Random dynamic analysis of an offshore single anchor of leg storage system

Random dynamic analysis of an offshore si

leg storage system

-R. S. LANGLEY and C. L. KIRK

Offshore Structures Group, Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL

A linearised spectral analysis is presented for the surge motion of a 240000 t tanker in head seas and the pivot and riser forces in the mooring system. The theoretical results are found to compare -favourably with those obtained from wave tank model tests, demonstrating the applicability of linearised spectral techniques in evaluating the combined high/low frequency response of moored

systems.

INTRODUCTION

The single anchor leg storage system (SALS) is a new

concept in the mooring of large vessel oil production

facilities or processing plant, with several systems being already installed or under construction.

The basic concept, shown in Fig. la was developed by Single Buoy Moorings. It consists of a mooring chain placed under high tension by a submerged buoyancy chamber, the vessel mooring forces being transmitted from the chain through a welded steel tubular yoke structure attached to the vessel by pivots.

A major requirement for a reliable design is to achieve sufficient initial chain tension to prevent it from becoming slack under the action of wave forces, thereby leading to large snatch loads. High tension is also needed to minimise vessel surge motion and the associated angle of rotation of

the yoke in order to limit pivot forces.

The chain or riser anchor base is generally in the form of a steel box containing ballast, but in weak soils drilled or driven piles may also be required. The major advantages of the SALS system is that it is quick to construct, easy to install and is reliable under the most severe environmental conditions in addition to which it can be used in almost any water depth.

The aim of the present paper is to give a linearised

spectral dynamic analysis for the determination of random motions and pivot forces due to direct wave induced hydro-dynamic loads and slowly varying second order wave forces. The parameters of main concern to the designer are riser

rotation and tension variation, pivot reactions and also

yoke and vessel surge motion which is an input to the

dynamic analysis of riser flexural response. The spectra of pivot and riser forces arc also important in fatigue analysis.

YOKE ol.imusitrAii MW L

11111WRear"""===.11=

0 i 2 , dt h_p 1/2 112 . RISER CYLINDRICAL BUOY

Figure Ia. Schematic of SA LS

0141-1187/82/0-10232-13 S2.00 TECHNISCHE UNVERSED' taboratorium vow Scheepahydromechanica Archtef

CD Deft

015-

nun

F+ F b

_Yw (M+ )4_ay) + SYS' -FY.;/.

Figure lb. Forces on tanker and yoke

The method of analysis involves a frequency domain solution. of a three degree of _freedom model of the SALS representing tanker slugs -heave and pitch

motions and

motion of the yoke system. Both direct wave induced

motions and slow drift motions are considered

simul-taneously in evaluating

nonlinear drag forces on the

buoyant chamber, which are linearized by the method of Tung and Wu.' Transfer functions are obtained for given sea states by an iterative procedure for linearising drag from which r.m.s..motions, yoke pivot reactions and riser force can be obtained.

The vessel selected for study has the following data:

D,t= 200000 tonne, length L = 310 m, beam B = 47 m,

draft dt= 18.9 m and 9.45 tn. Further dimensions arc given subsequently.

DYNAMIC RESPONSE ANALYSIS

In deriving the equations of motion it is assumed that the

SALS 'weather vanes' in the direction of the incident

random wave propagation which is assumed to be coplanar with current and wind forces. By neglecting the elastic

mode deformation of the vessel and assuming an

xtensible riser the motion of the system in the plane of /aye propagation is defined by the three rigid body degrees

freedom, heave y, pitch 0 and surge x.

I SSW/ iptions

The hydrodynamic forces are evaluated by linear wave heory with the following assumptions:

Forces on the buoyant chamber can be obtained from Morison's formula, the fluid motion being unaffected by the presence of the vessel. This is considered reasonable since the chamber is about 10 in in diameter and situated some 50 m from the vessel.

Forces on the small diameter yoke bracing

mem-bers can be neglected in comparison with the

forces on the buoyancy chamber.

Forces on the riser and mooring chain can be

neglected.

Only vessel heave, pitch and surge motions are considered with sway, roll and yaw being ignored. The vessel hull is symmetrical with the centre of gravity being amidships which is approximately true for a laden vessel.

The heave, pitch and surge exciting forces on the vessel can be adequately represented in terms of the Froude-Krylov forces neglecting diffraction for head seas.

(7) Vessel damping forces and moments and added mass can be evaluated by strip theoryusing Grim's data sheets.2

The hull is idealised as a box of length L, breadth B and draft dt.

The angles of rotation of the yoke and riser

during response are small.

Hydrodynamic forces are calculated at the static

equilibrium position of the system assuming a

vertical riser.

Potential damping forces on the cylinder at a

depth of 18 in for the very low frequencies of slow drift motion are negligible3 compared with the quadratic drag force damping

7oordinate system and geometry

The coordinate system is shown in Fig. la with the

Drigin 0 of the x-y frame centred at the undisturbed

:quilibrium position of the system. The general dimensions )f the vessel and yoke are given in Figs la, 2,3.

Figure 2. Dimensions of yoke

Figure 3. Coordinates of SA LS response

Yoke kinematic relationships

Figure 3 shows a line diagram of the deflected con-figuration of the system in which (x', y') are the coordi-nates of the (yoke/vessel) pivot and (x, 0, y) are the vessel coordinates in surge, pitch and heave. It can be seen that the following geometrical relationships hold,

x' = x + (1

cos 0) hp sin 0

2

sine y + hp(1 cos 0) 2

The angles of rotation of the riser and yoke, aand p are given to first order by

(2) P =

11 +

The displacements of the buoyant chamber, point q

(Fig. 3) are given by

x" = x'

/5(3

Y" =Y'1213

411

which on substituting equations (1) and (2) for small 0

gives /sLO isY ff

_{h 0}

Xa= X 2(l1+12) (11 + 12) L110

liy

2(11-1- 12) (11 + 12)

Similarly for the centre of gravity of the yoke, point p

161-0 16y ff XP= X 2 1 + 12) (1 1 + 2) (5) pff =

0 -

(1

---)y

. L 14 2 11+12/ + due Thus the displacements of the yoke points p and q to a general vessel displacement (x, 0,y) can be written

LO

qx= x

_ci(-2

-y)-hp0

LO)qy=

cs(y

x

hp0

a=

13+ d [(L12) 0 y] (1) (3) (4) (6)

Random dynamic analysis of an offshore single anchor leg storage systein: R. S. Langley and C L Kirk X IL

y)

hp0 (c) py= C2(Y (d) 2

where the constants ci are listed in Appendix A.

The displacements

in equation (6) will be used in

deriving the equations of motion of the SALS. Hydrodynwnic forces

The components of hYdrodynamic force are derived as follows:

Buoyant chamber

Inertia forces. The horizontal and vertical corn-ponents of fluid inertia force are calculated from Morison's formula

=

₍₇₎

4

vtb

where D and Lb are the diameter and length of the

chamber, u, v being calculated using linear wave theory at

the static equilibrium position of the cylinder at depth

yb= d +13 ls below mean water level. The origin of wave coordinates is taken at the static equilibrium position of the vessel centre of gravity G.

The components of the inertia force on the cylinder are thus

Ab.(w)H sin [k(12+ LI2) + wt] = Aby(w)H cos [k(12+ 1,12)+ wt1 where

sinh (ky b)

A ( )=- LbpD2C,w2

(9)

8 - sinh (kd)

Abx(w) is

obtained by substituting cosh(kyb) for

sinh(kyb) and k is calculated from the relationship (...)21g = k tanh (kV) where d = water depth.

Drag forces. The horizontal drag force due to

current velocity V,(y), wave induced fluid velocity u and buoy velocity 4x is given by

= pCDDL b(eix u

Vc)I 4x u

VI

and the vertical drag force by (10)

pCDDLb( v) I V I

To carry out a spectral analysis it is necessary to linearise the drag forces, including the influence of current using the

linearisation method of Tung and Wu' in which the

dynamic component of the horizontal drag force is given by = pCDDLo/R-Tr[a, exp ( Q212) +- -127-rV erf (Q)]

x (cix

u) = ax(elx u)

(11)

where Q = 11,1 our and cru,.= r.m.s. relative fluid velocity

in the absence of current, i.e. Lir= 4x u and ax is the

coefficient of relative velocity.

Writing V = 0 in equation (11) the vertical drag force is given by

Fly)= pCDDLb-Iffrra.,(ely

= ay( (I y v) (12)

where V, y V aryl ay = pCDDLbfiFTCr. vr.

(8)

In addition to the lincarised dynamic drag force the

linearisation procedure yields a static drag force

Ffs= pCDDLb[N/-2: V, exp (Q212)+(eur+ V,2) erf(Q)]

27r

Forces and moments on vessel (a) Direct forces and moments

(i) flleave fclre4 The vertical forces on the vessel are

calculated using strip theory by invoking the

Froude-Krylov hypothesis. Thus the force on a strip of hull is the sum of (1) the force due to the dynamic pressure p(y) and (2) the change in the hydrostatic pressure due to passage of a wave of amplitude n(x) equal to pgri(x). The heave force is then given by

L/2

Fyw = dx

p (y) cos(ny)dS + Bpg

n(x) dx

L/2

S

L/2

where p(y) is evaluated on a surface surrounding the total submerged volume; (ft,y) = angle between outward unit normal vector fi to dS and y axis.

By means of the Gauss divergence theorem the surface integral of p(y) is transformed to a volume integral thus

L/2 FYw ap

=

dV + Bpg

f n(x)dx

ay V

L/2

which. noting from linear wave theory that apmy =

pay/at gives

FYw = p

f

faIV + Bpg 710 dx

LI2

:

L/2

The first integral in equation (15) is equivalent to the average vertical acceleration of the fluid that would occupy the space taken by the vessel, an approach used by Wilson4 who included an inertial coefficient C.u, for the vessel.

Evaluating equation (15) gives

Fyw = HpgBsin (kL12){1

cosh k(d--d)l

.1

cosh (kd)

X COS Wt

Fx1

The heave force due to change of buoyancy is given by

Fyb = LBpg(dt y)

₍₁₇₎

Surge force.lBy integrating the dynamic pressure over the ends of the v ssel the surge force is obtained as

BpgH

Fx =

[sinh(kd) sinhk(dc11)]

P . k cosh (kd)

x sin (kW) sin wt (18)

(iii) \Pitch moment) Considering an element of hull dx and including the moment arm x in equation (15), after integration the pitch moment about G is given by

pgB

Mo = H

[sin (k1,12)(kL12) cos (kLI2)]

sin (kL,12)

0 =

pgH

cosh (kd)

(y + lig) cosh k(d + y) dy sin cot

P. . .

dt

- sin (kLI2) = pgH cosh

(kd)r

here hg cosh (kd)

(d,

hg)sinh

k(dd,)

I = sinh (kd)

1-k k2 cosh k(d dr) k2

(iv) awing forces and tnoments. Using strip theory

le wave damping force on an element clx of the hull due ) vertical velocity ,)", in still water has been evaluated by

any authors, see for example McCormick.s Thus pg2i2

Wyd

=

3 dx -co> 0 (21)

=here A = (generated wave ampl./heave ampl. of strip) rid is determined for beam/draft ratio of Bldr= 3.8 as a inction of 6.) from Grim's data2 for a rectangular hull :talon = 1.0).

Equation (21) is modified to account for the wave

iduced vertical velocity v by using the average velocity

ver the volume of the hull. This

gives the total heave amping force as

=

Pg2712L(f, It(k) sin wt)

Y w3

= By Fy.e (22)

where By = pg2/PLIco3 and Fy.fi=(pg2A2L/w3)-73(k) sin ca md

sin (kL/2)1 cosh k(d dt)1

.5(0= gH

1 (23)

ts.)kLd, cosh (kd)

introducing the moment arm x, the wave damping moment is given by

, fte2:4-2 [ L3

Med = ---

7(k) cos c...)t

= Boo M (24)

61

12

"

-where

Ag2

Bo = pg2d7I2L3/120 and = A2it cos ca (20)

Surge damping on the vessel is based on the experi-mental observations of Wichers and Sluijs6 for regular waves. Their results for a 200000 dwt VLCC (very large

B2(co) = [526)

39(2

12] tonne. s m-3 (26) for 0.3 < ca <1.0.

Since Bx in equation (25) is nonlinearlydependent on wave amplitude it was decided to replace Bx by its rims. value using the method derived by Pinksteri which gives

Bx=

a(B,)=I[Bo+

2 ST2(6.)) B2(o.))dc.of

(27) where si)(c..))= p.s.d. of incident wave amplitude. It can be

seen that a(B) contains contributions

from the direct

wave spectrum and the slow drift spectrum.

(b) Slow varying wave forces

The spectrum of slowly varying second order surge

forces is calculated by two methods. Pinkstee gives the p.s.d. as

Ssd(cok)= 2B2(pg)2 S n(w) S n(c..) + R4(co + (O kI2)dc.) G8a) in which R(w) is a relection coefficient obtained experi-mentally by Remcry and Hermane (Fig. 22).

The theoretical work of Bowers9 based on determination of the second order potential function for the slow drift forces gives

Ssd(cok)= 2B2(pg)2(1 d,Id)2 (coSn(co +

x sin2 [ (w, (.0k)] cico (28b) and

c(w, k) = [k(c...) + k(w)]

where k(w) is found from CJ2 = kg tanh (kd).

It should be noted that Pinkster's method is an approxi-mate solution

for slow drift forces and more recent

developments using three dimensional source/sink methods have now been developed see, for example, Pinkster and van Oortnierssen.'°

The static offset due to the second order wave forces is given by Pinkster7 as 00 OS ,,,k)Ilsrimsn(co+cooe(w+

T dw &o/c

00

Bpg

xs=

(13+ d)j Sn(co)R2(c...)) clw (29) To 0

gH [

cosh k(d kL kL

kLi

Sin cos wdtk2 1 cosh kd

_j

2 2 2 Cmv 1

cosh k(dd,

ca

(19)

crude carrier) of similar dimensions to the present vessel

are given by

Fsd= [Bo+ B2(w) n21 =

,(25)

cosh (kd)

The moment 1110j, about G due to dynamic pressure on

e ends of the hull is obtained by

reference to Fig. la.

-tus if the wavelength is large compared to the tapered ntion of an actual hull

where Bo= 20 tonne. s ni-1 is the still water damping coefficient which depends on the surge natural frequency of the moored vessel (cox.= 0.02 rad/s, T = 5.23 min) and

Random dynamic analysis of an offshore single anchor leg storage system: R. S. Langley and C L Kirk

EQUATIONS OF MOTION

The equations of motion of the three degree of freedom system are derived by considering the equilibrium of forces and moments on the vessel and yoke as shown in Fig. lb. The equations are as follows: _{. v}

sA,

+ Max) +

= Fs + Fxp

(30) where

M = Mass of vessel

= fluid added mass in surge

Bxx= surge damping coefficient (equation (27))

Rif = horizontal yoke mooring reaction (to be determined in equation (40))

Fs d = slow drift force (equation (28)) F = Froude-Krylov force (equation (18))xp

Heave

where

F = Froude-Krylov heave force (equation (16))Yw

F

Yb = buoyancy force (equation (17))

BOTy, = heave damping force (equation (22))

R0 = vertical

mooring reaction (to be determined in

equation (43))

May = fluid added mass in heave

Pitch

t

(I+ la)W + Bed Mei +MgGMO = Mow+ Mop

+ RL12 + hpRH (32)

= longitudinal metacentrie height of vessel = wave induced moments on vessel (equations

(19) and (20))

= Pitch damping moment (equation (24)) = moment of inertia of fluid added mass in

pitch

= structural moment of inertia of vessel The heave and pitch added mass coefficients are frequency dependent and were evaluated using Grim's data.2 The surge

added mass coefficient was taken as 0.05, as given by

Wichers and Sluijs.6

NATURAL PERIOD IN SURGE

The natural surge period of the system can only be defined in still water where the riser tension To remains constant. The natural period for small amplitude motion is given by

#1M + M ax)

Tx= 2ir (13 + d)

To

For To = 1250 tonne and d = 160 m, equation (33)

gives 7"; = 5.5 min at full draft and 3.878 min at half draft

(c 0.019 and c)x= 0.027).

+ May) j; +Mg +Bj

= Fyw F,bRo

where GM

1110w+Mop

Boe M0,3

(33)-EQUATIONS FOR PIVOT REACTIONS

Referring to Fig. lb, the forces on the yoke are defined as = M y133:

F2= Myg + My fiy,

F3 = FB + Ff,.11fyal1y

F4 =F

Fix)My

where

My = structural mass of yoke

M;= fluid added mass of yoke assumed equal

to mass displaced by buoyancy chamber

FB = buoyancy

_{force, FLy = fluid}

_inertia

_{forces on}

charnber (equation (8))

FxDa = drag forceson chamber (equation (10)).

In the following analysis,

in

order to carry out a

frequency domain solution of the equations of motion, it

is assumed that the riser

rension remains constant.

Although there is considerable tension variation it can be

shown that both the slow drift and direct

wave induced motions are independent of the mooring stiffness term for the following reasons:

the slow drift motion occurs primarily at the surge natural frequency 0.019 <w,,, <0.027 rad/s whereas tension variations take place in the range 0.3 < < 1.0 rad/s. Thus over a complete slow drift cycle the high frequency tension fluctuation will have approxi-mately zero mean value,

the direct wave induced motion of the SALS being at a much higher frequency than a.), is dominated by SALS inertia forces.

It can thus be seen that riser tension does not signifi-cantly affect the surge motions of the vessel. The subsequent analysis does, however, lead to the determination of tension variation which is an important design consideration. By taking moments of the forces about the vessel axis pivotwe have

T(11+ 12) cos a cos 13 + FIN cos 14 sin 13) + F2(14 cos

+16 sin (3) + F4(15 cos 0-12 sin (l) = F3(12 cos + I sin 13) (35) Assuming a and /3 are small, by using equations (2), (6) and (34) and neglecting second order terms, equation (35) gives the riser tension T in the form

(Lo

1, T(11+ 12) = myg14 My/4 c2 i ; 2 _{+ 12} . X(L-TO FA+ FR12+ F31,4

+ F 1

LO + Myals(.5!

+

(36) 1 2 P CiY LO +

Myle(2

+

_{c4j3) FA}

hpe 2

I.Lel+ FB15[(LO12)y]

in al

Y ₂

(4+12)

The components of linearised drag force in equation (36) (34)

are obtained from equations (6a), (6b), (11), (12) thus LO

Ff= ax I

ci hpO + ci9 (37) 2

(

Lo

F f,ay

---. C3i

--

(38) 2

Substituting equations (37) and (38) in equation (36) and collecting terms gives the variable tension as

T c14x + cue + c16Y _{C17i + Cis + Cisd) + C200}

+ c21Y. + ci2+ (12Fef,

_{l5F + 12C13v lscs01}

(11+ 12) ₍₃₉₎

where the ci constants are given in Appendix A. It is noted

that C22' _{To hence T T0= AT is the tension fluctuation.}

The horizontal pivot reaction R

H is found similarly by reference to Fig. lb. Since czAT is small compared with the drag and inertia forces,

RH i= Toa + Ff + Mylix + MY,,dx

(13+

d)(x

hp0)+csc,--hptj+co'ru)

2

To LO

(

+ Myi

hpO +c4Y)

(+

My. R

ci 0

hp0 + co,

From equilibrium of vertical forces on the yoke the vertical pivot reaction is obtained as

(

C3i Lc 3.u Rv= T + C13 v) + (MyC2 + /VIVO 2.

Le

x

(j;

+ Myg Fyr, 2

Substituting for T from equation (39) in equation (41) and making the substitution

Ro = To + Myg FB (42)

equation (41) becomes

Ro+ ci43e +C22 + C235; +c11i + C24j +c259 + c200

+ cny

(11P-1, + 15F,Ic + + c 5 u)I(11+ 12)

(43) Equations (40) and (43) are inserted in the equations of motion (30), (31), (32) to give the following final form,

S'urge

c&3ec7O + csji +c9 ciofj + c11S4c12x +c00 =F

(44) 'here Fx = Fsd Fxp+ F.,f+ csu (see equations (8), (18), :28)) in which the first two terms correspond to the slow 'rift and Froude-Krylov forces respectively, the remaining

elms being the inertial and linearised drag forces on the hamber. 'itch C28X 1- C29 C30; ÷ C32 6 c33)) c34x + crie e3sy LR012 = M0 (45) (41) where

Me = Mow+ Mop+ Mei

I

L liFyl + Is + 2hp( 1--2---12" F:1 + 11 ci3v

2(4+4)

- L

+

cs(1,+ 2h (11-1z 10).]

P L

(see equations (19), (20) and (24)). Heave

c14:i + C 0 + c26.i, +22 _{C24 + c2.0)+ c200 + c36Y}

+R0=Fy

(47)

where

Fy _{Fy, (l14 + 13F,1+ c13V + 15c4u)g11 +12)} (48) (see equations (16), (21) and (22).

STATIC EQUILIBRIUM

Since it has been assumed that the surge motion takes place relative to a static position in which the riser is vertical, equations (45) and (47) in the static case reduce to

C209 + c'36Y + R0 --= 0

(40) LR 0 (49)

c370 c3sy = 0 2

The solution of these equations will yield the static

deflection in heave (y0) and pitch 00. The constant Ro can then be removed from the equations of motion.

SOLUTION OF EQUATIONS OF MOTION

Equations (44), (45) and (47) are written in matrix form as

The mass, damping and stiffness matrices M, C and K are given in Appendix B.

In equation (50) the force vector F consists of the direct wave force FD and the slow drift force Fs. the latter having predominant frequency components in the region of the the natural surge frequency, where FD has components in the frequency range 0.3 <o. <1.0.

Refecence to the damping matrix which dominates the

resonant slow drift motion shows that the coefficients

depend on the r.m.s. relative velocity O(Ur) = a (.4 u). Since the yoke velocity 4x depends on the direct and slow

drift forces, the slow drift and direct motions will be

coupled through the nonlinear damping term. Solutions for the direct and slow drift motions AD and X, are then obtained as follows:

[MI AD + [C] AD + [K] XD = FD (52)

[M]),. + [CJ A+ [K]X=

(53)

in which FD = (Fx, Mo, Fy)T neglecting lid, and Fs= Fsdx. (46)

[M] + [C] A + [K] X F (50)

where

Random dynamic analysis of an offshore single anchor leg storage system: R. S. Langley and C L. Kirk

The damping matrix [CI contains the nonlinear drag

force coupling terms. The next step is to substitute in

equation (52), a solution of the form X0

X'D c't and

F = TIF OD e1. The complex transfer function for the

direct response is then obtained as

G,06))1 .VD(co) = Ifri 1-1

FOIII= +0(W)

(54)

Gy(ico)

The nine complex coefficients in A(i6.) are given in

Appendix C.

The solutions to equation (54) yield the steady state

surge, pitch and heave response due to direct wave forces in the form X'(io.)) = 72G (i w) and the mean square responses are obtained from

CO

=

low)12 STI(o.)) dw (55)

where G(ico) is the appropriate complex frequency response function .obtained from

equation (54) and

Sn(w)= p.s.d. of wave amplitude.

A similar solution to equation (53) for the slow drift response per unit slow drift force is obtained as

noting that only the slow drift corces in surge are

con-sidered, and the p.s.d. function is given by equations (28a) or (28b). The mean square slow drift surge is then found from

CO

02:= I I xis 0(.4)12 S,d(wk) dwk (57)

and the total mean square response is obtained by adding equations (55) and (57).

R.M.S. RELATIVE FLUID VELOCITY FOR CHAMBER

Equations (55) and (57) require the evaluation of the

r.m.s. horizontal and vertical fluid velocities ow. and (coefficients es, c13). The linearisation procedure of Tung

and Wu in the presence of current is based on calculating the r.m.s. velocities in the absence of current. Thus u,.= u, which on substituting from equation (6) is given by

u, =i

(Lei/2 + hp) + cip

u (58) It is now assumed that the relative velocity is given as the sum of the direct and slow drift terms

u,=u,D+ urs

(59)

and that the two velocity components are statitically in-dependent. For the direct

motionrr) . .

u =qxu

(L

O (60)

= iD

c4"hpp+

and for the slow drift motion.

u_{Fs S} (61)

0

Substitution of the transfer functions of equation (54) in equation (60) gives the transfer function for the direct relative velocity as urD06.0 71

.=

c1+

hp)Ge(io,) +eiGy(io.))1Cl(ico) 2 (62) where 14 = UN. The slow drift relative velocity is obtained from equation (56) as

-u,s= iwkVs(iwk) (63)

-The resultant mean square velocity is then given by

413(1w)

{1

74,(l(.4)= P064)]-1 0

' (56)

R.M.S. PIVOT REACTIONS

The horizontal pivot reaction RH, equation (40) was

de-termined for the direct and slow drift forces as RH=

RIM+ R113.

Substituting for x, 0, y and i, 6,3), and from equation (54) in equation (40) gives the T.F.

RH0(k...0 = 7114.} D(ico) (68). where

R4112)(6))= [c12 + c (My + Mya) co2]Gx0(.40

le1711p+ (c

les + h

)1u) e70o2 G0(io..)

2 P

Ft.

C80] Gy(ico) cs a

(69)

The mean square direct horizontal hinge reaction is then given by

01HD IRII0(iG.))12 S(o) do). (70)

CO

04=

:13(ici.))12S7,(6)) do.) +

0 0

wic I Vs° 03012

x S,d(wk) dwk

The vertical relative velocity is found from

(64) L V:D= Sly V = C33) C3 v 2 thus (65) r

o[Gy(iw)-- Go(ico)

kx3

WDOCO .

and the means square value is

-(66)

The slow drift horizontal reaction is obtained by con-ering only the surge motion in equation (40), thus

allis=

iRris(icok)12 Ssd(cok) dwk (71)

tere

RI.4(iwk) = [c12 + CS iWk (My + filYa)

QC,,k)

(72) d X(k) is obtained from equation (56).

Similarly the mean square components of the vertical Pot reaction are found using equation (44), neglecting , as

I

vD IR: DOco)12 S(w) dw

°2R (73)

a2Rvs= SIR:(ft...4)12 Ssd(cok) dcok (74)

le resultant mean square reactions were then obtained by ding equations (70), (71) and equations (73), (74), where

Viw) =

[c17

cia w2) G x(iw) + (c20 -I- C24 ICO C22 W2) G a (ico) + (c21 + C25 iCt) C23 (2) 1 x Gy(zw)

x{(114 +15Fxr +licoi3+

lscsit))1

R:diwk) =

c 14 w2) Xx' (iwk) (76) .M.S. RISER TENSION

aking moments about the (vessel/yoke) pivot axis, of

(Under fluid inertial forces, neglecting drag forces which -e small by comparison, the m.s. riser tension is obtained

_rbirpD2C,nr

f c..34 (A2 + B2) S 71(w) do.)

aT

4(11+12) sinh2(kd)

(77)

'here

=12 sinh ktP cos kt 1s cosh -14 sin kt = 12 sinh kly sin kt + is cosh -14 cos Ict = d + 13-15, t =12+ LI2.

iUMERICAL ANALYSIS

ince the r.m.s. relative velocities arc initially unknown, stimated values were inserted into equation (54) to give the omponent T.F.'s, Gx(ico), Go(ico), Gy(ico).

quation (56) gave the slow drift surge T.F., Xs(icok). The '.F.s were then used in equations (62), (64), (66), (67) o give new r.m.s. relative velocities, the iterative procedure

(75)

being repeated until satisfactory convergence was achieved for 40 frequencies covering the wave amplitude andslow drift force spectra. To increase accuracy, 20 frequency

intervals were concentrated near to the surge

natural frequency. Having achieved convergence the r.m.s.

ampli-tudes were

calculated by substituting the converged linearised T.F.'s into equations (55) and (57) for various

sea states defined by the JONSWAP and P-M spectra. With the known r.m.s. relative velocities, coefficients cs, c13 and all the related coefficients ci (Appendix A) are calculated. The r.m.s. pivot reactions arc then obtained from equations (70), (71), (73) and (74).

MODIFICATION OF BUOYANCY CHAMBER

An alternative design of buoyancy chamber consisting of three cylinders in the form of a U-shape is shown in Fig. 4.

The reason for this modification arose from

practical considerations involved in the welded construction of the large diameter single cylinder buoy (D = 9.86m) whereas the U-shape chamber cylinder diameters are smaller

(D1=

6 m, D2 7 m).

The dynamic analysis is almost identical with that given for the single cylinder except that the following additional forces are included on the chamber, (1) vertical dragand inertial 4orces. on the longitudinal cylinders evaluated at cylinder midpoint (2) dynamic pressure on the ends of the longitudinal cylinders.

NUMERICAL DATA

Riser initial static tension 1250 t. Tanker mass (t)

Structural mass M = 240000 (full draft) 120000 (half draft) Added mass: surge (12000; 6000)

The heave and pitch added misses were obtained using Grim's data2 for [3,7= 1.0, Bldt = 2.5, 5.0 as a function of frequency. co. Pitch moments of inertia were obtained from / =ML2/12:

.1

Figure 4. U-shaped chamber

X = (L/2 +12)

Random dynamic analysis of an offshore single anchor leg storage system: R. S. Langley and C. L Kirk

Vessel dimensions (,n).

Length L = 310, beam B= 47, draft dt = 18.9, h g = 5.7

(full draft) and -3.75 (half draft), hi, = 13, (full draft),

21.45 (half draft), GM = 423 (full draft), 841 [(half draft).

Yoke mass (t)

Cylinder, Afy = 1170, Mya = 3131 U-chamber, My = 1170, Mya = 1159

Yoke dimensions (m)

Hydrodynamic and wave data

Buoyancy chamber drag and inertia coefficients CD= 1.5, = 2.0.

The JONSWAP wave spectra were calculated using the parameters in Table 1, given by Spidsoe and SigbjOrnsson." The Pierson-Moskowitz spectra were found by substi-tuding y = 1.0 in the JONSWAP formula.

Model test data

Model tests carried out at the NSMB on behalf of

SBM(DBV) Ltd, for the tanker at half draft were

per-formed using the ISSC wave spectrum with Hs= 10.4m, Tar, = 12.4 s. Current speed was lie = 2.4 knots and wind

speed l,

64 knots at 30° off the bows. Water depth =

85m.

Heave damping coefficient

The curves in Fig. 21 represent the mean values of the heave damping coefficient obtained using equation (21) and Grirn's data, and 'theoretical and experiMental results given in refs 12, 13 and 15. In general there was agreement

to within about 10% between all the results over the

frequency range given.

DISCUSSION OF RESULTS

Table 2 gives the calculated r.m.s. riser deflection and

tension and horizontal and vertical pivot reactions. The

subscripts p and s denote the port and starboard pivot

reactions measured in the wave tank tests and the values

in brackets denote the most probable peak model test

results.

A video recording of the model tests showed that in addition to surge, heave and pitch of the tanker, yaw and

roll motions also existed which were neglected in the

Table!. Parameters for .10ArSIVAP spectrum"

Table 2. R.m.s. values

-theoretical analysis. This gives one reason why the model test results for r.m.s. pivot reactions are about 30% higher than the theoretical predictions. On the other hand the riser experimental tension is 28% lower than the theoretical value. This can probably be explained on the basis that riser tension is largely unaffected by tanker roll and yaw motion but dependent ca the fluid inertia forces on the bupyant chamber where a value of Cm = 2 was used. Thus a less conservative value of Cm = 1.5 would give closer agree-ment.

Surge motion is primarily due to slow drift resonant motion and is governed by nonlinear damping which is dependent on CD. Here the major difficulty is to choose a CD appropriate to the Keulegan-Carpenter number and Reynold's number for the combined direct wave induced motion and the slow drift motion, including current. A CD value lower than the 1.5 used would give larger surge motion which would agree more closely with the model tests. However, the small difference of about 10% would appear to indicate that CD= 1.5 was a reasonable value.

For the pivot forces the model tests gave an average number of zero crossing No = 160 and an average ratio r = (highest peak value/r.m.s. value) = 3.4 which compares well with the value r = N/2 loge N0 = 3.18. In the case of

the r.m.s. surge motion or riser angle a , No = 126 and

V2 loge N0 = 3.11 which agrees closely with the experi-mental value of 3.02.

A major design criterion for the SALS is that the riser tension remains positive to avoid snatch loads. The model tests indicate that the peak value of 800 t would not lead to a slack riser. The theoretical value of 4 x 286 = 1144 t implies that the riser is very near to becoming slack, but this result based on a Cm of 2.0 and a ratio of 4 is con-sidered to be an absolute upper bound.

A more detailed study of the theoretical results is now made. Figures 5, 6 and 7 compare the p.s.d. of slow drift force due to Pinkster and Bowers. It can be seen that for Hs= 15 in, Tay = 14 s. there is considerable difference but for the lower sea state H, = 4 m, T0,.,= 7 s the difference is much less. The disparity can be explained by reference to Bowers' paper9 where in evaluating the coefficient d,vAr , in

his equation (11), it

is assumed in order to simplify the second order potential function that tanh (k,Ard)= 1. Thus, noting that tanh (1.5) = 0.905, this approximation requires k N d > 1.5 or co,2,d g > 1.5, i.e. for d = 160 m, R.m.s. value Pinkster Bowers Model

Percentage error Pinkster/Bowers a (deg) 3.86 4.0 4.3(13) r=3.02 -11.0

-7.0

RH_ (tonne) 137.4 138 219 (828) -37.4 737.2 F _{r = 3.78} RH, 137.4 138 132 (463)

r= 33

3.78 4.2 Rvp 82.72 82.74 138 (498) -40.0

-40.0

= 3.6 RV, 82.72 82.74 114 (311) r = 2.72 -27.37 -27_35 286 286 223 (800) 28.0 28.0 r = 3.587 Average % error 24.6 24.0 113(m)

T(s)

Peak frequency cap rad/s 2 5 0.0064 0.9047 4.13 4 7 0.0067 0.666 4.24 6 8 0.0088 0.597 4.8 6.5 8.5 0.01 0.597 5.02 7.5 9 0.0087 0.534 4.77 9 ' 10 0.0084 0.4838 4.71 11 11 0.0085 0.4398 4.73 13 12.6 0.0082 0.4021 4.65 15 14 0.0057 0.3456 3.78 Cylinder, 11= 14.63, 12= 45.37, = 23.7, /6= 13, Lb= 40. 13=11.6, 14 = 39.6, U-chamber, 11=-14.63, 12= 38,

15=23.7,16= 13,LL=25,/4=40.

13= 11.6; 14 =39.6,

15 12-5 10 70 x104 60 50 40 30 20 10 2-5 2 1-5 0-5

0-00 x 104 x104

SLOW DRIFT FORCE P.S.0 BOWERS PINKSTER (b) (a) 0-2 JON SWAP RS. D. 15m Tay = 14 sec

SLOW DRIFT FORCE PS.D. BOWERS

PINKSTER

JONSWAP P.3.0.

Hs. 4m

Toy = 7 sec

> 0.42. The IONSWAP spectrum in Fig. 5 shows that tere is considerable wave energy below w = 0.42 thus ;counting for the order of magnitude between Pinkster and owers. Figures 6 and 7 show that as and Ii decrease, le assumption tanh (kNd)= I, becomes more acceptable

3 6.2 2 z1"E 1 4 I

7Z

Z

00

I----i

3-6 cc 72 ' 5-4 1-8 x104 JON SWAP Hs = 15m Toy = 14 sec HORIZONTAL VERTICAL (PINK STER) JONSWAP Hs 15m Tay= 14 sec Vc = 0 CYLINDER BUOY -011 _-022 -033 -044 , -055 . "1/ rod/sec

Figure 9. Transfer function (squared) of horizontal pivot reaction due to slow drift forces

and the two forms of slow drift force spectra agree more closely.

The linearised T.F.s of direct pivot reactions for the JONSWAP spectrum, 14= 15 in are given in Fig. S. The maxima and minima in this figure correspond to those in the tanker pitch T.F.s, Fig. 13.

The transfer T.F.s of the slow drift horizontal and

vertical pivot reactions are given in Figs. 9 and 10 and the T.F.s for slow drifting surge motion are given in Fig. 11, for the half draft case. It is seen that the peak occurs in each figure at the natural surge frequency of the SALS with the tanker at half draft,GI,= 0.027.

Figure 15 shows that for the Pinkster slow drift force spectrum, the r.m.s. surge amplitude increases with H. up to 6.5 m and then decreases. This behaviour is due to the reflection coefficient

given by Remery and Hermans8

increasing almost linearly with frequency up to a peak value at co = 0.65 rat/is (Fig. 22), thereafter remaining constant. With increasing Hs, the convolution of Sn(6.)) S n(c..) + in equation (28a) increases, but the reflection coefficient

R(w) decreases. The result for the Bowers' slow drift

force expression in Fig. 15 shows a continual increase with Hs, due to lack of validity of the analysis for high values of Hs and Tay. The close agreement between Pinkster and

0-0 0-2 0-4 0-6 0-8 1-0

w rad/sec. Igure 5. P.s.d. of wave amplitude and slow drift forces

JONSWAP P.SD.

Hs = 7-5m

-rev = 9 sec

SLOW DRIFT FORCE

(a) BOWERS P.S.D. ; 7-5 _(a) (b) PINKSTER (b) 2-5 0 0-0 0-2 0-4 0-6 0-8 1-0 co rod/sec

'igure 6. P.s.d. of wave amplitude and slow drift forces

1-0

0-4 0-6 0-8

rod/sec

igure 7. P.s.d. of wave amplitude and slow drift forces

0-9 1-0

0

02 0-3 0-4 0-5 0-6 0-7 0-8

CIRCULAR FREQUENCY

Figure 8. Transfer functions (squared) of pivot reactions due to direct wave forces

Random dynamic analysis of an offshore single anchor leg storage system: R. S. Langley and C L Kirk Elz 3 21 18 15 12 9 cc 6 6 3 25 x10"3 JON SWAP Hs= 15m Tau 14 sec Vc = 0 0 -011 X102 Hs = 15 m Toy =14 sec Yc = 0 -022 -033 CYLINDER BUOY 044 -055 (ak rad/sec Figure 10. Transfer function (squared) of verticalpivot reaction due to slow drift forces

CYLINDER BUOY

(-4 Hs= 15m

EIE Toy =14 sec

1-0 Vc = 0 3 12 X -5-5- 0-5 wo 10 8 X 00 0 2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-0 u ₆ co rod/sec

Figure 12. Transfer function (squared) of surge due to direct wave forces

2

Bowers for a in Table 2, at Hs= 10.4 m is

fortuitous, -being near to the 14 value at which the curves in Fig. 15

cross.

The total r.m.s. pivot reactions for direct and slowdrift

forces in Fig. 16 increase up to 14= 13 m and then fall

226 x10-5 22 20 18 16 14 12 10 6 4 2 0 02 0-3 0-4 0.5 0-6 0-7 0-8 0-9 1-0 Li rod/sec

Figure 13. Transfer function (squared) of

pitch due to

direct wave moments

0-75

0

02 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-0

co rod/sec

Figure 14. Transfer function (squared) of heave due to direct wave forces

0 PINKSTER SLOW DRIFT SURGE JON SWAP 1/2 dt U-CHAMBE d=160m Hs = 15m Tau =14sec vc.=O CYLINDER BUOY Hs =i 15 m Tav 14 sec Vc = 0 CYLINDER BUOY 0 2 4 6 8 10 12 14 Hs (m)

Figure 15.. R.m.s. motions significant wave height

3 wo z 1 0 -011 -022 -033 -04.4 -055 wk rod/sec

Figure 11. Transfer function (squared) of surge due to slow drift forces

2-0 CYLINDER BUOY 0-50 C-,' 015 3 s'.71

0

300 270 z 240 210 2 4 6 8 10 12 14 16 Hs(m)

Figure 16. Total r.m.s. pivot reactions significant wave ta

_z

heightz 0.7

0

I--300

0

270 6-2240 1-CJ ccw 210 g 120 90 ci. 60 30 HORIZONTAL (BOWERS) (P INKSTER ) BOWERS. DIRECT PINKST ER, DIRECT

JON SWAP 1/2 dt Li-CHAMBER d= 160m VERTICAL (BOWERS AND . PINKSTER) JON SWAP 112 dt BOWERS SLOW DRIFT PINK STER SLOW DRIFT 0 2 4 6 8 10 Hs (m)

Figure 17. Components of r.m.s. horizontal pivot reaction significant wave height

slightly because of the movement of the wave spectrum peak to values of co below the peaks in Fig. 8.

Figure 17 shows that for 14< 6.5 in the slow drift

tanker inertia forces give pivot reactions that are about

12

half those due to direct wave forces. For the higher values of 113, however, the direct wave forces give reactions that are about three times as great as those due to slow drift.

Variation of pivot reaction with water depth is very

small as shown in Fig. 18 where the U-shaped chamber

produces lower values than the cylinder. Furthermore

RH and are greatest at half draft With RH> Rv but with the U-shaped chamber the differences are less pronounced.

For 113= 15 m, the r.m.s. surge amplitude in Fig. 19. increases with water depth d. By taking d = 200 m and

d= 500 m it

can be shown that the surge amplitude

increases as Nfc-i the reason for this increase being explained in terms of the surge natural frequency cox which is

pro-portional to Ita Furthermore since for 1-1,.= 15 m the

spectrum of slow drift Fig. 5b (Pinkster) is almost like that of white noise and the analysis of a single degree of freedom system with quadratic damping and white noise excitation can be referred to14from which it can be shown

0-13

o

as I" 0'5 dt/2 U-SHAPE dt 04 ce 0.3 1--0-2 13 12 11 16 10 9 8 7

/dt/2

RH t ,rdt CYLINDER

/dt /2

.11, ,edt 01 0 100 200 300 . 400 500 WATER DEPTH Cm) Figure 18. Total r.m.s. pivot reactions water depth

JONSWAF: Hs=15m T =14sec av U cyl:-0 JONSWAP Hs= 15m Toy =14 sec PINKSTER 100 200 300 400 500 WATER DEPTH (m) Figure 19. Total rims. surge amplitude water depth

90 60 .30 ' 0 ' I i I 1 U-CHAMBER d = 160m 1 1 1 I 10 x 450 180 a. J 150

0

and buoy for Li-chamber and half draft Ve= 1.23 mls

Random dynamic analysis of an offshore single anchor leg storage system: R. S. Langley and C L Kit*

Tahle 3. Comparison of equivalent linear damping ratios for vessel

16 14 12

2

130 cox

Cm) vessel buoy Wave spectrum (rad/s) (m) 15 0.13 0.08 JONSWAP 0.027 160 10.4 0.08 0.057 1SSC 0.0373 85 JONSWAP 1/2 dt U- CHAMBER d = 160m

STATIC OFFSET DUE TO RANDOM

SLOW DRIFT FORCES (PINKSTER)

ED. 29.

0 1 1 t 1

0 2 4 6 8 10 12 14 116

Hs (m) Figure 20. Static offset significant wave height

that the r.m.s. displacement is proportional to 1/6.), which

is in

turn proportional to

Considering next the

damping of the tanker, the slow drift resonant motion can

be inferred from equation (30) as being inversely

pro-portional to w, and hence propro-portional to Va.

The effect of current was studied for V, = 1.23. mis

and 14= 15 m and was found to increase r.m.s. pivot

forces by 4% and decrease the r.m.s. surge amplitudes to the same extent. Thus for large 113 the effect of current on the drag force is small but will be more significant at lower H.s. where K. is of the same order as u.

It is clear that the resonant slow drift motion depends on the total linear and nonlinear damping produced by the vessel and the buoyant chamber. Table 3 compares the equivalent linear damping of the vessel and chamber where it is observed that fleq increases with Hs but not in propor-tion to it. It can be seen that although the basic purpose of the bupyant chamber is to pretension the riser, it makes a significant contribution to the damping.

The slow drift analysis in this paper is based on a re-flection

coefficient R(w) obtained experimentally by

Remery and Hermansg for a rectangular barge in regular

head waves. It has been, however, pointed out by Rye

that for slow drift model tests in random waves the measured amplitudes were considerably greater than those obtained using Pinkster's

method with R(w)

measured using a single wave. They discovered that much closer agreement was achieved when R(w) was measured using a wave group of two equal amplitude waves of slightly different frequency simulating a narrow band random sea.

It

is therefore recommended that R(w) be determined

wherever possible in random sea states.

In estimating upper bounds for the most probable peak values using the r.m.s. values it is suggested that for pivot reactions an upper bound ratio of 4 be used for a 12 hour storm. It is important to ensure during the forward surge of the tanker that the buoyancy chamber does not impact the riser. In most situations wind and current forces will give the SALS sufficient offset to prevent such impact occurring. In very exceptional circumstances where the resultant forces due to tidal and wind induced current and wind are zero with the riser vertical, the maximum surge displacement for no impact can be obtained as follows. Assuming that the yoke does not rotate during forward motion it can be shown from geometrical considerations in Fig. 2 that

x,,,<(13+

d)Ean-1(11113)

sin-1

+ /`

DI2 3-0 01 02 0-3 04 OS 06 0-7 08 0-9 10 1-1 12 44 rad/sec

Figure 21. Heave damping coefficient frequency

at.10m (EV .4) 41 . 0 .20m(81,.2) 0 I I I 0 0-25 0.5 0-75 1-0 63 rad/ sec Figure 22. Reflection coefficient for rectangular barges

400 300 r/71 0 I-200 LI cc 0 ui X. CC 100 (3) (5)

/

HORIZONTAURM) -

/

///

-.HORIZONTAL.(JONSWAP)

01

2 / I I I 0 4 6 8 10 12 - 14 16 Hs (m)

Figure 23. Total r.m.d. horizontal pivot reaction and riser tension significant wave height

Substituting 11= 14.63, 13= 11.6, /s = 23.7, D = 9.86, d = 160 it is found that xmax = 64.3 m but for d = 85 m

this is reduced to xn,a, = 34.8 m.

An additional factor which will reduce the probability

of riser/buoy impact is the mean or static drift due to

random waves, Fig. 20, which is in the sternward direction of the tanker.

CONCLUSIONS

(1) A linearised spectral method has been presonted for the coupled (direct wave induced/slow drift) motions of a single anchor leg storage system in head seas and current. The results yield linearised transfer

functions for motions and yoke pivot

reactions and the r.m.s. variation of various para-meters with significant wave height.

(2) Morison's formula was used for fluid drag and

inertia forces on the buoyant chamber and the

direct wave forces on the tanker were obtained

using strip theory and the Froude-Krylov hypothesis. Slow drift forces spectra were calculated using the methods of Pinksteri and Bowers.9

It is considered that Pinkster's formulation of slow drift forces is more realistic than that due to Bowers in which errors arise because of certain simplifying assumptions.

(4) Model test results of the randomly varying motions

and pivot reactions were on average about 25%

higher than the theoretical values, partly due to

yaw and roll of the tanker model which was neglec-ted in the theory.

Both the theoretical and experimental results show that in the most severe states, say .115> 10 m, the system is reliable in mooring the tanker and that the resultant riser tension due to buoyancy and fluid inertia

forces on the chamber, remains positive

hence avoiding snatch loads. For d= 160 m the

maximum theoretical r.m.s.

surge amplitude of

14 in occurred for lis= 6.5 in corresponding to an extreme value surge of about 56 m. The r.m.s. pivot

reactions increased up

to Ifs= 13 m and the

hofizontal component was greatest at 300 t with the theoretical expected peak value being 1200 t. Pivot reactions were found to be almost constant with water depth.

ACKNOWLEDGEMENT

The authors wish to thank Mr Roger Dyer of Single Buoy Moorings (DBV) Ltd, for his assistance in suggesting the dynamic analysis of the SALS system and providing design data and model test results.

REFERENCES

1 Wu, S. C. and Tung, C. C. Random response of structures to wave and current forces, Sea Grant Pub. UNC-SG-72-22, September 1975. Dept of Civ. Eng. North Carolina State

University

2 Grim, O. Oscillations of buoyant twodimensional bodies -Calculation of the hydro dynamic forces, Hamburgische

Schiffbau-Versuchsanstalt, Report No. 1171, September

1959. English translation by Alice Winzer, DL Note No. 578, March 1960

3 Frank, J. E. W. Oscillation of cylinders in or below the free surface of deep fluids, Naval Ship and Development Centre,

Washington, D.C. Report 2375,1967

4 Wilson, B. NV. Progress on the study of ships moored in waves, NATO Advanced Study Institute on Analytical treatment of

problems in the berthing and mooring of ships, held at Walling-ford, UK, 7-16 May 1973

5 McCormick, M. E. Ocean Engineering Wave Mechanics, John

Wiley and Sons, pp. 123-127

6 Wichers, J. E. W. and Sluijs, M. F. The influence of waves on the low frequency hydrodynamic coefficients of moored

vessels, Proc. of the 1979 Offshore Technology Conf. Houston, Texas, paper OTC 3625

7 Pinkster, J. A. Low frequency phenomena associated with vessels moored at sea, Society of Petroleum Engineers Paper No.4837,1975

8 Remery, G. F. M. and Hermans, A. J. The slow drift oscilla-tions of a moored object in random seas, Proc. of the 1971

Offshore Technology Con!. Houston, Texas, Paper OTC 1500 9 Bowers, E. C. Long period oscillations of moored ships subject

to short wave seas, R1NA paper no. W4,1975

10 Pinkster, J. A. and van Oortmerssen, G. Computation of the

first and second order wave forces on oscillating bodies in

regular waves, Proc. 2nd Int. Conf. on Numerical Ship Hydro-dynamics, Berkeley, California 1977. Published by NSNIB.

11 Spidsoe, N. and SigbjOrnsson, R. On the reliability of standard

wave spectra in structural response analysis. -in! of Engineering

Structures, 1980,2 (2), 000

12 Tasai, F. On the damping force and added ass of ships heaving and pitching. Jnl Zosen Kiokai, 1979,105,000

13 Vughts, J. H. The Hydrodynamic Coefficients of Swaying, Heaving and Rolling Cylinders in a Free Surface. Lobratorium voor Scheepsbpuwkunde, Technische Hogeschool, Delft,

Report No. 194,1968

14 Kirk, C. L. Random Vibration with Nonlinear Damping. Technical Note, The Journal of the Royal Aeronautical

Society, 1973,563

15. Rye, H. and Nloshagen, S. On the slow drift oscillations of

moored structures Proc. of the 1975 Offshore Technology

Conf. Houston, Texas, Paper OTC 2366

APPENDIX A

Notatiou for cylindrical chamber

c1 = 4/(l1-1- /2), Co =hprol(13+ d),c2= 1 14K/i 42/2),

c4= 161(11 + 12)

C5 = pCDDLaf-Fr[aur exP(Q2/2) + NfiTrIfc erf(Q)] = ax

U- CHAMBER HALF DRAFT

d .160m RISER TENSIONTM)

Random dynamic analysis of an offshore single anchor leg storage system: R. S. Langley and C L. Kirk

C6 = (Ai + Ma) + (My + MYa), C7 = (C4MY C1MY0)

2

+ hp(My+ Mya)

C8 _{c4My+c1iilyic9=Bx+ax, c10=Lc1c5/2+ hpcs} C11

C14

C L(c2M 1,14 c1Myals c4My16+ c3Myal2)/2(11 4- 12)

= CI CS, c12= T0/(13 +4 en= pCDDLb.4F rovr=ay = (111y16+Myals)1(11+12) = By C2S, C23= LC1412 + h p (My + Mra)

= I + 4

[h(c4my + c 01"K) C221 2 h2p(MY + Mira)

=Lc23/2 + hp(c4My ciMya), Csi = + hpes

= Bob+ 2(hp ci cs c24) + CS C33 = LC25/2 + hp CI CS,

. :

C27

It can be shown by substituting the various coefficients that the above matrices are symmetrical.

APPENDIX C

The complex coefficients A(iG)) are as follows:

A /3 = (.02 C6 + iWC9 + C12s A 13 = CO2 C8 + PAJCIls A22 -= W2 C29 + j0-)C32 + C37, A31 = CO2 c14+ 1WC17, A 33 C.42 C26 + kJ C27 + C36 C23 = C16 +Mycz+MYaC3, C24 = CO,

C2S = Ci9 +C3 Ci3,C26 =C23 +M±.Mya

A 12= 422 C7 0 C8 A 21 = (.02 C28 iC,JC31 C34 23 = C42 C30-- iCOC33 C35 A 32 = CO2C22 ks)C24 + C20 h p(111_{y16+ Al Ya15)/ (11+} 12)

C7

_C8

_{C9 C10}

cii= csisgli+

12) C16 [MI =

[C6.

C28

C29

C30

[C] = C31

C32

Cis =L(C3Cis 12 Cs 1s)/2 + 12) Cslshpgli.+ 12)

=

2(c."

+ 171 pc cs)1 c2o= FB C4gMYN C19 C14 C22 C26 C17 C24 x201+ /2), c2/=-2c2oiL C12 C9 0 c22 =(FB12efyi.ogi.+12)=To,C22 = C1S

[K] = c34

[

C37

C35

L(Myc2+My,c3)12 0 C20 C36

c=hc12

'Cm = LC21/2, C36 = C21 LBpg, c37=(g114. GM) Lc2012 + 11 pcx Co = 7 h pC12 APPENDIX B

The mass, damping and stiffness matrices are as follows:

C27 C29

C30