Stochastic response of compliant platforms to irregular waves

Ocean Engng, Vol. 10, No. 5, pp. 303-312, 1983. Printed in Great Britain.

STOCHASTIC

0029-8018/83 $3.00+.00 Pergamon Press Ltd.

RESPONSE OF COMPLIANT IRREGULAR WAVES

PLATFORMS

TO

M . AROCKIASAMY Faculty of Engineering & Applied Science, Memorial University of Newfoundland, St John's, Newfoundland, Canada

D.V. REDDY Department of Ocean Engineering, Florida Atlantic University, Boca Raton, Florida, U.S.A.

P.S. CHEEMA Dept of Applied Arts, College of Trades & Tech., St John's, Newfoundland, Canada and

H. EL-TAHAN Faculty of Engineering & Applied Science, Memorial University of Newfoundland, St John's, Newfoundland, Canada Abstract--Response of a compliant platform to irregular waves is determined using finite element method. The tower is idealized by 2-D beam elements with an elastic support at the guy lines location. The flexural characteristics of the beam correspond to the four corner members of the trusses. The guying system is modelled by an axial element with linear load deformation characteristics. A computer program based on the linearized Morison's equation and the linear (Airy) wave theory, is developed to calculate the total force based on the storm wave height data at different levels of the structure. The response of the structure to random waves is based on the spectral approach. The direct and cross spectral densities of the generalized wave forces are determined and used to obtain the spectral densities of the generalized modal coordinates and mean square response at each level. Possible extension of the method is indicated to compute the evolutionary response to nonstationary wave forces. INTRODUCTION

THE CONSIDEgABLE increase in material quantities required for large water depths restricts the conventional steel jacket and concrete gravity structures to relatively shallow water depths. The oil field developments in deep water areas such as North Sea and the Labrador coast necessitate the introduction of new drilling and production platforms. For water depths over 200 m, the compliant platform has an economic advantage over fixed platforms. The compliant platform rests on a vertical bearing foundation called a spud can and is held upright by several guylines as shown in Fig. 1. The guylines provide the upright moment and a softening restoring force to the platforms. The stochastic response of a compliant platform is determined due to random wave forces. The paper presents a dynamic analysis of a compliant platform subjected to irregular waves using the finite element method. A computer program is developed, based on the linearized Morison's equation and the linear (Airy) wave theory, and the total force is determined based on the storm wave height data at different levels of the structure. The response of 303

304

M. AROCKIASAMYet al.

FIG. I.

Compliant guyed tower (Smith et al., 1980).

the structure to random waves is based on the spectral approach. The direct and cross-spectral densities of the generalized wave forces are determined and used to obtain the spectral densities of response in general modal coordinates and the mean square response of the platform. REVIEW OF L I T E R A T U R E

The basic guyed-tower concept is described together with the design procedures to ensure that the compliant structure will safely withstand environmental forces (Finn, 1976). Salient features of the design, fabrication, and installation of the test guyed tower are described by Finn and Young (1978). The dynamic response of the tower agreed well with the theoretical predictions. Increased spreading of the wave energy reduced the maximum measured tower motions. Audibert et al. (1979) established the feasibility of the guyed tower concept in relatively deep water and made recommendations for more detailed investigations of spud can penetration, cyclic loading behaviour, tower settlements., clump weights, and anchors. Naess (1980) highlighted the results of model tests of an articulated loading platform both with and without a tanker moored to it. The tests were performed under wind, wave, and current loads on 1:70 scale model to obtain the response amplitude curves for forces and moments along the column together with the hawser tension. The effects of compliance on the behaviour of platforms in terms of the structural design parameters have been examined by Godfrey (1976). The dynamic behaviour of a compliant platform has been analysed by Smith et al. (1980) using stochastic linearization method. Reddy et al. (1981) obtained the nonstationary stochastic response of an offshore platform subjected to random forces treated as piece wise-separable. The frequency-independent modulating function is estimated by mean square minimization. STRU(~TURAL MODELLING

The structural model is based on the guyed-tower concept proposed for deep waters (Fig. 1). The tower is assumed as a four-leg trusswork with constant cross-section

Response of compliant platforms to irregular waves 1

q

305

P

2 3 4 5 6 7

I I II1~/

b/l/Ill

FIG. 2. Finite element model of the tower (Smith et al., 1980).

throughout the length providing moment compliance by means of a spud can or piles at the centre of the tower. Lateral restraint is provided by a system of guylines that provide the uprighting moment and reduce the moment effect of the lateral wave force by change in tension of the guylines. The tower is idealised as a beam with an elastic support (Fig. 2) at the guylines location with its flexural characteristics corresponding to the four corner members of the trusses. The cross-members are assumed to provide rigidity against lateral buckling. A lumped mass system is used for the dynamic analysis. A six-mass system is used (Fig. 2) with the added masses equivalent to the displaced volume. The deck mass is lumped along with the top tower mass. The guying system is replaced by a linear spring with characteristics corresponding to the restraint provided by the guylines for small excursions. Although the analysis technique presented is applicable to general motions involving all the degrees of freedom, the stochastic formulation is illustrated for motions only in one plane. IRREGULAR WAVE FORCES An analytical method is developed for computing the wave forces on cylindrical elements caused by a train of irregular waves. The method used the linearized Morison's equation and the linear (Airy) wave theory. The water surface elevation is assumed in a digitized form with N points at time interval At. The wave profile, Z(t), is decomposed into N/2+1 cosine waves and N/2 sine waves as follows: N/2

N/2

Z(t) = ~ A, cos (tOst) + ~ Bs sin (t%t) s=O

8--1

(1)

M. AROCKIASAMYet al.

306

where 2"R$

to, = N A t

' N-1

A, = 2/N ~ Z j . I cos (cost) , j=0 N-1

Bs = 2 / N 3" Zi+l sin(o~st) , /'=0

and s=0,1,2

N -~- .

.....

The computation of amplitudes A, and B, from the given values Zj is made using Fast Fourier Transform (FFT) of Cooley and Tukey (1965). The wave profile is first decomposed into sine and cosine waves using FFT and the force on a cylindrical element is evaluated from N/2

Fr(t) = Z C1~ cos (oJ,t) - Czs sin (~,t) s=O NI2

+ ~ Cx, sin (~,t) + Czs cos (o~,t)

(2)

where CDD a z C1 =

2X/'rr sinh (2 k d )

sinh (2 k d - 2 k Z 1 ) - sinh ( 2 k d - 2 k Z 2 )

+ 2k(Z2 - Zt)

"y C !

"rr D2a [sinh ( k d

-

C2

= 4 cosh ( k d )

~/

= specific weight of water;

CD

= drag coefficient;

C~

= inertia coefficient;

D

= pile diameter;

a

= wave amplitude;

k

-- 2"tr/L;

kZt)

-

sinh ( k d

-

kZ2)]

Response of compliant platforms to irregular waves

and

L

= wave length;

d

= water depth.

307

The wave force is assumed to be a zero-mean Gaussian and stationary process. The spectral and cross-spectral values of the forces at the nodes are obtained using the algorithm outlinea by Hallam et al. (1978). The raw estimate of the force power spectrum is obtained as Q~FpFp(f)----- =

hr,ff 2At m )" "i R% c o s , h=O, 1 . . . . m "IT r=0

(3a)

m

where 1

0
1/2

r--O, m

~-r= {

and g/--r

q~= RrF = -n--r =1 FqFq+r

r=0,1,2

.....

m

and smoothened by "hamming" as ~pF~ (f) = 0.54 Q ° 7 . 09 + 0.46 Q~.F. (D h h-1 h f'lh+ 1 SF.Fp Q') = 0.23 QFvF. (f) + 0.54 QF.Fp (f) + 0.23 ~ ~'.F,, (jr). 0 < h < m

and

(3b) SF.Fpm 09 = 0.54 QFrF . " (f) + 0.46 Q ( P.F. f)'-~

•

The raw cross-spectrum of the forces at different nodes is expressed as

]i 09 QhF,, (f) = Chr,,F, 09 + i EF,,F,,

(4a)

h where the co-spectrum, CV,,F,, (f) and the quadrature spectrum,

h

09

are given by r --r hrrr /, ~. ¢r (RF,,V,¢ + RFt,F,t) cos m CF,,F, (f) = A~rt r=~ '

"

h At ~ ~r (R%,,r,, - RT',r~,) sin hr'tr E F,,F,, (f) = --~ r~O ' ' m

(4b)

308

M. AROCKIASAMYet al.

The cross-covariances, R~,G and R~p~q in Equation (4b) are given by 1

, RGG

=

n~r

n - r j= 1

F.Fq,..

n--/"

~1 Fpi*" Fq, RF"G = --n - r /=

(4c)

and r=0,1,2

..... m .

The co-spectrum and quadrature spectrum are smoothened by "hamming" as shown in Equation (3b) to obtain S~-G Of). The direct spectral density SGG (f) and the cross-spectral density SGp (f) are used to determine the direct and cross-spectral densities of the generalized force~Sff, (f) (HaUam et al. 1978) as

p = l q=l

(5)

where SGG (f) = direct spectral density for p=q and cross-spectral density for p=#q at nodes p and q and ~b = normal mode vector. ANALYSIS The basic governing equation of motion for a multi-degree of freedom system is [M] {X} + [C] {,i'} + [KI {X} = {F}

(6)

where [M], [C], and [K] --- mass (sum of the structure and added water masses), damping (viscous equivalent of the structural damping), and stiffness matrices respectively, {X}, {X}, {A'}, and {F} = random displacement, velocity, acceleration, and load vectors respectively, and [c]

=

Response

of compliant

platforms

to irregular

waves

309

wherein a = a constant. The rth decoupled equation obtained by normal mode m e t h o d is

(7) where "qr = normal coordinates (r = 1, 2, ---, n), ~r = a fraction of critical damping ~r

=

{~Dr}T[c]{~)}

2tor to, = circular frequency of the rth mode; and F1,F2 . . . . , F,

=

random wave forces at nodes 1, 2 . . . . . n respectively.

The storm wave height in Fig. 3 (Chakrabarty and Cooley, 1980) has been digitized and the corresponding total wave forces computed at nodes 1, 2, 3, 4, and 5 using Equation (2) based on the linearized Morison's equation and linear (Airy) wave theory. The estimated values of the s m o o t h e n e d wave force direct power spectrum and cross-spectrum, obtained using Equations 3 and 4 respectively, are used to determine the generalised direct and cross-spectral densities from Equation 5.

30

o= ._

I

IIAlllli I AIIII l l t l l . Inll I0 i ~ , l l l l l l l l l f l l l t,1111111 ~ l l l l l l l i l Illli, 0 iIAIIII ]111111 lilt.Ill dll IIIIIII fl IIIIIII11111111111111h J I/'U Ill II II III fil 11/1211|111 ~ UIIIt'UI II I111111flll II Illld

UUlIIIIIV'II'tlIIIIIIIIlilIII'qI~III

.lC f ' l / ' t

" " I I / ! 1 ' I ~11

IM

-2°

I! I U I I! w I I I I I

'!!

I i

"

'

I I II '

I

F]c. 3.

"

~Jlnillll.,i

. I

: .i.il].i.

IIIl/lldl/ I|111, tnil I,, I, lr Ill I/I IJIMII~Il J ~ '! tl II U~111[.... II lip I I . ! ' V l " -

.301 0

filiii

I I

~ I.!u.!IU

i

i

I

1

2

3

4

5

6

7

8 9 Time in minutes

I0

11

12

13

14

Wave record for 1961 North Atlantic storm (Chakrabarti and Cooley, 1980).

15

M. AROCK1ASAbtYet

310

al.

The complex frequency response Hi(~o) relating the wave force spectrum and the platform response spectrum is given by 1

Hi(~)

(8)

The spectral and cross-spectral densities of the structure response in normal coordinates are given by (9)

5~,~1, (J~ = I~linjaf, fj (f)

and transformed into the original coordinate in the form as

Sxx (JO = L , i

~" ¢~,p~iqS~lp~l,, (ix)

p= 1 q~,=t

(lo)

Equations (10) can be rewritten in terms of only real numbers as

S~, (f) = L (b2ifl~m, (f) + 2 L p-1 Z ~bipd)ipRe{S~,, (f)} p=l

(11)

p=2 q = l

where Re{ } denotes the real part of the function. Neglecting the second term of Equation (11), the mean square response of the structure at the rth node is given by

X-2r

~o Sx~,Ogdf

(12)

NUMERICAL EXAMPLE The structure analysed is a compliant guyed tower (Fig. 1), described by Smith et al. (1980) and is modelled as lumped mass system (Fig. 2). Frequencies:

{o~} =

0.2696 1.3870 5.1870 11.5300 19.7700 27.4100

rad/sec

Response of compliant platforms to irregular waves

311

Mass matrix: 2.13140 0.39413 0.52550 0.52550 0.52550 0.52550

× 107 kg

Damping: = 0.06 . Generalized power spectral densities of the wave forces are obtained and the spectral densities of the displacements determined using Equation (9). The cross-spectral density values of the displacements are small compared to the direct spectral density values at the nodes and hence neglected in the computation of the mean square responses at the different levels (Fig. 4). 60

45

cO O. ~-

30

o

g o

0

0.04

009 Frequency

FIG. 4.

0 13

O. 18

022

(Hz)

Mean square response at levels 1, 2 and 6.

DISCUSSIONS AND CONCLUSIONS The complex cross-spectral densities take into account the spatial variation and phase differences of the exciting wave forces. The frequency of the waves lies in the range of 0.04 - 0.22 Hz. The wave record used (Fig. 3) has a duration of 15 min and the time

312

M. AROCKIASAMYet al.

interval of 2 sec for digitising of the record yields a maximum frequency of 0.25 Hz. The mean square responses determined at different levels, generally, indicate a gradual increase in the values up to 0.087 Hz and reach significantly large values in the range of 0.071 - 0.126 Hz. The maximum mean square response determined at level 1 is 56.8390 x 10 -4 m 2. In the higher frequency range of 0 . 1 3 4 - 0.25 Hz, the mean square response at all levels is negligible. Most of the actual wave height data have nonstationary characteristics. The method presented in this paper can be extended to compute the evolutionary response to nonstationary forces using the procedure outlined by Reddy et al. (1981). Acknowledgements--The authors would like to thank Dr G.R. Peters, Dean of the Faculty of Engineering and Applied Science and Dr I. Rusted, Vice-President, Memorial University of Newfoundland for the continued interest and encouragement. Appreciation is expressed to Dr K.K. Stevens, Chairman, Department of Ocean Engineering and Dr J.S. Tennant, Interim Dean of Engineering Florida Atlantic University, for encouraging the second author to complete work on this project and Mr O.S. Toope, Head of Applied Arts, College of Trades & Technology for encouraging the third author. The support of this investigation by the Natural Sciences and Engineering Research Council of Canada, (Operating Grant No. A-8119, and Strategic Grant No. G-0561 is gratefully acknowledged. REFERENCES

AUDIBERT,J.M.E., DOVER, A.R., THOMPSON,G.R. and HUBBARD,J.L. 1979. Geotechnical engineering for guyed tower offshore structures. Proc. Speciality Conf. Civil Eng. in the Ocean IV, San Francisco, California, Sept. 10--12, Vol. II, pp. 820-847. CHAKRABARTLS.K. and COOLEY,R.P. 1980. Ocean wave statistics for 1961 North Atlantic storm. Proc. ASCE, No. WWI. COOLEY, J.W. and TUKEY, J.W. 1965. An algorithm for the machine calculation of complex Fourier series. Maths Comput. 19 (90). FtNN, L.D. 1976. A new deepwater offshore platform - - The guyed tower. OTC 2688, Offshore Tech. Conf. Houston, Texas. FINN, L.D. and YOUNG, K.E. 1978. Field test of a guyed tower. OTC 3131, Offshore Tech. Conf. Houston. Texas. GODFREY, P.S. 1976. Compliant ardhng and production platforms, Proc. Conf. Design and construction of offshore structures, Oct. 27-28. Institution of Civil Engineers, London. HALLAM, M.G. HEAF, N.J. and WOOTON, L.R. 1978. Dynamics of Marine Structures: Methods of Calculating the Dynamic Response of Fixed Structures Subjected to Wave and Current Action. CIRIA Underwater Engineering Group, London, England. NAESS, A. 1980. Loads and motions of an articulated loading platforms with moored tanker. OTC 3841, Offshore Tech. Conf., Houston, Texas. REDDY, D.V. AROCKtASAMY,M. and CHEEMA, P.S. 1981. Response of offshore towers to nonstationary ice forces. Proc. POAC'81, Ouebec City, Quebec, Canada. SMITH, E., AAS-JAKOBSEN,A. and SmBJORNSSON, R. 1980. Nonlinear stochastic analysis of compliant platforms. OTC 3801, Offshore Tech. Conf., Houston, Texas.