Research Article An Analytical Approach for …downloads.hindawi.com/journals/mpe/2015/563097.pdf · Research Article An Analytical Approach for Deformation Shapes of a Cylindrical

Research ArticleAn Analytical Approach for Deformation Shapes ofa Cylindrical Shell with Internal Medium Subjected toLateral Contact Explosive Loading

Xiangyu Li Zhenduo Li and Minzu Liang

College of Science National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Xiangyu Li xiangyuleenudteducn

Received 27 February 2015 Revised 18 June 2015 Accepted 24 June 2015

Academic Editor Yuri Petryna

Copyright copy 2015 Xiangyu Li 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

An experimental investigation on deformation shape of a cylindrical shell with internal medium subjected to lateral contactexplosion was carried out briefly Deformation shapes at different covered width of lateral explosive were recovered experimentallyBased on the experimental results a corresponding analytical approach has been undertaken with rigid plastic hinge theory In theanalytical model the cylindrical shell is divided into end-to-end rigid square bars Deformation process of the cylindrical shell isdescribed by using the translations and rotations of all rigid square bars Expressions of the spring force buckling moment anddeflection angle between adjacent rigid square bars are conducted theoretically Given the structure parameters of the cylinderand the type of the lateral explosive charge deformation processes and shapes are reported and discussed using the analyticalapproach A good agreement has been obtained between calculated and experimental results and thus the analytical approachcan be considered as a valuable tool in understanding the deformation mechanism and predicting the deformation shapes of thecylindrical shell with internal medium subjected to lateral contact explosion Finally parametric studies are carried out to analyzethe effects of deformation shape including the covered width of the lateral explosive explosive charge material and distribution ofinitial velocity

1 Introduction

Cylindrical shells are used in a wide variety of engineer-ing applications from the containment pressure vessels ofnuclear reactors to the bracing elements of aerospace struc-tures Such structures may be subjected to a wide varietyof short duration transient loads throughout the course oftheir working life such as air blasts underwater explosionsand high velocity impact Accurate prediction of the dynamicplastic deformation and rupture of the cylindrical shellsubjected to high intensity transient loading is of greatimportance in many industrial applications

Early researches on the dynamic buckling and failureof cylinders were restricted to axisymmetric external radialpulse loading [1] and axial impact loading However thecorresponding analysis though presented in an elegant ana-lytical form is of limited applicability because axisymmetricdynamic loading seldom occurs in practice In real world

situations loading is usually applied to one side of the cylinderand is characterized by various degrees of locality It mayconsist of a projectile missile or mass impact standoffexplosion described by a pressure pulse or contact explosionoften approximated as an ideal impulsive loading

Depending on the load intensity and the special distribu-tion of contact pressures various forms of damage may resultranging from large amplitude lateral deflections to punch-through penetration fracture initiation at the base plateprogression of tearing fracture and finally massive structuraldamage Yakupov [2 3] studied the dynamic response ofthe cylindrical shell subjected to a planar plastic shock wavewith rigid plastic hinge theory and presented the residualdeformation of the cylindrical shell as a function of a planarwave pressure Gefken et al [4] extended the earlier analysisby Lindberg and Florence to one-side inward radial pressurethat varied as the cosine of the angular position around theshell and was uniform along length to identify the structural

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 563097 10 pageshttpdxdoiorg1011552015563097

2 Mathematical Problems in Engineering

response modes of thin cylindrical shell with and withoutinternal pressure subjected to external radial impulsive loadsFor unpressurized shells the response modes consisted ofdynamic pulse buckling followed by large inward deflectionsof the loaded surface In shell with high internal pressurethese response modes were followed by an outward motiondriven by the internal pressure Fatt and Wierzbicki [5] andWierzbicki and Fatt [6] investigated the large amplitudetransient response of plastic cylindrical shells using a string-on-foundation model and the model incorporated two mainload-resistance mechanisms in the shell stretching in thelongitudinal direction and bending in the circumferentialdirection Jiang and Olson [7] presented a numerical modelfor large deflection elastic-plastic analysis of the cylindricalshell structures under air blast loading condition based on atransversely curved finite strip formulation Li and Jones [8]studied the dynamic response of a ldquoshortrdquo cylindrical shellwhich is made from a rigid perfectly plastic material and theplastic behavior of thematerial is controlled by the transverseshear force as well as the circumferential membrane and thelongitudinal bending moment

In recent years increasing attention of both engineeringcommunities and government agencies has turned to thedynamic response of the cylindrical shell subjected to under-water explosion Pedron and Combescure [9] presented amodal method of analysis to determine the response ofan infinitely long stiffened cylindrical shell of revolutionlateral pressure produced by an underwater explosion andpropagating in an acoustic fluid Rajendran and Lee [10]studied comparative damage of air-backed and water-backedplates subjected to noncontact underwater explosion Hunget al [11] investigated the linear and nonlinear dynamicresponses of three cylindrical shell structures subjected tounderwater small charge explosion under different standoffdistances The three cylindrical shell structures were unstiff-ened internally stiffened and externally stiffened respec-tively Li and Rong [12] studied the dynamic response of thecylindrical shell structures subjected to underwater explosionby using experimental and numerical methods Li et al[13] investigated two kinds of the cylindrical shell modelswith the same geometry characteristics unfilled and mainhull sand-filled The main hull sand-filled cylindrical shellis more difficult to be damaged by the shock wave loadingthan the unfilled model Hu et al [14] studied the effects ofelastic modulus shell radius and thickness on the transientresponse characteristics of the cylindrical shell Yao et al [15]presented a new shock factor based on energy acting on thestructure to describe the loading of underwater explosionJama et al [16] reported an experimental and analyticalinvestigation of steel square hollow sections subjected totransverse blast load and the energy dissipated in the localdeformation is discussed In recent decades the research onthe performance of the cylindrical shell is still very limited Aseries of analyticalmodels have been developed to predict thedynamic response of a cylindrical shell subjected to a lateralshock loading or localized loading or underwater explosionBecause of complexities introduced by unsymmetric loadinglarge displacements and rotations of the shell amplifiedby material nonlinearities the problem does not lend itself

easily to an analytical treatment However very few studieshave been reported on cylindrical shell filled with mediumsubjected to contact explosion loading

To investigate the behavior of the cylindrical shell withinternal medium loaded by lateral contact explosion sev-eral experiments have been conducted and the experimen-tal results are presented and discussed in detail in thispaper Based on the experiments a corresponding analyticalapproach was conducted with rigid plastic hinge theoryDeformation processes of cylindrical shells are describedusing the translations and rotations of all rigid square barsIn this study an infinitely long cylindrical shell filled withmedium subject to lateral contact explosion is performedBecause the cylindrical shell has the unique deformationshape in the symmetric axis direction we take a ring repre-senting the cylinder Due to solving the inertiamoment of thering a unit height ring represents deformation of the infinitecylinder Because the ring has the same value of thickness andheight in the radial and axial direction the cross section ofthe ring in the circumferential direction is square so we callit ldquosquare barrdquo

Given the structural parameters of the ring and the typeof the explosive charge deformation processes and shapes arereported using the analytical approach A good agreementhas been obtained between calculated and experimentalresults Finally parametric studies are carried out to analyzethe effects of deformation shapes depending on the coveredwidth of the lateral explosive explosive charge material anddistribution of initial velocity

2 Experimental Procedure and Results

21 Experimental Procedure In order to investigate thedynamic response of the cylindrical shell with internalmedium subjected to lateral contact explosion loading someexperiments were carried out with respect to the differentcovered widths of the lateral explosive charge The photog-raphy and assembly schematics of experimental setups areshown in Figures 1 and 2 respectively

The experimental setup consists of a cylindrical shellwith internal medium (sand) a lateral explosive charge aconnecting rod and two endplates The thickness 119905 the outerradius 119903

1 and the axial length119867 of the cylindrical shell made

from 1020 Steel are 2mm 100mm and 220mm respectivelyThe radius of the cylindrical shell is more than 10 times thethickness The internal medium with a center hole reservedis local sand having a density of 175 gcm3 Two endplatesmade from LY12 Aluminium having a thickness of 10mmare fixed by the connecting rod so that fully closed conditionwill be simulated Material properties of 1020 Steel LY12Aluminum and sand are listed in Table 1

The lateral charge is an emulsion explosive (DL103-80)which is made from 75 PETN 20 emulsion and 5Pb3O4 and the density is 095 gcm3 The emulsion explosive

DL103-80 is a sort of mild and flexible material that can beeasily shapedThe inner radius the thickness and the coveredwidth of the lateral explosive charge are 100mm 5mm and120593 respectively The explosion of each test is initiated by anelectric detonator on the top of the lateral explosive charge

Mathematical Problems in Engineering 3

Table 1 Material properties of 1020 Steel LY12 Aluminium and sand

Material Density (gcm3) Yield stress (MPa) Youngrsquos modulus (GPa) Poissonrsquos ratio1020 Steel 785 275 210 029LY12 Aluminium 278 230 70 029Sand 175 423 001 026

Upperendplate

Lowerendplate

Emulsionexplosive

Electricdetonator

Casing

Figure 1 Photography of the experimental setups

22 Experimental Results Deformation shapes of the cylin-drical shell with internal medium at three different coveredwidths of the lateral explosive charge were recovered Defor-mation shapes after tests are shown in Figure 3 From thedeformation shapes recovered the angle of the lateral chargehas a significant effect on the deformation shape and thedeformation shapes are concave linear and convex whenthe covered widths of lateral charges are 45∘ 90∘ and 135∘respectively

3 Analytical Model

31 Basic Assumptions Deformation processes of the cylin-drical shell with internal medium subjected to lateral contactexplosion are a high nonlinear problem Due to complexitiesintroduced by unsymmetric loading large displacementsand rotations of the cylinder amplified by material non-linearities the problem does not lend itself easily to ananalytical treatmentHowever by introducing a suitable set ofassumptions a simple and realistic model can be establishedto describe the deformation processes of the cylindrical shellwith internal medium

Basic assumptions are as follows (1) the cylindrical shellis infinitely long and the axial thickness selected is equal tothe thickness of cylinder (2) the cylinder is divided into end-to-end rigid square bars along the circumferential direction(3) the square bars close to the lateral explosive chargeshave an instantaneous velocity pointing to the centre of thecylinder

During deformation processes of the cylindrical shellsome parameters of square bars may vary at differentmoments such as translational displacements translationalvelocities rotational displacements rotational angles andarea surrounded by square bars which affect the value of thespring force the bending moment and the deflection angleintensively

32 Analytical Approach Based on the above-mentionedassumptions the analytical model is established shown inFigure 4 where the origins for 119909 and 119910 are given and therange 119909 is from 0 to 2119903

119894 and the range 119910 is from minus119903

119894to 119903119894

The cylindrical shell is divided into end-to-end rigid squarebars along the circumferential direction The arc length 119897 ofeach square bar is 2120587119903

1119873 (119873 is the total number of square

bars)The relationships between two adjacent square bars areestablished by using a spring force a bending moment and adeflection angle

The spring force between adjacent square bars assumed asthe perfect elastic-plastic is described by a changeable springforce (Figure 5) The spring force between adjacent squarebars is expressed by the equation

119865 =

119864 sdot 119904 sdot 1199052

119904 le 1199040

120590119910sdot 119905

2119904 gt 1199040

(1)

where 119864 120590119910 and 119904

0are Youngrsquos modulus the yield stress

and the elastic limit displacement of bar respectively 119865 is thespring force between adjacent square bars and 119904 is the relativedisplacement between the end of the current bar and the headof the next bar

The relative displacements between the end of the currentbar and the head of the next bar are obtained by usingthe end displacement of the current square bar subtractingthe head displacement of the next square bar The relativedisplacements of adjacent square bars are described by theequation

119904119909(119894) = [119889119905 sdot V

119909(119894) minus

119897 sdot 120579 (119894)

2sin120572 (119894)]

minus [119889119905 sdot V119909(119894 + 1) minus 119897 sdot 120579 (119894 + 1)

2sin120572 (119894 + 1)]

119904119910(119894) = [119889119905 sdot V

119910(119894) +

119897 sdot 120579 (119894)

2cos120572 (119894)]

minus [119889119905 sdot V119910(119894 + 1) + 119897 sdot 120579 (119894 + 1)

2cos120572 (119894 + 1)]

(2)

where 119904119909(119894) and 119904

119910(119894) are relative displacements between two

adjacent square bars in the 119909-axis and 119910-axis directions


Casing

120593

t

r1Emulsionexplosive

Sand

(a) Section drawing

Sand

Casing

Emulsionexplosive

Lowerendplate

Upperendplate

Connectingrod

Electricdetonator

H

h1

h2

(b) Profile drawing

Figure 2 Assembly schematics of experimental setups (a) section drawing and (b) profile drawing

(a) 45∘ (b) 90∘ (c) 135∘

Figure 3 Experimental result of deformation shapes after tests at three different covered widths of the lateral explosive charge

respectively 119889119905 is a time step V119909(119894) and V

119910(119894) are translational

velocities of the current square at the 119909-axis and 119910-axisdirections respectively V

119909(119894+1) and V

119910(119894+1) are translational

velocities of the next square at the 119909-axis and 119910-axis direc-tions respectively 120579(119894) is the relative deflection angle betweentwo adjacent square bars The spring force between twoadjacent square bars is calculated by the following equationwhen 119904

119909(119894) and 119904

119910(119894) are less than 119904

0

119865119909(119894) = 119864 sdot 119905

2sdot 119904119909(119894)

119865119910(119894) = 119864 sdot 119905

2sdot 119904119910(119894)

(3)

where 119865119909(119894) and 119865

119910(119894) are spring forces between two adjacent

square bars in the 119909-axis and 119910-axis directions respectivelyBending moment and corresponding deflection angle

between two adjacent square bars are shown in Figure 6Based on the rigid plastic hinge theory the plastic hinges areachievedwhen the bendingmoment of the square bar reachesthe plastic ultimate bending moment During the defor-mation processes of the cylindrical shell the relationshipsbetween the bending moment 119872 and the deflection angle 120579are assumed to be linear when the bending moment is lessthan the plastic ultimate bending moment 119872

119901 Meanwhile

the deflection 120579 is set to the plastic limit deflection angle 120579119901

when the bending moment is greater than or equal to theplastic ultimate bending moment


Bar

l

x

y

O 120593120572 0

Sand

Figure 4 Analytical model

The relationship between the bending moment and thedeflection angle is expressed by the equation

120579 =

119872119897

119864119868119872 lt 119872

119901

120579119901

119872 ge 119872119901

(4)

where119872 is the bending moment 119868 is the moment of inertiawhich is equal to 119905

412 120579 is the deflection angle 119872

119901is the

plastic ultimate bending moment which is equal to 11990531205901199104

Moment of each square bar consists of two aspects themoment generated by the spring force and the momentresulting from the bar bending Moments of square bars aredescribed by the equation

119872ℎ(119894) = (

119897

2) sdot 119865119909(119894 minus 1) sin120572 (119894) minus ( 119897

2)

sdot 119865119910(119894 minus 1) cos120572 (119894) +119872 (119894)

119872119890(119894) = (

119897

2) sdot 119865119909(119894) sin120572 (119894) minus ( 119897

2)

sdot 119865119910(119894) cos120572 (119894) minus119872 (119894)

(5)

where 119872ℎ(119894) and119872

119890(119894) are the moment generated by the

spring force at the head and end of the current square barrespectively 119872(119894) is the moment generated by the relativebending between two adjacent square bars

The internal medium is compressed because of thetranslation and rotation of square bars During movementprocesses of all bars interactions between square bars andinternal medium are shown in Figure 7 where velocities ofprevious bar current bar and next bar are V(119894 minus 1) V(119894) andV(119894 + 1) respectively The resistance forces suffered by theinternal medium are 119891(119894 minus 1) 119891(119894) and 119891(119894 + 1) respectively

It is assumed that the processes of the internal mediumcompressed undergo two stages The first stage is that voidsof internal medium are compacted and the resistance forceof each square bar is equal to 120588

1198901199052(997888V sdot

997888119899 119897)

997888V where 120588119890is the

internal medium density The second stage is that medium

compacted suffers a shock compression and the relationshipbetween shock pressure and volume is assumed to be linear

119901 =120578 (119889119881 minus 1198950 sdot 1198810)

(1 minus 1198950) sdot 1198810 (6)

where 119901 is the shock pressure 120578 is a scale factor 1198810and 119889119881

are the initial volume and the compression volume of internalmedium respectively 119895

0is the compression ratio of medium

The resistance forces of bar generated by internalmediumduring the two stages are expressed by the equation

997888119891 =

1205881198901199052(997888V sdot

997888119899 119897)

997888V 119895 lt 1198950

119901119905 sdot997888119899 119897 119895 ge 1198950

(7)

By calculating the spring forces between adjacent squarebars and the resistance forces generated by the internalmedium translational accelerations of all square bars areobtained at corresponding time By calculating the bendingmoments of all square bars the rotational accelerationsare obtained at corresponding time The translational androtational accelerations of square bars are described by

997888119886 (119894) =

997888119891 (119894) +

997888119865 (119894 minus 1) + 997888

119865 (119894)

119898

120573 (119894) =119872ℎ(119894) + 119872

119890(119894)

119868

(8)

where 120572(119894) and 120573(119894) are the translational and rotationalaccelerations of the current square bar respectively

Utilizing (8) the translational and rotational accelera-tions are calculated which are the initial conditions at thenext time step

33 Initial Conditions Initial translational acceleration rota-tional acceleration and velocities of all square bars are set to0 and the distributions of initial velocities are as follows

997888V =

[V0 cos120572 V0 sin120572] minus120593

2le 120572 le

120593

20 120572 lt minus

120593

2 120572 gt

120593

2

(9)

where 120593 is the half angle covered by the lateral explosivecharge 120572 is the angle between the current square bar and 119909-axis and V

0is the initial velocity of cylindrical shell close to

the lateral explosive chargeAccording to the Gurney equations on contact explosion

the ring velocity can be obtained by using the equation [17]

V0 = radic2119864radic3

1 + 5119872119891119862119890+ 4 (119872

119891119862119890)2 (10)

where V0is the ring velocity (ms)radic2119864 is the Gurney energy

unit mass (ms) 119862119890is the mass of lateral explosive charge

(kg)119872119891is themass of the cylindrical shell close to the lateral

charge (kg) The Gurney equation is a classical method tosolve the metal velocity in recent decades the error betweenthe experimental and calculated results is about 5 and itsaccuracy is acceptable for our work


i i

ss = 0 i + 1

F F998400

i + 1

OO 120590

120590y

120576e 120576

F = F998400 = 0

Figure 5 Spring force between two adjacent square bars

120579

120579

120579p

Mp

M

120579 M

Figure 6 Bending moment and deflection angle between bars

f(i + 1)

f(i)

(i + 1)

(i)

f(i minus 1) (i minus 1)

Sand

Figure 7 Interactions between square bars and internal medium

4 Calculated Results

According to specific structure parameters and the type ofthe explosive charge the velocity of the ring can be obtainedwith (8) It is assumed that the ring made of 1020 Steel

has a thickness of 2mm and the explosivemade ofDL103-80has the thickness of 5mm Based on detonation parametersof DL103-80 substituting these parameters into the Gurneyequation the velocity is approximately 200ms If the explo-sive DL103-80 is replaced with RDX or HMX the velocity isgreater and the different initial velocity distribution can beobtained by varying the explosive charge material and theexplosive mass

Based on the analytical approach calculated results arereported and discussed Deformation processes of the cylin-drical shell consist of two stages stage I velocities of the cylin-drical shell obtained by the lateral contact explosion loadingand stage II interactions between the cylindrical shell andinternalmedium Figure 8 illustrates distribution of positionsand velocities of the cylindrical shell at five differentmomentsunder the covered width 45∘ and the initial velocity of thecylindrical shell 200ms respectively Figures 9 and 10 showdistributions of positions and velocities of the cylindricalshell at five classical times under the covered widths 90∘ and135∘ respectively The ultimate translational velocities of thecylindrical shell are 61ms 73ms and 109ms with thecovered widths 45∘ 90∘ and 135∘ respectively

From the results of deformation shapes a good agreementhas been obtained between calculated and experimentalresults and thus the analytical approach can be considered as


0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y

minus180 minus120 minus60

0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593

Figure 8 Distributions of positions and velocities of the cylindrical shell at different moments (120593 = 45∘)

0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5


a valuable tool in understanding the deformationmechanismand predicting the deformation shape of the cylindrical shellunder lateral contact explosion loading

5 Parametric Studies

Deformation shapes of the ring have a significant relation-ship with the covered width of lateral explosive explosivematerials and initial velocities distribution In order to betterunderstand the deformation mechanism parametric studiesare carried out for the deformation shapes and correspondingresults were discussed

51 Effect of Covered Width of Lateral Explosive From thecalculated and experimental results it is obvious that thecovered width of the lateral explosive charge is a key factorto the deformation shapes Deformation shapes of variouscovered widths of the lateral charge are shown in Figure 11where initial velocity of the ring equals 200ms Variousdeformation shapes can be achieved by changing the widthof lateral charge

52 Effect of Lateral Explosive Materials In order to inves-tigate the effect of lateral charge a series of calculatedresults are obtained by adjusting various initial velocities


t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876

Figure 11 Deformation shapes of the cylindrical shell at variouswidths of lateral explosive charge (V

0= 200ms)

of the cylindrical shell because higher initial velocities ofcylindrical shell represent greater power of charge

Figure 12 shows the deformation shapes of the cylindricalshell at various initial velocities with the covered width 60∘in which velocities are 50ms 100ms 150ms 200ms250ms and 300ms respectivelyThe compression capacityof the sand medium increases with initial velocities whileinitial velocities reach a certain extent there is little changein deformation shapes

0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms

Figure 12 Deformation shapes of the cylindrical shell at variousinitial velocities (120593 = 60∘)

53 Effect of Initial Velocities Distribution In general thelateral explosive charge has a uniform thickness in the cir-cumferential direction within the central angle and the ringclose to the lateral explosive has the same velocity value Thevelocity values varies with thickness of the circumferentialexplosive The simplest assumption is that the thickness ofthe lateral explosive is a linear distribution from one side to


0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

Figure 13 Distributions of positions and velocities of the cylindricalshell at different moments (V(120572) = V

0cos120572)

middle position and the thickness of one side is a half ofmiddle position

Due to the explosive mass reduction the initial velocitiesclose to the lateral charge are assumed to have a cosinedistribution Figure 13 illustrates distribution of positions andvelocities of the cylindrical shell at five classical times underthe coveredwidth 90∘ and the initial velocities having a cosinedistribution

The plotted data in Figures 8ndash13 are from mathematicalmodeling including deformation process different coveredwidth and initial velocity where there are several resultscompared with the experimental results For example inFigure 8 the deformation shape remains stationary at 2000microsecond moment which is in good agreement withthe experimental results From the calculated results of thevelocity distributions of the ring the velocity of each discretebar is zero almost at 2000 microsecond moment so it canrepresent the experimental result

6 Conclusions

This paper presents brief results of an experimental inves-tigation on the deformation process of the cylindrical shellwith internal medium under lateral contact explosion andthe deformation shapes were obtained Based on the exper-iments a corresponding analytical approach has been under-taken using the rigid plastic hinge theory

Given the structural parameters and explosive chargedeformation processes and shapes are reported using theanalytical approach A good agreement has been obtainedbetween calculated and experimental results and thus theanalytical approach can be considered as a valuable tool inunderstanding the deformation mechanism and predicting

the deformation shapes of the cylindrical shell with internalmedium subjected to lateral contact explosion Finally aparametric study is carried out to analyze the effects ofdeformation shapes depending on the covered width ofthe lateral explosive explosive materials and distribution ofinitial velocities Therefore an optimal deformation shapecan be achieved by adjusting the covered width of lateral andinitial velocities distribution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors wish to acknowledge with thanks the financialsupport from the China National Natural Science Fundingunder Grants nos 11202237 and 11132012

References

[1] V Hadavi J Z Ashani and A Mozaffari ldquoTheoretical cal-culation of the maximum radial deformation of a cylindricalshell under explosive forming by a new energy approachrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 226 no 3 pp576ndash584 2012

[2] R G Yakupov ldquoPlastic strains in a cylindrical shell under theeffect of a spherical blast waverdquo Strength of Materials vol 14no 1 pp 52ndash56 1982

[3] R G Yakupov ldquoPlastic deformations of a cylindrical shellunder the action of a planar explosion waverdquo Journal of AppliedMechanics and Technical Physics vol 23 no 4 pp 579ndash5841982

[4] P R Gefken S W Kirkpatrick and B S Holmes ldquoResponse ofimpulsively loaded cylindrical shellsrdquo International Journal ofImpact Engineering vol 7 no 2 pp 213ndash227 1988

[5] M S T Fatt and T Wierzbicki ldquoDamage of plastic cylindersunder localized pressure loadingrdquo International Journal ofMechanical Sciences vol 33 no 12 pp 999ndash1016 1991

[6] T Wierzbicki and M S H Fatt ldquoDamage assessment ofcylinders due to impact and explosive loadingrdquo InternationalJournal of Impact Engineering vol 13 no 2 pp 215ndash241 1993

[7] J Jiang and M D Olson ldquoNonlinear dynamic analysis of blastloaded cylindrical shell structuresrdquoComputersamp Structures vol41 no 1 pp 41ndash52 1991

[8] QM Li andN Jones ldquoBlast loading of a lsquoshortrsquo cylindrical shellwith transverse shear effectsrdquo International Journal of ImpactEngineering vol 16 no 2 pp 331ndash353 1995

[9] C Pedron and A Combescure ldquoDynamic buckling of stiffenedcylindrical shells of revolution under a transient lateral pressureshock waverdquoThin-Walled Structures vol 23 no 1ndash4 pp 85ndash1051995

[10] R Rajendran and JM Lee ldquoA comparative damage study of air-and water-backed plates subjected to non-contact underwaterexplosionrdquo International Journal of Modern Physics B vol 22no 9ndash11 pp 1311ndash1318 2008


[11] C F Hung B J Lin J J Hwang-Fuu and P Y Hsu ldquoDynamicresponse of cylindrical shell structures subjected to underwaterexplosionrdquoOcean Engineering vol 36 no 8 pp 564ndash577 2009

[12] J Li and J-L Rong ldquoExperimental and numerical investigationof the dynamic response of structures subjected to underwaterexplosionrdquo European Journal of Mechanics BFluids vol 32 no1 pp 59ndash69 2012

[13] L-J Li W-K Jiang and Y-H Ai ldquoExperimental study ondeformation and shock damage of cylindrical shell structuressubjected to underwater explosionrdquo Proceedings of the Insti-tution of Mechanical Engineers Part C Journal of MechanicalEngineering Science vol 224 no 11 pp 2505ndash2514 2010

[14] G Y Hu G Xia and J Li ldquoThe transient responses of two-layered cylindrical shells attacked by underwater explosiveshock wavesrdquo Composite Structures vol 92 no 7 pp 1551ndash15602010

[15] X-L Yao J Guo L-H Feng and A-M Zhang ldquoComparabilityresearch on impulsive response of double stiffened cylindricalshells subjected to underwater explosionrdquo International Journalof Impact Engineering vol 36 no 5 pp 754ndash762 2009

[16] H H Jama G N Nurick M R Bambach R H Grzebieta andX L Zhao ldquoSteel square hollow sections subjected to transverseblast loadsrdquoThin-Walled Structures vol 53 pp 109ndash122 2012

[17] M AMeyerDynamic Behavior ofMaterials JohnWiley amp SonNew York NY USA 1994

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


response modes of thin cylindrical shell with and withoutinternal pressure subjected to external radial impulsive loadsFor unpressurized shells the response modes consisted ofdynamic pulse buckling followed by large inward deflectionsof the loaded surface In shell with high internal pressurethese response modes were followed by an outward motiondriven by the internal pressure Fatt and Wierzbicki [5] andWierzbicki and Fatt [6] investigated the large amplitudetransient response of plastic cylindrical shells using a string-on-foundation model and the model incorporated two mainload-resistance mechanisms in the shell stretching in thelongitudinal direction and bending in the circumferentialdirection Jiang and Olson [7] presented a numerical modelfor large deflection elastic-plastic analysis of the cylindricalshell structures under air blast loading condition based on atransversely curved finite strip formulation Li and Jones [8]studied the dynamic response of a ldquoshortrdquo cylindrical shellwhich is made from a rigid perfectly plastic material and theplastic behavior of thematerial is controlled by the transverseshear force as well as the circumferential membrane and thelongitudinal bending moment

In recent years increasing attention of both engineeringcommunities and government agencies has turned to thedynamic response of the cylindrical shell subjected to under-water explosion Pedron and Combescure [9] presented amodal method of analysis to determine the response ofan infinitely long stiffened cylindrical shell of revolutionlateral pressure produced by an underwater explosion andpropagating in an acoustic fluid Rajendran and Lee [10]studied comparative damage of air-backed and water-backedplates subjected to noncontact underwater explosion Hunget al [11] investigated the linear and nonlinear dynamicresponses of three cylindrical shell structures subjected tounderwater small charge explosion under different standoffdistances The three cylindrical shell structures were unstiff-ened internally stiffened and externally stiffened respec-tively Li and Rong [12] studied the dynamic response of thecylindrical shell structures subjected to underwater explosionby using experimental and numerical methods Li et al[13] investigated two kinds of the cylindrical shell modelswith the same geometry characteristics unfilled and mainhull sand-filled The main hull sand-filled cylindrical shellis more difficult to be damaged by the shock wave loadingthan the unfilled model Hu et al [14] studied the effects ofelastic modulus shell radius and thickness on the transientresponse characteristics of the cylindrical shell Yao et al [15]presented a new shock factor based on energy acting on thestructure to describe the loading of underwater explosionJama et al [16] reported an experimental and analyticalinvestigation of steel square hollow sections subjected totransverse blast load and the energy dissipated in the localdeformation is discussed In recent decades the research onthe performance of the cylindrical shell is still very limited Aseries of analyticalmodels have been developed to predict thedynamic response of a cylindrical shell subjected to a lateralshock loading or localized loading or underwater explosionBecause of complexities introduced by unsymmetric loadinglarge displacements and rotations of the shell amplifiedby material nonlinearities the problem does not lend itself

easily to an analytical treatment However very few studieshave been reported on cylindrical shell filled with mediumsubjected to contact explosion loading

To investigate the behavior of the cylindrical shell withinternal medium loaded by lateral contact explosion sev-eral experiments have been conducted and the experimen-tal results are presented and discussed in detail in thispaper Based on the experiments a corresponding analyticalapproach was conducted with rigid plastic hinge theoryDeformation processes of cylindrical shells are describedusing the translations and rotations of all rigid square barsIn this study an infinitely long cylindrical shell filled withmedium subject to lateral contact explosion is performedBecause the cylindrical shell has the unique deformationshape in the symmetric axis direction we take a ring repre-senting the cylinder Due to solving the inertiamoment of thering a unit height ring represents deformation of the infinitecylinder Because the ring has the same value of thickness andheight in the radial and axial direction the cross section ofthe ring in the circumferential direction is square so we callit ldquosquare barrdquo

Given the structural parameters of the ring and the typeof the explosive charge deformation processes and shapes arereported using the analytical approach A good agreementhas been obtained between calculated and experimentalresults Finally parametric studies are carried out to analyzethe effects of deformation shapes depending on the coveredwidth of the lateral explosive explosive charge material anddistribution of initial velocity

2 Experimental Procedure and Results

21 Experimental Procedure In order to investigate thedynamic response of the cylindrical shell with internalmedium subjected to lateral contact explosion loading someexperiments were carried out with respect to the differentcovered widths of the lateral explosive charge The photog-raphy and assembly schematics of experimental setups areshown in Figures 1 and 2 respectively

The experimental setup consists of a cylindrical shellwith internal medium (sand) a lateral explosive charge aconnecting rod and two endplates The thickness 119905 the outerradius 119903

1 and the axial length119867 of the cylindrical shell made

from 1020 Steel are 2mm 100mm and 220mm respectivelyThe radius of the cylindrical shell is more than 10 times thethickness The internal medium with a center hole reservedis local sand having a density of 175 gcm3 Two endplatesmade from LY12 Aluminium having a thickness of 10mmare fixed by the connecting rod so that fully closed conditionwill be simulated Material properties of 1020 Steel LY12Aluminum and sand are listed in Table 1

The lateral charge is an emulsion explosive (DL103-80)which is made from 75 PETN 20 emulsion and 5Pb3O4 and the density is 095 gcm3 The emulsion explosive

DL103-80 is a sort of mild and flexible material that can beeasily shapedThe inner radius the thickness and the coveredwidth of the lateral explosive charge are 100mm 5mm and120593 respectively The explosion of each test is initiated by anelectric detonator on the top of the lateral explosive charge




Upperendplate

Lowerendplate

Emulsionexplosive

Electricdetonator

Casing



3 Analytical Model






119894to 119903119894





119865 =

119864 sdot 119904 sdot 1199052

119904 le 1199040

120590119910sdot 119905

2119904 gt 1199040

(1)

where 119864 120590119910 and 119904




119904119909(119894) = [119889119905 sdot V

119909(119894) minus

119897 sdot 120579 (119894)

2sin120572 (119894)]


2sin120572 (119894 + 1)]

119904119910(119894) = [119889119905 sdot V

119910(119894) +

119897 sdot 120579 (119894)

2cos120572 (119894)]

minus [119889119905 sdot V119910(119894 + 1) + 119897 sdot 120579 (119894 + 1)

2cos120572 (119894 + 1)]

(2)

where 119904119909(119894) and 119904




Casing

120593

t

r1Emulsionexplosive

Sand

(a) Section drawing

Sand

Casing

Emulsionexplosive

Lowerendplate

Upperendplate

Connectingrod

Electricdetonator

H

h1

h2

(b) Profile drawing


(a) 45∘ (b) 90∘ (c) 135∘





119909(119894+1) and V



119909(119894) and 119904

119910(119894) are less than 119904

0

119865119909(119894) = 119864 sdot 119905

2sdot 119904119909(119894)

119865119910(119894) = 119864 sdot 119905

2sdot 119904119910(119894)

(3)

where 119865119909(119894) and 119865




119901 Meanwhile




Bar

l

x

y

O 120593120572 0

Sand



120579 =

119872119897

119864119868119872 lt 119872

119901

120579119901

119872 ge 119872119901

(4)



119901is the



119872ℎ(119894) = (

119897


2)

sdot 119865119910(119894 minus 1) cos120572 (119894) +119872 (119894)

119872119890(119894) = (

119897

2) sdot 119865119909(119894) sin120572 (119894) minus ( 119897

2)

sdot 119865119910(119894) cos120572 (119894) minus119872 (119894)

(5)

where 119872ℎ(119894) and119872





1198901199052(997888V sdot

997888119899 119897)

997888V where 120588119890is the



119901 =120578 (119889119881 minus 1198950 sdot 1198810)

(1 minus 1198950) sdot 1198810 (6)





997888119891 =

1205881198901199052(997888V sdot

997888119899 119897)

997888V 119895 lt 1198950

119901119905 sdot997888119899 119897 119895 ge 1198950

(7)


997888119886 (119894) =

997888119891 (119894) +

997888119865 (119894 minus 1) + 997888

119865 (119894)

119898

120573 (119894) =119872ℎ(119894) + 119872

119890(119894)

119868

(8)




997888V =

[V0 cos120572 V0 sin120572] minus120593

2le 120572 le

120593

20 120572 lt minus

120593

2 120572 gt

120593

2

(9)






1 + 5119872119891119862119890+ 4 (119872

119891119862119890)2 (10)






i i

ss = 0 i + 1

F F998400

i + 1

OO 120590

120590y

120576e 120576

F = F998400 = 0


120579

120579

120579p

Mp

M

120579 M


f(i + 1)

f(i)

(i + 1)

(i)


Sand








0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y


0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593


0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5








t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Upperendplate

Lowerendplate

Emulsionexplosive

Electricdetonator

Casing



3 Analytical Model






119894to 119903119894





119865 =

119864 sdot 119904 sdot 1199052

119904 le 1199040

120590119910sdot 119905

2119904 gt 1199040

(1)

where 119864 120590119910 and 119904




119904119909(119894) = [119889119905 sdot V

119909(119894) minus

119897 sdot 120579 (119894)

2sin120572 (119894)]


2sin120572 (119894 + 1)]

119904119910(119894) = [119889119905 sdot V

119910(119894) +

119897 sdot 120579 (119894)

2cos120572 (119894)]

minus [119889119905 sdot V119910(119894 + 1) + 119897 sdot 120579 (119894 + 1)

2cos120572 (119894 + 1)]

(2)

where 119904119909(119894) and 119904




Casing

120593

t

r1Emulsionexplosive

Sand

(a) Section drawing

Sand

Casing

Emulsionexplosive

Lowerendplate

Upperendplate

Connectingrod

Electricdetonator

H

h1

h2

(b) Profile drawing


(a) 45∘ (b) 90∘ (c) 135∘





119909(119894+1) and V



119909(119894) and 119904

119910(119894) are less than 119904

0

119865119909(119894) = 119864 sdot 119905

2sdot 119904119909(119894)

119865119910(119894) = 119864 sdot 119905

2sdot 119904119910(119894)

(3)

where 119865119909(119894) and 119865




119901 Meanwhile




Bar

l

x

y

O 120593120572 0

Sand



120579 =

119872119897

119864119868119872 lt 119872

119901

120579119901

119872 ge 119872119901

(4)



119901is the



119872ℎ(119894) = (

119897


2)

sdot 119865119910(119894 minus 1) cos120572 (119894) +119872 (119894)

119872119890(119894) = (

119897

2) sdot 119865119909(119894) sin120572 (119894) minus ( 119897

2)

sdot 119865119910(119894) cos120572 (119894) minus119872 (119894)

(5)

where 119872ℎ(119894) and119872





1198901199052(997888V sdot

997888119899 119897)

997888V where 120588119890is the



119901 =120578 (119889119881 minus 1198950 sdot 1198810)

(1 minus 1198950) sdot 1198810 (6)





997888119891 =

1205881198901199052(997888V sdot

997888119899 119897)

997888V 119895 lt 1198950

119901119905 sdot997888119899 119897 119895 ge 1198950

(7)


997888119886 (119894) =

997888119891 (119894) +

997888119865 (119894 minus 1) + 997888

119865 (119894)

119898

120573 (119894) =119872ℎ(119894) + 119872

119890(119894)

119868

(8)




997888V =

[V0 cos120572 V0 sin120572] minus120593

2le 120572 le

120593

20 120572 lt minus

120593

2 120572 gt

120593

2

(9)






1 + 5119872119891119862119890+ 4 (119872

119891119862119890)2 (10)






i i

ss = 0 i + 1

F F998400

i + 1

OO 120590

120590y

120576e 120576

F = F998400 = 0


120579

120579

120579p

Mp

M

120579 M


f(i + 1)

f(i)

(i + 1)

(i)


Sand








0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y


0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593


0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5








t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Casing

120593

t

r1Emulsionexplosive

Sand

(a) Section drawing

Sand

Casing

Emulsionexplosive

Lowerendplate

Upperendplate

Connectingrod

Electricdetonator

H

h1

h2

(b) Profile drawing


(a) 45∘ (b) 90∘ (c) 135∘





119909(119894+1) and V



119909(119894) and 119904

119910(119894) are less than 119904

0

119865119909(119894) = 119864 sdot 119905

2sdot 119904119909(119894)

119865119910(119894) = 119864 sdot 119905

2sdot 119904119910(119894)

(3)

where 119865119909(119894) and 119865




119901 Meanwhile




Bar

l

x

y

O 120593120572 0

Sand



120579 =

119872119897

119864119868119872 lt 119872

119901

120579119901

119872 ge 119872119901

(4)



119901is the



119872ℎ(119894) = (

119897


2)

sdot 119865119910(119894 minus 1) cos120572 (119894) +119872 (119894)

119872119890(119894) = (

119897

2) sdot 119865119909(119894) sin120572 (119894) minus ( 119897

2)

sdot 119865119910(119894) cos120572 (119894) minus119872 (119894)

(5)

where 119872ℎ(119894) and119872





1198901199052(997888V sdot

997888119899 119897)

997888V where 120588119890is the



119901 =120578 (119889119881 minus 1198950 sdot 1198810)

(1 minus 1198950) sdot 1198810 (6)





997888119891 =

1205881198901199052(997888V sdot

997888119899 119897)

997888V 119895 lt 1198950

119901119905 sdot997888119899 119897 119895 ge 1198950

(7)


997888119886 (119894) =

997888119891 (119894) +

997888119865 (119894 minus 1) + 997888

119865 (119894)

119898

120573 (119894) =119872ℎ(119894) + 119872

119890(119894)

119868

(8)




997888V =

[V0 cos120572 V0 sin120572] minus120593

2le 120572 le

120593

20 120572 lt minus

120593

2 120572 gt

120593

2

(9)






1 + 5119872119891119862119890+ 4 (119872

119891119862119890)2 (10)






i i

ss = 0 i + 1

F F998400

i + 1

OO 120590

120590y

120576e 120576

F = F998400 = 0


120579

120579

120579p

Mp

M

120579 M


f(i + 1)

f(i)

(i + 1)

(i)


Sand








0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y


0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593


0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5








t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Bar

l

x

y

O 120593120572 0

Sand



120579 =

119872119897

119864119868119872 lt 119872

119901

120579119901

119872 ge 119872119901

(4)



119901is the



119872ℎ(119894) = (

119897


2)

sdot 119865119910(119894 minus 1) cos120572 (119894) +119872 (119894)

119872119890(119894) = (

119897

2) sdot 119865119909(119894) sin120572 (119894) minus ( 119897

2)

sdot 119865119910(119894) cos120572 (119894) minus119872 (119894)

(5)

where 119872ℎ(119894) and119872





1198901199052(997888V sdot

997888119899 119897)

997888V where 120588119890is the



119901 =120578 (119889119881 minus 1198950 sdot 1198810)

(1 minus 1198950) sdot 1198810 (6)





997888119891 =

1205881198901199052(997888V sdot

997888119899 119897)

997888V 119895 lt 1198950

119901119905 sdot997888119899 119897 119895 ge 1198950

(7)


997888119886 (119894) =

997888119891 (119894) +

997888119865 (119894 minus 1) + 997888

119865 (119894)

119898

120573 (119894) =119872ℎ(119894) + 119872

119890(119894)

119868

(8)




997888V =

[V0 cos120572 V0 sin120572] minus120593

2le 120572 le

120593

20 120572 lt minus

120593

2 120572 gt

120593

2

(9)






1 + 5119872119891119862119890+ 4 (119872

119891119862119890)2 (10)






i i

ss = 0 i + 1

F F998400

i + 1

OO 120590

120590y

120576e 120576

F = F998400 = 0


120579

120579

120579p

Mp

M

120579 M


f(i + 1)

f(i)

(i + 1)

(i)


Sand








0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y


0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593


0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5








t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










i i

ss = 0 i + 1

F F998400

i + 1

OO 120590

120590y

120576e 120576

F = F998400 = 0


120579

120579

120579p

Mp

M

120579 M


f(i + 1)

f(i)

(i + 1)

(i)


Sand








0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y


0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593


0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5








t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 60 120 180

0

50

100

150

200

0 5 10 15 20

0

5

10

y


0120583s50120583s200120583s

1000 120583s2000 120583s

(m

s)

minus10

minus5

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

120593


0 60 120 180

0

50

100

150

200

120593

0 5 10 15 20

0

5

10

y

x

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s

t = 0120583st = 50120583st = 200120583s

t = 1000 120583st = 2000 120583s


(m

s)

minus10

minus5








t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s

0 5 10 15 20

0

5

10

y

minus10

minus5

x

0 60 120 180

0

50

100

150

200


(m

s)

120593

0120583s50120583s

200120583s

100120583s1000 120583s2000 120583s


0 5 10 15 20

minus10

minus5

0

5

10

Initial

y

x

120593 = 21205873

120593 = 1205872

120593 = 1205873

120593 = 1205874

120593 = 1205876


0= 200ms)



0 5 10 15 20minus15

minus10

minus5

0

5

10

15

Initial

y

x

= 50ms = 100ms = 150ms

= 200ms = 250ms = 300ms




0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20minus15

minus10

minus5

0

5

10

15

y

x

t = 0120583st = 50120583st = 100120583s

t = 200120583st = 1000 120583st = 2000 120583s


0cos120572)




6 Conclusions






Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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