- Email: [email protected]
The dynamic response of a three-leg articulated tower
Ocean Engineering 27 (2000) 1455–1471
Technical Note
The dynamic response of a three-leg articulated tower K. Nagamani b
a ,*
, C. Ganapathy
b
a Structural Engineering Division, Anna University, Madras-600025, India Ocean Engineering Centre, Indian Institute of Technology, Madras-600036, India
Received 17 April 1999; accepted 23 June 1999
Abstract Articulated towers are a compliant type of platform particularly suited for deep water applications. In the design of articulated towers, it is very important that the motion characteristics include sufficient stability, less acceleration in the deck and the smallest possible loading on the articulated joint. The mass distribution along the tower also plays an important role in the motion characteristics of the tower. Multi-leg articulated towers with three or more towers (legs or shafts), which have been developed from the conventional single tower have reduced horizontal movements and have more deck area compared to the single-leg articulated towers. The experimental and analytical investigations on such towers are not available in the published literature. In this paper, both analytical treatment and an experimental program for a three-leg articulated tower model have been reported. The effect of mass distributions on the variations of the bending moment and the deck accelerations are also presented. The model has been tested in a 2 m wave flume for various wave frequencies and wave heights of regular waves. The model is also analysed using a computer program developed, and the comparison of theoretical results with the experimental results is presented. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Articulated; Tower; Compliant; Multileg; Dynamic; Response
* Corresponding author. E-mail address: [email protected] (K. Nagamani). 0029-8018/00/$ - see front matter. 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 9 9 ) 0 0 0 4 9 - 9
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Nomenclature Cφ t F F(i⫺1) Jφ J∗φ t KL t KNL t K t M t MA M R T0 U u(i) ⌬u(i) ∂2U/∂t2 W ZW ZB ρg ⵜ w0
Restoring moment [Nm2] Global internal force vector at time t [N] Internal force vector at iteration (i⫺1) [N] Mass moment of inertia [kgm2] Added mass moment of inertia [kgm2] Linear global stiffness matrix at time t [N/m] Nonlinear global stiffness matrix at time t [N/m] Total stiffness matrix at time t [N/m] Global mass matrix at time t [Ns2/m] Global added mass matrix at time t [Ns2/m] Total mass matrix [Ns2/m] Global external force vector at time t+⌬t [N] Natural period of the structure [s] Incremental nodal displacements at time t+⌬t [m] Displacement vector at iteration i [m] Incremental displacement vector at iteration i [m] Incremental nodal accelerations at time t+⌬t [m/s2] Weight [N] Centre of gravity from pivot [m] Centre of buoyancy from pivot [m] Weight density of water [N/m3] Displaced volume of the water [m3] Natural frequency of the tower in water [rad/s]
1. Introduction A typical single leg articulated tower as shown in Fig. 1(a) is used as mobile loading and storage system, which may have either a single universal joint or a number of joints in the intermediate level which can be used at very large water depths. An articulated tower with universal joints in the intermediate level is called a multi-hinged articulated tower [Fig. 1(b)]. The extension of the concept of the single-leg articulated tower led to the development of a new type of platform with several columns which are parallel to one another. The columns are connected by universal joints both to the deck and to the foundation. The use of universal joints ensures that the columns always remain parallel to one another and the deck remains in a horizontal position. There is no rotation about the vertical axis of the columns. This type of platform is called a multi-leg articulated tower which has three or more columns [Fig. 1(c)]. The advantage of this system is that the pay loads and deck areas are comparable with the conventional production platforms in moderate water depths and the sway or the horizontal dis-
K. Nagamani, C. Ganapathy / Ocean Engineering 27 (2000) 1455–1471
Fig. 1.
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Different types of articulated towers.
placements of the deck are considerably reduced compared to the single leg-articulated towers. In assessing the behaviour of articulated towers under environmental conditions, it is necessary to study the response characteristics. To predict the complex behaviour it is necessary to investigate the influence of different parameters by both analytical and experimental methods. A number of experimental investigations on single-leg articulated towers have been carried out in the past by Arge (1981), Chakrabarti and Cotter (1979, 1980) and Datta and Jain (1990) and Kirk and Jain (1977), however there have been no reports on multi-leg articulated towers. This paper presents an experimental study on a multi-leg articulated tower model in a wave flume. The primary aim of the present experimental investigation is to study the behaviour and also to verify the analytical method.
2. Three-leg articulated tower model The experimental investigations are carried out on a three-leg articulated tower model made of perspex in a 2 m wave flume. The model dimensions and the details of the testing are given below: 2.1. Details of the model The following model dimensions are used in this investigation. Model height =1.25 m Diameter of shafts = 0.09 m Water depth = 0.87 m
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Spacing between legs
= 0.24 m
The model dimensions are shown in Fig. 2. The wave characteristics of the testing are: Wave heights = 0.055 m–0.190 m Wave period = 1.15 s, 1.30 s, 1.45 s, 1.60 s, 1.75 s and 1.90 s 2.2. Mass distribution As the mass distribution along the tower affects substantially the motion characteristics of an articulated tower, to study the effect of mass distribution, the two mass distributions have been taken as shown in Fig. 3. The mass distributions have been obtained by ballasting with lead weights at the bottom of each leg. 2.2.1. Mass distribution I Weight of perspex tube = 13.33 N/m The restoring moment Cφ = rgⵜZB⫺WZW where ⵜ = Displaced volume of the water r = Mass density of the water Weight of deck Weight of ballast at bottom of 160 mm height Total weight W (each leg) Centre of gravity from pivot ZW Centre of buoyancy from pivot ZB Cφ Mass moment of inertia Jφ Added mass moment of inertia Jf∗ The natural frequency of the tower in water (w0) w0 Natural period of the structure T0 2.2.2. Mass distribution II Weight of the ballast at the bottom of 145 mm height Centre of gravity from pivot Centre of buoyancy from pivot Mass moment of inertia Added mass moment of inertia The natural frequency of the tower in water
= 18.0 N =13.0 N = 34.96 N = 0.564 m = 0.48 m =6.85 Nm2 = 1.732 kgm2 = 1.680 kgm2 = [Cf/(Jf+J∗f )]0.5 = 1.4 rad/s = 4.44 s
= 30.0 N ZW ZB Cφ Jφ Jf∗ (w0) w0
= = = = = = =
0.47 m 0.48 m 4.865 Nm2 1.753 kgm2 1.680 kgm2 [Cf/(Jf+J∗f )]0.5 1.2 rad/s
K. Nagamani, C. Ganapathy / Ocean Engineering 27 (2000) 1455–1471
Fig. 2.
Dimensions of the three-leg articulated tower model.
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Fig. 3.
(a) Mass distribution I (each leg). (b) Mass distribution II (each leg).
Natural period of the structure
T0
= 5.24 s
2.3. Model fabrication The fabrication of the model is briefly described below. Three 1.2 m lengths are cut from a 90 mm diameter perspex tube. The bottom of each tube is covered with a perspex plate and made water tight. Below the plate a universal joint is pasted using Araldite. The top of each leg is covered with a perspex plate and over that the ball and socket type joint is pasted. Then the three legs are connected to a steel plate at the bottom by welding the joint to the plate and the perspex deck is placed at the top of the legs. The three legs are placed in such a way that they form an equilateral triangle of sides 0.24 m in plan. Four pairs of strain gauges are fixed diagonally opposite at four points on each leg. After fixing the strain gauges, the water proofing paste is applied over the strain gauges. The model assembly and the universal joints fixed at the bottom are also shown in Fig. 4. 2.4. Wave flume The model has been tested in a 2 m wave flume. The maximum water depth of the flume is 1 m. The waves are generated by a wedge type plunger of 2 m width.
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Fig. 4.
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Three-leg articulated tower model being lifted from the 2 m wave flume.
The flume is 30 m long, 2 m wide and 1.5 m deep. It also has the facility to generate an underwater current. The far end of the flume is provided with a rubble stone beach. The wedge type wave generator will generate regular waves only. The maximum wave period that can be generated is 2 s and the maximum wave height is 200 mm. 2.5. Instrumentation The online recording instrumentation setup has been used for measuring the wave elevation, deck acceleration and bending strains along the tower. The following instruments have been used: 1. Wave probe 2. Accelerometer 3. Strain gauges
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4. Frequency amplifier 5. Wave monitor 6. Multi-channel recorder
2.6. Testing of the model The model was fixed to the base of the flume. It is oriented in the flume with the two legs facing the wave and the third leg on the rear side. The model in the 2 m wave flume with waves is shown in Fig. 5. The model test has been conducted for two different mass distributions and for regular waves of periods varying from 1.15 s to 1.9 s with wave heights ranging from 35 mm to 190 mm, for each period. The wave height and wave periods are recorded using the wave probes. The wave period is calculated by setting the paper speed of the recorders and measuring the distance of one cycle. The wave height is calculated by measuring the peak to peak value of the curve. The accelerometer is connected at the top of the deck in the direction of the wave propagation. The deck accelerations and wave heights are recorded simultaneously from which the location of the maximum acceleration can be found. The experiment is repeated for six wave periods of 1.15 s, 1.30 s, 1.45 s, 1.60 s, 1.75 s, 1.90 s and for each period five wave heights are generated. The above procedure is also repeated for a second mass distribution. The maximum bending
Fig. 5.
Three-leg articulated tower model in the 2 m wave flume.
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moment and acceleration values are given in Tables 1 and 2. The bending moment variation along the tower is presented in Figs. 6 to 7. The maximum displacement of the deck is 130 mm for a wave height of 136 mm and a wave period of 1.90 s, and it was found that the multi-leg articulated tower has a reduced motion and that the deck remains horizontal compared to a single leg articulated tower. 3. Finite element analysis In the nonlinear analysis of articulated towers, the geometric nonlinearity due to large displacements, large rotations and nonlinear drag force are the prime factors Table 1 Maximum bending moment for various wave periods and mass distributions Experimental results Wave period (s)
1.90 1.90 1.90 1.90 1.90 1.75 1.75 1.75 1.75 1.75 1.60 1.60 1.60 1.60 1.60 1.45 1.45 1.45 1.45 1.45 1.30 1.30 1.30 1.30 1.30 1.15 1.15 1.15 1.15 1.15
Wave height (m)
0.055 0.077 0.096 0.118 0.136 0.036 0.050 0.073 0.096 0.123 0.046 0.068 0.091 0.118 0.136 0.050 0.073 0.105 0.132 0.164 0.060 0.091 0.118 0.155 0.182 0.065 0.105 0.127 0.165 0.191
Maximum bending moment in Nm Mass distribution I
Mass distribution II
0.7350 0.9180 1.5290 1.5650 1.7130 0.6690 0.6000 1.7120 1.7120 1.8380 0.6890 1.1030 1.4700 1.5900 1.7760 1.0410 1.2250 1.2860 1.7760 1.8380 1.0410 1.3480 1.6540 1.7760 2.0210 1.0410 1.4090 1.7760 1.9600 2.0200
0.4350 0.6120 0.7560 1.0260 1.4050 0.3680 0.5940 1.0200 1.350 1.945 0.5320 0.8110 1.1890 1.4590 1.4500 0.4860 0.9670 1.2970 1.4590 1.6440 0.7250 1.1890 1.6210 1.7290 1.8910 0.7740 1.4050 1.6210 1.8910 2.1600
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Table 2 Maximum deck acceleration for various wave periods and mass distributions Experimental results Wave period (s)
1.90 1.90 1.90 1.90 1.90 1.75 1.75 1.75 1.75 1.75 1.60 1.60 1.60 1.60 1.60 1.45 1.45 1.45 1.45 1.45 1.30 1.30 1.30 1.30 1.30 1.15 1.15 1.15 1.15 1.15
Wave height (m)
0.055 0.077 0.096 0.118 0.136 0.036 0.050 0.073 0.096 0.123 0.046 0.068 0.091 0.118 0.136 0.050 0.073 0.105 0.132 0.164 0.060 0.091 0.118 0.155 0.182 0.065 0.105 0.127 0.165 0.191
Maximum deck acceleration in m/s2 Mass distribution I
Mass distribution II
0.5680 0.8250 0.9810 1.1870 1.4450 0.4130 0.5680 0.7740 1.0320 1.2390 0.5160 0.7740 1.0320 1.2900 1.6000 0.6190 0.8770 1.2390 1.4970 1.8580 0.7230 1.0320 1.3940 1.7550 2.0640 0.7740 1.2390 1.6520 2.0130 2.3740
0.3610 0.6710 0.8770 1.1350 1.3420 0.3100 0.5160 0.9290 1.3420 1.1870 0.4130 0.6710 0.9810 1.2900 1.4450 0.5160 0.9810 1.3420 1.6520 1.6000 0.6710 1.0840 1.4970 1.7030 1.6520 0.8770 1.3940 1.6520 1.9610 2.0130
which govern the formulation. The time dependant wave force has been considered as a drag component of the wave force which is a function of second order water particle velocity, hence the formulation nonlinearity due to the wave force has been included. 3.1. Incremental updated Lagrangian formulation Nonlinear beam element formulation has been adopted. The beam element has two nodes with six degrees of freedom per node, and can transmit an axial force, two shear forces, two bending moments and a torque.
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Fig. 6. (a) Bending moment variation for legs I and II (mass distribution I). (b) Bending moment variation for leg III (mass distribution I).
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Fig. 7. (a) Bending moment variation for legs I and II (mass distribution I). (b) Bending moment variation for leg III (mass distribution I).
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Fig. 8. (a) Bending moment variation for legs I and II (mass distribution I). (b) Bending moment variation for leg III (mass distribution I).
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Fig. 9. (a) Bending moment variation for legs I and II (mass distribution II). (b) Bending moment variation for leg III (mass distribution II).
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3.1.1. Incremental equilibrium equations The stiffness and mass matrices are obtained by computing all static and kinematic quantities in the element coordinate system and are then transformed to a global coordinate system. Hence the incremental equilibrium equations are: [tKL⫹tKNL]U⫽R⫺tF⫺[tM⫹tMA][∂2U/∂t2] where t KL t KNL U R t F t M t MA ∂2U/∂t2
(1)
=linear global stiffness matrix at time t =nonlinear global stiffness matrix at time t =incremental nodal displacements at time t+⌬t =global external force vector at time t+⌬t =global internal force vector at time t =global mass matrix at time t =global added mass matrix at time t =incremental nodal accelerations at time t+⌬t
3.1.2. Calculation of wave forces Wave forces are calculated for a randomly oriented element in space using a transformation matrix which is formed using the direction cosines. The wave particle velocities and accelerations are calculated in the global directions and velocities and accelerations are calculated normal to the axis of the element using the transformation matrix given by Chakrabarti (1987). The wave forces are calculated using Morison’s equation taking into account the relative velocities and accelerations of the fluid and the structure. 3.1.3. Solution of nonlinear equations The equations of motion are given by (1) for the system which is highly nonlinear due to the nonlinear drag force and large displacements of the tower. The equations are solved in time domain using the Newmark-b method or Wilson-q method in each time step. It is observed that both the above methods yield identical results. 3.1.3.1. Equilibrium iteration It is important to realize that the equations of motion given in Eq. (1) are only approximations to the actual equations to be solved, since the static and kinematic variables are evaluated from the last known configuration. This may introduce errors which ultimately result in solution instability. For this reason it may be necessary to iterate in each load step or time step until the required convergence is achieved. In the Updated Lagrangian formulation, the equation of motion is given by t
K⌬u(i)⫽R⫺F(i−1)⫺Mu(i); i⫽1,2,3
(2)
The step-by-step integration used for the solution of nonlinear static and dynamic equations are given by Bathe et al. (1975). The Newmark-b method and Wilson-q method are used in the above algorithm. A computer program has been developed
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based on the above formulation for the nonlinear dynamic analysis of articulated towers. The three-leg articulated tower model has been analysed using the above program and the results are compared in Fig. 10.
4. Summary and conclusions The investigation on the three-leg articulated tower model is carried out for a different range of frequencies and wave heights as described above. The tower responses and bending strains along each leg of the tower are also recorded for the two different mass distributions. It is found that the bending moment increases with wave height for all the legs. The experimental results agree closely with the theoretical results as shown in Fig. 10. Based on the present study the following conclusions are presented: 1. The theoretical results agree well with the experimental results. 2. The maximum bending moment along the legs increases with the wave frequency and decreases with the natural frequency of the tower.
Fig. 10. (a) Bending moment variation (mass distribution II). (b) Bending moment variation (mass distribution II).
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3. The bending moment increases with wave height for all three legs. 4. The deck acceleration increases with wave height and decreases with the natural frequency of the tower.
References Arge, C., 1981. The CONAT System, Report of an Industrial Presentation. London. Bathe, K.J., Ramm, E., Wilson, E.L., 1975. Finite element formulations for large deformation dynamic analysis. International Journal for Numerical Methods in Engineering 9, 353–386. Chakrabarti, S.K., Cotter, D.C., 1979. Motion analysis of articulated tower. Journal of the Waterway, Port, Coastal and Ocean Division, ASCE 105 (WW3), 281–292. Chakrabarti, S.K., Cotter, D.C., 1980. Transverse motion of articulated tower. Journal of the Waterway, Port, Coastal and Ocean Division, ASCE 106 (WW1), 65–78. Chakrabarti, S.K., 1987. Hydrodynamics of Offshore Structures. Computational Mechanics Publications. Datta, T.K., Jain, A.K., 1990. Response of articulated platforms to random wind and wave forces. Computers and Structures 34 (1), 137–144. Kirk, C.L. and Jain, R.K., Response of articulated towers to waves and current. In: Proc. Offshore Technology Conference, OTC 2798, Houston, TX, 1977. pp. 545–552.
Technical Note
The dynamic response of a three-leg articulated tower K. Nagamani b
a ,*
, C. Ganapathy
b
a Structural Engineering Division, Anna University, Madras-600025, India Ocean Engineering Centre, Indian Institute of Technology, Madras-600036, India
Received 17 April 1999; accepted 23 June 1999
Abstract Articulated towers are a compliant type of platform particularly suited for deep water applications. In the design of articulated towers, it is very important that the motion characteristics include sufficient stability, less acceleration in the deck and the smallest possible loading on the articulated joint. The mass distribution along the tower also plays an important role in the motion characteristics of the tower. Multi-leg articulated towers with three or more towers (legs or shafts), which have been developed from the conventional single tower have reduced horizontal movements and have more deck area compared to the single-leg articulated towers. The experimental and analytical investigations on such towers are not available in the published literature. In this paper, both analytical treatment and an experimental program for a three-leg articulated tower model have been reported. The effect of mass distributions on the variations of the bending moment and the deck accelerations are also presented. The model has been tested in a 2 m wave flume for various wave frequencies and wave heights of regular waves. The model is also analysed using a computer program developed, and the comparison of theoretical results with the experimental results is presented. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Articulated; Tower; Compliant; Multileg; Dynamic; Response
* Corresponding author. E-mail address: [email protected] (K. Nagamani). 0029-8018/00/$ - see front matter. 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 9 9 ) 0 0 0 4 9 - 9
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Nomenclature Cφ t F F(i⫺1) Jφ J∗φ t KL t KNL t K t M t MA M R T0 U u(i) ⌬u(i) ∂2U/∂t2 W ZW ZB ρg ⵜ w0
Restoring moment [Nm2] Global internal force vector at time t [N] Internal force vector at iteration (i⫺1) [N] Mass moment of inertia [kgm2] Added mass moment of inertia [kgm2] Linear global stiffness matrix at time t [N/m] Nonlinear global stiffness matrix at time t [N/m] Total stiffness matrix at time t [N/m] Global mass matrix at time t [Ns2/m] Global added mass matrix at time t [Ns2/m] Total mass matrix [Ns2/m] Global external force vector at time t+⌬t [N] Natural period of the structure [s] Incremental nodal displacements at time t+⌬t [m] Displacement vector at iteration i [m] Incremental displacement vector at iteration i [m] Incremental nodal accelerations at time t+⌬t [m/s2] Weight [N] Centre of gravity from pivot [m] Centre of buoyancy from pivot [m] Weight density of water [N/m3] Displaced volume of the water [m3] Natural frequency of the tower in water [rad/s]
1. Introduction A typical single leg articulated tower as shown in Fig. 1(a) is used as mobile loading and storage system, which may have either a single universal joint or a number of joints in the intermediate level which can be used at very large water depths. An articulated tower with universal joints in the intermediate level is called a multi-hinged articulated tower [Fig. 1(b)]. The extension of the concept of the single-leg articulated tower led to the development of a new type of platform with several columns which are parallel to one another. The columns are connected by universal joints both to the deck and to the foundation. The use of universal joints ensures that the columns always remain parallel to one another and the deck remains in a horizontal position. There is no rotation about the vertical axis of the columns. This type of platform is called a multi-leg articulated tower which has three or more columns [Fig. 1(c)]. The advantage of this system is that the pay loads and deck areas are comparable with the conventional production platforms in moderate water depths and the sway or the horizontal dis-
K. Nagamani, C. Ganapathy / Ocean Engineering 27 (2000) 1455–1471
Fig. 1.
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Different types of articulated towers.
placements of the deck are considerably reduced compared to the single leg-articulated towers. In assessing the behaviour of articulated towers under environmental conditions, it is necessary to study the response characteristics. To predict the complex behaviour it is necessary to investigate the influence of different parameters by both analytical and experimental methods. A number of experimental investigations on single-leg articulated towers have been carried out in the past by Arge (1981), Chakrabarti and Cotter (1979, 1980) and Datta and Jain (1990) and Kirk and Jain (1977), however there have been no reports on multi-leg articulated towers. This paper presents an experimental study on a multi-leg articulated tower model in a wave flume. The primary aim of the present experimental investigation is to study the behaviour and also to verify the analytical method.
2. Three-leg articulated tower model The experimental investigations are carried out on a three-leg articulated tower model made of perspex in a 2 m wave flume. The model dimensions and the details of the testing are given below: 2.1. Details of the model The following model dimensions are used in this investigation. Model height =1.25 m Diameter of shafts = 0.09 m Water depth = 0.87 m
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Spacing between legs
= 0.24 m
The model dimensions are shown in Fig. 2. The wave characteristics of the testing are: Wave heights = 0.055 m–0.190 m Wave period = 1.15 s, 1.30 s, 1.45 s, 1.60 s, 1.75 s and 1.90 s 2.2. Mass distribution As the mass distribution along the tower affects substantially the motion characteristics of an articulated tower, to study the effect of mass distribution, the two mass distributions have been taken as shown in Fig. 3. The mass distributions have been obtained by ballasting with lead weights at the bottom of each leg. 2.2.1. Mass distribution I Weight of perspex tube = 13.33 N/m The restoring moment Cφ = rgⵜZB⫺WZW where ⵜ = Displaced volume of the water r = Mass density of the water Weight of deck Weight of ballast at bottom of 160 mm height Total weight W (each leg) Centre of gravity from pivot ZW Centre of buoyancy from pivot ZB Cφ Mass moment of inertia Jφ Added mass moment of inertia Jf∗ The natural frequency of the tower in water (w0) w0 Natural period of the structure T0 2.2.2. Mass distribution II Weight of the ballast at the bottom of 145 mm height Centre of gravity from pivot Centre of buoyancy from pivot Mass moment of inertia Added mass moment of inertia The natural frequency of the tower in water
= 18.0 N =13.0 N = 34.96 N = 0.564 m = 0.48 m =6.85 Nm2 = 1.732 kgm2 = 1.680 kgm2 = [Cf/(Jf+J∗f )]0.5 = 1.4 rad/s = 4.44 s
= 30.0 N ZW ZB Cφ Jφ Jf∗ (w0) w0
= = = = = = =
0.47 m 0.48 m 4.865 Nm2 1.753 kgm2 1.680 kgm2 [Cf/(Jf+J∗f )]0.5 1.2 rad/s
K. Nagamani, C. Ganapathy / Ocean Engineering 27 (2000) 1455–1471
Fig. 2.
Dimensions of the three-leg articulated tower model.
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Fig. 3.
(a) Mass distribution I (each leg). (b) Mass distribution II (each leg).
Natural period of the structure
T0
= 5.24 s
2.3. Model fabrication The fabrication of the model is briefly described below. Three 1.2 m lengths are cut from a 90 mm diameter perspex tube. The bottom of each tube is covered with a perspex plate and made water tight. Below the plate a universal joint is pasted using Araldite. The top of each leg is covered with a perspex plate and over that the ball and socket type joint is pasted. Then the three legs are connected to a steel plate at the bottom by welding the joint to the plate and the perspex deck is placed at the top of the legs. The three legs are placed in such a way that they form an equilateral triangle of sides 0.24 m in plan. Four pairs of strain gauges are fixed diagonally opposite at four points on each leg. After fixing the strain gauges, the water proofing paste is applied over the strain gauges. The model assembly and the universal joints fixed at the bottom are also shown in Fig. 4. 2.4. Wave flume The model has been tested in a 2 m wave flume. The maximum water depth of the flume is 1 m. The waves are generated by a wedge type plunger of 2 m width.
K. Nagamani, C. Ganapathy / Ocean Engineering 27 (2000) 1455–1471
Fig. 4.
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Three-leg articulated tower model being lifted from the 2 m wave flume.
The flume is 30 m long, 2 m wide and 1.5 m deep. It also has the facility to generate an underwater current. The far end of the flume is provided with a rubble stone beach. The wedge type wave generator will generate regular waves only. The maximum wave period that can be generated is 2 s and the maximum wave height is 200 mm. 2.5. Instrumentation The online recording instrumentation setup has been used for measuring the wave elevation, deck acceleration and bending strains along the tower. The following instruments have been used: 1. Wave probe 2. Accelerometer 3. Strain gauges
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4. Frequency amplifier 5. Wave monitor 6. Multi-channel recorder
2.6. Testing of the model The model was fixed to the base of the flume. It is oriented in the flume with the two legs facing the wave and the third leg on the rear side. The model in the 2 m wave flume with waves is shown in Fig. 5. The model test has been conducted for two different mass distributions and for regular waves of periods varying from 1.15 s to 1.9 s with wave heights ranging from 35 mm to 190 mm, for each period. The wave height and wave periods are recorded using the wave probes. The wave period is calculated by setting the paper speed of the recorders and measuring the distance of one cycle. The wave height is calculated by measuring the peak to peak value of the curve. The accelerometer is connected at the top of the deck in the direction of the wave propagation. The deck accelerations and wave heights are recorded simultaneously from which the location of the maximum acceleration can be found. The experiment is repeated for six wave periods of 1.15 s, 1.30 s, 1.45 s, 1.60 s, 1.75 s, 1.90 s and for each period five wave heights are generated. The above procedure is also repeated for a second mass distribution. The maximum bending
Fig. 5.
Three-leg articulated tower model in the 2 m wave flume.
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moment and acceleration values are given in Tables 1 and 2. The bending moment variation along the tower is presented in Figs. 6 to 7. The maximum displacement of the deck is 130 mm for a wave height of 136 mm and a wave period of 1.90 s, and it was found that the multi-leg articulated tower has a reduced motion and that the deck remains horizontal compared to a single leg articulated tower. 3. Finite element analysis In the nonlinear analysis of articulated towers, the geometric nonlinearity due to large displacements, large rotations and nonlinear drag force are the prime factors Table 1 Maximum bending moment for various wave periods and mass distributions Experimental results Wave period (s)
1.90 1.90 1.90 1.90 1.90 1.75 1.75 1.75 1.75 1.75 1.60 1.60 1.60 1.60 1.60 1.45 1.45 1.45 1.45 1.45 1.30 1.30 1.30 1.30 1.30 1.15 1.15 1.15 1.15 1.15
Wave height (m)
0.055 0.077 0.096 0.118 0.136 0.036 0.050 0.073 0.096 0.123 0.046 0.068 0.091 0.118 0.136 0.050 0.073 0.105 0.132 0.164 0.060 0.091 0.118 0.155 0.182 0.065 0.105 0.127 0.165 0.191
Maximum bending moment in Nm Mass distribution I
Mass distribution II
0.7350 0.9180 1.5290 1.5650 1.7130 0.6690 0.6000 1.7120 1.7120 1.8380 0.6890 1.1030 1.4700 1.5900 1.7760 1.0410 1.2250 1.2860 1.7760 1.8380 1.0410 1.3480 1.6540 1.7760 2.0210 1.0410 1.4090 1.7760 1.9600 2.0200
0.4350 0.6120 0.7560 1.0260 1.4050 0.3680 0.5940 1.0200 1.350 1.945 0.5320 0.8110 1.1890 1.4590 1.4500 0.4860 0.9670 1.2970 1.4590 1.6440 0.7250 1.1890 1.6210 1.7290 1.8910 0.7740 1.4050 1.6210 1.8910 2.1600
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Table 2 Maximum deck acceleration for various wave periods and mass distributions Experimental results Wave period (s)
1.90 1.90 1.90 1.90 1.90 1.75 1.75 1.75 1.75 1.75 1.60 1.60 1.60 1.60 1.60 1.45 1.45 1.45 1.45 1.45 1.30 1.30 1.30 1.30 1.30 1.15 1.15 1.15 1.15 1.15
Wave height (m)
0.055 0.077 0.096 0.118 0.136 0.036 0.050 0.073 0.096 0.123 0.046 0.068 0.091 0.118 0.136 0.050 0.073 0.105 0.132 0.164 0.060 0.091 0.118 0.155 0.182 0.065 0.105 0.127 0.165 0.191
Maximum deck acceleration in m/s2 Mass distribution I
Mass distribution II
0.5680 0.8250 0.9810 1.1870 1.4450 0.4130 0.5680 0.7740 1.0320 1.2390 0.5160 0.7740 1.0320 1.2900 1.6000 0.6190 0.8770 1.2390 1.4970 1.8580 0.7230 1.0320 1.3940 1.7550 2.0640 0.7740 1.2390 1.6520 2.0130 2.3740
0.3610 0.6710 0.8770 1.1350 1.3420 0.3100 0.5160 0.9290 1.3420 1.1870 0.4130 0.6710 0.9810 1.2900 1.4450 0.5160 0.9810 1.3420 1.6520 1.6000 0.6710 1.0840 1.4970 1.7030 1.6520 0.8770 1.3940 1.6520 1.9610 2.0130
which govern the formulation. The time dependant wave force has been considered as a drag component of the wave force which is a function of second order water particle velocity, hence the formulation nonlinearity due to the wave force has been included. 3.1. Incremental updated Lagrangian formulation Nonlinear beam element formulation has been adopted. The beam element has two nodes with six degrees of freedom per node, and can transmit an axial force, two shear forces, two bending moments and a torque.
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Fig. 6. (a) Bending moment variation for legs I and II (mass distribution I). (b) Bending moment variation for leg III (mass distribution I).
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Fig. 7. (a) Bending moment variation for legs I and II (mass distribution I). (b) Bending moment variation for leg III (mass distribution I).
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Fig. 8. (a) Bending moment variation for legs I and II (mass distribution I). (b) Bending moment variation for leg III (mass distribution I).
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Fig. 9. (a) Bending moment variation for legs I and II (mass distribution II). (b) Bending moment variation for leg III (mass distribution II).
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3.1.1. Incremental equilibrium equations The stiffness and mass matrices are obtained by computing all static and kinematic quantities in the element coordinate system and are then transformed to a global coordinate system. Hence the incremental equilibrium equations are: [tKL⫹tKNL]U⫽R⫺tF⫺[tM⫹tMA][∂2U/∂t2] where t KL t KNL U R t F t M t MA ∂2U/∂t2
(1)
=linear global stiffness matrix at time t =nonlinear global stiffness matrix at time t =incremental nodal displacements at time t+⌬t =global external force vector at time t+⌬t =global internal force vector at time t =global mass matrix at time t =global added mass matrix at time t =incremental nodal accelerations at time t+⌬t
3.1.2. Calculation of wave forces Wave forces are calculated for a randomly oriented element in space using a transformation matrix which is formed using the direction cosines. The wave particle velocities and accelerations are calculated in the global directions and velocities and accelerations are calculated normal to the axis of the element using the transformation matrix given by Chakrabarti (1987). The wave forces are calculated using Morison’s equation taking into account the relative velocities and accelerations of the fluid and the structure. 3.1.3. Solution of nonlinear equations The equations of motion are given by (1) for the system which is highly nonlinear due to the nonlinear drag force and large displacements of the tower. The equations are solved in time domain using the Newmark-b method or Wilson-q method in each time step. It is observed that both the above methods yield identical results. 3.1.3.1. Equilibrium iteration It is important to realize that the equations of motion given in Eq. (1) are only approximations to the actual equations to be solved, since the static and kinematic variables are evaluated from the last known configuration. This may introduce errors which ultimately result in solution instability. For this reason it may be necessary to iterate in each load step or time step until the required convergence is achieved. In the Updated Lagrangian formulation, the equation of motion is given by t
K⌬u(i)⫽R⫺F(i−1)⫺Mu(i); i⫽1,2,3
(2)
The step-by-step integration used for the solution of nonlinear static and dynamic equations are given by Bathe et al. (1975). The Newmark-b method and Wilson-q method are used in the above algorithm. A computer program has been developed
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based on the above formulation for the nonlinear dynamic analysis of articulated towers. The three-leg articulated tower model has been analysed using the above program and the results are compared in Fig. 10.
4. Summary and conclusions The investigation on the three-leg articulated tower model is carried out for a different range of frequencies and wave heights as described above. The tower responses and bending strains along each leg of the tower are also recorded for the two different mass distributions. It is found that the bending moment increases with wave height for all the legs. The experimental results agree closely with the theoretical results as shown in Fig. 10. Based on the present study the following conclusions are presented: 1. The theoretical results agree well with the experimental results. 2. The maximum bending moment along the legs increases with the wave frequency and decreases with the natural frequency of the tower.
Fig. 10. (a) Bending moment variation (mass distribution II). (b) Bending moment variation (mass distribution II).
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3. The bending moment increases with wave height for all three legs. 4. The deck acceleration increases with wave height and decreases with the natural frequency of the tower.
References Arge, C., 1981. The CONAT System, Report of an Industrial Presentation. London. Bathe, K.J., Ramm, E., Wilson, E.L., 1975. Finite element formulations for large deformation dynamic analysis. International Journal for Numerical Methods in Engineering 9, 353–386. Chakrabarti, S.K., Cotter, D.C., 1979. Motion analysis of articulated tower. Journal of the Waterway, Port, Coastal and Ocean Division, ASCE 105 (WW3), 281–292. Chakrabarti, S.K., Cotter, D.C., 1980. Transverse motion of articulated tower. Journal of the Waterway, Port, Coastal and Ocean Division, ASCE 106 (WW1), 65–78. Chakrabarti, S.K., 1987. Hydrodynamics of Offshore Structures. Computational Mechanics Publications. Datta, T.K., Jain, A.K., 1990. Response of articulated platforms to random wind and wave forces. Computers and Structures 34 (1), 137–144. Kirk, C.L. and Jain, R.K., Response of articulated towers to waves and current. In: Proc. Offshore Technology Conference, OTC 2798, Houston, TX, 1977. pp. 545–552.