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Comparability research on impulsive response of double stiffened cylindrical shells subjected to und

Comparability research on impulsive response of double stiffened cylindrical shells subjected to underwater explosion

Xiong-liang Yao,Jun Guo *,Lin-han Feng,A-man Zhang

College of Shipbuilding Engineering,Harbin Engineering University,Harbin 150001,China

a r t i c l e i n f o

Article history:

Received 25January 2006Received in revised form 19September 2008

Accepted 26September 2008Available online xxx Keywords:

Underwater explosion Shock factor Similarity

Numerical experiment Dynamic response

a b s t r a c t

In order to evaluate the strength and comparability of impulsive environment of model and practical structure in the water when subjected to underwater explosion,a new shock factor based on energy acting on the structure is presented to describe the loading of underwater explosion.To test the validity of this new factor,numerical experiments of double stiffened cylindrical shells are carried out a series of cases designed by the new factor and two other standard shock factors respectively.The results of the cases designed by the new factor indicate that the kinetic energy,potential energy and shock response spectrums of the structures agree well with each other in different cases designed by the equal new shock factor.However,the results of the cases designed by the two other standard shock factors are rather diverse.The analysis considers that the old shock factors do not take the spherical characteristics of shock wave front and relative position between detonation and structure into account,which can hardly re?ect the similarity of underwater explosion loadings.The new shock factor can make up for such limitations.

Published by Elsevier Ltd.

1.Introduction

The dynamic response of submerged structures subjected to underwater explosions is increasingly valued.The accurate theo-retical research is dif?cult to execute due to its complexity,so model testing and numerical experiments become the main research methods.However,how to convert the results of model testing to real ship has not been completely solved,which makes model testing less signi?cant,and only qualitative results could be obtained.Therefore,in the developed countries,e.g.America,each ?rst-made ship must pass the quali?cation testing by the full scale underwater experiment.For lack of simple and effective parame-ters to describe underwater explosions,usually the standard shock factors are employed to describe work cases in military standards [1,2].However,these types of shock factors are lack of general meanings,the experimental results of one case cannot be popu-larized to other cases.Innumerable cases will crop out when a ship is attacked.There are only two methods to get a general cognition about the shock resistance of a ship through numerical experi-ments:one is to do a large number of numerical experiments considering (including)any possible cases;the other is to do a few of numerical tests and extend the results with certain method.

Obviously,the ?rst one is unrealistic so the second one is the unique way to solve the problem.This paper attempts to investigate the similarity of underwater explosion loadings from the viewpoint of energy,so as to get a whole cognition about the ability of a ship subjected to shock with the results of a few cases.

2.Calculating model and analytical method

To verify the validity of all kinds of shock factors and perform formula derivation,a double stiffened cylindrical shell is designed.The basic structure of the model is shown in Fig.1.

Except for the marked dimensions,the concrete dimensions of model are shown in Table 1.The ribs are ?xed at each frame on the inner structures,inner shells and outer shells.The brackets shown in the ?gure are ?xed at each frame between the inner shell and the outer shell.There is full of water between the inner shells and the outer shells.

The ?nite element code ABAQUS is used for dynamic response analysis of the double cylinder shells.The pre-processor of ABAQUS is used for three-dimensional mesh generation with the shell element for meshing the structure and the solid element for meshing surrounding water.The loading of underwater explosion is applied to the structure throughout the water around.The coupling between structure and water is simulated by the acoustic-structure interaction method in ABAQUS.

*Corresponding author.Fax:t8645182518296.

E-mail address:guojun9911228@https://www.wendangku.net/doc/0711653211.html, (J.

Guo).Contents lists available at ScienceDirect

International Journal of Impact Engineering

journal home page:

https://www.wendangku.net/doc/0711653211.html,/locate/ijimpeng

0734-743X/$–see front matter Published by Elsevier Ltd.doi:10.1016/j.ijimpeng.2008.09.009

International Journal of Impact Engineering xxx (2008)1–9

The experiments of the double stiffened cylindrical shells sub-jected to underwater explosions are simulated by the software ABAQUS.In this paper,the results are analyzed from the following two viewpoints:

(1)The time-histories of kinetic energy and elastic strain energy of

the whole structure,which re?ect the whole structural response.In ABAQUS,kinetic energy E k and elastic strain energy E E are de?ned as follows:

E K?Z

r*v$*v dV E E?

Z t

e1àd tTs u

_3el

A d s

where r is material density;n!is velocity vector;d t is material damage parameter,which is0in this paper when it is not involved in;s u is stress tensor of undamaged material;_3el is elastic strain rate.

(2)The shock spectrum analysis(Fig.2).Shock spectrum is a plot

of an analysis of a motion(transient motions due to explosions,earthquakes,package drops,railroad car bumping,vehicle collisions,etc.)that calculates the maximum response of many different frequencies damped single degree of free single degree of freedom(SDOFs).Based on the varieties of the horizontal coordinate,shock spectrum can be divided into several species:absolute acceleration spectrum,equivalent acceleration spectrum,equivalent velocity spectrum,etc. Equivalent velocity spectrum is applied in this paper.As is shown in Fig.3,horizontal coordinate is the frequency F(Hz)of the oscillator,and vertical coordinate is equivalent velocity,the product of relative displacement d and circle frequency u of the oscillator[3],where the relative displacement means the amplitude of vibration with frequency u.The equivalent velocity V exactly means peak relative displacement d,multi-plied by the natural frequency u in radianse

?????????

k=m

T,as shown in the following equation,

Fig.1.Structure chart of double stiffened cylindrical shells(unit:mm).a.Cross section of cabin1.b.Cross section of cabin2.c.Median longitudinal section of structured.d. Geometry model.

Table1

Fig.2.Sketch map of shock spectrum calculation.

X.-liang Yao et al./International Journal of Impact Engineering xxx(2008)1–9

V?ud

The shock spectrum algorithm?nds the peak relative displacement for a base excited SDOF.That’s the maximum energy stored in the elastic member during the transient event.

Since k/m?u2,consider the elastic energy stored in the spring:

U?k d2 2

Thus,

U m ?

k d2

eudT2

0ud?

???????

where ud is the equivalent velocity.Equivalent velocity is the square root of twice the peak energy per unit mass that is stored in the oscillator during the shock.

There are536nodes uniformly selected in the structure as is shown in Fig.1.For all the explosive locations designed along the negative direction of Y,the acceleration alongàY of each node is used to calculate shock spectrum.The structure impact response is analyzed by the average spectrum velocity which is the arithmet-ical average of all nodes.The equation of average spectrum velocity is:V?

P n

V i,where V is the average spectrum velocity,and V i is the velocity of the i th node.

In addition,International System of Units is applied to all the tables in the following text.

3.Shock factors of underwater explosion

To succinctly describe the ship shock environment,shock factors composed of charge weight,standoff distance and charge location should be considered.Hence,the shock factor C was de?ned. According to one same ship,the shock responses are regarded approximately equivalent if the shock factors of underwater explosions are equivalent.At present,two kinds of shock factors are in common use,explained hereinafter separately.

3.1.Shock factor based on hyper pressure of shock wave

The shock factor is given by:

C1?

??????

=R(2)

where C1is shock factor,W is charge weight(TNT),and R is the

shortest distance from the detonation points to the ship hull.C1is

keel shock factor if R is the distance between detonation to the keel,

and C1is gunwale shock factor if R is the shortest distance from

detonation to gunwale.The de?nition sketch is shown in Fig.4.

However,there is no much difference between the two kinds of C1

in far?eld explosions.

According to Cole’s empirical formula,the hyper pressure of

shock wave in free?eld is given by:

P m?53:3

?????

p!1:13

?53:3C1:13

(3)

If the shock factors C1of two different cases are equivalent,the

hyper pressure P m of the incident points observed is equal.This kind

of shock factor used to verify the survivability of submarine struc-

ture in NITO and pre-USSR[4].Though the shock factor is early used,

there are apparent de?ciencies in application.For the peak pressure

produced in the near?eld with less weight of charge may be higher

than that in the farther?eld explosions with lager weight of charge,

while the effects on the ship structure may not be so great.

In order to verify the validity of C1,several cases listed in Table2

are designed according to the de?nition of C1.

The explosive location of the cases shown in Table2located just

below the cylinder,where X?0,Z?0,R?Yàr,r?4.6m,R is the

standoff distance,r is radius of cylindrical of outer shell.The time-

histories of structural total kinetic energy and potential energy are

shown in Fig.5.

There are only three cases labeled in Fig.5,while the curves below

in sequence are case4,case3,case2and case1.With the equivalent

magnitude of C1in all cases,the structural total kinetic energy and

total potential energy signi?cantly increase with the increment of the

standoff distance.The maximum kinetic energy in case7is about20

times of that in case1,and the maximum potential energy is27times

of that in case1.The results indicate that the shock factor C1is not an

ideal index to re?ect the equivalency of structural explosive envi-

ronment induced by underwater explosion.

Fig.6shows the average spectrum velocity alongàY in veri?-

cation cases designed by C1.

While the magnitude of shock factor C1is constant,the shock

spectrum velocity of inner structure,inner shells and outer shells

dramatically increase with the increment of the standoff distance.

The maximum spectrum velocity of inner structure in case7is5.5

times of that in case1,4.5times of that in inner shells,and6times

of that in outer shells.There is similarity and smooth tendency in all

shapes of curves with the deduction of the standoff distance.

3.2.The shock factor based on plane wave assumption

The assumptions used in derivation procedure are shown as

following:

Fig.3.Equivalent velocity

spectrum.

Fig.4.De?nition of two different standoff distances.

X.-liang Yao et al./International Journal of Impact Engineering xxx(2008)1–93

1)Generally speaking,diffracted wave pressure acting on the

opposite surface of the structure is the secondary factor in explosive procedure [5],so the diffractiveness of shock wave has not been taken into consideration.

2)Though energy dissipation of shock wave is most severe around detonation in water [6],it becomes much slower at the distance outer than 25times of charge radius.Because the standoff distances of all the cases involved in this paper are more than 25times of charge radius,the research takes no account on energy dissipation.This shock factor is de?ned as:

C 2?

??????W p =R

(4)

The symbols in the equation are the same as that mentioned before.In fact,this de?nition is based on plane wave assumption and considered the equivalency of shock wave after structure shelter.De?ne r e is chemical energy per unit mass stored in charge (for TNT r e is 1060c/g),h e is the rate of conversion from chemical energy to shock wave energy (28%at the distance of 25times of charge radius).Hence the energy contained in shock wave during an underwater explosion is given by [7]

E ?W r e h e

(5)

In in?nite ?uid ?eld,the shock wave energy is uniformly distributed on the wave surface of the whole spherical blast wave.Assuming ratio of standoff distance R to structure characteristic parameter L is big enough,shock wave can be considered as plane wave for structure.The shock wave energy sheltered by structures is

E s ?E

S e 4p R (6)

where E is the total energy of shock wave,E s is shock wave energy after structure shelter,and S e is the structure shadow area on the plane front.Substituting Eqs.(5)and (4)into Eq.(6)results in

E s ?

r e h e S e C 224p

(7)

Assuming blast wave is plane wave,S e is constant on condition that explosive orientation is invariant and standoff distance is variable.The right side of Eq.(7)is constant except C 2,hence E s is proportional to C 2.

From equation (7)with the assumption of plane wave,the blast wave energy sheltered by structure at the same explosive orien-tation keep invariant as long as shock factor C 2is constant.Thus C 2is the shock factor based on plane wave assumption.

In order to verify the validity of C 2,a series of cases designed by C 2are listed in Table 3.

The detonation centers of cases in Table 3locate just below the cylinder,X ?0,Z ?0,R ?Y àr ,r ?4.6m,where R is the standoff distance and r is radius of cylinder of outer shell.The time-histories of the structural total kinetic energy and total potential energy are shown in Fig.7.

There are only three cases labeled in Fig.7,in which curves below in sequence are case 11,case 10,case 9and case 8.Though the shock factors corresponding to the curves in Fig.7keep invariant,they still discrete severely,and the structural total kinetic energy and total potential energy also increase dramatically with the increment of standoff distance.The maximum kinetic energy in case 14is 3.5times of that in case 11,and maximum potential energy is 4times.It is obvious that shock factor C 2is better than C 1by comparison,but its precision is not enough.

Fig.8shows the average spectrum velocity along àY in veri?-cation cases designed by C 2.

From Fig.8,it’s observed that on condition that shock factor C 2is equivalent in all cases,the shock spectrum velocity of inner structure,inner shells and outer shells signi?cantly increase with the increasing of the standoff distance.The maximum spectrum velocity of inner structure in case 14is 2times of that in case 8,1.8times of inner shells,and 2times of outer shells.Within the range of medium and low frequency below 400Hz,the curves discrete more severely,and the effect of shock factor C 2is worse;while within the range of high and medium frequency above 400Hz,the curves discrete less severely,and the shock factor C 2is more precise.

4.Equivalency of the spherical shock wave formed in underwater explosion

In this paper shock wave is treated as spherical wave,and the equivalency of underwater explosive loading is still studied from the viewpoint of shock wave energy after structure shelter.First of all,random curved facing with general meaning was deduced,and then the model in Section 2was

calculated.

Table 2

Fig.5.Time-history curve of total structural energy of veri?cation case C 1.a.Time-history curves of total structure kinetic energy,b.Time-history curves of total structural potential energy.

X.-liang Yao et al./International Journal of Impact Engineering xxx (2008)1–9

Assuming there is a random continuous curved face in space,slicing and smooth,no random ray from origin of coordinate will cross this curved surface twice,which represents the structural surface directly subjected to the shock wave sent out from origin of coordinate.In spherical coordinates,dS is area in?nitesimal,and the distance from a random point on the in?nitesimal to origin of coordinate (detonation position)is de?ned as local radius,R 0.Hence the projected area of dS onto spherical surface with radius R 0which is called dS R 0is

dS R 0?dS cos g

(8)

where g is the included angle between normal vector n !

and radius

vector m !

.By the de?nition of in?nitesimal area in spherical coordinates

dS R 0?R 0d 4R 0sin 4d q ?R 02

sin 4d 4d q

(9)

From the geometrical relationship,the projected area dS R of dS on the spherical surface of a random radius is

dS R ?R 2sin 4d 4d q

(10)

Thus

dS R 0

dS R

? R 0R 2

0dS R ?dS R 0

2(11)

The shelter ratio of shock wave sheltered by structure h is de?ned by

h ?

R R

dS R

4p R ?

R R

R 2R 02

R 02sin 4d 4d q 4p R ?1p

Z Z

sin 4d 4d q

(12)

It can be found in the expression of h

1)The structure shelter ratio for shock wave h is independent of the selection of projected spherical surface,namely radius-independent.Accordingly R is regarded as standoff distance in the deducing procedure in the following text.

2)For the different spatial curved face,only if the limits of inte-gration of 4and q are invariant,h keeps constant and inde-pendent of the concrete shape of spatial curved face.Apply Eq.(12)to calculate the double cylindrical shells structure showed in Section 2,and analyze the detonation position just below the cylinder in the interest of calculation convenience.The origin of coordinate locates at the detonation position.Figs.9and 10show coordinate system and the geometry sketch of derivation.The structure shelter ratio for shock wave is given by

h ?

Z 3p =2tb 3p =2àb

Z p =2ta

p =2àa

sin 4d 4d q ?

b sin a p

(13)

The range of integration of q at the head of the cylinder should be variable,and it is approximately treated as constant in order to simplify the derivation.Because the cylinder is slimness,this approximation won’t generate too much error.

Substituting the expressions of a and b which are easily obtained from Fig.10into Eq.(13)result

515253

Frequency(Hz)

A v e r a g e S p e c t r u m V e l o c i t y (m /s )

6m0. 24kg 9m0. 81kg 12m1. 92kg 15m3. 75kg

18m6. 48kg

24m15. 36kg

30m30kg

0515253

Frequency(Hz)

6m0. 24kg 9m0. 81kg 12m1. 92kg 15m3. 75kg

18m6. 48kg

24m15. 36kg

30m30kg

012345

Frequency(Hz)

9m0. 81kg 15m3. 75kg

18m6. 48kg

24m15. 36kg

30m30kg

Fig.6.Average shock spectrums of veri?cation case C 1.a.Average shock spectrums of inner structure.b.Average shock spectrums of inner shells.c.Average shock spectrums of outer

shells.

Table 3

X.-liang Yao et al./International Journal of Impact Engineering xxx (2008)1–9

h ?1

p arcsin r r tR sin

arctg

l 2R ?1p l ??????????????????4R 2tl 2p arcsin r

r tR

(14)

The shock wave energy after structure shelter is given by

E s ?W r e h e h ?

W r e h e

l ??????????????????4R 2tl 2

p arcsin

r r tR (15)

Substituting the de?nition of the shock factor C 2into Eq.(15)

results in

E s ?KR 2l ??????????????????4R 2tl 2

p arcsin

r tR (16)

where K ?C 22r e h

e p ,and when shock factor C 2is de?nite,K is

constant,E s is a single-variable function related to R .

The projected area of the structure on the front wave S e is:

S e ?4p R 2h ?4R 2l ??????????????????4R 2tl 2

p arcsin

r tR (17)

The shock wave energy sheltered by structure is:

Frequency(Hz)

A v e r a g e S p e c t r u m V e l o c i t y (m /s )

6m1. 2kg 9m2. 7kg

12m4. 8kg 15m7. 5kg

18m10. 8kg

24m19. 2kg

30m30kg

9m2. 7kg 12m4. 8kg 15m7. 5kg

18m10. 8kg 24m19. 2kg

30m30kg

Frequency(Hz)

9m2. 7kg 15m7. 5kg

18m10. 8kg 24m19. 2kg

30m30kg

A v e r a g e S p e c t r u m V e l o c i t y (m /s )

Fig.8.Average shock spectrums of veri?cation case C 2.a.Average shock spectrums of inner structure.b.Average shock spectrums of inner shells.c.Average shock spectrums of outer shells.

Fig.7.Time-history curve of total structural energy of veri?cation case C 2.a.Time-history curves of total structure kinetic energy,a.Time-history curves of total structure potential energy.

X.-liang Yao et al./International Journal of Impact Engineering xxx (2008)1–9

E s?W r e h e h?W r e h e

??????????????????

4R2tl2

p arcsin r(18)

Substitute parameters of the double cylindrical shells into the two equations above r?4.6m,l?18m,yields the graph of a rela-tion about E s/K and dimensionless standoff distance R/l shown in Fig.11.It is found in Fig.11that notwithstanding the equality of shock factor C2,the gap of the shock wave energy sheltered by structure are great with the standoff distance less than double of the cylinder length.The shorter of the standoff distance is the greater of the gap is,and the gap is zero when the standoff distance equals to zero.This is the reason why the value of the structural response mentioned in Section3.2is less in near?eld explosion.For plane wave,j trends to the extreme rl/2with the standoff distance greater than double of the cylinder length.

In the?nal analysis,the de?ciency of the shock factor C2is due to the neglect of the spherical blast characteristics.Obviously they should be considered especially in near?eld explosions,hence C2is not suitable for applying to describe the underwater explosion loading in near?eld https://www.wendangku.net/doc/0711653211.html,ually the length range of surface ship and submarine is about tens to hundreds meters.However experience indicates that the standoff distance within zero to one hundred or two hundreds meters possesses practical signi?cance, thus shock wave has to be treated as spherical wave in the research on underwater explosive problems of real ships.For example, assuming a submarine is100m long and its radius is6m,it can be dealt with as a cylindrical structure ignoring its detail structure. Substituting the parameters into Eq.(16),Fig.12will be got after calculation.It is obvious in the?gure that shock wave should be regarded as spherical wave if the standoff distance is within200m.5.Similarity parameters of shock wave

It can be seen from Eq.(7)that when the blast wave treated as plane wave,the projected area of ship hull normal to the propa-gation direction of plane wave S e is constant whatever the standoff distance changes.Here,the blast wave energy sheltered by struc-ture E s keeps invariant as long as shock factor C2is equal.

However,the blast wave is spherical wave in practical situation. For random spherical wave,the projected area S e of ship hull normal to the propagation direction of plane wave changes with the charge position.From equation(7)we can?nd that the shock wave energy E s sheltered by ship hull cannot be predicatively invariant only with C2keeping invariant.When S e C22keeps the same,the shock wave energy sheltered by ship hull E s can be invariant.Therefore,a new kind shock factor is de?ned in this paper,which is spherical wave shock factor C3:

C23?S e C22(19) Thus

C3?

?????

S e

C2(20) C3?

?????

S e

C2?

???????????????????????????????????????????????????

??????????????????

4R2tl2

p arcsin r rtR

(21)

It is found in Eq.(19)that C3is the modi?cation of C2.

Fig.9.Sketch map of derivation by formula.

Detonation Center Detonation Center Fig.10.Geometry sketch of derivation by formula.

024681012

R / l

Fig.11.Relationship between standoff distance and shock energy which acted on the cylinder.

00.51 1.52 2.53 3.54

100

150

200

250

300

R / l

Fig.12.Relationship between standoff distance and shock energy which acted on the submarine.

X.-liang Yao et al./International Journal of Impact Engineering xxx(2008)1–97

In order to verify the availability of C 3,the veri?cation cases based on the de?nition of C 3of equal shock factor are shown in Table 4.

The detonation centers of cases listed in Table 4locate just below the cylinder,X ?0,Z ?0,R ?Y àr and r ?4.6m,where R is the standoff distance and r is radius of cylinder of outer shell.The time-histories of the structural total kinetic energy and total potential energy are shown in Fig.13

As shown in Fig.13,the time-history of the structural total kinetic energy of every case is close to each other,while the difference of the structural total potential energy is slightly greater.In all the cases,the maximum structural potential energy is just 1.3times of the minimum.

Fig.14shows the average spectrum velocity along Y -axis of all the cases in Table 4.

As shown in Fig.14,while the shock factor C 3is equivalent,the average shock spectrums of inner structures of every case tally well,and only the shapes of curves have slight difference;the average shock spectrum of inner shells and outer shells similarly tally well with the shape of curves identical.Therefore,it is considered that C 3preferably re?ects the equivalency of underwater explosive loading,that is means,the average structure shock response of different cases will be considered to be the same if C 3is equivalent.6.Discussions on shock factor

As an underwater explosion similarity parameter,the shock factor should have two effects.

1)For an ascertain model,the shock factor should describe the shock environment accurately.No matter in what cases,the impulsive response of structure will be approximately the same,if the shock factor is equivalent.If the shock factor is great,the impulsive response will be great,the shock factor small,the impulsive response small.The shock factor satisfying this condition quite contributes to the numerical analysis and experiment study on the ship impulsive environment.It could wholly realize the impulsive response characteristics on ship structure requiring fewer cases,remarkably reducing the cost of

calculation.For trials,there is a great fatalness that the local structure may be distorted or damaged.In order to reduce this risk,we can put the charge in a safety distance to carry through the experiment and then to extend the results with explosive factor.The shock factor C 3de?ned in this paper can be used in the aspects mentioned above.But in this paper,a double stiffened cylindrical shell is discussed and the detonation position is just below the center of cylinder,so the appliance on ships needs a great deal of subsequent work.

2)The comparison and conversion between different structural models or between the real structure and reduction model.It is worthy to notice that the former shock factor C 1,C 2merely include information of charge weight and standoff distance,which are dif?cult to use in the comparison and conversion between different structure models or the real structure and reduction model.The shock factor C 3includes the information of structure dimensions consisted the parameters mentioned above,so it may be used in all the aspects mentioned above.However,the only solved problem by C 3is about load similarity.Besides,the similarity of structural dynamic problems needs to be solved,which will not be discussed deeply in this paper.The shock factor C 3is a reasonable index to show the impact criteria for vessels subjected to underwater explosion in near ?eld if the overall damage strength for hull girder is taken into consider-ation.Nevertheless,if the local structure strength for vessels is taken as the subject investigated,there may be some limitations for all the shock factors discussed above,including the shock factor C 3,for all of the shock factors are deduced in the viewpoint of overall response of vessels.Restated,the shock factor C 3is a more reasonable index to describe the whole vessels than C 1and C 2.7.Popularization of the relative formulas

It should be pointed out that the geometry model used in deducing formulas (13)and (16)is based on the hypothesis that the explosive position just below the center of the cylinder.In general case,assuming the subtract along axis-coordinate of explosive position and the end of cylinder is x (shown in Fig.10),the formula (14)should be changed as:

h ?

arcsin

r 0B @x ????????????????R 2tx 2p tl àx ???????????????????????????R 2tel àx T2

q 1

C A (22)

The rest also need change accordingly,but here we won’t

deduce

them.

Table 4

00.020.040.060.08

0246810x 104

Time S t r u c t u r e T o t a l K i n e t i c E n e r g y

024*********

Time

S t r u c t u r e T o t a l P o t e n t i a l E n e r g y

Fig.13.Time-history curve of total structural energy of veri?cation case C 3.a.Time-history curves of total structure kinetic energy.b.Time-history curves of total structure potential energy.

X.-liang Yao et al./International Journal of Impact Engineering xxx (2008)1–9

As in Section 4,formula (21)is also based on the assumption of just-below explosion,while in general case,the expression of C 3is given by

C 3???????????????????????????????????????????????????????????????????????????????????????????????????????W arcsin r r tR 0B @x lr ????????????????R 2tx 2p tl àx lr ???????????????????????????R 2tel àx T2

q 1C A v u u u u t (23)

8.Conclusions

In this study,a detailed analysis was performed of three shock factors.A new shock factor based on energy acting on the structure is presented to describe the loading of underwater explosion.This analysis also includes a series of numerical experiments designed by different shock factors.

Through the numerical results,it is indicated that the shock factor C 1based on the de?nition of hyper pressure of shock wave is dif?cult to re?ect the explosive environment of cylindrical shells correctly.When the shock factor is equivalent,the shock response becomes greater as the standoff distance increases.The shock factor C 2,de?ned from the perspective of structural shelter on the assumption of plane wave,can re?ect the similarity of the far ?eld underwater explosion.However,it can’t describe the shock envi-ronment in near ?eld explosion in that the plane wave assumption is not valid in that case.

The new shock factor C 3,de?ned from the angle of energy shel-tered by structure on the assumption of spherical shock blast,can

preferably re?ect the underwater explosive loading and can be used as similarity parameters in the analysis of the explosive environment.When C 3is equivalent,the time-history curves of the structural total kinetic energy and total potential energy tally well,and the structural response of the average shock spectrum is generally identical.The new shock factor can make up for such limitations.

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underwater explosion.International Journal of Impact Engineering

2005;31:151–68.

00.511.522.53Frequency (Hz)

A v e r a g e s p e c t r u m v e l o c i t y (m /s )

Frequency (Hz)Frequency (Hz)

6m3. 1kg 9m4. 7kg 12m6. 8kg 15m9. 4kg

18m12. 5kg 24m21. 7kg 30m30kg

6m3. 1kg 9m4. 7kg 12m6. 8kg 15m9. 4kg

18m12. 5kg

24m21. 7kg

30m30kg

6m3. 1kg 9m4. 7kg 12m6. 8kg 15m9. 4kg

24m21. 7kg 30m30kg

Fig.14.Average shock spectrums of veri?cation case C 3.a.Average shock spectrums of inner structure.b.Average shock spectrums of inner shells.c.Average shock spectrums of outer shells.

X.-liang Yao et al./International Journal of Impact Engineering xxx (2008)1–9