Qualitative considerations on statistical ship strength analysis, International Ship Structure Congress, ISSC73

Ship Structures Laboratory, Deift University of Technology, Mekeiweg 2, Deift,

The Netherlands. Report no. 177.

QUALITATIVE CONSIDERATIONS ON STATISTICAL SHIP STRENGTH ANALY.SIS'

by

R. Wereldsma

e

-Contribution for International Ship Structures Congress 1973 September 1973.

Dept. of Nay. Arh. Deift University of Technology, The Netherlands

Content:

Summary.

List of symbols. 1,) Introduction.

Fundamentals of the Normal Mode Method. Applications.

Bending Moment affected by hull flexibility. Lumped parameter analogy.

Statistics and standard wave loading. Quasi-static ship strength combined with wave generated hull vibrations.

Concluding -remarks. References.

-1-Summary.

By means ofthe Normal Mode Method an analysis is made of a ship structure as a spatially loadéd structure.

The ship is simplified to a long flexible beam supported by an elastic support. Criteria are obtained when the elastic deformation affects the wave generated loading., how the standard wave cän be statistically evalu-ated, and how dynamic amplification due. to resonance affects the quasi-static strength analysis. A procedure is indicated showing the effect of ship size and ship speed on the ratio of quasi-static and dynamic bending of the ship's girder, both treated by one type of analysis.

List of symbols.

A amplitude

a nth solution of the frequency equation of a vibrating beam C distance between centre of gravity and shear centre

e stiffness

EI bending stiffness F Foràe., wave fórce G = modulus of shear g centre of gravity

participation factor (generalized force, generalized wave-pattern) h wave .height

Ho = dynamic structural response funtion polar moïnent of inertia

'T = torsi.onal stiffness

I sectional moment of inertia

W constant factors - --£ ship length À. wave length M mass rn distributed mass

MB Bending Moment of ships girder

= damping distributed load S shear centre s power spectrum

XY

ign als x,y,z, coordinates nth modal deflection ---n

w - frequency

-of

encounter

natural frequency of nth node

i. introduction-.

-2--For many years ship's strength -analysis has been- basedon practical rules and experience. Modern computerized techniques have improved the reliábility and -accuracy of the. calculations, prticuiarly in the field of local strength problems. The Finite Element Technique is a powerful mean to analyze

corn-plicated structures, when boundary conditions can be well defined.

For the overall response ofthe ship in an irregular sea, however, it is necessary to gain knowledge and insight in connection with the

character-istic behaviour of the system: "ship in waves".

With the strong increase in dimensions of the ship, the fundamental change in construction (open containerships- almost without deck) and the application of other floating or standing maritime- structures such- as drilling platforms, working units at sea, -catamaranlike structures etc. it is necessary to re-consider the usual overall strength analyses and to try to base a new approach on the fundamentals of mechanics in order to overcome- unexpected structural reactions, such as critical vibrations, excited by short, high frequency, -- waves (springing) in e.g. vertical direction or combined horizontal-torsional

direction, for very wide ships- the flexural "membrane"- vibration in two -spatial dimensions of the wide- girder., the redudtion of wave excitations due to the increased sagging-hogging deflections of very long ships (rubber

- -boat effect).

A fundamental method may be found in the modal analysis as has been developed i-n the field of spatially loaded 3 dimensional structures.

This modal analysis opens the possibility to analyze the macroscopic behavi-our of structUres, from which a proper boundary condition can be deduced for a microscopic analysis of important details by means of e.g.. the Finite

Element Method. -

-The-purpose of this contribution is to give -some idea of the potentials of this method when applied on -ships or structures at sea.

-- 2. Fundamentals of-the- Normal Mode- Method.

For an analytical treatment of the cIassiò behaviour of a structure it is necessary to consider a strongly simplified model of the.reality. At a-first

gance. this -simplification may be too strong for a realisic analysis- but

- nevertheless interestiñg conclusions can -be drawn from this approach- and

refinements by means of lumped parameter appÑximations are possible in

order to-come closer to reality.- - - -

-An outline of the method will be given- with. a s-lender "uniförin" ship as -an

exainp1e----/1/, - /2/. - - -

-The ship is now -simplified to- a slender beam having one spatial coordinate ----oriented -along the longitudinal z-axis - of the ship and three degrees of free -- dom1 i.e. a vertical., a- horizöntal an-d a. torsional displacement, resp. X,

y and , (fig. i). - -

-Further we will base our considerations on a known spectrum of exòitation forces-, that are the result of the -wave pressures ad the acceleration forçes due to the distributed mass and ship motions This loading of the

ships girder is t-o hé Ñduced to a ditributed hòr.izontal and vertical load-

-Since the ship is, generally speaking, a symmetrical construction (referring. to starboard and port) there is the fortunate situation that the vertical deformation is independent of the horizontal and torsional deformations. The latter two are coupled because of the non-coincidence of the centre of gravity and the shèax centre.

For the vertical, behaviour of the hull the following equation holds:

a2 a2x .

{Ei(z) ---«-} + (z) - + C(z).x i- m(z)

_¡-

q (r, t)

This is the well known equation for a vibrating beam on an elastic support where shear and rotary inertia has been ñeglected.

The effect of'the water has been split into two contributions, viz.: The presence of the medium affects the dynamic properties the beam

i.e. added mass, damping and buoyancy.

- Thé wavy surface generates a spatially distributed time dependent mechanical loading represented by q (r, t).

The added mass is accounted for in m(z). The buoyancy is represented by c(z).

The damping.equals z) and: will be neglected.

For an analytical analysis a further simplification is necessary and will, be made by the assumption that the distribution of elasticity, mass and buoyancy

'îsuniform.

In that case the equation simplifies to.:

a4x

a2x

EI ---'+ Cx + m q, (z, t).

With similar simplifications the combined horizontal and torsional behavibur can be described 'by the following equations:

a2 EI

-4

+ m + mc

'-Ei-4 '

GIT

4

+ + IP + c (m + mc a2q t)

where y is the translation of the shèar centre S, is the rotation around

---the -shear.centre, C is the.stabiizing.torsional môment.

For the case that the stabili2ing moment and the St. Venant torsion can be " neglected _{'T '} O) the two formulae can be reduced to an identical form.

EI + rn - + mc q3 (z, t).,

(1)

(2)

where I is the total rotary inertia around the centre of shear (I i- mc2).

equals the horizontal distributed load, generated by the waves.

q, equals the torsional load of the waves around the shear centre, i.e. the integral of the pressure load on the section multiplied with the distance to the shear centre.

The neglect of the St. Venant torsion and its associated stiffness

'T is not

very well possible for the case that the cross section has closed cells. Many ship structures have closed cells or are closed sections in itself so that this simplificatiön is not allowed. (Examples of closed cells are double bottom, double hull, tanker structures).

When the shear centre coincides with the centre of gravity the equations become uncoupled (c O).

Solutions of equations (1), (2) and (3) can be obtained through the deter-mination of the eigenvalues and eigenfunctions of the homogeneous equations.

When the loading of the beam q(z,t), q7(z,t) and q(z,t) are also expanded in a series of eigenfunctions the solutions of the equations can be repre-sented by a series of mass-spring systems.

The total response of e.g. the vertical deformation of the ship girder is then .the sum of all the responses of the mass-spring systems in which the total system has been broken down, each having its particular' distribution of deformation (eigenfunction) along the z-axis.

This method is also known as the "normal mode method", (normal modes eigen-functions).

The breakdown of the beam loading q(zt), q(z,t) and _{(zt) into} eigen.-functions results in participation factors resp. r (t), r (t) and r (t)

n,x n,y n,4

where the subscript n stands for the n 'th eigenfunction /3/.

'This participation ftor is a tré funèt[òn of the time-ind

stial-distributed loading 'generated by the waves. It is a filter transferring those loadings that are responsible for the mode.

Because of the buoyncy C and the stabilizing torque C there aÑ also normal modes á'ssociated with ship motions (heave, pitch, roll). For a uniform ship

(i.e. all coefficients in the equation independent of z) these. normal modes do not contribute to the deformation of the ship structure. For 'realistic ..ships.,howéver,.thereis, also. a deformation.during these motions and

there-fore in that. case these modes need.aIso to be taken into consideration /2/. For our simplified model they can be omitted.

For. the vertical behaviour of a slender uniform ship the following analysis .in regular sinusoidal head waves, having different wave length, but constant

amplitude, can be made.

The first mode of inlerest is the well known 2-noded natural deflection and comes very close' to the sagging-hogging ondition of the ship. X, The second mode 'is the 3-noded deflection 'p ,,. (See fig. 3).

X,.

The corresponding participation factors r and .1' are 'given 'in the same

X1.j. X,

figure as a function of dimenionless 'wavelength . ' '

When the' ship 'speed is. thösen, the wavelength scale can be converted into a frequency scale (frequency of encounter' w)..

As has been outlined before, each of.these systems of deformation can now be treated as an independent 'mass-spring system and is it possible to draw

d3rn'ami9 transfer, functions Hi(w) and H2(w)' each having its characteristic natural frequency,. (w) _{and .(w0).2..}

-The product of the participation factor and the dynamic 'response function equals the total response of the particular mode under consideration dUe, to the sinusoidal wave excitation. ' ' '

A similar product is known in the .theory of ship motions under the name "Response Amplitude Operator". The separation of this operator into two terms opens the possibility to distinguish the static strength analysis and the dynamic amplification which is known nowadays as whipping or springing.

3. Applications.

a.) Vertical bending moment affected by.htill flexibility.

When the buoyancy of the ship, C, is taken into consideration there is the possibility to study the effect of the deflection on the wave excitation. An analytical solution of this problem can be obtained from the solution of the simplified equation:

2x

Ei + C.x t in q (z, t)

holding for the vertical deflection of the "uniform ship".

When the wave excitation q(z,t) is simplified to the Archimedes force pro-portional to the local draft of the ship we can state:

q(x,t)

Ch (z;,t), where h (z,t) equals the wave height.

The solution for the n-th normal mode deflection A due to a participation T of-averticai wave height can be--formulated_as.follows:

n

(Since ie refer to the vertical deflection, thé symbols are simplified).

where A C x n n -A C.2.-n X

i

-or -

-.

z.

r w n C.R. + _¡T i

-in the expression for w the termC9. is a correction on the natural frequency of the ship as a1eaxndue to the increaséd stiffness caused by the Archimedes force. A x,n n r x,n n

(w)

=w

oxin

-n

A

n_.

r

n

w n

Whether or not this correction is important depends on the magnitudes of the

EI

two terms

a

p

and

Two conditions

can

be distinguished, i.e.

'

«

C'9.

corresponding to a flexible ship

and

a4

- » C corresponding tó a rigid ship.

Fôr the flexible, ship we have

this natural frequency equals the natural frequency of

the rigid body motions.

The two-

and

higher-noded flexural vibrations of the ship are equal to zero

and simply not present because of the large flexibility.

The deflection of the hull due to wave excitation simplifies to

Ic

I!

X

w LI

n

vm

which means that, when the loading is treated in a quasisteady way <w + O)

the deformation is simply equal to thé wave surface. For the case the mass of the hull m is very small, we have the situation that w - and

-It

n

whch- -means-

that: -for-

a light: flexible. ship Le..g., an inflatable

n

rubberboat) the adaptation of the wave surface

and

hull deflection is always

: perfect., even when the frequency

w O.

For the 'rigid ship we have

21/EI

Wn_

V ;;jzr

which equals the

well

known

exression for vibrating beams. The effect of

the elastic support,. provided by the water, is simply not present.

The deflection equals . .

A

_{n_}

X i

k w

-A

Since C.9.. « a4 we can conclude that the deflection - O when

w w, and the ship behaves as a rigid body. n

For strength calculations it is interesting to calculate the bending moment

Mb..

Mb EI . A.

When we refer to the sagging-hogging condition of the ship, which is of major importance /1/ we can state that

= K EI.

It follows:

Mb Mb A2 K1EI.C.9.

ç-

ç

- CL

+ i

For the flexible ship we have

r2 EI

For the rigid ship it follows

M.

CL4

X

i

--K

L1

z.

r2 l.a2

1.-Under the assumption that the moment of inertia I is proportional to 2, we

have 1K.R3.

-2

When we restrict ourselves to static phenomena (w O) we can state that

1Mb].

K1C.L

Lr2istatic

i9:.

+P_

ÍMbi

[r

I

L 2-static

For large ship length:

Ir

I

L

2

static

K1K2 E2.3 K1 EI. (Flexible ship).

An illustration of the effect of the hull flexibility is given in fig. '-i.

Now when ships are growing longer and longer and the material of which ships are constructed is of an increased strength, which means that the allowable stresses and consequently the flexibility is increased we are approaching more and more towards the "flexible-boat-behaviourt' and for ships of more

than IWO meters of length i-t is questionable whether or not the conventional method of standard wave loading, or a conventionaÏ seakeepin'g test with a

simple wooden model will give proper reliable results. It might be neces-sary to design modified experiments or types of analysis based on system analysis of the fundamental mehanics.

An elucidation can be given by a mass-spring analogy representing the two-noded deflection dynamics of the hull girder.

Lumped parameter analogy of sagging-hogging deflection of 'a ship's hull. For the case the sagging-hogging deflection is assumed to be identical with the shape of the two-noded vertical hull vibration this deflection can be seen. as one of the normal coordinates of the system of hull dynamics and can be treated, independently of other deflection types, as a simple mass-spring system.

Simplifying the hydrodynamics again to static buoyancy forces.,. it is possible to have the entire system reduced to a mass-spring system as shown in fig - 5kj-where--X equals the-strength ofthe_participation of the wave force input,

C1 is the buoyancy of the hull, again referred to the mode shape, y is the deflection of the hull in the two-noded vibration shpe and C2 is the internal

1

stiffness of the hull beam. M is the generalized mass of the hull girder. The dynamics of this system are described by the block diagram given in fig. 5A and the transfer function -, being the strength of the 2-noded deflection over the wave fOrce input.,' equals:

C

Y.-

1 1

XMw

2

n

l-(--)

n

'where w is the natural frequency of the 2-noded hull vibration, w equals the excitation frequency and F is the net excitation, representing r2.

For the case hull_-vibrations by other sources such as propellers or engines are considered,, the diagram simplifies and is given in fig. SB..

In this case the feed back loop caused by the hull deflection ('adaptation of-.

-9-hull deformation to wave surface) is cancelled, and the transfer function equals:

.1 F c2.,

l-(-For the. case quasi-steady sagging-hogging calculations are considered, the hull dynamics are cancelled (w -' O).

We have (see fig. 5C):

if C1 » C2, for the case of a flexible boat, we obtain

i.e. the hull deflection is equal to wave surface..

If C2 » C1, a rigid ship,, the hull deflection will not affect the wave excitation ànd we obtain

= C1, which means that the ship girder loading is equal to the wave

height multiplied with the change in displacement, which refers to the, con-ventional sagging-hogging standard wave ]oad calculation.

Förthe analysis of springing and 'qùasistatic loading, however, we need to have the complete diagram of fig. .5A.

h) Statistics and standard wave loading.

For design purposes it .is possible to have the standard wave st'ength1cui. tion considered in connection with the participation factors, sagging-hogging deformation and wave statistics.. :

For many years the. standard wave method has been used successfully for the' primary strength calculation of the hull 'girder. The waveheight however was

subject to change.wheñ experience was gained.

Under the assumption that the sagging-hogging deformation for the' standard wave approach. can be sufficiently approximated by' the 2-noded normai deflec-tion and the irregular waves have a gaussian distribudeflec-tion, a more precise -analysis baséd on speetra'lanaiysis-:can 'be' made.

For this purpose we start with the calculation of the strength .of the two-noded deflection when the ship is sailing in waves, generating forces having a 'known .spect'r. When 'the .attention is focussed on the 2-noded deflection we can determine the participation function r. for. this deflection, see

.fig.6.

. .

'I'

4uitipiication of the square of the participation factOr and the 'spectrum of loading due to the waves results in the sagging-hogging load spectrum, Si2(w) (fig. 6). Depending òn the shape of the wave load spectrum and the length' of the ship the sagging-hogging load spectrum' 'has different appearances.

We obtain a shift in the participation factor due to a change in ship length as indicated in fig. 7A. The multiplication of r2 with the wave force spec-trum gives the 2-node excitation specspec-trum, having a nature as given in fig.

7B.

Jhe area of this spectrums

pi2

_{j SF}

(r1)2 dw,

lo

-equals the square of the standard deviation of the 2-noded deflection '1', and is,, through the

elastic properties of the- hull, also a measure for the. squared standard de-viation of the midship bending moment. MB. The area of this spectrum. will change with the ship length in a manner as shown in fig. 8. The shape of this curve can be verified by inspection of. fig. 7B. .

When we calcula-te also the sagging-hogging bending momeflt by means of a standard wave it is possible to. compare this bending moment and the standard deviation of the -irregular bending moment and for the conditions of our

cal-- culation (wave force spectrum, ship speed etc.) it is possible to indicate

a figure for the applied standard wave calculation., showing the probability -that the stndard bending.moment will be, exceeded. It is interesting to

study the effect of ship lengh in. this way-.

- The stndard wave calculation ¶results in bending moment lins, also given 'in fig. 8 where aso a comparison-can be made with the statistical approach.

it can be concluded that for increasing ship length the trend of the stand-ard wave caiculátion will result in a too large design bending moment.

---- This can be explained 'by the fact that the very long ships, when they meet a long wave having about ship length, the height of it will' be very much reduced due to the nature of the sea. The standard wave he'ight lines show still an increasing.height,withincreasing wave length, see 'fig. 9.

In the casê of very Ïong ships it might be. important -to consider also

the-- 3-noded and 4-noded deflection for longitudinal strength calculation,

refer-ing to shorter waves that might. have more significance.

Combination of -- wavegenerated-hull

vibrations..

-Beside' the' static sagging-hogging or 2-noded deflection, generated by the wave -surface there eists the possibility to, deal with .th.e two-noded

vibra-tibn- of the hull in asimilar manner. . - - '

-For that purpose we have to- determine the elastic dynamic response curve-of the hull as 'a girdér, where it is' possible to consider the dynamic ain plification due to resonance. - '- ' - - '

._i1níig. 10 is shown how the participation factör, the dynamic response and the spectrum of the wave- generated forces are to le -combined. '

In anaior with the theory of ship motions we can define the" product of

-- participation factor and dynamic response as the. Response Amplitude Operator

- (RAO) for the 2-noded deflection of the hull. -:

- -

-i

Multiplication of the spectrum of wave generated forces with the (RAO-)2

-result's n the spectrum of the amplitude of the 2-noded deflection. -- 'The -area of this spectrum is a measure for the. intensity of the 2-noded

deflection and also fr the midship bending moment.. '

-- in fig. il - is indicated how the combined 'bending can be broken down in a

it can be concluded, that for longer: ships. the springing will become more and more important..

A similar procedure can be applied for the case the ship speed is variable. In fig. 12 is shown how the ship speed. affects the amplitude of the sagging!

hogging deformation. . .

It can be qoncluded that for increasing speed the quasi-static bending is not affected and the springing phenomena will become of increasing significance..

12

-. Concluding Remarks.

For very long ships it might be necessary to consider not only the 2-noded natural deflection for wave generated strength and vibration studies but, in particular for the strength calculations, also the 3- and q-noded natural deflection, because the participation of the wave loading for these dflections will be more and more significant when ships are grow-ing longer and longer.

For a more realistiô analysis it is necessary to consider not only the vertical bending deflections but also the combined horizontal-torsional deformations. The latter types will also contribute to the longitudinal stresses of the ships girder. An addition of the various longitudinal stresses will open the possibility to give a spectral presentation and probabilities of exceeding a certain stress level, when the ship is sail-ing in irregular oblique seas /1/.

For wave generated vibrations it might be necessary to consider re-sonance phenomena, not only in the vertical direction (springing, whip-ping) but also in horizontal and torsional direction. Particularly the 1-noded torsional vibration of open container ships (associated with the yaw motioñ of the ship) might reach a critical value because of the large flexibility of these ship types in that direction.

LiP) For realistic ships it is necessary to include the effect of the non-uni-formity of the ships girder, which means that stripwise approximations and numerical approaches are necessary to analyse the ship's girder in the described way. It is then also necessary to consider the non-uniform loading of the girder due to water pressures and acceleration forces

caused by the rigid body ship motions.

5) Based on the "normal mode approach", combining ship strength and whipping phenomena, model experiments in seakeeping basins,measuring directly par-. ticipation factors for the various modes, as will be generated when

sail-ing in regular or irregular seas, can be designed.

The advantage of this technique can be found in the fact that distributed girder loadings (split into normal modes) are measured. The response of the elastic structure can then be calculated, including the effect of crit-ical-whipping phenomena. This in contradistinction with the regular sea-keeping tests where resulting bending moments in various cross sections are determined directly . A detailed description is given in ILl!.

References.

/1/ Wereldsma,, R.:

"Statistical approach to the analysis of longitudinal stresses in a simplified ship's girder, due. to a long crested irregular oblique sealoading".

Paper read before the symposium "Development in merchant shipbuilding 1972 Delft.

/2/ Bishop, R.E.D., Eatoch Taylor, R. and Jackson, K.L.: "On the structural dynamics of ship hulls in waves". The Royal Institute of Naval Architects, 1973.

.13/ Hurty, W.C. and Rubinstein., .M.F.:. "Dynamics f structures".

Prentice-Hall,, Inc. 1k! Wereldsma, R.:

"Normal mode approach for ship strength experiments". (To be published). . .

-List of figures.

Fig. 1. System of coordinates.

Fig. 2. Cross section of ships hull with centre of gravity G and shear centre S.

Fig. 3. Participation factor ro and dynamic amplification H for two normal modes.

(w)

Fig. i. Reduction of midship static sagging-hogging Bending Moment due to ship flexibility.

Fig. 5.. Block diagram of lumped parameter analor of the sagging-hogging loading of the ships girder.

Fig. 6. Spectral analysis of quasi-static 2-noded deflection.. Fig. 7. Effect of ship length on the quasi-static 2-noded

deflection. : .

Fig. 8. Midship Bending Moment according spectral analysis

(curve I) and standard wave technique (curve II and III). h height of standard wave.

Curve I indicates the Bending Moment MB having a probability of.exceedance of e.g.. 5%.

Fig. 9. Standard wave height h versus wave length A. Fig. 10... Combined static and dynamic ship bending.

Fig. 11.. . Effect of ship length on sagging-hogging and springing

Bending Moment.

O i:: bD bD bD ru u, w K1EI

J

reduction of bending moment due to adaptation of hull form to wave si.rface (fléxibility)

Flexible ship EI

CL

»

_p

moment of inertia for regular steel moment of inertia for High. Tensile Steel

wave bending moment determined by geometrical properties

-

.

-extra reduction of wave beflding moment due' to increased flexibility caused by application of high tensile steel

Fig. L Reduction of Midship tatic Sagging-Hogging Bending Moment due to Ship Flexibility..

Rigid ship EI

xØ-e-

cl

S#i(w)

SØW)

LONGER SHIPS ¡ t .$..

I

f 'i

'

/7.

/

WAVE FORCE SPECTRUM

PARTICIPATION FACTOR FOR THE 2-NODED DEFLECTION

SPECTRUM OF THE MAGÑITUDE OF THE TWO NODED DEFLECTION

...

_/

/

/

/

,STANDARD WAVE HEIGHT

I/I/I m/hi\tX

4- 50

-30.

w

-Fo.

.-..-.

- -...-.

_{h =3.75}

o _{200 4Ô0 SOD BOO lObo}

H#i(W)

(RAO)i

i

I

I

V

_{PARTICIPATION FACTOR}

i

«F

___/

/

\. .

...

-

--LONGER SHIPS LONGER SHIPS

tON GERSH IRS

t

-STATIC LOADING _SPRINGING

DYNAMIC AMPLIFICATION

i

Uil %%

I1'W-iLW.

2

w0i

w-

-LONGER SHIPS RESPONSE AMPLITUDE

LONGER SHIPS OPERATOR FOR THE

2-NODED DEFLECTION

L.\.

_,I'\

-POWER SPECTRUM OF WAVE GENERATED FORCES

SPECTRUM OF AMPLITUDE DF 2-NODED DEFLECTION

SF(W)I1'.fl(W), P1()2

A

b-

I-z

w

E

o

E

(D

z

o

z

w

(n Q-u'

o

z

SHORT MEOUH LONG

SF(W)

14* 1(W)

FASTER SHIPS -FAS-TER -SHIPS-'t

t

\

POWER SPECTRUM OF WAVE

GENERATED FORCES QUASI STEADY WAVE BENDING PARTICIPATION FACTOR HULL DYNAMICS

Ilj

'1 FASTER SHIPS SPRINGING

w

L6W HIGH .:SHIPSPEED MEDIUM