Parametric analysis of gravity offshore structures: Part 1

Parametric analysis of gravity offshore structures: Part 1 JAIME D O M I N G U E Z E,scuela Tecnica Superior de ln qeniero,s lndustriah, s de Secilla. Arda. Reimt Motet,des. Secilla. Spain and ENRIQUE ALARCON E T S In qenieros Industriales, Unirersidad Politecnica. Madrid 3. Spait7

INTRODUCTION The objective of this paper is to analyse the influence of the variation of some parameters used in the analysis of the dynamic response of offshore structures under the action of wind generated waves. The structural response has been obtained by stochastic methods using two discretizalion models. One with lumped parameters, using translational degrees of freedom {d.o.f.) and the other with one-dimensional finite elements. Using each of these methods the problem has been solved with several d.o.f., analysing the influence of the number of d.o.f, on the results.

to another matrix with translation and rotation degrees of freedom. In equations (2) to (5), K,,, and K, are the inertia and drag coefficients respectively: Mii is the structural mass at node.j: p is the water density: l~j is the volume of the structure lumped at ./: ('ij is the structural damping coefficient: Aj is the structural projected area in direction of flow: o-~i is the standard deviation of the relative velocities of water and structure: and ij is the water velocity at node./. if the discretization is done using one-dimensional finite elements, the terms of the structural element matrices can be written: l

SOLUTION METHOD To calculate the response by stochastic methods, the linearized equation of motion minimizing the error in the least square sense has been used~: [M.l],u~ + [ C ] , ,'~ u ~ + [ K ] ,i u l = , p,

t_~ - ~ !p 4,, = , ,P l

eli

=

Cij Jr- K,#~ "a,:i /8'Vr {C~i A2 j "~ '

i=/ i~j

Pli = K.,p l~}i:i A.

:

(2)

(3) (4)

°

P ,,i = K.p 5Ja,,/\,, ' 8,"~ r i

(5 )

The terms of [K] can be calculated by any of the known methods. Usually it is done applying static condensation 0141 1187/81/03098 0452.00 © 1981 CML Publications 98

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M., = '

"(K,,,-

(I

II

(6l d

(1)

where [M.I] is the mass matrix including the structural mass and the added mass introduced by the water: [C] is the damping matrix formed by the structural and the hydrodynamic damping; [K] is the stiffness matrix: {PI'~ and {PA~ are the inertia and drag force vectors respectively and {u} is the nodal displacement vector. If the discretization has been made using lumped parameters and considering only translational degrees of freedom the terms of the aforementioned matrices can be written: M Tii= Mi, + ( K . , - l),ot ~)

d

C~i= C~i+ j \.'8/r~K,#~ , Alx) . . d.\2 a"(x)'qi(xl'qi("~)

(7)

0

d

Pi=PI,+P~,=j

d

K,,,pV(x),qi(x)i!(xidx+ II

\ 8,rcKapx 0

(8)

.4(xl

.~ a,..(.\1.q~(x) dx

where {l(X) are the shape functions of thc structural elements, I their total length and the integrals between 0 and d extend along the submerged parts of the structural elements. To obtain some of the terms of the equations {7) and {8), it is necessary to know o-,~as a function of the various points of the structure. Thus, during the iterative process 1 required to calculate a~, it will be necessary to know the structural displacements at all points (u(x)). which can be obtained from the shape functions and the nodal displacements. Applying the nodal superposition method, equation ( 1) can be written in normal coordinates: [M*]"" ~), + [C']

'i.~,".' + [K*]',

_<,

= ',P*' ,

(9)

Parametric analysis of gravity off'shore structures (1): J. Dominguez and E. Alarcon [

~

11

~

9

10

7

8

5

6

3

4

11 -1

2 "-

I

/ / / / / / / / / a

Figure 1.

13

9._

~-]

11

/ /

b

7~ro models for discretization

In this equation, the damping matrix is not diagonal due to the hydrodynamic damping, However, to avoid a cumbersome formulation [C'] can be approximated by diagonal matrix [C*], minimizing the mean square error using the same method 1. After it, the new equation is: [M*]'~i;l. + [C*]{jJ + [K*]'O, l _-Cr,*,~_,

(lO)

Solving equation (10) by stochastic methods, the displacement spectral density function in normal coordinates can be written:

S,.,,(~o) = H*(cu)Hk(oJ)Srtp.(~u) = H*(cU)Hk(OJ)~.~puqL,,kSv,,,°(e~) |

ttt

(11)

absolute values of the response some simplifications which will equally affect both models have been made. The base and the deck of the structure are considered to be rigid and no difference in phase is taken into account for loads acting on both columns. The problem has been solved using lumped parameter models with 5, 7.9, 11, 13 and 15 d.o.f, and shape function models with 5, 9 and 13 d.o.f. In Fig. 1, two of them are shown one with 11 d.o.f, discretized using the first method and another with 13 d.o.f, discretized using the second one. Eace case has been solved taking for Km and K d the values of 1.8 and 0.5 respectively4. First, natural frequencies of each model were computed. In Fig. 2 the values of the frequencies for the 1st, 2nd, 3rd modes in each model are shown and compared with those obtained for the 13 d.o.f, discretization using one-dimensional finite elements. The calculated values of the 13 d.o.f, modal frequencies are 2.14, 7.98 and 8.72 rad/sec respectively. The Figure shows that the difference between values obtained for the first three modes using 9 or more d.o.f, is never greater than 2.5%. In a similar way a comparison has been made between the most probable extreme-values obtained at the deck, considering a fully developed sea with a power spectral density function proposed by Pierson-Moskowitz 5 and produced by a wind speed of 30 m/sec. A comparison of the computed values is showed in Fig. 3. In that case, the reference value is the response of the same model used before having a value of 27.7 cm. It can be verified that the difference with the reference value of the response of models with 9 or more d.o.f, is never greater than 3%. In order to study the influence of the wave regime on different models, the response of three of them to other regimes has been obtained. For this purpose the 5 and 11

where Hk(eJ) and H*(c~) are the complex frequency response function and its complex conjugate for modes k and j repsectively; ~0u and ~o,,k are terms of the modal matrix and Sj,a,,,is the force cross-spectral density function of the degrees of freedom I and m which can be calculated from the sea spectrum. Taking into account the transformation:

0-951 -

iuJ = [q~] O'J

0"90[

(12)

the response spectral density in global coordinates is:

/

1"10[

1.oo[

,i

(13)

,

:

--

I

]

5

1-10 1.05

S,,,~ = ~ ~ ~opj~qkSv,yj(O )

,

,

,

,

-~-

j___,a,.__

I

7

9

i

L

I

11

I

13

1-00

i

__~

0-95

RESULTS

For the analysis of results depending on the number of degrees of freedom and the discretization method used for modelling the system, the response of a gravity offshore structure with 4 columns in 150 m water depth has been studied. Since the structure is symmetrical and a wave propagating in the direction of the axis of symmetry is considered, only half of the structure has been studied. As we are going to study the difference in results but not the

0-90

1-10 1.05 1.00

~

]

15

17

i

2nd mode

k

From equations (11) and (13) and considering the wave process and the structural response as a zero mean Gaussian ergodic random process, a stochastic analysis of the response may be carried out 2"3.

,

1.051 3rd mode

I

~

!

I

5

7

9

11

]

I

I

I

I

1 I

13

15

I

I

17

1st mode I..__J

~ m m ~

0.95 0.90

I

I

I

I

i

I

5

7

9

11

13

15

17

N d.o.f.

Figure 2. Comparison of natural fi'equencies of platJbrm. One-dimensional Jinite element: - , lumped parameter

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Parametric amdvsis o/,qracity oglshorc strz~ctm'e.s Ilk J. Domilt~fm': am/ E. Alarcon

1.10

I

I

I

1

I

t

t

i

I

i

resonance peak in the response is higher for lower wind speeds: i.e. waves have dominant frequencies nearer the natural frequencies of the models. Thus the difference between their response is higher for low wind speeds. Simultaneously to the calculation of response, the accuracy achieved in the iterations was checked. In this way, it has been verified that only small improvements are obtained in the resulting accuracy during iteration. Even for wave regimes with a high value of Kculegan Carpenter number, differences between variances of velocities and displacements obtained in the first step, and those of the second step are usually less than 1",: and between second and third step differences arc less than 0.1",.

1"05

1"00

0'95

0.90!

I

I

I

I

I

5

7

9

11

13

15

N d.o,f.

CONCLUSIONS

Fiqure 3. Comparison otdeck response qt pla(lorm. One-dimensional finite element: , lumped parameter

According to the previous discussion, it can be said that in structures of the kind considered sufficient accuracy can be obtained through analysis with a relatively small number of d.o.f. In the selection of this number, attention should be paid to the fact that only a slight improvement of accuracy is reached when increasing the number of d.o.f, over a certain value with the disadvantage of more than linear increase of computer time and storage capacity. With the discretization methods, it can be said that both are accurate enough with a few d.o.f, and any of them

1.1o

1.o5 I

1.oo 0.95

J

0.90

I

lO

,j, !

t

I

""'1

f 1

0

i

i

1

30

40

50

Wind speed (m/sec) 30[

Fiqure 4. Comparison ot response q/ pla(lbrm to P.M. spectrum at dil.lerent wiml speeds. ,13, ,11:

a

26

, 5 d.o:ll

d.o.f, lumped parameter models and the system with 13 d.o.f, discretized by the shape functions method have been used for comparing their response to fully developed seas generated by wind velocities ranging from 10 to 50 m/sec. Figure 4 shows the deck response evolution in the aforementioned systems when the wind velocity changes within the specified limits. The values represented are the relations between the response of the 5 and 11 d.o.f. models and those obtained for the 13 d.o.f, case. The difference between the responses of 11 and 13 d.o.f. systems is never higher than 5',',; and decreases for higher sea states becoming almost zero for 50 m/sec. To ascertain the reason for these differences, the deck displacement spectral density for the three models for different wind speeds has been analysed. The spectral densities are almost equal for 11 and 13 d.o.f, except near the resonance peak. The 5 d.o.f, system presents higher differences than the 11 d.o.f, one at all frequencies. Figure 5 shows the spectral densities for the 10 and 30 m/sec wind speeds. The reason why the response in the 11 d.o.f, system at resonance is higher than the 13 d.o.f, one is the difference between their natural frequencies. Thus, as the 11 d.o.f. system natural frequency is a little lower, its response to waves with frequencies lower than the natural frequency will be a little higher than in the 13 d.o.f, model. In Fig. 5(a)it can be seen that the influence of the

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05

10

15

20

25

20

2.5

r ~K]/sec

tk 2O 18 ~6 ;

!4

~t2 08 or, 04 o2 i 05

h 10

I:S

Fi~lm'e 5. Spectral densities if/ response oj pla(lorm. ,5, , 11 d.o:/i (a) Wind speed = 10 m/see: (b) wiml speed = 36) m/see

Parametric analysis of gracity (~]2shore structures (1): J. Dominguez and E. Alarcon can be used successfully to solve the response problem. Furthermore the difference in computer time between the two methods is small and almost constant for any number of d.o.f. Finally, it can be said that for this kind of structure where the Keulegan Carpenter number Kc,¢15. it is possible to obtain sufficient accuracy without the iteration process. However, it is not possible to say the same for steel structures because in that case the Keulegan-Carpenter number is higher and drag forces become more important. Thus when the iterative process is not performed the error should be analysed depending on the Keulegan Carpenter number.

REFERENCES 1 2

3

4

5

Malhotra, A. K. and Penzien, J, Nondeterministic analysis of offshore structures, J. Era,t. Mech. Dir. (ASCE) Dec. 1970. 96, 985 Cartwright, D. E. and Longuet-Higgins, M. S. The statistical distributions of the maxima of a random function, Proc. R. Soc. (A) 1956, 237, 212 Davenport, A. G. Note on the distribution of the larguest value of a random function with application to gust loading, Proc, Inst. Civil Emt. 1964, 28, 187 Hogben, N., Miller, B. L., Searle, J. W. and Ward, G. Estimation of fluid loading on offshore structures, Proc. Inst. Cicil Enq. Sept. 1977, 63, 515 Pierson, W. J. and Moskowitz, L. A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii, J. Geophys. Res. 1964, 69, 5181

Parametric analysis of gravity offshore structures: Part 2 JAIME DOMINGUEZ

Escuela Tecnica Superior de lngenieros lndustriales de Secilla, Acda. Reina Mercedes, Seuilla, Spain and E N R I Q U E A L A R C O N

ETS lngenieros Industriales, Universidad Politecnica, Madrid 3, Spain

INTRODUCTION In Part 1 was presented an analysis of the influence of some parameters in the dynamic response of fixed offshore structures to forces generated by wind waves. There, different results were obtained depending on the number of degrees of freedom (d.o.f.) used for modelling the structure. Also, results were obtained and compared using two different discretization methods: the finite element method with one-dimensional elements and consistent matrices and another using lumped mass matrices (diagonal) with only translational degrees of freedom assuming static resultants for determining the nodal forces generated by the distributed wave loads. The object of this paper is to analyse the influence on response of the variation of other parameters. Thus, account is taken of the effect on the response of the inertia and drag coefficients (K,, and K~), in Morison's equation 1, taking into account the values proposed by different investigators 2-4. The importance of considering the difference of phase among the forces generated by the waves on the structure element will also be analysed. This difference in phase is a consequence of the spacing between the towers in the direction of wave propagation.

SOLUTION METHOD It is known from the previous paper that, if sufficient d.o.f. are used, the choice of discretization method has almost no influence on the results. Thus, in this paper the analysis will be performed using lumped mass matrices and applying the static resultants for determining the nodal forces. By this method solving the problem is easy because

0141 1187/81,/030101 0452.00 ©1981 CML Publications

of the use of diagonal mass matrices and because the complicated process for the linearization of the equation of motion is not needed. Before presenting the results of the analysis, some details of the solution process will be shown, mainly those in which the parameters considered have more influence. The form of Morison's equation, which expresses the hydrodynamic force per unit length for the case of flexible members in unsteady flow can be written: 1 -h)l?-~l +KmpA'i:-(K,,- 1)pA'/i (1) P(t)=2KapD(v which is valid for D/~.<~0.2, and where K,, and K a are the inertia and drag coefficients respectively; p is the water density; D is the diameter of the structural elements; A' is the cross-section area of the elements of the structure: and (i'-h) is the relative velocity of water and structure. Discretizing the structure and the forces acting on it and linearizing 5 the equation of motion of the system, it can be written:

[MT][ii I +[C]{ti] + [ K ] [ u ] = [PI} + 'tPAI = 're} (2) where [MT] and [P/} are the mass matrix and the inertia force vector respectively, which depend on K,,,; [C] and [PAl are the damping matrix and the drag force vector, which depend on Ka: and [K] is the stiffness matrix. This equation shows that the value of K,, has influence not only on the inertia forces acting on the structure, but also on the natural frequencies of the system. In the same way, any variation in the drag coefficient will affect the drag forces and the damping.

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