Analysis of the base-excited response of intake-outlet towers by a Green's function approach

Analysis of the base-excited response of intake-outlet towers by a Green's function approach A. N. w i m a m s

Department of Ovil and Environmental Engineering, Cullen College of Engineering, University of Houston, Houston, Texas 77204-4791, USA (Received December 1989; revised April 1990)

The dynamic response of a circular, intake-outlet tower structure subjected to high-frequency horizontal ground excitation is investigated theoretically. The tower is idealized as a one-dimensional beam of uniform flexural rigidity and mass per unit length, the fluid is assumed to be linearly compressible and to undergo small-amplitude irrotational motion. The presence of the fluid contained within, as well as that surrounding the structure, is considered in a consistent manner. Both the structural and fluid motions are expressed in terms of appropriate Green's functions leading to a pair of coupled line integral equations which are solved numerically for the fluid velocity potentials on the structure. It is found that the presence of both the interior and exterior fluids can signicantly influence the dynamic response, especially for those structures possessing larger diameter to water depth ratios (squatty towers).

Keywords: base-excited response, Green's function, intake-outlet towers

The continued provision of the water supply in a region following an earthquake will only be possible if, as well as the survival of the dams in the area, the inlet and outlet strictures associated with the reservoirs behind the dams maintain their structural integrity and remain operational. In the case of earth and rcckfill dams, the outlet works usually take the form of reinforced concrete towers located in the reservoir t. In some cases, these towers are free-standing, while in others they are con- nected to the s,jLrrounding land or to the dam itself by access bridges. It is recognized that any study of the dynamic reaponse of such stngtums to earthquake excitation may be significantly influenced by fluid structure interactions ~,3 and these effects should be considered in the design process 4'5. At the high frequencies typical of ~ excitation, surface wave effects are relatively unimpurtaat while fluid compressibility may become an important factor, especially for towers sing diameter to water depth ratios.

There are several solutions for the earthquake-induced hydrodynamic Im~mres o~.fi~xible, submerged, cylindrical structures surrounded by a compressible fluid. Results for: cylindrk~ towers of ciro~Aar cross-cross section, otmined by diffenmt techniques, have been p l by Goto and TokP, Liaw and Chopra 2, Mei et aL 7, Williams', and Tmmka and Hudspeth 9, and for

0141-0296/91/010043-11 © 1991 Butterworth-Heinemann Lid

cylinders of elliptic cross-section by Kotsubo 1°. The contribution of the fluid added mass towards changing the natural period of vibration of the structure and the significance of fluid compressibility and hydrodynamic radiation damping have been investigated. However, in the above studies no consideration was given to the dynamic effects which may be caused by the additional fluid present inside the tower. For slender towers, it has been proposed that an appropriate idealization is to treat the interior fluid as moving rigidly with the tower '.s. Recently, Goyal and Chopra " - t s have published a series of papers on the dynamics of axisymmetric intake- outlet towers of non-uniform cross-section in which both foundation interaction effe .'~ts and the influence of the exterior and interior fluids imve been considered. The interior and exterior fluids were considered as incom- pressible and the resulting fluid-structure-fonndation- soil system solved by a finite element approach. Goyal and Chopra have presented freqaency regxmse functions as well as actual earthquake respomes, have developed simplified design procedures to determine the maximum e a . ~ e forces on intake.o~et towers and have provided a simplified method to calculate the added mass due to the presence of the surrotmding and con- rained fluids.

In the present work, Williams s integral equation

Eng. Struct. 1991, Vol. 13, January 43

Analysis of base-excited response of intake-outlet towers: A. N. Williams

solution for the base-excited motion of a circular cylindrical tower surrounded by a compressible fluid is ex- tended to include the presence of the additional interior fluid and to study its influence on the dynamic characteristics of the system. In the present situation the integral equation approach may be expected to be a more efficient method of analysis than the more general finite-element method since only the boundaries of the fluid domains which are in contact with the structure need be discretized. Furthermore, it has been found that results obtained through the integral equation method do not exhibit the small oscillations sometimes encountered in analytical eigenfunction solutions 9. In the analysis presented here, the tower is idealized as a one- dimensional beam of uniform flexural rigidity and mass per unit length. The fluid is assumed to be inviscid and linearly compressible and "to undergo small-amplitude irrotational motion. Both the structural and fluid motions are expressed in terms of appropriate Green's functions, leading to a pair of line integral equations for the fluid velocity potential and its normal derivative on the structure. These two coupled equations are solved numerically. Explicit results are presented for the hydrodynamic force and dynamic displacement, and added mass and radiation damping profiles for several example structures.

Formulat ion

The tower-fluid system under consideration is shown in Figure 1. A vertical cylindrical tower of radius a and

Y

I I

2o

2

Z

0

< ..... > Uoe-i0 ~t

V V

Figure 7 Definition sketch

height h is surrounded by water of uniform depth d. Since the water level inside operating intake-outlet towers is typically within a few feet of that of the surrounding water, for simplicity in the present case the interior and exterior water levels will be taken to be equal. Unequal interior and exterior water elevations may be included in the problem formulation without undue dif- ficulty, but this aspect will not be pursued here.

The ground is subjected to horizontal motion of amplitude Uo and frexjuency ~0 directed along the x- axis, ie. Uo(t)= Re{Uoe-~'}. The tower is modelled as a one-dimensional beam of uniform flexural rigidity E1 and uniform mass per unit length ms. The equation of motion of the tower acted upon by both external (1) and internal (2) fluid pressure may be written in terms of its horizontal displacement U(z, t) = Re [ U(z)e-~'] a s

d4U E1 - - ~ - m,w20 = P(z) on r = a (1)

in which

P(z) = akoo (,I ,~ - cl 'e) cos (• - O)dO 0 < z < d

o

d < z < _ h

(2)

where cI,~(r, 0, z) and ~2 (r, 0, z) are the complex fluid velocity potentials in the exterior and interior domains, respectively, defined by ~ ( r , 0, z, t ) = { ~ ( r , 0, z)e-~'}, j = 1, 2, and p is the fluid density, The structural boundary conditions on 0(z) may be written as

dO / J=Oo and - - = 0 o n z = 0 (3)

dz

d20 d3/f/ - 0 and - - = 0 o n z = h (4)

dz 2 dz 3

The small-amplitude, i ~ fluid motions in the exterior and interior d o ~ s ate governed by

# 2 ~ + 1 #cI~ + 1 02cI~ #2~ 2 • - - - - + ~ + k , F = 0 (5) Or 2 r Or r 2 aO 2

for j = 1, 2, respectively, In equation (5), k = oJ/c, where c is the acoustic s ~ in water. The conditions on the ,l~jj = 1, 2, i n c l ~

~ = 0 o n z = d , I r t > a (6)

~ 2 = 0 o n z = d , I r l < a (7)

- 0 on z = O , I r l > a (8) ~)z

44 Eng. Struct. 1991, Vol 13, January


~I? 2 ..- = 0 o n z = 0 , I r l < a

8z (9) V6 ffi ,dVG == 0 on Z = 0 (18)

dz

8@ ~ M '2 . . . . io~O(Z) cos 0

~r ar

on r = a , 0 <_ z - d (10)

In addition, the pomntial @l(r, 0, z).is required to satisfy a. radiation condition as r - oo. For excitation frequ~cies o~ > ~ = xcl2d, the first cut-off frequency, this condition states that ~ must behave as an outward propasating wave at large radial distances from the structure. For o~ < ~o, the condition ~ -- 0 as r -- oo must be applied.

Timorett~ solution The fluid velocity potentials are now.assunmd expres- sible in the form

d2VG d3VG -- = 0 on z = h (19) dz 2 dz 3

where 6( ) is the Dimc delta f~nction. A suitable expression for V6(z; zo) has been given by Liu and Cheng ~6 as

1 { sinh a(z zo) - sin a(z - zo) } VG(Z; Zo) = 2a-"-" ~

+ 4a3(I + cosh ah cos ah)

X { [ C(zo)(sinh ah + sin ah)

~ ( r , O, Z)= ~l'~(r, Z)cosm0 (11) m s l

for j = I, 2, then by the change of variable

V(z) ffi 0 - Uo (12)

Equation (1) may be rewritten as

-S(zo)(cosh ah + cos ah) ]

× (cosh o~z - cos az)

+ [S(Zo)(sinh ah - sin ah)

- C(zo)(cosh oth + cos ah) ]

d4V

dz' - cx4V = ~ ( Z ) (13)

in which ot 4 = m/.o21EI and

~'~z) =

. Ia ,O ° .ai,~p I~l(a, z) - t,~(a, z)l E1

O < z < d

~,ot40o d < z < h

(14)

Similarly, tim structural ixmndary conditions may be wri t t~ in terms of V(z) as

dV V-- , , - 0 o n z - 0 (15)

dz

d2V d3V dz ~ = dz 3 = 0 on z ffi h .(16)

The Green's function to(z; z.) for the boundary value problem defined by equatior~ U-3), (15), and (16) is required to satisfy

d % dz ' ' a 4 VG ffi ~( z - z,,) 0 < z < h (17)

× (sinh ~ - sin ~ ) } (20)

for z > zo, in which

C(zo) = cosh a(h - z~) + cos a(h - zo)

(21) S(zo) '= sinh a(h - zo) + sin ot(h - zo)

The Green's function for z < zo may be obtained by interchanging z and zo on theright-hand side of equation (20).

Applying the one-dimensional Green's identity to V(z) and Va(z; zo) over [0, h] and invoking the appropriate boundary conditions leads to

O(zo) 0 o + z o ) ~ ( z ~ (22)

Thus, equation (10) may be rewritten as

a ~ (a, Zo) = a~l'2 Or ~ (a, Zo)

w21rap

_~2(=, z)l Vo(z; zo)dz- i=V,(zo)

(23)


Analysis o f base-excited response o f intake-out let towers: A. N. Williams

in which

v~.(Zo) = ~ - cosh aZo + coso~Zo

+

in which Jm and Hm are the Bessel function and Hankel function of the first kind, respectively; Im and K,~ are the modified Bessel functions of the first and second kinds, respectively; and

(2n - 1)It k~ = - - (30)

2d 1 + cosh c~h cos oth

× [sinh o~h sin c~h(cosh otzo - cos OtZo)

(24)

- (sinh oth cos cth + cosh oth sin oth)

× (sinh aZo - sin azo)]~ )

The fluid motions in both the interior and exterior domains may also be analysed by a Green's function approach. It is found that the same Green's function may be used in each domain, The fluid Green's function G(r, 0, z; re, 0o, Zo) is required to satisfy

0 2 G -~ 1 0 G + 1 O2G t92G

Or - ~ - " -r O--r r --~ O0 ---T + ~z 2 + k2G

4~r - - - 5(r - ro)b(O - Oo)b(z - Zo) (25)

r

throughout the fluid domain, together with the following boundary conditions

OG - - = 0 o n z = 0 ( 2 6 ) Oz

G = 0 on z = d (27)

and the appropriate radiation condition as r - ~ (exterior domain only), A suitable Green's function, developed on the basis of an eigenfunetion expansion procedure, has been given by Williams s as

(28)

a o

G(r, O, z; re, 0o, Zo)= E %Gin(r, z; re, Zo) m=O

× cos m(O - 0o)

where eo = 1, and ~. = 2 for m _> 1. In equation (28), Gin(r, z; re, Zo) is defined by

2ri ~ (Hm(h.r)J.,(3,.ro)'~ d .=, \ H.,(X.ro)J,.(X.r) /

× cos k,z cos knzo

G,.(r, z; ro, Zo) = - -

~k. = ( k 2 - k2 ) 1/2 n < N (31)

~k" = ( k 2 - k 2) 1/~ = i h n n > N (32)

where N is the largest integer satisfying the condition [k 2 - k 2 ] > 0. In equation (29), :the upper terms in the parentheses are to be used for r >_ re and the lower for r < r o.

Inspection of the expression for the fluid Green's function reveals that the inclusion of fluid compressi. bility in the formulation results in N p r a t i n g wave modes for excitation frequencies o~ > ~ Which provide physically for radiation damping in the solution. For o~ < &, these propagating modes are not present in the Green's function and, at these ~ u e n c i e s , ~ solution is equivalent to that obtained by neglecting "fluid compressibility where there is no radiation damping at any frequency, only added,mass. Application of Green's second identity to ~(r , O, z) and G(r, O, z; to, 0o. zo) over the exterior ( / = 1) and interior ( / = 2) fluid domains yields

s~ ¢I~(r, 0, Z) Onn (r, 0, Z; re, Zo)

-G(r, O, z; re, 0o, zo) ~ (r , 0 , z ) dS

= 2 r~( ro , 00, Zo) (33)

for (re, 0o, zo) on the surface Sj. The surface $1 consists of the immersed structural surface (So), the exterior free surface, the exterior ~ surface ~ a bounding cylinder in the fluid at infinity. ~ surface S~ consists of So plus the interior free surface ~ interior bottom surface. In equation (33), n is the unit normal to Sj at (re, 0o, zo) directed into the r e s ~ v e fluid domain, Restricting the point (re, 0o, ~ ) to lie on So and applying the various boundary conditions = results i n the following integral equa~ns

so 49(a' O, z) ~ r (a, O, Z; a, 0o, Zo)

- G ( a , O, z; a, 0o, zo) -~r (a, O, z)

4 + - - d

( g.(x r)t.(X:ro) ) .= , k, Km(X~ro)Im(h~r)

× cosk .zcosk .zo (29)

= ±2w~(a , 0o, Zo) (34)

where the upper sign is to be used .for j = 1 and the lower for j = 2, Substimtin8 ~ ~ forms equations (11) and (28) for ¢I,j and Ginto the above ~ and



integrating with respect to the angular coordinate analytic~y leads to an infinite set of integral equations for the Fourier ~ t s of ~j and. their normal derivations on So, munely

aI i [ t~a' z) aG-i~--t (a' z; a'

-C,(a, z; a, so) ~ (a, z) = ±t~(a, So)

(35)

For q = 0, I, 2 ..... In particular for q = I

oo a ~{(a, z) ~ r (a, z; a, 7.o)

1 -G:(a, z; a, ~) ~ (a, z) ~ = -t-'~(a, ~)

(36) Equations (23) and (36) now constitute three sinudtaneous integral equations for t ~, tL and ate/ c~r = at2m/ar---M~/ar on the immersed tower surface r = a, 0 ~ z ~ d. The numerical solution of these integral equations is accomp!i~gi by discretizing the contour of integration into a number of small elements and assuming the values of the potentials and derivatives within each element to be constant. This procedure replaces the integral equations by finite systems of algebraic equations which may be solved for the unknown quantities by standard matrix techniques. A detailed discussion of the numerical procedure may be found in Willkms s.

Once the nodal values of the potentials and their derivatives on the structural surface have been determined, the complex displacement amplitude may be determined from equation (10) which may be rewritten as

• i a t O ( z ) = - - - - ( a , z) (37)

ar

and the net hydrodynamic force F(O = R e | P e - = ' I a n d

o v e ~ _ ~ , moment about the tower base Me(t)= Referee -~} may be obtained from

I'I F(t) = -2ao ~ a~ (a, O, z, t) o o ( a t

04'--22 (a, 0, z, t)~ ~ cosOr - 0)dzd0 (38) at )

Me(t) = - 2 a o z ~ ' (a, O, z, t) 0 0 at •

~ 2

- at (a, 0, z,. t ) l cos0r - 0)dzd0 (39) J

giving

P= -io~omr 0{t :(a, z) - t l=(a, z)ldz (40)

I d

~o = -ioJomr {t~(a, z) - t~(a, z)lzdz 0

(41)

Following Tanaka and Hudspeth 9, depth-dependent added mass and hydrodynamic radiation damping terms, denoted by me and/~, respectively, may be defined in terms of the ground acceleration and velocity as

a2Uo aUo -me(Z, ~0) at--- T- -~(z , ,o) a---t-

I'~ a¢' (a, 0, z, t) - 0¢2 1 = -2ap J0( 0t ~ (a, 0, z, 0

cosbr - 0)(10

which reduces to

(42)

{w2ma(z, co)+ ic0/~(Z, oJ)} Oo = --ic0oa~:lt~(a, Z)

- t2(a, z)) (43)

From equation (43), dimensionless added mass and damping coefficients, defined by

Co(z, ~,)= me(z, o,____~) pTa2 (44)

Cd(z, o,) = ~(z, o,) pTra2~ ° (45)

may be determined.

Numerical mmmples

Numerical results are presented for example towers with geometries corresponding to ald=O.1, 0.25, and h/d-- 1.0 and 1.25 for both normal and zero interior fluid levels. The value aid = 0.1 is typical of many slender tower s~ctures, while aid = 0.25 corresponds to a somewhat squauy tower, this value of aid has been chosen for comparison purposes by several previous investigators. The results presented herein have been obtained with the contour of integration discretized into 40 elements, increasing the number of elemeats did not change the computed results by more than 1%, in- dicating that nmnerical convergeace had essentially been achieved. The example towers are taken to be uniform, elastic concrete structures (Ec = 34 x 106 kN/m 2) with a wall thickness 20% of the outer wall radius.

Figures 2 and 3 present the variation of the hydrodynamic force amplitude with dimensionless frequency 12 = 00/0 for slender and squatty towers, respectively, with and without the presence of interior fluid. The force amplitudes have beea ~lglimmgiot~ized by pra2d,o20o. It can be seen elm for both slender and squatty tO~-l"S the effect of the interior flllid is to lower the natural frequncies of the cmnbinzd symnn compared to that predicted by considmmion of the exterior fluid alone. This effect is most pronounced at the higher



12-

10

¢

7 8 ¢o

6 t J

2

0 a 0.0 0.5 1.0 1.5 2.0 2.5 3.0

12

10

5. E 8

._ 6

E

0 T, I..T , ~ , , , ~ , , , i ii~ i 0.0 0.5 1.0 1.5 2.0 2.5 3.0

b Dimensionless f requency

Figure 2 Var ia t ion o f hydrodynamic fo rce ampl i tude w i th d imen- sionless f requency ~1 for s lender towers . ( 13 ), w i t h inter ior f luid; ( I l L w i t h o u t inter ior f luid. (a), aid = 0.1 , hid = 1.0; (b), a/d = 0 . 1 ; h i d = 1.25

natural frequencies of the system. The presence of a 25 % freeboard is found to have a similar effect on the dynamic characteristics of the fluid-tower system, i.e. inclusion of the freeboard in the analysis leads to significantly lower predictions of the natural frequencies of the fluid-tower system, especially for the higher modes. Investigation of these fequency response curves also reveals that the hydrodynamic forces at resonant excitation frequencies below the fast cut-off frequency, i.e. ~ < 1.0, are unbounded wl~ile those at frequencies fl > 1.0 are bounded due to the presence of radiation damping in the solution. It ~ d he noted that if fluid compressibility were neglected in the formulation, all resonant peaks would be unbounded,

Attention is now focused on those of the above structures possessing a 25% freeboard. These structures were

selected for further analysis and the displacement magnitudes and phases of each of these example structures at selected dimensionless frequencies is shown in Figures 4 - 7. The displacement magnitudes are norma- lized by the amplitude of the ground motion. For each particular structure, the chosen fequencies occur in pairs located slightly lower and slightly higher than a resonant peak. The displacement profiles clearly indicate the influence of the vibrational modes of the structure on the dynamic response. Furthermore, as intuitively expected, it can be seen that for both the slender and squatty towers the inclusion of the interior fluid in the analysis does not change the fundamental vibrational characteristics of the tower. However, it clearly modifies the predicted response at a given excitation frequency

10

"O

~- 8

k,

6 .u E

>. -O o 4 "o > .

"1-

0

a

12

12!

0.0 0 .5 1.0 1.5 2 .0 2.5 3.0

10

¢

8

.~ 6 E

R3

e 4 > . . 'r

0 - . i : i =

0.0 0.5 1.0 " 1.5 2.0 2.5 3.0

b Dimensionless f requency

Figure 3 Var ia t ion of hydrodynamic fo roe ampl i tude w i th dimensionless f requency 0 fo r aqua t ty t o m . ( 13 ), w i t h inter ior f luid; ( I I ), w i t hou t inter ior f luid. (a), a /d = 0 .25 , h id = 1 .O; (b), a/d = 0 .25 ; hid = 1.25


Analysis of base.excited response o f intake-outlet towers: A. N. Williams

10 1 12 q 12

to ,~ to ~ - / / \ \ l / | # - 1 ~ to tou~ q " I \ l / % -2

i5 I - ~ i 15 i5 15 0 -3 O] , , , I l i , . l ~ l l l i l I , I l l i l , 1 0 0 LJ.i,i

0.00 0,25 0.50 0.75 1..00 1.25 0,0 0.25 0.50 0,75 1.00 1.25 0,00 0.25 0,50 0.75 1.00 1,25 z/d z/d z/d

,o, . . . . . . . ,,

-~ I ~ : 1 . 2 3 I "~ a : ' , . , " - - - - - - ' - - - - ' / 3 = .~

'°r I .." " ' " I / , , " \ ' ;" , : l i i l i ," - . i ,~ i / / , " ,, /, '1 i ° " p ,,, ~ ~,1~i \ i \ I f -o- ; - ~I I.-" I ~ " t7--4° ~ ~

o ~--" ~ " - . . . . . . . . 'o o , , , , , , , , , , , V , , . . . . 1-1 ol ;';/, , , , , , , , , ' i , , , , , , , , , , , -i 0.00 0,25 0.50 0,75 1.00 1,25 0,00 0,25 0.50 0.75 1,00 1,25 0,00 0,25 0,50 ,75 1,00 1.25

z/d z/d z/d

Figure 4 Displacement magnitude and p h H e at selected dimensionless frequencies for a slender tower without interior fluid, e/d ffi O. 1; h/d = 1.25; ( - - - ) , magnitude; ( ), phase

simply by shifting the location of the natural frequencies of the combined system. An illustration of this may be seen in Fibres 4 and ~. In Fibre 4, where interior fluid has been neglected, the response of the slender tower at

a dimensionless frequency fl = 1.12 is seen to occur predominantly in the second mode, this dimensionless fercluency being slightly below the second natural frequency of the fluid-tower system when the interior fluid

6 I 6

B

c E u to

i5

~ = 0 . 1 6 / /

/ /

/ /

/ / I

/ /

/ /

/ /

/ , I

0.o0 o.25 o.so 0.75 'i'.'oll' 'I', z/d

4,a

to

" ' 15 - I 5

12 /.7,, I'1=0.20

• ~ ~o

to E

6

I 4 i i to i i - 1 ~" 2 / .

0 "" / - .' 0 0.00 0.25 0.5,0 0.75 1.00 1.25

z ld

/

I I

I I

3

E 2

~=0 .90 5 -

2- / ~ ' ~ . I / I \ I /

. , , / \ I / • I~ \\I /

o ~ 0.00 0.25 0.50 0.75 1.00

z/d

2

0 .25

20 .---.~ 1 ~=2 .69

"B " 0

10 -1

/ \l ,"', I; 4 i / \ 1 ! ~1i I " " / ~ / ~ l i -t -2

/ ~ ~ i 1 0.00 0.25 0.51~ 0.75 1.00 1.25

zld

" 0

3 ° ~

i to

10

i 8 -

6 -

q -

2

0 0 .00

~=1.12 /

T /

I _ /

/ / _

q 20

f - -~ , /

" i / \ "J I I \ \ ,

u , ' . . . . . , . . . . . . \ d . , . . . . _ , 0.25 0.50 0.75 1,00 f725

z/d

~p

°- e- c

to 2 ~ E

o 15 ~ 15

10

r .... ~=2.79 j ~

J/J

/ \ i-I

zld

-1 1.25

RBum 5 Dk l l l~meme~ mIan i tude l ind p h ~ e at ~ I c t e d d i m e n s i o n l e ~ frequencies for a slanml~ tower with ~mdo¢ fluid; a/d -= O. 1; h/d ,= 1.2§; ( - - - ) , m l l ~ b u ~ ; ( ), p h u e

== fo .C CL

c E u to I == 15

==

O.

!

Eng. Struct. 1991 , Vol. 13, January 4 9


i - . - J " f i { I i I i I I I I i I I I I i i I I I

• 20

g E ~ ~o

m ~5

"E

E

u~

k5

' "7

~ 3

~ 2

~5

~=0.45 / / /

/ /

/ /

/ /

/ " / ,,,

/ /

/ /

/ /

/ /

/ /

/ /

/ , /

-1

9=0 .57

-2

-3

p- / /

/ /

/ / 2

/ /

/ /

- / /

/ /

/ /

/

" ~ ' l " t ~ ~ I I I i l i I i i I i I I l i I I 0

/ /

" /

! /

3 / !

/ ' ~ ' ~ . / - I / \ i / \

2 / \ / / \ /

/ \ / \

. 1 \ / \ ! \ / \.d

0 I I I I I I I I | I I I I I I I I I I I I I I

- - ~ = 3 . 1 3

_ j

i /

-', // , / °

/

\\ Y , o , , , , l , , , , , , , , , ~ , , , v , ~ , ' ; " ~ , - ' , - - , - ~ o.oo o.2s o.~o o . ~ ~.o0 ~.2s

Z l O

r

.if_ c

~ c

- ~

3

~ ' 1 0

k5 o_ ._m Q

~5 ~.

/ /

/ /

/ /

/ /

, / /

. /

/

/

/ /

/

/

5.

1 -L--4"1 I I J I J I I J I I I I J J = I I I I I - i

~ = 0 . 5 3

, /

/ /

/ /

/ /

/ /

/ /

/ /

/ /

J " r ~ . ~ J , I , i ~ t I = J i I t i = i i

6

/ \ / \

- / \ / \

2 / / \ /

- I / \ \ /

0 l I I I i l I I I 1 t L I [ I ~4/ I i [ J i I l

f l = 2 . 6 9 /

/ -

/ /

/ /

/ /

/ /

/ / \

/ \ / / \ /

/ \ /

/ \

0 I .__ I I l I i I i | i I I 1 1 1 I l I I I , i i : - I

0.00 0.25 0 .50 0 ,75 i ,00 1.25

zld

/ /

/ /

/ I,

t

-1

3

ro ,.¢ P_-

2 ~

O

/ / - 0

ro

I _~ I

/

I , I

- - - 2

J -3

to T~

0

Figure 6 D i s p l a c e m e n t m a g n i t u d e a n d p h u e a t s e l e c t e d d i m e n - s i o n l e s s f r e q u e n c i e s f o r • s q u e t t y t o w e r w i t h o u t i n t e r i o r f lu id; a/d = 0 . 2 5 ; h/d = 1 . 2 5 ; ( - - - ) , m a g n i t u d e ; ( . ), p h a s e

Figure 7 D i s p l a c e m e n t ~ t u d e ~ ~ a t i e l e c t e d d i ~ ~ f o r a ~ ~ w i t h i n t e r i o r f lu id; a/d = 0 . 2 5 ; h/d = 1 ; 2 5 ; ( - ~ - ) , ~ ; ( ''' ~ ) ,

5 0 Eng. S t r u c t . 1 9 9 1 , Vo l 13 , J a n u a r y

Analysis of base-excited response o f intake-outlet towers: A. N. Williams

is neglected. However, Figure 5, which presents the same data for a slender tower including both exterior and interior fluids in the analysis, reveals that the response at the same dimensionless frequency of O = 1.12 now occurs predominandy in the third mode,

s i ~ whenth¢ interior fluid is included in the analysis this dimensionless frequency is slightly above the second natural frequency of the fluid-tower system.

The depth-dependant, dimensionless added mass and radiation damping coefficients defined in t~quations (44)

1.0 , I 0.2 [

0.5

i ~ 0.0

-0 .5

-1 .0 0.0

1.0

,~ 0.5

~ 0 .0

t j ~ - 0 . 5

Q=0.19

, , , _ - . . . . . . . . . . . . ~ - ' ~ ....... - ~

i = i I I I I I I I i I I

0.2 0. u, 0.6 0.0

z/d .

f l=0.22

0.2

-0.1

, -0 .2

t f i = 1 1 2 I

0.,t oof 10., , ' 0.0 _ i -- 0.0

- - 0 . 1 / i I I I [ I , i l I I I i I J - - 0 . I

1.0 0.0 0.2 0.4 0.6 0.8 1.0

z/d 0.1

I ¢ ~ % ~ = 1 2 3 .

0.0 L \ //oo [3[3

i i , i I i i [ i i q | i I

0.0 0.2 0.~ 0.6 0.8

9=3 .13

0.0 0.2 0.U, 0.6 0.8 1.0

Z/C/ 0.1

~=3 .36

-1 .0 , , , , , , , , L , , , ~ , , , , -0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

z/d z/d z/d .0

Figure 8 Added mass and radiation demping coefficients at selected dimensionless frequencies for a slender tower without interior fluid; a/d = 0 .1 ; h/c/ = 1 .25 ; ( O ) C. (z) ; (11)

1 .0 , , 1 .0

O.S

0,0

~o -0 .5

-1 .0 0.0

1 .0 , -

~" 0.5 "o

tO

c 0 .0

LJ -0.5

9=0 .16

; . . . . . . . . Ui, l l l , ill / i [

-1 .0 0.0

i i L | i i i I I 1 I I ,

0.2 O.q 0.6 0.8

z/d

1.0

~=0 .20

m

L _ I:

I l l l [ l l l l t l l l

0 . 5

0.0

-0.S

-1 .0 , 0.0

0.1

0.0

I -0.1 f

-0 .2

" 0"q i ~ ~ l ~ ~

! ~= 0 . 9 0 0.2 E; = 2.69

~_;_. _--~-,,,,,n . . . . . " ' ' - ~ ' ' - - ~ m , , ~ ...... 0 .0

-0 .2

I I l l I I I I I l I I L . - - 0 , ~

0.2 0.q 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 .0

z/c/ z /d

• Q=1.12 ~ f l=2.79 d~= "

' % ~ o. 2

l ¢ ~ J ' - 0 . ~ 1 | I J J • ' D.2~ 0.2 0.1i 0.6 0.8 1.0 0.0 0.2 O.q 0.6 0.8 1.0 0.0 0.2 o.q 0.6 0.8 1.0

z/d z /d z /d

Figure 9 Added rnm~ and radiation damping coeff icients at selected dimensionless frequencies for I slender tower with interior fluid; e/d = 0 .1 ; h /d = 1 .25; ( D ) Ce(z); ( • ) Ca(z)

Eng . S t r u c t . 1 9 9 1 , V o l . 1 3 , J a n u a r y 5 1

Analysis of base-excited response of intake-outlet towers: A o N. Wilfiams

"c

L) "D c e~

k)

-% L}

c ¢o "c

0 . 5

0 . 0

-0 .5 I I 0 , 0

0 . 5

0 . 0

-0.5 v O.

0.06

f2= 0.45

I I I I I I I I I i I I 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

~ = 0 . 5 7 I

J

l I I l I I I I I I l I I I ) 0 . 2 0 . q 0 . 6 0 . 8 1 . 0

0.0q "c

0.02

o.oo

- 0 . 0 2 l I i I i I I i I i I i I i 0.0 0 . 2 0.4 0 . 6 0 . 8 1 . 0

~ = 3 . 1 3

0 . 0 2

0.00

F,,,-'- °o o° - 0 . 0 2

- 0 . 0 4

- 0 . 0 6

L>

0 O n 0 O 0 [] 0 0 0 0 O n

%%0oo00 °0

i i i i i l i i i i | 1 I i

• 0 0.2 0.4 0.6 0.8 l •

z/d

F i g u r e 1 0 A d d e d m a s s a n d r a d i a t i o n d a m p i n g c o e f f i c i e n t s a t s e l e c t e d d i m e n s i o n l e s s f r e q u e n c i e s f o r a s q u a t t y t o w e r w i t h o u t i n t e r i o r f l u i d ; a i d = 0 . 2 5 ; hid = 1 . 2 5 ; ( O ) C o ( z ) ; ( • ), C~(z)

0 . 5

% L.) "D c t~ 0.0

L

-0.5 ~ i 0.0

0.5

0.2 O.q 0,6 0.8 t ~0

% (j

aD c 0.0

-0.5 O.

0.2

i

~=0.53 }

I I i I L i I I I i I 1 P 0.2 0.4 0.6 0.8 1.0

~:2.46

-~ 0.1

n r l

~ 0 . 0

- 0 . 1 l I l l I I l I I I I I l I 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 2

~=2.69

~ 0 .0 ¢o

D

- 0 . 1

0

• %G G 0 OG -- Q D D ° o%_ ~va

- 0 , 2 I I [ I I 1 I I I I I I I

0 , 0 0 . 2 0 . 4 0 . 6 0 . 8 1

z/d

F i g u r e 17 A d d e d m a s s a n d r a O i a t i o n d a m p i n g c o e f f i c i e n t s a t s e l e c t e d d i m e n s i o n l e s s f r e q u e n c i e s f o r a s l e n d e r t o w e r w i t h i n - t e r i o r f l u i d ; a i d = 0 . 2 5 ; h / d = 1 . 2 5 ; ( O ) C j ( z ) ; ( • )

and (45) are shown for each of the above example structures, i.e. those possessing a 25 % freeboard, at the same dimensionless frequencies considered above in Figures 8 - 1 1 . Both the added mass and radiation ~ i n g coefficients are identically zero for z > d. Also, the radiation damping coefficients are zero at excitation frequencies corresponding'to f~ < 1.0, at these frequencies only

added mass effects are present and the solution is equivalent to that obtained by neglecting fluid compressibility. When fluid compressibility is neglected there is no radiation damping in the solution at any frequency, only added mass. In each of the figures, for both slender and squatty towers, whether the interior fluid is considered in the analysis or not, it can be seen



that the variation of the added mass and radiation damping coefficients with depth at these selected frequencies closely reflects the influence of the vibrational modes of the towers at those frequencies. An examination of Figures 8 and 9, reveals that fluid compressibility (radiation damping) effects are not significant for slender towers a t lower frequencies, however, for higher excitation frequencies, typically near the third resonant frequency in this case, it can be seen that the value of the radiation damping coefficient is not small. In Figure 8, where the interior fluid has been neglected, the added mass coefficient for a slender tower at' the dimensionless frequency f l = 1.12 is seen to be predominantly positive over the water column. How- ever, an investigation of Figure 9, which has been obtained by including both exterior and interior fluids in the analysis, reveals that the added mass coefficient at the same dimensionless frequency of fl = 1.12 is now negative over most of the water depth. This change in the behaviour of the added mass coefficient for a fixed dimensionless f requency is again related to the shift in the natural f requencies o f the tower-f lu id sys tem when the interior fluid is cons idered in the analysis. As far as squatty towers are concerned (Figures 10 and 11), the same general conclusions regarding the re la t ive impor - tance o f the added mass and radiat ion damping coefficients can be made although in this case the radiation damping coefficients are non-negligible compared to the added mass coefficients near the second natural frequency of the tower-fluid system.

Condns iens

An integral equation technique has been utilized to in- vestigate the dynamic response of an intake-tower structure of circular crosHection subjected to high- frequency horizontal ground motion. The presence of the fluid contained within, as well as that surrounding the structure, has been cons idered in a consistent manner. The methodo logy presented provides a computa - t ionally efficient a l ternat ive approach to the m o r e general f ini te-element analysis . Numer ica l results have been presented which il lustrate the hydrodynamic loads and dynamic response o f example s lender and squatty towers . Based on these numer ica l results it is found that the p r i ma ry effect o f including the interior fluid in the analysis is to reduce the natural f requencies o f the tower- fluid sys tem f r o m those values predic ted by considera- t ion o f the ex te r ior fluid alone. This downward shift in

the hatural frequencies of the tower-fluid system can result in a complete change in the characteristics of the dynamic response o f the tower at a selected excitation frequency. Finally, it has been verified that the inclusion of fluid compressibility in the formulation may have a significant influence on the hydrodynamic forces on and associated dynamic response of squatty towers at higher excitat ion frequencies.

References

10

11

12

13

14

15

16

1 Rea, D., Liaw, C-Y. and Chopra, A. K. Dynamic properties of San Bernardino intake tower. Pep No. EERC 75-7, Earthquake Engineer- ing Research Center, University of California, Berkeley, 1975

2 Liaw, C-Y. and Clmpra, A. K. Dynamics of towers surrounded by water. J Earthquake Engng and Struct. Dyn. 1974, 3 (1), 33-49

3 Liaw, C-Y. and Chopra, A. K. Earthquake analysis of axisymmetric towers partially submerged in water. J Earthquake Engng and Struct. Dyn. 1975, 3 (3), 233-248

4 Chopra, A. K. and Liaw, C-Y. Earthquake resistant design of intake- outlet towers. J. Struct. Div., ASCE 1975, 101 (7), 1349-1366

5 Chopra, A. K. and Fok, K. L. Evaluation of simplified earthquake analysis procedures for intake-outlet towers. Proc. Eighth World Conf. Earthquake Engng, San Francisco 1984, Vol. VII, pp. 467 -474

6 Goto, H. and Told, K. Vibrational characteristics and aseismic design of submerged bridge lflers. Proc. Third World Conf Earthquake Engng, New Zeala~, 1965, Vol. II, pp. 107-122

7 Mei, C. C., Foda, M.A. and Tong, P. Exact and hybrid element solutions for the vibration of a thin elastic structure on the sea-floor. Appl Ocean Res. 1979, 1 (2) 7 9 - 8 8

8 Williams, A. N. Earthquake response of submerged circular cylinder. J. Ocean Engng 1986, 13 (6), 569-585

9 Tanaka, Y. and Hndspeth, R. T. Restoring forces on vertical circular cylinders forced by earthquakes. J. Earthquake Engng and Struct. Dyn. 1988, 16 (1), 99-119 Kotsubo, S. Seismic effects on submerged bridge piers with elliptic cross.sections. Proc. Third World Conf. Earthquake Engn8, New Zealand, 1965, Vol. II, pp. 342-356 Goyal, A. and Clmpra, A. K. Earthquake mtalysis of intake-outlet towers i~cluding tower-waJer-fmmdation-soil imeraction. J. Earth- quake Engng and Struct. Dyn. 1989, 15 O), 325-344 Goyal, A. and Cholna, A. K. Hydrodynamic and foundation interaction effects in dynamics of intake towers: frequency response functions. J. 5truct. Dtv. A$CE 1989, 115 (6), 1371-1385 Goyal, A. and Chopra, A. I, Hydrodynamic and foundation interaction effects in dynamics of inlake towers: earthquake responses. J. Struct. Div. ASCE 1989, 115 (6), 1386-1395 Goyal, A. and Chopra, A. K. Simplified evaluation of hydrodynamic mass for intake towers. J. Eagng Mech. Div. ASCE. 1989, 115 (7), 1393-1412 Goyal, A. and Chopra, A. K. Earthquake response spectrum analysis of intake-cardet towers. J. Engng Mech. Div. ASCE 1989, 115 (7), 1413-1433 Liu, P. L-F. and Cheng, A. H-D. Boundary solutions for fluid- structure interaction. J. Hydraulics Div. ASCE 1984, 110 (1), 51-64

Eng. Struct. 1991, Vol, 13, January 53

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Analysis of the base-excited response of intake-outlet towers by a Green's function approach