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Dynamic buckling of stiffened cylindrical shells of revolution under a transient lateral pressure shock wave
r
Thin- Walled Structures 23 (1995) 85-105 Elsevier Science Limited Printed in Great Britain 0263-8231/95/$9.50
ELSEVIER
0263-8231(95)00006-2
Dynamic Buckling of Stiffened Cylindrical Shells of Revolution under a Transient Lateral Pressure Shock Wave
Christophe P6dron DCN Ing6nierie, 8 Bd Victor, 75732 Paris, France
& Alain Combescure CEA-Centre d'Etudes Nucl6aires de Saclay, 91191 Gif sur Yvette, France
ABSTRACT A modal method of analysis is used to determine the response of an infinitely long stiffened cylindrical shell of revolution to a transient lateral pressure produced by an underwater explosion and propagating in an acoustic fluid. The shell is initially immersed, hence prestressed by the external hydrostatic pressure. A theory of dynamic buckling is then developed for cylindrical shells subjected to transverse pressure pulses of different durations.
NOTATION
bo, b~ C E, Eo, Ex eo, e x
G, Go, Gx
Axial and circumferential stiffener spacing Sound velocity in the fluid Young's moduli o f the shell, the circumferential and the axial stiffeners Stiffener eccentricities Internal force vector Shear moduli 85
86
h
C. POdron, A. Combescure
Thickness of the shell 1 h2 12 R 2' bending rigidity of the shell
Io, lx
Moments of inertia of stiffeners about the middle surface of the shell Torsional constant of stiffeners about the middle surface JO, Jx of the shell Elastoplastic stiffness matrix [Kep(u)] Pressure stiffness matrix, due to P(t) [gp(t)] [Kp] Pressure stiffness matrix, due to PI Tangential stiffness matrix of the fundamental state [Ktg (t)] Stress stiffness matrix, due to a (u) Prestress stiffness matrix, due to a~ [K,,] Moo, Mxx, Mox Internal moments Mass matrix Noo, Nxx, Nox Internal resultants Cavitation pressure Pcav Peak pressure PMAX Critical static pressure Pstat Pl Hydrostatic pressure {P(t)} Variable pressure vector R Radius of the shell r Radial coordinate Cross-sectional area of stiffeners SO, Sx [S(u)] Area matrix T Time constant t Time Components of the displacement U of the middle Uo, Ux, U3 surface of the shell in cylindrical coordinates (u} Displacement vector of discrete variables of fundamental state {8} Acceleration vector x Axial coordinate F,O0, 8xx~ 130x
(zxu)
Membrane strains Membrane strain vector Linear membrane strain vector Quadratic membrane strain vector Displacement vector of discrete variables of perturbed motion
Dynamic buckling of cylindrical shells
0 /.z Po Ps
O" 1
ZOO,Zxx, Xox
{x}
(),x (..)
87
Perturbed acceleration vector Circumferential coordinate Poisson's ratio Fluid density Density of cylindrical shell Stress tensor between the initial prestressed configuration and the fundamental state Uniform stress, due to PI Bending strains Linear bending strain vector
o(
--, Ox 02 0 t 2(
)
partial differentiation with respect to x ), partial differentiation with respect to t
1 INTRODUCTION Although several studies exist in the literature on the response of submerged shells to a transverse shock wave, very few deal with the elastoplastic response and dynamic stability of stiffened cylinders. In this paper, the elastoplastic response of an infinitely long cylinder to a shock wave is investigated. The stiffened shell is initially immersed in an acoustic fluid and a shock wave, assumed to be plane, encounters the structure. The wave front is parallel to the axis of the shell and decays exponentially with time as follows:
P(t) = PMAXexp ( - t / T ) The structure in motion is then subjected to a dynamic pressure that is the sum of an incident plane wave and a radiated pressure created by the vibrating shell. If the incident shock wave is known exactly, the radiated pressure is determined by an approximate method. A modal method is used to determine the response of the structure, based on a decomposition in Fourier series of the displacement and pressure fields. The response of each mode can be analysed independently. The dynamic stability of the fundamental motion is examined by a perturbation method. The problem is to determine the incident dynamic loads that cause a perturbed bending mode to increase sufficiently to exceed a buckling criterion. The aim of this study is to check that a method, developed by Anderson and Lindberg,~ is well adapted for stif-
C. P$dron,A. Combescure
88
fened cylinders subjected to transverse pressure pulses, at least in elasticity, and for quasi-static and quasi-impulsive durations.
2 A STIFFENED CYLINDRICAL SHELL THEORY This study is limited to the case of stiffened cylindrical shells of revolution (Fig. 1). The shell theory is based on Kirchhoff-Love assumptions. For such a shell, the Green-Lagrange strain tenso~for a point at distance 03 from the middle surface in the displacement U is defined, in cylindrical coordinates, by: ~00 = -g1 (Uo, o + U3) + ~ 1
[(Uoo_~_U3)27L , Ux,2 o + (U3,0 + Uo) 2]
03 + - ~ (Uo, o + U3,oo) 1
2
03
~xx = Ux, x + ~ [V~o,x + Ux, x + U~,x] - - f i U~,xx ~ox = ~ - -
+2
+ Uo, x
(1)
)
l [Uo,o+ U3 Uo,x..l__~ Uxx..~(U3,0_ Uo)U3,0] R
--03(~ "----~x- 2U3,oo)
., Ux
/U0
U3 Fig. 1. Cylindrical shell of revolution.
89
Dynamic buckling of cylindrical shells
or 1
{13} = 031 } -+- ~ {/3q} q'-03{Z}
(2)
A Timoshenko beam theory is considered for the axial and circumferential stiffeners (Fig. 2). Moreover, stiffeners are located at regular intervals and their cross-sectional dimensions are small with respect to the radius of the shell. The rotation of a stiffener centroid is equal to the rotation of the middle surface of the shell. The displacement fields are continuous at each shell-stiffener interfacefl By using the virtual internal strain energy, the elastic stress resultants-strain relationship is obtained by integration over the thickness of the shell and is given by: 2
{S}-- [I, l Iol where
[B]
H + ExSx bx
#H
0
pH
Eo So H+ -bo
0
0
0
Gx S~ + Go So Gh+~
=
E~ S~
ex bx [K] =
0
0
0
Eo So eo bo
0
o
o
Gx Sx + eo Go So
ex b---£-x
ExI~ D+--
#D
0
pD
Eo Io D+-bo
0
bx
[ O ] --
0 and H-
Eh 1 -
--fro
lfl'
D _
Eh 3 12 (1 - p2)
D~-~+G"J~
"~+b~-oG°J°
90
C. P6dron, A. Combescure
.Y
eo~~
Fig. 2. Shell and stiffeners.
Nevertheless, local instabilities (Fig. 3) are excluded by this approach. The plastic flow of such a stiffened shell is treated by a multilayer approach. The stiffened shell is considered as a three layer material, one layer for the shell and two layers for the stiffeners (axial and circumferential). Perfect bonding at each interface is assumed; therefore, the displacement fields are continuous across each interface as in the elastic theory. The incremental plastic stress-strain relationship follows that of the classical simple flow theory of plasticity, Von Mises J2 yield surface and isotropic strain hardening. 3
3 EQUATIONS OF MOTION Initially, the shell is immersed in an acoustic fluid, hence subjected to the external hydrostatic pressure P~ in the fluid. The cylinder is subjected to a uniform prestress at due to the pressure Px.
\ Stiffener local instability
/
Local shell buckling
Fig. 3. Excluded instabilities.
Dynamic buckling of cylindrical shells
91
A lateral pressure shock wave, travelling in the fluid, then acts on the shell. The dynamic pressure on the shell, P(t), is the sum of an incident pressure plane wave and a radiated pressure created by the vibrating shell. This motion is called the fundamental motion defined by the displacement U of the middle surface of the shell. The stability of the fundamental solution is investigated by determining the response of an mfimteslmal perturbation A U with respect to the fundamental state (Fig. 4). By using a spatial discretization, the fundamental motion may be expressed in the discrete form: [M] {U} = -{Fint (U)} -- [K,, - Ke, ]{U} + [S(U)] {P(t)}
(4)
Considering a perturbed solution U + A T in (4), the perturbed motion, with respect to the fundamental state, is given by: [M] {A U} = - [ g e p (V) + K., + Ko(v) - Kp, - K.(,) ] {A U}
(5)
or [M] { A ~)} + [Ktg (t)] { A U} :
{0}
In the case of an infinitely long cylinder subjected to a transverse pressure pulse, the deflections Uo, Ux, U3 may be expressed in the form (Fig. 5): N
Uo (0, t) = ~
u. (t) sin nO
n=[ .-..+
U(0, t) =
(6)
Ux(0, t) = 0 N
U3 (0, t) ---- Wo(t) + Z
wn(t) cos nO
n=l
where N must be greater than the buckling mode.
U Initial configuration
Fundamental motion
Perturbed motion
Fig, 4. Initial, fundamental and perturbed motions.
C. P$dron. .4. Combescure
92 Z
I_
h
Incident pressure
~
U3
Fig. 5. Shock configuration. The incident pressure pulse wave pINC and the radiated pressure pRAD are also decomposed in Fourier series (Fig. 5): n
pInNC(r, t) cos nO
p1NC (0, r, t) = ~ n=0
pRAD (0, r, t) = ~
(7) nRAD/..n"tr, t) COS nO
n=0
For example, with small deformations and small displacement assumptions, the fundamental motion is governed by the following discrete equations: for n -- 0:
PsRhfvo(t)+
Eh
(1 --/~2)R
Wo(t) = R PIoNC(R, t) + R PoRAD (R, t)
f o r n >~1:
PsRh R - - (71 -'~
fvn nn2
rn2+,, n+n3,s]{Un}
(1 --~2) R Un+n3Is wu,n
= R
1 +n4Is
Wn
PlnNc (R, t) + pRAD (R, t)
(8)
Dynamic buckling of cylindrical shells
93
One obtains (2N + 1) equations of motion even though there are (3N + 2) unknowns, i.e. (2N + 1) displacement unknowns and (N + 1) radiated pressure unknowns. Indeed, if the decomposition in Fourier series o f P INc (0, R, t) is easy, the determination o f P RAD(0, R, t) needs special treatment.
4 INCIDENT PLANE WAVE The incident plane wave, travelling in the acoustic fluid, may be expressed as (Fig. 5): 4
pINC(O'r't)=PMAxeXp( R - Ct-rc°sO)
Ct- rcosO)
(lO)
where H is the Heaviside function defined by:
H(R-Ct-rcosO)= { 0l iif(R-Ct-rcosO)~O} f(R Ct rcosO)
(11)
Given that pINC(0, r, t) is decomposed in Fourier series as: N
piNC(0, r, t) = Z
PnINC (r. t) cos
nO
(12)
n=0
it follows that:
p~C(r,t)=Ii'PMAxexp(.RcTC_()exp(
rcos 0'~ •) cos
nOdO
(13)
with { 0t
=
}
(R~fCt_)R+r if t ~<
a cos
rt if t >
(14)
R+r C
In other terms: i
,,',INC," 2 r . tr, t) = - -
e,
PMAX
(.R ~ _C('~ i o , ( r ~ exp
k, CT J "\ CT/
(15)
with e0 = 2 and e, = 1 for n ~> 1 and where
r =~lIi' exp ( rcos0'~ 1o,(_~) --C--f .j cos nO dO
(16)
94
C. P~dron, A. Combescure
is the incomplete modified Bessel function of the first order, and becomes the complete function In(-r/CT) when 0t = ~r.4 Given that the radial component of the incident wave particle velocity is: wiNC(o, r, t) - -
PINC(o,r, t) c o s O PoC
(17)
then, for n = 0
PMAXexp(R C ZcCt I°2'(---~T)
W-oINC/n (o,r,t)-- - -
and for n/> l
¢v~¢(r,t)_PM^xexp(RcCt)(~t+ C ,
_ i n _ , ( -C--T))r
This radial velocity is useful; it will be used for the fluid-structure interaction equations.
5 RADIATED PRESSURE In an acoustic fluid, in cylindrical coordinates, the radiated fluid pressure PaAD(0, r, t) and the radiated particle velocity ffRAD(0, r, t) created by the vibrating shell are defined by the velocity potential ~(0, r, t): pRAD : po ~
(18)
wRAD = O~
Or
(19)
• (0, r, t) is expanded in the form: N
• (0, r, t) = E
~,(r, t) cos nO
(20)
n=0
and satisfies the hydrodynamic equation: 1
02~ (21)
A ~ -- C 2 0t 2
Using (20) and (21), it follows that 02~n
l°~I~n
Or--T 4 rOr
(n) 2 1 02~n -r (I~n -- ( 2 ~-~
(22)
Dynamic buckling of cylindrical shells
95
We now introduce the Laplace transform: exp (-st) ~.(r, t) dt
4~.(r, s) =
(23)
Using the Laplace transform, (22) becomes
~Or + r Or
~. =
q~.
(24)
A solution of this Bessel equation, appropriate to a transient diverging wave, is: s' 6
where K. is the modified Bessel function of the second kind. Hence,
Or
(r, s) = F. (s) s K' ( sr C "\-CI
(26)
and (25) and (26) lead to:
Ol d P n ( rd' s ) =Cs K r ~K. ' n ( c ) lOd pr ' ( r ' sr) = f * ( c )
r .(r,s)
(27,
where f * is defined by
f.* x
K'.
x--K-2(x)
with the following properties: 7
gn E N, l i m x ~ o f * ( x ) = 0 f*(x) = -x - ~ + 0
for x ~ ~
(28)
Vn E N*, d f * ( x = O) = n dx The problem to be solved is to find an approximate function g* o f f * , with the same properties, and which presents no difficulty when determining the inverse Laplace transform. The adopted approximations are: for n = 0: 1
g~(x)=-x-~-F
a2 x2+a2 witha=10
C. P~dron,A. Combescure
96 and for n >t 1: g* (x) : - x
1 2
1 (2n- 1)2 2(x+2n-1) 2
(29)
One then obtains by the inverse Laplace transform: for n = O: 0~o.. 1 • 1 [t CR . ( Ct (t Or ~ C ~ ° - 2 r r ~ ° + J o / - - 2 r 2sIn r - u )
) ~o(r,u) du
f o r n >~ l" Or
C
+
I~o(2n-1)3C2(u-t) 2r 2
exp
((2n-1)C(u-t))g2.(r,u)d u r
(30)
Therefore, one obtains, for each mode, a rather simple relation between the radiated particle velocity and the velocity potential.
6 FLUID-STRUCTURE INTERACTION Up to now, only ( 2 N + 1) discrete equations have been established, even though there are ( 3 N + 2) unknowns. The (N + 1) missing equations to solve the problem completely are obtained from the fluid-structure interaction equations: the radial velocity in the fluid is continuous, which means, to the first order (r = R), in each mode: fi~n(t) : w.. nINC/r~ ..RAD/n 0(I)n (R,t) ~..'x,t)+wn ~lX,t) = W.. ,INC/n t,,x,t)+--~r
(31)
The problem is now entirely determined with these (N + 1) extra equations combined with (29). This method is very similar to the residual potential method developed by Geers. 5 Moreover, a simple model of cavitation is introduced in the time integration algorithm: as soon as p~NC + pRAD ~< Pc'~v, pRAD is then defined by the relation pRAD = e c a v - pINC.
7 TIME I N T E G R A T I O N An explicit scheme, based on the central differences method, is used to compute the time integration.
Dynamicbucklingof cylindricalshells
97
8 C O M P A R I S O N WITH PLEXUS In order to validate the modal method described in this paper, the interaction of an incident lateral pressure pulse with an infinite unstiffened cylindrical shell was studied with the P L E X U S computer code. 8 This code, developed by CEA Saclay (France), is based on the finite element method. The results are examined in the case of an unstiffened elastic cylinder made of aluminium with a radius of 76.2 m m and a ratio R/h equal to 30. The incident pressure is characterized by a peak pressure PMAX-----5MPa and a time constant T = I OR/C. The response versus time of three points A, B, C of the shell (Fig. 6) is observed. Dimensionless time (Ct/R) is plotted on the x-axis and is shifted by 7 R/C. This time shift is caused, in the P L E X U S modelling, by the fluid mesh and represents the time the incident pressure needs to travel from a plane fluid boundary to point A. Very good agreement between the displacement curves (Figs 7a and 7b) was found. The m a x i m u m difference between the two simulations is less than 6%, for point C. The Von Mises stresses also agree well (Figs 8a and 8b). It means that the mode n = 1 and the breathing mode (n = 0) are computed with good accuracy. Nevertheless, the P L E X U S results oscillate more. These high frequency oscillations are caused by the discretization of the fluid by finite elements. The finer the mesh, the higher the frequencies.
9 DYNAMIC BUCKLING The dynamic buckling of the structure is analysed by a perturbation method. Given a solution of the fundamental motion (4), one follows the i
C
._A__ _ -~t)
I
Fig. 6. Locations of the points A, B and C.
Dynamic buckling of cylindrical shells 150 t (a)
99
- - V o n M i s e s Point A . . . . V o n M i s e s Point B - - - V o n M i s e s Point C
100
r~
50
............. 5
L~,
10
15
20
25
D i m e n s i o n l e s s t i m e (CffR)
Fig. 8a. Von Mises stresses (modal method).
150 - ( b ) 3 2 1 1-~......~_ 1 I/ ~X"~ 3
,--. lOO
V
¢.
0.0E+00
"-1
3 2.. 1--~
50
0.OE+0C
v~'~
I 5
2~
~ 1 - - J2 point A --2-- J2 point B - - 3 - - J2 point C
~
I 10
I 15 T i m e (Ct/R)
/3 2 3
I 20
I 25
Fig. 8b. Von Mises stresses (PLEXUS computation). with 2, (t) = R , ( t ) + j C , ( t ) , j2 = _ 1
(33)
2 . ( 0 is a complex eigenvalue of the characteristic equation o f the perturbed motion and {AU0}n, a vector which depends on the initial conditions at t = t0. At each time step t, 2.(t) is determined for each mode. As soon as (to, n) exists for which R,,(t) > 0, the perturbed solution may become unstable because the mode n may increase exponentially with time. At this time to,
C. POdron, A. Combescure
100
if it does exist, one considers an infinitesimal perturbation arbitrarily defined by:
1
{ A U ( t 0 ) } - 1000 { U ( t 0 ) } ::~ Vn,
{ Aun Unwn)) (to) (to) --
1000
(34)
Aw. (to) - , t o , 1000
Then, one examines the evolution, in each mode, of the perturbed radial deflection Awn(t) for t 1> to by solving, for each time step, eqn (4) and then eqn (5). One adopts the following dynamic buckling criterion: the buckling mode n is the first one for which:
Awn (t) >/fl Awn (to)
(35)
where fl is an amplification factor (fl = 100; 1000;...). This approach is very close to that of Anderson and Lindberg.1 According to their theory, the critical load that causes buckling is determined in the following logical way: given a structure and an incident pressure pulse characterized by PMAXand T, the computation of the equations of motion (4) and (5) is carried out until t = 3 T. At this time, the structure has been submitted to 95% of the initial impulse and it is considered that, if the dynamic buckling criterion (35) is not violated before t = 3 T, the structure will stay stable over this time and the computation is stopped. On the other hand, if tcr < 3 T and a mode n exists, such that Awn(tcr)
Awe(t0) the computation is stopped; the considered load is the critical one and n is the buckling mode. The authors then checked that the dynamic buckling theory developed in Ref. 2 is valid for the stiffened structures and lateral pulses considered, at least in the elastic range. The following results concern an infinite cylinder made of aluminium ( E = 70000MPa) with a radius of 76.2 mm. Following Ref. 2, at the critical time tcr, the relative amplification (Aw~(tcr)/Aw~(to)) of each perturbed bending mode is plotted versus n. An example is given in Fig. 9, obtained with R/h = 100, PMAX = 1.4MPa ( e s t a t : 0"018 MPa) and T = IOR/C. Figure 10 shows the buckled shape obtained with this critical load. The critical 'impulse-peak pressure' levels that cause the dynamic buckling are then plotted in a log-log diagram as shown in Fig. 11.
Dynamic buckling of cylindrical shells
101
1200 1000 800 o t~ 600 <
400 200
10 Mode n Fig. 9. A m p l i f i c a t i o n function.
y//
\
/ /
'/'\
7 ~\
'II ,I
'~Xx\
Fig. 10. B u c k l e d shape.
For quasi-static loads (long duration with regard to the shell response time), the dynamic buckling is characterized by the peak pressure. The critical curve, Fig. 1 l, presents an asymptote PMAX = Pstat and the buckling mode is the static one, mode 2 for an infinite cylinder. For quasi-impulsive loads (duration comparable to the shell response time), the dynamic buckling is governed by both the impulse and the peak pressure. The buckling modes are higher than the static one (Fig. 11). These results show very good agreement with the conclusions in Ref. 2. No results are presented for very impulsive loads (short duration) because the plastic flow has to be taken into account and only elastic shells are considered in this paper. The effect of the amplification factor is also examined. Figure 12 shows
102
C. Pddron, A. Combescure
10<>Bucklingmode
.o < 5~32 xe e~
0-
I
I
1
10
Impulse (MPa.ms) Fig. 11. Critical curve. the critical curves obtained with /~ = 100 and 13 = 1000. The m a x i m u m difference of impulse between the 100 and 1000 amplification curves appears in the quasi-static impulsive range with a factor 1.5, which is again very close to Ref. 1. Figure 13 shows the effect of R/h for an amplification of I000. The stiffened shell is characterized by ex = h, Sx = h 2 and bx = (R/10). The same curves are plotted in Fig. 14 in a dimensionless log-log diagram ((Impulse/Pstat. (R/C)), (PMAx/Pstat)). 10[]Amplification of 1000 <>Amplification of 100 eL~
e~
e~
o-
( I 1
Impulse (MPa.ms) Fig. 12. Effect of the amplification factor.
I
10
Dynamic buckling of cylindrical shells
103
100 -
<>R/h = 250 ZXR/h = 100 v R / h = 100 stiffened
10%.o
i::,. 0
0~-
I
0
I 1
I 10
I 100
Impulse (MPa.ms) Fig. 13. Effect o f
R/h.
1000 <>R/h = 250 A R/h = 100 ~R/h = 100 stiffened 100
.~
tO
1
100
lO0O Dimensionless impulse
i
10000
Fig. 14. D i m e n s i o n l e s s critical c u r v e s o f p r e s s u r e v e r s u s impulse.
10 DISCUSSION AND CONCLUSIONS Anderson and Lindberg I developed a modal method to analyse the dynamic buckling of: unstiffened elastic shells of finite length, --unstiffened plastic shells of infinite length (tangent modulus theory). --
104
C. P$dron, A. Combescure
They considered the case of shells: -
-
-
-
-
-
-
-
with inextensional bending modes, submitted to uniform pressure pulses decaying with time, with uniform plastic flow, and without fluid-structure interaction.
Their conclusions were: - - for quasi-static loads, the dynamic buckling is entirely characterized by the peak pressure. The problem becomes a classical static problem and the length of the cylinder is an important parameter of the physical phenomenon, - - for quasi-impulsive loads, both impulse and peak pressure govern the dynamic buckling, - - for impulsive loads, the dynamic buckling is characterized by the impulse; the cylinder length is not an important parameter because the wavelength of the buckling is short. The buckling occurs in the plastic range. Anderson and Lindberg compared their theory with experiments and found good agreement, mainly for quasi-static loads, when the buckling is dominated by the membrane stress, even for smoothly varying nonuniform pressures. Figure 15 shows a comparison between their theory and the modal method presented in this paper, for the same cylinder (R/h= 100, r = 7 6 . 2 m m , E=70000MPa). The critical curves from Ref. 1 show the influence of L/R where L is the cylinder length. The curve
10-
<>L / R c~L/R AL/R ~'L/R
= = = =
inf. 5 2 1
.o o~
0-
0
1
10
I 10o
Impulse (MPa.ms) Fig. 15. Peak pressure versus impulse -- a comparison
w i t h R e f . 2.
Dynamic buckling of cylindrical shells
105
obtained by the authors, characterized by L / R = c~, is in good agreement with Ref. 1. In conclusion, it seems that the theory developed by Anderson and Lindberg is well adapted for the structures and the pressure pulses studied here, at least for infinite elastic stiffened shells submitted to quasi-static or quasi,impulsive lateral loads. The work on plastic flow theory is in progress. The results should be different from those o f Ref. 2 for impulsive loads because the pressure pulses considered are not uniform; hence, the membrane stress and the plastic flow can not be considered uniform around the ring, especially for stiffened shells. Moreover, in this study, the fluid-structure interaction is taken into account, including cavitation.
REFERENCES 1. Anderson, D. L. & Lindberg, H. E., Dynamic pulse buckling of cylindrical shells under transient lateral pressures. AIAA J., 6(4) (1968) 589-98. 2. Bushnell, D., Analysis of buckling and vibration of ring-stiffened, segmented shells of revolution. Int. J. Solids Struct., 6 (1970) 157-81. 3. Bushnell, D., BOSOR 5 - Program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Computers and Structures, 6 (1976) 221-39. 4. Haxton, R. S., Haywood, J. H. & Hunter, I. T., Nonlinear inelastic response of an infinite cylindrical shell to underwater shock wave loading. In Advances in Marine Structures. Elsevier, Amsterdam, 1991, pp. 334-51. 5. Geers, T. L., Residual potential and approximate methods for three dimensional fluid-structure interaction problems. J. A coust. Soc. Amer., 49 (1971) 1505-10. 6. Pinsky, M. A., Partial Differential Equations and Boundary-Value Problems with Applications, Mathematics and Statistics Series, 2nd edn. McGraw-Hill, New York, 1991. 7. Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Ser. No. 55. Department of Commerce, Washington DC, 1964. 8. PLEXUS-CASTEM system, User's manual. DMT/SEMT, Centre d'Etudes Nucl~aires, 91191 Gif sur Yvette, France, January, 1994.
Thin- Walled Structures 23 (1995) 85-105 Elsevier Science Limited Printed in Great Britain 0263-8231/95/$9.50
ELSEVIER
0263-8231(95)00006-2
Dynamic Buckling of Stiffened Cylindrical Shells of Revolution under a Transient Lateral Pressure Shock Wave
Christophe P6dron DCN Ing6nierie, 8 Bd Victor, 75732 Paris, France
& Alain Combescure CEA-Centre d'Etudes Nucl6aires de Saclay, 91191 Gif sur Yvette, France
ABSTRACT A modal method of analysis is used to determine the response of an infinitely long stiffened cylindrical shell of revolution to a transient lateral pressure produced by an underwater explosion and propagating in an acoustic fluid. The shell is initially immersed, hence prestressed by the external hydrostatic pressure. A theory of dynamic buckling is then developed for cylindrical shells subjected to transverse pressure pulses of different durations.
NOTATION
bo, b~ C E, Eo, Ex eo, e x
G, Go, Gx
Axial and circumferential stiffener spacing Sound velocity in the fluid Young's moduli o f the shell, the circumferential and the axial stiffeners Stiffener eccentricities Internal force vector Shear moduli 85
86
h
C. POdron, A. Combescure
Thickness of the shell 1 h2 12 R 2' bending rigidity of the shell
Io, lx
Moments of inertia of stiffeners about the middle surface of the shell Torsional constant of stiffeners about the middle surface JO, Jx of the shell Elastoplastic stiffness matrix [Kep(u)] Pressure stiffness matrix, due to P(t) [gp(t)] [Kp] Pressure stiffness matrix, due to PI Tangential stiffness matrix of the fundamental state [Ktg (t)] Stress stiffness matrix, due to a (u) Prestress stiffness matrix, due to a~ [K,,] Moo, Mxx, Mox Internal moments Mass matrix Noo, Nxx, Nox Internal resultants Cavitation pressure Pcav Peak pressure PMAX Critical static pressure Pstat Pl Hydrostatic pressure {P(t)} Variable pressure vector R Radius of the shell r Radial coordinate Cross-sectional area of stiffeners SO, Sx [S(u)] Area matrix T Time constant t Time Components of the displacement U of the middle Uo, Ux, U3 surface of the shell in cylindrical coordinates (u} Displacement vector of discrete variables of fundamental state {8} Acceleration vector x Axial coordinate F,O0, 8xx~ 130x
(zxu)
Membrane strains Membrane strain vector Linear membrane strain vector Quadratic membrane strain vector Displacement vector of discrete variables of perturbed motion
Dynamic buckling of cylindrical shells
0 /.z Po Ps
O" 1
ZOO,Zxx, Xox
{x}
(),x (..)
87
Perturbed acceleration vector Circumferential coordinate Poisson's ratio Fluid density Density of cylindrical shell Stress tensor between the initial prestressed configuration and the fundamental state Uniform stress, due to PI Bending strains Linear bending strain vector
o(
--, Ox 02 0 t 2(
)
partial differentiation with respect to x ), partial differentiation with respect to t
1 INTRODUCTION Although several studies exist in the literature on the response of submerged shells to a transverse shock wave, very few deal with the elastoplastic response and dynamic stability of stiffened cylinders. In this paper, the elastoplastic response of an infinitely long cylinder to a shock wave is investigated. The stiffened shell is initially immersed in an acoustic fluid and a shock wave, assumed to be plane, encounters the structure. The wave front is parallel to the axis of the shell and decays exponentially with time as follows:
P(t) = PMAXexp ( - t / T ) The structure in motion is then subjected to a dynamic pressure that is the sum of an incident plane wave and a radiated pressure created by the vibrating shell. If the incident shock wave is known exactly, the radiated pressure is determined by an approximate method. A modal method is used to determine the response of the structure, based on a decomposition in Fourier series of the displacement and pressure fields. The response of each mode can be analysed independently. The dynamic stability of the fundamental motion is examined by a perturbation method. The problem is to determine the incident dynamic loads that cause a perturbed bending mode to increase sufficiently to exceed a buckling criterion. The aim of this study is to check that a method, developed by Anderson and Lindberg,~ is well adapted for stif-
C. P$dron,A. Combescure
88
fened cylinders subjected to transverse pressure pulses, at least in elasticity, and for quasi-static and quasi-impulsive durations.
2 A STIFFENED CYLINDRICAL SHELL THEORY This study is limited to the case of stiffened cylindrical shells of revolution (Fig. 1). The shell theory is based on Kirchhoff-Love assumptions. For such a shell, the Green-Lagrange strain tenso~for a point at distance 03 from the middle surface in the displacement U is defined, in cylindrical coordinates, by: ~00 = -g1 (Uo, o + U3) + ~ 1
[(Uoo_~_U3)27L , Ux,2 o + (U3,0 + Uo) 2]
03 + - ~ (Uo, o + U3,oo) 1
2
03
~xx = Ux, x + ~ [V~o,x + Ux, x + U~,x] - - f i U~,xx ~ox = ~ - -
+2
+ Uo, x
(1)
)
l [Uo,o+ U3 Uo,x..l__~ Uxx..~(U3,0_ Uo)U3,0] R
--03(~ "----~x- 2U3,oo)
., Ux
/U0
U3 Fig. 1. Cylindrical shell of revolution.
89
Dynamic buckling of cylindrical shells
or 1
{13} = 031 } -+- ~ {/3q} q'-03{Z}
(2)
A Timoshenko beam theory is considered for the axial and circumferential stiffeners (Fig. 2). Moreover, stiffeners are located at regular intervals and their cross-sectional dimensions are small with respect to the radius of the shell. The rotation of a stiffener centroid is equal to the rotation of the middle surface of the shell. The displacement fields are continuous at each shell-stiffener interfacefl By using the virtual internal strain energy, the elastic stress resultants-strain relationship is obtained by integration over the thickness of the shell and is given by: 2
{S}-- [I, l Iol where
[B]
H + ExSx bx
#H
0
pH
Eo So H+ -bo
0
0
0
Gx S~ + Go So Gh+~
=
E~ S~
ex bx [K] =
0
0
0
Eo So eo bo
0
o
o
Gx Sx + eo Go So
ex b---£-x
ExI~ D+--
#D
0
pD
Eo Io D+-bo
0
bx
[ O ] --
0 and H-
Eh 1 -
--fro
lfl'
D _
Eh 3 12 (1 - p2)
D~-~+G"J~
"~+b~-oG°J°
90
C. P6dron, A. Combescure
.Y
eo~~
Fig. 2. Shell and stiffeners.
Nevertheless, local instabilities (Fig. 3) are excluded by this approach. The plastic flow of such a stiffened shell is treated by a multilayer approach. The stiffened shell is considered as a three layer material, one layer for the shell and two layers for the stiffeners (axial and circumferential). Perfect bonding at each interface is assumed; therefore, the displacement fields are continuous across each interface as in the elastic theory. The incremental plastic stress-strain relationship follows that of the classical simple flow theory of plasticity, Von Mises J2 yield surface and isotropic strain hardening. 3
3 EQUATIONS OF MOTION Initially, the shell is immersed in an acoustic fluid, hence subjected to the external hydrostatic pressure P~ in the fluid. The cylinder is subjected to a uniform prestress at due to the pressure Px.
\ Stiffener local instability
/
Local shell buckling
Fig. 3. Excluded instabilities.
Dynamic buckling of cylindrical shells
91
A lateral pressure shock wave, travelling in the fluid, then acts on the shell. The dynamic pressure on the shell, P(t), is the sum of an incident pressure plane wave and a radiated pressure created by the vibrating shell. This motion is called the fundamental motion defined by the displacement U of the middle surface of the shell. The stability of the fundamental solution is investigated by determining the response of an mfimteslmal perturbation A U with respect to the fundamental state (Fig. 4). By using a spatial discretization, the fundamental motion may be expressed in the discrete form: [M] {U} = -{Fint (U)} -- [K,, - Ke, ]{U} + [S(U)] {P(t)}
(4)
Considering a perturbed solution U + A T in (4), the perturbed motion, with respect to the fundamental state, is given by: [M] {A U} = - [ g e p (V) + K., + Ko(v) - Kp, - K.(,) ] {A U}
(5)
or [M] { A ~)} + [Ktg (t)] { A U} :
{0}
In the case of an infinitely long cylinder subjected to a transverse pressure pulse, the deflections Uo, Ux, U3 may be expressed in the form (Fig. 5): N
Uo (0, t) = ~
u. (t) sin nO
n=[ .-..+
U(0, t) =
(6)
Ux(0, t) = 0 N
U3 (0, t) ---- Wo(t) + Z
wn(t) cos nO
n=l
where N must be greater than the buckling mode.
U Initial configuration
Fundamental motion
Perturbed motion
Fig, 4. Initial, fundamental and perturbed motions.
C. P$dron. .4. Combescure
92 Z
I_
h
Incident pressure
~
U3
Fig. 5. Shock configuration. The incident pressure pulse wave pINC and the radiated pressure pRAD are also decomposed in Fourier series (Fig. 5): n
pInNC(r, t) cos nO
p1NC (0, r, t) = ~ n=0
pRAD (0, r, t) = ~
(7) nRAD/..n"tr, t) COS nO
n=0
For example, with small deformations and small displacement assumptions, the fundamental motion is governed by the following discrete equations: for n -- 0:
PsRhfvo(t)+
Eh
(1 --/~2)R
Wo(t) = R PIoNC(R, t) + R PoRAD (R, t)
f o r n >~1:
PsRh R - - (71 -'~
fvn nn2
rn2+,, n+n3,s]{Un}
(1 --~2) R Un+n3Is wu,n
= R
1 +n4Is
Wn
PlnNc (R, t) + pRAD (R, t)
(8)
Dynamic buckling of cylindrical shells
93
One obtains (2N + 1) equations of motion even though there are (3N + 2) unknowns, i.e. (2N + 1) displacement unknowns and (N + 1) radiated pressure unknowns. Indeed, if the decomposition in Fourier series o f P INc (0, R, t) is easy, the determination o f P RAD(0, R, t) needs special treatment.
4 INCIDENT PLANE WAVE The incident plane wave, travelling in the acoustic fluid, may be expressed as (Fig. 5): 4
pINC(O'r't)=PMAxeXp( R - Ct-rc°sO)
Ct- rcosO)
(lO)
where H is the Heaviside function defined by:
H(R-Ct-rcosO)= { 0l iif(R-Ct-rcosO)~O} f(R Ct rcosO)
(11)
Given that pINC(0, r, t) is decomposed in Fourier series as: N
piNC(0, r, t) = Z
PnINC (r. t) cos
nO
(12)
n=0
it follows that:
p~C(r,t)=Ii'PMAxexp(.RcTC_()exp(
rcos 0'~ •) cos
nOdO
(13)
with { 0t
=
}
(R~fCt_)R+r if t ~<
a cos
rt if t >
(14)
R+r C
In other terms: i
,,',INC," 2 r . tr, t) = - -
e,
PMAX
(.R ~ _C('~ i o , ( r ~ exp
k, CT J "\ CT/
(15)
with e0 = 2 and e, = 1 for n ~> 1 and where
r =~lIi' exp ( rcos0'~ 1o,(_~) --C--f .j cos nO dO
(16)
94
C. P~dron, A. Combescure
is the incomplete modified Bessel function of the first order, and becomes the complete function In(-r/CT) when 0t = ~r.4 Given that the radial component of the incident wave particle velocity is: wiNC(o, r, t) - -
PINC(o,r, t) c o s O PoC
(17)
then, for n = 0
PMAXexp(R C ZcCt I°2'(---~T)
W-oINC/n (o,r,t)-- - -
and for n/> l
¢v~¢(r,t)_PM^xexp(RcCt)(~t+ C ,
_ i n _ , ( -C--T))r
This radial velocity is useful; it will be used for the fluid-structure interaction equations.
5 RADIATED PRESSURE In an acoustic fluid, in cylindrical coordinates, the radiated fluid pressure PaAD(0, r, t) and the radiated particle velocity ffRAD(0, r, t) created by the vibrating shell are defined by the velocity potential ~(0, r, t): pRAD : po ~
(18)
wRAD = O~
Or
(19)
• (0, r, t) is expanded in the form: N
• (0, r, t) = E
~,(r, t) cos nO
(20)
n=0
and satisfies the hydrodynamic equation: 1
02~ (21)
A ~ -- C 2 0t 2
Using (20) and (21), it follows that 02~n
l°~I~n
Or--T 4 rOr
(n) 2 1 02~n -r (I~n -- ( 2 ~-~
(22)
Dynamic buckling of cylindrical shells
95
We now introduce the Laplace transform: exp (-st) ~.(r, t) dt
4~.(r, s) =
(23)
Using the Laplace transform, (22) becomes
~Or + r Or
~. =
q~.
(24)
A solution of this Bessel equation, appropriate to a transient diverging wave, is: s' 6
where K. is the modified Bessel function of the second kind. Hence,
Or
(r, s) = F. (s) s K' ( sr C "\-CI
(26)
and (25) and (26) lead to:
Ol d P n ( rd' s ) =Cs K r ~K. ' n ( c ) lOd pr ' ( r ' sr) = f * ( c )
r .(r,s)
(27,
where f * is defined by
f.* x
K'.
x--K-2(x)
with the following properties: 7
gn E N, l i m x ~ o f * ( x ) = 0 f*(x) = -x - ~ + 0
for x ~ ~
(28)
Vn E N*, d f * ( x = O) = n dx The problem to be solved is to find an approximate function g* o f f * , with the same properties, and which presents no difficulty when determining the inverse Laplace transform. The adopted approximations are: for n = 0: 1
g~(x)=-x-~-F
a2 x2+a2 witha=10
C. P~dron,A. Combescure
96 and for n >t 1: g* (x) : - x
1 2
1 (2n- 1)2 2(x+2n-1) 2
(29)
One then obtains by the inverse Laplace transform: for n = O: 0~o.. 1 • 1 [t CR . ( Ct (t Or ~ C ~ ° - 2 r r ~ ° + J o / - - 2 r 2sIn r - u )
) ~o(r,u) du
f o r n >~ l" Or
C
+
I~o(2n-1)3C2(u-t) 2r 2
exp
((2n-1)C(u-t))g2.(r,u)d u r
(30)
Therefore, one obtains, for each mode, a rather simple relation between the radiated particle velocity and the velocity potential.
6 FLUID-STRUCTURE INTERACTION Up to now, only ( 2 N + 1) discrete equations have been established, even though there are ( 3 N + 2) unknowns. The (N + 1) missing equations to solve the problem completely are obtained from the fluid-structure interaction equations: the radial velocity in the fluid is continuous, which means, to the first order (r = R), in each mode: fi~n(t) : w.. nINC/r~ ..RAD/n 0(I)n (R,t) ~..'x,t)+wn ~lX,t) = W.. ,INC/n t,,x,t)+--~r
(31)
The problem is now entirely determined with these (N + 1) extra equations combined with (29). This method is very similar to the residual potential method developed by Geers. 5 Moreover, a simple model of cavitation is introduced in the time integration algorithm: as soon as p~NC + pRAD ~< Pc'~v, pRAD is then defined by the relation pRAD = e c a v - pINC.
7 TIME I N T E G R A T I O N An explicit scheme, based on the central differences method, is used to compute the time integration.
Dynamicbucklingof cylindricalshells
97
8 C O M P A R I S O N WITH PLEXUS In order to validate the modal method described in this paper, the interaction of an incident lateral pressure pulse with an infinite unstiffened cylindrical shell was studied with the P L E X U S computer code. 8 This code, developed by CEA Saclay (France), is based on the finite element method. The results are examined in the case of an unstiffened elastic cylinder made of aluminium with a radius of 76.2 m m and a ratio R/h equal to 30. The incident pressure is characterized by a peak pressure PMAX-----5MPa and a time constant T = I OR/C. The response versus time of three points A, B, C of the shell (Fig. 6) is observed. Dimensionless time (Ct/R) is plotted on the x-axis and is shifted by 7 R/C. This time shift is caused, in the P L E X U S modelling, by the fluid mesh and represents the time the incident pressure needs to travel from a plane fluid boundary to point A. Very good agreement between the displacement curves (Figs 7a and 7b) was found. The m a x i m u m difference between the two simulations is less than 6%, for point C. The Von Mises stresses also agree well (Figs 8a and 8b). It means that the mode n = 1 and the breathing mode (n = 0) are computed with good accuracy. Nevertheless, the P L E X U S results oscillate more. These high frequency oscillations are caused by the discretization of the fluid by finite elements. The finer the mesh, the higher the frequencies.
9 DYNAMIC BUCKLING The dynamic buckling of the structure is analysed by a perturbation method. Given a solution of the fundamental motion (4), one follows the i
C
._A__ _ -~t)
I
Fig. 6. Locations of the points A, B and C.
Dynamic buckling of cylindrical shells 150 t (a)
99
- - V o n M i s e s Point A . . . . V o n M i s e s Point B - - - V o n M i s e s Point C
100
r~
50
............. 5
L~,
10
15
20
25
D i m e n s i o n l e s s t i m e (CffR)
Fig. 8a. Von Mises stresses (modal method).
150 - ( b ) 3 2 1 1-~......~_ 1 I/ ~X"~ 3
,--. lOO
V
¢.
0.0E+00
"-1
3 2.. 1--~
50
0.OE+0C
v~'~
I 5
2~
~ 1 - - J2 point A --2-- J2 point B - - 3 - - J2 point C
~
I 10
I 15 T i m e (Ct/R)
/3 2 3
I 20
I 25
Fig. 8b. Von Mises stresses (PLEXUS computation). with 2, (t) = R , ( t ) + j C , ( t ) , j2 = _ 1
(33)
2 . ( 0 is a complex eigenvalue of the characteristic equation o f the perturbed motion and {AU0}n, a vector which depends on the initial conditions at t = t0. At each time step t, 2.(t) is determined for each mode. As soon as (to, n) exists for which R,,(t) > 0, the perturbed solution may become unstable because the mode n may increase exponentially with time. At this time to,
C. POdron, A. Combescure
100
if it does exist, one considers an infinitesimal perturbation arbitrarily defined by:
1
{ A U ( t 0 ) } - 1000 { U ( t 0 ) } ::~ Vn,
{ Aun Unwn)) (to) (to) --
1000
(34)
Aw. (to) - , t o , 1000
Then, one examines the evolution, in each mode, of the perturbed radial deflection Awn(t) for t 1> to by solving, for each time step, eqn (4) and then eqn (5). One adopts the following dynamic buckling criterion: the buckling mode n is the first one for which:
Awn (t) >/fl Awn (to)
(35)
where fl is an amplification factor (fl = 100; 1000;...). This approach is very close to that of Anderson and Lindberg.1 According to their theory, the critical load that causes buckling is determined in the following logical way: given a structure and an incident pressure pulse characterized by PMAXand T, the computation of the equations of motion (4) and (5) is carried out until t = 3 T. At this time, the structure has been submitted to 95% of the initial impulse and it is considered that, if the dynamic buckling criterion (35) is not violated before t = 3 T, the structure will stay stable over this time and the computation is stopped. On the other hand, if tcr < 3 T and a mode n exists, such that Awn(tcr)
Awe(t0) the computation is stopped; the considered load is the critical one and n is the buckling mode. The authors then checked that the dynamic buckling theory developed in Ref. 2 is valid for the stiffened structures and lateral pulses considered, at least in the elastic range. The following results concern an infinite cylinder made of aluminium ( E = 70000MPa) with a radius of 76.2 mm. Following Ref. 2, at the critical time tcr, the relative amplification (Aw~(tcr)/Aw~(to)) of each perturbed bending mode is plotted versus n. An example is given in Fig. 9, obtained with R/h = 100, PMAX = 1.4MPa ( e s t a t : 0"018 MPa) and T = IOR/C. Figure 10 shows the buckled shape obtained with this critical load. The critical 'impulse-peak pressure' levels that cause the dynamic buckling are then plotted in a log-log diagram as shown in Fig. 11.
Dynamic buckling of cylindrical shells
101
1200 1000 800 o t~ 600 <
400 200
10 Mode n Fig. 9. A m p l i f i c a t i o n function.
y//
\
/ /
'/'\
7 ~\
'II ,I
'~Xx\
Fig. 10. B u c k l e d shape.
For quasi-static loads (long duration with regard to the shell response time), the dynamic buckling is characterized by the peak pressure. The critical curve, Fig. 1 l, presents an asymptote PMAX = Pstat and the buckling mode is the static one, mode 2 for an infinite cylinder. For quasi-impulsive loads (duration comparable to the shell response time), the dynamic buckling is governed by both the impulse and the peak pressure. The buckling modes are higher than the static one (Fig. 11). These results show very good agreement with the conclusions in Ref. 2. No results are presented for very impulsive loads (short duration) because the plastic flow has to be taken into account and only elastic shells are considered in this paper. The effect of the amplification factor is also examined. Figure 12 shows
102
C. Pddron, A. Combescure
10<>Bucklingmode
.o < 5~32 xe e~
0-
I
I
1
10
Impulse (MPa.ms) Fig. 11. Critical curve. the critical curves obtained with /~ = 100 and 13 = 1000. The m a x i m u m difference of impulse between the 100 and 1000 amplification curves appears in the quasi-static impulsive range with a factor 1.5, which is again very close to Ref. 1. Figure 13 shows the effect of R/h for an amplification of I000. The stiffened shell is characterized by ex = h, Sx = h 2 and bx = (R/10). The same curves are plotted in Fig. 14 in a dimensionless log-log diagram ((Impulse/Pstat. (R/C)), (PMAx/Pstat)). 10[]Amplification of 1000 <>Amplification of 100 eL~
e~
e~
o-
( I 1
Impulse (MPa.ms) Fig. 12. Effect of the amplification factor.
I
10
Dynamic buckling of cylindrical shells
103
100 -
<>R/h = 250 ZXR/h = 100 v R / h = 100 stiffened
10%.o
i::,. 0
0~-
I
0
I 1
I 10
I 100
Impulse (MPa.ms) Fig. 13. Effect o f
R/h.
1000 <>R/h = 250 A R/h = 100 ~R/h = 100 stiffened 100
.~
tO
1
100
lO0O Dimensionless impulse
i
10000
Fig. 14. D i m e n s i o n l e s s critical c u r v e s o f p r e s s u r e v e r s u s impulse.
10 DISCUSSION AND CONCLUSIONS Anderson and Lindberg I developed a modal method to analyse the dynamic buckling of: unstiffened elastic shells of finite length, --unstiffened plastic shells of infinite length (tangent modulus theory). --
104
C. P$dron, A. Combescure
They considered the case of shells: -
-
-
-
-
-
-
-
with inextensional bending modes, submitted to uniform pressure pulses decaying with time, with uniform plastic flow, and without fluid-structure interaction.
Their conclusions were: - - for quasi-static loads, the dynamic buckling is entirely characterized by the peak pressure. The problem becomes a classical static problem and the length of the cylinder is an important parameter of the physical phenomenon, - - for quasi-impulsive loads, both impulse and peak pressure govern the dynamic buckling, - - for impulsive loads, the dynamic buckling is characterized by the impulse; the cylinder length is not an important parameter because the wavelength of the buckling is short. The buckling occurs in the plastic range. Anderson and Lindberg compared their theory with experiments and found good agreement, mainly for quasi-static loads, when the buckling is dominated by the membrane stress, even for smoothly varying nonuniform pressures. Figure 15 shows a comparison between their theory and the modal method presented in this paper, for the same cylinder (R/h= 100, r = 7 6 . 2 m m , E=70000MPa). The critical curves from Ref. 1 show the influence of L/R where L is the cylinder length. The curve
10-
<>L / R c~L/R AL/R ~'L/R
= = = =
inf. 5 2 1
.o o~
0-
0
1
10
I 10o
Impulse (MPa.ms) Fig. 15. Peak pressure versus impulse -- a comparison
w i t h R e f . 2.
Dynamic buckling of cylindrical shells
105
obtained by the authors, characterized by L / R = c~, is in good agreement with Ref. 1. In conclusion, it seems that the theory developed by Anderson and Lindberg is well adapted for the structures and the pressure pulses studied here, at least for infinite elastic stiffened shells submitted to quasi-static or quasi,impulsive lateral loads. The work on plastic flow theory is in progress. The results should be different from those o f Ref. 2 for impulsive loads because the pressure pulses considered are not uniform; hence, the membrane stress and the plastic flow can not be considered uniform around the ring, especially for stiffened shells. Moreover, in this study, the fluid-structure interaction is taken into account, including cavitation.
REFERENCES 1. Anderson, D. L. & Lindberg, H. E., Dynamic pulse buckling of cylindrical shells under transient lateral pressures. AIAA J., 6(4) (1968) 589-98. 2. Bushnell, D., Analysis of buckling and vibration of ring-stiffened, segmented shells of revolution. Int. J. Solids Struct., 6 (1970) 157-81. 3. Bushnell, D., BOSOR 5 - Program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Computers and Structures, 6 (1976) 221-39. 4. Haxton, R. S., Haywood, J. H. & Hunter, I. T., Nonlinear inelastic response of an infinite cylindrical shell to underwater shock wave loading. In Advances in Marine Structures. Elsevier, Amsterdam, 1991, pp. 334-51. 5. Geers, T. L., Residual potential and approximate methods for three dimensional fluid-structure interaction problems. J. A coust. Soc. Amer., 49 (1971) 1505-10. 6. Pinsky, M. A., Partial Differential Equations and Boundary-Value Problems with Applications, Mathematics and Statistics Series, 2nd edn. McGraw-Hill, New York, 1991. 7. Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Ser. No. 55. Department of Commerce, Washington DC, 1964. 8. PLEXUS-CASTEM system, User's manual. DMT/SEMT, Centre d'Etudes Nucl~aires, 91191 Gif sur Yvette, France, January, 1994.