A review of hydrodynamic load analysis for submerged structures excited by earthquakes

A review of hydrodynamic load analysis for submerged structures excited by earthquakes R. E a t o c k T a y l o r

Department of Mechanical Engineering, University College London, Torrington Place, London WCIE 7JE, UK (Received September 1980; revised November 1980)

Analysis of hydrodynamic loads on submerged structures is reviewed in the context of aseismic design of dams, intake towers and offshore platforms. The structures are considered to be flexible, and a linear fluidstructure interaction analysis is developed using the modes of vibration of a structure oscillating in the absence of a surrounding fluid. Particular attention is paid to the significance of fluid compressibility and free surface effects. As an illustration of the influence of these effects, a closed form solution and results are given for a simple rectangular reservoir-dam system. Numerical solutions are also discussed, based on finite element and boundary integral techniques for complex threedimensional problems. Introduction Man's need for energy has obliged him to undertake the construction of massive facilities in areas of high seismic risk. The design of any such facilities is highly challenging for the engineer, but in certain cases the problems go far beyond those associated with the aseismic design of landbased structures such as high rise buildings, towers, bridges and factories. A class of structures posing particular difficulties for the designer is that for which there are strong interactions between loading and response, whether through the foundation or a surrounding fluid medium. Submerged structures fall into this category: examples are dams, intake towers, and fixed and floating platforms associated with the exploitation of subsea hydrocarbon deposits or with the remote siting of nuclear power stations. In this paper we examine some of the analytical techniques available to designers of submerged structures which are to be placed in locations of high seismicity. It is inevitable that a short review such as this will be somewhat selective. Emphasis is directed towards structures founded on the ground (albeit underwater), although floating structures may also be investigated using some of the numerical methods discussed at the end of this paper. Indeed the theme behind this review of submerged structures is the importance of the dynamic interaction between fluid loading and structural response, which may be occasioned by * Based on a paper presented at the Reuniones T6cnicas 'lng. Luis Maria Machado', Departament de Estructuras, Universidad Nacional de Cordoba, Argentina, 29 July-1 August 1980 0141-0296/81/030131-09/$02.00 © IPC BuSiness Press

response in vibration modes, or by rigid body motions. It is only recently that a general theory has been applied to the analysis of these interactions. In what follows we first adopt a historical standpoint, considering methods of analysis for individual special cases, before reviewing numerical methods which may now be implemented for the general case. The types of submerged structure that have been most widely studied by civil engineers are reservoir structures (dams and intake towers) and offshore platforms. The following sections are therefore devoted to these in turn. Classical methods of dealing with fluid-structure interaction under earthquake excitation are reviewed. Simplifying assumptions are discussed, such as the neglect of compressibility or surface wave effects, and geometric idealizations. Finally it is suggested how arbitrary three-dimensional problems may be solved, subject mainly to assumptions about material behaviour rather than limitations in the fluid loading analysis. Dams

Historical overview Designers of dams understood at least 50 years ago that, during an earthquake, dynamic pressures on the dam by the retained mass of water form a most important load case. Thus in 1933 Westergaard 1 published a key paper which paved the way for our understanding of this phenomenon. His analysis considered a semi-infinite rectangular reservoir

Eng. Struct., 1981, Vol. 3, July

131

Hydrodynamic load analysis for submerged structures: R. Eatock Taylor bounded by a vertical upstream dam face; pressures were evaluated due to horizontal ground accelerations, neglecting viscosity and surface gravity waves in the water but including the influence of compressibility. The solution predicts unbounded pressures at the fundamental reservoir resonant frequency. But it was realised that deformability of the material enclosing the water would reduce the hydrodynamic pressures markedly. On the other hand, dynamic response of the dam itself would profoundly influence the fluid loading, through a complex interaction process. It was therefore essential to attempt to understand this dynamic interaction. Because of the complexity of the problem, the simple rectangular geometry adopted by Westergaard formed the basis for many subsequent analyses. The work prior to 1970 has been reviewed by Newmark and Rosenblueth. 2 Of particular importance were insights into the significance of compressibility and surface waves gained by Bustamente et al., 3 and the early studies by Chopra 4 of the influence of dam flexibility. The influence of reservoir length was also included in the investigation by Bustamente, 3 leading to the conclusion that if the length is greater than three times the dam height the semi-infinite model is adequate. Several attempts have been made to use incompressible ideal flow solutions, in view of their relative simplicity, yon Karman 5 showed how, for the vertical dam in a rectangular reservoir excited by horizontal earthquake motions, the ! appealing concept of an added mass of fluid could be derived from such a solution. This approach has recently been extended to sloping dams by Chwang 6 and Yang. 7 But parallel with these analyses have been attempts to quantify the ranges of validity of the incompressibility assumption. Some experiments by Selby and Severn 8 appeared to show that for vibrating plates in water the incompressible flow solution provided accurate predictions of measured pressures. Clarification of these results by Chopra 4,9 confirmed that it is the flexibility of the structure that governs the influence of compressibility. Based on an analysis including response of a dam in its first mode of vibration, Chopra concluded 4 that if the dam fundamental frequency (in the absence of water) is less than half of the reservoir fundamental frequency, then the water may be treated as incompressible. This important result was, however, based on an analysis which neglected surface waves, so it might be argued that certain questions were still left unanswered: this matter is considered in more detail in the next section. A more comprehensive analysis by Chakrabarti and Chopra l° extended the earlier results by Chopra 4 to include response in more than one mode of vibration of the dam in a rectangular reservoir. Studies of other dam geometries have also been reviewed by Newmark and Rosenblueth. 2 An important contribution is that of Kotsubo, examining cylindrical reservoirs of nonrectangular cross-section 11 and arch dams.12 More recently Sanchez-Sesma and Rosenblueth 13 have made a further investigation of a cylindrical reservoir with semi-circular cross-section, including compressibility effects but no free surface waves. A particular feature of this analysis is the derivation of impulse response functions for direct convolution in a time domain simulation. The more usual approach is to obtain frequency response functions, and then to use fast Fourier transform techniques if simulation of earthquake time histories is required in preference to use of ground motion spectra. All of these analyses of nonrectangular reservoirs are, however, restricted to the case of

132 Eng. Struct., 1981, Vol. 3, July

a rigid dam. It is clear that further progress in the analysis of complex geometries is dependent on the development of powerful numerical techniques. These must be capable of describing the vibrations of an arbitrary three-dimensional structure, including its interaction with the foundation, and the fluid pressures induced by a reservoir of quite general geometry. Such techniques have been developed in recent years, and they are of course applicable to a wide range of problems in addition to dams. Our consideration of these methods is therefore postponed to a later section of this paper. Interaction analysis

It is not suggested that the only way to gain further insight into the earthquake excited response of dams is through complex numerical models. Indeed the difficulties of interpreting the results of a sophisticated three-dimensional finite element simulation can act as a barrier to our understanding of the physical phenomena responsible for certain response characteristics. It is therefore considered instructive, before we dis¢uss numerical methods, to examine in more detail the simple two-dimensional case of the vertical flexible dam bounding a semi-infinite rectangular reservoir. The following analysis is very similar to that of Chopra, 4 including the choice of simplifying parameters to define the dam dynamic characteristics. Here, however, we have included the effects of free surface gravity waves in addition to compressibility, in an attempt to illustrate the regions of response where either one or both of these effects might be significant. The idealization of the problem is shown in Figure 1. The dam is assumed to be linearly elastic and the fluid is inviscid. A sinusoidally varying horizontal ground motion Uo is assumed to excite the base of the dam, and the objective of the analysis is to obtain frequency response functions for the hydrodynamic loads on the dam. These could then be used in the usual manner to evaluate the performance of the dam under prescribed earthquake ground motions. The modes of vibration of the dam, in the absence of the reservoir water, are assumed known (based on say a conventional finite element structural analysis). In the rth mode, the horizontal displacement of the upstream face is ~r(Y), where y is the vertical coordinate in a Cartesian system of axes with origin at the foot of the upstream face. The associated natural frequency is cot. The corresponding principal coordinate describing response in the rth mode is Pr(t), r = l, 2 . . . . . N, where N is the number of modes retained in the analysis (which may be much smaller than the total number of finite element degrees of freedom). If the hydrodynamic pressure on the upstream face of the dam is p(O, y ) , the equation of motion for the rth principal coordinate may be

/

/

liar l y )

i. X

Figure 1 Idealization of dam reservoir system

Hydrodynamic load analysis for submerged structures: R. Eatock Taylor written:

at the free surface (in water of depth H):

aO r(x, H )

602 -

ay

Mrr(Pr + 2~r60~r + 60~Pr) H

Or(x,~t)

g

(8)

at the reservoir bottom:

I

= f p(O, y, t) ~r(Y) dy - M r o (]o

(1)

0

The generalized structural inertia in the rth mode is Mrr, and the corresponding damping ratio (assuming classical viscous damping) is ~'r. The first term on the right-hand side of equation (1) is the generalized fluid pressure, integrated over the water depth H: this may be obtained by considering the principle of virtual work. The last term in the equation arises from the generaliz..ed inertia of the dam undergoing rigid body accelerations Uo, thus:

aOr(x, O) =0 3y

(9)

and on the dam face equation (5) must be satisfied. Additionally, a radiation condition must be met specifying that all disturbances propagate outwards (in direction of positive x). The required solution, obtained by separation of variables, is of the form: oo

Or(X,y ) = ~, (An coshotnX + B n sinhotnx ) cosh/3ny n=0

H0

(lO)

/i

Mro = ~ m(y) ~r(Y) dy

(2)

where the real or imaginary arguments an,/3n satisfy:

0

602

where the mass per unit height of the dam is re(y) distributed over a total height Ho. Since the excitation is harmonic at frequency 60, so also is the response of this linear system. We may write:

P,(t)

=

Re[frr e i¢°t ]

r = 1, 2 . . . . . N

p(x,y, t) Re[i0(x, y) e it°t]

(3a)

(3b)

=

It is convenient also to write: ~ro = -6o 2 Re[fro e iwt]

(3c)

thereby defining a parameter 15o. Using the concepts of modal analysis, we may then write the pressure as a linear superposition of effects due to the horizontal ground motion alone plus effects due to dynamic response in each retained vibration mode (up to N modes). Thus in a fluid of density p:

t~2n+/32n-

From equation (8) we fred that: 602

/3n tanh/3nH = - -

and it is convenient to defme the single real root as 13o= k, with the imaginary roots/3n - ikn (n = 1, 2 . . . . ). Then the real variables kn satisfy: --602

kn tankn -

Or(x, y) = A o exp [i(k z +

o)2/c 2)1/2] coshky

nl

+ y. An exp [i(-k~. + 602/c2)'/~x1 cosk.v n=l

(4)

o. + ~ An exp[-(k2n-eo2/c2)l/2x] coskny

where the unit pressures Or are specified by the boundary conditions at the dam face. These give:

n=n,+l

(14)

Here n i is the smallest integer value of n such that

ao,(o,y) - fr(Y)

r=0,

1 .....

N

(5)

H

k n > co/c. If n 1 = 1, then the first series disappears. The coefficients A n are prescribed by the boundary conditions on the dam, equation (5). Thus, for example, when nl = 1:

(and it is understood that Co(Y) = 1). Hence: H

p(0, y) ~r(Y) dy = 602p ~, frs s=O

0

- 4ik Os(O,Y) ~r(Y) dy

..I

0

60.02

~ Or

A 0 = (k 2 + co2/c)l/2(sinh 2kH + 2kH)

(6)

and the equations of motion may be solved provided that the integral on the right-hand-side of equation (6) may be found. If we take account of compressibility and surface wave effects, the boundary value problem for the fluid pressure and hence for the functions Or(x, y), is as follows:

V20r =

(13)

g

Making use of the radiation condition for each term of the series, we fred the final solution to be:

r=0

f

(12)

g

N

p(x, y) - 6020 ~ frrOr(x, y)

ax

(I1)

C2

(7)

H

(15a)

x f ~br(V) coshky dy 0

An -

- 4k n ( k~n_ 6o2/c2)1/2(sin 2knH + 2knH) H

(15b)

x f ~kr(y) cosknY dy where e is the acoustic wave velocity (K/p)l/2in fluid of bulk modulus K. The associated boundary conditions are :

0

n= 1,2,...

Eng. Struct., 1981, Vol. 3, July 133

Hydrodynamic load analysis for submerged structures: R. Eatock Taylor

5o~

These may more conveniently be expressed by:

An = anlrn

n = 0, 1, 2, ...

(16)

4o

where : 3o

H

lro f ~r(Y) coshky dy

(17a)

=

0

2o

10

~

Rigid (o'~ = co)

H

Irn = f ~r(Y) c°skny dy

n = 1,2,...

(17b)

0

0

10

20

30

5O

40

0 0

Figure 2 Nondimensional h y d r o d y n a m i c forces on rigid and

and a n follows accordingly from equations (14) and (15). We may now obtain the generalized forces in equation (6). We find: H

2.5

2O

II I I J I

15

n=O

0

//

r , s = 1,2 . . . . . N

Mrr(-60 2 + 2i~r60rw + 6ozr) r r N =c°2P E ~. anlrnlsnffs+602MroPo

• ""'"

I

" " " -4--

Ik//

05 0

J

I

l

I

I

01

02

03

04

]0 5

0

(19)

,v=0 rt=O

It should be recalled that/~o is the amplitude of sinusoidal ground displacement. Solution of equation (19) yields the induced dam responses. From these, the generalized hydrodynamic forces on the dam may be evaluated using equation (6). To illustrate the essential features of tiffs solution, we shall investigate a problem in which only the fundamental mode of vibration of the dam is assumed to make any significant contribution. Thus N = 1 and we have only to evaluate t51 to obtain the dam response. We will find the total horizontal force on the dam, given by equation (6) with r = 0. In this case equation (19) becomes: + 2i~'xr + 7 - 2 ) r l = C l o e o + Cll/-~1 +,)tri o

/" "

(18)

The equation of motion for the principal coordinates Pr, r = 1,2 . . . . . N, equation (1), becomes:

(-1

flexible dams, including surface wave effects

(20)

Figure 3 Nondimensional h y d r o d y n a m i c forces on rigid and flexible dams, w i t h and w i t h o u t surface wave effects ( - - ) , o L= ,2, with surface w a v e s ; ( . - - . ) , ¢ r ~ - O . 6 , with surface waves;( - - - - ) , cr~ = 0.2 w i t h surface waves; (--.--), o~ - ~, w i t h o u t surface waves

In spite of the apparent simplicity of these results, it is worth recalling that at least the first term of each inf'mite series is imaginary, whereas most (if not all) of the other terms are real. The first imaginary term corresponds to the radiation of energy away from the dam due to surface waves; any other such terms arise when co/c > kn, and correspond to acoustic wave radiation. We examine the influence of surface waves and compressibility by plotting the nondimensional hydrodynamic force magnitude : IP~I

(23)

f" - ~ah '2/)o

where : 7 = 6ol/6o

7 =

M10

Mu

and :

Panl~n

Cll= ~ - n=O Mll

oo POtnllnlon

Clo = ~ n=O

(21a)

~Jl(y) = a ( H ) + ( 1 - a ) ( ~ ) (21b)

mll

It is useful also to define :

~ POtnI~n Coo =

- n=O

(21c)

Mll

The total horizontal force on the dam is then, from equation (6):

i f , = 602M,, (Coofio + C,oP, ) in which ffl is obtained from equation (20).

134 Eng. Struct., 1981, Vol. 3, July

versus the nondimensional frequency 60H/c (defined as o). The results for a selection of cases are shown in Figures 2-4 corresponding to a dam very similar to that described by Chopra. 4 It has a triangular cross-section, with vertical upstream face, and the first mode horizontal displacement is approximated by the parabola: 2

(24)

T h e height of the dam is for simplicity taken equal to the water depth. We take a = O. 18, leading to 7 = 2.466, and we assume 5% of critical damping in this mode (S'l = 0.05). The other relevant parameters are H = 100 m, c = 1440 ms -1 and g = 9.807 ms -2. By considering dams of different stiffnesses, we are able to investigate a range of fundamental frequencies col, identified by the nondimensional parameter

O1 = (601H)/c. In Figure 2 are shown the dimensionless hydrodynamic (22)

forces for a flexible dam (ol = 0.6) and a rigid dam (Ol = oo). Above the frequency o = 0.2, the results are the same as those obtained by neglecting surface waves. The large values

Hydrodynamic load analysis for submerged structures: R. Eatock Taylor of force at o = 1.57 and o = 4.71 correspond to the reservoir resonances (at rr/2 and 31r/2 respectively, in the absence of surface waves). For the flexible dam, the structural resonance is identifiable by the large force fH = 2.10 at o --- 0.46. This frequency is well below the in vacuo natural frequency ol = 0.6, because of the added inertia effect of the reservoir water. At a frequency just above the resonance of the immersed flexible dam, the total force passes through a minumum (f/-/= 0.58): this arises from an interaction between the hydrodynamic force due to the ground motion alone (of constant phase), and the force due to resonant response of the dam (which undergoes a 180 ° phase change relative to the ground motions). To clarify the influence of surface waves, the range 0 ~< ~ ~< 0.5 has been plotted in Figure 3 for the cases ol = 0.6 and °°(rigid dam) as before, and for a more flexible dam for which ol = 0.2. Also shown is the result obtained for a rigid dam when surface wave effects are neglected (equivalent to taking g -~ 0). It is seen that as o ~ 0, the dimensionless force predicted by a theory neglecting surface waves is f n = 1.08. This corresponds to the (frequency independent) value obtained when in addition we neglect compressibility. Noting that when o > 0.2 surface wave effects are here negligible and fH is still very close to 1.08, we see that the influences of compressibility and surface waves are well separated: there is no mutual interaction. Turning to the results in Figure 3 for the flexible dams, (rl = 0.2 and 0.6, we find that below o = 0.05 these curves collapse on to the curve for a rigid dam, ol = ~ This has a minimum at o = 0.026, below which frequency the dimensionless force fH is seen to increase asyptotically to the line tr = 0. At these low frequencies the dam behaves as a rigid piston-type wave-maker: the force per unit piston stroke may be shown to be linearly proportional to frequency. Since, however, we are here plotting force per unit acceleration, in this range of frequency fH o~ o -1. This is shown in Figure 4, wherethe quantity f~r = fn(co2H/g) has been plotted (for t~l = oo). In the range 0 < o < 0.025 it is indeed found that f ~ varies almost linearly with o. It should, however, be noted that for the wave maker problem our chosen definition for the nondimensional frequency is not strictly appropriate. A more suitable parameter, based on Froude scaling, would be ~ = ~(H/g)U2: for the results shown in Figure 4, H w a s taken as 100 m, so that the value o = 0.026 corresponds to ~ = 1.2. Similarly we should use 0 in identifying the range beyond which surface waves are negligible. Taking the parameter o = 0.18 from Figure 2 we find the corresponding value 0 -- 8.3, which is equivalent

1.0

-~ o.5

o

I

o

Figure 4

0.01

I

0~02

I

(3"

0 03

[

0.04

N o n d i m e n s i o n a l f o r c e on a rigid piston w a v e m a k e r

0.05

to the criterion (H/T) > 4.2H u2 quoted by Bustamente 3 for a 5% maximum error in neglecting surface waves (where T is the response period in seconds, and H is in metres). Two conclusions may thus be drawn from such a simplified analysis of a reservoir/dam system, based on a twodimensional idealization and response in the first mode of vibration of the dam. Firstly, compressibility of the reservoir water has an insignificant effect on the earthquake excited dam response provided that the dam is flexible enough compared with the reservoir system (i.e. ol is sufficiently low). This is the conclusion reached by Chopra, 4 with the criterion that the dam fundamental frequency should be less than half of the first reservoir resonance: in the absence of surface waves this is equivalent to: lrc 4H The second conclusion is that surface waves are negligible provided that:

~1> 8.3 ~ (using the 5% error criterion given by Busamente). 3 These criteria only overlap when H > 1890 m, so that in practice the interaction between surface waves and compressibility may be safely neglected. This confirms the assumption that has been adopted historically for dams. Towers and offshore structures

Intake towers While much progress has been made in understanding the behaviour of dams, without resorting to highly complex numerical models, so have the dynamics of slender intake towers been clarified by application of analytical solutions for the fluid boundary problem. Liaw and Chopra 14 have used the solution for flow past a vertical circular cylinder, in an approach very similar to that outlined in the previous section. Their formulation included the effects of compressibility and surface waves, and examined the interaction between fluid loading and tower dynamic response. In fact the hydrodynamic analysis reduces in the limit to that of the previous section, as the tower radius/height ratio tends to infinity. As might be expected, the findings of Haw and Chopra regarding the influence of compressibility and surface waves are somewhat analogous to those discussed previously for dams. The compressibility effect was found to be negligible for slender towers but important for squat towers, because of the shorter natural periods of the latter. The fundamental period T 1 of a tower in a full reservoir of depth H was found to be only significantly affected by compressibility if cT~ < 4H (where c is the speed of sound in water). The influence of the fluid interaction is such as to increase T~ by at least 25 % (much more for a slender tower) over and above the fundamental period in an empty reservoir; if, however, the reservoir is only filled to half its working depth, the increase in T1 is negligible. The effect of surface waves was found to be significant only at very low frequencies (in fact the criterion given for the dam in the previous section is also appropriate here) and hence this is not practically o f importance because of the much higher tower fundamental frequency. There is therefore a considerable range over which neither compressibility nor

Eng. Struct., 1981, Vol. 3, July

135

Hydrodynamic load analysis for submerged structures: R. Eatock Taylor surface wave effects are significant. The fluid interaction may be represented by a sectional added mass coefficient, which depends on section elevation and tower mode shape. The results of Liaw and Chopra 14 indicate that it is clearly inadequate to rely on the added mass coefficient of unity, corresponding to rigid body sway of a tower in infinite water depth. Some rules 15 have, however, been suggested, based on simplified curves for added mass distributions, which may be used in preliminary design. Further use of the analytical solution which neglects compressibility has been made by Petrauskas. 16 Inclusion of surface wave effects leads to frequency-dependent added mass and radiation damping terms, which are expressed as generalized forces associated with vibration modes of the tower. At high frequencies these solutions are of course asymptotic to those which neglect surface waves. Warburton and Hutton 17 used the same analysis (the high frequency asymptotic solution) to quantify the influence of the surrounding water on the natural frequencies of slender and squat surface piercing cylinders. Their investigation, however, was directed more towards the dynamics of gravity platforms, which are considered next. Concrete gravity platforms

The aseismic design of offshore platforms is a much more recent requirement than that of dams, and serious interest in placing massive concrete gravity platforms in earthquake regions has been limited to about the last five years. It might be expected, therefore, that design studies for these platforms would employ certain of the newer numerical techniques described in the next section of this paper. A typical configuration is a massive cellular base, of approximately circular plan-form and height about onethird of the water depth, surmounted by one to four slender towers supporting a deck. Assessment of the dynamic interaction between such a structure and the surrounding fluid is clearly not amenable to a closed form analytical solution. Yet the investigations of such designs published by Penzien and Tseng,18 Watt et al. 19 and Watt et al. 2° do not in fact include a comprehensive fluid interaction analysis: since these were essentially feasibility studies, the effect of the fluid was simply bounded by a series of parameter studies, based on added mass coefficients for the base and towers undergoing rigid body motions. The influence of interaction has however been studied for an idealized gravity platform, consisting of a single flexible tower on a rigid base. 21 Free surface effects were included in the numerical solution, the emphasis of which, however, was response at wave frequencies. Further insight has also been gained from some experiments performed by Byrd and Nilrat, 22 using a model of solely the rigid base of a platform (which also simulates an underwater storage tank). The spring supported model was mounted on a shaking table and surrounded by water to a distance eight times the base radius. The test results were successfully correlated with a potential flow theoretical solution which included free surface effects, although for the chosen dimensions and excitation frequencies such effects were negligible. The resulting inertia coefficients showed strong departures from solutions which would be obtained from a two-dimensional flow representation. An interesting feature emanating from the study by Byrd and Nilrat 22 concerns the application of linear superposition and the relative motion hypothesis. The hydro-

136 Eng. Struct., 1981, Vol. 3, July

dynamic pressure on the spring-supported tank may be written as the sum of two components: the pressure due to the ground motions imparted to a rigidly supported tank, plus the pressure due to relative motions between tank and ground. In the case of horizontal ground accelerations, a single inertia coefficient may be used with the sum of foundation acceleration and relative acceleration to define the total horizontal force. Under vertical motions, however, this is not possible: the flow conditions during relative vertical motion without foundation motion are not the same as the flow conditions during foundation-structure motion without relative motion. In the latter case of no relative motion the structure would be subjected to the vertical inertia force from the total mass of the water column above it. F r a m e d platforms

Design of steel framed offshore structures for seismic loads has a somewhat longer history than that of concrete gravity platforms. Furthermore, although there are none of the latter type in regions of strong seismic risk, the second tallest framed platform installed anywhere to date is located in the active Santa Barbara Channel, offshore California. This is the Hondo platform in 259 m of water, the design of which has been described by Delflache et al. 23 Typical aspects of the aseismic design of such structures have been discussed by Marshall et al., 24 Arnold et al. 25 and Nair et al. 26 The American Petroleum Institute recommended practice for fixed offshore platforms 27 now includes provision for such structures, some of the background to which has been given by Bea. 28 Nevertheless, as far as the matter of prime concern to this paper is concerned, the interaction between fluid loading and dynamic response, it cannot be said with any confidence that the problem for framed structures is adequately understood. Penzien et al. 29,3° have developed an analytical procedure for analysis in the frequency domain, and examples of time domain analyses have been given by Burke and Tighe, 31 Godeau et al. 32 among others. References 29, 30 and 31 specifically include seismic analyses. In all of these studies, however, the fluid-structure interaction is represented by a modification to the Morison equation 29 having little basis on physical evidence. The problem is associated with the influence of viscosity on the fluid loading of framed structures. Whereas there is extensive (albeit still inadequate) evidence regarding the viability of the drag/ inertia decomposition of the Morison equation for wave forces, little of the experimental data is directly relevant to the earthquake excitation of a dynamically sensitive structure. A recent attempt to clarify this problem has been made by Moe and Verley. 33

General n u m e r i c a l m e t h o d s Interaction p r o b l e m s

We have emphasized throughout the importance of fluid-structure interaction, illustrating the phenomenon in some detail for the simple problem given earlier in the paper. Here we draw attention to numerical methods that have recently been developed to overcome the limitations inherent in the restriction to analytical solutions. The application of these methods has run parallel to developments in computer hardware, and some of the recent advances in both areas already make possible the solution

Hydrodynamic load analysis for submerged structures: R. Eatock Taylor of highly complex problems on medium size mini-computers. A key to this capability is obviously the minimization of the number of unknowns in the fluid-structure idealization, and it is this aspect which governs the manner in which the interaction problem is best formulated. Let us consider frequency domain analysis in which the structure and fluid are both idealized by f'mite elements (although the following discussion could be applied with only minor modifications to a boundary integral element approach). A typical formulation involving a compressible inviscid fluid has been described by Saini et al. 34 In a direct analysis this would require solution of coupled equations involving structural displacements and fluid displacements or pressures at the nodes of the structural and fluid regions respectively. The resulting equations are of large order and unsymmetric, and an entirely new solution for all unknowns (structural and fluid) is required at each frequency of interest. An improvement would be to solve a series of fluid boundary value problems corresponding to the imposition of unit velocities or accelerations at each of the degrees of freedom in the fluid-structure interface. This would lead to added mass and damping matrices (frequency dependent) associated with equations of motion in terms of the structural degrees of freedom. 'Wet' modes and frequencies could then be defined, corresponding to vibrations of the body in the surrounding fluid. The value of such modes is however questionable, since the equations of motion in terms of corresponding 'wet' principal coordinates would generally be coupled: non-proportional damping arising in the fluid (and possibly in the foundation) could lead to significant coupling terms. But for cases in which only undamped 'wet' frequencies and modes are required, an efficient method has been given by Dungar 3s which exploits features of the inverse iteration technique for obtaining eigenvalues. Consideration of the procedure described above suggests a method for the interaction analysis. In this the free vibration modes of the undamped structure oscillating in vacuo are used. The hydrodynamic actions are evaluated for responses in the lowest few such 'dry' modes. The resulting equations are coupled, but they are symmetric and may be solved without difficulty since only a small number of principal coordinates need be retained. The advantages of this 'dry' mode approach are that the number of fluid boundary value problems to be solved at each frequency is reduced to the small number of retained principal coordinates; and the evaluation of modes and natural frequencies need only be performed once and for all, since the 'dry' mass and stiffness matrices are assumed independent of frequency. The foregoing formulation in terms of 'dry' modes has been implemented by Chopra and Liaw is for the analysis of axisymmetric intake towers. Extending their study of cylindrical towers 14 (which used the analytical solution for the fluid boundary value problem), they have incorporated an axisymmetric fluid ffmite element analysis. In the context of offshore gravity platforms, a similar approach has been used by Eatock Taylor; 36 the fluid boundary value problem was, however, solved by a boundary integral idealization. Both of these studies emphasize that such a formulation, which is essentially an application of the method of substructures to fluid and structure respectively, leads to a very efficient interaction analysis for submerged structures. There is an extremely important additional complication, however, which we have not considered in the foregoing. Interaction between structure and foundation is not unique

to submerged structures, and it is therefore inappropriate to devote much space to the phenomenon in this brief review. But some considerations must be given to structure-foundation interaction, both for the sake of completeness and because of the similarity of the analysis to the above. It is found that, for this problem also, the method of substructures is extremely effective. 37 The structure is first analysed on a rigid foundation (that is to say its modes and natural frequencies are evaluated, together with the corresponding hydrodynamic actions). Displacements of the foundation are then imposed in each of its generalized coordinates, and the resulting soil impedances are obtained (from analytical solutions 38,39 or finite element analysis of the underlying soil strata). 4° Associated with each of these foundation displacements is a distortion (or rigid body motion) of the structure, for each of which a further set of hydrodynamic actions is evaluated. Finally the resulting soil impedances, fluid added mass and damping matrices, and structural genralized mass and stiffness matrices are combined and the resulting equations of motion solved. Although the procedure appears complex when summarized in this manner, it is in fact effective. A description of its application has been given by Eatock Taylor. 36 For problems of arbitrary structural configurations on complex foundations, the essential ingredients are efficient numerical techniques for the soil and for the fluid analyses. The former is not considered further here, but the latter is directly relevant to this paper: it is discussed in the following section. Analysis o f fluid loading

Numerical solution of the general three-dimensional flow past a body may in principle be obtained through some scheme of discretization throughout the fluid region, for example using finite difference or finite element techniques. In view of the ease of formulating the latter, even for regions of highly complex geometry, and because of its widespread use in analysis of civil engineering structures, the f'mite element method has been preferred for investigation of fluid loading on submerged structures. A third alternative, the boundary integral element method, has also attracted substantial interest and development in this context, because of its particularly efficient manner of dealing with infinite or semi-infmite domains. It is appropriate therefore briefly to review the advantages and disadvantages of the finite element and boundary integral element procedures. A major advantage of the f'mite element method for flow problems is the enormous effort that has already been invested in use of the method for structural analysis, and the consequential widespread availability of computer programs which may be adapted for the analysis of submerged structures. The method is well understood and extremely versatile. Thus for example the effects of compressibility and surface waves may readily be included in the governing finite element equations, which are based on a variational or weighted residual formulation (see for example Zienkiewicz). 41 Extensions may also be made to consider flow through porous media, 42 and nonlinear problems (e.g. inclusion of convective fluid accelerations) may in principle be tackled by incremental or iterative formulations. 43 The main difficulty in applying the method to flow past submerged structures arises from the extent of the fluid domain. Elements must be disposed sufficiently far from the body to be able to simulate an infinite domain and, for problems involving wave radiation (compression or surface

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Hydrodynamic load analysis for submerged structures: R. Eatock Taylor waves), the far boundary must be of the energy absorbing type. One way of overcoming this problem is the use of 'inf'mite' elements, developed by Bettess. 44 The analysis by Saini e t al. 34 of the dam reservoir interaction problem first discussed by Chopra 4 provides an instructive example of the application of these elements. Another effective approach is to use a localized finite element idealization close to the structure, coupled with an analytical solution in the far field, as described by Bai and Yeung. 4s The far field behaviour is represented by an eigenfunction series of wave solutions, the coefficients of which are constrained by a variational principle to provide continuity with the nodal values on the localized finite element boundary. A review of these adaptations of the finite element method has been given by Zienkiewicz e t al. 46 This difficulty of representing the infinite domain may, however, be avoided by a boundary integral element technique. In this the fluid boundary value problem is transformed, by use of Green's theorem, into an integral equation on the submerged surface of the structure. The exact integral equation is then approximated by a boundary element discretization, leading to a set of linear algebraic equations for the flow parameters on the surface or anywhere in the fluid region. The method was developed by Hess and Smith 47 for potential flow and has recently attracted much attention (reviewed for example by Brebbia and Walker, 48 and by Garrison 49 in the specific context of hydrodynamic loading) as an alternative to the use of finite elements. In contrast with the latter, only the submerged surface is discretized rather than a large volume of fluid; hence the order of resulting algebraic equations is significantly reduced. The boundary integral element procedure has recently been applied to the dynamic fluid-structure interaction problem for offshore structures. 21,36 The fluid was assumed incompressible, but the effects of surface waves were included. Results were obtained for an idealized concrete gravity platform founded on a uniform elastic halfspace. The substructuring procedure described above was used, based on a modal analysis. The structure had a single flexible tower on a large rigid base;thus the 'dry' modes incorporated in the interaction analysis were rigid body swaying and rocking about the seabed, and the first two distortion modes of the dry fixed based superstructure. The results were found to differ significantly from those based on a simple strip theory interaction analysis, in which two-dimensional added mass coefficients were used and surface wave effects were neglected. An aspect of this difference having particular relevance to seismic design is the higher fundamental vibration frequency of total fluidstructure systems, computed by the more sophisticated interaction analysis.

Future developments The preceding brief review suggests the feasibility of developing seismic analysis techniques for highly complex submerged structures. There are, however, several areas requiring further attention. One is the development of significantly more efficient numerical analyses of the fluid loading. A method holding out considerable promise 46, so involves a combination of finite element idealization in the fluid adjacent to the body, with a boundary integral element discretization to represent the far field behaviour. In this way advantages of the two methods may be effectively

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combined. A second area requiring extensive research, again based on numerical techniques, s t is the development of practical capabilities for nonlinear analyses. This is clearly of major importance if design criteria are to be defined in terms of controlled damaged but prevention of complete failure during the maximum credible earthquake. 52 A third area for research concerns the influence of non-rigid boundaries (for example the bottom of a reservoir) on seismic response. This is also related to the influence of phase differences between ground motions at different locations, and of course the distribution between ground motions at surface and bedrock levels. Much insight into this latter problem has been gained in recent years from seismic studies of nuclear power plants, sz There appears, however, to be another major requirement, complementing the above mentioned analytical procedures (and others not included here). This is the need for full scale data on the performance of structures during earthquakes and other forced excitation. In the context of this paper, instrumentation projects on dams, intake towers and offshore structures are relevant. Limited dynamic data have been measured on dams, both from forced vibration tests and during actual earthquakes, s3 56 On offshore structures more extensive programmes of instrumentation have recently been initiated, primarily with a view to developing techniques for monitoring integrity by observation of natural frequencies and mode s h a p e s , sT,s8 and for measuring damping. 59-61 Other limited data from offshore platforms have been used in assessing procedures for wave force prediction; but a full scale study of fluid-structure interaction comparing theoretical and experimental response transfer functions in waves or earthquakes has not yet been accomplished. A preliminary comparison of theory with full scale response data during six earthquake records has been given by Ueda and Shiraishi, 62 but this is concerned with a tanker terminal platform in only 22 m water depth. The first gap which should be filled in tire compilation of comprehensive full scale earthquake response data for submerged structures is that of responses to low level seismicity. It is assumed that under these conditions systems will behave essentially in a linear manner. Much harder to achieve will be a reliable collection of useful data pertaining to strong motion earthquakes, during which significantly nonlinear effects might be anticipated. It is in precisely this area that seismic analysis of submerged structures is currently deficient.

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