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On local buckling of thin shells
Inr. J Non-Lmear Mechaws. Prrnted I” Great Braam
Vol. 20. No
4. pp 249-259.
C020-7462/E SO3.00 + .oO Pergamon Press Ltd.
1985
ON LOCAL BUCKLING
OF THIN SHELLS
E. L. AXELRAD Universitlt
der BW, Munchen. Institut fuer Mechanik, Werner Heisenberg Weg 39. 8014 Neubiberg, West Germany (Received 14 May 1984; received for publication 7 March 1985)
Abstract-A class of problems is considered where the buckting initially starts only in a part of the shell--locally. The stability analysis is focused on the zone of initial buckling. This leads to radical simplification. First the basic hypothesis and stability equations are formulated. Closed-form stabihty criteria asymptotically exact for very thin shells are discussed. This gives sufficient conditions for the local character of budkling and for the adequacy of the asymptotic approximation. The analysis taking into account the variation of stresses and shape inside the buckling zone results in a check of stability by hand calculations or by simple coding of a desk-top computer. The adequacy of the simplest representation of the stress and strain variation in the buckling zone is tested.
1. INTRODUCTION
classic stability analysis studies bifurcation buckling which starts globally-over the entire shell including the edges. This approach appears to stem from Euler’s work on compressed columns. The following is concerned with local buckling (e.g. Fig. l), first of all for complicated, variable prebuckling shape and stress distributions [l-4]. The theoretical foundation of the local approach was recently built up by Koiter [5]. Of course, local buckling does not necessarily lead to an immediate failure of the entire structure. But when the prebuckling deformation is taken into account, the initial-buckling load should yield a conservative estimate of the collapse load. Several intrinsic reasons make it worth-while to employ the additional specialized instrument of local-buckling analysis. The effect of imperfections can be drastic. But the buckling analysis of an imperfect shell is not a simple issue. Even for the canonical case of a cylinder shell under axial compression the very problem of describing the problem that arises has proved to be a formidable one. The prebuckling deformation introduces severe complications for flexible sheI1.s. The global stability analysis involves an eigenvalue boundary problem for a non-linear high-order system of partial differential equations (e.g. of the Donnell-type). A rigorous solution of such a problem is at present hardly attainable. Experiences in computational mechanics are expressed in warnings [6,7] against overconfidence in numerics. But just those factors complicating the global stability analysis make the buckling local and the respective local-stability investigation effective and simple. Any variable with respect to surface coordinates stress and local shape presents in some part of the shell a conjunction more favourable for buckling. The initial buckling is confined to this zone. The corresponding mode is determined by a particular solution of the stability equations tending to zero away from the initial buckling zone. The boundary conditions as well as the stresses and geometry of the greater part of the shell need not be involved. The ultimate intent ofthe concept oflocal stability becomes more lucid in the limiting case: When the initial buckling zone is very small, the stress resultants IV,,, normal-section The
Fig. 1. A case of local buckling. Tube bend&. Ku!
20:4-D
249
E. L. AXELRAD
250
curvatures l/R,, , the wall-thickness h and the elasticity factors are constant in this zone. We can speak of a stability condition “at a point”
f(Nza,R,, h, E, v) 2 0
(1.1)
3
The asymptotic condition of this type stands in the same sort of relation to the stability analysis encompassing the entire shell as the condition of allowable stress (e.g. with the R.v.Mises plasticity condition) to the failure analysis of a structure as a whole. 2. HYPOTHESIS
The discussion is based on the following statement interpreting the fundamental feature of the local-buckling phenomenon in the form of hypothesis [4]: The buckling instability is determined by the stress state and the shape of the shell inside the zone of the initial buckle(s). Conditions outside the zone of initial buckling have a negligible effect. Functions describing the stress and shell-shape may be set (analytically extended) outside the buckling zone in any way helping to simplify the analysis. The only restriction: the analytical extension must not amount to assuming zones of the shell, which are more susceptible to buckling than the zone under consideration. The hypothesis directly leads to the corollary: In so far as in the domain of initial buckle(s) the stress resultants and the curvature parameters of the shell are approximately constant, they may be assumed constant in the stability analysis. A concept amounting to the last statement has been the nucleus of the local-stability approach. It can be traced back to the work of Staerman [ 11 on a conical shell under axial compression. The buckling is supposed in [ 1 ] to start at a point where the compressive stress N,/h first reaches the value N:/h: Eh
N: -_= h
(2.1)
RJm
where l/R, denotes the normal-section curvature at the “buckling point”. This concept of local stability was employed in 1965 [2] for the analysis of cylinders and initially curved tubes under bending (Fig. 1). In consequence of non-linear elastic deformations the stresses and the local shape vary in these cases with respect to both surface coordinates. The stability condition was stated in the form similar to (2.1) with the curvature l/R, of the deformed shell. The ground for this had been rendered by the result of Seide and Weingarten [S] for linear bending of a circular cylinder of radius R: the critical stress at the center of any initial buckles is nearly equal to that of uniform axial compression N,,,(O) = lo,,h,
ocl = Eh/[Rd%-=?].
(2.2)
The values of II are presented in Table 1. There is one rigorous solution of a shell stability problem which provides as direct a check of the hypothesis as ever possible. It is the explicit solution of the stability problem of a circular cylinder compressed inside a longitudinal strip of breadth cm,obtained by Hoff et al. Table 1. Buckling stress N,dh
N
R __ h
I
i 1 1 2 3 2
100 200 lOV0 loo 100 1000
1.015 1.009 1.003 -
PI
m 0.979 0.972 0.992 -
[?I 1.0138 1.0084 1.0027 I .0368 1.0652 1.0072
= iu,,cosNq
I
(5.9)
1.0150 1.0095 1.0033 1.0379 1.0649 1.0082
m
0.963 0.970 0.982 0.943 0.927 0.972
251
On local buckling of thin shells
[9]. When the breadth c, is not less then 3.5m, the critical stress is practically equal to that of uniform compression (to ccl). The stress state outside the buckling zone is of no influence. Turning to the available results of experiments, we start with an obvious observation: the conditions most unfavorable for local character of buckling are those of stress and curvature uniform over the shell. The experiments of Esslinger and Geier [lo] cover most comprehensively several cases ofjust this nature. They show beyond any doubt that even the utmost carefully produced circular cylinders under as uniform axial compression as ever possible buckle locally. The buckling starts with a single small buckle with both breadth and length having (for R/h = 100/0.251) approximate dimensions 2nR/18 g 7(hR)“’ 3. STABILITY
(2.3)
EQUATIONS
Any local buckling mode must by definition entail intensively varying stress and deformations with respect to the surface coordinates. These may be described by the Donnell-Mushtari-Vlassov-Koiter equations of quasi-shallow shells [ll 1: Eh3 v4w-g+2s~=q, 12(1 - vZ) &V4F
+ 2
x
+ f
y
- K,P+ - r2 = 0; Y (3.1)
v4=
(-$$+$.)2, ~~~+K” )...) ‘=
--
a2w
N
axay'
x= --
a? ay
2”“’
S
K,=
-$
)...)
a2F
=--
axar'
Here X and Y denote distances measured along the coordinate lines of the (shallow) reference surface from the point (O,O),the middle point of the buckling zone. The stress resultants N,, N, are positive when they act compressively. The functions Wand F determine the prebuckling state, q is the intensity of the normal load; R& R,” and R,, R, are normal-section radii of curvature before and after the deformation respectively. Consider the stability of equilibrium. An infinitely small deviation caused by a bifurcation instability makes the stress and strain functions equal to F + f* and W + w. The functions W + wand F + f* describe an equilibrium state and satisfy the equations (3.1) just as Wand F do. This leads to equations for w and f*. As the bifurcation increments w, f* are infinitely small all terms non-linear in w, f* are to be dropped. The stability equations presented in dimensionless form are
(a:+ a;)b+
&
(N,a,2 + Nyay2 - 2sa,a,)w + (kxa,2 + k,a: - 27Ra,a,)f =
0,
(3.2) (k&’ + k& -
- (a: + a,')'f = 0,
27Ra,a,)w
where it is denoted
a,=$
“y=y,
a
X XC--
c’
y=$
c=Rfi, (3.3)
h
ho =
k, = R/Rx,
k, = R/R,,
f = f*/(Ehh”R).
- RJ_’ The dimensional parameter R is to be chosen for each particular problem. In the following we will set R equal to the normal section radius R,(O, 0) at the center ofthe buckling zone (Fig. 1).
252
E. L. AXELRAD
The stability equations (3.2) remain valid also in the cases when the prebuckling state (N,, N,, S, R,, R, and z) cannot be determined with the aid of the Donnell-type equations. (For flexible shells the semi-membrane theory can be useful [4,11]). For the cases when the local shape of the shell varies much less intensively than the buckling mode, that is when
ai
(L1$ aif,
i,j = X,Y,
4
J
J
(3.4)
we can eliminate the function f from (3.2) and obtain ([3] p. 225; [5]): Dw=O, D =
ia; + a;)” + &(Nxa:
+ ~~ayZ- 2sa,a,)(~z, + ~;)2
(kxa; + k,a,2- 27Ra,a,)Z ;
+
(3.5)
Consider first the simplest limiting case of local buckling: when the basic issue of determining the initial-buckling zone and taking into account variations of the stress and shell-shape inside it does not arise. 4.
ASYMPTOTIC
APPROXIMATION.
LOCAL
BUCKLING
RESISTANCE
Consider local stability of a shell in an area around some point (0,O) of the middle surface assuming the hypothesis in the most restrictive form of the “corollary”: the resultants N,, N, and the curvature parameters k,, k, are assumed constant in the buckling zone-equal to the respective values at the point (0,O). In the following we consider the stress state and the shape of the buckling zone to be symmetric with respect to the lines x = 0, y = 0, which pass through the center of the zone. In particular S = 0, T = 0. According to the hypothesis we may set outside the zone any appropriate functions N,, N,, k,, k,. We set them equal to constants-their values at (0,O). The constant with respect to x and y stress-state and local shape suggest a periodic buckling mode. A simple mode of this kind is defined by w = A
Substitution A
cosmx cosny
into (3.5) where S,
(m2+ nz)4 -
7 = 0
(A, m,
n = const).
(4.1)
leads to (with R = R,(O,O), k, = I):
xm2 + N,n’)(m’
+ n2)2 + (k,n2 + m2):
1
= 0.
(4.2)
The buckling occurs, A # 0, when the expression in the brackets is equal to zero. This renders for the critical values N:, NT of the resultants the equation N, + N,H2 = Ehh” m2(1 + Hz)’ +
(1 + k,H2)2 m2(1 + H2)2
1 H=i. ’
(4.3)
proposed in [2] and [3] p. 228. (The equation was obtained independently in [.5]). The buckling load determined by equation (4.3) must be minimized with respect to m and n. Thus the length 2nc/m = c, and the breadth 2nc/n = c,, of a buckling-mode wave are found simultaneously with the critical stress-resultants N,* , NF. For any fixed ratios N,JN, and n/m the condition of extremum dN,/dm = 0 and (4.3) yield [3,11]: c, = 27cc/m, =
21cc(l + H2)/(1 + k,HZ)“2,
N: + N;H’
= 2Ehh”(l
+ k,H2)
cy = c,H-‘;
(4.4) (4.5)
On local buckling of thin shells
253
The equation (4.5) determines the ratio H which corresponds to the minimum critical load. But H is indirectly restricted through the relation (4.4) determining c,, cY. Namely the assumption of N,, N,, k,, k, being nearly constant inside the initial-buckling area holds usually only when this area is small, which requires (as illustrated in the following) that c, and c, be small compared to R, and R,. Now, let us apply the formulas (4.4), (4.5) for two canonical cases. Consider a spherical shell of radius R = R: = R,”loaded by a uniform normal pressure 4. For any part of the shell outside the edge-effect- zone one easily finds the membrane solution: R, = R, = R, N, = N, = qR/2. The formula (4.5) renders the critical pressure equal to the classical value 2N: 2Eh21R2 qc, = - R = Jm,
c, = 2nd
+ H2)“2.
(4.6)
The length-to-breadth ratio of a buckle H turns out to have no influence on qer(provided the initial buckling is not hindered by the edge effects). The smallest buckles correspond to H= 1: c,=c,,=$2lrc. Consider next a cylinder shell [kx = 0, k,, = R/R,@)] under the longitudinal stress N,/h (NY = 0). The stability condition (4.5) becomes identical to (2.1). In particular for circular shells, when R, = R, N: = oclh. This buckling stress corresponds to various buckling modes of the form (4.1) including the axisymmetric buckling with H = 0, c, = 2nc and the case of H = 1. when m = 0.5,
c, = c,, = 4nc x 7(hR)“2
(4.7)
The relations (4.4)-(4.7) provide a commentary on the well-known results of experiment and theory. These in turn throw some light on the applicability of the simplest solution (4.4), (4.5). The dimensions (4.7) of one wave of periodic buckling mode practically coincide with those (2.3) of a single buckle found experimentally. A similar situation can be observed for a circular cylinder with N,/h = amcosNY/R. The N:/h of (4.5) is approximately equal to the exact critical value of 6, of [S, 121: provided the variation of N, remains moderate inside the breadth c,/2, there we have ;1 x 1 (Table 1). So far the stress N,/h varying in the 4‘direction was discussed. For the stress NJh = a,(1 - X/L) varying along a circular cylinder with hinged edges Kabanov and Kurtsevich [ 131 computed the critical value to be am = l.O48a,,. As the length L of the cylinder was ten times the length c, = 2nc of a “free” axisymmetric buckle, such a buckle occurring in the zone of maximum stress, adjoining the edge X = 0. would have the center at X = 0.05L. At this point (NJh),, = l.O48a,, . 0.95 x a,l-almost exactly what it should be according to (4.5). The asymptotic solution is also useful when not the stress but the curvature (local shape) varies with respect to the surface coordinate. This is shown by comparison with the rigorous results of Volpe et al. [14], (p. 576) ] on uniform axial compression of oval cylinders defined by (2x1. is the perimeter of the cross-section) r -=
R,
1 - ~cosN+=
1 - rcosny
(0 < Y < 27rr, n = NC/r).
For allcylinders with t d 0.3, N = 2 thevalueofN,*/haccording to (4.5)or (2.l)is nearly equal to the a,, of [14]: in Table 2 i. x 1. Theexamplescalculatedinthefollowingsectionsprovideadefinitenotionofthedimensions of the zone of initial buckling. It has the length and breadth of the order of magnitude of one wave of an “unrestrained” buckling-mode determined in (4.4), that is, of the order of a, where R? is the radius of curvature of the section in which the predominant compression stress N,/h acts. This entails a perception of the relations (4.4) and (4.5) [ofthe type (l.l)] as asymptotically exact for very thin shells. Indeed in a general case the resultants N,, N, as well as the curvatures of a deformed shell l/R,, l/R, do not (outside the areas of edge effect or local loading) depend on the wall-thickness h. Thus inside a very small - JhR, buckling zone they
E. L. AXELRAD
254
Table 2. Uniform compression N,, N, of shells defined by (4.8) or (7.1). Critical stress N,Jh depends on R,(O,O), ky2 and N,; parameters m, n have a secondary influence
No.
m
In (7.1) N <
ky2
n
NY N,
10 20 30 40 50 60 70 80 90 10 110 12 13 14 15 16 17
0 0.3 0.0455 0.6 0.2720 0.9733 0.9 0.8 0.6412 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
-
0 0.15 0.15 0.3 0.3 0.6 0.6
2 6 2 4 2 2 4 8 10 10 10 10 10 10 10 10
0 0.0037 0.0037 0.0227 0.0227 0.5448 0.5448 0.5448 0.5448 0.3027 0.3027 0.3027 0.3027 0.3027 0.3027 0.3027 0.3027
0.1315 0.3945 0.1740 0.3479 0.1739 0.3479 0.5205 0.7808 0.778 0.778 0.778 0.778 0.778 0.778 0.778 0.778
0 0 0 0 0 0 0 0 0 0 0.2 0 0.2 0 0.2 0 0.2
2l~j\I,,,R,(O,O),Eh’ (5.9)
1.0357 1.1125 1.720 -
1.0001 1.029 1.029 1.097 1.097 1.708 1.708 1.708 1.708 -
(6.5) 1.0 1.028 1.026 1.096 1.092 1.700 1.707 1.705 1.646 1.467 1.215 1.517 1.211 1.579 1.259 1.673 1.394
are very nearly constant. With the same (in actual cases, better) accuracy the critical stresses are determined by the asymptotic relations (4.5). The asymptotic analysis renders a lower-limit estimate of the buckling load: The relations (4.4) and (4.5) serve to check the stability in the zones of the shell where the conjunction of the stresses and curvatures is the most susceptible of buckling. .The neighbouring parts of the shell are in a position which is better with respect to buckling than is assumed in the analysis. The values of N, and N, are determined by (4.5) as characteristics of resistance to buckling at a point of the shell. Consider now a more general analysis allowing for substantial variability of stress and shape inside the buckling zone. 5. BUCKLING
OF A FINITE
ZONE
OF A SHELL
The stress state and shape of a shell, in particular of an imperfect and deformed one, can be complicated and diverse, even locally. But for a local approximation of the relevant functions Ni(X, y), ki(x,y) a well-tested and simple device is provided by a power-series (Taylor) expansion. For instance, in the locality of the point (0,O) N,(x,Y)
=
N,(O,O)+ xN,,.x(O, 0) + YN,,,(O,O) + x2Nx,,,(OA+ . . . .
(5-l)
In this context the asymptotic solution of the previous section is based on the representation of the stress state and shape merely by the constant, Oth, terms of the expansions of N,(x, y), . . . , k,(x, y). This proves adequate enough for a substantial class of problems. Retaining also the next nonvanishing term of the expansions it appears possible to estimate the accuracy of the simplest approximation and to encompass in a most direct way problems with the situation substantially varying inside the buckling zone. To avoid complications of a formal nature we confine the analysis to problems symmetric with respect to the lines x = 0, y = 0 and S = 0, z = 0. We choose R = R,(O, 0) and represent the local shape and stress state by the following expressions (defining constant parameters kij, nij): .
(5.2)
255
On local buckling of thin shells
The terms with x, y and xy fall out in consequence of the symmetry. As a further restriction (dropped in the next section) we consider the stress and shape varying substantially only in the y-direction and the factors of N,, l/RX to be of secondary importance, setting:
[NX,;;(0,0,3
= [;
;“][$I,
k, = kxo,
N,IN,(O,O) = nyo.
(5.3)
This makes the buckling mode periodic with respect to x. The mode is local with respect to y. We seek this mode as an expansion in terms of the Bessel functions Ai (plotted in Fig. 2): W
Cpi=
&=
COSWIX Ai(
f p=.
=CCiCpi;
.’
(i +*p)!p!
(5.4)
i
--n2y2 4
p
I
(i,p = O,l,... ).
(5.5)
The functions Ai have been chosen on the ground of their similitude to the buckling modes found rigorously for a circular cylinder in [8,9]. Substitution of (5.4) into (3.5) and application of the Galerkin-method yields the linear equations for Cj: 4 DijCj = 0,
Dij = Sp, D(qj)qidxdy.
(5.6)
The non-zero solution of this system determines the buckling mode. The lowest eigenvalue I and the corresponding characteristics m and n are found by equating the determinant of the system (5.6) to zero: det(Dij) = 0,
Dij = Dij(L, m. H),
L = N,(O,0)/(2Ehh”)
(5.7)
When N, = 0, the 1 has the meaning of a correction factor to the asymptotical solution (4.5) and by h + 0, ,J + 1. By calculating the coefficients Dij the integration with respect to x need not go further than one period of cosmx. The functions Ai vary symmetrically with respect to y = 0 and diminish rapidly with the growth of y. Calculations show that an increase of the limit of integration over ny = 12 has little influence on the values of Dij. The formulas (5.6) with D from (3.5) determine the following expressions of Dij (written here briefly, without the indices ij by A, and I?,): Dij = m8(Ao - 4A2 + 6A4 - 4A6 + As) + m4(Ao + 2kY2Bo - 2k,oAz) - 2A[Ao - 2A2 + A4 + nx2(Bo - 2B2 + B4) + n,o(A2 - 2A4 + As)m-2]m6. 1.0 0.8 0.6 0.L 0.2 0 -0.2 -0.L 0
5 Fig. 2.
lfl
(5.8)
E. L.
256
AXELRAD
Consistent with the accuracy assumed in (5.2) the terms with products of the parameters ky2, kxO are neglected. The integrals of (5.8) can be computed conveniently with the help of the appropriately truncated series (5.5). The solution was programmed on the personal. computer HP-45. For the range of local shapes and stress distributions, represented in Fig. 3, the solution with only the AZ-term retained in the mode (5.4) leads to practically the same value of i; as the solution wtth up to eight terms in Cj cosmxAj (ny). Thus the simplest version of equation (5.7) Dj3 (1, m, H) = 0 may be used, providing the formula
/I=
m2F + me2(l + 6.869kY2nU2+ 0.1538kX0H2)
2 + 0.3077H2 + 0.0308H4 + nX2Un - 2 + nYOI/m- ’
F = 1 + 0.3077H2 + 0.0923H4 + 0.0181H6 + 0.0017H8,
U = 6.869 + 0.2606H2 - 0.2048H4,
(5.9)
I/ = 0.1538 + 0.0614H2 + 0.0091H4.
This relation determines also the values of m and n = Hm-those rendering a minimum value ofi. The results of the rigorous analysis [8,12,14], available for cylinders confirm those of the local-buckling investigation. This,is demonstrated in Tables 1 and 2. The i-values present the correction to the asymptotic relations (4.5) or (2.1) necessary for a finite wall-thickness. Formula (5.9) proves adequate even for ovals (4.8) with C:= 0.9-for the curvature l/R, varying from 0.1 jr to 1.9/r. The solution presented provides an explanation for the local character of buckling of circular cylinders under uniform compression, discovered in [lo] : The local buckling mode w = Ccosmx As(ny) leads to virtually the same critical stress (N,,,/h = 1.00010,~, Table 2) as does the mode (4.1). Moreover, even a single-buckle mode w = CA,(mx)A,(ny) (A, is plotted in Fig. 2) leads (through a relation differing from (5.9) only in the values of the coefficients) to N,,, = l.O15a,JI. 6. DOUBLE-FOURIER-SERIES
MODE
The restriction (3.4) on the intensity of variation of the prebuckling shape and the simplest description of the local shape and stress (5.2) have been tested so far only by comparison with the rigorous results available. Consider next a local-stability solution free of these restrictions. The buckling mode will be sought in the form
(6.1) Z=ML+M+L,k=l+i.p.
[i/(L+l)],I=i-(L+l)(k-1)
where i.p. [ ] denotes the integer part of the quantity in the brackets; and m and n are (constant) parameters of the buckling mode.
I,0
0
0.5
1
Fig. 3. Influence
1.5
2
of stress and curvature
6 variation.
8
257
On local buckling of thin shells
Substitute the expressions (6.1) into (3.2). Applying the Galerkin method or equating to zero the Fourier-coefficients of the left-hand side of each of equations (3.2) one obtains equations determining Ci, 6. This system can be composed in an easily programmable matrix-form [3,11] : (DX+D,)‘--L2N -K
K (DX + DY)2][;] w = {C,
= [:I’
Cl...C,},
[:]
= E:
f = {F,
(6.2)
:]E;]’
F,...F,}.
The matrices D, and D, represent the operators 8,’ and ai, respectively, in such a way that Dxw, D,f, D,w and D,f are column matrices of the Fourier coefficients of the functions - @w, - a,“f, - Jj?w and -a:f respectively. Specifically D, and D, are diagonal matrices determined by formulas for the elements of the line i and the column j: D,(i, i) = n212(i);D,(i,j) = D,(i,j) = 0, i # j
D,(i, i) = m2k2(i),
L determined by formulas (6.1). The matrices K,, K,, withk=1,2 ,..., M+ l,I=O,l,..., N,, N, are composed of the coefficients of double Fourier-series of the respective functions. For instance, the composition of the matrix K, assures that the column matrix K&w determines in (6.2) the Fourier-coefficients of the function - k$;w. For the expansion
k~(x,Y) = krO ii0 &
k,(k, 1)cosmkx cosnly; 60 = 2,6, = 1, a + 0;
(6.3)
it can be ascertained by means of the relation 2cos~cosy = cos(jI + y) + cos@ - y) that the elements of the matrix K, are determined by the formula
&(iJ) =
$ +
1
[k(lk - kjl, II - ljl) +
k(k + kj, II - ljl) +
k,(lk
- kjl, I + Ij)
+ kj, I + Lj)].
k,(k
(6.4)
Here k and 1 are rendered by the expressions (6.1), the integers kj and Ij are defined by the same expressions with j instead of i. Formulas similar to (6.4) determine the remaining matrices of the equations (6.2) K,, N, and N,. Eliminating f we obtain from (6.2) the system
[@x + DJ2 + K@x + D,)-‘K]-‘N.-
$
(6.5)
.
The largest eigenvalue 1/(2;1) of the matrix of equation (6.5) and the respective eigenvector w determine the critical load and the buckling mode w(x, y). They are effectively computed by means of the R.v.Mises matrix-iteration method. This solution does not assume the buckling to be local. It can be exact as far as the Donnelltype equations (3.2) are applicable; the series (6.1) fulfill the boundary conditions and the expansion (6.3), as well as the similar ones for k,, N,, N, describe the actual prebuckling situation. The algorithm was programmed on the desk-top computer HP-85. For a 30-term mode (6.1) the calculation requires seven minutes. 7. NONCIRCULAR
AND DOUBLE-CURVATURE-IMPERFECT SHELLS
CYLINDRICAL
The series solution of Section 6 allows a further check of the local-stability concept. The two issues to be discussed are: (a) to ascertain the initial-buckling zone, to which the localbuckling analysis may be restricted and (b) to test the sufficiency of the leading-term description (5.2) or (5.3) of the prebuckling stress and strain.
E.L.
258
AXELRAD
Consider first cylindrical shells-those defined by the formula (4.8). The results of calculations, presented in Table 2 (m = 0) and Fig. 4 display for a broad range of substantially different shell shapes nearly identical values of ,I and the modes w(x, y) which nearly coincide within the buckling zone. The value of i depends in these cases solely on the parameter ky2. The actual shape is (for a fixed ky2 -value) of no direct importance for any profiles (4.8) provided n d 0.5. [Recall that n = 0.5 corresponds to the unrestrained mode (4.1), (4.7).] The initial buckling is determined basically by the shape within the zone lyl < x-one half the wave-breadth c, of the unrestrained periodic mode (4.1), (4.7). This conclusion applies also to variations of stress and to non-cylindrical shells. However the simultaneous involvement of several parameters characterizing stress and shape in a more general case [there are ten in (5.2)] makes the influence of each of them less transparent. Further discussion is restricted to one type of shell and to the influence of two factors not discussed in the foregoing. Consider shells defined by a deviation W, = Acosmx . cosN Y/r from a circular cylinder of radius r setting A = 0.5h, h = O.Olr, N = 10. According to (3.1) r -=l-r~
4
R = R,(O,O) = 2r, v = 0.3 ky2 = tn2/(2 - 25) = 0.3027.
with
= 1 - ~cosmx . cosny, and
(3.3) we have
r = 0.5
(7.1)
c/r = 0.0778,
n = NC/r = 0.778,
The influence of the length 2rcc/m of the imperfection wave (7.1) and of the compression stress NY/h in the circumferential direction (represented by nyo) on the buckling stress N,/h is illustrated by Table 2 and Fig. 5. The m = 0 entries of Table 2 concern a cylindrical shell. The other lines of the Table indicate the critical stress to be higher the shorter the wave length 2nc/m of imperfection (7.1). But for any wave lengths not shorter than 2rrc/O.3 the critical stress differs only insignificantly from that of a cylindrical shell. The reduction of the longitudinal critical stress in consequence of the lateral compression is displayed by the lines 11-17 of Table 2. However the influence of the lateral compression caused by the normal
Fig. 4. Examples
6, 8 and 9 of Table 2.
Fig. 5. Examples
12 and
14 of Table 2.
On local buckling of thin shells
259
pressure q is not restricted to the local stability. It can lead to global buckling and a check is needed whether the shell can withstand this. 8. CONCLUDING
REMARKS
The possibility to simplify the analysis by focusing it on the zone of initial buckling does not depend on the symmetry of this zone. Extension to the unsymmetric case is direct. The local approach has rendered promising results in the analysis of shell vibration [15]. Acknowledgements-The
work was carried out with the support of the Deutsche Forschungsgemeinschaft. The author is indebted to Prof. F. A. Emmerling and the late Dr. B. 0. Almroth for valuable discussions and to Valery Axelrad for the programming. REFERENCES 1. I. S. Staerman, Stability of shells. Trudy Kieuskogo Aviats. Inst. 1 (1936) (In Russian). 2. E. L. Axelrad, Refinement of the upper critical load of pipe fiexure by considering precritical deformation. Zzuestija AN SSSR, OTN, Mekhanika i Mash. No. 4, 133-139 (1965) (In Russian). 3. E. L. Axelrad, Flexible Shells. (Gibkie Obolotshki). Moscow, Nauka (1976) (In Russian). 4. E. L. Axelrad, Flexible shells. In Proceedings oj rhe 15th International Congress on Theoretical and Applied Mechanics (1980), (Edited by F. P. J. Rimrott and B. Tabarrok) pp. 45-56. North-Holland, Amsterdam (1981). 5. W. T. Koiter, The application of the initiai postbuckling analysis to shells. In Buckling of Shells,Proceedings of a State-o$the-Arr CoNoquium (Edited by E. Ramm). Springer, Berlin (1982). 6. J. T. Oden and K. J. Bathe, A commentary on computational mechanics. Appl. Mech. Reoiews, J&1053-1058 (1978). 7. B. Brendel and E. Ramm, Nichtlineare Stabilit&suntersuchungen mit der Methode der Finiten Elemente. Ing.Arch. 51, 337-362 (1982). 8. P. Seide and V. I. Weingarten, On the buckling of circular cylindrical shells under pure bending. J. appl. Me& 28, 112-116 (1961). 9. N. J. Hoff, C. C. Chao and W. A. Madsen, Buckling of a thin-walled circular cylindrical shell heated along an axial strip. J. appl. Mech. 31, 253-258 (1964). 10. M. Esslinger and B. Geier, Postbuckling Behauiour of Srructures. Springer, Wien (1975). 11. E. L. Axelrad, Schalentheorie. B. G. Teubner, Stuttgart (1983). 12. A. Libai and D. Durban, Buckling of cylindrical shells subjected to nonuniform axial foads. J. uppl. Me& 44, 714-718 (1977). 13. V. V. Kabanov and G. I. Kurtsevich, Investigation of the stability of a cylindrical shell under axial compression nonuniform along the length. Mech. Solids 11, 171-174 (1976). 14. V. Volpe, Y. N. Chen and J. Kempner, Buckling of orthogonally stiffened finite oval cylindrical shells under axial compression. AIAA J. 18, 571-580 (1980). 15. D. Teichmann, Dynamik und Stabilitit offener Kegelschalen. Z. angew Math. Mech. 63, T109-T113 (1983).
Vol. 20. No
4. pp 249-259.
C020-7462/E SO3.00 + .oO Pergamon Press Ltd.
1985
ON LOCAL BUCKLING
OF THIN SHELLS
E. L. AXELRAD Universitlt
der BW, Munchen. Institut fuer Mechanik, Werner Heisenberg Weg 39. 8014 Neubiberg, West Germany (Received 14 May 1984; received for publication 7 March 1985)
Abstract-A class of problems is considered where the buckting initially starts only in a part of the shell--locally. The stability analysis is focused on the zone of initial buckling. This leads to radical simplification. First the basic hypothesis and stability equations are formulated. Closed-form stabihty criteria asymptotically exact for very thin shells are discussed. This gives sufficient conditions for the local character of budkling and for the adequacy of the asymptotic approximation. The analysis taking into account the variation of stresses and shape inside the buckling zone results in a check of stability by hand calculations or by simple coding of a desk-top computer. The adequacy of the simplest representation of the stress and strain variation in the buckling zone is tested.
1. INTRODUCTION
classic stability analysis studies bifurcation buckling which starts globally-over the entire shell including the edges. This approach appears to stem from Euler’s work on compressed columns. The following is concerned with local buckling (e.g. Fig. l), first of all for complicated, variable prebuckling shape and stress distributions [l-4]. The theoretical foundation of the local approach was recently built up by Koiter [5]. Of course, local buckling does not necessarily lead to an immediate failure of the entire structure. But when the prebuckling deformation is taken into account, the initial-buckling load should yield a conservative estimate of the collapse load. Several intrinsic reasons make it worth-while to employ the additional specialized instrument of local-buckling analysis. The effect of imperfections can be drastic. But the buckling analysis of an imperfect shell is not a simple issue. Even for the canonical case of a cylinder shell under axial compression the very problem of describing the problem that arises has proved to be a formidable one. The prebuckling deformation introduces severe complications for flexible sheI1.s. The global stability analysis involves an eigenvalue boundary problem for a non-linear high-order system of partial differential equations (e.g. of the Donnell-type). A rigorous solution of such a problem is at present hardly attainable. Experiences in computational mechanics are expressed in warnings [6,7] against overconfidence in numerics. But just those factors complicating the global stability analysis make the buckling local and the respective local-stability investigation effective and simple. Any variable with respect to surface coordinates stress and local shape presents in some part of the shell a conjunction more favourable for buckling. The initial buckling is confined to this zone. The corresponding mode is determined by a particular solution of the stability equations tending to zero away from the initial buckling zone. The boundary conditions as well as the stresses and geometry of the greater part of the shell need not be involved. The ultimate intent ofthe concept oflocal stability becomes more lucid in the limiting case: When the initial buckling zone is very small, the stress resultants IV,,, normal-section The
Fig. 1. A case of local buckling. Tube bend&. Ku!
20:4-D
249
E. L. AXELRAD
250
curvatures l/R,, , the wall-thickness h and the elasticity factors are constant in this zone. We can speak of a stability condition “at a point”
f(Nza,R,, h, E, v) 2 0
(1.1)
3
The asymptotic condition of this type stands in the same sort of relation to the stability analysis encompassing the entire shell as the condition of allowable stress (e.g. with the R.v.Mises plasticity condition) to the failure analysis of a structure as a whole. 2. HYPOTHESIS
The discussion is based on the following statement interpreting the fundamental feature of the local-buckling phenomenon in the form of hypothesis [4]: The buckling instability is determined by the stress state and the shape of the shell inside the zone of the initial buckle(s). Conditions outside the zone of initial buckling have a negligible effect. Functions describing the stress and shell-shape may be set (analytically extended) outside the buckling zone in any way helping to simplify the analysis. The only restriction: the analytical extension must not amount to assuming zones of the shell, which are more susceptible to buckling than the zone under consideration. The hypothesis directly leads to the corollary: In so far as in the domain of initial buckle(s) the stress resultants and the curvature parameters of the shell are approximately constant, they may be assumed constant in the stability analysis. A concept amounting to the last statement has been the nucleus of the local-stability approach. It can be traced back to the work of Staerman [ 11 on a conical shell under axial compression. The buckling is supposed in [ 1 ] to start at a point where the compressive stress N,/h first reaches the value N:/h: Eh
N: -_= h
(2.1)
RJm
where l/R, denotes the normal-section curvature at the “buckling point”. This concept of local stability was employed in 1965 [2] for the analysis of cylinders and initially curved tubes under bending (Fig. 1). In consequence of non-linear elastic deformations the stresses and the local shape vary in these cases with respect to both surface coordinates. The stability condition was stated in the form similar to (2.1) with the curvature l/R, of the deformed shell. The ground for this had been rendered by the result of Seide and Weingarten [S] for linear bending of a circular cylinder of radius R: the critical stress at the center of any initial buckles is nearly equal to that of uniform axial compression N,,,(O) = lo,,h,
ocl = Eh/[Rd%-=?].
(2.2)
The values of II are presented in Table 1. There is one rigorous solution of a shell stability problem which provides as direct a check of the hypothesis as ever possible. It is the explicit solution of the stability problem of a circular cylinder compressed inside a longitudinal strip of breadth cm,obtained by Hoff et al. Table 1. Buckling stress N,dh
N
R __ h
I
i 1 1 2 3 2
100 200 lOV0 loo 100 1000
1.015 1.009 1.003 -
PI
m 0.979 0.972 0.992 -
[?I 1.0138 1.0084 1.0027 I .0368 1.0652 1.0072
= iu,,cosNq
I
(5.9)
1.0150 1.0095 1.0033 1.0379 1.0649 1.0082
m
0.963 0.970 0.982 0.943 0.927 0.972
251
On local buckling of thin shells
[9]. When the breadth c, is not less then 3.5m, the critical stress is practically equal to that of uniform compression (to ccl). The stress state outside the buckling zone is of no influence. Turning to the available results of experiments, we start with an obvious observation: the conditions most unfavorable for local character of buckling are those of stress and curvature uniform over the shell. The experiments of Esslinger and Geier [lo] cover most comprehensively several cases ofjust this nature. They show beyond any doubt that even the utmost carefully produced circular cylinders under as uniform axial compression as ever possible buckle locally. The buckling starts with a single small buckle with both breadth and length having (for R/h = 100/0.251) approximate dimensions 2nR/18 g 7(hR)“’ 3. STABILITY
(2.3)
EQUATIONS
Any local buckling mode must by definition entail intensively varying stress and deformations with respect to the surface coordinates. These may be described by the Donnell-Mushtari-Vlassov-Koiter equations of quasi-shallow shells [ll 1: Eh3 v4w-g+2s~=q, 12(1 - vZ) &V4F
+ 2
x
+ f
y
- K,P+ - r2 = 0; Y (3.1)
v4=
(-$$+$.)2, ~~~+K” )...) ‘=
--
a2w
N
axay'
x= --
a? ay
2”“’
S
K,=
-$
)...)
a2F
=--
axar'
Here X and Y denote distances measured along the coordinate lines of the (shallow) reference surface from the point (O,O),the middle point of the buckling zone. The stress resultants N,, N, are positive when they act compressively. The functions Wand F determine the prebuckling state, q is the intensity of the normal load; R& R,” and R,, R, are normal-section radii of curvature before and after the deformation respectively. Consider the stability of equilibrium. An infinitely small deviation caused by a bifurcation instability makes the stress and strain functions equal to F + f* and W + w. The functions W + wand F + f* describe an equilibrium state and satisfy the equations (3.1) just as Wand F do. This leads to equations for w and f*. As the bifurcation increments w, f* are infinitely small all terms non-linear in w, f* are to be dropped. The stability equations presented in dimensionless form are
(a:+ a;)b+
&
(N,a,2 + Nyay2 - 2sa,a,)w + (kxa,2 + k,a: - 27Ra,a,)f =
0,
(3.2) (k&’ + k& -
- (a: + a,')'f = 0,
27Ra,a,)w
where it is denoted
a,=$
“y=y,
a
X XC--
c’
y=$
c=Rfi, (3.3)
h
ho =
k, = R/Rx,
k, = R/R,,
f = f*/(Ehh”R).
- RJ_’ The dimensional parameter R is to be chosen for each particular problem. In the following we will set R equal to the normal section radius R,(O, 0) at the center ofthe buckling zone (Fig. 1).
252
E. L. AXELRAD
The stability equations (3.2) remain valid also in the cases when the prebuckling state (N,, N,, S, R,, R, and z) cannot be determined with the aid of the Donnell-type equations. (For flexible shells the semi-membrane theory can be useful [4,11]). For the cases when the local shape of the shell varies much less intensively than the buckling mode, that is when
ai
(L1$ aif,
i,j = X,Y,
4
J
J
(3.4)
we can eliminate the function f from (3.2) and obtain ([3] p. 225; [5]): Dw=O, D =
ia; + a;)” + &(Nxa:
+ ~~ayZ- 2sa,a,)(~z, + ~;)2
(kxa; + k,a,2- 27Ra,a,)Z ;
+
(3.5)
Consider first the simplest limiting case of local buckling: when the basic issue of determining the initial-buckling zone and taking into account variations of the stress and shell-shape inside it does not arise. 4.
ASYMPTOTIC
APPROXIMATION.
LOCAL
BUCKLING
RESISTANCE
Consider local stability of a shell in an area around some point (0,O) of the middle surface assuming the hypothesis in the most restrictive form of the “corollary”: the resultants N,, N, and the curvature parameters k,, k, are assumed constant in the buckling zone-equal to the respective values at the point (0,O). In the following we consider the stress state and the shape of the buckling zone to be symmetric with respect to the lines x = 0, y = 0, which pass through the center of the zone. In particular S = 0, T = 0. According to the hypothesis we may set outside the zone any appropriate functions N,, N,, k,, k,. We set them equal to constants-their values at (0,O). The constant with respect to x and y stress-state and local shape suggest a periodic buckling mode. A simple mode of this kind is defined by w = A
Substitution A
cosmx cosny
into (3.5) where S,
(m2+ nz)4 -
7 = 0
(A, m,
n = const).
(4.1)
leads to (with R = R,(O,O), k, = I):
xm2 + N,n’)(m’
+ n2)2 + (k,n2 + m2):
1
= 0.
(4.2)
The buckling occurs, A # 0, when the expression in the brackets is equal to zero. This renders for the critical values N:, NT of the resultants the equation N, + N,H2 = Ehh” m2(1 + Hz)’ +
(1 + k,H2)2 m2(1 + H2)2
1 H=i. ’
(4.3)
proposed in [2] and [3] p. 228. (The equation was obtained independently in [.5]). The buckling load determined by equation (4.3) must be minimized with respect to m and n. Thus the length 2nc/m = c, and the breadth 2nc/n = c,, of a buckling-mode wave are found simultaneously with the critical stress-resultants N,* , NF. For any fixed ratios N,JN, and n/m the condition of extremum dN,/dm = 0 and (4.3) yield [3,11]: c, = 27cc/m, =
21cc(l + H2)/(1 + k,HZ)“2,
N: + N;H’
= 2Ehh”(l
+ k,H2)
cy = c,H-‘;
(4.4) (4.5)
On local buckling of thin shells
253
The equation (4.5) determines the ratio H which corresponds to the minimum critical load. But H is indirectly restricted through the relation (4.4) determining c,, cY. Namely the assumption of N,, N,, k,, k, being nearly constant inside the initial-buckling area holds usually only when this area is small, which requires (as illustrated in the following) that c, and c, be small compared to R, and R,. Now, let us apply the formulas (4.4), (4.5) for two canonical cases. Consider a spherical shell of radius R = R: = R,”loaded by a uniform normal pressure 4. For any part of the shell outside the edge-effect- zone one easily finds the membrane solution: R, = R, = R, N, = N, = qR/2. The formula (4.5) renders the critical pressure equal to the classical value 2N: 2Eh21R2 qc, = - R = Jm,
c, = 2nd
+ H2)“2.
(4.6)
The length-to-breadth ratio of a buckle H turns out to have no influence on qer(provided the initial buckling is not hindered by the edge effects). The smallest buckles correspond to H= 1: c,=c,,=$2lrc. Consider next a cylinder shell [kx = 0, k,, = R/R,@)] under the longitudinal stress N,/h (NY = 0). The stability condition (4.5) becomes identical to (2.1). In particular for circular shells, when R, = R, N: = oclh. This buckling stress corresponds to various buckling modes of the form (4.1) including the axisymmetric buckling with H = 0, c, = 2nc and the case of H = 1. when m = 0.5,
c, = c,, = 4nc x 7(hR)“2
(4.7)
The relations (4.4)-(4.7) provide a commentary on the well-known results of experiment and theory. These in turn throw some light on the applicability of the simplest solution (4.4), (4.5). The dimensions (4.7) of one wave of periodic buckling mode practically coincide with those (2.3) of a single buckle found experimentally. A similar situation can be observed for a circular cylinder with N,/h = amcosNY/R. The N:/h of (4.5) is approximately equal to the exact critical value of 6, of [S, 121: provided the variation of N, remains moderate inside the breadth c,/2, there we have ;1 x 1 (Table 1). So far the stress N,/h varying in the 4‘direction was discussed. For the stress NJh = a,(1 - X/L) varying along a circular cylinder with hinged edges Kabanov and Kurtsevich [ 131 computed the critical value to be am = l.O48a,,. As the length L of the cylinder was ten times the length c, = 2nc of a “free” axisymmetric buckle, such a buckle occurring in the zone of maximum stress, adjoining the edge X = 0. would have the center at X = 0.05L. At this point (NJh),, = l.O48a,, . 0.95 x a,l-almost exactly what it should be according to (4.5). The asymptotic solution is also useful when not the stress but the curvature (local shape) varies with respect to the surface coordinate. This is shown by comparison with the rigorous results of Volpe et al. [14], (p. 576) ] on uniform axial compression of oval cylinders defined by (2x1. is the perimeter of the cross-section) r -=
R,
1 - ~cosN+=
1 - rcosny
(0 < Y < 27rr, n = NC/r).
For allcylinders with t d 0.3, N = 2 thevalueofN,*/haccording to (4.5)or (2.l)is nearly equal to the a,, of [14]: in Table 2 i. x 1. Theexamplescalculatedinthefollowingsectionsprovideadefinitenotionofthedimensions of the zone of initial buckling. It has the length and breadth of the order of magnitude of one wave of an “unrestrained” buckling-mode determined in (4.4), that is, of the order of a, where R? is the radius of curvature of the section in which the predominant compression stress N,/h acts. This entails a perception of the relations (4.4) and (4.5) [ofthe type (l.l)] as asymptotically exact for very thin shells. Indeed in a general case the resultants N,, N, as well as the curvatures of a deformed shell l/R,, l/R, do not (outside the areas of edge effect or local loading) depend on the wall-thickness h. Thus inside a very small - JhR, buckling zone they
E. L. AXELRAD
254
Table 2. Uniform compression N,, N, of shells defined by (4.8) or (7.1). Critical stress N,Jh depends on R,(O,O), ky2 and N,; parameters m, n have a secondary influence
No.
m
In (7.1) N <
ky2
n
NY N,
10 20 30 40 50 60 70 80 90 10 110 12 13 14 15 16 17
0 0.3 0.0455 0.6 0.2720 0.9733 0.9 0.8 0.6412 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
-
0 0.15 0.15 0.3 0.3 0.6 0.6
2 6 2 4 2 2 4 8 10 10 10 10 10 10 10 10
0 0.0037 0.0037 0.0227 0.0227 0.5448 0.5448 0.5448 0.5448 0.3027 0.3027 0.3027 0.3027 0.3027 0.3027 0.3027 0.3027
0.1315 0.3945 0.1740 0.3479 0.1739 0.3479 0.5205 0.7808 0.778 0.778 0.778 0.778 0.778 0.778 0.778 0.778
0 0 0 0 0 0 0 0 0 0 0.2 0 0.2 0 0.2 0 0.2
2l~j\I,,,R,(O,O),Eh’ (5.9)
1.0357 1.1125 1.720 -
1.0001 1.029 1.029 1.097 1.097 1.708 1.708 1.708 1.708 -
(6.5) 1.0 1.028 1.026 1.096 1.092 1.700 1.707 1.705 1.646 1.467 1.215 1.517 1.211 1.579 1.259 1.673 1.394
are very nearly constant. With the same (in actual cases, better) accuracy the critical stresses are determined by the asymptotic relations (4.5). The asymptotic analysis renders a lower-limit estimate of the buckling load: The relations (4.4) and (4.5) serve to check the stability in the zones of the shell where the conjunction of the stresses and curvatures is the most susceptible of buckling. .The neighbouring parts of the shell are in a position which is better with respect to buckling than is assumed in the analysis. The values of N, and N, are determined by (4.5) as characteristics of resistance to buckling at a point of the shell. Consider now a more general analysis allowing for substantial variability of stress and shape inside the buckling zone. 5. BUCKLING
OF A FINITE
ZONE
OF A SHELL
The stress state and shape of a shell, in particular of an imperfect and deformed one, can be complicated and diverse, even locally. But for a local approximation of the relevant functions Ni(X, y), ki(x,y) a well-tested and simple device is provided by a power-series (Taylor) expansion. For instance, in the locality of the point (0,O) N,(x,Y)
=
N,(O,O)+ xN,,.x(O, 0) + YN,,,(O,O) + x2Nx,,,(OA+ . . . .
(5-l)
In this context the asymptotic solution of the previous section is based on the representation of the stress state and shape merely by the constant, Oth, terms of the expansions of N,(x, y), . . . , k,(x, y). This proves adequate enough for a substantial class of problems. Retaining also the next nonvanishing term of the expansions it appears possible to estimate the accuracy of the simplest approximation and to encompass in a most direct way problems with the situation substantially varying inside the buckling zone. To avoid complications of a formal nature we confine the analysis to problems symmetric with respect to the lines x = 0, y = 0 and S = 0, z = 0. We choose R = R,(O, 0) and represent the local shape and stress state by the following expressions (defining constant parameters kij, nij): .
(5.2)
255
On local buckling of thin shells
The terms with x, y and xy fall out in consequence of the symmetry. As a further restriction (dropped in the next section) we consider the stress and shape varying substantially only in the y-direction and the factors of N,, l/RX to be of secondary importance, setting:
[NX,;;(0,0,3
= [;
;“][$I,
k, = kxo,
N,IN,(O,O) = nyo.
(5.3)
This makes the buckling mode periodic with respect to x. The mode is local with respect to y. We seek this mode as an expansion in terms of the Bessel functions Ai (plotted in Fig. 2): W
Cpi=
&=
COSWIX Ai(
f p=.
=CCiCpi;
.’
(i +*p)!p!
(5.4)
i
--n2y2 4
p
I
(i,p = O,l,... ).
(5.5)
The functions Ai have been chosen on the ground of their similitude to the buckling modes found rigorously for a circular cylinder in [8,9]. Substitution of (5.4) into (3.5) and application of the Galerkin-method yields the linear equations for Cj: 4 DijCj = 0,
Dij = Sp, D(qj)qidxdy.
(5.6)
The non-zero solution of this system determines the buckling mode. The lowest eigenvalue I and the corresponding characteristics m and n are found by equating the determinant of the system (5.6) to zero: det(Dij) = 0,
Dij = Dij(L, m. H),
L = N,(O,0)/(2Ehh”)
(5.7)
When N, = 0, the 1 has the meaning of a correction factor to the asymptotical solution (4.5) and by h + 0, ,J + 1. By calculating the coefficients Dij the integration with respect to x need not go further than one period of cosmx. The functions Ai vary symmetrically with respect to y = 0 and diminish rapidly with the growth of y. Calculations show that an increase of the limit of integration over ny = 12 has little influence on the values of Dij. The formulas (5.6) with D from (3.5) determine the following expressions of Dij (written here briefly, without the indices ij by A, and I?,): Dij = m8(Ao - 4A2 + 6A4 - 4A6 + As) + m4(Ao + 2kY2Bo - 2k,oAz) - 2A[Ao - 2A2 + A4 + nx2(Bo - 2B2 + B4) + n,o(A2 - 2A4 + As)m-2]m6. 1.0 0.8 0.6 0.L 0.2 0 -0.2 -0.L 0
5 Fig. 2.
lfl
(5.8)
E. L.
256
AXELRAD
Consistent with the accuracy assumed in (5.2) the terms with products of the parameters ky2, kxO are neglected. The integrals of (5.8) can be computed conveniently with the help of the appropriately truncated series (5.5). The solution was programmed on the personal. computer HP-45. For the range of local shapes and stress distributions, represented in Fig. 3, the solution with only the AZ-term retained in the mode (5.4) leads to practically the same value of i; as the solution wtth up to eight terms in Cj cosmxAj (ny). Thus the simplest version of equation (5.7) Dj3 (1, m, H) = 0 may be used, providing the formula
/I=
m2F + me2(l + 6.869kY2nU2+ 0.1538kX0H2)
2 + 0.3077H2 + 0.0308H4 + nX2Un - 2 + nYOI/m- ’
F = 1 + 0.3077H2 + 0.0923H4 + 0.0181H6 + 0.0017H8,
U = 6.869 + 0.2606H2 - 0.2048H4,
(5.9)
I/ = 0.1538 + 0.0614H2 + 0.0091H4.
This relation determines also the values of m and n = Hm-those rendering a minimum value ofi. The results of the rigorous analysis [8,12,14], available for cylinders confirm those of the local-buckling investigation. This,is demonstrated in Tables 1 and 2. The i-values present the correction to the asymptotic relations (4.5) or (2.1) necessary for a finite wall-thickness. Formula (5.9) proves adequate even for ovals (4.8) with C:= 0.9-for the curvature l/R, varying from 0.1 jr to 1.9/r. The solution presented provides an explanation for the local character of buckling of circular cylinders under uniform compression, discovered in [lo] : The local buckling mode w = Ccosmx As(ny) leads to virtually the same critical stress (N,,,/h = 1.00010,~, Table 2) as does the mode (4.1). Moreover, even a single-buckle mode w = CA,(mx)A,(ny) (A, is plotted in Fig. 2) leads (through a relation differing from (5.9) only in the values of the coefficients) to N,,, = l.O15a,JI. 6. DOUBLE-FOURIER-SERIES
MODE
The restriction (3.4) on the intensity of variation of the prebuckling shape and the simplest description of the local shape and stress (5.2) have been tested so far only by comparison with the rigorous results available. Consider next a local-stability solution free of these restrictions. The buckling mode will be sought in the form
(6.1) Z=ML+M+L,k=l+i.p.
[i/(L+l)],I=i-(L+l)(k-1)
where i.p. [ ] denotes the integer part of the quantity in the brackets; and m and n are (constant) parameters of the buckling mode.
I,0
0
0.5
1
Fig. 3. Influence
1.5
2
of stress and curvature
6 variation.
8
257
On local buckling of thin shells
Substitute the expressions (6.1) into (3.2). Applying the Galerkin method or equating to zero the Fourier-coefficients of the left-hand side of each of equations (3.2) one obtains equations determining Ci, 6. This system can be composed in an easily programmable matrix-form [3,11] : (DX+D,)‘--L2N -K
K (DX + DY)2][;] w = {C,
= [:I’
Cl...C,},
[:]
= E:
f = {F,
(6.2)
:]E;]’
F,...F,}.
The matrices D, and D, represent the operators 8,’ and ai, respectively, in such a way that Dxw, D,f, D,w and D,f are column matrices of the Fourier coefficients of the functions - @w, - a,“f, - Jj?w and -a:f respectively. Specifically D, and D, are diagonal matrices determined by formulas for the elements of the line i and the column j: D,(i, i) = n212(i);D,(i,j) = D,(i,j) = 0, i # j
D,(i, i) = m2k2(i),
L determined by formulas (6.1). The matrices K,, K,, withk=1,2 ,..., M+ l,I=O,l,..., N,, N, are composed of the coefficients of double Fourier-series of the respective functions. For instance, the composition of the matrix K, assures that the column matrix K&w determines in (6.2) the Fourier-coefficients of the function - k$;w. For the expansion
k~(x,Y) = krO ii0 &
k,(k, 1)cosmkx cosnly; 60 = 2,6, = 1, a + 0;
(6.3)
it can be ascertained by means of the relation 2cos~cosy = cos(jI + y) + cos@ - y) that the elements of the matrix K, are determined by the formula
&(iJ) =
$ +
1
[k(lk - kjl, II - ljl) +
k(k + kj, II - ljl) +
k,(lk
- kjl, I + Ij)
+ kj, I + Lj)].
k,(k
(6.4)
Here k and 1 are rendered by the expressions (6.1), the integers kj and Ij are defined by the same expressions with j instead of i. Formulas similar to (6.4) determine the remaining matrices of the equations (6.2) K,, N, and N,. Eliminating f we obtain from (6.2) the system
[@x + DJ2 + K@x + D,)-‘K]-‘N.-
$
(6.5)
.
The largest eigenvalue 1/(2;1) of the matrix of equation (6.5) and the respective eigenvector w determine the critical load and the buckling mode w(x, y). They are effectively computed by means of the R.v.Mises matrix-iteration method. This solution does not assume the buckling to be local. It can be exact as far as the Donnelltype equations (3.2) are applicable; the series (6.1) fulfill the boundary conditions and the expansion (6.3), as well as the similar ones for k,, N,, N, describe the actual prebuckling situation. The algorithm was programmed on the desk-top computer HP-85. For a 30-term mode (6.1) the calculation requires seven minutes. 7. NONCIRCULAR
AND DOUBLE-CURVATURE-IMPERFECT SHELLS
CYLINDRICAL
The series solution of Section 6 allows a further check of the local-stability concept. The two issues to be discussed are: (a) to ascertain the initial-buckling zone, to which the localbuckling analysis may be restricted and (b) to test the sufficiency of the leading-term description (5.2) or (5.3) of the prebuckling stress and strain.
E.L.
258
AXELRAD
Consider first cylindrical shells-those defined by the formula (4.8). The results of calculations, presented in Table 2 (m = 0) and Fig. 4 display for a broad range of substantially different shell shapes nearly identical values of ,I and the modes w(x, y) which nearly coincide within the buckling zone. The value of i depends in these cases solely on the parameter ky2. The actual shape is (for a fixed ky2 -value) of no direct importance for any profiles (4.8) provided n d 0.5. [Recall that n = 0.5 corresponds to the unrestrained mode (4.1), (4.7).] The initial buckling is determined basically by the shape within the zone lyl < x-one half the wave-breadth c, of the unrestrained periodic mode (4.1), (4.7). This conclusion applies also to variations of stress and to non-cylindrical shells. However the simultaneous involvement of several parameters characterizing stress and shape in a more general case [there are ten in (5.2)] makes the influence of each of them less transparent. Further discussion is restricted to one type of shell and to the influence of two factors not discussed in the foregoing. Consider shells defined by a deviation W, = Acosmx . cosN Y/r from a circular cylinder of radius r setting A = 0.5h, h = O.Olr, N = 10. According to (3.1) r -=l-r~
4
R = R,(O,O) = 2r, v = 0.3 ky2 = tn2/(2 - 25) = 0.3027.
with
= 1 - ~cosmx . cosny, and
(3.3) we have
r = 0.5
(7.1)
c/r = 0.0778,
n = NC/r = 0.778,
The influence of the length 2rcc/m of the imperfection wave (7.1) and of the compression stress NY/h in the circumferential direction (represented by nyo) on the buckling stress N,/h is illustrated by Table 2 and Fig. 5. The m = 0 entries of Table 2 concern a cylindrical shell. The other lines of the Table indicate the critical stress to be higher the shorter the wave length 2nc/m of imperfection (7.1). But for any wave lengths not shorter than 2rrc/O.3 the critical stress differs only insignificantly from that of a cylindrical shell. The reduction of the longitudinal critical stress in consequence of the lateral compression is displayed by the lines 11-17 of Table 2. However the influence of the lateral compression caused by the normal
Fig. 4. Examples
6, 8 and 9 of Table 2.
Fig. 5. Examples
12 and
14 of Table 2.
On local buckling of thin shells
259
pressure q is not restricted to the local stability. It can lead to global buckling and a check is needed whether the shell can withstand this. 8. CONCLUDING
REMARKS
The possibility to simplify the analysis by focusing it on the zone of initial buckling does not depend on the symmetry of this zone. Extension to the unsymmetric case is direct. The local approach has rendered promising results in the analysis of shell vibration [15]. Acknowledgements-The
work was carried out with the support of the Deutsche Forschungsgemeinschaft. The author is indebted to Prof. F. A. Emmerling and the late Dr. B. 0. Almroth for valuable discussions and to Valery Axelrad for the programming. REFERENCES 1. I. S. Staerman, Stability of shells. Trudy Kieuskogo Aviats. Inst. 1 (1936) (In Russian). 2. E. L. Axelrad, Refinement of the upper critical load of pipe fiexure by considering precritical deformation. Zzuestija AN SSSR, OTN, Mekhanika i Mash. No. 4, 133-139 (1965) (In Russian). 3. E. L. Axelrad, Flexible Shells. (Gibkie Obolotshki). Moscow, Nauka (1976) (In Russian). 4. E. L. Axelrad, Flexible shells. In Proceedings oj rhe 15th International Congress on Theoretical and Applied Mechanics (1980), (Edited by F. P. J. Rimrott and B. Tabarrok) pp. 45-56. North-Holland, Amsterdam (1981). 5. W. T. Koiter, The application of the initiai postbuckling analysis to shells. In Buckling of Shells,Proceedings of a State-o$the-Arr CoNoquium (Edited by E. Ramm). Springer, Berlin (1982). 6. J. T. Oden and K. J. Bathe, A commentary on computational mechanics. Appl. Mech. Reoiews, J&1053-1058 (1978). 7. B. Brendel and E. Ramm, Nichtlineare Stabilit&suntersuchungen mit der Methode der Finiten Elemente. Ing.Arch. 51, 337-362 (1982). 8. P. Seide and V. I. Weingarten, On the buckling of circular cylindrical shells under pure bending. J. appl. Me& 28, 112-116 (1961). 9. N. J. Hoff, C. C. Chao and W. A. Madsen, Buckling of a thin-walled circular cylindrical shell heated along an axial strip. J. appl. Mech. 31, 253-258 (1964). 10. M. Esslinger and B. Geier, Postbuckling Behauiour of Srructures. Springer, Wien (1975). 11. E. L. Axelrad, Schalentheorie. B. G. Teubner, Stuttgart (1983). 12. A. Libai and D. Durban, Buckling of cylindrical shells subjected to nonuniform axial foads. J. uppl. Me& 44, 714-718 (1977). 13. V. V. Kabanov and G. I. Kurtsevich, Investigation of the stability of a cylindrical shell under axial compression nonuniform along the length. Mech. Solids 11, 171-174 (1976). 14. V. Volpe, Y. N. Chen and J. Kempner, Buckling of orthogonally stiffened finite oval cylindrical shells under axial compression. AIAA J. 18, 571-580 (1980). 15. D. Teichmann, Dynamik und Stabilitit offener Kegelschalen. Z. angew Math. Mech. 63, T109-T113 (1983).