Expected fatigue damage and expected extreme response for Morison-Type wave loading

EXTREME RESPONSE FOR MORISONTYPE WAVE LOADING

by

J.J.H. Brouwers & P.H.J. Verbeek

Koninklijke/Shell Exploratie & Produktie Laboratorium P.O. Box 60

2280 AB Rijswijk Z.H. The Netherlands

To be published in APPLIED OCEAN RESEARCH

Reproduction of this document in whole or in part is allowed if due acknowledgement is made to Shell Research B.V.

KONINKLIJKE/SHELL EXPLORATIE EN PRODUKTIE LABORATORIUM RIJSWIJK, THE NETHERLANDS

Abstract III

Introduction 1

Morison's equation 2

Probability distributions of peak force 3

Expected fatigue damage ₅

Expected extreme response ₈

Concluding remarks ₉

References ₁₂

Figures 1-3 Appendix A Appendix B

loading are used to indicate the effect of drag forces on the expected fatigue damage and the expected extreme response of quasi-statically responding (members of) offshore structures. Results are compared with those from commonly used equivalent

linear methods of analysis. It is found that the expected fatigue damage and the expected extreme response based on non-linear

methods are approximately equal to results from linear methods when inertia is the dominant force. However, in the event of the drag forces forming a considerable part of the total wave

loading, both fatigue damage and extreme response can

significantly exceed those predicted by linear methods. The difference between the two methods is quantified in terms of a drag-inertia parameter, which is directly related to the sea state under consideration.

EXPECTED FATIGUE DAMAGE AND EXPECTED EXTREME RESPONSE FOR MORISON-TYPE WAVE LOADING

INTRODUCTION

The environment of offshore structures is random in nature. Waves, wave loads and structural response vary randomly with time

and in a realistic model must be represented as stochastic variables quantified in terms of probability distributions. Accurate prediction of these distributions is important for assessing design parameters such as expected fatigue damage and expected extreme response.

A difficulty in the analysis of offshore structures in random seas is the prediction of the effect of non-linearities due to either environmental loading or non-linear structural

behaviour. For linear systems subject to Gaussian excitation, the response is also Gaussian, the properties of which can be

determined using spectral analysis techniques; see e.g. Bendat1. However, if the system contains non-linear elements, the response will no longer be Gaussian, and closed-form solutions _{can only be}

given in special cases;see e.g. Brouwers2. Some limited

information on the standard deviation of response may be obtained from equivalent linear solution methods; see e.g. Caughey3.

An example of a non-linear element is the drag term in

Morison's equation for the wave force on cylindrical members. For calculating the effect of this term on the probability

distributions of force, use can be made of methods of non-linear transformation of random variables: Pierson & Holmes4, Borgman5, Tung6, Tickell7 and Moe & Crandall8. Results thus obtained are of particular interest for the analysis of offshore structures whose response is linearly related to force and is quasi-static in

nature; e.g. the response of marine risers in the wave-active zone: Brouwers9, Verbeek10. In that case the probability

distributions of response, normalised with respect to standard deviation, are the same as those of the force.

In this paper, the non-linear probability distributions for Morison-type wave loading are used to indicate the effect of drag forces on expected fatigue damage and expected extreme response. As an extension to the numerical results previously

published7", analytical solutions are presented which reveal the non-linearity effect on expected fatigue damage and expected extreme response over the full force regime, i.e. from

inertia-dominant to drag-inertia-dominant, including the transition regime.

MORISON 'S EQUATION

Morison.'s equation is widely used to determine the wave force on members of offshore structures where reflection and diffraction effects are of secondary importance. For

unidirectional waves and zero current, the equation can be written as

F = 1/

PlTd CMV + PdCDv!vl, (1)

where

F = wave force per unit length acting on a member

v _{= wave-induced water-particle velocity normal to a member}

= wave-induced water-particle acceleration normal to a member p = water density

d = diameter of member (or equivalent diameter) CM = inertia coefficient (CM 2)

CD = drag coefficient (CD 1)

The first term on the right-hand side of equation (1) represents inertia force; the second term is the drag force, which is non-linear in water-particle velocity.

= 1

PROBABILITY DISTRIBUTIONS OF PEAK FORCE

For unidirectional linear Gaussian models of the sea, wave-induced water particle velocities and accelerations can be

represented as one-dimensional Gaussian variables of zero mean. The governing probability distributions of force, such as the distribution of peak values, can then be calculated from

equation (1 ) using methods of non-linear transformation of random variables48. For general wide-band representations of the

Gaussian variables, the formulae obtained from these methods can, in general, only be evaluated numerically by computer7.

Analytical descriptions can only be derived in special cases, e.g. by employing an approximate narrow-band model for the Gaussian variables,as done by Borgman5, or by disregarding the drag term or the inertia term in Morison's equation, as shown in Appendix A. In the subsequent paragraphs, the descriptions

obtained from Borgman's narrow-band model will be used to indicate the effect of non-linear drag forces _{on the peak}

distribution of force and on the expected fatigue damage and the expected extreme response. The accuracy of the solutions will be indicated by comparison with results obtained from the more

realistic wide-band model in typical cases.

The probability distribution of peak force P(F) derived from the narrow-band model can be expressed, in normalised form with respect to standard deviation, as

(3K2+1 )Fexp[1/3K2+1 )F2] when 0 F 1/2 KT1

(3K21)

1/2 (2) 1/ _1/ -1

1/3K21)

2 _{Kexp[1/3K2+1) 2K_1P+(SK2)} - 1/2 when 1/2 Cl (3K21) _{F* <}

In this equation, F is the peak value of force divided by the standard deviation of force5 and K is the drag-inertia parameter. The drag-inertia parameter compares the magnitude of the drag force with the magnitude of the inertia force and is defined by

2C a

_Dv

M my

where a and w _{are the standard deviation and the mean zero-up}

v my

crossing frequency of the wave-induced water-particle velocity normal to the member, respectively. It should be noted here that inherent in a narrow-band approximation, the central frequencies of sea surface elevation, water-particle velocity and

acceleration are assumed to be almost the same. Crandall and Moe's asymptotic analysis8 of a more realistic wide-band process has shown that the frequency of the higher force peaks is

determined by the mean zero-up crossing frequency o± the velocity process. The frequency in the drag-inertia parameter of eq. ₍₃₎ has been defined accordingly.

The standard deviation and frequency of water-particle velocity can be calculated for a given power spectrum of the waves. Assuming a deep-water representation for the wave

properties, the expression for the drag-inertia parameter near mean sea level can then be reduced to

2C

K D TI

4

msl - 1TdCM

where a _{is the standard deviation of sea the surface elevation}

TI

( significant wave height divided by four) and is the spectral width of sea surface elevation ( 0.7).

From equation (2) it can be verified that for K 0, i.e. when inertia forces dominate, the probability distribution of peak force reduces to that of a narrow-band G-aussian _process, i.e. the Rayleigh distribution

Fexp(h/2).

(5)

(3)

On the other hand, when K i.e. when drag forces dominate, equation (2) reduces to

pp(F*)

2 exp(- 2 *)

(6)

In Fig. 1 plots are presented of the cumulative probability

distribution of peak force,

PF(F*) =

I

PF(F*)dP*

(7)

0

as obtained from the above narrow-band solutions (2), on a

Rayleigh scale, for the inertia-dominant limit (K = 0), the drag-dominant limit (K _{+ )} and for K = 0.5. Furthermore, in this

figure plots are given of the cumulative peak distribution of

force obtained from a wide-band model for the inertia- and drag-dominant limit, respectively. Explicit descriptions o± _these wide-band distributions are given in Appendix A.

From Pig. 1 it can be seen that the effect of non-linear

drag forces is most pronounced in the tails of the distribution functions; the probability of exceedance of large peaks _increases with increasing value of the drag-inertia parameter K.

Furthermore, it will be noted that in the drag- and inertia-dominant limits, the narrow-band model yields results _{which are} conservative and close to those obtained from the _{corresponding} wide-band model. Therefore, the narrow-band model has been

employed throughout the following analysis of expected _fatigue damage and expected extreme response. Errors introduced _{in these} parameters due to band-width approximations _{are discussed in} Appendix B.

EXPECTED FATIGUE DAMAGE

Assume that fatigue damage accumulates according _{to Miner's} hypothesis. Furthermore, assume that for the i-th stress peak

with (single) amplitude Xj of a narrow-band stress history, fatigue damage can be written in the form

=

x,

₍₈₎

where and are material constants. A common method of

calculating the expected fatigue damage E[D] after n stress peaks is then12

E[D1 =

naf *(*)*

(9)

where is the standard deviation of stress response and

p(x) is the probability distribution of peak stress normalised with respect to the standard deviation (x

=

To indicate the effect of non-linear drag forces, we shall relate the above expression for the expected fatigue damage to that obtained from equivalent linear solution methods3. In

general, an equivalent linear method yields quite accurate values for the standard deviation of response3'5. The method, however, is incapable of predicting deviations from the Gaussian form in the probability distributions. In the equivalent linearisation approach the peak distribution of response, p(x*), is equal to

the peak distribution of a linear Gaussian

_{process, {p(x)}1..}

The expected fatigue damage based on non-linear distributions _can

thus be related to the expected fatigue damage obtained _from equivalent linear methods as

I xap(x)dx

E[D] 0 ELDJ -lin f x* 0

Assume now that stress response is linearly related to _wave force and is quasi-static in nature. The fatigue damage ratio can then be calculated from equation (10) by substituting for

P 630

p(x) the distribution given by equation (2) and for (px(x*)}iin the Rayleigh distribution given by equation (5). After some

algebraic manipulation, it can then be shown that

cx cx 1

2cx

1 1 E[D] -

(3K2+1)r1()[y(,)+2

K exp()r(cx,J],

(11) ELD]

un

8K2 8K 4K

where r(a) is the Gamma function and y(a,z) and r(a,z) are

Incomplete Gamma functions as defined by Abramowitz & Stegun13. Figure 2 presents plots of E[D]/E[D]u versus the drag-inertia parameter K for values of the slope in the SN-curve cx =

4.38, 4 and 3. Here, cx = 4.38 corresponds to the central part of

the AWS-X curve14, cx = 4 to Gurney's B-curve15 and cx = 3 to

Gurney's D-curve1 5.

From Fig. 2 it can be seen that the ratio of the expected fatigue damage based on non-linear probability distributions to the expected fatigue damage obtained from equivalent linear methods becomes equal to unity when K + 0 (inertia forces

dominant), and reaches a maximum value when K + (drag forces dominant). The ratio of the expected fatigue damages changes significantly with K in the region 0.1 < K < 10. Equivalent linear methods underestimate the expected fatigue damage when K > 0.3.It should be mentioned here that deviations are

significant for large values of the slope cx in the SN-curve. As appears from Pig. 1, non-linear drag forces lead to an increase in probability values in the tails of the peak distribution. Furthermore, with increasing cx, _{the value of the integral on the}

right-hand side of equation (9) becomes increasingly _{dependent on} -these tails of the peak distribution. Hence, deviations between

expected fatigue damage obtained from non-linear distributions, and expected fatigue damage obtained from linear distributions increase with increasing value of the slope cx of the SN-curve. As indicated in Appendix B, errors in the above fatigue damage _ratio due to non-zero spectral width are small. Futhermore, the _present

results are in qualitative agreement with numerical results

11

calculated in the lower (-regime (K < 0.5)

EXPECTED EXTREME RESPONSE

Consider a narrow-band wave force history of n peaks. Based on the assumption that peaks are mutually independent, the

probability distribution of the largest peak or extreme

j16

P(F) =

dP(P)

₍₁₂₎

dP

F=F/ap

where _{is standard deviation of force and P(P) is the} cumulative probability density of peak force normalised with

respect to the standard deviation. The expected value E[P] of _the extreme force is given by

dP(F)

E[F]

=

I

_dF

dP.

0

The integral on the right-hand side of this equation _{can be} calculated numerically when substituting for

_{p() the}

expressions given by equations (2) and (7). Results obtained for _{for a history of 1000 peaks (= approximately three} hours' sea state) are plotted in Pig. 3 versus the drag-inertia parameter K. It should be noted that these results _{are also valid} for variables (displacements and stresses) which are linearly and quasi-statically, related to force.

From Fig. 3 it can be seen that the ratio of the _expected extreme value to the standard deviation reaches a minimum value when K + 0 (inertia forces dominant) and a maximum value when K + _{(drag forces dominant). The ratio increases significantly} with K in the region 0.1 < K < _{10. The minimum value obtained}

(13)

when K + 0 corresponds to that of a linear Gaussian process, which would also be obtained, independent of K, from equivalent linear methods. Analytical expressions for this value as a

function of the number of peaks n have been given by Longuet-Higgins16. In the drag-dominant limit, i.e. for K + , it can be

shown that

E[F]/ap = -

{(n) +

+ (14)

where is the Digamma function as given by Abramowitz &

Stegun13, and y _{(= 0.57722) is Euler's constant. For large n, the}

simplification can be made that 1(n) = ln n with an error o± (2n)_1.

As indicated in Appendix B, errors in the above values for the expected extreme response due to non-zero spectral width are small. _{irthermore, the above results are in agreement with the} numerically calculated extreme values of Tickell7.

CONCLUDING REMARKS

The analysis has shown that non-linear drag forces _{can lead} to the expected fatigue damage and the expected extreme _response of quasi-statically responding (members of) offshore structures significantly exceeding those predicted by equivalent linear methods. The deviations depend on the value of the drag-inertia parameter K, which is proportional to the significant _{wave height} divided by the member diameter [of. eqs. _{(3) and (4)]. Deviations} in the expected fatigue damage and in the expected extreme

response become significant for drag-inertia ratios K beyond _0.5 and 0.2, respectively; see Figs. 2 and 3.

The ultimate effect of non-linear drag forces _{on fatigue} damage for varying sea states can be established as a summation of the damage caused in all the different sea states to which a

(member of a) structure is exposed. In general, the above

the evaluation of the total accumulated damage whenever the contribution of drag forces is of appreciable magnitude in

comparison with the inertia forces during a significant part of the life of the structure. Typical examples are small-diameter members of jacket structures in the wave-active zone, conductors and small-diameter risers in areas with severe wave conditions, e.g. the northern North Sea. The expected extreme response, _on the other hand, is mainly of interest for the analysis of

structures in extreme sea states, i.e. design events such as a 100 years storm condition. The effect of non-linearity _{of the} drag forces may then well be significant, even for larger diameter structures.

Results obtained can directly be used to assess the effect of non-linear drag forces on the quasi-static response of single-member structures (e.g. risers) and _{on the local response of}

multi-member structures (e.g. jackets) to direct wave loading. Apart from that, results can also be applied in the analysis of' the global response of multi-member structures. For _{such complex} structures numerical solution routines, based _{on equivalent}

linear methods of analysis, are often used to calculate _the standard deviation of response at various points in the

structure. Using this method, however, it is also _{possible to} determine the contribution of non-linear drag _{forces, as compared} to the contribution of linear inertia forces, to the _standard deviation response, thus enabling an effective K-value at any point in the structure to be assessed. The figures presented _in this paper can then be used to correct for the _{effect of} non-linear probability distributions, which _{are not taken into} account in equivalent linear methods of' _{analysis. In a similar} way, the present results can be used to correct, in _an

approximate manner, numerical results for _{response in cases of} multidirectional waves and current.

B.V. for permission to publish this paper. Furthermore they wish to thank Dr. J.H. Struik of Koninklijke/Shell Exploratie en

REFERENCES

Bendat, J.S., Principles and Applications o± Random Noise Theory.

John Wiley & Sons, New York (1955).

Brouwers, J.J.H., Response near Resonance of Non-Linearly Damped Systems subject to Random Excitation with

Application to Marine Risers.

Ocean Engng, 9 (1982), _{pp. 235-257.}

Caughey, T.K., Equivalent Linearisation Techniques. J. Acoust. Soc. Am. 35 (1963), _{pp. 1706-1711.}

Pierson, W.J. Jr & Holmes, P., Irregular Wave Forces on a Pile.

J. Waterways and Harbors, Proc. Amer. Soc. Civil Eng. ₉₁ (ww4) (1965), pp. 1-10.

Borgman, L.E., Statistical Models for Ocean Waves and Wave Forces.

Advances in Hydroscience (ed. V.T. Chow), Academic Press, New York (1972), Vol. 8.

Tung, C.C., Peak Distribution of Random Wave-Current Force. J. of the Engineering Mechanics Division, ASCE, EMS

(1974), pp. 873-884.

Tickell, R.G., Continuous Randoth Wave Loading on Structural Members.

The Structural Engineer 55 (1977), no. 5, _{pp. 209-222.}

Moe, . & Crandall, S.H., Extremes of Morison-Type Wave

Loading on a Single Pile.

Trans. ASME, J. Mech. Design 100 (1978), _{pp. 100-104.} Brouwers, J.J.H., Analytical Methods for Predicting _the

Response of Marine Risers.

Proc. Royal Netherlands Academy of Arts and Sciences, Series B 85 (1982), pp. 381-400.

Verbeek, P.H.J., Analysis of Riser Measurements _{in the North}

Sea.

Proc. 15th Annual Offshore Technology Conference, Paper 4562, Houston (1983).

Tickell, R.G., Holmes, P., Wave Loads on Ocean Structures - a Stochastic Model, ASCE Convention, Boston

_(1979).

Crandall, S.H., Zero-Crossings, Peaks and Other Statistical Measures of Random Responses.

J. Acoust. Soc. Am.

35 (1963), pp. 1693-1699.

Abramowitz, M. & Stegun, l.A., Handbook of Mathematical Functions.

Dover Publications, New York

_(1968).

American Welding Society, Structural Welding Code Steel.

AWSD 1.1-79,

Miami

(1979).

Gurney, T.R., Fatigue Design Rules for Welded Steel Structures.

The Welding Institute Research Bulletin,

_{17 (1976).}

Longuet-Higgins, M.S., On the Statistical Distribution of the Heights of Sea Waves.

7.0 6.0 5.0

z

Q I-4 > 4.0

4

z

U, -S w 4 2.0 1.0 0.0 -1.0 0.010 NARROW- BAND I /

-

WIDE-8AND(SPECTRAL WIDTH 0.7) /

K=DRAG-INERTIA PARAMETER K-.--aD

_/

/

/

I

/

/

/

/

/ I /

I

/

/

K0.5

/ /

/

/

/ I /

/

/

/

a

/

I

/

-a -a

-,,,,,,_,-a,-a-a-a

K0

0.500 0.900 0.990 PROBABILITY OF NON-EXCEEDANCE 0.999

WIDE - BAND AND NARROW- BAND CUMULAT1VEP.EAK DISTRIBUTIONS.. OF FORCE ON A RAYLEIGH SCALE

C, 2

a

w6

I-1L5

0

Ui

0

Ui a. >( Ui Ui

z

-J 00.01

RATIO OF EXPECTED FATIGUE DAMAGE BASED ON NON-LINEAR PROBABILITY DISTRIBUTIONS TO EXPECTED FATIGUE DAMAGE DERIVED FROM EQUIVALENT LINEAR METHODS VERSUS

DRAG-INERTiA PARAMETER K FOR SLOP3 OF THE SN CURVE a:4.3814 AND 3.

I I

III!

I I I

II

I I I I

III

0.1 1 10 100

z

0

I-<8

>

w

0

z

I-ci, U -J

u5

U

I-w4

0

U

0

U 001 I I I I I I 0.1 I I I DRAG-INERTIA PARAMETER K

RATIO OF EXPECTED EX-rREME VALUE TO STANDARD DEVIATION VERSUS DRAG-INERTIA

PARA-METER K FOR A HISTORY OF 1000 PEAKS.

1000 PEAKS

I I 111111 I I

It

1 10 100

in terms of the joint probability density of force and its first and second derivatives, p(F,F,P), as7

0.. .. .. + 0..

p(P)

_{= 5} Fp(F,0,P)dF/ _{5 5} Fp(F,0,F)dFdF. (Al)

Here, a description for p(F,P,F) can be derived by first constructing a inultivariate Gaussian distribution for the velocity and its derivatives and then transforming from the

velocity variables and its derivatives to the force variables and its derivatives using Morison's equation: e.g., see Tickell7. It is then found that in the drag dominated limit, i.e. _when

FD = _{1/2 PdODvlvL} (A2)

and in the inertia dominated limit, i.e. when

P1 = 1/4 P1d2CM1 _(A3)

the probability density of peak force can be expressed _{in terms} of known functions. In normalised form with respect to the

standard deviation of force we can write for the drag limit

2/2 F

1-c

* 21/2 /2 F

] erfc[-(

₂ ) lFl + PFD(P*) = (1-c ) exp[ 2 I *1

2/2"4

C 'F 11/2exPL-' I

2*

I 2c (A4)

In this equation, c is spectral width of water-particle velocity and erfc is the error function as defined by Abramowitz &

Stegun13. For the inertia limit ₍₌ Rice's distribution) we can write, 1/2

F2

2 1/2 1-c -= 1/1_c2) _F

_{exp[-J erfc[}

2 *] 2 exp{- _2j, (A5) (2)/2 2c

where c is spectral width of water-particle acceleration.

verified that the wide-band peak distributions of drag-dominant and inertia-dominant force reduce to the corresponding narrow-band solutions [cf. eqs. (5) and (6)] when + 0. Furthermore, it can be verified that the tails of the wide-band distribution of drag-dominant force,

1/ 1/ 1/

pF(F*)

2(i_2) 2exp()

when F + , (Bi)

differs by a factor (1_c2)l/2 _{from the corresponding narrow-band}

solution [cf. eq. (6)]. Exactly the same difference exists between the tails of the wide-band distribution of

inertia-dominant force (the Rice distribution) and the corresponding narrow-band solution (the Rayleigh distribution). Moreover, the tails of Tickell's7 wide-band distributions calculated

numerically for several K-values in the drag-inertia transition regime, also differ by a factor of approximately

(1_2) /2 _{from the present narrow-band solutions [cf. eq.}

(2)].

These observations indicate that the ratio of the expected

fatigue damage based on non-linear distributions to the expected fatigue damage based on linear distributions calculated from equation (10) using narrow-band solutions will be close to that obtained using wide-band solutions; in particular when the

slope in the SN-curve is large, in which case the value of the right hand side of equation (10) is primarily determined by the tails of the distribution functions. Errors due to non-zero spectral width in the given expected fatigue damage ratios can thus be expected to be small.

Deviations due to non-zero spectral width in the expected extreme values using narrow-band solutions (5) and (6) can be shown to be of relative magnitude (41n nYln(1-c2) and

(21n n) 1ln(1-2) in the inertia- and drag-dominant limit, respectively. As long as the number of peaks n is large, these deviations are small: e.g., for = 0.7 and n = 1000 the given extreme values reduce by 2.4 and 4.9 per cent for the inertia-and drag-dominant limit, respectively.