E L S E V I E R

Applied Ocean Research 15 (1994) 373-380 © 1994 Elsevier Science Limited

Printed in Great Britain. All fights reserved 0141 - I 187/94/$07.00

Technical Note

Extreme peak value vessel response combinations with wide band spectra

James Hamilton Brown and Root Ltd, 150 Broadway, Wimbledon, London, UK, SW19 1RX

(Received 5 April 1993; revised manuscript received 27 July 1993; accepted 28 September 1993)

An extension is proposed to the standard peak value distribution for time histories possessing wide banded spectra, which allows the distribution of an associated variable to be calculated at the instant that the peak value occurs, the theory is applicable to a wide range of practical circumstances, where it is desired to specify the value of associated response variables at the instant that the dominant response is a maximum. An example is given of the specification of coincident heave and roll of a barge for input to a structural load case analysis.

INTRODUCTION

The design of an offshore structure involves the estimation of the 'design' or 'characteristic' values of certain measures of strength (utilisation factors, limit states) under the action of the environment. These limit states are rarely linear functions of the dynamic stresses, although the dynamic stresses themselves are typically estimated by linear spectral methods, and the distri- bution of their extreme values can be obtained by standard methods, e.g. that of Ochi. l Thus, to calculate the limit state, typically the values of a number of dynamic stresses must be specified simultaneously.

One option is evidently to specify that all the individual stresses will be extreme (e.g. have a return period of the 3 h in the design storm); this, however, will be unduly conservative. This paper investigates the statistics of the value of one dynamic variable, given the value of another dynamic variable.

In the limit of a linear narrow banded response spectrum (i.e. when the response is nearly periodic), the relationship between response variables is deterministic and can be represented as a (constant) phase difference. In this case, even for limit states which are nonlinear functions of a number of first-order dynamic variables, it is a simple matter to establish the phase relationships between the components, and search for the worst utilisation factor by varying the phase of one of the variables through 360 ° .

For broad banded response spectra, however, the relationship between variables is not deterministic, and it is more difficult to decide which response com- binations should be used to test the limit state without being either unduly conservative or unduly uncon- servative. In particular, the individual extremes are often combined with the most onerous choice of sign (+, 0 or - ) to yield the characteristic value of a limit state, and this can be extremely conservative.

373

T H E O R E T I C A L D E V E L O P M E N T

The distribution of the peaks of a random process was first investigated by Rice 2 and subsequently by Cartwright and Longuet-Higgins, 3 Longuet-Higgins 4 and Ochi, 1 among others. The approach adopted here follows the notation of Ref. 5, and can be regarded as an extension to the work of Boccotti. 6

The following joint p.d.f. (probability density func- tion) of a Gaussian random process x(t) is used in this paper:

1 exp - i l l ~ H) (1) f (n )=(27r )3 /2 ( i z l ) l / 2 ( 1 T - 1

where H is the vector:

H = (x(t) xt( t ) Xn(t) ) T

and E is the covariance matrix obtained from the

374 J. Hamilton

spectral moments of 'x':

° 1 E = 0 m 2 0

--m 2 0 m 4

Assuming that x(t) is linearly related to the wave process:

[~ S(w)(ITxl) 2dw m2 = [~ S(w)(ITx])2w 2 dw m o j 0 J 0

S(oJ ) ( [Tx[ )2034 m4 = J o dw

where S(w) is the wave spectral energy density, and T x is the transfer function of x(t).

The average number of positive maxima (satisfying X > 0, X t = O, Xtt ( 0 ) , per unit time, with peak values larger than ~ is given by t

= E(N((, t)) = [xtt[f(x, O, xtt) dxtt dx OG

The cumulative distribution of the peaks is then approximated by

E(N({, t)) N¢ F(¢) = Pr (E > ¢) = E(N(O, t)) = -~o

(Note : E(N(O, t)) = No )

This has been evaluated, 4 using standard integrals, to be

2 F(rl, e) =

(1 + v q -

- ½ ( 1 - v/1--e2)+dg(~) . . . ~ r / )

where

rl = ~ o o ; ~ exp - du

is the cumulative normal, while the average rate of the peaks is

lo j ° -~o = ]x.[ f ( x , O, x . ) dxtt dx - - 0 0

1 ( ~ ° m ~ ) 1 C + 1v'i~2-e2~ mv~ ° =4--~ + =4-~ ~ ]

These are shown in Fig. 1.

DISTRIBUTION OF ASSOCIATED VARIABLES

The problem to be addressed is of determining the

probability distribution of a variable y (t), at the time that a peak x = Xma x occurs in x(t). A typical situation would be the transportation analysis of a barge, in which we wish to determine what values of heave acceleration (x(t), say) could reasonably be expected to occur coincident with the worst roll angle (y, say). This could form the basis of a load case specification for a structural analysis, or could be used in the reliability analysis of extreme load.

In general, we can calculate the correlation coefficient between x and y:

E(xy) = v/E(x2) (g )

where

E(xy) = I o S(w) ½ (VxTyy + Ty-Txx) dw

where the overbar here signifies the complex conjugate. It is convenient, however, to consider the variable z(t), given by

z(t) = y(t) - CxY~x cry

where

C r x = ~ ; c r . = ~

are the standard deviations, which has a zero correlation with x(t). If x and y are Gaussian processes as is assumed above, then z will also be Gaussian with a probability density function:

1 -'2--~ p z ( z ) - e x p d = (1 - 2

However, although z is uncorrelated with x, they are not necessarily independent processes. In the limit of a narrow spectrum, zero correlation will merely reflect a 90 ° phase lag, and the distribution of z, when x is a maximum will have a small (in the limit zero) standard deviation, even though crz may itself be large (crz = Cry when x and y are uncorrelated). Thus, it will only be with wide spectra that the probability density function Pz can made the basis of estimating the distribution of y when x is at a crest.

For the general case of processes with small, but nonzero spectral width, we can proceed by defining a new variable # which has zero correlations with x, xt, and xtt:

# = y -+ ( m 2 m y 2 - m y ° m 4 ) x - m y l x t

(morn4 - m~) m2 my2mo - myom2

+ (m0m 4 -- m 2) x t t

Extreme peak value vessel response combinations 375

R(rI,Q=d~FoI,~) d:

where

my 0 = E(xy) = I o

0 . 4

I I l i I I

¢:/ -\ "-\

/ ~ \'\ ///

/, //'

/, / ,

I 0.5 1

~ ~-.

1 . 5 2 2 . 5 3 3 . 5

Fig. 1. Probability density of positive peaks as a function of the bandwidth parameter e.

where f ( H ) is the original probability without #:

S(w) ½ (Tx~y + Ty'~x) dw

myl = E(xty) = S(w)~w(Tx(w)Ty(w)

-

too my2 :-E(xttY) = Jo S(w) l w2 ( Tx(w) ~y(W)

+

and Tx and Ty are the transfer functions for x and y, respectively

The joint p.d.f, then can be derived from

m 0 0 --m E i ] 0 m 2 0

~Z = --mE 0 m 4 0 0 0 muo

where

2 m2yo myl (my2mo -- my0m2) 2 mo m2 mo(m4mo-m~)

muo = E(# 2) ---- myy

myy = E(y2)

1 exp ( - 1ZTEzlZ ) f(Z) = (27r)2(lEzl)l/2

and Z is the state vector described by:

Z r = (x(t) xt(t ) xtt(t) #(t))

Since # is independent of H = (x, xt, xtt) the inversion gives the following result:

f (Z) = f u ( # ) f ( H )

f ( n ) =

distribution

(2~) 3/2 ~/(mo m4 - m2)m2

I x exp ~ (m 0 m 4 -- mE) m2

a n d

fu(#) = ~ exp 2

The value of '#' at the peak is Gaussian, and has a standard deviation given by

OfF, ~ v[m-#o

Again, since # is independent of H, the joint distribution of the peaks, and #, is simply the product of the two distributions:

fp(~, #) = R(~, e)f~,(/z)

However, since # is a function of x , as well as x (at the crest), it is necessary to use the joint height and period distribution of the peaks to determine the distribution of y. The average distribution can then be obtained by integrating over the distribution xu.

The joint p.d.f, of x and xu when xt = 0 can be obtained, using eqn (1), and renormalising:

f (x , xtt) = f (x , 0, xu)

I o IOoof(x' O'xtt) dzttdx

e x p [ - l ( x2m4+-2xx---gm2+'-x2tm°'~]mo m4 -- m2 ] J

27re(m0 m4 -- m 2)

376 J. Hami l ton

O.O8

- -D==2 .% 0.06

D=4.%

_ D = 1 6 % o o4

__ D=64.% 0.02

0.2 0.4 0.6 0.8 1 1.2

(0

frequency (radians/second)

Fig. 2. Simulated spectral energy density of barge roll for different damping rates (wS(w) against log(w)).

A t a pa r t i cu l a r value o f x, the d i s t r ibu t ion o f the second

der ivat ive xtt is given by

f ( x t t ) = exp

[ [(xt,+mZx~ 2 ] 1 \ mo / I

- 2 - - m'-'~2 [ j

which is normal , mean - m 2 x / m o , s t a n d a r d devia t ion = root(m4 - m22/mo) now

(m 2 my2 -- myo m4) . . . my2 mo - myo m2 # = y + - - - - - ~ - - ; .~--r

(mo m+ - m2) (mom4 - m 2) xt,

Thus, since E (#) = 0:

E ( y ) (m2 my2 - myo m4) x = - ( m o m 4 _ m 2)

+ my2 mo - myo m2 m 2 x x =- my 0 - -

(morn4 - m~) m0 m0

as expected. The s t anda rd dev ia t ion o f y at the peak value x is,

however , the s t a n d a r d devia t ion o f

my2 mo - myo m2 # ( m o m 4 _ m 2) xtt

which, since # and xtt are independent, is given by

4 / ' : E - m ~ ~00 = mu° + (m--o m--'4----" m~)m'-'-'~o

2 2 myo myl

= myy mo m2

Table 1. Simulated barge rob bandwidth parameter and correlation factors

(oA) --if-- %

0"5 0"414 -0"648 -0"024 0"761 11 "8 3"693 -2"393 0"296 0-019 1 0"407 -0"702 -0-044 0"711 11 "52 3"699 -2"597 0"287 0"036 2 0"388 -0"784 -0"076 0"616 11 "09 3'71 -2"908 0"267 0"064 4 0-358 -0"856 -0"13 0"501 10-67 3"72 -3 '183 0"24 0"115 8 0"327 -0"89 -0"228 0"395 10"35 3"728 -3"317 0"213 0"209 16 0"303 -0"866 -0"391 0"311 10"05 3"736 -3 '237 0"2 0"37 32 0 "291 -0" 749 -0" 608 0-264 9.74 3" 745 -2" 805 0"213 0- 595 64 0"299 -0" 536 -0"801 0"266 9.43 3'753 -2"012 0-251 0"809

Key: Ccrit : damping as a fraction of critical; ~ = bandwidth parameter, Cyx, C ,, = correlation factors of simulated surge (y) with . . . Y . . .

roll (x) and roll velocity (x,); Fyx = standard dewatlon of the combined value o f y (umts of %); T~ = zero crossing period of the roll, F x = maximum roll factor (Xmaxfirx), for a record length of 3 h; Cyx Fx is the expected value of surge, y (measured in standard deviations ~y) associated with Xmax; ~rm is the standard deviation of m; and C~y t is the correlation of x with Yr

E 2 - - , , / 1 - - =

3"36OO vm2 1 + 1CT-~-e 2 N = - ~ z F x = V / 2 1 n ( N )

1

g.

0 . 6 - I - -

- o .

° p m _ 0.4 ° y

t~yp

o.2 ° y

Extreme peak value vessel response combinations

I[HII// Fll

377

o . i i io ioo

D

%

Critical Damping

Fig. 3. Dependence of the bandwidth parameter, and correlation factors for barge roll on damping rate. (For key see Table 1.)

~m 2 2 myo myl O'yp= YY m---~ m---~2 ~F~x

where

F,x = x / 1 - -

For values of the bandwidth parameter E less than 0.9 we can use the approximate given by Longuet Higgins 4 and Ochi I for the most probable maximum value ofx(t) in N cycles:

Xmax = a x v ~ l n (N)

SAMPLE RESULTS

To provide a realistic simulation of these effects we consider the roll response of a barge, with transfer function approximated by the expression:

(2 sin (½kB) - kB cos (l kB) ) exp (-kD)

Tx(w' = [1,2iCcri t ~v ( ~ ) 2 ] w-":n - (k2B2)

This is the fundamental result of the paper. Note that, for a narrow spectrum, Cyx, and Cyo will correspond effectively to the cosine and the sine of the phase lag between y and x; thus, the variation of y(t) at the peak will be represented completely by its expected value (the first term). In this case, y will be proportional to x. Conversely, for a wide bandwidth spectrum, there will be a significant variation possible for y especially when Cxy is small.

4

I c~'Fx I - O -

I~ Fxl o~3Oy

Oy

i

0 0.1

.... tl ̧tilt :tt lttt

IIIlll Illll 1 IO lOO

D

%

Critical Damping

Fig. 4. Maximum and minimum bounds for barge surge at maximum barge roll, as a function of damping rate. (For key see Table 1, bounds drawn at + / - one standard deviation).

378 J. Hamilton

1 _ S g - - S

6y

Sx._L_ 1 2

f i x

S y ~ - O"y

I I I I

[ I J I L

i ̧

/ \ / \

0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2

O~r ~ p

frequency (radians/second)

Fig. 5. Comparison of normalised spectra for #,y, and x (D = 4%).

where

2 m k = - - ; g=9.807_s--V g

B is the beam, D is the draught, with natural roll of frequency wn, and a roll damping rate, equal to Ccrit of critical. We have chosen

B = 5 0 m T = 6 m 2~- - - = 15s 0 3 n

and calculated the moments and correlation functions for a range of critical damping values, and a mean Jonswap spectrum (3' = 3-3, Tp = 10 s).

As the damping changes the spectral energy density of x changes, and the bandwidth parameter changes value. Figure 2 shows the spectral energy density Sx(w) of the 'roll', x.

The response y(t), which we will compare with x(t), is that of barge sway, approximated by the transfer function:

Ty(w) = sin

It should be emphasised that these transfer functions are approximate (although based on simplistic physical reasoning) and have been used instead of more accurate values, so that the calculations presented can be more easily reproduced by the reader. The calculated bandwidth parameters and correlation coefficients are given in Table 1 and Fig. 3. A further point to notice is that the standard deviation of the primary peak values xmax and Yraax are themselves significant.

They can be estimated from the standard deviation of

the equivalent Gumbel distribution (Ref. 5, p. 534):

2 71" 0 ̀2 71" O" x

ff Fy -- V~ Ymax O'Fx -- ~ Xmax

Figure 4 gives the expected, and one standard deviation bounds for the 'sway' y(t), associated with the extreme value of roll, x(t), as functions of the damping rate. Note that, at the one standard deviation level, the upper bounds of y associated with Xma x are less than the individual maxima of y at the one standard deviation level. However, at the two or three standard deviation level this will not be the case

Figure 5 compares the normalised spectra for x ,y and #. Note that the spectral energy content of # is mostly at higher frequencies than that for x, y and #(t) should probably therefore be regarded as normally distributed noise, when calculating the combination of y and x.

- N m a ~ ,

z

~ m a x v

| ~ q - ~ )( -Zmax

i

X

Fig . 6. Illustration of domains for joint distributions and design event combinations. 'O' indicates a worst on worst event; 'X' indicates the expected c o m b i n a t i o n o f e v e n t s a t a n extreme. Note that x and z are assumed to be independent.

Extreme peak value vessel response combinations 379

Standard deviations

¢~y

T/l,ll Illlll Illlll

o o.i 1

D.

%

I1111 ' 11 lllll lllll IIIlll llll

IO ioo

Where Y(Pexc) satisfies

cnorm( Y ~ / - cnorm(-Fy [Cyx'Fxl ) \ ay /

cnorm(Fy - I Cyx.Fxl ) - enorm(-Fy - I Cyx.Fx[ )=Pexc

Fig. 7. Modified confidence limits for the maximum value of y to combine with Xma x = Fx.ax.

DISCUSSION

At first sight it would seem to be contradictory that it is possible for the value of y associated with the 'maximum' value of x can be larger than the 'maximum' value of y itself. This, however, would be misinterpreting the meaning of the values obtained.

Firstly, it is evidently inconsistent to add two standard deviations to the expected value of y, to give high confidence that the value used cannot be exceeded, when the same degree of certainty of exceeding the expected maximum values is not required of y (and x) when considered individually.

Secondly, Fig. 6 indicates the concepts involved when considering the extreme statistics of two independent variables, it is conventionally considered that both variables take values below their 'maxima' (actually 1/N exceedence levels; N = 1000 typically). Thus, the domain of interest would be a box as illustrated in Fig. 6.

0 _< Ix[ _< Xmax[ 0 ~ [z[ <__ Zmax

where z(t) is assumed to be independent of x(t). In fact the 1 /N level of reliability need not be achieved if the domain excluded some values of x less than Xmax. Possible domains 'guaranteeing' probabilities of exceedence of less than 1 /N would have to be either

0 <_ Ix I -< Xmax Izl < oo

or alternatively an ellipse could be drawn as shown in Fig. 6,

o< (±L(z] _ \~x/ \~z]

where FN is chosen appropriately. However, note that for both cases some values of z would be greater than 'Zmax', and x than Xma x.

For both the reasons given above, the results obtained are not inconsistent; however, they have to be carefully interpreted if they are to be used to set deterministic design combinations rather than in a more general reliability based approach to extreme loading.

Design events have traditionally been looked upon as the most probable extremes in the storm. Thus, there is a strong case for only considering the expected combination of the parameter y with Xmax. For reasons of convenience, it is, however, often appropriate to consider extreme rather than expected values of secondary parameters when it is not intended to confirm individually the load effect of all the com- binations Ymax, E (x I y = Ymax). However, from the above arguments, only values of y which do not exceed Yrnax should be considered to contribute to the distribution of y. The truncated distribution would not be Gaussian, and the extreme values and confidence limits therefore nonstandard. Figure 7 gives the modified 84 and 97.7% cumulative probability levels for Y (corresponding to one and two standard deviations of a Gaussian variable). Note that these are now less than Ymax, and are perhaps therefore more acceptable as a design combination than the simple one or two standard deviation bounds.

It should be emphasised that the preferred way to use these distributions would be in a reliability based calculation; however, in many practical cases, only spot checks can be made of particular environmental or response combinations, and in these cases the above discussion of design events may be useful.

380 J. Hamilton

Having carried out the spot checks, it is then possible to apply reliability methods to assess the overall probability of exceeding a particular load effect.

So far we have discussed only the case when one variable ('x') is dominant, it remains to discuss the procedure which would have to be adopted if there were a number of variables (xl, x2, x 3 . . . ) on which the limit state was dependent with no clear dominant component. In this case the correlations between the variables would be described by a fully populated matrix, containing the expected values of the products (the covariance matrix).

Cxi, j = E(xixj)

However, in principle the eigenmodes of this matrix could be determined, and the matrix diagonalised with respect to them, so that only diagonal terms would be nonzero:

C x i j : O i ~ j

Zero eigenvalues will occur when two variables are essentially completely correlated, and in these cases the effective number of independent variables can be

reduced. For the nonzero eigenvalues, the most probable maximum of each can be determined as indicated above, together with the standard deviations of confidence limits on the other eigenmodes (the expected values will be zero for the other modes, since the covariance is diagonalised).

REFERENCES

1. Ochi, M.K., On prediction of extreme values. J. Ship Res. 17 (1) (1973) 29-37.

2. Rice, O., Mathematical analysis of random noise. Bell System Tech. J. 23 (1944); 2,4 (1945) 46-156.

3. Cartwright, D.E. & Longuet Higgins, M.S., The statistical distribution of maxima of a random function. Proc Roy. Soc. of London, Set. A, 237 (1956) 212-37.

4. Longuet Higgins, M.S., On the statistical distribution of the height of sea waves, J. Marine Res., 11 (3) (1952) 245-66.

5. Sarpakaya, T. & Isaacson, M., Mechanics of Wave Forces on Offshore Structures, van Nostrand Reinhold, New York, 1981.

6. Boccotti, P., Some new results on statistical properties of wind waves: Appl. Ocean Res., 5 (3) (1983) 134-40.