Wave induced ship hull vibrations in stochastic seaways

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WAVE-INDUCED SHIP HULL VIBRATIONS IN STOCHASTIC SEAWAYS by.

J. Juncher Jensen and M. Dog liani

DEPARTMENT

OF

OCEAN ENGINEERING

Wave-induced Ship Hull Vibrations in Stochastic Seaways

by

j. Juncher Jensen

Department of Ocean Engineering

The Technical: University çf Denmark

and

M. Dogliani

Registro Italiano Navale

Genova, Italy

Abstract

A theoreticäl study is undertaken on the determination of wave-induced loads in flexible ship hulls.

The calculations are performed within the framework of a non-linear, quadratic strip theory formulated in the frequency domain. Included are non-linear effects due to changes in added mass, hydrodynamic damping and water line breadth with sectional immersion in waves. The study is limited to continuous excitations from the waves and thus transient, so-cafled

whipping vibrations due to slamming loads are not considered.

Because of the non-linearities the ship hull responses become non-Gaussian in stationary stochastic seaways. The statistical properties of a response is here described by the first four statistical moments through a Hermite senes approximation to the probability density function The peak value distributions of the low and high frequency responses are treated

independent-ly, due to the large separation between dominating wave frequencies and the lowest two-node

frequency of the hull beam Both extreme value predictions and fatigue damage are

considered.

For a fast container ship the rigid body and two-node (springing) vertical wave-induced

bending moments amidship are calculated in stationary and non-stationary seaways In the long term analysis due account is taken of speed redùction in heavy seas, different heading angles,. operational areas and clustering effects in the peak value Statistics.

The main result is that springing is relatively most pronounced in head or near head sea in lighter sea states where the zero-crossing penods are small Also it is found that the non-linear contributions to the spnnging response are at least as important as the linear contribution However, for the long term extreme peak responses the springing vibrations become less important. This indicates that a design wave bending moment probably can be denved without considering springing for normal merchant ship types For the example ship a factor of approximately two is found between the calculated sagging and hogging moments at the same probability level. This is in reasonable agreement With the cm eñt rule requirements for the wave bending moment.

1. Introduction

The rapid increase in speed and size of the ships constructed during the 1960's led to the realization thàt wave4nduced ship hull vibrations can give raise to significant stresses in the hull. The vibrations are usually classified as either whipping or springing, depending on whether the vibration mode is transient or steady state.

Vertical modes of vibrations have been the focus of interest, partly because the vertical wave loads are the largest and partly because the two-node vertical hull mode normally is associated with the lowest eigenfrequency of the hull. However, for ships with very low torsional rigidity,

also asymmetric springing coùld in principle, be expected, even if no experimental evidence of that has been found in the literature. In the present investigation only vertical springing' responses are considered.

Extensive reviews of the literature on springing can be found in [1] - [4]. Five main problems have been identified in the description and determination of the springing response in ships

[3]:

the calculation of thè exciting hydrodynamic forces on the hull, the magnitude of the structural and hydrodynamic danping, the importance of non-linearities in the excitation forces,

the combination of the springing and the wave-induced low frequency responses in a confused seaway,

(y) the form of the high frequency tail of the wave spectrum.

Although these problems were identified nearly IO years ago, they still remain unsolved to some extent.

In, the following; items (i)-(iv) will be dealt with in order to quantify their relative importance on the springing response.

The analysis will bé carried out using a quadratic strip theory which has been able to predict measured differences between the wave-induced hogging and sagging bending moments in a tanker [5] and a container ship [7].

The theoretical' background has been developed in [6]-[14] where also comparisons with available model tests and full scale measurements are shown.

Here only some of the advantages and limitations of the method will be mentioned:

- The method is formulated in the frequency dm,ain. This means that the correct

frequency dependence of the added mass an4 damping is used, also in a stochastic seaway [7].

- The full quadratic transfer functions are determined for a ship sailing in obligué

seaways. Thus the Newman approximation needs not to be used in the stochastic analysis, see e.g [15].

- The formulation is based on a semi-empirical generalization of the linear strip theory

approach Therefore the linear part of the solution is identical to that obtained from linear strip theoiy [7]. The linear theory can be either the Gtrritsma-Beukelman or the SalvesenTuck-Faltinsen strip theory.

Using a two-dimensional fonmilation implies an extremely fast calculation of the quadratic transfer functions This makes the procedure useful also in design studies The procedure is limited to the vertical responses (heave, pitch, vertiçal bending moments and vertical shear forces). Linear asymmetric responses can, however, be included [13].

Hull vibrations (springing) are easily included as the hull is modelled as a

non-prismatic Timoshenko beam [7], [8].

Statistical analysis in stationary, stochastic seaways can be done using both a Charlier senes representation with the full Joint distribution of the response and its time denvative and an approximate Hermite senes approach, [7], [12], [14] For the

Charlier series, an efficient procedure for calculating the coeffiòients in the series expansion is applied, [6] For more severe seastate where the Charher senes fails to converge (i e when negative probability densities are obtained), the Hermite senes approach seems useful, [12]-[14]. Also the formulation is directly applicabïe in the Kac-Siegert method [15] for extreme value predictions.

Whereas the statistical methods (Charlier and Hermite series) yield the statistical moments of the responses to any order, the extreme value predictions are based on the narròw-band approximation. This is reasonable as long as hull vibrations are of minor importance Otherwise, methods taking due account of the spectral density separation between the ngid and the elastic hull responses should be applied, [7], [8] The procedure allows for separation of the high frequency response from the wave

and slow dnft response Thereby the effect of spnnging is easily determined as

shown in the present study.

- Comparisons of the quadratic sectional loads in regular sea with model tests show reasonable agreement, [7], [12].

Comparisons of the statistical predictions in moderate, short term stochastic seaways with full scale measurements show that the method is able to predict the difference

between the hogging and sagging wave-induced bending moments with good

accuracy, [7], [8], [l2}

Comparisofls with computer-expensive time-simulation procedures show reasonable agreements in regular and irregular seas, [12].

- Fatigue damage predictions in stationary stochastic seaways can easily be included,

[9j-[l 1]. It is found that the non-linear contributiöns to the fatigue damage are

dominated by terms proportional to the kurtosis and to the skewneth squared. These terms seems to be of nearly equal importance, [11].

The short term statistical responses obtained from the quadratic strip theory will in the last

part of the paper be extended to cover long term predictions in non-sttionaiy seaway.

Different shIp speeds and headings will be accounted for..

As example the container ship described and used by Flokstra (16] in hjs thorough numerical and experimental analysis of linear ship responses will be used. The body plan is shown in Figure 1 and the main particulars in Table 1. From the data given for the fòu segments of the shi model used in the experimental set-up a mass distribution is derived and shown in Figure 8. This mass distribution will yield the crrect inertia contributions to the ectional forces at the intersections between the segments. These intersections are located at 0.24 L, 0.50 L and 0.75 L from the aft perpendicular.

Figure 1 Body plan of a container ship [161.

The analysis in [16] only concerned rigid body motions of the hull and thus no stiffness data were available nor relevant. Therefore, in the present analysis, the hull flexibility is modelled by constant shear and bending rigidities over the middle half length of the ship. These values

are chosen in accordance with the stiffnesses of eisdn i'Äeír decrease of these

values towards the ends of the ship is assumed with values

_{at dends equal to 10 per}

cent of the midship values.

It is noted that the calculated two-node natural frequency in water becomes 4.4 rad/s, which

is rather close to the value 5 rad/s quoted in the discussion by Hachjiajn to [7] for the

container ship "CTS Tokyo Express" of approximately the same hull form as the one given in Figure 1 and Table 1.

Table 1 Main Particulars

The structural damping is taken to be 0.001 times critical damping. This is not so important, [8], as the speed dependent hydrodynamic damping dominates for fast ships.

The response considered throughout in this study is the vertical bending moment amidship, but shear force responses could have been considered as well.

Length between perpendiculars (L) 270.0 m

Breadth amidship 32.2 m

Draught even keel 10.85 m

Block coefficient 0.598

LCG aft of amidships 10.12 m

Longitudinal radius of gyration 0.248 L Bending stiffness amidship 8.65 .1013 Nm2

LIEu!

+

ax L

2. A quadratiç strip theory

Some of the pertinent features of the quadratic strip theory Will be outlined below. For further details the references [7]-[14J could be consulted.

Equations of motions

The lower modes of ship hull vibrations can generally be determined quite accurately by modelling the hull as a non-prismatic Timoshenko beam. The equation of motions in the vertical plane then becomes

LIpGAI1

+11±-axL

L

at)Lax

L')±i+pGA(i +11;)(±:_cp1=m:r2_!

t) axj

_L

at)Lax

₎

at2

where Et(x) and .iGA(x) are the vertical bending and shear rigidities, respectively; p(,x) is the slope due to bending; w(t,x) is the total deflection; is an internal damping coefficient; and x is a longitudinal coordinate in axyz-coordinate system fixed with regard to the undisturbed

ship so that the z-axis is in the vertical direction Finally, m(x) is the hull mass per unit

length, m(x) is the mass moment òf inertia abOut the horizontal y-axis, andF(r,t) is the

external force per unit length which is nonlinear in w.

The boundary conditions to equations (1) express that the bending moments and shear forces are zero at the ends of the ship.

The solution to equations (1) arid the associated boundary conditions is approximated as a series in the form

ç(x,t)

_{= E}

u.(t) .(X)

w(X,t) = u.(t) v.(x)

hO

where u.(t) are coefficients to be determined and where (v(x), cç(x) j are the eigenfunctions

to the homogeneous, self-adjoining part of equations (1). These eigenfunctions are assumed orthogonalized and normalized so that

= m

.!:

- F(x,r)

at2

(1)

-:- kr2aa,

+ rnvv}dx ₌

where 3q is Kronecker's delta. Furthermore, the ejgenfunctions are assumed to beorderedSO

that (y0, a0) is the heave mode, (y1, a1J is the pitch mode, (y2, ct} is the two-node vibration

mode associated with the natural frequency Q, and so On..

If we substitute (2) in the differential equations (1), and use the orthonormality relations (3), we obtain the following set of equations of motipn:

uf +

ilç.

+ cJu1

= f v,F (x,t,E u1v1) dx

(4)

Forcing function

Calculation of the hydrodynamic forces will be based on the time derivative of the momentum of the added mass of water surrounding the hull. In addition to forces due to a change in momentum of the added mass of water, we will include a damping term and a restoring term, both dependent on the relative motion. Thus the force per unit length of the hull acting at position x is taken in the form

F(x,t)

= -

_ {rntix

.}

+ _N(2Çx)... +

fB(zx)

dz

z .w

where the difference between the absolute displacement of the ship in the vertical direction, w(x,t), and the surface of the Ocean, h(xt), corrected for the Smith effect is denoted by f(x,r). Furthermore, m is the added mass per unit length, and N is the damping. The operator D/Dt is the total derivative with respect to time t

Dt ät ¿3x

where V is the forward speed Of the ship. The breadth of the ship is denoted by B(z,x) and the draft by T(x). Finally, p is the Froude-Ktylov fluid pressure.

If we neglect the dependence in m, N and B, the force expression (5) corresponds to a linear strip theory. However, here we shall evaluate F(x,t) by a perturbational method, taking into

(3)

account linear and quadratic terms in the relative displacement i and thereby in the

displacement of the húll w and the wave surface elevation h. in order to do this we shall start by evaluating the waterline breadth B, the added mass m, and the damping coefficient N

around i = O:

where a are wave amplitudes, and

- k(x - Vt cos) - ot + O.

where k denotes the wave mimber in the x-diretiôn, and the wave fiuencyo is given byw i where g is the acceleration Of gravity. Finally, the ships heading angle is. , measured such

that head seas corresponds to 180 deg.

Similarly, the deflections of the ship hull are expressed as a sum of a linear part and a qUadratic part

w = (1)

+ (10)

These assumptions lead to

F(xJ,t) = F (') + F (11)

where F(» contains linear terms in the displacemen /» and the wave surface elevation hi", and F terms which are quadratic in these quantities as well as terms Untar in w and hdl)., The wave suzfäce elevation h and the pressure p are also expressed as sums of a linear term and a quadratic term so that we have, for instance

h(x,r) = h + h or h(x,t)

a cosp +

. + k.) cos(91. +

iI

¡=1 j1

k

B(i,x) B(O,x) + f B0(x) + 2(x,t)B1(x) (6)

-rn(z,x) m(O,x) + z

.am

E m0x) + zx,Om1x

+ r.nL)) + (ri.n

+ r.n,.n)r() - (rr.n

Here we have introduced the amplitude a1 and the phase lagO1 of the regtlar sea wave through the variables

and L given by

= a.cos61 and = a1 sin O

and the frequency of encounter

-

Ln:)Mi(4] cos (Ir),

_-

t

+ r.n:)M,(1)] Sfl_{(1k. +} t

-

,I(x)} sin (Ir),

_-

_{t }}

Resulting equations

Due to the form (11) of the exciting forces, we shall seek solutions to equations (4) in the form

u.(t) = uf» + Z (12)

This leads to two sets of generalized equations of motion; one set which governs the first-order motion and another set which governs the second-first-order contribution to the. motion. By determining the functions uf»(t) and u(r), the vertiçal displacement w(t), rotation angle

«r) and thereby also the wave-indúced bending moment M(x,t) become functiOns of the incoming waves. The expression takes the forrri

M(x,t) = MU)(x,r) + A1(x,r) + (13) where M '(x,t) = { [

-

r.nMrs(r)} t + [FMrÇt) + r.nMrC(1)1 sin ¿r) (14) and

M(x,t) =

{[(E1 r.nÌ*n) M,(x) (r:+,z +

,+1)Mj(4]

COS (1k + t

2

CL)1 - ....L. Vcos g

The analytical expressions for the coefficients M,, M,; a = c,s are given in [7].

To evaluate these coefficients, a suitable prócedure should be used to determine the sectional added mass ni and hydrodynainic damping N. Various procedures have been applied in the present investigation ranging from a very elaborated and computer-expensive boundaiy-element method t17] to the very simple Lewis transformation with the frequency variation calculated as described in [18]. Only minor differences in the linear, equation (14), and the

quadratic, equation (15), transfer functions were found. Therefore, it was considered

reasonable to tise the last mentioned method for the present analysis.

Also it should be mentioned that the linear transfer functions fòund are in perfect agreement with those given in [161 using. the GerritsrnaBeukelman strip theory.

3. Short term statistics

In. the present study only stochastic responses in long crested (unidirectional) seas are considered.

To model a stationary stochastic seaway, the wave amplitude a- and the phase lag O. are

chosen so that = cos O and = sin are jointly normally distributed with O, uniforthly distributed, and so that the half mean-squared amplitudes ½a12 are equal to the.wave energy in the associated range of frequencies.

The theory assumes a uniform discretization of the encounter frequency ¿, that is -co1_1 =

¿ for i 2,3,...,n. The independent variables and , (j = 1, 2,.., n) will then have zero

mean and a variance

=v.

(17)

J *11

where s() is the wave spectrum formulated in terms of the frequency of encounter. This

spectrum can be calculated from the ordinary wave spectrum S(co) by

where

i i g

=

= _Vcòs

In the calculation procedure th _{encounter frequency} is prescribed and for each of the corresponding wave frequencies co the linear and quadratic transfer functions are determined. Then in the statistical analysis thettansfer functions are multiplied by wave spectral densities

S() determined as

S() = S(o())/Iw/co0 -

fl (21)

using the associated solutiOn co to equation (19).

W,-Wo

i ± %/1

-co/wo

Figure 2 Relation betwéen encounter frequency ¿S and wave frequency co for near

following sea conditions (cos 1> O),

(19)

(20)

=

s(co(i)) . (18)

where q is the ntimbr of wave frequencies i» which coIsponds to a specific value of ¡ii. If cosq <O only One solütion ecist(q 1) but if cos4 > O up to three different wave frequencies co can correspond to the same encòúnter frequency . This is illustrated in Figure 2. For each

Now, on the basis of this stochastic description of the firstorder wave elevation, an expression for the wave-induced midship bending moment at time t = O can be written in the form

M(x,O)

+ e AJkXJXk

1

¡=1 ki

j.I

Here we have introduced as independent variables

xi = ci

and the dependent variable M(x,O) is normalized by the standard deviation ? of the first-order contribütion to the wave-induced bending moment. The second-order coefficients A? are assumed normalized by the nonlinearity parameter e so that

E

AJkAJL = i

j=ì k-i

Response spectraJ densities

From equation (22) th spectral density of the response can be calculated and divided into three contributions [8], the first being the usual liner response spectrum

SM(»(ii,) = [(Me)2 (,í,$)2]

) (23)

The next spectrum is assOciated with the slowly varying terms in equation (15). These terms are those involving difference frequencies ai, - ai. The specti-Im becomes

S (ai,)

=

4f,[(M:)2 + (M3)2]

-

.(ai) L ai (24)

where f, = i if r =.O andf, 2 if r O and where ai, = t.ai. Similarly, the spectral

density-function of the rapidly varying part takes the form

= E

4[CM:)

M)

}s

(ais)

('

2 2 V (25) -- V (22)

In Figure 3 these three spectral densities are shown for a specific sea state, characterized by a significant wave height H 6 m and a zero úperbssing period T 8.696 s in association with a Pierson-Moskowitch wave spectrtlm. Head sea and a relatively high forward speed

corresponding to a Froude number 0245 are assumed. This condition represents i rather

severe sea state and it is questionable if the ship can go that fast. It is clearly seen from the flgure that the linear response still gives the major contribution to the wave-inducçd bending

moment in the wave frequency range. However, for frequencies around the two-node frequency 4.4 rad/s the rapidly varying secOnd contribution is much higher than the linear part. This is because the quadratic response receives contributions from the wave spectrum at frequencies where some wave energy is present. Contrary to that the linear part only gets contributions from the high frequency tail of the wave spectrum where the energy is very small. 60 S()x1&6 [(Nm)2s] 55 (1 -' . ¡ ,.- (2)

H6m

V=12.Slmis

I=8.696s

4j:1800

-

Js()= 2.55 '1017(Nrn)2 2.68; 1016(Nm)2 2.62' 1&5(Nm 2 LF.WF HF p S'

1r

'I

/ \

-

.a'

1.8 2.7 3.6 &5 5.4 G3 [radis)

Figure 3 Linear and quadratic spectral densities for the mi4ship bending moment in a specific sea state.

Overall, the variañce of the linear response is seen to be an Order of magnitude greater than for the rapidly vaiying response, which again is an order of magnitude larger than the variance of the slowly varying terms.

From Figure 3 one important observation can be made. In the frequency range from about 2.7 rad/s to 3.6 md/s nearly no responses are found. Therefore, it seem plausible to divide the response into two regions. The flrst region, called LF+WF, contains the low frequency and wave frequency contributions. The linear part is dominating this region. However, the region

also includes the slowly varying quadratic part and. that part çf the:rapidly varying response which varies with about twice the frequency of the linear wave-induced response. The second region, cafledHF, is centered around the twO-node frequency and it gets contributions from both the linear response añd from the rapidly varying quadratic response.

Similar calculations have been performed for a large number of sea states and some results are given in Table 2 and Figure 4. It could be mentioned that the computer time on a standard 386 PC, 20MHZ jS about 2-5 minutes for each case depending on q i equation (18) and using 30 different frequencies of encounter (n = 30 in equation (22)). The results are the mean value p, the deviatin a, the skewness K3 and the kurtosis ic4 given by

p = K1 = Xc.

E

& = K2 X2(1 + 2c2)

= K3

=.X3I6CE

ii JI

E

X1A.JXJ + 8c3E¡.1

E E

j'1 k1 A.JA.kAkÎ]

(ic4 - 3) c = K = X4 48 c2

_[E

_{E E}

X A1 A1 X + c2

E E E E

A A A, A

i1

jt k1

i-1 j=1 k1 ¡àl

where the expressions for the cumulants K follow from equation (22). The datagiven in Table 2 is for the sagging bending moment amidship. The corresponding results for the hogging moments are obtained by simply changing the sign on the mean value and the skewness. Figure 4 shows the ratio between the standard deviation c?F of the springing response to that

0LF+WF _{of the low and Wave frequency part. The figure shows that springing is most}

pronounced in head or near head sea and increases with ship speed.

This trend was to be expected due to the increase in encounter frequency with these parameters. Also the sea state has a significant influence of the relative importance of

springing. If only a linear analysis was used, no dependence on the significant wave height H1 would be found in Figure 4 as both os" and a' then depend linearly on H,. The non-linear effects are seen to increase the relative importance of springing with the significant wave H. HowCv'er, for higher values of H1 also higher values of T Would be expected. Taldng. the one parameter 1TIC spectrum as an example, T1 is given by

V [nils] 5 . 12.6 'P [deg] [ml 2 2.3 3.8 .5.5 5.4 2.4 5.1 5.9 4.7 2.1 4.4. 7.9 . 8.7 2.3 58 8.6 7.5 2 36 ¡21 173 174 64 134 148 131 34 123 171

I0

60 134 146 128 2 .013 .084 .1.17 .106 .019 .149 .161 l33 008 .046 .126 .174 .000 066 .137 .162 2 003 014 022 017 003 038 (142 029 004 007 048 074 003 023 077 084 2 ' .176 .121 .106 .098 .163 .128 .1 I .106 .234 .164 .144 .133 .209 .164 .147 .137 2

-,

13 7 5 3 13 7 5 3 41 24 16 Il 45 25 16 II 2 : .209 .211 .562 .702 .132 .390 .593 .566 .099 .126 .243 .333 .072 .131 .265 .257 2

'JULI

.709 .705 .733 .746 .701 .7.18 .733 .733 . .706 .703 .709 .716 .700 .700 .709 .709 6 _____i 1 203 . 34.3 49.6 48.2 21.9 45.6 53.5 427 18.8 39.3 71.2 77.9 20.9 52.1 77.1 672 6 lilI 364 521 526 193 408 448 397 105 371 524 523 182 409 452 397 6 .041 .250 .347 .314 .056 .438 .475 .394 .031 .137 .372 .510 .002 .200 .407 .477 6 .026 .122 .195 .151 .028 .334 .337 .251 .041 05R .410 .630 .023 .201 .648 .705 6 .176 .125 .110 .101 .164 .136 .121 .1 II .233 .168 .156 .147 .211 .175 .167 .154 6 81 43 34 24 77 50 35 23 216 144 103 73 225 147 102 66 6 .810 .701 1.280 1.469 .692 1.206 1.406 1.336 .837 .612 .835 1.037 .700 .674 1035 1.040 6 .757 .740 .786 .807 .746 .780 .797 .793 .774 .743 .754 .771 .759 .743 .770 .772 TabLe 2

Statistical moments for the low and wave frequency (LF + WF) bending moment and for the high frequency (HF) springing bending moment.

p = mean value,

= standard deviation,

y0

= mean Uperossing rate,

K3

Figure 4 Changes in standard deviation of midship bending moment due to Springing. Arrows indicate T: according to the ITTC spectrum.

T₂ 11.12 I

-Thus for H: = 2 m a.nd H3 = 6 m, the zero mea uperossing period T becomes 5.02 s and 8.696 s, respectively. These vaines are highlighted with arrows on Figure 4. Now it appears that springing is relatively more important in the lower sea states where the mean uperossing periods are smaller. This ¡s álso what have been observed in the past, cf. [1]-[3].

Figure 4 shows only the standard deviation of the response. From Table 2 it is seen that the low and wave frequency responses have positive mean values p and skewnesses IC3 ithplyng

that the sagging bending moments are larger than the hogging bending moments at the saine probability level. Furthermore this difference increases with ship speed and sea state. With the Separation of the response in a LF + WF and a HF region, the high frequency responses get zero means and skewnesses and therefore the same probability distribution in hog and sag. The kurtosises are in all cases greater than 3. Therefore the tail distributiòns have more weights

7 9 11 5 7 9

tz(si

than in. a corresponding Gaussian response. The results regarding the low and wave frequency responses are in agreement with previous findings assuming a rigid hull beam, [7], [8], [12]. Finaily, the mean uperossing rates y0 are calculated as

v =

=

_L

+.!

(ç -

3)) (27)

where & is the standard deviation of the time derivative of the response, found from a formula analogous to equation (22). The joint probability density p(yj') of the. response y and its time derivative is given in [23] equation (27) is correct within the first four moments of the response, but neglects non-Gaussian effects in the time derivative of the response. In Table 2, mean uperossing rates v are given for both the low and wave frequency responses (LF + WF) and for the springing responses (HF). It is clearly seen the periods of the HF are much smaller than for the LF + WF response. This difference is included in the extreme value and fàtigue danage predictions considered in the next sections.

Extreme value predictions

Based On equation (22) the probability density function of the response can be expressed

exactly by for instance a GrarnCharlier series expansion. In [61 an efficient numerical procedure was given for calculating the coefficients in this expansion to any order. However, an inherent assumption in the Charher expansion is that the distribution does not deviate much from a Gaussian distribution. Otherwise, the series expansion diverges. Therefore, as shown by Winterstein [19], the Gram-Charlier expansion is of limited use for eveñ moderate non-linearities when extreme peak values are looked for. An alternative, although approximate, procedure is to apply a simple cubic Hermite polynomia transformation [19], where the first

four statistical moments of the response M are used to calculate the 4 determnistic

coefficients c; i = 0,1,2,3 in the approximation

M= Ec1

(28)

iO

of .the response M in terms of a standard Gaussian distributed variable U. Although much information is omitted in this approximation, numerical studies [.12], [13], [19] indicate that the probability distributions derived from eqúation (28) usúally are very .good approximations to the exact distributions or to the Gram-Charlier series expansions where applicable. Thus by equating the four lowest statistical moments from equation (28) with those found from

equation (22), seç equation (26), four non-linear algebraic equations with the four unknown coefficients c areobtained, [14], and solved by the Newton-Raphson method..

It should be mentioned that the transformation (28) only is moñotonic if the coefficients, c, satisfy 3c1c3 > c22. In a few cases, mostly in following sea this inequaiity Was slightly violated. In those cases just a marginal increase in the kurtosis ; or decrease in the skewness K3 was required to make the transformation monotonic. These corrections made no visible changes

in the peak value predictions, but anyway points at a restriction iii the Hermite series transformation.

From the coefficients c1 the probability Q(M) that individual peak values M of M exceed a level becomes [19]

Q(M,,)

P(M> )

P(u> u()) = ..L

=

ex(_!

u2(c)) (29)

where y is the uperossing rate of u(C). Here u© is the real solution to

=ci

(30)

which can be found ànaiyücally. The mean uperossing rate V0 is determined by equation (27). Of more interest for the design of ship structures are the extreme value predictions. For a total time period T, conventional Poisson uperossing yields

Q(M(fl)

P(max M> ) = i - exp(-v7')

(31)

where y is given by equation (29)

Equation (29) is only valid fòr a narrow-banded response as it is based on thC assumption that the distribution of maxima is píoportional to the expected number of uperossing the level u©. Furthermore, euation (31) requires a not too narrow-banded response. Otherwise, the

For the present analysis, it is evident from Figure 3 that the total response (LF + WF + HF) is too broad-banded fOr application of equation (29). HoWever, the assumption Seems justified for the LF + WF response and the HF response, treated individually.

Then, on the other hand,it is known that for narrow-banded processes, equation (31) can yield too high probabilities of exceedance, because the uperossings tend to occur in clumps, thereby violating the assumption of statistically independent peaks. A simple correction of equatiOn (31) to account for this clustering effect has been proposed by Vanrnarcke [20] whereas a more accurate but also more cómputaûonally expensive correction is given by Ditlevsen and Lindgren[21]. In both procedures the uperossing rate y. is replaced by the uperossing rate of

the envelope, process muhiplied by an estimation of the long run fraction of qualified excursions and corrected for the probability of having no excursions above the level in

question. Both procedures are derived for Gaussian processes and the corrections become insignificant for typical spectral bandwidths found in wave-induced responses if the peak level is higher than a few rimes the standard deviation, [22). In the analysis given in [20] and [21 the bandwidth is characterized by

rn0 rn2

2

rn1

(32)

wherern is the i'th spectral moment. A directly equivalent quantity to y for the present

non-Gaussian response is not at hand. Instead it has been jûdged acceptable to replace y by

V

7 =

-Vo

whereV0 is the mean uperossing rate given by equation (27) and where Vmis the expected rate

of local maxima. Ìn principle, see [23], Vmcan be calculated by a Pon-trivial extension of the.

present second-order formulation, but for the present purpose it seems legitimate to estimate Vm by

(33)

(34)

The standard deviation & is in the same way as & derived from an expression, analogous to equation (22).

As expected the corrections due to clustering effects are insignificant for all but the lowest peaks. Therefore, these corrections Will only be included in the results on long term sttistcs presented in the next chapter, However, a closer examination of the different approaches

accounting for clustering effects, [19]-[21], could be interesting in order to clarify exactly their range of applications for the present problem.

Now turning to the combination of the extreme values for LF + WF and the HF responses theñ, due to the clear separation between the spectral densities of these two responses, the

extreme values obtained from equation (31) for each of them could probably be added assuming the two reponses statistically independent. Alternatively, the procedure [24] proposed by Naess and Ness is promising as it includes the possibility of analyzing the combined linear and high frequency contributions. It is straightforward to transform equation (22) to the format needed to apply the method as only the eigenvalues and eigenvectors of the matrix A, in

equation (22) must be determined, see Appendix 1. Further studies on this method are

currently being done [25].

Figure 5 shows the extreme value predictions obtained, from equations (29)-(31) for two sea states. In both cases head sea, a ship speed of 12.61 rn/s and a total time dûration of 1O s are assumed. For the lowest sea state (H = 2 m, T 5.02 s) the springing (HF) response is larger than the ordinary wave-induced response (LF + WF). This was expected from Figure 4, where it was shown that in this case the standard deviatiön for the springingresponse was larger than the. low and wave induced response. It is also seen that whereas the non-linearities in the LF + WF response are small, the non-linear contribution to the HF response is significant. The total response level is of course low.

The other case in Figure 5 represents a much more severe sea state (H: = 6 m, T ,= 8.696 s). The non-linearities in the LF + WF response are evident and in accordance with previous results [7], [8], [12]. The springing response is higher than found for the lower sea state but now significantly below the LF .+ WF response level. Therefore, the springing response becomes .of decreasing importance with increasing severity of the sea state. Thus it is to be expected that the springing response only have a minor influence oti the extreme life time loads on the hull.

Fatigue damage

Springing can be of more significance for the fatigue damage Of the hull than for the extreme loads. This is so because the mean uperossing rate y0 of the HF response is much higher than for the LF + WF response, see Table 2..

Methods to include the non-Gaussian characteristics in springing response on the fatigue damage calculations can be found in [9]-[11], [19].

[MNm] 4000 3000 2000 1000 o:i 0.OÖ1 0.0001 Q(M(T=10's)) H,=6m 12 =8.696s - V =l2.6l',s =180° 0.1 0.01- 0.001 ÓOÖI O(MpCTIO'S))

Figure 5 Probability distributions for the extreme Values

f

the bendingmoment amidship during a time period T = 10,000 s.

For a symmetric, slightly non-Gaussian stress response the fatigue damage per cycle d can be estimated based on the Palmgren-Miner rule by, [19]

ni

d =

2) (

+ (m + m)(1c

-3")

(35)

/ 24

a I2/Ta)

where () is the Gamma functiOn, a and m the scale and slope parameters in the S-N curve and o and 1(4 the standard deviatioñ and kurtosis of the stress response A consistent analysis of non-symmetric responses, [il], shows that the fatigue damage also could depend on the skewness squared. However, due to lack of experimental proof of this, equation (35) will be used here. Equation (35) assumes a narrow-banded spectral density. As discussed before this is a reasonable assumption for the LF + WF and the HF response treated individually. Further-more, the total damage D during a time period T can be approximated by

D = T[(V0i) + (voc)"j (36)

as the springing response (HF) varies much faster than the LF + WF response. Iñ Figure 6 the relative increase in fatigue damage due to non-linear and springing contributions is shown for a specific condition (V = 12.61 m/s, = 180 deg.) and an exponent m = -3 in the S-N curve. The fatigue damage is normalized by the usual linear value obtained frOm equation (35) by taking 1(4 = 3 and o equal to linear standard dçviation omitting the springing contribution. It is seen that for the low sea state (H, = 2 m) the springing çontributes with about 80 per cent to the total fatigue damage assuming T.. = 5.02 m whereas fOr the more severe sea state (H, = 6 m) the springing contribution i.s negligible fôr th dominating zero crossing periods T, around 8 to 9 seconds. On the other hand the non-linearities in the wave frequency region plays no role in the low sea state but contributes with about 20 per cent to the fatigue damage in the more severe sea state. When comparing the fatigue damage in rwo different sea states it is for instance found that

DH,=2m,T,=502s

T1

D (H, = 6 m, T = 8.696 s) 658 T2

where T1 and T, are the rime the ship spends in sea state (2 m, 5.02 s) and (6 in, 8.696 s),

respectively. As T2 will be much less than T1 the lower sea state in which significant contributions from springing appear can be of more importance in the fatigue damage caIculations

Vd I(vd) LF.WF.HF -LF.WF.HF LF.WF L F. WV ---i.---9 115 7 9 Tz (s] Tz [s]

Figure 6 Relative impOrtance of non-linear and springing coniribttions to the fatigue

damage. Arrows indicate T according to the ITTC spectrum.

It is eniphazied that the present calculatiOns are carried out assuming uni-directional waves. For real short crested sea conditions springing will probably be of lesser importance as the springing contributions strongly depend on the heading angle, see Figure 4.

4. Long term predictions

In this chapter first some uefu1 fonnulas related to long term predictions in non-stationary

seaway are derived. The input to a long term analysis is results. for different short term conditions and the relative occurence of these conditions. As shown by equation (29)-(31) each of these short term responses is characterized uniquely by four coñstants, which here are

chosen as the mean value p, the variance &, the skewness K3 and the kurtosis K4. Sorne simplifiòätiöns cari be intrOduced because the Wave spectrum is proportional to the square of the significant wave height H. Within the framework of the quadratic strip theory approach, it is from equations (22), (26), easily seen that the cumulants K1 öf the response will depend

on H as

K1 = a.

fi22

+ b1 H i =' 2,3,..., (37)

Therefore, it is possible to calculate

&,

c3 and K4for any H, from the results obtained for oniy

two different values of H,. Denoting these values by H1, H,2 and the corresponding values of K1 by K and K11 it follows from (37) that

a=

K,K,1ii

and (H1H,,)212 (H,,: í) b1 K11H,r2 - K H,2 (38) (H11H,,)22 (H, H,)

where the valúes K11 and K are obtained from (26) lçno Wing p,G, K3 and ic for H,1 and H,. Finally, the mean value p is directly proportional to H, and the mean uperossing rate V0is

found from equatiön (27) using

=AH,2 BH,' (39) with and (H,1H,,)2 (H, - H,)

(HH)2

(z

_-

_H)

= _{2 (?i,)} = [1 + ¿ 3) H, (40) (41)

For the rate y,,, of local maxima, equation (34), formulas similar to equations (39)-(41) hold. Extreme value predictions

The long term analysis deals with the probability distribution of the extreme peaks taken over a period of typically 10-20 years. The basic assumption in the analysis is that the process can

A

H,

be modélled as a sequence of staüonaiy processes with independent peaks,. Thereby, the probability of exceethnce

Q(M(fl) =P max M> )

T

exp(-v«)7)

(42)

= i = [J

exp(=çfsTexp(-'h u.2«)))

-

expl -TE v fsexp(-½ u.2©)

using equation (29) and (31). Here the number of different stationary conditions, characterized by fixed values of significant Wave height H,, zero crossing period T, forward speed V and heading angle 4), is denoted by p and the time spent in the ¡'th stationery conditions by T. This period constitutes the fraction fs of the total time T. Thus specification of

S (He, T, V, 4), fs}1 , i

_',2,.-,p

(43)

together with calculation of

E y0 (Si)

u(S4)

from equation (27) and (30), respectively, yields the long term probability of exceedance. In the present paper the following values are used for the parameters in the sét S

V = (5 mIs, 12.61 m/s}

yielding totally 1200 different stationary conditions. However, only 160 different conditions need to be analyzed as the results for the remaining conditions fôllow from equation (7)-4l). This makes the calculation fast even on a standard 386 PC. Also different loading conditions

H3=jlm

;

j=1,2,...,15

T. = ji s

; j = 5,6,...,12

could have been included, but this has not been found necessaiy for the present example considering a container ship.

The fraction ft of time spent in a specific stationary condition is taken as

ft

= f,,,(H,,T2)

_{f(VIH) f,($)}

(44)

where fM(H$, T2) is the calculated froth, an assumed operational profile covering a number n of Marsden zones [26]

f(H5,T2) = PJfMJ(H$,TZ) (45)

The operational profile is thus characterized by the fraction P1 of time the ship is in Màrsden zone fj= 1,2...,n. Directionality in the sea states as well as annual variations are not included in equation (45) and fMJ(H3,T.) is taken from [26] as the all directions, annual average scatter diagram. Rather than using the discreûzed scatter diagrams, given in [26], one could choose a continuous analytical formulation based on the discretized scatter diagrams, [27]. Thereby, better extrapolations to extreme sea states become possible although some care has to be taken to avoid physically unrealistic conditions.

In severe sea states the. ship's master usually reduces the speed in order to avoid excessive

slamming and green water on deck. Therefore, the fractionf defining th use of the two forward speeds is made dependent n the significant wave heights H3

f(V=VIH) =

p

ifHH

3°

(46)

l-pifH3>H,

fvY=Vmml1s) = i _fy(V=VmIHs)

where p, p,,,

and H are to be Specified. Finally, the fraction f, for the different headings are chosen as

f, = (P0 P45' P90' P135' ¡' (47)

with either

po = P45 = Pgo = P13s = P180

Po P45 = p90 = 3p135 = 3p1

-The last choise constitutes an attempt to account for the short=crestednòss of the waves by reducing the time the ship spends in the headings yielding the largest responsés.

For the container ship several long term predictions have been performed. A typical

operational profile is given in. Table 3 and the corresponding probability of exceedances in Figure 7.

Table 3. Operational profile (T= 20 years)

8000 M [MN m] 7000 6000 5000 4000 3000 2000

1000 - clustering effects ircluded

clustering effects excluded

0.5 01 0.01 0.001

Q(Mp(T2oyears))

Figure. 7 Long term probability of exçeedance for the wave-induced bending moment

ami dship... Marsden zone Ï0 11 15 16 23 .25 33 35 48 Praction of time .083 .083 .083 .083 .083 .250 .083 .083 .167 Po P45 P90 P135 Piso .273 .273 ..273 .Ó9.1 .091 H30

6m

PIni4 1.0 Pinar . 0.9

From a design

point-of-view a realistic value _{for the probability} of exceedance

Q(M(T = 20 years)) could be 0.5, signifying a 50 per cent chance for the design value to be exceeded during 20 years service with the specified operational profile. From Figure 7 it is seen that especially the sagging bending moment is sensitive to this probability level Looking at the rigid body (LF + WF) wave-induced bending moment amidship it is clear from Figure

that the present ship will be subjected to sagging moments about twice as high as hogging moments at the same probability level. The standard linear result is found to be situated in

between the non-linear results. It should be mentioned that clustering effects have been

included using the procedute in [21]. Some reductions are' seen especially in the non-linear sagging bending moment, but the inclusion of clustering effects is not that important. Figure 7 also shows the long term prediciions for the extreme springing response. The design value at Q 0.5 is seen to be 22 per cent of the sagging bending moment and 48 per cent of the hogging bending moment. Thus if rigid body and springing responses are considered statistically independent, the springing response wifi only slightly change the design wave

bending moment. This indicates that springing probably needs not be considered in the extreme response calculations for normal merchant ships, but ftrther studies on this matter are needed.

The calculations performed for the extreme bending moments have so far been based on Poisson uperossings. HoWever,, often order statistics or individual peak tadstics have been used instead. Using order statistics, the long term probability of exceedance Q(M(T)) can be written

(

'1'«I)

v()

Q(MÇfl)=1

_ft11

v.

_o'

where the total number of peak N(T) during the time T is determined by

N(7) =Tv,,,fs

where v,,.d is the rate of local maxima for the i'th stationary condition.

P.U.: Poisson Úperossings O.S.: Order statistics

LP.: Individual peak statistics

All previous calculations have been performed with the mass distribution related to the ship model experiment by Flokstra [16]. This mass distribution is somewhat different from What to be expected for a real ship. Therefore, calculations have also been made with a fourth degree polynomial mass distribution giving the same displacement, longitudinal center of

Bending monient amidship

[]

Non-uniform headings:

P.U., linear 4.08 4.08

P.U., non-linear mean and variance 4.30 4.00

P.U., non-linear mean, variance, 1(3 and 1(4 6.15 2.73 O.S., non-linear mean, variance, 1(3 and; 6.12 2.77

LP., nonlinear mean, variance, 13 and 1(4 5.97 2.75 P.U., non-linear mean, variance, 1(3 and ;, clustering included

Uniform headings:

P.U., linear 4.19 4.19

RU., non-linear mean, variance, 1(3 and 1C4 clustering included 6.17 2.64

4. degree poIynbmia mass distribution:

P.U., linear 3.96 3.96

RU., non-linear mean, variance, 1(3 and ;, clusteriflg included 5.32 2.71

Q(M,,) =

Ê

v)

_{= Ñi)}

(50)

For even relatively high probabilities of exceedance, the formulas (42), (48) and (50) yield

nearly the same results. This is illustrated in Table 4 and Table 5. Here the design wave

bending moments, defined by Q(M( 20 years)) = 0.5, calculated under different assumptions have been compared. From Table 4 it follows for instance that the skewness is very important as it accounts for the major difference between the hogging and sagging wave bending moments.

Table 4 Extreme wave-induced rigid body bending moments amidship. 20 years service at sea. Probability of exceedancé = 0.5. Marsden zones and speed reduction as given in Table 3. The underlined terms are those found in Figure 7.

gravity, and longitudinal radius of gyration as befôre, see Table 1. This mass distribution is shown in Figure 8 and the main results are included in Table 4 arid 5.. These results are neaÈly the saille as for thé other mass distribution except for the rigid body sagging bending moment which has been reduced by approximately 10 per cent..

Table S Extreme wave-induced springing bending moments amidship. 20 years service at sea. Probability of exceedance = 0.5.: Marsden zones and speed reduction as given in Table 3.

P.U.:. Poisson uperossings O.S.: Order statistics

IP.:

Individual peak statistics

Other variations taken in the speed reduction ani operational prOfile have only given minor différences in the results. The only exceptiòns being that. the maximum (service) speed should not be applied in the mst severe sea states and the operational profile should include the

North Atlantic area (zones 15 and 16) but also other less severe areas. Otherwise, too

conservate design values are to be expected.

Bending morrjent amidship - [10e Nm]

Non-uniform headings:

P.U., linear 0.52

RU., non-linéar variance ' ' 1.00

P.U., non-linear variance and K4 . 1.51

O.S., non-linear variance and; 1.50

I.P., non-linear variance and; 1.45

RU., non-linear variance and K4, clustering included. ' 1.28

Uniform headings:

P.U, linear 0.54

P.U., non-linear variance and ;, clustering inch4ed 1.35

4. degree polynomia mass distribution:

P.U., linear.. . 0.50

4-kglm3] 3-.

r

J

-:

acc. to [16] - : polynomial approx. L...., --, L.. AP FP

Figtire 8 Mass distributions used in the long term extreme value calculations.

From a design pointofview, reasonable values of the wave-induced sagging and hogging

bending moments could be 5ì1O Nm and 2.71O Nm, respectively for the present ship. These válues could tentatively be compared to the present rule values of 4.3lO Nm and 2.8lO Nm fór the same ship.

The non-lineatities arise from the derivatives of the waterline breadth, the added mass and the hydredynamic damping, equa.'s (6)-(8), and from the use of a Second ordre wave theory, equa. (9) A numencal study has for the present ship shown that the denvative of the waterline

breadth is responsible for most of the non-linearities. ThereEore, care has to be taken when specifying these derivatives. In the preent study, these values are obtained from the body

plan, Figure 1, as the true denvatives at the mean waterline However, calculations using

numerical derivatives based on well separated waterlines may give a better modelling for the

X(m,e) = 0.926 + 0.033 m + (0.074 - 0.033m

where m is the (negative) slope in the S-N curve. The result can be written

D = D

D

and to avoid reference to a specific structural detail, the ratio

-DHF

=

D"

is calculated for the present ship using the operational profile given in Table 3 and, a slope m = -3 in, the S-N curve. The result becOmesT 0.036. In this case the springiiig contribution

to fatigue damage can thus be neglected, reflecting that the main contributions to the fatigue' damage arise from the stresses in relative severe sea states. This is so even if the number of

peaks in. the springing response is about 7 times higher than in the rigid body response.

Finally, the relative importáne. of the ñon-linearities on the fatigue damage can be seen from the folÏowing ratios

b

LW+WF - 1.14 Dwplinea,

[lJ

n

HF -'

_{_=8.30}

\-L587 -2.323 (52) Fatigue damage

In each stationary condition (43), the fatigue damage for details subjected to longitudinal Stresses can be estimate4 from the equations (35)-(36) The total fatigue damage can then be found as:

D = D. =

_TzfS(v0Xd)r

_{+ TEfs;v0Xd7'}

(51)

where d is given by equation (35) with a possible correction factor X for tainflow counting, [28],

The asymmetry in the sagging and hogging bending moments does not yield significant

changes in the fatigue damage predictions because the stress range is nearly the same, see Figure 7.

5. Conclusions

A consistent, non-linear procedure for determination of wave-induced loads in flexible ships has been presented.

In the procedure linear and quadratic transfer functions for vertical bending moments and

shear forces are calculated from a strip theOry formulation. These results are used in a

statistical analysis to yield extreme wave-induced loads in the ship hull during its lifetime. The quadratic transfer function leads to non-Gaussian stochastic responses. The extreme value predictions and fatigue damage estimates take into account these deviations from normality by a Hermite series approximation in which only the mean, variance, skewness and kurtosis of the response are used.

A consequence of the non-linearities is that the sagging and hogging bending moments are different at the same probability level.

The procedure has been applied to a fast container ship. The ectreme wave-induced vertical sagging bending during 20 years operation world wide turns out to be about twice as large as the corresponding hogging moment. Compared to the present rule values, the same hogging bending moment. is found Whereas the sagging bending. moment predicted by the quadratic strip theory is about 20 per cent larger than the rule value. This difference could in part be due to the use of only one loading condition, but also be a result of the approximation in the analysis that the added mass, waterline breadth and hydrodynamic damping coefficients vary linearly with the relative motion of the ship around its niean Waterline.

\ A parametric study has shOwn that the extreme value predictions are rather insensitive to

changes in the operational profile as long as unrealistic situations are avoided, i.e. full speed in head sea in the most severe sea states.

Whereas the non-linearities are found to be very important for the extreme values they are of minor importance for the fatigue damage. This is mainly because the stress range, i.e. sagging minus hogging is about the same as in the linear prediction.

The hull flexibility has been taken into account Such that wave-loads due to continuous wave excitation (springing) can be determined. FOr the present ship these stresses lead to a minor increase in the extreme values but are negligibic for the fatigue damage. För typical merchant ships it seems that springing needs not be taken intO account in the design.

The procedure builds on some assumptions. The most important are

- strip theory formulation for the transfer functions.

- Hermite series approximatiOn for the statistical predictions.

Generally, strip theory formulations, ci. [l6] yield predictions quite close to experimental

values, bt it could be interesting to see a comparison with results based on 3D diffiction methods, too.

As for the other assumption, the results in [12], [19] indicate that the Hermite series apprOach can yield a very accurate approximation to the true probability density. Further verifications, especially fôr the very low probabilities, would however, be welcomed.

Finally, similar calculations for other types of ships would help to identify the range of applicability of the present procedure.

Acknowledgements

The authors wish to thank Peter Friis Hansen and Francesco Baldi for many valuable discussions.

The financial support from the EEC contract BREU-CT-91-0501 is appreciated.

6

References

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ISSC'82, Committee 11.4, Chapter 4, "Wave excitation"., Ed. Majewski, Gdansk, Poland,

ISSC'85, Committee 11.4, Chapter 2, "Excitatioñ of Vibration-Waves", Eds. Spineffi & Merega1 Genova, Italy, 1985.

ISSC'88, Committee 1.2, "Environmental Forces", Eds. Pedersen & Jensen, Copenhagen, Denmark, 1988.

ISSC'91, Committee V.1, "Applied Design", Eds. Hsu & Wu, Wuxi, China, 1991. J. Juncher Jensen & P. Terndrup Pedersen: ''Ön the Calculation of the Joint Probability Density of Slightly Non-Linear Stochastiò .Processes', GAMM-Tagung, Lyngby 1977, ZAMM Vol. 58, T 481-T 484, 1978.

J. Juncher Jensen & P. Terndrup Pedersen: "Wave-induced Bending Moments in Ships -a Qu-adr-atic Theory", RINA, Supplement-ary P-apers, Vol. 121, pp. 151-165, 1979. J. Juncher Jensen & P. Terndrup Pedersen: "Bending Moments and Shear Forces in Ships Sailing in 1negular Waves", Jurnal of Ship Research, Vol. 24, No. 4, pp. 243-251, 1981.

A.H. Madsen & J. Juncher Jensen: "On a Non-Linear Stochastic Wave Theozy and

Morison's Fórmula", Proc. 8th Joint Ini Conf. on Offshore Mechanics and Polar Engineering (ÛMAE'89), the Hague, H011and, 19-23 March 1989.

J. Juncher Jensen: "Fatigue Damage Due to Non-Gaussian Responses", Journal of the Eng. Mech. Div. ASME, Vol. .116, No. 1, pp. 240-246, 1990.

J. Juncher Jensen: "Fatigue Analysis of Ship Hulls under Non-Gaussian Wave Loads", Marine Structures, Vol. 4, pp. 279-294, 1991.

1121 J. Juncher Jensen, J. Buus Petersen & P. .Terndrup Pedersen: "Prediction of Non-Linear Wave-induced Loads on Ships", Proc. TUTAM Symposium on the Dynamics of Marine Vehicles and Structures in Waves, Brune! University, London, 24-27 June, 1990.

[13] J. Juncher Jensen, P.

Temdrup Pedersen & J.

Buus Petersen: "Stresses in

Containerships", Proc. STG meeting, Lyngby, 10-11 June, 1992, Report Brite-Euram Project 4554, No. 2.1R-02(l), June 1992.

[14] J. luncher Jensen: "Dynamic Amplification of Offshore Steel Platform Responses Due to Non-Gaussian Wave LOads", accepted for publicationiri Marine Structures.

[151 Naess, A!: "Statistical Analysis of Second-Order Response of Marine Structures", J. Ship Research, Vol. 29, No. 4, pp. 270-284, 1985.

Flokstra, C.: "Comparison of Ship Motion TheOries with Experiments for a Container Ship", International Shipbuilding Progress,. VoL 21, June 1974

P. Andersen & W. He "On the Calculation of Two-Dimensional Added Mass and Damping Coefficients by Simple Green's Function 'technique", Ocean Engineering, Vol.

12, No. 5, pp. 425-451, 1.985.

F. Tasai: "On the Damping Force and Added Mass of Ships Heaving and Pitching", Reports of Research Institute for Applied Mechanics, Japan, VoL VII, No. 26, 1959. Winterstein, S.R.: 'Nonlinear Vibration Models fôr Extremes and Fatigue", ASCE, J. of Eng. Mech. Div., Vol. 114, No. 10, pp. 1772-1790, 198&

Vanmarcke, E.H.: "On the first-passage time for normal stationaiy random processes", Journal of Applied Mechanics, Trans ASME, pages 215-220, March 1975

Ditlevsen, and Lindgren, G.: "empty envelope excursions in stationary gaussian processes", Journal o. Sound and Vibration, Vol 122(3):511-587, 1988.

Mansour, A.E.:, "Extreme Value Distributions of Wave Loads and Their Application to Maxine Structures", Proceedings of Marine Structural Reliablity Symposium, Arlington, Virginia, 1987.

Longuet-Higgins, M.S.: "Modified Gaussian distribution for slightly non-linear variables", Radio Sci., National Bureau of Standards 68D(9), 1049-1062, 1964.

Naess, A. & Ness, G.M.: "Second-order Sum-frequency Response Statistics to Tethered Platforms in Random Waves", Applied Ocean Research, Vol. 14, No. 1, pp. 2332, 1992 Dogliani, M.: "Application of the Kac-Siegert Metho4 to Non-Linear Vertical Wave Bending Moment", Report Brite-Euram project 4554, No. 1.3R-03(A), June 1992.

where the constant C. can be chosen as

(J)

u uJ = C 6,

k=1

(A3a)

[26] Hogben,N., Dacunha, N.M.C. and 011ivér, G.F.: "Global Wave Statistics", Ed. British Maritime Technology, Uhwin Btothrs Ltd., UK., 1986.

[27] Cramer, E.H. and Friis-Hansen, P.:. "Stochastic Modeling of Long Term Wave Induced Responses of: Ship StructuÏes"! Súbmitted to Marine Structures, April 1992.

[28] Wirsching, P.H., and Light, M.C: "Fatigue Under Wide Band Random Stresses", Journal of the Structural Division, ASCE, Vol. 106, No. T7, July 1980.

Appendix .1

Transformation of equation (22) to the Kac-Siegert formulation

The quadratic equation (22) .

E cA (A.l)

i-1 ¡«1 j2

can be transformed to

MA _(IY, y2) (A.2)

where p. are the 2n eigenvalues (real), determined from the eigenvalue problem.

= 11

¡j = l,2,..,2n

and with uJ being the corresponding eigenvectors. As A ₌ X,, and A, real, uj" only contain real components and furthermore satisfy the following orthogonality conditions

and, if needéd, the reverse transformations become

xi

=Eí3k'i

;

= Epkui

¡A1 ;

p =

j1 k-1 k-I jI k-1

i = 1,2,...,2n (A.3b)

Without loose of generality. The transformed stochastic variables Y becomes

3

= (A.4)

i1

The original stochastic variables 2 were standard uncorrelated Gaussian variables. The

transformation (A.4) together with thé orthoriormality conditioñs (A.3a-b) ensure that Y, also are standärd uncorrelated Gaussian variables.

Finally

(k)

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