1957-Dynamic Pressure on Accelerated Fluid Containers

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DYNAMIC PRESSURES ON A C C E L E R A T E D F L U I D CONTAINERS
By G. W. I-IovsI~'ER
ABSTRACT
An analysis is presented of the hydrodynamic pressures developed when a fluid container is subjected to horizontal accelerations. Simplified formulas are given for containers having twofold
symmetry, for dams with sloping faces, and for flexibleretaining walls. The analysis includes both
impulsive and convectivefluid pressures.
INTRODUCTION
T~E DYNAMICfluid pressures developed during an earthquake are of importance in
the design of structures such as dams and tanks. The first solution of such a problem
was that by Westergaard (1933), who determined the pressures on a rectangular,
vertical dam subjected to horizontal acceleration. Jacobsen (1949) solved the corresponding problem for a cylindrical tank containing fluid and for a cylindrical pier
surrounded by fluid. Werner and Sundquist (1949) extended Jacobsen's work to
include a rectangular container, a semicircular trough, a triangular trough, and a
hemisphere. Graham and Rodriguez (1952) gave a very thorough analysis of the
impulsive and convective pressures in a rectangular container. Hoskins and Jacobsen (1934) determined impulsive fluid pressures experimentally, and Jacobsen and
Ayre (1951) gave the results of similar measurements. Zangar (1953) presented
the pressures on dam faces as measured on an electric analog.
The foregoing analyses were all carried out in the same fashion, which requires
finding a solution of La Place's equation that satisfies the boundary conditions. With
these known solutions as checks on accuracy, it is possible to derive satisfactory
solutions by an approximate method which avoids partial-differential equations and
infinite series and presents solutions in simple forms. The approximate method appeals to physical intuition and makes it easy to visualize the fluid motion, and it
thus seems particularly suitable for engineering applications. To introduce the
method, the problem of the rectangular tank is treated in some detail; applications
to other types of containers are treated more concisely.
The more exact analyses show that the pressures can be separated into impulsive
and convective parts. The impulsive pressures are those associated with the forces
of inertia produced by impulsive movements of the walls of the container, and the
pressures developed are directly proportional to the acceleration of the container
walls. The convective pressures are those produced by the oscillation of the fluid
and are thus the consequences of the impulsive pressures. In the following analysis
the impulsive and convective pressures are examined separately, the fluid is assumed
to be incompressible~ and fluid displacements are assumed to be small.
IMPULSIVE PRESSURES
Consider a container with vertical side walls and horizontal bottom that is symmetrical with respect to the vertical x-y and z-y planes. Let the walls of the container be given an impulsive acceleration ~0 in the x direction. This will generate
Manuscript received for publication November 17, 1955.

[15]

16

BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

fluid acceleration g, b in the x, y directions and may also generate an acceleration
component @ in the z direction. For a rectangular tank ~ is obviously zero, and
Jacobsen (1949) showed that for a cylindrical tank ~ is also zero. In what follows
it will be assumed that the ratio of ~b to ~ is either exactly zero or at least so small
that @ may be neglected. Physically, this is equivalent to haviag the fluid restrained
by thin, vertical membranes, spaced dz apart, which force the fluid motion to take

Fig. 1.

-1
7"

~u
Fig. 2.

u-x


Fig. 3.

place in the x, y plane only. It is then sufficient to consider the impulsive pressures
generated in a lamina of fluid.
Consider a lamina of fluid of unit thickness, figure 1, and let the walls be given a
horizontal acceleration it0. The initial effect of this acceleration is to impart a horizontal acceleration to the fluid and also a vertical component of acceleration. This
action of the fluid is similar to t h a t which would result if the horizontal component,
u, of fluid velocity were independent of the y eoSrdinate; that is, imagine the fluid
to be constrained by thin, massless, vertical membranes free to move in the x direction, and let the membranes be originally spaced a distance dx apart. When the
walls of the container are given an acceleration, the membranes will be accelerated
with the fluid, and fluid will also be squeezed vertically with respect to the membranes. As shown in figure 2, the fluid constrained between two adjacent mem-

DYNAMIC PRESSURES ON ACCELERATED FLUID CONTAINERS

17

branes is given a vertical velocity
v =

(h -

y)

du

(1)

Since the fluid is incompressible, the accelerations satisfy the same equation, so
d4
= (h - y) d~

(la)

The pressure in the fluid is then given by
(2)

Op = _ piJ
Oy

where p is the density of the fluid. The total horizontal force on one membrane is
h

P =

fo

(3)

p dy

These equations may be written

d~
b = (h p =

P=

--p

y) d z

fo y (h - y) dd~
x dy = - p h 2 ( y / h - ½(y/h) 2) dd~
x
h

--

oh 2

fo

(4)

dit
d4
( y / h -- ½(y/h) 2) dx dy = - Ph3/3 dx--

The acceleration ~ is determined from the horizontal motion of the fluid contained between two membranes. The slice of fluid shown in figure 2 will be accelerated in the x direction if the pressures on the two faces differ. The equation of
motion is
dR dx =

- ph d x

dx

Using the value of P from equation (4) gives
d2~

3

dx 2

h2

.

u = 0

(5)

and the solution of this equation is
X

6 = c1 oosh v/~ ~ + C2 smh V/g ~

(6)

Equations (4) and (6) determine the fluid pressures, and they are strictly applicable
only when the surface of the fluid is horizonthl, b u t if consideration is restricted to
small displacements of fluid the equations may be used even when the surface of the
fluid has been excited into motion, that is, equations (4) give the impulsive fluid
pressures, p(t), corresponding to arbitrary acceleration ~0(t).
If the container is slender, having h > 1.5i, somewhat better results are obtained

18

BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

- [-----

~e

i'

L

P.

j-

Fig. 4.
by applying equations (4) to the upper portion, h t = 1.5/, of the fluid only and considering the fluid below this point to move as a completely constrained fluid exerting a wall pressure p~ = olito (see fig. 3). At a depth of 1.5l the moment exerted on
the horizontal plane by the fluid above is approximately equal to the moment
(3 pill 3) exerted on the same plane by the constrained fluid below which implies
that the generation of fluid velocity is restricted essentially to the fluid in the upper
part of a slender container.
C O N N E C T I V E PRESSURES
When the walls of a fluid container are subjected to accelerations, the fluid itself
is excited into oscillations and this motion produces pressures on the walls and floor
of the container. To examine the first mode of vibration of the fluid consider constraints to be provided by horizontal, rigid membranes, free to rotate, as shown in
figure 4. Let u, v, w be the x, y, z components of fluid velocity, and describe the
constraints on the flow by the following equations:

O(ub) = _ b O Y
Ox
Oy
v =

oo
Oz

xO

(ou
-

-

~ +

(7)

19

DYNAMIC PRESSURES ON ACCELERATED FLUID CONTAINERS

where b and 0 are as shown in figure 4. These equations state, respectively, that the
fluid at a given x, y moves with a uniform u, that all the fluid at a given x, y moves
with the same v, and that continuity of flow is preserved. In a manner similar to
that of the preceding section the appropriate equations of motion could be written
for the particular shape of container under consideration. A general solution,
applicable to any shape (twofold symmetry) can be deduced as follows. From the
preceding equations

f;

1 O0
xb dx
b Oy -R

u =

~ = ~b'

'(8)

aO f f f xb dx

where b~ = db/dx. The total kinetic energy-is thus:

(o0)

fo f?I+ {

f:{

2 ( f _ R xb d x ) 2 ( 1 -Jr- Z 2 ( ~ - ) 2 ) } d x d y dz

ROy/)

where

Iz = f A x2 dA
K = 2

- . -b

+

-R

(o)

The potential energy of the fluid is

V = ½pgOh2 J x 2dxdz = ½pgOh~I~
By Hamilton's Principle
a

f t~(T
tl

- V) dt = 0
dt = 0

or

,

P

OY2/

, P

)

~Y h "1-" glzOh

~Oh dt = 0

This gives the two equations
028
Oy2

Ix 0 = 0
K

(lO)

o~ oo +g~oh
Ot2

h

=0

20

BULLETIN OF TI~IE SEIS~IOLOGICAL SOCIETY OF A2CIERICA

From which there is obtained by integration
sinh ~ / ~ y
0 ----0h

sin ~ot
sinh ~ / ~ h

(lOa)

These are the equations for the free oscillation and the natural frequency of the
fundamental mode of vibration. For a container of specified shape, such as rectangular, circular, elliptical, etc., it is necessary to evaluate only the integrals I,
and K.
The pressure in the fluid is given by
Op _
Oz

piv

p=-p~
Q =

f

Op _
Ox

-

p~

&+~Q

(11)

xb dx
R

Knowing p, the forces and moments on the walls and floor of the container can be
determined readily.
RECTANGULAR CONTAINER

For a rectangular container of unit width as shown in figure 1, the boundary conditions for the impulsive pressures are ~ = ~0 at x = 4-1, for which equation (6) gives
eosh ~ / 3

- ~0

X

(12)

cosh v/~ 1

Equations (4) then g i v e
sinh %/3
p = - o ~ o h ~Y3 ( y / h -

X

½(y/h)=)

cosh %//3 /
(13)
p =

h2 sinh V~3
-- pG 3

cosh ~/~

X

l

The wall acceleration, ~0, thus produces an increase of pressure on one wall and a
decrease of pressure on the opposite wall of

DYNAMIC PRESSURES ON ACCELERATED FLUID CONTAINERS

½(y/h) 2) v / 3 tanh V/3

pw = pitoh(y/h -

1

21
(14)

and produces a pressure on the bottom of the tank
pb =

-

p~0h ~ / ~

2

sinh

%/5 x

(15)

cosh x/3 l

The total force acting on one wall is
h2
P = p~0--tanh

%/3

1

(16)

and its resultant acts at a distance above the bottom
h0 = g h

~ 1.5

(17)

I t is seen that the over-all effect of the fluid on the walls of the container is the
same as if a fraction, 2 P + 21hpN, of the total mass of the fluid were fastened rigidly
to the walls of the container at a height 3/8 h above the bottom. The magnitude of
this equivalent mass, Mo, is
1
t a n h ~¢/3
M0 = M
(18)
1
where M is the total mass of the fluid.
The total moment exerted on the bottom of the tank is

xpb dx = - p~toh2l

1

1

(19)

Including this, the correct total moment on the tank is given when the equivalent
mass M0 is at an elevation above the bottom of
3

((

4

~//3

...
h0= h l+5\t .nhv/5

t

))

1

(20)

The accuracy of the preceding analysis can be judged by comparison with the
values computed b y Graham and Rodriguez (1952). Equation (18) gives an M0
slightly larger than that computed by these authors with maximum error less than
2.5 per cent, and equation (20) gives an h0 slightly smaller than theirs with a maximum error less than 2 per cent. I t m a y thus be concluded that for the rectangular
t a n k the errors introduced by the approximation of equation (1) are negligible so
far as engineering purposes are concerned.

22

B U L L E T I N OF Tt-IE SE,IS:IVf0LOGICAL SOCIETY OF A M E R I C A

In the case of free oscillations of the fluid in the fundamental mode for a rectangular tank of unit width, equations (9) are

L

=

K

f
2

+z
--Zz

2

x~dx

f[,2(f+ x )2

=

4 15
dx = ~o

~ dx

l

thus

-~ l 2

=

-l

1 and equations (lOa) are
1
sinh i ~
0 = O~

Y
l
sin ~t

sinh ~

h1

(21)
~2 =

~tanh ~2 l

The velocity at any point in the fluid is given by
U

l 2 --

x 2 d~

2

dy

--

v =t~x
The pressure in the fluid is given by
Op = _ p~t
Ox

(22)
P = -P5

-5

d-~

The pressure exerted on the wall of the container, (x = l), is

pw

=

p~
sinh i i 7

1 ~20h sin cot
h

(23)

The force exerted on one wall is
P = f0 h pw dy = p ~Ia ~20h sin ~t

(24)

The total force, 2P, exerted on the tank by the fluid is the same as would be produced by an equivalent mass M1 that is spring mounted as shown in figure 5. If M1

DYNAMIC PRESSURES ON

oscillates with displacement
the mass are as follows:

Xl

ACCELERATED FLUID CONTAINERS

23

the force against the tank and the kinetic energy of

xl = A1 sin ~t
- M ~ A l w ~ sin ~t
T = 1~ M ~ A 1 2~ 2 sin s ~t

(25)

F1 =

Comparing these with the corresponding equations for the oscillating fluid it is
seen that
h
A1 = 0h
tanh

1

(26)
Ml=M(l~-~/tanhdh)h

Fig. 5.
The elevation of M1 above the bottom of the tank is determined so that it produces the same moment as the fluid. Considering only the moment of the fluid
pressures on the walls (neglecting the pressures on the bottom), there is obtained
(27)

When the pressures exerted on the bottom are also taken into account the height is

hi = h

1 . . . . . . . .
/~ h .
/g
~
~-smh ~

(28)

24

BULLETIN Ot~ TtIE SEISMOLOGICAL SOCIETY OF A~[ERICA

Comparing with the exact solution of Graham and Rodriguez, it is found that
equation (21) gives a value for ¢02that is slightly too large with a maximum error
less than 1 per cent; equation (26) gives a value of M~ slightly too large with a
maximum error less than 2 per cent.
As shown in figure 5, the over-all effect of the fluid upon the container is the same
as a system consisting of the container, a fixed mass M0, and spring-mounted masses
M~, M~; etc. It will be noted that the formulas for the higher unsymmetrical
(n = 1, 3, 5 • • -) modes are the same as for the first mode if l is replaced by l/n.
The response of the system shown in Figure 5 when the container is subjected to
arbitrary horizontal acceleration can be computed readily. From the motion of M~,
the oscillation of the fluid in the fundamental mode can be determined from
equation (26), which gives the relation between A1 and 0h. The actual displacement
of the water surface is determined from equation (22), which at y = h gives

1)

ph = p ~

x / 1 - - ~ (x/1) 3 ~20hsin ~t

(29)

This pressure is produced by the weight and inertia force of the fluid above the
plane y = h. The depth d of water above this plane is thus
d-

p(e ph
-

(30)

CYLINDRICAL CONTAINER

Consider a cylindrical tank as shown in figure 6, subjected to a horizontal acceleration ~0 and let the fluid be constrained between fixed membranes parallel to the
x axis. Jacobsen (1949) has shown that an impulse ~0 does not generate a velocity
component ~ in the fluid so that in this ease the membranes do not actually introduce a constraint. Each slice of fluid may thus be treated as if it were a narrow
rectangular tank and the equations of the preceding section will apply. The pressure
exerted against the wall of the tank is, from equation (14),
pw = -p(toh(y/h - ½(y/h) 2) ~¢/3 tanh ( % / 3 hR-cos ¢)

(31)

The pressure on the bottom of the tank is
pb = -p

0h

sinh
--

2

x

(32)

cosh x/3

The preceding expressions are not convenient for calculating the total force exerted
by the fluid. The following modification gives very accurate values for R / h small
and somewhat overestimates the pressure when R / h is not small.
pw = --p(toh(y/h -- ½(y/h) 2) ~¢/5 cos ~ tanh ~¢/5 R
Ib

(31)

DYNA~IIC PRESSURES ON ACCELERATED FLUID CONTAINERS

25

From this expression the resultant force exerted on the wall is
h 2~

P ~ ( (

tanh x / 3 R

po cos ~ R d~ d~ = -p~0 ~ R2h

(33)

from which it is seen that the force exerted is the same as if an equivalent mass M0
were moving with the tank, where
-R
tanh %/3 ~M0 = M
(34)

Fig. 6.

Comparing with Jacobsen (1949), it is found that equation (34) overestimates M0
with a maximum error less than 4 per cent.
To exert a moment equal to that exerted by the fluid pressure on the wall, the
mass M0 should be at a height above the bottom

h0 = ~ h

~ 1.5

(35)

26

B U L L E T I N OF T H E S E I S M O L O G I C A L SOCIETY OF A M E R I C A

If the m o m e n t exerted b y the pressures on the t a n k b o t t o m are included, the equivalent mass, M0, m u s t be at a height

3

~0= ~

(

4

%/3 ~-

~ + ~ ,~nh ~/-~R - 1

(~ ~

1.5~
/

(36)

to produce the proper total m o m e n t on the tank. Comparing with Jacobsen (1949)
it is found t h a t equation (36) underestimates h0 with a m a x i m u m error less t h a n
6 per cent.
T h e free oscillations of the fluid (first mode) are determined from equations (21),
etc. For the cylindrical t a n k
I~

7rR~
4

K =

=

-

27 -~

R

R
sinh

y
- R

sinh

-- R

(37)

-- 8h

Comparing with the exact solution, L a m b (1932), it is found t h a t equation (37)
slightly overestimates ~2 with a m a x i m u m error less t h a n 1 per cent.
F r o m equations (11) the pressure in the fluid is given b y
P = --P3

Ix
4 R

cosh--)

--g

--

Oy

-R

Oy

(38)

0h~2 sin ~t

sinh

The pressure on the wall is
R3

P~=-PX~

0~( 1

c°s~¢3

si-~2-¢) cos4

(39)

T h e resultant horizontal force exerted on the wall is
11
P = -~r ~

,o~2R40hsin ~t

(40)
_ 12 M l g G sin ~t
11

DYNAMIC PRESSURES ON ACCELERATEDFLUID CONTAINERS

27

This force is the same as that produced by an equivalent mass M1 oscillating in a
horizontal plane with motion
x~ = Alsin ~t
h

M1 = M ~ \ 1 2 /

(41)
11

A1 = Oh

~

tanh

R

In order that M1 exert the same moment as the fluid pressure on the wall it should
be at an elevation above the bottom of

The pressure exerted on the bottom of the tank is
i

x

4 R
sinh

0~ sin ~t

(42)

- R

This exerts a moment about the z axis equal to
32-55~

~rRSp~2
sinh

R

Including this, the correct total moment on the tank is produced when
cosh ~

h

135

(43)
- - ~ slnh
ELLIPTICALTANK
Proceeding in the same way as for the cylindrical tank, the impulsive pressure on
the wall is given by equation (14)
1
pw = p(toh(y/h -- ½(y/h) 2) % / 3 tanh ~ / 5 ~
(44)
with a similar expression for acceleration in the direction of the y axis.

28

BULLETIN OF T H E SEISMOLOGICAL SOCIETY OF AMERICA

For oscillations of the fluid, equations (21) apply and for the first mode about the
minor axis
e2 = _g
a

54
2 tanh
15 ~- (b)

5+

54
h
(b) 2 a

where 2a is the major axis of the ellipse and 2b is the minor axis. For
reduces to
03

(45)

h/a small this

--

Comparing this with the exact solution, Jeffreys (1924), it is found that ~ is slightly
overestimated with a maximum error less than 1 per cent.
2
\

~o

i

Fig. 7.

Fig. 8.
' COMPOSITE TANKS

Symmetrical tanks formed of composite shapes such as that shown in figure 7 will
have impulsive pressures given by equation (14) and oscillations described by
equations (21). The tank shown in figure 7 has

(46)

K~ = RlS{O.233(R)5 + O.627(R)4 -~ l.3771R)3 + O.197(R) ~
~- 0.131

R

-t- 0.016 6~
}

I~ECTANGULAR D A M

For a dam with sloping rectangular face and constraints on the flow as shown in
figure 8, the impulsive pressures are given by the following equations:

DYNASTIC PRESSURES ON ACCELERATED FLUID CONTAINERS

29

du

v = (h-

y)~xx + u c ° s ¢

= 40 exp (-- ~¢/3 x / h )
Op ~
Oy

_ pi;

(47)

p~ = p40h

- ½

~ / 3 - ~ cos

cos°}
v'~

2

?/
Fig. 9.

The resultant horizontal force on the dam is

sin0

~/5

(48)

For 90 > ¢ > 55°, equation (48) overestimates Fh by 6.5 per cent; for ¢ < 55 ° the
accuracy of the preceding equation decreases and a different approximation must
be used, as given below,
When ¢ < 55° the fluid m a y be divided into two regions as shown in figure 9,
where a rigid membrane lies along the x axis and has a horizontal acceleration cito
such that the pressure force on each side of the membrane is the same. In the region
to the left of the x axis the following equations describe the flow:
OU

v = (h - - y) "~x q- CUo cos ¢

(49)
Op =
Oy

_ pb

30

BULLETIN

O F T H E S E I S M O L O G I C A L S O C I E T Y OF A M E R I C A

Applying Hamilton's Principle to the total kinetic energy in this region leads to
the equation
x2 d~it

dy

-t- 3x dx - 3 tan 2 ~ z~ = 3c u0 cos ~ tan ~

(50)

The appropriate solution is
c°s ~b~
¢ / ( / ) ~ + c tan
cos ~b)
~b~
= do ( ( 1 -- c tan
a = %/1 + 3 t a n ~ -

1

The pressure in the fluid is

-ll

1 - c ~-nnn~,] Ik-l]

-4- cy cos

(51)

In the region to the right of the x axis, figure 9,
v = (1 -

du
x) ~ + u cos

dd

p = --p(Ix--

½ z ~)~y + x ( ~ c o s ¢

P = p

+ ~u

~

cos~

(52)

)

Equating the pressure forces on the two sides of the membrane lying along the x
axis determines the value of the constant
I

2~ tan s ~ -/- cos ¢
~=(1

o/
COS qb

tang

oL

~-t-2

a + 2

cos¢)

The pressure on the inclined wall of the dam is thus

pw = - pdoH

-

½

t---anne

i -- c sin ¢ / + c y cos ~

(53)

DYNAMIC PRESSURES ON ACCELERATED FLUID CONTAINERS

31

where h0 is the horizontal acceleration of the inclined face. The resultant horizontal
force exerted against the inclined face is

F,, = - p a o t l

"(tan7

1 - c ~-~j

@ 7 c°s

(54)

Equation (54) overestimates Fh with a maximum error of 6 per cent at q~ = 25°.
FLEXIBLE RETAINING WALL
An approximate analysis may be made of the effect of wall flexibility on water
pressures. Suppose the fluid is retained by a vertical cantilever wall which is stiff
to the degree that wave propagation in the wall may be neglected. As shown in
figure 10, let the fluid be restrained by membranes whose shapes are proportional
to the deflected shape of the wall. For a sinusoidal vibration the horizontal velocity
of a fluid particle is u f ( y ) sin cot, and the vertical velocity is

ou~ h
v = Ox

(55)

f(y) dy sin cot

Applying Hamilton's Principle, there is obtained
d2(~
dx ~

A
+=--~

A =

J5

g

=

0

(56)
(f(y))2 dy

B =

f(y) dy

dy

The pressure on the wall is
pw = p(~oco~

f0T

f(y) dy dy sin cot

(57)

and the resultant force on the wall is
P = p~o~ 2

fTT
~O~O~y

f(y) dy dy dy sin cot

(58)

For a wall of uniform cross section, if we approximate the actual pressure by

32

BULLETIN O~ THE SEISMOLOGICALSOCIETY OF AIVIERICA

p0 sin (~r/2) ( y / h ) and compute the deflected shape of the wall, there is obtained
f ( y ) = ~0

(

1 - ~+~sin~

(59)
--

P.

h~

~0 ( ~ ) ~
~X

Fig.

10.

where P is the total force exerted on the wall. With this f ( y ) the pressure and force
are computed to be
Y 2

p ~ = ph~0~ %/~ ~ t i ~

\~/

p _ ph~0z~ ~ %/3

1.68~ + ~ 1 8 ~ (1 - o22~)

+ 2.4~ + 1.63fl 2

(6o)

The latter equation may be written

%/3 N =

N -~

~/~- ~

+ ~.1~ ¢~- o~)

+ 2.44f~ + 1.63B ~

fl

pw2h 5

For a given ~, t h a t is, a given N, this equation gives the appropriate value of ~.
Figure 11 gives a graph of N vs fl and also shows how the total force on the wall is
reduced by wall flexibility. For a rigid wall, E I = co, the preceding equation overestimates P by 6 per cent.

33

DYNA1V[IC PRESSURES ON ACCELERATED FLUID CONTAINERS

lO

8

\

4

\

!
0
0.~

0.4.

0.6

0.8

f.O

Fig. 11.

I~UMERICAL EXAMPLE

Consider a cylindrical tank of 40 ft. radius and 25 ft. depth of water. Suppose the
tank to be given a horizontal acceleration in the x direction of ~0 = 0.1g and let this
acceleration persist for 1/4 second and during the following 1/4 second let ~0 --0.1g, so that the tank comes to rest with a total displacement of 0.2 ft. During
this time the impulsive pressure on the wall of the tank is given by equation (31),
which for the present problem takes the form
2

40 cos ¢~
)

From equation (34) the total impulsive force exerted on the tank is
tanh ~/3 40
P=0.1W

40

- 0.36 W = 280,000 lbs.

34

BULLETIN

O~ T H E S E I S M O L O G I C A L SOCIETY OF A M E R I C A

The oscillations set up in the fluid are determined from equations (37), etc.
--

2~r

-

- 3.9 seconds period of vibration

co

The force exerted by the fluid during the oscillations is the same as that exerted by
a simple oscillator (figure 5), which has
MI = M 7

--

tanh

- 0.50 M

Since the natural period (3.9 see.) is long as compared with the 1/~ see. duration of
the ground acceleration, the net effect of the 0.1g acceleration is to generate an
initial displacement of 0.2 ft. of M~ relative to the tank wall and the consequent
motion of M1 is
Xl = 0.2 sin cot
The maximum xl is 0.2 ft., and hence from equation (41) the maximum 0h is
0h = 0 . 0 6 5 g1~ 1 -~-

0 tanh ~ 2 -5 4 0 -

0.008 radians

The pressure on the wall at this Ohis given by equations (37) and (39). The total
force exerted on the wall is given by equation (40).
12
P = l i (0.5 Mg)(0.008) sin cot = 33,000 sin cot
The amplitude of the water surface oscillations can be determined from equations
(30) and (39). The fluid pressure against the wall at y --- h and ¢ -- 0 is found from
equation (39) to be 17 lbs. per sq. ft., and from equation (30) the amplitude of the
water surface is
17
cl = p ( g _ X ~ h ) - - 0.28 ft.
In summary, during the first 1/~ second the fluid exerts a force of 280,000 Ibs.
against the tank wall, in the negative x direction; during the next ~ second the
280,000 lbs. force is in the nUx direction; following this the only force against the
wall is the oscillating force the amplitude of which is 30,000 lbs. and the period 3.9
sec. During this oscillation the maximum amplitude of the water surface is 0.28 ft.
above the horizontal position.

DYNASTIC PRESSURES ON ACCELERATED FLUID CONTAINERS

35

REFERENCES

H. Jeffreys (1924), "Free Oscillations of Water in an Elliptical Lake," Proc. London Math. Soc.,
Voh 23, 1924.
H. Lamb (1932), Hydrodynamics (Cambridge Univ. Press, 1932).
H. M. Westergaard (1933), "Water Pressures on Dams during Earthquakes," Trans. Am. Soc.
Civ. Eng., Vol. 98, 1933.
L. M. Hoskins and L. S. Jacobsen (1934), "Water Pressure in a Tank Caused by a Simulated
Earthquake, Bull. Seism. Soc. Am., Voh 24, 1934.
L. S. Jacobsen (1949), "Impulsive Hydrodynamics of Fluid Inside a Cylindrical Tank and of a
Fluid Surrounding a Cylindrical Pier," Bull. Seism. Soc. Am., Vol. 39, 1949.
P. W. Werner and K. J. Sundquist (1949), "On Hydrodynamic Earthquake Effects," Trans. Am.
Geophys. Union, Vol. 30, 1949.
L. S. Jacobsen and R. S. Ayre (1951), "Hydrodynamic Experiments with Rigid Cylindrical Tanks
Subjected to Transient Motions," Bull. Seism. Soc. Am., Voh 41, 1951.
E. W. Graham and A. M. Rodriguez (1952), "Characteristics of Fuel Motion Which Affect Air=
plane Dynamics," Jour. Applied Mechanics, Vol. 19, No. 3, 1952.
C. N. Zangar (1953), "Hydrodynamic Pressures on Dams Due to Horizontal Earthquakes," Proc.
Soc. Exper. Stress Analysis, Vol. 10, No. 2, 1953.
G. W. Housner (1954), Earthquake Pressures on Fluid Containers (California Institute of Technology, 1954).
DIVISION 01~ ENGINEER~NG~
CALIFORNIA INSTITUTE O~ TECItNOLOGY~

PASADENA~ ~ALIP.

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