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Seismic Design and Analysis of Concrete Liquid-Containing Tanks
Zhong (John) Liu, Ph.D., P.E., S.E.1
1
Kiewit Engineering Group Inc., 9401 Renner Blvd., Lenexa, KS 66219. E-mail:
John.Liu@kiewit.com
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ABSTRACT
Concrete tanks are used in high seismic regions for water, chemical storage, and treatment.
These tanks may be reinforced or prestressed concrete structures with rectangular or round
shapes and may locate above or underground. Based on governing codes, such as ACI 350-06,
ACI 373, the tanks should have strength, water tight, durability for cracking, and corrosion
control. ACI 350.3 also provides equations for seismic forces of rectangular and circular tanks.
However, in a design practice, we may still have issues that ACI codes don’t cover, such as
tsunami events, and uplifting requirement of underground tanks. The section forces of critical
load combinations are difficult to be found unless a 3-D computer model is created. After we
have a computer model, we may still doubt what is the impact of soil sub-grade on the base, side
soil spring for underground tanks because these data have a very big influence on dynamic
properties of tank system, such as the fundamental frequency. We may wander, without these
data, if the seismic analysis of tanks per ACI 350.3 is still reliable. Modeling of the complex
liquid-structure and soil-structure interaction are required for a reliable design. In this study, the
codes and phenomenon of concrete liquid tanks in high seismic regions will be reviewed and
summarized, and the seismic analysis based on ACI 350.3 will be illustrated. The finite element
methods of pseudo-static seismic analysis and responds analysis following original Houser’s
assumption of ACI 350.3 will be presented.
Keywords: Liquid-containing Concrete Tanks, Hydrodynamic Forces, Seismic analysis,
Convective Pressure, Impulsive Pressure, SAP 2000
1.0. INTRODUCTION
Concrete liquid containing tanks are used in high seismic regions for water, chemical storage
and treatment. See Figure 1, a concrete rectangular tank is buried underground as a back-washtank for Santa Barbara desalination project. Figure 2 is an overview of concrete tanks of
Carlsbad desalination project, in which, most tanks are rectangular above ground tanks.
Due to excessive damages reported on steel tanks, the concrete tanks have become
profoundly popular. Some tanks are essential part of society lifeline and must therefore be
maintained during emergencies. Some tanks store hazardous materials and should be always
avoided accidental leaking to nearby environment. These tanks may be reinforced or prestressed
concrete structures with rectangular or round shapes and may locate above or underground. For
design of such structures in seismic regions, following considerations should be included:
 Firstly, we need to consider the functions as a liquid containing structure. Based on
governing codes, such as ACI 350-06, ACI 373, ANSI/AWWA D115-06, the tanks
should have strength, water tight, durability for cracking and corrosion control.
 Secondly, we need to consider hydrodynamic seismic loads. The loads are the effect of
fluid interaction with structures on seismic response. The parameters of interactions
include fluid properties, natural frequencies of sloshing, hydrodynamic pressure
distribution on the wall, tank’s structural properties, and soil-structure interaction of
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foundations. Even though, the real performance of liquid containing tanks for seismic
response is very complicated, the design may follow ACI 350.3, ASCE 7 for simplified
calculation if tanks are rectangular or round shapes.
 Thirdly, we need to consider seismic related events which ACI codes don’t cover, such as
tsunami events, liquefaction of soil foundation, and floating uplifting of underground
tanks.
 In addition, we need to consider national or local building codes to look for any special
requirements of such structures.
In this paper, all above 3 considerations will be discussed. However, the analysis of
hydrodynamic seismic loads and design will be mainly addressed.
2.0 DESIGN OF LIQUID CONTAINING CONCRETE STRUCTURES
Concrete could be good for liquid containing structures, mostly because it’s controllable
permeable properties and corrosion resistance. Different from just ACI 318 for regular concrete
structures, ACI 350 will apply.
For watertight requirement, cracking width, reinforcement, concrete mix design, and
impervious protective coating may be considered. Proper expansion and construction joints with
waterstops are also important. Water leaking test is always required per ACI 350.
For corrosion requirement, following ACI 318 section and ACI 350 section, minimum
concrete strength is required for different types of exposure conditions. The maximum watercement ratio and concrete cover are also used for concrete expose conditions. Concrete mix
design, such as cement type, pozzolans (fly ash), air-entraining agents, chemical admixtures, and
aggregates are required per different exposure conditions.
Figure 1 BW Sump Tank for Santa Barbara Desalination Reactivation Project, Santa
Barbara, CA – Courtesy of Kiewit Corp.
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Figure 2 Overview of Carlsbad Desalination Project, San Diego, California – Courtesy of
Kiewit Corp.
3.0 SEISMIC ANALYSIS AND DESIGN PER CODES
The liquid sloshing interaction with concrete container subjected to seismic ground motion of
earthquakes can be described as two systems: the free liquid surface motion and the breathing
liquid container structure.
Figure 3 The Slosh Wave of Liquid
See Figure 3 for slosh wave of tank, the surface oscillating wave is a special phenomenon for
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the hydraulic structures. Such surface movement generally has frequencies much longer than the
fundamental frequency of the structures. Therefore, assumption described by Housner (1962) is
that the slosh wave can be separated from other dynamic motion as a single degree of freedom
system (SDOF). Laterally, such assumption was modified and used in different design Codes,
such as ACI 350.3R. The seismic response of surface movement is a convective pressure applied
to walls of containing structure. The portion of liquid can be called equivalent weight of the
convective component of stored liquid by ACI 350.3. The rest of liquid breath the surface
basically will be additional mass to be included in containing structure for impulsive vibration. It
is the equivalent weight of the impulsive component of stored liquid. The impulsive vibration
can also be simplified as a single degree of freedom system (SDOF). In which, the mass include
wall weight and impulsive liquid weight; and the stiffness is that of concrete containing structure.
Figure 4 Housner’s Model
See Figure 4, the code model is illustrated. The convective mass is connected to wall with
springs at the water level hc and the impulsive mass is connected to wall rigidly at the water level
hi.
Any structure system subjected to seismic ground motion of earthquakes can be described as
following equivalent matrix equation.
(1)
 M u’’   Cu’  K u    P
Where [M],[C], [K], [P] are defined as matrix of mass, damping, stiffness and force
respectively. The [u],[u’], [u”] are displacement, velocity, and acceleration respectively.
Figure 4 is a two single-degree-of freedom system, if we consider that the structure of
impulsive movement only depends on the fundamental frequency of foundation system; or a
multi-degree of freedom system if we consider that concrete containing structure is in a multiple
frequency domain. ACI 350.3 follows the simplified model of Figure 4 as a 2 single-degree-of
freedom system. In which, the convective and impulsive are two independent movements,
because the interaction of the low frequency of convective movement and high frequency of
impulsive movement can be ignored. Then, the results can be added per SRSS (the square root of
sum of square method). Such method is summarized in this section.
However, we could also analyze the model of Figure 4 directly by finite element and let
software to combine results of 2 or more frequency modes, if both impulsive and convective
liquid modes are included. Such method is summarized in Section 5. The procedures for
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rectangular and circular tanks are similar. However, to help understanding, the equations of
rectangular tanks are mainly illustrated in this paper.
For ACI 350.3, the effective weight of the impulsive component, the effective weight of
convective component can be calculated from equations or Figure 9.2.1 & 9.3.1 of ACI 350.3.
See Figure 5, rectangular tank mass factor Wc/WL, the convective component increases with the
length of liquid containing tank.
Figure 5 Rectangular Tank’s Factor of Weight – Courtesy of ACI
For rectangular tank, following equations are provided in this paper to explain the procedure.
In which, L = Length of tank; HL = Depth of stored liquid; WL = Total weight of the stored
liquid; Wi = The weight of impulsive component of the stored liquid; Wc = The weight of
convective component of the stored liquid; hi = Height above the base of the wall for impulsive
lateral force of the case excluding base pressure (EBP); h’i = Height above the base of the wall
for impulsive lateral force of the case including base pressure (IBP); hc = Height above the base
of the wall for convective lateral force of the case excluding base pressure (EBP); h’c = Height
above the base of the wall for convective lateral force of the case including base pressure (IBP);
Kc = The equivalent spring of stiffness for convective component; Tc = Nature period of the first
convective (sloshing) mode; g = Acceleration of gravity.
 L 
tanh[0.866 
]
H
Wi
 L
(2)

WL
 L 
0.866 

 HL 
 L 
Wc
 HL 
 0.264 
 tanh[3.16 
]
WL
 L 
 HL 
hi
L
 0.375, when
 1.333
HL
HL
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(3)
(4)
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hi
L
L
 0.5  0.09375( ), when
 1.333
HL
HL
HL

 H 
cosh 3.16  L    1
hc
 L 

 1
HL
H 
H 
3.16  L  *sinh[3.16  L ]
 L 
 L 
h 'i
L
 0.45, when
 0.75
HL
HL
 L 
0.866 

HL 
h 'i
1



HL
 L  8
2* tanh[0.866 
]
 HL 

 H 
cosh 3.16  L    2.01
h 'c
 L 

 1
HL
H 
H 
3.16  L  *sinh[3.16  L ]
 L 
 L 
For convective movement, the period of the convective component follows ACI 350.3
equation 9–14:
2
2
Tc 
( ) L
c

 HL 
]
 L 
  3.16 gtanh[3.16 
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Comparing with single degree of freedom system,
Wc
2
Tc 
 2
(12)
c
gK c
Therefore, the spring of stiffness Kc as shown in Figure 4 can be calculated as:
W
H 
(13)
Kc  3.16 c tanh[3.16  L ]
L
 L 
For impulsive movement, the dynamic properties of systems depend on weight and stiffness
of tank.
The total weight of wall and impulsive component of liquid = Ww + Wi. And Ww = weight of
wall. The effective weights should be calculated from the total weight as Wtotal = εWw + Wi. A
coefficient ɛ represents the ratio of the equivalent dynamic mass of tank shell to its total mass. It
is calculated per Equation 9-44 and 9-45 of ACI 350.3 respectively for rectangular and circular
tanks. For example, rectangular tank’s coefficient is:
2
 L 
 L 
(14)
  [0.0151
  0.19058 
  1.021]  1.0
H
H
 L
 L
We could simplify the tank as walls. For a fixed-base, free-top cantilever walls following
ACI 350.3 9.2.4,
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Ec tw 3
( )
(15)
48 h
In which, Ec =Modulus of elasticity of concrete; tw = Average wall thickness; h = The
equivalent height,
W h  Wi hi
(16)
h w w
Ww  Wi
In which, hw= Height from the base of the wall to the center of gravity of tank.
Similar to convective movement, the single degree of freedom system can be analyzed and
the fundamental period is,
Wtotal
2
Ti 
 2
(17)
i
gKi
Similar to building structures in ASCE 7 and IBC, the design response spectrum is shown as
Figure 6.
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Ki 
Figure 6 Design Response Spectrum – Courtesy of ACI
The dynamic forces are determined per equations 4-1 to 4-4 of ACI 350.3. The convective
dynamic force is
W
Pc  Cc I[ c ]
Rc
The important factor I and Response modification factor R of tanks are illustrated in Tables
4.1.1 (a) and 4.1.1.(b).
The location levels of dynamic seismic forces are shown in Figure 7 and 8. The forces of IBP
and EBP are different. The result of EBP is used for wall design and hold-downs for non-fixed
tanks; and the result of IBP is used for global analysis of tank.
The base shear of seismic forces shall be determined by combining shear of impulsive and
shear of convective modes with Square Root of Sum of Square (SRSS) as equations 4-5 of ACI
350.3. And the moments are combined similarly as equations 4-13 of ACI 350.3.
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Figure 7 Impulsive Forces on Tank
Figure 8 Convective Forces on Tank
Figure 9 Finite Element Model of a Tank
4.0 FINITE ELEMENT ANALYSIS AND DESIGN
Code’s design and analysis such as ACI 350.3 do not provide a detailed procedure for a three
dimensional modeling and analysis. A calculation may be used to estimate moment and axial
force in walls by using simplified tables based PCA publications for rectangular and circular
tanks. Because there is not any equation available to find stresses in walls of tanks, finite element
analysis based on software, such as SAP 2000, or STAAD is easier for engineers. See Figure 9, a
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concrete tank can be described with small shell elements in SAP 2000. Then, different load cases
could be established. For concrete tank design, we need to consider following critical cases:
Operating Status with Soil & Ground Water Pressure
Operating Status with Buoyancy of Ground Water Pressure
Leaking Test
Seismic Loads Along Two Perpendicular Directions
The seismic loads can be calculated following ACI 350.3. The results of seismic forces,
including impulsive and convective forces in Figures 7 and 8 can be added as static loads to
surface of shell elements. See Figure 10, through analysis, the stress distribution in tank structure
can be found for each case or enveloped for all cases. Such loads can be used in reinforcement
design.
However, such design and 3-D analysis are not a realistic 3-D dynamic analysis. For
example, the ACI 350.3 assumes tanks under cantilever action for walls, and the results of such
assumption still exists even though the tanks are modeled as elements of 3-D and the roof
structure is included.
Figure 10 Results of SAP 2000 Analysis
5.0 HYDRODYNAMIC SEISMIC RESPONSE ANALYSIS
See Figure 4, original Houser’s Model actually provided a simplified dynamic structural
model. The contained liquid is modeled as two mass components, the impulsive mass (Wi) is
rigidly connected with walls at level hi, and the convective mass (Wc) is connected to walls with
springs (Kc) at level hc. Where, the mass components are calculated per Equations (2) and (3);
the height of hc and hi are calculated per Equations (4) to (9); and the spring Kc is calculated per
Equation (13).
See Figure 11, a SAP 2000 model can be modified from the static analysis model previously.
The elastic links can be used to model the spring Kc to connect a point of convective mass at the
center of tank. Then, such model can be quickly analyzed by Response Spectrum Analysis of
SAP 2000 to find seismic response. By comparing with pseudo-static method, we could find that
concrete tank may be much stronger than the assumption of cantilever walls with Equation 15,
especially if concrete roof of tank is provided. Additionally, soil spring constants at the bottom
of base and side of walls for underground tanks are also factors that pseudo-static analysis could
not model. Moreover, the soil spring is critical to the fundamental frequency of a structure,
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which determines the seismic force per seismic response spectrum of Figure 6.
Figure 11 Kc, Mc, and Mi Are Added to a Tank Model
6.0 CONCLUSIONS
When finite element software, such as SAP 2000 is used to model the pseudo-static analysis
based on seismic loads of ACI 350.3, the internal SAP 2000 functions of time history, spectrum
analysis may be quickly established for additional dynamic analysis. Based on ACI 350.3,
hydrodynamic pressure distribution involves two components, the convective which depends on
the sloshing vibration near the liquid surface and the impulsive vibration underneath. Therefore,
adequate masses for the impulsive and convective components could be included and are
connected with concrete walls by springs. The dynamic analysis could consider soil sub-grade,
side soil springs, and adequate mass of side earth dynamic pressure for seismic analysis. To
summary the results, the Square Root Sum of the Squares method (SRSS) is internally
established by SAP 2000 as a type of load combination. Where appropriate, the results could be
checked for strain capability of the soil, dynamic masses and frequencies of components through
a simple, iterative process.
As stated before, proposed dynamic seismic analysis is an additional procedure in SAP 2000
design model based on ACI 350.3. Comparison of dynamic seismic results with pseudo-static
seismic results of ACI 350.3 could be done in any time. Pseudo-static seismic analysis directly
provides ACI 350.3 code design load; and new dynamic seismic analysis provides understanding
of performance of concrete tanks for uncertain parameters of soil, liquid, and dimensions of
structure. The analysis presented here can help to understand the dynamic liquid-structure and
soil-structure interactions and lead more reliable design, which is very important for concrete
tanks to have water tight, durability of cracking and corrosion control.
REFERENCES
ACI 318-11, Building Code Requirements for Structural Concrete and Commentary, American
Concrete Institute: Farmington Hills, MI.
ACI 350.1, Building Specification for Tightness Testing of Environmental Engineering Concrete
Containment Structures (ACI 350.1-10) and Commentary, American Concrete Institute:
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Farmington Hills, MI.
ACI 350.3, Building Seismic Design of Liquid Containing Concrete Structures, American
Concrete Institute: Farmington Hills, MI.
ASCE/SEI 7-10, Minimum Design Loads for Buildings and Other Structures, American Society
of Civil Engineers: Reston, VA.
Housner, G. W. (1963). The Dynamic Behavior of Water Tanks, Bulletin of the Seismological
Society of American.
Javeed A. Munshi (1994). Rectangular Concrete Tanks, Portland Cement Association, Skokle,
IL.
(1993) Circular Concrete Tanks without Prestressing, Portland Cement Association, Skokle, IL.
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