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A numerical study on the dynamic response of a floating offshore wind

turbine system due to resonance and nonlinear wave

Takeshi Ishihara1)

, Pham Van Phuc1)

, and Hiroyuki Sukegawa2)

1)Dept. of Civil Eng., The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, Japan

2)R&D Center, Tokyo Electric Power Company, 4-1 Egasaki-cho Tsurumi-ku, Yokohama, JAPAN

ABSTRACT: A FEM code was developed to predict the dynamic response of a floating offshore wind turbine

system in the time domain, employing the Morison’s equation and Srinivasan’s model to calculate the hydrody-

namic drag forces and inertia forces on the floating structure, and quasi-steady theory to calculate the aerody-

namic forces on wind turbines. The responses predicted by the proposed numerical model show good agreements

with experiments. Morison’s equation provides satisfactory prediction for the surge motion, but overestimates the

heave motion of the floating structure with vertical column. The drag force model proposed by Srinivasan et al.gives good predictions. Elastic deformation plays an important role in dynamic response of the floating structure.

The predicted response was underestimated when the elastic deformation is ignored. The nonlinearity of wave be-

comes dominant for the water depth less than 100m. The elastic modes might be resonant with the higher order

harmonic component of the nonlinear wave, resulting in the increase of the dynamic response of the floating

structure.

KEYWORDS: Floating offshore wind turbine system, Morison’s equation, Srinivasan’s model, hydrodynamic

damping, aerodynamic damping, elastic deformation, resonance, nonlinear wave.

1 INTRODUCTION

Accurate prediction of dynamic response of floating structures due to resonance and nonlinear wave is essen-

tial to evaluate the performance and to enlarge the applicability of the floating offshore wind turbine system when

the ultimate limit plays a dominant role in the design at the intermediate water depth. There are two approaches to

investigate the dynamic response of the floating structures. One is the application of the Morison’s equation [1],

and the other is the linear potential theory [2]. The linear potential theory is raised from the assumptions of zero

viscosity of the fluid, while the Morison’s equation considers the effect of the viscosity represented by the nonlin-

ear hydrodynamic drag force. After FLOAT project [3], several floating offshore wind turbine systems (see, [2],

[4], [5], [6]) have been proposed based on the concepts in the oil and gas industry such as semi-submersible, spar

and tension leg platform. Some preliminary studies have been done to assess the platform using the numerical

models that applied in the design of the platform used by oil and gas industry in the deep water. Hederson et al.

[6] used the Morison’s equation for the floating structures with large diameter sub-structures. They ignored the

hydrodynamic drag force and solve the linearized equation of motion in the frequency domain. This simplification

might lead to some loss in accuracy to predict the dynamic response of the floating structures with small diameter

sub-structures, such as the floater proposed by Ishihara et al. [7] for the economical design.

In the present study, a FEM code with Morison’s equation and Srinivasan’s model was developed in the time

domain to investigate the effects of the hydrodynamic damping and the nonlinear wave on the response of the

floating structures. The performance of the code for the dynamic response analysis of the semi-submersible float-

ing structures was verified by a model test.

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2 NUMERICAL METHOD OF FULL DYNAMIC SIMULATION

2.1 Governing equation

The general formulation of the differential equation of motion for a floating offshore wind turbine system can

be written as

[ ]{ } [ ]{ } [ ]{ } { } { } { } { }G R H W X C X K X F F F F + + = + + + (1)

where [ ] is a mass matrix, [ ]C is a damping matrix, [ ]K is a stiffness matrix of structure, X and its deriva-

tives are unknown vectors of 6 degree of freedom (3 translations and 3 rotations) and their derivatives. The terms

in right side of the equation (1) are the external force vectors acting on the system and typically varies with time,

where { }GF is the mooring force, { }RF is the hydrostatic restoring force, { }H F is the wave exciting force,

{ }W F is the aerodynamic force.

The mooring force is defined as follows,

{ } [ ]{ }G GF K X = −

(2)

where the mooring stiffness [ ]GK is determined by a result of catenary’s analysis from the steady forces, includ-

ing the tidal current force, wind force and wave drift force.

Under assumption of infinitesimal displacement theory, hydrostatic restoring force can be simplified by the

first-order hydrostatic restoring force coefficient [ ]RK [8] as follows.

{ } [ ]{ }R RF K X = − (3)

[ ]

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

w wR

X

Y

gAK

W GM

W GM

ρ

−

= − ×

− ×

(4)

where wρ is the density of water, g is the gravity acceleration, wA is the still surface area, W is the weight of

the model, X GM and Y GM are the meta-center height in X and Y direction, respectively.

In order to calculate the wave exciting force on the floating structure, the modified Morison’s equation by

Sarpkaya et al. [9] can be adapted as shown in the equation (5).

( )0.5H D EW EM w D w M aF F F F C A u X u X C Vu M X ρ ρ = + + = − − + − (5)

( )1a w M C V ρ = − (6)

where DF is drag force, wρ is water density; u is particle wave velocity, A and V are the area and volume

of the element, a is called as added mass coefficients, X is the velocity of the moving element,

DC and C are hydrodynamic drag and inertia coefficients, respectively.

The drag and inertia coefficients depend on the cross-sectional shape of the structure, which was commonly

given as the functions of the Keylegan-Carpenter number maxcK u T D= as shown by Offshore Standard DNV-

OS-J101[10], where maxu is the maximum wave particle velocity at still water level, ν is the kinematic viscosity

of water, T is the period of the waves,. Since the relative wave particle velocity as well as the first term of the

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wave exciting force contains the velocity of the moving element, the hydrodynamic damping is automatically

taken into account during the simulation.

Since the Morison’s equation cannot predict forces acting on the bottom of the vertical cylinder accurately, the

added inertia force acting on the bottom of the base floater proposed by Haslum [11] and the drag force proposed

by Srinivasan [12] are used, as shown in equations of (7) and (8),

( ) ( )3

1 2 3 2 , 2.0a w M M M C D C ρ π = − = (7)

( )3 3 3; 2 ; ( 20%)D ED ED aF C X C M M ζω ζ = − = + = (8)

where D is the diameter of the bottom of the base floater, 3X is the velocity of the moving element in the verti-

cal direction, ω is the angular frequency of heave mode, 3M and 3a are the structure mass and hydrody-

namic added mass of floater in the heave direction, respectively.

The quasi-steady aerodynamic theory is used in the calculation of the aerodynamic forces, in which the drag,

the lift force and the moment are estimated by using aerodynamic coefficients and the relative wind speed as fol-

lows.

{ } { } ( ) ( ) ( ){ }2 2 2 2, , 0.5 ,0.5 ,0.5W D L M D L M F F F F dC V dC V d C V ρ α ρ α ρ α = =

(9)

where ρ is the density of the air, d is a reference dimension of the wind turbine, DC is the aerodynamic drag co-

efficient, LC is aerodynamic lift coefficient, C is aerodynamic moment coefficient, α is angle of attack of the

relative wind speed V . Here, the relative wind speed with respect to the moving element can be written as

V U X = − (10)

where U is the wind velocity, X is the velocity of the moving element. Since the relative wind speed as well as

aerodynamic force terms contains the velocity of the moving element, the aerodynamic damping is automaticallytaken into account during the simulation.

2.2 Numerical scheme

In this study, the mooring force, hydrostatic restoring force and the added inertia force were moved to left side

of the equation to solve as follows,

[ ] [ ]( ){ } [ ] [ ]( ){ } [ ] [ ] [ ]( ){ } { } ( ) { }a D G R D EW W M X C C X K K K X F F F + + + + + + = + + (11)

A FEM code based on above equation was developed to predict the eigenperiods and dynamic responses of the

floating offshore wind turbine system. A brief description of the code is summarized in Table 1. The beam ele-ments were used for the discretization and the mass of each element was concentrated at its nodes constructing a

symmetrical lumped mass matrix.

The damping matrices is defined by Rayleigh damping method [13] and it can be written as follows,

[ ] [ ] [ ]C M K α β = + (12)

[ ] [ ] [ ]( ) [ ]1 1D aC M M K α β = + + (13)

where ,α β is the function of the eigen-periods and structural damping ratios, 1 1,α β is the function of the eigen-

periods and damping ratios of the system in the vertical direction.

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Table 1. A brief description of the FEM code

Dynamic analysis Direct numerical integration, the Newmark methodEigenvalue analysis Subspace iteration procedure

Element type Beam element

Formulation Total Lagrangian formulation

Damping Rayleigh damping

3 NUMERICAL RESULTS

A semi-submersible floating structure as shown in Figure 1 was modeled with beam elements including 186

nodes and 188 elements In order to evaluate the performance of the developed FEM code. Here, the mooring was

simplified by the longitudinal linear spring. The eigenvalue analysis was carried to calculate the mode of thefloating structure and the eigenperiods were compared with the natural periods obtained from the decay test [7].

The dynamic response analysis was also conducted in the same condition of the experiment and the predicted re-

sponses were compared with those from the water tank test.

Incident waveIncident wave

Figure 1 A semi-submersible FOWTS modeled with beam elements

In the analysis, the hydrodynamic drag and inertia coefficients were defined as the functions of Kevlegan-

Carpenter number recommended by Offshore Standard DNV-OS-J101 [10]. Here, these coefficients of the rec-

tangle connecting girders were modified by the ratio between the rectangle and the cylinder column described by

Motora et al. [8]. The Kevlegan-Carpenter number was simplified by the incident wave height and diameter of

column by the equation (14).

cK UT D H Dπ = = (14)

For the calculation of the aerodynamic force, the aerodynamic drag coefficient of 0.6, 0.6 and 1.3 was used for

the tower, nacelle and blades, respectively in the survival condition, and the rotor was modeled by the thrust act-

ing on the hub of the wind turbine with thrust coefficient 0.33t C = corresponding to the operating condition. The

wave particle velocities and accelerations were generated by the Airy theory for the linear wave and the stream

function for the nonlinear wave.

A test with 1:150 scale model was carried out under Froude similarity law in the water tank with wind tunnel

owed by the National Maritime Research Institute of Japan [7]. The motion of the central column and the strains

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of the horizontal brace were measured. The strains were used to calculate the excited bending moment on the

brace. Experimental conditions are shown in Table 2.

Table 2 Experimental conditions

Prototype 1:150 scale modelWind turbinecondition U(m/s) H (m) T (s) U (m/s) H (m) T (s)

Operating 3-25 1.5-4.6 5.5-9.6 0.2-2.0 1.0-3.1 0.4-0.8

Extreme 50 12.0 15.5 4.1 8.0 1.3

Experimental

condition0,25,50 3,6,12 7-37 0,2,4 2,4,8 0.6-3.0

Environmental condition (U: wind speed, H: wave height; T: wave period)

3.1 Eigenvalue analysis

A semi-submersible FOWTS was modeled with beam elements including 186 nodes and 188 elements. Theeigenvalue analysis was carried out to calculate the mode of the FOWTS and the results were compared with the

natural periods obtained from the decay test [7]. The eigen-periods in Table 3 show good agreement with the ex-

periment in the surge, heave and pitch direction. The higher mode corresponding to the elastic deformation was

also obtained.

Table 3. Comparison of measured and predicted natural periods

Mode 1 2 3 4

Natural period (s) 36.7 34.0 33.3 -

Eigen-period (s) 36.0 34.0 33.0 7.3

3.2 Hydrodynamic damping effect

Figure 2 presents the variations of normalized responses with the wave periods obtained from the numerical

simulation and the experiment. As expected, the responses of the surge motion vary linearly with the wave height

in the ranges of periods far from the eigen-periods, but they show significant peak around the eigen-period and

decrease significantly when wave heights increase. The predicted responses show good agreement with the ex-

periment in the nonresonant region. The Morison’s equation provides a good prediction for the surge motion, but

it overestimates the responses when the drag force is ignored in the resonant region. The heave motion is overes-

timated when the only Morison’s equation was applied, while the predicted heave motions by the Srinivasan’s

model show good agreement with experiments.

Figure 3 shows variations of responses and damping ratios with the wave height. The equivalent hydrody-

namic damping ratios increase with the increase of the wave height. This is why the surge and heave motions de-

crease for higher wave decrease.

0

4

8

Exp.(H=3m)Cal.(H=3m,F

D=0)

Cal.(H=3m,Morison)Exp.(H=12m)Cal.(H=12m,Morison)

2X/H

T(s)

6 12 18 24 32 360

2

4Exp.(H=3m)

Cal.(H=3m,Morison)

Cal.(H=3m,Srinivasan)

Exp.(H=12m)

Cal.(H=12m,Srinivasan)2Z/H

T(s)

6 12 18 24 32 360

4

8

Exp.(H=3m)Cal.(H=3m,F

D=0)

Cal.(H=3m,Morison)Exp.(H=12m)Cal.(H=12m,Morison)

2X/H

T(s)

6 12 18 24 32 360

2

4Exp.(H=3m)

Cal.(H=3m,Morison)

Cal.(H=3m,Srinivasan)

Exp.(H=12m)

Cal.(H=12m,Srinivasan)2Z/H

T(s)

6 12 18 24 32 36

(a) Surge (b) Heave

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Figure 2 Variations of predicted and measured responses with the wave period

0

4

8

0

4

8

D a m p i n g r a t i o ζ

h ( % )

H(m)

2X/H

3 6 9 12

Surge

Damping ratio (Morison)

0

2

4

20

30

40


h ( % )

2Z/H

H(m)3 6 9 12

Heave

Damping ratio (Srinivasan)

0

4

8

0

4

8


h ( % )

H(m)

2X/H

3 6 9 12

Surge

Damping ratio (Morison)

0

2

4

20

30

40


h ( % )

2Z/H

H(m)3 6 9 12

Heave

Damping ratio (Srinivasan)

(a) Surge (b) Heave

Figure 3 Variations of responses and damping ratios with the wave height

3.3 The aerodynamic damping effect

A dynamic response analysis was also carried out to investigate the effect of wind turbine on the floating

structure in the operating and the survival conditions. Figure 4(a) presents variations of responses of the surge

motion with the wave periods in the operating condition. As the experiment [7], the predicted peak responses

around the resonant period in the windy day were less than that in the calm day, and the effect of the aerody-

namic damping is more significant for the cases with smaller wave height, having lower hydrodynamic damping.

Figure 4(b) shows the variations of responses of the surge motion in the survival condition. There is little depend-

ency on the wind speed. This is because the aerodynamic dumping from the blade is much smaller than the hy-

drodynamic damping.

0

2

4

6

8

Exp.(H=3m)

Cal.(H=3m)

Exp.(H=3m;U=25m/s)

Cal.(H=3m;U=25m/s)2X/H

T(s)

6 12 18 24 32 36 0

2

4

6

8

Exp.(H=12m)

Cal.(H=12m)

Exp.(H=12m;U=50m/s)

Cal.(H=12m;U=50m/s)2X/H

T(s)

6 12 18 24 32 36

(a) In the operating condition (b) In the survival condition

Figure 4 Variations of responses of the surge motion with the wave periods

3.4 The effect of elastic deformation

To investigate the effect of elastic deformations to the dynamic response of the floating structure, two models

with the beam elements were constructed using the structural properties of the prototype floating structure includ-

ing the number of elements and nodes as the same as those used in the previous section. The SM570 steel material

defined by Architecture Institute of Japan [14] was used for the horizontal brace and others were rigid in elastic

model. The structural damping ratio of the horizontal brace was 0.8% for the steel material as mentioned by Bur-

ton et al. [15]. The dynamic response analyses for both models were performed for the regular wave periods from

1.0s~30.0s with the wave height of 12m in the survival condition. The incident wave direction was -90 degree.

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Figure 5 shows the variations of the excited bending moments with the wave periods for the rigid and elastic

models. The bending moments for both models show good agreements with the experiments in the nonresonant

region, but the rigid model underestimates the responses in the resonant region. The peaks of response found at

the period near 7 seconds correspond to the eigenperiods of the elastic model.

Exp.

Cal.(Transient response)

Cal.(Applied load)

MZ/H (kN)

4e+4

2e+4

0e+0

T(s)

6 12 18 24 32 36

Exp.


Cal.(Applied load)

MY/H (kN)

4e+4

2e+4

0e+0

T(s)

6 12 18 24 32 36

Exp.


Cal.(Applied load)

MZ/H (kN)

4e+4

2e+4

0e+0

T(s)

6 12 18 24 32 36

Exp.


Cal.(Applied load)

MY/H (kN)

4e+4

2e+4

0e+0

T(s)

6 12 18 24 32 36

(a) Surge (b) Heave

Figure 5 Variations of excited bending moments with the wave periods

3.5 The nonlinear wave effect

Waves at reference sites of water depth 50,100,200m with the maximum wave height H=22.23m and wave pe-

riod T=15.5s were selected to investigate the characteristics of the nonlinear wave. Figure 6 and Figure 7 show

regular wave theory selections and water elevations for several sea depths, respectively. The wave at the water

depth of 50m near water breaking limit requires to use the 9th order of the stream function. At 100-

200m sea depth, 1st harmonic component of the wave elevation is dominant, while about 25% of

the wave elevation is contributed by 2nd harmonic component at 50m depth.

0.0001

0.001

0.01

0.001 0.01 0.1

d=50m d=100m d=200m

H/gT

2

d/gT2

Shallow waterbreaking limitH/d=0.78

Hb

0.9Hb

Deep water breaking limitH/L=0.14

Stokes's 5th orstream function 3

Linear/Airy orstream function

Stream function

357911>11

-25

0

25

80 90 100 110 120

d=50m

d=100md=200m

Water elevation(m)

Time(s)

0.0001

0.001

0.01

0.001 0.01 0.1

d=50m d=100m d=200m

H/gT

2

d/gT2

Shallow waterbreaking limitH/d=0.78

Hb

0.9Hb

Deep water breaking limitH/L=0.14

Stokes's 5th orstream function 3

Linear/Airy orstream function

Stream function

357911>11

-25

0

25

80 90 100 110 120

d=50m

d=100md=200m

Water elevation(m)

Time(s)

Figure 6 Regular wave theory selections Figure 7 Water elevations for several sea depths.

A dynamic response analysis of the floating structure was carried out for these reference sites. The applied

load and surge-exciting bending moment on the horizontal brace was investigated. The applied load in surge di-

rection at 50m depth is larger than the other cases and has the second harmonic component as shown in Figure 8.

As a result, the surge-exciting bending moment at 50m depth is 1.8 times larger than the other cases, as shown

in Figure 9. The Fourier harmonic components of the surge-exciting bending moment indicate that the second

harmonic component for 50m sea depth has a large contribution to the response of the floating structure. This is

because the second harmonic component of wave (T=7.8s) resonates with the elastic mode (T=7.3s) of the float-

ing structure.

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-4 105

0

4 105

80 90 100 110 120

d=50md=100md=200m

MZ(kNm)

Time(s)

0

1 105

2 105

d=50md=100md=200m

MZ

(kNm)

15.5 7.8 5.2 3.9 3.1

Wave Period T(s)

-4 105

0

4 105

80 90 100 110 120

d=50md=100md=200m

MZ(kNm)

Time(s)

0

1 105

2 105

d=50md=100md=200m

MZ

(kNm)

15.5 7.8 5.2 3.9 3.1

Wave Period T(s)

(a) Time series (b) Fourier harmonics

Figure 8 Time series and Fourier harmonics of applied loads in the surge direction

0

1 105

2 105

d=50md=100md=200m

MZ(kNm)

15.5 7.8 5.2 3.9 3.1

Wave Period T(s)

Elastic mode’s period

-4 105

0

4 105

80 90 100 110 120

d=50m

d=100md=200m

MZ(kNm)

Time(s)

0

1 105

2 105

d=50md=100md=200m

MZ(kNm)

15.5 7.8 5.2 3.9 3.1

Wave Period T(s)

Elastic mode’s period

-4 105

0

4 105

80 90 100 110 120

d=50m

d=100md=200m

MZ(kNm)

Time(s)

(a) Time series (b) Fourier harmonics

Figure 9 Time series and Fourier harmonics of surge-exciting bending moment

4 SUMMARY AND CONCLUSIONS

A FEM code with the Morison’s equation and Srinivasan’s model was developed in the time domain to inves-

tigate the effects of the hydrodynamic damping and the nonlinear wave on the response of the floating structures.

A dynamic response analysis in the intermediate depth was also carried to clarify the effect of the nonlinear wave

on the response of the floating structure. The following results were obtained.

1) The predicted eigenperiods show good agreement with experiment in the surge, heave and pitch direction

2) The response is strongly influenced by hydrodynamic damping in the resonance region, but it is not affected

much in other regions. Morison’s equation provides satisfactory prediction of the surge motion, but overesti-

mates the heave motion of the floating structure with vertical column, while the drag force model proposed

by Srinivasan gives good prediction for the heave motion.

3) The dependency of the peak surge on the wind speed in the operating condition was observed due to the aero-

dynamic damping for the wind turbine, but it is not in the survival condition. It indicates that the surges

around the resonant period will be overestimated in the operating condition when the interaction between

wind turbines and floating structures are neglected.

4) Elastic deformation plays an important role in dynamic response of the floating structure. The predicted re-

sponse is underestimated when the elastic deformation is ignored.

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5) The nonlinearity of wave becomes dominant for the water depth less than 100m, and the elastic modes might

be resonant with the higher order harmonic component of nonlinear wave, resulting in the increase of dy-

namic response of the floating structure.

REFERENCES

[1] J.R Morison, et al. , The force exerted by surface waves on piles, Petroleum Transactions, AIME, Vol.18,

149–157, 1950.

[2] B.H. Bulder, M.Th. van Hees, A. Hendeson, R.H.M. Huijsmans, J.T.G. Pierik, E.J.B. Snijders, G.H.Wijnants,

and M.J.Woft, Study to feasibility of and boundary conditions for floating offshore wind turbines, Public re-

port 2002-CMC-R43, ECN, MARIN, MSC, Lagerwey the Windmaster, TNO, TUD, 2002.

[3] Tong KC, Quarton DC, Standing R, Float-a Floating Offshore Wind Turbine System, in Wind Energy Con-

version, Proceeding of the 1993 BWEA Wind Energy Conference. York, England. p. 407-413, 1993.

[4] Barltrop N. Multiple Floating Offshore Wind Farm (MUFOW). Wind Engineering, 17(4), 183-188, 1993.

[5] Bertacchi P, Di Monaco A, M. dG, Ferranti G. ELOMAR-A Moored Platform for Wind Turbines. Wind Engineer-

ing , 18(4),

189-198, 1994. [6] A. Henderson, M. Patel, Rigid-Body Motion of a floating offshore wind farm, Int. Journal of Ambient En-

ergy, Vol.19, No.3, pp 167-180, 1998.

[7] T. Ishihara, P.V. Phuc, H. Sukegawa, K. Shimada, T. Ohyama, A study on the dynamic response of a semi-

submersible floating offshore wind turbine system Part 1: water tank test with considering the effect of wind

load, ICWE12, Australia 2007.

[8] S.Z. Motora, T.O. Koyama, M.T. Fujino, H.A. Maeda, Dynamics of ships and offshore structures, 1997.[in

Japanese]

[9] T. Sarpkaya, M. and Isaacson, Mechanics of wave forces on offshore structures, Van Norstrand Reinhold,

1981.

[10] Offshore Standard DNV-OS-J101, Design of offshore wind turbine structures, 2004.

[11] H.A. Haslum, Alternative Shape of Spar Platforms for Use in Hostile Areas, Offshore Technology Confer-

ence, 1999.[12] Srinivasan, N., et al, Damping controlled response of a truss pontoon semi-submersible with heave plates,

Proceedings of the 24th International Conference of OMAE, Halkidiki,Greece, 2005.

[13] J.W. Tedesco, W.G. McDougal, C.A. Ross, Structural dynamic theory and application, Addison Wesley

Longman, Inc., pp 370-371, 1998.

[14] Architectural Institute of Japan, Design Standard for Steel Structures -Based on Allowable Stress Concept,

2005.

[15] T. Burton, D. Sharpe, N. Jenkins, E. Bossanyi, Wind Energy Handbook, Wiley, England, 2001.

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