Embed Size (px)
Citation preview
PARAMETRIC STUDY OF SEISMIC SOIL-TANK
INTERACTION. I: HORIZONTAL EXCITATION
By Medhat A. Haroun,' Member, ASCE, and Wajdi Abou-lzzeddine2
ABSTRACT: Dynamic soil-tank interaction under horizontal seismic excitations has a profound effect on the amplification of hydrodynamic forces and moments exerted on tank structure. Analytical mechanical models for flexible cylindrical tanks, which also account for the effect of rigid-base rocking motion and lateral translation, are used with a lumped-parameters idealization of foundation soil. A comprehensive parametric analysis is conducted to study the effects and relative importance of a few factors influencing the seismic response of tank-liquid-soil systems. Computer programs are implemented to evaluate the system response to ground earthquake motions. Results are presented in the form of graphs that give engineers a useful tool for understanding the seismic performance of the system and for developing effective design procedures.
INTRODUCTION
To formulate the dynamic response equations of a tank-liquid-soil system subjected to horizontal excitations, one should express the lateral force and overturning moment at the tank base in terms of the base motion. To avoid unnecessary complications, it is advantageous to represent the liquid by an equivalent mechanical model capable of producing the same liquid-exerted force and moment when subjected to the same ground motion. Housner first formulated an idealization for estimating liquid response in seismically excited rigid tanks (Housner 1957) and presented values for equivalent masses and their locations that would duplicate forces exerted by liquid on tank. The properties of this mechanical analog can be computed from tank geometry and characteristics of the contained liquid. Haroun and Housner (1981) modified Housner's rigid model to account for a rocking motion at its base. Later, Haroun and Ellaithy (1985a,b) presented an analytical mechanical model for flexible cylindrical tanks, which also accounts for the effects of rigid-base rocking motion and lateral translation. The model has been used to evaluate the maximum dynamic response of an elevated flexible cylindrical container supported on x-braced or pedestal towers (Haroun and Ellaithy 1985b). More recently, Veletsos and Tang (1987) presented an analysis of the dynamic response of upright circular cylindrical tanks subjected to a rocking base motion of arbitrary temporal variation. This study showed that the effects of base rocking may be approximately determined from available analyses (Veletsos and Yang 1977) of the responses of laterally excited fixed-base tanks. The present article is based on a comprehensive parametric study (Abou-lzzeddine 1989) of the effects and relative importance of a few factors influencing the seismic response of tank-liquid-soil systems. Graphs are provided as a tool for understanding the system performance and for developing effective design procedures.
'Prof, and Chair, Dept. of Civ. Engrg., Univ. of California, Irvine, CA 92717. 2Grad. Res. Asst., Dept. of Civ. Engrg. Univ. of California, Irvine, CA. Note. Discussion open until August 1, 1992. Separate discussions should be sub
mitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on September 6, 1990. This paper is part of the Journal of Structural Engineering, Vol. 118, No. 3, March, 1992. ©ASCE, ISSN 0733-9445/92/0003-0783/$1.00 + $.15 per page. Paper No. 437.
783
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
TANK GEOMETRY AND COORDINATE SYSTEM
The tank under consideration is a ground-supported, circular cylindrical, thin-walled liquid container of radius R, length L, and thickness hs, filled with liquid of mass density p; to a height H and subjected to a free-field horizontal ground displacement, G,,{t). A cylindrical coordinate system (r,9, and z) is used with the origin located at the center of the base, as shown in Fig. 1. The radial displacement of a point on shell middle surface is denoted by w(z,t).
ASSUMPTIONS
The following assumptions are made with regard to the tank-liquid-soil system and its motion:
1. Tank is completely filled with liquid (L = H) and undergoes a base translation, x(t), and a rigid-base rotation, a(t), about a horizontal axis passing through the center of its base.
2. Liquid free-surface sloshing is neglected since it barely affects soil-tank interaction.
3. Short-period pressure component due to wall flexibility is taken into consideration by assuming that tank wall vibrates with a prescribed displacement w(z,t) where
TTZ w(z,t) = i v ( 0 s i n — . (1)
4. Soil is represented by springs and dashpots that possess frequency-independent properties.
Rigid Base
Gh(t) Free Field Motion
Foundation Soil
FIG. 1. Tank Geometry and Coordinate System
784
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
HYDRODYNAMIC PRESSURE, LATERAL FORCE, AND OVERTURNING MOMENT
When a deformable tank is subjected to a ground displacement, Gh(t), it experiences hydrodynamic fluid pressures, pd, on its wall and base given by superposition of the long-period component contributed by convective fluid motion (however, sloshing is neglected in this study), of the impulsive liquid-pressure component, which varies in synchronism with horizontal ground acceleration, and of the short-period component contributed by vibrations of tank wall. The shearing force, Q(t), and the overturning moment, M(t), exerted on the tank are expressed as
Q(i) = \ \ pd(R,Q,z,t)R cos 0 dQ dz (2)
M ( 0 = L L Pd{R$,z,t)R cos 0 dd zdz
7 0 JO
0
2ir
pd(r,Q,0,t)r2 cos 6 dQ dr (3)
where the second term in (3) arises due to unsymmetric distribution of dynamic pressure on the tank base. Note that shell mass adds to the base shear and moment, but, in general, such a contribution is negligible.
MECHANICAL MODEL
A mechanical model representing a flexible cylindrical tank undergoing both translation and rotation (Haroun and Ellaithy 1985a) is depicted in Fig. 2. The model consists of an oscillating mass, mlt at height H±, and a mass, m0, having a central moment of inertia, I0, rigidly attached to the tank wall at height Ha. These heights include the effect of pressure variation on the base. The mass, mx, is associated with short-period hydrodynamic forces contributed by vibrations of the flexible walls relative to the base, and the mass, m0, is associated with impulsive hydrodynamic forces. The springs that attach the mass, mx, to the tank wall have a total equivalent stiffness, Ky, calculated from shell theory (Haroun 1980,1991) as
= (4Tr5Eshsm1 JS.y
\ m 0.0003351 ^ j - 0.00002l(^j
I rr\ I2
- 0.0163611 - + 0.065598 KRJ
(4)
where Es = the modulus of elasticity of shell material; and m = the total mass of liquid. This expression is obtained by curve fitting the numerical results calculated by the finite element program developed by Haroun (1980). Except for an error in the stiffness parameter published in Haroun and Ellaithy (1985a), all other parameters in that reference remain valid.
The generalized degrees of freedom of the mechanical model are y(f) ,x(f), and a(t), representing the relative motion of short-period mass to the base, the base translation, and the base rotation, respectively. The shearing force and overturning moment generated by the motion of the mechanical model are given by
785
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
FIG. 2. Mechanical Model for Tank-Liquid-Soil System
6(0 = mJijXf) + x\t) + HMO] + m0[x(t) + H0d(t)] (5)
M(t) = m1H1[y(t) + x(t) + H^t)] + m0H0[x(t) + H0a(t)] + I0a(t) . . . . (6)
By comparing the independent terms of (5) and (6) with those of (2) and (3), respectively, one obtains expressions for the parameters of the mechanical model. Detailed derivation and listing of these parameters can be found in Haroun and Ellaithy (1985a).
MODELING OF SOIL
Analysis of complicated soil-structure systems involving a large number of degrees of freedom can be greatly simplified by modeling the soil as a lumped-parameter system of frequency-independent characteristics. In this analysis, soil is modeled by two springs and two dashpots. Each set of a spring and dashpot corresponds to a degree of freedom: the rocking motion and the lateral translation. Fig. 2 depicts the soil model, the parameters of which were developed by Richart et al. (1970) as
Kr = 32(1 - vf)GfR
K„ — SGfR
3
3(1 - v,) (7)
0.3K„R
Cx = 0.576KXR 1% and Ca =
1 3(1 - vf)I
(8)
(8P/R5)
where pf, vf, and Gf ~ mass density, Poisson ratio, and shear modulus of
786
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
the foundation soil, respectively; and / = the mass moment of inertia for the system at foundation level.
GOVERNING EQUATIONS OF MOTION
Equations of motion of the system are derived from Hamilton's principle
8 | (T - V)dt + \ bWnc dt = 0 (9)
where the kinetic and potential energies (Tand V) of the mechanical analog shown in Fig. 2, when subjected to ground displacement, Gh(t), are given by
r (0 = \mQ(Gh +x + HQ&y + \mx(Gh + x + H1d + yf + ̂ /0(d)2 . . . . (10)
and
V(t) = i Kxx2 + \ Kaa
2 ~2Kyf (11)
whereas the work done by the nonconservative forces, ?>Wnc, is expressed as
SW„C = — Cxxbx — Caaha — Cyy§y (12)
Substitution of these expressions in (9) yields the matrix equation of motion in the form
m. ml + mQ
m1H1
m0H0 + m1H1
_m1H1 m0HQ + m1H1 mtHl + maHl + I0j l&(t)
C 0 0 Cx
0 0
0 0 c„
y(t)} x{t) +
U(0J
Ky
0 0
0 Kx
0
0 0
K„ \x(t)\ = -GA ( r )Mr.(13) U(0J
where M = the mass matrix of the system; r = represents the effect of excitation direction on the system loads; and Cy = the damping coefficient of shell vibration given by
Cy = 21-VK^ (14)
where £ = the damping ratio of fixed-base shell, and is taken as 0.02. The matrix equation of motion, (13), is solved by a step-by-step integration scheme using the trapezoidal method.
PARAMETRIC ANALYSIS
To isolate the effects of different factors on tank response, three basic parameters are examined: the aspect ratio of tank (H/R), the thickness ratio (hsIR), and the shear wave velocity of foundation soil cf = VG //p /. Furthermore, three different systems are analyzed: a fixed-base shell, rigid tank supported on deformable soil (two-degree-of-freedom system), and a flexible tank supported on deformable soil (three-degree-of-freedom system).
787
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
Throughout the study (Abou-Izzeddine 1989), the response of the tank and all parameters, if possible, are presented in a nondimensional form to facilitate the generalization of results. For example, the natural frequencies, a), will be reported in terms of nondimensional parameters, o>, such that
P/(l " v?).
where w = frequency parameter.
(15)
NATURAL FREQUENCY OF FIXED-BASE SHELL
In this section, the natural frequency of the fixed-base shell is computed neglecting interaction between shell and base and between base and soil; this analysis is mainly done as a reference for later comparisons. The natural frequency of the fixed-base shell, o)s, is represented (Haroun 1991) in terms of the frequency parameter, &s, where
(0, = 217* / ( l - V?) h\ H R \R
0.0003351 -
0.000021 'H vR
0.016361 0.065598 (16)
The variations of &s with the height-to-radius ratio and the thickness-to-radius ratio are presented in Fig. 3. It is observed that the frequency parameter, u>s, increases steadily to a value of (H/R) ~ 1.50, then it decreases as (H/R) is increased. If the aspect ratio, (H/R), is held fixed, then an increase in the ratio (hJR) produces an increase in the value of the frequency parameter, as expected.
hs/R - 0.001 hs/R - 0.002 hs/R - 0.003 hs/R - 0.004 hs/R - 0.005
Height-to-Radius ratio, H/R
FIG. 3. Variations of Frequency Parameter of Fixed-Base Shell, wsl with Geometric Properties
788
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
RIGID-TANK DEFORMABLE-SOIL SYSTEM
Equations of motion of the system are solved by a step-by-step integration to yield accurate results of the system displacements and forces. Yet, it was advantageous, from an engineering point of view, to calculate the natural frequencies of the soil-tank system to show the shifting trend in the system frequency, although these values may be somewhat inaccurate due to large values of damping exhibited by the soil. For numerical calculations of the system response, the following assumptions based on current design considerations are made: p, = pH, (i.e., liquid is water); p6 = 2.5p„, (concrete base); pf = 2pM, (typical soil density); ps = 7.8pvl, (shell is made of steel); and hb/H = 1/20. Unless otherwise stated, the earthquake record used is the 1940 El Centro earthquake (S 90 W component) and the thickness ratio (hJR) is taken as 0.003.
Effect of Foundation Soil on Natural Frequency Figs. 4 and 5 show variations of the fundamental frequency parameter,
u>, with changes in geometric properties of the tank as well as changes in soil stiffness. The second natural frequency of the system is not presented since the second mode is heavily damped, making the values of natural frequency irrelevant. It is observed that the fundamental frequency parameter increases as the ratio (H/R) is increased, reaching a maximum at (H/R) ~ 1.7; beyond that point the frequency parameter decreases as (H/R) is increased. If the value of (H/R) is held constant and the shear-wave velocity is increased, the value of a> increases. For example, at (H/R) = 2.0, cf = 600 ft/sec (183 m/s), and hJR = 0.003, one obtains w = 0.0224 compared with a value of 0.187 for cf = 5,000 ft/sec (1,524 m/s). Fig. 4 also demonstrates that the frequency parameter is insensitive to changes in (hJR) ratio in accordance with the assumption of a rigid tank. Fig. 5 indicates that the variation of the frequency parameter is linearly proportional to the shear-wave velocity, and the inclination of these lines depends on the value of (H/R) considered. Fig. 6 presents the variation of the first mode damping ratio, £,, with respect to changes in (H/R) ratio, from which it is observed that the damping ratio has its largest values for (H/R) < 1.0, and it decreases rapidly as (H/R) is increased. It should be noted also that the damping ratio does not vary with changes in shear-wave velocity of the soil.
Effect of Foundation Soil on Rocking Acceleration Variations of maximum acceleration response due to rocking, Ha, with
the stiffness of foundation soil are presented in Fig. 1(a); the acceleration values are normalized with respect to the maximum ground acceleration, (Gh)max. It is observed that at low shear-wave velocities (representative of soft soils), there is a magnification of rocking acceleration response. An increase in the shear-wave velocity (stiffer soils) causes a reduction in rocking acceleration that approaches zero as the shear-wave velocity takes on larger values, indicating that for very stiff soils, the rocking motion is minimal as expected. If shear-wave velocity is held constant, the rocking acceleration increases as the ratio (H/R) is increased, indicating that tall tanks are more prone to rocking than broad tanks.
Effect of Foundation Soil on Absolute Lateral Translational Acceleration
Fig. 1(b) illustrates the effect of foundation soil on the lateral acceleration. The maximum absolute lateral acceleration, (x + G,,)max, is also normalized
789
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
c, = 600 ft/sec c, = 5000 ft/sec
iiiiii ^
\ :N; o.ooi \» 0.002 • 0.003 0.004 0.005 0.006
Height-to-Radius ratio, H/R
E d
hs/R = 0.001 V --* hs/R = 0.002 >• • - -* hs/R = 0.003 - -< hs/B = 0,004 ---» / is/f l = 0 .005 —« hs/R - 0 .006
Height-to-Radius ratio, H/R
FIG. 4. Tank)
Variations of Fundamental Frequency with Soil and Tank Properties (Rigid
h /R = 0.003
1000 2000 3000 4000 5000 Shear Wave Velocity (ft/sec)
FIG. 5. Tank)
Variations of Fundamental Frequency with Shear-Wave Velocity (Rigid
with respect to the maximum ground acceleration, (Gh)max. The value of this ratio is typically greater than 1.0 for low shear-wave velocities of soil, and it approaches the value of 1.0 for larger shear-wave velocities. This is an indication that the relative acceleration, x, approaches zero for stiffer soils. The influence of (H/R) ratio on the maximum absolute translational acceleration is relatively small; curves representing the effect of different values of (H/R) follow a very close trend, in contrast to the behavior observed for rocking acceleration.
790
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
c, .= 1000 ft/sec
CO
E< CD
Q
Height-to-Radius ratio, H/R
FIG. 6. Variations of First-Mode Damping Ratio, £, with (H/R) Ratio (Rigid Tank)
-H
H/R - 1.0 -* H/R =2.0 - H/R = 3-0
+
0 WOO 2000 3000 4000 5000 0 WOO 2000 3000 4000 5000 <") Shear Wave Velocity (ft/sec) (b) Shear Wave Velocity (ft/sec)
FIG. 7. (a) Variations of IVIaximum Rocking Acceleration (Rigid Tank); (b) Variations of Maximum Absolute Lateral Translational Acceleration (Rigid Tank)
Effect of Foundation Soil on Base Shear and Overturning Moment The base shear and overturning moment for the system under consider
ation are
Qf = (mi + m0)(x + G„) + (m0H0 + mjl^d (17)
Mf = (m0HQ + m1H1)(x + Gh) + (mjil + m0H2
0 + 70)d (18)
If the foundation soil is assumed to be infinitely rigid, (x = a = 0), the shear and moment expressions are reduced to -
Qr = (mi + m0)Gh (19)
791
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
MK = (m0H0 + m^G,, (20)
Figs. 8(a) and 8(b) show the variations of the ratio of maximum shears (flexible soil/rigid soil) and the ratio of maximum moments (flexible soil/ rigid soil) with changes in soil stiffness. It is observed that both shear and moment are magnified for soils with low shear-wave velocities, and this magnification factor is further increased when the ratio (H/R) is increased. The values of these ratios are seen to approach 1.0 as the soil becomes stiffer, restating that the relative response of the tank with respect to soil vanishes.
FLEXIBLE-TANK DEFORMABLE-SOIL SYSTEM
Effect of Foundation Soil on Natural Frequency of System Fig. 9 shows the variation of the fundamental frequency parameter with
changes in tank geometric properties and soil conditions. The fundamental frequency parameter increases steadily to the ratio (H/R) of 1.5, then decays as (H/R) assumes larger values. By isolating a curve representative of a particular ratio (hJR) and varying the shear-wave velocity, it is noted that the frequency parameter increases as the shear-wave velocity is increased. Moreover, if one considers a particular value of (H/R) and varies (hJR), it is observed that the values of the frequency parameter become larger for higher values of (hJR), the difference being larger for stiffer soils (high shear-wave velocities). Therefore, one may conclude that for lower values of shear-wave velocity (softer soils), the soil dominates the fundamental frequency, and for higher shear-wave velocities the flexible shell response is dominant.
Effect of Foundation Soil on Maximum Relative Acceleration of Shell The maximum relative acceleration of a shell resting on a flexible foun
dation has been normalized with respect to the maximum ground acceleration, (G,,)max. Fig. 10 shows variations of this ratio as a result of changing
Shear Wave Velocity (ft/sec) (b) Shear Wave Velocity (ft/sec)
FIG. 8. (a) Variations of Maximum Base Shear (Rigid Tank); (b) Variations of Maximum Overturning Moment (Rigid Tank)
792
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
c, = 600 ft/sec
Height-to-Radius ratio, H/R Height-to-Radius ratio, H/R
FIG. 9. Variations of Fundamental Frequency Parameter (Flexible Tank)
0 WOO 2000 3000 4000 6000 Shear Wave Velocity (ft/sec)
FIG. 10. Variations of Maximum Wall Acceleration (Flexible Tank)
the tank geometric properties and the soil conditions; the ratio increases as shear-wave velocity is increased. If the shear-wave velocity is held constant, it is observed that the ratio of maximum accelerations is larger for tall tanks. It can be concluded that soil effects on the relative shell acceleration is beneficial; the softer the soil, the less the shell acceleration.
Effect of Foundation Soil on Rocking Acceleration The variations of maximum acceleration response due to rocking motion,
Ha, with foundation soil and tank geometric properties are depicted in Fig. 11(a). The acceleration values have been normalized with respect to peak ground acceleration, (G;,)max. It is observed that-at low shear-wave velocities, the ratio of maximum accelerations indicates a magnification of the rocking acceleration response. An increase in shear-wave velocity causes a reduction
793
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
0 WOO 2000 3000 4000 5000 0 WOO 2000 3000 4000 5000 (a) Shear Wave Velocity (ft/sec) <b) Shear Wave Velocity (ft/sec)
FIG. 11, (a) Variations of Maximum Rocking Acceleration (Flexible Tank); (b) Variations of Maximum Absolute Lateral Translational Acceleration (Flexible Tank)
in this ratio that approaches zero as shear-wave velocity assumes larger values. This indicates that for very stiff soils, (cf > 4,000 ft/sec; 1,219 m/s), the rocking motion is minimal as expected. If the shear-wave velocity is held constant for different values of (H/R), it is observed that the value of the ratio of maximum acceleration increases as (H/R) increases; the effect being more apparent for low shear-wave velocities. A comparison between Figs. 11(a) and 7(a) shows that the rocking responses of flexible and rigid tanks on soft soils are almost identical, whereas for stiff soils, the rocking response of a flexible tank is less than that of a rigid tank, although both asymptotically vanish for very stiff soils.
Effect of Foundation Soil on Absolute Lateral Translational Acceleration
Fig. 11(b) shows the effects that the foundation soil and the geometric properties of the tank have on the absolute lateral acceleration response. The value of the ratio of maximum accelerations is greater than 1.0 for shear-wave velocities representative of soft soils, and it approaches the value of 1.0 for shear-wave velocities representative of stiff soils. The effect of the ratio (H/R) on the ratio of maximum accelerations is relatively small.
Effect of Foundation Soil on Base Shear and Overturning Moment The base shear and overturning moment for the system under consider
ation are
Qf = mj + (m0 + m1)(x + Gh) + (m0H0 + trifled (21)
Mf = m1H1y + (m0H0 + m^H^x + Gh) + (m0H20 + mji\ + IQ)o. .... (22)
The maximum shear and maximum moment are normalized with respect to the values given by (19) and (20), respectively. Figs. 12(a) and 12(b) display the variations of the ratio of maximum shear (flexible soil/rigid soil) and the ratio of maximum moment (flexible soil/rigid soil) for changes in the geometric properties of the tank as well as changes in the stiffness of foun-
794
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
0 WOO 2000 3000 4000 5000 0 WOO 2000 3000 4000 5000 (a) Shear Wave Velocity (ft/sec) (b) Shear Wave Velocity (ft/sec)
FIG. 12. (a) Variations of Maximum Base Shear (Flexible Tank); Variations of Maximum Overturning Moment (Flexible Tank)
0 WOO 2000 3000 4000 5000 0 WOO 2000 3000 4000 5000 (a) Shear Wave Velocity (ft/sec) (b) Shear Wave Velocity (ft/sec)
FIG. 13. (a) Variations of Maximum Base Shear (Flexible Tank, San Fernando Record); (b) Variations of Maximum Overturning Moment (Flexible Tank, San Fernando Record)
dation soil. It is observed that at low shear-wave velocities, both the shear and the moment are magnified between 150 and 300% due to soil flexibility. For stiffer soils, the ratio of maximum shear and maximum moment shows a significant magnification due to shell flexibility. If this is compared with the behavior observed for a rigid tank-deformable soil system, one can easily see the significant effect that shell flexibility has on magnifying the response of tanks supported on stiff soils. Figs. 13(a) and-13(fr) illustrate the variations of the ratio of maximum shears and the ratio of maximum moments when the tank system is subjected to the horizontal component of ground motion
795
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
the 1971 San Fernando earthquake, recorded at Pacoima Dam. It is observed that the trend of response of the tank-soil system is the same as that observed for the 1940 El Centro earthquake.
CONCLUSIONS
The study demonstrates the relative importance of a number of parameters on the seismic response of ground-based tanks supported on deform-able soils and subjected to a horizontal component of ground excitation. The results clearly show that interaction of the tank and foundation soil magnifies the tank response, and that the magnification is a factor of both the shear-wave velocity of soil (higher magnifications for soft soils) as well as the geometric properties of tank (higher magnification for tall tanks). In addition, the results indicate that shell flexibility has a pronounced influence on the dynamic behavior of tanks; it contributes to the magnification of pressures developed in liquid and exerted on tank, thereby increasing the base shear and overturning moment, especially for stiff soils.
APPENDIX I. REFERENCES
Abou-Izzeddine, W. (1989). "Parametric study of seismic interaction of liquid storage tanks with foundation soil," thesis presented to the Univ. of California, at Irvine, Calif., in partial fulfillment of the requirements for the degree of master of science.
Haroun, M. A. (1991). "Assessment of seismic hazards to liquid storage tanks at port facilities." Proc. 1990 POLA Seismic Workshop.
Haroun, M. A. (1980). "Dynamic analyses of liquid storage tanks." EERL 80-4, California Inst, of Tech., Pasadena, Calif.
Haroun, M. A., and EUaithy, H. M. (1985a). "Model for flexible tanks undergoing rocking."/. Engrg. Mech., ASCE, 111(2), 143-157.
Haroun, M. A., and EUaithy, H. M. (1985b). "Seismically induced fluid forces on elevated tanks." /. Tech. Topics, ASCE, 111(1), 1-15.
Haroun, M. A., and Housner, G. W. (1981). "Dynamic interaction of liquid storage tanks and foundation soil." Proc. Second ASCEIEMD Specialty Conf. on Dynamic Response of Structures, ASCE, New York, N.Y., 346-360.
Housner, G. W. (1957). "Dynamic pressures on accelerated fluid containers." Bull. Seismol. Soc. Amer., 47(1), 15-35.
Richart, F. E., Woods, R. D., and Hall, J. R. (1970). Vibrations of soils and foundations. Prentice-Hall, Englewood Cliffs, N.J.
Veletsos, A. S., and Tang, Y. (1987). "Rocking response of liquid storage tanks." /. Engrg. Mech., ASCE, 113(11), 1774-1792.
Veletsos, A. S., and Yang, J. Y. (1977). "Earthquake response of liquid storage tanks." Proc. Annual EMD Specialty Conf, ASCE, New York, N.Y., 1-24.
APPENDIX II. NOTATION
the following symbols are used in this paper:
Cx, Ca = damper constants given by (8); Cy = shell damping coefficient; Cf = shear-wave velocity;
Es = Young's modulus of shell material; Gh = horizontal ground displacement; Gf = shear modulus of foundation soil; H = liquid depth;
H0, Hx = elevations of equivalent masses m0, mx;
796
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
hs, hb = shell and'base thickness; /, Ia = mass moments of inertia;
Ky = spring constant given by (4); Kx, Ka = spring constants given by (7);
L = shell length; M{t) = overturning moment;
M = mass matrix; Mf, Mr = overturning moments developed in tank supported on flex
ible and rigid soil; m = total mass of liquid;
m0, m1 = equivalent masses of mechanical analog; pd(r,Q,z,t) = hydrodynamic pressure;
Q{t) = base shearing force; Qf, Qr = shear forces developed in tank supported on flexible and rigid
soil; R = radius of tank; r = radial coordinate; r = excitation direction vector; T = kinetic energy; t = time;
V = potential energy; w(z,t) = radial displacement of shell;
w(t) = time-dependent generalized displacement; x(t), y(t) = generalized degrees of freedom;
z = axial coordinate; a(t) = rigid-base rotation of tank;
8 = variational operator; §Wnc = work done by nonconservative external loads;
9 = circumferential coordinate; Vp vs = Poisson's ratio of foundation soil and shell;
£ = damping ratio; p6, pf = mass density of base and foundation soil;
P/> PJ; P.v = mass density of liquid, shell, and water; co = natural frequency of vibration; w = frequency parameter; and
0)^,0^ = natural frequency and frequency parameter of fixed-base shell.
797
J. Struct. Eng. 1992.118:783-797.
Dow
nloa
ded
from
asc
elib
rary
.org
by
Uni
v. o
f A
laba
ma
At B
irm
ingh
am o
n 05
/11/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
