fischer,rammerstorfer - a refined analysis of sloshing effects in seismically excited tanks - 1999

8/10/2019 Fischer,Rammerstorfer - A Refined Analysis of Sloshing Effects in Seismically Excited Tanks - 1999

1/17

INTERHATIONAL/OURHALOf

ressu re

Vessels

nd Piping

ELSEVIER

International Journal of Pressure Vessels and Piping 76 (1999)

693-709

www.elsevier.comJIocate/ijpvp

A refined analysis

of

sloshing effects in seismically excited tanks

F.D. Fischera, F.G. Rammerstorfer

b

,

Institute o Mechanics, Montanuniversitt Leoben, Franz Josef-Strasse 18, A-8700 Leoben, Austria

bInstitute ofLightweight Structures and Aerospace Engineering, Vienna University ofTechnology, Gusshausstrasse 27-29/E317, A 1040 Vienna, Austri a

Received 18 March 1999; accepted 12 April 1999

Abstract

Sloshing in terms

of

liquid surface displacement in vertical liquid-filled cylindrical tanks under earthquake excitation is a weIl studied

phenomenon. Various design rules exist for liquid storage tanks to sustain the corresponding liquid pressure due to seismic excitation and to

take into account the necessary freeboard. However, usually the sloshing motion is considered under the assumption of a rigid tank with an

earthquake excitation at the base eircle. The arguments used

so

far in justifying this assumption are

of

rather qualitative but not of

quantitative nature. Since it is important to have a quantitative measure

of

that which is neglected, it is the intention of this paper to

show that this engineering approach is based

on

rigorous theoretical quantitative results. Therefore, in this paper coupling

of

sloshing with the

deformations of a flexible tank wall during earthquake excitation is investigated in a refined analysis. In contrast to former papers which have

studied the negligible influence

ofthe

wall deformations due to sloshing itself, in this paper the more important coupling including the wall

deformations caused by the impulsive effect ofthe contained liquid is taken into account. An analytical procedure is presented which allows

one to study explicitly the influence

of

the wall deformations on both the liquid pressure and the surface elevation for typical wall

deformation shapes, i.e. vibration modes. From the rather complex mathematical derivations a simple formula is drawn which enables

the engineer to get a quick guess ofthe magnitude

ofthe

infiuence ofthe wall deformations on the convective pressure contributions due to

sloshing and hence to decide whether or not the assumption

of

a rigid tank wall is suitable.

t is

shown that for tanks made

of

less stiff

materials, such as for instant polymers, this rigid wall assumption which is suitable for steel tanks may become questionable.

1999 Elsevier

Science Ltd. All rights reserved.

Keywords: Liquid sloshing; Hydrodynmnie pressure; Seismic excitation; Liquid storage tanks; Earthquake loading

1. Introduction

The response

of

vertical liquid storage tanks

to

earth

quakes has been one

of

the topics

in

the research

of

fluid

structure interaction in the last three decades. The reader is

referred to an extensive overview

of

the research results

until 1990 in the papers

of

the Austrian research group

[1,2]. Recently a comparison of design predictions due to

various codes and the recommendation [2] for unanchored

tanks was published by Hamdan in 1997 [3]. The Austrian

recommendation [2] gave good results with respect to field

observation in that paper. Finally it should be mentioned

that Eurocode 8 Part 4 spends 40 pages on tanks [4].

One of the main features in the theoretical treatments

discussed in numerous papers has been that the fluid pres

sure is split into four components:

fluid pressure P due to the ground acceleration (consid-

Corresponding author. Tel.:

+

43-1-58801-31700; fax:

+

43-1-

58801-31799.

E-mail address:[email protected]

(F.G. Rammerstorfer)

ering the tank wall

as

being rigid), named impulsive

pressure ;

fluid pressure P due to sloshing (liquid surface displace

ment) only, named convective pressure ;

fluid pressure P3 caused

by

the wall deformation relative

to the base circle due to the deformability

of

the tank

wall; and

fluid pressure

Pv

due to the vertical motion

of

the tank

(for details see Refs. [5] and [6]); in this case the radia

tion damping

of

the ground plays an important role.

The fluid pressure Pr due to the rocking motion

of

the

ground is not explicitly mentioned in this list since the

corresponding wall motion can be related to the wall motion

regime in the case

ofpz

andp3' Ofcourse, the ground motion

is different from the above mentioned cases. For details

regarding rocking motion the reader is referred to Ref. [7]

and to papers by Y. Tang and co-workers [8].

However, since the fluid motion due to the ground accel

eration (pressurePI and the wall deformation (pressure P3

produces a distribution of the displacement of the fluid

surface, a coupling exists between the pressure components

0308-0161199/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.

PU:

S0308-0161 99)00047-2


2/17

694

FD. Fischer,

FG.

Rammerstoifer / International Journal ofPressure Vessels

and

Piping

76 (1999) 693 709

H

/

/

/

/

/ - --

L_

,/

/

I

i - B

I

I

-=0

I at

I

I

I

I

I

I

I

I

I

I

- -----, B' B

7 -----._+

g _

0

/ at Bz

I

/

/

/

I

I

I

I

I

i -- : =x.(t) + v, x, t)

I

I

.i

x.(t)

I

I

~ ( - - - R - - - - 7 B$

= 0

Bz

Fig. I. Boundary conditions for the velocity potential.

Pb

P3 and the pressure P2 due to sloshing. To the knowledge

of the authors this coupling has not been studied explicitly,

even in the recent literature. Therefore, this paper deals with

the coupling between sloshing and the wall motion

of

a tanIe

This study should not be confused with the consideration of

the influence

ofthe

wall deformations due to sloshing only,

which was already investigated

by

various authors [9,10]

with the concIusion that the sloshing is hardly influenced

by the corresponding waIl deformation (which is very small

when compared with the surface eIevations).

It should be mentioned that the coupling between

Pb

P3

and

P2

is not incIuded in existing design mIes (see e.g. Ref.

[2] or some taler proposals [11]).

It

is interesting to note that the liquid sloshing is of

continuous interest.

An

inhomogeneous fluid [12], two

liquids [13], and layered liquids [14,15] were recently inves

tigated specifically with respect to sloshing, mainly

in

rigid

tanks. Usually sloshing is investigated taking into account

the linearized Bemoulli equation. In contrast to this, as a

new result the large amplitude sloshing in a rigid container

was investigated recently in Refs. [16,17], and in Ref. [18]

with a flexible bottom.

Since numerical methods are able to solve complicated

fluid-stmcture interaction problems, recently the finite

element method for the tank shelI and the boundary element

method for the liquid were applied to treat a fulIy coupled

system solid/tank/liquid, see e.g. Lay [19] or the work by Bo

and Tang [20] which investigates specificalIy the influence

of a base isolation on the sloshing behavior. A complete

finite element approach is described in Ref. [21]. The slosh

ing analysis can also be performed by an elasticity code

applying the analogy of the pressure wave equation and

the elasticity equations [22].

The main goal

of

this study, however, is to estimate the

overalI effect of the interaction between sloshing and the

wall motion; it is to be noted that here not just the walI

deformation due to sloshing itself [9,10] but also that due

to the action

ofthe

impulsive pressure is taken into account.

For this purpose the authors folIow an analytical technique.

The pressure wave equation can be solved by means of a

series expansion for various sets of boundary conditions.

Although the leading form

of

the differential equations

exist (see e.g. Flgge's exact equations in Ref. [23]), a

cIosed form analytical solution for the walI deformation is

not available, mainly because

ofthe

fact that in practice the

tank walI thickness varies over the tank's height. Therefore,

various types of the relative walI deformation shape are

assumed (see Ref. [24] and the text below Eq. (14)). This

assumption is based on practical observations and numerical

studies [2]. TalI tanks with the ratio

H/R)

> 1 (R radius of

the tank,

H

height of the tank) often show a more or less

linear variation

of

the deformation over the tank's height.

Broad tanks with a ratio

(HIR)

20 independent ofthe type Tl,T2,T3.

The solution of Eq. (I4c) can be found by fol 1owing

Duhamel s principle [28; Chap.

7],

takingln(O) = n (0) =


6/17

698

F.D.

Fischer, FG. Rammerstoifer / International Journal 0/Pressure Vessels and Piping

76 (1999)

693-709

0, as

(I5a)

Using now Eq. llc)

this leads to

l5b)

In

this relation the time history

x

g

r)

is given, namely the

ground velo city excitation by the earthquake. However, w(t)

is not known in advance. The time derivative

ofthe

integral

on the right hand side of Eq. (15b) can be obtained after

integration by parts with xg O) = w O) =

0

as

t s sin(wn(t - r[xg(r) Knw r)]

dr

= W

n

f

cos wn t - r[xg(r) Knw r)]

dr

=

s sin(wn(t -

r[xg(r) Knw r)]

dr.

(I5c)

Using Eqs. (2),

l la)

and

l5c)

the pressure distribution due

to sloshing can now be found for

cp =

0 as

(16)

It

is important to note that the hydrodynamic pressure due to

sloshing now reflects the influence

of

the wall displacement

by the term

K

n

w

r) in addition to

Xg t).

The sloshing pressure

reported in the literature contains only the

xg t)

term. This

additional term represents the substantial new contribution

to the research in the field

of

earthquake loaded liquid

storage tanks, and the question, whether or not it can be

disregarded, or under which conditions it might become

essential, is to be answered.

For the sake of brevity a parameter b; is introduced as

(I7a)

The parameter

b;

can easily be estimated using asymptotic

expansions. For small arguments ofthe Bessel function we

obtain b;

=

1.0 and for large arguments

I

b

i

=

TI

R

2i -1) 2 H - I

Finally, using Eqs. (9), (10) and (16) the pressure on the tank

wall at

cp =

0 can be calculated as

Po z;

t) = PI,O Pz,o P3,olr=R

{

00 2(- l i+

1

[ TI Z]

=

-2pR xg t) L . )

b

i

cos 2i

- 1

i=1 21 - I TI 2

H

+w(t) f ibiCOS[ 2i - ~

i=1

2

H

00 Wi

COSh

t . ..

+ L

Z _ I

H) f smwi(t

-

r)[xg(r)

i=1 I

cosh

A

0

R

l7b)

3.

Generalized

degrees

of

freedom system

In

order to find an approximate discretised system with a

finite number

of

generalized degrees

of

freedom for

areal

structure it is necessary that the motion

of

the structure can

be described by corresponding generalized coordinates. Let

us assume that the displacements

of

the tank wall can be

described, in accordance with Eq. (4b),

by

Ur z, cp; t) = xgCt)

cos cp

w t)ljJ z)

cos

cp

l8a)

U ,(z,

cp; t)

=

-xg(t)

sin

cp - u t)ljJ z)

sin

cp.

where

u

uip are the radial and tangential displacements,

respectively, of the tank wall 's midsurface. Here w t), u t)

are the generalized coordinates. Ovalizing

of

the initiaUy

circular cross section can be taken into account

by

an ovaliz

ing coefficient jL such that

u(t)

= jLw t).

(18b)

No ovalizing means jL = 1. The components

Ur

and uip are

assembled to the displacement vector u. The load vector is

given by the fluid pressure, Eq. l7b), as p

=

Po cos cpe

r

The inertia forces

ofthe

tank wall per unit midsurface can be

described

by

a vector

Prh z) - ) (Pr

is the mass density

of

the tank wall' s material,

h z)

is the thickness

of

tank waU)

and can be considered as a body force.

Let us now formulate the principle

of

virtual work in the

form

of

virtual displacements with respect to a coordinate

system fixed with the tank bottom. Here

(J

is the actual


7/17

F.D. Fischer, F.G. Rammerstor/er I International Journal 0/Pressure Vessels and Piping 76 (1999) 693-709

699

stress state in the tank wall due to pressure and inertia

forces. The corresponding displacement field fiT =

(w(t) cos

cp

-u(t) sin cp)t/J(z) represents a kinematically

admissible deformation state and is used as virtual displace

ments with the corresponding strain state E. Thus, the virtual

internal work VW

i

over the whole tank

(V

T

is the tank wall

volume) can be expressed

as

- VW

i

=

f

(T: E dV = k

l

w(t)2 k

2

w(t)u(t) k

3

u(t)2.

V

T

l9a)

The stiffness parameters

kJ, k

z

,

and

k

3

are described in

Appendix

B.

The virtual work

of

the pressure distribution

YW

is

VW

p

=

f

Po

cos

cp

e;fi

dA

T

w

f

7r

f

= R 0

cos

2

cp dcp w(t) 0 Pot/J(z) dz

l9b)

where Tw is the wetted area

of

the tank wall, and it is

assumed that the tank is completely filled. This along with

Eqs. (l7b) and (lOb) gives

{

2 -l i+1

VW

p

=

-2 ITR

2

Hw(t) xg(t) (2i _

I) IT

bii

+ w(t)

I b;/i;

i=1

+I

i=1

Wi Yi

A2=l (H)

cosh

Ai

R

l9c)

l9d)

The virtual work

of

the inertia force distribution YW

jn

is

YW

in

= -

f

pyh(z)

T

i

dA

T

w

f

7r

[ r

= - R 0

cos

2

cp dcp w(t) xg(t) J pTh(z) dz

+ w(t) f:

pTh(z)t/J(z) dz ] - R

f

r

sin

2

cp

dcp

u(t)

X [Xg(t) f pyh(z)dz (t)

f

pTh(z)t/J(z)

dz1

(lge)

After the introduction

of

a reference wall thickness h

o

, YW

jn

can be expressed as

{ f

hex)

VW

jn

= -R ITHhopy (w(t) u(Ixg(t) 0 h;dx

+ [w(t)w(t)

u(t)(I)] f

t/J(f} d

g

}

I9f)

We introduce the average relative wall thickness

w h o =

Iq

[h(fJ/h

o

] M and the weighted wall thickness hw/h

o

=

Io

[h(fJ/ho]t/J(f} dg. Taking Eq.

l8b)

into account and

assuming

/L

to be known (in the following

JL

=

1

is assumed

for the sake of simplicity) we are dealing with

just

one

generalized degree offreedom, e.g.

w(I),

which can be deter

mined by the principle of virtual work

(20a)

with the help

ofEqs. l9a),

l9c)

and (19f) after division by

w(t) and some re arrangement, the principle of virtual work

(20a) can be written as

w(l)

I

b;/i;

/L PT

[

1

2) h]

2

P R

fl

I

A

2

1

(KiH

sin(w;(1 -

rw(r)dr

,=1

cosh

A

0

R

+ w l) (k

l

JLk

/L2k

3

)

2 ITpR2

H

= -x (I) I -

b.

L

PT

[

2

I)i+1 I h]

g i=1

(2i

- I) IT ' , 2 P R

fl

- I A2 1

r sin(wi(t -

rxg(r)

dr.

1=1 I

cosh

A 0

IR

(20b)

Relation (20b) represents an integro-differential equation

for

W I).

In addition to the known eigenvalues Ai it contains

.. the ratios (H/R),

(hw/R), (hw/R)

and the set

b

i

(see Eq.

l7a,

Ai

and

wi,

respectively (see Eq.

lIc,

reflecting

the tank geometry;

..

the parameter sets

i

(see Eq. (lOb,

Yi

(see Eq.

l9d,

Ki

(see Eq. l4b, and

JL,

reflecting the wall deformation

type

t/J(f).

All the above parameters can be calculated in advance for

certain types

l

T4

of t/J(f}.

The coupling between the wall displacement and sloshing

is introduced

by

the Duhamel integral

g

sin wi l

rw( r)

dr. Since the time derivative

of

the Duhamel inte

gral is again a Duhamel integral, Eq. (20b) cannot be

reduced to a simple differential equation.

If

the influence

of

sloshing is ignored,

Eq.

(20b) reduces


8/17

700 F.D. Fischer, F.G. Rammerstoifer / International Journal ofPressure Vesse/s and Piping

76 (/999)

693-709

- - -TYPET1

0,50

-1 I -TYPET2

-A-TYPET3

0,45

-

.........

TYPET4

0,40

0,35

0,30

0,25

0,20

0,15

0,10

0,05

0,00

0,0

0,5

1,0 1,5 2,0 2,5

3,0

3,5 4,0

H/R

Fig.3.

The coefliceint c, evaluated numerically as a function of H/R for different types of .p fJ.

to the cIassieal ease of the differential equation:

w t) + w ~ w t )

= -

MPxg(t),

i:

2 - l r

l

b

l+ /LP r h

w

_1 (2i - l}rr 1 1

2

P R

MP= _

i:

bf + (1 +

L

z

)

Pr h

w

i=1 I 1 2 p R

w ~ =

_

_ _ ~ _ z _ _ / L _ z _ k c . . . 3 ~ ~

i: bz +

l

+

L

z

)

Pr h

w

.

i=1 1 1 2 p R

(2la)

2lb)

where MP is the cIassieal mode partieipation faetor.

The seeond terms in the numerator as weIl as in the

denominator of MP may be negleeted, and MP can be

expressed approximately by

00 2 - li+1

L

(2i -

1) lT

bii

i=1

00

P=

L

i ~

j=1 j=1

2le)

4.

Problem

solution

Some abbreviations are introdueed and eomments are

given in order to make the foIIowing derivations easier to

read.

The terms

CI =

2::1

bjf

as weIl

as

Cz =

2::1

[ 2 - l r

l

/(2i - l}rr]bii depend only on the ratio

(H/R) and the deformation type ifJ f) of the wall.

Diagrams ean be provided for the deformation types

mentioned below the Eq. (14e), see Figs. 3 and

4.

The sum

(22a)

depends on the tank geometry and the density ratio

PT/P

only.

Also the sum

l

+

/L)

Pr

h

w

H

Pr

)

Cz +

2 P R -

Cz

R P , ...

(22b)

depends only on these parameters.

The stiffness parameter on the l.h.s. of Eq. (20b)

is

denominated as

k

l

+

/Lk

z

+

/Lzk3

=

k

lTpR

z

H

The abbreviation

W j 'ri

AZ-l (H)

eosh Ai R

=

j

(22e)

(22d)

and the eoeffieients K;, see Eq. (l4b), depend only on the

ratio (H/R) and on the deformation type

of

the wall.

j

deereases rapidly with inereasing

i.

The first ten values

are presented in Tables 1 and 2 depending on the wall

deformation type

rfi f)

far the parameter

(H/R)

=

0.5 and


9/17

F.D. Fischer F.G. Rammerstorfer / International Journal 01 Pressure Vessels and Piping

76 (1999)

693-709

701

0 50

0 45

0 40

0 35

0 30

0 20

0 15

0 10

0 05

0 5

1 0

1 5

2 0

H R

2 5 3 0

3 5

- -TYPET1

1II TYPET2

--TYPET3

-e -TYPET4

4 0

Fig.4. Tbe coefficeint C2 evaluated numerical1y as a ftmction of H/R for different types of .p fJ.

2.5, respectively. Selected values

of

K; are depicted in

Figs. 2(a) and (b).

Now the integro-differential equation (20b) can be rewrit

ten

as

follows:

w(t) L a ~ K

0 - f1

;=1

CI

Tc

sin(w;(t - rw r) dr

-::-w(t)

CI

w t) is found by numerical integration using the Runge-

Kutta 415th order algorithrn with adaptive step size control

for the two dependent variables YI = w t) and Y2 = w(t),

respectively. For details see [29]. Several studies were

performed with a maximum allowed relative error E =

Tab1e I

Parameter i'i;Cg/R) 12 depending on i and on the wall deformation type for

H/R

=

0.5

Tl T2 T3

T4

0.255190852 0.203246168

0.152380775 0.381643076

2 0.025009425 0.020970448 0.017381462

0.031129397

3 0.008470108 0.00735051 0.006432547 0.009519165

4

0.004014605 0.003567162 0.003221356

0.004297134

5 0.002258492 0.00204178 0.001881557 0.002358797

6 0.001413388 0.001294673 0.001209846 0.001456306

7

0.000951755 0.000880803 0.000831443 0.000972719

8

0.000676164 0.0006309 0.000600078 0.000687454

9 0.000500407 0.00047003 0.000449702 0.000506958

10 0.000382445 0.000361218 0.000347214 0.000386476

10

-

3

and

10 6

as weIl as with a truncation

of

the series in

Eq. (23) after 50 and 90 terms, respectively. The studies

showed that a maximum relative error E =

10

-

3

and trunca

tion after 50 terms lead to sufficiently accurate results. Since

decreases significantly with increasing

i,

the number n =

50 may be even too high.

5. Examples

Two steel tanks are now investigated: a taU tank PI with

H/R)

= 2.5 showing a nearly linear wall deformation T2,

and a broad tank PlI with

H/R)

= 0.5, whose waU deforma

tion can be approximated by a sinusoidal deformation

pattern Tl. Both tanks have the same height of 10.0 m.

For the sake of simplicity a constant wall thickness

h

=

7.0 mm and

f L =

1.0 is assumed. The Y oung s modulus

Tab1e 2

Parameter i i/.gIR) 1 2 depending on i and on the wall deformation type for

HlR =

2.5

Tl

T2 T3 T4

0.111201858 0.097051194 0.85610264 0.123308584

2 0.006230354 0.005842951 0.005582628 0.006316884

3

0.001894689 0.001815691 0.001765495 0.001904954

4 0 .000856979 0.00083008 0.00081345 0.000859448

5

0.000470918 0.000459064 0.000451853

0.00047176

6

0.000290907 0.000284794 0.000281115

0.000291261

7

0.000194372 0.000190867 0.000188773

0.000194544

8 0.000137399

0.000135229

0.00013394 0.000137491

9

0.000101338 0.000999145 0.000990726 0.000101392

10 0.000772626 0.000762848 0.000757088

0.000077295


10/17

702

FD.

Fischer, FG. Rammerstoifer / International Journai ofPr essu re Vessels and Piping 76 (1999) 693-709

Table 3

Parameters determined for tbe

wo

sampIe tanks

Tank

H./R TYPE

Cl Cl

C2

CZ

Tc

s-z)

PI

2.5 T2 0.02 0.02683 0.04

0.05365

707.1

PII 0.5

TI

0.14

0.14472 0.21 0.21741 473.3

is E = 2.1

X

10

5

Nlmm

2

and Poisson's ratio

v=

0.3. The

values of the constants defined in Seetion 4 are listed in

Table

3.

The first term ofthe series is given in Table 4.

As an input for the earthquake the free field acceleration

xg t) measured in Tolmezzo (ltaly) during the 1976 Friuli

earthquake (Tolmezzo

1

N-S; see Ref. [30]) is used (see

Fig.5).

As results the time histories

of

the "cIassical" sloshing

pressure PsI,g(without coupling between sloshing and wall

deformation) as weIl as the "wall displacement" sloshing

pressurePsI,w, i.e. the contribution due to wall deformation,

are calculated by Eq. (17b) for z

= H

as

PsI,g = -2pR f A

2

1

f sinw/t

-

7)Xg(7) d7 (24a)

1=1 1

and

PsI,w = -2pR L i

K

; sinw;(t -

7)W(7)

d7,

0 l

;=1 Ai

- 1 0

(24b)

respectively.

The dimensionless entities

-PsI,g/ 2pgR)

=

PsI,g

and

-PsI,w/ 2pgR) = PsI,w

are discussed now. The maximum

elevation or lowering of the fluid near the wall (defining

the necessary "freeboard") can easily be calculated by

multiplying (sI,g sJ,w) with

2R.

The dimensionless pressure values PsI,g and PsI,w are

calculated in the time interval [0 :S t :S 20 s]. The results

are shown in Fig. 6 for tank PI and in Fig. 7 for tank PlI,

respectively. In these figures the vibration period of the

fundamental "cIassical" sloshing mode is also shown. One

can see that, in contrast to the common engineering

approach, the contributions from higher sloshing modes

might not be negligible, a fact which was already stated in

Ref. [33].

A comparison of the maximum values

of

the dimension

less pressure contributions, PsI,g and PsI,w shows that for

both typical tank geometries the maximum pressure values

due to "cIassical" sloshing (that means assuming a rigid tank

wall) are significantly higher than those due to the "wall

displacement" sloshing, see Table

5.

Table 4

Parameters determined for

the

wo sampIe tanks

Tank

PI

PII

HlR

2.5

0.5

Type

T2

TI

2.15

0.96

4.49722

0.83348

Consequently, the influence of the wall displacements,

caused by the individual contributions to the dynamically

activated pressure, on the sloshing pressure is rather small

and, roughly spoken, more or less negligible in the engineer

ing analysis

of

typical earthquake loaded steel tanks. This

finding corresponds with the widely accepted engineering

assumption that sloshing can be treated without taking wall

deformations into account [1; and the relevant papers cited

therein]. However, the arguments used so far in justifying

this assumption are of rather qualitative but not of quanti

tative nature. The results presented here show that the engi

neering approach is based on rigorous theoretical

quantitative findings. However, it should be mentioned

that the approach presented here is not restricted to typical

steel tanks, and for tanks made ofless stiffmaterial coupIing

effects can become more dominant, as will be shown beIow.

I t

can now be verified

if

a suitable estimation can be

derived by taking only one term

of

the series in Eq. (23)

into account. Following Appendix C, the coefficients

G, b,

C,

d, and the multiplier Klare given in Table 6, taking GI =

1.8412 and g = 9.81 r s

2

.

Simplified expressions for both pressure contributions

can be given by

C v, t) =

f

sin v t - 7))X

g

7)

d7.

The maximum absolute value

of

C v,

t

is denoted as

S,iv

and is called the spectral pseudo-velocity response

of

the ground motion

xi t

according to the circular

frequency v and no damping, see e.g. [28; Chap. 26-5].

Here Sp v) can be taken from the response spectrum related

to the investigated earthquake. Comparing the extreme

values of the two sloshing pressure components leads,

with

WIK1c/.jb) = WIKIC21jC;./k),

to

IMax1pslwi wIKlc2

1 Sklb

r

max

=

IMax1psI:gl

= jC; ./k Sp wl) .

(25)

An estimation of the magnitude of the various factors leads

to the following results:

(WIKlc21.,fC-;J =

0.6

S I

for tank PI and

OA S I

for tank

PlI'

" (l/./k) is 0.04 s for tank PI and 0.05 s for tank PlI;

26355

3270

1.99963

1.50228

5.71933

1.24687


11/17

FD.

Fischer, F.G. Rammerslorftr

/

International Journal 01 Pressure Vessels

and

Piping

76 (1999) 693-709 703

X

g

[m/sec

2

]

3

2

o

\ I I l L \ ~ I I . n , I I 1 1 I I

-1

-2

-3

o

2

4

max (3,1858)

min (2,6644)

6

8

10 12 14

16 18 20

t [sec]

Fig.5.

Free field acceleration during the 1976 Friuli earthquake (Tolmezzo #1, N-S).

U sing the response spectrum corresponding to the free

field acceleration shown in Fig. 5, the ratio

Sp .Jij)/Sp Wj) is around 0.1 for tank PI and around 10.

for tank PlI.

This

leads without

any numerical

integration to

esti

mations for r

m x

of

around 0.002 and 0.2, which means

that the contribution to the sloshing pressure resulting

from the wall deformation is very small in the case of

the tall tank (PI) and reasonably small in the case of the

broad tank (PlI), a fact which corresponds to the results

derived by numerical integration of Eq. (24b) after

solving the integro-differential-equation (23) (see Table

5).

From the above results one can conc1ude that for tanks

made of one and the same material the ratio Sp .Jij)/Sp WI)

appears to be the dominant factor for estimating the impor

tance of the coupling effects. It depends on the eigenfre

quencies

of

the tank wall vibrations and

of

the sloshing

vibrations, respectively, and on the response spectrum of

the earthquake. The other two ratios appearing in the r.h.s.

of

Eq. (25) can be combined to act as a factor

K R,h,H/R,p,E, ijJ f)) (see Appendix D). Hence it might

Table 5

Ratios of extreme values of the sloshing pressure contributions computed

by

Eq. (24)

Tank

PI

PII

0.015

0.181

IMinpsl.wI/IMinpsl gl

0.023

0.214

be practicable to calculate a rough guess

with

IMax1ps/ wl

=

KSp .Jij)

IMaxIps1 gI

Sp WI)

(26)

(27)

before deciding whether or not coupling effects should be

taken into account. Since the

factorfmainly

depends on the

H/R) ratio and the vibration mode shape ijJ f) (which both

are interconnected with each other) only, the influence

of

the size ofthe tank is a linear one, i.e. proportional to R, that

of

the density

of

the liquid is proportional to .JP and that of

the tank design is proportional to .j 1/Eh). This considera

tion allows the engineer to assess the importance

of

the

described coupling effects. In Appendix D a Sloshing

Sensitivity Parameter is defined which characterises the

above considerations.

6. Conclusion

n

extensive analytical and numerical procedure as weil

Table 6

Parameters determined for the two sampIe tanks

Tank

HIR

j

PI 2.5 0.78632 2.15

PII 0.5 0.66846 0.96

4.49722 26355 1.99963 5.71933

0.83348 3270 1.50228 1.24687


12/17

704

F D Fischer F G Rammerstoifer

/

International Journal ofPressure Vessels and Piping 76 (1999) 693 709

0,010

-

Psl,g

0,008

0,006

0,004

0,002

0,000

-0,002

-0,004

-0,006

-0,008

-0,010

....L L _.L > L _ 1 ......... . . . . . . l . ._ - - - - - - - - - - -L_ - - - - - - - - - - - -_ - - . . . l

(al

_ 0,00020

Psl ,w

0,00015

0,00010

0,00005

0,00000

-0,00005

-0,00010

-0,00015

o

2

4

6 8 10

PSLG T)

12 14 16 18

20

t [sec]

2

4

6 8 10 12 4

16 18 20

b)

PS LW T)

t

[sec]

Fig. 6. PsI,g and sI w calculated in the time interval (0 :5 t :5 20 s) fr tank PI.

as estimations are presented which allows one to compare

the maximum values of the classical sloshing pressure

with those of

an additional sloshing pressure contribution

due

to

the wall deformation. The latter contribution has not

been considered up to now in the various published resuIts

for calculating the earthquake response oftanks. The resuIts

presented here clearly shows that, it is difficuIt to estimate

this up to now ignored pressure contribution with sufficient

accuracy.

However, the order of magnitude ofthe ratio between the

extreme values of the new pressure component and those

of the classical sloshing pressure can be estimated in an

rather simple engineering approach. For typical liquid

storage steel tanks in the petrochemical industry this ratio

is

rather smalI. Therefore, it can be concluded, that for these

tanks the sloshing pressure component caused

by

the wall

deformation is of minor relevance and might in an engi

neering approach be neglected in relation to the classi

cal sloshing pressure. This holds true especially for

tall tanks for which the sloshing pressure due to the wall


13/17

FD. Fischer FG. Rammerst01ftr

/

International Journal 0/Pressure Vessels and Piping 76 (1999) 693 709

iOS

0,002

0,001

P

g

0,000

-0,001

-0,002

0 2 4

(al

0,00035

0,00030

Ps1 w 0,00025

0,00020

0,00015

0,00010

0,00005

0,00000

-0,00005

-0,00010

-0,00015

-0,00020

-0,00025

-0,00030

-0,00035

0 2 4

b)

6 8

6 8

10

PSLG T)

12

10

PSLW T)

12

14

16

14

16

18 20

t [sec]

18 20

t [sec]

Fig. 7. Psl g and PsI,w calculated in the time interval (0 oS t oS 20 s) for tank PlI.

deformation is very small in comparison to the maximum

cIassical sloshing pressure.

On the other hand, the results show that also for these

types

of

steeI

tanks-in

contrast to existing design

codes

the cIassical sloshing pressure should not be estimated by

just the fundamental sloshing mode but the contributions

from higher modes should also be taken into account. This

is especially important if the pressure rather than the over

turning moment is ofinterest, as for instance

ifthe

tank wall

buckling in the upper regions is considered.

By applying the derived estimation formula one can see,

that for large tanks made of more flexible material, as e.g.

aluminum or polymers, neglecting the coupling effects

between sloshing and wall deformation should be critically

checked.

Acknowledgements

The authors appreciate the cooperation with Prof.

V.

Poterasu, Tech. Univ. Iasii, Romania, who with his group

performed the numerical solution

of

Eq (23). Furthermore,


14/17

706 F.D.

Fischer, F.G. Rammerstorfer / International Journal ofPressure Vessels and Piping 76 (1999) 693-709

the authors would like to express their thanks to Prof. E.R.

Oberaigner, Institute

of

Meehanies, Montanuniversitt

Leoben, who solved the differential equation in Appendix

C by MATHEMATICA and helped to evaluate the various esti

mations, as well as to Dipl.Ing. W. Vonaeh, Institute

of

Lightweight Struetures and Aerospaee Engineering, Vienna

University ofTeehnology, who ealeulated the stiffuess para

meters

of

the tanks as explained in Appendix B.

Appendix

A. Eigenfunctions

J

1

(lIirIR)

and

their

integrals

The solution ofthe Laplaee equation is expressed in the

fonn = per,

z; t)

eos'P. per,

z; t)

is separated into

P(r,z; t) = F(r)G(z)j(t). Insertion in the Laplaee equation

delivers

[(F ~ F I

- : F)G FG ]

eos Pj(t)

=

O.

For j(t) eoscp 0;1= 0 it follows that

(F ~ F I - ~ F ) I F = -G IG=

- ~ r

where A is a eonstant.

The solution

of

the equation

1

I

1

A)2

+-F--F+-F=O

r r

R

is

F(r) = ClI Aj) DYI Aj),

where C and D are eonstants (see Ref. [26; Chap. 9] and

J

I

(A(rIR)) and Y

I

(A(rIR)) are the first order Bessel funetions

of the first and seeond kind, respeetiveIy.

To ensure that aPlat = 0 at r = 0, the funetion Y

I

(A(rlR))

is disearded, sinee it is singular at

r =

o.

The A-values are found by the seeond boundary eondition

at

r

=

R

leading to fl(A;) = 0 with AI = 1.8412,

A2

=

5.3314, ...

t

ean be shown that

F

i

= J

I

(Ai(rlR))

is the eigenfunetion

of

asymmetrie

and positive operator

A,

A = - - = - r ~ )

d r rdr r rdr dr r

X = A

i

2

R

t follows from integration by parts

of

(A(Fi))Fjr dr, that

I

R

SR( d

,

SRI

(A(F;))Fr

dr

= -

- r F ; )F dr - F F d r

o

) 0 dr } o r }

The boundary tenns disappear due

to Fjlr=R = O.

There

fore, only the two integrals over the interval

[O,R]

survive,

from whieh both the symmetry and positivity ean be seen.

The most important eonsequenee are the orthogonality

of

the eigenfunetions and the diserete speetrum of the positive

eigenvalues

Xi

S: FiFjr dr = 0

for

i 0;1= j.

This ean immediately be shown by inserting instead of F

i

=

(A(F;)/(X

i

) and Fj

=

(A(Fj)/X), respeetively:

f

R FiFJrdr

= l fR

(A(F;))Fjr dr

= .

fR Fi(A(Fj))r dr.

o

Ai

0

A

j

0

Due to the symmetry we have f ~ A F i ) ) F j r dr =

f ~ A F j ) ) F i r dr. Therefore, the integral FiFJr dr must

be 0 for Xi 0;1= Xj.With the dimensionless radial eoordinate

g

the integral

S

I 2 A;

- 1

2

o

F

i

g

dg = 2AT J

I

A;)

ean be taken from Ref. [26; Chap.

11,

Integral 11.4.5].

A further integral is needed in the main body

ofthe paper,

namely

S:FJI (7]g)g dg = S>I (Aig)/1 (7]g)g dg.

This integral ean be found in Ref. [26; Chap. 11, Integral

1l.3.29], taking into aeeount, however, that

1

I

z)

=

-iJI iz),h z) =

2

iz),

i = F l

t

follows that

S:

J

I

(Aig)/1

(7]g)g dg = A

i

J

2

A;)/1 (7])

7]JI

(Ai)h(

7]))/ A7

7]2).

Now the following identities are valid:

A

i

J

2

(Ai) = J

I

(Ai) - AiJi (Ai) = J

I

(A;),

sinee f

l

A;)

=

0, if Ai is an eigenvalue, and

7]h(7]) = 7]1 1 (7]) -

1

I

(7]).

With these identities the integral ean finally be written

as

S:

J

I

(Aig)/1 (7]g)g dg

=

7]JI (A;)1 1 (7])/(A; 7]2).

Appendix

B.

Stiffness

parameters in Eq.

(19a)

Eq. (19a) ean also be expressed in tenns

of

the strain

energy Uby

1 2 2

U(u,

w)

= 2(k

l

w k

2

wu kJu ).

Taking the strain energy expression from [31; Chap. 10.10,

Eq. (10.70)] and assuming a defonnation pattern aeeording


15/17

FD.

Fischer, F.G. Rammerstorfer

/

International Journal ofPressure Vesselsand Piping 76 1999) 693-709

707

to Eqs. 18) of the main body of this paper and

J.L

= 1,

comparing the coefficients in the above equation renders

for the deformation pattern Tl : /J(g) = sin[ 11/2)g], as

used for the tank PlI:

k Eh

3

H11 11

3

) Eh H11

I = 12(1 - V2) 2R

3

+

4HR

+ (1

- V2) 2R

k E H11

1

11

3

3 )

3

= 12 1 - V2) 2R3

+ - v 16HR

+

-

v

h (H11 R11

3

)

(1

-

V2)

2R ) 16H

and for the deformation pattern

T2: /J(g)

= g as used for

the tank PI:

k - Eh

3

(H11

+ 1 _

V 211)

+

Eh H11

I - 12(1 - V2) 3R

3

) HR (1 - V2) 3R

k - - Eh

3

(2H11

+

1 _ V 211)

2 -

12(1

- V2)

3R

3

) HR

Eh 2H11

1 - V2)

3R

Eh

3

(H11

1

11

)

k

3

= 12(1

- V2)

2R

3

+ -

v 2HR

+ Eh (H11 +

(1

_ V)R11)

(1

-

V2)

3R

2H

.

Appendix

C.

Solution of

an

integro-difIerential equation

of

type Eq. 23)

The integro-differential equation

w

+

a

f sin wI t - r))w(r) dr bw(t)

= -cXg t) - d

J:

sin wI t -

r))xg(r)dr

can easily be solved by applying the Laplace transformation

technique and, specifically, the convolution theorem.

With F(s) being the Laplace transformed L[((t)] and the

initial conditions

1(0)

= j(O) = 0, it follows

[fl Sin WI t-r))] r)dr]=

2 WI 2iF(s),

o s

+ W

I

and, therefore,

4 2 2 d

W

S +S WI+

w/c)

X

s = C

(s.

s4

+

s2 WT

+

aWI

+

b)

+

bwf)

g

The Laplace transformation ofthe both dimensionless pres

sure components Eqs. 24a,b) can be writ ten as

S4 + s2 wT + dWI/c)

E(s) = .

S2

+ wT) s4 +

s2 wT

+ aWI + b) + bwD

If the inverse e(t) of the Laplace transformed E(s) can be

found, then

Ipsl,w

can be presented explicitly.

A

study of the inversion of

E(s) by

MATHEMATICA [32]

brought out, that

e(t)

can be written as

e(t)

=

~ s n w l t

-

~

+ f1d)sin WI -

f1wI)t

acwI acwI

The entities

f1d, f1wI,

f1b

l

,

f1b

2

are very small compared to

a,

b,

c, d,

WI.

It should be kept in mind, that a, c,

d,

WI are of

the order of 1 and

b

of the order 10

3

_10

4

.

Some algebra, which is omitted here, leads to the follow

ing estimations:

f1d=

2wlc -d

2cb

The first two terms on the right side of e(t) can be combined

together leading to a beat motion and to the following final

expression:

e(t) = - 2 ~ b 2wlc - d)sinw1t - dWltcoswlt) + ~ s i n J b t

A beat with a low frequency

WI

and an amplitude ofroughly

[dwI/(2cb)]t is superposed by a high frequency vibration

with the amplitude

lIJb.

The absolute values

ofthe

extrema

ofthe beat can be approximated as

dk11/2cb

with

Wltk

=

k11

being the according time in the time interval in which xg(t)

has significant values. Therefore, the beat amplitudes and

the high frequency motion ofthe a p ~ 1 c relate as (dwI/2cb)tk

to (1/Jb).

For

the tanks investigated the value (1/J7j), being

the amplitude of the high frequency term, is usually much

higher than beat amplitudes. Therefore, only the high

frequency motion needs to be taken into account. Finally,

both pressure components IpsI,g and

Ipsl,w

can be estimated

by

the first term of their series development as

IpSIg t) = : 1 fl xg r)sin wI t -

r))

dr

,

g A

I

- 0


16/17

708 F.D. Fischer, F

G.

Rammerstoifer / International Journal ofPre8sure Vessels and Piping 76

(1999)

693 709

and

1

PsI,w(t) =

g A r ~

1) ) ( -

W ~ ~ ; t )

f

xgc

r)

X [sin(

Jb(t -

-

I fL 2wI

C -

d)sin(

WI (t

-

2cvb

-

dWI

(t - r)eos(wi

(t -

r

]

dr.

The seeond eontribution in the bracket term in the above

eonvolution integral usually ean be omitted as explained

above.

Appendix D The sloshing sensitivity parameter

The ratio r

rnax,

Eq. (25), eonsists of three multipliers,

WIKIC2 I SveJb

r =

nax

JCi

Jk S,, WI) .

The first two multipliers are now investigated with respeet

to their dependenee on the tank geometry, the liquid density

p,

the tank design

E, h, f>r),

the wall vibration mode

I/J f)

and

f.L:

WI> as

ean be seen from Eq.

(IIe),

depends weakly onH/

Rand ean be rewritten as WI = Jg7R jWl HIR);

CI>

C2>

as ean be seen from Eqs. (22a), (22b), depend only

on

HIR,

1 1 and very weakly on

f>rlp

and

hwlR;

KI depends on

HIR

and

I/J;

k, it follows from Eq. (22e), depends on

kJ

k

2

,

k

3

,

whieh

ean be written (see Appendix B) for k

l

as

k

l

=

HEhIR(kll(HlR) + h

2

IR

2

k

22

(HlR,

with

kll HIR),

k

22

HIR)

are funetions

ofHIR

with an order ofmagnitude

being

100.

Finally k ean be written as

k = p ) ~

h(HlR,

h

2

1R

2

,

f.L).

Sinee

h

2

1R

2

is

usually very small in relation to land fL of an

order

of

magnitude being

100,

k

follows as

-

E) h

= R2

p

Rh(HlR)

Inserting the above relations into r

rnax

yields

r = ~ pg j(HIR ,I,) sv Jb) .

rnax Eh

I

SvCwI)

The funetionj(HIR, 1/1 is of the order of magnitude being

10

0

_10

1

.

r

rnax

eonsists again of three multipliers; the first one,

f

SI=R

-

Eh

ean be eonsidered as the Sloshing Sensitivity Parameter.

If one eompares two tanks with the some value for

HIR

and a similar vibration mode 1/1, the following two faetors

may lead to a different sloshing response:

The sloshing sensitivity parameter SI representing the

tank desing

by

R, h, E and the fluid

by

pg.

Sinee usually

a minimum thiekness

h

min

must be seleeted (sometimes

independently of the tank material), an entity EIpg

becomes a charaeteristie length.

The earthquake itself, via its response speetrum, by

sveJb)/s (wI)

References

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[2] Fischer FD, Rammerstorfer FG, ScharfK. Earthquake resistant design

of

anchored and unanchored liquid storage tanks under three-dimen

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dynamics-recent

advances, BerIin: Springer, 1991. pp. 317-371.

[3]

Hamdan FH. Buckling

of

tanks during earthquakes: field observa

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Schneider P, editors. Proc. Carrying Capacity

of

Steel Shell Struc

tures, Brno, 1997, pp. 199-205.

[4] Eurocode 8: Design

of

structures for earthquake resistance. Part 4:

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Fischer FD, Seeber R. Dynamic response ofve rticall y excited liquid

storage tanks considering liquid-soi l' interaction. Earthquake Engng

Struct Dyn 1988;16:329-342.

[6]

Gupta

RK,

Hutchinson GL. Effects

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flexibility on the dynamic

response ofliquid storage tanks. Engng Struct 1991;13:253-266.

[7]

CasteIlani A, Boffi G.

On

the rotational components

of

seismic

motion. Earthquake Engng Struct Dyn 1989;18:785-797.

[8]

Veletsos AS, Tang

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fischer,rammerstorfer - a refined analysis of sloshing effects in seismically excited tanks - 1999

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