Dynamic axial plastic buckling of stringer stiffened cylindrical shells

Int. J. Mech. Sci. Vol. 24, No. I, pp. 1-20, 1982 Printed in Great Britain.

0020-7403/82/010001-20503.00/0 Pergamon Press Ltd.

DYNAMIC A X I A L PLASTIC B U C K L I N G OF STRINGER S T I F F E N E D C Y L I N D R I C A L SHELLS NORMAN JONESt Department of Mechanical Engineering, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX, England

and E. A. PAPAGEORGIOU Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

(Received 10 January 1981; in revised form 17 August 1981) Summary--A perturbation method of analysis has been used to examine the dynamic plastic buckling of a stringer-stiffened cylindrical shell subjected to an axial impact. It transpires that it is more efficient to place stiffeners on the outer shell surface rather than on the inner surface. Various results are presented to demonstrate the influence of the second moment of area, eccentricity, cross-sectional area and number of stiffeners on the dynamic plastic response.

NOTATION a an

b b. e e*

h h' m n nc

q S

t, tt U V

w, w! W O, W t X, Z X*

mean radius of cylindrical shell coefficient in Fourier series representation of dimensionless initial radial displacement imperfection (equation 43b) circumferential spacing on mid-surface of cylindrical shell between adjacent stiffeners coefficient in Fourier series representation of dimensionless initial radial velocity imperfection (equation 43c) eccentricity of a stiffener measured from mid-surface of cylindrical shell e/a, positive for outside stiffeners and negative for inside stiffeners thickness of a cylindrical shell A/b, additional shell thickness due to smearing of stiffeners defined by equation (21b) mass of cylindrical shell and stiffeners mode number (equation 43a) critical mode number defined by equation (8) number of stiffeners time, response duration axial displacement initial radial velocity of cylindrical shell immediately after impact total radial displacement, permanent dominant radial displacement initial radial displacement imperfection dominant and perturbed radial displacements axial and radial coordinates

x/L

A cross-sectional area of a stiffener A* A/bh E' tangent modulus (equation 13) P. defined by equation (41e) E.(O) displacement amplification function at tt F.(O) velocity amplification function at tl I Second moment of area of a stiffener about the mid-surface of a cylindrical shell I* I/a2bh K pV%, L length of cylindrical shell M mass on end of cylindrical shell as shown in Fig. 1 axial and circumferential dominant bending moments per unit length M,, M°/aah, M°/aah M~ total axial bending moment per unit length

u°t °

?Formerly Department of Ocean Engineering, M.I.T.

N. JONES and E. A. PAPAGEORGIOU

Mx Mxlo'ah N~°, No° axial and circumferential dominant membrane forces per unit length

Nx°, No° Nx°/~h, No°ltrh N~, No

N~, No V

total axial and circumferential membrane forces per unit length

NJcrh, Nd~rh initial axial velocity

a h/a B a/L y ~°l~rv g, 8 °, 8 if/a, w°la, w'la axial, circumferential and radial strains equivalent strain 0_ circumferential coordinate

*x, c0, ~,

,~, ~ E'l~r °, E'lcr~ mass per unit area defined by equation (46b)

l-tit I p ~rx, or0 ~r o'o ~ ~r°

(') (

)'

density of material axial and circumferential stresses equivalent stress defined by equation (2a) yield stress defined by equation (32c)

l+ M/m a( )~at, a( )la!~ perturbation terms

).A ).x. a( )lax, a( )lax* INTRODUCTION

Many theoretical and experimental studies have investigated the elastic buckling behaviour of stiffened cylindrical shells subjected to static axial loads ([1-5], etc.), while somewhat fewer have examined the behaviour in the plastic range ([6, 7], etc.). References [8-11] have investigated the elastic response when the axial load is applied dynamically but no articles appear to have been published for the corresponding plastic case, although Refs. [12-15] have employed a rigid-plastic method for the dynamic plastic buckling of unstiffened cylindrical shells subjected to axial loads. Recent interest in the dynamic axial buckling of structural members has arisen largely because of the worldwide attention on the crash-worthiness protection of aircraft, buses, cars, ships and trains. In fact, Fisher and Bert[16] have used their theoretical procedure for the dynamic axial buckling of an elastic cylindrical shell [11] to investigate the crashworthiness of light aircraft cabins. More recently, Tennyson et al.[17] have presented the results of some impact tests on scale model aluminium aircraft fuselage structures which in some instances failed catastrophically and involved extensive plastic behaviour. Other recent studies[18] have attempted to estimate the dynamic loads which act on nuclear reactors and chemical plant when impacted by various missiles some of which involve the dynamic axial buckling of cylindrical shells. It is the objective of this article to investigate the dynamic axial plastic buckling of stringer stiffened cylindrical shells in an attempt to lay some theoretical groundwork for future crashworthiness and energy absorption studies involving this structural member. The equilibrium and kinematic relations derived by Baruch and Singer[19] for stiffened cylindrical shells are used in the current study. The response of a cylindrical shell with an attached mass travelling with an initial velocity and striking a rigid wall is studied using a perturbation method of analysis which has been used successfully by Vaughan[13] for the unstiffened case and for the dynamic plastic buckling of various cylindrical[20, 21] and spherical[22] shell problems. DOMINANT MOTION Florence and Goodier[12] examined the response of an unstiffened tube impacted axially at one end with a heavy mass M travelling with a constant axial velocity V. The heavy mass was brought to rest after a finite time in the actual experiments but the buckling mode was selected early in the response which

Dynamic axial plastic buckling of stringer stiffened cylindrical shells provides some justification for the constant velocity assumption used in the theoretical analysis. Vaughan[13] relaxed this assumption and allowed the motion to cease in a finite time. He examined the impact against a rigid wall of an unstiffened shell mass m and attached mass M at one end travelling with an initial velocity V. Here the problem of Vanghan[13] is investigated except the shell is stiffened with stringers (axial stiffeners) as shown in Fig. 1. Baruch and Singer [19] smeared the cross-sectional area (A) of the axial stiffeners over the shell thickness (h) as indicated in Fig. 2 to produce a uniform shell thickness with a mean thickness of h + h', where h' = A / b and b is the circumferential spacing of the stiffener centres on the mid surface of the shell.' This simplified procedure has been used by many authors and found to provide an adequate idealisation of a stiffened shell. It may be shown that the equation of motion in the radial sense for the axisymmetric behaviour of a cylindrical shell made from a rigid-plastic material predicts )b° = v(1 - t/t/)

(1)2

Strir~ ers Cylindrical Shell

\

\

t

5,

v6

M -

-

rfi

I

L

L I

FIG. 1. Cylindrical shell shown with outside axial stiffeners (stringers) of total mass m and end mass M travelling with an initial velocity V immediately prior to impinging on a rigid wall.

Eccentricity sLe,,°f ifftener

Cross- sectionoL Areo A of ,_~/ Stringer h'

FIG. 2. Definition of parameters for a stringer-stiffened cylindrical shell and an equivalent shell.

mh' is the additional shell thickness only for thin shells when using b instead of b' which is the circumferential spacing at the mean radius of the additional shell thickness h'. It may be shown that b ' l b = (1 + a/2)[l + {I + 2A'a(1 + a/2)-2}lrz]/2 and b ' l b = (1 - a/2)[1 + {1 - 2A*a(l - al2)-2}112]12 for outside and inside'stiffeners, respectively. The eccentricity and moment of inertia of stiffeners is properly accounted for in these equations. 2Response is taken to be independent of the axial coordinate.

4

N. JONES and E. A. PAPAGEORGIOU

where v is the initial radial velocity at t = 0, (") = 0( )/Ot, t is time, w ° is the dominant radial displacement,

t! = a~zvI No

(2a) 3

is the response duration, /~ = p(h + h') and No = ooh (~ro is an average circumferential stress in the plastic range during the response 0 <- t <- t:). Equation (1) may be integrated to give the permanent dominant radial displacment,

w! = al~v2/2~oh.

(2b)

If the axial wave propagation effects are disregarded, then the axial velocity of the cylindrical shell is ~i° = V(1 - t/t/)(l - x/L),

(3)

where V is the initial axial velocity and L is the cylinder length. Now, the circumferential strain ~0 = wl(a + z) gives i0 = v(l - zla)(l - tltl)/a

(4)4

ix = - V(l - tltt)lL

(5)

~ = { VI L - v(1 - zIa)la} (1 - t[t/)

(6)

while equation (3) predicts

so that

according to the incompressibility requirement. The initial kinetic energy ( M + m)V2/2 is equal to the energy dissipated in a stiffened cylindrical shell,

foq f a x i ~ d V d t + fot/ ff~o~od~'Ot, where dl? is the volume of an element. Thus, if ~x acts over a cross-section with a thickness h + h', while ~0 act~ over a cross-section with a thickness h because only axial stiffeners are present, then 2q 3 + 2q2(1 + h'/2h) + 2q {1 + h'/h - (1 + MIm)(Lla) 2} - (1 + MIm)(L/a) 2 = 0

(7)5

q = id6~)z=0 = - Lvla V,

(8)

where

and ax and go are taken constant 6 and equal to the values at z = 0. The stresses ~r, and ~o have been eliminated from the energy equation using the Prandtl-Reuss equations for incremental plasticity which predict crx/cro = (2~x + io)/(2~o + ix), or

(o'Jo'o)z=o = (2 VIL - v/a)/( VIL - 2via).

(9)

q = - 1/2 + (3/4)(1 + h'/h)]{312 + h'lh - 2(1 + MIm)(L/a) 2}

(lO)

Equation (7) predicts

according to N e w t o n ' s method, and simplifies to q = - 1/2 - (3/8)(alL)2(1 + h'lh)l(1 + M/m)

(11)

when (Lla)2>> 1. The Prandtl-Reuss constitutive equations (cr~ = 2o'(2ix + io)/3i, etc.) together with equations (4)-(6), (8) and (1 1) predict crx = - ¢r {1 + zl3a - (1 + h'lh)(l - zla)(alL)214(l + MIm)}

(12a)

o'o = - o" {2zl3a - (1 + h'lh)(l - zla)(alL)212(1 + MIm)},

(12b)

and

3The reference radius of the stiffened shell is taken as a which is consistent with Refs [8, 19]. 4provided ( z / a ) 2 ~ 1. Equations (4) and (5) are identical to equations (1) in Ref. [13]. 5The trapezoidal shape of an element has not been considered in the calculations. q n order to be consistent with equation (2a) in which cro was taken constant.

Dynamic axial plastic buckling of stringer stiffened cylindrical shells

5

where the usual equivalent stress (or) with ~r~= 0 for shell theory is linearly related to the equivalent strain (e) in the following manner (13)

o = try + E'E,

try is uniaxial yield stress, E' is tangent modulus. The equivalent strain (14)

E = Vt(1 - tl2tt)lL

is found from ~ = f~i dt with • evaluated at the mid-plane (z = 0) (~ = V(I - t/t!)/L = - ~ with q = - ½from equation 11). The response duration is t! = pVL(I + M [ m ) h r

(15)

according to equations (2a), (8), (11) with ( L l a ) Z ~ 1 and (12b) for cro. If equations (13) and (14) are used to eliminate cr from equation (15), then t! = L~ry [{1 + 2E'pV2(I + M / m ) ( l

-

~2)]0"y2}1/2

--

II/E' V(1 - ~2),

(16a)

where (16b)

= 1 - t/t!.

Now, ~r given by equations (13) and (14) varies with time, whereas tr0 in equations (2) is taken to be constant. Thus, if the average stress during the interval 0_< t-< t!(l-> ~-> 0) is used, then E' in equation (16a) is replaced by E'I2 and t! = 2/_ary [{1 + E ' p V 2 ( I + M]m)hry2} '~ - I ] I E ' V

(17)7

when t = t / ( ~ = 0). Equation (17) with E ' ~ 0 (negligible strain hardening) reduces to equation (15) with ~r replaced by ~ry. The axial membrane force associated with the above dominant axisymmetric solution is th/2+h'

NO = J-~2

tr,(1 + z/a) dz,

which using trx = - tr(l + zl3a) from equation (12a) when ( L / a ) 2 -> 1 gives 1(1° = - 1 - A * - A*e*13

(18)

provided (hJa)2~ 1 and where 1V° = N~°hrh;

and

A*=Albh,

i = x, 0,

(19a,b)

e* = d a

(19c,d)

where e is the distance +of the centroid of a stiffener from the mid-surface of the main shell and is given by k the expression e = f~2 ' z d A I A . T h e corresponding circumferential membrane force is N ° = f~2 ire dz since the cylindrical shell is stiffened only with axial stiffeners. Thus substitution of equation (12b) gives /~o =/32 ~/2#

(20)

where [3 = a / L , h = 1 + h'[h,

and ~ = 1 + M l m .

(21a-c)

The associated axial bending moment defined as shown in Fig. 3 is

f

'hl2+h'

M ° = J-~2

trx(l + z l a ) z dz,

or

1~1° =

-

a2/9

-

A'e*

-

I*/3

(22)

when (/Ja) 2 ~ 1, where ~.o = M Ol~rah; a = h/a,

i = x, 0

I* = I[a2bh

(23a,b) (23c,d)

q n order to estimate the influence of strain hardening E' has been replaced by the value E'12 which is not quite the average value because ~r varies quadratically with time.

6

N. JONES and E. A, PAPAGEORGIOU Ne

Mx

/

z J

/~

ill

/

/

/

/ •No

,..J

,×

d~ FI~. 3. Definition of generalised forces and moments and displacements for stringer-stiffened cylindrical shells. and I = f ~ + h ' Z2 d A is the m o m e n t of inertia of a stiffener about the mid-plane at z = 0. Similarly, the circumferential bending m o m e n t is /~0° = - a2/18

(24)

when ( L / a ) 2,> 1. PERTURBED SOLUTION The theoretical solution in the previous section was developed for the axisymmetric uniform motion of a stiffened cylindrical shell with an attached mass M impacted axially against a rigid wall. Fig. I of Ref. [12] shows that the unstiffened tubes impacted axially by Florence and Goodier developed axisymmetric buckling waves. In order to examine the growth of buckling waves in the stiffened case, it is now assumed that the total radial displacement may be written, w = w°+ w'

(25)

where w ° is the dominant displacement considered in the last section and w' is the perturbed axisymmetric component. No perturbations of the axial displacement is considered so that u = u °.

(26)

~x = - V(1 - tltl)lL - z w ~

(27)

The axial strain rate ~x = uz - z w = becomes

where ( )~ = a( )lax, and when using equations (1), (3), (25) and (26). Similarly, the circumferential strain rate ~e = ~i,/(a + z) = ~i,(l - z / a ) / a is ~e = v(l - z[a)(l - t/tf)/a + w'(1 - z]a)a. Thus, ~ = - ~ x - ~ 0 for 4 ( ~ 2 + ~e2 + ~ 0 ) / 3 , or

an

incompressible

material

so

that

the

(28) equivalent

strain

rate

is

~2=

i2 = V1:[1 + z2/3a 2 _ (1 + h ' / h ) ( a / L ) 2 z / 2 a ( l + M / m ) + 4(2 + q ) z w ' 1 3 V~ - 4~i,' x {2q + 1 - (4q + l)z/a}/3a V~]

(29a)

where V~ = V(1 - t/tl)/L

(29b)

w h e n using equations (8), (11), (27) and (28), neglecting the squares of all perturbed terms, assuming (L/a)2,> 1, and ignoring (zla) ~ compared with unity in the perturbed quantities. Equation (29a) gives

= V~(I + z2/6a 2 - zft~2/4a~) + z C e " - 2~i,'(z/a - 3h~214~)/3a

(30)

when using 2 + q = 3/2, 4q + 1 = - 1 and equations (21). Now ~ = fd ~. dt, or = t V ( i - tl2tl)(1 + z2/6a 2 - BZflzJ4aff,)/L+ z ( w ' ~ - ff'~) + (B2h/2a~ - 2z/3a2)(w ' - if'),

(31)

Dynamic axial plastic buckling of stringer stiffened cylindrical shells where ~' is the initial radial displacement imperfection. Thus, equation (13) takes the form a = a ° + zE'(w~

- ~ " ~ ) + E ' ( h / i Z / 2 a d " - 2 z / 3 a 2 ) ( w ' - ~,),

(32a)

where 000 = 00~ + E'tV(I - t/2t/)(1 + z216a2)/L - E ' t V h [ ~ 2 z ( l - t / 2 t / ) 1 4 a ~ L

(32b)

which is simplified to 000 = ~ + E ' t V ( 1 - t / 2 t / ) l L

(32c)s

in the subsequent analysis. Equations (27), (28), (30) and (32a) with (32c) may be substituted into the Prandtl-Reuss constitutive equation (w, - 200(2d~ + i , ) / 3 i ) which when neglecting the higher order terms predicts 00, = - w°(1 + z / 3 a ) + 2oo°(1 - 2 z / a ) w ' / 3 a Vm - z E ' ( w ' ~ - ~ ' ~ ) + E ' ( 2 z / 3 a 2 - h ~ 2 / 2 a d ' ) ( w ' - ~ ' ) - z00°w'.~13 VI + E ' t V h / 3 2 ( 1 - t l 2 t / ) z ( 1 - 2~b'/3a VO/4aL6".

(33)

Similarly, ~s = - 2 z a ° / 3 a +

000]~/~2/2ff+ z E ' h~2( w ' ~ - ~ ) / 2 f f - 2 z E ' f l ~ 2 ( w ' - ff")/3a2~

- 2z00°g'~/3 V~ - 4 z a ° ~ , ' / 3 a 2 V~ + E ' / 3 ~ h 2 ( w ' - ~')/4a~ 2 + 400°w'/3a V~ - z E ' t V h ~ 2 ( l

- t/2t/)~i,'/3a 2 V ~ L ~

(34)

Thus, /'h/2+h'

N~ = J-|u2 a~(l + z / a ) d z

gives I~ = I~° + 2f:'13a Vl - A/~2(w '- f:')/2a~ + 2_f~'(A* - 2A*e*)/3a Vl - A A * e * a ( w ' ~ - ~'~) + AA*(2e*/3 - h~2/2~,)(w '- ~i,')/a- A*e*aw'~/3 V~,

(35)~-u

where (36)

= E,l~O

while No =

becomes

F

2

hi2

ao d z

I~0 = I~° + ~.g2~4(w' - f:')14ad2 + 4g,'13a Vj

(37) 9,12

Similarly, ~

=/~,o_ a2w,/18a Vl - A a h ( w ~ - ~ ) I 1 2 + A(- h~2~3124ff,h + a3/18hXw '- 6:) - ah~i,'~/36 Vi - A V t ~ 2 h ( l - t/2tl)( - a2148 + a 2 w ' 1 7 2 a V D I L d

+ 2(A'e* - 2 I * ) w ' 1 3 a V i - A a I * ( w ~ -

- ~'~) +

A( - 3 2 K A * e * I 2 ~ + 2 I * 1 3 ) ( w ' - ~ ' ) l a

I * a w ' ~ 1 3 VI - A V t ( l - t12t/)h[32~ - 1 " / 4 + I * w ' / 6 a V O / I - ~ .

(38)

The equation of motion for axisymmetric behaviour according to equations (55) and (56) of Ref. [23] is Mx.~ + (Nxwj)j

- N d a = ~fO

(39a) 13

~l'his is identical to equation (13) with ~ given by equation (14) for dominant motion. The ~_000 terms in equation (32a) are related to the perturbed solution. 9As remarked in the previous footnote a in equations (18) and (19) for the dominant solution is the same as 000 defined by equation (32c). re°In common with Refs. [8, 19] terms related to second moment of area of shell thickness h have not been retained in the membrane forces. "The last term in equation (33) can be written ( 0 0 ° - 0 0 2 ) h ~ 2 z ( l - 2 w ' / 3 a V O / 4 a d " and is negligible compared with other like terms when/32~ 1. I~'he last term in equation (34) can be written (o ° - o'y)~82hzg"/3a2Vmd " and is negligible compared with 4 0 0 ° w ' / 3 a V i when ~82,~ 1. wrhis equation should contain an additional term M o / a 2 to remain consistent with equation (4) according to the principle of virtual work. However, this term is not considered in Ref. [19] and is neglected here.

8

N. JONES and E. A. PAPAGEORGIOU

using the notation in Fig. 3, or M'~

+ N ° w ' ~ - N ~ a = p ( h + h'Db"

(39b) t4

when neglecting higher order terms and recognising that N ° = - p a ( h + h')f# ° which was used to obtain equation (1). Now substituting N, °, N~ and M" from equations (18), (37) and (38), respectively, gives Al~,:xo/ ~ + A2(8,x*x.x.x. - g,x.x.x.x') + A38,,.,. + m4(8,x.x* -/~,x*x*) + m s ~ , : : x , x * / ~ + A6(8 - ~) + A7~]~ + g = O,

(40) 15

where = ffp/a, x* = x / L ,

80= w°/a,

8 = w'/a

(41a-c) 16 (41d,e)

E, = E ' t : / p L ~

and A, = - a2~/18/~ + 2d(A*e* - 2I*)/3/~- a2//2(1 - o-y/or°)/72-//51"(1 - o,~hr°)/6 A: =/32/~(T* + a~/12)//~

(42a) 17 (42b)

A3 =/~ {1 + A * + A * e * / 3 - (1 - o r h r ° ) h / / 2 A * e * / 4 ~ } / A h

(42c) '7

A~ = E {A*e*//2h/2~, - aZ/18 - 2 I * / 3 } / h

(42d)

A5 = - ff,A J 3 f f ,

(42e)

As =//2J~h/4~2

(42f)

and

A7

=

--

4c7/3//:1~

(42g)

Equation (32c) is recast into the form 1+ ~ = 2(1- o'rlo'°)LIAVt for the purpose of transforming equation (39b) into equation (40). If 8(x*, ~:) = ~, 8.if) sin (nlrx*)

(43a)

n=l

with

8(x*, 1) = g = .=,~ 8.(1) sin (nzrx*) = .~ a. sin (nzrx*)

(43b) Is

8(x*, 1)= 8 = .~/~.(1) sin (nlrx*) = . ~ b. sin (n~-x*)

(43c) Is

and

is substituted into equation (40), then ,~;, - O.,~./~:- R28. =

S.a.

(44) 19

where Q,. = (nn.)2Al - (n,n.)4Ay..A7

(45a)

R. 2 = _ (n~r)4A2 + (ncr)2A3 + (ncr)2A4-A6 .

(45b)

and S. = (nrr)4A2 - (n,n')2A4 + As.

(45c)

Equation (44) is considerably simplified if ~r° given by equation (32c) is assumed constant which is t4Equations (37) and (38) may be written/V, =/~o + / ~ and Mx - - o - , ~5( " ) = a( )/at in equation (40) and all subsequent equations. - Mx + M~, respectively. ~s~,, w° and w' are initial, dominant and perturbation radial displacements, respectively. lTThe last two terms in equation (42a) and the last term in equation (42c) are very small when//2 ,~ 1 and were neglected in the numerical calculations. Is~: = 1 when t = 0 according to equation (16b). 19 " " An identical result is obtained for an equation similar to equation (43a) but with cos (nctx*).

Dynamic axial plastic buckling of stringer stiffened cylindrical shells

9

reasonable for materials with low strain hardening. Equation (44) then takes on the form of equation (24) in Ref. [13] or equation (15b) in Ref. [22] and has the solution 8. = ,e#{Clls(R.O + C2Ks(R.~)} - S.a,,/R.2

(46a)

when Rfl > 0 and ~ > 0, where =

(1

+

O.)12

(46b)

and Is is a modified Bessel function of the first kind of order ~, while K~ is a modified Bessel function of the second kind of order ~. The two constants of integration Cl and C~ are found from the initial conditions (43b) and (43c) which allows equation (46) to be cast in the final form

8, = E,(Oa, + F,(~)b,

(47a)

E.({D = R.~ s {K~-I(R.)I~(R.~) + I~_j(R.)Ks(R.~)} (I + S./R. 2) - S~/R. 2

(47b)2°

F, ( 0 = ~s {Ks(R, )Is(R,~) - Is(R. )Ks(R,O}.

(47c)~

where

and

E,(O and F,(O are the amplification or magnification functions of the initial dimensionless imperfections in the displacement and velocity fields, respectively. An asymptotic solution is now sought for the amplification functions at t = tf(~: = 0) when dominant motion ceases. It is evident from the standard expressions in Ref. [24] that tim {~:SKs(R,O} ~ 2~-'(~ - 1)!/R, s

lim {~Is(R,~:)} ~ 0, and

(R/2~2r+~=l

-

,_--'6r[(r + ~ - 1)["

These expressions allow equation (47b) with ~-*0 to be written E.(0) -

I+

(48a)

(~- I)! ?=__or!(r+ ~_ i)!

or

E'(0)-(I+RG~){I+(~-I)',ffi~r,(r+(R'2/4)" S'p-1),/-R-~ "

(48b)

which if ~ ~ 1 becomes E,(0)_(I+RS,){I+_~,

(R,2/4)'1

S,

(48c)

E,(0) - (1 + S,,]R, 2) exp (R,~/4~) - S,]R, ~.

(49)

provided the series converges fairly rapidly. Thus

Similarly, F.(0)

-

-

{exp (R,,2/4~)}/2f,

(50)

and therefore E,(0) = -2~(1 + SJR,2)F,(O)- S,JR, 2.

(51)

The foregoing theoretical analysis is valid for cylindrical shells with inside or outside longitudinal stiffeners. The dimensionless parameter e* defined by equation (19d) is positive for stiffeners on the outer shell surface and negative for ones on the inner surface. DISCUSSION The solution of equation (44) is considerably simplified when o-° given by equation (32c) is assumed constant which is reasonable for materials with low strain hardening as previously noted. Now equation 2°E,(I) = 1 a n d F , ( 1 ) -- 0 w h e n t -- 0(~ = 1).

10

N . JONES a n d E. A. PAPAGEORGIOU

(32c) with t = t/(~: = 0) gives tro~ _ trytro _ E,p V~ 6[2

(52)

= 0

when using equation (15) with tr replaced by tr0 (see footnotes 8 and 9) for the response duration tf. Equation (52) predicts y = {1 + (1 + 2AK6)~/2}/2,

(53)

where 3" = tro/try,

and

A = F/try

K = pValo'r

(54a-c)

Thus, 3' may be calculated for a given dimensionless initial velocity of mass M ( K ) , mass ratio (6 - 1), and ,~ which is the ratio between the tangent modulus E' and yield stress try for the m_aterial from which a stiffened cylindrical shell is made. ;t defined by equation (36) is then given by A/3' and E defined by equation (41e) becomes A K 6 2 / y 2. If a cylindrical shell has stiffeners on the outside with rectangular cross-sections (A), then it may be shown that the dimensionless second moment of area of a stiffener about the middle surface of a cylindrical shell is I* = A*(4e . 2 - e*a + a2/4)/3

(e* -> 0)

(55a)

while I* = A*(4e *: + e*a + a214)13

(55b)

when e* -< 0 for internal stiffeners, so that I* is fixed once A* and e* are specified. In the extreme case of a bulb type stiffeder with a negligible web between the bulb and the cylindrical shell, (56)

I* = A * e .2.

In order to simplify the presentation of the theoretical results the following values have been used in all the" calculations unless otherwise stated: a = 0.45 in. (11.43 ram), p = 0.098 Ib/in? _(2712.6 kg/m3), try = 42,0001b/in? (289.6MN/m2), /3=0.1, ,~=3 and 6 = 4 . 5 . The values of p, try and ,t are typical for the experimental results obtained from axial impact tests on aluminium 6061T6 unstiffened cylindrical shells reported in Table 1 of Ref. [13]. 21 The theoretical formulation contained herein may be used for unstiffened shells with A* = e* = I* = 0 which have been examined in Refs. [12-15]. The theoretical predictions of the present method for this case are compared in Table 1 with the experimental results reported in Refs. [12, 13] and the theoretical formulations by Vaughan (Table 1 of [13]) and Wojewodzki (Table 1 of [14] with 3' = oo for strain rate insensitive material). Vaughan[13] assumed tro= 50,0001b/in.2 (344.74 MN/m 2) for all the calculations in

0.2222, a = 0.45 in. (11.43 ram), )t = 3, try = 42,000 lb/in. 2 (289.6 MN/m2), p = 0.098 lb/in. 3 (2712.6 kg/m3), A * = e* = I * = 0 TABLE 1. COMPARISO_N OF THEORETICAL AND EXPERIMENTAL RESULTS FOR ot =

TUBE flUNBER

EXPERI~NTS [12, 13]

e

~"

PRESENT THEORY

VAUGHAN [13]

WOJEWODZKI [14]

K

[12,13]

o

nC

¢ xf

o e xf

nc for nc for EnC(O) FnC{O)

o ¢ xf

nc for FnC(O)

nc for EnC(O)

nc f o r FnC(O)

1,Z,3,

0.15

4.52

0.0958

0.168

8

0.172

II

9

0.182

9

12

lO

4

0.15

4.52

0.0958

0.183

8

0.172

II

9

0.182

9

12

lO

0.11Z5 3.35

0.0721

0.095

12

0.I04

15

II

0.I02

II

O.11Z5i 3.35

0.1017

O.l~

12

O.141

15

II

0.156

12

ZO

0.1125 3.35

0.1469

0.195

12

0.191

15

12

0.207

12

15

13

ZZ

0.075

4.95

0.0526

0.102

14

O.lll

22

17

0.109

16

23,24

0.075

4.95

0.0836

0.147

15

0.166

22

18

0.174

16

23

19

10 13,14,15

2~Presumably p = 0.98 lb/in. 3 is a misprint in Ref. [13]. 6 ranges from 3.35 to 4.95 and/3 varies from 0.15 to 0.075.

Dynamic axial plastic buckling of stringer stiffened cylindrical shells

11

Table 1 whereas equation (53) has been used here and gives different values of ¢0 as expected. For example, o ° = 60,833 lb/in.2 (419.4 MN/m 2) for tubes 1-4 and ~0= 60190 lb/in,z (415-03 MN/m 2) for tubes 23 and 24 in Table 1. Despite these differences the present predictions are very similar to Vaughan's theoretical predictions and the corresponding experimental results except for specimens 22-24 with the smallest values of /3. They appear to be slightly closer to the corresponding experimental values than Wojewodzki's and if material strain rate sensitivity had been considered in the present analysis then the theoretical predictions would lie even closer according to the trends in Ref. [14], although the strain rate sensitivity of aluminium 6061T6 is contentious [25]. Typical variations of the amplification functions E,(0) and F,(0) with harmonic n according to equations (49) and (50) for stiffened shells, are presented in Figs. 4 and 5, respectively. The peak of a curve gives the maximum value of the corresponding amplification function for a specified dimensionless initial velocity (K). The value of the associated mode is known as the critical mode number (n c) and varies as K increases from 20 to 19 in Fig. 4 and 16 to 17 in Fig. 5. Thus, the components of the initial imperfections which lie in the critical mode have the greatest propensity to grow so that the critical mode numbzr is most likely to dominate the buckled profile of a stiffened cylindrical shell. However, the curves in Figs. 4 and 5 indicate that a broad band of harmonics grow, especially for the lower values of K, so that some scatter might be anticipated in experimental tests. 22 Some further comments on this observation for cylindrical shells loaded impulsively may be found in Ref. [21]. It is evident from Figs. 4 and 5 and equation (47a) that initial imperfections in the initial displacement field are more deleterious than initial imperfections in the velocity field. 22 Thus, from a practical viewpoint, the behaviour of the displacement amplification function E,c(O) is more important than the velocity amplification function F,c(O). The variation of the peak values in Figs. 4 and 5 with K is shown in Fig. 6 and indicates that large changes in E,c(O) and F,c(O)occur for small changes in K for the larger values of K. This phenomenon has been observed for other structures and has led to the concept of a threshold impulse or critical velocity below which a structure is "safe" and above which unacceptably large deformations occur (also known as dynamic plastic buckling). Clearly, the definition of such a threshold impulse is arbitrary. However, the imperfections in the initial profile can be measured or estimated from the allowable manufacturing tolerances and an estimate of the maximum acceptable displacements could be made from a knowledge of

K = 0.175

30

En(O)

K=0.15

20 K= 0"125

K = 0.10 10

K= 0'075

o

'

'

2o'

'

'

n

FIG. 4. Displacement amplification function En(0) vs mode number n for various dimensionless velocities K and a = 0.1, s = 10, e* =0-075, A* = 0.25 and I* = 0.001458 for stiffeners with rectangular cross-sections. ~If ~,'=O(h), then a, =O(a) according to equation (43b). Similarly if Ow'/Ot=O(aV) at t=O then b,=-O(K05aly~) which gives bn=-O(3a) when K=O.1, /3=0.1, T=0(I.5) and 05=4.5. However, F.c(O) = - 0(10 -3 E~c(O)) according to Figs. 4 and 5 so that equation (47a) indicates that the influence of a, dominates b,,

o

'

lb

'

K-0"075

K-O.IO

:;o

'

R

3'0

70

F]o. 5. Velocity amplification f u n c t i o n /7,(0) vs m o d e n u m b e r n for various dimensionless velocities K and the same p a r a m e t e r s as in Fig. 4.

l}O0!

0010

0-015

0.029

0.025

-Fo (0)

0.030

0.035

K= O 175

o

f

/

/

!

!

/

i

,

olo5

/

(2o:

I

I

I

t

i

/

(201

o'-Io

~ z (191

K

I

'7

f, #

I

#

t

(9

~i I (

' /

I

I I I I

I

I

i

/ / I

i

I

!

/l/iiI/;/,i,,,;'/.I

/

/

/

F~(O)

i e': - 0 0 7 s I I

o'-15

19)

t

19l

e'= (}075

FIG. 6. D i s p l a c e m e n t and velocity amplification f u n c t i o n s vs d i m e n s i o n l e s s initial velocity K for same parameters as in Figs. 4 and 5. Also e* = - 0 . 0 7 5 case. N u m b e r s in p a r e n t h e s e s are critical m o d e n u m b e r s for E,c(O). O) unstiffened cylindrical shell with a = 0.1221, fl = 0-1012 and a = 0-4553 in. ( l l . 5 6 m m ) , and (~) unstiffened cylindrical shell with a = 0 . 1 2 5 , / 3 = 0 - 1 , a = 0.45 in. (11-43 mm).

0

10

15

20

-~O3Fo~(O)

25

30 Eoc(O)

35

.....

--E#(o)

i

I

ml ©

>

>

Z

Z

Dynamic axial plastic buckling of stringer stiffened cylindrical shells

13

the function of the structural member. The ratio of these two numbers gives an estimate for the largest acceptable value of Enc(0) which can then be used in Fig. 6 to find the associated threshold impulse, or, more generally, substituted into equation (49) to give the corresponding critical velocity (V c) from a transcendental equation. The critical mode number (n c) is not usually known, so to simplify the procedure for preliminary design purposes and to avoid a transcendental equation a trial value for the velocity V could be substituted-into equation (49) to give the corresponding value of E,(0). These calculations are then repeated for several values of n until the maximum value of En(0) is found. The associated value of the critical mode number n c is then assumed to be independent of the velocity V, which appears reasonable from the results reported in Figs. 6 and 7. Finally, equation (49) with n c can then be solved for several values of V until the critical or threshold velocity Vc is found which produces the largest acceptable value of E~c(O). If V~ is substantially different from that used in the initial calculations which were made to establish nfl then equation (49) with V = V~ could be solved for values of n either side of n c to establish the peak value. This whole process can be done quite quickly on a calculator and is therefore ideal for preliminary design purposes or for the checking of a particular design. The curves in Figs. 6 and 7 with e* > 0 correspond to rectangular shaped stiffeners on the outside surface of the shell with I* given by equation (55a), while the e* < 0 results correspond to identical stiffeners located on the inside surface. It is evident from these curves that it is much more efficient to place the stiffeners on the outside surface and this observation is in accord with the results presented in Refs [1, 2, 19] for the elastic buckling of stringer-stiffened cylindrical shells subjected to static axial loads. If e* < 0, then it is evident from equation (22) that the dimensionless resisting bending moment Mx° is smaller than the corresponding value when e* ~>0 for outer stiffeners. This is similar to the observation made by Baruch and Singer[19] to explain the difference in buckling loads for inside and outside stiffeners for the static elastic case. Now the mass of a stiffened shell is m = 21rahLp + sALp, or (57)~

m = 2trahLp(l + A*). I I

e w= - 0"15

I

- -

Eric(O)

.....

F.~ (0)

I ] (1/.)

I

40

I I

I I I I

t

I I

30

I

Eric(o)

/

I

I

-10s Fnqo)

:®

I

,'

I

/

/

20

/ /

I /

/ /

•

#'

/

/ /

.. e~-0.1S

/ I // /

113 I / J S

(1/.) 0

(13)

114;

0

0.1

0-2

0"3

Flo. 7. Displacement and velocity amplification functions vs dimensionless initial velocity K for a = 0 . 2 , s = l O , A*=0.25, e*=-+0.15, 1"=0.005833 for stiffeners with rectangular cross-sections. Numbers in parentheses are critical mode numbers for E,c(O). (]) unstiffened cylindrical shell with a ffi 0.2392,/] = 0.1022, and a = 0.46 in. (11.68 ram), and (~ unstiffened cylindrical shell with a = 0.25,/3 = 0.1, a = 0.45 in. (11-43 ram). 23It may be shown using equations (19c) and (21b) that 1 + A* =/~ M~ Vol. 24, No. I--B

14

N. JONES and E. A. PAPAGEORGIOU

The initial kinetic energy of a stiffened tube and end mass M travelling with an initial velocity V is ( M + re)V2/2, which using equations (57), (21c) and (54c) becomes K E = rrahLtry6"(1 + A*)K.

(58)

Consider two shells labelled l and 2 with the same masses and kinetic energies and same ~ry and p, but different a, h, L. 6, A* and K. Thus, alhlLl(1 + A'r) = a2h2L2(1 + A~)

(59a)

alhlLt6"l(l + A'r)KI = a2h2L262(1 + A~)K2

(59b)

61KI = 62K2

(59c)

and

which require

If we further assume L1 = L2 and take K, = / ( 2 , then equation (59c) demands 61 = 62, while equation (59a) gives athl/a2h2 = (1 + A~)/(I + A'I')

(59d)

or al/~12/0~2/~2 2 =

(1 + A~)/(1 + A'f).

(59e)

Thus, if shell 1 is an unstiffened shell with AT = 0, then the mass and initial kinetic energy are identical to a stiffened cylindrical shell 2 with same ~ry, p, L, 6 and K when a~/3, 2 = a2/322(1 + A~). One possibility is for /31 =/32 with OtI = a2(l + A~).

(60)

However, as pointed out in footnotes 1 and 3 and in Ref. [19], the mid-surface of a cylindrical shell is taken as the reference surface for both stiffened (2) and unstiffened (1) cylindrical shells. If the additional shell thickness h~ for a given stiffened shell is obtained using a mean radius a2 + hJ2 + h~/2, then h~/h2 = [{(2/a2 + 1)2 + 8A$/a2} I/2 - (2/a2 + 1)]/2

(61)

and therefore al = (h2 + h~)/(a2 + h~/2) for an equivalent unstiffened cylindrical shell becomes al = [2 {(1 + 0~2/2)2 + 2ot2A~) 112- 2 + a2]/[{1 + a2/2) 2 + 2a2A~} I/2 + 1 - a2/2].

(62) 24

This approach preserves the same total cross-sectional area of the shell and stiffeners so in order to maintain the same total mass and initial kinetic energy for the unstiffened case requires the same length L or LI = al/~l = a2/~2, which gives, El = (al/a2)/32,

(63a)

al = a2 + h~/2 = h2(1/a2 + h'~/2h2).

(63b)

//i = a2//2[{(2/a2 + 1)2 + 8A~/a2} jr2- 1 + 2/a2]/4

(64a)

al = h2[{(2/a2 + 1)2+ 8A~1~2} 112- 1 + 2/a2]/4

(64b)

where

Equations (63) may be written

and

respectively. Now, equation (60) for the particular case considered in Fig. 6 requires al = 0.125 with AT = 0, while equations (62) and (64) predict al = 0.1221,/31 = 0.1012 with al = 0.4553 in. (11.56 ram). It is evident from Fig. 6 that these two sets of results for equivalent unstiffened shells lie between the theoretical predictions for e* > 0 and e* < 0. All cylindrical shells have the same mass and therefore the same initial kinetic energy for a specified value of dimensionless initial velocity (K). Thus, using the unstiffened shells as a basis, it is more efficient to arrange the material as stiffeners on the outside surface of a shell rather than on the inside surface. In fact, this unstiffened shell absorbs the initial kinetic energy more efficiently than a shell having the same mass but with part of it arranged in the form of internal stiffeners. This situation is partly due to the expression trx = - tr(1 + z/3a) from equation (12a)with/32 ,~ 1 which for a given value of tr gives smaller axial stresses and axial bending m o m e n t s for inside stiffeners with z < 0. The first and second terms of equation (33) for trx act in a UThe reference surface for the equivalent unstiffened shell has a mean radius a: + h~/2.

Dynamic axial plastic buckling of stringer stiffened cylindrical shells

15

similar fashion, but the third, fourth and fifth terms do not. The foregoing observations are also confirmed for the thicker shell examined in Fig. 7. The variation of critical mode number (nO)and displacement amplification function (E,c(O)) with dimensionless area of stiffener (A*) is shown in Figs. 8 and 9 for stiffeners with rectangular cross-sections. In both cases the inside stiffeners exhibit an optimum response when A* = 5/16, while both n c and E,c(O) decrease with increase in A* for outside stiffeners. It is evident from equation (57) that the mass of a stiffened cylinder increases with A* for given a, h, L and p. However, the initial kinetic energy of the system also increases with A* according to equation (58) for specified values of a, h, L. ~r, ~ and K. Thus, the increase in material volume for the externally stiffened case is sufficient to absorb the additional kinetic energy, while it is not adequate for the internally stiffened case. The results in Figs. 8 and 9 also show that stiffeners (10 in Fig. 8 and 4~ in Fig. 9) with very small cross-sectional areas lead to dramatic reductions in the values of E,c(O) obtained for the unstiffened case. The theoretical predictions in Fig. 10 indicates that Emc(O)decreases with increase in the dimensionless second moment of area of a stiffener for the internal and external stiffened cases as expected. However, the greatest change for this particular case occurs for values of I* between approximately one and two times that of a rectangular cross-section, but little additional benefit is achieved for larger values of I*. E,c(O) decreases with increase in le*l in Fig. 11 for both inside and outside stiffeners with rectangular cross-sections on cylindrical shells. A* is constant and a = 0.1 for each of these curves so that I* according to equations (55) increases with e*. However, the two sets of curves in Fig. 11 have the same initial kinetic energy according to equation (58) since (1 + A*)K is the same for both but the masses are different (equation 57) because 1 + A* is 1.25 in one case and 1.5 in the other. The curves in Fig. 12 are for different values of a but both have the same initial kinetic energy and mass. Each of the curves in Figs. 8 and 9 is plotted for a constant value of e*, while each of those in Figs. 11 and 12 is plotted for a constant value of A*. Therefore the curves in all cases give the variation with A'e* by the simple expedient of altering the scales of the abscissae. The quantity A'e* is a dimensionless form of Ae which is the first moment of area of the stiffeners about the mid-surface of the shell.

I

25~)"

rf

o e'. 0.2

,k

x e%-.O. 2

20

I

0

0"5

'

1)0

'

(o)

A"

1~5

2!0

I

16

I I I I

12 Eric(O)

//

~,

t / e'--0.2

e%0-2 0

0

I

OJ'5

I

1!0

(b)

I

1~'5

I

210

A"

FIG. 8. Critical mode number (n c) and displacement amplification function (F~c(0)) vs A* for a =0.1, s = 10, e*=-+0-2, K = 0 . 1 7 5 and I* from equations (55) for stiffeners with rectangular cross-sections. [] denotes unstiffened case with a =0-1, K=0.175 and A* = e* = I* =0. ~s = 4 in an extreme case for a smeared theory.

0

2

'

'

0'-5

0!5

'

'

xI

I

I

I

i!

(b)

1.'0

,0.

/ e%-02

I

/

(o)

1!0

--

'

e%02

A"

'

A~

1.'5

1.'5

FIG. 9. Critical mode number (n c) and displacement plification function (E,c(0)) vs A* for a = 0 - 1 5 , s = 4 , +-0-2, K = 0 . 1 and I* from equation s (55) for stiffeners rectangular cross-sections. [] denotes unstiffened case a =0-15, K = 0 - 1 and A * = e * = I * = O .

Er,~(O)

0

5

0

ame*= with with

x e'= -0-2

o e"- 02

0

l.

8

12

16

0

_ _ _ L

I

~ _ _

C~002

I

I

e~=0-075 ~

0002

I

0-00~.

I

0004

I

I~

I

It

I

I

0.006

I

I

"... _

0006

I

l

o e'= o-o7s X el= -0075

FIG. 10. Critical mode number (n ~) and displacement amplification f u n c t i o n (E,c(O)) vs I * f o r a = 0 - I , s = I0, e* = -+ 0.075, A * = 0.25 and K=0-25. ( I * = 0 . 0 0 1 4 5 8 f o r stiffeners w i t h rectangular cross-sections.)

20

~.0

60

Eft(O)

8O

100

n~

2O

- -

©

0

r~ >

Z

Z

0

10

0

I

t I

®

®

o,

I

\ \ ,,\

0! 1

~

t I

Ii

II

II

(2),I

Ot

0..

~

:..

I

(b)

-

I

=

I

I

o~

_

-

_

-

I~*I

I

= _--= - : = =

e'
e~W>O

I

e~>o

03

I

= =

I

FIG. I I . Critical mode number (n ~) and displacement amplification f u n c t i o n (E,c(O)) vs le*l for a = 0.1, s = 10, and I* f o r stiffeners w i t h rectangular c r o s s - s e c t i o n s a c c o r d i n g to e q u a t i o n s (55) and (~) A* = 0.25, K = 0.15, ( ~ A* = 0.5, K = 0-125. ( X is v a l u e o f n ~ f o r e* = - 0 - 2 o n c u r v e (~).)

20

/,0

Enc(O) 6O

8O

100

n~

15

20 c

0

I.

o

0

'

I

o

!2

o

"12

I

~

(b}

l

(o)

I

'

o~

o~

l

'

i~oi

I'I

I

®

o6

!

06

|

FIG. 12. Critical mode number (n c) and displacement amplification function (E,c(O)) vs le*l for s = 10, K = 0.175 and I* for stiffeners with rectangular cross-sections according to equations (55) and (]) a = 0 - 1 , A * = 2 , (~) a = 0 . 2 , A * = 0 - 5 ( X is value of n c for e * = - 0 - 2 on curve ~)). rqa = 0-3, A* = 0.

0

End(Of

rl

=__

ch

or" e-

go

18

N.

JONES and E. A. PAPAGEORG1OU

If the dimensionless parameters A*, e*, I*, K and a are constant, then n c, E.c(O) and F,,c(O) are independent of the number of stiffeners (s) as shown in Fig. 13 which is a result of smearing the stiffeners to produce an additional shell thickness h'. Now, if

A~ = s2Aflsl

(65)

then two stiffened shells 1 and 2 with at = a2 and hm= h2 have stiffeners with the same cross-sectional area A. The variation of E,c(O) and n c with s is shown in Fig. 13 for shells which satisfy equation (65) and have the extreme bulb stiffeners defined by equation (56). However, it should be noted that the initial kinetic energy and masses are constant on the horizontal lines but they are not constant on the curves because A* varies with s according to equation (65) so that both mass and initial kinetic energy increase with s according to equations (57) and (58), respectively. Additional results are given in Fig. 14 for a shell stiffened with rectangular stiffeners. Both sets of curves conform to equation (65) but one set in addition has decreasing values of K with increase in s so as to maintain a constant initial kinetic energy according to equation (58). Despite this, the curve for E,c(O) with e* = - 0 . 2 rises with increases in s above s = 12, approximately. However, the permanent axial displacement up decreases for this particular case as s increases. Furthermore, m given by equation (57) increases with s so in order to maintain a constant value of ff = 1 + M/m, the end mass M must increase in the same proportion. Finally, although the dimensionless first moment of area (A'e*) and second moment of area (I*) of each stiffener increases with increase in s, the actual first and second moments of area are constant on a given curve. The type of dynamic plastic buckling examined in this article is related to "direct" dynamic buckling which is discussed in Ref. [26] for a simple elastic-plastic model. Another type of buckling known as "indirect" buckling was also found for the idealisation in Ref. [26] within a certain range of the parameters. "Indirect" buckling is characterised by a gradual build up of the buckling deflections over a number of vibration cycles, whereas "direct" buckling develops with an almost monotonic increase in displacements. The theoretical method in this article is not capable of dealing with "indirect" buckling which is a form of dynamic plastic buckling which has rarely been explored in the literature.

A'*" = 0"1

16

A° : 0.25 12

A* =O'S

13c

H

A*=I ----.0

era: 0.2

---x---

e':-0.2

I

I

I

I

,I

8

16

24

32

40

1o)

s

. . . . . . . . . . . . .

12

.-.)(

A':I

// / /

9

/

E.c (0)

//@=-O.2

7 / _ _ ~':o.~

XqX.

__x._

6

/

/

,~.,:_I " _ "-x: • -=

A' = 0 " 5

=..-.-~X, --o.~g.

A" :O1 A" =0-25

A'=I 0

0

I

1

1

8

16

2/.

S

I

I

32

~0

FIG. 13. Critical mode number (nO and displacement amplification function (E,c(O)) vs number(s) of extreme bulb type stiffeners governed by equations (56) and (65) for a = 0.1, K = 0 . 1 and e* =-+0.2.

Dynamic axial plastic buckling of stringer stiffened cylindrical shells

19

--e---e'=02 - - Y - - - e'=-0.2

12 n~

/, 0 0

Z

~

;2

(Q)

2'0

~

s

I /

2~ E.c(0)

/ s

/

/C)

I I I

18

/ i i I p

f

i j

..'~" @ = - 0 . 2 . ,

.xO s

X "~.-

- ~

~x ~

~ ~ C 0

4

1'2

8

S

o(9 o0 16

20

Ibl Flo. 14. Critical mode number (n c) and displacement amplification function (E,c(O)) vs number(s) of stiffeners with rectangular cross-sections governed by equations (55) and (65) for a = 0-2 and e* =-+ 0.2. (]) K = 0.2, (~ constant dimensionless kinetic energy ((1 + A * ) K = 0.24). The various perturbation terms in the analysis have been assumed small so that the theoretical results are only likely to be valid for the initiation or onset of dynamic plastic buckling. Nevertheless, a similar procedure for the unstiffened case has predicted reasonable agreement with the corresponding experimental results [13]. The theoretical procedure is strictly speaking only valid for closely spaced stiffeners for which smearing is appropriate. In this connection, it might be worthwhile to extend the theoretical procedure to allow for non-axisymmetric behaviour which could possibly become important for stiffened shells with values of a smaller than those considered here [27]. Finally, it is clear that a need exists for experimental tests on the dynamic plastic buckling of stiffened cylindrical shells impacted axially.

CONCLUSIONS A perturbation method of analysis has been used to examine the dynamic plastic buckling of a stringer-stiffened cylindrical shell subjected to an axial impact. The cylindrical shell and longitudinal stiffeners are made from a rigid-linear strain hardening material and the influence of initial imperfections is considered. The stiffeners are smeared over the shell surface but the eccentricities and second moments of area of the stiffeners are retained in the basic equations as proposed by Baruch and Singer for the static elastic behaviour. It is found that a relatively narrow band of harmonics relating to the axial variation of the radial displacement field is amplified and leads to the p h e n o m e n o n known as dynamic plastic buckling which has been studied previously for unstiffened shells. It transpires that it is more efficient to place stiffeners on the outer shell surface rather than on the inner surface. Various results are presented which demonstrate the

20

N. JONESand E. A. PAPAGEORGIOU

influence of the s e c o n d m o m e n t o f area, eccentricity, c r o s s - s e c t i o n a l area and n u m b e r of stiffeners on the d y n a m i c plastic r e s p o n s e . T h e theoretical predictions f o r the unstiffened case are c o m p a r e d with the e x p e r i m e n t a l results p r e s e n t e d in Refs. [12, 13] but no e x p e r i m e n t a l results a p p e a r to be available f o r the d y n a m i c plastic buckling of stringer-stiffened cylindrical shells. Acknowledgments--The authors wish to record their gratitude to Dr. N. Perrone and Dr. N. Basdekas of the Structural Mechanics Programme of O.N.R. (Washington, D.C.) (contract N00014-76-C-0195 Task NR 064 510) and the Department of Ocean Engineering at M.I.T. who supported the initial phase of this work. The later stage of this study was conducted in the Department of Mechanical Engineering at the University of Liverpool. The authors are indebted to the following staff members of the Department of Mechanical Engineering in the University of Liverpool: Mrs. M. White for her expert typing, Mr. J. Boyes for his assistance with the numerical calculations, and Mr. F. J. Cummins and Mrs. A. Green for their preparation of the drawings.

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