# Dynamic analyses and preliminary design of offshore triceratops in ultra-deep waters

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## Abstract

More adaptable geometric form of offshore platforms, counting on the benefits of form-dominant design, is effective to encounter various environmental loads. Offshore triceratops is a new-generation offshore platform, whose conceptual design showed good degree-of-adaptability to ultra-deep-water conditions. Deck is partially isolated from the buoyant legs by ball joints by allowing transfer of partial displacements of buoyant legs to deck but restraining transfer of rotational responses. Prior to the suitability assessment of triceratops for ultra-deep waters, detailed dynamic analysis on the preliminary geometric form is necessary as a proof of validation for design applications. Current study discusses a detailed numeric analysis of triceratops at 2400 m water depth under regular and irregular waves; preliminary design of both buoyant legs and the deck is also presented. Buoyant Legs are designed as stiffened cylinders and the deck is designed as the integrated truss system. In compliant structures, the role of tethers is of paramount importance. Hence, the stress analysis and fatigue analysis of the tethers are also carried out to assess the service life of the structure. Presented study shall aid offshore engineers and contractors to understand suitability of triceratops, in terms of design and dynamic response behaviour.

### Keywords

Triceratops Offshore structures Structural design Ultra-deep water## Introduction

Deck is connected to three groups of buoyant legs; each group is position-restrained by a set of taut-moored tethers, which are commissioned under high initial pre-tension. Buoyant legs are deep-draft structures, which are similar to that of a Spar buoy. In addition, excess buoyancy and position-restraint characteristics are similar to a tension leg platform (TLP). Excess buoyancy ensures high initial pre-tension of tethers [11]. Buoyant legs are designed as stiffened cylindrical shell structures, as they are found to be suitable for application in offshore structures [12]. Similar to that of TLPs, increase in cost of triceratops is only due to the mooring system and its installation in deep waters, not due to the geometric form. Experimental and numerical investigations carried out by previous researchers discussed the operational advantages of the structural configuration, reduced deck response, and good re-centering capability [7]. Experimental and analytical studies showed that the coupled responses of the deck in rotational degrees-of-freedom are lesser than that of the buoyant legs [13]. The presence of ball joints is advantageous even under the seismic action of triceratops [14]. Stiffened triceratops are the modified geometric forms, whose response behaviour showed improvement even under impact loads [7]; under impact loads, bursts are seen in heave response, but with no rapid built-ups. Beat phenomenon occurs in all degrees-of-freedom, but with a lower magnitude in sway and roll [15].

Prior to the suitability assessment of triceratops for ultra-deep waters, detailed dynamic analysis on the preliminary geometric form is necessary as a proof of validation for design applications. Current study discusses a detailed numeric analysis of triceratops at 2400 m water depth; preliminary design of both buoyant legs and the deck are also presented. Natural periods in different degrees-of-freedom are determined, followed by detailed dynamic analysis under both regular and irregular waves; stress analysis of tethers is also carried out for estimating possible fatigue failure. This study is a preliminary investigation for assessing suitability of triceratops for ultra-deep water. The coupling effect due to wind and waves on the platform is ignored. The study involves the modelling and analysis of the buoyant legs and deck only; effects of risers are not considered in the analysis.

## Preliminary design of buoyant legs

Details of triceratops

Description | Unit | Quantity |
---|---|---|

Water depth | m | 2400 |

Density of steel | kg/m | 7850 |

Density of sea water | kg/m | 1025 |

Geometric details | ||

Diameter of leg | m | 15 |

| m | 61.77 |

Length of the leg | m | 174.24 |

Freeboard | m | 20.24 |

Draft | m | 154 |

Tether length | m | 2246 |

Diameter of tether | m | 1.00 |

Vertical centre of gravity of BLS | m | − 112.74 |

Meta-centric height | m | 35.83 |

Load details | ||

Self-weight + payload | kN | 562,424 |

Buoyancy force | kN | 820,932 |

Total tether force | kN | 258,491 |

Structural properties | ||

Area of deck | m | 3933 |

Area of tether | m | 2.356 |

Stiffness of tethers | GN/m | 0.22 |

Preliminary design parameters

Description | Value |
---|---|

Diameter of buoyant leg | 15 m |

Length of the buoyant leg | 174.54 m |

Thickness of shell | 40 mm |

Stringer system | Flat bar (70 numbers) |

Ring frame system | Flat bars at 3 m apart |

Restraining system | 4 tethers per leg |

Topside weight | 97.4 MN |

Payload | 165 MN |

Buoyant leg and tether weight | 209 MN |

Ballast weight | 91.6 MN |

Total displacement | 562.4 MN |

Environmental criteria | |

Water depth | 2400 m |

Wave height | |

100 years | 23.7 m |

1 year | 7.9 m |

Peak period | |

100 years | 14.0 s |

1 year | 9.1 s |

### Shell buckling

*A*is reduced buckling coefficient, which depends upon the stress condition,

*E*is Modulus of elasticity of steel,

*v*is Poisson’s ratio (0.3), \( t_{\text{s}} \) is shell thickness, and \( s_{\text{s}} \) is spacing between longitudinal stiffeners. The characteristic buckling stress of shell is found to be 557 N/mm

^{2}, which is greater than the yield stress.

### Panel ring buckling

Panel buckling is avoided by proportioning the ring stiffeners properly. Design should satisfy the required cross-sectional area and moment of inertia, including the effective width of the shell plate. Buckling strength is then computed for the flat bar ring frame, by considering the effective supports of stiffened cylinder. In the present study, stiffened cylinder is assumed to have heavy ring frames at the ends. The buckling stress of longitudinally stiffened cylinder is 600 N/mm^{2}.

### Column buckling

^{2}) is found to be less than the characteristic buckling stress. Hence, the design criterion is satisfied.

## Preliminary design of deck

Design considerations of deck

Description | Value |
---|---|

Shape of the deck | Triangular deck |

Length of the deck | 95 m |

Number of decks | 3 |

Number of bays in the truss | 9 |

Length of each bay | 9.5 m |

Load details | |

Total topside weight | 97.4 MN |

Live load in process and drilling zone | 5 kN/m |

Live load in storage floors | 18 kN/m |

Sustained wind speed | 55.88 m/s |

Initial tether tension | 28,721 kN |

Design considerations | |

Factor of safety for deal and live load | 1.3 |

Type of steel | High strength steel |

Size of deck components | |

Thickness of deck plate | 100 mm |

Transverse beam | Wide flange beam W 27 × 114 |

Longitudinal beam | Back to back channel section stiffened with flange plates: Web = 700 × 30 mm Flange = 350 × 30 mm |

Open web joist type | k-series |

Depth of web joist | 710 mm |

Diameter of web joist members | 20 mm |

Main chord members of truss | Channels, back to back |

Diagonal members of truss | Tubular members |

Diameter and thickness of diagonal members | 800; 12 mm |

| 40 |

## Numerical analyses

*D*/

*L*< 0.2). Mass of buoyant legs and ballast loads are applied at the mass centre of each group of legs. The buoyant legs are defined as TUBE elements and Morison equation is used for calculating the wave force:

*A*and d

*V*are the exposed area and displaced volume per unit length, respectively. The drag and inertia coefficients are taken as 0.75 and 1.0, respectively. Three point Gaussian integration scheme is used for the calculation of hydrodynamic force. Weight of the deck and payload is applied at the mass centre of deck. Each group of buoyant legs is connected to the deck using ball joints. Tethers are modelled as cable elements with appropriate axial stiffness and stretched up to impart the desired initial tension. Tethers extend from the keel of the each buoyant leg group to the seafloor. Numerical model is shown in Fig. 3. Meshing of the developed model is done by three-dimensional panel method. Modelling is followed by assessing the critical hydrostatic parameters to assess its stability and operability in ultra-deep water. Table 4 summarizes the critical parameters of buoyant leg. By convolution integration technique, the following equation of motion is solved:

*F*(

*t*) are acceleration, velocity, displacement, and force vectors, [

*M*] is the structural mass matrix, \( M_{a} \) is the added mass matrix, [

*C*] is the damping matrix, and [

*K*] is the stiffness matrix.

Hydrostatic parameters of buoyant leg

Description | Value |
---|---|

Cut water plane properties | |

Cut water plane area | 176.7 m |

Principal second moment of inertia | 2487 m |

Small angle stability parameters | |

Distance between centre of gravity and centre of buoyancy | − 35.74 m |

Distance between metacentre and centre of buoyancy (BM) | 0.9131 m |

Meta-centric height (GM) | 35.83 m |

### Free oscillation studies

Natural period and damping of free-floating buoyant leg

Description | Natural period (s) | Damping ratio (%) |
---|---|---|

Heave | 19 | 0.24 |

Roll | 151.9 | 1.14 |

Pitch | 151.9 | 1.14 |

Natural period and damping of tethered triceratops

Degree-of-freedom | Present study | Chandrasekaran and Madhuri [20] | ||
---|---|---|---|---|

Natural period (s) | Damping ratio (%) | Natural period (s) | Damping ratio (%) | |

Surge | 215.0 | 5.84 | 88.4 | 8.15 |

Sway | 215.4 | 5.87 | 88.4 | 8.15 |

Heave | 4.3 | 0.94 | 1.8 | 1.08 |

Roll | 6.2 | 6.11 | 9.46 | 4.34 |

Pitch | 6.1 | 6.10 | 9.46 | 4.34 |

Yaw | 215.9 | 6.23 | – | – |

## Dynamic response under regular waves

Deck response under zero wave-heading angle for different wave heights

Wave period (s) | Wave height = 2 m | Wave height = 4 m | Wave height = 6 m | ||||||
---|---|---|---|---|---|---|---|---|---|

Sea state: moderate | Sea state: rough | Sea state: very rough | |||||||

Surge | Heave | Pitch | Surge | Heave | Pitch | Surge | Heave | Pitch | |

8 | 0.0376 | 0.00058 | 0.0036 | 0.0752 | 0.00116 | 0.0072 | 0.0756 | 0.0054 | 0.0048 |

9 | 0.078 | 0.00054 | 0.002 | 0.15612 | 0.00108 | 0.004 | 0.0612 | 0.0024 | 0.0102 |

10 | 0.1392 | 0.0005 | 0.0016 | 0.2784 | 0.001 | 0.0032 | 0.066 | 0.00078 | 0.0078 |

11 | 0.2012 | 0.0006 | 0.0018 | 0.4024 | 0.0012 | 0.0036 | 0.1074 | 0.00054 | 0.0036 |

12 | 0.2586 | 0.00036 | 0.0024 | 0.5172 | 0.00072 | 0.0048 | 0.3462 | 0.0003 | 0.0078 |

13 | 0.328 | 0.00028 | 0.0028 | 0.656 | 0.00056 | 0.0056 | 0.435 | 0.00012 | 0.0102 |

14 | 0.4162 | 0.00038 | 0.003 | 0.8324 | 0.00076 | 0.006 | 0.5106 | 0.00006 | 0.012 |

15 | 0.4916 | 0.00044 | 0.0034 | 0.9832 | 0.00088 | 0.0068 | 0.5874 | 0.00036 | 0.0132 |

16 | 0.5344 | 0.0004 | 0.0032 | 1.0688 | 0.0008 | 0.0064 | 0.6606 | 0.00024 | 0.015 |

17 | 0.5618 | 0.00062 | 0.0034 | 1.1236 | 0.00124 | 0.0068 | 0.7284 | 0.0048 | 0.0168 |

18 | 0.6096 | 0.0008 | 0.0038 | 1.2192 | 0.0016 | 0.0076 | 0.7998 | 0.0006 | 0.018 |

Deck response in surge degree-of-freedom influences all operational activities in the drilling platform. Though surge responses are not easily excited, the response is mainly due to coupling between surge and pitch degrees-of-freedom, resulting from differential tension variation in tethers. Deck response at wave period of 15 s under 120° is lower than that of 0° and 180° by 54 and 50%, respectively. Heave motion is responsible for the dynamic tether tension variation. As seen from the figures, heave response attains maximum value at the pitch natural period and then reduces gradually. The presence of ball joints reduces the deck response than that of the buoyant legs, which shows operational advantage of triceratops in ultra-deep waters. Since the amplitude of motion is very small, this will prevent the flexural yielding of the drilling risers. At a wave period of 15 s, heave response of the deck is 60, 76, and 81% lower than that of the buoyant legs, which is quite significant. Heave response of the deck under 120° is about 24 and 8% higher than that of the response at 0° and 180°, respectively. Reduced pitch response of the deck is attributed to the presence of ball joints. However, pitch response seen in the deck arise from the differential heave response of the buoyant legs, as transfer of heave motion is not completely restrained by the ball joints. It is seen from the figures that pitch response starts increasing after reduction in heave response, which is due to the coupling effect between different degrees-of-freedom. Pitch response in the buoyant legs under different wave heights increases with the increase in time period.

### Validation of results

Validation of results

Degrees-of-freedom | Present study | Chandrasekaran and Madhuri [20] |
---|---|---|

Surge (m/m) | 0.0979 | 0.38 |

Heave (m/m) | 0.00006 | 0.0025 |

Pitch (deg/m) | 0.0022 | 0.03 |

## Dynamic response under random waves

Assessment of response under random sea is necessary to determine the stress variation on members. Random sea state is defined by significant wave height (*H*_{s}) and zero crossing periods (*T*_{z}). Sea state is assumed to be zero-mean ergodic process and it is defined by the wave spectrum. Selection of the spectrum depends upon the frequency characteristics of the wave environment. Time-domain analysis is performed in the numerical model using the following wave spectra:

### Pierson Moskowitz spectrum (PM)

### JONSWAP spectrum

*π*/15). Numerical analysis is carried out under both the spectra, for significant wave height of 15 m and wave period of 15 s. Response of the deck in all active degrees-of-freedom is shown in Figs. 8 and 9, for P–M spectrum and JONSWAP spectrum, respectively. The power spectral density plot shows the energy content of the response [21]. As seen from the response under PM spectrum, maximum amplitude occurs at the neighbourhood of peak wave frequency; amplitude is also significantly higher than that of the PSD of JONSWAP. Furthermore, heave and pitch responses under JONSWAP spectrum have maximum amplitude at the neighbourhood of 0.8 rad/s, while maximum surge response occurs at the vicinity of peak wave frequency.

## Stress analysis of tethers

^{2}. One of the main advantages of using wired strands is to achieve high strength-to-weight ratio, low elongation, reduced rotation characteristics, and high endurance limit. Table 9 shows the structural properties of tethers used in the present study.

Properties of tethers

Item | Characteristics |
---|---|

Type of arrangement | Left lay-regular lay |

Number of tethers | 9 |

Length of the tethers | 2246 m |

Diameter of wire | 100 mm |

Number of wires | 52 |

Number of strands | 7 |

Area of tether | 2.356 m |

Stiffness of tether | 0.22 MN/m |

Yield stress of wires | 500 N/mm |

Breaking stress of wires | 1500 N/mm |

^{2}, which is much lower than the yield strength. Similarly, maximum tension variation observed in phenomenal sea state is observed as 19% of that of the initial pre-tension, which developed a maximum stress of about 28 N/mm

^{2}, which is only 6% of that of its yield stress. It is evident that the chosen configuration of tether is adequate under both the sea states and shall not cause tether pullout. Tether tension variations are found out to be periodic in nature, which is predominantly governed by the behaviour of regular waves. Hence, low-amplitude, large cycle effects such as fatigue may become important.

## Service life estimation of tethers

*S*–

*N*curves. The

*S*–

*N*relationship is given by

*N*is the number of allowable cycles,

*S*is the stress range, and

*A*and

*m*are the constants obtained from the

*S*–

*N*curves. Then, the fatigue damage is calculated by Palmgren–Miner’s Rule, which is given by

*D*is the fatigue damage,

*n*is the number of counts from the histogram, and

*N*is the number of allowable cycles from

*S*–

*N*relationship. The calculated fatigue life is extrapolated to get the service life. The service life of the structure under rough sea state is found out to be 20.03 years, whereas under phenomenal sea state (WH = 24 m,

*T*= 14 s), it is 13.02 years. The reduction in the service life is attributed to the increase in the number of stress cycles and the maximum stress range, compared to that of the normal case. The stress histogram for the extreme sea state is shown in Fig. 13.

## Conclusions

- 1.
The shift in the natural period of the structure from the wave period indicates the ease of installation under sea conditions.

- 2.
Heave response of deck is lesser compared to the surge response. This shows that the platform is stiff in heave degree-of-freedom, which is mandatory for deep-water offshore structures.

- 3.
The reduced pitch response in deck compared to the BLS unit shows the advantage of using the ball joints.

- 4.
The reduced pitch response even under higher amplitude waves will also be advantageous in operating with high deck loads under harsh environmental conditions. The smaller pitch response observed in the deck is due to differential heave.

This shows that the offshore triceratops is advantageous due to the chosen structural configuration, which is found to be effective for ultra-deep waters. The preliminary design of deck and the BLS units is also carried out, to understand the suitability of the Triceratops from the member-level design perspective. As tethers are the crucial components in the complaint platforms, the stress analysis of tethers is also presented. The stress variation in the tethers is comparatively lower than the yield stress. However, fatigue analysis is also presented as tethers are subjected to periodic tension variation. The service life of the tethers assessed from the fatigue analysis under rough sea state is approximately 20 years.

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