Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
Elastic-plastic Seismic Response Analysis of Tank Considering
Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
College of Civil and Architecture Engineering, Northeast Petroleum University, China
Heilongjiang Key Laboratory of Disaster Prevention, Mitigation and Protection Engineering,
Daqing, China, [email protected]
Abstract
It is less for seismic performance study of storage tank considering fluid-solid couple, based on
basic theory of the storage tank under seismic action, this paper has built 2000 m3 finite element
models of empty tank and tank with medium using ANSYS software. According to the specification,
time-history analysis of the tank was carried out under general earthquake by exerting 3 kinds of
seismic waves, and variation of displacement, acceleration response and equivalent stress for empty
and full tank is gotten. On this basis, time-history analysis is further developed for tank with medium
under rare earthquake, and destruction form and axial compressive stress time-history curves of tank
are acquired. The results show that the elephant foot buckling and diamond buckling occur near the
bottom of the tank wall, these can provide reference for seismic design of tanks in earthquake region.
Keywords: Seismic Performance, Fluid-Solid Couple, Elephant Foot Buckling, Diamond Buckling
1. Introduction
The tank is one of the most important equipments of petroleum processing and storage, and its good
seismic performance is a guarantee for safety production industry and processing of petroleum industry
[1-4]. If the tank is destroyed under the action of earthquake, serious economic losses will be caused,
and other secondary disasters will be produced such as fire, environmental pollution and so on. So the
research of tank shell fluid-solid coupling seismic performance has important significance. The
dynamic characteristics analysis of tank is researched by Jiangang Sun, considering the interaction
between tank and foundation, the finite element method is used to calculate the natural vibration
characteristics of tank structure which is floated on the foundation. ADINA finite element software is
used to research the dynamic response of vertical tank under the horizontal earthquake by Lijian Zhou.
Liquid height, tank geometry parameters and foundation stiffness are investigated on the effect of
seismic response, and anchorage tank and floating tank are compared on seismic response. ANSYS
software is used to research the seismic response of vertical cylindrical tank by Yan Zhang, and the
tank seismic effect under the condition of different liquid height is analyzed. The results show that
bottom deformation is larger with full or empty tank, and at the bottom of tank foot deformation
appears under rare earthquake. The static-dynamic numerical analysis of 15×104m3 floating tank is
researched by Xiaolei Zhao, and ADINA software is used for modal analysis. The results show that the
result of finite element is close to that of code. The vibration form of tank liquid solid coupling under
vibration low-frequency is rich with the beam-shape vibration of cosnθ and sinnθ, the vibration forms
of the liquid sloshing in low frequency is single.
Although the tank seismic method study is more [5], the application of the finite element analysis
software ANSYS on the dynamic analysis of tank shell fluid coupling dynamic analysis is less. The
software of ANSYS is used to carry out simulation analysis for 2000m3 tanks under two cases of empty
or full liquid tank in the article. Seismic behavior and failure mechanism of tank are got, and these can
provide the reference for the practical seismic method and design proposal of tank.
2. The basic theory
Kinetic motion equation of tank can be expressed by equation (1) under the action of earthquake [6]:
[ M ]{ut }  [C ]{u t }  [ K ]{u t }  {Pt }
Where [M] is quality matrix of tank, [K] is stiffness matrix of tank, [C] is damping matrix of tank,
[Pt] is earthquake load vector.
International Journal of Digital Content Technology and its Applications(JDCTA)
Volume6,Number12,July 2012
doi:10.4156/jdcta.vol6.issue12.8
64
Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
The damping of Rayleigh is used by tank and the linear combination of mass matrix and stiffness
matrix is shown in equation (2).
[C]=α [M] +β [K]
In equation (2), α and β represent the damping matrix coefficients respectively. Because the mass
matrix [M] and stiffness matrix [K] are consistent with the orthogonality condition of vibration mode of
structure, the damping matrix [C] is also consistent. Any of the two modes {φi} and {φj} is taken, and
the upper equation is transformed:
{ i }T [C ]{ i }   { i }T [ M ]{ i }   { i }T [ K ]{ i }
{ j }T [C ]{ j }   { j }T [ M ]{ j }   { j }T [ K ]{ j }
The both equations are divided by {φi} T [M] {φi} and {φj} T [M] {φj}, the relationship will be got:
{i }T [ K ]{i }   i2 {i }T [ M ]{i }
{ i }T [C ]{ i }  2 i  i { i }T [ M ]{ i }
2
2 i  i     i2 , 2 j j     j
The equations can be solved as follows:
2( j  j   i  i )
2 i  j ( i  j   j  i )
, 

2
2
 2j   i2
 j  i
3. The Selection of Earthquake Waves
It is well known that the earthquake is uncertain, and it is difficult to predict the grade of earthquake
which will be encounter for the tank equipment in using period. The time-history analysis method is
used in the paper, the first group acceleration time-history curve of artificial simulation and the both
group the preceding earthquake curves which are recorded are selected as the seismic wave load when
time-history analysis is carried on [7]. The tank is loaded respectively, and tank’s changes of stress are
researched. The maximum value of acceleration time-history curve is listed in Table 1.
Table1. The earthquake acceleration maximum value of time- history analysis
Earthquake effect
6 degree
7 degree
8 degree
9 degree
General earthquake
18
35(55)
70(110)
140
Rare earthquake
—
220(310)
400(510)
620
When the analysis is carried on, the seismic records are selected by following steps: (1) Login
seismic database, and download acceleration response spectrum and records which is same with site
category. (2) After downloading acceleration response spectrum, according to the earthquake influence
coefficient which is set in seismic design code, another curve will be got by putting the coefficient into
the spectrum. Comparing the two curves, the curve is selected as strong earthquake record which is
similar to the trend of revised curve platform. (3) The amplitude modulation processing is carried on
for selected seismic recording curve, and its maximum peak is adjusted to make its maximum peak
consistent with seismic acceleration time-history curve peak which is set by code.
The project is located in the eastern part of Daqing. The seismic fortification intensity is 8 degrees,
engineering ground soil category is class II, the selection of seismic acceleration is 0.30g in design, and
classification of earthquake design is the first group. According to the above parameters and seismic
design code, three seismic waves of El-centro wave, Taft wave and an artificial wave are loaded on the
tank respectively when time-history analysis is carried on.
4. Empty tank analysis under the action of earthquake
The finite element model of tank is taken node numbering, and numbering increases from the
bottom to the top which is shown in Figure1. The seismic wave is loaded along direction of X axis.
65
Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
Z
6
5
4
3
2
1
X
Figure1. Node numbering of finite element model of tank
Displacement, acceleration, the equivalent stress and the moment of peak value of empty tank under
the action of 3 kinds of seismic waves are shown in Table 2. In the absence of medium, when the tank
wall is under the action of seismic wave, its peak stress cloud chart is shown in Figure2. Displacement,
acceleration, and equivalent stress peak value distribution along the tank wall are shown from Figure3
to Figure5. Figure5 shows that tank wall stress distribution of empty tank under the action of 3 kinds of
seismic waves are basically identical. Tank wall exhibits shearing type deformation, the stress of the
bottom wall of tank along the direction of inputting seismic is the maximum, and gradually diffuses to
the above tank wall. The stress of tank top is zero.
Table2. Empty tank response peak value and time under three kinds of earthquake waves
Earthquake wave
Name
El-centro
Taft
Artificial
wave
Displacement
Acceleration
Equivalent stress
Peak
time(s)
3.74
2.14
Peak
(mm)
0.034
0.016
Time
(s)
5.62
6.58
Peak
(m·s-2)
0.047
0.011
Time
(s)
5.32
7.94
Peak
(MPa)
0.9975
0.8869
Time
(s)
5.62
6.58
10.88
0.034
10.88
-0.054
9.8
0.9972
10.88
(a)El-centro wave
(b) Taft wave
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Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
(c) Artificial wave
15
15
12
12
Tank height /m
Tank height/m
Figure2. The peak stress cloud chart of empty tank wall under the action of seismic waves
9
6
3
9
6
Taft wave
Artificial wave
0
0
0.01
0.02
0.03
El centro wave
3
El centro wave
Taft wave
artificial wave
0
0.04
0
0.01
Radial displacement/mm
Figure3. The maximum radial displacement
0.03
0.04
0.05
0.06
-2
Figure 4. The maximum radial acceleration
15
El centro wave
Taft wave
Artificial wave
12
Tank height /m
0.02
Radial acceleration /m.s
9
6
3
0
0
0.2
0.4
0.6
0.8
1
1.2
Effective stress /Mpa
Figure5. The maximum radial equivalent stress of empty tank wall
We can see the distribution of displacement, acceleration and equivalent stress of empty tank along
the tank wall height under 3 kinds of seismic wave from Figure3 to Figure5. The maximum
displacement in the X direction of tank wall increases gradually along the tank wall height, and
shearing type deformation is exhibited. The tank wall displacements under the action of El-Centro
67
Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
wave and artificial wave are very close. The displacement of tank under the action of Taft is the half of
the displacement of tank under the action of El-Centro wave and artificial wave, and the displacement
of the upper and inferior tank wall are same essentially. Artificial wave generates the largest radial
acceleration to the tank, followed by El-Centro wave, and Taft wave is the minimum, its value is only
25% of acceleration of artificial wave. The equivalent stress distributions of empty tank wall under the
action of 3 kinds of seismic waves are basically identical, and gradually decrease along the tank wall
height. The stress of top tank is near to zero. The equivalent stress under the effect of Taft wave is
smaller, which the maximum is 13% larger than the minimum.
5. Full tank analysis under the action of earthquake
Displacement, acceleration, the equivalent stress and the moment of peak value of full tank under
the action of 3 kinds of seismic waves are shown in Table 3. When the tank is filled with medium, its
peak stress cloud chart under the action of seismic wave is shown in Figure6. Displacement,
acceleration, and equivalent stress peak value distribution along the tank wall are shown from Figure 7
to Figure 9.
Table3. Full tank response peak value and the moment under three kinds of earthquake wave
earthquake wave
name
El-centro
Taft
Artificial
wave
Displacement
Acceleration
Equivalent stress
Peak time
(s)
3.74
2.14
Peak
(mm)
6.14
5.48
Time
(s)
17.76
7.12
Peak
(m·s-2)
4.825
-3.870
Time
(s)
18.02
7.12
Peak
(MPa)
120
113
Time
(s)
17.76
7.12
10.88
6.50
8.28
-5.440
9.84
126
9.84
(a) El-centro wave
(b) Taft wave
(c) Artificial wave
Figure6. The peak stress cloud chart of full tank wall under the action of seismic waves
68
Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
Figure6 shows that tank wall stress distribution of full tank under the action of 3 kinds of seismic
waves are basically identical. Tank wall exhibits two order beam deformation. For the hydraulic effect
of storage liquid, panel maximum stress occurred on the little up to the bottom of the tank along the
direction of inputting seismic wave, and it is also the place where occurs elephant-foot and diamond
failure. The stress of tank wall diffuses from the maximum place to surrounding, the stress value is
zero to the tank top [8].
We can see the distribution of displacement, acceleration and equivalent stress of full tank along the
tank wall height under 3 kinds of seismic wave from Figure7 to Figure9. The maximum displacement
in the X direction of tank wall increases gradually along the tank wall height firstly, and the maximum
value is reached in the height of 5m. Then the value reduced gradually, and the value is further reduced
in the height of storage liquid. The tank wall displacements under the action of three kinds of seismic
waves are very close. Artificial wave generates the largest displacement to the tank in the X direction,
followed by El-centro wave, and Taft wave is the minimum. The maximum acceleration in the X
direction of tank wall increases gradually along the tank wall height firstly, and the maximum value is
reached in the height of 9m. Then the value reduced gradually, and the acceleration values are basically
identical above the height of the liquid. The largest influence upon radial acceleration of the tank wall
is artificial wave. The least influence is Taft wave. The influence of El-centro wave is between the
above two wave. The equivalent stress distributions of empty tank wall under the action of 3 kinds of
seismic waves are basically identical, and gradually decrease along the tank wall height. The stress of
top tank is near to zero.
15
15
El centro wave
El centro wave
Taft wave
Artificial wave
Taft wave
Tank height /m
Tank height /m
12
Artificial wave
12
9
6
9
6
3
3
0
0
0
1
2
3
4
5
6
7
0
1
2
3
4
Radial acceleration /m.s
Radial displacement /mm
Figure7. The maximum radial displacement
5
6
-2
Figure 8. The maximum radial acceleration
15
El centro wave
Taft wave
Artificial wave
Tank height /m
12
9
6
3
0
0
20
40
60
80
100
Effective stress /Mpa
120
140
Figure9. The maximum radial equivalent stress of full tank wall
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Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
6. Failure analysis of full tank
When a strong rare earthquake happens, tank axial will get very big impact load and the tank wall is
led directly to cracking, bending deformation. Figure 10 shows the elephant-foot buckling failure and
rhombic buckling failure of 2000m3 tank in time-history analysis under rare earthquake.
We can see from Figure10, for the seismic loads, tank bottom bears very large axial compressive
stress, and there is very obvious stress concentration phenomenon. The elephant foot failure will be
produced at the position of maximum force of the tank. At the same time large circumferential
compression stress load will be born by tank wall in the lower part of the tank wall. So diamond
buckling deformation will be produced in the weak parts of the tank wall. Because of the interaction of
the two stresses together, buckling deformation may be produced in the weak parts of the other parts of
tank wall.
(a) Elephant-foot buckling under the action of El-centro wave
(b) Elephant-foot buckling under the action of Taft wave
(c) Elephant-foot buckling under the action of artificial wave
Figure10. The buckling failure of tank under the action of rare earthquake
70
Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
The greatest axial allowable stress which is born by the tank wall can be calculated by the following
formula [9, 10]:
[ cr ]  0.15 E
t1
d1
Where [σcr] is the greatest allowable stress of the tank wall (MPa).E is elastic modulus of tank wall
(MPa). t1 is calculation thickness of bottom tank wall (m). d1 is average diameter of bottom tank wall
(m).
According to the formula, the greatest axial allowable stress [σcr] of 2000m3 tank is obtained for
14.14MPa. ANSYS finite element software is used to calculate the maximum stress value and the
position under the three kinds of waves, they are respectively: the effect of El-centro wave is 54.2MPa
and it occurs in 2.16s.The effect of Taft wave is 46MPa and it occurs in 0.62s. The effect of artificial
wave is 50.52MPa and it occurs in 4.94s. It is visible that all the forces of 2000m3 full tank under rare
earthquake exceed the allowable stress. When the tank is subjected to seismic load, the axial stress
time-history curve in location of the tank buckling failure is shown in Figure11. So, it is very important
to carry out Seismic performance analysis of the tank under rare earthquake.
10
Axial stress
60
Axial stress
Axial stress/Mpa
40
Axial stress/Mpa
0
20
0
-20
-10
-20
-30
-40
-40
-60
-50
0.0
0.5
1.0
1.5
Time/s
2.0
0.0
2.5
0.2
0.4
0.6
0.8
Time/s
(a) The axial stress curve under El-centro wave
(b) The axial stress curve under Taft wave
60
Axial stress
Axial stress/Mpa
40
20
0
-20
-40
-60
0
1
2
3
Time/s
4
5
6
(c) The axial stress curve under artificial wave
Figure11. The axial stress time-history curve of the tank wall buckling
Through the above analysis results, when a rare earthquake occurs, the result of tank wall failure is
that the axial compressive stress applying on the tank exceeds wall material allowable stress values,
and these lead to the failure of tank deformation.
7. Summary
The tank is one of the most important equipments of petroleum processing and storage, and its good
seismic performance is a guarantee for safety production industry and processing of petroleum industry.
If the tank is destroyed under the action of earthquake, serious economic losses will be caused, and
other secondary disasters will be produced such as fire, environmental pollution and so on. So the
research of tank shell fluid-solid coupling seismic performance has important significance. The force
basic theory of tank under the action of earthquake is elaborated. According to the code requirement
71
Elastic-plastic Seismic Response Analysis of Tank Considering Fluid-solid Coupling
Jing Ji, Dan Yang , Wenfu Zhang, Liang Chang
and seismic intensity and soil parameters in the location of tank, three kinds of seismic waves are
selected. Referencing seismic acceleration time-history curve peak which is set by code, seismic wave
is carried on amplitude modulation corresponding. The finite element model of 2000m3 tank is carried
on time-history analysis under the action of earthquake, and the dynamic response and stress variation
of tank wall are got. Through the time-history analysis for the fluid-filled tank under rare earthquake,
the destruction form and the axial compressive stress time-history curve are got, and elephant-foot
buckling is occurred proximity to the bottom of tank. The tank subjects to very large circumferential
compression stress load under rare earthquake, tank occurs diamond buckling deformation in the lower
part of the tank.
8. Acknowledgements
The study described in this paper was supported by the Heilongjiang Provincial Department of
Education Science and technology research project (project number: 12511022) and The National
Natural Science Foundation of China (project number: 51178087).These supports are gratefully
acknowledged.
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[7] Ray.W.Clough, "Experimental seismic study of cylindrical tanks" Journal of the Structure
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