On jackup platforms and marine riser dynamics

RISER DYNAMICS

by

Dr. S. Spassov and

P. Spaarqaren.

Report nr. 793-M

May 1988

Deift Univerefty-of-Technulogy Ship Hydromechanics Laboratory Mekelweg 2

2628 CD Delft The Netherlands PhoneOl5-786882

1.. INTRODUCTION

2,. BASIC DESCRIPTION OF THE PROBLEM 2.1.. HYDRODYNANIC LOADS

22..

DYNAMIC MODELLING

2.3.

MARINE RISER RESIONSE,

3.

PRELIMINARY CALCULATIONS FOR MODEL TEST CONDITIONS

3.1.. CALCULATION OF HYDRODYNAMIC LOADS

3.1.1.

WATER PARTICLE KINEMATICS

3 .:1. 2. WAVE FORCE AND MOMENT ON VERi 1cAL CYLINDER

3.2.

ESTIMATION OF NATURAL FREQUENCY OF THE JACK-UP MODEL

3.3.

BUCKLING 4.. CONCLUSIONS REFERENCES. ACKNOWLEDGMENTS APPENDICES A.LISTINGS OF PROGRAMS

B.DRAWINCS: FOR 'MODEL,MODEL TEST EQUIPMENT.

Dr.S:.SPASSOVt P.SPAARGARENt

BSHC-Varna,Bulgaria,now a research fellow in Shiphydromechanics Laboratory ,De1f.t Untvers ity

of

technology

Deift University of Techno10 ,student assistant in Civil Engineering Faculty

Since any Offshore Structure exist in a sea, environmental hydrodynamic forces are an important subject of consideration. Among all the loads

imposed

on a

structure the fluid forces usually belong to be largest. A good deal of attention has therefore been paid to fluid forces and

corresponding to them dynamic response, but several respects adequate

understanding and knowledge is still lacking.. The more frequently building platforms-jack-up are being proposed for longer term use in deeper water and in more exposed locations. This implies that. the condition and resulting useful. life of such structures will have to be more carefully determined and checked than in past, when shortduration use of'. such equipment allowed thorough inspection. This are some of: the reasons, that,, in recent years,

the interest of dynamic behavior of jack-up platforms and production risers

are increasing.

In this report attention will be paid in an adequate description of the main hydrodynamic and dynamic problems in design process of jack-up platforms and drilling risers. After a briefly state of the of the

problem the efforts are concentrated on preliminary calculations for some dynamics characteristics of risers and j:ack-up corresponding to model tests conditions. This results can be used in real dynamical modeling of riser and jack-up model before hydrodynamic tests and for initial prediction of of

the value of hydrodynamic forces and displacement of single cylinder an4.

three leg platform.

The next purpose of this report is to be a good step in. studying and

investigation of jack-up platform dynamics for students and student

2. BASiC DESCRIPTION OF THE PROBLEX

At the first view the problems of marine risers dynamic and jack-up platform dynamic in operational conditions are different. But looking at the elements from which jack-ups are constructing and the general behaviour

of this main element(jack-up leg) it is seen. the similarity of the jack-up

legs dynamic and marine riser dynamic. One way of modelling of jack-ups is to use instead of frame works leg,a vertical cylinder. Then it is possible

to start the investigations of jack-up platform dynamics in experimental and numerical way with a single cylinder and continue with three cylinders for three leg platforms. SOme of the results for single cylinder can be used in marine riser analysis. A real next step will be to use a real frame made

jackup legs..

2.1. HYDRODYNAXIC LOADS

in any marine design procedure. an accurate specification of

hydrOdynamic forces is required.in some instance data for a new structure

will be obtained

by

a combination of model testing and analytical

calcul'ationFor some structures,particuiarly these with small diameter

components ,".scale effect" become important;in such instances prediction of

prototype forces from model results is, difficult with any degree of

certainty.I't is a modelling and numerical problem.However ,the. 'comparisons

between some experimental and numerical results show that 'and relatively

adequate formula for this purpose is Morison. formula.For applying of Morison formula is necessary to know:particle kinematics:; force coefficients and the

combination of element forces in a force for a general element of. jack-ups.

a)Waves and current'

A deterministic wave analysis' can

be

shown to be conservative by fact that it assumes that in the worst design storm'e.g. storm with 100 year return period ,.all the wave energy is concentrated into one regular long crested wave approaching the platform in the most unfavourable direction.

Quantitative comparisons between stochastic shortcrested analysis and

deterministic analysis for drag dominated platforms have shown the former

method may give only 60-70 percent of. the response of the latter method.

in figa,l is shown the well-known diagram for application areas of wave

theories

presented by Dean

in 1970.This diagram shows for which wave

height/water .depth regions the boundary conditions at the bottom and water

surface are best .satisfied.It .says nothing about method gives the best quantitative results..

0.I

'li/P (nil.t)

Fig.l Application areas for various wave theories.

Some results presented by .Dean. in 1974 shoved that S.tokes 5th order theory

may be

quite conservative.One example is shown in fig.2 .Furthermore1the

Stokes 5th order theory will approach a singularity solution at shallow. water depth.

I T.IJS S.._N.O.SUft

h-OIS7 I

$RTIcLE VEIOOTY tftIS.c), Fig.2 Comparison between wave theories and experiments

,w. ₀ I P 'S CNOIDAL 0%_I .

I

OA _STREAM rUNCIsON

-In tabl.1 , are presented the results

given by DnV

fit 1986(14]' for

different wave force predictions for vertical cylinder using 5th order

Stokes theory and Airy theory in different water depths.it is seen that after 100 meters difference rapidly diminishes with increasing of water

depth.

Water depth Cm)

'table 1

Wave 'force ratio Stokes

5th order/Airy wave, theory

70 1.3

90: 1.2

110 1.1.

130 ' 1.05

When a cross current passes over a slender member a wake is formed in the hydrodynamic shadow of the member.The eddIes are shed 'alternately from

either side of the cylinder producing an oscillating force at right angle to

the current.This l's so called lift force.This force is oscillatory and has' a

frequency equal to that at which eddies are shed.A smaller oscillatory drag force is induced at a frequency of' twice the 'lift force.When the natural

frequency of a cylindrical member coincides with the eddy pair frequency,,a

resonance will occur.If is the resonance frequency then the critical

current velocity Is defined :by

V S.f .D

C.

r

Similary for in line motion Vc2SfrD .The response of the structure to vortex Shedding excitation will generally be handled by nonlinear methods.

b)Breaking waves

It is only recent years,that the full importance of breaking, waves in design

has been fully recognized.Breaking waves can in fact occur in deep as well as shallow water .Since dIfferent wavelenghts have offering celerities non

breaking crests may converge to a point and form a wave of sufficient height

to cause overtopping'.The breaking process is an essential feature of energy

balance in both the deep oceanand coastal. waters.in shallow water high

waves tend to form. the character of a series of disconnected solitary waves

the distance of crest d above the still, water level being considerably greater then half of the wave height 11a which would be for a sinusoidal

surface profile.! It is generally considered that when.

_d

_{/ Ha}

the wave

level will begin to break.This formula is valid for intermediate depths to.'

Velocities, under breaking waves are. generally very much greater than these predicted .by _{standard. theories} for non breaking waves.For example

linear theory predicts that the maximum velocity at the crest of a wave of

limiting steepne's('height/wavelenghtl/7') (11/7)CO4SCVbr is,

where.

C the wave celerity equal to ,'Measurements in breaking waves shOw that in

nearly all cases the maximum velocity is close to C .Some laboratory measurements have indicated higher velocities, although these are generally

thought to be error.Since the drag force in Morison equation is proportional to velocity squared it is' seen that breaking Waves can expected to give forces of four times or' more those

which would be

calculated using nonbreaking wave. theory.Accelerations are at least g ,.which are normally obtained .This increase Morison force since the maximum acceleration.

predicted by linear theory is O.45g .Cocelet(1979)(6] has produced a

numerical, technique for computing, velocities and accelerations under breaking waves and 'his results show fair agreement with experiments.

Unfortunately research is no.t yet far enough advanced to allow firm

specifications to be given for a design breaking wave.. If breaking wave forces are thought to be significant it is safest to resort to model tests

(1) _{'F_pCDDIVxIVx +}

tanhkh -max velocity for breaking waves

c.) Slamming

Although the Morison force on submerged. member is increased when waves

break,the greatest force occurs at impact,this being known as the slaming

force..Slaming can occur whenever a member is intermittently iinmersed,whether or not the wave is breaking,but again the breaking wave Situation is by far the worst.Fig,3 shows how the force on the horizontal cylinder of diameter

varies

as

a function of time when slamming occurs

Siam

Drag + Inertia

D/C

Time

fig.3 Breaking wave force on a hOrizontal cylinder.

During the 'time interval tD/C the cylinder is only half immOrsed.This is

the slamming region.After this only the Morison force remains which is about one third of the slamming, force.In this diagram F is the horizontal force

normalized by dividing by its displacement weight in water.The slam force is proportional to the square of velocity i.e.for horizontal circular member:

F' - pV2LD

s

2s a

V5-slam water velocity L-the length of the member p-density of water

C -coefficient of slam

a

D-cyiinder diameter

Strictly speaking C is a function of a time but it is usually taken to have

a value

approximately 3.5. Slam forces are significantly reduced if waves impinge obliquely on a, horizontal structure and 'in fact the

probability of a Wave breaking parallel to a member will be fairly smaIl.The coefficients can also be reduced by about 20% if the' surface profile is

modulated 'by small amplitude waves.

d)Drag coefficient

Many designers have used a drag coefficients of 0.5 for full scale circular tubel irrespective of tube diameter and flow velocity and 1.2 for' model

scaie.This is obviously unconservative even when compared with perfectly smooth cylinders as shown in fig4

a

I

12

1.0

0.8

0.6

0.4:

:02

0

101 1'O _10' REYNOLD!$ NUMBER (N.)

fig.4 The variation of 'drag coeffitient with 'Reynolds

number for smooth cylinder in steady uniform flow,

For real legs ,the roughness caused by marine growth and attachments e.g. anodes will increase the actual drag further.For jackup platforms,heavy

marine growth is normally avoided by cleaning during rig move or because the suscRmcM. cnmcAL BUPERCNITI

NIGIME NEOIME

I.

S I CAL PO$TCRmcAI. SCOIME,

marine growth is wiped off during jacking up and down.Hówever ,,stiil it may

be assumed that moderate roughness will increase the drag coefficient of magnitude 25 percent Which typically will give a total, drag coefficient of

e)Leg calculation

The legs calculation involves the determination of the external forces acting on the jack-up system in elevated condition,in particular the hydrodynamic loads from waves and current.Furthermore ,it calculates the

support reaction 'forces 'and internal leg loads' that result from the external

loads plus the platform own weight and payload.These calculations are mostly regarded as the main part of jackup design.Traditionally the hydrodynamic

loads

acting on

the legs of the jackup are determined using a design wave approach' in appropriate drag 'and mass coefficient in ' Stokes-Morison formula.For truss type legs the drag and mass coefficients for total leg

are deriving using more or less standard values for individual

members,supplemented by test results for some leg and chord shapes.The

hydrodynamic forces-combined' from waves and current are calculated together-the current velocity 'should be combined with the wave particle velocity before the tOtal force is computed.Separately is estimated a pulsating (15')

wind' force acting on the platform in same directiop as the hydrodynamic forces.in present day much attention is given to the dynamic behavior of

j'ack-ups in elevated conditionWith increase in waterdepth the natural period of the platform approaches in deeper water the frequency of' the existing waves.When considering the jack-up as a simple-mass-spring system

dynamic amplification may expected of' excursion and forces' calculated in the

traditiona'l way.

In more detailed analysis effects ' such as directionality of the seas 1irregu'larity on the waves ,.nonlinear behaviour of the p]atform response,damping due. to spudcan-sea bottom interaction,etc.may have a strongly reducingeffect.in the past it was assumed that such effects would more then cancel the effect of dynamic calculations.Attempts now have been

initiated to allow a computational inclusion of such effects.Some

preliminary results seem not to suppàrt the old assumptions;'other suggest that the reducing effect. may be far more than had been assumed till now. Some full scale tests information exists to support or disregard the effect

of dynamic amplification in deep watr.The information that does exist is to be scrutinized.

f)Dynamic amplification factor

Nevertheless that dynamic effects should be included when .significant,static

analysis is stIll often used for jack-up analysis..For jack-up platforms

typical natural periods are:

-5-7 seconds for traditional jack-ups

-7-9 seconds for deep water jack-ups(90m and abo!e)

According to simplified dynamic analysis procedure DnV(l] presented that

typical dynamic amplification of the extreme structural wave induced

response for harsh environment jack-ups may be indicated in tabi. 2 below..

table 2

1.) _{3-legged jack-up with, lattice type legs.}

2) _{Dynamic amplification is only applied to the}

amplitude value.

Combined the various factors DnV presented. so called safety factor shown in tabl.3

1) 3 legged jack-up with lattice type legs. Water depth (rn) 1)Typical natural period 2)Typical dynamic. amplification Typical ratio Dynamic. Vs. Static response 70 90 110. .130 5 .7 ', .9 1]. ' 1.1 1.2 1.4 1.8 1.08 1.15 1.30 1.60 table 3 Water depth (rn) Analysis Assumption 70 90

hO

130 A. Regular wave 1.5 1,. 5 1.5 1.5

B. Wave theory Stokes 1.3 1.2' 1.1 1.05

5th Order C. Drag Coefficient 0.65 0 65 0.65 0.65 Cd=0.5 D. Static-Analysis 1) _0.93 0.87 0.77 0.63 Total. built in safety factor 1 ..2 1.0 0.85 0.65

It is seen that the "traditional"design procedure is satisfactory for water depths less then 90 m and unsafe for deeper water.To overcome these problems an alternative design procedure has been adapted which, explicitly accounts for dynamic effects,more accurate wave theory for relevant water depths(Airy or stream function) and drag coefficients for smooth and rough

cylinders accounting for tube slenderness..The method is described in. detail

in the Veritas Classification Note on Jack-ups.The gross implicit bias shown

in table 3 is eliminated for analysis assumption B and. D and improve from

065 to 0.80 for assumption C. with the overall implicit safety factor of

about 1.2 independent of water depth or the dynamic behavior.

The simplified response analysis is based on deterministic nonlinear static approach,adjus:ted for dynamic effects by a dynamic amplification factor(DAF).The main advantage of the method is that it is easy to establish instanteneous load distribution ,and it is possible to work with large detailed structural models.The effect of dynamic amplification is

may be

significant not only for natural period but also for waves with corresponding period to the longest natural periods of the structure.The

dynamic response of structures like jack-up platforms may be approximated by multiplying the static response of sideway deflection of the

barge by

a

dynamic amplification factor-DAF ,which for a single degree of freedom.

system is given by:

1

DAF

'4(l(T0/T))2 +(2pT0/T)2

T0-ñatural period

T-period of valuable load(wave period) p- damping ratio (percentage)

For elevated condition it is supposed that the three lowest natural

frequencies correspond .to surge,svay and yaw motion respectively. The

natural period is the inverse value of th.natural frequency and. is given by:

(DAF)

in order 'to account for the effect of irregular sea, A stochastic dyiarnic amplification factor. is defined

SDAF

1

(2) - 2wjI

f-natural frequency

e85t

stiffness of one leg

k -effective masS related to one leg

The damping ratio p to be used in evaluation of DAF' is the model damping

ratto.This. is quantity which depends on a number of varab'les:

C 2Jmk

rn,.c,k. -inass,damping,spring coefficients in equivalent

one-degree-of- freedom system.

It should be observed that the damping 'ratio ncreases with decreasing stiffness. This is important because the stlffness. of jack-up platforms. may

be an order of magnitude less then the stiffness of a corresponding Jacket .The total damping includes struturai damping,:hydrodyamic damping

and soil. damping.if the range comprises the natural period of the platform. we

will have

the extreme' response.The 'sharp peak of transfer function is

narrow compared to the width of realistic wave energy spectrurn.Only a fraction of the energy will then correspond to this peak.The value of DAF at, resonance is governed by' the'damping ratio: .

The SDAY was found to be rather intensive to variations in the natural period for the range 5-lOs.The relation between p and is given by:

0.65 pat .75P

This result is exclusive of effect of shortcrestedness.By use of an average relation between wave height and wave period it is possible to determine the shape of the long term distribution used in fatigue analysis,The results of a number Of computations are based on the following average vaveheight-period relation:

T-

max max

The. total damping ratIo was assumed to be 8 percent,, but the results were

2 .IDYNANIC MODELLING:.

Model testS are particularly invaluable when, analytical methods of'

prediction are inadequate as in some separated flow and dynamic response problems within the general area of fluid structure interaction..Their use

can result in considerable solution in helping avoid disastrous mistakes in prototype design. Good modelling before model tests is' an essential first step in order to clarify the phenomena which are significant and interpret

the reliability of experimental results obtained.Thus, even though a model test will invariable provide some results ,the relevance and reliability of

these must be carefully assessed.

The main difficulty in hydrodynamic modelling, is' Reynolds number.

'Several approaches which attempt to .by-pass this difficulty have been'

considered.in discussing these we first emphasize

that when no

flow separation occurs,or when it is localized so as not to' influence the overall

loads on structure,then differences in Reynolds number should be unimportant and the

effect may be

neglected" as was assumed in the diffraction analysis .It. i's known that in steady flow the effects of modest changes in

Reynolds number are reiattvely unimportant provided that the flow remains in the subcritical or post-critical range as the case may be.Sometimes a trip.

wires are used.'This technique has been made 'to apply to wave motion,but the

oscillatory nature of the separated flow and the complex vortex interaction with the structure suggest that such a procedure may be questionable.

When taking into account

of

the dynamic.response of elastic member,a number of. additional parameters are needed. to characterize its

behaviour. These includes density(or a' characteristic density) p3 ,modulus of

elasticity E ,.and damping ratio p (mentioned already

ratio represents the ratio

inertia force.in many' cases the whole structure i.e. where rn is the mass of the

properties ,conveniently characterized by dafliping

'in .2. l)or the logaritmic. decrement.The density

of structural inertia force (or weight) to fluid a suitably averaged density distribution over

a parameter m/pL may be used in place of

_p8/p,,

Structure and L its characteristic length.

The structural damping ratio p should also be held constant.,but very'

often accurate information concerning prototype values, is difficult to obtain and the damping ratio is only approximately düplicated.However it is

sometimes possible to accept an altered damping ,ratio,the intention being

that resonances will occur at approximately the correct frequencies although the response amplitudes will be alteredIf the model is known to be under

damped then the results should overpredct the response of the safe side.

The elasticity parameter E/pU2 describes the ratio of. 'structural

elastic to fluid' inertia forces and directly related to Caushy number.This parameter is. also difficult to hold constant in fully elastic model.in particular for same material:

k

-and. therefore the constancy of the elasticity parameter

indicates that we should have

In practice this is awkward to achieve:

k

'varies between '1/4 to 1. for most

metals and 'about 1/60 for plastics ,values' which are generally incompatible

with prescribed kL .The most usual ways of overcoming such a difficulty are by restoring to simpler sectional or linear mode mode'ls,or by:incorporating. a sectional distortion of complete model.These are nov briefly described 'in

turn.

In many problems' the most important effect of E lies in describing the structure' s(fundamental) natural frequency f ' ,and it is the convenient to addopt the alternative parameter fL/Uin place of E/pU2 .The related

parameter or UT/Lis. redUced velocity.The scale factor for frequency is given by

kf -k.'

The required natural frequency may be obtained by a mode].

which is ttself rigid but elastically mounted rather that by any flexibility in the mode ,itse1fThis is approach adopted with sectional model which is

k

_kE

rigid elastically mounted model representing only typical section of a

structtire.This technique is particularly useful for investigating two dimensional hydroelastic oscillations.For a single- degree-of-freedom system a linear mode model may be used.With this a

rigid canti1eer

elastically

mounted so as to pivot about its base to simulate a flexible cantilever oscillating in its fundamental. modeSuch models are relatively straight

forward to construct.The required natural frequency (stiffness) can be obtained by a spring arrangement fixed, externally to model so as not to obstruct significantly the incident flow. The necessary damping is generally introduced as a external, viscous damping:.

When a fully elastic model of the entire structure is required,the situation is more difficult but a sectional dIstortion of the structure may be used to achieve the necessary requlrements.As already mentioned,the single most important effect of E is the natural frequency f . Then the

reduced velocity may be

employed in place of E/pU2 . When the natural

frequency is associated with fiexural oscillations-as is often the case

-than we. generally rigidity of. the structure. Nàting that kf k, (since

S n

fL/u is constant) and one has fined:

kEl

-Ideally , we would require kEk. as already indicated ;and derives from the definition of I. But since 'E is taken to influence the problem, only

through bending ,E and I appear only in combination El and this provided

this compound variable is 'itself correctly scaled according to 'k_kL it may be acceptable that E and I each in isolation may not be.

This condition can generally be achieved by distorting the cross-section of the structural members relative to the length scale .Thus ,we

would' have kA,kL, where kA is a characteristic cross-sectional area of the

structure.This is most conveniently carried out' 'by disturbing the internal

dimensions of pipes or hollow structural elements and' leaving the external dimensions undistorted'. Flow patterns and added-mass effects should than be

model and prototype kEl we would required kn.kL .In the case of a thin

hollow cylinder of diameter D and wall thikness t ,ve have kLkt , and

therefore ye would require implying that the model cylinder would be

relatively thin.It is necessary to consider the effect required mass and density.The submerged weight to hydrodynamic force ratio should, be held' constant.

(pgAL/F)g - (1pgAL/F)f5

The required mass distribution is obtained by fitting the model to the above formula by the addition of local masseà around the structure,but taking care to keep the structural stiffness

Sometimes we need to adapting the plate and girder thickness so that the total stiffness is correct.This means that' for plastic models the

thickness has to be exaggerated and for 'metal models the material becomes

thin in proportión.The latter. appears to be dangerous as leading, to easy

buckling of members in the model, so in general plastic models are used.This means that the area has to 'be exaggerated proportionally to the lack of E.

This scale rule is the same for the axial deformations of the members.

Another possibility Was realized by applying..a thicker plate,representthg at the same time both the mass and the bending stiffness properly. With this design there was a relatively simple model with correct stiffness in

'horizontal. direction,the compromise lying in vertical gate stiffness 'being

too great in the. model.When the plate thickness has to be exaggerated additional mass is nee4ed.This is attached localy so'that the rigidity of

the model is not affected and that a Correct 'mass distribution is obtained A point to be considered is that too great plate thickness can in some cases

decrease the additional mass of wáter.in these cases extra added weight is

necessary.

The sensitivity to damping value mainly exists at resonance vibrations.As these have to be' prevented by a good design in preliminary

stages the damping of plastic is sufficiently to resonance vibration. When

means that strain gauges with watertight covers can localy influence the

ei'asticity.This needs not be serious for the total reproduction of

elasticity but it can affect the measurements .,This can be overcome by making

dummy parts

which can be

separately calibrated.Until

now no

elastic similarity models of drilling platforms have been made.. but they could be of

great help for the designer,especially when results are compared with

calculated forces.

2.3. MARINE RISER RESPONSE.

Marine riser analysis is based on the models,which involve descriptions of the sea,, the riser structure and interaction between' them.The

main problem is to estimate the force distribution and a bending moment.in

par.ticular,the standard relationship between the standard deviation ofriser, bending response , deflection and other several parameters should be

discussed.For this purpose it is necessary to calculate:

a)sea conditions:from a stochastic description,which is normally assumed to be gausian,the. water particle kinematics below sea' level can be

calculated using linear potential theory as was mentioned in p.2l.

.b)Forces-Morison formula.

FFI+FD - I1pR2(C_..

CA2)4pDCD(u-) I(u-),I

The possibility of forces transverse to the incident flow due to vortex shedding and forces dUe to wave slam in splash zone are non considered.

c)Response model:The riser is represented as an almost straight

vertical tensioned .bearn,which is subjected, to an axially, distributed hy4rodynamic force and a horizontal displacement excitation at the top,where

it is connected to a floating platform.The response of the riser can be

described by: . .

(3)

where:

I!

!xL

x(zt) -two-dimensional horizontal deflection

z -vertical distance from riser base t -time

E -modUlus of elasticity I moment of inertia

-riser tension,

m

-mass per unit length

F(z,t)-axternal

force per unit length.

The bottom end Of the riser is assumed to be fixed to the base: x-O at z..'O

and subject to rotational constraint according to

at z-O

where Cb is the rotational stiffness of the riser base.At the top,the riser is assumed to be connected by a hinge to a floating structure,expressed as

x-x0 at z-L

0 at z-L

where x°(t) is the time dependent horizontal displacement of the floating structure and L is the length of the rtser.The terms on the left-hand side

£4'

of the 'equation (3) represent bending forces .,'Ei(4)x ,tension forces,

,and inertia forces m(j.)x These forces are. balanced by

the transverse force F(z,t) , which is described as

FFH H(L+a(t)z)

where FH(z,t) is the hydrodynamic (Morison)force exerted by the the water

For solving this problem Brouwers and Verbeek(1985)(1O] presented some

formulae based on analytical results for the. response of hight].y tensioned risers in deep water.. The analytical approach has revealed three characteristics regions of response along the riser. (see flg.6)

MEAN SEA

LEVEL

'C

LB! BASE 1

fig.6 Regions of riser response

RISER MAIN SECTION .-'X, Xd,Xuv BOUNDARY LAYER BOUNDARY

/LAYER

'Xb

WAVE-ACTIVE ZONE

I

I

I 0 OULECtmN rnUR MAUI SCOTIOM -ONLY - -- OUANS4TAT*C - NCIONMT (1.d MOOt)

The effect of- direct wave loading on riser deflection angle and bending

moment in the wave active zone is presented in fig.7,8,9-.It is -seen that the

deflection and a ngle are generally small and bending moment response is located in wave active Zone near mean sea level.

111AM - lEA LEVEl.

r

I I I ANGIE SOUNOANG mICEMAIN SECIION OUAII.STATIC - RESONANT. (lad MODE) SOUNOAAY.LAYEN

fig.7 Typical vertical. dis- fig.8 Typical vertical die-tribution of riser deflection. tribution of riser angle.

I

0 BENDING MOMENT

fig.9 Typical vertical distribution of, riser bending moment

The results for standard deviation displacement and bending moment are presented in fig.lO.Static response are in close agreement while dynamic

response in main section are. slightly different.The. main reason is that the

natural frequency .does not coincide with the peak of the excitation 'Spectrum of floater. motion ,as it was assumed in the approximate solution.

'SO APPROXIMATE FORMULAE NUMERICAL SOLUTION ISO ,, -f -I I 1 I L

440...L--.

... ! I 40 L_ -i 0 0.0 0.2 0.4 0.6 00 .10 .12 .14 .10 .18 STANDARD DEVIATiON DISPLACEMENT (mJ .20 'II 4 ISO 160 1.0 rI2O 400 U U) I-60 40 20 0

DISTRIBU!ON ALONG RISER

PRODUCTION RISER IN N NORTH SEA Hs 2m TmTs

fig.lO Produàtion riser in North Sea - Ns-2m,Tm.7s

It should be noted ,however ,that longitudinal dynamic response,which may occur in ultra-deep water risers,cannot be established from this equation.Furthermore,terms accounting for torsional response and large

angular motions have been neglected..For iIustration of response bending moments sensitivity the some resultS are presented in fig. lll2,l3,.l4,i5;

WAVE. ACTIVE ZONE RISER MAIN SECTION BOUNDARY LAYER 0 2. 3 4 STANDARD DEVIATION BENDING MOMENT (kNm] Id 0 4 B

awl

2m

SUBSEA MANFOLO

fig.!1I Multibore production riser schematic

.1:20

W stàrmNN $10 $1,. .1199 Iw ZotrnNN $10 1.I23,. iIe.p.I 56

BALL JOINT (heave coITp.nsatad)

RISER TENSION (to MNI

15m E TENSION

Significant wave height. H1J3im)

2 mm section xx c.ntroIrIer,od:: 32nwn wt :22.2mm perlpherOIlbes.od: 8&9 mm wt: 6i35iun

ENTRAL EXPORT RISER

ISER CONNECTOR

RIPHERAL FLOW,SERVICE LINES E FUNNELS

-Jff

... 30 DIZ A20 0 10

-i:40

-

1'

1 0 10

Significont ave heght.H (m)

fig.12 Bemding moment :res fig.13 Bending moment

30

0)0

tX

.x'

.r,

a.

'E

20

0

C

0

0

I

0

10

15'

E0

z

C

x

2345678910

OOi 2345678910

Significant'

ve heht.. H1,3 (rn)

fig.14 Effect

of

C4,Cm on bending moment in upper

15

10

C

0

-

0"

£ I I A

012345678910 012457è9i0

Significant wa

:heght. H'1/3 (rn)

fig.15 Effect of Cd:,Cm on bending moE'Ont in lower

PRELIMINARY CALCULATIONS FOR MODEL TEST CONDITIONS

3,. 1 .CALCULATION OF HYDRODYNAMIC LOADS

The water action, on piles ond platforms can be devided on three parts:the

wave active zone at the top;the boundary layer at the bottom and a main

section between. in this paper main attention is focUsed on the results

obtained for forces and response in the wave .active zone which are in prime

importance in the assessment of the design parameters expected for fatigue

damage and extreme response.

3.1.1.WATER PARTICLE KINEMATICS.

As it was mentioned the model tests will be provided for small wave amplitude action.For calculation of hydrodynamic loads the linear boundary value problem for velocity potential have to be solved(fig.l6):

fig. 16 The potential boundary problem area.

The expression for the velocity potential may be writen.

('Kotchin[5J ,Kinsman[4]1) :

Following the linear dynamic boundary conditions on the free surface,we can receive the horizontal .and vertical partIcle diap]acement,their velocities

and acceleration.The complete range of water depths can relatively, be

dividOd, into the shallow water depth,intermediate water depth and deep water

ranges as follows (Kinsman[4],Sarpkayai7fl:

deep water depths:

ahál]ow water depths:

L>h'

20 L

hi

L 2

The most

common case is intermediate water conditions.Using some

approximations formulae for hyperbolic fUnctions. one can receive

corresponding kinematic values for deep and shallow water conditions.The horizontal and vertical particle displacement for intermediate water depths are:

sin(kx-wt);

jj sinhkv

2 sinhhk cos(kx-wt)

The corresponding velocities are:

- -, coshky

dt x 2. sinhkh '

(4)

- V -

si kx-wt

dt y 2 sinhkh n(

and corresponding accelerations are:

IL>

0.08

gT

0..0025

-dt - sli coshky2 sinhkh a n(I lcx v ),,

gH sinhky

dt a - 2 sinhkh cos(kx-wt)

where

y+h

2..WAVE FORCES AND NOMENTS ON VERTICAL CYLINDERS.

1

An approximate estimate of inline force and momentum acting on a pile may be

obtained through, the use 'of the linear wave theory,Morison equation and constant drag and inertia coeffitients.This equation can be used' if D/L<O.2.

Let F' is a wave force on unit lenght and Morison equation is presented.

(6) F' DV

_FVJ

+

p -water density,CD , c-drag and inertia coef'ficients,V

velocity and acceleration of water particles.'CD and Cm are

from Re

,KCmT

and roughness.tt difficult to estimate experimental or theoretical ways.Usually for:

(7) ' C 0.5 +' 1.2 C

-1.5 +

2.0 m a -horizontal x dependent mainly CD and C

F -

.0JYF'(y)dy

and the corresponding total moment

.

N - ,0JYyFI(y)dy

Substitute with (4).,(5),('6),(7) in. (8) it is recieved:

(10)

F +F

_{in res}

s!U

cosh ky

' kx

N 4 ° 2 siith n(

+ 2PCDDOSY ()

2

ub2 Icoswt1cos'(t) dy

0fYcosh2kydy .-s inhky coshky+ _{y inl*y+}

Then:' 2' 2 - aD fi sinhky in M 4 2k. sinhkh

pCD

F D 2sinbky +2ky

res

32k°

siniilch Icoswticoswt

in analogous way,the moment from the wave action on the pile is::

MaN +M

in res

PCDD'

Mres _64k2

Q1coswtcost,

n2

2k2 w2HQ2sinwt

2kysinh2ky-cosh2lcy+2(ky)2 1

icys inhky- coshky+1 s inhkh

From ('4) , (5;),(iO) and(II) it is seen that drag and inertia resistance

forces and moments have phase shift -when one has maximum,an other has

'minimum.

In fig.18,19,2O,21, 'are presented results for wave forces on vertical

'cylinders.

The corresponding conditions for calculating are:

flg.I'8:t -lmrn;p8 89OOkg/m3(copper);HO.25m;tubular pipe:;deep water. -fig.19:' T, -i.Osec;p3 -8,900kg/m3;t -lmm';tubuiar pipedeep water.'

-flg.20: T _l.Osec;t -1mm; D -l6mm;tubuIar pipe;deep water,;p3 -8900kg/m3

-fig.21: T _l.Osec;t -1mm; D -16mm; tubular pipe;X-O.8kg;deep water.

TaO.5 DIC

-9- 1-1.0 ,oc -ii- 1-1.5 g.e

.01' .02 0 .04 .05 , .06 07 .08 .09 .1'

Din meters

25 0 .01 .02- .03 .04 .05 .06 .07 .05 D In meters U. 1.3 L5 17 _!. 2* 2.3 I In meters H0.1 m.t.r 11-0.25 11-0.4 m.I.r

fig.19 Maximum foràe as a function of wave hieght and cylinder diameter

.l-- M0.2 kg

-.0-. U-0O kg

fig.2O Deflection as a function of lenght of the pile and the mass on

1.1 1.3 I5 I., 10 2.1 2.3

I In meters

-4- C1O Cli H1m2.

-e-- C-es £9 Il/m2.

-"- E-.O La N/m2

fig.21.Deflection as a function of the lenght. of the pIle and stiffness

From fig,. 18 it is seen that the influence of wave frequency is almost negligible ,but the wave height(fig.19) influence on wave forces acting on verticalcylinder is significant-the wave action force s' greater if. the wave is higher.The elasticity influence for different rnaterials-steei,,copper and aluminium is presented in fig..20..The results showed that one of the good

oportunities for jack-up model legs is copper..Fig.21 shows a possible

optimum mass on the top of the vertical cylinder, respectively on jack-up

To calculate natural frequency one have to simplify 'this model .The first

step is easy to make:

MASS:

This model represents a mass-spring system with three aprings.This model can be implified dividing the mass by three.

:3 .2.ESTXM&TION OF NaTURAL FREQUENCY OF THE' JACK-UP MODEL.

In general,modeling on naturalS frequency for single pile and jack-up platform follow the same procedure.In fig.22 a scetch of jack-up

Following some main properties in meàhanical engineering the displacement of previous model is the same as the displacement of next model:

OODT

I

This Is the dynamic model for estimation of natural frequency of jack-up platform. The calculation of these frequency is made using Rayleigh method.

This method calculates the maximum kinetic energy K and the maximum max

potential energy .The energy laW is:

K+Ucons:t

There is a time t on which the displacement is zero so

U OKK const

₀

max

Analogous for kinetik energy

0 U-U

conat

So

U K

_{max max}

Calculation of maximum potential energy can be done for the following case:

-

pAW

w20fl(lcos)8

A -area; p -Specific mass;

Discrete part:

Kd _l2w

max 2 max

So the total kinetic energy K is equal. to:

2 max

K

(pAl( + + M)

_2Wmax1(

-V The displacement V(x) can be described by

W(x) - Wmax(lcOS)

The maximum potential energy Uis:

U

-

EIwW2

140j1cos2

dx Computing the integral it is otained:

$iW2max if4

max - 64l

The calculation of is more difficult because there are two

parts:One continuum part and one discrete part.

The continuum part is equal.:

From these approximately values the natural frequency is equal to:.

2 3.03 E I w

18(0. 23pA1+M)

3.3. BUCKLING

As it was mentioned in p.2 one of the worst situation in model tests and in

full scale operations is buckling.

The model. for buckling investigation is the Same. Let the uniform, distributed

forces F0 are the hydrodynamical forces. The force _F is the weight on the top: (12)

a

x F V

This gives the differential equation: F

tot

1

Let's take one part out of the beam:

E, I

w

Thuck Fw M' +

W, +

While

X-EIW"andS --F

x V

EIW'"'+F W"F

_w

tot

To our stave the conditions are:

A

- JI/A

The solution of this equation gives the maximum force P (when

Zero) before buckling takes place. This force is given by Euler formula:

2E I

Fbuck 412

According to the rules of buckling(TGD-stee'l) one must calculate FbUCk,and Ma .From. there we can calculate;:

where is buckling iength( -21 here),, A is area, i is inertia

moment.For steel there are available information in handbooks ['13) for the relation between 'buàkling, stress and maxima stress for steei.According to that the condLtton(criterja) '

14 (13) A + n+1 Mres amax where: V0 is

%uck

£

is area

is resistance moment( for circular)

o

M 0

- 0

DO

xo - 0

0

For copper or PVC there are no regulations for buckling because we cannot calculate .The stress due to are altogether very small

inproportional

CONCLUSIONS

In this paragraph a resume is given of the main conclusions from this prelimynary study of dynamic behaviour of jack-up platforms and marine riser

and an attempt for prediction of some model tests results is made.Finaly a recomendations for calculation approach are made.

Studying conclusions:

a)For further jack-up platforms. investigation it is necessary to make an attempt to include a elastic modelling in numerical and model experiments. For model tegts it is necessary to use segmented model for tubular pipe or real frame made legs and to measure the force and moment distribution in

different Water depth.This will help in more correct solution of differential equations for estimation of jack-up platform and marine riser

dynamics in deep Water conditions..

b)A wider investigation aimed at establishing analytical methods or

approximate aproaChes for calculating, the statistics of riser response

and perhaps more importantly , developing a rational approach for

interpretation of these statistics

c) Dynamic response has to be investigated in time domain involving a nonlinearity of different forces espacially in drag force dominance

d)Buckling criteria or safety factor have to be approximately made for different from steel materials especially for model tests materials.

Main predictions:

a)The water loads on jackup platform and marine riser are drag force dominated.

b)An influence of different geometric and mass parameters are investigated.

The conclusions are made. in. 3.1.2..

c.)The most useful materials for model tests are PVC and Copper.For similary purposes the using of aluminium is not so bad.

d)Following a procedure for calculation of dynamic amplification factor has to be excpected that, a amplitude response will varying 1.5-2 times more than

statistical predictions.This conclUsion depends from wave load and geometry

.e)For deep water conditions using .a Airy theory is posible.For more detailed

of extreme situations a recomendàtions made for breaking waves and. .slaming

have to be used.

f)The loads, and the response of three legs jack-up platform perhaps will be less than three times one leg load and one leg. displasement.Thé reason for this (author Op'thion) is the phase difference between the legs. The reduction have to be deppend on the. loading direction.

g)The motions and loads due to certain, sea state can be influenced by choice of the platform' geometry - concerning the phase difference between legs.

REFERENCES

I..BOSWELL L.F.

"DYNAMIC OF JACK-UP PLATFORMS", 1987

2.J.J.H.BROUWERS

"ANALYTICAL METHODS FOR PREDICTING THE RESPONSE OF MARINE RISERS." PROC1 B

85

(4),DEC.,1982

3.J.E.:GORDON

"STRUCTERS" PLENUM PRESS, 1978

4.KINSMAN B.

"WIND WAVES"PRENTICE HALL,1965

5.KOCH1N,N.E. ,KIBEL,J.A. ,ROSE,N.V.

"THEORETICAL HYDRODYNAMICS"

(in Russian)

,MOSKOW,, 1955

6..SHAW

T.

"MECHANICS OF WAVE INDUCED FORCES ON CYLINDERS", PITMAN,1979

SARPKAYA T. ,ISAACSON N'.

"MECHANICS OF WAVE FORCES ON OFFSHORE STRUCTURES" 1981 SORTLAND BI.

"FORCE MEASUREMENTS IN OSCILLATING FLOW ON SHIP SECTIONS AND CIRCULAR CYLINDERS IN A U-TUBE WATER TANK"

UR-86-52,NT1,TRONDHE1M,NORWAY 9.TIMOSHENKO S.

"STRENCHT OF MATERIALS" ,VAN NOSTRAND ,1961

1O.VERBEEK P.H.J. ,BROUWERS J.J.H.

"APPROXIMATE FORMULAE FOR RESPONSE OF SLENDER RISERS IN DEEP WATER" ,PUBL,. 733 ,AUG. 1985, SHELL, RESEARCH B.V.

11.DYNAM1CA VAN CONSTRUCT]ES ,B9N,1987

12 .,PLANNINC ,DESICNING AND CONSTRUCTING FIXED OFFSHORE PLATFORMS,AMER. PETR. INST. 1977

13.TOEGAPASTE MECHANICA,BIIN, 1987

14.VERITAS CLASSIFICATION NOTE

31.

5,1984

"STRENGHT ANALYSIS OF MAIN STRUCTURES OF SELF-ELEVATING UNITS." ,DET NORSKE VERITAS.

15.LUGOVSKY

V.V.

ACKNOWLEDGMENTS

The author wanted to thank to the head of the laboratory

prof.Gerritsma for the opportunity to work of this problem.Special thanks to W.Massie ., B.Boon and J.Journee for the ideas and the support of this work.

C

C

CALCULATION OF HYDRODYNAMIC FORCE ACFINC

*

C

*

C

ON THE CIRCULAR CYLINDEfl USING MURIBON EQUATION

*

C

C

AS A FUNCTION OF DIAMETER, WAVEHEICHT , WAVCLI:NUHT OR

C

*

C WAVE FREQUENCY AND WATER DEPTH.

C

C

THE FORCE IS PRESENTED AS A INTECRATEL) VALI.JE AND

*

C

*

C

USING VELOCITY AND ACCELERATION DISTRIBUTION IN A

C

*

C

DEPTH AS DISTRIBUTED VALUES IN WATER OEP1H.

*

C

*

C

WALL rHICKNESS AND MASS ON FHE lOP OF THE P1L.E ARE

*

C C CALCULATED. C

*

C C C

TUDELFT

*

C

C SHiP HYDRODYNAMICS LABOTORY

C

C

TUTOR: PROF. J.CEAIUTSMA

C U

C

AUTOR: DR.SPASBOV S.A.

*

C C 31 C

INTERNAL U.SING

C C

LISI OF SYMBOLS

C

C CO

-DRAG (;OLr .[

I I ENT

C CM

-INERtIA COEFIT]:ENT

C DEPTH -WATER DEPTH

C OMEGA -WAVE FREQUENCY

C NUll -NUMAUI4 OF WAVE FAEQLffNCY

C OIAM

-OIAMFIER OF THE CYLiNDER

C NDIM

-NUMBER OF DIAMETERS

C

HEICHT

-WAVE HEIGHT

C NHA

-NUMBER OF WAVE HEICHIb

C AL

-HEADING ANGLE

C NANC -NEIMLIER UI HEAD INC ANCL.E1J

C WLEN -WAVE LE:NCIIT

C EE

-YOUNG MOL)ULE( 1L/\STICI I Y)

C RO

-SPECIFIC MASS

C

THICK

-WALL THICKNESS OF 1HE PILL

C AL

-LENGTH OF [HE CYLINDER

C FMA

-MAXiMUM S1ATIC DEFLECTION

C OME

-EXPECTING NAuRAL. FREQUENCY

C

C'' U **********U******

***38H

-1N3 *fl1 ***-***U

DIMENSION OIAM(5),HEICHT(5),FDC5,5,5,10),FM(5,5,5,10),GK(5)

DIMENSION HK(5),AK(h,5,10),F1(o,5,5,10),F2(5.5,5,1QJ,WF'EHti

DIMENSION WAK(5) ,FOACE(5,5,5, 1) ,OMEGA(5) ,ALC 10),WLEN(5)

C

DIMENSION F11(5,5,5,10),F22(5,5,5,1),Fr(5,5,5,10),VU.L(5.5,10,8)

I)[MENSION F3D(i,5,5,1)18),F3M(i,5,5,10,l3),I-T(5,5,5,1@,li),DEF'(I3)

DIMENSION ACC( 5,5, 10,0) , VEL (5,5, 10,8), Ht.L.( 5), FM/X( 5) , EC 3)

AEAO( AL, FMA, E:E , OME, RE) READ( 5, *) Cl), CM, OEP Fl-I

READ( 5, )( WPEFI( I) , I'1 ,NOM) READ(5,*)(DIAM(I),I"l,NOIM) AEAD(5,*)(HEICHI(I),I1,NHA) AEAD( 5, *) ( AL( r) , I 1, NANI3)

AEAD( 5, *)( WLEN( I) , I 1, NOM) C C INI1IAL CALCULATIC)NS C RHO 1 . 026 PI3. 1416 E)RAVO. 8065 00 79 I'1,N0M CK( I) 2. *PI/ I .66/( WPEI( I) **) 79 OMEGA(I)2.*PI/WPER(I)

DO 10 M1,NAN0

AL( M) AL( M) /57.3 00 10 ,J-1,ND1M DO 10 K-1,NHA

00 10 ]1,NOM

00 10 II1,NUL

OEP( 11) -DEPTH'II/NOL DEP 1 -OEPTH/NDL 1F(OMEGA( 1) .NE.0.) GO lU 50

OMEGA( I) -SQRTC 2. *pt*GAAV/WLEN( I))

50 CONTINUE

WAK( I) -OMEGA( I) *0MEGA( I) /GRAV

WK2-WAK( I)

16

WKWK2

Al TANFI( WK'I)EPTH)

WK2OME(3A( I) *0ME(/\( I) /CflAv/131

IF(A135(WK-WK2) .CT.0.001) CO II) 16

WAK( I) -'WK2

VLL( I,K,M,II)=OMEr.A( I)),IEI0FIT(K)/2.((1SiIl WAK( I) ( III ICIIT(KJ #OEP( Ii))) /S.tNH( WAK(

fl

-1)EP1 H) *1',fl$( Al.( M) )

WRITE(6,)VEI.(J,K,M,II)

I, K, M, 11) -( OMECA( fl **) *HE]:(HT( KS) /' . 3C0Il( wAI<( ii *( HEXUHT( K

II) /2. +OEP( II))) /HINH( WAK( I) 'DEPTH) *S)N( AL( M) ) HK( I) WAK( 1) *DEP 111

AK( I, K, M) -WAK( I) *( 0. 5*I-ILTCHT( K) CO( At.( N) 1 iDEP1 H) 1F CHK( I) .131.10.) 00 10 22

Iv(AK(I,K,M).cr.l0.) 001022

GO 10 33 22

HK(I)l0.

AK( I , K, N) -10. 33 CONFINUE C

C ESTIMATION OF DflAG,INEFITIA AHO TO1AL FOflCE C ON SINGLE PILE IN SIIALUW OR DEEP WATER C USING MORISON EQUATION

C

F1(]..J,K,M)RH0(UDJ.AM(JJ/(3.WAK( Ii) '(( I1M[GA( I)

HE1CH1tK))2)

#1CC C EXP( HKC I)) -EXP( -HK( 1))) /2.) 2)

FD( I , J, K, N) F 1( I, J K ,M) *A5S( COSt ALt Mi)) *COS( AL( Mi) *

#( C EXP( 2. AK( I ,K , Mi) --EXPI -2. *AK( I ,K , M) ) ) /2. .AK( I,K, N) )

# *21 *HEII3Il1( K) IC EXP( HK( I) ) -EXP( -HK( Ifl) *:.

FM( I, J , K,M) F2( I, J , K, N) *SIN( AL( Mi)*( EXP( AK( I ,K,M) /2.)

//-EXP( -AK( I, K, N) /2.)) /2.

FUACE( I, J , K ,M) 'FD( I, J , K, N) +FM( J ,1i ,K, Mi

F3o(I,J,g,M,II)_0.5*pHO*eu*01AMCJ)*0EP1*v11Js,M,St

yvEL(I,K,M,:r))

F3M( I, J,K, II, ii) -.25*UH0CM*PI*1).1AM( J) I)JAM( JJ 0 '1"

.(-L,K , 11,

#11)

C

OUTPUT DATA

C

OPEN( UNIT 6, FILE 'MInour:. OAT ')

WR[TE(6,2(H1) WRI rE( 6, 'ø 1)

00 11 J-1,ND(M

00 11 M1,NANG

DO 11 K'l,NHA

00 1L1 I1 , Null

WRITE( 6, 30Y1) DIAM( J) ,:HEICHI (K) ,OMECA( Ii

//,FOACE( I, J,K,M;) ,TNC I, J,K,M) ,.FlM( I, J,K, M)

I I CONTI NUE

C

OPEN( UNIT ,II!L.E 'DISTITIFiI.DAT')

DO 11. Ji,NDtM

00 71 M"l.,NANG D,O 7 i : 1 , NHA D1J 7 1 i It ,NOM

00 71 iii,NDL

WA1IE(b,4ø)U1AM(JJ,HEICHI(IJ,UMh1,A(1)

#,OEP(lI).F3D(I,J,K,M,T1),3M(t,.J,V,M,11),I-3I(],J,I,M,fl)

7 1 CO NT INUE C FIIJAMAT STATEMENTS

FOHMA:i( '. D1AM w.IILi.GHT

OMECA

FMAX THICKNESS

M/vs

201 FORMAT( ' N N

HAD/S

KN N KC ')

3ø FORMAT( 1FD.5')

STOP

0.02660

0.15000 7.65375 0.00045 0,110265 0.22204

0.02660

0.15000 6.28300 0.00083 0.00485 (1.41692 0.02660 '0. 15000 4.18867 0.00060 (1.0035(1 0.291W'1

0.02660

(1.15000 3. 14150 0.00059 0.00:346 0.29229

0.02660

0.20000 ?.853'?5 (3. (10080 (1. (30472 (1.41)458

0.02660 .0.200(10

6.28300 0.1101.80 tJ.111055 0.96723 0.02660 0.200(10 4. 10867 0.00.116 O.U0(.110 Ii,. 5'I'/!37

0.02660 0.20000 I.141S0 0.00110 0.00645 0.55464 0.02661.) 0.25000 7.85375 0.00125 0.00737 0.66192 0.02660 0.25000 6.28300 0.00343

0.fl01:5

2.04414 0.02660 0.25000 4.1886? 0.00196 0.01162 1.07'1H9 (1.02660 0.25000 3.14150 0.0(31.80 0.01057 0.96975 0. 02660 0. 30000 7. 85375 0. un 11131 nJ. 011Th 1 _0. 0.02660 0.30000 6.28:300 0.00604 0.03549 4.15(188 0.11266(1 0.30000 4.lUUf.7 0.00311 0.01829 1.02000 0.02660 0.30000 3.14150 0.00272 0.0159? 1.55200 0.03200 0.15000 7.85375 0.00054

0..00i83

U.2t1(I'I(; 0.03200 0.15000 6.20300 0.00099 0.00335

0.53'82

0.03200 0.15.000 4.18867 0.1J0072 0.111)242 _0.30216

0.03200

0.15000 3.14150 0.00071 0.00239

0.37)43

0.03200 0.2001)0 7.55375 0.IJUIJ'J'I 0.01)326 0.51766 0.1)321)0 0.20000 6.28300 0.00216 0.1)0779 I. 16745 0.11:3200 0.20000 4.18067

0.0Ui39. 0.00470

(1.9534,4 O.fl320O 0.20001] 3.14150 0.00132 0.00446 0.71323 0 03200 0 2500(1 7 85'37h

0 00th1

C) LI050 0 (31111

0.03200

0.25000 6.13300 0.00413 (1.01393 2.36258 0.032110 (1.25000 4.1886? 0.00230 O.U08U4 1.3:1305 0 032110 1) ?h000

3 14th0

0 00'1?

11 0117.30 1 1'1(131 (I. 0Z320L1 Il. 3000(1 9. 05395' 0. 01)2 1:') Of1113933 1 . 19471 .0.03200 0.30000 6.2831)0 0.00727 0.02453 4.42361 0.03200 0:.3(1(1lJU 4. 18067 0.00375 0.01264 7.12705

0.03200

(1.300(30 3' 14150 U..00327 0.011(14 1.54027 0.05000 0.15000 7.85375 0.000115 IJ.IJUI.V115 0.50094 0.050(10 0.15000 6.28300 0,00155 0.0(113') 0.0157'? 0.05000 0. 15000 4. .1:8869 (3. 00112 0. (10(11)9 0 66 1:114 0.0(301)0 0.15000 3.14150 0.00111 0.000'J8 0.6(3295 fJ.1l5t1D(.1 0.20000 7.853'15 0.011151 0.00133 L3.OIIIJ'JJ IL,1J5000 0.20000 6.28300 0.1Jfl33I3 11.00299 1.990(37 0.:05000 0.20000 4. 118867 0.002111 IJ..1111192 1 .2H631 0.05000 0.20001) 3. 14L50. 0.00206 0.00102

i .io3o

0.05000 0:. 25000 7. 85375 (1.. 00236 0. 00209

1 . :9:is

0.05000 0.25000 6.28300 (1.00645 0.0057(1 3,53371; 0.05001) 0.25000 4. th136'i 0.0(13,72 (1.00329 2.21)162

ti obbd Ii JUb

J 1U

HtiiJ

U dz'ii

0.05000 (1.31)000 7.85375 (1,00340 0.00300 2,009111 0.05000 0.30000 6.2830(1 0.01136 IJ.,011J05 6.79562 0.05000 0.3(1(100 4. 1886? 0.00565 OJ)05t8 3.49550 0.05000 0.311000 3.14150 11.00511 0.00452 3.03299 O 06301) (I. 15000 7, 85375 0. 00:1(17 0. (1004? 0. 6451:4 0.063011 0. 15000 6.28300 0.. 00196 0.0(3006 1 . 10586 0.06300 0.115000 4,118867 (1.00141 0.001)62 0.135531' 0.06300 0. 151100 3. 141150 0,00139 0.00O6 0.04489 0.063011 0.20000 9.85395 0.00190 0.1)1)0114 1. 1:5257 0 , 06300 0 .20000 6 . 20300 0, (11)425 I) . 13(11613 2 .5(31119 0 06300 0.20000 4, 181W? (1. 00214 0. 00121 1

0.06300

(1.201)110 '3.14150 0.00260' 0.00115 1 .S'1646 0. 0631)0 1), 25000 7. 85375 11. 0U20'/ (1. 01.312,1 1 . 1.111 t53 0.06300 0.25000

6283fl0

0.00013 0.00359 4.93666 0.0630(1 0. 25(100 4.. 18116'? 0.00468 : 0. (10211'/ 2. 1342(3(1 0.06300 0. 250013 3. 14150 0.00426 (J .0(1188 2 .513591) O (16300 (3 311000 1 1t 315 1) 01)4 'II 13 1)1)1 ui h'J 31

n,00aoo 0.30mm 6.28:3(10 0.01431

o.wis:j:i

,i.'iii'ao

0.06:400 0.30000 4. 1886-/ o.ocr)x? o..003;'G 4.4717:4 0.06300 Q..3LJ00U 3.14150 0.0(1644 0.00200 :3,11u22

1.0 CD=1.2\CM=1.5H=.25\T=1.5\E,=2\LA=2.5\KG=2*pI/(1.5:6*TA2) 20 DE=,04\.OM=2*PI/T\ DIM E(lO)\ DIM RO(i:0)\ DIM D(30)

30 REM

flu

voi..gb de materiaalinvoer

40 PRINT "hoevee]. verschillende sooren materialen"\ INPUT N

50 FOR 1=1 TO N

60 PRINT "E-moduius" INPUT E(I)

70 PRINT Sooxte1tjke massa"\ INPUT RO(I) 80 NEXT I

200. REM

flu

volgt het aanta]. diameter,s 210 PRINT "hoevee]. diamet,ers" INPUT Z 400 REM

flu

voi,gb de berek:ening

405 FOR 1=1 TO N

410 FOR 3=1 TO Z

411 PRINT "de diame.ter"\ INPUT D(3) 420 GOSUB 1000

430 NEXT 3 440 NEXT I

500 GOTO, 20000

1000 REM de subroubine beekening

1010 REM flu

voigt de berekening van de hydr. dyn. krachten

1015 CLEAR

1020 X=CM*1000*Pi*D(3)2*OMA2*H*(1_Exp(_KG*L))/(KG*8)

1030 Y=CD*1000*D(3)*OM"2*H2*(1-EXP(-z*KG*L))/(z*KG*8)

1040 TT=0\FH=0\DT=PI/200\MAX=0 1042 GOSUB 5:000 1045 FOR AI=1 TO 100 1046 FH=-X*COS(OM*TT)_Y*SIN(OM*TT)*AB3(S.IN(OM*,TT)) 1048 IF MAX(FH THEN GOSUB 3000

1049 TT=TT+DT

1.050 GOSUB 5048 .

1051 NEXT Al

1053PRINT "hoeveel tw's"

INPUT ZA

1054 FOR ZD=1 TO ZA

1055 REM

flu

volgt de berekening van de zakking

1056 PRINT "tw"\ INPUT PH

1060 A=CM*1000*PI*D(3)A2*OMAZ*H*COS(OM*TFMAX)*EXp(_KG*L),8

1070 B=CD*D)*,i000.AOM2*HA2*SINr,(OM*TFMAX)*ABS:(SIN(OM*TFMAX))*EXp(_2*G*L)/ 1080 C=A*EXP(KG*L)/KG+B*EXP(2*KG*L,)/(2*KG) 10.90 E=A*EXP(KG*L}/KGA2+B*Exp(2*KG*t,)/ (Z*KG)A2_C*L 1100 G=A/KGA3+B/(2*KG)A3 1110 Q=A/KGA4+B/:(2*KG)A4 1120 1130 IF TW=0 THEN EI=ABS((HULP+G*L+Q)/DE)

1131 IF TW>0 THEN DE=ABS((HULP+G*L+Q)/(E(I)*pI*(D(3)A4_(D(3)_2*Tw)*4)J54))

1.132 IF TW>0 THEN GOSUB 12030 1:133 IF TW>O THEN GOTO 1380 1200 REM

flu

volgb de wand'dikte

1:210 IW=EI/E(I) 1211. GOSUB 12.011 1.21.2 GOSUB 12030

1380 IF N1 THEN PRINT " de staáf kn:ikb"

1385: IF N<1 THEN PRINT D(J),,FKN1K,F,M,,IW

1390 IF NU THEN RETURN

1400 REM de afdrukopdrachten

1405 CLEAR

1.406. PRIN1 "de maximale kracht";MAX,"N"

1407 PRINT "Heb bijdstip" ; TFMAX ,":s"

1408 PRINT " da fasehoek i:n P1 rad.ialen";TFMAX*OM/PI,"PI radialen" 141.0 PRINT "E-moduius "E( I) , IN/mA 21"

1.415 PRINT de wanddikte t"; TW, "rn" 142,0 PRINT "diameter ";.D.(3) .

2.430 PRINT "stijfhejd hI;E(I)*IW,hINmA2u

1470 PRINT "KC

";PI *H/D'(0)

1480 PRINT "V water max vert. ";H*OM/Z,"m/s"

1482 PRINT "V water max hor.";Z*PI/KG,"m/s"

1485 BEOMA2/(2*Pi).2

1490 PRIN.T "V cyl. max

".;DE*OM,".m/s"

1500 PRINT "verse. M hu;(1+MNW*9.81*.5A3/(3*E(I)*1W))*DE*OM,hlm/sA2hs

1510 EiG=3.O3*E(I)*IW./(,L3*(.23*RO(I)*LA*pI*(D.()A2_(D)_2*TW).2)/4+I.1NW)):

151.1 PRINT "de werkeliikè e.igenperiode"2*PI/SQR(EIG),"s"

1520 PRINT "de massa van de paai,"R0(I).*LA*PI*(D(J)"Z(D(J)2*TW)'2)/4,."kg"

1530 PRINT hIpaa1massa/mhh;RO(I)*LA*PI.*(D(J)A2_(D(3)._2*TW)2)/(4*MNW)*100,9

1535 PRINT "de max.massa M";FKNIK/9.81,"kg"

1536 PRINT "de doorbuiging aan het oppervlak";DE.,"m"

1545 NEXT ZD

1700 RETURN 3000 )4AX=FH\TFMAX=TT 3010 RETURN

5000 SET VIEWPORT 0,1,0,.625

5001 SET WINDOW 1900,1985,0,15

50i0 PLOT (1900,0),

5.020 PLOT (1900,15)

5030 PLOT (1900,7.5),

5040 PLOT (1985,7.5)

5045 RETURN

5048 FHT=-X*C0S(OM*TT)-Y*SIN(OM*TT)*ABS(SIN(OM*TT))

5050 PLOT (50*(TT-DT)+1900.,.5*FH/1O+7.5),\ PLOT (50*TT+1900,5*FHT/10+7.5.),

5070 RETURN

12011 IF IW*64/PI>D(.7)'4 THEN PRINT "er i,s geen wanddikte te berekenen"

12012 IF IW*64/PI>D(7)A4 THEN PRINT "bij deze e-moduius"

12013 iF IW*64./PIS>Dt)'4 THEN RETURN

12020 TW=(D(3)_(fl()A4_iW*64/PI)A.25)/2

12025 RETURN

12030 IF TW>D(7) THEN PRINT "Mj deze E-modulus geen opiossing"

12040 IF TW>D(J) THEN PRINT E1(I),D(3)

12050 IF TW>D(7) THEN RETURN

1.3000 REM de eigenfrequentie

13001 IW=PI.*(D(3)A4_(D,(7)_z*TW;)A4),64

13010 KV=3.03*E(,I)*IW/(LA.3*4*PiA2,}

1.3020 M=KV

13030 I.? M(O THEN PRINT "massa kleinex dan nul"

13035 IF M<0 THEN RETURN

13340 REM knik

13050 F=M*9.82.

13060 FKNIK=PIA2*E(I)*IW/(4*LAA2)

13070 N=FKNIK/F

13080 RETURN

20000 END

10 CD=1.2\cMl.5\H.25\T=1.5\L2\LA2.5\KG=2*PI/(l

.56*TA2)

20 OM=2*PI/T\ DIM E(10)\ DIM R0(10)\ DIM D(30)\ DIM

M(10) 30 REM

flu

vo19t de mater jaalinuoer

40 PRINT "hoeveel ver schi 1.1 ende mater i aien"\ INPUT N

50 FOR 1=1 TO N

60 PRINT MEmodu1us\ INPUT E(I)

70 PRINT Ns.00rtelijke rnassa"\ INPUT R0(I) 80 NEXT I

200 REM nu volgen het aantal. d:iameters en massa's 210 PRINT H.hoeveel diameters"\ INPUT Z

220 PRINT hoevee1. rnas.sa's"\ INPUT HOEV

400 REM

flu volgt het eigenlijke programma

405 FOR 1=1 TO N

41OFORJ=1TOZ

415 PRINT N:dE, d:iarneter"\ INPUT D(J)

420 FOR ALP=1 TO HOEV

425 PRINT TMde massaN\ INPUT M(ALP) 430 GOSUB 1000

435 NEXT ALP 440 NEXT J 450 NEXT I 500 END

1000 REM de berekening van El uitgaande van rn(alp) en ei,genperlode 1010 EIM(ALP),*LAA3*4*P1A2/3,.03

1020 I;N=EI/E(I)

1030 PRINT TMhoeveel tw's"\ INPUT ZA

1040 FOR ZD=1 TO ZA

1050 PRFNT hItws\ INPUT TW

1060 IF TW=0 AND IW*64/PI<D(J)A4 THEF4 TW=(D(J)_(D(J)A4_IW*64/PI)A.25.)(2

1070 IF TN>0 THEN IW=PI*(D(J)'4(D(J)-2*TW)4)/64

1100 REM

flu

volgen fmax en tfmax

1110 X=CM*,i 000*PI *D( J) "2*OW2*i*(1EXP( -KG*L) )/(KG*8)

1120 YCD*1000*D(J)*OMA2*HA2*( 1-EXP( -2*KG*L) )/( 2*KGk8) 1130 TT=0\FH=0\DT=PI/200\MAX=0

1140 FOR AI=1 TO 100

1150 FH=X*C0.S(OM*TT)-Y*SlN(Oti*TT)*ABS(3iN(OM*TT))

11:60 IF MAX.<FH THEN GOSUB 3000

1170 TT=TT+DT 1180 NEXT Al

1500 REM nu voiSt de. verplaatsing aan het uiteinde

1510 A=_CM*10O0*PI*D(J)A2*OMA2*H*COS(OM.kTFMAX/8

1520 B=_CD*1000*D(J)*0N1A2*HA2*SLN(O*TFMX)kABS(9IN(0M*TFM<) )/8

1530 C=-A/KG-B/(2*KG) 1540 E=_A.kEXP(_KG.kL)/KGA2_B*EXP(_2*KG.kL)/(2kKG)A2

1550 Q=_A*EXP,(_KG*L)/KG4_B*EXP(_2*KG*L)/(2*KG)A4

1551 MINKL=_(A/KGA2+B/(2*KG) A2+c*L+E)

1560 G=_A/KGA3_B/(2*KG) A3_C*L2_E*L+MINKL*(LA_L)

1570 HULP=AtKGA4+B/( 2*KG) A4+C.kLA3/G+E.kLA2/2+G*L+Q 1580 DELTA=ABS( HULP)/CE( 1 )* IU)

1590 DWARS=_(A/KG*EXP(_KG*L)+B/(2*KG)*EXP(2*K0*L)+C)

1650 REM r.ik

1660 FKNI K=PV2*EC I )*IW/(4*LA2) 1670 FM=M(ALP)*9.81

1680 N=FKNIK/FM

1690 SIGMA=FM,(pi.k(D(j)A2_cD(J)_2*TW)A2,4))+W(N_1)*ABg(I1II1KL)/(IWk2/D(J))

2000 REM de afdrukopdrahten

3:Oj KDTHAX 2005 CLEAR

2010 PRINT 'FMAX

2020 PRINT

TFMAX

;1FMAX,NsU

2050 PRINT '"Specific rnassa

2060 PRINT "Diameter of the

pile

=";1000*D(J),"rnri."

2070 PRINT "Wall thickness

of the pile

=11 ;1000*TW,"rnrn"

2080 PRINT

E*Iw (1 paal)

2090 PRINT "'lw (i.,paal)

&I;.Iw,.drnA4hI

2100 'PRINT "Mass

(total on three legs)

;,3*M(ALP),Hk9u.

2110

PMS=R0(I)*LA*PI*(D(3Y2_(D(j)_2*Th)A2)'4

2111 PRINT "Mass of the pile

=";PMS,"kg"

2120

EIG3.O3*E(I),*IW,(LAA3.k,(.230(I)*Lj)4_(J)2ATW4)//4+M

2130 PRINT "Eigenpe.r'iOd of

the pile

2140 PRINT "Keulegan Carpenter

number

';PI'*H/D(J)

2150 PRINT' "Velocity water

horizontal

";l.56kT,"rA/S"

2170 PRINT "Velocity cylinder

=";,DELTA*OM,"m/s"

2180 MVS=(l+M*9.81*.53/(3*E(I)'IW))*DEITOM2

21,85 PRINT "Acceleration of mass M

;MVS,urr,/sA2li

2190 PRINT "The maxirnurn mass

M on thre,e"legs &';3*FKNIK/9.'8I,'"k'9"

2200 PRINT NThe displacement on the surf ace

";lOO*DELTA,"Cm"

221,0 PRINT "Fknik/F

=";N

2220 PRINT "The strews due to

forces

ES,IGMA*.O00001,IN/rAmA2N

2230' PRINT' "The bending moment

Minkl

=";MINKL,"Nm"

2240 PRINT "The hor I z,o.n'ta'l

fo;rce at' bottom

=", ;DWARS, "N"

2250 NEXT ZD

3000 MAXFH\TFMAXTT

3010 RETURN 20000 END

de fas:ehoek In P1 radlalen 1.46608

E-moduius

.21E+12

N/rn'2

de wanddikte t

.001

in N S

1.46608

N/rn" 2 in m Nm"2

kg/rn" 3

m"4

is nu .12277

P1 radialen

PI radlaien

kg

P1 radiaien

diameter

.01

m

st ii fheid

140

Nm" 2

soort .rnassa

7800 k gtrn" 3 tr . mOmén t

.66667E-0 9

m" 4

dè toe te passen massa is nu .508287

kg

KC

78.5398

V wat:er max vert. .523599

rn/s

V water max ho:r. 3.51

rn/s

V cyl. max

.419302E-02

rn/s

versn. M

.175898

m/s"2

de wer kel. like el genperl ode 1

S

de masse van de peal. .78

kg.

paalrnassa/m 153.457

de max.massa M 5.6340,2

kg

de doorbuiging aan bet opperviak

10 010 lE-Ol

Ready

de maximale kracht .660925

Het tiidstlp 1.09956

de fasehoek in P1 radlalen

E-modulus

.1E+12

de wan ddik te t

..00i

di arneter

.01

St ijfheid

66.6667

soort .rnassa

8900

tr .mornen t

666667E-09

de to.e te passen massa

KC 78.. 5398

V water max ver t. .523599

rn/s

V water max hor

3.51

rn/s

V cyl. max

.880535E-0i

rn/s

versn. K

.369115

rn/sA2

de werk,elijke eigenperlode 1

s

de massa van de paal .89

kg

paalmassa/m 724.932

de max.massa M 2.68287

kg

de doorbuiging aan het opperviak 210212E-01

Ready

de maximale kracht 2,79939

N

Het tijdstlp 1.00531

s

de fasehoek in P1 radialen 1.34041

E-moduius

3E+10

N/rn'2

de wan ddik te t

:O035 in

dicn,a1r

.U:.! rn

stijfhei.d

300.125

Nm"2

soort.massa

1400

kg/m'3

tr.rnoment

.100042E-OG

rn'4

de toe te passen massa is nu 1.07978

kg

KC

22.4399

V water max vert. .523599

rn/s

V water max hor, 3.51

rn/s

V cyl. max

.776763E-01

rn/s

versn. M

.325857

m/s"2

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