two-dimensional dynamic analysis of a floating hose string

Two-dimensional dynamic analysis of a floating hose string

M. J. B R O W N * and L. E L L I O T T

Department o f Applied Mathematical Studies, University o f Leeds, Leeds LS2 9JT, UK

A mathematical model is presented for the dynamic analysis of marine hoses. The hose-string is subject to both wave and buoy action. The accuracy and stability of the numerical method is demonstrated by comparison with analytic solutions arising from simplified problems. The model can be used to predict the stresses and motion of a realistic hose-strong attached to a buoy which undergoes heaving and pitching motions of various sea states.

INTRODUCTION

At a single point mooring (SPM) installation the marine hose-string plays a vital role in the transfer of oil between the tanker and the onshore facilities. The mooring facilities consist of a floating buoy secured to the sea-bed by chains. The tanker's hawser and surface hose-string are connected to a turntable on top of this buoy so that the tanker can weathervane about it in response to the prevailing wind, wave and current forces, as illustrated in Fig. 1. Oil is con- veyed from the buoy to the sub-sea manifold by an under- buoy submarine hose and then to the onshore facilities by a steel pipeline.

The surface hose.string consists of a large number of reinforced rubber hoses which are joined together by steel flanges at each end. The rubber sections of these hoses are constructed of several layers of wire or polyester embedded rubber. Additional reinforcement is provided by placing helical wires between these layers. Except for tire initial section of the first hose of the hose string, the hoses are enclosed by floatation covers. These floatation covers, the thickness of which is sufficient that each hose is individually byoyant, are applied in such a way that the change in radius of the hose between the sections with and without covers occurs very rapidly. All these features are clearly illustrated in Figs. 2 and 3.

The construction of the buoy manifold and the design of the first hose in the hose string cause this hose to act as a damper by absorbing small horizontal movements of the buoy. This generates extra forces there which are counter- acted by providing the unfloated section of tlfis hose with more reinforcement than is provided for the rest of the hoses. This reinforcement, like the reinforcement applied to all of the hoses, decreases gradually from the flanges towards the middle o f the hose so that the considerable forces induced by the position, inclination and motion of the buoy manifold are transferred into the main body of the hose.

The buoy manifold is usually designed such that even in the presence of wave action the initial section of first hose is at least partially submerged. This is necessary to avoid the generation of large internal forces within this hose due to it

Accepted March 1987. Discussion closes March 1988. * Now at Ferranti Computer Systems, Western Road, Bracknell, UK.

having to support much of its own weight. Therefore the initial displacement and slope of the axis of the hose at the buoy manifold will be treated as known, as is illustrated in Fig. 4, where 0o represents the manifold gradient, ho its depth below the mean water level, S the distance measured

along the axis of the hose string and u(S') the displacement of the hose below the mean water level (MWL).

Knowledge of the internal forces and moments within the hoses, which are dependent upon the hose design, the hose string configuration and environmental conditions arising at the SPM installation, is essential as magnitudes which exceed certain critical values will quickly cause failure with all its economic and ecological consequences.

Dunlop, 1 Bridgestone 2 and Brown 3 laave all presented mathematical models for calculating the profile and the stresses in a hose-string attached to a static buoy. The model developed in the latter case was such that

(i) tile bending stiffness could be discontinuous at the interfaces between the steel flanges and the reinforced rubber sections of the hose,

(ii) the initial section of the first hose could have no fioata- lion covers whilst the radius could change linearly at the transition between the unfloated and floated sections of tile hose,

(iii) the length of each hose or flange could be varied, as could the number of hoses in the string.

Tltree different approximations for the external fluid force were used and the resulting differential equations were integrated by the Runge-Kutta method.

The governing equations for an oil carrying hose subject to time.dependent loads, together with a suggested iterative procedure for their integration, are given by Ghosh. 4 Un- fortunately, no work has been conducted on these equations to substantiate whether this complex set of equations can be solved numerically as they stand. To obtain the results presented here, certain assumptions have been made to this highly non-linear problem. Nevertheless, all the important physical features of the hoses have been retained so that the stresses produced by particular hose designs or hose-string configurations can be compared, and hence this model is used to assist in the design of hoses with longer life-spans.

The motion of the hose has been assumed to result from both the movement of the buoy and wave forces. This permits the dynamic effects arising from changing the

20 Applied Ocean Research, 1988, Vol. 10, No. 1 �9 Computational Mechanics Publications

Two-dimeJnional dynamic attalysis o f a floatfltg hose string: M. J. Brown and L. Elliott

Figure 1. Sitigle point mooring bistallation

Figure 2. Hose string

_steel flanges

�9 rubber section �9 _ /. I-Ioatation cover

<-' " - . L . . ; ~ . , . ) ~ ' , ~ J Fi-si hose ol I ~ t ~ string

Figure 3. First hose of hose string

~.~ ~ . . . _ _ M w J _ j ~ x

. L eo u (S)

/ S-.~ . Buoy maNcl6

FigT~re 4. Boundary conditions at the buoy

hose's parameters to be investigated independently of any stresses arising due to wave-action. The hose string has been modelled as in ref. 3, but a fixed mesh finite-difference scheme used to integrate the equations instead of the Runge-Kutta method. This choice was made since the solution to the static problem involved both exponentially growing and decaying terms, and so the Runge-Kutta method, like any other numerical shooting method, experi- enced difficulties with convergence once the domain was such that the number of hoses within the hose string exceeded five. For the dynamic problem it was felt that a larger length of hosing than this would be required to ensure that the free end had reached its equilibrium state. Another reason for using a different method of integration was file large amount of computing time that the Runge- Kutta method required to integrate each time-step; and results for the present problem being required over several periods of the buoy's motion.

Simplified problems, for which analytic solutions have been derived, are considered so that the accuracy of the numerical integration can be shown by comparing its results with the corresponding analytic solution. These simplified problems have been restricted to periodic wave motions whereas the numerical model presented can also handle transient problems.

The effects of the extra stiffness within the initial section of the first hose of the hose string, the motion of the buoy and the steepness of the wave on the stresses within the hose string are all investigated with a view to

Applied Ocean Research, 1988, Vok 10, No. 1 21

Two-dimet~ional dynamic attalysis o.f a floathlg hose string: 3/. J. Brown and L. Elliott

minimising these stresses and so lessen the probability of hose-failure.

EQUATIONS OF MOTION FOR 2-D MOTION

Tile various assumptions include:

(i) neglecting tile effect of shear force on the curvature so curvature is dependent on bending moment only,

(ii) treating the hose as solid, with the bending stiffness constant over any cross-section perpendicular to the axis of the hose,

(iii) assuming the densities of rubber and steel flange sections, Pr and py-respectively, to be uniform,

(iv) expressing the curvature of the axis of tile hose as a2u/OS 2, illustrated in Fig. 4.

It should be noted that the restrictions imposed by (iii) and (iv) are not essential and are illustrated only to reduce the computing time.

I f the motion of the hose is restricted to the dkection perpendicular to the MWL then the equations of motion, relating, acceleration, curvature, internal force V and moment M, are as given by Bishop and Johnson, s namely

Oht E1 - - = - -h i (1)

aS 2

OM = V (2)

0S

2u 0 V m - - - I- F (3)

0t 2 3S

where m is mass of hose plus contents per unit length,E = Young's modulus, I = principal moment of inertia and F = external force per unit length, comprising weight of hose and hydrostatic forces generated by immersion of hose and by wave buoyancy. With the present analysis attempting mainly to highlight file response of the hose-string to different wave actions, t he drag forces arising from oil flow within the pipe, together with those from any current action have been omitted. Introduction of such forces could, however, have been included with minimum inconvenience.

In attempting to minimise the severe stresses that the lust hose of the hose-string is subject to, it is designed such that its initial section possesses extra reinforcement but not floatation cover, as illustrated in Fig. 3. For simplicity, at the transition between the unfloated and floated sections of the hose tile external radius of the hose is assumed to change linearly, so corresponding to the hose configuration in Fig. 5, and expressed mathematically as:

i Su ~ I I Sf

c

Figure 5. hfftial hose with linear change o f external radhts at transition between unfloated and floated sections o f hose

I r u 0 <<-S <~Su

r = ru+ (rf--ru) (S - -Su) Su<<-S<<-S f (4) Sr-S.

vr f S f < S <~L*

where L* = length of a hose and the external radii of the hose with and without floatation covers (floated and unfloated sections), are denoted by rf and r u respectively.

In Ref. 3 the nonlinear external force F was such that even for the simplified model of constant radius throughout the configuration no analytic solution was possible, hence prohibiting verification of the numerical solutions. To overcome this difficulty two alternative approximations to F were assumed, both permitting analytic solutions to be developed when the problem was reduced to hoses com- prised of steel and rubber sections but of the same constant radius.

The first approximation was to replace tile hydrostatic contribution to the external force, which is almost linear with depth for lu I < r, by that acquired when the hose is on an elastic foundation. Hence, if the hose is partially submerged a restoring force is generated propositional to the distance from the equilibrium depth. I f the axis of the hose-string is deeper than the equilibrium depth an upward or negative force is applied to tile hose, and if the axis lies above the equilibrium depth a downward or positive force is applied. When tile hose is totally out of the water the force equals the weight of the hose and if it is completely submerged tltis force is the weight minus tile maximum hydrostatic force Bmax, where Bma x = 7rgpw r2, Pw being the density of water. Hence the force for the elastic foundation approximation, in the absence of any wave action is

Ir u < - - r

F = W + K ( r + u ) l u l < r (5)

W - -Bma x u > r

where K = foundation stiffness parameter = - - (Bmax/2r), obtainable from the gradient of the straight line in Fig. 6 which expresses tile variation of force with depth when the hydrostatic component is modelled as if the hose is on an elastic foundation for I u [ < r.

A similar expression for the force can be derived by approximating the hydrostatic contribution to F via a tanh function. This was originally developed as it was thought that the discontinuities in the derivatives of the previous approximation at lu I = r might affect the ability of the numerical method to converge. The expression chosen was

F = lr189 + tanh/~u) (6)

where ~ is a constant chosen to minimise any error according to tile least squares method. This approximation to tile external force is illustrated in Fig. 7.

Although the equilibrium depth tte for the elastic foundation approximation, given by

IV u e . . . . r (7)

K

differs from that when tile tanh approximation is applied, when it

11 = - - In (8) tte 2/3 Bma x -- ltP

22 Applied Ocean Research, 1988, Vol. 10, No. 1

Two-dhnemional dynamic analysis o f a floating hose string: M. J. Brown and L. Elliott

4 W - B MAX

External force F

I �9 U ue"-- . . , r

Figure 6. External force approximated as i f on elastic foundation

Extemal force F

. !

- r

�9 u

Figure Z A #ernative approximation for external force

the errors between these approximations and the numerical value are small. For all these alternatives of the force F t h e modifications to the actual results were minimal, yet the elastic foundation and tanh approximations allowed analytic solutions to be developed by which the accuracy of the numerical technique could be verified.

In the presence of wave action only the elastic foundation approximation to the external force, from which analytic solutions could be derived for restricted types of hose and wave action, was considered, and in this situation

I Ir u --77 < - - r

F = l g + K ( r + u - - r l ) l u - - n l < r (9)

l IV - - Bmax It -- ~ > r

where 7/(x, t) is the equation of the sea-surface. A third order Stokes nonlinear wave, as given by Kinsman, 6 is taken to represent the sea-surface so that

r/(x, t) = a cos(Kx - - cot) - - -~a2~: cos2(~x - - wt)

+ ~sa 3 g2 cos3(Kx _ cot) (10)

where w 2 = Kg(1 + a2KZ), a = linear wave amplitude, K --- r = wave number, co = wave frequency and x ~ S is horizontal co-ordinate, from buoy manifold in a direction parallel to MWL. The higher order corrections to the sinusoidal wave in equation (I0) produce a more realistic representation of the sea-surface by generating steep crests and shallow troughs.

In the present analysis no attempt has been made to incorporate a damping term into the external force due to the motion of the hose. Such a contribution could have been introduced via Morison's equation. Hence the contri-

butions to F are restricted to those from the weight and hose immersion in the presence ofwaves.

The problem is non-dimensionalised by introducing the following variables

u = ~ L S = sL ( 1 1 )

t : rt' E1 = REImi n f = K f L f

where L = length of the hose string, t' = time scale, E l r n l n =

minimum bending stiffness along the hose and h ~ = the foundation stiffness parameter in equation (5) when r = rf, that is for a hose with full floatation covers. Substitution of equation (i 1) into equations (1) to (3), together with the elimination of V and M produces

a4~- a2~- R as---- ~ + b at--- ~- = cf (12)

where ~" = dimensionless hose displacement, R = EI/Elmin,

L4m L4K1, b - - - and c =

t'2EImin Elmi n

[The dimensionless external force f , as mentioned earlier, is assumed in the present model to be void of any damping.]

BOUNDARY CONDITIONS

The boundary conditions at the buoy are dependent on the design of the manifold and on the motion of the buoy. Operating experience has resulted in most buoys being designed such that their surface manifolds are bisecting the mean water level and slope into the water at an angle of 0o = 15 ~ as illustrated in Fig. 8. This design ensures that the first hose is not subjected to any unusual bending stresses when the system is in its neutral position and so prevents premature hose failure, as described in Dunlopfl

The pitching, heaving and surging motions of the buoy, wlfich are illustrated in Fig. 9, will vary according to the steepness of the waves. Graham 8 observed that in steep seas the buoy under-went pitching with no noticeable heave. However, as the wavelength X= 2zr/K was increased the buoy was able to follow the waves with greater ease. The effect of this was that the buoy underwent mainly heave and its motion tended to lag behind that of the waves.

~:~fo]d ~ ' ~ __. MWL

Figure & Standard position of buoy and hose string at equilibrium bt a cahn sea

Heave \ .S ge _ ~ ~ _ _ - . . . . . " MWL.

Buc~y Buoy manifold \Hose string Figure 9. lllustrathtg heaving, surging atzd pitching motion o f buoy

Applied Ocean Research, 1988, Vol. 10, No. 1 23

Two-dimensional dynamic attalysis o f a floathtg hose string: M. J. Bro~wz and L. Elliott

Consequently, the heave and pitch of the buoy, as well as the phase difference = v between the buoy and the waves, are dependent on the length and amplitude of the waves. Since the depth of the buoy manifold ho and its inclination 0o, illustrated in Fig. 4, require a complete solution to the combined problem of buoy, hose-string and waves, involving the interactions between them, they will be functions of a, co, K, t, v, together with certain of the properties of the buoy and the hose string. Expressible mathematically as

ho = ho(a, co, K, t, vii. �9 .) 03)

Oo=Oo(a, oa, K,t, vp. ..)

where VH and v~, are respectively the differences in phase between the heaving and pitching motions of the buoy and that of the waves.

Since no such complete solution is available it is the intention in the present analysis to apply similar expressions at the buoy to those of the waves, but with phase and amplitude variations included.

As such the boundary conditions applied at the buoy, S = 0, for t > 0 , are

displacement: u = h o (14)

0u slope: - - = 0 o (15)

~S or in a non-dimensionless form

ho =-- (16) d!Sp!acement: ~" L

slope: - - = 0o (17) 0s

In a calm sea the free end of a hose-string of constant radius will be fiat at its equilibrium depth, hb as illustrated in Fig. i0, whereas for a hose-string subject to wave action the free end will try to follow the wave profile. Its ability to do so is dependent on the steepness of the waves and on the stiffness and inertia of the hoses. As above in the problem regarding information concerning the position and orientation of the buoy manifold a complete solution of the entire problem with the free end subject to both internal force V= 0 a n d internal moment M = 0 is required. Such information is unavilable and the present approach seeks only to investigate the response of the hose string to various sea states and buoy motions. Hence it is assumed that for a sufficiently long time scale the conditions to be satisfied at the free end of the hose-string, S - ' x = L , for t ~> 0, will correspond to the local sea state (hose following waves), apart from aphase .difference = ~ and a change in amplitude from a to A, so

displacement: u = h l - - r h ( x , t , eo, tq.4, v-') (18)

~u O01(x,t, co, K,.~,k_ ) (19) slope: 0S ~--x "

MWL Free end of hose strhg(~ [h 1

Figure lO. Position o f free end ofhose string at equilibrium in a cahn string

and in non-dimensional form, when s = 1,

displacement: ~- = {hl -- rh(. . . )}/L (20)

slope: a~" A arl..___x 0s ax ('" ") (21)

where rh(X, t, oa, g,.4, v-') = rl(x, t, oo, K,a =.4, v = ~ . The two initial conditions are that the hose-string starts

from rest at its equilibrium position. Iffo(S) is the solution to the static problem as given in ref. 3 then the conditions at t = 0 for 0 ~< S ~< L are

au u = f0(S ) and ~ = 0 (22)

0t

and in non-dimensional form

~=Io(sL)lL and - - = 0 (23) Jr

r ,q R L as4Ji,i+l

where

METHOD OF SOLUTION

The Runge-Kutta Merson method was used in ref. 3 to integrate the governing equations for a static hose. However, the relatively large amount of computing time taken by this method in integrating these equations was felt to restrict its suitability for the dynamic case since results would be required over several time periods, Consequently, a finite-difference scheme over a uniform mesh is used to solve equation (12).

Standard finite-difference notation is used in this section to describe the method of solution. If there are N internal space intervals, the space and time step lengths are given by

1 6 s = h = 6 r = k

( N + i ) ' (24)

respectively, and the value of the non-dimensional hose displacement ~" at the non-dimensional position s and non- dimensional time r is denoted by

~ (s, r) = ~ (ih,/k) = ~i,/ (25)

Initially, a central difference was used for the second order time derivative so that equation (12) became

r 'q + Pel R L a s , Ai,i b LarZJi,l = cl"t,I (26)

where

[ i ~ 1 = ~t,]+a --2~i,1. + ~'i ,]- ,+ 0(k2 ) (27) ar= Ji,i k 2

It was found that this scheme was unstable and so a backward difference approximation was used and equation (12) taken as

+bill ar=4a+, = c4,/+i (28)

[ ~ 1 ~,a,, -2ha + ru-, + O(k) (29) arzJt,i+1 k s


75vo-dhnemional dynamic analysis o f a floating hose string: M. J. Brown and L. Elliott

This scheme is stable. Nevertheless, the accuracy of the backward difference approximation is an order lower than the central difference scheme. This suggested that it may be possible to improve the accuracy of the results by taking some account of the values obtained at the ]th time-level. This was achieved by introducing a weighting constant, ct, in order to weight the time derivative towards the central difference approximation. Therefore, the finite-difference scheme applied was

4 bra2~.] + (1 - -a )R [~ ' ] + = cf/,/+ i L~s4Ji,#+l t,/ L i~l-2Jl,j+l

(30)

where the fourth order space derivative in terms of central differences

[ 34[] ~'i+ 2 ,1- 4~'i+1,# + 6~'t,#-- 4~'t- 1,/+ ~i-2,# + 0(h 2) ~$4-J i, l h 4

(31)

For a hose with continuous derivatives, the substitution of equations (29) and (31) into equations (30) gave the following finite-difference equation at fi,#+ 1

e~l- •j=i-- 4e[i- i,/+l + 7~'t,1+1 -- 4e~i+l,j+l +e[i+ 2,1+1 =di,#+ 1 (32)

where

(1--a)R did+ 1 = h4 {~'t+ 2,j - - 4~ ' i+1 , / - - 4~ ' i - 1,j + ~t--2,j)

{ 6 ( l - - a ) R 2b} - - h 4 "~ ~i,# - - b ~ t , # - I + cfi,i+l (33)

aR e h4 (34)

6~R b 7 = 7 + k "-~ (35)

w i t h / = 1,2 . . . . ,Nand r = (] + 1). At s --- 0, ~" and a~/~s are prescribed by equations (16)

and (17), hence ~'o.i+l is known and

~i,j+i - - ~-- I,]+I : 2h0o(r) (36)

Similarly at s = 1, ~" and a~]bs are given by equations (20) and (21), hence ~N+LI+a is known and

= 2h Orb (37) ~'N+ 2,1+1 - - ~'N,/+ 1 ~X

These equations (32), (36) and (37) can now be expressed using a square matrix in the form

-1 0 1 0 0

0 1 0 0 0

e --4e 7 --4e e 0

0 e --4e 7 --4 e 0

�9 �9 . �9 * �9

0 e 4e

0

7 --4e e

0 1 0

--1 0 1

In the case when the external force Fis replaced by the elastic foundation approximation stated in equation (9), the term J],i+ 1 in equation (33) is available from

W L~i,j+l--r[<--r

W Lfi,#+l = -~f-- r(r + L~i,# +1 --*l)]rf

.-~f-- 2r2/r[

IL~i,i+l--r/l<r

r<L[i,i+l--7?

(39)

r<L[i,l--~l

(40)

Clearly the choice of ]~,i+a depends on ~'t,#+l, which is the required solution; hence it is necessary to make this choice based on the known value of~" from the previous time step, that is ~i,], so

I W_ L~i,# --T1 <--r

W Lfi,#+ l = -~f--r(r+ L~t,i+a--~l)r f IL~i,i--rll<r

[ W -~I - 2 r~ l r s

As f/,l+l contains a term involving ~'i,#+l when IL[l, 1-7/I <r , this contribution is transferred to the left hand side of equation (38) so modifying 7 in equation (35) to

6aR b r 7 = h---~+k--S--c-~r IL~i,l--r~l<r (41)

and J],#+l to

W fl.,]+l--.~--K~f--r(r--1l)[r f IL~t,#--lTl<r (42)

The quantities T, e and the di,#+ 1 are now all known since they contain values obtained from the previous two time- levels r = k(] -- 1) and ~" = k].

The finite difference relationship, equation (38), applies to all the N + 4 mesh points except when a point or its neighbour represents an interface between the steel flange and reinforced rubber. The relationships at such an interface are derived in the following section.

For a hose with a continuous bending stiffness, the (N + 4) square matrix has a half band width of three. This fact is utilised in solving equation (38) so as to reduce computing time. The solution is obtained by decomposing this square matrix into upper and lower triangular matrices by

-~-1,1+1

[o,1+1

~l,/+l

~2,1+1

~N,]+I ~'N+I,J+I

_ ~ ' N + 2 , ] + 1

"--2h00(0 , r) r)

dl,l+l

d2,1+l

dN,]+l ~'(1, T)

2hath(L, r)/ax

(38)


Two-dflnemional dynamic attalysis o f a floating hose string: 31. J. Brown and L. Eiliott

Gaussian elimination and then using backward substitution to obtain

~'/+1 --- (~'-1,/+1, ~'o,/+1 . . . . . ~'i,] . . . . . ~'N+I,i+I, ~'N+2j+I)

for each step, the method of solution being given in Wilkin- son. 9 On the Leeds University's Amdahl 470 V7 computer, this finite difference method required approximately 0.I 5 s of computing time per time step for N = 2000.

result at the interface. Substitution of equations (1) and (2) into equations (48) and (49) produces

aSur aSus - - - R - - (50) as s aS s

E/min a6Ur a:Fr'~ - . a6us a:Fs~ as~ + ) = R , -Td + / (51)

MODELLING DISCONTINUITIES IN HOSE STIFFNESS

The displacement, gradient and vertical acceleration of the hose, bending moment and shearing force are all continuous variables. The requirement of continuity of the bending moment in equation (1) imposes the constraint that a hose with flanges, for which the bending stiffness is discontinuous at an interface between a steel flange and a reinforced rubber section (illustrated in Fig. 11), has O~u[OS ~ discontinuous at the interface. Hence the following relationships apply at the interface

u r = u s (43)

art r au s - ( 4 4 )

as as

aZUr a~Us - - - - R ( 4 5 ) aS ~ aS ~

where the subscripts r and s indicate that the point is on the rubber or steel side of the interface. Equation (2) implies that aM/aS is continuous, and if the external force F also satisfies this restriction then equation (3) implies the continuity of a V[OS. These in turn require that the further relationships

aaUr aaZts aS----- S - R (46)

aS ~

~411r = R a4tls (47) aS s aS 4

apply at the interface. When the continuity of a3tl/a(20S, and that of the

higher derivatives of the external force with respect to S, together with the discontinuity in 34u[at2 aS: are applied to equation (3) the following relations

a~Vr a~vs aS ~ as ~ (48)

--ff-~ ) = R ~ ' - ~ + - - ~ j (49)

the latter of these equations being expressible as

a 6 U . . 06Ur = R~ ~ + (R -- 1) 0~F (52)

In non-dimensional variables equations (43)-(47) and (50) and (52) reduce to

~-, = ~, (53)

O~ r a~- s - (54)

as as

- - = R n = 2 . . . . . 5 ( 5 5 ) aS n aS n '

a6~r_ 2 a6~s + c(R azf as 6 - R as--- ~- -- 1) Os 2 (56)

The central difference formula representing 04[/as 4 in equation (27), full details of which can be found in Fox, 1~ assumes that ~ can be expanded as a Taylor series

~-(s, 0 = ~'(so, r) + h (So, r) + 2~ as - - i (so, 0 + . . . (57)

where h = s --So. To derive the finite difference approximation for the fourth derivative at an interface to the accuracy previously used requires that seven points are introduced, namely at s i = s o + i h , where i = - - 3 , --2, --1, 0, 1, 2, 3 with each being expanded as far as the sixth derivative, resulting in

a~" s h 2 a2~'s , h 6 06~'s = h (So, 0 +T., (So, 0 + . . - * (So, 0

(58)

~" a~ r I? a2~r. , h ~ ~6~- -, = o-h (So, 0 +T., tSo, O * �9 .. -g?(So , 0

(59)

= ~'o -- It (So, ~') + - - R ~ (So, to) + . . . (60) 2! as

Steel f ' e r ~ s Floatation cover

~i++~ i+ i , T ~'~, ii+~+U:::Z~i,;i~,[Reinforced :ii+ii',i',',iiiiiii+i++++ii++iii',i:iii~,ii~.i :. - i+++iiiiiiii ill;: rubber

l;iii+i+!+++!i;!!iiiiiiiiiiiiiiiiiiii!:~i " 1 l:~i!++iii{i{iiiiiiiiiiii~seclJoN

htedaees/' between \ steel flanges and re , fo rced rt123e" seclJor'~

Figure 11. btterfaces between steel flanges and reinforced rubber sectiom

and similar expressions for ~'-a, ~--2, ~'2 and ~'a where ~'i = ~-(st, 7"). Equations (53) to (56) a.re used in the above expressions to link the derivatives at the interface. If

h i 02~-s di = - - -----r (so, O, i = 1 . . . . ,6 (61)

i! 0s z

then the Taylor series of tile displacement of these points about s = So can be written as

~" = ~'o + P d ( 6 2 )


Two-dimemional dynamic analysis o f a floating hose string: M. J. Brown and L. Elliott

where

-~'-a

l " = ~'o

dl ~'o [

d2 ~'o l

da '- , ~oI d = ~'o = (63)

d4 ~'o1

ds ~'ol

d6 ~'o i

and P is a 6 x 6 matrix containing theknown coefficients of the Taylor series.

To determine each di, i = I , . . . ,6 explicitly in terms of the surrounding values requires that equation (62) is multi- plied by p-1. Hence

d = P-~(~" --~-o) (64)

from which the derivatives across the interface can be obtained in terms of the displacement at the surrounding points. This in turn results in the row in equation (38), corresponding to the point s = So at which the discontinuity in bending stiffness occurs, having to be modified.

A similar approach is required to establish the derivatives of the displacement ~', with respect to s at s = s-x and s = sl in terms of the displacement at the surrounding points. This results in the rows in equation (38) corresponding to those of the immediate neighbours of the interface also requiring modification.

COMPARISON OF ANALYTICAL AND NUMERICAL RESULTS

All of the important properties of a hose-string can be investigated by using a numerical rather than an analytical method for solving the equations. However, analytical solutions can be derived for much simplified problems and whilst they are of little practical use for modelling the realistic hose, they can be used to test the accuracy of the numerical integration. Therefore two problems are considered in order to test the accuracy of the basic finite difference equations and the equations at the interface, and to ensure that the method is stable when water wave action is included in the model.

The accuracy of the equation (38), modified to include tile equations arising from the discontinuity in bending stiffness at tile ends of each of the flanges and discussed in tile previous section, is tested by considering a hose-string in a calm sea which is forced into motion by the movement of the buoy.

The external force F in equation (5) can be written in the form

F = K u * - - r - - U e < U - - U e < r - - u e (65)

where u* = u-- t ie , and from equation (7) u e = ( - - I V / A ) - - r , so in this restricted domain the force is proportional to the displacement of the hose from its equilibrium depth. I f this domain restriction is removed so that regardless of its position the hose has an external force acting on it expressible as

F = Ku* = -- I K lu* (66)

then the governing equation for this hypothetical situation is given by the linear partial differential equation

a4U * a2t/* - - = - - I K l u * (67) E I - ~ + m at a

which in non-dimensional form is

a4i. . 2 �9 R + b 3 ~" = _ [c I~'* (68)

as 4 aT 2

where u* = u -- it e = ~*L. Further, when the hose is subjected to the following

spatial boundary conditions

u(0, t) = ha + al sin 6olt (69)

u(L, t) = h I (70)

Oft ~--~ (0, t) = 0o (71)

art ~-~ (L, t) = 0 (72)

which in the non-dimensionless form are

ha-- Ue ~ * ( 0 , r ) =

L

h 1 - - Ue ~ * ( l , r ) -- - -

L a~-* - - ( 0 , 0 = 0 o as

ag-* - - ( I , r ) = o 8s

+ al sin r (73) L

- 0 (74)

(75)

(76)

a solution can be found in the form

~*(s, r) = So(s) + Sl(s) sin r

where the non-dimensional time scale t ' = I/col and

S~(S, r) = efliS(A i cos fliS + Bi sin (ais)

q- e-(ais(C i cosfli$ q- D i sin/3is) i = 0 , 1

with

fl0 = 4~--]~ andfl,-- 4 ~ - ~

(78)

The analytical solution for one unflanged hose can be obtained by separating the spatial boundary condition into their static and dynamic parts so that the constants of integration for So and $1 can be found. When there are n hoses in the string the difference in bending stiffness at the interface between the steel flanges and the reinforced rubber sections causes the hose-string to be divided into (2n+ 1) regions. The constants of integration are specified by the four boundary conditions, equations (73)-(76), and by the relationships linking the derivatives across the discontinuities, equations (53)~ These relationships hold for both So and S~ so that two square matrices, each of order (8n + 4), have to be inverted to obtain the constants of integration.

Since the analytical solution at time t = 0 is given by So(s), the numerical integration is initiated by solving the static problem. To calculate the first-time-level requires that the displacements at the two previous time-levels are known. This is fulfilled by calculating the analytical solution for

Applied Ocean Research, 1988, Vol. 10, No. I 27

75vo-dimensional dynamic analysis o f a floating hose string: ill. J. Brown and L. Elliott

Table 1. Comparison of analytical and numerical results for ten flanged hoses for varying vahtes o f a

Analytic solution Numeric solution time = 0.0 time = 0.0 a = 1.0 after period u (m) u (m)

Numerical solut ion after one period

a = 0 . 8 u(m)

,~= 0.6 u (m)

0.00 0.100000e+ 01 0.100000e + 01 0.100000e+ 01 0.i00000e + 01 0.100000e + 01 0.20 0.658907e-03 0.660521e + 03 0.201311e-03 0.283323e--03 0.366729e--03 0.40 -0.570283e--02 -0.570293e--02 -0 .558733e-02 -0.561074e--02 --0.563684e--02 0.60 0.725192e--03 0.725185e--03 0.713912e--03 0.720154e--03 0.728542e--03 0.80 --0.606502e--04 --0.606479e--04 --0.618711e--04 --0.649326e--04 --0.702068e--04 1.00 0.000000e+ 00 0.000000e + 00 0.000000e + 00 0.000000e + 00 0.000000e + 00

r = - - S t and usifig this in tile numerical integration. After the lust time-level has been calculated the numerical integration is completely independent o f the analytical solution, with the analytical solution, equation (78), is given in Table approximations given earlier.

A comparison of the numerical solution of equation (68) with the analytical solution, equation (78), is given in Table 1. Since the solution is periodic, the analytical results after each period coincide with the initial results. These results are compared with the numerical values at time t = 0 and after one period for various values of tile time derivative weighting function, a. The numerical solution at time t = 0 is independent ofct since this time level is obtained by solving the static equations and so provides a measure of the truncation errors arising solely from the spatial finite- difference approximation. As can be seen from Table I , improved results are obtained by weighting the finite- difference equations towards the j th time-level. It is found that the integration is unstable when a is about 0.5.

The accuracy and stability of the finite difference equations for a hose subjected to wave motion is tested by Considering the external force F to be that from equation (9) With the domain restriction removed, resulting in

F = K(U* -- rl) (79)

If, i n addition, the sea surface is taken to be rep(esented by the linear wave so that only the linear term in equation (10) is required, then the governing equation is the linear partial differential equation

~4u* a2u * E l - d - ~ + m - - ~ t 2 = - - I K I {u* -- a c o s ( K S - - w t ) ) (80)

which in non-dimensional form is

a4~-* 0~-* R + b = --Icl ~* --a* cos(K's--7-)) (81)

~s 4 ~r 2

where

a K * = K L , ":=cot and a * = - -

L

I f equation (80) is subjected to

u*(0, t) = A cos(--cot) (82)

u*(L, t) = A cos( rL - - cot) (83) au '~ a--'S (0, t) = --AK s i n ( - - c o t ) (84)

~)u*

(L, t) = --AK sin(rL ,-- cot) (85) ~S

together with the initial conditions

u*(S,O) = A Cos(KS) and

Where

a I K I A =

IKI+ KaEI - mco 2

then its analytical solution is given by

u*(S, 0 = A cos(KS--cot)

c)U* (S, 0) = ACO sin(KS) (86)

0t

(87)

(88)

In non-dimensional form the boundary conditions in equations (82) to (85) become

~'*(0, 7") = A* cos(--7") (89)

~'*(1,7") = A* cos(K* -- r) (90)

a--s- (0, 7") = - - A ' K * cos ( - - r ) (91)

ag-* --~" (1, 7") = - - A ' r * sin(n* - - 7") (92)

and in equation (86) ~'*

~'*(s,0) = A * cos(K's) and ~ (s ,0) = A * sin(K's) (93)

where

A a * l c l A* = - - = (94)

L I c I + K * 4 R - b

resulting in the analytical solution

~'*(s, 7") = A* cos(K*s - - 7") (95)

Since the numerical integration requires that the solution be known at the two previous time levels, the analytical solutions at times --67- and --267- are used in calculating the numerical solution at time 7- = 0. After this time-level the numerical integration is again completely independent o f the analytical solution.

The numerical and analytical solutions of the above problem at times r = 0 and after ten periods (period =4.0, 57- = 0.1(7r/2) or (6t = 0.I since 7" = cot), for ct = 1.0 and a = 0.9 are given in Table 2. For the hose forced into motion by the movement of the buoy, the accuracy o f the numerical integration is found to be improved by weighting the derivatives towards the j th time level, see Table i. How- ever, as can be seen in Table 2, this does not seem to be the case w h e n the behaviour of the hose-string is dependent upon wave-motion. Consequently, the results given in the

28 Applied Ocean Research, 1988, 1(ol. 10, No. 1

75vo-dimelnional dynamic analysis o f a floating hose string: M. Jr. Brown and L. Elliott

next section have been obtained with the full backward finite-difference approximation o f the time derivative, i.e. ~ = 1 .

I t was felt that the correlation between the numerical and analytical results presented in these tables, and the stabili ty of the method as illustrated in Table 2, justified the use o f the method when the non-linearities arising from the hose parameters and the buoyancy forces were included and the problem became such that no analytical solution was obtainable.

NUMERICAL RESULTS

For reasons that were described earlier, the first hose off the buoy is designed differently from the rest of the hoses in the string. One o f these differences is the extra reinforcement which is built into it in order that the large stresses generated at the buoy are distributed throughout the main body of the hose. I f it is assumed that the stiffness o f the hose has a negligible effect upon the motion o f the buoy then the effect of the extra reinforcement upon the stresses occurring in the rubber sections o f each o f the hoses in the string is shown in Table 3. The parameters used in obtaining these, and subsequent results, are given in the appendix. The extra stiffness can be seen to have a significant effect

Table 2. Comparison of analytical and numerical solutions for one unflanged hose sTtb]ected to wave motion for a = 1.0 and ~ = 0.9

Analytical Numerical solutions, u On) solution,

u(m) Time = 0.0 s Time = 40.0 s t = 0 ,

s t = 4 0 s a = l . 0 a = 0 . 9 a = l . 0 a = 0 . 9

0.0 0.00000 0.00000 0.00000 0.00000 0.00000 0.1 1.74813 1.76960 1.75904 1.74340 1.79823 0.2 1.11221 1.10712 1.10897 1.07977 1.07052 0.3 -1.08433 -1.08873 -1.11998 -1.11085 -1.33719 0.4 -1.80210 -1.79970 -1.84262 -1.86734 -2.08376 0.5 0.00880 0.01463 0.01440 0.04564 0.06212 0.6 1.80770 1.80900 1.82060 1.80900 1.89274 0.7 1.07008 1.06502 1.01632 1.03857 1.03280 0.8 -1.12688 -1.13119 -1.16363 -1.15450 -1.37951 0.9 -1.74263 -1.74069 -1.78013 -1.74155 -1.94747 1.0 0.01817 0.01817 0.01817 0.01817 0.01817

Table 4. The effect of tile manifold design upon the internal stresses in the hose.string

0o

ho = - -0 .10m ho= 0.00m ho= 0.10m

Hose Maximum Maximum Maximum off moment shear moment shear moment shear the

buoy kN m kN kN m kN kN m kN

0.0 ~ 1 81.9 19.04 74.2 16.05 66.6 2 17.1 4.56 16.0 4.29 15.7 3 12.3 1.75 12.3 1.74 12.2 4 12.1 1.84 12.1 1.84 12.1 5 11.7 1.71 11.7 1.71 11.7

7.5* 1 103.5 14.01 111.2 15.52 118.0 2 15.8 5.35 17.0 5.08 18.0 3 12.5 1.83 12.4 1.81 12.3 4 12.1 1.82 12.1 1.83 12.1 5 11.7 1.71 11.7 1.69 11.7

15.0 ~ 1 179.3 26A2 185.8 28.16 192.3 2 23.6 8.10 24.1 8.37 24.6 3 12.7 2.15 12.5 2.22 12.4 4 i2.2 1.90 12.2 1.90 12.1 5 11.7 1.67 11.7 1.67 11.6

13.05 4.15 1.72 1.84 1.71

17.22 4.89 1.79 1.85 1.69

31.16 8.50 2.32 1.90 1.67

upon the stresses generated in the first two hoses of f the buoy. Whilst the first hose has extra reinforcement to cope with these addit ional stresses, the second hose has none and as such is more prone to failure than the rest o f the hoses in the string. This is found to be the case in practice.

I t has just been shown that the stresses in the hose-string are dependent upon the stiffness of the first hose off the buoy. These stresses will also be dependent upon the design o f the manifold and the forces restoring the hose-string to the surface. I t can be seen from Table 4 that the stresses produced are dependent upon the displacement and the slope o f the manifold, with the opt imum manifold design being that which will impose the least curvature upon the hose. Therefore, for the most common manifold-hose combination (the manifold sloping into the water at 15 ~ from the MWL and the first half o f the hose having no floatation covers), the benefits obtained b y reducing the axial loads through damping out the effects of horizontal movements of the buoy, have to be weighed against the increased bending moments and shear forces arising from the larger curvature.

Table 3. The effect upon the internal stresses of reinforcing the first hose off the buoy

R = I R = 2

Maximum Maximum moment shear moment shear

Hose off the buoy kN m kN kN m kN

R = 5

Maximum moment shear

kNm kN

R = 1 0


kN m kN

1 48.6 11.50 87.0 16.22 2 17.6 2.82 19.8 3.14 3 I 1.4 1.82 1 i .6 1.75 4 12.2 1.82 12.1 1.84 5 11.7 1.69 11.7 1.68 6 11.9 1.74 11.9 1.74 7 11.8 1.72 11.8 1.72 8 11.7 1.77 11.7 1.77 9 12.7 1.90 12.7 1.90

10 12.0 2.27 12.0 2.27

185.8 28.16 321.6 43.33 24.1 8.37 42.4 17.08 12.5 2.22 20.4 4.05 12.2 1.90 11.8 1.70 11.6 1.67 11.6 1.68 11.9 1.74 11.9 1.74 11.8 1.72 11.8 1.72 11.7 1.77 11.7 1.77 12.7 1.90 12.7 1.90 12.0 2.27 12.0 2.27

where R = bending stiffness of rubber section of first hose bending stiffness of rubber of subsequent hoses


75vo.dhnensional d y n a m i c analysis o f a floathtg hose string: N. J. Brown and L. Elliott

Table 5. The effect upon the bzternal stresses o f varying the wavelength o f the waves

h = 2 0 m h = 3 0 m h = 4 0 m h = 5 0 m

Maximum Maximum Maximum Maximum moment shear moment shear moment shear moment shear

Hose off the buoy kN m kN kN m kN kN m kN kN m kN

1 306.9 75.82 250.7 43.23 208.2 32.46 173.2 26.90 2 57.8 22.69 33.2 14.60 30.0 9.85 19.8 7.44 3 34.8 12.22 35.7 7.57 20.1 3.73 9.2 1.44 4 29.1 10.76 29.0 6.32 18.3 3.27 8.3 1.16 5 31.9 10.81 28.3 6.66 18.2 3.21 8.9 1.04 6 31.0 10.78 31.6 6.62 18.1 3.16 8.1 0.98 7 31.8 10.81 28.3 6.12 20.0 3.28 8.1 0.99 8 32.9 10.90 28.4 6.90 18.7 3.38 8.0 0.90 9 42.9 13.33 32.1 7.11 18.9 3.25 8.3 1.39

10 88.3 22.84 36.6 7.99 18.6 4.70 8.3 1.72

- , t / - ~176 o, - ~ ~ ~'~._-~-~_~ ~.-~

-. TIME= I.

i d i - \ T,~qE: 2, 56

Figure i2(a}. Hose position

-I p, T I ME=,-O~. 43

'nL , . / / ' ~ N ~ _ _ . ~ . . . _ . . ~ ~ , , ~

_ _~, /-,TIME= i . 7 ~ I I \ \ \ ," \ ..17<

q , together with wave profile - - -

I , / /~ ME= 2 , 15

. , . f - / ~ \

=- . , . ,

for one wave period, wave length 20 m

] TIME= O. O0 Z ,ooJ ~

O" I i / i i ---..-,.t-..-.---1 i ~" 04 s / l o ~s 2o 2s 3o ~s 4o I - 1 0 0 i I St.)

1-2ooI /

1 TIME= 0, 45 ioo

~- o~ I s,j ,'~-"---~-, ~-~ iS 20 25 30 35 40 - I O 0 I / S(,,i

2 -2oo~

! Io~I TIME= ~ ~= "~ ioo~ TIME= 1 .7~0

I n [ , i tn~. . -~ l~ * ~ ' - - ' r r * i 0 I i ~ * ~ * 7 ~ - - " ' - i i ~ - i o o 5 IS 20 25 30 35 40 ~" ~ 5 i(F--15 20 25 30 35 40

S (.) | - I 0 0 1 S I~i i -200 i 400 "I_

Sire) _100 I

-2 oo - I

~ -300

Figure 12(b),

IS 20 25 30 35 40 S (,,)

Bending momen t for one wave period, wavelength 20 m

1 TIME= O. 85 I00

. 0 -1 ~ ' ~ r , ~ " " ' T - ~ (~is~r~- 15 20 25 30 3s '0 i - I O0 5 G*)

4 o o - [ . , o o ~

~ ' - ~ , ~ _ . ~ . ~ 1 5 ~ ' - ~ 0 2S 30 3S 40

51~,)

-I TIME= 5. 41 } I~176 ~" o4 s/to is 20 2s ~o 3s ~o Z-ioo I / st.I 2 - 2 o o ' ~ " ~-~ooi


Two-dhnetuional dynamic analysis o f a f ioathtg hose string: M. J. Brown and L. Ell iott

>-'Y ~o'~ , ~ TIME=, ~0"00, , ~- ,o'~ ~ , /'~,"--.L_.~ ,"'~-~,TIME= O, /J) 'Ou). ~O|On~j] | | | / ' , ,,//~TIME = | ~0 " ~}5<

3 :ia~ / s,., :~oq ~ 2,,, :2oq 2,.,

so~ TIME= 1.28 - - 20 > ,o~ , , . ,_ , < - - . . . . _ ~ ,

.~ ,oLI,r 20 22 ~o ~2 ~o o -20-J 2 Ira)

~ -SO

~oZF'-"N TIME= 1.70 20

>, 'od , ," .^01 S 10 1"5-"20 25 30 35 /,0 : ~ ~ 2,.,

~o] TIME= 2.56 -~ ~o_I

r . . . - I . 2 0 _ j S (.1 -~~

: ~ o ~

~o~ TIME= 2.98

-30"20~1 2 (ml I~ -~o

-5o

Figure 12(c). Shear stress f o r one wave period, wavelength 20 m

3oq TIME= 2. 13 -: 20

O 1 / * ' , \ l j r ~ - - t . . __ j ~ . i 00"~ S I0 IS 2 O"-"~ 2 30 ~S 40

_ 2 0 " 1 S tml o -'~0

- t

~_ -~o I i~ -50

)oO~ TIME= 3.41

,o~ s'~)~ P'ts' 2o~'-zSY~o s' ,o' "~oZL.__./ 2 ,.)

In Tables 3 and 4, the stresses generated by the motion of the buoy, the manifold construction and the design of the first hose can be seen to have a significant effect upon only the first two hoses in the string. The stresses on the rest of the hose-string arise mainly from the motion generated by followint the wave profile. The ability of the hose to do this will depend upon its stiffness and inertia and upon the steepness of the waves. A comparison of the stresses generated by waves of different wavelengths is given in Table 5. For the shortest wave considered, X = 20 m, the steepness of the crests is such that the hose-string can not follow the wave prof'de, resulting in the hose being completely submerged by a crest. The crest produced by the wave, X = 30 m, is just sufficient to completely immerse

the hose whilst, for the longer wavelengths, the hose-string is always able to follow the wave. Examples of the motion of a hose-string for one wave period and the stresses which it has to withstand, are given in Figs. 12 and 13 for wavelengths of 20 m and 30 m respectively (only the first four hoses off the buoy have been displayed). The fact that the hose-string can not follow the shorter waves is highlighted by the increased stresses in the hoses near the end of the string. These unrealistic stresses arise because the boundary condition applied at the end of the hose, namely that the hose follows the wave-profile at its equilibrium depth beneath the surface, is inappropriate for this type of wave.

In obtaining the results presented in Table 5, the steepness of the waves is altered by keeping their amplitude

Lsq TIME= O. O0

=-u .~ .~ , .~_ / i 2(.) \ " - - - - J l ~,-~. o - k - / ~b-I. 5..J v

-2. 0

,.s7 TIME= 0. Sh ,.o-1 . . ~ "

g o. l , , /

e,-'. o-I\ v / ~-'.s-I -2. 0

i ] o, ~ o o- , , , , . , : . o 1 ~ ,o .~Ao' 2~' " ~ " ~ ' ,o S - l \ \ " - ~ " / s~.) \ v _ -

o:t x . G . . / ~11~ -2.0

, . q TA~- 1.61 I. oak / / A "~

E o . s h~, K / / ~ , "~ , . . . . .,,v.J.___ , , / / , / , r . . "-.K ,

F,-I. o-I \ / ~ - , . s - I

-2.0

• o]s.I \ "% / / ~ \.~ o.% t ~_o.~ \ . _ . . . ~ . .

,.s ~ . ~ TIME=~-~8 Lo-I" - ~ /P" "%,

I~ O.S ~ \ . ~ , / / ~ \ ~0 0 1 * ~ * \ . N . n , d / .'1 I \ *

:_o:d F~-I. o-I i~i* I, 5 , . I

-2. 0

l.s] ,.--. TIME= 3,.~nL ~ ) . o ~ / ~3~ / ' y "

0 o o [ . _ ~ ~ ~ ~ s , ,0, =-u. ~.. l ",,,5 i , - r _ 1 / E - i . o J N / ~ - I . S J v

-2. 0

, .q __\ T IME: 3.7~/ BI.O

-2.0

1. s~ ~.. TIME= 4.29

~ 0 . 0 I " = ' 7

-2.0

Figure 13(a). Hose position , together with wave profile - - - f o r one wave period, wave length 30 m

Appl ied Ocean Research, 1988, Vol. 10, No. 1 31

Two-dhnensional dynamic atmlysis o f a floating hose string: M. J. Bro~wz and L. Elliott

-so .-so 4 / s, . , ~-,oo'[ / 5,., .-sou / 5,., I-,sq / s ~oo~/ ~ a o o ~ ~aoo.. I ~ - a s o ] / . ~ - a s o _ l . i - ~ s o . l ~-~00 ~-~oo ~-~oo

~ I ~.-T--I-li~l. 61 x: ~0 a5 30 3S t,O

5 Ira) ~ - I 0 0 I J

200

"~-300

I T LIbel 5 = soO~ s ~o ~s ~o as 30 3s ~o u" - "-I 5 Ira,

c-IO0 ] ]-,so / ~-~oo. I

~-300

_I ~ T IUE--- 2.~8_, - - 0 I / i i ~ I = OFS I0 15 20 25 30 3S 40

?oOoq s,~ I : .o : I Caoo.. I .~-~soj d~-300

" I ~ TIME= ~,2,-?- -- I I / ' I I I ~ I I = ~ ,5 ao ~s ~o ~s ~o

. - S O . . . ~ 5 tin, ~ - ~ o o l

i--o ~-~oo.. I -'-aSO ..I ~ -300

S l s>-~,o ,s ao ~ ' ~o ~s ,o ,5 ao ~s so ,s ,o : -so.l / s (,, "~ s ~.~ c-IO0 r" ~-~so~ ~i-tso~ 2oo ~aoo_]

~aso .~-aSO.l m~_300 ~-300

Figure 13(b]. Bending moment for one wave period, wave length 30 m

, , , . ~ , , 0 . 0 0 ,

t -15 . J / -~ / ~-2s-t / ~_~5 ~

-I vT~I ~ L , , , . , , , , ,

> sol s ~o ~ao as 30 ~s ~o �9 sot s ~o ~s._..~o" as ~o ~s ~o

~ : ~ ~ ~:~o~

s i , , , Ti. > ~+..s' , 0 ' ~ , 0 ~ ,0 .-_,o ] ~ 5(., t'_15. ~ ~ - a o - 4

~ : ~ ~

~.-, 0 - 4 S (~d

-,S

I,-as-I ~ Z t s - I

.4 ~ T IME: 2 . /68 - _ s ! , . , , ,._ , , i , , > bl s / , o ,s 2o 2-~----.~3s ~o �9 -T6"I / s(., -IS -Lod-" ~ - a s . - I

-30 L ~ s ' t

.4 #----.~T[ME= 3 . 2 ~ ~ 51 , , . - - - . T ~ E = 3. rs - S.~,, , v ~ J ~ - . ~ , ~ / , "~ 0 ~ ~ ~ 11 ~ , ~ , �9 > ~crJ s l y Is 2o 2s 3 o - " ' ~ ~o > sot s , o ~ ' f s ao 2s 3o �9 "-~o S'--._/ s~., " .'&o~ so.,

--~ ~ - 2 o - I ~-~s-t ~.-2s-I -~o ~_3s- 1 ~.3s- ~ -3o

Figure 13(c). Shear stress for one wave period, wavelength 30 m

.4 ...--rrIl~= 4 . 2 9 s./ , , ,/"% -, -'v-.._~ ,

> ~ o l s io . v ~ 2 o 2s 30 3s~ro- ~ - ; ~ 3 , / s t , . ,

-IS

~ - 2 s 4 -30

~.~s "~

constant whilst varying the wavelength. The same effect can be obtained by varying the wave amplitude whilst keeping the wavelength constant. Results from performing such an operation are given in Table 6.

In all of the problems considered so far, the motion of the buoy has followed that of the waves. This will not always be the case since, as described in the boundary condition section, the motion of the buoy is dependent upon the type of waves present. The effect on the stresses in tile hose of a phase difference between the motion of the buoy and the waves is shown in Table 7. It can be seen tliat

the stresses in the hose-string are a function of the phase difference. Since very large stresses can cause serious damage to the hoses even if they are applied for only a short period of time, motion of the buoy out of phase with the waves will affect the life expectancy of the hoses.

CONCLUSION

A mathematical model of a surface hose-string subjected to wave-motion has been displayed together with the accuracy


T w o - d i m e m i o n a l dynamic amdysis o f a f loat ing hose string: 3L J. B r o w n and L. E l l io t t

and stabili ty of the numerical method. Tile model can be used to predict the stresses and the motion o f a hose-string at tached to a buoy, which can undergo heaving and pitching motions, for varying sea states.

I t has been shown that the crests o f very steep waves will completely submerge the hose-string. As their steepness decreases, either through increasing their wavelength or by decreasing their amplitude, the abil i ty o f the hose-string to follow tile waves increases with a consequent drop in tile internal stresses. An experimental investigation of a model system by Graham s confirmed the above findings.

The effect of building extra reinforcement into the first hose off the buoy is to increase the stresses in the subsequent hoses. I f the second hose off the buoy is of the same design as the rest of the hose-string, as normally is the case, then its life span is l ikely to be shorter than that o f the rest of the hose-string.

Phase difference between the mot ion of the buoy and the waves can lead to very high stresses being generated in the hoses nearest the buoy. This could have a serious effect upon their life expectancy.

Table 6. The effect upon the internal stresses o f varying the amplitude o f the waves

a = 0.75 m a = 1.00 m a = 1.25 m

Hose Maximum Maximum Maximum off moment shear moment shear moment shear the

buoy kN m kN kN m kN kN m kN

1 174.7 27.22 185.8 28.16 197.1 30.33 2 19.7 7.66 24.1 8.34 27.8 8.83 3 9.4 1.61 12.5 2.22 15.9 3.11 4 8.7 1.22 12.2 1.90 17.0 2.61 5 8.1 1.13 11.7 1.67 16.0 2.63 6 8.2 1.13 11.9 1.74 16.1 2.53 7 8.2 1.12 11.8 1.72 16.0 2.50 8 8.1 1.18 11.7 1.78 15.8 2.43 9 9.2 1.22 12.7 1.90 17.5 3.06

10 8.3 1.59 12.0 2.27 16.3 3.16

I t has also been shown that the benefits obtained by choosing a manifold design which reduces the axial load on the hose-string, have to be balanced again the addit ional bending moments and shear forces generated.

ACKNOWLEDGEMENTS

Both authors would like to thank Dr M. I. G. Bloor and Professor D. B. Ingham at Leeds University and Mr I. Brady and Dr P. Cox at Dunlop for all their help, information and insight w h i c h they provided through many useful discus- sions. Dr M. J. Brown also gratefully acknowledges the financial support provided by the Science and Engineering Council and Dunlop Ltd , Oil and Marine Division, through a CASE research studentship.

APPENDIX

Unless it has been otherwise indicated, either in the text or in tile tables, the following values have been used in obtaining tile presented results.

Ten hoses in tile string, each o f length i0 m Length o f steel flange = 1.000 m Radius without f loatation covers, ru = 0.365 m Radius with floatation covers, r / -= 0.469 m Floata t ion covers first applied, S u = 4.750 m Floata t ion covers fully applied, Sf = 5.250 m Manifold depth, h0 = 0.000 m Equilibrium depth, h~ = 0.236 m Weight in air per unit length, W = 5185 N m -I Density o f sea water, p w = 1024 k gm -3 Bending stiffness o f a steel flange = 40000 k N m 2 Bending stiffness o f reinforced first hose = 2000 k N m 2 Bending stiffness of rubber o f other hoses = 400 k N m 2 Wavelength, X = 50.00 m Wave period, 2rt[w = 5.61 s Wave ampli tude, a = 1.00 m Ampli tude o f buoy pitching mot ion = 1.00 m Ampli tude o f buoy heaving motion = 1.00 m Ampli tude of hose free end's mot ion = 1.00 m

Table 7. The effect upon the internal stresses o f out o f phase motion o f the buoy relative to the waves

v H = 0 ~ v H = 90* v H = 180 ~

Maximum Maximum Maximum moment shear moment shear moment shear

Hose off Up the buoy kN m kN kN m kN kN m kN

v H = 270 ~


kN m kN

0 ~ 1 185.8 28.16 167.7 21.99 144.8 32.86 157.8 22.12 2 24,1 8.37 26.7 8.33 27.1 10.04 26.4 9.10 3 12.5 2.22 12.6 2.17 15.7 2.61 12.4 2.58 4 12.2 1.90 12.2 2.00 12.2 1.86 12.2 1.84

90 ~ 1 161.4 22.00 137.9 18.18 168.8 49.06 174.0 27.25 2 18.8 8.80 19.2 8.00 32.1 1 0 . 0 2 27.8 10.62 3 i l . 2 3.05 15.4 2.98 16.6 4.06 18.8 4.36 4 12.2 1.85 12.1 1.96 12.3 1.78 - 12.3 1.85

180 ~ 1 201.0 28.16 200.1 26.73 160.3 32.86 183.7 31.13 2 26.4 12.65 16.7 9.35 29.6 10.86 27.7 11.89 3 19.9 4.17 14.4 2.37 17.3 4.02 23.6 4.71 4 12.3 1.87 12.2 1.85 12.4 1.80 i2.4 1.88

270 ~ 1 255.4 44.36 211.5 33.19 179.7 23.92 189.2 23.73 2 24.5 12A2 22.0 9.36 28.3 10.99 27.2 12.03 3 18.0 3.17 14.8 2.36 14.4 2.86 16.9 3.98 4 12.0 1.74 12.2 1.85 12.2 1.79 12.3 1.81

App l i ed Ocean Research, 1988, VoL 10, No. 1 33

Two-dflnetuional dynamic analysis o f a floating hose string: M. J. Brown and L. Elliott

Manifold slope, 0o = 15.00 ~ Heave phase difference, v n = 0.00 ~ Pitch phase difference, vp = 0.00 ~ Space mesh interval, ~s = 0.001 Time mesh interval, 5r = 0.I (zr/2)

REFERENCES

1 Dunlop, S. B. M. Floating hose configuration, Central Research and Development Division, Report No. PR 3408, 1976

2 Bridgestone, Study of causes of kinking in floating hoses at PetrobraslTefran terminal, Bridgestone Tyre Company of Japan, Report No. 6 YMT-O01, 1976

3 Brown, M. J.Ph.D. Thesis, Leeds, England, 1984 4 Ghosh, P. K. A numerical method for nonlinear transcent

analysis of oil carrying offshore hose, SRCMarine Technology Program, 1981

5 Bishop, R. E. D. and Johnson, D. C. The Mechanics of Vibra- tion, Cambridge University Press, p. 282, 1960

6 Kinsman, B. Wind leaves, Their Generation and Propagation on the Ocean Surface, Prentice-Hall, p. 295, 1965

7 Dunlop, Offshore Hose Manual, Dunlop Oil and Marine Divi- sion, England

8 Graham, tl. Conoco Tetney, installation of Safgard hose system, Dunlop Oil and Marine Division, Report No. V288/ MG/5, 1984

9 Wilkinson, J. H. and Reinsch, C. Handbook for Automatic Computation, Vol. II, Linear Algebra, Springer-Vedag, 1971

10 Fox, L. Two Point Boundary Vahte Problems, Oxford Univer- sity Press, 1957

LIST OF SYMBOLS

tl a* = tl/L

at

A A* =A/L A b nmax c e E E1 s

F h = ~ s ho hi 1 K = --Bmax/2r

k=(57" L L* !/'/

linear wave amplitude non-dimensional linear wave amplitude amplitude of forced motion of buoy in a calm sea alKI/(IKI + K 4 E I - mw 2) a*lcl/(Icl + K*4R--b) general amplitude of free end of hose L4m/t'2EImin zr gp w r2 L4K/EImIn otR /h 4 Young's modulus bending stiffness non-aimensional external force per unit length external force per unit length non-dimensional space step length depth of buoy manifold below MWL equilibrium depth of free end of hose principal moment inertia foundation stiffness parameter foundation stiffness parameter when r = rf, that is for hose with full floatation covers non-dimensional time step length length of hose string length of a hose mass of hose plus contents per unit length

M r rr

ru

R s = S / L

S sf

Su t t' u

Ue tt*

Ur us V Ir x-~-S

ot

~o

7 = u/I,

771

Oo K K* = KL X vn

pl Pr Pw r = t't

092= ~g(l +a2K 2) O31

internal moment external radius of hose external radius of hose with floatation covers external radius of hose without floatation covers EI/Elmin non-dimensional distance along hose string from buoy distance along hose string from buoy distance before complete floatation covers applied distance without any floatation covers time time scale displacement of hose below MWL equilibrium depth of hose u -- e e displacement on rubber side of interface displacement on steel side of interface internal force weight of hose per unit length in air horizontal co-ordinate, from buoy manifold in a direction parallel to MWL weighting constant constant, chosen to minimise error according to least squares ~/Icl/4R ~lc l - -b /4R 6od~/h 4 + b/k 2

non-dimensional displacement below MWL (U--Ue)/L equation of sea surface gradient of free end ofh0se = gradient of sea surface, but with amplitude A and phase ff manifold gradient wave number non-dimensional wave number wave length phase difference at the buoy between the waves and the heaving motion of the buoy phase difference at the buoy between the waves and tile pitching motion of the buoy phase difference at the free end of the hose string between the waves and the motion of the hose density of steel section density of rubber section density of water non-dimensional time (= cot for analytic solution) sea surface frequency frequency of forced motion of buoy in a calm sea


two-dimensional dynamic analysis of a floating hose string

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