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Plastic buckling of axially compressed circular cylindrical shells
Thin-Walled Structures I (1983) 139-163
Plastic Buckling of Axially Compressed Circular Cylindrical Shells Viggo T v e r g a a r d Department of Solid Mechanics The Technical University of Denmark, DK-2800 Lyngby, Denmark
A BSTRA CT For elastic-plastic cylindrical shells with initial axisymmetric imperfections bifurcation into a non-axisymmetric shape is analysed. The shell material is represented by a phenomenological plasticity theory that accounts for the formation of a vertex on subsequent yield surfaces. The influence of various geometric and material parameters is investigated for a wide range of radius-to-thickness ratios. It is shown that for the thicker shells bifurcation generally occurs beyond the maximum axial compressive load. A few analyses for shells with additional non-axisymmetric imperfections show the unstable postbifurcation behaviour and the sensitivity to imperfections of more general shapes.
1 INTRODUCTION The strong imperfection-sensitivity of axially compressed circular cylindrical shells in the elastic range results from interaction between several simultaneous buckling modes. This is shown asymptotically by Koiter's general post-buckling theory. 1 For the special case of sinusoidal axisymmetric imperfections Koiter 2 has obtained an upper bound to the load at which the axisymmetric deformation bifurcates into an asymmetric shape. This upper bound estimate can be considered a good approximation of the actual load carrying capacity, even for rather large imperfections, since it is based on a nonlinear pre-buckling solution. A more detailed investigation of the elastic bifurcation from an axisymmetric periodic mode, including an initial 139 Thin-Walled Structures 0263-8231/83/$03.00 (~) Applied Science Publishers Ltd, England, 1983. Printed in Great Britain
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Viggo Tvergaard
post-bifurcation analysis, has been carried out by Budiansky and Hutchinson. 3 For shells compressed into the plastic range results analogous with those of Koiter e have been obtained by Gellin,4 who considered a wide range of radius-to-thickness ratios and various degrees of strain hardenfng. These analyses were based o n J2 deformation theory, and thus the elastic-plastic material was represented as a nonlinear elastic material. It is well known that bifurcation calculations based on the deformation theory of plasticity generally agree much better with experimental buckling loads for plate and shell structures than similar calculations based on the simplest incremental plasticity theory. 5 As explained by Batdorf, 6 bifurcation predictions of deformation theory can be rigorously justified by appealing to a more sophisticated flow theory with a vertex on the yield surface. Such vertices are implied by physical models of polycrystalline metal plasticity, based on the concept of single crystal slip, 7'8 but experimental evidence is inconclusive. 9 However, the use of deformation theory moduli to analyse bifurcation at a vertex presumes that proportional or nearly proportional loading has taken place prior to bifurcation. For the cylindrical shell with axisymmetric imperfections the pre-bifurcation loading path is not always sufficiently near proportional loading, and thus a material stiffening effect is neglected in the deformation theory analysis. Also the possibility of elastic unloading prior to bifurcation is neglected by such nonlinear elastic analysis. The shell buckling behaviour in the plastic range is analysed here on the basis of a phenomenological corner theory of plasticity, which accounts for the material stiffening with increasing nonproportionality that was neglected in the deformation theory analyses. This so-called J: corner theory, proposed by Christoffersen and Hutchinson, t° has been used by Needleman and Tvergaard ~t to study the influence of yield surface vertex formation on the buckling behaviour of a cruciform column. For proportional loading the instantaneous moduli in J2 corner theory are chosen identical with those of J2 deformation theory so that bifurcation points predicted by the two theories on a proportional loading path are identical. For cruciform columns with initial imperfections deviations from proportional loading occur, and here the load carrying capacities predicted by J2 corner theory were found to exceed those of deformation theory,
Plastic buckling of axially compressed circular cylindrical shells
141
but to be lower than those of the simplest flow theory of plasticity with a smooth yield surface. In the present investigation for cylindrical shells the influence of differences in the vertex description, of various degrees of strain hardening and of various imperfection wavelengths are studied. In addition t o the bifurcation analyses a few computations for non-axisymmetric imperfections are included to illustrate postbifurcation behaviour and the sensitivity to imperfections of a more general shape.
2 SHELL E Q U A T I O N S AND CONSTITUTIVE R E L A T I O N S On the middle surface of the circular cylindrical shell with radius R a point is identified by the coordinates (x 1, x 2) = (z, R0), where z measures the distance along the cylinder axis and 0 is the circumferential angle. The displacement components are denoted u ~ on the surface base vectors and w on the outward surface normal. The strain measures used are the nonlinear membrane strain tensor co,13= 1/2(u~.1~+ u~.~,) - d~,~w + 1/2aYS(uy,ot + 1/2(w.~ + ~u~)(w~ + d~u~)
-
dy~w)(u6,[3 - dg3~514))
(1)
and the linear bending strain tensor specified by Koiter lz ~.~ = v : [ ( w , + ~ u ~ ) . ~ + ( w ~ + dffu~)., - 1/2~(u~.~ - u~.~) - l/:dff(u,.~ - u~.~)]
(2)
where u~,~ and d,~ are the metric tensor and the curvature tensor, respectively, of the undeformed middle surface, and ( ).~, denotes covariant differentiation. Greek indices range from 1 to 2, while Latin indices (to be employed subsequently) range from 1 to 3, and the summation convention is adopted for repeated indices. It is noted that strain measures proposed by Niordson 13 are identical with eqns (1) and (2), except for small differences in the bending strain measure of the order of dV~ev~. T h e three-dimensional constitutive relations are taken to be of the form 0 ij = L i j k l • k l
(3)
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142
where C j is the stress tensor, ~qkt is the strain tensor, L ~jk~ are the instantaneous moduli, and (') denotes an incremental quantity. Since the stress state in the shell is approximately plane, only the in-plane stresses enter into eqn (3). Thus, the constitutive relations can be written as Lc4~33L33V<",
daft = /_~,13y6/iv~' /~o,13v~= L,,,I~v~
L 3333
(4)
The in-plane components of the Lagrangian strain tensor at a distance x 3 outward from the shell middle surface are approximated by vl~13= ~ - x3r~
(5)
The membrane stress tensor N ~ and the moment tensor M ~ in a shell with thickness h are taken to be fh/2
N~a = j -h/2 (~°tl$dx3
'
MC'a "~
_ fh,2
J -hi20ctl$x3dx3
(6)
Then, from eqns (4), (5) and (6), incremental relations are obtained for ,~/~,13 and M~,I3 in terms of i~v~ and kv~. The requirement of equilibrium is specified in terms of the principle of virtual work
fA {N~fiSenf~+ Ma"8l%,g}dA= P~3up
(7)
where A is the middle surface area, P is the total axial load at one end of the cylinder, with corresponding axial displacement ur,, and at the other cylinder end zero axial displacement is prescribed. The elastic-plastic material behaviour is taken to be described by the phenomenological corner theory of plasticity, J2 corner theory, proposed by Christoffersen and Hutchinson. 1° In this theory the instantaneous moduli for nearly proportional loading are chosen equal to the J2 deformation theory moduli and for increasing deviation from proportional loading the moduli increase smoothly until they coincide with the elastic moduli for stress increments directed along or within the corner of the yield surface. W i t h Mffijk I denoting the deformation theory compliances, so that ~]i[ "~" M~ijk[~kl, and Mijkl denoting the linear elastic compliances, the plastic part of the compliances is Cqkt = M~ijkl -- Mijkl, The yield surface in the neighbourhood of the current loading point is taken to be a cone in stress deviator space with the cone axis in the direction kij = sil( cmnpqSmnSpq)- 1/2
(8)
Plastic buckling of axially compressed circular cylindrical shells
143
where s o = o ij - gSJ~k/3 is the stress deviator tensor and gu is the metric tensor in three-dimensional coordinates. The positive angular measure, 0, of the stress-rate direction relative to the cone axis is defined by
Cijkl~iJskl(fm.pqSmn~q)- I/2
COS 0 =
(9)
and a stress-rate potential at the vertex is formulated as 1
1
W = --Mijkt606 kt + -~f(O)CijktO'J6 kt 2
(10)
From this potential the strain-rate is obtained as 02W _ _
ilij - obijobkt
6d,t
= Mijkt( O)6 kl
(11)
with 0-dependent compliances. The plane-stress relationship (eqn (4)), with instantaneous moduli Lo,l~v,(0), is obtained by inverting eqn (11) numerically. The angle of the yield surface cone is specified by 0¢, so that the transition function, f(0), in eqn (10) is zero for 0¢ < 0 ~< ~x. In the total loading range, 0 ~< 0 ~< 00, f(0) is unity, and in the transition region, 00 ~< 0 ~< 0c, f(0) is chosen to smoothly merge the deformation theory moduli with the elastic moduli in a way ensures convexity of the incremental relation. The transition function to be used here is specified by riO) =
g(¢)[1 + 12(¢p)]
where l(q~)
O(¢p) = ¢p + arctan[l(cp)]
-~(dg/ddp)/g and, with 0~ = 0c - - - , 2 for 0 ~< ~ ~< 0o
1
g(q~) =
(12)
{1 - [ ( ¢ -
O o ) / ( O n - 0,,)] 3} - 2
(13)
forOo~
This transition function has been found ~° to rather closely duplicate moduli obtained using a self-consistent model of a polycrystalline aggregate. 8
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Viggo Tvergaard
T h e d e f o r m a t i o n t h e o r y compliance tensor used in the construction of the potential (10) has the form 12 M~i~k, = ~1 { v---~s (&kgi, + gi~gi,) - Vs gqgkt}
9(1 + 40-"-~
1)Sijb.kl Et
(14)
Es
w h e r e Es and E t are the ratio of stress and strain and the slope, respectively, for the uniaxial stress-strain curve, o~ = {3siisiJ/2} l/2 is the effective Mises stress, v is Poisson's ratio, and v~ = v E J E + (1 - EJE)/2. T h e angular measure, 0c, of the cone angle is taken to be given by tan0c = V ~ t a n 13c, a = (E/E~ - 1)/(E/E, - 1)
(15)
w h e r e , in terms of the initial yield stress Oy tan 13c = --Oy/(Oe 2 -- Oy) _2,V2, for
7t 2
~< [5c <~ ([5~)max
(16)
and otherwise 15c = ([5~)max. T h e uniaxial stress-strain curve is r e p r e s e n t e d by a piecewise p o w e r law with continuous tangent modulus
o
for o ~< Oy
E
E=
(17)
OYFl(&)n - - 1- + 11 E L t'/ \Oy/ ?'/
for o >
Oy
w h e r e n is the strain h a r d e n i n g exponent.
3 BIFURCATION ANALYSIS For a cylindrical shell with initial axisymmetric imperfections the
Plastic buckling of axially compressed circular cylindrical shells
145
possibility of bifurcation into an asymmetric shape will be investigated. Prior to bifurcation the solution remains axisymmetric. The equations governing bifurcation are obtained by assuming that there are at least two distinct incremental solutions (.)a and (')b corresponding to a given increment of the prescribed quantity. The difference between two such solutions is denoted by (-) = (')" - (')b, and using the incremental version of the principle of virtual work (eqn (7)) the following equation is obtained
+ (ff,~ + dV~av)(8w.~ + d~bu,)]}dA = 0
(18)
which must be satisfied by non-zero bifurcation solutions(-). Here, N ~'f~ are the current membrane stresses in the fundamental axisymmetric solution. To prove uniqueness of an incremental solution Hill ~4 makes use of the functional
+ (~,~ + dV~av)(~.13+ d~a,)]}dA
(19)
requiring I > 0 for any two solutions (')~ and (')b. At bifurcation I = 0, as is directly seen from eqn (18) by choosing the variation 8( ) = (-). In the case of a smooth yield surface and normality Hill la has introduced a comparison solid, identified by the plastic moduli in every material point currently on the yield surface and the elastic moduli elsewhere. For this comparison solid I is a quadratic functional, and the variational equation 81 = 0 is identical with eqn (18). The first critical bifurcation point of the comparison solid and that of the underlying elastic-plastic solid are, in most circumstances, identical and t.he comparison solid always gives a lower bound. If the fundamental solution based on J2 corner theory satisfies nearly proportional loading everywhere in the current plastic zone, so that 0 ~< 0o, the total loading (i.e. deformation theory) moduli can be used to define the comparison solid. The first critical bifurcation point corresponding to J2 corner theory is then identical with that predicted by deformation theory (see discussion by Needleman and Tvergaard 11 for J2 corner theory and by Hutchinson 5 for a pyramidal vertex). However, if the fundamental solution has 00 < 0 < 0c in some material
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Viggo Tvergaard
points, a comparison solid to be used in the solution of eqn (18) is not readily available. Using the deformation theory moduli may result in a rather poor lower bound, since bifurcation will be governed partly by the stiffer moduli of the transition range. On the other hand, using the instantaneous moduli associated with the current values of 0 on the fundamental solution gives an upper bound, which will be explained further, since these moduli are employed in the present investigation. A bifurcation solution based on the alternative comparison solid mentioned above (moduli corresponding to current 0-values in fundamental solution) is a possible solution for the underlying elastic-plastic solid. This follows from the fact that bifurcation stress-rate directions arbitrarily close to those of the fundamental solution can be enforced by superposing a sufficiently large fundamental solution increment on the e i g e n m o d e ( - ) . But earlier bifurcation is not excluded, since softer moduli resulting from a smaller angle 0 may reduce the value of the quantity under the integral in eqn (19) below that corresponding to the comparison solid. Determining the actual critical bifurcation point will be very complex, since the corresponding critical variation of the angular measure 0(x 1, x 2, x 3) must be determined simultaneously. It is expected, though, that the upper bound bifurcation predictions obtained by the alternative comparison solid will generally be rather good approximations of the actual bifurcation points. Eigenmodes are assumed of the form mx 2
/~1 m_ UI(X I) c o s -
R
mx 2
a2 = U2(xI) sin - R mx 2
v0 = ~,'(x I) c o s -
R
(20)
where the circumferential wave number, m, is an integer, and the amplitude functions UI, U2, W depend only on the axial coordinate x 1. The corresponding strain measures take the form
Plastic b u c k l i n g o f axially c o m p r e s s e d circular cylindrical shells mx 2
E,~ = E l l ( X m) C O S - -
m,x 2
,
Kll = g * * ( x * ) c o s - -
R
R
mx 2
"~12 = Et2(x 1) sin - R
mx 2
r12 = K't2(x1) sin - R
mx 2
E22 = E 2 2 ( x l ) c o s - -
147
mx 2
1(22 -- g 2 2 ( x 1) c o s -
(21)
R R The moduli of the comparison solid satisfy rotational symmetry, so that modes corresponding to different wave numbers, m, decouple when substituted into eqn (18). For each particular wave number the integration of eqn (18) in the circumferential direction is straightforward, after substituting eqns (20) and (21). Thus, in the reduced form the bifurcation eqn (18) depends o n l y on the one-dimensional amplitude functions U1, U2 and W. This equation is used to numerically determine the first critical bifurcation point for any mode number, m. The moduli governing the first critical bifurcation in the actual elastic-plastic solid do not necessarily satisfy rotational symmetry if part of the material has entered the transition region (00 < 0 < 0c) in the fundamental state. As mentioned above, the softer moduli corresponding to smaller 0-values than those of the comparison solid will be preferred by the critical bifurcation mode wherever possible. It is noted that the sinusoidal circumferential variation assumed in eqn (20) is no longer an exact bifurcation solution if the rotational symmetry of the relevent moduli is lost. For a circular cylindrical shell with no initial imperfections the first critical bifurcation mode in the plastic range is axisymmetric of the form ~-t,X1 /all =
B sin - - ,
7tX 1
ti2 ---=0,
v~ = C cos
tc
(22) tc
where B and C are constants. The critical stress, oc, and the corresponding critical half-wave length, lc, are given by 15
=
~2E1h' 61~-"~2
7t_R2h [--~'-2_] 1/4 r E2 LE1
/ ElZ \E1/.J
2---1/4
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148
where El, E 2 a n d E t 2 denote the physical values of the plastic moduli L 11:1, L 2222 and L 1~22, respectively, at the bifurcation point. In the elastic range the critical stress (eqn (23)) reduces to the well-known value oc = -{3(1 - v2)}-U2Eh/R, at which axisymmetric modes and several non-axisymmetric modes with circumferential wave numbers up to a certain value are critical simultaneously. In the plastic range bifurcation stresses corresponding to non-axisymmetric modes are slightly above the critical stress (eqn (23)) for an axisymmetric mode. :6 Bifurcation stresses of perfect circular cylindrical shells with o y / E = 0-0025 and v = 0.3 are shown in Fig. 1. For sufficiently thin shells, R/h > 242, bifurcation takes place in the elastic range, whereas for smaller radius-to-thickness ratios the critical stress (eqn (23)) is below the elastic value. The difference between the bifurcation predictions of J2 flow theory and of J2 deformation theory is shown in Fig. 1. The largest difference is found for a high hardening material (n = 5) and for rather thick-walled shells, where bifurcation occurs well into the plastic range. Bifurcation predictions of J2 Corner theory are identical with those of deformation theory in this case, where proportional loading (uniaxial stress) takes place prior to bifurcation. In Fig. 1 and in the
"\ [3(1_u2)l-1/2E h 2
J2 deformation theory
°o
~6o
260
R/h
300
Fig. I. Bifurcation stress versus radius-to-thickness ratio for perfect cylindrical shells (oy/E = 0"0025 and v = 0-3).
Plastic buckling of axially compressed circular cylindrical shells
149
following sections it is assumed that the length of the shell and the boundary conditions at the ends permit buckling modes with the critical wavelength for a long shell.
4 RESULTS FOR AXISYMMETRIC IMPERFECTIONS The geometrical imperfection in the stress-free shell is specified as an initial normal displacement, w, of the form ~-t,X1
w = - ~ h cos
(24)
l
where ~ is the imperfection amplitude relative to the shell thickness, and l is the axial half-wavelength. For the bifurcation mode eqn (20) an axial half-wavelength 21 is chosen a priori based on the physical considerations also discussed by Koiter 2 for elastic shells and by Gellin 4 for the deformation theory solution. Buckling in a short-wave mode in a circumferential direction will be stimulated in the axisymmetric wave-bottoms, where N 22 < 0, and impeded at the wave-tops, where N 22 > 0. Therefore the bifurcation mode peaks are expected at the axisymmetric wavebottoms, and here symmetry conditions are assumed at x ~ = 0, W,I
at x ~ = 2l
(25)
W,1
It follows from eqn (25) that the analyses in the present paper assume periodicity of the solution in the axial direction. In the numerical solution only half the interval of eqn (25) is considered, 0 ~< x I ~< 1, with symmetry of the pre-buckling solution and antisymmetry of the bifurcation solution prescribed at x 1 = l. A m o n g the many modes simultaneously critical in elastic perfect shells, the mode with an axial wavelength two-times that of the axisymmetric modes corresponds to approximately square buckles. In agreement with this, the critical circumferential wave numbers, m, found in the following are close to the value (:tR)/(21), particularly for small imperfection amplitudes ~. Bifurcation into modes with an axial wavelength longer than 2 / h a s been investigated for elastic shells by
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150
Budiansky and Hutchinson, 3 who found that such modes may be critical a little earlier. Pre-buckling solutions are obtained numerically by a linear incremental method. The current values of the field quantities N~ , M ~13, E,~, etc. are taken to express an approximate equilibrium state. By expanding the principle of virtual work (eqn (7)) about this state, using eqns (1) and (2), the equation governing the increments, fi¢~, M~I~, i ~ , etc. is found. To lowest order this incremental equation is fA
+
+
-
= ffbur,- [fa {N~'%e~'f~ + M'~%r~'"}dA - Pbur,]
(26)
The terms bracketed on the right-hand side of eqn (26) are included to prevent drifting of the solution away from the true equilibrium path; for sufficiently small increments these terms have little influence on the solutions found. For the axisymmetric solution, the integrations in the circumferential direction are trivial. In the reduced form eqn (26) depends only on the one-dimensional displacement increments til and ~i,, and the equation is solved approximately by the finite element method. The displacement increments are represented as Hermitian cubics within each element and, likewise, in the numerical solution of the bifurcation equation (eqn (18)) the mode amplitudes U1, U2 and W are approximated by Hermitian cubics in each element. Most of the results to be presented are obtained with only two elements over the interval 0 ~ x I <~ l; but computations based on four or six elements show good agreement with the more crude approximation. The integrals in the axial direction in eqns (26) and (18) are evaluated numerically, using four-point Gaussian quadrature within each element, while Simpson's rule with seven integration points is used through the thickness. The direction of the stress-rate, defining the angle 0 in the integration points, should, in principle, be determined by iteration at each incremental step; but instead the stress-rates computed in the previous increment are employed. Since small increments are used in the computations, this scheme leads to small errors. Figures 2 and 3 show curves of axial compressive load, P, versus
Plastic buckling of axially compressed circular cylindrical shells
m. ~ . a x ai ~~
1.0
~
151
"perfect"
P;Pc bifurcation
_
0.5 "
/ //
0
~=1.0
~
r
n
0
a
x
=100°
.... ~ .... Oo=e,2. C~c~mox.13~ =
/
0.002
O.OOZ,
AIr
0.006
0.008
Fig. 2. A x i a l c o m p r e s s i v e load versus a v e r a g e axial strain for cylindrical shells with = 100 a n d a x i s y m m e t r i c i m p e r f e c t i o n s (oy/E = 0.0025, v = 0-3, n = 10 a n d l = lc).
R/h
1.0
P/Pc
0.5
l /
_
maximum .I "*""~
~
~
m
m¥=I.
0
.ox
V 0
0
--- - ~"~" - - - --
0.004
0.008
ooo
JJ2 flow theory 0.012
,,/$
0.016
Fig. 3. A x i a l c o m p r e s s i v e load versus a v e r a g e axial strain for cylindrical shells with R/h = 50 a n d a x i s y m m e t r i c i m p e r f e c t i o n s (%,/E = 0-0025, v = 0.3, n = 10 a n d 1 = It),
average axial strain, A/l, for shells made of materials with oy/E = 0.0025, v = 0.3 and n = 10. Results for a material with a rather sharp vertex, (13c)max = 135 °, and a finite total loading range limited by 00 = 0n/2 are c o m p a r e d with results for a totally nonlinear material response, 0o = 0, and a less sharp vertex, (i3c)max = 100 °. Predictions of
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Viggo Tvergaard
J2 flow theory are also included for comparison. The load Pc used to normalise P in the figures is the critical load for J2 corner theory (deformation theory moduli) according to eqn (23). The halfwavelengths, l = lc, used are the critical values for each material. In Fig. 2, for R/h = 100, the curves predicted by corner theory differ little from those predicted by J2 flow theory, as would be expected based on the small difference between deformation theory and flow theory results for this case in Fig. 1. For the three larger imperfections considered, ~ = 1.0, 0-4 and 0-1, bifurcation occurs before the load maximum on the axisymmetric solution, whereas for smaller imperfections bifurcation takes place after the maximum. Bifurcation occurs first for (~c)max = 135°, then for ([3c)m~x = 100 ° and last for J2 flow theory. In fact, J2 flow theory is the limit of J2 corner theory, for 0o = 0 and (15c)m~x ~ 90 °. The total loading condition, 0 < 0o, is satisfied in the range considered for the sharp corner, at the three larger imperfections. Thus the corresponding three bifurcation points are identical with deformation theory predictions, and are lower bounds to corner theory predictions for all other values of 00 and ([3c)m~x. For the material with 00 = 0 exactly proportional loading would be required to obtain the same property, so generally here the bifurcation predictions are upper bounds, based on the alternative comparison solid discussed in Section 3. The behaviour of a perfect shell, as shown in Figs 2 and 3, is computed numerically as the behaviour of a shell with a very small initial imperfection, ~ = 10 -5. Such shells remain circular cylindrical up to the critical stress (eqn (23)), where bifurcation into the axisymmetric mode takes place. Subsequently on the post-bifurcation path the secondary bifurcation into a non-axisymmetric mode is found. It is noted that for elastic shells these two bifurcation points coincide. In Fig. 3 for the thicker shell, R/h = 50, larger differences are found between the predictions of the three material descriptions. Here, only = 0.4 and ~ = 1.0 give bifurcation before the load maximum. Also, for these two larger imperfections, the axisymmetric solutions corresponding to the two corner theory descriptions are indistinguishable from one another in the figure, and the solution for (13c)m.,,x= 135 ° satisfies total loading (0 < 0o) everywhere in the current plastic zone. The influence of the imperfection half-wave length, l, is investigated in Fig. 4. Here, and in all following results, we focus on the material with 0o = 0 and (~c)max = 100°. The two imperfection levels ~ = 0.02
Plastic buckling of axially compressed circular cylindrical shells
153
0.012 lair,)c
0.008
0.004
R/h:lO0. e=0.02
0
0.6
1.0
R/h=lO0~ = 1 ~ ~ 0
i
I
i
0.8
1.0 la)
1.2
../h=
...... um
tit c
1.4
so. ~= 0.02
load (bif. after mex.)
P/Pc
R/h=100. t
•
0.5
~bifurcation l o a d i
00.6
0.8
~R/h=100, ~=1.0 I
I
1.0 (b)
1.2
{It c
1.4
Fig. 4. Influence of the axisymmetric imperfection wavelength. (a) Average axial strain at bifurcation; (b) bifurcation load or m a x i m u m load (oy/E = 0.0025, v = 0.3, n = 10, 0o = 0 and (13c)max = 100°).
and ~ = 1.0 are considered for R/h = 100 and R/h = 50, respectively, in the range 0.6 ~< l/lc ~< 1.4. Figure 4(a) shows the average axial strains (A//)c at bifurcation. Figure 4(b) shows the bifurcation load for the larger imperfections and the maximum load for the smaller
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154
imperfections, where bifurcation occurs after the m a x i m u m (see also Figs 2 and 3). For ~ = 0-02 the critical wavelength, l/Ic = 1, corresponds rather closely to the smallest buckling load and the smallest critical axial strain but for ~ = 1.0 somewhat smaller buckling loads are found in the range l/lc < 1 and much smaller critical strains are found in the range //lc > 11 Such sensitivity to the imperfection wavelength was also found by Pedersen 17 for elastic shells. It is noted that the critical circumferential wave numbers m = 9 and m = 6 found for the small imperfection in the thinner and the thicker shell, respectively, correspond to approximately square buckles. The smaller critical wave numbers m = 6 and m -- 4, respectively, are found for the larger imperfection. Curves of buckling load versus imperfection amplitude are shown in Figs 5, 6 and 7 corresponding to various degrees of strain hardening n = 10, n = 5 and n = 20, respectively, for I = Ic. Such curves were a l s o shown by Gellin, 4 based on deformation theory. It is not clear from Gellin's results, though, that the bifurcations into a non-axisymmetric m o d e frequently occur after the maximum load, so that this loss of stability of the axisymmetric deformations may not always be relevant from the point of view of structural load carrying capacity. In Figs 5-7 a dashed curve is used to indicate the m a x i m u m load whenever this is
1.0
--
R/h = 12.5
0.5
n=10 A 0.5
Fig. $. Bifurcation load, or m a x i m u m load (dashed curves), versus axisymmetric imperfection amplitude (oy/E = 0.0025, v = 0-3, n = 10, 00 = 0, (~c)max = 100° and l = l~).
Plastic buckling of axially compressed circular cylindrical shells
1.0
155
x\\
0.5
=
~
elastic /
n=5 0
0
0.5
~
1.0
Fig. 6. Bifurcation load, or maximum load (dashed curves), versus axisymmetric imperfection amplitude (Oy/E = 0-0025, v = 0-3, n = 5, 0o = 0, (~c)m.x = 100° and l = l~). 1.0 ¢, \, '~'~, / R / h = 1 2 . 5
P/Pc
O.S
elastic /
n=20 0
0.5
~
1.0
Fig. 7. Bifurcation load, or maximum load (dashed curves), versus axisymmetric imperfection amplitude (oy/E = 0.0025, v = 0.3, n = 20, 00 = 0, ([Sc)max= 100° and l = to). reached before bifurcation, and solid curves give the bifurcation load. In all cases analysed a bifurcation point was found eventually. In the Galerkin solution of Gellin 4 the circumferential wave n u m b e r m was a s s u m e d so large that it could be treated as a c o n t i n u o u s variable. T h e n the equations were put in a non-dimensional form, in which the only material and geometric parameters are ~, n, v and Oy/a~, where ace = {3(1 - v2)}-l/2Eh/R. The results in the present
156
Viggo Tvergaard
paper, however, are based on treating the circumferential wave number as an integer, and in fact this integer is not always large (m = 3 or even rn = 2 is typical for the most thick-walled shells considered). Furthermore, the present results are based on specific choices of the parameters oy/E and v. It is noted that the shell theory used does not loose accuracy for small circumferential wave numbers, since the more accurate strain measures (eqns (1) and (2)) than those of shallow shell theory are employed, Figures 5-7 show that bifurcation occurs before the maximum for all imperfection levels, so long as the shells are not too thick-walled. In an intermediate range of somewhat smaller radius-to-thickness ratios bifurcation still precedes the maximum for larger imperfections (see also Figs 2 and 3). The high hardening material in Fig. 6 favours bifurcation prior to the maximum more than the low hardening material in Fig. 7. The tendency of the imperfection sensitivities found in Figs 5-7 is generally in good agreement with the results shown by Gellin, 4 but no direct comparison is made, since the uniaxial stress-strain law (eqn (17)) differs from the Ramberg-Osgood law used by Gellin. 4 The reduction in load carrying capacity due to small imperfections is clearly much less in the plastic range than in the elastic range. Furthermore, the imperfection amplitude relative to the shell thickness is likely to be smaller in thick-walled shells. The fact that a finite imperfection amplitude is required to bring the load carrying capacity below Pc is clearly seen for the thicker shells in Figs 5 and 6. 5 NON-AXISYMMETRIC IMPERFECTIONS The imperfection sensitivity curves shown in Figs 5, 6 and 7 imply that the load carrying capacity is exhausted, when bifurcation into a non-axisymmetric mode takes place. However, this requires knowledge of the post-bifurcation behaviour and the associated sensitivity to non-axisymmetric imperfections. For elastic shells the initial post-bifurcation behaviour has been investigated by Budiansky and Hutchinson, 3 who have found a transition from unstable to stable post-buckling behaviour as the imperfection amplitude increases. In the present paper a few numerical analyses will be used to get an impression of the
Plastic buckling of axially compressed circular cylindrical shells
157
post-bifurcation behaviour of the elastic-plastic shells and the sensitivity to imperfections of a more general shape. The numerical method applied is based on expanding the displacements in terms in trigonometric functions in the circumferential direction (analogous with the similar expansion in the axial direction .used previously is for oval cylindrical shells). Thus, the displacements are written in the form
U2
U](x1) cos jinx2
U (x
U1
=
R
U2(x') sin jmx2
0 I
W
Wo(x5
(27)
R
I
Wj(xb cos
jinx 2 R
where the integer m is the circumferential wave number of the mode to be studied. In the incremental finite element solution based on eqn (26) the one-dimensional amplitude functions U}, U~ and Wj are approximated by Hermitian cubics within each element. As in Section 4 the solutions are assumed to be periodic with an axial half-wavelength, 2l. The symmetry boundary conditions at x 1 = 0, 21 are given by eqn (25), and also here we need only consider half this interval, 0 ~< x I ~< l, with symmetry of the even-numbered functions and antisymmetry of the odd-numbered functions specified at x 1 = l. The solutions to be presented are based on using J = 2 in eqn (27) and two finite elements over the interval 0 ~< x I ~< I. In the circumferential direction integration over the interval 0 ~
~-t,X1
rH,X2
w = - ~ h cos - - + ~,,,h cos cos 1 21 R
(28)
Viggo Tvergaard
158
where ~ is the amplitude of the axisymmetric imperfection and ~m is the amplitude of a non-axisymmetric imperfection with circumferential wave number rn. Figure 8 shows load versus shortening curves obtained for shells with R/h = 100, for the material specified by n = 10, 0o = 0 and ([~c)max = 100°. The dashed curves are the axisymmetric solutions also given in Fig. 2. The wave number, m, of the non-axisymmetric imperfections is chosen equal to the critical wave number at the corresponding bifurcation point. These numbers are rn = 9 (approximately square buckles) for ~ = 0.1, m = 8 for ~0 = 0.4, and m = 6 for ~0 = 1-0. The computation for ~ = 0-1 and the extremely small nonaxisymmetric imperfection ~9 = 10 -5 in Fig. 8 is carried out to determine the post-bifurcation behaviour of the axisymmetric shell. Prior to bifurcation this solution coincides with the axisymmetric solution, but a maximum is reached immediately after the bifurcation point and subsequently the load drops off rapidly, showing the unstable post-bifurcation behaviour. Initially after bifurcation the post-bifurcation curve is nearly parallel with the fundamental solution somewhat similar to the smooth bifurcation found for the cruciform column, 11 when 0o = 0. It is noted that this computation for ~9 = 10-5 indicates good agreement with the bifurcation prediction based on the alternative comparison solid discussed in Section 3. Earlier bifurcation 1.0
P/Pc
maximum\
~°°°~\
.~=o.~ ~..,.-~-~"--_ 0
~o~o-s '
9
0.5 0 = 0.4 . ~8= 0 . 0 / .
0
Fig.
0
0.002
0.004
°i.~oiSyrnmetric
0.006
,x/,?, 0.008
8. A x i a l l o a d v e r s u s s h o r t e n i n g f o r c y l i n d r i c a l shells w i t h R/h = 100 a n d n o n - a x i s y m m e t r i c i m p e r f e c t i o n s ( o y / E = 0-0025, v = 0-3, n = 10, 01) = 0, ([~c)max = 100 ° a n d l = lc).
Plastic buckling of axially compressed circular cylindrical shells
159
from the axisymmetric solution is not excluded by this comparison solid, but the bifurcation result is supported by the solution with a small asymmetry which is expected to be unique. The results for ~ = 0-1 a n d ~9 = 0.01 and for ~o = ~9 = 0-1 in Fig. 8 illustrate the imperfection sensitivity associated with the unstable post-bifurcation behaviour. Also the curves for ~ = 0-4, ~8 = 0.04 and ~0 = 1.0, ~ = 0.1, respectively, show a reduction of the load carrying capacity relative to the corresponding bifurcation loads. It is noted that the results are very dependent on choosing the critical value of the circumferential wave number. A curve obtained for ~) = 1.0, ~ = 0.1 differs little from the corresponding axisymmetric solution in Fig. 8. Figure 9 shows similar results for a thinner shell, R/h = 200. The sharp load drop after the maximum at the small imperfection ~o = 0.1, ~13 = 0"01 is more like the unstable behaviour of elastic shells than the corresponding result in the previous figure. The curve for ~) = 1-0, ~9 = 0.1 falls below the axisymmetric solution, but here, at a relatively low value of P/Pc, the load carrying capacity is not reduced below the bifurcation load. For the thicker shell, R/h = 25 (Fig. 10), the smaller imperfection ~) = 0.1 represents a case in which bifurcation takes place after the maximum. Here, the additional imperfection ~ = 0.01 has nearly no effect on the maximum load, but does give a more rapidly decaying 1.0
•0-0.1
P/Pc
maximum.
J
0.5
/
axisymmetric
j bifurcation
g.o.1 00
0.002
0.00z.
a/(
0.006
Fig. 9. A x i a l l o a d v e r s u s s h o r t e n i n g for cylindrical shells w i t h R/h = 200 a n d n o n - a x i s y m m e t r i c i m p e r f e c t i o n s (oy/E = 0-0025, v = 0-3, n = 10, 0o = 0, (13c)max = 100 ° a n d l = lc).
160
Viggo Tvergaard 1.0
' ioifu rcation~' axisymmetric
0.5
0
maximum
0
.~. "~0 = 1.0
=
i
0.008
0.016
t./~.
0.02/.
Fig. 10. A x i a l l o a d v e r s u s s h o r t e n i n g for cylindrical shells w i t h R/h = 25 a n d n o n - a x i s y m m e t r i c i m p e r f e c t i o n s (oy/E = 0.0025, v = 0.3, n = 10, 0t~ = 0, (~c)max = 100 ° a n d l = lc).
load around the bifurcation point. For ~ = 1-0, where bifurcation occurs before the maximum, the additional imperfection ~3 = 0.1 results in a reduced load carrying capacity.
6 DISCUSSION The use of incremental elastic-plastic constitutive relations in the shell buckling analyses, rather than the nonlinear elastic approximation of deformation theory, has the advantage that it rules out speculations regarding the influence of neglecting elastic unloading, etc. Furthermore, the attractive quality of deformation theory that predicted bifurcation loads agree reasonably well with experimental buckling loads is incorporated in J2 corner theory, since the bifurcation predictions of the two theories are identical, at least for proportional loading. For shells with initial imperfections, where the loading path deviates from proportional loading, the general tendency is that J2 corner theory predicts a buckling load higher than that of J2 deformation theory, but lower than that of J2 flow theory. This agrees with results found for a cruciform column. ~1 However, apart from such quantitative differences the bifurcation results found in the present
Plastic buckling of axially compressed circular cylindrical shells
161
investigation, based on incremental plasticity theory, show the same trends found by Gellin 4 for deformation theory. In particular, bifurcation is found for all shells considered, also for very thick shells, which d o e s not agree with the experimental observation that thick shells (R/h smaller than about 30) buckle axisymmetrically. Among various possible reasons why the theory fails to predict this transition to purely axisymmetric buckling, Gellin 4 suggested that neglecting elastic unloading might be the most important--however, the present results show that this is not the case. A feature of the results found here that was not discussed by Gellin 4 is that the bifurcation point occurs after the maximum load point for the thicker shells, particularly the shells with small initial imperfections. This is obviously significant if the axial load is the prescribed quantity, since then stability is lost at the maximum. In cases where the overall axial shortening is the prescribed quantity it would seem that the bifurcation points beyond the load maxima are still relevant stability limits. However, recent studies by Tvergaard and Needleman,11, 19. 20 relating to a wide variety of structures with the common property that the applied load versus shortening curve achieves a maximum, have shown that localisation of the buckling pattern will take place at the load maximum. Thus the assumption of periodic solutions in the axial direction used in all analyses in the present paper (and in previous analyses) 2, 3.4. 17 is valid so long as no maximum has been reached, but after the maximum, localisation into one or a few buckles in the axial direction must be expected, even in cases where the axial shortening is prescribed. Such axisymmetric localisation, occurring mainly in the thicker shells, could be a cause of the experimentally observed transition to purely axisymmetric buckling. The influence of localisation on the buckling behaviour of circular cylindrical shells under axial compression is to be investigated in more detail by the author. The importance of a bifurcation point on a rising load versus shortening curve is associated w~th the expectation of an unstable post-bifurcation behaviour. However, elastic cylindrical shells with axisymmetric imperfections do exhibit a stable initial post-bifurcation behaviour in some ranges of the imperfection amplitude. 3 The analyses in the present paper with an additional non-axisymmetric imperfection indicate that the post-bifurcation behaviour is unstable for the elastic-plastic shells. Except for one case, in which the
162
Viggo Tvergaard
knock-down factor is already 1/4, the load carrying capacity is further reduced by the additional imperfections.
REFERENCES 1. Koiter, W. T., 'Over de stabiliteit van het elastisch evenwicht', Thesis, Delft, H. J. Paris, Amsterdam (1945). English translations (a) NASA TT-FIO (1967) 833, (b) AFFDL-TR-70-25 (1970). 2. Koiter, W. T., The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression, Proc. Kon. Ned. Ak. Wet., 66B (1963) 265-79. 3. Budiansky, B. and Hutchinson, J. W., Buckling of circular cylindrical shells under axial compression, Contributions to the Theory of Aircraft Structures, Delft University Press (1972) 239-59. 4. Gellin, S., Effect of an axisymmetric imperfection on the plastic buckling of an axially compressed cylindrical shell, J. Appl. Mech., 46(1979) 125-31. 5. Hutchinson, J. W., Plastic buckling, Advan. Appl. Mech., 14 (1974) 67-144. 6. Batdorf, S. B., Theories of plastic buckling, J. Aeronaut. Sci., 16(1949) 405-8. 7. Hill, R., Generalized constitutive relations for incremental deformation of metal crystals by multislip, J. Mech. Phys. Solids, 14 (1966) 95-102. 8. Hutchinson, J. W., Elastic-plastic behavior of polycrystalline metals and composites, Proc. Roy. Soc. London, A318 (1970)247-72. 9. Hecker, S. S., Experimental studies of yield phenomena in biaxially loaded metals, Constitutive Equations in Viscoplasticity, AMD-Vol. 20 ASME, New York (1976) 1-33. 10. Christoffersen, J. and Hutchinson, J. W., A class of phenomenoiogicai corner theories of plasticity, J. Mech. Phys. Solids, 27 (1979) 465-87. 11. Needleman, A. and Tvergaard, V., Aspects of plastic post-buckling behaviour, In: Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, Ed. by H. G. Hopkins & M. J. Sewell, London, Pergamon Press (1982) 453-98. 12. Koiter, W. T., On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Ak. Wet., B69 (1966) 1-54. 13. Niordson, F., Introduction to shell theory, Technical University of Denmark (1980). 14. Hill, R., A general theory of uniqueness and stability in elastic-plastic solids, J. Mech. Phys. Solids, 6 (1958) 236-49. 15. Tvergaard, V., Buckling of elastic-plastic oval cylindrical shells under axial compression, Int. J. Solids Structures, 12 (1976) 683-91. Errata, ibid. 14 (1978) 329.
Plastic buckling of axially compressed circular cylindrical shells
163
16. Tvergaard, V., Buckling of elastic-plastic cylindrical panel under axial compression, Int. J. Solids Structures, 13 (1977) 957-70. 17. Pedersen, P. T., Buckling of unstiffened and ring stiffened cylindrical shells under axial compression, Int. J. Solids Structures, 9 (1973) 671-91. 18. Tvergaard, V., Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells, J. Mech. Phys. Solids, 24 (1976) 291-304. 19. Tvergaard, V. and Needleman, A., On the localization of buckling patterns, J. Appl. Mech., 47 (1980) 613--19. 20. Tvergaard, V. and Needleman, A., On the development of localized buckling patterns, Danish Center for Appl. Math. and Mech., Report 243 (1982).
Plastic Buckling of Axially Compressed Circular Cylindrical Shells Viggo T v e r g a a r d Department of Solid Mechanics The Technical University of Denmark, DK-2800 Lyngby, Denmark
A BSTRA CT For elastic-plastic cylindrical shells with initial axisymmetric imperfections bifurcation into a non-axisymmetric shape is analysed. The shell material is represented by a phenomenological plasticity theory that accounts for the formation of a vertex on subsequent yield surfaces. The influence of various geometric and material parameters is investigated for a wide range of radius-to-thickness ratios. It is shown that for the thicker shells bifurcation generally occurs beyond the maximum axial compressive load. A few analyses for shells with additional non-axisymmetric imperfections show the unstable postbifurcation behaviour and the sensitivity to imperfections of more general shapes.
1 INTRODUCTION The strong imperfection-sensitivity of axially compressed circular cylindrical shells in the elastic range results from interaction between several simultaneous buckling modes. This is shown asymptotically by Koiter's general post-buckling theory. 1 For the special case of sinusoidal axisymmetric imperfections Koiter 2 has obtained an upper bound to the load at which the axisymmetric deformation bifurcates into an asymmetric shape. This upper bound estimate can be considered a good approximation of the actual load carrying capacity, even for rather large imperfections, since it is based on a nonlinear pre-buckling solution. A more detailed investigation of the elastic bifurcation from an axisymmetric periodic mode, including an initial 139 Thin-Walled Structures 0263-8231/83/$03.00 (~) Applied Science Publishers Ltd, England, 1983. Printed in Great Britain
140
Viggo Tvergaard
post-bifurcation analysis, has been carried out by Budiansky and Hutchinson. 3 For shells compressed into the plastic range results analogous with those of Koiter e have been obtained by Gellin,4 who considered a wide range of radius-to-thickness ratios and various degrees of strain hardenfng. These analyses were based o n J2 deformation theory, and thus the elastic-plastic material was represented as a nonlinear elastic material. It is well known that bifurcation calculations based on the deformation theory of plasticity generally agree much better with experimental buckling loads for plate and shell structures than similar calculations based on the simplest incremental plasticity theory. 5 As explained by Batdorf, 6 bifurcation predictions of deformation theory can be rigorously justified by appealing to a more sophisticated flow theory with a vertex on the yield surface. Such vertices are implied by physical models of polycrystalline metal plasticity, based on the concept of single crystal slip, 7'8 but experimental evidence is inconclusive. 9 However, the use of deformation theory moduli to analyse bifurcation at a vertex presumes that proportional or nearly proportional loading has taken place prior to bifurcation. For the cylindrical shell with axisymmetric imperfections the pre-bifurcation loading path is not always sufficiently near proportional loading, and thus a material stiffening effect is neglected in the deformation theory analysis. Also the possibility of elastic unloading prior to bifurcation is neglected by such nonlinear elastic analysis. The shell buckling behaviour in the plastic range is analysed here on the basis of a phenomenological corner theory of plasticity, which accounts for the material stiffening with increasing nonproportionality that was neglected in the deformation theory analyses. This so-called J: corner theory, proposed by Christoffersen and Hutchinson, t° has been used by Needleman and Tvergaard ~t to study the influence of yield surface vertex formation on the buckling behaviour of a cruciform column. For proportional loading the instantaneous moduli in J2 corner theory are chosen identical with those of J2 deformation theory so that bifurcation points predicted by the two theories on a proportional loading path are identical. For cruciform columns with initial imperfections deviations from proportional loading occur, and here the load carrying capacities predicted by J2 corner theory were found to exceed those of deformation theory,
Plastic buckling of axially compressed circular cylindrical shells
141
but to be lower than those of the simplest flow theory of plasticity with a smooth yield surface. In the present investigation for cylindrical shells the influence of differences in the vertex description, of various degrees of strain hardening and of various imperfection wavelengths are studied. In addition t o the bifurcation analyses a few computations for non-axisymmetric imperfections are included to illustrate postbifurcation behaviour and the sensitivity to imperfections of a more general shape.
2 SHELL E Q U A T I O N S AND CONSTITUTIVE R E L A T I O N S On the middle surface of the circular cylindrical shell with radius R a point is identified by the coordinates (x 1, x 2) = (z, R0), where z measures the distance along the cylinder axis and 0 is the circumferential angle. The displacement components are denoted u ~ on the surface base vectors and w on the outward surface normal. The strain measures used are the nonlinear membrane strain tensor co,13= 1/2(u~.1~+ u~.~,) - d~,~w + 1/2aYS(uy,ot + 1/2(w.~ + ~u~)(w~ + d~u~)
-
dy~w)(u6,[3 - dg3~514))
(1)
and the linear bending strain tensor specified by Koiter lz ~.~ = v : [ ( w , + ~ u ~ ) . ~ + ( w ~ + dffu~)., - 1/2~(u~.~ - u~.~) - l/:dff(u,.~ - u~.~)]
(2)
where u~,~ and d,~ are the metric tensor and the curvature tensor, respectively, of the undeformed middle surface, and ( ).~, denotes covariant differentiation. Greek indices range from 1 to 2, while Latin indices (to be employed subsequently) range from 1 to 3, and the summation convention is adopted for repeated indices. It is noted that strain measures proposed by Niordson 13 are identical with eqns (1) and (2), except for small differences in the bending strain measure of the order of dV~ev~. T h e three-dimensional constitutive relations are taken to be of the form 0 ij = L i j k l • k l
(3)
Viggo Tvergaard
142
where C j is the stress tensor, ~qkt is the strain tensor, L ~jk~ are the instantaneous moduli, and (') denotes an incremental quantity. Since the stress state in the shell is approximately plane, only the in-plane stresses enter into eqn (3). Thus, the constitutive relations can be written as Lc4~33L33V<",
daft = /_~,13y6/iv~' /~o,13v~= L,,,I~v~
L 3333
(4)
The in-plane components of the Lagrangian strain tensor at a distance x 3 outward from the shell middle surface are approximated by vl~13= ~ - x3r~
(5)
The membrane stress tensor N ~ and the moment tensor M ~ in a shell with thickness h are taken to be fh/2
N~a = j -h/2 (~°tl$dx3
'
MC'a "~
_ fh,2
J -hi20ctl$x3dx3
(6)
Then, from eqns (4), (5) and (6), incremental relations are obtained for ,~/~,13 and M~,I3 in terms of i~v~ and kv~. The requirement of equilibrium is specified in terms of the principle of virtual work
fA {N~fiSenf~+ Ma"8l%,g}dA= P~3up
(7)
where A is the middle surface area, P is the total axial load at one end of the cylinder, with corresponding axial displacement ur,, and at the other cylinder end zero axial displacement is prescribed. The elastic-plastic material behaviour is taken to be described by the phenomenological corner theory of plasticity, J2 corner theory, proposed by Christoffersen and Hutchinson. 1° In this theory the instantaneous moduli for nearly proportional loading are chosen equal to the J2 deformation theory moduli and for increasing deviation from proportional loading the moduli increase smoothly until they coincide with the elastic moduli for stress increments directed along or within the corner of the yield surface. W i t h Mffijk I denoting the deformation theory compliances, so that ~]i[ "~" M~ijk[~kl, and Mijkl denoting the linear elastic compliances, the plastic part of the compliances is Cqkt = M~ijkl -- Mijkl, The yield surface in the neighbourhood of the current loading point is taken to be a cone in stress deviator space with the cone axis in the direction kij = sil( cmnpqSmnSpq)- 1/2
(8)
Plastic buckling of axially compressed circular cylindrical shells
143
where s o = o ij - gSJ~k/3 is the stress deviator tensor and gu is the metric tensor in three-dimensional coordinates. The positive angular measure, 0, of the stress-rate direction relative to the cone axis is defined by
Cijkl~iJskl(fm.pqSmn~q)- I/2
COS 0 =
(9)
and a stress-rate potential at the vertex is formulated as 1
1
W = --Mijkt606 kt + -~f(O)CijktO'J6 kt 2
(10)
From this potential the strain-rate is obtained as 02W _ _
ilij - obijobkt
6d,t
= Mijkt( O)6 kl
(11)
with 0-dependent compliances. The plane-stress relationship (eqn (4)), with instantaneous moduli Lo,l~v,(0), is obtained by inverting eqn (11) numerically. The angle of the yield surface cone is specified by 0¢, so that the transition function, f(0), in eqn (10) is zero for 0¢ < 0 ~< ~x. In the total loading range, 0 ~< 0 ~< 00, f(0) is unity, and in the transition region, 00 ~< 0 ~< 0c, f(0) is chosen to smoothly merge the deformation theory moduli with the elastic moduli in a way ensures convexity of the incremental relation. The transition function to be used here is specified by riO) =
g(¢)[1 + 12(¢p)]
where l(q~)
O(¢p) = ¢p + arctan[l(cp)]
-~(dg/ddp)/g and, with 0~ = 0c - - - , 2 for 0 ~< ~ ~< 0o
1
g(q~) =
(12)
{1 - [ ( ¢ -
O o ) / ( O n - 0,,)] 3} - 2
(13)
forOo~
This transition function has been found ~° to rather closely duplicate moduli obtained using a self-consistent model of a polycrystalline aggregate. 8
144
Viggo Tvergaard
T h e d e f o r m a t i o n t h e o r y compliance tensor used in the construction of the potential (10) has the form 12 M~i~k, = ~1 { v---~s (&kgi, + gi~gi,) - Vs gqgkt}
9(1 + 40-"-~
1)Sijb.kl Et
(14)
Es
w h e r e Es and E t are the ratio of stress and strain and the slope, respectively, for the uniaxial stress-strain curve, o~ = {3siisiJ/2} l/2 is the effective Mises stress, v is Poisson's ratio, and v~ = v E J E + (1 - EJE)/2. T h e angular measure, 0c, of the cone angle is taken to be given by tan0c = V ~ t a n 13c, a = (E/E~ - 1)/(E/E, - 1)
(15)
w h e r e , in terms of the initial yield stress Oy tan 13c = --Oy/(Oe 2 -- Oy) _2,V2, for
7t 2
~< [5c <~ ([5~)max
(16)
and otherwise 15c = ([5~)max. T h e uniaxial stress-strain curve is r e p r e s e n t e d by a piecewise p o w e r law with continuous tangent modulus
o
for o ~< Oy
E
E=
(17)
OYFl(&)n - - 1- + 11 E L t'/ \Oy/ ?'/
for o >
Oy
w h e r e n is the strain h a r d e n i n g exponent.
3 BIFURCATION ANALYSIS For a cylindrical shell with initial axisymmetric imperfections the
Plastic buckling of axially compressed circular cylindrical shells
145
possibility of bifurcation into an asymmetric shape will be investigated. Prior to bifurcation the solution remains axisymmetric. The equations governing bifurcation are obtained by assuming that there are at least two distinct incremental solutions (.)a and (')b corresponding to a given increment of the prescribed quantity. The difference between two such solutions is denoted by (-) = (')" - (')b, and using the incremental version of the principle of virtual work (eqn (7)) the following equation is obtained
+ (ff,~ + dV~av)(8w.~ + d~bu,)]}dA = 0
(18)
which must be satisfied by non-zero bifurcation solutions(-). Here, N ~'f~ are the current membrane stresses in the fundamental axisymmetric solution. To prove uniqueness of an incremental solution Hill ~4 makes use of the functional
+ (~,~ + dV~av)(~.13+ d~a,)]}dA
(19)
requiring I > 0 for any two solutions (')~ and (')b. At bifurcation I = 0, as is directly seen from eqn (18) by choosing the variation 8( ) = (-). In the case of a smooth yield surface and normality Hill la has introduced a comparison solid, identified by the plastic moduli in every material point currently on the yield surface and the elastic moduli elsewhere. For this comparison solid I is a quadratic functional, and the variational equation 81 = 0 is identical with eqn (18). The first critical bifurcation point of the comparison solid and that of the underlying elastic-plastic solid are, in most circumstances, identical and t.he comparison solid always gives a lower bound. If the fundamental solution based on J2 corner theory satisfies nearly proportional loading everywhere in the current plastic zone, so that 0 ~< 0o, the total loading (i.e. deformation theory) moduli can be used to define the comparison solid. The first critical bifurcation point corresponding to J2 corner theory is then identical with that predicted by deformation theory (see discussion by Needleman and Tvergaard 11 for J2 corner theory and by Hutchinson 5 for a pyramidal vertex). However, if the fundamental solution has 00 < 0 < 0c in some material
146
Viggo Tvergaard
points, a comparison solid to be used in the solution of eqn (18) is not readily available. Using the deformation theory moduli may result in a rather poor lower bound, since bifurcation will be governed partly by the stiffer moduli of the transition range. On the other hand, using the instantaneous moduli associated with the current values of 0 on the fundamental solution gives an upper bound, which will be explained further, since these moduli are employed in the present investigation. A bifurcation solution based on the alternative comparison solid mentioned above (moduli corresponding to current 0-values in fundamental solution) is a possible solution for the underlying elastic-plastic solid. This follows from the fact that bifurcation stress-rate directions arbitrarily close to those of the fundamental solution can be enforced by superposing a sufficiently large fundamental solution increment on the e i g e n m o d e ( - ) . But earlier bifurcation is not excluded, since softer moduli resulting from a smaller angle 0 may reduce the value of the quantity under the integral in eqn (19) below that corresponding to the comparison solid. Determining the actual critical bifurcation point will be very complex, since the corresponding critical variation of the angular measure 0(x 1, x 2, x 3) must be determined simultaneously. It is expected, though, that the upper bound bifurcation predictions obtained by the alternative comparison solid will generally be rather good approximations of the actual bifurcation points. Eigenmodes are assumed of the form mx 2
/~1 m_ UI(X I) c o s -
R
mx 2
a2 = U2(xI) sin - R mx 2
v0 = ~,'(x I) c o s -
R
(20)
where the circumferential wave number, m, is an integer, and the amplitude functions UI, U2, W depend only on the axial coordinate x 1. The corresponding strain measures take the form
Plastic b u c k l i n g o f axially c o m p r e s s e d circular cylindrical shells mx 2
E,~ = E l l ( X m) C O S - -
m,x 2
,
Kll = g * * ( x * ) c o s - -
R
R
mx 2
"~12 = Et2(x 1) sin - R
mx 2
r12 = K't2(x1) sin - R
mx 2
E22 = E 2 2 ( x l ) c o s - -
147
mx 2
1(22 -- g 2 2 ( x 1) c o s -
(21)
R R The moduli of the comparison solid satisfy rotational symmetry, so that modes corresponding to different wave numbers, m, decouple when substituted into eqn (18). For each particular wave number the integration of eqn (18) in the circumferential direction is straightforward, after substituting eqns (20) and (21). Thus, in the reduced form the bifurcation eqn (18) depends o n l y on the one-dimensional amplitude functions U1, U2 and W. This equation is used to numerically determine the first critical bifurcation point for any mode number, m. The moduli governing the first critical bifurcation in the actual elastic-plastic solid do not necessarily satisfy rotational symmetry if part of the material has entered the transition region (00 < 0 < 0c) in the fundamental state. As mentioned above, the softer moduli corresponding to smaller 0-values than those of the comparison solid will be preferred by the critical bifurcation mode wherever possible. It is noted that the sinusoidal circumferential variation assumed in eqn (20) is no longer an exact bifurcation solution if the rotational symmetry of the relevent moduli is lost. For a circular cylindrical shell with no initial imperfections the first critical bifurcation mode in the plastic range is axisymmetric of the form ~-t,X1 /all =
B sin - - ,
7tX 1
ti2 ---=0,
v~ = C cos
tc
(22) tc
where B and C are constants. The critical stress, oc, and the corresponding critical half-wave length, lc, are given by 15
=
~2E1h' 61~-"~2
7t_R2h [--~'-2_] 1/4 r E2 LE1
/ ElZ \E1/.J
2---1/4
Viggo Tvergaard
148
where El, E 2 a n d E t 2 denote the physical values of the plastic moduli L 11:1, L 2222 and L 1~22, respectively, at the bifurcation point. In the elastic range the critical stress (eqn (23)) reduces to the well-known value oc = -{3(1 - v2)}-U2Eh/R, at which axisymmetric modes and several non-axisymmetric modes with circumferential wave numbers up to a certain value are critical simultaneously. In the plastic range bifurcation stresses corresponding to non-axisymmetric modes are slightly above the critical stress (eqn (23)) for an axisymmetric mode. :6 Bifurcation stresses of perfect circular cylindrical shells with o y / E = 0-0025 and v = 0.3 are shown in Fig. 1. For sufficiently thin shells, R/h > 242, bifurcation takes place in the elastic range, whereas for smaller radius-to-thickness ratios the critical stress (eqn (23)) is below the elastic value. The difference between the bifurcation predictions of J2 flow theory and of J2 deformation theory is shown in Fig. 1. The largest difference is found for a high hardening material (n = 5) and for rather thick-walled shells, where bifurcation occurs well into the plastic range. Bifurcation predictions of J2 Corner theory are identical with those of deformation theory in this case, where proportional loading (uniaxial stress) takes place prior to bifurcation. In Fig. 1 and in the
"\ [3(1_u2)l-1/2E h 2
J2 deformation theory
°o
~6o
260
R/h
300
Fig. I. Bifurcation stress versus radius-to-thickness ratio for perfect cylindrical shells (oy/E = 0"0025 and v = 0-3).
Plastic buckling of axially compressed circular cylindrical shells
149
following sections it is assumed that the length of the shell and the boundary conditions at the ends permit buckling modes with the critical wavelength for a long shell.
4 RESULTS FOR AXISYMMETRIC IMPERFECTIONS The geometrical imperfection in the stress-free shell is specified as an initial normal displacement, w, of the form ~-t,X1
w = - ~ h cos
(24)
l
where ~ is the imperfection amplitude relative to the shell thickness, and l is the axial half-wavelength. For the bifurcation mode eqn (20) an axial half-wavelength 21 is chosen a priori based on the physical considerations also discussed by Koiter 2 for elastic shells and by Gellin 4 for the deformation theory solution. Buckling in a short-wave mode in a circumferential direction will be stimulated in the axisymmetric wave-bottoms, where N 22 < 0, and impeded at the wave-tops, where N 22 > 0. Therefore the bifurcation mode peaks are expected at the axisymmetric wavebottoms, and here symmetry conditions are assumed at x ~ = 0, W,I
at x ~ = 2l
(25)
W,1
It follows from eqn (25) that the analyses in the present paper assume periodicity of the solution in the axial direction. In the numerical solution only half the interval of eqn (25) is considered, 0 ~< x I ~< 1, with symmetry of the pre-buckling solution and antisymmetry of the bifurcation solution prescribed at x 1 = l. A m o n g the many modes simultaneously critical in elastic perfect shells, the mode with an axial wavelength two-times that of the axisymmetric modes corresponds to approximately square buckles. In agreement with this, the critical circumferential wave numbers, m, found in the following are close to the value (:tR)/(21), particularly for small imperfection amplitudes ~. Bifurcation into modes with an axial wavelength longer than 2 / h a s been investigated for elastic shells by
Viggo Tvergaard
150
Budiansky and Hutchinson, 3 who found that such modes may be critical a little earlier. Pre-buckling solutions are obtained numerically by a linear incremental method. The current values of the field quantities N~ , M ~13, E,~, etc. are taken to express an approximate equilibrium state. By expanding the principle of virtual work (eqn (7)) about this state, using eqns (1) and (2), the equation governing the increments, fi¢~, M~I~, i ~ , etc. is found. To lowest order this incremental equation is fA
+
+
-
= ffbur,- [fa {N~'%e~'f~ + M'~%r~'"}dA - Pbur,]
(26)
The terms bracketed on the right-hand side of eqn (26) are included to prevent drifting of the solution away from the true equilibrium path; for sufficiently small increments these terms have little influence on the solutions found. For the axisymmetric solution, the integrations in the circumferential direction are trivial. In the reduced form eqn (26) depends only on the one-dimensional displacement increments til and ~i,, and the equation is solved approximately by the finite element method. The displacement increments are represented as Hermitian cubics within each element and, likewise, in the numerical solution of the bifurcation equation (eqn (18)) the mode amplitudes U1, U2 and W are approximated by Hermitian cubics in each element. Most of the results to be presented are obtained with only two elements over the interval 0 ~ x I <~ l; but computations based on four or six elements show good agreement with the more crude approximation. The integrals in the axial direction in eqns (26) and (18) are evaluated numerically, using four-point Gaussian quadrature within each element, while Simpson's rule with seven integration points is used through the thickness. The direction of the stress-rate, defining the angle 0 in the integration points, should, in principle, be determined by iteration at each incremental step; but instead the stress-rates computed in the previous increment are employed. Since small increments are used in the computations, this scheme leads to small errors. Figures 2 and 3 show curves of axial compressive load, P, versus
Plastic buckling of axially compressed circular cylindrical shells
m. ~ . a x ai ~~
1.0
~
151
"perfect"
P;Pc bifurcation
_
0.5 "
/ //
0
~=1.0
~
r
n
0
a
x
=100°
.... ~ .... Oo=e,2. C~c~mox.13~ =
/
0.002
O.OOZ,
AIr
0.006
0.008
Fig. 2. A x i a l c o m p r e s s i v e load versus a v e r a g e axial strain for cylindrical shells with = 100 a n d a x i s y m m e t r i c i m p e r f e c t i o n s (oy/E = 0.0025, v = 0-3, n = 10 a n d l = lc).
R/h
1.0
P/Pc
0.5
l /
_
maximum .I "*""~
~
~
m
m¥=I.
0
.ox
V 0
0
--- - ~"~" - - - --
0.004
0.008
ooo
JJ2 flow theory 0.012
,,/$
0.016
Fig. 3. A x i a l c o m p r e s s i v e load versus a v e r a g e axial strain for cylindrical shells with R/h = 50 a n d a x i s y m m e t r i c i m p e r f e c t i o n s (%,/E = 0-0025, v = 0.3, n = 10 a n d 1 = It),
average axial strain, A/l, for shells made of materials with oy/E = 0.0025, v = 0.3 and n = 10. Results for a material with a rather sharp vertex, (13c)max = 135 °, and a finite total loading range limited by 00 = 0n/2 are c o m p a r e d with results for a totally nonlinear material response, 0o = 0, and a less sharp vertex, (i3c)max = 100 °. Predictions of
152
Viggo Tvergaard
J2 flow theory are also included for comparison. The load Pc used to normalise P in the figures is the critical load for J2 corner theory (deformation theory moduli) according to eqn (23). The halfwavelengths, l = lc, used are the critical values for each material. In Fig. 2, for R/h = 100, the curves predicted by corner theory differ little from those predicted by J2 flow theory, as would be expected based on the small difference between deformation theory and flow theory results for this case in Fig. 1. For the three larger imperfections considered, ~ = 1.0, 0-4 and 0-1, bifurcation occurs before the load maximum on the axisymmetric solution, whereas for smaller imperfections bifurcation takes place after the maximum. Bifurcation occurs first for (~c)max = 135°, then for ([3c)m~x = 100 ° and last for J2 flow theory. In fact, J2 flow theory is the limit of J2 corner theory, for 0o = 0 and (15c)m~x ~ 90 °. The total loading condition, 0 < 0o, is satisfied in the range considered for the sharp corner, at the three larger imperfections. Thus the corresponding three bifurcation points are identical with deformation theory predictions, and are lower bounds to corner theory predictions for all other values of 00 and ([3c)m~x. For the material with 00 = 0 exactly proportional loading would be required to obtain the same property, so generally here the bifurcation predictions are upper bounds, based on the alternative comparison solid discussed in Section 3. The behaviour of a perfect shell, as shown in Figs 2 and 3, is computed numerically as the behaviour of a shell with a very small initial imperfection, ~ = 10 -5. Such shells remain circular cylindrical up to the critical stress (eqn (23)), where bifurcation into the axisymmetric mode takes place. Subsequently on the post-bifurcation path the secondary bifurcation into a non-axisymmetric mode is found. It is noted that for elastic shells these two bifurcation points coincide. In Fig. 3 for the thicker shell, R/h = 50, larger differences are found between the predictions of the three material descriptions. Here, only = 0.4 and ~ = 1.0 give bifurcation before the load maximum. Also, for these two larger imperfections, the axisymmetric solutions corresponding to the two corner theory descriptions are indistinguishable from one another in the figure, and the solution for (13c)m.,,x= 135 ° satisfies total loading (0 < 0o) everywhere in the current plastic zone. The influence of the imperfection half-wave length, l, is investigated in Fig. 4. Here, and in all following results, we focus on the material with 0o = 0 and (~c)max = 100°. The two imperfection levels ~ = 0.02
Plastic buckling of axially compressed circular cylindrical shells
153
0.012 lair,)c
0.008
0.004
R/h:lO0. e=0.02
0
0.6
1.0
R/h=lO0~ = 1 ~ ~ 0
i
I
i
0.8
1.0 la)
1.2
../h=
...... um
tit c
1.4
so. ~= 0.02
load (bif. after mex.)
P/Pc
R/h=100. t
•
0.5
~bifurcation l o a d i
00.6
0.8
~R/h=100, ~=1.0 I
I
1.0 (b)
1.2
{It c
1.4
Fig. 4. Influence of the axisymmetric imperfection wavelength. (a) Average axial strain at bifurcation; (b) bifurcation load or m a x i m u m load (oy/E = 0.0025, v = 0.3, n = 10, 0o = 0 and (13c)max = 100°).
and ~ = 1.0 are considered for R/h = 100 and R/h = 50, respectively, in the range 0.6 ~< l/lc ~< 1.4. Figure 4(a) shows the average axial strains (A//)c at bifurcation. Figure 4(b) shows the bifurcation load for the larger imperfections and the maximum load for the smaller
Viggo Tvergaard
154
imperfections, where bifurcation occurs after the m a x i m u m (see also Figs 2 and 3). For ~ = 0-02 the critical wavelength, l/Ic = 1, corresponds rather closely to the smallest buckling load and the smallest critical axial strain but for ~ = 1.0 somewhat smaller buckling loads are found in the range l/lc < 1 and much smaller critical strains are found in the range //lc > 11 Such sensitivity to the imperfection wavelength was also found by Pedersen 17 for elastic shells. It is noted that the critical circumferential wave numbers m = 9 and m = 6 found for the small imperfection in the thinner and the thicker shell, respectively, correspond to approximately square buckles. The smaller critical wave numbers m = 6 and m -- 4, respectively, are found for the larger imperfection. Curves of buckling load versus imperfection amplitude are shown in Figs 5, 6 and 7 corresponding to various degrees of strain hardening n = 10, n = 5 and n = 20, respectively, for I = Ic. Such curves were a l s o shown by Gellin, 4 based on deformation theory. It is not clear from Gellin's results, though, that the bifurcations into a non-axisymmetric m o d e frequently occur after the maximum load, so that this loss of stability of the axisymmetric deformations may not always be relevant from the point of view of structural load carrying capacity. In Figs 5-7 a dashed curve is used to indicate the m a x i m u m load whenever this is
1.0
--
R/h = 12.5
0.5
n=10 A 0.5
Fig. $. Bifurcation load, or m a x i m u m load (dashed curves), versus axisymmetric imperfection amplitude (oy/E = 0.0025, v = 0-3, n = 10, 00 = 0, (~c)max = 100° and l = l~).
Plastic buckling of axially compressed circular cylindrical shells
1.0
155
x\\
0.5
=
~
elastic /
n=5 0
0
0.5
~
1.0
Fig. 6. Bifurcation load, or maximum load (dashed curves), versus axisymmetric imperfection amplitude (Oy/E = 0-0025, v = 0-3, n = 5, 0o = 0, (~c)m.x = 100° and l = l~). 1.0 ¢, \, '~'~, / R / h = 1 2 . 5
P/Pc
O.S
elastic /
n=20 0
0.5
~
1.0
Fig. 7. Bifurcation load, or maximum load (dashed curves), versus axisymmetric imperfection amplitude (oy/E = 0.0025, v = 0.3, n = 20, 00 = 0, ([Sc)max= 100° and l = to). reached before bifurcation, and solid curves give the bifurcation load. In all cases analysed a bifurcation point was found eventually. In the Galerkin solution of Gellin 4 the circumferential wave n u m b e r m was a s s u m e d so large that it could be treated as a c o n t i n u o u s variable. T h e n the equations were put in a non-dimensional form, in which the only material and geometric parameters are ~, n, v and Oy/a~, where ace = {3(1 - v2)}-l/2Eh/R. The results in the present
156
Viggo Tvergaard
paper, however, are based on treating the circumferential wave number as an integer, and in fact this integer is not always large (m = 3 or even rn = 2 is typical for the most thick-walled shells considered). Furthermore, the present results are based on specific choices of the parameters oy/E and v. It is noted that the shell theory used does not loose accuracy for small circumferential wave numbers, since the more accurate strain measures (eqns (1) and (2)) than those of shallow shell theory are employed, Figures 5-7 show that bifurcation occurs before the maximum for all imperfection levels, so long as the shells are not too thick-walled. In an intermediate range of somewhat smaller radius-to-thickness ratios bifurcation still precedes the maximum for larger imperfections (see also Figs 2 and 3). The high hardening material in Fig. 6 favours bifurcation prior to the maximum more than the low hardening material in Fig. 7. The tendency of the imperfection sensitivities found in Figs 5-7 is generally in good agreement with the results shown by Gellin, 4 but no direct comparison is made, since the uniaxial stress-strain law (eqn (17)) differs from the Ramberg-Osgood law used by Gellin. 4 The reduction in load carrying capacity due to small imperfections is clearly much less in the plastic range than in the elastic range. Furthermore, the imperfection amplitude relative to the shell thickness is likely to be smaller in thick-walled shells. The fact that a finite imperfection amplitude is required to bring the load carrying capacity below Pc is clearly seen for the thicker shells in Figs 5 and 6. 5 NON-AXISYMMETRIC IMPERFECTIONS The imperfection sensitivity curves shown in Figs 5, 6 and 7 imply that the load carrying capacity is exhausted, when bifurcation into a non-axisymmetric mode takes place. However, this requires knowledge of the post-bifurcation behaviour and the associated sensitivity to non-axisymmetric imperfections. For elastic shells the initial post-bifurcation behaviour has been investigated by Budiansky and Hutchinson, 3 who have found a transition from unstable to stable post-buckling behaviour as the imperfection amplitude increases. In the present paper a few numerical analyses will be used to get an impression of the
Plastic buckling of axially compressed circular cylindrical shells
157
post-bifurcation behaviour of the elastic-plastic shells and the sensitivity to imperfections of a more general shape. The numerical method applied is based on expanding the displacements in terms in trigonometric functions in the circumferential direction (analogous with the similar expansion in the axial direction .used previously is for oval cylindrical shells). Thus, the displacements are written in the form
U2
U](x1) cos jinx2
U (x
U1
=
R
U2(x') sin jmx2
0 I
W
Wo(x5
(27)
R
I
Wj(xb cos
jinx 2 R
where the integer m is the circumferential wave number of the mode to be studied. In the incremental finite element solution based on eqn (26) the one-dimensional amplitude functions U}, U~ and Wj are approximated by Hermitian cubics within each element. As in Section 4 the solutions are assumed to be periodic with an axial half-wavelength, 2l. The symmetry boundary conditions at x 1 = 0, 21 are given by eqn (25), and also here we need only consider half this interval, 0 ~< x I ~< l, with symmetry of the even-numbered functions and antisymmetry of the odd-numbered functions specified at x 1 = l. The solutions to be presented are based on using J = 2 in eqn (27) and two finite elements over the interval 0 ~< x I ~< I. In the circumferential direction integration over the interval 0 ~
~-t,X1
rH,X2
w = - ~ h cos - - + ~,,,h cos cos 1 21 R
(28)
Viggo Tvergaard
158
where ~ is the amplitude of the axisymmetric imperfection and ~m is the amplitude of a non-axisymmetric imperfection with circumferential wave number rn. Figure 8 shows load versus shortening curves obtained for shells with R/h = 100, for the material specified by n = 10, 0o = 0 and ([~c)max = 100°. The dashed curves are the axisymmetric solutions also given in Fig. 2. The wave number, m, of the non-axisymmetric imperfections is chosen equal to the critical wave number at the corresponding bifurcation point. These numbers are rn = 9 (approximately square buckles) for ~ = 0.1, m = 8 for ~0 = 0.4, and m = 6 for ~0 = 1-0. The computation for ~ = 0-1 and the extremely small nonaxisymmetric imperfection ~9 = 10 -5 in Fig. 8 is carried out to determine the post-bifurcation behaviour of the axisymmetric shell. Prior to bifurcation this solution coincides with the axisymmetric solution, but a maximum is reached immediately after the bifurcation point and subsequently the load drops off rapidly, showing the unstable post-bifurcation behaviour. Initially after bifurcation the post-bifurcation curve is nearly parallel with the fundamental solution somewhat similar to the smooth bifurcation found for the cruciform column, 11 when 0o = 0. It is noted that this computation for ~9 = 10-5 indicates good agreement with the bifurcation prediction based on the alternative comparison solid discussed in Section 3. Earlier bifurcation 1.0
P/Pc
maximum\
~°°°~\
.~=o.~ ~..,.-~-~"--_ 0
~o~o-s '
9
0.5 0 = 0.4 . ~8= 0 . 0 / .
0
Fig.
0
0.002
0.004
°i.~oiSyrnmetric
0.006
,x/,?, 0.008
8. A x i a l l o a d v e r s u s s h o r t e n i n g f o r c y l i n d r i c a l shells w i t h R/h = 100 a n d n o n - a x i s y m m e t r i c i m p e r f e c t i o n s ( o y / E = 0-0025, v = 0-3, n = 10, 01) = 0, ([~c)max = 100 ° a n d l = lc).
Plastic buckling of axially compressed circular cylindrical shells
159
from the axisymmetric solution is not excluded by this comparison solid, but the bifurcation result is supported by the solution with a small asymmetry which is expected to be unique. The results for ~ = 0-1 a n d ~9 = 0.01 and for ~o = ~9 = 0-1 in Fig. 8 illustrate the imperfection sensitivity associated with the unstable post-bifurcation behaviour. Also the curves for ~ = 0-4, ~8 = 0.04 and ~0 = 1.0, ~ = 0.1, respectively, show a reduction of the load carrying capacity relative to the corresponding bifurcation loads. It is noted that the results are very dependent on choosing the critical value of the circumferential wave number. A curve obtained for ~) = 1.0, ~ = 0.1 differs little from the corresponding axisymmetric solution in Fig. 8. Figure 9 shows similar results for a thinner shell, R/h = 200. The sharp load drop after the maximum at the small imperfection ~o = 0.1, ~13 = 0"01 is more like the unstable behaviour of elastic shells than the corresponding result in the previous figure. The curve for ~) = 1-0, ~9 = 0.1 falls below the axisymmetric solution, but here, at a relatively low value of P/Pc, the load carrying capacity is not reduced below the bifurcation load. For the thicker shell, R/h = 25 (Fig. 10), the smaller imperfection ~) = 0.1 represents a case in which bifurcation takes place after the maximum. Here, the additional imperfection ~ = 0.01 has nearly no effect on the maximum load, but does give a more rapidly decaying 1.0
•0-0.1
P/Pc
maximum.
J
0.5
/
axisymmetric
j bifurcation
g.o.1 00
0.002
0.00z.
a/(
0.006
Fig. 9. A x i a l l o a d v e r s u s s h o r t e n i n g for cylindrical shells w i t h R/h = 200 a n d n o n - a x i s y m m e t r i c i m p e r f e c t i o n s (oy/E = 0-0025, v = 0-3, n = 10, 0o = 0, (13c)max = 100 ° a n d l = lc).
160
Viggo Tvergaard 1.0
' ioifu rcation~' axisymmetric
0.5
0
maximum
0
.~. "~0 = 1.0
=
i
0.008
0.016
t./~.
0.02/.
Fig. 10. A x i a l l o a d v e r s u s s h o r t e n i n g for cylindrical shells w i t h R/h = 25 a n d n o n - a x i s y m m e t r i c i m p e r f e c t i o n s (oy/E = 0.0025, v = 0.3, n = 10, 0t~ = 0, (~c)max = 100 ° a n d l = lc).
load around the bifurcation point. For ~ = 1-0, where bifurcation occurs before the maximum, the additional imperfection ~3 = 0.1 results in a reduced load carrying capacity.
6 DISCUSSION The use of incremental elastic-plastic constitutive relations in the shell buckling analyses, rather than the nonlinear elastic approximation of deformation theory, has the advantage that it rules out speculations regarding the influence of neglecting elastic unloading, etc. Furthermore, the attractive quality of deformation theory that predicted bifurcation loads agree reasonably well with experimental buckling loads is incorporated in J2 corner theory, since the bifurcation predictions of the two theories are identical, at least for proportional loading. For shells with initial imperfections, where the loading path deviates from proportional loading, the general tendency is that J2 corner theory predicts a buckling load higher than that of J2 deformation theory, but lower than that of J2 flow theory. This agrees with results found for a cruciform column. ~1 However, apart from such quantitative differences the bifurcation results found in the present
Plastic buckling of axially compressed circular cylindrical shells
161
investigation, based on incremental plasticity theory, show the same trends found by Gellin 4 for deformation theory. In particular, bifurcation is found for all shells considered, also for very thick shells, which d o e s not agree with the experimental observation that thick shells (R/h smaller than about 30) buckle axisymmetrically. Among various possible reasons why the theory fails to predict this transition to purely axisymmetric buckling, Gellin 4 suggested that neglecting elastic unloading might be the most important--however, the present results show that this is not the case. A feature of the results found here that was not discussed by Gellin 4 is that the bifurcation point occurs after the maximum load point for the thicker shells, particularly the shells with small initial imperfections. This is obviously significant if the axial load is the prescribed quantity, since then stability is lost at the maximum. In cases where the overall axial shortening is the prescribed quantity it would seem that the bifurcation points beyond the load maxima are still relevant stability limits. However, recent studies by Tvergaard and Needleman,11, 19. 20 relating to a wide variety of structures with the common property that the applied load versus shortening curve achieves a maximum, have shown that localisation of the buckling pattern will take place at the load maximum. Thus the assumption of periodic solutions in the axial direction used in all analyses in the present paper (and in previous analyses) 2, 3.4. 17 is valid so long as no maximum has been reached, but after the maximum, localisation into one or a few buckles in the axial direction must be expected, even in cases where the axial shortening is prescribed. Such axisymmetric localisation, occurring mainly in the thicker shells, could be a cause of the experimentally observed transition to purely axisymmetric buckling. The influence of localisation on the buckling behaviour of circular cylindrical shells under axial compression is to be investigated in more detail by the author. The importance of a bifurcation point on a rising load versus shortening curve is associated w~th the expectation of an unstable post-bifurcation behaviour. However, elastic cylindrical shells with axisymmetric imperfections do exhibit a stable initial post-bifurcation behaviour in some ranges of the imperfection amplitude. 3 The analyses in the present paper with an additional non-axisymmetric imperfection indicate that the post-bifurcation behaviour is unstable for the elastic-plastic shells. Except for one case, in which the
162
Viggo Tvergaard
knock-down factor is already 1/4, the load carrying capacity is further reduced by the additional imperfections.
REFERENCES 1. Koiter, W. T., 'Over de stabiliteit van het elastisch evenwicht', Thesis, Delft, H. J. Paris, Amsterdam (1945). English translations (a) NASA TT-FIO (1967) 833, (b) AFFDL-TR-70-25 (1970). 2. Koiter, W. T., The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression, Proc. Kon. Ned. Ak. Wet., 66B (1963) 265-79. 3. Budiansky, B. and Hutchinson, J. W., Buckling of circular cylindrical shells under axial compression, Contributions to the Theory of Aircraft Structures, Delft University Press (1972) 239-59. 4. Gellin, S., Effect of an axisymmetric imperfection on the plastic buckling of an axially compressed cylindrical shell, J. Appl. Mech., 46(1979) 125-31. 5. Hutchinson, J. W., Plastic buckling, Advan. Appl. Mech., 14 (1974) 67-144. 6. Batdorf, S. B., Theories of plastic buckling, J. Aeronaut. Sci., 16(1949) 405-8. 7. Hill, R., Generalized constitutive relations for incremental deformation of metal crystals by multislip, J. Mech. Phys. Solids, 14 (1966) 95-102. 8. Hutchinson, J. W., Elastic-plastic behavior of polycrystalline metals and composites, Proc. Roy. Soc. London, A318 (1970)247-72. 9. Hecker, S. S., Experimental studies of yield phenomena in biaxially loaded metals, Constitutive Equations in Viscoplasticity, AMD-Vol. 20 ASME, New York (1976) 1-33. 10. Christoffersen, J. and Hutchinson, J. W., A class of phenomenoiogicai corner theories of plasticity, J. Mech. Phys. Solids, 27 (1979) 465-87. 11. Needleman, A. and Tvergaard, V., Aspects of plastic post-buckling behaviour, In: Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, Ed. by H. G. Hopkins & M. J. Sewell, London, Pergamon Press (1982) 453-98. 12. Koiter, W. T., On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Ak. Wet., B69 (1966) 1-54. 13. Niordson, F., Introduction to shell theory, Technical University of Denmark (1980). 14. Hill, R., A general theory of uniqueness and stability in elastic-plastic solids, J. Mech. Phys. Solids, 6 (1958) 236-49. 15. Tvergaard, V., Buckling of elastic-plastic oval cylindrical shells under axial compression, Int. J. Solids Structures, 12 (1976) 683-91. Errata, ibid. 14 (1978) 329.
Plastic buckling of axially compressed circular cylindrical shells
163
16. Tvergaard, V., Buckling of elastic-plastic cylindrical panel under axial compression, Int. J. Solids Structures, 13 (1977) 957-70. 17. Pedersen, P. T., Buckling of unstiffened and ring stiffened cylindrical shells under axial compression, Int. J. Solids Structures, 9 (1973) 671-91. 18. Tvergaard, V., Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells, J. Mech. Phys. Solids, 24 (1976) 291-304. 19. Tvergaard, V. and Needleman, A., On the localization of buckling patterns, J. Appl. Mech., 47 (1980) 613--19. 20. Tvergaard, V. and Needleman, A., On the development of localized buckling patterns, Danish Center for Appl. Math. and Mech., Report 243 (1982).