vertical bending moments and axial forces in a floating marine hose-string

Vertical bending moments and axial forces in a floating marine hose-string Tom O'Donoghue and A. Roy Halliwell*

Department of Civil Engineering, Heriot-Watt University, Edinburgh, UK

The primary mechanisms by which vertical bending moments and axial forces are induced in a floating hose-string attached to a CALM buoy are quantitatively assessed by means of model tests and analytical study.

It is found that the dynamic axial forces on the hose-string are determined by the push-pull action exerted on the string by the surge response of the buoy in waves. If the surge characteristics of the buoy are known, dynamic axial forces along the hose-string may be estimated using the empirical model presented. The dynamic vertical bending moment on the hose at the buoy mostly depends on the heave response of the buoy in waves. Good estimates of the vertical bending at the buoy may be obtained using the analytical model presented.

Keywords: dynamic response, hose-string, single point mooring

When transporting oil from an offshore oil-field to a land-ba~d refinery, it is common to deploy a single point mooring (SPM) terminal system at the production location and/or, in the absence of adequate port facilities, at an offshore site close to the shore destination. Catenary anchor leg mooring (CALM) is the most common SPM and comprises a tanker moored to a buoy which is moored to the seabed by catenary chains, floating hose-strings for conveying oil between the tanker and the buoy and under-buoy flexible hoses for conveying oil between the buoy and a submarine pipeline.

The single point mooring dynamics, coupled with the prevailing sea conditions, subject the floating hose-string component to a complex system of motions and loads. The string is subjected to axial forces while it simultan- eously bends in a horizontal plane due to 'snaking' and in a vertical plane as it attempts to follow the profile of the waves.

The problem of quantifying the forces and moments induced in a hose-string has been considered by Brady et al. j and by Saito et al. 2, who report on measurements of bending, torsion and axial forces recorded at the hose -buoy manifolds of CALM buoys located in Nigeria and Tokyo Bay respectively. Brown 3 presents a mathematical model, based on a finite-difference scheme, for motion and bending in the vertical plane only, with no axial forces present.

The objective of research carried out at Heriot-Watt University (O'Donoghue 4) is to gain a quantitative understanding, based on hydraulic model tests and analytical study, of the mechanisms by which axial forces,

*Present address: Department of Civil Engineering. University of Technology, Lae, Papua New Guinea.

0141 0296/90/020124 - 10/$03.00 c~ 1990 Bunerworth & Co (Publishers) Ltd

124 Eng. Struct. 1990, Vol. 12, April

vertical bending moments and "snake'-induced horizontal bending moments are induced in a floating hose-string by the combined action of waves and buoy motions. The term 'snaking' was first used by Young et al. 5 in the context of floating hose-strings to describe the phenomenon whereby under certain conditions of waves and buoy motions the hose-string takes on a sine-wave shape close to the buoy, transverse to the direction of wave travel and in the plane of the water surface. The sine wave travels along the hose-string decaying in magnitude as it does so. Bree et ai. 6 have presented a detailed description of the snaking phenomenon. The present paper reports on the axial forces and vertical bending moments induced in the hose-string.

Set-up for hydraulic model tests

The prototype conditions were simplified in the set-up of the hydraulic model tests so that the main objectives could be realized: the waves were long-crested and travcllcd in the direction of hose-string alignment. The tests were conducted in a 9 m by 9 m wave basin facility with a working water depth of 0.9 m. The system was modcllcd to Froude scaling laws using a scale of 1:43.

Regular and irregular wave tests were conducted in which simultaneous measurements of the waves, buoy motions and bending moments or axial forces along the hose-string were recorded. Table I summarizes the properties of the irregular sea states used in the tests. The buoy motions were measured using a two-camera system focused on four light-emitting diodes attached to the top surface of the buoy. Forces and moments along the hose-string were measured using strain-gauged units fitted along the string. The waves were measured using a conductance wave probe.

Table 1

Spectrum type Peak spectral period (s) Significant wave height (m)

Bending moments and axial forces in marine hose-strings: T. O'Donoghue and A. R. Halliwell

Spectral properties of irregular seas used in model tests

SEA1 SEA2 SEA3

ISSC ISSC ISSC 606 7.25 14.28 2.1 1.7 1 4

Propcrties ~?[ttle hose-string mo,h'l

The important properties of the hose governing its loading and dynamic response in waves arc its bending stiffness, mass per unit length and displacement per unit length Iwhcrc displaccmcnt is the mass of water displaced when the ho,~e is totally submerged in water).

The bending still'ncss properties of proposed hose model samples wcrc studied using a test rig which simulates the so-called "bend test" to which prototype hose constructions arc subject. r i le hose sample is laid on supports which have PTFF. studs tittcd to their under-sides, as shown in Fi.t~m'c /. The table surface is also of I'TI"I- so that friction between the table and the hose supports is ntininlal. A known bending ntotncnt is applied by suspending weights as shown. The resulting curvature is recorded by photographing the dellcctcd shape, if no weights arc suspended at the sides, so that the compression inherent in the method of causing bending is not removed, thc test results may be compared with available bend lest results fi~r prototype floating hose. The inodcl eventually selected was ctmstructcd by I)unlop ()il and Marine specially fi~r the study and consists o r a low spccilic gravity rubber built onto a wire helix.

l h c higher bending stiffness usual in the lirst-off-thc- buoy hose in a prototype string was not modelled, rather the sanlc hose carcass was used for the tirst-off-the-buoy and mainline hose models. This does not affect the similarity between the dynanlic bchaviour of the model and prototype strings, though it does of course aft'oct the magnitude of bending moments in the lirst hose.

The target mass per unit length fi~r each hose was achieved by inserting a string with lead shot attached within the bore of the hose carcass. The target displacement per unit length was achieved by lilting Iloatation units along the length of the hose with sullicient space between the units to aw~id affecting the hose stiffness. The fioatation units were hand-bored from 12.5 mm thick sheets of closed-cell foam rubber, i"igur~" 2 shows the components of the hose model.

Figure I Test ng to study bending stiffness properties

II III Illllllll IIII Figure 2 Components of the hose model

Phosphor bronze units were used to model the steel flanges at tile end of each hose length and formed the basis of the bending moment and axial load transducers. The axial load transducers consist of a strain-gauged proving-ring littcd between arms onto which the hoses are attached; the essence of the bending moment transducers is a strain-gauged dccp plate which responds to bending in one plane only. All transducers were designed as full Wheatstonc bridge resistive transducers to maximize sensitivity and canccl differential tempera- ture ell'eels. The gauges were tixcd using a supcrglue and water-proofed by a coating of a mixture of two epoxy t o n i po ti n d s .

With the exception of the higher bending stiflhcss of tile lirst-off-the-buoy hose, the hose-string models the properties of a typical 24 in bore strmg. The prototype cquiwtlcnt of the model's bending stiffncss is 150 kNm z, outer radius of the mainline hoses is 0.5 m, outer radius of the hose al the buoy is 0.35 m and mass per unit length is 500 kg/m. The prototype equivalent of the total length of each hose in the string is 10.7 m.

Mothm characlerisli~'s of the (',-I L M houv

The ( 'AI ,M buoy model used in the model tests was constructed from PVC. It is divided into top and bottom sections with the top being capable of rotation about the bottom, moored section by virtue of a bldl-and-race bearing littcd between the sections. One end of the hosc-string manifi~ld is lixcd to the top section of the buoy and its other end intersects the still water level at an anglc of 15 to the horizontal. (It is common practice to design the manifl)ld with an angle of 15" to the horizontal so that tile lirst-off-the-buoy hose can absorb small surge movements of the buoy.)

For long-crested waves travclling in the direction of positive surge of the buoy, sway, roll and yaw motions of the buoy arc ncgligiblc. In addition, the pitch response of the tnoorcd buoy is low: the results of cross-spectral analysis between the measured waves and pitch response have ~hown that tile maxim'urn pitch response, at the pitch natural frequency, is of the order of 7 per metre wave amplitude.

The study has shown that the dynamic behaviour of the hose-string depends for the most part on the magnitude and period of the buoy's surgc and heave motions and on the phase difference between the heave

Eng. Struct. 1990, Vol. 12, April 125

Bending moments and axial forces in marine hose-strings: T O'Donoghue and A. R Halliwell

response and the waves. Accordingly, the frequency response functions established for the buoy from the cross-spectral analysis between the measured waves and measured heave and surge motions are presented as Figures 3 and 4, Only response function estimates at frequencies for which the coherence is greater than 0.8 and, therefore, for which the response is predominantly linear are presented.

The following is worth noting from Figures 3 and 4. The heave amplitude and out-of-phase values are a maximum at the heave natural frequency of about 0.12 Hz. As the frequency decreases from the heave natural frequency, the buoy tends to move in phase with the waves and with an amplitude somewhat greater than that of the waves. The heave response tends rapidly to a near-zero value for frequencies greater than the heave natural frequency. The surge amplitude per unit wave amplitude increases as the wave frequency decreases, indicative of the low surge natural frequency of the buoy.

3.01 i

i

2.0 ~-"

1 . 0 - -

0.0 0.0

Figure 4

- - - SEA 1 "~ . . . . SEA 2

k - - - - SeA3 , - ' x \ @ Re( ju lar

\ \ \ waves

'x,,', x

l I 1 1, l 0. I 0.2 0 .3

Frequency (Hz)

B u o y surge transfer function from model tests

MtKlel tt~t results

|i'rti('al h('tldinff ntomen/

Figure 5 presents the rms vertical bending nionient nlcasurcd along the hose-string in irregular waves with a peak spectral period of 6 s (SFAI) and the vertical bending moment amplitude per unit wave amplitude measured in regular waves with a period of 4 s. The vertical bending is a maximum at the hose buoy manifokt and dcc;iys rapidly Iv reach a near-constant value at hose positions greater than 2. (Position ! refers to the hose buoy nlanifold; a unit increase in position number corresponds to an increase in distance from the buoy of

q.0

3.0

2 .0 ¢-

l.O

0 . 0 ~

I00

0.0

-100 0-~

Figure 3

/

/ i

- - - - - SI-A t . . . . . SEA2

. . . . SEA3 R e g u l a r wav(25

@ L . . . . . . 1 1 - - 1 - - 1

_-~., , .,

~ . ' ~ ,~,, ( . , /,

I I '%' / / 1 1 I 0. I 0.2

Fret iuer, ~y ( t tz)

B u o y heave frequency response f rom model tests

0 .3

1 6 . 0

z 2~

8.0

t_

0.0

\ _ \

\

SEA1 i r r e~ juh l r waves

\

1 I I I I I I 3 4 5 6 7 t~ 9 10

16 .0

z

re

N \

8.0-- \

o.o I I 2 3

4 ; rut juI ,Jr WtlVt2S

~ ---e~ . ~ 1 / . 4 ~ " ~ .

I I I t I I 4 S 6 7 8 9

Pos i t ion on h o s e - s t r i n g

Figure 5 Vert ical bending moment along hose-string from model tests

one hose length.) It is appropriate to consider separately vertical bending at the buoy which depends on the buoy motions as well as the waves, and vertical bending at positions away from the buoy which depends on the waves only.

Bending moment at positions away from the huoy. For positions away from the buoy the wave-vertical bending moment mechanism may be considered as a single input/single output mechanism with the waves as input and the vertical bending moment as output. Estimates of the transfer function between the waves and vertical bending moment have been obtained from the cross- spectral analysis of the measured waves and bending moment at positions greater than 9 along the string. The envelope of results, covering the range of frequencies for which the coherence is greater than 0.8, is presented in Figure 6.

The scatter in the results at each wave frequency represents the difference between results obtained from

1 2 6 Eng. S t ruc t . 1 9 9 0 , Vo l . 12 , A p r i l

18.0

16.0


/ Envelope of

.~'x.L~..z~ model test results / /

12.0 --

Z .

8 . 0 - -

q . O - -

o . o I ~ I I I 0 .0 0 . 1 0 . 2 0 . 3

Frequency (Hz]

Figure 6 Transfer function for wave-vertical bending moment at positions away from "the buoy

the different locations along the string. The scatter is mainly attributed to two factors. Firstly, slight twisting of the string during a test can cause the plate of the bending transducer to move slightly out of the horizontal, resulting in an inaccurate measurement. Sccondly, the effect of snaking and axial forces varies along the hose-string.

The model test results presented in Figure 6 show a common underlying trend of increasing vertical bending moment with increasing wave frequency. This is as expected, since the string is subjcctcd to greater curvaturc in the steeper, higher frequency waves. However, the increase in bending as the wave frequency increases is limited by the maximum curvature the hose can attain. At a sufficiently high wave frequency, just above 0.3 Hz for the hose used in the model tests, the string will remain almost straight in the waves resulting in very low bending. At the low end of the frequency scale the vertical bending will tend to zero as the wave frequency and steepness tend to zero.

Bending moment at the hose-buoy manifold. Dynamic vertical bending of the hose-string at the buoy depends on the frequency-dependent response of the buoy to the waves, it is the sum of componcnts due to the heave, surge and pitch of the buoy. It is not possible, in tests of thc sort conducted, to separate the contributions of these motions to the total vertical bending moment at the buoy. (To establish the separate contributions, tests need to be conducted in which the buoy is constrained to move in heave only, surge only and pitch only.) Accordingly, cross-spectral analysis has been carricd out between the incident waves and the measured bending moment at the buoy, rather than between individual buoy responses and the bending.

The model test rcsults for the transfer function between the waves and the vertical bending at the buoy arc presented in Figure 7. There is general good agreement between individual sets of model test results.

The transfer function peaks at around 0.12 Hz, indicating a strong dependency on the buoy's heave response (Figure 3). This is not surprising since pitch motions are small and those surge motions which are

80.0

60.0 E Z J<

3= 4 0 . 0

Model test • results

"--"~ ] Analytical results r\

/ / o~

/ / ~ SEA3

/ / . . . . .

z0.0 - ~ . j

o.o I I I I I 0.0 0,1 0.2 0.3

Frequency (Hz)

Figure 7 Transfer function for wave-vertical bending moment at the hose~buoy manifold

large are also slow (Figure 4) and cause snaking and pushing of the hose-string. As the wave frequency decreases from 0.12 Hz the dynamic vertical bending decreases as the buoy tends to heave in phase and with the same amplitude as the waves. The heave response of the buoy in high frequency waves is small and the vertical bending at these frequencies is determined by the wave loading acting on the hose that is fixed to the almost stationary buoy.

Axial forces

The model test measurements showed that the variation of axial force at any position on the hose-string is in phase with the variation of axial force at every other position on the string. This means that the string is compressed and tensioned by the push-pull action exerted on it by the buoy's motions. Physical considerations lead to the conclusion that heave and pitch play little part in this push-pull action so that the mechanism for axial force may be considered as a single input/single output system with the buoy's surge as input.

Figure 8a presents the rms axial force divided by the rms surge of the buoy measured in irregular waves and Figure 8b the axial force amplitude divided by the buoy's surge amplitude measured in regular waves. The results reveal the main factors determining the axial force at a position on thc string: firstly, axial force increases as the wave and surge frequency increases because of the higher surge velocity and acceleration of the buoy; secondly, axial force generally increases with increasing proximity to the buoy or increasing distance from the free end of the hose-string. There is, however, a drop in axial force at position 2 which is primarily due to the non-zero hose-buoy angle.

Figur¢:~ 8a and 8b show that the axial force increases approximately linearly with distance from the free end, the rate of increase being dependent on the surge frequency. For the regular wave results the rate of increase is, approximately

= = f3 /a , L~ ( I )

where Fj/a, is the axial force amplitude at position 3 per


Bending moments and axial forces in marine hose-strings: T. O'Donoghue and A. R. HalliweU

g0 .0 ,

~ 20.0

.E

~- o. o

a

• SEA1 • SEA2

q l ' ~ • SEA3

-at: ~ ~ , ._ -~- - I

3 $ 7 9 11

frequency surge response in SEAl as well as the response at frequencies close to the wave frequencies.

The result presented in Figure 9 effectively acts as a surge-axial force transfer function normalized with respect to hose-string length. An empirical method for estimating the amplitude of the dynamic axial force F'~(k N) at a position on the string that is L~(m) from the free end is given as

[~ = aa, L~ (3)

80.0 • 6s • 8s

60o i ""' \ \ \" .,0s

20.0 ~ - ~ _ _ "--.,.

o.0 I I I I -"" ~"~ 3 5 7 9 11

b Position on hose-st r ing

Figure8 Axial forces along the hose-string measured in (a) irregular and (b) regular waves

0 o 8 ~

0.6

- j ~ 0 .~

, ' / O ° , /

0 . 2 - /

/

o.o I I I I 0.0 0.033 0.066 0.1 0.133

Buoy surge frequency (Hz)

Figure 9 Graph of a versus buoy surge frequency

• Regular wave ,." • Irregular wave y

I 0.166 0.2

where a is dependent on surge frequency as shown in Figure 9. Because of the small drop in axial force at position 2, for practical purposes the estimate of axial force for positions 1 and 2 should be taken as equal to that of position 3.

Analytical model f o r v e r t i c a l bending

To develop a simple analytical model for the vertical bending of a floating hose-string, the string's behaviour is simplified by considering the vertical bending as independent of the snake-induced horizontal bending and neglecting axial load effects. It is assumed that the Euler beam equation governs the bending of the hose-string. To facilitate the analytical solution, hydrodynamic damping forces which tend to reduce the motion of the string are omitted from the formulation. Hydrodynamic added mass forces are also omittcd.

Prohk, m definition

A hose-string of length L, outer diameter 2r, bending stiffness E1 and mass per unit length m, lies in still water with one end free and the other end attached to a CALM buoy such that its longitudinal axis intersects the still water at an angle 0o to the horizontal. The longitudinal axis of the string at positions far away from the buoy has a displacement do relative to the still water level. Reference x and y axes are set with x = 0 a t the hose-buoy manifold and x positive in the direction of the hose-string alignment; y, the vertical displacement axis, is positive vertically upwards and y = 0 at the still water level. Regular waves travel in the direction of hose-string alignment such that the water level at x along the string at time t is

metre surge amplitude, and L3 is the length of hose-string ahead of position 3. The corresponding value of ~, from the irregular wave tests is

= = (Fz),m,/(a,)rm , L3 (2)

where (F3)tm , is the rms axial force at position 3, and (a,),., is the rms surge of the buoy.

Figure 9 presents a graph of = versus surge frequency. For the irregular wave test results the surge frequency signature is the peak spectral frequency of the surge spectrum. Figure 9 shows that, for the range of frequencies covered by the test results, axial force is approximately linearly dependent on the buoy surge frequency. The deviation of the result at 0.14 Hz is due to poor surge frequency signature for the surge response in the high frequency sea SEAl (Table 1); there is a strong low

~l(x, t) = a,,cos(ot - kx) ( 4 )

where a,, = wave amplitude, o = wave circular frequency and k = wave number. (For linear, deep-water waves, k = ~z/g) .

The model test results have shown that the pitch response of the buoy is low and that vertical bending depends for the most part on the heave response of the buoy. For the purpose of the analytical model therefore, it is assumed the waves cause the buoy to heave only, with amplitude ah, frequency ¢ and a phase difference with the waves of Oh-

On the basis of the problem definition presented above, the analytical solution for vertical bending of the hose-string requires solving the equation of motion

E/ -5~x,~'Y + rn -~O'Y = p(x, t) (5)

128 Eng. Struct. 1990, Vol. 12, Apri l


subject to the boundary conditions substitution in equation (10) for mg and d(x, t) gives

y(O, t) = ahcos(ot + dp 0

0 Y (0, t) = t a n 0 t - 0o) 0x

(6) ~trZ pg p'(x, t)=

2r (7)

and a finite solution must exist at x equal to infinity. p(x, t) is the vertical force function per unit length.

Vertical force function

The vertical force on the hose-string is primarily due to the change in buoyancy force b(x, t) as waves travel along the string. If force is positive vertically upwards, then

p(x, t )=b (x , t ) - m g (8)

{,gx, t ) - y(x, t) -do} (13)

The linear force function p'(x, t) is a good approximation of p(x, t) when the hose is partially submerged. However, if the hose submerges fully and clears the water surface during its cycle of displacement the function p'(x, t) over-estimates the vertical force. A linear approximation p"(x, t) which under-estimates p(x, t) when the hose submerges fully and clears the water surface is illustrated by the chained line in Figure 10 and is given by

~r~pg p"(x, t )= 2r(1 + fl~ {t/(x, t ) - y(x, t ) - do} (14)

If d(x, t) is the instantaneous distance of the hose axis below the water level, then the force function is given by

In general a linear approximation of the vertical force function is given by

p(x, t) = 7trZpg -- mg for d(x, t) > r

d(x, t) p(x, t) = Pg ~rZ -- rZcos- t

r

+ d(x, t)~/r z - dZ(x, t) t - m g J

for --r <d(x, t) < r (9)

p(x, t)-- --mo for d ( x , t ) < - r

The solid curve in Figure I0 presents the force function as the hose clears the water surface by pr and is submerged by fir during its cycle of movement. Consider the linear approximation to the force function illustrated by the broken line in Figure !0. The approximation is given by

7trZ pg if(x, t) = ~ (d(x, t) + r) - m g (10)

2r

d o is the depth of the hose axis when the string floats in equilibrium in still water. Therefore

7rr'pg mg = 2r (do + r)

Since

(ll)

d(x, t )= ,7(x, t ) - y(x, t) (12)

,P

.nr 2 pg-mg

- - - v

I [....";.~" I ,~a g

Figure I0

=11-

[..--t q-l r r + B r

Vertical force versus depth of hose immersion

d

gx, t)= 7{,7(x, t)-y(x, t ) - do} (15)

where ? is the slope of the linear force function approximation.

Solution o f equation o f motion

! fp(x, t) in equation (5) is substituted by the approximation given in equation (15), the governing equation of motion can be arranged to read

0"y F I OZy . ,., Ox* "~ ~ Y + u y = Ga,,cos(ot - kx) - Gdo (16)

where c 2 = El /m and G = 7/El. Assuming a solution for vertical displacement of the

form

Y(x, t) = Ul(x)cosot + U2(x)sinot + Ua(x) (17)

the equation of motion is solved in the usual manner by substitution followed by application of the boundary conditions. The vertical bending moment is obtained using

M = - E1 O'y (18) c~x 2

and is

M(x, t) - E !

{2RZe- S=(ahCOS0h - - A a , , X s i n R x - cosRx)

- a , ,Ak 'coskx}cosot

+I2R 'e -a~{ - -ahs inc~hs inRx

+ (ahsin~h + Aawk "x )



where A is a dynamic amplification factor (equation (19)) and may be written in the form

4R'*= 7 az El c z A = I (23)

r 1+3-- 4S4 = E-"I

r/el A = k4 + 4R 4

and

(Positive bending moment implies tension on the topside of the hose with the convention adopted here.)

Static vertical bending moment

In still water a bending moment is induced on the hose-string by the method of fixing the string to the buoy. This static bending moment is given by the third term on the right-hand side of equation (19), i.e.,

(20)

f M,(x) = (El)tTte-SX~do(sinSx - cosSx)

I tan(n - Oo)cosSx~ m _

S )

For large values of x the values of Mr., tends to zero. This result is trivial, simply stating that at positions away from the buoy where the hose floats with its axis at the equilibrium depth do there is no imposed bending moment. Static vertical bending is a maximum at the hose-buoy manifold and is

M,(x = O) = (El)tTtdo + 2t(El)tTttan(n - 0o) (21)

Equation (21) shows that there are two separate contributions to the static vertical bending moment at the hose-buoy manifold. The first is due to the difference between the hose axis depth at the buoy (which in this case is equal to zero) and the hose axis depth when the hose lies in equilibrium in still water away from the buoy (i.e., do). This contribution lends linearly to zero as the difference in depth tends to zero. The second contribution is due to the angle of attachment of the hose to the manifold, which is zero when the hose is attached with its axis parallel to the still water surface (0 o = 0) and increases in magnitude as the angle increases.

Equation (20) may be used to show that stiffer hose-strings require a slightly longer length to reach equilibrium and greater stiffness results in greater bending moments. It may be shown that for most practical cases the hose-string reaches its equilibrium position within 20 m (two hose lengths) of the buoy.

Dynamic vertical bending moment away f rom the buoy

Consider values o fx greater than x0 where the hose at x o is sufficiently far from the buoy to be unaffected by conditions at the buoy. The dynamic vertical bending at these positions is

Ma(x > Xo, t) = E lk ZAa,,cos(at - kx) (22)

The dependence of the dynamic amplification factor on the wave frequency clearly illustrates the behaviour of the hose at positions away from the buoy. At low wave frequencies the value of A tends to unity as the hose moves in phase with, and with the same amplitude as, the waves. As the wave frequency increases the value of A increases, indicating a hose displacement greater than the wave amplitude. Dynamic amplification of the hose displacement is a maximum in waves with frequency Oo where

-(g'V a o - ~4cZ ) (24)

The value of A decreases rapidly as the wave frequency increases beyond no. At sufficiently high values of wave frequency, A has a near-zero value and the hose-string remains almost straight in the waves. For most practical cases A has a value of unity in waves with period greater than approximately 20 s; the maximum value of A is approximately !.1 and occurs in waves of between 5 s and 4 s. Accordingly, for most cases of practical application, the value of A may be approximated to unity. This further implies that the value of 7=~zrZpg/2r provides a good estimate in the applied load calculation.

The imposed vertical bending moment on the hose- string away from the buoy depends on the wave frequency, mainly because the wave length changes as the wave frequency changes rather than because the dynamic amplification varies with the wave frequency. At low wave frequencies the wave length is long and the imposed bending moment per unit wave amplitude is small. Indeed, for wave periods greater than about 10 s the dynamic vertical bending moment per unit wave amplitude is negligible (less than 0.5 kNm/m). As the wave frequency increases the wave length decreases and the imposed vertical bending moment increases rapidly (approximately in proportion to the cube of the wave frequency). The maximum dynamic vertical bending moment per unit wave amplitude occurs at a wave frequency that is slightly higher than a o, the frequency for maximum hose displacement. For hoses used in practice, the maximum dynamic vertical bending moment per unit wave amplitude at positions away from the buoy occurs in waves with period approximately equal to 3 s.

Dynamic vertical bending moment at the hose-buoy manifold

The dynamic vertical bending moment at the hose-buoy manifold follows from equation (19) and is

MD(x = 0, t) = -- El{{2RZ(a, , -- ahcos~h) -- awk2}costrt

z a,,k

130 Eng. Struct. 1990, Vol. 12, April


Equation (25) may be arranged to read moment must then lie between the two estimates. The

MD(x = O, t) = EIa,~2R 2 f a-~u cos(0-t + ~10 t aw

+ (4R" + k4)½cos(0-t + f~ + hi, J

where

2Rk Q = tan-t _ _ 2R z - k z

value of ~, the number of hose radii by which the hose at the buoy is submerged and clears the water surface, is given by

(26) os,,, l 0 r \ah/

(30)

//cannot be less than zero: if the value of the right-hand side of equation (30) is less than zero then l/must be set equal to zero, and the over.estimate and under-estimate of the force function is the same.

Noting from the previous section that the dynamic amplification factor

7/EI A 4R.t + k4 ~ 1.0 (27)

then

{(~, 0-2)~ a~ cos(0-t+,b~,) M d x = O, t) = El,,,, ~ 7 - ~- a-~

(28)

Equation (28) shows that the dynamic bending moment at the hose-buoy manifold is the sum of two separate contributions. The first results from the buoy's heave r~.~ponse, has amplitude Elah{(7/E! ) - (a'/c2)} t and is ~h out of phase with the wave at the buoy. The second results from the wave bending the hose-string in the region close to the buoy, is practically independent of the wave frequency, has amplitude a,(ElT) t and is (f~ + ~) out of phase with the wave. The ratio of the first contribution to the second is approximately equal to aJa, . The amplitude of the sum of these two dynamic components depends on the heave phase ~b~ and is

0-2

(29)

~4~x = 0) is approximately proportional to r t and (El) ~. Only in waves of long period, when the buoy heaves

in phase with the wave and with the same amplitude as the wave, does the hose at the buoy remain partially submerged; in such cases the linear force approximation with 7 = nrZpg/2r is a good approximation of the actual force function, in many cases, however, either as a result of the buoy's heave pulling thc hose clcar of the water and pushing it fully beneath the water or as a result of the buoy having a low heave response and the wave crest submerging the hose while the wave trough exposes the hose, the linear force function with 7 = 7rr~pg/2r over- estimates the forcing. In such cases a second linear force approximation with slope 7 = nr2pg/(I + ]I)2r (as described above under the heading 'Vertical force function') could be used to calculate an under-estimate of the bending moment. The actual value of the vertical bending

Comparison of analytical and model test results

Dynamic vertical bending moment away from the buoy

The analytical model prediction of the transfer function between the incident waves and the dynamic vertical bending moment at positions away from the buoy follows from equation (22) and may be applied to the hose-string used in the model tests ( E l = 150 kNm z, r=0 .5 m and m = 500 kg/m). The prediction and the model test results are presented for comparison in Figure 6. Clearly, although the underlying trend of increasing bending moment with increasing wave frequency is common to both results, there is poor agreement in terms of magnitude with the model test results being the greater at the lower wave frequencies and the analytical predictions exceeding the model test results by as much as a factor of two at high wave frequencies.

The differences between the analytical prediction and the model test results may be attributed to the physical simplifications that are fundamental to the formulation of the analytical model. In particular, displacement at a point on the hose-string is limited to one direction in the analytical model whereas, in practice, displacement occurs in three directions--vertically due to the wave, longitu- dinally due to the push-pull action of the buoy's surge and laterally due to hose-string snaking. It is noted that displacement in the latter two directions is small in higher frequency waves since the buoy's surge response is low in these waves. Also, the analytical model does not account for the hydrodynamic added mass and damping forces exerted on the hose as it movesthrough the water. It is expected that the damping force plays a significant role in reducing the vertical response of the string and that this effect becomes increasingly more significant as the wave frequency increases with the corresponding increase in the hose velocity. Finally, the nature of the hose-string construction, comprising rubber built onto a wire helix, may cause the bending stiffness to be dynamically dependent, leading to another possible source of diffcrence between thc model test results and the analytical prediction.

it is concluded that m waves likely to generate significant vertical bending of the hose-string at positions away from the buoy, conservative estimates of the bending moment are obtained using the analytical result presented here. It is noted that the results of a numerical model, such as that described by Brown 3, must be affected in the way described above if the model relies intrinsically on the same approximations used in the analytical model presented here.


Bending moments and axial forces in marine hose.strings:

Dynamic vertical bending moment at the hose-buoy manifold

The analytical prediction of the amplitude of the dynamic vertical bending moment at the hose-buoy manifold of the hose used in the model tests is obtained using the buoy heave response as described by Figure 3 as input to equation (29). (The first-off-the-buoy hose used in the model tests has r = 0.35 m.) Figure 7 presents the analytical results per unit wave amplitude (i.e., the transfer function) together with the model test results. Two analytical results are presented for each wave frequency: an upper estimate obtained using the linear force function with 7 = ~r2pg/2r and a lower estimate obtained using the linear force function with 7 = ~r2pg/(l +/~)2r, where /~ is obtained using equation (30). It is seen that there is general good agreement between the analytical predictions and the model test results in the sense that the model test results generally lie between the upper and lower analytical predictions.

The analytical prediction of the dynamic vertical bending moment at the buoy is not subject to the same level of inaccuracy as the prediction for positions away from the buoy. This is because the main factor determining the magnitude of the bending moment at the buoy--the buoy's heave response--is well defined from model test measurements. The cffccts of snaking, axial forces and hydrodynamic added mass and damping forces arc small compared with the effects of the buoy's heave response. Also, the dynamic bending moment at the buoy is less affected by dynamic changes in the bending stiffness than the dynamic bending moment away from the buoy, since the former depends approximately on the square root of E1 whereas the latter is linearly dependent on El.

Conclusions

The primary mechanisms by which vertical bending moments and axial forces arc induced in a floating hose-string attached to a CALM buoy have been quantitatively assessed by means of model tests and analytical study.

The dynamic axial forces oh the hose-string are determined by the push-pull action exerted on the string by the surge response of the buoy in waves, if the surge characteristics of the buoy are known then the dynamic axial forces along the hose-string may be estimated using the empirical model presented.

The complex interaction between vertical, longitudinal and lateral displacements of the hose-string as well as difficulties associated with incorporating the hydrodynamic added mass and damping forces in an analytical or numerical model makes it difficult to predict accurately the magnitudes of vertical bending moment at positions in the string away from the buoy. For wave frequencies for which the vertical bending moment away from the buoy is of practical concern, conservative estimates of the bending moments are obtained using the simple analytical result presented here.

The dynamic vertical bending moment on the hose at the buoy depends for the most part on the heave response of the buoy in waves. From the general good agreement between the model test results and the analytical predictions it is concluded that good estimates of the

T. O'Donoghue and A. R. Halliwell

vertical bending at the buoy may be obtained using the analytical model presented.

The results presented in this paper apply only to conditions in which the responses of the hose-string and the CALM buoy are linearly dependent on the wave height. The implication is that the results should only be applied in environments with waves of relatively low steepness. Where the wave conditions lead to buoy motions and hose displacements that are nonlinear, maximum bending moments and axial forces in the hose-string may be studied using statistical techniques applied to the results of model tests of long duration. Such a study is recommended as the logical progression from the work presented in this paper.

Notation

A ah a, aw b c 2

d do

E/ F F,

Frm~ f G g k L Ll M

Mn

M, m

P r t x

Y

P

7

0o P G

GO

dynamic amplification factor buoy heave amplitude buoy surge amplitude wave amplitude buoyancy force per unit hose length El/m distance of the hose axis below the water level distance of the hose axis below the still water level at positions away from the buoy hose bending stiffness axial force dynamic axial force amplitude at position i on string rms axial force linear frequency 7/El acceleration due to gravity wave number hose-string length length of string ahead of position i on the string vertical bending moment amplitude of dynamic vertical bending moment dynamic vertical bending moment rms vertical bending moment static vertical bending moment mass per unit length vertical force per unit length hose outer radius time distance along string vertical displacement multiplier for axial force number of hose radii by which hose is submerged and clears the water surface slope of linear force function approximation water surface elevation hose-buoy angle water density circular frequency wave circular frequency for which hose displacement at positions away from the buoy is maximum buoy heave phase difference w.r.t, wave

References Brady, l., Williams, S. and Golby, P. 'A study of the forces acting on hoses at a monobuoy due to environmental conditions', OTC 2136. 6th Ann. Offshore T¢ch. Conf., Houston, TX, 1974

132 Eng. Struct. 1990, VoL 12, April


2 Saito, H., Mochizuki, T., Fukai, T. and OkuL K. 'Actual measurements of external forces on marine hoses for SPM', 0TC3803. 12th Ann. Offshore Tech. Conf., Houston, TX, 1980

3 Brown, M. J. "Mathematical model of a marine hose-string at a buoy', 7th Polymodel Conf. Sunderland Polytechnic, UK, May 1984

4 O'Donoghue, T. "Dynamic behaviour of a surface hose attached to a CALM buoy', Ph.D. Thesis, Heriot-Watt University, Edinburgh,

UK, 1987 5 Young, R. A, Brogren" E. E. and Chakrabarti, S. K. 'lkhaviour of

loading hose models in laboratory waves and currents', OTC 3842. 12th Ann. Offshore Tech. Conf., Houston. TX, 1980

6 Bre¢, J., Halliwell, A. R. and O'Donoghue, T. "Snaking of floating marine oil hose attached to SPM buoy', ASCE, J. Engng Mech. 1989, 115, No. 2, 265--284


vertical bending moments and axial forces in a floating marine hose-string

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