Seismic Loads Influence Treatment on the Liquid Hydrocarbon Storage
Tanks Made of Nanocomposite Materials
OLENA SIERIKOVA *
Applied Mechanics and Environmental Protection Technologies Department
National University of Civil Defence of Ukraine
vul. Chernishevska, 94, Kharkiv, Ukraine, 61023, UKRAINE
ELENA STRELNIKOVA, KIRILL DEGTYAREV
Department of Hydroaeromechanics of Power Machines
A.M. Pidhorny Institute for Mechanical Engineering Problems NAS of Ukraine
vul. Pozharskoho, 2/10, Kharkiv, Ukraine 61046
UKRAINE
Abstract: - The liquid hydrocarbon storage tanks are the objects of environmental danger. It is necessary to
perform additional calculations and develop appropriate design solutions to minimize the risks of their accidents
in the event of the earthquake or explosions. The degree of damage to the environmentally hazardous object
during the earthquake depends not only on the seismic effects level, but also on the quality of seismic design and
construction. The possibility of exposure to smaller but more frequent and prolonged seismic loads caused by
technogenic and natural factors has not been sufficiently taken into account in tanks designing for the
environmentally hazardous liquids storage. The composite materials using with nanoinclusions in tanks for
storage liquid hydrocarbons, allows to increase the reliability of tanks under seismic loads and extend their service
life under the influence of natural and technogenic influences of various origin. The results of the calculations
have been shown that the use of composite materials with nanoinclusions in the steel spheres form is the best
option for environmentally friendly operation of tanks under seismic loads.
Key-Words: - liquid hydrocarbon, storage tank, earthquake, nanocomposites, environmental safety, seismic loads
Received: May 7, 2021. Revised: April 11, 2022. Accepted: May 12, 2022. Published: June 30, 2022.
1 Introduction
To design the liquid hydrocarbon storage tanks that
are objects of environmental danger, according to
current building standards, it is necessary to take into
account the 1% probability of exceeding the
estimated intensity of seismic impacts for 50 years.
This factor also significantly increases the risks of
trouble-free operation and, accordingly, the cost and
complexity of construction of these engineering
structures, as it is necessary to perform additional
calculations and develop appropriate design solutions
to minimize the risks of their accidents in the event
of the earthquake or explosions. The degree of
damage to the environmentally hazardous object
during the earthquake depends not only on the
seismic effects level, but also on the quality of
seismic design and construction. According to recent
seismological studies, it has been established that in
Ukraine, including its platform part, there is the
danger of local and strong subcortical
earthquakes with the magnitude more than 5 points
[1], [2], [3].
According to the existing standards [2], [3],
the foundation ring is calculated for the main
loads combination, and for construction sites
with seismicity of 7 points and above is for the
special loads combination. Thus, the possibility of
exposure to smaller but more frequent and
prolonged seismic loads caused by technogenic and
natural factors has not been sufficiently taken
into account in tanks designing for the
environmentally hazardous liquids storage.
2 Problem Formulation
Containers and tanks for environmentally hazardous
liquids storage are widely used in various
engineering practice fields, such as aircraft
construction, chemical and oil gas industry, energy
engineering, transport. These tanks are operating
under conditions of high technological loads and
filling with oil, flammable or toxic substances. As a
result of the sudden action of seismic loads, the liquid
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stored in the tanks begins to sense the intense
splashes.
Sloshing is a phenomenon in the number of
industrial facilities: in containers for storage of
liquefied gas, oil, fuel tanks, in the reservoirs of cargo
tanks. It is known that partially filled tanks are
exposed to particularly intense splashes. This could
lead to high pressure on the tank walls, destruction of
the structure or loss of stability and can cause the
release of environmentally hazardous contents into
the environment and lead to serious consequences
[4], [5], [6], [7], [8], [9].
The release of environmentally hazardous liquids,
especially liquid hydrocarbons from storage tanks to
the environment and their further spread to the
territory of settlements could cause mass poisoning
of people and animals, lead to environment pollution.
Liquid spills could lead to explosions and fires that
could spread to nearby reservoirs and surrounding
areas. Economic losses from accidents with tanks
destruction, leakage and fire of liquid hydrocarbons
include not only direct losses, but also the cost of
measures to restore the environment [10], [11].
To ensure the environmental safety of areas
adjacent to tanks filled with liquid hydrocarbons, it is
necessary to take into account the safe design of the
tank, tank material, forecasting the effects of natural
and technogenic factors on tanks. The set of natural
factors that must be taken into account are seismic
loads, groundwater level of the reservoir location and
others. Technogenic factors should include sudden
traffic accidents, industrial accidents, vibration,
seismic and artificial impacts, and so on.
2.1 Literature analysis
In the most research papers of Shevtsov A. A.
[11], Wilson S. [12] Islamovic F. [13], Godoy
L.A. [14], Jaca R.C. [15] the significance
estimation of tanks influences for liquid
hydrocarbons storage on environment and
monitoring of reservoirs tightness changes, the
destruction rate of their structure under the
technogenic and natural factors action have been
investigated.
The issues concerned with liquid sloshing in
tanks have been conducted in the works of
Ibrahim R.A. [16], [17]. It should be noted
the paper on sloshing liquid in cylindrical
tanks under the seismic loads action [18], [19],
[20].
The necessity of control and impact
assessment of nanomaterials on the environment
for safety and efficient use of nanotechnologies
has been substantiated in paper [7].
In the previous works of the authors [4],[5],
[6] the seismic loads on the reservoirs of oil
storages have been treated, the use of
nanocomposite materials has been proposed
to ensure the antistatic effect of
nanocomposite materials [21], [22]. In other
works [23], [24], [25] the mechanical
characteristics of materials with different
inclusions have been investigated. But the use
of nanocomposites as the reservoir material
to increase their strength characteristics has
not been studied.
3 Problem Solution
The problem of free and forced oscillations
of the elastic rotation shell, partially filled with
an ideal incompressible fluid has been
considered (Fig. 1.)
Fig. 1. Rotation shell, partially filled with liquid
Let S be the wetted surface of the shell, 0 is
the free surface of the liquid.
Suppose that the fluid is ideal,
incompressible, and its flow (induced by body
motion) is vortex-free. Denoting the velocity
components by
, the incompressibility
condition of the continuous medium will be
obtained from the following equality:
(1)
Since the flow is vortex-free, there is the
velocity potential that satisfies the harmonic
equation due to (1).
The equations system of motion is
symbolically written in the form
,
where L, М – operators of elastic and mass
forces;
U=(u1, u2, u3) – vector function of
displacements;
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P – the pressure of the liquid on the wetted
shell surface.
The small harmonic oscillations of the elastic
shell has been studied. There is represented the
vector U in the form U=ueit, where is the
frequency and u is the natural oscillations form
of the shell under consideration with the liquid.
To solve the free and forced oscillations
problem of shell structures with compartments
containing liquid, the method of given shapes
has been developed. Its essence is as follows.
The connected system of differential equations
to elastic displacements of the structure and fluid
pressure has been formed. Three sets of basic
functions have been used to build the
representation of system solutions. The first of
them are the oscillations natural forms of the
structure in the absence of aggregate used to
build hydroelastic movements.
The second and third basic functions sets are
used to represent the potential velocities and
fluid pressure on the wetted structure surface.
The velocity potential is described by the sum of
two "partial" potentials. One of them describes
the natural liquid oscillations in the rigid tank,
taking into account the forces of gravity. The
second set refers to the natural oscillations of the
elastic shell with the fluid without taking into
consideration gravitational forces. As basic
functions for solving problems concerning free
and forced oscillations of rotation shells,
partially filled with liquid, the natural oscillation
modes of the unfilled shell have been accepted.
The following presentation of the natural
oscillation modes of the shell with the liquid has
been used:
, (2.1)
where – displacement vector;
( , , ) ( , , ), ( , , ), ( , , )
k k k k
x y z u x y z v x y z w x y zu
– vector-function, which are natural oscillation
modes of the unfilled shell,
()
k
ct
– unknown coefficients that depend only
on time.
To determine
( , , )
kx y zu
it has been
assumed that
()
k
ct
=
it
e
,
0P
,
0
i
Q
and
it has been obtained the problem for determining
the natural oscillations frequencies and forms of
the unfilled shell.
The velocities potential will be determined.
To do this, it will be found the "partial" velocity
potentials that correspond to the natural
oscillations forms of the unfilled shell.
According to (1.1) there have
1
, , , , ,
m
kk
k
w x y z t w x y z c t
. (2.2)
Here the functions are normal components of
the natural oscillations forms of the unfilled
shell.
For the function it has been obtained the
following boundary value problem:
0
;
0
;
;
wMS
t
nM
t
0
0;
s
a t x g M
t
.
where
1
, , , , ,
m
kk
k
w x y z t w x y z c t
.
It has been proposed to present the velocity
potential as the sum of two potentials
12
.
To determine 1 it has been formulated the
following boundary value problem:
2
10
,
1,
wMS
nt
,
10, MS
t
. (2.3)
Note that from relation (1.2) and the second
of equations (1.3) it could be obtained the series
11
1
( , , , ) ( , , ) ( )
m
kk
k
x y z t x y z c t
. (2.4)
To determine the functions 1k it has been gained
m of the following boundary value problems:
2
10
k
,
1,
kk
w M S
n
,
10
0,
kM
. (2.5)
It could be note that problems (1.5)
correspond to zero acceleration of free fall. Also
important that (1.5) are mixed problems for the
Laplace equation, the solution condition for such
problems is not checked.
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To determine the potential 2, it has been
started with the auxiliary problem of fluid
oscillations in the rigid shell, taking into account
the forces of gravity. Such problem has been
formulated in the form:
20
,
1
0, MS
n
,
0
,MS
n
,
0
0,g M S
(2.6)
The last equation in (1.6) is the dynamic
condition (equality of atmospheric pressure) on
the free surface. Differentiation of this equation
by the variable t taking into account the third
relation in (1.6) gives:
0
0,gM
n
. (2.7)
The problem (1.7) - (1.8) as the eigenvalues
problem and representation of its solution in the
next form will be considered as
( , , , ) ( , , )
it
x y z t e x y z
.
For the function it is the following problem
about harmonic oscillations of the liquid in the
rigid tank:
20
,
0, MS
n
,
2
0
,M
ng
. (2.8)
Solving this problem gives the number of k
eigenvalues and their corresponding
eigenfunctions, which has been denoted by 2k.
Next, after solving this auxiliary problem, there
will be looking for the potential 2 in the form:
22
1
( , , , ) ( ) ( , , )
n
kk
k
x y z t d t x y z
. (2.9)
So, there are
12
,
where
11
1
( , , , ) ( ) ( , , )
m
kk
k
x y z t c t x y z
22
1
( , , , ) ( ) ( , , )
n
kk
k
x y z t d t x y z
.
First of all, it has been noted that the total
potential constructed in this way satisfies the
Laplace equation, i.e.
12
0
.
Next, on the wetted surface of the shell, the
condition of non-leakage is met, namely
12 ,
wMS
n n n t
.
The boundary conditions must be met on the
free surface:
0
,M
n
0
0,
s
g a t x M
.
From equations (1.8) and (1.9) it will be
obtained that the motion of the liquid free surface
determined by the ratio
21
11
, , , ,
nm
kk
kk
kk
x y z x y z
d t c t
nn
The advantage of the proposed approach has
been noted in determination of the free surface
shape that does not need to differentiate seismic
acceleration
s
at
over time, which is quite a
challenge.
Consider a dynamic boundary condition on a
free surface. From the last relation in (1.4) we
have. Therefore, the condition leads to
differential equations
Dynamic boundary condition on the free
surface has been considered. From the last
relation in (1.4) it has been gained
10
.
Therefore, the condition
0
s
g a t x
leads to differential equations
12
2
1 1 1
( , , ) ( , , )
( ) ( , , ) ( ) ( )
0.
n m n
jk
k k j k
k j k
s
x y z x y z
d t x y z g c t g d t
nn
a t x
(2.10)
There are no forms that correspond to the first
potential due to equality
10
0,
kM
.
In addition, there are expression for pressure
12
11
( ) ( , , ) ( ) ( , , )
nn
k k k k s
kk
l
pc t x y z d t x y z gz a t x
. (2.11)
Using the ratio 2k for functions
2
2
20
,
kk
kM
ng
,
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there are equation (1.10) takes the form
21
2
11
( , , )
( ) ( ) ( , , ) ( ) 0
nm
k
k k k k k s
kk
x y z
d t d t x y z g c t a t x
n
.
After performing the scalar product of this
relation on the functions 2l there the following
equations have been obtained (due to the
orthogonality of the oscillations natural forms of
the fluid in the rigid tank, at least for the shells
of rotation, 9):
21
2
1
22
( ) ( ) ( ) , , 0, 1,2..,
,
mk
l l l k l s l
k
ll
g
d t d t c t a t x l n
n
.
Defining the functions 1k and 2k, there the
expression for the pressure calculated by formula
(1.11) have been found. The scalar product of the
obtained equations on the proper form uj has
been also performed. The scalar product here is
in the sense:
,
k j k j k j k j
S
u u v v w w dS
uu
.
The differential operators from (1.15)
corresponding to the stiffness and mass matrices
as follows have been denoted:
11 12 13 2
21 22 33 2
31 32 33
1 0 0
; 0 1 0
0 0 1
L L L
L L L h t
L L L
LM
.
The condition of eigenforms orthogonality of
the unfilled shell on the matrix of masses have
been applied 48
2
k k k
Lu Mu
,
( , )
k j kj
Mu u
. (2.12)
where
k
– the natural frequency of the
corresponding unfilled shell k – that own form.
After performing the scalar product of the
first three equations in (1.15) on uj, taking into
account the relations (1.33) it has been obtained
2
1
1
2
1
,
, , , , , 1,
m
j j j l k k j
k
n
l i i j j s j j
i
c t c t c w
d w g z w a t x w Q j m
u
Finally, the system of ordinary second-order
differential equations with respect to unknown
coefficients has been gained
, 1,...,
k
c t k m
,
, 1,...,
k
d t k n
:
2
1
1
2
1
,
, , , , , 1,
m
j j j l k k j
k
n
l i i j j s j j
i
c t c t c w
d w g z w a t x w Q j m
u
(2.13)
21
2
1
22
( ) ( ) ( ) , , 0, 1,2..,
,
mk
l l l k l s l
k
ll
g
d t d t c t a t x l n
n
Since it has been assumed that at the initial
time (for example, before the earthquake or
explosion) the system "shell-liquid" was at rest,
zero initial conditions have been accepted
0 0 0, 1,...,
kk
c c k m
;
0 0 0, 1,...,
kk
d d k n
. (2.14)
Thus, the scheme of solving the related
dynamic problem for the rotation shell, partially
filled with fluid, contains several stages, each of
that has its own value. These stages are as
follows:
1. Frequencies and forms determination of
free oscillations of the unfilled shell by the finite
element method.
2. Frequencies and forms determination of
fluid oscillations in the rigid shell under the
action of gravity using the limiting elements
method.
3. Frequencies and forms determination of
oscillations of the elastic shell without taking
into account the action of gravity using the
limiting elements.method.
4. Solving the system of second-order
differential equations using the 4th and 5th order
Runge-Kutta method.
Provided calculations have been allowed to
build the necessary systems of basic functions
for the forced oscillations study, as well as the
study of the surface tension influence and
nonlinear effects on oscillations of shells with
fluid. First, the empty shell has been considered.
The Fig. 2 shows the oscillations forms of such
shell under the specified conditions of
attachment.
1 2
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3 4
5 6
Fig.2. Oscillations modes of the unfilled shell
The eigenfrequencies of the unfilled shell
have been given for different materials in Table
1.
Table 1. Eigenfrequencies of the unfilled shell,
Hz
Mat
eria
l/
Fre
que
ncy
nu
mbe
r
Aluminu
m
Comp
osite,
steel
bullets
Comp
osite,
steel
fibers
Comp
osite,
carbon
fiber
Comp
osite,
steel
sphere
s
1
90,573
85,429
85,789
85,071
84,674
2
90,575
85,43
85,791
85,073
84,676
3
100,36
94,567
94,974
93,599
93,644
4
100,37
94,578
94,985
93,61
93,654
5
103,52
97,751
98,153
97,874
96,986
6
103,52
97,753
98,154
97,876
96,988
Note that the oscillation shapes are the same
for shells made of different materials, while the
frequencies differ by about 5-7%. This allows
tuning from unwanted resonant frequencies,
including by selecting the appropriate material.
These calculations refer to the construction of
the basic functions first system according to
[18], [19].
Next, it has been formed the second system
construction of basic functions, for which
the liquid sloshing in the rigid tank has
been considered. The acoustic
approximation has been applied. The Fig. 3
shows the finite element grid for acoustic
calculation.
Fig. 3. Finite element grid for acoustic
calculation
The 31928 finite elements have been selected,
further increase in their number did not lead to
the significant change in the results. The method
of boundary elements has been also used
to compare the calculations [18], [19]. The
100 boundary elements along the cylindrical
wall, 100 elements along the bottom radius
and 120 elements along the free surface radius
have been selected. Fig. 4 shows the modes of
the liquid surface sloshing.
1 2
2 4
5 6
Fig. 4. The free surface sloshing modes
Table 2 shows the values of the frequencies of
free surface oscillations.
Table 2. Sloshing frequency of the liquid free
surface, Hz
Freque
ncy
1
2
2
4
5
6
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numbe
r
FЕM
0.65
965
0.65
965
0.86
931
0.86
931
0.97
542
1.02
16
BЕM
0.65
967
0.65
967
0.86
938
0.86
938
0.97
553
1.02
43
From the given data there has been drawn the
conclusion of the obtained results probability.
Note that the splash frequencies are the lowest,
they do not depend on the choice of material, at
least at the selected ratios between the geometric
characteristics of the shell.
Thus, the second system of basic functions
has been built.
The third system of basic functions definition
will be obtained. To do this, it has been
performed the calculation in the hydroelastic
formulation. Note that it has been proposed to
search for the velocities potential in the form of
the sum of two potentials = 1 + 2. In this
case, the potential 2 corresponds to the
definition of free surface splashes, i.e. it has been
found as the basic functions linear combination
of the second system. The potential 1
corresponds to the oscillations of the elastic shell
with the liquid, but without taking into account
the movements of the free surface and is depicted
as the linear combination of basic functions of
the third system. FEM and BEM could also be
used to define these basic functions.
Here the calculation has been performed with
the FЕM help. Fig. 5 shows the corresponding
oscillations forms.
1 2
2 4
5 6
Fig. 5. Shell oscillations modes taking into
account the walls elasticity
Table 3 shows the natural frequencies of shell
oscillation, taking into account the walls
elasticity.
Table 3. Eigenfrequencies of hydroelastic
oscillations of the shell, Hz
Mater
ial/
Frequ
ency
numb
er
Alumi
num
Comp
osite,
steel
bullets
Comp
osite,
steel
fibers
Comp
osite,
carbon
fiber
Comp
osite,
steel
sphere
s
1
48,02
9
50,565
50,625
43,828
43,692
2
48,05
5
50,593
50,653
43,85
43,716
3
51,60
7
54,672
54,722
47,343
47,014
4
51,61
7
54,682
54,732
47,349
47,023
5
54,90
2
57,646
57,723
49,792
49,851
6
54,94
8
57,694
57,722
49,829
49,891
From the above results the oscillations forms
of shells of different materials are the same, and
the difference in oscillation frequencies reaches
5-7 percent, which may be significant when
conducting resonance tuning.
4 Conclusion
The composite materials using with
nanoinclusions in tanks for storage liquid
hydrocarbons, allows to increase the reliability
of tanks under seismic loads and extend their
service life under the influence of natural and
technogenic influences of various origin. The
results of the calculations have been shown that
the use of composite materials with
nanoinclusions in the steel spheres form is the
best option for environmentally friendly
operation of tanks under seismic loads.
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Volume 17, 2022
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.9
Olena Sierikova,
Elena Strelnikova, Kirill Degtyarev
E-ISSN: 2224-3429
69
Volume 17, 2022
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Olena Sierikova: conceptualisation, data curation,
formal analysis, methodology.
Elena Strelnikova carried out the simulation and the
optimization.
Kirill Degtyarev: visualization, data curation.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.9
Olena Sierikova,
Elena Strelnikova, Kirill Degtyarev
E-ISSN: 2224-3429
70
Volume 17, 2022
Sources of Funding for Research Presented in
a Scientific Article or Scientific Article Itself
No funding was received for conducting this
study.
Conflicts of Interest
The authors have no conflicts of interest to
declare .
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