Paper No. T.3-1.2, pp. 1-5
The 6th PSU-UNS International Conference on Engineering and
Technology (ICET-2013), Novi Sad, Serbia, May 15-17, 2013
University of Novi Sad, Faculty of Technical Sciences
A SIMPLIFIED SEISMIC ANALYSIS OF
CIRCULAR LIQUID STORAGE TANKS
Vladimir Vukobratović*, Đorđe Lađinović
University of Novi Sad, Faculty of Technical Sciences, Novi Sad, Serbia
*Authors to correspondence should be addressed via email: vladavuk@uns.ac.rs
Abstract: Experience shows that inadequately designed
or detailed liquid storage tanks, which are common
structures in various facilities, suffered extensive damage
during past earthquakes. Therefore, the satisfactory
seismic response of such structures is very important.
This paper presents the results of a simplified analysis of
the total seismic response of ground-supported concrete
circular liquid storage tanks with various geometrical
properties. The analysis was conducted in accordance
with the provisions of Eurocode 8 and a comparative
analysis of the obtained results is presented. In addition,
a practice-oriented approach to the analysis of local
seismic response quantities of tank structures is also
presented.
Key Words: circular tanks, impulsive mode, convective
mode, Eurocode 8
1. INTRODUCTION
In various industrial facilities, such as petroleum or
wastewater treatment facilities, ground-supported liquid
storage tanks represent very common structures. The
satisfactory seismic response of such structures is crucial
since collapse usually results in heavy consequences
(fire, spilling of liquid, etc.). During a seismic action,
inertial forces are induced due to the acceleration of a
tank structure and hydrodynamic forces are induced due
to the acceleration of liquid. The liquid mass in the lower
part of a tank behaves like a mass that is rigidly
connected to the tank wall and it is termed impulsive
liquid mass. It accelerates along with the wall and
induces impulsive hydrodynamic pressure. The liquid
mass in the upper part of a tank undergoes sloshing
motion and it is termed convective liquid mass. It
induces convective hydrodynamic pressure. Obviously,
the response of a tank structure which is exposed to the
seismic action represents a complex problem. Therefore,
finding a simple, but accurate enough solution for such a
problem represents a big challenge for a designer.
2. A DYNAMIC MODEL USED IN THE ANALYSIS
The behaviour of a tank-liquid system exposed to a
seismic action can be accurately evaluated using various
1
powerful software tools. Nevertheless, the total seismic
response (base shear force and overturning moments) can
also be properly evaluated using some of the simplified
dynamic models defined in the past. Besides simplicity,
these models also provide a quick evaluation of the total
response, which makes them attractive for application.
The simplified dynamic model of a tank-liquid
system that is included in most of the design codes is
based on the model defined by Housner [4]. The
dynamic analysis of a tank structure can be carried out
using the concept of generalised single-degree-offreedom systems (SDOF), representing the impulsive and
convective modes of vibration, as shown in Fig. 1. The
seismic response of SDOF systems may be calculated
independently and then combined in order to provide the
total response of a structure. Only the first modes of
vibration need to be considered in the analysis.
It should be noted that concrete and steel tanks show
different behaviour under a seismic action. In the case of
concrete tanks, the wall may be taken as rigid, whereas
in the case of steel tanks, the wall may be taken as
flexible. In the following text, attention will be focused
on concrete tanks. A dynamic model which is suitable in
such cases is shown in Fig. 1. For steel tanks, a more
suitable model should be adopted, which is discussed
elsewhere [7].
Fig. 1. A dynamic model of a concrete circular tank
In the dynamic model presented in Fig. 1, mi and mc
denote impulsive and convective masses of liquid
respectively, hi is the height where the resultant of
impulsive pressure on the wall is located, while hi' is the
height where the resultant of impulsive pressure on the
wall and the base is located. Similarly, hc is the height
where the resultant of convective pressure on the wall is
located, while hc' is the height where the resultant of
convective pressure on the wall and the base is located.
The inner radius of a tank is denoted as R. All of the
heights are measured from the bottom of the tank wall.
The parameters of the dynamic model depend on
tank’s geometry. These parameters are different in cases
of tanks with rigid and flexible walls, although the
difference is not substantial. In the simplified procedure
proposed by Eurocode 8, the recommended design
values of dynamic parameters are originally defined by
Malhotra [6]. These values are related to the first
impulsive and convective modes of vibration and they
depend on the ratio H/R, where H denotes the height of
liquid. Additionally, ml denotes the total mass of liquid.
Besides already defined quantities, the coefficients Ci
and Cc also appear as the parameters of the dynamic
model and they are related to the natural periods of the
impulsive and convective modes of vibrations. This will
be shown below.
It should be noted that sometimes additional vertical
structural elements are present inside a tank. These
elements cause obstruction to the sloshing motion of
liquid. Unfortunately, no study is yet available to
quantify the effect of such obstructions, but it is
reasonable to expect that the impulsive component will
increase, while the convective component will decrease.
The overturning moment immediately above the base
plate (M) is calculated by the Eqn. 4,
(4)
M   mi hi  mw hw  mr hr  Se Ti   mc hc Se Tc 
where hw is the height of the center of gravity of the tank
wall and hr is the height of the center of gravity of the
tank roof.
The overturning moment immediately below the base
plate (M') is calculated by the Eqn. 5.
M '   mi hi'  mw hw  mr hr  Se Ti   mc hc' Se Tc  (5)
It is obvious from Eqns. 3-5 that Eurocode 8 suggests
the absolute summation rule to combine the response
from impulsive and convective modes. The basis for this
suggestion is that the natural period of the convective
mode is usually much greater than the natural period of
the impulsive mode. Therefore, it is expected that the
peak response of the impulsive and convective modes
will occur simultaneously, i.e. when the convective mode
response is near its peak. However, some recent studies
show that SRSS combination rule gives better results
than the absolute summation rule.
3.1. Parametric study
The seismic response of a ground-supported circular
concrete tank is primarily influenced by its geometrical
properties. According to Eurocode 8 standard, the ratio
of the liquid height and the inner radius of a tank (H/R)
defines the parameters of the dynamic model of a tank
structure. A comparative analysis of the seismic response
of tanks with various geometrical properties was
conducted through a parametric study and the results are
presented in the following text. All of the response
quantities introduced in the previous section were taken
into account in the study.
In the parametric study, the following input data were
used:
 The values of H/R from the Eurocode 8 were
taken into account in the analysis in order to be
consistent with the code [3].
 Water was considered as liquid (ρ=1000 kg/m³).
 The volume of water (V) was taken to be a
constant value for all tanks and it equaled 4000
m³, which means that the mass of water was also
a constant value (ml=4000 tons).
 Tank material was concrete C35/45 (E=34×109
N/m²).
 The thickness of the tank wall (s) was taken to be
constant along the wall height for all analysed
tanks and it equaled 0.40 m (it is clear that in the
case of tanks of great height, 0.40 m is not
enough from the concrete design aspect and that
wall thickness should be increased, but it does
not significantly affect this analysis).
 The analysis considered tanks without a roof (in
this way, analysis was significantly simplified
since the roof mass and the height of its center of
gravity did not need to be calculated).
 The height of a tank wall (Lw) was always taken
to be 1.0 m greater than the water height (in
tanks without a roof, spilling of liquid should be
prevented during a seismic action and although
the provisions of Eurocode 8 define sloshing
3. TOTAL SEISMIC RESPONSE
The total seismic response of a tank structure should
be analysed in terms of natural periods of vibrations,
base shear force and overturning moments. As already
mentioned, these quantities will be determined in
accordance with the provisions of Eurocode 8.
The modal properties of a tank structure should be
determined at the beginning of the analysis. The natural
periods of the impulsive and the convective modes are
calculated by the Eqns. 1 and 2, respectively,
H 
Ti  Ci
(1)
s/R E
Tc  Cc R
(2)
where s represents equivalent uniform thickness of the
tank wall (in the case of a wall with constant thickness, s
is equal to the wall thickness), E is the modulus of
elasticity of tank material and ρ is mass density of the
liquid.
The total base shear force is calculated by the Eqn. 3,
Q   mi  mw  mr  Se Ti   mc Se Tc 
(3)
where mw represents the mass of the tank wall, mr
represents the mass of the tank roof, Se(Ti) is the
impulsive spectral acceleration (obtained from a 5%
damped elastic response spectrum for the case of
reinforced concrete tanks, for both damage limitation and
ultimate limit state) and Se(Tc) is the convective spectral
acceleration (obtained from a 0.5% damped elastic
response spectrum in all cases).
2
wave height, for the purpose of this study, 1.0 m
was adopted as a conservative value which
satisfied all analysed tanks).
 In the analysis, Eurocode 8 elastic spectrum (type
1, soil type B) was used [2], as well as: reference
peak ground acceleration agR=0.20g, importance
factor for the structural class III γI=1.20 (which
gives design ground acceleration ag=0.24g),
damping ratio for the impulsive mode (for the
case of reinforced concrete tanks) ξi=5% and
damping ratio for the convective mode ξc=0.5%
(damping correction factor η=1.35).
The choice of a constant value of water volume
(mass) is quite logical, since the main idea of the study
was to investigate the influence of tank geometry on its
total response. This way, H and R values changed for
each selected H/R ratio.
Besides theoretical, this study also possesses practical
significance. Therefore, realistic input data were chosen
with the slight exception of the wall thickness and the
sloshing wave height, which has already been explained.
Based on the input data described above and H/R
values defined in Eurocode 8, the dynamic properties of
tanks with different H/R were calculated. Additionally,
the values of Lw, hw, and mw were also calculated.
The seismic response of the analysed tanks is
presented in the Tab. 1 and Figs. 2 and 3.
Table 1. The seismic response of the analysed tanks
Ti
Tc
Q
M
M’
H/R
[sec]
[sec]
[kN]
[kNm]
[kNm]
0.3 0.050
8.41
5837
14307
50018
0.5 0.053
6.44
8688
27556
68197
0.7 0.056
5.59 11259
43730
84042
1.0 0.061
5.00 14434
71732
105872
1.5 0.072
4.56 18476 123006
145998
2.0 0.085
4.34 21797 177814
192271
2.5 0.101
4.19 25078 237931
248973
3.0 0.117
4.05 28116 300816
309076
It is obvious from the results presented in Tab. 1 that
the periods of the impulsive and the convective modes
are rather distant which justifies the use of absolute
summation rule for the combination of responses from
impulsive and convective modes.
Fig. 3. The dependence of the base shear force (Q) and
the overturning moments (M, M') on the ratio H/R
All of the examined quantities except the convective
period of vibrations increase almost linearly with the
increase of the ratio H/R. The convective period of
vibrations is practically a constant value for the H/R>1.5.
Therefore, it can be concluded that the seismic response
of a circular tank structure mostly depends on the
response in the impulsive mode, although the convective
part of the response must not be neglected in order to
conduct an accurate seismic analysis.
The values of Q and M should be used for the
calculation of stresses and stress resultants in the tank
wall and at the connection to the base. The value of M'
should be used for the verification of its supporting
structure, base anchors or foundation. The increase of
base shear force and overturning moments controlled by
the increase of the ratio H/R directly influences the
detailing demands which are in certain cases hard to
fulfill. It should also be noted that with the increase of
the ratio H/R the difference between the overturning
moments tends to decrease, which is the fact that can be
of great practical significance.
Unfortunately, the geometry of a tank structure (i.e.
the ratio H/R) is in most cases pre-defined and it mainly
depends on technological demands inside the facility
which means that a structural designer practically has no
influence on its choice.
4. LOCAL SEISMIC RESPONSE
In order to obtain local response quantities (stresses,
strains, etc.) which occur in a tank structure during a
seismic action, a proper simulation of all individual
actions on the a structure must be made.
During lateral base excitation, the tank wall is
subjected to lateral hydrodynamic pressures and the tank
base is subjected to vertical hydrodynamic pressures.
Pressure due to wall inertia, which is in the case of walls
Fig. 2. The dependence of the impulsive (Ti) and the
convective (Tc) periods of vibrations on the ratio H/R
3

1


(9)
R  cos   cos3  
3
g 

where Qcw(y) is the coefficient of lateral convective
pressure on the wall defined by the Eqn. 10. The value of
Se(Tc) should be in units of g.
y 

cosh  3.674

2R 

(10)
Qcw  y   0.5625
H 

cosh  3.674

2R 

Vertical convective pressure on the tank base is
defined by the Eqn. 11,
with uniform thickness constant along the wall height,
should be added to the impulsive hydrodynamic
pressure. Since the provisions of Eurocode 8 regarding
the calculation of aforementioned pressure components
are informative, in the following text, simpler and more
practice-oriented expressions will be introduced. The
introduced expressions are essentially quite similar to
those proposed by Eurocode 8 and they are based on the
provisions of IITK-GSDMA Guidelines [5].
Lateral impulsive pressure exerted by the liquid on
the tank wall is defined by the Eqn. 6,
S T  
(6)
piw  Qiw  y  e i
H cos 
q g
where γ is the specific weight of liquid, φ is a
circumferential angle as shown in Fig. 4, y is a vertical
distance of a point on the tank wall measured from the
wall bottom, Qiw(y) is the coefficient of lateral impulsive
pressure on the wall defined by the Eqn. 7 and q is the
behaviour factor for the impulsive mode. The value of
Se(Ti) should be in units of g.
  y 2 
2R 

(7)
Qiw  y   0.866 1     tanh  0.866

H
H 

   
pcw  Qcw  y  Se Tc  2
pcb  Qcb  x  Se Tc  2

R
(11)
g
where Qcb(x) is the coefficient of vertical convective
pressure on the tank base defined by the Eqn. 12.
3
9 x 4 x  
H 

Qcb  x   
 
  sec h  3.674
 (12)
8  2 R 3  2 R  
2R 

As already said, pressure due to wall inertia should
be added to the impulsive hydrodynamic pressure. It
should be calculated by the Eqn. 13,
S T  
pww  e i s m
(13)
q
g
where ρm is the mass density and γm is the specific weight
of tank wall material. The value of Se(Ti) should be in
units of g.
It is obvious from the Eqns. 6-12 that in the case of
circular tanks, hydrodynamic pressure due to horizontal
excitation varies around the circumference of a tank as
well as along the height of the tank wall, which is
presented on Figs. 5 and 6.
It should be noted that Eurocode 8 defines that the
behaviour factor q should be equal to 1.0 in the case of
the damage limitation state. In the case of ultimate limit
state verifications, the use of q factors greater than 1.5 is
only allowed provided that the sources of energy
dissipation are explicitly identified and quantified and
that inelastic response is acceptable. Nevertheless, the
convective part of the liquid response shall always be
evaluated on the basis of elastic response (q=1.0). This
means that no energy dissipation can be associated with
the convective response of the liquid.
l’
R
x
φ
R
φ
Fig. 5. Idealized distribution of hydrodynamic pressure
around the circumference of a tank
Fig. 4. Geometric characteristics of a circular tank
Vertical impulsive pressure on the tank base on a
strip of length l' is defined by the Eqn. 8,
x

sinh 1.732 
Se Ti  
H

pib  0.866
H
(8)
q g

l' 
cosh  0.866 
H

where x is a horizontal distance of a point on the base of
a tank in the direction of seismic force, from the center
of the tank, as shown in Fig. 4.
Lateral convective pressure on the tank wall exerted
by the oscillating liquid is defined by the Eqn. 9,
impulsive
component
convective
component
Fig. 6. Idealized distribution of hydrodynamic pressure
along the height of the tank wall
4
In the absence of a more exact analysis, an equivalent
linear pressure distribution, which gives the same base
shear force and bending moments at the bottom of the
tank wall, can be assumed in a practical analysis.
During vertical base excitation, the effective weight
of liquid increases, which induces additional pressure on
the tank wall. Distribution of this pressure is similar to
hydrostatic pressure in the case of tanks with rigid walls.
The effective fluid pressure will be increased or
decreased due to effects of vertical acceleration. The
natural period of vibration of vertical motion of a tank
and the stored liquid can be determined by the Eqn. 14,
which is proposed by ACI 350.3 Standard [1].
Tv  2
 RH 2
(14)
gsE
Hydrodynamic pressure on the tank wall due to
vertical excitation can be calculated using the Eqn. 15,
S T   
y
(15)
pv  ve v
H 1  
qv g  H 
where Sve(Tv) is a vertical spectral acceleration value
(obtained from a 5% damped vertical elastic response
spectrum for the case of reinforced concrete tanks), in
units of g, and qv is the behaviour factor applied for the
vertical direction (it should be taken up to 1.5 in all
cases, unless justified through an appropriate analysis).
In the case of both horizontal and vertical excitation,
the maximum value of hydrodynamic pressure on the
tank wall should be obtained by combining pressures due
to both excitations by using the SRSS combination rule.
Most commonly, various mechanical equipment is
connected to a tank structure. Such equipment should be
adequately considered in a seismic design process, but
that issue is beyond the scope of this paper. Nevertheless,
it should be noted that sufficient flexibility of equipment
should be provided in order to prevent any damage in the
tank structure during a seismic action.
When a tank does not have a roof, a sufficient wall
height above the liquid level must be provided in order to
prevent spilling of liquid, which is particularly important
in the case of tanks containing toxic liquids. The sloshing
wave height is provided mainly by the first convective
mode. As already mentioned, the provisions of Eurocode
8 define maximum sloshing wave height (dmax), which
should be calculated by the Eqn. 16.
S T 
d max  0.84 R e c
(16)
g
5. CONCLUSIONS
A simple dynamic model for analysis of the seismic
response of ground-supported concrete circular liquid
storage tanks was presented. The model can be applied in
analysis of the total seismic response as well as for the
determination of local response quantities. All of the
obligatory provisions of Eurocode 8 were fully taken into
account, while some of the informative ones were
substituted with similar provisions defined by IITK-
5
GSDMA and ACI provisions. The effect of both the
impulsive and the convective component was considered
in the model and the influence of both horizontal and
vertical seismic action was taken into account. A
comparative analysis of the total seismic response of
tanks with various geometrical properties was conducted
through a parametric study. Since the seismic response of
ground-supported concrete circular tank is primarily
influenced by its geometrical properties, the ratio of the
liquid height and the inner radius of a (H/R) was the
main parameter in the analysis. The study showed that
the impulsive period of vibrations, base shear force and
overturning moments increase almost linearly with the
increase of the ratio H/R, whereas the convective period
of vibrations is practically a constant value for the
H/R>1.5. It was therefore concluded that the seismic
response of a tank structure mainly depends on the
response in the impulsive mode. In addition, it should be
noted that the increase of base shear force and
overturning moments controlled by the increase of the
ratio H/R directly influences the detailing demands.
6. ACKNOWLEDGMENTS
This research was supported by the Serbian Ministry
of Science and Technology, Grant No. 36043.
7. REFERENCES
[1] ACI 350.3, “Seismic Design of Liquid-Containing
Concrete Structures”, American Concrete Institute,
Farmington Hill, MI, USA, 2001.
[2] CEN Eurocode 8, “Design of structures for earthquake resistance. Part 1: General rules, seismic
actions and rules for buildings”, European Standard
EN 1998-1:2004, European Committee for Standardization, Brussels, 2004.
[3] CEN Eurocode 8, “Design of structures for earthquake resistance. Part 4: Silos, tanks and pipelines”,
European Standard EN 1998-4:2006, European
Committee for Standardization, Brussels, 2006.
[4] G. W. Housner, “The dynamic behaviour of water
tanks”, Bulletin of the Seismological Society of
America, 1963, Vol. 53, No. 2, pp. 381-387.
[5] IITK-GSDMA, “Guidelines for Seismic Design of
Liquid Storage Tanks – Provisions with Commentary
and Explanatory Examples”, Indian Institute of
Technology Kanpur, 2007.
[6] P. K. Malhotra, “Seismic response of Soil-Supported
Unanchored Liquid-Storage Tanks”, ASCE, Journal
of Structural Engineering, 1997, Vol. 123, No. 4, pp.
440-450.
[7] M. J. N. Priestley, J. H. Wood and B. J. Davidson,
“Seismic design of storage tanks”, Bulletin of the
New Zealand National Society for Earthquake
Engineering, 1986, Vol. 19, No. 4, pp. 272-284.