experimental and theoretical study of the natural
TRANSCRIPT
International Journal of Engineering & Technology IJET-IJENS Vol:15 No:05 24
I J E N S IJENS © October 2015 IJENS -IJET-3939-055154
Experimental and Theoretical Study of The Natural
Frequency for Unanchored Fuel Storage Tank Marwa Ghazi and Asst. Prof. Dr. Majid Habeeb
Abstract--- In this work, experimental and theoretical study are
focused on calculation natural frequencies for empty tank and
tanks with different filling ratio. A fuel storage tank which its
diameter (0.2m)and The (H/R) ratio was three values and these
values were (0.75, 1.0 and 1.25.The thickness of plate of were (1.5
, 3 and 4) mm. Theoretically, The finite element model of the
unanchored fuel storage tank used SHELL99 and SOLID187
Elements in order to described fluid and tank respectively. In this
mode, the tank was divided into cylinder, base and roof and the
cylinder was divided into several section depending on the
thickness of tank and/or the level of liquid. SHELL99 element
was used to mesh the liquid that was represented as a solid with
certain modulus of elasticity and certain density. Experimentally,
the vibration test used the roving hammer to calculate the
fundamental natural frequency of the tank. Comparison between
the experimental and theoretical results shows good agreement
and the maximum absolute error percentage was (15.6%) when
the thickness of plate was (4 mm) and the (H/R) ratio was (1.25).
Index Term--- Cylindrical tanks, natural frequency, finite
element method, vibration test.
I- INTRODACTION
The tank is one of the most important equipment of
petroleum processing and storage, and its seismic performance
must be good for safety production industry and processing of
petroleum industry [1-4]. When the tank is destroyed under
the action of earthquake, serious economic losses will be
caused, in additional to other secondary disasters such as fire,
environmental pollution and so on. Also, liquid storage tanks
are one of the essential structures in industrial facilities. The
tank structure is being utilized in a variety of configurations
such as ground supported, elevated, or partly buried. Due to
their vulnerability to earthquakes and their wide use,
expectedly a substantial number of incidents of damage to
tanks has occurred during past seismic events [5-8].
Several investigations are studied the evaluation of dynamic
behavior of cylindrical tanks [9-12]. Marchaj [13] studied the
tank behavior under vertical acceleration. He translated the
dynamically applied forces into equivalent static forces and
equated the work done by such forces to the stored elastic
energy of the shell. He considered a horizontal strip of the
tank wall in this study. Kumar [14] considered only the radial
motion of partially-filled tanks and assumed that the effect
of axial deformations.
Marwa Ghazi
Email: [email protected]
Asst. Prof. Dr. Majeed Habeeb
Email: [email protected].
Haroun and Tayel [15] evaluated the dynamic characteristics
of cylindrical tanks numerically. They presented the natural
frequencies , the mode shapes of both empty and partially
filled tanks, hydrodynamic pressure and stress distributions.
They, also, presented an analytical method for the
computation of the axisymmetrical dynamic characteristics of
partially-filled cylindrical tanks [16]. They assumed that the
liquid was in viscid and incompressible and the tank shell was
uniform thickness and its material to be linearly elastic. Under
these assumptions, the dynamic characteristics from analytical
solution were compared with those obtained from numerical
solution. Several analytical and numerical studies of the liquid
shell system were made [5, 11, 17, 18 ] and several studies
deal with vibration tests of reduced and full scale tanks were
done [4, 19]. These studies indicated that wall flexibility may
have significant effect on the dynamic behavior of fluid
containers and the hydrodynamic pressure exerted on the walls
of such deformable tanks can be estimated by superposition of
the impulsive and the convective pressure components to the
short period component contributed by the flexible wall. In
this work, the natural frequencies of the empty tanks and tanks
with different filling ratio were studied. Theoretically, the
natural frequencies were calculated by finite element method
(using ANSYS Software) and were measured experimentally.
The comparison between theoretical and experimental results
were made.
II- FINITE ELEMENT MODEL
According to the theoretical work described by Marwa and
Majid (under publishing) [20], the optimum dimensions of the
tank were:
A. The diameter of the tank was 0.2 m.
B. The (H/R) ratio was three values and these values
were (0.75, 1.0 and 1.25).
C. The thickness of plate of were (1.5 , 3 and 4) mm.
The finite element model of the unanchored fuel storage
tank was similar to that used in [20]. This model used
SHELL99 and SOLID187 Elements in order to described fluid
and tank respectively. In this mode, the tank was divided into
cylinder , base and roof and the cylinder was divided into
several section depending on the thickness of tank and/or the
level of liquid. SHELL99 element was used to mesh the liquid
that was represented as a solid with certain modulus of
elasticity and certain density [20].
III- EXPERIMENTAL WORK
The physical and mechanical properties of steel plate used
for manufacturing the tank are density and modulus of the
elasticity. The density of the plate measured using a classical
way (the mass and the volume were measured) and the value
of density was ( 7850 kg/m3 ). The modulus of elasticity of the
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I J E N S IJENS © October 2015 IJENS -IJET-3939-055154
plate was measured using the tensile test (according to ASTM
(A370-2012)) and was (206 GPas.) (see Fig.(1)) . In other
hand, the density of liquid used in vibration taste (Gasoline)
was
( 680 kg/m3 ) and the modulus of elasticity was (1.2 GPas.).
First of all, the reinforcement concrete foundation, with
dimensions (1m*1m*15 cm) was built. The circular pit, with
(0.2 m) diameter and (0.01 m) height, was done in the
reinforcement concrete foundation. This pit is used as a
support to prevent the tank to move in horizontal plane [see
Fig.(2-a)].
a- Tensile test machine used in this work.
b- Dimensions of tensile test sample.
Fig. 1. Tensile test according to ASTM number
(A370-2012).
The vibration test involves studying the fundamental natural
frequency for empty tanks and tanks with different filling ratio of
tanks and the dimensions of these tanks and the filling ratio using in
this work can be illustrated in table I.
TABLE I
THE DIMENSIONS OF TANKS AND HEIGHT OF LIQUID USED IN THIS WORK.
Tank Liquid
N
o H/R
Diameter
(m)
Height
(m)
Thickness
(mm) Filling Ratio
1
0.75 0.2 0.075
2 Empty,0.166
7, 0.3333,
0.5, 0.6667
and 0.8333
2 3
3 4
4
1.0 0.2 0.1
2 Empty,0.166
7, 0.3333,
0.5, 0.6667
and 0.8333 5 3
6 4
7
1.25 0.2 0.125
2 Empty,0.166
7, 0.3333,
0.5, 0.6667
and 0.8333
8 3
9 4
The rig used in vibration test consists of the following parts [see
Fig. (2b)]:
A. The Reinforcement Concrete Foundation.
B. Impact Hammer Instrument (model:086C01-PCB Piezotronic
vibration division)
C. The Accelerometer (model :353C68).
D. The Amplifier (model 480E09).
E. Digital Storage Oscilloscope ( model ADS 1202CL+ and serial
No.01020200300012 :
The reinforcement foundation the pit in the foundation
a- The reinforcement concrete foundation .
b- The rig components .
Fig. 2. The rig used in this work.
To calculate the fundamental natural frequency of the tank
,the impact hammer is used to generate the impulse force at a
certain point on the tank and then the response from
Frame
Support of
Sample Moving
Part Computer
Part
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accelerometer sensor will be measured using the amplifier and
digital storage oscilloscope . The certain point, that the impact
hummer hits the tank, was limited depending to the height of
liquid in the tank. There are three ways can be measured the
signal[1]. The method that used in this work used single
accelerometer that can be fixed to one position and then
impact our structure at several locations. This is known as a
'roving hammer' test. This uses the leas resources, but takes
longer to make several measurements. This is the most
common form of hammer impact testing.
IV- RESELTS AND DISCUSSIONS
The experimental results of this work were divided into
two groups. The first group are the experimental results of the
empty tanks with thickness of plates (1.5, 3 and 4) mm and the
second group are the experimental results of the tanks with
different filling ratio and with thickness of plates (1.5, 3 and
4) mm. For the empty tanks, table (2) shows the comparison
between the experimental results and two theoretical results.
From table (2), the maximum absolute error percentage are
(4.67, 6.72 and 7.22) % when the (H/R) ratio are (0.75, 1.0
and 1.25) respectively for the thickness of the plate is (1.5
mm). For the thickness of the plate is (3 mm), the maximum
absolute error percentage are (5.63, 3.95 and 6.91) % when
the (H/R) ratio are (0.75, 1.0 and 1.25) respectively. Finally,
for the thickness of the plate is (4 mm), the maximum absolute
error percentage are (8.164, 8.595 and 15.60) % when the
(H/R) ratio are (0.75, 1.0 and 1.25) respectively. From these
results, the maximum absolute error percentage increases
when the (H/R) ratio increases and the maximum absolute
error percentage increases when the thickness of plate
increases. For the second group of experimental results, the
comparison between the first four natural frequencies
measured experimentally and that calculated by ANSYS
Software due to the change in the filling ratio can be shown in
Fig.(3) - Fig.(5) when the (H/R) are (0.75), (1.0) and (1.25)
respectively and when the thickness of plate is (1.5 mm). For
the thickness of plate is (3 mm), the comparison between the
experimental and the theoretical natural frequencies can be
shown in Fig.(6) - Fig.(8). While Fig.(9) - Fig.(11) show the
same comparison when the (H/R) are (0.75), (1.0) and (1.25)
respectively and when the thickness of plate is (4 mm). From
these figures, the following points can be noted:
A- For the thickness of plate is (1.5 mm), the maximum
absolute error percentage, when the (H/R) ratio is (0.75), are
(1.16%) for the first mode, (8%) for the second mode, (20 %)
for the third mode and (54%) for the fourth mode. Also, the
absolute minimum error percentage for the same (H/R) ratio
are (0.7 %) for the first mode, (0.6 %) for the second mode,
(1.3%) for the third mode and (0.48 %) for the fourth mode.
For the same thickness of plate and when the (H/R) ratio is
(1.0), the absolute error percentage range are (0.77-1.15)% for
the first mode, (0.14-18.7) % for the second mode, (8.2-
TABLE II
THE COMPARISON BETWEEN THE EXPERIMENTAL AND THEORETICAL RESULTS OF THE EMPTY TANKS FOR
DIFFERENT (H/R) RATIO.
(1) The Thickness of Plate is (1.5) mm.
No. of
Mode
H/R=0.75 H/R=1.0 H/R=1.25
Exp. Freq.
(Hz)
Theo.
Freq. (Hz)
Error
%
Exp.
Freq.
(Hz)
Theo.
Freq. (Hz)
Error
%
Exp.
Freq.
(Hz)
Theo.
Freq.
(Hz)
Error
%
1 118.12 - - 52.3 - - 133.5 - -
2 347.32 347.49 0.048 347.61 347.5 -0.0316 347.73 347.51 -0.06
3 700.2 724.47 3.350 702.91 724.34 2.9585 698.8 724.24 3.512
4 1140.5 1182 3.510 1139.7 1182.7 3.6340 1097.7 1183.2 7.226
5 1549.6 1480.4 -4.674 1556.1 1458.1 -6.7210 1434.5 1443.1 0.595
(2) The Thickness of Plate is (3) mm.
No. of
Mode
H/R=0.75 H/R=1.0 H/R=1.25
Exp. Freq.
(Hz)
Theo.
Freq. (Hz)
Error
%
Exp.
Freq.
(Hz)
Theo.
Freq. (Hz)
Error
%
Exp.
Freq.
(Hz)
Theo.
Freq.
(Hz)
Error
%
1 77.45 - - 49.11 - - 73.3 - -
2 674.3 672.59 -0.254 674.62 671.83 -0.4152 647.82 671.76 3.563
3 826.17 821.62 -0.553 808.1 798.24 -1.2352 793.2 780.83 -1.58
4 1480.1 1470.6 -0.645 1387.4 1444.5 3.9529 1528.3 1429.4 -6.91
5 2713.2 2875 5.627 2860.2 2832.3 -0.985 2610.7 2719.3 3.993
(3) The Thickness of Plate is (4) mm.
No. of
Mode
H/R=0.75 H/R=1.0 H/R=1.25
Exp. Freq.
(Hz)
Theo.
Freq. (Hz)
Error
%
Exp.
Freq.
(Hz)
Theo.
Freq. (Hz)
Error
%
Exp.
Freq.
(Hz)
Theo.
Freq.
(Hz)
Error
%
1 317.1 1.3 - 61.27 1.12 - 40.33 - -
2 880.7 881.18 0.054 880.33 879.93 -0.045 887.89 879.4 -0.96
3 1685 1834.8 8.164 1367 1831.9 25.378 1944.2 1829 -6.29
4 2443.3 2930.5 16.62 2936 2937.6 0.0544 2477.5 2935.7 15.60
5 3816.5 3768.4 -1.27 4028.8 3709.9 -8.595 3203.5 3335.7 3.96
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-27.2)% for the third mode and (1.6-33)% for the fourth
mode. For the same thickness of plate and when the (H/R) ratio
is (1.25), the absolute error percentage range are (0.19-0.34)%
for the first mode, (2.1-10.35) % for the second mode, (0.515-
25.15)% for the third mode and (2.1-33.8)% for the fourth
mode.
B-For the thickness of plate is (3 mm) and when the (H/R)
ratio is (0.75), the absolute error percentage range is (0.06-
2.8)% for the first mode. The absolute error percentage
increases when the mode increases and the maximum absolute
error percentage reaches 80 % in the fourth mode . For the
same thickness of plate and when the (H/R) ratio are (1.0) and
(1.25), there is a very good agreement between experimental
and theoretical results for the first mode and the absolute error
percentage range increases with the number of mode. The
maximum absolute error percentage are (0.3)% and (0.4)%
for the (H/R) ratio are (1.0) and (1.25) respectively.
C-When the thickness of plate is (4 mm), there is a very good
agreement between experimental and theoretical results for the
first mode for the (H/R) ratio are (0.75, 1.0 and 1.25). The
maximum absolute error percentage are (0.3, 0.35 and 19) %
respectively. Also, the maximum absolute error percentage
increases when the number of mode increases.
a- The First Mode. b- The Second Mode.
c- The Third Mode. d- The Four Mode.
Fig. 3. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio changes and when the (h/r) ratio is 0.75 and the thickness of plate is (1.5 mm).
a- The first mode. b- The second mode.
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c- The third mode. d- The four mode.
Fig. 4. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio
changes and when the (h/r) ratio is 1.0 and the thickness of plate is (1.5 mm).
a- The first mode. b- The second mode.
c- The third mode. d- The four mode.
Fig. 5. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio
changes and when the (h/r) ratio is 1.25 and the thickness of plate is (1.5 mm).
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a- The first mode. b- The second mode.
c- The third mode. d- The four mode.
Fig. 6. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio
changes and when the (h/r) ratio is 0.75 and the thickness of plate is (3 mm).
a- The first mode. b- The second mode.
c- The third mode. d- The four mode.
Fig. 7. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio changes and when the (h/r) ratio is 1.0 and the thickness of plate is (3 mm).
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a- The first mode. b- The second mode.
c- The third mode. d- The four mode.
Fig. 8. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio
changes and when the (h/r) ratio is 1.25 and the thickness of plate is (3 mm).
a- The first mode. b- The second mode.
c- The third mode. d- The four mode.
Fig. 9. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio
changes and when the (h/r) ratio is 0.75 and the thickness of plate is (4 mm).
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a- The first mode. b- The second mode.
c- The third mode. d- the four mode.
Fig. 10. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio
changes and when the (h/r) ratio is 1.0 and the thickness of plate is (4 mm).
a- The first mode. b- The second mode.
c- The third mode. d- The four mode.
Fig. 11. The comparison between the experimental and theoretical natural frequencies of the first four modes for the filling ratio changes and when the (h/r) ratio is 1.25 and the thickness of plate is (4 mm).
V- CONCLUSION
The comparison between the experimental and ANSYS
results of this work. From the results of empty tank, the
maximum absolute error percentage increases when the (H/R)
ratio increases and the maximum absolute error percentage
increases when the thickness of plate increases. Generally,
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there is a very good agreement between the experimental and
theoretical results and the maximum absolute error
percentage was (15.6%) when the thickness of plate was (4
mm) and the (H/R) ratio was (1.25). From the results of the
tank with different filling ratio, there is a very good agreement
between experimental and theoretical results for the first mode
for different (H/R) ratio and different thickness of plate. But,
the maximum absolute error percentage increases when the
number of mode, thickness of plate and(H/R)ratio increase
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