experimental and theoretical study of the natural

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

mailto:[email protected]

mailto:[email protected]



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



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



-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.



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).




changes and when the (h/r) ratio is 1.25 and the thickness of plate is (1.5 mm).






changes and when the (h/r) ratio is 0.75 and the thickness of plate is (3 mm).



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).




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,



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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experimental and theoretical study of the natural

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