Statistical approach to the analysis of longitudinal stresse in a simplified ship’s girder due to a long crested irregular oblique sealoading

l)elft University of Technology

Ship Hydromechanics

_Laboratori

Library

Mekelweg 2, 2628 CD Deift

The Netherlands

Phone: ±31 15 2786873 - Fax: -t-31 15 2781836

"Statistical Approach to the Ana lysia

of

Longitudinal Stresses in a Silified Shíp 's Girder due to a Long Crested Irregular Oblique Sealcading"

BY

Prof.dr.ir. R. We.reldsrna

Deift University of Technology D elf t

The Netherlands

SYMPOSIUM 'TDEVELOPMENT IN MERCHANT SRII'BUILDING' 30.5.172 _{- 2.6.'72}

DELFL UNIVERSITY OF TECHNOLOGY

STATISTICAL APPROACH TO THE ANALYSIS OF LONGITUDINAL STRESSES IN A SIMPLIFIED SHIP'S GIRDER DUE TO A LONG CRESTED IRREGULAR CBLIQUE SEALOADING.

BY R. WERELDSMA CONTENT Abstract List of Symbols O. Introduction.

Equations of motion of the ship's girder. Excitation byobliaue regular waves.

Normal mode excitation by external harmonically distributed forces.

)-. _{Dynamic amplification and stress analysis for vertical modes.}

Horizontal and torsional modes.

Addition of horizontal bending-, vertical bending- and torsional

warping-stresses.

Spectral technique applied on the maximum stress.

Remarks.

ABSTRACT.

By means of the normal mode method, an analysis has been made of the stresses in a slender beam, induced by an oblique sea.

Three types of loading have been considered i.e. vertical bending, horizontal bending and longitudinal torsion. Participation factors for a long crested regu-lar sea are considered. Structural resonance has been taken into account and it is shown how 3 types of longitudinal stress distributions, _{caused by one unique} deterministic sea loading, appear.

Finally the summation of the various stress types is shown and an application of spectral analysis of the stresses in an irregular sea is indicated.

-LIST OF SYMBOLS.

b = beam of ship.

C = constant.

E = _{modulus of elasticity.}

E excitation of normal mode. e = _{extreme fiber.}

F _{force (torque) per unit of length in z-direction due to the waves in}

a direction x, y or , depending on subscript.

G = centre of gravity.

G shear modulus.

g = acceleration of gravity. H = dynamic response function.

h distance of G and S.

I = linear moment of inertia of cross-section.

I polar moment of inertia.

It = torsional rigidity.

K restoring force in z or direction.

L = length dimension.

1 shiplength.

M participation factor.

MB bending moment.

= mass of ship included entrained water, per unit of length in z-direc-tion.

p = characteristic root.

R = _{distance of shear centre to the resultant lateral wave force F.}

S = shear centre.

S = spectrum.

T draft of ship.

T = time dimension.

u,v,w, = displacements.

Ve = speed of encounter of waves.

V = ship speed.

= propagation speed of waves.

X,Y,Z, = coordinates.

= incident angle of waves.

X _{wave length.}

Xe = wave length of encounter.

= damping per unit of length.

P _{density of medium.}

O stress.

rotation due to torsion.

= _{normal mode deflection, function of x.} w = frequency.

sectorial coordinate. We frequency of encounter.

= natural frequency.

STATISTICAL APPROACH TO THE ANALYSIS OF LOIIGITUDINAL STRESSES IN A SIMPLIFIED SHIP'S GIRDER DUE TO A LONG CRESTED IRREGULAR OBLIQUE SEALOADING.

INTRODUCTION.

For long time longitudinal ship structural stresses have been analysed with a simple static load configuration (standard wave approach) and a fully detailed structure. The analysis is restricted to vertical bending loads

only. The combined horizontal and torsional loading has been taken into account by empirical means.

The vibration aspect has been dealt with fully inderendent of the _strength problem. This was acceptable, because vibration excitation was generated by

sources independent of the wave loading (propellers and engine) and the vibra-tory stress did not contribute significantly to the strength problem.

The increase in ship size and propulsion power, however, changed this situa-tion and an integrasitua-tion of txe strength and vibrasitua-tion problem has become

necessary. The magnitude of the excitation, generated by the propulsion system, is increased and out of resonance vibrations have grown to unacceptable levels from a point of view of habitability. Lcnger ships and application of high ten-sile steel have lead to a reduction of the natural frequencies of the hull de-flection modes. A critical 2-noded hull dede-flection, excited by high frequency waves has now become realistic and needs to be incorporated in the strength analysis. In this paper an outline is given of the application of the normal mode method, developed in statistical structural mechanics. This method opens

the possibility to combine the statical and vibrational aspects of the

strength problem /11*. The structure is strongly simplified but the applica-tion can be extended to important elements as vertical bending, combined

horizontal bending and warping of the ship's girder as accounted for in designs of ships with very large hatch openings.

In this approach linearity is assumed throughout the analysis, which implies small excitations and parallel sides of the hull.

1. EQUATIONS OF MOTIONS OF THE SHIP'S GIRDER;

For the purpose of an analytical treatment the ship's hull will be considered as a long slender thin-walled beam having a constant cross-section and mass

distribution.

z.w

*) These bracketed numbers refer to a list of references at the end _{of this}

paper.

--3-Fig. 7.

The structural effects of the fore- and aft-end of the real ship _{are neglected,} although the warping rigidity of these elements might. be of importance.

For a thin-walled section the following differential equations can be derived, /2/

The vertical plane is a plane of symmetry. The origin of the system of coordi-nates coincides with the shear centre S. G is the centre of gravity.

The external forces are reduced at the shear centre S and are functions of z

and the time t. (See Fig. i.)

For the equilibrium in vertical direction holds:

.2

4

aw

nz 2 + z at + EI-y 4 -- K w _{F(x,t) -} - - - (1)

at

where m = mass per unit of length of structure and environment.

C = damping per unit of length of the structure and environment. E = modulus of elasticity of structure material.

= moment of inertia around axis (through centre of gravity). K7 = hydrodynamic restoring force.

F(x,t) external wave loading in vertical direction (through S and G).

For the equilibrium in horizontal direction holds:

4 n

+C -+m .h

_+EI

av

_-F(x,t)

_- --

-(2)

y at2 y at y 2 z , at

For the torsional direction holds (motions around S):

2 4 2

a a

- (3)

o 2 at +

EI-4

GIt

at ax ax

where I _{= polair moment of inertia around S.} = moment of inertia around x-axis.

I( = sectorial moment of inertia.

= torsion constant.

G = shear modulus.

h = distance from shear centre to centre of gravity (see Fig. i).

Fy(Z,t) external loading in lateral direction.

F(z,t) = external torque loading in longitudinal direction.

These equations describe a mixture of the hydrodynamic phenomena and the elastic deformation of the ship's girder.

Because the attention is focussed on the structural behaviour of the girder, the hydrodynamic effects are oversimplified, a.o. by the neglect of the coup-ling terms.

For the solution of these equations use will be made of the independency _of equation (i). Equations (2) and (3) are coupled because of the different loca-tion of centre of gravity and shear centre.

For various shiptypes and loading conditions _{simplifications are possible and} in order to avoid complications of horizontal _{and torsional coupling the}

difference in location of the shear centre and _{centre o.f gravity has been} neg-lected in the further analysis. For particular _{applications however a mixture} of an open and a closed structure has to be considered.

2. EXCITATION BY OBLIQUE REGULAR WAVES.

The problem of analysis of the hydrodyriamic loading of a ship, sailing in regular or irregular oblique waves, is being studied _{for years by many ship} hydrodynanaicists. The aim of this paper is to show how the wave generated loading contributesto the stresses. For the _{sake of clarity the liberty is} taken to simplify the determination of the girder loading from the wave excita-tian to a static analysis with _{a correction factor. More adequate and realistic} methods are outlined

in

_{ref. /3,}

)/.

JVw

41 V

T

Fig. 2.

Shipsgirde in oblique waves.

5

It is assumed that the pressure on the hull, generated by the waves is a frac-tion of the height of local submergence of the hull. When the ship is sailing through a harmonic wave as indicated in Fig. 2 the _{following expressions can} be obtained for the distributed loading of the gir'er.

2v

2irVt

F(x,t)

p b P sin [ - - -. g . . . A 2v Vet Xe

_j

')i \r 4-eL 1-Xe r

F(xt)

2v T tg b p 2v x p . g . . . co s +

F(xt)

2v p. g b2 T.R Xe P.b tg cos Xe

2vx

Xe Xe

T

Ii

Where P is the pressure on the skin, generated by the waves.

Further simplifications, when a and the transverse ship dimensions are taken as a constant, and the wave height is small in comparison with the transverse dimensions of the ship, lead to excitations per unit of length as follows:

In vertical direction: F(x,t)

C,P.

.12X

Wet

In horizontal direction: In torsional direction:

F(x,t)

C.P. cos 2rTX +U)_et

I

F(x,t) Cq,.P. COSI ± Wet Xe

These excitations are related to one sinusoidal wave system and are therefore in the further analysis to be handled in mutual deDendency. It can be concluded that for the excitations F on one hand and Fr or F on the other hand, there

is a phase shift of 90°. All excitations are or small wave height and assumed to be proportional with the wave height.

3. NORMAL MODE EXCITATION BY EXTERNAL RAEMONICALLY DISTRIBUTED FORCES.

Eq»ations (i), (2), (3), control the vertical bending, horizontal bending and torsion of the ship's girder. The solution of these equations can be separated into two cases, one referring to the ship motions and another referring to gir-.

der deformations.

Since the elastic deformation of the ship's girder is very smaal in comparison with the wave height, and does not substantially affect the wave excitation

in a substantial way, separation is possible.

This means that in equations (i, 2, 3) the terms related to structure

deforma-tion,

24w a-v

and

' ax2

-6

) can be made zero.

Then the equations reduce to rigid body. motion equations.

On the other hand, for the elastic deformation it is possible to neglect the terms related to the hydraulic restoring forces. In that case equations for elastic beam vibration are obtained.

For very long ships, however, (e.g. over 500 meters) the girder defoxination will grow to a substantial fraction of the wave height arid both effects have

In the scope of this paper the mentioned separation is assumed to be allowed and the response of the girder on wave generated pressure fluctuations will be obtained through the normal mode method as illustrated below.

MODE MODE SHAPE VERTICAL

NUMBER

I

m

-L SI-II PSO RDR Fig-. 3.

First four orthogonal modes in three directions.

The solution of the homogeneous equations can be expanded in a set of orthogonal normal modes as shown in Fig. 3.

In this Figure the first four modes are shown. There are of course more func-tions (strictly speaking an infinite series) but the strongly simplified model of the ship (slenderness) does not allow to go beyond the fourth func-tion. Even the value of this latter function is questionable.

A number of functions refer to rigid body motions (5 types of ship motions), and do not contribute to the stresses in the girder material. Stresses are generated by the third and higher modes for bending and the higher modes for torsion. They are all related to the elastic deformation and do riot have

specific names. X:O x:½

ï

H E AV E ( w PITCH (Wx,i) SAGGING-HOGGING HORIZONTAL TORSIONAL

--WAVE PRESSURE TRAIN F tz); F7 (r.t); !if

i

Fig. 4.

Shipsgirder loaded by wave pressures.

ROLL',

2 MODEl W1)

The wave generated loading, a sinusoidal pressure distribution, expanded in the 3 directions as given in formulae (5), is moving along the longitudinal axis,

(see Fig. 14).

Each of these types can respectively be expanded in pure heave, pitch, ist and 2nd vertical mode excitations, in yaw, sway and ist and 2nd horizontal mode excitations and soll and ist and 2nd torsional mode excitations. This expansion in "participation factors" /5/, results in an elimination of the x-coordinate. Actually the x-coordinate is replaced by a series of normal modes. Each normal mode can now be treated as a simple time function.

The phase relation between the participation factors and the wave-pressure train can now be defined as the phase difference between the participation factor and the wave pressure excitation amidships (xo).

The shape of the normal mode and the harmonic pattern of the wave loading give rise to a trong wave length and frequency dependency of the participation

factor.

This factor E(t) for the ntn normal mode can be determined by the follo-wirg expression

En(t)

1

n2 dx

Fn(x,t) (x) dx

where F(x,t) is the wave loading in one of the three directions and P-1 equals the nth normal mode.

1 is a normalizing factor, in order to normalize

2(x) dx the assumed characteristic mode.

n

A dimensionless function M for the 0th participation factor of a sinusoidal wave pressure train can be obtained by the expression

iTi M (

n À

F(-t)

For the vertical, horizontal and torsional modes (respectively

i4,

and) the functions Mxr, Myri and M are shown in Fig. 5 as a function of

111

dimensionless length -i--- .

For the calculation of these functions acceptable approximations for the characteristic modes have been chosen. Improving the accuracy by taking the exact mathematical expression is assumed not to be necessary because of the strongly idealized model of the ship.

-o--W ( x , t)

lt

1. DYNAMIC AMPLIFICATION AND STEESS ANALYSIS FOE VEP.TICAL MODES.

When a speed, belonging to the considered ship, is selected and the incident angle of the waves is determined, the wave length can be converted in a fre-quency of encounter We and a dynamic analysis can be made. For this purpose, the transfer function H(w) of the normal mode deflection due to a normal mode excitation has to be introduced. Beside this function, covering the dynamic effects, a static response, characterised as a stiffness, has to be determined. The deformation of the girder for the th mode as a function of x and t, W0(x,t) can now be formulated as follows:

F(we) IT IN PHASE .zr_L X I

PITCH AND YAW

QUADRATURE M,, i lt i z I

(w)

e mw02 !Hn(w)

The factors of this expression can be explained as follows:

F( we) the local force per unit length along the ship's girder generated

F)

L

Mm( w) characteristic mode excitation as found by formula (6) and (7) (participation factor).

by the waves in vertical direction (dimension

9

Participation factors for various modes.

sin(wet + - - (8)

PHASE RELATION WITH EXCITATION MIDSHIPS

M r HEAVE SWAY AND ROLL Fig. 5.

M,, SAGGING AND HOGGING

M1, 21t IN PHASE

IL 3tt

3-NODED NORMAL MODE

QUADRATURE

1-NODED TORSIONAL MODE

QUADRATURE

lt _ITt

X

2-NODED TORSIONAL MODE

2 = a measure for the stiffness of the ship's girder and has the mw0

L2 dimension

H0(w ) = dimensionless transfer function, similar as that o± a second order

system, accounting for the dynamic amplification due to resonance. It results in a magnitude HI and a piase shift = arg. H0.

= the distribution of the deflection W(x,t) along the ship's girder, equal to the normal mode deflection.

sin(wt + ) = time function of the deflection (W0(x,t), where arg.H(w).

H()

The product M0(w)

2 is in the theory of ship motions known as the "response Amplitude Operator". Obviously a similar operator can also be applied

W(,t)

F

in ship strength analysis and could be defined as the quotient

In order to obtain the stresses it is necessary to determine the bending moment 2

and, as a consequence, the expression for

The product M(w) . F(x,t) = E0(t) . is independent of x as well as all the

other factors, except

.

2

th

The vertical bending moment for the n _{mode, MB, equals EI}

2 and it follows for the bending moment MBO:

M = EI . F( ) . M (w

Bn e n e

The stress at the extreme fiber due to the bendino moment of this deflection

mode ,

per unit of wave pressure equals:

e . M

* 2

JH(w)J

a2

* - - -

-- (o)

The section modulus (moment of inertia) can only be detected in the formula

2 EI - .

by the value of w0 = , substituting this expression we obtain:

ml

e.

I ri y

H(w)

mw02 H (w n e 2 i p °x n lo -2 )fl F ( we) sin(w t + ) - _(9) e - - -

-(ii)

a n F

For the 3rd mode (sagging - hogging condition) formuJa (ii) has been evaluated graphically in Figure

6.

XO Xr ---NORMAL MODE Wz.3

A

WAVE LOADING M

F()

Wo We WC PARTICIPATION FACTOR AS FUNCTION OF

-INTRODUCTION OF: SHIPSPEED

S H IPLENOTH

INCIDENT ANGLE OF WAVES PARTICIPATION FACTOR AS FUNCTION OF DYNAMIC STUCTURAL RESPONSE RESPONSE AMPLITUDE OPERATOR STANDARD WAVE SAGGING-HOGGING SPRINGING

From the last graph of this Figure it is possible to determine the maximum stress occurring in the extreme fiber, a distance e from the neutral axis,

in the midship section (for the sagging - hogging condition, 3rd normal mode) per unit of wave height as a function of encountered wave length.

The maximum value, encountered for the lower frequency, point A, represents the sagging - hogging condition and it is associated with the standard wave

approach. This maximum value occurs for xi

The second maximum (B) refers to the phenomenon springing, encountered fast long ships. In the general response functions, also in other directions, more of these resonance effects can be indicated.

Whether or not they are important depends on the excitation frequency from the waves and the frequency of resonance.

For the 4th mode a similar expression can be found.

As for the 3rd mode the maximum bending moment occurs amidships, for the 4th mode these extreme values are at about ¿ and of the ship length, see Figure

7.

Fig. 6.

Origin of wave to

stress trw2sfer function.

X3

d2

4,

x V2

MAXIMUM BENDING MOMENT

Fig. 7.

Fourth mode bending moment distribution.

The tensile and compressive longitudinal stress at midship, due to vertical bending) is not significantly influenced by the L+th mode) but is for the major part determined by the 3rd mode.

5. HORIZONTAL AND TORSIONAL NODES.

There are two other types of loadings generated by oblique waves i.e. the hori-zontal and the torsional loading.

When the shear centre and the centre of gravity do not coincide, a simul-taneous solution of equation (2) and (3) is necessary, (see section 1). For this purpose it is necessary to distinguish open and closed ship types.

For the closed ship types as e.g. tankers, it is .11owed to separated the two equations and to assume that the distance of the shear centre and centre of gravity is negligible.

The furhter analysis in this section is based on this assumption.

In a similar way an expression as (li) for the longitudinal stress due to horizontal bending can be obtained.

The torsional loading of the ship's girder is also given in (5). In this case, however, only the first mode refers to a rigid body motion (i.e. roll motion). The second (one-noded) mode contributes to the elastic deformation and results

:1 _{stresses. For the strongly simplified structure this mode results in shear}

stresses along the girder.

When a more realistic structure is considered, also normal longitudinal stresses are generated due to warping constrainst.

The 3rd torsional mode, however, shows warping stresses amidships. In a similar y as outlined in section 4 the transfer function of the wave loading to the :Jdship stresses can be found.

-6. ADDITION 0F HORIZONTAL BENDINO; VERTICAL BENDING-AND TORSIONAL

WARPING-STRESSES.

In Figure 8 the distribution over the cross-section is shown of the stresses belonging to each type of deformation (bending, warping).

MAX. LONGITUDINAL STRESS AT ¼ AND 3/i, OF SHIPLENGTH:

MAX. LONGITUDINAL STRESS AT 1/2 _{OF SHIPLENGTH:} =

Fig.

8.

Addition of

3 types

of girder stresses.

For one unique long crested wave system, however, we have to consider the relative phasing of horizontal and vertical bending as found by expressions

(1), (2) and (3).

When the stresses due to vertical and horizontal bending are to be added, this phase shift has to be taken into account arid a vectorial summation is

necessary.

Therefore, going back to formulae (5), we can state that the vertical, hori-zontal and torsional loadings result from one wave excitation function, P(x,t), having a wave length Xe and frequency We Each type of loading, however, has a particular phase, i.e. a 900 phase shift between the vertical loading F(z,t) and the horizontal loading F(y,t) or the torsional loading

( ,t).

The participation factors, (see Fig. 5), show also a phase relation between the wave loading and mode deflection. The general rule concerning the phase for the participation factors can be stated ás follows: for all the odd modes there is a phase shift of 900 and for the even modes there is _{no phase} shift. 13 -2

(---+) +(-+

+_)

VERTICAL BENDING IN PHASE HORIZONTAl, SEDiNG QUADRATURE TORSION QUADRATURE Z3 VY,L

---i

::

wç,3 -¶ 2VT

-iT-- -ç--i

,'Th

-=_-i Y2V

:-,.

-I z O N PHASE

V2'

QUADRATURE IN PHASE

--'

QUADRATURE QUADRATURE

2Lt

N PHASE

J

U) c

1I

This means that a vectorial summation of all the stresses is necessary. A scalar addition can be made for the stresses resulting from the odd verti-cal modes and the even horizontal and torsional modes and in reverse for the even vertical modes and the odd horizontal and torsional modes.

For the addition of all inphase components or all quadrature components it is sufficient to add simply the magnitude of the various contributions,

because, whatever the relative polarity of the various types of stresses may be, they will amplify each other in one of the corners of the rectangular

cross-section.

For the case that structural resonance has to he taken into account, the phase

of the mode vibrations can also be taken 00 or 1800 because in practice the damping has proved to be sufficiently small for this approximation.

It is necessary to consider the most severely loaded sections along the girder. For the 3rd mode deflection this turns out to be the midship section.

The 4th mode generates maximum deflections at and of the ship's length. The 3rd modes, however, contribute also stresses at these locations. Therefore it seams realistic to make a complete analysis of the normal stresses at the

three locations , and of the ship's length, for all important normal modes.

The total stress, obtained in this way, is given in Fig. 8.

7. SPECTRAL ANALYSIS.

When the phasing of each individual step is taken into account we can state that: S(o,1)

Ox,3/1+ 0y4

F 2 O + O + X,L

_y/12

/2 F 2 S(F) - (12)

where S(F) is the spectrum of the wave generated pressure fluctuation as encoun-tered by the sailing ship. (Only the 3rd and the 4th mode have been considered).

Similar expressions can be obtained for S(o1 ) ad S(a3 ).

Having knowledge of the spectrum of the stresses and having an insight in the

statistical distribution, conclusions can be drawn about the probability of extreme values, important for the judgement of the reliability of the structure.

8. REMARKS.

The method, outlined above, gives rise to a number of remarks.

a. The standard wave method for the strength calculation of the midsection of the ship is shown in point A of Fig. 4. Having a sinusoidal wave train, the sagging - hogging condition is covered by the 3rd mode vertical deformation and at point A the wave length equals the ship length. For trochiodal wave frorns the 2nd and the 3rd harmonics have also to be taken into account, which means that also point B and C have to be considered.

The "Response Amplitude Operator" is applied in the analysis of ship motions. The analogue version of the R.A.O. for elastic deflection is

w

x3

shown in Fig. 6 as the curve

F

Similar curves can also be found for horizontal and torsional deflections and for higher modes. The dynamic amplification, an essential effect in ship motions, has also in the field of elastic deformations become significant

during the last years. For the sagging - hogging mode it is known as the phenomenon "springing or whipping". /6/.

Similar critical conditions can also he indicated for the horizontal and torsional vibrations, although these effects have not shown to be important for today's ships.

The analytical approach, as given in this paper, has to be completed with or replaced by a numerical analysis for the case more realistic ships have to be analysed. The x-dependent mass and moment of inertia distributions and

elasticities necessitat3 a numerical approach.

Also from a point of view of structure- synthesis this approach is

neces-sary, because the analysis starts with a guess about the elasticity and

material distribution I)( I. I, when the dynamic structural response is considered. The stress, finally obtained will indicate whether or not a revi-sion of the distribution of the material is necessary.

In this way an optimisation procedure can be realised and computer aided structural design can be introduced.

The bending moment distribution along the ship's girder for one or another mode is given in formula (9). When the dynamic amplification Hn(We) is made equal to unity and independent of We the expression represents the bending moment distribution generated by the waves. Summation of all the modes will result in the moment distribution as has been obtained in seakeeping basins, where experiments are conducted in regular or irregular seas.

The results of these types of tests can be a start for the stress-analysis _as outlined in this paper, when this bending moment distribution along the

x-axis is expanded in normal modes.

The analysis in this paper is restricted to tensile or compressive longitu-dinal girder stresses. For a complete picture it is necessary also to consider the shear stresses as generated by the change of the bending moment.

This analysis of the shear stresses can be done in a way similar to that of the longitudinal stresses. A closer consideration learns that in that case also the , and points along the girder show maximum values. Since these

stresses are fully correlated with the bending, a deterministic analysis will be sufficient to complete the picture of the total stress for the longitudi-nal material.

In such a way it can be shown whether or not it is possible _{to reduce the} amount of material at and of the ship length, which, under control of the classification societies, is kept the same as that of the midship _section. In the introduction it has been shown that _{an independent treatment of} hori-zontal bending and torsion is only acceptable when the shear centre coincides with the centre of gravity. For many ship types, however, in particular contai-nerships the coincidence of shear centre and gravity _{centre does not exist.} Therefore it is necessary to develop the approach in such a way that also coupled horizontal- torsional normal modes _{can be handled. In ref. /2/ and} /5/ the fundamental aspects of these problems are outlined.

g. The indicated method of analysis can also he applied to structures which strongly deviate from ordinary ships such as extreme long or fast ships, drilling platforms. multiple hull vessels, etc.

The method enables the designer to meet the requirements of safety and reliability from a point of view of response and capability.

-LIST OF' REFERENCES.

/1/

Y.K. Lin:

"Probilistic Theory of Structural D3mamics".

McGraw Hill Book Cornpany 1967.

/2/ J.M. Cere and Y.K. Lin:

"Coupled Vibrations of Thin-Walled Beams of Open Cross-Section". Journal of Applied Mechanics, Sept. 1958.

/3/ J. Gerritsma and W. Beukelman:

"Analysis of the modified Strip Theory for the calculations of Ship Motions and Wave Bending Moments".

Report 96 Z, 1967 of the Netherlands Ship Research Centre

Deift.

l'il J.H. Vugts:

"The Hydrodynamic Forces and Ship Motions in Waves". Thesis, Deift University of Technology, 1970.

'5/ W.C. Hurty and H.P. Rubinstein:'

"Dynamics of Structures". Prentice - Hall, Inc. 196i.

/6/ F.F. van Gunsteren:

"Springing, Wave Induced Ship Vibrations".

International Shipbuilding Progress 197l page 333-347.