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Accepted Manuscript
The Similarity Law and Its Verification of Cylindrical Lattice Shell
Model under Internal Explosion
Shiqi Fu , Xuanneng Gao , Xin Chen
PII:
DOI:
Reference:
S0734-743X(18)30232-X
https://doi.org/10.1016/j.ijimpeng.2018.08.010
IE 3155
To appear in:
International Journal of Impact Engineering
Received date:
Revised date:
Accepted date:
14 March 2018
18 August 2018
19 August 2018
Please cite this article as: Shiqi Fu , Xuanneng Gao , Xin Chen , The Similarity Law and Its Verification
of Cylindrical Lattice Shell Model under Internal Explosion, International Journal of Impact Engineering
(2018), doi: https://doi.org/10.1016/j.ijimpeng.2018.08.010
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service
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Highlights
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? The similarity law of the cylindrical lattice shell structure under internal
explosion is deduced based on the dimensional analysis principle.
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? It is verified that the scale model test of the cylindrical lattice shell structure
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under internal explosion could accurately reflect the characteristics of the
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original model.
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? The reasonable and reliable scaling coefficients are proposed for the scale
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model tests of the cylindrical lattice shell structure under internal explosion.
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The Similarity Law and Its Verification of Cylindrical Lattice Shell
Model under Internal Explosion
3
Shiqi Fu a, Xuanneng Gao a,*, Xin Chen a,b
4
a. College of Civil Engineering, Huaqiao University, Xiamen, Fujian 361021, PR China
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b. College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002,
PR China
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Abstract: It is very important to ensure that the results of a scale model experiment of the large-span space steel
9
structure under internal explosion could accurately show the characteristics of its original model. For this purpose,
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a one-way inclined bar type of single layer cylindrical lattice shell was taken as an example in this paper, and the
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similarity law between the scale models and the original model under the internal explosion was deduced by the
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dimension analysis principle. Then, the original model and the scale models under internal explosions were
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simulated by ANSYS/LS-DYNA, and the correctness and the feasibility of the similarity law were compared and
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confirmed. The research results showed that: if the scale models conform to the similarity law, the propagation
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law and the overpressure distribution of shock waves to the scale models subjected to internal explosions were
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consistent with the original model; the phenomena of critical destruction of the scale models were similar to the
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experimental results. But the accuracy of the loading effects of blast (such as the overpressure peak of shock
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waves, the positive pressure time and the impulse etc.) and the structural dynamic responses obtained by the scale
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models was affected by the geometric incomplete similarity of the scale models and the strain rate effect of steel
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respectively, which would reduce with the decrease of the scaling coefficient. Especially, the research results also
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indicated that if the scaling coefficient was not less than 0.1, the maximum deviations of the overpressure peak of
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shock waves, the positive pressure time after conversion and the specific impulse after conversion would be 10%,
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4.78% and 29.8%, respectively, and if the scaling coefficient was less than 0.25, the stress responses and
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displacement responses of the scale model under internal explosion would not be reliable. Based on the research
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results, the reasonable scaling coefficients to the scale model experiment of cylindrical lattice shell under internal
26
explosion are proposed.
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Key words: Internal Explosion, Similarity Law, Cylindrical Lattice Shell, Dimensional Analysis, Scale Model
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*
Corresponding author. Xuanneng Gao
E-mail: gaoxn117@sina.com. Tel: (+86) 13599933263. ORCID ID:0000-0002-7905-7418.
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1
1 Introduction
It is well known that irreversible casualties, property damage and serious social panic would
3
occur if a terrorist attack happens in a large span space structure that is always built as a city
4
landmark. With the increase of terrorist attacks in recent years, the research of explosion prevention
5
and resistance (especially the experimental research) for protecting large-span space structure has
6
been paid increasingly attention [1-2]. Ishikawa and Beppu [3] summarized three explosion tests of
7
building envelope that were presided over by the late Professor A. Johoji, they considered that the
8
impact of blast should be taken into account in the design of building envelope. Remennikov and
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Uy [4] carried out the explosion test of hollow square steel tubes to study the dynamic responses of
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the ones in contact explosion and close explosion. Their study provided the experimental data for
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theoretical analysis and verification of the numerical simulation results. Dey and Nimje [5] adopted
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the method of combining experiment with numerical simulation to study the dynamic responses of
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metal sandwich plate subjected to explosive load. The results showed that the perfect coincidence
14
between the experimental and numerical simulation. If the detonation distance was 0.5m and the
15
mass of TNT was 8kg, the metal sandwich panel could consume the most explosive energy.
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Spranghers et al. [6] conducted the scale tests and verified the correctness of aluminum plate model
17
as well parameter selection under explosive loading. Li et al. [7] carried out sixteen original models
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to study explosion venting performance of non-reinforced brick masonry wall under internal
19
explosion. The results showed that the dynamic responses of masonry wall depended on the
20
overpressure peak of shock waves and the failure mode depended on typical yield curve. In
21
summary, researchers had focused on components or small-size concrete structures on the
22
experimental research of the explosion prevention and resistance, but very little research
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concentrated on the large-span space structures. Thus, there is a great significance to carry out the
2
model test of the large-span space structure under internal explosion.
Due to the actual constraints, the scale models are justifiably chosen by the model test of large
4
span space structure under internal explosion. The similar relationship between the scale models
5
and the original model should be conventionally defined before the the scale model test, namely the
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similarity law of the model under internal explosion should be determined, so that the test results
7
could really reflect the performances of the original model. Dimensional analysis is an appropriate
8
method for solving this problem. It is widely used in engineering tests, which is theoretically
9
applicable to all-type structures under any load [8-9]. For example, Coutinho et al. [10] reviewed
10
the similarity theory from rise to perfection and its application in major research areas, and pointed
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out the further development direction; Bakhtar et al. [11] established a series of tunnel scale models
12
based on the data of KLOTZ tunnel explosion test, and provided the basis for the explosion test of
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underground structure model with comparing original model and scale model test results; Li and
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Zhang [12] employed the dimensional analysis to sum up the similarity law of the model under the
15
impact load of bird. The results showed that the bird collision process followed the model similarity
16
law, and the strain rate effect would affect the similarity law of model; Yao and other scholars [13]
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built three groups of steel box models according to the geometric similarity law and implemented
18
corresponding internal explosion experiments. Then, the reason of the results deviation to
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traditional scale test was discussed and the size effect and the strain rate effect were considered on
20
the basis of test results account to amend the similarity law of steel box model under internal
21
explosion; Oshiro and Mazzariol et al. [14-15] addressed distortion of the scaling laws. For
22
instance, they suggested the density scaling factor to handle the generated incomplete similarity
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between the original model and the scale model in the structural model test subjected to dynamic
2
loads. And in order to solve material strain rate sensitivity, they set a scaling numbers that allowed
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perfect similarity between the original model and sensitive scale model. Yang and et al. [16-17]
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utilized experimental method to study the similarity law between the original model and the scale
5
model of the small-size confined concrete structure under internal explosion, and it was
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demonstrated that internal explosion damage effect of the original model could be predicted through
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the scale model with scaling coefficient ? ? 0.1 in accordance with load parameters and critical
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failure characteristics of concrete under internal explosion. Jiang et al. [18-19] used dimensional
9
analysis principle to deduce the similarity law of single layer spherical lattice shell model under
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impact loading, and took advantage of numerical simulation to validate related presumption while
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considering the factors of the strain rate effect, the geometric deviation and the gravity. Gao et al.
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[20] carried out internal explosion test of K6 single layer spherical lattice shell and proved scale
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model with scaling coefficient ? = 0.1 was similar to the original model, and the results of the
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internal explosion model test was able to characterize the original model.
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However, plenty of factors would seriously affect the accuracy of experimental results of the
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scale model under internal explosion. Firstly, the propagation law of shock waves under internal
17
explosion is pretty complicated because of numerous reflection phenomena and convergence effects
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[21-26]. Secondly, the strain rate effect and the strain hardening effect of steel under the high-speed
19
impact loading are remarkable [27]. Finally, the scale model could not be completely equivalent to
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the original model actually. In a word, the reliability of the results of internal explosion experiment
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would be reduced (sometimes even cause the failure of the test) unless the similarity law between
22
the original model and the scale model is verified.
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The one-way inclined bar type of single layer cylindrical lattice shell is a kind of typical large
2
span space structure [28]. In this paper, the similarity law of cylindrical lattice shell model under
3
internal explosion was deduced based on the dimensional analysis method. The dynamic analysis
4
software ANSYS / LS-DYNA was applied to simulate the original model and the scale models of
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cylindrical lattice shell, then correctness and feasibility of the similarity law were verified by
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comparing the calculation results. The purpose of all above is for the purpose of providing a
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reasonable and effective scaling coefficient to the internal explosion model test of the large span
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space structure.
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2 Theoretical derivations
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The loading effects and the structural dynamic responses are the main subjects to investigate
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into the scale model tests of the large-span space steel structure under internal explosion. In order to
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ensure that the research results of the scale model truly reflect the characteristics of its original
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model, it is required that the physical process of the characteristics (loading effects and responses of
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the internal explosion, etc.) of the scale models is the same as its original model. In other words,
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there is a confirmed scaling coefficient of each parameter involved in physical process between the
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original model and the scale model. The similarity criterion between the original model and the
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scale model can reflect their physical relationships and can further determine the scaling coefficient
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of each parameter. Therefore, the correct derivation of the similarity criterion is the key to extend
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the research results of the scale models to the original model.
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2.1 Loading effects of internal explosion
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Explosive impact load belongs to a kind of typical accidental load, and it has the characteristics
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of loads that are different from collision, fire and earthquake. At explosion, the shock waves rapidly
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change with time, instantly reach the pressure peak, gradually attenuate and eventually return to the
2
standard atmospheric pressure. In short, that procedure could be summarized as a high strength and
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a short pulse impact. Usually, the shock waves generated by both internal and external explosions
4
have the similar characteristics. According to this performance, the overpressure peak of shock
5
waves, the positive pressure time and the specific impulse were generally applied to quantitatively
6
describe the loading effects of blast waves in engineering design. The parameters that would ideally
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influence the overpressure peak of shock waves ?P (FL-2), the positive pressure time t+ (T) and the
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specific impulse i (FL-2T) were listed in Table 1, respectively.
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Table 1 The parameters involved in loading effects of internal explosion and their dimensions
Influence
Influence
Parameters
Dimensions
Parameters
b: Span of cylindrical lattice
-1 2
[FL T ]
[L]
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W: TNT mass
? e : Explosive density
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Explosive
Dimensions
factors
M
factors
shell
l: Length of cylindrical
-4 2
[FL T ]
[L]
lattice shell
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Ee : Releasing energy per
f: Vector height of
2 -2
[L T ]
Structure
unit mass
[L]
cylindrical lattice shell
h: Height of structure for
P0 : Initial pressure
[FL-2]
supporting cylindrical lattice
[L]
Air
shell
? 0 : Initial density
[FL-4T2]
R: Distance between
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[L]
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? 0 : Adiabatic coefficient
explosive center and
[1]
structural inner surface
According to the Table 1, the functional relationships of the overpressure peak of shock waves
2
?P, the positive pressure time t+ and the specific impulse i with each parameter could be expressed
3
as follows.
?P ? f (W , ?e , Ee , P0 , ?0 , b, l , f , h, R)
5
t ? ? g (W , ?e , Ee , P0 , ?0 , b, l , f , h, R)
6
i ? ? (W , ?e , Ee , P0 , ?0 , b, l , f , h, R)
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1
?1?
?2?
?3?
Now, taking the overpressure peak of shock waves ?P as an example, the similarity criteria
8
would be deduced based on the principle of dimensional harmony. From Eq.(1), the dimensional
9
relationship between the overpressure peak of shock waves ?P and each parameter could be
expressed as follows.
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[?P] ? [W ]x1 [ ?e ]x2 [ Ee ]x3 [ P0 ]x4 [ ?0 ]x5 [b]x6 [l ]x7 [ f ]x8 [h]x9 [ R]x10
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Substituting the dimensions in Table 1 into Eq.(4), the following equation was obtained.
[FL-2 ] ? [FL-1T 2 ]x1 [FL-4 T 2 ]x2 [L2T -2 ]x3 [FL-2 ]x4 [FL-4 T 2 ]x5 [L]x6 [L]x7 [L]x8 [L]x9 [L]x10
?5?
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?4?
where, x j ( j ? 1,2,3......10) was the dimensional index of each parameter.
According to the dimensional uniformity of the physical equation, the dimension index of the
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same parameters on both sides of the Eq.(5) should be equal. So, there were the following
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equations, corresponding to the dimensions F, L and T, respectively.
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x1 ? x 2 ? x 4 ? x5 ? 1
?
?
?? x1 ? 4 x 2 ? 2 x3 ? 2 x 4 ? 4 x5 ? x6 ? x7 ? x8 ? x9 ? x10 ? ?2
?
2 x1 ? 2 x 2 ? 2 x3 ? 2 x5 ? 0
?
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?6?
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follows.
1
0
1
1
?1
? ?1 ? 4 2 ? 2 ? 4
?
?? 2
2 ?2 0
2
3
4
5
6
7
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1?
? 2??
0 ??
After the finite elementary row transformations, it could be transformed into a row simplest
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The Eq.(6) was a system of linear equations, its coefficients could be written into a matrix, as
matrix, as below.
? 1
? 0
?
?? 0
0
1
0
0
0
1
0
1
1
0
1
0
1/ 3
1/ 3
1/ 3
1/ 3
1/ 3
?1/ 3 ?1/ 3 ?1/ 3 ?1/ 3 ?1/ 3
0
0
0
0
0
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0
1
1
?
?
?
??
Therefore, the Eq. (6) could be written as the Eq. (7), as follows.
? x1 ? ?1/ 3x6 ? 1/ 3x7 ? 1/ 3x8 ? 1/ 3x9 ? 1/ 3x10
?
? x2 ? ? x4 ? x5 ? 1/ 3x6 ? 1/ 3x7 ? 1/ 3x8 ? 1/ 3x9 ? 1/ 3x10 ? 1
?x ? ? x ? 1
4
? 3
?7?
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where, the x j ( j ? 4,5...10) was independent and arbitrary, could take any value but not all zero, and
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the x1 , x2 and x3 could be determined by x j ( j ? 4,5...10) . Substituting the Eq.(7) into Eq.(4), the
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following equation could be obtained.
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x4
x5
x6
x7
x8
x9
? ?P ? ? P0 ? ? ?0 ? ?
? ?
? ?
? ?
? ?
?
b
l
f
h
R
?
???
? ? ? ?
1/ 3 ? ?
1/ 3 ? ?
1/ 3 ? ?
1/ 3 ? ?
1/ 3 ?
? ?e Ee ? ? ?e Ee ? ? ?e ? ? (W / ?e ) ? ? (W / ?e ) ? ? (W / ?e ) ? ? (W / ?e ) ? ? (W / ?e ) ?
x10
?8?
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According to the ? theorem of explosion mechanics, the W, ? e and Ee would be taken as
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elementary variables. Taken the x j ( j ? 4,5...10) as equal 1, the mathematical relationships of the
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dimensionless term
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?P
with other seven dimensionless groups could be obtained, as follows.
? e Ee
P ?
?P
b
l
f
h
R
? f( 0 , 0,
,
,
,
,
)
1/ 3
1/ 3
1/ 3
1/ 3
?e Ee
?e Ee ?e (W / ?e ) (W / ?e ) (W / ?e ) (W / ?e ) (W / ?e )1/ 3
?9?
Similarly, the positive pressure time and the specific impulse could be expressed by the seven
dimensionless groups, as follows.
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t?
1
W
1/ 3
?e
-1 / 3
Ee
-1 / 2
? g(
P0 ?0
b
l
f
h
R
,
,
,
,
,
,
) ?10?
1/ 3
1/ 3
1/ 3
1/ 3
?e Ee ?e (W / ?e )
(W / ?e )
(W / ?e )
(W / ?e )
(W / ?e )1 / 3
2
i
P
?
b
l
f
h
R
? ?( 0 , 0 ,
,
,
,
,
) ?11?
2/3
1/ 2
?e Ee ?e (W / ?e )1 / 3 (W / ?e )1 / 3 (W / ?e )1 / 3 (W / ?e )1 / 3 (W / ?e )1 / 3
W 1 / 3 ?e Ee
3
The similarity criterion ? j ( j ? 1,2,3......10) could be obtained from Eqs.(9), (10) and (11), as
?P
?1 ?
? e Ee
5
?4 ?
6
7
?7 ?
l
(W / ? e )
1/ 3
?2 ?
t?
W
P0
? e Ee
?8 ?
1/ 3
?
?1 / 3
e
?0
?e
?5 ?
f
(W / ?e )
E
?1 / 2
e
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below.
?3 ?
?6 ?
i
W
1/ 3
?9 ?
1/ 3
? e2 / 3 Ee1/ 2
b
(W / ?e )1 / 3
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h
(W / ?e )
1/ 3
? 10 ?
R
(W / ? e )1/ 3
Thus it can be seen, that the similar conditions should be satisfied in the scale model design
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were ? jm ? ? jp (j = 1, 2......10). Here, the scale model and the original model were represented by
subscript m and subscript p, respectively.
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Assuming the scaling coefficient was a constant and no greater than 1, which could be
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represented by the ?, and the geometric relationship between the original model and the scale
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models would be easily obtained as follows.
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?b ? ?l ? ? f ? ?h ? ?
?12?
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If the explosive type was the same and the propagating medium did not change, the scaling
relation for the overpressure of shock waves and the explosive mass would be expressed as below.
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??P ? ?? ? 1
?13?
18
?W ? ?3
?14?
e
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According to the similar conditions above, the magnitude of overpressure in the scale model
2
should be equal to that of the original model and the amount of explosive should be cubic of scaling
3
coefficient.
5
Similarly, the scaling relation of the positive pressure time and the specific impulse should be
showed as follows.
6
?t ? ?
7
?i ? ?
2.2 Internal explosive responses
?15?
?16?
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?
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In fact, researchers tend to pay more attention to the motion and the deformation of lattice shell
10
structure subjected to the shock waves of internal explosion. However, the material parameters and
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the structural dynamic characteristics are different from ones of the static, because the structural
12
inertia and the propagation of the stress waves cannot be neglected under the short and strong loads.
13
Usually, the stress ? (t ) and displacement ? (t ) are the main parameters that represent the
14
structural dynamic responses. For the isotropic elastic-plastic materials, if the strain rate effect and
15
the strain hardening effect were neglected during material deformation process, the parameters
16
involved in the stress ? (t ) ([FL-2]) and the displacement ? (t ) ([L]) and their dimensions would
17
be shown in Table 2.
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Neglected the dimensionless parameters in Table 2, the functional relationships of the stress
and the displacement with each parameter were as follows.
20
? (t ) ? ? ( E, ? , t , ? s , P(t ), L(t ), E(t ))
?17?
21
?(t ) ? ? ( E, ? , t , ? s , P(t ), L(t ), E (t ))
?18?
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2
Taking the stress ? (t ) as an example, the dimension relation between it and each parameter
was as follows.
[? (t )] ? [ E] y1 [ ? ] y2 [t ] y3 [? s ] y4 [ P(t )] y5 [ L(t )] y6 [ E(t )] y7
3
(19)
Influence factors
parameters
dimensions
Time
t : Time
[T]
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Table2 The parameters involved in internal explosive responses and their dimensions
Influence factors
E : Elastic
[FL-2]
[FL T ]
density
[FL-2]
Time variable
Material
M
L(t ) : Structural
[1]
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? : Poisson ratio
[L]
dimensions
E (t ) : Tangential
-2
[FL-2]
[FL ]
modulus
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? s : Yield stress
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Substituting the dimensions in Table 2 into Eq.(19), the following equation was obtained.
[FL?2 ] ? [FL?2 ] y1 [FL?4T 2 ] y2 [T] y3 [FL?2 ] y4 [FL?2 ] y5 [L] y6 [FL?2 ] y7
(20)
where, y j ( j ? 1,2,3......7) was the dimensional index of each parameter.
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[1]
the unit area
-4 2
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? (t ) : Strains
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? : Material
5
dimensions
P(t ) : Pressure on
modulus
4
parameters
According to the dimensional uniformity of the physical equation, the dimension index of the
8
same parameters on both sides of the Eq. (20) should be equal. So, there were the following
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equations, corresponding to the dimensions F, L and T, respectively.
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y1 ? y2 ? y4 ? y5 ? y7 ? 1
?
?
?? 2 y1 ? 4 y2 ? 2 y4 ? 2 y5 ? y6 ? 2 y7 ? ?2
?
2 y 2 ? y3 ? 0
?
1
follows.
1
?1
?? 2 ? 4
?
?? 0
2
4
5
6
1
1
?2 ?2
0
0
0
1
0
1
1?
? 2 ? 2??
0
0 ??
After finite elementary row transformations, it would be transformed into a row simplest
matrix.
? 1
? 0
?
?? 0
7
0
1
0
0
0
1
1
0
0
1
0
0
1/ 2
?1/ 2
1
1
0
0
1
0
0
?
?
?
??
Therefore, the Eq. (21) could be written as the Eq. (22), as follows.
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8
0
0
1
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The Eq.(21) was a system of linear equations, its coefficients could be written into a matrix, as
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(21)
? y1 ? ? y4 ? y5 ? 1/ 2 y6 ? y7 ? 1
?
? y2 ? 1/ 2 y6
?y ? ?y
6
? 3
?22?
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where, the y j ( j ? 4,5...7) was independent and arbitrary, could take any value but not all zero, and
11
the y1 , y 2 and y3 could be determined by y j ( j ? 4,5...7) . Substituting the Eq.(22) into Eq.(19), the
12
following equation could be obtained.
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13
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10
y6
4
5
?? (t ) ? ?? s ? ? P(t ) ? ? L(t ) ? ? E (t ) ?
?
? E ? ? E ? ? E ? ? ( E / ? )1 / 2 t ? ? E ?
?
? ? ? ?
? ?
?
? ?
y
y
y7
?23?
According to the ? theorem of explosion mechanics, the E , ? and t would be taken as
15
elementary variables. Taken the y j ( j ? 4,5...7) as equal 1, the mathematical relationships of the
16
dimensionless term
17
? (t)
E
with other four dimensionless groups could be obtained, as follows.
? (t )
E
? ?(
? s P(t )
E
,
E
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,
L(t )
E (t )
,
)
1/ 2
(E / ? ) t E
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Similarly, the displacement could be expressed by four dimensionless groups, as follows.
2
? P(t )
? (t )
L(t )
E (t )
?? ( s ,
,
,
)
1/ 2
1/ 2
(E / ? ) t
E E (E / ? ) t E
3
The similar criterion ? j ( j ? 11,12,13......16) could be obtained from Eq. (24) and Eq. (25), as
below.
? 11 ?
5
? 14 ?
E
P(t )
E
? 12 ?
? (t )
( E / ? )1/ 2 t
? 15 ?
L(t )
( E / ? )1/ 2 t
? 13 ?
?s
? 16 ?
E
E (t )
E
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6
? (t)
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4
?25?
7
If the material types were consistent, namely ?E ? ?? ? 1 , and ? jm ? ? jp (j=11, 12......16)
8
could be satisfied, the scaling relation to the stress and the displacement would be expressed as
9
follows.
M
?? (t ) ? ?? ? ?P (t ) ? ?E (t ) ? ?E ? 1
10
s
?? (t ) ? ?L(t ) ? ?t
13
?L(0) ? ?? (t ) ? ?t ? ?
3 Verification of the similarity law
15
3.1 Numerical Model
?28?
AC
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14
16
?27?
At the t equal to zero, the scaling relation of the displacement would be as below.
PT
12
ED
11
?26?
A one-way inclined single-layer cylindrical lattice shell was selected as the original model. Its
17
geometric size was as follows: the span b =30 m, the rise to span ratio f / b = 1/5, the length to span
18
ratio l / b = 1.6 and the supporting structure height h = 8 m. The original model and the scale
19
models with ? = 0.5, 0.25, 0.1 and 0.05 were established by the ANSYS/LS-DYNA.
14
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The numerical model was shown in Figure 1 (b). The ground was considered rigid and the
2
bottom of supporting column was fixed to the ground. Besides, the major structure and building
3
envelope were connected by the connecting components. Among them, the major structure and the
4
connecting component were simulated by Beam161, and the building envelope and the ground were
5
stimulated by Shell163. The air of surrounding the entire structure model was set up and meshed,
6
which was simulated by Solid164 with 8 nodes and 6 faces, and the boundary was defined as the
7
transmission boundary to simulate the explosion in an infinite area. The TNT explosive was defined
8
by the volume fraction method in the cylindrical lattice shell structure, which was located at the
9
center above the ground 1.5m. In the whole process of explosion, the ALE (Arbitrarily
10
Lagrange-Euler) algorithm was adopted to consider the interaction between the air shock waves and
11
both the building envelope and the ground [29].
CE
PT
ED
M
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1
?b?Numerical model
12
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(a) Experimental model
Figure 1 Cylindrical lattice shell structure
The structural size, mesh size and TNT mass of the scale models were scaled down according
13
to Eqs.(12) and (14), and the element attributes and process of numerical calculation of the scale
14
models were the same as the original model. It was necessary to point out that material size of the
15
scale models theoretically should also satisfy the similarity law. However, when ? was small
15
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enough, the scaled-down material size would not be according with the true condition. Therefore,
2
according to the actual material size, when the scale models were established, the materials of
3
similar size should be selected as shown in Table 3. This performance that the geometric
4
characteristics of model or the form of construction could not be completely satisfied the same
5
proportion was defined as the geometric incomplete similarity of model [30-31].
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1
Table3 Material sizes of original model and scale model?unit: mm?
?
0.5
Latticed shell
?140�
?70�
(Seamless steel tube)
Supporting beam or column
General beam or column
ED
(square steel tube)
400�0�
Wall beam
PT
(rectangular steel tube)
200�0�
M
400�0�
200�0�
200�
CE
400�
0.25
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1
Material type
AC
3
0.05
?14�8
?7�4
(?14�
(?14�
?35�
100�0�
20��40��(20��
40��
20��
(40��2)
(20��
40�4
20�2
(40�
(20�
0.75
0.3
0.15
(1.2)
(1.2)
(1.2)
100�0�
100�
(angle iron)
Enclosure structure
0.1
1.5
(hot rolled steel plate)
Notes: The parameters in parentheses were the material type and the material size of the physical models.
6
In the original model and the scale models, the Q235 steel was selected and the modified
7
Johnson-Cook constitutive relation was used, which considered the strain rate effect and strain
16
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1
strengthening effect of steel subjected to high-speed impact loading [29]. Specifically, it was shown
2
in Eq.(29).
? ? ( A1 ? A2? n )(1 ? A3 ln ??* )(1 ? T *m )
?29?
t
3
where, ? , ? , ?? * and T* represented the equivalent flow stress, the equivalent plastic strain, the
5
relative strain rate and the relative temperature, respectively. A1, A2, A3, n and mt were undetermined
6
parameters that could be calibrated by experiment. Because the experiment and simulation were
7
carried out at room temperature (the T* is zero), the value of mt was arbitrary and A1, A2, n and A3
8
were shown in Table 4.
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4
Table 4 J-C constitutive model parameters of the Q235 [27,32]
10
320.7556�6
582.102�6
n
A3
0.3823
0.0255
M
A2
The explosive adopted the high energy explosive model (MAT_HIGH_EXPLOSIVE_BURN)
ED
9
A1
and the JWL (Jones-Wilkins-Lee) state control equation as shown in Eq.(30).
p ? A(1 ?
PT
11
?E0,exp
?
?
RV
RV
)e 1 exp ? B(1 ?
)e 2 exp ?
R1Vexp
R2Vexp
Vexp
?30?
where, p was the pressure of shock wave, A, B, R1, R2 and ? were input parameters, Vexp was the
13
relative volume of explosive. Besides, V0,exp and E0,exp represented the initial relative volume and
14
the initial internal energy of explosive, respectively. The parameters of the explosive were shown in
15
Table 5.
AC
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12
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Table 5 Material parameters of explosive
? exp /kg? m ?3
D /m? s ?1
PCJ /GPa
A/GPa
B/GPa
R1
R2
?
E0,exp /J? m ?3
V0,exp
1630
6713
18.5
540.9
9.4
4.5
1.1
0.35
8 ? 109
1.0
respectively.
The
air
was
simulated
by
the
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Notes: ? exp , D and PCJ were the density of the explosive, the detonation velocity and the detonation pressure,
MAT_NULL
EOS_LINEAR_POLYNOMIAL was adopted, as follows.
and
the
state
P0,air ? C0 ? C1? ? C2? 2 ? C3? 3 ? (C4 ? C5? ? C6? 2 ) Eair
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US
? ? 1 / Vair ? 1
equation
?31?
?32?
where, P0,air was the initial pressure of air, C0, C1, C2, C3, C4, C5 and C6 were the material
parameters of air, Eair was the internal energy per unit volume, Vair was the relative volume of air,
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? air was the density of air. Besides, V0,air and E0,air were the initial relative volume and the initial
ED
internal energy of air, respectively. The material parameters of air were shown in Table 6.
Table 6 Material parameters of air
C0
1.29
0
C1
C2
C3
C4
C5
C6
E0,air /J? m ?3
V0,air
0
0
0
0.4
0.4
0
2.5 ? 105
1.0
CE
PT
? air /kg? m ?3
AC
In order to verify the correctness and reliability of the numerical modeling method, the
numerical model of the air explosion was established by using the same method and the simulate
results were compared with ones of the air explosion tests. For the air explosion tests, the No. 2
rock emulsion explosive (its mass was 40g) were employed and the tests were carried out in an
open spaces. The overpressure of the shock waves were obtained by the dynamic pressure sensor
KD2009A that shown in Figure 2(a). The layout of the measuring points was shown in Figure 2(b).
18
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(a) Dynamic pressure sensor KD2009A
(b) Layout of the measuring points
Figure 2 The Layout of the measuring points and dynamic pressure sensor
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The comparison between experimental and simulated phenomenon at an explosion moment
was showed in Figure 3. The overpressures of shock waves obtained by the air explosion tests and
the simulation as well as their comparison results were showed in Table7. The compared results
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from the Table7 showed that the simulated results agreed well with ones of the air explosion tests in
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general. The errors between the simulation and the experiments were within 30%, the biggest error
was 27.32%. On the other hand, the existed literature also suggested that the simulated results
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agreed well with the empirical formula, were closest to the J. Henrych?s empirical formula [33-36].
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This indicated that the parameter selections of the numerical model were correct and the calculation
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results were reasonable and credible.
(a) The experimental phenomenon
(b) The simulated result
Figure 3 The comparisons between experimental and simulated results
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Table 7 The comparisons of overpressure peak values of shock waves between experiment and simulation
Measuring
Measuring
?Pexp /MPa
?Psim /MPa
?air / %
?Psim /MPa
?air / %
0.0391
0.0325
16.88
point No.
1
0.0703
0.0604
14.08
5
2
0.0388
0.0313
19.33
6
3
0.0831
0.0604
27.32
7
4
0.0240
0.0223
7.08
8
0.0232
0.0184
20.69
invalid
0.0325
/
invalid
0.0142
/
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Notes? ?Pexp were the overpressure peak values by air explosion tests.
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point No.
?Pexp /MPa
?Psim were the overpressure peak values by numerical simulation.
?air were the errors between ?Pexp and ?Psim , ? air ? (?Pexp ? ?Psim ) / ?Pexp .
3.2 Critical TNT mass and Critical failure phenomena
M
For the analysis of convenience, the critical TNT mass was here defined as the mass of the
initial venting of a cylindrical lattice shell structure subjected to internal shock waves. In the
ED
numerical simulation of the original model and scale models under internal explosion, gradually
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increasing the TNT mass until the cylindrical lattice shell structure occurred the first explosion
CE
venting. The corresponding TNT mass was the critical TNT mass. Then the explosion
characteristics of the original model and scale models could be compared and analyzed according to
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the phenomena of the initial venting.
The major structure and the building envelope were connected by the connecting components.
With the increase of the TNT mass, the tensile failure of the connecting components would
gradually occur. Therefore, the different stages of the structure subjected to internal shock waves
would be directly reflected by the failure ratio of the connecting components. The failure ratio of
the connecting components could be defined as the percentage of that the failing numbers of
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connecting components took up the total numbers of connecting components. Statistical results of
the failure ratio of the connecting components were shown in Table 8.
Table 8 The proportion of the connection component failure
? dam /%
1
? 6
15.35
0.5
? 6
13.39
0.25
? 6
12.99
0.1
? 6
12.99
0.05
? 6
Crucial TNT mass/kg
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? bef /%
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?
12.60
300
37.5
4.6875
0.3
0.0375
Notes: ? bef was the failure ratio of the connecting components before the initial venting.
M
? dam was the failure ratio of the connecting components at the initial venting.
? bef or ? dam were determined by the percentage of that the failing numbers of connecting components took
up the total numbers of connecting components.
ED
From the Table 8, it was clearly found that the failure ratios of connecting components of the
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original model (?=1) and the scale models (?=0.5, 0.25, 0.1 & 0.05) were all less than 6% before
explosion venting and rapidly increased up to more than 2 times once the explosion venting
CE
occurred. But in general, the multiplier value of the failure ratios decreased with the decrease of ?,
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and accordingly, the critical TNT mass of the scaled models decreased cubically with the decrease
of ?. The phenomena of the initial venting of the original model and scale models were showed in
Figure 4. As shown in Figure 4, the initial venting was occurred at the wall-shell junction of the
cylindrical lattice shell because of the gathering and enhanced effect of shock waves at the
structural corners. Remarkably, the positions appeared the initial venting of the original model and
the scale models were identical and then their venting phenomena were similar.
21
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?=1
?=0.25
?=0.05
PT
?=0.1
ED
M
?=0.5
CE
Figure 4 Initial venting phenomena of scale models under internal explosion
Following similarity law deduced by the Section 2, a physical model of the ?=0.1 cylindrical
AC
latticed shell was established. Its material type and material size were consistent with those in Table
3 parentheses. Remarkably, the roof panel steel and the major structure were welded with a small
steel bar. The wall panel and the major structure were connected with rivets. To ensure that the
experimental environment was consistent with the actual structure, the internal explosion
experiment of the cylindrical latticed shell was carried out in an open quarry. To ensure that the
explosion venting of the cylindrical latticed shell model could occur during the experiment, the
22
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internal explosion experiment used 1kg explosive. The failure process and the final failure
phenomena were shown in Figure 5. Figure 5 (a) and Figure 5 (b) demonstrated the failure process
of the physical model of the cylindrical latticed shell. Figure 5 (c) and Figure 5 (d) showed the final
failure phenomenon of the roof panel steel and wall panel, respectively. It was obviously found that
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the roof panel steel was torn locally due to the strong connection stiffness and the connection
component of wall was broken totally and the whole steel plate was separated from the major
structure due to the weak connection. Although the 1kg explosive was larger than the critical mass,
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it could be clearly seen from the final destruction that the initial venting area was at the joints of the
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ED
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wall-shell. Which all above was good agreement with the simulation results shown in Figure 4.
(b) Blasting moment
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(a) Scaled model before explosion
(c) Failure phenomenon of the roof panel steel
(d) Tear failure of the roof panel steel
Figure 5 The experimental phenomena under internal explosion
23
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3.3 Propagation law of shock wave
The propagation law of shock waves under internal explosion was different from that under
external explosion. Firstly, the shock waves reflected on the rigid ground and the reflected waves
and the incident waves stacked up to form the Maher waves. Subsequently, the incident waves and
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the Maher waves spread to the building envelope of roof and wall successively, and the shock
waves were gradually converged on the top of the cylindrical lattice shell and the joints of the
wall-shell owing to the obstruction of the building envelope. At the end, the overpressure of shock
AN
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waves reached the venting threshold and the cylindrical lattice shell began explosion venting. At
this moment the overpressure distribution of shock waves became disordered, because a part of the
shock waves were reflected back to inside and converged in the rigid ground center for starting over
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the reflections.
ED
Figure 6 showed the pressure contours of internal explosive shock waves in original model and
scale models under the critical TNT mass. It could be clearly observed that the impact of the Maher
PT
reflection, the convergence phenomena at the top of the cylindrical lattice shell and the junction of
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the wall-shell, and the second reflection of the shock waves. Remarkably, the internal explosive
phenomena in original model and scale models were identical at the same time point, which
AC
demonstrated that the propagation law of the shock waves in the scale models resembled that of the
original model under internal explosion.
t=8ms
t=19ms
t=26ms
?=1
24
t=58ms
ACCEPTED MANUSCRIPT
t=4ms
t=9.5ms
t=13ms
t=29ms
t=2ms
t=4.75ms
t=6.5ms
t=14.5ms
t=2.6ms
t=5.8ms
t=1.3ms
t=2.9ms
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t=1.9ms
AN
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?=0.25
t=0.8ms
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?=0.5
t=0.95ms
AC
CE
t=0.4ms
PT
ED
?=0.1
?=0.05
Figure 6 Propagation law of shock waves under internal explosion
3.4 overpressure distribution of shock waves
To study the similarity of the specific distribution of the internal explosive shock waves in the
model, according to the propagation law of the shock waves, the overpressure time history curves of
A, B, C, D and E points of the model (as shown in Figure 1) were chosen for analyzing the
characteristics of the blast waves in the internal surface. As shown in Figure 7, the point A was
25
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closest to the explosion position, and the shock waves reached first. Then the shock waves arrived
PT
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?=1
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at the two side walls.
?=0.25
AC
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?=0.5
?=0.1
Figure 7
?=0.05
The overpressure distribution under internal explosion
26
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It could be seen from the Figure 7 that the overpressure peaks of the points A, B and D,
whether in the original model or in the scale models, were the following relations: peak B larger
than peak D larger than peak A. That was reasonable because of the convergence effects of the
shock waves at the structural top and corners. And it could also be explained that the initial venting
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occurred at the joints of the wall-shell and some connection components of wall destroyed first as
shown in Figure 4. Since then the shock waves gradually spread to both end walls and the
overpressure constantly attenuated.
AN
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On the one hand, when the explosive waves arrived at the point E, the overpressure peak was
only half of that of the point A, which was not powerful enough to destroy the connecting
components of the end walls. On the other hand, the shock waves within the structure did not leak
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and congregated as shown in Figure 8, which made the second overpressure peak at the point E
ED
would be larger than the first overpressure peak.
The above comparative analysis presented that the overpressure time history curves of each
PT
characteristic point of the scale models were highly consistent with the original model. And it was
CE
implied that the overpressure distribution of each scale model was similar to that of the original
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model under internal explosion.
Figure 8
End effect of the structure under internal explosion
27
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4 Discussions
In summary, it could be concluded that the scale models of cylindrical lattice shell under
internal explosion obeyed the similarity law from the perspectives of the critical TNT mass, the
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phenomena of initial venting, the propagation law and the overpressure distribution of shock waves.
However, the loading effects and the structural dynamic responses under internal explosion were the
most concerned characteristic in practical engineering. In order to verify the similarity of the
loading effects and the explosion responses between the original model and the scale models and to
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explore the influence of scaling coefficient ? on similarity of the scale models, the data of the
loading effects and structural dynamic responses were obtained, and it was necessary to further
discuss.
M
4.1 Loading effects of internal explosion
ED
Theoretically, the overpressure of shock waves, the positive pressure time and the specific
impulse of the scale models should meet Eqs. (13), (15) and (16). For the sake of more intuitively
PT
explain the matter, the positive pressure time and the specific impulse divided by cube of the TNT
CE
mass was defined as the positive pressure time after conversion and the specific impulse after
conversion. From the data outside brackets of Table 9, it could be seen that the overpressure of
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shock waves, the positive pressure time after conversion and the specific impulse after conversion
of the scale models of the cylindrical lattice shell under internal explosion were similar to that of the
original model, yet the degree of the similarities were different with each other. In general, the
similarities reduced with the decrease of the scaling coefficient and the deviation was controlled
within -1.18%, 1.26% and 2.0%, respectively. However, due to the limitation of material modulus
on projects, the scale models with the scaling coefficient less than or equal to 0.25 existed the
28
ACCEPTED MANUSCRIPT
geometric incomplete similarity, which would evidently affect the similarity of the loading effects
of internal explosion.
Table 9 Loading effects of internal explosion
A
B
C
D
E
?
?
value
?
value
?
value
?
value
?
1
195528
0
247705
0
149632
0
218659
0
104996
0
0.5
195604
0.026
247135
-0.164
146675
-1.18
218032
-0.197
104749
-0.120
195661
0.045
246803
-0.259
146983
-1.06
217706
-0.299
104711
-0.139
(201267)
(1.94)
(258779)
(3.18)
(158783)
(3.67)
(224978)
(1.98)
(108295)
(1.61)
195866
0.114
246685
-0.293
147108
-1.01
217627
-0.324
104502
-0.241
(208007)
(4.22)
(273037)
(7.29)
(174708)
(10.0)
(235106)
(5.16)
(112075)
(3.45)
196058
0.179
246709
-0.286
147176
-0.984
217676
-0.308
105000
0.002
(210607)
(5.10)
(280174)
(9.34)
(181404)
(12.7)
(239796)
(6.63)
(113533)
(4.16)
1
2.09
0
2.84
0
3.88
0
2.39
0
3.14
0
0.5
2.09
0
2.84
0
3.88
0
2.42
1.26
3.14
0
2.09
0
2.84
0
3.88
0
2.39
0
3.14
0
(2.09)
(0)
(2.85)
(0.35)
(3.88)
(0)
(2.39)
(0)
(3.29)
(4.78)
2.09
0
2.84
0
3.88
0
2.39
0
3.14
0
(2.09)
(0)
(2.88)
(1.41)
(4.03)
(3.87)
(2.39)
(0)
(3.29)
(4.78)
2.09
0
2.84
0
3.88
0
2.39
0
3.17
0.955
(2.09)
(0)
(2.96)
(4.23)
(4.03)
(3.87)
(2.39)
(0)
(3.29)
(4.78)
0.25
?P / Pa
CE
ED
PT
0.05
M
0.1
CR
IP
T
value
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Item
0.25
+
-1/3
/
AC
t 稺
ms穔g-1/3
0.1
0.05
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ACCEPTED MANUSCRIPT
-1/3
i稺
/
1
173302
0
237026
0
175154
0
214425
0
130326
0
0.5
173345
0.025
237477
0.190
174943
-0.120
213713
-0.332
129833
-0.378
173451
0.086
236020
-0.424
176055
0.514
213452
-0.454
129856
-0.361
(180296)
(4.04)
(250640)
(5.74)
(193596)
(10.5)
(224385)
(4.64)
(137175)
(5.26)
173734
0.249
234914
-0.891
177885
1.56
213331
-0.510
129752
-0.440
(189323)
(9.24)
(275345)
(16.2)
(227283)
(29.8)
(238075)
(11.0)
(146328)
(12.3)
173879
0.333
234826
-0.928
178659
2.00
213315
-0.518
130259
-0.051
(192933)
(11.3)
(287512)
(21.3)
(244777)
(39.7)
(243505)
(13.6)
(149951)
(15.1)
0.25
kg
-1/3
0.1
AN
US
0.05
CR
IP
T
Pa穖s�
Notes: The values in parentheses were computed from the actual material size.
? represented the relative deviation between the scale model and the original model, expressed as percentile.
The geometric incomplete similarity of the scale models were taken into account from the data
M
inside brackets of Table 9. Compared with the original model, it could be informed that the
ED
similarities of the overpressure of shock waves, the positive pressure time after conversion and the
specific impulse after conversion obtained by the scale models still reduced with the decrease of the
PT
scaling coefficient but the deviations were further amplified. If the scaling coefficient equal to 0.1,
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the maximum deviation of the shock wave overpressure was 10.0% and the maximum deviation of
the specific impulse after conversion was 29.8%. The two above errors were still within the
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acceptable range. But if the scaling coefficient equal to 0.05, the maximum deviation of the shock
waves overpressure reached to 12.7% and the maximum deviation of the specific impulse after
conversion was 39.7%, which were not reasonable and credible to accurately present the
characteristics of the original model. It was indicated that the similarities of the loading effects of
internal explosion were reduced if the geometric incomplete similarity was extended, especially
presented to the scale models with the small scaling coefficients.
30
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It was known from the above analysis that if the scaling coefficient was not less than 0.1, the
loading effects of the scale models under internal explosion were similar to that of the original
model, namely the Eqs. (13), (15) and (16) were met. With the decrease of the scaling coefficient,
the influence of the geometric incomplete similarity of the scale models on the loading effects of
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internal explosion became bigger. However, this conclusion was not universally applicable, only
suitable for the cylindrical lattice shell under internal explosion which was discussed in this paper.
On the one hand, the degree of the similarity between the scale model and the original model also
AN
US
depends on different structure type, geometry and loading. On the other hand, in the practical
engineering, especially for large-span space structures with variety of structural types, the scale
model with a smaller scaled coefficient would be often adopted. Therefore, to get more accurate and
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more general conclusions, it is necessary to carry out more theoretical, experimental and numerical
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research on different structural types and loadings, to analyze the essential reasons for the errors
caused by the geometrically incomplete similarity model under internal explosion and to put
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forward the solutions.
4.2 Internal explosive responses
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The element stress time-history curves and node displacement time-history curves of the model
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center were showed in Figure 9 (a) and (b), respectively. It could be seen from Figure 9 (a) that the
maximum stress of element was more than 400MPa, much larger than the yield stress of the Q235.
This implied that the strain rate effect and the strain hardening effect of material of the cylindrical
lattice shell under internal explosive impact loading were significant. Similarly, it was clear stated
from Figure 9 (b) that the peak of displacement responses of the scale models was less than that of
the original model, and the deviation increased with the decrease of the scaling coefficients. For
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example, as ? ??0.25, the peak of displacement responses was only 5.56% less than that of the
original model, and as ? ??0.05, the peak of displacement responses was 44.31% less than that of
the original model. This implied that the scaling coefficient greatly affected the displacement
(a) Stress time-history curves in the middle point
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responses of the cylindrical lattice shell under internal explosion.
(b) Displacement time-history curves of the central member
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Figure 9 Influence of scaling coefficients on internal explosion responses
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The Existed literature indicated that the scale model subjected to the high-speed impact
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loading would cause the similar distortion because of the strain rate effect and the strain hardening
effect. And with the decrease of the scaling coefficient, the influence of the similar distortion of the
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scale models on the internal explosive responses became bigger [37-38]. In this paper, those
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opinions just had been verified by the above discussions because the Johnson-Cook constitutive
model could consider the influence of the strain rate effect and strain hardening of steel. However,
owing to spatial confined, the errors of internal explosive responses caused by the similar distortion
had not been discussed in this paper. In order to ensure the accuracy of the internal explosive
responses, the authors suggested that the scaling coefficient of the scale models was not less than
0.25, for dealing with the dynamic responses. For the purpose of getting better results, it is
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necessary to conduct further research on the similar distortion.
5 Conclusions
The similarity law of the cylindrical lattice shell model under internal explosion was deduced
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based on the principle of dimensional analysis, and the original model and the scale models with
different scaling coefficients were simulated by ANSYS/LS-DYNA. The correctness and the
feasibility of the similarity law were compared and confirmed. By studying the results of theoretical
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derivation and numerical simulation and by discussing the influence of scaling coefficient ? on
loading effects and structural dynamic responses, the conclusions were obtained as followings.
(1) The scale models and the original model of the cylindrical lattice shell under internal
explosion can satisfy the similarity law. Specifically, the propagation law and the overpressure
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distribution of explosive shock waves, the critical TNT mass and the initial venting phenomena of
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the scale models are similar to the ones of the original model.
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(2) In ideal status, the loading effects of the scale models of the cylindrical lattice shell under
internal explosion are similar to that of the original model, namely can meet the similarity law.
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(3) The influence of the geometric incomplete similarity of the scale models on the loading
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effects of internal explosion cannot be ignored with the scaling coefficient less than or equal to
0.25.
(4) The strain rate effect and the strain hardening effect of material have serious influence on
the accuracy of the stress and displacement responses of the cylindrical lattice shell model under
internal explosive impact loading. If the scaling coefficient is less than 0.25, the stress responses
and displacement responses of the scale models will not accurately reflect the characteristics of the
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original model.
(5) The scaling coefficient greatly affects the stress and displacement responses of the
cylindrical lattice shell under internal explosion. If the scaling coefficient is too small, the errors of
the stress responses and the displacement responses between the scale model and the original model
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will be too large to meet the similarity law. Therefore, for the scale models of the cylindrical lattice
shell affected by geometrical incomplete similarity and strain rate effect of material, the authors
suggest that the scaling coefficient should not be less than 0.1 for obtaining the loading effects of
obtaining the reliable dynamic responses.
Acknowledgments
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the structure under internal explosion by scale model test, and it should not be less than 0.25 for
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The authors are very grateful to the National Natural Science Foundation of China (Grant
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no.51278208), the Science and Technology Project of Fujian Province (Grant no.2018Y0063), and
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the Subsidized Project for Postgraduates? Innovative Fund in Scientific Research of Huaqiao
University (Grant no.17011086003) for the financial support of this work.
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References
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[1] Corley WG, Sozen MA, Thornton CH. The Oklahoma City Bombing: Analysis of blast damage to the Murrah
Building. Journal of Performance of Constructed Facilities 1998;12(3):113-119.
[2] Taveau J. The Buncefield explosion: Were the resulting overpressures really unforeseeable. Process Safety Progress
2012;31(1):55-71.
[3] Ishikawa N, Beppu M. Lessons from past explosive tests on protective structures in Japan. International Journal of
Impact Engineering 2007;34(9):1535-1545.
[4] Remennikov AM, Uy B. Explosive testing and modelling of square tubular steel columns for near-field detonations.
Journal of Constructional Steel Research 2014;101(101):290-303.
34
ACCEPTED MANUSCRIPT
[5] Dey C, Nimje S. Experimental and numerical study on response of sandwich plate subjected to blast load.
Experimental Techniques 2016;40(1):401-411.
[6] Spranghers K, Vasilakos I, Lecompte D, Sol H, Vantomme L. Numerical simulation and experimental validation of
the dynamic response of aluminum plates under free air explosions. International Journal of Impact Engineering
2013;54(4):83-95.
CR
IP
T
[7] Li Z, Chen L, Fang Q, Hao H, et al. Experimental and numerical study of unreinforced clay brick masonry walls
subjected to vented gas explosions. International Journal of Impact Engineering 2017;104:107-126.
[8] Kasivitamnuay J, Singhatanadgid P. Scaling laws for displacement of elastic beam by energy method. International
Journal of Mechanical Sciences 2017;128-129:361-367.
[9] Feng CK, Lee CY. The similarity analysis of vibrating membrane with its applications. International Journal of
AN
US
Mechanical Sciences 2005;47(6):961-981.
[10] Coutinho CP, Baptista AJ, Rodrigues JD. Reduced scale models based on similitude theory: A review up to 2015.
Engineering Structures 2016;119:81-94.
[11] Bakhtar K, Jenus J J. Comparison of full scale and scaled-model Klotz tunnel explosion test results. ADM000767,
M
Proceedings of the Twenty-Seventh DoD Explosives Safety Seminar, Las Vegas1996.
[12] Li Y, Zhang Y. Study of similarity law for bird impact on structure. Chinese Journal of Aeronautics (English
ED
Edition) 2008;21(6):512-517.
[13] Yao S, Zhang D, Lu F, Chen X, Zhao P. A combined experimental and numerical investigation on the scaling laws
PT
for steel box structures subjected to internal blast loading. International Journal of Impact Engineering 2016;102:36-46.
[14] Mazzariol L M, Oshiro R E, Alves M. A method to represent impacted structures using scaled models made of
CE
different materials. International Journal of Impact Engineering 2016;90:81-94.
[15] Alves M, Oshiro R E, Calle M A G, et al. Scaling and Structural Impact. Procedia Engineering 2017;173:391-396.
AC
[16] Yang YD, Li X, Wang X, Li Z, Fu W. Experimental study on similarity model of reinforced concrete structure
under internal explosion. Journal of Nanjing University of Science & Technology 2016;40(2):135-141. (in China)
[17] Yang YD, Li XD, Wang XM, Zhang LC, Zhang ML. Scale similarity model of internal explosion in closed field.
Journal of Vibration & Shock 2014;33(2):128-133+140. (in China)
[18] Jiang Z, Zhong Y, Shi K. Comparability rule and numerical simulation verification for impact dynamic responses
of single layer reticulated shells. Journal of Vibration & Shock 2016;35(21):143-149. (in China)
[19] Jiang ZR, Zhong YK, Shi KR, Luo B. Gravity-based impact comparability rule of single-layer reticulated shells
and its numerical verification. Journal of South China University of Technology 2016;44(10):43-48. (in China).
35
ACCEPTED MANUSCRIPT
[20] Wang WB, Gao XN, Le LH. Study of the similarities in scale models of a single-layer spherical lattice shell
structure under the effect of internal explosion. Shock and Vibration 2017;2017(4):1-13.
[21] Wang WB, Gao XN,. Propagation law of shock waves in single layer spherical lattice shell under internal explosion.
Journal of Interdisciplinary Mathematics 2016;19(3):527-547.
[22] Li YQ, Ma SZ. Explosion Mechanics; 1992.
CR
IP
T
[23] Orlienko. Explosion Physics; 2011.
[24] Ma JL. Blast loading and failure mechanism of single-layer reticulated dome subjected to interior blast. Harbin
Institute of Technology 2016. (in China)
[25] Ma JL, Wu CQ, Zhi XD, Fan F. Prediction of confined blast loading in single-layer lattice shells. Advances in
Structural Engineering 2014;17(7):1029-1044. (in China)
AN
US
[26] Xie WF, Young YL, Liu TG. Multiphase modeling of dynamic fluid-structure interaction during close-in explosion.
International Journal for Numerical Methods in Engineering 2010;74(6):1019-1043.
[27] Gao XN, Fu SQ. Influence of space height on the internal explosion response of single-layer spherical reticulated
shell. Journal of Civil, Architectrual & Environment Engineering 2017;39(4):07-114. (in China)
M
[28] Zhang YG. Structure of Large Span Space; 2005.
[29] Hallquist JO. LS-DYNA keyword user manual version 971; 2007.
2011;44:45-51. (in China)
ED
[30] Pan J. Error analysis of geometrical non-full-similar models for steel connections. China Civil Engineering Journal
PT
[31] Zhao P, Liu EH. Error analysis for steel plane frame non-full-similar model. Low Temperature Architecture
Technology 2016;38(11):86-88. (in China)
CE
[32] Li C. Failure Mechanism and explosion prevention method of cylindrical latticed shell under internal explosion.
Huaqiao University 2016. (in China)
AC
[33] Wu YJ. Study on method of explosion venting for large span cylindrical shell structures under internal explosion.
Huaqiao University 2014. (in China)
[34] Chen X, Gao XN. Numerical simulation and analysis of influence parameters for explosions near ground. Journal
of Huaqiao University (Natural Science Edition) 2014;35(5):570-575. (in China)
[35] Gao XN, Wu YJ. Numerical calculation and influence parameters for TNT explosion. Chinese Journal of
Explosives & Propellants 2015;38(3):32-39. (in China)
[36] Gao XN, Liu Y, Wang SP. Analysis of explosive shock wave pressure distribution on large-space cylindrical
reticulated shell based on LS-DYNA. Journal of Vibration & Shock 2011;30(9):70-75. (in China)
36
ACCEPTED MANUSCRIPT
[37] LI HT, Zhu X, Zhang ZH. Similarity law for dynamic response of hull girder subjected to underwater explosion in
near field. Journal of Vibration & Shock 2010;29(9):28-32. (in China)
[38] Zhang, ZH, Chen PY, Wan-Peng QI, Cheng W. Scaling law of dynamic response of stiffened plates for a ship
AC
CE
PT
ED
M
AN
US
CR
IP
T
subjected to under water shock. Journal of Vibration & Shock 2008;27(6):81-86. (in China)
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