i ". Since all load patterns are the result of the same wave they are perfectly correlated (assuming the wave does not change shape in passing through the structure). The m e m b e r failure probability, therefore, may be written as:
Pfj
=
max ( P [ Ei]}
(17)
1
The question of application and removal of each load pattern may now be ignored. The evaluation of terms such as eqn. (17) requires knowledge of the statistical properties of the m e m b e r strength as well as the loading acting on the member. For simplicity, the loading which the wave exerts on the structure will be assumed to be applied at the nodes only (see Fig. 2). The relative distribution of nodal forces makes up the applied load pattern. This will be assumed to be independent of the wave height. (The magnitude of the forces is not, of course, independent of wave height.) For a given sequence (A, B, C) of members to cause structural failure, the failure probability is given by:
pf=P[ANBAC]
(18)
where event A corresponds to failure of m e m b e r A and correspondingly for B and C. In principle, all possible combinations of load sequence applications need to be considered to evaluate eqn. (18). This is extremely tedious but can be done using a slight modification of existing techniques for structural systems reliability analysis [2,3]. The technique used in the present study is the Truncated Enumeration (T.E.) method described in detail elsewhere [11]. It is based on using an elastic structural analysis as an artifice to generate all failure modes for the
structure. It does this by gradually incrementing one or other of the applied loads; however, the load so used has no direct implications for the totality of failure modes generated and hence no direct influence on the calculated system reliability. This important aspect is often misunderstood. As considered in some detail in Chan and Melchers [12], the fact that the loading used for incrementation in the T.E. method is only an artifice, means that, because also of the dependent nature of all the load sequences, it is permissible to consider event A in eqn. (18) as that due to cycling the complete load sequence through the structure and selecting the critical sequence as in eqn. (17). Once this is done, the next member may be similarly examined. The process of considering a load sequence in this way would not be valid if complete dependence did not exist between them. A more concise argument is given in the Appendix. The complementary situation, with mutually independent load sequences has been considered by Wen and Chen [13].
4. E X A M P L E
A two-dimensional truss modified from the special truss considered by Bjerager [14] will be used for analysis (see Fig. 2). Member behaviour will be assumed to be elastic-plastic and the same for all members [11]. For the present, the mean wave height will be assumed 12.5 m, the wave period fixed at 15 s and the wave length 300 m. This corresponds to realistic storm conditions. Table 1 gives the numerical data for the truss members with Ri being the mean strength of the ith member, whose area is Ai and diameter is D~. For purposes of illustration, the coefficient of variation of the m e m b e r strengths will be assumed to be 0.15 for all members, and the coefficient 20.Om i~ Q7
~i Q~
M.s.L.
•/"
~-,
'o7
_
I
!~
39.1m
Fig. 2. Example offshore structure.
°
TABLE 1 Numerical data of truss tower Member i
Ri (kN)
Area, A~ (mE)
Diameter, (m)
1-6 7 8 9 1o- 11 12-13 14-15
51890.0 84400.0 18535.0 33600.0 6675.0 11535.0 18535.0
0.324 0.053 0.116 0.210 0.042 0.074 0.116
2.5 1.0 1.5 2.0 0.9 1.2 1.5
Di
of correlation between member strength 0.5. The drag and mass coefficients will be assumed to be Ca = 1.3 and C m = 2.0. The deck load Q7 acting vertically on the tower has a mean of /~07 = 22 x 103 k N with a coefficient of variation VQ7 = 0.1. This load will be assumed to be completely independent of the wave loading. Using the above data and those of Table 1, eqn. (10) together with eqns. (5)-(7) yielded a total mean horizontal load on one leg of /-to = 428.11 cos 0 Icos 0 1+ 5 4 8 . 5 4 sin 0
(19)
which has a maximum mean value of 603.82 k N at phase angle 0 = 36 o. This corresponds to the location of the maximum total wave force for the whole structure. (Here 0 is referenced to joint no. 1.) For the wave forces, a coefficient of variation VQ = 0.4 will be assumed in the absence of sufficient data with which to calculate it using the theory herein. To replicate the wave force profile with loading only at the nodes, the following mean nodal forces were used for each leg: QI=Q2=
15.56kN
03 = Q4 = 243.54 k N Q5 = Q6 = 242.41 kN The results from the reliability analysis are given in Tables 2 and 3. The "failure path" refers to the sequence of member failure, using the m e m b e r notation given in Fig. 2. It will be seen that the failure modes which dominate the system reliability index are closely similar, at least for the first few modes, and that the reliability indices are also similar. It also follows that the bounds on system reliability are similar. As found earlier [12], the use of only one load profile (i.e., at 0 = 36 °) overestimates the system reliability index (i.e., underestimates the system probability of failure). However, for the realistic loading magnitude of the present example, the error is not large, being about 1%. As seen from Table 3, this observation also appears to be valid for a range of the deck loads Q7The effect of shorter wavelength conditions on the system reliability index for a complete wave cycle was not detectable. Table 4 shows the b o u n d s on the reliability indices obtained for a range of wavelengths. Short wavelength waves cause a greater force on the leading leg compared with the trailing leg. However, at any one level these forces are additive and, as shown in Table 1, the horizontal member connecting pairs of nodes is not involved in critical failure paths. The critical
8 TABLE 2 D o m i n a n t failure modes a n d reliability index fl = - ~
l(pf) 0 = 00 = 36 o (where m a x i m u m horizontal load occurred)
C o m p l e t e wave cycle Failure path
Reliability index
Failure path
Bs 2-10 1-11 4-12 3 13 4-13 3-12
4.0762 4.0711 4.0814 4.2969 4.3340 4.3348
Bounds
3.8150 3.8101
Reliability Index
Bs 2-10 1-11 4-12 4-13 3-12 3-13
4.0762 4.0828 4.2814 4.3340 4.3404 4.3908 3.8308 3.8274
Note: Q7 = deck load = 22,000 kN, wavelength = 300 m, wave period = 15 s.
TABLE 3 Effect of deck load o n b o u n d s on system reliability index fls Deck load
40,000 kN 30,000 kN 22,000 kN
Complete wave cycle
0 = 36 °
Lowerbound
Upperbound
Lowerbound
Upperbound
1.4504 2.7218 3.8150
1.3460 2.6969 3.8101
1.4562 2.7357 3.8308
1.3662 2.7202 3.8274
Note: Wavelength = 300 m.
members are, realistically, the "verticals" and, to a lesser extent, the diagonals. In a more detailed modelling of the way loads are applied to the structure, allowing, for example, for lateral loading along the horizontals, it would be expected that wavelength will have some effect on the results for reliability index.
TABLE 4 Effect of wavelength on b o u n d s on system reliability index fls Wavelength
Bounds on Bs
(m) 350 300 250 200 150 100 Note: Q2 = 22,000 kN.
3.8150 3.8150 3.8149 3.8150 3.8145 3.8167
3.8102 3.8101 3.8101 3.8102 3.8101 3.8123
5. C O N C L U S I O N T w o d i f f e r e n t t e c h n i q u e s w e r e c o m p a r e d herein for the a s s e s s m e n t o f the reliability of s i m p l i f i e d m u l t i - m e m b e r e d j a c k e t - t y p e o f f s h o r e s t r u c t u r e s u n d e r the a c t i o n o f a fully d e v e l o p e d , e x t r e m e seastate. It was a s s u m e d t h a t for such l o a d i n g c o n d i t i o n s the w a v e p e r i o d is essentially constant. It w a s f o u n d t h a t the c o n s i d e r a b l y m o r e i n v o l v e d a n d t i m e - c o n s u m i n g c a l c u l a t i o n of the s y s t e m reliability i n d e x c o r r e s p o n d i n g to the c o m p l e t e l o a d i n g cycle led to a slightly l o w e r e s t i m a t e o f the s y s t e m reliability i n d e x o b t a i n e d f r o m a n e s t i m a t e b a s e d o n a n a r b i t r a r i l y selected, b u t realistically e x t r e m e p o s i t i o n for the w a v e p r o d u c i n g the m o s t severe t o t a l w a v e lateral force c o n d i t i o n . It is c o n c l u d e d t h a t for m o s t p r a c t i c a l p u r p o s e s the s i m p l e r m e t h o d is a d e q u a t e a n d t h a t cycling o f the c o m p l e t e w a v e t h r o u g h a j a c k e t - t y p e o f f s h o r e s t r u c t u r e is n o t necessary.
REFERENCES 1 P. Thoft-Christensen and M.J. Baker, Structural Reliability Theory and Its Applications, Springer-Verlag, Berlin,
1982. 2 0 . Ditlevsen and P. Bjerager, Methods of structural systems reliability, Structural Safety, 3 (1986) 195-229. 3 R.E. Melchers, Structural Reliability Analysis and Prediction, Ellis Horwood/J. Wiley, Chichester, UK, 1987. 4 Y.F. Guenard, Application of System Reliability Analysis to Offshore Structures, Report No. 71, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA, 1984. 5 Y. Murotsu, M. Kishi, H. Okada, Y. Ikeda and S. Matsuzaki, Probabilistic collapse analysis of offshore structures, in: J.S. Chung et al. (Eds.), Proc. 4th Int. Offshore Mechanics and Arctic Engineering Symposium, ASME, Vol. 1, 1985, pp. 250-258. 6 T. Sarpkaya and M. Isaacson, Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, 1982. 7 L.E. Borgman, Wave forces on piling for narrow-band spectra, J. Waterways Harbors Div., ASCE, 91(WW3) (1965) 65-90. 8 A.M. Nafday and J. Wang, Probabilistic model of wave forces on cylindrical piles, J. Waterway Port Ocean Eng., ASCE, 109(2) (May 1983). 9 R.L. Wiegel, Oceanographical Engineering, Prentice-Hall, Englewood Cliffs, N J, 1964. 10 M.S. Longuet-Higgins, On the Joint Distribution of the Periods and Amplitudes of Sea Waves, J. Geophys. Res., 80(18) (1975) 2688-2694. 11 R.E. Melchers and L.K. Tang, Dominant failure modes in stochastic structural systems, Structural Safety, 2 (1984) 127-142. 12 H.Y. Chan and R.E. Melchers Reliability of complex structures under wave loads, in: N.C. Lind (Ed.), Proc. 5th Int. Conf. Applications of Statistics and Probability in Soil and Structural Engineering, Vancouver, Canada, 1987, pp. 166-173. 13 Y.K. Wen and H.C. Chen, System reliability under multiple hazards, Struct. Res. Ser. No. 526, University of Illinois, Urbana, IL, 1986. 14 P. Bjerager, Reliability Analysis of Structural Systems, Department of Structural Engineering, Technical University of Denmark, Lyngby, Denmark, 1984.
APPENDIX C o n s i d e r a s i m p l e structure, with piles at l o c a t i o n s 1 a n d 2. L e t A r e p r e s e n t the e v e n t " f a i l u r e of m e m b e r A " a n d let A 1 r e p r e s e n t the e v e n t " f a i l u r e of m e m b e r A u n d e r the a p p l i e d l o a d at l o c a t i o n 1". L e t a n a l o g o u s n o t a t i o n a p p l y for m e m b e r s B a n d C for l o c a t i o n 2.
10
Member A will fail if it fails under the internal action (load effect) resulting in it from force being applied at 1 or at 2. Let these events be denoted A 1 and A 2 respectively, then
P ( A ) = P ( A 1 UA2)
(A.1)
However, since the actions result from the same wave, (which is deterministic in time, for simplicity), it follows that 2
P(A~ U AE) = m a x P ( A i )
(1.2)
i~l
and similarly for members B and C. The mode failure probability for the member failure sequence A, B, C then becomes Pf =
P ( A C~B ~ C)
= e[(A, u A )n (B, u B )n (C, uC ) = P[( AI f-) B, n CI) U ( A1N B, f') C2) U ( AI A B2 (3 C,) u (A, n B2 n C2)u (12 n B, n C , ) u (A2 n B, n C2) U ( A 2 n B 2 N C,) U (A 2 n B 2 n C2) ]
= m a x P [ A , NBjC3Ck], i,j,k
i, j , k = l , 2
(A.3) (A.4)
This shows that the probability of failure associated with the failure mode (A, B, C) is the maximum of all such probabilities for all possible load sequences. Expression (A.4) is significantly easier to evaluate than (A.3). As a result of eqn. (A.2) it may be seen that it is sufficient to run through all load cases in the load sequence to determine the next most likely m e m b e r to fail and so determine the load case under which this occurs. For the member failure sequence (A, B, C), this procedure fixes the indices in (A.4). That it is permissible to consider all load cases in the load sequence at each stage of the Truncated Enumeration analysis has been considered in a different m a n n e r in Chan and Melchers [12].