Seismic response of flexible cylindrical liquid storage tanks

Nuclear Engineeringand Design 52 (1979) 185-199 © North-HollandPublishing Company

SEISMIC RESPONSE OF FLEXIBLE CYLINDRICAL LIQUID STORAGE TANKS Daniel D. KANA Southwest Research Institute, San Antonio, Texas, USA

Received 10 November 1978 Liquid slosh and tank waftflexural vibrations are studied in a flexible model storage tank subject to simulatedearthquake environments. Emphasis is placed on determining the influence of wall flexural vibrations on induced strcemes.The approach is basically experimental, whereby slmil/tude considerationsare first pre,ented. Then, a series of scale model experiments are conducted, and preliminary observationsare evaluated. These evaluations allow formulation of an approximate analytical model for prediction of seismicallyinduced stress. Validity range for this model is established by comparison of various predicted responses with observedresults.

1. Introduction Inadequacy of existing procedures for aseismic design of storage tanks for hazardous liquids and safetyrelated liquids in nuclear plants has been recognized for some time, and has received special attention since the failure of several petrochemical tanks in the Alaska Earthquake of 1964 [1 ]. Most of these failures are suspected to have resulted from dynamic buckling caused by overturning moments of the seismicallyinduced liquid inertia and surface slosh waves, although actual details of the complete failure process are not well known. Undoubtedly, detailed design practices influence the failure resistance of a given tank. Various standard design practices are typical of petrochemical applications [2], but can be significantly different for conventional and nuclear power plant applications. In the past, a simple analytical model [3] has been used for aseismic design of this type structure. Generally, the model includes assumptions of a rigid container, and the effects of horizontal ground motion only. Some update of the model has been accom. plished in more recent work [4] wherein effects of both horizontal and vertical ground motion were studied for a rigid cylindrical tank. On the other hand, it is known that thin shells, with or without contained liquids, definitely do not display rigid characteristics, but typically exhibit a multitude of vibration modes, for which multiple flexural waves occur in patterns 185

that depend on the basic geometry. For cylindrical shells, the flexural waves appear along the longitudinal axis and around the circumference. Much of this knowledge has come from investigations performed for development of design procedures for liquid propellant responses in aerospace launch vehicles [5-8]. Recent analytical efforts [9-12] have probed the significance of tank flexibility for earthquake response to horizontal ground motion. An experimental inves. tigation [13] has also been reported for cylindrical tanks subject to horizontal excitation only. The results of this work, along with that for rigid tanks [3,4] indicate that prediction of seismic slosh responses, which occur at very low frequencies, can be predicted with reasonable accuracy with the simple analytical model concepts. However, the more dominant seismic pressure responses are of an inertial nature, so that they occur at higher frequencies, and may be strongly influenced by tank wall flexural responses. Thus, the design of flexible containers from a limiting stress point of view still suffers considerable uncertainty. The purpose of this paper is to determine further the influence of wall flexural response on aseismic design of a cylindrical liquid storage conta/ner, and explore an approximate analytical model for design use. Experiments are first conducted to provide a ba~s for development of the approximate analytical model, whose development then follows.

186

D.D. Kana /Seismic response o f flexible liquid tanks

2. Design of experiments 2.1. Similitude concepts Application of similitude concepts to the seismic slosh problem for a flexible cylinder has been outlined in previous work [4]. This work yielded a dimensional prediction equation for liquid slosh height rt, pressure p, and tank wall stress o, in the following general form:

(~ p , o ) (l R ' & g H p ~ g - = 9"1

h li ~, E Pt H ' R ' R ' plRg ' pt ' R '

u a x ay az gT2e w2R_] pgl/2R3/2 ' g ' g ' g ' R ' g / ' where the variables are defined as: R = tank radius (L), l = tank length (L), h = tank wall thickness (L), E : elastic modulus (ML -x T-2), o = stress response at some point (ML-1T-2), li = any other tank dimension (L), Pt = tank density (ML-3), f = structural damping ratio (nondimensional), H = liquid depth (L), 7? = vertical response relative to coordinates, r, 0, which are located at the quiescent liquid level, and move with the tank (L), Pl = liquid density (ML-3), # = viscisity (ML -1 T-l), p = pressure response at some point (ML -1 T-2), a x, ay, az = time-dependent components of base acceleration relative to a space-fixed coordinate system (LT-2), g = steady acceleration of gravity (LT -2) T e = earthquake duration (T), co = frequency (T-~). For a flexible cylindrical shell tank, there are two very significant nondimensional parameters that can be further formed [14] to replace the second and fifth parameters on the right side of the above equation. These two parameters result in the requirement that

This results from limitations on ihe ph5 sical size available experimental facilities, and the elastic mudutus ratio of most useful materials, ftowever, by co~'.ccn. trating on tile bending stress requirement, which wili provide the simulation of dominanl flexural molions experienced in wall responses, a useful scale m,.~det can be developed.

2.2. Apparatus and procedures A cylindrical aluminum tank with fiat rigid bottom and top was used for experimentation with water as a liquid. Characteristics of this tank, and location of various instrumentation transducers are shown in fig. l. This model was mounted on a biaxial seismic simulator which has the capability of simultaneons independent horizontal and vertical excitation. Details of this simulator have been described in ref. [15]. Initially, a preliminary series of experiments was conducted with sinusoidal excitation in order to determine the overall flexural mode characteristics of the model tank, and to measure typical transfer functions. These experiments were conducted only for horizontal excitation, simply for the sake of brevity. Similar experiments under longitudinal excitation !lad already been performed on this tank in earlier work, which was utilized in aerospace application experiments [8]. However, it is apparent that for a complete understanding of vibrational response in elastic Midwall pressure ~ \

-- liquid wave height

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,~- Midwall horizontal acceleration

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Fig. l . Location of i n s t r u m e n t a t i o n transducers.

D.D. Kana / Seismic responseof flexible liquid tanks cylinders under seismic excitation, it is desirable to obtain transfer functions for both lateral (horizontal) and longitudinal (vertical) excitation. Both liquid and shell wall responses were studied experimentally for 1/ 16 and 1/7 scale earthquake excitation. This was accomplished by producing a simulator table motion according to a realistic (although scaled) earthquake time history. The signals were synthesized by summing together multiple narrow band, transient random signals. The computed response spectrum of the actual table motion was compared with that prescribed by the US NRC [I 6] for typical ground motion. 3. Preliminary experimental observations 3.1. Harmonic excitation Some examples of observation for harmonic excitation in the horizontal direction only will first be described. Plots of natural frequencies for given modal patterns are shown in fig. 2 for two different liquid depths. These flexural modes have a radial wall displacement of the form w =Amn(t) cos(nO) sin(mlrz/l),

187

where 0 is the angular cylindrical coordinate, z is the axial coordinate, n is the number of circumferential waves, and m is the number of axial half-waves in a given modal pattern. Note that this mathematical form includes the assumption of a simply supported ring boundary at both the bottom and top. Bending of the tank would require a more complex description than that given above. It is obvious from fig. 2 that liquid depth has a significant influence on the frequencies of the flexural modes. This results from the difference in added masses for the respective modes in each case. More details of this type of liquid-elastic tank vibrational mode can be obtained from earlier work described in refs. [5,8]. A more graphical description of the modal pattern for the m = 1, n = 9 mode in a 3/4 full tank is given in fig. 3. Similar results occur for other modes, except of course, that a different number of circumferential and longitudinal waves are present. Two examples of transfer functions for horizontal excitation are shown in fig. 4. Only magnitudes are plotted as a function of frequency, for a fixed input acceleration. Midwall acceleration and pressure were measured at the respective positions identified in fig. 1. Note that wall response amplification can be quite large, relative to the rigid body amplitude line. Note

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D.D. Kana I Seismie response of flexible liquid tanks obtained for horizontal excitation. By meti~ods oi free decay, damping for these m o d e s was measured be~ = 0.001.

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3.2. Simulated earthquake excitation

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also that local pressure can also be amplified to much larger values than that predicted by the rigid body pressure prediction (the equation for this prediction. will be given later). However, it must be emphasized that all these results occur for single mode response to steady state sinusoidal excitation. Low frequency responses for the first several slosh modes were also

Simulated earthquake responses were obtained for 1/16 scale and 1/7 scale earthquakes at various magnitude levels. Two different time histories were developed for each scale condition, and independent time histories were also developed for the vertical axis in each case. As pointed out earlier, the table motion was tailored to match approximately a response spectrum given by the US NRC Regulatory Guide 1.60. An example of this parameter for the horizontal axis is shown in fig. 5 for the 1/16 scale earthquake at a magnitude of 0.3g. A similar plot occurs for 1/7 scale. although it is shifted lower in frequency by a factor equal to (7/16) I/2. Note also that the frequency range of various flexural modes for the model tank at 3/4 full liquid level is indicated on this figure, and should be compared with fig. 2. They can be seen to be in the upper part of the range where earthquake excitation occurs. This range for flexural modes is typical for tanks with I/2R ratios >1.0. For short tanks of large diaineter, the flexural modes will occur in even lower parts of the frequency range, and correspondingly more modes can be excited by typical earthquake motions. The vertical time history was tailored to provide a response spectrum of 2/3 that given for the horizontal axis. Figs. 6, 7, and 8 show typical time histories, respectively, for combined excitation, horizontal only, and vertical only excitation for 1/16 scale in a 3/4 full tank. Similar results occurred at other magnitude levels, and for the 1/7 scale case. It is important to recognize that peak pressures are mostly inertial, i.e.. they are related to the acceleration of the excitation, and dominate over those which result from the lower frequency slosh motion. (Note that in a short tank, the two are probably more nearly the same amplitude.) Furthermore, liquid slosh motions continue long after the earthquake event has subsided. From fig. 8, it can be seen that no slosh occurs for vertical only excitation, although midwall accelerations and pressures were found to depend very strongly on vertical ground motion. It was determined that the wall responses resulted principally from flexural modes in the 30 to

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D.D. Kana/Seismic responseof flexible liquid tanks I00 Hz frequency range (see fig. 2). The mechanism for such strong excitation of these modes by vertical excitation was not completely determined. However, the amplification of peak midwall response and pressures are nowhere near the values that occur for sinusoidal excitation, as described in fig. 4. The response reduction apparently occurs through the mechanism of destructive phasing of wall responses which occur in many modes that are excited simultaneously by the multifrequency earthquake excitation. Finally, it should be noted that strains in the wall appear to be influenced strongly by both higher frequency wall motions and inertial pressures, and lower frequency slosh pressures.

4. Analytical model developments The previous experimental observations form a basis for development of an approximate analytical model for predicting stress in the seismically excited container. This model will be based on modifications of existing slosh models that are form~ated for a rigid cylindrical tank. For convenience, a brief summary of two such existing models will now be given.

One of the frequently-used simple models for predicting seismic slosh in rigid cylinders subject to horizontal ground motion only was developed by Housner. Complete details for this model can be obtained from ref. [3]. Here, it will be useful for us to con. sider only the maximum slosh wave height predicted by eq. (6.22) of ref. [3]. In terms of the variables presented in section 2.1, the maximum slosh height is given as 0.408R coth(1.84H/R ) nm

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taken from a displacement response spectrum of the seismic excitation. If the response parameter is given as acceleration response spectrum a~, then

(4)

.

Prediction equations for other responses such as pres. sures, maximum forces, and overturning moment can be obtained from ref. [3].

4.2. Kana-Dodge rigid cylinder slosh model A more recently-developed simple model for predicting seismic inertial and slosh responses in rigid cylinders was developed by Kana and Dodge (ref. [4]). This model allows for both horizontal and vertical excitation, and allows for multimodal slosh responses. The analytical development is based on earlier aerospace fuel slosh applications. For this model, the maximum slosh wave height is given as a summation of contributions from ~/modes: ,Two,, =

(s) r/

where each modal wave amplitude is given as

rtn = [ ~

4.1. Housner rigid cylinder slosh model

(3)

Here, Xx is the spectral amplitude at frequency tea,

193

tanh(~nH/R1 Xn ,

(6)

and the spectral displacement response for each of the n modes is given as x . =a.lw

.

(7)

Here ~n is a corresponding root of the eigenvalue equation for a given mode (see ref. [4]), and an is the value of the acceleration response spectrum at frequency Wn, computed for the seismic excitation. Usually, such response spectra corresponding to liquid slosh are computed for a damping value of 0.0005 < /~< 0.001. That is, liquid slosh in a cylindrical tank is very lightly damped, unless some baffle design is incorporated. It will be shown that for the present study, eq. (5) provided more consistent wave response predictions than did eq. (1). Therefore, additional parameters predicted by the Kana-Dodge model will be included, and compared with experimental results. From the previously given experimental observe. tions, it can be seen that liquid pressure consists of two

194

D.D. Kana /Seismic response o f flexible liquid tanks

parts, that due to sloshing (which is usually small in amplitude), and that due to inertia (which is usually much larger in amplitude). Thus, one may investigate these pressure components independently. For the case of vertical only excitation, the bottom inertial pressure is given by (8)

p = plgHav,

where av is understood to be the peak vertical acceleration. The inertial pressure for horizontal excitation only is given by [4] PH = plgR [1 - ~

mn/mT] all,

(9)

n

where mT is the total liquid mass and the modal slosh mass is given by m n =mTI~n(~n22R- 1 ) H ] t a n h ( ~ n H / R ) .

(10)

In this case, a H is the peak horizontal acceleration component. Finally, the prediction equation for combined excitation is obtained by taking the square root of the sum of the squares (SRSS) for the respective horizontal and vertical values. 4.3. A p p r o x i m a t e m o d e l f o r liquM and wall response in flexible cylinders

The previously given equations are all subject to the restriction of a rigid cylinder. However, very little coupling exists in most applications between liquid slosh responses (which generally occur at very low frequencies), and liquid inertial responses (which generally occur at higher frequencies). Consequently, flexural tank modes (which also occur at higher frequencies) couple only with the inertial responses, as can be observed from the preliminary experimental observations. Thus, a simple approximate model can be postulated, and is based partly on concepts presented in ref. [3], although the prediction equations from section 4.2 are used instead. 4.3.1. Liquid response

In the approximate model for a flexible tank, eqs. (5) through (10) are used for prediction of liquid responses. However, the pressures from these

responses are considered to provide loading of the flexible tank. Care must be exercised in recognizing that the loading consists of two parts, the slosh pressure and the inertial pressure, and that the center of action for each is located at different points in tlae tank. Furthermore, vertical ground motion produces no slosh, but does produce inertial pressure, which adds in depen dently to that produced by horizontal motion. A means of accounting for this addition of inertial pressure will be described shortly. 4.3.2. Tank wall stress response

In the preliminary experimental observations it was established that the multiwave form of radial wall displacement occurs readily with many circumferential waves (see fig. 3). However, the midwall circumferential strains for higher wave numbers appear to suffer mutual phase cancellation. Thus, it is plausible to assume that the liquid loads described in section 4.3.1 produce greater stress in the wall modes having lowest circumferential wave number (i.e., n = 0, n = 1). With this assumption, the stresses and strains in the cylinder can be calculated by means of rather simple formulas which depend separately on the slosh and inertial loads, and for the latter, on each tile horizontal and the vertical excitation. Examples of such formulas will be applicable to calculation of strain only at the measurement points specified in fig. 1. ttowever, calculations for strain at other locations can readily be developed similarly. For vertical only excitation, the midwall circumferential strain is assumed to be given by the hoop strain equation: e0 = ( p l g z R / E h ) a v .

(11)

Note that in this case, pressure is assumed to be given by eq. (8), and the location is a distance z below the liquid surface. For horizontal only excitation, the strain is given by eo = (No - uNx)/Eh ,

(12)

where N o is the circumferential membrane stress per unit length given by NO = R p H .

v is Poisson's ratio, and N x is longitudinal membrane stress per unit length, which is approximated by the

D.D. Kana /Seismic response of flexible liquid tanks

beam equation (13)

Nx =MRh/I .

In this expression, I is the area moment of inertia of the cross section, and the effective momentM which acts at the midwall cross section is given by M=RPH(H-

~1).

(14)

In the above equations, PH is given by eq. (9). Longitudinal strain at the tank bottom is based on a bending beam approximation which results from peak inertiapressure loads. That is: (15)

e, = MIR/IE ,

where the longitudinal root bending moment is given from

195

will be accomplished by application of time averaged nonstationary random process theory. Furthermore, additional useful relationships between excitation and response will result from this development. Each time history of the excitation and response is considered to be a nonstationary random process, in which the time trends are very slow compared to the instantaneous fluctuations. Then, according to ref. [17], certain relationships between excitation and response can be developed as follows. For general responses of elastic systems, a transfer function, Hma(f), which depends only on frequency, can be defined. This function can b e developed under steady state sinusoidal excitation. For such systems excited by a nonstationary random process, a relationship between response and excitation can be written as

M, = moardo,

(16)

the rigid mass (too) part of the liquid is given [4] as me = roT(1 -- ~ m n / m T ) ,

m l 2R

me 2

m0 h

tanh(/~IH/2R ) .

IHma(f) 12Sa(f, t ) ,

(19)

(17)

(18)

Sm (,f) = IHma(f)12S,(f),

and its elevation in the tank is given [6] as mT H

=

where Sin(f, t) is the nonstationary power spectral density of the response, and Sa(f, t) is the nonstationary power spectral density of the excitation acceleration. If this expression is averaged over some suitable time period, it becomes

n

do -

SmOe, t)

In this expression, ~a is the hrst slosh mode eigenvalue (~l = 1.84) and the modal mass (rod for the first slosh mode is obtained by substitution of this value into eq. (10). Note that according to the above approximation, peak longitudinal root strain is produced only by lateral liquid inertia, since approximations for strain contributions due tO tank top inertia were negligible, and that due to low frequency slosh were also smaller. However, this condition is true only in the present tank, and must be checked in any given case. 4.3.3. Combination o f independent random responses

It is clear from figs. 6, 7, and 8 that wall accelerations and strains, and higher frequency pressure components are all dependent on both horizontal and vertical excitation. Since each of these excitations is independent, it is appropriate to consider combining their effects. One typical method is to use the square root of the sum of squares (SRSS) approach. However, use of this method needs to be justified. This

(20)

where the bar now denotes a time averaged power spectrum. Eq. (20) is particularly useful, as it involves parameters that can readily be measured in the laboratory. However, it will still be developed a little further, in order to allow the use of time averaged root mean square (RMS) quantities, which are even more convenient. One can stipulate that all samples of excitation to be considered have the identical normalized power spectral density shape (as a function of frequency), but the magnitude is proportional to the time-averaged mean square of the acceleration. This assertion is entirely realistic with the general nature of classes of earthquake ground motion signals, and in fact, is analogous to the response spectrum curves drawn in the US NRC Regulatory Guide 1.60 [16]. Thus, we can write ,-~a(f) = S',n(f) o,2 ,

(21)

where ~ is simply a number. By substituting eq. (21) into (20), integrating over frequency, and taking a

196

D.D. Kana /Seismic response of flexible liquid tanks

square root, one obtains (O'm)RMS = A (Oa)RM S

(22)

where A is an amplification factor. Eq. (22) is extremely useful for laboratory measurements, and will be used for comparison of further data hereafter. However, it also permits further development of concepts for combination of multiple processes. For example, for independent horizontal and vertical accelerations acting simultaneously, we can immediately write (O-a)RMS = (O-2H + O-2v)1'2 ,

(23)

which justifies the use of the SRSS method for combining the appropriate time averaged RMS parameters. If both the horizontal and vertical acceleration processes are described by the same time averaged probability density function, then the combination is also described by this density function, and plotting of SRSS peak values for the combined process is justified at the same probability of occurrence for a given peak level. However, one should expect that RMS values should be less sensitive to statistical sample variations than are peak values, and should, therefore, plot with less scatter. Furthermore, the above comments apply to all response parameters as well as tile excitation. One final comment should be made about eq. (19). If the response parameter is of inertial form, such as the bottom pressure response, then a time history expression can be written as

pB(t) =aa(t)

sought. Correlations in terms of peak responses and RMS responses will both be utilized,

5.1. Peak response correlations A summary of peak liquid wave response results is shown in fig. 9a, for simulated earthquakes of increasing magnitude. It is apparaent from these results that the Housner rigid cylinder slosh model (eq. 6.22, ref. [3], or eq. (1)) is not valid for describing the results for excitation amplitudes beyond XH/R = 0.03. Here x H is defined as peak horizontal ground displacement. On the other hand, the Kana-Dodge model (eq. (5)) continues to be valid for amplitudes as large as XH/R = 0.065. In view of these results, only the latter slosh model was incorporated into the approximate model for wall stresses, whose results will follow. Similar response plots for peak inertial pressures at the tank bottom are given in fig. 9b. For the case of 0.8

~

0.6

i

~

r

--/- . . . . . . ~ --Roofline

o Horizontal 1/7 Scal~ • Combined / H/R " 1.82 /

I

0.4

J

• /

J

.

/~Equation

(5i

~ 0.2 o_

f

.,~ j

~-Equation(6.221 RI eference 31

l

L

0.02 0.04 0.06 0.08 O.10 PEAKHORIZONTALGROUNDDISPLACEMENT, XHIR (a) WaveHeight

1.5

i

Experimental 0 Horiz.1 A Vert. j" 1/16Scale

=~ ,,.-

, Comb.J

i

i

J /& /

/ ~1" .<"

1.C

o Horiz.] / ,,! -, ~• Vert. [l/7Scale~ X'SRSS. Comb.} 0/7 / ./ Equation t P= PgHa v / j I " *

0.5

//7/D ~ " ~ E q u a t i o n

~

5. Comparison of results

,g)

/R = 1.82 0

A summary of several response parameters for the liquid/shell system will be presented in this section. Validity of the approximate analytical model will be

i

Experimental o Horizontal 1/16Scale • Combined

o=

(24)

where B is an inertial constant, and pB(t) simply becomes proportional to the nonstationary acceleration time history (see as an example the bottom pressure compared with vertical acceleration only in fig. 8). In this case, eq. (22) follows immediately by squaring both sides of eq. (24), integrating, and averaging over time.

--

0.5 1.0 1.5 SRSS PEAKACCELERATION.g (b) BottomPressure

2.0

Fig. 9. Peak liquid responses in model storage tank.

D.D. Kana / 5eismtc response of flexlbleliquid tan~

vertical only excitation, the pressure is predicted by eq. (8), when av is understood to be peak vertical acceleration. The peak pressure for horizontal only excitation is given by eq. (9), when ax is the peak horizontal acceleration component. Finally, the combined prediction is based on an SRSS of the respective horizontal and vertical values. These data indicate that the rigid tank model provides a good prediction for the bottom pressures under vertical excitation, but for horizontal excitation it is somewhat low. Results for midwall pressures are given in fig. 10a. Here, the predicted pressure due to horizontal excitation is the same as for fig. 9b, but for the vertical peak pressure, the liquid height z above the midwall point, replaces the total liquid height H, as used in eq. (8). In any event, it can be seen from fig. 10a that the vertical effects in midwall pressures appear to be predicted at lower values than actually occur. Apparently the fiexural wall motion increases the pressure. On the other hand, the horizontal motion effects appear to be predicted more correctly. For another comparison of predicted and measured i

1.(]

i

i

,

t

!

i

o Hociz.]

A• ~,o~nb.I 1116Scale 13 Horlz.~ Veer. ~' 1/7 Scale

~-EQUATION( 9 ) \

<>

• ~,b.j 0.s

-

_...~..~.

\

HIR -1.82

a

~

/

~

/ "Pw" PgZav

/SRSSI EQUATIION 0.4

0.8

I 1.2

t

197

results, the peak midwatl circumferential strain is given in fig. lOb. For vertical only excitation, the strain is assumed to be given by eq. (I 1), while for horizontal only excitation, the strain is given by eq. (12). Because of the scatter, it is difficult to assess a good comparison of the data. A further correlation of peak response data for the 3/4 full model tank is shown in fig. 11. The peak longitudinal strain at the tank bottom is plotted against peak acceleration. Predicted strains are based on eq. (15). It can be seen from fig. 11 that the beam approximation is much too conservative, and the strains are indeed significandy affected by the vertical motion. Thus, the simple mechanical model concept, along with simple beam theory, does not appear to be adequate for prediction of the longitudinal strain parameter for this case. 5.2. Time average RMS response correlations

Plotting of response data in terms of peak values necessarily tends to result in scatter because of the statistical character of the data. Plotting in terms of RMS averages, hopefully, should reduce this scatter. Some of the results for the 1/16 scale earthquake event will now be given in this form. Fig. 12a shows a plot of the bottom inertial pressure that corresponds to part of the data in fig. 9b. However, here the RMS acceleration values are used in the prediction equations. Thus, eqs. (8) and (9) take on the form of eq. (22). A similar trend of data exhibited as with the peak data. However, here the trend

1.6

SRSS PEAKACCELERATION, g la) Prusure .

.

.

.

:• :::t,,.so,. Comb.J

~20

- Horiz.1

.

/

/ ,

/

~,~

. E~UAT~ON ~

15]

SC/"

~'~"

"

/

,/I/~.

~. 20 _z"

,,

~

/

F..xl~imml~l o Horlz.l ~, Vert. 1116Scale

"

• comb.J

~ ~ ~

0

0.4

0.8

1.2

SRSS PEAKACCIEUERATION,g I b ) ClrcumlenmUal$1rMn Fig. IO. Mklwall respomma in model atorage tank.

0 1.6

0.4

Fig. 1 1 . L o a O t u d i m d storage tank.

<> Vert. ~ 117 Scale • Comb.J m H/R " |'SZ

I I | I 0.8 1.2 SRSS FT.AKACCELERATION, 9 strain resp(m~

at bottom

I

of

model

1.6

198

D.D. Kana / Seismic response o f flexible liquid tanks '

'

'

'

I

'

'

0.2

'

'

/ Equations t8,22) ~

S-"

0.I

8~

'

/"~

//

/

o'

5

/ /

/ t /

Vertical • Combined

4

-

3

i

/ / ' / / / /

A Z -

/ /

SRSS

//

c:c

%

I

H/R = 1.82 / A / //

o Horizontal A Vertical • Combined

i

4

Equation

Equatons Ill, 22I

~E

f

~R

i

0.3

0.I 0.2 AVERAGE ACCELERATION ( RMS ) , g

I

I

i

[

I

l

I

I

I

I

I

I

~

0.1 0.2 AVERAGEACCELERATIONIRMSI, g

!

03

(a) Circumferential Strain

la) BottomPressure

==

i

,

,

,

I

i

o Horizontal zx Vertical • Combined

01

i

i

i

I

i

i

i

H/R = 1.82

~:

•

I~.

:E

v

H,IR ; 1.82

~"

~ /-////'1"'/

0.6

A

m~

%

O Horizontal z~ Vertical • Combined

i

o/ /

1

I- i 4

Equations ( 9,2

0.1 0.2 AVERAGEACCELERATION ( RMS ), g

0.3

Ib) Midwall Pressure

Fig. 12. Time averaged pressure responses in 1/16-scale tank.

0

0.I 0.2 AVERAGEACCELERATION ( RMS ), g

0.3

(b) HorizontalAcceleration

Fig. 13. Time averaged midwall responses in 1/16-scale tank. is more clearly defined. Again, adequacy of the prediction for vertical motion is demonstrated, but actual bottom pressures exceed those predicted for horizontal only motion. A similar plot is shown for midwall pressure in fig. 12b, which corresponds to fig. 10a. Here a similar trend also exists, but pressures which result from vertical motion are now underpredicted. Results for circumferential strain are shown in fig. 13a. Much less data scatter occurs than in fig. 10b, and an evaluation can more readily be performed. Generally, the strain predictions appear to compare reasonably with the measured values. This seems to indicate that midwall circumferential strain is not so sensitive to local higher modal wave patterns as is the pressure, and is, therefore, more adequately predicted by the approximate analytical model, which is based on lower modes present. A final data correlation is shown in fig. 13b, where midwall acceleration response is given for average

acceleration excitation level. The solid lines in this figure merely represent best fit experimental curves, and do not correspond to any theory. These data clearly indicate that a relationship of the type of eq. (22) exists, and could be developed analytically in terms of transfer functions.

6. Concluding remarks The above results indicate that midwall circumferential strain is more dependent on hoop and lateral rigid body pressure, rather than on local variations caused by the shorter wavelength flexural wall motions. Apparently the response in a multitude of flexural modes tends to cause phase cancellation at the shorter wavelengths. Therefore, the approximate analytical model defined herein appears to be useful for

D.D. gana / Seismic response of flexible Ilqukt tanks prediction of some of the response parameters in an elastic tank of the type investigated. However, the longitudinal root strain is not predicted well (although it was conservative), and needs to be investigated further. Furthermore, the geometry of the present tank was such that inertial pressures were dominant over slosh pressures. In a shorter tank of greater diameter this may not be the case, and the two effects would need to be added directly. Sensitivity of lateral flexural wall motion to vertical excitation is a particularly interesting result. Some direct excitation undoubtedly occurs through longitudinal excitation of the shell and its top mass. How. ever, some lateral flexural responses may also result from parametric excitation in the combined liquid and shell system [18]. Verification of this assertion remains to be accomplished. In any event, phase cancellation apparently again results, and the dominant wall strain results from hoop (n = 0) stress. In the present model, bending modes of the cylinder (in which the lateral wall motion is proportional to cos nO with n = 1) fell well above the range of excitation, and therefore, were not a significant factor in the problem. However, for a long tank (i.e., I/R ~ 2) this type of mode could have a very pronounced effect on the results (see for example, ref. [12]). Consideration of earthquake excitations and associated structural responses as nonstationary random processes, and employment of corresponding time average parameters for their description, appears to provide an extremely useful approach. Of course, peak responses, rather than RMS responses are of most concem, but they can always be established from corresponding probability density functions. A more complete application of this approach to general seismic response problems appears to be appropriate.

Acknowledgements The author wishes to express his sincere appreciation for the efforts of Mr. Luis Vargas who aided with part of the preliminary experiments, and the faithful work of Mr. Dermis Scheidt, who performed the complex earthquake simulation experiments.

199

References [1] R.D. Hanson, The Great Alaska Earthquake of 1964, Engineering, Nat. Acad. Sci., Washington, DC (1973) p. 331. [2] Welded Steel Tanks for Oil Storage, API Standard 650, Am. Petrol. Inst., New York (1964). [3] Dynamic Pressure on Fluid Containers, Nuclear Reactors and Earthquakes, chapt. 6, TID-7024, US Atomic Energy Comm., Washington, DC (1963). [4] D.D. Kana and F.T. Dodge, proc. ASCE Conf. on Structural Design of Nuclear Plant Facilities, VoL IA (1975) p. 307. [5] D.D. Kana, U.S. Lindholm and H.N. Abramson, J. Aerospace Sci. 29 (1962) 1052. [6] The Dynamic Behavior of Liquids in MovingContainers, ed. H. Norman Abramson, NASA SP-106, NatL Aeton. Space Admin., Washington, DC (1966). [7l Slosh Design Handbook I, NASA CR-406, Natl. Aeron. Space Admin., Washington, DC (1966). [8] D.D. Kana and J.F. Gormiey, J. Spacecraft Rockets 4 (1967) 1585. [9] L.K. Liu, ASME Syrup. on SeismicAnalysis of Pressure Vessel and Piping Components (1971) p. 90. [10] R.L. Citerly and R.E. Ball, Paper No. 71-WA/PVP-2, ASME Winter Ann. Meeting, Washington, DC (1971). [ 11] W.A. Nash and S.H. Shaaban, Response of an Empty Cylindrical Ground Supported Liquid Storage Tank to Base Excitation, Rep. on NSF Grant GI 39644, Univ. Massachusetts, Amherst, MA (1975). [12] A.S. Veletsos and J.Y. Yang, Proc. ASCE Eng. Mech. Speciality Conf. Advances in Civil Engineering, North Carolina State Univ., Rayleigh, NC (1977). [13] D.P. Clough and R.W. Clough, NucL Eng. Des. 46 (1978) 367. [14] J.G. Berry and E. Reissner, J. Aerospace Sci., VoL 25 (1958). [15] D.D. Kana and D.C. Scheidt, Proc. 22nd Meeting of the Institute of Environmental Sciences, Philadelphia, PA (1976) p. 195. [16] Desi8~ Response Spectra for Seismic Design of Nuclear Power Plants, US Nucl. Regulatory Comm., Reg. Guide 1.60 (1973). [17] J.S. Bendat and A.G. Piersol, Measurement and Analysis of Random Data, Chap. 9 (John Wiley, New York, 1966). [18] D.D. Kana and R.R. Craig, J. Spacecraft and Rockets 5 (1968) 13.