Ship motions and loads, general

The calculation of design values of ship motions and loads caused by waves proceeds through the following steps:

Calculation of the 2-dimensional periodical flow around ship sections

Construction of the three-dimensional flow and the accompanying forces and moments

Determination of the short-term distribution of motions and loads in a natural stationary seaway

Calculation of the long-term distribution of motions and loads Calculation of the extreme value distribution.

Contrary to previous ISSC repors on these topics, much more atten-tion is given here to nonlinear effects as it is suspected that these effects have severe influence on extreme values of shear forces and bending moments and that they may dominate the springing behaviour and, of course, whipping and slarrOEning behaviour of ships.

1. Calculation of the 2-dimensional periodical flow around ship secticn Inspite of some effort in computing directly the 3-dimensional

po-tential flow around oscillating bodies (1,2,3,4,5), practical

computations for real ship forms with steady forward motion will be based at present on a solution of a series of 2-dimensional flow problems for the transverse sections of the ship. For the linear

case (small oscillations of the section in still water, fixed section in small waves) , singularity methods are pplied. Whilo for simple section forms in deep water, especially non-bulbous Lewis sections, and for moderate frequencies, the well-known older methods of Grim

(6,7) and Tasai (8) work very well, for other applications as, for instance, bulbous sections, sections inclined at the waterline, fully submerged sections, finite water depth or higher (springing) frequencies, various new methods have been developed (9,10,11,15). Usually, they need much more coxrputer time than the first-menticned ones, and most of these methods encònter numerical difficulties at certain ("irregular") frequencies. Methods to

difficulties are described in (57) and. (58).

ScheeshydrOmechfla .rchIet

Mekeweg 2, 2628 CD Dò!ft

For the non-linear case, the following methods may be applied: - Quasi-hydrostatic methods (13) are applicable until very high

orders (multiples of the fundamental frequency), but they under-estimate the nonlinearities and should at least be supplemented

(for vertical ralative motion between section and water) by the nonlinear terms of the force d (rn" (z) ) / dt, where m" (z) =

added mass of the section submerged to the draft z, and = rela-tive velocity.

- Singularity methods (14,16,18) reach only to the second order and lead to complicated formulae; their reliability range will have to be investigated further.

Variational principles similar to (17, 2) may be designed to de-liver second-order forces (not pressure distributions) from first-order potentials with comparatively simple and quick computations; however, that seems not yet to have been done. The same formulae may also be derived in a similar way as the Haskind relation or the Newman Theorem by application of Greens Theorem on the second-order term of the potential in the for-mula for the second-order force.

Grid methods transforming the differential equations to difference equations in a network of quadrilaterals with fixed (Eulerian) or floating (Lagrangeian) boundaries (3, 19, 20, 21, 22, 23) give reliable results to hi.gh order, however, in very considerable com-puter time only. These methods deserve much more attention by naval architects especially for the calculation of slamming

pres-sures and forces and may, on the 1onc run, become as important in hydrodynamics as finite-element-methods in solid mechanics.

2. Construction of the three-dimensional flow and the accompanying forces and moments

Strip methods, in the original form of Korvin-Kroukovski and Jakobs (24) or in the version independently developed by four authors (25, 26, 27, 28) still govern the field. The hydrodynamics mass and damping terms may also be determined by the method of Ogilvie and Tuck (29). Inspite of its derivation by considering the three-dimsional flow around the ship, the computing procedure is based

en-tirely, as in strip theories, on quantities aociated with the two-dimensional flow in the sectional planes. Similar methods for the calculation the exciting forces and moments on a non-os-cillating body in waves by Faltinsen (59) and Maruo (60) will have to be extended for practical application to the bow region or for ships with forward speed, respectively, and (both) for non-head waves.

Comparisons of motions and loads calculated by strip methods or the method (29) with results of model experiments show fairly good coincidence for wavelengths> 0.5 L and Froude numbers up to at least 0.3 for heaving and pitching motions; the coinciden-ce seems to be best for the Ogilvie-Tuck method (30). For

swaying and yawing motions and accompanying loads, however, large differences between calculations and model experiments are

en-countered (61, 62, 63). Completely wrong results will be

ob-tained in the vicinity of the ll resonance frequency lest empiri-cal roll damping coefficients are applied; difficulties arise also

in stern waves of low frequenca of encounter because of exces-sive surging motions.

For springing motion, at first very simple strip methods have been applied (65, 66). Principally, strip methods seem applica-ble for the determination of mass and damping terms, but - due to the short relevant wave-lengths - not for calculatiQn the

exciting quantities. In (64) linear springing exciting quantities are derived by means of the Newman-theorem (31) using a strip method for the determination of the motion-induced flow

poten-tial. However, investigations of Grim (yet to be published) have shown that linear springing exciting forces are insignificant as compared to the second-order excitation. In (64) there is

further shown that the damping of springing motions by wave-ma-king and by vortex generation at the bilge keels and at the tran-som if it is immersed, together amount to about the same damping as observed in full-scale measurements. The damping, however, is influenced severely by the change of the waterline height relative to the ship sections due to the low-frequency waves and ship mo-tions. As no computing methods exist which take into account

this time-dependant damping and which determine reliably the higher-order wave excitation, at the moment sprirgini amplitudes cannot be

predicted. It even seems uncertain if springing is really a

problem for the structural strength of ships (contrary to whip-ping, which, definitely, is very important).

The inclusion of nonlinear terms in strip theories may perhaps be no problem if the corresponding terms of the 2-dimensional

flow in the ship sections are knowi,Fr: analogy with added re-sistance in a seaway, one may suspect that in case of heave, pitch and bending moment, the nonlinear effects may be approxi-mated sufficiently by quasi-hydrostatic methods, as is done in

(36). For calculating extreme loads, inclusion of these effects appears essential (see Fig. 1).

For the author of this review, the most startling fact is that the work of Grim to overcome strip methods seems to have found no application or succession. The method is outlined in principle

in the well-known paper (6) and was further advanced and numeri-cally evaluated for a moving ship in (33) and (34). The method seems theoretically convincing and - contrary to the

above-men-tioned 3-dimensional methods (1,2,3,4,5) - practically applicable for ship motion problems.

3. DeterminaLion of the short-term distribution of motions and loads in a natural stationary seaway

For designing ship structures, the quantity of main interest is the cumulative distribution F(x) of the maxima of ship responses x. For responses depending linearly on the sea surface elevation, the cumulative distribution of maxima is (for a Gaussian seaway; see,

f. i., (38))

fX

)1

- --

_2(4-)

4i

i y (ti,1A) I

tif

H = response amplitude operator; = seaway spectrum;

= circular frequency;

= angle of wave encounter.

This distribution was verified in case of measured longitudinal stresses from combined low-frequency bending and springing by Miles (38).

If no elastic vibrations of the ship have to be considered, the response spectrum is narrow enough to calculate with the limit of the above formula for= 1, the Rayleigh distribution

f(x)

For non-linear responses, various methods have been developed; no single one solves adequately all problems associated with nonlinear ship responses, and some problems seem to be handled adequately by none of these methods. Such methods are:

Removing the nonlinearity. If a response is a nonlinear function y(x) of a single linear response x, the cumulative distribution function Y of y is Y(y(x)) = F(x), the function for linear re-sponse. An example are post-buckling stresses depending non-linearly on a conpressive load.

Integration of multi-dimensional Gauss distribution. If a re-sponse y is a nonlinear function y(x1, X2, ...) of a few linear responses x1, X2, ..., the distribution function Y(y) may directly be integrated from the multidimensional Gauss distribution (see, f. i., 37). An example is the quasistatic pressure of green water on the deck of a ship, depending on

a product of two nearly linear responses (relative motion surface - ship and vertical acceleration of the water on deck; see 39). Another well-known example is the slamming pressure (51) depending on relative position and relative velocity between ship and water surface. Here, the

2-dimen-sional Gauss distribution reduces to the product of two one-dimensional distributions as the position and the velocity are uncorrelated.

Perturbation analysis (see, f. i., 40 and 12). In this method, the nonlinear differential equations of motion are appro-ximated as a recursive set of linear equations. Convergence may be expected only for small nonlinearities.

Functional representation, using higher-order spectra (see, f. i., 40 and 42). This method seems to be applicable for relatively small nonlinearities, too.

Equivalent linearization (see, f. i., 41). Here, linear coeffi-cients are substituted such that the mean square of the error

term introduced by the linearization is minimized. It may he speculated that this technique, if applied not to mean square values, but to extreme values, underestimates the effect of the nonlinearities substantially if no special adaptions are applied.

Fokker-Plank Equation (see, f. i., 52). This method seems to be applicable in a simple way only for white-noise excitation;

for actual wave excitation, valuable results may be obtained if the response is confined to a relatively small frequency region (as is the case for ship rolling) over which the

exci-tation spectrum may be assumed to be constant.

Simulation. Here, the differential equations for the respective response are solved in the time domain by means of analog, hybrid or digital computation (see, f. i., 43). The

determina-tion of probability distribudetermina-tions usually takes very much com-puter time.

Besides these more generally applicable methods, a number of methods of taking into account nonlinearities in special cases have been deve-loped. It is supposed that some of these methods may be adapted to other problems: Newman (44) developed a method for calculating slowly-varying second-order forces; Grim (45) proposed a method for investi-gating the safety against capsizing which has also been applied by some other authors (46, 47); and he further developed a method for calculating the probability of extreme nonlinear surging and yawing motions (48, 53). Grim (to be published) ad Söding (64) developed a method to predict the response spectrum and probability distri-bution in case of linear mass, damping and restoring forces and

non--linear exciting forces for a narrow response spectrum-conditions prevailing in case of springing motions.

Besides the distribution of maxima, two other quantities of some interest in designing ship structures are:

The mean frequency of maxima,

i

maxi-mum responses during a given time. If m4 has not to be calcu-lated for other purposes, f may, for a narrow spectrum, also be approximated by the mean frequency of zero-up-crossings,

-10 írV ''c

(see, f. i.., 40).

The distribution function of range-pair-counted load or stress cycles is necessary for the determination of fatigue damage by means of damage accumulation hypotheses or crack propagation hypotheses. Appropriate formulae for linear (Gaussian) respon-ses are given in (49). In case of a narrow spectrum (no elastic vibrations), it may suffice in most cases to assume that the mean value of load cycles (maximum load + minimum load of a

cycle divided by 2) is ecrual to the still-water load (or, if nonlinearities are included, the mean load in waves) and that the cumulative distribution function of stress ranges s

(maxi-mum load - mini(maxi-mum load of a cycle) is equal to F(s/2), where F is the cumulative distribution function of maxima.

Calculation of the long-term distribution of motions and loads In previous ISSC reports, it was recommended to construct the long-term distribution of response maxima from short-long-term distributions computed with the rIISSC_gpectrumu by means of a superposition accor-ding to the long-term probability distribution of _{significant height,} period and angle of encounter

of

The systematic and random variations of spectral form may not be disregarded.

The mean spectra deviate substantially from the Pierson-Mos--kowitz form underlying the "ISSC-Spectrum".

The reduction of speed and change of course anqie in a severe seaway may not be disregarded for calculating extreme responses. It is hoped that a better procedure with regard to 1. and 2. is re-commended by Committee 1.1. Methods for taking into account 3. by means of adjusting speed and course angle in each short-term response

calculation with respect to frequency of slamming, deck wetness and propeller emergence exist (see, f. i., 50). However, they seem to be applied seldom, presumably because of uncertainties with regard to the proper limits of these effects and because of the very substantial increase in computer time.

5. Calculation of the extreme value distribution

ForN statistically independent response amplitudes distributed

according to the cumulative distribution function F(x), the probabi-lity that the greatest of these N amplitudes exceeds a value x is

(for 1-F(x)Zz1)

For N(1-F(x))<1, this reduces to N(1-F(x)). If F(x) is expl.icitely known, there are no reasons seen for applyina other formulae (as,

for instance, Gumbels distributions) which are based on special approximations of F(x). The influence of statistical dependence of amplitudes because of the finite number of severe gales encountered by a single ship may be taken into account by applying the correction factor of Fig. 2 (from 49).

For application to ship structures, the extreme-value distribution can only be determined in connection with the distribution of other types of loads, which, however, is the task of Committee 1.3.

Superposition of Loads

Dimensioning of ship structures is based in many cases on an analysis of several stresses at K different locations in or different

alternatives of the structure. Each is computed as a linear function of I different "loads" x.

- i

Q. )ç.

t

The "influence factors" a.k have to be determined by an elastomecha-nical analysis of the structure. The "loads" X. may be concentrated or simply distributed "elementary" loads (54) or actual load distri-butions occurring in regular waves of different frequency and angle of encounter in two different phases; the best choice depends mainly

on computational effort.

There has been much discussion about where and how to perform the superposition of loads to stresses in case of wave-induced loads, but it seems that this discussion has not fully clarified the matter. Sound and generally applicable methods are the following:

1. Application of the superposition formula between steps 2 and 3 of the abose description ("Transfer-Function Method"; see,

f. i., 54) : The complex transfer functions of stresses are

calculated from the comolex transfer functions "1' of loads:

xl

Superposition between steps 3 and 4 (Multiple-Input-Spectrum

Method"; 55): The variance R1 (called m0 in the previous section) of stresses is determined from the variances and covariances

R of the loads: x.x.

'J

irr

r

( Y*(,/)Y.(,/1) S(t,,,)

);

( indicates the conjugate complex value)

Superposition between steps 4 and 5 ("Square-Root Method", 56): An exact method seems not yet to have been developed, but may presumably be derived from the description in the previous ISSC-Report (Committee 3, 51-55).

It is supposed that, if K (number of stresses to be determined) and I (number of loads necessary for calculating these stresses) are in the same range, the first-mentioned method, containing no double-summation, takes the least computing effort.

For calculating equivalent stresses which depend nonlinearly on the normal and shear stresses, the former remarks regarding nonlineari-ties are applicable.

References

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s n

Sum of 1 and 2 Order

First-Order Moment Hogging

Moment, Sagging

1 2

Wave Amplitude, in

Fig. 1. Bending Moments of the Containership "Hauburg

Express" in regular head waves of different amplitudes. Wave length = Ship length. From ₍₃₆₎

u. 12 lu

N

Fig. Reduction of maximum wave response resuiting

from the corroation between ampitrdos The reduction

depends on the probability I - G with v'hich this maximum response is exceeded during 20 years of operation under

typical conditions for a scagoing merchant vessel From ₍₄₉₎ st nd un of i and 2 Order Moment, Hogging Mean Second-Order Mom ent

N