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Buckling of elastic-plastic cylindrical panel under axial compression
BUCKLING OF ELASTIC-PLASTIC CYLINDRICAL UNDER AXIAL ~O~P~SS~ON
PANEL
Dqwrtmcnt of Solid Mechanics,The Technical University of Denmark,Lyngby, Denmark (&cc&d 25 Jaauaty 1977) AMrxt-The bucklingb&a&our is investigatedfor an axially compressedeWic-plastic cylindricalpanel of the type occurringin stiffened shells. The ~~~~ stress is determinedanalytically and an asympto&a& exact expaoaion is obtained for the initial post-bjfurdn btiuviour in the plasticrange.For pa&s with SmalIinitial imperfectioes the behaviour is amdysed asymptotically on the basii of the ~~lastlc m, that r+ts from neglectingthe effect of elastic unloading.Tbc imperfection-sensitivity &tic&&c panel N also computednumericallyby a linearincrementalmethod,and the results are comparedwith the results of the asymptoticanalysis. For a low hardeningmaterialthe panel is found to be impufec&m+mitive in the whok range of curvaturesconsidered,whereas for a high hardeningmaterial the panel is only imp&e&n-sensitive if the curvatureexceeds a certain value. 1. INTRODUCTION
The type of axially compressed cylindrical panels to be considered here occur between stiffeners in Iongitudinally stiffened cylindrical shells. When the stiffeners on such cylindrical shells are su&iently close, the critical bifurcation mode is an overall mode with a circumferential wavelength much longer than the stiffener spacing. However, for larger spacing of the stifleners, ld buckling between stiffeners may be the critical mode. In the present paper the interest shall centre on this local buckling of the cylindrical panels between the stiffeners. The initial post-bu&ing behaviour of elastic cylindrical panels has been investigated by Koiter [l], on the basis of the general theory of elastic stability[2],This inves~tion shows that the postbuckling behaviour is stable for sufhciently Aatpanels in agreement with the behaviour of phues, whiie more curved panels have unstable post-buckling behaviour. The initial postbucklin8 analysis has been extended by Stephens [33to also account for the torsional rigidity of the stiffeners and for the effect of internal pressure, which tend to increase the bifurcation stress and to decrease the imperfection-sensitivity. Post-buckling analyses have also been carried out for orthotropic elastic panels with shear deformability[4] and for sandwich panels [S]. The present paper gives an. ~vest~tion of the ~st-buc~g behaviour and the imperfection-sensitivity of elastic-plastic cylindrical pan& compressed into the plastic range. The bifurcation behaviour at the tangent modulus load is determined analytically, and in cases where the bifurcation mode is unique Hutchinson’s asymptotic theory [6,7] is employed to obtain expressions for the initial post-bifurcation behaviour. For the effect of small initial imperfe&ions an asymptotic analysis is carried out, based on neglecting the infiuence of elastic unkuuiing. This type of hypoel~tic asymptotic analysis was deveioped by Hutchinson and B~sky[8] for the case of a cruciform column and has also been used by N~~ern~ and TvergaardR for an axially compressed square plate. The behaviour of imperfect cylindrical panels is also computed numerically by a linear incremental method, and the results are compared with the asymptotic results. 2. PROBLEM FORMULATION
The narrow cylindrical panel is taken to be part of a longitudinally stiffened circular cylindricaI shell under axial compression. Here the cylindrical panel occurs as a section of the shell bounded by two ne~~u~ng stiffeners, and the local buckling mode of interest is one in which the saeners remain straight, while the shell buckles in a short wave pattern between the stiffeners. The shell has the thickness h, the radius R, and the circumferential distance b between the equally spaced stiffeners (Fig. 1). The main effect of the stiffeners is to prevent waviness of 957
m
\R
with L* = L* = LW.Here, Latinindicesw to2.Theiw~ the other to eiaic The tbmy d
from 1 to 3, wbik Greek idw
ENxiuti 1;w have two bflmbes, unlotding.
raqa from 1
one CorrcopondiBo to pmic
ioadhg,
phticity cn&oycd hare is smaIkrain &%owWry with isotropichardening. In tke tkce ~~~ x’-coordii system with metric tensor &i the insets moduliare (2.3) where E and Y arc Young’smodulusand Poisson”sratio,respectively,and
E/Et-l 23 E/E,- fl - 2v)/3’
for a; = (u&x and& > 0
f(%) = 0,
for 0; c (u,)-
or c+#C 0.
(2.5)
Bucklingof ela$~-pia~~ cylindricalpanel under axial compression
959
Here the initialvalue of (cQ),,,,,~ is the yield stress a,, and the tangent modulus E, is the slope of the uniaxiai stress-strain curve, The representation of uniaxial stress-strain behaviour chosen is a piecewise power Iaw with ~~n~~~u§ tangent modulus
where it is the strain hardening exponent. Since the stress state in the shell is approximately plane, only the in-plane stresses enter into the constitutive relations, and we can write
where the tensor of in-plane moduh is given by
and the m
strain increrncnttensor at distance x3outward from the shell middle surface
is approximated by $4 = i&f -
x3&.
(2.9)
The membrane stress tensor i+l” and the moment tensor MM are taken to be
zfsing eqns (2.7)aud (2.9)in QIO) gives incremental relations for EiT*and &f‘@in terms of i* and & Now, within the context of ~~~~Mush~~asov shell theory, the incremental principle of virtual work takes the form (2.11) where {OVA * is the increment of the external virtual work. In the fohowing the ~te~tion area A shall be chosen according to the smallest repeatable intervals in the periodic deformation pattern. As the buckling pattern is periodic in c~cu~e~n~ directionl,due to the constant stiffener spacing, we need only consider a she&e&ion between the contres x’= 0, b of the two ne~~~~ cylindrical panels. Due to the symmetry of mode displacements about these panel centres the boundary conditions can be specified as (2.12) Across the line of attachment of a stiiTenerwe require continuity of aI1field quantities, except for the possible discontinuity of the transverse shear force resulting from the constraint l+W -“-i=O ax
at x2 = b12.
(2.13)
in axial direction the mode displacements are periodic, and the resultant axial load is spec&d
V. Tvmm
960
by the load parameter A, as b N
”
dx* = Au,’‘hb
(2.14)
where ai’ is a constant. 3. PLASTIC BIFURCATION
AND POST-BIFURCATIONBEHAVIOUR The stress state in the perfect cyMricai shell prior to bifurcation con&s of a constant uniaxkl stress u” = A# at every point of the shell. Thus, at any kad level the cosn~ta of tBe~~areduli~~~Coadrnt~~rhcll,pedt8ckmsrtbihpcrtiwlordCan
bedettrmincdasthatofanel~ticorthotropicsb&liwithmorfuli~totaeirutlMLyulw plastic moduli. In the followkIt we shall use the expressions E,, Ez,El2and& for the physical values of the plastic branches of t”“, l?, cnp and tu12, respectively, at the biMcatkn point, and we shah use 4 = E,h’/12, L?, = E2Jr3/12,D,2 = E12h3/12and & = E&/6. For a compkte unsti&ned cyli&kal shell, simply supported at the end& the modes are found of the form
(3.1)
wbcrc b = d/m
(m is the circumferential wave number); or in the form of an axkytnmetric sinus&al mode. The b&cation stress o, corresponding to (3.1) is given by
[~,a4+~~2+~~'+~(~~+E2~(ctT+~)-c,EnqjP a,= -
2
0.2)
Where
(3.3)
(3.4) An iterative procedure is used to determine the numerically smalkst critical stress u=, the correspondinB value(s) of the axial wavepammeter a and the inst8ntaneous moduli, for Biven material and g&en values of /,, b and R. In the elastic range it is well known that b&cation of a compkte unstiffened shell occurs at a, = -{3(1- y3)-‘%/& with several simuitaneow bu&litQ modea[2]. These include the axisymmetric modes and the modes with circumferential wave number m up to a certain limiting value, above which the bifurcation stress increases with increasing m. Fw 2 illustrates this for a rather thick shell, where the limit& vahre of m is between 12and13. In the plastic range the modulus E, decreases more rapidly than the other prebuckling moduh. Therefore, the pkstic bifurcation stress is smallest for the axisymmetric mode and increases with increasinB circumferential wave number. An example of this is also shown in Fig 2, for
u,JE=0.002.
Buckling of elastic-plastic
cyWrical
panel under axial compression
%1
0.008 -iit 0.006.
0’0 Fig. 2. Bifurcationswcss
terser
I 5
10
20
15
25
m
30
circumferential wave numberfor complete unstiffened cylindrical withR/h=XKl,n=lO, ~~0.3.
shell
In a stiffened shell with stringers located at x2= b/2 + kb (for integer k) the bifurcation modes (3.1) are still valid, provided that the only constraint along the stiffeners is the suppression of radial deflection waves. Thus, the bifurcation stress for the cylindrical panel is also given by the expression (3.2).Koiter[l] has found that for a sufficientlynarrow cylindrical panel in the elastic range the bifurcation stress (3.2) reduces to UC =
-E3(]
r*h* _ v2jb$l + @“I,
e
=
m2(1- y31 b 2a
v’(Rir)’
(3.5)
This elastic bifurcation stress is valid for 8 s 1, with only one critical wave parameter a = 1. For 8 > 1 the earlier mentioned critical stress for the complete elastic shell is valid, with two critical a-values for each circumferential wave number m. For a cylindrical panel that bifurcates in the plastic range we shall also use the parameter 8 to characterixe the ‘flstness” of the panel. Here again we find two critical a-v&es for each value of 8 in the range B> 1, and also in certain ranges of 8 values below unity, dependent on the values of CT,&,and n. In the present investigation the main interest is devoted to cases where only a single bifurcation mode (one a-value) is associated with the critical stress. Examples of such cases are shown later in Fiis. 3 and 4. An asymptotic theory of initial post-bifurcation behaviour in the plastic range has been developed by Hutc~son~6,7], which extends Hill’s general theory of b~~cation in elasticplastic solidsjll, 121.As in Koiter’s elastic post-buckling theory[2] the load in the vicinity of the bifurcation point is expanded in terms of the buckling mode amplitude, but in the plastic range the analysis is further complicated by the necessity to account for elastic unloading.~ In a case where the buckling mode is unique, the asymptotically exact expression for the load parameter A in terms of the buckling mode amplitude f is obtained of the form
for f 2: 0. Here, we normalize the buckling mode so that for 6 = 1 the maximum normal deflection w is equal to the shell thickness. The constant Ai is determined by the requirement that plastic loading takes place ~ou~out the current plastic zone, except in at least one point where neutral loading takes place. After bifurcation a region of elastic unloading spreads into the material from each of these neutral loading points, and the effect of these unloading regions is accounted for to lowest order in the third term of the expansion (3.6). For the cylindrical panels considered elastic unloading starts on the outside of the shell at the points of maximum outward deflection. In the limiting case of a flat plate (4 = 0) unloading starts s~ul~~ousIy at the points of m~imum outed- and inward deflection. The value of the parameter g depends on the shape of the unloading regions spreading into the material[7], and here we find g = l/3. The constant A2is found negative in all cases, so that the truncated expansion (3.6)can be used to estimate the maximum support load, which is slightly above the critical bifurcation load. The values of the constants o,, A, and AZare given in Table 1 for a few SS Vol. 13. No. 10-G
V. TVERGAMD
%2
Table 1. Constants in asymptotic post-bifurcationexpansion for various cylindricalpanels
0.025
0.025 o.ozs 0.02S
0.50 0.30 0.75 0.75
o.am o.Ou2O 0.0028 0.0028
0.3 0.3 0.3 0.3
10 3 10 3
1.104 1.162 1.033 1.052
0.643 0.884 0.804 0.878
-1.91 -3.59 -3.49 -4.84
l/3 l/3 l/3 l/3
examples of cylindrical panels that bifurcate in the plastic range with only a single bifurcation mode. It is known that bifurcation stresses obtained by &deformation theory are often in better agreement with experiments than the predictions of &flow theory[7]. However, for the cases considered in Table 1 bifurcation does not occur far into the plastic range, so the discrepancy is rather small with a maximum of 1.7%corresponding to the case 6 = 0.5, n = 10. 4. ASYMPTOTIC ANALYSIS WITH NO ELASTIC UNLOADING
The e&t of a small initial imperfection on a structure compressed into the plastic range has not yet been described by a simple asymptotic formula such as those obtained by Roi&r{2]for the elastic range. The main difficulty, is that an asymptotic expansion of the initial part of the equilibrium solution is only valid up to the point at which elastic unloading starts, while a second expansion that accounts for the growing elastic unloading region is required to represent the remaining part of the equilibrium path(l31. Also, the maximum support load of a perfect elastic-plastic structure is attained at a limit point after &rite buckling mode deflections and not at the bifurcation point as in the elastic range. Here, we shall analyse the imperfection-sensitivity on the basis of &flow theory, but with elastic unloading neglected. By using this hypoelastic theory the di&cuRies mcDtiQwdahove are avoided and an asymptotic estimate is obtahmd for the imperfect&se&&y of the hypoeWic cylindrical panel. This approach has been used earlier by Hutch&on and Bu&maky@] for the case of a cruciform column, and by Needleman and Tvergaa&W for the case of an axiaRycompressed square plate. The cruciform has the special propWty tlW&often, nostra$lratereverssJoccursbefonthemoximumlood,so~~tinlalodic~is completely just&d. For the plate strain rate reversrd does occur slightly before the maximum load. However, the results of Ref. [9] show that the hypoelastic expansion gives a reaao~& estimate of the imperfection-sensitivity of an elastic-plastic plate. Here, the asymptotic expansion based on hypoelastic theory shall be extended to cover shell behaviour, and Ml be used for the cylindrical panel. As the expansion has been described in detail for the plate[9], only the main steps are given here. The perfect shell made of hypoelastic material has the same bifurcation stress as the corresponding elastic-plastic shell, but instead of (3.6) the initial post-bifurcation expansion in the case of a unique buckling mode takes the form
A=A,+A,“~+A*k[*+-.
(4.1)
where the higher order contributions are taken to be orthogonal to the bifurcation mode
I Nc”g,$, A
dA = 0, for i > 1.
(4.31
The plastic branches of the &flow theory moduli are functions only of the stresses, and can be expanded in a Taylor series about the bifurcation point
Buckling of elastic-plastic
cylindrical
panel under axial compression
%3
Substituting (4.2)into (2.1)gives the expansion of the strains, and the similar expansion of the stresses is related to the strains by using (2.7) and (4.4). Expressrons for the parameters A,“, A21*etc. are obtained by substituting the expansions into the incremental principle of virtual work (2.11).Hutchinson 171has given the expression for Alk, which is zero for the cylindrical panel due to symmetry. The expression for Azk is considerably simphfmd if A,” = 0, in which case we find A2 hd =--
‘& D
(4.6)
(4.7) where V is the volume of shell material corresponding to the middle surface area A. The expressions (4.6) and (4.7) depend on the second order contributions to the asymptotic expansions, which are found by solution of a variationalquation obtained from the principle of virtual work. For AIk = 0 the variational equation takes the form (4.8)
(4.9) (4.10) CO
Q
where N” and &@ are obtained from (2.10)using (4.9)and (4.10).The solutions ?,, (&, g are periodic in axial direction, and are the same in all panels, as they grow proportionally with f2 while neighbouring panels have identical bifurcation modes with opposite signs of the amplitudes. Thus, the second order mode displacements are symmetric about the line of attachment of a stiffener, in addition to the symmetry about the panel centre lines. The load terms in eqn (4.8) suggest that the boundary value problem has a solution of the form
(4.11)
V.
964
TVERGAMD
where Z is a constant, and 32, @, 8i, & and i@are functions of x2. Furthermore, this solution satisfies the orthogonality condition (4.3). With the solution (4.11) the variational equation separates into two independent boundary value problems in the interval 0 s x2 s b/2,one for the (~)quantities and one for the huantities. Each of these problems is solved numerically by making a Lnite element approximation of the x2dtpendent functions in terms of Hermithm cubits. The (-)qu&ntities automatically satisfy the constraint (2.13) and thus have the boundary conditions (2.12) at both ends of the interval x2 = 0, b/2. For the o-
(4.12)
and k are undetermined by the analysis, and $=1+2
(4.13)
(4.14)
(4.15) Here, and in the following Q denotes the linear part of the membrane strain tensor Q. For an mtttritl, linttr or nonlinear, the parameter $ equals unity. In the regular perturbation analysis the lowest order effect of a small imperfection is de&mined for A
V&l=
v,” + cm,w=w"+G,
7jae=eze+&,
aOB=~o*+&~
(4.16)
where the solution for the perfect s&U at the current load level A is denoted by ( )‘, while (J denotes s&l per&&&m quantities that vanish at A = 0. AU equations are lhmari& with respect to the small pert&~&n- and imperfection quantities. Thus, the conatit&ve equgtion (2.7) y&da the line&& mlation
!!$L ~494$+!Lzl pdg?
(4.17)
0
and tht linearized principle of virtual work takes the form
I
{fi&b A
+ &‘%Q, + N,,@(O., + &.)Sw,}
dA = 0.
(4.18)
To solve eqn (4.18), we expand &, and G in terms of the set of eigenfunctions of the form (3.1), with any integer number of halfwaves along the length of the shell and over the width of the panel. We also use the fact that both toti and
Buckling of elastic-plastic
cylindrical
panel under axial compression
%5
at”@* d& au*"I 0 dA appearing in (4.17) are orthotropic fourth order tensors in the sense
This property is employed together with the fact that for each eigenfunction in the expansion also the corresponding e4 and K~ vary as sines or cosines, and the result is a complete decoupling of the equations for the eigenmode amplitudes. Finally we find that the variation of the amplitude f of the critical eigenmode (3.1) can be written on the form[9] (4.20) where 9 is given by (4.13) and p(A) is finite at A = A,. The function p(A), and in particular the value p(A,), is computed by using the eigenmode expansion in (4.18) and (4.17) and solving for increasing A in the interval 0 s A < AC(see Appendix). Matching of the expression (4.20) with the rising part of (4.12) in the vicinity of A = A, determines the values of the two constants c and k
c = -UGc)l”‘,
k
= I/$.
(4.21)
Now, according to the expression (4.12), with the values (4.21) substituted, the hypoelastic structure is imperfection-sensitive provided that A*‘”CO. In that case an imperfection of amplitude { converts the bifurcation buckling into snap buckling at a reduced load A, given by the asymptotic estimate (4.22)
(2#+ 1)(2$)-~2*+1)[p(AJW(2’+?
(4.23)
Numerically obtained values of the bifurcation stress a, and of the hypoelastic postbifurcation coefficient A2k are given in Figs. 3 and 4 as functions of the @tress parameter 8. The panels considered have a fixed thickness to width ratio h/b = 0.025, but various levels of yield stress are considered, with strain hardening exponent n = 10 in Fig. 3 and n = 3 in Fii. 4. The curves are only drawn in the ranges, where a single axial wave parameter a is associated with a,. Clearly, the change from one to two critical a-values occurs closer to the yield stress when 8 is closer to unity. For the elastic panel At&/A, reduces exactly to the corresponding elastic post-bifurcation coefficient, with the value &l - v2) in the plate case (6 = 0), and with the change from stable- to unstable post-bifurcation behaviour at 8 * 0.64, as obtained by Koiter[l]. For low strain hardening, Fig. 3, the material nonlinearities result in imperfection-sensitivity, even at small &values. At the larger B-values material nonlinearities and geometric non-linearities contribute about equally to the imperfection-sensitivity, so that A2kc is close to the corresponding elastic value. For the high hardening material, Fig. 4, a stable post-bifurcation behaviour is found for small e-values, but the change to unstable behaviour occurs at a smaller value of 8 than that found for the elastic panel. Both in Fig. 3 and Fig. 4 we note that the values of A2ikc, as functions of 0, are discontinuous at the points where a, equals the yield stress, because the moduliderivatives are discontinuous at the yield stress. Some values obtained for Az~, 4, p(A,) and ~1 are given in Table 2. For comparison, the value of p(A,) obtained in the elastic range (with linear prebuckling) is unity, so that with $ = 1 the expression (4.23) for ~1reduces to the well-known result for elastic bucklii. It is of interest to note that the values of ~1obtained in Table 2 are smaller (down to 25%) than those obtained
-4-
1
dY
-=-Y1
cl.WL.
0.003 elastic
zr~~ /
0.003t
?‘““;;’
i
eiastic
-----y’ _---
.----\/& _-_.w----
0.002~ '2
2~0.002
OS02
= 0.002
~-
! 0’
I 0
0.2
0.4
0.6
0.6 e
I OO
1.0
O.&t
0.2
0.L
0.6
0.8
0
I 1.0
---.-___
g%c --_‘-yf=y 0
-04 .
-0.8
T&k 2.Constaatn ia ayparkttic uymptotkadysis for cyliaQicaipontis with#b=O.@2Sadr=O.3 e 0.50 0.50 0.15 0.75
ujE
n
o.oea IO O.Otxm 3 0.0028to Om28 3
-ey 1.W 1.162 I.033 I.032
A2%, -0.414 -0.059 -0.430 -0.330
# 1.716 1.231 I.150 I.071
PU,) o.ws 0.421 0.278 0.628
P 0.344 0.489 0.638 0.978
by the elastic form&a, for the same values of AZ%, In the next section the values given in T&de 2 shall be used to compare the asymptotic estimate of ~~e~~-~~~vi~ with lmnkcal results. 5. NUMERICAL COMPUTATION OFIMPERFECTION-SENSITIVITY The behwiourof imperfect panels is here computed numerically by an incremental procedure based on the variational equation (2.11). As in the preceding section, the imperfections considered are taken in the shape of the critical in mode (3.1) with amplitude F In the numerical method employed to solve (2.11) the displacements are expanded in terms of trigonometric functions in the axial direction, while a one-dimensional finite element approximation is used in the circumferential direction.
Bucklingof elastic-plastic cylindrical panel under axial compression
967
(5.1)
Here, the functions #“(x2), v”‘(x2) and w(i)(x2) are approximated by Hermitian cubits within each finite element, and the wave parameter a is that corresponding to the critical bifurcation stress. This method has also been used for an axially compressed oval cylindrical shell[l4], within the context of a more accurate shell theory. The boundary conditions specified are the symmetry conditions (2.12) at the panel centres in addition to the stiffener constraint (2.13), which gives w’“(b/2) = 0 for i L 1. In the expansion (5.1) it is found that N = 2 giv6s good accuracy, with a half panel width divided into 2 elements. The integrals in eqn (2.11) are evaluated by 4 point Gaussian quadrature in circumferential direction within each element, while in the axial direction it is found sufficient to divide the interval -lb/a SX’ ~ib/a in two subintervals, using 4 point Gaussian quadrature within each subinterval. Through the thickness Simpson’s rule is used, with 7 points in (2.10). The active branch of the tensor of moduli (2.8) at an integration point is determined in each increment as follows. If the stress state at the integration point is on its current yield surface, the plastic branch is taken to be active. If S, for that integration point turns out to be negative, the elastic branch is taken to be active in the next loading increment. This .proeedure is su5ciently accurate if small increments are used and if the transition from loadii to unload& or vice versa, occurs only once or twice during the loading history. The results of numerical computations shown in Figs. S-8 give the load parameter A as a function of the de&&on contribution W, = w”‘(O), which is quite a good measure of the bifurcation mode growth. All the results are computed for h/b - 0.025 and v = 0.3, with strain hardening exponent n = 10 in the first two figures and n = 3 in the last two figares. The flatness parameter and the yield stress are taken as 8 = 0.5 and uJE = 0.002 in Figs. 5 and 7, while the values used in Fii. 6 and 8 are 8 = 6.75 and o,/E = 0.0028. The post-bifurcation behaviour of the perfect panel is computed numerically as the behaviour of a panel with a small init.@ imperfection $= 0.0002. In all four figures this behaviour
t
"Perfect"
1.0v
_v , :::-“ld , 0
0.2
0.L
0.6
0.6
Fig. s.
1.0 3
1 1.2
0
0.2
0.L
Q6
CM
1.0 3 h
Fii.6.
Fig. 5. Load versus mode defiection for cylindrical panelthat bifurcates in UK plastic raagc (h/b= 0.025, e=0.5,uJE-0.002,n= lO,v=O.3). Fig. 6. Load
cylindrical panel that bifurcates in the plastic range (&/b- 0.025, 8 = 0.75, OJE = o.ooi?8, n = 10, Y = 0.3).
versusmodedefkctionfor
1.;
968
V.
TVERGMRD
P
1cg
0.8
0.8
0.6
0.6
0.4 Y
maximum
0.2
0 0
0.2
OL
06
Fig.7.
08
I.0 "1 h
1.2
Fig. 8.
with the asymptotic expansion (3.6) in that bifurcation takes place under increasing load and in that a maximum is reached slightly above the b&ration load after a &de bifurcation mode d&e&n. As this characteristic behaviour results from the effect of eiastic unloading, it is not predict& by the hypoelastic its expansion (4.1). However, the hypoeMic expansion with the values obtained for A.z% (Tabk 2) does agree well with the general &en&of the numerically computed post-bifWcatioabehaviour in each of Fiis. 5-8, apwt from the maximum just after bifurcation. The shell conaider& in Fig. 5 is in the range of &values, where ekstic cyli&ical panels have a stabk ~st~c~g behaviour. The eiastic-pkstic _pl with A= IO is ckarly imperfection sensitive. For the largest bin considered, e = 1.0,a maxiumm isnotreached in the range cons&red, but the curve ikttcns out at a level of reduced support load. In Fii. 6, for a more curved panel with a higher yield stress, but still with the strain hardening exponent n = 10, the behaviour is quite similar to that obtained in Fig. 5. The sensitivity to small imperfections is slightly higher, but for the largest imperfection the support load is a iittle lessreduced. The cylindrical panel considered in Fig. 7 is identical to that of Fig. 5, exceptthat the material is high hardening with # = 3. For the perfect panel the post-bifurcation load after the maximum decreases to a value sliitly below the bifurcation load, in agreement with the small negative value obtained for A~~j‘lj(,, but then starts to increase again in the more advanced post-buckling range. The result is that even for the small imperfection, 5’= 0.01, no maximum is found before the support load exceeds the bif~c~n load. The panel in Fig. 8 is that of Fii. 6, with the strain hardening exponent changed to A= 3. The behaviour is much lie that found in Fig. 7, but the post-bifurcation ioad reaches a considerably lower level before again starting to increase in the advanced post-buckling range. In this case a local maximum is found for d = 0.01, but not for g = 0.1. A comparison between the numerical results and the asymptotic estimate (4.22) of the im~~ection-sensi~vi~ is given in Fig, 9. Here the curve(4.22)is drawn on the basis of the parameters + and F given in Table 2, and the numerically computed support loads are indicated by a dot for I= 0.0002,I= 0.01 and f= 0.1. In Figs. 9(a) and (b), corresponding to Fiis. 5 and 6, the agreement is very good for the range considered. However, for the large imperfection $= 1.0 the support load is underestimated by the asymptotic analysis, as one would expect. In Fig. 9(c) the agreement is also good for the smallest imperfection, but for $- 0.1 no maximum is reached in FJig.8, so the range of validity of the ~~ptoti~ expansion is smaller in is seen to agcc
this case. Such limited range of validity of the asymptotic results hL also been found for elastic
Bucklingof elastic-plasticcylindricalpanel under axial compression
959
Pig. 9. Comparisonof mimer&l rcswlts aud asymptotic l~ypoelasticpredictions for the imperfectionsensitivity of elastic-plastic cyst panels. (a) Panel with 8 = D.5,o;/E = O&02,II = IO.(bj Panel with 8 = 0.75, a#? = 0.0028, n = 10. (c) Panel with B= 0.73, KY@= 0.#28, n = 3.
buckling behaviour in a number of cases, in which the initially unstabk post-buckling behaviour changes to stable behaviour in the advanced post-buckling range[lS]. This also agrees with Koiter’s conjecture of the advanced ~st-buc~g behaviour for elastic cylindrical panekfl]. The asymptotic results ~o~es~~i~g to Fii 7 have not been plotted, as no maxima are obtained in this f&e for comparison. ‘%k value of g found for this case (Table 2) indicates a great deal of sensitivity to small ~~~e~~ns, but as the change to stable advanced postbuckling behaviour occurs ~~~e~iy earlier in Fig. 7 than in Fig. 8 an even smaller range of vahity of the asymptotic result must be expected here. Although the halite analysis does not account for the effect of elastic unloading, it is found that the asymptotic estimate (4.22}gives a good indication of the im~~ection-~nsi~v~y of an elastic-plastic cylindrical panel. This agrees with the results obtained for axially compressed square plates[I)].The range of validity of the asymptotic estimate is found to be smaller for a hii hardening material than for a low hardening material, because panels made of a high hardening material have the abihi to support loads above the bifurcation load in the advanced ~st-~~ range. As in Ref. 193we con&de that the a~~~~a of negkcting ehtstic unloading in the hypoekstic expansion seams to be less important than the appromo involved in co~si~~~ lowest order asymptotic results. It may be expected that also for other shell socks the h~i~tic analysis will prove valuable for iden~y~g parameter ranges, where the she&3wih exhiiit ~~~~on-~nsiti~ty in the piastic range. REFERENCES 1. W. T. Koiter, Buckling and post-buMttg Mavior of a cylindrical panel under axial compression. Nationaal lJtchrlwun L.l76onzrorf&m 26, Report S476, Amsterdam(19561. 2. W. T. Koiter, Over de stabiiit van bet elastisch cvenwicht. Thesis, Mft, H. J. Paris, Amsterdam(1943) English tmml. NASA ‘l-I’F-10,833 (1967);and AFFIX-TR-70.25(1970). 3. W. 8. Steph&s, Imp&&on se&ivity of axially compressedstringerreinforcedcyliidrical panels under intemal pressure. AlAA L 9, 1713-1719(197lf. 4.0. cl. Pope, The bi@itig bEbaviourin &al oompressioa of ayes panels, iecl&lg the e&t of sbenr deforr&ility. Jr&f. soiidr Srrr4crynu4,323,.34@(19681. 5. N. R. Bauld, Imperf&ion se&ivity of axially compressed stringerreiaforcul cylindricalsandwich panels. Jar. L solids ~~~~~ lb, l?83-m (1974). 6. J. W, Hymn, Peon behavior ia the plastic range. f, Me& Pays. &lids 21,163-190 (1973). 7. 1. W. Ham, Plastic bu&liq. A&races ie A@ied Mrckuaics (Edited by C. S. Yib}, Vol. 14, pp. 67-M. Academic Press, New York (1974). 8. f. W. Hut&moo aad 8. Budia&y+ Am&&al and mtmericalstudy of the effects of initial imperfectionson the . inclasti twkbg of s cruciformc&nm. Buck&g of Smc&rcs @lited by B. Budianskyj, pp. 98-103. SpriagtrVerlag, Be&i (1976). 9. A, Needkmaa and V. Tveqaard, An analysis of the imp&&o-sensitivity of squareeiastic-plasticplates under nxial compression.Int. 1 Solids &ucruns 12, 185-201(1976). 10. V. ‘I’vergaard and A. Necdkmaa, Bnoklingof ccccn~y stiffenedelastic-plastic panels on two simple supportsor multiply aupportul. Inr. J. SoUds Strucruns lL647-663 (1975). 11. R. Hill, A geueral theory of uniquenessand stability in elastic+astic solids. J, A4eck.Pkys. Solids 6,236249 (1958). 12. R Hill, Bi and uniquenessin non-linearmechanicsof continua. Pmblems of Can&mum MecMcs (S.I.A.M. Pales, ifs-164 fl%?). 13. J. w. Hut&inson, I~e~o~-~n~ti~ in the plastic range. 3. Me&. Pkys. salids 21, 191-204(19735 14. V. Tvcrgaard,Buckliag of elastic-plastic oval cylindricalshells underaxial compression.ia?. 1. Solids Srnrcrprnsff, 683-691 (1976). IS. V. Tvergaard,Buckling~bavio~ of plate and shell s~c~~s. Pruc. l&k Inf. Congr.Tlteor,or&Appf. Meek. @.diti by W. T. Kolter), pp. 233-247. No~~-Ho~, Amsterdam (1976).
970
V. TV~~GMRO APPENDIX
The functionp(A) occurriq in (4.20).and in particuultuthe value p(A,) to be used ia (4.21)and (4.23).ia dctemtinadby integmtiq the consthtive mhtioa (4.17) and suhtitutiq the m&t into tlte prhcipk of vittud wark. For the intep8th of (4.17) it is convenientto write
subatitutaiinto (4.17) tllia givea the diffemllthl quathns for p
(A4)
PANEL
Dqwrtmcnt of Solid Mechanics,The Technical University of Denmark,Lyngby, Denmark (&cc&d 25 Jaauaty 1977) AMrxt-The bucklingb&a&our is investigatedfor an axially compressedeWic-plastic cylindricalpanel of the type occurringin stiffened shells. The ~~~~ stress is determinedanalytically and an asympto&a& exact expaoaion is obtained for the initial post-bjfurdn btiuviour in the plasticrange.For pa&s with SmalIinitial imperfectioes the behaviour is amdysed asymptotically on the basii of the ~~lastlc m, that r+ts from neglectingthe effect of elastic unloading.Tbc imperfection-sensitivity &tic&&c panel N also computednumericallyby a linearincrementalmethod,and the results are comparedwith the results of the asymptoticanalysis. For a low hardeningmaterialthe panel is found to be impufec&m+mitive in the whok range of curvaturesconsidered,whereas for a high hardeningmaterial the panel is only imp&e&n-sensitive if the curvatureexceeds a certain value. 1. INTRODUCTION
The type of axially compressed cylindrical panels to be considered here occur between stiffeners in Iongitudinally stiffened cylindrical shells. When the stiffeners on such cylindrical shells are su&iently close, the critical bifurcation mode is an overall mode with a circumferential wavelength much longer than the stiffener spacing. However, for larger spacing of the stifleners, ld buckling between stiffeners may be the critical mode. In the present paper the interest shall centre on this local buckling of the cylindrical panels between the stiffeners. The initial post-bu&ing behaviour of elastic cylindrical panels has been investigated by Koiter [l], on the basis of the general theory of elastic stability[2],This inves~tion shows that the postbuckling behaviour is stable for sufhciently Aatpanels in agreement with the behaviour of phues, whiie more curved panels have unstable post-buckling behaviour. The initial postbucklin8 analysis has been extended by Stephens [33to also account for the torsional rigidity of the stiffeners and for the effect of internal pressure, which tend to increase the bifurcation stress and to decrease the imperfection-sensitivity. Post-buckling analyses have also been carried out for orthotropic elastic panels with shear deformability[4] and for sandwich panels [S]. The present paper gives an. ~vest~tion of the ~st-buc~g behaviour and the imperfection-sensitivity of elastic-plastic cylindrical pan& compressed into the plastic range. The bifurcation behaviour at the tangent modulus load is determined analytically, and in cases where the bifurcation mode is unique Hutchinson’s asymptotic theory [6,7] is employed to obtain expressions for the initial post-bifurcation behaviour. For the effect of small initial imperfe&ions an asymptotic analysis is carried out, based on neglecting the infiuence of elastic unkuuiing. This type of hypoel~tic asymptotic analysis was deveioped by Hutchinson and B~sky[8] for the case of a cruciform column and has also been used by N~~ern~ and TvergaardR for an axially compressed square plate. The behaviour of imperfect cylindrical panels is also computed numerically by a linear incremental method, and the results are compared with the asymptotic results. 2. PROBLEM FORMULATION
The narrow cylindrical panel is taken to be part of a longitudinally stiffened circular cylindricaI shell under axial compression. Here the cylindrical panel occurs as a section of the shell bounded by two ne~~u~ng stiffeners, and the local buckling mode of interest is one in which the saeners remain straight, while the shell buckles in a short wave pattern between the stiffeners. The shell has the thickness h, the radius R, and the circumferential distance b between the equally spaced stiffeners (Fig. 1). The main effect of the stiffeners is to prevent waviness of 957
m
\R
with L* = L* = LW.Here, Latinindicesw to2.Theiw~ the other to eiaic The tbmy d
from 1 to 3, wbik Greek idw
ENxiuti 1;w have two bflmbes, unlotding.
raqa from 1
one CorrcopondiBo to pmic
ioadhg,
phticity cn&oycd hare is smaIkrain &%owWry with isotropichardening. In tke tkce ~~~ x’-coordii system with metric tensor &i the insets moduliare (2.3) where E and Y arc Young’smodulusand Poisson”sratio,respectively,and
E/Et-l 23 E/E,- fl - 2v)/3’
for a; = (u&x and& > 0
f(%) = 0,
for 0; c (u,)-
or c+#C 0.
(2.5)
Bucklingof ela$~-pia~~ cylindricalpanel under axial compression
959
Here the initialvalue of (cQ),,,,,~ is the yield stress a,, and the tangent modulus E, is the slope of the uniaxiai stress-strain curve, The representation of uniaxial stress-strain behaviour chosen is a piecewise power Iaw with ~~n~~~u§ tangent modulus
where it is the strain hardening exponent. Since the stress state in the shell is approximately plane, only the in-plane stresses enter into the constitutive relations, and we can write
where the tensor of in-plane moduh is given by
and the m
strain increrncnttensor at distance x3outward from the shell middle surface
is approximated by $4 = i&f -
x3&.
(2.9)
The membrane stress tensor i+l” and the moment tensor MM are taken to be
zfsing eqns (2.7)aud (2.9)in QIO) gives incremental relations for EiT*and &f‘@in terms of i* and & Now, within the context of ~~~~Mush~~asov shell theory, the incremental principle of virtual work takes the form (2.11) where {OVA * is the increment of the external virtual work. In the fohowing the ~te~tion area A shall be chosen according to the smallest repeatable intervals in the periodic deformation pattern. As the buckling pattern is periodic in c~cu~e~n~ directionl,due to the constant stiffener spacing, we need only consider a she&e&ion between the contres x’= 0, b of the two ne~~~~ cylindrical panels. Due to the symmetry of mode displacements about these panel centres the boundary conditions can be specified as (2.12) Across the line of attachment of a stiiTenerwe require continuity of aI1field quantities, except for the possible discontinuity of the transverse shear force resulting from the constraint l+W -“-i=O ax
at x2 = b12.
(2.13)
in axial direction the mode displacements are periodic, and the resultant axial load is spec&d
V. Tvmm
960
by the load parameter A, as b N
”
dx* = Au,’‘hb
(2.14)
where ai’ is a constant. 3. PLASTIC BIFURCATION
AND POST-BIFURCATIONBEHAVIOUR The stress state in the perfect cyMricai shell prior to bifurcation con&s of a constant uniaxkl stress u” = A# at every point of the shell. Thus, at any kad level the cosn~ta of tBe~~areduli~~~Coadrnt~~rhcll,pedt8ckmsrtbihpcrtiwlordCan
bedettrmincdasthatofanel~ticorthotropicsb&liwithmorfuli~totaeirutlMLyulw plastic moduli. In the followkIt we shall use the expressions E,, Ez,El2and& for the physical values of the plastic branches of t”“, l?, cnp and tu12, respectively, at the biMcatkn point, and we shah use 4 = E,h’/12, L?, = E2Jr3/12,D,2 = E12h3/12and & = E&/6. For a compkte unsti&ned cyli&kal shell, simply supported at the end& the modes are found of the form
(3.1)
wbcrc b = d/m
(m is the circumferential wave number); or in the form of an axkytnmetric sinus&al mode. The b&cation stress o, corresponding to (3.1) is given by
[~,a4+~~2+~~'+~(~~+E2~(ctT+~)-c,EnqjP a,= -
2
0.2)
Where
(3.3)
(3.4) An iterative procedure is used to determine the numerically smalkst critical stress u=, the correspondinB value(s) of the axial wavepammeter a and the inst8ntaneous moduli, for Biven material and g&en values of /,, b and R. In the elastic range it is well known that b&cation of a compkte unstiffened shell occurs at a, = -{3(1- y3)-‘%/& with several simuitaneow bu&litQ modea[2]. These include the axisymmetric modes and the modes with circumferential wave number m up to a certain limiting value, above which the bifurcation stress increases with increasing m. Fw 2 illustrates this for a rather thick shell, where the limit& vahre of m is between 12and13. In the plastic range the modulus E, decreases more rapidly than the other prebuckling moduh. Therefore, the pkstic bifurcation stress is smallest for the axisymmetric mode and increases with increasinB circumferential wave number. An example of this is also shown in Fig 2, for
u,JE=0.002.
Buckling of elastic-plastic
cyWrical
panel under axial compression
%1
0.008 -iit 0.006.
0’0 Fig. 2. Bifurcationswcss
terser
I 5
10
20
15
25
m
30
circumferential wave numberfor complete unstiffened cylindrical withR/h=XKl,n=lO, ~~0.3.
shell
In a stiffened shell with stringers located at x2= b/2 + kb (for integer k) the bifurcation modes (3.1) are still valid, provided that the only constraint along the stiffeners is the suppression of radial deflection waves. Thus, the bifurcation stress for the cylindrical panel is also given by the expression (3.2).Koiter[l] has found that for a sufficientlynarrow cylindrical panel in the elastic range the bifurcation stress (3.2) reduces to UC =
-E3(]
r*h* _ v2jb$l + @“I,
e
=
m2(1- y31 b 2a
v’(Rir)’
(3.5)
This elastic bifurcation stress is valid for 8 s 1, with only one critical wave parameter a = 1. For 8 > 1 the earlier mentioned critical stress for the complete elastic shell is valid, with two critical a-values for each circumferential wave number m. For a cylindrical panel that bifurcates in the plastic range we shall also use the parameter 8 to characterixe the ‘flstness” of the panel. Here again we find two critical a-v&es for each value of 8 in the range B> 1, and also in certain ranges of 8 values below unity, dependent on the values of CT,&,and n. In the present investigation the main interest is devoted to cases where only a single bifurcation mode (one a-value) is associated with the critical stress. Examples of such cases are shown later in Fiis. 3 and 4. An asymptotic theory of initial post-bifurcation behaviour in the plastic range has been developed by Hutc~son~6,7], which extends Hill’s general theory of b~~cation in elasticplastic solidsjll, 121.As in Koiter’s elastic post-buckling theory[2] the load in the vicinity of the bifurcation point is expanded in terms of the buckling mode amplitude, but in the plastic range the analysis is further complicated by the necessity to account for elastic unloading.~ In a case where the buckling mode is unique, the asymptotically exact expression for the load parameter A in terms of the buckling mode amplitude f is obtained of the form
for f 2: 0. Here, we normalize the buckling mode so that for 6 = 1 the maximum normal deflection w is equal to the shell thickness. The constant Ai is determined by the requirement that plastic loading takes place ~ou~out the current plastic zone, except in at least one point where neutral loading takes place. After bifurcation a region of elastic unloading spreads into the material from each of these neutral loading points, and the effect of these unloading regions is accounted for to lowest order in the third term of the expansion (3.6). For the cylindrical panels considered elastic unloading starts on the outside of the shell at the points of maximum outward deflection. In the limiting case of a flat plate (4 = 0) unloading starts s~ul~~ousIy at the points of m~imum outed- and inward deflection. The value of the parameter g depends on the shape of the unloading regions spreading into the material[7], and here we find g = l/3. The constant A2is found negative in all cases, so that the truncated expansion (3.6)can be used to estimate the maximum support load, which is slightly above the critical bifurcation load. The values of the constants o,, A, and AZare given in Table 1 for a few SS Vol. 13. No. 10-G
V. TVERGAMD
%2
Table 1. Constants in asymptotic post-bifurcationexpansion for various cylindricalpanels
0.025
0.025 o.ozs 0.02S
0.50 0.30 0.75 0.75
o.am o.Ou2O 0.0028 0.0028
0.3 0.3 0.3 0.3
10 3 10 3
1.104 1.162 1.033 1.052
0.643 0.884 0.804 0.878
-1.91 -3.59 -3.49 -4.84
l/3 l/3 l/3 l/3
examples of cylindrical panels that bifurcate in the plastic range with only a single bifurcation mode. It is known that bifurcation stresses obtained by &deformation theory are often in better agreement with experiments than the predictions of &flow theory[7]. However, for the cases considered in Table 1 bifurcation does not occur far into the plastic range, so the discrepancy is rather small with a maximum of 1.7%corresponding to the case 6 = 0.5, n = 10. 4. ASYMPTOTIC ANALYSIS WITH NO ELASTIC UNLOADING
The e&t of a small initial imperfection on a structure compressed into the plastic range has not yet been described by a simple asymptotic formula such as those obtained by Roi&r{2]for the elastic range. The main difficulty, is that an asymptotic expansion of the initial part of the equilibrium solution is only valid up to the point at which elastic unloading starts, while a second expansion that accounts for the growing elastic unloading region is required to represent the remaining part of the equilibrium path(l31. Also, the maximum support load of a perfect elastic-plastic structure is attained at a limit point after &rite buckling mode deflections and not at the bifurcation point as in the elastic range. Here, we shall analyse the imperfection-sensitivity on the basis of &flow theory, but with elastic unloading neglected. By using this hypoelastic theory the di&cuRies mcDtiQwdahove are avoided and an asymptotic estimate is obtahmd for the imperfect&se&&y of the hypoeWic cylindrical panel. This approach has been used earlier by Hutch&on and Bu&maky@] for the case of a cruciform column, and by Needleman and Tvergaa&W for the case of an axiaRycompressed square plate. The cruciform has the special propWty tlW&often, nostra$lratereverssJoccursbefonthemoximumlood,so~~tinlalodic~is completely just&d. For the plate strain rate reversrd does occur slightly before the maximum load. However, the results of Ref. [9] show that the hypoelastic expansion gives a reaao~& estimate of the imperfection-sensitivity of an elastic-plastic plate. Here, the asymptotic expansion based on hypoelastic theory shall be extended to cover shell behaviour, and Ml be used for the cylindrical panel. As the expansion has been described in detail for the plate[9], only the main steps are given here. The perfect shell made of hypoelastic material has the same bifurcation stress as the corresponding elastic-plastic shell, but instead of (3.6) the initial post-bifurcation expansion in the case of a unique buckling mode takes the form
A=A,+A,“~+A*k[*+-.
(4.1)
where the higher order contributions are taken to be orthogonal to the bifurcation mode
I Nc”g,$, A
dA = 0, for i > 1.
(4.31
The plastic branches of the &flow theory moduli are functions only of the stresses, and can be expanded in a Taylor series about the bifurcation point
Buckling of elastic-plastic
cylindrical
panel under axial compression
%3
Substituting (4.2)into (2.1)gives the expansion of the strains, and the similar expansion of the stresses is related to the strains by using (2.7) and (4.4). Expressrons for the parameters A,“, A21*etc. are obtained by substituting the expansions into the incremental principle of virtual work (2.11).Hutchinson 171has given the expression for Alk, which is zero for the cylindrical panel due to symmetry. The expression for Azk is considerably simphfmd if A,” = 0, in which case we find A2 hd =--
‘& D
(4.6)
(4.7) where V is the volume of shell material corresponding to the middle surface area A. The expressions (4.6) and (4.7) depend on the second order contributions to the asymptotic expansions, which are found by solution of a variationalquation obtained from the principle of virtual work. For AIk = 0 the variational equation takes the form (4.8)
(4.9) (4.10) CO
Q
where N” and &@ are obtained from (2.10)using (4.9)and (4.10).The solutions ?,, (&, g are periodic in axial direction, and are the same in all panels, as they grow proportionally with f2 while neighbouring panels have identical bifurcation modes with opposite signs of the amplitudes. Thus, the second order mode displacements are symmetric about the line of attachment of a stiffener, in addition to the symmetry about the panel centre lines. The load terms in eqn (4.8) suggest that the boundary value problem has a solution of the form
(4.11)
V.
964
TVERGAMD
where Z is a constant, and 32, @, 8i, & and i@are functions of x2. Furthermore, this solution satisfies the orthogonality condition (4.3). With the solution (4.11) the variational equation separates into two independent boundary value problems in the interval 0 s x2 s b/2,one for the (~)quantities and one for the huantities. Each of these problems is solved numerically by making a Lnite element approximation of the x2dtpendent functions in terms of Hermithm cubits. The (-)qu&ntities automatically satisfy the constraint (2.13) and thus have the boundary conditions (2.12) at both ends of the interval x2 = 0, b/2. For the o-
(4.12)
and k are undetermined by the analysis, and $=1+2
(4.13)
(4.14)
(4.15) Here, and in the following Q denotes the linear part of the membrane strain tensor Q. For an mtttritl, linttr or nonlinear, the parameter $ equals unity. In the regular perturbation analysis the lowest order effect of a small imperfection is de&mined for A
V&l=
v,” + cm,w=w"+G,
7jae=eze+&,
aOB=~o*+&~
(4.16)
where the solution for the perfect s&U at the current load level A is denoted by ( )‘, while (J denotes s&l per&&&m quantities that vanish at A = 0. AU equations are lhmari& with respect to the small pert&~&n- and imperfection quantities. Thus, the conatit&ve equgtion (2.7) y&da the line&& mlation
!!$L ~494$+!Lzl pdg?
(4.17)
0
and tht linearized principle of virtual work takes the form
I
{fi&b A
+ &‘%Q, + N,,@(O., + &.)Sw,}
dA = 0.
(4.18)
To solve eqn (4.18), we expand &, and G in terms of the set of eigenfunctions of the form (3.1), with any integer number of halfwaves along the length of the shell and over the width of the panel. We also use the fact that both toti and
Buckling of elastic-plastic
cylindrical
panel under axial compression
%5
at”@* d& au*"I 0 dA appearing in (4.17) are orthotropic fourth order tensors in the sense
This property is employed together with the fact that for each eigenfunction in the expansion also the corresponding e4 and K~ vary as sines or cosines, and the result is a complete decoupling of the equations for the eigenmode amplitudes. Finally we find that the variation of the amplitude f of the critical eigenmode (3.1) can be written on the form[9] (4.20) where 9 is given by (4.13) and p(A) is finite at A = A,. The function p(A), and in particular the value p(A,), is computed by using the eigenmode expansion in (4.18) and (4.17) and solving for increasing A in the interval 0 s A < AC(see Appendix). Matching of the expression (4.20) with the rising part of (4.12) in the vicinity of A = A, determines the values of the two constants c and k
c = -UGc)l”‘,
k
= I/$.
(4.21)
Now, according to the expression (4.12), with the values (4.21) substituted, the hypoelastic structure is imperfection-sensitive provided that A*‘”CO. In that case an imperfection of amplitude { converts the bifurcation buckling into snap buckling at a reduced load A, given by the asymptotic estimate (4.22)
(2#+ 1)(2$)-~2*+1)[p(AJW(2’+?
(4.23)
Numerically obtained values of the bifurcation stress a, and of the hypoelastic postbifurcation coefficient A2k are given in Figs. 3 and 4 as functions of the @tress parameter 8. The panels considered have a fixed thickness to width ratio h/b = 0.025, but various levels of yield stress are considered, with strain hardening exponent n = 10 in Fig. 3 and n = 3 in Fii. 4. The curves are only drawn in the ranges, where a single axial wave parameter a is associated with a,. Clearly, the change from one to two critical a-values occurs closer to the yield stress when 8 is closer to unity. For the elastic panel At&/A, reduces exactly to the corresponding elastic post-bifurcation coefficient, with the value &l - v2) in the plate case (6 = 0), and with the change from stable- to unstable post-bifurcation behaviour at 8 * 0.64, as obtained by Koiter[l]. For low strain hardening, Fig. 3, the material nonlinearities result in imperfection-sensitivity, even at small &values. At the larger B-values material nonlinearities and geometric non-linearities contribute about equally to the imperfection-sensitivity, so that A2kc is close to the corresponding elastic value. For the high hardening material, Fig. 4, a stable post-bifurcation behaviour is found for small e-values, but the change to unstable behaviour occurs at a smaller value of 8 than that found for the elastic panel. Both in Fig. 3 and Fig. 4 we note that the values of A2ikc, as functions of 0, are discontinuous at the points where a, equals the yield stress, because the moduliderivatives are discontinuous at the yield stress. Some values obtained for Az~, 4, p(A,) and ~1 are given in Table 2. For comparison, the value of p(A,) obtained in the elastic range (with linear prebuckling) is unity, so that with $ = 1 the expression (4.23) for ~1reduces to the well-known result for elastic bucklii. It is of interest to note that the values of ~1obtained in Table 2 are smaller (down to 25%) than those obtained
-4-
1
dY
-=-Y1
cl.WL.
0.003 elastic
zr~~ /
0.003t
?‘““;;’
i
eiastic
-----y’ _---
.----\/& _-_.w----
0.002~ '2
2~0.002
OS02
= 0.002
~-
! 0’
I 0
0.2
0.4
0.6
0.6 e
I OO
1.0
O.&t
0.2
0.L
0.6
0.8
0
I 1.0
---.-___
g%c --_‘-yf=y 0
-04 .
-0.8
T&k 2.Constaatn ia ayparkttic uymptotkadysis for cyliaQicaipontis with#b=O.@2Sadr=O.3 e 0.50 0.50 0.15 0.75
ujE
n
o.oea IO O.Otxm 3 0.0028to Om28 3
-ey 1.W 1.162 I.033 I.032
A2%, -0.414 -0.059 -0.430 -0.330
# 1.716 1.231 I.150 I.071
PU,) o.ws 0.421 0.278 0.628
P 0.344 0.489 0.638 0.978
by the elastic form&a, for the same values of AZ%, In the next section the values given in T&de 2 shall be used to compare the asymptotic estimate of ~~e~~-~~~vi~ with lmnkcal results. 5. NUMERICAL COMPUTATION OFIMPERFECTION-SENSITIVITY The behwiourof imperfect panels is here computed numerically by an incremental procedure based on the variational equation (2.11). As in the preceding section, the imperfections considered are taken in the shape of the critical in mode (3.1) with amplitude F In the numerical method employed to solve (2.11) the displacements are expanded in terms of trigonometric functions in the axial direction, while a one-dimensional finite element approximation is used in the circumferential direction.
Bucklingof elastic-plastic cylindrical panel under axial compression
967
(5.1)
Here, the functions #“(x2), v”‘(x2) and w(i)(x2) are approximated by Hermitian cubits within each finite element, and the wave parameter a is that corresponding to the critical bifurcation stress. This method has also been used for an axially compressed oval cylindrical shell[l4], within the context of a more accurate shell theory. The boundary conditions specified are the symmetry conditions (2.12) at the panel centres in addition to the stiffener constraint (2.13), which gives w’“(b/2) = 0 for i L 1. In the expansion (5.1) it is found that N = 2 giv6s good accuracy, with a half panel width divided into 2 elements. The integrals in eqn (2.11) are evaluated by 4 point Gaussian quadrature in circumferential direction within each element, while in the axial direction it is found sufficient to divide the interval -lb/a SX’ ~ib/a in two subintervals, using 4 point Gaussian quadrature within each subinterval. Through the thickness Simpson’s rule is used, with 7 points in (2.10). The active branch of the tensor of moduli (2.8) at an integration point is determined in each increment as follows. If the stress state at the integration point is on its current yield surface, the plastic branch is taken to be active. If S, for that integration point turns out to be negative, the elastic branch is taken to be active in the next loading increment. This .proeedure is su5ciently accurate if small increments are used and if the transition from loadii to unload& or vice versa, occurs only once or twice during the loading history. The results of numerical computations shown in Figs. S-8 give the load parameter A as a function of the de&&on contribution W, = w”‘(O), which is quite a good measure of the bifurcation mode growth. All the results are computed for h/b - 0.025 and v = 0.3, with strain hardening exponent n = 10 in the first two figures and n = 3 in the last two figares. The flatness parameter and the yield stress are taken as 8 = 0.5 and uJE = 0.002 in Figs. 5 and 7, while the values used in Fii. 6 and 8 are 8 = 6.75 and o,/E = 0.0028. The post-bifurcation behaviour of the perfect panel is computed numerically as the behaviour of a panel with a small init.@ imperfection $= 0.0002. In all four figures this behaviour
t
"Perfect"
1.0v
_v , :::-“ld , 0
0.2
0.L
0.6
0.6
Fig. s.
1.0 3
1 1.2
0
0.2
0.L
Q6
CM
1.0 3 h
Fii.6.
Fig. 5. Load versus mode defiection for cylindrical panelthat bifurcates in UK plastic raagc (h/b= 0.025, e=0.5,uJE-0.002,n= lO,v=O.3). Fig. 6. Load
cylindrical panel that bifurcates in the plastic range (&/b- 0.025, 8 = 0.75, OJE = o.ooi?8, n = 10, Y = 0.3).
versusmodedefkctionfor
1.;
968
V.
TVERGMRD
P
1cg
0.8
0.8
0.6
0.6
0.4 Y
maximum
0.2
0 0
0.2
OL
06
Fig.7.
08
I.0 "1 h
1.2
Fig. 8.
with the asymptotic expansion (3.6) in that bifurcation takes place under increasing load and in that a maximum is reached slightly above the b&ration load after a &de bifurcation mode d&e&n. As this characteristic behaviour results from the effect of eiastic unloading, it is not predict& by the hypoelastic its expansion (4.1). However, the hypoeMic expansion with the values obtained for A.z% (Tabk 2) does agree well with the general &en&of the numerically computed post-bifWcatioabehaviour in each of Fiis. 5-8, apwt from the maximum just after bifurcation. The shell conaider& in Fig. 5 is in the range of &values, where ekstic cyli&ical panels have a stabk ~st~c~g behaviour. The eiastic-pkstic _pl with A= IO is ckarly imperfection sensitive. For the largest bin considered, e = 1.0,a maxiumm isnotreached in the range cons&red, but the curve ikttcns out at a level of reduced support load. In Fii. 6, for a more curved panel with a higher yield stress, but still with the strain hardening exponent n = 10, the behaviour is quite similar to that obtained in Fig. 5. The sensitivity to small imperfections is slightly higher, but for the largest imperfection the support load is a iittle lessreduced. The cylindrical panel considered in Fig. 7 is identical to that of Fig. 5, exceptthat the material is high hardening with # = 3. For the perfect panel the post-bifurcation load after the maximum decreases to a value sliitly below the bifurcation load, in agreement with the small negative value obtained for A~~j‘lj(,, but then starts to increase again in the more advanced post-buckling range. The result is that even for the small imperfection, 5’= 0.01, no maximum is found before the support load exceeds the bif~c~n load. The panel in Fig. 8 is that of Fii. 6, with the strain hardening exponent changed to A= 3. The behaviour is much lie that found in Fig. 7, but the post-bifurcation ioad reaches a considerably lower level before again starting to increase in the advanced post-buckling range. In this case a local maximum is found for d = 0.01, but not for g = 0.1. A comparison between the numerical results and the asymptotic estimate (4.22) of the im~~ection-sensi~vi~ is given in Fig, 9. Here the curve(4.22)is drawn on the basis of the parameters + and F given in Table 2, and the numerically computed support loads are indicated by a dot for I= 0.0002,I= 0.01 and f= 0.1. In Figs. 9(a) and (b), corresponding to Fiis. 5 and 6, the agreement is very good for the range considered. However, for the large imperfection $= 1.0 the support load is underestimated by the asymptotic analysis, as one would expect. In Fig. 9(c) the agreement is also good for the smallest imperfection, but for $- 0.1 no maximum is reached in FJig.8, so the range of validity of the ~~ptoti~ expansion is smaller in is seen to agcc
this case. Such limited range of validity of the asymptotic results hL also been found for elastic
Bucklingof elastic-plasticcylindricalpanel under axial compression
959
Pig. 9. Comparisonof mimer&l rcswlts aud asymptotic l~ypoelasticpredictions for the imperfectionsensitivity of elastic-plastic cyst panels. (a) Panel with 8 = D.5,o;/E = O&02,II = IO.(bj Panel with 8 = 0.75, a#? = 0.0028, n = 10. (c) Panel with B= 0.73, KY@= 0.#28, n = 3.
buckling behaviour in a number of cases, in which the initially unstabk post-buckling behaviour changes to stable behaviour in the advanced post-buckling range[lS]. This also agrees with Koiter’s conjecture of the advanced ~st-buc~g behaviour for elastic cylindrical panekfl]. The asymptotic results ~o~es~~i~g to Fii 7 have not been plotted, as no maxima are obtained in this f&e for comparison. ‘%k value of g found for this case (Table 2) indicates a great deal of sensitivity to small ~~~e~~ns, but as the change to stable advanced postbuckling behaviour occurs ~~~e~iy earlier in Fig. 7 than in Fig. 8 an even smaller range of vahity of the asymptotic result must be expected here. Although the halite analysis does not account for the effect of elastic unloading, it is found that the asymptotic estimate (4.22}gives a good indication of the im~~ection-~nsi~v~y of an elastic-plastic cylindrical panel. This agrees with the results obtained for axially compressed square plates[I)].The range of validity of the asymptotic estimate is found to be smaller for a hii hardening material than for a low hardening material, because panels made of a high hardening material have the abihi to support loads above the bifurcation load in the advanced ~st-~~ range. As in Ref. 193we con&de that the a~~~~a of negkcting ehtstic unloading in the hypoekstic expansion seams to be less important than the appromo involved in co~si~~~ lowest order asymptotic results. It may be expected that also for other shell socks the h~i~tic analysis will prove valuable for iden~y~g parameter ranges, where the she&3wih exhiiit ~~~~on-~nsiti~ty in the piastic range. REFERENCES 1. W. T. Koiter, Buckling and post-buMttg Mavior of a cylindrical panel under axial compression. Nationaal lJtchrlwun L.l76onzrorf&m 26, Report S476, Amsterdam(19561. 2. W. T. Koiter, Over de stabiiit van bet elastisch cvenwicht. Thesis, Mft, H. J. Paris, Amsterdam(1943) English tmml. NASA ‘l-I’F-10,833 (1967);and AFFIX-TR-70.25(1970). 3. W. 8. Steph&s, Imp&&on se&ivity of axially compressedstringerreinforcedcyliidrical panels under intemal pressure. AlAA L 9, 1713-1719(197lf. 4.0. cl. Pope, The bi@itig bEbaviourin &al oompressioa of ayes panels, iecl&lg the e&t of sbenr deforr&ility. Jr&f. soiidr Srrr4crynu4,323,.34@(19681. 5. N. R. Bauld, Imperf&ion se&ivity of axially compressed stringerreiaforcul cylindricalsandwich panels. Jar. L solids ~~~~~ lb, l?83-m (1974). 6. J. W, Hymn, Peon behavior ia the plastic range. f, Me& Pays. &lids 21,163-190 (1973). 7. 1. W. Ham, Plastic bu&liq. A&races ie A@ied Mrckuaics (Edited by C. S. Yib}, Vol. 14, pp. 67-M. Academic Press, New York (1974). 8. f. W. Hut&moo aad 8. Budia&y+ Am&&al and mtmericalstudy of the effects of initial imperfectionson the . inclasti twkbg of s cruciformc&nm. Buck&g of Smc&rcs @lited by B. Budianskyj, pp. 98-103. SpriagtrVerlag, Be&i (1976). 9. A, Needkmaa and V. Tveqaard, An analysis of the imp&&o-sensitivity of squareeiastic-plasticplates under nxial compression.Int. 1 Solids &ucruns 12, 185-201(1976). 10. V. ‘I’vergaard and A. Necdkmaa, Bnoklingof ccccn~y stiffenedelastic-plastic panels on two simple supportsor multiply aupportul. Inr. J. SoUds Strucruns lL647-663 (1975). 11. R. Hill, A geueral theory of uniquenessand stability in elastic+astic solids. J, A4eck.Pkys. Solids 6,236249 (1958). 12. R Hill, Bi and uniquenessin non-linearmechanicsof continua. Pmblems of Can&mum MecMcs (S.I.A.M. Pales, ifs-164 fl%?). 13. J. w. Hut&inson, I~e~o~-~n~ti~ in the plastic range. 3. Me&. Pkys. salids 21, 191-204(19735 14. V. Tvcrgaard,Buckliag of elastic-plastic oval cylindricalshells underaxial compression.ia?. 1. Solids Srnrcrprnsff, 683-691 (1976). IS. V. Tvergaard,Buckling~bavio~ of plate and shell s~c~~s. Pruc. l&k Inf. Congr.Tlteor,or&Appf. Meek. @.diti by W. T. Kolter), pp. 233-247. No~~-Ho~, Amsterdam (1976).
970
V. TV~~GMRO APPENDIX
The functionp(A) occurriq in (4.20).and in particuultuthe value p(A,) to be used ia (4.21)and (4.23).ia dctemtinadby integmtiq the consthtive mhtioa (4.17) and suhtitutiq the m&t into tlte prhcipk of vittud wark. For the intep8th of (4.17) it is convenientto write
subatitutaiinto (4.17) tllia givea the diffemllthl quathns for p
(A4)