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Localized buckling in long axially-loaded cylindrical shells
00-S5096/91 $3.00+0.00 C' 1991PergatnonPressplc
LOCALIZED
BUCKLING IN LONG AXIALLY-LOADED CYLINDRICAL SHELLS G.
W. HUNT and
E. LUCENA NET@3
Department of Civil Engineering, Imperial College London. London SW7 2BU, U.K t Institute Tecnol6gico dc Aeronciutica. 12225 %o Jest dos Campos-SP. Brazil
(Rrcei&
30 April 1990)
ABSTRACT DOIJISLE-SCALFperturbation analysis of a long elastic cylindrical shell under axial compression reveals a second-order non-linear differential equation for the buckleepattern amplitude in slow-space. Numerical solution then suggests that the most easily-triggered failure mode is localized along the length. The method is extended to include mode interaction, giving three coupled second-order non-linear differential equations in slow-space Localized solutions are again found, by combining features of the Lagrangian function with a systematic numerical search procedure. The predicted extent of the localization, about one-and-a-half axial wavelengths when fully developed. compares well with published experiments on long cylinders. Moreover, in contrast to the associated periodic solutions, “square” waves at the minimum critical load are denied ; the predominant waveform turns out to be long axially, again as seen experimentally.
1.
INTR~~NJCTION
THE buckling of a long elastic cylindrical shell under axial compression is a classic problem of structural mechanics. So much has been written that it is inappropriate to attempt a review, save to mention a few key contributions (DONNELL, 1934; VON KARMAN and TSIEN, 1941 ; KOITER, 1945 ; HOFF et ul., 1966). It is the case, however, that periodicity in the axial direction is often either explicitly or implicitly assumed in the process of formulation (see, for example, HANSEN, 1975 ; HUNT et al., 1986). This does not square with the experimental evidence, which indicates that, however long the cylinder, buckling occurs only over a range of about one-and-a-half wavelengths of the final diamond pattern (SENDELBECK et ul., 1967; YAMAKI, 1984). The localization of buckling along a length is a known phenomenon for long structures, notably railway lines and pipelines, and carries significant analytical implications. Successful formulations must be both non-linear and capable of admitting multiple solutions, since the localized buckle can occur anywhere along the length. Periodic solutions are also part of the overall picture, which is most usefully seen in phase-space (HUNT et al., 1989).
z Present address:
Department
ofCivil
Engineering, 881
Imperial
College London,
London
SW7 2BU, U.K.
882
G.
W. HUXTand E. LIJCENA NET~I
With a complex problem like the long cylinder. it is often the periodic solution that first catches the eye. The potential to localize, however, introduces significant extra ingredients. Differential equations of amplitude modulation replace simple algebraic coupling equations, and their solution can result in localized modes in differing combinations from their periodic counterparts. Also, localized solutions approach the pre-buckled state asymptotically, and standard structural boundary conditions hence carry little practical significance once the extent of the localization is down to the specimen length or less; the length of localization is thus a key feature, useful for distinguishing between “long” and “short” cylinders, for example. The most useful tool for the study of localized solutions is double-scale analysis, based on the underlying differential equation (SEGEI,, 1969 ; NEWEM and WHITEHEAD, 1969). For the pair of fourth-order von Kjrm~n~Donnell equations relevant to the axially-compressed cylinder, this supplies valuable new information. In particular. for a perfect cylinder, localized solutions are initiated for modes with wavelengths on the well-known Koiter circle, but not for “square” waves at the crown of the circle. This is a new effect. not found in periodic studies, but reflected in the experimental evidence. Indeed, comparisons with published experimental results of YAMAKI (1984) and SENDELBECKet N/. (1967) are most encouraging.
2.
LINEAR ANALYSIS
Based on the linear membrane equations (BRUSH and ALMROTH, 1975), prior to buckling the perfect axially-compressed cylinder is in a uniform membrane state N, = -P, with an associated
outward
N,. = N,, = 0,
(1)
radial displacement
where P is the applied compressive axial load per unit length; E and \’ are Young’s modulus and Poisson’s ratio; and p and t denote the shell curvature and thickness, respectively. The solution above can be considered as rigorous for infinitely long cylinders. As a result of buckling. contributions arise to the in-plane stress resultants and to the radial displacement. Using Eqs (1) and (2) these contributions are related to each other by von KLirm&--Donnell equations given as follows : (3)
(4) where .y and ~2 are the axial and circumferential coordinates; V” denotes the two4 is biharmonic operator: k’ = t’/l2(! -I*‘) ; ,i - P/Et: and the stress function
Localized
related
to the in-plane
buckling
stress resultant
in cylindrical
deviations
shells
from the uniform
883
membrane
state
by
The linear buckling
equations p~4~+~r72W
_pC?!? ax’
=
0
ax2
(6)
v4q5+p~=o
are obtained from the radial equilibrium equation (3) and the compatibility (4) by omitting the non-linear terms. If we assume
equation
w = A cos yx cos pyy
(8)
C#J = a cos yx cos pyy,
(9)
Eq. (7) yields
where A is a constant ; a = pd/y2 ; m = l/( 1-t fi’)’ ; y = p//l; fi is the modal aspect ratio (axial/circumferential wavelength) ; and n is the number of waves in the circumferential direction. Substitution of (8) and (9) into Eq. (6) leads to AA _ ‘;’
I “5 Y
(10)
as the condition of validity of the resulting equation for all values of x and y and non-zero values of A. %Ais the buckling parameter load ;1 corresponding to the mode (8). Similarly, if we assume the axisymmetric mode w = Bcos2yx the corresponding
The analytical
buckling
minimization
parameter
(11)
load is
of ILAwith respect to CIgives
(13) iA = 2pk corresponds to the classical solution and all the modes having this load are those on the well-known Koiter circle; tl = y2k/p is the required condition for the mode (8) to be on the Koiter circle.
xx4
‘3. W. HUNT and E. 3.
DOUBLE
LLCENA
Nrro
SCALE ANALYSIS
Equations (3) and (4) can be converted into an infinite set of linear partial differential equations by expanding the radial displacement, the stress function and the load parameter into power series in terms of an arbitrary parameter s (THOMPSON and HUNT, 1973) H’(.Y,X. J) = W,(S. X,),)X+ W,(.U, X._r).VZ+ ll’,(.Y. X, J).V + &.Y,X.J)
= ~~,(.L.,X,~‘)~Y+41)~(.\~,X,~).~~+~1(.~,X,~’).s1+“’ i = ~:‘+i,.s+R_,.s’+i,.s’+~~~
IV, (b and j. are expanded as can easily be seen by attempt to get not only variable X (LANGI: and defined by
(14)
about the critical point corresponding to the Koiter circle, substituting .c = 0 into the above expanded forms. As an periodic solutions but also localized ones, the slow-space NF.WEIJ.. 1971 ; POTIER-FERRY, 1983 ; HUNT et ul., 1989)
X = ss
(15)
is incorporated in H‘and 4, Eqs (l4), by assuming that the amplitudes II’, , H‘!, etc., depend on this variable. Considering Eq. (I 5), derivatives with respect to x become
of the modes
(2 ^ (’ + ,:A, S . = t .Y is (2’ (7.P
= ?’ +2 ,$s+ i.Y’
$.x2.
etc. Then, from the first series of Eqs (14). we can write
(16)
Localized
buckling
in cylindrical
shells
885
where a prime denotes partial differentiation with respect to X. Similar expressions can be written for 4 by simpIy replacing w with (b in above equations. Since s was taken to be an arbitrary parameter, the stipulation that a power series in s vanishes requires that each coefficient of the power series vanishes as well. If Eqs (14) and (i7) are substituted into Eqs (3) and (4), the requirement that each coefficient in the power series vanishes leads to the following linear equations, which are the first three of an infinite set : Order s
(19) Order s’
Order s3
(22)
886
G. W. HIJN-and E. LUCENA NETO
The order sn of this sequence
of equations
can be written
as
where F,,(x, X, JJ) and G,,(x, X, y) are determined from the solutions of the previous (n- I) set of equations. Terms proportional to the solution of the homogeneous equations arising in F,(x, X, y) or G,(s, X, J>) must be eliminated because they lead to unbounded particular solution (that is, secular). The procedure employed of eliminating the secular terms is similar to that outiined by AMAZIGO and FRASER (1971) and is given as follows : suppose that the homogeneous solution of Eqs (24), that we wili be concerned with, is M‘,,= A,, cos yx cos j?YJl,
C/I,,= a,, cos gx cos p74’,
(25)
with a,, = h-A,?. Suppose that F,,, and G,, are the coefficients of the terms proportional to cos ^;s cos &JJ arising in F,(r, X, I:) and G,,(x. X, .Y), respectively. Writing Eqs (24) with these terms only on the right-hand side, we have
A solution
to (26) in the form (25) gives (pa,, - pkA,)
cos yx cos /I$jy = F,, i cos y.r cos /$y
(pffn - pk.4,) cos 7.Xcos j?$;y = kc,, Dividing
cos 3’” cos p,y.
(27) (28)
(27) by (28) gives F,, -kG,,
= 0.
(29)
This condition of suppressing secular terms arising from a nonhomogeneous expression in the form (25) holds for any secular terms arising from trigonometric expressions in the form of modes which are on the Koiter circle.
Localized
3.1. First-order Equations
buckling
in cylindrical
887
shells
equations
(18) and (19) have solution H’,
=
A(X)
similar
to Eqs (6) and (7)
$I = a(X) cos yx cos Bi~y,
cos yx cos /$jy,
where a = kA and A is no longer a constant,
After substitution of (30) and reduction sums, Eqs (20) and (21) become
but depends
of products
(30)
on X.
of trigonometric
functions
+[2ypk-44y3k2(1+~2)]A’sinyxcos~yy+~’~4kA’(cos2y.~+cos2f3~~~~)
Elimination of secular (29) provides
terms arising
from cos yx cos fi>~+~ by applying
IL,
to
(31)
the condition
=o.
(33)
The solution of (31) and (32) contains no secular terms arising from sin yx cos /$JJ because the coefficients of these terms on the right-hand side of these equations satisfy condition (29) identically. If we assume the particular solution 11’2= B(X) cos 2yx+ C(X) cos 2/$JJ’,
(34)
Eq. (32) provides 42 = b(X) cos 2yx+ e(X) cos 2~~~+d~~)
sin yx cos &y,
(35)
where P
h = #?Substituting
P” 3ZA2,
Eqs (33)-(36)
I C= ---A’,
W--l)5 N+B*)
.
(36)
into (3 1) gives
B = ~____ pB2(l + 8~) A 2, 32a
with
d=
c=
1 -A*, 16B*k
(37)
XXX
G. W. HUNT and E. LUCENA NETO D = 2-H~ i,‘,
3.3. Seculm
terrm in third-order equations
The right-hand side of Eqs (22) and (23) can be determined by substituting previous results given by (30) and (33) (37). Terms proportional to cos y.v cos P;,y. which arise in both equations, must be eliminated to avoid unbounded particular solutions. Let F3, and G?, be the coefficients of these terms in (22) and (23), respectively. Then II,
F3,
=
-fk’(l
+3[~4)A"+]J'i.2A
+
4
i6 1-/14+~~~~(1+4~)(1+8+ i
G?, = -i,3h.(~4-8/~Z+3)A”Now, substituting
,6
(1+8x)
1
A’.
(39) into (29) we have
4(1 -[~~)‘k’A”+&A+
,‘;
$’
(1 +8a)‘+y’(3-/I14)
A3 = 0. 1
(40)
The periodic solution is recovered by setting A” = 0. In this case, we get the same solution as that obtained by conventional post-buckling theory, with mode A represented by an active, and mode B a passive, coordinate (THOMPSON and HUNT, 1984). But localized solutions also exist. If we set the perturbation parameter s to %/(A, ~ j_). so that, on equating coefficients in (14), jb2 = - I for L < j_“, we retrieve the well-known amplitude modulation equation of NEWELL and WHITEHEAD (I 969) and SEC;EI. (1969), with signs suitable for the localized form (POTIER-FERRY, 1983 ; HUNT et al., 1989). As aspect ratio [j is varied, the position of mode A moves around the Koiter circle, with the square waveform represented by /? = 1 at the crown. Runge Kutta solutions to Eq. (40), from the same starting state of A,, = IO-‘, A;, = 0, are as shown in Fig. 1. For clarity, the curves have been drawn with different initial value X0. We see that the curves flatten to the X-axis as p + 1. Here critical loads for A and B coincide, so cr = 0 and the coefficient of A ’ is infinite while, simultaneously, that of A” is zero ; the latter forces periodicity, and the former suggests that the amplitude is 7ero. We also note that solution of the third-order perturbation equations would involve terms of the form cos 37.~ cos ~?JZ in pi’?. These are of particular interest, since at [j = ,,/3 they also lie on the Koiter circle. Our single-mode view, which results from the inclusion just of mode A in the solution (30) for II’,, is thus called into question specifically at /j = 1 and /j = J3. The possibility of localization combined with mode interaction occurs for each, the first case involving two modes, A and B, and the second. three.
4.
LOCALIZATION AND MODE INTERACTION
Let us take the new solution (30)
to the first-order
perturbation
equations
to replace
Locaiized
buckling
in cylindrical
shells
889
I
b
?
I-
O
Slow-space Flc;. 1. Rung+Kutla
wj = A(X)
solutions
variable,
to Eq. (40), for representative
X modes on the Koiler circle.
cos “ix cos p/4’+ B(X) cos (I ffi’)y.x+ C(X) cos p’yx cos ,oi’3?
4, = n(X) cos yx cos fly4 + h(X) cos (I+
p’);Lx+ c(X) cos p2yxcos p1’4’,
(41)
where a = kA, h = kB, c = kC and, as before, k = PM/~‘. All three modes appear on the Koiter circle, as seen in Fig. 2. The same analytical sequence is now followed; this time, however, to account for the differing influence of non-linear terms, slow-space is defined by
mode A
mode C
ZrrR/oxiol
wovelength
FIG. 2. Modal interactions and the Koiter circle. As the post-buckling develops, it is observed ex~rimentalIy that there is a drift to within the circie, via a sequence of secondary bifurcations (see Fig. 4).
G. W. HUNT and E. LUCENANETO
890
x = .y”7_y
(42)
to replace the linear relation (15). The remainder of the analysis proceeds much as above. No secular terms are found to arise in the solution of ~~~’ level equations. A particular solution is given by il’j,l = 0 $3,2 = cl(X) sin yx cos /$v+e(X)
sin (1 +/?‘)yx+f’(X)
sin a2;,x cos &y,
(43)
where ‘l
=
_
741-a’>k
A’
’
y(l+fll)
2k P = - i’(l +mjTj B’,
2( 1 -/P)k .f‘= j,/j2(
Suppression of secular terms at the s2 level then leads to the three coupled order differential equations 8(1 -[~2)‘k’A”+2i,A+3p/?BC 16(1 ff12)‘k’B”+4(l
+/~‘)‘~,B+3/$‘AC
8(1 -/~‘)‘k’C”+2/?4i,,C+3pfl’AB
(44) second-
= 0 = 0 = 0
(45)
and we can set s = 1,” -& so that i, = - 1. We note that the coefficient of A” again approaches zero as /j’ -+ 1. For the interaction at the crown of the Koiter circle, we thus conclude that no localized solutions exist. For the remaining values of fl, the possibility of localized interactive solutions arises. In the search for these, one useful technique is to borrow from the work of TOLAND (1986) and others, and take this as a system of equations associated with the Lagrangian Y= where we can identify
a “kinetic
energy”
T-V, part
T=4k’[(l-/?2)‘A’Z+2(l+~2)ZB’Z+(l-~~’)ZC’1] and a “potential”
(46)
(47)
part V=
-AA’-2(l+/?2)‘B’-j14C’+3pp2ABC.
We note that the quadratic form for T is positive-definite Hamiltonian system, progress in X is along lines .X = Tf
V = constant.
(48) for fi # 1. This being
a
(49)
In the fundamental unbuckled state we can take T = V = 0, all amplitudes and first derivatives vanishing. The symmetric section, through which localized solutions must pass (HUNT et al., 1989), is defined by T = A’ = B’ = C’ = 0. The energy contour V = 0 then carries a particular significance; points on this locus in ABC space represent states on the symmetric section with just enough “energy” to arrive at the flat fundamental state with zero kinetic and potential parts. In the search for localized solutions, this is valuable extra information.
891
Localized buckling in cylindrical shells
Slow-space FIG. 3. Runge-Kutta
variable,
X
solutions to Eqs (45), showing relative modulated amplitudes to the modes of (41). at p = 2.
Using a systematic numerical search in two dimensions, from starts both on the symmetric section and zero energy contour, apparent localizations at p = J3 and /I = 2 have been found; solutions exist for which -4, B and C all simultaneously decrease from finite amplitude and drop monotonically towards zero, as seen in Fig. 3. We note that the modal amplitudes found here appear in different proportions from the related periodic solutions and fully anticipate that such solutions exist for other values of p # 1. Further examination is left for later publication.
5.
COMPARISON WITH EXPERIMENTS
Initial buckling of the long thin axially-loaded cylinder is known to be violently unstable, with a sharp restabilization once the load has dropped to some fraction (perhaps one-fifth) of the classical critical value. Our first-order analyses pick up only the initial destabilization, not the subsequent restabilization. It is therefore possible to compare against published experiments only the wavelengths and predicted extent of the localization at about the correct load level, not the restabilizing load itself. It is hoped eventually to extend the analysis to include the restabilization, as has been the case with periodic solutions (HUNT et al., 1986). It is a further experimental observation (YAMAKI, 1984) that the equilibrium path between destabilization and restabilization is punctuated by a sequence of secondary bifurcations into longer circumferential waves (see Fig. 4). Thus, although buckling starts on the Koiter circle of Fig. 2, post-buckling is marked by a vertical drop to within the circle, as shown. Because of the severity of the initial instability, even under
Ci. W. flr!ur
E. LWI:NA N~:ro
md
(a)
Cd)
8.1525°
1360
0
0 1 I
0
Q2
Q4
Q6
Cla
~
,
-IO
2
1.2 8 ,rn”$
(b)
06,
P
8 01
-4
8
i
-2 -4
L
m
(cl
1.0 ’
09
d-+%+%-as
b 3=210*
1845’
4
0
4
w tmm,
Localized buckling in cylindrical shells
893
rigid loading, the triggering bifurcation and its corresponding wavelengths are unlikely to be seen experimentally. As an example, let us consider the longest of the experimental specimens of YAMAKI (19X4), for which the shell radius R = 100mm, t = 0.247mm, 5’= 0.3, and the effective length L is chosen such that the Batdorf parameter Z = J(l - ?)L'/Rt= 1000. Substituting for the known values and rounding to the nearest integer then gives the typical circumferential wavenumbers for positions on the Koiter circle “p_,
= 18,
np -: \- 3 = n/j= ,,,.‘? = 16,
H,j& = n,j= 1.2= 15.
(50)
Experimental results are given in Fig. 4, taken directly from Fig. 3.52e of YAMAKI (19X4). Two types of deflection pattern are observed, symmetric and asymmetric ; both are reproducible by the above analysis, the latter by replacing the cosine functions of Eq. (30) with equally valid sine functions. For the symmetric case, the highest circumferential wavenumber observed experimentally is 9. As the post-buckling develops. the sequence of secondary bifurcations is seen to bottom-out at M = 6; deflections at IZ= 7 are shown at the bottom-right of Fig. 4, and crude wavelength measurements indicate that here the diamond wavepattern is close to being of the “square” form p = 1. By projecting upwards from the appropriate point on Fig. 2, tracing in reverse the sequence of secondary bifurcations, the point of initial instability on the Koiter circle can be approximated; this suggests a triggering bifurcation at TV= 14 and /I = 2.2. If we assume a post-buckling load of about one-fifth of the classical critical load, the range of localization seen in Figs 1 and 3 is in each case over about one to one-and-a-half axial wavelengths ; although no account is taken of the restabilization, nor the sequence of secondary bifurcations, it is encouraging to note that this is exactly as reported experimentally (SENDELBECK etul., 1967;YAMAKI, 1984).
6.
CONC’LUDING REMARKS
We offer in this paper two complementary analyses for the initial post-buckling of long thin axially-loaded cylinders. the first assuming distinct critical behaviour (THOMPSON and HUNT. 1973, 19X4), and the second allowing interaction between modes. The approach is distinguishable from earlier perturbation analyses by the fact that amplitudes are left free to modulate, which then opens the way for localized solutions. Some general conclusions can be drawn. The most significant feature, which is reflected in published experiments on long cylinders (see Fig. 4), is that localization forces the critical modes away from the crown of the Koiter circle. It has been realized for some time that the initial triggering buckle pattern comprises long axial waves, but adequate explanations apparently have been lacking. The answer is of course found in the form of Eqs (40) and (45). Extension of the analysis to find the minimum localized post-buckling load. via the sequence of secondary bifurcations, certainly seems possible. &WELL (1986) develops a perturbation approach for the interaction between the periodic crown mode and its axially-symmetric counterpart, which in one step includes both initial destabilization and subsequent restabilization of the post-buckling path. A similar approach, but
G. W. HUNT and E. LLCENA NATO
894
allowing prove
for modulating
most
amplitudes
and interactions
away from the crown,
could
rewarding.
REFERENCES A,VAZIGO. J. C. and
1971
I/71. J. Solids Structures 7, 883.
FKASKR,W. B. BRUSHY,D. 0. and
1975 1986 1934 1975 1966
Buckliny of’Bclrs, Plrrtrs, rrnd Shdls. McGraw-Hill. New York. QV?. Stub. Sl’sterns 1, 115. Tram AS’ME Aero. Engng AER-56-12, 795. ht. J. Solids Structurrs II, 1223. AIAA J/4, 126.
1986
Irlt. J. Solids Str.wtures
HUNT, G. W.. BOLT. H. M. and TIIOMPSON, J. M. T. Korrw. W. T. LANGE. C. G. and NIWI:LL, A. C. N~WIZLL, A. C. and WHITEHEAD. J. A. POTIER-FEKKY, M.
1989
Proc. R. Sot. Land. A425, 245.
1945 1971
Dissertation, Dclft. The Ncthcrlands. SIAM J. Appl. Math. 21, 605.
1969
J. Fluid Mech. 38, 279.
1983
SEGEL. L. A.
1969 1967
C’olltrpsc~: tlw Bwkling of’ Structums ’ in Thcor~~ rml Pructiw (IUTAM Symposium, London. 1982)(editedbyJ.M.T.T~0~~~0~andG. W. HUXT), p. 149. Cambridge University Press. Cambridge. J. Fluid Mec~h. 38, 203. Report SUDAAR 318. Dcpt of Aeronautics and Astronautics. Stanford University.
ALMROTI-I, B. 0. COWFLI., R. G. DONNELL. L. H. HANXN. J. S. HOFF. N. J., MADSEN, W. A. and MAYPRS, J. HUNT, G. W., WILLIAMS. K. A. J
and COWIX.
R. L., CARLSON, R. L. and HOFF, N. J. THOMPSON, J. M. T. and HUNT, G. W. THOMPSON, J. M. T. and SI!NDIiLBh(‘K,
197.1 1984
HUNT, G. W. T~I.ANII. .I. F.
1986
VON KAKMAN, T. and
1941
TSIEN, H. S. YAMAU. N.
22, 150 1.
R. G.
1984
A Gmrral Thcorj, of Elastic Stubili/)~. John Wiley. London. Elastic Instability Phenonwnu. John Wiley. Chichestcr. Am. Muth. Sot., Proc. S~wzp. Pwr Mnth. 45, Part 2. 447. .I. Awe. Sci. 8, 303. Eltrstic Stubilit~~ of C’irculur C~~lindricul Shc1l.s. Elsevicr. Amsterdam.
LOCALIZED
BUCKLING IN LONG AXIALLY-LOADED CYLINDRICAL SHELLS G.
W. HUNT and
E. LUCENA NET@3
Department of Civil Engineering, Imperial College London. London SW7 2BU, U.K t Institute Tecnol6gico dc Aeronciutica. 12225 %o Jest dos Campos-SP. Brazil
(Rrcei&
30 April 1990)
ABSTRACT DOIJISLE-SCALFperturbation analysis of a long elastic cylindrical shell under axial compression reveals a second-order non-linear differential equation for the buckleepattern amplitude in slow-space. Numerical solution then suggests that the most easily-triggered failure mode is localized along the length. The method is extended to include mode interaction, giving three coupled second-order non-linear differential equations in slow-space Localized solutions are again found, by combining features of the Lagrangian function with a systematic numerical search procedure. The predicted extent of the localization, about one-and-a-half axial wavelengths when fully developed. compares well with published experiments on long cylinders. Moreover, in contrast to the associated periodic solutions, “square” waves at the minimum critical load are denied ; the predominant waveform turns out to be long axially, again as seen experimentally.
1.
INTR~~NJCTION
THE buckling of a long elastic cylindrical shell under axial compression is a classic problem of structural mechanics. So much has been written that it is inappropriate to attempt a review, save to mention a few key contributions (DONNELL, 1934; VON KARMAN and TSIEN, 1941 ; KOITER, 1945 ; HOFF et ul., 1966). It is the case, however, that periodicity in the axial direction is often either explicitly or implicitly assumed in the process of formulation (see, for example, HANSEN, 1975 ; HUNT et al., 1986). This does not square with the experimental evidence, which indicates that, however long the cylinder, buckling occurs only over a range of about one-and-a-half wavelengths of the final diamond pattern (SENDELBECK et ul., 1967; YAMAKI, 1984). The localization of buckling along a length is a known phenomenon for long structures, notably railway lines and pipelines, and carries significant analytical implications. Successful formulations must be both non-linear and capable of admitting multiple solutions, since the localized buckle can occur anywhere along the length. Periodic solutions are also part of the overall picture, which is most usefully seen in phase-space (HUNT et al., 1989).
z Present address:
Department
ofCivil
Engineering, 881
Imperial
College London,
London
SW7 2BU, U.K.
882
G.
W. HUXTand E. LIJCENA NET~I
With a complex problem like the long cylinder. it is often the periodic solution that first catches the eye. The potential to localize, however, introduces significant extra ingredients. Differential equations of amplitude modulation replace simple algebraic coupling equations, and their solution can result in localized modes in differing combinations from their periodic counterparts. Also, localized solutions approach the pre-buckled state asymptotically, and standard structural boundary conditions hence carry little practical significance once the extent of the localization is down to the specimen length or less; the length of localization is thus a key feature, useful for distinguishing between “long” and “short” cylinders, for example. The most useful tool for the study of localized solutions is double-scale analysis, based on the underlying differential equation (SEGEI,, 1969 ; NEWEM and WHITEHEAD, 1969). For the pair of fourth-order von Kjrm~n~Donnell equations relevant to the axially-compressed cylinder, this supplies valuable new information. In particular. for a perfect cylinder, localized solutions are initiated for modes with wavelengths on the well-known Koiter circle, but not for “square” waves at the crown of the circle. This is a new effect. not found in periodic studies, but reflected in the experimental evidence. Indeed, comparisons with published experimental results of YAMAKI (1984) and SENDELBECKet N/. (1967) are most encouraging.
2.
LINEAR ANALYSIS
Based on the linear membrane equations (BRUSH and ALMROTH, 1975), prior to buckling the perfect axially-compressed cylinder is in a uniform membrane state N, = -P, with an associated
outward
N,. = N,, = 0,
(1)
radial displacement
where P is the applied compressive axial load per unit length; E and \’ are Young’s modulus and Poisson’s ratio; and p and t denote the shell curvature and thickness, respectively. The solution above can be considered as rigorous for infinitely long cylinders. As a result of buckling. contributions arise to the in-plane stress resultants and to the radial displacement. Using Eqs (1) and (2) these contributions are related to each other by von KLirm&--Donnell equations given as follows : (3)
(4) where .y and ~2 are the axial and circumferential coordinates; V” denotes the two4 is biharmonic operator: k’ = t’/l2(! -I*‘) ; ,i - P/Et: and the stress function
Localized
related
to the in-plane
buckling
stress resultant
in cylindrical
deviations
shells
from the uniform
883
membrane
state
by
The linear buckling
equations p~4~+~r72W
_pC?!? ax’
=
0
ax2
(6)
v4q5+p~=o
are obtained from the radial equilibrium equation (3) and the compatibility (4) by omitting the non-linear terms. If we assume
equation
w = A cos yx cos pyy
(8)
C#J = a cos yx cos pyy,
(9)
Eq. (7) yields
where A is a constant ; a = pd/y2 ; m = l/( 1-t fi’)’ ; y = p//l; fi is the modal aspect ratio (axial/circumferential wavelength) ; and n is the number of waves in the circumferential direction. Substitution of (8) and (9) into Eq. (6) leads to AA _ ‘;’
I “5 Y
(10)
as the condition of validity of the resulting equation for all values of x and y and non-zero values of A. %Ais the buckling parameter load ;1 corresponding to the mode (8). Similarly, if we assume the axisymmetric mode w = Bcos2yx the corresponding
The analytical
buckling
minimization
parameter
(11)
load is
of ILAwith respect to CIgives
(13) iA = 2pk corresponds to the classical solution and all the modes having this load are those on the well-known Koiter circle; tl = y2k/p is the required condition for the mode (8) to be on the Koiter circle.
xx4
‘3. W. HUNT and E. 3.
DOUBLE
LLCENA
Nrro
SCALE ANALYSIS
Equations (3) and (4) can be converted into an infinite set of linear partial differential equations by expanding the radial displacement, the stress function and the load parameter into power series in terms of an arbitrary parameter s (THOMPSON and HUNT, 1973) H’(.Y,X. J) = W,(S. X,),)X+ W,(.U, X._r).VZ+ ll’,(.Y. X, J).V + &.Y,X.J)
= ~~,(.L.,X,~‘)~Y+41)~(.\~,X,~).~~+~1(.~,X,~’).s1+“’ i = ~:‘+i,.s+R_,.s’+i,.s’+~~~
IV, (b and j. are expanded as can easily be seen by attempt to get not only variable X (LANGI: and defined by
(14)
about the critical point corresponding to the Koiter circle, substituting .c = 0 into the above expanded forms. As an periodic solutions but also localized ones, the slow-space NF.WEIJ.. 1971 ; POTIER-FERRY, 1983 ; HUNT et ul., 1989)
X = ss
(15)
is incorporated in H‘and 4, Eqs (l4), by assuming that the amplitudes II’, , H‘!, etc., depend on this variable. Considering Eq. (I 5), derivatives with respect to x become
of the modes
(2 ^ (’ + ,:A, S . = t .Y is (2’ (7.P
= ?’ +2 ,$s+ i.Y’
$.x2.
etc. Then, from the first series of Eqs (14). we can write
(16)
Localized
buckling
in cylindrical
shells
885
where a prime denotes partial differentiation with respect to X. Similar expressions can be written for 4 by simpIy replacing w with (b in above equations. Since s was taken to be an arbitrary parameter, the stipulation that a power series in s vanishes requires that each coefficient of the power series vanishes as well. If Eqs (14) and (i7) are substituted into Eqs (3) and (4), the requirement that each coefficient in the power series vanishes leads to the following linear equations, which are the first three of an infinite set : Order s
(19) Order s’
Order s3
(22)
886
G. W. HIJN-and E. LUCENA NETO
The order sn of this sequence
of equations
can be written
as
where F,,(x, X, JJ) and G,,(x, X, y) are determined from the solutions of the previous (n- I) set of equations. Terms proportional to the solution of the homogeneous equations arising in F,(x, X, y) or G,(s, X, J>) must be eliminated because they lead to unbounded particular solution (that is, secular). The procedure employed of eliminating the secular terms is similar to that outiined by AMAZIGO and FRASER (1971) and is given as follows : suppose that the homogeneous solution of Eqs (24), that we wili be concerned with, is M‘,,= A,, cos yx cos j?YJl,
C/I,,= a,, cos gx cos p74’,
(25)
with a,, = h-A,?. Suppose that F,,, and G,, are the coefficients of the terms proportional to cos ^;s cos &JJ arising in F,(r, X, I:) and G,,(x. X, .Y), respectively. Writing Eqs (24) with these terms only on the right-hand side, we have
A solution
to (26) in the form (25) gives (pa,, - pkA,)
cos yx cos /I$jy = F,, i cos y.r cos /$y
(pffn - pk.4,) cos 7.Xcos j?$;y = kc,, Dividing
cos 3’” cos p,y.
(27) (28)
(27) by (28) gives F,, -kG,,
= 0.
(29)
This condition of suppressing secular terms arising from a nonhomogeneous expression in the form (25) holds for any secular terms arising from trigonometric expressions in the form of modes which are on the Koiter circle.
Localized
3.1. First-order Equations
buckling
in cylindrical
887
shells
equations
(18) and (19) have solution H’,
=
A(X)
similar
to Eqs (6) and (7)
$I = a(X) cos yx cos Bi~y,
cos yx cos /$jy,
where a = kA and A is no longer a constant,
After substitution of (30) and reduction sums, Eqs (20) and (21) become
but depends
of products
(30)
on X.
of trigonometric
functions
+[2ypk-44y3k2(1+~2)]A’sinyxcos~yy+~’~4kA’(cos2y.~+cos2f3~~~~)
Elimination of secular (29) provides
terms arising
from cos yx cos fi>~+~ by applying
IL,
to
(31)
the condition
=o.
(33)
The solution of (31) and (32) contains no secular terms arising from sin yx cos /$JJ because the coefficients of these terms on the right-hand side of these equations satisfy condition (29) identically. If we assume the particular solution 11’2= B(X) cos 2yx+ C(X) cos 2/$JJ’,
(34)
Eq. (32) provides 42 = b(X) cos 2yx+ e(X) cos 2~~~+d~~)
sin yx cos &y,
(35)
where P
h = #?Substituting
P” 3ZA2,
Eqs (33)-(36)
I C= ---A’,
W--l)5 N+B*)
.
(36)
into (3 1) gives
B = ~____ pB2(l + 8~) A 2, 32a
with
d=
c=
1 -A*, 16B*k
(37)
XXX
G. W. HUNT and E. LUCENA NETO D = 2-H~ i,‘,
3.3. Seculm
terrm in third-order equations
The right-hand side of Eqs (22) and (23) can be determined by substituting previous results given by (30) and (33) (37). Terms proportional to cos y.v cos P;,y. which arise in both equations, must be eliminated to avoid unbounded particular solutions. Let F3, and G?, be the coefficients of these terms in (22) and (23), respectively. Then II,
F3,
=
-fk’(l
+3[~4)A"+]J'i.2A
+
4
i6 1-/14+~~~~(1+4~)(1+8+ i
G?, = -i,3h.(~4-8/~Z+3)A”Now, substituting
,6
(1+8x)
1
A’.
(39) into (29) we have
4(1 -[~~)‘k’A”+&A+
,‘;
$’
(1 +8a)‘+y’(3-/I14)
A3 = 0. 1
(40)
The periodic solution is recovered by setting A” = 0. In this case, we get the same solution as that obtained by conventional post-buckling theory, with mode A represented by an active, and mode B a passive, coordinate (THOMPSON and HUNT, 1984). But localized solutions also exist. If we set the perturbation parameter s to %/(A, ~ j_). so that, on equating coefficients in (14), jb2 = - I for L < j_“, we retrieve the well-known amplitude modulation equation of NEWELL and WHITEHEAD (I 969) and SEC;EI. (1969), with signs suitable for the localized form (POTIER-FERRY, 1983 ; HUNT et al., 1989). As aspect ratio [j is varied, the position of mode A moves around the Koiter circle, with the square waveform represented by /? = 1 at the crown. Runge Kutta solutions to Eq. (40), from the same starting state of A,, = IO-‘, A;, = 0, are as shown in Fig. 1. For clarity, the curves have been drawn with different initial value X0. We see that the curves flatten to the X-axis as p + 1. Here critical loads for A and B coincide, so cr = 0 and the coefficient of A ’ is infinite while, simultaneously, that of A” is zero ; the latter forces periodicity, and the former suggests that the amplitude is 7ero. We also note that solution of the third-order perturbation equations would involve terms of the form cos 37.~ cos ~?JZ in pi’?. These are of particular interest, since at [j = ,,/3 they also lie on the Koiter circle. Our single-mode view, which results from the inclusion just of mode A in the solution (30) for II’,, is thus called into question specifically at /j = 1 and /j = J3. The possibility of localization combined with mode interaction occurs for each, the first case involving two modes, A and B, and the second. three.
4.
LOCALIZATION AND MODE INTERACTION
Let us take the new solution (30)
to the first-order
perturbation
equations
to replace
Locaiized
buckling
in cylindrical
shells
889
I
b
?
I-
O
Slow-space Flc;. 1. Rung+Kutla
wj = A(X)
solutions
variable,
to Eq. (40), for representative
X modes on the Koiler circle.
cos “ix cos p/4’+ B(X) cos (I ffi’)y.x+ C(X) cos p’yx cos ,oi’3?
4, = n(X) cos yx cos fly4 + h(X) cos (I+
p’);Lx+ c(X) cos p2yxcos p1’4’,
(41)
where a = kA, h = kB, c = kC and, as before, k = PM/~‘. All three modes appear on the Koiter circle, as seen in Fig. 2. The same analytical sequence is now followed; this time, however, to account for the differing influence of non-linear terms, slow-space is defined by
mode A
mode C
ZrrR/oxiol
wovelength
FIG. 2. Modal interactions and the Koiter circle. As the post-buckling develops, it is observed ex~rimentalIy that there is a drift to within the circie, via a sequence of secondary bifurcations (see Fig. 4).
G. W. HUNT and E. LUCENANETO
890
x = .y”7_y
(42)
to replace the linear relation (15). The remainder of the analysis proceeds much as above. No secular terms are found to arise in the solution of ~~~’ level equations. A particular solution is given by il’j,l = 0 $3,2 = cl(X) sin yx cos /$v+e(X)
sin (1 +/?‘)yx+f’(X)
sin a2;,x cos &y,
(43)
where ‘l
=
_
741-a’>k
A’
’
y(l+fll)
2k P = - i’(l +mjTj B’,
2( 1 -/P)k .f‘= j,/j2(
Suppression of secular terms at the s2 level then leads to the three coupled order differential equations 8(1 -[~2)‘k’A”+2i,A+3p/?BC 16(1 ff12)‘k’B”+4(l
+/~‘)‘~,B+3/$‘AC
8(1 -/~‘)‘k’C”+2/?4i,,C+3pfl’AB
(44) second-
= 0 = 0 = 0
(45)
and we can set s = 1,” -& so that i, = - 1. We note that the coefficient of A” again approaches zero as /j’ -+ 1. For the interaction at the crown of the Koiter circle, we thus conclude that no localized solutions exist. For the remaining values of fl, the possibility of localized interactive solutions arises. In the search for these, one useful technique is to borrow from the work of TOLAND (1986) and others, and take this as a system of equations associated with the Lagrangian Y= where we can identify
a “kinetic
energy”
T-V, part
T=4k’[(l-/?2)‘A’Z+2(l+~2)ZB’Z+(l-~~’)ZC’1] and a “potential”
(46)
(47)
part V=
-AA’-2(l+/?2)‘B’-j14C’+3pp2ABC.
We note that the quadratic form for T is positive-definite Hamiltonian system, progress in X is along lines .X = Tf
V = constant.
(48) for fi # 1. This being
a
(49)
In the fundamental unbuckled state we can take T = V = 0, all amplitudes and first derivatives vanishing. The symmetric section, through which localized solutions must pass (HUNT et al., 1989), is defined by T = A’ = B’ = C’ = 0. The energy contour V = 0 then carries a particular significance; points on this locus in ABC space represent states on the symmetric section with just enough “energy” to arrive at the flat fundamental state with zero kinetic and potential parts. In the search for localized solutions, this is valuable extra information.
891
Localized buckling in cylindrical shells
Slow-space FIG. 3. Runge-Kutta
variable,
X
solutions to Eqs (45), showing relative modulated amplitudes to the modes of (41). at p = 2.
Using a systematic numerical search in two dimensions, from starts both on the symmetric section and zero energy contour, apparent localizations at p = J3 and /I = 2 have been found; solutions exist for which -4, B and C all simultaneously decrease from finite amplitude and drop monotonically towards zero, as seen in Fig. 3. We note that the modal amplitudes found here appear in different proportions from the related periodic solutions and fully anticipate that such solutions exist for other values of p # 1. Further examination is left for later publication.
5.
COMPARISON WITH EXPERIMENTS
Initial buckling of the long thin axially-loaded cylinder is known to be violently unstable, with a sharp restabilization once the load has dropped to some fraction (perhaps one-fifth) of the classical critical value. Our first-order analyses pick up only the initial destabilization, not the subsequent restabilization. It is therefore possible to compare against published experiments only the wavelengths and predicted extent of the localization at about the correct load level, not the restabilizing load itself. It is hoped eventually to extend the analysis to include the restabilization, as has been the case with periodic solutions (HUNT et al., 1986). It is a further experimental observation (YAMAKI, 1984) that the equilibrium path between destabilization and restabilization is punctuated by a sequence of secondary bifurcations into longer circumferential waves (see Fig. 4). Thus, although buckling starts on the Koiter circle of Fig. 2, post-buckling is marked by a vertical drop to within the circle, as shown. Because of the severity of the initial instability, even under
Ci. W. flr!ur
E. LWI:NA N~:ro
md
(a)
Cd)
8.1525°
1360
0
0 1 I
0
Q2
Q4
Q6
Cla
~
,
-IO
2
1.2 8 ,rn”$
(b)
06,
P
8 01
-4
8
i
-2 -4
L
m
(cl
1.0 ’
09
d-+%+%-as
b 3=210*
1845’
4
0
4
w tmm,
Localized buckling in cylindrical shells
893
rigid loading, the triggering bifurcation and its corresponding wavelengths are unlikely to be seen experimentally. As an example, let us consider the longest of the experimental specimens of YAMAKI (19X4), for which the shell radius R = 100mm, t = 0.247mm, 5’= 0.3, and the effective length L is chosen such that the Batdorf parameter Z = J(l - ?)L'/Rt= 1000. Substituting for the known values and rounding to the nearest integer then gives the typical circumferential wavenumbers for positions on the Koiter circle “p_,
= 18,
np -: \- 3 = n/j= ,,,.‘? = 16,
H,j& = n,j= 1.2= 15.
(50)
Experimental results are given in Fig. 4, taken directly from Fig. 3.52e of YAMAKI (19X4). Two types of deflection pattern are observed, symmetric and asymmetric ; both are reproducible by the above analysis, the latter by replacing the cosine functions of Eq. (30) with equally valid sine functions. For the symmetric case, the highest circumferential wavenumber observed experimentally is 9. As the post-buckling develops. the sequence of secondary bifurcations is seen to bottom-out at M = 6; deflections at IZ= 7 are shown at the bottom-right of Fig. 4, and crude wavelength measurements indicate that here the diamond wavepattern is close to being of the “square” form p = 1. By projecting upwards from the appropriate point on Fig. 2, tracing in reverse the sequence of secondary bifurcations, the point of initial instability on the Koiter circle can be approximated; this suggests a triggering bifurcation at TV= 14 and /I = 2.2. If we assume a post-buckling load of about one-fifth of the classical critical load, the range of localization seen in Figs 1 and 3 is in each case over about one to one-and-a-half axial wavelengths ; although no account is taken of the restabilization, nor the sequence of secondary bifurcations, it is encouraging to note that this is exactly as reported experimentally (SENDELBECK etul., 1967;YAMAKI, 1984).
6.
CONC’LUDING REMARKS
We offer in this paper two complementary analyses for the initial post-buckling of long thin axially-loaded cylinders. the first assuming distinct critical behaviour (THOMPSON and HUNT. 1973, 19X4), and the second allowing interaction between modes. The approach is distinguishable from earlier perturbation analyses by the fact that amplitudes are left free to modulate, which then opens the way for localized solutions. Some general conclusions can be drawn. The most significant feature, which is reflected in published experiments on long cylinders (see Fig. 4), is that localization forces the critical modes away from the crown of the Koiter circle. It has been realized for some time that the initial triggering buckle pattern comprises long axial waves, but adequate explanations apparently have been lacking. The answer is of course found in the form of Eqs (40) and (45). Extension of the analysis to find the minimum localized post-buckling load. via the sequence of secondary bifurcations, certainly seems possible. &WELL (1986) develops a perturbation approach for the interaction between the periodic crown mode and its axially-symmetric counterpart, which in one step includes both initial destabilization and subsequent restabilization of the post-buckling path. A similar approach, but
G. W. HUNT and E. LLCENA NATO
894
allowing prove
for modulating
most
amplitudes
and interactions
away from the crown,
could
rewarding.
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1971
I/71. J. Solids Structures 7, 883.
FKASKR,W. B. BRUSHY,D. 0. and
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Buckliny of’Bclrs, Plrrtrs, rrnd Shdls. McGraw-Hill. New York. QV?. Stub. Sl’sterns 1, 115. Tram AS’ME Aero. Engng AER-56-12, 795. ht. J. Solids Structurrs II, 1223. AIAA J/4, 126.
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Irlt. J. Solids Str.wtures
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Dissertation, Dclft. The Ncthcrlands. SIAM J. Appl. Math. 21, 605.
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J. Fluid Mech. 38, 279.
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SEGEL. L. A.
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C’olltrpsc~: tlw Bwkling of’ Structums ’ in Thcor~~ rml Pructiw (IUTAM Symposium, London. 1982)(editedbyJ.M.T.T~0~~~0~andG. W. HUXT), p. 149. Cambridge University Press. Cambridge. J. Fluid Mec~h. 38, 203. Report SUDAAR 318. Dcpt of Aeronautics and Astronautics. Stanford University.
ALMROTI-I, B. 0. COWFLI., R. G. DONNELL. L. H. HANXN. J. S. HOFF. N. J., MADSEN, W. A. and MAYPRS, J. HUNT, G. W., WILLIAMS. K. A. J
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R. L., CARLSON, R. L. and HOFF, N. J. THOMPSON, J. M. T. and HUNT, G. W. THOMPSON, J. M. T. and SI!NDIiLBh(‘K,
197.1 1984
HUNT, G. W. T~I.ANII. .I. F.
1986
VON KAKMAN, T. and
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TSIEN, H. S. YAMAU. N.
22, 150 1.
R. G.
1984
A Gmrral Thcorj, of Elastic Stubili/)~. John Wiley. London. Elastic Instability Phenonwnu. John Wiley. Chichestcr. Am. Muth. Sot., Proc. S~wzp. Pwr Mnth. 45, Part 2. 447. .I. Awe. Sci. 8, 303. Eltrstic Stubilit~~ of C’irculur C~~lindricul Shc1l.s. Elsevicr. Amsterdam.