Ocean Engng, Vol. 17, No 3, pp. 235-261, 1990. 0029-8018/90 $3.00 + .00 Printed in Great Britain. Pergamon Press plc

MODEL STUDIES OF THE MOTION RESPONSE OF A DAMAGED FOUR COLUMN SEMISUBMERSIBLE IN

REGULAR AND IRREGULAR WAVES

B. M. STONE, M. A. SULLIVAN, V. M. ARUNACHALAM and D. B. MUGGERIDGE Ocean Engineering Research Centre, Memorial University of Newfoundland, St John's, Newfoundland

A1B 3X5, Canada

Abstract--A brief review of the stability requirements of semisubmersibles is presented. Results of the dynamic response of a damaged, twin pontoon, four column semisubmersible at a scale of 1:100 are discussed. The tests involved both regular and irregular waves with the model oriented in head, beam and quartering directions for both intact as well as damage conditions. Four damage conditions representing partial damage to one column were simulated: two in windward (positive) direction and two in leeward (negative). The "light damage" condition represented about 9% flooding of the damaged column, while for the "moderate damage" the flooding was equivalent to about 18%. For moderate damage conditions, regular wave studies showed that the motions are essentially nonlinear, although for light damage conditions this cannot be said with certainty. Model tests showed certain asymmetry in the motions of the damaged semisuhmersible with respect to the position of the damaged column. Moderate damage conditions seemed to produce significant subharmonic response of the vessel in a frequency range which is twice the natural frequency of the vessel in heave. These observations were confirmed from the results of irregular wave studies. Irregular wave studies showed that the quartering sea pitch and roll motions in windward damage conditions are as significant as those in the leeward conditions which was not the case for regular wave studies. The energy due to the motion of the semisubmersible was concentrated in the frequency of 0.7-1.3 Hz, which corresponds to the energetic range of the normal sea state. The natural frequencies of the vessel in damaged condition in pitch, roll and heave are higher than the corresponding frequencies in the intact condition of the vessel. The natural frequency in heave, for both intact and damage conditions, is higher than either those of pitch and roll in similar conditions. These natural frequencies in pitch and roll begin to approach that of the natural frequency in heave as the damage condition increases. This is true irrespective of the position of the damaged column or the sea state. The value of the natural frequency in heave itself increased much more slowly with increase in damage condition. It was inferred that the nonlinear wave pressure term played only a minor role in the asymmetry of motions of the vessel, while the mooring characteristics had a more dominant influence.

1. INTRODUCTION

SEMISUB~ERSIaLES are becoming attractive as production units for developing marginal and deep water oil fields. The functions and requirements of these production units are more demanding than the drilling units. Hence, it is recognized that consistent with risk criteria associated with fixed production systems, a more rigorous design criterion needs to be applied to floating production systems than is presently being applied to mobile offshore drilling units (MODUs). An important design criterion for floating units is the stability criterion. One of the fundamental stability criteria for MODUs has been the intact area ratio which has been applied by regulatory bodies since its formal introduction two decades ago by the American Bureau of Shipping (ABS). This original rule was derived on an empirical basis from experiences gained over many years in real world stability situations and accidents encountered by ships. The International

235

236 B.M. ST~)N~ et ~/.

Maritime Organization (IMO) has adopted an extension of ABS rules for world-wide service of MODUs. It has since been recognized that the IMO code is inadequate for harsh environments such as the North Sea and Eastern Canada (Praught et al., 1985: Springett and Praught, 1986). A critical review of the existing and proposed requirements on the stability conditions for semisubmersibles is presented by Kuo et al., (1983), Havig (1983), Hammett (1983), Dahle (1985), Mowatt and Allen (1986), Martinovich and Praught (1986), Springett and Praught (1986) and Pawlowski (1987). Most of the above propose that there is a need to incorporate as much dynamic information as possible for dealing with the stability requirements of semisubmersibles. At the same time, industry is concerned that many changes, both proposed and enacted, are not based on any demonstrated weaknesses in the present rules.

Stability criteria developed for ships are based on the hydrostatic restoring capabilities of the floating vessel subjected to a mean wind force. The principle behind the intact ratio criterion is that the structure must be able to absorb the energy from the heeling moment when inclined. It may be noted that the assumption involved in this principle, namely the heeling moment determined by the horizontal wind force and the horizontal hydrostatic reaction force acting through the centre of lateral resistance of the underwater part, is not always true for a moored structure. Semisubmersibles are also different compared to ships in their shape of displacement volume. In spite of these facts, the stability criteria for semisubmersibles, as mentioned earlier, have their origins in traditional naval architecture. The problem of stability of MODUs is even more complex when other environmental forces are simultaneously acting along with the wind. In fact, Kuo et al. (1983) and Takarada el al. (1986) go so far as to report that the existing stability criteria considering only unmoored vessels subject only to wind moments is not sufficient to prevent dangerous situations in violent seas.

In the early part of the last decade there were two major marine disasters involving semisubmersibles. In the first case, the Alexander L. Kielland, in 1980, capsized due to structural damage resulting in the loss of a vital buoyancy element. In the later case, in 1982, the Ocean Ranger capsized as an intact unit (Mogridge et al., 1986). These incidences were perceived to indicate an apparent deficiency in both intact as well as damaged stability criteria for semisubmersibles. As a response to the above marine disasters, attempts have been made to improve the understanding of the physical processes involved in stability. The Norwegian Maritime Directorate (NMD) and the U.K. Department of Energy commissioned Mobile Platform Stability (MOPS) projects and the results have been made public in recent years. ABS has developed a ,loint Industry Project (,liP) to address similar topics. MOPS and ,liP programs were developed to study the effect of wind, wave and current acting either alone or in combination on the stability of a semisubmersible. The effect of wind alone on the static stability of a semisubmersible was studied by Macha and Reid (1984) as a part of SNAME panel program.

Early experiments to assess intact stability criteria of semisubmersibles were started by Numata et al. (1976) and Kuo et al. (1977) even before these marine disasters as part of SNAME panel program. After extensive testing of typical semisubmersible models under wind and wave conditions, they concluded that no capsizing tendency is possible for semisubmersibles in a condition of normal intact stability under regular

Motion response of a damaged four column semisubmersible 237

wave conditions. However, they questioned any such possibility for a combined wave and wind condition due to the influence of steepness of waves and steady tilting of the vessel. They have also emphasized the inappropriateness of the area ratio criterion for semisubmersible design.

The study of the dynamic behaviour of semisubmersibles in a condition of normal intact stability is well documented (Numata et al., 1976; Kuo et al., 1977), but the study of the behaviour of a damaged unit is very limited. In fact, outside of the programs such as MOPS (Naess and Hoff, 1984; Herfjord, 1984) and JIP (Collins and Grove, 1988; Stiansen et al., 1988), there are few published works on this topic. The relevant passage (section 3.17.5) of ABS (1988) on this topic reads: "Based on authoritative wind tunnel tests as in section 3.17.4 and behaviour tests of a representative model in waves, or by proven calculation methods, alternative stability criteria may be considered for approval."

In this paper we present the outcome of model studies of an intact as well as a damaged, four column, twin pontoon semisubmersible unit subjected to both regular (Stone, 1986) and irregular waves. The simulated damage conditions represent partial loss of buoyancy to one column.

2. EXPERIMENTAL PROGRAM

Experiments were conducted in a wave-tow tank (length 58.3 m; width 4.6 m; depth 3.0 m) at the Memorial University of Newfoundland. The tank is equipped with an MTS servohydraulic piston-type wave generator at one end of the tank which can be programmed to generate regular and irregular waves of desired characteristics. A detailed description of the tank has been given by Muggeridge and Murray (1981) and Little (1985).

2.1. Model construction and its properties

The model semisubmersible is a twin pontoon four column unit constructed to a scale of 1:100, and is considered similar in geometry and mass properties to the GVA 4000 design. Details of model construction and the measurement of physical and experimental characteristics of the model are given by Stone (1986). Only a brief mention of the various aspects of the model is given here.

The model was constructed entirely of polyvinyl chloride (pvc). For the deck structure and pontoons, pvc sheets of 0.318 mm thickness were used, while machined thickwail pvc tubing and machined pvc rods were used for columns and bracings, respectively. Pontoons were equipped with two ballast tubes running parallel through the length of the pontoons with provisions for ballast weights. Each column was equipped with a threaded rod which allowed for adjustment of ballast. Underwater joints were hot air welded while glue joints were employed for the box deck structure. The profile and general configuration of the model with all major dimensions is shown in Fig. 1. The constructed model in the final stages of completion is shown in Fig. 2. These arrangements provided control of the position of horizontal and vertical centres of gravity, draft and metacentric height of the model, all of which were adjusted to correspond to the simulated values of the prototype.

238 B .M. STONY. et al.

,U - - 1290

1292 ! 54 72 - - 80.56

PROFILE

J 3300

1

750

~ - 0.00 1292

I I

16.00 ~J~

-- 33 O0

20 50

I 1 20 - - 750

- . .72 - - -~ . - , . .oo -~- - . . . . 70.72

FRONT VIEW

54,72

SECTION A-A - PONTOONS

1 I o

MAIN DECK

FIG. 1. General arrangement of the semisubmersible model.

2.2. Measurement of static stability and natural periods The natural periods of the model in pitch and roll for both free-floating and moored

conditions were measured using a level sensor. The heave natural period for the free- floating condition was obtained by an acccelerometer while a linear rotary potentiometer was used to obtain the corresponding period in the moored condition. In all instances, a digital signal analyzer with an accuracy of 0.01 sec was used to process the transducer signal. The physical properties and measured experimental characteristics of the model are given in Table 1 and show good agreement with the prototype values except pitch and roll which are slightly lower for the moored condition. A simple inclining experiment was carried out to obtain the static stability characteristics of the model at the operating draft. To this end, an external heeling moment was applied to the model using equal weights attached to the eye bolts installed at equal distance from the axis of rotation. The angle of inclination was measured using a two axis electrolytic level sensor. The resulting static stability curve agreed well with the computed values (Stone, 1986).

2.3. Modelling the mooring system The spread mooring system used in earlier model tests (Lundgren and Berg, 1982;

Mathisen et al., 1982) of the same semisubmersible unit (GVA 4000) along with the

FI~. 2. Model semisubmersible (1:100 scale) in the final phase of construction.

239

FIG. 6. Model undergoing tests under simulated moderate damage condition in regular waves.

240

Motion response of a damaged four column semisubmersible 241

information collected on the prototype mooring from Naval Architect (1981), Price and Wu (1983) and Grtaverken Arendal (1984) formed a basis for our model (Table 2). The model mooring system consisted of an 8-point system deployed in a 45 symmetrical pattern. Due to limited width of the wave tank (4.6 m) it was not possible to model the mooring system on the basis of weight per unit length. Hence, the mooring system was simulated by compound springs. To obtain the geometric configuration (segmental length, angle of inclination) and stiffness characteristics (horizontal and vertical tension and their rate of change) of the prototype mooring system a static analysis following the traditional catenary equation (Korkut and Hebert, 1970; Rothwell, 1979) was performed for slack and taut modes of the mooring line. Figure 3 shows the mooring line tension for the horizontal excursion while Fig. 4 shows the same for vertical excursion. Using these data the stiffness of the springs, permissible stretch and initial attachment angle of the mooring system were selected to correctly model horizontal and vertical stiffnesses as a function of the horizontal displacements over the whole range. Given the dominant influence of the mooring system in providing horizontal restoring force relative to that in the vertical direction (which is mostly determined by hydrostatic characteristics) the model mooring system stiffness was largely based on the horizontal stiffness. The resulting system provided close agreement for horizontal excursions while it approximated the less important vertical excursions (Figs 3 and 4).

2.4. Simulated damage condition Damage conditions were simulated by adding weights to a column at the height of

its vertical centre of gravity, which inclined the model with approximately equal amounts of heel and trim. This method of damage simulation may be representative of the flooding of the vessel. In the first case, referred to in this paper as the "light damage" condition, the inclination of the damaged column was 12 - 0.4 and the deck was entirely above the water surface while the pontoons were completely immersed in the water. In the other case, referred to as the "moderate damage" condition, a larger angle of inclination of 20 --- 0.5 of the column was simulated. In this case, the pontoons and the deck both pierced the water surface. The weights had an equivalent displacement volume of 487.8 and 975.6 cm 3 which correspond to column lengths of 3.73 and 7.46 cm. In terms of the percentage of buoyancy, the simulated damage conditions were equivalent to 9.1 and 18.2% of total flooding of the damaged column from its keel to main deck.

2.5. Instrumenting the model Initially, the relative positions of the model, mooring touch-down (at the tank wall)

and mooring termination points were established for all three orientations (head, beam and quartering directions) of the model with respect to the direction of wave advance. The model was rigidly held at these pre-established positions, the compound spring mooring assembly connected into the mooring line just below the fairleader, and the required pretension (127.5 g) applied. The top end of the spring assembly was connected to a 0.6 mm nylon coated stainless steel cable run via the fairleader to a rigid attachment under the main deck. Similarly, the lower end of the spring assembly ran through a pulley (located at the touch-down point on the tank wall) to a cantilever beam load cell mounted on the tank wall (Fig. 5).

242 B. M, SroNiet ~tl.

TABI,E 1. PtIYSICAL DIMENSIONS AND EXPERIMIN1AI. PROPERI'IES OF MODFA. AND PROIOFYPE

Designation

1. Pontoons Length (m) Beam (m) Height (m) Bilge radius (m)

2. Columns Diameter (m)

Spacing (center-center) Longitudinal (m) Beam (m)

3. Braces Diameter (m) Height of center (above keel) (m)

4. Deck Lower deck

Length (m) Beam (m)

Tween deck Length (m) Beam (m)

Main deck Length (m) Beam (m)

5. Height Keel to lower deck (m) Keel to main deck (m)

6. Center of gravity Vertical (above keel) (m) Longitudinal (from midship) (m) Transverse (from center line) (m)

7. Radius of gyration Pitch Roll

8. Natural period (frequency)

Heave, sec (Hz) Pitch, sec (Hz) Roll, sec (Hz) Surge, sec (Hz) Sway, sec (Hz) Yaw, sec (Hz)

9. Displacement Operational (m s)

Model (1:100) Prototype

0.806 80.56 0.16t) 16.00 0.075 7.5/) 11.013 1.35

0.130 12 .9(1

0.547 54.72 0.547 54.72

0.021 2.06 0.112 11.20

t).547 54.72 1t.547 54.72

0.623 62.32 0.547 54.72

0.670 67.00 0.575 57.50

I).33 33.00 0.41 41.00

0.210 20.97 0.0 0.0 0.0 1/.0

11.273 27.80 0.296 29.20

Free Moored Free Moored 2.1 (0.480) 21. (0.480) 21 (0.0480) 21 (0.0480) 4.1 (0.245) 3.6 (0.286) 41 (0.0245) 37 (0.0285) 5.0 (0.200) 4.3 (0.235) 52 (0.0195) 46 (0.0224) - - (--) 7.3 (0.140) - - (--) - - (--) - - (--) 8.9 (0.119) - - ( - - ) - - (--) - - ( - - ) - - ( - - ) - - ( - - )

0.024 24,368

Motion response of a damaged four column semisubmersible 243

TABLE 2. PROTOTYPE MOORING SPECIFICATIONS FORMING THE BASIS OF MODEL MOORING SYSTEM

Designation Prototype values

1. Linear weight 1.323 kN/m 2. Proof load 4730 kN 3. Breaking load 6010 kN 4. Total chain length 900 m 5. Pretension 1275 kN 6. Water depth 195 m 7. Fairleader depth 5.33 m 8. Anchor chain 76 mm (grade K4)

HODEL Tx --X"- HODEL TM

PROTOTYPE Tx --X"-- PROTOTYPE TU

o t - /

i

o. g - - - - L ~ * . . . . ~0 ~$ ~0 - IS -10 ~ 5 10 15 20

HORIZONTRL OISPLRCEMENT (a)

FIG. 3. Moof ing l inetens ion as a ~nct ion ofhor izontalexcurs ion o f the vessel(comparison of mode land prototype data).

Four light emitting diodes (LEDs) were mounted noncolinearly at the four corners on the deck of the vessel along with a control unit. An electrolytic two axis level sensor was mounted on the longitudinal centreline at the stern of the vessel. Two electronic cameras with photo-sensitive detectors were mounted 90 apart on custom mounts at the tank wall along with an administration unit. These cameras provided the angular displacements of LEDs from the origin of its focal planes. The initial x, y and z coordinates of the LEDs and the inclination of the cameras were determined in the tank coordinate system using precision survey instruments. Using these data as input, the system software calculates two transformation matrices (one for each camera). This

244 B .M. SIoN~ et al.

z

==.*

t/J o I--O =

.

o- -25 -20 -15 -10 -S

MODEL Tx - -X - - HODEL T u

PROTOTTPE Tx

--~c-- PROTOTYPE TW

10 15 20 25 VERTICRL DISPLFICEMENT (m)

FtG. 4. Mooring line tension as a function of vertical excursion of the vessel (comparison of model and prototype data).

enables measurements made by the cameras to be transformed to the tank coordinate system enabling the six degrees-of-freedom motion response of any vessel to be monitored in real time. Laurich (1984) provides an in-depth description of the selective spot (SELSPOT) recognition system and associated software being used at this facility.

A strain gauge conditioner and amplifier system connected to a digital multimeter was used to establish and measure the mooring line tension. All mooring line load cells were calibrated in situ prior to each series of tests. Wave profiles were measured at two locations in the tank using twin wire linear resistance wave probes. The wave probes were calibrated before and after each series of experiments. A schematic representation showing relative positions of the vessel, SELSPOT system, load cells, wave probes and mooring system used in the experimental arrangement is given in Fig. 5.

3. TEST PROGRAM

Model tests were carried out with three different orientations of the vessel (head, beam and quartering) with respect to wave direction for even keel and simulated damage conditions at operating draft. These tests were carried out for both regular and irregular waves. A summary of the test conditions is provided in Table 3. The model undergoing test in regular waves under moderate damage conditions is shown in Fig. 6.

Motion response of a damaged four column semisubmersible 245

SELSPOT CAMERA

oWAVE PROBE

:O ~7 ~LED

WAVE PROBE

o

LEVEL SENSOR'~

246 B. M. Stoxt: et al.

TABI.E 3. SUMMAR't O1" MOI)I-I I I{SI (ONDI I IONS

Orientat ion of vessel

(A) Head

(B) Beam

(C) Quarter ing

Column with loss Loss of buoyancy Angle of inclination of buoyancy volume, cm ~ of column in

degrees-:-

Heel tangle in degrees*

3-4 0.0 0.6 + 0.4 7-8 487.8 + 1 /.6 8.2 7-8 975.6 + 19.5 13.5 5-6 487.8 - 12.3 - 8.3 5-6 975.6 - 19.5 - 13.11

34 11.0 + 0.4 + 0.1 1-2 487.8 + 12.4 + 0.89 1-2 975.6 +20.5 + 14,5 7-8 487.8 12.11 - 8.6 7-8 975.6 - 19.9 -- 14.2

3-4 0.0 -+ 11.4 + tl.2 1-2 487.8 +12.2 + 9./1 1-2 975.6 +20.2 + 14.6 5-6 487.8 -13 .1 - 8.5 5-6 975.6 -21/.2 - 13.5

Tr im angle in degrees ~

- 0.5 - 8.2 -13 .8 + 9.11 --14.3

+ 11.5 8.6

-13 .9 - 8.3 -13 .6

+ 0.3 - - 8.2

13.6 +9.9

+14.7

*Positive angles correspond to windward damage tSign convention follows right-hand rule.

Draft Water depth Vessel orientation Wave conditions:

Regular waves Wave periods Wave height

Irregular waves Spectrum used Significant height Significant period

and negative angles correspond

211.5 cm (operating draft). 1.95 m. Head, beam and quartering seas.

From 11.711 to 2.50 sec in steps of 0.10 sec. 6.0 -+ 1.0 cm.

ISSC spcctrum. 3.0 cm 5.0 cm 7.5 cm 9.5 cm. I).81 sec 0.92 see 1.03 sec 1.19 scc.

to leeward damage.

3.1. Regular waves

For each vessel orientation and damaged condition, the vessel was subjected to 19 wave periods ranging from 0.7 to 2.5 sec in steps of 0.1 sec. The wave heights were approximately 6 -+ 1.0 cm (Table 3).

3.2. Irregular waves In order to understand the behaviour of the vessel in irregular waves and to see

what, if any, difference there would be from that of the regular wave field the tests were repeated using an International Ship Structures Congress (ISSC) spectrum. For each vessel orientation and simulated damaged conditions, four spectra were run with significant wave periods of 0.81, 0.92, 1.03 and 1.19 sec. The corresponding significant wave heights were 3.0, 5.0, 7.5 and 9.5 cm. In all, the vessel was subjected to 60 spectra in this study (Table 3).

3.3. Data recording and analysis Time histories of wave profiles measured by both the probes were recorded on an

instrumentation tape recorder. SELSPOT data for the corresponding time interval were

Motion response of a damaged four column semisubmersible 247

digitized and transferred to computer compatible magnetic tapes. From SELSPOT data recordings, amplitudes of motions of the semisubmersible in surge, heave, sway, pitch, roll and yaw were obtained.

4. RESULTS AND DISCUSSIONS

4.1. Regular waves

Comparison of even keel results. The motions of the vessel under even keel conditions were compared with those of Lundgren and Berg (1982) for all available data. The work of Lundgren and Berg was carried out at a scale of 1:65 with even keel conditions for both regular and irregular sea states. Such a comparison for the regular sea even keel response amplitude operator (RAO) for heave showed a good agreement throughout the entire frequency range, except near the resonant frequency of the vessel (0.50 Hz) where there was some marginal discrepancy. Such variations are to be expected, given the effect of damping near the resonant condition, for any small shifts in the experimental wave conditions and model natural frequency. Comparison of surge motion under even keel conditions showed acceptable levels of agreement. Pitch motion RAO also showed a good agreement for wave frequencies higher than 0.50 Hz. However, for wave frequencies less than 0.50 Hz, it was found that our results were consistently lower than those of Lundgren and Berg (1982). To ensure that the present studies did provide the correct motions, tests were repeated for wave frequencies between 0.25 and 0.50 Hz. In these extended studies it was observed that the pitch motion attained a sharp peak at about 0.27 Hz, which closely agreed with the measured natural frequency of the vessel in pitch. Hence, it could be inferred that the large pitch RAO observed in the work of Lundgren and Berg (1982) for wave frequencies less than 0.50 Hz is possible due to a slow receding part of the RAO after attaining the peak value at or about 0.27 Hz, instead of exhibiting a sharp peak as observed in the present study.

Comparison of the heave RAO in beam seas showed good agreement as in the case of head seas. The roll motion agreement in beam sea was quite good throughout the entire frequency range except near 0.45 Hz where the results of Lundgren and Berg (1982) showed a peak value. However, the actual value of peak was much less than that exhibited for pitch motion RAO in head seas. The sway motion, in our studies, was slightly larger than that of Lundgren and Berg (1982). This change might be due to the sensitivity of horizontal motions to changes in the pretensions as shown by Price and Wu (1983). However, it is believed that our compound mooring system provided the correct restoring forces over the entire range of excursion as shown in Figs 3 and 4.

4.1.1. Damaged condition: head seas

Heave motion. Heave motion RAO for the light damage condition showed a similar trend as in the case of even keel. This observation was applicable for both windward and leeward positions of the damaged column. The RAO for the light damage condition was almost the same as for the even keel case except near the resonant frequency and near 0.90 Hz. The resonant condition for moderate damage seemed to occur at a

248 B .M. STOYI: et al,

frequency slightly higher than the natural frequency. The moderate damage produced a slightly larger peak value of RAO near its resonant frequency. Near 0.90 Hz, under the moderate damage condition, there was found to be another peak of reduced value for both windward and leeward positions. Symmetry of the RAO with respect to windward and leeward direction as reported by Huang et al. (1982) and Huang and Naess (1983) was observed only to a certain extent in our studies.

Surge motion. A comparison of the surge RAO for windward light damage conditions showed that it was almost the same as for even keel condition. For windward moderate damage, there was only a marginal reduction. For the leeward orientation of the damaged column for both light and moderate damage conditions there was an equally marginal reduction in the surge RAO compared with even keel values. For all conditions, there is a progressive reduction in surge RAO with increase in the incident wave frequency from 1.0 at 0.4 Hz to 0.1 at 1.2 Hz.

Sway motion. No significant sway motion was observed either in windward or leeward position under any damage conditions, with the RAO typically being less than 0.05 except near 0.85 Hz. Near 0.85 Hz, the windward sway RAO was about 0.10 while for the leeward damage it was 0.20. The even keel values were even more reduced in magnitude throughout the whole frequency domain.

Yaw motion. It was observed that for the light damage condition, yaw was almost zero for all wave frequencies as in the case of even keel. For the moderate damage condition, there appeared to be some yaw motion, near 0.80 Hz. These observations were true for both windward and leeward positions of the damaged column, but as the values were typically 0.05, there was no measurable yaw motion.

Pitch motion. The pitch motions for damage conditions, shown in Fig. 7, indicated that for both windward and leeward positions, light damage produced almost the same motion as for the even keel situation throughout the entire range of frequency. For moderate damage, the pitch motion is quite different for both windward and leeward positions of the damaged column, which exhibits well-defined peak values near 0.85 Hz and 0.45 Hz. Leeward moderate damage condition produced a peak RAO of 2.5 while the corresponding windward value was about 1.5 at about 0.85 Hz, The other peak at 0.45 Hz in both windward and leeward damage conditions were about 40% less than their corresponding larger peak value.

Huang et al. (1982) and Chen et al. (1986) reported the existence of significant subharmonic motion in heave at wave frequencies near twice the natural frequency of the vessel in heave. In our studies, this phenomenon occurred in the moderate damage condition for both windward as well as leeward conditions when the pontoons and the deck pierced the water surface. In the studies of the above authors, this phenomenon occurred only in the leeward damage condition.

Roll motion. Roll motions shown in However, the corresponding peaks are windward and leeward) produced an

Fig. 8 exhibit similar trends as the pitch motion. much reduced. Light damage conditions (both almost uniform RAO throughout the whole

Motion response of a damaged four column semisubmersible 249

0

OJ

o~ qrt

E u

o o

o

0 .4 .< nr

r" U

O. 113

0

,/ \\ / \ /

\ /

(8) leve l

. . . . . . . . +iO" (Windwlrd)

/ \ ,2o" (**naw=rd)

/ \ / \

/ \ \ \

\

I I I

E u

tO O . -

2 0

n - O

=3 -

0

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r \ (.) I I \ l ,v ,1

\ - l o " (Leewaral

/ \ -20" (Lee*I l l 'd) --

/ \ / \ I /\ \ \ / \

\ / \ \ I \

j , \ ~ \ \

I ! I 0.3 0 .6 0 .9 t .2 i .5

Frequency (Hz]

FZG. 7. Pitch RAO for head sea regular wave condition.

frequency range and its magnitude was not significant, the value being about 0.10. For moderate damage two peaks were produced for both windward and leeward damage conditions, as in the case of pitch motion. The largest peak RAO was attained in leeward moderate damage near 0.85 Hz and the magnitude was about 1.2. The windward moderate damage condition peak RAO near 0.90 Hz was about 0.50.

The foregoing discussions on heave, pitch and roll motions for damage conditions (Figs 7 and 8) show that moderate damage at the leeward column produced larger peak motions than for other conditions. These peak values are observed not only near the region of natural frequency of the vessel in heave (0.49 Hz) but also near the frequency

250 B.M. StuNt: el al.

g]

o

o

nr LO

'-' o - o

E o

~

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~r kO

0 ~r

0

o

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. . . . . . . . . +20" (windward)

h / / ' \

I I I

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. . . . . . . . . . @O' (Leeward)

/ \ I \ / \

/ \ / \ . .

i 01 # 0 .3 0 .6 .9 i .2 ~.5

Frequency (Hz)

FIG. 8. Roll RAO for head sea regular wave condition.

range 0.90 Hz where subharmonic motions have been observed. Again, leeward moderate damage produced well-defined peak RAOs in yaw and sway near 0.45 Hz. For windward moderate damage, these observations were valid but the magnitude of the peaks were reduced. This shows the effect of coupling as discussed by Mathisen et al. (1982). This coupling effect is particularly understandable for vessels with symmetry in the xz and yz planes. It may be pointed out that the results of the time domain simulation of large amplitude wave effects on four, six and eight column semisubmersibles by Naess et al. (1985), Chen et al. (1986) and Matsuura and lkegami (1986) showed a similar large nonlinear roll motion considered to be subharmonic in nature with a frequency near one-half the wave exciting frequency. This motion is caused predominantly by nonlinear time dependent restoring characteristics of the vessel when the draft is shallow. The wave exciting frequency at which this subharmonic motion of the vessel can be induced was, as discussed before, about twice the natural frequency of the vessel in heave.

4.1.2. Damaged condit ion: beam seas

The translational (surge, sway and heave) motion RAOs and the rotational (pitch, roll and yaw) motion RAOs, for beam orientation of the rig, indicated that the

Motion response of a damaged four column semisubmersible 251

discussions brought out for head sea conditions could be directly applied to beam sea results as well since similar trends were maintained for corresponding damage conditions. It may be recalled that the motions are referred with respect to the fixed tank coordinate system.

4.1.3. Damaged condition: quartering seas

The RAO for surge and sway in quartering seas were almost identical. This is to be expected for a semisubmersible of this type, having an almost square column configuration. However, the pitch (Fig. 9) and roll (Fig. 10) do not exhibit such a response. Except for this difference, the behaviour of the semisubmersible in quartering

If)" ,~ (a)

leve l .

u -. o ,, /\ o . - +20" (Windward) i / / \ / \ - \ / o \ . , \ / " - - \ / -J u o \ ~- .~ /

~ / \

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r r

e- U

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\ I \ \ N I \

\.._J \ \

\

I I 0.6 0 .9 ~ .2 1 .5

Frequency (Hz)

FIG. 9. Pitch RAO for quartering sea regular wave condition.

252 B .M. STONE el tl/.

E .~o

O

Motion response of a damaged four column semisubmersible 253

0 0

a I "0 0

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a I

0 O_

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0 O

O

0 0

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1.5

FIG. 11. FFT of time series from pitch motion response (head sea--moderate damage--regular wave data).

2. Second-order effects, due to incident conditions, motion and mooring characteristics of the vessel;

3. The asymmetry of the motion of the vessel with respect to the wave direction, particularly when the damage is moderate;

4. The asymmetry is independent of the direction of sea state with respect to the vessel orientation.

The coupled motions of the vessel and the factors influencing it were discussed earlier. The second-order effects and the influence on the motion asymmetry will be discussed now. Figure 11 showed that the asymmetry was noticeable both in the first- order as well as the second-order motions. It should be noted that the incident wave height at 0.90 Hz used for the tests with leeward side damage of the vessel was lower than those with windward side damage of the vessel (5.60 cm and 6.40 cm). In spite of this reduced wave height for the leeward damage condition, the first- and second-order motions were higher. The first-order motion at 0.40 Hz was at least 60% more for the leeward condition than for the windward damage condition in spite of the wave height being almost equal. At 0.50 Hz, although the wave height for leeward direction was small, the reduction in first-order motion was not proportional to the reduction in wave height. The motion was much more reduced for windward direction. Naess and Hoff

254 B .M. SmN~ e/ a/.

(1984) and Huse and Nedrelid (1985) computed the effect of the increased wave height (from 2 cm to 4 cm) in understanding the asymmetry of the motions with respect t~ wave direction for a twin pontoon (eight column) vessel listed at about 12.5 . Their results showed that the asymmetry at this angle of list (equivalent to the light damage condition in this study) was not influenced by the wave steepness. This showed that the second-order effect due to nonlinear pressure terms on the asymmetry of the motions is negligible. In addition, if one looked at the graphs of Naess and Hoff (1984) and Huse and Nedrelid (1985) on the influence of draft on the motion of the vessel at a list angle of 15 the asymmetry in motion with respect to wave direction was found to be reduced when the draft was increased beyond 24 m (in full scale) where both the pontoons were well below the water surface. That is, when the effect of the waves were not directly felt by the inclined pontoons near the free surface. As observed by other workers, and confirmed in our present study, when the pontoons come closer to the water surface either partly or wholly, the hydrodynamic phenomenon is quite different than when they are well below the water surface since there will be additional diffraction as well as wave generation effects. However, these effects should be independent of the position of the damaged column. The only difference that might be contributing to forces when the damaged column is on the opposite side of the weather will be the interference effects due to the columns being at varying lengths below the water surface. However, they cannot be so much different as to explain the observed asymmetry of motions, when the damaged column is on the windward side or on the leeward side.

Thus, it implies that the motion in the leeward direction is influenced by additional phenomena other than the wave effect alone. This leaves us to at least explore whether this asymmetry could be possibly due to any drastically changed mooring characteristics of the vessel. If one takes into account the fact that the natural frequency of the vessel is shifted towards higher frequency particularly in pitch and roll (this is clearly demonstrated for irregular waves), then this seems a possibility. A moored body under the action of waves oscillates about the mean position it attains after the drifting. Since the net horizontal drifting force is in the direction of wave advance, the windward mooring lines are held taut in comparison to the leeward mooring lines irrespective of the position of the damaged column. Hence, the leeward side mooring line has sufficient freedom to oscillate about its equilibrium position, as compared to the windward line.

4.2. Irregular waves

Although the experiments were carried out using four different spectra, complete analysis will be given only to one spectrum (0.81 sec) in head sea condition. To conserve the length of the paper, the results from other spectra will be restricted to bringing out only the very salient features.

4.2.1. Damaged condition: head seas

Heave motion. The heave magnitude spectrum showed that most of the energy due to motion is concentrated within the frequency range of 0.60 Hz and 1.10 Hz for even keel and all conditions of damage. The resonance phenomenon under even keel condition occurred, as in the case of regular waves, near 0.50 Hz which corresponded to the natural frequency of the vessel in heave. As the damage condition increased from light to moderate, the resonant condition seemed to occur at a higher frequency

Motion response of a damaged four column semisubmersible 255

than the natural frequency of the unit. This was quite evident for moderate leeward damage while for other damage conditions it was slightly discernable. The resonant peak value in windward damage was less than the corresponding peak in the high energy frequency range (0.6-1.10 Hz). However, it was not true for leeward damage. As in the case of regular waves, resonant heave motion was higher for leeward damage, the peak value increasing with severity of damage. Moderate windward damage produced at resonance an amount only equal to the even keel values.

Surge motion. From the surge magnitude spectrum for windward and leeward positions of the damaged column it was observed that most of the energy due to the motion, as in heave, is concentrated in the frequency range of 0.60--1.10 Hz. The maximum peak value of surge RAO for windward light and moderate damage was less than the corresponding even keel value. In other frequency ranges, the magnitude for even keel and damage conditions was almost the same. The leeward damage produced larger peak values in surge motion which was slightly larger than for windward or even keel positions at the frequency of about 0.9 Hz. In both windward and leeward damage conditions, the peak value increased with increase in the severity of damage. One interesting phenomenon in surge motion was the presence of an additional higher frequency secondary component of considerable surge motion in the frequency range of 1.3-1.5 Hz for even keel. The maximum value in this range remained almost the same for all the damage conditions as for the even keel position.

Sway motion. The magnitude spectrum for sway motion showed that throughout the entire frequency range, the sway motion at light and moderate damage for both leeward and windward positions was slightly more than the corresponding even keel values. However, the sway motion under even keel was insignificant when compared with the corresponding surge and heave motions.

Pitch motion. The pitch motion of the vessel in head sea orientation is presented in Fig. 12 for even keel values and for both windward and leeward position of the damaged column. As in surge and heave motions, most of the energy is concentrated around the frequency range of 0.70-1.20 Hz. The peak value for windward damage in this frequency range for both light and moderate damages is slightly less than even keel values, while the corresponding values for leeward damages are slightly larger. Apart from this, there is a well-recognizable peak value of about 0.12 for leeward moderate damage near the frequency of 0.45 Hz. For windward moderate damage this peak is not well-pronounced at this frequency. For light damage in both windward and leeward condition, looking at the shape of the graph, it could be said that there appears to be a recognizable peak near the frequency of 0.2 Hz.

Roll motion. The magnitude spectrum for roll motion is given in Fig. 13. The discussions presented above for the pitch motion is applicable for the roll motion as well, although the magnitude of the peaks in the roll motion is much reduced when compared with the pitch motion. The peak value in leeward moderate damage was about 0.06.

256 B .M. STONE et al.

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: l eve l

. . . . . . . . . +$0" (Windward)

- - - - +20" (Windward)

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Frequency (Hz.)

FiG. 12. Pitch response for head sea in irregular wave condition. (ISSC spectrum--significant period = 0.81 sec.)

Yaw motion. The yaw magnitude spectrum showed insignificant motion for the entire frequency range and all damage conditions. There appeared to be a peak value near 0.45 Hz for the leeward damage condition, the magnitude of which was about 0.10. In other frequency ranges for all damage and even keel conditions, the magnitude was less than 0.05.

4.2.2. Damaged condition: beam seas

The magnitude spectra for all six degrees-of-freedom in beam seas showed similar trends for all damaged conditions and the values were almost nearly equal to the corresponding head sea values. The motion energy was again concentrated in the frequency range of 0.70-1.10 Hz.

4.2.3. Damaged condition: quartering seas

The heave magnitude spectrum for quartering seas showed that the response was similar to the head sea and beam sea conditions. The motion energy was again concentrated in the frequency range of 0.60-1.20 Hz. The peak values were slightly lower than the corresponding values for head sea. However, the shift in the resonant

Motion response of a damaged four column semisubmersible 257

frequency position as the damage was increased was more well-defined than in the head sea.

The pitch motion magnitude spectrum shown in Fig. 14 indicated that the values are significantly higher than the head sea values (Fig. 12). Leeward damage again produced increased motion compared with the windward damage. A careful look at the graphs in Fig. 14 also shows a shift in the resonant frequency for both light and moderate damage conditions.

The roll motion magnitude spectrum for quartering sea showed a similar trend as that of the head and beam sea conditions. The magnitudes were almost equal to that of the corresponding head sea conditions.

4.3. Effect of higher sea states The effect of increased sea states on the motion of the vessel with moderate damage

is shown in Fig. 15 for head sea orientation and windward position of the damaged column. As expected, the magnitude spectrum shows higher values of motion with an increase in the sea state. The energy due to motion in higher sea states seems to be

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FIG. 13. Roll response for head sea in irregular wave condition. (ISSC spectrum---significant period = 0.81 sec.)

258 B.M. S]oN[ et al.

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Fie,. 14. Pitch response for quartering sea in irregular wave condition. (ISSC spectrum--significant period = 0.81 sec.)

concentrated at a lower frequency range than for the lower sea states as seen in the heave and surge spectrum.

5. CONCLUSION

Model results of the dynamic response of a partially damaged, twin pontoon, four column semisubmersible showed that moderate (and higher) damage conditions seemed to produced significant subharmonic motion response of the vessel. These subharmonic motions occur at a frequency that is twice the natural frequency of the vessel in heave (intact condition), which lies within the normal range of wave excitation frequencies. Model tests also showed asymmetry in the motions of the damaged semisubmersible with respect to the position of the damaged column. These observations were confirmed from the results of both regular and irregular wave studies. Additionally, irregular wave studies showed that the energy due to the motion of the wave was concentrated in the frequency domain of 0.7-1.3 Hz, which corresponds to the energetic range of the normal sea state. It appears from both regular and irregular wave studies that the natural frequency of the vessel in pitch and roll tends to increase faster from its intact value, and approaches the value of the natural frequency in heave as the damage

Motion response of a damaged four column semisubmersible 259

0

o

Q Q

o 0

o

o

0 r~ m 13 0

m . - "O 0

4J 4.0 m g i

o 4-)

0 o

0

(s) t . 03 seconds

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~ o. . I seconds -

I I I

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condition is increased. This is true irrespective of the position of the damaged column or the orientation of the vessel with respect to the wave direction. The value of the natural frequency in heave itself slowly increases with increase in damage condition. The asymmetry in the motion of the damaged vessel with respect to the wave direction was equally true whether the vessel was in head, beam or quartering orientation. It appeared that nonlinear wave effects have only a minor role when compared to the influence of mooring characteristics in predicting the motions of the damaged vessel.

Acknowledgements--This research was supported financially by the National Sciences and Engineering Research Council of Canada to Dr D. B. Muggeridge through grant A4885.

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26(I B.M. SIONE et al.

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