Research Article Plastic Deformation of Metal Tubes Subjected to Lateral Blast Loads Kejian Song, 1 Yuan Long, 1 Chong Ji, 1,2 and Fuyin Gao 1 1 College of Field Engineering, PLA University of Science and Technology, Nanjing 210007, China 2 National Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 10081, China Correspondence should be addressed to Chong Ji; [email protected] Received 20 February 2014; Revised 8 November 2014; Accepted 9 November 2014; Published 20 November 2014 Academic Editor: Igor Andrianov Copyright © 2014 Kejian Song et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. When subjected to the dynamic load, the behavior of the structures is complex and makes it difficult to describe the process of the deformation. In the paper, an analytical model is presented to analyze the plastic deformation of the steel circular tubes. e aim of the research is to calculate the deflection and the deformation angle of the tubes. A series of assumptions are made to achieve the objective. During the research, we build a mathematical model for simply supported thin-walled metal tubes with finite length. At a specified distance above the tube, a TNT charge explodes and generates a plastic shock wave. e wave can be seen as uniformly distributed over the upper semicircle of the cross-section. e simplified Tresca yield domain can be used to describe the plastic flow of the circular tube. e yield domain together with the plastic flow law and other assumptions can finally lead to the solving of the deflection. In the end, tubes with different dimensions subjected to blast wave induced by the TNT charge are observed in experiments. Comparison shows that the numerical results agree well with experiment observations. 1. Introduction Pipeline systems are the main elements of many types of eng- ineering structures, such as aerospace, marine, petrochemical, nuclear, and power generation. Suffering impact or blast loads, the structures will undergo large deflection or even damage. Due to the increased security needs and the occur- rence of accidental or intentional explosions, the subjects of structures suffering transverse blast load are of interest all over the world [1–6]. In most engineer applications, structural elements can be geometrically classified as being beams or plates. As for tubes, they can be classified as being hollow beams or thin wall cylindrical shells. e subject plays an important role in the industrial systems, so there have been many experimental and analytical investigations focus on the structures behaviors when subjected to transverse blast load. e previous inves- tigations are mostly on beams, including blast tests on solid steel beams done by Humphreys [7] and Jones et al. [8]. Symonds and Jones [9] also did experimental research on fully clamped beams. e analytical investigations of rigid- plastic beams subjected to transverse impulsive loads were done by Lee and Symonds [10], Symonds [11], and Jones [12]. e methods they used are generally based on the exact maxi- mum normal stress bending-membrane yield curve shown in Figure 1, which can be simplified using the approximate square yield curves also shown in Figure 1. e square yield curve can provide reasonable bounds to the exact solution and good agreement with tests in references [8, 9, 13]. e above experimental and analytical investigations are mainly on the behaviors of solid beam section, while the hollow sections provide an additional complication for the localized cross-section distortions in addition to the beam-bending deflections. e research on the steel circular hollow beams mostly focuses on the quasistatic transverse loads, which result in both local indentations and global beam bending [14–16]. However, only a small number of studies on steel hollow section beam subjected to local transverse impact load can be found. Jones et al. reported an extensive series of exper- iments on clamped circular tubes under transverse impact loads at different points along the span [17]. Analytical studies by Jones et al. provided rigid-plastic solutions using exact parabolic interaction curves of the locally deformed sections. Some other papers also study the effects of internal pressure Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 250379, 10 pages http://dx.doi.org/10.1155/2014/250379
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Research ArticlePlastic Deformation of Metal Tubes Subjected toLateral Blast Loads
Kejian Song1 Yuan Long1 Chong Ji12 and Fuyin Gao1
1 College of Field Engineering PLA University of Science and Technology Nanjing 210007 China2National Key Laboratory of Explosion Science and Technology Beijing Institute of Technology Beijing 10081 China
Correspondence should be addressed to Chong Ji blastingcaptain163com
Received 20 February 2014 Revised 8 November 2014 Accepted 9 November 2014 Published 20 November 2014
Academic Editor Igor Andrianov
Copyright copy 2014 Kejian Song et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
When subjected to the dynamic load the behavior of the structures is complex and makes it difficult to describe the process of thedeformation In the paper an analytical model is presented to analyze the plastic deformation of the steel circular tubesThe aim ofthe research is to calculate the deflection and the deformation angle of the tubes A series of assumptions are made to achieve theobjective During the research we build a mathematical model for simply supported thin-walled metal tubes with finite length Ata specified distance above the tube a TNT charge explodes and generates a plastic shock wave The wave can be seen as uniformlydistributed over the upper semicircle of the cross-section The simplified Tresca yield domain can be used to describe the plasticflow of the circular tube The yield domain together with the plastic flow law and other assumptions can finally lead to the solvingof the deflection In the end tubes with different dimensions subjected to blast wave induced by the TNT charge are observed inexperiments Comparison shows that the numerical results agree well with experiment observations
1 Introduction
Pipeline systems are the main elements of many types of eng-ineering structures such as aerospacemarine petrochemicalnuclear and power generation Suffering impact or blastloads the structures will undergo large deflection or evendamage Due to the increased security needs and the occur-rence of accidental or intentional explosions the subjects ofstructures suffering transverse blast load are of interest allover the world [1ndash6]
In most engineer applications structural elements can begeometrically classified as being beams or plates As for tubesthey can be classified as being hollow beams or thin wallcylindrical shells The subject plays an important role in theindustrial systems so there have beenmany experimental andanalytical investigations focus on the structures behaviorswhen subjected to transverse blast load The previous inves-tigations are mostly on beams including blast tests on solidsteel beams done by Humphreys [7] and Jones et al [8]Symonds and Jones [9] also did experimental research onfully clamped beams The analytical investigations of rigid-plastic beams subjected to transverse impulsive loads were
done by Lee and Symonds [10] Symonds [11] and Jones [12]Themethods they used are generally based on the exactmaxi-mumnormal stress bending-membrane yield curve shown inFigure 1 which can be simplified using the approximatesquare yield curves also shown in Figure 1 The square yieldcurve can provide reasonable bounds to the exact solutionand good agreement with tests in references [8 9 13] Theabove experimental and analytical investigations are mainlyon the behaviors of solid beam section while the hollowsections provide an additional complication for the localizedcross-section distortions in addition to the beam-bendingdeflections The research on the steel circular hollow beamsmostly focuses on the quasistatic transverse loads whichresult in both local indentations and global beam bending[14ndash16] However only a small number of studies on steelhollow section beam subjected to local transverse impact loadcan be found Jones et al reported an extensive series of exper-iments on clamped circular tubes under transverse impactloads at different points along the span [17] Analytical studiesby Jones et al provided rigid-plastic solutions using exactparabolic interaction curves of the locally deformed sectionsSome other papers also study the effects of internal pressure
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 250379 10 pageshttpdxdoiorg1011552014250379
2 Mathematical Problems in Engineering
1
10 0618
Exact maximum normal stress yield curve
Upper bound-circumscribing
Lower bound-inscribing
minus1
minus1
M
M0
N
N0
Figure 1 Exact and approximate (square) moment axial yieldcurves
on the transverse impact response of circular hollow sectionbeams [18 19] There is a very strong tendency that thefinal expressions of the dynamic plastic deformation shouldbe presented in a dimensionless form [20 21] In Zhaorsquospaper he suggested a dimensionless number which is namedldquoResponse Number Rnrdquo and widely used for the characteriz-ing of the dynamic plastic deformation of various structures[21]
From the above discussion we can see that mostresearches idealize the material behavior as being elastic-perfectly plastic or rigid-perfectly plastic Such a simpli-fication permits the study of the main characteristics ofstructures with relative simplicity and without great loss ofaccuracy In fact the dynamic structural response can besatisfactorily predicted by a rigid-plasticmodel [22] in whichthe governing equations of the structure can be greatly sim-plified by neglecting elastic effects on the plastic deformation
As is mentioned in Yakupovrsquos paper [23] suffering exter-nal loads lots of the pipelines used in the underground oiltraveling industry or the offshore industry will experiencelocal deformation Yakupov made a series of researches onthe local plastic deformations of cylindrical shell under theaction of the explosive Based on Yakupovrsquos theory this paperintroduces an analytical method to study the dynamic plasticdeformation of metal tubes subjected to lateral blast loadsand also reveal the relationships between the structures andthe shock waves During the research we set up a calculationmodel to study the dynamic response of steel tubes shown inFigure 2 The thin-walled tubes with finite length are simplysupported at the two ends At a specified distance above thetube a TNT charge explodes and generates a plastic shockwaveThe shockwave can lead to the local plastic deformationof the tube as shown in Figure 3 Based on the differentintensity of the wave we can divide the plastic deformationprogress into two modes Mechanism I and Mechanism IIAccording to the plastic dynamic theories we can assumethe tube material to be ideally rigid-plastic and use thepiecewise-linear Tresca yield condition to simulate the plas-ticity condition of the tubes The yield conditions together
Charge
Length of the tube
Various standoff distance
Wall thickness
Outer diameter
z
Shock wave
y0
Figure 2 Analysis model and the coordinate system
Initial shape
Deformed portion
R1205791
w0
Figure 3 Schematic representation of the local deformation
with the plastic flow law and initiation conditions can leadto the final plastic deformation of the thin-walled tubes
2 Theoretical Analysis Model
21 Shock Wave Load The theoretical analysis model isshown in Figure 2 The shape of the explosive charge isassumed to be cuboid so the shockwave can be seen as planarwaves at the middle spot of the tube The shock wave pro-duced by the detonation causes the air to move When thewave reaches the surface of the tube the interaction betweenthe moving air and the tube impedes the air velocity andcauses a significant increase in the load on the tube surfaceThe pressure on the surface rises very quickly to the peakreflected pressure instead of the incident peak pressureThenthe reflected pressure decays to zero within 119905
0 The typical
pressure-time history curve is shown in Figure 4 In order tosimplify the calculation we use the decaying shapes shown inFigure 4(b)
As the pressure only acts on the upper part of the circularsection so we can use the ldquoisolated element principlerdquo [23]and define the wave pressure in the following form
119875 (119905) =
1198751(1 minus
119905
1199050
) cos 120579 minus120587
2le 120579 le
120587
2
0 minus120587
2le 120579 le
120587
2
(1)
Mathematical Problems in Engineering 3
Pres
sure
Time0
Pmax
t0
P0
(a)
Pres
sure
Time0
Pmax
t0
P0
(b)
Figure 4 Different decaying shapes of the shock wave
M
M0
N
N0
( 12 1)
(a) Uniform thickness
M
M0
N
N0
( 12 1)
(b) Ideal sandwich
A
B C
D
M
M0
N
N0
(1 1)
(c) Simplest approximation
Figure 5 Yield domains
where 1198751is the peak pressure of the reflected wave and 119875
1=
1198750(1 + radic119899) 119899 is the compression exponent 119905
0is the period
over which the pressure acts and 120579 is the angular coordinate
22 Governing Equation The dynamic behavior of the thin-walled cylindrical shell can be expressed in the followingform [22]
1205972119872119910
1205971199102
+ ((1
119877) +
1205972120596
1205971199102)119873119910+ 119875 minus 120588119867
1205972120596
1205971199052= 0
120597119879
120597119910=
120597119873119910
120597119910= 0
(2)
where119872119910is the bendingmoment in the peripheral direction
119879 and 119873119910are the tangent and normal stresses in the mean
surface 119877 and119867 are the radius and thickness of the tube 120588 isthe density of the material 119875 is the pressure defined in (1)
The prime denotes differentiation with respect to 119910 and thedot with respect to 119905
23 Yield Condition According to Hodge and Normanrsquostheories the phenomena which actually take place under adynamic loading will be complex but certain simplificationsmay be made in order to get the satisfying result The presentpaper is concerned with a load which is greater than the staticcollapse load so that the elastic strains both static anddynamic may reasonably be neglected For a perfectly plasticmaterial the reduced stress resultants must satisfy certaininequalities which depend upon the yield condition and thecross-section of the tube This may be expressed in geomet-rical terms by stating that the stress point must lie within acertain bounded domain called the yield domain For theparticular case of Trescarsquos (maximum shearing stress) yieldcondition Drucker has found the yield domain of Figure 5(a)for a uniform section and Hodge has found the curve ofFigure 5(b) for an ideal sandwich section In the analysis to
4 Mathematical Problems in Engineering
Table 1 Expressions for the generalized stresses and deformationrates
Plasticmode
Stress Vector deformation rate1198731198730
1198721198720
10158401015840
AD minus1 le 1198731198730le 1 minus1 = 0
10158401015840ge 0
A minus1 minus1 ge 0 10158401015840ge 0
AB minus1 minus1 le 1198721198720le 1 ge 0
10158401015840 = 0B minus1 1 ge 0
10158401015840le 0
follow a simplified yield domain as shown in Figure 5(c) willbe considered The simplified yield condition has been usedin many research papers
For a perfectly plastic material the stress profile mustlie in or on the yield domain Points of the tube where thestress profile lies within the yield domain must remain rigidwhile at a boundary point a certain strain rate vector must benormal to the boundary FromHodgewe can see in this paperonly the sides AD and AB and the vertices A and B of theyield domain are appropriate and these will be denoted byplastic regime AD plastic regime A and so forth The result-ing stresses and strain-rates are listed in Table 1 together withthe equations and inequalities which derive from them
3 Mechanism of Local Plastic Deformation
31 Mechanism I For rigid-plastic structures undergoinglarge displacements the rate of external work and the rate ofinternal energy dissipation are fully defined by prescribingsuitable velocity and displacement fields The accuracy ofthe approximate solutions depended on the right choice ofthe velocity and displacement fields During this mechanismwhen the incident wave pressure is greater than the static col-lapse pressure of the tube there appear three plastic joints atthe points 120579 = 0 plusmn120579
1(Figure 6) The middle plastic joint is
moving while the other two are static onesFrom the above discussion we know that the segments
between the two static joints are in a plastic state ADTheflowlaw corresponding to this state can be written in the form
120576119910 119910= minus1 0 (3)
where 120576119910and
119910are generalized deformation velocities
corresponding to generalized stress119873119910and119872
119910 FromTable 1
we can acquire the following equation
119910= 10158401015840= 0 (4)
Thus the flow law which is a solution of the aboveequation can be defined as (Figure 5)
0= 1198881119910 + 1198882 (5)
where 1198881and 1198882are constants and satisfying the conditions
0= 1
(119910 = 0)
0= 0 (119910 = 119910
1)
(6)
0
1205791
Figure 6 Velocity profile of Mechanism I
Applying the initial conditions to the velocity equationwe can acquire the following velocity equations
0= 1(1 minus
119910
1199101
) (0 le 119910 le 1199101)
0= 1(1 +
119910
1199101
) (minus1199101le 119910 le 0)
(7)
From the above discussion and using the stress field andflow law we can transform (2) to the following form
1205972119872119910
1205971199102
minus1198730
119877+ 119875 minus 119898
0
1205972120596
1205971199052= 0 (8)
The solution of (8) must also satisfy the boundaryconditions
119872(0 119905) = 1198720
1198721015840(0 119905) = 0
119872 (plusmn1205791 119905) = minus119872
0
1198721015840(plusmn1205791 119905) = 0
(9)
If we denote by 1198721the bending moment of the region
0 le 119910 le 1199101 substituting in (8) the equation (1) and the first
equation of (7) and satisfying the first two equations of (9)we can get the following equation
1198721= (
1198730
119877+ 11989801)1199102
2minus 1198751(1 minus
119905
1199050
) (1 minus cos 120579)
minus 11989801
1199103
61199101
+1198720
(10)
Applying the last two equations of (9) to (10) and trans-form 119910 to the angular coordinate we get
11989801205792
1
31= 1198751(1 minus
119905
1199050
) (1 minus cos 1205791) minus
21198720
1198772
minus11987301205792
1
2119877
11989801205791
21= 1198751(1 minus
119905
1199050
) sin 1205791minus11987301205791
119877
(11)
Mathematical Problems in Engineering 5
0
1205791
1205790
Figure 7 Velocity profile of Mechanism II
If we set 1= 119905 = 0 in (11) we can get the expressions for
the limiting static pressure in the following forms
119875119904=
11987301198771205792
119904+ 41198720
21198772(1 minus cos 120579
119904)
119875119904=
1198730120579119904
119877 sin 120579119904
(12)
From (11)we can also get the expression for the coordinate1205791as a function of wave pressure at the time of reflection
1198751=
121198720minus 11987301198771205792
1
21198772[3 (1 minus cos 120579
119904) minus 2120579
1sin 1205791] (13)
As (10) must obey the yield condition 119872101584010158401(0 119905) le 0
11987210158401015840
1(plusmn1205791 119905) ge 0 we can acquire the following conditions for
1198750of Mechanism I
1198751le
11987301205791
119877 (2 sin 1205791minus 1205791) 119875
1le
1198730
119877 cos 1205791
(14)
32 Mechanism II When the shock wave pressure is muchgreater than that in Mechanism I there appear four plasticjoints at the points 120579 = plusmn120579
0 plusmn1205791(Figure 7) During this
mechanism region I (minus1205790le 120579 le 120579
0) is in a plastic state A
while region II is in a plastic state AD (Figure 5(c))According to the plastic flow law we can get velocity field
in the following form (Figure 6)
= 0
(0 le 120579 le 1205790)
= 0(
119910 minus 1198771205791
119877 (1205790minus 1205791)) (120579
0le 120579 le 120579
1)
(15)
Based on Jonesrsquos theory we can get the equation ofmotionfor region I
11989800= 1198751(1 minus
119905
1199050
) cos 120579 minus1198730
119877 (16)
As for region II analysis is analogous to that in Mecha-nism I We can substitute in (8) the second expression of (15)The solution must satisfy the following conditions
119872(1205790 119905) = 119872
0
119872 (1205791 119905) = minus119872
0
1198721015840(1205790 119905) = 0
1198721015840(1205791 119905) = 0
(17)
Applying the above conditions we can get
11989801198911
3(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198912minus1198730
21198771198912
1minus21198720
1198772
1198980
2(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198913minus1198730
1198771198911
(18)
where 1198911= 1205791minus 1205790 1198912= cos 120579
0minus cos 120579
1minus (1205791minus 1205790) sin 120579
0
1198913= sin 120579
1minus sin 120579
0
The expression for the coordinate 1205790as a function of wave
pressure has the form
1198751=(1198730119877) 1198912
1minus 12119872
01198772
411989111198913minus 61198912
(19)
From the yield condition 119872101584010158402(1205790 119905) le 0 11987210158401015840
2(1205791 119905) ge 0
we can acquire the following conditions for 1198750of Mechanism
II
1198751le
11987301198771198912
1+ 12119872
0
21198772(31198912minus 1198912
1cos 1205790) 119875
1le
1198730
119877 cos 1205791
(20)
In Mechanism II the motion of the tube can be dividedinto three stages The first is the drive stage During the firststage the plastic joint 120579
0remains static At the end of the
first stage the velocity reaches its highest value and theacceleration velocity decays to zero In the second stage theplastic joint 120579
0begins tomove from 120579
0to 0The velocity of the
moving joint can be calculated from (16) Analysis of the thirdstage is analogous to that in Mechanism I At the end of thethird stage the velocity decays to zero
4 Numerical Examples
Consider a steel circular tube with the dimensions 119877 =
50mmand119867 = 275mmThe yield point of thematerial120590119904=
320Nmm2 and the density of the tube 120588 = 7800 kgm3 Theduration of the load 119905
0 as in Chen et al [24] is taken as 15ms
From (12) we can get the value of angular coordinate 120579119904=
08832 Meanwhile from the same equation we can acquiredifferent values of 120579
119904depending on different values of 119867119877
as shown in Figure 8From (12) we can get the limiting static pressure for
Mechanism I119875119904= 232MPa If the pressurewhich acts on the
surface of the tube is smaller than 232MPa no motion willtake place If the pressure is greater than 119875
119904 the tube begins
6 Mathematical Problems in EngineeringIn
itial
def
orm
atio
n an
gle (
rad)
Thickness to radius
11
10
09
08
07
06
05012010008006004002000
Figure 8 The initial deformation angles curves
Pres
sure
wav
es (M
Pa)
The values of the angle 1205791
P1P01P02
80
70
60
50
40
30
20
131211100908
Figure 9 Yield conditions for Mechanism I
to experience accelerated plastic flow From (13) and (14) wecan draw three different curves as shown in Figure 9 As forMechanism I the tube model must obey the yield conditionof (14) so from Figure 9 we get the condition 119875
1le 323MPa
From the above discussion we get a conclusion that Mech-anism I is realized in the pressure range 232MPa le 119875
1le
323MPa The deflection of the tube depending on thepressure wave can be obtained from (11) and the result isshown in Figure 10
For Mechanism II the discussion will be a little morecomplex Firstly from (19) we get the coordinate 120579
0as a
function of wave pressure 1198751 as shown in Figure 11 where 120579
0
commences from pressure 1198751gt 323MPa
According to the yield condition for Mechanism IIwe can also draw three curves of wave pressure from(20) as shown in Figure 12 From Figure 12 and (13) wecan get the pressure range for Mechanism II which is
Defl
ectio
n (c
m)
Pressure (MPa)
6
5
4
3
2
1
0
323028262422
Figure 10 The residual deflection as a function of pressure wave
Def
orm
atio
n an
gle (
rad)
Pressure wave (MPa)
12057911205792
14
12
10
08
06
04
02
00
807060504030
Figure 11 The values of the deformation angles as a function ofpressure wave
Pres
sure
wav
es (M
Pa)
The values of the deformation angle 1205791
P1P03P04
160
140
120
100
80
60
40
20145140135130125120115110105
Figure 12 Yield conditions for Mechanism II
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
1
10 0618
Exact maximum normal stress yield curve
Upper bound-circumscribing
Lower bound-inscribing
minus1
minus1
M
M0
N
N0
Figure 1 Exact and approximate (square) moment axial yieldcurves
on the transverse impact response of circular hollow sectionbeams [18 19] There is a very strong tendency that thefinal expressions of the dynamic plastic deformation shouldbe presented in a dimensionless form [20 21] In Zhaorsquospaper he suggested a dimensionless number which is namedldquoResponse Number Rnrdquo and widely used for the characteriz-ing of the dynamic plastic deformation of various structures[21]
From the above discussion we can see that mostresearches idealize the material behavior as being elastic-perfectly plastic or rigid-perfectly plastic Such a simpli-fication permits the study of the main characteristics ofstructures with relative simplicity and without great loss ofaccuracy In fact the dynamic structural response can besatisfactorily predicted by a rigid-plasticmodel [22] in whichthe governing equations of the structure can be greatly sim-plified by neglecting elastic effects on the plastic deformation
As is mentioned in Yakupovrsquos paper [23] suffering exter-nal loads lots of the pipelines used in the underground oiltraveling industry or the offshore industry will experiencelocal deformation Yakupov made a series of researches onthe local plastic deformations of cylindrical shell under theaction of the explosive Based on Yakupovrsquos theory this paperintroduces an analytical method to study the dynamic plasticdeformation of metal tubes subjected to lateral blast loadsand also reveal the relationships between the structures andthe shock waves During the research we set up a calculationmodel to study the dynamic response of steel tubes shown inFigure 2 The thin-walled tubes with finite length are simplysupported at the two ends At a specified distance above thetube a TNT charge explodes and generates a plastic shockwaveThe shockwave can lead to the local plastic deformationof the tube as shown in Figure 3 Based on the differentintensity of the wave we can divide the plastic deformationprogress into two modes Mechanism I and Mechanism IIAccording to the plastic dynamic theories we can assumethe tube material to be ideally rigid-plastic and use thepiecewise-linear Tresca yield condition to simulate the plas-ticity condition of the tubes The yield conditions together
Charge
Length of the tube
Various standoff distance
Wall thickness
Outer diameter
z
Shock wave
y0
Figure 2 Analysis model and the coordinate system
Initial shape
Deformed portion
R1205791
w0
Figure 3 Schematic representation of the local deformation
with the plastic flow law and initiation conditions can leadto the final plastic deformation of the thin-walled tubes
2 Theoretical Analysis Model
21 Shock Wave Load The theoretical analysis model isshown in Figure 2 The shape of the explosive charge isassumed to be cuboid so the shockwave can be seen as planarwaves at the middle spot of the tube The shock wave pro-duced by the detonation causes the air to move When thewave reaches the surface of the tube the interaction betweenthe moving air and the tube impedes the air velocity andcauses a significant increase in the load on the tube surfaceThe pressure on the surface rises very quickly to the peakreflected pressure instead of the incident peak pressureThenthe reflected pressure decays to zero within 119905
0 The typical
pressure-time history curve is shown in Figure 4 In order tosimplify the calculation we use the decaying shapes shown inFigure 4(b)
As the pressure only acts on the upper part of the circularsection so we can use the ldquoisolated element principlerdquo [23]and define the wave pressure in the following form
119875 (119905) =
1198751(1 minus
119905
1199050
) cos 120579 minus120587
2le 120579 le
120587
2
0 minus120587
2le 120579 le
120587
2
(1)
Mathematical Problems in Engineering 3
Pres
sure
Time0
Pmax
t0
P0
(a)
Pres
sure
Time0
Pmax
t0
P0
(b)
Figure 4 Different decaying shapes of the shock wave
M
M0
N
N0
( 12 1)
(a) Uniform thickness
M
M0
N
N0
( 12 1)
(b) Ideal sandwich
A
B C
D
M
M0
N
N0
(1 1)
(c) Simplest approximation
Figure 5 Yield domains
where 1198751is the peak pressure of the reflected wave and 119875
1=
1198750(1 + radic119899) 119899 is the compression exponent 119905
0is the period
over which the pressure acts and 120579 is the angular coordinate
22 Governing Equation The dynamic behavior of the thin-walled cylindrical shell can be expressed in the followingform [22]
1205972119872119910
1205971199102
+ ((1
119877) +
1205972120596
1205971199102)119873119910+ 119875 minus 120588119867
1205972120596
1205971199052= 0
120597119879
120597119910=
120597119873119910
120597119910= 0
(2)
where119872119910is the bendingmoment in the peripheral direction
119879 and 119873119910are the tangent and normal stresses in the mean
surface 119877 and119867 are the radius and thickness of the tube 120588 isthe density of the material 119875 is the pressure defined in (1)
The prime denotes differentiation with respect to 119910 and thedot with respect to 119905
23 Yield Condition According to Hodge and Normanrsquostheories the phenomena which actually take place under adynamic loading will be complex but certain simplificationsmay be made in order to get the satisfying result The presentpaper is concerned with a load which is greater than the staticcollapse load so that the elastic strains both static anddynamic may reasonably be neglected For a perfectly plasticmaterial the reduced stress resultants must satisfy certaininequalities which depend upon the yield condition and thecross-section of the tube This may be expressed in geomet-rical terms by stating that the stress point must lie within acertain bounded domain called the yield domain For theparticular case of Trescarsquos (maximum shearing stress) yieldcondition Drucker has found the yield domain of Figure 5(a)for a uniform section and Hodge has found the curve ofFigure 5(b) for an ideal sandwich section In the analysis to
4 Mathematical Problems in Engineering
Table 1 Expressions for the generalized stresses and deformationrates
Plasticmode
Stress Vector deformation rate1198731198730
1198721198720
10158401015840
AD minus1 le 1198731198730le 1 minus1 = 0
10158401015840ge 0
A minus1 minus1 ge 0 10158401015840ge 0
AB minus1 minus1 le 1198721198720le 1 ge 0
10158401015840 = 0B minus1 1 ge 0
10158401015840le 0
follow a simplified yield domain as shown in Figure 5(c) willbe considered The simplified yield condition has been usedin many research papers
For a perfectly plastic material the stress profile mustlie in or on the yield domain Points of the tube where thestress profile lies within the yield domain must remain rigidwhile at a boundary point a certain strain rate vector must benormal to the boundary FromHodgewe can see in this paperonly the sides AD and AB and the vertices A and B of theyield domain are appropriate and these will be denoted byplastic regime AD plastic regime A and so forth The result-ing stresses and strain-rates are listed in Table 1 together withthe equations and inequalities which derive from them
3 Mechanism of Local Plastic Deformation
31 Mechanism I For rigid-plastic structures undergoinglarge displacements the rate of external work and the rate ofinternal energy dissipation are fully defined by prescribingsuitable velocity and displacement fields The accuracy ofthe approximate solutions depended on the right choice ofthe velocity and displacement fields During this mechanismwhen the incident wave pressure is greater than the static col-lapse pressure of the tube there appear three plastic joints atthe points 120579 = 0 plusmn120579
1(Figure 6) The middle plastic joint is
moving while the other two are static onesFrom the above discussion we know that the segments
between the two static joints are in a plastic state ADTheflowlaw corresponding to this state can be written in the form
120576119910 119910= minus1 0 (3)
where 120576119910and
119910are generalized deformation velocities
corresponding to generalized stress119873119910and119872
119910 FromTable 1
we can acquire the following equation
119910= 10158401015840= 0 (4)
Thus the flow law which is a solution of the aboveequation can be defined as (Figure 5)
0= 1198881119910 + 1198882 (5)
where 1198881and 1198882are constants and satisfying the conditions
0= 1
(119910 = 0)
0= 0 (119910 = 119910
1)
(6)
0
1205791
Figure 6 Velocity profile of Mechanism I
Applying the initial conditions to the velocity equationwe can acquire the following velocity equations
0= 1(1 minus
119910
1199101
) (0 le 119910 le 1199101)
0= 1(1 +
119910
1199101
) (minus1199101le 119910 le 0)
(7)
From the above discussion and using the stress field andflow law we can transform (2) to the following form
1205972119872119910
1205971199102
minus1198730
119877+ 119875 minus 119898
0
1205972120596
1205971199052= 0 (8)
The solution of (8) must also satisfy the boundaryconditions
119872(0 119905) = 1198720
1198721015840(0 119905) = 0
119872 (plusmn1205791 119905) = minus119872
0
1198721015840(plusmn1205791 119905) = 0
(9)
If we denote by 1198721the bending moment of the region
0 le 119910 le 1199101 substituting in (8) the equation (1) and the first
equation of (7) and satisfying the first two equations of (9)we can get the following equation
1198721= (
1198730
119877+ 11989801)1199102
2minus 1198751(1 minus
119905
1199050
) (1 minus cos 120579)
minus 11989801
1199103
61199101
+1198720
(10)
Applying the last two equations of (9) to (10) and trans-form 119910 to the angular coordinate we get
11989801205792
1
31= 1198751(1 minus
119905
1199050
) (1 minus cos 1205791) minus
21198720
1198772
minus11987301205792
1
2119877
11989801205791
21= 1198751(1 minus
119905
1199050
) sin 1205791minus11987301205791
119877
(11)
Mathematical Problems in Engineering 5
0
1205791
1205790
Figure 7 Velocity profile of Mechanism II
If we set 1= 119905 = 0 in (11) we can get the expressions for
the limiting static pressure in the following forms
119875119904=
11987301198771205792
119904+ 41198720
21198772(1 minus cos 120579
119904)
119875119904=
1198730120579119904
119877 sin 120579119904
(12)
From (11)we can also get the expression for the coordinate1205791as a function of wave pressure at the time of reflection
1198751=
121198720minus 11987301198771205792
1
21198772[3 (1 minus cos 120579
119904) minus 2120579
1sin 1205791] (13)
As (10) must obey the yield condition 119872101584010158401(0 119905) le 0
11987210158401015840
1(plusmn1205791 119905) ge 0 we can acquire the following conditions for
1198750of Mechanism I
1198751le
11987301205791
119877 (2 sin 1205791minus 1205791) 119875
1le
1198730
119877 cos 1205791
(14)
32 Mechanism II When the shock wave pressure is muchgreater than that in Mechanism I there appear four plasticjoints at the points 120579 = plusmn120579
0 plusmn1205791(Figure 7) During this
mechanism region I (minus1205790le 120579 le 120579
0) is in a plastic state A
while region II is in a plastic state AD (Figure 5(c))According to the plastic flow law we can get velocity field
in the following form (Figure 6)
= 0
(0 le 120579 le 1205790)
= 0(
119910 minus 1198771205791
119877 (1205790minus 1205791)) (120579
0le 120579 le 120579
1)
(15)
Based on Jonesrsquos theory we can get the equation ofmotionfor region I
11989800= 1198751(1 minus
119905
1199050
) cos 120579 minus1198730
119877 (16)
As for region II analysis is analogous to that in Mecha-nism I We can substitute in (8) the second expression of (15)The solution must satisfy the following conditions
119872(1205790 119905) = 119872
0
119872 (1205791 119905) = minus119872
0
1198721015840(1205790 119905) = 0
1198721015840(1205791 119905) = 0
(17)
Applying the above conditions we can get
11989801198911
3(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198912minus1198730
21198771198912
1minus21198720
1198772
1198980
2(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198913minus1198730
1198771198911
(18)
where 1198911= 1205791minus 1205790 1198912= cos 120579
0minus cos 120579
1minus (1205791minus 1205790) sin 120579
0
1198913= sin 120579
1minus sin 120579
0
The expression for the coordinate 1205790as a function of wave
pressure has the form
1198751=(1198730119877) 1198912
1minus 12119872
01198772
411989111198913minus 61198912
(19)
From the yield condition 119872101584010158402(1205790 119905) le 0 11987210158401015840
2(1205791 119905) ge 0
we can acquire the following conditions for 1198750of Mechanism
II
1198751le
11987301198771198912
1+ 12119872
0
21198772(31198912minus 1198912
1cos 1205790) 119875
1le
1198730
119877 cos 1205791
(20)
In Mechanism II the motion of the tube can be dividedinto three stages The first is the drive stage During the firststage the plastic joint 120579
0remains static At the end of the
first stage the velocity reaches its highest value and theacceleration velocity decays to zero In the second stage theplastic joint 120579
0begins tomove from 120579
0to 0The velocity of the
moving joint can be calculated from (16) Analysis of the thirdstage is analogous to that in Mechanism I At the end of thethird stage the velocity decays to zero
4 Numerical Examples
Consider a steel circular tube with the dimensions 119877 =
50mmand119867 = 275mmThe yield point of thematerial120590119904=
320Nmm2 and the density of the tube 120588 = 7800 kgm3 Theduration of the load 119905
0 as in Chen et al [24] is taken as 15ms
From (12) we can get the value of angular coordinate 120579119904=
08832 Meanwhile from the same equation we can acquiredifferent values of 120579
119904depending on different values of 119867119877
as shown in Figure 8From (12) we can get the limiting static pressure for
Mechanism I119875119904= 232MPa If the pressurewhich acts on the
surface of the tube is smaller than 232MPa no motion willtake place If the pressure is greater than 119875
119904 the tube begins
6 Mathematical Problems in EngineeringIn
itial
def
orm
atio
n an
gle (
rad)
Thickness to radius
11
10
09
08
07
06
05012010008006004002000
Figure 8 The initial deformation angles curves
Pres
sure
wav
es (M
Pa)
The values of the angle 1205791
P1P01P02
80
70
60
50
40
30
20
131211100908
Figure 9 Yield conditions for Mechanism I
to experience accelerated plastic flow From (13) and (14) wecan draw three different curves as shown in Figure 9 As forMechanism I the tube model must obey the yield conditionof (14) so from Figure 9 we get the condition 119875
1le 323MPa
From the above discussion we get a conclusion that Mech-anism I is realized in the pressure range 232MPa le 119875
1le
323MPa The deflection of the tube depending on thepressure wave can be obtained from (11) and the result isshown in Figure 10
For Mechanism II the discussion will be a little morecomplex Firstly from (19) we get the coordinate 120579
0as a
function of wave pressure 1198751 as shown in Figure 11 where 120579
0
commences from pressure 1198751gt 323MPa
According to the yield condition for Mechanism IIwe can also draw three curves of wave pressure from(20) as shown in Figure 12 From Figure 12 and (13) wecan get the pressure range for Mechanism II which is
Defl
ectio
n (c
m)
Pressure (MPa)
6
5
4
3
2
1
0
323028262422
Figure 10 The residual deflection as a function of pressure wave
Def
orm
atio
n an
gle (
rad)
Pressure wave (MPa)
12057911205792
14
12
10
08
06
04
02
00
807060504030
Figure 11 The values of the deformation angles as a function ofpressure wave
Pres
sure
wav
es (M
Pa)
The values of the deformation angle 1205791
P1P03P04
160
140
120
100
80
60
40
20145140135130125120115110105
Figure 12 Yield conditions for Mechanism II
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Pres
sure
Time0
Pmax
t0
P0
(a)
Pres
sure
Time0
Pmax
t0
P0
(b)
Figure 4 Different decaying shapes of the shock wave
M
M0
N
N0
( 12 1)
(a) Uniform thickness
M
M0
N
N0
( 12 1)
(b) Ideal sandwich
A
B C
D
M
M0
N
N0
(1 1)
(c) Simplest approximation
Figure 5 Yield domains
where 1198751is the peak pressure of the reflected wave and 119875
1=
1198750(1 + radic119899) 119899 is the compression exponent 119905
0is the period
over which the pressure acts and 120579 is the angular coordinate
22 Governing Equation The dynamic behavior of the thin-walled cylindrical shell can be expressed in the followingform [22]
1205972119872119910
1205971199102
+ ((1
119877) +
1205972120596
1205971199102)119873119910+ 119875 minus 120588119867
1205972120596
1205971199052= 0
120597119879
120597119910=
120597119873119910
120597119910= 0
(2)
where119872119910is the bendingmoment in the peripheral direction
119879 and 119873119910are the tangent and normal stresses in the mean
surface 119877 and119867 are the radius and thickness of the tube 120588 isthe density of the material 119875 is the pressure defined in (1)
The prime denotes differentiation with respect to 119910 and thedot with respect to 119905
23 Yield Condition According to Hodge and Normanrsquostheories the phenomena which actually take place under adynamic loading will be complex but certain simplificationsmay be made in order to get the satisfying result The presentpaper is concerned with a load which is greater than the staticcollapse load so that the elastic strains both static anddynamic may reasonably be neglected For a perfectly plasticmaterial the reduced stress resultants must satisfy certaininequalities which depend upon the yield condition and thecross-section of the tube This may be expressed in geomet-rical terms by stating that the stress point must lie within acertain bounded domain called the yield domain For theparticular case of Trescarsquos (maximum shearing stress) yieldcondition Drucker has found the yield domain of Figure 5(a)for a uniform section and Hodge has found the curve ofFigure 5(b) for an ideal sandwich section In the analysis to
4 Mathematical Problems in Engineering
Table 1 Expressions for the generalized stresses and deformationrates
Plasticmode
Stress Vector deformation rate1198731198730
1198721198720
10158401015840
AD minus1 le 1198731198730le 1 minus1 = 0
10158401015840ge 0
A minus1 minus1 ge 0 10158401015840ge 0
AB minus1 minus1 le 1198721198720le 1 ge 0
10158401015840 = 0B minus1 1 ge 0
10158401015840le 0
follow a simplified yield domain as shown in Figure 5(c) willbe considered The simplified yield condition has been usedin many research papers
For a perfectly plastic material the stress profile mustlie in or on the yield domain Points of the tube where thestress profile lies within the yield domain must remain rigidwhile at a boundary point a certain strain rate vector must benormal to the boundary FromHodgewe can see in this paperonly the sides AD and AB and the vertices A and B of theyield domain are appropriate and these will be denoted byplastic regime AD plastic regime A and so forth The result-ing stresses and strain-rates are listed in Table 1 together withthe equations and inequalities which derive from them
3 Mechanism of Local Plastic Deformation
31 Mechanism I For rigid-plastic structures undergoinglarge displacements the rate of external work and the rate ofinternal energy dissipation are fully defined by prescribingsuitable velocity and displacement fields The accuracy ofthe approximate solutions depended on the right choice ofthe velocity and displacement fields During this mechanismwhen the incident wave pressure is greater than the static col-lapse pressure of the tube there appear three plastic joints atthe points 120579 = 0 plusmn120579
1(Figure 6) The middle plastic joint is
moving while the other two are static onesFrom the above discussion we know that the segments
between the two static joints are in a plastic state ADTheflowlaw corresponding to this state can be written in the form
120576119910 119910= minus1 0 (3)
where 120576119910and
119910are generalized deformation velocities
corresponding to generalized stress119873119910and119872
119910 FromTable 1
we can acquire the following equation
119910= 10158401015840= 0 (4)
Thus the flow law which is a solution of the aboveequation can be defined as (Figure 5)
0= 1198881119910 + 1198882 (5)
where 1198881and 1198882are constants and satisfying the conditions
0= 1
(119910 = 0)
0= 0 (119910 = 119910
1)
(6)
0
1205791
Figure 6 Velocity profile of Mechanism I
Applying the initial conditions to the velocity equationwe can acquire the following velocity equations
0= 1(1 minus
119910
1199101
) (0 le 119910 le 1199101)
0= 1(1 +
119910
1199101
) (minus1199101le 119910 le 0)
(7)
From the above discussion and using the stress field andflow law we can transform (2) to the following form
1205972119872119910
1205971199102
minus1198730
119877+ 119875 minus 119898
0
1205972120596
1205971199052= 0 (8)
The solution of (8) must also satisfy the boundaryconditions
119872(0 119905) = 1198720
1198721015840(0 119905) = 0
119872 (plusmn1205791 119905) = minus119872
0
1198721015840(plusmn1205791 119905) = 0
(9)
If we denote by 1198721the bending moment of the region
0 le 119910 le 1199101 substituting in (8) the equation (1) and the first
equation of (7) and satisfying the first two equations of (9)we can get the following equation
1198721= (
1198730
119877+ 11989801)1199102
2minus 1198751(1 minus
119905
1199050
) (1 minus cos 120579)
minus 11989801
1199103
61199101
+1198720
(10)
Applying the last two equations of (9) to (10) and trans-form 119910 to the angular coordinate we get
11989801205792
1
31= 1198751(1 minus
119905
1199050
) (1 minus cos 1205791) minus
21198720
1198772
minus11987301205792
1
2119877
11989801205791
21= 1198751(1 minus
119905
1199050
) sin 1205791minus11987301205791
119877
(11)
Mathematical Problems in Engineering 5
0
1205791
1205790
Figure 7 Velocity profile of Mechanism II
If we set 1= 119905 = 0 in (11) we can get the expressions for
the limiting static pressure in the following forms
119875119904=
11987301198771205792
119904+ 41198720
21198772(1 minus cos 120579
119904)
119875119904=
1198730120579119904
119877 sin 120579119904
(12)
From (11)we can also get the expression for the coordinate1205791as a function of wave pressure at the time of reflection
1198751=
121198720minus 11987301198771205792
1
21198772[3 (1 minus cos 120579
119904) minus 2120579
1sin 1205791] (13)
As (10) must obey the yield condition 119872101584010158401(0 119905) le 0
11987210158401015840
1(plusmn1205791 119905) ge 0 we can acquire the following conditions for
1198750of Mechanism I
1198751le
11987301205791
119877 (2 sin 1205791minus 1205791) 119875
1le
1198730
119877 cos 1205791
(14)
32 Mechanism II When the shock wave pressure is muchgreater than that in Mechanism I there appear four plasticjoints at the points 120579 = plusmn120579
0 plusmn1205791(Figure 7) During this
mechanism region I (minus1205790le 120579 le 120579
0) is in a plastic state A
while region II is in a plastic state AD (Figure 5(c))According to the plastic flow law we can get velocity field
in the following form (Figure 6)
= 0
(0 le 120579 le 1205790)
= 0(
119910 minus 1198771205791
119877 (1205790minus 1205791)) (120579
0le 120579 le 120579
1)
(15)
Based on Jonesrsquos theory we can get the equation ofmotionfor region I
11989800= 1198751(1 minus
119905
1199050
) cos 120579 minus1198730
119877 (16)
As for region II analysis is analogous to that in Mecha-nism I We can substitute in (8) the second expression of (15)The solution must satisfy the following conditions
119872(1205790 119905) = 119872
0
119872 (1205791 119905) = minus119872
0
1198721015840(1205790 119905) = 0
1198721015840(1205791 119905) = 0
(17)
Applying the above conditions we can get
11989801198911
3(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198912minus1198730
21198771198912
1minus21198720
1198772
1198980
2(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198913minus1198730
1198771198911
(18)
where 1198911= 1205791minus 1205790 1198912= cos 120579
0minus cos 120579
1minus (1205791minus 1205790) sin 120579
0
1198913= sin 120579
1minus sin 120579
0
The expression for the coordinate 1205790as a function of wave
pressure has the form
1198751=(1198730119877) 1198912
1minus 12119872
01198772
411989111198913minus 61198912
(19)
From the yield condition 119872101584010158402(1205790 119905) le 0 11987210158401015840
2(1205791 119905) ge 0
we can acquire the following conditions for 1198750of Mechanism
II
1198751le
11987301198771198912
1+ 12119872
0
21198772(31198912minus 1198912
1cos 1205790) 119875
1le
1198730
119877 cos 1205791
(20)
In Mechanism II the motion of the tube can be dividedinto three stages The first is the drive stage During the firststage the plastic joint 120579
0remains static At the end of the
first stage the velocity reaches its highest value and theacceleration velocity decays to zero In the second stage theplastic joint 120579
0begins tomove from 120579
0to 0The velocity of the
moving joint can be calculated from (16) Analysis of the thirdstage is analogous to that in Mechanism I At the end of thethird stage the velocity decays to zero
4 Numerical Examples
Consider a steel circular tube with the dimensions 119877 =
50mmand119867 = 275mmThe yield point of thematerial120590119904=
320Nmm2 and the density of the tube 120588 = 7800 kgm3 Theduration of the load 119905
0 as in Chen et al [24] is taken as 15ms
From (12) we can get the value of angular coordinate 120579119904=
08832 Meanwhile from the same equation we can acquiredifferent values of 120579
119904depending on different values of 119867119877
as shown in Figure 8From (12) we can get the limiting static pressure for
Mechanism I119875119904= 232MPa If the pressurewhich acts on the
surface of the tube is smaller than 232MPa no motion willtake place If the pressure is greater than 119875
119904 the tube begins
6 Mathematical Problems in EngineeringIn
itial
def
orm
atio
n an
gle (
rad)
Thickness to radius
11
10
09
08
07
06
05012010008006004002000
Figure 8 The initial deformation angles curves
Pres
sure
wav
es (M
Pa)
The values of the angle 1205791
P1P01P02
80
70
60
50
40
30
20
131211100908
Figure 9 Yield conditions for Mechanism I
to experience accelerated plastic flow From (13) and (14) wecan draw three different curves as shown in Figure 9 As forMechanism I the tube model must obey the yield conditionof (14) so from Figure 9 we get the condition 119875
1le 323MPa
From the above discussion we get a conclusion that Mech-anism I is realized in the pressure range 232MPa le 119875
1le
323MPa The deflection of the tube depending on thepressure wave can be obtained from (11) and the result isshown in Figure 10
For Mechanism II the discussion will be a little morecomplex Firstly from (19) we get the coordinate 120579
0as a
function of wave pressure 1198751 as shown in Figure 11 where 120579
0
commences from pressure 1198751gt 323MPa
According to the yield condition for Mechanism IIwe can also draw three curves of wave pressure from(20) as shown in Figure 12 From Figure 12 and (13) wecan get the pressure range for Mechanism II which is
Defl
ectio
n (c
m)
Pressure (MPa)
6
5
4
3
2
1
0
323028262422
Figure 10 The residual deflection as a function of pressure wave
Def
orm
atio
n an
gle (
rad)
Pressure wave (MPa)
12057911205792
14
12
10
08
06
04
02
00
807060504030
Figure 11 The values of the deformation angles as a function ofpressure wave
Pres
sure
wav
es (M
Pa)
The values of the deformation angle 1205791
P1P03P04
160
140
120
100
80
60
40
20145140135130125120115110105
Figure 12 Yield conditions for Mechanism II
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Table 1 Expressions for the generalized stresses and deformationrates
Plasticmode
Stress Vector deformation rate1198731198730
1198721198720
10158401015840
AD minus1 le 1198731198730le 1 minus1 = 0
10158401015840ge 0
A minus1 minus1 ge 0 10158401015840ge 0
AB minus1 minus1 le 1198721198720le 1 ge 0
10158401015840 = 0B minus1 1 ge 0
10158401015840le 0
follow a simplified yield domain as shown in Figure 5(c) willbe considered The simplified yield condition has been usedin many research papers
For a perfectly plastic material the stress profile mustlie in or on the yield domain Points of the tube where thestress profile lies within the yield domain must remain rigidwhile at a boundary point a certain strain rate vector must benormal to the boundary FromHodgewe can see in this paperonly the sides AD and AB and the vertices A and B of theyield domain are appropriate and these will be denoted byplastic regime AD plastic regime A and so forth The result-ing stresses and strain-rates are listed in Table 1 together withthe equations and inequalities which derive from them
3 Mechanism of Local Plastic Deformation
31 Mechanism I For rigid-plastic structures undergoinglarge displacements the rate of external work and the rate ofinternal energy dissipation are fully defined by prescribingsuitable velocity and displacement fields The accuracy ofthe approximate solutions depended on the right choice ofthe velocity and displacement fields During this mechanismwhen the incident wave pressure is greater than the static col-lapse pressure of the tube there appear three plastic joints atthe points 120579 = 0 plusmn120579
1(Figure 6) The middle plastic joint is
moving while the other two are static onesFrom the above discussion we know that the segments
between the two static joints are in a plastic state ADTheflowlaw corresponding to this state can be written in the form
120576119910 119910= minus1 0 (3)
where 120576119910and
119910are generalized deformation velocities
corresponding to generalized stress119873119910and119872
119910 FromTable 1
we can acquire the following equation
119910= 10158401015840= 0 (4)
Thus the flow law which is a solution of the aboveequation can be defined as (Figure 5)
0= 1198881119910 + 1198882 (5)
where 1198881and 1198882are constants and satisfying the conditions
0= 1
(119910 = 0)
0= 0 (119910 = 119910
1)
(6)
0
1205791
Figure 6 Velocity profile of Mechanism I
Applying the initial conditions to the velocity equationwe can acquire the following velocity equations
0= 1(1 minus
119910
1199101
) (0 le 119910 le 1199101)
0= 1(1 +
119910
1199101
) (minus1199101le 119910 le 0)
(7)
From the above discussion and using the stress field andflow law we can transform (2) to the following form
1205972119872119910
1205971199102
minus1198730
119877+ 119875 minus 119898
0
1205972120596
1205971199052= 0 (8)
The solution of (8) must also satisfy the boundaryconditions
119872(0 119905) = 1198720
1198721015840(0 119905) = 0
119872 (plusmn1205791 119905) = minus119872
0
1198721015840(plusmn1205791 119905) = 0
(9)
If we denote by 1198721the bending moment of the region
0 le 119910 le 1199101 substituting in (8) the equation (1) and the first
equation of (7) and satisfying the first two equations of (9)we can get the following equation
1198721= (
1198730
119877+ 11989801)1199102
2minus 1198751(1 minus
119905
1199050
) (1 minus cos 120579)
minus 11989801
1199103
61199101
+1198720
(10)
Applying the last two equations of (9) to (10) and trans-form 119910 to the angular coordinate we get
11989801205792
1
31= 1198751(1 minus
119905
1199050
) (1 minus cos 1205791) minus
21198720
1198772
minus11987301205792
1
2119877
11989801205791
21= 1198751(1 minus
119905
1199050
) sin 1205791minus11987301205791
119877
(11)
Mathematical Problems in Engineering 5
0
1205791
1205790
Figure 7 Velocity profile of Mechanism II
If we set 1= 119905 = 0 in (11) we can get the expressions for
the limiting static pressure in the following forms
119875119904=
11987301198771205792
119904+ 41198720
21198772(1 minus cos 120579
119904)
119875119904=
1198730120579119904
119877 sin 120579119904
(12)
From (11)we can also get the expression for the coordinate1205791as a function of wave pressure at the time of reflection
1198751=
121198720minus 11987301198771205792
1
21198772[3 (1 minus cos 120579
119904) minus 2120579
1sin 1205791] (13)
As (10) must obey the yield condition 119872101584010158401(0 119905) le 0
11987210158401015840
1(plusmn1205791 119905) ge 0 we can acquire the following conditions for
1198750of Mechanism I
1198751le
11987301205791
119877 (2 sin 1205791minus 1205791) 119875
1le
1198730
119877 cos 1205791
(14)
32 Mechanism II When the shock wave pressure is muchgreater than that in Mechanism I there appear four plasticjoints at the points 120579 = plusmn120579
0 plusmn1205791(Figure 7) During this
mechanism region I (minus1205790le 120579 le 120579
0) is in a plastic state A
while region II is in a plastic state AD (Figure 5(c))According to the plastic flow law we can get velocity field
in the following form (Figure 6)
= 0
(0 le 120579 le 1205790)
= 0(
119910 minus 1198771205791
119877 (1205790minus 1205791)) (120579
0le 120579 le 120579
1)
(15)
Based on Jonesrsquos theory we can get the equation ofmotionfor region I
11989800= 1198751(1 minus
119905
1199050
) cos 120579 minus1198730
119877 (16)
As for region II analysis is analogous to that in Mecha-nism I We can substitute in (8) the second expression of (15)The solution must satisfy the following conditions
119872(1205790 119905) = 119872
0
119872 (1205791 119905) = minus119872
0
1198721015840(1205790 119905) = 0
1198721015840(1205791 119905) = 0
(17)
Applying the above conditions we can get
11989801198911
3(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198912minus1198730
21198771198912
1minus21198720
1198772
1198980
2(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198913minus1198730
1198771198911
(18)
where 1198911= 1205791minus 1205790 1198912= cos 120579
0minus cos 120579
1minus (1205791minus 1205790) sin 120579
0
1198913= sin 120579
1minus sin 120579
0
The expression for the coordinate 1205790as a function of wave
pressure has the form
1198751=(1198730119877) 1198912
1minus 12119872
01198772
411989111198913minus 61198912
(19)
From the yield condition 119872101584010158402(1205790 119905) le 0 11987210158401015840
2(1205791 119905) ge 0
we can acquire the following conditions for 1198750of Mechanism
II
1198751le
11987301198771198912
1+ 12119872
0
21198772(31198912minus 1198912
1cos 1205790) 119875
1le
1198730
119877 cos 1205791
(20)
In Mechanism II the motion of the tube can be dividedinto three stages The first is the drive stage During the firststage the plastic joint 120579
0remains static At the end of the
first stage the velocity reaches its highest value and theacceleration velocity decays to zero In the second stage theplastic joint 120579
0begins tomove from 120579
0to 0The velocity of the
moving joint can be calculated from (16) Analysis of the thirdstage is analogous to that in Mechanism I At the end of thethird stage the velocity decays to zero
4 Numerical Examples
Consider a steel circular tube with the dimensions 119877 =
50mmand119867 = 275mmThe yield point of thematerial120590119904=
320Nmm2 and the density of the tube 120588 = 7800 kgm3 Theduration of the load 119905
0 as in Chen et al [24] is taken as 15ms
From (12) we can get the value of angular coordinate 120579119904=
08832 Meanwhile from the same equation we can acquiredifferent values of 120579
119904depending on different values of 119867119877
as shown in Figure 8From (12) we can get the limiting static pressure for
Mechanism I119875119904= 232MPa If the pressurewhich acts on the
surface of the tube is smaller than 232MPa no motion willtake place If the pressure is greater than 119875
119904 the tube begins
6 Mathematical Problems in EngineeringIn
itial
def
orm
atio
n an
gle (
rad)
Thickness to radius
11
10
09
08
07
06
05012010008006004002000
Figure 8 The initial deformation angles curves
Pres
sure
wav
es (M
Pa)
The values of the angle 1205791
P1P01P02
80
70
60
50
40
30
20
131211100908
Figure 9 Yield conditions for Mechanism I
to experience accelerated plastic flow From (13) and (14) wecan draw three different curves as shown in Figure 9 As forMechanism I the tube model must obey the yield conditionof (14) so from Figure 9 we get the condition 119875
1le 323MPa
From the above discussion we get a conclusion that Mech-anism I is realized in the pressure range 232MPa le 119875
1le
323MPa The deflection of the tube depending on thepressure wave can be obtained from (11) and the result isshown in Figure 10
For Mechanism II the discussion will be a little morecomplex Firstly from (19) we get the coordinate 120579
0as a
function of wave pressure 1198751 as shown in Figure 11 where 120579
0
commences from pressure 1198751gt 323MPa
According to the yield condition for Mechanism IIwe can also draw three curves of wave pressure from(20) as shown in Figure 12 From Figure 12 and (13) wecan get the pressure range for Mechanism II which is
Defl
ectio
n (c
m)
Pressure (MPa)
6
5
4
3
2
1
0
323028262422
Figure 10 The residual deflection as a function of pressure wave
Def
orm
atio
n an
gle (
rad)
Pressure wave (MPa)
12057911205792
14
12
10
08
06
04
02
00
807060504030
Figure 11 The values of the deformation angles as a function ofpressure wave
Pres
sure
wav
es (M
Pa)
The values of the deformation angle 1205791
P1P03P04
160
140
120
100
80
60
40
20145140135130125120115110105
Figure 12 Yield conditions for Mechanism II
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
0
1205791
1205790
Figure 7 Velocity profile of Mechanism II
If we set 1= 119905 = 0 in (11) we can get the expressions for
the limiting static pressure in the following forms
119875119904=
11987301198771205792
119904+ 41198720
21198772(1 minus cos 120579
119904)
119875119904=
1198730120579119904
119877 sin 120579119904
(12)
From (11)we can also get the expression for the coordinate1205791as a function of wave pressure at the time of reflection
1198751=
121198720minus 11987301198771205792
1
21198772[3 (1 minus cos 120579
119904) minus 2120579
1sin 1205791] (13)
As (10) must obey the yield condition 119872101584010158401(0 119905) le 0
11987210158401015840
1(plusmn1205791 119905) ge 0 we can acquire the following conditions for
1198750of Mechanism I
1198751le
11987301205791
119877 (2 sin 1205791minus 1205791) 119875
1le
1198730
119877 cos 1205791
(14)
32 Mechanism II When the shock wave pressure is muchgreater than that in Mechanism I there appear four plasticjoints at the points 120579 = plusmn120579
0 plusmn1205791(Figure 7) During this
mechanism region I (minus1205790le 120579 le 120579
0) is in a plastic state A
while region II is in a plastic state AD (Figure 5(c))According to the plastic flow law we can get velocity field
in the following form (Figure 6)
= 0
(0 le 120579 le 1205790)
= 0(
119910 minus 1198771205791
119877 (1205790minus 1205791)) (120579
0le 120579 le 120579
1)
(15)
Based on Jonesrsquos theory we can get the equation ofmotionfor region I
11989800= 1198751(1 minus
119905
1199050
) cos 120579 minus1198730
119877 (16)
As for region II analysis is analogous to that in Mecha-nism I We can substitute in (8) the second expression of (15)The solution must satisfy the following conditions
119872(1205790 119905) = 119872
0
119872 (1205791 119905) = minus119872
0
1198721015840(1205790 119905) = 0
1198721015840(1205791 119905) = 0
(17)
Applying the above conditions we can get
11989801198911
3(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198912minus1198730
21198771198912
1minus21198720
1198772
1198980
2(11989110+ 12057900) = 1198751(1 minus
119905
1199050
)1198913minus1198730
1198771198911
(18)
where 1198911= 1205791minus 1205790 1198912= cos 120579
0minus cos 120579
1minus (1205791minus 1205790) sin 120579
0
1198913= sin 120579
1minus sin 120579
0
The expression for the coordinate 1205790as a function of wave
pressure has the form
1198751=(1198730119877) 1198912
1minus 12119872
01198772
411989111198913minus 61198912
(19)
From the yield condition 119872101584010158402(1205790 119905) le 0 11987210158401015840
2(1205791 119905) ge 0
we can acquire the following conditions for 1198750of Mechanism
II
1198751le
11987301198771198912
1+ 12119872
0
21198772(31198912minus 1198912
1cos 1205790) 119875
1le
1198730
119877 cos 1205791
(20)
In Mechanism II the motion of the tube can be dividedinto three stages The first is the drive stage During the firststage the plastic joint 120579
0remains static At the end of the
first stage the velocity reaches its highest value and theacceleration velocity decays to zero In the second stage theplastic joint 120579
0begins tomove from 120579
0to 0The velocity of the
moving joint can be calculated from (16) Analysis of the thirdstage is analogous to that in Mechanism I At the end of thethird stage the velocity decays to zero
4 Numerical Examples
Consider a steel circular tube with the dimensions 119877 =
50mmand119867 = 275mmThe yield point of thematerial120590119904=
320Nmm2 and the density of the tube 120588 = 7800 kgm3 Theduration of the load 119905
0 as in Chen et al [24] is taken as 15ms
From (12) we can get the value of angular coordinate 120579119904=
08832 Meanwhile from the same equation we can acquiredifferent values of 120579
119904depending on different values of 119867119877
as shown in Figure 8From (12) we can get the limiting static pressure for
Mechanism I119875119904= 232MPa If the pressurewhich acts on the
surface of the tube is smaller than 232MPa no motion willtake place If the pressure is greater than 119875
119904 the tube begins
6 Mathematical Problems in EngineeringIn
itial
def
orm
atio
n an
gle (
rad)
Thickness to radius
11
10
09
08
07
06
05012010008006004002000
Figure 8 The initial deformation angles curves
Pres
sure
wav
es (M
Pa)
The values of the angle 1205791
P1P01P02
80
70
60
50
40
30
20
131211100908
Figure 9 Yield conditions for Mechanism I
to experience accelerated plastic flow From (13) and (14) wecan draw three different curves as shown in Figure 9 As forMechanism I the tube model must obey the yield conditionof (14) so from Figure 9 we get the condition 119875
1le 323MPa
From the above discussion we get a conclusion that Mech-anism I is realized in the pressure range 232MPa le 119875
1le
323MPa The deflection of the tube depending on thepressure wave can be obtained from (11) and the result isshown in Figure 10
For Mechanism II the discussion will be a little morecomplex Firstly from (19) we get the coordinate 120579
0as a
function of wave pressure 1198751 as shown in Figure 11 where 120579
0
commences from pressure 1198751gt 323MPa
According to the yield condition for Mechanism IIwe can also draw three curves of wave pressure from(20) as shown in Figure 12 From Figure 12 and (13) wecan get the pressure range for Mechanism II which is
Defl
ectio
n (c
m)
Pressure (MPa)
6
5
4
3
2
1
0
323028262422
Figure 10 The residual deflection as a function of pressure wave
Def
orm
atio
n an
gle (
rad)
Pressure wave (MPa)
12057911205792
14
12
10
08
06
04
02
00
807060504030
Figure 11 The values of the deformation angles as a function ofpressure wave
Pres
sure
wav
es (M
Pa)
The values of the deformation angle 1205791
P1P03P04
160
140
120
100
80
60
40
20145140135130125120115110105
Figure 12 Yield conditions for Mechanism II
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in EngineeringIn
itial
def
orm
atio
n an
gle (
rad)
Thickness to radius
11
10
09
08
07
06
05012010008006004002000
Figure 8 The initial deformation angles curves
Pres
sure
wav
es (M
Pa)
The values of the angle 1205791
P1P01P02
80
70
60
50
40
30
20
131211100908
Figure 9 Yield conditions for Mechanism I
to experience accelerated plastic flow From (13) and (14) wecan draw three different curves as shown in Figure 9 As forMechanism I the tube model must obey the yield conditionof (14) so from Figure 9 we get the condition 119875
1le 323MPa
From the above discussion we get a conclusion that Mech-anism I is realized in the pressure range 232MPa le 119875
1le
323MPa The deflection of the tube depending on thepressure wave can be obtained from (11) and the result isshown in Figure 10
For Mechanism II the discussion will be a little morecomplex Firstly from (19) we get the coordinate 120579
0as a
function of wave pressure 1198751 as shown in Figure 11 where 120579
0
commences from pressure 1198751gt 323MPa
According to the yield condition for Mechanism IIwe can also draw three curves of wave pressure from(20) as shown in Figure 12 From Figure 12 and (13) wecan get the pressure range for Mechanism II which is
Defl
ectio
n (c
m)
Pressure (MPa)
6
5
4
3
2
1
0
323028262422
Figure 10 The residual deflection as a function of pressure wave
Def
orm
atio
n an
gle (
rad)
Pressure wave (MPa)
12057911205792
14
12
10
08
06
04
02
00
807060504030
Figure 11 The values of the deformation angles as a function ofpressure wave
Pres
sure
wav
es (M
Pa)
The values of the deformation angle 1205791
P1P03P04
160
140
120
100
80
60
40
20145140135130125120115110105
Figure 12 Yield conditions for Mechanism II
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Table 2 The result of the experiments
Run Charge mass (g) Tube thickness (mm) Distance (cm) Deflection (cm) Deformation angle1 75 275 8 24 0742 75 275 6 42 0963 75 275 4 57 1044 75 275 2 75 1425 75 20 8 625 1396 75 20 6 86 167 200 275 16 78 158 200 275 14 Damage
r
D
H
Bracket
TNT charge
Thin wall tubes
Figure 13 Sketches of experiment setup
323MPa le 1198751
le 164MPa The residual deflection forMechanism II can be obtained from (18)
5 Experimental Phenomena and Results
In order to get a visual impression of themetal tubes impactedby shock waves we set up a list of experiments to observe thephenomenaThematerial of the tubes used in the experimentis Q235 with the length of 1m and the outer diameter of01m The tubes have two types of thickness 0002m and000275mThe setup of the experiment is shown in Figure 13The explosives used in the test are 75 g column TNT chargeswith a dimension of 30mm times 70mm and 200 g bulk TNTcharge with a dimension of 100mm times 50mm times 25mmrespectively The explosive is fixed at a certain distance abovethe tube The ends of the tube are simply supported
The typical results of the tube with a thickness of 275mmare shown in Figures 14 and 15 The explosives used are 75 gcolumn TNT charges In Figure 14 results of four differentdistances between the explosive and the tube surface whichare 8 cm 6 cm 4 cm and 2 cm are shown The deformationof the cross-section in the middle of the tube is shown in
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 2 cm
Figure 14 75 g TNT charges at different distances
Figure 15 With the distance decreasing from 8 cm to 2 cmthe deflection becomes larger
Some of the deflection curves of the cross-section areshown in Figure 16 Figure 16(a) shows the deflection curvesof the tubewith a thickness of 000275mand distance of 8 cmThe largest deflection at the midpoint is 24 cm and the angleof deformation 120579
119904= 0942 Figure 15(b) shows the deflection
curves of the same tube with a distance of 6 cm The largestdeflection at the midpoint in this situation is 24 cm and theangle of deformation 120579
119904= 104
The results of the other experiments are shown in Table 2As the distance between the explosive charge and the tube
is very small and the pressure sensor we used at present istoo big to use in the experiment so it is nearly impossiblefor us to acquire the applied pressure waves acting onthe surface of the tube But the numerical simulationmethodscan solve the problem quite well So we carry out the numer-ical simulations in order to acquire the pressure waves on thetube The numerical simulations are performed with thecommercial finite element code LS-DYNA Three differentmaterial types are involved in the finite element model theTNT explosive charge the air and the metal tube TheJones-Wilkens-Lee (JWL) equation of state [25] is used tomodel the action of the TNT explosive charge The C-Jpressure of the TNT explosions 119875CJ = 210GPa The mass
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
(a) 8 cm (b) 6 cm
(c) 4 cm (d) 2 cm
Figure 15 Deformation of the cross-section
Distance from mid-point (cm)
Defl
ectio
n (c
m)
8 cm
25
20
15
10
05
00
3530252015100500minus05
(a)
Distance from mid-point (cm)
Defl
ectio
n (c
m)
6 cm
5
4
3
2
1
0
543210
(b)
Figure 16 Deflection curves of the tubes subjected to different intensities of the pressure waves
density 120588 and detonation velocity 119863 of TNT explosions are1610 kgm3 and 6930ms respectively Material Type 9 of LS-DYNA (lowastMAT NULL) is used to model the behavior of theair Air mass density 120588
0and initial internal energy 119890
0are
129 kgm3and 025 Jcm3 respectivelyThe Johnson-Cook (J-C) model [26] is used to study the dynamic mechanicalbehavior of the metal tubes
Four different geometric parameters are modelednamely distances with 2 cm 4 cm 6 cm and 8 cm Due to thesymmetry of the problem only 14 model is used as shownin Figure 17
Typical results are presented in Figure 18 From the figurewe can get the peak pressures caused by 75 g TNT charges atthe distance of 2 cm 4 cm 6 cm and 8 cmThrough using the
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
TNT charge
Air
Metal tube
Figure 17 Finite element model
Pres
sure
(MPa
)
Time (ms)
2 cm4 cm
6 cm8 cm
25
20
15
10
5
0
200150100500
Figure 18 Pressure-time curves at different distances
methods discussed in Sections 3 and 4 we can get the residualdeflection
The pressures are tabulated along with the theoreticaldeflections calculated with these pressures for comparisonwith the tests The result is shown in Table 3
The comparison shows that the results of numerical simu-lation agreewell with the experimentThus it seems relativelyadequate to use this method in the forthcoming optimizationstudy of the metal tubes subjected to the explosion impactloading
6 Conclusions
In this study an analytical model is presented which predictsthe transverse displacement and the deformation angle ofthe steel circular tubes subjected to a blast load A series ofassumptions is made during the analysis The form of theload is simplified to be a linearly decaying pulse shape and
Table 3 Comparison between numerical simulation and experi-ment
Distance(cm)
Numerical simulation Experiment Error1198750(MPa) Deflection (cm) Deflection (cm)
2 2213 798 751 +634 1842 592 570 +386 1226 402 422 minus478 1068 226 243 minus69
distributed uniformly over the upper semicircle the tubematerial is thought to be ideally rigid-plastic use the simpli-fied Tresca yield domain to describe the plastic flow of thecircular tubeOn the basis of these assumptions the followingresults are obtained
(1) Two different deformationmechanisms are presenteddepending on the intensity of the wave Mechanism Iand Mechanism II The pressure range of each mech-anism is calculated depending on the yield conditionThe residual deflection can also be obtained from theequation
(2) A series of experiments are made to observe thephenomena of the circular steel tubes subjected to theblast load From the experiment we can see that thetubes experience local and global deformations Forthe local deformation the residual deflection of exp-eriment agrees well with that of the numerical calcu-lation
(3) The method we provided can predict the local defor-mation of the circular tubes and can be used in thefield of security of the oil traveling industry
Conflict of Interests
The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted
Acknowledgment
This research was financially supported by the NationalNature Science Foundation of China nos 11102233 and51178460
References
[1] P G Hodge ldquoImpact pressure loading of rigid-plastic cylindri-cal shellsrdquo Journal of the Mechanics and Physics of Solids vol 3pp 176ndash188 1955
[2] N Jones ldquoCombined distributed loads on rigid-plastic circularplates with large deflectionsrdquo International Journal of Solids andStructures vol 5 no 1 pp 51ndash64 1969
[3] W J Stronge D Shu and V P-W Shim ldquoDynamic modes ofplastic deformation for suddenly loaded curved beamsrdquo Inter-national Journal of Impact Engineering vol 9 no 1 pp 1ndash181990
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
[4] M R Bambach H Jama X L Zhao and R H Grzebieta ldquoHol-low and concrete filled steel hollow sections under transverseimpact loadsrdquo Engineering Structures vol 30 no 10 pp 2859ndash2870 2008
[5] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquo Ocean Engineering vol 65 pp 177ndash191 2012
[6] K Micallef A S Fallah D J Pope and L A Louca ldquoThedynamic performance of simply-supported rigid-plastic circu-lar steel plates subjected to localised blast loadingrdquo InternationalJournal of Mechanical Sciences vol 65 no 1 pp 177ndash191 2012
[7] J S Humphreys ldquoPlastic deformation of impulsively loadedstraight clamped beamsrdquo Transactions of the ASME Journal ofApplied Mechanics vol 32 no 1 pp 7ndash10 1965
[8] N Jones R N Griffin and R E van Duzer ldquoAn experimentalstudy into the dynamic plastic behaviour of wide beams andrectangular platesrdquo International Journal ofMechanical Sciencesvol 13 no 8 pp 721ndash735 1971
[9] P S Symonds andN Jones ldquoImpulsive loading of fully clampedbeams with finite plastic deflections and strain-rate sensitivityrdquoInternational Journal of Mechanical Sciences vol 14 no 1 pp49ndash69 1972
[10] E H Lee and P S Symonds ldquoLarge plastic deformations ofbeams under transverse impactrdquo Journal of Applied Mechanicsvol 19 pp 308ndash314 1952
[11] P S Symonds ldquoPlastic shear deformations in dynamic loadproblemsrdquo in Engineering Plasticity pp 647ndash664 CambridgeUniversity Press Cambridge UK 1968
[12] N Jones ldquoInfluence of strain-hardening and strain-rate sen-sitivity on the permanent deformation of impulsively loadedrigid-plastic beamsrdquo International Journal of Mechanical Sci-ences vol 9 no 12 pp 777ndash796 1967
[13] N Jones ldquoA theoretical study of the dynamic plastic behavior ofbeams and plates with finite-deflectionsrdquo International Journalof Solids and Structures vol 7 no 8 pp 1007ndash1029 1971
[14] T Wierzbicki and M S Suh ldquoIndentation of tubes under com-bined loadingrdquo International Journal ofMechanical Sciences vol30 no 3-4 pp 229ndash248 1988
[15] A R Watson S R Reid and W Johnson ldquoLarge deformationsof thin-walled circular tubes under transverse loadingmdashIIIfurther experiments on the bending of simply supported tubesrdquoInternational Journal ofMechanical Sciences vol 18 no 9-10 pp501ndash502 1976
[16] A R Watson S R Reid W Johnson and S GThomas ldquoLargedeformations of thin-walled circular tubes under transverseloadingmdashII Experimental study of the crushing of circulartubes by centrally applied opposed wedge-shaped indentersrdquoInternational Journal of Mechanical Sciences vol 18 no 7-8 pp387ndashIN14 1976
[17] N Jones S E Birch R S Birch L Zhu and M Brown ldquoAnexperimental study on the lateral impact of clamped mild steelpipesrdquo Proceedings of the Institution of Mechanical EngineersPart E Journal of Process Mechanical Engineering vol 206 pp111ndash127 1992
[18] W Q Shen and D W Shu ldquoA theoretical analysis on the failureof unpressurized and pressurized pipelinesrdquo Proceedings of theInstitution of Mechanical Engineers Part E Journal of ProcessMechanical Engineering vol 216 no 3 pp 151ndash165 2002
[19] C S Ng andWQ Shen ldquoEffect of lateral impact loads on failureof pressurized pipelines supported by foundationrdquo Proceedingsof the Institution of Mechanical Engineers Part E Journal of
Process Mechanical Engineering vol 220 no 4 pp 193ndash2062006
[20] A S Fallah and K Micallef ldquoDynamic response of Dyne-emaHB26 plates to localised blast loadingrdquo InternationalJournal of Impact Engineering vol 73 pp 91ndash100 2014
[21] Y P Zhao ldquoSuggestion of a new dimensionless number fordynamic plastic response of beams and platesrdquo Archive ofApplied Mechanics vol 68 no 7-8 pp 524ndash538 1998
[22] N Jones Structural Impact Cambridge University Press Cam-bridge UK 1997
[23] R G Yakupov ldquoPlastic deformation of a cylindrical shell underthe action of a planar explosion waverdquo Zhurnal PrikladnoiMekhaniki I Tekhnicheskoi Fiziki no 4 pp 127ndash132 1982
[24] H Chen J Zhou H Fan et al ldquoDynamic responses of buriedarch structure subjected to subsurface localized impulsive load-ing experimental studyrdquo International Journal of Impact Engi-neering vol 65 pp 89ndash101 2014
[25] Livermore Software Technology Corporation (LSTC) LS-DYNA Keyword Userrsquos Manual Version 960 Livermore CalifUSA 2003
[26] G R Johnson and W H Cook ldquoA constitutive model and datafor metals subjected to large strain high strain rates and hightemperaturerdquo in Proceedings of the 7th International Symposiumon Ballistics pp 541ndash548 The Hague The Netherlands 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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