Dynamic Behavior and Response
Analysis
of Fluid/Tank Systems
He Liu, Ph.D., P.E.
University Alaska Anchorage
Daniel H. Schubert, P.E.
Dept. of Environmental
Health & Engineering
ANTHC
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Tanks Rupture: A water tank was lifted off a gravel base six to ten inches during an
earthquake. After the earthquake, the tank was resting on the gravel base about212
inches lower (due to buckle) and shifted about an inch to the west.
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Courtesy U.C. Berkley
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Courtesy U.C. Berkley
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Courtesy U.C. Berkley
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Courtesy U.C. Berkley
 Tank seismic Design Code is mainly
based on simplified Rigid Tank
assumption, and NO fluid/structure
interaction included
 Theoretical solutions are available
only for rigid tanks and no interaction
 An approximate “Cantilever Beam”
approach needs numerical proof.
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The Approximate
“Cantilever-Beam” Approach
 For approximate frequencies can be calculated
by equation
 f’0 is the natural frequency of the fluid tank
system with roof mass
 fo is the natural frequency without the roof
 fFB and fSB is the natural frequency of an
empty tank with the mass of the roof
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Purposes of This Study
Use the Finite Element Analysis (FEA)
 To evaluate the performance of steel
water tanks due to earthquake
excitation
 To compare results with Design Code
of AWWA’s simplified formula’s
 To verify the approximate
“Cantilever-Beam” approach.
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z
water level
Ht
H
x
tb
y
r
Radius =16 feet

Height = 26 feet
x
R
Water Depth = 24 feet
=0o
ts
Wall Thickness =0.315in
Tank Geometry
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Modeling Approach I -Tank
 Tank wall/roof/base
- shell and beam elements
 Fluid
- Fluid 3-D Contained Fluid Element
 Material Properties
- Steel E = 29,000 ksi
- Steel density = 7.34x104 lb-sec2/in4
- Fluid density = 0.9345x104 lb-sec2/in4
- Fluid bulk modulus = 30x104 lb/in211
Modeling Approach II 3-D Contained Fluid Elements




8 nodes - 3 DOF
Free surface - added spring
Bulk modulus = 30 x 104 lb/in2
Fluid elements do not attached at tank
wall and base
 Coincident nodes coupled normal to
the interface to allow fluid relative
movement in tangential and vertical
directions
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 Free horizontal movement at base
Modeling Approach III
- Meshing FEA Models
 Because of the system symmetry, one half
of the tank is modeled.
 Fluid elements are rectangular-brick
shaped whenever possible
 Number of Fluid Elements in ANSYS
Model:
- 640, 1280, 2112, 3072
- Based on accuracy and efficiency, 1280
fluid elements was chosen
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 partially filled
with a near
incompressible
water
 water-contained
fluid elements
 tank--shell and
beam elements
 interaction
between water and
tank wall is
included
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How to verify the
numerical solution
from ANSYS FEA
models?
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ANSYS
Rigid Tanks
ANSYS
Theory
Rigid Tanks
Yes
Approximate
Flexible Tanks
Flexible Tanks
Comparison
&
Conclusion
AWWA
Simplified “Rigid”
Tank Method
ANSYS
Unanchored
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1st Natural Mode Shape – Water Sloshing
One-Cosine Type Sloshing Mode
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2nd Natural Mode Shape – Water Sloshing
Two-Cosine Type Sloshing Mode
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3rd Natural Mode Shape – Water Sloshing
Three-Cosine Type Sloshing Mode
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4th Natural Mode Shape – Water Sloshing
Four-Cosine Type Sloshing Mode
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Modal Analysis Results:
Comparison of Convective Frequencies
No. of Fluid Elements in ANSYS Model
640
1280
2112
3072
Theory
(Units: Hz)
1st mode 0.299
0.298 0.298 0.298
0.305
2nd mode 0.477
0.476 0.475 0.474
0.521
3rd mode 0.562
0.555
0.660
0.552 0.551
Note: Compared with results of linear theory, the first mode
differs by 1.7%. Differences may be related to limitations on
the linear theory, with nonlinear theory closer to FEA values.21
For Rigid Tanks
ANSYS Results
Modeling approach
is acceptable
Theoretical Results
Flexible Tank Analysis
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Modal Analysis for Flexible Tanks

A total of 54 geometric variations, with
and without roofs, were analyzed.
 Tank/fluid variables were represented by
three basic parameters:
 Tank geometric aspect ratios, as represented
by the tank height to radius (H/R)
 Tank shell wall thickness ratio represented
by the wall thickness to tank radius (ts/R)
 Liquid depth ratio (h/R)
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1st Natural Mode Shape – Full Tank
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2nd Natural Mode Shape – Full Tank
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3rd Natural Mode Shape – Full Tank
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1st Natural Mode Shape - Partially Full Tank
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2nd Natural Mode Shape – Partially Full Tank
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3rd Natural Mode Shape – Partially Full Tank
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1st Natural Mode Shape – Tall-Full Tank
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1st Natural Mode Shape – Short-Full Tank
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1st Natural Mode Shape – Tall-Partial-Full Tank
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2nd Natural Mode Shape – Tall-Partial-Full Tank
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Modal Frequency Comparison
1st Natural Frequency (H/R=0.667)
Frequency(Hz)
14
12
10
8
6
4
2
0
0
t/R=.0005
Theory
0.2
h/R 0.4
t/R=.001
Theory
0.6
0.8
t/R=.00139
Theory
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Modal Frequency Comparison
1st Natural Frequency (H/R=1.5)
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Frequency(Hz)
25
20
15
10
5
0
0
0.5
t/R=.001
Theory
h/R
1
t/R=.002
Theory
1.5
t/R=.0025
Theory 35
Modal Frequency Comparison
Frequency(Hz)
1st Natural Frequency (H/R=3)
20
15
10
5
0
0
t/R=.001
Theory
1
h/R
t/R=.002
Theory
2
3
t/R=.0035
Theory
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Modal Frequency Comparison
2nd Natural Frequency (H/R=0.667)
Frequency(Hz)
20
15
10
5
0
0
t/R=.0005
Theory
0.2
h/R 0.4
t/R=.001
Theory
0.6
0.8
t/R=.00139
Theory 37
Modal Frequency Comparison
Frequency(Hz)
2nd Natural Frequency (H/R=1.5)
50
40
30
20
10
0
0
t/R=.001
Theory
0.5
h/R
t/R=.002
Theory
1
1.5
t/R=.0025
Theory 38
Modal Frequency Comparison
2nd Natural Frequency (H/R=3)
Frequency(Hz)
25
20
15
10
5
0
0
t/R=.001
Theory
1
h/R
t/R=.002
Theory
2
3
t/R=.0035
Theory 39
For Flexible Tanks
Approximate
“Cantilever-Beam”
Results
Acceptable
ANSYS Results
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(ANSYS Results)
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Earthquake Ground Input:
El Centro N-S Adjusted 0.4g

Acceleration
Acceleration, g
0.4
max = 0.4 g
0.2
0
-0.2
-0.4

Velocity
Velocity, in/sec
0
Displacement, in
Displacement
4
6
30
20
10
8
10
max = 21.3 in/sec
0
-10
-20
-30
0

2
2
4
6
10
8
10
max.= 14.5 in
0
-10
-20
0
2
4
6
Time, seconds
8
10
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Pressure Time History (3 ft. from Base)
Pressure, (ksi)
0.016
0.012
0.008
0.004
0
0
1
2
3
4
5
Time (seconds)
6
7
8
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0.000
0.005
0.010
0.015
Depth of Fluid (ft)
-4
Pressure
distribution
along wall
at  = 0o and
T=3.21 Sec.
-9
-14
-19
-24
Pressure (ksi)
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Depth of fluid, ft.
20
Compare with
Rigid Tank
solutions:
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Pressure
distribution
along wall
at  = 0o
10
5
0
0
0.005
0.01
Pressure, ksi
0.015
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Courtesy U.C. Berkley
Stress Time History Results

Hoop and
Axial Stress
Z=-21 ft
at  = 0o
Hoop and
Axial Stress
Z=-21 ft
at  = 180o
10
5
0
-5
0
2
4
Time (sec)
Hoop Right
Stress (ksi)

Stress (ksi)
15
6
8
Axial Right
12
10
8
6
4
2
0
-2
-4
0
2
Hoop Left
4
Time (sec)
6
Axial Right 47
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Water Surface Displacement Time History
Displacement (inch)
40
20
 =180o
0
-20
0
5
 =0o
Left
side
Right
side
-40
Time (seconds)
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Water Surface Profile at Time=5.09sec.
Displacement, (inch)
(Maximum Water Surface Displacement = 33 inches)
40
20
0
0
6
12
18
24
30
-20
-40
Distance, (ft)
49
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von Mises Stress Distribution at T=4.63 sec.
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Courtesy U.C. Berkley
Base Shear Time History Results
0.013
Pressure (ksi)
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0
1
2
3
4
Time (second)
5
6
7
8
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Overturning Moment Time History
0.013
Pressure (ksi)
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0
1
2
3
4
Time (second)
5
6
7
8
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Comparison of ANSYS Results
with that from AWWA D100
Method
Base Shear
(kips)
Comparison
(%)
FEA
ANSYS
620
100
Approximate
Approach
591
95.3
Simplified
in AWWA
487
78.6
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Spectrum Analysis Comparison
Spectral A cceleration (g's)
90
H /R=.67, h/R=.6, t/R=.001
1.2
80
70
1
600.8
East
West
North
500.6
400.4
30
0.2
20
0
10 0.001
0
0.01
0.1
1
Period (sec)
Spectrum
W ith Roof
1st Qtr Design
2nd
Qtr
3rd
Qtr
4thCode
Qtr
Theoretical
Design
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Conclusions
 FEA method allows for a more complete
evaluation of seismic loading conditions
on fluid/tank systems.
 Rigid tank assumptions provide
un-conservative solutions.
 “Cantilever-beam” approach provide
very good approximations for design
purpose.
 Further refinements in standard design
procedures should permit performance
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based designs.
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